LMFDB - Embedded newform 2156.2.a.j.1.3

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:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.85577$ 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)

$f(q)$	$=$	$q+2.85577 q^{3} +4.15544 q^{5} +5.15544 q^{9} +1.00000 q^{11} -5.29966 q^{13} +11.8670 q^{15} -4.41188 q^{17} +5.71155 q^{19} -3.55611 q^{23} +12.2677 q^{25} +6.15544 q^{27} -0.599328 q^{29} -4.56732 q^{31} +2.85577 q^{33} +6.15544 q^{37} -15.1346 q^{39} +4.41188 q^{41} +3.71155 q^{43} +21.4231 q^{45} -5.01121 q^{47} -12.5993 q^{51} +8.31087 q^{53} +4.15544 q^{55} +16.3109 q^{57} +5.14423 q^{59} -13.6105 q^{61} -22.0224 q^{65} -0.443892 q^{67} -10.1554 q^{69} +4.75476 q^{71} +8.41188 q^{73} +35.0336 q^{75} -8.02242 q^{79} +2.11222 q^{81} -8.82376 q^{83} -18.3333 q^{85} -1.71155 q^{87} -7.57853 q^{89} -13.0432 q^{93} +23.7340 q^{95} -13.8670 q^{97} +5.15544 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} + 4 q^{9} + 3 q^{11} - 12 q^{13} + 9 q^{15} - 2 q^{17} + 2 q^{19} - 7 q^{23} + 18 q^{25} + 7 q^{27} + 6 q^{29} + 9 q^{31} + q^{33} + 7 q^{37} + 2 q^{41} - 4 q^{43} + 34 q^{45} + 4 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q + q^3 + q^5 + 4 * q^9 + 3 * q^11 - 12 * q^13 + 9 * q^15 - 2 * q^17 + 2 * q^19 - 7 * q^23 + 18 * q^25 + 7 * q^27 + 6 * q^29 + 9 * q^31 + q^33 + 7 * q^37 + 2 * q^41 - 4 * q^43 + 34 * q^45 + 4 * q^47 - 30 * q^51 + 2 * q^53 + q^55 + 26 * q^57 + 23 * q^59 - 14 * q^61 - 28 * q^65 - 5 * q^67 - 19 * q^69 - 5 * q^71 + 14 * q^73 + 48 * q^75 + 14 * q^79 - q^81 - 4 * q^83 + 6 * q^85 + 10 * q^87 + 19 * q^89 - 35 * q^93 + 18 * q^95 - 15 * q^97 + 4 * 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
$3$	2.85577	1.64878	0.824391	−	0.566021i	$-0.191518\pi$
$3$	2.85577	1.64878	0.824391	+	0.566021i	$0.191518\pi$
$4$	0	0
$5$	4.15544	1.85837	0.929184	−	0.369618i	$-0.120511\pi$
$5$	4.15544	1.85837	0.929184	+	0.369618i	$0.120511\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.15544	1.71848
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.29966	−1.46986	−0.734931	−	0.678142i	$-0.762786\pi$
$13$	−5.29966	−1.46986	−0.734931	+	0.678142i	$0.762786\pi$
$14$	0	0
$15$	11.8670	3.06404
$16$	0	0
$17$	−4.41188	−1.07004	−0.535019	−	0.844840i	$-0.679696\pi$
$17$	−4.41188	−1.07004	−0.535019	+	0.844840i	$0.679696\pi$
$18$	0	0
$19$	5.71155	1.31032	0.655159	−	0.755491i	$-0.272602\pi$
$19$	5.71155	1.31032	0.655159	+	0.755491i	$0.272602\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.55611	−0.741500	−0.370750	−	0.928733i	$-0.620899\pi$
$23$	−3.55611	−0.741500	−0.370750	+	0.928733i	$0.620899\pi$
$24$	0	0
$25$	12.2677	2.45353
$26$	0	0
$27$	6.15544	1.18461
$28$	0	0
$29$	−0.599328	−0.111292	−0.0556462	−	0.998451i	$-0.517722\pi$
$29$	−0.599328	−0.111292	−0.0556462	+	0.998451i	$0.517722\pi$
$30$	0	0
$31$	−4.56732	−0.820314	−0.410157	−	0.912015i	$-0.634526\pi$
$31$	−4.56732	−0.820314	−0.410157	+	0.912015i	$0.634526\pi$
$32$	0	0
$33$	2.85577	0.497126
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.15544	1.01195	0.505974	−	0.862549i	$-0.331133\pi$
$37$	6.15544	1.01195	0.505974	+	0.862549i	$0.331133\pi$
$38$	0	0
$39$	−15.1346	−2.42348
$40$	0	0
$41$	4.41188	0.689020	0.344510	−	0.938783i	$-0.388045\pi$
$41$	4.41188	0.689020	0.344510	+	0.938783i	$0.388045\pi$
$42$	0	0
$43$	3.71155	0.566005	0.283003	−	0.959119i	$-0.408669\pi$
$43$	3.71155	0.566005	0.283003	+	0.959119i	$0.408669\pi$
$44$	0	0
$45$	21.4231	3.19357
$46$	0	0
$47$	−5.01121	−0.730960	−0.365480	−	0.930819i	$-0.619095\pi$
$47$	−5.01121	−0.730960	−0.365480	+	0.930819i	$0.619095\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−12.5993	−1.76426
$52$	0	0
$53$	8.31087	1.14159	0.570793	−	0.821094i	$-0.306636\pi$
$53$	8.31087	1.14159	0.570793	+	0.821094i	$0.306636\pi$
$54$	0	0
$55$	4.15544	0.560319
$56$	0	0
$57$	16.3109	2.16043
$58$	0	0
$59$	5.14423	0.669721	0.334861	−	0.942268i	$-0.391311\pi$
$59$	5.14423	0.669721	0.334861	+	0.942268i	$0.391311\pi$
$60$	0	0
$61$	−13.6105	−1.74265	−0.871325	−	0.490706i	$-0.836739\pi$
$61$	−13.6105	−1.74265	−0.871325	+	0.490706i	$0.836739\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−22.0224	−2.73154
$66$	0	0
$67$	−0.443892	−0.0542300	−0.0271150	−	0.999632i	$-0.508632\pi$
$67$	−0.443892	−0.0542300	−0.0271150	+	0.999632i	$0.508632\pi$
$68$	0	0
$69$	−10.1554	−1.22257
$70$	0	0
$71$	4.75476	0.564287	0.282143	−	0.959372i	$-0.408955\pi$
$71$	4.75476	0.564287	0.282143	+	0.959372i	$0.408955\pi$
$72$	0	0
$73$	8.41188	0.984536	0.492268	−	0.870444i	$-0.336168\pi$
$73$	8.41188	0.984536	0.492268	+	0.870444i	$0.336168\pi$
$74$	0	0
$75$	35.0336	4.04533
