LMFDB - Embedded newform 4338.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4338,2,Mod(1,4338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4338 = 2 \cdot 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4338.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:	$34.6391043968$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 20x^{5} + 26x^{4} + 95x^{3} - 121x^{2} - 126x + 158 $ x^7 - x^6 - 20x^5 + 26x^4 + 95x^3 - 121x^2 - 126*x + 158
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1446)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.18498$ of defining polynomial
Character	$\chi$	$=$	4338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.57869 q^{5} -3.07991 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.57869 q^{5} -3.07991 q^{7} +1.00000 q^{8} -3.57869 q^{10} +1.80420 q^{11} -5.32909 q^{13} -3.07991 q^{14} +1.00000 q^{16} -2.72230 q^{17} +4.24918 q^{19} -3.57869 q^{20} +1.80420 q^{22} +0.644837 q^{23} +7.80702 q^{25} -5.32909 q^{26} -3.07991 q^{28} -6.88411 q^{29} -0.939691 q^{31} +1.00000 q^{32} -2.72230 q^{34} +11.0220 q^{35} +4.72230 q^{37} +4.24918 q^{38} -3.57869 q^{40} -9.02205 q^{41} +9.94016 q^{43} +1.80420 q^{44} +0.644837 q^{46} -3.66924 q^{47} +2.48586 q^{49} +7.80702 q^{50} -5.32909 q^{52} +3.50160 q^{53} -6.45666 q^{55} -3.07991 q^{56} -6.88411 q^{58} +1.92905 q^{59} -0.306833 q^{61} -0.939691 q^{62} +1.00000 q^{64} +19.0712 q^{65} -0.761571 q^{67} -2.72230 q^{68} +11.0220 q^{70} +9.40656 q^{71} +3.71277 q^{73} +4.72230 q^{74} +4.24918 q^{76} -5.55676 q^{77} +13.3724 q^{79} -3.57869 q^{80} -9.02205 q^{82} +12.5811 q^{83} +9.74228 q^{85} +9.94016 q^{86} +1.80420 q^{88} -4.87243 q^{89} +16.4131 q^{91} +0.644837 q^{92} -3.66924 q^{94} -15.2065 q^{95} +9.44564 q^{97} +2.48586 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) $ 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8	$ 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8 - 5 * q^10 - 4 * q^11 + 14 * q^13 + 7 * q^14 + 7 * q^16 - 4 * q^17 + 7 * q^19 - 5 * q^20 - 4 * q^22 + q^23 + 14 * q^25 + 14 * q^26 + 7 * q^28 - 3 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^34 + 17 * q^35 + 18 * q^37 + 7 * q^38 - 5 * q^40 - 3 * q^41 + 11 * q^43 - 4 * q^44 + q^46 + 18 * q^47 + 20 * q^49 + 14 * q^50 + 14 * q^52 + 9 * q^53 - 4 * q^55 + 7 * q^56 - 3 * q^58 - 6 * q^59 + 31 * q^61 + 8 * q^62 + 7 * q^64 + 6 * q^65 + 4 * q^67 - 4 * q^68 + 17 * q^70 + 3 * q^71 + 16 * q^73 + 18 * q^74 + 7 * q^76 + 34 * q^79 - 5 * q^80 - 3 * q^82 - 10 * q^83 + 34 * q^85 + 11 * q^86 - 4 * q^88 - 24 * q^89 + 40 * q^91 + q^92 + 18 * q^94 + q^95 + 27 * q^97 + 20 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.57869	−1.60044	−0.800219	−	0.599707i	$-0.795284\pi$
$5$	−3.57869	−1.60044	−0.800219	+	0.599707i	$0.795284\pi$
$6$	0	0
$7$	−3.07991	−1.16410	−0.582049	−	0.813154i	$-0.697749\pi$
$7$	−3.07991	−1.16410	−0.582049	+	0.813154i	$0.697749\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.57869	−1.13168
$11$	1.80420	0.543986	0.271993	−	0.962299i	$-0.412317\pi$
$11$	1.80420	0.543986	0.271993	+	0.962299i	$0.412317\pi$
$12$	0	0
$13$	−5.32909	−1.47802	−0.739012	−	0.673692i	$-0.764707\pi$
$13$	−5.32909	−1.47802	−0.739012	+	0.673692i	$0.764707\pi$
$14$	−3.07991	−0.823141
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.72230	−0.660256	−0.330128	−	0.943936i	$-0.607092\pi$
$17$	−2.72230	−0.660256	−0.330128	+	0.943936i	$0.607092\pi$
$18$	0	0
$19$	4.24918	0.974829	0.487414	−	0.873171i	$-0.337940\pi$
$19$	4.24918	0.974829	0.487414	+	0.873171i	$0.337940\pi$
$20$	−3.57869	−0.800219
$21$	0	0
$22$	1.80420	0.384656
$23$	0.644837	0.134458	0.0672289	−	0.997738i	$-0.478584\pi$
$23$	0.644837	0.134458	0.0672289	+	0.997738i	$0.478584\pi$
$24$	0	0
$25$	7.80702	1.56140
$26$	−5.32909	−1.04512
$27$	0	0
$28$	−3.07991	−0.582049
$29$	−6.88411	−1.27835	−0.639173	−	0.769063i	$-0.720723\pi$
$29$	−6.88411	−1.27835	−0.639173	+	0.769063i	$0.720723\pi$
$30$	0	0
$31$	−0.939691	−0.168774	−0.0843868	−	0.996433i	$-0.526893\pi$
$31$	−0.939691	−0.168774	−0.0843868	+	0.996433i	$0.526893\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.72230	−0.466871
$35$	11.0220	1.86307
$36$	0	0
$37$	4.72230	0.776342	0.388171	−	0.921587i	$-0.373107\pi$
$37$	4.72230	0.776342	0.388171	+	0.921587i	$0.373107\pi$
$38$	4.24918	0.689308
$39$	0	0
$40$	−3.57869	−0.565840
$41$	−9.02205	−1.40901	−0.704504	−	0.709700i	$-0.748830\pi$
$41$	−9.02205	−1.40901	−0.704504	+	0.709700i	$0.748830\pi$
$42$	0	0
$43$	9.94016	1.51586	0.757930	−	0.652336i	$-0.226211\pi$
$43$	9.94016	1.51586	0.757930	+	0.652336i	$0.226211\pi$
$44$	1.80420	0.271993
$45$	0	0
$46$	0.644837	0.0950760
$47$	−3.66924	−0.535214	−0.267607	−	0.963528i	$-0.586233\pi$
$47$	−3.66924	−0.535214	−0.267607	+	0.963528i	$0.586233\pi$
$48$	0	0
$49$	2.48586	0.355122
$50$	7.80702	1.10408
$51$	0	0
$52$	−5.32909	−0.739012
$53$	3.50160	0.480982	0.240491	−	0.970651i	$-0.422692\pi$
$53$	3.50160	0.480982	0.240491	+	0.970651i	$0.422692\pi$
$54$	0	0
$55$	−6.45666	−0.870616
$56$	−3.07991	−0.411571
$57$	0	0
$58$	−6.88411	−0.903928
