LMFDB - Embedded newform 950.2.b.e.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("950.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	950.b (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:	$7.58578819202$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	950.799
Dual form			950.2.b.e.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} -1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} -1.00000 q^{19} -1.00000 q^{21} +3.00000i q^{23} -1.00000 q^{24} -1.00000 q^{26} +5.00000i q^{27} -1.00000i q^{28} +3.00000 q^{29} +2.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} -2.00000 q^{36} +10.0000i q^{37} +1.00000i q^{38} +1.00000 q^{39} +6.00000 q^{41} +1.00000i q^{42} +2.00000i q^{43} +3.00000 q^{46} +1.00000i q^{48} +6.00000 q^{49} -3.00000 q^{51} +1.00000i q^{52} +3.00000i q^{53} +5.00000 q^{54} -1.00000 q^{56} -1.00000i q^{57} -3.00000i q^{58} -3.00000 q^{59} +8.00000 q^{61} -2.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +7.00000i q^{67} -3.00000i q^{68} -3.00000 q^{69} +12.0000 q^{71} +2.00000i q^{72} -13.0000i q^{73} +10.0000 q^{74} +1.00000 q^{76} -1.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +6.00000i q^{83} +1.00000 q^{84} +2.00000 q^{86} +3.00000i q^{87} -6.00000 q^{89} +1.00000 q^{91} -3.00000i q^{92} +2.00000i q^{93} +1.00000 q^{96} +10.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} + 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{21} - 2 q^{24} - 2 q^{26} + 6 q^{29} + 4 q^{31} + 6 q^{34} - 4 q^{36} + 2 q^{39} + 12 q^{41} + 6 q^{46} + 12 q^{49} - 6 q^{51} + 10 q^{54} - 2 q^{56} - 6 q^{59} + 16 q^{61} - 2 q^{64} - 6 q^{69} + 24 q^{71} + 20 q^{74} + 2 q^{76} - 28 q^{79} + 2 q^{81} + 2 q^{84} + 4 q^{86} - 12 q^{89} + 2 q^{91} + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 + 2 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^21 - 2 * q^24 - 2 * q^26 + 6 * q^29 + 4 * q^31 + 6 * q^34 - 4 * q^36 + 2 * q^39 + 12 * q^41 + 6 * q^46 + 12 * q^49 - 6 * q^51 + 10 * q^54 - 2 * q^56 - 6 * q^59 + 16 * q^61 - 2 * q^64 - 6 * q^69 + 24 * q^71 + 20 * q^74 + 2 * q^76 - 28 * q^79 + 2 * q^81 + 2 * q^84 + 4 * q^86 - 12 * q^89 + 2 * q^91 + 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	1.00000i	0.353553i
$9$	2.00000	0.666667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	− 2.00000i	− 0.471405i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	5.00000i	0.962250i
$28$	− 1.00000i	− 0.188982i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	1.00000i	0.162221i
$39$	1.00000	0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000i	0.154303i
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	3.00000	0.442326
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	6.00000	0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	1.00000i	0.138675i
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−1.00000	−0.133631
$57$	− 1.00000i	− 0.132453i
$58$	− 3.00000i	− 0.393919i
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	− 2.00000i	− 0.254000i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	− 3.00000i	− 0.363803i
$69$	−3.00000	−0.361158
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	2.00000i	0.235702i
$73$	− 13.0000i	− 1.52153i	−0.649025	−	0.760767i	$-0.724823\pi$
$73$	− 13.0000i	− 1.52153i	0.649025	−	0.760767i	$-0.275177\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	− 1.00000i	− 0.113228i
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	2.00000	0.215666
$87$	3.00000i	0.321634i
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	− 3.00000i	− 0.312772i
$93$	2.00000i	0.207390i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 6.00000i	− 0.606092i
