LMFDB - Embedded newform 9405.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$75.0993031010$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.284630$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.22871 q^{2} +2.96714 q^{4} -1.00000 q^{5} -4.42518 q^{7} -2.15546 q^{8} +O(q^{10})$	\(q-2.22871 q^{2} +2.96714 q^{4} -1.00000 q^{5} -4.42518 q^{7} -2.15546 q^{8} +2.22871 q^{10} +1.00000 q^{11} -3.92613 q^{13} +9.86243 q^{14} -1.13037 q^{16} +1.61520 q^{17} +1.00000 q^{19} -2.96714 q^{20} -2.22871 q^{22} -0.113248 q^{23} +1.00000 q^{25} +8.75019 q^{26} -13.1301 q^{28} -1.08612 q^{29} -4.17296 q^{31} +6.83020 q^{32} -3.59980 q^{34} +4.42518 q^{35} -5.75406 q^{37} -2.22871 q^{38} +2.15546 q^{40} -4.26750 q^{41} +3.62271 q^{43} +2.96714 q^{44} +0.252396 q^{46} +6.39567 q^{47} +12.5822 q^{49} -2.22871 q^{50} -11.6494 q^{52} -12.0154 q^{53} -1.00000 q^{55} +9.53832 q^{56} +2.42065 q^{58} -0.883911 q^{59} +3.77712 q^{61} +9.30032 q^{62} -12.9618 q^{64} +3.92613 q^{65} -12.2192 q^{67} +4.79251 q^{68} -9.86243 q^{70} -4.28400 q^{71} -1.92676 q^{73} +12.8241 q^{74} +2.96714 q^{76} -4.42518 q^{77} -7.81004 q^{79} +1.13037 q^{80} +9.51102 q^{82} +3.33410 q^{83} -1.61520 q^{85} -8.07397 q^{86} -2.15546 q^{88} +11.9145 q^{89} +17.3738 q^{91} -0.336021 q^{92} -14.2541 q^{94} -1.00000 q^{95} -13.3478 q^{97} -28.0421 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8	$ 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31} + 2 q^{32} + 4 q^{34} + 11 q^{35} - 9 q^{37} + 3 q^{38} + 3 q^{40} - 15 q^{41} - 13 q^{43} + 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{49} + 3 q^{50} + q^{52} + 5 q^{53} - 5 q^{55} - 33 q^{58} + 17 q^{59} + 3 q^{61} - 14 q^{62} - 17 q^{64} - q^{65} - 28 q^{67} + 25 q^{68} + 6 q^{71} - 16 q^{73} + 21 q^{74} + 5 q^{76} - 11 q^{77} + 3 q^{79} + 3 q^{80} + 2 q^{82} + 33 q^{83} - 3 q^{85} - 10 q^{86} - 3 q^{88} + 16 q^{89} - 22 q^{91} + 19 q^{92} - 10 q^{94} - 5 q^{95} - 14 q^{97} - 10 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.22871	−1.57593	−0.787967	−	0.615717i	$-0.788866\pi$
$2$	−2.22871	−1.57593	−0.787967	+	0.615717i	$0.788866\pi$
$3$	0	0
$3$	0	0
$4$	2.96714	1.48357
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.42518	−1.67256	−0.836281	−	0.548302i	$-0.815275\pi$
$7$	−4.42518	−1.67256	−0.836281	+	0.548302i	$0.815275\pi$
$8$	−2.15546	−0.762072
$9$	0	0
$10$	2.22871	0.704779
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.92613	−1.08891	−0.544456	−	0.838789i	$-0.683264\pi$
$13$	−3.92613	−1.08891	−0.544456	+	0.838789i	$0.683264\pi$
$14$	9.86243	2.63585
$15$	0	0
$16$	−1.13037	−0.282593
$17$	1.61520	0.391743	0.195872	−	0.980630i	$-0.437246\pi$
$17$	1.61520	0.391743	0.195872	+	0.980630i	$0.437246\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.96714	−0.663472
$21$	0	0
$22$	−2.22871	−0.475162
$23$	−0.113248	−0.0236138	−0.0118069	−	0.999930i	$-0.503758\pi$
$23$	−0.113248	−0.0236138	−0.0118069	+	0.999930i	$0.503758\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	8.75019	1.71605
$27$	0	0
$28$	−13.1301	−2.48136
$29$	−1.08612	−0.201688	−0.100844	−	0.994902i	$-0.532154\pi$
$29$	−1.08612	−0.201688	−0.100844	+	0.994902i	$0.532154\pi$
$30$	0	0
$31$	−4.17296	−0.749487	−0.374743	−	0.927129i	$-0.622269\pi$
$31$	−4.17296	−0.749487	−0.374743	+	0.927129i	$0.622269\pi$
$32$	6.83020	1.20742
$33$	0	0
$34$	−3.59980	−0.617361
$35$	4.42518	0.747992
$36$	0	0
$37$	−5.75406	−0.945962	−0.472981	−	0.881073i	$-0.656822\pi$
$37$	−5.75406	−0.945962	−0.472981	+	0.881073i	$0.656822\pi$
$38$	−2.22871	−0.361544
$39$	0	0
$40$	2.15546	0.340809
$41$	−4.26750	−0.666472	−0.333236	−	0.942843i	$-0.608141\pi$
$41$	−4.26750	−0.666472	−0.333236	+	0.942843i	$0.608141\pi$
$42$	0	0
$43$	3.62271	0.552459	0.276229	−	0.961092i	$-0.410915\pi$
$43$	3.62271	0.552459	0.276229	+	0.961092i	$0.410915\pi$
$44$	2.96714	0.447313
$45$	0	0
$46$	0.252396	0.0372138
$47$	6.39567	0.932904	0.466452	−	0.884547i	$-0.345532\pi$
$47$	6.39567	0.932904	0.466452	+	0.884547i	$0.345532\pi$
$48$	0	0
$49$	12.5822	1.79746
$50$	−2.22871	−0.315187
$51$	0	0
$52$	−11.6494	−1.61548
$53$	−12.0154	−1.65044	−0.825219	−	0.564812i	$-0.808949\pi$
$53$	−12.0154	−1.65044	−0.825219	+	0.564812i	$0.808949\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	9.53832	1.27461
$57$	0	0
$58$	2.42065	0.317847
$59$	−0.883911	−0.115075	−0.0575377	−	0.998343i	$-0.518325\pi$
$59$	−0.883911	−0.115075	−0.0575377	+	0.998343i	$0.518325\pi$
$60$	0	0
$61$	3.77712	0.483611	0.241805	−	0.970325i	$-0.422260\pi$
$61$	3.77712	0.483611	0.241805	+	0.970325i	$0.422260\pi$
$62$	9.30032	1.18114
$63$	0	0
$64$	−12.9618	−1.62022
$65$	3.92613	0.486976
$66$	0	0
$67$	−12.2192	−1.49282	−0.746409	−	0.665487i	$-0.768224\pi$
$67$	−12.2192	−1.49282	−0.746409	+	0.665487i	$0.768224\pi$
$68$	4.79251	0.581178
$69$	0	0
$70$	−9.86243	−1.17879
$71$	−4.28400	−0.508417	−0.254209	−	0.967149i	$-0.581815\pi$
$71$	−4.28400	−0.508417	−0.254209	+	0.967149i	$0.581815\pi$
$72$	0	0
