LMFDB - Embedded newform 3675.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} +0.732051 q^{6} +2.53590 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} +0.732051 q^{6} +2.53590 q^{8} +1.00000 q^{9} -2.73205 q^{11} +1.46410 q^{12} +5.73205 q^{13} +1.07180 q^{16} +6.73205 q^{17} -0.732051 q^{18} +2.46410 q^{19} +2.00000 q^{22} +1.26795 q^{23} -2.53590 q^{24} -4.19615 q^{26} -1.00000 q^{27} +6.19615 q^{29} -6.46410 q^{31} -5.85641 q^{32} +2.73205 q^{33} -4.92820 q^{34} -1.46410 q^{36} -7.19615 q^{37} -1.80385 q^{38} -5.73205 q^{39} -2.73205 q^{41} +7.19615 q^{43} +4.00000 q^{44} -0.928203 q^{46} +2.00000 q^{47} -1.07180 q^{48} -6.73205 q^{51} -8.39230 q^{52} +8.39230 q^{53} +0.732051 q^{54} -2.46410 q^{57} -4.53590 q^{58} -10.1962 q^{59} -4.00000 q^{61} +4.73205 q^{62} +2.14359 q^{64} -2.00000 q^{66} -2.66025 q^{67} -9.85641 q^{68} -1.26795 q^{69} -4.19615 q^{71} +2.53590 q^{72} -4.66025 q^{73} +5.26795 q^{74} -3.60770 q^{76} +4.19615 q^{78} +13.3923 q^{79} +1.00000 q^{81} +2.00000 q^{82} -9.12436 q^{83} -5.26795 q^{86} -6.19615 q^{87} -6.92820 q^{88} +9.12436 q^{89} -1.85641 q^{92} +6.46410 q^{93} -1.46410 q^{94} +5.85641 q^{96} +1.07180 q^{97} -2.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} + 12 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^6 + 12 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} + 12 q^{8} + 2 q^{9} - 2 q^{11} - 4 q^{12} + 8 q^{13} + 16 q^{16} + 10 q^{17} + 2 q^{18} - 2 q^{19} + 4 q^{22} + 6 q^{23} - 12 q^{24} + 2 q^{26} - 2 q^{27} + 2 q^{29} - 6 q^{31} + 16 q^{32} + 2 q^{33} + 4 q^{34} + 4 q^{36} - 4 q^{37} - 14 q^{38} - 8 q^{39} - 2 q^{41} + 4 q^{43} + 8 q^{44} + 12 q^{46} + 4 q^{47} - 16 q^{48} - 10 q^{51} + 4 q^{52} - 4 q^{53} - 2 q^{54} + 2 q^{57} - 16 q^{58} - 10 q^{59} - 8 q^{61} + 6 q^{62} + 32 q^{64} - 4 q^{66} + 12 q^{67} + 8 q^{68} - 6 q^{69} + 2 q^{71} + 12 q^{72} + 8 q^{73} + 14 q^{74} - 28 q^{76} - 2 q^{78} + 6 q^{79} + 2 q^{81} + 4 q^{82} + 6 q^{83} - 14 q^{86} - 2 q^{87} - 6 q^{89} + 24 q^{92} + 6 q^{93} + 4 q^{94} - 16 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^6 + 12 * q^8 + 2 * q^9 - 2 * q^11 - 4 * q^12 + 8 * q^13 + 16 * q^16 + 10 * q^17 + 2 * q^18 - 2 * q^19 + 4 * q^22 + 6 * q^23 - 12 * q^24 + 2 * q^26 - 2 * q^27 + 2 * q^29 - 6 * q^31 + 16 * q^32 + 2 * q^33 + 4 * q^34 + 4 * q^36 - 4 * q^37 - 14 * q^38 - 8 * q^39 - 2 * q^41 + 4 * q^43 + 8 * q^44 + 12 * q^46 + 4 * q^47 - 16 * q^48 - 10 * q^51 + 4 * q^52 - 4 * q^53 - 2 * q^54 + 2 * q^57 - 16 * q^58 - 10 * q^59 - 8 * q^61 + 6 * q^62 + 32 * q^64 - 4 * q^66 + 12 * q^67 + 8 * q^68 - 6 * q^69 + 2 * q^71 + 12 * q^72 + 8 * q^73 + 14 * q^74 - 28 * q^76 - 2 * q^78 + 6 * q^79 + 2 * q^81 + 4 * q^82 + 6 * q^83 - 14 * q^86 - 2 * q^87 - 6 * q^89 + 24 * q^92 + 6 * q^93 + 4 * q^94 - 16 * q^96 + 16 * q^97 - 2 * q^99

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 $

$2$	−0.732051	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	−0.732051	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.46410	−0.732051
$5$	0	0
$5$	0	0
$6$	0.732051	0.298858
$7$	0	0
$7$	0	0
$8$	2.53590	0.896575
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.73205	−0.823744	−0.411872	−	0.911242i	$-0.635125\pi$
$11$	−2.73205	−0.823744	−0.411872	+	0.911242i	$0.635125\pi$
$12$	1.46410	0.422650
$13$	5.73205	1.58978	0.794892	−	0.606750i	$-0.207527\pi$
$13$	5.73205	1.58978	0.794892	+	0.606750i	$0.207527\pi$
$14$	0	0
$15$	0	0
$16$	1.07180	0.267949
$17$	6.73205	1.63276	0.816381	−	0.577514i	$-0.195977\pi$
$17$	6.73205	1.63276	0.816381	+	0.577514i	$0.195977\pi$
$18$	−0.732051	−0.172546
$19$	2.46410	0.565304	0.282652	−	0.959223i	$-0.408786\pi$
$19$	2.46410	0.565304	0.282652	+	0.959223i	$0.408786\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	1.26795	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$23$	1.26795	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$24$	−2.53590	−0.517638
$25$	0	0
$26$	−4.19615	−0.822933
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.19615	1.15060	0.575298	−	0.817944i	$-0.304886\pi$
$29$	6.19615	1.15060	0.575298	+	0.817944i	$0.304886\pi$
$30$	0	0
$31$	−6.46410	−1.16099	−0.580493	−	0.814265i	$-0.697140\pi$
$31$	−6.46410	−1.16099	−0.580493	+	0.814265i	$0.697140\pi$
$32$	−5.85641	−1.03528
$33$	2.73205	0.475589
$34$	−4.92820	−0.845180
$35$	0	0
$36$	−1.46410	−0.244017
$37$	−7.19615	−1.18304	−0.591520	−	0.806290i	$-0.701472\pi$
$37$	−7.19615	−1.18304	−0.591520	+	0.806290i	$0.701472\pi$
$38$	−1.80385	−0.292623
$39$	−5.73205	−0.917863
$40$	0	0
$41$	−2.73205	−0.426675	−0.213337	−	0.976979i	$-0.568433\pi$
$41$	−2.73205	−0.426675	−0.213337	+	0.976979i	$0.568433\pi$
$42$	0	0
$43$	7.19615	1.09740	0.548701	−	0.836018i	$-0.315122\pi$
$43$	7.19615	1.09740	0.548701	+	0.836018i	$0.315122\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−0.928203	−0.136856
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	−1.07180	−0.154701
$49$	0	0
$50$	0	0
$51$	−6.73205	−0.942676
$52$	−8.39230	−1.16380
$53$	8.39230	1.15277	0.576386	−	0.817178i	$-0.304463\pi$
$53$	8.39230	1.15277	0.576386	+	0.817178i	$0.304463\pi$
$54$	0.732051	0.0996195
$55$	0	0
$56$	0	0
$57$	−2.46410	−0.326378
$58$	−4.53590	−0.595593
$59$	−10.1962	−1.32743	−0.663713	−	0.747987i	$-0.731020\pi$
