LMFDB - Embedded newform 3136.2.b.g.1569.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(1569,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3136.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.b (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:	$25.0410860739$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 448)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3136.1569
Dual form			3136.2.b.g.1569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205i q^{3} +2.73205i q^{5} -4.46410 q^{9} +O(q^{10})$	$q+2.73205i q^{3} +2.73205i q^{5} -4.46410 q^{9} -5.46410i q^{11} -6.73205i q^{13} -7.46410 q^{15} -2.00000 q^{17} -1.26795i q^{19} +3.46410 q^{23} -2.46410 q^{25} -4.00000i q^{27} -1.46410i q^{29} +4.00000 q^{31} +14.9282 q^{33} +1.46410i q^{37} +18.3923 q^{39} +2.00000 q^{41} -5.46410i q^{43} -12.1962i q^{45} +2.92820 q^{47} -5.46410i q^{51} -12.0000i q^{53} +14.9282 q^{55} +3.46410 q^{57} -9.66025i q^{59} +11.1244i q^{61} +18.3923 q^{65} -8.00000i q^{67} +9.46410i q^{69} -2.92820 q^{71} -12.9282 q^{73} -6.73205i q^{75} +10.9282 q^{79} -2.46410 q^{81} -5.66025i q^{83} -5.46410i q^{85} +4.00000 q^{87} -11.8564 q^{89} +10.9282i q^{93} +3.46410 q^{95} -8.92820 q^{97} +24.3923i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 16 q^{15} - 8 q^{17} + 4 q^{25} + 16 q^{31} + 32 q^{33} + 32 q^{39} + 8 q^{41} - 16 q^{47} + 32 q^{55} + 32 q^{65} + 16 q^{71} - 24 q^{73} + 16 q^{79} + 4 q^{81} + 16 q^{87} + 8 q^{89} - 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 - 16 * q^15 - 8 * q^17 + 4 * q^25 + 16 * q^31 + 32 * q^33 + 32 * q^39 + 8 * q^41 - 16 * q^47 + 32 * q^55 + 32 * q^65 + 16 * q^71 - 24 * q^73 + 16 * q^79 + 4 * q^81 + 16 * q^87 + 8 * q^89 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.73205i	1.57735i	0.614810	+	0.788675i	$0.289233\pi$
$3$	2.73205i	1.57735i	−0.614810	+	0.788675i	$0.710767\pi$
$4$	0	0
$5$	2.73205i	1.22181i	0.791704	+	0.610905i	$0.209194\pi$
$5$	2.73205i	1.22181i	−0.791704	+	0.610905i	$0.790806\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−4.46410	−1.48803
$10$	0	0
$11$	− 5.46410i	− 1.64749i	−0.566961	−	0.823744i	$-0.691881\pi$
$11$	− 5.46410i	− 1.64749i	0.566961	−	0.823744i	$-0.308119\pi$
$12$	0	0
$13$	− 6.73205i	− 1.86713i	−0.358402	−	0.933567i	$-0.616678\pi$
$13$	− 6.73205i	− 1.86713i	0.358402	−	0.933567i	$-0.383322\pi$
$14$	0	0
$15$	−7.46410	−1.92722
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 1.26795i	− 0.290887i	−0.989367	−	0.145444i	$-0.953539\pi$
$19$	− 1.26795i	− 0.290887i	0.989367	−	0.145444i	$-0.0464610\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	−2.46410	−0.492820
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	− 1.46410i	− 0.271877i	−0.990717	−	0.135938i	$-0.956595\pi$
$29$	− 1.46410i	− 0.271877i	0.990717	−	0.135938i	$-0.0434049\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	14.9282	2.59867
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.46410i	0.240697i	0.992732	+	0.120348i	$0.0384012\pi$
$37$	1.46410i	0.240697i	−0.992732	+	0.120348i	$0.961599\pi$
$38$	0	0
$39$	18.3923	2.94513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 5.46410i	− 0.833268i	−0.909074	−	0.416634i	$-0.863210\pi$
$43$	− 5.46410i	− 0.833268i	0.909074	−	0.416634i	$-0.136790\pi$
$44$	0	0
$45$	− 12.1962i	− 1.81810i
$46$	0	0
$47$	2.92820	0.427122	0.213561	−	0.976930i	$-0.431494\pi$
$47$	2.92820	0.427122	0.213561	+	0.976930i	$0.431494\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 5.46410i	− 0.765127i
$52$	0	0
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	14.9282	2.01292
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	− 9.66025i	− 1.25766i	−0.777544	−	0.628829i	$-0.783535\pi$
$59$	− 9.66025i	− 1.25766i	0.777544	−	0.628829i	$-0.216465\pi$
$60$	0	0
$61$	11.1244i	1.42433i	0.702013	+	0.712164i	$0.252285\pi$
$61$	11.1244i	1.42433i	−0.702013	+	0.712164i	$0.747715\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.3923	2.28128
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	9.46410i	1.13934i
$70$	0	0
$71$	−2.92820	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$71$	−2.92820	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$72$	0	0
$73$	−12.9282	−1.51313	−0.756566	−	0.653917i	$-0.773124\pi$
$73$	−12.9282	−1.51313	−0.756566	+	0.653917i	$0.773124\pi$
$74$	0	0
$75$	− 6.73205i	− 0.777350i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.9282	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	− 5.66025i	− 0.621294i	−0.950525	−	0.310647i	$-0.899454\pi$
