LMFDB - Embedded newform 4815.2.a.u.1.9

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			1.9
Root			$-1.37578$ 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)

$f(q)$	$=$	$q+1.37578 q^{2} -0.107224 q^{4} -1.00000 q^{5} -1.29286 q^{7} -2.89908 q^{8} +O(q^{10})$	\(q+1.37578 q^{2} -0.107224 q^{4} -1.00000 q^{5} -1.29286 q^{7} -2.89908 q^{8} -1.37578 q^{10} +5.18674 q^{11} -1.56082 q^{13} -1.77869 q^{14} -3.77405 q^{16} +5.67618 q^{17} +0.785965 q^{19} +0.107224 q^{20} +7.13582 q^{22} -9.21903 q^{23} +1.00000 q^{25} -2.14735 q^{26} +0.138626 q^{28} -9.36714 q^{29} +9.91330 q^{31} +0.605886 q^{32} +7.80919 q^{34} +1.29286 q^{35} +4.37397 q^{37} +1.08132 q^{38} +2.89908 q^{40} +0.756558 q^{41} -3.28845 q^{43} -0.556144 q^{44} -12.6834 q^{46} +4.30876 q^{47} -5.32852 q^{49} +1.37578 q^{50} +0.167358 q^{52} -6.56151 q^{53} -5.18674 q^{55} +3.74810 q^{56} -12.8871 q^{58} -3.51171 q^{59} -0.507656 q^{61} +13.6385 q^{62} +8.38168 q^{64} +1.56082 q^{65} +11.1427 q^{67} -0.608624 q^{68} +1.77869 q^{70} +4.08780 q^{71} +14.7644 q^{73} +6.01762 q^{74} -0.0842745 q^{76} -6.70572 q^{77} +7.34598 q^{79} +3.77405 q^{80} +1.04086 q^{82} +12.1115 q^{83} -5.67618 q^{85} -4.52419 q^{86} -15.0368 q^{88} +11.1117 q^{89} +2.01792 q^{91} +0.988504 q^{92} +5.92792 q^{94} -0.785965 q^{95} +11.6889 q^{97} -7.33088 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.37578	0.972825	0.486412	−	0.873729i	$-0.338305\pi$
$2$	1.37578	0.972825	0.486412	+	0.873729i	$0.338305\pi$
$3$	0	0
$3$	0	0
$4$	−0.107224	−0.0536121
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.29286	−0.488655	−0.244327	−	0.969693i	$-0.578567\pi$
$7$	−1.29286	−0.488655	−0.244327	+	0.969693i	$0.578567\pi$
$8$	−2.89908	−1.02498
$9$	0	0
$10$	−1.37578	−0.435060
$11$	5.18674	1.56386	0.781931	−	0.623365i	$-0.214235\pi$
$11$	5.18674	1.56386	0.781931	+	0.623365i	$0.214235\pi$
$12$	0	0
$13$	−1.56082	−0.432894	−0.216447	−	0.976294i	$-0.569447\pi$
$13$	−1.56082	−0.432894	−0.216447	+	0.976294i	$0.569447\pi$
$14$	−1.77869	−0.475375
$15$	0	0
$16$	−3.77405	−0.943514
$17$	5.67618	1.37668	0.688338	−	0.725390i	$-0.258341\pi$
$17$	5.67618	1.37668	0.688338	+	0.725390i	$0.258341\pi$
$18$	0	0
$19$	0.785965	0.180313	0.0901564	−	0.995928i	$-0.471263\pi$
$19$	0.785965	0.180313	0.0901564	+	0.995928i	$0.471263\pi$
$20$	0.107224	0.0239761
$21$	0	0
$22$	7.13582	1.52136
$23$	−9.21903	−1.92230	−0.961150	−	0.276025i	$-0.910983\pi$
$23$	−9.21903	−1.92230	−0.961150	+	0.276025i	$0.910983\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.14735	−0.421130
$27$	0	0
$28$	0.138626	0.0261978
$29$	−9.36714	−1.73944	−0.869718	−	0.493550i	$-0.835699\pi$
$29$	−9.36714	−1.73944	−0.869718	+	0.493550i	$0.835699\pi$
$30$	0	0
$31$	9.91330	1.78048	0.890241	−	0.455490i	$-0.150536\pi$
$31$	9.91330	1.78048	0.890241	+	0.455490i	$0.150536\pi$
$32$	0.605886	0.107107
$33$	0	0
$34$	7.80919	1.33926
$35$	1.29286	0.218533
$36$	0	0
$37$	4.37397	0.719075	0.359538	−	0.933131i	$-0.382934\pi$
$37$	4.37397	0.719075	0.359538	+	0.933131i	$0.382934\pi$
$38$	1.08132	0.175413
$39$	0	0
$40$	2.89908	0.458385
$41$	0.756558	0.118154	0.0590772	−	0.998253i	$-0.481184\pi$
$41$	0.756558	0.118154	0.0590772	+	0.998253i	$0.481184\pi$
$42$	0	0
$43$	−3.28845	−0.501484	−0.250742	−	0.968054i	$-0.580675\pi$
$43$	−3.28845	−0.501484	−0.250742	+	0.968054i	$0.580675\pi$
$44$	−0.556144	−0.0838419
$45$	0	0
$46$	−12.6834	−1.87006
$47$	4.30876	0.628498	0.314249	−	0.949341i	$-0.398247\pi$
$47$	4.30876	0.628498	0.314249	+	0.949341i	$0.398247\pi$
$48$	0	0
$49$	−5.32852	−0.761217
$50$	1.37578	0.194565
$51$	0	0
$52$	0.167358	0.0232084
$53$	−6.56151	−0.901293	−0.450646	−	0.892703i	$-0.648806\pi$
$53$	−6.56151	−0.901293	−0.450646	+	0.892703i	$0.648806\pi$
$54$	0	0
$55$	−5.18674	−0.699380
$56$	3.74810	0.500861
$57$	0	0
$58$	−12.8871	−1.69217
$59$	−3.51171	−0.457186	−0.228593	−	0.973522i	$-0.573413\pi$
$59$	−3.51171	−0.457186	−0.228593	+	0.973522i	$0.573413\pi$
$60$	0	0
$61$	−0.507656	−0.0649987	−0.0324994	−	0.999472i	$-0.510347\pi$
$61$	−0.507656	−0.0649987	−0.0324994	+	0.999472i	$0.510347\pi$
$62$	13.6385	1.73210
$63$	0	0
$64$	8.38168	1.04771
$65$	1.56082	0.193596
$66$	0	0
$67$	11.1427	1.36130	0.680651	−	0.732607i	$-0.261697\pi$
$67$	11.1427	1.36130	0.680651	+	0.732607i	$0.261697\pi$
$68$	−0.608624	−0.0738065
$69$	0	0
$70$	1.77869	0.212594
$71$	4.08780	0.485133	0.242567	−	0.970135i	$-0.422011\pi$
$71$	4.08780	0.485133	0.242567	+	0.970135i	$0.422011\pi$
$72$	0	0
$73$	14.7644	1.72805	0.864024	−	0.503451i	$-0.167936\pi$
$73$	14.7644	1.72805	0.864024	+	0.503451i	$0.167936\pi$
$74$	6.01762	0.699534
$75$	0	0
$76$	−0.0842745	−0.00966695
$77$	−6.70572	−0.764188
$78$	0	0
$79$	7.34598	0.826488	0.413244	−	0.910620i	$-0.364396\pi$
$79$	7.34598	0.826488	0.413244	+	0.910620i	$0.364396\pi$
$80$	3.77405	0.421952
$81$	0	0
$82$	1.04086	0.114944
