LMFDB - Embedded newform 6762.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$53.9948418468$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.41421 q^{10} +2.00000 q^{11} +1.00000 q^{12} +1.41421 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -5.41421 q^{19} +1.41421 q^{20} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -3.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} -1.41421 q^{30} +0.242641 q^{31} -1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +5.41421 q^{38} -1.41421 q^{40} -4.24264 q^{41} -1.65685 q^{43} +2.00000 q^{44} +1.41421 q^{45} -1.00000 q^{46} -11.0711 q^{47} +1.00000 q^{48} +3.00000 q^{50} -4.00000 q^{51} +13.3137 q^{53} -1.00000 q^{54} +2.82843 q^{55} -5.41421 q^{57} +3.65685 q^{58} -9.65685 q^{59} +1.41421 q^{60} -4.24264 q^{61} -0.242641 q^{62} +1.00000 q^{64} -2.00000 q^{66} -9.65685 q^{67} -4.00000 q^{68} +1.00000 q^{69} -15.6569 q^{71} -1.00000 q^{72} +9.89949 q^{73} -2.00000 q^{74} -3.00000 q^{75} -5.41421 q^{76} -2.34315 q^{79} +1.41421 q^{80} +1.00000 q^{81} +4.24264 q^{82} +3.07107 q^{83} -5.65685 q^{85} +1.65685 q^{86} -3.65685 q^{87} -2.00000 q^{88} -14.8284 q^{89} -1.41421 q^{90} +1.00000 q^{92} +0.242641 q^{93} +11.0711 q^{94} -7.65685 q^{95} -1.00000 q^{96} -9.17157 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 4 q^{22} + 2 q^{23} - 2 q^{24} - 6 q^{25} + 2 q^{27} + 4 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{33} + 8 q^{34} + 2 q^{36} + 4 q^{37} + 8 q^{38} + 8 q^{43} + 4 q^{44} - 2 q^{46} - 8 q^{47} + 2 q^{48} + 6 q^{50} - 8 q^{51} + 4 q^{53} - 2 q^{54} - 8 q^{57} - 4 q^{58} - 8 q^{59} + 8 q^{62} + 2 q^{64} - 4 q^{66} - 8 q^{67} - 8 q^{68} + 2 q^{69} - 20 q^{71} - 2 q^{72} - 4 q^{74} - 6 q^{75} - 8 q^{76} - 16 q^{79} + 2 q^{81} - 8 q^{83} - 8 q^{86} + 4 q^{87} - 4 q^{88} - 24 q^{89} + 2 q^{92} - 8 q^{93} + 8 q^{94} - 4 q^{95} - 2 q^{96} - 24 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 2 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 4 * q^22 + 2 * q^23 - 2 * q^24 - 6 * q^25 + 2 * q^27 + 4 * q^29 - 8 * q^31 - 2 * q^32 + 4 * q^33 + 8 * q^34 + 2 * q^36 + 4 * q^37 + 8 * q^38 + 8 * q^43 + 4 * q^44 - 2 * q^46 - 8 * q^47 + 2 * q^48 + 6 * q^50 - 8 * q^51 + 4 * q^53 - 2 * q^54 - 8 * q^57 - 4 * q^58 - 8 * q^59 + 8 * q^62 + 2 * q^64 - 4 * q^66 - 8 * q^67 - 8 * q^68 + 2 * q^69 - 20 * q^71 - 2 * q^72 - 4 * q^74 - 6 * q^75 - 8 * q^76 - 16 * q^79 + 2 * q^81 - 8 * q^83 - 8 * q^86 + 4 * q^87 - 4 * q^88 - 24 * q^89 + 2 * q^92 - 8 * q^93 + 8 * q^94 - 4 * q^95 - 2 * q^96 - 24 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.41421	−0.447214
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	−5.41421	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$19$	−5.41421	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$20$	1.41421	0.316228
$21$	0	0
$22$	−2.00000	−0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−3.00000	−0.600000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	−1.41421	−0.258199
$31$	0.242641	0.0435796	0.0217898	−	0.999763i	$-0.493064\pi$
$31$	0.242641	0.0435796	0.0217898	+	0.999763i	$0.493064\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	5.41421	0.878301
$39$	0	0
$40$	−1.41421	−0.223607
$41$	−4.24264	−0.662589	−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264	−0.662589	−0.331295	+	0.943527i	$0.607485\pi$
$42$	0	0
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	2.00000	0.301511
$45$	1.41421	0.210819
$46$	−1.00000	−0.147442
$47$	−11.0711	−1.61488	−0.807441	−	0.589949i	$-0.799148\pi$
$47$	−11.0711	−1.61488	−0.807441	+	0.589949i	$0.799148\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	3.00000	0.424264
$51$	−4.00000	−0.560112
$52$	0	0
$53$	13.3137	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$53$	13.3137	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$54$	−1.00000	−0.136083
$55$	2.82843	0.381385
$56$	0	0
$57$	−5.41421	−0.717130
$58$	3.65685	0.480168
$59$	−9.65685	−1.25722	−0.628608	−	0.777723i	$-0.716375\pi$
$59$	−9.65685	−1.25722	−0.628608	+	0.777723i	$0.716375\pi$
$60$	1.41421	0.182574
$61$	−4.24264	−0.543214	−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264	−0.543214	−0.271607	+	0.962408i	$0.587555\pi$
$62$	−0.242641	−0.0308154
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	−4.00000	−0.485071
$69$	1.00000	0.120386
$70$	0	0
$71$	−15.6569	−1.85813	−0.929063	−	0.369921i	$-0.879385\pi$
$71$	−15.6569	−1.85813	−0.929063	+	0.369921i	$0.879385\pi$
$72$	−1.00000	−0.117851
$73$	9.89949	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$73$	9.89949	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$74$	−2.00000	−0.232495
$75$	−3.00000	−0.346410
$76$	−5.41421	−0.621053
$77$	0	0
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	1.41421	0.158114
$81$	1.00000	0.111111
$82$	4.24264	0.468521
$83$	3.07107	0.337093	0.168547	−	0.985694i	$-0.446093\pi$
$83$	3.07107	0.337093	0.168547	+	0.985694i	$0.446093\pi$
$84$	0	0
$85$	−5.65685	−0.613572
$86$	1.65685	0.178663
$87$	−3.65685	−0.392056
$88$	−2.00000	−0.213201
