LMFDB - Embedded newform 4004.2.m.c.2157.35

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			2157.35
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.36

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.38076 q^{3} +0.456146i q^{5} +1.00000i q^{7} +8.42952 q^{9} +O(q^{10})$	$q+3.38076 q^{3} +0.456146i q^{5} +1.00000i q^{7} +8.42952 q^{9} -1.00000i q^{11} +(-3.09402 + 1.85123i) q^{13} +1.54212i q^{15} -0.655725 q^{17} +4.48109i q^{19} +3.38076i q^{21} +2.36260 q^{23} +4.79193 q^{25} +18.3559 q^{27} +3.97545 q^{29} -6.41458i q^{31} -3.38076i q^{33} -0.456146 q^{35} -5.80729i q^{37} +(-10.4601 + 6.25854i) q^{39} +4.28642i q^{41} +1.05655 q^{43} +3.84509i q^{45} -2.20785i q^{47} -1.00000 q^{49} -2.21685 q^{51} +2.63216 q^{53} +0.456146 q^{55} +15.1495i q^{57} +4.87701i q^{59} -1.09790 q^{61} +8.42952i q^{63} +(-0.844429 - 1.41133i) q^{65} +9.64890i q^{67} +7.98738 q^{69} +5.47883i q^{71} +16.3757i q^{73} +16.2004 q^{75} +1.00000 q^{77} +9.07736 q^{79} +36.7683 q^{81} +0.226954i q^{83} -0.299107i q^{85} +13.4400 q^{87} +13.9962i q^{89} +(-1.85123 - 3.09402i) q^{91} -21.6861i q^{93} -2.04403 q^{95} -15.7812i q^{97} -8.42952i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.38076			1.95188			0.975941	−	0.218036i	$-0.0699651\pi$
$3$	3.38076			1.95188			0.975941	+	0.218036i	$0.0699651\pi$
$4$		0			0
$5$			0.456146i			0.203995i	0.994785	+	0.101997i	$0.0325233\pi$
$5$			0.456146i			0.203995i	−0.994785	+	0.101997i	$0.967477\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	8.42952			2.80984
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−3.09402	+	1.85123i	−0.858127	+	0.513438i
$13$	−3.09402	+	1.85123i	−0.858127	+	0.513438i
$14$		0			0
$15$			1.54212i			0.398174i
$16$		0			0
$17$	−0.655725			−0.159037			−0.0795184	−	0.996833i	$-0.525338\pi$
$17$	−0.655725			−0.159037			−0.0795184	+	0.996833i	$0.525338\pi$
$18$		0			0
$19$			4.48109i			1.02803i	0.857780	+	0.514016i	$0.171843\pi$
$19$			4.48109i			1.02803i	−0.857780	+	0.514016i	$0.828157\pi$
$20$		0			0
$21$			3.38076i			0.737742i
$22$		0			0
$23$	2.36260			0.492636			0.246318	−	0.969189i	$-0.420779\pi$
$23$	2.36260			0.492636			0.246318	+	0.969189i	$0.420779\pi$
$24$		0			0
$25$	4.79193			0.958386
$26$		0			0
$27$	18.3559			3.53259
$28$		0			0
$29$	3.97545			0.738223			0.369111	−	0.929385i	$-0.379662\pi$
$29$	3.97545			0.738223			0.369111	+	0.929385i	$0.379662\pi$
$30$		0			0
$31$		−	6.41458i		−	1.15209i	−0.817417	−	0.576046i	$-0.804595\pi$
$31$		−	6.41458i		−	1.15209i	0.817417	−	0.576046i	$-0.195405\pi$
$32$		0			0
$33$		−	3.38076i		−	0.588514i
$34$		0			0
$35$	−0.456146			−0.0771028
$36$		0			0
$37$		−	5.80729i		−	0.954712i	−0.878710	−	0.477356i	$-0.841595\pi$
$37$		−	5.80729i		−	0.954712i	0.878710	−	0.477356i	$-0.158405\pi$
$38$		0			0
$39$	−10.4601	+	6.25854i	−1.67496	+	1.00217i
$40$		0			0
$41$			4.28642i			0.669427i	0.942320	+	0.334713i	$0.108639\pi$
$41$			4.28642i			0.669427i	−0.942320	+	0.334713i	$0.891361\pi$
$42$		0			0
$43$	1.05655			0.161122			0.0805612	−	0.996750i	$-0.474329\pi$
$43$	1.05655			0.161122			0.0805612	+	0.996750i	$0.474329\pi$
$44$		0			0
$45$			3.84509i			0.573193i
$46$		0			0
$47$		−	2.20785i		−	0.322049i	−0.986950	−	0.161024i	$-0.948520\pi$
$47$		−	2.20785i		−	0.322049i	0.986950	−	0.161024i	$-0.0514798\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−2.21685			−0.310421
$52$		0			0
$53$	2.63216			0.361555			0.180778	−	0.983524i	$-0.442139\pi$
$53$	2.63216			0.361555			0.180778	+	0.983524i	$0.442139\pi$
$54$		0			0
$55$	0.456146			0.0615067
$56$		0			0
$57$			15.1495i			2.00660i
$58$		0			0
$59$			4.87701i			0.634933i	0.948269	+	0.317466i	$0.102832\pi$
$59$			4.87701i			0.634933i	−0.948269	+	0.317466i	$0.897168\pi$
$60$		0			0
$61$	−1.09790			−0.140572			−0.0702859	−	0.997527i	$-0.522391\pi$
$61$	−1.09790			−0.140572			−0.0702859	+	0.997527i	$0.522391\pi$
$62$		0			0
$63$			8.42952i			1.06202i
$64$		0			0
$65$	−0.844429	−	1.41133i	−0.104739	−	0.175053i
$66$		0			0
$67$			9.64890i			1.17880i	0.807841	+	0.589400i	$0.200636\pi$
$67$			9.64890i			1.17880i	−0.807841	+	0.589400i	$0.799364\pi$
$68$		0			0
$69$	7.98738			0.961567
$70$		0			0
$71$			5.47883i			0.650217i	0.945677	+	0.325109i	$0.105401\pi$
$71$			5.47883i			0.650217i	−0.945677	+	0.325109i	$0.894599\pi$
$72$		0			0
$73$			16.3757i			1.91663i	0.285707	+	0.958317i	$0.407771\pi$
$73$			16.3757i			1.91663i	−0.285707	+	0.958317i	$0.592229\pi$
$74$		0			0
$75$	16.2004			1.87066
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	9.07736			1.02128			0.510641	−	0.859794i	$-0.329408\pi$
$79$	9.07736			1.02128			0.510641	+	0.859794i	$0.329408\pi$
$80$		0			0
$81$	36.7683			4.08536
$82$		0			0
$83$			0.226954i			0.0249115i	0.999922	+	0.0124557i	$0.00396489\pi$
$83$			0.226954i			0.0249115i	−0.999922	+	0.0124557i	$0.996035\pi$
$84$		0			0
$85$		−	0.299107i		−	0.0324427i
$86$		0			0
$87$	13.4400			1.44092
$88$		0			0
$89$			13.9962i			1.48359i	0.670624	+	0.741797i	$0.266026\pi$
$89$			13.9962i			1.48359i	−0.670624	+	0.741797i	$0.733974\pi$
$90$		0			0
$91$	−1.85123	−	3.09402i	−0.194061	−	0.324341i
