LMFDB - Embedded newform 4004.2.m.c.2157.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("4004.2157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.m (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.12
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08723 q^{3} -3.34996i q^{5} -1.00000i q^{7} -1.81792 q^{9} +O(q^{10})$	$q-1.08723 q^{3} -3.34996i q^{5} -1.00000i q^{7} -1.81792 q^{9} +1.00000i q^{11} +(-3.52800 - 0.743786i) q^{13} +3.64219i q^{15} +4.31834 q^{17} +7.13566i q^{19} +1.08723i q^{21} +1.49079 q^{23} -6.22222 q^{25} +5.23821 q^{27} +2.80792 q^{29} +6.91308i q^{31} -1.08723i q^{33} -3.34996 q^{35} -3.56372i q^{37} +(3.83576 + 0.808669i) q^{39} +8.88418i q^{41} -6.06590 q^{43} +6.08996i q^{45} +4.61283i q^{47} -1.00000 q^{49} -4.69505 q^{51} +12.9549 q^{53} +3.34996 q^{55} -7.75813i q^{57} -9.89042i q^{59} +3.97309 q^{61} +1.81792i q^{63} +(-2.49165 + 11.8187i) q^{65} +12.7654i q^{67} -1.62084 q^{69} +6.59867i q^{71} -12.0334i q^{73} +6.76501 q^{75} +1.00000 q^{77} +9.43515 q^{79} -0.241394 q^{81} -0.795337i q^{83} -14.4663i q^{85} -3.05286 q^{87} +1.02531i q^{89} +(-0.743786 + 3.52800i) q^{91} -7.51614i q^{93} +23.9041 q^{95} -12.9734i q^{97} -1.81792i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) $ 36 * q - 4 * q^3 + 40 * q^9	$ 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) $ 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.08723			−0.627715			−0.313857	−	0.949470i	$-0.601621\pi$
$3$	−1.08723			−0.627715			−0.313857	+	0.949470i	$0.601621\pi$
$4$		0			0
$5$		−	3.34996i		−	1.49815i	−0.662487	−	0.749073i	$-0.730499\pi$
$5$		−	3.34996i		−	1.49815i	0.662487	−	0.749073i	$-0.269501\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.81792			−0.605974
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−3.52800	−	0.743786i	−0.978491	−	0.206289i
$13$	−3.52800	−	0.743786i	−0.978491	−	0.206289i
$14$		0			0
$15$			3.64219i			0.940409i
$16$		0			0
$17$	4.31834			1.04735			0.523676	−	0.851918i	$-0.324560\pi$
$17$	4.31834			1.04735			0.523676	+	0.851918i	$0.324560\pi$
$18$		0			0
$19$			7.13566i			1.63703i	0.574484	+	0.818516i	$0.305203\pi$
$19$			7.13566i			1.63703i	−0.574484	+	0.818516i	$0.694797\pi$
$20$		0			0
$21$			1.08723i			0.237254i
$22$		0			0
$23$	1.49079			0.310851			0.155426	−	0.987848i	$-0.450325\pi$
$23$	1.49079			0.310851			0.155426	+	0.987848i	$0.450325\pi$
$24$		0			0
$25$	−6.22222			−1.24444
$26$		0			0
$27$	5.23821			1.00809
$28$		0			0
$29$	2.80792			0.521417			0.260708	−	0.965418i	$-0.416044\pi$
$29$	2.80792			0.521417			0.260708	+	0.965418i	$0.416044\pi$
$30$		0			0
$31$			6.91308i			1.24163i	0.783959	+	0.620813i	$0.213197\pi$
$31$			6.91308i			1.24163i	−0.783959	+	0.620813i	$0.786803\pi$
$32$		0			0
$33$		−	1.08723i		−	0.189263i
$34$		0			0
$35$	−3.34996			−0.566246
$36$		0			0
$37$		−	3.56372i		−	0.585873i	−0.956132	−	0.292936i	$-0.905368\pi$
$37$		−	3.56372i		−	0.585873i	0.956132	−	0.292936i	$-0.0946324\pi$
$38$		0			0
$39$	3.83576	+	0.808669i	0.614213	+	0.129491i
$40$		0			0
$41$			8.88418i			1.38748i	0.720227	+	0.693738i	$0.244037\pi$
$41$			8.88418i			1.38748i	−0.720227	+	0.693738i	$0.755963\pi$
$42$		0			0
$43$	−6.06590			−0.925040			−0.462520	−	0.886609i	$-0.653055\pi$
$43$	−6.06590			−0.925040			−0.462520	+	0.886609i	$0.653055\pi$
$44$		0			0
$45$			6.08996i			0.907838i
$46$		0			0
$47$			4.61283i			0.672850i	0.941710	+	0.336425i	$0.109218\pi$
$47$			4.61283i			0.672850i	−0.941710	+	0.336425i	$0.890782\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−4.69505			−0.657438
$52$		0			0
$53$	12.9549			1.77950			0.889749	−	0.456451i	$-0.150879\pi$
$53$	12.9549			1.77950			0.889749	+	0.456451i	$0.150879\pi$
$54$		0			0
$55$	3.34996			0.451708
$56$		0			0
$57$		−	7.75813i		−	1.02759i
$58$		0			0
$59$		−	9.89042i		−	1.28762i	−0.765184	−	0.643811i	$-0.777352\pi$
$59$		−	9.89042i		−	1.28762i	0.765184	−	0.643811i	$-0.222648\pi$
$60$		0			0
$61$	3.97309			0.508701			0.254351	−	0.967112i	$-0.418138\pi$
$61$	3.97309			0.508701			0.254351	+	0.967112i	$0.418138\pi$
$62$		0			0
$63$			1.81792i			0.229037i
$64$		0			0
$65$	−2.49165	+	11.8187i	−0.309051	+	1.46592i
$66$		0			0
$67$			12.7654i			1.55954i	0.626065	+	0.779771i	$0.284664\pi$
$67$			12.7654i			1.55954i	−0.626065	+	0.779771i	$0.715336\pi$
$68$		0			0
$69$	−1.62084			−0.195126
$70$		0			0
$71$			6.59867i			0.783118i	0.920153	+	0.391559i	$0.128064\pi$
$71$			6.59867i			0.783118i	−0.920153	+	0.391559i	$0.871936\pi$
$72$		0			0
$73$		−	12.0334i		−	1.40841i	−0.709998	−	0.704204i	$-0.751304\pi$
$73$		−	12.0334i		−	1.40841i	0.709998	−	0.704204i	$-0.248696\pi$
$74$		0			0
$75$	6.76501			0.781156
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	9.43515			1.06154			0.530769	−	0.847517i	$-0.321903\pi$
$79$	9.43515			1.06154			0.530769	+	0.847517i	$0.321903\pi$
$80$		0			0
$81$	−0.241394			−0.0268215
$82$		0			0
$83$		−	0.795337i		−	0.0872995i	−0.999047	−	0.0436498i	$-0.986101\pi$
$83$		−	0.795337i		−	0.0872995i	0.999047	−	0.0436498i	$-0.0138986\pi$
$84$		0			0
$85$		−	14.4663i		−	1.56909i
$86$		0			0
$87$	−3.05286			−0.327301
$88$		0			0
