LMFDB - Embedded newform 9405.2.a.y.1.4

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.20413244.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 11x^{3} + 7x^{2} - 11x - 2 $ x^6 - 2x^5 - 6x^4 + 11x^3 + 7x^2 - 11*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3135)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.31008$ 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)

$f(q)$	$=$	$q+1.31008 q^{2} -0.283703 q^{4} +1.00000 q^{5} -0.0459382 q^{7} -2.99182 q^{8} +O(q^{10})$	\(q+1.31008 q^{2} -0.283703 q^{4} +1.00000 q^{5} -0.0459382 q^{7} -2.99182 q^{8} +1.31008 q^{10} -1.00000 q^{11} -0.670359 q^{13} -0.0601825 q^{14} -3.35211 q^{16} +7.26175 q^{17} -1.00000 q^{19} -0.283703 q^{20} -1.31008 q^{22} -8.99941 q^{23} +1.00000 q^{25} -0.878220 q^{26} +0.0130328 q^{28} +5.76592 q^{29} +3.47642 q^{31} +1.59213 q^{32} +9.51344 q^{34} -0.0459382 q^{35} +0.401213 q^{37} -1.31008 q^{38} -2.99182 q^{40} -7.75821 q^{41} -7.81186 q^{43} +0.283703 q^{44} -11.7899 q^{46} +3.18756 q^{47} -6.99789 q^{49} +1.31008 q^{50} +0.190183 q^{52} +3.31513 q^{53} -1.00000 q^{55} +0.137439 q^{56} +7.55379 q^{58} -5.14894 q^{59} -4.01303 q^{61} +4.55437 q^{62} +8.79003 q^{64} -0.670359 q^{65} -1.48182 q^{67} -2.06018 q^{68} -0.0601825 q^{70} +12.6022 q^{71} +3.55938 q^{73} +0.525619 q^{74} +0.283703 q^{76} +0.0459382 q^{77} -13.4225 q^{79} -3.35211 q^{80} -10.1638 q^{82} -13.0039 q^{83} +7.26175 q^{85} -10.2341 q^{86} +2.99182 q^{88} -1.94409 q^{89} +0.0307951 q^{91} +2.55316 q^{92} +4.17594 q^{94} -1.00000 q^{95} +6.31493 q^{97} -9.16776 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 - 7 * q^7 + 3 * q^8	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} - 7 q^{7} + 3 q^{8} + 2 q^{10} - 6 q^{11} - 3 q^{13} - 4 q^{16} + 15 q^{17} - 6 q^{19} + 4 q^{20} - 2 q^{22} - 6 q^{23} + 6 q^{25} - 9 q^{26} - 19 q^{28} + 3 q^{29} - 10 q^{31} + 3 q^{32} - 9 q^{34} - 7 q^{35} - 11 q^{37} - 2 q^{38} + 3 q^{40} + 15 q^{41} - 22 q^{43} - 4 q^{44} - 23 q^{46} - 4 q^{47} + 7 q^{49} + 2 q^{50} - 13 q^{52} - 2 q^{53} - 6 q^{55} - 13 q^{56} + 5 q^{58} - 7 q^{59} - 5 q^{61} + 35 q^{62} - q^{64} - 3 q^{65} - 17 q^{67} - 12 q^{68} - 14 q^{71} - 38 q^{73} - 5 q^{74} - 4 q^{76} + 7 q^{77} - 15 q^{79} - 4 q^{80} - 26 q^{82} + 24 q^{83} + 15 q^{85} - 9 q^{86} - 3 q^{88} - 16 q^{89} - 23 q^{91} + 11 q^{92} + 18 q^{94} - 6 q^{95} - 8 q^{97} - 27 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 - 7 * q^7 + 3 * q^8 + 2 * q^10 - 6 * q^11 - 3 * q^13 - 4 * q^16 + 15 * q^17 - 6 * q^19 + 4 * q^20 - 2 * q^22 - 6 * q^23 + 6 * q^25 - 9 * q^26 - 19 * q^28 + 3 * q^29 - 10 * q^31 + 3 * q^32 - 9 * q^34 - 7 * q^35 - 11 * q^37 - 2 * q^38 + 3 * q^40 + 15 * q^41 - 22 * q^43 - 4 * q^44 - 23 * q^46 - 4 * q^47 + 7 * q^49 + 2 * q^50 - 13 * q^52 - 2 * q^53 - 6 * q^55 - 13 * q^56 + 5 * q^58 - 7 * q^59 - 5 * q^61 + 35 * q^62 - q^64 - 3 * q^65 - 17 * q^67 - 12 * q^68 - 14 * q^71 - 38 * q^73 - 5 * q^74 - 4 * q^76 + 7 * q^77 - 15 * q^79 - 4 * q^80 - 26 * q^82 + 24 * q^83 + 15 * q^85 - 9 * q^86 - 3 * q^88 - 16 * q^89 - 23 * q^91 + 11 * q^92 + 18 * q^94 - 6 * q^95 - 8 * q^97 - 27 * 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 $

$2$	1.31008	0.926363	0.463182	−	0.886263i	$-0.346708\pi$
$2$	1.31008	0.926363	0.463182	+	0.886263i	$0.346708\pi$
$3$	0	0
$3$	0	0
$4$	−0.283703	−0.141852
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.0459382	−0.0173630	−0.00868150	−	0.999962i	$-0.502763\pi$
$7$	−0.0459382	−0.0173630	−0.00868150	+	0.999962i	$0.502763\pi$
$8$	−2.99182	−1.05777
$9$	0	0
$10$	1.31008	0.414282
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.670359	−0.185924	−0.0929620	−	0.995670i	$-0.529634\pi$
$13$	−0.670359	−0.185924	−0.0929620	+	0.995670i	$0.529634\pi$
$14$	−0.0601825	−0.0160844
$15$	0	0
$16$	−3.35211	−0.838027
$17$	7.26175	1.76123	0.880617	−	0.473829i	$-0.157128\pi$
$17$	7.26175	1.76123	0.880617	+	0.473829i	$0.157128\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.283703	−0.0634380
$21$	0	0
$22$	−1.31008	−0.279309
$23$	−8.99941	−1.87651	−0.938254	−	0.345948i	$-0.887557\pi$
$23$	−8.99941	−1.87651	−0.938254	+	0.345948i	$0.887557\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.878220	−0.172233
$27$	0	0
$28$	0.0130328	0.00246297
$29$	5.76592	1.07070	0.535352	−	0.844629i	$-0.320179\pi$
$29$	5.76592	1.07070	0.535352	+	0.844629i	$0.320179\pi$
$30$	0	0
$31$	3.47642	0.624384	0.312192	−	0.950019i	$-0.398937\pi$
$31$	3.47642	0.624384	0.312192	+	0.950019i	$0.398937\pi$
$32$	1.59213	0.281452
$33$	0	0
$34$	9.51344	1.63154
$35$	−0.0459382	−0.00776497
$36$	0	0
$37$	0.401213	0.0659590	0.0329795	−	0.999456i	$-0.489500\pi$
$37$	0.401213	0.0659590	0.0329795	+	0.999456i	$0.489500\pi$
$38$	−1.31008	−0.212522
$39$	0	0
$40$	−2.99182	−0.473049
$41$	−7.75821	−1.21163	−0.605815	−	0.795606i	$-0.707153\pi$
$41$	−7.75821	−1.21163	−0.605815	+	0.795606i	$0.707153\pi$
$42$	0	0
$43$	−7.81186	−1.19130	−0.595649	−	0.803245i	$-0.703105\pi$
$43$	−7.81186	−1.19130	−0.595649	+	0.803245i	$0.703105\pi$
$44$	0.283703	0.0427699
$45$	0	0
$46$	−11.7899	−1.73833
$47$	3.18756	0.464953	0.232476	−	0.972602i	$-0.425317\pi$
$47$	3.18756	0.464953	0.232476	+	0.972602i	$0.425317\pi$
$48$	0	0
$49$	−6.99789	−0.999699
$50$	1.31008	0.185273
$51$	0	0
