LMFDB - Embedded newform 2850.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(1,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, 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("2850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.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:	$22.7573645761$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} +4.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -4.00000 q^{21} +6.00000 q^{23} +1.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -1.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} +4.00000 q^{42} +4.00000 q^{43} -6.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +6.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -1.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +14.0000 q^{61} -2.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -8.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} -1.00000 q^{72} -14.0000 q^{73} -4.00000 q^{74} +1.00000 q^{76} +4.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{86} -6.00000 q^{87} -6.00000 q^{89} +16.0000 q^{91} +6.00000 q^{92} -2.00000 q^{93} +6.00000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	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
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−1.00000	−0.162221
$39$	−4.00000	−0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	4.00000	0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	4.00000	0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	−1.00000	−0.132453
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−2.00000	−0.254000
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−6.00000	−0.727607
$69$	−6.00000	−0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	4.00000	0.452911
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−4.00000	−0.431331
$87$	−6.00000	−0.643268
$88$	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
$91$	16.0000	1.67726
$92$	6.00000	0.625543
$93$	−2.00000	−0.207390
$94$	6.00000	0.618853
$95$	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	$0.330506\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−6.00000	−0.594089
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	4.00000	0.377964
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	6.00000	0.557086
$117$	4.00000	0.369800
$118$	12.0000	1.10469
$119$	−24.0000	−2.20008
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−14.0000	−1.26750
$123$	−6.00000	−0.541002
$124$	2.00000	0.179605
$125$	0	0
$126$	−4.00000	−0.356348
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	8.00000	0.691095
$135$	0	0
$136$	6.00000	0.514496
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	6.00000	0.510754
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	−9.00000	−0.742307
$148$	4.00000	0.328798
$149$	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	$0.633388\pi$
$152$	−1.00000	−0.0811107
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	10.0000	0.795557
$159$	6.00000	0.475831
$160$	0	0
$161$	24.0000	1.89146
$162$	−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	$0.449930\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	4.00000	0.308607
$169$	3.00000	0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	0	0
$177$	12.0000	0.901975
$178$	6.00000	0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	−16.0000	−1.18600
$183$	−14.0000	−1.03491
$184$	−6.00000	−0.442326
$185$	0	0
$186$	2.00000	0.146647
$187$	0	0
$188$	−6.00000	−0.437595
$189$	−4.00000	−0.290957
$190$	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	$0.382808\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	24.0000	1.68447
$204$	6.00000	0.420084
$205$	0	0
$206$	−10.0000	−0.696733
$207$	6.00000	0.417029
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	16.0000	1.08366
$219$	14.0000	0.946032
$220$	0	0
$221$	−24.0000	−1.61441
$222$	4.00000	0.268462
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−18.0000	−1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−1.00000	−0.0662266
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−12.0000	−0.781133
$237$	10.0000	0.649570
$238$	24.0000	1.55569
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	11.0000	0.707107
$243$	−1.00000	−0.0641500
$244$	14.0000	0.896258
$245$	0	0
$246$	6.00000	0.382546
$247$	4.00000	0.254514
$248$	−2.00000	−0.127000
$249$	−12.0000	−0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	4.00000	0.249029
$259$	16.0000	0.994192
$260$	0	0
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	6.00000	0.367194
$268$	−8.00000	−0.488678
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−6.00000	−0.363803
$273$	−16.0000	−0.968364
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−6.00000	−0.361158
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−20.0000	−1.19952
$279$	2.00000	0.119737
