LMFDB - Embedded newform 4004.2.m.b.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:	$30$
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.b.2157.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.705440 q^{3} +2.65594i q^{5} -1.00000i q^{7} -2.50236 q^{9} +O(q^{10})$	$q+0.705440 q^{3} +2.65594i q^{5} -1.00000i q^{7} -2.50236 q^{9} -1.00000i q^{11} +(3.54183 - 0.674887i) q^{13} +1.87360i q^{15} -0.198331 q^{17} -5.34779i q^{19} -0.705440i q^{21} +6.13987 q^{23} -2.05400 q^{25} -3.88158 q^{27} +2.90461 q^{29} +5.55972i q^{31} -0.705440i q^{33} +2.65594 q^{35} -3.81890i q^{37} +(2.49854 - 0.476092i) q^{39} +0.304576i q^{41} -2.09415 q^{43} -6.64610i q^{45} +3.81034i q^{47} -1.00000 q^{49} -0.139911 q^{51} +4.43329 q^{53} +2.65594 q^{55} -3.77254i q^{57} -14.5529i q^{59} +9.96003 q^{61} +2.50236i q^{63} +(1.79246 + 9.40687i) q^{65} +6.86965i q^{67} +4.33130 q^{69} -14.7769i q^{71} +11.4976i q^{73} -1.44897 q^{75} -1.00000 q^{77} -4.78198 q^{79} +4.76885 q^{81} -15.0172i q^{83} -0.526755i q^{85} +2.04903 q^{87} +7.28061i q^{89} +(-0.674887 - 3.54183i) q^{91} +3.92204i q^{93} +14.2034 q^{95} +13.8099i q^{97} +2.50236i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 18 q^{9}+O(q^{10}) $ 30 * q + 18 * q^9	$ 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) $ 30 * q + 18 * q^9 + 4 * q^13 + 12 * q^17 + 6 * q^23 - 6 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 30 * q^49 + 60 * q^51 - 38 * q^53 - 2 * q^55 - 36 * q^61 + 10 * q^65 + 36 * q^69 - 20 * q^75 - 30 * q^77 - 10 * q^79 - 42 * q^81 + 36 * q^87 + 6 * q^91 - 2 * 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
$3$	0.705440			0.407286			0.203643	−	0.979045i	$-0.434722\pi$
$3$	0.705440			0.407286			0.203643	+	0.979045i	$0.434722\pi$
$4$		0			0
$5$			2.65594i			1.18777i	0.804549	+	0.593886i	$0.202407\pi$
$5$			2.65594i			1.18777i	−0.804549	+	0.593886i	$0.797593\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−2.50236			−0.834118
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	3.54183	−	0.674887i	0.982326	−	0.187180i
$13$	3.54183	−	0.674887i	0.982326	−	0.187180i
$14$		0			0
$15$			1.87360i			0.483762i
$16$		0			0
$17$	−0.198331			−0.0481024			−0.0240512	−	0.999711i	$-0.507656\pi$
$17$	−0.198331			−0.0481024			−0.0240512	+	0.999711i	$0.507656\pi$
$18$		0			0
$19$		−	5.34779i		−	1.22687i	−0.789746	−	0.613434i	$-0.789788\pi$
$19$		−	5.34779i		−	1.22687i	0.789746	−	0.613434i	$-0.210212\pi$
$20$		0			0
$21$		−	0.705440i		−	0.153940i
$22$		0			0
$23$	6.13987			1.28025			0.640125	−	0.768270i	$-0.278882\pi$
$23$	6.13987			1.28025			0.640125	+	0.768270i	$0.278882\pi$
$24$		0			0
$25$	−2.05400			−0.410800
$26$		0			0
$27$	−3.88158			−0.747010
$28$		0			0
$29$	2.90461			0.539373			0.269686	−	0.962948i	$-0.413080\pi$
$29$	2.90461			0.539373			0.269686	+	0.962948i	$0.413080\pi$
$30$		0			0
$31$			5.55972i			0.998555i	0.866442	+	0.499277i	$0.166401\pi$
$31$			5.55972i			0.998555i	−0.866442	+	0.499277i	$0.833599\pi$
$32$		0			0
$33$		−	0.705440i		−	0.122801i
$34$		0			0
$35$	2.65594			0.448935
$36$		0			0
$37$		−	3.81890i		−	0.627824i	−0.949452	−	0.313912i	$-0.898360\pi$
$37$		−	3.81890i		−	0.627824i	0.949452	−	0.313912i	$-0.101640\pi$
$38$		0			0
$39$	2.49854	−	0.476092i	0.400087	−	0.0762358i
$40$		0			0
$41$			0.304576i			0.0475668i	0.999717	+	0.0237834i	$0.00757121\pi$
$41$			0.304576i			0.0475668i	−0.999717	+	0.0237834i	$0.992429\pi$
$42$		0			0
$43$	−2.09415			−0.319355			−0.159677	−	0.987169i	$-0.551045\pi$
$43$	−2.09415			−0.319355			−0.159677	+	0.987169i	$0.551045\pi$
$44$		0			0
$45$		−	6.64610i		−	0.990742i
$46$		0			0
$47$			3.81034i			0.555795i	0.960611	+	0.277897i	$0.0896375\pi$
$47$			3.81034i			0.555795i	−0.960611	+	0.277897i	$0.910363\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−0.139911			−0.0195914
$52$		0			0
$53$	4.43329			0.608959			0.304479	−	0.952519i	$-0.401518\pi$
$53$	4.43329			0.608959			0.304479	+	0.952519i	$0.401518\pi$
$54$		0			0
$55$	2.65594			0.358126
$56$		0			0
$57$		−	3.77254i		−	0.499686i
$58$		0			0
$59$		−	14.5529i		−	1.89463i	−0.320309	−	0.947313i	$-0.603787\pi$
$59$		−	14.5529i		−	1.89463i	0.320309	−	0.947313i	$-0.396213\pi$
$60$		0			0
$61$	9.96003			1.27525			0.637626	−	0.770346i	$-0.279917\pi$
$61$	9.96003			1.27525			0.637626	+	0.770346i	$0.279917\pi$
$62$		0			0
$63$			2.50236i			0.315267i
$64$		0			0
$65$	1.79246	+	9.40687i	0.222327	+	1.16678i
$66$		0			0
$67$			6.86965i			0.839261i	0.907695	+	0.419630i	$0.137840\pi$
$67$			6.86965i			0.839261i	−0.907695	+	0.419630i	$0.862160\pi$
$68$		0			0
$69$	4.33130			0.521428
$70$		0			0
$71$		−	14.7769i		−	1.75369i	−0.480772	−	0.876845i	$-0.659644\pi$
$71$		−	14.7769i		−	1.75369i	0.480772	−	0.876845i	$-0.340356\pi$
$72$		0			0
$73$			11.4976i			1.34570i	0.739781	+	0.672848i	$0.234929\pi$
$73$			11.4976i			1.34570i	−0.739781	+	0.672848i	$0.765071\pi$
$74$		0			0
$75$	−1.44897			−0.167313
$76$		0			0
$77$	−1.00000			−0.113961
$78$		0			0
$79$	−4.78198			−0.538015			−0.269007	−	0.963138i	$-0.586696\pi$
$79$	−4.78198			−0.538015			−0.269007	+	0.963138i	$0.586696\pi$
$80$		0			0
$81$	4.76885			0.529872
$82$		0			0
$83$		−	15.0172i		−	1.64836i	−0.566331	−	0.824178i	$-0.691637\pi$
