LMFDB - Embedded newform 2325.2.a.t.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1,2325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("2325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,2,0,0,4,6,3,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.5652184699$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.21432 q^{2} -1.00000 q^{3} +2.90321 q^{4} -2.21432 q^{6} -0.903212 q^{7} +2.00000 q^{8} +1.00000 q^{9} -4.59210 q^{11} -2.90321 q^{12} +4.49532 q^{13} -2.00000 q^{14} -1.37778 q^{16} -6.73975 q^{17} +2.21432 q^{18} -3.62222 q^{19} +0.903212 q^{21} -10.1684 q^{22} -8.39853 q^{23} -2.00000 q^{24} +9.95407 q^{26} -1.00000 q^{27} -2.62222 q^{28} -1.68889 q^{29} -1.00000 q^{31} -7.05086 q^{32} +4.59210 q^{33} -14.9240 q^{34} +2.90321 q^{36} +8.68889 q^{37} -8.02074 q^{38} -4.49532 q^{39} +4.02074 q^{41} +2.00000 q^{42} +11.4795 q^{43} -13.3319 q^{44} -18.5970 q^{46} -1.18421 q^{47} +1.37778 q^{48} -6.18421 q^{49} +6.73975 q^{51} +13.0509 q^{52} +0.873100 q^{53} -2.21432 q^{54} -1.80642 q^{56} +3.62222 q^{57} -3.73975 q^{58} -11.2652 q^{59} +12.8573 q^{61} -2.21432 q^{62} -0.903212 q^{63} -12.8573 q^{64} +10.1684 q^{66} -10.0049 q^{67} -19.5669 q^{68} +8.39853 q^{69} -13.0509 q^{71} +2.00000 q^{72} +8.42864 q^{73} +19.2400 q^{74} -10.5161 q^{76} +4.14764 q^{77} -9.95407 q^{78} -1.11753 q^{79} +1.00000 q^{81} +8.90321 q^{82} -3.96989 q^{83} +2.62222 q^{84} +25.4193 q^{86} +1.68889 q^{87} -9.18421 q^{88} -9.50961 q^{89} -4.06022 q^{91} -24.3827 q^{92} +1.00000 q^{93} -2.62222 q^{94} +7.05086 q^{96} -17.1891 q^{97} -13.6938 q^{98} -4.59210 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{4} + 4 q^{7} + 6 q^{8} + 3 q^{9} - 7 q^{11} - 2 q^{12} - 6 q^{14} - 4 q^{16} - 7 q^{17} - 11 q^{19} - 4 q^{21} - 4 q^{22} - 5 q^{23} - 6 q^{24} + 10 q^{26} - 3 q^{27} - 8 q^{28} - 5 q^{29}+ \cdots - 7 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^4 + 4 * q^7 + 6 * q^8 + 3 * q^9 - 7 * q^11 - 2 * q^12 - 6 * q^14 - 4 * q^16 - 7 * q^17 - 11 * q^19 - 4 * q^21 - 4 * q^22 - 5 * q^23 - 6 * q^24 + 10 * q^26 - 3 * q^27 - 8 * q^28 - 5 * q^29 - 3 * q^31 - 8 * q^32 + 7 * q^33 - 18 * q^34 + 2 * q^36 + 26 * q^37 - 4 * q^38 - 8 * q^41 + 6 * q^42 + 8 * q^43 - 20 * q^44 - 16 * q^46 + 10 * q^47 + 4 * q^48 - 5 * q^49 + 7 * q^51 + 26 * q^52 - 11 * q^53 + 8 * q^56 + 11 * q^57 + 2 * q^58 - 14 * q^59 + 12 * q^61 + 4 * q^63 - 12 * q^64 + 4 * q^66 + 3 * q^67 - 12 * q^68 + 5 * q^69 - 26 * q^71 + 6 * q^72 + 12 * q^73 - 2 * q^74 + 2 * q^76 + 6 * q^77 - 10 * q^78 + 10 * q^79 + 3 * q^81 + 20 * q^82 - 5 * q^83 + 8 * q^84 + 36 * q^86 + 5 * q^87 - 14 * q^88 - 9 * q^89 - 26 * q^91 - 40 * q^92 + 3 * q^93 - 8 * q^94 + 8 * q^96 - 5 * q^97 - 8 * q^98 - 7 * 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 $

$2$	2.21432	1.56576	0.782880	−	0.622172i	$-0.213750\pi$
$2$	2.21432	1.56576	0.782880	+	0.622172i	$0.213750\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.90321	1.45161
$5$	0	0
$5$	0	0
$6$	−2.21432	−0.903992
$7$	−0.903212	−0.341382	−0.170691	−	0.985325i	$-0.554600\pi$
$7$	−0.903212	−0.341382	−0.170691	+	0.985325i	$0.554600\pi$
$8$	2.00000	0.707107
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.59210	−1.38457	−0.692286	−	0.721623i	$-0.743396\pi$
$11$	−4.59210	−1.38457	−0.692286	+	0.721623i	$0.743396\pi$
$12$	−2.90321	−0.838085
$13$	4.49532	1.24678	0.623388	−	0.781913i	$-0.285756\pi$
$13$	4.49532	1.24678	0.623388	+	0.781913i	$0.285756\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.37778	−0.344446
$17$	−6.73975	−1.63463	−0.817314	−	0.576192i	$-0.804538\pi$
$17$	−6.73975	−1.63463	−0.817314	+	0.576192i	$0.804538\pi$
$18$	2.21432	0.521920
$19$	−3.62222	−0.830993	−0.415497	−	0.909595i	$-0.636392\pi$
$19$	−3.62222	−0.830993	−0.415497	+	0.909595i	$0.636392\pi$
$20$	0	0
$21$	0.903212	0.197097
$22$	−10.1684	−2.16791
$23$	−8.39853	−1.75121	−0.875607	−	0.483024i	$-0.839538\pi$
$23$	−8.39853	−1.75121	−0.875607	+	0.483024i	$0.839538\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	9.95407	1.95215
$27$	−1.00000	−0.192450
$28$	−2.62222	−0.495552
$29$	−1.68889	−0.313619	−0.156810	−	0.987629i	$-0.550121\pi$
$29$	−1.68889	−0.313619	−0.156810	+	0.987629i	$0.550121\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−7.05086	−1.24643
$33$	4.59210	0.799383
$34$	−14.9240	−2.55944
$35$	0	0
$36$	2.90321	0.483869
$37$	8.68889	1.42845	0.714223	−	0.699919i	$-0.246780\pi$
$37$	8.68889	1.42845	0.714223	+	0.699919i	$0.246780\pi$
$38$	−8.02074	−1.30114
$39$	−4.49532	−0.719827
$40$	0	0
$41$	4.02074	0.627935	0.313967	−	0.949434i	$-0.398342\pi$
$41$	4.02074	0.627935	0.313967	+	0.949434i	$0.398342\pi$
$42$	2.00000	0.308607
$43$	11.4795	1.75061	0.875303	−	0.483574i	$-0.160662\pi$
$43$	11.4795	1.75061	0.875303	+	0.483574i	$0.160662\pi$
$44$	−13.3319	−2.00985
$45$	0	0
$46$	−18.5970	−2.74198
$47$	−1.18421	−0.172735	−0.0863673	−	0.996263i	$-0.527526\pi$
$47$	−1.18421	−0.172735	−0.0863673	+	0.996263i	$0.527526\pi$
$48$	1.37778	0.198866
$49$	−6.18421	−0.883458
$50$	0	0
$51$	6.73975	0.943753
$52$	13.0509	1.80983
$53$	0.873100	0.119930	0.0599648	−	0.998200i	$-0.480901\pi$
$53$	0.873100	0.119930	0.0599648	+	0.998200i	$0.480901\pi$
$54$	−2.21432	−0.301331
$55$	0	0
$56$	−1.80642	−0.241394
$57$	3.62222	0.479774
$58$	−3.73975	−0.491053
$59$	−11.2652	−1.46660	−0.733300	−	0.679905i	$-0.762021\pi$
$59$	−11.2652	−1.46660	−0.733300	+	0.679905i	$0.762021\pi$