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−6.00000	−0.357295
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−14.0000	−0.819288
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−4.00000	−0.232495
$297$	0	0
$298$	−12.0000	−0.695141
$299$	24.0000	1.38796
$300$	0	0
$301$	16.0000	0.922225
$302$	10.0000	0.575435
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	6.00000	0.342997
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	4.00000	0.226455
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	−24.0000	−1.33747
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	16.0000	0.884802
$328$	−6.00000	−0.331295
$329$	−24.0000	−1.32316
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	12.0000	0.658586
$333$	4.00000	0.219199
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−4.00000	−0.218218
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−3.00000	−0.163178
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	−1.00000	−0.0540738
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	18.0000	0.967686
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	−6.00000	−0.321634
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	24.0000	1.27021
$358$	−12.0000	−0.634220
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	16.0000	0.840941
$363$	11.0000	0.577350
$364$	16.0000	0.838628
$365$	0	0
$366$	14.0000	0.731792
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	6.00000	0.312772
$369$	6.00000	0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	−2.00000	−0.103695
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	24.0000	1.23606
$378$	4.00000	0.205738
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	−18.0000	−0.920960
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	4.00000	0.203331
$388$	10.0000	0.507673
$389$	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
$391$	−36.0000	−1.82060
$392$	−9.00000	−0.454569
$393$	−12.0000	−0.605320
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−20.0000	−1.00251
$399$	−4.00000	−0.200250
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−8.00000	−0.399004
$403$	8.00000	0.398508
$404$	0	0
$405$	0	0
$406$	−24.0000	−1.19110
$407$	0	0
$408$	−6.00000	−0.297044
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	10.0000	0.492665
$413$	−48.0000	−2.36193
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−4.00000	−0.196116
$417$	−20.0000	−0.979404
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	4.00000	0.194717
$423$	−6.00000	−0.291730
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	56.0000	2.71003
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−1.00000	−0.0481125
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−16.0000	−0.766261
$437$	6.00000	0.287019
$438$	−14.0000	−0.668946
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	24.0000	1.14156
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−22.0000	−1.04173
$447$	−12.0000	−0.567581
$448$	4.00000	0.188982
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	18.0000	0.846649
$453$	10.0000	0.469841
$454$	12.0000	0.563188
$455$	0	0
$456$	1.00000	0.0468293
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	10.0000	0.467269
$459$	6.00000	0.280056
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	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
$466$	−6.00000	−0.277945
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	4.00000	0.184900
$469$	−32.0000	−1.47762
$470$	0	0
$471$	2.00000	0.0921551
$472$	12.0000	0.552345
$473$	0	0
$474$	−10.0000	−0.459315
$475$	0	0
$476$	−24.0000	−1.10004
$477$	−6.00000	−0.274721
$478$	18.0000	0.823301
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	−14.0000	−0.637683
$483$	−24.0000	−1.09204
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	−14.0000	−0.633750
$489$	−4.00000	−0.180886
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	−6.00000	−0.270501
$493$	−36.0000	−1.62136
$494$	−4.00000	−0.179969
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	12.0000	0.537733
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	−12.0000	−0.535586
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	−3.00000	−0.133235
$508$	−2.00000	−0.0887357
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	18.0000	0.793946
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	−16.0000	−0.703000
$519$	18.0000	0.790112
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−6.00000	−0.262613
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−18.0000	−0.784837
$527$	−12.0000	−0.522728
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−12.0000	−0.520756
$532$	4.00000	0.173422
$533$	24.0000	1.03956
$534$	−6.00000	−0.259645
$535$	0	0
$536$	8.00000	0.345547
$537$	−12.0000	−0.517838
$538$	6.00000	0.258678
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−20.0000	−0.859074
$543$	16.0000	0.686626
$544$	6.00000	0.257248
$545$	0	0
$546$	16.0000	0.684737
