LMFDB - Embedded newform 2850.2.a.v.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 570)
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -4.00000 q^{21} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -6.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} +1.00000 q^{57} +6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +12.0000 q^{67} +6.00000 q^{68} +4.00000 q^{69} -16.0000 q^{71} +1.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} -1.00000 q^{76} -16.0000 q^{77} -6.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{84} +8.00000 q^{86} -6.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} +24.0000 q^{91} -4.00000 q^{92} +8.00000 q^{93} -12.0000 q^{94} -1.00000 q^{96} -14.0000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$7$	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
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.00000	−0.872872
$22$	−4.00000	−0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	−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
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.00000	−0.162221
$39$	−6.00000	−0.960769
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	−4.00000	−0.617213
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	6.00000	0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	1.00000	0.132453
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−8.00000	−1.01600
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	6.00000	0.727607
$69$	4.00000	0.481543
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	1.00000	0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−16.0000	−1.82337
$78$	−6.00000	−0.679366
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	8.00000	0.862662
$87$	−6.00000	−0.643268
$88$	−4.00000	−0.426401
$89$	−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
$91$	24.0000	2.51588
$92$	−4.00000	−0.417029
$93$	8.00000	0.829561
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	9.00000	0.909137
$99$	−4.00000	−0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	4.00000	0.377964
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	−4.00000	−0.368230
$119$	24.0000	2.20008
$120$	0	0
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	−10.0000	−0.901670
$124$	−8.00000	−0.718421
$125$	0	0
$126$	4.00000	0.356348
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000	0.348155
$133$	−4.00000	−0.346844
$134$	12.0000	1.03664
$135$	0	0
$136$	6.00000	0.514496
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	4.00000	0.340503
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	−16.0000	−1.34269
$143$	−24.0000	−2.00698
$144$	1.00000	0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	−9.00000	−0.742307
$148$	−2.00000	−0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−1.00000	−0.0811107
$153$	6.00000	0.485071
$154$	−16.0000	−1.28932
$155$	0	0
$156$	−6.00000	−0.480384
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	8.00000	0.636446
$159$	2.00000	0.158610
$160$	0	0
$161$	−16.0000	−1.26098
$162$	1.00000	0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−4.00000	−0.308607
$169$	23.0000	1.76923
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	8.00000	0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	4.00000	0.300658
$178$	−6.00000	−0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	24.0000	1.77900
$183$	2.00000	0.147844
$184$	−4.00000	−0.294884
$185$	0	0
$186$	8.00000	0.586588
$187$	−24.0000	−1.75505
$188$	−12.0000	−0.875190
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−4.00000	−0.284268
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	6.00000	0.422159
$203$	24.0000	1.68447
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	−4.00000	−0.278019
$208$	6.00000	0.416025
$209$	4.00000	0.276686
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−2.00000	−0.137361
$213$	16.0000	1.09630
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−32.0000	−2.17230
$218$	6.00000	0.406371
$219$	−14.0000	−0.946032
$220$	0	0
$221$	36.0000	2.42162
$222$	2.00000	0.134231
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	10.0000	0.665190
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	1.00000	0.0662266
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	6.00000	0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−4.00000	−0.260378
$237$	−8.00000	−0.519656
$238$	24.0000	1.55569
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−10.0000	−0.637577
$247$	−6.00000	−0.381771
$248$	−8.00000	−0.508001
$249$	0	0
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	4.00000	0.251976
$253$	16.0000	1.00591
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−8.00000	−0.498058
$259$	−8.00000	−0.497096
$260$	0	0
$261$	6.00000	0.371391
$262$	12.0000	0.741362
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	−4.00000	−0.245256
$267$	6.00000	0.367194
$268$	12.0000	0.733017
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	6.00000	0.363803
$273$	−24.0000	−1.45255
$274$	−2.00000	−0.120824
$275$	0	0
$276$	4.00000	0.240772
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	12.0000	0.719712
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	12.0000	0.714590
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−24.0000	−1.41915
$287$	40.0000	2.36113
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	14.0000	0.820695
