LMFDB - Embedded newform 1200.4.f.v.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.v.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +4.43560i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +4.43560i q^{7} -9.00000 q^{9} +3.43560 q^{11} -78.7424i q^{13} +53.1780i q^{17} +20.4356 q^{19} -13.3068 q^{21} +118.307i q^{23} -27.0000i q^{27} -168.049 q^{29} +61.0492 q^{31} +10.3068i q^{33} -246.614i q^{37} +236.227 q^{39} +422.663 q^{41} +362.436i q^{43} +170.515i q^{47} +323.325 q^{49} -159.534 q^{51} +546.049i q^{53} +61.3068i q^{57} -216.970 q^{59} +130.902 q^{61} -39.9204i q^{63} +614.890i q^{67} -354.920 q^{69} -324.822 q^{71} +88.8712i q^{73} +15.2389i q^{77} -1137.42 q^{79} +81.0000 q^{81} -758.909i q^{83} -504.148i q^{87} -195.681 q^{89} +349.269 q^{91} +183.148i q^{93} +521.000i q^{97} -30.9204 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9}+O(q^{10}) $ 4 * q - 36 * q^9	$ 4 q - 36 q^{9} - 56 q^{11} + 12 q^{19} + 156 q^{21} - 184 q^{29} - 244 q^{31} + 108 q^{39} + 784 q^{41} - 520 q^{49} + 408 q^{51} + 248 q^{59} + 1500 q^{61} - 792 q^{69} - 1648 q^{71} - 1760 q^{79} + 324 q^{81} + 1728 q^{89} + 4396 q^{91} + 504 q^{99}+O(q^{100}) $ 4 * q - 36 * q^9 - 56 * q^11 + 12 * q^19 + 156 * q^21 - 184 * q^29 - 244 * q^31 + 108 * q^39 + 784 * q^41 - 520 * q^49 + 408 * q^51 + 248 * q^59 + 1500 * q^61 - 792 * q^69 - 1648 * q^71 - 1760 * q^79 + 324 * q^81 + 1728 * q^89 + 4396 * q^91 + 504 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.43560i	0.239500i	0.992804	+	0.119750i	$0.0382092\pi$
$7$	4.43560i	0.239500i	−0.992804	+	0.119750i	$0.961791\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	3.43560	0.0941701	0.0470851	−	0.998891i	$-0.485007\pi$
$11$	3.43560	0.0941701	0.0470851	+	0.998891i	$0.485007\pi$
$12$	0	0
$13$	− 78.7424i	− 1.67994i	−0.542634	−	0.839970i	$-0.682573\pi$
$13$	− 78.7424i	− 1.67994i	0.542634	−	0.839970i	$-0.317427\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	53.1780i	0.758680i	0.925257	+	0.379340i	$0.123849\pi$
$17$	53.1780i	0.758680i	−0.925257	+	0.379340i	$0.876151\pi$
$18$	0	0
$19$	20.4356	0.246750	0.123375	−	0.992360i	$-0.460628\pi$
$19$	20.4356	0.246750	0.123375	+	0.992360i	$0.460628\pi$
$20$	0	0
$21$	−13.3068	−0.138275
$22$	0	0
$23$	118.307i	1.07255i	0.844043	+	0.536275i	$0.180169\pi$
$23$	118.307i	1.07255i	−0.844043	+	0.536275i	$0.819831\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	−168.049	−1.07607	−0.538034	−	0.842923i	$-0.680833\pi$
$29$	−168.049	−1.07607	−0.538034	+	0.842923i	$0.680833\pi$
$30$	0	0
$31$	61.0492	0.353702	0.176851	−	0.984238i	$-0.443409\pi$
$31$	61.0492	0.353702	0.176851	+	0.984238i	$0.443409\pi$
$32$	0	0
$33$	10.3068i	0.0543691i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 246.614i	− 1.09576i	−0.836558	−	0.547879i	$-0.815436\pi$
$37$	− 246.614i	− 1.09576i	0.836558	−	0.547879i	$-0.184564\pi$
$38$	0	0
$39$	236.227	0.969913
$40$	0	0
$41$	422.663	1.60997	0.804986	−	0.593294i	$-0.202173\pi$
$41$	422.663	1.60997	0.804986	+	0.593294i	$0.202173\pi$
$42$	0	0
$43$	362.436i	1.28537i	0.766131	+	0.642685i	$0.222180\pi$
$43$	362.436i	1.28537i	−0.766131	+	0.642685i	$0.777820\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	170.515i	0.529196i	0.964359	+	0.264598i	$0.0852392\pi$
$47$	170.515i	0.529196i	−0.964359	+	0.264598i	$0.914761\pi$
$48$	0	0
$49$	323.325	0.942640
$50$	0	0
$51$	−159.534	−0.438024
$52$	0	0
$53$	546.049i	1.41520i	0.706613	+	0.707600i	$0.250222\pi$
$53$	546.049i	1.41520i	−0.706613	+	0.707600i	$0.749778\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	61.3068i	0.142461i
$58$	0	0
$59$	−216.970	−0.478763	−0.239382	−	0.970926i	$-0.576945\pi$
$59$	−216.970	−0.478763	−0.239382	+	0.970926i	$0.576945\pi$
$60$	0	0
$61$	130.902	0.274758	0.137379	−	0.990519i	$-0.456132\pi$
$61$	130.902	0.274758	0.137379	+	0.990519i	$0.456132\pi$
$62$	0	0
$63$	− 39.9204i	− 0.0798332i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	614.890i	1.12121i	0.828085	+	0.560603i	$0.189430\pi$
$67$	614.890i	1.12121i	−0.828085	+	0.560603i	$0.810570\pi$
$68$	0	0
$69$	−354.920	−0.619238
$70$	0	0
$71$	−324.822	−0.542948	−0.271474	−	0.962446i	$-0.587511\pi$
$71$	−324.822	−0.542948	−0.271474	+	0.962446i	$0.587511\pi$
$72$	0	0
$73$	88.8712i	0.142487i	0.997459	+	0.0712437i	$0.0226968\pi$
$73$	88.8712i	0.142487i	−0.997459	+	0.0712437i	$0.977303\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.2389i	0.0225537i
$78$	0	0
$79$	−1137.42	−1.61988	−0.809938	−	0.586516i	$-0.800499\pi$
$79$	−1137.42	−1.61988	−0.809938	+	0.586516i	$0.800499\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 758.909i	− 1.00363i	−0.864976	−	0.501813i	$-0.832666\pi$
