LMFDB - Embedded newform 4004.2.m.c.2157.18

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			2157.18
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.722868 q^{3} -0.383791i q^{5} +1.00000i q^{7} -2.47746 q^{9} +O(q^{10})$	$q-0.722868 q^{3} -0.383791i q^{5} +1.00000i q^{7} -2.47746 q^{9} -1.00000i q^{11} +(3.37304 - 1.27380i) q^{13} +0.277430i q^{15} -5.43236 q^{17} -5.21859i q^{19} -0.722868i q^{21} +2.81438 q^{23} +4.85270 q^{25} +3.95948 q^{27} +1.99936 q^{29} +2.19118i q^{31} +0.722868i q^{33} +0.383791 q^{35} +8.80912i q^{37} +(-2.43827 + 0.920790i) q^{39} -5.89210i q^{41} -9.25357 q^{43} +0.950828i q^{45} +8.28744i q^{47} -1.00000 q^{49} +3.92688 q^{51} -0.632174 q^{53} -0.383791 q^{55} +3.77235i q^{57} +13.2980i q^{59} -13.3069 q^{61} -2.47746i q^{63} +(-0.488874 - 1.29454i) q^{65} -7.84695i q^{67} -2.03442 q^{69} -8.46182i q^{71} -13.6678i q^{73} -3.50786 q^{75} +1.00000 q^{77} -13.9841 q^{79} +4.57021 q^{81} -13.5504i q^{83} +2.08489i q^{85} -1.44528 q^{87} +3.98493i q^{89} +(1.27380 + 3.37304i) q^{91} -1.58393i q^{93} -2.00285 q^{95} -4.98484i q^{97} +2.47746i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.722868			−0.417348			−0.208674	−	0.977985i	$-0.566915\pi$
$3$	−0.722868			−0.417348			−0.208674	+	0.977985i	$0.566915\pi$
$4$		0			0
$5$		−	0.383791i		−	0.171637i	−0.996311	−	0.0858183i	$-0.972650\pi$
$5$		−	0.383791i		−	0.171637i	0.996311	−	0.0858183i	$-0.0273505\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−2.47746			−0.825821
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	3.37304	−	1.27380i	0.935514	−	0.353289i
$13$	3.37304	−	1.27380i	0.935514	−	0.353289i
$14$		0			0
$15$			0.277430i			0.0716322i
$16$		0			0
$17$	−5.43236			−1.31754			−0.658771	−	0.752344i	$-0.728923\pi$
$17$	−5.43236			−1.31754			−0.658771	+	0.752344i	$0.728923\pi$
$18$		0			0
$19$		−	5.21859i		−	1.19723i	−0.801038	−	0.598614i	$-0.795718\pi$
$19$		−	5.21859i		−	1.19723i	0.801038	−	0.598614i	$-0.204282\pi$
$20$		0			0
$21$		−	0.722868i		−	0.157743i
$22$		0			0
$23$	2.81438			0.586838			0.293419	−	0.955984i	$-0.405207\pi$
$23$	2.81438			0.586838			0.293419	+	0.955984i	$0.405207\pi$
$24$		0			0
$25$	4.85270			0.970541
$26$		0			0
$27$	3.95948			0.762002
$28$		0			0
$29$	1.99936			0.371272			0.185636	−	0.982619i	$-0.440565\pi$
$29$	1.99936			0.371272			0.185636	+	0.982619i	$0.440565\pi$
$30$		0			0
$31$			2.19118i			0.393548i	0.980449	+	0.196774i	$0.0630465\pi$
$31$			2.19118i			0.393548i	−0.980449	+	0.196774i	$0.936954\pi$
$32$		0			0
$33$			0.722868i			0.125835i
$34$		0			0
$35$	0.383791			0.0648726
$36$		0			0
$37$			8.80912i			1.44821i	0.689690	+	0.724105i	$0.257747\pi$
$37$			8.80912i			1.44821i	−0.689690	+	0.724105i	$0.742253\pi$
$38$		0			0
$39$	−2.43827	+	0.920790i	−0.390435	+	0.147444i
$40$		0			0
$41$		−	5.89210i		−	0.920191i	−0.887869	−	0.460096i	$-0.847815\pi$
$41$		−	5.89210i		−	0.920191i	0.887869	−	0.460096i	$-0.152185\pi$
$42$		0			0
$43$	−9.25357			−1.41116			−0.705578	−	0.708632i	$-0.749313\pi$
$43$	−9.25357			−1.41116			−0.705578	+	0.708632i	$0.749313\pi$
$44$		0			0
$45$			0.950828i			0.141741i
$46$		0			0
$47$			8.28744i			1.20885i	0.796663	+	0.604424i	$0.206597\pi$
$47$			8.28744i			1.20885i	−0.796663	+	0.604424i	$0.793403\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	3.92688			0.549873
$52$		0			0
$53$	−0.632174			−0.0868358			−0.0434179	−	0.999057i	$-0.513825\pi$
$53$	−0.632174			−0.0868358			−0.0434179	+	0.999057i	$0.513825\pi$
$54$		0			0
$55$	−0.383791			−0.0517504
$56$		0			0
$57$			3.77235i			0.499660i
$58$		0			0
$59$			13.2980i			1.73125i	0.500694	+	0.865625i	$0.333078\pi$
$59$			13.2980i			1.73125i	−0.500694	+	0.865625i	$0.666922\pi$
$60$		0			0
$61$	−13.3069			−1.70377			−0.851884	−	0.523731i	$-0.824540\pi$
$61$	−13.3069			−1.70377			−0.851884	+	0.523731i	$0.824540\pi$
$62$		0			0
$63$		−	2.47746i		−	0.312131i
$64$		0			0
$65$	−0.488874	−	1.29454i	−0.0606373	−	0.160569i
$66$		0			0
$67$		−	7.84695i		−	0.958657i	−0.877636	−	0.479329i	$-0.840880\pi$
$67$		−	7.84695i		−	0.958657i	0.877636	−	0.479329i	$-0.159120\pi$
$68$		0			0
$69$	−2.03442			−0.244916
$70$		0			0
$71$		−	8.46182i		−	1.00423i	−0.864800	−	0.502116i	$-0.832555\pi$
$71$		−	8.46182i		−	1.00423i	0.864800	−	0.502116i	$-0.167445\pi$
$72$		0			0
$73$		−	13.6678i		−	1.59970i	−0.600203	−	0.799848i	$-0.704914\pi$
$73$		−	13.6678i		−	1.59970i	0.600203	−	0.799848i	$-0.295086\pi$
$74$		0			0
$75$	−3.50786			−0.405053
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	−13.9841			−1.57333			−0.786666	−	0.617379i	$-0.788195\pi$
$79$	−13.9841			−1.57333			−0.786666	+	0.617379i	$0.788195\pi$
$80$		0			0
$81$	4.57021			0.507801
$82$		0			0
$83$		−	13.5504i		−	1.48735i	−0.668540	−	0.743677i	$-0.733080\pi$
$83$		−	13.5504i		−	1.48735i	0.668540	−	0.743677i	$-0.266920\pi$
$84$		0			0
$85$			2.08489i			0.226138i
$86$		0			0
$87$	−1.44528			−0.154950
$88$		0			0
$89$			3.98493i			0.422402i	0.977443	+	0.211201i	$0.0677374\pi$
$89$			3.98493i			0.422402i	−0.977443	+	0.211201i	$0.932263\pi$
$90$		0			0
$91$	1.27380	+	3.37304i	0.133531	+	0.353591i
$92$		0			0