$83$		−	15.0172i		−	1.64836i	0.566331	−	0.824178i	$-0.308363\pi$
$84$		0			0
$85$		−	0.526755i		−	0.0571346i
$86$		0			0
$87$	2.04903			0.219679
$88$		0			0
$89$			7.28061i			0.771743i	0.922553	+	0.385871i	$0.126099\pi$
$89$			7.28061i			0.771743i	−0.922553	+	0.385871i	$0.873901\pi$
$90$		0			0
$91$	−0.674887	−	3.54183i	−0.0707474	−	0.371284i
$92$		0			0
$93$			3.92204i			0.406697i
$94$		0			0
$95$	14.2034			1.45724
$96$		0			0
$97$			13.8099i			1.40218i	0.713071	+	0.701092i	$0.247304\pi$
$97$			13.8099i			1.40218i	−0.713071	+	0.701092i	$0.752696\pi$
$98$		0			0
$99$			2.50236i			0.251496i
$100$		0			0
$101$	10.0844			1.00344			0.501718	−	0.865031i	$-0.332701\pi$
$101$	10.0844			1.00344			0.501718	+	0.865031i	$0.332701\pi$
$102$		0			0
$103$	15.0334			1.48128			0.740642	−	0.671900i	$-0.234521\pi$
$103$	15.0334			1.48128			0.740642	+	0.671900i	$0.234521\pi$
$104$		0			0
$105$	1.87360			0.182845
$106$		0			0
$107$	5.99038			0.579112			0.289556	−	0.957161i	$-0.406492\pi$
$107$	5.99038			0.579112			0.289556	+	0.957161i	$0.406492\pi$
$108$		0			0
$109$		−	3.11708i		−	0.298562i	−0.988795	−	0.149281i	$-0.952304\pi$
$109$		−	3.11708i		−	0.298562i	0.988795	−	0.149281i	$-0.0476959\pi$
$110$		0			0
$111$		−	2.69401i		−	0.255704i
$112$		0			0
$113$	−7.77719			−0.731616			−0.365808	−	0.930690i	$-0.619207\pi$
$113$	−7.77719			−0.731616			−0.365808	+	0.930690i	$0.619207\pi$
$114$		0			0
$115$			16.3071i			1.52064i
$116$		0			0
$117$	−8.86290	+	1.68881i	−0.819376	+	0.156130i
$118$		0			0
$119$			0.198331i			0.0181810i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			0.214860i			0.0193733i
$124$		0			0
$125$			7.82438i			0.699834i
$126$		0			0
$127$	19.8017			1.75712			0.878560	−	0.477632i	$-0.158505\pi$
$127$	19.8017			1.75712			0.878560	+	0.477632i	$0.158505\pi$
$128$		0			0
$129$	−1.47729			−0.130069
$130$		0			0
$131$	5.57293			0.486909			0.243454	−	0.969912i	$-0.421719\pi$
$131$	5.57293			0.486909			0.243454	+	0.969912i	$0.421719\pi$
$132$		0			0
$133$	−5.34779			−0.463712
$134$		0			0
$135$		−	10.3092i		−	0.887277i
$136$		0			0
$137$			12.9336i			1.10500i	0.833514	+	0.552498i	$0.186325\pi$
$137$			12.9336i			1.10500i	−0.833514	+	0.552498i	$0.813675\pi$
$138$		0			0
$139$	19.2437			1.63223			0.816113	−	0.577893i	$-0.196125\pi$
$139$	19.2437			1.63223			0.816113	+	0.577893i	$0.196125\pi$
$140$		0			0
$141$			2.68796i			0.226367i
$142$		0			0
$143$	−0.674887	−	3.54183i	−0.0564369	−	0.296182i
$144$		0			0
$145$			7.71447i			0.640652i
$146$		0			0
$147$	−0.705440			−0.0581837
$148$		0			0
$149$		−	5.62913i		−	0.461157i	−0.973054	−	0.230578i	$-0.925938\pi$
$149$		−	5.62913i		−	0.461157i	0.973054	−	0.230578i	$-0.0740618\pi$
$150$		0			0
$151$			15.7528i			1.28195i	0.767563	+	0.640974i	$0.221469\pi$
$151$			15.7528i			1.28195i	−0.767563	+	0.640974i	$0.778531\pi$
$152$		0			0
$153$	0.496295			0.0401231
$154$		0			0
$155$	−14.7663			−1.18605
$156$		0			0
$157$	−17.5648			−1.40182			−0.700912	−	0.713248i	$-0.747223\pi$
$157$	−17.5648			−1.40182			−0.700912	+	0.713248i	$0.747223\pi$
$158$		0			0
$159$	3.12741			0.248020
$160$		0			0
$161$		−	6.13987i		−	0.483889i
$162$		0			0
$163$			3.54171i			0.277409i	0.990334	+	0.138704i	$0.0442937\pi$
$163$			3.54171i			0.277409i	−0.990334	+	0.138704i	$0.955706\pi$
$164$		0			0
$165$	1.87360			0.145860
$166$		0			0
$167$		−	9.04934i		−	0.700259i	−0.936701	−	0.350130i	$-0.886138\pi$
$167$		−	9.04934i		−	0.700259i	0.936701	−	0.350130i	$-0.113862\pi$
$168$		0			0
$169$	12.0891	−	4.78067i	0.929927	−	0.367744i
$170$		0			0
$171$			13.3821i			1.02335i
$172$		0			0
$173$	19.0848			1.45099			0.725495	−	0.688227i	$-0.241611\pi$
$173$	19.0848			1.45099			0.725495	+	0.688227i	$0.241611\pi$
$174$		0			0
$175$			2.05400i			0.155268i
$176$		0			0
$177$		−	10.2662i		−	0.771654i
$178$		0			0
$179$	−22.6188			−1.69061			−0.845303	−	0.534287i	$-0.820580\pi$
$179$	−22.6188			−1.69061			−0.845303	+	0.534287i	$0.820580\pi$
$180$		0			0
$181$	17.4861			1.29973			0.649866	−	0.760049i	$-0.274825\pi$
$181$	17.4861			1.29973			0.649866	+	0.760049i	$0.274825\pi$
$182$		0			0
$183$	7.02620			0.519392
$184$		0			0
$185$	10.1428			0.745711
$186$		0			0
$187$			0.198331i			0.0145034i
$188$		0			0
$189$			3.88158i			0.282343i
$190$		0			0
$191$	13.6134			0.985028			0.492514	−	0.870305i	$-0.336078\pi$
$191$	13.6134			0.985028			0.492514	+	0.870305i	$0.336078\pi$
$192$		0			0
$193$		−	5.46264i		−	0.393209i	−0.980483	−	0.196605i	$-0.937008\pi$
$193$		−	5.46264i		−	0.393209i	0.980483	−	0.196605i	$-0.0629915\pi$
$194$		0			0
$195$	1.26447	+	6.63598i	0.0905507	+	0.475212i
$196$		0			0
$197$		−	3.67116i		−	0.261560i	−0.991411	−	0.130780i	$-0.958252\pi$
$197$		−	3.67116i		−	0.261560i	0.991411	−	0.130780i	$-0.0417481\pi$
$198$		0			0
$199$	−17.9167			−1.27008			−0.635042	−	0.772478i	$-0.719017\pi$
$199$	−17.9167			−1.27008			−0.635042	+	0.772478i	$0.719017\pi$
$200$		0			0
$201$			4.84612i			0.341819i
$202$		0			0
$203$		−	2.90461i		−	0.203864i
$204$		0			0
$205$	−0.808935			−0.0564985
$206$		0			0
$207$	−15.3641			−1.06788
$208$		0			0
$209$	−5.34779			−0.369915
$210$		0			0
$211$	25.3108			1.74247			0.871233	−	0.490869i	$-0.163321\pi$
$211$	25.3108			1.74247			0.871233	+	0.490869i	$0.163321\pi$