$89$			1.02531i			0.108683i	0.998522	+	0.0543415i	$0.0173060\pi$
$89$			1.02531i			0.108683i	−0.998522	+	0.0543415i	$0.982694\pi$
$90$		0			0
$91$	−0.743786	+	3.52800i	−0.0779700	+	0.369835i
$92$		0			0
$93$		−	7.51614i		−	0.779387i
$94$		0			0
$95$	23.9041			2.45251
$96$		0			0
$97$		−	12.9734i		−	1.31725i	−0.752472	−	0.658625i	$-0.771139\pi$
$97$		−	12.9734i		−	1.31725i	0.752472	−	0.658625i	$-0.228861\pi$
$98$		0			0
$99$		−	1.81792i		−	0.182708i
$100$		0			0
$101$	2.21979			0.220877			0.110439	−	0.993883i	$-0.464774\pi$
$101$	2.21979			0.220877			0.110439	+	0.993883i	$0.464774\pi$
$102$		0			0
$103$	−8.76051			−0.863199			−0.431600	−	0.902065i	$-0.642051\pi$
$103$	−8.76051			−0.863199			−0.431600	+	0.902065i	$0.642051\pi$
$104$		0			0
$105$	3.64219			0.355441
$106$		0			0
$107$	5.48980			0.530719			0.265360	−	0.964150i	$-0.414509\pi$
$107$	5.48980			0.530719			0.265360	+	0.964150i	$0.414509\pi$
$108$		0			0
$109$		−	4.16288i		−	0.398731i	−0.979925	−	0.199366i	$-0.936112\pi$
$109$		−	4.16288i		−	0.398731i	0.979925	−	0.199366i	$-0.0638882\pi$
$110$		0			0
$111$			3.87460i			0.367761i
$112$		0			0
$113$	−11.4716			−1.07916			−0.539578	−	0.841936i	$-0.681416\pi$
$113$	−11.4716			−1.07916			−0.539578	+	0.841936i	$0.681416\pi$
$114$		0			0
$115$		−	4.99408i		−	0.465701i
$116$		0			0
$117$	6.41363	+	1.35214i	0.592940	+	0.125006i
$118$		0			0
$119$		−	4.31834i		−	0.395862i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	9.65918i		−	0.870939i
$124$		0			0
$125$			4.09438i			0.366212i
$126$		0			0
$127$	−4.72215			−0.419023			−0.209511	−	0.977806i	$-0.567187\pi$
$127$	−4.72215			−0.419023			−0.209511	+	0.977806i	$0.567187\pi$
$128$		0			0
$129$	6.59505			0.580662
$130$		0			0
$131$	1.01648			0.0888100			0.0444050	−	0.999014i	$-0.485861\pi$
$131$	1.01648			0.0888100			0.0444050	+	0.999014i	$0.485861\pi$
$132$		0			0
$133$	7.13566			0.618740
$134$		0			0
$135$		−	17.5478i		−	1.51027i
$136$		0			0
$137$		−	2.51956i		−	0.215261i	−0.994191	−	0.107630i	$-0.965674\pi$
$137$		−	2.51956i		−	0.215261i	0.994191	−	0.107630i	$-0.0343263\pi$
$138$		0			0
$139$	9.39901			0.797213			0.398607	−	0.917122i	$-0.369494\pi$
$139$	9.39901			0.797213			0.398607	+	0.917122i	$0.369494\pi$
$140$		0			0
$141$		−	5.01523i		−	0.422358i
$142$		0			0
$143$	0.743786	−	3.52800i	0.0621985	−	0.295026i
$144$		0			0
$145$		−	9.40640i		−	0.781159i
$146$		0			0
$147$	1.08723			0.0896736
$148$		0			0
$149$			8.40917i			0.688906i	0.938804	+	0.344453i	$0.111936\pi$
$149$			8.40917i			0.688906i	−0.938804	+	0.344453i	$0.888064\pi$
$150$		0			0
$151$			4.61043i			0.375192i	0.982246	+	0.187596i	$0.0600695\pi$
$151$			4.61043i			0.375192i	−0.982246	+	0.187596i	$0.939930\pi$
$152$		0			0
$153$	−7.85041			−0.634668
$154$		0			0
$155$	23.1585			1.86014
$156$		0			0
$157$	−17.0879			−1.36376			−0.681880	−	0.731464i	$-0.738838\pi$
$157$	−17.0879			−1.36376			−0.681880	+	0.731464i	$0.738838\pi$
$158$		0			0
$159$	−14.0850			−1.11702
$160$		0			0
$161$		−	1.49079i		−	0.117491i
$162$		0			0
$163$		−	22.8787i		−	1.79200i	−0.444057	−	0.895999i	$-0.646461\pi$
$163$		−	22.8787i		−	1.79200i	0.444057	−	0.895999i	$-0.353539\pi$
$164$		0			0
$165$	−3.64219			−0.283544
$166$		0			0
$167$		−	17.2131i		−	1.33199i	−0.745957	−	0.665994i	$-0.768008\pi$
$167$		−	17.2131i		−	1.33199i	0.745957	−	0.665994i	$-0.231992\pi$
$168$		0			0
$169$	11.8936	+	5.24815i	0.914890	+	0.403704i
$170$		0			0
$171$		−	12.9721i		−	0.991999i
$172$		0			0
$173$	−11.9899			−0.911575			−0.455788	−	0.890089i	$-0.650642\pi$
$173$	−11.9899			−0.911575			−0.455788	+	0.890089i	$0.650642\pi$
$174$		0			0
$175$			6.22222i			0.470355i
$176$		0			0
$177$			10.7532i			0.808260i
$178$		0			0
$179$	19.8321			1.48232			0.741162	−	0.671327i	$-0.234275\pi$
$179$	19.8321			1.48232			0.741162	+	0.671327i	$0.234275\pi$
$180$		0			0
$181$	−19.6548			−1.46093			−0.730464	−	0.682951i	$-0.760696\pi$
$181$	−19.6548			−1.46093			−0.730464	+	0.682951i	$0.760696\pi$
$182$		0			0
$183$	−4.31967			−0.319319
$184$		0			0
$185$	−11.9383			−0.877723
$186$		0			0
$187$			4.31834i			0.315788i
$188$		0			0
$189$		−	5.23821i		−	0.381024i
$190$		0			0
$191$	13.7595			0.995601			0.497800	−	0.867292i	$-0.334141\pi$
$191$	13.7595			0.995601			0.497800	+	0.867292i	$0.334141\pi$
$192$		0			0
$193$			16.3099i			1.17401i	0.809582	+	0.587007i	$0.199694\pi$
$193$			16.3099i			1.17401i	−0.809582	+	0.587007i	$0.800306\pi$
$194$		0			0
$195$	2.70901	−	12.8496i	0.193996	−	0.920182i
$196$		0			0
$197$			18.3513i			1.30748i	0.756719	+	0.653740i	$0.226801\pi$
$197$			18.3513i			1.30748i	−0.756719	+	0.653740i	$0.773199\pi$
$198$		0			0
$199$	20.4478			1.44950			0.724752	−	0.689010i	$-0.241954\pi$
$199$	20.4478			1.44950			0.724752	+	0.689010i	$0.241954\pi$
$200$		0			0
$201$		−	13.8790i		−	0.978948i
$202$		0			0
$203$		−	2.80792i		−	0.197077i
$204$		0			0
$205$	29.7616			2.07864
$206$		0			0
$207$	−2.71014			−0.188368
$208$		0			0
$209$	−7.13566			−0.493584
$210$		0			0
$211$	13.8352			0.952454			0.476227	−	0.879322i	$-0.342004\pi$
$211$	13.8352			0.952454			0.476227	+	0.879322i	$0.342004\pi$
$212$		0			0
$213$		−	7.17430i		−	0.491575i
