LMFDB - Embedded newform 2475.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.792287$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.792287 q^{2} -1.37228 q^{4} +3.46410 q^{7} +2.67181 q^{8} +1.00000 q^{11} -2.74456 q^{14} +0.627719 q^{16} +5.04868 q^{17} +4.00000 q^{19} -0.792287 q^{22} +2.52434 q^{23} -4.75372 q^{28} +2.74456 q^{29} -2.37228 q^{31} -5.84096 q^{32} -4.00000 q^{34} -11.0371 q^{37} -3.16915 q^{38} +2.74456 q^{41} +3.46410 q^{43} -1.37228 q^{44} -2.00000 q^{46} -6.63325 q^{47} +5.00000 q^{49} +3.16915 q^{53} +9.25544 q^{56} -2.17448 q^{58} +1.62772 q^{59} +10.7446 q^{61} +1.87953 q^{62} +3.37228 q^{64} -0.644810 q^{67} -6.92820 q^{68} -7.11684 q^{71} -6.92820 q^{73} +8.74456 q^{74} -5.48913 q^{76} +3.46410 q^{77} +12.7446 q^{79} -2.17448 q^{82} +6.63325 q^{83} -2.74456 q^{86} +2.67181 q^{88} +4.37228 q^{89} -3.46410 q^{92} +5.25544 q^{94} -4.10891 q^{97} -3.96143 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 4 q^{11} + 12 q^{14} + 14 q^{16} + 16 q^{19} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 12 q^{41} + 6 q^{44} - 8 q^{46} + 20 q^{49} + 60 q^{56} + 18 q^{59} + 20 q^{61} + 2 q^{64} + 6 q^{71} + 12 q^{74}+ \cdots + 44 q^{94}+O(q^{100}) $ 4 * q + 6 * q^4 + 4 * q^11 + 12 * q^14 + 14 * q^16 + 16 * q^19 - 12 * q^29 + 2 * q^31 - 16 * q^34 - 12 * q^41 + 6 * q^44 - 8 * q^46 + 20 * q^49 + 60 * q^56 + 18 * q^59 + 20 * q^61 + 2 * q^64 + 6 * q^71 + 12 * q^74 + 24 * q^76 + 28 * q^79 + 12 * q^86 + 6 * q^89 + 44 * q^94

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 $

$2$	−0.792287	−0.560232	−0.280116	−	0.959966i	$-0.590373\pi$
$2$	−0.792287	−0.560232	−0.280116	+	0.959966i	$0.590373\pi$
$3$	0	0
$3$	0	0
$4$	−1.37228	−0.686141
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	2.67181	0.944629
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−2.74456	−0.733515
$15$	0	0
$16$	0.627719	0.156930
$17$	5.04868	1.22448	0.612242	−	0.790671i	$-0.290268\pi$
$17$	5.04868	1.22448	0.612242	+	0.790671i	$0.290268\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	−0.792287	−0.168916
$23$	2.52434	0.526361	0.263180	−	0.964747i	$-0.415229\pi$
$23$	2.52434	0.526361	0.263180	+	0.964747i	$0.415229\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	−4.75372	−0.898369
$29$	2.74456	0.509652	0.254826	−	0.966987i	$-0.417982\pi$
$29$	2.74456	0.509652	0.254826	+	0.966987i	$0.417982\pi$
$30$	0	0
$31$	−2.37228	−0.426074	−0.213037	−	0.977044i	$-0.568336\pi$
$31$	−2.37228	−0.426074	−0.213037	+	0.977044i	$0.568336\pi$
$32$	−5.84096	−1.03255
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	−11.0371	−1.81449	−0.907245	−	0.420602i	$-0.861819\pi$
$37$	−11.0371	−1.81449	−0.907245	+	0.420602i	$0.861819\pi$
$38$	−3.16915	−0.514104
$39$	0	0
$40$	0	0
$41$	2.74456	0.428629	0.214314	−	0.976765i	$-0.431248\pi$
$41$	2.74456	0.428629	0.214314	+	0.976765i	$0.431248\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	−1.37228	−0.206879
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−6.63325	−0.967559	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	−6.63325	−0.967559	−0.483779	+	0.875190i	$0.660736\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.16915	0.435316	0.217658	−	0.976025i	$-0.430158\pi$
$53$	3.16915	0.435316	0.217658	+	0.976025i	$0.430158\pi$
$54$	0	0
$55$	0	0
$56$	9.25544	1.23681
$57$	0	0
$58$	−2.17448	−0.285523
$59$	1.62772	0.211911	0.105955	−	0.994371i	$-0.466210\pi$
$59$	1.62772	0.211911	0.105955	+	0.994371i	$0.466210\pi$
$60$	0	0
$61$	10.7446	1.37570	0.687850	−	0.725853i	$-0.258555\pi$
$61$	10.7446	1.37570	0.687850	+	0.725853i	$0.258555\pi$
$62$	1.87953	0.238700
$63$	0	0
$64$	3.37228	0.421535
$65$	0	0
$66$	0	0
$67$	−0.644810	−0.0787761	−0.0393880	−	0.999224i	$-0.512541\pi$
$67$	−0.644810	−0.0787761	−0.0393880	+	0.999224i	$0.512541\pi$
$68$	−6.92820	−0.840168
$69$	0	0
$70$	0	0
$71$	−7.11684	−0.844614	−0.422307	−	0.906453i	$-0.638780\pi$
$71$	−7.11684	−0.844614	−0.422307	+	0.906453i	$0.638780\pi$
$72$	0	0
$73$	−6.92820	−0.810885	−0.405442	−	0.914121i	$-0.632883\pi$
$73$	−6.92820	−0.810885	−0.405442	+	0.914121i	$0.632883\pi$
$74$	8.74456	1.01653
$75$	0	0
$76$	−5.48913	−0.629646
$77$	3.46410	0.394771
$78$	0	0
$79$	12.7446	1.43388	0.716938	−	0.697137i	$-0.245543\pi$
$79$	12.7446	1.43388	0.716938	+	0.697137i	$0.245543\pi$
$80$	0	0
$81$	0	0
$82$	−2.17448	−0.240131
$83$	6.63325	0.728094	0.364047	−	0.931381i	$-0.381395\pi$
$83$	6.63325	0.728094	0.364047	+	0.931381i	$0.381395\pi$
$84$	0	0
$85$	0	0
$86$	−2.74456	−0.295954
$87$	0	0
$88$	2.67181	0.284816
$89$	4.37228	0.463461	0.231730	−	0.972780i	$-0.425561\pi$
$89$	4.37228	0.463461	0.231730	+	0.972780i	$0.425561\pi$
$90$	0	0
$91$	0	0
$92$	−3.46410	−0.361158
