LMFDB - Embedded newform 9036.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9036,2,Mod(1,9036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9036 = 2^{2} \cdot 3^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9036.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:	$72.1528232664$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 $ x^7 - 3x^6 - 6x^5 + 18x^4 + 8x^3 - 17x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.85375$ of defining polynomial
Character	$\chi$	$=$	9036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.454452 q^{5} -2.18653 q^{7} +O(q^{10})$	$q-0.454452 q^{5} -2.18653 q^{7} +1.15877 q^{11} +6.03215 q^{13} -0.0535246 q^{17} -2.08924 q^{19} -3.22849 q^{23} -4.79347 q^{25} +1.62034 q^{29} -2.01992 q^{31} +0.993671 q^{35} -3.49384 q^{37} -2.85650 q^{41} +2.78023 q^{43} +9.03663 q^{47} -2.21910 q^{49} +4.09037 q^{53} -0.526603 q^{55} +4.76605 q^{59} -14.6219 q^{61} -2.74132 q^{65} -4.98740 q^{67} +2.04056 q^{71} -0.0937424 q^{73} -2.53367 q^{77} -1.92521 q^{79} -7.96981 q^{83} +0.0243243 q^{85} +16.7462 q^{89} -13.1895 q^{91} +0.949460 q^{95} +3.00301 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q + 2 * q^5 - 6 * q^7	$ 7 q + 2 q^{5} - 6 q^{7} + 5 q^{11} - q^{13} + 8 q^{17} - 15 q^{19} + 5 q^{23} - 9 q^{25} - 21 q^{31} + 7 q^{35} - q^{37} + 10 q^{41} - 23 q^{43} + 10 q^{47} - 13 q^{49} + q^{53} - 23 q^{55} + 4 q^{59} + 3 q^{61} - 4 q^{65} - 28 q^{67} + 18 q^{71} - 7 q^{73} - 6 q^{77} - 30 q^{79} - 13 q^{83} + q^{85} - 18 q^{91} - 2 q^{97}+O(q^{100}) $ 7 * q + 2 * q^5 - 6 * q^7 + 5 * q^11 - q^13 + 8 * q^17 - 15 * q^19 + 5 * q^23 - 9 * q^25 - 21 * q^31 + 7 * q^35 - q^37 + 10 * q^41 - 23 * q^43 + 10 * q^47 - 13 * q^49 + q^53 - 23 * q^55 + 4 * q^59 + 3 * q^61 - 4 * q^65 - 28 * q^67 + 18 * q^71 - 7 * q^73 - 6 * q^77 - 30 * q^79 - 13 * q^83 + q^85 - 18 * q^91 - 2 * 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
$5$	−0.454452	−0.203237	−0.101618	−	0.994823i	$-0.532402\pi$
$5$	−0.454452	−0.203237	−0.101618	+	0.994823i	$0.532402\pi$
$6$	0	0
$7$	−2.18653	−0.826429	−0.413215	−	0.910634i	$-0.635594\pi$
$7$	−2.18653	−0.826429	−0.413215	+	0.910634i	$0.635594\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.15877	0.349381	0.174691	−	0.984623i	$-0.444107\pi$
$11$	1.15877	0.349381	0.174691	+	0.984623i	$0.444107\pi$
$12$	0	0
$13$	6.03215	1.67302	0.836509	−	0.547953i	$-0.184593\pi$
$13$	6.03215	1.67302	0.836509	+	0.547953i	$0.184593\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.0535246	−0.0129816	−0.00649081	−	0.999979i	$-0.502066\pi$
$17$	−0.0535246	−0.0129816	−0.00649081	+	0.999979i	$0.502066\pi$
$18$	0	0
$19$	−2.08924	−0.479305	−0.239653	−	0.970859i	$-0.577034\pi$
$19$	−2.08924	−0.479305	−0.239653	+	0.970859i	$0.577034\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.22849	−0.673187	−0.336593	−	0.941650i	$-0.609275\pi$
$23$	−3.22849	−0.673187	−0.336593	+	0.941650i	$0.609275\pi$
$24$	0	0
$25$	−4.79347	−0.958695
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.62034	0.300889	0.150445	−	0.988618i	$-0.451929\pi$
$29$	1.62034	0.300889	0.150445	+	0.988618i	$0.451929\pi$
$30$	0	0
$31$	−2.01992	−0.362788	−0.181394	−	0.983411i	$-0.558061\pi$
$31$	−2.01992	−0.362788	−0.181394	+	0.983411i	$0.558061\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.993671	0.167961
$36$	0	0
$37$	−3.49384	−0.574383	−0.287192	−	0.957873i	$-0.592722\pi$
$37$	−3.49384	−0.574383	−0.287192	+	0.957873i	$0.592722\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.85650	−0.446111	−0.223055	−	0.974806i	$-0.571603\pi$
$41$	−2.85650	−0.446111	−0.223055	+	0.974806i	$0.571603\pi$
$42$	0	0
$43$	2.78023	0.423980	0.211990	−	0.977272i	$-0.432005\pi$
$43$	2.78023	0.423980	0.211990	+	0.977272i	$0.432005\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.03663	1.31813	0.659064	−	0.752087i	$-0.270953\pi$
$47$	9.03663	1.31813	0.659064	+	0.752087i	$0.270953\pi$
$48$	0	0
$49$	−2.21910	−0.317014
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.09037	0.561855	0.280927	−	0.959729i	$-0.409358\pi$
$53$	4.09037	0.561855	0.280927	+	0.959729i	$0.409358\pi$
$54$	0	0
$55$	−0.526603	−0.0710072
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.76605	0.620487	0.310244	−	0.950657i	$-0.399589\pi$
$59$	4.76605	0.620487	0.310244	+	0.950657i	$0.399589\pi$
$60$	0	0
$61$	−14.6219	−1.87214	−0.936068	−	0.351819i	$-0.885563\pi$
$61$	−14.6219	−1.87214	−0.936068	+	0.351819i	$0.885563\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.74132	−0.340019
$66$	0	0
$67$	−4.98740	−0.609308	−0.304654	−	0.952463i	$-0.598541\pi$