$83$	− 5.66025i	− 0.621294i	0.950525	−	0.310647i	$-0.100546\pi$
$84$	0	0
$85$	− 5.46410i	− 0.592665i
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	−11.8564	−1.25678	−0.628388	−	0.777900i	$-0.716285\pi$
$89$	−11.8564	−1.25678	−0.628388	+	0.777900i	$0.716285\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	10.9282i	1.13320i
$94$	0	0
$95$	3.46410	0.355409
$96$	0	0
$97$	−8.92820	−0.906522	−0.453261	−	0.891378i	$-0.649739\pi$
$97$	−8.92820	−0.906522	−0.453261	+	0.891378i	$0.649739\pi$
$98$	0	0
$99$	24.3923i	2.45152i
$100$	0	0
$101$	− 1.66025i	− 0.165201i	−0.996583	−	0.0826007i	$-0.973677\pi$
$101$	− 1.66025i	− 0.165201i	0.996583	−	0.0826007i	$-0.0263226\pi$
$102$	0	0
$103$	9.85641	0.971181	0.485590	−	0.874187i	$-0.338605\pi$
$103$	9.85641	0.971181	0.485590	+	0.874187i	$0.338605\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.07180i	− 0.490309i	−0.969484	−	0.245155i	$-0.921161\pi$
$107$	− 5.07180i	− 0.490309i	0.969484	−	0.245155i	$-0.0788387\pi$
$108$	0	0
$109$	12.3923i	1.18697i	0.804846	+	0.593484i	$0.202248\pi$
$109$	12.3923i	1.18697i	−0.804846	+	0.593484i	$0.797752\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−6.53590	−0.614846	−0.307423	−	0.951573i	$-0.599467\pi$
$113$	−6.53590	−0.614846	−0.307423	+	0.951573i	$0.599467\pi$
$114$	0	0
$115$	9.46410i	0.882532i
$116$	0	0
$117$	30.0526i	2.77836i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−18.8564	−1.71422
$122$	0	0
$123$	5.46410i	0.492681i
$124$	0	0
$125$	6.92820i	0.619677i
$126$	0	0
$127$	11.4641	1.01727	0.508637	−	0.860981i	$-0.330149\pi$
$127$	11.4641	1.01727	0.508637	+	0.860981i	$0.330149\pi$
$128$	0	0
$129$	14.9282	1.31436
$130$	0	0
$131$	− 10.7321i	− 0.937664i	−0.883287	−	0.468832i	$-0.844675\pi$
$131$	− 10.7321i	− 0.937664i	0.883287	−	0.468832i	$-0.155325\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	10.9282	0.940550
$136$	0	0
$137$	19.8564	1.69645	0.848224	−	0.529638i	$-0.177672\pi$
$137$	19.8564	1.69645	0.848224	+	0.529638i	$0.177672\pi$
$138$	0	0
$139$	− 9.26795i	− 0.786097i	−0.919518	−	0.393049i	$-0.871420\pi$
$139$	− 9.26795i	− 0.786097i	0.919518	−	0.393049i	$-0.128580\pi$
$140$	0	0
$141$	8.00000i	0.673722i
$142$	0	0
$143$	−36.7846	−3.07608
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1.07180i	− 0.0878050i	−0.999036	−	0.0439025i	$-0.986021\pi$
$149$	− 1.07180i	− 0.0878050i	0.999036	−	0.0439025i	$-0.0139791\pi$
$150$	0	0
$151$	2.39230	0.194683	0.0973415	−	0.995251i	$-0.468966\pi$
$151$	2.39230	0.194683	0.0973415	+	0.995251i	$0.468966\pi$
$152$	0	0
$153$	8.92820	0.721802
$154$	0	0
$155$	10.9282i	0.877774i
$156$	0	0
$157$	12.1962i	0.973359i	0.873581	+	0.486679i	$0.161792\pi$
$157$	12.1962i	0.973359i	−0.873581	+	0.486679i	$0.838208\pi$
$158$	0	0
$159$	32.7846	2.59999
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.4641i	1.05459i	0.849682	+	0.527295i	$0.176794\pi$
$163$	13.4641i	1.05459i	−0.849682	+	0.527295i	$0.823206\pi$
$164$	0	0
$165$	40.7846i	3.17508i
$166$	0	0
$167$	−5.07180	−0.392467	−0.196234	−	0.980557i	$-0.562871\pi$
$167$	−5.07180	−0.392467	−0.196234	+	0.980557i	$0.562871\pi$
$168$	0	0
$169$	−32.3205	−2.48619
$170$	0	0
$171$	5.66025i	0.432850i
$172$	0	0
$173$	− 3.80385i	− 0.289201i	−0.989490	−	0.144601i	$-0.953810\pi$
$173$	− 3.80385i	− 0.289201i	0.989490	−	0.144601i	$-0.0461897\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	26.3923	1.98377
$178$	0	0
$179$	2.92820i	0.218864i	0.993994	+	0.109432i	$0.0349032\pi$
$179$	2.92820i	0.218864i	−0.993994	+	0.109432i	$0.965097\pi$
$180$	0	0
$181$	− 10.7321i	− 0.797707i	−0.917015	−	0.398854i	$-0.869408\pi$
$181$	− 10.7321i	− 0.797707i	0.917015	−	0.398854i	$-0.130592\pi$
$182$	0	0
$183$	−30.3923	−2.24666
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	10.9282i	0.799149i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	6.53590	0.470464	0.235232	−	0.971939i	$-0.424415\pi$
$193$	6.53590	0.470464	0.235232	+	0.971939i	$0.424415\pi$
$194$	0	0
$195$	50.2487i	3.59838i
$196$	0	0
$197$	− 25.8564i	− 1.84219i	−0.389335	−	0.921096i	$-0.627295\pi$
$197$	− 25.8564i	− 1.84219i	0.389335	−	0.921096i	$-0.372705\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	21.8564	1.54163
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5.46410i	0.381629i
$206$	0	0
$207$	−15.4641	−1.07483
$208$	0	0
$209$	−6.92820	−0.479234
$210$	0	0
$211$	− 16.7846i	− 1.15550i	−0.816214	−	0.577750i	$-0.803931\pi$
$211$	− 16.7846i	− 1.15550i	0.816214	−	0.577750i	$-0.196069\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	14.9282	1.01810
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 35.3205i	− 2.38674i