$92$		0			0
$93$		−	21.6861i		−	2.24875i
$94$		0			0
$95$	−2.04403			−0.209713
$96$		0			0
$97$		−	15.7812i		−	1.60233i	−0.598441	−	0.801167i	$-0.704213\pi$
$97$		−	15.7812i		−	1.60233i	0.598441	−	0.801167i	$-0.295787\pi$
$98$		0			0
$99$		−	8.42952i		−	0.847199i
$100$		0			0
$101$	−15.8170			−1.57385			−0.786924	−	0.617050i	$-0.788328\pi$
$101$	−15.8170			−1.57385			−0.786924	+	0.617050i	$0.788328\pi$
$102$		0			0
$103$	−15.3678			−1.51424			−0.757118	−	0.653278i	$-0.773393\pi$
$103$	−15.3678			−1.51424			−0.757118	+	0.653278i	$0.773393\pi$
$104$		0			0
$105$	−1.54212			−0.150495
$106$		0			0
$107$	12.0461			1.16454			0.582271	−	0.812994i	$-0.302164\pi$
$107$	12.0461			1.16454			0.582271	+	0.812994i	$0.302164\pi$
$108$		0			0
$109$		−	9.50668i		−	0.910575i	−0.890344	−	0.455288i	$-0.849536\pi$
$109$		−	9.50668i		−	0.910575i	0.890344	−	0.455288i	$-0.150464\pi$
$110$		0			0
$111$		−	19.6330i		−	1.86348i
$112$		0			0
$113$	−2.85514			−0.268589			−0.134294	−	0.990941i	$-0.542877\pi$
$113$	−2.85514			−0.268589			−0.134294	+	0.990941i	$0.542877\pi$
$114$		0			0
$115$			1.07769i			0.100495i
$116$		0			0
$117$	−26.0811	+	15.6049i	−2.41120	+	1.44268i
$118$		0			0
$119$		−	0.655725i		−	0.0601102i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			14.4914i			1.30664i
$124$		0			0
$125$			4.46655i			0.399501i
$126$		0			0
$127$	4.79378			0.425379			0.212690	−	0.977120i	$-0.431778\pi$
$127$	4.79378			0.425379			0.212690	+	0.977120i	$0.431778\pi$
$128$		0			0
$129$	3.57194			0.314492
$130$		0			0
$131$	−15.1497			−1.32364			−0.661818	−	0.749665i	$-0.730215\pi$
$131$	−15.1497			−1.32364			−0.661818	+	0.749665i	$0.730215\pi$
$132$		0			0
$133$	−4.48109			−0.388560
$134$		0			0
$135$			8.37297i			0.720631i
$136$		0			0
$137$		−	17.9967i		−	1.53756i	−0.639515	−	0.768779i	$-0.720865\pi$
$137$		−	17.9967i		−	1.53756i	0.639515	−	0.768779i	$-0.279135\pi$
$138$		0			0
$139$	9.36382			0.794229			0.397114	−	0.917769i	$-0.370012\pi$
$139$	9.36382			0.794229			0.397114	+	0.917769i	$0.370012\pi$
$140$		0			0
$141$		−	7.46422i		−	0.628601i
$142$		0			0
$143$	1.85123	+	3.09402i	0.154807	+	0.258735i
$144$		0			0
$145$			1.81339i			0.150594i
$146$		0			0
$147$	−3.38076			−0.278840
$148$		0			0
$149$			0.225212i			0.0184501i	0.999957	+	0.00922503i	$0.00293646\pi$
$149$			0.225212i			0.0184501i	−0.999957	+	0.00922503i	$0.997064\pi$
$150$		0			0
$151$		−	6.32260i		−	0.514526i	−0.966341	−	0.257263i	$-0.917179\pi$
$151$		−	6.32260i		−	0.514526i	0.966341	−	0.257263i	$-0.0828206\pi$
$152$		0			0
$153$	−5.52745			−0.446868
$154$		0			0
$155$	2.92599			0.235021
$156$		0			0
$157$	0.745119			0.0594670			0.0297335	−	0.999558i	$-0.490534\pi$
$157$	0.745119			0.0594670			0.0297335	+	0.999558i	$0.490534\pi$
$158$		0			0
$159$	8.89870			0.705713
$160$		0			0
$161$			2.36260i			0.186199i
$162$		0			0
$163$		−	10.8938i		−	0.853270i	−0.904424	−	0.426635i	$-0.859699\pi$
$163$		−	10.8938i		−	0.853270i	0.904424	−	0.426635i	$-0.140301\pi$
$164$		0			0
$165$	1.54212			0.120054
$166$		0			0
$167$		−	24.1884i		−	1.87176i	−0.352325	−	0.935878i	$-0.614609\pi$
$167$		−	24.1884i		−	1.87176i	0.352325	−	0.935878i	$-0.385391\pi$
$168$		0			0
$169$	6.14593	−	11.4555i	0.472764	−	0.881189i
$170$		0			0
$171$			37.7735i			2.88861i
$172$		0			0
$173$	−8.15059			−0.619678			−0.309839	−	0.950789i	$-0.600275\pi$
$173$	−8.15059			−0.619678			−0.309839	+	0.950789i	$0.600275\pi$
$174$		0			0
$175$			4.79193i			0.362236i
$176$		0			0
$177$			16.4880i			1.23931i
$178$		0			0
$179$	−11.0317			−0.824550			−0.412275	−	0.911060i	$-0.635266\pi$
$179$	−11.0317			−0.824550			−0.412275	+	0.911060i	$0.635266\pi$
$180$		0			0
$181$	5.01127			0.372485			0.186242	−	0.982504i	$-0.440369\pi$
$181$	5.01127			0.372485			0.186242	+	0.982504i	$0.440369\pi$
$182$		0			0
$183$	−3.71174			−0.274380
$184$		0			0
$185$	2.64897			0.194756
$186$		0			0
$187$			0.655725i			0.0479514i
$188$		0			0
$189$			18.3559i			1.33519i
$190$		0			0
$191$	3.68489			0.266630			0.133315	−	0.991074i	$-0.457438\pi$
$191$	3.68489			0.266630			0.133315	+	0.991074i	$0.457438\pi$
$192$		0			0
$193$		−	6.22453i		−	0.448052i	−0.974583	−	0.224026i	$-0.928080\pi$
$193$		−	6.22453i		−	0.448052i	0.974583	−	0.224026i	$-0.0719200\pi$
$194$		0			0
$195$	−2.85481	−	4.77135i	−0.204437	−	0.341683i
$196$		0			0
$197$		−	4.68227i		−	0.333598i	−0.985991	−	0.166799i	$-0.946657\pi$
$197$		−	4.68227i		−	0.333598i	0.985991	−	0.166799i	$-0.0533431\pi$
$198$		0			0
$199$	−21.8158			−1.54648			−0.773240	−	0.634113i	$-0.781365\pi$
$199$	−21.8158			−1.54648			−0.773240	+	0.634113i	$0.781365\pi$
$200$		0			0
$201$			32.6206i			2.30088i
$202$		0			0
$203$			3.97545i			0.279022i
$204$		0			0
$205$	−1.95524			−0.136560
$206$		0			0
$207$	19.9156			1.38423
$208$		0			0
$209$	4.48109			0.309964
$210$		0			0
$211$	1.85820			0.127923			0.0639617	−	0.997952i	$-0.479626\pi$
$211$	1.85820			0.127923			0.0639617	+	0.997952i	$0.479626\pi$
$212$		0			0
$213$			18.5226i			1.26915i
$214$		0			0
$215$			0.481941i			0.0328681i
$216$		0			0
$217$	6.41458			0.435450
$218$		0			0
$219$			55.3624i			3.74104i
