LMFDB - Embedded newform 4650.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.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:	$37.1304369399$
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	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +5.00000 q^{13} +5.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +5.00000 q^{21} +1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -5.00000 q^{26} -1.00000 q^{27} -5.00000 q^{28} +6.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} -5.00000 q^{39} +7.00000 q^{41} -5.00000 q^{42} -11.0000 q^{43} -1.00000 q^{44} -4.00000 q^{46} -11.0000 q^{47} -1.00000 q^{48} +18.0000 q^{49} +4.00000 q^{51} +5.00000 q^{52} +3.00000 q^{53} +1.00000 q^{54} +5.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} +9.00000 q^{61} +1.00000 q^{62} -5.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +10.0000 q^{67} -4.00000 q^{68} -4.00000 q^{69} -1.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -1.00000 q^{74} -2.00000 q^{76} +5.00000 q^{77} +5.00000 q^{78} +1.00000 q^{81} -7.00000 q^{82} +15.0000 q^{83} +5.00000 q^{84} +11.0000 q^{86} -6.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -25.0000 q^{91} +4.00000 q^{92} +1.00000 q^{93} +11.0000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -18.0000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	5.00000	1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	1.00000	0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−5.00000	−0.980581
$27$	−1.00000	−0.192450
$28$	−5.00000	−0.944911
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	2.00000	0.324443
$39$	−5.00000	−0.800641
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	−5.00000	−0.771517
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	−1.00000	−0.144338
$49$	18.0000	2.57143
$50$	0	0
$51$	4.00000	0.560112
$52$	5.00000	0.693375
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	5.00000	0.668153
$57$	2.00000	0.264906
$58$	−6.00000	−0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	9.00000	1.15233	0.576166	−	0.817333i	$-0.304548\pi$
$61$	9.00000	1.15233	0.576166	+	0.817333i	$0.304548\pi$
$62$	1.00000	0.127000
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	−4.00000	−0.485071
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−2.00000	−0.229416
$77$	5.00000	0.569803
$78$	5.00000	0.566139
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	5.00000	0.545545
$85$	0	0
$86$	11.0000	1.18616
$87$	−6.00000	−0.643268
$88$	1.00000	0.106600
$89$	−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
$91$	−25.0000	−2.62071
$92$	4.00000	0.417029
$93$	1.00000	0.103695
$94$	11.0000	1.13456
$95$	0	0
$96$	1.00000	0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−18.0000	−1.81827
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	−4.00000	−0.396059
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	−5.00000	−0.490290
$105$	0	0
$106$	−3.00000	−0.291386
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	−5.00000	−0.472456
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	6.00000	0.557086
$117$	5.00000	0.462250
$118$	−6.00000	−0.552345
$119$	20.0000	1.83340
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−9.00000	−0.814822
$123$	−7.00000	−0.631169
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	5.00000	0.445435
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	11.0000	0.968496
$130$	0	0
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	1.00000	0.0870388
$133$	10.0000	0.867110
$134$	−10.0000	−0.863868
$135$	0	0
$136$	4.00000	0.342997
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	4.00000	0.340503
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	11.0000	0.926367
$142$	1.00000	0.0839181