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	18.0000	0.768922
$549$	14.0000	0.597505
$550$	0	0
$551$	6.00000	0.255609
$552$	6.00000	0.255377
$553$	−40.0000	−1.70097
$554$	−10.0000	−0.424859
$555$	0	0
$556$	20.0000	0.848189
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	−2.00000	−0.0846668
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	−4.00000	−0.168133
$567$	4.00000	0.167984
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	−24.0000	−1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−19.0000	−0.790296
$579$	−10.0000	−0.415586
$580$	0	0
$581$	48.0000	1.99138
$582$	10.0000	0.414513
$583$	0	0
$584$	14.0000	0.579324
$585$	0	0
$586$	−6.00000	−0.247858
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−9.00000	−0.371154
$589$	2.00000	0.0824086
$590$	0	0
$591$	12.0000	0.493614
$592$	4.00000	0.164399
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	−20.0000	−0.818546
$598$	−24.0000	−0.981433
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	−16.0000	−0.652111
$603$	−8.00000	−0.325785
$604$	−10.0000	−0.406894
$605$	0	0
$606$	0	0
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−1.00000	−0.0405554
$609$	−24.0000	−0.972529
$610$	0	0
$611$	−24.0000	−0.970936
$612$	−6.00000	−0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	10.0000	0.402259
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	−30.0000	−1.20289
$623$	−24.0000	−0.961540
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−24.0000	−0.956943
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	10.0000	0.397779
$633$	4.00000	0.158986
$634$	18.0000	0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	36.0000	1.42637
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	−12.0000	−0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	6.00000	0.236067
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	4.00000	0.156652
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	−16.0000	−0.625650
$655$	0	0
$656$	6.00000	0.234261
$657$	−14.0000	−0.546192
$658$	24.0000	0.935617
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	−8.00000	−0.310929
$663$	24.0000	0.932083
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−4.00000	−0.154997
$667$	36.0000	1.39393
$668$	12.0000	0.464294
$669$	−22.0000	−0.850569
$670$	0	0
$671$	0	0
$672$	4.00000	0.154303
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	3.00000	0.115385
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	18.0000	0.691286
$679$	40.0000	1.53506
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−8.00000	−0.305441
$687$	10.0000	0.381524
$688$	4.00000	0.152499
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	24.0000	0.911028
$695$	0	0
$696$	6.00000	0.227429
$697$	−36.0000	−1.36360
$698$	−2.00000	−0.0757011
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	4.00000	0.150970
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	12.0000	0.450988
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	6.00000	0.224860
$713$	12.0000	0.449404
$714$	−24.0000	−0.898177
$715$	0	0
$716$	12.0000	0.448461
$717$	18.0000	0.672222
$718$	6.00000	0.223918
$719$	−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
$721$	40.0000	1.48968
$722$	−1.00000	−0.0372161
$723$	−14.0000	−0.520666
$724$	−16.0000	−0.594635
$725$	0	0
$726$	−11.0000	−0.408248
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−16.0000	−0.592999
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	−14.0000	−0.517455
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−6.00000	−0.221163
$737$	0	0
$738$	−6.00000	−0.220863
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	24.0000	0.881068
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−4.00000	−0.146450
$747$	12.0000	0.439057
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	−6.00000	−0.218797
$753$	−12.0000	−0.437304
$754$	−24.0000	−0.874028
$755$	0	0
$756$	−4.00000	−0.145479
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−2.00000	−0.0724524
$763$	−64.0000	−2.31696
$764$	18.0000	0.651217
$765$	0	0
$766$	24.0000	0.867155
$767$	−48.0000	−1.73318
$768$	−1.00000	−0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	10.0000	0.359908
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−10.0000	−0.358979
$777$	−16.0000	−0.573997
$778$	−24.0000	−0.860442
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	36.0000	1.28736
$783$	−6.00000	−0.214423
$784$	9.00000	0.321429
$785$	0	0
$786$	12.0000	0.428026
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	−12.0000	−0.427482
$789$	−18.0000	−0.640817
$790$	0	0
$791$	72.0000	2.56003
$792$	0	0
$793$	56.0000	1.98862
$794$	26.0000	0.922705
$795$	0	0
$796$	20.0000	0.708881
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	4.00000	0.141598
$799$	36.0000	1.27359
$800$	0	0
$801$	−6.00000	−0.212000
$802$	−30.0000	−1.05934
$803$	0	0
$804$	8.00000	0.282138
$805$	0	0
