LMFDB - Embedded newform 3312.3.g.b.737.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,3,Mod(737,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3312.737");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3312.g (of order $2$, degree $1$, not 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:	$90.2454635561$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 $ x^8 - 2x^7 + 19x^6 - 88x^5 + 301x^4 - 1010x^3 + 2713x^2 - 7044*x + 9558
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 414)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			737.8
Root			$2.43130 - 0.475563i$ of defining polynomial
Character	$\chi$	$=$	3312.737
Dual form			3312.3.g.b.737.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.87676i q^{5} +5.43723 q^{7} +O(q^{10})$	$q+6.87676i q^{5} +5.43723 q^{7} -7.68941i q^{11} -5.83945 q^{13} -17.8019i q^{17} +10.8176 q^{19} +4.79583i q^{23} -22.2899 q^{25} +4.08109i q^{29} +16.2549 q^{31} +37.3906i q^{35} -5.67890 q^{37} -54.0221i q^{41} +55.7885 q^{43} -36.2529i q^{47} -19.4365 q^{49} -46.4465i q^{53} +52.8783 q^{55} -83.4972i q^{59} -4.39115 q^{61} -40.1565i q^{65} -37.8058 q^{67} -30.7203i q^{71} -141.252 q^{73} -41.8091i q^{77} -69.9060 q^{79} -69.1749i q^{83} +122.419 q^{85} -101.485i q^{89} -31.7505 q^{91} +74.3902i q^{95} +67.1358 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^7	$ 8 q - 16 q^{7} - 8 q^{13} + 48 q^{19} + 32 q^{31} + 32 q^{37} - 32 q^{43} + 80 q^{49} - 32 q^{55} + 48 q^{61} + 16 q^{67} - 432 q^{73} + 416 q^{79} + 584 q^{85} - 368 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q - 16 * q^7 - 8 * q^13 + 48 * q^19 + 32 * q^31 + 32 * q^37 - 32 * q^43 + 80 * q^49 - 32 * q^55 + 48 * q^61 + 16 * q^67 - 432 * q^73 + 416 * q^79 + 584 * q^85 - 368 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3312\mathbb{Z}\right)^\times$.

$n$	$415$	$2305$	$2485$	$2945$
$\chi(n)$	$1$	$1$	$1$	$-1$

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	0
$3$	0	0
$4$	0	0
$5$	6.87676i	1.37535i	0.726017	+	0.687676i	$0.241369\pi$
$5$	6.87676i	1.37535i	−0.726017	+	0.687676i	$0.758631\pi$
$6$	0	0
$7$	5.43723	0.776748	0.388374	−	0.921502i	$-0.373037\pi$
$7$	5.43723	0.776748	0.388374	+	0.921502i	$0.373037\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 7.68941i	− 0.699037i	−0.936929	−	0.349519i	$-0.886345\pi$
$11$	− 7.68941i	− 0.699037i	0.936929	−	0.349519i	$-0.113655\pi$
$12$	0	0
$13$	−5.83945	−0.449188	−0.224594	−	0.974452i	$-0.572106\pi$
$13$	−5.83945	−0.449188	−0.224594	+	0.974452i	$0.572106\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 17.8019i	− 1.04717i	−0.851974	−	0.523584i	$-0.824595\pi$
$17$	− 17.8019i	− 1.04717i	0.851974	−	0.523584i	$-0.175405\pi$
$18$	0	0
$19$	10.8176	0.569348	0.284674	−	0.958624i	$-0.408115\pi$
$19$	10.8176	0.569348	0.284674	+	0.958624i	$0.408115\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	−22.2899	−0.891595
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.08109i	0.140727i	0.997521	+	0.0703636i	$0.0224159\pi$
$29$	4.08109i	0.140727i	−0.997521	+	0.0703636i	$0.977584\pi$
$30$	0	0
$31$	16.2549	0.524350	0.262175	−	0.965020i	$-0.415560\pi$
$31$	16.2549	0.524350	0.262175	+	0.965020i	$0.415560\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	37.3906i	1.06830i
$36$	0	0
$37$	−5.67890	−0.153484	−0.0767418	−	0.997051i	$-0.524452\pi$
$37$	−5.67890	−0.153484	−0.0767418	+	0.997051i	$0.524452\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 54.0221i	− 1.31761i	−0.752313	−	0.658806i	$-0.771062\pi$
$41$	− 54.0221i	− 1.31761i	0.752313	−	0.658806i	$-0.228938\pi$
$42$	0	0
$43$	55.7885	1.29741	0.648704	−	0.761041i	$-0.275312\pi$
$43$	55.7885	1.29741	0.648704	+	0.761041i	$0.275312\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 36.2529i	− 0.771339i	−0.922637	−	0.385670i	$-0.873970\pi$
$47$	− 36.2529i	− 0.771339i	0.922637	−	0.385670i	$-0.126030\pi$
$48$	0	0
$49$	−19.4365	−0.396663
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 46.4465i	− 0.876350i	−0.898890	−	0.438175i	$-0.855625\pi$
$53$	− 46.4465i	− 0.876350i	0.898890	−	0.438175i	$-0.144375\pi$
$54$	0	0
$55$	52.8783	0.961423
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 83.4972i	− 1.41521i	−0.706610	−	0.707604i	$-0.749776\pi$
$59$	− 83.4972i	− 1.41521i	0.706610	−	0.707604i	$-0.250224\pi$
$60$	0	0
$61$	−4.39115	−0.0719860	−0.0359930	−	0.999352i	$-0.511459\pi$
$61$	−4.39115	−0.0719860	−0.0359930	+	0.999352i	$0.511459\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 40.1565i	− 0.617792i
$66$	0	0
$67$	−37.8058	−0.564266	−0.282133	−	0.959375i	$-0.591042\pi$