$83$	− 758.909i	− 1.00363i	0.864976	−	0.501813i	$-0.167334\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 504.148i	− 0.621268i
$88$	0	0
$89$	−195.681	−0.233058	−0.116529	−	0.993187i	$-0.537177\pi$
$89$	−195.681	−0.233058	−0.116529	+	0.993187i	$0.537177\pi$
$90$	0	0
$91$	349.269	0.402345
$92$	0	0
$93$	183.148i	0.204210i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	521.000i	0.545356i	0.962105	+	0.272678i	$0.0879094\pi$
$97$	521.000i	0.545356i	−0.962105	+	0.272678i	$0.912091\pi$
$98$	0	0
$99$	−30.9204	−0.0313900
$100$	0	0
$101$	660.920	0.651129	0.325565	−	0.945520i	$-0.394446\pi$
$101$	660.920	0.651129	0.325565	+	0.945520i	$0.394446\pi$
$102$	0	0
$103$	1530.75i	1.46436i	0.681110	+	0.732181i	$0.261497\pi$
$103$	1530.75i	1.46436i	−0.681110	+	0.732181i	$0.738503\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	264.625i	0.239087i	0.992829	+	0.119543i	$0.0381431\pi$
$107$	264.625i	0.239087i	−0.992829	+	0.119543i	$0.961857\pi$
$108$	0	0
$109$	−1117.61	−0.982091	−0.491046	−	0.871134i	$-0.663385\pi$
$109$	−1117.61	−0.982091	−0.491046	+	0.871134i	$0.663385\pi$
$110$	0	0
$111$	739.841	0.632636
$112$	0	0
$113$	934.061i	0.777602i	0.921322	+	0.388801i	$0.127111\pi$
$113$	934.061i	0.777602i	−0.921322	+	0.388801i	$0.872889\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	708.681i	0.559980i
$118$	0	0
$119$	−235.876	−0.181704
$120$	0	0
$121$	−1319.20	−0.991132
$122$	0	0
$123$	1267.99i	0.929517i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 630.356i	− 0.440433i	−0.975451	−	0.220217i	$-0.929324\pi$
$127$	− 630.356i	− 0.440433i	0.975451	−	0.220217i	$-0.0706765\pi$
$128$	0	0
$129$	−1087.31	−0.742109
$130$	0	0
$131$	2163.06	1.44265	0.721325	−	0.692597i	$-0.243534\pi$
$131$	2163.06	1.44265	0.721325	+	0.692597i	$0.243534\pi$
$132$	0	0
$133$	90.6440i	0.0590965i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1118.61i	0.697588i	0.937199	+	0.348794i	$0.113409\pi$
$137$	1118.61i	0.697588i	−0.937199	+	0.348794i	$0.886591\pi$
$138$	0	0
$139$	−166.478	−0.101586	−0.0507930	−	0.998709i	$-0.516175\pi$
$139$	−166.478	−0.101586	−0.0507930	+	0.998709i	$0.516175\pi$
$140$	0	0
$141$	−511.546	−0.305531
$142$	0	0
$143$	− 270.527i	− 0.158200i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	969.976i	0.544233i
$148$	0	0
$149$	653.143	0.359111	0.179555	−	0.983748i	$-0.442534\pi$
$149$	653.143	0.359111	0.179555	+	0.983748i	$0.442534\pi$
$150$	0	0
$151$	1929.38	1.03981	0.519903	−	0.854225i	$-0.325968\pi$
$151$	1929.38	1.03981	0.519903	+	0.854225i	$0.325968\pi$
$152$	0	0
$153$	− 478.602i	− 0.252893i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2169.75i	1.10296i	0.834188	+	0.551480i	$0.185937\pi$
$157$	2169.75i	1.10296i	−0.834188	+	0.551480i	$0.814063\pi$
$158$	0	0
$159$	−1638.15	−0.817066
$160$	0	0
$161$	−524.761	−0.256876
$162$	0	0
$163$	− 763.738i	− 0.366997i	−0.983020	−	0.183499i	$-0.941258\pi$
$163$	− 763.738i	− 0.366997i	0.983020	−	0.183499i	$-0.0587423\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2564.28i	1.18820i	0.804389	+	0.594102i	$0.202493\pi$
$167$	2564.28i	1.18820i	−0.804389	+	0.594102i	$0.797507\pi$
$168$	0	0
$169$	−4003.36	−1.82220
$170$	0	0
$171$	−183.920	−0.0822500
$172$	0	0
$173$	− 51.8290i	− 0.0227774i	−0.999935	−	0.0113887i	$-0.996375\pi$
$173$	− 51.8290i	− 0.0227774i	0.999935	−	0.0113887i	$-0.00362521\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 650.909i	− 0.276414i
$178$	0	0
$179$	−3956.63	−1.65214	−0.826068	−	0.563571i	$-0.809427\pi$
$179$	−3956.63	−1.65214	−0.826068	+	0.563571i	$0.809427\pi$
$180$	0	0
$181$	1804.04	0.740848	0.370424	−	0.928863i	$-0.379212\pi$
$181$	1804.04	0.740848	0.370424	+	0.928863i	$0.379212\pi$
$182$	0	0
$183$	392.705i	0.158632i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	182.698i	0.0714449i
$188$	0	0
$189$	119.761	0.0460917
$190$	0	0
$191$	−3666.75	−1.38909	−0.694547	−	0.719448i	$-0.744395\pi$
$191$	−3666.75	−1.38909	−0.694547	+	0.719448i	$0.744395\pi$
$192$	0	0
$193$	2716.98i	1.01333i	0.862144	+	0.506664i	$0.169121\pi$
$193$	2716.98i	1.01333i	−0.862144	+	0.506664i	$0.830879\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2034.30i	0.735723i	0.929881	+	0.367862i	$0.119910\pi$
$197$	2034.30i	0.735723i	−0.929881	+	0.367862i	$0.880090\pi$
$198$	0	0
$199$	−1551.27	−0.552596	−0.276298	−	0.961072i	$-0.589108\pi$
$199$	−1551.27	−0.552596	−0.276298	+	0.961072i	$0.589108\pi$
$200$	0	0
$201$	−1844.67	−0.647328
$202$	0	0
$203$	− 745.398i	− 0.257718i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1064.76i	− 0.357517i
$208$	0	0
$209$	70.2084	0.0232365
$210$	0	0
$211$	−3192.51	−1.04162	−0.520809	−	0.853673i	$-0.674370\pi$
$211$	−3192.51	−1.04162	−0.520809	+	0.853673i	$0.674370\pi$
$212$	0	0