$93$	0	0
$94$	5.25544	0.542057
$95$	0	0
$96$	0	0
$97$	−4.10891	−0.417197	−0.208598	−	0.978001i	$-0.566890\pi$
$97$	−4.10891	−0.417197	−0.208598	+	0.978001i	$0.566890\pi$
$98$	−3.96143	−0.400165
$99$	0	0
$100$	0	0
$101$	−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
$103$	−10.3923	−1.02398	−0.511992	−	0.858990i	$-0.671092\pi$
$103$	−10.3923	−1.02398	−0.511992	+	0.858990i	$0.671092\pi$
$104$	0	0
$105$	0	0
$106$	−2.51087	−0.243878
$107$	−6.63325	−0.641260	−0.320630	−	0.947204i	$-0.603895\pi$
$107$	−6.63325	−0.641260	−0.320630	+	0.947204i	$0.603895\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	2.17448	0.205469
$113$	−16.0858	−1.51322	−0.756612	−	0.653864i	$-0.773147\pi$
$113$	−16.0858	−1.51322	−0.756612	+	0.653864i	$0.773147\pi$
$114$	0	0
$115$	0	0
$116$	−3.76631	−0.349693
$117$	0	0
$118$	−1.28962	−0.118719
$119$	17.4891	1.60323
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.51278	−0.770711
$123$	0	0
$124$	3.25544	0.292347
$125$	0	0
$126$	0	0
$127$	11.6819	1.03660	0.518302	−	0.855198i	$-0.326564\pi$
$127$	11.6819	1.03660	0.518302	+	0.855198i	$0.326564\pi$
$128$	9.01011	0.796389
$129$	0	0
$130$	0	0
$131$	−8.74456	−0.764016	−0.382008	−	0.924159i	$-0.624767\pi$
$131$	−8.74456	−0.764016	−0.382008	+	0.924159i	$0.624767\pi$
$132$	0	0
$133$	13.8564	1.20150
$134$	0.510875	0.0441329
$135$	0	0
$136$	13.4891	1.15668
$137$	2.22938	0.190469	0.0952346	−	0.995455i	$-0.469640\pi$
$137$	2.22938	0.190469	0.0952346	+	0.995455i	$0.469640\pi$
$138$	0	0
$139$	18.2337	1.54656	0.773281	−	0.634064i	$-0.218614\pi$
$139$	18.2337	1.54656	0.773281	+	0.634064i	$0.218614\pi$
$140$	0	0
$141$	0	0
$142$	5.63858	0.473179
$143$	0	0
$144$	0	0
$145$	0	0
$146$	5.48913	0.454283
$147$	0	0
$148$	15.1460	1.24500
$149$	−11.4891	−0.941226	−0.470613	−	0.882340i	$-0.655967\pi$
$149$	−11.4891	−0.941226	−0.470613	+	0.882340i	$0.655967\pi$
$150$	0	0
$151$	22.2337	1.80935	0.904676	−	0.426100i	$-0.140113\pi$
$151$	22.2337	1.80935	0.904676	+	0.426100i	$0.140113\pi$
$152$	10.6873	0.866851
$153$	0	0
$154$	−2.74456	−0.221163
$155$	0	0
$156$	0	0
$157$	5.39853	0.430850	0.215425	−	0.976520i	$-0.430886\pi$
$157$	5.39853	0.430850	0.215425	+	0.976520i	$0.430886\pi$
$158$	−10.0974	−0.803302
$159$	0	0
$160$	0	0
$161$	8.74456	0.689168
$162$	0	0
$163$	−3.46410	−0.271329	−0.135665	−	0.990755i	$-0.543317\pi$
$163$	−3.46410	−0.271329	−0.135665	+	0.990755i	$0.543317\pi$
$164$	−3.76631	−0.294100
$165$	0	0
$166$	−5.25544	−0.407901
$167$	22.3692	1.73098	0.865490	−	0.500927i	$-0.167007\pi$
$167$	22.3692	1.73098	0.865490	+	0.500927i	$0.167007\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	−4.75372	−0.362468
$173$	1.87953	0.142898	0.0714489	−	0.997444i	$-0.477238\pi$
$173$	1.87953	0.142898	0.0714489	+	0.997444i	$0.477238\pi$
$174$	0	0
$175$	0	0
$176$	0.627719	0.0473161
$177$	0	0
$178$	−3.46410	−0.259645
$179$	15.8614	1.18554	0.592769	−	0.805373i	$-0.298035\pi$
$179$	15.8614	1.18554	0.592769	+	0.805373i	$0.298035\pi$
$180$	0	0
$181$	6.88316	0.511621	0.255810	−	0.966727i	$-0.417658\pi$
$181$	6.88316	0.511621	0.255810	+	0.966727i	$0.417658\pi$
$182$	0	0
$183$	0	0
$184$	6.74456	0.497216
$185$	0	0
$186$	0	0
$187$	5.04868	0.369196
$188$	9.10268	0.663881
$189$	0	0
$190$	0	0
$191$	13.6277	0.986067	0.493034	−	0.870010i	$-0.335888\pi$
$191$	13.6277	0.986067	0.493034	+	0.870010i	$0.335888\pi$
$192$	0	0
$193$	23.3639	1.68177	0.840883	−	0.541216i	$-0.182036\pi$
$193$	23.3639	1.68177	0.840883	+	0.541216i	$0.182036\pi$
$194$	3.25544	0.233727
$195$	0	0
$196$	−6.86141	−0.490100
$197$	−1.87953	−0.133911	−0.0669554	−	0.997756i	$-0.521329\pi$
$197$	−1.87953	−0.133911	−0.0669554	+	0.997756i	$0.521329\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	4.75372	0.334471
$203$	9.50744	0.667292
$204$	0	0
$205$	0	0
$206$	8.23369	0.573668
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−21.4891	−1.47937	−0.739686	−	0.672952i	$-0.765026\pi$
$211$	−21.4891	−1.47937	−0.739686	+	0.672952i	$0.765026\pi$
$212$	−4.34896	−0.298688
$213$	0	0
$214$	5.25544	0.359254
$215$	0	0
$216$	0	0
$217$	−8.21782	−0.557862
$218$	−7.92287	−0.536604
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.57301	0.507126	0.253563	−	0.967319i	$-0.418397\pi$
$223$	7.57301	0.507126	0.253563	+	0.967319i	$0.418397\pi$
$224$	−20.2337	−1.35192
$225$	0	0
$226$	12.7446	0.847756
$227$	9.80240	0.650608	0.325304	−	0.945609i	$-0.394533\pi$
$227$	9.80240	0.650608	0.325304	+	0.945609i	$0.394533\pi$
$228$	0	0
$229$	20.3723	1.34624	0.673119	−	0.739534i	$-0.264954\pi$
$229$	20.3723	1.34624	0.673119	+	0.739534i	$0.264954\pi$
$230$	0	0
$231$	0	0
$232$	7.33296	0.481433
$233$	17.0256	1.11538	0.557691	−	0.830049i	$-0.311688\pi$
$233$	17.0256	1.11538	0.557691	+	0.830049i	$0.311688\pi$