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.02242	−0.902593	−0.451296	−	0.892374i	$-0.649038\pi$
$79$	−8.02242	−0.902593	−0.451296	+	0.892374i	$0.649038\pi$
$80$	0	0
$81$	2.11222	0.234691
$82$	0	0
$83$	−8.82376	−0.968534	−0.484267	−	0.874920i	$-0.660914\pi$
$83$	−8.82376	−0.968534	−0.484267	+	0.874920i	$0.660914\pi$
$84$	0	0
$85$	−18.3333	−1.98852
$86$	0	0
$87$	−1.71155	−0.183497
$88$	0	0
$89$	−7.57853	−0.803322	−0.401661	−	0.915788i	$-0.631567\pi$
$89$	−7.57853	−0.803322	−0.401661	+	0.915788i	$0.631567\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−13.0432	−1.35252
$94$	0	0
$95$	23.7340	2.43505
$96$	0	0
$97$	−13.8670	−1.40798	−0.703989	−	0.710211i	$-0.748600\pi$
$97$	−13.8670	−1.40798	−0.703989	+	0.710211i	$0.748600\pi$
$98$	0	0
$99$	5.15544	0.518141
$100$	0	0
$101$	2.70034	0.268693	0.134347	−	0.990934i	$-0.457106\pi$
$101$	2.70034	0.268693	0.134347	+	0.990934i	$0.457106\pi$
$102$	0	0
$103$	−6.41188	−0.631781	−0.315891	−	0.948796i	$-0.602303\pi$
$103$	−6.41188	−0.631781	−0.315891	+	0.948796i	$0.602303\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.31087	0.803442	0.401721	−	0.915762i	$-0.368412\pi$
$107$	8.31087	0.803442	0.401721	+	0.915762i	$0.368412\pi$
$108$	0	0
$109$	−15.1346	−1.44964	−0.724818	−	0.688941i	$-0.758076\pi$
$109$	−15.1346	−1.44964	−0.724818	+	0.688941i	$0.758076\pi$
$110$	0	0
$111$	17.5785	1.66848
$112$	0	0
$113$	−15.8894	−1.49475	−0.747375	−	0.664403i	$-0.768686\pi$
$113$	−15.8894	−1.49475	−0.747375	+	0.664403i	$0.768686\pi$
$114$	0	0
$115$	−14.7772	−1.37798
$116$	0	0
$117$	−27.3221	−2.52593
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.5993	1.13604
$124$	0	0
$125$	30.2003	2.70119
$126$	0	0
$127$	15.1122	1.34099	0.670496	−	0.741913i	$-0.266081\pi$
$127$	15.1122	1.34099	0.670496	+	0.741913i	$0.266081\pi$
$128$	0	0
$129$	10.5993	0.933219
$130$	0	0
$131$	3.68913	0.322321	0.161160	−	0.986928i	$-0.448476\pi$
$131$	3.68913	0.322321	0.161160	+	0.986928i	$0.448476\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	25.5785	2.20145
$136$	0	0
$137$	3.04322	0.260000	0.130000	−	0.991514i	$-0.458502\pi$
$137$	3.04322	0.260000	0.130000	+	0.991514i	$0.458502\pi$
$138$	0	0
$139$	−10.9102	−0.925391	−0.462696	−	0.886517i	$-0.653118\pi$
$139$	−10.9102	−0.925391	−0.462696	+	0.886517i	$0.653118\pi$
$140$	0	0
$141$	−14.3109	−1.20519
$142$	0	0
$143$	−5.29966	−0.443180
$144$	0	0
$145$	−2.49047	−0.206822
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.19866	0.589737	0.294868	−	0.955538i	$-0.404724\pi$
$149$	7.19866	0.589737	0.294868	+	0.955538i	$0.404724\pi$
$150$	0	0
$151$	−8.02242	−0.652855	−0.326428	−	0.945222i	$-0.605845\pi$
$151$	−8.02242	−0.652855	−0.326428	+	0.945222i	$0.605845\pi$
$152$	0	0
$153$	−22.7452	−1.83884
$154$	0	0
$155$	−18.9792	−1.52445
$156$	0	0
$157$	−3.84456	−0.306830	−0.153415	−	0.988162i	$-0.549027\pi$
$157$	−3.84456	−0.306830	−0.153415	+	0.988162i	$0.549027\pi$
$158$	0	0
$159$	23.7340	1.88223
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−14.0224	−1.09832	−0.549160	−	0.835717i	$-0.685052\pi$
$163$	−14.0224	−1.09832	−0.549160	+	0.835717i	$0.685052\pi$
$164$	0	0
$165$	11.8670	0.923843
$166$	0	0
$167$	−19.1122	−1.47895	−0.739474	−	0.673185i	$-0.764926\pi$
$167$	−19.1122	−1.47895	−0.739474	+	0.673185i	$0.764926\pi$
$168$	0	0
$169$	15.0864	1.16050
$170$	0	0
$171$	29.4455	2.25175
$172$	0	0
$173$	−19.3221	−1.46903	−0.734515	−	0.678592i	$-0.762590\pi$
$173$	−19.3221	−1.46903	−0.734515	+	0.678592i	$0.762590\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.6907	1.10422
$178$	0	0
$179$	6.73235	0.503199	0.251600	−	0.967831i	$-0.419043\pi$
$179$	6.73235	0.503199	0.251600	+	0.967831i	$0.419043\pi$
$180$	0	0
$181$	7.64255	0.568066	0.284033	−	0.958814i	$-0.408327\pi$
$181$	7.64255	0.568066	0.284033	+	0.958814i	$0.408327\pi$
$182$	0	0
$183$	−38.8686	−2.87325
$184$	0	0
$185$	25.5785	1.88057
$186$	0	0
$187$	−4.41188	−0.322629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.84456	−0.133468	−0.0667340	−	0.997771i	$-0.521258\pi$
$191$	−1.84456	−0.133468	−0.0667340	+	0.997771i	$0.521258\pi$
$192$	0	0
$193$	−14.6217	−1.05250	−0.526248	−	0.850331i	$-0.676402\pi$
$193$	−14.6217	−1.05250	−0.526248	+	0.850331i	$0.676402\pi$
$194$	0	0
$195$	−62.8910	−4.50372
$196$	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
$199$	10.2099	0.723758	0.361879	−	0.932225i	$-0.382135\pi$
$199$	10.2099	0.723758	0.361879	+	0.932225i	$0.382135\pi$
$200$	0	0
$201$	−1.26765	−0.0894134
$202$	0	0
$203$	0	0
$204$	0	0
$205$	18.3333	1.28045
$206$	0	0
$207$	−18.3333	−1.27425
$208$	0	0
$209$	5.71155	0.395076
$210$	0	0
$211$	16.3109	1.12289	0.561443	−	0.827515i	$-0.310246\pi$
