LMFDB - Embedded newform 4900.2.e.u.2549.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(2549,4900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4900.2549"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16, 0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.3
Root			$0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.u.2549.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.874032i q^{3} +2.23607 q^{9} +3.47214 q^{11} +2.28825i q^{13} +1.74806i q^{17} -0.333851 q^{19} -5.47214i q^{23} -4.57649i q^{27} +4.23607 q^{29} +1.20788 q^{31} -3.03476i q^{33} -0.236068i q^{37} +2.00000 q^{39} -1.95440 q^{41} -8.23607i q^{43} +7.73877i q^{47} +1.52786 q^{51} +1.70820i q^{53} +0.291796i q^{57} -5.11667 q^{59} +14.6035 q^{61} +3.94427i q^{67} -4.78282 q^{69} +3.29180 q^{71} +14.6035i q^{73} +2.52786 q^{79} +2.70820 q^{81} +9.02546i q^{83} -3.70246i q^{87} +15.3500 q^{89} -1.05573i q^{93} +12.1089i q^{97} +7.76393 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{11} + 16 q^{29} + 16 q^{39} + 48 q^{51} + 80 q^{71} + 56 q^{79} - 32 q^{81} + 80 q^{99}+O(q^{100}) $ 8 * q - 8 * q^11 + 16 * q^29 + 16 * q^39 + 48 * q^51 + 80 * q^71 + 56 * q^79 - 32 * q^81 + 80 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 0.874032i	− 0.504623i	−0.967646	−	0.252311i	$-0.918809\pi$
$3$	− 0.874032i	− 0.504623i	0.967646	−	0.252311i	$-0.0811907\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.23607	0.745356
$10$	0	0
$11$	3.47214	1.04689	0.523444	−	0.852060i	$-0.324647\pi$
$11$	3.47214	1.04689	0.523444	+	0.852060i	$0.324647\pi$
$12$	0	0
$13$	2.28825i	0.634645i	0.948318	+	0.317323i	$0.102784\pi$
$13$	2.28825i	0.634645i	−0.948318	+	0.317323i	$0.897216\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.74806i	0.423968i	0.977273	+	0.211984i	$0.0679924\pi$
$17$	1.74806i	0.423968i	−0.977273	+	0.211984i	$0.932008\pi$
$18$	0	0
$19$	−0.333851	−0.0765906	−0.0382953	−	0.999266i	$-0.512193\pi$
$19$	−0.333851	−0.0765906	−0.0382953	+	0.999266i	$0.512193\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.47214i	− 1.14102i	−0.821291	−	0.570510i	$-0.806746\pi$
$23$	− 5.47214i	− 1.14102i	0.821291	−	0.570510i	$-0.193254\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.57649i	− 0.880746i
$28$	0	0
$29$	4.23607	0.786618	0.393309	−	0.919406i	$-0.371330\pi$
$29$	4.23607	0.786618	0.393309	+	0.919406i	$0.371330\pi$
$30$	0	0
$31$	1.20788	0.216942	0.108471	−	0.994100i	$-0.465405\pi$
$31$	1.20788	0.216942	0.108471	+	0.994100i	$0.465405\pi$
$32$	0	0
$33$	− 3.03476i	− 0.528284i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.236068i	− 0.0388093i	−0.999812	−	0.0194047i	$-0.993823\pi$
$37$	− 0.236068i	− 0.0388093i	0.999812	−	0.0194047i	$-0.00617709\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−1.95440	−0.305225	−0.152613	−	0.988286i	$-0.548769\pi$
$41$	−1.95440	−0.305225	−0.152613	+	0.988286i	$0.548769\pi$
$42$	0	0
$43$	− 8.23607i	− 1.25599i	−0.778218	−	0.627994i	$-0.783876\pi$
$43$	− 8.23607i	− 1.25599i	0.778218	−	0.627994i	$-0.216124\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.73877i	1.12882i	0.825496	+	0.564408i	$0.190895\pi$
$47$	7.73877i	1.12882i	−0.825496	+	0.564408i	$0.809105\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.52786	0.213944
$52$	0	0
$53$	1.70820i	0.234640i	0.993094	+	0.117320i	$0.0374303\pi$
$53$	1.70820i	0.234640i	−0.993094	+	0.117320i	$0.962570\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.291796i	0.0386493i
$58$	0	0
$59$	−5.11667	−0.666134	−0.333067	−	0.942903i	$-0.608084\pi$
$59$	−5.11667	−0.666134	−0.333067	+	0.942903i	$0.608084\pi$
$60$	0	0
$61$	14.6035	1.86979	0.934894	−	0.354928i	$-0.115495\pi$
$61$	14.6035	1.86979	0.934894	+	0.354928i	$0.115495\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.94427i	0.481870i	0.970541	+	0.240935i	$0.0774540\pi$
$67$	3.94427i	0.481870i	−0.970541	+	0.240935i	$0.922546\pi$
$68$	0	0
$69$	−4.78282	−0.575784
$70$	0	0
$71$	3.29180	0.390664	0.195332	−	0.980737i	$-0.437422\pi$
$71$	3.29180	0.390664	0.195332	+	0.980737i	$0.437422\pi$
$72$	0	0
$73$	14.6035i	1.70921i	0.519278	+	0.854606i	$0.326201\pi$
$73$	14.6035i	1.70921i	−0.519278	+	0.854606i	$0.673799\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.52786	0.284407	0.142203	−	0.989837i	$-0.454581\pi$
$79$	2.52786	0.284407	0.142203	+	0.989837i	$0.454581\pi$
$80$	0	0
$81$	2.70820	0.300912
$82$	0	0
$83$	9.02546i	0.990673i	0.868701	+	0.495337i	$0.164955\pi$
