LMFDB - Embedded newform 9025.2.a.cv.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9025,2,Mod(1,9025)] 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("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [40,0,0,48,0,20,0,0,52,0,20,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.66900 q^{2} -1.76244 q^{3} +5.12357 q^{4} +4.70394 q^{6} -2.20993 q^{7} -8.33681 q^{8} +0.106178 q^{9} +2.86854 q^{11} -9.02996 q^{12} -4.83675 q^{13} +5.89830 q^{14} +12.0038 q^{16} +2.06280 q^{17} -0.283389 q^{18} +3.89485 q^{21} -7.65613 q^{22} -4.40865 q^{23} +14.6931 q^{24} +12.9093 q^{26} +5.10017 q^{27} -11.3227 q^{28} -8.08729 q^{29} -3.97776 q^{31} -15.3646 q^{32} -5.05561 q^{33} -5.50563 q^{34} +0.544010 q^{36} +9.05780 q^{37} +8.52446 q^{39} -1.21984 q^{41} -10.3954 q^{42} +5.13320 q^{43} +14.6972 q^{44} +11.7667 q^{46} -2.50131 q^{47} -21.1559 q^{48} -2.11623 q^{49} -3.63556 q^{51} -24.7814 q^{52} +1.22600 q^{53} -13.6124 q^{54} +18.4237 q^{56} +21.5850 q^{58} -0.244718 q^{59} +0.498155 q^{61} +10.6166 q^{62} -0.234645 q^{63} +17.0004 q^{64} +13.4934 q^{66} -7.09797 q^{67} +10.5689 q^{68} +7.76996 q^{69} +9.59683 q^{71} -0.885185 q^{72} -0.287288 q^{73} -24.1753 q^{74} -6.33926 q^{77} -22.7518 q^{78} -15.8865 q^{79} -9.30726 q^{81} +3.25576 q^{82} -5.70094 q^{83} +19.9555 q^{84} -13.7005 q^{86} +14.2533 q^{87} -23.9144 q^{88} -16.6834 q^{89} +10.6889 q^{91} -22.5880 q^{92} +7.01055 q^{93} +6.67600 q^{94} +27.0791 q^{96} -0.202690 q^{97} +5.64821 q^{98} +0.304575 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 48 q^{4} + 20 q^{6} + 52 q^{9} + 20 q^{11} + 40 q^{16} + 92 q^{24} + 76 q^{26} + 156 q^{36} + 80 q^{39} + 48 q^{44} + 72 q^{49} + 32 q^{54} + 80 q^{61} + 72 q^{64} + 16 q^{66} + 100 q^{74} + 40 q^{81}+ \cdots + 128 q^{99}+O(q^{100}) $ 40 * q + 48 * q^4 + 20 * q^6 + 52 * q^9 + 20 * q^11 + 40 * q^16 + 92 * q^24 + 76 * q^26 + 156 * q^36 + 80 * q^39 + 48 * q^44 + 72 * q^49 + 32 * q^54 + 80 * q^61 + 72 * q^64 + 16 * q^66 + 100 * q^74 + 40 * q^81 + 380 * q^96 + 128 * q^99

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 $

$2$	−2.66900	−1.88727	−0.943634	−	0.330989i	$-0.892618\pi$
$2$	−2.66900	−1.88727	−0.943634	+	0.330989i	$0.892618\pi$
$3$	−1.76244	−1.01754	−0.508771	−	0.860902i	$-0.669900\pi$
$3$	−1.76244	−1.01754	−0.508771	+	0.860902i	$0.669900\pi$
$4$	5.12357	2.56178
$5$	0	0
$5$	0	0
$6$	4.70394	1.92038
$7$	−2.20993	−0.835274	−0.417637	−	0.908614i	$-0.637142\pi$
$7$	−2.20993	−0.835274	−0.417637	+	0.908614i	$0.637142\pi$
$8$	−8.33681	−2.94751
$9$	0.106178	0.0353926
$10$	0	0
$11$	2.86854	0.864897	0.432448	−	0.901659i	$-0.357650\pi$
$11$	2.86854	0.864897	0.432448	+	0.901659i	$0.357650\pi$
$12$	−9.02996	−2.60672
$13$	−4.83675	−1.34147	−0.670736	−	0.741696i	$-0.734022\pi$
$13$	−4.83675	−1.34147	−0.670736	+	0.741696i	$0.734022\pi$
$14$	5.89830	1.57639
$15$	0	0
$16$	12.0038	3.00095
$17$	2.06280	0.500304	0.250152	−	0.968207i	$-0.419519\pi$
$17$	2.06280	0.500304	0.250152	+	0.968207i	$0.419519\pi$
$18$	−0.283389	−0.0667954
$19$	0	0
$19$	0	0
$20$	0	0
$21$	3.89485	0.849926
$22$	−7.65613	−1.63229
$23$	−4.40865	−0.919267	−0.459634	−	0.888109i	$-0.652019\pi$
$23$	−4.40865	−0.919267	−0.459634	+	0.888109i	$0.652019\pi$
$24$	14.6931	2.99921
$25$	0	0
$26$	12.9093	2.53172
$27$	5.10017	0.981529
$28$	−11.3227	−2.13979
$29$	−8.08729	−1.50177	−0.750886	−	0.660432i	$-0.770373\pi$
$29$	−8.08729	−1.50177	−0.750886	+	0.660432i	$0.770373\pi$
$30$	0	0
$31$	−3.97776	−0.714427	−0.357213	−	0.934023i	$-0.616273\pi$
$31$	−3.97776	−0.714427	−0.357213	+	0.934023i	$0.616273\pi$
$32$	−15.3646	−2.71610
$33$	−5.05561	−0.880069
$34$	−5.50563	−0.944207
$35$	0	0
$36$	0.544010	0.0906683
$37$	9.05780	1.48909	0.744546	−	0.667571i	$-0.232666\pi$
$37$	9.05780	1.48909	0.744546	+	0.667571i	$0.232666\pi$
$38$	0	0
$39$	8.52446	1.36501
$40$	0	0
$41$	−1.21984	−0.190507	−0.0952537	−	0.995453i	$-0.530366\pi$
$41$	−1.21984	−0.190507	−0.0952537	+	0.995453i	$0.530366\pi$
$42$	−10.3954	−1.60404
$43$	5.13320	0.782806	0.391403	−	0.920219i	$-0.371990\pi$
$43$	5.13320	0.782806	0.391403	+	0.920219i	$0.371990\pi$
$44$	14.6972	2.21568
$45$	0	0
$46$	11.7667	1.73490
$47$	−2.50131	−0.364854	−0.182427	−	0.983219i	$-0.558395\pi$
$47$	−2.50131	−0.364854	−0.182427	+	0.983219i	$0.558395\pi$
$48$	−21.1559	−3.05360
$49$	−2.11623	−0.302318
$50$	0	0
$51$	−3.63556	−0.509080
$52$	−24.7814	−3.43656
$53$	1.22600	0.168404	0.0842018	−	0.996449i	$-0.473166\pi$
$53$	1.22600	0.168404	0.0842018	+	0.996449i	$0.473166\pi$
$54$	−13.6124	−1.85241
$55$	0	0
$56$	18.4237	2.46197
$57$	0	0
$58$	21.5850	2.83425
$59$	−0.244718	−0.0318596	−0.0159298	−	0.999873i	$-0.505071\pi$
$59$	−0.244718	−0.0318596	−0.0159298	+	0.999873i	$0.505071\pi$
$60$	0	0
$61$	0.498155	0.0637823	0.0318911	−	0.999491i	$-0.489847\pi$
$61$	0.498155	0.0637823	0.0318911	+	0.999491i	$0.489847\pi$
$62$	10.6166	1.34832
$63$	−0.234645	−0.0295625
$64$	17.0004	2.12506
$65$	0	0
$66$	13.4934	1.66093
$67$	−7.09797	−0.867155	−0.433578	−	0.901116i	$-0.642749\pi$
$67$	−7.09797	−0.867155	−0.433578	+	0.901116i	$0.642749\pi$
$68$	10.5689	1.28167
$69$	7.76996	0.935393
$70$	0	0
$71$	9.59683	1.13893	0.569467	−	0.822014i	$-0.307150\pi$