$220$		0			0
$221$	2.02883	−	1.21390i	0.136474	−	0.0816554i
$222$		0			0
$223$		−	13.1462i		−	0.880335i	−0.897916	−	0.440168i	$-0.854919\pi$
$223$		−	13.1462i		−	0.880335i	0.897916	−	0.440168i	$-0.145081\pi$
$224$		0			0
$225$	40.3937			2.69291
$226$		0			0
$227$		−	5.46659i		−	0.362830i	−0.983407	−	0.181415i	$-0.941932\pi$
$227$		−	5.46659i		−	0.362830i	0.983407	−	0.181415i	$-0.0580678\pi$
$228$		0			0
$229$		−	2.05175i		−	0.135584i	−0.997699	−	0.0677918i	$-0.978405\pi$
$229$		−	2.05175i		−	0.135584i	0.997699	−	0.0677918i	$-0.0215954\pi$
$230$		0			0
$231$	3.38076			0.222438
$232$		0			0
$233$	−7.44365			−0.487650			−0.243825	−	0.969819i	$-0.578402\pi$
$233$	−7.44365			−0.487650			−0.243825	+	0.969819i	$0.578402\pi$
$234$		0			0
$235$	1.00710			0.0656963
$236$		0			0
$237$	30.6884			1.99342
$238$		0			0
$239$			5.19698i			0.336165i	0.985773	+	0.168082i	$0.0537575\pi$
$239$			5.19698i			0.336165i	−0.985773	+	0.168082i	$0.946243\pi$
$240$		0			0
$241$			6.79141i			0.437473i	0.975784	+	0.218737i	$0.0701936\pi$
$241$			6.79141i			0.437473i	−0.975784	+	0.218737i	$0.929806\pi$
$242$		0			0
$243$	69.2369			4.44155
$244$		0			0
$245$		−	0.456146i		−	0.0291421i
$246$		0			0
$247$	−8.29551	−	13.8646i	−0.527831	−	0.882183i
$248$		0			0
$249$			0.767277i			0.0486242i
$250$		0			0
$251$	−5.21260			−0.329016			−0.164508	−	0.986376i	$-0.552604\pi$
$251$	−5.21260			−0.329016			−0.164508	+	0.986376i	$0.552604\pi$
$252$		0			0
$253$		−	2.36260i		−	0.148535i
$254$		0			0
$255$		−	1.01121i		−	0.0633242i
$256$		0			0
$257$	−28.3305			−1.76721			−0.883604	−	0.468235i	$-0.844890\pi$
$257$	−28.3305			−1.76721			−0.883604	+	0.468235i	$0.844890\pi$
$258$		0			0
$259$	5.80729			0.360847
$260$		0			0
$261$	33.5111			2.07429
$262$		0			0
$263$	−15.6993			−0.968057			−0.484029	−	0.875052i	$-0.660827\pi$
$263$	−15.6993			−0.968057			−0.484029	+	0.875052i	$0.660827\pi$
$264$		0			0
$265$			1.20065i			0.0737553i
$266$		0			0
$267$			47.3178i			2.89580i
$268$		0			0
$269$	15.0682			0.918725			0.459362	−	0.888249i	$-0.348078\pi$
$269$	15.0682			0.918725			0.459362	+	0.888249i	$0.348078\pi$
$270$		0			0
$271$		−	11.6612i		−	0.708366i	−0.935176	−	0.354183i	$-0.884759\pi$
$271$		−	11.6612i		−	0.708366i	0.935176	−	0.354183i	$-0.115241\pi$
$272$		0			0
$273$	−6.25854	−	10.4601i	−0.378784	−	0.633076i
$274$		0			0
$275$		−	4.79193i		−	0.288964i
$276$		0			0
$277$	4.52019			0.271592			0.135796	−	0.990737i	$-0.456641\pi$
$277$	4.52019			0.271592			0.135796	+	0.990737i	$0.456641\pi$
$278$		0			0
$279$		−	54.0718i		−	3.23720i
$280$		0			0
$281$			18.6799i			1.11435i	0.830395	+	0.557175i	$0.188115\pi$
$281$			18.6799i			1.11435i	−0.830395	+	0.557175i	$0.811885\pi$
$282$		0			0
$283$	−8.82250			−0.524443			−0.262222	−	0.965008i	$-0.584455\pi$
$283$	−8.82250			−0.524443			−0.262222	+	0.965008i	$0.584455\pi$
$284$		0			0
$285$	−6.91038			−0.409335
$286$		0			0
$287$	−4.28642			−0.253020
$288$		0			0
$289$	−16.5700			−0.974707
$290$		0			0
$291$		−	53.3523i		−	3.12757i
$292$		0			0
$293$		−	10.8649i		−	0.634733i	−0.948303	−	0.317367i	$-0.897201\pi$
$293$		−	10.8649i		−	0.634733i	0.948303	−	0.317367i	$-0.102799\pi$
$294$		0			0
$295$	−2.22463			−0.129523
$296$		0			0
$297$		−	18.3559i		−	1.06512i
$298$		0			0
$299$	−7.30993	+	4.37371i	−0.422744	+	0.252938i
$300$		0			0
$301$			1.05655i			0.0608986i
$302$		0			0
$303$	−53.4734			−3.07196
$304$		0			0
$305$		−	0.500803i		−	0.0286759i
$306$		0			0
$307$			21.7037i			1.23870i	0.785116	+	0.619349i	$0.212603\pi$
$307$			21.7037i			1.23870i	−0.785116	+	0.619349i	$0.787397\pi$
$308$		0			0
$309$	−51.9549			−2.95561
$310$		0			0
$311$	−23.6042			−1.33847			−0.669237	−	0.743049i	$-0.733379\pi$
$311$	−23.6042			−1.33847			−0.669237	+	0.743049i	$0.733379\pi$
$312$		0			0
$313$	12.9347			0.731110			0.365555	−	0.930790i	$-0.380879\pi$
$313$	12.9347			0.731110			0.365555	+	0.930790i	$0.380879\pi$
$314$		0			0
$315$	−3.84509			−0.216647
$316$		0			0
$317$		−	8.17911i		−	0.459384i	−0.973263	−	0.229692i	$-0.926228\pi$
$317$		−	8.17911i		−	0.459384i	0.973263	−	0.229692i	$-0.0737720\pi$
$318$		0			0
$319$		−	3.97545i		−	0.222583i
$320$		0			0
$321$	40.7250			2.27305
$322$		0			0
$323$		−	2.93836i		−	0.163495i
$324$		0			0
$325$	−14.8263	+	8.87094i	−0.822417	+	0.492071i
$326$		0			0
$327$		−	32.1398i		−	1.77733i
$328$		0			0
$329$	2.20785			0.121723
$330$		0			0
$331$		−	20.2596i		−	1.11357i	−0.830658	−	0.556784i	$-0.812035\pi$
$331$		−	20.2596i		−	1.11357i	0.830658	−	0.556784i	$-0.187965\pi$
$332$		0			0
$333$		−	48.9527i		−	2.68259i
$334$		0			0
$335$	−4.40131			−0.240469
$336$		0			0
$337$	17.6072			0.959127			0.479564	−	0.877507i	$-0.340795\pi$
$337$	17.6072			0.959127			0.479564	+	0.877507i	$0.340795\pi$
$338$		0			0
$339$	−9.65253			−0.524254
$340$		0			0
$341$	−6.41458			−0.347369
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			3.64341i			0.196155i
$346$		0			0
$347$	6.18515			0.332036			0.166018	−	0.986123i	$-0.446909\pi$
$347$	6.18515			0.332036			0.166018	+	0.986123i	$0.446909\pi$
$348$		0			0
$349$		−	19.6517i		−	1.05193i	−0.850506	−	0.525965i	$-0.823704\pi$