$52$	0.190183	0.0263736
$53$	3.31513	0.455368	0.227684	−	0.973735i	$-0.426885\pi$
$53$	3.31513	0.455368	0.227684	+	0.973735i	$0.426885\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0.137439	0.0183660
$57$	0	0
$58$	7.55379	0.991861
$59$	−5.14894	−0.670335	−0.335167	−	0.942159i	$-0.608793\pi$
$59$	−5.14894	−0.670335	−0.335167	+	0.942159i	$0.608793\pi$
$60$	0	0
$61$	−4.01303	−0.513816	−0.256908	−	0.966436i	$-0.582704\pi$
$61$	−4.01303	−0.513816	−0.256908	+	0.966436i	$0.582704\pi$
$62$	4.55437	0.578406
$63$	0	0
$64$	8.79003	1.09875
$65$	−0.670359	−0.0831478
$66$	0	0
$67$	−1.48182	−0.181033	−0.0905165	−	0.995895i	$-0.528852\pi$
$67$	−1.48182	−0.181033	−0.0905165	+	0.995895i	$0.528852\pi$
$68$	−2.06018	−0.249834
$69$	0	0
$70$	−0.0601825	−0.00719318
$71$	12.6022	1.49561	0.747803	−	0.663921i	$-0.231109\pi$
$71$	12.6022	1.49561	0.747803	+	0.663921i	$0.231109\pi$
$72$	0	0
$73$	3.55938	0.416594	0.208297	−	0.978066i	$-0.433208\pi$
$73$	3.55938	0.416594	0.208297	+	0.978066i	$0.433208\pi$
$74$	0.525619	0.0611019
$75$	0	0
$76$	0.283703	0.0325430
$77$	0.0459382	0.00523514
$78$	0	0
$79$	−13.4225	−1.51015	−0.755077	−	0.655636i	$-0.772401\pi$
$79$	−13.4225	−1.51015	−0.755077	+	0.655636i	$0.772401\pi$
$80$	−3.35211	−0.374777
$81$	0	0
$82$	−10.1638	−1.12241
$83$	−13.0039	−1.42736	−0.713681	−	0.700471i	$-0.752973\pi$
$83$	−13.0039	−1.42736	−0.713681	+	0.700471i	$0.752973\pi$
$84$	0	0
$85$	7.26175	0.787648
$86$	−10.2341	−1.10357
$87$	0	0
$88$	2.99182	0.318929
$89$	−1.94409	−0.206073	−0.103036	−	0.994678i	$-0.532856\pi$
$89$	−1.94409	−0.206073	−0.103036	+	0.994678i	$0.532856\pi$
$90$	0	0
$91$	0.0307951	0.00322820
$92$	2.55316	0.266186
$93$	0	0
$94$	4.17594	0.430715
$95$	−1.00000	−0.102598
$96$	0	0
$97$	6.31493	0.641184	0.320592	−	0.947217i	$-0.396118\pi$
$97$	6.31493	0.641184	0.320592	+	0.947217i	$0.396118\pi$
$98$	−9.16776	−0.926084
$99$	0	0
$100$	−0.283703	−0.0283703
$101$	−9.90601	−0.985685	−0.492843	−	0.870118i	$-0.664042\pi$
$101$	−9.90601	−0.985685	−0.492843	+	0.870118i	$0.664042\pi$
$102$	0	0
$103$	−9.43048	−0.929213	−0.464607	−	0.885517i	$-0.653804\pi$
$103$	−9.43048	−0.929213	−0.464607	+	0.885517i	$0.653804\pi$
$104$	2.00559	0.196665
$105$	0	0
$106$	4.34307	0.421836
$107$	15.7919	1.52667	0.763333	−	0.646006i	$-0.223562\pi$
$107$	15.7919	1.52667	0.763333	+	0.646006i	$0.223562\pi$
$108$	0	0
$109$	19.4517	1.86314	0.931570	−	0.363563i	$-0.118440\pi$
$109$	19.4517	1.86314	0.931570	+	0.363563i	$0.118440\pi$
$110$	−1.31008	−0.124911
$111$	0	0
$112$	0.153990	0.0145507
$113$	−7.24273	−0.681338	−0.340669	−	0.940183i	$-0.610654\pi$
$113$	−7.24273	−0.681338	−0.340669	+	0.940183i	$0.610654\pi$
$114$	0	0
$115$	−8.99941	−0.839200
$116$	−1.63581	−0.151881
$117$	0	0
$118$	−6.74549	−0.620973
$119$	−0.333592	−0.0305803
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.25737	−0.475980
$123$	0	0
$124$	−0.986272	−0.0885698
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.0073	−0.976744	−0.488372	−	0.872635i	$-0.662409\pi$
$127$	−11.0073	−0.976744	−0.488372	+	0.872635i	$0.662409\pi$
$128$	8.33133	0.736392
$129$	0	0
$130$	−0.878220	−0.0770250
$131$	2.34467	0.204855	0.102427	−	0.994740i	$-0.467339\pi$
$131$	2.34467	0.204855	0.102427	+	0.994740i	$0.467339\pi$
$132$	0	0
$133$	0.0459382	0.00398334
$134$	−1.94130	−0.167702
$135$	0	0
$136$	−21.7259	−1.86298
$137$	−9.57007	−0.817626	−0.408813	−	0.912618i	$-0.634057\pi$
$137$	−9.57007	−0.817626	−0.408813	+	0.912618i	$0.634057\pi$
$138$	0	0
$139$	−20.3810	−1.72870	−0.864349	−	0.502893i	$-0.832269\pi$
$139$	−20.3810	−1.72870	−0.864349	+	0.502893i	$0.832269\pi$
$140$	0.0130328	0.00110147
$141$	0	0
$142$	16.5098	1.38547
$143$	0.670359	0.0560582
$144$	0	0
$145$	5.76592	0.478834
$146$	4.66306	0.385917
$147$	0	0
$148$	−0.113825	−0.00935638
$149$	15.8754	1.30056	0.650282	−	0.759693i	$-0.274651\pi$
$149$	15.8754	1.30056	0.650282	+	0.759693i	$0.274651\pi$
$150$	0	0
$151$	−1.61303	−0.131267	−0.0656333	−	0.997844i	$-0.520907\pi$
$151$	−1.61303	−0.131267	−0.0656333	+	0.997844i	$0.520907\pi$
$152$	2.99182	0.242669
$153$	0	0
$154$	0.0601825	0.00484964
$155$	3.47642	0.279233
$156$	0	0
$157$	−21.1965	−1.69167	−0.845833	−	0.533448i	$-0.820896\pi$
$157$	−21.1965	−1.69167	−0.845833	+	0.533448i	$0.820896\pi$
$158$	−17.5845	−1.39895
$159$	0	0
$160$	1.59213	0.125869
$161$	0.413417	0.0325818
$162$	0	0
$163$	−15.4960	−1.21374	−0.606871	−	0.794800i	$-0.707576\pi$
$163$	−15.4960	−1.21374	−0.606871	+	0.794800i	$0.707576\pi$
$164$	2.20103	0.171872
$165$	0	0
$166$	−17.0361	−1.32225
$167$	11.8791	0.919231	0.459615	−	0.888118i	$-0.347987\pi$
$167$	11.8791	0.919231	0.459615	+	0.888118i	$0.347987\pi$
$168$	0	0
$169$	−12.5506	−0.965432
$170$	9.51344	0.729648
$171$	0	0
$172$	2.21625	0.168987
$173$	−6.40058	−0.486627	−0.243314	−	0.969948i	$-0.578234\pi$
$173$	−6.40058	−0.486627	−0.243314	+	0.969948i	$0.578234\pi$
$174$	0	0
$175$	−0.0459382	−0.00347260
$176$	3.35211	0.252675
$177$	0	0
$178$	−2.54690	−0.190898
$179$	1.92995	0.144251	0.0721257	−	0.997396i	$-0.477022\pi$
$179$	1.92995	0.144251	0.0721257	+	0.997396i	$0.477022\pi$
$180$	0	0
$181$	−1.53419	−0.114035	−0.0570177	−	0.998373i	$-0.518159\pi$
$181$	−1.53419	−0.114035	−0.0570177	+	0.998373i	$0.518159\pi$
$182$	0.0403438	0.00299048
$183$	0	0
$184$	26.9247	1.98491