$71$	9.59683	1.13893	0.569467	+	0.822014i	$0.307150\pi$
$72$	−0.885185	−0.104320
$73$	−0.287288	−0.0336245	−0.0168122	−	0.999859i	$-0.505352\pi$
$73$	−0.287288	−0.0336245	−0.0168122	+	0.999859i	$0.505352\pi$
$74$	−24.1753	−2.81032
$75$	0	0
$76$	0	0
$77$	−6.33926	−0.722425
$78$	−22.7518	−2.57613
$79$	−15.8865	−1.78737	−0.893687	−	0.448691i	$-0.851890\pi$
$79$	−15.8865	−1.78737	−0.893687	+	0.448691i	$0.851890\pi$
$80$	0	0
$81$	−9.30726	−1.03414
$82$	3.25576	0.359539
$83$	−5.70094	−0.625759	−0.312879	−	0.949793i	$-0.601294\pi$
$83$	−5.70094	−0.625759	−0.312879	+	0.949793i	$0.601294\pi$
$84$	19.9555	2.17733
$85$	0	0
$86$	−13.7005	−1.47737
$87$	14.2533	1.52812
$88$	−23.9144	−2.54929
$89$	−16.6834	−1.76844	−0.884220	−	0.467072i	$-0.845309\pi$
$89$	−16.6834	−1.76844	−0.884220	+	0.467072i	$0.845309\pi$
$90$	0	0
$91$	10.6889	1.12050
$92$	−22.5880	−2.35496
$93$	7.01055	0.726960
$94$	6.67600	0.688577
$95$	0	0
$96$	27.0791	2.76375
$97$	−0.202690	−0.0205801	−0.0102900	−	0.999947i	$-0.503275\pi$
$97$	−0.202690	−0.0205801	−0.0102900	+	0.999947i	$0.503275\pi$
$98$	5.64821	0.570556
$99$	0.304575	0.0306110
$100$	0	0
$101$	2.36522	0.235349	0.117674	−	0.993052i	$-0.462456\pi$
$101$	2.36522	0.235349	0.117674	+	0.993052i	$0.462456\pi$
$102$	9.70331	0.960771
$103$	−1.07422	−0.105846	−0.0529231	−	0.998599i	$-0.516854\pi$
$103$	−1.07422	−0.105846	−0.0529231	+	0.998599i	$0.516854\pi$
$104$	40.3230	3.95400
$105$	0	0
$106$	−3.27219	−0.317823
$107$	0.186188	0.0179994	0.00899972	−	0.999960i	$-0.497135\pi$
$107$	0.186188	0.0179994	0.00899972	+	0.999960i	$0.497135\pi$
$108$	26.1311	2.51446
$109$	13.1387	1.25846	0.629229	−	0.777220i	$-0.283371\pi$
$109$	13.1387	1.25846	0.629229	+	0.777220i	$0.283371\pi$
$110$	0	0
$111$	−15.9638	−1.51522
$112$	−26.5275	−2.50662
$113$	−4.36458	−0.410586	−0.205293	−	0.978701i	$-0.565815\pi$
$113$	−4.36458	−0.410586	−0.205293	+	0.978701i	$0.565815\pi$
$114$	0	0
$115$	0	0
$116$	−41.4358	−3.84721
$117$	−0.513556	−0.0474783
$118$	0.653154	0.0601277
$119$	−4.55864	−0.417890
$120$	0	0
$121$	−2.77149	−0.251953
$122$	−1.32958	−0.120374
$123$	2.14989	0.193849
$124$	−20.3803	−1.83021
$125$	0	0
$126$	0.626269	0.0557925
$127$	−3.40581	−0.302217	−0.151108	−	0.988517i	$-0.548284\pi$
$127$	−3.40581	−0.302217	−0.151108	+	0.988517i	$0.548284\pi$
$128$	−14.6451	−1.29445
$129$	−9.04694	−0.796538
$130$	0	0
$131$	0.573140	0.0500755	0.0250378	−	0.999687i	$-0.492029\pi$
$131$	0.573140	0.0500755	0.0250378	+	0.999687i	$0.492029\pi$
$132$	−25.9028	−2.25455
$133$	0	0
$134$	18.9445	1.63655
$135$	0	0
$136$	−17.1972	−1.47465
$137$	−4.82207	−0.411977	−0.205988	−	0.978554i	$-0.566041\pi$
$137$	−4.82207	−0.411977	−0.205988	+	0.978554i	$0.566041\pi$
$138$	−20.7380	−1.76534
$139$	−14.9925	−1.27164	−0.635822	−	0.771836i	$-0.719339\pi$
$139$	−14.9925	−1.27164	−0.635822	+	0.771836i	$0.719339\pi$
$140$	0	0
$141$	4.40840	0.371254
$142$	−25.6139	−2.14947
$143$	−13.8744	−1.16024
$144$	1.27454	0.106212
$145$	0	0
$146$	0.766771	0.0634584
$147$	3.72971	0.307622
$148$	46.4082	3.81473
$149$	12.6439	1.03583	0.517914	−	0.855433i	$-0.326709\pi$
$149$	12.6439	1.03583	0.517914	+	0.855433i	$0.326709\pi$
$150$	0	0
$151$	−11.0266	−0.897328	−0.448664	−	0.893701i	$-0.648100\pi$
$151$	−11.0266	−0.897328	−0.448664	+	0.893701i	$0.648100\pi$
$152$	0	0
$153$	0.219024	0.0177071
$154$	16.9195	1.36341
$155$	0	0
$156$	43.6756	3.49685
$157$	−22.4216	−1.78944	−0.894718	−	0.446632i	$-0.852623\pi$
$157$	−22.4216	−1.78944	−0.894718	+	0.446632i	$0.852623\pi$
$158$	42.4011	3.37325
$159$	−2.16074	−0.171358
$160$	0	0
$161$	9.74279	0.767840
$162$	24.8411	1.95170
$163$	−1.74791	−0.136907	−0.0684535	−	0.997654i	$-0.521806\pi$
$163$	−1.74791	−0.136907	−0.0684535	+	0.997654i	$0.521806\pi$
$164$	−6.24994	−0.488039
$165$	0	0
$166$	15.2158	1.18098
$167$	21.0484	1.62877	0.814386	−	0.580324i	$-0.197074\pi$
$167$	21.0484	1.62877	0.814386	+	0.580324i	$0.197074\pi$
$168$	−32.4706	−2.50516
$169$	10.3941	0.799549
$170$	0	0
$171$	0	0
$172$	26.3003	2.00538
$173$	11.6088	0.882602	0.441301	−	0.897359i	$-0.354517\pi$
$173$	11.6088	0.882602	0.441301	+	0.897359i	$0.354517\pi$
$174$	−38.0421	−2.88397
$175$	0	0
$176$	34.4334	2.59551
$177$	0.431300	0.0324185
$178$	44.5281	3.33752
$179$	3.98091	0.297547	0.148774	−	0.988871i	$-0.452467\pi$
$179$	3.98091	0.297547	0.148774	+	0.988871i	$0.452467\pi$
$180$	0	0
$181$	−2.81859	−0.209504	−0.104752	−	0.994498i	$-0.533405\pi$
$181$	−2.81859	−0.209504	−0.104752	+	0.994498i	$0.533405\pi$
$182$	−28.5286	−2.11468
$183$	−0.877967	−0.0649012
$184$	36.7541	2.70955
$185$	0	0
$186$	−18.7112	−1.37197
$187$	5.91723	0.432711
$188$	−12.8156	−0.934676
$189$	−11.2710	−0.819845
$190$	0	0
$191$	−19.8105	−1.43344	−0.716719	−	0.697362i	$-0.754357\pi$
$191$	−19.8105	−1.43344	−0.716719	+	0.697362i	$0.754357\pi$
$192$	−29.9622	−2.16233
$193$	−5.52399	−0.397626	−0.198813	−	0.980037i	$-0.563709\pi$
$193$	−5.52399	−0.397626	−0.198813	+	0.980037i	$0.563709\pi$
$194$	0.540981	0.0388402
$195$	0	0
$196$	−10.8426	−0.774474
$197$	20.4993	1.46051	0.730257	−	0.683173i	$-0.239400\pi$
$197$	20.4993	1.46051	0.730257	+	0.683173i	$0.239400\pi$
$198$	−0.812912	−0.0577712
$199$	−17.2575	−1.22335	−0.611677	−	0.791107i	$-0.709505\pi$