$59$	1.92905	0.251141	0.125570	−	0.992085i	$-0.459924\pi$
$59$	1.92905	0.251141	0.125570	+	0.992085i	$0.459924\pi$
$60$	0	0
$61$	−0.306833	−0.0392859	−0.0196430	−	0.999807i	$-0.506253\pi$
$61$	−0.306833	−0.0392859	−0.0196430	+	0.999807i	$0.506253\pi$
$62$	−0.939691	−0.119341
$63$	0	0
$64$	1.00000	0.125000
$65$	19.0712	2.36549
$66$	0	0
$67$	−0.761571	−0.0930408	−0.0465204	−	0.998917i	$-0.514813\pi$
$67$	−0.761571	−0.0930408	−0.0465204	+	0.998917i	$0.514813\pi$
$68$	−2.72230	−0.330128
$69$	0	0
$70$	11.0220	1.31739
$71$	9.40656	1.11635	0.558177	−	0.829722i	$-0.311501\pi$
$71$	9.40656	1.11635	0.558177	+	0.829722i	$0.311501\pi$
$72$	0	0
$73$	3.71277	0.434547	0.217273	−	0.976111i	$-0.430284\pi$
$73$	3.71277	0.434547	0.217273	+	0.976111i	$0.430284\pi$
$74$	4.72230	0.548957
$75$	0	0
$76$	4.24918	0.487414
$77$	−5.55676	−0.633252
$78$	0	0
$79$	13.3724	1.50451	0.752257	−	0.658870i	$-0.228965\pi$
$79$	13.3724	1.50451	0.752257	+	0.658870i	$0.228965\pi$
$80$	−3.57869	−0.400110
$81$	0	0
$82$	−9.02205	−0.996318
$83$	12.5811	1.38095	0.690477	−	0.723354i	$-0.257401\pi$
$83$	12.5811	1.38095	0.690477	+	0.723354i	$0.257401\pi$
$84$	0	0
$85$	9.74228	1.05670
$86$	9.94016	1.07187
$87$	0	0
$88$	1.80420	0.192328
$89$	−4.87243	−0.516477	−0.258238	−	0.966081i	$-0.583142\pi$
$89$	−4.87243	−0.516477	−0.258238	+	0.966081i	$0.583142\pi$
$90$	0	0
$91$	16.4131	1.72056
$92$	0.644837	0.0672289
$93$	0	0
$94$	−3.66924	−0.378453
$95$	−15.2065	−1.56015
$96$	0	0
$97$	9.44564	0.959059	0.479530	−	0.877526i	$-0.340807\pi$
$97$	9.44564	0.959059	0.479530	+	0.877526i	$0.340807\pi$
$98$	2.48586	0.251109
$99$	0	0
$100$	7.80702	0.780702
$101$	−17.0368	−1.69522	−0.847611	−	0.530619i	$-0.821960\pi$
$101$	−17.0368	−1.69522	−0.847611	+	0.530619i	$0.821960\pi$
$102$	0	0
$103$	11.9895	1.18136	0.590679	−	0.806906i	$-0.298860\pi$
$103$	11.9895	1.18136	0.590679	+	0.806906i	$0.298860\pi$
$104$	−5.32909	−0.522560
$105$	0	0
$106$	3.50160	0.340105
$107$	1.77487	0.171583	0.0857916	−	0.996313i	$-0.472658\pi$
$107$	1.77487	0.171583	0.0857916	+	0.996313i	$0.472658\pi$
$108$	0	0
$109$	13.1912	1.26348	0.631742	−	0.775179i	$-0.282340\pi$
$109$	13.1912	1.26348	0.631742	+	0.775179i	$0.282340\pi$
$110$	−6.45666	−0.615618
$111$	0	0
$112$	−3.07991	−0.291024
$113$	16.9233	1.59201	0.796006	−	0.605289i	$-0.206942\pi$
$113$	16.9233	1.59201	0.796006	+	0.605289i	$0.206942\pi$
$114$	0	0
$115$	−2.30767	−0.215192
$116$	−6.88411	−0.639173
$117$	0	0
$118$	1.92905	0.177583
$119$	8.38446	0.768602
$120$	0	0
$121$	−7.74488	−0.704080
$122$	−0.306833	−0.0277793
$123$	0	0
$124$	−0.939691	−0.0843868
$125$	−10.0454	−0.898492
$126$	0	0
$127$	3.66626	0.325328	0.162664	−	0.986682i	$-0.447991\pi$
$127$	3.66626	0.325328	0.162664	+	0.986682i	$0.447991\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	19.0712	1.67265
$131$	0.502637	0.0439156	0.0219578	−	0.999759i	$-0.493010\pi$
$131$	0.502637	0.0439156	0.0219578	+	0.999759i	$0.493010\pi$
$132$	0	0
$133$	−13.0871	−1.13480
$134$	−0.761571	−0.0657897
$135$	0	0
$136$	−2.72230	−0.233436
$137$	−4.12041	−0.352030	−0.176015	−	0.984387i	$-0.556321\pi$
$137$	−4.12041	−0.352030	−0.176015	+	0.984387i	$0.556321\pi$
$138$	0	0
$139$	7.49715	0.635900	0.317950	−	0.948107i	$-0.397006\pi$
$139$	7.49715	0.635900	0.317950	+	0.948107i	$0.397006\pi$
$140$	11.0220	0.931533
$141$	0	0
$142$	9.40656	0.789381
$143$	−9.61473	−0.804024
$144$	0	0
$145$	24.6361	2.04592
$146$	3.71277	0.307271
$147$	0	0
$148$	4.72230	0.388171
$149$	−17.3802	−1.42384	−0.711920	−	0.702260i	$-0.752174\pi$
$149$	−17.3802	−1.42384	−0.711920	+	0.702260i	$0.752174\pi$
$150$	0	0
$151$	16.8328	1.36983	0.684915	−	0.728623i	$-0.259839\pi$
$151$	16.8328	1.36983	0.684915	+	0.728623i	$0.259839\pi$
$152$	4.24918	0.344654
$153$	0	0
$154$	−5.55676	−0.447777
$155$	3.36286	0.270112
$156$	0	0
$157$	4.56737	0.364516	0.182258	−	0.983251i	$-0.441659\pi$
$157$	4.56737	0.364516	0.182258	+	0.983251i	$0.441659\pi$
$158$	13.3724	1.06385
$159$	0	0
$160$	−3.57869	−0.282920
$161$	−1.98604	−0.156522
$162$	0	0
$163$	−0.808621	−0.0633361	−0.0316680	−	0.999498i	$-0.510082\pi$
$163$	−0.808621	−0.0633361	−0.0316680	+	0.999498i	$0.510082\pi$
$164$	−9.02205	−0.704504
$165$	0	0
$166$	12.5811	0.976482
$167$	19.4239	1.50307	0.751533	−	0.659696i	$-0.229315\pi$
$167$	19.4239	1.50307	0.751533	+	0.659696i	$0.229315\pi$
$168$	0	0
$169$	15.3992	1.18455
$170$	9.74228	0.747199
$171$	0	0
$172$	9.94016	0.757930
$173$	−14.9037	−1.13311	−0.566554	−	0.824025i	$-0.691724\pi$
$173$	−14.9037	−1.13311	−0.566554	+	0.824025i	$0.691724\pi$
$174$	0	0
$175$	−24.0449	−1.81763
$176$	1.80420	0.135996
$177$	0	0
$178$	−4.87243	−0.365204
$179$	10.1042	0.755226	0.377613	−	0.925963i	$-0.376745\pi$
$179$	10.1042	0.755226	0.377613	+	0.925963i	$0.376745\pi$
$180$	0	0
$181$	−2.89882	−0.215468	−0.107734	−	0.994180i	$-0.534359\pi$
$181$	−2.89882	−0.215468	−0.107734	+	0.994180i	$0.534359\pi$
$182$	16.4131	1.21662
$183$	0	0
$184$	0.644837	0.0475380
$185$	−16.8997	−1.24249
$186$	0	0
$187$	−4.91157	−0.359170
$188$	−3.66924	−0.267607
$189$	0	0
$190$	−15.2065	−1.10320
