LMFDB - Embedded newform 3025.2.a.ba.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3025,2,Mod(1,3025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,6,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.1547466114$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
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			1.3
Root			$0.792287$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.792287 q^{2} -2.52434 q^{3} -1.37228 q^{4} -2.00000 q^{6} +3.46410 q^{7} -2.67181 q^{8} +3.37228 q^{9} +3.46410 q^{12} +2.74456 q^{14} +0.627719 q^{16} -5.04868 q^{17} +2.67181 q^{18} -4.00000 q^{19} -8.74456 q^{21} +2.52434 q^{23} +6.74456 q^{24} -0.939764 q^{27} -4.75372 q^{28} +2.74456 q^{29} -2.37228 q^{31} +5.84096 q^{32} -4.00000 q^{34} -4.62772 q^{36} +11.0371 q^{37} -3.16915 q^{38} +2.74456 q^{41} -6.92820 q^{42} +3.46410 q^{43} +2.00000 q^{46} -6.63325 q^{47} -1.58457 q^{48} +5.00000 q^{49} +12.7446 q^{51} +3.16915 q^{53} -0.744563 q^{54} -9.25544 q^{56} +10.0974 q^{57} +2.17448 q^{58} -1.62772 q^{59} -10.7446 q^{61} -1.87953 q^{62} +11.6819 q^{63} +3.37228 q^{64} +0.644810 q^{67} +6.92820 q^{68} -6.37228 q^{69} +7.11684 q^{71} -9.01011 q^{72} -6.92820 q^{73} +8.74456 q^{74} +5.48913 q^{76} -12.7446 q^{79} -7.74456 q^{81} +2.17448 q^{82} -6.63325 q^{83} +12.0000 q^{84} +2.74456 q^{86} -6.92820 q^{87} -4.37228 q^{89} -3.46410 q^{92} +5.98844 q^{93} -5.25544 q^{94} -14.7446 q^{96} +4.10891 q^{97} +3.96143 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} - 8 q^{6} + 2 q^{9} - 12 q^{14} + 14 q^{16} - 16 q^{19} - 12 q^{21} + 4 q^{24} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 30 q^{36} - 12 q^{41} + 8 q^{46} + 20 q^{49} + 28 q^{51} + 20 q^{54} - 60 q^{56}+ \cdots - 36 q^{96}+O(q^{100}) $ 4 * q + 6 * q^4 - 8 * q^6 + 2 * q^9 - 12 * q^14 + 14 * q^16 - 16 * q^19 - 12 * q^21 + 4 * q^24 - 12 * q^29 + 2 * q^31 - 16 * q^34 - 30 * q^36 - 12 * q^41 + 8 * q^46 + 20 * q^49 + 28 * q^51 + 20 * q^54 - 60 * q^56 - 18 * q^59 - 20 * q^61 + 2 * q^64 - 14 * q^69 - 6 * q^71 + 12 * q^74 - 24 * q^76 - 28 * q^79 - 8 * q^81 + 48 * q^84 - 12 * q^86 - 6 * q^89 - 44 * q^94 - 36 * q^96

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	$-0.409627\pi$
$2$	0.792287	0.560232	0.280116	+	0.959966i	$0.409627\pi$
$3$	−2.52434	−1.45743	−0.728714	−	0.684819i	$-0.759881\pi$
$3$	−2.52434	−1.45743	−0.728714	+	0.684819i	$0.759881\pi$
$4$	−1.37228	−0.686141
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	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
$9$	3.37228	1.12409
$10$	0	0
$11$	0	0
$11$	0	0
$12$	3.46410	1.00000
$13$	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	$-0.709732\pi$
$17$	−5.04868	−1.22448	−0.612242	+	0.790671i	$0.709732\pi$
$18$	2.67181	0.629753
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−8.74456	−1.90822
$22$	0	0
$23$	2.52434	0.526361	0.263180	−	0.964747i	$-0.415229\pi$
$23$	2.52434	0.526361	0.263180	+	0.964747i	$0.415229\pi$
$24$	6.74456	1.37673
$25$	0	0
$26$	0	0
$27$	−0.939764	−0.180858
$28$	−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
$36$	−4.62772	−0.771286
$37$	11.0371	1.81449	0.907245	−	0.420602i	$-0.138181\pi$
$37$	11.0371	1.81449	0.907245	+	0.420602i	$0.138181\pi$
$38$	−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	$0.431248\pi$
$42$	−6.92820	−1.06904
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	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	$0.660736\pi$
$48$	−1.58457	−0.228714
$49$	5.00000	0.714286
$50$	0	0
$51$	12.7446	1.78460
$52$	0	0
$53$	3.16915	0.435316	0.217658	−	0.976025i	$-0.430158\pi$
$53$	3.16915	0.435316	0.217658	+	0.976025i	$0.430158\pi$
$54$	−0.744563	−0.101322
$55$	0	0
$56$	−9.25544	−1.23681
$57$	10.0974	1.33743
$58$	2.17448	0.285523
$59$	−1.62772	−0.211911	−0.105955	−	0.994371i	$-0.533790\pi$
$59$	−1.62772	−0.211911	−0.105955	+	0.994371i	$0.533790\pi$
$60$	0	0
$61$	−10.7446	−1.37570	−0.687850	−	0.725853i	$-0.741445\pi$
$61$	−10.7446	−1.37570	−0.687850	+	0.725853i	$0.741445\pi$
$62$	−1.87953	−0.238700
$63$	11.6819	1.47178
$64$	3.37228	0.421535
$65$	0	0
$66$	0	0
$67$	0.644810	0.0787761	0.0393880	−	0.999224i	$-0.487459\pi$
$67$	0.644810	0.0787761	0.0393880	+	0.999224i	$0.487459\pi$