$67$	−37.8058	−0.564266	−0.282133	+	0.959375i	$0.591042\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 30.7203i	− 0.432681i	−0.976318	−	0.216340i	$-0.930588\pi$
$71$	− 30.7203i	− 0.432681i	0.976318	−	0.216340i	$-0.0694121\pi$
$72$	0	0
$73$	−141.252	−1.93496	−0.967480	−	0.252947i	$-0.918600\pi$
$73$	−141.252	−1.93496	−0.967480	+	0.252947i	$0.918600\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 41.8091i	− 0.542976i
$78$	0	0
$79$	−69.9060	−0.884886	−0.442443	−	0.896797i	$-0.645888\pi$
$79$	−69.9060	−0.884886	−0.442443	+	0.896797i	$0.645888\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 69.1749i	− 0.833432i	−0.909037	−	0.416716i	$-0.863181\pi$
$83$	− 69.1749i	− 0.833432i	0.909037	−	0.416716i	$-0.136819\pi$
$84$	0	0
$85$	122.419	1.44023
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 101.485i	− 1.14028i	−0.821547	−	0.570141i	$-0.806889\pi$
$89$	− 101.485i	− 1.14028i	0.821547	−	0.570141i	$-0.193111\pi$
$90$	0	0
$91$	−31.7505	−0.348906
$92$	0	0
$93$	0	0
$94$	0	0
$95$	74.3902i	0.783055i
$96$	0	0
$97$	67.1358	0.692121	0.346061	−	0.938212i	$-0.387519\pi$
$97$	67.1358	0.692121	0.346061	+	0.938212i	$0.387519\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	168.433i	1.66765i	0.552025	+	0.833827i	$0.313855\pi$
$101$	168.433i	1.66765i	−0.552025	+	0.833827i	$0.686145\pi$
$102$	0	0
$103$	−16.6856	−0.161996	−0.0809979	−	0.996714i	$-0.525811\pi$
$103$	−16.6856	−0.161996	−0.0809979	+	0.996714i	$0.525811\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 141.806i	− 1.32529i	−0.748935	−	0.662644i	$-0.769434\pi$
$107$	− 141.806i	− 1.32529i	0.748935	−	0.662644i	$-0.230566\pi$
$108$	0	0
$109$	4.31466	0.0395841	0.0197920	−	0.999804i	$-0.493700\pi$
$109$	4.31466	0.0395841	0.0197920	+	0.999804i	$0.493700\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	23.3559i	0.206689i	0.994646	+	0.103345i	$0.0329545\pi$
$113$	23.3559i	0.206689i	−0.994646	+	0.103345i	$0.967046\pi$
$114$	0	0
$115$	−32.9798	−0.286781
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 96.7929i	− 0.813386i
$120$	0	0
$121$	61.8730	0.511347
$122$	0	0
$123$	0	0
$124$	0	0
$125$	18.6369i	0.149095i
$126$	0	0
$127$	217.702	1.71419	0.857095	−	0.515159i	$-0.172267\pi$
$127$	217.702	1.71419	0.857095	+	0.515159i	$0.172267\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 202.955i	− 1.54927i	−0.632406	−	0.774637i	$-0.717933\pi$
$131$	− 202.955i	− 1.54927i	0.632406	−	0.774637i	$-0.282067\pi$
$132$	0	0
$133$	58.8179	0.442240
$134$	0	0
$135$	0	0
$136$	0	0
$137$	196.599i	1.43503i	0.696542	+	0.717516i	$0.254721\pi$
$137$	196.599i	1.43503i	−0.696542	+	0.717516i	$0.745279\pi$
$138$	0	0
$139$	196.342	1.41253	0.706265	−	0.707948i	$-0.250379\pi$
$139$	196.342	1.41253	0.706265	+	0.707948i	$0.250379\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	44.9019i	0.313999i
$144$	0	0
$145$	−28.0647	−0.193549
$146$	0	0
$147$	0	0
$148$	0	0
$149$	257.166i	1.72595i	0.505251	+	0.862973i	$0.331400\pi$
$149$	257.166i	1.72595i	−0.505251	+	0.862973i	$0.668600\pi$
$150$	0	0
$151$	−94.8135	−0.627904	−0.313952	−	0.949439i	$-0.601653\pi$
$151$	−94.8135	−0.627904	−0.313952	+	0.949439i	$0.601653\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	111.781i	0.721166i
$156$	0	0
$157$	37.0443	0.235951	0.117975	−	0.993017i	$-0.462360\pi$
$157$	37.0443	0.235951	0.117975	+	0.993017i	$0.462360\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	26.0761i	0.161963i
$162$	0	0
$163$	115.075	0.705981	0.352990	−	0.935627i	$-0.385165\pi$
$163$	115.075	0.705981	0.352990	+	0.935627i	$0.385165\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.06703i	0.0303415i	0.999885	+	0.0151707i	$0.00482918\pi$
$167$	5.06703i	0.0303415i	−0.999885	+	0.0151707i	$0.995171\pi$
$168$	0	0
$169$	−134.901	−0.798230
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 257.245i	− 1.48697i	−0.668755	−	0.743483i	$-0.733173\pi$
$173$	− 257.245i	− 1.48697i	0.668755	−	0.743483i	$-0.266827\pi$
$174$	0	0
$175$	−121.195	−0.692544
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 32.7967i	− 0.183222i	−0.995795	−	0.0916110i	$-0.970798\pi$
$179$	− 32.7967i	− 0.183222i	0.995795	−	0.0916110i	$-0.0292016\pi$
$180$	0	0
$181$	183.605	1.01439	0.507195	−	0.861831i	$-0.330682\pi$
$181$	183.605	1.01439	0.507195	+	0.861831i	$0.330682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 39.0524i	− 0.211094i
$186$	0	0
$187$	−136.886	−0.732010
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.7258i	0.0718627i	0.999354	+	0.0359313i	$0.0114398\pi$
$191$	13.7258i	0.0718627i	−0.999354	+	0.0359313i	$0.988560\pi$
$192$	0	0
$193$	106.204	0.550278	0.275139	−	0.961404i	$-0.411276\pi$