$199$	−17.2575	−1.22335	−0.611677	+	0.791107i	$0.709505\pi$
$200$	0	0
$201$	12.5097	0.882367
$202$	−6.31279	−0.444166
$203$	17.8723	1.25439
$204$	−18.6270	−1.30415
$205$	0	0
$206$	2.86710	0.199760
$207$	−0.468101	−0.0325353
$208$	−58.0594	−4.02570
$209$	0	0
$210$	0	0
$211$	21.2379	1.46207	0.731037	−	0.682337i	$-0.239036\pi$
$211$	21.2379	1.46207	0.731037	+	0.682337i	$0.239036\pi$
$212$	6.28148	0.431414
$213$	−16.9138	−1.15891
$214$	−0.496935	−0.0339698
$215$	0	0
$216$	−42.5192	−2.89306
$217$	8.79056	0.596742
$218$	−35.0672	−2.37505
$219$	0.506326	0.0342143
$220$	0	0
$221$	−9.97726	−0.671143
$222$	42.6074	2.85962
$223$	−28.8555	−1.93231	−0.966154	−	0.257967i	$-0.916947\pi$
$223$	−28.8555	−1.93231	−0.966154	+	0.257967i	$0.916947\pi$
$224$	33.9546	2.26869
$225$	0	0
$226$	11.6491	0.774885
$227$	−8.42533	−0.559209	−0.279604	−	0.960115i	$-0.590203\pi$
$227$	−8.42533	−0.559209	−0.279604	+	0.960115i	$0.590203\pi$
$228$	0	0
$229$	24.6658	1.62996	0.814981	−	0.579487i	$-0.196747\pi$
$229$	24.6658	1.62996	0.814981	+	0.579487i	$0.196747\pi$
$230$	0	0
$231$	11.1725	0.735099
$232$	67.4221	4.42648
$233$	−20.7240	−1.35768	−0.678839	−	0.734287i	$-0.737516\pi$
$233$	−20.7240	−1.35768	−0.678839	+	0.734287i	$0.737516\pi$
$234$	1.37068	0.0896042
$235$	0	0
$236$	−1.25383	−0.0816175
$237$	27.9990	1.81873
$238$	12.1670	0.788671
$239$	10.4341	0.674924	0.337462	−	0.941339i	$-0.390431\pi$
$239$	10.4341	0.674924	0.337462	+	0.941339i	$0.390431\pi$
$240$	0	0
$241$	20.0493	1.29149	0.645744	−	0.763554i	$-0.276548\pi$
$241$	20.0493	1.29149	0.645744	+	0.763554i	$0.276548\pi$
$242$	7.39710	0.475504
$243$	1.10292	0.0707524
$244$	2.55233	0.163396
$245$	0	0
$246$	−5.73807	−0.365846
$247$	0	0
$248$	33.1618	2.10578
$249$	10.0475	0.636736
$250$	0	0
$251$	−0.614289	−0.0387736	−0.0193868	−	0.999812i	$-0.506171\pi$
$251$	−0.614289	−0.0387736	−0.0193868	+	0.999812i	$0.506171\pi$
$252$	−1.20222	−0.0757328
$253$	−12.6464	−0.795071
$254$	9.09012	0.570365
$255$	0	0
$256$	5.08679	0.317924
$257$	10.1007	0.630064	0.315032	−	0.949081i	$-0.397985\pi$
$257$	10.1007	0.630064	0.315032	+	0.949081i	$0.397985\pi$
$258$	24.1463	1.50328
$259$	−20.0171	−1.24380
$260$	0	0
$261$	−0.858691	−0.0531517
$262$	−1.52971	−0.0945060
$263$	6.14445	0.378883	0.189441	−	0.981892i	$-0.439332\pi$
$263$	6.14445	0.378883	0.189441	+	0.981892i	$0.439332\pi$
$264$	42.1477	2.59401
$265$	0	0
$266$	0	0
$267$	29.4034	1.79946
$268$	−36.3669	−2.22146
$269$	23.0791	1.40716	0.703579	−	0.710617i	$-0.251584\pi$
$269$	23.0791	1.40716	0.703579	+	0.710617i	$0.251584\pi$
$270$	0	0
$271$	13.7769	0.836885	0.418443	−	0.908243i	$-0.362576\pi$
$271$	13.7769	0.836885	0.418443	+	0.908243i	$0.362576\pi$
$272$	24.7615	1.50139
$273$	−18.8384	−1.14015
$274$	12.8701	0.777511
$275$	0	0
$276$	39.8099	2.39628
$277$	−18.2471	−1.09636	−0.548181	−	0.836360i	$-0.684679\pi$
$277$	−18.2471	−1.09636	−0.548181	+	0.836360i	$0.684679\pi$
$278$	40.0149	2.39993
$279$	−0.422350	−0.0252855
$280$	0	0
$281$	−20.3990	−1.21690	−0.608451	−	0.793591i	$-0.708209\pi$
$281$	−20.3990	−1.21690	−0.608451	+	0.793591i	$0.708209\pi$
$282$	−11.7660	−0.700656
$283$	−19.5454	−1.16186	−0.580928	−	0.813955i	$-0.697310\pi$
$283$	−19.5454	−1.16186	−0.580928	+	0.813955i	$0.697310\pi$
$284$	49.1700	2.91770
$285$	0	0
$286$	37.0308	2.18968
$287$	2.69576	0.159126
$288$	−1.63138	−0.0961299
$289$	−12.7448	−0.749696
$290$	0	0
$291$	0.357229	0.0209411
$292$	−1.47194	−0.0861386
$293$	9.22402	0.538873	0.269436	−	0.963018i	$-0.413163\pi$
$293$	9.22402	0.538873	0.269436	+	0.963018i	$0.413163\pi$
$294$	−9.95461	−0.580565
$295$	0	0
$296$	−75.5131	−4.38911
$297$	14.6300	0.848921
$298$	−33.7466	−1.95489
$299$	21.3235	1.23317
$300$	0	0
$301$	−11.3440	−0.653857
$302$	29.4299	1.69350
$303$	−4.16855	−0.239477
$304$	0	0
$305$	0	0
$306$	−0.584576	−0.0334180
$307$	7.75219	0.442441	0.221220	−	0.975224i	$-0.428996\pi$
$307$	7.75219	0.442441	0.221220	+	0.975224i	$0.428996\pi$
$308$	−32.4796	−1.85070
$309$	1.89325	0.107703
$310$	0	0
$311$	24.8427	1.40870	0.704350	−	0.709853i	$-0.251239\pi$
$311$	24.8427	1.40870	0.704350	+	0.709853i	$0.251239\pi$
$312$	−71.0667	−4.02336
$313$	−5.79256	−0.327415	−0.163707	−	0.986509i	$-0.552345\pi$
$313$	−5.79256	−0.327415	−0.163707	+	0.986509i	$0.552345\pi$
$314$	59.8432	3.37715
$315$	0	0
$316$	−81.3957	−4.57886
$317$	−4.59924	−0.258319	−0.129159	−	0.991624i	$-0.541228\pi$
$317$	−4.59924	−0.258319	−0.129159	+	0.991624i	$0.541228\pi$
$318$	5.76702	0.323398
$319$	−23.1987	−1.29888
$320$	0	0
$321$	−0.328144	−0.0183152
$322$	−26.0035	−1.44912
$323$	0	0
$324$	−47.6864	−2.64924
$325$	0	0
$326$	4.66518	0.258380
$327$	−23.1561	−1.28053
$328$	10.1696	0.561522
$329$	5.52771	0.304753
$330$	0	0
$331$	7.39548	0.406493	0.203246	−	0.979128i	$-0.434851\pi$
$331$	7.39548	0.406493	0.203246	+	0.979128i	$0.434851\pi$
$332$	−29.2091	−1.60306
$333$	0.961738	0.0527029
$334$	−56.1781	−3.07393
$335$	0	0
$336$	46.7531	2.55059
$337$	−16.6765	−0.908425	−0.454212	−	0.890893i	$-0.650079\pi$
$337$	−16.6765	−0.908425	−0.454212	+	0.890893i	$0.650079\pi$
$338$	−27.7420	−1.50896
$339$	7.69230	0.417788
$340$	0	0
$341$	−11.4104	−0.617906
$342$	0	0
$343$	20.1462	1.08779
