LMFDB - Embedded newform 8046.2.a.p.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
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} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$3.31930$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.31930 q^{5} -3.41918 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.31930 q^{5} -3.41918 q^{7} +1.00000 q^{8} +3.31930 q^{10} +3.88326 q^{11} -0.659818 q^{13} -3.41918 q^{14} +1.00000 q^{16} +1.70659 q^{17} +2.86730 q^{19} +3.31930 q^{20} +3.88326 q^{22} +3.18162 q^{23} +6.01774 q^{25} -0.659818 q^{26} -3.41918 q^{28} +1.15585 q^{29} -0.368522 q^{31} +1.00000 q^{32} +1.70659 q^{34} -11.3493 q^{35} +5.75882 q^{37} +2.86730 q^{38} +3.31930 q^{40} -6.68785 q^{41} +11.1569 q^{43} +3.88326 q^{44} +3.18162 q^{46} -11.2875 q^{47} +4.69080 q^{49} +6.01774 q^{50} -0.659818 q^{52} -2.43546 q^{53} +12.8897 q^{55} -3.41918 q^{56} +1.15585 q^{58} +5.82474 q^{59} -0.592765 q^{61} -0.368522 q^{62} +1.00000 q^{64} -2.19013 q^{65} -1.28183 q^{67} +1.70659 q^{68} -11.3493 q^{70} +1.63956 q^{71} -0.306499 q^{73} +5.75882 q^{74} +2.86730 q^{76} -13.2776 q^{77} -4.61890 q^{79} +3.31930 q^{80} -6.68785 q^{82} +6.08908 q^{83} +5.66467 q^{85} +11.1569 q^{86} +3.88326 q^{88} -10.1345 q^{89} +2.25604 q^{91} +3.18162 q^{92} -11.2875 q^{94} +9.51741 q^{95} -8.75741 q^{97} +4.69080 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.31930	1.48444	0.742218	−	0.670159i	$-0.233774\pi$
$5$	3.31930	1.48444	0.742218	+	0.670159i	$0.233774\pi$
$6$	0	0
$7$	−3.41918	−1.29233	−0.646165	−	0.763198i	$-0.723628\pi$
$7$	−3.41918	−1.29233	−0.646165	+	0.763198i	$0.723628\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.31930	1.04965
$11$	3.88326	1.17085	0.585423	−	0.810728i	$-0.300928\pi$
$11$	3.88326	1.17085	0.585423	+	0.810728i	$0.300928\pi$
$12$	0	0
$13$	−0.659818	−0.183001	−0.0915003	−	0.995805i	$-0.529166\pi$
$13$	−0.659818	−0.183001	−0.0915003	+	0.995805i	$0.529166\pi$
$14$	−3.41918	−0.913815
$15$	0	0
$16$	1.00000	0.250000
$17$	1.70659	0.413908	0.206954	−	0.978351i	$-0.433645\pi$
$17$	1.70659	0.413908	0.206954	+	0.978351i	$0.433645\pi$
$18$	0	0
$19$	2.86730	0.657803	0.328901	−	0.944364i	$-0.393322\pi$
$19$	2.86730	0.657803	0.328901	+	0.944364i	$0.393322\pi$
$20$	3.31930	0.742218
$21$	0	0
$22$	3.88326	0.827913
$23$	3.18162	0.663414	0.331707	−	0.943382i	$-0.392375\pi$
$23$	3.18162	0.663414	0.331707	+	0.943382i	$0.392375\pi$
$24$	0	0
$25$	6.01774	1.20355
$26$	−0.659818	−0.129401
$27$	0	0
$28$	−3.41918	−0.646165
$29$	1.15585	0.214636	0.107318	−	0.994225i	$-0.465774\pi$
$29$	1.15585	0.214636	0.107318	+	0.994225i	$0.465774\pi$
$30$	0	0
$31$	−0.368522	−0.0661885	−0.0330943	−	0.999452i	$-0.510536\pi$
$31$	−0.368522	−0.0661885	−0.0330943	+	0.999452i	$0.510536\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.70659	0.292677
$35$	−11.3493	−1.91838
$36$	0	0
$37$	5.75882	0.946744	0.473372	−	0.880863i	$-0.343037\pi$
$37$	5.75882	0.946744	0.473372	+	0.880863i	$0.343037\pi$
$38$	2.86730	0.465137
$39$	0	0
$40$	3.31930	0.524827
$41$	−6.68785	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$41$	−6.68785	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$42$	0	0
$43$	11.1569	1.70142	0.850708	−	0.525638i	$-0.176173\pi$
$43$	11.1569	1.70142	0.850708	+	0.525638i	$0.176173\pi$
$44$	3.88326	0.585423
$45$	0	0
$46$	3.18162	0.469105
$47$	−11.2875	−1.64646	−0.823229	−	0.567710i	$-0.807830\pi$
$47$	−11.2875	−1.64646	−0.823229	+	0.567710i	$0.807830\pi$
$48$	0	0
$49$	4.69080	0.670115
$50$	6.01774	0.851037
$51$	0	0
$52$	−0.659818	−0.0915003
$53$	−2.43546	−0.334537	−0.167268	−	0.985911i	$-0.553495\pi$
$53$	−2.43546	−0.334537	−0.167268	+	0.985911i	$0.553495\pi$
$54$	0	0
$55$	12.8897	1.73805
$56$	−3.41918	−0.456907
$57$	0	0
$58$	1.15585	0.151771
$59$	5.82474	0.758317	0.379158	−	0.925332i	$-0.376214\pi$
$59$	5.82474	0.758317	0.379158	+	0.925332i	$0.376214\pi$
$60$	0	0
$61$	−0.592765	−0.0758958	−0.0379479	−	0.999280i	$-0.512082\pi$
$61$	−0.592765	−0.0758958	−0.0379479	+	0.999280i	$0.512082\pi$
$62$	−0.368522	−0.0468024
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.19013	−0.271653
$66$	0	0
$67$	−1.28183	−0.156600	−0.0783001	−	0.996930i	$-0.524949\pi$
$67$	−1.28183	−0.156600	−0.0783001	+	0.996930i	$0.524949\pi$
$68$	1.70659	0.206954
$69$	0	0
$70$	−11.3493	−1.35650
$71$	1.63956	0.194580	0.0972900	−	0.995256i	$-0.468983\pi$
$71$	1.63956	0.194580	0.0972900	+	0.995256i	$0.468983\pi$
$72$	0	0
$73$	−0.306499	−0.0358730	−0.0179365	−	0.999839i	$-0.505710\pi$
$73$	−0.306499	−0.0358730	−0.0179365	+	0.999839i	$0.505710\pi$
$74$	5.75882	0.669449
$75$	0	0
$76$	2.86730	0.328901
$77$	−13.2776	−1.51312
$78$	0	0
$79$	−4.61890	−0.519667	−0.259833	−	0.965653i	$-0.583668\pi$
$79$	−4.61890	−0.519667	−0.259833	+	0.965653i	$0.583668\pi$
$80$	3.31930	0.371109