$234$	0	0
$235$	0	0
$236$	−2.23369	−0.145401
$237$	0	0
$238$	−13.8564	−0.898177
$239$	3.25544	0.210577	0.105288	−	0.994442i	$-0.466423\pi$
$239$	3.25544	0.210577	0.105288	+	0.994442i	$0.466423\pi$
$240$	0	0
$241$	5.25544	0.338532	0.169266	−	0.985570i	$-0.445860\pi$
$241$	5.25544	0.338532	0.169266	+	0.985570i	$0.445860\pi$
$242$	−0.792287	−0.0509301
$243$	0	0
$244$	−14.7446	−0.943924
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−6.33830	−0.402482
$249$	0	0
$250$	0	0
$251$	4.88316	0.308222	0.154111	−	0.988054i	$-0.450749\pi$
$251$	4.88316	0.308222	0.154111	+	0.988054i	$0.450749\pi$
$252$	0	0
$253$	2.52434	0.158704
$254$	−9.25544	−0.580738
$255$	0	0
$256$	−13.8832	−0.867697
$257$	−23.9538	−1.49419	−0.747097	−	0.664715i	$-0.768553\pi$
$257$	−23.9538	−1.49419	−0.747097	+	0.664715i	$0.768553\pi$
$258$	0	0
$259$	−38.2337	−2.37573
$260$	0	0
$261$	0	0
$262$	6.92820	0.428026
$263$	−14.1514	−0.872610	−0.436305	−	0.899799i	$-0.643713\pi$
$263$	−14.1514	−0.872610	−0.436305	+	0.899799i	$0.643713\pi$
$264$	0	0
$265$	0	0
$266$	−10.9783	−0.673120
$267$	0	0
$268$	0.884861	0.0540515
$269$	−11.4891	−0.700504	−0.350252	−	0.936655i	$-0.613904\pi$
$269$	−11.4891	−0.700504	−0.350252	+	0.936655i	$0.613904\pi$
$270$	0	0
$271$	−9.48913	−0.576423	−0.288212	−	0.957567i	$-0.593061\pi$
$271$	−9.48913	−0.576423	−0.288212	+	0.957567i	$0.593061\pi$
$272$	3.16915	0.192158
$273$	0	0
$274$	−1.76631	−0.106707
$275$	0	0
$276$	0	0
$277$	8.21782	0.493761	0.246881	−	0.969046i	$-0.420594\pi$
$277$	8.21782	0.493761	0.246881	+	0.969046i	$0.420594\pi$
$278$	−14.4463	−0.866432
$279$	0	0
$280$	0	0
$281$	23.4891	1.40124	0.700622	−	0.713533i	$-0.252906\pi$
$281$	23.4891	1.40124	0.700622	+	0.713533i	$0.252906\pi$
$282$	0	0
$283$	−4.75372	−0.282579	−0.141290	−	0.989968i	$-0.545125\pi$
$283$	−4.75372	−0.282579	−0.141290	+	0.989968i	$0.545125\pi$
$284$	9.76631	0.579524
$285$	0	0
$286$	0	0
$287$	9.50744	0.561207
$288$	0	0
$289$	8.48913	0.499360
$290$	0	0
$291$	0	0
$292$	9.50744	0.556381
$293$	10.0974	0.589894	0.294947	−	0.955514i	$-0.404698\pi$
$293$	10.0974	0.589894	0.294947	+	0.955514i	$0.404698\pi$
$294$	0	0
$295$	0	0
$296$	−29.4891	−1.71402
$297$	0	0
$298$	9.10268	0.527304
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	−17.6155	−1.01366
$303$	0	0
$304$	2.51087	0.144009
$305$	0	0
$306$	0	0
$307$	−28.1176	−1.60475	−0.802377	−	0.596817i	$-0.796432\pi$
$307$	−28.1176	−1.60475	−0.802377	+	0.596817i	$0.796432\pi$
$308$	−4.75372	−0.270868
$309$	0	0
$310$	0	0
$311$	17.4891	0.991717	0.495859	−	0.868403i	$-0.334853\pi$
$311$	17.4891	0.991717	0.495859	+	0.868403i	$0.334853\pi$
$312$	0	0
$313$	−31.8217	−1.79867	−0.899335	−	0.437260i	$-0.855949\pi$
$313$	−31.8217	−1.79867	−0.899335	+	0.437260i	$0.855949\pi$
$314$	−4.27719	−0.241376
$315$	0	0
$316$	−17.4891	−0.983840
$317$	3.51900	0.197647	0.0988235	−	0.995105i	$-0.468492\pi$
$317$	3.51900	0.197647	0.0988235	+	0.995105i	$0.468492\pi$
$318$	0	0
$319$	2.74456	0.153666
$320$	0	0
$321$	0	0
$322$	−6.92820	−0.386094
$323$	20.1947	1.12366
$324$	0	0
$325$	0	0
$326$	2.74456	0.152007
$327$	0	0
$328$	7.33296	0.404895
$329$	−22.9783	−1.26683
$330$	0	0
$331$	3.11684	0.171317	0.0856586	−	0.996325i	$-0.472701\pi$
$331$	3.11684	0.171317	0.0856586	+	0.996325i	$0.472701\pi$
$332$	−9.10268	−0.499575
$333$	0	0
$334$	−17.7228	−0.969749
$335$	0	0
$336$	0	0
$337$	−12.5668	−0.684556	−0.342278	−	0.939599i	$-0.611199\pi$
$337$	−12.5668	−0.684556	−0.342278	+	0.939599i	$0.611199\pi$
$338$	10.2997	0.560232
$339$	0	0
$340$	0	0
$341$	−2.37228	−0.128466
$342$	0	0
$343$	−6.92820	−0.374088
$344$	9.25544	0.499020
$345$	0	0
$346$	−1.48913	−0.0800559
$347$	29.2974	1.57277	0.786383	−	0.617739i	$-0.211951\pi$
$347$	29.2974	1.57277	0.786383	+	0.617739i	$0.211951\pi$
$348$	0	0
$349$	−7.48913	−0.400884	−0.200442	−	0.979706i	$-0.564238\pi$
$349$	−7.48913	−0.400884	−0.200442	+	0.979706i	$0.564238\pi$
$350$	0	0
$351$	0	0
$352$	−5.84096	−0.311324
$353$	−21.7244	−1.15627	−0.578136	−	0.815941i	$-0.696220\pi$
$353$	−21.7244	−1.15627	−0.578136	+	0.815941i	$0.696220\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−12.5668	−0.664175
$359$	−6.51087	−0.343631	−0.171815	−	0.985129i	$-0.554963\pi$
$359$	−6.51087	−0.343631	−0.171815	+	0.985129i	$0.554963\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−5.45343	−0.286626
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−24.0087	−1.25324	−0.626621	−	0.779324i	$-0.715563\pi$
$367$	−24.0087	−1.25324	−0.626621	+	0.779324i	$0.715563\pi$
$368$	1.58457	0.0826016
$369$	0	0
$370$	0	0
$371$	10.9783	0.569962
$372$	0	0
$373$	8.21782	0.425503	0.212751	−	0.977106i	$-0.431758\pi$