$349$		−	19.6517i		−	1.05193i	0.850506	−	0.525965i	$-0.176296\pi$
$350$		0			0
$351$	−56.7935	+	33.9809i	−3.03141	+	1.81377i
$352$		0			0
$353$		−	12.4479i		−	0.662535i	−0.943537	−	0.331267i	$-0.892524\pi$
$353$		−	12.4479i		−	0.662535i	0.943537	−	0.331267i	$-0.107476\pi$
$354$		0			0
$355$	−2.49915			−0.132641
$356$		0			0
$357$		−	2.21685i		−	0.117328i
$358$		0			0
$359$		−	13.8762i		−	0.732357i	−0.930545	−	0.366179i	$-0.880666\pi$
$359$		−	13.8762i		−	0.732357i	0.930545	−	0.366179i	$-0.119334\pi$
$360$		0			0
$361$	−1.08017			−0.0568513
$362$		0			0
$363$	−3.38076			−0.177444
$364$		0			0
$365$	−7.46972			−0.390983
$366$		0			0
$367$	5.66931			0.295936			0.147968	−	0.988992i	$-0.452727\pi$
$367$	5.66931			0.295936			0.147968	+	0.988992i	$0.452727\pi$
$368$		0			0
$369$			36.1325i			1.88098i
$370$		0			0
$371$			2.63216i			0.136655i
$372$		0			0
$373$	8.40040			0.434956			0.217478	−	0.976065i	$-0.430217\pi$
$373$	8.40040			0.434956			0.217478	+	0.976065i	$0.430217\pi$
$374$		0			0
$375$			15.1003i			0.779778i
$376$		0			0
$377$	−12.3001	+	7.35946i	−0.633489	+	0.379031i
$378$		0			0
$379$		−	6.77517i		−	0.348017i	−0.984744	−	0.174009i	$-0.944328\pi$
$379$		−	6.77517i		−	0.348017i	0.984744	−	0.174009i	$-0.0556721\pi$
$380$		0			0
$381$	16.2066			0.830289
$382$		0			0
$383$		−	4.43051i		−	0.226389i	−0.993573	−	0.113194i	$-0.963892\pi$
$383$		−	4.43051i		−	0.226389i	0.993573	−	0.113194i	$-0.0361083\pi$
$384$		0			0
$385$			0.456146i			0.0232474i
$386$		0			0
$387$	8.90621			0.452728
$388$		0			0
$389$	17.3672			0.880552			0.440276	−	0.897862i	$-0.354881\pi$
$389$	17.3672			0.880552			0.440276	+	0.897862i	$0.354881\pi$
$390$		0			0
$391$	−1.54922			−0.0783473
$392$		0			0
$393$	−51.2175			−2.58358
$394$		0			0
$395$			4.14060i			0.208336i
$396$		0			0
$397$			32.8355i			1.64796i	0.566616	+	0.823982i	$0.308252\pi$
$397$			32.8355i			1.64796i	−0.566616	+	0.823982i	$0.691748\pi$
$398$		0			0
$399$	−15.1495			−0.758423
$400$		0			0
$401$			19.9817i			0.997838i	0.866649	+	0.498919i	$0.166269\pi$
$401$			19.9817i			0.997838i	−0.866649	+	0.498919i	$0.833731\pi$
$402$		0			0
$403$	11.8748	+	19.8468i	0.591527	+	0.988641i
$404$		0			0
$405$			16.7717i			0.833393i
$406$		0			0
$407$	−5.80729			−0.287857
$408$		0			0
$409$		−	34.8303i		−	1.72225i	−0.508396	−	0.861123i	$-0.669762\pi$
$409$		−	34.8303i		−	1.72225i	0.508396	−	0.861123i	$-0.330238\pi$
$410$		0			0
$411$		−	60.8423i		−	3.00113i
$412$		0			0
$413$	−4.87701			−0.239982
$414$		0			0
$415$	−0.103524			−0.00508181
$416$		0			0
$417$	31.6568			1.55024
$418$		0			0
$419$	−11.7118			−0.572158			−0.286079	−	0.958206i	$-0.592352\pi$
$419$	−11.7118			−0.572158			−0.286079	+	0.958206i	$0.592352\pi$
$420$		0			0
$421$			25.1005i			1.22332i	0.791119	+	0.611662i	$0.209499\pi$
$421$			25.1005i			1.22332i	−0.791119	+	0.611662i	$0.790501\pi$
$422$		0			0
$423$		−	18.6112i		−	0.904906i
$424$		0			0
$425$	−3.14219			−0.152419
$426$		0			0
$427$		−	1.09790i		−	0.0531312i
$428$		0			0
$429$	6.25854	+	10.4601i	0.302165	+	0.505020i
$430$		0			0
$431$		−	32.1392i		−	1.54809i	−0.633130	−	0.774045i	$-0.718230\pi$
$431$		−	32.1392i		−	1.54809i	0.633130	−	0.774045i	$-0.281770\pi$
$432$		0			0
$433$	−15.5762			−0.748544			−0.374272	−	0.927319i	$-0.622107\pi$
$433$	−15.5762			−0.748544			−0.374272	+	0.927319i	$0.622107\pi$
$434$		0			0
$435$			6.13062i			0.293941i
$436$		0			0
$437$			10.5870i			0.506446i
$438$		0			0
$439$	−32.1100			−1.53253			−0.766264	−	0.642526i	$-0.777886\pi$
$439$	−32.1100			−1.53253			−0.766264	+	0.642526i	$0.777886\pi$
$440$		0			0
$441$	−8.42952			−0.401406
$442$		0			0
$443$	−14.1610			−0.672810			−0.336405	−	0.941717i	$-0.609211\pi$
$443$	−14.1610			−0.672810			−0.336405	+	0.941717i	$0.609211\pi$
$444$		0			0
$445$	−6.38431			−0.302645
$446$		0			0
$447$			0.761386i			0.0360123i
$448$		0			0
$449$		−	21.3001i		−	1.00521i	−0.864515	−	0.502607i	$-0.832374\pi$
$449$		−	21.3001i		−	1.00521i	0.864515	−	0.502607i	$-0.167626\pi$
$450$		0			0
$451$	4.28642			0.201840
$452$		0			0
$453$		−	21.3752i		−	1.00429i
$454$		0			0
$455$	1.41133	−	0.844429i	0.0661640	−	0.0395875i
$456$		0			0
$457$			11.0839i			0.518482i	0.965813	+	0.259241i	$0.0834723\pi$
$457$			11.0839i			0.518482i	−0.965813	+	0.259241i	$0.916528\pi$
$458$		0			0
$459$	−12.0364			−0.561812
$460$		0			0
$461$		−	3.16306i		−	0.147319i	−0.997283	−	0.0736593i	$-0.976532\pi$
$461$		−	3.16306i		−	0.147319i	0.997283	−	0.0736593i	$-0.0234677\pi$
$462$		0			0
$463$			30.9664i			1.43913i	0.694424	+	0.719566i	$0.255659\pi$
$463$			30.9664i			1.43913i	−0.694424	+	0.719566i	$0.744341\pi$
$464$		0			0
$465$	9.89205			0.458733
$466$		0			0
$467$	36.2888			1.67925			0.839623	−	0.543170i	$-0.182776\pi$
$467$	36.2888			1.67925			0.839623	+	0.543170i	$0.182776\pi$
$468$		0			0
$469$	−9.64890			−0.445545
$470$		0			0
$471$	2.51907			0.116073
$472$		0			0
$473$		−	1.05655i		−	0.0485802i
$474$		0			0
$475$			21.4731i			0.985252i
$476$		0			0
$477$	22.1879			1.01591
$478$		0			0
$479$		−	38.3762i		−	1.75345i	−0.480990	−	0.876726i	$-0.659723\pi$