$212$		0			0
$213$		−	10.4242i		−	0.714253i
$214$		0			0
$215$		−	5.56193i		−	0.379320i
$216$		0			0
$217$	5.55972			0.377418
$218$		0			0
$219$			8.11088i			0.548083i
$220$		0			0
$221$	−0.702454	+	0.133851i	−0.0472522	+	0.00900380i
$222$		0			0
$223$		−	6.91130i		−	0.462815i	−0.972857	−	0.231408i	$-0.925667\pi$
$223$		−	6.91130i		−	0.462815i	0.972857	−	0.231408i	$-0.0743331\pi$
$224$		0			0
$225$	5.13984			0.342656
$226$		0			0
$227$			23.8006i			1.57970i	0.613300	+	0.789850i	$0.289842\pi$
$227$			23.8006i			1.57970i	−0.613300	+	0.789850i	$0.710158\pi$
$228$		0			0
$229$			27.8953i			1.84338i	0.387931	+	0.921688i	$0.373190\pi$
$229$			27.8953i			1.84338i	−0.387931	+	0.921688i	$0.626810\pi$
$230$		0			0
$231$	−0.705440			−0.0464145
$232$		0			0
$233$	−0.570395			−0.0373678			−0.0186839	−	0.999825i	$-0.505948\pi$
$233$	−0.570395			−0.0373678			−0.0186839	+	0.999825i	$0.505948\pi$
$234$		0			0
$235$	−10.1200			−0.660157
$236$		0			0
$237$	−3.37340			−0.219126
$238$		0			0
$239$			10.4981i			0.679068i	0.940594	+	0.339534i	$0.110269\pi$
$239$			10.4981i			0.679068i	−0.940594	+	0.339534i	$0.889731\pi$
$240$		0			0
$241$		−	2.27105i		−	0.146291i	−0.997321	−	0.0731457i	$-0.976696\pi$
$241$		−	2.27105i		−	0.146291i	0.997321	−	0.0731457i	$-0.0233038\pi$
$242$		0			0
$243$	15.0089			0.962819
$244$		0			0
$245$		−	2.65594i		−	0.169682i
$246$		0			0
$247$	−3.60916	−	18.9409i	−0.229645	−	1.20518i
$248$		0			0
$249$		−	10.5938i		−	0.671352i
$250$		0			0
$251$	−15.0831			−0.952036			−0.476018	−	0.879436i	$-0.657920\pi$
$251$	−15.0831			−0.952036			−0.476018	+	0.879436i	$0.657920\pi$
$252$		0			0
$253$		−	6.13987i		−	0.386010i
$254$		0			0
$255$		−	0.371594i		−	0.0232701i
$256$		0			0
$257$	−26.4790			−1.65172			−0.825859	−	0.563877i	$-0.809309\pi$
$257$	−26.4790			−1.65172			−0.825859	+	0.563877i	$0.809309\pi$
$258$		0			0
$259$	−3.81890			−0.237295
$260$		0			0
$261$	−7.26837			−0.449901
$262$		0			0
$263$	−13.1052			−0.808103			−0.404052	−	0.914736i	$-0.632398\pi$
$263$	−13.1052			−0.808103			−0.404052	+	0.914736i	$0.632398\pi$
$264$		0			0
$265$			11.7745i			0.723303i
$266$		0			0
$267$			5.13603i			0.314320i
$268$		0			0
$269$	−11.8567			−0.722914			−0.361457	−	0.932389i	$-0.617720\pi$
$269$	−11.8567			−0.722914			−0.361457	+	0.932389i	$0.617720\pi$
$270$		0			0
$271$			22.1867i			1.34775i	0.738847	+	0.673873i	$0.235371\pi$
$271$			22.1867i			1.34775i	−0.738847	+	0.673873i	$0.764629\pi$
$272$		0			0
$273$	−0.476092	−	2.49854i	−0.0288144	−	0.151219i
$274$		0			0
$275$			2.05400i			0.123861i
$276$		0			0
$277$	−3.53738			−0.212541			−0.106270	−	0.994337i	$-0.533891\pi$
$277$	−3.53738			−0.212541			−0.106270	+	0.994337i	$0.533891\pi$
$278$		0			0
$279$		−	13.9124i		−	0.832913i
$280$		0			0
$281$		−	5.06393i		−	0.302089i	−0.988527	−	0.151044i	$-0.951736\pi$
$281$		−	5.06393i		−	0.302089i	0.988527	−	0.151044i	$-0.0482636\pi$
$282$		0			0
$283$	0.366079			0.0217612			0.0108806	−	0.999941i	$-0.496537\pi$
$283$	0.366079			0.0217612			0.0108806	+	0.999941i	$0.496537\pi$
$284$		0			0
$285$	10.0196			0.593512
$286$		0			0
$287$	0.304576			0.0179786
$288$		0			0
$289$	−16.9607			−0.997686
$290$		0			0
$291$			9.74205i			0.571089i
$292$		0			0
$293$		−	14.8379i		−	0.866842i	−0.901192	−	0.433421i	$-0.857306\pi$
$293$		−	14.8379i		−	0.866842i	0.901192	−	0.433421i	$-0.142694\pi$
$294$		0			0
$295$	38.6516			2.25038
$296$		0			0
$297$			3.88158i			0.225232i
$298$		0			0
$299$	21.7463	−	4.14372i	1.25762	−	0.239637i
$300$		0			0
$301$			2.09415i			0.120705i
$302$		0			0
$303$	7.11395			0.408686
$304$		0			0
$305$			26.4532i			1.51471i
$306$		0			0
$307$		−	15.7430i		−	0.898503i	−0.893405	−	0.449251i	$-0.851691\pi$
$307$		−	15.7430i		−	0.898503i	0.893405	−	0.449251i	$-0.148309\pi$
$308$		0			0
$309$	10.6051			0.603306
$310$		0			0
$311$	16.3495			0.927097			0.463549	−	0.886071i	$-0.346576\pi$
$311$	16.3495			0.927097			0.463549	+	0.886071i	$0.346576\pi$
$312$		0			0
$313$	19.9006			1.12485			0.562424	−	0.826849i	$-0.309869\pi$
$313$	19.9006			1.12485			0.562424	+	0.826849i	$0.309869\pi$
$314$		0			0
$315$	−6.64610			−0.374465
$316$		0			0
$317$			1.86991i			0.105024i	0.998620	+	0.0525122i	$0.0167228\pi$
$317$			1.86991i			0.105024i	−0.998620	+	0.0525122i	$0.983277\pi$
$318$		0			0
$319$		−	2.90461i		−	0.162627i
$320$		0			0
$321$	4.22585			0.235864
$322$		0			0
$323$			1.06063i			0.0590152i
$324$		0			0
$325$	−7.27492	+	1.38622i	−0.403540	+	0.0768937i
$326$		0			0
$327$		−	2.19891i		−	0.121600i
$328$		0			0
$329$	3.81034			0.210071
$330$		0			0
$331$		−	17.4955i		−	0.961641i	−0.876819	−	0.480821i	$-0.840339\pi$
$331$		−	17.4955i		−	0.961641i	0.876819	−	0.480821i	$-0.159661\pi$
$332$		0			0
$333$			9.55625i			0.523679i
$334$		0			0
$335$	−18.2453			−0.996850
$336$		0			0
$337$	−11.2181			−0.611091			−0.305545	−	0.952178i	$-0.598839\pi$
$337$	−11.2181			−0.611091			−0.305545	+	0.952178i	$0.598839\pi$
$338$		0			0
$339$	−5.48633			−0.297977
$340$		0			0
$341$	5.55972			0.301076
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			11.5037i			0.619337i
$346$		0			0
$347$	−29.5357			−1.58556			−0.792780	−	0.609508i	$-0.791367\pi$
$347$	−29.5357			−1.58556			−0.792780	+	0.609508i	$0.791367\pi$