$83$	12.1115	1.32941	0.664703	−	0.747108i	$-0.268558\pi$
$83$	12.1115	1.32941	0.664703	+	0.747108i	$0.268558\pi$
$84$	0	0
$85$	−5.67618	−0.615668
$86$	−4.52419	−0.487856
$87$	0	0
$88$	−15.0368	−1.60293
$89$	11.1117	1.17783	0.588917	−	0.808194i	$-0.299555\pi$
$89$	11.1117	1.17783	0.588917	+	0.808194i	$0.299555\pi$
$90$	0	0
$91$	2.01792	0.211536
$92$	0.988504	0.103059
$93$	0	0
$94$	5.92792	0.611418
$95$	−0.785965	−0.0806383
$96$	0	0
$97$	11.6889	1.18683	0.593416	−	0.804896i	$-0.297779\pi$
$97$	11.6889	1.18683	0.593416	+	0.804896i	$0.297779\pi$
$98$	−7.33088	−0.740530
$99$	0	0
$100$	−0.107224	−0.0107224
$101$	2.11771	0.210720	0.105360	−	0.994434i	$-0.466400\pi$
$101$	2.11771	0.210720	0.105360	+	0.994434i	$0.466400\pi$
$102$	0	0
$103$	19.3063	1.90231	0.951154	−	0.308717i	$-0.0998997\pi$
$103$	19.3063	1.90231	0.951154	+	0.308717i	$0.0998997\pi$
$104$	4.52495	0.443708
$105$	0	0
$106$	−9.02721	−0.876800
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	0	0
$109$	3.75798	0.359950	0.179975	−	0.983671i	$-0.442398\pi$
$109$	3.75798	0.359950	0.179975	+	0.983671i	$0.442398\pi$
$110$	−7.13582	−0.680374
$111$	0	0
$112$	4.87932	0.461052
$113$	13.7716	1.29553	0.647763	−	0.761842i	$-0.275705\pi$
$113$	13.7716	1.29553	0.647763	+	0.761842i	$0.275705\pi$
$114$	0	0
$115$	9.21903	0.859679
$116$	1.00438	0.0932548
$117$	0	0
$118$	−4.83135	−0.444762
$119$	−7.33850	−0.672719
$120$	0	0
$121$	15.9023	1.44566
$122$	−0.698424	−0.0632324
$123$	0	0
$124$	−1.06295	−0.0954554
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.71645	0.862196	0.431098	−	0.902305i	$-0.358126\pi$
$127$	9.71645	0.862196	0.431098	+	0.902305i	$0.358126\pi$
$128$	10.3196	0.912131
$129$	0	0
$130$	2.14735	0.188335
$131$	−7.43178	−0.649317	−0.324659	−	0.945831i	$-0.605249\pi$
$131$	−7.43178	−0.649317	−0.324659	+	0.945831i	$0.605249\pi$
$132$	0	0
$133$	−1.01614	−0.0881107
$134$	15.3300	1.32431
$135$	0	0
$136$	−16.4557	−1.41107
$137$	−13.9411	−1.19107	−0.595533	−	0.803330i	$-0.703059\pi$
$137$	−13.9411	−1.19107	−0.595533	+	0.803330i	$0.703059\pi$
$138$	0	0
$139$	−17.8433	−1.51345	−0.756725	−	0.653734i	$-0.773202\pi$
$139$	−17.8433	−1.51345	−0.756725	+	0.653734i	$0.773202\pi$
$140$	−0.138626	−0.0117160
$141$	0	0
$142$	5.62393	0.471949
$143$	−8.09558	−0.676986
$144$	0	0
$145$	9.36714	0.777899
$146$	20.3127	1.68109
$147$	0	0
$148$	−0.468995	−0.0385512
$149$	−14.3146	−1.17270	−0.586348	−	0.810059i	$-0.699435\pi$
$149$	−14.3146	−1.17270	−0.586348	+	0.810059i	$0.699435\pi$
$150$	0	0
$151$	−0.452621	−0.0368338	−0.0184169	−	0.999830i	$-0.505863\pi$
$151$	−0.452621	−0.0368338	−0.0184169	+	0.999830i	$0.505863\pi$
$152$	−2.27858	−0.184817
$153$	0	0
$154$	−9.22561	−0.743421
$155$	−9.91330	−0.796256
$156$	0	0
$157$	17.6813	1.41112	0.705562	−	0.708648i	$-0.250695\pi$
$157$	17.6813	1.41112	0.705562	+	0.708648i	$0.250695\pi$
$158$	10.1065	0.804028
$159$	0	0
$160$	−0.605886	−0.0478995
$161$	11.9189	0.939341
$162$	0	0
$163$	12.2108	0.956424	0.478212	−	0.878244i	$-0.341285\pi$
$163$	12.2108	0.956424	0.478212	+	0.878244i	$0.341285\pi$
$164$	−0.0811213	−0.00633451
$165$	0	0
$166$	16.6627	1.29328
$167$	0.255788	0.0197935	0.00989674	−	0.999951i	$-0.496850\pi$
$167$	0.255788	0.0197935	0.00989674	+	0.999951i	$0.496850\pi$
$168$	0	0
$169$	−10.5638	−0.812603
$170$	−7.80919	−0.598937
$171$	0	0
$172$	0.352601	0.0268856
$173$	1.67384	0.127259	0.0636297	−	0.997974i	$-0.479732\pi$
$173$	1.67384	0.127259	0.0636297	+	0.997974i	$0.479732\pi$
$174$	0	0
$175$	−1.29286	−0.0977309
$176$	−19.5750	−1.47552
$177$	0	0
$178$	15.2872	1.14583
$179$	14.4146	1.07740	0.538698	−	0.842499i	$-0.318916\pi$
$179$	14.4146	1.07740	0.538698	+	0.842499i	$0.318916\pi$
$180$	0	0
$181$	−7.82843	−0.581883	−0.290941	−	0.956741i	$-0.593968\pi$
$181$	−7.82843	−0.581883	−0.290941	+	0.956741i	$0.593968\pi$
$182$	2.77622	0.205787
$183$	0	0
$184$	26.7267	1.97032
$185$	−4.37397	−0.321580
$186$	0	0
$187$	29.4409	2.15293
$188$	−0.462004	−0.0336951
$189$	0	0
$190$	−1.08132	−0.0784470
$191$	12.7680	0.923859	0.461929	−	0.886917i	$-0.347157\pi$
$191$	12.7680	0.923859	0.461929	+	0.886917i	$0.347157\pi$
$192$	0	0
$193$	−7.36412	−0.530081	−0.265040	−	0.964237i	$-0.585385\pi$
$193$	−7.36412	−0.530081	−0.265040	+	0.964237i	$0.585385\pi$
$194$	16.0814	1.15458
$195$	0	0
$196$	0.571346	0.0408104
$197$	10.0463	0.715768	0.357884	−	0.933766i	$-0.383498\pi$
$197$	10.0463	0.715768	0.357884	+	0.933766i	$0.383498\pi$
$198$	0	0
$199$	15.0806	1.06904	0.534518	−	0.845157i	$-0.320493\pi$
$199$	15.0806	1.06904	0.534518	+	0.845157i	$0.320493\pi$
$200$	−2.89908	−0.204996
$201$	0	0
$202$	2.91351	0.204994
$203$	12.1104	0.849983
$204$	0	0
$205$	−0.756558	−0.0528403
$206$	26.5613	1.85061
$207$	0	0
$208$	5.89063	0.408441
$209$	4.07660	0.281984
$210$	0	0
$211$	−9.03224	−0.621805	−0.310902	−	0.950442i	$-0.600631\pi$