$191$	3.79665	0.274716	0.137358	−	0.990521i	$-0.456139\pi$
$191$	3.79665	0.274716	0.137358	+	0.990521i	$0.456139\pi$
$192$	0	0
$193$	−1.49200	−0.107396	−0.0536981	−	0.998557i	$-0.517101\pi$
$193$	−1.49200	−0.107396	−0.0536981	+	0.998557i	$0.517101\pi$
$194$	9.44564	0.678157
$195$	0	0
$196$	2.48586	0.177561
$197$	−10.4870	−0.747168	−0.373584	−	0.927596i	$-0.621871\pi$
$197$	−10.4870	−0.747168	−0.373584	+	0.927596i	$0.621871\pi$
$198$	0	0
$199$	7.20710	0.510898	0.255449	−	0.966823i	$-0.417777\pi$
$199$	7.20710	0.510898	0.255449	+	0.966823i	$0.417777\pi$
$200$	7.80702	0.552040
$201$	0	0
$202$	−17.0368	−1.19870
$203$	21.2024	1.48812
$204$	0	0
$205$	32.2871	2.25503
$206$	11.9895	0.835347
$207$	0	0
$208$	−5.32909	−0.369506
$209$	7.66635	0.530293
$210$	0	0
$211$	−25.6998	−1.76925	−0.884625	−	0.466304i	$-0.845585\pi$
$211$	−25.6998	−1.76925	−0.884625	+	0.466304i	$0.845585\pi$
$212$	3.50160	0.240491
$213$	0	0
$214$	1.77487	0.121328
$215$	−35.5727	−2.42604
$216$	0	0
$217$	2.89417	0.196469
$218$	13.1912	0.893418
$219$	0	0
$220$	−6.45666	−0.435308
$221$	14.5074	0.975874
$222$	0	0
$223$	−25.9950	−1.74076	−0.870378	−	0.492384i	$-0.836125\pi$
$223$	−25.9950	−1.74076	−0.870378	+	0.492384i	$0.836125\pi$
$224$	−3.07991	−0.205785
$225$	0	0
$226$	16.9233	1.12572
$227$	12.2373	0.812218	0.406109	−	0.913825i	$-0.366885\pi$
$227$	12.2373	0.812218	0.406109	+	0.913825i	$0.366885\pi$
$228$	0	0
$229$	−25.1563	−1.66238	−0.831188	−	0.555991i	$-0.812339\pi$
$229$	−25.1563	−1.66238	−0.831188	+	0.555991i	$0.812339\pi$
$230$	−2.30767	−0.152163
$231$	0	0
$232$	−6.88411	−0.451964
$233$	6.99284	0.458116	0.229058	−	0.973413i	$-0.426435\pi$
$233$	6.99284	0.458116	0.229058	+	0.973413i	$0.426435\pi$
$234$	0	0
$235$	13.1311	0.856577
$236$	1.92905	0.125570
$237$	0	0
$238$	8.38446	0.543484
$239$	−5.46724	−0.353646	−0.176823	−	0.984243i	$-0.556582\pi$
$239$	−5.46724	−0.353646	−0.176823	+	0.984243i	$0.556582\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−7.74488	−0.497860
$243$	0	0
$244$	−0.306833	−0.0196430
$245$	−8.89611	−0.568351
$246$	0	0
$247$	−22.6443	−1.44082
$248$	−0.939691	−0.0596705
$249$	0	0
$250$	−10.0454	−0.635330
$251$	−19.3959	−1.22426	−0.612128	−	0.790759i	$-0.709686\pi$
$251$	−19.3959	−1.22426	−0.612128	+	0.790759i	$0.709686\pi$
$252$	0	0
$253$	1.16341	0.0731431
$254$	3.66626	0.230041
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.28440	0.142497	0.0712485	−	0.997459i	$-0.477302\pi$
$257$	2.28440	0.142497	0.0712485	+	0.997459i	$0.477302\pi$
$258$	0	0
$259$	−14.5443	−0.903738
$260$	19.0712	1.18274
$261$	0	0
$262$	0.502637	0.0310530
$263$	−9.32886	−0.575242	−0.287621	−	0.957744i	$-0.592864\pi$
$263$	−9.32886	−0.575242	−0.287621	+	0.957744i	$0.592864\pi$
$264$	0	0
$265$	−12.5311	−0.769782
$266$	−13.0871	−0.802421
$267$	0	0
$268$	−0.761571	−0.0465204
$269$	25.7767	1.57163	0.785816	−	0.618460i	$-0.212243\pi$
$269$	25.7767	1.57163	0.785816	+	0.618460i	$0.212243\pi$
$270$	0	0
$271$	27.2910	1.65781	0.828905	−	0.559389i	$-0.188964\pi$
$271$	27.2910	1.65781	0.828905	+	0.559389i	$0.188964\pi$
$272$	−2.72230	−0.165064
$273$	0	0
$274$	−4.12041	−0.248923
$275$	14.0854	0.849381
$276$	0	0
$277$	17.1849	1.03254	0.516270	−	0.856426i	$-0.327320\pi$
$277$	17.1849	1.03254	0.516270	+	0.856426i	$0.327320\pi$
$278$	7.49715	0.449649
$279$	0	0
$280$	11.0220	0.658693
$281$	14.8497	0.885859	0.442930	−	0.896556i	$-0.353939\pi$
$281$	14.8497	0.885859	0.442930	+	0.896556i	$0.353939\pi$
$282$	0	0
$283$	−0.279529	−0.0166163	−0.00830814	−	0.999965i	$-0.502645\pi$
$283$	−0.279529	−0.0166163	−0.00830814	+	0.999965i	$0.502645\pi$
$284$	9.40656	0.558177
$285$	0	0
$286$	−9.61473	−0.568531
$287$	27.7871	1.64022
$288$	0	0
$289$	−9.58906	−0.564062
$290$	24.6361	1.44668
$291$	0	0
$292$	3.71277	0.217273
$293$	11.6084	0.678170	0.339085	−	0.940756i	$-0.389883\pi$
$293$	11.6084	0.678170	0.339085	+	0.940756i	$0.389883\pi$
$294$	0	0
$295$	−6.90347	−0.401936
$296$	4.72230	0.274478
$297$	0	0
$298$	−17.3802	−1.00681
$299$	−3.43640	−0.198732
$300$	0	0
$301$	−30.6148	−1.76461
$302$	16.8328	0.968617
$303$	0	0
$304$	4.24918	0.243707
$305$	1.09806	0.0628747
$306$	0	0
$307$	12.0151	0.685739	0.342869	−	0.939383i	$-0.388601\pi$
$307$	12.0151	0.685739	0.342869	+	0.939383i	$0.388601\pi$
$308$	−5.55676	−0.316626
$309$	0	0
$310$	3.36286	0.190998
$311$	−18.4138	−1.04415	−0.522076	−	0.852899i	$-0.674842\pi$
$311$	−18.4138	−1.04415	−0.522076	+	0.852899i	$0.674842\pi$
$312$	0	0
$313$	19.2139	1.08603	0.543016	−	0.839723i	$-0.317282\pi$
$313$	19.2139	1.08603	0.543016	+	0.839723i	$0.317282\pi$
$314$	4.56737	0.257752
$315$	0	0
$316$	13.3724	0.752257
$317$	−5.73748	−0.322249	−0.161125	−	0.986934i	$-0.551512\pi$
$317$	−5.73748	−0.322249	−0.161125	+	0.986934i	$0.551512\pi$
$318$	0	0
$319$	−12.4203	−0.695402
$320$	−3.57869	−0.200055
$321$	0	0
$322$	−1.98604	−0.110678
$323$	−11.5676	−0.643636
$324$	0	0
$325$	−41.6043	−2.30779
$326$	−0.808621	−0.0447854
$327$	0	0
$328$	−9.02205	−0.498159
$329$	11.3009	0.623041
$330$	0	0
$331$	1.64835	0.0906014	0.0453007	−	0.998973i	$-0.485575\pi$