$81$	0	0
$82$	−6.68785	−0.738549
$83$	6.08908	0.668363	0.334182	−	0.942509i	$-0.391540\pi$
$83$	6.08908	0.668363	0.334182	+	0.942509i	$0.391540\pi$
$84$	0	0
$85$	5.66467	0.614419
$86$	11.1569	1.20308
$87$	0	0
$88$	3.88326	0.413957
$89$	−10.1345	−1.07425	−0.537126	−	0.843502i	$-0.680490\pi$
$89$	−10.1345	−1.07425	−0.537126	+	0.843502i	$0.680490\pi$
$90$	0	0
$91$	2.25604	0.236497
$92$	3.18162	0.331707
$93$	0	0
$94$	−11.2875	−1.16422
$95$	9.51741	0.976466
$96$	0	0
$97$	−8.75741	−0.889180	−0.444590	−	0.895734i	$-0.646651\pi$
$97$	−8.75741	−0.889180	−0.444590	+	0.895734i	$0.646651\pi$
$98$	4.69080	0.473843
$99$	0	0
$100$	6.01774	0.601774
$101$	5.92280	0.589340	0.294670	−	0.955599i	$-0.404790\pi$
$101$	5.92280	0.589340	0.294670	+	0.955599i	$0.404790\pi$
$102$	0	0
$103$	9.78315	0.963962	0.481981	−	0.876182i	$-0.339917\pi$
$103$	9.78315	0.963962	0.481981	+	0.876182i	$0.339917\pi$
$104$	−0.659818	−0.0647005
$105$	0	0
$106$	−2.43546	−0.236553
$107$	−6.34665	−0.613554	−0.306777	−	0.951781i	$-0.599251\pi$
$107$	−6.34665	−0.613554	−0.306777	+	0.951781i	$0.599251\pi$
$108$	0	0
$109$	8.77757	0.840739	0.420369	−	0.907353i	$-0.361900\pi$
$109$	8.77757	0.840739	0.420369	+	0.907353i	$0.361900\pi$
$110$	12.8897	1.22898
$111$	0	0
$112$	−3.41918	−0.323082
$113$	14.1873	1.33463	0.667316	−	0.744775i	$-0.267443\pi$
$113$	14.1873	1.33463	0.667316	+	0.744775i	$0.267443\pi$
$114$	0	0
$115$	10.5608	0.984796
$116$	1.15585	0.107318
$117$	0	0
$118$	5.82474	0.536211
$119$	−5.83513	−0.534905
$120$	0	0
$121$	4.07969	0.370881
$122$	−0.592765	−0.0536665
$123$	0	0
$124$	−0.368522	−0.0330943
$125$	3.37819	0.302155
$126$	0	0
$127$	10.8566	0.963368	0.481684	−	0.876345i	$-0.340025\pi$
$127$	10.8566	0.963368	0.481684	+	0.876345i	$0.340025\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.19013	−0.192087
$131$	2.05050	0.179153	0.0895766	−	0.995980i	$-0.471449\pi$
$131$	2.05050	0.179153	0.0895766	+	0.995980i	$0.471449\pi$
$132$	0	0
$133$	−9.80381	−0.850098
$134$	−1.28183	−0.110733
$135$	0	0
$136$	1.70659	0.146339
$137$	4.50917	0.385245	0.192622	−	0.981273i	$-0.438301\pi$
$137$	4.50917	0.385245	0.192622	+	0.981273i	$0.438301\pi$
$138$	0	0
$139$	−19.9103	−1.68877	−0.844385	−	0.535736i	$-0.820034\pi$
$139$	−19.9103	−1.68877	−0.844385	+	0.535736i	$0.820034\pi$
$140$	−11.3493	−0.959190
$141$	0	0
$142$	1.63956	0.137589
$143$	−2.56224	−0.214266
$144$	0	0
$145$	3.83662	0.318614
$146$	−0.306499	−0.0253660
$147$	0	0
$148$	5.75882	0.473372
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	21.0877	1.71609	0.858046	−	0.513574i	$-0.171679\pi$
$151$	21.0877	1.71609	0.858046	+	0.513574i	$0.171679\pi$
$152$	2.86730	0.232568
$153$	0	0
$154$	−13.2776	−1.06994
$155$	−1.22324	−0.0982526
$156$	0	0
$157$	−24.5164	−1.95662	−0.978312	−	0.207136i	$-0.933586\pi$
$157$	−24.5164	−1.95662	−0.978312	+	0.207136i	$0.933586\pi$
$158$	−4.61890	−0.367460
$159$	0	0
$160$	3.31930	0.262414
$161$	−10.8785	−0.857350
$162$	0	0
$163$	22.6400	1.77330	0.886651	−	0.462440i	$-0.153026\pi$
$163$	22.6400	1.77330	0.886651	+	0.462440i	$0.153026\pi$
$164$	−6.68785	−0.522233
$165$	0	0
$166$	6.08908	0.472604
$167$	5.77970	0.447246	0.223623	−	0.974676i	$-0.428212\pi$
$167$	5.77970	0.447246	0.223623	+	0.974676i	$0.428212\pi$
$168$	0	0
$169$	−12.5646	−0.966511
$170$	5.66467	0.434460
$171$	0	0
$172$	11.1569	0.850708
$173$	−14.0919	−1.07139	−0.535694	−	0.844412i	$-0.679950\pi$
$173$	−14.0919	−1.07139	−0.535694	+	0.844412i	$0.679950\pi$
$174$	0	0
$175$	−20.5758	−1.55538
$176$	3.88326	0.292712
$177$	0	0
$178$	−10.1345	−0.759610
$179$	14.7201	1.10023	0.550117	−	0.835088i	$-0.314583\pi$
$179$	14.7201	1.10023	0.550117	+	0.835088i	$0.314583\pi$
$180$	0	0
$181$	22.4918	1.67180	0.835901	−	0.548881i	$-0.184946\pi$
$181$	22.4918	1.67180	0.835901	+	0.548881i	$0.184946\pi$
$182$	2.25604	0.167229
$183$	0	0
$184$	3.18162	0.234552
$185$	19.1152	1.40538
$186$	0	0
$187$	6.62711	0.484622
$188$	−11.2875	−0.823229
$189$	0	0
$190$	9.51741	0.690466
$191$	14.8180	1.07219	0.536097	−	0.844157i	$-0.319898\pi$
$191$	14.8180	1.07219	0.536097	+	0.844157i	$0.319898\pi$
$192$	0	0
$193$	−2.79106	−0.200905	−0.100452	−	0.994942i	$-0.532029\pi$
$193$	−2.79106	−0.200905	−0.100452	+	0.994942i	$0.532029\pi$
$194$	−8.75741	−0.628745
$195$	0	0
$196$	4.69080	0.335057
$197$	7.30166	0.520221	0.260111	−	0.965579i	$-0.416241\pi$
$197$	7.30166	0.520221	0.260111	+	0.965579i	$0.416241\pi$
$198$	0	0
$199$	19.1422	1.35696	0.678479	−	0.734620i	$-0.262640\pi$
$199$	19.1422	1.35696	0.678479	+	0.734620i	$0.262640\pi$
$200$	6.01774	0.425519
$201$	0	0
$202$	5.92280	0.416726
$203$	−3.95207	−0.277381
$204$	0	0
$205$	−22.1990	−1.55044
$206$	9.78315	0.681624
$207$	0	0
$208$	−0.659818	−0.0457501
$209$	11.1345	0.770186
$210$	0	0
$211$	3.83647	0.264113	0.132057	−	0.991242i	$-0.457842\pi$