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	3.00000i	0.297044i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	3.00000	0.291386
$107$	− 15.0000i	− 1.45010i	−0.688694	−	0.725052i	$-0.741816\pi$
$107$	− 15.0000i	− 1.45010i	0.688694	−	0.725052i	$-0.258184\pi$
$108$	− 5.00000i	− 0.481125i
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	1.00000i	0.0944911i
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−3.00000	−0.278543
$117$	− 2.00000i	− 0.184900i
$118$	3.00000i	0.276172i
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 8.00000i	− 0.724286i
$123$	6.00000i	0.541002i
$124$	−2.00000	−0.179605
$125$	0	0
$126$	2.00000	0.178174
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	1.00000i	0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 1.00000i	− 0.0867110i
$134$	7.00000	0.604708
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 9.00000i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$137$	− 9.00000i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$138$	3.00000i	0.255377i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	− 12.0000i	− 1.00702i
$143$	0	0
$144$	2.00000	0.166667
$145$	0	0
$146$	−13.0000	−1.07589
$147$	6.00000i	0.494872i
$148$	− 10.0000i	− 0.821995i
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	14.0000i	1.11378i
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−3.00000	−0.236433
$162$	− 1.00000i	− 0.0785674i
$163$	− 22.0000i	− 1.72317i	−0.507611	−	0.861586i	$-0.669471\pi$
$163$	− 22.0000i	− 1.72317i	0.507611	−	0.861586i	$-0.330529\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	6.00000	0.465690
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	− 1.00000i	− 0.0771517i
$169$	12.0000	0.923077
$170$	0	0
$171$	−2.00000	−0.152944
$172$	− 2.00000i	− 0.152499i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	0	0
$177$	− 3.00000i	− 0.225494i
$178$	6.00000i	0.449719i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 1.00000i	− 0.0741249i
$183$	8.00000i	0.591377i
$184$	−3.00000	−0.221163
$185$	0	0
$186$	2.00000	0.146647
$187$	0	0
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−27.0000	−1.95365	−0.976826	−	0.214036i	$-0.931339\pi$
$191$	−27.0000	−1.95365	−0.976826	+	0.214036i	$0.931339\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	19.0000	1.34687	0.673437	−	0.739244i	$-0.264817\pi$
$199$	19.0000	1.34687	0.673437	+	0.739244i	$0.264817\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	− 12.0000i	− 0.844317i
$203$	3.00000i	0.210559i
$204$	3.00000	0.210042
$205$	0	0
$206$	8.00000	0.557386
$207$	6.00000i	0.417029i
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	− 3.00000i	− 0.206041i
$213$	12.0000i	0.822226i
$214$	−15.0000	−1.02538
$215$	0	0
$216$	−5.00000	−0.340207
$217$	2.00000i	0.135769i
$218$	11.0000i	0.745014i
$219$	13.0000	0.878459
$220$	0	0
$221$	3.00000	0.201802
$222$	10.0000i	0.671156i
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−12.0000	−0.798228
$227$	15.0000i	0.995585i	0.867296	+	0.497792i	$0.165856\pi$
$227$	15.0000i	0.995585i	−0.867296	+	0.497792i	$0.834144\pi$
$228$	1.00000i	0.0662266i
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	3.00000	0.195283
$237$	− 14.0000i	− 0.909398i
$238$	3.00000i	0.194461i
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	11.0000i	0.707107i
$243$	16.0000i	1.02640i
$244$	−8.00000	−0.512148
$245$	0	0
$246$	6.00000	0.382546
$247$	1.00000i	0.0636285i
$248$	2.00000i	0.127000i
$249$	−6.00000	−0.380235
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	2.00000i	0.124515i
$259$	−10.0000	−0.621370
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−1.00000	−0.0613139
$267$	− 6.00000i	− 0.367194i
$268$	− 7.00000i	− 0.427593i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	3.00000i	0.181902i