$479$		−	38.3762i		−	1.75345i	0.480990	−	0.876726i	$-0.340277\pi$
$480$		0			0
$481$	10.7506	+	17.9679i	0.490185	+	0.819264i
$482$		0			0
$483$			7.98738i			0.363438i
$484$		0			0
$485$	7.19852			0.326868
$486$		0			0
$487$			32.4691i			1.47131i	0.677355	+	0.735657i	$0.263126\pi$
$487$			32.4691i			1.47131i	−0.677355	+	0.735657i	$0.736874\pi$
$488$		0			0
$489$		−	36.8294i		−	1.66548i
$490$		0			0
$491$	−9.36144			−0.422476			−0.211238	−	0.977435i	$-0.567749\pi$
$491$	−9.36144			−0.422476			−0.211238	+	0.977435i	$0.567749\pi$
$492$		0			0
$493$	−2.60680			−0.117405
$494$		0			0
$495$	3.84509			0.172824
$496$		0			0
$497$	−5.47883			−0.245759
$498$		0			0
$499$		−	7.83730i		−	0.350846i	−0.984493	−	0.175423i	$-0.943871\pi$
$499$		−	7.83730i		−	0.350846i	0.984493	−	0.175423i	$-0.0561293\pi$
$500$		0			0
$501$		−	81.7752i		−	3.65344i
$502$		0			0
$503$	20.7892			0.926945			0.463472	−	0.886111i	$-0.346603\pi$
$503$	20.7892			0.926945			0.463472	+	0.886111i	$0.346603\pi$
$504$		0			0
$505$		−	7.21485i		−	0.321057i
$506$		0			0
$507$	20.7779	−	38.7281i	0.922779	−	1.71998i
$508$		0			0
$509$		−	22.8424i		−	1.01247i	−0.862394	−	0.506237i	$-0.831036\pi$
$509$		−	22.8424i		−	1.01247i	0.862394	−	0.506237i	$-0.168964\pi$
$510$		0			0
$511$	−16.3757			−0.724419
$512$		0			0
$513$			82.2544i			3.63162i
$514$		0			0
$515$		−	7.00997i		−	0.308896i
$516$		0			0
$517$	−2.20785			−0.0971013
$518$		0			0
$519$	−27.5552			−1.20954
$520$		0			0
$521$	19.3892			0.849457			0.424728	−	0.905321i	$-0.360370\pi$
$521$	19.3892			0.849457			0.424728	+	0.905321i	$0.360370\pi$
$522$		0			0
$523$	35.2253			1.54030			0.770148	−	0.637865i	$-0.220182\pi$
$523$	35.2253			1.54030			0.770148	+	0.637865i	$0.220182\pi$
$524$		0			0
$525$			16.2004i			0.707041i
$526$		0			0
$527$			4.20620i			0.183225i
$528$		0			0
$529$	−17.4181			−0.757310
$530$		0			0
$531$			41.1109i			1.78406i
$532$		0			0
$533$	−7.93513	−	13.2623i	−0.343709	−	0.574453i
$534$		0			0
$535$			5.49479i			0.237561i
$536$		0			0
$537$	−37.2956			−1.60942
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$			3.52622i			0.151604i	0.997123	+	0.0758021i	$0.0241517\pi$
$541$			3.52622i			0.151604i	−0.997123	+	0.0758021i	$0.975848\pi$
$542$		0			0
$543$	16.9419			0.727046
$544$		0			0
$545$	4.33644			0.185753
$546$		0			0
$547$	−21.4917			−0.918918			−0.459459	−	0.888199i	$-0.651957\pi$
$547$	−21.4917			−0.918918			−0.459459	+	0.888199i	$0.651957\pi$
$548$		0			0
$549$	−9.25478			−0.394984
$550$		0			0
$551$			17.8144i			0.758917i
$552$		0			0
$553$			9.07736i			0.386009i
$554$		0			0
$555$	8.95553			0.380141
$556$		0			0
$557$			28.7472i			1.21806i	0.793149	+	0.609028i	$0.208440\pi$
$557$			28.7472i			1.21806i	−0.793149	+	0.609028i	$0.791560\pi$
$558$		0			0
$559$	−3.26899	+	1.95591i	−0.138263	+	0.0827263i
$560$		0			0
$561$			2.21685i			0.0935954i
$562$		0			0
$563$	9.77731			0.412064			0.206032	−	0.978545i	$-0.433945\pi$
$563$	9.77731			0.412064			0.206032	+	0.978545i	$0.433945\pi$
$564$		0			0
$565$		−	1.30236i		−	0.0547907i
$566$		0			0
$567$			36.7683i			1.54412i
$568$		0			0
$569$	−33.6136			−1.40915			−0.704577	−	0.709627i	$-0.748863\pi$
$569$	−33.6136			−1.40915			−0.704577	+	0.709627i	$0.748863\pi$
$570$		0			0
$571$	26.3116			1.10111			0.550554	−	0.834800i	$-0.314416\pi$
$571$	26.3116			1.10111			0.550554	+	0.834800i	$0.314416\pi$
$572$		0			0
$573$	12.4577			0.520429
$574$		0			0
$575$	11.3214			0.472136
$576$		0			0
$577$		−	29.6664i		−	1.23503i	−0.786559	−	0.617515i	$-0.788140\pi$
$577$		−	29.6664i		−	1.23503i	0.786559	−	0.617515i	$-0.211860\pi$
$578$		0			0
$579$		−	21.0436i		−	0.874544i
$580$		0			0
$581$	−0.226954			−0.00941565
$582$		0			0
$583$		−	2.63216i		−	0.109013i
$584$		0			0
$585$	−7.11814	−	11.8968i	−0.294299	−	0.491872i
$586$		0			0
$587$		−	31.7190i		−	1.30918i	−0.755982	−	0.654592i	$-0.772840\pi$
$587$		−	31.7190i		−	1.30918i	0.755982	−	0.654592i	$-0.227160\pi$
$588$		0			0
$589$	28.7443			1.18439
$590$		0			0
$591$		−	15.8296i		−	0.651144i
$592$		0			0
$593$		−	42.9680i		−	1.76449i	−0.470795	−	0.882243i	$-0.656033\pi$
$593$		−	42.9680i		−	1.76449i	0.470795	−	0.882243i	$-0.343967\pi$
$594$		0			0
$595$	0.299107			0.0122622
$596$		0			0
$597$	−73.7539			−3.01855
$598$		0			0
$599$	1.13129			0.0462231			0.0231115	−	0.999733i	$-0.492643\pi$
$599$	1.13129			0.0462231			0.0231115	+	0.999733i	$0.492643\pi$
$600$		0			0
$601$	−40.9623			−1.67089			−0.835443	−	0.549576i	$-0.814789\pi$
$601$	−40.9623			−1.67089			−0.835443	+	0.549576i	$0.814789\pi$
$602$		0			0
$603$			81.3356i			3.31224i
$604$		0			0
$605$		−	0.456146i		−	0.0185450i
$606$		0			0
$607$	14.1122			0.572797			0.286399	−	0.958111i	$-0.407542\pi$
$607$	14.1122			0.572797			0.286399	+	0.958111i	$0.407542\pi$
$608$		0			0
$609$			13.4400i			0.544618i
$610$		0			0
$611$	4.08724	+	6.83115i	0.165352	+	0.276359i
$612$		0			0
$613$			15.1329i			0.611212i	0.952158	+	0.305606i	$0.0988591\pi$
$613$			15.1329i			0.611212i	−0.952158	+	0.305606i	$0.901141\pi$
$614$		0			0
$615$	−6.61018			−0.266548
$616$		0			0
$617$		−	26.0703i		−	1.04955i	−0.851240	−	0.524776i	$-0.824149\pi$