$67$	−4.98740	−0.609308	−0.304654	+	0.952463i	$0.598541\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.04056	0.242170	0.121085	−	0.992642i	$-0.461363\pi$
$71$	2.04056	0.242170	0.121085	+	0.992642i	$0.461363\pi$
$72$	0	0
$73$	−0.0937424	−0.0109717	−0.00548586	−	0.999985i	$-0.501746\pi$
$73$	−0.0937424	−0.0109717	−0.00548586	+	0.999985i	$0.501746\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.53367	−0.288739
$78$	0	0
$79$	−1.92521	−0.216603	−0.108302	−	0.994118i	$-0.534541\pi$
$79$	−1.92521	−0.216603	−0.108302	+	0.994118i	$0.534541\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.96981	−0.874801	−0.437400	−	0.899267i	$-0.644101\pi$
$83$	−7.96981	−0.874801	−0.437400	+	0.899267i	$0.644101\pi$
$84$	0	0
$85$	0.0243243	0.00263834
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.7462	1.77510	0.887549	−	0.460713i	$-0.152406\pi$
$89$	16.7462	1.77510	0.887549	+	0.460713i	$0.152406\pi$
$90$	0	0
$91$	−13.1895	−1.38263
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.949460	0.0974125
$96$	0	0
$97$	3.00301	0.304910	0.152455	−	0.988310i	$-0.451282\pi$
$97$	3.00301	0.304910	0.152455	+	0.988310i	$0.451282\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.55655	0.353890	0.176945	−	0.984221i	$-0.443379\pi$
$101$	3.55655	0.353890	0.176945	+	0.984221i	$0.443379\pi$
$102$	0	0
$103$	−3.10568	−0.306011	−0.153006	−	0.988225i	$-0.548895\pi$
$103$	−3.10568	−0.306011	−0.153006	+	0.988225i	$0.548895\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.8110	−1.04514	−0.522571	−	0.852596i	$-0.675027\pi$
$107$	−10.8110	−1.04514	−0.522571	+	0.852596i	$0.675027\pi$
$108$	0	0
$109$	7.53872	0.722078	0.361039	−	0.932551i	$-0.382422\pi$
$109$	7.53872	0.722078	0.361039	+	0.932551i	$0.382422\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.30774	0.311166	0.155583	−	0.987823i	$-0.450274\pi$
$113$	3.30774	0.311166	0.155583	+	0.987823i	$0.450274\pi$
$114$	0	0
$115$	1.46719	0.136816
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.117033	0.0107284
$120$	0	0
$121$	−9.65726	−0.877933
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.45066	0.398079
$126$	0	0
$127$	−13.1951	−1.17087	−0.585437	−	0.810718i	$-0.699077\pi$
$127$	−13.1951	−1.17087	−0.585437	+	0.810718i	$0.699077\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.18286	−0.365458	−0.182729	−	0.983163i	$-0.558493\pi$
$131$	−4.18286	−0.365458	−0.182729	+	0.983163i	$0.558493\pi$
$132$	0	0
$133$	4.56819	0.396112
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.2951	1.05044	0.525219	−	0.850967i	$-0.323983\pi$
$137$	12.2951	1.05044	0.525219	+	0.850967i	$0.323983\pi$
$138$	0	0
$139$	6.49640	0.551018	0.275509	−	0.961299i	$-0.411154\pi$
$139$	6.49640	0.551018	0.275509	+	0.961299i	$0.411154\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.98985	0.584521
$144$	0	0
$145$	−0.736365	−0.0611518
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.0170	−0.902548	−0.451274	−	0.892385i	$-0.649030\pi$
$149$	−11.0170	−0.902548	−0.451274	+	0.892385i	$0.649030\pi$
$150$	0	0
$151$	−21.4320	−1.74411	−0.872055	−	0.489408i	$-0.837213\pi$
$151$	−21.4320	−1.74411	−0.872055	+	0.489408i	$0.837213\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.917954	0.0737318
$156$	0	0
$157$	11.7026	0.933967	0.466984	−	0.884266i	$-0.345341\pi$
$157$	11.7026	0.933967	0.466984	+	0.884266i	$0.345341\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.05918	0.556341
$162$	0	0
$163$	−3.32371	−0.260333	−0.130167	−	0.991492i	$-0.541551\pi$
$163$	−3.32371	−0.260333	−0.130167	+	0.991492i	$0.541551\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.4924	−1.81789	−0.908947	−	0.416913i	$-0.863112\pi$
$167$	−23.4924	−1.81789	−0.908947	+	0.416913i	$0.863112\pi$
$168$	0	0
$169$	23.3869	1.79899
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.48627	0.341085	0.170543	−	0.985350i	$-0.445448\pi$
$173$	4.48627	0.341085	0.170543	+	0.985350i	$0.445448\pi$
$174$	0	0
$175$	10.4811	0.792294
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.40995	0.479102	0.239551	−	0.970884i	$-0.423000\pi$
$179$	6.40995	0.479102	0.239551	+	0.970884i	$0.423000\pi$
$180$	0	0
$181$	6.25690	0.465072	0.232536	−	0.972588i	$-0.425298\pi$
$181$	6.25690	0.465072	0.232536	+	0.972588i	$0.425298\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.58778	0.116736
$186$	0	0
$187$	−0.0620225	−0.00453553
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.8680	0.858738	0.429369	−	0.903129i	$-0.358736\pi$
$191$	11.8680	0.858738	0.429369	+	0.903129i	$0.358736\pi$
$192$	0	0
$193$	7.71261	0.555166	0.277583	−	0.960702i	$-0.410467\pi$