$331$	1.64835	0.0906014	0.0453007	+	0.998973i	$0.485575\pi$
$332$	12.5811	0.690477
$333$	0	0
$334$	19.4239	1.06283
$335$	2.72543	0.148906
$336$	0	0
$337$	11.1048	0.604916	0.302458	−	0.953163i	$-0.402193\pi$
$337$	11.1048	0.604916	0.302458	+	0.953163i	$0.402193\pi$
$338$	15.3992	0.837607
$339$	0	0
$340$	9.74228	0.528349
$341$	−1.69539	−0.0918104
$342$	0	0
$343$	13.9032	0.750700
$344$	9.94016	0.535937
$345$	0	0
$346$	−14.9037	−0.801228
$347$	19.7747	1.06156	0.530782	−	0.847508i	$-0.321898\pi$
$347$	19.7747	1.06156	0.530782	+	0.847508i	$0.321898\pi$
$348$	0	0
$349$	−10.3089	−0.551823	−0.275911	−	0.961183i	$-0.588980\pi$
$349$	−10.3089	−0.551823	−0.275911	+	0.961183i	$0.588980\pi$
$350$	−24.0449	−1.28526
$351$	0	0
$352$	1.80420	0.0961640
$353$	−5.50455	−0.292978	−0.146489	−	0.989212i	$-0.546797\pi$
$353$	−5.50455	−0.292978	−0.146489	+	0.989212i	$0.546797\pi$
$354$	0	0
$355$	−33.6632	−1.78665
$356$	−4.87243	−0.258238
$357$	0	0
$358$	10.1042	0.534026
$359$	4.16661	0.219905	0.109953	−	0.993937i	$-0.464930\pi$
$359$	4.16661	0.219905	0.109953	+	0.993937i	$0.464930\pi$
$360$	0	0
$361$	−0.944474	−0.0497091
$362$	−2.89882	−0.152359
$363$	0	0
$364$	16.4131	0.860282
$365$	−13.2869	−0.695466
$366$	0	0
$367$	8.81489	0.460134	0.230067	−	0.973175i	$-0.426106\pi$
$367$	8.81489	0.460134	0.230067	+	0.973175i	$0.426106\pi$
$368$	0.644837	0.0336145
$369$	0	0
$370$	−16.8997	−0.878572
$371$	−10.7846	−0.559910
$372$	0	0
$373$	−25.9674	−1.34454	−0.672270	−	0.740306i	$-0.734680\pi$
$373$	−25.9674	−1.34454	−0.672270	+	0.740306i	$0.734680\pi$
$374$	−4.91157	−0.253971
$375$	0	0
$376$	−3.66924	−0.189227
$377$	36.6860	1.88943
$378$	0	0
$379$	−8.49387	−0.436301	−0.218150	−	0.975915i	$-0.570002\pi$
$379$	−8.49387	−0.436301	−0.218150	+	0.975915i	$0.570002\pi$
$380$	−15.2065	−0.780077
$381$	0	0
$382$	3.79665	0.194253
$383$	−37.7753	−1.93023	−0.965114	−	0.261829i	$-0.915674\pi$
$383$	−37.7753	−1.93023	−0.965114	+	0.261829i	$0.915674\pi$
$384$	0	0
$385$	19.8859	1.01348
$386$	−1.49200	−0.0759406
$387$	0	0
$388$	9.44564	0.479530
$389$	−20.2617	−1.02731	−0.513654	−	0.857997i	$-0.671709\pi$
$389$	−20.2617	−1.02731	−0.513654	+	0.857997i	$0.671709\pi$
$390$	0	0
$391$	−1.75544	−0.0887766
$392$	2.48586	0.125555
$393$	0	0
$394$	−10.4870	−0.528328
$395$	−47.8557	−2.40788
$396$	0	0
$397$	−25.0338	−1.25641	−0.628205	−	0.778047i	$-0.716210\pi$
$397$	−25.0338	−1.25641	−0.628205	+	0.778047i	$0.716210\pi$
$398$	7.20710	0.361259
$399$	0	0
$400$	7.80702	0.390351
$401$	26.1843	1.30758	0.653791	−	0.756675i	$-0.273177\pi$
$401$	26.1843	1.30758	0.653791	+	0.756675i	$0.273177\pi$
$402$	0	0
$403$	5.00770	0.249451
$404$	−17.0368	−0.847611
$405$	0	0
$406$	21.2024	1.05226
$407$	8.51996	0.422319
$408$	0	0
$409$	−38.0178	−1.87986	−0.939929	−	0.341369i	$-0.889110\pi$
$409$	−38.0178	−1.87986	−0.939929	+	0.341369i	$0.889110\pi$
$410$	32.2871	1.59455
$411$	0	0
$412$	11.9895	0.590679
$413$	−5.94130	−0.292352
$414$	0	0
$415$	−45.0238	−2.21013
$416$	−5.32909	−0.261280
$417$	0	0
$418$	7.66635	0.374974
$419$	35.8523	1.75150	0.875749	−	0.482767i	$-0.160368\pi$
$419$	35.8523	1.75150	0.875749	+	0.482767i	$0.160368\pi$
$420$	0	0
$421$	23.1239	1.12699	0.563495	−	0.826119i	$-0.309456\pi$
$421$	23.1239	1.12699	0.563495	+	0.826119i	$0.309456\pi$
$422$	−25.6998	−1.25105
$423$	0	0
$424$	3.50160	0.170053
$425$	−21.2531	−1.03093
$426$	0	0
$427$	0.945018	0.0457326
$428$	1.77487	0.0857916
$429$	0	0
$430$	−35.5727	−1.71547
$431$	−30.0470	−1.44731	−0.723656	−	0.690160i	$-0.757540\pi$
$431$	−30.0470	−1.44731	−0.723656	+	0.690160i	$0.757540\pi$
$432$	0	0
$433$	18.5938	0.893559	0.446780	−	0.894644i	$-0.352571\pi$
$433$	18.5938	0.893559	0.446780	+	0.894644i	$0.352571\pi$
$434$	2.89417	0.138924
$435$	0	0
$436$	13.1912	0.631742
$437$	2.74003	0.131073
$438$	0	0
$439$	22.6285	1.08000	0.539999	−	0.841666i	$-0.318425\pi$
$439$	22.6285	1.08000	0.539999	+	0.841666i	$0.318425\pi$
$440$	−6.45666	−0.307809
$441$	0	0
$442$	14.5074	0.690047
$443$	−39.7144	−1.88689	−0.943444	−	0.331531i	$-0.892435\pi$
$443$	−39.7144	−1.88689	−0.943444	+	0.331531i	$0.892435\pi$
$444$	0	0
$445$	17.4369	0.826590
$446$	−25.9950	−1.23090
$447$	0	0
$448$	−3.07991	−0.145512
$449$	−39.2262	−1.85120	−0.925599	−	0.378506i	$-0.876438\pi$
$449$	−39.2262	−1.85120	−0.925599	+	0.378506i	$0.876438\pi$
$450$	0	0
$451$	−16.2775	−0.766480
$452$	16.9233	0.796006
$453$	0	0
$454$	12.2373	0.574325
$455$	−58.7375	−2.75366
$456$	0	0
$457$	−34.4192	−1.61006	−0.805032	−	0.593232i	$-0.797852\pi$
$457$	−34.4192	−1.61006	−0.805032	+	0.593232i	$0.797852\pi$
$458$	−25.1563	−1.17548
$459$	0	0
$460$	−2.30767	−0.107596
$461$	17.9984	0.838270	0.419135	−	0.907924i	$-0.362333\pi$
$461$	17.9984	0.838270	0.419135	+	0.907924i	$0.362333\pi$
$462$	0	0
$463$	−5.85010	−0.271877	−0.135939	−	0.990717i	$-0.543405\pi$
$463$	−5.85010	−0.271877	−0.135939	+	0.990717i	$0.543405\pi$
$464$	−6.88411	−0.319587
$465$	0	0
$466$	6.99284	0.323937
$467$	10.1514	0.469749	0.234875	−	0.972026i	$-0.424532\pi$
$467$	10.1514	0.469749	0.234875	+	0.972026i	$0.424532\pi$
$468$	0	0