$273$	1.00000i	0.0605228i
$274$	−9.00000	−0.543710
$275$	0	0
$276$	3.00000	0.180579
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	8.00000i	0.479808i
$279$	4.00000	0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	− 2.00000i	− 0.117851i
$289$	8.00000	0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	13.0000i	0.760767i
$293$	21.0000i	1.22683i	0.789760	+	0.613417i	$0.210205\pi$
$293$	21.0000i	1.22683i	−0.789760	+	0.613417i	$0.789795\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	−10.0000	−0.581238
$297$	0	0
$298$	− 12.0000i	− 0.695141i
$299$	3.00000	0.173494
$300$	0	0
$301$	−2.00000	−0.115278
$302$	10.0000i	0.575435i
$303$	12.0000i	0.689382i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	6.00000	0.342997
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	1.00000i	0.0566139i
$313$	− 1.00000i	− 0.0565233i	−0.999601	−	0.0282617i	$-0.991003\pi$
$313$	− 1.00000i	− 0.0565233i	0.999601	−	0.0282617i	$-0.00899717\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	14.0000	0.787562
$317$	− 9.00000i	− 0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$	− 9.00000i	− 0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$	3.00000i	0.168232i
$319$	0	0
$320$	0	0
$321$	15.0000	0.837218
$322$	3.00000i	0.167183i
$323$	− 3.00000i	− 0.166924i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−22.0000	−1.21847
$327$	− 11.0000i	− 0.608301i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	− 6.00000i	− 0.329293i
$333$	20.0000i	1.09599i
$334$	−18.0000	−0.984916
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	− 12.0000i	− 0.652714i
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	2.00000i	0.108148i
$343$	13.0000i	0.701934i
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−6.00000	−0.322562
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	− 3.00000i	− 0.160817i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	− 33.0000i	− 1.75641i	−0.478282	−	0.878206i	$-0.658740\pi$
$353$	− 33.0000i	− 1.75641i	0.478282	−	0.878206i	$-0.341260\pi$
$354$	−3.00000	−0.159448
$355$	0	0
$356$	6.00000	0.317999
$357$	− 3.00000i	− 0.158777i
$358$	0	0
$359$	33.0000	1.74167	0.870837	−	0.491572i	$-0.163578\pi$
$359$	33.0000	1.74167	0.870837	+	0.491572i	$0.163578\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 2.00000i	− 0.105118i
$363$	− 11.0000i	− 0.577350i
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	8.00000	0.418167
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	3.00000i	0.156386i
$369$	12.0000	0.624695
$370$	0	0
$371$	−3.00000	−0.155752
$372$	− 2.00000i	− 0.103695i
$373$	− 31.0000i	− 1.60512i	−0.596572	−	0.802560i	$-0.703471\pi$
$373$	− 31.0000i	− 1.60512i	0.596572	−	0.802560i	$-0.296529\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.00000i	− 0.154508i
$378$	5.00000i	0.257172i
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	27.0000i	1.38144i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	4.00000i	0.203331i
$388$	− 10.0000i	− 0.507673i
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	6.00000i	0.303046i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	− 8.00000i	− 0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$	− 8.00000i	− 0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$	− 19.0000i	− 0.952384i
$399$	1.00000	0.0500626
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	7.00000i	0.349128i
$403$	− 2.00000i	− 0.0996271i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	3.00000	0.148888
$407$	0	0
$408$	− 3.00000i	− 0.148522i
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	− 8.00000i	− 0.394132i
$413$	− 3.00000i	− 0.147620i
$414$	6.00000	0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	− 8.00000i	− 0.391762i