$214$		0			0
$215$			20.3205i			1.38585i
$216$		0			0
$217$	6.91308			0.469291
$218$		0			0
$219$			13.0832i			0.884079i
$220$		0			0
$221$	−15.2351	−	3.21192i	−1.02482	−	0.216057i
$222$		0			0
$223$			11.0590i			0.740568i	0.928919	+	0.370284i	$0.120740\pi$
$223$			11.0590i			0.740568i	−0.928919	+	0.370284i	$0.879260\pi$
$224$		0			0
$225$	11.3115			0.754100
$226$		0			0
$227$		−	16.2606i		−	1.07925i	−0.841904	−	0.539627i	$-0.818565\pi$
$227$		−	16.2606i		−	1.07925i	0.841904	−	0.539627i	$-0.181435\pi$
$228$		0			0
$229$			17.6808i			1.16838i	0.811617	+	0.584189i	$0.198587\pi$
$229$			17.6808i			1.16838i	−0.811617	+	0.584189i	$0.801413\pi$
$230$		0			0
$231$	−1.08723			−0.0715348
$232$		0			0
$233$	17.8756			1.17107			0.585535	−	0.810647i	$-0.300885\pi$
$233$	17.8756			1.17107			0.585535	+	0.810647i	$0.300885\pi$
$234$		0			0
$235$	15.4528			1.00803
$236$		0			0
$237$	−10.2582			−0.666343
$238$		0			0
$239$			10.3242i			0.667819i	0.942605	+	0.333909i	$0.108368\pi$
$239$			10.3242i			0.667819i	−0.942605	+	0.333909i	$0.891632\pi$
$240$		0			0
$241$			0.774848i			0.0499124i	0.999689	+	0.0249562i	$0.00794462\pi$
$241$			0.774848i			0.0499124i	−0.999689	+	0.0249562i	$0.992055\pi$
$242$		0			0
$243$	−15.4522			−0.991258
$244$		0			0
$245$			3.34996i			0.214021i
$246$		0			0
$247$	5.30740	−	25.1746i	0.337702	−	1.60182i
$248$		0			0
$249$			0.864717i			0.0547992i
$250$		0			0
$251$	−5.64264			−0.356161			−0.178080	−	0.984016i	$-0.556989\pi$
$251$	−5.64264			−0.356161			−0.178080	+	0.984016i	$0.556989\pi$
$252$		0			0
$253$			1.49079i			0.0937251i
$254$		0			0
$255$			15.7282i			0.984939i
$256$		0			0
$257$	−11.7966			−0.735853			−0.367926	−	0.929855i	$-0.619932\pi$
$257$	−11.7966			−0.735853			−0.367926	+	0.929855i	$0.619932\pi$
$258$		0			0
$259$	−3.56372			−0.221439
$260$		0			0
$261$	−5.10457			−0.315965
$262$		0			0
$263$	3.17206			0.195598			0.0977988	−	0.995206i	$-0.468820\pi$
$263$	3.17206			0.195598			0.0977988	+	0.995206i	$0.468820\pi$
$264$		0			0
$265$		−	43.3985i		−	2.66595i
$266$		0			0
$267$		−	1.11476i		−	0.0682220i
$268$		0			0
$269$	10.9042			0.664839			0.332420	−	0.943132i	$-0.392135\pi$
$269$	10.9042			0.664839			0.332420	+	0.943132i	$0.392135\pi$
$270$		0			0
$271$			11.9374i			0.725147i	0.931955	+	0.362574i	$0.118102\pi$
$271$			11.9374i			0.725147i	−0.931955	+	0.362574i	$0.881898\pi$
$272$		0			0
$273$	0.808669	−	3.83576i	0.0489429	−	0.232151i
$274$		0			0
$275$		−	6.22222i		−	0.375214i
$276$		0			0
$277$	25.8415			1.55267			0.776334	−	0.630322i	$-0.217077\pi$
$277$	25.8415			1.55267			0.776334	+	0.630322i	$0.217077\pi$
$278$		0			0
$279$		−	12.5674i		−	0.752393i
$280$		0			0
$281$		−	8.45300i		−	0.504264i	−0.967693	−	0.252132i	$-0.918868\pi$
$281$		−	8.45300i		−	0.504264i	0.967693	−	0.252132i	$-0.0811317\pi$
$282$		0			0
$283$	29.9613			1.78102			0.890508	−	0.454968i	$-0.150349\pi$
$283$	29.9613			1.78102			0.890508	+	0.454968i	$0.150349\pi$
$284$		0			0
$285$	−25.9894			−1.53948
$286$		0			0
$287$	8.88418			0.524417
$288$		0			0
$289$	1.64806			0.0969449
$290$		0			0
$291$			14.1051i			0.826857i
$292$		0			0
$293$			4.86174i			0.284026i	0.989865	+	0.142013i	$0.0453574\pi$
$293$			4.86174i			0.284026i	−0.989865	+	0.142013i	$0.954643\pi$
$294$		0			0
$295$	−33.1325			−1.92905
$296$		0			0
$297$			5.23821i			0.303952i
$298$		0			0
$299$	−5.25950	−	1.10883i	−0.304165	−	0.0641252i
$300$		0			0
$301$			6.06590i			0.349632i
$302$		0			0
$303$	−2.41343			−0.138648
$304$		0			0
$305$		−	13.3097i		−	0.762109i
$306$		0			0
$307$		−	33.5920i		−	1.91719i	−0.284765	−	0.958597i	$-0.591916\pi$
$307$		−	33.5920i		−	1.91719i	0.284765	−	0.958597i	$-0.408084\pi$
$308$		0			0
$309$	9.52473			0.541843
$310$		0			0
$311$	−0.887922			−0.0503494			−0.0251747	−	0.999683i	$-0.508014\pi$
$311$	−0.887922			−0.0503494			−0.0251747	+	0.999683i	$0.508014\pi$
$312$		0			0
$313$	25.4993			1.44131			0.720654	−	0.693295i	$-0.243842\pi$
$313$	25.4993			1.44131			0.720654	+	0.693295i	$0.243842\pi$
$314$		0			0
$315$	6.08996			0.343130
$316$		0			0
$317$		−	16.0285i		−	0.900252i	−0.892965	−	0.450126i	$-0.851379\pi$
$317$		−	16.0285i		−	0.900252i	0.892965	−	0.450126i	$-0.148621\pi$
$318$		0			0
$319$			2.80792i			0.157213i
$320$		0			0
$321$	−5.96870			−0.333140
$322$		0			0
$323$			30.8142i			1.71455i
$324$		0			0
$325$	21.9520	+	4.62800i	1.21768	+	0.256715i
$326$		0			0
$327$			4.52602i			0.250290i
$328$		0			0
$329$	4.61283			0.254314
$330$		0			0
$331$			26.4189i			1.45211i	0.687634	+	0.726057i	$0.258649\pi$
$331$			26.4189i			1.45211i	−0.687634	+	0.726057i	$0.741351\pi$
$332$		0			0
$333$			6.47857i			0.355024i
$334$		0			0
$335$	42.7635			2.33642
$336$		0			0
$337$	5.97919			0.325707			0.162854	−	0.986650i	$-0.447930\pi$
$337$	5.97919			0.325707			0.162854	+	0.986650i	$0.447930\pi$
$338$		0			0
$339$	12.4723			0.677402
$340$		0			0
$341$	−6.91308			−0.374364
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			5.42974i			0.292327i
$346$		0			0
$347$	15.6757			0.841517			0.420758	−	0.907173i	$-0.361764\pi$
$347$	15.6757			0.841517			0.420758	+	0.907173i	$0.361764\pi$