$68$	6.92820	0.840168
$69$	−6.37228	−0.767133
$70$	0	0
$71$	7.11684	0.844614	0.422307	−	0.906453i	$-0.361220\pi$
$71$	7.11684	0.844614	0.422307	+	0.906453i	$0.361220\pi$
$72$	−9.01011	−1.06185
$73$	−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
$77$	0	0
$78$	0	0
$79$	−12.7446	−1.43388	−0.716938	−	0.697137i	$-0.754457\pi$
$79$	−12.7446	−1.43388	−0.716938	+	0.697137i	$0.754457\pi$
$80$	0	0
$81$	−7.74456	−0.860507
$82$	2.17448	0.240131
$83$	−6.63325	−0.728094	−0.364047	−	0.931381i	$-0.618605\pi$
$83$	−6.63325	−0.728094	−0.364047	+	0.931381i	$0.618605\pi$
$84$	12.0000	1.30931
$85$	0	0
$86$	2.74456	0.295954
$87$	−6.92820	−0.742781
$88$	0	0
$89$	−4.37228	−0.463461	−0.231730	−	0.972780i	$-0.574439\pi$
$89$	−4.37228	−0.463461	−0.231730	+	0.972780i	$0.574439\pi$
$90$	0	0
$91$	0	0
$92$	−3.46410	−0.361158
$93$	5.98844	0.620972
$94$	−5.25544	−0.542057
$95$	0	0
$96$	−14.7446	−1.50486
$97$	4.10891	0.417197	0.208598	−	0.978001i	$-0.433110\pi$
$97$	4.10891	0.417197	0.208598	+	0.978001i	$0.433110\pi$
$98$	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	$0.596490\pi$
$102$	10.0974	0.999787
$103$	10.3923	1.02398	0.511992	−	0.858990i	$-0.328908\pi$
$103$	10.3923	1.02398	0.511992	+	0.858990i	$0.328908\pi$
$104$	0	0
$105$	0	0
$106$	2.51087	0.243878
$107$	6.63325	0.641260	0.320630	−	0.947204i	$-0.396105\pi$
$107$	6.63325	0.641260	0.320630	+	0.947204i	$0.396105\pi$
$108$	1.28962	0.124094
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−27.8614	−2.64449
$112$	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	$0.773147\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	−3.76631	−0.349693
$117$	0	0
$118$	−1.28962	−0.118719
$119$	−17.4891	−1.60323
$120$	0	0
$121$	0	0
$122$	−8.51278	−0.770711
$123$	−6.92820	−0.624695
$124$	3.25544	0.292347
$125$	0	0
$126$	9.25544	0.824540
$127$	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
$129$	−8.74456	−0.769916
$130$	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	$0.469640\pi$
$138$	−5.04868	−0.429772
$139$	−18.2337	−1.54656	−0.773281	−	0.634064i	$-0.781386\pi$
$139$	−18.2337	−1.54656	−0.773281	+	0.634064i	$0.781386\pi$
$140$	0	0
$141$	16.7446	1.41015
$142$	5.63858	0.473179
$143$	0	0
$144$	2.11684	0.176404
$145$	0	0
$146$	−5.48913	−0.454283
$147$	−12.6217	−1.04102
$148$	−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	$-0.859887\pi$
$151$	−22.2337	−1.80935	−0.904676	+	0.426100i	$0.859887\pi$
$152$	10.6873	0.866851
$153$	−17.0256	−1.37643
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.39853	−0.430850	−0.215425	−	0.976520i	$-0.569114\pi$
$157$	−5.39853	−0.430850	−0.215425	+	0.976520i	$0.569114\pi$
$158$	−10.0974	−0.803302
$159$	−8.00000	−0.634441
$160$	0	0
$161$	8.74456	0.689168
$162$	−6.13592	−0.482083
$163$	3.46410	0.271329	0.135665	−	0.990755i	$-0.456683\pi$
$163$	3.46410	0.271329	0.135665	+	0.990755i	$0.456683\pi$
$164$	−3.76631	−0.294100
$165$	0	0
$166$	−5.25544	−0.407901
$167$	−22.3692	−1.73098	−0.865490	−	0.500927i	$-0.832993\pi$
$167$	−22.3692	−1.73098	−0.865490	+	0.500927i	$0.832993\pi$
$168$	23.3639	1.80256
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−13.4891	−1.03154
$172$	−4.75372	−0.362468
$173$	−1.87953	−0.142898	−0.0714489	−	0.997444i	$-0.522762\pi$
$173$	−1.87953	−0.142898	−0.0714489	+	0.997444i	$0.522762\pi$
$174$	−5.48913	−0.416130
$175$	0	0
$176$	0	0
$177$	4.10891	0.308845
$178$	−3.46410	−0.259645
$179$	−15.8614	−1.18554	−0.592769	−	0.805373i	$-0.701965\pi$
$179$	−15.8614	−1.18554	−0.592769	+	0.805373i	$0.701965\pi$
$180$	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
$183$	27.1229	2.00498
$184$	−6.74456	−0.497216
$185$	0	0
$186$	4.74456	0.347888
$187$	0	0
$188$	9.10268	0.663881
$189$	−3.25544	−0.236798
$190$	0	0
$191$	−13.6277	−0.986067	−0.493034	−	0.870010i	$-0.664112\pi$
$191$	−13.6277	−0.986067	−0.493034	+	0.870010i	$0.664112\pi$
$192$	−8.51278	−0.614357
$193$	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	$-0.478671\pi$
$197$	1.87953	0.133911	0.0669554	+	0.997756i	$0.478671\pi$
$198$	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
$201$	−1.62772	−0.114810
$202$	−4.75372	−0.334471
$203$	9.50744	0.667292
$204$	−17.4891	−1.22448
$205$	0	0
