LMFDB - Embedded newform 2156.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(1,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, 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("2156.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.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:	$17.2157466758$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.301088.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 13x^{2} + 18 $ x^4 - 13*x^2 + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.38000$ of defining polynomial
Character	$\chi$	$=$	2156.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.38000 q^{3} -1.60486 q^{5} +8.42443 q^{9} +1.00000 q^{11} -4.98486 q^{13} +5.42443 q^{15} -1.77515 q^{17} +6.76001 q^{19} -1.42443 q^{23} -2.42443 q^{25} -18.3346 q^{27} -6.00000 q^{29} -3.38000 q^{31} -3.38000 q^{33} -5.42443 q^{37} +16.8489 q^{39} +8.19458 q^{41} -8.84886 q^{43} -13.5200 q^{45} +1.77515 q^{47} +6.00000 q^{51} -10.8489 q^{53} -1.60486 q^{55} -22.8489 q^{57} +0.170287 q^{59} +11.7449 q^{61} +8.00000 q^{65} +13.4244 q^{67} +4.81458 q^{69} +5.42443 q^{71} -5.32544 q^{73} +8.19458 q^{75} +6.00000 q^{79} +36.6977 q^{81} -3.55029 q^{83} +2.84886 q^{85} +20.2800 q^{87} -11.9152 q^{89} +11.4244 q^{93} -10.8489 q^{95} +8.36487 q^{97} +8.42443 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{9} + 4 q^{11} + 2 q^{15} + 14 q^{23} + 10 q^{25} - 24 q^{29} - 2 q^{37} + 28 q^{39} + 4 q^{43} + 24 q^{51} - 4 q^{53} - 52 q^{57} + 32 q^{65} + 34 q^{67} + 2 q^{71} + 24 q^{79} + 68 q^{81} - 28 q^{85}+ \cdots + 14 q^{99}+O(q^{100}) $ 4 * q + 14 * q^9 + 4 * q^11 + 2 * q^15 + 14 * q^23 + 10 * q^25 - 24 * q^29 - 2 * q^37 + 28 * q^39 + 4 * q^43 + 24 * q^51 - 4 * q^53 - 52 * q^57 + 32 * q^65 + 34 * q^67 + 2 * q^71 + 24 * q^79 + 68 * q^81 - 28 * q^85 + 26 * q^93 - 4 * q^95 + 14 * q^99

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
$3$	−3.38000	−1.95145	−0.975723	−	0.219007i	$-0.929718\pi$
$3$	−3.38000	−1.95145	−0.975723	+	0.219007i	$0.929718\pi$
$4$	0	0
$5$	−1.60486	−0.717715	−0.358857	−	0.933392i	$-0.616834\pi$
$5$	−1.60486	−0.717715	−0.358857	+	0.933392i	$0.616834\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.42443	2.80814
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.98486	−1.38255	−0.691276	−	0.722591i	$-0.742951\pi$
$13$	−4.98486	−1.38255	−0.691276	+	0.722591i	$0.742951\pi$
$14$	0	0
$15$	5.42443	1.40058
$16$	0	0
$17$	−1.77515	−0.430536	−0.215268	−	0.976555i	$-0.569063\pi$
$17$	−1.77515	−0.430536	−0.215268	+	0.976555i	$0.569063\pi$
$18$	0	0
$19$	6.76001	1.55085	0.775426	−	0.631438i	$-0.217535\pi$
$19$	6.76001	1.55085	0.775426	+	0.631438i	$0.217535\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.42443	−0.297014	−0.148507	−	0.988911i	$-0.547447\pi$
$23$	−1.42443	−0.297014	−0.148507	+	0.988911i	$0.547447\pi$
$24$	0	0
$25$	−2.42443	−0.484886
$26$	0	0
$27$	−18.3346	−3.52849
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−3.38000	−0.607067	−0.303533	−	0.952821i	$-0.598166\pi$
$31$	−3.38000	−0.607067	−0.303533	+	0.952821i	$0.598166\pi$
$32$	0	0
$33$	−3.38000	−0.588383
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.42443	−0.891771	−0.445885	−	0.895090i	$-0.647111\pi$
$37$	−5.42443	−0.891771	−0.445885	+	0.895090i	$0.647111\pi$
$38$	0	0
$39$	16.8489	2.69798
$40$	0	0
$41$	8.19458	1.27978	0.639889	−	0.768467i	$-0.278980\pi$
$41$	8.19458	1.27978	0.639889	+	0.768467i	$0.278980\pi$
$42$	0	0
$43$	−8.84886	−1.34944	−0.674719	−	0.738075i	$-0.735735\pi$
$43$	−8.84886	−1.34944	−0.674719	+	0.738075i	$0.735735\pi$
$44$	0	0
$45$	−13.5200	−2.01545
$46$	0	0
$47$	1.77515	0.258932	0.129466	−	0.991584i	$-0.458674\pi$
$47$	1.77515	0.258932	0.129466	+	0.991584i	$0.458674\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	−10.8489	−1.49021	−0.745103	−	0.666950i	$-0.767600\pi$
$53$	−10.8489	−1.49021	−0.745103	+	0.666950i	$0.767600\pi$
$54$	0	0
$55$	−1.60486	−0.216399
$56$	0	0
$57$	−22.8489	−3.02641
$58$	0	0
$59$	0.170287	0.0221695	0.0110847	−	0.999939i	$-0.496472\pi$
$59$	0.170287	0.0221695	0.0110847	+	0.999939i	$0.496472\pi$
$60$	0	0
$61$	11.7449	1.50378	0.751888	−	0.659290i	$-0.229143\pi$
$61$	11.7449	1.50378	0.751888	+	0.659290i	$0.229143\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	13.4244	1.64006	0.820028	−	0.572324i	$-0.193958\pi$
$67$	13.4244	1.64006	0.820028	+	0.572324i	$0.193958\pi$
$68$	0	0
$69$	4.81458	0.579607