$193$	7.71261	0.555166	0.277583	+	0.960702i	$0.410467\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.2830	−0.803881	−0.401941	−	0.915666i	$-0.631664\pi$
$197$	−11.2830	−0.803881	−0.401941	+	0.915666i	$0.631664\pi$
$198$	0	0
$199$	−4.78352	−0.339095	−0.169547	−	0.985522i	$-0.554231\pi$
$199$	−4.78352	−0.339095	−0.169547	+	0.985522i	$0.554231\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.54291	−0.248664
$204$	0	0
$205$	1.29814	0.0906661
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.42094	−0.167460
$210$	0	0
$211$	12.7879	0.880358	0.440179	−	0.897910i	$-0.354915\pi$
$211$	12.7879	0.880358	0.440179	+	0.897910i	$0.354915\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.26348	−0.0861685
$216$	0	0
$217$	4.41660	0.299818
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.322868	−0.0217185
$222$	0	0
$223$	−22.5770	−1.51187	−0.755933	−	0.654649i	$-0.772817\pi$
$223$	−22.5770	−1.51187	−0.755933	+	0.654649i	$0.772817\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.7505	−1.57637	−0.788186	−	0.615437i	$-0.788980\pi$
$227$	−23.7505	−1.57637	−0.788186	+	0.615437i	$0.788980\pi$
$228$	0	0
$229$	−6.30407	−0.416585	−0.208292	−	0.978067i	$-0.566791\pi$
$229$	−6.30407	−0.416585	−0.208292	+	0.978067i	$0.566791\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.4538	1.07793	0.538963	−	0.842329i	$-0.318816\pi$
$233$	16.4538	1.07793	0.538963	+	0.842329i	$0.318816\pi$
$234$	0	0
$235$	−4.10671	−0.267892
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.59054	0.102884	0.0514418	−	0.998676i	$-0.483618\pi$
$239$	1.59054	0.102884	0.0514418	+	0.998676i	$0.483618\pi$
$240$	0	0
$241$	3.38944	0.218333	0.109167	−	0.994023i	$-0.465182\pi$
$241$	3.38944	0.218333	0.109167	+	0.994023i	$0.465182\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00847	0.0644291
$246$	0	0
$247$	−12.6026	−0.801886
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−3.74107	−0.235199
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.8858	−1.67709	−0.838546	−	0.544831i	$-0.816594\pi$
$257$	−26.8858	−1.67709	−0.838546	+	0.544831i	$0.816594\pi$
$258$	0	0
$259$	7.63937	0.474687
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.39336	−0.270906	−0.135453	−	0.990784i	$-0.543249\pi$
$263$	−4.39336	−0.270906	−0.135453	+	0.990784i	$0.543249\pi$
$264$	0	0
$265$	−1.85887	−0.114190
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−22.6845	−1.38310	−0.691549	−	0.722329i	$-0.743072\pi$
$269$	−22.6845	−1.38310	−0.691549	+	0.722329i	$0.743072\pi$
$270$	0	0
$271$	−15.0589	−0.914763	−0.457382	−	0.889271i	$-0.651213\pi$
$271$	−15.0589	−0.914763	−0.457382	+	0.889271i	$0.651213\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.55452	−0.334950
$276$	0	0
$277$	−10.7918	−0.648419	−0.324210	−	0.945985i	$-0.605098\pi$
$277$	−10.7918	−0.648419	−0.324210	+	0.945985i	$0.605098\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.9382	1.30872	0.654362	−	0.756182i	$-0.272937\pi$
$281$	21.9382	1.30872	0.654362	+	0.756182i	$0.272937\pi$
$282$	0	0
$283$	−12.1997	−0.725198	−0.362599	−	0.931945i	$-0.618111\pi$
$283$	−12.1997	−0.725198	−0.362599	+	0.931945i	$0.618111\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.24582	0.368679
$288$	0	0
$289$	−16.9971	−0.999831
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−29.4675	−1.72151	−0.860756	−	0.509017i	$-0.830009\pi$
$293$	−29.4675	−1.72151	−0.860756	+	0.509017i	$0.830009\pi$
$294$	0	0
$295$	−2.16594	−0.126106
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−19.4747	−1.12625
$300$	0	0
$301$	−6.07904	−0.350390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.64492	0.380487
$306$	0	0
$307$	−21.3436	−1.21814	−0.609072	−	0.793115i	$-0.708458\pi$
$307$	−21.3436	−1.21814	−0.609072	+	0.793115i	$0.708458\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.9771	1.75655	0.878275	−	0.478155i	$-0.158694\pi$
$311$	30.9771	1.75655	0.878275	+	0.478155i	$0.158694\pi$
$312$	0	0
$313$	−16.2372	−0.917783	−0.458892	−	0.888492i	$-0.651753\pi$
$313$	−16.2372	−0.917783	−0.458892	+	0.888492i	$0.651753\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.5305	1.54627	0.773133	−	0.634244i	$-0.218688\pi$
$317$	27.5305	1.54627	0.773133	+	0.634244i	$0.218688\pi$
$318$	0	0
$319$	1.87759	0.105125
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.111826	0.00622216
$324$	0	0
$325$	−28.9150	−1.60391
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−19.7588	−1.08934
$330$	0	0
$331$	−31.9237	−1.75469	−0.877344	−	0.479862i	$-0.840687\pi$
$331$	−31.9237	−1.75469	−0.877344	+	0.479862i	$0.840687\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.26653	0.123834
$336$	0	0