$193$	106.204	0.550278	0.275139	+	0.961404i	$0.411276\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 35.3539i	− 0.179462i	−0.995966	−	0.0897308i	$-0.971399\pi$
$197$	− 35.3539i	− 0.179462i	0.995966	−	0.0897308i	$-0.0286007\pi$
$198$	0	0
$199$	268.863	1.35107	0.675535	−	0.737328i	$-0.263913\pi$
$199$	268.863	1.35107	0.675535	+	0.737328i	$0.263913\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	22.1898i	0.109309i
$204$	0	0
$205$	371.497	1.81218
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 83.1811i	− 0.397996i
$210$	0	0
$211$	129.399	0.613266	0.306633	−	0.951828i	$-0.400798\pi$
$211$	129.399	0.613266	0.306633	+	0.951828i	$0.400798\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	383.644i	1.78439i
$216$	0	0
$217$	88.3814	0.407288
$218$	0	0
$219$	0	0
$220$	0	0
$221$	103.953i	0.470376i
$222$	0	0
$223$	−389.713	−1.74759	−0.873797	−	0.486291i	$-0.838349\pi$
$223$	−389.713	−1.74759	−0.873797	+	0.486291i	$0.838349\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	306.520i	1.35031i	0.737677	+	0.675154i	$0.235923\pi$
$227$	306.520i	1.35031i	−0.737677	+	0.675154i	$0.764077\pi$
$228$	0	0
$229$	366.794	1.60172	0.800861	−	0.598851i	$-0.204376\pi$
$229$	366.794	1.60172	0.800861	+	0.598851i	$0.204376\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 241.335i	− 1.03577i	−0.855449	−	0.517887i	$-0.826719\pi$
$233$	− 241.335i	− 1.03577i	0.855449	−	0.517887i	$-0.173281\pi$
$234$	0	0
$235$	249.303	1.06086
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 183.710i	− 0.768661i	−0.923196	−	0.384330i	$-0.874432\pi$
$239$	− 183.710i	− 0.768661i	0.923196	−	0.384330i	$-0.125568\pi$
$240$	0	0
$241$	−326.918	−1.35651	−0.678253	−	0.734828i	$-0.737263\pi$
$241$	−326.918	−1.35651	−0.678253	+	0.734828i	$0.737263\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 133.660i	− 0.545551i
$246$	0	0
$247$	−63.1689	−0.255745
$248$	0	0
$249$	0	0
$250$	0	0
$251$	70.7716i	0.281958i	0.990013	+	0.140979i	$0.0450251\pi$
$251$	70.7716i	0.281958i	−0.990013	+	0.140979i	$0.954975\pi$
$252$	0	0
$253$	36.8771	0.145759
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 88.0745i	− 0.342702i	−0.985210	−	0.171351i	$-0.945187\pi$
$257$	− 88.0745i	− 0.342702i	0.985210	−	0.171351i	$-0.0548132\pi$
$258$	0	0
$259$	−30.8775	−0.119218
$260$	0	0
$261$	0	0
$262$	0	0
$263$	235.795i	0.896561i	0.893893	+	0.448280i	$0.147963\pi$
$263$	235.795i	0.896561i	−0.893893	+	0.448280i	$0.852037\pi$
$264$	0	0
$265$	319.402	1.20529
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.44214i	0.0165135i	0.999966	+	0.00825677i	$0.00262824\pi$
$269$	4.44214i	0.0165135i	−0.999966	+	0.00825677i	$0.997372\pi$
$270$	0	0
$271$	258.331	0.953252	0.476626	−	0.879106i	$-0.341860\pi$
$271$	258.331	0.953252	0.476626	+	0.879106i	$0.341860\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	171.396i	0.623258i
$276$	0	0
$277$	−150.711	−0.544084	−0.272042	−	0.962285i	$-0.587699\pi$
$277$	−150.711	−0.544084	−0.272042	+	0.962285i	$0.587699\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 461.687i	− 1.64301i	−0.570199	−	0.821507i	$-0.693134\pi$
$281$	− 461.687i	− 1.64301i	0.570199	−	0.821507i	$-0.306866\pi$
$282$	0	0
$283$	335.124	1.18418	0.592092	−	0.805870i	$-0.298302\pi$
$283$	335.124	1.18418	0.592092	+	0.805870i	$0.298302\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 293.731i	− 1.02345i
$288$	0	0
$289$	−27.9063	−0.0965616
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 331.756i	− 1.13227i	−0.824312	−	0.566136i	$-0.808438\pi$
$293$	− 331.756i	− 1.13227i	0.824312	−	0.566136i	$-0.191562\pi$
$294$	0	0
$295$	574.191	1.94641
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 28.0050i	− 0.0936622i
$300$	0	0
$301$	303.335	1.00776
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 30.1969i	− 0.0990061i
$306$	0	0
$307$	246.452	0.802775	0.401387	−	0.915908i	$-0.368528\pi$
$307$	246.452	0.802775	0.401387	+	0.915908i	$0.368528\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 347.854i	− 1.11850i	−0.828999	−	0.559250i	$-0.811089\pi$
$311$	− 347.854i	− 1.11850i	0.828999	−	0.559250i	$-0.188911\pi$
$312$	0	0
$313$	262.326	0.838101	0.419051	−	0.907963i	$-0.362363\pi$
$313$	262.326	0.838101	0.419051	+	0.907963i	$0.362363\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	496.740i	1.56700i	0.621391	+	0.783501i	$0.286568\pi$
$317$	496.740i	1.56700i	−0.621391	+	0.783501i	$0.713432\pi$
$318$	0	0
$319$	31.3811	0.0983735
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 192.574i	− 0.596203i
$324$	0	0
$325$	130.161	0.400494
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 197.116i	− 0.599136i
$330$	0	0
