LMFDB - Embedded newform 2160.4.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.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:	$127.444125612$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 540)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} -12.7477 q^{7} +O(q^{10})$	$q+5.00000 q^{5} -12.7477 q^{7} +34.7477 q^{11} -75.7386 q^{13} +84.2341 q^{17} +83.7477 q^{19} +172.982 q^{23} +25.0000 q^{25} -88.2250 q^{29} -205.711 q^{31} -63.7386 q^{35} -286.000 q^{37} +321.702 q^{41} +168.766 q^{43} -390.468 q^{47} -180.495 q^{49} -91.6932 q^{53} +173.739 q^{55} +122.766 q^{59} -878.405 q^{61} -378.693 q^{65} +1037.40 q^{67} +605.216 q^{71} -13.8114 q^{73} -442.955 q^{77} +573.298 q^{79} -627.936 q^{83} +421.170 q^{85} +1013.11 q^{89} +965.495 q^{91} +418.739 q^{95} +1155.82 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 10 * q^5 + 2 * q^7	$ 2 q + 10 q^{5} + 2 q^{7} + 42 q^{11} - 14 q^{13} - 24 q^{17} + 140 q^{19} + 126 q^{23} + 50 q^{25} + 126 q^{29} + 56 q^{31} + 10 q^{35} - 572 q^{37} + 66 q^{41} + 530 q^{43} - 396 q^{47} - 306 q^{49} + 504 q^{53} + 210 q^{55} + 438 q^{59} - 602 q^{61} - 70 q^{65} + 920 q^{67} + 798 q^{71} - 770 q^{73} - 336 q^{77} + 1724 q^{79} - 486 q^{83} - 120 q^{85} + 294 q^{89} + 1876 q^{91} + 700 q^{95} + 112 q^{97}+O(q^{100}) $ 2 * q + 10 * q^5 + 2 * q^7 + 42 * q^11 - 14 * q^13 - 24 * q^17 + 140 * q^19 + 126 * q^23 + 50 * q^25 + 126 * q^29 + 56 * q^31 + 10 * q^35 - 572 * q^37 + 66 * q^41 + 530 * q^43 - 396 * q^47 - 306 * q^49 + 504 * q^53 + 210 * q^55 + 438 * q^59 - 602 * q^61 - 70 * q^65 + 920 * q^67 + 798 * q^71 - 770 * q^73 - 336 * q^77 + 1724 * q^79 - 486 * q^83 - 120 * q^85 + 294 * q^89 + 1876 * q^91 + 700 * q^95 + 112 * 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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−12.7477	−0.688313	−0.344156	−	0.938912i	$-0.611835\pi$
$7$	−12.7477	−0.688313	−0.344156	+	0.938912i	$0.611835\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	34.7477	0.952439	0.476220	−	0.879326i	$-0.342007\pi$
$11$	34.7477	0.952439	0.476220	+	0.879326i	$0.342007\pi$
$12$	0	0
$13$	−75.7386	−1.61586	−0.807928	−	0.589282i	$-0.799411\pi$
$13$	−75.7386	−1.61586	−0.807928	+	0.589282i	$0.799411\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	84.2341	1.20175	0.600876	−	0.799343i	$-0.294819\pi$
$17$	84.2341	1.20175	0.600876	+	0.799343i	$0.294819\pi$
$18$	0	0
$19$	83.7477	1.01121	0.505606	−	0.862764i	$-0.331269\pi$
$19$	83.7477	1.01121	0.505606	+	0.862764i	$0.331269\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	172.982	1.56823	0.784113	−	0.620618i	$-0.213118\pi$
$23$	172.982	1.56823	0.784113	+	0.620618i	$0.213118\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−88.2250	−0.564930	−0.282465	−	0.959278i	$-0.591152\pi$
$29$	−88.2250	−0.564930	−0.282465	+	0.959278i	$0.591152\pi$
$30$	0	0
$31$	−205.711	−1.19183	−0.595917	−	0.803046i	$-0.703211\pi$
$31$	−205.711	−1.19183	−0.595917	+	0.803046i	$0.703211\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−63.7386	−0.307823
$36$	0	0
$37$	−286.000	−1.27076	−0.635380	−	0.772200i	$-0.719156\pi$
$37$	−286.000	−1.27076	−0.635380	+	0.772200i	$0.719156\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	321.702	1.22540	0.612701	−	0.790315i	$-0.290083\pi$
$41$	321.702	1.22540	0.612701	+	0.790315i	$0.290083\pi$
$42$	0	0
$43$	168.766	0.598525	0.299262	−	0.954171i	$-0.403259\pi$
$43$	168.766	0.598525	0.299262	+	0.954171i	$0.403259\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−390.468	−1.21182	−0.605911	−	0.795532i	$-0.707191\pi$
$47$	−390.468	−1.21182	−0.605911	+	0.795532i	$0.707191\pi$
$48$	0	0
$49$	−180.495	−0.526226
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−91.6932	−0.237642	−0.118821	−	0.992916i	$-0.537911\pi$
$53$	−91.6932	−0.237642	−0.118821	+	0.992916i	$0.537911\pi$
$54$	0	0
$55$	173.739	0.425944
$56$	0	0
$57$	0	0
$58$	0	0
$59$	122.766	0.270894	0.135447	−	0.990785i	$-0.456753\pi$
$59$	122.766	0.270894	0.135447	+	0.990785i	$0.456753\pi$
$60$	0	0
$61$	−878.405	−1.84374	−0.921870	−	0.387499i	$-0.873339\pi$
$61$	−878.405	−1.84374	−0.921870	+	0.387499i	$0.873339\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−378.693	−0.722632
$66$	0	0
$67$	1037.40	1.89163	0.945814	−	0.324708i	$-0.105266\pi$
$67$	1037.40	1.89163	0.945814	+	0.324708i	$0.105266\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	605.216	1.01163	0.505816	−	0.862641i	$-0.331191\pi$
$71$	605.216	1.01163	0.505816	+	0.862641i	$0.331191\pi$
$72$	0	0
$73$	−13.8114	−0.0221438	−0.0110719	−	0.999939i	$-0.503524\pi$
