LMFDB - Embedded newform 1950.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$15.5708283941$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} +1.00000 q^{21} +5.00000 q^{22} -1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -7.00000 q^{29} -9.00000 q^{31} -1.00000 q^{32} -5.00000 q^{33} +5.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +1.00000 q^{39} -2.00000 q^{41} -1.00000 q^{42} -8.00000 q^{43} -5.00000 q^{44} +9.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -5.00000 q^{51} +1.00000 q^{52} -11.0000 q^{53} -1.00000 q^{54} -1.00000 q^{56} +7.00000 q^{58} +1.00000 q^{59} -7.00000 q^{61} +9.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} +15.0000 q^{67} -5.00000 q^{68} -8.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} -8.00000 q^{74} -5.00000 q^{77} -1.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +9.00000 q^{83} +1.00000 q^{84} +8.00000 q^{86} -7.00000 q^{87} +5.00000 q^{88} +16.0000 q^{89} +1.00000 q^{91} -9.00000 q^{93} -9.00000 q^{94} -1.00000 q^{96} -2.00000 q^{97} +6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	5.00000	1.06600
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−1.00000	−0.176777
$33$	−5.00000	−0.870388
$34$	5.00000	0.857493
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−1.00000	−0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−5.00000	−0.700140
$52$	1.00000	0.138675
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	7.00000	0.919145
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	9.00000	1.14300
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	5.00000	0.615457
$67$	15.0000	1.83254	0.916271	−	0.400559i	$-0.131184\pi$
$67$	15.0000	1.83254	0.916271	+	0.400559i	$0.131184\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	0	0
$77$	−5.00000	−0.569803
$78$	−1.00000	−0.113228
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	8.00000	0.862662
$87$	−7.00000	−0.750479
$88$	5.00000	0.533002
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−9.00000	−0.933257
$94$	−9.00000	−0.928279
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	6.00000	0.606092
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	5.00000	0.495074
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	11.0000	1.06841
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	1.00000	0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	1.00000	0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	−7.00000	−0.649934
$117$	1.00000	0.0924500
$118$	−1.00000	−0.0920575
$119$	−5.00000	−0.458349
$120$	0	0
$121$	14.0000	1.27273
$122$	7.00000	0.633750
$123$	−2.00000	−0.180334
$124$	−9.00000	−0.808224
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	−5.00000	−0.435194
$133$	0	0
$134$	−15.0000	−1.29580
$135$	0	0
$136$	5.00000	0.428746
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	8.00000	0.671345
$143$	−5.00000	−0.418121
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	−6.00000	−0.494872
$148$	8.00000	0.657596
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−21.0000	−1.70896	−0.854478	−	0.519488i	$-0.826123\pi$
$151$	−21.0000	−1.70896	−0.854478	+	0.519488i	$0.826123\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	5.00000	0.402911
$155$	0	0
$156$	1.00000	0.0800641
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	4.00000	0.318223
$159$	−11.0000	−0.872357
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−9.00000	−0.698535
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	−1.00000	−0.0771517
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	7.00000	0.530669
$175$	0	0
$176$	−5.00000	−0.376889
$177$	1.00000	0.0751646
$178$	−16.0000	−1.19925
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	−1.00000	−0.0741249
$183$	−7.00000	−0.517455
$184$	0	0
$185$	0	0
$186$	9.00000	0.659912
$187$	25.0000	1.82818
$188$	9.00000	0.656392
$189$	1.00000	0.0727393
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	1.00000	0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	5.00000	0.355335
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	15.0000	1.05802