$373$	8.21782	0.425503	0.212751	+	0.977106i	$0.431758\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−17.7228	−0.913984
$377$	0	0
$378$	0	0
$379$	−6.37228	−0.327322	−0.163661	−	0.986517i	$-0.552330\pi$
$379$	−6.37228	−0.327322	−0.163661	+	0.986517i	$0.552330\pi$
$380$	0	0
$381$	0	0
$382$	−10.7971	−0.552426
$383$	−5.69349	−0.290924	−0.145462	−	0.989364i	$-0.546467\pi$
$383$	−5.69349	−0.290924	−0.145462	+	0.989364i	$0.546467\pi$
$384$	0	0
$385$	0	0
$386$	−18.5109	−0.942179
$387$	0	0
$388$	5.63858	0.286256
$389$	9.86141	0.499993	0.249997	−	0.968247i	$-0.419571\pi$
$389$	9.86141	0.499993	0.249997	+	0.968247i	$0.419571\pi$
$390$	0	0
$391$	12.7446	0.644520
$392$	13.3591	0.674735
$393$	0	0
$394$	1.48913	0.0750210
$395$	0	0
$396$	0	0
$397$	23.3639	1.17260	0.586299	−	0.810095i	$-0.300584\pi$
$397$	23.3639	1.17260	0.586299	+	0.810095i	$0.300584\pi$
$398$	6.33830	0.317710
$399$	0	0
$400$	0	0
$401$	−11.4891	−0.573740	−0.286870	−	0.957970i	$-0.592615\pi$
$401$	−11.4891	−0.573740	−0.286870	+	0.957970i	$0.592615\pi$
$402$	0	0
$403$	0	0
$404$	8.23369	0.409641
$405$	0	0
$406$	−7.53262	−0.373838
$407$	−11.0371	−0.547089
$408$	0	0
$409$	4.51087	0.223048	0.111524	−	0.993762i	$-0.464427\pi$
$409$	4.51087	0.223048	0.111524	+	0.993762i	$0.464427\pi$
$410$	0	0
$411$	0	0
$412$	14.2612	0.702597
$413$	5.63858	0.277457
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−3.16915	−0.155008
$419$	−22.9783	−1.12256	−0.561280	−	0.827626i	$-0.689691\pi$
$419$	−22.9783	−1.12256	−0.561280	+	0.827626i	$0.689691\pi$
$420$	0	0
$421$	31.4891	1.53469	0.767343	−	0.641237i	$-0.221578\pi$
$421$	31.4891	1.53469	0.767343	+	0.641237i	$0.221578\pi$
$422$	17.0256	0.828791
$423$	0	0
$424$	8.46738	0.411212
$425$	0	0
$426$	0	0
$427$	37.2203	1.80121
$428$	9.10268	0.439995
$429$	0	0
$430$	0	0
$431$	−31.7228	−1.52803	−0.764017	−	0.645196i	$-0.776776\pi$
$431$	−31.7228	−1.52803	−0.764017	+	0.645196i	$0.776776\pi$
$432$	0	0
$433$	−20.5446	−0.987308	−0.493654	−	0.869658i	$-0.664339\pi$
$433$	−20.5446	−0.987308	−0.493654	+	0.869658i	$0.664339\pi$
$434$	6.51087	0.312532
$435$	0	0
$436$	−13.7228	−0.657204
$437$	10.0974	0.483022
$438$	0	0
$439$	−1.48913	−0.0710721	−0.0355360	−	0.999368i	$-0.511314\pi$
$439$	−1.48913	−0.0710721	−0.0355360	+	0.999368i	$0.511314\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.0911	0.717001	0.358500	−	0.933530i	$-0.383288\pi$
$443$	15.0911	0.717001	0.358500	+	0.933530i	$0.383288\pi$
$444$	0	0
$445$	0	0
$446$	−6.00000	−0.284108
$447$	0	0
$448$	11.6819	0.551919
$449$	21.8614	1.03170	0.515852	−	0.856678i	$-0.327476\pi$
$449$	21.8614	1.03170	0.515852	+	0.856678i	$0.327476\pi$
$450$	0	0
$451$	2.74456	0.129236
$452$	22.0742	1.03828
$453$	0	0
$454$	−7.76631	−0.364491
$455$	0	0
$456$	0	0
$457$	−20.7846	−0.972263	−0.486132	−	0.873886i	$-0.661592\pi$
$457$	−20.7846	−0.972263	−0.486132	+	0.873886i	$0.661592\pi$
$458$	−16.1407	−0.754205
$459$	0	0
$460$	0	0
$461$	32.2337	1.50127	0.750636	−	0.660716i	$-0.229747\pi$
$461$	32.2337	1.50127	0.750636	+	0.660716i	$0.229747\pi$
$462$	0	0
$463$	−20.1398	−0.935976	−0.467988	−	0.883735i	$-0.655021\pi$
$463$	−20.1398	−0.935976	−0.467988	+	0.883735i	$0.655021\pi$
$464$	1.72281	0.0799796
$465$	0	0
$466$	−13.4891	−0.624872
$467$	4.40387	0.203787	0.101893	−	0.994795i	$-0.467510\pi$
$467$	4.40387	0.203787	0.101893	+	0.994795i	$0.467510\pi$
$468$	0	0
$469$	−2.23369	−0.103142
$470$	0	0
$471$	0	0
$472$	4.34896	0.200177
$473$	3.46410	0.159280
$474$	0	0
$475$	0	0
$476$	−24.0000	−1.10004
$477$	0	0
$478$	−2.57924	−0.117972
$479$	−17.4891	−0.799099	−0.399549	−	0.916712i	$-0.630833\pi$
$479$	−17.4891	−0.799099	−0.399549	+	0.916712i	$0.630833\pi$
$480$	0	0
$481$	0	0
$482$	−4.16381	−0.189657
$483$	0	0
$484$	−1.37228	−0.0623764
$485$	0	0
$486$	0	0
$487$	22.7190	1.02950	0.514749	−	0.857341i	$-0.327885\pi$
$487$	22.7190	1.02950	0.514749	+	0.857341i	$0.327885\pi$
$488$	28.7075	1.29953
$489$	0	0
$490$	0	0
$491$	−29.4891	−1.33083	−0.665413	−	0.746476i	$-0.731744\pi$
$491$	−29.4891	−1.33083	−0.665413	+	0.746476i	$0.731744\pi$
$492$	0	0
$493$	13.8564	0.624061
$494$	0	0
$495$	0	0
$496$	−1.48913	−0.0668637
$497$	−24.6535	−1.10586
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	−3.86886	−0.172676
$503$	−0.294954	−0.0131513	−0.00657567	−	0.999978i	$-0.502093\pi$
$503$	−0.294954	−0.0131513	−0.00657567	+	0.999978i	$0.502093\pi$
$504$	0	0
$505$	0	0
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	−16.0309	−0.711256
$509$	−28.3723	−1.25758	−0.628790	−	0.777575i	$-0.716449\pi$
$509$	−28.3723	−1.25758	−0.628790	+	0.777575i	$0.716449\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	−7.02078	−0.310277
$513$	0	0
$514$	18.9783	0.837095
$515$	0	0
$516$	0	0
$517$	−6.63325	−0.291730
$518$	30.2921	1.33096
