LMFDB - Embedded newform 6100.2.c.f.4149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6100,2,Mod(4149,6100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6100.4149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6100 = 2^{2} \cdot 5^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6100.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.7087452330$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4149.1
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	6100.4149
Dual form			6100.2.c.f.4149.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24698i q^{3} +1.35690i q^{7} -2.04892 q^{9} +O(q^{10})$	$q-2.24698i q^{3} +1.35690i q^{7} -2.04892 q^{9} +4.40581 q^{11} -3.55496i q^{13} +4.80194i q^{17} -1.96077 q^{19} +3.04892 q^{21} +1.82908i q^{23} -2.13706i q^{27} +3.58211 q^{29} +3.24698 q^{31} -9.89977i q^{33} +1.42327i q^{37} -7.98792 q^{39} -4.07069 q^{41} +2.13706i q^{43} +5.43296i q^{47} +5.15883 q^{49} +10.7899 q^{51} -8.85086i q^{53} +4.40581i q^{57} +10.3448 q^{59} -1.00000 q^{61} -2.78017i q^{63} +7.67994i q^{67} +4.10992 q^{69} -1.71379 q^{71} +0.652793i q^{73} +5.97823i q^{77} +6.71917 q^{79} -10.9487 q^{81} -12.6189i q^{83} -8.04892i q^{87} +5.11960 q^{89} +4.82371 q^{91} -7.29590i q^{93} +17.7899i q^{97} -9.02715 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^9	$ 6 q + 6 q^{9} + 14 q^{19} + 10 q^{29} + 10 q^{31} - 10 q^{39} + 14 q^{49} + 18 q^{51} + 16 q^{59} - 6 q^{61} + 26 q^{69} + 6 q^{71} + 18 q^{79} - 2 q^{81} - 12 q^{89} + 14 q^{91} - 42 q^{99}+O(q^{100}) $ 6 * q + 6 * q^9 + 14 * q^19 + 10 * q^29 + 10 * q^31 - 10 * q^39 + 14 * q^49 + 18 * q^51 + 16 * q^59 - 6 * q^61 + 26 * q^69 + 6 * q^71 + 18 * q^79 - 2 * q^81 - 12 * q^89 + 14 * q^91 - 42 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6100\mathbb{Z}\right)^\times$.

$n$	$977$	$3051$	$3601$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 2.24698i	− 1.29729i	−0.761089	−	0.648647i	$-0.775335\pi$
$3$	− 2.24698i	− 1.29729i	0.761089	−	0.648647i	$-0.224665\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.35690i	0.512858i	0.966563	+	0.256429i	$0.0825461\pi$
$7$	1.35690i	0.512858i	−0.966563	+	0.256429i	$0.917454\pi$
$8$	0	0
$9$	−2.04892	−0.682972
$10$	0	0
$11$	4.40581	1.32840	0.664201	−	0.747554i	$-0.268772\pi$
$11$	4.40581	1.32840	0.664201	+	0.747554i	$0.268772\pi$
$12$	0	0
$13$	− 3.55496i	− 0.985968i	−0.870038	−	0.492984i	$-0.835906\pi$
$13$	− 3.55496i	− 0.985968i	0.870038	−	0.492984i	$-0.164094\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.80194i	1.16464i	0.812959	+	0.582320i	$0.197855\pi$
$17$	4.80194i	1.16464i	−0.812959	+	0.582320i	$0.802145\pi$
$18$	0	0
$19$	−1.96077	−0.449832	−0.224916	−	0.974378i	$-0.572211\pi$
$19$	−1.96077	−0.449832	−0.224916	+	0.974378i	$0.572211\pi$
$20$	0	0
$21$	3.04892	0.665328
$22$	0	0
$23$	1.82908i	0.381391i	0.981649	+	0.190695i	$0.0610742\pi$
$23$	1.82908i	0.381391i	−0.981649	+	0.190695i	$0.938926\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.13706i	− 0.411278i
$28$	0	0
$29$	3.58211	0.665180	0.332590	−	0.943071i	$-0.392077\pi$
$29$	3.58211	0.665180	0.332590	+	0.943071i	$0.392077\pi$
$30$	0	0
$31$	3.24698	0.583175	0.291587	−	0.956544i	$-0.405817\pi$
$31$	3.24698	0.583175	0.291587	+	0.956544i	$0.405817\pi$
$32$	0	0
$33$	− 9.89977i	− 1.72333i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.42327i	0.233984i	0.993133	+	0.116992i	$0.0373253\pi$
$37$	1.42327i	0.233984i	−0.993133	+	0.116992i	$0.962675\pi$
$38$	0	0
$39$	−7.98792	−1.27909
$40$	0	0
$41$	−4.07069	−0.635735	−0.317867	−	0.948135i	$-0.602967\pi$
$41$	−4.07069	−0.635735	−0.317867	+	0.948135i	$0.602967\pi$
$42$	0	0
$43$	2.13706i	0.325899i	0.986634	+	0.162950i	$0.0521008\pi$
$43$	2.13706i	0.325899i	−0.986634	+	0.162950i	$0.947899\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.43296i	0.792479i	0.918147	+	0.396239i	$0.129685\pi$
$47$	5.43296i	0.792479i	−0.918147	+	0.396239i	$0.870315\pi$
$48$	0	0
$49$	5.15883	0.736976
$50$	0	0
$51$	10.7899	1.51088
$52$	0	0
$53$	− 8.85086i	− 1.21576i	−0.794030	−	0.607879i	$-0.792020\pi$
$53$	− 8.85086i	− 1.21576i	0.794030	−	0.607879i	$-0.207980\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.40581i	0.583564i
$58$	0	0
$59$	10.3448	1.34678	0.673390	−	0.739287i	$-0.264837\pi$
$59$	10.3448	1.34678	0.673390	+	0.739287i	$0.264837\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	− 2.78017i	− 0.350268i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.67994i	0.938254i	0.883131	+	0.469127i	$0.155431\pi$
$67$	7.67994i	0.938254i	−0.883131	+	0.469127i	$0.844569\pi$
$68$	0	0
$69$	4.10992	0.494776
$70$	0	0
$71$	−1.71379	−0.203390	−0.101695	−	0.994816i	$-0.532427\pi$
$71$	−1.71379	−0.203390	−0.101695	+	0.994816i	$0.532427\pi$
