LMFDB - Embedded newform 1287.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, 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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.89122$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46793 q^{2} +4.09069 q^{4} +0.776183 q^{5} +2.69175 q^{7} -5.15968 q^{8} +O(q^{10})$	\(q-2.46793 q^{2} +4.09069 q^{4} +0.776183 q^{5} +2.69175 q^{7} -5.15968 q^{8} -1.91557 q^{10} +1.00000 q^{11} +1.00000 q^{13} -6.64306 q^{14} +4.55237 q^{16} -2.62136 q^{17} +6.69175 q^{19} +3.17512 q^{20} -2.46793 q^{22} +8.02655 q^{23} -4.39754 q^{25} -2.46793 q^{26} +11.0111 q^{28} +8.09695 q^{29} -6.34106 q^{31} -0.915567 q^{32} +6.46933 q^{34} +2.08929 q^{35} -0.398941 q^{37} -16.5148 q^{38} -4.00486 q^{40} -6.87313 q^{41} +7.23786 q^{43} +4.09069 q^{44} -19.8090 q^{46} +4.53692 q^{47} +0.245516 q^{49} +10.8528 q^{50} +4.09069 q^{52} +2.75448 q^{53} +0.776183 q^{55} -13.8886 q^{56} -19.9827 q^{58} -10.8104 q^{59} -5.02796 q^{61} +15.6493 q^{62} -6.84517 q^{64} +0.776183 q^{65} -4.95756 q^{67} -10.7232 q^{68} -5.15623 q^{70} +5.96382 q^{71} +3.79648 q^{73} +0.984558 q^{74} +27.3739 q^{76} +2.69175 q^{77} +2.14564 q^{79} +3.53347 q^{80} +16.9624 q^{82} -3.73375 q^{83} -2.03465 q^{85} -17.8625 q^{86} -5.15968 q^{88} -2.37724 q^{89} +2.69175 q^{91} +32.8341 q^{92} -11.1968 q^{94} +5.19402 q^{95} -17.8355 q^{97} -0.605916 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7	$ 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7} - 2 q^{10} + 4 q^{11} + 4 q^{13} - 12 q^{14} + 12 q^{16} + 8 q^{17} + 18 q^{19} + 10 q^{20} + 2 q^{22} + 4 q^{25} + 2 q^{26} - 6 q^{28} + 10 q^{29} + 12 q^{31} + 2 q^{32} + 36 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} + 20 q^{40} - 2 q^{41} + 28 q^{43} + 8 q^{44} - 30 q^{46} - 6 q^{47} + 8 q^{49} + 36 q^{50} + 8 q^{52} + 4 q^{53} - 48 q^{56} - 6 q^{58} - 16 q^{59} - 10 q^{61} + 34 q^{62} - 12 q^{64} + 34 q^{68} - 58 q^{70} - 10 q^{71} - 6 q^{73} - 14 q^{74} + 26 q^{76} + 2 q^{77} - 8 q^{79} + 48 q^{80} + 12 q^{82} + 8 q^{83} - 18 q^{85} + 8 q^{86} - 6 q^{89} + 2 q^{91} + 28 q^{92} - 46 q^{94} - 22 q^{95} + 10 q^{97} + 34 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7 - 2 * q^10 + 4 * q^11 + 4 * q^13 - 12 * q^14 + 12 * q^16 + 8 * q^17 + 18 * q^19 + 10 * q^20 + 2 * q^22 + 4 * q^25 + 2 * q^26 - 6 * q^28 + 10 * q^29 + 12 * q^31 + 2 * q^32 + 36 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 + 20 * q^40 - 2 * q^41 + 28 * q^43 + 8 * q^44 - 30 * q^46 - 6 * q^47 + 8 * q^49 + 36 * q^50 + 8 * q^52 + 4 * q^53 - 48 * q^56 - 6 * q^58 - 16 * q^59 - 10 * q^61 + 34 * q^62 - 12 * q^64 + 34 * q^68 - 58 * q^70 - 10 * q^71 - 6 * q^73 - 14 * q^74 + 26 * q^76 + 2 * q^77 - 8 * q^79 + 48 * q^80 + 12 * q^82 + 8 * q^83 - 18 * q^85 + 8 * q^86 - 6 * q^89 + 2 * q^91 + 28 * q^92 - 46 * q^94 - 22 * q^95 + 10 * q^97 + 34 * 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 $

