LMFDB - Embedded newform 3822.2.c.k.883.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3822,2,Mod(883,3822)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3822, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3822.883"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,6,-6,0,0,0,0,6,0,0,-6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.5188236525$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.9144576.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 4 $ x^6 + 12x^4 + 36x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.5
Root			$-0.339877i$ of defining polynomial
Character	$\chi$	$=$	3822.883
Dual form			3822.2.c.k.883.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -0.339877i q^{5} +1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +0.339877 q^{10} -0.660123i q^{11} -1.00000 q^{12} +(0.660123 - 3.54461i) q^{13} -0.339877i q^{15} +1.00000 q^{16} -6.54461 q^{17} +1.00000i q^{18} +6.08921i q^{19} +0.339877i q^{20} +0.660123 q^{22} -7.42909 q^{23} -1.00000i q^{24} +4.88448 q^{25} +(3.54461 + 0.660123i) q^{26} +1.00000 q^{27} -3.56424 q^{29} +0.339877 q^{30} +4.00000i q^{31} +1.00000i q^{32} -0.660123i q^{33} -6.54461i q^{34} -1.00000 q^{36} +3.54461i q^{37} -6.08921 q^{38} +(0.660123 - 3.54461i) q^{39} -0.339877 q^{40} +0.864853i q^{41} -5.08921 q^{43} +0.660123i q^{44} -0.339877i q^{45} -7.42909i q^{46} +9.44872i q^{47} +1.00000 q^{48} +4.88448i q^{50} -6.54461 q^{51} +(-0.660123 + 3.54461i) q^{52} +3.00000 q^{53} +1.00000i q^{54} -0.224361 q^{55} +6.08921i q^{57} -3.56424i q^{58} -3.90411i q^{59} +0.339877i q^{60} +1.86485 q^{61} -4.00000 q^{62} -1.00000 q^{64} +(-1.20473 - 0.224361i) q^{65} +0.660123 q^{66} +7.44872i q^{67} +6.54461 q^{68} -7.42909 q^{69} -0.884484i q^{71} -1.00000i q^{72} +10.0696i q^{73} -3.54461 q^{74} +4.88448 q^{75} -6.08921i q^{76} +(3.54461 + 0.660123i) q^{78} -12.4487 q^{79} -0.339877i q^{80} +1.00000 q^{81} -0.864853 q^{82} +16.2939i q^{83} +2.22436i q^{85} -5.08921i q^{86} -3.56424 q^{87} -0.660123 q^{88} -4.45539i q^{89} +0.339877 q^{90} +7.42909 q^{92} +4.00000i q^{93} -9.44872 q^{94} +2.06958 q^{95} +1.00000i q^{96} +16.1088i q^{97} -0.660123i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{12} + 6 q^{13} + 6 q^{16} - 18 q^{17} + 6 q^{22} + 6 q^{25} + 6 q^{27} + 6 q^{29} - 6 q^{36} + 6 q^{38} + 6 q^{39} + 12 q^{43} + 6 q^{48} - 18 q^{51} - 6 q^{52}+ \cdots - 24 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^12 + 6 * q^13 + 6 * q^16 - 18 * q^17 + 6 * q^22 + 6 * q^25 + 6 * q^27 + 6 * q^29 - 6 * q^36 + 6 * q^38 + 6 * q^39 + 12 * q^43 + 6 * q^48 - 18 * q^51 - 6 * q^52 + 18 * q^53 + 24 * q^55 - 6 * q^61 - 24 * q^62 - 6 * q^64 + 12 * q^65 + 6 * q^66 + 18 * q^68 + 6 * q^75 - 24 * q^79 + 6 * q^81 + 12 * q^82 + 6 * q^87 - 6 * q^88 - 6 * q^94 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times$.

$n$	$1471$	$2549$	$3433$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−1.00000			−0.500000
$5$		−	0.339877i		−	0.151998i	−0.997108	−	0.0759988i	$-0.975785\pi$
$5$		−	0.339877i		−	0.151998i	0.997108	−	0.0759988i	$-0.0242145\pi$
$6$			1.00000i			0.408248i
$7$		0			0
$7$		0			0
$8$		−	1.00000i		−	0.353553i
$9$	1.00000			0.333333
$10$	0.339877			0.107479
$11$		−	0.660123i		−	0.199035i	−0.995036	−	0.0995173i	$-0.968270\pi$
$11$		−	0.660123i		−	0.199035i	0.995036	−	0.0995173i	$-0.0317299\pi$
$12$	−1.00000			−0.288675
$13$	0.660123	−	3.54461i	0.183085	−	0.983097i
$13$	0.660123	−	3.54461i	0.183085	−	0.983097i
$14$		0			0
$15$		−	0.339877i		−	0.0877558i
$16$	1.00000			0.250000
$17$	−6.54461			−1.58730			−0.793650	−	0.608374i	$-0.791822\pi$
$17$	−6.54461			−1.58730			−0.793650	+	0.608374i	$0.791822\pi$
$18$			1.00000i			0.235702i
$19$			6.08921i			1.39696i	0.715629	+	0.698481i	$0.246140\pi$
$19$			6.08921i			1.39696i	−0.715629	+	0.698481i	$0.753860\pi$
$20$			0.339877i			0.0759988i
$21$		0			0
$22$	0.660123			0.140739
$23$	−7.42909			−1.54907			−0.774536	−	0.632530i	$-0.782017\pi$
$23$	−7.42909			−1.54907			−0.774536	+	0.632530i	$0.782017\pi$
$24$		−	1.00000i		−	0.204124i
$25$	4.88448			0.976897
$26$	3.54461	+	0.660123i	0.695155	+	0.129461i
$27$	1.00000			0.192450
$28$		0			0
$29$	−3.56424			−0.661862			−0.330931	−	0.943655i	$-0.607363\pi$
$29$	−3.56424			−0.661862			−0.330931	+	0.943655i	$0.607363\pi$
$30$	0.339877			0.0620527
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$			1.00000i			0.176777i
$33$		−	0.660123i		−	0.114913i
$34$		−	6.54461i		−	1.12239i
$35$		0			0
$36$	−1.00000			−0.166667
$37$			3.54461i			0.582730i	0.956612	+	0.291365i	$0.0941094\pi$
$37$			3.54461i			0.582730i	−0.956612	+	0.291365i	$0.905891\pi$
$38$	−6.08921			−0.987801
$39$	0.660123	−	3.54461i	0.105704	−	0.567591i
$40$	−0.339877			−0.0537393
$41$			0.864853i			0.135067i	0.997717	+	0.0675337i	$0.0215130\pi$
$41$			0.864853i			0.135067i	−0.997717	+	0.0675337i	$0.978487\pi$
$42$		0			0
$43$	−5.08921			−0.776098			−0.388049	−	0.921639i	$-0.626851\pi$
$43$	−5.08921			−0.776098			−0.388049	+	0.921639i	$0.626851\pi$
$44$			0.660123i			0.0995173i
$45$		−	0.339877i		−	0.0506659i
$46$		−	7.42909i		−	1.09536i
$47$			9.44872i			1.37824i	0.724648	+	0.689119i	$0.242002\pi$
$47$			9.44872i			1.37824i	−0.724648	+	0.689119i	$0.757998\pi$
$48$	1.00000			0.144338
$49$		0			0
$50$			4.88448i			0.690770i
$51$	−6.54461			−0.916428
$52$	−0.660123	+	3.54461i	−0.0915426	+	0.491549i
$53$	3.00000			0.412082			0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000			0.412082			0.206041	+	0.978543i	$0.433942\pi$
$54$			1.00000i			0.136083i
$55$	−0.224361			−0.0302528
$56$		0			0
$57$			6.08921i			0.806536i
$58$		−	3.56424i		−	0.468007i
$59$		−	3.90411i		−	0.508272i	−0.967168	−	0.254136i	$-0.918209\pi$
$59$		−	3.90411i		−	0.508272i	0.967168	−	0.254136i	$-0.0817911\pi$
$60$			0.339877i			0.0438779i
$61$	1.86485			0.238770			0.119385	−	0.992848i	$-0.461908\pi$
$61$	1.86485			0.238770			0.119385	+	0.992848i	$0.461908\pi$
$62$	−4.00000			−0.508001
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−1.20473	−	0.224361i	−0.149428	−	0.0278285i
$66$	0.660123			0.0812555
$67$			7.44872i			0.910006i	0.890490	+	0.455003i	$0.150362\pi$
$67$			7.44872i			0.910006i	−0.890490	+	0.455003i	$0.849638\pi$
$68$	6.54461			0.793650
$69$	−7.42909			−0.894357
$70$		0			0
$71$		−	0.884484i		−	0.104969i	−0.998622	−	0.0524845i	$-0.983286\pi$