$617$		−	26.0703i		−	1.04955i	0.851240	−	0.524776i	$-0.175851\pi$
$618$		0			0
$619$		−	10.8334i		−	0.435433i	−0.976012	−	0.217717i	$-0.930139\pi$
$619$		−	10.8334i		−	0.435433i	0.976012	−	0.217717i	$-0.0698608\pi$
$620$		0			0
$621$	43.3676			1.74028
$622$		0			0
$623$	−13.9962			−0.560746
$624$		0			0
$625$	21.9223			0.876890
$626$		0			0
$627$	15.1495			0.605012
$628$		0			0
$629$			3.80799i			0.151834i
$630$		0			0
$631$			31.4735i			1.25294i	0.779445	+	0.626471i	$0.215501\pi$
$631$			31.4735i			1.25294i	−0.779445	+	0.626471i	$0.784499\pi$
$632$		0			0
$633$	6.28211			0.249691
$634$		0			0
$635$			2.18666i			0.0867751i
$636$		0			0
$637$	3.09402	−	1.85123i	0.122590	−	0.0733482i
$638$		0			0
$639$			46.1839i			1.82701i
$640$		0			0
$641$	27.5808			1.08937			0.544687	−	0.838639i	$-0.316648\pi$
$641$	27.5808			1.08937			0.544687	+	0.838639i	$0.316648\pi$
$642$		0			0
$643$			26.0732i			1.02823i	0.857722	+	0.514114i	$0.171879\pi$
$643$			26.0732i			1.02823i	−0.857722	+	0.514114i	$0.828121\pi$
$644$		0			0
$645$			1.62933i			0.0641547i
$646$		0			0
$647$	46.0960			1.81222			0.906111	−	0.423039i	$-0.139037\pi$
$647$	46.0960			1.81222			0.906111	+	0.423039i	$0.139037\pi$
$648$		0			0
$649$	4.87701			0.191439
$650$		0			0
$651$	21.6861			0.849947
$652$		0			0
$653$	−12.7968			−0.500779			−0.250390	−	0.968145i	$-0.580559\pi$
$653$	−12.7968			−0.500779			−0.250390	+	0.968145i	$0.580559\pi$
$654$		0			0
$655$		−	6.91048i		−	0.270015i
$656$		0			0
$657$			138.040i			5.38544i
$658$		0			0
$659$	−22.6078			−0.880674			−0.440337	−	0.897833i	$-0.645141\pi$
$659$	−22.6078			−0.880674			−0.440337	+	0.897833i	$0.645141\pi$
$660$		0			0
$661$			0.497129i			0.0193361i	0.999953	+	0.00966803i	$0.00307748\pi$
$661$			0.497129i			0.0193361i	−0.999953	+	0.00966803i	$0.996923\pi$
$662$		0			0
$663$	6.85897	−	4.10389i	0.266381	−	0.159382i
$664$		0			0
$665$		−	2.04403i		−	0.0792642i
$666$		0			0
$667$	9.39240			0.363675
$668$		0			0
$669$		−	44.4442i		−	1.71831i
$670$		0			0
$671$			1.09790i			0.0423840i
$672$		0			0
$673$	10.6672			0.411189			0.205594	−	0.978637i	$-0.434087\pi$
$673$	10.6672			0.411189			0.205594	+	0.978637i	$0.434087\pi$
$674$		0			0
$675$	87.9602			3.38559
$676$		0			0
$677$	2.49001			0.0956990			0.0478495	−	0.998855i	$-0.484763\pi$
$677$	2.49001			0.0956990			0.0478495	+	0.998855i	$0.484763\pi$
$678$		0			0
$679$	15.7812			0.605625
$680$		0			0
$681$		−	18.4812i		−	0.708202i
$682$		0			0
$683$		−	24.9416i		−	0.954366i	−0.878804	−	0.477183i	$-0.841658\pi$
$683$		−	24.9416i		−	0.954366i	0.878804	−	0.477183i	$-0.158342\pi$
$684$		0			0
$685$	8.20910			0.313654
$686$		0			0
$687$		−	6.93648i		−	0.264643i
$688$		0			0
$689$	−8.14396	+	4.87272i	−0.310260	+	0.185636i
$690$		0			0
$691$			46.1063i			1.75397i	0.480521	+	0.876983i	$0.340448\pi$
$691$			46.1063i			1.75397i	−0.480521	+	0.876983i	$0.659552\pi$
$692$		0			0
$693$	8.42952			0.320211
$694$		0			0
$695$			4.27127i			0.162019i
$696$		0			0
$697$		−	2.81072i		−	0.106463i
$698$		0			0
$699$	−25.1652			−0.951834
$700$		0			0
$701$	−50.9743			−1.92527			−0.962636	−	0.270797i	$-0.912713\pi$
$701$	−50.9743			−1.92527			−0.962636	+	0.270797i	$0.912713\pi$
$702$		0			0
$703$	26.0230			0.981475
$704$		0			0
$705$	3.40478			0.128231
$706$		0			0
$707$		−	15.8170i		−	0.594859i
$708$		0			0
$709$		−	0.666894i		−	0.0250457i	−0.999922	−	0.0125229i	$-0.996014\pi$
$709$		−	0.666894i		−	0.0250457i	0.999922	−	0.0125229i	$-0.00398626\pi$
$710$		0			0
$711$	76.5178			2.86964
$712$		0			0
$713$		−	15.1551i		−	0.567562i
$714$		0			0
$715$	−1.41133	+	0.844429i	−0.0527806	+	0.0315799i
$716$		0			0
$717$			17.5697i			0.656154i
$718$		0			0
$719$	−32.4874			−1.21157			−0.605787	−	0.795627i	$-0.707142\pi$
$719$	−32.4874			−1.21157			−0.605787	+	0.795627i	$0.707142\pi$
$720$		0			0
$721$		−	15.3678i		−	0.572328i
$722$		0			0
$723$			22.9601i			0.853896i
$724$		0			0
$725$	19.0501			0.707502
$726$		0			0
$727$	25.5538			0.947738			0.473869	−	0.880595i	$-0.342857\pi$
$727$	25.5538			0.947738			0.473869	+	0.880595i	$0.342857\pi$
$728$		0			0
$729$	123.768			4.58402
$730$		0			0
$731$	−0.692807			−0.0256244
$732$		0			0
$733$			24.4959i			0.904776i	0.891821	+	0.452388i	$0.149428\pi$
$733$			24.4959i			0.904776i	−0.891821	+	0.452388i	$0.850572\pi$
$734$		0			0
$735$		−	1.54212i		−	0.0568819i
$736$		0			0
$737$	9.64890			0.355422
$738$		0			0
$739$		−	4.96701i		−	0.182714i	−0.995818	−	0.0913572i	$-0.970879\pi$
$739$		−	4.96701i		−	0.182714i	0.995818	−	0.0913572i	$-0.0291205\pi$
$740$		0			0
$741$	−28.0451	−	46.8728i	−1.03026	−	1.72192i
$742$		0			0
$743$			52.3091i			1.91903i	0.281652	+	0.959517i	$0.409118\pi$
$743$			52.3091i			1.91903i	−0.281652	+	0.959517i	$0.590882\pi$
$744$		0			0
$745$	−0.102729			−0.00376372
$746$		0			0
$747$			1.91312i			0.0699972i
$748$		0			0
$749$			12.0461i			0.440156i
$750$		0			0
$751$	23.4003			0.853889			0.426944	−	0.904278i	$-0.359590\pi$
$751$	23.4003			0.853889			0.426944	+	0.904278i	$0.359590\pi$
$752$		0			0
$753$	−17.6225			−0.642201
$754$		0			0
$755$	2.88403			0.104961
$756$		0			0