$185$	0.401213	0.0294977
$186$	0	0
$187$	−7.26175	−0.531032
$188$	−0.904320	−0.0659543
$189$	0	0
$190$	−1.31008	−0.0950428
$191$	−24.5650	−1.77746	−0.888730	−	0.458431i	$-0.848412\pi$
$191$	−24.5650	−1.77746	−0.888730	+	0.458431i	$0.848412\pi$
$192$	0	0
$193$	−21.8972	−1.57620	−0.788099	−	0.615548i	$-0.788935\pi$
$193$	−21.8972	−1.57620	−0.788099	+	0.615548i	$0.788935\pi$
$194$	8.27303	0.593969
$195$	0	0
$196$	1.98532	0.141809
$197$	−12.1614	−0.866463	−0.433231	−	0.901283i	$-0.642627\pi$
$197$	−12.1614	−0.866463	−0.433231	+	0.901283i	$0.642627\pi$
$198$	0	0
$199$	−23.4031	−1.65900	−0.829501	−	0.558506i	$-0.811375\pi$
$199$	−23.4031	−1.65900	−0.829501	+	0.558506i	$0.811375\pi$
$200$	−2.99182	−0.211554
$201$	0	0
$202$	−12.9776	−0.913102
$203$	−0.264876	−0.0185906
$204$	0	0
$205$	−7.75821	−0.541857
$206$	−12.3546	−0.860789
$207$	0	0
$208$	2.24711	0.155809
$209$	1.00000	0.0691714
$210$	0	0
$211$	−3.86320	−0.265954	−0.132977	−	0.991119i	$-0.542454\pi$
$211$	−3.86320	−0.265954	−0.132977	+	0.991119i	$0.542454\pi$
$212$	−0.940513	−0.0645947
$213$	0	0
$214$	20.6886	1.41425
$215$	−7.81186	−0.532764
$216$	0	0
$217$	−0.159700	−0.0108412
$218$	25.4833	1.72594
$219$	0	0
$220$	0.283703	0.0191273
$221$	−4.86798	−0.327456
$222$	0	0
$223$	−13.6672	−0.915223	−0.457612	−	0.889152i	$-0.651295\pi$
$223$	−13.6672	−0.915223	−0.457612	+	0.889152i	$0.651295\pi$
$224$	−0.0731397	−0.00488685
$225$	0	0
$226$	−9.48852	−0.631167
$227$	−6.58470	−0.437042	−0.218521	−	0.975832i	$-0.570123\pi$
$227$	−6.58470	−0.437042	−0.218521	+	0.975832i	$0.570123\pi$
$228$	0	0
$229$	12.5156	0.827051	0.413526	−	0.910492i	$-0.364297\pi$
$229$	12.5156	0.827051	0.413526	+	0.910492i	$0.364297\pi$
$230$	−11.7899	−0.777404
$231$	0	0
$232$	−17.2506	−1.13256
$233$	26.0657	1.70762	0.853811	−	0.520584i	$-0.174286\pi$
$233$	26.0657	1.70762	0.853811	+	0.520584i	$0.174286\pi$
$234$	0	0
$235$	3.18756	0.207933
$236$	1.46077	0.0950880
$237$	0	0
$238$	−0.437030	−0.0283285
$239$	−18.4855	−1.19573	−0.597863	−	0.801598i	$-0.703983\pi$
$239$	−18.4855	−1.19573	−0.597863	+	0.801598i	$0.703983\pi$
$240$	0	0
$241$	−27.7704	−1.78885	−0.894425	−	0.447218i	$-0.852415\pi$
$241$	−27.7704	−1.78885	−0.894425	+	0.447218i	$0.852415\pi$
$242$	1.31008	0.0842148
$243$	0	0
$244$	1.13851	0.0728856
$245$	−6.99789	−0.447079
$246$	0	0
$247$	0.670359	0.0426539
$248$	−10.4008	−0.660454
$249$	0	0
$250$	1.31008	0.0828564
$251$	20.7247	1.30813	0.654065	−	0.756438i	$-0.273062\pi$
$251$	20.7247	1.30813	0.654065	+	0.756438i	$0.273062\pi$
$252$	0	0
$253$	8.99941	0.565788
$254$	−14.4205	−0.904820
$255$	0	0
$256$	−6.66539	−0.416587
$257$	29.2153	1.82240	0.911201	−	0.411961i	$-0.135156\pi$
$257$	29.2153	1.82240	0.911201	+	0.411961i	$0.135156\pi$
$258$	0	0
$259$	−0.0184310	−0.00114525
$260$	0.190183	0.0117946
$261$	0	0
$262$	3.07169	0.189770
$263$	4.20016	0.258993	0.129496	−	0.991580i	$-0.458664\pi$
$263$	4.20016	0.258993	0.129496	+	0.991580i	$0.458664\pi$
$264$	0	0
$265$	3.31513	0.203647
$266$	0.0601825	0.00369002
$267$	0	0
$268$	0.420397	0.0256798
$269$	−1.37742	−0.0839830	−0.0419915	−	0.999118i	$-0.513370\pi$
$269$	−1.37742	−0.0839830	−0.0419915	+	0.999118i	$0.513370\pi$
$270$	0	0
$271$	2.44349	0.148431	0.0742156	−	0.997242i	$-0.476355\pi$
$271$	2.44349	0.148431	0.0742156	+	0.997242i	$0.476355\pi$
$272$	−24.3422	−1.47596
$273$	0	0
$274$	−12.5375	−0.757419
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−12.7285	−0.764784	−0.382392	−	0.924000i	$-0.624900\pi$
$277$	−12.7285	−0.764784	−0.382392	+	0.924000i	$0.624900\pi$
$278$	−26.7007	−1.60140
$279$	0	0
$280$	0.137439	0.00821354
$281$	−4.08159	−0.243487	−0.121743	−	0.992562i	$-0.538849\pi$
$281$	−4.08159	−0.243487	−0.121743	+	0.992562i	$0.538849\pi$
$282$	0	0
$283$	5.84982	0.347736	0.173868	−	0.984769i	$-0.444373\pi$
$283$	5.84982	0.347736	0.173868	+	0.984769i	$0.444373\pi$
$284$	−3.57528	−0.212154
$285$	0	0
$286$	0.878220	0.0519303
$287$	0.356398	0.0210375
$288$	0	0
$289$	35.7331	2.10195
$290$	7.55379	0.443574
$291$	0	0
$292$	−1.00981	−0.0590945
$293$	2.77452	0.162089	0.0810445	−	0.996710i	$-0.474174\pi$
$293$	2.77452	0.162089	0.0810445	+	0.996710i	$0.474174\pi$
$294$	0	0
$295$	−5.14894	−0.299783
$296$	−1.20036	−0.0697693
$297$	0	0
$298$	20.7980	1.20479
$299$	6.03284	0.348888
$300$	0	0
$301$	0.358862	0.0206845
$302$	−2.11319	−0.121601
$303$	0	0
$304$	3.35211	0.192256
$305$	−4.01303	−0.229786
$306$	0	0
$307$	−12.6532	−0.722154	−0.361077	−	0.932536i	$-0.617591\pi$
$307$	−12.6532	−0.722154	−0.361077	+	0.932536i	$0.617591\pi$
$308$	−0.0130328	−0.000742613 0
$309$	0	0
$310$	4.55437	0.258671
$311$	15.3789	0.872056	0.436028	−	0.899933i	$-0.356385\pi$
$311$	15.3789	0.872056	0.436028	+	0.899933i	$0.356385\pi$
$312$	0	0
$313$	13.9974	0.791180	0.395590	−	0.918427i	$-0.370540\pi$
$313$	13.9974	0.791180	0.395590	+	0.918427i	$0.370540\pi$
$314$	−27.7690	−1.56710
$315$	0	0
$316$	3.80802	0.214218
$317$	−13.2141	−0.742181	−0.371090	−	0.928597i	$-0.621016\pi$
$317$	−13.2141	−0.742181	−0.371090	+	0.928597i	$0.621016\pi$
$318$	0	0
$319$	−5.76592	−0.322830
$320$	8.79003	0.491378
$321$	0	0
$322$	0.541607	0.0301826
$323$	−7.26175	−0.404055
$324$	0	0
$325$	−0.670359	−0.0371848
$326$	−20.3010	−1.12437
$327$	0	0
$328$	23.2112	1.28162
$329$	−0.146431	−0.00807298