$213$	− 974.466i	− 0.313471i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	270.789i	0.0847115i
$218$	0	0
$219$	−266.614	−0.0822652
$220$	0	0
$221$	4187.36	1.27454
$222$	0	0
$223$	− 1555.55i	− 0.467120i	−0.972342	−	0.233560i	$-0.924963\pi$
$223$	− 1555.55i	− 0.467120i	0.972342	−	0.233560i	$-0.0750374\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6206.86i	1.81482i	0.420248	+	0.907409i	$0.361943\pi$
$227$	6206.86i	1.81482i	−0.420248	+	0.907409i	$0.638057\pi$
$228$	0	0
$229$	−4679.51	−1.35035	−0.675176	−	0.737657i	$-0.735932\pi$
$229$	−4679.51	−1.35035	−0.675176	+	0.737657i	$0.735932\pi$
$230$	0	0
$231$	−45.7167	−0.0130214
$232$	0	0
$233$	3244.53i	0.912259i	0.889913	+	0.456129i	$0.150765\pi$
$233$	3244.53i	0.912259i	−0.889913	+	0.456129i	$0.849235\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3412.27i	− 0.935236i
$238$	0	0
$239$	−3658.62	−0.990193	−0.495097	−	0.868838i	$-0.664867\pi$
$239$	−3658.62	−0.990193	−0.495097	+	0.868838i	$0.664867\pi$
$240$	0	0
$241$	1931.01	0.516131	0.258065	−	0.966127i	$-0.416915\pi$
$241$	1931.01	0.516131	0.258065	+	0.966127i	$0.416915\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1609.15i	− 0.414525i
$248$	0	0
$249$	2276.73	0.579444
$250$	0	0
$251$	5843.34	1.46944	0.734718	−	0.678373i	$-0.237315\pi$
$251$	5843.34	1.46944	0.734718	+	0.678373i	$0.237315\pi$
$252$	0	0
$253$	406.454i	0.101002i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4506.11i	1.09371i	0.837227	+	0.546855i	$0.184175\pi$
$257$	4506.11i	1.09371i	−0.837227	+	0.546855i	$0.815825\pi$
$258$	0	0
$259$	1093.88	0.262434
$260$	0	0
$261$	1512.44	0.358689
$262$	0	0
$263$	5340.16i	1.25205i	0.779804	+	0.626024i	$0.215319\pi$
$263$	5340.16i	1.25205i	−0.779804	+	0.626024i	$0.784681\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 587.044i	− 0.134556i
$268$	0	0
$269$	−2809.79	−0.636863	−0.318431	−	0.947946i	$-0.603156\pi$
$269$	−2809.79	−0.636863	−0.318431	+	0.947946i	$0.603156\pi$
$270$	0	0
$271$	−3102.95	−0.695537	−0.347769	−	0.937580i	$-0.613060\pi$
$271$	−3102.95	−0.695537	−0.347769	+	0.937580i	$0.613060\pi$
$272$	0	0
$273$	1047.81i	0.232294i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4598.93i	− 0.997555i	−0.866730	−	0.498777i	$-0.833783\pi$
$277$	− 4598.93i	− 0.997555i	0.866730	−	0.498777i	$-0.166217\pi$
$278$	0	0
$279$	−549.443	−0.117901
$280$	0	0
$281$	2571.83	0.545987	0.272994	−	0.962016i	$-0.411986\pi$
$281$	2571.83	0.545987	0.272994	+	0.962016i	$0.411986\pi$
$282$	0	0
$283$	5575.31i	1.17109i	0.810641	+	0.585544i	$0.199119\pi$
$283$	5575.31i	1.17109i	−0.810641	+	0.585544i	$0.800881\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1874.76i	0.385588i
$288$	0	0
$289$	2085.10	0.424405
$290$	0	0
$291$	−1563.00	−0.314861
$292$	0	0
$293$	− 5794.27i	− 1.15531i	−0.816282	−	0.577654i	$-0.803968\pi$
$293$	− 5794.27i	− 1.15531i	0.816282	−	0.577654i	$-0.196032\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 92.7611i	− 0.0181230i
$298$	0	0
$299$	9315.76	1.80182
$300$	0	0
$301$	−1607.62	−0.307846
$302$	0	0
$303$	1982.76i	0.375930i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1404.47i	0.261099i	0.991442	+	0.130550i	$0.0416742\pi$
$307$	1404.47i	0.261099i	−0.991442	+	0.130550i	$0.958326\pi$
$308$	0	0
$309$	−4592.25	−0.845449
$310$	0	0
$311$	4096.75	0.746963	0.373481	−	0.927638i	$-0.378164\pi$
$311$	4096.75	0.746963	0.373481	+	0.927638i	$0.378164\pi$
$312$	0	0
$313$	− 974.611i	− 0.176001i	−0.996120	−	0.0880004i	$-0.971952\pi$
$313$	− 974.611i	− 0.176001i	0.996120	−	0.0880004i	$-0.0280477\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2071.69i	− 0.367058i	−0.983014	−	0.183529i	$-0.941248\pi$
$317$	− 2071.69i	− 0.367058i	0.983014	−	0.183529i	$-0.0587522\pi$
$318$	0	0
$319$	−577.349	−0.101333
$320$	0	0
$321$	−793.876	−0.138037
$322$	0	0
$323$	1086.72i	0.187204i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3352.84i	− 0.567011i
$328$	0	0
$329$	−756.337	−0.126742
$330$	0	0
$331$	6159.17	1.02278	0.511388	−	0.859350i	$-0.329132\pi$
$331$	6159.17	1.02278	0.511388	+	0.859350i	$0.329132\pi$
$332$	0	0
$333$	2219.52i	0.365252i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2791.26i	0.451186i	0.974222	+	0.225593i	$0.0724320\pi$
$337$	2791.26i	0.451186i	−0.974222	+	0.225593i	$0.927568\pi$
$338$	0	0
$339$	−2802.18	−0.448949
$340$	0	0
$341$	209.740	0.0333081
$342$	0	0
$343$	2955.55i	0.465262i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 940.848i	− 0.145554i	−0.997348	−	0.0727772i	$-0.976814\pi$
$347$	− 940.848i	− 0.145554i	0.997348	−	0.0727772i	$-0.0231862\pi$
$348$	0	0
$349$	3519.62	0.539831	0.269915	−	0.962884i	$-0.413004\pi$
$349$	3519.62	0.539831	0.269915	+	0.962884i	$0.413004\pi$
$350$	0	0
$351$	−2126.04	−0.323304
$352$	0	0