$89$	−14.8284	−1.57181	−0.785905	−	0.618347i	$-0.787803\pi$
$89$	−14.8284	−1.57181	−0.785905	+	0.618347i	$0.787803\pi$
$90$	−1.41421	−0.149071
$91$	0	0
$92$	1.00000	0.104257
$93$	0.242641	0.0251607
$94$	11.0711	1.14189
$95$	−7.65685	−0.785577
$96$	−1.00000	−0.102062
$97$	−9.17157	−0.931232	−0.465616	−	0.884987i	$-0.654167\pi$
$97$	−9.17157	−0.931232	−0.465616	+	0.884987i	$0.654167\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	−3.00000	−0.300000
$101$	16.4853	1.64035	0.820173	−	0.572115i	$-0.193877\pi$
$101$	16.4853	1.64035	0.820173	+	0.572115i	$0.193877\pi$
$102$	4.00000	0.396059
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	−13.3137	−1.29314
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	1.00000	0.0962250
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−2.82843	−0.269680
$111$	2.00000	0.189832
$112$	0	0
$113$	4.34315	0.408569	0.204284	−	0.978912i	$-0.434513\pi$
$113$	4.34315	0.408569	0.204284	+	0.978912i	$0.434513\pi$
$114$	5.41421	0.507088
$115$	1.41421	0.131876
$116$	−3.65685	−0.339530
$117$	0	0
$118$	9.65685	0.888985
$119$	0	0
$120$	−1.41421	−0.129099
$121$	−7.00000	−0.636364
$122$	4.24264	0.384111
$123$	−4.24264	−0.382546
$124$	0.242641	0.0217898
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−11.6569	−1.03438	−0.517189	−	0.855871i	$-0.673022\pi$
$127$	−11.6569	−1.03438	−0.517189	+	0.855871i	$0.673022\pi$
$128$	−1.00000	−0.0883883
$129$	−1.65685	−0.145878
$130$	0	0
$131$	17.6569	1.54269	0.771343	−	0.636419i	$-0.219585\pi$
$131$	17.6569	1.54269	0.771343	+	0.636419i	$0.219585\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	9.65685	0.834225
$135$	1.41421	0.121716
$136$	4.00000	0.342997
$137$	0.343146	0.0293169	0.0146585	−	0.999893i	$-0.495334\pi$
$137$	0.343146	0.0293169	0.0146585	+	0.999893i	$0.495334\pi$
$138$	−1.00000	−0.0851257
$139$	12.4853	1.05899	0.529494	−	0.848314i	$-0.322382\pi$
$139$	12.4853	1.05899	0.529494	+	0.848314i	$0.322382\pi$
$140$	0	0
$141$	−11.0711	−0.932352
$142$	15.6569	1.31389
$143$	0	0
$144$	1.00000	0.0833333
$145$	−5.17157	−0.429476
$146$	−9.89949	−0.819288
$147$	0	0
$148$	2.00000	0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	3.00000	0.244949
$151$	−8.34315	−0.678956	−0.339478	−	0.940614i	$-0.610250\pi$
$151$	−8.34315	−0.678956	−0.339478	+	0.940614i	$0.610250\pi$
$152$	5.41421	0.439151
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0.343146	0.0275621
$156$	0	0
$157$	−7.07107	−0.564333	−0.282166	−	0.959366i	$-0.591053\pi$
$157$	−7.07107	−0.564333	−0.282166	+	0.959366i	$0.591053\pi$
$158$	2.34315	0.186411
$159$	13.3137	1.05585
$160$	−1.41421	−0.111803
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	16.9706	1.32924	0.664619	−	0.747183i	$-0.268594\pi$
$163$	16.9706	1.32924	0.664619	+	0.747183i	$0.268594\pi$
$164$	−4.24264	−0.331295
$165$	2.82843	0.220193
$166$	−3.07107	−0.238361
$167$	0.242641	0.0187761	0.00938805	−	0.999956i	$-0.497012\pi$
$167$	0.242641	0.0187761	0.00938805	+	0.999956i	$0.497012\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	5.65685	0.433861
$171$	−5.41421	−0.414035
$172$	−1.65685	−0.126334
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	3.65685	0.277225
$175$	0	0
$176$	2.00000	0.150756
$177$	−9.65685	−0.725854
$178$	14.8284	1.11144
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	1.41421	0.105409
$181$	−17.8995	−1.33046	−0.665229	−	0.746639i	$-0.731666\pi$
$181$	−17.8995	−1.33046	−0.665229	+	0.746639i	$0.731666\pi$
$182$	0	0
$183$	−4.24264	−0.313625
$184$	−1.00000	−0.0737210
$185$	2.82843	0.207950
$186$	−0.242641	−0.0177913
$187$	−8.00000	−0.585018
$188$	−11.0711	−0.807441
$189$	0	0
$190$	7.65685	0.555487
$191$	27.3137	1.97635	0.988175	−	0.153328i	$-0.0489992\pi$
$191$	27.3137	1.97635	0.988175	+	0.153328i	$0.0489992\pi$
$192$	1.00000	0.0721688
$193$	7.31371	0.526452	0.263226	−	0.964734i	$-0.415213\pi$
$193$	7.31371	0.526452	0.263226	+	0.964734i	$0.415213\pi$
$194$	9.17157	0.658481
$195$	0	0
$196$	0	0
$197$	−17.3137	−1.23355	−0.616775	−	0.787139i	$-0.711561\pi$
$197$	−17.3137	−1.23355	−0.616775	+	0.787139i	$0.711561\pi$
$198$	−2.00000	−0.142134
$199$	−2.82843	−0.200502	−0.100251	−	0.994962i	$-0.531965\pi$
$199$	−2.82843	−0.200502	−0.100251	+	0.994962i	$0.531965\pi$
$200$	3.00000	0.212132
$201$	−9.65685	−0.681142
$202$	−16.4853	−1.15990
$203$	0	0
$204$	−4.00000	−0.280056
$205$	−6.00000	−0.419058
$206$	2.82843	0.197066
$207$	1.00000	0.0695048
$208$	0	0
$209$	−10.8284	−0.749018
$210$	0	0
$211$	4.68629	0.322618	0.161309	−	0.986904i	$-0.448428\pi$
$211$	4.68629	0.322618	0.161309	+	0.986904i	$0.448428\pi$
$212$	13.3137	0.914389
$213$	−15.6569	−1.07279
$214$	−2.00000	−0.136717
$215$	−2.34315	−0.159801
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	8.00000	0.541828
$219$	9.89949	0.668946
$220$	2.82843	0.190693
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−13.4142	−0.898282	−0.449141	−	0.893461i	$-0.648270\pi$
$223$	−13.4142	−0.898282	−0.449141	+	0.893461i	$0.648270\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	−4.34315	−0.288902
$227$	11.0711	0.734813	0.367406	−	0.930061i	$-0.380246\pi$