$211$	16.3109	1.12289	0.561443	+	0.827515i	$0.310246\pi$
$212$	0	0
$213$	13.5785	0.930385
$214$	0	0
$215$	15.4231	1.05185
$216$	0	0
$217$	0	0
$218$	0	0
$219$	24.0224	1.62328
$220$	0	0
$221$	23.3815	1.57281
$222$	0	0
$223$	13.0802	0.875915	0.437958	−	0.898996i	$-0.355702\pi$
$223$	13.0802	0.875915	0.437958	+	0.898996i	$0.355702\pi$
$224$	0	0
$225$	63.2451	4.21634
$226$	0	0
$227$	16.6217	1.10322	0.551612	−	0.834101i	$-0.314013\pi$
$227$	16.6217	1.10322	0.551612	+	0.834101i	$0.314013\pi$
$228$	0	0
$229$	−22.1779	−1.46555	−0.732777	−	0.680469i	$-0.761776\pi$
$229$	−22.1779	−1.46555	−0.732777	+	0.680469i	$0.761776\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.5993	−0.825409	−0.412705	−	0.910865i	$-0.635416\pi$
$233$	−12.5993	−0.825409	−0.412705	+	0.910865i	$0.635416\pi$
$234$	0	0
$235$	−20.8238	−1.35839
$236$	0	0
$237$	−22.9102	−1.48818
$238$	0	0
$239$	14.2244	0.920102	0.460051	−	0.887892i	$-0.347831\pi$
$239$	14.2244	0.920102	0.460051	+	0.887892i	$0.347831\pi$
$240$	0	0
$241$	0.101008	0.00650648	0.00325324	−	0.999995i	$-0.498964\pi$
$241$	0.101008	0.00650648	0.00325324	+	0.999995i	$0.498964\pi$
$242$	0	0
$243$	−12.4343	−0.797661
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−30.2693	−1.92599
$248$	0	0
$249$	−25.1987	−1.59690
$250$	0	0
$251$	25.9680	1.63908	0.819542	−	0.573018i	$-0.194228\pi$
$251$	25.9680	1.63908	0.819542	+	0.573018i	$0.194228\pi$
$252$	0	0
$253$	−3.55611	−0.223571
$254$	0	0
$255$	−52.3557	−3.27864
$256$	0	0
$257$	−2.57691	−0.160743	−0.0803716	−	0.996765i	$-0.525611\pi$
$257$	−2.57691	−0.160743	−0.0803716	+	0.996765i	$0.525611\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.08980	−0.191254
$262$	0	0
$263$	20.3109	1.25242	0.626211	−	0.779654i	$-0.284605\pi$
$263$	20.3109	1.25242	0.626211	+	0.779654i	$0.284605\pi$
$264$	0	0
$265$	34.5353	2.12149
$266$	0	0
$267$	−21.6425	−1.32450
$268$	0	0
$269$	−2.82376	−0.172168	−0.0860839	−	0.996288i	$-0.527435\pi$
$269$	−2.82376	−0.172168	−0.0860839	+	0.996288i	$0.527435\pi$
$270$	0	0
$271$	−4.82376	−0.293023	−0.146511	−	0.989209i	$-0.546804\pi$
$271$	−4.82376	−0.293023	−0.146511	+	0.989209i	$0.546804\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.2677	0.739767
$276$	0	0
$277$	13.7340	0.825194	0.412597	−	0.910914i	$-0.364622\pi$
$277$	13.7340	0.825194	0.412597	+	0.910914i	$0.364622\pi$
$278$	0	0
$279$	−23.5465	−1.40969
$280$	0	0
$281$	−20.9102	−1.24740	−0.623699	−	0.781665i	$-0.714371\pi$
$281$	−20.9102	−1.24740	−0.623699	+	0.781665i	$0.714371\pi$
$282$	0	0
$283$	5.19866	0.309028	0.154514	−	0.987991i	$-0.450619\pi$
$283$	5.19866	0.309028	0.154514	+	0.987991i	$0.450619\pi$
$284$	0	0
$285$	67.7788	4.01487
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.46469	0.144982
$290$	0	0
$291$	−39.6009	−2.32145
$292$	0	0
$293$	17.8574	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$293$	17.8574	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$294$	0	0
$295$	21.3765	1.24459
$296$	0	0
$297$	6.15544	0.357175
$298$	0	0
$299$	18.8462	1.08990
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.71155	0.443017
$304$	0	0
$305$	−56.5577	−3.23849
$306$	0	0
$307$	32.6217	1.86182	0.930911	−	0.365247i	$-0.119015\pi$
$307$	32.6217	1.86182	0.930911	+	0.365247i	$0.119015\pi$
$308$	0	0
$309$	−18.3109	−1.04167
$310$	0	0
$311$	25.0560	1.42080	0.710399	−	0.703799i	$-0.248515\pi$
$311$	25.0560	1.42080	0.710399	+	0.703799i	$0.248515\pi$
$312$	0	0
$313$	19.8894	1.12422	0.562108	−	0.827064i	$-0.309991\pi$
$313$	19.8894	1.12422	0.562108	+	0.827064i	$0.309991\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.04322	−0.507918	−0.253959	−	0.967215i	$-0.581733\pi$
$317$	−9.04322	−0.507918	−0.253959	+	0.967215i	$0.581733\pi$
$318$	0	0
$319$	−0.599328	−0.0335559
$320$	0	0
$321$	23.7340	1.32470
$322$	0	0
$323$	−25.1987	−1.40209
$324$	0	0
$325$	−65.0144	−3.60635
$326$	0	0
$327$	−43.2211	−2.39013
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.8894	1.64287	0.821435	−	0.570302i	$-0.193174\pi$
$331$	29.8894	1.64287	0.821435	+	0.570302i	$0.193174\pi$
$332$	0	0
$333$	31.7340	1.73901
$334$	0	0
$335$	−1.84456	−0.100779
$336$	0	0
$337$	3.97758	0.216673	0.108336	−	0.994114i	$-0.465448\pi$
$337$	3.97758	0.216673	0.108336	+	0.994114i	$0.465448\pi$
$338$	0	0
$339$	−45.3765	−2.46451
$340$	0	0
$341$	−4.56732	−0.247334
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−42.2003	−2.27199
$346$	0	0
$347$	−2.82376	−0.151587	−0.0757937	−	0.997124i	$-0.524149\pi$
$347$	−2.82376	−0.151587	−0.0757937	+	0.997124i	$0.524149\pi$
$348$	0	0
$349$	12.7868	0.684460	0.342230	−	0.939616i	$-0.388818\pi$
$349$	12.7868	0.684460	0.342230	+	0.939616i	$0.388818\pi$