$71$		−	0.884484i		−	0.104969i	0.998622	−	0.0524845i	$-0.0167140\pi$
$72$		−	1.00000i		−	0.117851i
$73$			10.0696i			1.17856i	0.807931	+	0.589278i	$0.200588\pi$
$73$			10.0696i			1.17856i	−0.807931	+	0.589278i	$0.799412\pi$
$74$	−3.54461			−0.412052
$75$	4.88448			0.564012
$76$		−	6.08921i		−	0.698481i
$77$		0			0
$78$	3.54461	+	0.660123i	0.401348	+	0.0747442i
$79$	−12.4487			−1.40059			−0.700295	−	0.713853i	$-0.746948\pi$
$79$	−12.4487			−1.40059			−0.700295	+	0.713853i	$0.746948\pi$
$80$		−	0.339877i		−	0.0379994i
$81$	1.00000			0.111111
$82$	−0.864853			−0.0955070
$83$			16.2939i			1.78849i	0.447575	+	0.894246i	$0.352288\pi$
$83$			16.2939i			1.78849i	−0.447575	+	0.894246i	$0.647712\pi$
$84$		0			0
$85$			2.22436i			0.241266i
$86$		−	5.08921i		−	0.548784i
$87$	−3.56424			−0.382126
$88$	−0.660123			−0.0703694
$89$		−	4.45539i		−	0.472271i	−0.971720	−	0.236135i	$-0.924119\pi$
$89$		−	4.45539i		−	0.472271i	0.971720	−	0.236135i	$-0.0758809\pi$
$90$	0.339877			0.0358262
$91$		0			0
$92$	7.42909			0.774536
$93$			4.00000i			0.414781i
$94$	−9.44872			−0.974561
$95$	2.06958			0.212335
$96$			1.00000i			0.102062i
$97$			16.1088i			1.63561i	0.575499	+	0.817803i	$0.304808\pi$
$97$			16.1088i			1.63561i	−0.575499	+	0.817803i	$0.695192\pi$
$98$		0			0
$99$		−	0.660123i		−	0.0663449i
$100$	−4.88448			−0.488448
$101$	2.79527			0.278140			0.139070	−	0.990283i	$-0.455589\pi$
$101$	2.79527			0.278140			0.139070	+	0.990283i	$0.455589\pi$
$102$		−	6.54461i		−	0.648013i
$103$	−7.53793			−0.742735			−0.371367	−	0.928486i	$-0.621111\pi$
$103$	−7.53793			−0.742735			−0.371367	+	0.928486i	$0.621111\pi$
$104$	−3.54461	−	0.660123i	−0.347577	−	0.0647304i
$105$		0			0
$106$			3.00000i			0.291386i
$107$	−3.72971			−0.360564			−0.180282	−	0.983615i	$-0.557701\pi$
$107$	−3.72971			−0.360564			−0.180282	+	0.983615i	$0.557701\pi$
$108$	−1.00000			−0.0962250
$109$			14.4095i			1.38018i	0.723725	+	0.690088i	$0.242428\pi$
$109$			14.4095i			1.38018i	−0.723725	+	0.690088i	$0.757572\pi$
$110$		−	0.224361i		−	0.0213919i
$111$			3.54461i			0.336439i
$112$		0			0
$113$	15.5576			1.46353			0.731766	−	0.681556i	$-0.238696\pi$
$113$	15.5576			1.46353			0.731766	+	0.681556i	$0.238696\pi$
$114$	−6.08921			−0.570307
$115$			2.52498i			0.235455i
$116$	3.56424			0.330931
$117$	0.660123	−	3.54461i	0.0610284	−	0.327699i
$118$	3.90411			0.359403
$119$		0			0
$120$	−0.339877			−0.0310264
$121$	10.5642			0.960385
$122$			1.86485i			0.168836i
$123$			0.864853i			0.0779812i
$124$		−	4.00000i		−	0.359211i
$125$		−	3.35951i		−	0.300483i
$126$		0			0
$127$	14.4028			1.27804			0.639020	−	0.769190i	$-0.279340\pi$
$127$	14.4028			1.27804			0.639020	+	0.769190i	$0.279340\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−5.08921			−0.448080
$130$	0.224361	−	1.20473i	0.0196777	−	0.105662i
$131$	−20.6731			−1.80622			−0.903108	−	0.429414i	$-0.858720\pi$
$131$	−20.6731			−1.80622			−0.903108	+	0.429414i	$0.858720\pi$
$132$			0.660123i			0.0574563i
$133$		0			0
$134$	−7.44872			−0.643472
$135$		−	0.339877i		−	0.0292519i
$136$			6.54461i			0.561195i
$137$		−	14.2244i		−	1.21527i	−0.794217	−	0.607635i	$-0.792119\pi$
$137$		−	14.2244i		−	1.21527i	0.794217	−	0.607635i	$-0.207881\pi$
$138$		−	7.42909i		−	0.632406i
$139$	9.42909			0.799765			0.399883	−	0.916566i	$-0.369051\pi$
$139$	9.42909			0.799765			0.399883	+	0.916566i	$0.369051\pi$
$140$		0			0
$141$			9.44872i			0.795726i
$142$	0.884484			0.0742242
$143$	−2.33988	−	0.435763i	−0.195670	−	0.0364403i
$144$	1.00000			0.0833333
$145$			1.21140i			0.100601i
$146$	−10.0696			−0.833365
$147$		0			0
$148$		−	3.54461i		−	0.291365i
$149$		−	8.10884i		−	0.664302i	−0.943226	−	0.332151i	$-0.892226\pi$
$149$		−	8.10884i		−	0.664302i	0.943226	−	0.332151i	$-0.107774\pi$
$150$			4.88448i			0.398816i
$151$		−	5.30062i		−	0.431358i	−0.976464	−	0.215679i	$-0.930804\pi$
$151$		−	5.30062i		−	0.431358i	0.976464	−	0.215679i	$-0.0691965\pi$
$152$	6.08921			0.493900
$153$	−6.54461			−0.529100
$154$		0			0
$155$	1.35951			0.109198
$156$	−0.660123	+	3.54461i	−0.0528521	+	0.283796i
$157$	−23.0562			−1.84009			−0.920044	−	0.391815i	$-0.871847\pi$
$157$	−23.0562			−1.84009			−0.920044	+	0.391815i	$0.871847\pi$
$158$		−	12.4487i		−	0.990367i
$159$	3.00000			0.237915
$160$	0.339877			0.0268696
$161$		0			0
$162$			1.00000i			0.0785674i
$163$		−	13.5642i		−	1.06243i	−0.847236	−	0.531217i	$-0.821735\pi$
$163$		−	13.5642i		−	1.06243i	0.847236	−	0.531217i	$-0.178265\pi$
$164$		−	0.864853i		−	0.0675337i
$165$	−0.224361			−0.0174664
$166$	−16.2939			−1.26466
$167$			15.7034i			1.21517i	0.794256	+	0.607583i	$0.207861\pi$
$167$			15.7034i			1.21517i	−0.794256	+	0.607583i	$0.792139\pi$
$168$		0			0
$169$	−12.1285	−	4.67975i	−0.932960	−	0.359981i
$170$	−2.22436			−0.170601
$171$			6.08921i			0.465654i
$172$	5.08921			0.388049
$173$	13.6035			1.03425			0.517127	−	0.855908i	$-0.327001\pi$
$173$	13.6035			1.03425			0.517127	+	0.855908i	$0.327001\pi$
$174$		−	3.56424i		−	0.270204i
$175$		0			0
$176$		−	0.660123i		−	0.0497587i
$177$		−	3.90411i		−	0.293451i
$178$	4.45539			0.333946
$179$	0.448721			0.0335390			0.0167695	−	0.999859i	$-0.494662\pi$
$179$	0.448721			0.0335390			0.0167695	+	0.999859i	$0.494662\pi$
$180$			0.339877i			0.0253329i
$181$	−15.9041			−1.18214			−0.591072	−	0.806619i	$-0.701295\pi$
$181$	−15.9041			−1.18214			−0.591072	+	0.806619i	$0.701295\pi$
$182$		0			0
$183$	1.86485			0.137854
$184$			7.42909i			0.547680i
$185$	1.20473			0.0885735
$186$	−4.00000			−0.293294
$187$			4.32025i			0.315928i
$188$		−	9.44872i		−	0.689119i
$189$		0			0
$190$			2.06958i			0.150143i
$191$	−10.4095			−0.753202			−0.376601	−	0.926376i	$-0.622907\pi$
$191$	−10.4095			−0.753202			−0.376601	+	0.926376i	$0.622907\pi$
$192$	−1.00000			−0.0721688
$193$		−	7.69938i		−	0.554214i	−0.960839	−	0.277107i	$-0.910624\pi$
$193$		−	7.69938i		−	0.554214i	0.960839	−	0.277107i	$-0.0893756\pi$
$194$	−16.1088			−1.15655
$195$	−1.20473	−	0.224361i	−0.0862725	−	0.0160668i
$196$		0			0
$197$		−	8.98037i		−	0.639825i	−0.947447	−	0.319912i	$-0.896347\pi$
$197$		−	8.98037i		−	0.639825i	0.947447	−	0.319912i	$-0.103653\pi$
$198$	0.660123			0.0469129
$199$	−20.5946			−1.45991			−0.729955	−	0.683495i	$-0.760459\pi$
$199$	−20.5946			−1.45991			−0.729955	+	0.683495i	$0.760459\pi$
$200$		−	4.88448i		−	0.345385i