$206$	8.23369	0.573668
$207$	8.51278	0.591679
$208$	0	0
$209$	0	0
$210$	0	0
$211$	21.4891	1.47937	0.739686	−	0.672952i	$-0.234974\pi$
$211$	21.4891	1.47937	0.739686	+	0.672952i	$0.234974\pi$
$212$	−4.34896	−0.298688
$213$	−17.9653	−1.23096
$214$	5.25544	0.359254
$215$	0	0
$216$	2.51087	0.170843
$217$	−8.21782	−0.557862
$218$	−7.92287	−0.536604
$219$	17.4891	1.18181
$220$	0	0
$221$	0	0
$222$	−22.0742	−1.48153
$223$	−7.57301	−0.507126	−0.253563	−	0.967319i	$-0.581603\pi$
$223$	−7.57301	−0.507126	−0.253563	+	0.967319i	$0.581603\pi$
$224$	20.2337	1.35192
$225$	0	0
$226$	−12.7446	−0.847756
$227$	−9.80240	−0.650608	−0.325304	−	0.945609i	$-0.605467\pi$
$227$	−9.80240	−0.650608	−0.325304	+	0.945609i	$0.605467\pi$
$228$	−13.8564	−0.917663
$229$	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	$-0.688312\pi$
$233$	−17.0256	−1.11538	−0.557691	+	0.830049i	$0.688312\pi$
$234$	0	0
$235$	0	0
$236$	2.23369	0.145401
$237$	32.1716	2.08977
$238$	−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	$-0.554140\pi$
$241$	−5.25544	−0.338532	−0.169266	+	0.985570i	$0.554140\pi$
$242$	0	0
$243$	22.3692	1.43498
$244$	14.7446	0.943924
$245$	0	0
$246$	−5.48913	−0.349974
$247$	0	0
$248$	6.33830	0.402482
$249$	16.7446	1.06114
$250$	0	0
$251$	−4.88316	−0.308222	−0.154111	−	0.988054i	$-0.549251\pi$
$251$	−4.88316	−0.308222	−0.154111	+	0.988054i	$0.549251\pi$
$252$	−16.0309	−1.00985
$253$	0	0
$254$	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	$0.768553\pi$
$258$	−6.92820	−0.431331
$259$	38.2337	2.37573
$260$	0	0
$261$	9.25544	0.572897
$262$	−6.92820	−0.428026
$263$	14.1514	0.872610	0.436305	−	0.899799i	$-0.356287\pi$
$263$	14.1514	0.872610	0.436305	+	0.899799i	$0.356287\pi$
$264$	0	0
$265$	0	0
$266$	−10.9783	−0.673120
$267$	11.0371	0.675460
$268$	−0.884861	−0.0540515
$269$	11.4891	0.700504	0.350252	−	0.936655i	$-0.386096\pi$
$269$	11.4891	0.700504	0.350252	+	0.936655i	$0.386096\pi$
$270$	0	0
$271$	9.48913	0.576423	0.288212	−	0.957567i	$-0.406939\pi$
$271$	9.48913	0.576423	0.288212	+	0.957567i	$0.406939\pi$
$272$	−3.16915	−0.192158
$273$	0	0
$274$	1.76631	0.106707
$275$	0	0
$276$	8.74456	0.526361
$277$	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
$279$	−8.00000	−0.478947
$280$	0	0
$281$	23.4891	1.40124	0.700622	−	0.713533i	$-0.252906\pi$
$281$	23.4891	1.40124	0.700622	+	0.713533i	$0.252906\pi$
$282$	13.2665	0.790009
$283$	−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
$288$	19.6974	1.16068
$289$	8.48913	0.499360
$290$	0	0
$291$	−10.3723	−0.608034
$292$	9.50744	0.556381
$293$	−10.0974	−0.589894	−0.294947	−	0.955514i	$-0.595302\pi$
$293$	−10.0974	−0.589894	−0.294947	+	0.955514i	$0.595302\pi$
$294$	−10.0000	−0.583212
$295$	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
$303$	15.1460	0.870117
$304$	−2.51087	−0.144009
$305$	0	0
$306$	−13.4891	−0.771122
$307$	−28.1176	−1.60475	−0.802377	−	0.596817i	$-0.796432\pi$
$307$	−28.1176	−1.60475	−0.802377	+	0.596817i	$0.796432\pi$
$308$	0	0
$309$	−26.2337	−1.49238
$310$	0	0
$311$	−17.4891	−0.991717	−0.495859	−	0.868403i	$-0.665147\pi$
$311$	−17.4891	−0.991717	−0.495859	+	0.868403i	$0.665147\pi$
$312$	0	0
$313$	31.8217	1.79867	0.899335	−	0.437260i	$-0.144051\pi$
$313$	31.8217	1.79867	0.899335	+	0.437260i	$0.144051\pi$
$314$	−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	$0.468492\pi$
$318$	−6.33830	−0.355434
$319$	0	0
$320$	0	0
$321$	−16.7446	−0.934590
$322$	6.92820	0.386094
$323$	20.1947	1.12366
$324$	10.6277	0.590429
$325$	0	0
$326$	2.74456	0.152007
$327$	25.2434	1.39596
$328$	−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
$333$	37.2203	2.03966
$334$	−17.7228	−0.969749
$335$	0	0
$336$	−5.48913	−0.299456
$337$	−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
$339$	40.6060	2.20541
$340$	0	0
$341$	0	0
$342$	−10.6873	−0.577901
$343$	−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	$-0.788049\pi$
$347$	−29.2974	−1.57277	−0.786383	+	0.617739i	$0.788049\pi$
$348$	9.50744	0.509652
$349$	7.48913	0.400884	0.200442	−	0.979706i	$-0.435762\pi$
$349$	7.48913	0.400884	0.200442	+	0.979706i	$0.435762\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.7244	−1.15627	−0.578136	−	0.815941i	$-0.696220\pi$