$220$	0	0
$221$	13.4641i	0.905693i
$222$	0	0
$223$	−6.92820	−0.463947	−0.231973	−	0.972722i	$-0.574518\pi$
$223$	−6.92820	−0.463947	−0.231973	+	0.972722i	$0.574518\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	3.80385i	0.252470i	0.992000	+	0.126235i	$0.0402894\pi$
$227$	3.80385i	0.252470i	−0.992000	+	0.126235i	$0.959711\pi$
$228$	0	0
$229$	− 17.2679i	− 1.14110i	−0.821263	−	0.570549i	$-0.806730\pi$
$229$	− 17.2679i	− 1.14110i	0.821263	−	0.570549i	$-0.193270\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	8.00000i	0.521862i
$236$	0	0
$237$	29.8564i	1.93938i
$238$	0	0
$239$	12.5359	0.810880	0.405440	−	0.914122i	$-0.367118\pi$
$239$	12.5359	0.810880	0.405440	+	0.914122i	$0.367118\pi$
$240$	0	0
$241$	18.7846	1.21002	0.605012	−	0.796217i	$-0.293168\pi$
$241$	18.7846	1.21002	0.605012	+	0.796217i	$0.293168\pi$
$242$	0	0
$243$	− 18.7321i	− 1.20166i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.53590	−0.543126
$248$	0	0
$249$	15.4641	0.979998
$250$	0	0
$251$	− 15.8038i	− 0.997530i	−0.866737	−	0.498765i	$-0.833787\pi$
$251$	− 15.8038i	− 0.997530i	0.866737	−	0.498765i	$-0.166213\pi$
$252$	0	0
$253$	− 18.9282i	− 1.19001i
$254$	0	0
$255$	14.9282	0.934840
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.53590i	0.404562i
$262$	0	0
$263$	−10.9282	−0.673862	−0.336931	−	0.941529i	$-0.609389\pi$
$263$	−10.9282	−0.673862	−0.336931	+	0.941529i	$0.609389\pi$
$264$	0	0
$265$	32.7846	2.01394
$266$	0	0
$267$	− 32.3923i	− 1.98238i
$268$	0	0
$269$	− 12.5885i	− 0.767532i	−0.923430	−	0.383766i	$-0.874627\pi$
$269$	− 12.5885i	− 0.767532i	0.923430	−	0.383766i	$-0.125373\pi$
$270$	0	0
$271$	14.9282	0.906824	0.453412	−	0.891301i	$-0.350207\pi$
$271$	14.9282	0.906824	0.453412	+	0.891301i	$0.350207\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.4641i	0.811916i
$276$	0	0
$277$	4.00000i	0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$	4.00000i	0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$	0	0
$279$	−17.8564	−1.06904
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	16.5885i	0.986081i	0.870006	+	0.493041i	$0.164115\pi$
$283$	16.5885i	0.986081i	−0.870006	+	0.493041i	$0.835885\pi$
$284$	0	0
$285$	9.46410i	0.560605i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 24.3923i	− 1.42990i
$292$	0	0
$293$	12.1962i	0.712507i	0.934389	+	0.356253i	$0.115946\pi$
$293$	12.1962i	0.712507i	−0.934389	+	0.356253i	$0.884054\pi$
$294$	0	0
$295$	26.3923	1.53662
$296$	0	0
$297$	−21.8564	−1.26824
$298$	0	0
$299$	− 23.3205i	− 1.34866i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.53590	0.260581
$304$	0	0
$305$	−30.3923	−1.74026
$306$	0	0
$307$	− 8.58846i	− 0.490169i	−0.969502	−	0.245085i	$-0.921184\pi$
$307$	− 8.58846i	− 0.490169i	0.969502	−	0.245085i	$-0.0788157\pi$
$308$	0	0
$309$	26.9282i	1.53189i
$310$	0	0
$311$	−5.85641	−0.332086	−0.166043	−	0.986118i	$-0.553099\pi$
$311$	−5.85641	−0.332086	−0.166043	+	0.986118i	$0.553099\pi$
$312$	0	0
$313$	−19.8564	−1.12235	−0.561175	−	0.827697i	$-0.689651\pi$
$313$	−19.8564	−1.12235	−0.561175	+	0.827697i	$0.689651\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.7846i	1.61670i	0.588699	+	0.808352i	$0.299640\pi$
$317$	28.7846i	1.61670i	−0.588699	+	0.808352i	$0.700360\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	13.8564	0.773389
$322$	0	0
$323$	2.53590i	0.141101i
$324$	0	0
$325$	16.5885i	0.920162i
$326$	0	0
$327$	−33.8564	−1.87226
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.3923i	0.901003i	0.892776	+	0.450501i	$0.148755\pi$
$331$	16.3923i	0.901003i	−0.892776	+	0.450501i	$0.851245\pi$
$332$	0	0
$333$	− 6.53590i	− 0.358165i
$334$	0	0
$335$	21.8564	1.19414
$336$	0	0
$337$	16.3923	0.892946	0.446473	−	0.894797i	$-0.352680\pi$
$337$	16.3923	0.892946	0.446473	+	0.894797i	$0.352680\pi$
$338$	0	0
$339$	− 17.8564i	− 0.969827i
$340$	0	0
$341$	− 21.8564i	− 1.18359i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−25.8564	−1.39206
$346$	0	0
$347$	35.3205i	1.89610i	0.318115	+	0.948052i	$0.396950\pi$
$347$	35.3205i	1.89610i	−0.318115	+	0.948052i	$0.603050\pi$
$348$	0	0
$349$	− 14.0526i	− 0.752216i	−0.926576	−	0.376108i	$-0.877262\pi$
$349$	− 14.0526i	− 0.752216i	0.926576	−	0.376108i	$-0.122738\pi$
$350$	0	0
$351$	−26.9282	−1.43732
$352$	0	0
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	− 8.00000i	− 0.424596i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.4641	1.44950	0.724750	−	0.689012i	$-0.241955\pi$
$359$	27.4641	1.44950	0.724750	+	0.689012i	$0.241955\pi$
$360$	0	0
$361$	17.3923	0.915384
$362$	0	0