$73$	−13.8114	−0.0221438	−0.0110719	+	0.999939i	$0.503524\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−442.955	−0.655576
$78$	0	0
$79$	573.298	0.816469	0.408234	−	0.912877i	$-0.366145\pi$
$79$	573.298	0.816469	0.408234	+	0.912877i	$0.366145\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−627.936	−0.830421	−0.415211	−	0.909725i	$-0.636292\pi$
$83$	−627.936	−0.830421	−0.415211	+	0.909725i	$0.636292\pi$
$84$	0	0
$85$	421.170	0.537439
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1013.11	1.20662	0.603310	−	0.797507i	$-0.293848\pi$
$89$	1013.11	1.20662	0.603310	+	0.797507i	$0.293848\pi$
$90$	0	0
$91$	965.495	1.11221
$92$	0	0
$93$	0	0
$94$	0	0
$95$	418.739	0.452228
$96$	0	0
$97$	1155.82	1.20985	0.604926	−	0.796282i	$-0.293203\pi$
$97$	1155.82	1.20985	0.604926	+	0.796282i	$0.293203\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	853.873	0.841223	0.420611	−	0.907241i	$-0.361816\pi$
$101$	853.873	0.841223	0.420611	+	0.907241i	$0.361816\pi$
$102$	0	0
$103$	1029.95	0.985277	0.492639	−	0.870234i	$-0.336032\pi$
$103$	1029.95	0.985277	0.492639	+	0.870234i	$0.336032\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	514.845	0.465159	0.232579	−	0.972577i	$-0.425283\pi$
$107$	514.845	0.465159	0.232579	+	0.972577i	$0.425283\pi$
$108$	0	0
$109$	−656.873	−0.577220	−0.288610	−	0.957447i	$-0.593193\pi$
$109$	−656.873	−0.577220	−0.288610	+	0.957447i	$0.593193\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1701.44	−1.41644	−0.708222	−	0.705990i	$-0.750502\pi$
$113$	−1701.44	−1.41644	−0.708222	+	0.705990i	$0.750502\pi$
$114$	0	0
$115$	864.909	0.701332
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1073.79	−0.827180
$120$	0	0
$121$	−123.595	−0.0928591
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−345.405	−0.241336	−0.120668	−	0.992693i	$-0.538504\pi$
$127$	−345.405	−0.241336	−0.120668	+	0.992693i	$0.538504\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2827.87	1.88605	0.943024	−	0.332724i	$-0.107968\pi$
$131$	2827.87	1.88605	0.943024	+	0.332724i	$0.107968\pi$
$132$	0	0
$133$	−1067.59	−0.696031
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1227.84	0.765703	0.382852	−	0.923810i	$-0.374942\pi$
$137$	1227.84	0.765703	0.382852	+	0.923810i	$0.374942\pi$
$138$	0	0
$139$	40.1091	0.0244749	0.0122374	−	0.999925i	$-0.496105\pi$
$139$	40.1091	0.0244749	0.0122374	+	0.999925i	$0.496105\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2631.75	−1.53900
$144$	0	0
$145$	−441.125	−0.252644
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−480.730	−0.264315	−0.132157	−	0.991229i	$-0.542190\pi$
$149$	−480.730	−0.264315	−0.132157	+	0.991229i	$0.542190\pi$
$150$	0	0
$151$	76.0000	0.0409589	0.0204794	−	0.999790i	$-0.493481\pi$
$151$	76.0000	0.0409589	0.0204794	+	0.999790i	$0.493481\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1028.56	−0.533004
$156$	0	0
$157$	−438.602	−0.222957	−0.111479	−	0.993767i	$-0.535559\pi$
$157$	−438.602	−0.222957	−0.111479	+	0.993767i	$0.535559\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2205.12	−1.07943
$162$	0	0
$163$	−1755.87	−0.843746	−0.421873	−	0.906655i	$-0.638627\pi$
$163$	−1755.87	−0.843746	−0.421873	+	0.906655i	$0.638627\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2787.55	−1.29166	−0.645831	−	0.763481i	$-0.723489\pi$
$167$	−2787.55	−1.29166	−0.645831	+	0.763481i	$0.723489\pi$
$168$	0	0
$169$	3539.34	1.61099
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1216.36	0.534556	0.267278	−	0.963619i	$-0.413876\pi$
$173$	1216.36	0.534556	0.267278	+	0.963619i	$0.413876\pi$
$174$	0	0
$175$	−318.693	−0.137663
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2531.98	1.05726	0.528628	−	0.848854i	$-0.322707\pi$
$179$	2531.98	1.05726	0.528628	+	0.848854i	$0.322707\pi$
$180$	0	0
$181$	3444.82	1.41465	0.707324	−	0.706889i	$-0.249902\pi$
$181$	3444.82	1.41465	0.707324	+	0.706889i	$0.249902\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1430.00	−0.568301
$186$	0	0
$187$	2926.94	1.14460
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1702.57	0.644994	0.322497	−	0.946571i	$-0.395478\pi$
$191$	1702.57	0.644994	0.322497	+	0.946571i	$0.395478\pi$
$192$	0	0
$193$	685.489	0.255661	0.127830	−	0.991796i	$-0.459199\pi$
$193$	685.489	0.255661	0.127830	+	0.991796i	$0.459199\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4362.95	1.57791	0.788953	−	0.614454i	$-0.210623\pi$
$197$	4362.95	1.57791	0.788953	+	0.614454i	$0.210623\pi$
$198$	0	0
$199$	2920.20	1.04024	0.520119	−	0.854094i	$-0.325888\pi$
$199$	2920.20	1.04024	0.520119	+	0.854094i	$0.325888\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1124.67	0.388848