$348$		0			0
$349$			21.0977i			1.12933i	0.825319	+	0.564667i	$0.190995\pi$
$349$			21.0977i			1.12933i	−0.825319	+	0.564667i	$0.809005\pi$
$350$		0			0
$351$	−13.7479	+	2.61963i	−0.733807	+	0.139825i
$352$		0			0
$353$		−	18.3734i		−	0.977916i	−0.872307	−	0.488958i	$-0.837377\pi$
$353$		−	18.3734i		−	0.977916i	0.872307	−	0.488958i	$-0.162623\pi$
$354$		0			0
$355$	39.2464			2.08298
$356$		0			0
$357$			0.139911i			0.00740485i
$358$		0			0
$359$		−	2.68579i		−	0.141750i	−0.997485	−	0.0708752i	$-0.977421\pi$
$359$		−	2.68579i		−	0.141750i	0.997485	−	0.0708752i	$-0.0225792\pi$
$360$		0			0
$361$	−9.59889			−0.505205
$362$		0			0
$363$	−0.705440			−0.0370260
$364$		0			0
$365$	−30.5370			−1.59838
$366$		0			0
$367$	16.0522			0.837920			0.418960	−	0.908005i	$-0.362395\pi$
$367$	16.0522			0.837920			0.418960	+	0.908005i	$0.362395\pi$
$368$		0			0
$369$		−	0.762158i		−	0.0396764i
$370$		0			0
$371$		−	4.43329i		−	0.230165i
$372$		0			0
$373$	−26.9851			−1.39723			−0.698617	−	0.715496i	$-0.746201\pi$
$373$	−26.9851			−1.39723			−0.698617	+	0.715496i	$0.746201\pi$
$374$		0			0
$375$			5.51963i			0.285032i
$376$		0			0
$377$	10.2876	−	1.96029i	0.529840	−	0.100960i
$378$		0			0
$379$		−	14.0542i		−	0.721917i	−0.932582	−	0.360959i	$-0.882450\pi$
$379$		−	14.0542i		−	0.721917i	0.932582	−	0.360959i	$-0.117550\pi$
$380$		0			0
$381$	13.9689			0.715650
$382$		0			0
$383$		−	34.4515i		−	1.76039i	−0.474611	−	0.880196i	$-0.657411\pi$
$383$		−	34.4515i		−	1.76039i	0.474611	−	0.880196i	$-0.342589\pi$
$384$		0			0
$385$		−	2.65594i		−	0.135359i
$386$		0			0
$387$	5.24030			0.266380
$388$		0			0
$389$	22.3692			1.13416			0.567082	−	0.823661i	$-0.308072\pi$
$389$	22.3692			1.13416			0.567082	+	0.823661i	$0.308072\pi$
$390$		0			0
$391$	−1.21773			−0.0615831
$392$		0			0
$393$	3.93136			0.198311
$394$		0			0
$395$		−	12.7006i		−	0.639038i
$396$		0			0
$397$		−	4.67112i		−	0.234437i	−0.993106	−	0.117218i	$-0.962602\pi$
$397$		−	4.67112i		−	0.234437i	0.993106	−	0.117218i	$-0.0373978\pi$
$398$		0			0
$399$	−3.77254			−0.188863
$400$		0			0
$401$		−	29.8732i		−	1.49179i	−0.666061	−	0.745897i	$-0.732021\pi$
$401$		−	29.8732i		−	1.49179i	0.666061	−	0.745897i	$-0.267979\pi$
$402$		0			0
$403$	3.75218	+	19.6916i	0.186910	+	0.980906i
$404$		0			0
$405$			12.6658i			0.629366i
$406$		0			0
$407$	−3.81890			−0.189296
$408$		0			0
$409$		−	29.9253i		−	1.47971i	−0.672767	−	0.739855i	$-0.734894\pi$
$409$		−	29.9253i		−	1.47971i	0.672767	−	0.739855i	$-0.265106\pi$
$410$		0			0
$411$			9.12391i			0.450049i
$412$		0			0
$413$	−14.5529			−0.716101
$414$		0			0
$415$	39.8849			1.95787
$416$		0			0
$417$	13.5752			0.664782
$418$		0			0
$419$	16.6971			0.815709			0.407855	−	0.913047i	$-0.366277\pi$
$419$	16.6971			0.815709			0.407855	+	0.913047i	$0.366277\pi$
$420$		0			0
$421$		−	15.4641i		−	0.753676i	−0.926279	−	0.376838i	$-0.877011\pi$
$421$		−	15.4641i		−	0.753676i	0.926279	−	0.376838i	$-0.122989\pi$
$422$		0			0
$423$		−	9.53481i		−	0.463598i
$424$		0			0
$425$	0.407373			0.0197605
$426$		0			0
$427$		−	9.96003i		−	0.482000i
$428$		0			0
$429$	−0.476092	−	2.49854i	−0.0229860	−	0.120631i
$430$		0			0
$431$			10.9153i			0.525772i	0.964827	+	0.262886i	$0.0846744\pi$
$431$			10.9153i			0.525772i	−0.964827	+	0.262886i	$0.915326\pi$
$432$		0			0
$433$	25.6461			1.23247			0.616237	−	0.787561i	$-0.288656\pi$
$433$	25.6461			1.23247			0.616237	+	0.787561i	$0.288656\pi$
$434$		0			0
$435$			5.44209i			0.260928i
$436$		0			0
$437$		−	32.8347i		−	1.57070i
$438$		0			0
$439$	−17.9269			−0.855606			−0.427803	−	0.903872i	$-0.640712\pi$
$439$	−17.9269			−0.855606			−0.427803	+	0.903872i	$0.640712\pi$
$440$		0			0
$441$	2.50236			0.119160
$442$		0			0
$443$	−26.2777			−1.24849			−0.624245	−	0.781229i	$-0.714593\pi$
$443$	−26.2777			−1.24849			−0.624245	+	0.781229i	$0.714593\pi$
$444$		0			0
$445$	−19.3368			−0.916654
$446$		0			0
$447$		−	3.97101i		−	0.187822i
$448$		0			0
$449$		−	19.1211i		−	0.902382i	−0.892427	−	0.451191i	$-0.850999\pi$
$449$		−	19.1211i		−	0.902382i	0.892427	−	0.451191i	$-0.149001\pi$
$450$		0			0
$451$	0.304576			0.0143419
$452$		0			0
$453$			11.1127i			0.522119i
$454$		0			0
$455$	9.40687	−	1.79246i	0.441001	−	0.0840318i
$456$		0			0
$457$		−	4.77591i		−	0.223408i	−0.993742	−	0.111704i	$-0.964369\pi$
$457$		−	4.77591i		−	0.223408i	0.993742	−	0.111704i	$-0.0356308\pi$
$458$		0			0
$459$	0.769838			0.0359330
$460$		0			0
$461$		−	23.5341i		−	1.09609i	−0.836448	−	0.548046i	$-0.815372\pi$
$461$		−	23.5341i		−	1.09609i	0.836448	−	0.548046i	$-0.184628\pi$
$462$		0			0
$463$			4.55074i			0.211491i	0.994393	+	0.105745i	$0.0337229\pi$
$463$			4.55074i			0.211491i	−0.994393	+	0.105745i	$0.966277\pi$
$464$		0			0
$465$	−10.4167			−0.483063
$466$		0			0
$467$	25.5871			1.18403			0.592016	−	0.805926i	$-0.298332\pi$
$467$	25.5871			1.18403			0.592016	+	0.805926i	$0.298332\pi$
$468$		0			0
$469$	6.86965			0.317211
$470$		0			0
$471$	−12.3909			−0.570943
$472$		0			0
$473$			2.09415i			0.0962890i
$474$		0			0
$475$			10.9844i			0.503998i
$476$		0			0
$477$	−11.0937			−0.507943
$478$		0			0
$479$		−	16.5055i		−	0.754154i	−0.926182	−	0.377077i	$-0.876929\pi$