$73$	−1.92676	−0.225510	−0.112755	−	0.993623i	$-0.535968\pi$
$73$	−1.92676	−0.225510	−0.112755	+	0.993623i	$0.535968\pi$
$74$	12.8241	1.49077
$75$	0	0
$76$	2.96714	0.340354
$77$	−4.42518	−0.504296
$78$	0	0
$79$	−7.81004	−0.878698	−0.439349	−	0.898316i	$-0.644791\pi$
$79$	−7.81004	−0.878698	−0.439349	+	0.898316i	$0.644791\pi$
$80$	1.13037	0.126380
$81$	0	0
$82$	9.51102	1.05032
$83$	3.33410	0.365964	0.182982	−	0.983116i	$-0.441425\pi$
$83$	3.33410	0.365964	0.182982	+	0.983116i	$0.441425\pi$
$84$	0	0
$85$	−1.61520	−0.175193
$86$	−8.07397	−0.870639
$87$	0	0
$88$	−2.15546	−0.229773
$89$	11.9145	1.26293	0.631466	−	0.775403i	$-0.282453\pi$
$89$	11.9145	1.26293	0.631466	+	0.775403i	$0.282453\pi$
$90$	0	0
$91$	17.3738	1.82127
$92$	−0.336021	−0.0350327
$93$	0	0
$94$	−14.2541	−1.47019
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−13.3478	−1.35526	−0.677631	−	0.735402i	$-0.736993\pi$
$97$	−13.3478	−1.35526	−0.677631	+	0.735402i	$0.736993\pi$
$98$	−28.0421	−2.83268
$99$	0	0
$100$	2.96714	0.296714
$101$	−3.72067	−0.370221	−0.185110	−	0.982718i	$-0.559264\pi$
$101$	−3.72067	−0.370221	−0.185110	+	0.982718i	$0.559264\pi$
$102$	0	0
$103$	−18.7975	−1.85218	−0.926088	−	0.377308i	$-0.876850\pi$
$103$	−18.7975	−1.85218	−0.926088	+	0.377308i	$0.876850\pi$
$104$	8.46263	0.829829
$105$	0	0
$106$	26.7788	2.60098
$107$	5.33910	0.516150	0.258075	−	0.966125i	$-0.416912\pi$
$107$	5.33910	0.516150	0.258075	+	0.966125i	$0.416912\pi$
$108$	0	0
$109$	16.7781	1.60706	0.803528	−	0.595268i	$-0.202954\pi$
$109$	16.7781	1.60706	0.803528	+	0.595268i	$0.202954\pi$
$110$	2.22871	0.212499
$111$	0	0
$112$	5.00211	0.472655
$113$	−9.64670	−0.907485	−0.453742	−	0.891133i	$-0.649911\pi$
$113$	−9.64670	−0.907485	−0.453742	+	0.891133i	$0.649911\pi$
$114$	0	0
$115$	0.113248	0.0105604
$116$	−3.22268	−0.299218
$117$	0	0
$118$	1.96998	0.181351
$119$	−7.14754	−0.655214
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.41809	−0.762138
$123$	0	0
$124$	−12.3818	−1.11191
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.515860	0.0457752	0.0228876	−	0.999738i	$-0.492714\pi$
$127$	0.515860	0.0457752	0.0228876	+	0.999738i	$0.492714\pi$
$128$	15.2276	1.34594
$129$	0	0
$130$	−8.75019	−0.767442
$131$	2.12758	0.185888	0.0929439	−	0.995671i	$-0.470372\pi$
$131$	2.12758	0.185888	0.0929439	+	0.995671i	$0.470372\pi$
$132$	0	0
$133$	−4.42518	−0.383712
$134$	27.2331	2.35258
$135$	0	0
$136$	−3.48150	−0.298536
$137$	−3.76917	−0.322022	−0.161011	−	0.986953i	$-0.551475\pi$
$137$	−3.76917	−0.322022	−0.161011	+	0.986953i	$0.551475\pi$
$138$	0	0
$139$	−11.2237	−0.951981	−0.475990	−	0.879451i	$-0.657910\pi$
$139$	−11.2237	−0.951981	−0.475990	+	0.879451i	$0.657910\pi$
$140$	13.1301	1.10970
$141$	0	0
$142$	9.54778	0.801232
$143$	−3.92613	−0.328319
$144$	0	0
$145$	1.08612	0.0901976
$146$	4.29418	0.355389
$147$	0	0
$148$	−17.0731	−1.40340
$149$	−21.0620	−1.72546	−0.862732	−	0.505662i	$-0.831248\pi$
$149$	−21.0620	−1.72546	−0.862732	+	0.505662i	$0.831248\pi$
$150$	0	0
$151$	7.46759	0.607704	0.303852	−	0.952719i	$-0.401727\pi$
$151$	7.46759	0.607704	0.303852	+	0.952719i	$0.401727\pi$
$152$	−2.15546	−0.174831
$153$	0	0
$154$	9.86243	0.794738
$155$	4.17296	0.335181
$156$	0	0
$157$	−5.84405	−0.466406	−0.233203	−	0.972428i	$-0.574921\pi$
$157$	−5.84405	−0.466406	−0.233203	+	0.972428i	$0.574921\pi$
$158$	17.4063	1.38477
$159$	0	0
$160$	−6.83020	−0.539975
$161$	0.501142	0.0394955
$162$	0	0
$163$	−20.3077	−1.59062	−0.795311	−	0.606202i	$-0.792692\pi$
$163$	−20.3077	−1.59062	−0.795311	+	0.606202i	$0.792692\pi$
$164$	−12.6623	−0.988757
$165$	0	0
$166$	−7.43072	−0.576736
$167$	−10.6177	−0.821625	−0.410812	−	0.911720i	$-0.634755\pi$
$167$	−10.6177	−0.821625	−0.410812	+	0.911720i	$0.634755\pi$
$168$	0	0
$169$	2.41448	0.185729
$170$	3.59980	0.276092
$171$	0	0
$172$	10.7491	0.819610
$173$	−24.7343	−1.88051	−0.940257	−	0.340466i	$-0.889415\pi$
$173$	−24.7343	−1.88051	−0.940257	+	0.340466i	$0.889415\pi$
$174$	0	0
$175$	−4.42518	−0.334512
$176$	−1.13037	−0.0852051
$177$	0	0
$178$	−26.5539	−1.99030
$179$	14.9732	1.11915	0.559574	−	0.828781i	$-0.310965\pi$
$179$	14.9732	1.11915	0.559574	+	0.828781i	$0.310965\pi$
$180$	0	0
$181$	−11.2782	−0.838298	−0.419149	−	0.907917i	$-0.637672\pi$
$181$	−11.2782	−0.838298	−0.419149	+	0.907917i	$0.637672\pi$
$182$	−38.7212	−2.87020
$183$	0	0
$184$	0.244101	0.0179954
$185$	5.75406	0.423047
$186$	0	0
$187$	1.61520	0.118115
$188$	18.9768	1.38403
$189$	0	0
$190$	2.22871	0.161687
$191$	−22.3038	−1.61385	−0.806923	−	0.590656i	$-0.798869\pi$
$191$	−22.3038	−1.61385	−0.806923	+	0.590656i	$0.798869\pi$
$192$	0	0
$193$	25.1845	1.81282	0.906412	−	0.422395i	$-0.138811\pi$
$193$	25.1845	1.81282	0.906412	+	0.422395i	$0.138811\pi$
$194$	29.7483	2.13580
$195$	0	0
$196$	37.3332	2.66666
$197$	10.4483	0.744413	0.372207	−	0.928150i	$-0.378601\pi$
$197$	10.4483	0.744413	0.372207	+	0.928150i	$0.378601\pi$
$198$	0	0
$199$	−6.40716	−0.454191	−0.227096	−	0.973872i	$-0.572923\pi$