$363$	− 51.5167i	− 2.70392i
$364$	0	0
$365$	− 35.3205i	− 1.84876i
$366$	0	0
$367$	20.7846	1.08495	0.542474	−	0.840073i	$-0.317488\pi$
$367$	20.7846	1.08495	0.542474	+	0.840073i	$0.317488\pi$
$368$	0	0
$369$	−8.92820	−0.464784
$370$	0	0
$371$	0	0
$372$	0	0
$373$	30.9282i	1.60140i	0.599064	+	0.800701i	$0.295539\pi$
$373$	30.9282i	1.60140i	−0.599064	+	0.800701i	$0.704461\pi$
$374$	0	0
$375$	−18.9282	−0.977448
$376$	0	0
$377$	−9.85641	−0.507631
$378$	0	0
$379$	14.2487i	0.731907i	0.930633	+	0.365954i	$0.119257\pi$
$379$	14.2487i	0.731907i	−0.930633	+	0.365954i	$0.880743\pi$
$380$	0	0
$381$	31.3205i	1.60460i
$382$	0	0
$383$	−32.7846	−1.67522	−0.837608	−	0.546272i	$-0.816046\pi$
$383$	−32.7846	−1.67522	−0.837608	+	0.546272i	$0.816046\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.3923i	1.23993i
$388$	0	0
$389$	− 23.3205i	− 1.18240i	−0.806526	−	0.591198i	$-0.798655\pi$
$389$	− 23.3205i	− 1.18240i	0.806526	−	0.591198i	$-0.201345\pi$
$390$	0	0
$391$	−6.92820	−0.350374
$392$	0	0
$393$	29.3205	1.47902
$394$	0	0
$395$	29.8564i	1.50224i
$396$	0	0
$397$	4.87564i	0.244702i	0.992487	+	0.122351i	$0.0390433\pi$
$397$	4.87564i	0.244702i	−0.992487	+	0.122351i	$0.960957\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.4641	0.672365	0.336183	−	0.941797i	$-0.390864\pi$
$401$	13.4641	0.672365	0.336183	+	0.941797i	$0.390864\pi$
$402$	0	0
$403$	− 26.9282i	− 1.34139i
$404$	0	0
$405$	− 6.73205i	− 0.334518i
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	3.07180	0.151891	0.0759453	−	0.997112i	$-0.475803\pi$
$409$	3.07180	0.151891	0.0759453	+	0.997112i	$0.475803\pi$
$410$	0	0
$411$	54.2487i	2.67589i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	15.4641	0.759103
$416$	0	0
$417$	25.3205	1.23995
$418$	0	0
$419$	23.8038i	1.16289i	0.813584	+	0.581447i	$0.197513\pi$
$419$	23.8038i	1.16289i	−0.813584	+	0.581447i	$0.802487\pi$
$420$	0	0
$421$	25.8564i	1.26016i	0.776529	+	0.630082i	$0.216979\pi$
$421$	25.8564i	1.26016i	−0.776529	+	0.630082i	$0.783021\pi$
$422$	0	0
$423$	−13.0718	−0.635573
$424$	0	0
$425$	4.92820	0.239053
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 100.497i	− 4.85206i
$430$	0	0
$431$	35.4641	1.70825	0.854123	−	0.520071i	$-0.174095\pi$
$431$	35.4641	1.70825	0.854123	+	0.520071i	$0.174095\pi$
$432$	0	0
$433$	−15.8564	−0.762010	−0.381005	−	0.924573i	$-0.624422\pi$
$433$	−15.8564	−0.762010	−0.381005	+	0.924573i	$0.624422\pi$
$434$	0	0
$435$	10.9282i	0.523967i
$436$	0	0
$437$	− 4.39230i	− 0.210112i
$438$	0	0
$439$	30.9282	1.47612	0.738061	−	0.674734i	$-0.235742\pi$
$439$	30.9282	1.47612	0.738061	+	0.674734i	$0.235742\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	− 32.3923i	− 1.53554i
$446$	0	0
$447$	2.92820	0.138499
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	− 10.9282i	− 0.514589i
$452$	0	0
$453$	6.53590i	0.307083i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.3205	0.716663	0.358332	−	0.933594i	$-0.383346\pi$
$457$	15.3205	0.716663	0.358332	+	0.933594i	$0.383346\pi$
$458$	0	0
$459$	8.00000i	0.373408i
$460$	0	0
$461$	− 24.1962i	− 1.12693i	−0.826141	−	0.563464i	$-0.809469\pi$
$461$	− 24.1962i	− 1.12693i	0.826141	−	0.563464i	$-0.190531\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	−29.8564	−1.38456
$466$	0	0
$467$	0.875644i	0.0405200i	0.999795	+	0.0202600i	$0.00644940\pi$
$467$	0.875644i	0.0405200i	−0.999795	+	0.0202600i	$0.993551\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−33.3205	−1.53533
$472$	0	0
$473$	−29.8564	−1.37280
$474$	0	0
$475$	3.12436i	0.143355i
$476$	0	0
$477$	53.5692i	2.45277i
$478$	0	0
$479$	−9.85641	−0.450351	−0.225175	−	0.974318i	$-0.572295\pi$
$479$	−9.85641	−0.450351	−0.225175	+	0.974318i	$0.572295\pi$
$480$	0	0
$481$	9.85641	0.449413
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 24.3923i	− 1.10760i
$486$	0	0
$487$	21.6077	0.979138	0.489569	−	0.871965i	$-0.337154\pi$
$487$	21.6077	0.979138	0.489569	+	0.871965i	$0.337154\pi$
$488$	0	0
$489$	−36.7846	−1.66346
$490$	0	0
$491$	− 32.7846i	− 1.47955i	−0.672855	−	0.739774i	$-0.734932\pi$
$491$	− 32.7846i	− 1.47955i	0.672855	−	0.739774i	$-0.265068\pi$
$492$	0	0
$493$	2.92820i	0.131880i
$494$	0	0
$495$	−66.6410	−2.99529
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13.0718i	0.585174i	0.956239	+	0.292587i	$0.0945161\pi$
$499$	13.0718i	0.585174i	−0.956239	+	0.292587i	$0.905484\pi$
$500$	0	0
$501$	− 13.8564i	− 0.619059i
$502$	0	0
$503$	30.9282	1.37902	0.689510	−	0.724276i	$-0.257826\pi$
$503$	30.9282	1.37902	0.689510	+	0.724276i	$0.257826\pi$
$504$	0	0
$505$	4.53590	0.201845
$506$	0	0
