LMFDB - Embedded newform 2175.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(1,2175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,-1,-1,0,1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{21} -2.00000 q^{23} -3.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +1.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{39} +10.0000 q^{41} +2.00000 q^{42} +2.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} +4.00000 q^{52} +12.0000 q^{53} +1.00000 q^{54} +6.00000 q^{56} -1.00000 q^{58} +4.00000 q^{59} +2.00000 q^{61} -4.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{67} +2.00000 q^{68} +2.00000 q^{69} -8.00000 q^{71} +3.00000 q^{72} -14.0000 q^{73} +2.00000 q^{74} -4.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} +6.00000 q^{83} +2.00000 q^{84} -1.00000 q^{87} +10.0000 q^{89} -8.00000 q^{91} +2.00000 q^{92} -4.00000 q^{93} +12.0000 q^{94} +5.00000 q^{96} -10.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	3.00000	1.06066
$9$	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	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	2.00000	0.342997
$35$	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	$0.552568\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	2.00000	0.308607
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	2.00000	0.294884
$47$	−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
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	4.00000	0.554700
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	−1.00000	−0.131306
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−4.00000	−0.508001
$63$	2.00000	0.251976
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	2.00000	0.242536
$69$	2.00000	0.240772
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	3.00000	0.353553
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	0	0
$78$	−4.00000	−0.452911
$79$	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
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	2.00000	0.208514
$93$	−4.00000	−0.414781
$94$	12.0000	1.23771
$95$	0	0
$96$	5.00000	0.510310
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−2.00000	−0.198030
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	−12.0000	−1.17670
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	1.00000	0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	−2.00000	−0.188982
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	−4.00000	−0.369800
$118$	−4.00000	−0.368230
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−2.00000	−0.181071
$123$	−10.0000	−0.901670
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	−6.00000	−0.514496
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−2.00000	−0.170251
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	8.00000	0.671345
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	3.00000	0.247436
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−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
$159$	−12.0000	−0.951662
$160$	0	0
$161$	−4.00000	−0.315244
$162$	−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	$0.449930\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	−6.00000	−0.462910
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	0	0
$177$	−4.00000	−0.300658
$178$	−10.0000	−0.749532
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	8.00000	0.592999
$183$	−2.00000	−0.147844
$184$	−6.00000	−0.442326
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	12.0000	0.875190
$189$	−2.00000	−0.145479
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−7.00000	−0.505181
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	3.00000	0.214286
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	−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
$201$	2.00000	0.141069
$202$	−2.00000	−0.140720
$203$	2.00000	0.140372
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−2.00000	−0.139347
$207$	−2.00000	−0.139010
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−12.0000	−0.824163
$213$	8.00000	0.548151
$214$	6.00000	0.410152
$215$	0	0
$216$	−3.00000	−0.204124
$217$	8.00000	0.543075
$218$	14.0000	0.948200
$219$	14.0000	0.946032
$220$	0	0
$221$	8.00000	0.538138
$222$	−2.00000	−0.134231
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	14.0000	0.931266
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	−20.0000	−1.31024	−0.655122	−	0.755523i	$-0.727383\pi$
$233$	−20.0000	−1.31024	−0.655122	+	0.755523i	$0.727383\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−4.00000	−0.260378
$237$	−8.00000	−0.519656
$238$	4.00000	0.259281
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	30.0000	1.93247	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	11.0000	0.707107
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	10.0000	0.637577
$247$	0	0
$248$	12.0000	0.762001
$249$	−6.00000	−0.380235
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	12.0000	0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	1.00000	0.0618984
$262$	12.0000	0.741362
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−10.0000	−0.611990
$268$	2.00000	0.122169
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	2.00000	0.121268
