LMFDB - Embedded newform 9405.2.a.z.1.5

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:	$6$
Coefficient field:	6.6.7281497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 2*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			1.5
Root			$-1.79049$ 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)

$f(q)$	$=$	$q+2.23198 q^{2} +2.98176 q^{4} +1.00000 q^{5} +4.13295 q^{7} +2.19127 q^{8} +O(q^{10})$	\(q+2.23198 q^{2} +2.98176 q^{4} +1.00000 q^{5} +4.13295 q^{7} +2.19127 q^{8} +2.23198 q^{10} +1.00000 q^{11} -0.110474 q^{13} +9.22468 q^{14} -1.07264 q^{16} +6.99924 q^{17} -1.00000 q^{19} +2.98176 q^{20} +2.23198 q^{22} -8.36934 q^{23} +1.00000 q^{25} -0.246577 q^{26} +12.3235 q^{28} +3.28289 q^{29} -0.202971 q^{31} -6.77665 q^{32} +15.6222 q^{34} +4.13295 q^{35} -0.683573 q^{37} -2.23198 q^{38} +2.19127 q^{40} +4.97522 q^{41} +8.52642 q^{43} +2.98176 q^{44} -18.6802 q^{46} -3.85948 q^{47} +10.0813 q^{49} +2.23198 q^{50} -0.329408 q^{52} +10.1554 q^{53} +1.00000 q^{55} +9.05639 q^{56} +7.32737 q^{58} +10.4857 q^{59} -3.53161 q^{61} -0.453029 q^{62} -12.9801 q^{64} -0.110474 q^{65} -7.84063 q^{67} +20.8700 q^{68} +9.22468 q^{70} +1.58930 q^{71} +8.41595 q^{73} -1.52572 q^{74} -2.98176 q^{76} +4.13295 q^{77} +2.80598 q^{79} -1.07264 q^{80} +11.1046 q^{82} +5.52580 q^{83} +6.99924 q^{85} +19.0308 q^{86} +2.19127 q^{88} -0.654537 q^{89} -0.456585 q^{91} -24.9553 q^{92} -8.61430 q^{94} -1.00000 q^{95} -10.8858 q^{97} +22.5013 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + 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 $

$2$	2.23198	1.57825	0.789126	−	0.614232i	$-0.210534\pi$
$2$	2.23198	1.57825	0.789126	+	0.614232i	$0.210534\pi$
$3$	0	0
$3$	0	0
$4$	2.98176	1.49088
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.13295	1.56211	0.781054	−	0.624463i	$-0.214682\pi$
$7$	4.13295	1.56211	0.781054	+	0.624463i	$0.214682\pi$
$8$	2.19127	0.774729
$9$	0	0
$10$	2.23198	0.705816
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.110474	−0.0306401	−0.0153200	−	0.999883i	$-0.504877\pi$
$13$	−0.110474	−0.0306401	−0.0153200	+	0.999883i	$0.504877\pi$
$14$	9.22468	2.46540
$15$	0	0
$16$	−1.07264	−0.268160
$17$	6.99924	1.69756	0.848782	−	0.528743i	$-0.177336\pi$
$17$	6.99924	1.69756	0.848782	+	0.528743i	$0.177336\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.98176	0.666741
$21$	0	0
$22$	2.23198	0.475861
$23$	−8.36934	−1.74513	−0.872564	−	0.488500i	$-0.837544\pi$
$23$	−8.36934	−1.74513	−0.872564	+	0.488500i	$0.837544\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.246577	−0.0483577
$27$	0	0
$28$	12.3235	2.32891
$29$	3.28289	0.609618	0.304809	−	0.952414i	$-0.401407\pi$
$29$	3.28289	0.609618	0.304809	+	0.952414i	$0.401407\pi$
$30$	0	0
$31$	−0.202971	−0.0364547	−0.0182274	−	0.999834i	$-0.505802\pi$
$31$	−0.202971	−0.0364547	−0.0182274	+	0.999834i	$0.505802\pi$
$32$	−6.77665	−1.19795
$33$	0	0
$34$	15.6222	2.67918
$35$	4.13295	0.698596
$36$	0	0
$37$	−0.683573	−0.112379	−0.0561893	−	0.998420i	$-0.517895\pi$
$37$	−0.683573	−0.112379	−0.0561893	+	0.998420i	$0.517895\pi$
$38$	−2.23198	−0.362076
$39$	0	0
$40$	2.19127	0.346470
$41$	4.97522	0.776999	0.388499	−	0.921449i	$-0.372994\pi$
$41$	4.97522	0.776999	0.388499	+	0.921449i	$0.372994\pi$
$42$	0	0
$43$	8.52642	1.30027	0.650134	−	0.759820i	$-0.274713\pi$
$43$	8.52642	1.30027	0.650134	+	0.759820i	$0.274713\pi$
$44$	2.98176	0.449517
$45$	0	0
$46$	−18.6802	−2.75425
$47$	−3.85948	−0.562963	−0.281482	−	0.959567i	$-0.590826\pi$
$47$	−3.85948	−0.562963	−0.281482	+	0.959567i	$0.590826\pi$
$48$	0	0
$49$	10.0813	1.44018
$50$	2.23198	0.315650
$51$	0	0
$52$	−0.329408	−0.0456806
$53$	10.1554	1.39496	0.697478	−	0.716607i	$-0.254306\pi$
$53$	10.1554	1.39496	0.697478	+	0.716607i	$0.254306\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	9.05639	1.21021
$57$	0	0
$58$	7.32737	0.962131
$59$	10.4857	1.36512	0.682561	−	0.730829i	$-0.260866\pi$
$59$	10.4857	1.36512	0.682561	+	0.730829i	$0.260866\pi$
$60$	0	0
$61$	−3.53161	−0.452177	−0.226088	−	0.974107i	$-0.572594\pi$
$61$	−3.53161	−0.452177	−0.226088	+	0.974107i	$0.572594\pi$
$62$	−0.453029	−0.0575347
$63$	0	0
$64$	−12.9801	−1.62251
$65$	−0.110474	−0.0137027
$66$	0	0
$67$	−7.84063	−0.957886	−0.478943	−	0.877846i	$-0.658980\pi$
$67$	−7.84063	−0.957886	−0.478943	+	0.877846i	$0.658980\pi$
$68$	20.8700	2.53086
$69$	0	0
$70$	9.22468	1.10256
$71$	1.58930	0.188616	0.0943078	−	0.995543i	$-0.469936\pi$
$71$	1.58930	0.188616	0.0943078	+	0.995543i	$0.469936\pi$
$72$	0	0
$73$	8.41595	0.985012	0.492506	−	0.870309i	$-0.336081\pi$
$73$	8.41595	0.985012	0.492506	+	0.870309i	$0.336081\pi$
$74$	−1.52572	−0.177362
$75$	0	0
$76$	−2.98176	−0.342031
$77$	4.13295	0.470993
$78$	0	0
$79$	2.80598	0.315697	0.157848	−	0.987463i	$-0.449544\pi$
$79$	2.80598	0.315697	0.157848	+	0.987463i	$0.449544\pi$
$80$	−1.07264	−0.119925