$507$	− 88.3013i	− 3.92160i
$508$	0	0
$509$	− 3.12436i	− 0.138485i	−0.997600	−	0.0692423i	$-0.977942\pi$
$509$	− 3.12436i	− 0.138485i	0.997600	−	0.0692423i	$-0.0220582\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.07180	−0.223925
$514$	0	0
$515$	26.9282i	1.18660i
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	10.3923	0.456172
$520$	0	0
$521$	14.7846	0.647726	0.323863	−	0.946104i	$-0.395018\pi$
$521$	14.7846	0.647726	0.323863	+	0.946104i	$0.395018\pi$
$522$	0	0
$523$	3.41154i	0.149176i	0.997214	+	0.0745882i	$0.0237642\pi$
$523$	3.41154i	0.149176i	−0.997214	+	0.0745882i	$0.976236\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	43.1244i	1.87144i
$532$	0	0
$533$	− 13.4641i	− 0.583195i
$534$	0	0
$535$	13.8564	0.599065
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.00000i	0.171973i	0.996296	+	0.0859867i	$0.0274043\pi$
$541$	4.00000i	0.171973i	−0.996296	+	0.0859867i	$0.972596\pi$
$542$	0	0
$543$	29.3205	1.25826
$544$	0	0
$545$	−33.8564	−1.45025
$546$	0	0
$547$	46.2487i	1.97745i	0.149736	+	0.988726i	$0.452158\pi$
$547$	46.2487i	1.97745i	−0.149736	+	0.988726i	$0.547842\pi$
$548$	0	0
$549$	− 49.6603i	− 2.11945i
$550$	0	0
$551$	−1.85641	−0.0790856
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 10.9282i	− 0.463876i
$556$	0	0
$557$	− 23.7128i	− 1.00474i	−0.864652	−	0.502372i	$-0.832461\pi$
$557$	− 23.7128i	− 1.00474i	0.864652	−	0.502372i	$-0.167539\pi$
$558$	0	0
$559$	−36.7846	−1.55582
$560$	0	0
$561$	−29.8564	−1.26054
$562$	0	0
$563$	34.0526i	1.43514i	0.696484	+	0.717572i	$0.254747\pi$
$563$	34.0526i	1.43514i	−0.696484	+	0.717572i	$0.745253\pi$
$564$	0	0
$565$	− 17.8564i	− 0.751225i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.46410	−0.229067	−0.114534	−	0.993419i	$-0.536537\pi$
$569$	−5.46410	−0.229067	−0.114534	+	0.993419i	$0.536537\pi$
$570$	0	0
$571$	− 25.1769i	− 1.05362i	−0.849983	−	0.526811i	$-0.823388\pi$
$571$	− 25.1769i	− 1.05362i	0.849983	−	0.526811i	$-0.176612\pi$
$572$	0	0
$573$	− 43.7128i	− 1.82613i
$574$	0	0
$575$	−8.53590	−0.355972
$576$	0	0
$577$	31.0718	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$577$	31.0718	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$578$	0	0
$579$	17.8564i	0.742087i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−65.5692	−2.71560
$584$	0	0
$585$	−82.1051	−3.39463
$586$	0	0
$587$	− 33.2679i	− 1.37312i	−0.727075	−	0.686558i	$-0.759121\pi$
$587$	− 33.2679i	− 1.37312i	0.727075	−	0.686558i	$-0.240879\pi$
$588$	0	0
$589$	− 5.07180i	− 0.208980i
$590$	0	0
$591$	70.6410	2.90578
$592$	0	0
$593$	−12.1436	−0.498678	−0.249339	−	0.968416i	$-0.580213\pi$
$593$	−12.1436	−0.498678	−0.249339	+	0.968416i	$0.580213\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10.9282i	− 0.447262i
$598$	0	0
$599$	−27.7128	−1.13231	−0.566157	−	0.824297i	$-0.691571\pi$
$599$	−27.7128	−1.13231	−0.566157	+	0.824297i	$0.691571\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	35.7128i	1.45434i
$604$	0	0
$605$	− 51.5167i	− 2.09445i
$606$	0	0
$607$	−27.7128	−1.12483	−0.562414	−	0.826856i	$-0.690127\pi$
$607$	−27.7128	−1.12483	−0.562414	+	0.826856i	$0.690127\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 19.7128i	− 0.797495i
$612$	0	0
$613$	− 39.3205i	− 1.58814i	−0.607826	−	0.794070i	$-0.707958\pi$
$613$	− 39.3205i	− 1.58814i	0.607826	−	0.794070i	$-0.292042\pi$
$614$	0	0
$615$	−14.9282	−0.601963
$616$	0	0
$617$	−24.3923	−0.981997	−0.490999	−	0.871160i	$-0.663368\pi$
$617$	−24.3923	−0.981997	−0.490999	+	0.871160i	$0.663368\pi$
$618$	0	0
$619$	− 24.1962i	− 0.972525i	−0.873813	−	0.486263i	$-0.838360\pi$
$619$	− 24.1962i	− 0.972525i	0.873813	−	0.486263i	$-0.161640\pi$
$620$	0	0
$621$	− 13.8564i	− 0.556038i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	− 18.9282i	− 0.755920i
$628$	0	0
$629$	− 2.92820i	− 0.116755i
$630$	0	0
$631$	49.5692	1.97332	0.986660	−	0.162796i	$-0.0520513\pi$
$631$	49.5692	1.97332	0.986660	+	0.162796i	$0.0520513\pi$
$632$	0	0
$633$	45.8564	1.82263
$634$	0	0
$635$	31.3205i	1.24292i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.0718	0.517112
$640$	0	0
$641$	−22.2487	−0.878771	−0.439386	−	0.898299i	$-0.644804\pi$
$641$	−22.2487	−0.878771	−0.439386	+	0.898299i	$0.644804\pi$
$642$	0	0
$643$	14.3397i	0.565504i	0.959193	+	0.282752i	$0.0912474\pi$
$643$	14.3397i	0.565504i	−0.959193	+	0.282752i	$0.908753\pi$
$644$	0	0
$645$	40.7846i	1.60589i
$646$	0	0
$647$	−1.85641	−0.0729829	−0.0364914	−	0.999334i	$-0.511618\pi$
$647$	−1.85641	−0.0729829	−0.0364914	+	0.999334i	$0.511618\pi$
$648$	0	0
$649$	−52.7846	−2.07198