$353$	−21.7244	−1.15627	−0.578136	+	0.815941i	$0.696220\pi$
$354$	3.25544	0.173025
$355$	0	0
$356$	6.00000	0.317999
$357$	44.1485	2.33658
$358$	−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
$366$	21.4891	1.12325
$367$	24.0087	1.25324	0.626621	−	0.779324i	$-0.284437\pi$
$367$	24.0087	1.25324	0.626621	+	0.779324i	$0.284437\pi$
$368$	1.58457	0.0826016
$369$	9.25544	0.481819
$370$	0	0
$371$	10.9783	0.569962
$372$	−8.21782	−0.426074
$373$	8.21782	0.425503	0.212751	−	0.977106i	$-0.431758\pi$
$373$	8.21782	0.425503	0.212751	+	0.977106i	$0.431758\pi$
$374$	0	0
$375$	0	0
$376$	17.7228	0.913984
$377$	0	0
$378$	−2.57924	−0.132662
$379$	−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
$381$	−29.4891	−1.51077
$382$	−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	$0.546467\pi$
$384$	22.7446	1.16068
$385$	0	0
$386$	18.5109	0.942179
$387$	11.6819	0.593826
$388$	−5.63858	−0.286256
$389$	−9.86141	−0.499993	−0.249997	−	0.968247i	$-0.580429\pi$
$389$	−9.86141	−0.499993	−0.249997	+	0.968247i	$0.580429\pi$
$390$	0	0
$391$	−12.7446	−0.644520
$392$	−13.3591	−0.674735
$393$	22.0742	1.11350
$394$	1.48913	0.0750210
$395$	0	0
$396$	0	0
$397$	−23.3639	−1.17260	−0.586299	−	0.810095i	$-0.699416\pi$
$397$	−23.3639	−1.17260	−0.586299	+	0.810095i	$0.699416\pi$
$398$	−6.33830	−0.317710
$399$	34.9783	1.75110
$400$	0	0
$401$	11.4891	0.573740	0.286870	−	0.957970i	$-0.407385\pi$
$401$	11.4891	0.573740	0.286870	+	0.957970i	$0.407385\pi$
$402$	−1.28962	−0.0643204
$403$	0	0
$404$	8.23369	0.409641
$405$	0	0
$406$	7.53262	0.373838
$407$	0	0
$408$	−34.0511	−1.68578
$409$	−4.51087	−0.223048	−0.111524	−	0.993762i	$-0.535573\pi$
$409$	−4.51087	−0.223048	−0.111524	+	0.993762i	$0.535573\pi$
$410$	0	0
$411$	−5.62772	−0.277595
$412$	−14.2612	−0.702597
$413$	−5.63858	−0.277457
$414$	6.74456	0.331477
$415$	0	0
$416$	0	0
$417$	46.0280	2.25400
$418$	0	0
$419$	22.9783	1.12256	0.561280	−	0.827626i	$-0.310309\pi$
$419$	22.9783	1.12256	0.561280	+	0.827626i	$0.310309\pi$
$420$	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
$423$	−22.3692	−1.08763
$424$	−8.46738	−0.411212
$425$	0	0
$426$	−14.2337	−0.689624
$427$	−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	$0.776776\pi$
$432$	−0.589907	−0.0283819
$433$	20.5446	0.987308	0.493654	−	0.869658i	$-0.335661\pi$
$433$	20.5446	0.987308	0.493654	+	0.869658i	$0.335661\pi$
$434$	−6.51087	−0.312532
$435$	0	0
$436$	13.7228	0.657204
$437$	−10.0974	−0.483022
$438$	13.8564	0.662085
$439$	1.48913	0.0710721	0.0355360	−	0.999368i	$-0.488686\pi$
$439$	1.48913	0.0710721	0.0355360	+	0.999368i	$0.488686\pi$
$440$	0	0
$441$	16.8614	0.802924
$442$	0	0
$443$	15.0911	0.717001	0.358500	−	0.933530i	$-0.383288\pi$
$443$	15.0911	0.717001	0.358500	+	0.933530i	$0.383288\pi$
$444$	38.2337	1.81449
$445$	0	0
$446$	−6.00000	−0.284108
$447$	29.0024	1.37177
$448$	11.6819	0.551919
$449$	−21.8614	−1.03170	−0.515852	−	0.856678i	$-0.672524\pi$
$449$	−21.8614	−1.03170	−0.515852	+	0.856678i	$0.672524\pi$
$450$	0	0
$451$	0	0
$452$	22.0742	1.03828
$453$	56.1253	2.63700
$454$	−7.76631	−0.364491
$455$	0	0
$456$	−26.9783	−1.26337
$457$	−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
$459$	4.74456	0.221457
$460$	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	$-0.344979\pi$
$463$	20.1398	0.935976	0.467988	+	0.883735i	$0.344979\pi$
$464$	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
$471$	13.6277	0.627932
$472$	4.34896	0.200177
$473$	0	0
$474$	25.4891	1.17075
$475$	0	0
$476$	24.0000	1.10004
$477$	10.6873	0.489336
$478$	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
$483$	−22.0742	−1.00441
$484$	0	0
$485$	0	0
$486$	17.7228	0.803923
$487$	−22.7190	−1.02950	−0.514749	−	0.857341i	$-0.672115\pi$
$487$	−22.7190	−1.02950	−0.514749	+	0.857341i	$0.672115\pi$
$488$	28.7075	1.29953
$489$	−8.74456	−0.395443
$490$	0	0
$491$	−29.4891	−1.33083	−0.665413	−	0.746476i	$-0.731744\pi$
$491$	−29.4891	−1.33083	−0.665413	+	0.746476i	$0.731744\pi$
$492$	9.50744	0.428629
$493$	−13.8564	−0.624061
$494$	0	0
$495$	0	0
$496$	−1.48913	−0.0668637
$497$	24.6535	1.10586
$498$	13.2665	0.594486
$499$	−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
$501$	56.4674	2.52278
$502$	−3.86886	−0.172676