$81$	0	0
$82$	11.1046	1.22630
$83$	5.52580	0.606535	0.303267	−	0.952905i	$-0.401922\pi$
$83$	5.52580	0.606535	0.303267	+	0.952905i	$0.401922\pi$
$84$	0	0
$85$	6.99924	0.759174
$86$	19.0308	2.05215
$87$	0	0
$88$	2.19127	0.233590
$89$	−0.654537	−0.0693808	−0.0346904	−	0.999398i	$-0.511045\pi$
$89$	−0.654537	−0.0693808	−0.0346904	+	0.999398i	$0.511045\pi$
$90$	0	0
$91$	−0.456585	−0.0478631
$92$	−24.9553	−2.60177
$93$	0	0
$94$	−8.61430	−0.888498
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−10.8858	−1.10529	−0.552643	−	0.833418i	$-0.686381\pi$
$97$	−10.8858	−1.10529	−0.552643	+	0.833418i	$0.686381\pi$
$98$	22.5013	2.27297
$99$	0	0
$100$	2.98176	0.298176
$101$	−9.08470	−0.903961	−0.451980	−	0.892028i	$-0.649282\pi$
$101$	−9.08470	−0.903961	−0.451980	+	0.892028i	$0.649282\pi$
$102$	0	0
$103$	−3.59063	−0.353795	−0.176898	−	0.984229i	$-0.556606\pi$
$103$	−3.59063	−0.353795	−0.176898	+	0.984229i	$0.556606\pi$
$104$	−0.242079	−0.0237378
$105$	0	0
$106$	22.6668	2.20159
$107$	7.13069	0.689350	0.344675	−	0.938722i	$-0.387989\pi$
$107$	7.13069	0.689350	0.344675	+	0.938722i	$0.387989\pi$
$108$	0	0
$109$	−15.1412	−1.45026	−0.725131	−	0.688611i	$-0.758221\pi$
$109$	−15.1412	−1.45026	−0.725131	+	0.688611i	$0.758221\pi$
$110$	2.23198	0.212811
$111$	0	0
$112$	−4.43317	−0.418896
$113$	2.67686	0.251818	0.125909	−	0.992042i	$-0.459815\pi$
$113$	2.67686	0.251818	0.125909	+	0.992042i	$0.459815\pi$
$114$	0	0
$115$	−8.36934	−0.780445
$116$	9.78879	0.908866
$117$	0	0
$118$	23.4039	2.15451
$119$	28.9275	2.65178
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.88251	−0.713649
$123$	0	0
$124$	−0.605211	−0.0543496
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.0799	−1.16065	−0.580326	−	0.814384i	$-0.697075\pi$
$127$	−13.0799	−1.16065	−0.580326	+	0.814384i	$0.697075\pi$
$128$	−15.4181	−1.36278
$129$	0	0
$130$	−0.246577	−0.0216262
$131$	8.33532	0.728260	0.364130	−	0.931348i	$-0.381366\pi$
$131$	8.33532	0.728260	0.364130	+	0.931348i	$0.381366\pi$
$132$	0	0
$133$	−4.13295	−0.358372
$134$	−17.5002	−1.51179
$135$	0	0
$136$	15.3372	1.31515
$137$	13.0803	1.11753	0.558765	−	0.829326i	$-0.311276\pi$
$137$	13.0803	1.11753	0.558765	+	0.829326i	$0.311276\pi$
$138$	0	0
$139$	−18.2803	−1.55052	−0.775259	−	0.631644i	$-0.782381\pi$
$139$	−18.2803	−1.55052	−0.775259	+	0.631644i	$0.782381\pi$
$140$	12.3235	1.04152
$141$	0	0
$142$	3.54730	0.297683
$143$	−0.110474	−0.00923833
$144$	0	0
$145$	3.28289	0.272629
$146$	18.7843	1.55460
$147$	0	0
$148$	−2.03825	−0.167543
$149$	−18.3204	−1.50087	−0.750435	−	0.660945i	$-0.770156\pi$
$149$	−18.3204	−1.50087	−0.750435	+	0.660945i	$0.770156\pi$
$150$	0	0
$151$	8.66581	0.705213	0.352607	−	0.935772i	$-0.385295\pi$
$151$	8.66581	0.705213	0.352607	+	0.935772i	$0.385295\pi$
$152$	−2.19127	−0.177735
$153$	0	0
$154$	9.22468	0.743346
$155$	−0.202971	−0.0163030
$156$	0	0
$157$	−18.7512	−1.49651	−0.748254	−	0.663412i	$-0.769108\pi$
$157$	−18.7512	−1.49651	−0.748254	+	0.663412i	$0.769108\pi$
$158$	6.26290	0.498249
$159$	0	0
$160$	−6.77665	−0.535741
$161$	−34.5901	−2.72608
$162$	0	0
$163$	19.8023	1.55103	0.775517	−	0.631327i	$-0.217489\pi$
$163$	19.8023	1.55103	0.775517	+	0.631327i	$0.217489\pi$
$164$	14.8349	1.15841
$165$	0	0
$166$	12.3335	0.957265
$167$	−1.91597	−0.148262	−0.0741312	−	0.997248i	$-0.523618\pi$
$167$	−1.91597	−0.148262	−0.0741312	+	0.997248i	$0.523618\pi$
$168$	0	0
$169$	−12.9878	−0.999061
$170$	15.6222	1.19817
$171$	0	0
$172$	25.4237	1.93854
$173$	3.30818	0.251516	0.125758	−	0.992061i	$-0.459864\pi$
$173$	3.30818	0.251516	0.125758	+	0.992061i	$0.459864\pi$
$174$	0	0
$175$	4.13295	0.312422
$176$	−1.07264	−0.0808534
$177$	0	0
$178$	−1.46092	−0.109500
$179$	21.9873	1.64341	0.821703	−	0.569915i	$-0.193024\pi$
$179$	21.9873	1.64341	0.821703	+	0.569915i	$0.193024\pi$
$180$	0	0
$181$	−17.0245	−1.26542	−0.632709	−	0.774389i	$-0.718057\pi$
$181$	−17.0245	−1.26542	−0.632709	+	0.774389i	$0.718057\pi$
$182$	−1.01909	−0.0755400
$183$	0	0
$184$	−18.3394	−1.35200
$185$	−0.683573	−0.0502573
$186$	0	0
$187$	6.99924	0.511835
$188$	−11.5080	−0.839310
$189$	0	0
$190$	−2.23198	−0.161925
$191$	12.8748	0.931589	0.465794	−	0.884893i	$-0.345769\pi$
$191$	12.8748	0.931589	0.465794	+	0.884893i	$0.345769\pi$
$192$	0	0
$193$	2.49248	0.179412	0.0897062	−	0.995968i	$-0.471407\pi$
$193$	2.49248	0.179412	0.0897062	+	0.995968i	$0.471407\pi$
$194$	−24.2970	−1.74442
$195$	0	0
$196$	30.0599	2.14714
$197$	−0.604202	−0.0430476	−0.0215238	−	0.999768i	$-0.506852\pi$
$197$	−0.604202	−0.0430476	−0.0215238	+	0.999768i	$0.506852\pi$
$198$	0	0
$199$	−17.6373	−1.25027	−0.625137	−	0.780515i	$-0.714957\pi$
$199$	−17.6373	−1.25027	−0.625137	+	0.780515i	$0.714957\pi$
$200$	2.19127	0.154946
$201$	0	0
$202$	−20.2769	−1.42668
$203$	13.5680	0.952289
$204$	0	0
$205$	4.97522	0.347484
$206$	−8.01423	−0.558378
$207$	0	0