$337$	−18.7610	−1.02198	−0.510990	−	0.859587i	$-0.670721\pi$
$337$	−18.7610	−1.02198	−0.510990	+	0.859587i	$0.670721\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.34061	−0.126751
$342$	0	0
$343$	20.1578	1.08842
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.43093	0.237865	0.118932	−	0.992902i	$-0.462053\pi$
$347$	4.43093	0.237865	0.118932	+	0.992902i	$0.462053\pi$
$348$	0	0
$349$	−13.3203	−0.713018	−0.356509	−	0.934292i	$-0.616033\pi$
$349$	−13.3203	−0.713018	−0.356509	+	0.934292i	$0.616033\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.3691	−1.13736	−0.568681	−	0.822558i	$-0.692546\pi$
$353$	−21.3691	−1.13736	−0.568681	+	0.822558i	$0.692546\pi$
$354$	0	0
$355$	−0.927334	−0.0492178
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.72114	0.354728	0.177364	−	0.984145i	$-0.443243\pi$
$359$	6.72114	0.354728	0.177364	+	0.984145i	$0.443243\pi$
$360$	0	0
$361$	−14.6351	−0.770267
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.0426014	0.00222986
$366$	0	0
$367$	4.60106	0.240174	0.120087	−	0.992763i	$-0.461683\pi$
$367$	4.60106	0.240174	0.120087	+	0.992763i	$0.461683\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.94369	−0.464333
$372$	0	0
$373$	−25.0550	−1.29730	−0.648650	−	0.761087i	$-0.724666\pi$
$373$	−25.0550	−1.29730	−0.648650	+	0.761087i	$0.724666\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.77412	0.503393
$378$	0	0
$379$	−10.3050	−0.529333	−0.264666	−	0.964340i	$-0.585262\pi$
$379$	−10.3050	−0.529333	−0.264666	+	0.964340i	$0.585262\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.92110	−0.506944	−0.253472	−	0.967343i	$-0.581573\pi$
$383$	−9.92110	−0.506944	−0.253472	+	0.967343i	$0.581573\pi$
$384$	0	0
$385$	1.15143	0.0586824
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.3777	1.69232	0.846158	−	0.532932i	$-0.178910\pi$
$389$	33.3777	1.69232	0.846158	+	0.532932i	$0.178910\pi$
$390$	0	0
$391$	0.172804	0.00873905
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.874915	0.0440218
$396$	0	0
$397$	−2.43725	−0.122322	−0.0611611	−	0.998128i	$-0.519480\pi$
$397$	−2.43725	−0.122322	−0.0611611	+	0.998128i	$0.519480\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.0399	−0.551306	−0.275653	−	0.961257i	$-0.588894\pi$
$401$	−11.0399	−0.551306	−0.275653	+	0.961257i	$0.588894\pi$
$402$	0	0
$403$	−12.1844	−0.606950
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.04854	−0.200679
$408$	0	0
$409$	−7.10281	−0.351211	−0.175606	−	0.984461i	$-0.556188\pi$
$409$	−7.10281	−0.351211	−0.175606	+	0.984461i	$0.556188\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.4211	−0.512789
$414$	0	0
$415$	3.62189	0.177792
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.4900	−1.09871	−0.549354	−	0.835590i	$-0.685126\pi$
$419$	−22.4900	−1.09871	−0.549354	+	0.835590i	$0.685126\pi$
$420$	0	0
$421$	−23.3162	−1.13636	−0.568180	−	0.822904i	$-0.692352\pi$
$421$	−23.3162	−1.13636	−0.568180	+	0.822904i	$0.692352\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.256569	0.0124454
$426$	0	0
$427$	31.9711	1.54719
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.63794	−0.271570	−0.135785	−	0.990738i	$-0.543356\pi$
$431$	−5.63794	−0.271570	−0.135785	+	0.990738i	$0.543356\pi$
$432$	0	0
$433$	26.5522	1.27602	0.638009	−	0.770029i	$-0.279758\pi$
$433$	26.5522	1.27602	0.638009	+	0.770029i	$0.279758\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.74510	0.322662
$438$	0	0
$439$	37.8832	1.80807	0.904034	−	0.427462i	$-0.140592\pi$
$439$	37.8832	1.80807	0.904034	+	0.427462i	$0.140592\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.3614	−1.77509	−0.887547	−	0.460717i	$-0.847592\pi$
$443$	−37.3614	−1.77509	−0.887547	+	0.460717i	$0.847592\pi$
$444$	0	0
$445$	−7.61036	−0.360766
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.53079	−0.166628	−0.0833141	−	0.996523i	$-0.526550\pi$
$449$	−3.53079	−0.166628	−0.0833141	+	0.996523i	$0.526550\pi$
$450$	0	0
$451$	−3.31002	−0.155863
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.99397	0.281002
$456$	0	0
$457$	19.8957	0.930681	0.465341	−	0.885132i	$-0.345932\pi$
$457$	19.8957	0.930681	0.465341	+	0.885132i	$0.345932\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.9530	−1.20875	−0.604375	−	0.796700i	$-0.706577\pi$
$461$	−25.9530	−1.20875	−0.604375	+	0.796700i	$0.706577\pi$
$462$	0	0
$463$	−23.5757	−1.09566	−0.547828	−	0.836591i	$-0.684545\pi$
$463$	−23.5757	−1.09566	−0.547828	+	0.836591i	$0.684545\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.93839	−0.182247	−0.0911234	−	0.995840i	$-0.529046\pi$
$467$	−3.93839	−0.182247	−0.0911234	+	0.995840i	$0.529046\pi$
$468$	0	0