$83$	9.02546i	0.990673i	−0.868701	+	0.495337i	$0.835045\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.70246i	− 0.396945i
$88$	0	0
$89$	15.3500	1.62710	0.813549	−	0.581496i	$-0.197532\pi$
$89$	15.3500	1.62710	0.813549	+	0.581496i	$0.197532\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 1.05573i	− 0.109474i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.1089i	1.22948i	0.788732	+	0.614738i	$0.210738\pi$
$97$	12.1089i	1.22948i	−0.788732	+	0.614738i	$0.789262\pi$
$98$	0	0
$99$	7.76393	0.780305
$100$	0	0
$101$	−6.53089	−0.649847	−0.324924	−	0.945740i	$-0.605339\pi$
$101$	−6.53089	−0.649847	−0.324924	+	0.945740i	$0.605339\pi$
$102$	0	0
$103$	− 18.3848i	− 1.81151i	−0.423806	−	0.905753i	$-0.639306\pi$
$103$	− 18.3848i	− 1.81151i	0.423806	−	0.905753i	$-0.360694\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.52786i	0.341051i	0.985353	+	0.170526i	$0.0545466\pi$
$107$	3.52786i	0.341051i	−0.985353	+	0.170526i	$0.945453\pi$
$108$	0	0
$109$	−10.4164	−0.997711	−0.498855	−	0.866685i	$-0.666246\pi$
$109$	−10.4164	−0.997711	−0.498855	+	0.866685i	$0.666246\pi$
$110$	0	0
$111$	−0.206331	−0.0195841
$112$	0	0
$113$	− 18.2361i	− 1.71550i	−0.514063	−	0.857752i	$-0.671860\pi$
$113$	− 18.2361i	− 1.71550i	0.514063	−	0.857752i	$-0.328140\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.11667i	0.473037i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.05573	0.0959753
$122$	0	0
$123$	1.70820i	0.154024i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.1803i	0.992095i	0.868295	+	0.496047i	$0.165216\pi$
$127$	11.1803i	0.992095i	−0.868295	+	0.496047i	$0.834784\pi$
$128$	0	0
$129$	−7.19859	−0.633800
$130$	0	0
$131$	14.8098	1.29394	0.646971	−	0.762515i	$-0.276036\pi$
$131$	14.8098	1.29394	0.646971	+	0.762515i	$0.276036\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	−11.7751	−0.998749	−0.499375	−	0.866386i	$-0.666437\pi$
$139$	−11.7751	−0.998749	−0.499375	+	0.866386i	$0.666437\pi$
$140$	0	0
$141$	6.76393	0.569626
$142$	0	0
$143$	7.94510i	0.664403i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.70820	0.549557	0.274779	−	0.961507i	$-0.411395\pi$
$149$	6.70820	0.549557	0.274779	+	0.961507i	$0.411395\pi$
$150$	0	0
$151$	3.76393	0.306304	0.153152	−	0.988203i	$-0.451058\pi$
$151$	3.76393	0.306304	0.153152	+	0.988203i	$0.451058\pi$
$152$	0	0
$153$	3.90879i	0.316007i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 12.1877i	− 0.972688i	−0.873768	−	0.486344i	$-0.838330\pi$
$157$	− 12.1877i	− 0.972688i	0.873768	−	0.486344i	$-0.161670\pi$
$158$	0	0
$159$	1.49302	0.118405
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 20.1803i	− 1.58065i	−0.612690	−	0.790323i	$-0.709913\pi$
$163$	− 20.1803i	− 1.58065i	0.612690	−	0.790323i	$-0.290087\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 19.9265i	− 1.54196i	−0.636860	−	0.770980i	$-0.719767\pi$
$167$	− 19.9265i	− 1.54196i	0.636860	−	0.770980i	$-0.280233\pi$
$168$	0	0
$169$	7.76393	0.597226
$170$	0	0
$171$	−0.746512	−0.0570872
$172$	0	0
$173$	− 18.3848i	− 1.39777i	−0.715235	−	0.698884i	$-0.753680\pi$
$173$	− 18.3848i	− 1.39777i	0.715235	−	0.698884i	$-0.246320\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.47214i	0.336146i
$178$	0	0
$179$	11.7082	0.875112	0.437556	−	0.899191i	$-0.355844\pi$
$179$	11.7082	0.875112	0.437556	+	0.899191i	$0.355844\pi$
$180$	0	0
$181$	−24.8369	−1.84611	−0.923054	−	0.384670i	$-0.874315\pi$
$181$	−24.8369	−1.84611	−0.923054	+	0.384670i	$0.874315\pi$
$182$	0	0
$183$	− 12.7639i	− 0.943537i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.06952i	0.443847i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.1803	0.736624	0.368312	−	0.929702i	$-0.379936\pi$
$191$	10.1803	0.736624	0.368312	+	0.929702i	$0.379936\pi$
$192$	0	0
$193$	− 15.6525i	− 1.12669i	−0.826222	−	0.563345i	$-0.809514\pi$
$193$	− 15.6525i	− 1.12669i	0.826222	−	0.563345i	$-0.190486\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.5279i	0.750079i	0.927009	+	0.375040i	$0.122371\pi$
$197$	10.5279i	0.750079i	−0.927009	+	0.375040i	$0.877629\pi$
$198$	0	0
$199$	0.461370	0.0327057	0.0163528	−	0.999866i	$-0.494795\pi$
$199$	0.461370	0.0327057	0.0163528	+	0.999866i	$0.494795\pi$
$200$	0	0
$201$	3.44742	0.243162
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.2361i	− 0.850466i
$208$	0	0
$209$	−1.15917	−0.0801818
$210$	0	0
$211$	18.4721	1.27167	0.635837	−	0.771823i	$-0.280655\pi$
$211$	18.4721	1.27167	0.635837	+	0.771823i	$0.280655\pi$