$59$	−10.1962	−1.32743	−0.663713	+	0.747987i	$0.731020\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	4.73205	0.600971
$63$	0	0
$64$	2.14359	0.267949
$65$	0	0
$66$	−2.00000	−0.246183
$67$	−2.66025	−0.325002	−0.162501	−	0.986708i	$-0.551956\pi$
$67$	−2.66025	−0.325002	−0.162501	+	0.986708i	$0.551956\pi$
$68$	−9.85641	−1.19526
$69$	−1.26795	−0.152643
$70$	0	0
$71$	−4.19615	−0.497992	−0.248996	−	0.968505i	$-0.580101\pi$
$71$	−4.19615	−0.497992	−0.248996	+	0.968505i	$0.580101\pi$
$72$	2.53590	0.298858
$73$	−4.66025	−0.545441	−0.272721	−	0.962093i	$-0.587924\pi$
$73$	−4.66025	−0.545441	−0.272721	+	0.962093i	$0.587924\pi$
$74$	5.26795	0.612387
$75$	0	0
$76$	−3.60770	−0.413831
$77$	0	0
$78$	4.19615	0.475121
$79$	13.3923	1.50675	0.753376	−	0.657590i	$-0.228424\pi$
$79$	13.3923	1.50675	0.753376	+	0.657590i	$0.228424\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−9.12436	−1.00153	−0.500764	−	0.865584i	$-0.666948\pi$
$83$	−9.12436	−1.00153	−0.500764	+	0.865584i	$0.666948\pi$
$84$	0	0
$85$	0	0
$86$	−5.26795	−0.568058
$87$	−6.19615	−0.664297
$88$	−6.92820	−0.738549
$89$	9.12436	0.967180	0.483590	−	0.875295i	$-0.339333\pi$
$89$	9.12436	0.967180	0.483590	+	0.875295i	$0.339333\pi$
$90$	0	0
$91$	0	0
$92$	−1.85641	−0.193544
$93$	6.46410	0.670296
$94$	−1.46410	−0.151011
$95$	0	0
$96$	5.85641	0.597717
$97$	1.07180	0.108824	0.0544122	−	0.998519i	$-0.482671\pi$
$97$	1.07180	0.108824	0.0544122	+	0.998519i	$0.482671\pi$
$98$	0	0
$99$	−2.73205	−0.274581
$100$	0	0
$101$	10.7321	1.06788	0.533939	−	0.845523i	$-0.320711\pi$
$101$	10.7321	1.06788	0.533939	+	0.845523i	$0.320711\pi$
$102$	4.92820	0.487965
$103$	1.19615	0.117860	0.0589302	−	0.998262i	$-0.481231\pi$
$103$	1.19615	0.117860	0.0589302	+	0.998262i	$0.481231\pi$
$104$	14.5359	1.42536
$105$	0	0
$106$	−6.14359	−0.596719
$107$	8.19615	0.792352	0.396176	−	0.918175i	$-0.370337\pi$
$107$	8.19615	0.792352	0.396176	+	0.918175i	$0.370337\pi$
$108$	1.46410	0.140883
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	7.19615	0.683029
$112$	0	0
$113$	4.92820	0.463606	0.231803	−	0.972763i	$-0.425537\pi$
$113$	4.92820	0.463606	0.231803	+	0.972763i	$0.425537\pi$
$114$	1.80385	0.168946
$115$	0	0
$116$	−9.07180	−0.842295
$117$	5.73205	0.529928
$118$	7.46410	0.687126
$119$	0	0
$120$	0	0
$121$	−3.53590	−0.321445
$122$	2.92820	0.265107
$123$	2.73205	0.246341
$124$	9.46410	0.849901
$125$	0	0
$126$	0	0
$127$	−15.1962	−1.34844	−0.674220	−	0.738530i	$-0.735520\pi$
$127$	−15.1962	−1.34844	−0.674220	+	0.738530i	$0.735520\pi$
$128$	10.1436	0.896575
$129$	−7.19615	−0.633586
$130$	0	0
$131$	−8.53590	−0.745785	−0.372892	−	0.927875i	$-0.621634\pi$
$131$	−8.53590	−0.745785	−0.372892	+	0.927875i	$0.621634\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	1.94744	0.168233
$135$	0	0
$136$	17.0718	1.46389
$137$	8.19615	0.700245	0.350122	−	0.936704i	$-0.386140\pi$
$137$	8.19615	0.700245	0.350122	+	0.936704i	$0.386140\pi$
$138$	0.928203	0.0790139
$139$	7.92820	0.672461	0.336231	−	0.941780i	$-0.390848\pi$
$139$	7.92820	0.672461	0.336231	+	0.941780i	$0.390848\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	3.07180	0.257779
$143$	−15.6603	−1.30958
$144$	1.07180	0.0893164
$145$	0	0
$146$	3.41154	0.282341
$147$	0	0
$148$	10.5359	0.866046
$149$	−21.8564	−1.79055	−0.895273	−	0.445517i	$-0.853020\pi$
$149$	−21.8564	−1.79055	−0.895273	+	0.445517i	$0.853020\pi$
$150$	0	0
$151$	4.92820	0.401051	0.200526	−	0.979688i	$-0.435735\pi$
$151$	4.92820	0.401051	0.200526	+	0.979688i	$0.435735\pi$
$152$	6.24871	0.506837
$153$	6.73205	0.544254
$154$	0	0
$155$	0	0
$156$	8.39230	0.671922
$157$	−14.3923	−1.14863	−0.574315	−	0.818634i	$-0.694732\pi$
$157$	−14.3923	−1.14863	−0.574315	+	0.818634i	$0.694732\pi$
$158$	−9.80385	−0.779952
$159$	−8.39230	−0.665553
$160$	0	0
$161$	0	0
$162$	−0.732051	−0.0575153
$163$	5.85641	0.458709	0.229355	−	0.973343i	$-0.426338\pi$
$163$	5.85641	0.458709	0.229355	+	0.973343i	$0.426338\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	6.67949	0.518429
$167$	−0.339746	−0.0262903	−0.0131452	−	0.999914i	$-0.504184\pi$
$167$	−0.339746	−0.0262903	−0.0131452	+	0.999914i	$0.504184\pi$
$168$	0	0
$169$	19.8564	1.52742
$170$	0	0
$171$	2.46410	0.188435
$172$	−10.5359	−0.803355
$173$	21.4641	1.63189	0.815943	−	0.578133i	$-0.196218\pi$
$173$	21.4641	1.63189	0.815943	+	0.578133i	$0.196218\pi$
$174$	4.53590	0.343866
$175$	0	0
$176$	−2.92820	−0.220722
$177$	10.1962	0.766390
$178$	−6.67949	−0.500649
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−10.3205	−0.767117	−0.383559	−	0.923517i	$-0.625302\pi$
$181$	−10.3205	−0.767117	−0.383559	+	0.923517i	$0.625302\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	3.21539	0.237042
$185$	0	0
$186$	−4.73205	−0.346971
$187$	−18.3923	−1.34498
$188$	−2.92820	−0.213561
$189$	0	0
$190$	0	0
$191$	4.92820	0.356592	0.178296	−	0.983977i	$-0.442941\pi$
$191$	4.92820	0.356592	0.178296	+	0.983977i	$0.442941\pi$
$192$	−2.14359	−0.154701
$193$	9.19615	0.661954	0.330977	−	0.943639i	$-0.392622\pi$
$193$	9.19615	0.661954	0.330977	+	0.943639i	$0.392622\pi$
$194$	−0.784610	−0.0563317
$195$	0	0
$196$	0	0
$197$	17.6603	1.25824	0.629121	−	0.777308i	$-0.283415\pi$
$197$	17.6603	1.25824	0.629121	+	0.777308i	$0.283415\pi$
$198$	2.00000	0.142134
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	0	0
$201$	2.66025	0.187640