$201$			7.44872i			0.525392i
$202$			2.79527i			0.196675i
$203$		0			0
$204$	6.54461			0.458214
$205$	0.293944			0.0205299
$206$		−	7.53793i		−	0.525193i
$207$	−7.42909			−0.516357
$208$	0.660123	−	3.54461i	0.0457713	−	0.245774i
$209$	4.01963			0.278044
$210$		0			0
$211$	15.3898			1.05948			0.529740	−	0.848160i	$-0.322290\pi$
$211$	15.3898			1.05948			0.529740	+	0.848160i	$0.322290\pi$
$212$	−3.00000			−0.206041
$213$		−	0.884484i		−	0.0606038i
$214$		−	3.72971i		−	0.254957i
$215$			1.72971i			0.117965i
$216$		−	1.00000i		−	0.0680414i
$217$		0			0
$218$	−14.4095			−0.975932
$219$			10.0696i			0.680439i
$220$	0.224361			0.0151264
$221$	−4.32025	+	23.1981i	−0.290611	+	1.56047i
$222$	−3.54461			−0.237898
$223$		−	18.0196i		−	1.20668i	−0.797483	−	0.603342i	$-0.793835\pi$
$223$		−	18.0196i		−	1.20668i	0.797483	−	0.603342i	$-0.206165\pi$
$224$		0			0
$225$	4.88448			0.325632
$226$			15.5576i			1.03487i
$227$			0.231033i			0.0153342i	0.999971	+	0.00766709i	$0.00244053\pi$
$227$			0.231033i			0.0153342i	−0.999971	+	0.00766709i	$0.997559\pi$
$228$		−	6.08921i		−	0.403268i
$229$			6.64049i			0.438816i	0.975633	+	0.219408i	$0.0704126\pi$
$229$			6.64049i			0.438816i	−0.975633	+	0.219408i	$0.929587\pi$
$230$	−2.52498			−0.166492
$231$		0			0
$232$			3.56424i			0.234004i
$233$	−29.5183			−1.93381			−0.966904	−	0.255140i	$-0.917879\pi$
$233$	−29.5183			−1.93381			−0.966904	+	0.255140i	$0.917879\pi$
$234$	3.54461	+	0.660123i	0.231718	+	0.0431536i
$235$	3.21140			0.209489
$236$			3.90411i			0.254136i
$237$	−12.4487			−0.808631
$238$		0			0
$239$		−	15.0236i		−	0.971799i	−0.874015	−	0.485900i	$-0.838492\pi$
$239$		−	15.0236i		−	0.971799i	0.874015	−	0.485900i	$-0.161508\pi$
$240$		−	0.339877i		−	0.0219390i
$241$		−	4.44872i		−	0.286567i	−0.989682	−	0.143284i	$-0.954234\pi$
$241$		−	4.44872i		−	0.286567i	0.989682	−	0.143284i	$-0.0457661\pi$
$242$			10.5642i			0.679095i
$243$	1.00000			0.0641500
$244$	−1.86485			−0.119385
$245$		0			0
$246$	−0.864853			−0.0551410
$247$	21.5839	+	4.01963i	1.37335	+	0.255763i
$248$	4.00000			0.254000
$249$			16.2939i			1.03259i
$250$	3.35951			0.212474
$251$	−24.3961			−1.53987			−0.769935	−	0.638123i	$-0.779711\pi$
$251$	−24.3961			−1.53987			−0.769935	+	0.638123i	$0.779711\pi$
$252$		0			0
$253$			4.90411i			0.308319i
$254$			14.4028i			0.903711i
$255$			2.22436i			0.139295i
$256$	1.00000			0.0625000
$257$	18.4487			1.15080			0.575400	−	0.817872i	$-0.304846\pi$
$257$	18.4487			1.15080			0.575400	+	0.817872i	$0.304846\pi$
$258$		−	5.08921i		−	0.316841i
$259$		0			0
$260$	1.20473	+	0.224361i	0.0747142	+	0.0139143i
$261$	−3.56424			−0.220621
$262$		−	20.6731i		−	1.27719i
$263$	−6.30062			−0.388513			−0.194256	−	0.980951i	$-0.562229\pi$
$263$	−6.30062			−0.388513			−0.194256	+	0.980951i	$0.562229\pi$
$264$	−0.660123			−0.0406278
$265$		−	1.01963i		−	0.0626354i
$266$		0			0
$267$		−	4.45539i		−	0.272666i
$268$		−	7.44872i		−	0.455003i
$269$	−9.89517			−0.603319			−0.301660	−	0.953416i	$-0.597541\pi$
$269$	−9.89517			−0.603319			−0.301660	+	0.953416i	$0.597541\pi$
$270$	0.339877			0.0206842
$271$		−	12.6994i		−	0.771433i	−0.922617	−	0.385716i	$-0.873954\pi$
$271$		−	12.6994i		−	0.771433i	0.922617	−	0.385716i	$-0.126046\pi$
$272$	−6.54461			−0.396825
$273$		0			0
$274$	14.2244			0.859325
$275$		−	3.22436i		−	0.194436i
$276$	7.42909			0.447179
$277$	28.0433			1.68496			0.842479	−	0.538730i	$-0.181096\pi$
$277$	28.0433			1.68496			0.842479	+	0.538730i	$0.181096\pi$
$278$			9.42909i			0.565519i
$279$			4.00000i			0.239474i
$280$		0			0
$281$			1.82157i			0.108666i	0.998523	+	0.0543330i	$0.0173032\pi$
$281$			1.82157i			0.108666i	−0.998523	+	0.0543330i	$0.982697\pi$
$282$	−9.44872			−0.562663
$283$	8.98037			0.533828			0.266914	−	0.963720i	$-0.413996\pi$
$283$	8.98037			0.533828			0.266914	+	0.963720i	$0.413996\pi$
$284$			0.884484i			0.0524845i
$285$	2.06958			0.122592
$286$	0.435763	−	2.33988i	0.0257672	−	0.138360i
$287$		0			0
$288$			1.00000i			0.0589256i
$289$	25.8319			1.51952
$290$	−1.21140			−0.0711360
$291$			16.1088i			0.944317i
$292$		−	10.0696i		−	0.589278i
$293$			2.80194i			0.163691i	0.996645	+	0.0818456i	$0.0260814\pi$
$293$			2.80194i			0.163691i	−0.996645	+	0.0818456i	$0.973919\pi$
$294$		0			0
$295$	−1.32692			−0.0772562
$296$	3.54461			0.206026
$297$		−	0.660123i		−	0.0383042i
$298$	8.10884			0.469733
$299$	−4.90411	+	26.3332i	−0.283612	+	1.52289i
$300$	−4.88448			−0.282006
$301$		0			0
$302$	5.30062			0.305016
$303$	2.79527			0.160584
$304$			6.08921i			0.349240i
$305$		−	0.633820i		−	0.0362925i
$306$		−	6.54461i		−	0.374130i
$307$			9.21769i			0.526081i	0.964785	+	0.263041i	$0.0847253\pi$
$307$			9.21769i			0.526081i	−0.964785	+	0.263041i	$0.915275\pi$
$308$		0			0
$309$	−7.53793			−0.428818
$310$			1.35951i			0.0772148i
$311$	−11.4684			−0.650311			−0.325155	−	0.945661i	$-0.605417\pi$
$311$	−11.4684			−0.650311			−0.325155	+	0.945661i	$0.605417\pi$
$312$	−3.54461	−	0.660123i	−0.200674	−	0.0373721i
$313$	−22.5642			−1.27541			−0.637703	−	0.770282i	$-0.720115\pi$
$313$	−22.5642			−1.27541			−0.637703	+	0.770282i	$0.720115\pi$
$314$		−	23.0562i		−	1.30114i
$315$		0			0
$316$	12.4487			0.700295
$317$		−	12.6271i		−	0.709211i	−0.935016	−	0.354606i	$-0.884615\pi$
$317$		−	12.6271i		−	0.709211i	0.935016	−	0.354606i	$-0.115385\pi$
$318$			3.00000i			0.168232i
$319$			2.35284i			0.131733i
$320$			0.339877i			0.0189997i
$321$	−3.72971			−0.208172
$322$		0			0
$323$		−	39.8515i		−	2.21740i
$324$	−1.00000			−0.0555556
$325$	3.22436	−	17.3136i	0.178855	−	0.960384i
$326$	13.5642			0.751254
$327$			14.4095i			0.796845i
$328$	0.864853			0.0477535
$329$		0			0
$330$		−	0.224361i		−	0.0123506i
$331$			26.8212i			1.47423i	0.675770	+	0.737113i	$0.263811\pi$
$331$			26.8212i			1.47423i	−0.675770	+	0.737113i	$0.736189\pi$
$332$		−	16.2939i		−	0.894246i
$333$			3.54461i			0.194243i
$334$	−15.7034			−0.859252
$335$	2.53165			0.138319
$336$		0			0
$337$	9.81892			0.534871			0.267435	−	0.963576i	$-0.413824\pi$
$337$	9.81892			0.534871			0.267435	+	0.963576i	$0.413824\pi$
$338$	4.67975	−	12.1285i	0.254545	−	0.659702i
$339$	15.5576			0.844971
$340$		−	2.22436i		−	0.120633i
$341$	2.64049			0.142991
$342$	−6.08921			−0.329267
$343$		0			0
$344$			5.08921i			0.274392i
$345$			2.52498i			0.135940i
$346$			13.6035i			0.731329i
