LMFDB - Embedded newform 2299.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2299,2,Mod(1,2299)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2299, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2299.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2299.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:	$18.3576074247$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2299.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-2.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{12} +4.00000 q^{13} -6.00000 q^{15} +4.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} -6.00000 q^{20} -2.00000 q^{21} +4.00000 q^{25} +4.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +3.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} -8.00000 q^{39} +6.00000 q^{41} +1.00000 q^{43} +3.00000 q^{45} -3.00000 q^{47} -8.00000 q^{48} -6.00000 q^{49} -6.00000 q^{51} -8.00000 q^{52} +12.0000 q^{53} +2.00000 q^{57} -6.00000 q^{59} +12.0000 q^{60} +1.00000 q^{61} +1.00000 q^{63} -8.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +6.00000 q^{71} +7.00000 q^{73} -8.00000 q^{75} +2.00000 q^{76} -8.00000 q^{79} +12.0000 q^{80} -11.0000 q^{81} -12.0000 q^{83} +4.00000 q^{84} +9.00000 q^{85} +12.0000 q^{87} +12.0000 q^{89} +4.00000 q^{91} +8.00000 q^{93} -3.00000 q^{95} +8.00000 q^{97} +O(q^{100})$

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−2.00000	−1.00000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	4.00000	1.15470
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−6.00000	−1.54919
$16$	4.00000	1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−6.00000	−1.34164
$21$	−2.00000	−0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	4.00000	0.769800
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.00000	0.507093
$36$	−2.00000	−0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−8.00000	−1.15470
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	−8.00000	−1.10940
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	12.0000	1.54919
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	12.0000	1.48842
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	−8.00000	−0.923760
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	12.0000	1.34164
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	4.00000	0.436436
$85$	9.00000	0.976187
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	−8.00000	−0.800000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	−8.00000	−0.769800
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	12.0000	1.11417
$117$	4.00000	0.369800
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−12.0000	−1.08200
$124$	8.00000	0.718421
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	12.0000	1.03280
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	−6.00000	−0.507093
$141$	6.00000	0.505291
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	−18.0000	−1.49482
$146$	0	0
$147$	12.0000	0.989743
$148$	−4.00000	−0.328798
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	0	0
$155$	−12.0000	−0.963863
$156$	16.0000	1.28103
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	−24.0000	−1.90332
$160$	0	0
$161$	0	0
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−2.00000	−0.152499
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	−6.00000	−0.447214
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	6.00000	0.437595
$189$	4.00000	0.290957
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	16.0000	1.15470
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	−24.0000	−1.71868
$196$	12.0000	0.857143
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	−6.00000	−0.421117
$204$	12.0000	0.840168
$205$	18.0000	1.25717
$206$	0	0
$207$	0	0
$208$	16.0000	1.10940
$209$	0	0
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	−24.0000	−1.64833
$213$	−12.0000	−0.822226
$214$	0	0
$215$	3.00000	0.204598
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	12.0000	0.781133
$237$	16.0000	1.03931
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	−24.0000	−1.54919