$757$	−9.80188			−0.356255			−0.178128	−	0.984007i	$-0.557004\pi$
$757$	−9.80188			−0.356255			−0.178128	+	0.984007i	$0.557004\pi$
$758$		0			0
$759$		−	7.98738i		−	0.289923i
$760$		0			0
$761$		−	14.6053i		−	0.529443i	−0.964325	−	0.264721i	$-0.914720\pi$
$761$		−	14.6053i		−	0.529443i	0.964325	−	0.264721i	$-0.0852800\pi$
$762$		0			0
$763$	9.50668			0.344165
$764$		0			0
$765$		−	2.52133i		−	0.0911587i
$766$		0			0
$767$	−9.02845	−	15.0896i	−0.325998	−	0.544853i
$768$		0			0
$769$		−	48.1601i		−	1.73670i	−0.495955	−	0.868348i	$-0.665182\pi$
$769$		−	48.1601i		−	1.73670i	0.495955	−	0.868348i	$-0.334818\pi$
$770$		0			0
$771$	−95.7786			−3.44938
$772$		0			0
$773$		−	3.64755i		−	0.131193i	−0.997846	−	0.0655966i	$-0.979105\pi$
$773$		−	3.64755i		−	0.131193i	0.997846	−	0.0655966i	$-0.0208951\pi$
$774$		0			0
$775$		−	30.7382i		−	1.10415i
$776$		0			0
$777$	19.6330			0.704331
$778$		0			0
$779$	−19.2078			−0.688193
$780$		0			0
$781$	5.47883			0.196048
$782$		0			0
$783$	72.9730			2.60784
$784$		0			0
$785$			0.339883i			0.0121310i
$786$		0			0
$787$		−	25.1756i		−	0.897412i	−0.893679	−	0.448706i	$-0.851885\pi$
$787$		−	25.1756i		−	0.897412i	0.893679	−	0.448706i	$-0.148115\pi$
$788$		0			0
$789$	−53.0754			−1.88953
$790$		0			0
$791$		−	2.85514i		−	0.101517i
$792$		0			0
$793$	3.39693	−	2.03246i	0.120628	−	0.0721749i
$794$		0			0
$795$			4.05911i			0.143962i
$796$		0			0
$797$	−43.8111			−1.55187			−0.775934	−	0.630814i	$-0.782721\pi$
$797$	−43.8111			−1.55187			−0.775934	+	0.630814i	$0.782721\pi$
$798$		0			0
$799$			1.44775i			0.0512176i
$800$		0			0
$801$			117.981i			4.16866i
$802$		0			0
$803$	16.3757			0.577887
$804$		0			0
$805$	−1.07769			−0.0379836
$806$		0			0
$807$	50.9419			1.79324
$808$		0			0
$809$	−44.8785			−1.57784			−0.788922	−	0.614494i	$-0.789360\pi$
$809$	−44.8785			−1.57784			−0.788922	+	0.614494i	$0.789360\pi$
$810$		0			0
$811$			32.7316i			1.14936i	0.818377	+	0.574682i	$0.194874\pi$
$811$			32.7316i			1.14936i	−0.818377	+	0.574682i	$0.805126\pi$
$812$		0			0
$813$		−	39.4236i		−	1.38265i
$814$		0			0
$815$	4.96918			0.174063
$816$		0			0
$817$			4.73450i			0.165639i
$818$		0			0
$819$	−15.6049	−	26.0811i	−0.545281	−	0.911348i
$820$		0			0
$821$			53.5597i			1.86925i	0.355637	+	0.934624i	$0.384264\pi$
$821$			53.5597i			1.86925i	−0.355637	+	0.934624i	$0.615736\pi$
$822$		0			0
$823$	−22.6065			−0.788013			−0.394007	−	0.919108i	$-0.628911\pi$
$823$	−22.6065			−0.788013			−0.394007	+	0.919108i	$0.628911\pi$
$824$		0			0
$825$		−	16.2004i		−	0.564024i
$826$		0			0
$827$			22.6431i			0.787379i	0.919244	+	0.393689i	$0.128801\pi$
$827$			22.6431i			0.787379i	−0.919244	+	0.393689i	$0.871199\pi$
$828$		0			0
$829$	−28.5791			−0.992592			−0.496296	−	0.868153i	$-0.665307\pi$
$829$	−28.5791			−0.992592			−0.496296	+	0.868153i	$0.665307\pi$
$830$		0			0
$831$	15.2817			0.530115
$832$		0			0
$833$	0.655725			0.0227195
$834$		0			0
$835$	11.0335			0.381828
$836$		0			0
$837$		−	117.745i		−	4.06987i
$838$		0			0
$839$			42.4305i			1.46486i	0.680840	+	0.732432i	$0.261615\pi$
$839$			42.4305i			1.46486i	−0.680840	+	0.732432i	$0.738385\pi$
$840$		0			0
$841$	−13.1958			−0.455027
$842$		0			0
$843$			63.1522i			2.17508i
$844$		0			0
$845$	5.22536	+	2.80344i	0.179758	+	0.0964413i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	−29.8267			−1.02365
$850$		0			0
$851$		−	13.7203i		−	0.470326i
$852$		0			0
$853$		−	20.4165i		−	0.699049i	−0.936927	−	0.349525i	$-0.886343\pi$
$853$		−	20.4165i		−	0.699049i	0.936927	−	0.349525i	$-0.113657\pi$
$854$		0			0
$855$	−17.2302			−0.589261
$856$		0			0
$857$	−53.0681			−1.81277			−0.906386	−	0.422451i	$-0.861170\pi$
$857$	−53.0681			−1.81277			−0.906386	+	0.422451i	$0.861170\pi$
$858$		0			0
$859$	29.8281			1.01772			0.508861	−	0.860849i	$-0.330067\pi$
$859$	29.8281			1.01772			0.508861	+	0.860849i	$0.330067\pi$
$860$		0			0
$861$	−14.4914			−0.493864
$862$		0			0
$863$			7.54283i			0.256761i	0.991725	+	0.128380i	$0.0409779\pi$
$863$			7.54283i			0.256761i	−0.991725	+	0.128380i	$0.959022\pi$
$864$		0			0
$865$		−	3.71786i		−	0.126411i
$866$		0			0
$867$	−56.0192			−1.90251
$868$		0			0
$869$		−	9.07736i		−	0.307928i
$870$		0			0
$871$	−17.8623	−	29.8539i	−0.605241	−	1.01156i
$872$		0			0
$873$		−	133.028i		−	4.50230i
$874$		0			0
$875$	−4.46655			−0.150997
$876$		0			0
$877$			53.5889i			1.80957i	0.425870	+	0.904785i	$0.359968\pi$
$877$			53.5889i			1.80957i	−0.425870	+	0.904785i	$0.640032\pi$
$878$		0			0
$879$		−	36.7315i		−	1.23892i
$880$		0			0
$881$	48.1140			1.62100			0.810501	−	0.585737i	$-0.199195\pi$
$881$	48.1140			1.62100			0.810501	+	0.585737i	$0.199195\pi$
$882$		0			0
$883$	50.5078			1.69972			0.849862	−	0.527006i	$-0.176685\pi$
$883$	50.5078			1.69972			0.849862	+	0.527006i	$0.176685\pi$
$884$		0			0
$885$	−7.52094			−0.252814
$886$		0			0
$887$	−39.3396			−1.32089			−0.660447	−	0.750873i	$-0.729633\pi$
$887$	−39.3396			−1.32089			−0.660447	+	0.750873i	$0.729633\pi$
$888$		0			0
$889$			4.79378i			0.160778i
$890$		0			0
$891$		−	36.7683i		−	1.23178i
$892$		0			0
$893$	9.89360			0.331077