$211$	−9.03224	−0.621805	−0.310902	+	0.950442i	$0.600631\pi$
$212$	0.703553	0.0483202
$213$	0	0
$214$	−1.37578	−0.0940465
$215$	3.28845	0.224270
$216$	0	0
$217$	−12.8165	−0.870041
$218$	5.17017	0.350168
$219$	0	0
$220$	0.556144	0.0374952
$221$	−8.85951	−0.595955
$222$	0	0
$223$	21.9892	1.47250	0.736252	−	0.676708i	$-0.236594\pi$
$223$	21.9892	1.47250	0.736252	+	0.676708i	$0.236594\pi$
$224$	−0.783325	−0.0523381
$225$	0	0
$226$	18.9468	1.26032
$227$	−25.9768	−1.72414	−0.862070	−	0.506790i	$-0.830832\pi$
$227$	−25.9768	−1.72414	−0.862070	+	0.506790i	$0.830832\pi$
$228$	0	0
$229$	8.53416	0.563953	0.281976	−	0.959421i	$-0.409010\pi$
$229$	8.53416	0.563953	0.281976	+	0.959421i	$0.409010\pi$
$230$	12.6834	0.836317
$231$	0	0
$232$	27.1561	1.78289
$233$	27.1213	1.77678	0.888389	−	0.459091i	$-0.151825\pi$
$233$	27.1213	1.77678	0.888389	+	0.459091i	$0.151825\pi$
$234$	0	0
$235$	−4.30876	−0.281073
$236$	0.376541	0.0245107
$237$	0	0
$238$	−10.0962	−0.654438
$239$	−25.6285	−1.65777	−0.828886	−	0.559417i	$-0.811025\pi$
$239$	−25.6285	−1.65777	−0.828886	+	0.559417i	$0.811025\pi$
$240$	0	0
$241$	−20.6681	−1.33135	−0.665676	−	0.746241i	$-0.731857\pi$
$241$	−20.6681	−1.33135	−0.665676	+	0.746241i	$0.731857\pi$
$242$	21.8781	1.40638
$243$	0	0
$244$	0.0544331	0.00348472
$245$	5.32852	0.340426
$246$	0	0
$247$	−1.22675	−0.0780563
$248$	−28.7395	−1.82496
$249$	0	0
$250$	−1.37578	−0.0870121
$251$	−15.3902	−0.971419	−0.485710	−	0.874120i	$-0.661439\pi$
$251$	−15.3902	−0.971419	−0.485710	+	0.874120i	$0.661439\pi$
$252$	0	0
$253$	−47.8167	−3.00621
$254$	13.3677	0.838765
$255$	0	0
$256$	−2.56585	−0.160366
$257$	19.0856	1.19053	0.595264	−	0.803530i	$-0.297047\pi$
$257$	19.0856	1.19053	0.595264	+	0.803530i	$0.297047\pi$
$258$	0	0
$259$	−5.65492	−0.351380
$260$	−0.167358	−0.0103791
$261$	0	0
$262$	−10.2245	−0.631672
$263$	−1.71026	−0.105459	−0.0527296	−	0.998609i	$-0.516792\pi$
$263$	−1.71026	−0.105459	−0.0527296	+	0.998609i	$0.516792\pi$
$264$	0	0
$265$	6.56151	0.403070
$266$	−1.39799	−0.0857163
$267$	0	0
$268$	−1.19477	−0.0729823
$269$	8.42120	0.513450	0.256725	−	0.966485i	$-0.417357\pi$
$269$	8.42120	0.513450	0.256725	+	0.966485i	$0.417357\pi$
$270$	0	0
$271$	−20.5279	−1.24698	−0.623491	−	0.781830i	$-0.714286\pi$
$271$	−20.5279	−1.24698	−0.623491	+	0.781830i	$0.714286\pi$
$272$	−21.4222	−1.29891
$273$	0	0
$274$	−19.1799	−1.15870
$275$	5.18674	0.312772
$276$	0	0
$277$	−12.4087	−0.745568	−0.372784	−	0.927918i	$-0.621597\pi$
$277$	−12.4087	−0.745568	−0.372784	+	0.927918i	$0.621597\pi$
$278$	−24.5485	−1.47232
$279$	0	0
$280$	−3.74810	−0.223992
$281$	−1.56318	−0.0932516	−0.0466258	−	0.998912i	$-0.514847\pi$
$281$	−1.56318	−0.0932516	−0.0466258	+	0.998912i	$0.514847\pi$
$282$	0	0
$283$	8.97776	0.533672	0.266836	−	0.963742i	$-0.414022\pi$
$283$	8.97776	0.533672	0.266836	+	0.963742i	$0.414022\pi$
$284$	−0.438312	−0.0260090
$285$	0	0
$286$	−11.1378	−0.658589
$287$	−0.978122	−0.0577367
$288$	0	0
$289$	15.2190	0.895237
$290$	12.8871	0.756759
$291$	0	0
$292$	−1.58311	−0.0926443
$293$	3.15706	0.184437	0.0922187	−	0.995739i	$-0.470604\pi$
$293$	3.15706	0.184437	0.0922187	+	0.995739i	$0.470604\pi$
$294$	0	0
$295$	3.51171	0.204460
$296$	−12.6805	−0.737038
$297$	0	0
$298$	−19.6937	−1.14083
$299$	14.3893	0.832153
$300$	0	0
$301$	4.25150	0.245052
$302$	−0.622708	−0.0358328
$303$	0	0
$304$	−2.96628	−0.170128
$305$	0.507656	0.0290683
$306$	0	0
$307$	14.9338	0.852318	0.426159	−	0.904648i	$-0.359866\pi$
$307$	14.9338	0.852318	0.426159	+	0.904648i	$0.359866\pi$
$308$	0.719016	0.0409697
$309$	0	0
$310$	−13.6385	−0.774617
$311$	−13.1905	−0.747964	−0.373982	−	0.927436i	$-0.622008\pi$
$311$	−13.1905	−0.747964	−0.373982	+	0.927436i	$0.622008\pi$
$312$	0	0
$313$	34.8029	1.96718	0.983588	−	0.180429i	$-0.0577486\pi$
$313$	34.8029	1.96718	0.983588	+	0.180429i	$0.0577486\pi$
$314$	24.3257	1.37278
$315$	0	0
$316$	−0.787668	−0.0443098
$317$	−24.5194	−1.37715	−0.688574	−	0.725166i	$-0.741763\pi$
$317$	−24.5194	−1.37715	−0.688574	+	0.725166i	$0.741763\pi$
$318$	0	0
$319$	−48.5850	−2.72024
$320$	−8.38168	−0.468550
$321$	0	0
$322$	16.3978	0.913814
$323$	4.46128	0.248232
$324$	0	0
$325$	−1.56082	−0.0865788
$326$	16.7994	0.930433
$327$	0	0
$328$	−2.19332	−0.121106
$329$	−5.57062	−0.307118
$330$	0	0
$331$	−2.92043	−0.160521	−0.0802607	−	0.996774i	$-0.525575\pi$
$331$	−2.92043	−0.160521	−0.0802607	+	0.996774i	$0.525575\pi$
$332$	−1.29864	−0.0712723
$333$	0	0
$334$	0.351909	0.0192556
$335$	−11.1427	−0.608793
$336$	0	0
$337$	−20.2923	−1.10539	−0.552697	−	0.833383i	$-0.686401\pi$
$337$	−20.2923	−1.10539	−0.552697	+	0.833383i	$0.686401\pi$
$338$	−14.5335	−0.790520
$339$	0	0
$340$	0.608624	0.0330073
$341$	51.4177	2.78443
$342$	0	0
$343$	15.9390	0.860627
$344$	9.53348	0.514011
$345$	0	0
$346$	2.30283	0.123801
$347$	−16.9121	−0.907892	−0.453946	−	0.891029i	$-0.649984\pi$