$208$	0.118499	0.00821645
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	19.9281	1.37190	0.685952	−	0.727646i	$-0.259386\pi$
$211$	19.9281	1.37190	0.685952	+	0.727646i	$0.259386\pi$
$212$	30.2810	2.07971
$213$	0	0
$214$	15.9156	1.08797
$215$	8.52642	0.581497
$216$	0	0
$217$	−0.838870	−0.0569462
$218$	−33.7949	−2.28888
$219$	0	0
$220$	2.98176	0.201030
$221$	−0.773236	−0.0520135
$222$	0	0
$223$	12.6225	0.845263	0.422631	−	0.906302i	$-0.361106\pi$
$223$	12.6225	0.845263	0.422631	+	0.906302i	$0.361106\pi$
$224$	−28.0076	−1.87133
$225$	0	0
$226$	5.97472	0.397432
$227$	−7.57012	−0.502447	−0.251223	−	0.967929i	$-0.580833\pi$
$227$	−7.57012	−0.502447	−0.251223	+	0.967929i	$0.580833\pi$
$228$	0	0
$229$	−8.20266	−0.542047	−0.271023	−	0.962573i	$-0.587362\pi$
$229$	−8.20266	−0.542047	−0.271023	+	0.962573i	$0.587362\pi$
$230$	−18.6802	−1.23174
$231$	0	0
$232$	7.19369	0.472289
$233$	−16.9904	−1.11308	−0.556538	−	0.830822i	$-0.687871\pi$
$233$	−16.9904	−1.11308	−0.556538	+	0.830822i	$0.687871\pi$
$234$	0	0
$235$	−3.85948	−0.251765
$236$	31.2658	2.03523
$237$	0	0
$238$	64.5657	4.18517
$239$	15.2540	0.986699	0.493350	−	0.869831i	$-0.335772\pi$
$239$	15.2540	0.986699	0.493350	+	0.869831i	$0.335772\pi$
$240$	0	0
$241$	−1.53642	−0.0989698	−0.0494849	−	0.998775i	$-0.515758\pi$
$241$	−1.53642	−0.0989698	−0.0494849	+	0.998775i	$0.515758\pi$
$242$	2.23198	0.143477
$243$	0	0
$244$	−10.5304	−0.674141
$245$	10.0813	0.644069
$246$	0	0
$247$	0.110474	0.00702931
$248$	−0.444764	−0.0282425
$249$	0	0
$250$	2.23198	0.141163
$251$	−24.6721	−1.55729	−0.778646	−	0.627464i	$-0.784093\pi$
$251$	−24.6721	−1.55729	−0.778646	+	0.627464i	$0.784093\pi$
$252$	0	0
$253$	−8.36934	−0.526176
$254$	−29.1941	−1.83180
$255$	0	0
$256$	−8.45273	−0.528296
$257$	−11.9052	−0.742624	−0.371312	−	0.928508i	$-0.621092\pi$
$257$	−11.9052	−0.742624	−0.371312	+	0.928508i	$0.621092\pi$
$258$	0	0
$259$	−2.82517	−0.175548
$260$	−0.329408	−0.0204290
$261$	0	0
$262$	18.6043	1.14938
$263$	0.849505	0.0523827	0.0261914	−	0.999657i	$-0.491662\pi$
$263$	0.849505	0.0523827	0.0261914	+	0.999657i	$0.491662\pi$
$264$	0	0
$265$	10.1554	0.623843
$266$	−9.22468	−0.565602
$267$	0	0
$268$	−23.3789	−1.42809
$269$	16.6805	1.01703	0.508514	−	0.861054i	$-0.330195\pi$
$269$	16.6805	1.01703	0.508514	+	0.861054i	$0.330195\pi$
$270$	0	0
$271$	−13.2403	−0.804291	−0.402146	−	0.915576i	$-0.631735\pi$
$271$	−13.2403	−0.804291	−0.402146	+	0.915576i	$0.631735\pi$
$272$	−7.50767	−0.455219
$273$	0	0
$274$	29.1951	1.76374
$275$	1.00000	0.0603023
$276$	0	0
$277$	−17.2124	−1.03419	−0.517096	−	0.855927i	$-0.672987\pi$
$277$	−17.2124	−1.03419	−0.517096	+	0.855927i	$0.672987\pi$
$278$	−40.8014	−2.44711
$279$	0	0
$280$	9.05639	0.541223
$281$	−9.32448	−0.556252	−0.278126	−	0.960545i	$-0.589713\pi$
$281$	−9.32448	−0.556252	−0.278126	+	0.960545i	$0.589713\pi$
$282$	0	0
$283$	12.8642	0.764698	0.382349	−	0.924018i	$-0.375115\pi$
$283$	12.8642	0.764698	0.382349	+	0.924018i	$0.375115\pi$
$284$	4.73891	0.281203
$285$	0	0
$286$	−0.246577	−0.0145804
$287$	20.5623	1.21376
$288$	0	0
$289$	31.9893	1.88172
$290$	7.32737	0.430278
$291$	0	0
$292$	25.0943	1.46853
$293$	14.4092	0.841796	0.420898	−	0.907108i	$-0.361715\pi$
$293$	14.4092	0.841796	0.420898	+	0.907108i	$0.361715\pi$
$294$	0	0
$295$	10.4857	0.610501
$296$	−1.49789	−0.0870630
$297$	0	0
$298$	−40.8910	−2.36875
$299$	0.924597	0.0534708
$300$	0	0
$301$	35.2393	2.03116
$302$	19.3419	1.11300
$303$	0	0
$304$	1.07264	0.0615202
$305$	−3.53161	−0.202220
$306$	0	0
$307$	−28.2838	−1.61424	−0.807122	−	0.590385i	$-0.798976\pi$
$307$	−28.2838	−1.61424	−0.807122	+	0.590385i	$0.798976\pi$
$308$	12.3235	0.702194
$309$	0	0
$310$	−0.453029	−0.0257303
$311$	12.3942	0.702809	0.351404	−	0.936224i	$-0.385704\pi$
$311$	12.3942	0.702809	0.351404	+	0.936224i	$0.385704\pi$
$312$	0	0
$313$	−33.0560	−1.86844	−0.934219	−	0.356701i	$-0.883901\pi$
$313$	−33.0560	−1.86844	−0.934219	+	0.356701i	$0.883901\pi$
$314$	−41.8524	−2.36187
$315$	0	0
$316$	8.36674	0.470666
$317$	−9.95795	−0.559294	−0.279647	−	0.960103i	$-0.590217\pi$
$317$	−9.95795	−0.559294	−0.279647	+	0.960103i	$0.590217\pi$
$318$	0	0
$319$	3.28289	0.183807
$320$	−12.9801	−0.725610
$321$	0	0
$322$	−77.2045	−4.30244
$323$	−6.99924	−0.389448
$324$	0	0
$325$	−0.110474	−0.00612801
$326$	44.1984	2.44792
$327$	0	0
$328$	10.9020	0.601964
$329$	−15.9510	−0.879409
$330$	0	0
$331$	29.9575	1.64661	0.823307	−	0.567596i	$-0.192126\pi$
$331$	29.9575	1.64661	0.823307	+	0.567596i	$0.192126\pi$
$332$	16.4766	0.904270
$333$	0	0
$334$	−4.27642	−0.233995
$335$	−7.84063	−0.428380
$336$	0	0
$337$	11.7426	0.639662	0.319831	−	0.947475i	$-0.396374\pi$
$337$	11.7426	0.639662	0.319831	+	0.947475i	$0.396374\pi$
$338$	−28.9886	−1.57677
$339$	0	0
$340$	20.8700	1.13184
$341$	−0.202971	−0.0109915
$342$	0	0
$343$	12.7348	0.687613