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	25.0000i	1.21698i
$423$	0	0
$424$	−3.00000	−0.145693
$425$	0	0
$426$	12.0000	0.581402
$427$	8.00000i	0.387147i
$428$	15.0000i	0.725052i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	5.00000i	0.240563i
$433$	− 4.00000i	− 0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$	− 4.00000i	− 0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	11.0000	0.526804
$437$	− 3.00000i	− 0.143509i
$438$	− 13.0000i	− 0.621164i
$439$	−38.0000	−1.81364	−0.906821	−	0.421517i	$-0.861498\pi$
$439$	−38.0000	−1.81364	−0.906821	+	0.421517i	$0.861498\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	− 3.00000i	− 0.142695i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	14.0000	0.662919
$447$	12.0000i	0.567581i
$448$	− 1.00000i	− 0.0472456i
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	12.0000i	0.564433i
$453$	− 10.0000i	− 0.469841i
$454$	15.0000	0.703985
$455$	0	0
$456$	1.00000	0.0468293
$457$	− 35.0000i	− 1.63723i	−0.574342	−	0.818615i	$-0.694742\pi$
$457$	− 35.0000i	− 1.63723i	0.574342	−	0.818615i	$-0.305258\pi$
$458$	26.0000i	1.21490i
$459$	−15.0000	−0.700140
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	2.00000i	0.0924500i
$469$	−7.00000	−0.323230
$470$	0	0
$471$	−10.0000	−0.460776
$472$	− 3.00000i	− 0.138086i
$473$	0	0
$474$	−14.0000	−0.643041
$475$	0	0
$476$	3.00000	0.137505
$477$	6.00000i	0.274721i
$478$	− 15.0000i	− 0.686084i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	4.00000i	0.182195i
$483$	− 3.00000i	− 0.136505i
$484$	11.0000	0.500000
$485$	0	0
$486$	16.0000	0.725775
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	8.00000i	0.362143i
$489$	22.0000	0.994874
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	− 6.00000i	− 0.270501i
$493$	9.00000i	0.405340i
$494$	1.00000	0.0449921
$495$	0	0
$496$	2.00000	0.0898027
$497$	12.0000i	0.538274i
$498$	6.00000i	0.268866i
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	0	0
$503$	39.0000i	1.73892i	0.494000	+	0.869462i	$0.335534\pi$
$503$	39.0000i	1.73892i	−0.494000	+	0.869462i	$0.664466\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	12.0000i	0.532939i
$508$	2.00000i	0.0887357i
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	13.0000	0.575086
$512$	− 1.00000i	− 0.0441942i
$513$	− 5.00000i	− 0.220755i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	10.0000i	0.439375i
$519$	6.00000	0.263371
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	− 6.00000i	− 0.262613i
$523$	11.0000i	0.480996i	0.970650	+	0.240498i	$0.0773108\pi$
$523$	11.0000i	0.480996i	−0.970650	+	0.240498i	$0.922689\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.00000i	0.261364i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	−6.00000	−0.260378
$532$	1.00000i	0.0433555i
$533$	− 6.00000i	− 0.259889i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−7.00000	−0.302354
$537$	0	0
$538$	18.0000i	0.776035i
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	7.00000i	0.300676i
$543$	2.00000i	0.0858282i
$544$	3.00000	0.128624
$545$	0	0
$546$	1.00000	0.0427960
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	9.00000i	0.384461i
$549$	16.0000	0.682863
$550$	0	0
$551$	−3.00000	−0.127804
$552$	− 3.00000i	− 0.127688i
$553$	− 14.0000i	− 0.595341i
$554$	−8.00000	−0.339887
$555$	0	0
$556$	8.00000	0.339276
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	− 4.00000i	− 0.169334i
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	− 18.0000i	− 0.759284i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	14.0000	0.588464
$567$	1.00000i	0.0419961i
$568$	12.0000i	0.503509i
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	− 27.0000i	− 1.12794i
$574$	6.00000	0.250435