$347$	−16.9121	−0.907892	−0.453946	+	0.891029i	$0.649984\pi$
$348$	0	0
$349$	11.0357	0.590729	0.295365	−	0.955385i	$-0.404559\pi$
$349$	11.0357	0.590729	0.295365	+	0.955385i	$0.404559\pi$
$350$	−1.77869	−0.0950751
$351$	0	0
$352$	3.14257	0.167500
$353$	16.3137	0.868291	0.434146	−	0.900843i	$-0.357050\pi$
$353$	16.3137	0.868291	0.434146	+	0.900843i	$0.357050\pi$
$354$	0	0
$355$	−4.08780	−0.216958
$356$	−1.19144	−0.0631461
$357$	0	0
$358$	19.8313	1.04812
$359$	15.4892	0.817488	0.408744	−	0.912649i	$-0.365967\pi$
$359$	15.4892	0.817488	0.408744	+	0.912649i	$0.365967\pi$
$360$	0	0
$361$	−18.3823	−0.967487
$362$	−10.7702	−0.566070
$363$	0	0
$364$	−0.216370	−0.0113409
$365$	−14.7644	−0.772807
$366$	0	0
$367$	5.49447	0.286809	0.143404	−	0.989664i	$-0.454195\pi$
$367$	5.49447	0.286809	0.143404	+	0.989664i	$0.454195\pi$
$368$	34.7931	1.81372
$369$	0	0
$370$	−6.01762	−0.312841
$371$	8.48311	0.440421
$372$	0	0
$373$	−6.38304	−0.330501	−0.165251	−	0.986252i	$-0.552843\pi$
$373$	−6.38304	−0.330501	−0.165251	+	0.986252i	$0.552843\pi$
$374$	40.5042	2.09442
$375$	0	0
$376$	−12.4915	−0.644198
$377$	14.6204	0.752991
$378$	0	0
$379$	−25.3489	−1.30209	−0.651043	−	0.759041i	$-0.725668\pi$
$379$	−25.3489	−1.30209	−0.651043	+	0.759041i	$0.725668\pi$
$380$	0.0842745	0.00432319
$381$	0	0
$382$	17.5660	0.898753
$383$	−27.6697	−1.41386	−0.706929	−	0.707284i	$-0.749920\pi$
$383$	−27.6697	−1.41386	−0.706929	+	0.707284i	$0.749920\pi$
$384$	0	0
$385$	6.70572	0.341755
$386$	−10.1314	−0.515676
$387$	0	0
$388$	−1.25334	−0.0636285
$389$	−27.3305	−1.38571	−0.692855	−	0.721077i	$-0.743647\pi$
$389$	−27.3305	−1.38571	−0.692855	+	0.721077i	$0.743647\pi$
$390$	0	0
$391$	−52.3289	−2.64639
$392$	15.4478	0.780232
$393$	0	0
$394$	13.8215	0.696317
$395$	−7.34598	−0.369617
$396$	0	0
$397$	−1.93421	−0.0970754	−0.0485377	−	0.998821i	$-0.515456\pi$
$397$	−1.93421	−0.0970754	−0.0485377	+	0.998821i	$0.515456\pi$
$398$	20.7476	1.03998
$399$	0	0
$400$	−3.77405	−0.188703
$401$	22.4305	1.12013	0.560063	−	0.828450i	$-0.310777\pi$
$401$	22.4305	1.12013	0.560063	+	0.828450i	$0.310777\pi$
$402$	0	0
$403$	−15.4729	−0.770760
$404$	−0.227070	−0.0112972
$405$	0	0
$406$	16.6613	0.826885
$407$	22.6866	1.12453
$408$	0	0
$409$	−9.89267	−0.489161	−0.244581	−	0.969629i	$-0.578650\pi$
$409$	−9.89267	−0.489161	−0.244581	+	0.969629i	$0.578650\pi$
$410$	−1.04086	−0.0514043
$411$	0	0
$412$	−2.07010	−0.101987
$413$	4.54015	0.223406
$414$	0	0
$415$	−12.1115	−0.594529
$416$	−0.945680	−0.0463658
$417$	0	0
$418$	5.60851	0.274321
$419$	−2.10990	−0.103075	−0.0515376	−	0.998671i	$-0.516412\pi$
$419$	−2.10990	−0.103075	−0.0515376	+	0.998671i	$0.516412\pi$
$420$	0	0
$421$	−21.1115	−1.02891	−0.514456	−	0.857517i	$-0.672006\pi$
$421$	−21.1115	−1.02891	−0.514456	+	0.857517i	$0.672006\pi$
$422$	−12.4264	−0.604907
$423$	0	0
$424$	19.0223	0.923807
$425$	5.67618	0.275335
$426$	0	0
$427$	0.656328	0.0317619
$428$	0.107224	0.00518288
$429$	0	0
$430$	4.52419	0.218176
$431$	5.81994	0.280337	0.140168	−	0.990128i	$-0.455236\pi$
$431$	5.81994	0.280337	0.140168	+	0.990128i	$0.455236\pi$
$432$	0	0
$433$	−12.5241	−0.601868	−0.300934	−	0.953645i	$-0.597298\pi$
$433$	−12.5241	−0.601868	−0.300934	+	0.953645i	$0.597298\pi$
$434$	−17.6327	−0.846397
$435$	0	0
$436$	−0.402947	−0.0192977
$437$	−7.24584	−0.346615
$438$	0	0
$439$	−23.8382	−1.13774	−0.568868	−	0.822429i	$-0.692619\pi$
$439$	−23.8382	−1.13774	−0.568868	+	0.822429i	$0.692619\pi$
$440$	15.0368	0.716851
$441$	0	0
$442$	−12.1887	−0.579760
$443$	−8.63521	−0.410271	−0.205136	−	0.978734i	$-0.565764\pi$
$443$	−8.63521	−0.410271	−0.205136	+	0.978734i	$0.565764\pi$
$444$	0	0
$445$	−11.1117	−0.526743
$446$	30.2523	1.43249
$447$	0	0
$448$	−10.8363	−0.511968
$449$	25.7021	1.21296	0.606479	−	0.795099i	$-0.292581\pi$
$449$	25.7021	1.21296	0.606479	+	0.795099i	$0.292581\pi$
$450$	0	0
$451$	3.92407	0.184777
$452$	−1.47665	−0.0694559
$453$	0	0
$454$	−35.7384	−1.67729
$455$	−2.01792	−0.0946017
$456$	0	0
$457$	−24.2914	−1.13630	−0.568152	−	0.822924i	$-0.692341\pi$
$457$	−24.2914	−1.13630	−0.568152	+	0.822924i	$0.692341\pi$
$458$	11.7411	0.548627
$459$	0	0
$460$	−0.988504	−0.0460892
$461$	−14.8142	−0.689965	−0.344982	−	0.938609i	$-0.612115\pi$
$461$	−14.8142	−0.689965	−0.344982	+	0.938609i	$0.612115\pi$
$462$	0	0
$463$	−23.0000	−1.06890	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000	−1.06890	−0.534450	+	0.845200i	$0.679481\pi$
$464$	35.3521	1.64118
$465$	0	0
$466$	37.3130	1.72849
$467$	34.0153	1.57404	0.787020	−	0.616928i	$-0.211623\pi$
$467$	34.0153	1.57404	0.787020	+	0.616928i	$0.211623\pi$
$468$	0	0
$469$	−14.4060	−0.665207
$470$	−5.92792	−0.273435
$471$	0	0
$472$	10.1807	0.468607
$473$	−17.0563	−0.784251
$474$	0	0
$475$	0.785965	0.0360626
$476$	0.786865	0.0360659
$477$	0	0
$478$	−35.2593	−1.61272