$347$	19.0959			1.02512			0.512560	−	0.858651i	$-0.328697\pi$
$347$	19.0959			1.02512			0.512560	+	0.858651i	$0.328697\pi$
$348$	3.56424			0.191063
$349$		−	0.416132i		−	0.0222750i	−0.999938	−	0.0111375i	$-0.996455\pi$
$349$		−	0.416132i		−	0.0222750i	0.999938	−	0.0111375i	$-0.00354525\pi$
$350$		0			0
$351$	0.660123	−	3.54461i	0.0352348	−	0.189197i
$352$	0.660123			0.0351847
$353$			16.4880i			0.877567i	0.898593	+	0.438783i	$0.144590\pi$
$353$			16.4880i			0.877567i	−0.898593	+	0.438783i	$0.855410\pi$
$354$	3.90411			0.207501
$355$	−0.300616			−0.0159550
$356$			4.45539i			0.236135i
$357$		0			0
$358$			0.448721i			0.0237157i
$359$			24.4724i			1.29160i	0.763506	+	0.645801i	$0.223477\pi$
$359$			24.4724i			1.29160i	−0.763506	+	0.645801i	$0.776523\pi$
$360$	−0.339877			−0.0179131
$361$	−18.0785			−0.951501
$362$		−	15.9041i		−	0.835902i
$363$	10.5642			0.554479
$364$		0			0
$365$	3.42242			0.179138
$366$			1.86485i			0.0974774i
$367$	−12.4880			−0.651867			−0.325934	−	0.945393i	$-0.605679\pi$
$367$	−12.4880			−0.651867			−0.325934	+	0.945393i	$0.605679\pi$
$368$	−7.42909			−0.387268
$369$			0.864853i			0.0450225i
$370$			1.20473i			0.0626309i
$371$		0			0
$372$		−	4.00000i		−	0.207390i
$373$	−7.82786			−0.405311			−0.202656	−	0.979250i	$-0.564957\pi$
$373$	−7.82786			−0.405311			−0.202656	+	0.979250i	$0.564957\pi$
$374$	−4.32025			−0.223395
$375$		−	3.35951i		−	0.173484i
$376$	9.44872			0.487281
$377$	−2.35284	+	12.6338i	−0.121177	+	0.650675i
$378$		0			0
$379$			8.67975i			0.445849i	0.974836	+	0.222925i	$0.0715603\pi$
$379$			8.67975i			0.445849i	−0.974836	+	0.222925i	$0.928440\pi$
$380$	−2.06958			−0.106167
$381$	14.4028			0.737877
$382$		−	10.4095i		−	0.532594i
$383$		−	23.4224i		−	1.19683i	−0.801186	−	0.598415i	$-0.795797\pi$
$383$		−	23.4224i		−	1.19683i	0.801186	−	0.598415i	$-0.204203\pi$
$384$		−	1.00000i		−	0.0510310i
$385$		0			0
$386$	7.69938			0.391888
$387$	−5.08921			−0.258699
$388$		−	16.1088i		−	0.817803i
$389$	−30.1914			−1.53076			−0.765382	−	0.643576i	$-0.777450\pi$
$389$	−30.1914			−1.53076			−0.765382	+	0.643576i	$0.777450\pi$
$390$	0.224361	−	1.20473i	0.0113609	−	0.0610039i
$391$	48.6205			2.45884
$392$		0			0
$393$	−20.6731			−1.04282
$394$	8.98037			0.452424
$395$			4.23103i			0.212886i
$396$			0.660123i			0.0331724i
$397$			8.49465i			0.426334i	0.977016	+	0.213167i	$0.0683779\pi$
$397$			8.49465i			0.426334i	−0.977016	+	0.213167i	$0.931622\pi$
$398$		−	20.5946i		−	1.03231i
$399$		0			0
$400$	4.88448			0.244224
$401$		−	20.1718i		−	1.00733i	−0.863899	−	0.503665i	$-0.831985\pi$
$401$		−	20.1718i		−	1.00733i	0.863899	−	0.503665i	$-0.168015\pi$
$402$	−7.44872			−0.371508
$403$	14.1784	+	2.64049i	0.706278	+	0.131532i
$404$	−2.79527			−0.139070
$405$		−	0.339877i		−	0.0168886i
$406$		0			0
$407$	2.33988			0.115983
$408$			6.54461i			0.324006i
$409$			4.30062i			0.212652i	0.994331	+	0.106326i	$0.0339087\pi$
$409$			4.30062i			0.212652i	−0.994331	+	0.106326i	$0.966091\pi$
$410$			0.293944i			0.0145168i
$411$		−	14.2244i		−	0.701636i
$412$	7.53793			0.371367
$413$		0			0
$414$		−	7.42909i		−	0.365120i
$415$	5.53793			0.271847
$416$	3.54461	+	0.660123i	0.173789	+	0.0323652i
$417$	9.42909			0.461745
$418$			4.01963i			0.196607i
$419$	−32.8974			−1.60715			−0.803573	−	0.595207i	$-0.797070\pi$
$419$	−32.8974			−1.60715			−0.803573	+	0.595207i	$0.797070\pi$
$420$		0			0
$421$			21.7230i			1.05872i	0.848399	+	0.529358i	$0.177567\pi$
$421$			21.7230i			1.05872i	−0.848399	+	0.529358i	$0.822433\pi$
$422$			15.3898i			0.749165i
$423$			9.44872i			0.459413i
$424$		−	3.00000i		−	0.145693i
$425$	−31.9670			−1.55063
$426$	0.884484			0.0428534
$427$		0			0
$428$	3.72971			0.180282
$429$	−2.33988	−	0.435763i	−0.112970	−	0.0210388i
$430$	−1.72971			−0.0834138
$431$		−	6.06291i		−	0.292040i	−0.989282	−	0.146020i	$-0.953354\pi$
$431$		−	6.06291i		−	0.292040i	0.989282	−	0.146020i	$-0.0466464\pi$
$432$	1.00000			0.0481125
$433$	11.1392			0.535314			0.267657	−	0.963514i	$-0.413751\pi$
$433$	11.1392			0.535314			0.267657	+	0.963514i	$0.413751\pi$
$434$		0			0
$435$			1.21140i			0.0580823i
$436$		−	14.4095i		−	0.690088i
$437$		−	45.2373i		−	2.16399i
$438$	−10.0696			−0.481143
$439$	11.6231			0.554742			0.277371	−	0.960763i	$-0.410537\pi$
$439$	11.6231			0.554742			0.277371	+	0.960763i	$0.410537\pi$
$440$			0.224361i			0.0106960i
$441$		0			0
$442$	−23.1981	−	4.32025i	−1.10342	−	0.205493i
$443$	−4.67308			−0.222025			−0.111012	−	0.993819i	$-0.535409\pi$
$443$	−4.67308			−0.222025			−0.111012	+	0.993819i	$0.535409\pi$
$444$		−	3.54461i		−	0.168220i
$445$	−1.51429			−0.0717840
$446$	18.0196			0.853254
$447$		−	8.10884i		−	0.383535i
$448$		0			0
$449$		−	36.6271i		−	1.72854i	−0.503026	−	0.864271i	$-0.667780\pi$
$449$		−	36.6271i		−	1.72854i	0.503026	−	0.864271i	$-0.332220\pi$
$450$			4.88448i			0.230257i
$451$	0.570909			0.0268831
$452$	−15.5576			−0.731766
$453$		−	5.30062i		−	0.249045i
$454$	−0.231033			−0.0108429
$455$		0			0
$456$	6.08921			0.285154
$457$		−	31.3372i		−	1.46589i	−0.680286	−	0.732947i	$-0.738144\pi$
$457$		−	31.3372i		−	1.46589i	0.680286	−	0.732947i	$-0.261856\pi$
$458$	−6.64049			−0.310290
$459$	−6.54461			−0.305476
$460$		−	2.52498i		−	0.117728i
$461$		−	0.601231i		−	0.0280021i	−0.999902	−	0.0140011i	$-0.995543\pi$
$461$		−	0.601231i		−	0.0280021i	0.999902	−	0.0140011i	$-0.00445682\pi$
$462$		0			0
$463$			10.5013i			0.488038i	0.969770	+	0.244019i	$0.0784659\pi$
$463$			10.5013i			0.488038i	−0.969770	+	0.244019i	$0.921534\pi$
$464$	−3.56424			−0.165466
$465$	1.35951			0.0630457
$466$		−	29.5183i		−	1.36741i
$467$	27.2217			1.25967			0.629835	−	0.776729i	$-0.283122\pi$
$467$	27.2217			1.25967			0.629835	+	0.776729i	$0.283122\pi$
$468$	−0.660123	+	3.54461i	−0.0305142	+	0.163850i
$469$		0			0
$470$			3.21140i			0.148131i
$471$	−23.0562			−1.06238
$472$	−3.90411			−0.179701
$473$			3.35951i			0.154470i
$474$		−	12.4487i		−	0.571789i
$475$			29.7427i			1.36469i
$476$		0			0
$477$	3.00000			0.137361
$478$	15.0236			0.687166
$479$			28.8711i			1.31916i	0.751636	+	0.659578i	$0.229265\pi$
$479$			28.8711i			1.31916i	−0.751636	+	0.659578i	$0.770735\pi$
$480$	0.339877			0.0155132
$481$	12.5642	+	2.33988i	0.572880	+	0.106689i
$482$	4.44872			0.202634
$483$		0			0
$484$	−10.5642			−0.480193
$485$	5.47502			0.248608
$486$			1.00000i			0.0453609i