$70$	0	0
$71$	5.42443	0.643761	0.321881	−	0.946780i	$-0.395685\pi$
$71$	5.42443	0.643761	0.321881	+	0.946780i	$0.395685\pi$
$72$	0	0
$73$	−5.32544	−0.623295	−0.311648	−	0.950198i	$-0.600881\pi$
$73$	−5.32544	−0.623295	−0.311648	+	0.950198i	$0.600881\pi$
$74$	0	0
$75$	8.19458	0.946229
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	36.6977	4.07752
$82$	0	0
$83$	−3.55029	−0.389695	−0.194848	−	0.980834i	$-0.562421\pi$
$83$	−3.55029	−0.389695	−0.194848	+	0.980834i	$0.562421\pi$
$84$	0	0
$85$	2.84886	0.309002
$86$	0	0
$87$	20.2800	2.17425
$88$	0	0
$89$	−11.9152	−1.26300	−0.631502	−	0.775374i	$-0.717561\pi$
$89$	−11.9152	−1.26300	−0.631502	+	0.775374i	$0.717561\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	11.4244	1.18466
$94$	0	0
$95$	−10.8489	−1.11307
$96$	0	0
$97$	8.36487	0.849324	0.424662	−	0.905352i	$-0.360393\pi$
$97$	8.36487	0.849324	0.424662	+	0.905352i	$0.360393\pi$
$98$	0	0
$99$	8.42443	0.846687
$100$	0	0
$101$	−11.4043	−1.13477	−0.567385	−	0.823453i	$-0.692045\pi$
$101$	−11.4043	−1.13477	−0.567385	+	0.823453i	$0.692045\pi$
$102$	0	0
$103$	−1.77515	−0.174910	−0.0874552	−	0.996168i	$-0.527873\pi$
$103$	−1.77515	−0.174910	−0.0874552	+	0.996168i	$0.527873\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.8489	1.43549	0.717747	−	0.696304i	$-0.245174\pi$
$107$	14.8489	1.43549	0.717747	+	0.696304i	$0.245174\pi$
$108$	0	0
$109$	−12.8489	−1.23070	−0.615349	−	0.788255i	$-0.710985\pi$
$109$	−12.8489	−1.23070	−0.615349	+	0.788255i	$0.710985\pi$
$110$	0	0
$111$	18.3346	1.74024
$112$	0	0
$113$	0.575571	0.0541452	0.0270726	−	0.999633i	$-0.491381\pi$
$113$	0.575571	0.0541452	0.0270726	+	0.999633i	$0.491381\pi$
$114$	0	0
$115$	2.28601	0.213171
$116$	0	0
$117$	−41.9946	−3.88240
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−27.6977	−2.49742
$124$	0	0
$125$	11.9152	1.06572
$126$	0	0
$127$	18.8489	1.67257	0.836283	−	0.548298i	$-0.184724\pi$
$127$	18.8489	1.67257	0.836283	+	0.548298i	$0.184724\pi$
$128$	0	0
$129$	29.9092	2.63336
$130$	0	0
$131$	10.3103	0.900815	0.450408	−	0.892823i	$-0.351279\pi$
$131$	10.3103	0.900815	0.450408	+	0.892823i	$0.351279\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	29.4244	2.53245
$136$	0	0
$137$	17.4244	1.48867	0.744335	−	0.667807i	$-0.232767\pi$
$137$	17.4244	1.48867	0.744335	+	0.667807i	$0.232767\pi$
$138$	0	0
$139$	6.76001	0.573376	0.286688	−	0.958024i	$-0.407446\pi$
$139$	6.76001	0.573376	0.286688	+	0.958024i	$0.407446\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−4.98486	−0.416855
$144$	0	0
$145$	9.62915	0.799658
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.6977	−1.61370	−0.806850	−	0.590757i	$-0.798829\pi$
$149$	−19.6977	−1.61370	−0.806850	+	0.590757i	$0.798829\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	0	0
$153$	−14.9546	−1.20901
$154$	0	0
$155$	5.42443	0.435701
$156$	0	0
$157$	8.02429	0.640408	0.320204	−	0.947349i	$-0.396248\pi$
$157$	8.02429	0.640408	0.320204	+	0.947349i	$0.396248\pi$
$158$	0	0
$159$	36.6692	2.90806
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	5.42443	0.422291
$166$	0	0
$167$	23.8303	1.84405	0.922023	−	0.387136i	$-0.126536\pi$
$167$	23.8303	1.84405	0.922023	+	0.387136i	$0.126536\pi$
$168$	0	0
$169$	11.8489	0.911451
$170$	0	0
$171$	56.9492	4.35502
$172$	0	0
$173$	−7.85401	−0.597129	−0.298565	−	0.954389i	$-0.596508\pi$
$173$	−7.85401	−0.597129	−0.298565	+	0.954389i	$0.596508\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.575571	−0.0432626
$178$	0	0
$179$	4.27329	0.319400	0.159700	−	0.987166i	$-0.448947\pi$
$179$	4.27329	0.319400	0.159700	+	0.987166i	$0.448947\pi$
$180$	0	0
$181$	15.4654	1.14954	0.574769	−	0.818316i	$-0.305092\pi$
$181$	15.4654	1.14954	0.574769	+	0.818316i	$0.305092\pi$
$182$	0	0
$183$	−39.6977	−2.93454
$184$	0	0
$185$	8.70544	0.640037
$186$	0	0
$187$	−1.77515	−0.129812
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.4244	−0.971358	−0.485679	−	0.874137i	$-0.661428\pi$
$191$	−13.4244	−0.971358	−0.485679	+	0.874137i	$0.661428\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	−27.0400	−1.93638
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−15.2952	−1.08425	−0.542123	−	0.840299i	$-0.682379\pi$
$199$	−15.2952	−1.08425	−0.542123	+	0.840299i	$0.682379\pi$
$200$	0	0
$201$	−45.3746	−3.20048
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−13.1511	−0.918516
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	6.76001	0.467600
