LMFDB - Embedded newform 4815.2.a.u.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4815,2,Mod(1,4815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4815, 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("4815.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4815 = 3^{2} \cdot 5 \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4815.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:	$38.4479685732$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} + \cdots - 6 $ x^12 - 3x^11 - 15x^10 + 49x^9 + 71x^8 - 278x^7 - 92x^6 + 649x^5 - 127x^4 - 529x^3 + 267x^2 + 15*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.142275$ of defining polynomial
Character	$\chi$	$=$	4815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.142275 q^{2} -1.97976 q^{4} -1.00000 q^{5} -1.55684 q^{7} +0.566220 q^{8} +O(q^{10})$	\(q-0.142275 q^{2} -1.97976 q^{4} -1.00000 q^{5} -1.55684 q^{7} +0.566220 q^{8} +0.142275 q^{10} -1.06174 q^{11} -6.73897 q^{13} +0.221499 q^{14} +3.87896 q^{16} -4.71226 q^{17} +3.03705 q^{19} +1.97976 q^{20} +0.151059 q^{22} -5.80958 q^{23} +1.00000 q^{25} +0.958786 q^{26} +3.08217 q^{28} -4.16204 q^{29} -8.70959 q^{31} -1.68432 q^{32} +0.670437 q^{34} +1.55684 q^{35} +6.57240 q^{37} -0.432096 q^{38} -0.566220 q^{40} -7.77249 q^{41} +10.7312 q^{43} +2.10199 q^{44} +0.826558 q^{46} -10.0921 q^{47} -4.57624 q^{49} -0.142275 q^{50} +13.3415 q^{52} +1.17807 q^{53} +1.06174 q^{55} -0.881514 q^{56} +0.592154 q^{58} +3.58301 q^{59} +0.168963 q^{61} +1.23916 q^{62} -7.51828 q^{64} +6.73897 q^{65} +0.691226 q^{67} +9.32914 q^{68} -0.221499 q^{70} -6.38099 q^{71} -10.3711 q^{73} -0.935087 q^{74} -6.01262 q^{76} +1.65296 q^{77} -0.977121 q^{79} -3.87896 q^{80} +1.10583 q^{82} +12.6407 q^{83} +4.71226 q^{85} -1.52678 q^{86} -0.601179 q^{88} -17.7527 q^{89} +10.4915 q^{91} +11.5016 q^{92} +1.43586 q^{94} -3.03705 q^{95} +19.4157 q^{97} +0.651085 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8} + 3 q^{10} - 4 q^{11} + 13 q^{13} - 4 q^{14} + 13 q^{16} + 4 q^{17} + 14 q^{19} - 15 q^{20} + 15 q^{22} - 11 q^{23} + 12 q^{25} + 8 q^{26} + 16 q^{28} + 7 q^{29} + 4 q^{31} - 4 q^{32} + q^{34} - 7 q^{35} + 24 q^{37} + 11 q^{38} + 3 q^{40} - 13 q^{41} + 25 q^{43} - 10 q^{44} - 22 q^{46} - 19 q^{47} + 9 q^{49} - 3 q^{50} + 20 q^{52} - 11 q^{53} + 4 q^{55} + 37 q^{56} - 2 q^{58} - 8 q^{59} + 7 q^{61} + 11 q^{62} - 19 q^{64} - 13 q^{65} + 33 q^{67} + 24 q^{68} + 4 q^{70} + 34 q^{73} + 27 q^{74} - 9 q^{76} + 29 q^{77} - 13 q^{80} + q^{82} + 24 q^{83} - 4 q^{85} + 36 q^{86} - 6 q^{88} + 10 q^{89} + 30 q^{91} + 28 q^{92} - 8 q^{94} - 14 q^{95} + 16 q^{97} + 36 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8 + 3 * q^10 - 4 * q^11 + 13 * q^13 - 4 * q^14 + 13 * q^16 + 4 * q^17 + 14 * q^19 - 15 * q^20 + 15 * q^22 - 11 * q^23 + 12 * q^25 + 8 * q^26 + 16 * q^28 + 7 * q^29 + 4 * q^31 - 4 * q^32 + q^34 - 7 * q^35 + 24 * q^37 + 11 * q^38 + 3 * q^40 - 13 * q^41 + 25 * q^43 - 10 * q^44 - 22 * q^46 - 19 * q^47 + 9 * q^49 - 3 * q^50 + 20 * q^52 - 11 * q^53 + 4 * q^55 + 37 * q^56 - 2 * q^58 - 8 * q^59 + 7 * q^61 + 11 * q^62 - 19 * q^64 - 13 * q^65 + 33 * q^67 + 24 * q^68 + 4 * q^70 + 34 * q^73 + 27 * q^74 - 9 * q^76 + 29 * q^77 - 13 * q^80 + q^82 + 24 * q^83 - 4 * q^85 + 36 * q^86 - 6 * q^88 + 10 * q^89 + 30 * q^91 + 28 * q^92 - 8 * q^94 - 14 * q^95 + 16 * q^97 + 36 * 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 $

$2$	−0.142275	−0.100604	−0.0503018	−	0.998734i	$-0.516018\pi$
$2$	−0.142275	−0.100604	−0.0503018	+	0.998734i	$0.516018\pi$
$3$	0	0
$3$	0	0
$4$	−1.97976	−0.989879
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.55684	−0.588431	−0.294215	−	0.955739i	$-0.595058\pi$
$7$	−1.55684	−0.588431	−0.294215	+	0.955739i	$0.595058\pi$
$8$	0.566220	0.200189
$9$	0	0
$10$	0.142275	0.0449913
$11$	−1.06174	−0.320127	−0.160064	−	0.987107i	$-0.551170\pi$
$11$	−1.06174	−0.320127	−0.160064	+	0.987107i	$0.551170\pi$
$12$	0	0
$13$	−6.73897	−1.86905	−0.934527	−	0.355893i	$-0.884177\pi$
$13$	−6.73897	−1.86905	−0.934527	+	0.355893i	$0.884177\pi$
$14$	0.221499	0.0591982
$15$	0	0
$16$	3.87896	0.969739
$17$	−4.71226	−1.14289	−0.571446	−	0.820640i	$-0.693617\pi$
$17$	−4.71226	−1.14289	−0.571446	+	0.820640i	$0.693617\pi$
$18$	0	0
$19$	3.03705	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$19$	3.03705	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$20$	1.97976	0.442687
$21$	0	0
$22$	0.151059	0.0322059
$23$	−5.80958	−1.21138	−0.605691	−	0.795700i	$-0.707103\pi$
$23$	−5.80958	−1.21138	−0.605691	+	0.795700i	$0.707103\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.958786	0.188033
$27$	0	0
$28$	3.08217	0.582475
$29$	−4.16204	−0.772872	−0.386436	−	0.922316i	$-0.626294\pi$
$29$	−4.16204	−0.772872	−0.386436	+	0.922316i	$0.626294\pi$
$30$	0	0
$31$	−8.70959	−1.56429	−0.782145	−	0.623097i	$-0.785874\pi$
$31$	−8.70959	−1.56429	−0.782145	+	0.623097i	$0.785874\pi$
$32$	−1.68432	−0.297748
$33$	0	0
$34$	0.670437	0.114979
$35$	1.55684	0.263154
$36$	0	0
$37$	6.57240	1.08050	0.540248	−	0.841506i	$-0.318330\pi$
$37$	6.57240	1.08050	0.540248	+	0.841506i	$0.318330\pi$
$38$	−0.432096	−0.0700952
$39$	0	0
$40$	−0.566220	−0.0895272
$41$	−7.77249	−1.21386	−0.606929	−	0.794756i	$-0.707599\pi$
$41$	−7.77249	−1.21386	−0.606929	+	0.794756i	$0.707599\pi$