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	−2.00000	−0.128037
$245$	−18.0000	−1.14998
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	24.0000	1.52094
$250$	0	0
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	0	0
$255$	−18.0000	−1.12720
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	−24.0000	−1.48842
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	0	0
$267$	−24.0000	−1.46878
$268$	8.00000	0.488678
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	12.0000	0.727607
$273$	−8.00000	−0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	−12.0000	−0.712069
$285$	6.00000	0.355409
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−16.0000	−0.937937
$292$	−14.0000	−0.819288
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	16.0000	0.923760
$301$	1.00000	0.0576390
$302$	0	0
$303$	12.0000	0.689382
$304$	−4.00000	−0.229416
$305$	3.00000	0.171780
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−28.0000	−1.59286
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	16.0000	0.900070
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	−24.0000	−1.34164
$321$	−36.0000	−2.00932
$322$	0	0
$323$	−3.00000	−0.166924
$324$	22.0000	1.22222
$325$	16.0000	0.887520
$326$	0	0
$327$	−32.0000	−1.76960
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	24.0000	1.31717
$333$	2.00000	0.109599
$334$	0	0
$335$	−12.0000	−0.655630
$336$	−8.00000	−0.436436
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	−18.0000	−0.976187
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.0000	−1.12734	−0.563670	−	0.826000i	$-0.690611\pi$
$347$	−21.0000	−1.12734	−0.563670	+	0.826000i	$0.690611\pi$
$348$	−24.0000	−1.28654
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	−24.0000	−1.27200
$357$	−6.00000	−0.317554
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	−8.00000	−0.419314
$365$	21.0000	1.09919
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	12.0000	0.623009
$372$	−16.0000	−0.829561
$373$	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
$375$	6.00000	0.309839
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	6.00000	0.307794
$381$	4.00000	0.204926
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.00000	0.0508329
$388$	−16.0000	−0.812277
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−30.0000	−1.51330
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	16.0000	0.800000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	12.0000	0.597022
$405$	−33.0000	−1.63978
$406$	0	0
$407$	0	0
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−28.0000	−1.37946
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−36.0000	−1.76717
$416$	0	0
$417$	−26.0000	−1.27323
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	12.0000	0.585540
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	1.00000	0.0483934
$428$	−36.0000	−1.74013
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	16.0000	0.769800
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	36.0000	1.72607
$436$	−32.0000	−1.53252
$437$	0	0
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−3.00000	−0.142534	−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000	−0.142534	−0.0712672	+	0.997457i	$0.522704\pi$
$444$	8.00000	0.379663
$445$	36.0000	1.70656
$446$	0	0
$447$	42.0000	1.98653
$448$	−8.00000	−0.377964
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	−12.0000	−0.564433
$453$	−20.0000	−0.939682
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	37.0000	1.73079	0.865393	−	0.501093i	$-0.167069\pi$
$457$	37.0000	1.73079	0.865393	+	0.501093i	$0.167069\pi$
$458$	0	0
$459$	12.0000	0.560112
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	−24.0000	−1.11417
$465$	24.0000	1.11297
$466$	0	0
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	−8.00000	−0.369800
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	−6.00000	−0.275010
$477$	12.0000	0.549442
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.0000	1.08978
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−40.0000	−1.80886