$199$	−6.40716	−0.454191	−0.227096	+	0.973872i	$0.572923\pi$
$200$	−2.15546	−0.152414
$201$	0	0
$202$	8.29229	0.583444
$203$	4.80629	0.337336
$204$	0	0
$205$	4.26750	0.298055
$206$	41.8942	2.91891
$207$	0	0
$208$	4.43799	0.307719
$209$	1.00000	0.0691714
$210$	0	0
$211$	7.96892	0.548603	0.274302	−	0.961644i	$-0.411553\pi$
$211$	7.96892	0.548603	0.274302	+	0.961644i	$0.411553\pi$
$212$	−35.6513	−2.44854
$213$	0	0
$214$	−11.8993	−0.813418
$215$	−3.62271	−0.247067
$216$	0	0
$217$	18.4661	1.25356
$218$	−37.3936	−2.53261
$219$	0	0
$220$	−2.96714	−0.200044
$221$	−6.34147	−0.426574
$222$	0	0
$223$	−23.9096	−1.60111	−0.800553	−	0.599262i	$-0.795461\pi$
$223$	−23.9096	−1.60111	−0.800553	+	0.599262i	$0.795461\pi$
$224$	−30.2249	−2.01948
$225$	0	0
$226$	21.4997	1.43014
$227$	10.0128	0.664575	0.332287	−	0.943178i	$-0.392180\pi$
$227$	10.0128	0.664575	0.332287	+	0.943178i	$0.392180\pi$
$228$	0	0
$229$	10.4511	0.690629	0.345314	−	0.938487i	$-0.387772\pi$
$229$	10.4511	0.690629	0.345314	+	0.938487i	$0.387772\pi$
$230$	−0.252396	−0.0166425
$231$	0	0
$232$	2.34110	0.153701
$233$	4.56505	0.299066	0.149533	−	0.988757i	$-0.452223\pi$
$233$	4.56505	0.299066	0.149533	+	0.988757i	$0.452223\pi$
$234$	0	0
$235$	−6.39567	−0.417207
$236$	−2.62268	−0.170722
$237$	0	0
$238$	15.9298	1.03257
$239$	15.3794	0.994808	0.497404	−	0.867519i	$-0.334287\pi$
$239$	15.3794	0.994808	0.497404	+	0.867519i	$0.334287\pi$
$240$	0	0
$241$	17.8737	1.15135	0.575673	−	0.817680i	$-0.304740\pi$
$241$	17.8737	1.15135	0.575673	+	0.817680i	$0.304740\pi$
$242$	−2.22871	−0.143267
$243$	0	0
$244$	11.2072	0.717469
$245$	−12.5822	−0.803849
$246$	0	0
$247$	−3.92613	−0.249814
$248$	8.99468	0.571163
$249$	0	0
$250$	2.22871	0.140956
$251$	10.0425	0.633878	0.316939	−	0.948446i	$-0.397345\pi$
$251$	10.0425	0.633878	0.316939	+	0.948446i	$0.397345\pi$
$252$	0	0
$253$	−0.113248	−0.00711982
$254$	−1.14970	−0.0721387
$255$	0	0
$256$	−8.01431	−0.500895
$257$	10.7636	0.671416	0.335708	−	0.941966i	$-0.391025\pi$
$257$	10.7636	0.671416	0.335708	+	0.941966i	$0.391025\pi$
$258$	0	0
$259$	25.4628	1.58218
$260$	11.6494	0.722463
$261$	0	0
$262$	−4.74176	−0.292947
$263$	−22.9691	−1.41634	−0.708169	−	0.706043i	$-0.750478\pi$
$263$	−22.9691	−1.41634	−0.708169	+	0.706043i	$0.750478\pi$
$264$	0	0
$265$	12.0154	0.738099
$266$	9.86243	0.604705
$267$	0	0
$268$	−36.2562	−2.21470
$269$	20.5868	1.25520	0.627600	−	0.778536i	$-0.284037\pi$
$269$	20.5868	1.25520	0.627600	+	0.778536i	$0.284037\pi$
$270$	0	0
$271$	−4.27052	−0.259416	−0.129708	−	0.991552i	$-0.541404\pi$
$271$	−4.27052	−0.259416	−0.129708	+	0.991552i	$0.541404\pi$
$272$	−1.82578	−0.110704
$273$	0	0
$274$	8.40038	0.507485
$275$	1.00000	0.0603023
$276$	0	0
$277$	−14.6746	−0.881712	−0.440856	−	0.897578i	$-0.645325\pi$
$277$	−14.6746	−0.881712	−0.440856	+	0.897578i	$0.645325\pi$
$278$	25.0143	1.50026
$279$	0	0
$280$	−9.53832	−0.570024
$281$	24.1329	1.43965	0.719824	−	0.694157i	$-0.244222\pi$
$281$	24.1329	1.43965	0.719824	+	0.694157i	$0.244222\pi$
$282$	0	0
$283$	0.197192	0.0117218	0.00586092	−	0.999983i	$-0.498134\pi$
$283$	0.197192	0.0117218	0.00586092	+	0.999983i	$0.498134\pi$
$284$	−12.7112	−0.754272
$285$	0	0
$286$	8.75019	0.517410
$287$	18.8845	1.11472
$288$	0	0
$289$	−14.3911	−0.846537
$290$	−2.42065	−0.142146
$291$	0	0
$292$	−5.71695	−0.334559
$293$	−2.73471	−0.159763	−0.0798817	−	0.996804i	$-0.525454\pi$
$293$	−2.73471	−0.159763	−0.0798817	+	0.996804i	$0.525454\pi$
$294$	0	0
$295$	0.883911	0.0514633
$296$	12.4027	0.720891
$297$	0	0
$298$	46.9410	2.71922
$299$	0.444625	0.0257133
$300$	0	0
$301$	−16.0312	−0.924021
$302$	−16.6431	−0.957702
$303$	0	0
$304$	−1.13037	−0.0648313
$305$	−3.77712	−0.216277
$306$	0	0
$307$	−16.7313	−0.954905	−0.477452	−	0.878658i	$-0.658440\pi$
$307$	−16.7313	−0.954905	−0.477452	+	0.878658i	$0.658440\pi$
$308$	−13.1301	−0.748158
$309$	0	0
$310$	−9.30032	−0.528222
$311$	14.8941	0.844565	0.422283	−	0.906464i	$-0.361229\pi$
$311$	14.8941	0.844565	0.422283	+	0.906464i	$0.361229\pi$
$312$	0	0
$313$	8.89539	0.502797	0.251399	−	0.967884i	$-0.419109\pi$
$313$	8.89539	0.502797	0.251399	+	0.967884i	$0.419109\pi$
$314$	13.0247	0.735026
$315$	0	0
$316$	−23.1735	−1.30361
$317$	−6.16319	−0.346159	−0.173079	−	0.984908i	$-0.555372\pi$
$317$	−6.16319	−0.346159	−0.173079	+	0.984908i	$0.555372\pi$
$318$	0	0
$319$	−1.08612	−0.0608112
$320$	12.9618	0.724585
$321$	0	0
$322$	−1.11690	−0.0622423
$323$	1.61520	0.0898720
$324$	0	0
$325$	−3.92613	−0.217782
$326$	45.2599	2.50671
$327$	0	0
$328$	9.19846	0.507900
$329$	−28.3020	−1.56034
$330$	0	0
$331$	17.5144	0.962681	0.481340	−	0.876534i	$-0.340150\pi$
$331$	17.5144	0.962681	0.481340	+	0.876534i	$0.340150\pi$
$332$	9.89272	0.542933
$333$	0	0
$334$	23.6638	1.29483
$335$	12.2192	0.667609
$336$	0	0
$337$	−34.0733	−1.85609	−0.928045	−	0.372467i	$-0.878512\pi$
$337$	−34.0733	−1.85609	−0.928045	+	0.372467i	$0.878512\pi$
$338$	−5.38117	−0.292697
$339$	0	0
$340$	−4.79251	−0.259911
$341$	−4.17296	−0.225979
$342$	0	0