$353$	− 5021.60i	− 0.757147i	−0.925571	−	0.378573i	$-0.876415\pi$
$353$	− 5021.60i	− 0.757147i	0.925571	−	0.378573i	$-0.123585\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 707.628i	− 0.104907i
$358$	0	0
$359$	6811.99	1.00146	0.500728	−	0.865604i	$-0.333066\pi$
$359$	6811.99	1.00146	0.500728	+	0.865604i	$0.333066\pi$
$360$	0	0
$361$	−6441.39	−0.939115
$362$	0	0
$363$	− 3957.59i	− 0.572230i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3748.07i	− 0.533099i	−0.963821	−	0.266550i	$-0.914116\pi$
$367$	− 3748.07i	− 0.533099i	0.963821	−	0.266550i	$-0.0858836\pi$
$368$	0	0
$369$	−3803.96	−0.536657
$370$	0	0
$371$	−2422.05	−0.338940
$372$	0	0
$373$	898.302i	0.124698i	0.998054	+	0.0623489i	$0.0198592\pi$
$373$	898.302i	0.124698i	−0.998054	+	0.0623489i	$0.980141\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13232.6i	1.80773i
$378$	0	0
$379$	−9378.99	−1.27115	−0.635576	−	0.772038i	$-0.719237\pi$
$379$	−9378.99	−1.27115	−0.635576	+	0.772038i	$0.719237\pi$
$380$	0	0
$381$	1891.07	0.254284
$382$	0	0
$383$	− 9446.29i	− 1.26027i	−0.776486	−	0.630134i	$-0.783000\pi$
$383$	− 9446.29i	− 1.26027i	0.776486	−	0.630134i	$-0.217000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3261.92i	− 0.428457i
$388$	0	0
$389$	7643.23	0.996214	0.498107	−	0.867116i	$-0.334029\pi$
$389$	7643.23	0.996214	0.498107	+	0.867116i	$0.334029\pi$
$390$	0	0
$391$	−6291.32	−0.813723
$392$	0	0
$393$	6489.17i	0.832914i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12013.6i	− 1.51876i	−0.650650	−	0.759378i	$-0.725503\pi$
$397$	− 12013.6i	− 1.51876i	0.650650	−	0.759378i	$-0.274497\pi$
$398$	0	0
$399$	−271.932	−0.0341194
$400$	0	0
$401$	−8538.51	−1.06332	−0.531662	−	0.846957i	$-0.678432\pi$
$401$	−8538.51	−1.06332	−0.531662	+	0.846957i	$0.678432\pi$
$402$	0	0
$403$	− 4807.16i	− 0.594197i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 847.265i	− 0.103188i
$408$	0	0
$409$	12267.6	1.48312	0.741558	−	0.670889i	$-0.234087\pi$
$409$	12267.6	1.48312	0.741558	+	0.670889i	$0.234087\pi$
$410$	0	0
$411$	−3355.84	−0.402753
$412$	0	0
$413$	− 962.389i	− 0.114664i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 499.433i	− 0.0586508i
$418$	0	0
$419$	15493.0	1.80641	0.903204	−	0.429212i	$-0.141209\pi$
$419$	15493.0	1.80641	0.903204	+	0.429212i	$0.141209\pi$
$420$	0	0
$421$	7510.67	0.869472	0.434736	−	0.900558i	$-0.356842\pi$
$421$	7510.67	0.869472	0.434736	+	0.900558i	$0.356842\pi$
$422$	0	0
$423$	− 1534.64i	− 0.176399i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	580.627i	0.0658045i
$428$	0	0
$429$	811.581	0.0913368
$430$	0	0
$431$	15675.3	1.75186	0.875932	−	0.482434i	$-0.160247\pi$
$431$	15675.3	1.75186	0.875932	+	0.482434i	$0.160247\pi$
$432$	0	0
$433$	9604.78i	1.06600i	0.846117	+	0.532998i	$0.178935\pi$
$433$	9604.78i	1.06600i	−0.846117	+	0.532998i	$0.821065\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2417.67i	0.264652i
$438$	0	0
$439$	−6362.06	−0.691673	−0.345837	−	0.938295i	$-0.612405\pi$
$439$	−6362.06	−0.691673	−0.345837	+	0.938295i	$0.612405\pi$
$440$	0	0
$441$	−2909.93	−0.314213
$442$	0	0
$443$	931.658i	0.0999196i	0.998751	+	0.0499598i	$0.0159093\pi$
$443$	931.658i	0.0999196i	−0.998751	+	0.0499598i	$0.984091\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1959.43i	0.207333i
$448$	0	0
$449$	18684.1	1.96383	0.981914	−	0.189329i	$-0.0606314\pi$
$449$	18684.1	1.96383	0.981914	+	0.189329i	$0.0606314\pi$
$450$	0	0
$451$	1452.10	0.151611
$452$	0	0
$453$	5788.14i	0.600333i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11565.9i	− 1.18387i	−0.805985	−	0.591936i	$-0.798364\pi$
$457$	− 11565.9i	− 1.18387i	0.805985	−	0.591936i	$-0.201636\pi$
$458$	0	0
$459$	1435.81	0.146008
$460$	0	0
$461$	19401.0	1.96008	0.980039	−	0.198806i	$-0.0637062\pi$
$461$	19401.0	1.96008	0.980039	+	0.198806i	$0.0637062\pi$
$462$	0	0
$463$	1576.28i	0.158220i	0.996866	+	0.0791099i	$0.0252078\pi$
$463$	1576.28i	0.158220i	−0.996866	+	0.0791099i	$0.974792\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3256.55i	− 0.322687i	−0.986898	−	0.161344i	$-0.948417\pi$
$467$	− 3256.55i	− 0.322687i	0.986898	−	0.161344i	$-0.0515827\pi$
$468$	0	0
$469$	−2727.40	−0.268528
$470$	0	0
$471$	−6509.25	−0.636795
$472$	0	0
$473$	1245.18i	0.121043i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4914.44i	− 0.471733i
$478$	0	0
$479$	8291.59	0.790924	0.395462	−	0.918482i	$-0.370585\pi$
$479$	8291.59	0.790924	0.395462	+	0.918482i	$0.370585\pi$
$480$	0	0
$481$	−19418.9	−1.84081
$482$	0	0
$483$	− 1574.28i	− 0.148307i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4758.55i	− 0.442773i	−0.975186	−	0.221387i	$-0.928942\pi$
$487$	− 4758.55i	− 0.442773i	0.975186	−	0.221387i	$-0.0710583\pi$
$488$	0	0
$489$	2291.21	0.211886