$348$		0			0
$349$			12.5860i			0.673711i	0.941556	+	0.336855i	$0.109363\pi$
$349$			12.5860i			0.673711i	−0.941556	+	0.336855i	$0.890637\pi$
$350$		0			0
$351$	−18.4804	−	3.89611i	−0.986411	−	0.207959i
$352$		0			0
$353$			19.0812i			1.01559i	0.861478	+	0.507794i	$0.169539\pi$
$353$			19.0812i			1.01559i	−0.861478	+	0.507794i	$0.830461\pi$
$354$		0			0
$355$	22.1053			1.17323
$356$		0			0
$357$			4.69505i			0.248488i
$358$		0			0
$359$			18.2818i			0.964879i	0.875929	+	0.482439i	$0.160249\pi$
$359$			18.2818i			0.964879i	−0.875929	+	0.482439i	$0.839751\pi$
$360$		0			0
$361$	−31.9176			−1.67987
$362$		0			0
$363$	1.08723			0.0570650
$364$		0			0
$365$	−40.3115			−2.11000
$366$		0			0
$367$	33.7377			1.76109			0.880547	−	0.473959i	$-0.157175\pi$
$367$	33.7377			1.76109			0.880547	+	0.473959i	$0.157175\pi$
$368$		0			0
$369$		−	16.1507i		−	0.840774i
$370$		0			0
$371$		−	12.9549i		−	0.672587i
$372$		0			0
$373$	31.7608			1.64451			0.822256	−	0.569118i	$-0.192715\pi$
$373$	31.7608			1.64451			0.822256	+	0.569118i	$0.192715\pi$
$374$		0			0
$375$		−	4.45155i		−	0.229877i
$376$		0			0
$377$	−9.90633	−	2.08849i	−0.510202	−	0.107563i
$378$		0			0
$379$			7.77706i			0.399481i	0.979849	+	0.199740i	$0.0640099\pi$
$379$			7.77706i			0.399481i	−0.979849	+	0.199740i	$0.935990\pi$
$380$		0			0
$381$	5.13408			0.263027
$382$		0			0
$383$			20.9064i			1.06827i	0.845400	+	0.534134i	$0.179362\pi$
$383$			20.9064i			1.06827i	−0.845400	+	0.534134i	$0.820638\pi$
$384$		0			0
$385$		−	3.34996i		−	0.170730i
$386$		0			0
$387$	11.0273			0.560550
$388$		0			0
$389$	28.0133			1.42033			0.710166	−	0.704034i	$-0.248620\pi$
$389$	28.0133			1.42033			0.710166	+	0.704034i	$0.248620\pi$
$390$		0			0
$391$	6.43774			0.325570
$392$		0			0
$393$	−1.10515			−0.0557474
$394$		0			0
$395$		−	31.6074i		−	1.59034i
$396$		0			0
$397$			16.5116i			0.828694i	0.910119	+	0.414347i	$0.135990\pi$
$397$			16.5116i			0.828694i	−0.910119	+	0.414347i	$0.864010\pi$
$398$		0			0
$399$	−7.75813			−0.388392
$400$		0			0
$401$			22.1648i			1.10686i	0.832896	+	0.553430i	$0.186681\pi$
$401$			22.1648i			1.10686i	−0.832896	+	0.553430i	$0.813319\pi$
$402$		0			0
$403$	5.14185	−	24.3894i	0.256134	−	1.21492i
$404$		0			0
$405$			0.808658i			0.0401826i
$406$		0			0
$407$	3.56372			0.176647
$408$		0			0
$409$		−	0.356646i		−	0.0176350i	−0.999961	−	0.00881751i	$-0.997193\pi$
$409$		−	0.356646i		−	0.0176350i	0.999961	−	0.00881751i	$-0.00280674\pi$
$410$		0			0
$411$			2.73935i			0.135122i
$412$		0			0
$413$	−9.89042			−0.486676
$414$		0			0
$415$	−2.66434			−0.130788
$416$		0			0
$417$	−10.2189			−0.500423
$418$		0			0
$419$	33.5854			1.64076			0.820378	−	0.571821i	$-0.193763\pi$
$419$	33.5854			1.64076			0.820378	+	0.571821i	$0.193763\pi$
$420$		0			0
$421$			1.02335i			0.0498752i	0.999689	+	0.0249376i	$0.00793870\pi$
$421$			1.02335i			0.0498752i	−0.999689	+	0.0249376i	$0.992061\pi$
$422$		0			0
$423$		−	8.38576i		−	0.407730i
$424$		0			0
$425$	−26.8697			−1.30337
$426$		0			0
$427$		−	3.97309i		−	0.192271i
$428$		0			0
$429$	−0.808669	+	3.83576i	−0.0390429	+	0.185192i
$430$		0			0
$431$			17.8561i			0.860100i	0.902805	+	0.430050i	$0.141504\pi$
$431$			17.8561i			0.860100i	−0.902805	+	0.430050i	$0.858496\pi$
$432$		0			0
$433$	−17.2041			−0.826775			−0.413388	−	0.910555i	$-0.635655\pi$
$433$	−17.2041			−0.826775			−0.413388	+	0.910555i	$0.635655\pi$
$434$		0			0
$435$			10.2270i			0.490345i
$436$		0			0
$437$			10.6378i			0.508873i
$438$		0			0
$439$	−15.3419			−0.732229			−0.366114	−	0.930570i	$-0.619312\pi$
$439$	−15.3419			−0.732229			−0.366114	+	0.930570i	$0.619312\pi$
$440$		0			0
$441$	1.81792			0.0865677
$442$		0			0
$443$	−35.0634			−1.66591			−0.832955	−	0.553340i	$-0.813353\pi$
$443$	−35.0634			−1.66591			−0.832955	+	0.553340i	$0.813353\pi$
$444$		0			0
$445$	3.43476			0.162823
$446$		0			0
$447$		−	9.14273i		−	0.432436i
$448$		0			0
$449$		−	6.70661i		−	0.316504i	−0.987399	−	0.158252i	$-0.949414\pi$
$449$		−	6.70661i		−	0.316504i	0.987399	−	0.158252i	$-0.0505859\pi$
$450$		0			0
$451$	−8.88418			−0.418340
$452$		0			0
$453$		−	5.01262i		−	0.235513i
$454$		0			0
$455$	11.8187	+	2.49165i	0.554067	+	0.116810i
$456$		0			0
$457$			37.6775i			1.76248i	0.472668	+	0.881240i	$0.343291\pi$
$457$			37.6775i			1.76248i	−0.472668	+	0.881240i	$0.656709\pi$
$458$		0			0
$459$	22.6204			1.05583
$460$		0			0
$461$		−	40.6991i		−	1.89555i	−0.318947	−	0.947773i	$-0.603329\pi$
$461$		−	40.6991i		−	1.89555i	0.318947	−	0.947773i	$-0.396671\pi$
$462$		0			0
$463$		−	2.98402i		−	0.138679i	−0.997593	−	0.0693396i	$-0.977911\pi$
$463$		−	2.98402i		−	0.138679i	0.997593	−	0.0693396i	$-0.0220892\pi$
$464$		0			0
$465$	−25.1787			−1.16764
$466$		0			0
$467$	−26.4650			−1.22465			−0.612327	−	0.790605i	$-0.709766\pi$
$467$	−26.4650			−1.22465			−0.612327	+	0.790605i	$0.709766\pi$
$468$		0			0
$469$	12.7654			0.589451
$470$		0			0
$471$	18.5785			0.856053
$472$		0			0
$473$		−	6.06590i		−	0.278910i
$474$		0			0
$475$		−	44.3996i		−	2.03719i
$476$		0			0
$477$	−23.5511			−1.07833
$478$		0			0
$479$			11.4762i			0.524361i	0.965019	+	0.262180i	$0.0844416\pi$