$487$		−	3.91746i		−	0.177517i	−0.996053	−	0.0887585i	$-0.971710\pi$
$487$		−	3.91746i		−	0.177517i	0.996053	−	0.0887585i	$-0.0282899\pi$
$488$		−	1.86485i		−	0.0844179i
$489$		−	13.5642i		−	0.613396i
$490$		0			0
$491$	−13.4202			−0.605643			−0.302821	−	0.953047i	$-0.597929\pi$
$491$	−13.4202			−0.605643			−0.302821	+	0.953047i	$0.597929\pi$
$492$		−	0.864853i		−	0.0389906i
$493$	23.3265			1.05057
$494$	−4.01963	+	21.5839i	−0.180852	+	0.971104i
$495$	−0.224361			−0.0100843
$496$			4.00000i			0.179605i
$497$		0			0
$498$	−16.2939			−0.730149
$499$		−	32.7819i		−	1.46752i	−0.679408	−	0.733760i	$-0.737764\pi$
$499$		−	32.7819i		−	1.46752i	0.679408	−	0.733760i	$-0.262236\pi$
$500$			3.35951i			0.150242i
$501$			15.7034i			0.701576i
$502$		−	24.3961i		−	1.08885i
$503$	22.8582			1.01920			0.509598	−	0.860413i	$-0.329794\pi$
$503$	22.8582			1.01920			0.509598	+	0.860413i	$0.329794\pi$
$504$		0			0
$505$		−	0.950048i		−	0.0422766i
$506$	−4.90411			−0.218014
$507$	−12.1285	−	4.67975i	−0.538644	−	0.207835i
$508$	−14.4028			−0.639020
$509$		−	32.8448i		−	1.45582i	−0.685672	−	0.727911i	$-0.740491\pi$
$509$		−	32.8448i		−	1.45582i	0.685672	−	0.727911i	$-0.259509\pi$
$510$	−2.22436			−0.0984963
$511$		0			0
$512$			1.00000i			0.0441942i
$513$			6.08921i			0.268845i
$514$			18.4487i			0.813738i
$515$			2.56197i			0.112894i
$516$	5.08921			0.224040
$517$	6.23732			0.274317
$518$		0			0
$519$	13.6035			0.597127
$520$	−0.224361	+	1.20473i	−0.00983886	+	0.0528309i
$521$	2.81892			0.123499			0.0617496	−	0.998092i	$-0.480332\pi$
$521$	2.81892			0.123499			0.0617496	+	0.998092i	$0.480332\pi$
$522$		−	3.56424i		−	0.156002i
$523$	−5.62980			−0.246174			−0.123087	−	0.992396i	$-0.539279\pi$
$523$	−5.62980			−0.246174			−0.123087	+	0.992396i	$0.539279\pi$
$524$	20.6731			0.903108
$525$		0			0
$526$		−	6.30062i		−	0.274720i
$527$		−	26.1784i		−	1.14035i
$528$		−	0.660123i		−	0.0287282i
$529$	32.1914			1.39963
$530$	1.01963			0.0442899
$531$		−	3.90411i		−	0.169424i
$532$		0			0
$533$	3.06556	+	0.570909i	0.132784	+	0.0247288i
$534$	4.45539			0.192804
$535$			1.26764i			0.0548049i
$536$	7.44872			0.321736
$537$	0.448721			0.0193637
$538$		−	9.89517i		−	0.426611i
$539$		0			0
$540$			0.339877i			0.0146260i
$541$			24.7123i			1.06247i	0.847226	+	0.531233i	$0.178271\pi$
$541$			24.7123i			1.06247i	−0.847226	+	0.531233i	$0.821729\pi$
$542$	12.6994			0.545485
$543$	−15.9041			−0.682511
$544$		−	6.54461i		−	0.280598i
$545$	4.89744			0.209783
$546$		0			0
$547$	18.1874			0.777636			0.388818	−	0.921315i	$-0.372884\pi$
$547$	18.1874			0.777636			0.388818	+	0.921315i	$0.372884\pi$
$548$			14.2244i			0.607635i
$549$	1.86485			0.0795900
$550$	3.22436			0.137487
$551$		−	21.7034i		−	0.924596i
$552$			7.42909i			0.316203i
$553$		0			0
$554$			28.0433i			1.19144i
$555$	1.20473			0.0511379
$556$	−9.42909			−0.399883
$557$		−	26.6405i		−	1.12879i	−0.825504	−	0.564397i	$-0.809109\pi$
$557$		−	26.6405i		−	1.12879i	0.825504	−	0.564397i	$-0.190891\pi$
$558$	−4.00000			−0.169334
$559$	−3.35951	+	18.0393i	−0.142092	+	0.762979i
$560$		0			0
$561$			4.32025i			0.182401i
$562$	−1.82157			−0.0768384
$563$	27.6771			1.16645			0.583225	−	0.812310i	$-0.301790\pi$
$563$	27.6771			1.16645			0.583225	+	0.812310i	$0.301790\pi$
$564$		−	9.44872i		−	0.397863i
$565$		−	5.28766i		−	0.222453i
$566$			8.98037i			0.377473i
$567$		0			0
$568$	−0.884484			−0.0371121
$569$	−20.3898			−0.854786			−0.427393	−	0.904066i	$-0.640568\pi$
$569$	−20.3898			−0.854786			−0.427393	+	0.904066i	$0.640568\pi$
$570$			2.06958i			0.0866853i
$571$	38.2043			1.59880			0.799401	−	0.600798i	$-0.205150\pi$
$571$	38.2043			1.59880			0.799401	+	0.600798i	$0.205150\pi$
$572$	2.33988	+	0.435763i	0.0978352	+	0.0182201i
$573$	−10.4095			−0.434861
$574$		0			0
$575$	−36.2873			−1.51328
$576$	−1.00000			−0.0416667
$577$			11.3595i			0.472902i	0.971643	+	0.236451i	$0.0759844\pi$
$577$			11.3595i			0.472902i	−0.971643	+	0.236451i	$0.924016\pi$
$578$			25.8319i			1.07446i
$579$		−	7.69938i		−	0.319975i
$580$		−	1.21140i		−	0.0503007i
$581$		0			0
$582$	−16.1088			−0.667733
$583$		−	1.98037i		−	0.0820185i
$584$	10.0696			0.416682
$585$	−1.20473	−	0.224361i	−0.0498095	−	0.00927617i
$586$	−2.80194			−0.115747
$587$			37.2347i			1.53684i	0.639946	+	0.768420i	$0.278957\pi$
$587$			37.2347i			1.53684i	−0.639946	+	0.768420i	$0.721043\pi$
$588$		0			0
$589$	−24.3569			−1.00361
$590$		−	1.32692i		−	0.0546284i
$591$		−	8.98037i		−	0.369403i
$592$			3.54461i			0.145682i
$593$			24.2110i			0.994227i	0.867685	+	0.497114i	$0.165607\pi$
$593$			24.2110i			0.994227i	−0.867685	+	0.497114i	$0.834393\pi$
$594$	0.660123			0.0270852
$595$		0			0
$596$			8.10884i			0.332151i
$597$	−20.5946			−0.842879
$598$	−26.3332	−	4.90411i	−1.07684	−	0.200544i
$599$	−6.47904			−0.264727			−0.132363	−	0.991201i	$-0.542257\pi$
$599$	−6.47904			−0.264727			−0.132363	+	0.991201i	$0.542257\pi$
$600$		−	4.88448i		−	0.199408i
$601$	41.4354			1.69018			0.845092	−	0.534621i	$-0.179546\pi$
$601$	41.4354			1.69018			0.845092	+	0.534621i	$0.179546\pi$
$602$		0			0
$603$			7.44872i			0.303335i
$604$			5.30062i			0.215679i
$605$		−	3.59054i		−	0.145976i
$606$			2.79527i			0.113550i
$607$	−8.40279			−0.341059			−0.170529	−	0.985353i	$-0.554548\pi$
$607$	−8.40279			−0.341059			−0.170529	+	0.985353i	$0.554548\pi$
$608$	−6.08921			−0.246950
$609$		0			0
$610$	0.633820			0.0256626
$611$	33.4920	+	6.23732i	1.35494	+	0.252335i
$612$	6.54461			0.264550
$613$			46.8974i			1.89417i	0.320983	+	0.947085i	$0.395987\pi$
$613$			46.8974i			1.89417i	−0.320983	+	0.947085i	$0.604013\pi$
$614$	−9.21769			−0.371996
$615$	0.293944			0.0118529
$616$		0			0
$617$			32.6271i			1.31352i	0.754100	+	0.656760i	$0.228073\pi$
$617$			32.6271i			1.31352i	−0.754100	+	0.656760i	$0.771927\pi$
$618$		−	7.53793i		−	0.303220i
$619$		−	5.46168i		−	0.219523i	−0.993958	−	0.109762i	$-0.964991\pi$
$619$		−	5.46168i		−	0.219523i	0.993958	−	0.109762i	$-0.0350088\pi$
$620$	−1.35951			−0.0545991
$621$	−7.42909			−0.298119
$622$		−	11.4684i		−	0.459839i
$623$		0			0
$624$	0.660123	−	3.54461i	0.0264261	−	0.141898i
$625$	23.2806			0.931224
$626$		−	22.5642i		−	0.901848i
$627$	4.01963			0.160529
$628$	23.0562			0.920044
$629$		−	23.1981i		−	0.924967i
$630$		0			0
$631$		−	41.8845i		−	1.66739i	−0.552221	−	0.833697i	$-0.686220\pi$
$631$		−	41.8845i		−	1.66739i	0.552221	−	0.833697i	$-0.313780\pi$