$344$	18.6837	1.00736
$345$	0	0
$346$	7.38381	0.396956
$347$	−7.01661	−0.376671	−0.188336	−	0.982105i	$-0.560309\pi$
$347$	−7.01661	−0.376671	−0.188336	+	0.982105i	$0.560309\pi$
$348$	0	0
$349$	4.66034	0.249462	0.124731	−	0.992191i	$-0.460193\pi$
$349$	4.66034	0.249462	0.124731	+	0.992191i	$0.460193\pi$
$350$	9.22468	0.493080
$351$	0	0
$352$	−6.77665	−0.361197
$353$	36.5878	1.94737	0.973687	−	0.227889i	$-0.0731823\pi$
$353$	36.5878	1.94737	0.973687	+	0.227889i	$0.0731823\pi$
$354$	0	0
$355$	1.58930	0.0843514
$356$	−1.95167	−0.103438
$357$	0	0
$358$	49.0753	2.59371
$359$	−1.09396	−0.0577369	−0.0288684	−	0.999583i	$-0.509190\pi$
$359$	−1.09396	−0.0577369	−0.0288684	+	0.999583i	$0.509190\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−37.9984	−1.99715
$363$	0	0
$364$	−1.36143	−0.0713581
$365$	8.41595	0.440511
$366$	0	0
$367$	−12.3334	−0.643801	−0.321900	−	0.946774i	$-0.604322\pi$
$367$	−12.3334	−0.643801	−0.321900	+	0.946774i	$0.604322\pi$
$368$	8.97730	0.467974
$369$	0	0
$370$	−1.52572	−0.0793186
$371$	41.9719	2.17907
$372$	0	0
$373$	−9.20094	−0.476407	−0.238203	−	0.971215i	$-0.576558\pi$
$373$	−9.20094	−0.476407	−0.238203	+	0.971215i	$0.576558\pi$
$374$	15.6222	0.807804
$375$	0	0
$376$	−8.45715	−0.436144
$377$	−0.362675	−0.0186787
$378$	0	0
$379$	36.7806	1.88929	0.944645	−	0.328093i	$-0.106406\pi$
$379$	36.7806	1.88929	0.944645	+	0.328093i	$0.106406\pi$
$380$	−2.98176	−0.152961
$381$	0	0
$382$	28.7364	1.47028
$383$	−23.8111	−1.21669	−0.608345	−	0.793673i	$-0.708166\pi$
$383$	−23.8111	−1.21669	−0.608345	+	0.793673i	$0.708166\pi$
$384$	0	0
$385$	4.13295	0.210635
$386$	5.56317	0.283158
$387$	0	0
$388$	−32.4588	−1.64785
$389$	−3.05680	−0.154986	−0.0774928	−	0.996993i	$-0.524691\pi$
$389$	−3.05680	−0.154986	−0.0774928	+	0.996993i	$0.524691\pi$
$390$	0	0
$391$	−58.5790	−2.96247
$392$	22.0908	1.11575
$393$	0	0
$394$	−1.34857	−0.0679400
$395$	2.80598	0.141184
$396$	0	0
$397$	6.65411	0.333960	0.166980	−	0.985960i	$-0.446598\pi$
$397$	6.65411	0.333960	0.166980	+	0.985960i	$0.446598\pi$
$398$	−39.3661	−1.97325
$399$	0	0
$400$	−1.07264	−0.0536321
$401$	−18.0655	−0.902148	−0.451074	−	0.892487i	$-0.648959\pi$
$401$	−18.0655	−0.902148	−0.451074	+	0.892487i	$0.648959\pi$
$402$	0	0
$403$	0.0224231	0.00111698
$404$	−27.0883	−1.34770
$405$	0	0
$406$	30.2836	1.50295
$407$	−0.683573	−0.0338834
$408$	0	0
$409$	30.6190	1.51401	0.757005	−	0.653409i	$-0.226662\pi$
$409$	30.6190	1.51401	0.757005	+	0.653409i	$0.226662\pi$
$410$	11.1046	0.548418
$411$	0	0
$412$	−10.7064	−0.527465
$413$	43.3369	2.13247
$414$	0	0
$415$	5.52580	0.271251
$416$	0.748646	0.0367054
$417$	0	0
$418$	−2.23198	−0.109170
$419$	38.4886	1.88029	0.940145	−	0.340774i	$-0.110689\pi$
$419$	38.4886	1.88029	0.940145	+	0.340774i	$0.110689\pi$
$420$	0	0
$421$	−33.2827	−1.62210	−0.811049	−	0.584979i	$-0.801103\pi$
$421$	−33.2827	−1.62210	−0.811049	+	0.584979i	$0.801103\pi$
$422$	44.4791	2.16521
$423$	0	0
$424$	22.2532	1.08071
$425$	6.99924	0.339513
$426$	0	0
$427$	−14.5960	−0.706349
$428$	21.2620	1.02774
$429$	0	0
$430$	19.0308	0.917749
$431$	−4.84543	−0.233396	−0.116698	−	0.993167i	$-0.537231\pi$
$431$	−4.84543	−0.233396	−0.116698	+	0.993167i	$0.537231\pi$
$432$	0	0
$433$	29.2320	1.40480	0.702400	−	0.711783i	$-0.252112\pi$
$433$	29.2320	1.40480	0.702400	+	0.711783i	$0.252112\pi$
$434$	−1.87235	−0.0898755
$435$	0	0
$436$	−45.1473	−2.16216
$437$	8.36934	0.400360
$438$	0	0
$439$	−12.2196	−0.583211	−0.291605	−	0.956539i	$-0.594189\pi$
$439$	−12.2196	−0.583211	−0.291605	+	0.956539i	$0.594189\pi$
$440$	2.19127	0.104464
$441$	0	0
$442$	−1.72585	−0.0820903
$443$	24.1325	1.14657	0.573284	−	0.819357i	$-0.305669\pi$
$443$	24.1325	1.14657	0.573284	+	0.819357i	$0.305669\pi$
$444$	0	0
$445$	−0.654537	−0.0310280
$446$	28.1732	1.33404
$447$	0	0
$448$	−53.6461	−2.53454
$449$	8.77225	0.413988	0.206994	−	0.978342i	$-0.433632\pi$
$449$	8.77225	0.413988	0.206994	+	0.978342i	$0.433632\pi$
$450$	0	0
$451$	4.97522	0.234274
$452$	7.98176	0.375430
$453$	0	0
$454$	−16.8964	−0.792987
$455$	−0.456585	−0.0214050
$456$	0	0
$457$	−6.85848	−0.320826	−0.160413	−	0.987050i	$-0.551283\pi$
$457$	−6.85848	−0.320826	−0.160413	+	0.987050i	$0.551283\pi$
$458$	−18.3082	−0.855487
$459$	0	0
$460$	−24.9553	−1.16355
$461$	24.9007	1.15974	0.579870	−	0.814709i	$-0.303104\pi$
$461$	24.9007	1.15974	0.579870	+	0.814709i	$0.303104\pi$
$462$	0	0
$463$	−2.79250	−0.129778	−0.0648892	−	0.997892i	$-0.520669\pi$
$463$	−2.79250	−0.129778	−0.0648892	+	0.997892i	$0.520669\pi$
$464$	−3.52137	−0.163475
$465$	0	0
$466$	−37.9222	−1.75671
$467$	7.55358	0.349538	0.174769	−	0.984609i	$-0.444082\pi$
$467$	7.55358	0.349538	0.174769	+	0.984609i	$0.444082\pi$
$468$	0	0
$469$	−32.4050	−1.49632
$470$	−8.61430	−0.397348
$471$	0	0
$472$	22.9770	1.05760
$473$	8.52642	0.392045
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	86.2547	3.95348