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	24.0000	1.08200
$493$	−18.0000	−0.810679
$494$	0	0
$495$	0	0
$496$	−16.0000	−0.718421
$497$	6.00000	0.269137
$498$	0	0
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	6.00000	0.268328
$501$	−36.0000	−1.60836
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	−6.00000	−0.266469
$508$	4.00000	0.177471
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	42.0000	1.85074
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	−36.0000	−1.58022
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	−30.0000	−1.31056
$525$	−8.00000	−0.349149
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	2.00000	0.0867110
$533$	24.0000	1.03956
$534$	0	0
$535$	54.0000	2.33462
$536$	0	0
$537$	36.0000	1.55351
$538$	0	0
$539$	0	0
$540$	−24.0000	−1.03280
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	0	0
$543$	−4.00000	−0.171656
$544$	0	0
$545$	48.0000	2.05609
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	6.00000	0.256307
$549$	1.00000	0.0426790
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	−12.0000	−0.509372
$556$	−26.0000	−1.10265
$557$	−21.0000	−0.889799	−0.444899	−	0.895581i	$-0.646761\pi$
$557$	−21.0000	−0.889799	−0.444899	+	0.895581i	$0.646761\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	12.0000	0.507093
$561$	0	0
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	−12.0000	−0.505291
$565$	18.0000	0.757266
$566$	0	0
$567$	−11.0000	−0.461957
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	0	0
$579$	−8.00000	−0.332469
$580$	36.0000	1.49482
$581$	−12.0000	−0.497844
$582$	0	0
$583$	0	0
$584$	0	0
$585$	12.0000	0.496139
$586$	0	0
$587$	45.0000	1.85735	0.928674	−	0.370896i	$-0.120949\pi$
$587$	45.0000	1.85735	0.928674	+	0.370896i	$0.120949\pi$
$588$	−24.0000	−0.989743
$589$	4.00000	0.164817
$590$	0	0
$591$	36.0000	1.48084
$592$	8.00000	0.328798
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	42.0000	1.72039
$597$	−22.0000	−0.900400
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−20.0000	−0.813788
$605$	0	0
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−12.0000	−0.485468
$612$	−6.00000	−0.242536
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	0	0
$615$	−36.0000	−1.45166
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	24.0000	0.963863
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	−32.0000	−1.28103
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	−28.0000	−1.11732
$629$	6.00000	0.239236
$630$	0	0
$631$	11.0000	0.437903	0.218952	−	0.975736i	$-0.429736\pi$
$631$	11.0000	0.437903	0.218952	+	0.975736i	$0.429736\pi$
$632$	0	0
$633$	28.0000	1.11290
$634$	0	0
$635$	−6.00000	−0.238103
$636$	48.0000	1.90332
$637$	−24.0000	−0.950915
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−13.0000	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	8.00000	0.313545
$652$	−40.0000	−1.56652
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	0	0
$655$	45.0000	1.75830
$656$	24.0000	0.937043
$657$	7.00000	0.273096
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	0	0
$668$	−36.0000	−1.39288
$669$	20.0000	0.773245
$670$	0	0
$671$	0	0
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	16.0000	0.615840
$676$	−6.00000	−0.230769
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	2.00000	0.0764719
$685$	−9.00000	−0.343872
$686$	0	0
$687$	−10.0000	−0.381524
$688$	4.00000	0.152499
$689$	48.0000	1.82865
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−36.0000	−1.36851
$693$	0	0
$694$	0	0
$695$	39.0000	1.47935
$696$	0	0
$697$	18.0000	0.681799
$698$	0	0
$699$	−42.0000	−1.58859
$700$	−8.00000	−0.302372
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	18.0000	0.677919
$706$	0	0
$707$	−6.00000	−0.225653
$708$	−24.0000	−0.901975
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	36.0000	1.34538
$717$	30.0000	1.12037
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	12.0000	0.447214
$721$	14.0000	0.521387
$722$	0	0
$723$	−20.0000	−0.743808
$724$	−4.00000	−0.148659