$330$	0	0
$331$	17.2209	0.946545	0.473273	−	0.880916i	$-0.343073\pi$
$331$	17.2209	0.946545	0.473273	+	0.880916i	$0.343073\pi$
$332$	3.68924	0.202473
$333$	0	0
$334$	15.5625	0.851541
$335$	−1.48182	−0.0809605
$336$	0	0
$337$	−5.51617	−0.300485	−0.150242	−	0.988649i	$-0.548005\pi$
$337$	−5.51617	−0.300485	−0.150242	+	0.988649i	$0.548005\pi$
$338$	−16.4423	−0.894341
$339$	0	0
$340$	−2.06018	−0.111729
$341$	−3.47642	−0.188259
$342$	0	0
$343$	0.643037	0.0347208
$344$	23.3717	1.26012
$345$	0	0
$346$	−8.38524	−0.450793
$347$	14.0377	0.753584	0.376792	−	0.926298i	$-0.377027\pi$
$347$	14.0377	0.753584	0.376792	+	0.926298i	$0.377027\pi$
$348$	0	0
$349$	1.11525	0.0596978	0.0298489	−	0.999554i	$-0.490497\pi$
$349$	1.11525	0.0596978	0.0298489	+	0.999554i	$0.490497\pi$
$350$	−0.0601825	−0.00321689
$351$	0	0
$352$	−1.59213	−0.0848610
$353$	28.6187	1.52322	0.761609	−	0.648037i	$-0.224410\pi$
$353$	28.6187	1.52322	0.761609	+	0.648037i	$0.224410\pi$
$354$	0	0
$355$	12.6022	0.668855
$356$	0.551544	0.0292318
$357$	0	0
$358$	2.52838	0.133629
$359$	−11.3401	−0.598505	−0.299253	−	0.954174i	$-0.596737\pi$
$359$	−11.3401	−0.598505	−0.299253	+	0.954174i	$0.596737\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−2.00990	−0.105638
$363$	0	0
$364$	−0.00873665	−0.000457925 0
$365$	3.55938	0.186307
$366$	0	0
$367$	14.6528	0.764872	0.382436	−	0.923982i	$-0.375085\pi$
$367$	14.6528	0.764872	0.382436	+	0.923982i	$0.375085\pi$
$368$	30.1670	1.57256
$369$	0	0
$370$	0.525619	0.0273256
$371$	−0.152291	−0.00790656
$372$	0	0
$373$	−28.2032	−1.46031	−0.730154	−	0.683282i	$-0.760552\pi$
$373$	−28.2032	−1.46031	−0.730154	+	0.683282i	$0.760552\pi$
$374$	−9.51344	−0.491928
$375$	0	0
$376$	−9.53660	−0.491813
$377$	−3.86523	−0.199070
$378$	0	0
$379$	−13.7448	−0.706023	−0.353011	−	0.935619i	$-0.614842\pi$
$379$	−13.7448	−0.706023	−0.353011	+	0.935619i	$0.614842\pi$
$380$	0.283703	0.0145537
$381$	0	0
$382$	−32.1820	−1.64657
$383$	19.9867	1.02127	0.510637	−	0.859796i	$-0.329409\pi$
$383$	19.9867	1.02127	0.510637	+	0.859796i	$0.329409\pi$
$384$	0	0
$385$	0.0459382	0.00234123
$386$	−28.6870	−1.46013
$387$	0	0
$388$	−1.79157	−0.0909530
$389$	−19.4723	−0.987284	−0.493642	−	0.869665i	$-0.664335\pi$
$389$	−19.4723	−0.987284	−0.493642	+	0.869665i	$0.664335\pi$
$390$	0	0
$391$	−65.3515	−3.30497
$392$	20.9364	1.05745
$393$	0	0
$394$	−15.9323	−0.802659
$395$	−13.4225	−0.675362
$396$	0	0
$397$	−18.9970	−0.953431	−0.476716	−	0.879058i	$-0.658173\pi$
$397$	−18.9970	−0.953431	−0.476716	+	0.879058i	$0.658173\pi$
$398$	−30.6598	−1.53684
$399$	0	0
$400$	−3.35211	−0.167605
$401$	5.48110	0.273713	0.136856	−	0.990591i	$-0.456300\pi$
$401$	5.48110	0.273713	0.136856	+	0.990591i	$0.456300\pi$
$402$	0	0
$403$	−2.33045	−0.116088
$404$	2.81037	0.139821
$405$	0	0
$406$	−0.347007	−0.0172217
$407$	−0.401213	−0.0198874
$408$	0	0
$409$	15.1666	0.749938	0.374969	−	0.927037i	$-0.377653\pi$
$409$	15.1666	0.749938	0.374969	+	0.927037i	$0.377653\pi$
$410$	−10.1638	−0.501956
$411$	0	0
$412$	2.67546	0.131810
$413$	0.236533	0.0116390
$414$	0	0
$415$	−13.0039	−0.638335
$416$	−1.06730	−0.0523287
$417$	0	0
$418$	1.31008	0.0640779
$419$	−10.3184	−0.504089	−0.252045	−	0.967716i	$-0.581103\pi$
$419$	−10.3184	−0.504089	−0.252045	+	0.967716i	$0.581103\pi$
$420$	0	0
$421$	16.3990	0.799240	0.399620	−	0.916681i	$-0.369142\pi$
$421$	16.3990	0.799240	0.399620	+	0.916681i	$0.369142\pi$
$422$	−5.06109	−0.246370
$423$	0	0
$424$	−9.91828	−0.481674
$425$	7.26175	0.352247
$426$	0	0
$427$	0.184351	0.00892139
$428$	−4.48023	−0.216560
$429$	0	0
$430$	−10.2341	−0.493533
$431$	15.4812	0.745702	0.372851	−	0.927891i	$-0.378380\pi$
$431$	15.4812	0.745702	0.372851	+	0.927891i	$0.378380\pi$
$432$	0	0
$433$	−4.34940	−0.209019	−0.104509	−	0.994524i	$-0.533327\pi$
$433$	−4.34940	−0.209019	−0.104509	+	0.994524i	$0.533327\pi$
$434$	−0.209220	−0.0100429
$435$	0	0
$436$	−5.51852	−0.264289
$437$	8.99941	0.430500
$438$	0	0
$439$	−14.2170	−0.678541	−0.339270	−	0.940689i	$-0.610180\pi$
$439$	−14.2170	−0.678541	−0.339270	+	0.940689i	$0.610180\pi$
$440$	2.99182	0.142630
$441$	0	0
$442$	−6.37742	−0.303343
$443$	−27.0252	−1.28400	−0.642002	−	0.766703i	$-0.721896\pi$
$443$	−27.0252	−1.28400	−0.642002	+	0.766703i	$0.721896\pi$
$444$	0	0
$445$	−1.94409	−0.0921586
$446$	−17.9051	−0.847829
$447$	0	0
$448$	−0.403798	−0.0190777
$449$	−29.5700	−1.39550	−0.697748	−	0.716344i	$-0.745814\pi$
$449$	−29.5700	−1.39550	−0.697748	+	0.716344i	$0.745814\pi$
$450$	0	0
$451$	7.75821	0.365320
$452$	2.05478	0.0966489
$453$	0	0
$454$	−8.62645	−0.404859
$455$	0.0307951	0.00144369
$456$	0	0
$457$	−40.0064	−1.87142	−0.935709	−	0.352772i	$-0.885239\pi$
$457$	−40.0064	−1.87142	−0.935709	+	0.352772i	$0.885239\pi$
$458$	16.3963	0.766150
$459$	0	0
$460$	2.55316	0.119042
$461$	8.87332	0.413272	0.206636	−	0.978418i	$-0.433748\pi$
$461$	8.87332	0.413272	0.206636	+	0.978418i	$0.433748\pi$
$462$	0	0
$463$	−14.3604	−0.667383	−0.333692	−	0.942682i	$-0.608294\pi$
$463$	−14.3604	−0.667383	−0.333692	+	0.942682i	$0.608294\pi$
$464$	−19.3280	−0.897279
$465$	0	0
$466$	34.1480	1.58188
$467$	−18.2237	−0.843290	−0.421645	−	0.906761i	$-0.638547\pi$
$467$	−18.2237	−0.843290	−0.421645	+	0.906761i	$0.638547\pi$
$468$	0	0
$469$	0.0680721	0.00314328
$470$	4.17594	0.192622