$202$	−7.85641	−0.552775
$203$	0	0
$204$	9.85641	0.690086
$205$	0	0
$206$	−0.875644	−0.0610090
$207$	1.26795	0.0881286
$208$	6.14359	0.425982
$209$	−6.73205	−0.465666
$210$	0	0
$211$	20.9282	1.44076	0.720378	−	0.693581i	$-0.243968\pi$
$211$	20.9282	1.44076	0.720378	+	0.693581i	$0.243968\pi$
$212$	−12.2872	−0.843887
$213$	4.19615	0.287516
$214$	−6.00000	−0.410152
$215$	0	0
$216$	−2.53590	−0.172546
$217$	0	0
$218$	−8.05256	−0.545388
$219$	4.66025	0.314911
$220$	0	0
$221$	38.5885	2.59574
$222$	−5.26795	−0.353562
$223$	0.392305	0.0262707	0.0131353	−	0.999914i	$-0.495819\pi$
$223$	0.392305	0.0262707	0.0131353	+	0.999914i	$0.495819\pi$
$224$	0	0
$225$	0	0
$226$	−3.60770	−0.239980
$227$	15.6603	1.03941	0.519704	−	0.854347i	$-0.326042\pi$
$227$	15.6603	1.03941	0.519704	+	0.854347i	$0.326042\pi$
$228$	3.60770	0.238925
$229$	3.00000	0.198246	0.0991228	−	0.995075i	$-0.468396\pi$
$229$	3.00000	0.198246	0.0991228	+	0.995075i	$0.468396\pi$
$230$	0	0
$231$	0	0
$232$	15.7128	1.03160
$233$	17.3205	1.13470	0.567352	−	0.823475i	$-0.307968\pi$
$233$	17.3205	1.13470	0.567352	+	0.823475i	$0.307968\pi$
$234$	−4.19615	−0.274311
$235$	0	0
$236$	14.9282	0.971743
$237$	−13.3923	−0.869924
$238$	0	0
$239$	20.9282	1.35373	0.676866	−	0.736106i	$-0.263337\pi$
$239$	20.9282	1.35373	0.676866	+	0.736106i	$0.263337\pi$
$240$	0	0
$241$	−6.53590	−0.421014	−0.210507	−	0.977592i	$-0.567512\pi$
$241$	−6.53590	−0.421014	−0.210507	+	0.977592i	$0.567512\pi$
$242$	2.58846	0.166392
$243$	−1.00000	−0.0641500
$244$	5.85641	0.374918
$245$	0	0
$246$	−2.00000	−0.127515
$247$	14.1244	0.898711
$248$	−16.3923	−1.04091
$249$	9.12436	0.578233
$250$	0	0
$251$	−6.58846	−0.415860	−0.207930	−	0.978144i	$-0.566673\pi$
$251$	−6.58846	−0.415860	−0.207930	+	0.978144i	$0.566673\pi$
$252$	0	0
$253$	−3.46410	−0.217786
$254$	11.1244	0.698004
$255$	0	0
$256$	−11.7128	−0.732051
$257$	11.6603	0.727347	0.363673	−	0.931527i	$-0.381522\pi$
$257$	11.6603	0.727347	0.363673	+	0.931527i	$0.381522\pi$
$258$	5.26795	0.327968
$259$	0	0
$260$	0	0
$261$	6.19615	0.383532
$262$	6.24871	0.386047
$263$	12.3923	0.764142	0.382071	−	0.924133i	$-0.375211\pi$
$263$	12.3923	0.764142	0.382071	+	0.924133i	$0.375211\pi$
$264$	6.92820	0.426401
$265$	0	0
$266$	0	0
$267$	−9.12436	−0.558401
$268$	3.89488	0.237918
$269$	19.4641	1.18675	0.593374	−	0.804927i	$-0.297796\pi$
$269$	19.4641	1.18675	0.593374	+	0.804927i	$0.297796\pi$
$270$	0	0
$271$	−16.9282	−1.02832	−0.514158	−	0.857696i	$-0.671895\pi$
$271$	−16.9282	−1.02832	−0.514158	+	0.857696i	$0.671895\pi$
$272$	7.21539	0.437497
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	1.85641	0.111743
$277$	−2.66025	−0.159839	−0.0799196	−	0.996801i	$-0.525466\pi$
$277$	−2.66025	−0.159839	−0.0799196	+	0.996801i	$0.525466\pi$
$278$	−5.80385	−0.348092
$279$	−6.46410	−0.386996
$280$	0	0
$281$	−13.8564	−0.826604	−0.413302	−	0.910594i	$-0.635625\pi$
$281$	−13.8564	−0.826604	−0.413302	+	0.910594i	$0.635625\pi$
$282$	1.46410	0.0871860
$283$	0.124356	0.00739218	0.00369609	−	0.999993i	$-0.498823\pi$
$283$	0.124356	0.00739218	0.00369609	+	0.999993i	$0.498823\pi$
$284$	6.14359	0.364555
$285$	0	0
$286$	11.4641	0.677887
$287$	0	0
$288$	−5.85641	−0.345092
$289$	28.3205	1.66591
$290$	0	0
$291$	−1.07180	−0.0628298
$292$	6.82309	0.399291
$293$	5.07180	0.296298	0.148149	−	0.988965i	$-0.452669\pi$
$293$	5.07180	0.296298	0.148149	+	0.988965i	$0.452669\pi$
$294$	0	0
$295$	0	0
$296$	−18.2487	−1.06068
$297$	2.73205	0.158530
$298$	16.0000	0.926855
$299$	7.26795	0.420316
$300$	0	0
$301$	0	0
$302$	−3.60770	−0.207600
$303$	−10.7321	−0.616540
$304$	2.64102	0.151473
$305$	0	0
$306$	−4.92820	−0.281727
$307$	−7.87564	−0.449487	−0.224743	−	0.974418i	$-0.572154\pi$
$307$	−7.87564	−0.449487	−0.224743	+	0.974418i	$0.572154\pi$
$308$	0	0
$309$	−1.19615	−0.0680467
$310$	0	0
$311$	15.1244	0.857624	0.428812	−	0.903394i	$-0.358932\pi$
$311$	15.1244	0.857624	0.428812	+	0.903394i	$0.358932\pi$
$312$	−14.5359	−0.822933
$313$	4.66025	0.263413	0.131707	−	0.991289i	$-0.457954\pi$
$313$	4.66025	0.263413	0.131707	+	0.991289i	$0.457954\pi$
$314$	10.5359	0.594575
$315$	0	0
$316$	−19.6077	−1.10302
$317$	−30.4449	−1.70995	−0.854977	−	0.518666i	$-0.826429\pi$
$317$	−30.4449	−1.70995	−0.854977	+	0.518666i	$0.826429\pi$
$318$	6.14359	0.344516
$319$	−16.9282	−0.947797
$320$	0	0
$321$	−8.19615	−0.457465
$322$	0	0
$323$	16.5885	0.923006
$324$	−1.46410	−0.0813390
$325$	0	0
$326$	−4.28719	−0.237445
$327$	−11.0000	−0.608301
$328$	−6.92820	−0.382546
$329$	0	0
$330$	0	0
$331$	21.9282	1.20528	0.602642	−	0.798012i	$-0.294115\pi$
$331$	21.9282	1.20528	0.602642	+	0.798012i	$0.294115\pi$
$332$	13.3590	0.733169
$333$	−7.19615	−0.394347
$334$	0.248711	0.0136089
$335$	0	0
$336$	0	0
$337$	33.9808	1.85105	0.925525	−	0.378686i	$-0.123624\pi$
$337$	33.9808	1.85105	0.925525	+	0.378686i	$0.123624\pi$
$338$	−14.5359	−0.790649
$339$	−4.92820	−0.267663
$340$	0	0
$341$	17.6603	0.956356
$342$	−1.80385	−0.0975409
$343$	0	0
$344$	18.2487	0.983905
$345$	0	0
$346$	−15.7128	−0.844726
$347$	34.9282	1.87504	0.937522	−	0.347926i	$-0.113114\pi$
$347$	34.9282	1.87504	0.937522	+	0.347926i	$0.113114\pi$
$348$	9.07180	0.486299
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	−5.73205	−0.305954
$352$	16.0000	0.852803
$353$	21.1244	1.12434	0.562168	−	0.827023i	$-0.309967\pi$