$344$	−42.7945	−2.30733
$345$	0	0
$346$	−30.9840	−1.66571
$347$	−36.0300	−1.93419	−0.967095	−	0.254417i	$-0.918116\pi$
$347$	−36.0300	−1.93419	−0.967095	+	0.254417i	$0.918116\pi$
$348$	73.0278	3.91470
$349$	−23.6370	−1.26526	−0.632629	−	0.774455i	$-0.718024\pi$
$349$	−23.6370	−1.26526	−0.632629	+	0.774455i	$0.718024\pi$
$350$	0	0
$351$	−24.6683	−1.31669
$352$	−44.0739	−2.34914
$353$	−33.9305	−1.80594	−0.902968	−	0.429708i	$-0.858617\pi$
$353$	−33.9305	−1.80594	−0.902968	+	0.429708i	$0.858617\pi$
$354$	−1.15114	−0.0611825
$355$	0	0
$356$	−85.4786	−4.53036
$357$	8.03432	0.425221
$358$	−10.6251	−0.561551
$359$	−9.28731	−0.490165	−0.245083	−	0.969502i	$-0.578815\pi$
$359$	−9.28731	−0.490165	−0.245083	+	0.969502i	$0.578815\pi$
$360$	0	0
$361$	0	0
$362$	7.52282	0.395391
$363$	4.88457	0.256373
$364$	54.7651	2.87047
$365$	0	0
$366$	2.34329	0.122486
$367$	11.4283	0.596555	0.298277	−	0.954479i	$-0.403588\pi$
$367$	11.4283	0.596555	0.298277	+	0.954479i	$0.403588\pi$
$368$	−52.9206	−2.75868
$369$	−0.129520	−0.00674256
$370$	0	0
$371$	−2.70936	−0.140663
$372$	35.9190	1.86231
$373$	−25.7467	−1.33312	−0.666558	−	0.745454i	$-0.732233\pi$
$373$	−25.7467	−1.33312	−0.666558	+	0.745454i	$0.732233\pi$
$374$	−15.7931	−0.816642
$375$	0	0
$376$	20.8529	1.07541
$377$	39.1162	2.01459
$378$	30.0823	1.54727
$379$	−20.3628	−1.04597	−0.522984	−	0.852343i	$-0.675181\pi$
$379$	−20.3628	−1.04597	−0.522984	+	0.852343i	$0.675181\pi$
$380$	0	0
$381$	6.00252	0.307519
$382$	52.8742	2.70528
$383$	−18.1004	−0.924886	−0.462443	−	0.886649i	$-0.653027\pi$
$383$	−18.1004	−0.924886	−0.462443	+	0.886649i	$0.653027\pi$
$384$	25.8110	1.31716
$385$	0	0
$386$	14.7435	0.750426
$387$	0.545033	0.0277056
$388$	−1.03850	−0.0527217
$389$	−30.3016	−1.53635	−0.768175	−	0.640239i	$-0.778835\pi$
$389$	−30.3016	−1.53635	−0.768175	+	0.640239i	$0.778835\pi$
$390$	0	0
$391$	−9.09418	−0.459913
$392$	17.6426	0.891084
$393$	−1.01012	−0.0509540
$394$	−54.7126	−2.75638
$395$	0	0
$396$	1.56051	0.0784187
$397$	3.57059	0.179203	0.0896015	−	0.995978i	$-0.471441\pi$
$397$	3.57059	0.179203	0.0896015	+	0.995978i	$0.471441\pi$
$398$	46.0604	2.30880
$399$	0	0
$400$	0	0
$401$	−0.553079	−0.0276194	−0.0138097	−	0.999905i	$-0.504396\pi$
$401$	−0.553079	−0.0276194	−0.0138097	+	0.999905i	$0.504396\pi$
$402$	−33.3884	−1.66526
$403$	19.2394	0.958384
$404$	12.1184	0.602912
$405$	0	0
$406$	−47.7012	−2.36737
$407$	25.9826	1.28791
$408$	30.3089	1.50052
$409$	−2.56270	−0.126718	−0.0633588	−	0.997991i	$-0.520181\pi$
$409$	−2.56270	−0.126718	−0.0633588	+	0.997991i	$0.520181\pi$
$410$	0	0
$411$	8.49858	0.419204
$412$	−5.50385	−0.271155
$413$	0.540810	0.0266115
$414$	1.24936	0.0614028
$415$	0	0
$416$	74.3146	3.64357
$417$	26.4232	1.29395
$418$	0	0
$419$	−9.01138	−0.440235	−0.220117	−	0.975473i	$-0.570644\pi$
$419$	−9.01138	−0.440235	−0.220117	+	0.975473i	$0.570644\pi$
$420$	0	0
$421$	−8.61610	−0.419923	−0.209962	−	0.977710i	$-0.567334\pi$
$421$	−8.61610	−0.419923	−0.209962	+	0.977710i	$0.567334\pi$
$422$	−56.6839	−2.75933
$423$	−0.265584	−0.0129131
$424$	−10.2209	−0.496371
$425$	0	0
$426$	45.1429	2.18718
$427$	−1.10089	−0.0532756
$428$	0.953945	0.0461107
$429$	24.4527	1.18059
$430$	0	0
$431$	5.21526	0.251210	0.125605	−	0.992080i	$-0.459913\pi$
$431$	5.21526	0.251210	0.125605	+	0.992080i	$0.459913\pi$
$432$	61.2215	2.94552
$433$	5.17272	0.248585	0.124293	−	0.992246i	$-0.460334\pi$
$433$	5.17272	0.248585	0.124293	+	0.992246i	$0.460334\pi$
$434$	−23.4620	−1.12621
$435$	0	0
$436$	67.3170	3.22390
$437$	0	0
$438$	−1.35138	−0.0645716
$439$	−16.1031	−0.768560	−0.384280	−	0.923217i	$-0.625550\pi$
$439$	−16.1031	−0.768560	−0.384280	+	0.923217i	$0.625550\pi$
$440$	0	0
$441$	−0.224697	−0.0106998
$442$	26.6293	1.26663
$443$	12.1169	0.575693	0.287846	−	0.957677i	$-0.407061\pi$
$443$	12.1169	0.575693	0.287846	+	0.957677i	$0.407061\pi$
$444$	−81.7915	−3.88165
$445$	0	0
$446$	77.0154	3.64678
$447$	−22.2841	−1.05400
$448$	−37.5697	−1.77500
$449$	−7.07384	−0.333835	−0.166918	−	0.985971i	$-0.553381\pi$
$449$	−7.07384	−0.333835	−0.166918	+	0.985971i	$0.553381\pi$
$450$	0	0
$451$	−3.49916	−0.164769
$452$	−22.3622	−1.05183
$453$	19.4336	0.913069
$454$	22.4872	1.05538
$455$	0	0
$456$	0	0
$457$	−26.3810	−1.23405	−0.617027	−	0.786942i	$-0.711663\pi$
$457$	−26.3810	−1.23405	−0.617027	+	0.786942i	$0.711663\pi$
$458$	−65.8331	−3.07618
$459$	10.5207	0.491062
$460$	0	0
$461$	−9.59005	−0.446653	−0.223327	−	0.974744i	$-0.571692\pi$
$461$	−9.59005	−0.446653	−0.223327	+	0.974744i	$0.571692\pi$
$462$	−29.8195	−1.38733
$463$	6.50235	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$463$	6.50235	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$464$	−97.0783	−4.50674
$465$	0	0
$466$	55.3125	2.56230
$467$	8.71364	0.403219	0.201610	−	0.979466i	$-0.435383\pi$
$467$	8.71364	0.403219	0.201610	+	0.979466i	$0.435383\pi$
$468$	−2.63124	−0.121629
$469$	15.6860	0.724312
$470$	0	0
$471$	39.5165	1.82083
$472$	2.04017	0.0939064
$473$	14.7248	0.677047
$474$	−74.7293	−3.43243
$475$	0	0
$476$	−23.3565	−1.07054
$477$	0.130174	0.00596025
$478$	−27.8486	−1.27376
$479$	27.3003	1.24738	0.623692	−	0.781670i	$-0.285632\pi$
$479$	27.3003	1.24738	0.623692	+	0.781670i	$0.285632\pi$