$212$	0	0
$213$	− 2.87714i	− 0.197138i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.7639	0.862507
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	15.0162i	1.00556i	0.864415	+	0.502778i	$0.167689\pi$
$223$	15.0162i	1.00556i	−0.864415	+	0.502778i	$0.832311\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.0525i	1.26456i	0.774741	+	0.632279i	$0.217880\pi$
$227$	19.0525i	1.26456i	−0.774741	+	0.632279i	$0.782120\pi$
$228$	0	0
$229$	19.7202	1.30315	0.651573	−	0.758586i	$-0.274109\pi$
$229$	19.7202	1.30315	0.651573	+	0.758586i	$0.274109\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 19.4721i	− 1.27566i	−0.770176	−	0.637831i	$-0.779832\pi$
$233$	− 19.4721i	− 1.27566i	0.770176	−	0.637831i	$-0.220168\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2.20943i	− 0.143518i
$238$	0	0
$239$	22.6525	1.46527	0.732633	−	0.680623i	$-0.238291\pi$
$239$	22.6525	1.46527	0.732633	+	0.680623i	$0.238291\pi$
$240$	0	0
$241$	1.41421	0.0910975	0.0455488	−	0.998962i	$-0.485496\pi$
$241$	1.41421	0.0910975	0.0455488	+	0.998962i	$0.485496\pi$
$242$	0	0
$243$	− 16.0965i	− 1.03259i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.763932i	− 0.0486078i
$248$	0	0
$249$	7.88854	0.499916
$250$	0	0
$251$	4.57649	0.288866	0.144433	−	0.989515i	$-0.453864\pi$
$251$	4.57649	0.288866	0.144433	+	0.989515i	$0.453864\pi$
$252$	0	0
$253$	− 19.0000i	− 1.19452i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.20788i	0.0753456i	0.999290	+	0.0376728i	$0.0119945\pi$
$257$	1.20788i	0.0753456i	−0.999290	+	0.0376728i	$0.988006\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.47214	0.586310
$262$	0	0
$263$	− 2.70820i	− 0.166995i	−0.996508	−	0.0834975i	$-0.973391\pi$
$263$	− 2.70820i	− 0.166995i	0.996508	−	0.0834975i	$-0.0266091\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 13.4164i	− 0.821071i
$268$	0	0
$269$	2.54328	0.155067	0.0775334	−	0.996990i	$-0.475296\pi$
$269$	2.54328	0.155067	0.0775334	+	0.996990i	$0.475296\pi$
$270$	0	0
$271$	−3.62365	−0.220121	−0.110060	−	0.993925i	$-0.535104\pi$
$271$	−3.62365	−0.220121	−0.110060	+	0.993925i	$0.535104\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.05573i	0.183601i	0.995777	+	0.0918005i	$0.0292622\pi$
$277$	3.05573i	0.183601i	−0.995777	+	0.0918005i	$0.970738\pi$
$278$	0	0
$279$	2.70091	0.161699
$280$	0	0
$281$	16.1246	0.961914	0.480957	−	0.876744i	$-0.340289\pi$
$281$	16.1246	0.961914	0.480957	+	0.876744i	$0.340289\pi$
$282$	0	0
$283$	− 2.08191i	− 0.123757i	−0.998084	−	0.0618785i	$-0.980291\pi$
$283$	− 2.08191i	− 0.123757i	0.998084	−	0.0618785i	$-0.0197091\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.9443	0.820251
$290$	0	0
$291$	10.5836	0.620421
$292$	0	0
$293$	21.2619i	1.24213i	0.783757	+	0.621067i	$0.213301\pi$
$293$	21.2619i	1.24213i	−0.783757	+	0.621067i	$0.786699\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 15.8902i	− 0.922043i
$298$	0	0
$299$	12.5216	0.724142
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.70820i	0.327928i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.3941i	− 0.707367i	−0.935365	−	0.353684i	$-0.884929\pi$
$307$	− 12.3941i	− 0.707367i	0.935365	−	0.353684i	$-0.115071\pi$
$308$	0	0
$309$	−16.0689	−0.914127
$310$	0	0
$311$	−34.8639	−1.97695	−0.988474	−	0.151389i	$-0.951625\pi$
$311$	−34.8639	−1.97695	−0.988474	+	0.151389i	$0.951625\pi$
$312$	0	0
$313$	− 16.2241i	− 0.917038i	−0.888685	−	0.458519i	$-0.848380\pi$
$313$	− 16.2241i	− 0.917038i	0.888685	−	0.458519i	$-0.151620\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.4164i	1.48369i	0.670570	+	0.741847i	$0.266050\pi$
$317$	26.4164i	1.48369i	−0.670570	+	0.741847i	$0.733950\pi$
$318$	0	0
$319$	14.7082	0.823501
$320$	0	0
$321$	3.08347	0.172102
$322$	0	0
$323$	− 0.583592i	− 0.0324719i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.10427i	0.503468i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	27.3607	1.50388	0.751939	−	0.659232i	$-0.229119\pi$
$331$	27.3607	1.50388	0.751939	+	0.659232i	$0.229119\pi$
$332$	0	0
$333$	− 0.527864i	− 0.0289268i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.1246i	1.25968i	0.776726	+	0.629839i	$0.216879\pi$
$337$	23.1246i	1.25968i	−0.776726	+	0.629839i	$0.783121\pi$
$338$	0	0
$339$	−15.9389	−0.865683
$340$	0	0
$341$	4.19393	0.227114
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 19.3607i	− 1.03934i	−0.854368	−	0.519668i	$-0.826056\pi$
$347$	− 19.3607i	− 1.03934i	0.854368	−	0.519668i	$-0.173944\pi$