$2$	−2.46793	−1.74509	−0.872546	−	0.488532i	$-0.837532\pi$
$2$	−2.46793	−1.74509	−0.872546	+	0.488532i	$0.837532\pi$
$3$	0	0
$3$	0	0
$4$	4.09069	2.04535
$5$	0.776183	0.347120	0.173560	−	0.984823i	$-0.444473\pi$
$5$	0.776183	0.347120	0.173560	+	0.984823i	$0.444473\pi$
$6$	0	0
$7$	2.69175	1.01739	0.508693	−	0.860948i	$-0.330129\pi$
$7$	2.69175	1.01739	0.508693	+	0.860948i	$0.330129\pi$
$8$	−5.15968	−1.82422
$9$	0	0
$10$	−1.91557	−0.605755
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−6.64306	−1.77543
$15$	0	0
$16$	4.55237	1.13809
$17$	−2.62136	−0.635773	−0.317886	−	0.948129i	$-0.602973\pi$
$17$	−2.62136	−0.635773	−0.317886	+	0.948129i	$0.602973\pi$
$18$	0	0
$19$	6.69175	1.53519	0.767596	−	0.640934i	$-0.221453\pi$
$19$	6.69175	1.53519	0.767596	+	0.640934i	$0.221453\pi$
$20$	3.17512	0.709979
$21$	0	0
$22$	−2.46793	−0.526165
$23$	8.02655	1.67365	0.836826	−	0.547469i	$-0.184408\pi$
$23$	8.02655	1.67365	0.836826	+	0.547469i	$0.184408\pi$
$24$	0	0
$25$	−4.39754	−0.879508
$26$	−2.46793	−0.484001
$27$	0	0
$28$	11.0111	2.08090
$29$	8.09695	1.50357	0.751783	−	0.659411i	$-0.229194\pi$
$29$	8.09695	1.50357	0.751783	+	0.659411i	$0.229194\pi$
$30$	0	0
$31$	−6.34106	−1.13889	−0.569444	−	0.822030i	$-0.692842\pi$
$31$	−6.34106	−1.13889	−0.569444	+	0.822030i	$0.692842\pi$
$32$	−0.915567	−0.161851
$33$	0	0
$34$	6.46933	1.10948
$35$	2.08929	0.353154
$36$	0	0
$37$	−0.398941	−0.0655854	−0.0327927	−	0.999462i	$-0.510440\pi$
$37$	−0.398941	−0.0655854	−0.0327927	+	0.999462i	$0.510440\pi$
$38$	−16.5148	−2.67905
$39$	0	0
$40$	−4.00486	−0.633223
$41$	−6.87313	−1.07340	−0.536701	−	0.843772i	$-0.680330\pi$
$41$	−6.87313	−1.07340	−0.536701	+	0.843772i	$0.680330\pi$
$42$	0	0
$43$	7.23786	1.10376	0.551882	−	0.833923i	$-0.313910\pi$
$43$	7.23786	1.10376	0.551882	+	0.833923i	$0.313910\pi$
$44$	4.09069	0.616695
$45$	0	0
$46$	−19.8090	−2.92068
$47$	4.53692	0.661778	0.330889	−	0.943670i	$-0.392651\pi$
$47$	4.53692	0.661778	0.330889	+	0.943670i	$0.392651\pi$
$48$	0	0
$49$	0.245516	0.0350737
$50$	10.8528	1.53482
$51$	0	0
$52$	4.09069	0.567277
$53$	2.75448	0.378358	0.189179	−	0.981943i	$-0.439417\pi$
$53$	2.75448	0.378358	0.189179	+	0.981943i	$0.439417\pi$
$54$	0	0
$55$	0.776183	0.104660
$56$	−13.8886	−1.85594
$57$	0	0
$58$	−19.9827	−2.62386
$59$	−10.8104	−1.40739	−0.703697	−	0.710500i	$-0.748469\pi$
$59$	−10.8104	−1.40739	−0.703697	+	0.710500i	$0.748469\pi$
$60$	0	0
$61$	−5.02796	−0.643764	−0.321882	−	0.946780i	$-0.604315\pi$
$61$	−5.02796	−0.643764	−0.321882	+	0.946780i	$0.604315\pi$
$62$	15.6493	1.98746
$63$	0	0
$64$	−6.84517	−0.855647
$65$	0.776183	0.0962736
$66$	0	0
$67$	−4.95756	−0.605663	−0.302831	−	0.953044i	$-0.597932\pi$
$67$	−4.95756	−0.605663	−0.302831	+	0.953044i	$0.597932\pi$
$68$	−10.7232	−1.30037
$69$	0	0
$70$	−5.15623	−0.616287
$71$	5.96382	0.707775	0.353888	−	0.935288i	$-0.384860\pi$
$71$	5.96382	0.707775	0.353888	+	0.935288i	$0.384860\pi$
$72$	0	0
$73$	3.79648	0.444344	0.222172	−	0.975007i	$-0.428685\pi$