$93$		−	1.58393i		−	0.164246i
$94$		0			0
$95$	−2.00285			−0.205488
$96$		0			0
$97$		−	4.98484i		−	0.506134i	−0.967449	−	0.253067i	$-0.918561\pi$
$97$		−	4.98484i		−	0.506134i	0.967449	−	0.253067i	$-0.0814393\pi$
$98$		0			0
$99$			2.47746i			0.248994i
$100$		0			0
$101$	−0.966532			−0.0961735			−0.0480867	−	0.998843i	$-0.515312\pi$
$101$	−0.966532			−0.0961735			−0.0480867	+	0.998843i	$0.515312\pi$
$102$		0			0
$103$	−11.4249			−1.12573			−0.562864	−	0.826550i	$-0.690300\pi$
$103$	−11.4249			−1.12573			−0.562864	+	0.826550i	$0.690300\pi$
$104$		0			0
$105$	−0.277430			−0.0270744
$106$		0			0
$107$	12.2340			1.18271			0.591353	−	0.806413i	$-0.298594\pi$
$107$	12.2340			1.18271			0.591353	+	0.806413i	$0.298594\pi$
$108$		0			0
$109$			1.65127i			0.158163i	0.996868	+	0.0790814i	$0.0251987\pi$
$109$			1.65127i			0.158163i	−0.996868	+	0.0790814i	$0.974801\pi$
$110$		0			0
$111$		−	6.36783i		−	0.604407i
$112$		0			0
$113$	−7.25772			−0.682749			−0.341375	−	0.939927i	$-0.610892\pi$
$113$	−7.25772			−0.682749			−0.341375	+	0.939927i	$0.610892\pi$
$114$		0			0
$115$		−	1.08013i		−	0.100723i
$116$		0			0
$117$	−8.35659	+	3.15579i	−0.772567	+	0.291753i
$118$		0			0
$119$		−	5.43236i		−	0.497984i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			4.25921i			0.384040i
$124$		0			0
$125$		−	3.78138i		−	0.338217i
$126$		0			0
$127$	−8.03568			−0.713052			−0.356526	−	0.934285i	$-0.616039\pi$
$127$	−8.03568			−0.713052			−0.356526	+	0.934285i	$0.616039\pi$
$128$		0			0
$129$	6.68911			0.588943
$130$		0			0
$131$	14.5041			1.26723			0.633616	−	0.773648i	$-0.281570\pi$
$131$	14.5041			1.26723			0.633616	+	0.773648i	$0.281570\pi$
$132$		0			0
$133$	5.21859			0.452509
$134$		0			0
$135$		−	1.51961i		−	0.130788i
$136$		0			0
$137$			4.28927i			0.366457i	0.983070	+	0.183228i	$0.0586548\pi$
$137$			4.28927i			0.366457i	−0.983070	+	0.183228i	$0.941345\pi$
$138$		0			0
$139$	−16.6819			−1.41494			−0.707469	−	0.706745i	$-0.750163\pi$
$139$	−16.6819			−1.41494			−0.707469	+	0.706745i	$0.750163\pi$
$140$		0			0
$141$		−	5.99073i		−	0.504510i
$142$		0			0
$143$	−1.27380	−	3.37304i	−0.106521	−	0.282068i
$144$		0			0
$145$		−	0.767338i		−	0.0637240i
$146$		0			0
$147$	0.722868			0.0596211
$148$		0			0
$149$			19.6040i			1.60602i	0.595967	+	0.803009i	$0.296769\pi$
$149$			19.6040i			1.60602i	−0.595967	+	0.803009i	$0.703231\pi$
$150$		0			0
$151$		−	2.47295i		−	0.201246i	−0.994925	−	0.100623i	$-0.967916\pi$
$151$		−	2.47295i		−	0.201246i	0.994925	−	0.100623i	$-0.0320835\pi$
$152$		0			0
$153$	13.4585			1.08805
$154$		0			0
$155$	0.840956			0.0675472
$156$		0			0
$157$	−10.1146			−0.807234			−0.403617	−	0.914928i	$-0.632247\pi$
$157$	−10.1146			−0.807234			−0.403617	+	0.914928i	$0.632247\pi$
$158$		0			0
$159$	0.456978			0.0362407
$160$		0			0
$161$			2.81438i			0.221804i
$162$		0			0
$163$		−	22.9361i		−	1.79650i	−0.439488	−	0.898248i	$-0.644840\pi$
$163$		−	22.9361i		−	1.79650i	0.439488	−	0.898248i	$-0.355160\pi$
$164$		0			0
$165$	0.277430			0.0215979
$166$		0			0
$167$		−	20.6241i		−	1.59594i	−0.602699	−	0.797969i	$-0.705908\pi$
$167$		−	20.6241i		−	1.59594i	0.602699	−	0.797969i	$-0.294092\pi$
$168$		0			0
$169$	9.75486	−	8.59318i	0.750374	−	0.661014i
$170$		0			0
$171$			12.9289i			0.988695i
$172$		0			0
$173$	−5.66630			−0.430801			−0.215400	−	0.976526i	$-0.569106\pi$
$173$	−5.66630			−0.430801			−0.215400	+	0.976526i	$0.569106\pi$
$174$		0			0
$175$			4.85270i			0.366830i
$176$		0			0
$177$		−	9.61268i		−	0.722533i
$178$		0			0
$179$	−12.5552			−0.938418			−0.469209	−	0.883087i	$-0.655461\pi$
$179$	−12.5552			−0.938418			−0.469209	+	0.883087i	$0.655461\pi$
$180$		0			0
$181$	−14.9079			−1.10810			−0.554049	−	0.832484i	$-0.686918\pi$
$181$	−14.9079			−1.10810			−0.554049	+	0.832484i	$0.686918\pi$
$182$		0			0
$183$	9.61909			0.711064
$184$		0			0
$185$	3.38086			0.248566
$186$		0			0
$187$			5.43236i			0.397254i
$188$		0			0
$189$			3.95948i			0.288010i
$190$		0			0
$191$	−17.7347			−1.28324			−0.641618	−	0.767025i	$-0.721736\pi$
$191$	−17.7347			−1.28324			−0.641618	+	0.767025i	$0.721736\pi$
$192$		0			0
$193$			15.4927i			1.11519i	0.830115	+	0.557593i	$0.188275\pi$
$193$			15.4927i			1.11519i	−0.830115	+	0.557593i	$0.811725\pi$
$194$		0			0
$195$	0.353391	+	0.935785i	0.0253069	+	0.0670129i
$196$		0			0
$197$		−	5.02539i		−	0.358044i	−0.983845	−	0.179022i	$-0.942707\pi$
$197$		−	5.02539i		−	0.358044i	0.983845	−	0.179022i	$-0.0572934\pi$
$198$		0			0
$199$	−16.5290			−1.17171			−0.585855	−	0.810416i	$-0.699241\pi$
$199$	−16.5290			−1.17171			−0.585855	+	0.810416i	$0.699241\pi$
$200$		0			0
$201$			5.67231i			0.400094i
$202$		0			0
$203$			1.99936i			0.140328i
$204$		0			0
$205$	−2.26134			−0.157939
$206$		0			0
$207$	−6.97251			−0.484623
$208$		0			0
$209$	−5.21859			−0.360978
$210$		0			0
$211$	−23.0461			−1.58656			−0.793279	−	0.608859i	$-0.791628\pi$
$211$	−23.0461			−1.58656			−0.793279	+	0.608859i	$0.791628\pi$
$212$		0			0
$213$			6.11677i			0.419114i
$214$		0			0
$215$			3.55144i			0.242206i
$216$		0			0
$217$	−2.19118			−0.148747
$218$		0			0
$219$			9.88001i			0.667630i