$72$	0	0
$73$	0.652793i	0.0764036i	0.999270	+	0.0382018i	$0.0121630\pi$
$73$	0.652793i	0.0764036i	−0.999270	+	0.0382018i	$0.987837\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.97823i	0.681283i
$78$	0	0
$79$	6.71917	0.755966	0.377983	−	0.925813i	$-0.376618\pi$
$79$	6.71917	0.755966	0.377983	+	0.925813i	$0.376618\pi$
$80$	0	0
$81$	−10.9487	−1.21652
$82$	0	0
$83$	− 12.6189i	− 1.38511i	−0.721366	−	0.692554i	$-0.756485\pi$
$83$	− 12.6189i	− 1.38511i	0.721366	−	0.692554i	$-0.243515\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 8.04892i	− 0.862935i
$88$	0	0
$89$	5.11960	0.542677	0.271339	−	0.962484i	$-0.412534\pi$
$89$	5.11960	0.542677	0.271339	+	0.962484i	$0.412534\pi$
$90$	0	0
$91$	4.82371	0.505662
$92$	0	0
$93$	− 7.29590i	− 0.756549i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.7899i	1.80629i	0.429340	+	0.903143i	$0.358746\pi$
$97$	17.7899i	1.80629i	−0.429340	+	0.903143i	$0.641254\pi$
$98$	0	0
$99$	−9.02715	−0.907262
$100$	0	0
$101$	17.2325	1.71470	0.857349	−	0.514735i	$-0.172110\pi$
$101$	17.2325	1.71470	0.857349	+	0.514735i	$0.172110\pi$
$102$	0	0
$103$	12.7995i	1.26118i	0.776118	+	0.630588i	$0.217186\pi$
$103$	12.7995i	1.26118i	−0.776118	+	0.630588i	$0.782814\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.05429i	− 0.778638i	−0.921103	−	0.389319i	$-0.872710\pi$
$107$	− 8.05429i	− 0.778638i	0.921103	−	0.389319i	$-0.127290\pi$
$108$	0	0
$109$	14.8442	1.42181	0.710906	−	0.703287i	$-0.248285\pi$
$109$	14.8442	1.42181	0.710906	+	0.703287i	$0.248285\pi$
$110$	0	0
$111$	3.19806	0.303547
$112$	0	0
$113$	5.33273i	0.501661i	0.968031	+	0.250831i	$0.0807037\pi$
$113$	5.33273i	0.501661i	−0.968031	+	0.250831i	$0.919296\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.28382i	0.673389i
$118$	0	0
$119$	−6.51573	−0.597296
$120$	0	0
$121$	8.41119	0.764654
$122$	0	0
$123$	9.14675i	0.824735i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.35690i	− 0.830290i	−0.909755	−	0.415145i	$-0.863731\pi$
$127$	− 9.35690i	− 0.830290i	0.909755	−	0.415145i	$-0.136269\pi$
$128$	0	0
$129$	4.80194	0.422787
$130$	0	0
$131$	−15.2717	−1.33430	−0.667149	−	0.744924i	$-0.732485\pi$
$131$	−15.2717	−1.33430	−0.667149	+	0.744924i	$0.732485\pi$
$132$	0	0
$133$	− 2.66056i	− 0.230700i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 11.9095i	− 1.01749i	−0.860916	−	0.508747i	$-0.830109\pi$
$137$	− 11.9095i	− 1.01749i	0.860916	−	0.508747i	$-0.169891\pi$
$138$	0	0
$139$	19.2784	1.63518	0.817588	−	0.575804i	$-0.195311\pi$
$139$	19.2784	1.63518	0.817588	+	0.575804i	$0.195311\pi$
$140$	0	0
$141$	12.2078	1.02808
$142$	0	0
$143$	− 15.6625i	− 1.30976i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 11.5918i	− 0.956075i
$148$	0	0
$149$	−0.170915	−0.0140019	−0.00700096	−	0.999975i	$-0.502228\pi$
$149$	−0.170915	−0.0140019	−0.00700096	+	0.999975i	$0.502228\pi$
$150$	0	0
$151$	0.951083	0.0773980	0.0386990	−	0.999251i	$-0.487679\pi$
$151$	0.951083	0.0773980	0.0386990	+	0.999251i	$0.487679\pi$
$152$	0	0
$153$	− 9.83877i	− 0.795418i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.24027i	− 0.577837i	−0.957354	−	0.288918i	$-0.906704\pi$
$157$	− 7.24027i	− 0.577837i	0.957354	−	0.288918i	$-0.0932956\pi$
$158$	0	0
$159$	−19.8877	−1.57720
$160$	0	0
$161$	−2.48188	−0.195599
$162$	0	0
$163$	− 18.6136i	− 1.45793i	−0.684553	−	0.728963i	$-0.740002\pi$
$163$	− 18.6136i	− 1.45793i	0.684553	−	0.728963i	$-0.259998\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.8092i	1.30074i	0.759619	+	0.650369i	$0.225386\pi$
$167$	16.8092i	1.30074i	−0.759619	+	0.650369i	$0.774614\pi$
$168$	0	0
$169$	0.362273	0.0278671
$170$	0	0
$171$	4.01746	0.307223
$172$	0	0
$173$	− 9.39373i	− 0.714192i	−0.934068	−	0.357096i	$-0.883767\pi$
$173$	− 9.39373i	− 0.714192i	0.934068	−	0.357096i	$-0.116233\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 23.2446i	− 1.74717i
$178$	0	0
$179$	−12.8562	−0.960920	−0.480460	−	0.877017i	$-0.659530\pi$
$179$	−12.8562	−0.960920	−0.480460	+	0.877017i	$0.659530\pi$
$180$	0	0
$181$	−1.46681	−0.109027	−0.0545136	−	0.998513i	$-0.517361\pi$
$181$	−1.46681	−0.109027	−0.0545136	+	0.998513i	$0.517361\pi$
$182$	0	0
$183$	2.24698i	0.166102i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	21.1564i	1.54711i
$188$	0	0
$189$	2.89977	0.210927
$190$	0	0
$191$	19.8485	1.43618	0.718092	−	0.695948i	$-0.245016\pi$
$191$	19.8485	1.43618	0.718092	+	0.695948i	$0.245016\pi$
$192$	0	0
$193$	5.69740i	0.410108i	0.978751	+	0.205054i	$0.0657370\pi$
$193$	5.69740i	0.410108i	−0.978751	+	0.205054i	$0.934263\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.5036i	− 0.819601i	−0.912175	−	0.409800i	$-0.865598\pi$
$197$	− 11.5036i	− 0.819601i	0.912175	−	0.409800i	$-0.134402\pi$