$503$	0.294954	0.0131513	0.00657567	−	0.999978i	$-0.497907\pi$
$503$	0.294954	0.0131513	0.00657567	+	0.999978i	$0.497907\pi$
$504$	−31.2119	−1.39029
$505$	0	0
$506$	0	0
$507$	32.8164	1.45743
$508$	−16.0309	−0.711256
$509$	28.3723	1.25758	0.628790	−	0.777575i	$-0.283551\pi$
$509$	28.3723	1.25758	0.628790	+	0.777575i	$0.283551\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	7.02078	0.310277
$513$	3.75906	0.165966
$514$	−18.9783	−0.837095
$515$	0	0
$516$	12.0000	0.528271
$517$	0	0
$518$	30.2921	1.33096
$519$	4.74456	0.208263
$520$	0	0
$521$	18.6060	0.815142	0.407571	−	0.913173i	$-0.366376\pi$
$521$	18.6060	0.815142	0.407571	+	0.913173i	$0.366376\pi$
$522$	7.33296	0.320955
$523$	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
$531$	−5.48913	−0.238208
$532$	19.0149	0.824400
$533$	0	0
$534$	8.74456	0.378414
$535$	0	0
$536$	−1.72281	−0.0744142
$537$	40.0395	1.72783
$538$	9.10268	0.392445
$539$	0	0
$540$	0	0
$541$	0.233688	0.0100470	0.00502351	−	0.999987i	$-0.498401\pi$
$541$	0.233688	0.0100470	0.00502351	+	0.999987i	$0.498401\pi$
$542$	7.51811	0.322930
$543$	−17.3754	−0.745650
$544$	−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
$549$	−36.2337	−1.54642
$550$	0	0
$551$	−10.9783	−0.467689
$552$	17.0256	0.724656
$553$	−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	$-0.261295\pi$
$557$	32.1716	1.36315	0.681577	+	0.731747i	$0.261295\pi$
$558$	−6.33830	−0.268321
$559$	0	0
$560$	0	0
$561$	0	0
$562$	18.6101	0.785021
$563$	−12.2718	−0.517196	−0.258598	−	0.965985i	$-0.583261\pi$
$563$	−12.2718	−0.517196	−0.258598	+	0.965985i	$0.583261\pi$
$564$	−22.9783	−0.967559
$565$	0	0
$566$	−3.76631	−0.158310
$567$	−26.8280	−1.12667
$568$	−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	$-0.351550\pi$
$571$	21.4891	0.899292	0.449646	+	0.893207i	$0.351550\pi$
$572$	0	0
$573$	34.4010	1.43712
$574$	7.53262	0.314406
$575$	0	0
$576$	11.3723	0.473845
$577$	−31.8217	−1.32476	−0.662378	−	0.749170i	$-0.730453\pi$
$577$	−31.8217	−1.32476	−0.662378	+	0.749170i	$0.730453\pi$
$578$	6.72582	0.279757
$579$	−58.9783	−2.45105
$580$	0	0
$581$	−22.9783	−0.953298
$582$	−8.21782	−0.340640
$583$	0	0
$584$	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	$0.303833\pi$
$588$	17.3205	0.714286
$589$	9.48913	0.390993
$590$	0	0
$591$	−4.74456	−0.195165
$592$	6.92820	0.284747
$593$	43.5586	1.78874	0.894368	−	0.447333i	$-0.147626\pi$
$593$	43.5586	1.78874	0.894368	+	0.447333i	$0.147626\pi$
$594$	0	0
$595$	0	0
$596$	15.7663	0.645813
$597$	20.1947	0.826514
$598$	0	0
$599$	−34.9783	−1.42917	−0.714586	−	0.699547i	$-0.753385\pi$
$599$	−34.9783	−1.42917	−0.714586	+	0.699547i	$0.753385\pi$
$600$	0	0
$601$	−30.4674	−1.24279	−0.621395	−	0.783497i	$-0.713434\pi$
$601$	−30.4674	−1.24279	−0.621395	+	0.783497i	$0.713434\pi$
$602$	9.50744	0.387494
$603$	2.17448	0.0885517
$604$	30.5109	1.24147
$605$	0	0
$606$	12.0000	0.487467
$607$	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
$609$	−24.0000	−0.972529
$610$	0	0
$611$	0	0
$612$	23.3639	0.944428
$613$	−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
$616$	0	0
$617$	3.75906	0.151334	0.0756669	−	0.997133i	$-0.475891\pi$
$617$	3.75906	0.151334	0.0756669	+	0.997133i	$0.475891\pi$
$618$	−20.7846	−0.836080
$619$	−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
$621$	−2.37228	−0.0951964
$622$	−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
$633$	−54.2458	−2.15608
$634$	2.78806	0.110728
$635$	0	0
$636$	10.9783	0.435316
$637$	0	0
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	−19.6277	−0.775248	−0.387624	−	0.921818i	$-0.626704\pi$
$641$	−19.6277	−0.775248	−0.387624	+	0.921818i	$0.626704\pi$
$642$	−13.2665	−0.523587
$643$	−39.1547	−1.54411	−0.772055	−	0.635556i	$-0.780771\pi$
$643$	−39.1547	−1.54411	−0.772055	+	0.635556i	$0.780771\pi$
$644$	−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	$0.798701\pi$
$648$	20.6920	0.812860
$649$	0	0
$650$	0	0
$651$	20.7446	0.813044
$652$	−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	$0.666950\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	1.72281	0.0672646
$657$	−23.3639	−0.911511
$658$	−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
$666$	29.4891	1.14268
$667$	6.92820	0.268261