$42$	0	0
$43$	10.7312	1.63649	0.818247	−	0.574866i	$-0.194946\pi$
$43$	10.7312	1.63649	0.818247	+	0.574866i	$0.194946\pi$
$44$	2.10199	0.316887
$45$	0	0
$46$	0.826558	0.121869
$47$	−10.0921	−1.47209	−0.736045	−	0.676932i	$-0.763309\pi$
$47$	−10.0921	−1.47209	−0.736045	+	0.676932i	$0.763309\pi$
$48$	0	0
$49$	−4.57624	−0.653749
$50$	−0.142275	−0.0201207
$51$	0	0
$52$	13.3415	1.85014
$53$	1.17807	0.161821	0.0809103	−	0.996721i	$-0.474217\pi$
$53$	1.17807	0.161821	0.0809103	+	0.996721i	$0.474217\pi$
$54$	0	0
$55$	1.06174	0.143165
$56$	−0.881514	−0.117797
$57$	0	0
$58$	0.592154	0.0777536
$59$	3.58301	0.466468	0.233234	−	0.972421i	$-0.425069\pi$
$59$	3.58301	0.466468	0.233234	+	0.972421i	$0.425069\pi$
$60$	0	0
$61$	0.168963	0.0216336	0.0108168	−	0.999941i	$-0.496557\pi$
$61$	0.168963	0.0216336	0.0108168	+	0.999941i	$0.496557\pi$
$62$	1.23916	0.157373
$63$	0	0
$64$	−7.51828	−0.939785
$65$	6.73897	0.835866
$66$	0	0
$67$	0.691226	0.0844467	0.0422234	−	0.999108i	$-0.486556\pi$
$67$	0.691226	0.0844467	0.0422234	+	0.999108i	$0.486556\pi$
$68$	9.32914	1.13132
$69$	0	0
$70$	−0.221499	−0.0264742
$71$	−6.38099	−0.757285	−0.378642	−	0.925543i	$-0.623609\pi$
$71$	−6.38099	−0.757285	−0.378642	+	0.925543i	$0.623609\pi$
$72$	0	0
$73$	−10.3711	−1.21384	−0.606921	−	0.794762i	$-0.707596\pi$
$73$	−10.3711	−1.21384	−0.606921	+	0.794762i	$0.707596\pi$
$74$	−0.935087	−0.108702
$75$	0	0
$76$	−6.01262	−0.689695
$77$	1.65296	0.188373
$78$	0	0
$79$	−0.977121	−0.109935	−0.0549674	−	0.998488i	$-0.517505\pi$
$79$	−0.977121	−0.109935	−0.0549674	+	0.998488i	$0.517505\pi$
$80$	−3.87896	−0.433681
$81$	0	0
$82$	1.10583	0.122118
$83$	12.6407	1.38750	0.693748	−	0.720218i	$-0.255958\pi$
$83$	12.6407	1.38750	0.693748	+	0.720218i	$0.255958\pi$
$84$	0	0
$85$	4.71226	0.511117
$86$	−1.52678	−0.164637
$87$	0	0
$88$	−0.601179	−0.0640859
$89$	−17.7527	−1.88178	−0.940890	−	0.338714i	$-0.890008\pi$
$89$	−17.7527	−1.88178	−0.940890	+	0.338714i	$0.890008\pi$
$90$	0	0
$91$	10.4915	1.09981
$92$	11.5016	1.19912
$93$	0	0
$94$	1.43586	0.148097
$95$	−3.03705	−0.311595
$96$	0	0
$97$	19.4157	1.97136	0.985681	−	0.168623i	$-0.0539320\pi$
$97$	19.4157	1.97136	0.985681	+	0.168623i	$0.0539320\pi$
$98$	0.651085	0.0657695
$99$	0	0
$100$	−1.97976	−0.197976
$101$	−6.93092	−0.689652	−0.344826	−	0.938667i	$-0.612062\pi$
$101$	−6.93092	−0.689652	−0.344826	+	0.938667i	$0.612062\pi$
$102$	0	0
$103$	−2.52510	−0.248805	−0.124403	−	0.992232i	$-0.539701\pi$
$103$	−2.52510	−0.248805	−0.124403	+	0.992232i	$0.539701\pi$
$104$	−3.81574	−0.374164
$105$	0	0
$106$	−0.167610	−0.0162797
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	0	0
$109$	1.42277	0.136277	0.0681383	−	0.997676i	$-0.478294\pi$
$109$	1.42277	0.136277	0.0681383	+	0.997676i	$0.478294\pi$
$110$	−0.151059	−0.0144029
$111$	0	0
$112$	−6.03892	−0.570624
$113$	1.19401	0.112323	0.0561614	−	0.998422i	$-0.482114\pi$
$113$	1.19401	0.112323	0.0561614	+	0.998422i	$0.482114\pi$
$114$	0	0
$115$	5.80958	0.541747
$116$	8.23984	0.765050
$117$	0	0
$118$	−0.509772	−0.0469283
$119$	7.33625	0.672513
$120$	0	0
$121$	−9.87270	−0.897518
$122$	−0.0240393	−0.00217641
$123$	0	0
$124$	17.2429	1.54846
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.1137	1.16365	0.581827	−	0.813313i	$-0.302338\pi$
$127$	13.1137	1.16365	0.581827	+	0.813313i	$0.302338\pi$
$128$	4.43830	0.392294
$129$	0	0
$130$	−0.958786	−0.0840911
$131$	−0.904026	−0.0789851	−0.0394926	−	0.999220i	$-0.512574\pi$
$131$	−0.904026	−0.0789851	−0.0394926	+	0.999220i	$0.512574\pi$
$132$	0	0
$133$	−4.72821	−0.409987
$134$	−0.0983441	−0.00849564
$135$	0	0
$136$	−2.66818	−0.228794
$137$	9.47331	0.809359	0.404680	−	0.914459i	$-0.367383\pi$
$137$	9.47331	0.809359	0.404680	+	0.914459i	$0.367383\pi$
$138$	0	0
$139$	6.62526	0.561947	0.280974	−	0.959715i	$-0.409343\pi$
$139$	6.62526	0.561947	0.280974	+	0.959715i	$0.409343\pi$
$140$	−3.08217	−0.260491
$141$	0	0
$142$	0.907855	0.0761855
$143$	7.15505	0.598335
$144$	0	0
$145$	4.16204	0.345639
$146$	1.47554	0.122117
$147$	0	0
$148$	−13.0118	−1.06956
$149$	11.6577	0.955038	0.477519	−	0.878621i	$-0.341536\pi$
$149$	11.6577	0.955038	0.477519	+	0.878621i	$0.341536\pi$
$150$	0	0
$151$	−21.6173	−1.75919	−0.879597	−	0.475719i	$-0.842188\pi$
$151$	−21.6173	−1.75919	−0.879597	+	0.475719i	$0.842188\pi$
$152$	1.71964	0.139481
$153$	0	0
$154$	−0.235175	−0.0189510
$155$	8.70959	0.699571
$156$	0	0
$157$	15.9515	1.27306	0.636532	−	0.771250i	$-0.280368\pi$
$157$	15.9515	1.27306	0.636532	+	0.771250i	$0.280368\pi$
$158$	0.139020	0.0110598
$159$	0	0
$160$	1.68432	0.133157
$161$	9.04460	0.712814
$162$	0	0
$163$	13.7452	1.07661	0.538304	−	0.842751i	$-0.319065\pi$
$163$	13.7452	1.07661	0.538304	+	0.842751i	$0.319065\pi$
$164$	15.3876	1.20157
$165$	0	0
$166$	−1.79845	−0.139587
$167$	−2.85448	−0.220886	−0.110443	−	0.993882i	$-0.535227\pi$
$167$	−2.85448	−0.220886	−0.110443	+	0.993882i	$0.535227\pi$
$168$	0	0
$169$	32.4137	2.49336
$170$	−0.670437	−0.0514202
$171$	0	0
$172$	−21.2452	−1.61993
$173$	−13.8358	−1.05192	−0.525959	−	0.850510i	$-0.676293\pi$
$173$	−13.8358	−1.05192	−0.525959	+	0.850510i	$0.676293\pi$
$174$	0	0
$175$	−1.55684	−0.117686
$176$	−4.11845	−0.310440
$177$	0	0
$178$	2.52576	0.189314
$179$	13.7364	1.02671	0.513355	−	0.858177i	$-0.328403\pi$