$60$	0	0
$61$	12.8573	1.64621	0.823103	−	0.567892i	$-0.192241\pi$
$61$	12.8573	1.64621	0.823103	+	0.567892i	$0.192241\pi$
$62$	−2.21432	−0.281219
$63$	−0.903212	−0.113794
$64$	−12.8573	−1.60716
$65$	0	0
$66$	10.1684	1.25164
$67$	−10.0049	−1.22230	−0.611148	−	0.791516i	$-0.709292\pi$
$67$	−10.0049	−1.22230	−0.611148	+	0.791516i	$0.709292\pi$
$68$	−19.5669	−2.37284
$69$	8.39853	1.01106
$70$	0	0
$71$	−13.0509	−1.54885	−0.774426	−	0.632665i	$-0.781961\pi$
$71$	−13.0509	−1.54885	−0.774426	+	0.632665i	$0.781961\pi$
$72$	2.00000	0.235702
$73$	8.42864	0.986498	0.493249	−	0.869888i	$-0.335809\pi$
$73$	8.42864	0.986498	0.493249	+	0.869888i	$0.335809\pi$
$74$	19.2400	2.23660
$75$	0	0
$76$	−10.5161	−1.20627
$77$	4.14764	0.472668
$78$	−9.95407	−1.12708
$79$	−1.11753	−0.125732	−0.0628661	−	0.998022i	$-0.520024\pi$
$79$	−1.11753	−0.125732	−0.0628661	+	0.998022i	$0.520024\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.90321	0.983195
$83$	−3.96989	−0.435752	−0.217876	−	0.975976i	$-0.569913\pi$
$83$	−3.96989	−0.435752	−0.217876	+	0.975976i	$0.569913\pi$
$84$	2.62222	0.286107
$85$	0	0
$86$	25.4193	2.74103
$87$	1.68889	0.181068
$88$	−9.18421	−0.979040
$89$	−9.50961	−1.00802	−0.504008	−	0.863699i	$-0.668142\pi$
$89$	−9.50961	−1.00802	−0.504008	+	0.863699i	$0.668142\pi$
$90$	0	0
$91$	−4.06022	−0.425627
$92$	−24.3827	−2.54207
$93$	1.00000	0.103695
$94$	−2.62222	−0.270461
$95$	0	0
$96$	7.05086	0.719625
$97$	−17.1891	−1.74529	−0.872646	−	0.488354i	$-0.837598\pi$
$97$	−17.1891	−1.74529	−0.872646	+	0.488354i	$0.837598\pi$
$98$	−13.6938	−1.38328
$99$	−4.59210	−0.461524
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	14.9240	1.47769
$103$	0.673071	0.0663196	0.0331598	−	0.999450i	$-0.489443\pi$
$103$	0.673071	0.0663196	0.0331598	+	0.999450i	$0.489443\pi$
$104$	8.99063	0.881604
$105$	0	0
$106$	1.93332	0.187781
$107$	9.13182	0.882807	0.441403	−	0.897309i	$-0.354481\pi$
$107$	9.13182	0.882807	0.441403	+	0.897309i	$0.354481\pi$
$108$	−2.90321	−0.279362
$109$	−13.0049	−1.24565	−0.622823	−	0.782363i	$-0.714014\pi$
$109$	−13.0049	−1.24565	−0.622823	+	0.782363i	$0.714014\pi$
$110$	0	0
$111$	−8.68889	−0.824713
$112$	1.24443	0.117588
$113$	14.0415	1.32091	0.660456	−	0.750865i	$-0.270363\pi$
$113$	14.0415	1.32091	0.660456	+	0.750865i	$0.270363\pi$
$114$	8.02074	0.751211
$115$	0	0
$116$	−4.90321	−0.455252
$117$	4.49532	0.415592
$118$	−24.9447	−2.29635
$119$	6.08742	0.558033
$120$	0	0
$121$	10.0874	0.917038
$122$	28.4701	2.57756
$123$	−4.02074	−0.362538
$124$	−2.90321	−0.260716
$125$	0	0
$126$	−2.00000	−0.178174
$127$	0.755569	0.0670459	0.0335229	−	0.999438i	$-0.489327\pi$
$127$	0.755569	0.0670459	0.0335229	+	0.999438i	$0.489327\pi$
$128$	−14.3684	−1.27000
$129$	−11.4795	−1.01071
$130$	0	0
$131$	10.3160	0.901316	0.450658	−	0.892697i	$-0.351189\pi$
$131$	10.3160	0.901316	0.450658	+	0.892697i	$0.351189\pi$
$132$	13.3319	1.16039
$133$	3.27163	0.283686
$134$	−22.1541	−1.91382
$135$	0	0
$136$	−13.4795	−1.15586
$137$	−10.1778	−0.869544	−0.434772	−	0.900540i	$-0.643171\pi$
$137$	−10.1778	−0.869544	−0.434772	+	0.900540i	$0.643171\pi$
$138$	18.5970	1.58308
$139$	9.35260	0.793277	0.396638	−	0.917975i	$-0.370177\pi$
$139$	9.35260	0.793277	0.396638	+	0.917975i	$0.370177\pi$
$140$	0	0
$141$	1.18421	0.0997283
$142$	−28.8988	−2.42513
$143$	−20.6430	−1.72625
$144$	−1.37778	−0.114815
$145$	0	0
$146$	18.6637	1.54462
$147$	6.18421	0.510065
$148$	25.2257	2.07354
$149$	6.44938	0.528354	0.264177	−	0.964474i	$-0.414900\pi$
$149$	6.44938	0.528354	0.264177	+	0.964474i	$0.414900\pi$
$150$	0	0
$151$	−23.9081	−1.94562	−0.972808	−	0.231612i	$-0.925600\pi$
$151$	−23.9081	−1.94562	−0.972808	+	0.231612i	$0.925600\pi$
$152$	−7.24443	−0.587601
$153$	−6.73975	−0.544876
$154$	9.18421	0.740085
$155$	0	0
$156$	−13.0509	−1.04490
$157$	−5.19358	−0.414493	−0.207246	−	0.978289i	$-0.566450\pi$
$157$	−5.19358	−0.414493	−0.207246	+	0.978289i	$0.566450\pi$
$158$	−2.47457	−0.196866
$159$	−0.873100	−0.0692414
$160$	0	0
$161$	7.58565	0.597833
$162$	2.21432	0.173973
$163$	20.2257	1.58420	0.792099	−	0.610392i	$-0.208988\pi$
$163$	20.2257	1.58420	0.792099	+	0.610392i	$0.208988\pi$
$164$	11.6731	0.911514
$165$	0	0
$166$	−8.79060	−0.682283
$167$	19.0923	1.47741	0.738705	−	0.674029i	$-0.235438\pi$
$167$	19.0923	1.47741	0.738705	+	0.674029i	$0.235438\pi$
$168$	1.80642	0.139369
$169$	7.20787	0.554451
$170$	0	0
$171$	−3.62222	−0.276998
$172$	33.3274	2.54119
$173$	5.20495	0.395725	0.197863	−	0.980230i	$-0.436600\pi$
$173$	5.20495	0.395725	0.197863	+	0.980230i	$0.436600\pi$
$174$	3.73975	0.283510
$175$	0	0
$176$	6.32693	0.476910
$177$	11.2652	0.846742
$178$	−21.0573	−1.57831
$179$	19.2400	1.43806	0.719032	−	0.694977i	$-0.244585\pi$
$179$	19.2400	1.43806	0.719032	+	0.694977i	$0.244585\pi$
$180$	0	0
$181$	4.06668	0.302274	0.151137	−	0.988513i	$-0.451707\pi$
$181$	4.06668	0.302274	0.151137	+	0.988513i	$0.451707\pi$
$182$	−8.99063	−0.666430
$183$	−12.8573	−0.950437
$184$	−16.7971	−1.23830
$185$	0	0
$186$	2.21432	0.162362
$187$	30.9496	2.26326
$188$	−3.43801	−0.250742
$189$	0.903212	0.0656990
$190$	0	0
$191$	8.12245	0.587720	0.293860	−	0.955848i	$-0.405060\pi$
$191$	8.12245	0.587720	0.293860	+	0.955848i	$0.405060\pi$
$192$	12.8573	0.927894
$193$	9.01429	0.648863	0.324431	−	0.945909i	$-0.394827\pi$
$193$	9.01429	0.648863	0.324431	+	0.945909i	$0.394827\pi$
$194$	−38.0622	−2.73271
$195$	0	0