$220$		0			0
$221$	−18.3236	+	6.91975i	−1.23258	+	0.465473i
$222$		0			0
$223$		−	3.12458i		−	0.209237i	−0.994512	−	0.104619i	$-0.966638\pi$
$223$		−	3.12458i		−	0.209237i	0.994512	−	0.104619i	$-0.0333622\pi$
$224$		0			0
$225$	−12.0224			−0.801493
$226$		0			0
$227$		−	11.5176i		−	0.764449i	−0.924069	−	0.382225i	$-0.875158\pi$
$227$		−	11.5176i		−	0.764449i	0.924069	−	0.382225i	$-0.124842\pi$
$228$		0			0
$229$			3.82300i			0.252631i	0.991990	+	0.126315i	$0.0403151\pi$
$229$			3.82300i			0.252631i	−0.991990	+	0.126315i	$0.959685\pi$
$230$		0			0
$231$	−0.722868			−0.0475612
$232$		0			0
$233$	−4.54901			−0.298016			−0.149008	−	0.988836i	$-0.547608\pi$
$233$	−4.54901			−0.298016			−0.149008	+	0.988836i	$0.547608\pi$
$234$		0			0
$235$	3.18065			0.207483
$236$		0			0
$237$	10.1086			0.656627
$238$		0			0
$239$		−	12.3313i		−	0.797644i	−0.917028	−	0.398822i	$-0.869419\pi$
$239$		−	12.3313i		−	0.797644i	0.917028	−	0.398822i	$-0.130581\pi$
$240$		0			0
$241$		−	21.7442i		−	1.40067i	−0.713815	−	0.700335i	$-0.753034\pi$
$241$		−	21.7442i		−	1.40067i	0.713815	−	0.700335i	$-0.246966\pi$
$242$		0			0
$243$	−15.1821			−0.973932
$244$		0			0
$245$			0.383791i			0.0245195i
$246$		0			0
$247$	−6.64745	−	17.6025i	−0.422967	−	1.12002i
$248$		0			0
$249$			9.79517i			0.620744i
$250$		0			0
$251$	15.3695			0.970115			0.485057	−	0.874482i	$-0.338799\pi$
$251$	15.3695			0.970115			0.485057	+	0.874482i	$0.338799\pi$
$252$		0			0
$253$		−	2.81438i		−	0.176938i
$254$		0			0
$255$		−	1.50710i		−	0.0943783i
$256$		0			0
$257$	12.7666			0.796358			0.398179	−	0.917308i	$-0.369642\pi$
$257$	12.7666			0.796358			0.398179	+	0.917308i	$0.369642\pi$
$258$		0			0
$259$	−8.80912			−0.547372
$260$		0			0
$261$	−4.95335			−0.306605
$262$		0			0
$263$	−27.6129			−1.70268			−0.851342	−	0.524610i	$-0.824211\pi$
$263$	−27.6129			−1.70268			−0.851342	+	0.524610i	$0.824211\pi$
$264$		0			0
$265$			0.242623i			0.0149042i
$266$		0			0
$267$		−	2.88058i		−	0.176289i
$268$		0			0
$269$	28.1801			1.71817			0.859084	−	0.511835i	$-0.171034\pi$
$269$	28.1801			1.71817			0.859084	+	0.511835i	$0.171034\pi$
$270$		0			0
$271$			3.96706i			0.240981i	0.992714	+	0.120491i	$0.0384468\pi$
$271$			3.96706i			0.240981i	−0.992714	+	0.120491i	$0.961553\pi$
$272$		0			0
$273$	−0.920790	−	2.43827i	−0.0557287	−	0.147571i
$274$		0			0
$275$		−	4.85270i		−	0.292629i
$276$		0			0
$277$	25.0315			1.50400			0.751998	−	0.659166i	$-0.229090\pi$
$277$	25.0315			1.50400			0.751998	+	0.659166i	$0.229090\pi$
$278$		0			0
$279$		−	5.42857i		−	0.325000i
$280$		0			0
$281$		−	3.59437i		−	0.214422i	−0.994236	−	0.107211i	$-0.965808\pi$
$281$		−	3.59437i		−	0.214422i	0.994236	−	0.107211i	$-0.0341920\pi$
$282$		0			0
$283$	−11.2143			−0.666618			−0.333309	−	0.942818i	$-0.608165\pi$
$283$	−11.2143			−0.666618			−0.333309	+	0.942818i	$0.608165\pi$
$284$		0			0
$285$	1.44780			0.0857600
$286$		0			0
$287$	5.89210			0.347800
$288$		0			0
$289$	12.5105			0.735914
$290$		0			0
$291$			3.60338i			0.211234i
$292$		0			0
$293$			10.2580i			0.599279i	0.954052	+	0.299640i	$0.0968665\pi$
$293$			10.2580i			0.599279i	−0.954052	+	0.299640i	$0.903134\pi$
$294$		0			0
$295$	5.10365			0.297146
$296$		0			0
$297$		−	3.95948i		−	0.229752i
$298$		0			0
$299$	9.49302	−	3.58496i	0.548995	−	0.207323i
$300$		0			0
$301$		−	9.25357i		−	0.533367i
$302$		0			0
$303$	0.698674			0.0401378
$304$		0			0
$305$			5.10705i			0.292429i
$306$		0			0
$307$			18.1345i			1.03499i	0.855687	+	0.517494i	$0.173135\pi$
$307$			18.1345i			1.03499i	−0.855687	+	0.517494i	$0.826865\pi$
$308$		0			0
$309$	8.25868			0.469820
$310$		0			0
$311$	−21.3181			−1.20884			−0.604420	−	0.796666i	$-0.706595\pi$
$311$	−21.3181			−1.20884			−0.604420	+	0.796666i	$0.706595\pi$
$312$		0			0
$313$	−0.529307			−0.0299182			−0.0149591	−	0.999888i	$-0.504762\pi$
$313$	−0.529307			−0.0299182			−0.0149591	+	0.999888i	$0.504762\pi$
$314$		0			0
$315$	−0.950828			−0.0535731
$316$		0			0
$317$		−	3.03911i		−	0.170693i	−0.996351	−	0.0853467i	$-0.972800\pi$
$317$		−	3.03911i		−	0.170693i	0.996351	−	0.0853467i	$-0.0271998\pi$
$318$		0			0
$319$		−	1.99936i		−	0.111943i
$320$		0			0
$321$	−8.84356			−0.493600
$322$		0			0
$323$			28.3493i			1.57740i
$324$		0			0
$325$	16.3684	−	6.18138i	0.907955	−	0.342881i
$326$		0			0
$327$		−	1.19365i		−	0.0660089i
$328$		0			0
$329$	−8.28744			−0.456902
$330$		0			0
$331$			15.8844i			0.873087i	0.899683	+	0.436543i	$0.143797\pi$
$331$			15.8844i			0.873087i	−0.899683	+	0.436543i	$0.856203\pi$
$332$		0			0
$333$		−	21.8243i		−	1.19596i
$334$		0			0
$335$	−3.01159			−0.164541
$336$		0			0
$337$	16.1118			0.877667			0.438833	−	0.898568i	$-0.355392\pi$
$337$	16.1118			0.877667			0.438833	+	0.898568i	$0.355392\pi$
$338$		0			0
$339$	5.24637			0.284944
$340$		0			0
$341$	2.19118			0.118659
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			0.780793i			0.0420365i
$346$		0			0
$347$	−2.13479			−0.114602			−0.0573008	−	0.998357i	$-0.518249\pi$
$347$	−2.13479			−0.114602			−0.0573008	+	0.998357i	$0.518249\pi$
$348$		0			0
$349$			30.3659i			1.62545i	0.582647	+	0.812725i	$0.302017\pi$