$179$	13.7364	1.02671	0.513355	+	0.858177i	$0.328403\pi$
$180$	0	0
$181$	−22.3133	−1.65853	−0.829266	−	0.558854i	$-0.811241\pi$
$181$	−22.3133	−1.65853	−0.829266	+	0.558854i	$0.811241\pi$
$182$	−1.49268	−0.110645
$183$	0	0
$184$	−3.28950	−0.242505
$185$	−6.57240	−0.483212
$186$	0	0
$187$	5.00321	0.365871
$188$	19.9800	1.45719
$189$	0	0
$190$	0.432096	0.0313475
$191$	−8.38921	−0.607022	−0.303511	−	0.952828i	$-0.598159\pi$
$191$	−8.38921	−0.607022	−0.303511	+	0.952828i	$0.598159\pi$
$192$	0	0
$193$	−14.9915	−1.07911	−0.539556	−	0.841950i	$-0.681408\pi$
$193$	−14.9915	−1.07911	−0.539556	+	0.841950i	$0.681408\pi$
$194$	−2.76236	−0.198326
$195$	0	0
$196$	9.05986	0.647133
$197$	13.4039	0.954986	0.477493	−	0.878635i	$-0.341546\pi$
$197$	13.4039	0.954986	0.477493	+	0.878635i	$0.341546\pi$
$198$	0	0
$199$	−20.6913	−1.46677	−0.733383	−	0.679815i	$-0.762060\pi$
$199$	−20.6913	−1.46677	−0.733383	+	0.679815i	$0.762060\pi$
$200$	0.566220	0.0400378
$201$	0	0
$202$	0.986095	0.0693814
$203$	6.47964	0.454782
$204$	0	0
$205$	7.77249	0.542854
$206$	0.359258	0.0250307
$207$	0	0
$208$	−26.1402	−1.81249
$209$	−3.22457	−0.223048
$210$	0	0
$211$	18.4519	1.27028	0.635140	−	0.772397i	$-0.280942\pi$
$211$	18.4519	1.27028	0.635140	+	0.772397i	$0.280942\pi$
$212$	−2.33230	−0.160183
$213$	0	0
$214$	0.142275	0.00972571
$215$	−10.7312	−0.731863
$216$	0	0
$217$	13.5595	0.920476
$218$	−0.202424	−0.0137099
$219$	0	0
$220$	−2.10199	−0.141716
$221$	31.7558	2.13613
$222$	0	0
$223$	6.67543	0.447020	0.223510	−	0.974702i	$-0.428249\pi$
$223$	6.67543	0.447020	0.223510	+	0.974702i	$0.428249\pi$
$224$	2.62221	0.175204
$225$	0	0
$226$	−0.169877	−0.0113001
$227$	−16.3863	−1.08760	−0.543798	−	0.839216i	$-0.683014\pi$
$227$	−16.3863	−1.08760	−0.543798	+	0.839216i	$0.683014\pi$
$228$	0	0
$229$	2.54717	0.168322	0.0841609	−	0.996452i	$-0.473179\pi$
$229$	2.54717	0.168322	0.0841609	+	0.996452i	$0.473179\pi$
$230$	−0.826558	−0.0545016
$231$	0	0
$232$	−2.35663	−0.154720
$233$	27.0246	1.77044	0.885221	−	0.465170i	$-0.154007\pi$
$233$	27.0246	1.77044	0.885221	+	0.465170i	$0.154007\pi$
$234$	0	0
$235$	10.0921	0.658339
$236$	−7.09349	−0.461747
$237$	0	0
$238$	−1.04376	−0.0676572
$239$	5.72385	0.370245	0.185123	−	0.982715i	$-0.440732\pi$
$239$	5.72385	0.370245	0.185123	+	0.982715i	$0.440732\pi$
$240$	0	0
$241$	10.2480	0.660135	0.330067	−	0.943957i	$-0.392929\pi$
$241$	10.2480	0.660135	0.330067	+	0.943957i	$0.392929\pi$
$242$	1.40464	0.0902935
$243$	0	0
$244$	−0.334507	−0.0214146
$245$	4.57624	0.292366
$246$	0	0
$247$	−20.4666	−1.30226
$248$	−4.93154	−0.313153
$249$	0	0
$250$	0.142275	0.00899825
$251$	−5.95225	−0.375703	−0.187851	−	0.982197i	$-0.560152\pi$
$251$	−5.95225	−0.375703	−0.187851	+	0.982197i	$0.560152\pi$
$252$	0	0
$253$	6.16828	0.387797
$254$	−1.86575	−0.117068
$255$	0	0
$256$	14.4051	0.900319
$257$	−11.8724	−0.740580	−0.370290	−	0.928916i	$-0.620742\pi$
$257$	−11.8724	−0.740580	−0.370290	+	0.928916i	$0.620742\pi$
$258$	0	0
$259$	−10.2322	−0.635797
$260$	−13.3415	−0.827406
$261$	0	0
$262$	0.128620	0.00794618
$263$	18.3835	1.13358	0.566788	−	0.823864i	$-0.308186\pi$
$263$	18.3835	1.13358	0.566788	+	0.823864i	$0.308186\pi$
$264$	0	0
$265$	−1.17807	−0.0723684
$266$	0.672705	0.0412462
$267$	0	0
$268$	−1.36846	−0.0835920
$269$	29.4693	1.79677	0.898387	−	0.439205i	$-0.144740\pi$
$269$	29.4693	1.79677	0.898387	+	0.439205i	$0.144740\pi$
$270$	0	0
$271$	4.88944	0.297013	0.148506	−	0.988911i	$-0.452553\pi$
$271$	4.88944	0.297013	0.148506	+	0.988911i	$0.452553\pi$
$272$	−18.2787	−1.10831
$273$	0	0
$274$	−1.34781	−0.0814244
$275$	−1.06174	−0.0640255
$276$	0	0
$277$	−7.74025	−0.465066	−0.232533	−	0.972588i	$-0.574701\pi$
$277$	−7.74025	−0.465066	−0.232533	+	0.972588i	$0.574701\pi$
$278$	−0.942608	−0.0565339
$279$	0	0
$280$	0.881514	0.0526805
$281$	−13.5427	−0.807888	−0.403944	−	0.914784i	$-0.632361\pi$
$281$	−13.5427	−0.807888	−0.403944	+	0.914784i	$0.632361\pi$
$282$	0	0
$283$	27.0328	1.60693	0.803466	−	0.595351i	$-0.202987\pi$
$283$	27.0328	1.60693	0.803466	+	0.595351i	$0.202987\pi$
$284$	12.6328	0.749620
$285$	0	0
$286$	−1.01798	−0.0601946
$287$	12.1005	0.714272
$288$	0	0
$289$	5.20544	0.306202
$290$	−0.592154	−0.0347725
$291$	0	0
$292$	20.5322	1.20156
$293$	−10.7737	−0.629406	−0.314703	−	0.949190i	$-0.601905\pi$
$293$	−10.7737	−0.629406	−0.314703	+	0.949190i	$0.601905\pi$
$294$	0	0
$295$	−3.58301	−0.208611
$296$	3.72142	0.216303
$297$	0	0
$298$	−1.65860	−0.0960802
$299$	39.1506	2.26414
$300$	0	0
$301$	−16.7068	−0.962964
$302$	3.07560	0.176981
$303$	0	0
$304$	11.7806	0.675663
$305$	−0.168963	−0.00967482
$306$	0	0
$307$	22.6034	1.29004	0.645022	−	0.764164i	$-0.276848\pi$
$307$	22.6034	1.29004	0.645022	+	0.764164i	$0.276848\pi$
$308$	−3.27247	−0.186466
$309$	0	0
$310$	−1.23916	−0.0703793
$311$	6.74810	0.382650	0.191325	−	0.981527i	$-0.438722\pi$
$311$	6.74810	0.382650	0.191325	+	0.981527i	$0.438722\pi$
$312$	0	0
$313$	−18.4587	−1.04335	−0.521675	−	0.853145i	$-0.674692\pi$
$313$	−18.4587	−1.04335	−0.521675	+	0.853145i	$0.674692\pi$
$314$	−2.26949	−0.128075
$315$	0	0
$316$	1.93446	0.108822
$317$	−5.42724	−0.304824	−0.152412	−	0.988317i	$-0.548704\pi$
$317$	−5.42724	−0.304824	−0.152412	+	0.988317i	$0.548704\pi$
$318$	0	0
$319$	4.41902	0.247418
$320$	7.51828	0.420285
$321$	0	0
$322$	−1.28682	−0.0717116