$210$	0	0
$211$	−6.84886	−0.471495	−0.235747	−	0.971814i	$-0.575754\pi$
$211$	−6.84886	−0.471495	−0.235747	+	0.971814i	$0.575754\pi$
$212$	0	0
$213$	−18.3346	−1.25627
$214$	0	0
$215$	14.2012	0.968511
$216$	0	0
$217$	0	0
$218$	0	0
$219$	18.0000	1.21633
$220$	0	0
$221$	8.84886	0.595239
$222$	0	0
$223$	6.93030	0.464087	0.232043	−	0.972705i	$-0.425459\pi$
$223$	6.93030	0.464087	0.232043	+	0.972705i	$0.425459\pi$
$224$	0	0
$225$	−20.4244	−1.36163
$226$	0	0
$227$	−7.10058	−0.471282	−0.235641	−	0.971840i	$-0.575719\pi$
$227$	−7.10058	−0.471282	−0.235641	+	0.971840i	$0.575719\pi$
$228$	0	0
$229$	−1.26428	−0.0835463	−0.0417731	−	0.999127i	$-0.513301\pi$
$229$	−1.26428	−0.0835463	−0.0417731	+	0.999127i	$0.513301\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.6977	1.29044	0.645220	−	0.763997i	$-0.276766\pi$
$233$	19.6977	1.29044	0.645220	+	0.763997i	$0.276766\pi$
$234$	0	0
$235$	−2.84886	−0.185839
$236$	0	0
$237$	−20.2800	−1.31733
$238$	0	0
$239$	−21.6977	−1.40351	−0.701754	−	0.712419i	$-0.747600\pi$
$239$	−21.6977	−1.40351	−0.701754	+	0.712419i	$0.747600\pi$
$240$	0	0
$241$	−8.53515	−0.549798	−0.274899	−	0.961473i	$-0.588644\pi$
$241$	−8.53515	−0.549798	−0.274899	+	0.961473i	$0.588644\pi$
$242$	0	0
$243$	−69.0347	−4.42858
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−33.6977	−2.14413
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−6.58972	−0.415940	−0.207970	−	0.978135i	$-0.566686\pi$
$251$	−6.58972	−0.415940	−0.207970	+	0.978135i	$0.566686\pi$
$252$	0	0
$253$	−1.42443	−0.0895531
$254$	0	0
$255$	−9.62915	−0.603001
$256$	0	0
$257$	19.9395	1.24379	0.621894	−	0.783101i	$-0.286363\pi$
$257$	19.9395	1.24379	0.621894	+	0.783101i	$0.286363\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−50.5466	−3.12875
$262$	0	0
$263$	−5.15114	−0.317633	−0.158817	−	0.987308i	$-0.550768\pi$
$263$	−5.15114	−0.317633	−0.158817	+	0.987308i	$0.550768\pi$
$264$	0	0
$265$	17.4109	1.06954
$266$	0	0
$267$	40.2733	2.46469
$268$	0	0
$269$	10.6509	0.649395	0.324698	−	0.945818i	$-0.394737\pi$
$269$	10.6509	0.649395	0.324698	+	0.945818i	$0.394737\pi$
$270$	0	0
$271$	−9.96973	−0.605618	−0.302809	−	0.953051i	$-0.597924\pi$
$271$	−9.96973	−0.605618	−0.302809	+	0.953051i	$0.597924\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.42443	−0.146199
$276$	0	0
$277$	24.8489	1.49302	0.746512	−	0.665372i	$-0.231727\pi$
$277$	24.8489	1.49302	0.746512	+	0.665372i	$0.231727\pi$
$278$	0	0
$279$	−28.4746	−1.70473
$280$	0	0
$281$	4.84886	0.289259	0.144629	−	0.989486i	$-0.453801\pi$
$281$	4.84886	0.289259	0.144629	+	0.989486i	$0.453801\pi$
$282$	0	0
$283$	12.8389	0.763192	0.381596	−	0.924329i	$-0.375375\pi$
$283$	12.8389	0.763192	0.381596	+	0.924329i	$0.375375\pi$
$284$	0	0
$285$	36.6692	2.17210
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.8489	−0.814639
$290$	0	0
$291$	−28.2733	−1.65741
$292$	0	0
$293$	18.1643	1.06117	0.530585	−	0.847632i	$-0.321972\pi$
$293$	18.1643	1.06117	0.530585	+	0.847632i	$0.321972\pi$
$294$	0	0
$295$	−0.273287	−0.0159114
$296$	0	0
$297$	−18.3346	−1.06388
$298$	0	0
$299$	7.10058	0.410637
$300$	0	0
$301$	0	0
$302$	0	0
$303$	38.5466	2.21444
$304$	0	0
$305$	−18.8489	−1.07928
$306$	0	0
$307$	−0.681148	−0.0388752	−0.0194376	−	0.999811i	$-0.506188\pi$
$307$	−0.681148	−0.0388752	−0.0194376	+	0.999811i	$0.506188\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	−22.3957	−1.26995	−0.634973	−	0.772534i	$-0.718989\pi$
$311$	−22.3957	−1.26995	−0.634973	+	0.772534i	$0.718989\pi$
$312$	0	0
$313$	18.3346	1.03633	0.518166	−	0.855280i	$-0.326615\pi$
$313$	18.3346	1.03633	0.518166	+	0.855280i	$0.326615\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.42443	−0.416997	−0.208499	−	0.978023i	$-0.566858\pi$
$317$	−7.42443	−0.416997	−0.208499	+	0.978023i	$0.566858\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−50.1892	−2.80129
$322$	0	0
$323$	−12.0000	−0.667698
$324$	0	0
$325$	12.0854	0.670380
$326$	0	0
$327$	43.4292	2.40164
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.4244	0.737873	0.368937	−	0.929455i	$-0.379722\pi$
$331$	13.4244	0.737873	0.368937	+	0.929455i	$0.379722\pi$
$332$	0	0
$333$	−45.6977	−2.50422
$334$	0	0
$335$	−21.5443	−1.17709
$336$	0	0
$337$	15.6977	0.855109	0.427555	−	0.903990i	$-0.359375\pi$
$337$	15.6977	0.855109	0.427555	+	0.903990i	$0.359375\pi$
$338$	0	0
$339$	−1.94543	−0.105661
$340$	0	0
$341$	−3.38000	−0.183037