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.5359i	1.50803i	0.656859	+	0.754013i	$0.271885\pi$
$653$	38.5359i	1.50803i	−0.656859	+	0.754013i	$0.728115\pi$
$654$	0	0
$655$	29.3205	1.14565
$656$	0	0
$657$	57.7128	2.25159
$658$	0	0
$659$	− 3.32051i	− 0.129349i	−0.997906	−	0.0646743i	$-0.979399\pi$
$659$	− 3.32051i	− 0.129349i	0.997906	−	0.0646743i	$-0.0206009\pi$
$660$	0	0
$661$	28.9808i	1.12722i	0.826041	+	0.563611i	$0.190588\pi$
$661$	28.9808i	1.12722i	−0.826041	+	0.563611i	$0.809412\pi$
$662$	0	0
$663$	−36.7846	−1.42860
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5.07180i	− 0.196381i
$668$	0	0
$669$	− 18.9282i	− 0.731807i
$670$	0	0
$671$	60.7846	2.34656
$672$	0	0
$673$	4.14359	0.159724	0.0798619	−	0.996806i	$-0.474552\pi$
$673$	4.14359	0.159724	0.0798619	+	0.996806i	$0.474552\pi$
$674$	0	0
$675$	9.85641i	0.379373i
$676$	0	0
$677$	− 14.4449i	− 0.555161i	−0.960702	−	0.277581i	$-0.910467\pi$
$677$	− 14.4449i	− 0.555161i	0.960702	−	0.277581i	$-0.0895326\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−10.3923	−0.398234
$682$	0	0
$683$	− 30.6410i	− 1.17245i	−0.810150	−	0.586223i	$-0.800614\pi$
$683$	− 30.6410i	− 1.17245i	0.810150	−	0.586223i	$-0.199386\pi$
$684$	0	0
$685$	54.2487i	2.07274i
$686$	0	0
$687$	47.1769	1.79991
$688$	0	0
$689$	−80.7846	−3.07765
$690$	0	0
$691$	11.1244i	0.423190i	0.977357	+	0.211595i	$0.0678658\pi$
$691$	11.1244i	0.423190i	−0.977357	+	0.211595i	$0.932134\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	25.3205	0.960462
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	16.3923i	0.620014i
$700$	0	0
$701$	19.6077i	0.740572i	0.928918	+	0.370286i	$0.120740\pi$
$701$	19.6077i	0.740572i	−0.928918	+	0.370286i	$0.879260\pi$
$702$	0	0
$703$	1.85641	0.0700157
$704$	0	0
$705$	−21.8564	−0.823160
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 48.1051i	− 1.80663i	−0.428982	−	0.903313i	$-0.641128\pi$
$709$	− 48.1051i	− 1.80663i	0.428982	−	0.903313i	$-0.358872\pi$
$710$	0	0
$711$	−48.7846	−1.82957
$712$	0	0
$713$	13.8564	0.518927
$714$	0	0
$715$	− 100.497i	− 3.75839i
$716$	0	0
$717$	34.2487i	1.27904i
$718$	0	0
$719$	21.0718	0.785845	0.392923	−	0.919572i	$-0.371464\pi$
$719$	21.0718	0.785845	0.392923	+	0.919572i	$0.371464\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	51.3205i	1.90863i
$724$	0	0
$725$	3.60770i	0.133986i
$726$	0	0
$727$	10.9282	0.405305	0.202652	−	0.979251i	$-0.435044\pi$
$727$	10.9282	0.405305	0.202652	+	0.979251i	$0.435044\pi$
$728$	0	0
$729$	43.7846	1.62165
$730$	0	0
$731$	10.9282i	0.404194i
$732$	0	0
$733$	− 13.6603i	− 0.504553i	−0.967655	−	0.252276i	$-0.918821\pi$
$733$	− 13.6603i	− 0.504553i	0.967655	−	0.252276i	$-0.0811792\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.7128	−1.61018
$738$	0	0
$739$	3.32051i	0.122147i	0.998133	+	0.0610734i	$0.0194524\pi$
$739$	3.32051i	0.122147i	−0.998133	+	0.0610734i	$0.980548\pi$
$740$	0	0
$741$	− 23.3205i	− 0.856700i
$742$	0	0
$743$	−32.2487	−1.18309	−0.591545	−	0.806272i	$-0.701482\pi$
$743$	−32.2487	−1.18309	−0.591545	+	0.806272i	$0.701482\pi$
$744$	0	0
$745$	2.92820	0.107281
$746$	0	0
$747$	25.2679i	0.924506i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.2487	−0.884848	−0.442424	−	0.896806i	$-0.645881\pi$
$751$	−24.2487	−0.884848	−0.442424	+	0.896806i	$0.645881\pi$
$752$	0	0
$753$	43.1769	1.57345
$754$	0	0
$755$	6.53590i	0.237866i
$756$	0	0
$757$	10.2487i	0.372496i	0.982503	+	0.186248i	$0.0596327\pi$
$757$	10.2487i	0.372496i	−0.982503	+	0.186248i	$0.940367\pi$
$758$	0	0
$759$	51.7128	1.87706
$760$	0	0
$761$	21.7128	0.787089	0.393544	−	0.919306i	$-0.371249\pi$
$761$	21.7128	0.787089	0.393544	+	0.919306i	$0.371249\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	24.3923i	0.881906i
$766$	0	0
$767$	−65.0333	−2.34822
$768$	0	0
$769$	10.7846	0.388903	0.194451	−	0.980912i	$-0.437707\pi$
$769$	10.7846	0.388903	0.194451	+	0.980912i	$0.437707\pi$
$770$	0	0
$771$	16.3923i	0.590354i
$772$	0	0
$773$	20.8756i	0.750845i	0.926854	+	0.375422i	$0.122502\pi$
$773$	20.8756i	0.750845i	−0.926854	+	0.375422i	$0.877498\pi$
$774$	0	0
$775$	−9.85641	−0.354053
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.53590i	− 0.0908580i
$780$	0	0
$781$	16.0000i	0.572525i
$782$	0	0
$783$	−5.85641	−0.209291
$784$	0	0
$785$	−33.3205	−1.18926
$786$	0	0
$787$	20.5885i	0.733899i	0.930241	+	0.366950i	$0.119598\pi$
$787$	20.5885i	0.733899i	−0.930241	+	0.366950i	$0.880402\pi$
$788$	0	0
$789$	− 29.8564i	− 1.06292i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	74.8897	2.65941
$794$	0	0
$795$	89.5692i	3.17669i