$143$	−5.00000	−0.418121
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	−18.0000	−1.48461
$148$	1.00000	0.0821995
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	2.00000	0.162221
$153$	−4.00000	−0.323381
$154$	−5.00000	−0.402911
$155$	0	0
$156$	−5.00000	−0.400320
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−20.0000	−1.57622
$162$	−1.00000	−0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	−15.0000	−1.16423
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	−5.00000	−0.385758
$169$	12.0000	0.923077
$170$	0	0
$171$	−2.00000	−0.152944
$172$	−11.0000	−0.838742
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−6.00000	−0.450988
$178$	6.00000	0.449719
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	25.0000	1.85312
$183$	−9.00000	−0.665299
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	4.00000	0.292509
$188$	−11.0000	−0.802257
$189$	5.00000	0.363696
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	−1.00000	−0.0721688
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	18.0000	1.28571
$197$	−17.0000	−1.21120	−0.605600	−	0.795769i	$-0.707067\pi$
$197$	−17.0000	−1.21120	−0.605600	+	0.795769i	$0.707067\pi$
$198$	1.00000	0.0710669
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	8.00000	0.562878
$203$	−30.0000	−2.10559
$204$	4.00000	0.280056
$205$	0	0
$206$	13.0000	0.905753
$207$	4.00000	0.278019
$208$	5.00000	0.346688
$209$	2.00000	0.138343
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	3.00000	0.206041
$213$	1.00000	0.0685189
$214$	−4.00000	−0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	5.00000	0.339422
$218$	10.0000	0.677285
$219$	2.00000	0.135147
$220$	0	0
$221$	−20.0000	−1.34535
$222$	1.00000	0.0671156
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	14.0000	0.931266
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	2.00000	0.132453
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	−6.00000	−0.393919
$233$	−7.00000	−0.458585	−0.229293	−	0.973358i	$-0.573641\pi$
$233$	−7.00000	−0.458585	−0.229293	+	0.973358i	$0.573641\pi$
$234$	−5.00000	−0.326860
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	−20.0000	−1.29641
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	10.0000	0.642824
$243$	−1.00000	−0.0641500
$244$	9.00000	0.576166
$245$	0	0
$246$	7.00000	0.446304
$247$	−10.0000	−0.636285
$248$	1.00000	0.0635001
$249$	−15.0000	−0.950586
$250$	0	0
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	−5.00000	−0.314970
$253$	−4.00000	−0.251478
$254$	−4.00000	−0.250982
$255$	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	$0.818685\pi$
$258$	−11.0000	−0.684830
$259$	−5.00000	−0.310685
$260$	0	0
$261$	6.00000	0.371391
$262$	2.00000	0.123560
$263$	−22.0000	−1.35658	−0.678289	−	0.734795i	$-0.737278\pi$
$263$	−22.0000	−1.35658	−0.678289	+	0.734795i	$0.737278\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	−10.0000	−0.613139
$267$	6.00000	0.367194
$268$	10.0000	0.610847
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	−4.00000	−0.242536
$273$	25.0000	1.51307
$274$	−4.00000	−0.241649
$275$	0	0
$276$	−4.00000	−0.240772
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	11.0000	0.659736
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−25.0000	−1.49137	−0.745687	−	0.666296i	$-0.767879\pi$
$281$	−25.0000	−1.49137	−0.745687	+	0.666296i	$0.767879\pi$
$282$	−11.0000	−0.655040
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−1.00000	−0.0593391
$285$	0	0
$286$	5.00000	0.295656
$287$	−35.0000	−2.06598
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−2.00000	−0.117041
$293$	20.0000	1.16841	0.584206	−	0.811605i	$-0.301406\pi$
$293$	20.0000	1.16841	0.584206	+	0.811605i	$0.301406\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	1.00000	0.0580259