$471$	0	0
$472$	15.4047	0.709059
$473$	7.81186	0.359190
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0.0946410	0.00433786
$477$	0	0
$478$	−24.2174	−1.10768
$479$	−29.5781	−1.35146	−0.675729	−	0.737150i	$-0.736171\pi$
$479$	−29.5781	−1.35146	−0.675729	+	0.737150i	$0.736171\pi$
$480$	0	0
$481$	−0.268956	−0.0122634
$482$	−36.3813	−1.65712
$483$	0	0
$484$	−0.283703	−0.0128956
$485$	6.31493	0.286746
$486$	0	0
$487$	−41.1765	−1.86588	−0.932942	−	0.360026i	$-0.882768\pi$
$487$	−41.1765	−1.86588	−0.932942	+	0.360026i	$0.882768\pi$
$488$	12.0063	0.543499
$489$	0	0
$490$	−9.16776	−0.414157
$491$	−29.3336	−1.32381	−0.661904	−	0.749588i	$-0.730251\pi$
$491$	−29.3336	−1.32381	−0.661904	+	0.749588i	$0.730251\pi$
$492$	0	0
$493$	41.8707	1.88576
$494$	0.878220	0.0395130
$495$	0	0
$496$	−11.6533	−0.523250
$497$	−0.578922	−0.0259682
$498$	0	0
$499$	19.6484	0.879584	0.439792	−	0.898100i	$-0.355052\pi$
$499$	19.6484	0.879584	0.439792	+	0.898100i	$0.355052\pi$
$500$	−0.283703	−0.0126876
$501$	0	0
$502$	27.1509	1.21180
$503$	−14.7316	−0.656850	−0.328425	−	0.944530i	$-0.606518\pi$
$503$	−14.7316	−0.656850	−0.328425	+	0.944530i	$0.606518\pi$
$504$	0	0
$505$	−9.90601	−0.440812
$506$	11.7899	0.524125
$507$	0	0
$508$	3.12282	0.138553
$509$	15.6584	0.694044	0.347022	−	0.937857i	$-0.387193\pi$
$509$	15.6584	0.694044	0.347022	+	0.937857i	$0.387193\pi$
$510$	0	0
$511$	−0.163511	−0.00723332
$512$	−25.3948	−1.12230
$513$	0	0
$514$	38.2743	1.68821
$515$	−9.43048	−0.415557
$516$	0	0
$517$	−3.18756	−0.140189
$518$	−0.0241460	−0.00106091
$519$	0	0
$520$	2.00559	0.0879511
$521$	−11.5177	−0.504598	−0.252299	−	0.967649i	$-0.581187\pi$
$521$	−11.5177	−0.504598	−0.252299	+	0.967649i	$0.581187\pi$
$522$	0	0
$523$	30.8368	1.34840	0.674199	−	0.738550i	$-0.264489\pi$
$523$	30.8368	1.34840	0.674199	+	0.738550i	$0.264489\pi$
$524$	−0.665190	−0.0290589
$525$	0	0
$526$	5.50252	0.239921
$527$	25.2449	1.09969
$528$	0	0
$529$	57.9895	2.52128
$530$	4.34307	0.188651
$531$	0	0
$532$	−0.0130328	−0.000565044 0
$533$	5.20079	0.225271
$534$	0	0
$535$	15.7919	0.682745
$536$	4.43334	0.191491
$537$	0	0
$538$	−1.80453	−0.0777987
$539$	6.99789	0.301420
$540$	0	0
$541$	−12.7813	−0.549510	−0.274755	−	0.961514i	$-0.588597\pi$
$541$	−12.7813	−0.549510	−0.274755	+	0.961514i	$0.588597\pi$
$542$	3.20115	0.137501
$543$	0	0
$544$	11.5617	0.495703
$545$	19.4517	0.833221
$546$	0	0
$547$	45.8536	1.96056	0.980279	−	0.197621i	$-0.0633214\pi$
$547$	45.8536	1.96056	0.980279	+	0.197621i	$0.0633214\pi$
$548$	2.71506	0.115982
$549$	0	0
$550$	−1.31008	−0.0558618
$551$	−5.76592	−0.245636
$552$	0	0
$553$	0.616607	0.0262208
$554$	−16.6753	−0.708468
$555$	0	0
$556$	5.78216	0.245218
$557$	0.119732	0.00507322	0.00253661	−	0.999997i	$-0.499193\pi$
$557$	0.119732	0.00507322	0.00253661	+	0.999997i	$0.499193\pi$
$558$	0	0
$559$	5.23675	0.221491
$560$	0.153990	0.00650725
$561$	0	0
$562$	−5.34718	−0.225557
$563$	43.9151	1.85080	0.925400	−	0.378992i	$-0.123729\pi$
$563$	43.9151	1.85080	0.925400	+	0.378992i	$0.123729\pi$
$564$	0	0
$565$	−7.24273	−0.304704
$566$	7.66371	0.322130
$567$	0	0
$568$	−37.7035	−1.58201
$569$	3.35766	0.140760	0.0703802	−	0.997520i	$-0.477579\pi$
$569$	3.35766	0.140760	0.0703802	+	0.997520i	$0.477579\pi$
$570$	0	0
$571$	−15.4088	−0.644840	−0.322420	−	0.946597i	$-0.604496\pi$
$571$	−15.4088	−0.644840	−0.322420	+	0.946597i	$0.604496\pi$
$572$	−0.190183	−0.00795195
$573$	0	0
$574$	0.466908	0.0194884
$575$	−8.99941	−0.375302
$576$	0	0
$577$	25.3587	1.05570	0.527849	−	0.849338i	$-0.322999\pi$
$577$	25.3587	1.05570	0.527849	+	0.849338i	$0.322999\pi$
$578$	46.8130	1.94716
$579$	0	0
$580$	−1.63581	−0.0679233
$581$	0.597375	0.0247833
$582$	0	0
$583$	−3.31513	−0.137299
$584$	−10.6490	−0.440660
$585$	0	0
$586$	3.63482	0.150153
$587$	−14.2122	−0.586601	−0.293301	−	0.956020i	$-0.594754\pi$
$587$	−14.2122	−0.586601	−0.293301	+	0.956020i	$0.594754\pi$
$588$	0	0
$589$	−3.47642	−0.143243
$590$	−6.74549	−0.277708
$591$	0	0
$592$	−1.34491	−0.0552754
$593$	16.3696	0.672218	0.336109	−	0.941823i	$-0.390889\pi$
$593$	16.3696	0.672218	0.336109	+	0.941823i	$0.390889\pi$
$594$	0	0
$595$	−0.333592	−0.0136759
$596$	−4.50390	−0.184487
$597$	0	0
$598$	7.90347	0.323197
$599$	−36.2534	−1.48127	−0.740637	−	0.671905i	$-0.765476\pi$
$599$	−36.2534	−1.48127	−0.740637	+	0.671905i	$0.765476\pi$
$600$	0	0
$601$	13.0869	0.533828	0.266914	−	0.963720i	$-0.413996\pi$
$601$	13.0869	0.533828	0.266914	+	0.963720i	$0.413996\pi$
$602$	0.470137	0.0191613
$603$	0	0
$604$	0.457622	0.0186204
$605$	1.00000	0.0406558
$606$	0	0
$607$	42.8327	1.73853	0.869263	−	0.494351i	$-0.164594\pi$
$607$	42.8327	1.73853	0.869263	+	0.494351i	$0.164594\pi$
$608$	−1.59213	−0.0645696
$609$	0	0
$610$	−5.25737	−0.212865
$611$	−2.13681	−0.0864459
$612$	0	0
$613$	−12.8152	−0.517602	−0.258801	−	0.965931i	$-0.583327\pi$
$613$	−12.8152	−0.517602	−0.258801	+	0.965931i	$0.583327\pi$
$614$	−16.5766	−0.668977
$615$	0	0
$616$	−0.137439	−0.00553757
$617$	36.1671	1.45603	0.728017	−	0.685559i	$-0.240442\pi$
$617$	36.1671	1.45603	0.728017	+	0.685559i	$0.240442\pi$
$618$	0	0
$619$	38.1619	1.53386	0.766928	−	0.641733i	$-0.221784\pi$
$619$	38.1619	1.53386	0.766928	+	0.641733i	$0.221784\pi$
$620$	−0.986272	−0.0396096
$621$	0	0
$622$	20.1475	0.807841