$204$	0	0
$205$	1608.51	0.548016
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2910.04	0.963119
$210$	0	0
$211$	213.384	0.0696207	0.0348103	−	0.999394i	$-0.488917\pi$
$211$	213.384	0.0696207	0.0348103	+	0.999394i	$0.488917\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	843.830	0.267668
$216$	0	0
$217$	2622.35	0.820354
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6379.78	−1.94186
$222$	0	0
$223$	−2091.55	−0.628073	−0.314036	−	0.949411i	$-0.601681\pi$
$223$	−2091.55	−0.628073	−0.314036	+	0.949411i	$0.601681\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4483.60	−1.31095	−0.655477	−	0.755215i	$-0.727532\pi$
$227$	−4483.60	−1.31095	−0.655477	+	0.755215i	$0.727532\pi$
$228$	0	0
$229$	4931.09	1.42295	0.711475	−	0.702712i	$-0.248028\pi$
$229$	4931.09	1.42295	0.711475	+	0.702712i	$0.248028\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	773.782	0.217563	0.108781	−	0.994066i	$-0.465305\pi$
$233$	773.782	0.217563	0.108781	+	0.994066i	$0.465305\pi$
$234$	0	0
$235$	−1952.34	−0.541943
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4412.40	1.19420	0.597101	−	0.802166i	$-0.296319\pi$
$239$	4412.40	1.19420	0.597101	+	0.802166i	$0.296319\pi$
$240$	0	0
$241$	6143.41	1.64204	0.821021	−	0.570898i	$-0.193405\pi$
$241$	6143.41	1.64204	0.821021	+	0.570898i	$0.193405\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−902.477	−0.235335
$246$	0	0
$247$	−6342.94	−1.63397
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−746.895	−0.187823	−0.0939116	−	0.995581i	$-0.529937\pi$
$251$	−746.895	−0.187823	−0.0939116	+	0.995581i	$0.529937\pi$
$252$	0	0
$253$	6010.72	1.49364
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7546.62	−1.83169	−0.915847	−	0.401528i	$-0.868479\pi$
$257$	−7546.62	−1.83169	−0.915847	+	0.401528i	$0.868479\pi$
$258$	0	0
$259$	3645.85	0.874680
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1383.99	−0.324488	−0.162244	−	0.986751i	$-0.551873\pi$
$263$	−1383.99	−0.324488	−0.162244	+	0.986751i	$0.551873\pi$
$264$	0	0
$265$	−458.466	−0.106277
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6987.28	1.58372	0.791862	−	0.610700i	$-0.209112\pi$
$269$	6987.28	1.58372	0.791862	+	0.610700i	$0.209112\pi$
$270$	0	0
$271$	3600.27	0.807014	0.403507	−	0.914977i	$-0.367791\pi$
$271$	3600.27	0.807014	0.403507	+	0.914977i	$0.367791\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	868.693	0.190488
$276$	0	0
$277$	1803.74	0.391251	0.195625	−	0.980679i	$-0.437326\pi$
$277$	1803.74	0.391251	0.195625	+	0.980679i	$0.437326\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3712.97	−0.788245	−0.394123	−	0.919058i	$-0.628952\pi$
$281$	−3712.97	−0.788245	−0.394123	+	0.919058i	$0.628952\pi$
$282$	0	0
$283$	828.961	0.174122	0.0870612	−	0.996203i	$-0.472252\pi$
$283$	828.961	0.174122	0.0870612	+	0.996203i	$0.472252\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4100.97	−0.843459
$288$	0	0
$289$	2182.38	0.444206
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1686.32	0.336232	0.168116	−	0.985767i	$-0.446232\pi$
$293$	1686.32	0.336232	0.168116	+	0.985767i	$0.446232\pi$
$294$	0	0
$295$	613.830	0.121148
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13101.4	−2.53403
$300$	0	0
$301$	−2151.38	−0.411972
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4392.02	−0.824546
$306$	0	0
$307$	5760.81	1.07097	0.535483	−	0.844546i	$-0.320129\pi$
$307$	5760.81	1.07097	0.535483	+	0.844546i	$0.320129\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6887.96	1.25588	0.627942	−	0.778260i	$-0.283897\pi$
$311$	6887.96	1.25588	0.627942	+	0.778260i	$0.283897\pi$
$312$	0	0
$313$	−3056.66	−0.551989	−0.275995	−	0.961159i	$-0.589007\pi$
$313$	−3056.66	−0.551989	−0.275995	+	0.961159i	$0.589007\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9656.47	1.71092	0.855459	−	0.517870i	$-0.173275\pi$
$317$	9656.47	1.71092	0.855459	+	0.517870i	$0.173275\pi$
$318$	0	0
$319$	−3065.62	−0.538062
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7054.41	1.21523
$324$	0	0
$325$	−1893.47	−0.323171
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4977.58	0.834112
$330$	0	0
$331$	−8879.07	−1.47443	−0.737217	−	0.675656i	$-0.763861\pi$
$331$	−8879.07	−1.47443	−0.737217	+	0.675656i	$0.763861\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5187.02	0.845962
$336$	0	0
$337$	2610.17	0.421914	0.210957	−	0.977495i	$-0.432342\pi$
$337$	2610.17	0.421914	0.210957	+	0.977495i	$0.432342\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7148.00	−1.13515
$342$	0	0
$343$	6673.38	1.05052
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3806.48	−0.588884	−0.294442	−	0.955669i	$-0.595134\pi$