$469$	2.34557	0.108308
$470$	13.1311	0.605692
$471$	0	0
$472$	1.92905	0.0887917
$473$	17.9340	0.824606
$474$	0	0
$475$	33.1734	1.52210
$476$	8.38446	0.384301
$477$	0	0
$478$	−5.46724	−0.250066
$479$	17.9493	0.820125	0.410063	−	0.912057i	$-0.365507\pi$
$479$	17.9493	0.820125	0.410063	+	0.912057i	$0.365507\pi$
$480$	0	0
$481$	−25.1656	−1.14745
$482$	1.00000	0.0455488
$483$	0	0
$484$	−7.74488	−0.352040
$485$	−33.8030	−1.53492
$486$	0	0
$487$	−17.7582	−0.804700	−0.402350	−	0.915486i	$-0.631806\pi$
$487$	−17.7582	−0.804700	−0.402350	+	0.915486i	$0.631806\pi$
$488$	−0.306833	−0.0138897
$489$	0	0
$490$	−8.89611	−0.401885
$491$	31.6628	1.42892	0.714461	−	0.699675i	$-0.246672\pi$
$491$	31.6628	1.42892	0.714461	+	0.699675i	$0.246672\pi$
$492$	0	0
$493$	18.7406	0.844036
$494$	−22.6443	−1.01881
$495$	0	0
$496$	−0.939691	−0.0421934
$497$	−28.9714	−1.29954
$498$	0	0
$499$	27.4259	1.22775	0.613877	−	0.789402i	$-0.289609\pi$
$499$	27.4259	1.22775	0.613877	+	0.789402i	$0.289609\pi$
$500$	−10.0454	−0.449246
$501$	0	0
$502$	−19.3959	−0.865680
$503$	−16.4039	−0.731413	−0.365707	−	0.930730i	$-0.619173\pi$
$503$	−16.4039	−0.731413	−0.365707	+	0.930730i	$0.619173\pi$
$504$	0	0
$505$	60.9693	2.71310
$506$	1.16341	0.0517200
$507$	0	0
$508$	3.66626	0.162664
$509$	12.1281	0.537569	0.268785	−	0.963200i	$-0.413378\pi$
$509$	12.1281	0.537569	0.268785	+	0.963200i	$0.413378\pi$
$510$	0	0
$511$	−11.4350	−0.505855
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.28440	0.100761
$515$	−42.9066	−1.89069
$516$	0	0
$517$	−6.62003	−0.291149
$518$	−14.5443	−0.639039
$519$	0	0
$520$	19.0712	0.836326
$521$	3.27604	0.143526	0.0717629	−	0.997422i	$-0.477137\pi$
$521$	3.27604	0.143526	0.0717629	+	0.997422i	$0.477137\pi$
$522$	0	0
$523$	−10.1517	−0.443902	−0.221951	−	0.975058i	$-0.571243\pi$
$523$	−10.1517	−0.443902	−0.221951	+	0.975058i	$0.571243\pi$
$524$	0.502637	0.0219578
$525$	0	0
$526$	−9.32886	−0.406758
$527$	2.55813	0.111434
$528$	0	0
$529$	−22.5842	−0.981921
$530$	−12.5311	−0.544318
$531$	0	0
$532$	−13.0871	−0.567398
$533$	48.0793	2.08255
$534$	0	0
$535$	−6.35171	−0.274608
$536$	−0.761571	−0.0328949
$537$	0	0
$538$	25.7767	1.11131
$539$	4.48497	0.193181
$540$	0	0
$541$	−3.97812	−0.171033	−0.0855163	−	0.996337i	$-0.527254\pi$
$541$	−3.97812	−0.171033	−0.0855163	+	0.996337i	$0.527254\pi$
$542$	27.2910	1.17225
$543$	0	0
$544$	−2.72230	−0.116718
$545$	−47.2070	−2.02213
$546$	0	0
$547$	30.3081	1.29588	0.647940	−	0.761691i	$-0.275631\pi$
$547$	30.3081	1.29588	0.647940	+	0.761691i	$0.275631\pi$
$548$	−4.12041	−0.176015
$549$	0	0
$550$	14.0854	0.600603
$551$	−29.2518	−1.24617
$552$	0	0
$553$	−41.1858	−1.75140
$554$	17.1849	0.730117
$555$	0	0
$556$	7.49715	0.317950
$557$	20.8625	0.883971	0.441985	−	0.897022i	$-0.354274\pi$
$557$	20.8625	0.883971	0.441985	+	0.897022i	$0.354274\pi$
$558$	0	0
$559$	−52.9720	−2.24048
$560$	11.0220	0.465767
$561$	0	0
$562$	14.8497	0.626397
$563$	18.0694	0.761534	0.380767	−	0.924671i	$-0.375660\pi$
$563$	18.0694	0.761534	0.380767	+	0.924671i	$0.375660\pi$
$564$	0	0
$565$	−60.5633	−2.54792
$566$	−0.279529	−0.0117495
$567$	0	0
$568$	9.40656	0.394690
$569$	2.12719	0.0891765	0.0445883	−	0.999005i	$-0.485802\pi$
$569$	2.12719	0.0891765	0.0445883	+	0.999005i	$0.485802\pi$
$570$	0	0
$571$	38.6732	1.61842	0.809212	−	0.587517i	$-0.199894\pi$
$571$	38.6732	1.61842	0.809212	+	0.587517i	$0.199894\pi$
$572$	−9.61473	−0.402012
$573$	0	0
$574$	27.7871	1.15981
$575$	5.03426	0.209943
$576$	0	0
$577$	2.01228	0.0837723	0.0418861	−	0.999122i	$-0.486663\pi$
$577$	2.01228	0.0837723	0.0418861	+	0.999122i	$0.486663\pi$
$578$	−9.58906	−0.398852
$579$	0	0
$580$	24.6361	1.02296
$581$	−38.7487	−1.60757
$582$	0	0
$583$	6.31757	0.261647
$584$	3.71277	0.153636
$585$	0	0
$586$	11.6084	0.479538
$587$	39.4697	1.62909	0.814545	−	0.580101i	$-0.196987\pi$
$587$	39.4697	1.62909	0.814545	+	0.580101i	$0.196987\pi$
$588$	0	0
$589$	−3.99292	−0.164525
$590$	−6.90347	−0.284211
$591$	0	0
$592$	4.72230	0.194086
$593$	23.3097	0.957214	0.478607	−	0.878029i	$-0.341142\pi$
$593$	23.3097	0.957214	0.478607	+	0.878029i	$0.341142\pi$
$594$	0	0
$595$	−30.0054	−1.23010
$596$	−17.3802	−0.711920
$597$	0	0
$598$	−3.43640	−0.140525
$599$	6.13708	0.250754	0.125377	−	0.992109i	$-0.459986\pi$
$599$	6.13708	0.250754	0.125377	+	0.992109i	$0.459986\pi$
$600$	0	0
$601$	−18.6623	−0.761253	−0.380626	−	0.924729i	$-0.624292\pi$
$601$	−18.6623	−0.761253	−0.380626	+	0.924729i	$0.624292\pi$
$602$	−30.6148	−1.24777
$603$	0	0
$604$	16.8328	0.684915
$605$	27.7165	1.12684
$606$	0	0
$607$	−16.0797	−0.652656	−0.326328	−	0.945257i	$-0.605811\pi$
$607$	−16.0797	−0.652656	−0.326328	+	0.945257i	$0.605811\pi$
$608$	4.24918	0.172327
$609$	0	0
$610$	1.09806	0.0444591
$611$	19.5537	0.791059
$612$	0	0
$613$	43.1591	1.74318	0.871591	−	0.490234i	$-0.163089\pi$
$613$	43.1591	1.74318	0.871591	+	0.490234i	$0.163089\pi$
$614$	12.0151	0.484891
$615$	0	0
$616$	−5.55676	−0.223888
$617$	28.0103	1.12765	0.563826	−	0.825894i	$-0.309329\pi$
$617$	28.0103	1.12765	0.563826	+	0.825894i	$0.309329\pi$
$618$	0	0