$519$	0	0
$520$	0	0
$521$	−18.6060	−0.815142	−0.407571	−	0.913173i	$-0.633624\pi$
$521$	−18.6060	−0.815142	−0.407571	+	0.913173i	$0.633624\pi$
$522$	0	0
$523$	9.10268	0.398033	0.199016	−	0.979996i	$-0.436225\pi$
$523$	9.10268	0.398033	0.199016	+	0.979996i	$0.436225\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	11.2119	0.488864
$527$	−11.9769	−0.521721
$528$	0	0
$529$	−16.6277	−0.722944
$530$	0	0
$531$	0	0
$532$	−19.0149	−0.824400
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−1.72281	−0.0744142
$537$	0	0
$538$	9.10268	0.392445
$539$	5.00000	0.215365
$540$	0	0
$541$	−0.233688	−0.0100470	−0.00502351	−	0.999987i	$-0.501599\pi$
$541$	−0.233688	−0.0100470	−0.00502351	+	0.999987i	$0.501599\pi$
$542$	7.51811	0.322930
$543$	0	0
$544$	−29.4891	−1.26434
$545$	0	0
$546$	0	0
$547$	−9.10268	−0.389203	−0.194601	−	0.980882i	$-0.562341\pi$
$547$	−9.10268	−0.389203	−0.194601	+	0.980882i	$0.562341\pi$
$548$	−3.05934	−0.130689
$549$	0	0
$550$	0	0
$551$	10.9783	0.467689
$552$	0	0
$553$	44.1485	1.87738
$554$	−6.51087	−0.276621
$555$	0	0
$556$	−25.0217	−1.06116
$557$	−32.1716	−1.36315	−0.681577	−	0.731747i	$-0.738705\pi$
$557$	−32.1716	−1.36315	−0.681577	+	0.731747i	$0.738705\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−18.6101	−0.785021
$563$	12.2718	0.517196	0.258598	−	0.965985i	$-0.416739\pi$
$563$	12.2718	0.517196	0.258598	+	0.965985i	$0.416739\pi$
$564$	0	0
$565$	0	0
$566$	3.76631	0.158310
$567$	0	0
$568$	−19.0149	−0.797847
$569$	38.7446	1.62426	0.812128	−	0.583479i	$-0.198309\pi$
$569$	38.7446	1.62426	0.812128	+	0.583479i	$0.198309\pi$
$570$	0	0
$571$	−21.4891	−0.899292	−0.449646	−	0.893207i	$-0.648450\pi$
$571$	−21.4891	−0.899292	−0.449646	+	0.893207i	$0.648450\pi$
$572$	0	0
$573$	0	0
$574$	−7.53262	−0.314406
$575$	0	0
$576$	0	0
$577$	31.8217	1.32476	0.662378	−	0.749170i	$-0.269547\pi$
$577$	31.8217	1.32476	0.662378	+	0.749170i	$0.269547\pi$
$578$	−6.72582	−0.279757
$579$	0	0
$580$	0	0
$581$	22.9783	0.953298
$582$	0	0
$583$	3.16915	0.131253
$584$	−18.5109	−0.765985
$585$	0	0
$586$	−8.00000	−0.330477
$587$	28.0078	1.15600	0.578002	−	0.816035i	$-0.303833\pi$
$587$	28.0078	1.15600	0.578002	+	0.816035i	$0.303833\pi$
$588$	0	0
$589$	−9.48913	−0.390993
$590$	0	0
$591$	0	0
$592$	−6.92820	−0.284747
$593$	−43.5586	−1.78874	−0.894368	−	0.447333i	$-0.852374\pi$
$593$	−43.5586	−1.78874	−0.894368	+	0.447333i	$0.852374\pi$
$594$	0	0
$595$	0	0
$596$	15.7663	0.645813
$597$	0	0
$598$	0	0
$599$	34.9783	1.42917	0.714586	−	0.699547i	$-0.246615\pi$
$599$	34.9783	1.42917	0.714586	+	0.699547i	$0.246615\pi$
$600$	0	0
$601$	30.4674	1.24279	0.621395	−	0.783497i	$-0.286566\pi$
$601$	30.4674	1.24279	0.621395	+	0.783497i	$0.286566\pi$
$602$	−9.50744	−0.387494
$603$	0	0
$604$	−30.5109	−1.24147
$605$	0	0
$606$	0	0
$607$	3.46410	0.140604	0.0703018	−	0.997526i	$-0.477604\pi$
$607$	3.46410	0.140604	0.0703018	+	0.997526i	$0.477604\pi$
$608$	−23.3639	−0.947529
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−44.1485	−1.78314	−0.891570	−	0.452883i	$-0.850395\pi$
$613$	−44.1485	−1.78314	−0.891570	+	0.452883i	$0.850395\pi$
$614$	22.2772	0.899034
$615$	0	0
$616$	9.25544	0.372912
$617$	3.75906	0.151334	0.0756669	−	0.997133i	$-0.475891\pi$
$617$	3.75906	0.151334	0.0756669	+	0.997133i	$0.475891\pi$
$618$	0	0
$619$	−3.11684	−0.125277	−0.0626383	−	0.998036i	$-0.519951\pi$
$619$	−3.11684	−0.125277	−0.0626383	+	0.998036i	$0.519951\pi$
$620$	0	0
$621$	0	0
$622$	−13.8564	−0.555591
$623$	15.1460	0.606813
$624$	0	0
$625$	0	0
$626$	25.2119	1.00767
$627$	0	0
$628$	−7.40830	−0.295624
$629$	−55.7228	−2.22181
$630$	0	0
$631$	−16.6060	−0.661073	−0.330537	−	0.943793i	$-0.607230\pi$
$631$	−16.6060	−0.661073	−0.330537	+	0.943793i	$0.607230\pi$
$632$	34.0511	1.35448
$633$	0	0
$634$	−2.78806	−0.110728
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−2.17448	−0.0860885
$639$	0	0
$640$	0	0
$641$	19.6277	0.775248	0.387624	−	0.921818i	$-0.373296\pi$
$641$	19.6277	0.775248	0.387624	+	0.921818i	$0.373296\pi$
$642$	0	0
$643$	39.1547	1.54411	0.772055	−	0.635556i	$-0.219229\pi$
$643$	39.1547	1.54411	0.772055	+	0.635556i	$0.219229\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−16.0000	−0.629512
$647$	−41.0342	−1.61322	−0.806611	−	0.591083i	$-0.798701\pi$
$647$	−41.0342	−1.61322	−0.806611	+	0.591083i	$0.798701\pi$
$648$	0	0
$649$	1.62772	0.0638935
$650$	0	0
$651$	0	0
$652$	4.75372	0.186170
$653$	−25.5932	−1.00154	−0.500770	−	0.865580i	$-0.666950\pi$
$653$	−25.5932	−1.00154	−0.500770	+	0.865580i	$0.666950\pi$
$654$	0	0
$655$	0	0
$656$	1.72281	0.0672646
$657$	0	0
$658$	18.2054	0.709719
$659$	−32.7446	−1.27555	−0.637774	−	0.770224i	$-0.720144\pi$
$659$	−32.7446	−1.27555	−0.637774	+	0.770224i	$0.720144\pi$