$477$	0	0
$478$	34.0467	1.55726
$479$	−28.2961	−1.29288	−0.646440	−	0.762965i	$-0.723743\pi$
$479$	−28.2961	−1.29288	−0.646440	+	0.762965i	$0.723743\pi$
$480$	0	0
$481$	0.0755172	0.00344329
$482$	−3.42928	−0.156199
$483$	0	0
$484$	2.98176	0.135534
$485$	−10.8858	−0.494299
$486$	0	0
$487$	−22.0840	−1.00072	−0.500361	−	0.865817i	$-0.666799\pi$
$487$	−22.0840	−1.00072	−0.500361	+	0.865817i	$0.666799\pi$
$488$	−7.73870	−0.350315
$489$	0	0
$490$	22.5013	1.01650
$491$	−34.1421	−1.54081	−0.770405	−	0.637555i	$-0.779946\pi$
$491$	−34.1421	−1.54081	−0.770405	+	0.637555i	$0.779946\pi$
$492$	0	0
$493$	22.9777	1.03487
$494$	0.246577	0.0110940
$495$	0	0
$496$	0.217715	0.00977571
$497$	6.56851	0.294638
$498$	0	0
$499$	−9.33688	−0.417976	−0.208988	−	0.977918i	$-0.567017\pi$
$499$	−9.33688	−0.417976	−0.208988	+	0.977918i	$0.567017\pi$
$500$	2.98176	0.133348
$501$	0	0
$502$	−55.0678	−2.45780
$503$	−16.8794	−0.752616	−0.376308	−	0.926495i	$-0.622807\pi$
$503$	−16.8794	−0.752616	−0.376308	+	0.926495i	$0.622807\pi$
$504$	0	0
$505$	−9.08470	−0.404264
$506$	−18.6802	−0.830438
$507$	0	0
$508$	−39.0010	−1.73039
$509$	−13.2971	−0.589383	−0.294691	−	0.955592i	$-0.595217\pi$
$509$	−13.2971	−0.589383	−0.294691	+	0.955592i	$0.595217\pi$
$510$	0	0
$511$	34.7827	1.53870
$512$	11.9698	0.528995
$513$	0	0
$514$	−26.5722	−1.17205
$515$	−3.59063	−0.158222
$516$	0	0
$517$	−3.85948	−0.169740
$518$	−6.30574	−0.277058
$519$	0	0
$520$	−0.242079	−0.0106158
$521$	20.5200	0.898997	0.449499	−	0.893281i	$-0.351603\pi$
$521$	20.5200	0.898997	0.449499	+	0.893281i	$0.351603\pi$
$522$	0	0
$523$	19.6848	0.860757	0.430378	−	0.902649i	$-0.358380\pi$
$523$	19.6848	0.860757	0.430378	+	0.902649i	$0.358380\pi$
$524$	24.8539	1.08575
$525$	0	0
$526$	1.89608	0.0826731
$527$	−1.42064	−0.0618842
$528$	0	0
$529$	47.0458	2.04547
$530$	22.6668	0.984581
$531$	0	0
$532$	−12.3235	−0.534289
$533$	−0.549634	−0.0238073
$534$	0	0
$535$	7.13069	0.308287
$536$	−17.1809	−0.742102
$537$	0	0
$538$	37.2306	1.60513
$539$	10.0813	0.434231
$540$	0	0
$541$	−15.9758	−0.686852	−0.343426	−	0.939180i	$-0.611587\pi$
$541$	−15.9758	−0.686852	−0.343426	+	0.939180i	$0.611587\pi$
$542$	−29.5522	−1.26937
$543$	0	0
$544$	−47.4314	−2.03360
$545$	−15.1412	−0.648577
$546$	0	0
$547$	−41.5489	−1.77650	−0.888252	−	0.459357i	$-0.848080\pi$
$547$	−41.5489	−1.77650	−0.888252	+	0.459357i	$0.848080\pi$
$548$	39.0024	1.66610
$549$	0	0
$550$	2.23198	0.0951722
$551$	−3.28289	−0.139856
$552$	0	0
$553$	11.5970	0.493153
$554$	−38.4178	−1.63222
$555$	0	0
$556$	−54.5075	−2.31163
$557$	−30.0766	−1.27439	−0.637194	−	0.770703i	$-0.719905\pi$
$557$	−30.0766	−1.27439	−0.637194	+	0.770703i	$0.719905\pi$
$558$	0	0
$559$	−0.941951	−0.0398403
$560$	−4.43317	−0.187336
$561$	0	0
$562$	−20.8121	−0.877906
$563$	−21.3804	−0.901079	−0.450539	−	0.892757i	$-0.648768\pi$
$563$	−21.3804	−0.901079	−0.450539	+	0.892757i	$0.648768\pi$
$564$	0	0
$565$	2.67686	0.112617
$566$	28.7127	1.20689
$567$	0	0
$568$	3.48258	0.146126
$569$	−11.5873	−0.485767	−0.242883	−	0.970055i	$-0.578093\pi$
$569$	−11.5873	−0.485767	−0.242883	+	0.970055i	$0.578093\pi$
$570$	0	0
$571$	−25.1618	−1.05299	−0.526494	−	0.850179i	$-0.676494\pi$
$571$	−25.1618	−1.05299	−0.526494	+	0.850179i	$0.676494\pi$
$572$	−0.329408	−0.0137732
$573$	0	0
$574$	45.8948	1.91561
$575$	−8.36934	−0.349026
$576$	0	0
$577$	−1.55228	−0.0646222	−0.0323111	−	0.999478i	$-0.510287\pi$
$577$	−1.55228	−0.0646222	−0.0323111	+	0.999478i	$0.510287\pi$
$578$	71.3996	2.96983
$579$	0	0
$580$	9.78879	0.406457
$581$	22.8378	0.947473
$582$	0	0
$583$	10.1554	0.420595
$584$	18.4416	0.763118
$585$	0	0
$586$	32.1612	1.32857
$587$	5.32002	0.219581	0.109790	−	0.993955i	$-0.464982\pi$
$587$	5.32002	0.219581	0.109790	+	0.993955i	$0.464982\pi$
$588$	0	0
$589$	0.202971	0.00836329
$590$	23.4039	0.963525
$591$	0	0
$592$	0.733228	0.0301355
$593$	−36.2723	−1.48952	−0.744762	−	0.667331i	$-0.767437\pi$
$593$	−36.2723	−1.48952	−0.744762	+	0.667331i	$0.767437\pi$
$594$	0	0
$595$	28.9275	1.18591
$596$	−54.6271	−2.23761
$597$	0	0
$598$	2.06369	0.0843904
$599$	40.0043	1.63453	0.817266	−	0.576261i	$-0.195489\pi$
$599$	40.0043	1.63453	0.817266	+	0.576261i	$0.195489\pi$
$600$	0	0
$601$	36.6257	1.49400	0.746998	−	0.664827i	$-0.231495\pi$
$601$	36.6257	1.49400	0.746998	+	0.664827i	$0.231495\pi$
$602$	78.6535	3.20568
$603$	0	0
$604$	25.8393	1.05139
$605$	1.00000	0.0406558
$606$	0	0
$607$	23.7724	0.964893	0.482446	−	0.875926i	$-0.339748\pi$
$607$	23.7724	0.964893	0.482446	+	0.875926i	$0.339748\pi$
$608$	6.77665	0.274829
$609$	0	0
$610$	−7.88251	−0.319153
$611$	0.426374	0.0172492
$612$	0	0
$613$	−2.77756	−0.112184	−0.0560922	−	0.998426i	$-0.517864\pi$
$613$	−2.77756	−0.112184	−0.0560922	+	0.998426i	$0.517864\pi$
$614$	−63.1291	−2.54768
$615$	0	0
$616$	9.05639	0.364892