$198$	0	0
$199$	−10.3013	−0.730238	−0.365119	−	0.930961i	$-0.618972\pi$
$199$	−10.3013	−0.730238	−0.365119	+	0.930961i	$0.618972\pi$
$200$	0	0
$201$	17.2567	1.21719
$202$	0	0
$203$	4.86054i	0.341143i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3.74764i	− 0.260479i
$208$	0	0
$209$	−8.63879	−0.597558
$210$	0	0
$211$	11.4179	0.786040	0.393020	−	0.919530i	$-0.371430\pi$
$211$	11.4179	0.786040	0.393020	+	0.919530i	$0.371430\pi$
$212$	0	0
$213$	3.85086i	0.263856i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.40581i	0.299086i
$218$	0	0
$219$	1.46681	0.0991180
$220$	0	0
$221$	17.0707	1.14830
$222$	0	0
$223$	− 16.1153i	− 1.07916i	−0.841934	−	0.539580i	$-0.818583\pi$
$223$	− 16.1153i	− 1.07916i	0.841934	−	0.539580i	$-0.181417\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.30260i	− 0.351946i	−0.984395	−	0.175973i	$-0.943693\pi$
$227$	− 5.30260i	− 0.351946i	0.984395	−	0.175973i	$-0.0563071\pi$
$228$	0	0
$229$	2.70410	0.178692	0.0893461	−	0.996001i	$-0.471522\pi$
$229$	2.70410	0.178692	0.0893461	+	0.996001i	$0.471522\pi$
$230$	0	0
$231$	13.4330	0.883824
$232$	0	0
$233$	17.2392i	1.12938i	0.825304	+	0.564689i	$0.191004\pi$
$233$	17.2392i	1.12938i	−0.825304	+	0.564689i	$0.808996\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 15.0978i	− 0.980710i
$238$	0	0
$239$	−11.3351	−0.733208	−0.366604	−	0.930377i	$-0.619480\pi$
$239$	−11.3351	−0.733208	−0.366604	+	0.930377i	$0.619480\pi$
$240$	0	0
$241$	1.40581	0.0905564	0.0452782	−	0.998974i	$-0.485583\pi$
$241$	1.40581	0.0905564	0.0452782	+	0.998974i	$0.485583\pi$
$242$	0	0
$243$	18.1903i	1.16691i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.97046i	0.443520i
$248$	0	0
$249$	−28.3545	−1.79689
$250$	0	0
$251$	−10.1400	−0.640034	−0.320017	−	0.947412i	$-0.603689\pi$
$251$	−10.1400	−0.640034	−0.320017	+	0.947412i	$0.603689\pi$
$252$	0	0
$253$	8.05861i	0.506640i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.7603i	− 0.920723i	−0.887731	−	0.460362i	$-0.847720\pi$
$257$	− 14.7603i	− 0.920723i	0.887731	−	0.460362i	$-0.152280\pi$
$258$	0	0
$259$	−1.93123	−0.120001
$260$	0	0
$261$	−7.33944	−0.454300
$262$	0	0
$263$	10.1860i	0.628094i	0.949407	+	0.314047i	$0.101685\pi$
$263$	10.1860i	0.628094i	−0.949407	+	0.314047i	$0.898315\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 11.5036i	− 0.704012i
$268$	0	0
$269$	−0.902165	−0.0550060	−0.0275030	−	0.999622i	$-0.508756\pi$
$269$	−0.902165	−0.0550060	−0.0275030	+	0.999622i	$0.508756\pi$
$270$	0	0
$271$	13.0881	0.795048	0.397524	−	0.917592i	$-0.369869\pi$
$271$	13.0881	0.795048	0.397524	+	0.917592i	$0.369869\pi$
$272$	0	0
$273$	− 10.8388i	− 0.655992i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.76271i	0.226079i	0.993590	+	0.113040i	$0.0360587\pi$
$277$	3.76271i	0.226079i	−0.993590	+	0.113040i	$0.963941\pi$
$278$	0	0
$279$	−6.65279	−0.398292
$280$	0	0
$281$	31.6450	1.88778	0.943892	−	0.330255i	$-0.107135\pi$
$281$	31.6450	1.88778	0.943892	+	0.330255i	$0.107135\pi$
$282$	0	0
$283$	− 19.0858i	− 1.13453i	−0.823535	−	0.567265i	$-0.808001\pi$
$283$	− 19.0858i	− 1.13453i	0.823535	−	0.567265i	$-0.191999\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5.52350i	− 0.326042i
$288$	0	0
$289$	−6.05861	−0.356389
$290$	0	0
$291$	39.9734	2.34328
$292$	0	0
$293$	− 6.55927i	− 0.383197i	−0.981473	−	0.191598i	$-0.938633\pi$
$293$	− 6.55927i	− 0.383197i	0.981473	−	0.191598i	$-0.0613671\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 9.41550i	− 0.546343i
$298$	0	0
$299$	6.50232	0.376039
$300$	0	0
$301$	−2.89977	−0.167140
$302$	0	0
$303$	− 38.7211i	− 2.22447i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 13.2935i	− 0.758700i	−0.925253	−	0.379350i	$-0.876148\pi$
$307$	− 13.2935i	− 0.758700i	0.925253	−	0.379350i	$-0.123852\pi$
$308$	0	0
$309$	28.7603	1.63612
$310$	0	0
$311$	3.82371	0.216823	0.108411	−	0.994106i	$-0.465424\pi$
$311$	3.82371	0.216823	0.108411	+	0.994106i	$0.465424\pi$
$312$	0	0
$313$	− 5.81833i	− 0.328872i	−0.986388	−	0.164436i	$-0.947420\pi$
$313$	− 5.81833i	− 0.328872i	0.986388	−	0.164436i	$-0.0525803\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.8810i	1.39746i	0.715388	+	0.698728i	$0.246250\pi$
$317$	24.8810i	1.39746i	−0.715388	+	0.698728i	$0.753750\pi$
$318$	0	0
$319$	15.7821	0.883627
$320$	0	0
$321$	−18.0978	−1.01012
$322$	0	0
$323$	− 9.41550i	− 0.523893i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 33.3545i	− 1.84451i
$328$	0	0
$329$	−7.37196	−0.406429
$330$	0	0
$331$	−16.1957	−0.890194	−0.445097	−	0.895482i	$-0.646831\pi$
$331$	−16.1957	−0.890194	−0.445097	+	0.895482i	$0.646831\pi$
$332$	0	0
$333$	− 2.91617i	− 0.159805i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.90946i	0.376382i	0.982132	+	0.188191i	$0.0602625\pi$