$668$	30.6968	1.18770
$669$	19.1168	0.739100
$670$	0	0
$671$	0	0
$672$	−51.0767	−1.97033
$673$	−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	$-0.185505\pi$
$677$	43.4487	1.66987	0.834935	+	0.550348i	$0.185505\pi$
$678$	32.1716	1.23554
$679$	14.2337	0.546239
$680$	0	0
$681$	24.7446	0.948214
$682$	0	0
$683$	−44.4434	−1.70058	−0.850290	−	0.526314i	$-0.823573\pi$
$683$	−44.4434	−1.70058	−0.850290	+	0.526314i	$0.823573\pi$
$684$	18.5109	0.707781
$685$	0	0
$686$	−5.48913	−0.209576
$687$	−51.4265	−1.96204
$688$	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
$696$	18.5109	0.701653
$697$	−13.8564	−0.524849
$698$	5.93354	0.224588
$699$	42.9783	1.62559
$700$	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
$704$	0	0
$705$	0	0
$706$	−17.2119	−0.647780
$707$	−20.7846	−0.781686
$708$	−5.63858	−0.211911
$709$	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
$711$	−42.9783	−1.61181
$712$	11.6819	0.437799
$713$	−5.98844	−0.224269
$714$	34.9783	1.30903
$715$	0	0
$716$	21.7663	0.813445
$717$	−8.21782	−0.306900
$718$	−5.15848	−0.192513
$719$	21.3505	0.796240	0.398120	−	0.917333i	$-0.369663\pi$
$719$	21.3505	0.796240	0.398120	+	0.917333i	$0.369663\pi$
$720$	0	0
$721$	36.0000	1.34071
$722$	−2.37686	−0.0884576
$723$	13.2665	0.493386
$724$	−9.44563	−0.351044
$725$	0	0
$726$	0	0
$727$	15.7908	0.585650	0.292825	−	0.956166i	$-0.405405\pi$
$727$	15.7908	0.585650	0.292825	+	0.956166i	$0.405405\pi$
$728$	0	0
$729$	−33.2337	−1.23088
$730$	0	0
$731$	−17.4891	−0.646859
$732$	−37.2203	−1.37570
$733$	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	0
$738$	7.33296	0.269930
$739$	−0.744563	−0.0273892	−0.0136946	−	0.999906i	$-0.504359\pi$
$739$	−0.744563	−0.0273892	−0.0136946	+	0.999906i	$0.504359\pi$
$740$	0	0
$741$	0	0
$742$	8.69793	0.319311
$743$	−21.7793	−0.799004	−0.399502	−	0.916732i	$-0.630817\pi$
$743$	−21.7793	−0.799004	−0.399502	+	0.916732i	$0.630817\pi$
$744$	−16.0000	−0.586588
$745$	0	0
$746$	6.51087	0.238380
$747$	−22.3692	−0.818446
$748$	0	0
$749$	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
$753$	12.3267	0.449211
$754$	0	0
$755$	0	0
$756$	4.46738	0.162477
$757$	39.7995	1.44654	0.723269	−	0.690567i	$-0.242639\pi$
$757$	39.7995	1.44654	0.723269	+	0.690567i	$0.242639\pi$
$758$	−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	$0.625886\pi$
$762$	−23.3639	−0.846383
$763$	−34.6410	−1.25409
$764$	18.7011	0.676581
$765$	0	0
$766$	−4.51087	−0.162985
$767$	0	0
$768$	35.0458	1.26461
$769$	51.2119	1.84675	0.923375	−	0.383900i	$-0.125419\pi$
$769$	51.2119	1.84675	0.923375	+	0.383900i	$0.125419\pi$
$770$	0	0
$771$	60.4674	2.17768
$772$	−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	$0.312575\pi$
$774$	9.25544	0.332680
$775$	0	0
$776$	−10.9783	−0.394096
$777$	−96.5147	−3.46245
$778$	−7.81306	−0.280112
$779$	−10.9783	−0.393337
$780$	0	0
$781$	0	0
$782$	−10.0974	−0.361081
$783$	−2.57924	−0.0921745
$784$	3.13859	0.112093
$785$	0	0
$786$	17.4891	0.623816
$787$	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
$789$	−35.7228	−1.27177
$790$	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	$0.313428\pi$
$798$	27.7128	0.981023
$799$	33.4891	1.18476
$800$	0	0
$801$	−14.7446	−0.520974
$802$	9.10268	0.321427
$803$	0	0
$804$	2.23369	0.0787761
$805$	0	0
$806$	0	0
$807$	−29.0024	−1.02093
$808$	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	$-0.705252\pi$
$811$	−34.2337	−1.20211	−0.601054	+	0.799209i	$0.705252\pi$
$812$	−13.0469	−0.457856
$813$	−23.9538	−0.840095
$814$	0	0
$815$	0	0
$816$	8.00000	0.280056
$817$	−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	$0.398297\pi$
$822$	−4.45877	−0.155517
$823$	33.5161	1.16830	0.584149	−	0.811646i	$-0.301428\pi$
$823$	33.5161	1.16830	0.584149	+	0.811646i	$0.301428\pi$
$824$	−27.7663	−0.967285
$825$	0	0
$826$	−4.46738	−0.155440
$827$	−18.0202	−0.626624	−0.313312	−	0.949650i	$-0.601439\pi$
$827$	−18.0202	−0.626624	−0.313312	+	0.949650i	$0.601439\pi$
$828$	−11.6819	−0.405975
$829$	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
$831$	−20.7446	−0.719621
$832$	0	0
$833$	−25.2434	−0.874631
$834$	36.4674	1.26276
$835$	0	0
$836$	0	0
$837$	2.22938	0.0770588
$838$	18.2054	0.628894
