LMFDB - Embedded newform 3312.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,2,Mod(1,3312)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3312.a (trivial)

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:	yes
Analytic conductor:	$26.4464531494$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.44688.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 6x + 2 $ x^4 - 2x^3 - 5x^2 + 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1656)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.92812$ of defining polynomial
Character	$\chi$	$=$	3312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.64575 q^{5} -3.17320 q^{7} +O(q^{10})$	$q+3.64575 q^{5} -3.17320 q^{7} -0.962718 q^{11} +5.85623 q^{13} +2.68303 q^{17} +5.50198 q^{19} +1.00000 q^{23} +8.29150 q^{25} -1.85623 q^{29} +1.85623 q^{31} -11.5687 q^{35} -6.67519 q^{37} -9.14774 q^{41} -7.57655 q^{43} +8.34640 q^{47} +3.06920 q^{49} -5.99215 q^{53} -3.50983 q^{55} +8.80133 q^{59} +1.03728 q^{61} +21.3504 q^{65} +1.08102 q^{67} +13.1477 q^{71} +13.2223 q^{73} +3.05490 q^{77} +4.53927 q^{79} +3.03728 q^{83} +9.78167 q^{85} -3.10400 q^{89} -18.5830 q^{91} +20.0589 q^{95} +11.6379 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 2 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 2 * q^7	$ 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55} + 16 q^{59} + 6 q^{61} + 4 q^{65} - 8 q^{67} + 12 q^{71} + 16 q^{73} + 12 q^{77} - 10 q^{79} + 14 q^{83} + 16 q^{85} + 14 q^{89} - 32 q^{91} + 20 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 2 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 12 * q^25 + 12 * q^29 - 12 * q^31 + 12 * q^35 + 14 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 28 * q^49 + 8 * q^53 - 16 * q^55 + 16 * q^59 + 6 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 16 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 16 * q^85 + 14 * q^89 - 32 * q^91 + 20 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$5$	3.64575	1.63043	0.815215	−	0.579159i	$-0.196619\pi$
$5$	3.64575	1.63043	0.815215	+	0.579159i	$0.196619\pi$
$6$	0	0
$7$	−3.17320	−1.19936	−0.599679	−	0.800241i	$-0.704705\pi$
$7$	−3.17320	−1.19936	−0.599679	+	0.800241i	$0.704705\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.962718	−0.290271	−0.145135	−	0.989412i	$-0.546362\pi$
$11$	−0.962718	−0.290271	−0.145135	+	0.989412i	$0.546362\pi$
$12$	0	0
$13$	5.85623	1.62423	0.812113	−	0.583500i	$-0.198317\pi$
$13$	5.85623	1.62423	0.812113	+	0.583500i	$0.198317\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.68303	0.650731	0.325366	−	0.945588i	$-0.394513\pi$
$17$	2.68303	0.650731	0.325366	+	0.945588i	$0.394513\pi$
$18$	0	0
$19$	5.50198	1.26224	0.631121	−	0.775684i	$-0.282595\pi$
$19$	5.50198	1.26224	0.631121	+	0.775684i	$0.282595\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	8.29150	1.65830
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.85623	−0.344694	−0.172347	−	0.985036i	$-0.555135\pi$
$29$	−1.85623	−0.344694	−0.172347	+	0.985036i	$0.555135\pi$
$30$	0	0
$31$	1.85623	0.333389	0.166695	−	0.986009i	$-0.446691\pi$
$31$	1.85623	0.333389	0.166695	+	0.986009i	$0.446691\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−11.5687	−1.95547
$36$	0	0
$37$	−6.67519	−1.09739	−0.548697	−	0.836021i	$-0.684876\pi$
$37$	−6.67519	−1.09739	−0.548697	+	0.836021i	$0.684876\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.14774	−1.42864	−0.714318	−	0.699821i	$-0.753263\pi$
$41$	−9.14774	−1.42864	−0.714318	+	0.699821i	$0.753263\pi$
$42$	0	0
$43$	−7.57655	−1.15541	−0.577706	−	0.816245i	$-0.696052\pi$
$43$	−7.57655	−1.15541	−0.577706	+	0.816245i	$0.696052\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.34640	1.21745	0.608724	−	0.793382i	$-0.291682\pi$
$47$	8.34640	1.21745	0.608724	+	0.793382i	$0.291682\pi$
$48$	0	0
$49$	3.06920	0.438458
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.99215	−0.823086	−0.411543	−	0.911390i	$-0.635010\pi$
$53$	−5.99215	−0.823086	−0.411543	+	0.911390i	$0.635010\pi$
$54$	0	0
$55$	−3.50983	−0.473266
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.80133	1.14584	0.572918	−	0.819613i	$-0.305811\pi$
$59$	8.80133	1.14584	0.572918	+	0.819613i	$0.305811\pi$
$60$	0	0
$61$	1.03728	0.132810	0.0664051	−	0.997793i	$-0.478847\pi$
$61$	1.03728	0.132810	0.0664051	+	0.997793i	$0.478847\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21.3504	2.64819
$66$	0	0
$67$	1.08102	0.132068	0.0660338	−	0.997817i	$-0.478965\pi$