$353$	21.1244	1.12434	0.562168	+	0.827023i	$0.309967\pi$
$354$	−7.46410	−0.396713
$355$	0	0
$356$	−13.3590	−0.708025
$357$	0	0
$358$	7.32051	0.386901
$359$	−4.73205	−0.249748	−0.124874	−	0.992173i	$-0.539853\pi$
$359$	−4.73205	−0.249748	−0.124874	+	0.992173i	$0.539853\pi$
$360$	0	0
$361$	−12.9282	−0.680432
$362$	7.55514	0.397089
$363$	3.53590	0.185587
$364$	0	0
$365$	0	0
$366$	−2.92820	−0.153060
$367$	0.803848	0.0419605	0.0209803	−	0.999780i	$-0.493321\pi$
$367$	0.803848	0.0419605	0.0209803	+	0.999780i	$0.493321\pi$
$368$	1.35898	0.0708419
$369$	−2.73205	−0.142225
$370$	0	0
$371$	0	0
$372$	−9.46410	−0.490691
$373$	18.5167	0.958756	0.479378	−	0.877608i	$-0.340862\pi$
$373$	18.5167	0.958756	0.479378	+	0.877608i	$0.340862\pi$
$374$	13.4641	0.696212
$375$	0	0
$376$	5.07180	0.261558
$377$	35.5167	1.82920
$378$	0	0
$379$	−28.3205	−1.45473	−0.727363	−	0.686253i	$-0.759255\pi$
$379$	−28.3205	−1.45473	−0.727363	+	0.686253i	$0.759255\pi$
$380$	0	0
$381$	15.1962	0.778522
$382$	−3.60770	−0.184586
$383$	11.3205	0.578451	0.289225	−	0.957261i	$-0.406602\pi$
$383$	11.3205	0.578451	0.289225	+	0.957261i	$0.406602\pi$
$384$	−10.1436	−0.517638
$385$	0	0
$386$	−6.73205	−0.342652
$387$	7.19615	0.365801
$388$	−1.56922	−0.0796650
$389$	36.5885	1.85511	0.927554	−	0.373689i	$-0.121907\pi$
$389$	36.5885	1.85511	0.927554	+	0.373689i	$0.121907\pi$
$390$	0	0
$391$	8.53590	0.431679
$392$	0	0
$393$	8.53590	0.430579
$394$	−12.9282	−0.651313
$395$	0	0
$396$	4.00000	0.201008
$397$	−20.8038	−1.04412	−0.522058	−	0.852910i	$-0.674835\pi$
$397$	−20.8038	−1.04412	−0.522058	+	0.852910i	$0.674835\pi$
$398$	16.1051	0.807277
$399$	0	0
$400$	0	0
$401$	4.39230	0.219341	0.109671	−	0.993968i	$-0.465020\pi$
$401$	4.39230	0.219341	0.109671	+	0.993968i	$0.465020\pi$
$402$	−1.94744	−0.0971295
$403$	−37.0526	−1.84572
$404$	−15.7128	−0.781742
$405$	0	0
$406$	0	0
$407$	19.6603	0.974523
$408$	−17.0718	−0.845180
$409$	−30.8564	−1.52575	−0.762876	−	0.646545i	$-0.776213\pi$
$409$	−30.8564	−1.52575	−0.762876	+	0.646545i	$0.776213\pi$
$410$	0	0
$411$	−8.19615	−0.404286
$412$	−1.75129	−0.0862798
$413$	0	0
$414$	−0.928203	−0.0456187
$415$	0	0
$416$	−33.5692	−1.64587
$417$	−7.92820	−0.388246
$418$	4.92820	0.241046
$419$	28.5359	1.39407	0.697035	−	0.717037i	$-0.254502\pi$
$419$	28.5359	1.39407	0.697035	+	0.717037i	$0.254502\pi$
$420$	0	0
$421$	13.9282	0.678819	0.339410	−	0.940639i	$-0.389773\pi$
$421$	13.9282	0.678819	0.339410	+	0.940639i	$0.389773\pi$
$422$	−15.3205	−0.745791
$423$	2.00000	0.0972433
$424$	21.2820	1.03355
$425$	0	0
$426$	−3.07180	−0.148829
$427$	0	0
$428$	−12.0000	−0.580042
$429$	15.6603	0.756084
$430$	0	0
$431$	−17.3205	−0.834300	−0.417150	−	0.908838i	$-0.636971\pi$
$431$	−17.3205	−0.834300	−0.417150	+	0.908838i	$0.636971\pi$
$432$	−1.07180	−0.0515668
$433$	4.80385	0.230858	0.115429	−	0.993316i	$-0.463176\pi$
$433$	4.80385	0.230858	0.115429	+	0.993316i	$0.463176\pi$
$434$	0	0
$435$	0	0
$436$	−16.1051	−0.771295
$437$	3.12436	0.149458
$438$	−3.41154	−0.163010
$439$	−7.46410	−0.356242	−0.178121	−	0.984009i	$-0.557002\pi$
$439$	−7.46410	−0.356242	−0.178121	+	0.984009i	$0.557002\pi$
$440$	0	0
$441$	0	0
$442$	−28.2487	−1.34365
$443$	−2.53590	−0.120484	−0.0602421	−	0.998184i	$-0.519187\pi$
$443$	−2.53590	−0.120484	−0.0602421	+	0.998184i	$0.519187\pi$
$444$	−10.5359	−0.500012
$445$	0	0
$446$	−0.287187	−0.0135987
$447$	21.8564	1.03377
$448$	0	0
$449$	−8.14359	−0.384320	−0.192160	−	0.981364i	$-0.561549\pi$
$449$	−8.14359	−0.384320	−0.192160	+	0.981364i	$0.561549\pi$
$450$	0	0
$451$	7.46410	0.351471
$452$	−7.21539	−0.339383
$453$	−4.92820	−0.231547
$454$	−11.4641	−0.538037
$455$	0	0
$456$	−6.24871	−0.292623
$457$	−0.660254	−0.0308854	−0.0154427	−	0.999881i	$-0.504916\pi$
$457$	−0.660254	−0.0308854	−0.0154427	+	0.999881i	$0.504916\pi$
$458$	−2.19615	−0.102619
$459$	−6.73205	−0.314225
$460$	0	0
$461$	34.9808	1.62922	0.814608	−	0.580012i	$-0.196952\pi$
$461$	34.9808	1.62922	0.814608	+	0.580012i	$0.196952\pi$
$462$	0	0
$463$	−22.2679	−1.03488	−0.517440	−	0.855720i	$-0.673115\pi$
$463$	−22.2679	−1.03488	−0.517440	+	0.855720i	$0.673115\pi$
$464$	6.64102	0.308301
$465$	0	0
$466$	−12.6795	−0.587366
$467$	−27.8564	−1.28904	−0.644520	−	0.764587i	$-0.722943\pi$
$467$	−27.8564	−1.28904	−0.644520	+	0.764587i	$0.722943\pi$
$468$	−8.39230	−0.387934
$469$	0	0
$470$	0	0
$471$	14.3923	0.663162
$472$	−25.8564	−1.19014
$473$	−19.6603	−0.903979
$474$	9.80385	0.450306
$475$	0	0
$476$	0	0
$477$	8.39230	0.384257
$478$	−15.3205	−0.700744
$479$	32.7846	1.49797	0.748984	−	0.662589i	$-0.230542\pi$
$479$	32.7846	1.49797	0.748984	+	0.662589i	$0.230542\pi$
$480$	0	0
$481$	−41.2487	−1.88078
$482$	4.78461	0.217933
$483$	0	0
$484$	5.17691	0.235314
$485$	0	0
$486$	0.732051	0.0332065
$487$	31.5885	1.43141	0.715705	−	0.698403i	$-0.246106\pi$
$487$	31.5885	1.43141	0.715705	+	0.698403i	$0.246106\pi$
$488$	−10.1436	−0.459179
$489$	−5.85641	−0.264836
$490$	0	0
$491$	10.2487	0.462518	0.231259	−	0.972892i	$-0.425716\pi$
$491$	10.2487	0.462518	0.231259	+	0.972892i	$0.425716\pi$
$492$	−4.00000	−0.180334
$493$	41.7128	1.87865
$494$	−10.3397	−0.465207
$495$	0	0
$496$	−6.92820	−0.311086
$497$	0	0
$498$	−6.67949	−0.299315
$499$	−20.4641	−0.916099	−0.458050	−	0.888927i	$-0.651452\pi$