$298$	−16.0000	−0.926855
$299$	20.0000	1.15663
$300$	0	0
$301$	55.0000	3.17015
$302$	8.00000	0.460348
$303$	8.00000	0.459588
$304$	−2.00000	−0.114708
$305$	0	0
$306$	4.00000	0.228665
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	5.00000	0.284901
$309$	13.0000	0.739544
$310$	0	0
$311$	−25.0000	−1.41762	−0.708810	−	0.705399i	$-0.750768\pi$
$311$	−25.0000	−1.41762	−0.708810	+	0.705399i	$0.750768\pi$
$312$	5.00000	0.283069
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	0	0
$317$	28.0000	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$317$	28.0000	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$318$	3.00000	0.168232
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−4.00000	−0.223258
$322$	20.0000	1.11456
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	−2.00000	−0.110770
$327$	10.0000	0.553001
$328$	−7.00000	−0.386510
$329$	55.0000	3.03225
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	15.0000	0.823232
$333$	1.00000	0.0547997
$334$	4.00000	0.218870
$335$	0	0
$336$	5.00000	0.272772
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	−12.0000	−0.652714
$339$	14.0000	0.760376
$340$	0	0
$341$	1.00000	0.0541530
$342$	2.00000	0.108148
$343$	−55.0000	−2.96972
$344$	11.0000	0.593080
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−5.00000	−0.268414	−0.134207	−	0.990953i	$-0.542849\pi$
$347$	−5.00000	−0.268414	−0.134207	+	0.990953i	$0.542849\pi$
$348$	−6.00000	−0.321634
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	1.00000	0.0533002
$353$	32.0000	1.70319	0.851594	−	0.524202i	$-0.175636\pi$
$353$	32.0000	1.70319	0.851594	+	0.524202i	$0.175636\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−20.0000	−1.05851
$358$	3.00000	0.158555
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	19.0000	0.998618
$363$	10.0000	0.524864
$364$	−25.0000	−1.31036
$365$	0	0
$366$	9.00000	0.470438
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	4.00000	0.208514
$369$	7.00000	0.364405
$370$	0	0
$371$	−15.0000	−0.778761
$372$	1.00000	0.0518476
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	11.0000	0.567282
$377$	30.0000	1.54508
$378$	−5.00000	−0.257172
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	−4.00000	−0.204658
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−5.00000	−0.254493
$387$	−11.0000	−0.559161
$388$	10.0000	0.507673
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	−18.0000	−0.909137
$393$	2.00000	0.100887
$394$	17.0000	0.856448
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−38.0000	−1.90717	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000	−1.90717	−0.953583	+	0.301131i	$0.902636\pi$
$398$	14.0000	0.701757
$399$	−10.0000	−0.500626
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	10.0000	0.498755
$403$	−5.00000	−0.249068
$404$	−8.00000	−0.398015
$405$	0	0
$406$	30.0000	1.48888
$407$	−1.00000	−0.0495682
$408$	−4.00000	−0.198030
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	−13.0000	−0.640464
$413$	−30.0000	−1.47620
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−5.00000	−0.245145
$417$	11.0000	0.538672
$418$	−2.00000	−0.0978232
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−4.00000	−0.194717
$423$	−11.0000	−0.534838
$424$	−3.00000	−0.145693
$425$	0	0
$426$	−1.00000	−0.0484502
$427$	−45.0000	−2.17770
$428$	4.00000	0.193347
$429$	5.00000	0.241402
$430$	0	0
$431$	23.0000	1.10787	0.553936	−	0.832560i	$-0.313125\pi$
$431$	23.0000	1.10787	0.553936	+	0.832560i	$0.313125\pi$
$432$	−1.00000	−0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−8.00000	−0.382692
$438$	−2.00000	−0.0955637
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	20.0000	0.951303
$443$	−10.0000	−0.475114	−0.237557	−	0.971374i	$-0.576347\pi$