$323$	−14.3114	−0.796307
$324$	0	0
$325$	−6.73897	−0.373811
$326$	−1.95560	−0.108310
$327$	0	0
$328$	−4.40093	−0.243001
$329$	15.7119	0.866223
$330$	0	0
$331$	−17.8512	−0.981190	−0.490595	−	0.871388i	$-0.663221\pi$
$331$	−17.8512	−0.981190	−0.490595	+	0.871388i	$0.663221\pi$
$332$	−25.0255	−1.37345
$333$	0	0
$334$	0.406120	0.0222219
$335$	−0.691226	−0.0377657
$336$	0	0
$337$	−18.8163	−1.02499	−0.512494	−	0.858691i	$-0.671278\pi$
$337$	−18.8163	−1.02499	−0.512494	+	0.858691i	$0.671278\pi$
$338$	−4.61165	−0.250841
$339$	0	0
$340$	−9.32914	−0.505944
$341$	9.24735	0.500772
$342$	0	0
$343$	18.0224	0.973117
$344$	6.07622	0.327608
$345$	0	0
$346$	1.96849	0.105827
$347$	29.3905	1.57776	0.788881	−	0.614546i	$-0.210661\pi$
$347$	29.3905	1.57776	0.788881	+	0.614546i	$0.210661\pi$
$348$	0	0
$349$	1.62710	0.0870966	0.0435483	−	0.999051i	$-0.486134\pi$
$349$	1.62710	0.0870966	0.0435483	+	0.999051i	$0.486134\pi$
$350$	0.221499	0.0118396
$351$	0	0
$352$	1.78831	0.0953173
$353$	1.64201	0.0873954	0.0436977	−	0.999045i	$-0.486086\pi$
$353$	1.64201	0.0873954	0.0436977	+	0.999045i	$0.486086\pi$
$354$	0	0
$355$	6.38099	0.338668
$356$	35.1460	1.86273
$357$	0	0
$358$	−1.95435	−0.103291
$359$	21.1076	1.11402	0.557008	−	0.830507i	$-0.311949\pi$
$359$	21.1076	1.11402	0.557008	+	0.830507i	$0.311949\pi$
$360$	0	0
$361$	−9.77633	−0.514544
$362$	3.17462	0.166854
$363$	0	0
$364$	−20.7706	−1.08868
$365$	10.3711	0.542847
$366$	0	0
$367$	−8.02117	−0.418702	−0.209351	−	0.977841i	$-0.567135\pi$
$367$	−8.02117	−0.418702	−0.209351	+	0.977841i	$0.567135\pi$
$368$	−22.5351	−1.17472
$369$	0	0
$370$	0.935087	0.0486129
$371$	−1.83407	−0.0952203
$372$	0	0
$373$	−8.44980	−0.437514	−0.218757	−	0.975779i	$-0.570200\pi$
$373$	−8.44980	−0.437514	−0.218757	+	0.975779i	$0.570200\pi$
$374$	−0.711831	−0.0368079
$375$	0	0
$376$	−5.71437	−0.294696
$377$	28.0479	1.44454
$378$	0	0
$379$	21.6351	1.11132	0.555660	−	0.831410i	$-0.312466\pi$
$379$	21.6351	1.11132	0.555660	+	0.831410i	$0.312466\pi$
$380$	6.01262	0.308441
$381$	0	0
$382$	1.19357	0.0610686
$383$	11.1306	0.568749	0.284375	−	0.958713i	$-0.408214\pi$
$383$	11.1306	0.568749	0.284375	+	0.958713i	$0.408214\pi$
$384$	0	0
$385$	−1.65296	−0.0842429
$386$	2.13292	0.108563
$387$	0	0
$388$	−38.4383	−1.95141
$389$	39.1986	1.98745	0.993723	−	0.111868i	$-0.0356834\pi$
$389$	39.1986	1.98745	0.993723	+	0.111868i	$0.0356834\pi$
$390$	0	0
$391$	27.3763	1.38448
$392$	−2.59116	−0.130873
$393$	0	0
$394$	−1.90703	−0.0960750
$395$	0.977121	0.0491643
$396$	0	0
$397$	−2.75063	−0.138050	−0.0690251	−	0.997615i	$-0.521989\pi$
$397$	−2.75063	−0.138050	−0.0690251	+	0.997615i	$0.521989\pi$
$398$	2.94385	0.147562
$399$	0	0
$400$	3.87896	0.193948
$401$	6.48570	0.323880	0.161940	−	0.986801i	$-0.448225\pi$
$401$	6.48570	0.323880	0.161940	+	0.986801i	$0.448225\pi$
$402$	0	0
$403$	58.6937	2.92374
$404$	13.7215	0.682672
$405$	0	0
$406$	−0.921890	−0.0457526
$407$	−6.97820	−0.345896
$408$	0	0
$409$	−0.223489	−0.0110508	−0.00552541	−	0.999985i	$-0.501759\pi$
$409$	−0.223489	−0.0110508	−0.00552541	+	0.999985i	$0.501759\pi$
$410$	−1.10583	−0.0546130
$411$	0	0
$412$	4.99908	0.246287
$413$	−5.57818	−0.274484
$414$	0	0
$415$	−12.6407	−0.620507
$416$	11.3506	0.556507
$417$	0	0
$418$	0.458775	0.0224394
$419$	−15.7962	−0.771694	−0.385847	−	0.922563i	$-0.626091\pi$
$419$	−15.7962	−0.771694	−0.385847	+	0.922563i	$0.626091\pi$
$420$	0	0
$421$	−20.7419	−1.01090	−0.505449	−	0.862856i	$-0.668673\pi$
$421$	−20.7419	−1.01090	−0.505449	+	0.862856i	$0.668673\pi$
$422$	−2.62524	−0.127795
$423$	0	0
$424$	0.667047	0.0323947
$425$	−4.71226	−0.228578
$426$	0	0
$427$	−0.263049	−0.0127298
$428$	1.97976	0.0956952
$429$	0	0
$430$	1.52678	0.0736280
$431$	−14.7526	−0.710608	−0.355304	−	0.934751i	$-0.615623\pi$
$431$	−14.7526	−0.710608	−0.355304	+	0.934751i	$0.615623\pi$
$432$	0	0
$433$	8.48565	0.407794	0.203897	−	0.978992i	$-0.434639\pi$
$433$	8.48565	0.407794	0.203897	+	0.978992i	$0.434639\pi$
$434$	−1.92917	−0.0926031
$435$	0	0
$436$	−2.81674	−0.134897
$437$	−17.6440	−0.844027
$438$	0	0
$439$	−8.30354	−0.396306	−0.198153	−	0.980171i	$-0.563494\pi$
$439$	−8.30354	−0.396306	−0.198153	+	0.980171i	$0.563494\pi$
$440$	0.601179	0.0286601
$441$	0	0
$442$	−4.51805	−0.214902
$443$	−28.9430	−1.37512	−0.687562	−	0.726126i	$-0.741319\pi$
$443$	−28.9430	−1.37512	−0.687562	+	0.726126i	$0.741319\pi$
$444$	0	0
$445$	17.7527	0.841557
$446$	−0.949746	−0.0449718
$447$	0	0
$448$	11.7048	0.552998
$449$	−4.64509	−0.219215	−0.109608	−	0.993975i	$-0.534959\pi$
$449$	−4.64509	−0.219215	−0.109608	+	0.993975i	$0.534959\pi$
$450$	0	0
$451$	8.25238	0.388589
$452$	−2.36384	−0.111186
$453$	0	0
$454$	2.33136	0.109416
$455$	−10.4915	−0.491849
$456$	0	0
$457$	−11.1424	−0.521219	−0.260609	−	0.965444i	$-0.583923\pi$
$457$	−11.1424	−0.521219	−0.260609	+	0.965444i	$0.583923\pi$
$458$	−0.362398	−0.0169338
$459$	0	0
$460$	−11.5016	−0.536263
$461$	35.2378	1.64119	0.820594	−	0.571511i	$-0.193643\pi$
$461$	35.2378	1.64119	0.820594	+	0.571511i	$0.193643\pi$
$462$	0	0
$463$	−1.22425	−0.0568959	−0.0284480	−	0.999595i	$-0.509056\pi$
$463$	−1.22425	−0.0568959	−0.0284480	+	0.999595i	$0.509056\pi$
$464$	−16.1444	−0.749484
$465$	0	0
$466$	−3.84493	−0.178113
$467$	−25.1605	−1.16429	−0.582144	−	0.813086i	$-0.697786\pi$
$467$	−25.1605	−1.16429	−0.582144	+	0.813086i	$0.697786\pi$