$499$	−20.4641	−0.916099	−0.458050	+	0.888927i	$0.651452\pi$
$500$	0	0
$501$	0.339746	0.0151787
$502$	4.82309	0.215265
$503$	−6.39230	−0.285019	−0.142509	−	0.989793i	$-0.545517\pi$
$503$	−6.39230	−0.285019	−0.142509	+	0.989793i	$0.545517\pi$
$504$	0	0
$505$	0	0
$506$	2.53590	0.112734
$507$	−19.8564	−0.881854
$508$	22.2487	0.987127
$509$	−11.4641	−0.508137	−0.254069	−	0.967186i	$-0.581769\pi$
$509$	−11.4641	−0.508137	−0.254069	+	0.967186i	$0.581769\pi$
$510$	0	0
$511$	0	0
$512$	−11.7128	−0.517638
$513$	−2.46410	−0.108793
$514$	−8.53590	−0.376502
$515$	0	0
$516$	10.5359	0.463817
$517$	−5.46410	−0.240311
$518$	0	0
$519$	−21.4641	−0.942169
$520$	0	0
$521$	−1.46410	−0.0641435	−0.0320717	−	0.999486i	$-0.510210\pi$
$521$	−1.46410	−0.0641435	−0.0320717	+	0.999486i	$0.510210\pi$
$522$	−4.53590	−0.198531
$523$	−24.2679	−1.06116	−0.530582	−	0.847634i	$-0.678026\pi$
$523$	−24.2679	−1.06116	−0.530582	+	0.847634i	$0.678026\pi$
$524$	12.4974	0.545952
$525$	0	0
$526$	−9.07180	−0.395549
$527$	−43.5167	−1.89562
$528$	2.92820	0.127434
$529$	−21.3923	−0.930100
$530$	0	0
$531$	−10.1962	−0.442475
$532$	0	0
$533$	−15.6603	−0.678321
$534$	6.67949	0.289050
$535$	0	0
$536$	−6.74613	−0.291389
$537$	10.0000	0.431532
$538$	−14.2487	−0.614306
$539$	0	0
$540$	0	0
$541$	−35.7846	−1.53850	−0.769250	−	0.638948i	$-0.779370\pi$
$541$	−35.7846	−1.53850	−0.769250	+	0.638948i	$0.779370\pi$
$542$	12.3923	0.532295
$543$	10.3205	0.442895
$544$	−39.4256	−1.69036
$545$	0	0
$546$	0	0
$547$	−22.2487	−0.951286	−0.475643	−	0.879638i	$-0.657785\pi$
$547$	−22.2487	−0.951286	−0.475643	+	0.879638i	$0.657785\pi$
$548$	−12.0000	−0.512615
$549$	−4.00000	−0.170716
$550$	0	0
$551$	15.2679	0.650437
$552$	−3.21539	−0.136856
$553$	0	0
$554$	1.94744	0.0827388
$555$	0	0
$556$	−11.6077	−0.492276
$557$	26.7846	1.13490	0.567450	−	0.823408i	$-0.307930\pi$
$557$	26.7846	1.13490	0.567450	+	0.823408i	$0.307930\pi$
$558$	4.73205	0.200324
$559$	41.2487	1.74463
$560$	0	0
$561$	18.3923	0.776524
$562$	10.1436	0.427882
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	2.92820	0.123300
$565$	0	0
$566$	−0.0910347	−0.00382647
$567$	0	0
$568$	−10.6410	−0.446487
$569$	−26.4449	−1.10863	−0.554313	−	0.832308i	$-0.687019\pi$
$569$	−26.4449	−1.10863	−0.554313	+	0.832308i	$0.687019\pi$
$570$	0	0
$571$	39.3923	1.64852	0.824258	−	0.566214i	$-0.191592\pi$
$571$	39.3923	1.64852	0.824258	+	0.566214i	$0.191592\pi$
$572$	22.9282	0.958676
$573$	−4.92820	−0.205879
$574$	0	0
$575$	0	0
$576$	2.14359	0.0893164
$577$	11.3397	0.472080	0.236040	−	0.971743i	$-0.424150\pi$
$577$	11.3397	0.472080	0.236040	+	0.971743i	$0.424150\pi$
$578$	−20.7321	−0.862340
$579$	−9.19615	−0.382179
$580$	0	0
$581$	0	0
$582$	0.784610	0.0325231
$583$	−22.9282	−0.949589
$584$	−11.8179	−0.489029
$585$	0	0
$586$	−3.71281	−0.153375
$587$	37.2679	1.53821	0.769106	−	0.639121i	$-0.220702\pi$
$587$	37.2679	1.53821	0.769106	+	0.639121i	$0.220702\pi$
$588$	0	0
$589$	−15.9282	−0.656310
$590$	0	0
$591$	−17.6603	−0.726446
$592$	−7.71281	−0.316995
$593$	−37.9090	−1.55673	−0.778367	−	0.627809i	$-0.783952\pi$
$593$	−37.9090	−1.55673	−0.778367	+	0.627809i	$0.783952\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	32.0000	1.31077
$597$	22.0000	0.900400
$598$	−5.32051	−0.217572
$599$	10.2487	0.418751	0.209375	−	0.977835i	$-0.432857\pi$
$599$	10.2487	0.418751	0.209375	+	0.977835i	$0.432857\pi$
$600$	0	0
$601$	13.9282	0.568143	0.284072	−	0.958803i	$-0.408315\pi$
$601$	13.9282	0.568143	0.284072	+	0.958803i	$0.408315\pi$
$602$	0	0
$603$	−2.66025	−0.108334
$604$	−7.21539	−0.293590
$605$	0	0
$606$	7.85641	0.319145
$607$	−7.19615	−0.292083	−0.146041	−	0.989278i	$-0.546653\pi$
$607$	−7.19615	−0.292083	−0.146041	+	0.989278i	$0.546653\pi$
$608$	−14.4308	−0.585245
$609$	0	0
$610$	0	0
$611$	11.4641	0.463788
$612$	−9.85641	−0.398422
$613$	13.0718	0.527965	0.263982	−	0.964527i	$-0.414964\pi$
$613$	13.0718	0.527965	0.263982	+	0.964527i	$0.414964\pi$
$614$	5.76537	0.232671
$615$	0	0
$616$	0	0
$617$	−12.2487	−0.493115	−0.246557	−	0.969128i	$-0.579299\pi$
$617$	−12.2487	−0.493115	−0.246557	+	0.969128i	$0.579299\pi$
$618$	0.875644	0.0352236
$619$	43.9282	1.76562	0.882812	−	0.469727i	$-0.155648\pi$
$619$	43.9282	1.76562	0.882812	+	0.469727i	$0.155648\pi$
$620$	0	0
$621$	−1.26795	−0.0508810
$622$	−11.0718	−0.443939
$623$	0	0
$624$	−6.14359	−0.245941
$625$	0	0
$626$	−3.41154	−0.136353
$627$	6.73205	0.268852
$628$	21.0718	0.840856
$629$	−48.4449	−1.93162
$630$	0	0
$631$	7.21539	0.287240	0.143620	−	0.989633i	$-0.454126\pi$
$631$	7.21539	0.287240	0.143620	+	0.989633i	$0.454126\pi$
$632$	33.9615	1.35092
$633$	−20.9282	−0.831821
$634$	22.2872	0.885137
$635$	0	0
$636$	12.2872	0.487219
$637$	0	0
$638$	12.3923	0.490616
$639$	−4.19615	−0.165997
$640$	0	0
$641$	14.1962	0.560714	0.280357	−	0.959896i	$-0.409547\pi$
$641$	14.1962	0.560714	0.280357	+	0.959896i	$0.409547\pi$
$642$	6.00000	0.236801
$643$	−40.5167	−1.59782	−0.798911	−	0.601450i	$-0.794590\pi$
$643$	−40.5167	−1.59782	−0.798911	+	0.601450i	$0.794590\pi$
$644$	0	0
$645$	0	0
$646$	−12.1436	−0.477783
$647$	37.9090	1.49036	0.745178	−	0.666866i	$-0.232365\pi$
$647$	37.9090	1.49036	0.745178	+	0.666866i	$0.232365\pi$
$648$	2.53590	0.0996195
$649$	27.8564	1.09346
$650$	0	0
$651$	0	0
$652$	−8.57437	−0.335798
$653$	13.4115	0.524834	0.262417	−	0.964955i	$-0.415480\pi$