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	7.00000i	0.291414i	0.989328	+	0.145707i	$0.0465456\pi$
$577$	7.00000i	0.291414i	−0.989328	+	0.145707i	$0.953454\pi$
$578$	− 8.00000i	− 0.332756i
$579$	22.0000	0.914289
$580$	0	0
$581$	−6.00000	−0.248922
$582$	10.0000i	0.414513i
$583$	0	0
$584$	13.0000	0.537944
$585$	0	0
$586$	21.0000	0.867502
$587$	− 30.0000i	− 1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$	− 30.0000i	− 1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	− 6.00000i	− 0.247436i
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	6.00000	0.246807
$592$	10.0000i	0.410997i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	19.0000i	0.777618i
$598$	− 3.00000i	− 0.122679i
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	2.00000i	0.0815139i
$603$	14.0000i	0.570124i
$604$	10.0000	0.406894
$605$	0	0
$606$	12.0000	0.487467
$607$	10.0000i	0.405887i	0.979190	+	0.202944i	$0.0650509\pi$
$607$	10.0000i	0.405887i	−0.979190	+	0.202944i	$0.934949\pi$
$608$	1.00000i	0.0405554i
$609$	−3.00000	−0.121566
$610$	0	0
$611$	0	0
$612$	− 6.00000i	− 0.242536i
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	8.00000i	0.321807i
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	3.00000i	0.120289i
$623$	− 6.00000i	− 0.240385i
$624$	1.00000	0.0400320
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	−30.0000	−1.19618
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	− 14.0000i	− 0.556890i
$633$	− 25.0000i	− 0.993661i
$634$	−9.00000	−0.357436
$635$	0	0
$636$	3.00000	0.118958
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	− 15.0000i	− 0.592003i
$643$	50.0000i	1.97181i	0.167313	+	0.985904i	$0.446491\pi$
$643$	50.0000i	1.97181i	−0.167313	+	0.985904i	$0.553509\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	−3.00000	−0.118033
$647$	33.0000i	1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$	33.0000i	1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	22.0000i	0.861586i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−11.0000	−0.430134
$655$	0	0
$656$	6.00000	0.234261
$657$	− 26.0000i	− 1.01436i
$658$	0	0
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	−49.0000	−1.90588	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−49.0000	−1.90588	−0.952940	+	0.303160i	$0.901958\pi$
$662$	− 5.00000i	− 0.194331i
$663$	3.00000i	0.116510i
$664$	−6.00000	−0.232845
$665$	0	0
$666$	20.0000	0.774984
$667$	9.00000i	0.348481i
$668$	18.0000i	0.696441i
$669$	−14.0000	−0.541271
$670$	0	0
$671$	0	0
$672$	1.00000i	0.0385758i
$673$	20.0000i	0.770943i	0.922720	+	0.385472i	$0.125961\pi$
$673$	20.0000i	0.770943i	−0.922720	+	0.385472i	$0.874039\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−12.0000	−0.461538
$677$	15.0000i	0.576497i	0.957556	+	0.288248i	$0.0930729\pi$
$677$	15.0000i	0.576497i	−0.957556	+	0.288248i	$0.906927\pi$
$678$	− 12.0000i	− 0.460857i
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−15.0000	−0.574801
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	13.0000	0.496342
$687$	− 26.0000i	− 0.991962i
$688$	2.00000i	0.0762493i
$689$	3.00000	0.114291
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	30.0000	1.13878
$695$	0	0
$696$	−3.00000	−0.113715
$697$	18.0000i	0.681799i
$698$	14.0000i	0.529908i
$699$	6.00000	0.226941
$700$	0	0
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	− 5.00000i	− 0.188713i
$703$	− 10.0000i	− 0.377157i
$704$	0	0
$705$	0	0
$706$	−33.0000	−1.24197
$707$	12.0000i	0.451306i
$708$	3.00000i	0.112747i
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−28.0000	−1.05008
$712$	− 6.00000i	− 0.224860i
$713$	6.00000i	0.224702i
$714$	−3.00000	−0.112272
$715$	0	0
$716$	0	0
$717$	15.0000i	0.560185i
$718$	− 33.0000i	− 1.23155i