$468$	0	0
$469$	−1.07613	−0.0496910
$470$	−1.43586	−0.0662312
$471$	0	0
$472$	2.02877	0.0933817
$473$	−11.3938	−0.523887
$474$	0	0
$475$	3.03705	0.139349
$476$	−14.5240	−0.665706
$477$	0	0
$478$	−0.814360	−0.0372480
$479$	−14.1959	−0.648627	−0.324313	−	0.945950i	$-0.605133\pi$
$479$	−14.1959	−0.648627	−0.324313	+	0.945950i	$0.605133\pi$
$480$	0	0
$481$	−44.2912	−2.01950
$482$	−1.45804	−0.0664119
$483$	0	0
$484$	19.5456	0.888435
$485$	−19.4157	−0.881620
$486$	0	0
$487$	−2.50122	−0.113341	−0.0566707	−	0.998393i	$-0.518049\pi$
$487$	−2.50122	−0.113341	−0.0566707	+	0.998393i	$0.518049\pi$
$488$	0.0956704	0.00433080
$489$	0	0
$490$	−0.651085	−0.0294130
$491$	−29.3243	−1.32339	−0.661694	−	0.749774i	$-0.730162\pi$
$491$	−29.3243	−1.32339	−0.661694	+	0.749774i	$0.730162\pi$
$492$	0	0
$493$	19.6126	0.883309
$494$	2.91188	0.131012
$495$	0	0
$496$	−33.7841	−1.51695
$497$	9.93420	0.445610
$498$	0	0
$499$	7.46040	0.333973	0.166987	−	0.985959i	$-0.446596\pi$
$499$	7.46040	0.333973	0.166987	+	0.985959i	$0.446596\pi$
$500$	1.97976	0.0885375
$501$	0	0
$502$	0.846855	0.0377970
$503$	−40.2235	−1.79348	−0.896738	−	0.442562i	$-0.854070\pi$
$503$	−40.2235	−1.79348	−0.896738	+	0.442562i	$0.854070\pi$
$504$	0	0
$505$	6.93092	0.308422
$506$	−0.877592	−0.0390137
$507$	0	0
$508$	−25.9620	−1.15188
$509$	−31.3381	−1.38904	−0.694518	−	0.719476i	$-0.744382\pi$
$509$	−31.3381	−1.38904	−0.694518	+	0.719476i	$0.744382\pi$
$510$	0	0
$511$	16.1461	0.714263
$512$	−10.9261	−0.482869
$513$	0	0
$514$	1.68914	0.0745050
$515$	2.52510	0.111269
$516$	0	0
$517$	10.7153	0.471257
$518$	1.45578	0.0639634
$519$	0	0
$520$	3.81574	0.167331
$521$	−19.1001	−0.836792	−0.418396	−	0.908265i	$-0.637408\pi$
$521$	−19.1001	−0.836792	−0.418396	+	0.908265i	$0.637408\pi$
$522$	0	0
$523$	−33.3927	−1.46016	−0.730080	−	0.683361i	$-0.760517\pi$
$523$	−33.3927	−1.46016	−0.730080	+	0.683361i	$0.760517\pi$
$524$	1.78975	0.0781857
$525$	0	0
$526$	−2.61551	−0.114042
$527$	41.0419	1.78781
$528$	0	0
$529$	10.7513	0.467446
$530$	0.167610	0.00728052
$531$	0	0
$532$	9.36070	0.405838
$533$	52.3785	2.26877
$534$	0	0
$535$	1.00000	0.0432338
$536$	0.391386	0.0169053
$537$	0	0
$538$	−4.19274	−0.180762
$539$	4.85879	0.209283
$540$	0	0
$541$	0.664311	0.0285610	0.0142805	−	0.999898i	$-0.495454\pi$
$541$	0.664311	0.0285610	0.0142805	+	0.999898i	$0.495454\pi$
$542$	−0.695645	−0.0298805
$543$	0	0
$544$	7.93695	0.340294
$545$	−1.42277	−0.0609447
$546$	0	0
$547$	25.9800	1.11082	0.555412	−	0.831575i	$-0.312561\pi$
$547$	25.9800	1.11082	0.555412	+	0.831575i	$0.312561\pi$
$548$	−18.7549	−0.801168
$549$	0	0
$550$	0.151059	0.00644119
$551$	−12.6403	−0.538496
$552$	0	0
$553$	1.52122	0.0646890
$554$	1.10124	0.0467873
$555$	0	0
$556$	−13.1164	−0.556260
$557$	6.76018	0.286438	0.143219	−	0.989691i	$-0.454255\pi$
$557$	6.76018	0.286438	0.143219	+	0.989691i	$0.454255\pi$
$558$	0	0
$559$	−72.3173	−3.05870
$560$	6.03892	0.255191
$561$	0	0
$562$	1.92678	0.0812763
$563$	40.6699	1.71403	0.857017	−	0.515288i	$-0.172315\pi$
$563$	40.6699	1.71403	0.857017	+	0.515288i	$0.172315\pi$
$564$	0	0
$565$	−1.19401	−0.0502322
$566$	−3.84608	−0.161663
$567$	0	0
$568$	−3.61304	−0.151600
$569$	27.8866	1.16907	0.584534	−	0.811369i	$-0.301277\pi$
$569$	27.8866	1.16907	0.584534	+	0.811369i	$0.301277\pi$
$570$	0	0
$571$	−40.7635	−1.70590	−0.852950	−	0.521993i	$-0.825189\pi$
$571$	−40.7635	−1.70590	−0.852950	+	0.521993i	$0.825189\pi$
$572$	−14.1653	−0.592280
$573$	0	0
$574$	−1.72160	−0.0718582
$575$	−5.80958	−0.242276
$576$	0	0
$577$	36.8117	1.53249	0.766245	−	0.642549i	$-0.222123\pi$
$577$	36.8117	1.53249	0.766245	+	0.642549i	$0.222123\pi$
$578$	−0.740603	−0.0308050
$579$	0	0
$580$	−8.23984	−0.342141
$581$	−19.6795	−0.816445
$582$	0	0
$583$	−1.25081	−0.0518032
$584$	−5.87231	−0.242998
$585$	0	0
$586$	1.53283	0.0633205
$587$	−31.9926	−1.32048	−0.660238	−	0.751057i	$-0.729544\pi$
$587$	−31.9926	−1.32048	−0.660238	+	0.751057i	$0.729544\pi$
$588$	0	0
$589$	−26.4515	−1.08991
$590$	0.509772	0.0209870
$591$	0	0
$592$	25.4941	1.04780
$593$	−15.9962	−0.656886	−0.328443	−	0.944524i	$-0.606524\pi$
$593$	−15.9962	−0.656886	−0.328443	+	0.944524i	$0.606524\pi$
$594$	0	0
$595$	−7.33625	−0.300757
$596$	−23.0795	−0.945372
$597$	0	0
$598$	−5.57015	−0.227780
$599$	11.0507	0.451518	0.225759	−	0.974183i	$-0.427514\pi$
$599$	11.0507	0.451518	0.225759	+	0.974183i	$0.427514\pi$
$600$	0	0
$601$	−32.6921	−1.33354	−0.666770	−	0.745264i	$-0.732324\pi$
$601$	−32.6921	−1.33354	−0.666770	+	0.745264i	$0.732324\pi$
$602$	2.37696	0.0968775
$603$	0	0
$604$	42.7971	1.74139
$605$	9.87270	0.401382
$606$	0	0
$607$	−41.3189	−1.67708	−0.838542	−	0.544837i	$-0.816591\pi$
$607$	−41.3189	−1.67708	−0.838542	+	0.544837i	$0.816591\pi$
$608$	−5.11535	−0.207455
$609$	0	0
$610$	0.0240393	0.000973321 0
$611$	68.0106	2.75142
$612$	0	0
$613$	−20.3601	−0.822339	−0.411169	−	0.911559i	$-0.634880\pi$
$613$	−20.3601	−0.822339	−0.411169	+	0.911559i	$0.634880\pi$
$614$	−3.21589	−0.129783
$615$	0	0
$616$	0.935941	0.0377101
$617$	21.7536	0.875766	0.437883	−	0.899032i	$-0.355728\pi$
$617$	21.7536	0.875766	0.437883	+	0.899032i	$0.355728\pi$
$618$	0	0
$619$	−13.5462	−0.544469	−0.272235	−	0.962231i	$-0.587763\pi$
$619$	−13.5462	−0.544469	−0.272235	+	0.962231i	$0.587763\pi$