$469$	10.9051	0.503550
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.22163	0.148131
$474$	0	0
$475$	10.0147	0.459507
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17.7826	−0.812506	−0.406253	−	0.913761i	$-0.633165\pi$
$479$	−17.7826	−0.812506	−0.406253	+	0.913761i	$0.633165\pi$
$480$	0	0
$481$	−21.0754	−0.960954
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.36472	−0.0619689
$486$	0	0
$487$	38.4530	1.74247	0.871237	−	0.490863i	$-0.163319\pi$
$487$	38.4530	1.74247	0.871237	+	0.490863i	$0.163319\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.93938	−0.222911	−0.111456	−	0.993769i	$-0.535551\pi$
$491$	−4.93938	−0.222911	−0.111456	+	0.993769i	$0.535551\pi$
$492$	0	0
$493$	−0.0867279	−0.00390603
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.46173	−0.200136
$498$	0	0
$499$	−9.78960	−0.438243	−0.219121	−	0.975698i	$-0.570319\pi$
$499$	−9.78960	−0.438243	−0.219121	+	0.975698i	$0.570319\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.9870	−0.757412	−0.378706	−	0.925517i	$-0.623631\pi$
$503$	−16.9870	−0.757412	−0.378706	+	0.925517i	$0.623631\pi$
$504$	0	0
$505$	−1.61628	−0.0719235
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.2558	−0.897822	−0.448911	−	0.893577i	$-0.648188\pi$
$509$	−20.2558	−0.897822	−0.448911	+	0.893577i	$0.648188\pi$
$510$	0	0
$511$	0.204970	0.00906735
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.41138	0.0621928
$516$	0	0
$517$	10.4713	0.460529
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.9138	0.653384	0.326692	−	0.945131i	$-0.394066\pi$
$521$	14.9138	0.653384	0.326692	+	0.945131i	$0.394066\pi$
$522$	0	0
$523$	−30.4438	−1.33121	−0.665607	−	0.746302i	$-0.731827\pi$
$523$	−30.4438	−1.33121	−0.665607	+	0.746302i	$0.731827\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.108115	0.00470957
$528$	0	0
$529$	−12.5768	−0.546820
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17.2308	−0.746351
$534$	0	0
$535$	4.91309	0.212412
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.57142	−0.110759
$540$	0	0
$541$	2.65322	0.114071	0.0570353	−	0.998372i	$-0.481835\pi$
$541$	2.65322	0.114071	0.0570353	+	0.998372i	$0.481835\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.42598	−0.146753
$546$	0	0
$547$	18.8391	0.805500	0.402750	−	0.915310i	$-0.368054\pi$
$547$	18.8391	0.805500	0.402750	+	0.915310i	$0.368054\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.38528	−0.144218
$552$	0	0
$553$	4.20952	0.179007
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.30321	0.139962	0.0699808	−	0.997548i	$-0.477706\pi$
$557$	3.30321	0.139962	0.0699808	+	0.997548i	$0.477706\pi$
$558$	0	0
$559$	16.7707	0.709327
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.19333	−0.345308	−0.172654	−	0.984983i	$-0.555234\pi$
$563$	−8.19333	−0.345308	−0.172654	+	0.984983i	$0.555234\pi$
$564$	0	0
$565$	−1.50321	−0.0632405
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.3058	1.10280	0.551398	−	0.834242i	$-0.314094\pi$
$569$	26.3058	1.10280	0.551398	+	0.834242i	$0.314094\pi$
$570$	0	0
$571$	−29.5821	−1.23797	−0.618987	−	0.785401i	$-0.712457\pi$
$571$	−29.5821	−1.23797	−0.618987	+	0.785401i	$0.712457\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.4757	0.645381
$576$	0	0
$577$	22.7949	0.948964	0.474482	−	0.880265i	$-0.342635\pi$
$577$	22.7949	0.948964	0.474482	+	0.880265i	$0.342635\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	17.4262	0.722961
$582$	0	0
$583$	4.73978	0.196302
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.3149	0.879761	0.439881	−	0.898056i	$-0.355021\pi$
$587$	21.3149	0.879761	0.439881	+	0.898056i	$0.355021\pi$
$588$	0	0
$589$	4.22009	0.173886
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.84036	0.198770	0.0993850	−	0.995049i	$-0.468312\pi$
$593$	4.84036	0.198770	0.0993850	+	0.995049i	$0.468312\pi$
$594$	0	0
$595$	−0.0531858	−0.00218040
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−43.7104	−1.78596	−0.892978	−	0.450100i	$-0.851388\pi$
$599$	−43.7104	−1.78596	−0.892978	+	0.450100i	$0.851388\pi$
$600$	0	0
$601$	−16.0174	−0.653365	−0.326682	−	0.945134i	$-0.605931\pi$
$601$	−16.0174	−0.653365	−0.326682	+	0.945134i	$0.605931\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.38876	0.178428
$606$	0	0
$607$	−3.55146	−0.144149	−0.0720747	−	0.997399i	$-0.522962\pi$
$607$	−3.55146	−0.144149	−0.0720747	+	0.997399i	$0.522962\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	54.5103	2.20525
$612$	0	0
$613$	20.6687	0.834801	0.417400	−	0.908723i	$-0.362941\pi$
$613$	20.6687	0.834801	0.417400	+	0.908723i	$0.362941\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.0094	1.36917	0.684583	−	0.728935i	$-0.259984\pi$