$211$	3.83647	0.264113	0.132057	+	0.991242i	$0.457842\pi$
$212$	−2.43546	−0.167268
$213$	0	0
$214$	−6.34665	−0.433848
$215$	37.0332	2.52564
$216$	0	0
$217$	1.26004	0.0855374
$218$	8.77757	0.594492
$219$	0	0
$220$	12.8897	0.869023
$221$	−1.12604	−0.0757454
$222$	0	0
$223$	−9.67951	−0.648188	−0.324094	−	0.946025i	$-0.605059\pi$
$223$	−9.67951	−0.648188	−0.324094	+	0.946025i	$0.605059\pi$
$224$	−3.41918	−0.228454
$225$	0	0
$226$	14.1873	0.943728
$227$	4.30896	0.285996	0.142998	−	0.989723i	$-0.454326\pi$
$227$	4.30896	0.285996	0.142998	+	0.989723i	$0.454326\pi$
$228$	0	0
$229$	19.2781	1.27393	0.636965	−	0.770892i	$-0.280189\pi$
$229$	19.2781	1.27393	0.636965	+	0.770892i	$0.280189\pi$
$230$	10.5608	0.696356
$231$	0	0
$232$	1.15585	0.0758854
$233$	13.5941	0.890579	0.445289	−	0.895387i	$-0.353101\pi$
$233$	13.5941	0.890579	0.445289	+	0.895387i	$0.353101\pi$
$234$	0	0
$235$	−37.4667	−2.44406
$236$	5.82474	0.379158
$237$	0	0
$238$	−5.83513	−0.378235
$239$	−15.4852	−1.00165	−0.500827	−	0.865548i	$-0.666971\pi$
$239$	−15.4852	−1.00165	−0.500827	+	0.865548i	$0.666971\pi$
$240$	0	0
$241$	−7.72783	−0.497793	−0.248897	−	0.968530i	$-0.580068\pi$
$241$	−7.72783	−0.497793	−0.248897	+	0.968530i	$0.580068\pi$
$242$	4.07969	0.262252
$243$	0	0
$244$	−0.592765	−0.0379479
$245$	15.5702	0.994742
$246$	0	0
$247$	−1.89189	−0.120378
$248$	−0.368522	−0.0234012
$249$	0	0
$250$	3.37819	0.213656
$251$	18.1567	1.14604	0.573021	−	0.819540i	$-0.305771\pi$
$251$	18.1567	1.14604	0.573021	+	0.819540i	$0.305771\pi$
$252$	0	0
$253$	12.3551	0.776756
$254$	10.8566	0.681204
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.1905	−0.822802	−0.411401	−	0.911454i	$-0.634960\pi$
$257$	−13.1905	−0.822802	−0.411401	+	0.911454i	$0.634960\pi$
$258$	0	0
$259$	−19.6904	−1.22351
$260$	−2.19013	−0.135826
$261$	0	0
$262$	2.05050	0.126681
$263$	8.87215	0.547080	0.273540	−	0.961861i	$-0.411805\pi$
$263$	8.87215	0.547080	0.273540	+	0.961861i	$0.411805\pi$
$264$	0	0
$265$	−8.08403	−0.496598
$266$	−9.80381	−0.601110
$267$	0	0
$268$	−1.28183	−0.0783001
$269$	−1.58090	−0.0963894	−0.0481947	−	0.998838i	$-0.515347\pi$
$269$	−1.58090	−0.0963894	−0.0481947	+	0.998838i	$0.515347\pi$
$270$	0	0
$271$	−7.84822	−0.476745	−0.238373	−	0.971174i	$-0.576614\pi$
$271$	−7.84822	−0.476745	−0.238373	+	0.971174i	$0.576614\pi$
$272$	1.70659	0.103477
$273$	0	0
$274$	4.50917	0.272409
$275$	23.3684	1.40917
$276$	0	0
$277$	10.4077	0.625340	0.312670	−	0.949862i	$-0.398777\pi$
$277$	10.4077	0.625340	0.312670	+	0.949862i	$0.398777\pi$
$278$	−19.9103	−1.19414
$279$	0	0
$280$	−11.3493	−0.678249
$281$	−3.11318	−0.185717	−0.0928585	−	0.995679i	$-0.529600\pi$
$281$	−3.11318	−0.185717	−0.0928585	+	0.995679i	$0.529600\pi$
$282$	0	0
$283$	−4.20364	−0.249880	−0.124940	−	0.992164i	$-0.539874\pi$
$283$	−4.20364	−0.249880	−0.124940	+	0.992164i	$0.539874\pi$
$284$	1.63956	0.0972900
$285$	0	0
$286$	−2.56224	−0.151509
$287$	22.8670	1.34979
$288$	0	0
$289$	−14.0876	−0.828680
$290$	3.83662	0.225294
$291$	0	0
$292$	−0.306499	−0.0179365
$293$	−8.09004	−0.472625	−0.236313	−	0.971677i	$-0.575939\pi$
$293$	−8.09004	−0.472625	−0.236313	+	0.971677i	$0.575939\pi$
$294$	0	0
$295$	19.3341	1.12567
$296$	5.75882	0.334725
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−2.09929	−0.121405
$300$	0	0
$301$	−38.1476	−2.19879
$302$	21.0877	1.21346
$303$	0	0
$304$	2.86730	0.164451
$305$	−1.96757	−0.112662
$306$	0	0
$307$	7.01307	0.400257	0.200129	−	0.979770i	$-0.435864\pi$
$307$	7.01307	0.400257	0.200129	+	0.979770i	$0.435864\pi$
$308$	−13.2776	−0.756559
$309$	0	0
$310$	−1.22324	−0.0694751
$311$	−16.7912	−0.952143	−0.476071	−	0.879407i	$-0.657940\pi$
$311$	−16.7912	−0.952143	−0.476071	+	0.879407i	$0.657940\pi$
$312$	0	0
$313$	−18.6910	−1.05648	−0.528239	−	0.849096i	$-0.677148\pi$
$313$	−18.6910	−1.05648	−0.528239	+	0.849096i	$0.677148\pi$
$314$	−24.5164	−1.38354
$315$	0	0
$316$	−4.61890	−0.259833
$317$	17.5842	0.987625	0.493813	−	0.869568i	$-0.335603\pi$
$317$	17.5842	0.987625	0.493813	+	0.869568i	$0.335603\pi$
$318$	0	0
$319$	4.48847	0.251306
$320$	3.31930	0.185554
$321$	0	0
$322$	−10.8785	−0.606238
$323$	4.89329	0.272270
$324$	0	0
$325$	−3.97062	−0.220250
$326$	22.6400	1.25391
$327$	0	0
$328$	−6.68785	−0.369275
$329$	38.5942	2.12776
$330$	0	0
$331$	−14.8810	−0.817933	−0.408966	−	0.912549i	$-0.634111\pi$
$331$	−14.8810	−0.817933	−0.408966	+	0.912549i	$0.634111\pi$
$332$	6.08908	0.334182
$333$	0	0
$334$	5.77970	0.316251
$335$	−4.25477	−0.232463
$336$	0	0
$337$	−6.28946	−0.342609	−0.171304	−	0.985218i	$-0.554798\pi$
$337$	−6.28946	−0.342609	−0.171304	+	0.985218i	$0.554798\pi$
$338$	−12.5646	−0.683426
$339$	0	0
$340$	5.66467	0.307210
$341$	−1.43107	−0.0774966
$342$	0	0
$343$	7.89557	0.426321
$344$	11.1569	0.601542
$345$	0	0
$346$	−14.0919	−0.757586