$479$		−	16.5055i		−	0.754154i	0.926182	−	0.377077i	$-0.123071\pi$
$480$		0			0
$481$	−2.57733	−	13.5259i	−0.117516	−	0.616727i
$482$		0			0
$483$		−	4.33130i		−	0.197081i
$484$		0			0
$485$	−36.6782			−1.66547
$486$		0			0
$487$			9.40067i			0.425985i	0.977054	+	0.212992i	$0.0683210\pi$
$487$			9.40067i			0.425985i	−0.977054	+	0.212992i	$0.931679\pi$
$488$		0			0
$489$			2.49847i			0.112985i
$490$		0			0
$491$	35.1695			1.58718			0.793589	−	0.608454i	$-0.208210\pi$
$491$	35.1695			1.58718			0.793589	+	0.608454i	$0.208210\pi$
$492$		0			0
$493$	−0.576075			−0.0259451
$494$		0			0
$495$	−6.64610			−0.298720
$496$		0			0
$497$	−14.7769			−0.662833
$498$		0			0
$499$			16.4953i			0.738432i	0.929344	+	0.369216i	$0.120374\pi$
$499$			16.4953i			0.738432i	−0.929344	+	0.369216i	$0.879626\pi$
$500$		0			0
$501$		−	6.38376i		−	0.285206i
$502$		0			0
$503$	18.6269			0.830532			0.415266	−	0.909700i	$-0.363689\pi$
$503$	18.6269			0.830532			0.415266	+	0.909700i	$0.363689\pi$
$504$		0			0
$505$			26.7836i			1.19185i
$506$		0			0
$507$	8.52810	−	3.37247i	0.378746	−	0.149777i
$508$		0			0
$509$			4.45845i			0.197617i	0.995106	+	0.0988086i	$0.0315032\pi$
$509$			4.45845i			0.197617i	−0.995106	+	0.0988086i	$0.968497\pi$
$510$		0			0
$511$	11.4976			0.508625
$512$		0			0
$513$			20.7579i			0.916483i
$514$		0			0
$515$			39.9277i			1.75943i
$516$		0			0
$517$	3.81034			0.167578
$518$		0			0
$519$	13.4632			0.590968
$520$		0			0
$521$	36.3517			1.59260			0.796298	−	0.604905i	$-0.206789\pi$
$521$	36.3517			1.59260			0.796298	+	0.604905i	$0.206789\pi$
$522$		0			0
$523$	−30.2262			−1.32170			−0.660849	−	0.750519i	$-0.729804\pi$
$523$	−30.2262			−1.32170			−0.660849	+	0.750519i	$0.729804\pi$
$524$		0			0
$525$			1.44897i			0.0632384i
$526$		0			0
$527$		−	1.10267i		−	0.0480328i
$528$		0			0
$529$	14.6980			0.639042
$530$		0			0
$531$			36.4165i			1.58034i
$532$		0			0
$533$	0.205555	+	1.07876i	0.00890356	+	0.0467261i
$534$		0			0
$535$			15.9101i			0.687852i
$536$		0			0
$537$	−15.9562			−0.688560
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$			5.08391i			0.218574i	0.994010	+	0.109287i	$0.0348568\pi$
$541$			5.08391i			0.218574i	−0.994010	+	0.109287i	$0.965143\pi$
$542$		0			0
$543$	12.3354			0.529362
$544$		0			0
$545$	8.27876			0.354623
$546$		0			0
$547$	−31.4442			−1.34446			−0.672228	−	0.740344i	$-0.734663\pi$
$547$	−31.4442			−1.34446			−0.672228	+	0.740344i	$0.734663\pi$
$548$		0			0
$549$	−24.9235			−1.06371
$550$		0			0
$551$		−	15.5333i		−	0.661739i
$552$		0			0
$553$			4.78198i			0.203350i
$554$		0			0
$555$	7.15511			0.303717
$556$		0			0
$557$			15.5675i			0.659618i	0.944048	+	0.329809i	$0.106984\pi$
$557$			15.5675i			0.659618i	−0.944048	+	0.329809i	$0.893016\pi$
$558$		0			0
$559$	−7.41711	+	1.41331i	−0.313710	+	0.0597768i
$560$		0			0
$561$			0.139911i			0.00590703i
$562$		0			0
$563$	20.5965			0.868038			0.434019	−	0.900904i	$-0.357095\pi$
$563$	20.5965			0.868038			0.434019	+	0.900904i	$0.357095\pi$
$564$		0			0
$565$		−	20.6557i		−	0.868993i
$566$		0			0
$567$		−	4.76885i		−	0.200273i
$568$		0			0
$569$	8.21354			0.344329			0.172165	−	0.985068i	$-0.444924\pi$
$569$	8.21354			0.344329			0.172165	+	0.985068i	$0.444924\pi$
$570$		0			0
$571$	−5.26806			−0.220461			−0.110231	−	0.993906i	$-0.535159\pi$
$571$	−5.26806			−0.220461			−0.110231	+	0.993906i	$0.535159\pi$
$572$		0			0
$573$	9.60340			0.401188
$574$		0			0
$575$	−12.6113			−0.525928
$576$		0			0
$577$		−	28.6436i		−	1.19245i	−0.802817	−	0.596225i	$-0.796667\pi$
$577$		−	28.6436i		−	1.19245i	0.802817	−	0.596225i	$-0.203333\pi$
$578$		0			0
$579$		−	3.85356i		−	0.160149i
$580$		0			0
$581$	−15.0172			−0.623020
$582$		0			0
$583$		−	4.43329i		−	0.183608i
$584$		0			0
$585$	−4.48537	−	23.5393i	−0.185447	−	0.973231i
$586$		0			0
$587$		−	32.8645i		−	1.35646i	−0.734848	−	0.678232i	$-0.762747\pi$
$587$		−	32.8645i		−	1.35646i	0.734848	−	0.678232i	$-0.237253\pi$
$588$		0			0
$589$	29.7322			1.22509
$590$		0			0
$591$		−	2.58978i		−	0.106530i
$592$		0			0
$593$			18.6661i			0.766523i	0.923640	+	0.383262i	$0.125199\pi$
$593$			18.6661i			0.766523i	−0.923640	+	0.383262i	$0.874801\pi$
$594$		0			0
$595$	−0.526755			−0.0215948
$596$		0			0
$597$	−12.6392			−0.517287
$598$		0			0
$599$	−6.46110			−0.263993			−0.131997	−	0.991250i	$-0.542139\pi$
$599$	−6.46110			−0.263993			−0.131997	+	0.991250i	$0.542139\pi$
$600$		0			0
$601$	−29.2597			−1.19353			−0.596765	−	0.802416i	$-0.703547\pi$
$601$	−29.2597			−1.19353			−0.596765	+	0.802416i	$0.703547\pi$
$602$		0			0
$603$		−	17.1903i		−	0.700043i
$604$		0			0
$605$		−	2.65594i		−	0.107979i
$606$		0			0
$607$	−24.7736			−1.00553			−0.502765	−	0.864423i	$-0.667684\pi$
$607$	−24.7736			−1.00553			−0.502765	+	0.864423i	$0.667684\pi$
$608$		0			0
$609$		−	2.04903i		−	0.0830308i
$610$		0			0
$611$	2.57155	+	13.4955i	0.104034	+	0.545971i
$612$		0			0
$613$			29.4688i			1.19023i	0.803639	+	0.595117i	$0.202894\pi$
$613$			29.4688i			1.19023i	−0.803639	+	0.595117i	$0.797106\pi$
$614$		0			0
$615$	−0.570655			−0.0230110
$616$		0			0
$617$		−	12.8453i		−	0.517134i	−0.965993	−	0.258567i	$-0.916750\pi$
$617$		−	12.8453i		−	0.517134i	0.965993	−	0.258567i	$-0.0832502\pi$
$618$		0			0