$660$	0	0
$661$	35.3505	1.37498	0.687488	−	0.726196i	$-0.258713\pi$
$661$	35.3505	1.37498	0.687488	+	0.726196i	$0.258713\pi$
$662$	−2.46943	−0.0959773
$663$	0	0
$664$	17.7228	0.687779
$665$	0	0
$666$	0	0
$667$	6.92820	0.268261
$668$	−30.6968	−1.18770
$669$	0	0
$670$	0	0
$671$	10.7446	0.414789
$672$	0	0
$673$	−1.28962	−0.0497112	−0.0248556	−	0.999691i	$-0.507913\pi$
$673$	−1.28962	−0.0497112	−0.0248556	+	0.999691i	$0.507913\pi$
$674$	9.95650	0.383510
$675$	0	0
$676$	17.8397	0.686141
$677$	−43.4487	−1.66987	−0.834935	−	0.550348i	$-0.814495\pi$
$677$	−43.4487	−1.66987	−0.834935	+	0.550348i	$0.814495\pi$
$678$	0	0
$679$	−14.2337	−0.546239
$680$	0	0
$681$	0	0
$682$	1.87953	0.0719708
$683$	−44.4434	−1.70058	−0.850290	−	0.526314i	$-0.823573\pi$
$683$	−44.4434	−1.70058	−0.850290	+	0.526314i	$0.823573\pi$
$684$	0	0
$685$	0	0
$686$	5.48913	0.209576
$687$	0	0
$688$	2.17448	0.0829013
$689$	0	0
$690$	0	0
$691$	16.1386	0.613941	0.306971	−	0.951719i	$-0.400685\pi$
$691$	16.1386	0.613941	0.306971	+	0.951719i	$0.400685\pi$
$692$	−2.57924	−0.0980480
$693$	0	0
$694$	−23.2119	−0.881113
$695$	0	0
$696$	0	0
$697$	13.8564	0.524849
$698$	5.93354	0.224588
$699$	0	0
$700$	0	0
$701$	35.4891	1.34041	0.670203	−	0.742178i	$-0.266207\pi$
$701$	35.4891	1.34041	0.670203	+	0.742178i	$0.266207\pi$
$702$	0	0
$703$	−44.1485	−1.66509
$704$	3.37228	0.127098
$705$	0	0
$706$	17.2119	0.647780
$707$	−20.7846	−0.781686
$708$	0	0
$709$	41.1168	1.54418	0.772088	−	0.635516i	$-0.219213\pi$
$709$	41.1168	1.54418	0.772088	+	0.635516i	$0.219213\pi$
$710$	0	0
$711$	0	0
$712$	11.6819	0.437799
$713$	−5.98844	−0.224269
$714$	0	0
$715$	0	0
$716$	−21.7663	−0.813445
$717$	0	0
$718$	5.15848	0.192513
$719$	−21.3505	−0.796240	−0.398120	−	0.917333i	$-0.630337\pi$
$719$	−21.3505	−0.796240	−0.398120	+	0.917333i	$0.630337\pi$
$720$	0	0
$721$	−36.0000	−1.34071
$722$	2.37686	0.0884576
$723$	0	0
$724$	−9.44563	−0.351044
$725$	0	0
$726$	0	0
$727$	−15.7908	−0.585650	−0.292825	−	0.956166i	$-0.594595\pi$
$727$	−15.7908	−0.585650	−0.292825	+	0.956166i	$0.594595\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.4891	0.646859
$732$	0	0
$733$	30.2921	1.11886	0.559431	−	0.828877i	$-0.311020\pi$
$733$	30.2921	1.11886	0.559431	+	0.828877i	$0.311020\pi$
$734$	19.0217	0.702106
$735$	0	0
$736$	−14.7446	−0.543492
$737$	−0.644810	−0.0237519
$738$	0	0
$739$	0.744563	0.0273892	0.0136946	−	0.999906i	$-0.495641\pi$
$739$	0.744563	0.0273892	0.0136946	+	0.999906i	$0.495641\pi$
$740$	0	0
$741$	0	0
$742$	−8.69793	−0.319311
$743$	21.7793	0.799004	0.399502	−	0.916732i	$-0.369183\pi$
$743$	21.7793	0.799004	0.399502	+	0.916732i	$0.369183\pi$
$744$	0	0
$745$	0	0
$746$	−6.51087	−0.238380
$747$	0	0
$748$	−6.92820	−0.253320
$749$	−22.9783	−0.839607
$750$	0	0
$751$	21.6277	0.789207	0.394603	−	0.918852i	$-0.370882\pi$
$751$	21.6277	0.789207	0.394603	+	0.918852i	$0.370882\pi$
$752$	−4.16381	−0.151839
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−39.7995	−1.44654	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	−39.7995	−1.44654	−0.723269	+	0.690567i	$0.757361\pi$
$758$	5.04868	0.183376
$759$	0	0
$760$	0	0
$761$	−21.2554	−0.770509	−0.385255	−	0.922810i	$-0.625886\pi$
$761$	−21.2554	−0.770509	−0.385255	+	0.922810i	$0.625886\pi$
$762$	0	0
$763$	34.6410	1.25409
$764$	−18.7011	−0.676581
$765$	0	0
$766$	4.51087	0.162985
$767$	0	0
$768$	0	0
$769$	−51.2119	−1.84675	−0.923375	−	0.383900i	$-0.874581\pi$
$769$	−51.2119	−1.84675	−0.923375	+	0.383900i	$0.874581\pi$
$770$	0	0
$771$	0	0
$772$	−32.0618	−1.15393
$773$	30.8820	1.11075	0.555373	−	0.831601i	$-0.312575\pi$
$773$	30.8820	1.11075	0.555373	+	0.831601i	$0.312575\pi$
$774$	0	0
$775$	0	0
$776$	−10.9783	−0.394096
$777$	0	0
$778$	−7.81306	−0.280112
$779$	10.9783	0.393337
$780$	0	0
$781$	−7.11684	−0.254661
$782$	−10.0974	−0.361081
$783$	0	0
$784$	3.13859	0.112093
$785$	0	0
$786$	0	0
$787$	4.75372	0.169452	0.0847259	−	0.996404i	$-0.472999\pi$
$787$	4.75372	0.169452	0.0847259	+	0.996404i	$0.472999\pi$
$788$	2.57924	0.0918816
$789$	0	0
$790$	0	0
$791$	−55.7228	−1.98128
$792$	0	0
$793$	0	0
$794$	−18.5109	−0.656926
$795$	0	0
$796$	10.9783	0.389114
$797$	31.2318	1.10629	0.553144	−	0.833086i	$-0.313428\pi$
$797$	31.2318	1.10629	0.553144	+	0.833086i	$0.313428\pi$
$798$	0	0
$799$	−33.4891	−1.18476
$800$	0	0
$801$	0	0
$802$	9.10268	0.321427
$803$	−6.92820	−0.244491
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−16.0309	−0.563965
$809$	21.2554	0.747301	0.373651	−	0.927569i	$-0.378106\pi$
$809$	21.2554	0.747301	0.373651	+	0.927569i	$0.378106\pi$
$810$	0	0
$811$	34.2337	1.20211	0.601054	−	0.799209i	$-0.294748\pi$
$811$	34.2337	1.20211	0.601054	+	0.799209i	$0.294748\pi$