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.72671	−0.415992
$346$	0	0
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	11.0637	0.592228	0.296114	−	0.955153i	$-0.404309\pi$
$349$	11.0637	0.592228	0.296114	+	0.955153i	$0.404309\pi$
$350$	0	0
$351$	91.3954	4.87833
$352$	0	0
$353$	−35.7455	−1.90254	−0.951270	−	0.308360i	$-0.900220\pi$
$353$	−35.7455	−1.90254	−0.951270	+	0.308360i	$0.900220\pi$
$354$	0	0
$355$	−8.70544	−0.462037
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.84886	0.255913	0.127956	−	0.991780i	$-0.459158\pi$
$359$	4.84886	0.255913	0.127956	+	0.991780i	$0.459158\pi$
$360$	0	0
$361$	26.6977	1.40514
$362$	0	0
$363$	−3.38000	−0.177404
$364$	0	0
$365$	8.54657	0.447348
$366$	0	0
$367$	−23.6600	−1.23504	−0.617522	−	0.786554i	$-0.711863\pi$
$367$	−23.6600	−1.23504	−0.617522	+	0.786554i	$0.711863\pi$
$368$	0	0
$369$	69.0347	3.59380
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8.84886	0.458176	0.229088	−	0.973406i	$-0.426426\pi$
$373$	8.84886	0.458176	0.229088	+	0.973406i	$0.426426\pi$
$374$	0	0
$375$	−40.2733	−2.07970
$376$	0	0
$377$	29.9092	1.54040
$378$	0	0
$379$	26.5756	1.36510	0.682548	−	0.730841i	$-0.260872\pi$
$379$	26.5756	1.36510	0.682548	+	0.730841i	$0.260872\pi$
$380$	0	0
$381$	−63.7092	−3.26392
$382$	0	0
$383$	6.58972	0.336719	0.168360	−	0.985726i	$-0.446153\pi$
$383$	6.58972	0.336719	0.168360	+	0.985726i	$0.446153\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−74.5466	−3.78942
$388$	0	0
$389$	−30.2733	−1.53492	−0.767458	−	0.641099i	$-0.778479\pi$
$389$	−30.2733	−1.53492	−0.767458	+	0.641099i	$0.778479\pi$
$390$	0	0
$391$	2.52857	0.127875
$392$	0	0
$393$	−34.8489	−1.75789
$394$	0	0
$395$	−9.62915	−0.484495
$396$	0	0
$397$	−13.5200	−0.678550	−0.339275	−	0.940687i	$-0.610182\pi$
$397$	−13.5200	−0.678550	−0.339275	+	0.940687i	$0.610182\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.6977	1.18341	0.591704	−	0.806156i	$-0.298456\pi$
$401$	23.6977	1.18341	0.591704	+	0.806156i	$0.298456\pi$
$402$	0	0
$403$	16.8489	0.839301
$404$	0	0
$405$	−58.8946	−2.92650
$406$	0	0
$407$	−5.42443	−0.268879
$408$	0	0
$409$	21.7146	1.07372	0.536859	−	0.843672i	$-0.319611\pi$
$409$	21.7146	1.07372	0.536859	+	0.843672i	$0.319611\pi$
$410$	0	0
$411$	−58.8946	−2.90506
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.69772	0.279690
$416$	0	0
$417$	−22.8489	−1.11891
$418$	0	0
$419$	−8.87573	−0.433608	−0.216804	−	0.976215i	$-0.569563\pi$
$419$	−8.87573	−0.433608	−0.216804	+	0.976215i	$0.569563\pi$
$420$	0	0
$421$	17.1511	0.835896	0.417948	−	0.908471i	$-0.362750\pi$
$421$	17.1511	0.835896	0.417948	+	0.908471i	$0.362750\pi$
$422$	0	0
$423$	14.9546	0.727117
$424$	0	0
$425$	4.30371	0.208761
$426$	0	0
$427$	0	0
$428$	0	0
$429$	16.8489	0.813471
$430$	0	0
$431$	8.84886	0.426234	0.213117	−	0.977027i	$-0.431638\pi$
$431$	8.84886	0.426234	0.213117	+	0.977027i	$0.431638\pi$
$432$	0	0
$433$	1.94543	0.0934915	0.0467458	−	0.998907i	$-0.485115\pi$
$433$	1.94543	0.0934915	0.0467458	+	0.998907i	$0.485115\pi$
$434$	0	0
$435$	−32.5466	−1.56049
$436$	0	0
$437$	−9.62915	−0.460625
$438$	0	0
$439$	40.9006	1.95208	0.976040	−	0.217590i	$-0.0698197\pi$
$439$	40.9006	1.95208	0.976040	+	0.217590i	$0.0698197\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.5756	0.502461	0.251230	−	0.967927i	$-0.419165\pi$
$443$	10.5756	0.502461	0.251230	+	0.967927i	$0.419165\pi$
$444$	0	0
$445$	19.1221	0.906477
$446$	0	0
$447$	66.5784	3.14905
$448$	0	0
$449$	7.72671	0.364646	0.182323	−	0.983239i	$-0.441638\pi$
$449$	7.72671	0.364646	0.182323	+	0.983239i	$0.441638\pi$
$450$	0	0
$451$	8.19458	0.385868
$452$	0	0
$453$	−47.3201	−2.22329
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.1511	−0.708741	−0.354370	−	0.935105i	$-0.615305\pi$
$457$	−15.1511	−0.708741	−0.354370	+	0.935105i	$0.615305\pi$
$458$	0	0
$459$	32.5466	1.51914
$460$	0	0
$461$	−31.6843	−1.47569	−0.737843	−	0.674972i	$-0.764156\pi$
$461$	−31.6843	−1.47569	−0.737843	+	0.674972i	$0.764156\pi$
$462$	0	0
$463$	17.4244	0.809782	0.404891	−	0.914365i	$-0.367309\pi$
$463$	17.4244	0.809782	0.404891	+	0.914365i	$0.367309\pi$
$464$	0	0
$465$	−18.3346	−0.850246
$466$	0	0
$467$	3.38000	0.156408	0.0782040	−	0.996937i	$-0.475081\pi$
$467$	3.38000	0.156408	0.0782040	+	0.996937i	$0.475081\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−27.1221	−1.24972
$472$	0	0
$473$	−8.84886	−0.406871
$474$	0	0