$347$	−26.2007	−1.40653	−0.703264	−	0.710929i	$-0.748275\pi$
$347$	−26.2007	−1.40653	−0.703264	+	0.710929i	$0.748275\pi$
$348$	0	0
$349$	3.92916	0.210323	0.105162	−	0.994455i	$-0.466464\pi$
$349$	3.92916	0.210323	0.105162	+	0.994455i	$0.466464\pi$
$350$	−20.5758	−1.09982
$351$	0	0
$352$	3.88326	0.206978
$353$	−17.5437	−0.933756	−0.466878	−	0.884322i	$-0.654621\pi$
$353$	−17.5437	−0.933756	−0.466878	+	0.884322i	$0.654621\pi$
$354$	0	0
$355$	5.44219	0.288841
$356$	−10.1345	−0.537126
$357$	0	0
$358$	14.7201	0.777983
$359$	−2.68279	−0.141592	−0.0707962	−	0.997491i	$-0.522554\pi$
$359$	−2.68279	−0.141592	−0.0707962	+	0.997491i	$0.522554\pi$
$360$	0	0
$361$	−10.7786	−0.567295
$362$	22.4918	1.18214
$363$	0	0
$364$	2.25604	0.118248
$365$	−1.01736	−0.0532511
$366$	0	0
$367$	9.47959	0.494830	0.247415	−	0.968910i	$-0.420419\pi$
$367$	9.47959	0.494830	0.247415	+	0.968910i	$0.420419\pi$
$368$	3.18162	0.165854
$369$	0	0
$370$	19.1152	0.993754
$371$	8.32729	0.432331
$372$	0	0
$373$	−10.8247	−0.560481	−0.280241	−	0.959930i	$-0.590414\pi$
$373$	−10.8247	−0.560481	−0.280241	+	0.959930i	$0.590414\pi$
$374$	6.62711	0.342680
$375$	0	0
$376$	−11.2875	−0.582111
$377$	−0.762652	−0.0392786
$378$	0	0
$379$	−9.55464	−0.490789	−0.245394	−	0.969423i	$-0.578917\pi$
$379$	−9.55464	−0.490789	−0.245394	+	0.969423i	$0.578917\pi$
$380$	9.51741	0.488233
$381$	0	0
$382$	14.8180	0.758155
$383$	−24.7637	−1.26537	−0.632684	−	0.774410i	$-0.718047\pi$
$383$	−24.7637	−1.26537	−0.632684	+	0.774410i	$0.718047\pi$
$384$	0	0
$385$	−44.0722	−2.24613
$386$	−2.79106	−0.142061
$387$	0	0
$388$	−8.75741	−0.444590
$389$	0.0617531	0.00313100	0.00156550	−	0.999999i	$-0.499502\pi$
$389$	0.0617531	0.00313100	0.00156550	+	0.999999i	$0.499502\pi$
$390$	0	0
$391$	5.42971	0.274592
$392$	4.69080	0.236921
$393$	0	0
$394$	7.30166	0.367852
$395$	−15.3315	−0.771412
$396$	0	0
$397$	−29.1702	−1.46401	−0.732005	−	0.681300i	$-0.761415\pi$
$397$	−29.1702	−1.46401	−0.732005	+	0.681300i	$0.761415\pi$
$398$	19.1422	0.959514
$399$	0	0
$400$	6.01774	0.300887
$401$	0.235736	0.0117721	0.00588606	−	0.999983i	$-0.498126\pi$
$401$	0.235736	0.0117721	0.00588606	+	0.999983i	$0.498126\pi$
$402$	0	0
$403$	0.243158	0.0121125
$404$	5.92280	0.294670
$405$	0	0
$406$	−3.95207	−0.196138
$407$	22.3630	1.10849
$408$	0	0
$409$	8.07427	0.399247	0.199623	−	0.979873i	$-0.436028\pi$
$409$	8.07427	0.399247	0.199623	+	0.979873i	$0.436028\pi$
$410$	−22.1990	−1.09633
$411$	0	0
$412$	9.78315	0.481981
$413$	−19.9158	−0.979995
$414$	0	0
$415$	20.2115	0.992142
$416$	−0.659818	−0.0323502
$417$	0	0
$418$	11.1345	0.544604
$419$	14.4174	0.704334	0.352167	−	0.935937i	$-0.385445\pi$
$419$	14.4174	0.704334	0.352167	+	0.935937i	$0.385445\pi$
$420$	0	0
$421$	14.2632	0.695145	0.347573	−	0.937653i	$-0.387006\pi$
$421$	14.2632	0.695145	0.347573	+	0.937653i	$0.387006\pi$
$422$	3.83647	0.186756
$423$	0	0
$424$	−2.43546	−0.118277
$425$	10.2698	0.498158
$426$	0	0
$427$	2.02677	0.0980824
$428$	−6.34665	−0.306777
$429$	0	0
$430$	37.0332	1.78590
$431$	−15.6491	−0.753789	−0.376895	−	0.926256i	$-0.623008\pi$
$431$	−15.6491	−0.753789	−0.376895	+	0.926256i	$0.623008\pi$
$432$	0	0
$433$	−2.88858	−0.138816	−0.0694081	−	0.997588i	$-0.522111\pi$
$433$	−2.88858	−0.138816	−0.0694081	+	0.997588i	$0.522111\pi$
$434$	1.26004	0.0604841
$435$	0	0
$436$	8.77757	0.420369
$437$	9.12266	0.436396
$438$	0	0
$439$	−22.8344	−1.08983	−0.544914	−	0.838492i	$-0.683438\pi$
$439$	−22.8344	−1.08983	−0.544914	+	0.838492i	$0.683438\pi$
$440$	12.8897	0.614492
$441$	0	0
$442$	−1.12604	−0.0535601
$443$	−37.3167	−1.77297	−0.886486	−	0.462756i	$-0.846861\pi$
$443$	−37.3167	−1.77297	−0.886486	+	0.462756i	$0.846861\pi$
$444$	0	0
$445$	−33.6393	−1.59466
$446$	−9.67951	−0.458338
$447$	0	0
$448$	−3.41918	−0.161541
$449$	16.2084	0.764923	0.382462	−	0.923971i	$-0.375076\pi$
$449$	16.2084	0.764923	0.382462	+	0.923971i	$0.375076\pi$
$450$	0	0
$451$	−25.9706	−1.22291
$452$	14.1873	0.667316
$453$	0	0
$454$	4.30896	0.202230
$455$	7.48846	0.351065
$456$	0	0
$457$	1.76518	0.0825718	0.0412859	−	0.999147i	$-0.486855\pi$
$457$	1.76518	0.0825718	0.0412859	+	0.999147i	$0.486855\pi$
$458$	19.2781	0.900805
$459$	0	0
$460$	10.5608	0.492398
$461$	10.2936	0.479419	0.239709	−	0.970845i	$-0.422948\pi$
$461$	10.2936	0.479419	0.239709	+	0.970845i	$0.422948\pi$
$462$	0	0
$463$	11.7978	0.548291	0.274146	−	0.961688i	$-0.411605\pi$
$463$	11.7978	0.548291	0.274146	+	0.961688i	$0.411605\pi$
$464$	1.15585	0.0536591
$465$	0	0
$466$	13.5941	0.629734
$467$	−29.3142	−1.35650	−0.678250	−	0.734831i	$-0.737261\pi$
$467$	−29.3142	−1.35650	−0.678250	+	0.734831i	$0.737261\pi$
$468$	0	0
$469$	4.38280	0.202379
$470$	−37.4667	−1.72821
$471$	0	0
$472$	5.82474	0.268105
$473$	43.3252	1.99210
$474$	0	0
$475$	17.2547	0.791698
$476$	−5.83513	−0.267453
$477$	0	0