$331$	−157.807	−0.476757	−0.238378	−	0.971172i	$-0.576616\pi$
$331$	−157.807	−0.476757	−0.238378	+	0.971172i	$0.576616\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 259.981i	− 0.776064i
$336$	0	0
$337$	387.069	1.14857	0.574286	−	0.818655i	$-0.305280\pi$
$337$	387.069	1.14857	0.574286	+	0.818655i	$0.305280\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 124.990i	− 0.366540i
$342$	0	0
$343$	−372.105	−1.08485
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 226.549i	− 0.652879i	−0.945218	−	0.326440i	$-0.894151\pi$
$347$	− 226.549i	− 0.652879i	0.945218	−	0.326440i	$-0.105849\pi$
$348$	0	0
$349$	−85.2057	−0.244142	−0.122071	−	0.992521i	$-0.538954\pi$
$349$	−85.2057	−0.244142	−0.122071	+	0.992521i	$0.538954\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 101.019i	− 0.286172i	−0.989710	−	0.143086i	$-0.954297\pi$
$353$	− 101.019i	− 0.286172i	0.989710	−	0.143086i	$-0.0457026\pi$
$354$	0	0
$355$	211.257	0.595089
$356$	0	0
$357$	0	0
$358$	0	0
$359$	473.121i	1.31789i	0.752193	+	0.658943i	$0.228996\pi$
$359$	473.121i	1.31789i	−0.752193	+	0.658943i	$0.771004\pi$
$360$	0	0
$361$	−243.979	−0.675843
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 971.357i	− 2.66125i
$366$	0	0
$367$	−378.938	−1.03253	−0.516264	−	0.856429i	$-0.672678\pi$
$367$	−378.938	−1.03253	−0.516264	+	0.856429i	$0.672678\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 252.541i	− 0.680703i
$372$	0	0
$373$	21.7047	0.0581894	0.0290947	−	0.999577i	$-0.490738\pi$
$373$	21.7047	0.0581894	0.0290947	+	0.999577i	$0.490738\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.8313i	− 0.0632130i
$378$	0	0
$379$	379.508	1.00134	0.500670	−	0.865638i	$-0.333087\pi$
$379$	379.508	1.00134	0.500670	+	0.865638i	$0.333087\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	198.586i	0.518502i	0.965810	+	0.259251i	$0.0834756\pi$
$383$	198.586i	0.518502i	−0.965810	+	0.259251i	$0.916524\pi$
$384$	0	0
$385$	287.512	0.746783
$386$	0	0
$387$	0	0
$388$	0	0
$389$	336.782i	0.865763i	0.901451	+	0.432881i	$0.142503\pi$
$389$	336.782i	0.865763i	−0.901451	+	0.432881i	$0.857497\pi$
$390$	0	0
$391$	85.3747	0.218350
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 480.727i	− 1.21703i
$396$	0	0
$397$	233.940	0.589269	0.294634	−	0.955610i	$-0.404802\pi$
$397$	233.940	0.589269	0.294634	+	0.955610i	$0.404802\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 49.1903i	− 0.122669i	−0.998117	−	0.0613345i	$-0.980464\pi$
$401$	− 49.1903i	− 0.122669i	0.998117	−	0.0613345i	$-0.0195356\pi$
$402$	0	0
$403$	−94.9194	−0.235532
$404$	0	0
$405$	0	0
$406$	0	0
$407$	43.6674i	0.107291i
$408$	0	0
$409$	−615.803	−1.50563	−0.752815	−	0.658232i	$-0.771305\pi$
$409$	−615.803	−1.50563	−0.752815	+	0.658232i	$0.771305\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 453.994i	− 1.09926i
$414$	0	0
$415$	475.699	1.14626
$416$	0	0
$417$	0	0
$418$	0	0
$419$	274.066i	0.654096i	0.945008	+	0.327048i	$0.106054\pi$
$419$	274.066i	0.654096i	−0.945008	+	0.327048i	$0.893946\pi$
$420$	0	0
$421$	−256.187	−0.608521	−0.304260	−	0.952589i	$-0.598409\pi$
$421$	−256.187	−0.608521	−0.304260	+	0.952589i	$0.598409\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	396.801i	0.933650i
$426$	0	0
$427$	−23.8757	−0.0559150
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 651.526i	− 1.51166i	−0.654768	−	0.755830i	$-0.727233\pi$
$431$	− 651.526i	− 1.51166i	0.654768	−	0.755830i	$-0.272767\pi$
$432$	0	0
$433$	−717.465	−1.65696	−0.828481	−	0.560017i	$-0.810795\pi$
$433$	−717.465	−1.65696	−0.828481	+	0.560017i	$0.810795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	51.8795i	0.118717i
$438$	0	0
$439$	273.747	0.623569	0.311784	−	0.950153i	$-0.399073\pi$
$439$	273.747	0.623569	0.311784	+	0.950153i	$0.399073\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 219.891i	− 0.496369i	−0.968713	−	0.248184i	$-0.920166\pi$
$443$	− 219.891i	− 0.496369i	0.968713	−	0.248184i	$-0.0798339\pi$
$444$	0	0
$445$	697.889	1.56829
$446$	0	0
$447$	0	0
$448$	0	0
$449$	235.399i	0.524275i	0.965031	+	0.262137i	$0.0844274\pi$
$449$	235.399i	0.524275i	−0.965031	+	0.262137i	$0.915573\pi$
$450$	0	0
$451$	−415.398	−0.921059
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 218.340i	− 0.479869i
$456$	0	0
$457$	464.879	1.01724	0.508621	−	0.860991i	$-0.330156\pi$
$457$	464.879	1.01724	0.508621	+	0.860991i	$0.330156\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	705.324i	1.52999i	0.644038	+	0.764993i	$0.277258\pi$
$461$	705.324i	1.52999i	−0.644038	+	0.764993i	$0.722742\pi$
$462$	0	0
$463$	−249.636	−0.539170	−0.269585	−	0.962977i	$-0.586886\pi$