$479$			11.4762i			0.524361i	−0.965019	+	0.262180i	$0.915558\pi$
$480$		0			0
$481$	−2.65065	+	12.5728i	−0.120859	+	0.573271i
$482$		0			0
$483$			1.62084i			0.0737506i
$484$		0			0
$485$	−43.4603			−1.97343
$486$		0			0
$487$			18.7141i			0.848018i	0.905658	+	0.424009i	$0.139378\pi$
$487$			18.7141i			0.848018i	−0.905658	+	0.424009i	$0.860622\pi$
$488$		0			0
$489$			24.8745i			1.12486i
$490$		0			0
$491$	−9.55959			−0.431418			−0.215709	−	0.976458i	$-0.569206\pi$
$491$	−9.55959			−0.431418			−0.215709	+	0.976458i	$0.569206\pi$
$492$		0			0
$493$	12.1255			0.546107
$494$		0			0
$495$	−6.08996			−0.273723
$496$		0			0
$497$	6.59867			0.295991
$498$		0			0
$499$			18.2878i			0.818672i	0.912384	+	0.409336i	$0.134240\pi$
$499$			18.2878i			0.818672i	−0.912384	+	0.409336i	$0.865760\pi$
$500$		0			0
$501$			18.7147i			0.836109i
$502$		0			0
$503$	−22.0398			−0.982705			−0.491352	−	0.870961i	$-0.663497\pi$
$503$	−22.0398			−0.982705			−0.491352	+	0.870961i	$0.663497\pi$
$504$		0			0
$505$		−	7.43620i		−	0.330907i
$506$		0			0
$507$	−12.9311	−	5.70597i	−0.574290	−	0.253411i
$508$		0			0
$509$			23.5200i			1.04250i	0.853403	+	0.521252i	$0.174535\pi$
$509$			23.5200i			1.04250i	−0.853403	+	0.521252i	$0.825465\pi$
$510$		0			0
$511$	−12.0334			−0.532328
$512$		0			0
$513$			37.3781i			1.65028i
$514$		0			0
$515$			29.3474i			1.29320i
$516$		0			0
$517$	−4.61283			−0.202872
$518$		0			0
$519$	13.0358			0.572209
$520$		0			0
$521$	−7.16958			−0.314105			−0.157052	−	0.987590i	$-0.550199\pi$
$521$	−7.16958			−0.314105			−0.157052	+	0.987590i	$0.550199\pi$
$522$		0			0
$523$	43.9276			1.92082			0.960410	−	0.278590i	$-0.0898670\pi$
$523$	43.9276			1.92082			0.960410	+	0.278590i	$0.0898670\pi$
$524$		0			0
$525$		−	6.76501i		−	0.295249i
$526$		0			0
$527$			29.8530i			1.30042i
$528$		0			0
$529$	−20.7775			−0.903372
$530$		0			0
$531$			17.9800i			0.780266i
$532$		0			0
$533$	6.60793	−	31.3434i	0.286221	−	1.35763i
$534$		0			0
$535$		−	18.3906i		−	0.795095i
$536$		0			0
$537$	−21.5622			−0.930476
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$			5.02702i			0.216129i	0.994144	+	0.108064i	$0.0344652\pi$
$541$			5.02702i			0.216129i	−0.994144	+	0.108064i	$0.965535\pi$
$542$		0			0
$543$	21.3694			0.917047
$544$		0			0
$545$	−13.9455			−0.597358
$546$		0			0
$547$	29.2289			1.24974			0.624868	−	0.780730i	$-0.285153\pi$
$547$	29.2289			1.24974			0.624868	+	0.780730i	$0.285153\pi$
$548$		0			0
$549$	−7.22276			−0.308260
$550$		0			0
$551$			20.0363i			0.853576i
$552$		0			0
$553$		−	9.43515i		−	0.401223i
$554$		0			0
$555$	12.9798			0.550960
$556$		0			0
$557$			17.8653i			0.756979i	0.925606	+	0.378489i	$0.123556\pi$
$557$			17.8653i			0.756979i	−0.925606	+	0.378489i	$0.876444\pi$
$558$		0			0
$559$	21.4005	+	4.51173i	0.905144	+	0.190826i
$560$		0			0
$561$		−	4.69505i		−	0.198225i
$562$		0			0
$563$	−0.379493			−0.0159937			−0.00799685	−	0.999968i	$-0.502546\pi$
$563$	−0.379493			−0.0159937			−0.00799685	+	0.999968i	$0.502546\pi$
$564$		0			0
$565$			38.4293i			1.61673i
$566$		0			0
$567$			0.241394i			0.0101376i
$568$		0			0
$569$	35.5588			1.49070			0.745352	−	0.666672i	$-0.232282\pi$
$569$	35.5588			1.49070			0.745352	+	0.666672i	$0.232282\pi$
$570$		0			0
$571$	−41.3720			−1.73136			−0.865682	−	0.500595i	$-0.833115\pi$
$571$	−41.3720			−1.73136			−0.865682	+	0.500595i	$0.833115\pi$
$572$		0			0
$573$	−14.9598			−0.624954
$574$		0			0
$575$	−9.27602			−0.386837
$576$		0			0
$577$		−	37.7093i		−	1.56986i	−0.619585	−	0.784929i	$-0.712699\pi$
$577$		−	37.7093i		−	1.56986i	0.619585	−	0.784929i	$-0.287301\pi$
$578$		0			0
$579$		−	17.7327i		−	0.736946i
$580$		0			0
$581$	−0.795337			−0.0329961
$582$		0			0
$583$			12.9549i			0.536539i
$584$		0			0
$585$	4.52963	−	21.4854i	0.187277	−	0.888311i
$586$		0			0
$587$		−	32.3211i		−	1.33404i	−0.745042	−	0.667018i	$-0.767571\pi$
$587$		−	32.3211i		−	1.33404i	0.745042	−	0.667018i	$-0.232429\pi$
$588$		0			0
$589$	−49.3294			−2.03258
$590$		0			0
$591$		−	19.9522i		−	0.820724i
$592$		0			0
$593$			31.2125i			1.28174i	0.767647	+	0.640872i	$0.221427\pi$
$593$			31.2125i			1.28174i	−0.767647	+	0.640872i	$0.778573\pi$
$594$		0			0
$595$	−14.4663			−0.593059
$596$		0			0
$597$	−22.2315			−0.909875
$598$		0			0
$599$	−28.7899			−1.17633			−0.588163	−	0.808743i	$-0.700149\pi$
$599$	−28.7899			−1.17633			−0.588163	+	0.808743i	$0.700149\pi$
$600$		0			0
$601$	−2.90195			−0.118373			−0.0591866	−	0.998247i	$-0.518851\pi$
$601$	−2.90195			−0.118373			−0.0591866	+	0.998247i	$0.518851\pi$
$602$		0			0
$603$		−	23.2065i		−	0.945042i
$604$		0			0
$605$			3.34996i			0.136195i
$606$		0			0
$607$	−6.59972			−0.267875			−0.133937	−	0.990990i	$-0.542762\pi$
$607$	−6.59972			−0.267875			−0.133937	+	0.990990i	$0.542762\pi$
$608$		0			0
$609$			3.05286i			0.123708i
$610$		0			0
$611$	3.43096	−	16.2741i	0.138802	−	0.658378i
$612$		0			0
$613$		−	32.0862i		−	1.29595i	−0.761662	−	0.647975i	$-0.775616\pi$
$613$		−	32.0862i		−	1.29595i	0.761662	−	0.647975i	$-0.224384\pi$
$614$		0			0
$615$	−32.3579			−1.30479
$616$		0			0
$617$		−	42.3420i		−	1.70462i	−0.523034	−	0.852312i	$-0.675200\pi$