$348$	0	0
$349$	−15.9690	−0.854802	−0.427401	−	0.904062i	$-0.640571\pi$
$349$	−15.9690	−0.854802	−0.427401	+	0.904062i	$0.640571\pi$
$350$	0	0
$351$	10.4721	0.558961
$352$	0	0
$353$	10.3609i	0.551453i	0.961236	+	0.275727i	$0.0889184\pi$
$353$	10.3609i	0.551453i	−0.961236	+	0.275727i	$0.911082\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.0689	−1.53420	−0.767099	−	0.641529i	$-0.778300\pi$
$359$	−29.0689	−1.53420	−0.767099	+	0.641529i	$0.778300\pi$
$360$	0	0
$361$	−18.8885	−0.994134
$362$	0	0
$363$	− 0.922740i	− 0.0484313i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10.0270i	− 0.523406i	−0.965148	−	0.261703i	$-0.915716\pi$
$367$	− 10.0270i	− 0.523406i	0.965148	−	0.261703i	$-0.0842841\pi$
$368$	0	0
$369$	−4.37016	−0.227501
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.1246i	0.524233i	0.965036	+	0.262116i	$0.0844204\pi$
$373$	10.1246i	0.524233i	−0.965036	+	0.262116i	$0.915580\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.69316i	0.499223i
$378$	0	0
$379$	30.1246	1.54740	0.773699	−	0.633554i	$-0.218404\pi$
$379$	30.1246	1.54740	0.773699	+	0.633554i	$0.218404\pi$
$380$	0	0
$381$	9.77198	0.500633
$382$	0	0
$383$	− 25.8384i	− 1.32028i	−0.751142	−	0.660140i	$-0.770497\pi$
$383$	− 25.8384i	− 1.32028i	0.751142	−	0.660140i	$-0.229503\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 18.4164i	− 0.936159i
$388$	0	0
$389$	−19.1803	−0.972482	−0.486241	−	0.873825i	$-0.661632\pi$
$389$	−19.1803	−0.972482	−0.486241	+	0.873825i	$0.661632\pi$
$390$	0	0
$391$	9.56564	0.483755
$392$	0	0
$393$	− 12.9443i	− 0.652952i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.91034i	0.246443i	0.992379	+	0.123221i	$0.0393226\pi$
$397$	4.91034i	0.246443i	−0.992379	+	0.123221i	$0.960677\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.7082	−0.634617	−0.317309	−	0.948322i	$-0.602779\pi$
$401$	−12.7082	−0.634617	−0.317309	+	0.948322i	$0.602779\pi$
$402$	0	0
$403$	2.76393i	0.137681i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 0.819660i	− 0.0406290i
$408$	0	0
$409$	9.89949	0.489499	0.244749	−	0.969586i	$-0.421294\pi$
$409$	9.89949	0.489499	0.244749	+	0.969586i	$0.421294\pi$
$410$	0	0
$411$	8.74032	0.431128
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.2918i	0.503991i
$418$	0	0
$419$	12.9343	0.631880	0.315940	−	0.948779i	$-0.397680\pi$
$419$	12.9343	0.631880	0.315940	+	0.948779i	$0.397680\pi$
$420$	0	0
$421$	−18.5967	−0.906350	−0.453175	−	0.891422i	$-0.649709\pi$
$421$	−18.5967	−0.906350	−0.453175	+	0.891422i	$0.649709\pi$
$422$	0	0
$423$	17.3044i	0.841369i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.94427	0.335273
$430$	0	0
$431$	−16.6525	−0.802122	−0.401061	−	0.916051i	$-0.631358\pi$
$431$	−16.6525	−0.802122	−0.401061	+	0.916051i	$0.631358\pi$
$432$	0	0
$433$	− 16.5579i	− 0.795722i	−0.917446	−	0.397861i	$-0.869753\pi$
$433$	− 16.5579i	− 0.795722i	0.917446	−	0.397861i	$-0.130247\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.82688i	0.0873913i
$438$	0	0
$439$	11.0286	0.526365	0.263182	−	0.964746i	$-0.415228\pi$
$439$	11.0286	0.526365	0.263182	+	0.964746i	$0.415228\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.12461i	0.0534319i	0.999643	+	0.0267160i	$0.00850497\pi$
$443$	1.12461i	0.0534319i	−0.999643	+	0.0267160i	$0.991495\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.86319i	− 0.277319i
$448$	0	0
$449$	−14.2361	−0.671842	−0.335921	−	0.941890i	$-0.609047\pi$
$449$	−14.2361	−0.671842	−0.335921	+	0.941890i	$0.609047\pi$
$450$	0	0
$451$	−6.78593	−0.319537
$452$	0	0
$453$	− 3.28980i	− 0.154568i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.65248i	0.264412i	0.991222	+	0.132206i	$0.0422060\pi$
$457$	5.65248i	0.264412i	−0.991222	+	0.132206i	$0.957794\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−24.0903	−1.12200	−0.560999	−	0.827816i	$-0.689583\pi$
$461$	−24.0903	−1.12200	−0.560999	+	0.827816i	$0.689583\pi$
$462$	0	0
$463$	13.4164i	0.623513i	0.950162	+	0.311757i	$0.100917\pi$
$463$	13.4164i	0.623513i	−0.950162	+	0.311757i	$0.899083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20.3879i	− 0.943439i	−0.881749	−	0.471719i	$-0.843634\pi$
$467$	− 20.3879i	− 0.943439i	0.881749	−	0.471719i	$-0.156366\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.6525	−0.490840
$472$	0	0
$473$	− 28.5967i	− 1.31488i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.81966i	0.174890i
$478$	0	0
$479$	−30.5725	−1.39689	−0.698447	−	0.715662i	$-0.746125\pi$