$227$	11.0711	0.734813	0.367406	+	0.930061i	$0.380246\pi$
$228$	−5.41421	−0.358565
$229$	−10.3848	−0.686245	−0.343123	−	0.939291i	$-0.611485\pi$
$229$	−10.3848	−0.686245	−0.343123	+	0.939291i	$0.611485\pi$
$230$	−1.41421	−0.0932505
$231$	0	0
$232$	3.65685	0.240084
$233$	21.3137	1.39631	0.698154	−	0.715948i	$-0.254005\pi$
$233$	21.3137	1.39631	0.698154	+	0.715948i	$0.254005\pi$
$234$	0	0
$235$	−15.6569	−1.02134
$236$	−9.65685	−0.628608
$237$	−2.34315	−0.152204
$238$	0	0
$239$	−18.9706	−1.22710	−0.613552	−	0.789654i	$-0.710260\pi$
$239$	−18.9706	−1.22710	−0.613552	+	0.789654i	$0.710260\pi$
$240$	1.41421	0.0912871
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	−4.24264	−0.271607
$245$	0	0
$246$	4.24264	0.270501
$247$	0	0
$248$	−0.242641	−0.0154077
$249$	3.07107	0.194621
$250$	11.3137	0.715542
$251$	−11.0711	−0.698800	−0.349400	−	0.936974i	$-0.613614\pi$
$251$	−11.0711	−0.698800	−0.349400	+	0.936974i	$0.613614\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	11.6569	0.731416
$255$	−5.65685	−0.354246
$256$	1.00000	0.0625000
$257$	−0.443651	−0.0276742	−0.0138371	−	0.999904i	$-0.504405\pi$
$257$	−0.443651	−0.0276742	−0.0138371	+	0.999904i	$0.504405\pi$
$258$	1.65685	0.103151
$259$	0	0
$260$	0	0
$261$	−3.65685	−0.226354
$262$	−17.6569	−1.09084
$263$	−4.97056	−0.306498	−0.153249	−	0.988188i	$-0.548974\pi$
$263$	−4.97056	−0.306498	−0.153249	+	0.988188i	$0.548974\pi$
$264$	−2.00000	−0.123091
$265$	18.8284	1.15662
$266$	0	0
$267$	−14.8284	−0.907485
$268$	−9.65685	−0.589886
$269$	14.1421	0.862261	0.431131	−	0.902290i	$-0.358115\pi$
$269$	14.1421	0.862261	0.431131	+	0.902290i	$0.358115\pi$
$270$	−1.41421	−0.0860663
$271$	0.727922	0.0442181	0.0221091	−	0.999756i	$-0.492962\pi$
$271$	0.727922	0.0442181	0.0221091	+	0.999756i	$0.492962\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−0.343146	−0.0207302
$275$	−6.00000	−0.361814
$276$	1.00000	0.0601929
$277$	2.97056	0.178484	0.0892419	−	0.996010i	$-0.471556\pi$
$277$	2.97056	0.178484	0.0892419	+	0.996010i	$0.471556\pi$
$278$	−12.4853	−0.748817
$279$	0.242641	0.0145265
$280$	0	0
$281$	−4.34315	−0.259090	−0.129545	−	0.991574i	$-0.541352\pi$
$281$	−4.34315	−0.259090	−0.129545	+	0.991574i	$0.541352\pi$
$282$	11.0711	0.659272
$283$	−2.58579	−0.153709	−0.0768545	−	0.997042i	$-0.524488\pi$
$283$	−2.58579	−0.153709	−0.0768545	+	0.997042i	$0.524488\pi$
$284$	−15.6569	−0.929063
$285$	−7.65685	−0.453553
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	5.17157	0.303685
$291$	−9.17157	−0.537647
$292$	9.89949	0.579324
$293$	4.72792	0.276208	0.138104	−	0.990418i	$-0.455899\pi$
$293$	4.72792	0.276208	0.138104	+	0.990418i	$0.455899\pi$
$294$	0	0
$295$	−13.6569	−0.795133
$296$	−2.00000	−0.116248
$297$	2.00000	0.116052
$298$	0	0
$299$	0	0
$300$	−3.00000	−0.173205
$301$	0	0
$302$	8.34315	0.480094
$303$	16.4853	0.947055
$304$	−5.41421	−0.310526
$305$	−6.00000	−0.343559
$306$	4.00000	0.228665
$307$	−11.5147	−0.657180	−0.328590	−	0.944473i	$-0.606573\pi$
$307$	−11.5147	−0.657180	−0.328590	+	0.944473i	$0.606573\pi$
$308$	0	0
$309$	−2.82843	−0.160904
$310$	−0.343146	−0.0194894
$311$	−11.5563	−0.655300	−0.327650	−	0.944799i	$-0.606257\pi$
$311$	−11.5563	−0.655300	−0.327650	+	0.944799i	$0.606257\pi$
$312$	0	0
$313$	17.6569	0.998024	0.499012	−	0.866595i	$-0.333696\pi$
$313$	17.6569	0.998024	0.499012	+	0.866595i	$0.333696\pi$
$314$	7.07107	0.399043
$315$	0	0
$316$	−2.34315	−0.131812
$317$	6.97056	0.391506	0.195753	−	0.980653i	$-0.437285\pi$
$317$	6.97056	0.391506	0.195753	+	0.980653i	$0.437285\pi$
$318$	−13.3137	−0.746596
$319$	−7.31371	−0.409489
$320$	1.41421	0.0790569
$321$	2.00000	0.111629
$322$	0	0
$323$	21.6569	1.20502
$324$	1.00000	0.0555556
$325$	0	0
$326$	−16.9706	−0.939913
$327$	−8.00000	−0.442401
$328$	4.24264	0.234261
$329$	0	0
$330$	−2.82843	−0.155700
$331$	−9.65685	−0.530789	−0.265394	−	0.964140i	$-0.585502\pi$
$331$	−9.65685	−0.530789	−0.265394	+	0.964140i	$0.585502\pi$
$332$	3.07107	0.168547
$333$	2.00000	0.109599
$334$	−0.242641	−0.0132767
$335$	−13.6569	−0.746154
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	13.0000	0.707107
$339$	4.34315	0.235887
$340$	−5.65685	−0.306786
$341$	0.485281	0.0262795
$342$	5.41421	0.292767
$343$	0	0
$344$	1.65685	0.0893316
$345$	1.41421	0.0761387
$346$	2.82843	0.152057
$347$	32.9706	1.76995	0.884976	−	0.465636i	$-0.154174\pi$
$347$	32.9706	1.76995	0.884976	+	0.465636i	$0.154174\pi$
$348$	−3.65685	−0.196028
$349$	18.8284	1.00786	0.503931	−	0.863744i	$-0.331886\pi$
$349$	18.8284	1.00786	0.503931	+	0.863744i	$0.331886\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	−23.0711	−1.22795	−0.613975	−	0.789326i	$-0.710430\pi$
$353$	−23.0711	−1.22795	−0.613975	+	0.789326i	$0.710430\pi$
$354$	9.65685	0.513256
$355$	−22.1421	−1.17518
$356$	−14.8284	−0.785905
$357$	0	0
$358$	−23.3137	−1.23217
$359$	12.9706	0.684560	0.342280	−	0.939598i	$-0.388801\pi$
$359$	12.9706	0.684560	0.342280	+	0.939598i	$0.388801\pi$
$360$	−1.41421	−0.0745356
$361$	10.3137	0.542827
$362$	17.8995	0.940777
$363$	−7.00000	−0.367405
$364$	0	0
$365$	14.0000	0.732793