$463$	−249.636	−0.539170	−0.269585	+	0.962977i	$0.586886\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	342.795i	0.734035i	0.930214	+	0.367018i	$0.119621\pi$
$467$	342.795i	0.734035i	−0.930214	+	0.367018i	$0.880379\pi$
$468$	0	0
$469$	−205.559	−0.438292
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 428.981i	− 0.906936i
$474$	0	0
$475$	−241.123	−0.507628
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 710.409i	− 1.48311i	−0.670892	−	0.741555i	$-0.734089\pi$
$479$	− 710.409i	− 1.48311i	0.670892	−	0.741555i	$-0.265911\pi$
$480$	0	0
$481$	33.1616	0.0689431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	461.677i	0.951911i
$486$	0	0
$487$	332.013	0.681752	0.340876	−	0.940108i	$-0.389276\pi$
$487$	332.013	0.681752	0.340876	+	0.940108i	$0.389276\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 475.304i	− 0.968034i	−0.875059	−	0.484017i	$-0.839177\pi$
$491$	− 475.304i	− 0.968034i	0.875059	−	0.484017i	$-0.160823\pi$
$492$	0	0
$493$	72.6509	0.147365
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 167.034i	− 0.336084i
$498$	0	0
$499$	−764.152	−1.53137	−0.765683	−	0.643218i	$-0.777599\pi$
$499$	−764.152	−1.53137	−0.765683	+	0.643218i	$0.777599\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 209.165i	− 0.415834i	−0.978146	−	0.207917i	$-0.933332\pi$
$503$	− 209.165i	− 0.415834i	0.978146	−	0.207917i	$-0.0666684\pi$
$504$	0	0
$505$	−1158.27	−2.29361
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 784.506i	− 1.54127i	−0.637277	−	0.770635i	$-0.719939\pi$
$509$	− 784.506i	− 1.54127i	0.637277	−	0.770635i	$-0.280061\pi$
$510$	0	0
$511$	−768.021	−1.50298
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 114.743i	− 0.222801i
$516$	0	0
$517$	−278.764	−0.539195
$518$	0	0
$519$	0	0
$520$	0	0
$521$	473.326i	0.908495i	0.890876	+	0.454247i	$0.150092\pi$
$521$	473.326i	0.908495i	−0.890876	+	0.454247i	$0.849908\pi$
$522$	0	0
$523$	−635.254	−1.21463	−0.607317	−	0.794459i	$-0.707754\pi$
$523$	−635.254	−1.21463	−0.607317	+	0.794459i	$0.707754\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 289.367i	− 0.549083i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	315.459i	0.591856i
$534$	0	0
$535$	975.165	1.82274
$536$	0	0
$537$	0	0
$538$	0	0
$539$	149.455i	0.277282i
$540$	0	0
$541$	1011.63	1.86993	0.934967	−	0.354734i	$-0.115429\pi$
$541$	1011.63	1.86993	0.934967	+	0.354734i	$0.115429\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	29.6709i	0.0544421i
$546$	0	0
$547$	−567.388	−1.03727	−0.518636	−	0.854995i	$-0.673560\pi$
$547$	−567.388	−1.03727	−0.518636	+	0.854995i	$0.673560\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	44.1476i	0.0801227i
$552$	0	0
$553$	−380.095	−0.687333
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 449.285i	− 0.806615i	−0.915064	−	0.403308i	$-0.867860\pi$
$557$	− 449.285i	− 0.806615i	0.915064	−	0.403308i	$-0.132140\pi$
$558$	0	0
$559$	−325.774	−0.582780
$560$	0	0
$561$	0	0
$562$	0	0
$563$	762.535i	1.35441i	0.735792	+	0.677207i	$0.236810\pi$
$563$	762.535i	1.35441i	−0.735792	+	0.677207i	$0.763190\pi$
$564$	0	0
$565$	−160.613	−0.284271
$566$	0	0
$567$	0	0
$568$	0	0
$569$	504.828i	0.887221i	0.896220	+	0.443610i	$0.146303\pi$
$569$	504.828i	0.887221i	−0.896220	+	0.443610i	$0.853697\pi$
$570$	0	0
$571$	1046.44	1.83264	0.916320	−	0.400447i	$-0.131145\pi$
$571$	1046.44	1.83264	0.916320	+	0.400447i	$0.131145\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 106.898i	− 0.185910i
$576$	0	0
$577$	1088.56	1.88659	0.943296	−	0.331953i	$-0.107708\pi$
$577$	1088.56	1.88659	0.943296	+	0.331953i	$0.107708\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 376.120i	− 0.647367i
$582$	0	0
$583$	−357.147	−0.612601
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 720.312i	− 1.22711i	−0.789653	−	0.613553i	$-0.789740\pi$
$587$	− 720.312i	− 1.22711i	0.789653	−	0.613553i	$-0.210260\pi$
$588$	0	0
$589$	175.839	0.298538
$590$	0	0
$591$	0	0
$592$	0	0
$593$	132.133i	0.222821i	0.993774	+	0.111411i	$0.0355369\pi$
$593$	132.133i	0.222821i	−0.993774	+	0.111411i	$0.964463\pi$
$594$	0	0
$595$	665.622	1.11869
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 506.024i	− 0.844782i	−0.906414	−	0.422391i	$-0.861191\pi$
$599$	− 506.024i	− 0.844782i	0.906414	−	0.422391i	$-0.138809\pi$
$600$	0	0
$601$	511.932	0.851800	0.425900	−	0.904770i	$-0.359958\pi$
$601$	511.932	0.851800	0.425900	+	0.904770i	$0.359958\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	425.486i	0.703282i
$606$	0	0
$607$	515.170	0.848716	0.424358	−	0.905495i	$-0.360500\pi$
$607$	515.170	0.848716	0.424358	+	0.905495i	$0.360500\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	211.697i	0.346477i