$632$			12.4487i			0.495184i
$633$	15.3898			0.611691
$634$	12.6271			0.501488
$635$		−	4.89517i		−	0.194259i
$636$	−3.00000			−0.118958
$637$		0			0
$638$	−2.35284			−0.0931497
$639$		−	0.884484i		−	0.0349896i
$640$	−0.339877			−0.0134348
$641$	6.61684			0.261350			0.130675	−	0.991425i	$-0.458286\pi$
$641$	6.61684			0.261350			0.130675	+	0.991425i	$0.458286\pi$
$642$		−	3.72971i		−	0.147200i
$643$		−	23.7427i		−	0.936319i	−0.883644	−	0.468160i	$-0.844917\pi$
$643$		−	23.7427i		−	0.936319i	0.883644	−	0.468160i	$-0.155083\pi$
$644$		0			0
$645$			1.72971i			0.0681071i
$646$	39.8515			1.56794
$647$	39.1285			1.53830			0.769150	−	0.639069i	$-0.220680\pi$
$647$	39.1285			1.53830			0.769150	+	0.639069i	$0.220680\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	−2.57720			−0.101164
$650$	17.3136	+	3.22436i	0.679094	+	0.126470i
$651$		0			0
$652$			13.5642i			0.531217i
$653$	5.51429			0.215791			0.107895	−	0.994162i	$-0.465589\pi$
$653$	5.51429			0.215791			0.107895	+	0.994162i	$0.465589\pi$
$654$	−14.4095			−0.563454
$655$			7.02630i			0.274540i
$656$			0.864853i			0.0337668i
$657$			10.0696i			0.392852i
$658$		0			0
$659$	−49.2217			−1.91741			−0.958703	−	0.284410i	$-0.908202\pi$
$659$	−49.2217			−1.91741			−0.958703	+	0.284410i	$0.908202\pi$
$660$	0.224361			0.00873322
$661$		−	12.3243i		−	0.479358i	−0.970852	−	0.239679i	$-0.922958\pi$
$661$		−	12.3243i		−	0.479358i	0.970852	−	0.239679i	$-0.0770423\pi$
$662$	−26.8212			−1.04244
$663$	−4.32025	+	23.1981i	−0.167784	+	0.900938i
$664$	16.2939			0.632328
$665$		0			0
$666$	−3.54461			−0.137351
$667$	26.4790			1.02527
$668$		−	15.7034i		−	0.607583i
$669$		−	18.0196i		−	0.696679i
$670$			2.53165i			0.0978061i
$671$		−	1.23103i		−	0.0475235i
$672$		0			0
$673$	6.79527			0.261938			0.130969	−	0.991386i	$-0.458191\pi$
$673$	6.79527			0.261938			0.130969	+	0.991386i	$0.458191\pi$
$674$			9.81892i			0.378211i
$675$	4.88448			0.188004
$676$	12.1285	+	4.67975i	0.466480	+	0.179991i
$677$	16.6927			0.641553			0.320777	−	0.947155i	$-0.396056\pi$
$677$	16.6927			0.641553			0.320777	+	0.947155i	$0.396056\pi$
$678$			15.5576i			0.597485i
$679$		0			0
$680$	2.22436			0.0853003
$681$			0.231033i			0.00885319i
$682$			2.64049i			0.101110i
$683$		−	21.7164i		−	0.830954i	−0.909604	−	0.415477i	$-0.863615\pi$
$683$		−	21.7164i		−	0.830954i	0.909604	−	0.415477i	$-0.136385\pi$
$684$		−	6.08921i		−	0.232827i
$685$	−4.83453			−0.184718
$686$		0			0
$687$			6.64049i			0.253351i
$688$	−5.08921			−0.194024
$689$	1.98037	−	10.6338i	0.0754461	−	0.405116i
$690$	−2.52498			−0.0961242
$691$			46.9341i			1.78546i	0.450597	+	0.892728i	$0.351211\pi$
$691$			46.9341i			1.78546i	−0.450597	+	0.892728i	$0.648789\pi$
$692$	−13.6035			−0.517127
$693$		0			0
$694$			19.0959i			0.724870i
$695$		−	3.20473i		−	0.121562i
$696$			3.56424i			0.135102i
$697$		−	5.66012i		−	0.214392i
$698$	0.416132			0.0157508
$699$	−29.5183			−1.11648
$700$		0			0
$701$	−7.32251			−0.276568			−0.138284	−	0.990393i	$-0.544159\pi$
$701$	−7.32251			−0.276568			−0.138284	+	0.990393i	$0.544159\pi$
$702$	3.54461	+	0.660123i	0.133783	+	0.0249147i
$703$	−21.5839			−0.814051
$704$			0.660123i			0.0248793i
$705$	3.21140			0.120948
$706$	−16.4880			−0.620533
$707$		0			0
$708$			3.90411i			0.146726i
$709$		−	6.64716i		−	0.249640i	−0.992179	−	0.124820i	$-0.960165\pi$
$709$		−	6.64716i		−	0.249640i	0.992179	−	0.124820i	$-0.0398353\pi$
$710$		−	0.300616i		−	0.0112819i
$711$	−12.4487			−0.466864
$712$	−4.45539			−0.166973
$713$		−	29.7164i		−	1.11289i
$714$		0			0
$715$	−0.148106	+	0.795270i	−0.00553884	+	0.0297414i
$716$	−0.448721			−0.0167695
$717$		−	15.0236i		−	0.561068i
$718$	−24.4724			−0.913301
$719$	−4.87153			−0.181677			−0.0908386	−	0.995866i	$-0.528955\pi$
$719$	−4.87153			−0.181677			−0.0908386	+	0.995866i	$0.528955\pi$
$720$		−	0.339877i		−	0.0126665i
$721$		0			0
$722$		−	18.0785i		−	0.672813i
$723$		−	4.44872i		−	0.165450i
$724$	15.9041			0.591072
$725$	−17.4095			−0.646571
$726$			10.5642i			0.392076i
$727$	−4.98931			−0.185043			−0.0925216	−	0.995711i	$-0.529493\pi$
$727$	−4.98931			−0.185043			−0.0925216	+	0.995711i	$0.529493\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$			3.42242i			0.126669i
$731$	33.3069			1.23190
$732$	−1.86485			−0.0689270
$733$			7.46608i			0.275766i	0.990449	+	0.137883i	$0.0440298\pi$
$733$			7.46608i			0.275766i	−0.990449	+	0.137883i	$0.955970\pi$
$734$		−	12.4880i		−	0.460940i
$735$		0			0
$736$		−	7.42909i		−	0.273840i
$737$	4.91707			0.181123
$738$	−0.864853			−0.0318357
$739$			1.46168i			0.0537688i	0.999639	+	0.0268844i	$0.00855860\pi$
$739$			1.46168i			0.0537688i	−0.999639	+	0.0268844i	$0.991441\pi$
$740$	−1.20473			−0.0442868
$741$	21.5839	+	4.01963i	0.792903	+	0.147665i
$742$		0			0
$743$			5.67975i			0.208370i	0.994558	+	0.104185i	$0.0332234\pi$
$743$			5.67975i			0.208370i	−0.994558	+	0.104185i	$0.966777\pi$
$744$	4.00000			0.146647
$745$	−2.75601			−0.100972
$746$		−	7.82786i		−	0.286598i
$747$			16.2939i			0.596164i
$748$		−	4.32025i		−	0.157964i
$749$		0			0
$750$	3.35951			0.122672
$751$	4.39612			0.160417			0.0802083	−	0.996778i	$-0.474441\pi$
$751$	4.39612			0.160417			0.0802083	+	0.996778i	$0.474441\pi$
$752$			9.44872i			0.344559i
$753$	−24.3961			−0.889044
$754$	−12.6338	−	2.35284i	−0.460097	−	0.0856852i
$755$	−1.80156			−0.0655654
$756$		0			0
$757$	21.0326			0.764442			0.382221	−	0.924071i	$-0.375159\pi$
$757$	21.0326			0.764442			0.382221	+	0.924071i	$0.375159\pi$
$758$	−8.67975			−0.315263
$759$			4.90411i			0.178008i
$760$		−	2.06958i		−	0.0750717i
$761$		−	12.6012i		−	0.456794i	−0.973568	−	0.228397i	$-0.926652\pi$
$761$		−	12.6012i		−	0.456794i	0.973568	−	0.228397i	$-0.0733485\pi$
$762$			14.4028i			0.521758i
$763$		0			0
$764$	10.4095			0.376601
$765$			2.22436i			0.0804219i
$766$	23.4224			0.846286
$767$	−13.8386	−	2.57720i	−0.499681	−	0.0930572i
$768$	1.00000			0.0360844
$769$		−	34.4398i		−	1.24193i	−0.783838	−	0.620965i	$-0.786741\pi$
$769$		−	34.4398i		−	1.24193i	0.783838	−	0.620965i	$-0.213259\pi$
$770$		0			0
$771$	18.4487			0.664414
$772$			7.69938i			0.277107i
$773$			16.7975i			0.604165i	0.953282	+	0.302083i	$0.0976819\pi$
$773$			16.7975i			0.604165i	−0.953282	+	0.302083i	$0.902318\pi$
$774$		−	5.08921i		−	0.182928i
$775$			19.5379i			0.701823i