$343$	−24.7024	−1.33380
$344$	−7.80863	−0.421013
$345$	0	0
$346$	55.1255	2.96357
$347$	−14.8065	−0.794856	−0.397428	−	0.917633i	$-0.630097\pi$
$347$	−14.8065	−0.794856	−0.397428	+	0.917633i	$0.630097\pi$
$348$	0	0
$349$	−20.2032	−1.08145	−0.540727	−	0.841198i	$-0.681851\pi$
$349$	−20.2032	−1.08145	−0.540727	+	0.841198i	$0.681851\pi$
$350$	9.86243	0.527169
$351$	0	0
$352$	6.83020	0.364051
$353$	34.8488	1.85481	0.927407	−	0.374053i	$-0.122032\pi$
$353$	34.8488	1.85481	0.927407	+	0.374053i	$0.122032\pi$
$354$	0	0
$355$	4.28400	0.227371
$356$	35.3519	1.87365
$357$	0	0
$358$	−33.3708	−1.76370
$359$	25.8010	1.36173	0.680863	−	0.732410i	$-0.261605\pi$
$359$	25.8010	1.36173	0.680863	+	0.732410i	$0.261605\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	25.1357	1.32110
$363$	0	0
$364$	51.5505	2.70198
$365$	1.92676	0.100851
$366$	0	0
$367$	12.6344	0.659510	0.329755	−	0.944066i	$-0.393034\pi$
$367$	12.6344	0.659510	0.329755	+	0.944066i	$0.393034\pi$
$368$	0.128012	0.00667310
$369$	0	0
$370$	−12.8241	−0.666694
$371$	53.1702	2.76046
$372$	0	0
$373$	31.3623	1.62388	0.811940	−	0.583741i	$-0.198412\pi$
$373$	31.3623	1.62388	0.811940	+	0.583741i	$0.198412\pi$
$374$	−3.59980	−0.186141
$375$	0	0
$376$	−13.7856	−0.710940
$377$	4.26426	0.219620
$378$	0	0
$379$	−17.2730	−0.887256	−0.443628	−	0.896211i	$-0.646309\pi$
$379$	−17.2730	−0.887256	−0.443628	+	0.896211i	$0.646309\pi$
$380$	−2.96714	−0.152211
$381$	0	0
$382$	49.7086	2.54332
$383$	−1.96340	−0.100325	−0.0501625	−	0.998741i	$-0.515974\pi$
$383$	−1.96340	−0.100325	−0.0501625	+	0.998741i	$0.515974\pi$
$384$	0	0
$385$	4.42518	0.225528
$386$	−56.1290	−2.85689
$387$	0	0
$388$	−39.6047	−2.01062
$389$	−27.2460	−1.38143	−0.690714	−	0.723128i	$-0.742704\pi$
$389$	−27.2460	−1.38143	−0.690714	+	0.723128i	$0.742704\pi$
$390$	0	0
$391$	−0.182917	−0.00925054
$392$	−27.1206	−1.36979
$393$	0	0
$394$	−23.2863	−1.17315
$395$	7.81004	0.392966
$396$	0	0
$397$	24.1194	1.21052	0.605258	−	0.796029i	$-0.293070\pi$
$397$	24.1194	1.21052	0.605258	+	0.796029i	$0.293070\pi$
$398$	14.2797	0.715776
$399$	0	0
$400$	−1.13037	−0.0565187
$401$	18.6773	0.932699	0.466349	−	0.884601i	$-0.345569\pi$
$401$	18.6773	0.932699	0.466349	+	0.884601i	$0.345569\pi$
$402$	0	0
$403$	16.3836	0.816125
$404$	−11.0397	−0.549248
$405$	0	0
$406$	−10.7118	−0.531619
$407$	−5.75406	−0.285218
$408$	0	0
$409$	−19.9442	−0.986176	−0.493088	−	0.869979i	$-0.664132\pi$
$409$	−19.9442	−0.986176	−0.493088	+	0.869979i	$0.664132\pi$
$410$	−9.51102	−0.469716
$411$	0	0
$412$	−55.7748	−2.74783
$413$	3.91146	0.192471
$414$	0	0
$415$	−3.33410	−0.163664
$416$	−26.8162	−1.31477
$417$	0	0
$418$	−2.22871	−0.109010
$419$	−22.0961	−1.07946	−0.539731	−	0.841837i	$-0.681474\pi$
$419$	−22.0961	−1.07946	−0.539731	+	0.841837i	$0.681474\pi$
$420$	0	0
$421$	−25.4045	−1.23814	−0.619070	−	0.785336i	$-0.712490\pi$
$421$	−25.4045	−1.23814	−0.619070	+	0.785336i	$0.712490\pi$
$422$	−17.7604	−0.864563
$423$	0	0
$424$	25.8987	1.25775
$425$	1.61520	0.0783486
$426$	0	0
$427$	−16.7144	−0.808868
$428$	15.8418	0.765744
$429$	0	0
$430$	8.07397	0.389361
$431$	3.51614	0.169367	0.0846833	−	0.996408i	$-0.473012\pi$
$431$	3.51614	0.169367	0.0846833	+	0.996408i	$0.473012\pi$
$432$	0	0
$433$	−9.71737	−0.466987	−0.233493	−	0.972358i	$-0.575016\pi$
$433$	−9.71737	−0.466987	−0.233493	+	0.972358i	$0.575016\pi$
$434$	−41.1556	−1.97553
$435$	0	0
$436$	49.7831	2.38418
$437$	−0.113248	−0.00541737
$438$	0	0
$439$	0.888955	0.0424275	0.0212138	−	0.999775i	$-0.493247\pi$
$439$	0.888955	0.0424275	0.0212138	+	0.999775i	$0.493247\pi$
$440$	2.15546	0.102758
$441$	0	0
$442$	14.1333	0.672252
$443$	−5.44113	−0.258516	−0.129258	−	0.991611i	$-0.541260\pi$
$443$	−5.44113	−0.258516	−0.129258	+	0.991611i	$0.541260\pi$
$444$	0	0
$445$	−11.9145	−0.564801
$446$	53.2875	2.52324
$447$	0	0
$448$	57.3582	2.70992
$449$	−21.6076	−1.01972	−0.509862	−	0.860256i	$-0.670303\pi$
$449$	−21.6076	−1.01972	−0.509862	+	0.860256i	$0.670303\pi$
$450$	0	0
$451$	−4.26750	−0.200949
$452$	−28.6231	−1.34632
$453$	0	0
$454$	−22.3157	−1.04733
$455$	−17.3738	−0.814498
$456$	0	0
$457$	−29.1996	−1.36590	−0.682950	−	0.730465i	$-0.739303\pi$
$457$	−29.1996	−1.36590	−0.682950	+	0.730465i	$0.739303\pi$
$458$	−23.2925	−1.08839
$459$	0	0
$460$	0.336021	0.0156671
$461$	32.2335	1.50126	0.750631	−	0.660722i	$-0.229750\pi$
$461$	32.2335	1.50126	0.750631	+	0.660722i	$0.229750\pi$
$462$	0	0
$463$	−11.8543	−0.550917	−0.275459	−	0.961313i	$-0.588830\pi$
$463$	−11.8543	−0.550917	−0.275459	+	0.961313i	$0.588830\pi$
$464$	1.22772	0.0569957
$465$	0	0
$466$	−10.1742	−0.471309
$467$	31.4900	1.45718	0.728592	−	0.684948i	$-0.240175\pi$
$467$	31.4900	1.45718	0.728592	+	0.684948i	$0.240175\pi$
$468$	0	0
$469$	54.0724	2.49683
$470$	14.2541	0.657491
$471$	0	0
$472$	1.90524	0.0876957
$473$	3.62271	0.166573
$474$	0	0
$475$	1.00000	0.0458831
$476$	−21.2077	−0.972055
$477$	0	0
$478$	−34.2761	−1.56775
$479$	−15.1565	−0.692519	−0.346259	−	0.938139i	$-0.612548\pi$