$612$	0	0
$613$	261.653	0.426840	0.213420	−	0.976961i	$-0.431540\pi$
$613$	261.653	0.426840	0.213420	+	0.976961i	$0.431540\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	619.954i	1.00479i	0.864639	+	0.502394i	$0.167547\pi$
$617$	619.954i	1.00479i	−0.864639	+	0.502394i	$0.832453\pi$
$618$	0	0
$619$	−775.178	−1.25231	−0.626154	−	0.779700i	$-0.715372\pi$
$619$	−775.178	−1.25231	−0.626154	+	0.779700i	$0.715372\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 551.798i	− 0.885712i
$624$	0	0
$625$	−685.408	−1.09665
$626$	0	0
$627$	0	0
$628$	0	0
$629$	101.095i	0.160723i
$630$	0	0
$631$	−175.311	−0.277831	−0.138915	−	0.990304i	$-0.544362\pi$
$631$	−175.311	−0.277831	−0.138915	+	0.990304i	$0.544362\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1497.09i	2.35762i
$636$	0	0
$637$	113.498	0.178176
$638$	0	0
$639$	0	0
$640$	0	0
$641$	205.237i	0.320183i	0.987102	+	0.160091i	$0.0511789\pi$
$641$	205.237i	0.320183i	−0.987102	+	0.160091i	$0.948821\pi$
$642$	0	0
$643$	614.006	0.954908	0.477454	−	0.878657i	$-0.341560\pi$
$643$	614.006	0.954908	0.477454	+	0.878657i	$0.341560\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 10.3908i	− 0.0160599i	−0.999968	−	0.00802997i	$-0.997444\pi$
$647$	− 10.3908i	− 0.0160599i	0.999968	−	0.00802997i	$-0.00255605\pi$
$648$	0	0
$649$	−642.045	−0.989283
$650$	0	0
$651$	0	0
$652$	0	0
$653$	435.069i	0.666261i	0.942881	+	0.333131i	$0.108105\pi$
$653$	435.069i	0.666261i	−0.942881	+	0.333131i	$0.891895\pi$
$654$	0	0
$655$	1395.67	2.13080
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 907.782i	− 1.37751i	−0.724992	−	0.688757i	$-0.758157\pi$
$659$	− 907.782i	− 1.37751i	0.724992	−	0.688757i	$-0.241843\pi$
$660$	0	0
$661$	856.923	1.29640	0.648202	−	0.761468i	$-0.275521\pi$
$661$	856.923	1.29640	0.648202	+	0.761468i	$0.275521\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	404.477i	0.608236i
$666$	0	0
$667$	−19.5722	−0.0293436
$668$	0	0
$669$	0	0
$670$	0	0
$671$	33.7653i	0.0503209i
$672$	0	0
$673$	−958.152	−1.42370	−0.711851	−	0.702330i	$-0.752143\pi$
$673$	−958.152	−1.42370	−0.711851	+	0.702330i	$0.752143\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 848.650i	− 1.25355i	−0.779202	−	0.626773i	$-0.784375\pi$
$677$	− 848.650i	− 1.25355i	0.779202	−	0.626773i	$-0.215625\pi$
$678$	0	0
$679$	365.033	0.537604
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1140.08i	1.66923i	0.550837	+	0.834613i	$0.314309\pi$
$683$	1140.08i	1.66923i	−0.550837	+	0.834613i	$0.685691\pi$
$684$	0	0
$685$	−1351.97	−1.97367
$686$	0	0
$687$	0	0
$688$	0	0
$689$	271.222i	0.393646i
$690$	0	0
$691$	905.691	1.31070	0.655348	−	0.755327i	$-0.272522\pi$
$691$	905.691	1.31070	0.655348	+	0.755327i	$0.272522\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1350.19i	1.94273i
$696$	0	0
$697$	−961.693	−1.37976
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 493.120i	− 0.703453i	−0.936103	−	0.351726i	$-0.885595\pi$
$701$	− 493.120i	− 0.703453i	0.936103	−	0.351726i	$-0.114405\pi$
$702$	0	0
$703$	−61.4321	−0.0873857
$704$	0	0
$705$	0	0
$706$	0	0
$707$	915.810i	1.29535i
$708$	0	0
$709$	283.873	0.400385	0.200192	−	0.979757i	$-0.435843\pi$
$709$	283.873	0.400385	0.200192	+	0.979757i	$0.435843\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	77.9555i	0.109335i
$714$	0	0
$715$	−308.780	−0.431860
$716$	0	0
$717$	0	0
$718$	0	0
$719$	646.489i	0.899150i	0.893243	+	0.449575i	$0.148425\pi$
$719$	646.489i	0.899150i	−0.893243	+	0.449575i	$0.851575\pi$
$720$	0	0
$721$	−90.7233	−0.125830
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 90.9669i	− 0.125472i
$726$	0	0
$727$	371.314	0.510748	0.255374	−	0.966842i	$-0.417801\pi$
$727$	371.314	0.510748	0.255374	+	0.966842i	$0.417801\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 993.139i	− 1.35860i
$732$	0	0
$733$	−290.264	−0.395995	−0.197997	−	0.980203i	$-0.563444\pi$
$733$	−290.264	−0.395995	−0.197997	+	0.980203i	$0.563444\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	290.704i	0.394443i
$738$	0	0
$739$	532.129	0.720066	0.360033	−	0.932940i	$-0.382765\pi$
$739$	532.129	0.720066	0.360033	+	0.932940i	$0.382765\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1288.86i	1.73466i	0.497730	+	0.867332i	$0.334167\pi$
$743$	1288.86i	1.73466i	−0.497730	+	0.867332i	$0.665833\pi$
$744$	0	0
$745$	−1768.47	−2.37378
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 771.031i	− 1.02941i
$750$	0	0
$751$	−578.243	−0.769964	−0.384982	−	0.922924i	$-0.625792\pi$
$751$	−578.243	−0.769964	−0.384982	+	0.922924i	$0.625792\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 652.010i	− 0.863590i