$479$	−0.220254	−0.0100637	−0.00503184	−	0.999987i	$-0.501602\pi$
$479$	−0.220254	−0.0100637	−0.00503184	+	0.999987i	$0.501602\pi$
$480$	0	0
$481$	−6.82698	−0.311284
$482$	−28.4348	−1.29517
$483$	0	0
$484$	−1.70511	−0.0775050
$485$	−11.6889	−0.530767
$486$	0	0
$487$	13.3357	0.604298	0.302149	−	0.953261i	$-0.402296\pi$
$487$	13.3357	0.604298	0.302149	+	0.953261i	$0.402296\pi$
$488$	1.47174	0.0666224
$489$	0	0
$490$	7.33088	0.331175
$491$	40.1922	1.81385	0.906924	−	0.421295i	$-0.138424\pi$
$491$	40.1922	1.81385	0.906924	+	0.421295i	$0.138424\pi$
$492$	0	0
$493$	−53.1696	−2.39464
$494$	−1.68774	−0.0759351
$495$	0	0
$496$	−37.4133	−1.67991
$497$	−5.28495	−0.237063
$498$	0	0
$499$	9.90868	0.443573	0.221787	−	0.975095i	$-0.428811\pi$
$499$	9.90868	0.443573	0.221787	+	0.975095i	$0.428811\pi$
$500$	0.107224	0.00479521
$501$	0	0
$502$	−21.1735	−0.945020
$503$	−5.90437	−0.263263	−0.131631	−	0.991299i	$-0.542022\pi$
$503$	−5.90437	−0.263263	−0.131631	+	0.991299i	$0.542022\pi$
$504$	0	0
$505$	−2.11771	−0.0942370
$506$	−65.7854	−2.92452
$507$	0	0
$508$	−1.04184	−0.0462241
$509$	−8.73646	−0.387237	−0.193618	−	0.981077i	$-0.562022\pi$
$509$	−8.73646	−0.387237	−0.193618	+	0.981077i	$0.562022\pi$
$510$	0	0
$511$	−19.0883	−0.844419
$512$	−24.1692	−1.06814
$513$	0	0
$514$	26.2577	1.15818
$515$	−19.3063	−0.850738
$516$	0	0
$517$	22.3485	0.982884
$518$	−7.77994	−0.341831
$519$	0	0
$520$	−4.52495	−0.198432
$521$	−0.189571	−0.00830524	−0.00415262	−	0.999991i	$-0.501322\pi$
$521$	−0.189571	−0.00830524	−0.00415262	+	0.999991i	$0.501322\pi$
$522$	0	0
$523$	30.8578	1.34932	0.674659	−	0.738129i	$-0.264291\pi$
$523$	30.8578	1.34932	0.674659	+	0.738129i	$0.264291\pi$
$524$	0.796867	0.0348113
$525$	0	0
$526$	−2.35295	−0.102593
$527$	56.2697	2.45115
$528$	0	0
$529$	61.9905	2.69524
$530$	9.02721	0.392117
$531$	0	0
$532$	0.108955	0.00472380
$533$	−1.18085	−0.0511484
$534$	0	0
$535$	1.00000	0.0432338
$536$	−32.3037	−1.39531
$537$	0	0
$538$	11.5857	0.499497
$539$	−27.6376	−1.19044
$540$	0	0
$541$	32.4654	1.39580	0.697899	−	0.716197i	$-0.254119\pi$
$541$	32.4654	1.39580	0.697899	+	0.716197i	$0.254119\pi$
$542$	−28.2420	−1.21310
$543$	0	0
$544$	3.43912	0.147451
$545$	−3.75798	−0.160974
$546$	0	0
$547$	26.7396	1.14330	0.571651	−	0.820497i	$-0.306303\pi$
$547$	26.7396	1.14330	0.571651	+	0.820497i	$0.306303\pi$
$548$	1.49482	0.0638556
$549$	0	0
$550$	7.13582	0.304273
$551$	−7.36225	−0.313642
$552$	0	0
$553$	−9.49732	−0.403867
$554$	−17.0717	−0.725307
$555$	0	0
$556$	1.91324	0.0811392
$557$	6.95257	0.294590	0.147295	−	0.989093i	$-0.452943\pi$
$557$	6.95257	0.294590	0.147295	+	0.989093i	$0.452943\pi$
$558$	0	0
$559$	5.13268	0.217089
$560$	−4.87932	−0.206189
$561$	0	0
$562$	−2.15060	−0.0907175
$563$	−32.8101	−1.38278	−0.691390	−	0.722482i	$-0.743001\pi$
$563$	−32.8101	−1.38278	−0.691390	+	0.722482i	$0.743001\pi$
$564$	0	0
$565$	−13.7716	−0.579377
$566$	12.3514	0.519170
$567$	0	0
$568$	−11.8509	−0.497252
$569$	−32.7831	−1.37434	−0.687168	−	0.726498i	$-0.741147\pi$
$569$	−32.7831	−1.37434	−0.687168	+	0.726498i	$0.741147\pi$
$570$	0	0
$571$	9.61675	0.402448	0.201224	−	0.979545i	$-0.435508\pi$
$571$	9.61675	0.402448	0.201224	+	0.979545i	$0.435508\pi$
$572$	0.868042	0.0362947
$573$	0	0
$574$	−1.34568	−0.0561677
$575$	−9.21903	−0.384460
$576$	0	0
$577$	3.14542	0.130945	0.0654727	−	0.997854i	$-0.479144\pi$
$577$	3.14542	0.130945	0.0654727	+	0.997854i	$0.479144\pi$
$578$	20.9381	0.870909
$579$	0	0
$580$	−1.00438	−0.0417048
$581$	−15.6584	−0.649621
$582$	0	0
$583$	−34.0329	−1.40950
$584$	−42.8033	−1.77121
$585$	0	0
$586$	4.34343	0.179425
$587$	8.31983	0.343396	0.171698	−	0.985150i	$-0.445075\pi$
$587$	8.31983	0.343396	0.171698	+	0.985150i	$0.445075\pi$
$588$	0	0
$589$	7.79151	0.321044
$590$	4.83135	0.198904
$591$	0	0
$592$	−16.5076	−0.678457
$593$	−23.2391	−0.954315	−0.477157	−	0.878818i	$-0.658333\pi$
$593$	−23.2391	−0.954315	−0.477157	+	0.878818i	$0.658333\pi$
$594$	0	0
$595$	7.33850	0.300849
$596$	1.53487	0.0628707
$597$	0	0
$598$	19.7965	0.809539
$599$	36.4894	1.49092	0.745459	−	0.666551i	$-0.232230\pi$
$599$	36.4894	1.49092	0.745459	+	0.666551i	$0.232230\pi$
$600$	0	0
$601$	3.67854	0.150051	0.0750254	−	0.997182i	$-0.476096\pi$
$601$	3.67854	0.150051	0.0750254	+	0.997182i	$0.476096\pi$
$602$	5.84914	0.238393
$603$	0	0
$604$	0.0485320	0.00197474
$605$	−15.9023	−0.646520
$606$	0	0
$607$	−12.5719	−0.510279	−0.255140	−	0.966904i	$-0.582121\pi$
$607$	−12.5719	−0.510279	−0.255140	+	0.966904i	$0.582121\pi$
$608$	0.476205	0.0193127
$609$	0	0
$610$	0.698424	0.0282784
$611$	−6.72521	−0.272073
$612$	0	0
$613$	−24.5055	−0.989767	−0.494883	−	0.868959i	$-0.664789\pi$
$613$	−24.5055	−0.989767	−0.494883	+	0.868959i	$0.664789\pi$
$614$	20.5457	0.829156
$615$	0	0
$616$	19.4404	0.783278
$617$	−4.56364	−0.183725	−0.0918626	−	0.995772i	$-0.529282\pi$