$67$	1.08102	0.132068	0.0660338	+	0.997817i	$0.478965\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.1477	1.56035	0.780175	−	0.625562i	$-0.215130\pi$
$71$	13.1477	1.56035	0.780175	+	0.625562i	$0.215130\pi$
$72$	0	0
$73$	13.2223	1.54755	0.773777	−	0.633459i	$-0.218365\pi$
$73$	13.2223	1.54755	0.773777	+	0.633459i	$0.218365\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.05490	0.348138
$78$	0	0
$79$	4.53927	0.510707	0.255354	−	0.966848i	$-0.417808\pi$
$79$	4.53927	0.510707	0.255354	+	0.966848i	$0.417808\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.03728	0.333385	0.166692	−	0.986009i	$-0.446691\pi$
$83$	3.03728	0.333385	0.166692	+	0.986009i	$0.446691\pi$
$84$	0	0
$85$	9.78167	1.06097
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.10400	−0.329023	−0.164512	−	0.986375i	$-0.552605\pi$
$89$	−3.10400	−0.329023	−0.164512	+	0.986375i	$0.552605\pi$
$90$	0	0
$91$	−18.5830	−1.94803
$92$	0	0
$93$	0	0
$94$	0	0
$95$	20.0589	2.05800
$96$	0	0
$97$	11.6379	1.18165	0.590825	−	0.806800i	$-0.298802\pi$
$97$	11.6379	1.18165	0.590825	+	0.806800i	$0.298802\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.49017	0.844803	0.422402	−	0.906409i	$-0.361187\pi$
$101$	8.49017	0.844803	0.422402	+	0.906409i	$0.361187\pi$
$102$	0	0
$103$	−9.09864	−0.896515	−0.448258	−	0.893904i	$-0.647955\pi$
$103$	−9.09864	−0.896515	−0.448258	+	0.893904i	$0.647955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.6006	1.41150	0.705748	−	0.708463i	$-0.250611\pi$
$107$	14.6006	1.41150	0.705748	+	0.708463i	$0.250611\pi$
$108$	0	0
$109$	−9.54572	−0.914315	−0.457157	−	0.889386i	$-0.651132\pi$
$109$	−9.54572	−0.914315	−0.457157	+	0.889386i	$0.651132\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.26207	−0.212797	−0.106399	−	0.994324i	$-0.533932\pi$
$113$	−2.26207	−0.212797	−0.106399	+	0.994324i	$0.533932\pi$
$114$	0	0
$115$	3.64575	0.339968
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.51380	−0.780459
$120$	0	0
$121$	−10.0732	−0.915743
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−20.9294	−1.85718	−0.928592	−	0.371102i	$-0.878980\pi$
$127$	−20.9294	−1.85718	−0.928592	+	0.371102i	$0.878980\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.63790	−0.492586	−0.246293	−	0.969195i	$-0.579213\pi$
$131$	−5.63790	−0.492586	−0.246293	+	0.969195i	$0.579213\pi$
$132$	0	0
$133$	−17.4589	−1.51388
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.7616	1.68834	0.844172	−	0.536072i	$-0.180092\pi$
$137$	19.7616	1.68834	0.844172	+	0.536072i	$0.180092\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.63790	−0.471465
$144$	0	0
$145$	−6.76737	−0.561999
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.0864	0.908232	0.454116	−	0.890943i	$-0.349955\pi$
$149$	11.0864	0.908232	0.454116	+	0.890943i	$0.349955\pi$
$150$	0	0
$151$	22.7856	1.85427	0.927135	−	0.374729i	$-0.122264\pi$
$151$	22.7856	1.85427	0.927135	+	0.374729i	$0.122264\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.76737	0.543568
$156$	0	0
$157$	−3.90782	−0.311878	−0.155939	−	0.987767i	$-0.549840\pi$
$157$	−3.90782	−0.311878	−0.155939	+	0.987767i	$0.549840\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.17320	−0.250083
$162$	0	0
$163$	−15.1424	−1.18604	−0.593021	−	0.805187i	$-0.702065\pi$
$163$	−15.1424	−1.18604	−0.593021	+	0.805187i	$0.702065\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.41560	0.186925	0.0934626	−	0.995623i	$-0.470206\pi$
$167$	2.41560	0.186925	0.0934626	+	0.995623i	$0.470206\pi$
$168$	0	0
$169$	21.2955	1.63811
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.7071	1.65036	0.825180	−	0.564869i	$-0.191073\pi$
$173$	21.7071	1.65036	0.825180	+	0.564869i	$0.191073\pi$
$174$	0	0
$175$	−26.3106	−1.98889
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.7071	−0.875030	−0.437515	−	0.899211i	$-0.644141\pi$
$179$	−11.7071	−0.875030	−0.437515	+	0.899211i	$0.644141\pi$
$180$	0	0
$181$	−12.6006	−0.936597	−0.468298	−	0.883570i	$-0.655133\pi$
$181$	−12.6006	−0.936597	−0.468298	+	0.883570i	$0.655133\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−24.3361	−1.78922
$186$	0	0
$187$	−2.58301	−0.188888
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.0549	0.799904	0.399952	−	0.916536i	$-0.369027\pi$
$191$	11.0549	0.799904	0.399952	+	0.916536i	$0.369027\pi$
$192$	0	0
$193$	26.2263	1.88781	0.943904	−	0.330220i	$-0.107123\pi$
$193$	26.2263	1.88781	0.943904	+	0.330220i	$0.107123\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.4941	1.38890	0.694450	−	0.719541i	$-0.255648\pi$
$197$	19.4941	1.38890	0.694450	+	0.719541i	$0.255648\pi$
$198$	0	0
$199$	−7.55893	−0.535838	−0.267919	−	0.963441i	$-0.586336\pi$