$479$	−30.5725	−1.39689	−0.698447	+	0.715662i	$0.746125\pi$
$480$	0	0
$481$	0.540182	0.0246302
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 22.5279i	− 1.02083i	−0.859927	−	0.510417i	$-0.829491\pi$
$487$	− 22.5279i	− 1.02083i	0.859927	−	0.510417i	$-0.170509\pi$
$488$	0	0
$489$	−17.6383	−0.797630
$490$	0	0
$491$	6.23607	0.281430	0.140715	−	0.990050i	$-0.455060\pi$
$491$	6.23607	0.281430	0.140715	+	0.990050i	$0.455060\pi$
$492$	0	0
$493$	7.40492i	0.333501i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.05573	0.405390	0.202695	−	0.979242i	$-0.435030\pi$
$499$	9.05573	0.405390	0.202695	+	0.979242i	$0.435030\pi$
$500$	0	0
$501$	−17.4164	−0.778108
$502$	0	0
$503$	− 6.65841i	− 0.296884i	−0.988921	−	0.148442i	$-0.952574\pi$
$503$	− 6.65841i	− 0.296884i	0.988921	−	0.148442i	$-0.0474258\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.78593i	− 0.301374i
$508$	0	0
$509$	11.8539	0.525414	0.262707	−	0.964876i	$-0.415385\pi$
$509$	11.8539	0.525414	0.262707	+	0.964876i	$0.415385\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.52786i	0.0674568i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	26.8701i	1.18174i
$518$	0	0
$519$	−16.0689	−0.705346
$520$	0	0
$521$	3.41732	0.149715	0.0748577	−	0.997194i	$-0.476150\pi$
$521$	3.41732	0.149715	0.0748577	+	0.997194i	$0.476150\pi$
$522$	0	0
$523$	10.2333i	0.447473i	0.974650	+	0.223736i	$0.0718255\pi$
$523$	10.2333i	0.447473i	−0.974650	+	0.223736i	$0.928175\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.11146i	0.0919765i
$528$	0	0
$529$	−6.94427	−0.301925
$530$	0	0
$531$	−11.4412	−0.496507
$532$	0	0
$533$	− 4.47214i	− 0.193710i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 10.2333i	− 0.441601i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.70820	0.116435	0.0582174	−	0.998304i	$-0.481458\pi$
$541$	2.70820	0.116435	0.0582174	+	0.998304i	$0.481458\pi$
$542$	0	0
$543$	21.7082i	0.931588i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.7082i	− 0.714391i	−0.934030	−	0.357196i	$-0.883733\pi$
$547$	− 16.7082i	− 0.714391i	0.934030	−	0.357196i	$-0.116267\pi$
$548$	0	0
$549$	32.6544	1.39366
$550$	0	0
$551$	−1.41421	−0.0602475
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.6525i	1.93436i	0.254097	+	0.967179i	$0.418222\pi$
$557$	45.6525i	1.93436i	−0.254097	+	0.967179i	$0.581778\pi$
$558$	0	0
$559$	18.8461	0.797107
$560$	0	0
$561$	5.30495	0.223975
$562$	0	0
$563$	− 29.0308i	− 1.22350i	−0.791051	−	0.611751i	$-0.790466\pi$
$563$	− 29.0308i	− 1.22350i	0.791051	−	0.611751i	$-0.209534\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.05573	−0.253869	−0.126935	−	0.991911i	$-0.540514\pi$
$569$	−6.05573	−0.253869	−0.126935	+	0.991911i	$0.540514\pi$
$570$	0	0
$571$	35.6525	1.49201	0.746005	−	0.665941i	$-0.231970\pi$
$571$	35.6525	1.49201	0.746005	+	0.665941i	$0.231970\pi$
$572$	0	0
$573$	− 8.89794i	− 0.371717i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 14.2209i	− 0.592026i	−0.955184	−	0.296013i	$-0.904343\pi$
$577$	− 14.2209i	− 0.592026i	0.955184	−	0.296013i	$-0.0956571\pi$
$578$	0	0
$579$	−13.6808	−0.568553
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5.93112i	0.245642i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 21.0857i	− 0.870299i	−0.900358	−	0.435150i	$-0.856695\pi$
$587$	− 21.0857i	− 0.870299i	0.900358	−	0.435150i	$-0.143305\pi$
$588$	0	0
$589$	−0.403252	−0.0166157
$590$	0	0
$591$	9.20169	0.378507
$592$	0	0
$593$	39.0277i	1.60268i	0.598212	+	0.801338i	$0.295878\pi$
$593$	39.0277i	1.60268i	−0.598212	+	0.801338i	$0.704122\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 0.403252i	− 0.0165040i
$598$	0	0
$599$	−17.4721	−0.713892	−0.356946	−	0.934125i	$-0.616182\pi$
$599$	−17.4721	−0.713892	−0.356946	+	0.934125i	$0.616182\pi$
$600$	0	0
$601$	−7.14988	−0.291650	−0.145825	−	0.989310i	$-0.546584\pi$
$601$	−7.14988	−0.291650	−0.145825	+	0.989310i	$0.546584\pi$
$602$	0	0
$603$	8.81966i	0.359164i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.4226i	0.950696i	0.879798	+	0.475348i	$0.157678\pi$
$607$	23.4226i	0.950696i	−0.879798	+	0.475348i	$0.842322\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17.7082	−0.716397
$612$	0	0
$613$	8.88854i	0.359005i	0.983758	+	0.179502i	$0.0574488\pi$
$613$	8.88854i	0.359005i	−0.983758	+	0.179502i	$0.942551\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.3607i	1.18202i	0.806666	+	0.591008i	$0.201270\pi$