$478$	−15.4852	−0.708276
$479$	14.2253	0.649968	0.324984	−	0.945719i	$-0.394641\pi$
$479$	14.2253	0.649968	0.324984	+	0.945719i	$0.394641\pi$
$480$	0	0
$481$	−3.79977	−0.173255
$482$	−7.72783	−0.351993
$483$	0	0
$484$	4.07969	0.185440
$485$	−29.0685	−1.31993
$486$	0	0
$487$	−11.8354	−0.536314	−0.268157	−	0.963375i	$-0.586415\pi$
$487$	−11.8354	−0.536314	−0.268157	+	0.963375i	$0.586415\pi$
$488$	−0.592765	−0.0268332
$489$	0	0
$490$	15.5702	0.703389
$491$	43.0587	1.94321	0.971606	−	0.236603i	$-0.0760340\pi$
$491$	43.0587	1.94321	0.971606	+	0.236603i	$0.0760340\pi$
$492$	0	0
$493$	1.97256	0.0888396
$494$	−1.89189	−0.0851203
$495$	0	0
$496$	−0.368522	−0.0165471
$497$	−5.60595	−0.251461
$498$	0	0
$499$	6.66301	0.298277	0.149139	−	0.988816i	$-0.452350\pi$
$499$	6.66301	0.298277	0.149139	+	0.988816i	$0.452350\pi$
$500$	3.37819	0.151077
$501$	0	0
$502$	18.1567	0.810375
$503$	7.56797	0.337439	0.168720	−	0.985664i	$-0.446037\pi$
$503$	7.56797	0.337439	0.168720	+	0.985664i	$0.446037\pi$
$504$	0	0
$505$	19.6595	0.874837
$506$	12.3551	0.549249
$507$	0	0
$508$	10.8566	0.481684
$509$	−0.275753	−0.0122225	−0.00611126	−	0.999981i	$-0.501945\pi$
$509$	−0.275753	−0.0122225	−0.00611126	+	0.999981i	$0.501945\pi$
$510$	0	0
$511$	1.04798	0.0463597
$512$	1.00000	0.0441942
$513$	0	0
$514$	−13.1905	−0.581809
$515$	32.4732	1.43094
$516$	0	0
$517$	−43.8324	−1.92775
$518$	−19.6904	−0.865149
$519$	0	0
$520$	−2.19013	−0.0960437
$521$	16.1975	0.709627	0.354814	−	0.934937i	$-0.384544\pi$
$521$	16.1975	0.709627	0.354814	+	0.934937i	$0.384544\pi$
$522$	0	0
$523$	14.5612	0.636718	0.318359	−	0.947970i	$-0.396868\pi$
$523$	14.5612	0.636718	0.318359	+	0.947970i	$0.396868\pi$
$524$	2.05050	0.0895766
$525$	0	0
$526$	8.87215	0.386844
$527$	−0.628915	−0.0273960
$528$	0	0
$529$	−12.8773	−0.559882
$530$	−8.08403	−0.351148
$531$	0	0
$532$	−9.80381	−0.425049
$533$	4.41276	0.191138
$534$	0	0
$535$	−21.0664	−0.910781
$536$	−1.28183	−0.0553665
$537$	0	0
$538$	−1.58090	−0.0681576
$539$	18.2156	0.784601
$540$	0	0
$541$	8.14359	0.350120	0.175060	−	0.984558i	$-0.443988\pi$
$541$	8.14359	0.350120	0.175060	+	0.984558i	$0.443988\pi$
$542$	−7.84822	−0.337110
$543$	0	0
$544$	1.70659	0.0731693
$545$	29.1354	1.24802
$546$	0	0
$547$	−16.9550	−0.724942	−0.362471	−	0.931995i	$-0.618067\pi$
$547$	−16.9550	−0.724942	−0.362471	+	0.931995i	$0.618067\pi$
$548$	4.50917	0.192622
$549$	0	0
$550$	23.3684	0.996434
$551$	3.31417	0.141188
$552$	0	0
$553$	15.7929	0.671581
$554$	10.4077	0.442182
$555$	0	0
$556$	−19.9103	−0.844385
$557$	10.9033	0.461988	0.230994	−	0.972955i	$-0.425802\pi$
$557$	10.9033	0.461988	0.230994	+	0.972955i	$0.425802\pi$
$558$	0	0
$559$	−7.36155	−0.311360
$560$	−11.3493	−0.479595
$561$	0	0
$562$	−3.11318	−0.131322
$563$	−18.6667	−0.786710	−0.393355	−	0.919387i	$-0.628686\pi$
$563$	−18.6667	−0.786710	−0.393355	+	0.919387i	$0.628686\pi$
$564$	0	0
$565$	47.0920	1.98118
$566$	−4.20364	−0.176692
$567$	0	0
$568$	1.63956	0.0687944
$569$	−14.0651	−0.589638	−0.294819	−	0.955553i	$-0.595259\pi$
$569$	−14.0651	−0.589638	−0.294819	+	0.955553i	$0.595259\pi$
$570$	0	0
$571$	9.09335	0.380545	0.190273	−	0.981731i	$-0.439063\pi$
$571$	9.09335	0.380545	0.190273	+	0.981731i	$0.439063\pi$
$572$	−2.56224	−0.107133
$573$	0	0
$574$	22.8670	0.954449
$575$	19.1462	0.798451
$576$	0	0
$577$	6.05339	0.252006	0.126003	−	0.992030i	$-0.459785\pi$
$577$	6.05339	0.252006	0.126003	+	0.992030i	$0.459785\pi$
$578$	−14.0876	−0.585965
$579$	0	0
$580$	3.83662	0.159307
$581$	−20.8197	−0.863745
$582$	0	0
$583$	−9.45753	−0.391691
$584$	−0.306499	−0.0126830
$585$	0	0
$586$	−8.09004	−0.334197
$587$	−43.7558	−1.80600	−0.902998	−	0.429645i	$-0.858639\pi$
$587$	−43.7558	−1.80600	−0.902998	+	0.429645i	$0.858639\pi$
$588$	0	0
$589$	−1.05666	−0.0435390
$590$	19.3341	0.795971
$591$	0	0
$592$	5.75882	0.236686
$593$	14.7155	0.604292	0.302146	−	0.953262i	$-0.402297\pi$
$593$	14.7155	0.604292	0.302146	+	0.953262i	$0.402297\pi$
$594$	0	0
$595$	−19.3685	−0.794032
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−2.09929	−0.0858464
$599$	−13.5421	−0.553316	−0.276658	−	0.960968i	$-0.589227\pi$
$599$	−13.5421	−0.553316	−0.276658	+	0.960968i	$0.589227\pi$
$600$	0	0
$601$	−41.5841	−1.69625	−0.848125	−	0.529796i	$-0.822268\pi$
$601$	−41.5841	−1.69625	−0.848125	+	0.529796i	$0.822268\pi$
$602$	−38.1476	−1.55478
$603$	0	0
$604$	21.0877	0.858046
$605$	13.5417	0.550549
$606$	0	0
$607$	22.3605	0.907585	0.453792	−	0.891107i	$-0.350071\pi$
$607$	22.3605	0.907585	0.453792	+	0.891107i	$0.350071\pi$
$608$	2.86730	0.116284
$609$	0	0
$610$	−1.96757	−0.0796644
$611$	7.44772	0.301303
$612$	0	0
$613$	36.2628	1.46464	0.732321	−	0.680960i	$-0.238437\pi$
$613$	36.2628	1.46464	0.732321	+	0.680960i	$0.238437\pi$
$614$	7.01307	0.283025
$615$	0	0
$616$	−13.2776	−0.534968