$480$	0	0
$481$	−43.8103	−1.99758
$482$	−53.5115	−2.43738
$483$	−17.1710	−0.781309
$484$	−14.1999	−0.645450
$485$	0	0
$486$	−2.94370	−0.133529
$487$	−35.5582	−1.61129	−0.805647	−	0.592395i	$-0.798182\pi$
$487$	−35.5582	−1.61129	−0.805647	+	0.592395i	$0.798182\pi$
$488$	−4.15302	−0.187999
$489$	3.08058	0.139309
$490$	0	0
$491$	41.5813	1.87654	0.938269	−	0.345906i	$-0.112428\pi$
$491$	41.5813	1.87654	0.938269	+	0.345906i	$0.112428\pi$
$492$	11.0151	0.496600
$493$	−16.6825	−0.751342
$494$	0	0
$495$	0	0
$496$	−47.7483	−2.14396
$497$	−21.2083	−0.951321
$498$	−26.8169	−1.20169
$499$	−27.1248	−1.21427	−0.607136	−	0.794598i	$-0.707682\pi$
$499$	−27.1248	−1.21427	−0.607136	+	0.794598i	$0.707682\pi$
$500$	0	0
$501$	−37.0964	−1.65734
$502$	1.63954	0.0731762
$503$	34.8198	1.55254	0.776269	−	0.630401i	$-0.217110\pi$
$503$	34.8198	1.55254	0.776269	+	0.630401i	$0.217110\pi$
$504$	1.95619	0.0871358
$505$	0	0
$506$	33.7532	1.50051
$507$	−18.3190	−0.813575
$508$	−17.4499	−0.774215
$509$	41.2656	1.82907	0.914533	−	0.404512i	$-0.132559\pi$
$509$	41.2656	1.82907	0.914533	+	0.404512i	$0.132559\pi$
$510$	0	0
$511$	0.634884	0.0280856
$512$	15.7135	0.694443
$513$	0	0
$514$	−26.9588	−1.18910
$515$	0	0
$516$	−46.3526	−2.04056
$517$	−7.17511	−0.315561
$518$	53.4256	2.34738
$519$	−20.4598	−0.898085
$520$	0	0
$521$	−33.5554	−1.47009	−0.735045	−	0.678018i	$-0.762839\pi$
$521$	−33.5554	−1.47009	−0.735045	+	0.678018i	$0.762839\pi$
$522$	2.29185	0.100311
$523$	−2.43983	−0.106687	−0.0533433	−	0.998576i	$-0.516988\pi$
$523$	−2.43983	−0.106687	−0.0533433	+	0.998576i	$0.516988\pi$
$524$	2.93652	0.128283
$525$	0	0
$526$	−16.3995	−0.715054
$527$	−8.20534	−0.357430
$528$	−60.6866	−2.64105
$529$	−3.56380	−0.154948
$530$	0	0
$531$	−0.0259837	−0.00112760
$532$	0	0
$533$	5.90007	0.255560
$534$	−78.4778	−3.39607
$535$	0	0
$536$	59.1744	2.55594
$537$	−7.01609	−0.302767
$538$	−61.5981	−2.65568
$539$	−6.07048	−0.261474
$540$	0	0
$541$	−27.7635	−1.19365	−0.596823	−	0.802373i	$-0.703571\pi$
$541$	−27.7635	−1.19365	−0.596823	+	0.802373i	$0.703571\pi$
$542$	−36.7705	−1.57943
$543$	4.96758	0.213179
$544$	−31.6941	−1.35887
$545$	0	0
$546$	50.2798	2.15177
$547$	25.2149	1.07811	0.539056	−	0.842270i	$-0.318781\pi$
$547$	25.2149	1.07811	0.539056	+	0.842270i	$0.318781\pi$
$548$	−24.7062	−1.05540
$549$	0.0528931	0.00225742
$550$	0	0
$551$	0	0
$552$	−64.7767	−2.75708
$553$	35.1080	1.49295
$554$	48.7015	2.06913
$555$	0	0
$556$	−76.8148	−3.25767
$557$	−18.0110	−0.763151	−0.381576	−	0.924338i	$-0.624618\pi$
$557$	−18.0110	−0.763151	−0.381576	+	0.924338i	$0.624618\pi$
$558$	1.12725	0.0477205
$559$	−24.8280	−1.05011
$560$	0	0
$561$	−10.4287	−0.440302
$562$	54.4450	2.29662
$563$	8.99696	0.379177	0.189588	−	0.981864i	$-0.439285\pi$
$563$	8.99696	0.379177	0.189588	+	0.981864i	$0.439285\pi$
$564$	22.5867	0.951073
$565$	0	0
$566$	52.1668	2.19273
$567$	20.5684	0.863790
$568$	−80.0069	−3.35701
$569$	25.4166	1.06552	0.532759	−	0.846267i	$-0.321155\pi$
$569$	25.4166	1.06552	0.532759	+	0.846267i	$0.321155\pi$
$570$	0	0
$571$	16.9476	0.709233	0.354617	−	0.935012i	$-0.384611\pi$
$571$	16.9476	0.709233	0.354617	+	0.935012i	$0.384611\pi$
$572$	−71.0864	−2.97227
$573$	34.9147	1.45858
$574$	−7.19499	−0.300313
$575$	0	0
$576$	1.80507	0.0752113
$577$	25.1670	1.04772	0.523859	−	0.851805i	$-0.324492\pi$
$577$	25.1670	1.04772	0.523859	+	0.851805i	$0.324492\pi$
$578$	34.0160	1.41488
$579$	9.73568	0.404601
$580$	0	0
$581$	12.5986	0.522680
$582$	−0.953444	−0.0395215
$583$	3.51682	0.145652
$584$	2.39506	0.0991083
$585$	0	0
$586$	−24.6189	−1.01700
$587$	−1.13849	−0.0469903	−0.0234952	−	0.999724i	$-0.507479\pi$
$587$	−1.13849	−0.0469903	−0.0234952	+	0.999724i	$0.507479\pi$
$588$	19.1094	0.788060
$589$	0	0
$590$	0	0
$591$	−36.1287	−1.48613
$592$	108.728	4.46870
$593$	−5.56477	−0.228518	−0.114259	−	0.993451i	$-0.536449\pi$
$593$	−5.56477	−0.228518	−0.114259	+	0.993451i	$0.536449\pi$
$594$	−39.0476	−1.60214
$595$	0	0
$596$	64.7819	2.65357
$597$	30.4153	1.24482
$598$	−56.9125	−2.32733
$599$	−10.5181	−0.429758	−0.214879	−	0.976641i	$-0.568936\pi$
$599$	−10.5181	−0.429758	−0.214879	+	0.976641i	$0.568936\pi$
$600$	0	0
$601$	36.9336	1.50655	0.753277	−	0.657703i	$-0.228472\pi$
$601$	36.9336	1.50655	0.753277	+	0.657703i	$0.228472\pi$
$602$	30.2771	1.23400
$603$	−0.753648	−0.0306909
$604$	−56.4953	−2.29876
$605$	0	0
$606$	11.1259	0.451958
$607$	31.6955	1.28648	0.643240	−	0.765665i	$-0.277590\pi$
$607$	31.6955	1.28648	0.643240	+	0.765665i	$0.277590\pi$
$608$	0	0
$609$	−31.4988	−1.27640
$610$	0	0
$611$	12.0982	0.489441
$612$	1.12219	0.0453617
$613$	−24.7004	−0.997641	−0.498820	−	0.866705i	$-0.666233\pi$
$613$	−24.7004	−0.997641	−0.498820	+	0.866705i	$0.666233\pi$
$614$	−20.6906	−0.835004
$615$	0	0
$616$	52.8492	2.12935
$617$	−17.9984	−0.724589	−0.362294	−	0.932064i	$-0.618006\pi$
$617$	−17.9984	−0.724589	−0.362294	+	0.932064i	$0.618006\pi$
$618$	−5.05308	−0.203265
$619$	32.0927	1.28991	0.644957	−	0.764219i	$-0.276875\pi$
$619$	32.0927	1.28991	0.644957	+	0.764219i	$0.276875\pi$
$620$	0	0
$621$	−22.4849	−0.902287
$622$	−66.3052	−2.65860
$623$	36.8691	1.47713
$624$	102.326	4.09632
$625$	0	0