$349$			30.3659i			1.62545i	−0.582647	+	0.812725i	$0.697983\pi$
$350$		0			0
$351$	13.3555	−	5.04359i	0.712864	−	0.269207i
$352$		0			0
$353$			2.02651i			0.107860i	0.998545	+	0.0539301i	$0.0171748\pi$
$353$			2.02651i			0.107860i	−0.998545	+	0.0539301i	$0.982825\pi$
$354$		0			0
$355$	−3.24757			−0.172363
$356$		0			0
$357$			3.92688i			0.207832i
$358$		0			0
$359$			11.8666i			0.626295i	0.949705	+	0.313147i	$0.101383\pi$
$359$			11.8666i			0.626295i	−0.949705	+	0.313147i	$0.898617\pi$
$360$		0			0
$361$	−8.23372			−0.433354
$362$		0			0
$363$	0.722868			0.0379407
$364$		0			0
$365$	−5.24558			−0.274566
$366$		0			0
$367$	−21.2420			−1.10882			−0.554412	−	0.832242i	$-0.687057\pi$
$367$	−21.2420			−1.10882			−0.554412	+	0.832242i	$0.687057\pi$
$368$		0			0
$369$			14.5975i			0.759913i
$370$		0			0
$371$		−	0.632174i		−	0.0328209i
$372$		0			0
$373$	−17.6162			−0.912131			−0.456066	−	0.889946i	$-0.650742\pi$
$373$	−17.6162			−0.912131			−0.456066	+	0.889946i	$0.650742\pi$
$374$		0			0
$375$			2.73344i			0.141154i
$376$		0			0
$377$	6.74394	−	2.54679i	0.347331	−	0.131166i
$378$		0			0
$379$		−	16.5983i		−	0.852595i	−0.904583	−	0.426298i	$-0.859818\pi$
$379$		−	16.5983i		−	0.852595i	0.904583	−	0.426298i	$-0.140182\pi$
$380$		0			0
$381$	5.80874			0.297591
$382$		0			0
$383$		−	18.4942i		−	0.945008i	−0.881329	−	0.472504i	$-0.843350\pi$
$383$		−	18.4942i		−	0.945008i	0.881329	−	0.472504i	$-0.156650\pi$
$384$		0			0
$385$		−	0.383791i		−	0.0195598i
$386$		0			0
$387$	22.9254			1.16536
$388$		0			0
$389$	−17.1720			−0.870653			−0.435327	−	0.900273i	$-0.643367\pi$
$389$	−17.1720			−0.870653			−0.435327	+	0.900273i	$0.643367\pi$
$390$		0			0
$391$	−15.2887			−0.773183
$392$		0			0
$393$	−10.4846			−0.528877
$394$		0			0
$395$			5.36697i			0.270041i
$396$		0			0
$397$			1.69929i			0.0852847i	0.999090	+	0.0426424i	$0.0135776\pi$
$397$			1.69929i			0.0852847i	−0.999090	+	0.0426424i	$0.986422\pi$
$398$		0			0
$399$	−3.77235			−0.188854
$400$		0			0
$401$		−	16.0941i		−	0.803703i	−0.915705	−	0.401851i	$-0.868367\pi$
$401$		−	16.0941i		−	0.803703i	0.915705	−	0.401851i	$-0.131633\pi$
$402$		0			0
$403$	2.79113	+	7.39095i	0.139036	+	0.368169i
$404$		0			0
$405$		−	1.75400i		−	0.0871572i
$406$		0			0
$407$	8.80912			0.436652
$408$		0			0
$409$		−	23.5297i		−	1.16347i	−0.813379	−	0.581734i	$-0.802374\pi$
$409$		−	23.5297i		−	1.16347i	0.813379	−	0.581734i	$-0.197626\pi$
$410$		0			0
$411$		−	3.10057i		−	0.152940i
$412$		0			0
$413$	−13.2980			−0.654351
$414$		0			0
$415$	−5.20054			−0.255284
$416$		0			0
$417$	12.0588			0.590521
$418$		0			0
$419$	6.71087			0.327847			0.163924	−	0.986473i	$-0.447585\pi$
$419$	6.71087			0.327847			0.163924	+	0.986473i	$0.447585\pi$
$420$		0			0
$421$			12.7391i			0.620864i	0.950596	+	0.310432i	$0.100474\pi$
$421$			12.7391i			0.620864i	−0.950596	+	0.310432i	$0.899526\pi$
$422$		0			0
$423$		−	20.5318i		−	0.998292i
$424$		0			0
$425$	−26.3616			−1.27873
$426$		0			0
$427$		−	13.3069i		−	0.643964i
$428$		0			0
$429$	0.920790	+	2.43827i	0.0444562	+	0.117721i
$430$		0			0
$431$			8.34857i			0.402137i	0.979577	+	0.201068i	$0.0644413\pi$
$431$			8.34857i			0.402137i	−0.979577	+	0.201068i	$0.935559\pi$
$432$		0			0
$433$	−3.00531			−0.144426			−0.0722130	−	0.997389i	$-0.523006\pi$
$433$	−3.00531			−0.144426			−0.0722130	+	0.997389i	$0.523006\pi$
$434$		0			0
$435$			0.554684i			0.0265951i
$436$		0			0
$437$		−	14.6871i		−	0.702579i
$438$		0			0
$439$	15.5404			0.741704			0.370852	−	0.928692i	$-0.379066\pi$
$439$	15.5404			0.741704			0.370852	+	0.928692i	$0.379066\pi$
$440$		0			0
$441$	2.47746			0.117974
$442$		0			0
$443$	−25.6406			−1.21822			−0.609111	−	0.793085i	$-0.708473\pi$
$443$	−25.6406			−1.21822			−0.609111	+	0.793085i	$0.708473\pi$
$444$		0			0
$445$	1.52938			0.0724996
$446$		0			0
$447$		−	14.1711i		−	0.670268i
$448$		0			0
$449$		−	9.03275i		−	0.426282i	−0.977021	−	0.213141i	$-0.931631\pi$
$449$		−	9.03275i		−	0.426282i	0.977021	−	0.213141i	$-0.0683694\pi$
$450$		0			0
$451$	−5.89210			−0.277448
$452$		0			0
$453$			1.78761i			0.0839894i
$454$		0			0
$455$	1.29454	−	0.488874i	0.0606892	−	0.0229188i
$456$		0			0
$457$			10.1393i			0.474295i	0.971474	+	0.237148i	$0.0762125\pi$
$457$			10.1393i			0.474295i	−0.971474	+	0.237148i	$0.923787\pi$
$458$		0			0
$459$	−21.5093			−1.00397
$460$		0			0
$461$		−	37.1766i		−	1.73149i	−0.500486	−	0.865745i	$-0.666845\pi$
$461$		−	37.1766i		−	1.73149i	0.500486	−	0.865745i	$-0.333155\pi$
$462$		0			0
$463$			18.4199i			0.856046i	0.903768	+	0.428023i	$0.140790\pi$
$463$			18.4199i			0.856046i	−0.903768	+	0.428023i	$0.859210\pi$
$464$		0			0
$465$	−0.607900			−0.0281907
$466$		0			0
$467$	−18.7425			−0.867298			−0.433649	−	0.901082i	$-0.642774\pi$
$467$	−18.7425			−0.867298			−0.433649	+	0.901082i	$0.642774\pi$
$468$		0			0
$469$	7.84695			0.362338
$470$		0			0
$471$	7.31153			0.336897
$472$		0			0
$473$			9.25357i			0.425480i
$474$		0			0
$475$		−	25.3243i		−	1.16196i
$476$		0			0
$477$	1.56619			0.0717108
$478$		0			0
$479$		−	10.2727i		−	0.469373i	−0.972071	−	0.234686i	$-0.924594\pi$