$273$	8.00000	0.484182
$274$	−14.0000	−0.845771
$275$	0	0
$276$	−2.00000	−0.120386
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	4.00000	0.239904
$279$	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	$0.557275\pi$
$282$	−12.0000	−0.714590
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	20.0000	1.18056
$288$	−5.00000	−0.294628
$289$	−13.0000	−0.764706
$290$	0	0
$291$	10.0000	0.586210
$292$	14.0000	0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	6.00000	0.347571
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	24.0000	1.38104
$303$	−2.00000	−0.114897
$304$	0	0
$305$	0	0
$306$	2.00000	0.114332
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	−2.00000	−0.113776
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	12.0000	0.679366
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	−8.00000	−0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	12.0000	0.672927
$319$	0	0
$320$	0	0
$321$	6.00000	0.334887
$322$	4.00000	0.222911
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	14.0000	0.774202
$328$	30.0000	1.65647
$329$	−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	$0.429440\pi$
$332$	−6.00000	−0.329293
$333$	−2.00000	−0.109599
$334$	2.00000	0.109435
$335$	0	0
$336$	2.00000	0.109109
$337$	−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
$339$	14.0000	0.760376
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	1.00000	0.0536056
$349$	−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
$351$	4.00000	0.213504
$352$	0	0
$353$	−28.0000	−1.49029	−0.745145	−	0.666903i	$-0.767620\pi$
$353$	−28.0000	−1.49029	−0.745145	+	0.666903i	$0.767620\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	−10.0000	−0.529999
$357$	4.00000	0.211702
$358$	20.0000	1.05703
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−2.00000	−0.105118
$363$	11.0000	0.577350
$364$	8.00000	0.419314
$365$	0	0
$366$	2.00000	0.104542
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	2.00000	0.104257
$369$	10.0000	0.520579
$370$	0	0
$371$	24.0000	1.24602
$372$	4.00000	0.207390
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	−36.0000	−1.85656
$377$	−4.00000	−0.206010
$378$	2.00000	0.102869
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	−12.0000	−0.613973
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	6.00000	0.305392
$387$	0	0
$388$	10.0000	0.507673
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	−9.00000	−0.454569
$393$	12.0000	0.605320
$394$	−8.00000	−0.403034
$395$	0	0
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	0	0
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	−2.00000	−0.0997509
$403$	−16.0000	−0.797017
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	6.00000	0.297044
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	−2.00000	−0.0985329
$413$	8.00000	0.393654
$414$	2.00000	0.0982946
$415$	0	0
$416$	20.0000	0.980581
$417$	4.00000	0.195881
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	12.0000	0.584151
$423$	−12.0000	−0.583460
$424$	36.0000	1.74831
$425$	0	0
$426$	−8.00000	−0.387601
$427$	4.00000	0.193574
$428$	6.00000	0.290021
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	−14.0000	−0.668946
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−8.00000	−0.380521
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−18.0000	−0.852325
$447$	6.00000	0.283790
$448$	14.0000	0.661438
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	0	0
$452$	14.0000	0.658505
$453$	24.0000	1.12762
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	22.0000	1.02799
$459$	2.00000	0.0933520
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	0	0
$463$	30.0000	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	20.0000	0.926482
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	4.00000	0.184900
$469$	−4.00000	−0.184703
$470$	0	0
$471$	22.0000	1.01371
$472$	12.0000	0.552345
$473$	0	0
$474$	8.00000	0.367452
$475$	0	0
$476$	4.00000	0.183340
$477$	12.0000	0.549442
$478$	−24.0000	−1.09773
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	−30.0000	−1.36646
$483$	4.00000	0.182006
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	−14.0000	−0.634401	−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000	−0.634401	−0.317200	+	0.948359i	$0.602743\pi$
$488$	6.00000	0.271607
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	10.0000	0.450835
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−16.0000	−0.717698
$498$	6.00000	0.268866
$499$	−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
$501$	2.00000	0.0893534
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	0	0
$507$	−3.00000	−0.133235
$508$	12.0000	0.532414
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	11.0000	0.486136
$513$	0	0
$514$	8.00000	0.352865
$515$	0	0
$516$	0	0
$517$	0	0
$518$	4.00000	0.175750
$519$	0	0
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	−1.00000	−0.0437688
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	16.0000	0.697633
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−40.0000	−1.73259
$534$	10.0000	0.432742
$535$	0	0
$536$	−6.00000	−0.259161
$537$	20.0000	0.863064
$538$	−26.0000	−1.12094