$719$	−33.0000	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$719$	−33.0000	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 1.00000i	− 0.0372161i
$723$	− 4.00000i	− 0.148762i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	13.0000i	0.482143i	0.970507	+	0.241072i	$0.0774989\pi$
$727$	13.0000i	0.482143i	−0.970507	+	0.241072i	$0.922501\pi$
$728$	1.00000i	0.0370625i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−6.00000	−0.221918
$732$	− 8.00000i	− 0.295689i
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	3.00000	0.110581
$737$	0	0
$738$	− 12.0000i	− 0.441726i
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	3.00000i	0.110133i
$743$	18.0000i	0.660356i	0.943919	+	0.330178i	$0.107109\pi$
$743$	18.0000i	0.660356i	−0.943919	+	0.330178i	$0.892891\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	−31.0000	−1.13499
$747$	12.0000i	0.439057i
$748$	0	0
$749$	15.0000	0.548088
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	0	0
$753$	0	0
$754$	−3.00000	−0.109254
$755$	0	0
$756$	5.00000	0.181848
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	35.0000i	1.27126i
$759$	0	0
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	− 2.00000i	− 0.0724524i
$763$	− 11.0000i	− 0.398227i
$764$	27.0000	0.976826
$765$	0	0
$766$	−24.0000	−0.867155
$767$	3.00000i	0.108324i
$768$	1.00000i	0.0360844i
$769$	55.0000	1.98335	0.991675	−	0.128763i	$-0.0411007\pi$
$769$	55.0000	1.98335	0.991675	+	0.128763i	$0.0411007\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	22.0000i	0.791797i
$773$	21.0000i	0.755318i	0.925945	+	0.377659i	$0.123271\pi$
$773$	21.0000i	0.755318i	−0.925945	+	0.377659i	$0.876729\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−10.0000	−0.358979
$777$	− 10.0000i	− 0.358748i
$778$	− 24.0000i	− 0.860442i
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	9.00000i	0.321839i
$783$	15.0000i	0.536056i
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	− 5.00000i	− 0.178231i	−0.996021	−	0.0891154i	$-0.971596\pi$
$787$	− 5.00000i	− 0.178231i	0.996021	−	0.0891154i	$-0.0284040\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	− 8.00000i	− 0.284088i
$794$	−8.00000	−0.283909
$795$	0	0
$796$	−19.0000	−0.673437
$797$	33.0000i	1.16892i	0.811423	+	0.584460i	$0.198694\pi$
$797$	33.0000i	1.16892i	−0.811423	+	0.584460i	$0.801306\pi$
$798$	− 1.00000i	− 0.0353996i
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	− 6.00000i	− 0.211867i
$803$	0	0
$804$	7.00000	0.246871
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	− 18.0000i	− 0.633630i
$808$	12.0000i	0.422159i
$809$	−33.0000	−1.16022	−0.580109	−	0.814539i	$-0.696990\pi$
$809$	−33.0000	−1.16022	−0.580109	+	0.814539i	$0.696990\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	− 3.00000i	− 0.105279i
$813$	− 7.00000i	− 0.245501i
$814$	0	0
$815$	0	0
$816$	−3.00000	−0.105021
$817$	− 2.00000i	− 0.0699711i
$818$	− 22.0000i	− 0.769212i
$819$	2.00000	0.0698857
$820$	0	0
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	− 9.00000i	− 0.313911i
$823$	− 31.0000i	− 1.08059i	−0.841475	−	0.540296i	$-0.818312\pi$
$823$	− 31.0000i	− 1.08059i	0.841475	−	0.540296i	$-0.181688\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−3.00000	−0.104383
$827$	− 33.0000i	− 1.14752i	−0.819023	−	0.573761i	$-0.805484\pi$
$827$	− 33.0000i	− 1.14752i	0.819023	−	0.573761i	$-0.194516\pi$
$828$	− 6.00000i	− 0.208514i
$829$	13.0000	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$829$	13.0000	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	1.00000i	0.0346688i
$833$	18.0000i	0.623663i
$834$	−8.00000	−0.277017
$835$	0	0
$836$	0	0
$837$	10.0000i	0.345651i
$838$	− 30.0000i	− 1.03633i
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 17.0000i	− 0.585859i