$292$	14.0000	0.819288
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	−10.0000	−0.579284
$299$	−24.0000	−1.38796
$300$	0	0
$301$	32.0000	1.84445
$302$	8.00000	0.460348
$303$	−6.00000	−0.344691
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	6.00000	0.342997
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	−16.0000	−0.911685
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−6.00000	−0.339683
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	8.00000	0.450035
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	2.00000	0.112154
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−12.0000	−0.669775
$322$	−16.0000	−0.891645
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	−6.00000	−0.331801
$328$	10.0000	0.552158
$329$	−48.0000	−2.64633
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	−4.00000	−0.218218
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	23.0000	1.25104
$339$	−10.0000	−0.543125
$340$	0	0
$341$	32.0000	1.73290
$342$	−1.00000	−0.0540738
$343$	8.00000	0.431959
$344$	8.00000	0.431331
$345$	0	0
$346$	6.00000	0.322562
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	−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
$351$	−6.00000	−0.320256
$352$	−4.00000	−0.213201
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−24.0000	−1.27021
$358$	−12.0000	−0.634220
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−2.00000	−0.105118
$363$	−5.00000	−0.262432
$364$	24.0000	1.25794
$365$	0	0
$366$	2.00000	0.104542
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−4.00000	−0.208514
$369$	10.0000	0.520579
$370$	0	0
$371$	−8.00000	−0.415339
$372$	8.00000	0.414781
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	−12.0000	−0.618853
$377$	36.0000	1.85409
$378$	−4.00000	−0.205738
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−8.00000	−0.409316
$383$	−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
$386$	2.00000	0.101797
$387$	8.00000	0.406663
$388$	−14.0000	−0.710742
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	9.00000	0.454569
$393$	−12.0000	−0.605320
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−4.00000	−0.201008
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−16.0000	−0.802008
$399$	4.00000	0.200250
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	−12.0000	−0.598506
$403$	−48.0000	−2.39105
$404$	6.00000	0.298511
$405$	0	0
$406$	24.0000	1.19110
$407$	8.00000	0.396545
$408$	−6.00000	−0.297044
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	−8.00000	−0.394132
$413$	−16.0000	−0.787309
$414$	−4.00000	−0.196589
$415$	0	0
$416$	6.00000	0.294174
$417$	−12.0000	−0.587643
$418$	4.00000	0.195646
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−20.0000	−0.973585
$423$	−12.0000	−0.583460
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	16.0000	0.775203
$427$	−8.00000	−0.387147
$428$	12.0000	0.580042
$429$	24.0000	1.15873
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	−1.00000	−0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	6.00000	0.287348
$437$	4.00000	0.191346
$438$	−14.0000	−0.668946
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	36.0000	1.71235
$443$	−8.00000	−0.380091	−0.190046	−	0.981775i	$-0.560864\pi$
$443$	−8.00000	−0.380091	−0.190046	+	0.981775i	$0.560864\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	16.0000	0.757622
$447$	10.0000	0.472984
$448$	4.00000	0.188982
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	10.0000	0.470360
$453$	−8.00000	−0.375873
$454$	4.00000	0.187729
$455$	0	0
$456$	1.00000	0.0468293
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	6.00000	0.280362
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	16.0000	0.744387
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−10.0000	−0.463241
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	6.00000	0.277350
$469$	48.0000	2.21643
$470$	0	0
$471$	22.0000	1.01371
$472$	−4.00000	−0.184115
$473$	−32.0000	−1.47136
$474$	−8.00000	−0.367452
$475$	0	0
$476$	24.0000	1.10004
$477$	−2.00000	−0.0915737
$478$	16.0000	0.731823
$479$	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
$481$	−12.0000	−0.547153
$482$	−22.0000	−1.00207
$483$	16.0000	0.728025
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−2.00000	−0.0905357
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−10.0000	−0.450835
$493$	36.0000	1.62136
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−64.0000	−2.87079
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	0	0
$502$	4.00000	0.178529
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	16.0000	0.711287
$507$	−23.0000	−1.02147
$508$	16.0000	0.709885
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	56.0000	2.47729
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	18.0000	0.793946
$515$	0	0
$516$	−8.00000	−0.352180
$517$	48.0000	2.11104
$518$	−8.00000	−0.351500
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	6.00000	0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−20.0000	−0.872041
$527$	−48.0000	−2.09091
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−4.00000	−0.173585
$532$	−4.00000	−0.173422
$533$	60.0000	2.59889
$534$	6.00000	0.259645
$535$	0	0
$536$	12.0000	0.518321
$537$	12.0000	0.517838
$538$	14.0000	0.603583
$539$	−36.0000	−1.55063
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−8.00000	−0.343629
$543$	2.00000	0.0858282