$350$	0	0
$351$	−32.6217	−1.74122
$352$	0	0
$353$	−33.9758	−1.80835	−0.904176	−	0.427161i	$-0.859514\pi$
$353$	−33.9758	−1.80835	−0.904176	+	0.427161i	$0.859514\pi$
$354$	0	0
$355$	19.7581	1.04865
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.1346	−0.587664	−0.293832	−	0.955857i	$-0.594931\pi$
$359$	−11.1346	−0.587664	−0.293832	+	0.955857i	$0.594931\pi$
$360$	0	0
$361$	13.6217	0.716934
$362$	0	0
$363$	2.85577	0.149889
$364$	0	0
$365$	34.9550	1.82963
$366$	0	0
$367$	−24.3653	−1.27186	−0.635929	−	0.771747i	$-0.719383\pi$
$367$	−24.3653	−1.27186	−0.635929	+	0.771747i	$0.719383\pi$
$368$	0	0
$369$	22.7452	1.18407
$370$	0	0
$371$	0	0
$372$	0	0
$373$	21.1122	1.09315	0.546575	−	0.837410i	$-0.315931\pi$
$373$	21.1122	1.09315	0.546575	+	0.837410i	$0.315931\pi$
$374$	0	0
$375$	86.2451	4.45368
$376$	0	0
$377$	3.17624	0.163585
$378$	0	0
$379$	10.0914	0.518361	0.259181	−	0.965829i	$-0.416548\pi$
$379$	10.0914	0.518361	0.259181	+	0.965829i	$0.416548\pi$
$380$	0	0
$381$	43.1571	2.21100
$382$	0	0
$383$	−37.4359	−1.91289	−0.956443	−	0.291919i	$-0.905706\pi$
$383$	−37.4359	−1.91289	−0.956443	+	0.291919i	$0.905706\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	19.1346	0.972668
$388$	0	0
$389$	17.3541	0.879887	0.439944	−	0.898025i	$-0.354998\pi$
$389$	17.3541	0.879887	0.439944	+	0.898025i	$0.354998\pi$
$390$	0	0
$391$	15.6891	0.793433
$392$	0	0
$393$	10.5353	0.531436
$394$	0	0
$395$	−33.3367	−1.67735
$396$	0	0
$397$	18.0448	0.905644	0.452822	−	0.891601i	$-0.350417\pi$
$397$	18.0448	0.905644	0.452822	+	0.891601i	$0.350417\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.57691	0.128685	0.0643424	−	0.997928i	$-0.479505\pi$
$401$	2.57691	0.128685	0.0643424	+	0.997928i	$0.479505\pi$
$402$	0	0
$403$	24.2052	1.20575
$404$	0	0
$405$	8.77718	0.436142
$406$	0	0
$407$	6.15544	0.305114
$408$	0	0
$409$	25.0336	1.23783	0.618917	−	0.785457i	$-0.287572\pi$
$409$	25.0336	1.23783	0.618917	+	0.785457i	$0.287572\pi$
$410$	0	0
$411$	8.69074	0.428683
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−36.6666	−1.79989
$416$	0	0
$417$	−31.1571	−1.52577
$418$	0	0
$419$	−3.56570	−0.174196	−0.0870979	−	0.996200i	$-0.527759\pi$
$419$	−3.56570	−0.174196	−0.0870979	+	0.996200i	$0.527759\pi$
$420$	0	0
$421$	4.31087	0.210099	0.105050	−	0.994467i	$-0.466500\pi$
$421$	4.31087	0.210099	0.105050	+	0.994467i	$0.466500\pi$
$422$	0	0
$423$	−25.8350	−1.25614
$424$	0	0
$425$	−54.1234	−2.62537
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−15.1346	−0.730707
$430$	0	0
$431$	14.5577	0.701221	0.350610	−	0.936521i	$-0.385974\pi$
$431$	14.5577	0.701221	0.350610	+	0.936521i	$0.385974\pi$
$432$	0	0
$433$	0.357452	0.0171780	0.00858902	−	0.999963i	$-0.497266\pi$
$433$	0.357452	0.0171780	0.00858902	+	0.999963i	$0.497266\pi$
$434$	0	0
$435$	−7.11222	−0.341005
$436$	0	0
$437$	−20.3109	−0.971601
$438$	0	0
$439$	14.2885	0.681951	0.340975	−	0.940072i	$-0.389243\pi$
$439$	14.2885	0.681951	0.340975	+	0.940072i	$0.389243\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.95678	0.425549	0.212775	−	0.977101i	$-0.431750\pi$
$443$	8.95678	0.425549	0.212775	+	0.977101i	$0.431750\pi$
$444$	0	0
$445$	−31.4921	−1.49287
$446$	0	0
$447$	20.5577	0.972347
$448$	0	0
$449$	0.379870	0.0179272	0.00896359	−	0.999960i	$-0.497147\pi$
$449$	0.379870	0.0179272	0.00896359	+	0.999960i	$0.497147\pi$
$450$	0	0
$451$	4.41188	0.207747
$452$	0	0
$453$	−22.9102	−1.07642
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.3557	−1.23287	−0.616434	−	0.787407i	$-0.711423\pi$
$457$	−26.3557	−1.23287	−0.616434	+	0.787407i	$0.711423\pi$
$458$	0	0
$459$	−27.1571	−1.26758
$460$	0	0
$461$	−38.4343	−1.79006	−0.895032	−	0.446002i	$-0.852847\pi$
$461$	−38.4343	−1.79006	−0.895032	+	0.446002i	$0.852847\pi$
$462$	0	0
$463$	−1.64255	−0.0763357	−0.0381678	−	0.999271i	$-0.512152\pi$
$463$	−1.64255	−0.0763357	−0.0381678	+	0.999271i	$0.512152\pi$
$464$	0	0
$465$	−54.2003	−2.51348
$466$	0	0
$467$	−5.65712	−0.261780	−0.130890	−	0.991397i	$-0.541783\pi$
$467$	−5.65712	−0.261780	−0.130890	+	0.991397i	$0.541783\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.9792	−0.505895
$472$	0	0
$473$	3.71155	0.170657
$474$	0	0
$475$	70.0673	3.21491
$476$	0	0
$477$	42.8462	1.96179
$478$	0	0
$479$	−32.8238	−1.49976	−0.749878	−	0.661576i	$-0.769888\pi$
$479$	−32.8238	−1.49976	−0.749878	+	0.661576i	$0.769888\pi$
$480$	0	0
$481$	−32.6217	−1.48742
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−57.6234	−2.61654
$486$	0	0
$487$	−6.53033	−0.295918	−0.147959	−	0.988994i	$-0.547270\pi$
$487$	−6.53033	−0.295918	−0.147959	+	0.988994i	$0.547270\pi$
$488$	0	0