$347$	−3806.48	−0.588884	−0.294442	+	0.955669i	$0.595134\pi$
$348$	0	0
$349$	4080.99	0.625932	0.312966	−	0.949764i	$-0.398677\pi$
$349$	4080.99	0.625932	0.312966	+	0.949764i	$0.398677\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5016.64	0.756399	0.378200	−	0.925724i	$-0.376543\pi$
$353$	5016.64	0.756399	0.378200	+	0.925724i	$0.376543\pi$
$354$	0	0
$355$	3026.08	0.452416
$356$	0	0
$357$	0	0
$358$	0	0
$359$	515.859	0.0758384	0.0379192	−	0.999281i	$-0.487927\pi$
$359$	515.859	0.0758384	0.0379192	+	0.999281i	$0.487927\pi$
$360$	0	0
$361$	154.682	0.0225517
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−69.0568	−0.00990301
$366$	0	0
$367$	−4562.29	−0.648909	−0.324455	−	0.945901i	$-0.605181\pi$
$367$	−4562.29	−0.648909	−0.324455	+	0.945901i	$0.605181\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1168.88	0.163572
$372$	0	0
$373$	10438.1	1.44897	0.724485	−	0.689290i	$-0.242078\pi$
$373$	10438.1	1.44897	0.724485	+	0.689290i	$0.242078\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6682.04	0.912845
$378$	0	0
$379$	−7375.64	−0.999634	−0.499817	−	0.866131i	$-0.666599\pi$
$379$	−7375.64	−0.999634	−0.499817	+	0.866131i	$0.666599\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7277.13	0.970872	0.485436	−	0.874272i	$-0.338661\pi$
$383$	7277.13	0.970872	0.485436	+	0.874272i	$0.338661\pi$
$384$	0	0
$385$	−2214.77	−0.293183
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11099.4	−1.44669	−0.723344	−	0.690488i	$-0.757396\pi$
$389$	−11099.4	−1.44669	−0.723344	+	0.690488i	$0.757396\pi$
$390$	0	0
$391$	14571.0	1.88462
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2866.49	0.365136
$396$	0	0
$397$	5.73405	0.000724896 0	0.000362448	−	1.00000i	$-0.499885\pi$
$397$	5.73405	0.000724896 0	0.000362448		1.00000i	$0.499885\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6078.65	−0.756991	−0.378496	−	0.925603i	$-0.623559\pi$
$401$	−6078.65	−0.756991	−0.378496	+	0.925603i	$0.623559\pi$
$402$	0	0
$403$	15580.3	1.92583
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9937.85	−1.21032
$408$	0	0
$409$	−11154.5	−1.34855	−0.674273	−	0.738482i	$-0.735543\pi$
$409$	−11154.5	−1.34855	−0.674273	+	0.738482i	$0.735543\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1564.99	−0.186460
$414$	0	0
$415$	−3139.68	−0.371376
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6718.81	0.783378	0.391689	−	0.920098i	$-0.371891\pi$
$419$	6718.81	0.783378	0.391689	+	0.920098i	$0.371891\pi$
$420$	0	0
$421$	−9299.47	−1.07655	−0.538276	−	0.842768i	$-0.680924\pi$
$421$	−9299.47	−1.07655	−0.538276	+	0.842768i	$0.680924\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2105.85	0.240350
$426$	0	0
$427$	11197.7	1.26907
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2338.96	0.261400	0.130700	−	0.991422i	$-0.458277\pi$
$431$	2338.96	0.261400	0.130700	+	0.991422i	$0.458277\pi$
$432$	0	0
$433$	11685.4	1.29691	0.648457	−	0.761251i	$-0.275414\pi$
$433$	11685.4	1.29691	0.648457	+	0.761251i	$0.275414\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14486.8	1.58581
$438$	0	0
$439$	16103.8	1.75078	0.875388	−	0.483421i	$-0.160606\pi$
$439$	16103.8	1.75078	0.875388	+	0.483421i	$0.160606\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11263.9	−1.20804	−0.604021	−	0.796968i	$-0.706436\pi$
$443$	−11263.9	−1.20804	−0.604021	+	0.796968i	$0.706436\pi$
$444$	0	0
$445$	5065.53	0.539617
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10612.8	1.11548	0.557739	−	0.830016i	$-0.311669\pi$
$449$	10612.8	1.11548	0.557739	+	0.830016i	$0.311669\pi$
$450$	0	0
$451$	11178.4	1.16712
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4827.48	0.497397
$456$	0	0
$457$	2493.02	0.255183	0.127592	−	0.991827i	$-0.459275\pi$
$457$	2493.02	0.255183	0.127592	+	0.991827i	$0.459275\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15411.4	−1.55701	−0.778506	−	0.627637i	$-0.784022\pi$
$461$	−15411.4	−1.55701	−0.778506	+	0.627637i	$0.784022\pi$
$462$	0	0
$463$	8831.79	0.886497	0.443248	−	0.896399i	$-0.353826\pi$
$463$	8831.79	0.886497	0.443248	+	0.896399i	$0.353826\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8430.83	−0.835401	−0.417701	−	0.908585i	$-0.637164\pi$
$467$	−8430.83	−0.835401	−0.417701	+	0.908585i	$0.637164\pi$
$468$	0	0
$469$	−13224.5	−1.30203
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5864.23	0.570058
$474$	0	0
$475$	2093.69	0.202243
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8626.05	0.822827	0.411414	−	0.911449i	$-0.365035\pi$
$479$	8626.05	0.822827	0.411414	+	0.911449i	$0.365035\pi$
$480$	0	0