$366$	4.24264	0.221766
$367$	−13.1716	−0.687551	−0.343775	−	0.939052i	$-0.611706\pi$
$367$	−13.1716	−0.687551	−0.343775	+	0.939052i	$0.611706\pi$
$368$	1.00000	0.0521286
$369$	−4.24264	−0.220863
$370$	−2.82843	−0.147043
$371$	0	0
$372$	0.242641	0.0125803
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	8.00000	0.413670
$375$	−11.3137	−0.584237
$376$	11.0711	0.570947
$377$	0	0
$378$	0	0
$379$	−17.3137	−0.889345	−0.444673	−	0.895693i	$-0.646680\pi$
$379$	−17.3137	−0.889345	−0.444673	+	0.895693i	$0.646680\pi$
$380$	−7.65685	−0.392788
$381$	−11.6569	−0.597199
$382$	−27.3137	−1.39749
$383$	−19.3137	−0.986884	−0.493442	−	0.869779i	$-0.664262\pi$
$383$	−19.3137	−0.986884	−0.493442	+	0.869779i	$0.664262\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−7.31371	−0.372258
$387$	−1.65685	−0.0842226
$388$	−9.17157	−0.465616
$389$	19.3137	0.979244	0.489622	−	0.871935i	$-0.337135\pi$
$389$	19.3137	0.979244	0.489622	+	0.871935i	$0.337135\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	17.6569	0.890670
$394$	17.3137	0.872252
$395$	−3.31371	−0.166731
$396$	2.00000	0.100504
$397$	−16.9706	−0.851728	−0.425864	−	0.904787i	$-0.640030\pi$
$397$	−16.9706	−0.851728	−0.425864	+	0.904787i	$0.640030\pi$
$398$	2.82843	0.141776
$399$	0	0
$400$	−3.00000	−0.150000
$401$	5.31371	0.265354	0.132677	−	0.991159i	$-0.457643\pi$
$401$	5.31371	0.265354	0.132677	+	0.991159i	$0.457643\pi$
$402$	9.65685	0.481640
$403$	0	0
$404$	16.4853	0.820173
$405$	1.41421	0.0702728
$406$	0	0
$407$	4.00000	0.198273
$408$	4.00000	0.198030
$409$	20.7279	1.02493	0.512465	−	0.858708i	$-0.328732\pi$
$409$	20.7279	1.02493	0.512465	+	0.858708i	$0.328732\pi$
$410$	6.00000	0.296319
$411$	0.343146	0.0169261
$412$	−2.82843	−0.139347
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	4.34315	0.213197
$416$	0	0
$417$	12.4853	0.611407
$418$	10.8284	0.529636
$419$	0.242641	0.0118538	0.00592689	−	0.999982i	$-0.498113\pi$
$419$	0.242641	0.0118538	0.00592689	+	0.999982i	$0.498113\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−4.68629	−0.228125
$423$	−11.0711	−0.538294
$424$	−13.3137	−0.646571
$425$	12.0000	0.582086
$426$	15.6569	0.758577
$427$	0	0
$428$	2.00000	0.0966736
$429$	0	0
$430$	2.34315	0.112997
$431$	31.3137	1.50833	0.754164	−	0.656686i	$-0.228042\pi$
$431$	31.3137	1.50833	0.754164	+	0.656686i	$0.228042\pi$
$432$	1.00000	0.0481125
$433$	−17.1716	−0.825213	−0.412607	−	0.910909i	$-0.635382\pi$
$433$	−17.1716	−0.825213	−0.412607	+	0.910909i	$0.635382\pi$
$434$	0	0
$435$	−5.17157	−0.247958
$436$	−8.00000	−0.383131
$437$	−5.41421	−0.258997
$438$	−9.89949	−0.473016
$439$	0.727922	0.0347418	0.0173709	−	0.999849i	$-0.494470\pi$
$439$	0.727922	0.0347418	0.0173709	+	0.999849i	$0.494470\pi$
$440$	−2.82843	−0.134840
$441$	0	0
$442$	0	0
$443$	4.68629	0.222652	0.111326	−	0.993784i	$-0.464490\pi$
$443$	4.68629	0.222652	0.111326	+	0.993784i	$0.464490\pi$
$444$	2.00000	0.0949158
$445$	−20.9706	−0.994100
$446$	13.4142	0.635181
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	3.00000	0.141421
$451$	−8.48528	−0.399556
$452$	4.34315	0.204284
$453$	−8.34315	−0.391995
$454$	−11.0711	−0.519591
$455$	0	0
$456$	5.41421	0.253544
$457$	22.9706	1.07452	0.537259	−	0.843417i	$-0.319460\pi$
$457$	22.9706	1.07452	0.537259	+	0.843417i	$0.319460\pi$
$458$	10.3848	0.485249
$459$	−4.00000	−0.186704
$460$	1.41421	0.0659380
$461$	−10.8284	−0.504330	−0.252165	−	0.967684i	$-0.581143\pi$
$461$	−10.8284	−0.504330	−0.252165	+	0.967684i	$0.581143\pi$
$462$	0	0
$463$	11.3137	0.525793	0.262896	−	0.964824i	$-0.415322\pi$
$463$	11.3137	0.525793	0.262896	+	0.964824i	$0.415322\pi$
$464$	−3.65685	−0.169765
$465$	0.343146	0.0159130
$466$	−21.3137	−0.987338
$467$	−7.27208	−0.336512	−0.168256	−	0.985743i	$-0.553813\pi$
$467$	−7.27208	−0.336512	−0.168256	+	0.985743i	$0.553813\pi$
$468$	0	0
$469$	0	0
$470$	15.6569	0.722197
$471$	−7.07107	−0.325818
$472$	9.65685	0.444493
$473$	−3.31371	−0.152364
$474$	2.34315	0.107624
$475$	16.2426	0.745263
$476$	0	0
$477$	13.3137	0.609593
$478$	18.9706	0.867693
$479$	17.4558	0.797578	0.398789	−	0.917043i	$-0.369431\pi$
$479$	17.4558	0.797578	0.398789	+	0.917043i	$0.369431\pi$
$480$	−1.41421	−0.0645497
$481$	0	0
$482$	−12.0000	−0.546585
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−12.9706	−0.588963
$486$	−1.00000	−0.0453609
$487$	−3.31371	−0.150158	−0.0750792	−	0.997178i	$-0.523921\pi$
$487$	−3.31371	−0.150158	−0.0750792	+	0.997178i	$0.523921\pi$
$488$	4.24264	0.192055
$489$	16.9706	0.767435
$490$	0	0
$491$	37.9411	1.71226	0.856130	−	0.516761i	$-0.172863\pi$
$491$	37.9411	1.71226	0.856130	+	0.516761i	$0.172863\pi$
$492$	−4.24264	−0.191273
$493$	14.6274	0.658786
$494$	0	0
$495$	2.82843	0.127128
$496$	0.242641	0.0108949
$497$	0	0
$498$	−3.07107	−0.137618
$499$	11.3137	0.506471	0.253236	−	0.967405i	$-0.418505\pi$
$499$	11.3137	0.506471	0.253236	+	0.967405i	$0.418505\pi$
$500$	−11.3137	−0.505964
$501$	0.242641	0.0108404
$502$	11.0711	0.494126
$503$	−24.9706	−1.11338	−0.556691	−	0.830720i	$-0.687929\pi$
$503$	−24.9706	−1.11338	−0.556691	+	0.830720i	$0.687929\pi$