$812$	−13.0469	−0.457856
$813$	0	0
$814$	8.74456	0.306497
$815$	0	0
$816$	0	0
$817$	13.8564	0.484774
$818$	−3.57391	−0.124959
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−33.5161	−1.16830	−0.584149	−	0.811646i	$-0.698572\pi$
$823$	−33.5161	−1.16830	−0.584149	+	0.811646i	$0.698572\pi$
$824$	−27.7663	−0.967285
$825$	0	0
$826$	−4.46738	−0.155440
$827$	18.0202	0.626624	0.313312	−	0.949650i	$-0.398561\pi$
$827$	18.0202	0.626624	0.313312	+	0.949650i	$0.398561\pi$
$828$	0	0
$829$	31.3505	1.08885	0.544424	−	0.838810i	$-0.316748\pi$
$829$	31.3505	1.08885	0.544424	+	0.838810i	$0.316748\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	25.2434	0.874631
$834$	0	0
$835$	0	0
$836$	−5.48913	−0.189845
$837$	0	0
$838$	18.2054	0.628894
$839$	−7.11684	−0.245701	−0.122850	−	0.992425i	$-0.539204\pi$
$839$	−7.11684	−0.245701	−0.122850	+	0.992425i	$0.539204\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	−24.9484	−0.859779
$843$	0	0
$844$	29.4891	1.01506
$845$	0	0
$846$	0	0
$847$	3.46410	0.119028
$848$	1.98933	0.0683140
$849$	0	0
$850$	0	0
$851$	−27.8614	−0.955077
$852$	0	0
$853$	−24.6535	−0.844119	−0.422059	−	0.906568i	$-0.638693\pi$
$853$	−24.6535	−0.844119	−0.422059	+	0.906568i	$0.638693\pi$
$854$	−29.4891	−1.00910
$855$	0	0
$856$	−17.7228	−0.605753
$857$	10.6873	0.365070	0.182535	−	0.983199i	$-0.441570\pi$
$857$	10.6873	0.365070	0.182535	+	0.983199i	$0.441570\pi$
$858$	0	0
$859$	11.1168	0.379302	0.189651	−	0.981852i	$-0.439264\pi$
$859$	11.1168	0.379302	0.189651	+	0.981852i	$0.439264\pi$
$860$	0	0
$861$	0	0
$862$	25.1336	0.856053
$863$	−23.6588	−0.805355	−0.402678	−	0.915342i	$-0.631920\pi$
$863$	−23.6588	−0.805355	−0.402678	+	0.915342i	$0.631920\pi$
$864$	0	0
$865$	0	0
$866$	16.2772	0.553121
$867$	0	0
$868$	11.2772	0.382772
$869$	12.7446	0.432330
$870$	0	0
$871$	0	0
$872$	26.7181	0.904791
$873$	0	0
$874$	−8.00000	−0.270604
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	1.17981	0.0398168
$879$	0	0
$880$	0	0
$881$	−21.8614	−0.736530	−0.368265	−	0.929721i	$-0.620048\pi$
$881$	−21.8614	−0.736530	−0.368265	+	0.929721i	$0.620048\pi$
$882$	0	0
$883$	24.2487	0.816034	0.408017	−	0.912974i	$-0.366220\pi$
$883$	24.2487	0.816034	0.408017	+	0.912974i	$0.366220\pi$
$884$	0	0
$885$	0	0
$886$	−11.9565	−0.401687
$887$	14.1514	0.475156	0.237578	−	0.971368i	$-0.423646\pi$
$887$	14.1514	0.475156	0.237578	+	0.971368i	$0.423646\pi$
$888$	0	0
$889$	40.4674	1.35723
$890$	0	0
$891$	0	0
$892$	−10.3923	−0.347960
$893$	−26.5330	−0.887893
$894$	0	0
$895$	0	0
$896$	31.2119	1.04272
$897$	0	0
$898$	−17.3205	−0.577993
$899$	−6.51087	−0.217150
$900$	0	0
$901$	16.0000	0.533037
$902$	−2.17448	−0.0724023
$903$	0	0
$904$	−42.9783	−1.42944
$905$	0	0
$906$	0	0
$907$	19.8997	0.660760	0.330380	−	0.943848i	$-0.392823\pi$
$907$	19.8997	0.660760	0.330380	+	0.943848i	$0.392823\pi$
$908$	−13.4516	−0.446409
$909$	0	0
$910$	0	0
$911$	30.5109	1.01087	0.505435	−	0.862865i	$-0.331332\pi$
$911$	30.5109	1.01087	0.505435	+	0.862865i	$0.331332\pi$
$912$	0	0
$913$	6.63325	0.219529
$914$	16.4674	0.544692
$915$	0	0
$916$	−27.9565	−0.923709
$917$	−30.2921	−1.00033
$918$	0	0
$919$	6.23369	0.205630	0.102815	−	0.994700i	$-0.467215\pi$
$919$	6.23369	0.205630	0.102815	+	0.994700i	$0.467215\pi$
$920$	0	0
$921$	0	0
$922$	−25.5383	−0.841060
$923$	0	0
$924$	0	0
$925$	0	0
$926$	15.9565	0.524363
$927$	0	0
$928$	−16.0309	−0.526240
$929$	52.9783	1.73816	0.869080	−	0.494672i	$-0.164712\pi$
$929$	52.9783	1.73816	0.869080	+	0.494672i	$0.164712\pi$
$930$	0	0
$931$	20.0000	0.655474
$932$	−23.3639	−0.765308
$933$	0	0
$934$	−3.48913	−0.114168
$935$	0	0
$936$	0	0
$937$	−25.9431	−0.847524	−0.423762	−	0.905774i	$-0.639291\pi$
$937$	−25.9431	−0.847524	−0.423762	+	0.905774i	$0.639291\pi$
$938$	1.76972	0.0577835
$939$	0	0
$940$	0	0
$941$	−10.4674	−0.341227	−0.170613	−	0.985338i	$-0.554575\pi$
$941$	−10.4674	−0.341227	−0.170613	+	0.985338i	$0.554575\pi$
$942$	0	0
$943$	6.92820	0.225613
$944$	1.02175	0.0332551
$945$	0	0
$946$	−2.74456	−0.0892334
$947$	−56.1802	−1.82561	−0.912806	−	0.408393i	$-0.866089\pi$
$947$	−56.1802	−1.82561	−0.912806	+	0.408393i	$0.866089\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	46.7277	1.51445
$953$	41.6790	1.35012	0.675058	−	0.737765i	$-0.264119\pi$
$953$	41.6790	1.35012	0.675058	+	0.737765i	$0.264119\pi$
$954$	0	0
$955$	0	0
$956$	−4.46738	−0.144485
$957$	0	0
$958$	13.8564	0.447680
$959$	7.72281	0.249383
$960$	0	0
$961$	−25.3723	−0.818461
$962$	0	0
$963$	0	0
$964$	−7.21194	−0.232281
$965$	0	0
$966$	0	0
$967$	−46.3229	−1.48965	−0.744823	−	0.667262i	$-0.767466\pi$
$967$	−46.3229	−1.48965	−0.744823	+	0.667262i	$0.767466\pi$
$968$	2.67181	0.0858754
$969$	0	0
$970$	0	0