$653$	13.4115	0.524834	0.262417	+	0.964955i	$0.415480\pi$
$654$	8.05256	0.314880
$655$	0	0
$656$	−2.92820	−0.114327
$657$	−4.66025	−0.181814
$658$	0	0
$659$	−10.9282	−0.425702	−0.212851	−	0.977085i	$-0.568275\pi$
$659$	−10.9282	−0.425702	−0.212851	+	0.977085i	$0.568275\pi$
$660$	0	0
$661$	−3.53590	−0.137531	−0.0687653	−	0.997633i	$-0.521906\pi$
$661$	−3.53590	−0.137531	−0.0687653	+	0.997633i	$0.521906\pi$
$662$	−16.0526	−0.623900
$663$	−38.5885	−1.49865
$664$	−23.1384	−0.897946
$665$	0	0
$666$	5.26795	0.204129
$667$	7.85641	0.304201
$668$	0.497423	0.0192459
$669$	−0.392305	−0.0151674
$670$	0	0
$671$	10.9282	0.421879
$672$	0	0
$673$	44.6603	1.72153	0.860763	−	0.509006i	$-0.169987\pi$
$673$	44.6603	1.72153	0.860763	+	0.509006i	$0.169987\pi$
$674$	−24.8756	−0.958174
$675$	0	0
$676$	−29.0718	−1.11815
$677$	−8.87564	−0.341119	−0.170559	−	0.985347i	$-0.554557\pi$
$677$	−8.87564	−0.341119	−0.170559	+	0.985347i	$0.554557\pi$
$678$	3.60770	0.138553
$679$	0	0
$680$	0	0
$681$	−15.6603	−0.600102
$682$	−12.9282	−0.495046
$683$	−10.0526	−0.384650	−0.192325	−	0.981331i	$-0.561603\pi$
$683$	−10.0526	−0.384650	−0.192325	+	0.981331i	$0.561603\pi$
$684$	−3.60770	−0.137944
$685$	0	0
$686$	0	0
$687$	−3.00000	−0.114457
$688$	7.71281	0.294048
$689$	48.1051	1.83266
$690$	0	0
$691$	18.8564	0.717332	0.358666	−	0.933466i	$-0.383232\pi$
$691$	18.8564	0.717332	0.358666	+	0.933466i	$0.383232\pi$
$692$	−31.4256	−1.19462
$693$	0	0
$694$	−25.5692	−0.970594
$695$	0	0
$696$	−15.7128	−0.595593
$697$	−18.3923	−0.696658
$698$	16.1051	0.609588
$699$	−17.3205	−0.655122
$700$	0	0
$701$	22.5885	0.853154	0.426577	−	0.904451i	$-0.359719\pi$
$701$	22.5885	0.853154	0.426577	+	0.904451i	$0.359719\pi$
$702$	4.19615	0.158374
$703$	−17.7321	−0.668777
$704$	−5.85641	−0.220722
$705$	0	0
$706$	−15.4641	−0.581999
$707$	0	0
$708$	−14.9282	−0.561036
$709$	−14.9282	−0.560640	−0.280320	−	0.959907i	$-0.590441\pi$
$709$	−14.9282	−0.560640	−0.280320	+	0.959907i	$0.590441\pi$
$710$	0	0
$711$	13.3923	0.502251
$712$	23.1384	0.867150
$713$	−8.19615	−0.306948
$714$	0	0
$715$	0	0
$716$	14.6410	0.547160
$717$	−20.9282	−0.781578
$718$	3.46410	0.129279
$719$	27.4641	1.02424	0.512119	−	0.858914i	$-0.328861\pi$
$719$	27.4641	1.02424	0.512119	+	0.858914i	$0.328861\pi$
$720$	0	0
$721$	0	0
$722$	9.46410	0.352217
$723$	6.53590	0.243073
$724$	15.1103	0.561569
$725$	0	0
$726$	−2.58846	−0.0960667
$727$	−30.6603	−1.13713	−0.568563	−	0.822640i	$-0.692500\pi$
$727$	−30.6603	−1.13713	−0.568563	+	0.822640i	$0.692500\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	48.4449	1.79180
$732$	−5.85641	−0.216459
$733$	−18.6603	−0.689232	−0.344616	−	0.938744i	$-0.611991\pi$
$733$	−18.6603	−0.689232	−0.344616	+	0.938744i	$0.611991\pi$
$734$	−0.588457	−0.0217204
$735$	0	0
$736$	−7.42563	−0.273712
$737$	7.26795	0.267718
$738$	2.00000	0.0736210
$739$	−13.7846	−0.507075	−0.253538	−	0.967326i	$-0.581594\pi$
$739$	−13.7846	−0.507075	−0.253538	+	0.967326i	$0.581594\pi$
$740$	0	0
$741$	−14.1244	−0.518871
$742$	0	0
$743$	−49.9090	−1.83098	−0.915491	−	0.402338i	$-0.868198\pi$
$743$	−49.9090	−1.83098	−0.915491	+	0.402338i	$0.868198\pi$
$744$	16.3923	0.600971
$745$	0	0
$746$	−13.5551	−0.496289
$747$	−9.12436	−0.333843
$748$	26.9282	0.984593
$749$	0	0
$750$	0	0
$751$	31.9282	1.16508	0.582538	−	0.812803i	$-0.302060\pi$
$751$	31.9282	1.16508	0.582538	+	0.812803i	$0.302060\pi$
$752$	2.14359	0.0781688
$753$	6.58846	0.240097
$754$	−26.0000	−0.946864
$755$	0	0
$756$	0	0
$757$	0.143594	0.00521900	0.00260950	−	0.999997i	$-0.499169\pi$
$757$	0.143594	0.00521900	0.00260950	+	0.999997i	$0.499169\pi$
$758$	20.7321	0.753022
$759$	3.46410	0.125739
$760$	0	0
$761$	−43.2679	−1.56846	−0.784231	−	0.620469i	$-0.786942\pi$
$761$	−43.2679	−1.56846	−0.784231	+	0.620469i	$0.786942\pi$
$762$	−11.1244	−0.402993
$763$	0	0
$764$	−7.21539	−0.261044
$765$	0	0
$766$	−8.28719	−0.299428
$767$	−58.4449	−2.11032
$768$	11.7128	0.422650
$769$	−17.6795	−0.637539	−0.318769	−	0.947832i	$-0.603270\pi$
$769$	−17.6795	−0.637539	−0.318769	+	0.947832i	$0.603270\pi$
$770$	0	0
$771$	−11.6603	−0.419934
$772$	−13.4641	−0.484584
$773$	−1.51666	−0.0545505	−0.0272752	−	0.999628i	$-0.508683\pi$
$773$	−1.51666	−0.0545505	−0.0272752	+	0.999628i	$0.508683\pi$
$774$	−5.26795	−0.189353
$775$	0	0
$776$	2.71797	0.0975694
$777$	0	0
$778$	−26.7846	−0.960275
$779$	−6.73205	−0.241201
$780$	0	0
$781$	11.4641	0.410218
$782$	−6.24871	−0.223453
$783$	−6.19615	−0.221432
$784$	0	0
$785$	0	0
$786$	−6.24871	−0.222884
$787$	6.53590	0.232980	0.116490	−	0.993192i	$-0.462836\pi$
$787$	6.53590	0.232980	0.116490	+	0.993192i	$0.462836\pi$
$788$	−25.8564	−0.921096
$789$	−12.3923	−0.441178
$790$	0	0
$791$	0	0
$792$	−6.92820	−0.246183
$793$	−22.9282	−0.814204
$794$	15.2295	0.540474
$795$	0	0
$796$	32.2102	1.14166
$797$	−42.0526	−1.48958	−0.744789	−	0.667300i	$-0.767450\pi$
$797$	−42.0526	−1.48958	−0.744789	+	0.667300i	$0.767450\pi$
$798$	0	0
$799$	13.4641	0.476326
$800$	0	0
$801$	9.12436	0.322393
$802$	−3.21539	−0.113539
$803$	12.7321	0.449304
$804$	−3.89488	−0.137362
$805$	0	0
$806$	27.1244	0.955415
$807$	−19.4641	−0.685169
$808$	27.2154	0.957434
$809$	−29.7128	−1.04465	−0.522323	−	0.852747i	$-0.674935\pi$
$809$	−29.7128	−1.04465	−0.522323	+	0.852747i	$0.674935\pi$