$617$	−4.56364	−0.183725	−0.0918626	+	0.995772i	$0.529282\pi$
$618$	0	0
$619$	12.2757	0.493401	0.246700	−	0.969092i	$-0.420654\pi$
$619$	12.2757	0.493401	0.246700	+	0.969092i	$0.420654\pi$
$620$	1.06295	0.0426890
$621$	0	0
$622$	−18.1472	−0.727638
$623$	−14.3658	−0.575554
$624$	0	0
$625$	1.00000	0.0400000
$626$	47.8812	1.91372
$627$	0	0
$628$	−1.89587	−0.0756534
$629$	24.8274	0.989934
$630$	0	0
$631$	30.2099	1.20264	0.601319	−	0.799009i	$-0.294642\pi$
$631$	30.2099	1.20264	0.601319	+	0.799009i	$0.294642\pi$
$632$	−21.2966	−0.847133
$633$	0	0
$634$	−33.7334	−1.33972
$635$	−9.71645	−0.385586
$636$	0	0
$637$	8.31686	0.329526
$638$	−66.8423	−2.64631
$639$	0	0
$640$	−10.3196	−0.407917
$641$	37.3717	1.47609	0.738047	−	0.674749i	$-0.235748\pi$
$641$	37.3717	1.47609	0.738047	+	0.674749i	$0.235748\pi$
$642$	0	0
$643$	−25.5582	−1.00792	−0.503958	−	0.863728i	$-0.668124\pi$
$643$	−25.5582	−1.00792	−0.503958	+	0.863728i	$0.668124\pi$
$644$	−1.27800	−0.0503601
$645$	0	0
$646$	6.13775	0.241487
$647$	−17.8098	−0.700176	−0.350088	−	0.936717i	$-0.613848\pi$
$647$	−17.8098	−0.700176	−0.350088	+	0.936717i	$0.613848\pi$
$648$	0	0
$649$	−18.2144	−0.714976
$650$	−2.14735	−0.0842260
$651$	0	0
$652$	−1.30929	−0.0512759
$653$	−2.36879	−0.0926979	−0.0463489	−	0.998925i	$-0.514759\pi$
$653$	−2.36879	−0.0926979	−0.0463489	+	0.998925i	$0.514759\pi$
$654$	0	0
$655$	7.43178	0.290384
$656$	−2.85529	−0.111480
$657$	0	0
$658$	−7.66396	−0.298772
$659$	5.22016	0.203349	0.101674	−	0.994818i	$-0.467580\pi$
$659$	5.22016	0.203349	0.101674	+	0.994818i	$0.467580\pi$
$660$	0	0
$661$	42.3940	1.64893	0.824467	−	0.565910i	$-0.191475\pi$
$661$	42.3940	1.64893	0.824467	+	0.565910i	$0.191475\pi$
$662$	−4.01787	−0.156159
$663$	0	0
$664$	−35.1121	−1.36261
$665$	1.01614	0.0394043
$666$	0	0
$667$	86.3560	3.34372
$668$	−0.0274267	−0.00106117
$669$	0	0
$670$	−15.3300	−0.592249
$671$	−2.63308	−0.101649
$672$	0	0
$673$	11.2210	0.432536	0.216268	−	0.976334i	$-0.430611\pi$
$673$	11.2210	0.432536	0.216268	+	0.976334i	$0.430611\pi$
$674$	−27.9178	−1.07535
$675$	0	0
$676$	1.13270	0.0435654
$677$	33.8810	1.30215	0.651075	−	0.759013i	$-0.274318\pi$
$677$	33.8810	1.30215	0.651075	+	0.759013i	$0.274318\pi$
$678$	0	0
$679$	−15.1121	−0.579951
$680$	16.4557	0.631048
$681$	0	0
$682$	70.7396	2.70876
$683$	51.9618	1.98826	0.994132	−	0.108171i	$-0.0344994\pi$
$683$	51.9618	1.98826	0.994132	+	0.108171i	$0.0344994\pi$
$684$	0	0
$685$	13.9411	0.532661
$686$	21.9286	0.837239
$687$	0	0
$688$	12.4108	0.473157
$689$	10.2413	0.390164
$690$	0	0
$691$	−18.4024	−0.700060	−0.350030	−	0.936739i	$-0.613829\pi$
$691$	−18.4024	−0.700060	−0.350030	+	0.936739i	$0.613829\pi$
$692$	−0.179476	−0.00682265
$693$	0	0
$694$	−23.2674	−0.883219
$695$	17.8433	0.676835
$696$	0	0
$697$	4.29436	0.162660
$698$	15.1828	0.574676
$699$	0	0
$700$	0.138626	0.00523956
$701$	49.2148	1.85882	0.929408	−	0.369054i	$-0.120318\pi$
$701$	49.2148	1.85882	0.929408	+	0.369054i	$0.120318\pi$
$702$	0	0
$703$	3.43778	0.129659
$704$	43.4736	1.63847
$705$	0	0
$706$	22.4441	0.844695
$707$	−2.73790	−0.102969
$708$	0	0
$709$	−18.6015	−0.698595	−0.349297	−	0.937012i	$-0.613580\pi$
$709$	−18.6015	−0.698595	−0.349297	+	0.937012i	$0.613580\pi$
$710$	−5.62393	−0.211062
$711$	0	0
$712$	−32.2136	−1.20726
$713$	−91.3910	−3.42262
$714$	0	0
$715$	8.09558	0.302758
$716$	−1.54559	−0.0577615
$717$	0	0
$718$	21.3098	0.795273
$719$	34.5054	1.28683	0.643417	−	0.765516i	$-0.277516\pi$
$719$	34.5054	1.28683	0.643417	+	0.765516i	$0.277516\pi$
$720$	0	0
$721$	−24.9603	−0.929571
$722$	−25.2900	−0.941196
$723$	0	0
$724$	0.839398	0.0311960
$725$	−9.36714	−0.347887
$726$	0	0
$727$	−9.29638	−0.344784	−0.172392	−	0.985028i	$-0.555150\pi$
$727$	−9.29638	−0.344784	−0.172392	+	0.985028i	$0.555150\pi$
$728$	−5.85012	−0.216820
$729$	0	0
$730$	−20.3127	−0.751805
$731$	−18.6658	−0.690381
$732$	0	0
$733$	37.2005	1.37403	0.687017	−	0.726642i	$-0.258920\pi$
$733$	37.2005	1.37403	0.687017	+	0.726642i	$0.258920\pi$
$734$	7.55919	0.279015
$735$	0	0
$736$	−5.58568	−0.205891
$737$	57.7945	2.12889
$738$	0	0
$739$	6.41527	0.235990	0.117995	−	0.993014i	$-0.462353\pi$
$739$	6.41527	0.235990	0.117995	+	0.993014i	$0.462353\pi$
$740$	0.468995	0.0172406
$741$	0	0
$742$	11.6709	0.428452
$743$	−31.2008	−1.14465	−0.572323	−	0.820028i	$-0.693958\pi$
$743$	−31.2008	−1.14465	−0.572323	+	0.820028i	$0.693958\pi$
$744$	0	0
$745$	14.3146	0.524445
$746$	−8.78168	−0.321520
$747$	0	0
$748$	−3.15678	−0.115423
$749$	1.29286	0.0472400
$750$	0	0
$751$	18.5714	0.677679	0.338840	−	0.940844i	$-0.389966\pi$
$751$	18.5714	0.677679	0.338840	+	0.940844i	$0.389966\pi$
$752$	−16.2615	−0.592996
$753$	0	0
$754$	20.1145	0.732528
$755$	0.452621	0.0164726
$756$	0	0
$757$	−33.9894	−1.23536	−0.617682	−	0.786428i	$-0.711928\pi$
$757$	−33.9894	−1.23536	−0.617682	+	0.786428i	$0.711928\pi$
$758$	−34.8745	−1.26670