$619$		−	32.2058i		−	1.29446i	−0.762295	−	0.647230i	$-0.775927\pi$
$619$		−	32.2058i		−	1.29446i	0.762295	−	0.647230i	$-0.224073\pi$
$620$		0			0
$621$	−23.8324			−0.956360
$622$		0			0
$623$	7.28061			0.291691
$624$		0			0
$625$	−31.0511			−1.24204
$626$		0			0
$627$	−3.77254			−0.150661
$628$		0			0
$629$			0.757407i			0.0301998i
$630$		0			0
$631$		−	12.1313i		−	0.482940i	−0.970408	−	0.241470i	$-0.922370\pi$
$631$		−	12.1313i		−	0.482940i	0.970408	−	0.241470i	$-0.0776296\pi$
$632$		0			0
$633$	17.8552			0.709682
$634$		0			0
$635$			52.5922i			2.08706i
$636$		0			0
$637$	−3.54183	+	0.674887i	−0.140332	+	0.0267400i
$638$		0			0
$639$			36.9770i			1.46279i
$640$		0			0
$641$	−39.3335			−1.55358			−0.776789	−	0.629761i	$-0.783153\pi$
$641$	−39.3335			−1.55358			−0.776789	+	0.629761i	$0.783153\pi$
$642$		0			0
$643$			33.0164i			1.30204i	0.759061	+	0.651020i	$0.225659\pi$
$643$			33.0164i			1.30204i	−0.759061	+	0.651020i	$0.774341\pi$
$644$		0			0
$645$		−	3.92360i		−	0.154492i
$646$		0			0
$647$	−34.6704			−1.36303			−0.681517	−	0.731803i	$-0.738679\pi$
$647$	−34.6704			−1.36303			−0.681517	+	0.731803i	$0.738679\pi$
$648$		0			0
$649$	−14.5529			−0.571251
$650$		0			0
$651$	3.92204			0.153717
$652$		0			0
$653$	3.40662			0.133311			0.0666557	−	0.997776i	$-0.478767\pi$
$653$	3.40662			0.133311			0.0666557	+	0.997776i	$0.478767\pi$
$654$		0			0
$655$			14.8013i			0.578336i
$656$		0			0
$657$		−	28.7711i		−	1.12247i
$658$		0			0
$659$	25.2434			0.983342			0.491671	−	0.870781i	$-0.336386\pi$
$659$	25.2434			0.983342			0.491671	+	0.870781i	$0.336386\pi$
$660$		0			0
$661$		−	43.2666i		−	1.68288i	−0.540354	−	0.841438i	$-0.681710\pi$
$661$		−	43.2666i		−	1.68288i	0.540354	−	0.841438i	$-0.318290\pi$
$662$		0			0
$663$	−0.495539	+	0.0944239i	−0.0192451	+	0.00366712i
$664$		0			0
$665$		−	14.2034i		−	0.550784i
$666$		0			0
$667$	17.8339			0.690533
$668$		0			0
$669$		−	4.87551i		−	0.188498i
$670$		0			0
$671$		−	9.96003i		−	0.384503i
$672$		0			0
$673$	−27.5991			−1.06387			−0.531934	−	0.846786i	$-0.678534\pi$
$673$	−27.5991			−1.06387			−0.531934	+	0.846786i	$0.678534\pi$
$674$		0			0
$675$	7.97277			0.306872
$676$		0			0
$677$	−47.2797			−1.81711			−0.908554	−	0.417768i	$-0.862813\pi$
$677$	−47.2797			−1.81711			−0.908554	+	0.417768i	$0.862813\pi$
$678$		0			0
$679$	13.8099			0.529976
$680$		0			0
$681$			16.7899i			0.643389i
$682$		0			0
$683$			25.2893i			0.967670i	0.875159	+	0.483835i	$0.160757\pi$
$683$			25.2893i			0.967670i	−0.875159	+	0.483835i	$0.839243\pi$
$684$		0			0
$685$	−34.3510			−1.31248
$686$		0			0
$687$			19.6785i			0.750781i
$688$		0			0
$689$	15.7019	−	2.99197i	0.598196	−	0.113985i
$690$		0			0
$691$		−	29.5772i		−	1.12517i	−0.826740	−	0.562584i	$-0.809807\pi$
$691$		−	29.5772i		−	1.12517i	0.826740	−	0.562584i	$-0.190193\pi$
$692$		0			0
$693$	2.50236			0.0950566
$694$		0			0
$695$			51.1099i			1.93871i
$696$		0			0
$697$		−	0.0604069i		−	0.00228808i
$698$		0			0
$699$	−0.402379			−0.0152194
$700$		0			0
$701$	−35.2769			−1.33239			−0.666196	−	0.745777i	$-0.732079\pi$
$701$	−35.2769			−1.33239			−0.666196	+	0.745777i	$0.732079\pi$
$702$		0			0
$703$	−20.4227			−0.770257
$704$		0			0
$705$	−7.13906			−0.268872
$706$		0			0
$707$		−	10.0844i		−	0.379264i
$708$		0			0
$709$			15.6266i			0.586870i	0.955979	+	0.293435i	$0.0947984\pi$
$709$			15.6266i			0.586870i	−0.955979	+	0.293435i	$0.905202\pi$
$710$		0			0
$711$	11.9662			0.448768
$712$		0			0
$713$			34.1359i			1.27840i
$714$		0			0
$715$	9.40687	−	1.79246i	0.351797	−	0.0670341i
$716$		0			0
$717$			7.40581i			0.276575i
$718$		0			0
$719$	−20.4518			−0.762722			−0.381361	−	0.924426i	$-0.624545\pi$
$719$	−20.4518			−0.762722			−0.381361	+	0.924426i	$0.624545\pi$
$720$		0			0
$721$		−	15.0334i		−	0.559873i
$722$		0			0
$723$		−	1.60209i		−	0.0595824i
$724$		0			0
$725$	−5.96608			−0.221575
$726$		0			0
$727$	−18.2659			−0.677444			−0.338722	−	0.940887i	$-0.609995\pi$
$727$	−18.2659			−0.677444			−0.338722	+	0.940887i	$0.609995\pi$
$728$		0			0
$729$	−3.71869			−0.137729
$730$		0			0
$731$	0.415335			0.0153617
$732$		0			0
$733$		−	20.8353i		−	0.769569i	−0.923006	−	0.384785i	$-0.874276\pi$
$733$		−	20.8353i		−	0.769569i	0.923006	−	0.384785i	$-0.125724\pi$
$734$		0			0
$735$		−	1.87360i		−	0.0691089i
$736$		0			0
$737$	6.86965			0.253047
$738$		0			0
$739$			46.7097i			1.71824i	0.511772	+	0.859122i	$0.328989\pi$
$739$			46.7097i			1.71824i	−0.511772	+	0.859122i	$0.671011\pi$
$740$		0			0
$741$	−2.54604	−	13.3617i	−0.0935312	−	0.490854i
$742$		0			0
$743$		−	14.7362i		−	0.540620i	−0.962773	−	0.270310i	$-0.912874\pi$
$743$		−	14.7362i		−	0.540620i	0.962773	−	0.270310i	$-0.0871261\pi$
$744$		0			0
$745$	14.9506			0.547749
$746$		0			0
$747$			37.5785i			1.37492i
$748$		0			0
$749$		−	5.99038i		−	0.218884i
$750$		0			0
$751$	29.8710			1.09001			0.545004	−	0.838433i	$-0.316528\pi$
$751$	29.8710			1.09001			0.545004	+	0.838433i	$0.316528\pi$
$752$		0			0
$753$	−10.6402			−0.387750
$754$		0			0
$755$	−41.8385			−1.52266
$756$		0			0
$757$	−18.1350			−0.659129			−0.329564	−	0.944133i	$-0.606902\pi$
$757$	−18.1350			−0.659129			−0.329564	+	0.944133i	$0.606902\pi$
$758$		0			0
$759$		−	4.33130i		−	0.157216i
$760$		0			0