$479$	−15.1565	−0.692519	−0.346259	+	0.938139i	$0.612548\pi$
$480$	0	0
$481$	22.5912	1.03007
$482$	−39.8353	−1.81445
$483$	0	0
$484$	2.96714	0.134870
$485$	13.3478	0.606092
$486$	0	0
$487$	28.0154	1.26950	0.634749	−	0.772718i	$-0.281103\pi$
$487$	28.0154	1.26950	0.634749	+	0.772718i	$0.281103\pi$
$488$	−8.14145	−0.368546
$489$	0	0
$490$	28.0421	1.26681
$491$	1.56411	0.0705873	0.0352937	−	0.999377i	$-0.488763\pi$
$491$	1.56411	0.0705873	0.0352937	+	0.999377i	$0.488763\pi$
$492$	0	0
$493$	−1.75430	−0.0790099
$494$	8.75019	0.393690
$495$	0	0
$496$	4.71701	0.211800
$497$	18.9575	0.850359
$498$	0	0
$499$	15.4416	0.691263	0.345631	−	0.938370i	$-0.387665\pi$
$499$	15.4416	0.691263	0.345631	+	0.938370i	$0.387665\pi$
$500$	−2.96714	−0.132694
$501$	0	0
$502$	−22.3818	−0.998950
$503$	23.3185	1.03972	0.519861	−	0.854251i	$-0.325984\pi$
$503$	23.3185	1.03972	0.519861	+	0.854251i	$0.325984\pi$
$504$	0	0
$505$	3.72067	0.165568
$506$	0.252396	0.0112204
$507$	0	0
$508$	1.53063	0.0679106
$509$	−32.7538	−1.45178	−0.725892	−	0.687808i	$-0.758573\pi$
$509$	−32.7538	−1.45178	−0.725892	+	0.687808i	$0.758573\pi$
$510$	0	0
$511$	8.52625	0.377179
$512$	−12.5936	−0.556565
$513$	0	0
$514$	−23.9889	−1.05811
$515$	18.7975	0.828318
$516$	0	0
$517$	6.39567	0.281281
$518$	−56.7490	−2.49341
$519$	0	0
$520$	−8.46263	−0.371111
$521$	−38.5714	−1.68984	−0.844922	−	0.534889i	$-0.820353\pi$
$521$	−38.5714	−1.68984	−0.844922	+	0.534889i	$0.820353\pi$
$522$	0	0
$523$	34.9926	1.53012	0.765060	−	0.643959i	$-0.222709\pi$
$523$	34.9926	1.53012	0.765060	+	0.643959i	$0.222709\pi$
$524$	6.31283	0.275777
$525$	0	0
$526$	51.1915	2.23205
$527$	−6.74016	−0.293606
$528$	0	0
$529$	−22.9872	−0.999442
$530$	−26.7788	−1.16319
$531$	0	0
$532$	−13.1301	−0.569263
$533$	16.7548	0.725730
$534$	0	0
$535$	−5.33910	−0.230829
$536$	26.3382	1.13764
$537$	0	0
$538$	−45.8820	−1.97811
$539$	12.5822	0.541955
$540$	0	0
$541$	35.1455	1.51102	0.755511	−	0.655136i	$-0.227389\pi$
$541$	35.1455	1.51102	0.755511	+	0.655136i	$0.227389\pi$
$542$	9.51774	0.408822
$543$	0	0
$544$	11.0321	0.472999
$545$	−16.7781	−0.718697
$546$	0	0
$547$	2.79933	0.119691	0.0598454	−	0.998208i	$-0.480939\pi$
$547$	2.79933	0.119691	0.0598454	+	0.998208i	$0.480939\pi$
$548$	−11.1836	−0.477741
$549$	0	0
$550$	−2.22871	−0.0950324
$551$	−1.08612	−0.0462704
$552$	0	0
$553$	34.5608	1.46968
$554$	32.7054	1.38952
$555$	0	0
$556$	−33.3022	−1.41233
$557$	27.8118	1.17843	0.589213	−	0.807978i	$-0.299438\pi$
$557$	27.8118	1.17843	0.589213	+	0.807978i	$0.299438\pi$
$558$	0	0
$559$	−14.2232	−0.601579
$560$	−5.00211	−0.211378
$561$	0	0
$562$	−53.7852	−2.26879
$563$	−26.4265	−1.11374	−0.556872	−	0.830598i	$-0.687999\pi$
$563$	−26.4265	−1.11374	−0.556872	+	0.830598i	$0.687999\pi$
$564$	0	0
$565$	9.64670	0.405840
$566$	−0.439483	−0.0184728
$567$	0	0
$568$	9.23401	0.387451
$569$	−15.1625	−0.635643	−0.317822	−	0.948151i	$-0.602951\pi$
$569$	−15.1625	−0.635643	−0.317822	+	0.948151i	$0.602951\pi$
$570$	0	0
$571$	−11.4144	−0.477677	−0.238838	−	0.971059i	$-0.576767\pi$
$571$	−11.4144	−0.477677	−0.238838	+	0.971059i	$0.576767\pi$
$572$	−11.6494	−0.487084
$573$	0	0
$574$	−42.0880	−1.75672
$575$	−0.113248	−0.00472276
$576$	0	0
$577$	−9.33855	−0.388769	−0.194385	−	0.980925i	$-0.562271\pi$
$577$	−9.33855	−0.388769	−0.194385	+	0.980925i	$0.562271\pi$
$578$	32.0736	1.33409
$579$	0	0
$580$	3.22268	0.133814
$581$	−14.7540	−0.612098
$582$	0	0
$583$	−12.0154	−0.497626
$584$	4.15306	0.171855
$585$	0	0
$586$	6.09486	0.251776
$587$	−9.38542	−0.387378	−0.193689	−	0.981063i	$-0.562045\pi$
$587$	−9.38542	−0.387378	−0.193689	+	0.981063i	$0.562045\pi$
$588$	0	0
$589$	−4.17296	−0.171944
$590$	−1.96998	−0.0811027
$591$	0	0
$592$	6.50424	0.267322
$593$	35.6523	1.46406	0.732032	−	0.681270i	$-0.238572\pi$
$593$	35.6523	1.46406	0.732032	+	0.681270i	$0.238572\pi$
$594$	0	0
$595$	7.14754	0.293021
$596$	−62.4937	−2.55984
$597$	0	0
$598$	−0.990939	−0.0405225
$599$	−37.1404	−1.51751	−0.758757	−	0.651373i	$-0.774193\pi$
$599$	−37.1404	−1.51751	−0.758757	+	0.651373i	$0.774193\pi$
$600$	0	0
$601$	12.2448	0.499474	0.249737	−	0.968314i	$-0.419656\pi$
$601$	12.2448	0.499474	0.249737	+	0.968314i	$0.419656\pi$
$602$	35.7288	1.45620
$603$	0	0
$604$	22.1574	0.901571
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	31.3652	1.27307	0.636536	−	0.771247i	$-0.280366\pi$
$607$	31.3652	1.27307	0.636536	+	0.771247i	$0.280366\pi$
$608$	6.83020	0.277001
$609$	0	0
$610$	8.41809	0.340839
$611$	−25.1102	−1.01585
$612$	0	0
$613$	−5.48484	−0.221531	−0.110765	−	0.993847i	$-0.535330\pi$
$613$	−5.48484	−0.221531	−0.110765	+	0.993847i	$0.535330\pi$
$614$	37.2891	1.50487
$615$	0	0
$616$	9.53832	0.384310
$617$	32.6913	1.31610	0.658052	−	0.752973i	$-0.271381\pi$
$617$	32.6913	1.31610	0.658052	+	0.752973i	$0.271381\pi$
$618$	0	0
$619$	1.71800	0.0690522	0.0345261	−	0.999404i	$-0.489008\pi$
$619$	1.71800	0.0690522	0.0345261	+	0.999404i	$0.489008\pi$
$620$	12.3818	0.497263
$621$	0	0
$622$	−33.1945	−1.33098