$475$	−16.3892	−0.751986
$476$	0	0
$477$	−91.3954	−4.18471
$478$	0	0
$479$	3.55029	0.162217	0.0811085	−	0.996705i	$-0.474154\pi$
$479$	3.55029	0.162217	0.0811085	+	0.996705i	$0.474154\pi$
$480$	0	0
$481$	27.0400	1.23292
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.4244	−0.609572
$486$	0	0
$487$	24.2733	1.09993	0.549964	−	0.835188i	$-0.314642\pi$
$487$	24.2733	1.09993	0.549964	+	0.835188i	$0.314642\pi$
$488$	0	0
$489$	−40.5601	−1.83419
$490$	0	0
$491$	3.69772	0.166876	0.0834378	−	0.996513i	$-0.473410\pi$
$491$	3.69772	0.166876	0.0834378	+	0.996513i	$0.473410\pi$
$492$	0	0
$493$	10.6509	0.479691
$494$	0	0
$495$	−13.5200	−0.607680
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−80.5466	−3.59856
$502$	0	0
$503$	30.5903	1.36396	0.681978	−	0.731373i	$-0.261120\pi$
$503$	30.5903	1.36396	0.681978	+	0.731373i	$0.261120\pi$
$504$	0	0
$505$	18.3023	0.814441
$506$	0	0
$507$	−40.0492	−1.77865
$508$	0	0
$509$	41.8243	1.85383	0.926916	−	0.375270i	$-0.122450\pi$
$509$	41.8243	1.85383	0.926916	+	0.375270i	$0.122450\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−123.942	−5.47217
$514$	0	0
$515$	2.84886	0.125536
$516$	0	0
$517$	1.77515	0.0780708
$518$	0	0
$519$	26.5466	1.16527
$520$	0	0
$521$	31.5140	1.38065	0.690327	−	0.723497i	$-0.257466\pi$
$521$	31.5140	1.38065	0.690327	+	0.723497i	$0.257466\pi$
$522$	0	0
$523$	6.41943	0.280702	0.140351	−	0.990102i	$-0.455177\pi$
$523$	6.41943	0.280702	0.140351	+	0.990102i	$0.455177\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	−20.9710	−0.911783
$530$	0	0
$531$	1.43457	0.0622551
$532$	0	0
$533$	−40.8489	−1.76936
$534$	0	0
$535$	−23.8303	−1.03027
$536$	0	0
$537$	−14.4437	−0.623293
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.1511	−0.479425	−0.239713	−	0.970844i	$-0.577053\pi$
$541$	−11.1511	−0.479425	−0.239713	+	0.970844i	$0.577053\pi$
$542$	0	0
$543$	−52.2733	−2.24326
$544$	0	0
$545$	20.6206	0.883289
$546$	0	0
$547$	9.69772	0.414644	0.207322	−	0.978273i	$-0.433525\pi$
$547$	9.69772	0.414644	0.207322	+	0.978273i	$0.433525\pi$
$548$	0	0
$549$	98.9438	4.22282
$550$	0	0
$551$	−40.5601	−1.72792
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−29.4244	−1.24900
$556$	0	0
$557$	27.6977	1.17359	0.586795	−	0.809736i	$-0.300389\pi$
$557$	27.6977	1.17359	0.586795	+	0.809736i	$0.300389\pi$
$558$	0	0
$559$	44.1103	1.86567
$560$	0	0
$561$	6.00000	0.253320
$562$	0	0
$563$	−30.5903	−1.28923	−0.644614	−	0.764508i	$-0.722982\pi$
$563$	−30.5903	−1.28923	−0.644614	+	0.764508i	$0.722982\pi$
$564$	0	0
$565$	−0.923710	−0.0388608
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	33.6977	1.41021	0.705103	−	0.709105i	$-0.250901\pi$
$571$	33.6977	1.41021	0.705103	+	0.709105i	$0.250901\pi$
$572$	0	0
$573$	45.3746	1.89555
$574$	0	0
$575$	3.45343	0.144018
$576$	0	0
$577$	1.94543	0.0809894	0.0404947	−	0.999180i	$-0.487107\pi$
$577$	1.94543	0.0809894	0.0404947	+	0.999180i	$0.487107\pi$
$578$	0	0
$579$	−33.8000	−1.40468
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−10.8489	−0.449314
$584$	0	0
$585$	67.3954	2.78646
$586$	0	0
$587$	−31.6843	−1.30775	−0.653876	−	0.756602i	$-0.726858\pi$
$587$	−31.6843	−1.30775	−0.653876	+	0.756602i	$0.726858\pi$
$588$	0	0
$589$	−22.8489	−0.941471
$590$	0	0
$591$	20.2800	0.834209
$592$	0	0
$593$	−28.8152	−1.18330	−0.591649	−	0.806196i	$-0.701523\pi$
$593$	−28.8152	−1.18330	−0.591649	+	0.806196i	$0.701523\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	51.6977	2.11585
$598$	0	0
$599$	−25.6977	−1.04998	−0.524990	−	0.851108i	$-0.675931\pi$
$599$	−25.6977	−1.04998	−0.524990	+	0.851108i	$0.675931\pi$
$600$	0	0
$601$	−14.2734	−0.582226	−0.291113	−	0.956689i	$-0.594026\pi$
$601$	−14.2734	−0.582226	−0.291113	+	0.956689i	$0.594026\pi$
$602$	0	0
$603$	113.093	4.60551
$604$	0	0
$605$	−1.60486	−0.0652468
$606$	0	0
$607$	3.89087	0.157925	0.0789627	−	0.996878i	$-0.474839\pi$
$607$	3.89087	0.157925	0.0789627	+	0.996878i	$0.474839\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.84886	−0.357986
$612$	0	0
$613$	−8.84886	−0.357402	−0.178701	−	0.983903i	$-0.557189\pi$
$613$	−8.84886	−0.357402	−0.178701	+	0.983903i	$0.557189\pi$
$614$	0	0
$615$	44.4509	1.79243
$616$	0	0
$617$	−14.8489	−0.597793	−0.298896	−	0.954286i	$-0.596618\pi$
$617$	−14.8489	−0.597793	−0.298896	+	0.954286i	$0.596618\pi$
$618$	0	0
$619$	−9.45886	−0.380184	−0.190092	−	0.981766i	$-0.560879\pi$
$619$	−9.45886	−0.380184	−0.190092	+	0.981766i	$0.560879\pi$