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	3.00000	0.110959
$732$	4.00000	0.147844
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	36.0000	1.32788
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	−12.0000	−0.441129
$741$	8.00000	0.293887
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−63.0000	−2.30814
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−12.0000	−0.437595
$753$	−42.0000	−1.53057
$754$	0	0
$755$	30.0000	1.09181
$756$	−8.00000	−0.290957
$757$	−25.0000	−0.908640	−0.454320	−	0.890838i	$-0.650118\pi$
$757$	−25.0000	−0.908640	−0.454320	+	0.890838i	$0.650118\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	−6.00000	−0.217072
$765$	9.00000	0.325396
$766$	0	0
$767$	−24.0000	−0.866590
$768$	−32.0000	−1.15470
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	0	0
$772$	−8.00000	−0.287926
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	−6.00000	−0.214972
$780$	48.0000	1.71868
$781$	0	0
$782$	0	0
$783$	−24.0000	−0.857690
$784$	−24.0000	−0.857143
$785$	42.0000	1.49904
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	36.0000	1.28245
$789$	18.0000	0.640817
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	−72.0000	−2.55358
$796$	−22.0000	−0.779769
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	0	0
$804$	−16.0000	−0.564276
$805$	0	0
$806$	0	0
$807$	−48.0000	−1.68968
$808$	0	0
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	12.0000	0.421117
$813$	−32.0000	−1.12229
$814$	0	0
$815$	60.0000	2.10171
$816$	−24.0000	−0.840168
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	4.00000	0.139771
$820$	−36.0000	−1.25717
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	0	0
$823$	−49.0000	−1.70803	−0.854016	−	0.520246i	$-0.825840\pi$
$823$	−49.0000	−1.70803	−0.854016	+	0.520246i	$0.825840\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−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
$831$	−38.0000	−1.31821
$832$	−32.0000	−1.10940
$833$	−18.0000	−0.623663
$834$	0	0
$835$	54.0000	1.86875
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	12.0000	0.413302
$844$	28.0000	0.963800
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	48.0000	1.64833
$849$	−26.0000	−0.892318
$850$	0	0
$851$	0	0
$852$	24.0000	0.822226
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	0	0
$857$	−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
$859$	−49.0000	−1.67186	−0.835929	−	0.548837i	$-0.815071\pi$
$859$	−49.0000	−1.67186	−0.835929	+	0.548837i	$0.815071\pi$
$860$	−6.00000	−0.204598
$861$	−12.0000	−0.408959
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	54.0000	1.83606
$866$	0	0
$867$	16.0000	0.543388
$868$	8.00000	0.271538
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	−3.00000	−0.101419
$876$	28.0000	0.946032
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	47.0000	1.58168	0.790838	−	0.612026i	$-0.209645\pi$
$883$	47.0000	1.58168	0.790838	+	0.612026i	$0.209645\pi$
$884$	−24.0000	−0.807207
$885$	36.0000	1.21013
$886$	0	0
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	0	0
$892$	20.0000	0.669650
$893$	3.00000	0.100391
$894$	0	0
$895$	−54.0000	−1.80502
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000	0.800445
$900$	−8.00000	−0.266667
$901$	36.0000	1.19933
$902$	0	0
$903$	−2.00000	−0.0665558
$904$	0	0
$905$	6.00000	0.199447
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	24.0000	0.796468
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	8.00000	0.264906
$913$	0	0
$914$	0	0
$915$	−6.00000	−0.198354
$916$	−10.0000	−0.330409
$917$	15.0000	0.495344
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	40.0000	1.31804
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	14.0000	0.459820
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−42.0000	−1.37576
$933$	6.00000	0.196431
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	18.0000	0.587095
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	0	0
$944$	−24.0000	−0.781133
$945$	12.0000	0.390360
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−32.0000	−1.03931