$619$	29.1482	1.17157	0.585783	−	0.810468i	$-0.300787\pi$
$619$	29.1482	1.17157	0.585783	+	0.810468i	$0.300787\pi$
$620$	3.36286	0.135056
$621$	0	0
$622$	−18.4138	−0.738326
$623$	15.0067	0.601229
$624$	0	0
$625$	−3.08556	−0.123422
$626$	19.2139	0.767940
$627$	0	0
$628$	4.56737	0.182258
$629$	−12.8555	−0.512584
$630$	0	0
$631$	−41.9581	−1.67033	−0.835163	−	0.550003i	$-0.814627\pi$
$631$	−41.9581	−1.67033	−0.835163	+	0.550003i	$0.814627\pi$
$632$	13.3724	0.531926
$633$	0	0
$634$	−5.73748	−0.227865
$635$	−13.1204	−0.520667
$636$	0	0
$637$	−13.2474	−0.524879
$638$	−12.4203	−0.491724
$639$	0	0
$640$	−3.57869	−0.141460
$641$	−22.1343	−0.874251	−0.437126	−	0.899400i	$-0.644004\pi$
$641$	−22.1343	−0.874251	−0.437126	+	0.899400i	$0.644004\pi$
$642$	0	0
$643$	14.6985	0.579651	0.289825	−	0.957080i	$-0.406403\pi$
$643$	14.6985	0.579651	0.289825	+	0.957080i	$0.406403\pi$
$644$	−1.98604	−0.0782610
$645$	0	0
$646$	−11.5676	−0.455120
$647$	5.66019	0.222525	0.111262	−	0.993791i	$-0.464511\pi$
$647$	5.66019	0.222525	0.111262	+	0.993791i	$0.464511\pi$
$648$	0	0
$649$	3.48038	0.136617
$650$	−41.6043	−1.63186
$651$	0	0
$652$	−0.808621	−0.0316680
$653$	31.6478	1.23847	0.619237	−	0.785204i	$-0.287442\pi$
$653$	31.6478	1.23847	0.619237	+	0.785204i	$0.287442\pi$
$654$	0	0
$655$	−1.79878	−0.0702842
$656$	−9.02205	−0.352252
$657$	0	0
$658$	11.3009	0.440557
$659$	−28.2101	−1.09891	−0.549455	−	0.835523i	$-0.685165\pi$
$659$	−28.2101	−1.09891	−0.549455	+	0.835523i	$0.685165\pi$
$660$	0	0
$661$	−23.0156	−0.895202	−0.447601	−	0.894233i	$-0.647721\pi$
$661$	−23.0156	−0.895202	−0.447601	+	0.894233i	$0.647721\pi$
$662$	1.64835	0.0640648
$663$	0	0
$664$	12.5811	0.488241
$665$	46.8347	1.81617
$666$	0	0
$667$	−4.43913	−0.171884
$668$	19.4239	0.751533
$669$	0	0
$670$	2.72543	0.105292
$671$	−0.553587	−0.0213710
$672$	0	0
$673$	27.0817	1.04392	0.521961	−	0.852970i	$-0.325201\pi$
$673$	27.0817	1.04392	0.521961	+	0.852970i	$0.325201\pi$
$674$	11.1048	0.427740
$675$	0	0
$676$	15.3992	0.592277
$677$	40.4462	1.55448	0.777238	−	0.629207i	$-0.216620\pi$
$677$	40.4462	1.55448	0.777238	+	0.629207i	$0.216620\pi$
$678$	0	0
$679$	−29.0917	−1.11644
$680$	9.74228	0.373599
$681$	0	0
$682$	−1.69539	−0.0649197
$683$	27.3048	1.04479	0.522395	−	0.852704i	$-0.325039\pi$
$683$	27.3048	1.04479	0.522395	+	0.852704i	$0.325039\pi$
$684$	0	0
$685$	14.7457	0.563403
$686$	13.9032	0.530825
$687$	0	0
$688$	9.94016	0.378965
$689$	−18.6603	−0.710903
$690$	0	0
$691$	26.9506	1.02525	0.512625	−	0.858613i	$-0.328673\pi$
$691$	26.9506	1.02525	0.512625	+	0.858613i	$0.328673\pi$
$692$	−14.9037	−0.566554
$693$	0	0
$694$	19.7747	0.750639
$695$	−26.8300	−1.01772
$696$	0	0
$697$	24.5608	0.930305
$698$	−10.3089	−0.390198
$699$	0	0
$700$	−24.0449	−0.908813
$701$	−19.2985	−0.728895	−0.364448	−	0.931224i	$-0.618742\pi$
$701$	−19.2985	−0.728895	−0.364448	+	0.931224i	$0.618742\pi$
$702$	0	0
$703$	20.0659	0.756800
$704$	1.80420	0.0679982
$705$	0	0
$706$	−5.50455	−0.207167
$707$	52.4717	1.97340
$708$	0	0
$709$	−0.218928	−0.00822203	−0.00411101	−	0.999992i	$-0.501309\pi$
$709$	−0.218928	−0.00822203	−0.00411101	+	0.999992i	$0.501309\pi$
$710$	−33.6632	−1.26336
$711$	0	0
$712$	−4.87243	−0.182602
$713$	−0.605948	−0.0226929
$714$	0	0
$715$	34.4081	1.28679
$716$	10.1042	0.377613
$717$	0	0
$718$	4.16661	0.155497
$719$	44.2817	1.65143	0.825714	−	0.564089i	$-0.190773\pi$
$719$	44.2817	1.65143	0.825714	+	0.564089i	$0.190773\pi$
$720$	0	0
$721$	−36.9265	−1.37522
$722$	−0.944474	−0.0351497
$723$	0	0
$724$	−2.89882	−0.107734
$725$	−53.7444	−1.99602
$726$	0	0
$727$	−30.0599	−1.11486	−0.557431	−	0.830223i	$-0.688213\pi$
$727$	−30.0599	−1.11486	−0.557431	+	0.830223i	$0.688213\pi$
$728$	16.4131	0.608311
$729$	0	0
$730$	−13.2869	−0.491768
$731$	−27.0601	−1.00086
$732$	0	0
$733$	40.9180	1.51134	0.755670	−	0.654953i	$-0.227311\pi$
$733$	40.9180	1.51134	0.755670	+	0.654953i	$0.227311\pi$
$734$	8.81489	0.325364
$735$	0	0
$736$	0.644837	0.0237690
$737$	−1.37402	−0.0506128
$738$	0	0
$739$	−14.6794	−0.539991	−0.269995	−	0.962862i	$-0.587022\pi$
$739$	−14.6794	−0.539991	−0.269995	+	0.962862i	$0.587022\pi$
$740$	−16.8997	−0.621244
$741$	0	0
$742$	−10.7846	−0.395916
$743$	−6.00489	−0.220298	−0.110149	−	0.993915i	$-0.535133\pi$
$743$	−6.00489	−0.220298	−0.110149	+	0.993915i	$0.535133\pi$
$744$	0	0
$745$	62.1983	2.27877
$746$	−25.9674	−0.950733
$747$	0	0
$748$	−4.91157	−0.179585
$749$	−5.46645	−0.199740
$750$	0	0
$751$	−51.1790	−1.86755	−0.933775	−	0.357861i	$-0.883506\pi$
$751$	−51.1790	−1.86755	−0.933775	+	0.357861i	$0.883506\pi$
$752$	−3.66924	−0.133804
$753$	0	0
$754$	36.6860	1.33603
$755$	−60.2392	−2.19233
$756$	0	0
$757$	29.0634	1.05633	0.528163	−	0.849143i	$-0.322881\pi$
$757$	29.0634	1.05633	0.528163	+	0.849143i	$0.322881\pi$
$758$	−8.49387	−0.308511
$759$	0	0
$760$	−15.2065	−0.551598
$761$	−37.9206	−1.37462	−0.687310	−	0.726364i	$-0.741208\pi$
$761$	−37.9206	−1.37462	−0.687310	+	0.726364i	$0.741208\pi$
$762$	0	0
$763$	−40.6276	−1.47082
$764$	3.79665	0.137358
$765$	0	0
$766$	−37.7753	−1.36488
$767$	−10.2801	−0.371192