$337$	6.90946i	0.376382i	−0.982132	+	0.188191i	$0.939738\pi$
$338$	0	0
$339$	11.9825	0.650802
$340$	0	0
$341$	14.3056	0.774691
$342$	0	0
$343$	16.4983i	0.890823i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.5633i	− 0.889166i	−0.895738	−	0.444583i	$-0.853352\pi$
$347$	− 16.5633i	− 0.889166i	0.895738	−	0.444583i	$-0.146648\pi$
$348$	0	0
$349$	10.0978	0.540525	0.270262	−	0.962787i	$-0.412890\pi$
$349$	10.0978	0.540525	0.270262	+	0.962787i	$0.412890\pi$
$350$	0	0
$351$	−7.59717	−0.405507
$352$	0	0
$353$	4.36765i	0.232467i	0.993222	+	0.116233i	$0.0370820\pi$
$353$	4.36765i	0.232467i	−0.993222	+	0.116233i	$0.962918\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14.6407i	0.774869i
$358$	0	0
$359$	−7.69633	−0.406197	−0.203098	−	0.979158i	$-0.565101\pi$
$359$	−7.69633	−0.406197	−0.203098	+	0.979158i	$0.565101\pi$
$360$	0	0
$361$	−15.1554	−0.797651
$362$	0	0
$363$	− 18.8998i	− 0.991981i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.0489i	− 0.889946i	−0.895544	−	0.444973i	$-0.853213\pi$
$367$	− 17.0489i	− 0.889946i	0.895544	−	0.444973i	$-0.146787\pi$
$368$	0	0
$369$	8.34050	0.434189
$370$	0	0
$371$	12.0097	0.623512
$372$	0	0
$373$	− 19.3284i	− 1.00079i	−0.865798	−	0.500394i	$-0.833189\pi$
$373$	− 19.3284i	− 1.00079i	0.865798	−	0.500394i	$-0.166811\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.7342i	− 0.655846i
$378$	0	0
$379$	−21.1812	−1.08800	−0.544002	−	0.839084i	$-0.683092\pi$
$379$	−21.1812	−1.08800	−0.544002	+	0.839084i	$0.683092\pi$
$380$	0	0
$381$	−21.0248	−1.07713
$382$	0	0
$383$	− 19.1153i	− 0.976746i	−0.872635	−	0.488373i	$-0.837591\pi$
$383$	− 19.1153i	− 0.976746i	0.872635	−	0.488373i	$-0.162409\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.37867i	− 0.222580i
$388$	0	0
$389$	8.25236	0.418411	0.209206	−	0.977872i	$-0.432912\pi$
$389$	8.25236	0.418411	0.209206	+	0.977872i	$0.432912\pi$
$390$	0	0
$391$	−8.78315	−0.444183
$392$	0	0
$393$	34.3153i	1.73098i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.5405i	1.43241i	0.697892	+	0.716203i	$0.254121\pi$
$397$	28.5405i	1.43241i	−0.697892	+	0.716203i	$0.745879\pi$
$398$	0	0
$399$	−5.97823	−0.299286
$400$	0	0
$401$	−15.7614	−0.787086	−0.393543	−	0.919306i	$-0.628751\pi$
$401$	−15.7614	−0.787086	−0.393543	+	0.919306i	$0.628751\pi$
$402$	0	0
$403$	− 11.5429i	− 0.574992i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.27067i	0.310826i
$408$	0	0
$409$	−24.0170	−1.18756	−0.593782	−	0.804626i	$-0.702366\pi$
$409$	−24.0170	−1.18756	−0.593782	+	0.804626i	$0.702366\pi$
$410$	0	0
$411$	−26.7603	−1.31999
$412$	0	0
$413$	14.0368i	0.690708i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 43.3183i	− 2.12130i
$418$	0	0
$419$	−9.95108	−0.486142	−0.243071	−	0.970008i	$-0.578155\pi$
$419$	−9.95108	−0.486142	−0.243071	+	0.970008i	$0.578155\pi$
$420$	0	0
$421$	−29.9312	−1.45876	−0.729379	−	0.684109i	$-0.760191\pi$
$421$	−29.9312	−1.45876	−0.729379	+	0.684109i	$0.760191\pi$
$422$	0	0
$423$	− 11.1317i	− 0.541241i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.35690i	− 0.0656648i
$428$	0	0
$429$	−35.1933	−1.69915
$430$	0	0
$431$	−22.8552	−1.10089	−0.550447	−	0.834870i	$-0.685543\pi$
$431$	−22.8552	−1.10089	−0.550447	+	0.834870i	$0.685543\pi$
$432$	0	0
$433$	0.882314i	0.0424013i	0.999775	+	0.0212007i	$0.00674888\pi$
$433$	0.882314i	0.0424013i	−0.999775	+	0.0212007i	$0.993251\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.58642i	− 0.171562i
$438$	0	0
$439$	−8.78554	−0.419311	−0.209656	−	0.977775i	$-0.567234\pi$
$439$	−8.78554	−0.419311	−0.209656	+	0.977775i	$0.567234\pi$
$440$	0	0
$441$	−10.5700	−0.503334
$442$	0	0
$443$	− 15.2664i	− 0.725327i	−0.931920	−	0.362663i	$-0.881868\pi$
$443$	− 15.2664i	− 0.725327i	0.931920	−	0.362663i	$-0.118132\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.384043i	0.0181646i
$448$	0	0
$449$	19.3588	0.913599	0.456799	−	0.889570i	$-0.348996\pi$
$449$	19.3588	0.913599	0.456799	+	0.889570i	$0.348996\pi$
$450$	0	0
$451$	−17.9347	−0.844512
$452$	0	0
$453$	− 2.13706i	− 0.100408i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.34913i	0.437334i	0.975800	+	0.218667i	$0.0701708\pi$
$457$	9.34913i	0.437334i	−0.975800	+	0.218667i	$0.929829\pi$
$458$	0	0
$459$	10.2620	0.478991
$460$	0	0
$461$	10.2101	0.475534	0.237767	−	0.971322i	$-0.423585\pi$
$461$	10.2101	0.475534	0.237767	+	0.971322i	$0.423585\pi$
$462$	0	0
$463$	− 2.49396i	− 0.115904i	−0.998319	−	0.0579521i	$-0.981543\pi$
$463$	− 2.49396i	− 0.115904i	0.998319	−	0.0579521i	$-0.0184571\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.89008i	0.272561i	0.990670	+	0.136280i	$0.0435148\pi$
$467$	5.89008i	0.272561i	−0.990670	+	0.136280i	$0.956485\pi$