$839$	7.11684	0.245701	0.122850	−	0.992425i	$-0.460796\pi$
$839$	7.11684	0.245701	0.122850	+	0.992425i	$0.460796\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	24.9484	0.859779
$843$	−59.2945	−2.04221
$844$	−29.4891	−1.01506
$845$	0	0
$846$	−17.7228	−0.609323
$847$	0	0
$848$	1.98933	0.0683140
$849$	12.0000	0.411839
$850$	0	0
$851$	27.8614	0.955077
$852$	24.6535	0.844614
$853$	−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	$-0.558430\pi$
$857$	−10.6873	−0.365070	−0.182535	+	0.983199i	$0.558430\pi$
$858$	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
$861$	−24.0000	−0.817918
$862$	−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	$0.631920\pi$
$864$	−5.48913	−0.186744
$865$	0	0
$866$	16.2772	0.553121
$867$	−21.4294	−0.727781
$868$	11.2772	0.382772
$869$	0	0
$870$	0	0
$871$	0	0
$872$	26.7181	0.904791
$873$	13.8564	0.468968
$874$	−8.00000	−0.270604
$875$	0	0
$876$	−24.0000	−0.810885
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	1.17981	0.0398168
$879$	25.4891	0.859727
$880$	0	0
$881$	21.8614	0.736530	0.368265	−	0.929721i	$-0.379952\pi$
$881$	21.8614	0.736530	0.368265	+	0.929721i	$0.379952\pi$
$882$	13.3591	0.449823
$883$	−24.2487	−0.816034	−0.408017	−	0.912974i	$-0.633780\pi$
$883$	−24.2487	−0.816034	−0.408017	+	0.912974i	$0.633780\pi$
$884$	0	0
$885$	0	0
$886$	11.9565	0.401687
$887$	−14.1514	−0.475156	−0.237578	−	0.971368i	$-0.576354\pi$
$887$	−14.1514	−0.475156	−0.237578	+	0.971368i	$0.576354\pi$
$888$	74.4405	2.49806
$889$	40.4674	1.35723
$890$	0	0
$891$	0	0
$892$	10.3923	0.347960
$893$	26.5330	0.887893
$894$	22.9783	0.768508
$895$	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
$902$	0	0
$903$	−30.2921	−1.00806
$904$	42.9783	1.42944
$905$	0	0
$906$	44.4674	1.47733
$907$	−19.8997	−0.660760	−0.330380	−	0.943848i	$-0.607177\pi$
$907$	−19.8997	−0.660760	−0.330380	+	0.943848i	$0.607177\pi$
$908$	13.4516	0.446409
$909$	−20.2337	−0.671109
$910$	0	0
$911$	−30.5109	−1.01087	−0.505435	−	0.862865i	$-0.668668\pi$
$911$	−30.5109	−1.01087	−0.505435	+	0.862865i	$0.668668\pi$
$912$	6.33830	0.209882
$913$	0	0
$914$	−16.4674	−0.544692
$915$	0	0
$916$	−27.9565	−0.923709
$917$	−30.2921	−1.00033
$918$	3.75906	0.124067
$919$	−6.23369	−0.205630	−0.102815	−	0.994700i	$-0.532785\pi$
$919$	−6.23369	−0.205630	−0.102815	+	0.994700i	$0.532785\pi$
$920$	0	0
$921$	70.9783	2.33881
$922$	25.5383	0.841060
$923$	0	0
$924$	0	0
$925$	0	0
$926$	15.9565	0.524363
$927$	35.0458	1.15105
$928$	16.0309	0.526240
$929$	−52.9783	−1.73816	−0.869080	−	0.494672i	$-0.835288\pi$
$929$	−52.9783	−1.73816	−0.869080	+	0.494672i	$0.835288\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	23.3639	0.765308
$933$	44.1485	1.44536
$934$	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
$939$	−80.3288	−2.62143
$940$	0	0
$941$	−10.4674	−0.341227	−0.170613	−	0.985338i	$-0.554575\pi$
$941$	−10.4674	−0.341227	−0.170613	+	0.985338i	$0.554575\pi$
$942$	10.7971	0.351787
$943$	6.92820	0.225613
$944$	−1.02175	−0.0332551
$945$	0	0
$946$	0	0
$947$	−56.1802	−1.82561	−0.912806	−	0.408393i	$-0.866089\pi$
$947$	−56.1802	−1.82561	−0.912806	+	0.408393i	$0.866089\pi$
$948$	−44.1485	−1.43388
$949$	0	0
$950$	0	0
$951$	−8.88316	−0.288056
$952$	46.7277	1.51445
$953$	−41.6790	−1.35012	−0.675058	−	0.737765i	$-0.735881\pi$
$953$	−41.6790	−1.35012	−0.675058	+	0.737765i	$0.735881\pi$
$954$	8.46738	0.274141
$955$	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
$963$	22.3692	0.720837
$964$	7.21194	0.232281
$965$	0	0
$966$	−17.4891	−0.562703
$967$	−46.3229	−1.48965	−0.744823	−	0.667262i	$-0.767466\pi$
$967$	−46.3229	−1.48965	−0.744823	+	0.667262i	$0.767466\pi$
$968$	0	0
$969$	−50.9783	−1.63766
$970$	0	0
$971$	54.0951	1.73599	0.867997	−	0.496569i	$-0.165407\pi$
$971$	54.0951	1.73599	0.867997	+	0.496569i	$0.165407\pi$
$972$	−30.6968	−0.984601
$973$	−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	$0.355790\pi$
$978$	−6.92820	−0.221540
$979$	0	0
$980$	0	0
$981$	−33.7228	−1.07669
$982$	−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	$0.458445\pi$
$984$	18.5109	0.590105
$985$	0	0
$986$	−10.9783	−0.349619
$987$	58.0049	1.84632
$988$	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
$993$	−7.86797	−0.249682