$481$	21661.2	2.05336
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5779.09	0.541062
$486$	0	0
$487$	12182.8	1.13358	0.566792	−	0.823861i	$-0.308184\pi$
$487$	12182.8	1.13358	0.566792	+	0.823861i	$0.308184\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21020.9	1.93210	0.966049	−	0.258357i	$-0.0831812\pi$
$491$	21020.9	1.93210	0.966049	+	0.258357i	$0.0831812\pi$
$492$	0	0
$493$	−7431.55	−0.678905
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7715.13	−0.696319
$498$	0	0
$499$	8890.70	0.797600	0.398800	−	0.917038i	$-0.369427\pi$
$499$	8890.70	0.797600	0.398800	+	0.917038i	$0.369427\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2433.17	−0.215686	−0.107843	−	0.994168i	$-0.534394\pi$
$503$	−2433.17	−0.215686	−0.107843	+	0.994168i	$0.534394\pi$
$504$	0	0
$505$	4269.36	0.376206
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15257.6	1.32865	0.664324	−	0.747445i	$-0.268719\pi$
$509$	15257.6	1.32865	0.664324	+	0.747445i	$0.268719\pi$
$510$	0	0
$511$	176.064	0.0152419
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5149.73	0.440629
$516$	0	0
$517$	−13567.9	−1.15419
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12991.2	−1.09242	−0.546212	−	0.837647i	$-0.683931\pi$
$521$	−12991.2	−1.09242	−0.546212	+	0.837647i	$0.683931\pi$
$522$	0	0
$523$	−1501.60	−0.125545	−0.0627727	−	0.998028i	$-0.519994\pi$
$523$	−1501.60	−0.125545	−0.0627727	+	0.998028i	$0.519994\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17327.9	−1.43229
$528$	0	0
$529$	17755.7	1.45933
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−24365.3	−1.98007
$534$	0	0
$535$	2574.23	0.208025
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6271.81	−0.501198
$540$	0	0
$541$	−18683.1	−1.48475	−0.742375	−	0.669985i	$-0.766301\pi$
$541$	−18683.1	−1.48475	−0.742375	+	0.669985i	$0.766301\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3284.36	−0.258141
$546$	0	0
$547$	2830.89	0.221280	0.110640	−	0.993861i	$-0.464710\pi$
$547$	2830.89	0.221280	0.110640	+	0.993861i	$0.464710\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7388.64	−0.571265
$552$	0	0
$553$	−7308.24	−0.561986
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2065.19	0.157100	0.0785500	−	0.996910i	$-0.474971\pi$
$557$	2065.19	0.157100	0.0785500	+	0.996910i	$0.474971\pi$
$558$	0	0
$559$	−12782.1	−0.967129
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20528.0	−1.53668	−0.768341	−	0.640040i	$-0.778918\pi$
$563$	−20528.0	−1.53668	−0.768341	+	0.640040i	$0.778918\pi$
$564$	0	0
$565$	−8507.20	−0.633453
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17758.5	1.30839	0.654195	−	0.756326i	$-0.273008\pi$
$569$	17758.5	1.30839	0.654195	+	0.756326i	$0.273008\pi$
$570$	0	0
$571$	−12609.1	−0.924126	−0.462063	−	0.886847i	$-0.652891\pi$
$571$	−12609.1	−0.924126	−0.462063	+	0.886847i	$0.652891\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4324.55	0.313645
$576$	0	0
$577$	1266.62	0.0913863	0.0456932	−	0.998956i	$-0.485450\pi$
$577$	1266.62	0.0913863	0.0456932	+	0.998956i	$0.485450\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8004.76	0.571589
$582$	0	0
$583$	−3186.13	−0.226340
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8943.52	0.628856	0.314428	−	0.949281i	$-0.398187\pi$
$587$	8943.52	0.628856	0.314428	+	0.949281i	$0.398187\pi$
$588$	0	0
$589$	−17227.9	−1.20520
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16200.0	1.12185	0.560924	−	0.827868i	$-0.310446\pi$
$593$	16200.0	1.12185	0.560924	+	0.827868i	$0.310446\pi$
$594$	0	0
$595$	−5368.97	−0.369926
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19630.1	−1.33900	−0.669501	−	0.742811i	$-0.733492\pi$
$599$	−19630.1	−1.33900	−0.669501	+	0.742811i	$0.733492\pi$
$600$	0	0
$601$	11537.5	0.783066	0.391533	−	0.920164i	$-0.371945\pi$
$601$	11537.5	0.783066	0.391533	+	0.920164i	$0.371945\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−617.977	−0.0415279
$606$	0	0
$607$	6346.35	0.424366	0.212183	−	0.977230i	$-0.431943\pi$
$607$	6346.35	0.424366	0.212183	+	0.977230i	$0.431943\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29573.5	1.95813
$612$	0	0
$613$	−12007.3	−0.791144	−0.395572	−	0.918435i	$-0.629454\pi$
$613$	−12007.3	−0.791144	−0.395572	+	0.918435i	$0.629454\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8650.85	−0.564457	−0.282229	−	0.959347i	$-0.591074\pi$
$617$	−8650.85	−0.564457	−0.282229	+	0.959347i	$0.591074\pi$
$618$	0	0
$619$	−11528.8	−0.748597	−0.374299	−	0.927308i	$-0.622117\pi$
$619$	−11528.8	−0.748597	−0.374299	+	0.927308i	$0.622117\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12914.8	−0.830531