$617$	34.0094	1.36917	0.684583	+	0.728935i	$0.259984\pi$
$618$	0	0
$619$	1.90390	0.0765243	0.0382621	−	0.999268i	$-0.487818\pi$
$619$	1.90390	0.0765243	0.0382621	+	0.999268i	$0.487818\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−36.6161	−1.46699
$624$	0	0
$625$	21.9448	0.877790
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.187006	0.00745642
$630$	0	0
$631$	11.4055	0.454048	0.227024	−	0.973889i	$-0.427100\pi$
$631$	11.4055	0.454048	0.227024	+	0.973889i	$0.427100\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.99652	0.237965
$636$	0	0
$637$	−13.3860	−0.530371
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.81401	−0.0716491	−0.0358245	−	0.999358i	$-0.511406\pi$
$641$	−1.81401	−0.0716491	−0.0358245	+	0.999358i	$0.511406\pi$
$642$	0	0
$643$	−10.4605	−0.412523	−0.206261	−	0.978497i	$-0.566130\pi$
$643$	−10.4605	−0.412523	−0.206261	+	0.978497i	$0.566130\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.7209	0.893249	0.446624	−	0.894722i	$-0.352626\pi$
$647$	22.7209	0.893249	0.446624	+	0.894722i	$0.352626\pi$
$648$	0	0
$649$	5.52274	0.216787
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.8539	−0.503013	−0.251507	−	0.967856i	$-0.580926\pi$
$653$	−12.8539	−0.503013	−0.251507	+	0.967856i	$0.580926\pi$
$654$	0	0
$655$	1.90091	0.0742746
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−33.7873	−1.31617	−0.658083	−	0.752946i	$-0.728632\pi$
$659$	−33.7873	−1.31617	−0.658083	+	0.752946i	$0.728632\pi$
$660$	0	0
$661$	27.6933	1.07714	0.538572	−	0.842580i	$-0.318964\pi$
$661$	27.6933	1.07714	0.538572	+	0.842580i	$0.318964\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.07602	−0.0805046
$666$	0	0
$667$	−5.23124	−0.202555
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16.9433	−0.654089
$672$	0	0
$673$	17.1280	0.660237	0.330118	−	0.943940i	$-0.392911\pi$
$673$	17.1280	0.660237	0.330118	+	0.943940i	$0.392911\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.9570	−0.421111	−0.210555	−	0.977582i	$-0.567527\pi$
$677$	−10.9570	−0.421111	−0.210555	+	0.977582i	$0.567527\pi$
$678$	0	0
$679$	−6.56617	−0.251986
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.6594	1.32620	0.663102	−	0.748529i	$-0.269240\pi$
$683$	34.6594	1.32620	0.663102	+	0.748529i	$0.269240\pi$
$684$	0	0
$685$	−5.58752	−0.213488
$686$	0	0
$687$	0	0
$688$	0	0
$689$	24.6737	0.939993
$690$	0	0
$691$	9.23393	0.351275	0.175638	−	0.984455i	$-0.443801\pi$
$691$	9.23393	0.351275	0.175638	+	0.984455i	$0.443801\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.95230	−0.111987
$696$	0	0
$697$	0.152893	0.00579123
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.2605	−0.500842	−0.250421	−	0.968137i	$-0.580569\pi$
$701$	−13.2605	−0.500842	−0.250421	+	0.968137i	$0.580569\pi$
$702$	0	0
$703$	7.29947	0.275305
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.77649	−0.292465
$708$	0	0
$709$	46.0334	1.72882	0.864410	−	0.502787i	$-0.167692\pi$
$709$	46.0334	1.72882	0.864410	+	0.502787i	$0.167692\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.52128	0.244224
$714$	0	0
$715$	−3.17655	−0.118796
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.81518	0.0676949	0.0338474	−	0.999427i	$-0.489224\pi$
$719$	1.81518	0.0676949	0.0338474	+	0.999427i	$0.489224\pi$
$720$	0	0
$721$	6.79064	0.252897
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.76705	−0.288461
$726$	0	0
$727$	−34.6521	−1.28518	−0.642588	−	0.766212i	$-0.722139\pi$
$727$	−34.6521	−1.28518	−0.642588	+	0.766212i	$0.722139\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.148810	−0.00550395
$732$	0	0
$733$	26.5655	0.981220	0.490610	−	0.871379i	$-0.336774\pi$
$733$	26.5655	0.981220	0.490610	+	0.871379i	$0.336774\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.77923	−0.212881
$738$	0	0
$739$	14.3106	0.526424	0.263212	−	0.964738i	$-0.415218\pi$
$739$	14.3106	0.526424	0.263212	+	0.964738i	$0.415218\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.40007	0.344855	0.172428	−	0.985022i	$-0.444839\pi$
$743$	9.40007	0.344855	0.172428	+	0.985022i	$0.444839\pi$
$744$	0	0
$745$	5.00670	0.183431
$746$	0	0
$747$	0	0
$748$	0	0
$749$	23.6386	0.863736
$750$	0	0
$751$	21.4083	0.781198	0.390599	−	0.920561i	$-0.372268\pi$
$751$	21.4083	0.781198	0.390599	+	0.920561i	$0.372268\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9.73980	0.354468
$756$	0	0
$757$	−19.4330	−0.706304	−0.353152	−	0.935566i	$-0.614890\pi$
$757$	−19.4330	−0.706304	−0.353152	+	0.935566i	$0.614890\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−36.1110	−1.30902	−0.654511	−	0.756053i	$-0.727125\pi$