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	24.0000	1.03089
$543$	−2.00000	−0.0858282
$544$	10.0000	0.428746
$545$	0	0
$546$	−8.00000	−0.342368
$547$	46.0000	1.96682	0.983409	−	0.181402i	$-0.0580636\pi$
$547$	46.0000	1.96682	0.983409	+	0.181402i	$0.0580636\pi$
$548$	−14.0000	−0.598050
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	6.00000	0.255377
$553$	16.0000	0.680389
$554$	−8.00000	−0.339887
$555$	0	0
$556$	4.00000	0.169638
$557$	20.0000	0.847427	0.423714	−	0.905796i	$-0.360726\pi$
$557$	20.0000	0.847427	0.423714	+	0.905796i	$0.360726\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−10.0000	−0.420331
$567$	2.00000	0.0839921
$568$	−24.0000	−1.00702
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	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
$573$	−12.0000	−0.501307
$574$	−20.0000	−0.834784
$575$	0	0
$576$	7.00000	0.291667
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	13.0000	0.540729
$579$	6.00000	0.249351
$580$	0	0
$581$	12.0000	0.497844
$582$	−10.0000	−0.414513
$583$	0	0
$584$	−42.0000	−1.73797
$585$	0	0
$586$	6.00000	0.247858
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	−3.00000	−0.123718
$589$	0	0
$590$	0	0
$591$	−8.00000	−0.329076
$592$	2.00000	0.0821995
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	16.0000	0.654836
$598$	−8.00000	−0.327144
$599$	−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
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	24.0000	0.976546
$605$	0	0
$606$	2.00000	0.0812444
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	48.0000	1.94187
$612$	2.00000	0.0808452
$613$	−40.0000	−1.61558	−0.807792	−	0.589467i	$-0.799338\pi$
$613$	−40.0000	−1.61558	−0.807792	+	0.589467i	$0.799338\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	2.00000	0.0804518
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	−8.00000	−0.320771
$623$	20.0000	0.801283
$624$	−4.00000	−0.160128
$625$	0	0
$626$	28.0000	1.11911
$627$	0	0
$628$	22.0000	0.877896
$629$	4.00000	0.159490
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	24.0000	0.954669
$633$	12.0000	0.476957
$634$	−6.00000	−0.238290
$635$	0	0
$636$	12.0000	0.475831
$637$	12.0000	0.475457
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	50.0000	1.97488	0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000	1.97488	0.987441	+	0.157991i	$0.0505015\pi$
$642$	−6.00000	−0.236801
$643$	10.0000	0.394362	0.197181	−	0.980367i	$-0.436821\pi$
$643$	10.0000	0.394362	0.197181	+	0.980367i	$0.436821\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	3.00000	0.117851
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	−4.00000	−0.156652
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	−10.0000	−0.390434
$657$	−14.0000	−0.546192
$658$	24.0000	0.935617
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−8.00000	−0.310929
$663$	−8.00000	−0.310694
$664$	18.0000	0.698535
$665$	0	0
$666$	2.00000	0.0774984
$667$	−2.00000	−0.0774403
$668$	2.00000	0.0773823
$669$	−18.0000	−0.695920
$670$	0	0
$671$	0	0
$672$	10.0000	0.385758
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−3.00000	−0.115385
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	−14.0000	−0.537667
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	22.0000	0.839352
$688$	0	0
$689$	−48.0000	−1.82865
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	0	0
$693$	0	0
$694$	−6.00000	−0.227757
$695$	0	0
$696$	−3.00000	−0.113715
$697$	−20.0000	−0.757554
$698$	2.00000	0.0757011
$699$	20.0000	0.756469
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−4.00000	−0.150970
$703$	0	0
$704$	0	0
$705$	0	0
$706$	28.0000	1.05379
$707$	4.00000	0.150435
$708$	4.00000	0.150329
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	30.0000	1.12430
$713$	−8.00000	−0.299602
$714$	−4.00000	−0.149696
$715$	0	0
$716$	20.0000	0.747435
$717$	−24.0000	−0.896296
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	19.0000	0.707107
$723$	−30.0000	−1.11571
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	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	$0.125402\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	10.0000	0.368605
$737$	0	0
$738$	−10.0000	−0.368105
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	−24.0000	−0.881068
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	−12.0000	−0.439941
$745$	0	0
$746$	4.00000	0.146450
$747$	6.00000	0.219529
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	2.00000	0.0727393
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	−12.0000	−0.434714
$763$	−28.0000	−1.01367
$764$	−12.0000	−0.434145
$765$	0	0
$766$	18.0000	0.650366
$767$	−16.0000	−0.577727
$768$	17.0000	0.613435
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	6.00000	0.215945
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	0	0
$776$	−30.0000	−1.07694
$777$	4.00000	0.143499
$778$	−2.00000	−0.0717035
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−4.00000	−0.143040
$783$	−1.00000	−0.0357371
$784$	3.00000	0.107143
$785$	0	0
$786$	−12.0000	−0.428026
$787$	34.0000	1.21197	0.605985	−	0.795476i	$-0.292779\pi$
$787$	34.0000	1.21197	0.605985	+	0.795476i	$0.292779\pi$
$788$	−8.00000	−0.284988