$620$	0	0
$621$	26.1163	1.04801
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.00000	−0.280000
$626$	0	0
$627$	−22.8489	−0.912495
$628$	0	0
$629$	9.62915	0.383939
$630$	0	0
$631$	−20.2733	−0.807067	−0.403533	−	0.914965i	$-0.632218\pi$
$631$	−20.2733	−0.807067	−0.403533	+	0.914965i	$0.632218\pi$
$632$	0	0
$633$	23.1492	0.920097
$634$	0	0
$635$	−30.2498	−1.20042
$636$	0	0
$637$	0	0
$638$	0	0
$639$	45.6977	1.80777
$640$	0	0
$641$	−13.4244	−0.530233	−0.265117	−	0.964216i	$-0.585410\pi$
$641$	−13.4244	−0.530233	−0.265117	+	0.964216i	$0.585410\pi$
$642$	0	0
$643$	26.1886	1.03278	0.516389	−	0.856354i	$-0.327276\pi$
$643$	26.1886	1.03278	0.516389	+	0.856354i	$0.327276\pi$
$644$	0	0
$645$	−48.0000	−1.89000
$646$	0	0
$647$	49.3378	1.93967	0.969834	−	0.243767i	$-0.0783831\pi$
$647$	49.3378	1.93967	0.969834	+	0.243767i	$0.0783831\pi$
$648$	0	0
$649$	0.170287	0.00668435
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.2733	0.871621	0.435811	−	0.900038i	$-0.356462\pi$
$653$	22.2733	0.871621	0.435811	+	0.900038i	$0.356462\pi$
$654$	0	0
$655$	−16.5466	−0.646528
$656$	0	0
$657$	−44.8638	−1.75030
$658$	0	0
$659$	0.302284	0.0117753	0.00588766	−	0.999983i	$-0.498126\pi$
$659$	0.302284	0.0117753	0.00588766	+	0.999983i	$0.498126\pi$
$660$	0	0
$661$	14.4437	0.561796	0.280898	−	0.959738i	$-0.409368\pi$
$661$	14.4437	0.561796	0.280898	+	0.959738i	$0.409368\pi$
$662$	0	0
$663$	−29.9092	−1.16158
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.54657	0.330925
$668$	0	0
$669$	−23.4244	−0.905641
$670$	0	0
$671$	11.7449	0.453406
$672$	0	0
$673$	15.1511	0.584034	0.292017	−	0.956413i	$-0.405674\pi$
$673$	15.1511	0.584034	0.292017	+	0.956413i	$0.405674\pi$
$674$	0	0
$675$	44.4509	1.71092
$676$	0	0
$677$	31.6843	1.21773	0.608864	−	0.793275i	$-0.291626\pi$
$677$	31.6843	1.21773	0.608864	+	0.793275i	$0.291626\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	−27.9637	−1.06844
$686$	0	0
$687$	4.27329	0.163036
$688$	0	0
$689$	54.0801	2.06029
$690$	0	0
$691$	2.35828	0.0897133	0.0448566	−	0.998993i	$-0.485717\pi$
$691$	2.35828	0.0897133	0.0448566	+	0.998993i	$0.485717\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.8489	−0.411521
$696$	0	0
$697$	−14.5466	−0.550991
$698$	0	0
$699$	−66.5784	−2.51822
$700$	0	0
$701$	22.5466	0.851572	0.425786	−	0.904824i	$-0.359998\pi$
$701$	22.5466	0.851572	0.425786	+	0.904824i	$0.359998\pi$
$702$	0	0
$703$	−36.6692	−1.38300
$704$	0	0
$705$	9.62915	0.362655
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14.2733	0.536045	0.268022	−	0.963413i	$-0.413630\pi$
$709$	14.2733	0.536045	0.268022	+	0.963413i	$0.413630\pi$
$710$	0	0
$711$	50.5466	1.89564
$712$	0	0
$713$	4.81458	0.180307
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	73.3384	2.73887
$718$	0	0
$719$	13.0092	0.485160	0.242580	−	0.970131i	$-0.422006\pi$
$719$	13.0092	0.485160	0.242580	+	0.970131i	$0.422006\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	28.8489	1.07290
$724$	0	0
$725$	14.5466	0.540246
$726$	0	0
$727$	−42.5778	−1.57912	−0.789561	−	0.613672i	$-0.789692\pi$
$727$	−42.5778	−1.57912	−0.789561	+	0.613672i	$0.789692\pi$
$728$	0	0
$729$	123.244	4.56460
$730$	0	0
$731$	15.7080	0.580982
$732$	0	0
$733$	−44.8638	−1.65708	−0.828540	−	0.559929i	$-0.810828\pi$
$733$	−44.8638	−1.65708	−0.828540	+	0.559929i	$0.810828\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.4244	0.494495
$738$	0	0
$739$	22.8489	0.840509	0.420254	−	0.907406i	$-0.361941\pi$
$739$	22.8489	0.840509	0.420254	+	0.907406i	$0.361941\pi$
$740$	0	0
$741$	113.898	4.18416
$742$	0	0
$743$	−5.69772	−0.209029	−0.104514	−	0.994523i	$-0.533329\pi$
$743$	−5.69772	−0.209029	−0.104514	+	0.994523i	$0.533329\pi$
$744$	0	0
$745$	31.6120	1.15818
$746$	0	0
$747$	−29.9092	−1.09432
$748$	0	0
$749$	0	0
$750$	0	0
$751$	37.4244	1.36564	0.682818	−	0.730588i	$-0.260754\pi$
$751$	37.4244	1.36564	0.682818	+	0.730588i	$0.260754\pi$
$752$	0	0
$753$	22.2733	0.811684
$754$	0	0
$755$	−22.4680	−0.817695
$756$	0	0
$757$	11.6977	0.425161	0.212580	−	0.977144i	$-0.431813\pi$
$757$	11.6977	0.425161	0.212580	+	0.977144i	$0.431813\pi$
$758$	0	0
$759$	4.81458	0.174758
$760$	0	0
$761$	24.9243	0.903506	0.451753	−	0.892143i	$-0.350799\pi$
$761$	24.9243	0.903506	0.451753	+	0.892143i	$0.350799\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	24.0000	0.867722
$766$	0	0
$767$	−0.848858	−0.0306505
$768$	0	0