$623$	−52.7238	−2.11233
$624$	0	0
$625$	1.00000	0.0400000
$626$	−19.8252	−0.792375
$627$	0	0
$628$	−17.3401	−0.691946
$629$	−9.29395	−0.370574
$630$	0	0
$631$	39.9355	1.58981	0.794903	−	0.606737i	$-0.207522\pi$
$631$	39.9355	1.58981	0.794903	+	0.606737i	$0.207522\pi$
$632$	16.8343	0.669631
$633$	0	0
$634$	13.7359	0.545524
$635$	−0.515860	−0.0204713
$636$	0	0
$637$	−49.3994	−1.95728
$638$	2.42065	0.0958345
$639$	0	0
$640$	−15.2276	−0.601924
$641$	3.50123	0.138290	0.0691452	−	0.997607i	$-0.477973\pi$
$641$	3.50123	0.138290	0.0691452	+	0.997607i	$0.477973\pi$
$642$	0	0
$643$	6.59740	0.260176	0.130088	−	0.991502i	$-0.458474\pi$
$643$	6.59740	0.260176	0.130088	+	0.991502i	$0.458474\pi$
$644$	1.48696	0.0585943
$645$	0	0
$646$	−3.59980	−0.141632
$647$	24.8389	0.976518	0.488259	−	0.872699i	$-0.337632\pi$
$647$	24.8389	0.976518	0.488259	+	0.872699i	$0.337632\pi$
$648$	0	0
$649$	−0.883911	−0.0346965
$650$	8.75019	0.343211
$651$	0	0
$652$	−60.2557	−2.35980
$653$	17.5186	0.685555	0.342778	−	0.939417i	$-0.388632\pi$
$653$	17.5186	0.685555	0.342778	+	0.939417i	$0.388632\pi$
$654$	0	0
$655$	−2.12758	−0.0831315
$656$	4.82387	0.188341
$657$	0	0
$658$	63.0768	2.45899
$659$	−3.56748	−0.138969	−0.0694846	−	0.997583i	$-0.522135\pi$
$659$	−3.56748	−0.138969	−0.0694846	+	0.997583i	$0.522135\pi$
$660$	0	0
$661$	−48.5242	−1.88737	−0.943687	−	0.330841i	$-0.892668\pi$
$661$	−48.5242	−1.88737	−0.943687	+	0.330841i	$0.892668\pi$
$662$	−39.0346	−1.51712
$663$	0	0
$664$	−7.18652	−0.278891
$665$	4.42518	0.171601
$666$	0	0
$667$	0.123001	0.00476262
$668$	−31.5043	−1.21894
$669$	0	0
$670$	−27.2331	−1.05211
$671$	3.77712	0.145814
$672$	0	0
$673$	−0.476829	−0.0183804	−0.00919020	−	0.999958i	$-0.502925\pi$
$673$	−0.476829	−0.0183804	−0.00919020	+	0.999958i	$0.502925\pi$
$674$	75.9394	2.92508
$675$	0	0
$676$	7.16409	0.275542
$677$	−3.68052	−0.141454	−0.0707269	−	0.997496i	$-0.522532\pi$
$677$	−3.68052	−0.141454	−0.0707269	+	0.997496i	$0.522532\pi$
$678$	0	0
$679$	59.0664	2.26676
$680$	3.48150	0.133510
$681$	0	0
$682$	9.30032	0.356128
$683$	−0.701724	−0.0268507	−0.0134254	−	0.999910i	$-0.504274\pi$
$683$	−0.701724	−0.0268507	−0.0134254	+	0.999910i	$0.504274\pi$
$684$	0	0
$685$	3.76917	0.144013
$686$	55.0544	2.10199
$687$	0	0
$688$	−4.09502	−0.156121
$689$	47.1739	1.79718
$690$	0	0
$691$	3.65425	0.139014	0.0695071	−	0.997581i	$-0.477857\pi$
$691$	3.65425	0.139014	0.0695071	+	0.997581i	$0.477857\pi$
$692$	−73.3900	−2.78987
$693$	0	0
$694$	32.9994	1.25264
$695$	11.2237	0.425739
$696$	0	0
$697$	−6.89287	−0.261086
$698$	45.0271	1.70430
$699$	0	0
$700$	−13.1301	−0.496272
$701$	45.3852	1.71417	0.857087	−	0.515171i	$-0.172272\pi$
$701$	45.3852	1.71417	0.857087	+	0.515171i	$0.172272\pi$
$702$	0	0
$703$	−5.75406	−0.217019
$704$	−12.9618	−0.488515
$705$	0	0
$706$	−77.6678	−2.92307
$707$	16.4647	0.619217
$708$	0	0
$709$	16.6965	0.627051	0.313525	−	0.949580i	$-0.398490\pi$
$709$	16.6965	0.627051	0.313525	+	0.949580i	$0.398490\pi$
$710$	−9.54778	−0.358322
$711$	0	0
$712$	−25.6813	−0.962446
$713$	0.472579	0.0176982
$714$	0	0
$715$	3.92613	0.146829
$716$	44.4274	1.66033
$717$	0	0
$718$	−57.5030	−2.14599
$719$	−16.1346	−0.601718	−0.300859	−	0.953669i	$-0.597273\pi$
$719$	−16.1346	−0.601718	−0.300859	+	0.953669i	$0.597273\pi$
$720$	0	0
$721$	83.1825	3.09788
$722$	−2.22871	−0.0829439
$723$	0	0
$724$	−33.4638	−1.24367
$725$	−1.08612	−0.0403376
$726$	0	0
$727$	−21.9546	−0.814251	−0.407126	−	0.913372i	$-0.633469\pi$
$727$	−21.9546	−0.814251	−0.407126	+	0.913372i	$0.633469\pi$
$728$	−37.4487	−1.38794
$729$	0	0
$730$	−4.29418	−0.158935
$731$	5.85140	0.216422
$732$	0	0
$733$	−47.5004	−1.75447	−0.877234	−	0.480063i	$-0.840614\pi$
$733$	−47.5004	−1.75447	−0.877234	+	0.480063i	$0.840614\pi$
$734$	−28.1584	−1.03935
$735$	0	0
$736$	−0.773505	−0.0285118
$737$	−12.2192	−0.450102
$738$	0	0
$739$	29.9890	1.10316	0.551581	−	0.834122i	$-0.314025\pi$
$739$	29.9890	1.10316	0.551581	+	0.834122i	$0.314025\pi$
$740$	17.0731	0.627619
$741$	0	0
$742$	−118.501	−4.35030
$743$	−2.24383	−0.0823181	−0.0411590	−	0.999153i	$-0.513105\pi$
$743$	−2.24383	−0.0823181	−0.0411590	+	0.999153i	$0.513105\pi$
$744$	0	0
$745$	21.0620	0.771651
$746$	−69.8974	−2.55913
$747$	0	0
$748$	4.79251	0.175232
$749$	−23.6265	−0.863293
$750$	0	0
$751$	29.0275	1.05923	0.529614	−	0.848239i	$-0.322337\pi$
$751$	29.0275	1.05923	0.529614	+	0.848239i	$0.322337\pi$
$752$	−7.22949	−0.263632
$753$	0	0
$754$	−9.50378	−0.346107
$755$	−7.46759	−0.271774
$756$	0	0
$757$	47.0009	1.70828	0.854138	−	0.520046i	$-0.174085\pi$
$757$	47.0009	1.70828	0.854138	+	0.520046i	$0.174085\pi$
$758$	38.4965	1.39826
$759$	0	0
$760$	2.15546	0.0781869
$761$	−46.6332	−1.69045	−0.845226	−	0.534409i	$-0.820534\pi$
$761$	−46.6332	−1.69045	−0.845226	+	0.534409i	$0.820534\pi$
$762$	0	0
$763$	−74.2463	−2.68790
$764$	−66.1784	−2.39425
$765$	0	0
$766$	4.37584	0.158106
$767$	3.47035	0.125307
$768$	0	0
$769$	−22.6169	−0.815585	−0.407793	−	0.913075i	$-0.633701\pi$