$789$	16.0000	0.569615
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	−8.00000	−0.284088
$794$	20.0000	0.709773
$795$	0	0
$796$	16.0000	0.567105
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	10.0000	0.353333
$802$	34.0000	1.20058
$803$	0	0
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	16.0000	0.563576
$807$	−26.0000	−0.915243
$808$	6.00000	0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	−2.00000	−0.0701862
$813$	24.0000	0.841717
$814$	0	0
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	22.0000	0.769212
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	14.0000	0.488306
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	2.00000	0.0695048
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	−28.0000	−0.970725
$833$	6.00000	0.207888
$834$	−4.00000	−0.138509
$835$	0	0
$836$	0	0
$837$	−4.00000	−0.138260
$838$	−12.0000	−0.414533
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	22.0000	0.758170
$843$	6.00000	0.206651
$844$	12.0000	0.413057
$845$	0	0
$846$	12.0000	0.412568
$847$	−22.0000	−0.755929
$848$	−12.0000	−0.412082
$849$	−10.0000	−0.343199
$850$	0	0
$851$	4.00000	0.137118
$852$	−8.00000	−0.274075
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−32.0000	−1.09310	−0.546550	−	0.837427i	$-0.684059\pi$
$857$	−32.0000	−1.09310	−0.546550	+	0.837427i	$0.684059\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	−20.0000	−0.681598
$862$	0	0
$863$	−50.0000	−1.70202	−0.851010	−	0.525150i	$-0.824009\pi$
$863$	−50.0000	−1.70202	−0.851010	+	0.525150i	$0.824009\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−14.0000	−0.475739
$867$	13.0000	0.441503
$868$	−8.00000	−0.271538
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	−42.0000	−1.42230
$873$	−10.0000	−0.338449
$874$	0	0
$875$	0	0
$876$	−14.0000	−0.473016
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	16.0000	0.539974
$879$	6.00000	0.202375
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	3.00000	0.101015
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−44.0000	−1.47738	−0.738688	−	0.674048i	$-0.764554\pi$
$887$	−44.0000	−1.47738	−0.738688	+	0.674048i	$0.764554\pi$
$888$	6.00000	0.201347
$889$	−24.0000	−0.804934
$890$	0	0
$891$	0	0
$892$	−18.0000	−0.602685
$893$	0	0
$894$	−6.00000	−0.200670
$895$	0	0
$896$	6.00000	0.200446
$897$	−8.00000	−0.267112
$898$	34.0000	1.13459
$899$	4.00000	0.133407
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	0	0
$904$	−42.0000	−1.39690
$905$	0	0
$906$	−24.0000	−0.797347
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−6.00000	−0.199117
$909$	2.00000	0.0663358
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	0	0
$914$	−8.00000	−0.264616
$915$	0	0
$916$	22.0000	0.726900
$917$	−24.0000	−0.792550
$918$	−2.00000	−0.0660098
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	−26.0000	−0.856264
$923$	32.0000	1.05329
$924$	0	0
$925$	0	0
$926$	−30.0000	−0.985861
$927$	2.00000	0.0656886
$928$	−5.00000	−0.164133
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	0	0
$932$	20.0000	0.655122
$933$	−8.00000	−0.261908
$934$	0	0
$935$	0	0
$936$	−12.0000	−0.392232
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	4.00000	0.130605
$939$	28.0000	0.913745
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	−22.0000	−0.716799
$943$	−20.0000	−0.651290
$944$	−4.00000	−0.130189
$945$	0	0
$946$	0	0
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	8.00000	0.259828
$949$	56.0000	1.81784
$950$	0	0
$951$	−6.00000	−0.194563
$952$	−12.0000	−0.388922
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−36.0000	−1.16311
$959$	28.0000	0.904167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−8.00000	−0.257930
$963$	−6.00000	−0.193347
$964$	−30.0000	−0.966235
$965$	0	0
$966$	−4.00000	−0.128698
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	1.00000	0.0320750
$973$	−8.00000	−0.256468
$974$	14.0000	0.448589
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	32.0000	1.02377	0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000	1.02377	0.511885	+	0.859054i	$0.328947\pi$
$978$	4.00000	0.127906
$979$	0	0
$980$	0	0
$981$	−14.0000	−0.446986
$982$	4.00000	0.127645
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	−30.0000	−0.956365
$985$	0	0
$986$	2.00000	0.0636930
$987$	24.0000	0.763928
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	−20.0000	−0.635001
$993$	−8.00000	−0.253872
$994$	16.0000	0.507489
$995$	0	0
$996$	6.00000	0.190117
$997$	−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
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.c.1.1		1
3.2	odd	2		6525.2.a.k.1.1		1
5.2	odd	4		435.2.c.b.349.1	✓	2
5.3	odd	4		435.2.c.b.349.2	yes	2
5.4	even	2		2175.2.a.i.1.1		1
15.2	even	4		1305.2.c.b.784.2		2
15.8	even	4		1305.2.c.b.784.1		2
15.14	odd	2		6525.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.b.349.1	✓	2	5.2	odd	4
435.2.c.b.349.2	yes	2	5.3	odd	4
1305.2.c.b.784.1		2	15.8	even	4
1305.2.c.b.784.2		2	15.2	even	4
2175.2.a.c.1.1		1	1.1	even	1	trivial
2175.2.a.i.1.1		1	5.4	even	2
6525.2.a.c.1.1		1	15.14	odd	2
6525.2.a.k.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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