$489$	−40.0448	−1.81089
$490$	0	0
$491$	31.4455	1.41912	0.709558	−	0.704647i	$-0.248895\pi$
$491$	31.4455	1.41912	0.709558	+	0.704647i	$0.248895\pi$
$492$	0	0
$493$	2.64416	0.119087
$494$	0	0
$495$	21.4231	0.962896
$496$	0	0
$497$	0	0
$498$	0	0
$499$	23.6699	1.05961	0.529806	−	0.848119i	$-0.322265\pi$
$499$	23.6699	1.05961	0.529806	+	0.848119i	$0.322265\pi$
$500$	0	0
$501$	−54.5801	−2.43846
$502$	0	0
$503$	−5.57355	−0.248512	−0.124256	−	0.992250i	$-0.539654\pi$
$503$	−5.57355	−0.248512	−0.124256	+	0.992250i	$0.539654\pi$
$504$	0	0
$505$	11.2211	0.499331
$506$	0	0
$507$	43.0834	1.91340
$508$	0	0
$509$	9.82215	0.435359	0.217679	−	0.976020i	$-0.430151\pi$
$509$	9.82215	0.435359	0.217679	+	0.976020i	$0.430151\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	35.1571	1.55222
$514$	0	0
$515$	−26.6442	−1.17408
$516$	0	0
$517$	−5.01121	−0.220393
$518$	0	0
$519$	−55.1795	−2.42211
$520$	0	0
$521$	−40.1363	−1.75840	−0.879201	−	0.476452i	$-0.841923\pi$
$521$	−40.1363	−1.75840	−0.879201	+	0.476452i	$0.841923\pi$
$522$	0	0
$523$	18.2244	0.796899	0.398449	−	0.917190i	$-0.369548\pi$
$523$	18.2244	0.796899	0.398449	+	0.917190i	$0.369548\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.1505	0.877768
$528$	0	0
$529$	−10.3541	−0.450178
$530$	0	0
$531$	26.5207	1.15090
$532$	0	0
$533$	−23.3815	−1.01276
$534$	0	0
$535$	34.5353	1.49309
$536$	0	0
$537$	19.2261	0.829665
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.68913	−0.0726212	−0.0363106	−	0.999341i	$-0.511561\pi$
$541$	−1.68913	−0.0726212	−0.0363106	+	0.999341i	$0.511561\pi$
$542$	0	0
$543$	21.8254	0.936617
$544$	0	0
$545$	−62.8910	−2.69396
$546$	0	0
$547$	−25.0706	−1.07194	−0.535971	−	0.844236i	$-0.680054\pi$
$547$	−25.0706	−1.07194	−0.535971	+	0.844236i	$0.680054\pi$
$548$	0	0
$549$	−70.1683	−2.99471
$550$	0	0
$551$	−3.42309	−0.145829
$552$	0	0
$553$	0	0
$554$	0	0
$555$	73.0465	3.10065
$556$	0	0
$557$	26.0448	1.10356	0.551778	−	0.833991i	$-0.313950\pi$
$557$	26.0448	1.10356	0.551778	+	0.833991i	$0.313950\pi$
$558$	0	0
$559$	−19.6699	−0.831950
$560$	0	0
$561$	−12.5993	−0.531944
$562$	0	0
$563$	37.4455	1.57814	0.789070	−	0.614303i	$-0.210563\pi$
$563$	37.4455	1.57814	0.789070	+	0.614303i	$0.210563\pi$
$564$	0	0
$565$	−66.0274	−2.77779
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.97758	−0.334438	−0.167219	−	0.985920i	$-0.553479\pi$
$569$	−7.97758	−0.334438	−0.167219	+	0.985920i	$0.553479\pi$
$570$	0	0
$571$	9.19866	0.384952	0.192476	−	0.981302i	$-0.438348\pi$
$571$	9.19866	0.384952	0.192476	+	0.981302i	$0.438348\pi$
$572$	0	0
$573$	−5.26765	−0.220059
$574$	0	0
$575$	−43.6251	−1.81929
$576$	0	0
$577$	2.75476	0.114682	0.0573412	−	0.998355i	$-0.481738\pi$
$577$	2.75476	0.114682	0.0573412	+	0.998355i	$0.481738\pi$
$578$	0	0
$579$	−41.7564	−1.73534
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.31087	0.344201
$584$	0	0
$585$	−113.535	−4.69410
$586$	0	0
$587$	−6.20987	−0.256309	−0.128154	−	0.991754i	$-0.540905\pi$
$587$	−6.20987	−0.256309	−0.128154	+	0.991754i	$0.540905\pi$
$588$	0	0
$589$	−26.0864	−1.07487
$590$	0	0
$591$	−17.1346	−0.704825
$592$	0	0
$593$	25.2356	1.03630	0.518152	−	0.855289i	$-0.326620\pi$
$593$	25.2356	1.03630	0.518152	+	0.855289i	$0.326620\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.1571	1.19332
$598$	0	0
$599$	−14.6442	−0.598344	−0.299172	−	0.954199i	$-0.596710\pi$
$599$	−14.6442	−0.598344	−0.299172	+	0.954199i	$0.596710\pi$
$600$	0	0
$601$	9.87657	0.402874	0.201437	−	0.979501i	$-0.435439\pi$
$601$	9.87657	0.402874	0.201437	+	0.979501i	$0.435439\pi$
$602$	0	0
$603$	−2.28845	−0.0931931
$604$	0	0
$605$	4.15544	0.168943
$606$	0	0
$607$	−47.1571	−1.91405	−0.957023	−	0.290012i	$-0.906341\pi$
$607$	−47.1571	−1.91405	−0.957023	+	0.290012i	$0.906341\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.5577	1.07441
$612$	0	0
$613$	6.10886	0.246734	0.123367	−	0.992361i	$-0.460631\pi$
$613$	6.10886	0.246734	0.123367	+	0.992361i	$0.460631\pi$
$614$	0	0
$615$	52.3557	2.11119
$616$	0	0
$617$	−29.3815	−1.18285	−0.591427	−	0.806359i	$-0.701435\pi$
$617$	−29.3815	−1.18285	−0.591427	+	0.806359i	$0.701435\pi$
$618$	0	0
$619$	17.4551	0.701580	0.350790	−	0.936454i	$-0.385913\pi$
$619$	17.4551	0.701580	0.350790	+	0.936454i	$0.385913\pi$
$620$	0	0
$621$	−21.8894	−0.878391
$622$	0	0
$623$	0	0
$624$	0	0
$625$	64.1571	2.56628
$626$	0	0
$627$	16.3109	0.651394
$628$	0	0
$629$	−27.1571	−1.08282
$630$	0	0
$631$	−49.6234	−1.97547	−0.987737	−	0.156124i	$-0.950100\pi$
$631$	−49.6234	−1.97547	−0.987737	+	0.156124i	$0.950100\pi$
$632$	0	0
$633$	46.5801	1.85139
$634$	0	0
$635$	62.7979	2.49206
$636$	0	0
$637$	0	0
$638$	0	0
$639$	24.5129	0.969715