$202$	7.00000	0.492518
$203$	−7.00000	−0.491304
$204$	−5.00000	−0.350070
$205$	0	0
$206$	6.00000	0.418040
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−11.0000	−0.755483
$213$	−8.00000	−0.548151
$214$	6.00000	0.410152
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−9.00000	−0.610960
$218$	−6.00000	−0.406371
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−5.00000	−0.336336
$222$	−8.00000	−0.536925
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	−29.0000	−1.92480	−0.962399	−	0.271640i	$-0.912434\pi$
$227$	−29.0000	−1.92480	−0.962399	+	0.271640i	$0.912434\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	7.00000	0.459573
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	1.00000	0.0650945
$237$	−4.00000	−0.259828
$238$	5.00000	0.324102
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−14.0000	−0.899954
$243$	1.00000	0.0641500
$244$	−7.00000	−0.448129
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	9.00000	0.571501
$249$	9.00000	0.570352
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	8.00000	0.498058
$259$	8.00000	0.497096
$260$	0	0
$261$	−7.00000	−0.433289
$262$	2.00000	0.123560
$263$	−26.0000	−1.60323	−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000	−1.60323	−0.801614	+	0.597841i	$0.796025\pi$
$264$	5.00000	0.307729
$265$	0	0
$266$	0	0
$267$	16.0000	0.979184
$268$	15.0000	0.916271
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	−5.00000	−0.303170
$273$	1.00000	0.0605228
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	2.00000	0.119952
$279$	−9.00000	−0.538816
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−9.00000	−0.535942
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	5.00000	0.295656
$287$	−2.00000	−0.118056
$288$	−1.00000	−0.0589256
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	−4.00000	−0.234082
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	−8.00000	−0.464991
$297$	−5.00000	−0.290129
$298$	22.0000	1.27443
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	21.0000	1.20841
$303$	−7.00000	−0.402139
$304$	0	0
$305$	0	0
$306$	5.00000	0.285831
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−5.00000	−0.284901
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−1.00000	−0.0566139
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	−4.00000	−0.225018
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	11.0000	0.616849
$319$	35.0000	1.95962
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	6.00000	0.331801
$328$	2.00000	0.110432
$329$	9.00000	0.496186
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	9.00000	0.493939
$333$	8.00000	0.438397
$334$	−4.00000	−0.218870
$335$	0	0
$336$	1.00000	0.0545545
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−1.00000	−0.0543928
$339$	−2.00000	−0.108625
$340$	0	0
$341$	45.0000	2.43689
$342$	0	0
$343$	−13.0000	−0.701934
$344$	8.00000	0.431331
$345$	0	0
$346$	−11.0000	−0.591364
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−7.00000	−0.375239
$349$	32.0000	1.71292	0.856460	−	0.516213i	$-0.172659\pi$
$349$	32.0000	1.71292	0.856460	+	0.516213i	$0.172659\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	5.00000	0.266501
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	−1.00000	−0.0531494
$355$	0	0
$356$	16.0000	0.847998
$357$	−5.00000	−0.264628
$358$	−2.00000	−0.105703
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	11.0000	0.578147
$363$	14.0000	0.734809
$364$	1.00000	0.0524142
$365$	0	0
$366$	7.00000	0.365896
$367$	36.0000	1.87918	0.939592	−	0.342296i	$-0.111204\pi$
$367$	36.0000	1.87918	0.939592	+	0.342296i	$0.111204\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−11.0000	−0.571092
$372$	−9.00000	−0.466628
$373$	17.0000	0.880227	0.440113	−	0.897942i	$-0.354938\pi$
$373$	17.0000	0.880227	0.440113	+	0.897942i	$0.354938\pi$
$374$	−25.0000	−1.29272
$375$	0	0
$376$	−9.00000	−0.464140
$377$	−7.00000	−0.360518
$378$	−1.00000	−0.0514344
$379$	−15.0000	−0.770498	−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000	−0.770498	−0.385249	+	0.922813i	$0.625884\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	−18.0000	−0.920960