$769$	14.2734	0.514713	0.257357	−	0.966316i	$-0.417148\pi$
$769$	14.2734	0.514713	0.257357	+	0.966316i	$0.417148\pi$
$770$	0	0
$771$	−67.3954	−2.42719
$772$	0	0
$773$	17.0703	0.613976	0.306988	−	0.951713i	$-0.400679\pi$
$773$	17.0703	0.613976	0.306988	+	0.951713i	$0.400679\pi$
$774$	0	0
$775$	8.19458	0.294358
$776$	0	0
$777$	0	0
$778$	0	0
$779$	55.3954	1.98475
$780$	0	0
$781$	5.42443	0.194101
$782$	0	0
$783$	110.008	3.93135
$784$	0	0
$785$	−12.8779	−0.459630
$786$	0	0
$787$	−26.0183	−0.927453	−0.463726	−	0.885978i	$-0.653488\pi$
$787$	−26.0183	−0.927453	−0.463726	+	0.885978i	$0.653488\pi$
$788$	0	0
$789$	17.4109	0.619844
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−58.5466	−2.07905
$794$	0	0
$795$	−58.8489	−2.08715
$796$	0	0
$797$	31.1735	1.10422	0.552110	−	0.833771i	$-0.313823\pi$
$797$	31.1735	1.10422	0.552110	+	0.833771i	$0.313823\pi$
$798$	0	0
$799$	−3.15114	−0.111479
$800$	0	0
$801$	−100.378	−3.54670
$802$	0	0
$803$	−5.32544	−0.187931
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−36.0000	−1.26726
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	−23.1492	−0.812877	−0.406439	−	0.913678i	$-0.633230\pi$
$811$	−23.1492	−0.812877	−0.406439	+	0.913678i	$0.633230\pi$
$812$	0	0
$813$	33.6977	1.18183
$814$	0	0
$815$	−19.2583	−0.674589
$816$	0	0
$817$	−59.8184	−2.09278
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.3954	1.02591	0.512954	−	0.858416i	$-0.328551\pi$
$821$	29.3954	1.02591	0.512954	+	0.858416i	$0.328551\pi$
$822$	0	0
$823$	41.4244	1.44396	0.721982	−	0.691911i	$-0.243231\pi$
$823$	41.4244	1.44396	0.721982	+	0.691911i	$0.243231\pi$
$824$	0	0
$825$	8.19458	0.285299
$826$	0	0
$827$	−32.8489	−1.14227	−0.571133	−	0.820857i	$-0.693496\pi$
$827$	−32.8489	−1.14227	−0.571133	+	0.820857i	$0.693496\pi$
$828$	0	0
$829$	−22.2255	−0.771922	−0.385961	−	0.922515i	$-0.626130\pi$
$829$	−22.2255	−0.771922	−0.385961	+	0.922515i	$0.626130\pi$
$830$	0	0
$831$	−83.9892	−2.91355
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−38.2443	−1.32350
$836$	0	0
$837$	61.9710	2.14203
$838$	0	0
$839$	−12.6686	−0.437368	−0.218684	−	0.975796i	$-0.570176\pi$
$839$	−12.6686	−0.437368	−0.218684	+	0.975796i	$0.570176\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−16.3892	−0.564473
$844$	0	0
$845$	−19.0157	−0.654161
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−43.3954	−1.48933
$850$	0	0
$851$	7.72671	0.264868
$852$	0	0
$853$	5.32544	0.182339	0.0911697	−	0.995835i	$-0.470939\pi$
$853$	5.32544	0.182339	0.0911697	+	0.995835i	$0.470939\pi$
$854$	0	0
$855$	−91.3954	−3.12566
$856$	0	0
$857$	11.4043	0.389563	0.194782	−	0.980847i	$-0.437600\pi$
$857$	11.4043	0.389563	0.194782	+	0.980847i	$0.437600\pi$
$858$	0	0
$859$	21.1315	0.720996	0.360498	−	0.932760i	$-0.382607\pi$
$859$	21.1315	0.720996	0.360498	+	0.932760i	$0.382607\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.30228	0.0783707	0.0391853	−	0.999232i	$-0.487524\pi$
$863$	2.30228	0.0783707	0.0391853	+	0.999232i	$0.487524\pi$
$864$	0	0
$865$	12.6046	0.428568
$866$	0	0
$867$	46.8092	1.58972
$868$	0	0
$869$	6.00000	0.203536
$870$	0	0
$871$	−66.9189	−2.26746
$872$	0	0
$873$	70.4692	2.38502
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	−61.3954	−2.07082
$880$	0	0
$881$	10.8934	0.367009	0.183505	−	0.983019i	$-0.441256\pi$
$881$	10.8934	0.367009	0.183505	+	0.983019i	$0.441256\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	0.923710	0.0310502
$886$	0	0
$887$	19.5989	0.658066	0.329033	−	0.944318i	$-0.393277\pi$
$887$	19.5989	0.658066	0.329033	+	0.944318i	$0.393277\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	36.6977	1.22942
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	−6.85802	−0.229238
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	20.2800	0.676377
$900$	0	0
$901$	19.2583	0.641587
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.8199	−0.825040
$906$	0	0
$907$	−19.3954	−0.644015	−0.322007	−	0.946737i	$-0.604358\pi$
$907$	−19.3954	−0.644015	−0.322007	+	0.946737i	$0.604358\pi$
$908$	0	0
$909$	−96.0747	−3.18660
$910$	0	0
$911$	−49.6977	−1.64656	−0.823279	−	0.567636i	$-0.807858\pi$
$911$	−49.6977	−1.64656	−0.823279	+	0.567636i	$0.807858\pi$
$912$	0	0
$913$	−3.55029	−0.117497
$914$	0	0
$915$	63.7092	2.10616
$916$	0	0
$917$	0	0
$918$	0	0
$919$	10.3023	0.339841	0.169921	−	0.985458i	$-0.445649\pi$
$919$	10.3023	0.339841	0.169921	+	0.985458i	$0.445649\pi$
$920$	0	0