$479$		−	10.2727i		−	0.469373i	0.972071	−	0.234686i	$-0.0754063\pi$
$480$		0			0
$481$	11.2211	+	29.7135i	0.511636	+	1.35482i
$482$		0			0
$483$		−	2.03442i		−	0.0925694i
$484$		0			0
$485$	−1.91314			−0.0868711
$486$		0			0
$487$			39.0239i			1.76834i	0.467162	+	0.884172i	$0.345276\pi$
$487$			39.0239i			1.76834i	−0.467162	+	0.884172i	$0.654724\pi$
$488$		0			0
$489$			16.5798i			0.749764i
$490$		0			0
$491$	4.02434			0.181616			0.0908079	−	0.995868i	$-0.471055\pi$
$491$	4.02434			0.181616			0.0908079	+	0.995868i	$0.471055\pi$
$492$		0			0
$493$	−10.8613			−0.489167
$494$		0			0
$495$	0.950828			0.0427365
$496$		0			0
$497$	8.46182			0.379564
$498$		0			0
$499$			11.7324i			0.525212i	0.964903	+	0.262606i	$0.0845820\pi$
$499$			11.7324i			0.525212i	−0.964903	+	0.262606i	$0.915418\pi$
$500$		0			0
$501$			14.9085i			0.666061i
$502$		0			0
$503$	9.52353			0.424633			0.212317	−	0.977201i	$-0.431899\pi$
$503$	9.52353			0.424633			0.212317	+	0.977201i	$0.431899\pi$
$504$		0			0
$505$			0.370946i			0.0165069i
$506$		0			0
$507$	−7.05147	+	6.21173i	−0.313167	+	0.275873i
$508$		0			0
$509$		−	4.56408i		−	0.202299i	−0.994871	−	0.101150i	$-0.967748\pi$
$509$		−	4.56408i		−	0.202299i	0.994871	−	0.101150i	$-0.0322521\pi$
$510$		0			0
$511$	13.6678			0.604628
$512$		0			0
$513$		−	20.6629i		−	0.912290i
$514$		0			0
$515$			4.38477i			0.193216i
$516$		0			0
$517$	8.28744			0.364481
$518$		0			0
$519$	4.09599			0.179794
$520$		0			0
$521$	27.7904			1.21752			0.608759	−	0.793355i	$-0.291668\pi$
$521$	27.7904			1.21752			0.608759	+	0.793355i	$0.291668\pi$
$522$		0			0
$523$	−30.8738			−1.35002			−0.675009	−	0.737809i	$-0.735860\pi$
$523$	−30.8738			−1.35002			−0.675009	+	0.737809i	$0.735860\pi$
$524$		0			0
$525$		−	3.50786i		−	0.153096i
$526$		0			0
$527$		−	11.9033i		−	0.518515i
$528$		0			0
$529$	−15.0793			−0.655621
$530$		0			0
$531$		−	32.9452i		−	1.42970i
$532$		0			0
$533$	−7.50536	−	19.8743i	−0.325093	−	0.860852i
$534$		0			0
$535$		−	4.69530i		−	0.202996i
$536$		0			0
$537$	9.07573			0.391647
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$			25.6566i			1.10306i	0.834154	+	0.551532i	$0.185957\pi$
$541$			25.6566i			1.10306i	−0.834154	+	0.551532i	$0.814043\pi$
$542$		0			0
$543$	10.7765			0.462462
$544$		0			0
$545$	0.633742			0.0271465
$546$		0			0
$547$	−17.7340			−0.758251			−0.379126	−	0.925345i	$-0.623775\pi$
$547$	−17.7340			−0.758251			−0.379126	+	0.925345i	$0.623775\pi$
$548$		0			0
$549$	32.9672			1.40701
$550$		0			0
$551$		−	10.4339i		−	0.444498i
$552$		0			0
$553$		−	13.9841i		−	0.594664i
$554$		0			0
$555$	−2.44392			−0.103738
$556$		0			0
$557$		−	4.60249i		−	0.195014i	−0.995235	−	0.0975069i	$-0.968913\pi$
$557$		−	4.60249i		−	0.195014i	0.995235	−	0.0975069i	$-0.0310868\pi$
$558$		0			0
$559$	−31.2127	+	11.7872i	−1.32016	+	0.498546i
$560$		0			0
$561$		−	3.92688i		−	0.165793i
$562$		0			0
$563$	15.6375			0.659041			0.329521	−	0.944148i	$-0.393113\pi$
$563$	15.6375			0.659041			0.329521	+	0.944148i	$0.393113\pi$
$564$		0			0
$565$			2.78545i			0.117185i
$566$		0			0
$567$			4.57021i			0.191931i
$568$		0			0
$569$	27.8266			1.16655			0.583276	−	0.812274i	$-0.301771\pi$
$569$	27.8266			1.16655			0.583276	+	0.812274i	$0.301771\pi$
$570$		0			0
$571$	−32.8460			−1.37456			−0.687280	−	0.726392i	$-0.741196\pi$
$571$	−32.8460			−1.37456			−0.687280	+	0.726392i	$0.741196\pi$
$572$		0			0
$573$	12.8198			0.535555
$574$		0			0
$575$	13.6573			0.569550
$576$		0			0
$577$			13.7664i			0.573102i	0.958065	+	0.286551i	$0.0925088\pi$
$577$			13.7664i			0.573102i	−0.958065	+	0.286551i	$0.907491\pi$
$578$		0			0
$579$		−	11.1991i		−	0.465420i
$580$		0			0
$581$	13.5504			0.562167
$582$		0			0
$583$			0.632174i			0.0261820i
$584$		0			0
$585$	1.21117	+	3.20719i	0.0500756	+	0.132601i
$586$		0			0
$587$		−	19.0750i		−	0.787310i	−0.919258	−	0.393655i	$-0.871211\pi$
$587$		−	19.0750i		−	0.787310i	0.919258	−	0.393655i	$-0.128789\pi$
$588$		0			0
$589$	11.4349			0.471166
$590$		0			0
$591$			3.63269i			0.149429i
$592$		0			0
$593$			21.8115i			0.895690i	0.894111	+	0.447845i	$0.147808\pi$
$593$			21.8115i			0.895690i	−0.894111	+	0.447845i	$0.852192\pi$
$594$		0			0
$595$	−2.08489			−0.0854722
$596$		0			0
$597$	11.9483			0.489011
$598$		0			0
$599$	1.30419			0.0532880			0.0266440	−	0.999645i	$-0.491518\pi$
$599$	1.30419			0.0532880			0.0266440	+	0.999645i	$0.491518\pi$
$600$		0			0
$601$	44.8904			1.83112			0.915559	−	0.402183i	$-0.131748\pi$
$601$	44.8904			1.83112			0.915559	+	0.402183i	$0.131748\pi$
$602$		0			0
$603$			19.4405i			0.791679i
$604$		0			0
$605$			0.383791i			0.0156033i
$606$		0			0
$607$	−4.20928			−0.170849			−0.0854247	−	0.996345i	$-0.527225\pi$
$607$	−4.20928			−0.170849			−0.0854247	+	0.996345i	$0.527225\pi$
$608$		0			0
$609$		−	1.44528i		−	0.0585655i
$610$		0			0
$611$	10.5566	+	27.9539i	0.427073	+	1.13089i
$612$		0			0
$613$			18.2079i			0.735411i	0.929942	+	0.367706i	$0.119857\pi$
$613$			18.2079i			0.735411i	−0.929942	+	0.367706i	$0.880143\pi$
$614$		0			0
$615$	1.63465			0.0659153
$616$		0			0
$617$			14.2443i			0.573455i	0.958012	+	0.286727i	$0.0925673\pi$