$761$	−36.1110	−1.30902	−0.654511	+	0.756053i	$0.727125\pi$
$762$	0	0
$763$	−16.4836	−0.596747
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.7496	1.03809
$768$	0	0
$769$	32.4205	1.16911	0.584556	−	0.811353i	$-0.301269\pi$
$769$	32.4205	1.16911	0.584556	+	0.811353i	$0.301269\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.6367	0.922089	0.461045	−	0.887377i	$-0.347475\pi$
$773$	25.6367	0.922089	0.461045	+	0.887377i	$0.347475\pi$
$774$	0	0
$775$	9.68241	0.347803
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.96792	0.213823
$780$	0	0
$781$	2.36453	0.0846095
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.31826	−0.189817
$786$	0	0
$787$	2.24434	0.0800022	0.0400011	−	0.999200i	$-0.487264\pi$
$787$	2.24434	0.0800022	0.0400011	+	0.999200i	$0.487264\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.23247	−0.257157
$792$	0	0
$793$	−88.2012	−3.13212
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.7390	0.734613	0.367307	−	0.930100i	$-0.380280\pi$
$797$	20.7390	0.734613	0.367307	+	0.930100i	$0.380280\pi$
$798$	0	0
$799$	−0.483682	−0.0171114
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.108626	−0.00383331
$804$	0	0
$805$	−3.20806	−0.113069
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.7671	0.835606	0.417803	−	0.908538i	$-0.362800\pi$
$809$	23.7671	0.835606	0.417803	+	0.908538i	$0.362800\pi$
$810$	0	0
$811$	−6.35066	−0.223002	−0.111501	−	0.993764i	$-0.535566\pi$
$811$	−6.35066	−0.223002	−0.111501	+	0.993764i	$0.535566\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.51047	0.0529093
$816$	0	0
$817$	−5.80857	−0.203216
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.6403	−0.371350	−0.185675	−	0.982611i	$-0.559447\pi$
$821$	−10.6403	−0.371350	−0.185675	+	0.982611i	$0.559447\pi$
$822$	0	0
$823$	31.4709	1.09701	0.548503	−	0.836149i	$-0.315198\pi$
$823$	31.4709	1.09701	0.548503	+	0.836149i	$0.315198\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.2833	−0.392360	−0.196180	−	0.980568i	$-0.562854\pi$
$827$	−11.2833	−0.392360	−0.196180	+	0.980568i	$0.562854\pi$
$828$	0	0
$829$	2.17919	0.0756864	0.0378432	−	0.999284i	$-0.487951\pi$
$829$	2.17919	0.0756864	0.0378432	+	0.999284i	$0.487951\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.118776	0.00411536
$834$	0	0
$835$	10.6761	0.369463
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.1253	−1.41980	−0.709901	−	0.704301i	$-0.751260\pi$
$839$	−41.1253	−1.41980	−0.709901	+	0.704301i	$0.751260\pi$
$840$	0	0
$841$	−26.3745	−0.909466
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.6282	−0.365621
$846$	0	0
$847$	21.1159	0.725549
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.2798	0.386667
$852$	0	0
$853$	20.1056	0.688403	0.344202	−	0.938896i	$-0.388150\pi$
$853$	20.1056	0.688403	0.344202	+	0.938896i	$0.388150\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.75122	0.162299	0.0811493	−	0.996702i	$-0.474141\pi$
$857$	4.75122	0.162299	0.0811493	+	0.996702i	$0.474141\pi$
$858$	0	0
$859$	27.2044	0.928203	0.464102	−	0.885782i	$-0.346377\pi$
$859$	27.2044	0.928203	0.464102	+	0.885782i	$0.346377\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−43.0961	−1.46701	−0.733505	−	0.679684i	$-0.762117\pi$
$863$	−43.0961	−1.46701	−0.733505	+	0.679684i	$0.762117\pi$
$864$	0	0
$865$	−2.03879	−0.0693211
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.23087	−0.0756771
$870$	0	0
$871$	−30.0848	−1.01938
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.73149	−0.328984
$876$	0	0
$877$	−37.2416	−1.25756	−0.628780	−	0.777583i	$-0.716445\pi$
$877$	−37.2416	−1.25756	−0.628780	+	0.777583i	$0.716445\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.0353	−0.776079	−0.388039	−	0.921643i	$-0.626848\pi$
$881$	−23.0353	−0.776079	−0.388039	+	0.921643i	$0.626848\pi$
$882$	0	0
$883$	13.7702	0.463403	0.231701	−	0.972787i	$-0.425571\pi$
$883$	13.7702	0.463403	0.231701	+	0.972787i	$0.425571\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.3219	0.883804	0.441902	−	0.897063i	$-0.354304\pi$
$887$	26.3219	0.883804	0.441902	+	0.897063i	$0.354304\pi$
$888$	0	0
$889$	28.8514	0.967644
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−18.8797	−0.631785
$894$	0	0
$895$	−2.91301	−0.0973712
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.27295	−0.109159
$900$	0	0
$901$	−0.218935	−0.00729378
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.84346	−0.0945198
$906$	0	0
$907$	−32.6153	−1.08297	−0.541486	−	0.840710i	$-0.682138\pi$