$73$	3.79648	0.444344	0.222172	+	0.975007i	$0.428685\pi$
$74$	0.984558	0.114453
$75$	0	0
$76$	27.3739	3.14000
$77$	2.69175	0.306753
$78$	0	0
$79$	2.14564	0.241403	0.120702	−	0.992689i	$-0.461486\pi$
$79$	2.14564	0.241403	0.120702	+	0.992689i	$0.461486\pi$
$80$	3.53347	0.395054
$81$	0	0
$82$	16.9624	1.87319
$83$	−3.73375	−0.409832	−0.204916	−	0.978780i	$-0.565692\pi$
$83$	−3.73375	−0.409832	−0.204916	+	0.978780i	$0.565692\pi$
$84$	0	0
$85$	−2.03465	−0.220689
$86$	−17.8625	−1.92617
$87$	0	0
$88$	−5.15968	−0.550024
$89$	−2.37724	−0.251987	−0.125994	−	0.992031i	$-0.540212\pi$
$89$	−2.37724	−0.251987	−0.125994	+	0.992031i	$0.540212\pi$
$90$	0	0
$91$	2.69175	0.282172
$92$	32.8341	3.42320
$93$	0	0
$94$	−11.1968	−1.15486
$95$	5.19402	0.532895
$96$	0	0
$97$	−17.8355	−1.81093	−0.905463	−	0.424426i	$-0.860476\pi$
$97$	−17.8355	−1.81093	−0.905463	+	0.424426i	$0.860476\pi$
$98$	−0.605916	−0.0612067
$99$	0	0
$100$	−17.9890	−1.79890
$101$	11.1096	1.10545	0.552723	−	0.833365i	$-0.313589\pi$
$101$	11.1096	1.10545	0.552723	+	0.833365i	$0.313589\pi$
$102$	0	0
$103$	10.1814	1.00320	0.501601	−	0.865099i	$-0.332745\pi$
$103$	10.1814	1.00320	0.501601	+	0.865099i	$0.332745\pi$
$104$	−5.15968	−0.505948
$105$	0	0
$106$	−6.79788	−0.660269
$107$	−3.06133	−0.295950	−0.147975	−	0.988991i	$-0.547276\pi$
$107$	−3.06133	−0.295950	−0.147975	+	0.988991i	$0.547276\pi$
$108$	0	0
$109$	13.1800	1.26241	0.631207	−	0.775615i	$-0.282560\pi$
$109$	13.1800	1.26241	0.631207	+	0.775615i	$0.282560\pi$
$110$	−1.91557	−0.182642
$111$	0	0
$112$	12.2538	1.15788
$113$	−4.50074	−0.423394	−0.211697	−	0.977335i	$-0.567899\pi$
$113$	−4.50074	−0.423394	−0.211697	+	0.977335i	$0.567899\pi$
$114$	0	0
$115$	6.23007	0.580957
$116$	33.1221	3.07531
$117$	0	0
$118$	26.6793	2.45603
$119$	−7.05604	−0.646826
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.4087	1.12343
$123$	0	0
$124$	−25.9393	−2.32942
$125$	−7.29421	−0.652414
$126$	0	0
$127$	19.9687	1.77193	0.885967	−	0.463749i	$-0.153496\pi$
$127$	19.9687	1.77193	0.885967	+	0.463749i	$0.153496\pi$
$128$	18.7246	1.65503
$129$	0	0
$130$	−1.91557	−0.168006
$131$	−4.92335	−0.430155	−0.215078	−	0.976597i	$-0.569000\pi$
$131$	−4.92335	−0.430155	−0.215078	+	0.976597i	$0.569000\pi$
$132$	0	0
$133$	18.0125	1.56188
$134$	12.2349	1.05694
$135$	0	0
$136$	13.5254	1.15979
$137$	13.1876	1.12670	0.563348	−	0.826220i	$-0.309513\pi$
$137$	13.1876	1.12670	0.563348	+	0.826220i	$0.309513\pi$
$138$	0	0
$139$	11.0203	0.934729	0.467365	−	0.884065i	$-0.345203\pi$
$139$	11.0203	0.934729	0.467365	+	0.884065i	$0.345203\pi$
$140$	8.54664	0.722323
$141$	0	0
$142$	−14.7183	−1.23513
$143$	1.00000	0.0836242
$144$	0	0
$145$	6.28471	0.521917
$146$	−9.36946	−0.775422
$147$	0	0
$148$	−1.63194	−0.134145
$149$	−8.25663	−0.676409	−0.338205	−	0.941073i	$-0.609820\pi$
$149$	−8.25663	−0.676409	−0.338205	+	0.941073i	$0.609820\pi$
$150$	0	0
$151$	−3.61510	−0.294193	−0.147096	−	0.989122i	$-0.546993\pi$