$196$	−17.9541	−1.28243
$197$	7.71456	0.549639	0.274820	−	0.961496i	$-0.411382\pi$
$197$	7.71456	0.549639	0.274820	+	0.961496i	$0.411382\pi$
$198$	−10.1684	−0.722636
$199$	0.790602	0.0560443	0.0280222	−	0.999607i	$-0.491079\pi$
$199$	0.790602	0.0560443	0.0280222	+	0.999607i	$0.491079\pi$
$200$	0	0
$201$	10.0049	0.705693
$202$	−17.7146	−1.24639
$203$	1.52543	0.107064
$204$	19.5669	1.36996
$205$	0	0
$206$	1.49039	0.103841
$207$	−8.39853	−0.583738
$208$	−6.19358	−0.429447
$209$	16.6336	1.15057
$210$	0	0
$211$	−15.9304	−1.09669	−0.548347	−	0.836251i	$-0.684743\pi$
$211$	−15.9304	−1.09669	−0.548347	+	0.836251i	$0.684743\pi$
$212$	2.53480	0.174090
$213$	13.0509	0.894230
$214$	20.2208	1.38226
$215$	0	0
$216$	−2.00000	−0.136083
$217$	0.903212	0.0613140
$218$	−28.7971	−1.95038
$219$	−8.42864	−0.569555
$220$	0	0
$221$	−30.2973	−2.03802
$222$	−19.2400	−1.29130
$223$	5.86665	0.392860	0.196430	−	0.980518i	$-0.437065\pi$
$223$	5.86665	0.392860	0.196430	+	0.980518i	$0.437065\pi$
$224$	6.36842	0.425508
$225$	0	0
$226$	31.0923	2.06823
$227$	−15.1526	−1.00571	−0.502856	−	0.864370i	$-0.667717\pi$
$227$	−15.1526	−1.00571	−0.502856	+	0.864370i	$0.667717\pi$
$228$	10.5161	0.696443
$229$	1.67952	0.110986	0.0554930	−	0.998459i	$-0.482327\pi$
$229$	1.67952	0.110986	0.0554930	+	0.998459i	$0.482327\pi$
$230$	0	0
$231$	−4.14764	−0.272895
$232$	−3.37778	−0.221762
$233$	13.2444	0.867672	0.433836	−	0.900992i	$-0.357160\pi$
$233$	13.2444	0.867672	0.433836	+	0.900992i	$0.357160\pi$
$234$	9.95407	0.650718
$235$	0	0
$236$	−32.7052	−2.12893
$237$	1.11753	0.0725915
$238$	13.4795	0.873746
$239$	−3.36196	−0.217467	−0.108734	−	0.994071i	$-0.534680\pi$
$239$	−3.36196	−0.217467	−0.108734	+	0.994071i	$0.534680\pi$
$240$	0	0
$241$	3.63804	0.234347	0.117173	−	0.993111i	$-0.462617\pi$
$241$	3.63804	0.234347	0.117173	+	0.993111i	$0.462617\pi$
$242$	22.3368	1.43586
$243$	−1.00000	−0.0641500
$244$	37.3274	2.38964
$245$	0	0
$246$	−8.90321	−0.567648
$247$	−16.2830	−1.03606
$248$	−2.00000	−0.127000
$249$	3.96989	0.251581
$250$	0	0
$251$	−14.3526	−0.905928	−0.452964	−	0.891529i	$-0.649633\pi$
$251$	−14.3526	−0.905928	−0.452964	+	0.891529i	$0.649633\pi$
$252$	−2.62222	−0.165184
$253$	38.5669	2.42468
$254$	1.67307	0.104978
$255$	0	0
$256$	−6.10171	−0.381357
$257$	6.25581	0.390227	0.195113	−	0.980781i	$-0.437493\pi$
$257$	6.25581	0.390227	0.195113	+	0.980781i	$0.437493\pi$
$258$	−25.4193	−1.58253
$259$	−7.84791	−0.487645
$260$	0	0
$261$	−1.68889	−0.104540
$262$	22.8430	1.41124
$263$	30.8385	1.90159	0.950793	−	0.309827i	$-0.100271\pi$
$263$	30.8385	1.90159	0.950793	+	0.309827i	$0.100271\pi$
$264$	9.18421	0.565249
$265$	0	0
$266$	7.24443	0.444185
$267$	9.50961	0.581978
$268$	−29.0464	−1.77429
$269$	−11.5526	−0.704376	−0.352188	−	0.935929i	$-0.614562\pi$
$269$	−11.5526	−0.704376	−0.352188	+	0.935929i	$0.614562\pi$
$270$	0	0
$271$	−18.0667	−1.09747	−0.548736	−	0.835996i	$-0.684891\pi$
$271$	−18.0667	−1.09747	−0.548736	+	0.835996i	$0.684891\pi$
$272$	9.28592	0.563042
$273$	4.06022	0.245736
$274$	−22.5368	−1.36150
$275$	0	0
$276$	24.3827	1.46767
$277$	14.7239	0.884675	0.442337	−	0.896849i	$-0.354149\pi$
$277$	14.7239	0.884675	0.442337	+	0.896849i	$0.354149\pi$
$278$	20.7096	1.24208
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	9.61285	0.573454	0.286727	−	0.958012i	$-0.407433\pi$
$281$	9.61285	0.573454	0.286727	+	0.958012i	$0.407433\pi$
$282$	2.62222	0.156151
$283$	3.88892	0.231172	0.115586	−	0.993297i	$-0.463125\pi$
$283$	3.88892	0.231172	0.115586	+	0.993297i	$0.463125\pi$
$284$	−37.8894	−2.24832
$285$	0	0
$286$	−45.7101	−2.70290
$287$	−3.63158	−0.214366
$288$	−7.05086	−0.415476
$289$	28.4242	1.67201
$290$	0	0
$291$	17.1891	1.00764
$292$	24.4701	1.43201
$293$	−3.22369	−0.188330	−0.0941649	−	0.995557i	$-0.530018\pi$
$293$	−3.22369	−0.188330	−0.0941649	+	0.995557i	$0.530018\pi$
$294$	13.6938	0.798639
$295$	0	0
$296$	17.3778	1.01006
$297$	4.59210	0.266461
$298$	14.2810	0.827276
$299$	−37.7540	−2.18337
$300$	0	0
$301$	−10.3684	−0.597626
$302$	−52.9403	−3.04637
$303$	8.00000	0.459588
$304$	4.99063	0.286232
$305$	0	0
$306$	−14.9240	−0.853146
$307$	−11.0049	−0.628084	−0.314042	−	0.949409i	$-0.601683\pi$
$307$	−11.0049	−0.628084	−0.314042	+	0.949409i	$0.601683\pi$
$308$	12.0415	0.686127
$309$	−0.673071	−0.0382897
$310$	0	0
$311$	10.9699	0.622045	0.311023	−	0.950402i	$-0.399328\pi$
$311$	10.9699	0.622045	0.311023	+	0.950402i	$0.399328\pi$
$312$	−8.99063	−0.508994
$313$	−1.81288	−0.102470	−0.0512349	−	0.998687i	$-0.516316\pi$
$313$	−1.81288	−0.102470	−0.0512349	+	0.998687i	$0.516316\pi$
$314$	−11.5002	−0.648996
$315$	0	0
$316$	−3.24443	−0.182514
$317$	−24.9699	−1.40245	−0.701224	−	0.712941i	$-0.747363\pi$
$317$	−24.9699	−1.40245	−0.701224	+	0.712941i	$0.747363\pi$
$318$	−1.93332	−0.108415
$319$	7.75557	0.434228
$320$	0	0
$321$	−9.13182	−0.509689
$322$	16.7971	0.936063
$323$	24.4128	1.35837
$324$	2.90321	0.161290
$325$	0	0
$326$	44.7862	2.48048
$327$	13.0049	0.719174
$328$	8.04149	0.444017
$329$	1.06959	0.0589685
$330$	0	0
$331$	−29.1590	−1.60272	−0.801362	−	0.598179i	$-0.795891\pi$
$331$	−29.1590	−1.60272	−0.801362	+	0.598179i	$0.795891\pi$
$332$	−11.5254	−0.632540
$333$	8.68889	0.476148
$334$	42.2766	2.31327
$335$	0	0
$336$	−1.24443	−0.0678893
$337$	−7.51759	−0.409509	−0.204755	−	0.978813i	$-0.565640\pi$
$337$	−7.51759	−0.409509	−0.204755	+	0.978813i	$0.565640\pi$
$338$	15.9605	0.868138