$617$			14.2443i			0.573455i	−0.958012	+	0.286727i	$0.907433\pi$
$618$		0			0
$619$		−	30.1836i		−	1.21318i	−0.795014	−	0.606591i	$-0.792537\pi$
$619$		−	30.1836i		−	1.21318i	0.795014	−	0.606591i	$-0.207463\pi$
$620$		0			0
$621$	11.1435			0.447172
$622$		0			0
$623$	−3.98493			−0.159653
$624$		0			0
$625$	22.8123			0.912490
$626$		0			0
$627$	3.77235			0.150653
$628$		0			0
$629$		−	47.8543i		−	1.90808i
$630$		0			0
$631$			1.93512i			0.0770358i	0.999258	+	0.0385179i	$0.0122637\pi$
$631$			1.93512i			0.0770358i	−0.999258	+	0.0385179i	$0.987736\pi$
$632$		0			0
$633$	16.6593			0.662146
$634$		0			0
$635$			3.08402i			0.122386i
$636$		0			0
$637$	−3.37304	+	1.27380i	−0.133645	+	0.0504698i
$638$		0			0
$639$			20.9638i			0.829316i
$640$		0			0
$641$	45.5002			1.79715			0.898574	−	0.438821i	$-0.144604\pi$
$641$	45.5002			1.79715			0.898574	+	0.438821i	$0.144604\pi$
$642$		0			0
$643$		−	18.6451i		−	0.735292i	−0.929966	−	0.367646i	$-0.880164\pi$
$643$		−	18.6451i		−	0.735292i	0.929966	−	0.367646i	$-0.119836\pi$
$644$		0			0
$645$		−	2.56722i		−	0.101084i
$646$		0			0
$647$	27.6333			1.08638			0.543189	−	0.839611i	$-0.317217\pi$
$647$	27.6333			1.08638			0.543189	+	0.839611i	$0.317217\pi$
$648$		0			0
$649$	13.2980			0.521991
$650$		0			0
$651$	1.58393			0.0620793
$652$		0			0
$653$	10.6935			0.418468			0.209234	−	0.977866i	$-0.432903\pi$
$653$	10.6935			0.418468			0.209234	+	0.977866i	$0.432903\pi$
$654$		0			0
$655$		−	5.56656i		−	0.217503i
$656$		0			0
$657$			33.8615i			1.32106i
$658$		0			0
$659$	−11.1509			−0.434376			−0.217188	−	0.976130i	$-0.569688\pi$
$659$	−11.1509			−0.434376			−0.217188	+	0.976130i	$0.569688\pi$
$660$		0			0
$661$		−	6.18156i		−	0.240435i	−0.992748	−	0.120217i	$-0.961641\pi$
$661$		−	6.18156i		−	0.240435i	0.992748	−	0.120217i	$-0.0383591\pi$
$662$		0			0
$663$	13.2455	−	5.00206i	0.514414	−	0.194264i
$664$		0			0
$665$		−	2.00285i		−	0.0776672i
$666$		0			0
$667$	5.62696			0.217877
$668$		0			0
$669$			2.25866i			0.0873247i
$670$		0			0
$671$			13.3069i			0.513705i
$672$		0			0
$673$	24.1526			0.931015			0.465507	−	0.885044i	$-0.345872\pi$
$673$	24.1526			0.931015			0.465507	+	0.885044i	$0.345872\pi$
$674$		0			0
$675$	19.2142			0.739554
$676$		0			0
$677$	25.1643			0.967143			0.483571	−	0.875305i	$-0.339339\pi$
$677$	25.1643			0.967143			0.483571	+	0.875305i	$0.339339\pi$
$678$		0			0
$679$	4.98484			0.191301
$680$		0			0
$681$			8.32569i			0.319041i
$682$		0			0
$683$			38.7812i			1.48392i	0.670443	+	0.741961i	$0.266104\pi$
$683$			38.7812i			1.48392i	−0.670443	+	0.741961i	$0.733896\pi$
$684$		0			0
$685$	1.64618			0.0628974
$686$		0			0
$687$		−	2.76352i		−	0.105435i
$688$		0			0
$689$	−2.13235	+	0.805265i	−0.0812362	+	0.0306781i
$690$		0			0
$691$		−	3.27158i		−	0.124457i	−0.998062	−	0.0622283i	$-0.980179\pi$
$691$		−	3.27158i		−	0.124457i	0.998062	−	0.0622283i	$-0.0198207\pi$
$692$		0			0
$693$	−2.47746			−0.0941110
$694$		0			0
$695$			6.40235i			0.242855i
$696$		0			0
$697$			32.0080i			1.21239i
$698$		0			0
$699$	3.28833			0.124376
$700$		0			0
$701$	−48.2566			−1.82263			−0.911313	−	0.411715i	$-0.864930\pi$
$701$	−48.2566			−1.82263			−0.911313	+	0.411715i	$0.864930\pi$
$702$		0			0
$703$	45.9712			1.73384
$704$		0			0
$705$	−2.29919			−0.0865924
$706$		0			0
$707$		−	0.966532i		−	0.0363502i
$708$		0			0
$709$			47.1828i			1.77199i	0.463698	+	0.885993i	$0.346522\pi$
$709$			47.1828i			1.77199i	−0.463698	+	0.885993i	$0.653478\pi$
$710$		0			0
$711$	34.6450			1.29929
$712$		0			0
$713$			6.16681i			0.230949i
$714$		0			0
$715$	−1.29454	+	0.488874i	−0.0484132	+	0.0182828i
$716$		0			0
$717$			8.91388i			0.332895i
$718$		0			0
$719$	−3.56804			−0.133066			−0.0665328	−	0.997784i	$-0.521194\pi$
$719$	−3.56804			−0.133066			−0.0665328	+	0.997784i	$0.521194\pi$
$720$		0			0
$721$		−	11.4249i		−	0.425485i
$722$		0			0
$723$			15.7182i			0.584566i
$724$		0			0
$725$	9.70232			0.360335
$726$		0			0
$727$	27.6748			1.02640			0.513202	−	0.858268i	$-0.328459\pi$
$727$	27.6748			1.02640			0.513202	+	0.858268i	$0.328459\pi$
$728$		0			0
$729$	−2.73597			−0.101332
$730$		0			0
$731$	50.2687			1.85926
$732$		0			0
$733$		−	37.6774i		−	1.39165i	−0.718212	−	0.695824i	$-0.755039\pi$
$733$		−	37.6774i		−	1.39165i	0.718212	−	0.695824i	$-0.244961\pi$
$734$		0			0
$735$		−	0.277430i		−	0.0102332i
$736$		0			0
$737$	−7.84695			−0.289046
$738$		0			0
$739$		−	24.3642i		−	0.896252i	−0.893970	−	0.448126i	$-0.852092\pi$
$739$		−	24.3642i		−	0.896252i	0.893970	−	0.448126i	$-0.147908\pi$
$740$		0			0
$741$	4.80523	+	12.7243i	0.176524	+	0.467439i
$742$		0			0
$743$			41.4463i			1.52052i	0.649622	+	0.760258i	$0.274927\pi$
$743$			41.4463i			1.52052i	−0.649622	+	0.760258i	$0.725073\pi$
$744$		0			0
$745$	7.52382			0.275652
$746$		0			0
$747$			33.5707i			1.22829i
$748$		0			0
$749$			12.2340i			0.447021i
$750$		0			0
$751$	−36.2124			−1.32141			−0.660705	−	0.750645i	$-0.729743\pi$
$751$	−36.2124			−1.32141			−0.660705	+	0.750645i	$0.729743\pi$
$752$		0			0
$753$	−11.1101			−0.404875
$754$		0			0
$755$	−0.949095			−0.0345411