$617$	29.3607i	1.18202i	−0.806666	+	0.591008i	$0.798730\pi$
$618$	0	0
$619$	27.5865	1.10879	0.554397	−	0.832252i	$-0.312949\pi$
$619$	27.5865	1.10879	0.554397	+	0.832252i	$0.312949\pi$
$620$	0	0
$621$	−25.0432	−1.00495
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.01316i	0.0404615i
$628$	0	0
$629$	0.412662	0.0164539
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	0	0
$633$	− 16.1452i	− 0.641716i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.36068	0.291184
$640$	0	0
$641$	5.36068	0.211734	0.105867	−	0.994380i	$-0.466238\pi$
$641$	5.36068	0.211734	0.105867	+	0.994380i	$0.466238\pi$
$642$	0	0
$643$	17.8933i	0.705643i	0.935691	+	0.352821i	$0.114778\pi$
$643$	17.8933i	0.705643i	−0.935691	+	0.352821i	$0.885222\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.0957i	1.73358i	0.498674	+	0.866790i	$0.333821\pi$
$647$	44.0957i	1.73358i	−0.498674	+	0.866790i	$0.666179\pi$
$648$	0	0
$649$	−17.7658	−0.697368
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 27.3050i	− 1.06853i	−0.845319	−	0.534263i	$-0.820589\pi$
$653$	− 27.3050i	− 1.06853i	0.845319	−	0.534263i	$-0.179411\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.6544i	1.27397i
$658$	0	0
$659$	−7.52786	−0.293244	−0.146622	−	0.989193i	$-0.546840\pi$
$659$	−7.52786	−0.293244	−0.146622	+	0.989193i	$0.546840\pi$
$660$	0	0
$661$	12.4729	0.485139	0.242569	−	0.970134i	$-0.422010\pi$
$661$	12.4729	0.485139	0.242569	+	0.970134i	$0.422010\pi$
$662$	0	0
$663$	3.49613i	0.135778i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 23.1803i	− 0.897546i
$668$	0	0
$669$	13.1246	0.507427
$670$	0	0
$671$	50.7054	1.95746
$672$	0	0
$673$	21.7082i	0.836790i	0.908265	+	0.418395i	$0.137407\pi$
$673$	21.7082i	0.836790i	−0.908265	+	0.418395i	$0.862593\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 37.1034i	− 1.42600i	−0.701164	−	0.713000i	$-0.747336\pi$
$677$	− 37.1034i	− 1.42600i	0.701164	−	0.713000i	$-0.252664\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	16.6525	0.638124
$682$	0	0
$683$	− 0.527864i	− 0.0201982i	−0.999949	−	0.0100991i	$-0.996785\pi$
$683$	− 0.527864i	− 0.0201982i	0.999949	−	0.0100991i	$-0.00321469\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 17.2361i	− 0.657597i
$688$	0	0
$689$	−3.90879	−0.148913
$690$	0	0
$691$	−36.5632	−1.39093	−0.695465	−	0.718560i	$-0.744802\pi$
$691$	−36.5632	−1.39093	−0.695465	+	0.718560i	$0.744802\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3.41641i	− 0.129406i
$698$	0	0
$699$	−17.0193	−0.643728
$700$	0	0
$701$	−46.2492	−1.74681	−0.873405	−	0.486995i	$-0.838093\pi$
$701$	−46.2492	−1.74681	−0.873405	+	0.486995i	$0.838093\pi$
$702$	0	0
$703$	0.0788114i	0.00297243i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−19.8885	−0.746930	−0.373465	−	0.927644i	$-0.621830\pi$
$709$	−19.8885	−0.746930	−0.373465	+	0.927644i	$0.621830\pi$
$710$	0	0
$711$	5.65248	0.211984
$712$	0	0
$713$	− 6.60970i	− 0.247535i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 19.7990i	− 0.739407i
$718$	0	0
$719$	−36.4056	−1.35770	−0.678850	−	0.734277i	$-0.737521\pi$
$719$	−36.4056	−1.35770	−0.678850	+	0.734277i	$0.737521\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 1.23607i	− 0.0459699i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37.9473i	− 1.40739i	−0.710503	−	0.703694i	$-0.751532\pi$
$727$	− 37.9473i	− 1.40739i	0.710503	−	0.703694i	$-0.248468\pi$
$728$	0	0
$729$	−5.94427	−0.220158
$730$	0	0
$731$	14.3972	0.532499
$732$	0	0
$733$	31.7804i	1.17384i	0.809646	+	0.586918i	$0.199659\pi$
$733$	31.7804i	1.17384i	−0.809646	+	0.586918i	$0.800341\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.6950i	0.504464i
$738$	0	0
$739$	35.9443	1.32223	0.661116	−	0.750284i	$-0.270083\pi$
$739$	35.9443	1.32223	0.661116	+	0.750284i	$0.270083\pi$
$740$	0	0
$741$	−0.667701	−0.0245286
$742$	0	0
$743$	23.8197i	0.873859i	0.899496	+	0.436929i	$0.143934\pi$
$743$	23.8197i	0.873859i	−0.899496	+	0.436929i	$0.856066\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.1815i	0.738404i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.9443	−0.472343	−0.236172	−	0.971711i	$-0.575893\pi$
$751$	−12.9443	−0.472343	−0.236172	+	0.971711i	$0.575893\pi$
$752$	0	0
$753$	− 4.00000i	− 0.145768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 39.7214i	− 1.44370i	−0.692051	−	0.721849i	$-0.743293\pi$
$757$	− 39.7214i	− 1.44370i	0.692051	−	0.721849i	$-0.256707\pi$
$758$	0	0
$759$	−16.6066	−0.602782
$760$	0	0