$490$	0	0
$491$	−3906.46	−0.359055	−0.179528	−	0.983753i	$-0.557457\pi$
$491$	−3906.46	−0.359055	−0.179528	+	0.983753i	$0.557457\pi$
$492$	0	0
$493$	− 8936.52i	− 0.816390i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1440.78i	− 0.130036i
$498$	0	0
$499$	−3093.31	−0.277506	−0.138753	−	0.990327i	$-0.544309\pi$
$499$	−3093.31	−0.277506	−0.138753	+	0.990327i	$0.544309\pi$
$500$	0	0
$501$	−7692.85	−0.686010
$502$	0	0
$503$	− 18153.9i	− 1.60923i	−0.593796	−	0.804616i	$-0.702371\pi$
$503$	− 18153.9i	− 1.60923i	0.593796	−	0.804616i	$-0.297629\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12010.1i	− 1.05204i
$508$	0	0
$509$	−2281.32	−0.198660	−0.0993298	−	0.995055i	$-0.531670\pi$
$509$	−2281.32	−0.198660	−0.0993298	+	0.995055i	$0.531670\pi$
$510$	0	0
$511$	−394.197	−0.0341257
$512$	0	0
$513$	− 551.761i	− 0.0474870i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	585.821i	0.0498344i
$518$	0	0
$519$	155.487	0.0131505
$520$	0	0
$521$	−16691.9	−1.40362	−0.701809	−	0.712366i	$-0.747624\pi$
$521$	−16691.9	−1.40362	−0.701809	+	0.712366i	$0.747624\pi$
$522$	0	0
$523$	− 17090.4i	− 1.42889i	−0.699690	−	0.714446i	$-0.746679\pi$
$523$	− 17090.4i	− 1.42889i	0.699690	−	0.714446i	$-0.253321\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3246.47i	0.268346i
$528$	0	0
$529$	−1829.50	−0.150365
$530$	0	0
$531$	1952.73	0.159588
$532$	0	0
$533$	− 33281.5i	− 2.70465i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 11869.9i	− 0.953861i
$538$	0	0
$539$	1110.82	0.0887685
$540$	0	0
$541$	5271.40	0.418919	0.209459	−	0.977817i	$-0.432830\pi$
$541$	5271.40	0.418919	0.209459	+	0.977817i	$0.432830\pi$
$542$	0	0
$543$	5412.13i	0.427729i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15182.2i	− 1.18673i	−0.804933	−	0.593366i	$-0.797799\pi$
$547$	− 15182.2i	− 1.18673i	0.804933	−	0.593366i	$-0.202201\pi$
$548$	0	0
$549$	−1178.11	−0.0915860
$550$	0	0
$551$	−3434.18	−0.265519
$552$	0	0
$553$	− 5045.15i	− 0.387960i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12241.2i	0.931198i	0.884996	+	0.465599i	$0.154161\pi$
$557$	12241.2i	0.931198i	−0.884996	+	0.465599i	$0.845839\pi$
$558$	0	0
$559$	28539.0	2.15934
$560$	0	0
$561$	−548.094	−0.0412488
$562$	0	0
$563$	14196.4i	1.06271i	0.847149	+	0.531355i	$0.178317\pi$
$563$	14196.4i	1.06271i	−0.847149	+	0.531355i	$0.821683\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	359.283i	0.0266111i
$568$	0	0
$569$	−9150.05	−0.674148	−0.337074	−	0.941478i	$-0.609437\pi$
$569$	−9150.05	−0.674148	−0.337074	+	0.941478i	$0.609437\pi$
$570$	0	0
$571$	−23582.1	−1.72833	−0.864167	−	0.503206i	$-0.832154\pi$
$571$	−23582.1	−1.72833	−0.864167	+	0.503206i	$0.832154\pi$
$572$	0	0
$573$	− 11000.3i	− 0.801993i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3906.22i	0.281834i	0.990021	+	0.140917i	$0.0450050\pi$
$577$	3906.22i	0.281834i	−0.990021	+	0.140917i	$0.954995\pi$
$578$	0	0
$579$	−8150.93	−0.585045
$580$	0	0
$581$	3366.21	0.240368
$582$	0	0
$583$	1876.00i	0.133270i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25938.0i	1.82381i	0.410401	+	0.911905i	$0.365389\pi$
$587$	25938.0i	1.82381i	−0.410401	+	0.911905i	$0.634611\pi$
$588$	0	0
$589$	1247.58	0.0872759
$590$	0	0
$591$	−6102.89	−0.424770
$592$	0	0
$593$	− 1908.23i	− 0.132145i	−0.997815	−	0.0660723i	$-0.978953\pi$
$593$	− 1908.23i	− 0.132145i	0.997815	−	0.0660723i	$-0.0210468\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4653.81i	− 0.319041i
$598$	0	0
$599$	−3495.41	−0.238429	−0.119214	−	0.992869i	$-0.538038\pi$
$599$	−3495.41	−0.238429	−0.119214	+	0.992869i	$0.538038\pi$
$600$	0	0
$601$	−18267.2	−1.23983	−0.619913	−	0.784671i	$-0.712832\pi$
$601$	−18267.2	−1.23983	−0.619913	+	0.784671i	$0.712832\pi$
$602$	0	0
$603$	− 5534.01i	− 0.373735i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11538.2i	− 0.771534i	−0.922596	−	0.385767i	$-0.873937\pi$
$607$	− 11538.2i	− 0.771534i	0.922596	−	0.385767i	$-0.126063\pi$
$608$	0	0
$609$	2236.19	0.148793
$610$	0	0
$611$	13426.8	0.889017
$612$	0	0
$613$	− 21136.9i	− 1.39268i	−0.717713	−	0.696340i	$-0.754811\pi$
$613$	− 21136.9i	− 1.39268i	0.717713	−	0.696340i	$-0.245189\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15673.2i	1.02266i	0.859385	+	0.511329i	$0.170847\pi$
$617$	15673.2i	1.02266i	−0.859385	+	0.511329i	$0.829153\pi$
$618$	0	0
$619$	22923.7	1.48850	0.744249	−	0.667902i	$-0.232807\pi$
$619$	22923.7	1.48850	0.744249	+	0.667902i	$0.232807\pi$
$620$	0	0
$621$	3194.28	0.206413
$622$	0	0
$623$	− 867.964i	− 0.0558174i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	210.625i	0.0134156i
$628$	0	0
$629$	13114.4	0.831329
$630$	0	0
$631$	−9108.23	−0.574632	−0.287316	−	0.957836i	$-0.592763\pi$