$468$	0	0
$469$	−10.4209	−0.481191
$470$	0	0
$471$	−16.2687	−0.749624
$472$	0	0
$473$	9.41550i	0.432925i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.1347i	0.830330i
$478$	0	0
$479$	−38.2857	−1.74932	−0.874660	−	0.484737i	$-0.838915\pi$
$479$	−38.2857	−1.74932	−0.874660	+	0.484737i	$0.838915\pi$
$480$	0	0
$481$	5.05967	0.230701
$482$	0	0
$483$	5.57673i	0.253750i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 36.2653i	− 1.64334i	−0.569966	−	0.821669i	$-0.693043\pi$
$487$	− 36.2653i	− 1.64334i	0.569966	−	0.821669i	$-0.306957\pi$
$488$	0	0
$489$	−41.8243	−1.89136
$490$	0	0
$491$	−3.95002	−0.178262	−0.0891309	−	0.996020i	$-0.528409\pi$
$491$	−3.95002	−0.178262	−0.0891309	+	0.996020i	$0.528409\pi$
$492$	0	0
$493$	17.2010i	0.774696i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.32544i	− 0.104310i
$498$	0	0
$499$	20.4179	0.914031	0.457015	−	0.889459i	$-0.348918\pi$
$499$	20.4179	0.914031	0.457015	+	0.889459i	$0.348918\pi$
$500$	0	0
$501$	37.7700	1.68744
$502$	0	0
$503$	4.39134i	0.195800i	0.995196	+	0.0979001i	$0.0312126\pi$
$503$	4.39134i	0.195800i	−0.995196	+	0.0979001i	$0.968787\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.814019i	− 0.0361519i
$508$	0	0
$509$	6.48619	0.287495	0.143748	−	0.989614i	$-0.454085\pi$
$509$	6.48619	0.287495	0.143748	+	0.989614i	$0.454085\pi$
$510$	0	0
$511$	−0.885772	−0.0391842
$512$	0	0
$513$	4.19029i	0.185006i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.9366i	1.05273i
$518$	0	0
$519$	−21.1075	−0.926517
$520$	0	0
$521$	−43.0036	−1.88402	−0.942010	−	0.335584i	$-0.891066\pi$
$521$	−43.0036	−1.88402	−0.942010	+	0.335584i	$0.891066\pi$
$522$	0	0
$523$	− 31.6329i	− 1.38321i	−0.722275	−	0.691606i	$-0.756903\pi$
$523$	− 31.6329i	− 1.38321i	0.722275	−	0.691606i	$-0.243097\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.5918i	0.679189i
$528$	0	0
$529$	19.6544	0.854541
$530$	0	0
$531$	−21.1957	−0.919814
$532$	0	0
$533$	14.4711i	0.626814i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	28.8877i	1.24660i
$538$	0	0
$539$	22.7289	0.979001
$540$	0	0
$541$	28.4650	1.22381	0.611903	−	0.790933i	$-0.290404\pi$
$541$	28.4650	1.22381	0.611903	+	0.790933i	$0.290404\pi$
$542$	0	0
$543$	3.29590i	0.141440i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 17.0713i	− 0.729915i	−0.931024	−	0.364958i	$-0.881083\pi$
$547$	− 17.0713i	− 0.729915i	0.931024	−	0.364958i	$-0.118917\pi$
$548$	0	0
$549$	2.04892	0.0874457
$550$	0	0
$551$	−7.02369	−0.299219
$552$	0	0
$553$	9.11721i	0.387703i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.29483i	0.393835i	0.980420	+	0.196917i	$0.0630931\pi$
$557$	9.29483i	0.393835i	−0.980420	+	0.196917i	$0.936907\pi$
$558$	0	0
$559$	7.59717	0.321326
$560$	0	0
$561$	47.5381	2.00706
$562$	0	0
$563$	22.7928i	0.960604i	0.877103	+	0.480302i	$0.159473\pi$
$563$	22.7928i	0.960604i	−0.877103	+	0.480302i	$0.840527\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 14.8562i	− 0.623903i
$568$	0	0
$569$	−4.99223	−0.209285	−0.104643	−	0.994510i	$-0.533370\pi$
$569$	−4.99223	−0.209285	−0.104643	+	0.994510i	$0.533370\pi$
$570$	0	0
$571$	−5.59419	−0.234109	−0.117055	−	0.993125i	$-0.537345\pi$
$571$	−5.59419	−0.234109	−0.117055	+	0.993125i	$0.537345\pi$
$572$	0	0
$573$	− 44.5991i	− 1.86315i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 29.0374i	− 1.20884i	−0.796664	−	0.604422i	$-0.793404\pi$
$577$	− 29.0374i	− 1.20884i	0.796664	−	0.604422i	$-0.206596\pi$
$578$	0	0
$579$	12.8019	0.532030
$580$	0	0
$581$	17.1226	0.710365
$582$	0	0
$583$	− 38.9952i	− 1.61502i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 42.1400i	− 1.73931i	−0.493664	−	0.869653i	$-0.664343\pi$
$587$	− 42.1400i	− 1.73931i	0.493664	−	0.869653i	$-0.335657\pi$
$588$	0	0
$589$	−6.36658	−0.262331
$590$	0	0
$591$	−25.8485	−1.06326
$592$	0	0
$593$	4.59419i	0.188661i	0.995541	+	0.0943303i	$0.0300710\pi$
$593$	4.59419i	0.188661i	−0.995541	+	0.0943303i	$0.969929\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.1468i	0.947333i
$598$	0	0
$599$	20.4838	0.836945	0.418473	−	0.908229i	$-0.362566\pi$
$599$	20.4838	0.836945	0.418473	+	0.908229i	$0.362566\pi$
$600$	0	0
$601$	−31.8431	−1.29891	−0.649453	−	0.760402i	$-0.725002\pi$
$601$	−31.8431	−1.29891	−0.649453	+	0.760402i	$0.725002\pi$
$602$	0	0
$603$	− 15.7356i	− 0.640802i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 29.9845i	− 1.21703i	−0.793542	−	0.608516i	$-0.791765\pi$
$607$	− 29.9845i	− 1.21703i	0.793542	−	0.608516i	$-0.208235\pi$
$608$	0	0
$609$	10.9215	0.442563
$610$	0	0
$611$	19.3139	0.781359
$612$	0	0
$613$	7.43594i	0.300335i	0.988661	+	0.150167i	$0.0479813\pi$
$613$	7.43594i	0.300335i	−0.988661	+	0.150167i	$0.952019\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.7205i	0.793917i	0.917836	+	0.396959i	$0.129934\pi$