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24090.9	−1.52714
$630$	0	0
$631$	4739.76	0.299028	0.149514	−	0.988760i	$-0.452229\pi$
$631$	4739.76	0.299028	0.149514	+	0.988760i	$0.452229\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1727.02	−0.107929
$636$	0	0
$637$	13670.5	0.850305
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29213.8	−1.80012	−0.900060	−	0.435766i	$-0.856478\pi$
$641$	−29213.8	−1.80012	−0.900060	+	0.435766i	$0.856478\pi$
$642$	0	0
$643$	−23444.5	−1.43789	−0.718943	−	0.695069i	$-0.755374\pi$
$643$	−23444.5	−1.43789	−0.718943	+	0.695069i	$0.755374\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27783.6	1.68823	0.844116	−	0.536160i	$-0.180126\pi$
$647$	27783.6	1.68823	0.844116	+	0.536160i	$0.180126\pi$
$648$	0	0
$649$	4265.84	0.258010
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8533.05	−0.511369	−0.255684	−	0.966760i	$-0.582301\pi$
$653$	−8533.05	−0.511369	−0.255684	+	0.966760i	$0.582301\pi$
$654$	0	0
$655$	14139.4	0.843467
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1085.28	0.0641522	0.0320761	−	0.999485i	$-0.489788\pi$
$659$	1085.28	0.0641522	0.0320761	+	0.999485i	$0.489788\pi$
$660$	0	0
$661$	−403.736	−0.0237572	−0.0118786	−	0.999929i	$-0.503781\pi$
$661$	−403.736	−0.0237572	−0.0118786	+	0.999929i	$0.503781\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5337.97	−0.311274
$666$	0	0
$667$	−15261.3	−0.885938
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−30522.6	−1.75605
$672$	0	0
$673$	−25125.0	−1.43907	−0.719537	−	0.694454i	$-0.755646\pi$
$673$	−25125.0	−1.43907	−0.719537	+	0.694454i	$0.755646\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4361.67	0.247611	0.123805	−	0.992307i	$-0.460490\pi$
$677$	4361.67	0.247611	0.123805	+	0.992307i	$0.460490\pi$
$678$	0	0
$679$	−14734.1	−0.832756
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14945.1	−0.837275	−0.418637	−	0.908153i	$-0.637492\pi$
$683$	−14945.1	−0.837275	−0.418637	+	0.908153i	$0.637492\pi$
$684$	0	0
$685$	6139.19	0.342433
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6944.72	0.383995
$690$	0	0
$691$	−21499.1	−1.18359	−0.591797	−	0.806087i	$-0.701581\pi$
$691$	−21499.1	−1.18359	−0.591797	+	0.806087i	$0.701581\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	200.545	0.0109455
$696$	0	0
$697$	27098.3	1.47263
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28878.4	1.55595	0.777977	−	0.628293i	$-0.216246\pi$
$701$	28878.4	1.55595	0.777977	+	0.628293i	$0.216246\pi$
$702$	0	0
$703$	−23951.8	−1.28501
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10884.9	−0.579024
$708$	0	0
$709$	−19945.5	−1.05651	−0.528257	−	0.849084i	$-0.677154\pi$
$709$	−19945.5	−1.05651	−0.528257	+	0.849084i	$0.677154\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−35584.3	−1.86907
$714$	0	0
$715$	−13158.7	−0.688264
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18280.0	0.948160	0.474080	−	0.880482i	$-0.342781\pi$
$719$	18280.0	0.948160	0.474080	+	0.880482i	$0.342781\pi$
$720$	0	0
$721$	−13129.5	−0.678179
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2205.62	−0.112986
$726$	0	0
$727$	−17999.1	−0.918225	−0.459112	−	0.888378i	$-0.651833\pi$
$727$	−17999.1	−0.918225	−0.459112	+	0.888378i	$0.651833\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14215.8	0.719278
$732$	0	0
$733$	9899.87	0.498854	0.249427	−	0.968394i	$-0.419758\pi$
$733$	9899.87	0.498854	0.249427	+	0.968394i	$0.419758\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36047.4	1.80166
$738$	0	0
$739$	16358.7	0.814294	0.407147	−	0.913363i	$-0.366524\pi$
$739$	16358.7	0.814294	0.407147	+	0.913363i	$0.366524\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−756.028	−0.0373297	−0.0186648	−	0.999826i	$-0.505942\pi$
$743$	−756.028	−0.0373297	−0.0186648	+	0.999826i	$0.505942\pi$
$744$	0	0
$745$	−2403.65	−0.118205
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6563.11	−0.320175
$750$	0	0
$751$	31368.4	1.52417	0.762083	−	0.647480i	$-0.224177\pi$
$751$	31368.4	1.52417	0.762083	+	0.647480i	$0.224177\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	380.000	0.0183174
$756$	0	0
$757$	−17559.4	−0.843072	−0.421536	−	0.906812i	$-0.638509\pi$
$757$	−17559.4	−0.843072	−0.421536	+	0.906812i	$0.638509\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23502.4	1.11953	0.559765	−	0.828651i	$-0.310891\pi$
$761$	23502.4	1.11953	0.559765	+	0.828651i	$0.310891\pi$
$762$	0	0
$763$	8373.63	0.397308
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9298.12	−0.437726
$768$	0	0
$769$	32276.0	1.51352	0.756762	−	0.653690i	$-0.226780\pi$