$756$	0	0
$757$	903.264	1.19322	0.596608	−	0.802533i	$-0.296515\pi$
$757$	903.264	1.19322	0.596608	+	0.802533i	$0.296515\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	771.153i	1.01334i	0.862139	+	0.506671i	$0.169124\pi$
$761$	771.153i	1.01334i	−0.862139	+	0.506671i	$0.830876\pi$
$762$	0	0
$763$	23.4598	0.0307468
$764$	0	0
$765$	0	0
$766$	0	0
$767$	487.578i	0.635695i
$768$	0	0
$769$	282.553	0.367429	0.183714	−	0.982980i	$-0.441188\pi$
$769$	282.553	0.367429	0.183714	+	0.982980i	$0.441188\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 871.210i	− 1.12705i	−0.826099	−	0.563525i	$-0.809445\pi$
$773$	− 871.210i	− 1.12705i	0.826099	−	0.563525i	$-0.190555\pi$
$774$	0	0
$775$	−362.319	−0.467508
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 584.390i	− 0.750180i
$780$	0	0
$781$	−236.221	−0.302460
$782$	0	0
$783$	0	0
$784$	0	0
$785$	254.745i	0.324515i
$786$	0	0
$787$	−485.671	−0.617117	−0.308559	−	0.951205i	$-0.599847\pi$
$787$	−485.671	−0.617117	−0.308559	+	0.951205i	$0.599847\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	126.992i	0.160546i
$792$	0	0
$793$	25.6419	0.0323353
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1068.29i	− 1.34038i	−0.742188	−	0.670192i	$-0.766212\pi$
$797$	− 1068.29i	− 1.34038i	0.742188	−	0.670192i	$-0.233788\pi$
$798$	0	0
$799$	−645.370	−0.807722
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1086.15i	1.35261i
$804$	0	0
$805$	−179.319	−0.222756
$806$	0	0
$807$	0	0
$808$	0	0
$809$	44.3081i	0.0547690i	0.999625	+	0.0273845i	$0.00871784\pi$
$809$	44.3081i	0.0547690i	−0.999625	+	0.0273845i	$0.991282\pi$
$810$	0	0
$811$	−940.317	−1.15945	−0.579727	−	0.814811i	$-0.696841\pi$
$811$	−940.317	−1.15945	−0.579727	+	0.814811i	$0.696841\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	791.343i	0.970973i
$816$	0	0
$817$	603.499	0.738676
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 159.843i	− 0.194693i	−0.995251	−	0.0973463i	$-0.968965\pi$
$821$	− 159.843i	− 0.194693i	0.995251	−	0.0973463i	$-0.0310354\pi$
$822$	0	0
$823$	−757.411	−0.920305	−0.460153	−	0.887840i	$-0.652205\pi$
$823$	−757.411	−0.920305	−0.460153	+	0.887840i	$0.652205\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 867.464i	− 1.04893i	−0.851432	−	0.524464i	$-0.824266\pi$
$827$	− 867.464i	− 1.04893i	0.851432	−	0.524464i	$-0.175734\pi$
$828$	0	0
$829$	−623.021	−0.751533	−0.375766	−	0.926714i	$-0.622620\pi$
$829$	−623.021	−0.751533	−0.375766	+	0.926714i	$0.622620\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	346.006i	0.415373i
$834$	0	0
$835$	−34.8447	−0.0417302
$836$	0	0
$837$	0	0
$838$	0	0
$839$	344.509i	0.410618i	0.978697	+	0.205309i	$0.0658200\pi$
$839$	344.509i	0.410618i	−0.978697	+	0.205309i	$0.934180\pi$
$840$	0	0
$841$	824.345	0.980196
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 927.681i	− 1.09785i
$846$	0	0
$847$	336.418	0.397187
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 27.2350i	− 0.0320036i
$852$	0	0
$853$	−570.907	−0.669294	−0.334647	−	0.942344i	$-0.608617\pi$
$853$	−570.907	−0.669294	−0.334647	+	0.942344i	$0.608617\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1531.64i	1.78721i	0.448857	+	0.893603i	$0.351831\pi$
$857$	1531.64i	1.78721i	−0.448857	+	0.893603i	$0.648169\pi$
$858$	0	0
$859$	245.268	0.285528	0.142764	−	0.989757i	$-0.454401\pi$
$859$	245.268	0.285528	0.142764	+	0.989757i	$0.454401\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 258.768i	− 0.299847i	−0.988698	−	0.149924i	$-0.952097\pi$
$863$	− 258.768i	− 0.299847i	0.988698	−	0.149924i	$-0.0479028\pi$
$864$	0	0
$865$	1769.01	2.04510
$866$	0	0
$867$	0	0
$868$	0	0
$869$	537.536i	0.618568i
$870$	0	0
$871$	220.765	0.253462
$872$	0	0
$873$	0	0
$874$	0	0
$875$	101.333i	0.115809i
$876$	0	0
$877$	986.840	1.12525	0.562623	−	0.826714i	$-0.309792\pi$
$877$	986.840	1.12525	0.562623	+	0.826714i	$0.309792\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	834.817i	0.947579i	0.880638	+	0.473789i	$0.157114\pi$
$881$	834.817i	0.947579i	−0.880638	+	0.473789i	$0.842886\pi$
$882$	0	0
$883$	1217.61	1.37894	0.689472	−	0.724313i	$-0.257843\pi$
$883$	1217.61	1.37894	0.689472	+	0.724313i	$0.257843\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 740.177i	− 0.834472i	−0.908798	−	0.417236i	$-0.862999\pi$
$887$	− 740.177i	− 0.834472i	0.908798	−	0.417236i	$-0.137001\pi$
$888$	0	0
$889$	1183.70	1.33149
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 392.170i	− 0.439161i
$894$	0	0
$895$	225.535	0.251995
$896$	0	0
$897$	0	0
$898$	0	0
$899$	66.3374i	0.0737903i
$900$	0	0
$901$	−826.835	−0.917686