$199$	−7.55893	−0.535838	−0.267919	+	0.963441i	$0.586336\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.89020	0.413411
$204$	0	0
$205$	−33.3504	−2.32929
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.29686	−0.366392
$210$	0	0
$211$	19.9843	1.37578	0.687888	−	0.725817i	$-0.258538\pi$
$211$	19.9843	1.37578	0.687888	+	0.725817i	$0.258538\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−27.6222	−1.88382
$216$	0	0
$217$	−5.89020	−0.399853
$218$	0	0
$219$	0	0
$220$	0	0
$221$	15.7125	1.05693
$222$	0	0
$223$	1.61963	0.108458	0.0542292	−	0.998529i	$-0.482730\pi$
$223$	1.61963	0.108458	0.0542292	+	0.998529i	$0.482730\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	29.1836	1.93699	0.968493	−	0.249042i	$-0.0801157\pi$
$227$	29.1836	1.93699	0.968493	+	0.249042i	$0.0801157\pi$
$228$	0	0
$229$	13.5457	0.895127	0.447563	−	0.894252i	$-0.352292\pi$
$229$	13.5457	0.895127	0.447563	+	0.894252i	$0.352292\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.6928	−1.35563	−0.677815	−	0.735232i	$-0.737073\pi$
$233$	−20.6928	−1.35563	−0.677815	+	0.735232i	$0.737073\pi$
$234$	0	0
$235$	30.4289	1.98496
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.6928	−0.821029	−0.410515	−	0.911854i	$-0.634651\pi$
$239$	−12.6928	−0.821029	−0.410515	+	0.911854i	$0.634651\pi$
$240$	0	0
$241$	−16.7674	−1.08008	−0.540041	−	0.841639i	$-0.681591\pi$
$241$	−16.7674	−1.08008	−0.540041	+	0.841639i	$0.681591\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11.1896	0.714874
$246$	0	0
$247$	32.2209	2.05017
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.2582	−0.963088	−0.481544	−	0.876422i	$-0.659924\pi$
$251$	−15.2582	−0.963088	−0.481544	+	0.876422i	$0.659924\pi$
$252$	0	0
$253$	−0.962718	−0.0605256
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.1674	−1.50752	−0.753761	−	0.657149i	$-0.771762\pi$
$257$	−24.1674	−1.50752	−0.753761	+	0.657149i	$0.771762\pi$
$258$	0	0
$259$	21.1817	1.31617
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.7125	−0.968872	−0.484436	−	0.874827i	$-0.660975\pi$
$263$	−15.7125	−0.968872	−0.484436	+	0.874827i	$0.660975\pi$
$264$	0	0
$265$	−21.8459	−1.34198
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.6522	−0.649477	−0.324738	−	0.945804i	$-0.605276\pi$
$269$	−10.6522	−0.649477	−0.324738	+	0.945804i	$0.605276\pi$
$270$	0	0
$271$	6.34640	0.385516	0.192758	−	0.981246i	$-0.438257\pi$
$271$	6.34640	0.385516	0.192758	+	0.981246i	$0.438257\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.98238	−0.481356
$276$	0	0
$277$	−12.0746	−0.725490	−0.362745	−	0.931888i	$-0.618160\pi$
$277$	−12.0746	−0.725490	−0.362745	+	0.931888i	$0.618160\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.1915	−1.14487	−0.572434	−	0.819951i	$-0.694001\pi$
$281$	−19.1915	−1.14487	−0.572434	+	0.819951i	$0.694001\pi$
$282$	0	0
$283$	−18.1948	−1.08157	−0.540784	−	0.841162i	$-0.681872\pi$
$283$	−18.1948	−1.08157	−0.540784	+	0.841162i	$0.681872\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	29.0276	1.71345
$288$	0	0
$289$	−9.80133	−0.576549
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.9216	−0.871727	−0.435863	−	0.900013i	$-0.643557\pi$
$293$	−14.9216	−0.871727	−0.435863	+	0.900013i	$0.643557\pi$
$294$	0	0
$295$	32.0875	1.86821
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.85623	0.338675
$300$	0	0
$301$	24.0419	1.38575
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.78167	0.216538
$306$	0	0
$307$	−18.7164	−1.06820	−0.534102	−	0.845420i	$-0.679350\pi$
$307$	−18.7164	−1.06820	−0.534102	+	0.845420i	$0.679350\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	34.6601	1.96540	0.982698	−	0.185213i	$-0.0592976\pi$
$311$	34.6601	1.96540	0.982698	+	0.185213i	$0.0592976\pi$
$312$	0	0
$313$	−28.3307	−1.60135	−0.800673	−	0.599101i	$-0.795525\pi$
$313$	−28.3307	−1.60135	−0.800673	+	0.599101i	$0.795525\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.6982	1.49952	0.749759	−	0.661711i	$-0.230169\pi$
$317$	26.6982	1.49952	0.749759	+	0.661711i	$0.230169\pi$
$318$	0	0
$319$	1.78703	0.100054
$320$	0	0
$321$	0	0
$322$	0	0
$323$	14.7620	0.821380
$324$	0	0
$325$	48.5570	2.69346
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−26.4848	−1.46015
$330$	0	0
$331$	−5.87451	−0.322892	−0.161446	−	0.986882i	$-0.551616\pi$
$331$	−5.87451	−0.322892	−0.161446	+	0.986882i	$0.551616\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.94113	0.215327
$336$	0	0
$337$	18.2561	0.994476	0.497238	−	0.867614i	$-0.334348\pi$