$617$		−	42.3420i		−	1.70462i	0.523034	−	0.852312i	$-0.324800\pi$
$618$		0			0
$619$		−	18.0020i		−	0.723562i	−0.932263	−	0.361781i	$-0.882169\pi$
$619$		−	18.0020i		−	0.723562i	0.932263	−	0.361781i	$-0.117831\pi$
$620$		0			0
$621$	7.80907			0.313367
$622$		0			0
$623$	1.02531			0.0410783
$624$		0			0
$625$	−17.3951			−0.695804
$626$		0			0
$627$	7.75813			0.309830
$628$		0			0
$629$		−	15.3894i		−	0.613614i
$630$		0			0
$631$		−	27.0601i		−	1.07725i	−0.842547	−	0.538623i	$-0.818945\pi$
$631$		−	27.0601i		−	1.07725i	0.842547	−	0.538623i	$-0.181055\pi$
$632$		0			0
$633$	−15.0421			−0.597870
$634$		0			0
$635$			15.8190i			0.627758i
$636$		0			0
$637$	3.52800	+	0.743786i	0.139784	+	0.0294699i
$638$		0			0
$639$		−	11.9959i		−	0.474549i
$640$		0			0
$641$	−27.9694			−1.10472			−0.552362	−	0.833604i	$-0.686273\pi$
$641$	−27.9694			−1.10472			−0.552362	+	0.833604i	$0.686273\pi$
$642$		0			0
$643$			8.99767i			0.354834i	0.984136	+	0.177417i	$0.0567741\pi$
$643$			8.99767i			0.354834i	−0.984136	+	0.177417i	$0.943226\pi$
$644$		0			0
$645$		−	22.0931i		−	0.869916i
$646$		0			0
$647$	−38.2166			−1.50245			−0.751225	−	0.660046i	$-0.770537\pi$
$647$	−38.2166			−1.50245			−0.751225	+	0.660046i	$0.770537\pi$
$648$		0			0
$649$	9.89042			0.388233
$650$		0			0
$651$	−7.51614			−0.294581
$652$		0			0
$653$	36.7247			1.43715			0.718574	−	0.695451i	$-0.244795\pi$
$653$	36.7247			1.43715			0.718574	+	0.695451i	$0.244795\pi$
$654$		0			0
$655$		−	3.40516i		−	0.133050i
$656$		0			0
$657$			21.8759i			0.853459i
$658$		0			0
$659$	16.5319			0.643992			0.321996	−	0.946741i	$-0.395646\pi$
$659$	16.5319			0.643992			0.321996	+	0.946741i	$0.395646\pi$
$660$		0			0
$661$			14.2119i			0.552778i	0.961046	+	0.276389i	$0.0891378\pi$
$661$			14.2119i			0.552778i	−0.961046	+	0.276389i	$0.910862\pi$
$662$		0			0
$663$	16.5641	+	3.49211i	0.643297	+	0.135622i
$664$		0			0
$665$		−	23.9041i		−	0.926963i
$666$		0			0
$667$	4.18601			0.162083
$668$		0			0
$669$		−	12.0238i		−	0.464866i
$670$		0			0
$671$			3.97309i			0.153379i
$672$		0			0
$673$	11.9327			0.459973			0.229987	−	0.973194i	$-0.426132\pi$
$673$	11.9327			0.459973			0.229987	+	0.973194i	$0.426132\pi$
$674$		0			0
$675$	−32.5933			−1.25452
$676$		0			0
$677$	16.7278			0.642902			0.321451	−	0.946926i	$-0.395830\pi$
$677$	16.7278			0.642902			0.321451	+	0.946926i	$0.395830\pi$
$678$		0			0
$679$	−12.9734			−0.497873
$680$		0			0
$681$			17.6791i			0.677464i
$682$		0			0
$683$			5.48026i			0.209696i	0.994488	+	0.104848i	$0.0334356\pi$
$683$			5.48026i			0.209696i	−0.994488	+	0.104848i	$0.966564\pi$
$684$		0			0
$685$	−8.44043			−0.322492
$686$		0			0
$687$		−	19.2231i		−	0.733409i
$688$		0			0
$689$	−45.7050	−	9.63570i	−1.74122	−	0.367091i
$690$		0			0
$691$		−	41.8095i		−	1.59051i	−0.606275	−	0.795255i	$-0.707337\pi$
$691$		−	41.8095i		−	1.59051i	0.606275	−	0.795255i	$-0.292663\pi$
$692$		0			0
$693$	−1.81792			−0.0690571
$694$		0			0
$695$		−	31.4863i		−	1.19434i
$696$		0			0
$697$			38.3649i			1.45317i
$698$		0			0
$699$	−19.4350			−0.735098
$700$		0			0
$701$	−0.566554			−0.0213984			−0.0106992	−	0.999943i	$-0.503406\pi$
$701$	−0.566554			−0.0213984			−0.0106992	+	0.999943i	$0.503406\pi$
$702$		0			0
$703$	25.4295			0.959092
$704$		0			0
$705$	−16.8008			−0.632754
$706$		0			0
$707$		−	2.21979i		−	0.0834838i
$708$		0			0
$709$		−	8.88377i		−	0.333637i	−0.985988	−	0.166819i	$-0.946651\pi$
$709$		−	8.88377i		−	0.333637i	0.985988	−	0.166819i	$-0.0533494\pi$
$710$		0			0
$711$	−17.1524			−0.643264
$712$		0			0
$713$			10.3059i			0.385961i
$714$		0			0
$715$	−11.8187	−	2.49165i	−0.441992	−	0.0931825i
$716$		0			0
$717$		−	11.2249i		−	0.419200i
$718$		0			0
$719$	43.4041			1.61870			0.809350	−	0.587326i	$-0.199819\pi$
$719$	43.4041			1.61870			0.809350	+	0.587326i	$0.199819\pi$
$720$		0			0
$721$			8.76051i			0.326259i
$722$		0			0
$723$		−	0.842442i		−	0.0313307i
$724$		0			0
$725$	−17.4715			−0.648874
$726$		0			0
$727$	15.0438			0.557943			0.278972	−	0.960299i	$-0.410006\pi$
$727$	15.0438			0.557943			0.278972	+	0.960299i	$0.410006\pi$
$728$		0			0
$729$	17.5243			0.649049
$730$		0			0
$731$	−26.1946			−0.968842
$732$		0			0
$733$		−	5.47061i		−	0.202062i	−0.994883	−	0.101031i	$-0.967786\pi$
$733$		−	5.47061i		−	0.202062i	0.994883	−	0.101031i	$-0.0322141\pi$
$734$		0			0
$735$		−	3.64219i		−	0.134344i
$736$		0			0
$737$	−12.7654			−0.470220
$738$		0			0
$739$		−	14.8229i		−	0.545268i	−0.962118	−	0.272634i	$-0.912105\pi$
$739$		−	14.8229i		−	0.545268i	0.962118	−	0.272634i	$-0.0878947\pi$
$740$		0			0
$741$	−5.77039	+	27.3707i	−0.211980	+	1.00549i
$742$		0			0
$743$			34.1196i			1.25173i	0.779933	+	0.625863i	$0.215253\pi$
$743$			34.1196i			1.25173i	−0.779933	+	0.625863i	$0.784747\pi$
$744$		0			0
$745$	28.1704			1.03208
$746$		0			0
$747$			1.44586i			0.0529013i
$748$		0			0
$749$		−	5.48980i		−	0.200593i
$750$		0			0
$751$	34.1659			1.24673			0.623366	−	0.781930i	$-0.285765\pi$
$751$	34.1659			1.24673			0.623366	+	0.781930i	$0.285765\pi$
$752$		0			0
$753$	6.13487			0.223567
$754$		0			0
$755$	15.4448			0.562092
$756$		0			0