$626$	15.4603	0.617920
$627$	0	0
$628$	−114.878	−4.58415
$629$	18.6845	0.744998
$630$	0	0
$631$	−6.97644	−0.277728	−0.138864	−	0.990311i	$-0.544345\pi$
$631$	−6.97644	−0.277728	−0.138864	+	0.990311i	$0.544345\pi$
$632$	132.443	5.26829
$633$	−37.4304	−1.48772
$634$	12.2754	0.487517
$635$	0	0
$636$	−11.0707	−0.438982
$637$	10.2357	0.405551
$638$	61.9173	2.45133
$639$	1.01897	0.0403099
$640$	0	0
$641$	−27.8888	−1.10154	−0.550770	−	0.834657i	$-0.685666\pi$
$641$	−27.8888	−1.10154	−0.550770	+	0.834657i	$0.685666\pi$
$642$	0.875816	0.0345657
$643$	6.24974	0.246466	0.123233	−	0.992378i	$-0.460674\pi$
$643$	6.24974	0.246466	0.123233	+	0.992378i	$0.460674\pi$
$644$	49.9179	1.96704
$645$	0	0
$646$	0	0
$647$	22.1840	0.872142	0.436071	−	0.899912i	$-0.356370\pi$
$647$	22.1840	0.872142	0.436071	+	0.899912i	$0.356370\pi$
$648$	77.5928	3.04813
$649$	−0.701984	−0.0275553
$650$	0	0
$651$	−15.4928	−0.607210
$652$	−8.95554	−0.350726
$653$	11.3373	0.443664	0.221832	−	0.975085i	$-0.428796\pi$
$653$	11.3373	0.443664	0.221832	+	0.975085i	$0.428796\pi$
$654$	61.8036	2.41671
$655$	0	0
$656$	−14.6428	−0.571704
$657$	−0.0305036	−0.00119006
$658$	−14.7535	−0.575150
$659$	26.6149	1.03677	0.518385	−	0.855147i	$-0.326533\pi$
$659$	26.6149	1.03677	0.518385	+	0.855147i	$0.326533\pi$
$660$	0	0
$661$	−45.7240	−1.77846	−0.889228	−	0.457465i	$-0.848758\pi$
$661$	−45.7240	−1.77846	−0.889228	+	0.457465i	$0.848758\pi$
$662$	−19.7386	−0.767161
$663$	17.5843	0.682917
$664$	47.5276	1.84443
$665$	0	0
$666$	−2.56688	−0.0994646
$667$	35.6540	1.38053
$668$	107.843	4.17256
$669$	50.8560	1.96620
$670$	0	0
$671$	1.42898	0.0551651
$672$	−59.8427	−2.30848
$673$	25.6435	0.988484	0.494242	−	0.869324i	$-0.335446\pi$
$673$	25.6435	0.988484	0.494242	+	0.869324i	$0.335446\pi$
$674$	44.5095	1.71444
$675$	0	0
$676$	53.2550	2.04827
$677$	7.17287	0.275676	0.137838	−	0.990455i	$-0.455985\pi$
$677$	7.17287	0.275676	0.137838	+	0.990455i	$0.455985\pi$
$678$	−20.5307	−0.788479
$679$	0.447931	0.0171900
$680$	0	0
$681$	14.8491	0.569019
$682$	30.4543	1.16615
$683$	−6.25807	−0.239458	−0.119729	−	0.992807i	$-0.538203\pi$
$683$	−6.25807	−0.239458	−0.119729	+	0.992807i	$0.538203\pi$
$684$	0	0
$685$	0	0
$686$	−53.7702	−2.05296
$687$	−43.4719	−1.65856
$688$	61.6180	2.34916
$689$	−5.92984	−0.225909
$690$	0	0
$691$	−19.4240	−0.738925	−0.369463	−	0.929246i	$-0.620458\pi$
$691$	−19.4240	−0.738925	−0.369463	+	0.929246i	$0.620458\pi$
$692$	59.4786	2.26104
$693$	−0.673089	−0.0255685
$694$	96.1640	3.65034
$695$	0	0
$696$	−118.827	−4.50413
$697$	−2.51630	−0.0953115
$698$	63.0872	2.38788
$699$	36.5248	1.38149
$700$	0	0
$701$	9.49134	0.358483	0.179241	−	0.983805i	$-0.442636\pi$
$701$	9.49134	0.358483	0.179241	+	0.983805i	$0.442636\pi$
$702$	65.8396	2.48496
$703$	0	0
$704$	48.7664	1.83795
$705$	0	0
$706$	90.5604	3.40829
$707$	−5.22697	−0.196580
$708$	2.20980	0.0830492
$709$	−21.9473	−0.824246	−0.412123	−	0.911128i	$-0.635213\pi$
$709$	−21.9473	−0.824246	−0.412123	+	0.911128i	$0.635213\pi$
$710$	0	0
$711$	−1.68680	−0.0632599
$712$	139.086	5.21248
$713$	17.5366	0.656749
$714$	−21.4436	−0.802507
$715$	0	0
$716$	20.3965	0.762251
$717$	−18.3894	−0.686764
$718$	24.7878	0.925074
$719$	−3.61331	−0.134754	−0.0673768	−	0.997728i	$-0.521463\pi$
$719$	−3.61331	−0.134754	−0.0673768	+	0.997728i	$0.521463\pi$
$720$	0	0
$721$	2.37395	0.0884105
$722$	0	0
$723$	−35.3355	−1.31414
$724$	−14.4412	−0.536704
$725$	0	0
$726$	−13.0369	−0.483845
$727$	−41.1295	−1.52541	−0.762705	−	0.646747i	$-0.776129\pi$
$727$	−41.1295	−1.52541	−0.762705	+	0.646747i	$0.776129\pi$
$728$	−89.1109	−3.30267
$729$	25.9780	0.962146
$730$	0	0
$731$	10.5888	0.391641
$732$	−4.49832	−0.166263
$733$	26.4052	0.975297	0.487648	−	0.873040i	$-0.337855\pi$
$733$	26.4052	0.975297	0.487648	+	0.873040i	$0.337855\pi$
$734$	−30.5023	−1.12586
$735$	0	0
$736$	67.7370	2.49682
$737$	−20.3608	−0.750000
$738$	0.345690	0.0127250
$739$	−16.0961	−0.592106	−0.296053	−	0.955172i	$-0.595670\pi$
$739$	−16.0961	−0.592106	−0.296053	+	0.955172i	$0.595670\pi$
$740$	0	0
$741$	0	0
$742$	7.23129	0.265469
$743$	51.0206	1.87176	0.935882	−	0.352315i	$-0.114605\pi$
$743$	51.0206	1.87176	0.935882	+	0.352315i	$0.114605\pi$
$744$	−58.4456	−2.14272
$745$	0	0
$746$	68.7181	2.51595
$747$	−0.605314	−0.0221473
$748$	30.3173	1.10851
$749$	−0.411461	−0.0150345
$750$	0	0
$751$	−7.88161	−0.287604	−0.143802	−	0.989606i	$-0.545933\pi$
$751$	−7.88161	−0.287604	−0.143802	+	0.989606i	$0.545933\pi$
$752$	−30.0253	−1.09491
$753$	1.08264	0.0394538
$754$	−104.401	−3.80206
$755$	0	0
$756$	−57.7478	−2.10027
$757$	31.9261	1.16037	0.580186	−	0.814484i	$-0.302980\pi$
$757$	31.9261	1.16037	0.580186	+	0.814484i	$0.302980\pi$
$758$	54.3484	1.97402
$759$	22.2884	0.809019
$760$	0	0
$761$	21.6563	0.785039	0.392519	−	0.919744i	$-0.371604\pi$
$761$	21.6563	0.785039	0.392519	+	0.919744i	$0.371604\pi$
$762$	−16.0207	−0.580370
$763$	−29.0355	−1.05116
$764$	−101.500	−3.67216
$765$	0	0
$766$	48.3099	1.74551
$767$	1.18364	0.0427388
$768$	−8.96514	−0.323502
$769$	14.8767	0.536467	0.268233	−	0.963354i	$-0.413560\pi$
$769$	14.8767	0.536467	0.268233	+	0.963354i	$0.413560\pi$
$770$	0	0
$771$	−17.8018	−0.641117
$772$	−28.3025	−1.01863