$337$	18.2561	0.994476	0.497238	+	0.867614i	$0.334348\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.78703	−0.0967731
$342$	0	0
$343$	12.4732	0.673490
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−33.1714	−1.78073	−0.890366	−	0.455245i	$-0.849552\pi$
$347$	−33.1714	−1.78073	−0.890366	+	0.455245i	$0.849552\pi$
$348$	0	0
$349$	4.76201	0.254904	0.127452	−	0.991845i	$-0.459320\pi$
$349$	4.76201	0.254904	0.127452	+	0.991845i	$0.459320\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.0589	0.535380	0.267690	−	0.963505i	$-0.413740\pi$
$353$	10.0589	0.535380	0.267690	+	0.963505i	$0.413740\pi$
$354$	0	0
$355$	47.9334	2.54404
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.657568	0.0347051	0.0173526	−	0.999849i	$-0.494476\pi$
$359$	0.657568	0.0347051	0.0173526	+	0.999849i	$0.494476\pi$
$360$	0	0
$361$	11.2718	0.593255
$362$	0	0
$363$	0	0
$364$	0	0
$365$	48.2052	2.52318
$366$	0	0
$367$	5.95626	0.310914	0.155457	−	0.987843i	$-0.450315\pi$
$367$	5.95626	0.310914	0.155457	+	0.987843i	$0.450315\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	19.0143	0.987173
$372$	0	0
$373$	13.4073	0.694205	0.347102	−	0.937827i	$-0.387166\pi$
$373$	13.4073	0.694205	0.347102	+	0.937827i	$0.387166\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.8705	−0.559861
$378$	0	0
$379$	−32.9269	−1.69134	−0.845671	−	0.533704i	$-0.820800\pi$
$379$	−32.9269	−1.69134	−0.845671	+	0.533704i	$0.820800\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.71247	0.189698	0.0948491	−	0.995492i	$-0.469763\pi$
$383$	3.71247	0.189698	0.0948491	+	0.995492i	$0.469763\pi$
$384$	0	0
$385$	11.1374	0.567615
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−26.0904	−1.32283	−0.661417	−	0.750019i	$-0.730044\pi$
$389$	−26.0904	−1.32283	−0.661417	+	0.750019i	$0.730044\pi$
$390$	0	0
$391$	2.68303	0.135687
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.5490	0.832672
$396$	0	0
$397$	−20.2955	−1.01860	−0.509300	−	0.860589i	$-0.670096\pi$
$397$	−20.2955	−1.01860	−0.509300	+	0.860589i	$0.670096\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.68303	0.133984	0.0669921	−	0.997754i	$-0.478660\pi$
$401$	2.68303	0.133984	0.0669921	+	0.997754i	$0.478660\pi$
$402$	0	0
$403$	10.8705	0.541500
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.42632	0.318541
$408$	0	0
$409$	−15.6325	−0.772980	−0.386490	−	0.922294i	$-0.626313\pi$
$409$	−15.6325	−0.772980	−0.386490	+	0.922294i	$0.626313\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−27.9284	−1.37427
$414$	0	0
$415$	11.0732	0.543561
$416$	0	0
$417$	0	0
$418$	0	0
$419$	25.0452	1.22354	0.611770	−	0.791036i	$-0.290458\pi$
$419$	25.0452	1.22354	0.611770	+	0.791036i	$0.290458\pi$
$420$	0	0
$421$	−14.0176	−0.683177	−0.341588	−	0.939850i	$-0.610965\pi$
$421$	−14.0176	−0.683177	−0.341588	+	0.939850i	$0.610965\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	22.2464	1.07911
$426$	0	0
$427$	−3.29150	−0.159287
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.3700	−1.36654	−0.683268	−	0.730167i	$-0.739442\pi$
$431$	−28.3700	−1.36654	−0.683268	+	0.730167i	$0.739442\pi$
$432$	0	0
$433$	−32.6655	−1.56980	−0.784902	−	0.619620i	$-0.787287\pi$
$433$	−32.6655	−1.56980	−0.784902	+	0.619620i	$0.787287\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.50198	0.263196
$438$	0	0
$439$	−19.2758	−0.919984	−0.459992	−	0.887923i	$-0.652148\pi$
$439$	−19.2758	−0.919984	−0.459992	+	0.887923i	$0.652148\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.8705	−0.516475	−0.258237	−	0.966081i	$-0.583142\pi$
$443$	−10.8705	−0.516475	−0.258237	+	0.966081i	$0.583142\pi$
$444$	0	0
$445$	−11.3164	−0.536449
$446$	0	0
$447$	0	0
$448$	0	0
$449$	31.1557	1.47033	0.735164	−	0.677890i	$-0.237105\pi$
$449$	31.1557	1.47033	0.735164	+	0.677890i	$0.237105\pi$
$450$	0	0
$451$	8.80669	0.414691
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−67.7490	−3.17612
$456$	0	0
$457$	−16.3621	−0.765387	−0.382693	−	0.923875i	$-0.625003\pi$
$457$	−16.3621	−0.765387	−0.382693	+	0.923875i	$0.625003\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.7727	0.594885	0.297443	−	0.954740i	$-0.403866\pi$
$461$	12.7727	0.594885	0.297443	+	0.954740i	$0.403866\pi$
$462$	0	0
$463$	36.1177	1.67853	0.839267	−	0.543720i	$-0.182985\pi$
$463$	36.1177	1.67853	0.839267	+	0.543720i	$0.182985\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.2216	0.704374	0.352187	−	0.935930i	$-0.385438\pi$
$467$	15.2216	0.704374	0.352187	+	0.935930i	$0.385438\pi$
$468$	0	0
$469$	−3.43029	−0.158396
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.29408	0.335382
$474$	0	0