$971$	−54.0951	−1.73599	−0.867997	−	0.496569i	$-0.834593\pi$
$971$	−54.0951	−1.73599	−0.867997	+	0.496569i	$0.834593\pi$
$972$	0	0
$973$	63.1633	2.02492
$974$	−18.0000	−0.576757
$975$	0	0
$976$	6.74456	0.215888
$977$	27.3630	0.875419	0.437709	−	0.899117i	$-0.355790\pi$
$977$	27.3630	0.875419	0.437709	+	0.899117i	$0.355790\pi$
$978$	0	0
$979$	4.37228	0.139739
$980$	0	0
$981$	0	0
$982$	23.3639	0.745570
$983$	8.16292	0.260357	0.130178	−	0.991491i	$-0.458445\pi$
$983$	8.16292	0.260357	0.130178	+	0.991491i	$0.458445\pi$
$984$	0	0
$985$	0	0
$986$	−10.9783	−0.349619
$987$	0	0
$988$	0	0
$989$	8.74456	0.278061
$990$	0	0
$991$	−26.9783	−0.856992	−0.428496	−	0.903544i	$-0.640956\pi$
$991$	−26.9783	−0.856992	−0.428496	+	0.903544i	$0.640956\pi$
$992$	13.8564	0.439941
$993$	0	0
$994$	19.5326	0.619537
$995$	0	0
$996$	0	0
$997$	22.0742	0.699098	0.349549	−	0.936918i	$-0.386335\pi$
$997$	22.0742	0.699098	0.349549	+	0.936918i	$0.386335\pi$
$998$	15.8457	0.501588
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.bi.1.2		4
3.2	odd	2		275.2.a.h.1.3		4
5.2	odd	4		495.2.c.a.199.2		4
5.3	odd	4		495.2.c.a.199.3		4
5.4	even	2	inner	2475.2.a.bi.1.3		4
12.11	even	2		4400.2.a.cc.1.1		4
15.2	even	4		55.2.b.a.34.3	yes	4
15.8	even	4		55.2.b.a.34.2	✓	4
15.14	odd	2		275.2.a.h.1.2		4
33.32	even	2		3025.2.a.ba.1.2		4
60.23	odd	4		880.2.b.h.529.1		4
60.47	odd	4		880.2.b.h.529.4		4
60.59	even	2		4400.2.a.cc.1.4		4
165.2	odd	20		605.2.j.j.444.2		16
165.8	odd	20		605.2.j.j.9.3		16
165.17	odd	20		605.2.j.j.124.3		16
165.32	odd	4		605.2.b.c.364.2		4
165.38	even	20		605.2.j.i.124.3		16
165.47	even	20		605.2.j.i.9.3		16
165.53	even	20		605.2.j.i.444.2		16
165.62	odd	20		605.2.j.j.269.3		16
165.68	odd	20		605.2.j.j.444.3		16
165.83	odd	20		605.2.j.j.124.2		16
165.92	even	20		605.2.j.i.269.2		16
165.98	odd	4		605.2.b.c.364.3		4
165.107	odd	20		605.2.j.j.9.2		16
165.113	even	20		605.2.j.i.9.2		16
165.128	odd	20		605.2.j.j.269.2		16
165.137	even	20		605.2.j.i.124.2		16
165.152	even	20		605.2.j.i.444.3		16
165.158	even	20		605.2.j.i.269.3		16
165.164	even	2		3025.2.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.b.a.34.2	✓	4	15.8	even	4
55.2.b.a.34.3	yes	4	15.2	even	4
275.2.a.h.1.2		4	15.14	odd	2
275.2.a.h.1.3		4	3.2	odd	2
495.2.c.a.199.2		4	5.2	odd	4
495.2.c.a.199.3		4	5.3	odd	4
605.2.b.c.364.2		4	165.32	odd	4
605.2.b.c.364.3		4	165.98	odd	4
605.2.j.i.9.2		16	165.113	even	20
605.2.j.i.9.3		16	165.47	even	20
605.2.j.i.124.2		16	165.137	even	20
605.2.j.i.124.3		16	165.38	even	20
605.2.j.i.269.2		16	165.92	even	20
605.2.j.i.269.3		16	165.158	even	20
605.2.j.i.444.2		16	165.53	even	20
605.2.j.i.444.3		16	165.152	even	20
605.2.j.j.9.2		16	165.107	odd	20
605.2.j.j.9.3		16	165.8	odd	20
605.2.j.j.124.2		16	165.83	odd	20
605.2.j.j.124.3		16	165.17	odd	20
605.2.j.j.269.2		16	165.128	odd	20
605.2.j.j.269.3		16	165.62	odd	20
605.2.j.j.444.2		16	165.2	odd	20
605.2.j.j.444.3		16	165.68	odd	20
880.2.b.h.529.1		4	60.23	odd	4
880.2.b.h.529.4		4	60.47	odd	4
2475.2.a.bi.1.2		4	1.1	even	1	trivial
2475.2.a.bi.1.3		4	5.4	even	2	inner
3025.2.a.ba.1.2		4	33.32	even	2
3025.2.a.ba.1.3		4	165.164	even	2
4400.2.a.cc.1.1		4	12.11	even	2
4400.2.a.cc.1.4		4	60.59	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-0.792287 q^{2} -1.37228 q^{4} +3.46410 q^{7} +2.67181 q^{8} +1.00000 q^{11} -2.74456 q^{14} +0.627719 q^{16} +5.04868 q^{17} +4.00000 q^{19} -0.792287 q^{22} +2.52434 q^{23} -4.75372 q^{28} +2.74456 q^{29} -2.37228 q^{31} -5.84096 q^{32} -4.00000 q^{34} -11.0371 q^{37} -3.16915 q^{38} +2.74456 q^{41} +3.46410 q^{43} -1.37228 q^{44} -2.00000 q^{46} -6.63325 q^{47} +5.00000 q^{49} +3.16915 q^{53} +9.25544 q^{56} -2.17448 q^{58} +1.62772 q^{59} +10.7446 q^{61} +1.87953 q^{62} +3.37228 q^{64} -0.644810 q^{67} -6.92820 q^{68} -7.11684 q^{71} -6.92820 q^{73} +8.74456 q^{74} -5.48913 q^{76} +3.46410 q^{77} +12.7446 q^{79} -2.17448 q^{82} +6.63325 q^{83} -2.74456 q^{86} +2.67181 q^{88} +4.37228 q^{89} -3.46410 q^{92} +5.25544 q^{94} -4.10891 q^{97} -3.96143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 4 q^{11} + 12 q^{14} + 14 q^{16} + 16 q^{19} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 12 q^{41} + 6 q^{44} - 8 q^{46} + 20 q^{49} + 60 q^{56} + 18 q^{59} + 20 q^{61} + 2 q^{64} + 6 q^{71} + 12 q^{74}+ \cdots + 44 q^{94}+O(q^{100}) \) 4 * q + 6 * q^4 + 4 * q^11 + 12 * q^14 + 14 * q^16 + 16 * q^19 - 12 * q^29 + 2 * q^31 - 16 * q^34 - 12 * q^41 + 6 * q^44 - 8 * q^46 + 20 * q^49 + 60 * q^56 + 18 * q^59 + 20 * q^61 + 2 * q^64 + 6 * q^71 + 12 * q^74 + 24 * q^76 + 28 * q^79 + 12 * q^86 + 6 * q^89 + 44 * q^94

Introduction

L-functions

Modular forms

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