$544$	6.00000	0.257248
$545$	0	0
$546$	−24.0000	−1.02711
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	−2.00000	−0.0854358
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−6.00000	−0.255609
$552$	4.00000	0.170251
$553$	32.0000	1.36078
$554$	2.00000	0.0849719
$555$	0	0
$556$	12.0000	0.508913
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	−8.00000	−0.338667
$559$	48.0000	2.03018
$560$	0	0
$561$	24.0000	1.01328
$562$	−22.0000	−0.928014
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	16.0000	0.672530
$567$	4.00000	0.167984
$568$	−16.0000	−0.671345
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−24.0000	−1.00349
$573$	8.00000	0.334205
$574$	40.0000	1.66957
$575$	0	0
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	19.0000	0.790296
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	14.0000	0.580319
$583$	8.00000	0.331326
$584$	14.0000	0.579324
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	−9.00000	−0.371154
$589$	8.00000	0.329634
$590$	0	0
$591$	6.00000	0.246807
$592$	−2.00000	−0.0821995
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	16.0000	0.654836
$598$	−24.0000	−0.981433
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	32.0000	1.30422
$603$	12.0000	0.488678
$604$	8.00000	0.325515
$605$	0	0
$606$	−6.00000	−0.243733
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	−1.00000	−0.0405554
$609$	−24.0000	−0.972529
$610$	0	0
$611$	−72.0000	−2.91281
$612$	6.00000	0.242536
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	−16.0000	−0.644658
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	8.00000	0.321807
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−8.00000	−0.320771
$623$	−24.0000	−0.961540
$624$	−6.00000	−0.240192
$625$	0	0
$626$	6.00000	0.239808
$627$	−4.00000	−0.159745
$628$	−22.0000	−0.877896
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	8.00000	0.318223
$633$	20.0000	0.794929
$634$	14.0000	0.556011
$635$	0	0
$636$	2.00000	0.0793052
$637$	54.0000	2.13956
$638$	−24.0000	−0.950169
$639$	−16.0000	−0.632950
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	−12.0000	−0.473602
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−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
$649$	16.0000	0.628055
$650$	0	0
$651$	32.0000	1.25418
$652$	16.0000	0.626608
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	10.0000	0.390434
$657$	14.0000	0.546192
$658$	−48.0000	−1.87123
$659$	−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
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000	0.155464
$663$	−36.0000	−1.39812
$664$	0	0
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−24.0000	−0.929284
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	8.00000	0.308837
$672$	−4.00000	−0.154303
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	23.0000	0.884615
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	−10.0000	−0.384048
$679$	−56.0000	−2.14908
$680$	0	0
$681$	−4.00000	−0.153280
$682$	32.0000	1.22534
$683$	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
$687$	−6.00000	−0.228914
$688$	8.00000	0.304997
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	6.00000	0.228086
$693$	−16.0000	−0.607790
$694$	−32.0000	−1.21470
$695$	0	0
$696$	−6.00000	−0.227429
$697$	60.0000	2.27266
$698$	−2.00000	−0.0757011
$699$	10.0000	0.378235
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−6.00000	−0.226455
$703$	2.00000	0.0754314
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−26.0000	−0.978523
$707$	24.0000	0.902613
$708$	4.00000	0.150329
$709$	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
$711$	8.00000	0.300023
$712$	−6.00000	−0.224860
$713$	32.0000	1.19841
$714$	−24.0000	−0.898177
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−16.0000	−0.597531
$718$	−16.0000	−0.597115
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	1.00000	0.0372161
$723$	22.0000	0.818189
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−5.00000	−0.185567
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	24.0000	0.889499
$729$	1.00000	0.0370370
$730$	0	0
$731$	48.0000	1.77534
$732$	2.00000	0.0739221
$733$	50.0000	1.84679	0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000	1.84679	0.923396	+	0.383849i	$0.125402\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−48.0000	−1.76810
$738$	10.0000	0.368105
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	−8.00000	−0.293689
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−10.0000	−0.366126
$747$	0	0
$748$	−24.0000	−0.877527
$749$	48.0000	1.75388
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−12.0000	−0.437595
$753$	−4.00000	−0.145768
$754$	36.0000	1.31104
$755$	0	0
$756$	−4.00000	−0.145479
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	4.00000	0.145287
$759$	−16.0000	−0.580763
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−16.0000	−0.579619
$763$	24.0000	0.868858
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−24.0000	−0.867155
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	2.00000	0.0719816
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	−14.0000	−0.502571
$777$	8.00000	0.286998
$778$	−26.0000	−0.932145
$779$	−10.0000	−0.358287
$780$	0	0
$781$	64.0000	2.29010
$782$	−24.0000	−0.858238
$783$	−6.00000	−0.214423
$784$	9.00000	0.321429
$785$	0	0
$786$	−12.0000	−0.428026