$796$	0	0
$797$	12.9808i	0.459802i	0.973214	+	0.229901i	$0.0738403\pi$
$797$	12.9808i	0.459802i	−0.973214	+	0.229901i	$0.926160\pi$
$798$	0	0
$799$	−5.85641	−0.207185
$800$	0	0
$801$	52.9282	1.87013
$802$	0	0
$803$	70.6410i	2.49287i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	34.3923	1.21067
$808$	0	0
$809$	23.3205	0.819905	0.409953	−	0.912107i	$-0.365545\pi$
$809$	23.3205	0.819905	0.409953	+	0.912107i	$0.365545\pi$
$810$	0	0
$811$	− 27.1244i	− 0.952465i	−0.879319	−	0.476232i	$-0.842002\pi$
$811$	− 27.1244i	− 0.952465i	0.879319	−	0.476232i	$-0.157998\pi$
$812$	0	0
$813$	40.7846i	1.43038i
$814$	0	0
$815$	−36.7846	−1.28851
$816$	0	0
$817$	−6.92820	−0.242387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.92820i	0.241796i	0.992665	+	0.120898i	$0.0385774\pi$
$821$	6.92820i	0.241796i	−0.992665	+	0.120898i	$0.961423\pi$
$822$	0	0
$823$	2.92820	0.102071	0.0510354	−	0.998697i	$-0.483748\pi$
$823$	2.92820	0.102071	0.0510354	+	0.998697i	$0.483748\pi$
$824$	0	0
$825$	−36.7846	−1.28068
$826$	0	0
$827$	13.8564i	0.481834i	0.970546	+	0.240917i	$0.0774482\pi$
$827$	13.8564i	0.481834i	−0.970546	+	0.240917i	$0.922552\pi$
$828$	0	0
$829$	40.9808i	1.42332i	0.702524	+	0.711660i	$0.252056\pi$
$829$	40.9808i	1.42332i	−0.702524	+	0.711660i	$0.747944\pi$
$830$	0	0
$831$	−10.9282	−0.379095
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 13.8564i	− 0.479521i
$836$	0	0
$837$	− 16.0000i	− 0.553041i
$838$	0	0
$839$	−39.7128	−1.37104	−0.685519	−	0.728054i	$-0.740425\pi$
$839$	−39.7128	−1.37104	−0.685519	+	0.728054i	$0.740425\pi$
$840$	0	0
$841$	26.8564	0.926083
$842$	0	0
$843$	5.46410i	0.188194i
$844$	0	0
$845$	− 88.3013i	− 3.03766i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−45.3205	−1.55540
$850$	0	0
$851$	5.07180i	0.173859i
$852$	0	0
$853$	− 4.98076i	− 0.170538i	−0.996358	−	0.0852690i	$-0.972825\pi$
$853$	− 4.98076i	− 0.170538i	0.996358	−	0.0852690i	$-0.0271750\pi$
$854$	0	0
$855$	−15.4641	−0.528861
$856$	0	0
$857$	30.7846	1.05158	0.525791	−	0.850614i	$-0.323769\pi$
$857$	30.7846	1.05158	0.525791	+	0.850614i	$0.323769\pi$
$858$	0	0
$859$	− 30.4449i	− 1.03877i	−0.854542	−	0.519383i	$-0.826162\pi$
$859$	− 30.4449i	− 1.03877i	0.854542	−	0.519383i	$-0.173838\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	10.3923	0.353349
$866$	0	0
$867$	− 35.5167i	− 1.20621i
$868$	0	0
$869$	− 59.7128i	− 2.02562i
$870$	0	0
$871$	−53.8564	−1.82485
$872$	0	0
$873$	39.8564	1.34893
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 20.3923i	− 0.688599i	−0.938860	−	0.344300i	$-0.888116\pi$
$877$	− 20.3923i	− 0.688599i	0.938860	−	0.344300i	$-0.111884\pi$
$878$	0	0
$879$	−33.3205	−1.12387
$880$	0	0
$881$	50.7846	1.71098	0.855488	−	0.517822i	$-0.173257\pi$
$881$	50.7846	1.71098	0.855488	+	0.517822i	$0.173257\pi$
$882$	0	0
$883$	− 5.07180i	− 0.170680i	−0.996352	−	0.0853398i	$-0.972802\pi$
$883$	− 5.07180i	− 0.170680i	0.996352	−	0.0853398i	$-0.0271976\pi$
$884$	0	0
$885$	72.1051i	2.42379i
$886$	0	0
$887$	−46.6410	−1.56605	−0.783026	−	0.621989i	$-0.786325\pi$
$887$	−46.6410	−1.56605	−0.783026	+	0.621989i	$0.786325\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	13.4641i	0.451064i
$892$	0	0
$893$	− 3.71281i	− 0.124245i
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	63.7128	2.12731
$898$	0	0
$899$	− 5.85641i	− 0.195322i
$900$	0	0
$901$	24.0000i	0.799556i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	29.3205	0.974647
$906$	0	0
$907$	43.7128i	1.45146i	0.687980	+	0.725730i	$0.258498\pi$
$907$	43.7128i	1.45146i	−0.687980	+	0.725730i	$0.741502\pi$
$908$	0	0
$909$	7.41154i	0.245825i
$910$	0	0
$911$	4.53590	0.150281	0.0751405	−	0.997173i	$-0.476059\pi$
$911$	4.53590	0.150281	0.0751405	+	0.997173i	$0.476059\pi$
$912$	0	0
$913$	−30.9282	−1.02357
$914$	0	0
$915$	− 83.0333i	− 2.74500i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	7.21539	0.238014	0.119007	−	0.992893i	$-0.462029\pi$
$919$	7.21539	0.238014	0.119007	+	0.992893i	$0.462029\pi$
$920$	0	0
$921$	23.4641	0.773168
$922$	0	0
$923$	19.7128i	0.648855i
$924$	0	0
$925$	− 3.60770i	− 0.118620i
$926$	0	0
$927$	−44.0000	−1.44515
$928$	0	0
$929$	−37.7128	−1.23732	−0.618659	−	0.785660i	$-0.712324\pi$
$929$	−37.7128	−1.23732	−0.618659	+	0.785660i	$0.712324\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 16.0000i	− 0.523816i
$934$	0	0
$935$	−29.8564	−0.976409
$936$	0	0
$937$	−8.64102	−0.282290	−0.141145	−	0.989989i	$-0.545078\pi$
$937$	−8.64102	−0.282290	−0.141145	+	0.989989i	$0.545078\pi$
$938$	0	0
$939$	− 54.2487i	− 1.77034i
$940$	0	0