$504$	0	0
$505$	23.3137	1.03745
$506$	−2.00000	−0.0889108
$507$	−13.0000	−0.577350
$508$	−11.6569	−0.517189
$509$	−0.485281	−0.0215097	−0.0107549	−	0.999942i	$-0.503423\pi$
$509$	−0.485281	−0.0215097	−0.0107549	+	0.999942i	$0.503423\pi$
$510$	5.65685	0.250490
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−5.41421	−0.239043
$514$	0.443651	0.0195686
$515$	−4.00000	−0.176261
$516$	−1.65685	−0.0729389
$517$	−22.1421	−0.973810
$518$	0	0
$519$	−2.82843	−0.124154
$520$	0	0
$521$	−1.17157	−0.0513275	−0.0256638	−	0.999671i	$-0.508170\pi$
$521$	−1.17157	−0.0513275	−0.0256638	+	0.999671i	$0.508170\pi$
$522$	3.65685	0.160056
$523$	−38.8701	−1.69967	−0.849834	−	0.527050i	$-0.823298\pi$
$523$	−38.8701	−1.69967	−0.849834	+	0.527050i	$0.823298\pi$
$524$	17.6569	0.771343
$525$	0	0
$526$	4.97056	0.216727
$527$	−0.970563	−0.0422784
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	−18.8284	−0.817855
$531$	−9.65685	−0.419072
$532$	0	0
$533$	0	0
$534$	14.8284	0.641689
$535$	2.82843	0.122284
$536$	9.65685	0.417113
$537$	23.3137	1.00606
$538$	−14.1421	−0.609711
$539$	0	0
$540$	1.41421	0.0608581
$541$	0.343146	0.0147530	0.00737649	−	0.999973i	$-0.497652\pi$
$541$	0.343146	0.0147530	0.00737649	+	0.999973i	$0.497652\pi$
$542$	−0.727922	−0.0312669
$543$	−17.8995	−0.768141
$544$	4.00000	0.171499
$545$	−11.3137	−0.484626
$546$	0	0
$547$	−24.9706	−1.06766	−0.533832	−	0.845591i	$-0.679249\pi$
$547$	−24.9706	−1.06766	−0.533832	+	0.845591i	$0.679249\pi$
$548$	0.343146	0.0146585
$549$	−4.24264	−0.181071
$550$	6.00000	0.255841
$551$	19.7990	0.843465
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−2.97056	−0.126207
$555$	2.82843	0.120060
$556$	12.4853	0.529494
$557$	−4.00000	−0.169485	−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000	−0.169485	−0.0847427	+	0.996403i	$0.527007\pi$
$558$	−0.242641	−0.0102718
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	4.34315	0.183205
$563$	−40.2426	−1.69603	−0.848013	−	0.529976i	$-0.822201\pi$
$563$	−40.2426	−1.69603	−0.848013	+	0.529976i	$0.822201\pi$
$564$	−11.0711	−0.466176
$565$	6.14214	0.258402
$566$	2.58579	0.108689
$567$	0	0
$568$	15.6569	0.656947
$569$	21.3137	0.893517	0.446759	−	0.894655i	$-0.352578\pi$
$569$	21.3137	0.893517	0.446759	+	0.894655i	$0.352578\pi$
$570$	7.65685	0.320710
$571$	17.6569	0.738916	0.369458	−	0.929247i	$-0.379543\pi$
$571$	17.6569	0.738916	0.369458	+	0.929247i	$0.379543\pi$
$572$	0	0
$573$	27.3137	1.14105
$574$	0	0
$575$	−3.00000	−0.125109
$576$	1.00000	0.0416667
$577$	−42.8701	−1.78470	−0.892352	−	0.451340i	$-0.850946\pi$
$577$	−42.8701	−1.78470	−0.892352	+	0.451340i	$0.850946\pi$
$578$	1.00000	0.0415945
$579$	7.31371	0.303947
$580$	−5.17157	−0.214738
$581$	0	0
$582$	9.17157	0.380174
$583$	26.6274	1.10279
$584$	−9.89949	−0.409644
$585$	0	0
$586$	−4.72792	−0.195309
$587$	0.201010	0.00829658	0.00414829	−	0.999991i	$-0.498680\pi$
$587$	0.201010	0.00829658	0.00414829	+	0.999991i	$0.498680\pi$
$588$	0	0
$589$	−1.31371	−0.0541304
$590$	13.6569	0.562244
$591$	−17.3137	−0.712191
$592$	2.00000	0.0821995
$593$	7.07107	0.290374	0.145187	−	0.989404i	$-0.453622\pi$
$593$	7.07107	0.290374	0.145187	+	0.989404i	$0.453622\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	0	0
$597$	−2.82843	−0.115760
$598$	0	0
$599$	4.68629	0.191477	0.0957383	−	0.995407i	$-0.469479\pi$
$599$	4.68629	0.191477	0.0957383	+	0.995407i	$0.469479\pi$
$600$	3.00000	0.122474
$601$	18.3848	0.749931	0.374965	−	0.927039i	$-0.377655\pi$
$601$	18.3848	0.749931	0.374965	+	0.927039i	$0.377655\pi$
$602$	0	0
$603$	−9.65685	−0.393258
$604$	−8.34315	−0.339478
$605$	−9.89949	−0.402472
$606$	−16.4853	−0.669669
$607$	−35.0711	−1.42349	−0.711745	−	0.702438i	$-0.752095\pi$
$607$	−35.0711	−1.42349	−0.711745	+	0.702438i	$0.752095\pi$
$608$	5.41421	0.219575
$609$	0	0
$610$	6.00000	0.242933
$611$	0	0
$612$	−4.00000	−0.161690
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	11.5147	0.464696
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−29.3137	−1.18013	−0.590063	−	0.807357i	$-0.700897\pi$
$617$	−29.3137	−1.18013	−0.590063	+	0.807357i	$0.700897\pi$
$618$	2.82843	0.113776
$619$	−24.7279	−0.993899	−0.496950	−	0.867779i	$-0.665547\pi$
$619$	−24.7279	−0.993899	−0.496950	+	0.867779i	$0.665547\pi$
$620$	0.343146	0.0137811
$621$	1.00000	0.0401286
$622$	11.5563	0.463367
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	−17.6569	−0.705710
$627$	−10.8284	−0.432446
$628$	−7.07107	−0.282166
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−20.9706	−0.834825	−0.417412	−	0.908717i	$-0.637063\pi$
$631$	−20.9706	−0.834825	−0.417412	+	0.908717i	$0.637063\pi$
$632$	2.34315	0.0932053
$633$	4.68629	0.186263
$634$	−6.97056	−0.276836
$635$	−16.4853	−0.654198
$636$	13.3137	0.527923
$637$	0	0
$638$	7.31371	0.289552
$639$	−15.6569	−0.619376
$640$	−1.41421	−0.0559017
$641$	−0.627417	−0.0247815	−0.0123907	−	0.999923i	$-0.503944\pi$
$641$	−0.627417	−0.0247815	−0.0123907	+	0.999923i	$0.503944\pi$
$642$	−2.00000	−0.0789337
$643$	−48.7279	−1.92164	−0.960821	−	0.277170i	$-0.910603\pi$