$631$	−9108.23	−0.574632	−0.287316	+	0.957836i	$0.592763\pi$
$632$	0	0
$633$	− 9577.53i	− 0.601379i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 25459.4i	− 1.58358i
$638$	0	0
$639$	2923.40	0.180983
$640$	0	0
$641$	20103.5	1.23875	0.619375	−	0.785095i	$-0.287386\pi$
$641$	20103.5	1.23875	0.619375	+	0.785095i	$0.287386\pi$
$642$	0	0
$643$	5934.92i	0.363997i	0.983299	+	0.181999i	$0.0582567\pi$
$643$	5934.92i	0.363997i	−0.983299	+	0.181999i	$0.941743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14193.7i	0.862460i	0.902242	+	0.431230i	$0.141920\pi$
$647$	14193.7i	0.862460i	−0.902242	+	0.431230i	$0.858080\pi$
$648$	0	0
$649$	−745.420	−0.0450852
$650$	0	0
$651$	−812.368	−0.0489082
$652$	0	0
$653$	4795.80i	0.287403i	0.989621	+	0.143701i	$0.0459005\pi$
$653$	4795.80i	0.287403i	−0.989621	+	0.143701i	$0.954100\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 799.841i	− 0.0474958i
$658$	0	0
$659$	4399.57	0.260065	0.130032	−	0.991510i	$-0.458492\pi$
$659$	4399.57	0.260065	0.130032	+	0.991510i	$0.458492\pi$
$660$	0	0
$661$	24096.0	1.41789	0.708945	−	0.705263i	$-0.249171\pi$
$661$	24096.0	1.41789	0.708945	+	0.705263i	$0.249171\pi$
$662$	0	0
$663$	12562.1i	0.735853i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 19881.4i	− 1.15414i
$668$	0	0
$669$	4666.66	0.269692
$670$	0	0
$671$	449.725	0.0258740
$672$	0	0
$673$	27648.3i	1.58360i	0.610781	+	0.791800i	$0.290856\pi$
$673$	27648.3i	1.58360i	−0.610781	+	0.791800i	$0.709144\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27605.5i	− 1.56716i	−0.621292	−	0.783580i	$-0.713392\pi$
$677$	− 27605.5i	− 1.56716i	0.621292	−	0.783580i	$-0.286608\pi$
$678$	0	0
$679$	−2310.95	−0.130613
$680$	0	0
$681$	−18620.6	−1.04779
$682$	0	0
$683$	− 14949.4i	− 0.837513i	−0.908099	−	0.418756i	$-0.862466\pi$
$683$	− 14949.4i	− 0.837513i	0.908099	−	0.418756i	$-0.137534\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 14038.5i	− 0.779626i
$688$	0	0
$689$	42997.2	2.37745
$690$	0	0
$691$	−8884.30	−0.489110	−0.244555	−	0.969635i	$-0.578642\pi$
$691$	−8884.30	−0.489110	−0.244555	+	0.969635i	$0.578642\pi$
$692$	0	0
$693$	− 137.150i	− 0.00751790i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	22476.4i	1.22145i
$698$	0	0
$699$	−9733.59	−0.526693
$700$	0	0
$701$	10556.9	0.568798	0.284399	−	0.958706i	$-0.408206\pi$
$701$	10556.9	0.568798	0.284399	+	0.958706i	$0.408206\pi$
$702$	0	0
$703$	− 5039.70i	− 0.270378i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2931.58i	0.155945i
$708$	0	0
$709$	−25351.9	−1.34289	−0.671445	−	0.741055i	$-0.734326\pi$
$709$	−25351.9	−1.34289	−0.671445	+	0.741055i	$0.734326\pi$
$710$	0	0
$711$	10236.8	0.539959
$712$	0	0
$713$	7222.53i	0.379363i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10975.8i	− 0.571688i
$718$	0	0
$719$	9719.94	0.504162	0.252081	−	0.967706i	$-0.418885\pi$
$719$	9719.94	0.504162	0.252081	+	0.967706i	$0.418885\pi$
$720$	0	0
$721$	−6789.79	−0.350714
$722$	0	0
$723$	5793.04i	0.297988i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27509.3i	1.40339i	0.712479	+	0.701694i	$0.247572\pi$
$727$	27509.3i	1.40339i	−0.712479	+	0.701694i	$0.752428\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−19273.6	−0.975184
$732$	0	0
$733$	7240.49i	0.364848i	0.983220	+	0.182424i	$0.0583944\pi$
$733$	7240.49i	0.364848i	−0.983220	+	0.182424i	$0.941606\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2112.51i	0.105584i
$738$	0	0
$739$	−15875.3	−0.790234	−0.395117	−	0.918631i	$-0.629296\pi$
$739$	−15875.3	−0.790234	−0.395117	+	0.918631i	$0.629296\pi$
$740$	0	0
$741$	4827.44	0.239326
$742$	0	0
$743$	− 25714.3i	− 1.26967i	−0.772647	−	0.634836i	$-0.781068\pi$
$743$	− 25714.3i	− 1.26967i	0.772647	−	0.634836i	$-0.218932\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6830.18i	0.334542i
$748$	0	0
$749$	−1173.77	−0.0572612
$750$	0	0
$751$	9709.09	0.471757	0.235879	−	0.971783i	$-0.424203\pi$
$751$	9709.09	0.471757	0.235879	+	0.971783i	$0.424203\pi$
$752$	0	0
$753$	17530.0i	0.848379i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 9567.13i	− 0.459344i	−0.973268	−	0.229672i	$-0.926235\pi$
$757$	− 9567.13i	− 0.459344i	0.973268	−	0.229672i	$-0.0737653\pi$
$758$	0	0
$759$	−1219.36	−0.0583137
$760$	0	0
$761$	−12322.5	−0.586980	−0.293490	−	0.955962i	$-0.594817\pi$
$761$	−12322.5	−0.586980	−0.293490	+	0.955962i	$0.594817\pi$
$762$	0	0
$763$	− 4957.28i	− 0.235211i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17084.7i	0.804293i
$768$	0	0
$769$	−2575.56	−0.120776	−0.0603881	−	0.998175i	$-0.519234\pi$
$769$	−2575.56	−0.120776	−0.0603881	+	0.998175i	$0.519234\pi$
$770$	0	0
$771$	−13518.3	−0.631454
$772$	0	0
$773$	6606.23i	0.307386i	0.988119	+	0.153693i	$0.0491167\pi$