$907$	−32.6153	−1.08297	−0.541486	+	0.840710i	$0.682138\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.5936	−1.57685	−0.788423	−	0.615133i	$-0.789102\pi$
$911$	−47.5936	−1.57685	−0.788423	+	0.615133i	$0.789102\pi$
$912$	0	0
$913$	−9.23515	−0.305639
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.14593	0.302025
$918$	0	0
$919$	−16.4448	−0.542463	−0.271232	−	0.962514i	$-0.587431\pi$
$919$	−16.4448	−0.542463	−0.271232	+	0.962514i	$0.587431\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.3089	0.405154
$924$	0	0
$925$	16.7476	0.550658
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−58.6216	−1.92331	−0.961657	−	0.274256i	$-0.911569\pi$
$929$	−58.6216	−1.92331	−0.961657	+	0.274256i	$0.911569\pi$
$930$	0	0
$931$	4.63624	0.151947
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.0281862	0.000921788 0
$936$	0	0
$937$	12.5654	0.410494	0.205247	−	0.978710i	$-0.434200\pi$
$937$	12.5654	0.410494	0.205247	+	0.978710i	$0.434200\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.27635	0.139405	0.0697025	−	0.997568i	$-0.477795\pi$
$941$	4.27635	0.139405	0.0697025	+	0.997568i	$0.477795\pi$
$942$	0	0
$943$	9.22219	0.300316
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.9545	−0.940895	−0.470447	−	0.882428i	$-0.655907\pi$
$947$	−28.9545	−0.940895	−0.470447	+	0.882428i	$0.655907\pi$
$948$	0	0
$949$	−0.565469	−0.0183559
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.0253	0.357144	0.178572	−	0.983927i	$-0.442852\pi$
$953$	11.0253	0.357144	0.178572	+	0.983927i	$0.442852\pi$
$954$	0	0
$955$	−5.39343	−0.174527
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−26.8835	−0.868114
$960$	0	0
$961$	−26.9199	−0.868385
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.50501	−0.112830
$966$	0	0
$967$	−29.3057	−0.942409	−0.471205	−	0.882024i	$-0.656181\pi$
$967$	−29.3057	−0.942409	−0.471205	+	0.882024i	$0.656181\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.75260	−0.0562437	−0.0281219	−	0.999605i	$-0.508953\pi$
$971$	−1.75260	−0.0562437	−0.0281219	+	0.999605i	$0.508953\pi$
$972$	0	0
$973$	−14.2046	−0.455377
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.9928	−0.639626	−0.319813	−	0.947481i	$-0.603620\pi$
$977$	−19.9928	−0.639626	−0.319813	+	0.947481i	$0.603620\pi$
$978$	0	0
$979$	19.4050	0.620186
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33.6346	1.07278	0.536389	−	0.843971i	$-0.319788\pi$
$983$	33.6346	1.07278	0.536389	+	0.843971i	$0.319788\pi$
$984$	0	0
$985$	5.12758	0.163378
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.97593	−0.285418
$990$	0	0
$991$	54.2527	1.72339	0.861696	−	0.507424i	$-0.169402\pi$
$991$	54.2527	1.72339	0.861696	+	0.507424i	$0.169402\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.17388	0.0689166
$996$	0	0
$997$	15.3787	0.487048	0.243524	−	0.969895i	$-0.421697\pi$
$997$	15.3787	0.487048	0.243524	+	0.969895i	$0.421697\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9036.2.a.i.1.3		7
3.2	odd	2		1004.2.a.a.1.1	✓	7
12.11	even	2		4016.2.a.g.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.a.1.1	✓	7	3.2	odd	2
4016.2.a.g.1.7		7	12.11	even	2
9036.2.a.i.1.3		7	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 9036 = 2^{2} \cdot 3^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9036.a (trivial)

\(f(q)\)	\(=\)	\(q-0.454452 q^{5} -2.18653 q^{7} +O(q^{10})\)	\(q-0.454452 q^{5} -2.18653 q^{7} +1.15877 q^{11} +6.03215 q^{13} -0.0535246 q^{17} -2.08924 q^{19} -3.22849 q^{23} -4.79347 q^{25} +1.62034 q^{29} -2.01992 q^{31} +0.993671 q^{35} -3.49384 q^{37} -2.85650 q^{41} +2.78023 q^{43} +9.03663 q^{47} -2.21910 q^{49} +4.09037 q^{53} -0.526603 q^{55} +4.76605 q^{59} -14.6219 q^{61} -2.74132 q^{65} -4.98740 q^{67} +2.04056 q^{71} -0.0937424 q^{73} -2.53367 q^{77} -1.92521 q^{79} -7.96981 q^{83} +0.0243243 q^{85} +16.7462 q^{89} -13.1895 q^{91} +0.949460 q^{95} +3.00301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q + 2 * q^5 - 6 * q^7	\( 7 q + 2 q^{5} - 6 q^{7} + 5 q^{11} - q^{13} + 8 q^{17} - 15 q^{19} + 5 q^{23} - 9 q^{25} - 21 q^{31} + 7 q^{35} - q^{37} + 10 q^{41} - 23 q^{43} + 10 q^{47} - 13 q^{49} + q^{53} - 23 q^{55} + 4 q^{59} + 3 q^{61} - 4 q^{65} - 28 q^{67} + 18 q^{71} - 7 q^{73} - 6 q^{77} - 30 q^{79} - 13 q^{83} + q^{85} - 18 q^{91} - 2 q^{97}+O(q^{100}) \) 7 * q + 2 * q^5 - 6 * q^7 + 5 * q^11 - q^13 + 8 * q^17 - 15 * q^19 + 5 * q^23 - 9 * q^25 - 21 * q^31 + 7 * q^35 - q^37 + 10 * q^41 - 23 * q^43 + 10 * q^47 - 13 * q^49 + q^53 - 23 * q^55 + 4 * q^59 + 3 * q^61 - 4 * q^65 - 28 * q^67 + 18 * q^71 - 7 * q^73 - 6 * q^77 - 30 * q^79 - 13 * q^83 + q^85 - 18 * q^91 - 2 * q^97

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