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−8.00000	−0.406663
$388$	−2.00000	−0.101535
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	0	0
$392$	6.00000	0.303046
$393$	−2.00000	−0.100887
$394$	6.00000	0.302276
$395$	0	0
$396$	−5.00000	−0.251259
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	−18.0000	−0.902258
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	−15.0000	−0.748132
$403$	−9.00000	−0.448322
$404$	−7.00000	−0.348263
$405$	0	0
$406$	7.00000	0.347404
$407$	−40.0000	−1.98273
$408$	5.00000	0.247537
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−6.00000	−0.295599
$413$	1.00000	0.0492068
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	−2.00000	−0.0977064	−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000	−0.0977064	−0.0488532	+	0.998806i	$0.515557\pi$
$420$	0	0
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	−16.0000	−0.778868
$423$	9.00000	0.437595
$424$	11.0000	0.534207
$425$	0	0
$426$	8.00000	0.387601
$427$	−7.00000	−0.338754
$428$	−6.00000	−0.290021
$429$	−5.00000	−0.241402
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	1.00000	0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	9.00000	0.432014
$435$	0	0
$436$	6.00000	0.287348
$437$	0	0
$438$	4.00000	0.191127
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	5.00000	0.237826
$443$	32.0000	1.52037	0.760183	−	0.649709i	$-0.225109\pi$
$443$	32.0000	1.52037	0.760183	+	0.649709i	$0.225109\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	16.0000	0.757622
$447$	−22.0000	−1.04056
$448$	1.00000	0.0472456
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	−2.00000	−0.0940721
$453$	−21.0000	−0.986666
$454$	29.0000	1.36104
$455$	0	0
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	2.00000	0.0934539
$459$	−5.00000	−0.233380
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	5.00000	0.232621
$463$	11.0000	0.511213	0.255607	−	0.966781i	$-0.417725\pi$
$463$	11.0000	0.511213	0.255607	+	0.966781i	$0.417725\pi$
$464$	−7.00000	−0.324967
$465$	0	0
$466$	6.00000	0.277945
$467$	32.0000	1.48078	0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000	1.48078	0.740392	+	0.672176i	$0.234640\pi$
$468$	1.00000	0.0462250
$469$	15.0000	0.692636
$470$	0	0
$471$	5.00000	0.230388
$472$	−1.00000	−0.0460287
$473$	40.0000	1.83920
$474$	4.00000	0.183726
$475$	0	0
$476$	−5.00000	−0.229175
$477$	−11.0000	−0.503655
$478$	−1.00000	−0.0457389
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	22.0000	1.00207
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	1.00000	0.0453143	0.0226572	−	0.999743i	$-0.492787\pi$
$487$	1.00000	0.0453143	0.0226572	+	0.999743i	$0.492787\pi$
$488$	7.00000	0.316875
$489$	4.00000	0.180886
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	−2.00000	−0.0901670
$493$	35.0000	1.57632
$494$	0	0
$495$	0	0
$496$	−9.00000	−0.404112
$497$	−8.00000	−0.358849
$498$	−9.00000	−0.403300
$499$	11.0000	0.492428	0.246214	−	0.969216i	$-0.420813\pi$
$499$	11.0000	0.492428	0.246214	+	0.969216i	$0.420813\pi$
$500$	0	0
$501$	4.00000	0.178707
$502$	6.00000	0.267793
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	8.00000	0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	27.0000	1.19092
$515$	0	0
$516$	−8.00000	−0.352180
$517$	−45.0000	−1.97910
$518$	−8.00000	−0.351500
$519$	11.0000	0.482846
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	7.00000	0.306382
$523$	18.0000	0.787085	0.393543	−	0.919306i	$-0.371249\pi$
$523$	18.0000	0.787085	0.393543	+	0.919306i	$0.371249\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	26.0000	1.13365
$527$	45.0000	1.96023
$528$	−5.00000	−0.217597
$529$	−23.0000	−1.00000
$530$	0	0
$531$	1.00000	0.0433963
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	−16.0000	−0.692388
$535$	0	0
$536$	−15.0000	−0.647901
$537$	2.00000	0.0863064
$538$	−9.00000	−0.388018
$539$	30.0000	1.29219
$540$	0	0
$541$	28.0000	1.20381	0.601907	−	0.798566i	$-0.294408\pi$
$541$	28.0000	1.20381	0.601907	+	0.798566i	$0.294408\pi$
$542$	−7.00000	−0.300676
$543$	−11.0000	−0.472055