$769$	−22.6169	−0.815585	−0.407793	+	0.913075i	$0.633701\pi$
$770$	−9.86243	−0.355417
$771$	0	0
$772$	74.7260	2.68945
$773$	−16.1951	−0.582499	−0.291249	−	0.956647i	$-0.594071\pi$
$773$	−16.1951	−0.582499	−0.291249	+	0.956647i	$0.594071\pi$
$774$	0	0
$775$	−4.17296	−0.149897
$776$	28.7707	1.03281
$777$	0	0
$778$	60.7234	2.17704
$779$	−4.26750	−0.152899
$780$	0	0
$781$	−4.28400	−0.153294
$782$	0.407670	0.0145782
$783$	0	0
$784$	−14.2226	−0.507950
$785$	5.84405	0.208583
$786$	0	0
$787$	−27.6814	−0.986736	−0.493368	−	0.869821i	$-0.664234\pi$
$787$	−27.6814	−0.986736	−0.493368	+	0.869821i	$0.664234\pi$
$788$	31.0016	1.10439
$789$	0	0
$790$	−17.4063	−0.619288
$791$	42.6884	1.51782
$792$	0	0
$793$	−14.8295	−0.526609
$794$	−53.7550	−1.90769
$795$	0	0
$796$	−19.0109	−0.673824
$797$	4.49477	0.159213	0.0796065	−	0.996826i	$-0.474634\pi$
$797$	4.49477	0.159213	0.0796065	+	0.996826i	$0.474634\pi$
$798$	0	0
$799$	10.3303	0.365459
$800$	6.83020	0.241484
$801$	0	0
$802$	−41.6262	−1.46987
$803$	−1.92676	−0.0679938
$804$	0	0
$805$	−0.501142	−0.0176629
$806$	−36.5142	−1.28616
$807$	0	0
$808$	8.01978	0.282135
$809$	−48.0468	−1.68924	−0.844618	−	0.535369i	$-0.820172\pi$
$809$	−48.0468	−1.68924	−0.844618	+	0.535369i	$0.820172\pi$
$810$	0	0
$811$	20.9156	0.734448	0.367224	−	0.930133i	$-0.380308\pi$
$811$	20.9156	0.734448	0.367224	+	0.930133i	$0.380308\pi$
$812$	14.2609	0.500460
$813$	0	0
$814$	12.8241	0.449485
$815$	20.3077	0.711347
$816$	0	0
$817$	3.62271	0.126743
$818$	44.4497	1.55415
$819$	0	0
$820$	12.6623	0.442186
$821$	−10.7228	−0.374227	−0.187114	−	0.982338i	$-0.559913\pi$
$821$	−10.7228	−0.374227	−0.187114	+	0.982338i	$0.559913\pi$
$822$	0	0
$823$	42.5971	1.48484	0.742420	−	0.669935i	$-0.233678\pi$
$823$	42.5971	1.48484	0.742420	+	0.669935i	$0.233678\pi$
$824$	40.5174	1.41149
$825$	0	0
$826$	−8.71751	−0.303321
$827$	15.3377	0.533343	0.266672	−	0.963787i	$-0.414076\pi$
$827$	15.3377	0.533343	0.266672	+	0.963787i	$0.414076\pi$
$828$	0	0
$829$	−9.78901	−0.339986	−0.169993	−	0.985445i	$-0.554375\pi$
$829$	−9.78901	−0.339986	−0.169993	+	0.985445i	$0.554375\pi$
$830$	7.43072	0.257924
$831$	0	0
$832$	50.8896	1.76428
$833$	20.3228	0.704143
$834$	0	0
$835$	10.6177	0.367442
$836$	2.96714	0.102621
$837$	0	0
$838$	49.2456	1.70116
$839$	48.1417	1.66204	0.831019	−	0.556244i	$-0.187758\pi$
$839$	48.1417	1.66204	0.831019	+	0.556244i	$0.187758\pi$
$840$	0	0
$841$	−27.8203	−0.959322
$842$	56.6192	1.95123
$843$	0	0
$844$	23.6449	0.813891
$845$	−2.41448	−0.0830606
$846$	0	0
$847$	−4.42518	−0.152051
$848$	13.5819	0.466403
$849$	0	0
$850$	−3.59980	−0.123472
$851$	0.651634	0.0223377
$852$	0	0
$853$	−41.0884	−1.40684	−0.703420	−	0.710775i	$-0.748345\pi$
$853$	−41.0884	−1.40684	−0.703420	+	0.710775i	$0.748345\pi$
$854$	37.2516	1.27472
$855$	0	0
$856$	−11.5082	−0.393343
$857$	30.0628	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$857$	30.0628	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$858$	0	0
$859$	9.99982	0.341190	0.170595	−	0.985341i	$-0.445431\pi$
$859$	9.99982	0.341190	0.170595	+	0.985341i	$0.445431\pi$
$860$	−10.7491	−0.366541
$861$	0	0
$862$	−7.83646	−0.266911
$863$	−18.3253	−0.623800	−0.311900	−	0.950115i	$-0.600965\pi$
$863$	−18.3253	−0.623800	−0.311900	+	0.950115i	$0.600965\pi$
$864$	0	0
$865$	24.7343	0.840991
$866$	21.6572	0.735940
$867$	0	0
$868$	54.7915	1.85975
$869$	−7.81004	−0.264937
$870$	0	0
$871$	47.9743	1.62555
$872$	−36.1647	−1.22469
$873$	0	0
$874$	0.252396	0.00853742
$875$	4.42518	0.149598
$876$	0	0
$877$	54.2556	1.83208	0.916040	−	0.401086i	$-0.131367\pi$
$877$	54.2556	1.83208	0.916040	+	0.401086i	$0.131367\pi$
$878$	−1.98122	−0.0668629
$879$	0	0
$880$	1.13037	0.0381049
$881$	20.9741	0.706636	0.353318	−	0.935503i	$-0.385053\pi$
$881$	20.9741	0.706636	0.353318	+	0.935503i	$0.385053\pi$
$882$	0	0
$883$	−16.2916	−0.548256	−0.274128	−	0.961693i	$-0.588389\pi$
$883$	−16.2916	−0.548256	−0.274128	+	0.961693i	$0.588389\pi$
$884$	−18.8160	−0.632851
$885$	0	0
$886$	12.1267	0.407404
$887$	−41.2926	−1.38647	−0.693234	−	0.720712i	$-0.743815\pi$
$887$	−41.2926	−1.38647	−0.693234	+	0.720712i	$0.743815\pi$
$888$	0	0
$889$	−2.28277	−0.0765618
$890$	26.5539	0.890089
$891$	0	0
$892$	−70.9431	−2.37535
$893$	6.39567	0.214023
$894$	0	0
$895$	−14.9732	−0.500498
$896$	−67.3849	−2.25117
$897$	0	0
$898$	48.1569	1.60702
$899$	4.53235	0.151162
$900$	0	0
$901$	−19.4072	−0.646548
$902$	9.51102	0.316682
$903$	0	0
$904$	20.7931	0.691569
$905$	11.2782	0.374898
$906$	0	0
$907$	35.5669	1.18098	0.590490	−	0.807045i	$-0.298935\pi$
$907$	35.5669	1.18098	0.590490	+	0.807045i	$0.298935\pi$
$908$	29.7094	0.985942
$909$	0	0
$910$	38.7212	1.28359
$911$	−26.1812	−0.867422	−0.433711	−	0.901052i	$-0.642796\pi$
$911$	−26.1812	−0.867422	−0.433711	+	0.901052i	$0.642796\pi$
$912$	0	0
$913$	3.33410	0.110342
$914$	65.0774	2.15257
$915$	0	0
$916$	31.0099	1.02459
$917$	−9.41494	−0.310909
$918$	0	0
$919$	0.328420	0.0108336	0.00541678	−	0.999985i	$-0.498276\pi$