$810$	0	0
$811$	−3.46410	−0.121641	−0.0608205	−	0.998149i	$-0.519372\pi$
$811$	−3.46410	−0.121641	−0.0608205	+	0.998149i	$0.519372\pi$
$812$	0	0
$813$	16.9282	0.593698
$814$	−14.3923	−0.504450
$815$	0	0
$816$	−7.21539	−0.252589
$817$	17.7321	0.620366
$818$	22.5885	0.789787
$819$	0	0
$820$	0	0
$821$	19.5167	0.681136	0.340568	−	0.940220i	$-0.389381\pi$
$821$	19.5167	0.681136	0.340568	+	0.940220i	$0.389381\pi$
$822$	6.00000	0.209274
$823$	23.1769	0.807896	0.403948	−	0.914782i	$-0.367638\pi$
$823$	23.1769	0.807896	0.403948	+	0.914782i	$0.367638\pi$
$824$	3.03332	0.105671
$825$	0	0
$826$	0	0
$827$	52.2487	1.81687	0.908433	−	0.418031i	$-0.137280\pi$
$827$	52.2487	1.81687	0.908433	+	0.418031i	$0.137280\pi$
$828$	−1.85641	−0.0645146
$829$	25.3923	0.881911	0.440956	−	0.897529i	$-0.354640\pi$
$829$	25.3923	0.881911	0.440956	+	0.897529i	$0.354640\pi$
$830$	0	0
$831$	2.66025	0.0922832
$832$	12.2872	0.425982
$833$	0	0
$834$	5.80385	0.200971
$835$	0	0
$836$	9.85641	0.340891
$837$	6.46410	0.223432
$838$	−20.8897	−0.721624
$839$	40.4449	1.39631	0.698156	−	0.715946i	$-0.254004\pi$
$839$	40.4449	1.39631	0.698156	+	0.715946i	$0.254004\pi$
$840$	0	0
$841$	9.39230	0.323873
$842$	−10.1962	−0.351383
$843$	13.8564	0.477240
$844$	−30.6410	−1.05471
$845$	0	0
$846$	−1.46410	−0.0503369
$847$	0	0
$848$	8.99485	0.308884
$849$	−0.124356	−0.00426787
$850$	0	0
$851$	−9.12436	−0.312779
$852$	−6.14359	−0.210476
$853$	19.9808	0.684128	0.342064	−	0.939677i	$-0.388874\pi$
$853$	19.9808	0.684128	0.342064	+	0.939677i	$0.388874\pi$
$854$	0	0
$855$	0	0
$856$	20.7846	0.710403
$857$	4.87564	0.166549	0.0832744	−	0.996527i	$-0.473462\pi$
$857$	4.87564	0.166549	0.0832744	+	0.996527i	$0.473462\pi$
$858$	−11.4641	−0.391378
$859$	−0.535898	−0.0182846	−0.00914231	−	0.999958i	$-0.502910\pi$
$859$	−0.535898	−0.0182846	−0.00914231	+	0.999958i	$0.502910\pi$
$860$	0	0
$861$	0	0
$862$	12.6795	0.431865
$863$	6.39230	0.217597	0.108798	−	0.994064i	$-0.465300\pi$
$863$	6.39230	0.217597	0.108798	+	0.994064i	$0.465300\pi$
$864$	5.85641	0.199239
$865$	0	0
$866$	−3.51666	−0.119501
$867$	−28.3205	−0.961815
$868$	0	0
$869$	−36.5885	−1.24118
$870$	0	0
$871$	−15.2487	−0.516683
$872$	27.8949	0.944640
$873$	1.07180	0.0362748
$874$	−2.28719	−0.0773653
$875$	0	0
$876$	−6.82309	−0.230531
$877$	31.8564	1.07571	0.537857	−	0.843036i	$-0.319234\pi$
$877$	31.8564	1.07571	0.537857	+	0.843036i	$0.319234\pi$
$878$	5.46410	0.184404
$879$	−5.07180	−0.171067
$880$	0	0
$881$	17.8564	0.601598	0.300799	−	0.953688i	$-0.402747\pi$
$881$	17.8564	0.601598	0.300799	+	0.953688i	$0.402747\pi$
$882$	0	0
$883$	−22.4115	−0.754208	−0.377104	−	0.926171i	$-0.623080\pi$
$883$	−22.4115	−0.754208	−0.377104	+	0.926171i	$0.623080\pi$
$884$	−56.4974	−1.90021
$885$	0	0
$886$	1.85641	0.0623672
$887$	−28.7321	−0.964728	−0.482364	−	0.875971i	$-0.660222\pi$
$887$	−28.7321	−0.964728	−0.482364	+	0.875971i	$0.660222\pi$
$888$	18.2487	0.612387
$889$	0	0
$890$	0	0
$891$	−2.73205	−0.0915271
$892$	−0.574374	−0.0192315
$893$	4.92820	0.164916
$894$	−16.0000	−0.535120
$895$	0	0
$896$	0	0
$897$	−7.26795	−0.242670
$898$	5.96152	0.198939
$899$	−40.0526	−1.33583
$900$	0	0
$901$	56.4974	1.88220
$902$	−5.46410	−0.181935
$903$	0	0
$904$	12.4974	0.415658
$905$	0	0
$906$	3.60770	0.119858
$907$	−2.41154	−0.0800740	−0.0400370	−	0.999198i	$-0.512748\pi$
$907$	−2.41154	−0.0800740	−0.0400370	+	0.999198i	$0.512748\pi$
$908$	−22.9282	−0.760899
$909$	10.7321	0.355960
$910$	0	0
$911$	−11.2679	−0.373324	−0.186662	−	0.982424i	$-0.559767\pi$
$911$	−11.2679	−0.373324	−0.186662	+	0.982424i	$0.559767\pi$
$912$	−2.64102	−0.0874528
$913$	24.9282	0.825003
$914$	0.483340	0.0159874
$915$	0	0
$916$	−4.39230	−0.145126
$917$	0	0
$918$	4.92820	0.162655
$919$	−3.14359	−0.103698	−0.0518488	−	0.998655i	$-0.516511\pi$
$919$	−3.14359	−0.103698	−0.0518488	+	0.998655i	$0.516511\pi$
$920$	0	0
$921$	7.87564	0.259511
$922$	−25.6077	−0.843345
$923$	−24.0526	−0.791700
$924$	0	0
$925$	0	0
$926$	16.3013	0.535693
$927$	1.19615	0.0392868
$928$	−36.2872	−1.19119
$929$	−6.44486	−0.211449	−0.105725	−	0.994395i	$-0.533716\pi$
$929$	−6.44486	−0.211449	−0.105725	+	0.994395i	$0.533716\pi$
$930$	0	0
$931$	0	0
$932$	−25.3590	−0.830661
$933$	−15.1244	−0.495149
$934$	20.3923	0.667257
$935$	0	0
$936$	14.5359	0.475121
$937$	28.2679	0.923474	0.461737	−	0.887017i	$-0.347226\pi$
$937$	28.2679	0.923474	0.461737	+	0.887017i	$0.347226\pi$
$938$	0	0
$939$	−4.66025	−0.152082
$940$	0	0
$941$	−8.05256	−0.262506	−0.131253	−	0.991349i	$-0.541900\pi$
$941$	−8.05256	−0.262506	−0.131253	+	0.991349i	$0.541900\pi$
$942$	−10.5359	−0.343278
$943$	−3.46410	−0.112807
$944$	−10.9282	−0.355683
$945$	0	0
$946$	14.3923	0.467934
$947$	−11.6603	−0.378907	−0.189454	−	0.981890i	$-0.560672\pi$
$947$	−11.6603	−0.378907	−0.189454	+	0.981890i	$0.560672\pi$
$948$	19.6077	0.636828
$949$	−26.7128	−0.867135
$950$	0	0
$951$	30.4449	0.987242
$952$	0	0
$953$	−40.1051	−1.29913	−0.649566	−	0.760305i	$-0.725049\pi$
$953$	−40.1051	−1.29913	−0.649566	+	0.760305i	$0.725049\pi$
$954$	−6.14359	−0.198906
$955$	0	0
$956$	−30.6410	−0.991001
$957$	16.9282	0.547211
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	10.7846	0.347891
$962$	30.1962	0.973563
$963$	8.19615	0.264117
$964$	9.56922	0.308204
$965$	0	0
$966$	0	0