$769$	32276.0	1.51352	0.756762	+	0.653690i	$0.226780\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8693.38	−0.404501	−0.202251	−	0.979334i	$-0.564826\pi$
$773$	−8693.38	−0.404501	−0.202251	+	0.979334i	$0.564826\pi$
$774$	0	0
$775$	−5142.78	−0.238367
$776$	0	0
$777$	0	0
$778$	0	0
$779$	26941.8	1.23914
$780$	0	0
$781$	21029.9	0.963519
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2193.01	−0.0997095
$786$	0	0
$787$	−20817.0	−0.942881	−0.471440	−	0.881898i	$-0.656266\pi$
$787$	−20817.0	−0.942881	−0.471440	+	0.881898i	$0.656266\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	21689.5	0.974956
$792$	0	0
$793$	66529.2	2.97922
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42466.0	1.88736	0.943678	−	0.330865i	$-0.107341\pi$
$797$	42466.0	1.88736	0.943678	+	0.330865i	$0.107341\pi$
$798$	0	0
$799$	−32890.7	−1.45631
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−479.914	−0.0210906
$804$	0	0
$805$	−11025.6	−0.482736
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32955.4	1.43220	0.716101	−	0.697997i	$-0.245925\pi$
$809$	32955.4	1.43220	0.716101	+	0.697997i	$0.245925\pi$
$810$	0	0
$811$	3396.82	0.147076	0.0735379	−	0.997292i	$-0.476571\pi$
$811$	3396.82	0.147076	0.0735379	+	0.997292i	$0.476571\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8779.36	−0.377335
$816$	0	0
$817$	14133.8	0.605236
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13138.9	−0.558529	−0.279265	−	0.960214i	$-0.590091\pi$
$821$	−13138.9	−0.558529	−0.279265	+	0.960214i	$0.590091\pi$
$822$	0	0
$823$	−24645.2	−1.04384	−0.521918	−	0.852996i	$-0.674783\pi$
$823$	−24645.2	−1.04384	−0.521918	+	0.852996i	$0.674783\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14508.9	0.610064	0.305032	−	0.952342i	$-0.401333\pi$
$827$	14508.9	0.610064	0.305032	+	0.952342i	$0.401333\pi$
$828$	0	0
$829$	28408.6	1.19019	0.595097	−	0.803654i	$-0.297113\pi$
$829$	28408.6	1.19019	0.595097	+	0.803654i	$0.297113\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15203.9	−0.632392
$834$	0	0
$835$	−13937.8	−0.577649
$836$	0	0
$837$	0	0
$838$	0	0
$839$	14950.0	0.615173	0.307587	−	0.951520i	$-0.400479\pi$
$839$	14950.0	0.615173	0.307587	+	0.951520i	$0.400479\pi$
$840$	0	0
$841$	−16605.3	−0.680854
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17696.7	0.720456
$846$	0	0
$847$	1575.56	0.0639161
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−49472.8	−1.99284
$852$	0	0
$853$	−47855.4	−1.92091	−0.960456	−	0.278432i	$-0.910185\pi$
$853$	−47855.4	−1.92091	−0.960456	+	0.278432i	$0.910185\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25938.2	−1.03388	−0.516939	−	0.856022i	$-0.672929\pi$
$857$	−25938.2	−1.03388	−0.516939	+	0.856022i	$0.672929\pi$
$858$	0	0
$859$	7594.80	0.301666	0.150833	−	0.988559i	$-0.451804\pi$
$859$	7594.80	0.301666	0.150833	+	0.988559i	$0.451804\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20686.0	0.815943	0.407972	−	0.912995i	$-0.366236\pi$
$863$	20686.0	0.815943	0.407972	+	0.912995i	$0.366236\pi$
$864$	0	0
$865$	6081.81	0.239061
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19920.8	0.777637
$870$	0	0
$871$	−78571.6	−3.05660
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1593.47	−0.0615645
$876$	0	0
$877$	−40023.1	−1.54103	−0.770515	−	0.637422i	$-0.780001\pi$
$877$	−40023.1	−1.54103	−0.770515	+	0.637422i	$0.780001\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21863.6	0.836100	0.418050	−	0.908424i	$-0.362714\pi$
$881$	21863.6	0.836100	0.418050	+	0.908424i	$0.362714\pi$
$882$	0	0
$883$	−25239.1	−0.961906	−0.480953	−	0.876746i	$-0.659709\pi$
$883$	−25239.1	−0.961906	−0.480953	+	0.876746i	$0.659709\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42160.4	1.59595	0.797974	−	0.602691i	$-0.205905\pi$
$887$	42160.4	1.59595	0.797974	+	0.602691i	$0.205905\pi$
$888$	0	0
$889$	4403.12	0.166115
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−32700.8	−1.22541
$894$	0	0
$895$	12659.9	0.472819
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18148.9	0.673303
$900$	0	0
$901$	−7723.69	−0.285587
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17224.1	0.632650
$906$	0	0
$907$	9290.99	0.340135	0.170067	−	0.985432i	$-0.445601\pi$
$907$	9290.99	0.340135	0.170067	+	0.985432i	$0.445601\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48448.8	−1.76200	−0.881000	−	0.473116i	$-0.843129\pi$
$911$	−48448.8	−1.76200	−0.881000	+	0.473116i	$0.843129\pi$
$912$	0	0
$913$	−21819.4	−0.790926
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−36048.9	−1.29819
$918$	0	0
$919$	39905.6	1.43239	0.716193	−	0.697902i	$-0.245883\pi$