$620$	−17.2429	−0.692491
$621$	0	0
$622$	−0.960085	−0.0384959
$623$	27.6381	1.10730
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.62621	0.104965
$627$	0	0
$628$	−31.5800	−1.26018
$629$	−30.9709	−1.23489
$630$	0	0
$631$	13.4983	0.537359	0.268680	−	0.963230i	$-0.413413\pi$
$631$	13.4983	0.537359	0.268680	+	0.963230i	$0.413413\pi$
$632$	−0.553265	−0.0220077
$633$	0	0
$634$	0.772160	0.0306664
$635$	−13.1137	−0.520402
$636$	0	0
$637$	30.8392	1.22189
$638$	−0.628715	−0.0248911
$639$	0	0
$640$	−4.43830	−0.175439
$641$	−1.54694	−0.0611005	−0.0305502	−	0.999533i	$-0.509726\pi$
$641$	−1.54694	−0.0611005	−0.0305502	+	0.999533i	$0.509726\pi$
$642$	0	0
$643$	21.3540	0.842121	0.421060	−	0.907033i	$-0.361658\pi$
$643$	21.3540	0.842121	0.421060	+	0.907033i	$0.361658\pi$
$644$	−17.9061	−0.705600
$645$	0	0
$646$	2.03615	0.0801112
$647$	39.3135	1.54557	0.772787	−	0.634666i	$-0.218862\pi$
$647$	39.3135	1.54557	0.772787	+	0.634666i	$0.218862\pi$
$648$	0	0
$649$	−3.80423	−0.149329
$650$	0.958786	0.0376067
$651$	0	0
$652$	−27.2122	−1.06571
$653$	−17.5061	−0.685067	−0.342534	−	0.939506i	$-0.611285\pi$
$653$	−17.5061	−0.685067	−0.342534	+	0.939506i	$0.611285\pi$
$654$	0	0
$655$	0.904026	0.0353232
$656$	−30.1491	−1.17713
$657$	0	0
$658$	−2.23540	−0.0871451
$659$	9.80484	0.381942	0.190971	−	0.981596i	$-0.438836\pi$
$659$	9.80484	0.381942	0.190971	+	0.981596i	$0.438836\pi$
$660$	0	0
$661$	−7.71103	−0.299924	−0.149962	−	0.988692i	$-0.547915\pi$
$661$	−7.71103	−0.299924	−0.149962	+	0.988692i	$0.547915\pi$
$662$	2.53978	0.0987112
$663$	0	0
$664$	7.15740	0.277761
$665$	4.72821	0.183352
$666$	0	0
$667$	24.1797	0.936243
$668$	5.65117	0.218650
$669$	0	0
$670$	0.0983441	0.00379936
$671$	−0.179396	−0.00692549
$672$	0	0
$673$	27.7833	1.07097	0.535484	−	0.844546i	$-0.320129\pi$
$673$	27.7833	1.07097	0.535484	+	0.844546i	$0.320129\pi$
$674$	2.67709	0.103117
$675$	0	0
$676$	−64.1713	−2.46813
$677$	−28.9529	−1.11275	−0.556375	−	0.830932i	$-0.687808\pi$
$677$	−28.9529	−1.11275	−0.556375	+	0.830932i	$0.687808\pi$
$678$	0	0
$679$	−30.2271	−1.16001
$680$	2.66818	0.102320
$681$	0	0
$682$	−1.31566	−0.0503794
$683$	−12.4082	−0.474786	−0.237393	−	0.971414i	$-0.576293\pi$
$683$	−12.4082	−0.474786	−0.237393	+	0.971414i	$0.576293\pi$
$684$	0	0
$685$	−9.47331	−0.361956
$686$	−2.56413	−0.0978990
$687$	0	0
$688$	41.6259	1.58697
$689$	−7.93899	−0.302451
$690$	0	0
$691$	31.1489	1.18496	0.592479	−	0.805586i	$-0.298149\pi$
$691$	31.1489	1.18496	0.592479	+	0.805586i	$0.298149\pi$
$692$	27.3916	1.04127
$693$	0	0
$694$	−4.18152	−0.158728
$695$	−6.62526	−0.251311
$696$	0	0
$697$	36.6260	1.38731
$698$	−0.231495	−0.00876222
$699$	0	0
$700$	3.08217	0.116495
$701$	−39.2234	−1.48145	−0.740724	−	0.671809i	$-0.765517\pi$
$701$	−39.2234	−1.48145	−0.740724	+	0.671809i	$0.765517\pi$
$702$	0	0
$703$	19.9607	0.752832
$704$	7.98248	0.300851
$705$	0	0
$706$	−0.233617	−0.00879229
$707$	10.7903	0.405812
$708$	0	0
$709$	−11.1968	−0.420504	−0.210252	−	0.977647i	$-0.567428\pi$
$709$	−11.1968	−0.420504	−0.210252	+	0.977647i	$0.567428\pi$
$710$	−0.907855	−0.0340712
$711$	0	0
$712$	−10.0519	−0.376711
$713$	50.5991	1.89495
$714$	0	0
$715$	−7.15505	−0.267584
$716$	−27.1948	−1.01632
$717$	0	0
$718$	−3.00308	−0.112074
$719$	−29.7770	−1.11049	−0.555247	−	0.831685i	$-0.687376\pi$
$719$	−29.7770	−1.11049	−0.555247	+	0.831685i	$0.687376\pi$
$720$	0	0
$721$	3.93118	0.146405
$722$	1.39093	0.0517649
$723$	0	0
$724$	44.1749	1.64175
$725$	−4.16204	−0.154574
$726$	0	0
$727$	−3.80006	−0.140936	−0.0704682	−	0.997514i	$-0.522449\pi$
$727$	−3.80006	−0.140936	−0.0704682	+	0.997514i	$0.522449\pi$
$728$	5.94049	0.220169
$729$	0	0
$730$	−1.47554	−0.0546123
$731$	−50.5683	−1.87034
$732$	0	0
$733$	21.4965	0.793990	0.396995	−	0.917821i	$-0.370053\pi$
$733$	21.4965	0.793990	0.396995	+	0.917821i	$0.370053\pi$
$734$	1.14121	0.0421229
$735$	0	0
$736$	9.78518	0.360687
$737$	−0.733904	−0.0270337
$738$	0	0
$739$	−8.12176	−0.298764	−0.149382	−	0.988780i	$-0.547728\pi$
$739$	−8.12176	−0.298764	−0.149382	+	0.988780i	$0.547728\pi$
$740$	13.0118	0.478322
$741$	0	0
$742$	0.260942	0.00957949
$743$	−43.6385	−1.60094	−0.800470	−	0.599373i	$-0.795417\pi$
$743$	−43.6385	−1.60094	−0.800470	+	0.599373i	$0.795417\pi$
$744$	0	0
$745$	−11.6577	−0.427106
$746$	1.20219	0.0440155
$747$	0	0
$748$	−9.90515	−0.362168
$749$	1.55684	0.0568857
$750$	0	0
$751$	−8.94047	−0.326242	−0.163121	−	0.986606i	$-0.552156\pi$
$751$	−8.94047	−0.326242	−0.163121	+	0.986606i	$0.552156\pi$
$752$	−39.1470	−1.42754
$753$	0	0
$754$	−3.99051	−0.145326
$755$	21.6173	0.786736
$756$	0	0
$757$	39.2445	1.42637	0.713183	−	0.700978i	$-0.247253\pi$
$757$	39.2445	1.42637	0.713183	+	0.700978i	$0.247253\pi$
$758$	−3.07813	−0.111803
$759$	0	0
$760$	−1.71964	−0.0623778
$761$	−37.1537	−1.34682	−0.673411	−	0.739269i	$-0.735171\pi$
$761$	−37.1537	−1.34682	−0.673411	+	0.739269i	$0.735171\pi$
$762$	0	0
$763$	−2.21503	−0.0801893
$764$	16.6086	0.600878
$765$	0	0
$766$	−1.58361	−0.0572182
$767$	−24.1458	−0.871853
$768$	0	0
$769$	23.3452	0.841848	0.420924	−	0.907096i	$-0.361706\pi$
$769$	23.3452	0.841848	0.420924	+	0.907096i	$0.361706\pi$
$770$	0.235175	0.00847513
$771$	0	0
$772$	29.6796	1.06819
$773$	−36.8715	−1.32618	−0.663088	−	0.748542i	$-0.730754\pi$
$773$	−36.8715	−1.32618	−0.663088	+	0.748542i	$0.730754\pi$