$773$	1.66685	0.0599524	0.0299762	−	0.999551i	$-0.490457\pi$
$773$	1.66685	0.0599524	0.0299762	+	0.999551i	$0.490457\pi$
$774$	−1.45469	−0.0522879
$775$	0	0
$776$	1.68979	0.0606599
$777$	35.2788	1.26562
$778$	80.8749	2.89951
$779$	0	0
$780$	0	0
$781$	27.5289	0.985060
$782$	24.2724	0.867979
$783$	−41.2466	−1.47403
$784$	−25.4028	−0.907242
$785$	0	0
$786$	2.69602	0.0961638
$787$	4.38273	0.156228	0.0781138	−	0.996944i	$-0.475110\pi$
$787$	4.38273	0.156228	0.0781138	+	0.996944i	$0.475110\pi$
$788$	105.029	3.74152
$789$	−10.8292	−0.385530
$790$	0	0
$791$	9.64541	0.342951
$792$	−2.53919	−0.0902261
$793$	−2.40945	−0.0855621
$794$	−9.52992	−0.338204
$795$	0	0
$796$	−88.4202	−3.13397
$797$	9.86857	0.349563	0.174781	−	0.984607i	$-0.444078\pi$
$797$	9.86857	0.349563	0.174781	+	0.984607i	$0.444078\pi$
$798$	0	0
$799$	−5.15972	−0.182538
$800$	0	0
$801$	−1.77141	−0.0625897
$802$	1.47617	0.0521253
$803$	−0.824096	−0.0290817
$804$	64.0944	2.26043
$805$	0	0
$806$	−51.3501	−1.80873
$807$	−40.6754	−1.43184
$808$	−19.7184	−0.693691
$809$	−40.4522	−1.42222	−0.711112	−	0.703079i	$-0.751808\pi$
$809$	−40.4522	−1.42222	−0.711112	+	0.703079i	$0.751808\pi$
$810$	0	0
$811$	−28.9125	−1.01526	−0.507628	−	0.861576i	$-0.669478\pi$
$811$	−28.9125	−1.01526	−0.507628	+	0.861576i	$0.669478\pi$
$812$	91.5700	3.21348
$813$	−24.2808	−0.851566
$814$	−69.3477	−2.43064
$815$	0	0
$816$	−43.6406	−1.52773
$817$	0	0
$818$	6.83986	0.239150
$819$	1.13492	0.0396573
$820$	0	0
$821$	−13.2912	−0.463866	−0.231933	−	0.972732i	$-0.574505\pi$
$821$	−13.2912	−0.463866	−0.231933	+	0.972732i	$0.574505\pi$
$822$	−22.6827	−0.791151
$823$	47.8247	1.66706	0.833532	−	0.552471i	$-0.186315\pi$
$823$	47.8247	1.66706	0.833532	+	0.552471i	$0.186315\pi$
$824$	8.95558	0.311982
$825$	0	0
$826$	−1.44342	−0.0502230
$827$	51.7917	1.80098	0.900488	−	0.434882i	$-0.143210\pi$
$827$	51.7917	1.80098	0.900488	+	0.434882i	$0.143210\pi$
$828$	−2.39835	−0.0833484
$829$	8.42434	0.292589	0.146295	−	0.989241i	$-0.453265\pi$
$829$	8.42434	0.292589	0.146295	+	0.989241i	$0.453265\pi$
$830$	0	0
$831$	32.1593	1.11559
$832$	−82.2269	−2.85070
$833$	−4.36536	−0.151251
$834$	−70.5236	−2.44203
$835$	0	0
$836$	0	0
$837$	−20.2873	−0.701231
$838$	24.0514	0.830841
$839$	23.7821	0.821051	0.410525	−	0.911849i	$-0.365345\pi$
$839$	23.7821	0.821051	0.410525	+	0.911849i	$0.365345\pi$
$840$	0	0
$841$	36.4042	1.25532
$842$	22.9964	0.792508
$843$	35.9519	1.23825
$844$	108.814	3.74552
$845$	0	0
$846$	0.708844	0.0243706
$847$	6.12478	0.210450
$848$	14.7166	0.505371
$849$	34.4476	1.18224
$850$	0	0
$851$	−39.9327	−1.36887
$852$	−86.6589	−2.96889
$853$	−27.3430	−0.936205	−0.468102	−	0.883674i	$-0.655062\pi$
$853$	−27.3430	−0.936205	−0.468102	+	0.883674i	$0.655062\pi$
$854$	2.93827	0.100545
$855$	0	0
$856$	−1.55221	−0.0530534
$857$	30.7007	1.04872	0.524358	−	0.851498i	$-0.324305\pi$
$857$	30.7007	1.04872	0.524358	+	0.851498i	$0.324305\pi$
$858$	−65.2644	−2.22809
$859$	−36.8449	−1.25713	−0.628567	−	0.777756i	$-0.716358\pi$
$859$	−36.8449	−1.25713	−0.628567	+	0.777756i	$0.716358\pi$
$860$	0	0
$861$	−4.75110	−0.161917
$862$	−13.9195	−0.474102
$863$	−4.42788	−0.150727	−0.0753634	−	0.997156i	$-0.524012\pi$
$863$	−4.42788	−0.150727	−0.0753634	+	0.997156i	$0.524012\pi$
$864$	−78.3620	−2.66593
$865$	0	0
$866$	−13.8060	−0.469147
$867$	22.4620	0.762848
$868$	45.0390	1.52872
$869$	−45.5711	−1.54589
$870$	0	0
$871$	34.3311	1.16326
$872$	−109.535	−3.70931
$873$	−0.0215212	−0.000728384 0
$874$	0	0
$875$	0	0
$876$	2.59419	0.0876497
$877$	0.177859	0.00600587	0.00300294	−	0.999995i	$-0.499044\pi$
$877$	0.177859	0.00600587	0.00300294	+	0.999995i	$0.499044\pi$
$878$	42.9792	1.45048
$879$	−16.2567	−0.548326
$880$	0	0
$881$	35.4916	1.19574	0.597871	−	0.801593i	$-0.296014\pi$
$881$	35.4916	1.19574	0.597871	+	0.801593i	$0.296014\pi$
$882$	0.599715	0.0201935
$883$	44.6775	1.50352	0.751759	−	0.659438i	$-0.229206\pi$
$883$	44.6775	1.50352	0.751759	+	0.659438i	$0.229206\pi$
$884$	−51.1192	−1.71932
$885$	0	0
$886$	−32.3401	−1.08649
$887$	34.5396	1.15973	0.579863	−	0.814714i	$-0.303106\pi$
$887$	34.5396	1.15973	0.579863	+	0.814714i	$0.303106\pi$
$888$	133.087	4.46611
$889$	7.52659	0.252434
$890$	0	0
$891$	−26.6982	−0.894424
$892$	−147.843	−4.95015
$893$	0	0
$894$	59.4762	1.98918
$895$	0	0
$896$	32.3645	1.08122
$897$	−37.5813	−1.25480
$898$	18.8801	0.630037
$899$	32.1693	1.07291
$900$	0	0
$901$	2.52899	0.0842529
$902$	9.33927	0.310964
$903$	19.9931	0.665327
$904$	36.3867	1.21020
$905$	0	0
$906$	−51.8683	−1.72321
$907$	30.4093	1.00972	0.504862	−	0.863200i	$-0.331543\pi$
$907$	30.4093	1.00972	0.504862	+	0.863200i	$0.331543\pi$
$908$	−43.1677	−1.43257
$909$	0.251135	0.00832961
$910$	0	0
$911$	48.5898	1.60985	0.804926	−	0.593375i	$-0.202205\pi$
$911$	48.5898	1.60985	0.804926	+	0.593375i	$0.202205\pi$
$912$	0	0
$913$	−16.3534	−0.541217
$914$	70.4110	2.32899
$915$	0	0
$916$	126.377	4.17561
$917$	−1.26660	−0.0418268
$918$	−28.0797	−0.926767
$919$	46.0057	1.51759	0.758794	−	0.651331i	$-0.225789\pi$
$919$	46.0057	1.51759	0.758794	+	0.651331i	$0.225789\pi$
$920$	0	0
$921$	−13.6627	−0.450202
$922$	25.5958	0.842954
$923$	−46.4174	−1.52785
$924$	57.2432	1.88316