$617$	−18.8477	−0.758780	−0.379390	−	0.925237i	$-0.623866\pi$
$617$	−18.8477	−0.758780	−0.379390	+	0.925237i	$0.623866\pi$
$618$	0	0
$619$	−26.0949	−1.04884	−0.524422	−	0.851459i	$-0.675718\pi$
$619$	−26.0949	−1.04884	−0.524422	+	0.851459i	$0.675718\pi$
$620$	−1.22324	−0.0491263
$621$	0	0
$622$	−16.7912	−0.673267
$623$	34.6516	1.38829
$624$	0	0
$625$	−18.8755	−0.755019
$626$	−18.6910	−0.747043
$627$	0	0
$628$	−24.5164	−0.978312
$629$	9.82792	0.391865
$630$	0	0
$631$	−22.0978	−0.879700	−0.439850	−	0.898071i	$-0.644968\pi$
$631$	−22.0978	−0.879700	−0.439850	+	0.898071i	$0.644968\pi$
$632$	−4.61890	−0.183730
$633$	0	0
$634$	17.5842	0.698356
$635$	36.0363	1.43006
$636$	0	0
$637$	−3.09508	−0.122631
$638$	4.48847	0.177700
$639$	0	0
$640$	3.31930	0.131207
$641$	−8.33616	−0.329259	−0.164629	−	0.986356i	$-0.552643\pi$
$641$	−8.33616	−0.329259	−0.164629	+	0.986356i	$0.552643\pi$
$642$	0	0
$643$	1.93815	0.0764331	0.0382165	−	0.999269i	$-0.487832\pi$
$643$	1.93815	0.0764331	0.0382165	+	0.999269i	$0.487832\pi$
$644$	−10.8785	−0.428675
$645$	0	0
$646$	4.89329	0.192524
$647$	19.5053	0.766834	0.383417	−	0.923575i	$-0.374747\pi$
$647$	19.5053	0.766834	0.383417	+	0.923575i	$0.374747\pi$
$648$	0	0
$649$	22.6190	0.887872
$650$	−3.97062	−0.155740
$651$	0	0
$652$	22.6400	0.886651
$653$	−16.9339	−0.662676	−0.331338	−	0.943512i	$-0.607500\pi$
$653$	−16.9339	−0.662676	−0.331338	+	0.943512i	$0.607500\pi$
$654$	0	0
$655$	6.80623	0.265942
$656$	−6.68785	−0.261117
$657$	0	0
$658$	38.5942	1.50456
$659$	48.8736	1.90384	0.951922	−	0.306341i	$-0.0991049\pi$
$659$	48.8736	1.90384	0.951922	+	0.306341i	$0.0991049\pi$
$660$	0	0
$661$	−10.4517	−0.406525	−0.203262	−	0.979124i	$-0.565154\pi$
$661$	−10.4517	−0.406525	−0.203262	+	0.979124i	$0.565154\pi$
$662$	−14.8810	−0.578366
$663$	0	0
$664$	6.08908	0.236302
$665$	−32.5418	−1.26192
$666$	0	0
$667$	3.67748	0.142393
$668$	5.77970	0.223623
$669$	0	0
$670$	−4.25477	−0.164376
$671$	−2.30186	−0.0888624
$672$	0	0
$673$	−15.5901	−0.600953	−0.300476	−	0.953789i	$-0.597146\pi$
$673$	−15.5901	−0.600953	−0.300476	+	0.953789i	$0.597146\pi$
$674$	−6.28946	−0.242261
$675$	0	0
$676$	−12.5646	−0.483255
$677$	−36.9325	−1.41943	−0.709715	−	0.704488i	$-0.751177\pi$
$677$	−36.9325	−1.41943	−0.709715	+	0.704488i	$0.751177\pi$
$678$	0	0
$679$	29.9432	1.14911
$680$	5.66467	0.217230
$681$	0	0
$682$	−1.43107	−0.0547984
$683$	−12.3594	−0.472918	−0.236459	−	0.971641i	$-0.575987\pi$
$683$	−12.3594	−0.472918	−0.236459	+	0.971641i	$0.575987\pi$
$684$	0	0
$685$	14.9673	0.571871
$686$	7.89557	0.301454
$687$	0	0
$688$	11.1569	0.425354
$689$	1.60696	0.0612204
$690$	0	0
$691$	−48.1519	−1.83179	−0.915893	−	0.401423i	$-0.868516\pi$
$691$	−48.1519	−1.83179	−0.915893	+	0.401423i	$0.868516\pi$
$692$	−14.0919	−0.535694
$693$	0	0
$694$	−26.2007	−0.994565
$695$	−66.0883	−2.50687
$696$	0	0
$697$	−11.4134	−0.432313
$698$	3.92916	0.148721
$699$	0	0
$700$	−20.5758	−0.777690
$701$	14.1269	0.533567	0.266784	−	0.963756i	$-0.414039\pi$
$701$	14.1269	0.533567	0.266784	+	0.963756i	$0.414039\pi$
$702$	0	0
$703$	16.5122	0.622771
$704$	3.88326	0.146356
$705$	0	0
$706$	−17.5437	−0.660265
$707$	−20.2511	−0.761621
$708$	0	0
$709$	−18.4274	−0.692055	−0.346027	−	0.938224i	$-0.612470\pi$
$709$	−18.4274	−0.692055	−0.346027	+	0.938224i	$0.612470\pi$
$710$	5.44219	0.204242
$711$	0	0
$712$	−10.1345	−0.379805
$713$	−1.17250	−0.0439104
$714$	0	0
$715$	−8.50485	−0.318063
$716$	14.7201	0.550117
$717$	0	0
$718$	−2.68279	−0.100121
$719$	19.1056	0.712517	0.356259	−	0.934387i	$-0.384052\pi$
$719$	19.1056	0.712517	0.356259	+	0.934387i	$0.384052\pi$
$720$	0	0
$721$	−33.4503	−1.24576
$722$	−10.7786	−0.401138
$723$	0	0
$724$	22.4918	0.835901
$725$	6.95562	0.258325
$726$	0	0
$727$	33.3288	1.23610	0.618048	−	0.786140i	$-0.287924\pi$
$727$	33.3288	1.23610	0.618048	+	0.786140i	$0.287924\pi$
$728$	2.25604	0.0836143
$729$	0	0
$730$	−1.01736	−0.0376542
$731$	19.0403	0.704230
$732$	0	0
$733$	−8.24031	−0.304363	−0.152181	−	0.988353i	$-0.548630\pi$
$733$	−8.24031	−0.304363	−0.152181	+	0.988353i	$0.548630\pi$
$734$	9.47959	0.349898
$735$	0	0
$736$	3.18162	0.117276
$737$	−4.97767	−0.183355
$738$	0	0
$739$	3.26374	0.120059	0.0600293	−	0.998197i	$-0.480881\pi$
$739$	3.26374	0.120059	0.0600293	+	0.998197i	$0.480881\pi$
$740$	19.1152	0.702690
$741$	0	0
$742$	8.32729	0.305704
$743$	34.8426	1.27825	0.639126	−	0.769102i	$-0.279296\pi$
$743$	34.8426	1.27825	0.639126	+	0.769102i	$0.279296\pi$
$744$	0	0
$745$	−3.31930	−0.121610
$746$	−10.8247	−0.396320
$747$	0	0
$748$	6.62711	0.242311
$749$	21.7004	0.792914
$750$	0	0
$751$	44.2179	1.61353	0.806767	−	0.590870i	$-0.201215\pi$
$751$	44.2179	1.61353	0.806767	+	0.590870i	$0.201215\pi$
$752$	−11.2875	−0.411614
$753$	0	0
$754$	−0.762652	−0.0277741
$755$	69.9963	2.54743
$756$	0	0