$773$	6606.23i	0.307386i	−0.988119	+	0.153693i	$0.950883\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3281.63i	0.151516i
$778$	0	0
$779$	8637.37	0.397260
$780$	0	0
$781$	−1115.96	−0.0511294
$782$	0	0
$783$	4537.33i	0.207089i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 16417.0i	− 0.743587i	−0.928315	−	0.371793i	$-0.878743\pi$
$787$	− 16417.0i	− 0.743587i	0.928315	−	0.371793i	$-0.121257\pi$
$788$	0	0
$789$	−16020.5	−0.722870
$790$	0	0
$791$	−4143.12	−0.186235
$792$	0	0
$793$	− 10307.5i	− 0.461577i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3944.19i	− 0.175295i	−0.996152	−	0.0876477i	$-0.972065\pi$
$797$	− 3944.19i	− 0.175295i	0.996152	−	0.0876477i	$-0.0279350\pi$
$798$	0	0
$799$	−9067.66	−0.401490
$800$	0	0
$801$	1761.13	0.0776861
$802$	0	0
$803$	305.325i	0.0134181i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 8429.37i	− 0.367693i
$808$	0	0
$809$	17960.7	0.780549	0.390275	−	0.920699i	$-0.372380\pi$
$809$	17960.7	0.780549	0.390275	+	0.920699i	$0.372380\pi$
$810$	0	0
$811$	13162.5	0.569912	0.284956	−	0.958541i	$-0.408021\pi$
$811$	13162.5	0.569912	0.284956	+	0.958541i	$0.408021\pi$
$812$	0	0
$813$	− 9308.84i	− 0.401569i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7406.59i	0.317165i
$818$	0	0
$819$	−3143.42	−0.134115
$820$	0	0
$821$	26502.4	1.12660	0.563302	−	0.826251i	$-0.309531\pi$
$821$	26502.4	1.12660	0.563302	+	0.826251i	$0.309531\pi$
$822$	0	0
$823$	6937.86i	0.293850i	0.989148	+	0.146925i	$0.0469376\pi$
$823$	6937.86i	0.293850i	−0.989148	+	0.146925i	$0.953062\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 41197.9i	− 1.73228i	−0.499805	−	0.866138i	$-0.666595\pi$
$827$	− 41197.9i	− 1.73228i	0.499805	−	0.866138i	$-0.333405\pi$
$828$	0	0
$829$	693.324	0.0290472	0.0145236	−	0.999895i	$-0.495377\pi$
$829$	693.324	0.0290472	0.0145236	+	0.999895i	$0.495377\pi$
$830$	0	0
$831$	13796.8	0.575938
$832$	0	0
$833$	17193.8i	0.715162i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1648.33i	− 0.0680699i
$838$	0	0
$839$	−6491.28	−0.267108	−0.133554	−	0.991042i	$-0.542639\pi$
$839$	−6491.28	−0.267108	−0.133554	+	0.991042i	$0.542639\pi$
$840$	0	0
$841$	3851.52	0.157921
$842$	0	0
$843$	7715.49i	0.315226i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5851.42i	− 0.237376i
$848$	0	0
$849$	−16725.9	−0.676128
$850$	0	0
$851$	29176.1	1.17526
$852$	0	0
$853$	1116.68i	0.0448233i	0.999749	+	0.0224117i	$0.00713445\pi$
$853$	1116.68i	0.0448233i	−0.999749	+	0.0224117i	$0.992866\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44383.9i	1.76911i	0.466438	+	0.884554i	$0.345537\pi$
$857$	44383.9i	1.76911i	−0.466438	+	0.884554i	$0.654463\pi$
$858$	0	0
$859$	−25579.3	−1.01601	−0.508006	−	0.861354i	$-0.669617\pi$
$859$	−25579.3	−1.01601	−0.508006	+	0.861354i	$0.669617\pi$
$860$	0	0
$861$	−5624.28	−0.222619
$862$	0	0
$863$	− 11194.8i	− 0.441570i	−0.975323	−	0.220785i	$-0.929138\pi$
$863$	− 11194.8i	− 0.441570i	0.975323	−	0.220785i	$-0.0708619\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6255.31i	0.245030i
$868$	0	0
$869$	−3907.73	−0.152544
$870$	0	0
$871$	48417.9	1.88356
$872$	0	0
$873$	− 4689.00i	− 0.181785i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5721.75i	− 0.220308i	−0.993915	−	0.110154i	$-0.964866\pi$
$877$	− 5721.75i	− 0.220308i	0.993915	−	0.110154i	$-0.0351344\pi$
$878$	0	0
$879$	17382.8	0.667017
$880$	0	0
$881$	−34682.8	−1.32633	−0.663163	−	0.748475i	$-0.730786\pi$
$881$	−34682.8	−1.32633	−0.663163	+	0.748475i	$0.730786\pi$
$882$	0	0
$883$	37990.4i	1.44788i	0.689862	+	0.723941i	$0.257671\pi$
$883$	37990.4i	1.44788i	−0.689862	+	0.723941i	$0.742329\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28299.0i	1.07124i	0.844460	+	0.535618i	$0.179921\pi$
$887$	28299.0i	1.07124i	−0.844460	+	0.535618i	$0.820079\pi$
$888$	0	0
$889$	2796.00	0.105484
$890$	0	0
$891$	278.283	0.0104633
$892$	0	0
$893$	3484.58i	0.130579i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	27947.3i	1.04028i
$898$	0	0
$899$	−10259.3	−0.380607
$900$	0	0
$901$	−29037.8	−1.07368
$902$	0	0
$903$	− 4822.85i	− 0.177735i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	17388.0i	0.636559i	0.947997	+	0.318280i	$0.103105\pi$
$907$	17388.0i	0.636559i	−0.947997	+	0.318280i	$0.896895\pi$
$908$	0	0
$909$	−5948.28	−0.217043
$910$	0	0
$911$	−23555.3	−0.856663	−0.428332	−	0.903622i	$-0.640899\pi$
$911$	−23555.3	−0.856663	−0.428332	+	0.903622i	$0.640899\pi$
$912$	0	0
$913$	− 2607.30i	− 0.0945117i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9594.44i	0.345514i
$918$	0	0
$919$	−5983.09	−0.214760	−0.107380	−	0.994218i	$-0.534246\pi$
$919$	−5983.09	−0.214760	−0.107380	+	0.994218i	$0.534246\pi$
$920$	0	0
$921$	−4213.42	−0.150746
$922$	0	0