$759$	0	0
$760$	2.27858	0.0826527
$761$	−21.2251	−0.769410	−0.384705	−	0.923040i	$-0.625697\pi$
$761$	−21.2251	−0.769410	−0.384705	+	0.923040i	$0.625697\pi$
$762$	0	0
$763$	−4.85854	−0.175891
$764$	−1.36904	−0.0495300
$765$	0	0
$766$	−38.0675	−1.37544
$767$	5.48116	0.197913
$768$	0	0
$769$	53.1763	1.91759	0.958793	−	0.284107i	$-0.0916971\pi$
$769$	53.1763	1.91759	0.958793	+	0.284107i	$0.0916971\pi$
$770$	9.22561	0.332468
$771$	0	0
$772$	0.789612	0.0284188
$773$	−21.3776	−0.768899	−0.384449	−	0.923146i	$-0.625609\pi$
$773$	−21.3776	−0.768899	−0.384449	+	0.923146i	$0.625609\pi$
$774$	0	0
$775$	9.91330	0.356096
$776$	−33.8872	−1.21648
$777$	0	0
$778$	−37.6007	−1.34805
$779$	0.594628	0.0213048
$780$	0	0
$781$	21.2024	0.758681
$782$	−71.9931	−2.57447
$783$	0	0
$784$	20.1101	0.718218
$785$	−17.6813	−0.631074
$786$	0	0
$787$	−6.24888	−0.222748	−0.111374	−	0.993779i	$-0.535525\pi$
$787$	−6.24888	−0.222748	−0.111374	+	0.993779i	$0.535525\pi$
$788$	−1.07721	−0.0383739
$789$	0	0
$790$	−10.1065	−0.359572
$791$	−17.8048	−0.633065
$792$	0	0
$793$	0.792361	0.0281376
$794$	−2.66106	−0.0944373
$795$	0	0
$796$	−1.61701	−0.0573133
$797$	37.8159	1.33951	0.669753	−	0.742584i	$-0.266400\pi$
$797$	37.8159	1.33951	0.669753	+	0.742584i	$0.266400\pi$
$798$	0	0
$799$	24.4573	0.865238
$800$	0.605886	0.0214213
$801$	0	0
$802$	30.8595	1.08969
$803$	76.5794	2.70243
$804$	0	0
$805$	−11.9189	−0.420086
$806$	−21.2873	−0.749814
$807$	0	0
$808$	−6.13942	−0.215984
$809$	−12.0344	−0.423108	−0.211554	−	0.977366i	$-0.567852\pi$
$809$	−12.0344	−0.423108	−0.211554	+	0.977366i	$0.567852\pi$
$810$	0	0
$811$	17.3896	0.610633	0.305316	−	0.952251i	$-0.401238\pi$
$811$	17.3896	0.610633	0.305316	+	0.952251i	$0.401238\pi$
$812$	−1.29853	−0.0455694
$813$	0	0
$814$	31.2119	1.09397
$815$	−12.2108	−0.427726
$816$	0	0
$817$	−2.58461	−0.0904240
$818$	−13.6102	−0.475868
$819$	0	0
$820$	0.0811213	0.00283288
$821$	30.3564	1.05945	0.529723	−	0.848171i	$-0.322296\pi$
$821$	30.3564	1.05945	0.529723	+	0.848171i	$0.322296\pi$
$822$	0	0
$823$	20.2579	0.706147	0.353074	−	0.935596i	$-0.385136\pi$
$823$	20.2579	0.706147	0.353074	+	0.935596i	$0.385136\pi$
$824$	−55.9706	−1.94983
$825$	0	0
$826$	6.24626	0.217335
$827$	−28.2739	−0.983179	−0.491589	−	0.870827i	$-0.663584\pi$
$827$	−28.2739	−0.983179	−0.491589	+	0.870827i	$0.663584\pi$
$828$	0	0
$829$	−17.3221	−0.601623	−0.300811	−	0.953684i	$-0.597257\pi$
$829$	−17.3221	−0.601623	−0.300811	+	0.953684i	$0.597257\pi$
$830$	−16.6627	−0.578372
$831$	0	0
$832$	−13.0823	−0.453547
$833$	−30.2456	−1.04795
$834$	0	0
$835$	−0.255788	−0.00885192
$836$	−0.437110	−0.0151178
$837$	0	0
$838$	−2.90276	−0.100274
$839$	−12.1522	−0.419539	−0.209769	−	0.977751i	$-0.567271\pi$
$839$	−12.1522	−0.419539	−0.209769	+	0.977751i	$0.567271\pi$
$840$	0	0
$841$	58.7434	2.02563
$842$	−29.0448	−1.00095
$843$	0	0
$844$	0.968475	0.0333363
$845$	10.5638	0.363407
$846$	0	0
$847$	−20.5594	−0.706430
$848$	24.7635	0.850382
$849$	0	0
$850$	7.80919	0.267853
$851$	−40.3237	−1.38228
$852$	0	0
$853$	−42.6561	−1.46052	−0.730258	−	0.683172i	$-0.760600\pi$
$853$	−42.6561	−1.46052	−0.730258	+	0.683172i	$0.760600\pi$
$854$	0.902964	0.0308988
$855$	0	0
$856$	2.89908	0.0990885
$857$	11.7238	0.400478	0.200239	−	0.979747i	$-0.435828\pi$
$857$	11.7238	0.400478	0.200239	+	0.979747i	$0.435828\pi$
$858$	0	0
$859$	−39.2770	−1.34011	−0.670057	−	0.742310i	$-0.733730\pi$
$859$	−39.2770	−1.34011	−0.670057	+	0.742310i	$0.733730\pi$
$860$	−0.352601	−0.0120236
$861$	0	0
$862$	8.00696	0.272718
$863$	17.9138	0.609793	0.304896	−	0.952386i	$-0.401378\pi$
$863$	17.9138	0.609793	0.304896	+	0.952386i	$0.401378\pi$
$864$	0	0
$865$	−1.67384	−0.0569121
$866$	−17.2304	−0.585512
$867$	0	0
$868$	1.37424	0.0466447
$869$	38.1017	1.29251
$870$	0	0
$871$	−17.3918	−0.589300
$872$	−10.8947	−0.368941
$873$	0	0
$874$	−9.96869	−0.337196
$875$	1.29286	0.0437066
$876$	0	0
$877$	−21.9915	−0.742602	−0.371301	−	0.928513i	$-0.621088\pi$
$877$	−21.9915	−0.742602	−0.371301	+	0.928513i	$0.621088\pi$
$878$	−32.7962	−1.10682
$879$	0	0
$880$	19.5750	0.659875
$881$	−16.4690	−0.554855	−0.277427	−	0.960747i	$-0.589482\pi$
$881$	−16.4690	−0.554855	−0.277427	+	0.960747i	$0.589482\pi$
$882$	0	0
$883$	52.1294	1.75430	0.877148	−	0.480221i	$-0.159444\pi$
$883$	52.1294	1.75430	0.877148	+	0.480221i	$0.159444\pi$
$884$	0.949954	0.0319504
$885$	0	0
$886$	−11.8802	−0.399122
$887$	−46.6500	−1.56635	−0.783177	−	0.621798i	$-0.786402\pi$
$887$	−46.6500	−1.56635	−0.783177	+	0.621798i	$0.786402\pi$
$888$	0	0
$889$	−12.5620	−0.421316
$890$	−15.2872	−0.512429
$891$	0	0
$892$	−2.35777	−0.0789440
$893$	3.38654	0.113326
$894$	0	0
$895$	−14.4146	−0.481826
$896$	−13.3418	−0.445717
$897$	0	0
$898$	35.3605	1.18000
$899$	−92.8593	−3.09703
$900$	0	0
$901$	−37.2443	−1.24079
$902$	5.39866	0.179756
$903$	0	0