$617$	−39.0259	−1.57112	−0.785562	−	0.618783i	$-0.787626\pi$
$617$	−39.0259	−1.57112	−0.785562	+	0.618783i	$0.787626\pi$
$618$	0	0
$619$	45.6890	1.83639	0.918197	−	0.396123i	$-0.129645\pi$
$619$	45.6890	1.83639	0.918197	+	0.396123i	$0.129645\pi$
$620$	−0.605211	−0.0243059
$621$	0	0
$622$	27.6636	1.10921
$623$	−2.70517	−0.108380
$624$	0	0
$625$	1.00000	0.0400000
$626$	−73.7806	−2.94886
$627$	0	0
$628$	−55.9115	−2.23111
$629$	−4.78448	−0.190770
$630$	0	0
$631$	−20.2465	−0.806000	−0.403000	−	0.915200i	$-0.632032\pi$
$631$	−20.2465	−0.806000	−0.403000	+	0.915200i	$0.632032\pi$
$632$	6.14864	0.244580
$633$	0	0
$634$	−22.2260	−0.882707
$635$	−13.0799	−0.519059
$636$	0	0
$637$	−1.11372	−0.0441273
$638$	7.32737	0.290093
$639$	0	0
$640$	−15.4181	−0.609453
$641$	−22.9945	−0.908229	−0.454115	−	0.890943i	$-0.650044\pi$
$641$	−22.9945	−0.908229	−0.454115	+	0.890943i	$0.650044\pi$
$642$	0	0
$643$	−36.6452	−1.44514	−0.722572	−	0.691295i	$-0.757040\pi$
$643$	−36.6452	−1.44514	−0.722572	+	0.691295i	$0.757040\pi$
$644$	−103.139	−4.06425
$645$	0	0
$646$	−15.6222	−0.614647
$647$	−36.1179	−1.41994	−0.709970	−	0.704232i	$-0.751291\pi$
$647$	−36.1179	−1.41994	−0.709970	+	0.704232i	$0.751291\pi$
$648$	0	0
$649$	10.4857	0.411600
$650$	−0.246577	−0.00967155
$651$	0	0
$652$	59.0455	2.31240
$653$	−28.1710	−1.10242	−0.551208	−	0.834368i	$-0.685833\pi$
$653$	−28.1710	−1.10242	−0.551208	+	0.834368i	$0.685833\pi$
$654$	0	0
$655$	8.33532	0.325688
$656$	−5.33663	−0.208360
$657$	0	0
$658$	−35.6025	−1.38793
$659$	46.4859	1.81083	0.905416	−	0.424525i	$-0.139559\pi$
$659$	46.4859	1.81083	0.905416	+	0.424525i	$0.139559\pi$
$660$	0	0
$661$	−37.2017	−1.44698	−0.723489	−	0.690336i	$-0.757463\pi$
$661$	−37.2017	−1.44698	−0.723489	+	0.690336i	$0.757463\pi$
$662$	66.8648	2.59877
$663$	0	0
$664$	12.1085	0.469900
$665$	−4.13295	−0.160269
$666$	0	0
$667$	−27.4756	−1.06386
$668$	−5.71296	−0.221041
$669$	0	0
$670$	−17.5002	−0.676091
$671$	−3.53161	−0.136336
$672$	0	0
$673$	31.3183	1.20723	0.603616	−	0.797276i	$-0.293726\pi$
$673$	31.3183	1.20723	0.603616	+	0.797276i	$0.293726\pi$
$674$	26.2094	1.00955
$675$	0	0
$676$	−38.7264	−1.48948
$677$	−44.6151	−1.71470	−0.857349	−	0.514736i	$-0.827890\pi$
$677$	−44.6151	−1.71470	−0.857349	+	0.514736i	$0.827890\pi$
$678$	0	0
$679$	−44.9905	−1.72658
$680$	15.3372	0.588154
$681$	0	0
$682$	−0.453029	−0.0173474
$683$	−11.0815	−0.424024	−0.212012	−	0.977267i	$-0.568002\pi$
$683$	−11.0815	−0.424024	−0.212012	+	0.977267i	$0.568002\pi$
$684$	0	0
$685$	13.0803	0.499774
$686$	28.4238	1.08523
$687$	0	0
$688$	−9.14579	−0.348680
$689$	−1.12191	−0.0427415
$690$	0	0
$691$	−28.3303	−1.07773	−0.538867	−	0.842391i	$-0.681148\pi$
$691$	−28.3303	−1.07773	−0.538867	+	0.842391i	$0.681148\pi$
$692$	9.86419	0.374980
$693$	0	0
$694$	−15.6610	−0.594482
$695$	−18.2803	−0.693412
$696$	0	0
$697$	34.8227	1.31900
$698$	10.4018	0.393714
$699$	0	0
$700$	12.3235	0.465783
$701$	−41.4446	−1.56534	−0.782671	−	0.622435i	$-0.786143\pi$
$701$	−41.4446	−1.56534	−0.782671	+	0.622435i	$0.786143\pi$
$702$	0	0
$703$	0.683573	0.0257814
$704$	−12.9801	−0.489206
$705$	0	0
$706$	81.6635	3.07345
$707$	−37.5466	−1.41208
$708$	0	0
$709$	−39.1483	−1.47025	−0.735123	−	0.677933i	$-0.762876\pi$
$709$	−39.1483	−1.47025	−0.735123	+	0.677933i	$0.762876\pi$
$710$	3.54730	0.133128
$711$	0	0
$712$	−1.43426	−0.0537513
$713$	1.69874	0.0636182
$714$	0	0
$715$	−0.110474	−0.00413151
$716$	65.5607	2.45012
$717$	0	0
$718$	−2.44170	−0.0911233
$719$	−27.5810	−1.02860	−0.514299	−	0.857611i	$-0.671948\pi$
$719$	−27.5810	−1.02860	−0.514299	+	0.857611i	$0.671948\pi$
$720$	0	0
$721$	−14.8399	−0.552666
$722$	2.23198	0.0830659
$723$	0	0
$724$	−50.7628	−1.88659
$725$	3.28289	0.121924
$726$	0	0
$727$	44.7807	1.66083	0.830413	−	0.557149i	$-0.188105\pi$
$727$	44.7807	1.66083	0.830413	+	0.557149i	$0.188105\pi$
$728$	−1.00050	−0.0370810
$729$	0	0
$730$	18.7843	0.695237
$731$	59.6784	2.20729
$732$	0	0
$733$	−15.0617	−0.556316	−0.278158	−	0.960535i	$-0.589724\pi$
$733$	−15.0617	−0.556316	−0.278158	+	0.960535i	$0.589724\pi$
$734$	−27.5281	−1.01608
$735$	0	0
$736$	56.7161	2.09058
$737$	−7.84063	−0.288813
$738$	0	0
$739$	−16.7492	−0.616128	−0.308064	−	0.951366i	$-0.599681\pi$
$739$	−16.7492	−0.616128	−0.308064	+	0.951366i	$0.599681\pi$
$740$	−2.03825	−0.0749274
$741$	0	0
$742$	93.6806	3.43912
$743$	−38.4706	−1.41135	−0.705674	−	0.708537i	$-0.749356\pi$
$743$	−38.4706	−1.41135	−0.705674	+	0.708537i	$0.749356\pi$
$744$	0	0
$745$	−18.3204	−0.671209
$746$	−20.5364	−0.751890
$747$	0	0
$748$	20.8700	0.763083
$749$	29.4708	1.07684
$750$	0	0
$751$	−38.7951	−1.41565	−0.707827	−	0.706386i	$-0.750324\pi$
$751$	−38.7951	−1.41565	−0.707827	+	0.706386i	$0.750324\pi$
$752$	4.13984	0.150964
$753$	0	0
$754$	−0.809486	−0.0294797
$755$	8.66581	0.315381
$756$	0	0
$757$	49.3111	1.79224	0.896120	−	0.443811i	$-0.146374\pi$