$757$	−19.3431	−0.703039	−0.351519	−	0.936181i	$-0.614335\pi$
$757$	−19.3431	−0.703039	−0.351519	+	0.936181i	$0.614335\pi$
$758$	−9.55464	−0.347040
$759$	0	0
$760$	9.51741	0.345233
$761$	17.2280	0.624516	0.312258	−	0.949997i	$-0.398915\pi$
$761$	17.2280	0.624516	0.312258	+	0.949997i	$0.398915\pi$
$762$	0	0
$763$	−30.0121	−1.08651
$764$	14.8180	0.536097
$765$	0	0
$766$	−24.7637	−0.894750
$767$	−3.84327	−0.138772
$768$	0	0
$769$	−16.5955	−0.598447	−0.299224	−	0.954183i	$-0.596728\pi$
$769$	−16.5955	−0.598447	−0.299224	+	0.954183i	$0.596728\pi$
$770$	−44.0722	−1.58825
$771$	0	0
$772$	−2.79106	−0.100452
$773$	−50.4823	−1.81572	−0.907862	−	0.419269i	$-0.862286\pi$
$773$	−50.4823	−1.81572	−0.907862	+	0.419269i	$0.862286\pi$
$774$	0	0
$775$	−2.21767	−0.0796611
$776$	−8.75741	−0.314373
$777$	0	0
$778$	0.0617531	0.00221395
$779$	−19.1760	−0.687053
$780$	0	0
$781$	6.36683	0.227823
$782$	5.42971	0.194166
$783$	0	0
$784$	4.69080	0.167529
$785$	−81.3774	−2.90448
$786$	0	0
$787$	−15.9661	−0.569129	−0.284564	−	0.958657i	$-0.591849\pi$
$787$	−15.9661	−0.569129	−0.284564	+	0.958657i	$0.591849\pi$
$788$	7.30166	0.260111
$789$	0	0
$790$	−15.3315	−0.545471
$791$	−48.5091	−1.72478
$792$	0	0
$793$	0.391117	0.0138890
$794$	−29.1702	−1.03521
$795$	0	0
$796$	19.1422	0.678479
$797$	−6.43798	−0.228045	−0.114022	−	0.993478i	$-0.536374\pi$
$797$	−6.43798	−0.228045	−0.114022	+	0.993478i	$0.536374\pi$
$798$	0	0
$799$	−19.2632	−0.681482
$800$	6.01774	0.212759
$801$	0	0
$802$	0.235736	0.00832414
$803$	−1.19021	−0.0420017
$804$	0	0
$805$	−36.1091	−1.27268
$806$	0.243158	0.00856486
$807$	0	0
$808$	5.92280	0.208363
$809$	−55.7479	−1.95999	−0.979995	−	0.199020i	$-0.936224\pi$
$809$	−55.7479	−1.95999	−0.979995	+	0.199020i	$0.936224\pi$
$810$	0	0
$811$	2.04955	0.0719696	0.0359848	−	0.999352i	$-0.488543\pi$
$811$	2.04955	0.0719696	0.0359848	+	0.999352i	$0.488543\pi$
$812$	−3.95207	−0.138690
$813$	0	0
$814$	22.3630	0.783822
$815$	75.1489	2.63235
$816$	0	0
$817$	31.9902	1.11920
$818$	8.07427	0.282310
$819$	0	0
$820$	−22.1990	−0.775222
$821$	−21.0785	−0.735644	−0.367822	−	0.929896i	$-0.619896\pi$
$821$	−21.0785	−0.735644	−0.367822	+	0.929896i	$0.619896\pi$
$822$	0	0
$823$	48.7089	1.69789	0.848943	−	0.528485i	$-0.177240\pi$
$823$	48.7089	1.69789	0.848943	+	0.528485i	$0.177240\pi$
$824$	9.78315	0.340812
$825$	0	0
$826$	−19.9158	−0.692961
$827$	−5.73574	−0.199451	−0.0997255	−	0.995015i	$-0.531796\pi$
$827$	−5.73574	−0.199451	−0.0997255	+	0.995015i	$0.531796\pi$
$828$	0	0
$829$	50.3262	1.74790	0.873951	−	0.486014i	$-0.161550\pi$
$829$	50.3262	1.74790	0.873951	+	0.486014i	$0.161550\pi$
$830$	20.2115	0.701550
$831$	0	0
$832$	−0.659818	−0.0228751
$833$	8.00525	0.277366
$834$	0	0
$835$	19.1845	0.663908
$836$	11.1345	0.385093
$837$	0	0
$838$	14.4174	0.498039
$839$	−41.9377	−1.44785	−0.723925	−	0.689878i	$-0.757664\pi$
$839$	−41.9377	−1.44785	−0.723925	+	0.689878i	$0.757664\pi$
$840$	0	0
$841$	−27.6640	−0.953931
$842$	14.2632	0.491542
$843$	0	0
$844$	3.83647	0.132057
$845$	−41.7058	−1.43472
$846$	0	0
$847$	−13.9492	−0.479300
$848$	−2.43546	−0.0836341
$849$	0	0
$850$	10.2698	0.352251
$851$	18.3224	0.628084
$852$	0	0
$853$	−34.8144	−1.19202	−0.596011	−	0.802976i	$-0.703249\pi$
$853$	−34.8144	−1.19202	−0.596011	+	0.802976i	$0.703249\pi$
$854$	2.02677	0.0693547
$855$	0	0
$856$	−6.34665	−0.216924
$857$	−5.93225	−0.202642	−0.101321	−	0.994854i	$-0.532307\pi$
$857$	−5.93225	−0.202642	−0.101321	+	0.994854i	$0.532307\pi$
$858$	0	0
$859$	−8.43959	−0.287955	−0.143978	−	0.989581i	$-0.545989\pi$
$859$	−8.43959	−0.287955	−0.143978	+	0.989581i	$0.545989\pi$
$860$	37.0332	1.26282
$861$	0	0
$862$	−15.6491	−0.533009
$863$	−13.9186	−0.473794	−0.236897	−	0.971535i	$-0.576130\pi$
$863$	−13.9186	−0.473794	−0.236897	+	0.971535i	$0.576130\pi$
$864$	0	0
$865$	−46.7752	−1.59041
$866$	−2.88858	−0.0981579
$867$	0	0
$868$	1.26004	0.0427687
$869$	−17.9364	−0.608450
$870$	0	0
$871$	0.845773	0.0286579
$872$	8.77757	0.297246
$873$	0	0
$874$	9.12266	0.308578
$875$	−11.5507	−0.390483
$876$	0	0
$877$	−23.1221	−0.780779	−0.390390	−	0.920650i	$-0.627660\pi$
$877$	−23.1221	−0.780779	−0.390390	+	0.920650i	$0.627660\pi$
$878$	−22.8344	−0.770625
$879$	0	0
$880$	12.8897	0.434511
$881$	−21.5462	−0.725911	−0.362955	−	0.931807i	$-0.618232\pi$
$881$	−21.5462	−0.725911	−0.362955	+	0.931807i	$0.618232\pi$
$882$	0	0
$883$	−16.1827	−0.544590	−0.272295	−	0.962214i	$-0.587783\pi$
$883$	−16.1827	−0.544590	−0.272295	+	0.962214i	$0.587783\pi$
$884$	−1.12604	−0.0378727
$885$	0	0
$886$	−37.3167	−1.25368
$887$	32.8691	1.10364	0.551818	−	0.833965i	$-0.313934\pi$
$887$	32.8691	1.10364	0.551818	+	0.833965i	$0.313934\pi$
$888$	0	0
$889$	−37.1207	−1.24499
$890$	−33.6393	−1.12759
$891$	0	0
$892$	−9.67951	−0.324094
$893$	−32.3647	−1.08304
$894$	0	0
$895$	48.8605	1.63323
$896$	−3.41918	−0.114227