$643$	−48.7279	−1.92164	−0.960821	+	0.277170i	$0.910603\pi$
$644$	0	0
$645$	−2.34315	−0.0922613
$646$	−21.6569	−0.852078
$647$	7.75736	0.304973	0.152487	−	0.988306i	$-0.451272\pi$
$647$	7.75736	0.304973	0.152487	+	0.988306i	$0.451272\pi$
$648$	−1.00000	−0.0392837
$649$	−19.3137	−0.758129
$650$	0	0
$651$	0	0
$652$	16.9706	0.664619
$653$	−43.2548	−1.69269	−0.846346	−	0.532633i	$-0.821203\pi$
$653$	−43.2548	−1.69269	−0.846346	+	0.532633i	$0.821203\pi$
$654$	8.00000	0.312825
$655$	24.9706	0.975681
$656$	−4.24264	−0.165647
$657$	9.89949	0.386216
$658$	0	0
$659$	25.6569	0.999449	0.499725	−	0.866184i	$-0.333435\pi$
$659$	25.6569	0.999449	0.499725	+	0.866184i	$0.333435\pi$
$660$	2.82843	0.110096
$661$	48.5269	1.88748	0.943739	−	0.330691i	$-0.107282\pi$
$661$	48.5269	1.88748	0.943739	+	0.330691i	$0.107282\pi$
$662$	9.65685	0.375324
$663$	0	0
$664$	−3.07107	−0.119181
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−3.65685	−0.141594
$668$	0.242641	0.00938805
$669$	−13.4142	−0.518623
$670$	13.6569	0.527610
$671$	−8.48528	−0.327571
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	14.0000	0.539260
$675$	−3.00000	−0.115470
$676$	−13.0000	−0.500000
$677$	30.1838	1.16006	0.580028	−	0.814596i	$-0.303042\pi$
$677$	30.1838	1.16006	0.580028	+	0.814596i	$0.303042\pi$
$678$	−4.34315	−0.166798
$679$	0	0
$680$	5.65685	0.216930
$681$	11.0711	0.424244
$682$	−0.485281	−0.0185824
$683$	−25.9411	−0.992610	−0.496305	−	0.868148i	$-0.665310\pi$
$683$	−25.9411	−0.992610	−0.496305	+	0.868148i	$0.665310\pi$
$684$	−5.41421	−0.207018
$685$	0.485281	0.0185416
$686$	0	0
$687$	−10.3848	−0.396204
$688$	−1.65685	−0.0631670
$689$	0	0
$690$	−1.41421	−0.0538382
$691$	−26.1421	−0.994494	−0.497247	−	0.867609i	$-0.665656\pi$
$691$	−26.1421	−0.994494	−0.497247	+	0.867609i	$0.665656\pi$
$692$	−2.82843	−0.107521
$693$	0	0
$694$	−32.9706	−1.25155
$695$	17.6569	0.669763
$696$	3.65685	0.138613
$697$	16.9706	0.642806
$698$	−18.8284	−0.712666
$699$	21.3137	0.806158
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−10.8284	−0.408402
$704$	2.00000	0.0753778
$705$	−15.6569	−0.589671
$706$	23.0711	0.868291
$707$	0	0
$708$	−9.65685	−0.362927
$709$	−24.6274	−0.924902	−0.462451	−	0.886645i	$-0.653030\pi$
$709$	−24.6274	−0.924902	−0.462451	+	0.886645i	$0.653030\pi$
$710$	22.1421	0.830980
$711$	−2.34315	−0.0878748
$712$	14.8284	0.555719
$713$	0.242641	0.00908697
$714$	0	0
$715$	0	0
$716$	23.3137	0.871274
$717$	−18.9706	−0.708469
$718$	−12.9706	−0.484057
$719$	−7.27208	−0.271203	−0.135601	−	0.990763i	$-0.543297\pi$
$719$	−7.27208	−0.271203	−0.135601	+	0.990763i	$0.543297\pi$
$720$	1.41421	0.0527046
$721$	0	0
$722$	−10.3137	−0.383836
$723$	12.0000	0.446285
$724$	−17.8995	−0.665229
$725$	10.9706	0.407436
$726$	7.00000	0.259794
$727$	28.7696	1.06700	0.533502	−	0.845799i	$-0.320876\pi$
$727$	28.7696	1.06700	0.533502	+	0.845799i	$0.320876\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−14.0000	−0.518163
$731$	6.62742	0.245124
$732$	−4.24264	−0.156813
$733$	−18.3848	−0.679057	−0.339529	−	0.940596i	$-0.610268\pi$
$733$	−18.3848	−0.679057	−0.339529	+	0.940596i	$0.610268\pi$
$734$	13.1716	0.486172
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−19.3137	−0.711430
$738$	4.24264	0.156174
$739$	−33.6569	−1.23809	−0.619044	−	0.785357i	$-0.712480\pi$
$739$	−33.6569	−1.23809	−0.619044	+	0.785357i	$0.712480\pi$
$740$	2.82843	0.103975
$741$	0	0
$742$	0	0
$743$	−5.65685	−0.207530	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685	−0.207530	−0.103765	+	0.994602i	$0.533089\pi$
$744$	−0.242641	−0.00889564
$745$	0	0
$746$	8.00000	0.292901
$747$	3.07107	0.112364
$748$	−8.00000	−0.292509
$749$	0	0
$750$	11.3137	0.413118
$751$	−22.3431	−0.815313	−0.407656	−	0.913135i	$-0.633654\pi$
$751$	−22.3431	−0.815313	−0.407656	+	0.913135i	$0.633654\pi$
$752$	−11.0711	−0.403720
$753$	−11.0711	−0.403452
$754$	0	0
$755$	−11.7990	−0.429409
$756$	0	0
$757$	53.2548	1.93558	0.967790	−	0.251759i	$-0.0810091\pi$
$757$	53.2548	1.93558	0.967790	+	0.251759i	$0.0810091\pi$
$758$	17.3137	0.628862
$759$	2.00000	0.0725954
$760$	7.65685	0.277743
$761$	−13.6152	−0.493551	−0.246776	−	0.969073i	$-0.579371\pi$
$761$	−13.6152	−0.493551	−0.246776	+	0.969073i	$0.579371\pi$
$762$	11.6569	0.422283
$763$	0	0
$764$	27.3137	0.988175
$765$	−5.65685	−0.204524
$766$	19.3137	0.697833
$767$	0	0
$768$	1.00000	0.0360844
$769$	−17.6569	−0.636722	−0.318361	−	0.947969i	$-0.603132\pi$
$769$	−17.6569	−0.636722	−0.318361	+	0.947969i	$0.603132\pi$
$770$	0	0
$771$	−0.443651	−0.0159777
$772$	7.31371	0.263226
$773$	29.2132	1.05073	0.525363	−	0.850878i	$-0.323930\pi$
$773$	29.2132	1.05073	0.525363	+	0.850878i	$0.323930\pi$
$774$	1.65685	0.0595544
$775$	−0.727922	−0.0261477
$776$	9.17157	0.329240
$777$	0	0
$778$	−19.3137	−0.692430
$779$	22.9706	0.823006
$780$	0	0
$781$	−31.3137	−1.12049
$782$	4.00000	0.143040
$783$	−3.65685	−0.130685
$784$	0	0
$785$	−10.0000	−0.356915
$786$	−17.6569	−0.629799
$787$	−35.0711	−1.25015	−0.625074	−	0.780565i	$-0.714931\pi$
$787$	−35.0711	−1.25015	−0.625074	+	0.780565i	$0.714931\pi$