$757$	49.3111	1.79224	0.896120	+	0.443811i	$0.146374\pi$
$758$	82.0937	2.98178
$759$	0	0
$760$	−2.19127	−0.0794856
$761$	7.21992	0.261722	0.130861	−	0.991401i	$-0.458226\pi$
$761$	7.21992	0.261722	0.130861	+	0.991401i	$0.458226\pi$
$762$	0	0
$763$	−62.5777	−2.26547
$764$	38.3896	1.38889
$765$	0	0
$766$	−53.1460	−1.92024
$767$	−1.15840	−0.0418274
$768$	0	0
$769$	15.7986	0.569714	0.284857	−	0.958570i	$-0.408054\pi$
$769$	15.7986	0.569714	0.284857	+	0.958570i	$0.408054\pi$
$770$	9.22468	0.332434
$771$	0	0
$772$	7.43196	0.267482
$773$	−13.1031	−0.471287	−0.235643	−	0.971840i	$-0.575720\pi$
$773$	−13.1031	−0.471287	−0.235643	+	0.971840i	$0.575720\pi$
$774$	0	0
$775$	−0.202971	−0.00729095
$776$	−23.8537	−0.856298
$777$	0	0
$778$	−6.82272	−0.244606
$779$	−4.97522	−0.178256
$780$	0	0
$781$	1.58930	0.0568697
$782$	−130.747	−4.67552
$783$	0	0
$784$	−10.8136	−0.386200
$785$	−18.7512	−0.669259
$786$	0	0
$787$	−4.60737	−0.164235	−0.0821176	−	0.996623i	$-0.526168\pi$
$787$	−4.60737	−0.164235	−0.0821176	+	0.996623i	$0.526168\pi$
$788$	−1.80158	−0.0641787
$789$	0	0
$790$	6.26290	0.222824
$791$	11.0633	0.393367
$792$	0	0
$793$	0.390153	0.0138547
$794$	14.8519	0.527073
$795$	0	0
$796$	−52.5901	−1.86401
$797$	−12.6952	−0.449687	−0.224844	−	0.974395i	$-0.572187\pi$
$797$	−12.6952	−0.449687	−0.224844	+	0.974395i	$0.572187\pi$
$798$	0	0
$799$	−27.0134	−0.955666
$800$	−6.77665	−0.239591
$801$	0	0
$802$	−40.3219	−1.42382
$803$	8.41595	0.296992
$804$	0	0
$805$	−34.5901	−1.21914
$806$	0.0500481	0.00176287
$807$	0	0
$808$	−19.9070	−0.700325
$809$	3.22295	0.113313	0.0566564	−	0.998394i	$-0.481956\pi$
$809$	3.22295	0.113313	0.0566564	+	0.998394i	$0.481956\pi$
$810$	0	0
$811$	18.0724	0.634609	0.317304	−	0.948324i	$-0.397222\pi$
$811$	18.0724	0.634609	0.317304	+	0.948324i	$0.397222\pi$
$812$	40.4566	1.41975
$813$	0	0
$814$	−1.52572	−0.0534766
$815$	19.8023	0.693643
$816$	0	0
$817$	−8.52642	−0.298302
$818$	68.3410	2.38949
$819$	0	0
$820$	14.8349	0.518057
$821$	−0.572887	−0.0199939	−0.00999694	−	0.999950i	$-0.503182\pi$
$821$	−0.572887	−0.0199939	−0.00999694	+	0.999950i	$0.503182\pi$
$822$	0	0
$823$	−37.8486	−1.31932	−0.659660	−	0.751564i	$-0.729300\pi$
$823$	−37.8486	−1.31932	−0.659660	+	0.751564i	$0.729300\pi$
$824$	−7.86802	−0.274095
$825$	0	0
$826$	96.7273	3.36557
$827$	10.9219	0.379792	0.189896	−	0.981804i	$-0.439185\pi$
$827$	10.9219	0.379792	0.189896	+	0.981804i	$0.439185\pi$
$828$	0	0
$829$	49.6425	1.72415	0.862077	−	0.506777i	$-0.169163\pi$
$829$	49.6425	1.72415	0.862077	+	0.506777i	$0.169163\pi$
$830$	12.3335	0.428102
$831$	0	0
$832$	1.43397	0.0497139
$833$	70.5612	2.44480
$834$	0	0
$835$	−1.91597	−0.0663050
$836$	−2.98176	−0.103126
$837$	0	0
$838$	85.9060	2.96757
$839$	43.5296	1.50281	0.751404	−	0.659842i	$-0.229377\pi$
$839$	43.5296	1.50281	0.751404	+	0.659842i	$0.229377\pi$
$840$	0	0
$841$	−18.2226	−0.628366
$842$	−74.2864	−2.56008
$843$	0	0
$844$	59.4206	2.04534
$845$	−12.9878	−0.446794
$846$	0	0
$847$	4.13295	0.142010
$848$	−10.8931	−0.374072
$849$	0	0
$850$	15.6222	0.535837
$851$	5.72105	0.196115
$852$	0	0
$853$	24.4635	0.837616	0.418808	−	0.908075i	$-0.362448\pi$
$853$	24.4635	0.837616	0.418808	+	0.908075i	$0.362448\pi$
$854$	−32.5780	−1.11480
$855$	0	0
$856$	15.6252	0.534060
$857$	22.5611	0.770672	0.385336	−	0.922776i	$-0.374086\pi$
$857$	22.5611	0.770672	0.385336	+	0.922776i	$0.374086\pi$
$858$	0	0
$859$	−14.6892	−0.501189	−0.250595	−	0.968092i	$-0.580626\pi$
$859$	−14.6892	−0.501189	−0.250595	+	0.968092i	$0.580626\pi$
$860$	25.4237	0.866941
$861$	0	0
$862$	−10.8149	−0.368357
$863$	41.8413	1.42429	0.712147	−	0.702031i	$-0.247723\pi$
$863$	41.8413	1.42429	0.712147	+	0.702031i	$0.247723\pi$
$864$	0	0
$865$	3.30818	0.112482
$866$	65.2453	2.21713
$867$	0	0
$868$	−2.50131	−0.0848999
$869$	2.80598	0.0951862
$870$	0	0
$871$	0.866189	0.0293497
$872$	−33.1784	−1.12356
$873$	0	0
$874$	18.6802	0.631868
$875$	4.13295	0.139719
$876$	0	0
$877$	−54.9892	−1.85685	−0.928426	−	0.371516i	$-0.878838\pi$
$877$	−54.9892	−1.85685	−0.928426	+	0.371516i	$0.878838\pi$
$878$	−27.2740	−0.920453
$879$	0	0
$880$	−1.07264	−0.0361587
$881$	30.1002	1.01410	0.507050	−	0.861917i	$-0.330736\pi$
$881$	30.1002	1.01410	0.507050	+	0.861917i	$0.330736\pi$
$882$	0	0
$883$	54.9963	1.85077	0.925386	−	0.379026i	$-0.123741\pi$
$883$	54.9963	1.85077	0.925386	+	0.379026i	$0.123741\pi$
$884$	−2.30560	−0.0775458
$885$	0	0
$886$	53.8633	1.80957
$887$	−32.8266	−1.10221	−0.551105	−	0.834436i	$-0.685794\pi$
$887$	−32.8266	−1.10221	−0.551105	+	0.834436i	$0.685794\pi$
$888$	0	0
$889$	−54.0585	−1.81306
$890$	−1.46092	−0.0489700
$891$	0	0
$892$	37.6371	1.26018
$893$	3.85948	0.129153
$894$	0	0
$895$	21.9873	0.734954
$896$	−63.7222	−2.12881
$897$	0	0
$898$	19.5795	0.653377
$899$	−0.666333	−0.0222235
$900$	0	0
$901$	71.0802	2.36803
$902$	11.1046	0.369743
$903$	0	0