$761$			3.35917i			0.121770i	0.998145	+	0.0608850i	$0.0193923\pi$
$761$			3.35917i			0.121770i	−0.998145	+	0.0608850i	$0.980608\pi$
$762$		0			0
$763$	−3.11708			−0.112846
$764$		0			0
$765$			1.31813i			0.0476570i
$766$		0			0
$767$	−9.82157	−	51.5438i	−0.354636	−	1.86114i
$768$		0			0
$769$		−	21.8495i		−	0.787914i	−0.919129	−	0.393957i	$-0.871106\pi$
$769$		−	21.8495i		−	0.787914i	0.919129	−	0.393957i	$-0.128894\pi$
$770$		0			0
$771$	−18.6794			−0.672721
$772$		0			0
$773$			30.6298i			1.10168i	0.834611	+	0.550839i	$0.185692\pi$
$773$			30.6298i			1.10168i	−0.834611	+	0.550839i	$0.814308\pi$
$774$		0			0
$775$		−	11.4197i		−	0.410207i
$776$		0			0
$777$	−2.69401			−0.0966469
$778$		0			0
$779$	1.62881			0.0583582
$780$		0			0
$781$	−14.7769			−0.528758
$782$		0			0
$783$	−11.2745			−0.402917
$784$		0			0
$785$		−	46.6510i		−	1.66505i
$786$		0			0
$787$		−	21.7694i		−	0.775995i	−0.921660	−	0.387997i	$-0.873167\pi$
$787$		−	21.7694i		−	0.775995i	0.921660	−	0.387997i	$-0.126833\pi$
$788$		0			0
$789$	−9.24495			−0.329129
$790$		0			0
$791$			7.77719i			0.276525i
$792$		0			0
$793$	35.2767	−	6.72190i	1.25271	−	0.238702i
$794$		0			0
$795$			8.30622i			0.294591i
$796$		0			0
$797$	2.37738			0.0842113			0.0421056	−	0.999113i	$-0.486593\pi$
$797$	2.37738			0.0842113			0.0421056	+	0.999113i	$0.486593\pi$
$798$		0			0
$799$		−	0.755708i		−	0.0267350i
$800$		0			0
$801$		−	18.2187i		−	0.643725i
$802$		0			0
$803$	11.4976			0.405742
$804$		0			0
$805$	16.3071			0.574750
$806$		0			0
$807$	−8.36416			−0.294432
$808$		0			0
$809$	−9.24207			−0.324934			−0.162467	−	0.986714i	$-0.551945\pi$
$809$	−9.24207			−0.324934			−0.162467	+	0.986714i	$0.551945\pi$
$810$		0			0
$811$			24.0516i			0.844567i	0.906464	+	0.422284i	$0.138771\pi$
$811$			24.0516i			0.844567i	−0.906464	+	0.422284i	$0.861229\pi$
$812$		0			0
$813$			15.6514i			0.548918i
$814$		0			0
$815$	−9.40657			−0.329498
$816$		0			0
$817$			11.1991i			0.391806i
$818$		0			0
$819$	1.68881	+	8.86290i	0.0590117	+	0.309695i
$820$		0			0
$821$		−	3.30719i		−	0.115422i	−0.998333	−	0.0577109i	$-0.981620\pi$
$821$		−	3.30719i		−	0.115422i	0.998333	−	0.0577109i	$-0.0183802\pi$
$822$		0			0
$823$	46.1081			1.60723			0.803613	−	0.595152i	$-0.202908\pi$
$823$	46.1081			1.60723			0.803613	+	0.595152i	$0.202908\pi$
$824$		0			0
$825$			1.44897i			0.0504468i
$826$		0			0
$827$			40.8890i			1.42185i	0.703268	+	0.710925i	$0.251723\pi$
$827$			40.8890i			1.42185i	−0.703268	+	0.710925i	$0.748277\pi$
$828$		0			0
$829$	−50.6020			−1.75748			−0.878740	−	0.477301i	$-0.841615\pi$
$829$	−50.6020			−1.75748			−0.878740	+	0.477301i	$0.841615\pi$
$830$		0			0
$831$	−2.49541			−0.0865649
$832$		0			0
$833$	0.198331			0.00687177
$834$		0			0
$835$	24.0345			0.831748
$836$		0			0
$837$		−	21.5805i		−	0.745931i
$838$		0			0
$839$			10.4839i			0.361945i	0.983488	+	0.180972i	$0.0579244\pi$
$839$			10.4839i			0.361945i	−0.983488	+	0.180972i	$0.942076\pi$
$840$		0			0
$841$	−20.5632			−0.709077
$842$		0			0
$843$		−	3.57229i		−	0.123036i
$844$		0			0
$845$	12.6972	+	32.1078i	0.436795	+	1.10454i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	0.258247			0.00886301
$850$		0			0
$851$		−	23.4476i		−	0.803772i
$852$		0			0
$853$			20.9651i			0.717832i	0.933370	+	0.358916i	$0.116854\pi$
$853$			20.9651i			0.717832i	−0.933370	+	0.358916i	$0.883146\pi$
$854$		0			0
$855$	−35.5420			−1.21551
$856$		0			0
$857$	24.6410			0.841721			0.420861	−	0.907125i	$-0.361728\pi$
$857$	24.6410			0.841721			0.420861	+	0.907125i	$0.361728\pi$
$858$		0			0
$859$	46.6433			1.59145			0.795724	−	0.605659i	$-0.207091\pi$
$859$	46.6433			1.59145			0.795724	+	0.605659i	$0.207091\pi$
$860$		0			0
$861$	0.214860			0.00732241
$862$		0			0
$863$		−	34.4036i		−	1.17111i	−0.810632	−	0.585555i	$-0.800876\pi$
$863$		−	34.4036i		−	1.17111i	0.810632	−	0.585555i	$-0.199124\pi$
$864$		0			0
$865$			50.6880i			1.72345i
$866$		0			0
$867$	−11.9647			−0.406343
$868$		0			0
$869$			4.78198i			0.162218i
$870$		0			0
$871$	4.63624	+	24.3311i	0.157093	+	0.824427i
$872$		0			0
$873$		−	34.5573i		−	1.16959i
$874$		0			0
$875$	7.82438			0.264512
$876$		0			0
$877$		−	1.57270i		−	0.0531064i	−0.999647	−	0.0265532i	$-0.991547\pi$
$877$		−	1.57270i		−	0.0531064i	0.999647	−	0.0265532i	$-0.00845313\pi$
$878$		0			0
$879$		−	10.4673i		−	0.353052i
$880$		0			0
$881$	−6.18142			−0.208257			−0.104129	−	0.994564i	$-0.533205\pi$
$881$	−6.18142			−0.208257			−0.104129	+	0.994564i	$0.533205\pi$
$882$		0			0
$883$	39.8498			1.34105			0.670527	−	0.741885i	$-0.266068\pi$
$883$	39.8498			1.34105			0.670527	+	0.741885i	$0.266068\pi$
$884$		0			0
$885$	27.2664			0.916549
$886$		0			0
$887$	−21.1687			−0.710776			−0.355388	−	0.934719i	$-0.615651\pi$
$887$	−21.1687			−0.710776			−0.355388	+	0.934719i	$0.615651\pi$
$888$		0			0
$889$		−	19.8017i		−	0.664129i
$890$		0			0
$891$		−	4.76885i		−	0.159762i
$892$		0			0
$893$	20.3769			0.681886
$894$		0			0
$895$		−	60.0740i		−	2.00805i
$896$		0			0
$897$	15.3407	−	2.92314i	0.512212	−	0.0976009i
$898$		0			0
$899$			16.1488i			0.538593i
$900$		0			0
$901$	−0.879258			−0.0292923
$902$		0			0
$903$			1.47729i			0.0491613i
$904$		0			0
$905$			46.4420i			1.54378i
$906$		0			0