$756$		0			0
$757$	40.6886			1.47885			0.739426	−	0.673238i	$-0.235097\pi$
$757$	40.6886			1.47885			0.739426	+	0.673238i	$0.235097\pi$
$758$		0			0
$759$			2.03442i			0.0738448i
$760$		0			0
$761$		−	30.1176i		−	1.09176i	−0.837863	−	0.545881i	$-0.816195\pi$
$761$		−	30.1176i		−	1.09176i	0.837863	−	0.545881i	$-0.183805\pi$
$762$		0			0
$763$	−1.65127			−0.0597799
$764$		0			0
$765$		−	5.16524i		−	0.186750i
$766$		0			0
$767$	16.9390	+	44.8547i	0.611631	+	1.61961i
$768$		0			0
$769$		−	26.4775i		−	0.954803i	−0.878685	−	0.477401i	$-0.841579\pi$
$769$		−	26.4775i		−	0.954803i	0.878685	−	0.477401i	$-0.158421\pi$
$770$		0			0
$771$	−9.22856			−0.332358
$772$		0			0
$773$		−	24.1869i		−	0.869944i	−0.900444	−	0.434972i	$-0.856758\pi$
$773$		−	24.1869i		−	0.869944i	0.900444	−	0.434972i	$-0.143242\pi$
$774$		0			0
$775$			10.6332i			0.381954i
$776$		0			0
$777$	6.36783			0.228444
$778$		0			0
$779$	−30.7485			−1.10168
$780$		0			0
$781$	−8.46182			−0.302788
$782$		0			0
$783$	7.91644			0.282911
$784$		0			0
$785$			3.88190i			0.138551i
$786$		0			0
$787$			2.31299i			0.0824493i	0.999150	+	0.0412247i	$0.0131259\pi$
$787$			2.31299i			0.0824493i	−0.999150	+	0.0412247i	$0.986874\pi$
$788$		0			0
$789$	19.9605			0.710612
$790$		0			0
$791$		−	7.25772i		−	0.258055i
$792$		0			0
$793$	−44.8846	+	16.9503i	−1.59390	+	0.601922i
$794$		0			0
$795$		−	0.175384i		−	0.00622024i
$796$		0			0
$797$	3.77307			0.133649			0.0668245	−	0.997765i	$-0.478713\pi$
$797$	3.77307			0.133649			0.0668245	+	0.997765i	$0.478713\pi$
$798$		0			0
$799$		−	45.0204i		−	1.59271i
$800$		0			0
$801$		−	9.87252i		−	0.348828i
$802$		0			0
$803$	−13.6678			−0.482326
$804$		0			0
$805$	1.08013			0.0380697
$806$		0			0
$807$	−20.3704			−0.717074
$808$		0			0
$809$	25.8667			0.909425			0.454712	−	0.890638i	$-0.349742\pi$
$809$	25.8667			0.909425			0.454712	+	0.890638i	$0.349742\pi$
$810$		0			0
$811$		−	48.4744i		−	1.70217i	−0.525030	−	0.851084i	$-0.675946\pi$
$811$		−	48.4744i		−	1.70217i	0.525030	−	0.851084i	$-0.324054\pi$
$812$		0			0
$813$		−	2.86766i		−	0.100573i
$814$		0			0
$815$	−8.80269			−0.308345
$816$		0			0
$817$			48.2906i			1.68948i
$818$		0			0
$819$	−3.15579	−	8.35659i	−0.110272	−	0.292003i
$820$		0			0
$821$			51.5115i			1.79776i	0.438191	+	0.898882i	$0.355619\pi$
$821$			51.5115i			1.79776i	−0.438191	+	0.898882i	$0.644381\pi$
$822$		0			0
$823$	−10.5098			−0.366350			−0.183175	−	0.983080i	$-0.558638\pi$
$823$	−10.5098			−0.366350			−0.183175	+	0.983080i	$0.558638\pi$
$824$		0			0
$825$			3.50786i			0.122128i
$826$		0			0
$827$		−	32.7724i		−	1.13961i	−0.821781	−	0.569804i	$-0.807019\pi$
$827$		−	32.7724i		−	1.13961i	0.821781	−	0.569804i	$-0.192981\pi$
$828$		0			0
$829$	−29.0578			−1.00922			−0.504609	−	0.863348i	$-0.668363\pi$
$829$	−29.0578			−1.00922			−0.504609	+	0.863348i	$0.668363\pi$
$830$		0			0
$831$	−18.0944			−0.627689
$832$		0			0
$833$	5.43236			0.188220
$834$		0			0
$835$	−7.91533			−0.273921
$836$		0			0
$837$			8.67594i			0.299884i
$838$		0			0
$839$			32.8360i			1.13362i	0.823847	+	0.566812i	$0.191823\pi$
$839$			32.8360i			1.13362i	−0.823847	+	0.566812i	$0.808177\pi$
$840$		0			0
$841$	−25.0025			−0.862157
$842$		0			0
$843$			2.59825i			0.0894886i
$844$		0			0
$845$	−3.29799	−	3.74383i	−0.113454	−	0.128792i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	8.10642			0.278212
$850$		0			0
$851$			24.7922i			0.849865i
$852$		0			0
$853$		−	12.7512i		−	0.436592i	−0.975883	−	0.218296i	$-0.929950\pi$
$853$		−	12.7512i		−	0.436592i	0.975883	−	0.218296i	$-0.0700498\pi$
$854$		0			0
$855$	4.96199			0.169696
$856$		0			0
$857$	−27.6957			−0.946068			−0.473034	−	0.881044i	$-0.656841\pi$
$857$	−27.6957			−0.946068			−0.473034	+	0.881044i	$0.656841\pi$
$858$		0			0
$859$	17.3531			0.592080			0.296040	−	0.955176i	$-0.404334\pi$
$859$	17.3531			0.592080			0.296040	+	0.955176i	$0.404334\pi$
$860$		0			0
$861$	−4.25921			−0.145153
$862$		0			0
$863$			30.9484i			1.05350i	0.850022	+	0.526748i	$0.176589\pi$
$863$			30.9484i			1.05350i	−0.850022	+	0.526748i	$0.823411\pi$
$864$		0			0
$865$			2.17468i			0.0739412i
$866$		0			0
$867$	−9.04347			−0.307132
$868$		0			0
$869$			13.9841i			0.474378i
$870$		0			0
$871$	−9.99545	−	26.4681i	−0.338683	−	0.896838i
$872$		0			0
$873$			12.3498i			0.417976i
$874$		0			0
$875$	3.78138			0.127834
$876$		0			0
$877$			3.39890i			0.114773i	0.998352	+	0.0573863i	$0.0182767\pi$
$877$			3.39890i			0.114773i	−0.998352	+	0.0573863i	$0.981723\pi$
$878$		0			0
$879$		−	7.41519i		−	0.250108i
$880$		0			0
$881$	7.97084			0.268544			0.134272	−	0.990944i	$-0.457130\pi$
$881$	7.97084			0.268544			0.134272	+	0.990944i	$0.457130\pi$
$882$		0			0
$883$	8.74983			0.294455			0.147228	−	0.989103i	$-0.452965\pi$
$883$	8.74983			0.294455			0.147228	+	0.989103i	$0.452965\pi$
$884$		0			0
$885$	−3.68926			−0.124013
$886$		0			0
$887$	−17.8797			−0.600341			−0.300170	−	0.953886i	$-0.597043\pi$
$887$	−17.8797			−0.600341			−0.300170	+	0.953886i	$0.597043\pi$
$888$		0			0
$889$		−	8.03568i		−	0.269508i
$890$		0			0
$891$		−	4.57021i		−	0.153108i
$892$		0			0
$893$	43.2488			1.44727