$544$	5.00000	0.214373
$545$	0	0
$546$	−1.00000	−0.0427960
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−12.0000	−0.512615
$549$	−7.00000	−0.298753
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.00000	−0.170097
$554$	26.0000	1.10463
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	9.00000	0.381000
$559$	−8.00000	−0.338364
$560$	0	0
$561$	25.0000	1.05550
$562$	−30.0000	−1.26547
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	−22.0000	−0.924729
$567$	1.00000	0.0419961
$568$	8.00000	0.335673
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	−5.00000	−0.209061
$573$	18.0000	0.751961
$574$	2.00000	0.0834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	−8.00000	−0.332756
$579$	10.0000	0.415586
$580$	0	0
$581$	9.00000	0.373383
$582$	2.00000	0.0829027
$583$	55.0000	2.27787
$584$	4.00000	0.165521
$585$	0	0
$586$	−14.0000	−0.578335
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	8.00000	0.328798
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	−22.0000	−0.901155
$597$	18.0000	0.736691
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	8.00000	0.326056
$603$	15.0000	0.610847
$604$	−21.0000	−0.854478
$605$	0	0
$606$	7.00000	0.284356
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	0	0
$609$	−7.00000	−0.283654
$610$	0	0
$611$	9.00000	0.364101
$612$	−5.00000	−0.202113
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	5.00000	0.201456
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	6.00000	0.241355
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	18.0000	0.721734
$623$	16.0000	0.641026
$624$	1.00000	0.0400320
$625$	0	0
$626$	21.0000	0.839329
$627$	0	0
$628$	5.00000	0.199522
$629$	−40.0000	−1.59490
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	4.00000	0.159111
$633$	16.0000	0.635943
$634$	0	0
$635$	0	0
$636$	−11.0000	−0.436178
$637$	−6.00000	−0.237729
$638$	−35.0000	−1.38566
$639$	−8.00000	−0.316475
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	6.00000	0.236801
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	−1.00000	−0.0392837
$649$	−5.00000	−0.196267
$650$	0	0
$651$	−9.00000	−0.352738
$652$	4.00000	0.156652
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	−4.00000	−0.156055
$658$	−9.00000	−0.350857
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	20.0000	0.777322
$663$	−5.00000	−0.194184
$664$	−9.00000	−0.349268
$665$	0	0
$666$	−8.00000	−0.309994
$667$	0	0
$668$	4.00000	0.154765
$669$	−16.0000	−0.618596
$670$	0	0
$671$	35.0000	1.35116
$672$	−1.00000	−0.0385758
$673$	−11.0000	−0.424019	−0.212009	−	0.977268i	$-0.568001\pi$
$673$	−11.0000	−0.424019	−0.212009	+	0.977268i	$0.568001\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	1.00000	0.0384615
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	2.00000	0.0768095
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−29.0000	−1.11128
$682$	−45.0000	−1.72314
$683$	−19.0000	−0.727015	−0.363507	−	0.931591i	$-0.618421\pi$
$683$	−19.0000	−0.727015	−0.363507	+	0.931591i	$0.618421\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	−2.00000	−0.0763048
$688$	−8.00000	−0.304997
$689$	−11.0000	−0.419067
$690$	0	0
$691$	7.00000	0.266293	0.133146	−	0.991096i	$-0.457492\pi$
$691$	7.00000	0.266293	0.133146	+	0.991096i	$0.457492\pi$
$692$	11.0000	0.418157
$693$	−5.00000	−0.189934
$694$	−6.00000	−0.227757
$695$	0	0
$696$	7.00000	0.265334
$697$	10.0000	0.378777
$698$	−32.0000	−1.21122
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−25.0000	−0.944237	−0.472118	−	0.881535i	$-0.656511\pi$
$701$	−25.0000	−0.944237	−0.472118	+	0.881535i	$0.656511\pi$
$702$	−1.00000	−0.0377426
$703$	0	0
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	−7.00000	−0.263262
$708$	1.00000	0.0375823
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−16.0000	−0.599625
$713$	0	0
$714$	5.00000	0.187120
$715$	0	0
$716$	2.00000	0.0747435
$717$	1.00000	0.0373457
$718$	−27.0000	−1.00763
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	19.0000	0.707107
$723$	−22.0000	−0.818189
$724$	−11.0000	−0.408812
$725$	0	0
$726$	−14.0000	−0.519589