$776$	16.1088			0.578274
$777$		0			0
$778$		−	30.1914i		−	1.08241i
$779$	−5.26627			−0.188684
$780$	1.20473	+	0.224361i	0.0431363	+	0.00803340i
$781$	−0.583868			−0.0208924
$782$			48.6205i			1.73866i
$783$	−3.56424			−0.127375
$784$		0			0
$785$			7.83628i			0.279689i
$786$		−	20.6731i		−	0.737384i
$787$			36.9474i			1.31703i	0.752567	+	0.658516i	$0.228816\pi$
$787$			36.9474i			1.31703i	−0.752567	+	0.658516i	$0.771184\pi$
$788$			8.98037i			0.319912i
$789$	−6.30062			−0.224308
$790$	−4.23103			−0.150533
$791$		0			0
$792$	−0.660123			−0.0234565
$793$	1.23103	−	6.61017i	0.0437152	−	0.234734i
$794$	−8.49465			−0.301464
$795$		−	1.01963i		−	0.0361626i
$796$	20.5946			0.729955
$797$	38.7819			1.37373			0.686863	−	0.726787i	$-0.258987\pi$
$797$	38.7819			1.37373			0.686863	+	0.726787i	$0.258987\pi$
$798$		0			0
$799$		−	61.8382i		−	2.18768i
$800$			4.88448i			0.172693i
$801$		−	4.45539i		−	0.157424i
$802$	20.1718			0.712289
$803$	6.64716			0.234573
$804$		−	7.44872i		−	0.262696i
$805$		0			0
$806$	−2.64049	+	14.1784i	−0.0930074	+	0.499414i
$807$	−9.89517			−0.348327
$808$		−	2.79527i		−	0.0983373i
$809$	2.12180			0.0745986			0.0372993	−	0.999304i	$-0.488125\pi$
$809$	2.12180			0.0745986			0.0372993	+	0.999304i	$0.488125\pi$
$810$	0.339877			0.0119421
$811$		−	3.66680i		−	0.128759i	−0.997926	−	0.0643793i	$-0.979493\pi$
$811$		−	3.66680i		−	0.128759i	0.997926	−	0.0643793i	$-0.0205067\pi$
$812$		0			0
$813$		−	12.6994i		−	0.445387i
$814$			2.33988i			0.0820126i
$815$	−4.61017			−0.161487
$816$	−6.54461			−0.229107
$817$		−	30.9893i		−	1.08418i
$818$	−4.30062			−0.150367
$819$		0			0
$820$	−0.293944			−0.0102650
$821$		−	21.5076i		−	0.750621i	−0.926899	−	0.375310i	$-0.877536\pi$
$821$		−	21.5076i		−	0.750621i	0.926899	−	0.375310i	$-0.122464\pi$
$822$	14.2244			0.496132
$823$	−35.9541			−1.25328			−0.626640	−	0.779309i	$-0.715570\pi$
$823$	−35.9541			−1.25328			−0.626640	+	0.779309i	$0.715570\pi$
$824$			7.53793i			0.262596i
$825$		−	3.22436i		−	0.112258i
$826$		0			0
$827$			32.8778i			1.14327i	0.820507	+	0.571637i	$0.193691\pi$
$827$			32.8778i			1.14327i	−0.820507	+	0.571637i	$0.806309\pi$
$828$	7.42909			0.258179
$829$	−9.82559			−0.341257			−0.170628	−	0.985335i	$-0.554580\pi$
$829$	−9.82559			−0.341257			−0.170628	+	0.985335i	$0.554580\pi$
$830$			5.53793i			0.192225i
$831$	28.0433			0.972811
$832$	−0.660123	+	3.54461i	−0.0228857	+	0.122887i
$833$		0			0
$834$			9.42909i			0.326503i
$835$	5.33722			0.184702
$836$	−4.01963			−0.139022
$837$			4.00000i			0.138260i
$838$		−	32.8974i		−	1.13642i
$839$			42.6008i			1.47074i	0.677663	+	0.735372i	$0.262993\pi$
$839$			42.6008i			1.47074i	−0.677663	+	0.735372i	$0.737007\pi$
$840$		0			0
$841$	−16.2962			−0.561938
$842$	−21.7230			−0.748625
$843$			1.82157i			0.0627383i
$844$	−15.3898			−0.529740
$845$	−1.59054	+	4.12219i	−0.0547162	+	0.141808i
$846$	−9.44872			−0.324854
$847$		0			0
$848$	3.00000			0.103020
$849$	8.98037			0.308205
$850$		−	31.9670i		−	1.09646i
$851$		−	26.3332i		−	0.902691i
$852$			0.884484i			0.0303019i
$853$		−	47.7490i		−	1.63489i	−0.576005	−	0.817446i	$-0.695389\pi$
$853$		−	47.7490i		−	1.63489i	0.576005	−	0.817446i	$-0.304611\pi$
$854$		0			0
$855$	2.06958			0.0707782
$856$			3.72971i			0.127479i
$857$	−6.89342			−0.235475			−0.117737	−	0.993045i	$-0.537564\pi$
$857$	−6.89342			−0.235475			−0.117737	+	0.993045i	$0.537564\pi$
$858$	0.435763	−	2.33988i	0.0148767	−	0.0798821i
$859$	28.7797			0.981949			0.490975	−	0.871174i	$-0.336641\pi$
$859$	28.7797			0.981949			0.490975	+	0.871174i	$0.336641\pi$
$860$		−	1.72971i		−	0.0589825i
$861$		0			0
$862$	6.06291			0.206504
$863$			48.5745i			1.65350i	0.562572	+	0.826748i	$0.309812\pi$
$863$			48.5745i			1.65350i	−0.562572	+	0.826748i	$0.690188\pi$
$864$			1.00000i			0.0340207i
$865$		−	4.62351i		−	0.157204i
$866$			11.1392i			0.378524i
$867$	25.8319			0.877297
$868$		0			0
$869$			8.21769i			0.278766i
$870$	−1.21140			−0.0410704
$871$	26.4028	+	4.91707i	0.894624	+	0.166609i
$872$	14.4095			0.487966
$873$			16.1088i			0.545202i
$874$	45.2373			1.53018
$875$		0			0
$876$		−	10.0696i		−	0.340220i
$877$		−	9.23505i		−	0.311846i	−0.987769	−	0.155923i	$-0.950165\pi$
$877$		−	9.23505i		−	0.311846i	0.987769	−	0.155923i	$-0.0498351\pi$
$878$			11.6231i			0.392262i
$879$			2.80194i			0.0945072i
$880$	−0.224361			−0.00756319
$881$	−14.9108			−0.502357			−0.251179	−	0.967941i	$-0.580818\pi$
$881$	−14.9108			−0.502357			−0.251179	+	0.967941i	$0.580818\pi$
$882$		0			0
$883$	−37.9171			−1.27601			−0.638006	−	0.770032i	$-0.720240\pi$
$883$	−37.9171			−1.27601			−0.638006	+	0.770032i	$0.720240\pi$
$884$	4.32025	−	23.1981i	0.145306	−	0.780235i
$885$	−1.32692			−0.0446039
$886$		−	4.67308i		−	0.156995i
$887$	11.1285			0.373658			0.186829	−	0.982392i	$-0.440179\pi$
$887$	11.1285			0.373658			0.186829	+	0.982392i	$0.440179\pi$
$888$	3.54461			0.118949
$889$		0			0
$890$		−	1.51429i		−	0.0507590i
$891$		−	0.660123i		−	0.0221150i
$892$			18.0196i			0.603342i
$893$	−57.5353			−1.92534
$894$	8.10884			0.271200
$895$		−	0.152510i		−	0.00509785i
$896$		0			0
$897$	−4.90411	+	26.3332i	−0.163744	+	0.879240i
$898$	36.6271			1.22226
$899$		−	14.2569i		−	0.475496i
$900$	−4.88448			−0.162816
$901$	−19.6338			−0.654097
$902$			0.570909i			0.0190092i
$903$		0			0
$904$		−	15.5576i		−	0.517437i
$905$			5.40544i			0.179683i
$906$	5.30062			0.176101
$907$	−12.1088			−0.402068			−0.201034	−	0.979584i	$-0.564430\pi$
$907$	−12.1088			−0.402068			−0.201034	+	0.979584i	$0.564430\pi$
$908$		−	0.231033i		−	0.00766709i
$909$	2.79527			0.0927133
$910$		0			0
$911$	−46.4354			−1.53847			−0.769236	−	0.638964i	$-0.779363\pi$
$911$	−46.4354			−1.53847			−0.769236	+	0.638964i	$0.779363\pi$
$912$			6.08921i			0.201634i
$913$	10.7560			0.355972
$914$	31.3372			1.03654
$915$		−	0.633820i		−	0.0209535i
$916$		−	6.64049i		−	0.219408i
$917$		0			0
$918$		−	6.54461i		−	0.216004i
$919$	36.7797			1.21325			0.606624	−	0.794989i	$-0.292523\pi$
$919$	36.7797			1.21325			0.606624	+	0.794989i	$0.292523\pi$
$920$	2.52498			0.0832460
$921$			9.21769i			0.303733i
$922$	0.601231			0.0198005
$923$	−3.13515	−	0.583868i	−0.103195	−	0.0192183i
$924$		0			0
$925$			17.3136i			0.569267i
$926$	−10.5013			−0.345095
$927$	−7.53793			−0.247578
$928$		−	3.56424i		−	0.117002i
$929$			4.13917i			0.135802i	0.997692	+	0.0679008i	$0.0216301\pi$