$806$	−8.00000	−0.281788
$807$	6.00000	0.211210
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	24.0000	0.842235
$813$	−20.0000	−0.701431
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	4.00000	0.139942
$818$	−14.0000	−0.489499
$819$	16.0000	0.559085
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	18.0000	0.627822
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−10.0000	−0.348367
$825$	0	0
$826$	48.0000	1.67013
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	6.00000	0.208514
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	4.00000	0.138675
$833$	−54.0000	−1.87099
$834$	20.0000	0.692543
$835$	0	0
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	28.0000	0.964944
$843$	18.0000	0.619953
$844$	−4.00000	−0.137686
$845$	0	0
$846$	6.00000	0.206284
$847$	−44.0000	−1.51186
$848$	−6.00000	−0.206041
$849$	−4.00000	−0.137280
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	−56.0000	−1.91628
$855$	0	0
$856$	12.0000	0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	12.0000	0.408722
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−10.0000	−0.339814
$867$	−19.0000	−0.645274
$868$	8.00000	0.271538
$869$	0	0
$870$	0	0
$871$	−32.0000	−1.08428
$872$	16.0000	0.541828
$873$	10.0000	0.338449
$874$	−6.00000	−0.202953
$875$	0	0
$876$	14.0000	0.473016
$877$	−44.0000	−1.48577	−0.742887	−	0.669417i	$-0.766544\pi$
$877$	−44.0000	−1.48577	−0.742887	+	0.669417i	$0.766544\pi$
$878$	−26.0000	−0.877457
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−9.00000	−0.303046
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	24.0000	0.806296
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	4.00000	0.134231
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	22.0000	0.736614
$893$	−6.00000	−0.200782
$894$	12.0000	0.401340
$895$	0	0
$896$	−4.00000	−0.133631
$897$	−24.0000	−0.801337
$898$	30.0000	1.00111
$899$	12.0000	0.400222
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	−16.0000	−0.532447
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−10.0000	−0.332228
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	2.00000	0.0661541
$915$	0	0
$916$	−10.0000	−0.330409
$917$	48.0000	1.58510
$918$	−6.00000	−0.198030
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	−12.0000	−0.395199
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	10.0000	0.328443
$928$	−6.00000	−0.196960
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	6.00000	0.196537
$933$	−30.0000	−0.982156
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−4.00000	−0.130744
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	32.0000	1.04484
$939$	−10.0000	−0.326338
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−2.00000	−0.0651635
$943$	36.0000	1.17232
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	10.0000	0.324785
$949$	−56.0000	−1.81784
$950$	0	0
$951$	18.0000	0.583690
$952$	24.0000	0.777844
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−18.0000	−0.582162
$957$	0	0
$958$	6.00000	0.193851
$959$	72.0000	2.32500
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−16.0000	−0.515861
$963$	−12.0000	−0.386695
$964$	14.0000	0.450910
$965$	0	0
$966$	24.0000	0.772187
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	11.0000	0.353553
$969$	6.00000	0.192748
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−1.00000	−0.0320750
$973$	80.0000	2.56468
$974$	38.0000	1.21760
$975$	0	0
$976$	14.0000	0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	4.00000	0.127906
$979$	0	0
$980$	0	0
$981$	−16.0000	−0.510841
$982$	−36.0000	−1.14881
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	36.0000	1.14647
$987$	24.0000	0.763928
$988$	4.00000	0.127257
$989$	24.0000	0.763156
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	−2.00000	−0.0635001
$993$	−8.00000	−0.253872
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	4.00000	0.126618
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.a.g.1.1		1
3.2	odd	2		8550.2.a.bj.1.1		1
5.2	odd	4		2850.2.d.p.799.1		2
5.3	odd	4		2850.2.d.p.799.2		2
5.4	even	2		114.2.a.c.1.1	✓	1
15.14	odd	2		342.2.a.c.1.1		1
20.19	odd	2		912.2.a.c.1.1		1
35.34	odd	2		5586.2.a.u.1.1		1
40.19	odd	2		3648.2.a.bc.1.1		1
40.29	even	2		3648.2.a.i.1.1		1
60.59	even	2		2736.2.a.o.1.1		1
95.94	odd	2		2166.2.a.a.1.1		1
285.284	even	2		6498.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.a.c.1.1	✓	1	5.4	even	2
342.2.a.c.1.1		1	15.14	odd	2
912.2.a.c.1.1		1	20.19	odd	2
2166.2.a.a.1.1		1	95.94	odd	2
2736.2.a.o.1.1		1	60.59	even	2
2850.2.a.g.1.1		1	1.1	even	1	trivial
2850.2.d.p.799.1		2	5.2	odd	4
2850.2.d.p.799.2		2	5.3	odd	4
3648.2.a.i.1.1		1	40.29	even	2
3648.2.a.bc.1.1		1	40.19	odd	2
5586.2.a.u.1.1		1	35.34	odd	2
6498.2.a.t.1.1		1	285.284	even	2
8550.2.a.bj.1.1		1	3.2	odd	2

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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more