$727$	38.0000	1.40934	0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000	1.40934	0.704671	+	0.709534i	$0.251095\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	−7.00000	−0.258727
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	−36.0000	−1.32878
$735$	0	0
$736$	0	0
$737$	−75.0000	−2.76266
$738$	2.00000	0.0736210
$739$	3.00000	0.110357	0.0551784	−	0.998477i	$-0.482427\pi$
$739$	3.00000	0.110357	0.0551784	+	0.998477i	$0.482427\pi$
$740$	0	0
$741$	0	0
$742$	11.0000	0.403823
$743$	−39.0000	−1.43077	−0.715386	−	0.698730i	$-0.753749\pi$
$743$	−39.0000	−1.43077	−0.715386	+	0.698730i	$0.753749\pi$
$744$	9.00000	0.329956
$745$	0	0
$746$	−17.0000	−0.622414
$747$	9.00000	0.329293
$748$	25.0000	0.914091
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	9.00000	0.328196
$753$	−6.00000	−0.218652
$754$	7.00000	0.254925
$755$	0	0
$756$	1.00000	0.0363696
$757$	−33.0000	−1.19941	−0.599703	−	0.800223i	$-0.704714\pi$
$757$	−33.0000	−1.19941	−0.599703	+	0.800223i	$0.704714\pi$
$758$	15.0000	0.544825
$759$	0	0
$760$	0	0
$761$	−8.00000	−0.290000	−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000	−0.290000	−0.145000	+	0.989432i	$0.546318\pi$
$762$	−8.00000	−0.289809
$763$	6.00000	0.217215
$764$	18.0000	0.651217
$765$	0	0
$766$	0	0
$767$	1.00000	0.0361079
$768$	1.00000	0.0360844
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	10.0000	0.359908
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	2.00000	0.0717958
$777$	8.00000	0.286998
$778$	−2.00000	−0.0717035
$779$	0	0
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	−7.00000	−0.250160
$784$	−6.00000	−0.214286
$785$	0	0
$786$	2.00000	0.0713376
$787$	5.00000	0.178231	0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000	0.178231	0.0891154	+	0.996021i	$0.471596\pi$
$788$	−6.00000	−0.213741
$789$	−26.0000	−0.925625
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	5.00000	0.177667
$793$	−7.00000	−0.248577
$794$	−34.0000	−1.20661
$795$	0	0
$796$	18.0000	0.637993
$797$	13.0000	0.460484	0.230242	−	0.973133i	$-0.426048\pi$
$797$	13.0000	0.460484	0.230242	+	0.973133i	$0.426048\pi$
$798$	0	0
$799$	−45.0000	−1.59199
$800$	0	0
$801$	16.0000	0.565332
$802$	0	0
$803$	20.0000	0.705785
$804$	15.0000	0.529009
$805$	0	0
$806$	9.00000	0.317011
$807$	9.00000	0.316815
$808$	7.00000	0.246259
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	−29.0000	−1.01833	−0.509164	−	0.860670i	$-0.670045\pi$
$811$	−29.0000	−1.01833	−0.509164	+	0.860670i	$0.670045\pi$
$812$	−7.00000	−0.245652
$813$	7.00000	0.245501
$814$	40.0000	1.40200
$815$	0	0
$816$	−5.00000	−0.175035
$817$	0	0
$818$	14.0000	0.489499
$819$	1.00000	0.0349428
$820$	0	0
$821$	−16.0000	−0.558404	−0.279202	−	0.960232i	$-0.590070\pi$
$821$	−16.0000	−0.558404	−0.279202	+	0.960232i	$0.590070\pi$
$822$	12.0000	0.418548
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−1.00000	−0.0347945
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	0	0
$829$	−37.0000	−1.28506	−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000	−1.28506	−0.642532	+	0.766259i	$0.722116\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	1.00000	0.0346688
$833$	30.0000	1.03944
$834$	2.00000	0.0692543
$835$	0	0
$836$	0	0
$837$	−9.00000	−0.311086
$838$	2.00000	0.0690889
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−28.0000	−0.964944
$843$	30.0000	1.03325
$844$	16.0000	0.550743
$845$	0	0
$846$	−9.00000	−0.309426
$847$	14.0000	0.481046
$848$	−11.0000	−0.377742
$849$	22.0000	0.755038
$850$	0	0
$851$	0	0
$852$	−8.00000	−0.274075
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	7.00000	0.239535
$855$	0	0
$856$	6.00000	0.205076
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	5.00000	0.170697
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	−12.0000	−0.408722
$863$	−27.0000	−0.919091	−0.459545	−	0.888154i	$-0.651988\pi$
$863$	−27.0000	−0.919091	−0.459545	+	0.888154i	$0.651988\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	34.0000	1.15537
$867$	8.00000	0.271694
$868$	−9.00000	−0.305480
$869$	20.0000	0.678454
$870$	0	0
$871$	15.0000	0.508256
$872$	−6.00000	−0.203186