$929$			4.13917i			0.135802i	−0.997692	+	0.0679008i	$0.978370\pi$
$930$			1.35951i			0.0445800i
$931$		0			0
$932$	29.5183			0.966904
$933$	−11.4684			−0.375457
$934$			27.2217i			0.890721i
$935$	1.46835			0.0480202
$936$	−3.54461	−	0.660123i	−0.115859	−	0.0215768i
$937$	45.2939			1.47969			0.739844	−	0.672778i	$-0.234899\pi$
$937$	45.2939			1.47969			0.739844	+	0.672778i	$0.234899\pi$
$938$		0			0
$939$	−22.5642			−0.736356
$940$	−3.21140			−0.104744
$941$			47.8948i			1.56133i	0.624953	+	0.780663i	$0.285118\pi$
$941$			47.8948i			1.56133i	−0.624953	+	0.780663i	$0.714882\pi$
$942$		−	23.0562i		−	0.751213i
$943$		−	6.42507i		−	0.209229i
$944$		−	3.90411i		−	0.127068i
$945$		0			0
$946$	−3.35951			−0.109227
$947$			21.4554i			0.697207i	0.937270	+	0.348603i	$0.113344\pi$
$947$			21.4554i			0.697207i	−0.937270	+	0.348603i	$0.886656\pi$
$948$	12.4487			0.404316
$949$	35.6927	+	6.64716i	1.15863	+	0.215776i
$950$	−29.7427			−0.964979
$951$		−	12.6271i		−	0.409463i
$952$		0			0
$953$	48.0433			1.55627			0.778137	−	0.628094i	$-0.216165\pi$
$953$	48.0433			1.55627			0.778137	+	0.628094i	$0.216165\pi$
$954$			3.00000i			0.0971286i
$955$			3.53793i			0.114485i
$956$			15.0236i			0.485900i
$957$			2.35284i			0.0760564i
$958$	−28.8711			−0.932784
$959$		0			0
$960$			0.339877i			0.0109695i
$961$	15.0000			0.483871
$962$	−2.33988	+	12.5642i	−0.0754407	+	0.405087i
$963$	−3.72971			−0.120188
$964$			4.44872i			0.143284i
$965$	−2.61684			−0.0842392
$966$		0			0
$967$		−	4.91481i		−	0.158049i	−0.996873	−	0.0790247i	$-0.974819\pi$
$967$		−	4.91481i		−	0.158049i	0.996873	−	0.0790247i	$-0.0251806\pi$
$968$		−	10.5642i		−	0.339547i
$969$		−	39.8515i		−	1.28021i
$970$			5.47502i			0.175792i
$971$	30.1584			0.967829			0.483915	−	0.875115i	$-0.339214\pi$
$971$	30.1584			0.967829			0.483915	+	0.875115i	$0.339214\pi$
$972$	−1.00000			−0.0320750
$973$		0			0
$974$	3.91746			0.125523
$975$	3.22436	−	17.3136i	0.103262	−	0.554478i
$976$	1.86485			0.0596925
$977$			34.4095i			1.10086i	0.834883	+	0.550428i	$0.185535\pi$
$977$			34.4095i			1.10086i	−0.834883	+	0.550428i	$0.814465\pi$
$978$	13.5642			0.433736
$979$	−2.94111			−0.0939982
$980$		0			0
$981$			14.4095i			0.460059i
$982$		−	13.4202i		−	0.428254i
$983$		−	19.9500i		−	0.636308i	−0.948039	−	0.318154i	$-0.896937\pi$
$983$		−	19.9500i		−	0.636308i	0.948039	−	0.318154i	$-0.103063\pi$
$984$	0.864853			0.0275705
$985$	−3.05222			−0.0972518
$986$			23.3265i			0.742868i
$987$		0			0
$988$	−21.5839	−	4.01963i	−0.686674	−	0.127881i
$989$	37.8082			1.20223
$990$		−	0.224361i		−	0.00713065i
$991$	−23.8948			−0.759043			−0.379521	−	0.925183i	$-0.623911\pi$
$991$	−23.8948			−0.759043			−0.379521	+	0.925183i	$0.623911\pi$
$992$	−4.00000			−0.127000
$993$			26.8212i			0.851145i
$994$		0			0
$995$			6.99961i			0.221903i
$996$		−	16.2939i		−	0.516293i
$997$	6.14545			0.194628			0.0973142	−	0.995254i	$-0.468975\pi$
$997$	6.14545			0.194628			0.0973142	+	0.995254i	$0.468975\pi$
$998$	32.7819			1.03769
$999$			3.54461i			0.112146i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.c.k.883.5		6
7.2	even	3		546.2.bk.b.25.5	yes	12
7.4	even	3		546.2.bk.b.415.2	yes	12
7.6	odd	2		3822.2.c.j.883.5		6
13.12	even	2	inner	3822.2.c.k.883.2		6
21.2	odd	6		1638.2.dm.c.1117.2		12
21.11	odd	6		1638.2.dm.c.415.5		12
91.25	even	6		546.2.bk.b.415.5	yes	12
91.51	even	6		546.2.bk.b.25.2	✓	12
91.90	odd	2		3822.2.c.j.883.2		6
273.116	odd	6		1638.2.dm.c.415.2		12
273.233	odd	6		1638.2.dm.c.1117.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.bk.b.25.2	✓	12	91.51	even	6
546.2.bk.b.25.5	yes	12	7.2	even	3
546.2.bk.b.415.2	yes	12	7.4	even	3
546.2.bk.b.415.5	yes	12	91.25	even	6
1638.2.dm.c.415.2		12	273.116	odd	6
1638.2.dm.c.415.5		12	21.11	odd	6
1638.2.dm.c.1117.2		12	21.2	odd	6
1638.2.dm.c.1117.5		12	273.233	odd	6
3822.2.c.j.883.2		6	91.90	odd	2
3822.2.c.j.883.5		6	7.6	odd	2
3822.2.c.k.883.2		6	13.12	even	2	inner
3822.2.c.k.883.5		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -0.339877i q^{5} +1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +0.339877 q^{10} -0.660123i q^{11} -1.00000 q^{12} +(0.660123 - 3.54461i) q^{13} -0.339877i q^{15} +1.00000 q^{16} -6.54461 q^{17} +1.00000i q^{18} +6.08921i q^{19} +0.339877i q^{20} +0.660123 q^{22} -7.42909 q^{23} -1.00000i q^{24} +4.88448 q^{25} +(3.54461 + 0.660123i) q^{26} +1.00000 q^{27} -3.56424 q^{29} +0.339877 q^{30} +4.00000i q^{31} +1.00000i q^{32} -0.660123i q^{33} -6.54461i q^{34} -1.00000 q^{36} +3.54461i q^{37} -6.08921 q^{38} +(0.660123 - 3.54461i) q^{39} -0.339877 q^{40} +0.864853i q^{41} -5.08921 q^{43} +0.660123i q^{44} -0.339877i q^{45} -7.42909i q^{46} +9.44872i q^{47} +1.00000 q^{48} +4.88448i q^{50} -6.54461 q^{51} +(-0.660123 + 3.54461i) q^{52} +3.00000 q^{53} +1.00000i q^{54} -0.224361 q^{55} +6.08921i q^{57} -3.56424i q^{58} -3.90411i q^{59} +0.339877i q^{60} +1.86485 q^{61} -4.00000 q^{62} -1.00000 q^{64} +(-1.20473 - 0.224361i) q^{65} +0.660123 q^{66} +7.44872i q^{67} +6.54461 q^{68} -7.42909 q^{69} -0.884484i q^{71} -1.00000i q^{72} +10.0696i q^{73} -3.54461 q^{74} +4.88448 q^{75} -6.08921i q^{76} +(3.54461 + 0.660123i) q^{78} -12.4487 q^{79} -0.339877i q^{80} +1.00000 q^{81} -0.864853 q^{82} +16.2939i q^{83} +2.22436i q^{85} -5.08921i q^{86} -3.56424 q^{87} -0.660123 q^{88} -4.45539i q^{89} +0.339877 q^{90} +7.42909 q^{92} +4.00000i q^{93} -9.44872 q^{94} +2.06958 q^{95} +1.00000i q^{96} +16.1088i q^{97} -0.660123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{12} + 6 q^{13} + 6 q^{16} - 18 q^{17} + 6 q^{22} + 6 q^{25} + 6 q^{27} + 6 q^{29} - 6 q^{36} + 6 q^{38} + 6 q^{39} + 12 q^{43} + 6 q^{48} - 18 q^{51} - 6 q^{52}+ \cdots - 24 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^12 + 6 * q^13 + 6 * q^16 - 18 * q^17 + 6 * q^22 + 6 * q^25 + 6 * q^27 + 6 * q^29 - 6 * q^36 + 6 * q^38 + 6 * q^39 + 12 * q^43 + 6 * q^48 - 18 * q^51 - 6 * q^52 + 18 * q^53 + 24 * q^55 - 6 * q^61 - 24 * q^62 - 6 * q^64 + 12 * q^65 + 6 * q^66 + 18 * q^68 + 6 * q^75 - 24 * q^79 + 6 * q^81 + 12 * q^82 + 6 * q^87 - 6 * q^88 - 6 * q^94 - 24 * q^95

Introduction

L-functions

Modular forms

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Representations

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Database

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