$617$	19.7205i	0.793917i	−0.917836	+	0.396959i	$0.870066\pi$
$618$	0	0
$619$	29.7942	1.19753	0.598764	−	0.800925i	$-0.295659\pi$
$619$	29.7942	1.19753	0.598764	+	0.800925i	$0.295659\pi$
$620$	0	0
$621$	3.90887	0.156858
$622$	0	0
$623$	6.94677i	0.278316i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	19.4112i	0.775208i
$628$	0	0
$629$	−6.83446	−0.272508
$630$	0	0
$631$	7.72827	0.307657	0.153829	−	0.988098i	$-0.450840\pi$
$631$	7.72827	0.307657	0.153829	+	0.988098i	$0.450840\pi$
$632$	0	0
$633$	− 25.6558i	− 1.01973i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.3394i	− 0.726635i
$638$	0	0
$639$	3.51142	0.138910
$640$	0	0
$641$	−18.9041	−0.746666	−0.373333	−	0.927697i	$-0.621785\pi$
$641$	−18.9041	−0.746666	−0.373333	+	0.927697i	$0.621785\pi$
$642$	0	0
$643$	− 2.01315i	− 0.0793907i	−0.999212	−	0.0396954i	$-0.987361\pi$
$643$	− 2.01315i	− 0.0793907i	0.999212	−	0.0396954i	$-0.0126388\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.56273i	0.179379i	0.995970	+	0.0896897i	$0.0285875\pi$
$647$	4.56273i	0.179379i	−0.995970	+	0.0896897i	$0.971412\pi$
$648$	0	0
$649$	45.5773	1.78907
$650$	0	0
$651$	9.89977	0.388003
$652$	0	0
$653$	39.4077i	1.54214i	0.636748	+	0.771072i	$0.280279\pi$
$653$	39.4077i	1.54214i	−0.636748	+	0.771072i	$0.719721\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.33752i	− 0.0521816i
$658$	0	0
$659$	−25.2379	−0.983128	−0.491564	−	0.870841i	$-0.663575\pi$
$659$	−25.2379	−0.983128	−0.491564	+	0.870841i	$0.663575\pi$
$660$	0	0
$661$	−31.7004	−1.23300	−0.616501	−	0.787354i	$-0.711450\pi$
$661$	−31.7004	−1.23300	−0.616501	+	0.787354i	$0.711450\pi$
$662$	0	0
$663$	− 38.3575i	− 1.48968i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.55197i	0.253693i
$668$	0	0
$669$	−36.2107	−1.39999
$670$	0	0
$671$	−4.40581	−0.170085
$672$	0	0
$673$	47.5978i	1.83476i	0.398014	+	0.917379i	$0.369700\pi$
$673$	47.5978i	1.83476i	−0.398014	+	0.917379i	$0.630300\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.01447i	0.346454i	0.984882	+	0.173227i	$0.0554195\pi$
$677$	9.01447i	0.346454i	−0.984882	+	0.173227i	$0.944581\pi$
$678$	0	0
$679$	−24.1390	−0.926369
$680$	0	0
$681$	−11.9148	−0.456578
$682$	0	0
$683$	11.4547i	0.438303i	0.975691	+	0.219152i	$0.0703289\pi$
$683$	11.4547i	0.438303i	−0.975691	+	0.219152i	$0.929671\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 6.07606i	− 0.231816i
$688$	0	0
$689$	−31.4644	−1.19870
$690$	0	0
$691$	−12.8556	−0.489052	−0.244526	−	0.969643i	$-0.578632\pi$
$691$	−12.8556	−0.489052	−0.244526	+	0.969643i	$0.578632\pi$
$692$	0	0
$693$	− 12.2489i	− 0.465297i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 19.5472i	− 0.740403i
$698$	0	0
$699$	38.7362	1.46514
$700$	0	0
$701$	39.8509	1.50515	0.752573	−	0.658509i	$-0.228812\pi$
$701$	39.8509	1.50515	0.752573	+	0.658509i	$0.228812\pi$
$702$	0	0
$703$	− 2.79071i	− 0.105254i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23.3827i	0.879398i
$708$	0	0
$709$	4.01879	0.150929	0.0754643	−	0.997149i	$-0.475956\pi$
$709$	4.01879	0.150929	0.0754643	+	0.997149i	$0.475956\pi$
$710$	0	0
$711$	−13.7670	−0.516304
$712$	0	0
$713$	5.93900i	0.222417i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.4698i	0.951187i
$718$	0	0
$719$	50.4687	1.88217	0.941083	−	0.338176i	$-0.109810\pi$
$719$	50.4687	1.88217	0.941083	+	0.338176i	$0.109810\pi$
$720$	0	0
$721$	−17.3676	−0.646805
$722$	0	0
$723$	− 3.15883i	− 0.117478i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 49.7493i	− 1.84510i	−0.385878	−	0.922550i	$-0.626102\pi$
$727$	− 49.7493i	− 1.84510i	0.385878	−	0.922550i	$-0.373898\pi$
$728$	0	0
$729$	8.02715	0.297302
$730$	0	0
$731$	−10.2620	−0.379555
$732$	0	0
$733$	− 20.7396i	− 0.766035i	−0.923741	−	0.383018i	$-0.874885\pi$
$733$	− 20.7396i	− 0.766035i	0.923741	−	0.383018i	$-0.125115\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.8364i	1.24638i
$738$	0	0
$739$	31.7429	1.16768	0.583840	−	0.811869i	$-0.301550\pi$
$739$	31.7429	1.16768	0.583840	+	0.811869i	$0.301550\pi$
$740$	0	0
$741$	15.6625	0.575376
$742$	0	0
$743$	36.9965i	1.35727i	0.734475	+	0.678636i	$0.237428\pi$
$743$	36.9965i	1.35727i	−0.734475	+	0.678636i	$0.762572\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	25.8552i	0.945991i
$748$	0	0
$749$	10.9288	0.399331
$750$	0	0
$751$	29.0887	1.06146	0.530732	−	0.847540i	$-0.321917\pi$
$751$	29.0887	1.06146	0.530732	+	0.847540i	$0.321917\pi$
$752$	0	0
$753$	22.7845i	0.830313i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	11.4813i	0.417294i	0.977991	+	0.208647i	$0.0669060\pi$
$757$	11.4813i	0.417294i	−0.977991	+	0.208647i	$0.933094\pi$
$758$	0	0
$759$	18.1075	0.657261