$907$	33.3216			1.10643			0.553213	−	0.833040i	$-0.313401\pi$
$907$	33.3216			1.10643			0.553213	+	0.833040i	$0.313401\pi$
$908$		0			0
$909$	−25.2348			−0.836985
$910$		0			0
$911$	6.98570			0.231447			0.115723	−	0.993281i	$-0.463081\pi$
$911$	6.98570			0.231447			0.115723	+	0.993281i	$0.463081\pi$
$912$		0			0
$913$	−15.0172			−0.496998
$914$		0			0
$915$			18.6611i			0.616919i
$916$		0			0
$917$		−	5.57293i		−	0.184034i
$918$		0			0
$919$	−11.4737			−0.378481			−0.189240	−	0.981931i	$-0.560603\pi$
$919$	−11.4737			−0.378481			−0.189240	+	0.981931i	$0.560603\pi$
$920$		0			0
$921$		−	11.1058i		−	0.365947i
$922$		0			0
$923$	−9.97272	−	52.3371i	−0.328256	−	1.72270i
$924$		0			0
$925$			7.84404i			0.257910i
$926$		0			0
$927$	−37.6189			−1.23557
$928$		0			0
$929$		−	30.6879i		−	1.00684i	−0.864043	−	0.503418i	$-0.832076\pi$
$929$		−	30.6879i		−	1.00684i	0.864043	−	0.503418i	$-0.167924\pi$
$930$		0			0
$931$			5.34779i			0.175267i
$932$		0			0
$933$	11.5336			0.377594
$934$		0			0
$935$	−0.526755			−0.0172267
$936$		0			0
$937$	−27.3527			−0.893574			−0.446787	−	0.894640i	$-0.647432\pi$
$937$	−27.3527			−0.893574			−0.446787	+	0.894640i	$0.647432\pi$
$938$		0			0
$939$	14.0387			0.458134
$940$		0			0
$941$		−	12.9241i		−	0.421312i	−0.977560	−	0.210656i	$-0.932440\pi$
$941$		−	12.9241i		−	0.421312i	0.977560	−	0.210656i	$-0.0675600\pi$
$942$		0			0
$943$			1.87006i			0.0608975i
$944$		0			0
$945$	−10.3092			−0.335359
$946$		0			0
$947$			20.5595i			0.668093i	0.942557	+	0.334046i	$0.108414\pi$
$947$			20.5595i			0.668093i	−0.942557	+	0.334046i	$0.891586\pi$
$948$		0			0
$949$	7.75960	+	40.7226i	0.251887	+	1.32191i
$950$		0			0
$951$			1.31911i			0.0427749i
$952$		0			0
$953$	−13.2123			−0.427988			−0.213994	−	0.976835i	$-0.568647\pi$
$953$	−13.2123			−0.427988			−0.213994	+	0.976835i	$0.568647\pi$
$954$		0			0
$955$			36.1562i			1.16999i
$956$		0			0
$957$		−	2.04903i		−	0.0662357i
$958$		0			0
$959$	12.9336			0.417649
$960$		0			0
$961$	0.0895349			0.00288822
$962$		0			0
$963$	−14.9901			−0.483048
$964$		0			0
$965$	14.5084			0.467043
$966$		0			0
$967$			40.0154i			1.28681i	0.765526	+	0.643405i	$0.222479\pi$
$967$			40.0154i			1.28681i	−0.765526	+	0.643405i	$0.777521\pi$
$968$		0			0
$969$			0.748213i			0.0240361i
$970$		0			0
$971$	−21.6080			−0.693435			−0.346717	−	0.937970i	$-0.612704\pi$
$971$	−21.6080			−0.693435			−0.346717	+	0.937970i	$0.612704\pi$
$972$		0			0
$973$		−	19.2437i		−	0.616923i
$974$		0			0
$975$	−5.13201	+	0.977895i	−0.164356	+	0.0313177i
$976$		0			0
$977$			54.7924i			1.75296i	0.481434	+	0.876482i	$0.340116\pi$
$977$			54.7924i			1.75296i	−0.481434	+	0.876482i	$0.659884\pi$
$978$		0			0
$979$	7.28061			0.232689
$980$		0			0
$981$			7.80003i			0.249036i
$982$		0			0
$983$			30.4918i			0.972538i	0.873809	+	0.486269i	$0.161642\pi$
$983$			30.4918i			0.972538i	−0.873809	+	0.486269i	$0.838358\pi$
$984$		0			0
$985$	9.75038			0.310673
$986$		0			0
$987$	2.68796			0.0855587
$988$		0			0
$989$	−12.8578			−0.408854
$990$		0			0
$991$	9.46454			0.300651			0.150326	−	0.988637i	$-0.451968\pi$
$991$	9.46454			0.300651			0.150326	+	0.988637i	$0.451968\pi$
$992$		0			0
$993$		−	12.3420i		−	0.391663i
$994$		0			0
$995$		−	47.5857i		−	1.50857i
$996$		0			0
$997$	−3.85518			−0.122095			−0.0610473	−	0.998135i	$-0.519444\pi$
$997$	−3.85518			−0.122095			−0.0610473	+	0.998135i	$0.519444\pi$
$998$		0			0
$999$			14.8234i			0.468991i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.b.2157.18	yes	30
13.12	even	2	inner	4004.2.m.b.2157.17	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.b.2157.17	✓	30	13.12	even	2	inner
4004.2.m.b.2157.18	yes	30	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)

\(f(q)\)	\(=\)	\(q+0.705440 q^{3} +2.65594i q^{5} -1.00000i q^{7} -2.50236 q^{9} +O(q^{10})\)	\(q+0.705440 q^{3} +2.65594i q^{5} -1.00000i q^{7} -2.50236 q^{9} -1.00000i q^{11} +(3.54183 - 0.674887i) q^{13} +1.87360i q^{15} -0.198331 q^{17} -5.34779i q^{19} -0.705440i q^{21} +6.13987 q^{23} -2.05400 q^{25} -3.88158 q^{27} +2.90461 q^{29} +5.55972i q^{31} -0.705440i q^{33} +2.65594 q^{35} -3.81890i q^{37} +(2.49854 - 0.476092i) q^{39} +0.304576i q^{41} -2.09415 q^{43} -6.64610i q^{45} +3.81034i q^{47} -1.00000 q^{49} -0.139911 q^{51} +4.43329 q^{53} +2.65594 q^{55} -3.77254i q^{57} -14.5529i q^{59} +9.96003 q^{61} +2.50236i q^{63} +(1.79246 + 9.40687i) q^{65} +6.86965i q^{67} +4.33130 q^{69} -14.7769i q^{71} +11.4976i q^{73} -1.44897 q^{75} -1.00000 q^{77} -4.78198 q^{79} +4.76885 q^{81} -15.0172i q^{83} -0.526755i q^{85} +2.04903 q^{87} +7.28061i q^{89} +(-0.674887 - 3.54183i) q^{91} +3.92204i q^{93} +14.2034 q^{95} +13.8099i q^{97} +2.50236i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 18 q^{9}+O(q^{10}) \) 30 * q + 18 * q^9	\( 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) \) 30 * q + 18 * q^9 + 4 * q^13 + 12 * q^17 + 6 * q^23 - 6 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 30 * q^49 + 60 * q^51 - 38 * q^53 - 2 * q^55 - 36 * q^61 + 10 * q^65 + 36 * q^69 - 20 * q^75 - 30 * q^77 - 10 * q^79 - 42 * q^81 + 36 * q^87 + 6 * q^91 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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