$949$	28.0000	0.908918
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	0	0
$955$	9.00000	0.291233
$956$	30.0000	0.970269
$957$	0	0
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	48.0000	1.54919
$961$	−15.0000	−0.483871
$962$	0	0
$963$	18.0000	0.580042
$964$	−20.0000	−0.644157
$965$	12.0000	0.386294
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	−20.0000	−0.641500
$973$	13.0000	0.416761
$974$	0	0
$975$	−32.0000	−1.02482
$976$	4.00000	0.128037
$977$	24.0000	0.767828	0.383914	−	0.923369i	$-0.374576\pi$
$977$	24.0000	0.767828	0.383914	+	0.923369i	$0.374576\pi$
$978$	0	0
$979$	0	0
$980$	36.0000	1.14998
$981$	16.0000	0.510841
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	−54.0000	−1.72058
$986$	0	0
$987$	6.00000	0.190982
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	0	0
$993$	56.0000	1.77711
$994$	0	0
$995$	33.0000	1.04617
$996$	−48.0000	−1.52094
$997$	−17.0000	−0.538395	−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000	−0.538395	−0.269198	+	0.963085i	$0.586759\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2299.2.a.b.1.1		1
11.10	odd	2		19.2.a.a.1.1	✓	1
33.32	even	2		171.2.a.b.1.1		1
44.43	even	2		304.2.a.f.1.1		1
55.32	even	4		475.2.b.a.324.2		2
55.43	even	4		475.2.b.a.324.1		2
55.54	odd	2		475.2.a.b.1.1		1
77.10	even	6		931.2.f.b.324.1		2
77.32	odd	6		931.2.f.c.324.1		2
77.54	even	6		931.2.f.b.704.1		2
77.65	odd	6		931.2.f.c.704.1		2
77.76	even	2		931.2.a.a.1.1		1
88.21	odd	2		1216.2.a.o.1.1		1
88.43	even	2		1216.2.a.b.1.1		1
132.131	odd	2		2736.2.a.c.1.1		1
143.142	odd	2		3211.2.a.a.1.1		1
165.164	even	2		4275.2.a.i.1.1		1
187.186	odd	2		5491.2.a.b.1.1		1
209.10	even	18		361.2.e.e.62.1		6
209.21	even	18		361.2.e.e.99.1		6
209.32	even	18		361.2.e.e.245.1		6
209.43	odd	18		361.2.e.d.234.1		6
209.54	odd	18		361.2.e.d.28.1		6
209.65	even	6		361.2.c.a.292.1		2
209.87	odd	6		361.2.c.c.292.1		2
209.98	even	18		361.2.e.e.28.1		6
209.109	even	18		361.2.e.e.234.1		6
209.120	odd	18		361.2.e.d.245.1		6
209.131	odd	18		361.2.e.d.99.1		6
209.142	odd	18		361.2.e.d.62.1		6
209.164	even	6		361.2.c.a.68.1		2
209.175	odd	18		361.2.e.d.54.1		6
209.186	even	18		361.2.e.e.54.1		6
209.197	odd	6		361.2.c.c.68.1		2
209.208	even	2		361.2.a.b.1.1		1
220.219	even	2		7600.2.a.c.1.1		1
231.230	odd	2		8379.2.a.j.1.1		1
627.626	odd	2		3249.2.a.d.1.1		1
836.835	odd	2		5776.2.a.c.1.1		1
1045.1044	even	2		9025.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.a.a.1.1	✓	1	11.10	odd	2
171.2.a.b.1.1		1	33.32	even	2
304.2.a.f.1.1		1	44.43	even	2
361.2.a.b.1.1		1	209.208	even	2
361.2.c.a.68.1		2	209.164	even	6
361.2.c.a.292.1		2	209.65	even	6
361.2.c.c.68.1		2	209.197	odd	6
361.2.c.c.292.1		2	209.87	odd	6
361.2.e.d.28.1		6	209.54	odd	18
361.2.e.d.54.1		6	209.175	odd	18
361.2.e.d.62.1		6	209.142	odd	18
361.2.e.d.99.1		6	209.131	odd	18
361.2.e.d.234.1		6	209.43	odd	18
361.2.e.d.245.1		6	209.120	odd	18
361.2.e.e.28.1		6	209.98	even	18
361.2.e.e.54.1		6	209.186	even	18
361.2.e.e.62.1		6	209.10	even	18
361.2.e.e.99.1		6	209.21	even	18
361.2.e.e.234.1		6	209.109	even	18
361.2.e.e.245.1		6	209.32	even	18
475.2.a.b.1.1		1	55.54	odd	2
475.2.b.a.324.1		2	55.43	even	4
475.2.b.a.324.2		2	55.32	even	4
931.2.a.a.1.1		1	77.76	even	2
931.2.f.b.324.1		2	77.10	even	6
931.2.f.b.704.1		2	77.54	even	6
931.2.f.c.324.1		2	77.32	odd	6
931.2.f.c.704.1		2	77.65	odd	6
1216.2.a.b.1.1		1	88.43	even	2
1216.2.a.o.1.1		1	88.21	odd	2
2299.2.a.b.1.1		1	1.1	even	1	trivial
2736.2.a.c.1.1		1	132.131	odd	2
3211.2.a.a.1.1		1	143.142	odd	2
3249.2.a.d.1.1		1	627.626	odd	2
4275.2.a.i.1.1		1	165.164	even	2
5491.2.a.b.1.1		1	187.186	odd	2
5776.2.a.c.1.1		1	836.835	odd	2
7600.2.a.c.1.1		1	220.219	even	2
8379.2.a.j.1.1		1	231.230	odd	2
9025.2.a.d.1.1		1	1045.1044	even	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:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2299.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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Database

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