$443$	−10.0000	−0.475114	−0.237557	+	0.971374i	$0.576347\pi$
$444$	−1.00000	−0.0474579
$445$	0	0
$446$	−18.0000	−0.852325
$447$	−16.0000	−0.756774
$448$	−5.00000	−0.236228
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	−7.00000	−0.329617
$452$	−14.0000	−0.658505
$453$	8.00000	0.375873
$454$	−4.00000	−0.187729
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	22.0000	1.02799
$459$	4.00000	0.186704
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	5.00000	0.232621
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	7.00000	0.324269
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	5.00000	0.231125
$469$	−50.0000	−2.30879
$470$	0	0
$471$	14.0000	0.645086
$472$	−6.00000	−0.276172
$473$	11.0000	0.505781
$474$	0	0
$475$	0	0
$476$	20.0000	0.916698
$477$	3.00000	0.137361
$478$	−18.0000	−0.823301
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	26.0000	1.18427
$483$	20.0000	0.910032
$484$	−10.0000	−0.454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−9.00000	−0.407411
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−7.00000	−0.315584
$493$	−24.0000	−1.08091
$494$	10.0000	0.449921
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	5.00000	0.224281
$498$	15.0000	0.672166
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	4.00000	0.178707
$502$	−23.0000	−1.02654
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	5.00000	0.222718
$505$	0	0
$506$	4.00000	0.177822
$507$	−12.0000	−0.532939
$508$	4.00000	0.177471
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	27.0000	1.19092
$515$	0	0
$516$	11.0000	0.484248
$517$	11.0000	0.483779
$518$	5.00000	0.219687
$519$	−18.0000	−0.790112
$520$	0	0
$521$	35.0000	1.53338	0.766689	−	0.642019i	$-0.221903\pi$
$521$	35.0000	1.53338	0.766689	+	0.642019i	$0.221903\pi$
$522$	−6.00000	−0.262613
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	22.0000	0.959246
$527$	4.00000	0.174243
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	10.0000	0.433555
$533$	35.0000	1.51602
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−10.0000	−0.431934
$537$	3.00000	0.129460
$538$	−10.0000	−0.431131
$539$	−18.0000	−0.775315
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	12.0000	0.515444
$543$	19.0000	0.815368
$544$	4.00000	0.171499
$545$	0	0
$546$	−25.0000	−1.06990
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	4.00000	0.170872
$549$	9.00000	0.384111
$550$	0	0
$551$	−12.0000	−0.511217
$552$	4.00000	0.170251
$553$	0	0
$554$	−6.00000	−0.254916
$555$	0	0
$556$	−11.0000	−0.466504
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	1.00000	0.0423334
$559$	−55.0000	−2.32625
$560$	0	0
$561$	−4.00000	−0.168880
$562$	25.0000	1.05456
$563$	−16.0000	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$564$	11.0000	0.463184
$565$	0	0
$566$	−12.0000	−0.504398
$567$	−5.00000	−0.209980
$568$	1.00000	0.0419591
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	−21.0000	−0.878823	−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000	−0.878823	−0.439411	+	0.898286i	$0.644813\pi$
$572$	−5.00000	−0.209061
$573$	−4.00000	−0.167102
$574$	35.0000	1.46087
$575$	0	0
$576$	1.00000	0.0416667
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	1.00000	0.0415945
$579$	−5.00000	−0.207793
$580$	0	0
$581$	−75.0000	−3.11152
$582$	10.0000	0.414513
$583$	−3.00000	−0.124247
$584$	2.00000	0.0827606
$585$	0	0
$586$	−20.0000	−0.826192
$587$	45.0000	1.85735	0.928674	−	0.370896i	$-0.120949\pi$
$587$	45.0000	1.85735	0.928674	+	0.370896i	$0.120949\pi$
$588$	−18.0000	−0.742307
$589$	2.00000	0.0824086
$590$	0	0
$591$	17.0000	0.699287
$592$	1.00000	0.0410997
$593$	−1.00000	−0.0410651	−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000	−0.0410651	−0.0205325	+	0.999789i	$0.506536\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	16.0000	0.655386