$339$	−14.0415	−0.762629
$340$	0	0
$341$	4.59210	0.248676
$342$	−8.02074	−0.433712
$343$	11.9081	0.642979
$344$	22.9590	1.23787
$345$	0	0
$346$	11.5254	0.619611
$347$	23.8938	1.28269	0.641344	−	0.767253i	$-0.278377\pi$
$347$	23.8938	1.28269	0.641344	+	0.767253i	$0.278377\pi$
$348$	4.90321	0.262840
$349$	−0.377784	−0.0202223	−0.0101112	−	0.999949i	$-0.503219\pi$
$349$	−0.377784	−0.0202223	−0.0101112	+	0.999949i	$0.503219\pi$
$350$	0	0
$351$	−4.49532	−0.239942
$352$	32.3783	1.72577
$353$	0.0444016	0.00236326	0.00118163	−	0.999999i	$-0.499624\pi$
$353$	0.0444016	0.00236326	0.00118163	+	0.999999i	$0.499624\pi$
$354$	24.9447	1.32580
$355$	0	0
$356$	−27.6084	−1.46324
$357$	−6.08742	−0.322180
$358$	42.6035	2.25166
$359$	−24.1146	−1.27272	−0.636360	−	0.771392i	$-0.719561\pi$
$359$	−24.1146	−1.27272	−0.636360	+	0.771392i	$0.719561\pi$
$360$	0	0
$361$	−5.87955	−0.309450
$362$	9.00492	0.473288
$363$	−10.0874	−0.529452
$364$	−11.7877	−0.617843
$365$	0	0
$366$	−28.4701	−1.48816
$367$	−30.0098	−1.56650	−0.783251	−	0.621706i	$-0.786440\pi$
$367$	−30.0098	−1.56650	−0.783251	+	0.621706i	$0.786440\pi$
$368$	11.5714	0.603199
$369$	4.02074	0.209312
$370$	0	0
$371$	−0.788595	−0.0409418
$372$	2.90321	0.150525
$373$	−19.8113	−1.02579	−0.512896	−	0.858451i	$-0.671427\pi$
$373$	−19.8113	−1.02579	−0.512896	+	0.858451i	$0.671427\pi$
$374$	68.5324	3.54372
$375$	0	0
$376$	−2.36842	−0.122142
$377$	−7.59210	−0.391013
$378$	2.00000	0.102869
$379$	−28.8113	−1.47994	−0.739970	−	0.672640i	$-0.765160\pi$
$379$	−28.8113	−1.47994	−0.739970	+	0.672640i	$0.765160\pi$
$380$	0	0
$381$	−0.755569	−0.0387090
$382$	17.9857	0.920229
$383$	−30.4543	−1.55614	−0.778071	−	0.628176i	$-0.783802\pi$
$383$	−30.4543	−1.55614	−0.778071	+	0.628176i	$0.783802\pi$
$384$	14.3684	0.733235
$385$	0	0
$386$	19.9605	1.01596
$387$	11.4795	0.583536
$388$	−49.9037	−2.53348
$389$	14.8573	0.753294	0.376647	−	0.926357i	$-0.377077\pi$
$389$	14.8573	0.753294	0.376647	+	0.926357i	$0.377077\pi$
$390$	0	0
$391$	56.6040	2.86259
$392$	−12.3684	−0.624699
$393$	−10.3160	−0.520375
$394$	17.0825	0.860604
$395$	0	0
$396$	−13.3319	−0.669951
$397$	−8.09679	−0.406366	−0.203183	−	0.979141i	$-0.565129\pi$
$397$	−8.09679	−0.406366	−0.203183	+	0.979141i	$0.565129\pi$
$398$	1.75065	0.0877520
$399$	−3.27163	−0.163786
$400$	0	0
$401$	10.9605	0.547342	0.273671	−	0.961823i	$-0.411762\pi$
$401$	10.9605	0.547342	0.273671	+	0.961823i	$0.411762\pi$
$402$	22.1541	1.10495
$403$	−4.49532	−0.223928
$404$	−23.2257	−1.15552
$405$	0	0
$406$	3.37778	0.167637
$407$	−39.9003	−1.97778
$408$	13.4795	0.667334
$409$	−37.7748	−1.86784	−0.933921	−	0.357479i	$-0.883637\pi$
$409$	−37.7748	−1.86784	−0.933921	+	0.357479i	$0.883637\pi$
$410$	0	0
$411$	10.1778	0.502032
$412$	1.95407	0.0962700
$413$	10.1748	0.500671
$414$	−18.5970	−0.913994
$415$	0	0
$416$	−31.6958	−1.55402
$417$	−9.35260	−0.457999
$418$	36.8321	1.80152
$419$	−8.42864	−0.411766	−0.205883	−	0.978577i	$-0.566007\pi$
$419$	−8.42864	−0.411766	−0.205883	+	0.978577i	$0.566007\pi$
$420$	0	0
$421$	−29.5254	−1.43898	−0.719491	−	0.694502i	$-0.755625\pi$
$421$	−29.5254	−1.43898	−0.719491	+	0.694502i	$0.755625\pi$
$422$	−35.2750	−1.71716
$423$	−1.18421	−0.0575782
$424$	1.74620	0.0848030
$425$	0	0
$426$	28.8988	1.40015
$427$	−11.6128	−0.561985
$428$	26.5116	1.28149
$429$	20.6430	0.996651
$430$	0	0
$431$	36.2766	1.74738	0.873690	−	0.486483i	$-0.161720\pi$
$431$	36.2766	1.74738	0.873690	+	0.486483i	$0.161720\pi$
$432$	1.37778	0.0662887
$433$	−19.7560	−0.949415	−0.474707	−	0.880144i	$-0.657446\pi$
$433$	−19.7560	−0.949415	−0.474707	+	0.880144i	$0.657446\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	−37.7560	−1.80819
$437$	30.4213	1.45525
$438$	−18.6637	−0.891786
$439$	−12.8528	−0.613432	−0.306716	−	0.951801i	$-0.599230\pi$
$439$	−12.8528	−0.613432	−0.306716	+	0.951801i	$0.599230\pi$
$440$	0	0
$441$	−6.18421	−0.294486
$442$	−67.0879	−3.19105
$443$	−1.62375	−0.0771465	−0.0385733	−	0.999256i	$-0.512281\pi$
$443$	−1.62375	−0.0771465	−0.0385733	+	0.999256i	$0.512281\pi$
$444$	−25.2257	−1.19716
$445$	0	0
$446$	12.9906	0.615124
$447$	−6.44938	−0.305045
$448$	11.6128	0.548655
$449$	−6.75404	−0.318743	−0.159371	−	0.987219i	$-0.550947\pi$
$449$	−6.75404	−0.318743	−0.159371	+	0.987219i	$0.550947\pi$
$450$	0	0
$451$	−18.4637	−0.869420
$452$	40.7654	1.91744
$453$	23.9081	1.12330
$454$	−33.5526	−1.57470
$455$	0	0
$456$	7.24443	0.339252
$457$	20.4351	0.955913	0.477957	−	0.878383i	$-0.341378\pi$
$457$	20.4351	0.955913	0.477957	+	0.878383i	$0.341378\pi$
$458$	3.71900	0.173778
$459$	6.73975	0.314584
$460$	0	0
$461$	−8.20495	−0.382143	−0.191071	−	0.981576i	$-0.561196\pi$
$461$	−8.20495	−0.382143	−0.191071	+	0.981576i	$0.561196\pi$
$462$	−9.18421	−0.427288
$463$	−26.2766	−1.22117	−0.610587	−	0.791949i	$-0.709067\pi$
$463$	−26.2766	−1.22117	−0.610587	+	0.791949i	$0.709067\pi$
$464$	2.32693	0.108025
$465$	0	0
$466$	29.3274	1.35857
$467$	5.38763	0.249310	0.124655	−	0.992200i	$-0.460218\pi$
$467$	5.38763	0.249310	0.124655	+	0.992200i	$0.460218\pi$
$468$	13.0509	0.603276
$469$	9.03657	0.417270
$470$	0	0
$471$	5.19358	0.239307
$472$	−22.5303	−1.03704
$473$	−52.7150	−2.42384
$474$	2.47457	0.113661
$475$	0	0
$476$	17.6731	0.810044
$477$	0.873100	0.0399765
$478$	−7.44446	−0.340502
$479$	1.61285	0.0736929	0.0368464	−	0.999321i	$-0.488269\pi$
$479$	1.61285	0.0736929	0.0368464	+	0.999321i	$0.488269\pi$