$843$	18.0000i	0.619953i
$844$	25.0000	0.860535
$845$	0	0
$846$	0	0
$847$	− 11.0000i	− 0.377964i
$848$	3.00000i	0.103020i
$849$	−14.0000	−0.480479
$850$	0	0
$851$	−30.0000	−1.02839
$852$	− 12.0000i	− 0.411113i
$853$	50.0000i	1.71197i	0.517003	+	0.855984i	$0.327048\pi$
$853$	50.0000i	1.71197i	−0.517003	+	0.855984i	$0.672952\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	15.0000	0.512689
$857$	− 48.0000i	− 1.63965i	−0.572615	−	0.819824i	$-0.694071\pi$
$857$	− 48.0000i	− 1.63965i	0.572615	−	0.819824i	$-0.305929\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	− 24.0000i	− 0.817443i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−4.00000	−0.135926
$867$	8.00000i	0.271694i
$868$	− 2.00000i	− 0.0678844i
$869$	0	0
$870$	0	0
$871$	7.00000	0.237186
$872$	− 11.0000i	− 0.372507i
$873$	20.0000i	0.676897i
$874$	−3.00000	−0.101477
$875$	0	0
$876$	−13.0000	−0.439229
$877$	31.0000i	1.04680i	0.852088	+	0.523398i	$0.175336\pi$
$877$	31.0000i	1.04680i	−0.852088	+	0.523398i	$0.824664\pi$
$878$	38.0000i	1.28244i
$879$	−21.0000	−0.708312
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	− 12.0000i	− 0.404061i
$883$	38.0000i	1.27880i	0.768874	+	0.639401i	$0.220818\pi$
$883$	38.0000i	1.27880i	−0.768874	+	0.639401i	$0.779182\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−24.0000	−0.806296
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	− 10.0000i	− 0.335578i
$889$	2.00000	0.0670778
$890$	0	0
$891$	0	0
$892$	− 14.0000i	− 0.468755i
$893$	0	0
$894$	12.0000	0.401340
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	3.00000i	0.100167i
$898$	− 30.0000i	− 1.00111i
$899$	6.00000	0.200111
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	− 2.00000i	− 0.0665558i
$904$	12.0000	0.399114
$905$	0	0
$906$	−10.0000	−0.332228
$907$	− 35.0000i	− 1.16216i	−0.813848	−	0.581078i	$-0.802631\pi$
$907$	− 35.0000i	− 1.16216i	0.813848	−	0.581078i	$-0.197369\pi$
$908$	− 15.0000i	− 0.497792i
$909$	24.0000	0.796030
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	−35.0000	−1.15770
$915$	0	0
$916$	26.0000	0.859064
$917$	0	0
$918$	15.0000i	0.495074i
$919$	1.00000	0.0329870	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	1.00000	0.0329870	0.0164935	+	0.999864i	$0.494750\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	− 6.00000i	− 0.197599i
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	16.0000i	0.525509i
$928$	− 3.00000i	− 0.0984798i
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	6.00000i	0.196537i
$933$	− 3.00000i	− 0.0982156i
$934$	−12.0000	−0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	− 47.0000i	− 1.53542i	−0.640796	−	0.767712i	$-0.721395\pi$
$937$	− 47.0000i	− 1.53542i	0.640796	−	0.767712i	$-0.278605\pi$
$938$	7.00000i	0.228558i
$939$	1.00000	0.0326338
$940$	0	0
$941$	45.0000	1.46696	0.733479	−	0.679712i	$-0.237895\pi$
$941$	45.0000	1.46696	0.733479	+	0.679712i	$0.237895\pi$
$942$	10.0000i	0.325818i
$943$	18.0000i	0.586161i
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	0	0
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	14.0000i	0.454699i
$949$	−13.0000	−0.421998
$950$	0	0
$951$	9.00000	0.291845
$952$	− 3.00000i	− 0.0972306i
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−15.0000	−0.485135
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	9.00000	0.290625
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 10.0000i	− 0.322413i
$963$	− 30.0000i	− 0.966736i
$964$	4.00000	0.128831
$965$	0	0
$966$	−3.00000	−0.0965234
$967$	− 44.0000i	− 1.41494i	−0.706741	−	0.707472i	$-0.749835\pi$
$967$	− 44.0000i	− 1.41494i	0.706741	−	0.707472i	$-0.250165\pi$
$968$	− 11.0000i	− 0.353553i
$969$	3.00000	0.0963739
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 16.0000i	− 0.513200i