$760$	0	0
$761$	15.5060	0.562094	0.281047	−	0.959694i	$-0.409318\pi$
$761$	15.5060	0.562094	0.281047	+	0.959694i	$0.409318\pi$
$762$	0	0
$763$	20.1420i	0.729188i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 36.7754i	− 1.32788i
$768$	0	0
$769$	15.0049	0.541091	0.270545	−	0.962707i	$-0.412796\pi$
$769$	15.0049	0.541091	0.270545	+	0.962707i	$0.412796\pi$
$770$	0	0
$771$	−33.1661	−1.19445
$772$	0	0
$773$	14.0978i	0.507064i	0.967327	+	0.253532i	$0.0815923\pi$
$773$	14.0978i	0.507064i	−0.967327	+	0.253532i	$0.918408\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.33944i	0.155676i
$778$	0	0
$779$	7.98169	0.285974
$780$	0	0
$781$	−7.55065	−0.270183
$782$	0	0
$783$	− 7.65519i	− 0.273574i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.51679i	0.232299i	0.993232	+	0.116149i	$0.0370551\pi$
$787$	6.51679i	0.232299i	−0.993232	+	0.116149i	$0.962945\pi$
$788$	0	0
$789$	22.8877	0.814823
$790$	0	0
$791$	−7.23596	−0.257281
$792$	0	0
$793$	3.55496i	0.126240i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.3502i	1.11048i	0.831690	+	0.555240i	$0.187374\pi$
$797$	31.3502i	1.11048i	−0.831690	+	0.555240i	$0.812626\pi$
$798$	0	0
$799$	−26.0887	−0.922953
$800$	0	0
$801$	−10.4896	−0.370633
$802$	0	0
$803$	2.87608i	0.101495i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.02715i	0.0713590i
$808$	0	0
$809$	36.4196	1.28044	0.640222	−	0.768190i	$-0.278843\pi$
$809$	36.4196	1.28044	0.640222	+	0.768190i	$0.278843\pi$
$810$	0	0
$811$	24.1148	0.846786	0.423393	−	0.905946i	$-0.360839\pi$
$811$	24.1148	0.846786	0.423393	+	0.905946i	$0.360839\pi$
$812$	0	0
$813$	− 29.4088i	− 1.03141i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.19029i	− 0.146600i
$818$	0	0
$819$	−9.88338	−0.345353
$820$	0	0
$821$	46.7385	1.63119	0.815593	−	0.578626i	$-0.196411\pi$
$821$	46.7385	1.63119	0.815593	+	0.578626i	$0.196411\pi$
$822$	0	0
$823$	3.35988i	0.117118i	0.998284	+	0.0585590i	$0.0186506\pi$
$823$	3.35988i	0.117118i	−0.998284	+	0.0585590i	$0.981349\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.8616i	0.968843i	0.874835	+	0.484422i	$0.160970\pi$
$827$	27.8616i	0.968843i	−0.874835	+	0.484422i	$0.839030\pi$
$828$	0	0
$829$	24.2502	0.842245	0.421123	−	0.907004i	$-0.361636\pi$
$829$	24.2502	0.842245	0.421123	+	0.907004i	$0.361636\pi$
$830$	0	0
$831$	8.45473	0.293291
$832$	0	0
$833$	24.7724i	0.858313i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 6.93900i	− 0.239847i
$838$	0	0
$839$	−33.3836	−1.15253	−0.576264	−	0.817263i	$-0.695490\pi$
$839$	−33.3836	−1.15253	−0.576264	+	0.817263i	$0.695490\pi$
$840$	0	0
$841$	−16.1685	−0.557535
$842$	0	0
$843$	− 71.1057i	− 2.44901i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.4131i	0.392159i
$848$	0	0
$849$	−42.8853	−1.47182
$850$	0	0
$851$	−2.60328	−0.0892394
$852$	0	0
$853$	10.3515i	0.354429i	0.984172	+	0.177215i	$0.0567087\pi$
$853$	10.3515i	0.354429i	−0.984172	+	0.177215i	$0.943291\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	56.6896i	1.93648i	0.250021	+	0.968240i	$0.419562\pi$
$857$	56.6896i	1.93648i	−0.250021	+	0.968240i	$0.580438\pi$
$858$	0	0
$859$	−10.4534	−0.356665	−0.178333	−	0.983970i	$-0.557070\pi$
$859$	−10.4534	−0.356665	−0.178333	+	0.983970i	$0.557070\pi$
$860$	0	0
$861$	−12.4112	−0.422972
$862$	0	0
$863$	19.8890i	0.677030i	0.940961	+	0.338515i	$0.109925\pi$
$863$	19.8890i	0.677030i	−0.940961	+	0.338515i	$0.890075\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.6136i	0.462341i
$868$	0	0
$869$	29.6034	1.00423
$870$	0	0
$871$	27.3019	0.925088
$872$	0	0
$873$	− 36.4499i	− 1.23364i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.8256i	0.703231i	0.936144	+	0.351616i	$0.114368\pi$
$877$	20.8256i	0.703231i	−0.936144	+	0.351616i	$0.885632\pi$
$878$	0	0
$879$	−14.7385	−0.497119
$880$	0	0
$881$	45.6655	1.53851	0.769254	−	0.638943i	$-0.220628\pi$
$881$	45.6655	1.53851	0.769254	+	0.638943i	$0.220628\pi$
$882$	0	0
$883$	− 2.48081i	− 0.0834860i	−0.999128	−	0.0417430i	$-0.986709\pi$
$883$	− 2.48081i	− 0.0834860i	0.999128	−	0.0417430i	$-0.0132911\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.94869i	0.132584i	0.997800	+	0.0662920i	$0.0211169\pi$
$887$	3.94869i	0.132584i	−0.997800	+	0.0662920i	$0.978883\pi$
$888$	0	0
$889$	12.6963	0.425821
$890$	0	0
$891$	−48.2379	−1.61603
$892$	0	0
$893$	− 10.6528i	− 0.356482i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 14.6106i	− 0.487833i
$898$	0	0
$899$	11.6310	0.387916
$900$	0	0
$901$	42.5013	1.41592
$902$	0	0
$903$	6.51573i	0.216830i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	35.7881i	1.18832i	0.804346	+	0.594161i	$0.202516\pi$
$907$	35.7881i	1.18832i	−0.804346	+	0.594161i	$0.797484\pi$
$908$	0	0
$909$	−35.3080	−1.17109