$475$	45.6197	2.09318
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.15807	0.144296	0.0721480	−	0.997394i	$-0.477015\pi$
$479$	3.15807	0.144296	0.0721480	+	0.997394i	$0.477015\pi$
$480$	0	0
$481$	−39.0914	−1.78242
$482$	0	0
$483$	0	0
$484$	0	0
$485$	42.4289	1.92660
$486$	0	0
$487$	−33.8588	−1.53429	−0.767145	−	0.641474i	$-0.778323\pi$
$487$	−33.8588	−1.53429	−0.767145	+	0.641474i	$0.778323\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.14377	−0.367523	−0.183762	−	0.982971i	$-0.558827\pi$
$491$	−8.14377	−0.367523	−0.183762	+	0.982971i	$0.558827\pi$
$492$	0	0
$493$	−4.98034	−0.224303
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−41.7204	−1.87142
$498$	0	0
$499$	−6.46029	−0.289202	−0.144601	−	0.989490i	$-0.546190\pi$
$499$	−6.46029	−0.289202	−0.144601	+	0.989490i	$0.546190\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.9165	−0.486742	−0.243371	−	0.969933i	$-0.578253\pi$
$503$	−10.9165	−0.486742	−0.243371	+	0.969933i	$0.578253\pi$
$504$	0	0
$505$	30.9530	1.37739
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.30978	0.146703	0.0733516	−	0.997306i	$-0.476630\pi$
$509$	3.30978	0.146703	0.0733516	+	0.997306i	$0.476630\pi$
$510$	0	0
$511$	−41.9570	−1.85607
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−33.1714	−1.46171
$516$	0	0
$517$	−8.03523	−0.353389
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.5418	1.42568	0.712842	−	0.701325i	$-0.247408\pi$
$521$	32.5418	1.42568	0.712842	+	0.701325i	$0.247408\pi$
$522$	0	0
$523$	−10.7346	−0.469392	−0.234696	−	0.972069i	$-0.575409\pi$
$523$	−10.7346	−0.469392	−0.234696	+	0.972069i	$0.575409\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.98034	0.216947
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−53.5713	−2.32043
$534$	0	0
$535$	53.2302	2.30134
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.95478	−0.127271
$540$	0	0
$541$	29.9387	1.28717	0.643583	−	0.765376i	$-0.277447\pi$
$541$	29.9387	1.28717	0.643583	+	0.765376i	$0.277447\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−34.8013	−1.49073
$546$	0	0
$547$	45.4378	1.94278	0.971391	−	0.237486i	$-0.0763232\pi$
$547$	45.4378	1.94278	0.971391	+	0.237486i	$0.0763232\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.2130	−0.435087
$552$	0	0
$553$	−14.4040	−0.612521
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.7046	0.919654	0.459827	−	0.888008i	$-0.347911\pi$
$557$	21.7046	0.919654	0.459827	+	0.888008i	$0.347911\pi$
$558$	0	0
$559$	−44.3700	−1.87665
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.0452	−1.56127	−0.780635	−	0.624987i	$-0.785104\pi$
$563$	−37.0452	−1.56127	−0.780635	+	0.624987i	$0.785104\pi$
$564$	0	0
$565$	−8.24694	−0.346951
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.3758	−0.476900	−0.238450	−	0.971155i	$-0.576639\pi$
$569$	−11.3758	−0.476900	−0.238450	+	0.971155i	$0.576639\pi$
$570$	0	0
$571$	−5.08102	−0.212634	−0.106317	−	0.994332i	$-0.533906\pi$
$571$	−5.08102	−0.212634	−0.106317	+	0.994332i	$0.533906\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.29150	0.345780
$576$	0	0
$577$	−19.6968	−0.819987	−0.409994	−	0.912088i	$-0.634469\pi$
$577$	−19.6968	−0.819987	−0.409994	+	0.912088i	$0.634469\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.63790	−0.399848
$582$	0	0
$583$	5.76876	0.238917
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.0825	0.828894	0.414447	−	0.910074i	$-0.363975\pi$
$587$	20.0825	0.828894	0.414447	+	0.910074i	$0.363975\pi$
$588$	0	0
$589$	10.2130	0.420818
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19.5816	−0.804121	−0.402060	−	0.915613i	$-0.631706\pi$
$593$	−19.5816	−0.804121	−0.402060	+	0.915613i	$0.631706\pi$
$594$	0	0
$595$	−31.0392	−1.27248
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−38.5830	−1.57646	−0.788229	−	0.615381i	$-0.789002\pi$
$599$	−38.5830	−1.57646	−0.788229	+	0.615381i	$0.789002\pi$
$600$	0	0
$601$	−32.8628	−1.34050	−0.670250	−	0.742135i	$-0.733813\pi$
$601$	−32.8628	−1.34050	−0.670250	+	0.742135i	$0.733813\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−36.7243	−1.49305
$606$	0	0
$607$	0.639294	0.0259481	0.0129741	−	0.999916i	$-0.495870\pi$
$607$	0.639294	0.0259481	0.0129741	+	0.999916i	$0.495870\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	48.8785	1.97741
$612$	0	0
$613$	11.2503	0.454393	0.227197	−	0.973849i	$-0.427044\pi$
$613$	11.2503	0.454393	0.227197	+	0.973849i	$0.427044\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.2335	−0.975602	−0.487801	−	0.872955i	$-0.662201\pi$
$617$	−24.2335	−0.975602	−0.487801	+	0.872955i	$0.662201\pi$