$973$	− 8.00000i	− 0.256468i
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	8.00000	0.256074
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	− 22.0000i	− 0.703482i
$979$	0	0
$980$	0	0
$981$	−22.0000	−0.702406
$982$	− 30.0000i	− 0.957338i
$983$	54.0000i	1.72233i	0.508323	+	0.861166i	$0.330265\pi$
$983$	54.0000i	1.72233i	−0.508323	+	0.861166i	$0.669735\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	9.00000	0.286618
$987$	0	0
$988$	− 1.00000i	− 0.0318142i
$989$	−6.00000	−0.190789
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	5.00000i	0.158670i
$994$	12.0000	0.380617
$995$	0	0
$996$	6.00000	0.190117
$997$	46.0000i	1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$	46.0000i	1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	− 10.0000i	− 0.316544i
$999$	−50.0000	−1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.b.e.799.1		2
5.2	odd	4		190.2.a.c.1.1	✓	1
5.3	odd	4		950.2.a.a.1.1		1
5.4	even	2	inner	950.2.b.e.799.2		2
15.2	even	4		1710.2.a.d.1.1		1
15.8	even	4		8550.2.a.bd.1.1		1
20.3	even	4		7600.2.a.m.1.1		1
20.7	even	4		1520.2.a.d.1.1		1
35.27	even	4		9310.2.a.o.1.1		1
40.27	even	4		6080.2.a.p.1.1		1
40.37	odd	4		6080.2.a.h.1.1		1
95.37	even	4		3610.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.c.1.1	✓	1	5.2	odd	4
950.2.a.a.1.1		1	5.3	odd	4
950.2.b.e.799.1		2	1.1	even	1	trivial
950.2.b.e.799.2		2	5.4	even	2	inner
1520.2.a.d.1.1		1	20.7	even	4
1710.2.a.d.1.1		1	15.2	even	4
3610.2.a.b.1.1		1	95.37	even	4
6080.2.a.h.1.1		1	40.37	odd	4
6080.2.a.p.1.1		1	40.27	even	4
7600.2.a.m.1.1		1	20.3	even	4
8550.2.a.bd.1.1		1	15.8	even	4
9310.2.a.o.1.1		1	35.27	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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} -1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} -1.00000 q^{19} -1.00000 q^{21} +3.00000i q^{23} -1.00000 q^{24} -1.00000 q^{26} +5.00000i q^{27} -1.00000i q^{28} +3.00000 q^{29} +2.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} -2.00000 q^{36} +10.0000i q^{37} +1.00000i q^{38} +1.00000 q^{39} +6.00000 q^{41} +1.00000i q^{42} +2.00000i q^{43} +3.00000 q^{46} +1.00000i q^{48} +6.00000 q^{49} -3.00000 q^{51} +1.00000i q^{52} +3.00000i q^{53} +5.00000 q^{54} -1.00000 q^{56} -1.00000i q^{57} -3.00000i q^{58} -3.00000 q^{59} +8.00000 q^{61} -2.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +7.00000i q^{67} -3.00000i q^{68} -3.00000 q^{69} +12.0000 q^{71} +2.00000i q^{72} -13.0000i q^{73} +10.0000 q^{74} +1.00000 q^{76} -1.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +6.00000i q^{83} +1.00000 q^{84} +2.00000 q^{86} +3.00000i q^{87} -6.00000 q^{89} +1.00000 q^{91} -3.00000i q^{92} +2.00000i q^{93} +1.00000 q^{96} +10.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} + 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{21} - 2 q^{24} - 2 q^{26} + 6 q^{29} + 4 q^{31} + 6 q^{34} - 4 q^{36} + 2 q^{39} + 12 q^{41} + 6 q^{46} + 12 q^{49} - 6 q^{51} + 10 q^{54} - 2 q^{56} - 6 q^{59} + 16 q^{61} - 2 q^{64} - 6 q^{69} + 24 q^{71} + 20 q^{74} + 2 q^{76} - 28 q^{79} + 2 q^{81} + 2 q^{84} + 4 q^{86} - 12 q^{89} + 2 q^{91} + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 + 2 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^21 - 2 * q^24 - 2 * q^26 + 6 * q^29 + 4 * q^31 + 6 * q^34 - 4 * q^36 + 2 * q^39 + 12 * q^41 + 6 * q^46 + 12 * q^49 - 6 * q^51 + 10 * q^54 - 2 * q^56 - 6 * q^59 + 16 * q^61 - 2 * q^64 - 6 * q^69 + 24 * q^71 + 20 * q^74 + 2 * q^76 - 28 * q^79 + 2 * q^81 + 2 * q^84 + 4 * q^86 - 12 * q^89 + 2 * q^91 + 2 * q^96

Introduction

L-functions

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Database

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