$768$	0	0
$769$	−32.9236	−1.18725	−0.593627	−	0.804740i	$-0.702304\pi$
$769$	−32.9236	−1.18725	−0.593627	+	0.804740i	$0.702304\pi$
$770$	19.8859	0.716639
$771$	0	0
$772$	−1.49200	−0.0536981
$773$	41.6100	1.49661	0.748303	−	0.663357i	$-0.230869\pi$
$773$	41.6100	1.49661	0.748303	+	0.663357i	$0.230869\pi$
$774$	0	0
$775$	−7.33619	−0.263524
$776$	9.44564	0.339079
$777$	0	0
$778$	−20.2617	−0.726417
$779$	−38.3363	−1.37354
$780$	0	0
$781$	16.9713	0.607280
$782$	−1.75544	−0.0627745
$783$	0	0
$784$	2.48586	0.0887806
$785$	−16.3452	−0.583385
$786$	0	0
$787$	31.7435	1.13153	0.565767	−	0.824565i	$-0.308580\pi$
$787$	31.7435	1.13153	0.565767	+	0.824565i	$0.308580\pi$
$788$	−10.4870	−0.373584
$789$	0	0
$790$	−47.8557	−1.70263
$791$	−52.1223	−1.85326
$792$	0	0
$793$	1.63514	0.0580655
$794$	−25.0338	−0.888417
$795$	0	0
$796$	7.20710	0.255449
$797$	−26.2587	−0.930129	−0.465065	−	0.885277i	$-0.653969\pi$
$797$	−26.2587	−0.930129	−0.465065	+	0.885277i	$0.653969\pi$
$798$	0	0
$799$	9.98879	0.353378
$800$	7.80702	0.276020
$801$	0	0
$802$	26.1843	0.924600
$803$	6.69857	0.236387
$804$	0	0
$805$	7.10743	0.250504
$806$	5.00770	0.176389
$807$	0	0
$808$	−17.0368	−0.599351
$809$	−13.2896	−0.467236	−0.233618	−	0.972328i	$-0.575056\pi$
$809$	−13.2896	−0.467236	−0.233618	+	0.972328i	$0.575056\pi$
$810$	0	0
$811$	12.2477	0.430073	0.215037	−	0.976606i	$-0.431013\pi$
$811$	12.2477	0.430073	0.215037	+	0.976606i	$0.431013\pi$
$812$	21.2024	0.744060
$813$	0	0
$814$	8.51996	0.298625
$815$	2.89380	0.101365
$816$	0	0
$817$	42.2375	1.47770
$818$	−38.0178	−1.32926
$819$	0	0
$820$	32.2871	1.12751
$821$	52.1673	1.82065	0.910326	−	0.413892i	$-0.135831\pi$
$821$	52.1673	1.82065	0.910326	+	0.413892i	$0.135831\pi$
$822$	0	0
$823$	42.5896	1.48458	0.742291	−	0.670078i	$-0.233739\pi$
$823$	42.5896	1.48458	0.742291	+	0.670078i	$0.233739\pi$
$824$	11.9895	0.417673
$825$	0	0
$826$	−5.94130	−0.206724
$827$	−37.0521	−1.28843	−0.644214	−	0.764846i	$-0.722815\pi$
$827$	−37.0521	−1.28843	−0.644214	+	0.764846i	$0.722815\pi$
$828$	0	0
$829$	24.9792	0.867564	0.433782	−	0.901018i	$-0.357179\pi$
$829$	24.9792	0.867564	0.433782	+	0.901018i	$0.357179\pi$
$830$	−45.0238	−1.56280
$831$	0	0
$832$	−5.32909	−0.184753
$833$	−6.76726	−0.234472
$834$	0	0
$835$	−69.5120	−2.40556
$836$	7.66635	0.265146
$837$	0	0
$838$	35.8523	1.23850
$839$	−16.3061	−0.562951	−0.281475	−	0.959568i	$-0.590824\pi$
$839$	−16.3061	−0.562951	−0.281475	+	0.959568i	$0.590824\pi$
$840$	0	0
$841$	18.3909	0.634170
$842$	23.1239	0.796903
$843$	0	0
$844$	−25.6998	−0.884625
$845$	−55.1090	−1.89581
$846$	0	0
$847$	23.8535	0.819617
$848$	3.50160	0.120245
$849$	0	0
$850$	−21.2531	−0.728975
$851$	3.04512	0.104385
$852$	0	0
$853$	2.20738	0.0755792	0.0377896	−	0.999286i	$-0.487968\pi$
$853$	2.20738	0.0755792	0.0377896	+	0.999286i	$0.487968\pi$
$854$	0.945018	0.0323379
$855$	0	0
$856$	1.77487	0.0606638
$857$	43.5204	1.48663	0.743314	−	0.668942i	$-0.233253\pi$
$857$	43.5204	1.48663	0.743314	+	0.668942i	$0.233253\pi$
$858$	0	0
$859$	36.2344	1.23630	0.618151	−	0.786059i	$-0.287882\pi$
$859$	36.2344	1.23630	0.618151	+	0.786059i	$0.287882\pi$
$860$	−35.5727	−1.21302
$861$	0	0
$862$	−30.0470	−1.02340
$863$	50.1579	1.70739	0.853697	−	0.520771i	$-0.174355\pi$
$863$	50.1579	1.70739	0.853697	+	0.520771i	$0.174355\pi$
$864$	0	0
$865$	53.3358	1.81347
$866$	18.5938	0.631842
$867$	0	0
$868$	2.89417	0.0982344
$869$	24.1264	0.818434
$870$	0	0
$871$	4.05848	0.137516
$872$	13.1912	0.446709
$873$	0	0
$874$	2.74003	0.0926828
$875$	30.9391	1.04593
$876$	0	0
$877$	−10.8562	−0.366587	−0.183294	−	0.983058i	$-0.558676\pi$
$877$	−10.8562	−0.366587	−0.183294	+	0.983058i	$0.558676\pi$
$878$	22.6285	0.763674
$879$	0	0
$880$	−6.45666	−0.217654
$881$	32.8827	1.10785	0.553923	−	0.832568i	$-0.313130\pi$
$881$	32.8827	1.10785	0.553923	+	0.832568i	$0.313130\pi$
$882$	0	0
$883$	45.4993	1.53117	0.765587	−	0.643333i	$-0.222449\pi$
$883$	45.4993	1.53117	0.765587	+	0.643333i	$0.222449\pi$
$884$	14.5074	0.487937
$885$	0	0
$886$	−39.7144	−1.33423
$887$	19.8616	0.666889	0.333444	−	0.942770i	$-0.391789\pi$
$887$	19.8616	0.666889	0.333444	+	0.942770i	$0.391789\pi$
$888$	0	0
$889$	−11.2917	−0.378713
$890$	17.4369	0.584487
$891$	0	0
$892$	−25.9950	−0.870378
$893$	−15.5913	−0.521742
$894$	0	0
$895$	−36.1599	−1.20869
$896$	−3.07991	−0.102893
$897$	0	0
$898$	−39.2262	−1.30899
$899$	6.46894	0.215751
$900$	0	0
$901$	−9.53242	−0.317571
$902$	−16.2775	−0.541983
$903$	0	0
$904$	16.9233	0.562861
$905$	10.3740	0.344843
$906$	0	0
$907$	−43.4376	−1.44232	−0.721161	−	0.692767i	$-0.756391\pi$
$907$	−43.4376	−1.44232	−0.721161	+	0.692767i	$0.756391\pi$
$908$	12.2373	0.406109
$909$	0	0
$910$	−58.7375	−1.94713
$911$	19.6939	0.652487	0.326244	−	0.945286i	$-0.394217\pi$
$911$	19.6939	0.652487	0.326244	+	0.945286i	$0.394217\pi$
$912$	0	0
$913$	22.6988	0.751219
$914$	−34.4192	−1.13849
$915$	0	0
$916$	−25.1563	−0.831188
$917$	−1.54808	−0.0511220
$918$	0	0
$919$	21.6435	0.713952	0.356976	−	0.934114i	$-0.383808\pi$
$919$	21.6435	0.713952	0.356976	+	0.934114i	$0.383808\pi$