$618$	0	0
$619$	−40.3568	−1.62208	−0.811039	−	0.584992i	$-0.801098\pi$
$619$	−40.3568	−1.62208	−0.811039	+	0.584992i	$0.801098\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.84961	0.394616
$624$	0	0
$625$	2.29150	0.0916601
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−17.9097	−0.714108
$630$	0	0
$631$	2.83971	0.113047	0.0565236	−	0.998401i	$-0.481998\pi$
$631$	2.83971	0.113047	0.0565236	+	0.998401i	$0.481998\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−76.3034	−3.02801
$636$	0	0
$637$	17.9740	0.712155
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.2857	0.643247	0.321623	−	0.946868i	$-0.395772\pi$
$641$	16.2857	0.643247	0.321623	+	0.946868i	$0.395772\pi$
$642$	0	0
$643$	−31.5765	−1.24526	−0.622629	−	0.782517i	$-0.713935\pi$
$643$	−31.5765	−1.24526	−0.622629	+	0.782517i	$0.713935\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.9191	−0.822414	−0.411207	−	0.911542i	$-0.634893\pi$
$647$	−20.9191	−0.822414	−0.411207	+	0.911542i	$0.634893\pi$
$648$	0	0
$649$	−8.47321	−0.332602
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.0535	0.628222	0.314111	−	0.949386i	$-0.398294\pi$
$653$	16.0535	0.628222	0.314111	+	0.949386i	$0.398294\pi$
$654$	0	0
$655$	−20.5544	−0.803127
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.7117	1.58590	0.792952	−	0.609284i	$-0.208543\pi$
$659$	40.7117	1.58590	0.792952	+	0.609284i	$0.208543\pi$
$660$	0	0
$661$	7.10112	0.276202	0.138101	−	0.990418i	$-0.455900\pi$
$661$	7.10112	0.276202	0.138101	+	0.990418i	$0.455900\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−63.6508	−2.46827
$666$	0	0
$667$	−1.85623	−0.0718737
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.998610	−0.0385509
$672$	0	0
$673$	−11.9843	−0.461961	−0.230981	−	0.972958i	$-0.574193\pi$
$673$	−11.9843	−0.461961	−0.230981	+	0.972958i	$0.574193\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.2051	1.08401	0.542005	−	0.840375i	$-0.317665\pi$
$677$	28.2051	1.08401	0.542005	+	0.840375i	$0.317665\pi$
$678$	0	0
$679$	−36.9294	−1.41722
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.806695	−0.0308673	−0.0154337	−	0.999881i	$-0.504913\pi$
$683$	−0.806695	−0.0308673	−0.0154337	+	0.999881i	$0.504913\pi$
$684$	0	0
$685$	72.0458	2.75273
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−35.0914	−1.33688
$690$	0	0
$691$	−10.0236	−0.381317	−0.190659	−	0.981656i	$-0.561062\pi$
$691$	−10.0236	−0.381317	−0.190659	+	0.981656i	$0.561062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.5830	−0.553165
$696$	0	0
$697$	−24.5437	−0.929658
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.02751	0.189886	0.0949432	−	0.995483i	$-0.469733\pi$
$701$	5.02751	0.189886	0.0949432	+	0.995483i	$0.469733\pi$
$702$	0	0
$703$	−36.7268	−1.38518
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.9410	−1.01322
$708$	0	0
$709$	11.1118	0.417314	0.208657	−	0.977989i	$-0.433091\pi$
$709$	11.1118	0.417314	0.208657	+	0.977989i	$0.433091\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.85623	0.0695165
$714$	0	0
$715$	−20.5544	−0.768691
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.6433	−0.657983	−0.328991	−	0.944333i	$-0.606709\pi$
$719$	−17.6433	−0.657983	−0.328991	+	0.944333i	$0.606709\pi$
$720$	0	0
$721$	28.8718	1.07524
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.3910	−0.571606
$726$	0	0
$727$	31.5080	1.16857	0.584284	−	0.811550i	$-0.301376\pi$
$727$	31.5080	1.16857	0.584284	+	0.811550i	$0.301376\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−20.3281	−0.751863
$732$	0	0
$733$	9.71937	0.358993	0.179496	−	0.983759i	$-0.442553\pi$
$733$	9.71937	0.358993	0.179496	+	0.983759i	$0.442553\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.04072	−0.0383353
$738$	0	0
$739$	−5.05899	−0.186098	−0.0930490	−	0.995662i	$-0.529661\pi$
$739$	−5.05899	−0.186098	−0.0930490	+	0.995662i	$0.529661\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.0473	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$743$	36.0473	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$744$	0	0
$745$	40.4182	1.48081
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−46.3307	−1.69289
$750$	0	0
$751$	−30.9575	−1.12965	−0.564827	−	0.825210i	$-0.691057\pi$
$751$	−30.9575	−1.12965	−0.564827	+	0.825210i	$0.691057\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	83.0708	3.02326
$756$	0	0
$757$	−31.3680	−1.14009	−0.570044	−	0.821614i	$-0.693074\pi$
$757$	−31.3680	−1.14009	−0.570044	+	0.821614i	$0.693074\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−27.0392	−0.980170	−0.490085	−	0.871675i	$-0.663034\pi$