$597$	14.0000	0.572982
$598$	−20.0000	−0.817861
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	−55.0000	−2.24163
$603$	10.0000	0.407231
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−8.00000	−0.324978
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	2.00000	0.0811107
$609$	30.0000	1.21566
$610$	0	0
$611$	−55.0000	−2.22506
$612$	−4.00000	−0.161690
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	−5.00000	−0.201456
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	−13.0000	−0.522937
$619$	−43.0000	−1.72832	−0.864158	−	0.503221i	$-0.832148\pi$
$619$	−43.0000	−1.72832	−0.864158	+	0.503221i	$0.832148\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	25.0000	1.00241
$623$	30.0000	1.20192
$624$	−5.00000	−0.200160
$625$	0	0
$626$	34.0000	1.35891
$627$	−2.00000	−0.0798723
$628$	−14.0000	−0.558661
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	−28.0000	−1.11202
$635$	0	0
$636$	−3.00000	−0.118958
$637$	90.0000	3.56593
$638$	6.00000	0.237542
$639$	−1.00000	−0.0395594
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000	0.157867
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	−20.0000	−0.788110
$645$	0	0
$646$	−8.00000	−0.314756
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−1.00000	−0.0392837
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−5.00000	−0.195965
$652$	2.00000	0.0783260
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	7.00000	0.273304
$657$	−2.00000	−0.0780274
$658$	−55.0000	−2.14412
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	17.0000	0.660724
$663$	20.0000	0.776736
$664$	−15.0000	−0.582113
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	24.0000	0.929284
$668$	−4.00000	−0.154765
$669$	−18.0000	−0.695920
$670$	0	0
$671$	−9.00000	−0.347441
$672$	−5.00000	−0.192879
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−30.0000	−1.15556
$675$	0	0
$676$	12.0000	0.461538
$677$	−49.0000	−1.88322	−0.941611	−	0.336701i	$-0.890689\pi$
$677$	−49.0000	−1.88322	−0.941611	+	0.336701i	$0.890689\pi$
$678$	−14.0000	−0.537667
$679$	−50.0000	−1.91882
$680$	0	0
$681$	−4.00000	−0.153280
$682$	−1.00000	−0.0382920
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	55.0000	2.09991
$687$	22.0000	0.839352
$688$	−11.0000	−0.419371
$689$	15.0000	0.571454
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	18.0000	0.684257
$693$	5.00000	0.189934
$694$	5.00000	0.189797
$695$	0	0
$696$	6.00000	0.227429
$697$	−28.0000	−1.06058
$698$	2.00000	0.0757011
$699$	7.00000	0.264764
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	5.00000	0.188713
$703$	−2.00000	−0.0754314
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−32.0000	−1.20434
$707$	40.0000	1.50435
$708$	−6.00000	−0.225494
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	−4.00000	−0.149801
$714$	20.0000	0.748481
$715$	0	0
$716$	−3.00000	−0.112115
$717$	−18.0000	−0.672222
$718$	−20.0000	−0.746393
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	65.0000	2.42073
$722$	15.0000	0.558242
$723$	26.0000	0.966950
$724$	−19.0000	−0.706129
$725$	0	0
$726$	−10.0000	−0.371135
$727$	−27.0000	−1.00137	−0.500687	−	0.865628i	$-0.666919\pi$
$727$	−27.0000	−1.00137	−0.500687	+	0.865628i	$0.666919\pi$
$728$	25.0000	0.926562
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.0000	1.62740
$732$	−9.00000	−0.332650
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	24.0000	0.885856
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−10.0000	−0.368355
$738$	−7.00000	−0.257674
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	15.0000	0.550667
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	8.00000	0.292901
$747$	15.0000	0.548821
$748$	4.00000	0.146254