$480$	0	0
$481$	39.0593	1.78095
$482$	8.05578	0.366931
$483$	−7.58565	−0.345159
$484$	29.2859	1.33118
$485$	0	0
$486$	−2.21432	−0.100444
$487$	24.6287	1.11603	0.558016	−	0.829830i	$-0.311563\pi$
$487$	24.6287	1.11603	0.558016	+	0.829830i	$0.311563\pi$
$488$	25.7146	1.16404
$489$	−20.2257	−0.914638
$490$	0	0
$491$	43.0721	1.94382	0.971908	−	0.235362i	$-0.0756276\pi$
$491$	43.0721	1.94382	0.971908	+	0.235362i	$0.0756276\pi$
$492$	−11.6731	−0.526263
$493$	11.3827	0.512651
$494$	−36.0558	−1.62223
$495$	0	0
$496$	1.37778	0.0618643
$497$	11.7877	0.528750
$498$	8.79060	0.393916
$499$	27.7496	1.24224	0.621121	−	0.783715i	$-0.286677\pi$
$499$	27.7496	1.24224	0.621121	+	0.783715i	$0.286677\pi$
$500$	0	0
$501$	−19.0923	−0.852983
$502$	−31.7812	−1.41847
$503$	−29.2543	−1.30438	−0.652192	−	0.758054i	$-0.726150\pi$
$503$	−29.2543	−1.30438	−0.652192	+	0.758054i	$0.726150\pi$
$504$	−1.80642	−0.0804645
$505$	0	0
$506$	85.3995	3.79647
$507$	−7.20787	−0.320113
$508$	2.19358	0.0973242
$509$	−18.1778	−0.805715	−0.402857	−	0.915263i	$-0.631983\pi$
$509$	−18.1778	−0.805715	−0.402857	+	0.915263i	$0.631983\pi$
$510$	0	0
$511$	−7.61285	−0.336773
$512$	15.2257	0.672887
$513$	3.62222	0.159925
$514$	13.8524	0.611001
$515$	0	0
$516$	−33.3274	−1.46716
$517$	5.43801	0.239163
$518$	−17.3778	−0.763536
$519$	−5.20495	−0.228472
$520$	0	0
$521$	−17.4084	−0.762675	−0.381337	−	0.924436i	$-0.624536\pi$
$521$	−17.4084	−0.762675	−0.381337	+	0.924436i	$0.624536\pi$
$522$	−3.73975	−0.163684
$523$	24.8671	1.08736	0.543682	−	0.839291i	$-0.317030\pi$
$523$	24.8671	1.08736	0.543682	+	0.839291i	$0.317030\pi$
$524$	29.9496	1.30836
$525$	0	0
$526$	68.2864	2.97743
$527$	6.73975	0.293588
$528$	−6.32693	−0.275344
$529$	47.5353	2.06675
$530$	0	0
$531$	−11.2652	−0.488867
$532$	9.49823	0.411801
$533$	18.0745	0.782894
$534$	21.0573	0.911239
$535$	0	0
$536$	−20.0098	−0.864294
$537$	−19.2400	−0.830267
$538$	−25.5812	−1.10288
$539$	28.3985	1.22321
$540$	0	0
$541$	15.9131	0.684156	0.342078	−	0.939672i	$-0.388869\pi$
$541$	15.9131	0.684156	0.342078	+	0.939672i	$0.388869\pi$
$542$	−40.0054	−1.71838
$543$	−4.06668	−0.174518
$544$	47.5210	2.03745
$545$	0	0
$546$	8.99063	0.384764
$547$	25.1704	1.07621	0.538104	−	0.842878i	$-0.319141\pi$
$547$	25.1704	1.07621	0.538104	+	0.842878i	$0.319141\pi$
$548$	−29.5482	−1.26224
$549$	12.8573	0.548735
$550$	0	0
$551$	6.11753	0.260616
$552$	16.7971	0.714930
$553$	1.00937	0.0429227
$554$	32.6035	1.38519
$555$	0	0
$556$	27.1526	1.15153
$557$	−44.4499	−1.88340	−0.941700	−	0.336452i	$-0.890773\pi$
$557$	−44.4499	−1.88340	−0.941700	+	0.336452i	$0.890773\pi$
$558$	−2.21432	−0.0937396
$559$	51.6040	2.18261
$560$	0	0
$561$	−30.9496	−1.30669
$562$	21.2859	0.897892
$563$	35.9605	1.51556	0.757778	−	0.652513i	$-0.226285\pi$
$563$	35.9605	1.51556	0.757778	+	0.652513i	$0.226285\pi$
$564$	3.43801	0.144766
$565$	0	0
$566$	8.61132	0.361961
$567$	−0.903212	−0.0379313
$568$	−26.1017	−1.09520
$569$	5.92840	0.248532	0.124266	−	0.992249i	$-0.460342\pi$
$569$	5.92840	0.248532	0.124266	+	0.992249i	$0.460342\pi$
$570$	0	0
$571$	8.97773	0.375706	0.187853	−	0.982197i	$-0.439847\pi$
$571$	8.97773	0.375706	0.187853	+	0.982197i	$0.439847\pi$
$572$	−59.9309	−2.50584
$573$	−8.12245	−0.339320
$574$	−8.04149	−0.335645
$575$	0	0
$576$	−12.8573	−0.535720
$577$	−7.19850	−0.299677	−0.149839	−	0.988710i	$-0.547875\pi$
$577$	−7.19850	−0.299677	−0.149839	+	0.988710i	$0.547875\pi$
$578$	62.9403	2.61797
$579$	−9.01429	−0.374621
$580$	0	0
$581$	3.58565	0.148758
$582$	38.0622	1.57773
$583$	−4.00937	−0.166051
$584$	16.8573	0.697559
$585$	0	0
$586$	−7.13828	−0.294879
$587$	−41.3259	−1.70570	−0.852851	−	0.522155i	$-0.825128\pi$
$587$	−41.3259	−1.70570	−0.852851	+	0.522155i	$0.825128\pi$
$588$	17.9541	0.740413
$589$	3.62222	0.149251
$590$	0	0
$591$	−7.71456	−0.317335
$592$	−11.9714	−0.492022
$593$	−8.44938	−0.346975	−0.173487	−	0.984836i	$-0.555504\pi$
$593$	−8.44938	−0.346975	−0.173487	+	0.984836i	$0.555504\pi$
$594$	10.1684	0.417214
$595$	0	0
$596$	18.7239	0.766962
$597$	−0.790602	−0.0323572
$598$	−83.5995	−3.41864
$599$	−4.83654	−0.197615	−0.0988077	−	0.995107i	$-0.531503\pi$
$599$	−4.83654	−0.197615	−0.0988077	+	0.995107i	$0.531503\pi$
$600$	0	0
$601$	26.5368	1.08246	0.541229	−	0.840875i	$-0.317959\pi$
$601$	26.5368	1.08246	0.541229	+	0.840875i	$0.317959\pi$
$602$	−22.9590	−0.935739
$603$	−10.0049	−0.407432
$604$	−69.4104	−2.82427
$605$	0	0
$606$	17.7146	0.719605
$607$	−31.0272	−1.25936	−0.629678	−	0.776857i	$-0.716813\pi$
$607$	−31.0272	−1.25936	−0.629678	+	0.776857i	$0.716813\pi$
$608$	25.5397	1.03577
$609$	−1.52543	−0.0618134
$610$	0	0
$611$	−5.32339	−0.215361
$612$	−19.5669	−0.790946
$613$	23.0509	0.931015	0.465508	−	0.885044i	$-0.345872\pi$
$613$	23.0509	0.931015	0.465508	+	0.885044i	$0.345872\pi$
$614$	−24.3684	−0.983429
$615$	0	0
$616$	8.29529	0.334227
$617$	−2.50961	−0.101033	−0.0505165	−	0.998723i	$-0.516087\pi$
$617$	−2.50961	−0.101033	−0.0505165	+	0.998723i	$0.516087\pi$
$618$	−1.49039	−0.0599524
$619$	−37.6543	−1.51346	−0.756728	−	0.653730i	$-0.773203\pi$
$619$	−37.6543	−1.51346	−0.756728	+	0.653730i	$0.773203\pi$
$620$	0	0
$621$	8.39853	0.337021
$622$	24.2908	0.973974
$623$	8.58919	0.344119
$624$	6.19358	0.247941
$625$	0	0
$626$	−4.01429	−0.160443
$627$	−16.6336	−0.664282
$628$	−15.0781	−0.601680
$629$	−58.5609	−2.33498
$630$	0	0
$631$	38.2701	1.52351	0.761754	−	0.647866i	$-0.224338\pi$