$919$	39905.6	1.43239	0.716193	+	0.697902i	$0.245883\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−45838.2	−1.63465
$924$	0	0
$925$	−7150.00	−0.254152
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34488.4	−1.21800	−0.609002	−	0.793169i	$-0.708430\pi$
$929$	−34488.4	−1.21800	−0.609002	+	0.793169i	$0.708430\pi$
$930$	0	0
$931$	−15116.1	−0.532126
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14634.7	0.511878
$936$	0	0
$937$	−57056.3	−1.98927	−0.994635	−	0.103443i	$-0.967014\pi$
$937$	−57056.3	−1.98927	−0.994635	+	0.103443i	$0.967014\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−52473.4	−1.81784	−0.908919	−	0.416974i	$-0.863091\pi$
$941$	−52473.4	−1.81784	−0.908919	+	0.416974i	$0.863091\pi$
$942$	0	0
$943$	55648.6	1.92171
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19062.2	0.654107	0.327053	−	0.945006i	$-0.393944\pi$
$947$	19062.2	0.654107	0.327053	+	0.945006i	$0.393944\pi$
$948$	0	0
$949$	1046.05	0.0357812
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36107.0	−1.22730	−0.613652	−	0.789577i	$-0.710300\pi$
$953$	−36107.0	−1.22730	−0.613652	+	0.789577i	$0.710300\pi$
$954$	0	0
$955$	8512.86	0.288450
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15652.2	−0.527043
$960$	0	0
$961$	12526.2	0.420468
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3427.44	0.114335
$966$	0	0
$967$	−42749.6	−1.42165	−0.710824	−	0.703369i	$-0.751678\pi$
$967$	−42749.6	−1.42165	−0.710824	+	0.703369i	$0.751678\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42208.9	−1.39500	−0.697501	−	0.716583i	$-0.745705\pi$
$971$	−42208.9	−1.39500	−0.697501	+	0.716583i	$0.745705\pi$
$972$	0	0
$973$	−511.300	−0.0168464
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6217.51	0.203599	0.101799	−	0.994805i	$-0.467540\pi$
$977$	6217.51	0.203599	0.101799	+	0.994805i	$0.467540\pi$
$978$	0	0
$979$	35203.2	1.14923
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−40932.0	−1.32811	−0.664053	−	0.747686i	$-0.731165\pi$
$983$	−40932.0	−1.32811	−0.664053	+	0.747686i	$0.731165\pi$
$984$	0	0
$985$	21814.8	0.705661
$986$	0	0
$987$	0	0
$988$	0	0
$989$	29193.4	0.938622
$990$	0	0
$991$	−13878.5	−0.444870	−0.222435	−	0.974947i	$-0.571401\pi$
$991$	−13878.5	−0.444870	−0.222435	+	0.974947i	$0.571401\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14601.0	0.465208
$996$	0	0
$997$	6733.80	0.213903	0.106952	−	0.994264i	$-0.465891\pi$
$997$	6733.80	0.213903	0.106952	+	0.994264i	$0.465891\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bc.1.1		2
3.2	odd	2		2160.4.a.x.1.1		2
4.3	odd	2		540.4.a.h.1.2	yes	2
12.11	even	2		540.4.a.e.1.2	✓	2
36.7	odd	6		1620.4.i.o.1081.1		4
36.11	even	6		1620.4.i.r.1081.1		4
36.23	even	6		1620.4.i.r.541.1		4
36.31	odd	6		1620.4.i.o.541.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.e.1.2	✓	2	12.11	even	2
540.4.a.h.1.2	yes	2	4.3	odd	2
1620.4.i.o.541.1		4	36.31	odd	6
1620.4.i.o.1081.1		4	36.7	odd	6
1620.4.i.r.541.1		4	36.23	even	6
1620.4.i.r.1081.1		4	36.11	even	6
2160.4.a.x.1.1		2	3.2	odd	2
2160.4.a.bc.1.1		2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -12.7477 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} -12.7477 q^{7} +34.7477 q^{11} -75.7386 q^{13} +84.2341 q^{17} +83.7477 q^{19} +172.982 q^{23} +25.0000 q^{25} -88.2250 q^{29} -205.711 q^{31} -63.7386 q^{35} -286.000 q^{37} +321.702 q^{41} +168.766 q^{43} -390.468 q^{47} -180.495 q^{49} -91.6932 q^{53} +173.739 q^{55} +122.766 q^{59} -878.405 q^{61} -378.693 q^{65} +1037.40 q^{67} +605.216 q^{71} -13.8114 q^{73} -442.955 q^{77} +573.298 q^{79} -627.936 q^{83} +421.170 q^{85} +1013.11 q^{89} +965.495 q^{91} +418.739 q^{95} +1155.82 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 10 * q^5 + 2 * q^7	\( 2 q + 10 q^{5} + 2 q^{7} + 42 q^{11} - 14 q^{13} - 24 q^{17} + 140 q^{19} + 126 q^{23} + 50 q^{25} + 126 q^{29} + 56 q^{31} + 10 q^{35} - 572 q^{37} + 66 q^{41} + 530 q^{43} - 396 q^{47} - 306 q^{49} + 504 q^{53} + 210 q^{55} + 438 q^{59} - 602 q^{61} - 70 q^{65} + 920 q^{67} + 798 q^{71} - 770 q^{73} - 336 q^{77} + 1724 q^{79} - 486 q^{83} - 120 q^{85} + 294 q^{89} + 1876 q^{91} + 700 q^{95} + 112 q^{97}+O(q^{100}) \) 2 * q + 10 * q^5 + 2 * q^7 + 42 * q^11 - 14 * q^13 - 24 * q^17 + 140 * q^19 + 126 * q^23 + 50 * q^25 + 126 * q^29 + 56 * q^31 + 10 * q^35 - 572 * q^37 + 66 * q^41 + 530 * q^43 - 396 * q^47 - 306 * q^49 + 504 * q^53 + 210 * q^55 + 438 * q^59 - 602 * q^61 - 70 * q^65 + 920 * q^67 + 798 * q^71 - 770 * q^73 - 336 * q^77 + 1724 * q^79 - 486 * q^83 - 120 * q^85 + 294 * q^89 + 1876 * q^91 + 700 * q^95 + 112 * q^97

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