$910$	0	0
$911$	33.8909	1.12286	0.561428	−	0.827525i	$-0.310252\pi$
$911$	33.8909	1.12286	0.561428	+	0.827525i	$0.310252\pi$
$912$	0	0
$913$	− 55.5967i	− 1.83998i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 20.7222i	− 0.684306i
$918$	0	0
$919$	48.9812	1.61574	0.807871	−	0.589360i	$-0.200620\pi$
$919$	48.9812	1.61574	0.807871	+	0.589360i	$0.200620\pi$
$920$	0	0
$921$	−29.8702	−0.984258
$922$	0	0
$923$	6.09246i	0.200536i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 26.2252i	− 0.861349i
$928$	0	0
$929$	−42.8049	−1.40438	−0.702192	−	0.711988i	$-0.747795\pi$
$929$	−42.8049	−1.40438	−0.702192	+	0.711988i	$0.747795\pi$
$930$	0	0
$931$	−10.1153	−0.331515
$932$	0	0
$933$	− 8.59179i	− 0.281283i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51.8283i	1.69316i	0.532263	+	0.846579i	$0.321342\pi$
$937$	51.8283i	1.69316i	−0.532263	+	0.846579i	$0.678658\pi$
$938$	0	0
$939$	−13.0737	−0.426643
$940$	0	0
$941$	49.4631	1.61245	0.806225	−	0.591609i	$-0.201507\pi$
$941$	49.4631	1.61245	0.806225	+	0.591609i	$0.201507\pi$
$942$	0	0
$943$	− 7.44563i	− 0.242463i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.0116i	− 1.04024i	−0.854094	−	0.520119i	$-0.825888\pi$
$947$	− 32.0116i	− 1.04024i	0.854094	−	0.520119i	$-0.174112\pi$
$948$	0	0
$949$	2.32065	0.0753315
$950$	0	0
$951$	55.9071	1.81291
$952$	0	0
$953$	56.1280i	1.81816i	0.416618	+	0.909082i	$0.363215\pi$
$953$	56.1280i	1.81816i	−0.416618	+	0.909082i	$0.636785\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 35.4620i	− 1.14632i
$958$	0	0
$959$	16.1599	0.521830
$960$	0	0
$961$	−20.4571	−0.659907
$962$	0	0
$963$	16.5026i	0.531788i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.93182i	0.255070i	0.991834	+	0.127535i	$0.0407065\pi$
$967$	7.93182i	0.255070i	−0.991834	+	0.127535i	$0.959293\pi$
$968$	0	0
$969$	−21.1564	−0.679643
$970$	0	0
$971$	22.6872	0.728068	0.364034	−	0.931386i	$-0.381399\pi$
$971$	22.6872	0.728068	0.364034	+	0.931386i	$0.381399\pi$
$972$	0	0
$973$	26.1588i	0.838614i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.8340i	0.794510i	0.917708	+	0.397255i	$0.130037\pi$
$977$	24.8340i	0.794510i	−0.917708	+	0.397255i	$0.869963\pi$
$978$	0	0
$979$	22.5560	0.720894
$980$	0	0
$981$	−30.4144	−0.971058
$982$	0	0
$983$	33.7670i	1.07700i	0.842625	+	0.538500i	$0.181009\pi$
$983$	33.7670i	1.07700i	−0.842625	+	0.538500i	$0.818991\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.5646i	0.527259i
$988$	0	0
$989$	−3.90887	−0.124295
$990$	0	0
$991$	22.1196	0.702652	0.351326	−	0.936253i	$-0.385731\pi$
$991$	22.1196	0.702652	0.351326	+	0.936253i	$0.385731\pi$
$992$	0	0
$993$	36.3913i	1.15484i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 40.7157i	− 1.28948i	−0.764402	−	0.644740i	$-0.776966\pi$
$997$	− 40.7157i	− 1.28948i	0.764402	−	0.644740i	$-0.223034\pi$
$998$	0	0
$999$	3.04162	0.0962326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6100.2.c.f.4149.1		6
5.2	odd	4		1220.2.a.c.1.1	✓	3
5.3	odd	4		6100.2.a.h.1.3		3
5.4	even	2	inner	6100.2.c.f.4149.6		6
20.7	even	4		4880.2.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1220.2.a.c.1.1	✓	3	5.2	odd	4
4880.2.a.q.1.3		3	20.7	even	4
6100.2.a.h.1.3		3	5.3	odd	4
6100.2.c.f.4149.1		6	1.1	even	1	trivial
6100.2.c.f.4149.6		6	5.4	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6100 = 2^{2} \cdot 5^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6100.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.24698i q^{3} +1.35690i q^{7} -2.04892 q^{9} +O(q^{10})\)	\(q-2.24698i q^{3} +1.35690i q^{7} -2.04892 q^{9} +4.40581 q^{11} -3.55496i q^{13} +4.80194i q^{17} -1.96077 q^{19} +3.04892 q^{21} +1.82908i q^{23} -2.13706i q^{27} +3.58211 q^{29} +3.24698 q^{31} -9.89977i q^{33} +1.42327i q^{37} -7.98792 q^{39} -4.07069 q^{41} +2.13706i q^{43} +5.43296i q^{47} +5.15883 q^{49} +10.7899 q^{51} -8.85086i q^{53} +4.40581i q^{57} +10.3448 q^{59} -1.00000 q^{61} -2.78017i q^{63} +7.67994i q^{67} +4.10992 q^{69} -1.71379 q^{71} +0.652793i q^{73} +5.97823i q^{77} +6.71917 q^{79} -10.9487 q^{81} -12.6189i q^{83} -8.04892i q^{87} +5.11960 q^{89} +4.82371 q^{91} -7.29590i q^{93} +17.7899i q^{97} -9.02715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^9	\( 6 q + 6 q^{9} + 14 q^{19} + 10 q^{29} + 10 q^{31} - 10 q^{39} + 14 q^{49} + 18 q^{51} + 16 q^{59} - 6 q^{61} + 26 q^{69} + 6 q^{71} + 18 q^{79} - 2 q^{81} - 12 q^{89} + 14 q^{91} - 42 q^{99}+O(q^{100}) \) 6 * q + 6 * q^9 + 14 * q^19 + 10 * q^29 + 10 * q^31 - 10 * q^39 + 14 * q^49 + 18 * q^51 + 16 * q^59 - 6 * q^61 + 26 * q^69 + 6 * q^71 + 18 * q^79 - 2 * q^81 - 12 * q^89 + 14 * q^91 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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