$921$	2.30228	0.0758629
$922$	0	0
$923$	−27.0400	−0.890034
$924$	0	0
$925$	13.1511	0.432407
$926$	0	0
$927$	−14.9546	−0.491173
$928$	0	0
$929$	−9.28858	−0.304748	−0.152374	−	0.988323i	$-0.548692\pi$
$929$	−9.28858	−0.304748	−0.152374	+	0.988323i	$0.548692\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	75.6977	2.47823
$934$	0	0
$935$	2.84886	0.0931676
$936$	0	0
$937$	−12.7666	−0.417066	−0.208533	−	0.978015i	$-0.566869\pi$
$937$	−12.7666	−0.417066	−0.208533	+	0.978015i	$0.566869\pi$
$938$	0	0
$939$	−61.9710	−2.02235
$940$	0	0
$941$	−15.9763	−0.520813	−0.260406	−	0.965499i	$-0.583857\pi$
$941$	−15.9763	−0.520813	−0.260406	+	0.965499i	$0.583857\pi$
$942$	0	0
$943$	−11.6726	−0.380112
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.42443	−0.0462877	−0.0231439	−	0.999732i	$-0.507368\pi$
$947$	−1.42443	−0.0462877	−0.0231439	+	0.999732i	$0.507368\pi$
$948$	0	0
$949$	26.5466	0.861738
$950$	0	0
$951$	25.0946	0.813748
$952$	0	0
$953$	−4.84886	−0.157070	−0.0785350	−	0.996911i	$-0.525024\pi$
$953$	−4.84886	−0.157070	−0.0785350	+	0.996911i	$0.525024\pi$
$954$	0	0
$955$	21.5443	0.697158
$956$	0	0
$957$	20.2800	0.655560
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19.5756	−0.631470
$962$	0	0
$963$	125.093	4.03107
$964$	0	0
$965$	−16.0486	−0.516622
$966$	0	0
$967$	26.0000	0.836104	0.418052	−	0.908423i	$-0.362713\pi$
$967$	26.0000	0.836104	0.418052	+	0.908423i	$0.362713\pi$
$968$	0	0
$969$	40.5601	1.30298
$970$	0	0
$971$	42.5778	1.36639	0.683193	−	0.730238i	$-0.260591\pi$
$971$	42.5778	1.36639	0.683193	+	0.730238i	$0.260591\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−40.8489	−1.30821
$976$	0	0
$977$	−0.575571	−0.0184142	−0.00920708	−	0.999958i	$-0.502931\pi$
$977$	−0.575571	−0.0184142	−0.00920708	+	0.999958i	$0.502931\pi$
$978$	0	0
$979$	−11.9152	−0.380810
$980$	0	0
$981$	−108.244	−3.45597
$982$	0	0
$983$	−3.72058	−0.118668	−0.0593340	−	0.998238i	$-0.518898\pi$
$983$	−3.72058	−0.118668	−0.0593340	+	0.998238i	$0.518898\pi$
$984$	0	0
$985$	9.62915	0.306810
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.6046	0.400802
$990$	0	0
$991$	43.3954	1.37850	0.689251	−	0.724523i	$-0.257940\pi$
$991$	43.3954	1.37850	0.689251	+	0.724523i	$0.257940\pi$
$992$	0	0
$993$	−45.3746	−1.43992
$994$	0	0
$995$	24.5466	0.778179
$996$	0	0
$997$	32.0249	1.01424	0.507119	−	0.861876i	$-0.330710\pi$
$997$	32.0249	1.01424	0.507119	+	0.861876i	$0.330710\pi$
$998$	0	0
$999$	99.4547	3.14661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.a.m.1.1	✓	4
4.3	odd	2		8624.2.a.cx.1.4		4
7.2	even	3		2156.2.i.n.1145.4		8
7.3	odd	6		2156.2.i.n.177.1		8
7.4	even	3		2156.2.i.n.177.4		8
7.5	odd	6		2156.2.i.n.1145.1		8
7.6	odd	2	inner	2156.2.a.m.1.4	yes	4
28.27	even	2		8624.2.a.cx.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2156.2.a.m.1.1	✓	4	1.1	even	1	trivial
2156.2.a.m.1.4	yes	4	7.6	odd	2	inner
2156.2.i.n.177.1		8	7.3	odd	6
2156.2.i.n.177.4		8	7.4	even	3
2156.2.i.n.1145.1		8	7.5	odd	6
2156.2.i.n.1145.4		8	7.2	even	3
8624.2.a.cx.1.1		4	28.27	even	2
8624.2.a.cx.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.a (trivial)

\(f(q)\)	\(=\)	\(q-3.38000 q^{3} -1.60486 q^{5} +8.42443 q^{9} +1.00000 q^{11} -4.98486 q^{13} +5.42443 q^{15} -1.77515 q^{17} +6.76001 q^{19} -1.42443 q^{23} -2.42443 q^{25} -18.3346 q^{27} -6.00000 q^{29} -3.38000 q^{31} -3.38000 q^{33} -5.42443 q^{37} +16.8489 q^{39} +8.19458 q^{41} -8.84886 q^{43} -13.5200 q^{45} +1.77515 q^{47} +6.00000 q^{51} -10.8489 q^{53} -1.60486 q^{55} -22.8489 q^{57} +0.170287 q^{59} +11.7449 q^{61} +8.00000 q^{65} +13.4244 q^{67} +4.81458 q^{69} +5.42443 q^{71} -5.32544 q^{73} +8.19458 q^{75} +6.00000 q^{79} +36.6977 q^{81} -3.55029 q^{83} +2.84886 q^{85} +20.2800 q^{87} -11.9152 q^{89} +11.4244 q^{93} -10.8489 q^{95} +8.36487 q^{97} +8.42443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{9} + 4 q^{11} + 2 q^{15} + 14 q^{23} + 10 q^{25} - 24 q^{29} - 2 q^{37} + 28 q^{39} + 4 q^{43} + 24 q^{51} - 4 q^{53} - 52 q^{57} + 32 q^{65} + 34 q^{67} + 2 q^{71} + 24 q^{79} + 68 q^{81} - 28 q^{85}+ \cdots + 14 q^{99}+O(q^{100}) \) 4 * q + 14 * q^9 + 4 * q^11 + 2 * q^15 + 14 * q^23 + 10 * q^25 - 24 * q^29 - 2 * q^37 + 28 * q^39 + 4 * q^43 + 24 * q^51 - 4 * q^53 - 52 * q^57 + 32 * q^65 + 34 * q^67 + 2 * q^71 + 24 * q^79 + 68 * q^81 - 28 * q^85 + 26 * q^93 - 4 * q^95 + 14 * q^99

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