$920$	−2.30767	−0.0760817
$921$	0	0
$922$	17.9984	0.592746
$923$	−50.1284	−1.65000
$924$	0	0
$925$	36.8671	1.21218
$926$	−5.85010	−0.192246
$927$	0	0
$928$	−6.88411	−0.225982
$929$	2.82226	0.0925953	0.0462977	−	0.998928i	$-0.485258\pi$
$929$	2.82226	0.0925953	0.0462977	+	0.998928i	$0.485258\pi$
$930$	0	0
$931$	10.5628	0.346183
$932$	6.99284	0.229058
$933$	0	0
$934$	10.1514	0.332163
$935$	17.5770	0.574829
$936$	0	0
$937$	51.3450	1.67737	0.838684	−	0.544618i	$-0.183326\pi$
$937$	51.3450	1.67737	0.838684	+	0.544618i	$0.183326\pi$
$938$	2.34557	0.0765857
$939$	0	0
$940$	13.1311	0.428289
$941$	7.68275	0.250450	0.125225	−	0.992128i	$-0.460035\pi$
$941$	7.68275	0.250450	0.125225	+	0.992128i	$0.460035\pi$
$942$	0	0
$943$	−5.81775	−0.189452
$944$	1.92905	0.0627852
$945$	0	0
$946$	17.9340	0.583084
$947$	33.6126	1.09226	0.546131	−	0.837700i	$-0.316100\pi$
$947$	33.6126	1.09226	0.546131	+	0.837700i	$0.316100\pi$
$948$	0	0
$949$	−19.7857	−0.642271
$950$	33.1734	1.07629
$951$	0	0
$952$	8.38446	0.271742
$953$	31.7220	1.02758	0.513788	−	0.857917i	$-0.328242\pi$
$953$	31.7220	1.02758	0.513788	+	0.857917i	$0.328242\pi$
$954$	0	0
$955$	−13.5870	−0.439666
$956$	−5.46724	−0.176823
$957$	0	0
$958$	17.9493	0.579916
$959$	12.6905	0.409797
$960$	0	0
$961$	−30.1170	−0.971515
$962$	−25.1656	−0.811371
$963$	0	0
$964$	1.00000	0.0322078
$965$	5.33939	0.171881
$966$	0	0
$967$	−26.3494	−0.847340	−0.423670	−	0.905817i	$-0.639258\pi$
$967$	−26.3494	−0.847340	−0.423670	+	0.905817i	$0.639258\pi$
$968$	−7.74488	−0.248930
$969$	0	0
$970$	−33.8030	−1.08535
$971$	−26.8117	−0.860427	−0.430214	−	0.902727i	$-0.641562\pi$
$971$	−26.8117	−0.860427	−0.430214	+	0.902727i	$0.641562\pi$
$972$	0	0
$973$	−23.0906	−0.740250
$974$	−17.7582	−0.569009
$975$	0	0
$976$	−0.306833	−0.00982148
$977$	46.4034	1.48458	0.742288	−	0.670081i	$-0.233741\pi$
$977$	46.4034	1.48458	0.742288	+	0.670081i	$0.233741\pi$
$978$	0	0
$979$	−8.79083	−0.280956
$980$	−8.89611	−0.284176
$981$	0	0
$982$	31.6628	1.01040
$983$	−40.3010	−1.28540	−0.642701	−	0.766117i	$-0.722186\pi$
$983$	−40.3010	−1.28540	−0.642701	+	0.766117i	$0.722186\pi$
$984$	0	0
$985$	37.5297	1.19580
$986$	18.7406	0.596823
$987$	0	0
$988$	−22.6443	−0.720410
$989$	6.40978	0.203819
$990$	0	0
$991$	2.46568	0.0783249	0.0391625	−	0.999233i	$-0.487531\pi$
$991$	2.46568	0.0783249	0.0391625	+	0.999233i	$0.487531\pi$
$992$	−0.939691	−0.0298352
$993$	0	0
$994$	−28.9714	−0.918916
$995$	−25.7920	−0.817661
$996$	0	0
$997$	22.3526	0.707913	0.353957	−	0.935262i	$-0.384836\pi$
$997$	22.3526	0.707913	0.353957	+	0.935262i	$0.384836\pi$
$998$	27.4259	0.868153
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4338.2.a.w.1.2		7
3.2	odd	2		1446.2.a.n.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1446.2.a.n.1.6	✓	7	3.2	odd	2
4338.2.a.w.1.2		7	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:	\( N \)	\(=\)	\( 4338 = 2 \cdot 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4338.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.57869 q^{5} -3.07991 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.57869 q^{5} -3.07991 q^{7} +1.00000 q^{8} -3.57869 q^{10} +1.80420 q^{11} -5.32909 q^{13} -3.07991 q^{14} +1.00000 q^{16} -2.72230 q^{17} +4.24918 q^{19} -3.57869 q^{20} +1.80420 q^{22} +0.644837 q^{23} +7.80702 q^{25} -5.32909 q^{26} -3.07991 q^{28} -6.88411 q^{29} -0.939691 q^{31} +1.00000 q^{32} -2.72230 q^{34} +11.0220 q^{35} +4.72230 q^{37} +4.24918 q^{38} -3.57869 q^{40} -9.02205 q^{41} +9.94016 q^{43} +1.80420 q^{44} +0.644837 q^{46} -3.66924 q^{47} +2.48586 q^{49} +7.80702 q^{50} -5.32909 q^{52} +3.50160 q^{53} -6.45666 q^{55} -3.07991 q^{56} -6.88411 q^{58} +1.92905 q^{59} -0.306833 q^{61} -0.939691 q^{62} +1.00000 q^{64} +19.0712 q^{65} -0.761571 q^{67} -2.72230 q^{68} +11.0220 q^{70} +9.40656 q^{71} +3.71277 q^{73} +4.72230 q^{74} +4.24918 q^{76} -5.55676 q^{77} +13.3724 q^{79} -3.57869 q^{80} -9.02205 q^{82} +12.5811 q^{83} +9.74228 q^{85} +9.94016 q^{86} +1.80420 q^{88} -4.87243 q^{89} +16.4131 q^{91} +0.644837 q^{92} -3.66924 q^{94} -15.2065 q^{95} +9.44564 q^{97} +2.48586 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) \) 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8	\( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8 - 5 * q^10 - 4 * q^11 + 14 * q^13 + 7 * q^14 + 7 * q^16 - 4 * q^17 + 7 * q^19 - 5 * q^20 - 4 * q^22 + q^23 + 14 * q^25 + 14 * q^26 + 7 * q^28 - 3 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^34 + 17 * q^35 + 18 * q^37 + 7 * q^38 - 5 * q^40 - 3 * q^41 + 11 * q^43 - 4 * q^44 + q^46 + 18 * q^47 + 20 * q^49 + 14 * q^50 + 14 * q^52 + 9 * q^53 - 4 * q^55 + 7 * q^56 - 3 * q^58 - 6 * q^59 + 31 * q^61 + 8 * q^62 + 7 * q^64 + 6 * q^65 + 4 * q^67 - 4 * q^68 + 17 * q^70 + 3 * q^71 + 16 * q^73 + 18 * q^74 + 7 * q^76 + 34 * q^79 - 5 * q^80 - 3 * q^82 - 10 * q^83 + 34 * q^85 + 11 * q^86 - 4 * q^88 - 24 * q^89 + 40 * q^91 + q^92 + 18 * q^94 + q^95 + 27 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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