$925$	0	0
$926$	−17.3548	−0.570313
$927$	−0.114059	−0.00374618
$928$	124.258	4.07896
$929$	47.6797	1.56432	0.782161	−	0.623077i	$-0.214118\pi$
$929$	47.6797	1.56432	0.782161	+	0.623077i	$0.214118\pi$
$930$	0	0
$931$	0	0
$932$	−106.181	−3.47808
$933$	−43.7836	−1.43341
$934$	−23.2567	−0.760983
$935$	0	0
$936$	4.28142	0.139942
$937$	−32.4440	−1.05990	−0.529950	−	0.848029i	$-0.677789\pi$
$937$	−32.4440	−1.05990	−0.529950	+	0.848029i	$0.677789\pi$
$938$	−41.8659	−1.36697
$939$	10.2090	0.333158
$940$	0	0
$941$	22.9865	0.749339	0.374669	−	0.927159i	$-0.377756\pi$
$941$	22.9865	0.749339	0.374669	+	0.927159i	$0.377756\pi$
$942$	−105.470	−3.43639
$943$	5.37786	0.175127
$944$	−2.93755	−0.0956092
$945$	0	0
$946$	−39.3005	−1.27777
$947$	−19.5969	−0.636814	−0.318407	−	0.947954i	$-0.603148\pi$
$947$	−19.5969	−0.636814	−0.318407	+	0.947954i	$0.603148\pi$
$948$	143.455	4.65919
$949$	1.38954	0.0451063
$950$	0	0
$951$	8.10586	0.262850
$952$	38.0045	1.23173
$953$	21.8067	0.706387	0.353194	−	0.935550i	$-0.385096\pi$
$953$	21.8067	0.706387	0.353194	+	0.935550i	$0.385096\pi$
$954$	−0.347434	−0.0112486
$955$	0	0
$956$	53.4597	1.72901
$957$	40.8862	1.32166
$958$	−72.8646	−2.35415
$959$	10.6564	0.344113
$960$	0	0
$961$	−15.1774	−0.489594
$962$	116.930	3.76997
$963$	0.0197690	0.000637048 0
$964$	102.724	3.30851
$965$	0	0
$966$	45.8295	1.47454
$967$	−30.6010	−0.984063	−0.492031	−	0.870577i	$-0.663745\pi$
$967$	−30.6010	−0.984063	−0.492031	+	0.870577i	$0.663745\pi$
$968$	23.1054	0.742634
$969$	0	0
$970$	0	0
$971$	−24.6525	−0.791136	−0.395568	−	0.918437i	$-0.629452\pi$
$971$	−24.6525	−0.791136	−0.395568	+	0.918437i	$0.629452\pi$
$972$	5.65089	0.181252
$973$	33.1322	1.06217
$974$	94.9048	3.04095
$975$	0	0
$976$	5.97976	0.191408
$977$	−40.0301	−1.28067	−0.640337	−	0.768094i	$-0.721205\pi$
$977$	−40.0301	−1.28067	−0.640337	+	0.768094i	$0.721205\pi$
$978$	−8.22207	−0.262913
$979$	−47.8570	−1.52952
$980$	0	0
$981$	1.39504	0.0445402
$982$	−110.981	−3.54153
$983$	−18.0067	−0.574323	−0.287162	−	0.957882i	$-0.592712\pi$
$983$	−18.0067	−0.574323	−0.287162	+	0.957882i	$0.592712\pi$
$984$	−17.9232	−0.571372
$985$	0	0
$986$	44.5256	1.41798
$987$	−9.74224	−0.310099
$988$	0	0
$989$	−22.6305	−0.719608
$990$	0	0
$991$	−47.2626	−1.50135	−0.750674	−	0.660673i	$-0.770271\pi$
$991$	−47.2626	−1.50135	−0.750674	+	0.660673i	$0.770271\pi$
$992$	61.1166	1.94045
$993$	−13.0341	−0.413623
$994$	56.6049	1.79540
$995$	0	0
$996$	51.4792	1.63118
$997$	30.7551	0.974024	0.487012	−	0.873395i	$-0.338087\pi$
$997$	30.7551	0.974024	0.487012	+	0.873395i	$0.338087\pi$
$998$	72.3961	2.29166
$999$	46.1964	1.46159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cv.1.1		40
5.2	odd	4		1805.2.b.m.1084.1	✓	40
5.3	odd	4		1805.2.b.m.1084.40	yes	40
5.4	even	2	inner	9025.2.a.cv.1.40		40
19.18	odd	2	inner	9025.2.a.cv.1.39		40
95.18	even	4		1805.2.b.m.1084.2	yes	40
95.37	even	4		1805.2.b.m.1084.39	yes	40
95.94	odd	2	inner	9025.2.a.cv.1.2		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.m.1084.1	✓	40	5.2	odd	4
1805.2.b.m.1084.2	yes	40	95.18	even	4
1805.2.b.m.1084.39	yes	40	95.37	even	4
1805.2.b.m.1084.40	yes	40	5.3	odd	4
9025.2.a.cv.1.1		40	1.1	even	1	trivial
9025.2.a.cv.1.2		40	95.94	odd	2	inner
9025.2.a.cv.1.39		40	19.18	odd	2	inner
9025.2.a.cv.1.40		40	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66900 q^{2} -1.76244 q^{3} +5.12357 q^{4} +4.70394 q^{6} -2.20993 q^{7} -8.33681 q^{8} +0.106178 q^{9} +2.86854 q^{11} -9.02996 q^{12} -4.83675 q^{13} +5.89830 q^{14} +12.0038 q^{16} +2.06280 q^{17} -0.283389 q^{18} +3.89485 q^{21} -7.65613 q^{22} -4.40865 q^{23} +14.6931 q^{24} +12.9093 q^{26} +5.10017 q^{27} -11.3227 q^{28} -8.08729 q^{29} -3.97776 q^{31} -15.3646 q^{32} -5.05561 q^{33} -5.50563 q^{34} +0.544010 q^{36} +9.05780 q^{37} +8.52446 q^{39} -1.21984 q^{41} -10.3954 q^{42} +5.13320 q^{43} +14.6972 q^{44} +11.7667 q^{46} -2.50131 q^{47} -21.1559 q^{48} -2.11623 q^{49} -3.63556 q^{51} -24.7814 q^{52} +1.22600 q^{53} -13.6124 q^{54} +18.4237 q^{56} +21.5850 q^{58} -0.244718 q^{59} +0.498155 q^{61} +10.6166 q^{62} -0.234645 q^{63} +17.0004 q^{64} +13.4934 q^{66} -7.09797 q^{67} +10.5689 q^{68} +7.76996 q^{69} +9.59683 q^{71} -0.885185 q^{72} -0.287288 q^{73} -24.1753 q^{74} -6.33926 q^{77} -22.7518 q^{78} -15.8865 q^{79} -9.30726 q^{81} +3.25576 q^{82} -5.70094 q^{83} +19.9555 q^{84} -13.7005 q^{86} +14.2533 q^{87} -23.9144 q^{88} -16.6834 q^{89} +10.6889 q^{91} -22.5880 q^{92} +7.01055 q^{93} +6.67600 q^{94} +27.0791 q^{96} -0.202690 q^{97} +5.64821 q^{98} +0.304575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 48 q^{4} + 20 q^{6} + 52 q^{9} + 20 q^{11} + 40 q^{16} + 92 q^{24} + 76 q^{26} + 156 q^{36} + 80 q^{39} + 48 q^{44} + 72 q^{49} + 32 q^{54} + 80 q^{61} + 72 q^{64} + 16 q^{66} + 100 q^{74} + 40 q^{81}+ \cdots + 128 q^{99}+O(q^{100}) \) 40 * q + 48 * q^4 + 20 * q^6 + 52 * q^9 + 20 * q^11 + 40 * q^16 + 92 * q^24 + 76 * q^26 + 156 * q^36 + 80 * q^39 + 48 * q^44 + 72 * q^49 + 32 * q^54 + 80 * q^61 + 72 * q^64 + 16 * q^66 + 100 * q^74 + 40 * q^81 + 380 * q^96 + 128 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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