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−6.00000	−0.213741
$789$	20.0000	0.712019
$790$	0	0
$791$	40.0000	1.42224
$792$	−4.00000	−0.142134
$793$	−12.0000	−0.426132
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−16.0000	−0.567105
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	4.00000	0.141598
$799$	−72.0000	−2.54718
$800$	0	0
$801$	−6.00000	−0.212000
$802$	−22.0000	−0.776847
$803$	−56.0000	−1.97620
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−48.0000	−1.69073
$807$	−14.0000	−0.492823
$808$	6.00000	0.211079
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	24.0000	0.842235
$813$	8.00000	0.280572
$814$	8.00000	0.280400
$815$	0	0
$816$	−6.00000	−0.210042
$817$	−8.00000	−0.279885
$818$	−6.00000	−0.209785
$819$	24.0000	0.838628
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	2.00000	0.0697580
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−16.0000	−0.556711
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−4.00000	−0.139010
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	6.00000	0.208013
$833$	54.0000	1.87099
$834$	−12.0000	−0.415526
$835$	0	0
$836$	4.00000	0.138343
$837$	8.00000	0.276520
$838$	20.0000	0.690889
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−26.0000	−0.896019
$843$	22.0000	0.757720
$844$	−20.0000	−0.688428
$845$	0	0
$846$	−12.0000	−0.412568
$847$	20.0000	0.687208
$848$	−2.00000	−0.0686803
$849$	−16.0000	−0.549119
$850$	0	0
$851$	8.00000	0.274236
$852$	16.0000	0.548151
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	12.0000	0.410152
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	24.0000	0.819346
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	−40.0000	−1.36320
$862$	−32.0000	−1.08992
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−14.0000	−0.475739
$867$	−19.0000	−0.645274
$868$	−32.0000	−1.08615
$869$	−32.0000	−1.08553
$870$	0	0
$871$	72.0000	2.43963
$872$	6.00000	0.203186
$873$	−14.0000	−0.473828
$874$	4.00000	0.135302
$875$	0	0
$876$	−14.0000	−0.473016
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	32.0000	1.07995
$879$	2.00000	0.0674583
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	9.00000	0.303046
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	36.0000	1.21081
$885$	0	0
$886$	−8.00000	−0.268765
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	2.00000	0.0671156
$889$	64.0000	2.14649
$890$	0	0
$891$	−4.00000	−0.134005
$892$	16.0000	0.535720
$893$	12.0000	0.401565
$894$	10.0000	0.334450
$895$	0	0
$896$	4.00000	0.133631
$897$	24.0000	0.801337
$898$	10.0000	0.333704
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−12.0000	−0.399778
$902$	−40.0000	−1.33185
$903$	−32.0000	−1.06489
$904$	10.0000	0.332595
$905$	0	0
$906$	−8.00000	−0.265782
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	4.00000	0.132745
$909$	6.00000	0.199007
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	1.00000	0.0331133
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	6.00000	0.198246
$917$	48.0000	1.58510
$918$	−6.00000	−0.198030
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	−18.0000	−0.592798
$923$	−96.0000	−3.15988
$924$	16.0000	0.526361
$925$	0	0
$926$	−20.0000	−0.657241
$927$	−8.00000	−0.262754
$928$	6.00000	0.196960
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	−10.0000	−0.327561
$933$	8.00000	0.261908
$934$	8.00000	0.261768
$935$	0	0
$936$	6.00000	0.196116
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	48.0000	1.56726
$939$	−6.00000	−0.195803
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	22.0000	0.716799
$943$	−40.0000	−1.30258
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−32.0000	−1.04041
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−8.00000	−0.259828
$949$	84.0000	2.72676
$950$	0	0
$951$	−14.0000	−0.453981
$952$	24.0000	0.777844
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	16.0000	0.517477
$957$	24.0000	0.775810
$958$	24.0000	0.775405
$959$	−8.00000	−0.258333
$960$	0	0
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	−22.0000	−0.708572
$965$	0	0
$966$	16.0000	0.514792
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	5.00000	0.160706
$969$	6.00000	0.192748
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	−1.00000	−0.0320750
$973$	48.0000	1.53881
$974$	16.0000	0.512673
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	−16.0000	−0.511624
$979$	24.0000	0.767043
$980$	0	0
$981$	6.00000	0.191565
$982$	−12.0000	−0.382935
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	36.0000	1.14647
$987$	48.0000	1.52786
$988$	−6.00000	−0.190885
$989$	−32.0000	−1.01754
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−8.00000	−0.254000
$993$	−4.00000	−0.126936
$994$	−64.0000	−2.02996
$995$	0	0
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	−28.0000	−0.886325
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.a.v.1.1		1
3.2	odd	2		8550.2.a.r.1.1		1
5.2	odd	4		2850.2.d.a.799.2		2
5.3	odd	4		2850.2.d.a.799.1		2
5.4	even	2		570.2.a.e.1.1	✓	1
15.14	odd	2		1710.2.a.l.1.1		1
20.19	odd	2		4560.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.e.1.1	✓	1	5.4	even	2
1710.2.a.l.1.1		1	15.14	odd	2
2850.2.a.v.1.1		1	1.1	even	1	trivial
2850.2.d.a.799.1		2	5.3	odd	4
2850.2.d.a.799.2		2	5.2	odd	4
4560.2.a.q.1.1		1	20.19	odd	2
8550.2.a.r.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

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