$788$	−17.3137	−0.616775
$789$	−4.97056	−0.176957
$790$	3.31371	0.117896
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	0	0
$794$	16.9706	0.602263
$795$	18.8284	0.667775
$796$	−2.82843	−0.100251
$797$	−9.89949	−0.350658	−0.175329	−	0.984510i	$-0.556099\pi$
$797$	−9.89949	−0.350658	−0.175329	+	0.984510i	$0.556099\pi$
$798$	0	0
$799$	44.2843	1.56666
$800$	3.00000	0.106066
$801$	−14.8284	−0.523937
$802$	−5.31371	−0.187634
$803$	19.7990	0.698691
$804$	−9.65685	−0.340571
$805$	0	0
$806$	0	0
$807$	14.1421	0.497827
$808$	−16.4853	−0.579950
$809$	18.6274	0.654905	0.327453	−	0.944868i	$-0.393810\pi$
$809$	18.6274	0.654905	0.327453	+	0.944868i	$0.393810\pi$
$810$	−1.41421	−0.0496904
$811$	36.9706	1.29821	0.649106	−	0.760698i	$-0.275143\pi$
$811$	36.9706	1.29821	0.649106	+	0.760698i	$0.275143\pi$
$812$	0	0
$813$	0.727922	0.0255293
$814$	−4.00000	−0.140200
$815$	24.0000	0.840683
$816$	−4.00000	−0.140028
$817$	8.97056	0.313840
$818$	−20.7279	−0.724735
$819$	0	0
$820$	−6.00000	−0.209529
$821$	0.343146	0.0119759	0.00598793	−	0.999982i	$-0.498094\pi$
$821$	0.343146	0.0119759	0.00598793	+	0.999982i	$0.498094\pi$
$822$	−0.343146	−0.0119686
$823$	−13.0294	−0.454178	−0.227089	−	0.973874i	$-0.572921\pi$
$823$	−13.0294	−0.454178	−0.227089	+	0.973874i	$0.572921\pi$
$824$	2.82843	0.0985329
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−0.627417	−0.0218174	−0.0109087	−	0.999940i	$-0.503472\pi$
$827$	−0.627417	−0.0218174	−0.0109087	+	0.999940i	$0.503472\pi$
$828$	1.00000	0.0347524
$829$	−29.6569	−1.03003	−0.515013	−	0.857183i	$-0.672213\pi$
$829$	−29.6569	−1.03003	−0.515013	+	0.857183i	$0.672213\pi$
$830$	−4.34315	−0.150753
$831$	2.97056	0.103048
$832$	0	0
$833$	0	0
$834$	−12.4853	−0.432330
$835$	0.343146	0.0118750
$836$	−10.8284	−0.374509
$837$	0.242641	0.00838689
$838$	−0.242641	−0.00838188
$839$	−5.65685	−0.195296	−0.0976481	−	0.995221i	$-0.531132\pi$
$839$	−5.65685	−0.195296	−0.0976481	+	0.995221i	$0.531132\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	−22.0000	−0.758170
$843$	−4.34315	−0.149586
$844$	4.68629	0.161309
$845$	−18.3848	−0.632456
$846$	11.0711	0.380631
$847$	0	0
$848$	13.3137	0.457195
$849$	−2.58579	−0.0887440
$850$	−12.0000	−0.411597
$851$	2.00000	0.0685591
$852$	−15.6569	−0.536395
$853$	7.02944	0.240683	0.120342	−	0.992733i	$-0.461601\pi$
$853$	7.02944	0.240683	0.120342	+	0.992733i	$0.461601\pi$
$854$	0	0
$855$	−7.65685	−0.261859
$856$	−2.00000	−0.0683586
$857$	−49.8995	−1.70453	−0.852267	−	0.523107i	$-0.824773\pi$
$857$	−49.8995	−1.70453	−0.852267	+	0.523107i	$0.824773\pi$
$858$	0	0
$859$	−48.7696	−1.66400	−0.831998	−	0.554779i	$-0.812803\pi$
$859$	−48.7696	−1.66400	−0.831998	+	0.554779i	$0.812803\pi$
$860$	−2.34315	−0.0799006
$861$	0	0
$862$	−31.3137	−1.06655
$863$	−50.9706	−1.73506	−0.867529	−	0.497386i	$-0.834293\pi$
$863$	−50.9706	−1.73506	−0.867529	+	0.497386i	$0.834293\pi$
$864$	−1.00000	−0.0340207
$865$	−4.00000	−0.136004
$866$	17.1716	0.583514
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−4.68629	−0.158972
$870$	5.17157	0.175333
$871$	0	0
$872$	8.00000	0.270914
$873$	−9.17157	−0.310411
$874$	5.41421	0.183139
$875$	0	0
$876$	9.89949	0.334473
$877$	−34.9706	−1.18087	−0.590436	−	0.807084i	$-0.701044\pi$
$877$	−34.9706	−1.18087	−0.590436	+	0.807084i	$0.701044\pi$
$878$	−0.727922	−0.0245662
$879$	4.72792	0.159469
$880$	2.82843	0.0953463
$881$	−37.9411	−1.27827	−0.639134	−	0.769095i	$-0.720707\pi$
$881$	−37.9411	−1.27827	−0.639134	+	0.769095i	$0.720707\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	−13.6569	−0.459070
$886$	−4.68629	−0.157439
$887$	15.7574	0.529080	0.264540	−	0.964375i	$-0.414780\pi$
$887$	15.7574	0.529080	0.264540	+	0.964375i	$0.414780\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	20.9706	0.702935
$891$	2.00000	0.0670025
$892$	−13.4142	−0.449141
$893$	59.9411	2.00585
$894$	0	0
$895$	32.9706	1.10208
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.887302	−0.0295932
$900$	−3.00000	−0.100000
$901$	−53.2548	−1.77418
$902$	8.48528	0.282529
$903$	0	0
$904$	−4.34315	−0.144451
$905$	−25.3137	−0.841456
$906$	8.34315	0.277182
$907$	0.627417	0.0208330	0.0104165	−	0.999946i	$-0.496684\pi$
$907$	0.627417	0.0208330	0.0104165	+	0.999946i	$0.496684\pi$
$908$	11.0711	0.367406
$909$	16.4853	0.546782
$910$	0	0
$911$	−48.2843	−1.59973	−0.799865	−	0.600180i	$-0.795095\pi$
$911$	−48.2843	−1.59973	−0.799865	+	0.600180i	$0.795095\pi$
$912$	−5.41421	−0.179283
$913$	6.14214	0.203275
$914$	−22.9706	−0.759799
$915$	−6.00000	−0.198354
$916$	−10.3848	−0.343123
$917$	0	0
$918$	4.00000	0.132020
$919$	−50.9117	−1.67942	−0.839711	−	0.543034i	$-0.817276\pi$
$919$	−50.9117	−1.67942	−0.839711	+	0.543034i	$0.817276\pi$
$920$	−1.41421	−0.0466252
$921$	−11.5147	−0.379423
$922$	10.8284	0.356615
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−11.3137	−0.371792
$927$	−2.82843	−0.0928977
$928$	3.65685	0.120042
$929$	16.4437	0.539499	0.269749	−	0.962931i	$-0.413059\pi$
$929$	16.4437	0.539499	0.269749	+	0.962931i	$0.413059\pi$
$930$	−0.343146	−0.0112522
$931$	0	0
$932$	21.3137	0.698154