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1262.60i	1.39514i
$906$	0	0
$907$	−1607.94	−1.77282	−0.886408	−	0.462905i	$-0.846807\pi$
$907$	−1607.94	−1.77282	−0.886408	+	0.462905i	$0.846807\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	474.080i	0.520395i	0.965555	+	0.260198i	$0.0837877\pi$
$911$	474.080i	0.520395i	−0.965555	+	0.260198i	$0.916212\pi$
$912$	0	0
$913$	−531.914	−0.582600
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1103.51i	− 1.20339i
$918$	0	0
$919$	−253.043	−0.275346	−0.137673	−	0.990478i	$-0.543962\pi$
$919$	−253.043	−0.275346	−0.137673	+	0.990478i	$0.543962\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	179.390i	0.194355i
$924$	0	0
$925$	126.582	0.136845
$926$	0	0
$927$	0	0
$928$	0	0
$929$	867.351i	0.933640i	0.884352	+	0.466820i	$0.154600\pi$
$929$	867.351i	0.933640i	−0.884352	+	0.466820i	$0.845400\pi$
$930$	0	0
$931$	−210.256	−0.225839
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 941.331i	− 1.00677i
$936$	0	0
$937$	−225.411	−0.240567	−0.120283	−	0.992740i	$-0.538380\pi$
$937$	−225.411	−0.240567	−0.120283	+	0.992740i	$0.538380\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 85.7975i	− 0.0911769i	−0.998960	−	0.0455885i	$-0.985484\pi$
$941$	− 85.7975i	− 0.0911769i	0.998960	−	0.0455885i	$-0.0145163\pi$
$942$	0	0
$943$	259.081	0.274741
$944$	0	0
$945$	0	0
$946$	0	0
$947$	545.858i	0.576408i	0.957569	+	0.288204i	$0.0930581\pi$
$947$	545.858i	0.576408i	−0.957569	+	0.288204i	$0.906942\pi$
$948$	0	0
$949$	824.834	0.869162
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1734.12i	− 1.81965i	−0.414996	−	0.909823i	$-0.636217\pi$
$953$	− 1734.12i	− 1.81965i	0.414996	−	0.909823i	$-0.363783\pi$
$954$	0	0
$955$	−94.3889	−0.0988365
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1068.96i	1.11466i
$960$	0	0
$961$	−696.780	−0.725057
$962$	0	0
$963$	0	0
$964$	0	0
$965$	730.337i	0.756826i
$966$	0	0
$967$	−1020.38	−1.05520	−0.527598	−	0.849494i	$-0.676907\pi$
$967$	−1020.38	−1.05520	−0.527598	+	0.849494i	$0.676907\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 927.378i	− 0.955075i	−0.878611	−	0.477538i	$-0.841529\pi$
$971$	− 927.378i	− 0.955075i	0.878611	−	0.477538i	$-0.158471\pi$
$972$	0	0
$973$	1067.56	1.09718
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 20.6108i	− 0.0210960i	−0.999944	−	0.0105480i	$-0.996642\pi$
$977$	− 20.6108i	− 0.0210960i	0.999944	−	0.0105480i	$-0.00335759\pi$
$978$	0	0
$979$	−780.361	−0.797100
$980$	0	0
$981$	0	0
$982$	0	0
$983$	489.658i	0.498126i	0.968487	+	0.249063i	$0.0801226\pi$
$983$	489.658i	0.498126i	−0.968487	+	0.249063i	$0.919877\pi$
$984$	0	0
$985$	243.121	0.246823
$986$	0	0
$987$	0	0
$988$	0	0
$989$	267.552i	0.270528i
$990$	0	0
$991$	−1258.48	−1.26991	−0.634957	−	0.772548i	$-0.718982\pi$
$991$	−1258.48	−1.26991	−0.634957	+	0.772548i	$0.718982\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1848.91i	1.85820i
$996$	0	0
$997$	−628.704	−0.630595	−0.315298	−	0.948993i	$-0.602104\pi$
$997$	−628.704	−0.630595	−0.315298	+	0.948993i	$0.602104\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.3.g.b.737.8		8
3.2	odd	2	inner	3312.3.g.b.737.1		8
4.3	odd	2		414.3.c.b.323.8	yes	8
12.11	even	2		414.3.c.b.323.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.c.b.323.1	✓	8	12.11	even	2
414.3.c.b.323.8	yes	8	4.3	odd	2
3312.3.g.b.737.1		8	3.2	odd	2	inner
3312.3.g.b.737.8		8	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:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3312.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.87676i q^{5} +5.43723 q^{7} +O(q^{10})\)	\(q+6.87676i q^{5} +5.43723 q^{7} -7.68941i q^{11} -5.83945 q^{13} -17.8019i q^{17} +10.8176 q^{19} +4.79583i q^{23} -22.2899 q^{25} +4.08109i q^{29} +16.2549 q^{31} +37.3906i q^{35} -5.67890 q^{37} -54.0221i q^{41} +55.7885 q^{43} -36.2529i q^{47} -19.4365 q^{49} -46.4465i q^{53} +52.8783 q^{55} -83.4972i q^{59} -4.39115 q^{61} -40.1565i q^{65} -37.8058 q^{67} -30.7203i q^{71} -141.252 q^{73} -41.8091i q^{77} -69.9060 q^{79} -69.1749i q^{83} +122.419 q^{85} -101.485i q^{89} -31.7505 q^{91} +74.3902i q^{95} +67.1358 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^7	\( 8 q - 16 q^{7} - 8 q^{13} + 48 q^{19} + 32 q^{31} + 32 q^{37} - 32 q^{43} + 80 q^{49} - 32 q^{55} + 48 q^{61} + 16 q^{67} - 432 q^{73} + 416 q^{79} + 584 q^{85} - 368 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q - 16 * q^7 - 8 * q^13 + 48 * q^19 + 32 * q^31 + 32 * q^37 - 32 * q^43 + 80 * q^49 - 32 * q^55 + 48 * q^61 + 16 * q^67 - 432 * q^73 + 416 * q^79 + 584 * q^85 - 368 * q^91 - 8 * q^97

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