$749$	−20.0000	−0.730784
$750$	0	0
$751$	−33.0000	−1.20419	−0.602094	−	0.798426i	$-0.705667\pi$
$751$	−33.0000	−1.20419	−0.602094	+	0.798426i	$0.705667\pi$
$752$	−11.0000	−0.401129
$753$	−23.0000	−0.838167
$754$	−30.0000	−1.09254
$755$	0	0
$756$	5.00000	0.181848
$757$	11.0000	0.399802	0.199901	−	0.979816i	$-0.435938\pi$
$757$	11.0000	0.399802	0.199901	+	0.979816i	$0.435938\pi$
$758$	34.0000	1.23494
$759$	4.00000	0.145191
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	4.00000	0.144905
$763$	50.0000	1.81012
$764$	4.00000	0.144715
$765$	0	0
$766$	20.0000	0.722629
$767$	30.0000	1.08324
$768$	−1.00000	−0.0360844
$769$	43.0000	1.55062	0.775310	−	0.631581i	$-0.217594\pi$
$769$	43.0000	1.55062	0.775310	+	0.631581i	$0.217594\pi$
$770$	0	0
$771$	27.0000	0.972381
$772$	5.00000	0.179954
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	11.0000	0.395387
$775$	0	0
$776$	−10.0000	−0.358979
$777$	5.00000	0.179374
$778$	6.00000	0.215110
$779$	−14.0000	−0.501602
$780$	0	0
$781$	1.00000	0.0357828
$782$	16.0000	0.572159
$783$	−6.00000	−0.214423
$784$	18.0000	0.642857
$785$	0	0
$786$	−2.00000	−0.0713376
$787$	35.0000	1.24762	0.623808	−	0.781578i	$-0.285585\pi$
$787$	35.0000	1.24762	0.623808	+	0.781578i	$0.285585\pi$
$788$	−17.0000	−0.605600
$789$	22.0000	0.783221
$790$	0	0
$791$	70.0000	2.48891
$792$	1.00000	0.0355335
$793$	45.0000	1.59800
$794$	38.0000	1.34857
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	10.0000	0.353996
$799$	44.0000	1.55661
$800$	0	0
$801$	−6.00000	−0.212000
$802$	24.0000	0.847469
$803$	2.00000	0.0705785
$804$	−10.0000	−0.352673
$805$	0	0
$806$	5.00000	0.176117
$807$	−10.0000	−0.352017
$808$	8.00000	0.281439
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	18.0000	0.632065	0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000	0.632065	0.316033	+	0.948748i	$0.397649\pi$
$812$	−30.0000	−1.05279
$813$	12.0000	0.420858
$814$	1.00000	0.0350500
$815$	0	0
$816$	4.00000	0.140028
$817$	22.0000	0.769683
$818$	−34.0000	−1.18878
$819$	−25.0000	−0.873571
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	4.00000	0.139516
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	30.0000	1.04383
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	4.00000	0.139010
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	5.00000	0.173344
$833$	−72.0000	−2.49465
$834$	−11.0000	−0.380899
$835$	0	0
$836$	2.00000	0.0691714
$837$	1.00000	0.0345651
$838$	−36.0000	−1.24360
$839$	−5.00000	−0.172619	−0.0863096	−	0.996268i	$-0.527507\pi$
$839$	−5.00000	−0.172619	−0.0863096	+	0.996268i	$0.527507\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−2.00000	−0.0689246
$843$	25.0000	0.861046
$844$	4.00000	0.137686
$845$	0	0
$846$	11.0000	0.378188
$847$	50.0000	1.71802
$848$	3.00000	0.103020
$849$	−12.0000	−0.411839
$850$	0	0
$851$	4.00000	0.137118
$852$	1.00000	0.0342594
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	45.0000	1.53987
$855$	0	0
$856$	−4.00000	−0.136717
$857$	−53.0000	−1.81045	−0.905223	−	0.424937i	$-0.860296\pi$
$857$	−53.0000	−1.81045	−0.905223	+	0.424937i	$0.860296\pi$
$858$	−5.00000	−0.170697
$859$	−15.0000	−0.511793	−0.255897	−	0.966704i	$-0.582371\pi$
$859$	−15.0000	−0.511793	−0.255897	+	0.966704i	$0.582371\pi$
$860$	0	0
$861$	35.0000	1.19280
$862$	−23.0000	−0.783383
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−16.0000	−0.543702
$867$	1.00000	0.0339618
$868$	5.00000	0.169711
$869$	0	0
$870$	0	0
$871$	50.0000	1.69419
$872$	10.0000	0.338643
$873$	10.0000	0.338449
$874$	8.00000	0.270604
$875$	0	0
$876$	2.00000	0.0675737
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	24.0000	0.809961
$879$	−20.0000	−0.674583
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	−18.0000	−0.606092