$897$	0	0
$898$	16.2084	0.540882
$899$	−0.425957	−0.0142065
$900$	0	0
$901$	−4.15633	−0.138467
$902$	−25.9706	−0.864728
$903$	0	0
$904$	14.1873	0.471864
$905$	74.6570	2.48168
$906$	0	0
$907$	1.56753	0.0520491	0.0260245	−	0.999661i	$-0.491715\pi$
$907$	1.56753	0.0520491	0.0260245	+	0.999661i	$0.491715\pi$
$908$	4.30896	0.142998
$909$	0	0
$910$	7.48846	0.248240
$911$	52.9858	1.75550	0.877748	−	0.479122i	$-0.159045\pi$
$911$	52.9858	1.75550	0.877748	+	0.479122i	$0.159045\pi$
$912$	0	0
$913$	23.6455	0.782551
$914$	1.76518	0.0583871
$915$	0	0
$916$	19.2781	0.636965
$917$	−7.01104	−0.231525
$918$	0	0
$919$	15.1962	0.501276	0.250638	−	0.968081i	$-0.419360\pi$
$919$	15.1962	0.501276	0.250638	+	0.968081i	$0.419360\pi$
$920$	10.5608	0.348178
$921$	0	0
$922$	10.2936	0.339000
$923$	−1.08181	−0.0356083
$924$	0	0
$925$	34.6551	1.13945
$926$	11.7978	0.387701
$927$	0	0
$928$	1.15585	0.0379427
$929$	−26.8630	−0.881345	−0.440673	−	0.897668i	$-0.645260\pi$
$929$	−26.8630	−0.881345	−0.440673	+	0.897668i	$0.645260\pi$
$930$	0	0
$931$	13.4499	0.440803
$932$	13.5941	0.445289
$933$	0	0
$934$	−29.3142	−0.959191
$935$	21.9974	0.719391
$936$	0	0
$937$	−43.3195	−1.41519	−0.707594	−	0.706619i	$-0.750219\pi$
$937$	−43.3195	−1.41519	−0.707594	+	0.706619i	$0.750219\pi$
$938$	4.38280	0.143104
$939$	0	0
$940$	−37.4667	−1.22203
$941$	27.5766	0.898972	0.449486	−	0.893287i	$-0.351607\pi$
$941$	27.5766	0.898972	0.449486	+	0.893287i	$0.351607\pi$
$942$	0	0
$943$	−21.2782	−0.692914
$944$	5.82474	0.189579
$945$	0	0
$946$	43.3252	1.40863
$947$	26.9871	0.876962	0.438481	−	0.898741i	$-0.355517\pi$
$947$	26.9871	0.876962	0.438481	+	0.898741i	$0.355517\pi$
$948$	0	0
$949$	0.202233	0.00656478
$950$	17.2547	0.559815
$951$	0	0
$952$	−5.83513	−0.189118
$953$	−10.5600	−0.342072	−0.171036	−	0.985265i	$-0.554711\pi$
$953$	−10.5600	−0.342072	−0.171036	+	0.985265i	$0.554711\pi$
$954$	0	0
$955$	49.1854	1.59160
$956$	−15.4852	−0.500827
$957$	0	0
$958$	14.2253	0.459597
$959$	−15.4177	−0.497863
$960$	0	0
$961$	−30.8642	−0.995619
$962$	−3.79977	−0.122510
$963$	0	0
$964$	−7.72783	−0.248897
$965$	−9.26436	−0.298230
$966$	0	0
$967$	10.1106	0.325135	0.162567	−	0.986697i	$-0.448023\pi$
$967$	10.1106	0.325135	0.162567	+	0.986697i	$0.448023\pi$
$968$	4.07969	0.131126
$969$	0	0
$970$	−29.0685	−0.933332
$971$	4.21511	0.135269	0.0676347	−	0.997710i	$-0.478455\pi$
$971$	4.21511	0.135269	0.0676347	+	0.997710i	$0.478455\pi$
$972$	0	0
$973$	68.0770	2.18245
$974$	−11.8354	−0.379231
$975$	0	0
$976$	−0.592765	−0.0189740
$977$	1.11624	0.0357115	0.0178558	−	0.999841i	$-0.494316\pi$
$977$	1.11624	0.0357115	0.0178558	+	0.999841i	$0.494316\pi$
$978$	0	0
$979$	−39.3547	−1.25778
$980$	15.5702	0.497371
$981$	0	0
$982$	43.0587	1.37406
$983$	2.80526	0.0894740	0.0447370	−	0.998999i	$-0.485755\pi$
$983$	2.80526	0.0894740	0.0447370	+	0.998999i	$0.485755\pi$
$984$	0	0
$985$	24.2364	0.772235
$986$	1.97256	0.0628191
$987$	0	0
$988$	−1.89189	−0.0601892
$989$	35.4972	1.12874
$990$	0	0
$991$	−42.0052	−1.33434	−0.667170	−	0.744906i	$-0.732495\pi$
$991$	−42.0052	−1.33434	−0.667170	+	0.744906i	$0.732495\pi$
$992$	−0.368522	−0.0117006
$993$	0	0
$994$	−5.60595	−0.177810
$995$	63.5388	2.01432
$996$	0	0
$997$	−22.2907	−0.705953	−0.352977	−	0.935632i	$-0.614830\pi$
$997$	−22.2907	−0.705953	−0.352977	+	0.935632i	$0.614830\pi$
$998$	6.66301	0.210914
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.11	yes	12
3.2	odd	2		8046.2.a.i.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.2	✓	12	3.2	odd	2
8046.2.a.p.1.11	yes	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.31930 q^{5} -3.41918 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.31930 q^{5} -3.41918 q^{7} +1.00000 q^{8} +3.31930 q^{10} +3.88326 q^{11} -0.659818 q^{13} -3.41918 q^{14} +1.00000 q^{16} +1.70659 q^{17} +2.86730 q^{19} +3.31930 q^{20} +3.88326 q^{22} +3.18162 q^{23} +6.01774 q^{25} -0.659818 q^{26} -3.41918 q^{28} +1.15585 q^{29} -0.368522 q^{31} +1.00000 q^{32} +1.70659 q^{34} -11.3493 q^{35} +5.75882 q^{37} +2.86730 q^{38} +3.31930 q^{40} -6.68785 q^{41} +11.1569 q^{43} +3.88326 q^{44} +3.18162 q^{46} -11.2875 q^{47} +4.69080 q^{49} +6.01774 q^{50} -0.659818 q^{52} -2.43546 q^{53} +12.8897 q^{55} -3.41918 q^{56} +1.15585 q^{58} +5.82474 q^{59} -0.592765 q^{61} -0.368522 q^{62} +1.00000 q^{64} -2.19013 q^{65} -1.28183 q^{67} +1.70659 q^{68} -11.3493 q^{70} +1.63956 q^{71} -0.306499 q^{73} +5.75882 q^{74} +2.86730 q^{76} -13.2776 q^{77} -4.61890 q^{79} +3.31930 q^{80} -6.68785 q^{82} +6.08908 q^{83} +5.66467 q^{85} +11.1569 q^{86} +3.88326 q^{88} -10.1345 q^{89} +2.25604 q^{91} +3.18162 q^{92} -11.2875 q^{94} +9.51741 q^{95} -8.75741 q^{97} +4.69080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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