$623$	0.0893078	0.00357804
$624$	0	0
$625$	1.00000	0.0400000
$626$	18.3376	0.732919
$627$	0	0
$628$	6.01352	0.239965
$629$	2.91351	0.116169
$630$	0	0
$631$	30.1987	1.20219	0.601095	−	0.799177i	$-0.294731\pi$
$631$	30.1987	1.20219	0.601095	+	0.799177i	$0.294731\pi$
$632$	40.1579	1.59740
$633$	0	0
$634$	−17.3115	−0.687529
$635$	−11.0073	−0.436813
$636$	0	0
$637$	4.69110	0.185868
$638$	−7.55379	−0.299057
$639$	0	0
$640$	8.33133	0.329325
$641$	8.69574	0.343461	0.171731	−	0.985144i	$-0.445064\pi$
$641$	8.69574	0.343461	0.171731	+	0.985144i	$0.445064\pi$
$642$	0	0
$643$	−38.9210	−1.53489	−0.767447	−	0.641112i	$-0.778473\pi$
$643$	−38.9210	−1.53489	−0.767447	+	0.641112i	$0.778473\pi$
$644$	−0.117288	−0.00462178
$645$	0	0
$646$	−9.51344	−0.374301
$647$	18.5600	0.729670	0.364835	−	0.931072i	$-0.381125\pi$
$647$	18.5600	0.729670	0.364835	+	0.931072i	$0.381125\pi$
$648$	0	0
$649$	5.14894	0.202113
$650$	−0.878220	−0.0344466
$651$	0	0
$652$	4.39627	0.172171
$653$	−8.12148	−0.317818	−0.158909	−	0.987293i	$-0.550798\pi$
$653$	−8.12148	−0.317818	−0.158909	+	0.987293i	$0.550798\pi$
$654$	0	0
$655$	2.34467	0.0916138
$656$	26.0064	1.01538
$657$	0	0
$658$	−0.191835	−0.00747851
$659$	12.7803	0.497852	0.248926	−	0.968523i	$-0.419922\pi$
$659$	12.7803	0.497852	0.248926	+	0.968523i	$0.419922\pi$
$660$	0	0
$661$	48.7774	1.89722	0.948610	−	0.316446i	$-0.102490\pi$
$661$	48.7774	1.89722	0.948610	+	0.316446i	$0.102490\pi$
$662$	22.5606	0.876844
$663$	0	0
$664$	38.9053	1.50982
$665$	0.0459382	0.00178141
$666$	0	0
$667$	−51.8899	−2.00918
$668$	−3.37013	−0.130394
$669$	0	0
$670$	−1.94130	−0.0749988
$671$	4.01303	0.154921
$672$	0	0
$673$	−26.4225	−1.01851	−0.509255	−	0.860615i	$-0.670079\pi$
$673$	−26.4225	−1.01851	−0.509255	+	0.860615i	$0.670079\pi$
$674$	−7.22660	−0.278358
$675$	0	0
$676$	3.56065	0.136948
$677$	−3.80976	−0.146421	−0.0732104	−	0.997317i	$-0.523324\pi$
$677$	−3.80976	−0.146421	−0.0732104	+	0.997317i	$0.523324\pi$
$678$	0	0
$679$	−0.290096	−0.0111329
$680$	−21.7259	−0.833150
$681$	0	0
$682$	−4.55437	−0.174396
$683$	−25.8187	−0.987924	−0.493962	−	0.869484i	$-0.664452\pi$
$683$	−25.8187	−0.987924	−0.493962	+	0.869484i	$0.664452\pi$
$684$	0	0
$685$	−9.57007	−0.365654
$686$	0.842427	0.0321640
$687$	0	0
$688$	26.1862	0.998339
$689$	−2.22233	−0.0846639
$690$	0	0
$691$	27.0739	1.02994	0.514969	−	0.857209i	$-0.327803\pi$
$691$	27.0739	1.02994	0.514969	+	0.857209i	$0.327803\pi$
$692$	1.81586	0.0690288
$693$	0	0
$694$	18.3905	0.698093
$695$	−20.3810	−0.773097
$696$	0	0
$697$	−56.3382	−2.13396
$698$	1.46106	0.0553018
$699$	0	0
$700$	0.0130328	0.000492594 0
$701$	−32.3535	−1.22198	−0.610988	−	0.791640i	$-0.709228\pi$
$701$	−32.3535	−1.22198	−0.610988	+	0.791640i	$0.709228\pi$
$702$	0	0
$703$	−0.401213	−0.0151320
$704$	−8.79003	−0.331287
$705$	0	0
$706$	37.4926	1.41105
$707$	0.455064	0.0171144
$708$	0	0
$709$	7.78682	0.292440	0.146220	−	0.989252i	$-0.453289\pi$
$709$	7.78682	0.292440	0.146220	+	0.989252i	$0.453289\pi$
$710$	16.5098	0.619603
$711$	0	0
$712$	5.81637	0.217978
$713$	−31.2858	−1.17166
$714$	0	0
$715$	0.670359	0.0250700
$716$	−0.547534	−0.0204623
$717$	0	0
$718$	−14.8563	−0.554433
$719$	−11.1846	−0.417116	−0.208558	−	0.978010i	$-0.566877\pi$
$719$	−11.1846	−0.417116	−0.208558	+	0.978010i	$0.566877\pi$
$720$	0	0
$721$	0.433219	0.0161339
$722$	1.31008	0.0487559
$723$	0	0
$724$	0.435254	0.0161761
$725$	5.76592	0.214141
$726$	0	0
$727$	−33.5929	−1.24589	−0.622945	−	0.782265i	$-0.714064\pi$
$727$	−33.5929	−1.24589	−0.622945	+	0.782265i	$0.714064\pi$
$728$	−0.0921333	−0.00341469
$729$	0	0
$730$	4.66306	0.172588
$731$	−56.7278	−2.09815
$732$	0	0
$733$	−35.2861	−1.30332	−0.651660	−	0.758511i	$-0.725927\pi$
$733$	−35.2861	−1.30332	−0.651660	+	0.758511i	$0.725927\pi$
$734$	19.1963	0.708549
$735$	0	0
$736$	−14.3283	−0.528147
$737$	1.48182	0.0545835
$738$	0	0
$739$	51.0315	1.87722	0.938612	−	0.344974i	$-0.112112\pi$
$739$	51.0315	1.87722	0.938612	+	0.344974i	$0.112112\pi$
$740$	−0.113825	−0.00418430
$741$	0	0
$742$	−0.199513	−0.00732434
$743$	7.59972	0.278807	0.139403	−	0.990236i	$-0.455482\pi$
$743$	7.59972	0.278807	0.139403	+	0.990236i	$0.455482\pi$
$744$	0	0
$745$	15.8754	0.581630
$746$	−36.9484	−1.35278
$747$	0	0
$748$	2.06018	0.0753277
$749$	−0.725453	−0.0265075
$750$	0	0
$751$	4.60241	0.167944	0.0839722	−	0.996468i	$-0.473239\pi$
$751$	4.60241	0.167944	0.0839722	+	0.996468i	$0.473239\pi$
$752$	−10.6850	−0.389643
$753$	0	0
$754$	−5.06375	−0.184411
$755$	−1.61303	−0.0587042
$756$	0	0
$757$	14.3750	0.522468	0.261234	−	0.965275i	$-0.415871\pi$
$757$	14.3750	0.522468	0.261234	+	0.965275i	$0.415871\pi$
$758$	−18.0067	−0.654034
$759$	0	0
$760$	2.99182	0.108525
$761$	50.3207	1.82412	0.912062	−	0.410053i	$-0.134490\pi$
$761$	50.3207	1.82412	0.912062	+	0.410053i	$0.134490\pi$
$762$	0	0
$763$	−0.893578	−0.0323497
$764$	6.96916	0.252135
$765$	0	0
$766$	26.1841	0.946071
$767$	3.45163	0.124631
$768$	0	0
$769$	24.8105	0.894690	0.447345	−	0.894361i	$-0.352370\pi$
$769$	24.8105	0.894690	0.447345	+	0.894361i	$0.352370\pi$
$770$	0.0601825	0.00216883
$771$	0	0
$772$	6.21232	0.223586
$773$	6.44996	0.231989	0.115994	−	0.993250i	$-0.462995\pi$
$773$	6.44996	0.231989	0.115994	+	0.993250i	$0.462995\pi$
$774$	0	0
$775$	3.47642	0.124877