$919$	0.328420	0.0108336	0.00541678	+	0.999985i	$0.498276\pi$
$920$	−0.244101	−0.00804779
$921$	0	0
$922$	−71.8390	−2.36589
$923$	16.8195	0.553622
$924$	0	0
$925$	−5.75406	−0.189192
$926$	26.4198	0.868210
$927$	0	0
$928$	−7.41844	−0.243522
$929$	−0.642511	−0.0210801	−0.0105401	−	0.999944i	$-0.503355\pi$
$929$	−0.642511	−0.0210801	−0.0105401	+	0.999944i	$0.503355\pi$
$930$	0	0
$931$	12.5822	0.412366
$932$	13.5451	0.443685
$933$	0	0
$934$	−70.1820	−2.29643
$935$	−1.61520	−0.0528226
$936$	0	0
$937$	19.2863	0.630055	0.315027	−	0.949083i	$-0.397986\pi$
$937$	19.2863	0.630055	0.315027	+	0.949083i	$0.397986\pi$
$938$	−120.512	−3.93484
$939$	0	0
$940$	−18.9768	−0.618955
$941$	−19.4348	−0.633556	−0.316778	−	0.948500i	$-0.602601\pi$
$941$	−19.4348	−0.633556	−0.316778	+	0.948500i	$0.602601\pi$
$942$	0	0
$943$	0.483285	0.0157379
$944$	0.999149	0.0325195
$945$	0	0
$946$	−8.07397	−0.262507
$947$	61.1924	1.98849	0.994243	−	0.107147i	$-0.0341716\pi$
$947$	61.1924	1.98849	0.994243	+	0.107147i	$0.0341716\pi$
$948$	0	0
$949$	7.56470	0.245560
$950$	−2.22871	−0.0723088
$951$	0	0
$952$	15.4063	0.499320
$953$	−3.19828	−0.103602	−0.0518012	−	0.998657i	$-0.516496\pi$
$953$	−3.19828	−0.103602	−0.0518012	+	0.998657i	$0.516496\pi$
$954$	0	0
$955$	22.3038	0.721734
$956$	45.6327	1.47587
$957$	0	0
$958$	33.7794	1.09136
$959$	16.6793	0.538601
$960$	0	0
$961$	−13.5864	−0.438270
$962$	−50.3491	−1.62332
$963$	0	0
$964$	53.0337	1.70810
$965$	−25.1845	−0.810719
$966$	0	0
$967$	45.2536	1.45526	0.727629	−	0.685970i	$-0.240622\pi$
$967$	45.2536	1.45526	0.727629	+	0.685970i	$0.240622\pi$
$968$	−2.15546	−0.0692793
$969$	0	0
$970$	−29.7483	−0.955160
$971$	49.4103	1.58565	0.792826	−	0.609448i	$-0.208609\pi$
$971$	49.4103	1.58565	0.792826	+	0.609448i	$0.208609\pi$
$972$	0	0
$973$	49.6668	1.59225
$974$	−62.4381	−2.00065
$975$	0	0
$976$	−4.26955	−0.136665
$977$	−48.8753	−1.56366	−0.781830	−	0.623492i	$-0.785714\pi$
$977$	−48.8753	−1.56366	−0.781830	+	0.623492i	$0.785714\pi$
$978$	0	0
$979$	11.9145	0.380789
$980$	−37.3332	−1.19257
$981$	0	0
$982$	−3.48594	−0.111241
$983$	−16.7325	−0.533684	−0.266842	−	0.963740i	$-0.585980\pi$
$983$	−16.7325	−0.533684	−0.266842	+	0.963740i	$0.585980\pi$
$984$	0	0
$985$	−10.4483	−0.332912
$986$	3.90983	0.124514
$987$	0	0
$988$	−11.6494	−0.370615
$989$	−0.410264	−0.0130456
$990$	0	0
$991$	−46.6422	−1.48164	−0.740820	−	0.671704i	$-0.765563\pi$
$991$	−46.6422	−1.48164	−0.740820	+	0.671704i	$0.765563\pi$
$992$	−28.5022	−0.904945
$993$	0	0
$994$	−42.2507	−1.34011
$995$	6.40716	0.203121
$996$	0	0
$997$	30.3768	0.962043	0.481021	−	0.876709i	$-0.340266\pi$
$997$	30.3768	0.962043	0.481021	+	0.876709i	$0.340266\pi$
$998$	−34.4149	−1.08938
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.v.1.1		5
3.2	odd	2		1045.2.a.d.1.5	✓	5
15.14	odd	2		5225.2.a.j.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.5	✓	5	3.2	odd	2
5225.2.a.j.1.1		5	15.14	odd	2
9405.2.a.v.1.1		5	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22871 q^{2} +2.96714 q^{4} -1.00000 q^{5} -4.42518 q^{7} -2.15546 q^{8} +O(q^{10})\)	\(q-2.22871 q^{2} +2.96714 q^{4} -1.00000 q^{5} -4.42518 q^{7} -2.15546 q^{8} +2.22871 q^{10} +1.00000 q^{11} -3.92613 q^{13} +9.86243 q^{14} -1.13037 q^{16} +1.61520 q^{17} +1.00000 q^{19} -2.96714 q^{20} -2.22871 q^{22} -0.113248 q^{23} +1.00000 q^{25} +8.75019 q^{26} -13.1301 q^{28} -1.08612 q^{29} -4.17296 q^{31} +6.83020 q^{32} -3.59980 q^{34} +4.42518 q^{35} -5.75406 q^{37} -2.22871 q^{38} +2.15546 q^{40} -4.26750 q^{41} +3.62271 q^{43} +2.96714 q^{44} +0.252396 q^{46} +6.39567 q^{47} +12.5822 q^{49} -2.22871 q^{50} -11.6494 q^{52} -12.0154 q^{53} -1.00000 q^{55} +9.53832 q^{56} +2.42065 q^{58} -0.883911 q^{59} +3.77712 q^{61} +9.30032 q^{62} -12.9618 q^{64} +3.92613 q^{65} -12.2192 q^{67} +4.79251 q^{68} -9.86243 q^{70} -4.28400 q^{71} -1.92676 q^{73} +12.8241 q^{74} +2.96714 q^{76} -4.42518 q^{77} -7.81004 q^{79} +1.13037 q^{80} +9.51102 q^{82} +3.33410 q^{83} -1.61520 q^{85} -8.07397 q^{86} -2.15546 q^{88} +11.9145 q^{89} +17.3738 q^{91} -0.336021 q^{92} -14.2541 q^{94} -1.00000 q^{95} -13.3478 q^{97} -28.0421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8	\( 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31} + 2 q^{32} + 4 q^{34} + 11 q^{35} - 9 q^{37} + 3 q^{38} + 3 q^{40} - 15 q^{41} - 13 q^{43} + 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{49} + 3 q^{50} + q^{52} + 5 q^{53} - 5 q^{55} - 33 q^{58} + 17 q^{59} + 3 q^{61} - 14 q^{62} - 17 q^{64} - q^{65} - 28 q^{67} + 25 q^{68} + 6 q^{71} - 16 q^{73} + 21 q^{74} + 5 q^{76} - 11 q^{77} + 3 q^{79} + 3 q^{80} + 2 q^{82} + 33 q^{83} - 3 q^{85} - 10 q^{86} - 3 q^{88} + 16 q^{89} - 22 q^{91} + 19 q^{92} - 10 q^{94} - 5 q^{95} - 14 q^{97} - 10 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

Introduction

L-functions

Modular forms

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Database

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