$904$	5.86572	0.195091
$905$	−17.0245	−0.565913
$906$	0	0
$907$	14.9631	0.496841	0.248421	−	0.968652i	$-0.420088\pi$
$907$	14.9631	0.496841	0.248421	+	0.968652i	$0.420088\pi$
$908$	−22.5723	−0.749087
$909$	0	0
$910$	−1.01909	−0.0337825
$911$	−26.0121	−0.861818	−0.430909	−	0.902395i	$-0.641807\pi$
$911$	−26.0121	−0.861818	−0.430909	+	0.902395i	$0.641807\pi$
$912$	0	0
$913$	5.52580	0.182877
$914$	−15.3080	−0.506344
$915$	0	0
$916$	−24.4583	−0.808126
$917$	34.4495	1.13762
$918$	0	0
$919$	40.8366	1.34707	0.673537	−	0.739153i	$-0.264774\pi$
$919$	40.8366	1.34707	0.673537	+	0.739153i	$0.264774\pi$
$920$	−18.3394	−0.604633
$921$	0	0
$922$	55.5779	1.83036
$923$	−0.175577	−0.00577919
$924$	0	0
$925$	−0.683573	−0.0224757
$926$	−6.23281	−0.204823
$927$	0	0
$928$	−22.2470	−0.730294
$929$	−20.3088	−0.666309	−0.333155	−	0.942872i	$-0.608113\pi$
$929$	−20.3088	−0.666309	−0.333155	+	0.942872i	$0.608113\pi$
$930$	0	0
$931$	−10.0813	−0.330400
$932$	−50.6611	−1.65946
$933$	0	0
$934$	16.8595	0.551659
$935$	6.99924	0.228899
$936$	0	0
$937$	30.8199	1.00684	0.503422	−	0.864041i	$-0.332074\pi$
$937$	30.8199	1.00684	0.503422	+	0.864041i	$0.332074\pi$
$938$	−72.3274	−2.36157
$939$	0	0
$940$	−11.5080	−0.375351
$941$	18.2874	0.596153	0.298077	−	0.954542i	$-0.403655\pi$
$941$	18.2874	0.596153	0.298077	+	0.954542i	$0.403655\pi$
$942$	0	0
$943$	−41.6393	−1.35596
$944$	−11.2474	−0.366072
$945$	0	0
$946$	19.0308	0.618746
$947$	53.3021	1.73209	0.866043	−	0.499970i	$-0.166656\pi$
$947$	53.3021	1.73209	0.866043	+	0.499970i	$0.166656\pi$
$948$	0	0
$949$	−0.929746	−0.0301808
$950$	−2.23198	−0.0724152
$951$	0	0
$952$	63.3878	2.05441
$953$	−24.4980	−0.793568	−0.396784	−	0.917912i	$-0.629874\pi$
$953$	−24.4980	−0.793568	−0.396784	+	0.917912i	$0.629874\pi$
$954$	0	0
$955$	12.8748	0.416619
$956$	45.4837	1.47105
$957$	0	0
$958$	−63.1564	−2.04049
$959$	54.0604	1.74570
$960$	0	0
$961$	−30.9588	−0.998671
$962$	0.168553	0.00543438
$963$	0	0
$964$	−4.58124	−0.147552
$965$	2.49248	0.0802357
$966$	0	0
$967$	29.9164	0.962047	0.481023	−	0.876708i	$-0.340265\pi$
$967$	29.9164	0.962047	0.481023	+	0.876708i	$0.340265\pi$
$968$	2.19127	0.0704299
$969$	0	0
$970$	−24.2970	−0.780128
$971$	−42.2300	−1.35522	−0.677612	−	0.735419i	$-0.736985\pi$
$971$	−42.2300	−1.35522	−0.677612	+	0.735419i	$0.736985\pi$
$972$	0	0
$973$	−75.5517	−2.42208
$974$	−49.2912	−1.57939
$975$	0	0
$976$	3.78816	0.121256
$977$	−25.2891	−0.809070	−0.404535	−	0.914523i	$-0.632567\pi$
$977$	−25.2891	−0.809070	−0.404535	+	0.914523i	$0.632567\pi$
$978$	0	0
$979$	−0.654537	−0.0209191
$980$	30.0599	0.960229
$981$	0	0
$982$	−76.2046	−2.43179
$983$	9.20537	0.293606	0.146803	−	0.989166i	$-0.453102\pi$
$983$	9.20537	0.293606	0.146803	+	0.989166i	$0.453102\pi$
$984$	0	0
$985$	−0.604202	−0.0192515
$986$	51.2860	1.63328
$987$	0	0
$988$	0.329408	0.0104799
$989$	−71.3605	−2.26913
$990$	0	0
$991$	32.9139	1.04554	0.522772	−	0.852472i	$-0.324898\pi$
$991$	32.9139	1.04554	0.522772	+	0.852472i	$0.324898\pi$
$992$	1.37547	0.0436711
$993$	0	0
$994$	14.6608	0.465013
$995$	−17.6373	−0.559139
$996$	0	0
$997$	−6.45140	−0.204318	−0.102159	−	0.994768i	$-0.532575\pi$
$997$	−6.45140	−0.204318	−0.102159	+	0.994768i	$0.532575\pi$
$998$	−20.8398	−0.659671
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.z.1.5		6
3.2	odd	2		1045.2.a.f.1.2	✓	6
15.14	odd	2		5225.2.a.l.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.2	✓	6	3.2	odd	2
5225.2.a.l.1.5		6	15.14	odd	2
9405.2.a.z.1.5		6	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)

\(f(q)\)	\(=\)	\(q+2.23198 q^{2} +2.98176 q^{4} +1.00000 q^{5} +4.13295 q^{7} +2.19127 q^{8} +O(q^{10})\)	\(q+2.23198 q^{2} +2.98176 q^{4} +1.00000 q^{5} +4.13295 q^{7} +2.19127 q^{8} +2.23198 q^{10} +1.00000 q^{11} -0.110474 q^{13} +9.22468 q^{14} -1.07264 q^{16} +6.99924 q^{17} -1.00000 q^{19} +2.98176 q^{20} +2.23198 q^{22} -8.36934 q^{23} +1.00000 q^{25} -0.246577 q^{26} +12.3235 q^{28} +3.28289 q^{29} -0.202971 q^{31} -6.77665 q^{32} +15.6222 q^{34} +4.13295 q^{35} -0.683573 q^{37} -2.23198 q^{38} +2.19127 q^{40} +4.97522 q^{41} +8.52642 q^{43} +2.98176 q^{44} -18.6802 q^{46} -3.85948 q^{47} +10.0813 q^{49} +2.23198 q^{50} -0.329408 q^{52} +10.1554 q^{53} +1.00000 q^{55} +9.05639 q^{56} +7.32737 q^{58} +10.4857 q^{59} -3.53161 q^{61} -0.453029 q^{62} -12.9801 q^{64} -0.110474 q^{65} -7.84063 q^{67} +20.8700 q^{68} +9.22468 q^{70} +1.58930 q^{71} +8.41595 q^{73} -1.52572 q^{74} -2.98176 q^{76} +4.13295 q^{77} +2.80598 q^{79} -1.07264 q^{80} +11.1046 q^{82} +5.52580 q^{83} +6.99924 q^{85} +19.0308 q^{86} +2.19127 q^{88} -0.654537 q^{89} -0.456585 q^{91} -24.9553 q^{92} -8.61430 q^{94} -1.00000 q^{95} -10.8858 q^{97} +22.5013 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + q^98

Introduction

L-functions

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