$894$		0			0
$895$			4.81857i			0.161067i
$896$		0			0
$897$	−6.86220	+	2.59145i	−0.229122	+	0.0865260i
$898$		0			0
$899$			4.38097i			0.146113i
$900$		0			0
$901$	3.43420			0.114410
$902$		0			0
$903$			6.68911i			0.222600i
$904$		0			0
$905$			5.72153i			0.190190i
$906$		0			0
$907$	−19.3274			−0.641757			−0.320879	−	0.947120i	$-0.603978\pi$
$907$	−19.3274			−0.641757			−0.320879	+	0.947120i	$0.603978\pi$
$908$		0			0
$909$	2.39455			0.0794221
$910$		0			0
$911$	25.0603			0.830286			0.415143	−	0.909756i	$-0.363732\pi$
$911$	25.0603			0.830286			0.415143	+	0.909756i	$0.363732\pi$
$912$		0			0
$913$	−13.5504			−0.448454
$914$		0			0
$915$		−	3.69172i		−	0.122045i
$916$		0			0
$917$			14.5041i			0.478969i
$918$		0			0
$919$	46.5914			1.53691			0.768454	−	0.639905i	$-0.221026\pi$
$919$	46.5914			1.53691			0.768454	+	0.639905i	$0.221026\pi$
$920$		0			0
$921$		−	13.1088i		−	0.431950i
$922$		0			0
$923$	−10.7787	−	28.5421i	−0.354784	−	0.939474i
$924$		0			0
$925$			42.7480i			1.40555i
$926$		0			0
$927$	28.3047			0.929649
$928$		0			0
$929$			53.2838i			1.74818i	0.485760	+	0.874092i	$0.338543\pi$
$929$			53.2838i			1.74818i	−0.485760	+	0.874092i	$0.661457\pi$
$930$		0			0
$931$			5.21859i			0.171032i
$932$		0			0
$933$	15.4102			0.504506
$934$		0			0
$935$	2.08489			0.0681833
$936$		0			0
$937$	5.72228			0.186939			0.0934693	−	0.995622i	$-0.470204\pi$
$937$	5.72228			0.186939			0.0934693	+	0.995622i	$0.470204\pi$
$938$		0			0
$939$	0.382619			0.0124863
$940$		0			0
$941$			10.7898i			0.351739i	0.984414	+	0.175869i	$0.0562736\pi$
$941$			10.7898i			0.351739i	−0.984414	+	0.175869i	$0.943726\pi$
$942$		0			0
$943$		−	16.5826i		−	0.540003i
$944$		0			0
$945$	1.51961			0.0494330
$946$		0			0
$947$		−	5.73552i		−	0.186379i	−0.995648	−	0.0931896i	$-0.970294\pi$
$947$		−	5.73552i		−	0.186379i	0.995648	−	0.0931896i	$-0.0297063\pi$
$948$		0			0
$949$	−17.4101	−	46.1021i	−0.565155	−	1.49654i
$950$		0			0
$951$			2.19687i			0.0712385i
$952$		0			0
$953$	30.9541			1.00270			0.501351	−	0.865244i	$-0.332836\pi$
$953$	30.9541			1.00270			0.501351	+	0.865244i	$0.332836\pi$
$954$		0			0
$955$			6.80641i			0.220250i
$956$		0			0
$957$			1.44528i			0.0467191i
$958$		0			0
$959$	−4.28927			−0.138508
$960$		0			0
$961$	26.1987			0.845120
$962$		0			0
$963$	−30.3093			−0.976703
$964$		0			0
$965$	5.94594			0.191407
$966$		0			0
$967$			6.27307i			0.201728i	0.994900	+	0.100864i	$0.0321608\pi$
$967$			6.27307i			0.201728i	−0.994900	+	0.100864i	$0.967839\pi$
$968$		0			0
$969$		−	20.4928i		−	0.658323i
$970$		0			0
$971$	17.2312			0.552974			0.276487	−	0.961018i	$-0.410830\pi$
$971$	17.2312			0.552974			0.276487	+	0.961018i	$0.410830\pi$
$972$		0			0
$973$		−	16.6819i		−	0.534796i
$974$		0			0
$975$	−11.8322	+	4.46832i	−0.378933	+	0.143101i
$976$		0			0
$977$			3.94006i			0.126054i	0.998012	+	0.0630269i	$0.0200754\pi$
$977$			3.94006i			0.126054i	−0.998012	+	0.0630269i	$0.979925\pi$
$978$		0			0
$979$	3.98493			0.127359
$980$		0			0
$981$		−	4.09095i		−	0.130614i
$982$		0			0
$983$		−	36.4023i		−	1.16105i	−0.814241	−	0.580527i	$-0.802847\pi$
$983$		−	36.4023i		−	1.16105i	0.814241	−	0.580527i	$-0.197153\pi$
$984$		0			0
$985$	−1.92870			−0.0614535
$986$		0			0
$987$	5.99073			0.190687
$988$		0			0
$989$	−26.0430			−0.828121
$990$		0			0
$991$	40.8466			1.29754			0.648768	−	0.760986i	$-0.275285\pi$
$991$	40.8466			1.29754			0.648768	+	0.760986i	$0.275285\pi$
$992$		0			0
$993$		−	11.4823i		−	0.364381i
$994$		0			0
$995$			6.34368i			0.201108i
$996$		0			0
$997$	−36.3419			−1.15096			−0.575480	−	0.817816i	$-0.695185\pi$
$997$	−36.3419			−1.15096			−0.575480	+	0.817816i	$0.695185\pi$
$998$		0			0
$999$			34.8795i			1.10354i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.18	yes	36
13.12	even	2	inner	4004.2.m.c.2157.17	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.17	✓	36	13.12	even	2	inner
4004.2.m.c.2157.18	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)

\(f(q)\)	\(=\)	\(q-0.722868 q^{3} -0.383791i q^{5} +1.00000i q^{7} -2.47746 q^{9} +O(q^{10})\)	\(q-0.722868 q^{3} -0.383791i q^{5} +1.00000i q^{7} -2.47746 q^{9} -1.00000i q^{11} +(3.37304 - 1.27380i) q^{13} +0.277430i q^{15} -5.43236 q^{17} -5.21859i q^{19} -0.722868i q^{21} +2.81438 q^{23} +4.85270 q^{25} +3.95948 q^{27} +1.99936 q^{29} +2.19118i q^{31} +0.722868i q^{33} +0.383791 q^{35} +8.80912i q^{37} +(-2.43827 + 0.920790i) q^{39} -5.89210i q^{41} -9.25357 q^{43} +0.950828i q^{45} +8.28744i q^{47} -1.00000 q^{49} +3.92688 q^{51} -0.632174 q^{53} -0.383791 q^{55} +3.77235i q^{57} +13.2980i q^{59} -13.3069 q^{61} -2.47746i q^{63} +(-0.488874 - 1.29454i) q^{65} -7.84695i q^{67} -2.03442 q^{69} -8.46182i q^{71} -13.6678i q^{73} -3.50786 q^{75} +1.00000 q^{77} -13.9841 q^{79} +4.57021 q^{81} -13.5504i q^{83} +2.08489i q^{85} -1.44528 q^{87} +3.98493i q^{89} +(1.27380 + 3.37304i) q^{91} -1.58393i q^{93} -2.00285 q^{95} -4.98484i q^{97} +2.47746i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 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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