$923$	25577.3i	0.912119i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 13776.7i	− 0.488120i
$928$	0	0
$929$	−20576.7	−0.726694	−0.363347	−	0.931654i	$-0.618366\pi$
$929$	−20576.7	−0.726694	−0.363347	+	0.931654i	$0.618366\pi$
$930$	0	0
$931$	6607.35	0.232596
$932$	0	0
$933$	12290.2i	0.431259i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 11228.6i	− 0.391485i	−0.980655	−	0.195743i	$-0.937288\pi$
$937$	− 11228.6i	− 0.391485i	0.980655	−	0.195743i	$-0.0627117\pi$
$938$	0	0
$939$	2923.83	0.101614
$940$	0	0
$941$	38567.6	1.33610	0.668049	−	0.744118i	$-0.267130\pi$
$941$	38567.6	1.33610	0.668049	+	0.744118i	$0.267130\pi$
$942$	0	0
$943$	50003.9i	1.72678i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4606.17i	− 0.158057i	−0.996872	−	0.0790287i	$-0.974818\pi$
$947$	− 4606.17i	− 0.158057i	0.996872	−	0.0790287i	$-0.0251819\pi$
$948$	0	0
$949$	6997.93	0.239370
$950$	0	0
$951$	6215.06	0.211921
$952$	0	0
$953$	25559.7i	0.868795i	0.900721	+	0.434397i	$0.143039\pi$
$953$	25559.7i	0.868795i	−0.900721	+	0.434397i	$0.856961\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1732.05i	− 0.0585048i
$958$	0	0
$959$	−4961.72	−0.167072
$960$	0	0
$961$	−26064.0	−0.874895
$962$	0	0
$963$	− 2381.63i	− 0.0796956i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 37895.8i	− 1.26023i	−0.776500	−	0.630117i	$-0.783007\pi$
$967$	− 37895.8i	− 1.26023i	0.776500	−	0.630117i	$-0.216993\pi$
$968$	0	0
$969$	−3260.17	−0.108082
$970$	0	0
$971$	46761.0	1.54545	0.772726	−	0.634740i	$-0.218893\pi$
$971$	46761.0	1.54545	0.772726	+	0.634740i	$0.218893\pi$
$972$	0	0
$973$	− 738.428i	− 0.0243298i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3070.29i	0.100540i	0.998736	+	0.0502698i	$0.0160081\pi$
$977$	3070.29i	0.100540i	−0.998736	+	0.0502698i	$0.983992\pi$
$978$	0	0
$979$	−672.282	−0.0219471
$980$	0	0
$981$	10058.5	0.327364
$982$	0	0
$983$	16319.0i	0.529498i	0.964317	+	0.264749i	$0.0852891\pi$
$983$	16319.0i	0.529498i	−0.964317	+	0.264749i	$0.914711\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2269.01i	− 0.0731747i
$988$	0	0
$989$	−42878.6	−1.37862
$990$	0	0
$991$	−5105.79	−0.163664	−0.0818319	−	0.996646i	$-0.526077\pi$
$991$	−5105.79	−0.163664	−0.0818319	+	0.996646i	$0.526077\pi$
$992$	0	0
$993$	18477.5i	0.590500i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7206.97i	0.228934i	0.993427	+	0.114467i	$0.0365160\pi$
$997$	7206.97i	0.228934i	−0.993427	+	0.114467i	$0.963484\pi$
$998$	0	0
$999$	−6658.57	−0.210879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.v.49.4		4
4.3	odd	2		75.4.b.c.49.1		4
5.2	odd	4		1200.4.a.bu.1.1		2
5.3	odd	4		1200.4.a.bl.1.2		2
5.4	even	2	inner	1200.4.f.v.49.1		4
12.11	even	2		225.4.b.h.199.4		4
20.3	even	4		75.4.a.d.1.1	✓	2
20.7	even	4		75.4.a.e.1.2	yes	2
20.19	odd	2		75.4.b.c.49.4		4
60.23	odd	4		225.4.a.n.1.2		2
60.47	odd	4		225.4.a.j.1.1		2
60.59	even	2		225.4.b.h.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.1	✓	2	20.3	even	4
75.4.a.e.1.2	yes	2	20.7	even	4
75.4.b.c.49.1		4	4.3	odd	2
75.4.b.c.49.4		4	20.19	odd	2
225.4.a.j.1.1		2	60.47	odd	4
225.4.a.n.1.2		2	60.23	odd	4
225.4.b.h.199.1		4	60.59	even	2
225.4.b.h.199.4		4	12.11	even	2
1200.4.a.bl.1.2		2	5.3	odd	4
1200.4.a.bu.1.1		2	5.2	odd	4
1200.4.f.v.49.1		4	5.4	even	2	inner
1200.4.f.v.49.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +4.43560i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +4.43560i q^{7} -9.00000 q^{9} +3.43560 q^{11} -78.7424i q^{13} +53.1780i q^{17} +20.4356 q^{19} -13.3068 q^{21} +118.307i q^{23} -27.0000i q^{27} -168.049 q^{29} +61.0492 q^{31} +10.3068i q^{33} -246.614i q^{37} +236.227 q^{39} +422.663 q^{41} +362.436i q^{43} +170.515i q^{47} +323.325 q^{49} -159.534 q^{51} +546.049i q^{53} +61.3068i q^{57} -216.970 q^{59} +130.902 q^{61} -39.9204i q^{63} +614.890i q^{67} -354.920 q^{69} -324.822 q^{71} +88.8712i q^{73} +15.2389i q^{77} -1137.42 q^{79} +81.0000 q^{81} -758.909i q^{83} -504.148i q^{87} -195.681 q^{89} +349.269 q^{91} +183.148i q^{93} +521.000i q^{97} -30.9204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9}+O(q^{10}) \) 4 * q - 36 * q^9	\( 4 q - 36 q^{9} - 56 q^{11} + 12 q^{19} + 156 q^{21} - 184 q^{29} - 244 q^{31} + 108 q^{39} + 784 q^{41} - 520 q^{49} + 408 q^{51} + 248 q^{59} + 1500 q^{61} - 792 q^{69} - 1648 q^{71} - 1760 q^{79} + 324 q^{81} + 1728 q^{89} + 4396 q^{91} + 504 q^{99}+O(q^{100}) \) 4 * q - 36 * q^9 - 56 * q^11 + 12 * q^19 + 156 * q^21 - 184 * q^29 - 244 * q^31 + 108 * q^39 + 784 * q^41 - 520 * q^49 + 408 * q^51 + 248 * q^59 + 1500 * q^61 - 792 * q^69 - 1648 * q^71 - 1760 * q^79 + 324 * q^81 + 1728 * q^89 + 4396 * q^91 + 504 * q^99

Introduction

L-functions

Modular forms

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