$640$	0	0
$641$	22.1554	0.875087	0.437544	−	0.899197i	$-0.355849\pi$
$641$	22.1554	0.875087	0.437544	+	0.899197i	$0.355849\pi$
$642$	0	0
$643$	−2.07685	−0.0819029	−0.0409514	−	0.999161i	$-0.513039\pi$
$643$	−2.07685	−0.0819029	−0.0409514	+	0.999161i	$0.513039\pi$
$644$	0	0
$645$	44.0448	1.73426
$646$	0	0
$647$	−3.67953	−0.144657	−0.0723287	−	0.997381i	$-0.523043\pi$
$647$	−3.67953	−0.144657	−0.0723287	+	0.997381i	$0.523043\pi$
$648$	0	0
$649$	5.14423	0.201929
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.0016	−1.36972	−0.684860	−	0.728675i	$-0.740136\pi$
$653$	−35.0016	−1.36972	−0.684860	+	0.728675i	$0.740136\pi$
$654$	0	0
$655$	15.3299	0.598990
$656$	0	0
$657$	43.3669	1.69190
$658$	0	0
$659$	48.0673	1.87243	0.936217	−	0.351422i	$-0.114302\pi$
$659$	48.0673	1.87243	0.936217	+	0.351422i	$0.114302\pi$
$660$	0	0
$661$	41.3989	1.61023	0.805116	−	0.593118i	$-0.202103\pi$
$661$	41.3989	1.61023	0.805116	+	0.593118i	$0.202103\pi$
$662$	0	0
$663$	66.7722	2.59322
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.13128	0.0825233
$668$	0	0
$669$	37.3541	1.44419
$670$	0	0
$671$	−13.6105	−0.525429
$672$	0	0
$673$	17.9360	0.691381	0.345691	−	0.938349i	$-0.387645\pi$
$673$	17.9360	0.691381	0.345691	+	0.938349i	$0.387645\pi$
$674$	0	0
$675$	75.5128	2.90649
$676$	0	0
$677$	−18.6363	−0.716252	−0.358126	−	0.933673i	$-0.616584\pi$
$677$	−18.6363	−0.716252	−0.358126	+	0.933673i	$0.616584\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	47.4679	1.81897
$682$	0	0
$683$	−1.19866	−0.0458653	−0.0229327	−	0.999737i	$-0.507300\pi$
$683$	−1.19866	−0.0458653	−0.0229327	+	0.999737i	$0.507300\pi$
$684$	0	0
$685$	12.6459	0.483175
$686$	0	0
$687$	−63.3349	−2.41638
$688$	0	0
$689$	−44.0448	−1.67797
$690$	0	0
$691$	47.7884	1.81796	0.908978	−	0.416844i	$-0.136864\pi$
$691$	47.7884	1.81796	0.908978	+	0.416844i	$0.136864\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−45.3367	−1.71972
$696$	0	0
$697$	−19.4647	−0.737278
$698$	0	0
$699$	−35.9808	−1.36092
$700$	0	0
$701$	−6.51289	−0.245988	−0.122994	−	0.992407i	$-0.539250\pi$
$701$	−6.51289	−0.245988	−0.122994	+	0.992407i	$0.539250\pi$
$702$	0	0
$703$	35.1571	1.32597
$704$	0	0
$705$	−59.4679	−2.23969
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−20.1554	−0.756953	−0.378477	−	0.925611i	$-0.623552\pi$
$709$	−20.1554	−0.756953	−0.378477	+	0.925611i	$0.623552\pi$
$710$	0	0
$711$	−41.3591	−1.55109
$712$	0	0
$713$	16.2419	0.608263
$714$	0	0
$715$	−22.0224	−0.823592
$716$	0	0
$717$	40.6217	1.51705
$718$	0	0
$719$	27.6795	1.03227	0.516136	−	0.856507i	$-0.327370\pi$
$719$	27.6795	1.03227	0.516136	+	0.856507i	$0.327370\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0.288455	0.0107278
$724$	0	0
$725$	−7.35235	−0.273059
$726$	0	0
$727$	−14.3429	−0.531948	−0.265974	−	0.963980i	$-0.585694\pi$
$727$	−14.3429	−0.531948	−0.265974	+	0.963980i	$0.585694\pi$
$728$	0	0
$729$	−41.8462	−1.54986
$730$	0	0
$731$	−16.3749	−0.605647
$732$	0	0
$733$	6.16826	0.227830	0.113915	−	0.993491i	$-0.463661\pi$
$733$	6.16826	0.227830	0.113915	+	0.993491i	$0.463661\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.443892	−0.0163510
$738$	0	0
$739$	−25.3367	−0.932024	−0.466012	−	0.884778i	$-0.654310\pi$
$739$	−25.3367	−0.932024	−0.466012	+	0.884778i	$0.654310\pi$
$740$	0	0
$741$	−86.4421	−3.17553
$742$	0	0
$743$	33.2435	1.21959	0.609793	−	0.792561i	$-0.291253\pi$
$743$	33.2435	1.21959	0.609793	+	0.792561i	$0.291253\pi$
$744$	0	0
$745$	29.9136	1.09595
$746$	0	0
$747$	−45.4903	−1.66440
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.7548	0.465428	0.232714	−	0.972545i	$-0.425239\pi$
$751$	12.7548	0.465428	0.232714	+	0.972545i	$0.425239\pi$
$752$	0	0
$753$	74.1587	2.70249
$754$	0	0
$755$	−33.3367	−1.21324
$756$	0	0
$757$	−21.4679	−0.780265	−0.390133	−	0.920759i	$-0.627571\pi$
$757$	−21.4679	−0.780265	−0.390133	+	0.920759i	$0.627571\pi$
$758$	0	0
$759$	−10.1554	−0.368619
$760$	0	0
$761$	40.1458	1.45529	0.727643	−	0.685956i	$-0.240616\pi$
$761$	40.1458	1.45529	0.727643	+	0.685956i	$0.240616\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−94.5161	−3.41724
$766$	0	0
$767$	−27.2627	−0.984398
$768$	0	0
$769$	−0.145844	−0.00525927	−0.00262964	−	0.999997i	$-0.500837\pi$
$769$	−0.145844	−0.00525927	−0.00262964	+	0.999997i	$0.500837\pi$
$770$	0	0
$771$	−7.35907	−0.265030
$772$	0	0
$773$	20.4713	0.736301	0.368150	−	0.929766i	$-0.379991\pi$
$773$	20.4713	0.736301	0.368150	+	0.929766i	$0.379991\pi$
$774$	0	0
$775$	−56.0303	−2.01267
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.1987	0.902836
$780$	0	0
$781$	4.75476	0.170139
$782$	0	0
$783$	−3.68913	−0.131839