$761$	−27.0392	−0.980170	−0.490085	+	0.871675i	$0.663034\pi$
$762$	0	0
$763$	30.2905	1.09659
$764$	0	0
$765$	0	0
$766$	0	0
$767$	51.5427	1.86110
$768$	0	0
$769$	14.4719	0.521870	0.260935	−	0.965356i	$-0.415969\pi$
$769$	14.4719	0.521870	0.260935	+	0.965356i	$0.415969\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.9086	0.752032	0.376016	−	0.926613i	$-0.377294\pi$
$773$	20.9086	0.752032	0.376016	+	0.926613i	$0.377294\pi$
$774$	0	0
$775$	15.3910	0.552860
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−50.3307	−1.80328
$780$	0	0
$781$	−12.6576	−0.452923
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−14.2469	−0.508495
$786$	0	0
$787$	8.93189	0.318388	0.159194	−	0.987247i	$-0.449111\pi$
$787$	8.93189	0.318388	0.159194	+	0.987247i	$0.449111\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.17800	0.255220
$792$	0	0
$793$	6.07456	0.215714
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.7359	0.521972	0.260986	−	0.965343i	$-0.415952\pi$
$797$	14.7359	0.521972	0.260986	+	0.965343i	$0.415952\pi$
$798$	0	0
$799$	22.3937	0.792231
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−12.7294	−0.449209
$804$	0	0
$805$	−11.5687	−0.407743
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−33.6536	−1.18320	−0.591599	−	0.806233i	$-0.701503\pi$
$809$	−33.6536	−1.18320	−0.591599	+	0.806233i	$0.701503\pi$
$810$	0	0
$811$	−51.2708	−1.80036	−0.900181	−	0.435515i	$-0.856566\pi$
$811$	−51.2708	−1.80036	−0.900181	+	0.435515i	$0.856566\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−55.2053	−1.93376
$816$	0	0
$817$	−41.6861	−1.45841
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.3607	−0.675693	−0.337847	−	0.941201i	$-0.609699\pi$
$821$	−19.3607	−0.675693	−0.337847	+	0.941201i	$0.609699\pi$
$822$	0	0
$823$	−22.0054	−0.767059	−0.383529	−	0.923529i	$-0.625291\pi$
$823$	−22.0054	−0.767059	−0.383529	+	0.923529i	$0.625291\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.8243	−1.28051	−0.640253	−	0.768164i	$-0.721171\pi$
$827$	−36.8243	−1.28051	−0.640253	+	0.768164i	$0.721171\pi$
$828$	0	0
$829$	10.2577	0.356263	0.178132	−	0.984007i	$-0.442995\pi$
$829$	10.2577	0.356263	0.178132	+	0.984007i	$0.442995\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.23477	0.285318
$834$	0	0
$835$	8.80669	0.304768
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.7035	−0.990956	−0.495478	−	0.868621i	$-0.665007\pi$
$839$	−28.7035	−0.990956	−0.495478	+	0.868621i	$0.665007\pi$
$840$	0	0
$841$	−25.5544	−0.881186
$842$	0	0
$843$	0	0
$844$	0	0
$845$	77.6380	2.67083
$846$	0	0
$847$	31.9642	1.09830
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.67519	−0.228822
$852$	0	0
$853$	−18.2561	−0.625078	−0.312539	−	0.949905i	$-0.601180\pi$
$853$	−18.2561	−0.625078	−0.312539	+	0.949905i	$0.601180\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.60267	−0.191384	−0.0956918	−	0.995411i	$-0.530506\pi$
$857$	−5.60267	−0.191384	−0.0956918	+	0.995411i	$0.530506\pi$
$858$	0	0
$859$	−27.1374	−0.925916	−0.462958	−	0.886380i	$-0.653212\pi$
$859$	−27.1374	−0.925916	−0.462958	+	0.886380i	$0.653212\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.5335	0.664927	0.332463	−	0.943116i	$-0.392120\pi$
$863$	19.5335	0.664927	0.332463	+	0.943116i	$0.392120\pi$
$864$	0	0
$865$	79.1387	2.69080
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.37004	−0.148243
$870$	0	0
$871$	6.33071	0.214508
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−38.0784	−1.28729
$876$	0	0
$877$	51.7556	1.74766	0.873832	−	0.486228i	$-0.161628\pi$
$877$	51.7556	1.74766	0.873832	+	0.486228i	$0.161628\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−31.7773	−1.07060	−0.535302	−	0.844661i	$-0.679802\pi$
$881$	−31.7773	−1.07060	−0.535302	+	0.844661i	$0.679802\pi$
$882$	0	0
$883$	14.3112	0.481609	0.240805	−	0.970574i	$-0.422589\pi$
$883$	14.3112	0.481609	0.240805	+	0.970574i	$0.422589\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.7606	1.33503	0.667515	−	0.744596i	$-0.267358\pi$
$887$	39.7606	1.33503	0.667515	+	0.744596i	$0.267358\pi$
$888$	0	0
$889$	66.4132	2.22743
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45.9218	1.53671
$894$	0	0
$895$	−42.6812	−1.42668
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.44560	−0.114917
$900$	0	0
$901$	−16.0771	−0.535607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−45.9387	−1.52706
$906$	0	0
$907$	−18.6680	−0.619861	−0.309930	−	0.950759i	$-0.600306\pi$