$757$	34.0261			1.23670			0.618350	−	0.785903i	$-0.287801\pi$
$757$	34.0261			1.23670			0.618350	+	0.785903i	$0.287801\pi$
$758$		0			0
$759$		−	1.62084i		−	0.0588327i
$760$		0			0
$761$			19.9598i			0.723544i	0.932267	+	0.361772i	$0.117828\pi$
$761$			19.9598i			0.723544i	−0.932267	+	0.361772i	$0.882172\pi$
$762$		0			0
$763$	−4.16288			−0.150706
$764$		0			0
$765$			26.2985i			0.950825i
$766$		0			0
$767$	−7.35635	+	34.8934i	−0.265623	+	1.25993i
$768$		0			0
$769$		−	53.3918i		−	1.92536i	−0.270649	−	0.962678i	$-0.587238\pi$
$769$		−	53.3918i		−	1.92536i	0.270649	−	0.962678i	$-0.412762\pi$
$770$		0			0
$771$	12.8257			0.461906
$772$		0			0
$773$			9.47357i			0.340741i	0.985380	+	0.170370i	$0.0544964\pi$
$773$			9.47357i			0.340741i	−0.985380	+	0.170370i	$0.945504\pi$
$774$		0			0
$775$		−	43.0147i		−	1.54513i
$776$		0			0
$777$	3.87460			0.139001
$778$		0			0
$779$	−63.3945			−2.27134
$780$		0			0
$781$	−6.59867			−0.236119
$782$		0			0
$783$	14.7085			0.525637
$784$		0			0
$785$			57.2437i			2.04311i
$786$		0			0
$787$			43.0640i			1.53507i	0.641009	+	0.767533i	$0.278516\pi$
$787$			43.0640i			1.53507i	−0.641009	+	0.767533i	$0.721484\pi$
$788$		0			0
$789$	−3.44877			−0.122779
$790$		0			0
$791$			11.4716i			0.407882i
$792$		0			0
$793$	−14.0170	−	2.95513i	−0.497760	−	0.104940i
$794$		0			0
$795$			47.1843i			1.67346i
$796$		0			0
$797$	−11.3563			−0.402260			−0.201130	−	0.979565i	$-0.564461\pi$
$797$	−11.3563			−0.402260			−0.201130	+	0.979565i	$0.564461\pi$
$798$		0			0
$799$			19.9198i			0.704711i
$800$		0			0
$801$		−	1.86394i		−	0.0658591i
$802$		0			0
$803$	12.0334			0.424651
$804$		0			0
$805$	−4.99408			−0.176018
$806$		0			0
$807$	−11.8554			−0.417330
$808$		0			0
$809$	8.27264			0.290850			0.145425	−	0.989369i	$-0.453545\pi$
$809$	8.27264			0.290850			0.145425	+	0.989369i	$0.453545\pi$
$810$		0			0
$811$		−	24.0311i		−	0.843845i	−0.906632	−	0.421922i	$-0.861356\pi$
$811$		−	24.0311i		−	0.843845i	0.906632	−	0.421922i	$-0.138644\pi$
$812$		0			0
$813$		−	12.9788i		−	0.455186i
$814$		0			0
$815$	−76.6427			−2.68468
$816$		0			0
$817$		−	43.2841i		−	1.51432i
$818$		0			0
$819$	1.35214	−	6.41363i	0.0472478	−	0.224110i
$820$		0			0
$821$		−	49.6505i		−	1.73281i	−0.499339	−	0.866407i	$-0.666424\pi$
$821$		−	49.6505i		−	1.73281i	0.499339	−	0.866407i	$-0.333576\pi$
$822$		0			0
$823$	−10.3234			−0.359850			−0.179925	−	0.983680i	$-0.557586\pi$
$823$	−10.3234			−0.359850			−0.179925	+	0.983680i	$0.557586\pi$
$824$		0			0
$825$			6.76501i			0.235527i
$826$		0			0
$827$		−	26.7564i		−	0.930413i	−0.885202	−	0.465206i	$-0.845980\pi$
$827$		−	26.7564i		−	0.930413i	0.885202	−	0.465206i	$-0.154020\pi$
$828$		0			0
$829$	−31.2214			−1.08436			−0.542182	−	0.840261i	$-0.682402\pi$
$829$	−31.2214			−1.08436			−0.542182	+	0.840261i	$0.682402\pi$
$830$		0			0
$831$	−28.0958			−0.974633
$832$		0			0
$833$	−4.31834			−0.149622
$834$		0			0
$835$	−57.6631			−1.99551
$836$		0			0
$837$			36.2122i			1.25168i
$838$		0			0
$839$			21.8707i			0.755061i	0.925997	+	0.377531i	$0.123227\pi$
$839$			21.8707i			0.755061i	−0.925997	+	0.377531i	$0.876773\pi$
$840$		0			0
$841$	−21.1156			−0.728124
$842$		0			0
$843$			9.19039i			0.316534i
$844$		0			0
$845$	17.5811	−	39.8429i	0.604808	−	1.37064i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	−32.5750			−1.11797
$850$		0			0
$851$		−	5.31276i		−	0.182119i
$852$		0			0
$853$		−	21.0838i		−	0.721895i	−0.932586	−	0.360948i	$-0.882453\pi$
$853$		−	21.0838i		−	0.721895i	0.932586	−	0.360948i	$-0.117547\pi$
$854$		0			0
$855$	−43.4559			−1.48616
$856$		0			0
$857$	−34.7184			−1.18596			−0.592979	−	0.805218i	$-0.702048\pi$
$857$	−34.7184			−1.18596			−0.592979	+	0.805218i	$0.702048\pi$
$858$		0			0
$859$	13.4087			0.457500			0.228750	−	0.973485i	$-0.426536\pi$
$859$	13.4087			0.457500			0.228750	+	0.973485i	$0.426536\pi$
$860$		0			0
$861$	−9.65918			−0.329184
$862$		0			0
$863$			26.7141i			0.909359i	0.890655	+	0.454680i	$0.150246\pi$
$863$			26.7141i			0.909359i	−0.890655	+	0.454680i	$0.849754\pi$
$864$		0			0
$865$			40.1657i			1.36567i
$866$		0			0
$867$	−1.79183			−0.0608538
$868$		0			0
$869$			9.43515i			0.320066i
$870$		0			0
$871$	9.49473	−	45.0363i	0.321716	−	1.52600i
$872$		0			0
$873$			23.5846i			0.798219i
$874$		0			0
$875$	4.09438			0.138415
$876$		0			0
$877$		−	6.42868i		−	0.217081i	−0.994092	−	0.108541i	$-0.965382\pi$
$877$		−	6.42868i		−	0.217081i	0.994092	−	0.108541i	$-0.0346177\pi$
$878$		0			0
$879$		−	5.28585i		−	0.178287i
$880$		0			0
$881$	−31.8595			−1.07337			−0.536687	−	0.843782i	$-0.680324\pi$
$881$	−31.8595			−1.07337			−0.536687	+	0.843782i	$0.680324\pi$
$882$		0			0
$883$	1.88272			0.0633586			0.0316793	−	0.999498i	$-0.489914\pi$
$883$	1.88272			0.0633586			0.0316793	+	0.999498i	$0.489914\pi$
$884$		0			0
$885$	36.0228			1.21089
$886$		0			0
$887$	0.605266			0.0203228			0.0101614	−	0.999948i	$-0.496765\pi$
$887$	0.605266			0.0203228			0.0101614	+	0.999948i	$0.496765\pi$
$888$		0			0
$889$			4.72215i			0.158376i
$890$		0			0
$891$		−	0.241394i		−	0.00808699i
$892$		0			0
$893$	−32.9156			−1.10148
$894$		0			0