$631$	38.2701	1.52351	0.761754	+	0.647866i	$0.224338\pi$
$632$	−2.23506	−0.0889060
$633$	15.9304	0.633177
$634$	−55.2913	−2.19590
$635$	0	0
$636$	−2.53480	−0.100511
$637$	−27.8000	−1.10147
$638$	17.1733	0.679898
$639$	−13.0509	−0.516284
$640$	0	0
$641$	−17.7304	−0.700308	−0.350154	−	0.936692i	$-0.613871\pi$
$641$	−17.7304	−0.700308	−0.350154	+	0.936692i	$0.613871\pi$
$642$	−20.2208	−0.798050
$643$	−33.0355	−1.30279	−0.651397	−	0.758737i	$-0.725817\pi$
$643$	−33.0355	−1.30279	−0.651397	+	0.758737i	$0.725817\pi$
$644$	22.0228	0.867818
$645$	0	0
$646$	54.0578	2.12688
$647$	7.58274	0.298108	0.149054	−	0.988829i	$-0.452377\pi$
$647$	7.58274	0.298108	0.149054	+	0.988829i	$0.452377\pi$
$648$	2.00000	0.0785674
$649$	51.7309	2.03061
$650$	0	0
$651$	−0.903212	−0.0353997
$652$	58.7195	2.29963
$653$	−25.8687	−1.01232	−0.506159	−	0.862440i	$-0.668935\pi$
$653$	−25.8687	−1.01232	−0.506159	+	0.862440i	$0.668935\pi$
$654$	28.7971	1.12605
$655$	0	0
$656$	−5.53972	−0.216290
$657$	8.42864	0.328833
$658$	2.36842	0.0923305
$659$	12.8287	0.499735	0.249868	−	0.968280i	$-0.419613\pi$
$659$	12.8287	0.499735	0.249868	+	0.968280i	$0.419613\pi$
$660$	0	0
$661$	−9.04593	−0.351846	−0.175923	−	0.984404i	$-0.556291\pi$
$661$	−9.04593	−0.351846	−0.175923	+	0.984404i	$0.556291\pi$
$662$	−64.5674	−2.50948
$663$	30.2973	1.17665
$664$	−7.93978	−0.308123
$665$	0	0
$666$	19.2400	0.745534
$667$	14.1842	0.549215
$668$	55.4291	2.14462
$669$	−5.86665	−0.226818
$670$	0	0
$671$	−59.0420	−2.27929
$672$	−6.36842	−0.245667
$673$	7.56491	0.291606	0.145803	−	0.989314i	$-0.453423\pi$
$673$	7.56491	0.291606	0.145803	+	0.989314i	$0.453423\pi$
$674$	−16.6464	−0.641193
$675$	0	0
$676$	20.9260	0.804845
$677$	0.699791	0.0268952	0.0134476	−	0.999910i	$-0.495719\pi$
$677$	0.699791	0.0268952	0.0134476	+	0.999910i	$0.495719\pi$
$678$	−31.0923	−1.19409
$679$	15.5254	0.595811
$680$	0	0
$681$	15.1526	0.580648
$682$	10.1684	0.389368
$683$	−3.29481	−0.126072	−0.0630362	−	0.998011i	$-0.520078\pi$
$683$	−3.29481	−0.126072	−0.0630362	+	0.998011i	$0.520078\pi$
$684$	−10.5161	−0.402092
$685$	0	0
$686$	26.3684	1.00675
$687$	−1.67952	−0.0640778
$688$	−15.8163	−0.602990
$689$	3.92486	0.149525
$690$	0	0
$691$	10.3876	0.395164	0.197582	−	0.980286i	$-0.436691\pi$
$691$	10.3876	0.395164	0.197582	+	0.980286i	$0.436691\pi$
$692$	15.1111	0.574437
$693$	4.14764	0.157556
$694$	52.9086	2.00838
$695$	0	0
$696$	3.37778	0.128035
$697$	−27.0988	−1.02644
$698$	−0.836535	−0.0316633
$699$	−13.2444	−0.500950
$700$	0	0
$701$	−28.3892	−1.07224	−0.536122	−	0.844141i	$-0.680111\pi$
$701$	−28.3892	−1.07224	−0.536122	+	0.844141i	$0.680111\pi$
$702$	−9.95407	−0.375692
$703$	−31.4730	−1.18703
$704$	59.0420	2.22523
$705$	0	0
$706$	0.0983194	0.00370030
$707$	7.22570	0.271750
$708$	32.7052	1.22914
$709$	7.43155	0.279098	0.139549	−	0.990215i	$-0.455435\pi$
$709$	7.43155	0.279098	0.139549	+	0.990215i	$0.455435\pi$
$710$	0	0
$711$	−1.11753	−0.0419107
$712$	−19.0192	−0.712775
$713$	8.39853	0.314527
$714$	−13.4795	−0.504457
$715$	0	0
$716$	55.8578	2.08750
$717$	3.36196	0.125555
$718$	−53.3975	−1.99278
$719$	21.3363	0.795710	0.397855	−	0.917448i	$-0.369755\pi$
$719$	21.3363	0.795710	0.397855	+	0.917448i	$0.369755\pi$
$720$	0	0
$721$	−0.607926	−0.0226403
$722$	−13.0192	−0.484525
$723$	−3.63804	−0.135300
$724$	11.8064	0.438782
$725$	0	0
$726$	−22.3368	−0.828995
$727$	−18.4844	−0.685549	−0.342775	−	0.939418i	$-0.611367\pi$
$727$	−18.4844	−0.685549	−0.342775	+	0.939418i	$0.611367\pi$
$728$	−8.12045	−0.300964
$729$	1.00000	0.0370370
$730$	0	0
$731$	−77.3689	−2.86159
$732$	−37.3274	−1.37966
$733$	34.6731	1.28068	0.640340	−	0.768092i	$-0.278794\pi$
$733$	34.6731	1.28068	0.640340	+	0.768092i	$0.278794\pi$
$734$	−66.4514	−2.45277
$735$	0	0
$736$	59.2168	2.18276
$737$	45.9436	1.69236
$738$	8.90321	0.327732
$739$	−27.1590	−0.999061	−0.499530	−	0.866296i	$-0.666494\pi$
$739$	−27.1590	−0.999061	−0.499530	+	0.866296i	$0.666494\pi$
$740$	0	0
$741$	16.2830	0.598171
$742$	−1.74620	−0.0641050
$743$	−25.6430	−0.940749	−0.470374	−	0.882467i	$-0.655881\pi$
$743$	−25.6430	−0.940749	−0.470374	+	0.882467i	$0.655881\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−43.8687	−1.60615
$747$	−3.96989	−0.145251
$748$	89.8533	3.28536
$749$	−8.24797	−0.301374
$750$	0	0
$751$	0.925487	0.0337715	0.0168857	−	0.999857i	$-0.494625\pi$
$751$	0.925487	0.0337715	0.0168857	+	0.999857i	$0.494625\pi$
$752$	1.63158	0.0594977
$753$	14.3526	0.523038
$754$	−16.8113	−0.612233
$755$	0	0
$756$	2.62222	0.0953691
$757$	−26.4351	−0.960800	−0.480400	−	0.877050i	$-0.659508\pi$
$757$	−26.4351	−0.960800	−0.480400	+	0.877050i	$0.659508\pi$
$758$	−63.7975	−2.31723
$759$	−38.5669	−1.39989
$760$	0	0
$761$	−17.5768	−0.637157	−0.318579	−	0.947896i	$-0.603205\pi$
$761$	−17.5768	−0.637157	−0.318579	+	0.947896i	$0.603205\pi$
$762$	−1.67307	−0.0606090
$763$	11.7462	0.425241
$764$	23.5812	0.853138
$765$	0	0
$766$	−67.4356	−2.43655
$767$	−50.6405	−1.82852
$768$	6.10171	0.220177
$769$	−2.14717	−0.0774288	−0.0387144	−	0.999250i	$-0.512326\pi$
$769$	−2.14717	−0.0774288	−0.0387144	+	0.999250i	$0.512326\pi$
$770$	0	0
$771$	−6.25581	−0.225297
$772$	26.1704	0.941893
$773$	−51.6055	−1.85612	−0.928060	−	0.372430i	$-0.878525\pi$
$773$	−51.6055	−1.85612	−0.928060	+	0.372430i	$0.878525\pi$
$774$	25.4193	0.913677
$775$	0	0
$776$	−34.3783	−1.23411
$777$	7.84791	0.281542