$907$	−18.6680	−0.619861	−0.309930	+	0.950759i	$0.600306\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.1816	1.13249	0.566243	−	0.824238i	$-0.308396\pi$
$911$	34.1816	1.13249	0.566243	+	0.824238i	$0.308396\pi$
$912$	0	0
$913$	−2.92405	−0.0967718
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.8902	0.590787
$918$	0	0
$919$	31.7641	1.04780	0.523901	−	0.851779i	$-0.324476\pi$
$919$	31.7641	1.04780	0.523901	+	0.851779i	$0.324476\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	76.9962	2.53436
$924$	0	0
$925$	−55.3473	−1.81981
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−18.6717	−0.612600	−0.306300	−	0.951935i	$-0.599091\pi$
$929$	−18.6717	−0.612600	−0.306300	+	0.951935i	$0.599091\pi$
$930$	0	0
$931$	16.8867	0.553440
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.41699	−0.307969
$936$	0	0
$937$	−6.84193	−0.223516	−0.111758	−	0.993735i	$-0.535648\pi$
$937$	−6.84193	−0.223516	−0.111758	+	0.993735i	$0.535648\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−36.7243	−1.19718	−0.598589	−	0.801057i	$-0.704272\pi$
$941$	−36.7243	−1.19718	−0.598589	+	0.801057i	$0.704272\pi$
$942$	0	0
$943$	−9.14774	−0.297891
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.7230	−1.29082	−0.645412	−	0.763835i	$-0.723314\pi$
$947$	−39.7230	−1.29082	−0.645412	+	0.763835i	$0.723314\pi$
$948$	0	0
$949$	77.4329	2.51358
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.4464	0.727111	0.363556	−	0.931573i	$-0.381563\pi$
$953$	22.4464	0.727111	0.363556	+	0.931573i	$0.381563\pi$
$954$	0	0
$955$	40.3034	1.30419
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−62.7074	−2.02493
$960$	0	0
$961$	−27.5544	−0.888852
$962$	0	0
$963$	0	0
$964$	0	0
$965$	95.6145	3.07794
$966$	0	0
$967$	−1.01430	−0.0326178	−0.0163089	−	0.999867i	$-0.505192\pi$
$967$	−1.01430	−0.0326178	−0.0163089	+	0.999867i	$0.505192\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30.8323	−0.989454	−0.494727	−	0.869048i	$-0.664732\pi$
$971$	−30.8323	−0.989454	−0.494727	+	0.869048i	$0.664732\pi$
$972$	0	0
$973$	12.6928	0.406913
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.54285	0.177332	0.0886658	−	0.996061i	$-0.471740\pi$
$977$	5.54285	0.177332	0.0886658	+	0.996061i	$0.471740\pi$
$978$	0	0
$979$	2.98828	0.0955057
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.7753	0.662629	0.331315	−	0.943520i	$-0.392508\pi$
$983$	20.7753	0.662629	0.331315	+	0.943520i	$0.392508\pi$
$984$	0	0
$985$	71.0708	2.26450
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.57655	−0.240920
$990$	0	0
$991$	−11.1348	−0.353709	−0.176855	−	0.984237i	$-0.556592\pi$
$991$	−11.1348	−0.353709	−0.176855	+	0.984237i	$0.556592\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−27.5580	−0.873647
$996$	0	0
$997$	9.82100	0.311034	0.155517	−	0.987833i	$-0.450296\pi$
$997$	9.82100	0.311034	0.155517	+	0.987833i	$0.450296\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.bh.1.3		4
3.2	odd	2		3312.2.a.bg.1.1		4
4.3	odd	2		1656.2.a.p.1.4	yes	4
12.11	even	2		1656.2.a.o.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1656.2.a.o.1.2	✓	4	12.11	even	2
1656.2.a.p.1.4	yes	4	4.3	odd	2
3312.2.a.bg.1.1		4	3.2	odd	2
3312.2.a.bh.1.3		4	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.64575 q^{5} -3.17320 q^{7} +O(q^{10})\)	\(q+3.64575 q^{5} -3.17320 q^{7} -0.962718 q^{11} +5.85623 q^{13} +2.68303 q^{17} +5.50198 q^{19} +1.00000 q^{23} +8.29150 q^{25} -1.85623 q^{29} +1.85623 q^{31} -11.5687 q^{35} -6.67519 q^{37} -9.14774 q^{41} -7.57655 q^{43} +8.34640 q^{47} +3.06920 q^{49} -5.99215 q^{53} -3.50983 q^{55} +8.80133 q^{59} +1.03728 q^{61} +21.3504 q^{65} +1.08102 q^{67} +13.1477 q^{71} +13.2223 q^{73} +3.05490 q^{77} +4.53927 q^{79} +3.03728 q^{83} +9.78167 q^{85} -3.10400 q^{89} -18.5830 q^{91} +20.0589 q^{95} +11.6379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 2 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 2 * q^7	\( 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55} + 16 q^{59} + 6 q^{61} + 4 q^{65} - 8 q^{67} + 12 q^{71} + 16 q^{73} + 12 q^{77} - 10 q^{79} + 14 q^{83} + 16 q^{85} + 14 q^{89} - 32 q^{91} + 20 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 2 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 12 * q^25 + 12 * q^29 - 12 * q^31 + 12 * q^35 + 14 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 28 * q^49 + 8 * q^53 - 16 * q^55 + 16 * q^59 + 6 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 16 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 16 * q^85 + 14 * q^89 - 32 * q^91 + 20 * q^95 + 4 * q^97

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