$761$	38.6150	1.39979	0.699897	−	0.714244i	$-0.253229\pi$
$761$	38.6150	1.39979	0.699897	+	0.714244i	$0.253229\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 11.7082i	− 0.422759i
$768$	0	0
$769$	18.0996	0.652689	0.326345	−	0.945251i	$-0.394183\pi$
$769$	18.0996	0.652689	0.326345	+	0.945251i	$0.394183\pi$
$770$	0	0
$771$	1.05573	0.0380211
$772$	0	0
$773$	− 22.7549i	− 0.818438i	−0.912436	−	0.409219i	$-0.865801\pi$
$773$	− 22.7549i	− 0.818438i	0.912436	−	0.409219i	$-0.134199\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.652476	0.0233774
$780$	0	0
$781$	11.4296	0.408982
$782$	0	0
$783$	− 19.3863i	− 0.692811i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.2449i	1.22070i	0.792133	+	0.610349i	$0.208971\pi$
$787$	34.2449i	1.22070i	−0.792133	+	0.610349i	$0.791029\pi$
$788$	0	0
$789$	−2.36706	−0.0842695
$790$	0	0
$791$	0	0
$792$	0	0
$793$	33.4164i	1.18665i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.2256i	0.610162i	0.952326	+	0.305081i	$0.0986836\pi$
$797$	17.2256i	0.610162i	−0.952326	+	0.305081i	$0.901316\pi$
$798$	0	0
$799$	−13.5279	−0.478581
$800$	0	0
$801$	34.3237	1.21277
$802$	0	0
$803$	50.7054i	1.78935i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.22291i	− 0.0782502i
$808$	0	0
$809$	−52.1246	−1.83260	−0.916302	−	0.400488i	$-0.868841\pi$
$809$	−52.1246	−1.83260	−0.916302	+	0.400488i	$0.868841\pi$
$810$	0	0
$811$	21.4682	0.753852	0.376926	−	0.926243i	$-0.376981\pi$
$811$	21.4682	0.753852	0.376926	+	0.926243i	$0.376981\pi$
$812$	0	0
$813$	3.16718i	0.111078i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.74962i	0.0961969i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.4721	−0.993684	−0.496842	−	0.867841i	$-0.665507\pi$
$821$	−28.4721	−0.993684	−0.496842	+	0.867841i	$0.665507\pi$
$822$	0	0
$823$	33.9443i	1.18322i	0.806223	+	0.591611i	$0.201508\pi$
$823$	33.9443i	1.18322i	−0.806223	+	0.591611i	$0.798492\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 16.3475i	− 0.568459i	−0.958756	−	0.284230i	$-0.908262\pi$
$827$	− 16.3475i	− 0.568459i	0.958756	−	0.284230i	$-0.0917378\pi$
$828$	0	0
$829$	−0.922740	−0.0320481	−0.0160240	−	0.999872i	$-0.505101\pi$
$829$	−0.922740	−0.0320481	−0.0160240	+	0.999872i	$0.505101\pi$
$830$	0	0
$831$	2.67080	0.0926492
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.52786i	− 0.191071i
$838$	0	0
$839$	45.7162	1.57830	0.789149	−	0.614201i	$-0.210522\pi$
$839$	45.7162	1.57830	0.789149	+	0.614201i	$0.210522\pi$
$840$	0	0
$841$	−11.0557	−0.381232
$842$	0	0
$843$	− 14.0934i	− 0.485403i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.81966	−0.0624506
$850$	0	0
$851$	−1.29180	−0.0442822
$852$	0	0
$853$	− 31.3677i	− 1.07401i	−0.843579	−	0.537005i	$-0.819555\pi$
$853$	− 31.3677i	− 1.07401i	0.843579	−	0.537005i	$-0.180445\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.3948i	1.41402i	0.707205	+	0.707009i	$0.249956\pi$
$857$	41.3948i	1.41402i	−0.707205	+	0.707009i	$0.750044\pi$
$858$	0	0
$859$	−52.8360	−1.80274	−0.901370	−	0.433049i	$-0.857438\pi$
$859$	−52.8360	−1.80274	−0.901370	+	0.433049i	$0.857438\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50.1246i	1.70626i	0.521697	+	0.853131i	$0.325299\pi$
$863$	50.1246i	1.70626i	−0.521697	+	0.853131i	$0.674701\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 12.1877i	− 0.413917i
$868$	0	0
$869$	8.77709	0.297742
$870$	0	0
$871$	−9.02546	−0.305816
$872$	0	0
$873$	27.0764i	0.916397i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 58.4296i	− 1.97303i	−0.163682	−	0.986513i	$-0.552337\pi$
$877$	− 58.4296i	− 1.97303i	0.163682	−	0.986513i	$-0.447663\pi$
$878$	0	0
$879$	18.5836	0.626809
$880$	0	0
$881$	−6.60970	−0.222686	−0.111343	−	0.993782i	$-0.535515\pi$
$881$	−6.60970	−0.222686	−0.111343	+	0.993782i	$0.535515\pi$
$882$	0	0
$883$	23.9443i	0.805789i	0.915246	+	0.402894i	$0.131996\pi$
$883$	23.9443i	0.805789i	−0.915246	+	0.402894i	$0.868004\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.03165i	− 0.0346396i	−0.999850	−	0.0173198i	$-0.994487\pi$
$887$	− 1.03165i	− 0.0346396i	0.999850	−	0.0173198i	$-0.00551334\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.40325	0.315021
$892$	0	0
$893$	− 2.58359i	− 0.0864566i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 10.9443i	− 0.365419i
$898$	0	0
$899$	5.11667	0.170651
$900$	0	0
$901$	−2.98605	−0.0994797
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 43.7771i	− 1.45359i	−0.686852	−	0.726797i	$-0.741008\pi$