$776$	−18.8932	−0.678225
$777$	0	0
$778$	−25.5102	−0.914584
$779$	7.75821	0.277967
$780$	0	0
$781$	−12.6022	−0.450942
$782$	−85.6154	−3.06160
$783$	0	0
$784$	23.4577	0.837774
$785$	−21.1965	−0.756536
$786$	0	0
$787$	−3.35379	−0.119550	−0.0597749	−	0.998212i	$-0.519038\pi$
$787$	−3.35379	−0.119550	−0.0597749	+	0.998212i	$0.519038\pi$
$788$	3.45022	0.122909
$789$	0	0
$790$	−17.5845	−0.625630
$791$	0.332718	0.0118301
$792$	0	0
$793$	2.69017	0.0955308
$794$	−24.8875	−0.883224
$795$	0	0
$796$	6.63953	0.235332
$797$	7.51135	0.266066	0.133033	−	0.991112i	$-0.457528\pi$
$797$	7.51135	0.266066	0.133033	+	0.991112i	$0.457528\pi$
$798$	0	0
$799$	23.1473	0.818891
$800$	1.59213	0.0562904
$801$	0	0
$802$	7.18065	0.253557
$803$	−3.55938	−0.125608
$804$	0	0
$805$	0.413417	0.0145710
$806$	−3.05306	−0.107540
$807$	0	0
$808$	29.6370	1.04263
$809$	43.8080	1.54021	0.770104	−	0.637918i	$-0.220204\pi$
$809$	43.8080	1.54021	0.770104	+	0.637918i	$0.220204\pi$
$810$	0	0
$811$	−41.4737	−1.45634	−0.728170	−	0.685396i	$-0.759629\pi$
$811$	−41.4737	−1.45634	−0.728170	+	0.685396i	$0.759629\pi$
$812$	0.0751461	0.00263711
$813$	0	0
$814$	−0.525619	−0.0184229
$815$	−15.4960	−0.542802
$816$	0	0
$817$	7.81186	0.273302
$818$	19.8693	0.694715
$819$	0	0
$820$	2.20103	0.0768633
$821$	30.9218	1.07918	0.539590	−	0.841928i	$-0.318579\pi$
$821$	30.9218	1.07918	0.539590	+	0.841928i	$0.318579\pi$
$822$	0	0
$823$	33.6565	1.17319	0.586597	−	0.809879i	$-0.300467\pi$
$823$	33.6565	1.17319	0.586597	+	0.809879i	$0.300467\pi$
$824$	28.2143	0.982893
$825$	0	0
$826$	0.309876	0.0107820
$827$	−53.0019	−1.84306	−0.921528	−	0.388312i	$-0.873058\pi$
$827$	−53.0019	−1.84306	−0.921528	+	0.388312i	$0.873058\pi$
$828$	0	0
$829$	48.8099	1.69524	0.847618	−	0.530607i	$-0.178036\pi$
$829$	48.8099	1.69524	0.847618	+	0.530607i	$0.178036\pi$
$830$	−17.0361	−0.591330
$831$	0	0
$832$	−5.89247	−0.204285
$833$	−50.8170	−1.76070
$834$	0	0
$835$	11.8791	0.411092
$836$	−0.283703	−0.00981208
$837$	0	0
$838$	−13.5179	−0.466970
$839$	−41.1240	−1.41976	−0.709879	−	0.704323i	$-0.751251\pi$
$839$	−41.1240	−1.41976	−0.709879	+	0.704323i	$0.751251\pi$
$840$	0	0
$841$	4.24582	0.146408
$842$	21.4840	0.740387
$843$	0	0
$844$	1.09600	0.0377260
$845$	−12.5506	−0.431754
$846$	0	0
$847$	−0.0459382	−0.00157845
$848$	−11.1127	−0.381611
$849$	0	0
$850$	9.51344	0.326308
$851$	−3.61068	−0.123772
$852$	0	0
$853$	1.34400	0.0460176	0.0230088	−	0.999735i	$-0.492675\pi$
$853$	1.34400	0.0460176	0.0230088	+	0.999735i	$0.492675\pi$
$854$	0.241514	0.00826445
$855$	0	0
$856$	−47.2467	−1.61486
$857$	−32.6082	−1.11388	−0.556938	−	0.830554i	$-0.688024\pi$
$857$	−32.6082	−1.11388	−0.556938	+	0.830554i	$0.688024\pi$
$858$	0	0
$859$	−29.4158	−1.00365	−0.501827	−	0.864968i	$-0.667339\pi$
$859$	−29.4158	−1.00365	−0.501827	+	0.864968i	$0.667339\pi$
$860$	2.21625	0.0755734
$861$	0	0
$862$	20.2815	0.690791
$863$	−24.9710	−0.850022	−0.425011	−	0.905188i	$-0.639730\pi$
$863$	−24.9710	−0.850022	−0.425011	+	0.905188i	$0.639730\pi$
$864$	0	0
$865$	−6.40058	−0.217626
$866$	−5.69804	−0.193627
$867$	0	0
$868$	0.0453075	0.00153784
$869$	13.4225	0.455329
$870$	0	0
$871$	0.993351	0.0336584
$872$	−58.1962	−1.97077
$873$	0	0
$874$	11.7899	0.398800
$875$	−0.0459382	−0.00155299
$876$	0	0
$877$	−13.8555	−0.467868	−0.233934	−	0.972252i	$-0.575160\pi$
$877$	−13.8555	−0.467868	−0.233934	+	0.972252i	$0.575160\pi$
$878$	−18.6253	−0.628575
$879$	0	0
$880$	3.35211	0.112999
$881$	−23.3533	−0.786794	−0.393397	−	0.919369i	$-0.628700\pi$
$881$	−23.3533	−0.786794	−0.393397	+	0.919369i	$0.628700\pi$
$882$	0	0
$883$	39.0324	1.31354	0.656772	−	0.754089i	$-0.271921\pi$
$883$	39.0324	1.31354	0.656772	+	0.754089i	$0.271921\pi$
$884$	1.38106	0.0464501
$885$	0	0
$886$	−35.4050	−1.18945
$887$	17.3827	0.583655	0.291828	−	0.956471i	$-0.405737\pi$
$887$	17.3827	0.583655	0.291828	+	0.956471i	$0.405737\pi$
$888$	0	0
$889$	0.505657	0.0169592
$890$	−2.54690	−0.0853723
$891$	0	0
$892$	3.87743	0.129826
$893$	−3.18756	−0.106668
$894$	0	0
$895$	1.92995	0.0645112
$896$	−0.382726	−0.0127860
$897$	0	0
$898$	−38.7389	−1.29274
$899$	20.0448	0.668530
$900$	0	0
$901$	24.0737	0.802010
$902$	10.1638	0.338419
$903$	0	0
$904$	21.6690	0.720699
$905$	−1.53419	−0.0509981
$906$	0	0
$907$	−22.7667	−0.755955	−0.377978	−	0.925815i	$-0.623380\pi$
$907$	−22.7667	−0.755955	−0.377978	+	0.925815i	$0.623380\pi$
$908$	1.86810	0.0619951
$909$	0	0
$910$	0.0403438	0.00133739
$911$	−12.1324	−0.401964	−0.200982	−	0.979595i	$-0.564413\pi$
$911$	−12.1324	−0.401964	−0.200982	+	0.979595i	$0.564413\pi$
$912$	0	0
$913$	13.0039	0.430366
$914$	−52.4113	−1.73361
$915$	0	0
$916$	−3.55070	−0.117319
$917$	−0.107710	−0.00355689
$918$	0	0
$919$	−42.1151	−1.38925	−0.694625	−	0.719372i	$-0.744430\pi$
$919$	−42.1151	−1.38925	−0.694625	+	0.719372i	$0.744430\pi$
$920$	26.9247	0.887679
$921$	0	0
$922$	11.6247	0.382840
$923$	−8.44799	−0.278069
$924$	0	0
$925$	0.401213	0.0131918
$926$	−18.8132	−0.618239
$927$	0	0
$928$	9.18012	0.301352
$929$	5.58087	0.183102	0.0915512	−	0.995800i	$-0.470817\pi$
$929$	5.58087	0.183102	0.0915512	+	0.995800i	$0.470817\pi$
$930$	0	0
$931$	6.99789	0.229347
$932$	−7.39492	−0.242229
$933$	0	0
$934$	−23.8744	−0.781193
$935$	−7.26175	−0.237485
$936$	0	0