$904$	−39.9251	−1.32789
$905$	7.82843	0.260226
$906$	0	0
$907$	31.3316	1.04035	0.520174	−	0.854060i	$-0.325867\pi$
$907$	31.3316	1.04035	0.520174	+	0.854060i	$0.325867\pi$
$908$	2.78534	0.0924348
$909$	0	0
$910$	−2.77622	−0.0920308
$911$	−3.59385	−0.119069	−0.0595347	−	0.998226i	$-0.518962\pi$
$911$	−3.59385	−0.119069	−0.0595347	+	0.998226i	$0.518962\pi$
$912$	0	0
$913$	62.8191	2.07901
$914$	−33.4196	−1.10542
$915$	0	0
$916$	−0.915068	−0.0302347
$917$	9.60824	0.317292
$918$	0	0
$919$	−22.3591	−0.737559	−0.368780	−	0.929517i	$-0.620224\pi$
$919$	−22.3591	−0.737559	−0.368780	+	0.929517i	$0.620224\pi$
$920$	−26.7267	−0.881154
$921$	0	0
$922$	−20.3811	−0.671215
$923$	−6.38033	−0.210011
$924$	0	0
$925$	4.37397	0.143815
$926$	−31.6430	−1.03985
$927$	0	0
$928$	−5.67542	−0.186305
$929$	38.1453	1.25151	0.625754	−	0.780021i	$-0.284792\pi$
$929$	38.1453	1.25151	0.625754	+	0.780021i	$0.284792\pi$
$930$	0	0
$931$	−4.18803	−0.137257
$932$	−2.90806	−0.0952568
$933$	0	0
$934$	46.7976	1.53126
$935$	−29.4409	−0.962820
$936$	0	0
$937$	−45.9282	−1.50041	−0.750205	−	0.661206i	$-0.770045\pi$
$937$	−45.9282	−1.50041	−0.750205	+	0.661206i	$0.770045\pi$
$938$	−19.8195	−0.647130
$939$	0	0
$940$	0.462004	0.0150689
$941$	3.90156	0.127187	0.0635936	−	0.997976i	$-0.479744\pi$
$941$	3.90156	0.127187	0.0635936	+	0.997976i	$0.479744\pi$
$942$	0	0
$943$	−6.97473	−0.227128
$944$	13.2534	0.431361
$945$	0	0
$946$	−23.4658	−0.762939
$947$	28.3789	0.922189	0.461095	−	0.887351i	$-0.347457\pi$
$947$	28.3789	0.922189	0.461095	+	0.887351i	$0.347457\pi$
$948$	0	0
$949$	−23.0447	−0.748062
$950$	1.08132	0.0350825
$951$	0	0
$952$	21.2749	0.689524
$953$	−28.2289	−0.914425	−0.457212	−	0.889358i	$-0.651152\pi$
$953$	−28.2289	−0.914425	−0.457212	+	0.889358i	$0.651152\pi$
$954$	0	0
$955$	−12.7680	−0.413162
$956$	2.74800	0.0888767
$957$	0	0
$958$	−0.303022	−0.00979020
$959$	18.0238	0.582020
$960$	0	0
$961$	67.2736	2.17012
$962$	−9.39244	−0.302824
$963$	0	0
$964$	2.21612	0.0713765
$965$	7.36412	0.237059
$966$	0	0
$967$	−13.2130	−0.424902	−0.212451	−	0.977172i	$-0.568145\pi$
$967$	−13.2130	−0.424902	−0.212451	+	0.977172i	$0.568145\pi$
$968$	−46.1020	−1.48178
$969$	0	0
$970$	−16.0814	−0.516343
$971$	−52.4953	−1.68465	−0.842327	−	0.538967i	$-0.818815\pi$
$971$	−52.4953	−1.68465	−0.842327	+	0.538967i	$0.818815\pi$
$972$	0	0
$973$	23.0689	0.739554
$974$	18.3470	0.587876
$975$	0	0
$976$	1.91592	0.0613272
$977$	30.6472	0.980491	0.490245	−	0.871584i	$-0.336907\pi$
$977$	30.6472	0.980491	0.490245	+	0.871584i	$0.336907\pi$
$978$	0	0
$979$	57.6333	1.84197
$980$	−0.571346	−0.0182510
$981$	0	0
$982$	55.2957	1.76456
$983$	34.4922	1.10013	0.550066	−	0.835121i	$-0.314603\pi$
$983$	34.4922	1.10013	0.550066	+	0.835121i	$0.314603\pi$
$984$	0	0
$985$	−10.0463	−0.320101
$986$	−73.1498	−2.32956
$987$	0	0
$988$	0.131538	0.00418477
$989$	30.3163	0.964003
$990$	0	0
$991$	11.9433	0.379393	0.189696	−	0.981843i	$-0.439250\pi$
$991$	11.9433	0.379393	0.189696	+	0.981843i	$0.439250\pi$
$992$	6.00633	0.190701
$993$	0	0
$994$	−7.27094	−0.230620
$995$	−15.0806	−0.478087
$996$	0	0
$997$	−29.1180	−0.922176	−0.461088	−	0.887354i	$-0.652541\pi$
$997$	−29.1180	−0.922176	−0.461088	+	0.887354i	$0.652541\pi$
$998$	13.6322	0.431519
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4815.2.a.u.1.9		12
3.2	odd	2		1605.2.a.n.1.4	✓	12
15.14	odd	2		8025.2.a.bf.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.a.n.1.4	✓	12	3.2	odd	2
4815.2.a.u.1.9		12	1.1	even	1	trivial
8025.2.a.bf.1.9		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)

\(f(q)\)	\(=\)	\(q+1.37578 q^{2} -0.107224 q^{4} -1.00000 q^{5} -1.29286 q^{7} -2.89908 q^{8} +O(q^{10})\)	\(q+1.37578 q^{2} -0.107224 q^{4} -1.00000 q^{5} -1.29286 q^{7} -2.89908 q^{8} -1.37578 q^{10} +5.18674 q^{11} -1.56082 q^{13} -1.77869 q^{14} -3.77405 q^{16} +5.67618 q^{17} +0.785965 q^{19} +0.107224 q^{20} +7.13582 q^{22} -9.21903 q^{23} +1.00000 q^{25} -2.14735 q^{26} +0.138626 q^{28} -9.36714 q^{29} +9.91330 q^{31} +0.605886 q^{32} +7.80919 q^{34} +1.29286 q^{35} +4.37397 q^{37} +1.08132 q^{38} +2.89908 q^{40} +0.756558 q^{41} -3.28845 q^{43} -0.556144 q^{44} -12.6834 q^{46} +4.30876 q^{47} -5.32852 q^{49} +1.37578 q^{50} +0.167358 q^{52} -6.56151 q^{53} -5.18674 q^{55} +3.74810 q^{56} -12.8871 q^{58} -3.51171 q^{59} -0.507656 q^{61} +13.6385 q^{62} +8.38168 q^{64} +1.56082 q^{65} +11.1427 q^{67} -0.608624 q^{68} +1.77869 q^{70} +4.08780 q^{71} +14.7644 q^{73} +6.01762 q^{74} -0.0842745 q^{76} -6.70572 q^{77} +7.34598 q^{79} +3.77405 q^{80} +1.04086 q^{82} +12.1115 q^{83} -5.67618 q^{85} -4.52419 q^{86} -15.0368 q^{88} +11.1117 q^{89} +2.01792 q^{91} +0.988504 q^{92} +5.92792 q^{94} -0.785965 q^{95} +11.6889 q^{97} -7.33088 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) \) 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