$774$	0	0
$775$	−8.70959	−0.312858
$776$	10.9935	0.394645
$777$	0	0
$778$	−5.57697	−0.199944
$779$	−23.6054	−0.845752
$780$	0	0
$781$	6.77497	0.242428
$782$	−3.89496	−0.139283
$783$	0	0
$784$	−17.7511	−0.633966
$785$	−15.9515	−0.569332
$786$	0	0
$787$	−26.6509	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$787$	−26.6509	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$788$	−26.5364	−0.945321
$789$	0	0
$790$	−0.139020	−0.00494610
$791$	−1.85888	−0.0660941
$792$	0	0
$793$	−1.13864	−0.0404343
$794$	0.391346	0.0138883
$795$	0	0
$796$	40.9637	1.45192
$797$	35.5604	1.25961	0.629806	−	0.776752i	$-0.283134\pi$
$797$	35.5604	1.25961	0.629806	+	0.776752i	$0.283134\pi$
$798$	0	0
$799$	47.5569	1.68244
$800$	−1.68432	−0.0595496
$801$	0	0
$802$	−0.922752	−0.0325835
$803$	11.0114	0.388584
$804$	0	0
$805$	−9.04460	−0.318780
$806$	−8.35063	−0.294139
$807$	0	0
$808$	−3.92442	−0.138061
$809$	32.6025	1.14624	0.573122	−	0.819470i	$-0.305732\pi$
$809$	32.6025	1.14624	0.573122	+	0.819470i	$0.305732\pi$
$810$	0	0
$811$	7.72477	0.271253	0.135627	−	0.990760i	$-0.456695\pi$
$811$	7.72477	0.271253	0.135627	+	0.990760i	$0.456695\pi$
$812$	−12.8281	−0.450179
$813$	0	0
$814$	0.992822	0.0347984
$815$	−13.7452	−0.481473
$816$	0	0
$817$	32.5912	1.14022
$818$	0.0317968	0.00111175
$819$	0	0
$820$	−15.3876	−0.537360
$821$	0.0137056	0.000478328 0	0.000239164	−	1.00000i	$-0.499924\pi$
$821$	0.0137056	0.000478328 0	0.000239164		1.00000i	$0.499924\pi$
$822$	0	0
$823$	31.8601	1.11057	0.555287	−	0.831659i	$-0.312608\pi$
$823$	31.8601	1.11057	0.555287	+	0.831659i	$0.312608\pi$
$824$	−1.42976	−0.0498080
$825$	0	0
$826$	0.793634	0.0276141
$827$	37.5382	1.30533	0.652666	−	0.757646i	$-0.273650\pi$
$827$	37.5382	1.30533	0.652666	+	0.757646i	$0.273650\pi$
$828$	0	0
$829$	−39.6458	−1.37696	−0.688478	−	0.725257i	$-0.741721\pi$
$829$	−39.6458	−1.37696	−0.688478	+	0.725257i	$0.741721\pi$
$830$	1.79845	0.0624252
$831$	0	0
$832$	50.6654	1.75651
$833$	21.5645	0.747165
$834$	0	0
$835$	2.85448	0.0987832
$836$	6.38386	0.220790
$837$	0	0
$838$	2.24740	0.0776351
$839$	−3.78235	−0.130581	−0.0652906	−	0.997866i	$-0.520797\pi$
$839$	−3.78235	−0.130581	−0.0652906	+	0.997866i	$0.520797\pi$
$840$	0	0
$841$	−11.6774	−0.402669
$842$	2.95105	0.101700
$843$	0	0
$844$	−36.5303	−1.25742
$845$	−32.4137	−1.11506
$846$	0	0
$847$	15.3702	0.528127
$848$	4.56969	0.156924
$849$	0	0
$850$	0.670437	0.0229958
$851$	−38.1829	−1.30889
$852$	0	0
$853$	6.62965	0.226995	0.113498	−	0.993538i	$-0.463795\pi$
$853$	6.62965	0.226995	0.113498	+	0.993538i	$0.463795\pi$
$854$	0.0374253	0.00128067
$855$	0	0
$856$	−0.566220	−0.0193530
$857$	−40.1097	−1.37012	−0.685060	−	0.728486i	$-0.740224\pi$
$857$	−40.1097	−1.37012	−0.685060	+	0.728486i	$0.740224\pi$
$858$	0	0
$859$	28.8359	0.983867	0.491933	−	0.870633i	$-0.336290\pi$
$859$	28.8359	0.983867	0.491933	+	0.870633i	$0.336290\pi$
$860$	21.2452	0.724455
$861$	0	0
$862$	2.09892	0.0714897
$863$	−36.9702	−1.25848	−0.629240	−	0.777211i	$-0.716634\pi$
$863$	−36.9702	−1.25848	−0.629240	+	0.777211i	$0.716634\pi$
$864$	0	0
$865$	13.8358	0.470432
$866$	−1.20729	−0.0410256
$867$	0	0
$868$	−26.8444	−0.911160
$869$	1.03745	0.0351931
$870$	0	0
$871$	−4.65815	−0.157835
$872$	0.805600	0.0272810
$873$	0	0
$874$	2.51030	0.0849121
$875$	1.55684	0.0526308
$876$	0	0
$877$	55.6787	1.88014	0.940068	−	0.340988i	$-0.110762\pi$
$877$	55.6787	1.88014	0.940068	+	0.340988i	$0.110762\pi$
$878$	1.18138	0.0398698
$879$	0	0
$880$	4.11845	0.138833
$881$	34.1323	1.14995	0.574974	−	0.818172i	$-0.305012\pi$
$881$	34.1323	1.14995	0.574974	+	0.818172i	$0.305012\pi$
$882$	0	0
$883$	−47.9156	−1.61249	−0.806244	−	0.591584i	$-0.798503\pi$
$883$	−47.9156	−1.61249	−0.806244	+	0.591584i	$0.798503\pi$
$884$	−62.8688	−2.11451
$885$	0	0
$886$	4.11786	0.138342
$887$	10.0289	0.336736	0.168368	−	0.985724i	$-0.446150\pi$
$887$	10.0289	0.336736	0.168368	+	0.985724i	$0.446150\pi$
$888$	0	0
$889$	−20.4160	−0.684730
$890$	−2.52576	−0.0846636
$891$	0	0
$892$	−13.2157	−0.442495
$893$	−30.6503	−1.02567
$894$	0	0
$895$	−13.7364	−0.459158
$896$	−6.90972	−0.230838
$897$	0	0
$898$	0.660879	0.0220538
$899$	36.2497	1.20900
$900$	0	0
$901$	−5.55139	−0.184944
$902$	−1.17411	−0.0390935
$903$	0	0
$904$	0.676070	0.0224858
$905$	22.3133	0.741718
$906$	0	0
$907$	17.2245	0.571930	0.285965	−	0.958240i	$-0.407686\pi$
$907$	17.2245	0.571930	0.285965	+	0.958240i	$0.407686\pi$
$908$	32.4409	1.07659
$909$	0	0
$910$	1.49268	0.0494818
$911$	20.9699	0.694762	0.347381	−	0.937724i	$-0.387071\pi$
$911$	20.9699	0.694762	0.347381	+	0.937724i	$0.387071\pi$
$912$	0	0
$913$	−13.4212	−0.444175
$914$	1.58528	0.0524364
$915$	0	0
$916$	−5.04278	−0.166618
$917$	1.40743	0.0464773
$918$	0	0
$919$	−22.0040	−0.725844	−0.362922	−	0.931819i	$-0.618221\pi$
$919$	−22.0040	−0.725844	−0.362922	+	0.931819i	$0.618221\pi$
$920$	3.28950	0.108452
$921$	0	0
$922$	−5.01346	−0.165109
$923$	43.0013	1.41541
$924$	0	0
$925$	6.57240	0.216099
$926$	0.174181	0.00572393
$927$	0	0
$928$	7.01020	0.230121
$929$	10.1196	0.332014	0.166007	−	0.986125i	$-0.446913\pi$
$929$	10.1196	0.332014	0.166007	+	0.986125i	$0.446913\pi$
$930$	0	0
$931$	−13.8983	−0.455498
$932$	−53.5022	−1.75252
$933$	0	0
$934$	3.57970	0.117131
$935$	−5.00321	−0.163623
$936$	0	0
$937$	4.47128	0.146070	0.0730352	−	0.997329i	$-0.476731\pi$