$937$	31.0344	1.01385	0.506925	−	0.861990i	$-0.330782\pi$
$937$	31.0344	1.01385	0.506925	+	0.861990i	$0.330782\pi$
$938$	0.0891796	0.00291182
$939$	0	0
$940$	−0.904320	−0.0294957
$941$	−47.5393	−1.54974	−0.774868	−	0.632123i	$-0.782184\pi$
$941$	−47.5393	−1.54974	−0.774868	+	0.632123i	$0.782184\pi$
$942$	0	0
$943$	69.8194	2.27363
$944$	17.2598	0.561758
$945$	0	0
$946$	10.2341	0.332740
$947$	−8.06096	−0.261946	−0.130973	−	0.991386i	$-0.541810\pi$
$947$	−8.06096	−0.261946	−0.130973	+	0.991386i	$0.541810\pi$
$948$	0	0
$949$	−2.38606	−0.0774549
$950$	−1.31008	−0.0425045
$951$	0	0
$952$	0.998047	0.0323469
$953$	35.1400	1.13830	0.569149	−	0.822235i	$-0.307273\pi$
$953$	35.1400	1.13830	0.569149	+	0.822235i	$0.307273\pi$
$954$	0	0
$955$	−24.5650	−0.794904
$956$	5.24439	0.169616
$957$	0	0
$958$	−38.7495	−1.25194
$959$	0.439632	0.0141964
$960$	0	0
$961$	−18.9145	−0.610145
$962$	−0.352353	−0.0113603
$963$	0	0
$964$	7.87856	0.253751
$965$	−21.8972	−0.704897
$966$	0	0
$967$	−16.0851	−0.517263	−0.258632	−	0.965976i	$-0.583272\pi$
$967$	−16.0851	−0.517263	−0.258632	+	0.965976i	$0.583272\pi$
$968$	−2.99182	−0.0961608
$969$	0	0
$970$	8.27303	0.265631
$971$	50.2083	1.61126	0.805631	−	0.592418i	$-0.201827\pi$
$971$	50.2083	1.61126	0.805631	+	0.592418i	$0.201827\pi$
$972$	0	0
$973$	0.936268	0.0300154
$974$	−53.9443	−1.72849
$975$	0	0
$976$	13.4521	0.430592
$977$	−1.00917	−0.0322861	−0.0161431	−	0.999870i	$-0.505139\pi$
$977$	−1.00917	−0.0322861	−0.0161431	+	0.999870i	$0.505139\pi$
$978$	0	0
$979$	1.94409	0.0621333
$980$	1.98532	0.0634188
$981$	0	0
$982$	−38.4293	−1.22633
$983$	25.2319	0.804772	0.402386	−	0.915470i	$-0.368181\pi$
$983$	25.2319	0.804772	0.402386	+	0.915470i	$0.368181\pi$
$984$	0	0
$985$	−12.1614	−0.387494
$986$	54.8537	1.74690
$987$	0	0
$988$	−0.190183	−0.00605052
$989$	70.3021	2.23548
$990$	0	0
$991$	−30.8482	−0.979925	−0.489962	−	0.871744i	$-0.662989\pi$
$991$	−30.8482	−0.979925	−0.489962	+	0.871744i	$0.662989\pi$
$992$	5.53493	0.175734
$993$	0	0
$994$	−0.758431	−0.0240560
$995$	−23.4031	−0.741928
$996$	0	0
$997$	−40.1424	−1.27132	−0.635662	−	0.771967i	$-0.719273\pi$
$997$	−40.1424	−1.27132	−0.635662	+	0.771967i	$0.719273\pi$
$998$	25.7409	0.814814
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.y.1.4		6
3.2	odd	2		3135.2.a.o.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3135.2.a.o.1.3	✓	6	3.2	odd	2
9405.2.a.y.1.4		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)

\(f(q)\)	\(=\)	\(q+1.31008 q^{2} -0.283703 q^{4} +1.00000 q^{5} -0.0459382 q^{7} -2.99182 q^{8} +O(q^{10})\)	\(q+1.31008 q^{2} -0.283703 q^{4} +1.00000 q^{5} -0.0459382 q^{7} -2.99182 q^{8} +1.31008 q^{10} -1.00000 q^{11} -0.670359 q^{13} -0.0601825 q^{14} -3.35211 q^{16} +7.26175 q^{17} -1.00000 q^{19} -0.283703 q^{20} -1.31008 q^{22} -8.99941 q^{23} +1.00000 q^{25} -0.878220 q^{26} +0.0130328 q^{28} +5.76592 q^{29} +3.47642 q^{31} +1.59213 q^{32} +9.51344 q^{34} -0.0459382 q^{35} +0.401213 q^{37} -1.31008 q^{38} -2.99182 q^{40} -7.75821 q^{41} -7.81186 q^{43} +0.283703 q^{44} -11.7899 q^{46} +3.18756 q^{47} -6.99789 q^{49} +1.31008 q^{50} +0.190183 q^{52} +3.31513 q^{53} -1.00000 q^{55} +0.137439 q^{56} +7.55379 q^{58} -5.14894 q^{59} -4.01303 q^{61} +4.55437 q^{62} +8.79003 q^{64} -0.670359 q^{65} -1.48182 q^{67} -2.06018 q^{68} -0.0601825 q^{70} +12.6022 q^{71} +3.55938 q^{73} +0.525619 q^{74} +0.283703 q^{76} +0.0459382 q^{77} -13.4225 q^{79} -3.35211 q^{80} -10.1638 q^{82} -13.0039 q^{83} +7.26175 q^{85} -10.2341 q^{86} +2.99182 q^{88} -1.94409 q^{89} +0.0307951 q^{91} +2.55316 q^{92} +4.17594 q^{94} -1.00000 q^{95} +6.31493 q^{97} -9.16776 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 - 7 * q^7 + 3 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} - 7 q^{7} + 3 q^{8} + 2 q^{10} - 6 q^{11} - 3 q^{13} - 4 q^{16} + 15 q^{17} - 6 q^{19} + 4 q^{20} - 2 q^{22} - 6 q^{23} + 6 q^{25} - 9 q^{26} - 19 q^{28} + 3 q^{29} - 10 q^{31} + 3 q^{32} - 9 q^{34} - 7 q^{35} - 11 q^{37} - 2 q^{38} + 3 q^{40} + 15 q^{41} - 22 q^{43} - 4 q^{44} - 23 q^{46} - 4 q^{47} + 7 q^{49} + 2 q^{50} - 13 q^{52} - 2 q^{53} - 6 q^{55} - 13 q^{56} + 5 q^{58} - 7 q^{59} - 5 q^{61} + 35 q^{62} - q^{64} - 3 q^{65} - 17 q^{67} - 12 q^{68} - 14 q^{71} - 38 q^{73} - 5 q^{74} - 4 q^{76} + 7 q^{77} - 15 q^{79} - 4 q^{80} - 26 q^{82} + 24 q^{83} + 15 q^{85} - 9 q^{86} - 3 q^{88} - 16 q^{89} - 23 q^{91} + 11 q^{92} + 18 q^{94} - 6 q^{95} - 8 q^{97} - 27 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 - 7 * q^7 + 3 * q^8 + 2 * q^10 - 6 * q^11 - 3 * q^13 - 4 * q^16 + 15 * q^17 - 6 * q^19 + 4 * q^20 - 2 * q^22 - 6 * q^23 + 6 * q^25 - 9 * q^26 - 19 * q^28 + 3 * q^29 - 10 * q^31 + 3 * q^32 - 9 * q^34 - 7 * q^35 - 11 * q^37 - 2 * q^38 + 3 * q^40 + 15 * q^41 - 22 * q^43 - 4 * q^44 - 23 * q^46 - 4 * q^47 + 7 * q^49 + 2 * q^50 - 13 * q^52 - 2 * q^53 - 6 * q^55 - 13 * q^56 + 5 * q^58 - 7 * q^59 - 5 * q^61 + 35 * q^62 - q^64 - 3 * q^65 - 17 * q^67 - 12 * q^68 - 14 * q^71 - 38 * q^73 - 5 * q^74 - 4 * q^76 + 7 * q^77 - 15 * q^79 - 4 * q^80 - 26 * q^82 + 24 * q^83 + 15 * q^85 - 9 * q^86 - 3 * q^88 - 16 * q^89 - 23 * q^91 + 11 * q^92 + 18 * q^94 - 6 * q^95 - 8 * q^97 - 27 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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