$941$	− 44.9808i	− 1.46633i	−0.680050	−	0.733165i	$-0.738042\pi$
$941$	− 44.9808i	− 1.46633i	0.680050	−	0.733165i	$-0.261958\pi$
$942$	0	0
$943$	6.92820	0.225613
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.3923i	0.792643i	0.918112	+	0.396322i	$0.129714\pi$
$947$	24.3923i	0.792643i	−0.918112	+	0.396322i	$0.870286\pi$
$948$	0	0
$949$	87.0333i	2.82522i
$950$	0	0
$951$	−78.6410	−2.55011
$952$	0	0
$953$	−19.8564	−0.643212	−0.321606	−	0.946874i	$-0.604223\pi$
$953$	−19.8564	−0.643212	−0.321606	+	0.946874i	$0.604223\pi$
$954$	0	0
$955$	− 43.7128i	− 1.41451i
$956$	0	0
$957$	− 21.8564i	− 0.706517i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	22.6410i	0.729597i
$964$	0	0
$965$	17.8564i	0.574818i
$966$	0	0
$967$	15.1769	0.488057	0.244028	−	0.969768i	$-0.421531\pi$
$967$	15.1769	0.488057	0.244028	+	0.969768i	$0.421531\pi$
$968$	0	0
$969$	−6.92820	−0.222566
$970$	0	0
$971$	− 55.5167i	− 1.78161i	−0.454381	−	0.890807i	$-0.650140\pi$
$971$	− 55.5167i	− 1.78161i	0.454381	−	0.890807i	$-0.349860\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−45.3205	−1.45142
$976$	0	0
$977$	−28.1436	−0.900393	−0.450197	−	0.892929i	$-0.648646\pi$
$977$	−28.1436	−0.900393	−0.450197	+	0.892929i	$0.648646\pi$
$978$	0	0
$979$	64.7846i	2.07053i
$980$	0	0
$981$	− 55.3205i	− 1.76625i
$982$	0	0
$983$	−47.7128	−1.52180	−0.760901	−	0.648868i	$-0.775243\pi$
$983$	−47.7128	−1.52180	−0.760901	+	0.648868i	$0.775243\pi$
$984$	0	0
$985$	70.6410	2.25081
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 18.9282i	− 0.601882i
$990$	0	0
$991$	−24.7846	−0.787309	−0.393655	−	0.919258i	$-0.628789\pi$
$991$	−24.7846	−0.787309	−0.393655	+	0.919258i	$0.628789\pi$
$992$	0	0
$993$	−44.7846	−1.42120
$994$	0	0
$995$	− 10.9282i	− 0.346447i
$996$	0	0
$997$	− 5.26795i	− 0.166838i	−0.996515	−	0.0834188i	$-0.973416\pi$
$997$	− 5.26795i	− 0.166838i	0.996515	−	0.0834188i	$-0.0265839\pi$
$998$	0	0
$999$	5.85641	0.185289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.b.g.1569.4		4
4.3	odd	2		3136.2.b.h.1569.1		4
7.6	odd	2		448.2.b.c.225.1	✓	4
8.3	odd	2		3136.2.b.h.1569.4		4
8.5	even	2	inner	3136.2.b.g.1569.1		4
21.20	even	2		4032.2.c.k.2017.4		4
28.27	even	2		448.2.b.d.225.4	yes	4
56.13	odd	2		448.2.b.c.225.4	yes	4
56.27	even	2		448.2.b.d.225.1	yes	4
84.83	odd	2		4032.2.c.n.2017.4		4
112.13	odd	4		1792.2.a.q.1.2		2
112.27	even	4		1792.2.a.s.1.2		2
112.69	odd	4		1792.2.a.k.1.1		2
112.83	even	4		1792.2.a.i.1.1		2
168.83	odd	2		4032.2.c.n.2017.1		4
168.125	even	2		4032.2.c.k.2017.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.b.c.225.1	✓	4	7.6	odd	2
448.2.b.c.225.4	yes	4	56.13	odd	2
448.2.b.d.225.1	yes	4	56.27	even	2
448.2.b.d.225.4	yes	4	28.27	even	2
1792.2.a.i.1.1		2	112.83	even	4
1792.2.a.k.1.1		2	112.69	odd	4
1792.2.a.q.1.2		2	112.13	odd	4
1792.2.a.s.1.2		2	112.27	even	4
3136.2.b.g.1569.1		4	8.5	even	2	inner
3136.2.b.g.1569.4		4	1.1	even	1	trivial
3136.2.b.h.1569.1		4	4.3	odd	2
3136.2.b.h.1569.4		4	8.3	odd	2
4032.2.c.k.2017.1		4	168.125	even	2
4032.2.c.k.2017.4		4	21.20	even	2
4032.2.c.n.2017.1		4	168.83	odd	2
4032.2.c.n.2017.4		4	84.83	odd	2

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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.73205i q^{3} +2.73205i q^{5} -4.46410 q^{9} +O(q^{10})\)	\(q+2.73205i q^{3} +2.73205i q^{5} -4.46410 q^{9} -5.46410i q^{11} -6.73205i q^{13} -7.46410 q^{15} -2.00000 q^{17} -1.26795i q^{19} +3.46410 q^{23} -2.46410 q^{25} -4.00000i q^{27} -1.46410i q^{29} +4.00000 q^{31} +14.9282 q^{33} +1.46410i q^{37} +18.3923 q^{39} +2.00000 q^{41} -5.46410i q^{43} -12.1962i q^{45} +2.92820 q^{47} -5.46410i q^{51} -12.0000i q^{53} +14.9282 q^{55} +3.46410 q^{57} -9.66025i q^{59} +11.1244i q^{61} +18.3923 q^{65} -8.00000i q^{67} +9.46410i q^{69} -2.92820 q^{71} -12.9282 q^{73} -6.73205i q^{75} +10.9282 q^{79} -2.46410 q^{81} -5.66025i q^{83} -5.46410i q^{85} +4.00000 q^{87} -11.8564 q^{89} +10.9282i q^{93} +3.46410 q^{95} -8.92820 q^{97} +24.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 16 q^{15} - 8 q^{17} + 4 q^{25} + 16 q^{31} + 32 q^{33} + 32 q^{39} + 8 q^{41} - 16 q^{47} + 32 q^{55} + 32 q^{65} + 16 q^{71} - 24 q^{73} + 16 q^{79} + 4 q^{81} + 16 q^{87} + 8 q^{89} - 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 - 16 * q^15 - 8 * q^17 + 4 * q^25 + 16 * q^31 + 32 * q^33 + 32 * q^39 + 8 * q^41 - 16 * q^47 + 32 * q^55 + 32 * q^65 + 16 * q^71 - 24 * q^73 + 16 * q^79 + 4 * q^81 + 16 * q^87 + 8 * q^89 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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