$778$	32.8988	1.17948
$779$	−14.5640	−0.521809
$780$	0	0
$781$	59.9309	2.14450
$782$	125.339	4.48212
$783$	1.68889	0.0603561
$784$	8.52051	0.304304
$785$	0	0
$786$	−22.8430	−0.814782
$787$	13.0509	0.465213	0.232606	−	0.972571i	$-0.425275\pi$
$787$	13.0509	0.465213	0.232606	+	0.972571i	$0.425275\pi$
$788$	22.3970	0.797860
$789$	−30.8385	−1.09788
$790$	0	0
$791$	−12.6824	−0.450936
$792$	−9.18421	−0.326347
$793$	57.7975	2.05245
$794$	−17.9289	−0.636272
$795$	0	0
$796$	2.29529	0.0813543
$797$	−46.5892	−1.65027	−0.825137	−	0.564933i	$-0.808902\pi$
$797$	−46.5892	−1.65027	−0.825137	+	0.564933i	$0.808902\pi$
$798$	−7.24443	−0.256450
$799$	7.98126	0.282357
$800$	0	0
$801$	−9.50961	−0.336005
$802$	24.2701	0.857007
$803$	−38.7052	−1.36588
$804$	29.0464	1.02439
$805$	0	0
$806$	−9.95407	−0.350617
$807$	11.5526	0.406672
$808$	−16.0000	−0.562878
$809$	30.1907	1.06145	0.530724	−	0.847545i	$-0.321920\pi$
$809$	30.1907	1.06145	0.530724	+	0.847545i	$0.321920\pi$
$810$	0	0
$811$	−0.0173528	−0.000609338 0	−0.000304669	−	1.00000i	$-0.500097\pi$
$811$	−0.0173528	−0.000609338 0	−0.000304669		1.00000i	$0.500097\pi$
$812$	4.42864	0.155415
$813$	18.0667	0.633626
$814$	−88.3520	−3.09674
$815$	0	0
$816$	−9.28592	−0.325072
$817$	−41.5812	−1.45474
$818$	−83.6454	−2.92459
$819$	−4.06022	−0.141876
$820$	0	0
$821$	−37.2371	−1.29958	−0.649791	−	0.760113i	$-0.725144\pi$
$821$	−37.2371	−1.29958	−0.649791	+	0.760113i	$0.725144\pi$
$822$	22.5368	0.786061
$823$	−18.0228	−0.628234	−0.314117	−	0.949384i	$-0.601708\pi$
$823$	−18.0228	−0.628234	−0.314117	+	0.949384i	$0.601708\pi$
$824$	1.34614	0.0468951
$825$	0	0
$826$	22.5303	0.783931
$827$	−3.45383	−0.120101	−0.0600507	−	0.998195i	$-0.519126\pi$
$827$	−3.45383	−0.120101	−0.0600507	+	0.998195i	$0.519126\pi$
$828$	−24.3827	−0.847358
$829$	−30.0163	−1.04251	−0.521255	−	0.853401i	$-0.674536\pi$
$829$	−30.0163	−1.04251	−0.521255	+	0.853401i	$0.674536\pi$
$830$	0	0
$831$	−14.7239	−0.510767
$832$	−57.7975	−2.00377
$833$	41.6800	1.44413
$834$	−20.7096	−0.717116
$835$	0	0
$836$	48.2908	1.67017
$837$	1.00000	0.0345651
$838$	−18.6637	−0.644727
$839$	2.10171	0.0725591	0.0362795	−	0.999342i	$-0.488449\pi$
$839$	2.10171	0.0725591	0.0362795	+	0.999342i	$0.488449\pi$
$840$	0	0
$841$	−26.1476	−0.901643
$842$	−65.3787	−2.25310
$843$	−9.61285	−0.331084
$844$	−46.2494	−1.59197
$845$	0	0
$846$	−2.62222	−0.0901536
$847$	−9.11108	−0.313060
$848$	−1.20294	−0.0413093
$849$	−3.88892	−0.133467
$850$	0	0
$851$	−72.9739	−2.50151
$852$	37.8894	1.29807
$853$	−26.9032	−0.921148	−0.460574	−	0.887621i	$-0.652356\pi$
$853$	−26.9032	−0.921148	−0.460574	+	0.887621i	$0.652356\pi$
$854$	−25.7146	−0.879934
$855$	0	0
$856$	18.2636	0.624238
$857$	40.3763	1.37923	0.689613	−	0.724178i	$-0.257781\pi$
$857$	40.3763	1.37923	0.689613	+	0.724178i	$0.257781\pi$
$858$	45.7101	1.56052
$859$	2.53035	0.0863344	0.0431672	−	0.999068i	$-0.486255\pi$
$859$	2.53035	0.0863344	0.0431672	+	0.999068i	$0.486255\pi$
$860$	0	0
$861$	3.63158	0.123764
$862$	80.3279	2.73598
$863$	−0.828699	−0.0282092	−0.0141046	−	0.999901i	$-0.504490\pi$
$863$	−0.828699	−0.0282092	−0.0141046	+	0.999901i	$0.504490\pi$
$864$	7.05086	0.239875
$865$	0	0
$866$	−43.7462	−1.48656
$867$	−28.4242	−0.965336
$868$	2.62222	0.0890038
$869$	5.13182	0.174085
$870$	0	0
$871$	−44.9753	−1.52393
$872$	−26.0098	−0.880804
$873$	−17.1891	−0.581764
$874$	67.3624	2.27857
$875$	0	0
$876$	−24.4701	−0.826769
$877$	−8.44738	−0.285248	−0.142624	−	0.989777i	$-0.545554\pi$
$877$	−8.44738	−0.285248	−0.142624	+	0.989777i	$0.545554\pi$
$878$	−28.4603	−0.960488
$879$	3.22369	0.108732
$880$	0	0
$881$	0.580728	0.0195652	0.00978262	−	0.999952i	$-0.496886\pi$
$881$	0.580728	0.0195652	0.00978262	+	0.999952i	$0.496886\pi$
$882$	−13.6938	−0.461095
$883$	40.4449	1.36108	0.680540	−	0.732711i	$-0.261745\pi$
$883$	40.4449	1.36108	0.680540	+	0.732711i	$0.261745\pi$
$884$	−87.9595	−2.95840
$885$	0	0
$886$	−3.59549	−0.120793
$887$	−5.19204	−0.174332	−0.0871659	−	0.996194i	$-0.527781\pi$
$887$	−5.19204	−0.174332	−0.0871659	+	0.996194i	$0.527781\pi$
$888$	−17.3778	−0.583160
$889$	−0.682439	−0.0228883
$890$	0	0
$891$	−4.59210	−0.153841
$892$	17.0321	0.570278
$893$	4.28946	0.143541
$894$	−14.2810	−0.477628
$895$	0	0
$896$	12.9777	0.433555
$897$	37.7540	1.26057
$898$	−14.9556	−0.499075
$899$	1.68889	0.0563277
$900$	0	0
$901$	−5.88448	−0.196040
$902$	−40.8845	−1.36130
$903$	10.3684	0.345039
$904$	28.0830	0.934026
$905$	0	0
$906$	52.9403	1.75882
$907$	−31.6865	−1.05213	−0.526066	−	0.850444i	$-0.676333\pi$
$907$	−31.6865	−1.05213	−0.526066	+	0.850444i	$0.676333\pi$
$908$	−43.9911	−1.45990
$909$	−8.00000	−0.265343
$910$	0	0
$911$	24.8415	0.823034	0.411517	−	0.911402i	$-0.364999\pi$
$911$	24.8415	0.823034	0.411517	+	0.911402i	$0.364999\pi$
$912$	−4.99063	−0.165256
$913$	18.2301	0.603330
$914$	45.2498	1.49673
$915$	0	0
$916$	4.87601	0.161108
$917$	−9.31756	−0.307693
$918$	14.9240	0.492564
$919$	−7.95359	−0.262365	−0.131182	−	0.991358i	$-0.541877\pi$
$919$	−7.95359	−0.262365	−0.131182	+	0.991358i	$0.541877\pi$
$920$	0	0
$921$	11.0049	0.362625
$922$	−18.1684	−0.598344
$923$	−58.6677	−1.93107
$924$	−12.0415	−0.396136
$925$	0	0
$926$	−58.1847	−1.91207
$927$	0.673071	0.0221065
$928$	11.9081	0.390904
$929$	12.5462	0.411627	0.205813	−	0.978591i	$-0.434016\pi$
$929$	12.5462	0.411627	0.205813	+	0.978591i	$0.434016\pi$
$930$	0	0
$931$	22.4005	0.734148