$895$		−	66.4368i		−	2.22074i
$896$		0			0
$897$	5.71831	+	1.20556i	0.190929	+	0.0402523i
$898$		0			0
$899$			19.4114i			0.647405i
$900$		0			0
$901$	55.9438			1.86376
$902$		0			0
$903$		−	6.59505i		−	0.219469i
$904$		0			0
$905$			65.8427i			2.18869i
$906$		0			0
$907$	−19.4191			−0.644801			−0.322400	−	0.946603i	$-0.604490\pi$
$907$	−19.4191			−0.644801			−0.322400	+	0.946603i	$0.604490\pi$
$908$		0			0
$909$	−4.03540			−0.133846
$910$		0			0
$911$	10.5051			0.348050			0.174025	−	0.984741i	$-0.444323\pi$
$911$	10.5051			0.348050			0.174025	+	0.984741i	$0.444323\pi$
$912$		0			0
$913$	0.795337			0.0263218
$914$		0			0
$915$			14.4707i			0.478387i
$916$		0			0
$917$		−	1.01648i		−	0.0335670i
$918$		0			0
$919$	−5.23099			−0.172555			−0.0862773	−	0.996271i	$-0.527497\pi$
$919$	−5.23099			−0.172555			−0.0862773	+	0.996271i	$0.527497\pi$
$920$		0			0
$921$			36.5223i			1.20345i
$922$		0			0
$923$	4.90800	−	23.2801i	0.161549	−	0.766274i
$924$		0			0
$925$			22.1743i			0.729085i
$926$		0			0
$927$	15.9259			0.523076
$928$		0			0
$929$			0.236803i			0.00776924i	0.999992	+	0.00388462i	$0.00123652\pi$
$929$			0.236803i			0.00776924i	−0.999992	+	0.00388462i	$0.998763\pi$
$930$		0			0
$931$		−	7.13566i		−	0.233862i
$932$		0			0
$933$	0.965379			0.0316051
$934$		0			0
$935$	14.4663			0.473097
$936$		0			0
$937$	−60.0746			−1.96255			−0.981276	−	0.192607i	$-0.938306\pi$
$937$	−60.0746			−1.96255			−0.981276	+	0.192607i	$0.938306\pi$
$938$		0			0
$939$	−27.7237			−0.904730
$940$		0			0
$941$		−	12.6019i		−	0.410811i	−0.978677	−	0.205406i	$-0.934149\pi$
$941$		−	12.6019i		−	0.410811i	0.978677	−	0.205406i	$-0.0658513\pi$
$942$		0			0
$943$			13.2444i			0.431298i
$944$		0			0
$945$	−17.5478			−0.570829
$946$		0			0
$947$		−	18.8695i		−	0.613176i	−0.951842	−	0.306588i	$-0.900813\pi$
$947$		−	18.8695i		−	0.613176i	0.951842	−	0.306588i	$-0.0991874\pi$
$948$		0			0
$949$	−8.95031	+	42.4540i	−0.290539	+	1.37811i
$950$		0			0
$951$			17.4268i			0.565102i
$952$		0			0
$953$	12.2874			0.398027			0.199014	−	0.979997i	$-0.436226\pi$
$953$	12.2874			0.398027			0.199014	+	0.979997i	$0.436226\pi$
$954$		0			0
$955$		−	46.0937i		−	1.49156i
$956$		0			0
$957$		−	3.05286i		−	0.0986850i
$958$		0			0
$959$	−2.51956			−0.0813609
$960$		0			0
$961$	−16.7907			−0.541636
$962$		0			0
$963$	−9.98003			−0.321602
$964$		0			0
$965$	54.6375			1.75884
$966$		0			0
$967$			54.8891i			1.76511i	0.470205	+	0.882557i	$0.344180\pi$
$967$			54.8891i			1.76511i	−0.470205	+	0.882557i	$0.655820\pi$
$968$		0			0
$969$		−	33.5022i		−	1.07625i
$970$		0			0
$971$	−1.77490			−0.0569591			−0.0284796	−	0.999594i	$-0.509067\pi$
$971$	−1.77490			−0.0569591			−0.0284796	+	0.999594i	$0.509067\pi$
$972$		0			0
$973$		−	9.39901i		−	0.301318i
$974$		0			0
$975$	−23.8669	−	5.03172i	−0.764354	−	0.161144i
$976$		0			0
$977$			52.7699i			1.68826i	0.536139	+	0.844130i	$0.319882\pi$
$977$			52.7699i			1.68826i	−0.536139	+	0.844130i	$0.680118\pi$
$978$		0			0
$979$	−1.02531			−0.0327692
$980$		0			0
$981$			7.56779i			0.241621i
$982$		0			0
$983$			38.3002i			1.22159i	0.791790	+	0.610794i	$0.209150\pi$
$983$			38.3002i			1.22159i	−0.791790	+	0.610794i	$0.790850\pi$
$984$		0			0
$985$	61.4762			1.95880
$986$		0			0
$987$	−5.01523			−0.159636
$988$		0			0
$989$	−9.04297			−0.287550
$990$		0			0
$991$	44.7153			1.42043			0.710214	−	0.703986i	$-0.248598\pi$
$991$	44.7153			1.42043			0.710214	+	0.703986i	$0.248598\pi$
$992$		0			0
$993$		−	28.7235i		−	0.911514i
$994$		0			0
$995$		−	68.4991i		−	2.17157i
$996$		0			0
$997$	13.1786			0.417372			0.208686	−	0.977983i	$-0.433081\pi$
$997$	13.1786			0.417372			0.208686	+	0.977983i	$0.433081\pi$
$998$		0			0
$999$		−	18.6675i		−	0.590615i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.12	yes	36
13.12	even	2	inner	4004.2.m.c.2157.11	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.11	✓	36	13.12	even	2	inner
4004.2.m.c.2157.12	yes	36	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.08723 q^{3} -3.34996i q^{5} -1.00000i q^{7} -1.81792 q^{9} +O(q^{10})\)	\(q-1.08723 q^{3} -3.34996i q^{5} -1.00000i q^{7} -1.81792 q^{9} +1.00000i q^{11} +(-3.52800 - 0.743786i) q^{13} +3.64219i q^{15} +4.31834 q^{17} +7.13566i q^{19} +1.08723i q^{21} +1.49079 q^{23} -6.22222 q^{25} +5.23821 q^{27} +2.80792 q^{29} +6.91308i q^{31} -1.08723i q^{33} -3.34996 q^{35} -3.56372i q^{37} +(3.83576 + 0.808669i) q^{39} +8.88418i q^{41} -6.06590 q^{43} +6.08996i q^{45} +4.61283i q^{47} -1.00000 q^{49} -4.69505 q^{51} +12.9549 q^{53} +3.34996 q^{55} -7.75813i q^{57} -9.89042i q^{59} +3.97309 q^{61} +1.81792i q^{63} +(-2.49165 + 11.8187i) q^{65} +12.7654i q^{67} -1.62084 q^{69} +6.59867i q^{71} -12.0334i q^{73} +6.76501 q^{75} +1.00000 q^{77} +9.43515 q^{79} -0.241394 q^{81} -0.795337i q^{83} -14.4663i q^{85} -3.05286 q^{87} +1.02531i q^{89} +(-0.743786 + 3.52800i) q^{91} -7.51614i q^{93} +23.9041 q^{95} -12.9734i q^{97} -1.81792i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 36 * q - 4 * q^3 + 40 * q^9	\( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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