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	−4.00000	−0.134993
$879$	14.0000	0.472208
$880$	0	0
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	6.00000	0.202031
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	−5.00000	−0.168168
$885$	0	0
$886$	−32.0000	−1.07506
$887$	−4.00000	−0.134307	−0.0671534	−	0.997743i	$-0.521392\pi$
$887$	−4.00000	−0.134307	−0.0671534	+	0.997743i	$0.521392\pi$
$888$	−8.00000	−0.268462
$889$	8.00000	0.268311
$890$	0	0
$891$	−5.00000	−0.167506
$892$	−16.0000	−0.535720
$893$	0	0
$894$	22.0000	0.735790
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−36.0000	−1.20134
$899$	63.0000	2.10117
$900$	0	0
$901$	55.0000	1.83232
$902$	−10.0000	−0.332964
$903$	−8.00000	−0.266223
$904$	2.00000	0.0665190
$905$	0	0
$906$	21.0000	0.697678
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	−29.0000	−0.962399
$909$	−7.00000	−0.232175
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	−45.0000	−1.48928
$914$	2.00000	0.0661541
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	−2.00000	−0.0660458
$918$	5.00000	0.165025
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	6.00000	0.197599
$923$	−8.00000	−0.263323
$924$	−5.00000	−0.164488
$925$	0	0
$926$	−11.0000	−0.361482
$927$	−6.00000	−0.197066
$928$	7.00000	0.229786
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−18.0000	−0.589294
$934$	−32.0000	−1.04707
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	−51.0000	−1.66610	−0.833049	−	0.553200i	$-0.813407\pi$
$937$	−51.0000	−1.66610	−0.833049	+	0.553200i	$0.813407\pi$
$938$	−15.0000	−0.489767
$939$	−21.0000	−0.685309
$940$	0	0
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	−5.00000	−0.162909
$943$	0	0
$944$	1.00000	0.0325472
$945$	0	0
$946$	−40.0000	−1.30051
$947$	41.0000	1.33232	0.666160	−	0.745808i	$-0.267937\pi$
$947$	41.0000	1.33232	0.666160	+	0.745808i	$0.267937\pi$
$948$	−4.00000	−0.129914
$949$	−4.00000	−0.129845
$950$	0	0
$951$	0	0
$952$	5.00000	0.162051
$953$	−51.0000	−1.65205	−0.826026	−	0.563632i	$-0.809404\pi$
$953$	−51.0000	−1.65205	−0.826026	+	0.563632i	$0.809404\pi$
$954$	11.0000	0.356138
$955$	0	0
$956$	1.00000	0.0323423
$957$	35.0000	1.13139
$958$	−29.0000	−0.936947
$959$	−12.0000	−0.387500
$960$	0	0
$961$	50.0000	1.61290
$962$	−8.00000	−0.257930
$963$	−6.00000	−0.193347
$964$	−22.0000	−0.708572
$965$	0	0
$966$	0	0
$967$	17.0000	0.546683	0.273342	−	0.961917i	$-0.411871\pi$
$967$	17.0000	0.546683	0.273342	+	0.961917i	$0.411871\pi$
$968$	−14.0000	−0.449977
$969$	0	0
$970$	0	0
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	1.00000	0.0320750
$973$	−2.00000	−0.0641171
$974$	−1.00000	−0.0320421
$975$	0	0
$976$	−7.00000	−0.224065
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	−4.00000	−0.127906
$979$	−80.0000	−2.55681
$980$	0	0
$981$	6.00000	0.191565
$982$	−30.0000	−0.957338
$983$	31.0000	0.988746	0.494373	−	0.869250i	$-0.335398\pi$
$983$	31.0000	0.988746	0.494373	+	0.869250i	$0.335398\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−35.0000	−1.11463
$987$	9.00000	0.286473
$988$	0	0
$989$	0	0
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	9.00000	0.285750
$993$	−20.0000	−0.634681
$994$	8.00000	0.253745
$995$	0	0
$996$	9.00000	0.285176
$997$	11.0000	0.348373	0.174187	−	0.984713i	$-0.444270\pi$
$997$	11.0000	0.348373	0.174187	+	0.984713i	$0.444270\pi$
$998$	−11.0000	−0.348199
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.l.1.1	✓	1
3.2	odd	2		5850.2.a.bu.1.1		1
5.2	odd	4		1950.2.e.a.1249.1		2
5.3	odd	4		1950.2.e.a.1249.2		2
5.4	even	2		1950.2.a.p.1.1	yes	1
15.2	even	4		5850.2.e.be.5149.2		2
15.8	even	4		5850.2.e.be.5149.1		2
15.14	odd	2		5850.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.l.1.1	✓	1	1.1	even	1	trivial
1950.2.a.p.1.1	yes	1	5.4	even	2
1950.2.e.a.1249.1		2	5.2	odd	4
1950.2.e.a.1249.2		2	5.3	odd	4
5850.2.a.k.1.1		1	15.14	odd	2
5850.2.a.bu.1.1		1	3.2	odd	2
5850.2.e.be.5149.1		2	15.8	even	4
5850.2.e.be.5149.2		2	15.2	even	4

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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more