$932$	38.4514	1.25952
$933$	−10.9699	−0.359138
$934$	11.9299	0.390359
$935$	0	0
$936$	8.99063	0.293868
$937$	1.87601	0.0612867	0.0306434	−	0.999530i	$-0.490244\pi$
$937$	1.87601	0.0612867	0.0306434	+	0.999530i	$0.490244\pi$
$938$	20.0098	0.653345
$939$	1.81288	0.0591610
$940$	0	0
$941$	12.6207	0.411423	0.205711	−	0.978613i	$-0.434049\pi$
$941$	12.6207	0.411423	0.205711	+	0.978613i	$0.434049\pi$
$942$	11.5002	0.374698
$943$	−33.7683	−1.09965
$944$	15.5210	0.505165
$945$	0	0
$946$	−116.728	−3.79515
$947$	46.8785	1.52335	0.761673	−	0.647961i	$-0.224378\pi$
$947$	46.8785	1.52335	0.761673	+	0.647961i	$0.224378\pi$
$948$	3.24443	0.105374
$949$	37.8894	1.22994
$950$	0	0
$951$	24.9699	0.809704
$952$	12.1748	0.394589
$953$	−55.5308	−1.79882	−0.899410	−	0.437106i	$-0.856003\pi$
$953$	−55.5308	−1.79882	−0.899410	+	0.437106i	$0.856003\pi$
$954$	1.93332	0.0625937
$955$	0	0
$956$	−9.76049	−0.315677
$957$	−7.75557	−0.250702
$958$	3.57136	0.115385
$959$	9.19267	0.296847
$960$	0	0
$961$	1.00000	0.0322581
$962$	86.4898	2.78854
$963$	9.13182	0.294269
$964$	10.5620	0.340179
$965$	0	0
$966$	−16.7971	−0.540436
$967$	58.1595	1.87028	0.935142	−	0.354274i	$-0.115272\pi$
$967$	58.1595	1.87028	0.935142	+	0.354274i	$0.115272\pi$
$968$	20.1748	0.648444
$969$	−24.4128	−0.784253
$970$	0	0
$971$	−50.0721	−1.60689	−0.803445	−	0.595379i	$-0.797002\pi$
$971$	−50.0721	−1.60689	−0.803445	+	0.595379i	$0.797002\pi$
$972$	−2.90321	−0.0931206
$973$	−8.44738	−0.270810
$974$	54.5357	1.74744
$975$	0	0
$976$	−17.7146	−0.567029
$977$	5.70271	0.182446	0.0912229	−	0.995831i	$-0.470922\pi$
$977$	5.70271	0.182446	0.0912229	+	0.995831i	$0.470922\pi$
$978$	−44.7862	−1.43210
$979$	43.6691	1.39567
$980$	0	0
$981$	−13.0049	−0.415215
$982$	95.3753	3.04355
$983$	4.87310	0.155428	0.0777139	−	0.996976i	$-0.475238\pi$
$983$	4.87310	0.155428	0.0777139	+	0.996976i	$0.475238\pi$
$984$	−8.04149	−0.256353
$985$	0	0
$986$	25.2050	0.802689
$987$	−1.06959	−0.0340455
$988$	−47.2730	−1.50395
$989$	−96.4109	−3.06569
$990$	0	0
$991$	−11.8415	−0.376156	−0.188078	−	0.982154i	$-0.560226\pi$
$991$	−11.8415	−0.376156	−0.188078	+	0.982154i	$0.560226\pi$
$992$	7.05086	0.223865
$993$	29.1590	0.925333
$994$	26.1017	0.827896
$995$	0	0
$996$	11.5254	0.365197
$997$	−36.5067	−1.15618	−0.578089	−	0.815974i	$-0.696201\pi$
$997$	−36.5067	−1.15618	−0.578089	+	0.815974i	$0.696201\pi$
$998$	61.4465	1.94505
$999$	−8.68889	−0.274904

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.t.1.3	✓	3
3.2	odd	2		6975.2.a.bd.1.1		3
5.2	odd	4		2325.2.c.m.1024.6		6
5.3	odd	4		2325.2.c.m.1024.1		6
5.4	even	2		2325.2.a.u.1.1	yes	3
15.14	odd	2		6975.2.a.bc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.t.1.3	✓	3	1.1	even	1	trivial
2325.2.a.u.1.1	yes	3	5.4	even	2
2325.2.c.m.1024.1		6	5.3	odd	4
2325.2.c.m.1024.6		6	5.2	odd	4
6975.2.a.bc.1.3		3	15.14	odd	2
6975.2.a.bd.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21432 q^{2} -1.00000 q^{3} +2.90321 q^{4} -2.21432 q^{6} -0.903212 q^{7} +2.00000 q^{8} +1.00000 q^{9} -4.59210 q^{11} -2.90321 q^{12} +4.49532 q^{13} -2.00000 q^{14} -1.37778 q^{16} -6.73975 q^{17} +2.21432 q^{18} -3.62222 q^{19} +0.903212 q^{21} -10.1684 q^{22} -8.39853 q^{23} -2.00000 q^{24} +9.95407 q^{26} -1.00000 q^{27} -2.62222 q^{28} -1.68889 q^{29} -1.00000 q^{31} -7.05086 q^{32} +4.59210 q^{33} -14.9240 q^{34} +2.90321 q^{36} +8.68889 q^{37} -8.02074 q^{38} -4.49532 q^{39} +4.02074 q^{41} +2.00000 q^{42} +11.4795 q^{43} -13.3319 q^{44} -18.5970 q^{46} -1.18421 q^{47} +1.37778 q^{48} -6.18421 q^{49} +6.73975 q^{51} +13.0509 q^{52} +0.873100 q^{53} -2.21432 q^{54} -1.80642 q^{56} +3.62222 q^{57} -3.73975 q^{58} -11.2652 q^{59} +12.8573 q^{61} -2.21432 q^{62} -0.903212 q^{63} -12.8573 q^{64} +10.1684 q^{66} -10.0049 q^{67} -19.5669 q^{68} +8.39853 q^{69} -13.0509 q^{71} +2.00000 q^{72} +8.42864 q^{73} +19.2400 q^{74} -10.5161 q^{76} +4.14764 q^{77} -9.95407 q^{78} -1.11753 q^{79} +1.00000 q^{81} +8.90321 q^{82} -3.96989 q^{83} +2.62222 q^{84} +25.4193 q^{86} +1.68889 q^{87} -9.18421 q^{88} -9.50961 q^{89} -4.06022 q^{91} -24.3827 q^{92} +1.00000 q^{93} -2.62222 q^{94} +7.05086 q^{96} -17.1891 q^{97} -13.6938 q^{98} -4.59210 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{4} + 4 q^{7} + 6 q^{8} + 3 q^{9} - 7 q^{11} - 2 q^{12} - 6 q^{14} - 4 q^{16} - 7 q^{17} - 11 q^{19} - 4 q^{21} - 4 q^{22} - 5 q^{23} - 6 q^{24} + 10 q^{26} - 3 q^{27} - 8 q^{28} - 5 q^{29}+ \cdots - 7 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^4 + 4 * q^7 + 6 * q^8 + 3 * q^9 - 7 * q^11 - 2 * q^12 - 6 * q^14 - 4 * q^16 - 7 * q^17 - 11 * q^19 - 4 * q^21 - 4 * q^22 - 5 * q^23 - 6 * q^24 + 10 * q^26 - 3 * q^27 - 8 * q^28 - 5 * q^29 - 3 * q^31 - 8 * q^32 + 7 * q^33 - 18 * q^34 + 2 * q^36 + 26 * q^37 - 4 * q^38 - 8 * q^41 + 6 * q^42 + 8 * q^43 - 20 * q^44 - 16 * q^46 + 10 * q^47 + 4 * q^48 - 5 * q^49 + 7 * q^51 + 26 * q^52 - 11 * q^53 + 8 * q^56 + 11 * q^57 + 2 * q^58 - 14 * q^59 + 12 * q^61 + 4 * q^63 - 12 * q^64 + 4 * q^66 + 3 * q^67 - 12 * q^68 + 5 * q^69 - 26 * q^71 + 6 * q^72 + 12 * q^73 - 2 * q^74 + 2 * q^76 + 6 * q^77 - 10 * q^78 + 10 * q^79 + 3 * q^81 + 20 * q^82 - 5 * q^83 + 8 * q^84 + 36 * q^86 + 5 * q^87 - 14 * q^88 - 9 * q^89 - 26 * q^91 - 40 * q^92 + 3 * q^93 - 8 * q^94 + 8 * q^96 - 5 * q^97 - 8 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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