$967$	−14.1244	−0.454209	−0.227104	−	0.973870i	$-0.572926\pi$
$967$	−14.1244	−0.454209	−0.227104	+	0.973870i	$0.572926\pi$
$968$	−8.96668	−0.288200
$969$	−16.5885	−0.532898
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	1.46410	0.0469611
$973$	0	0
$974$	−23.1244	−0.740952
$975$	0	0
$976$	−4.28719	−0.137230
$977$	−14.5885	−0.466726	−0.233363	−	0.972390i	$-0.574973\pi$
$977$	−14.5885	−0.466726	−0.233363	+	0.972390i	$0.574973\pi$
$978$	4.28719	0.137089
$979$	−24.9282	−0.796709
$980$	0	0
$981$	11.0000	0.351203
$982$	−7.50258	−0.239417
$983$	20.1962	0.644157	0.322079	−	0.946713i	$-0.395618\pi$
$983$	20.1962	0.644157	0.322079	+	0.946713i	$0.395618\pi$
$984$	6.92820	0.220863
$985$	0	0
$986$	−30.5359	−0.972461
$987$	0	0
$988$	−20.6795	−0.657902
$989$	9.12436	0.290138
$990$	0	0
$991$	55.1051	1.75047	0.875236	−	0.483696i	$-0.160706\pi$
$991$	55.1051	1.75047	0.875236	+	0.483696i	$0.160706\pi$
$992$	37.8564	1.20194
$993$	−21.9282	−0.695870
$994$	0	0
$995$	0	0
$996$	−13.3590	−0.423296
$997$	−4.01924	−0.127291	−0.0636453	−	0.997973i	$-0.520273\pi$
$997$	−4.01924	−0.127291	−0.0636453	+	0.997973i	$0.520273\pi$
$998$	14.9808	0.474208
$999$	7.19615	0.227676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.be.1.1		2
5.4	even	2		735.2.a.h.1.2		2
7.3	odd	6		525.2.i.f.226.2		4
7.5	odd	6		525.2.i.f.151.2		4
7.6	odd	2		3675.2.a.bg.1.1		2
15.14	odd	2		2205.2.a.ba.1.1		2
35.3	even	12		525.2.r.a.499.2		4
35.4	even	6		735.2.i.l.226.1		4
35.9	even	6		735.2.i.l.361.1		4
35.12	even	12		525.2.r.a.424.2		4
35.17	even	12		525.2.r.f.499.1		4
35.19	odd	6		105.2.i.d.46.1	yes	4
35.24	odd	6		105.2.i.d.16.1	✓	4
35.33	even	12		525.2.r.f.424.1		4
35.34	odd	2		735.2.a.g.1.2		2
105.59	even	6		315.2.j.c.226.2		4
105.89	even	6		315.2.j.c.46.2		4
105.104	even	2		2205.2.a.z.1.1		2
140.19	even	6		1680.2.bg.o.1201.2		4
140.59	even	6		1680.2.bg.o.961.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.i.d.16.1	✓	4	35.24	odd	6
105.2.i.d.46.1	yes	4	35.19	odd	6
315.2.j.c.46.2		4	105.89	even	6
315.2.j.c.226.2		4	105.59	even	6
525.2.i.f.151.2		4	7.5	odd	6
525.2.i.f.226.2		4	7.3	odd	6
525.2.r.a.424.2		4	35.12	even	12
525.2.r.a.499.2		4	35.3	even	12
525.2.r.f.424.1		4	35.33	even	12
525.2.r.f.499.1		4	35.17	even	12
735.2.a.g.1.2		2	35.34	odd	2
735.2.a.h.1.2		2	5.4	even	2
735.2.i.l.226.1		4	35.4	even	6
735.2.i.l.361.1		4	35.9	even	6
1680.2.bg.o.961.2		4	140.59	even	6
1680.2.bg.o.1201.2		4	140.19	even	6
2205.2.a.z.1.1		2	105.104	even	2
2205.2.a.ba.1.1		2	15.14	odd	2
3675.2.a.be.1.1		2	1.1	even	1	trivial
3675.2.a.bg.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} +0.732051 q^{6} +2.53590 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} +0.732051 q^{6} +2.53590 q^{8} +1.00000 q^{9} -2.73205 q^{11} +1.46410 q^{12} +5.73205 q^{13} +1.07180 q^{16} +6.73205 q^{17} -0.732051 q^{18} +2.46410 q^{19} +2.00000 q^{22} +1.26795 q^{23} -2.53590 q^{24} -4.19615 q^{26} -1.00000 q^{27} +6.19615 q^{29} -6.46410 q^{31} -5.85641 q^{32} +2.73205 q^{33} -4.92820 q^{34} -1.46410 q^{36} -7.19615 q^{37} -1.80385 q^{38} -5.73205 q^{39} -2.73205 q^{41} +7.19615 q^{43} +4.00000 q^{44} -0.928203 q^{46} +2.00000 q^{47} -1.07180 q^{48} -6.73205 q^{51} -8.39230 q^{52} +8.39230 q^{53} +0.732051 q^{54} -2.46410 q^{57} -4.53590 q^{58} -10.1962 q^{59} -4.00000 q^{61} +4.73205 q^{62} +2.14359 q^{64} -2.00000 q^{66} -2.66025 q^{67} -9.85641 q^{68} -1.26795 q^{69} -4.19615 q^{71} +2.53590 q^{72} -4.66025 q^{73} +5.26795 q^{74} -3.60770 q^{76} +4.19615 q^{78} +13.3923 q^{79} +1.00000 q^{81} +2.00000 q^{82} -9.12436 q^{83} -5.26795 q^{86} -6.19615 q^{87} -6.92820 q^{88} +9.12436 q^{89} -1.85641 q^{92} +6.46410 q^{93} -1.46410 q^{94} +5.85641 q^{96} +1.07180 q^{97} -2.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^6 + 12 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} + 12 q^{8} + 2 q^{9} - 2 q^{11} - 4 q^{12} + 8 q^{13} + 16 q^{16} + 10 q^{17} + 2 q^{18} - 2 q^{19} + 4 q^{22} + 6 q^{23} - 12 q^{24} + 2 q^{26} - 2 q^{27} + 2 q^{29} - 6 q^{31} + 16 q^{32} + 2 q^{33} + 4 q^{34} + 4 q^{36} - 4 q^{37} - 14 q^{38} - 8 q^{39} - 2 q^{41} + 4 q^{43} + 8 q^{44} + 12 q^{46} + 4 q^{47} - 16 q^{48} - 10 q^{51} + 4 q^{52} - 4 q^{53} - 2 q^{54} + 2 q^{57} - 16 q^{58} - 10 q^{59} - 8 q^{61} + 6 q^{62} + 32 q^{64} - 4 q^{66} + 12 q^{67} + 8 q^{68} - 6 q^{69} + 2 q^{71} + 12 q^{72} + 8 q^{73} + 14 q^{74} - 28 q^{76} - 2 q^{78} + 6 q^{79} + 2 q^{81} + 4 q^{82} + 6 q^{83} - 14 q^{86} - 2 q^{87} - 6 q^{89} + 24 q^{92} + 6 q^{93} + 4 q^{94} - 16 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^6 + 12 * q^8 + 2 * q^9 - 2 * q^11 - 4 * q^12 + 8 * q^13 + 16 * q^16 + 10 * q^17 + 2 * q^18 - 2 * q^19 + 4 * q^22 + 6 * q^23 - 12 * q^24 + 2 * q^26 - 2 * q^27 + 2 * q^29 - 6 * q^31 + 16 * q^32 + 2 * q^33 + 4 * q^34 + 4 * q^36 - 4 * q^37 - 14 * q^38 - 8 * q^39 - 2 * q^41 + 4 * q^43 + 8 * q^44 + 12 * q^46 + 4 * q^47 - 16 * q^48 - 10 * q^51 + 4 * q^52 - 4 * q^53 - 2 * q^54 + 2 * q^57 - 16 * q^58 - 10 * q^59 - 8 * q^61 + 6 * q^62 + 32 * q^64 - 4 * q^66 + 12 * q^67 + 8 * q^68 - 6 * q^69 + 2 * q^71 + 12 * q^72 + 8 * q^73 + 14 * q^74 - 28 * q^76 - 2 * q^78 + 6 * q^79 + 2 * q^81 + 4 * q^82 + 6 * q^83 - 14 * q^86 - 2 * q^87 - 6 * q^89 + 24 * q^92 + 6 * q^93 + 4 * q^94 - 16 * q^96 + 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more