$994$	19.5326	0.619537
$995$	0	0
$996$	−22.9783	−0.728094
$997$	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
$999$	−10.3723	−0.328164

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.ba.1.3		4
5.2	odd	4		605.2.b.c.364.3		4
5.3	odd	4		605.2.b.c.364.2		4
5.4	even	2	inner	3025.2.a.ba.1.2		4
11.10	odd	2		275.2.a.h.1.2		4
33.32	even	2		2475.2.a.bi.1.3		4
44.43	even	2		4400.2.a.cc.1.4		4
55.2	even	20		605.2.j.i.444.2		16
55.3	odd	20		605.2.j.j.9.2		16
55.7	even	20		605.2.j.i.269.3		16
55.8	even	20		605.2.j.i.9.3		16
55.13	even	20		605.2.j.i.444.3		16
55.17	even	20		605.2.j.i.124.3		16
55.18	even	20		605.2.j.i.269.2		16
55.27	odd	20		605.2.j.j.124.2		16
55.28	even	20		605.2.j.i.124.2		16
55.32	even	4		55.2.b.a.34.2	✓	4
55.37	odd	20		605.2.j.j.269.2		16
55.38	odd	20		605.2.j.j.124.3		16
55.42	odd	20		605.2.j.j.444.3		16
55.43	even	4		55.2.b.a.34.3	yes	4
55.47	odd	20		605.2.j.j.9.3		16
55.48	odd	20		605.2.j.j.269.3		16
55.52	even	20		605.2.j.i.9.2		16
55.53	odd	20		605.2.j.j.444.2		16
55.54	odd	2		275.2.a.h.1.3		4
165.32	odd	4		495.2.c.a.199.3		4
165.98	odd	4		495.2.c.a.199.2		4
165.164	even	2		2475.2.a.bi.1.2		4
220.43	odd	4		880.2.b.h.529.4		4
220.87	odd	4		880.2.b.h.529.1		4
220.219	even	2		4400.2.a.cc.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.b.a.34.2	✓	4	55.32	even	4
55.2.b.a.34.3	yes	4	55.43	even	4
275.2.a.h.1.2		4	11.10	odd	2
275.2.a.h.1.3		4	55.54	odd	2
495.2.c.a.199.2		4	165.98	odd	4
495.2.c.a.199.3		4	165.32	odd	4
605.2.b.c.364.2		4	5.3	odd	4
605.2.b.c.364.3		4	5.2	odd	4
605.2.j.i.9.2		16	55.52	even	20
605.2.j.i.9.3		16	55.8	even	20
605.2.j.i.124.2		16	55.28	even	20
605.2.j.i.124.3		16	55.17	even	20
605.2.j.i.269.2		16	55.18	even	20
605.2.j.i.269.3		16	55.7	even	20
605.2.j.i.444.2		16	55.2	even	20
605.2.j.i.444.3		16	55.13	even	20
605.2.j.j.9.2		16	55.3	odd	20
605.2.j.j.9.3		16	55.47	odd	20
605.2.j.j.124.2		16	55.27	odd	20
605.2.j.j.124.3		16	55.38	odd	20
605.2.j.j.269.2		16	55.37	odd	20
605.2.j.j.269.3		16	55.48	odd	20
605.2.j.j.444.2		16	55.53	odd	20
605.2.j.j.444.3		16	55.42	odd	20
880.2.b.h.529.1		4	220.87	odd	4
880.2.b.h.529.4		4	220.43	odd	4
2475.2.a.bi.1.2		4	165.164	even	2
2475.2.a.bi.1.3		4	33.32	even	2
3025.2.a.ba.1.2		4	5.4	even	2	inner
3025.2.a.ba.1.3		4	1.1	even	1	trivial
4400.2.a.cc.1.1		4	220.219	even	2
4400.2.a.cc.1.4		4	44.43	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.792287 q^{2} -2.52434 q^{3} -1.37228 q^{4} -2.00000 q^{6} +3.46410 q^{7} -2.67181 q^{8} +3.37228 q^{9} +3.46410 q^{12} +2.74456 q^{14} +0.627719 q^{16} -5.04868 q^{17} +2.67181 q^{18} -4.00000 q^{19} -8.74456 q^{21} +2.52434 q^{23} +6.74456 q^{24} -0.939764 q^{27} -4.75372 q^{28} +2.74456 q^{29} -2.37228 q^{31} +5.84096 q^{32} -4.00000 q^{34} -4.62772 q^{36} +11.0371 q^{37} -3.16915 q^{38} +2.74456 q^{41} -6.92820 q^{42} +3.46410 q^{43} +2.00000 q^{46} -6.63325 q^{47} -1.58457 q^{48} +5.00000 q^{49} +12.7446 q^{51} +3.16915 q^{53} -0.744563 q^{54} -9.25544 q^{56} +10.0974 q^{57} +2.17448 q^{58} -1.62772 q^{59} -10.7446 q^{61} -1.87953 q^{62} +11.6819 q^{63} +3.37228 q^{64} +0.644810 q^{67} +6.92820 q^{68} -6.37228 q^{69} +7.11684 q^{71} -9.01011 q^{72} -6.92820 q^{73} +8.74456 q^{74} +5.48913 q^{76} -12.7446 q^{79} -7.74456 q^{81} +2.17448 q^{82} -6.63325 q^{83} +12.0000 q^{84} +2.74456 q^{86} -6.92820 q^{87} -4.37228 q^{89} -3.46410 q^{92} +5.98844 q^{93} -5.25544 q^{94} -14.7446 q^{96} +4.10891 q^{97} +3.96143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} - 8 q^{6} + 2 q^{9} - 12 q^{14} + 14 q^{16} - 16 q^{19} - 12 q^{21} + 4 q^{24} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 30 q^{36} - 12 q^{41} + 8 q^{46} + 20 q^{49} + 28 q^{51} + 20 q^{54} - 60 q^{56}+ \cdots - 36 q^{96}+O(q^{100}) \) 4 * q + 6 * q^4 - 8 * q^6 + 2 * q^9 - 12 * q^14 + 14 * q^16 - 16 * q^19 - 12 * q^21 + 4 * q^24 - 12 * q^29 + 2 * q^31 - 16 * q^34 - 30 * q^36 - 12 * q^41 + 8 * q^46 + 20 * q^49 + 28 * q^51 + 20 * q^54 - 60 * q^56 - 18 * q^59 - 20 * q^61 + 2 * q^64 - 14 * q^69 - 6 * q^71 + 12 * q^74 - 24 * q^76 - 28 * q^79 - 8 * q^81 + 48 * q^84 - 12 * q^86 - 6 * q^89 - 44 * q^94 - 36 * q^96

Introduction

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