$784$	0	0
$785$	−15.9758	−0.570202
$786$	0	0
$787$	−5.75638	−0.205193	−0.102596	−	0.994723i	$-0.532715\pi$
$787$	−5.75638	−0.205193	−0.102596	+	0.994723i	$0.532715\pi$
$788$	0	0
$789$	58.0032	2.06497
$790$	0	0
$791$	0	0
$792$	0	0
$793$	72.1313	2.56146
$794$	0	0
$795$	98.6250	3.49787
$796$	0	0
$797$	−33.0432	−1.17045	−0.585225	−	0.810871i	$-0.698994\pi$
$797$	−33.0432	−1.17045	−0.585225	+	0.810871i	$0.698994\pi$
$798$	0	0
$799$	22.1089	0.782155
$800$	0	0
$801$	−39.0706	−1.38049
$802$	0	0
$803$	8.41188	0.296849
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.06402	−0.283867
$808$	0	0
$809$	−29.4679	−1.03604	−0.518019	−	0.855369i	$-0.673330\pi$
$809$	−29.4679	−1.03604	−0.518019	+	0.855369i	$0.673330\pi$
$810$	0	0
$811$	13.5095	0.474384	0.237192	−	0.971463i	$-0.423773\pi$
$811$	13.5095	0.474384	0.237192	+	0.971463i	$0.423773\pi$
$812$	0	0
$813$	−13.7756	−0.483130
$814$	0	0
$815$	−58.2693	−2.04108
$816$	0	0
$817$	21.1987	0.741647
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.5960	−0.753705	−0.376852	−	0.926273i	$-0.622994\pi$
$821$	−21.5960	−0.753705	−0.376852	+	0.926273i	$0.622994\pi$
$822$	0	0
$823$	8.64591	0.301377	0.150689	−	0.988581i	$-0.451851\pi$
$823$	8.64591	0.301377	0.150689	+	0.988581i	$0.451851\pi$
$824$	0	0
$825$	35.0336	1.21971
$826$	0	0
$827$	−10.1089	−0.351519	−0.175760	−	0.984433i	$-0.556238\pi$
$827$	−10.1089	−0.351519	−0.175760	+	0.984433i	$0.556238\pi$
$828$	0	0
$829$	−20.4663	−0.710824	−0.355412	−	0.934710i	$-0.615659\pi$
$829$	−20.4663	−0.710824	−0.355412	+	0.934710i	$0.615659\pi$
$830$	0	0
$831$	39.2211	1.36056
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−79.4196	−2.74843
$836$	0	0
$837$	−28.1138	−0.971756
$838$	0	0
$839$	27.6795	0.955604	0.477802	−	0.878468i	$-0.341434\pi$
$839$	27.6795	0.955604	0.477802	+	0.878468i	$0.341434\pi$
$840$	0	0
$841$	−28.6408	−0.987614
$842$	0	0
$843$	−59.7148	−2.05669
$844$	0	0
$845$	62.6907	2.15663
$846$	0	0
$847$	0	0
$848$	0	0
$849$	14.8462	0.509520
$850$	0	0
$851$	−21.8894	−0.750359
$852$	0	0
$853$	55.1009	1.88662	0.943309	−	0.331915i	$-0.107695\pi$
$853$	55.1009	1.88662	0.943309	+	0.331915i	$0.107695\pi$
$854$	0	0
$855$	122.359	4.18459
$856$	0	0
$857$	43.5689	1.48829	0.744143	−	0.668020i	$-0.232858\pi$
$857$	43.5689	1.48829	0.744143	+	0.668020i	$0.232858\pi$
$858$	0	0
$859$	−9.45510	−0.322604	−0.161302	−	0.986905i	$-0.551569\pi$
$859$	−9.45510	−0.322604	−0.161302	+	0.986905i	$0.551569\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.77557	−0.332764	−0.166382	−	0.986061i	$-0.553209\pi$
$863$	−9.77557	−0.332764	−0.166382	+	0.986061i	$0.553209\pi$
$864$	0	0
$865$	−80.2917	−2.73000
$866$	0	0
$867$	7.03860	0.239043
$868$	0	0
$869$	−8.02242	−0.272142
$870$	0	0
$871$	2.35248	0.0797106
$872$	0	0
$873$	−71.4903	−2.41958
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.44551	0.116347	0.0581733	−	0.998307i	$-0.481472\pi$
$877$	3.44551	0.116347	0.0581733	+	0.998307i	$0.481472\pi$
$878$	0	0
$879$	50.9966	1.72007
$880$	0	0
$881$	11.7805	0.396897	0.198448	−	0.980111i	$-0.436410\pi$
$881$	11.7805	0.396897	0.198448	+	0.980111i	$0.436410\pi$
$882$	0	0
$883$	16.0448	0.539952	0.269976	−	0.962867i	$-0.412984\pi$
$883$	16.0448	0.539952	0.269976	+	0.962867i	$0.412984\pi$
$884$	0	0
$885$	61.0465	2.05205
$886$	0	0
$887$	−46.2885	−1.55421	−0.777107	−	0.629368i	$-0.783314\pi$
$887$	−46.2885	−1.55421	−0.777107	+	0.629368i	$0.783314\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.11222	0.0707619
$892$	0	0
$893$	−28.6217	−0.957790
$894$	0	0
$895$	27.9758	0.935129
$896$	0	0
$897$	53.8204	1.79701
$898$	0	0
$899$	2.73732	0.0912948
$900$	0	0
$901$	−36.6666	−1.22154
$902$	0	0
$903$	0	0
$904$	0	0
$905$	31.7581	1.05568
$906$	0	0
$907$	−8.37489	−0.278084	−0.139042	−	0.990286i	$-0.544402\pi$
$907$	−8.37489	−0.278084	−0.139042	+	0.990286i	$0.544402\pi$
$908$	0	0
$909$	13.9214	0.461744
$910$	0	0
$911$	−36.2469	−1.20091	−0.600456	−	0.799658i	$-0.705014\pi$
$911$	−36.2469	−1.20091	−0.600456	+	0.799658i	$0.705014\pi$
$912$	0	0
$913$	−8.82376	−0.292024
$914$	0	0
$915$	−161.516	−5.33955
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.60269	0.184816	0.0924078	−	0.995721i	$-0.470544\pi$
$919$	5.60269	0.184816	0.0924078	+	0.995721i	$0.470544\pi$
$920$	0	0
$921$	93.1603	3.06974
$922$	0	0
$923$	−25.1987	−0.829424
$924$	0	0
$925$	75.5128	2.48284
$926$	0	0
$927$	−33.0560	−1.08570
$928$	0	0
$929$	−25.7980	−0.846404	−0.423202	−	0.906035i	$-0.639094\pi$
$929$	−25.7980	−0.846404	−0.423202	+	0.906035i	$0.639094\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	71.5544	2.34258
$934$	0	0
$935$	−18.3333	−0.599563
$936$	0	0