$937$	4.47128	0.146070	0.0730352	+	0.997329i	$0.476731\pi$
$938$	0.153106	0.00499909
$939$	0	0
$940$	−19.9800	−0.651676
$941$	38.3096	1.24886	0.624429	−	0.781082i	$-0.285332\pi$
$941$	38.3096	1.24886	0.624429	+	0.781082i	$0.285332\pi$
$942$	0	0
$943$	45.1549	1.47045
$944$	13.8983	0.452352
$945$	0	0
$946$	1.62105	0.0527049
$947$	−43.5569	−1.41541	−0.707705	−	0.706508i	$-0.750269\pi$
$947$	−43.5569	−1.41541	−0.707705	+	0.706508i	$0.750269\pi$
$948$	0	0
$949$	69.8904	2.26874
$950$	−0.432096	−0.0140190
$951$	0	0
$952$	4.15393	0.134630
$953$	59.4936	1.92719	0.963594	−	0.267370i	$-0.0861545\pi$
$953$	59.4936	1.92719	0.963594	+	0.267370i	$0.0861545\pi$
$954$	0	0
$955$	8.38921	0.271469
$956$	−11.3318	−0.366498
$957$	0	0
$958$	2.01972	0.0652542
$959$	−14.7484	−0.476252
$960$	0	0
$961$	44.8570	1.44700
$962$	6.30152	0.203169
$963$	0	0
$964$	−20.2887	−0.653453
$965$	14.9915	0.482594
$966$	0	0
$967$	1.05878	0.0340481	0.0170241	−	0.999855i	$-0.494581\pi$
$967$	1.05878	0.0340481	0.0170241	+	0.999855i	$0.494581\pi$
$968$	−5.59012	−0.179673
$969$	0	0
$970$	2.76236	0.0886940
$971$	−54.4742	−1.74816	−0.874080	−	0.485782i	$-0.838535\pi$
$971$	−54.4742	−1.74816	−0.874080	+	0.485782i	$0.838535\pi$
$972$	0	0
$973$	−10.3145	−0.330667
$974$	0.355861	0.0114025
$975$	0	0
$976$	0.655402	0.0209789
$977$	−36.5070	−1.16796	−0.583981	−	0.811767i	$-0.698506\pi$
$977$	−36.5070	−1.16796	−0.583981	+	0.811767i	$0.698506\pi$
$978$	0	0
$979$	18.8488	0.602409
$980$	−9.05986	−0.289406
$981$	0	0
$982$	4.17211	0.133138
$983$	−52.9790	−1.68977	−0.844883	−	0.534951i	$-0.820330\pi$
$983$	−52.9790	−1.68977	−0.844883	+	0.534951i	$0.820330\pi$
$984$	0	0
$985$	−13.4039	−0.427083
$986$	−2.79039	−0.0888640
$987$	0	0
$988$	40.5189	1.28908
$989$	−62.3439	−1.98242
$990$	0	0
$991$	−25.4774	−0.809317	−0.404658	−	0.914468i	$-0.632610\pi$
$991$	−25.4774	−0.809317	−0.404658	+	0.914468i	$0.632610\pi$
$992$	14.6697	0.465764
$993$	0	0
$994$	−1.41339	−0.0448299
$995$	20.6913	0.655958
$996$	0	0
$997$	−7.50346	−0.237637	−0.118818	−	0.992916i	$-0.537911\pi$
$997$	−7.50346	−0.237637	−0.118818	+	0.992916i	$0.537911\pi$
$998$	−1.06143	−0.0335989
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4815.2.a.u.1.7		12
3.2	odd	2		1605.2.a.n.1.6	✓	12
15.14	odd	2		8025.2.a.bf.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.a.n.1.6	✓	12	3.2	odd	2
4815.2.a.u.1.7		12	1.1	even	1	trivial
8025.2.a.bf.1.7		12	15.14	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:	\( N \)	\(=\)	\( 4815 = 3^{2} \cdot 5 \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4815.a (trivial)

\(f(q)\)	\(=\)	\(q-0.142275 q^{2} -1.97976 q^{4} -1.00000 q^{5} -1.55684 q^{7} +0.566220 q^{8} +O(q^{10})\)	\(q-0.142275 q^{2} -1.97976 q^{4} -1.00000 q^{5} -1.55684 q^{7} +0.566220 q^{8} +0.142275 q^{10} -1.06174 q^{11} -6.73897 q^{13} +0.221499 q^{14} +3.87896 q^{16} -4.71226 q^{17} +3.03705 q^{19} +1.97976 q^{20} +0.151059 q^{22} -5.80958 q^{23} +1.00000 q^{25} +0.958786 q^{26} +3.08217 q^{28} -4.16204 q^{29} -8.70959 q^{31} -1.68432 q^{32} +0.670437 q^{34} +1.55684 q^{35} +6.57240 q^{37} -0.432096 q^{38} -0.566220 q^{40} -7.77249 q^{41} +10.7312 q^{43} +2.10199 q^{44} +0.826558 q^{46} -10.0921 q^{47} -4.57624 q^{49} -0.142275 q^{50} +13.3415 q^{52} +1.17807 q^{53} +1.06174 q^{55} -0.881514 q^{56} +0.592154 q^{58} +3.58301 q^{59} +0.168963 q^{61} +1.23916 q^{62} -7.51828 q^{64} +6.73897 q^{65} +0.691226 q^{67} +9.32914 q^{68} -0.221499 q^{70} -6.38099 q^{71} -10.3711 q^{73} -0.935087 q^{74} -6.01262 q^{76} +1.65296 q^{77} -0.977121 q^{79} -3.87896 q^{80} +1.10583 q^{82} +12.6407 q^{83} +4.71226 q^{85} -1.52678 q^{86} -0.601179 q^{88} -17.7527 q^{89} +10.4915 q^{91} +11.5016 q^{92} +1.43586 q^{94} -3.03705 q^{95} +19.4157 q^{97} +0.651085 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8} + 3 q^{10} - 4 q^{11} + 13 q^{13} - 4 q^{14} + 13 q^{16} + 4 q^{17} + 14 q^{19} - 15 q^{20} + 15 q^{22} - 11 q^{23} + 12 q^{25} + 8 q^{26} + 16 q^{28} + 7 q^{29} + 4 q^{31} - 4 q^{32} + q^{34} - 7 q^{35} + 24 q^{37} + 11 q^{38} + 3 q^{40} - 13 q^{41} + 25 q^{43} - 10 q^{44} - 22 q^{46} - 19 q^{47} + 9 q^{49} - 3 q^{50} + 20 q^{52} - 11 q^{53} + 4 q^{55} + 37 q^{56} - 2 q^{58} - 8 q^{59} + 7 q^{61} + 11 q^{62} - 19 q^{64} - 13 q^{65} + 33 q^{67} + 24 q^{68} + 4 q^{70} + 34 q^{73} + 27 q^{74} - 9 q^{76} + 29 q^{77} - 13 q^{80} + q^{82} + 24 q^{83} - 4 q^{85} + 36 q^{86} - 6 q^{88} + 10 q^{89} + 30 q^{91} + 28 q^{92} - 8 q^{94} - 14 q^{95} + 16 q^{97} + 36 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8 + 3 * q^10 - 4 * q^11 + 13 * q^13 - 4 * q^14 + 13 * q^16 + 4 * q^17 + 14 * q^19 - 15 * q^20 + 15 * q^22 - 11 * q^23 + 12 * q^25 + 8 * q^26 + 16 * q^28 + 7 * q^29 + 4 * q^31 - 4 * q^32 + q^34 - 7 * q^35 + 24 * q^37 + 11 * q^38 + 3 * q^40 - 13 * q^41 + 25 * q^43 - 10 * q^44 - 22 * q^46 - 19 * q^47 + 9 * q^49 - 3 * q^50 + 20 * q^52 - 11 * q^53 + 4 * q^55 + 37 * q^56 - 2 * q^58 - 8 * q^59 + 7 * q^61 + 11 * q^62 - 19 * q^64 - 13 * q^65 + 33 * q^67 + 24 * q^68 + 4 * q^70 + 34 * q^73 + 27 * q^74 - 9 * q^76 + 29 * q^77 - 13 * q^80 + q^82 + 24 * q^83 - 4 * q^85 + 36 * q^86 - 6 * q^88 + 10 * q^89 + 30 * q^91 + 28 * q^92 - 8 * q^94 - 14 * q^95 + 16 * q^97 + 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more