$894$		0			0
$895$		−	5.03208i		−	0.168204i
$896$		0			0
$897$	−24.7131	+	14.7864i	−0.825147	+	0.493705i
$898$		0			0
$899$		−	25.5008i		−	0.850501i
$900$		0			0
$901$	−1.72597			−0.0575005
$902$		0			0
$903$			3.57194i			0.118867i
$904$		0			0
$905$			2.28587i			0.0759849i
$906$		0			0
$907$	38.7952			1.28817			0.644086	−	0.764953i	$-0.277238\pi$
$907$	38.7952			1.28817			0.644086	+	0.764953i	$0.277238\pi$
$908$		0			0
$909$	−133.330			−4.42226
$910$		0			0
$911$	43.6231			1.44530			0.722649	−	0.691215i	$-0.242924\pi$
$911$	43.6231			1.44530			0.722649	+	0.691215i	$0.242924\pi$
$912$		0			0
$913$	0.226954			0.00751109
$914$		0			0
$915$		−	1.69309i		−	0.0559720i
$916$		0			0
$917$		−	15.1497i		−	0.500287i
$918$		0			0
$919$	−53.6079			−1.76836			−0.884181	−	0.467144i	$-0.845283\pi$
$919$	−53.6079			−1.76836			−0.884181	+	0.467144i	$0.845283\pi$
$920$		0			0
$921$			73.3751i			2.41779i
$922$		0			0
$923$	−10.1425	−	16.9516i	−0.333846	−	0.557969i
$924$		0			0
$925$		−	27.8281i		−	0.914983i
$926$		0			0
$927$	−129.543			−4.25476
$928$		0			0
$929$			45.3649i			1.48838i	0.667970	+	0.744188i	$0.267163\pi$
$929$			45.3649i			1.48838i	−0.667970	+	0.744188i	$0.732837\pi$
$930$		0			0
$931$		−	4.48109i		−	0.146862i
$932$		0			0
$933$	−79.8002			−2.61254
$934$		0			0
$935$	−0.299107			−0.00978183
$936$		0			0
$937$	27.2739			0.891000			0.445500	−	0.895282i	$-0.353026\pi$
$937$	27.2739			0.891000			0.445500	+	0.895282i	$0.353026\pi$
$938$		0			0
$939$	43.7290			1.42704
$940$		0			0
$941$		−	41.3396i		−	1.34763i	−0.738900	−	0.673815i	$-0.764654\pi$
$941$		−	41.3396i		−	1.34763i	0.738900	−	0.673815i	$-0.235346\pi$
$942$		0			0
$943$			10.1271i			0.329784i
$944$		0			0
$945$	−8.37297			−0.272373
$946$		0			0
$947$		−	14.8185i		−	0.481536i	−0.970583	−	0.240768i	$-0.922601\pi$
$947$		−	14.8185i		−	0.481536i	0.970583	−	0.240768i	$-0.0773993\pi$
$948$		0			0
$949$	−30.3152	−	50.6668i	−0.984072	−	1.64472i
$950$		0			0
$951$		−	27.6516i		−	0.896664i
$952$		0			0
$953$	−55.9767			−1.81326			−0.906631	−	0.421925i	$-0.861355\pi$
$953$	−55.9767			−1.81326			−0.906631	+	0.421925i	$0.861355\pi$
$954$		0			0
$955$			1.68085i			0.0543910i
$956$		0			0
$957$		−	13.4400i		−	0.434455i
$958$		0			0
$959$	17.9967			0.581142
$960$		0			0
$961$	−10.1468			−0.327317
$962$		0			0
$963$	101.543			3.27218
$964$		0			0
$965$	2.83930			0.0914002
$966$		0			0
$967$			23.9473i			0.770093i	0.922897	+	0.385046i	$0.125815\pi$
$967$			23.9473i			0.770093i	−0.922897	+	0.385046i	$0.874185\pi$
$968$		0			0
$969$		−	9.93390i		−	0.319123i
$970$		0			0
$971$	−36.1151			−1.15899			−0.579494	−	0.814976i	$-0.696750\pi$
$971$	−36.1151			−1.15899			−0.579494	+	0.814976i	$0.696750\pi$
$972$		0			0
$973$			9.36382i			0.300190i
$974$		0			0
$975$	−50.1242	+	29.9905i	−1.60526	+	0.960465i
$976$		0			0
$977$			9.79248i			0.313289i	0.987655	+	0.156645i	$0.0500677\pi$
$977$			9.79248i			0.313289i	−0.987655	+	0.156645i	$0.949932\pi$
$978$		0			0
$979$	13.9962			0.447320
$980$		0			0
$981$		−	80.1368i		−	2.55857i
$982$		0			0
$983$			29.1905i			0.931033i	0.885039	+	0.465517i	$0.154131\pi$
$983$			29.1905i			0.931033i	−0.885039	+	0.465517i	$0.845869\pi$
$984$		0			0
$985$	2.13580			0.0680523
$986$		0			0
$987$	7.46422			0.237589
$988$		0			0
$989$	2.49621			0.0793747
$990$		0			0
$991$	−1.38624			−0.0440353			−0.0220176	−	0.999758i	$-0.507009\pi$
$991$	−1.38624			−0.0440353			−0.0220176	+	0.999758i	$0.507009\pi$
$992$		0			0
$993$		−	68.4927i		−	2.17355i
$994$		0			0
$995$		−	9.95119i		−	0.315474i
$996$		0			0
$997$	−48.4955			−1.53587			−0.767934	−	0.640529i	$-0.778715\pi$
$997$	−48.4955			−1.53587			−0.767934	+	0.640529i	$0.778715\pi$
$998$		0			0
$999$		−	106.598i		−	3.37261i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.35	✓	36
13.12	even	2	inner	4004.2.m.c.2157.36	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.35	✓	36	1.1	even	1	trivial
4004.2.m.c.2157.36	yes	36	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q+3.38076 q^{3} +0.456146i q^{5} +1.00000i q^{7} +8.42952 q^{9} +O(q^{10})\)	\(q+3.38076 q^{3} +0.456146i q^{5} +1.00000i q^{7} +8.42952 q^{9} -1.00000i q^{11} +(-3.09402 + 1.85123i) q^{13} +1.54212i q^{15} -0.655725 q^{17} +4.48109i q^{19} +3.38076i q^{21} +2.36260 q^{23} +4.79193 q^{25} +18.3559 q^{27} +3.97545 q^{29} -6.41458i q^{31} -3.38076i q^{33} -0.456146 q^{35} -5.80729i q^{37} +(-10.4601 + 6.25854i) q^{39} +4.28642i q^{41} +1.05655 q^{43} +3.84509i q^{45} -2.20785i q^{47} -1.00000 q^{49} -2.21685 q^{51} +2.63216 q^{53} +0.456146 q^{55} +15.1495i q^{57} +4.87701i q^{59} -1.09790 q^{61} +8.42952i q^{63} +(-0.844429 - 1.41133i) q^{65} +9.64890i q^{67} +7.98738 q^{69} +5.47883i q^{71} +16.3757i q^{73} +16.2004 q^{75} +1.00000 q^{77} +9.07736 q^{79} +36.7683 q^{81} +0.226954i q^{83} -0.299107i q^{85} +13.4400 q^{87} +13.9962i q^{89} +(-1.85123 - 3.09402i) q^{91} -21.6861i q^{93} -2.04403 q^{95} -15.7812i q^{97} -8.42952i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 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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