$937$	−60.8124	−1.98666	−0.993328	−	0.115326i	$-0.963209\pi$
$937$	−60.8124	−1.98666	−0.993328	+	0.115326i	$0.963209\pi$
$938$	0	0
$939$	56.7996	1.85358
$940$	0	0
$941$	−42.2771	−1.37819	−0.689097	−	0.724669i	$-0.741993\pi$
$941$	−42.2771	−1.37819	−0.689097	+	0.724669i	$0.741993\pi$
$942$	0	0
$943$	−15.6891	−0.510908
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.4405	−0.956689	−0.478344	−	0.878172i	$-0.658763\pi$
$947$	−29.4405	−0.956689	−0.478344	+	0.878172i	$0.658763\pi$
$948$	0	0
$949$	−44.5801	−1.44713
$950$	0	0
$951$	−25.8254	−0.837445
$952$	0	0
$953$	−24.5801	−0.796229	−0.398114	−	0.917336i	$-0.630335\pi$
$953$	−24.5801	−0.796229	−0.398114	+	0.917336i	$0.630335\pi$
$954$	0	0
$955$	−7.66497	−0.248032
$956$	0	0
$957$	−1.71155	−0.0553264
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−10.1396	−0.327084
$962$	0	0
$963$	42.8462	1.38070
$964$	0	0
$965$	−60.7597	−1.95593
$966$	0	0
$967$	−12.6442	−0.406609	−0.203304	−	0.979116i	$-0.565168\pi$
$967$	−12.6442	−0.406609	−0.203304	+	0.979116i	$0.565168\pi$
$968$	0	0
$969$	−71.9616	−2.31174
$970$	0	0
$971$	39.6795	1.27338	0.636688	−	0.771121i	$-0.280304\pi$
$971$	39.6795	1.27338	0.636688	+	0.771121i	$0.280304\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−185.666	−5.94609
$976$	0	0
$977$	47.1745	1.50925	0.754623	−	0.656159i	$-0.227820\pi$
$977$	47.1745	1.50925	0.754623	+	0.656159i	$0.227820\pi$
$978$	0	0
$979$	−7.57853	−0.242211
$980$	0	0
$981$	−78.0257	−2.49117
$982$	0	0
$983$	51.6795	1.64832	0.824161	−	0.566356i	$-0.191647\pi$
$983$	51.6795	1.64832	0.824161	+	0.566356i	$0.191647\pi$
$984$	0	0
$985$	−24.9326	−0.794419
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.1987	−0.419693
$990$	0	0
$991$	55.2211	1.75416	0.877078	−	0.480349i	$-0.159490\pi$
$991$	55.2211	1.75416	0.877078	+	0.480349i	$0.159490\pi$
$992$	0	0
$993$	85.3573	2.70873
$994$	0	0
$995$	42.4264	1.34501
$996$	0	0
$997$	46.5432	1.47404	0.737018	−	0.675873i	$-0.236233\pi$
$997$	46.5432	1.47404	0.737018	+	0.675873i	$0.236233\pi$
$998$	0	0
$999$	37.8894	1.19877

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.a.j.1.3		3
4.3	odd	2		8624.2.a.cj.1.1		3
7.2	even	3		2156.2.i.k.1145.1		6
7.3	odd	6		2156.2.i.m.177.3		6
7.4	even	3		2156.2.i.k.177.1		6
7.5	odd	6		2156.2.i.m.1145.3		6
7.6	odd	2		308.2.a.c.1.1	✓	3
21.20	even	2		2772.2.a.s.1.3		3
28.27	even	2		1232.2.a.r.1.3		3
35.13	even	4		7700.2.e.p.1849.1		6
35.27	even	4		7700.2.e.p.1849.6		6
35.34	odd	2		7700.2.a.y.1.3		3
56.13	odd	2		4928.2.a.ca.1.3		3
56.27	even	2		4928.2.a.bx.1.1		3
77.76	even	2		3388.2.a.o.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.a.c.1.1	✓	3	7.6	odd	2
1232.2.a.r.1.3		3	28.27	even	2
2156.2.a.j.1.3		3	1.1	even	1	trivial
2156.2.i.k.177.1		6	7.4	even	3
2156.2.i.k.1145.1		6	7.2	even	3
2156.2.i.m.177.3		6	7.3	odd	6
2156.2.i.m.1145.3		6	7.5	odd	6
2772.2.a.s.1.3		3	21.20	even	2
3388.2.a.o.1.1		3	77.76	even	2
4928.2.a.bx.1.1		3	56.27	even	2
4928.2.a.ca.1.3		3	56.13	odd	2
7700.2.a.y.1.3		3	35.34	odd	2
7700.2.e.p.1849.1		6	35.13	even	4
7700.2.e.p.1849.6		6	35.27	even	4
8624.2.a.cj.1.1		3	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)

\(f(q)\)	\(=\)	\(q+2.85577 q^{3} +4.15544 q^{5} +5.15544 q^{9} +1.00000 q^{11} -5.29966 q^{13} +11.8670 q^{15} -4.41188 q^{17} +5.71155 q^{19} -3.55611 q^{23} +12.2677 q^{25} +6.15544 q^{27} -0.599328 q^{29} -4.56732 q^{31} +2.85577 q^{33} +6.15544 q^{37} -15.1346 q^{39} +4.41188 q^{41} +3.71155 q^{43} +21.4231 q^{45} -5.01121 q^{47} -12.5993 q^{51} +8.31087 q^{53} +4.15544 q^{55} +16.3109 q^{57} +5.14423 q^{59} -13.6105 q^{61} -22.0224 q^{65} -0.443892 q^{67} -10.1554 q^{69} +4.75476 q^{71} +8.41188 q^{73} +35.0336 q^{75} -8.02242 q^{79} +2.11222 q^{81} -8.82376 q^{83} -18.3333 q^{85} -1.71155 q^{87} -7.57853 q^{89} -13.0432 q^{93} +23.7340 q^{95} -13.8670 q^{97} +5.15544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} + 4 q^{9} + 3 q^{11} - 12 q^{13} + 9 q^{15} - 2 q^{17} + 2 q^{19} - 7 q^{23} + 18 q^{25} + 7 q^{27} + 6 q^{29} + 9 q^{31} + q^{33} + 7 q^{37} + 2 q^{41} - 4 q^{43} + 34 q^{45} + 4 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q + q^3 + q^5 + 4 * q^9 + 3 * q^11 - 12 * q^13 + 9 * q^15 - 2 * q^17 + 2 * q^19 - 7 * q^23 + 18 * q^25 + 7 * q^27 + 6 * q^29 + 9 * q^31 + q^33 + 7 * q^37 + 2 * q^41 - 4 * q^43 + 34 * q^45 + 4 * q^47 - 30 * q^51 + 2 * q^53 + q^55 + 26 * q^57 + 23 * q^59 - 14 * q^61 - 28 * q^65 - 5 * q^67 - 19 * q^69 - 5 * q^71 + 14 * q^73 + 48 * q^75 + 14 * q^79 - q^81 - 4 * q^83 + 6 * q^85 + 10 * q^87 + 19 * q^89 - 35 * q^93 + 18 * q^95 - 15 * q^97 + 4 * q^99

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