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	−20.0000	−0.672673
$885$	0	0
$886$	10.0000	0.335957
$887$	13.0000	0.436497	0.218249	−	0.975893i	$-0.429966\pi$
$887$	13.0000	0.436497	0.218249	+	0.975893i	$0.429966\pi$
$888$	1.00000	0.0335578
$889$	−20.0000	−0.670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	18.0000	0.602685
$893$	22.0000	0.736202
$894$	16.0000	0.535120
$895$	0	0
$896$	5.00000	0.167038
$897$	−20.0000	−0.667781
$898$	34.0000	1.13459
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−12.0000	−0.399778
$902$	7.00000	0.233075
$903$	−55.0000	−1.83029
$904$	14.0000	0.465633
$905$	0	0
$906$	−8.00000	−0.265782
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	4.00000	0.132745
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−2.00000	−0.0662630	−0.0331315	−	0.999451i	$-0.510548\pi$
$911$	−2.00000	−0.0662630	−0.0331315	+	0.999451i	$0.510548\pi$
$912$	2.00000	0.0662266
$913$	−15.0000	−0.496428
$914$	0	0
$915$	0	0
$916$	−22.0000	−0.726900
$917$	10.0000	0.330229
$918$	−4.00000	−0.132020
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	21.0000	0.691598
$923$	−5.00000	−0.164577
$924$	−5.00000	−0.164488
$925$	0	0
$926$	−4.00000	−0.131448
$927$	−13.0000	−0.426976
$928$	−6.00000	−0.196960
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	−7.00000	−0.229293
$933$	25.0000	0.818463
$934$	−18.0000	−0.588978
$935$	0	0
$936$	−5.00000	−0.163430
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	50.0000	1.63256
$939$	34.0000	1.10955
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−14.0000	−0.456145
$943$	28.0000	0.911805
$944$	6.00000	0.195283
$945$	0	0
$946$	−11.0000	−0.357641
$947$	−27.0000	−0.877382	−0.438691	−	0.898638i	$-0.644558\pi$
$947$	−27.0000	−0.877382	−0.438691	+	0.898638i	$0.644558\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	−28.0000	−0.907962
$952$	−20.0000	−0.648204
$953$	−16.0000	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$954$	−3.00000	−0.0971286
$955$	0	0
$956$	18.0000	0.582162
$957$	6.00000	0.193952
$958$	16.0000	0.516937
$959$	−20.0000	−0.645834
$960$	0	0
$961$	1.00000	0.0322581
$962$	−5.00000	−0.161206
$963$	4.00000	0.128898
$964$	−26.0000	−0.837404
$965$	0	0
$966$	−20.0000	−0.643489
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	10.0000	0.321412
$969$	−8.00000	−0.256997
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	55.0000	1.76322
$974$	4.00000	0.128168
$975$	0	0
$976$	9.00000	0.288083
$977$	−35.0000	−1.11975	−0.559875	−	0.828577i	$-0.689151\pi$
$977$	−35.0000	−1.11975	−0.559875	+	0.828577i	$0.689151\pi$
$978$	2.00000	0.0639529
$979$	6.00000	0.191761
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−12.0000	−0.382935
$983$	58.0000	1.84991	0.924956	−	0.380073i	$-0.124101\pi$
$983$	58.0000	1.84991	0.924956	+	0.380073i	$0.124101\pi$
$984$	7.00000	0.223152
$985$	0	0
$986$	24.0000	0.764316
$987$	−55.0000	−1.75067
$988$	−10.0000	−0.318142
$989$	−44.0000	−1.39912
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	1.00000	0.0317500
$993$	17.0000	0.539479
$994$	−5.00000	−0.158590
$995$	0	0
$996$	−15.0000	−0.475293
$997$	−24.0000	−0.760088	−0.380044	−	0.924968i	$-0.624091\pi$
$997$	−24.0000	−0.760088	−0.380044	+	0.924968i	$0.624091\pi$
$998$	32.0000	1.01294
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.a.1.1	✓	1
5.2	odd	4		4650.2.d.s.3349.1		2
5.3	odd	4		4650.2.d.s.3349.2		2
5.4	even	2		4650.2.a.bx.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4650.2.a.a.1.1	✓	1	1.1	even	1	trivial
4650.2.a.bx.1.1	yes	1	5.4	even	2
4650.2.d.s.3349.1		2	5.2	odd	4
4650.2.d.s.3349.2		2	5.3	odd	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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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