$933$	−11.5563	−0.378338
$934$	7.27208	0.237950
$935$	−11.3137	−0.369998
$936$	0	0
$937$	49.6569	1.62222	0.811109	−	0.584895i	$-0.198864\pi$
$937$	49.6569	1.62222	0.811109	+	0.584895i	$0.198864\pi$
$938$	0	0
$939$	17.6569	0.576210
$940$	−15.6569	−0.510670
$941$	8.92893	0.291075	0.145537	−	0.989353i	$-0.453509\pi$
$941$	8.92893	0.291075	0.145537	+	0.989353i	$0.453509\pi$
$942$	7.07107	0.230388
$943$	−4.24264	−0.138159
$944$	−9.65685	−0.314304
$945$	0	0
$946$	3.31371	0.107738
$947$	32.2843	1.04910	0.524549	−	0.851380i	$-0.324234\pi$
$947$	32.2843	1.04910	0.524549	+	0.851380i	$0.324234\pi$
$948$	−2.34315	−0.0761018
$949$	0	0
$950$	−16.2426	−0.526981
$951$	6.97056	0.226036
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−13.3137	−0.431047
$955$	38.6274	1.24995
$956$	−18.9706	−0.613552
$957$	−7.31371	−0.236419
$958$	−17.4558	−0.563973
$959$	0	0
$960$	1.41421	0.0456435
$961$	−30.9411	−0.998101
$962$	0	0
$963$	2.00000	0.0644491
$964$	12.0000	0.386494
$965$	10.3431	0.332958
$966$	0	0
$967$	−30.6274	−0.984911	−0.492456	−	0.870338i	$-0.663901\pi$
$967$	−30.6274	−0.984911	−0.492456	+	0.870338i	$0.663901\pi$
$968$	7.00000	0.224989
$969$	21.6569	0.695718
$970$	12.9706	0.416460
$971$	−4.44365	−0.142604	−0.0713018	−	0.997455i	$-0.522715\pi$
$971$	−4.44365	−0.142604	−0.0713018	+	0.997455i	$0.522715\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	3.31371	0.106178
$975$	0	0
$976$	−4.24264	−0.135804
$977$	33.5980	1.07489	0.537447	−	0.843297i	$-0.319389\pi$
$977$	33.5980	1.07489	0.537447	+	0.843297i	$0.319389\pi$
$978$	−16.9706	−0.542659
$979$	−29.6569	−0.947837
$980$	0	0
$981$	−8.00000	−0.255420
$982$	−37.9411	−1.21075
$983$	21.6569	0.690746	0.345373	−	0.938465i	$-0.387752\pi$
$983$	21.6569	0.690746	0.345373	+	0.938465i	$0.387752\pi$
$984$	4.24264	0.135250
$985$	−24.4853	−0.780166
$986$	−14.6274	−0.465832
$987$	0	0
$988$	0	0
$989$	−1.65685	−0.0526849
$990$	−2.82843	−0.0898933
$991$	8.34315	0.265029	0.132514	−	0.991181i	$-0.457695\pi$
$991$	8.34315	0.265029	0.132514	+	0.991181i	$0.457695\pi$
$992$	−0.242641	−0.00770385
$993$	−9.65685	−0.306451
$994$	0	0
$995$	−4.00000	−0.126809
$996$	3.07107	0.0973105
$997$	−8.00000	−0.253363	−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000	−0.253363	−0.126681	+	0.991943i	$0.540433\pi$
$998$	−11.3137	−0.358129
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.by.1.2	yes	2
7.6	odd	2		6762.2.a.bp.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6762.2.a.bp.1.1	✓	2	7.6	odd	2
6762.2.a.by.1.2	yes	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.41421 q^{10} +2.00000 q^{11} +1.00000 q^{12} +1.41421 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -5.41421 q^{19} +1.41421 q^{20} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -3.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} -1.41421 q^{30} +0.242641 q^{31} -1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +5.41421 q^{38} -1.41421 q^{40} -4.24264 q^{41} -1.65685 q^{43} +2.00000 q^{44} +1.41421 q^{45} -1.00000 q^{46} -11.0711 q^{47} +1.00000 q^{48} +3.00000 q^{50} -4.00000 q^{51} +13.3137 q^{53} -1.00000 q^{54} +2.82843 q^{55} -5.41421 q^{57} +3.65685 q^{58} -9.65685 q^{59} +1.41421 q^{60} -4.24264 q^{61} -0.242641 q^{62} +1.00000 q^{64} -2.00000 q^{66} -9.65685 q^{67} -4.00000 q^{68} +1.00000 q^{69} -15.6569 q^{71} -1.00000 q^{72} +9.89949 q^{73} -2.00000 q^{74} -3.00000 q^{75} -5.41421 q^{76} -2.34315 q^{79} +1.41421 q^{80} +1.00000 q^{81} +4.24264 q^{82} +3.07107 q^{83} -5.65685 q^{85} +1.65685 q^{86} -3.65685 q^{87} -2.00000 q^{88} -14.8284 q^{89} -1.41421 q^{90} +1.00000 q^{92} +0.242641 q^{93} +11.0711 q^{94} -7.65685 q^{95} -1.00000 q^{96} -9.17157 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 4 q^{22} + 2 q^{23} - 2 q^{24} - 6 q^{25} + 2 q^{27} + 4 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{33} + 8 q^{34} + 2 q^{36} + 4 q^{37} + 8 q^{38} + 8 q^{43} + 4 q^{44} - 2 q^{46} - 8 q^{47} + 2 q^{48} + 6 q^{50} - 8 q^{51} + 4 q^{53} - 2 q^{54} - 8 q^{57} - 4 q^{58} - 8 q^{59} + 8 q^{62} + 2 q^{64} - 4 q^{66} - 8 q^{67} - 8 q^{68} + 2 q^{69} - 20 q^{71} - 2 q^{72} - 4 q^{74} - 6 q^{75} - 8 q^{76} - 16 q^{79} + 2 q^{81} - 8 q^{83} - 8 q^{86} + 4 q^{87} - 4 q^{88} - 24 q^{89} + 2 q^{92} - 8 q^{93} + 8 q^{94} - 4 q^{95} - 2 q^{96} - 24 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 2 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 4 * q^22 + 2 * q^23 - 2 * q^24 - 6 * q^25 + 2 * q^27 + 4 * q^29 - 8 * q^31 - 2 * q^32 + 4 * q^33 + 8 * q^34 + 2 * q^36 + 4 * q^37 + 8 * q^38 + 8 * q^43 + 4 * q^44 - 2 * q^46 - 8 * q^47 + 2 * q^48 + 6 * q^50 - 8 * q^51 + 4 * q^53 - 2 * q^54 - 8 * q^57 - 4 * q^58 - 8 * q^59 + 8 * q^62 + 2 * q^64 - 4 * q^66 - 8 * q^67 - 8 * q^68 + 2 * q^69 - 20 * q^71 - 2 * q^72 - 4 * q^74 - 6 * q^75 - 8 * q^76 - 16 * q^79 + 2 * q^81 - 8 * q^83 - 8 * q^86 + 4 * q^87 - 4 * q^88 - 24 * q^89 + 2 * q^92 - 8 * q^93 + 8 * q^94 - 4 * q^95 - 2 * q^96 - 24 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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