$907$	− 43.7771i	− 1.45359i	0.686852	−	0.726797i	$-0.258992\pi$
$908$	0	0
$909$	−14.6035	−0.484368
$910$	0	0
$911$	−42.8885	−1.42096	−0.710480	−	0.703717i	$-0.751522\pi$
$911$	−42.8885	−1.42096	−0.710480	+	0.703717i	$0.751522\pi$
$912$	0	0
$913$	31.3376i	1.03712i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.29180	−0.174560	−0.0872801	−	0.996184i	$-0.527818\pi$
$919$	−5.29180	−0.174560	−0.0872801	+	0.996184i	$0.527818\pi$
$920$	0	0
$921$	−10.8328	−0.356953
$922$	0	0
$923$	7.53244i	0.247933i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 41.1096i	− 1.35022i
$928$	0	0
$929$	6.78593	0.222639	0.111319	−	0.993785i	$-0.464492\pi$
$929$	6.78593	0.222639	0.111319	+	0.993785i	$0.464492\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	30.4721i	0.997613i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33.4798i	1.09374i	0.837219	+	0.546868i	$0.184180\pi$
$937$	33.4798i	1.09374i	−0.837219	+	0.546868i	$0.815820\pi$
$938$	0	0
$939$	−14.1803	−0.462758
$940$	0	0
$941$	−27.9991	−0.912746	−0.456373	−	0.889789i	$-0.650852\pi$
$941$	−27.9991	−0.912746	−0.456373	+	0.889789i	$0.650852\pi$
$942$	0	0
$943$	10.6947i	0.348268i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	55.7771i	1.81251i	0.422730	+	0.906256i	$0.361072\pi$
$947$	55.7771i	1.81251i	−0.422730	+	0.906256i	$0.638928\pi$
$948$	0	0
$949$	−33.4164	−1.08474
$950$	0	0
$951$	23.0888	0.748705
$952$	0	0
$953$	55.9017i	1.81083i	0.424524	+	0.905417i	$0.360442\pi$
$953$	55.9017i	1.81083i	−0.424524	+	0.905417i	$0.639558\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 12.8554i	− 0.415557i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.5410	−0.952936
$962$	0	0
$963$	7.88854i	0.254205i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 31.4164i	− 1.01028i	−0.863036	−	0.505142i	$-0.831440\pi$
$967$	− 31.4164i	− 1.01028i	0.863036	−	0.505142i	$-0.168560\pi$
$968$	0	0
$969$	−0.510078	−0.0163861
$970$	0	0
$971$	−44.0957	−1.41510	−0.707549	−	0.706665i	$-0.750199\pi$
$971$	−44.0957	−1.41510	−0.707549	+	0.706665i	$0.750199\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.5967i	1.36279i	0.731916	+	0.681395i	$0.238627\pi$
$977$	42.5967i	1.36279i	−0.731916	+	0.681395i	$0.761373\pi$
$978$	0	0
$979$	53.2974	1.70339
$980$	0	0
$981$	−23.2918	−0.743650
$982$	0	0
$983$	− 1.36551i	− 0.0435529i	−0.999763	−	0.0217764i	$-0.993068\pi$
$983$	− 1.36551i	− 0.0435529i	0.999763	−	0.0217764i	$-0.00693220\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−45.0689	−1.43311
$990$	0	0
$991$	35.6525	1.13254	0.566269	−	0.824220i	$-0.308386\pi$
$991$	35.6525	1.13254	0.566269	+	0.824220i	$0.308386\pi$
$992$	0	0
$993$	− 23.9141i	− 0.758891i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 21.8809i	− 0.692975i	−0.938055	−	0.346488i	$-0.887374\pi$
$997$	− 21.8809i	− 0.692975i	0.938055	−	0.346488i	$-0.112626\pi$
$998$	0	0
$999$	−1.08036	−0.0341812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.u.2549.3		8
5.2	odd	4		4900.2.a.bg.1.2	✓	4
5.3	odd	4		4900.2.a.bi.1.3	yes	4
5.4	even	2	inner	4900.2.e.u.2549.5		8
7.6	odd	2	inner	4900.2.e.u.2549.6		8
35.13	even	4		4900.2.a.bi.1.2	yes	4
35.27	even	4		4900.2.a.bg.1.3	yes	4
35.34	odd	2	inner	4900.2.e.u.2549.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4900.2.a.bg.1.2	✓	4	5.2	odd	4
4900.2.a.bg.1.3	yes	4	35.27	even	4
4900.2.a.bi.1.2	yes	4	35.13	even	4
4900.2.a.bi.1.3	yes	4	5.3	odd	4
4900.2.e.u.2549.3		8	1.1	even	1	trivial
4900.2.e.u.2549.4		8	35.34	odd	2	inner
4900.2.e.u.2549.5		8	5.4	even	2	inner
4900.2.e.u.2549.6		8	7.6	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.874032i q^{3} +2.23607 q^{9} +3.47214 q^{11} +2.28825i q^{13} +1.74806i q^{17} -0.333851 q^{19} -5.47214i q^{23} -4.57649i q^{27} +4.23607 q^{29} +1.20788 q^{31} -3.03476i q^{33} -0.236068i q^{37} +2.00000 q^{39} -1.95440 q^{41} -8.23607i q^{43} +7.73877i q^{47} +1.52786 q^{51} +1.70820i q^{53} +0.291796i q^{57} -5.11667 q^{59} +14.6035 q^{61} +3.94427i q^{67} -4.78282 q^{69} +3.29180 q^{71} +14.6035i q^{73} +2.52786 q^{79} +2.70820 q^{81} +9.02546i q^{83} -3.70246i q^{87} +15.3500 q^{89} -1.05573i q^{93} +12.1089i q^{97} +7.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{11} + 16 q^{29} + 16 q^{39} + 48 q^{51} + 80 q^{71} + 56 q^{79} - 32 q^{81} + 80 q^{99}+O(q^{100}) \) 8 * q - 8 * q^11 + 16 * q^29 + 16 * q^39 + 48 * q^51 + 80 * q^71 + 56 * q^79 - 32 * q^81 + 80 * q^99

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