$151$	−3.61510	−0.294193	−0.147096	+	0.989122i	$0.546993\pi$
$152$	−34.5273	−2.80053
$153$	0	0
$154$	−6.64306	−0.535313
$155$	−4.92182	−0.395330
$156$	0	0
$157$	20.0672	1.60153	0.800766	−	0.598977i	$-0.204426\pi$
$157$	20.0672	1.60153	0.800766	+	0.598977i	$0.204426\pi$
$158$	−5.29530	−0.421271
$159$	0	0
$160$	−0.710647	−0.0561816
$161$	21.6055	1.70275
$162$	0	0
$163$	17.9827	1.40852	0.704258	−	0.709945i	$-0.251280\pi$
$163$	17.9827	1.40852	0.704258	+	0.709945i	$0.251280\pi$
$164$	−28.1158	−2.19548
$165$	0	0
$166$	9.21463	0.715194
$167$	−9.74626	−0.754188	−0.377094	−	0.926175i	$-0.623077\pi$
$167$	−9.74626	−0.754188	−0.377094	+	0.926175i	$0.623077\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	5.02138	0.385123
$171$	0	0
$172$	29.6078	2.25758
$173$	14.9841	1.13922	0.569611	−	0.821915i	$-0.307094\pi$
$173$	14.9841	1.13922	0.569611	+	0.821915i	$0.307094\pi$
$174$	0	0
$175$	−11.8371	−0.894799
$176$	4.55237	0.343147
$177$	0	0
$178$	5.86687	0.439741
$179$	14.5273	1.08582	0.542911	−	0.839790i	$-0.317322\pi$
$179$	14.5273	1.08582	0.542911	+	0.839790i	$0.317322\pi$
$180$	0	0
$181$	0.909310	0.0675885	0.0337942	−	0.999429i	$-0.489241\pi$
$181$	0.909310	0.0675885	0.0337942	+	0.999429i	$0.489241\pi$
$182$	−6.64306	−0.492416
$183$	0	0
$184$	−41.4145	−3.05312
$185$	−0.309651	−0.0227660
$186$	0	0
$187$	−2.62136	−0.191693
$188$	18.5592	1.35357
$189$	0	0
$190$	−12.8185	−0.929951
$191$	−12.4448	−0.900477	−0.450238	−	0.892908i	$-0.648661\pi$
$191$	−12.4448	−0.900477	−0.450238	+	0.892908i	$0.648661\pi$
$192$	0	0
$193$	18.3406	1.32019	0.660093	−	0.751184i	$-0.270517\pi$
$193$	18.3406	1.32019	0.660093	+	0.751184i	$0.270517\pi$
$194$	44.0169	3.16023
$195$	0	0
$196$	1.00433	0.0717377
$197$	7.96535	0.567508	0.283754	−	0.958897i	$-0.408420\pi$
$197$	7.96535	0.567508	0.283754	+	0.958897i	$0.408420\pi$
$198$	0	0
$199$	−14.6415	−1.03791	−0.518955	−	0.854801i	$-0.673679\pi$
$199$	−14.6415	−1.03791	−0.518955	+	0.854801i	$0.673679\pi$
$200$	22.6899	1.60442
$201$	0	0
$202$	−27.4177	−1.92910
$203$	21.7950	1.52971
$204$	0	0
$205$	−5.33481	−0.372599
$206$	−25.1270	−1.75068
$207$	0	0
$208$	4.55237	0.315650
$209$	6.69175	0.462878
$210$	0	0
$211$	14.9716	1.03069	0.515344	−	0.856983i	$-0.327664\pi$
$211$	14.9716	1.03069	0.515344	+	0.856983i	$0.327664\pi$
$212$	11.2677	0.773872
$213$	0	0
$214$	7.55517	0.516461
$215$	5.61790	0.383138
$216$	0	0
$217$	−17.0686	−1.15869
$218$	−32.5273	−2.20303
$219$	0	0
$220$	3.17512	0.214067
$221$	−2.62136	−0.176332
$222$	0	0
$223$	−26.2003	−1.75450	−0.877250	−	0.480033i	$-0.840625\pi$
$223$	−26.2003	−1.75450	−0.877250	+	0.480033i	$0.840625\pi$
$224$	−2.46448	−0.164665
$225$	0	0
$226$	11.1075	0.738862
$227$	−24.2441	−1.60914	−0.804569	−	0.593859i	$-0.797604\pi$
$227$	−24.2441	−1.60914	−0.804569	+	0.593859i	$0.797604\pi$
$228$	0	0
$229$	−6.43954	−0.425537	−0.212768	−	0.977103i	$-0.568248\pi$
$229$	−6.43954	−0.425537	−0.212768	+	0.977103i	$0.568248\pi$
$230$	−15.3754	−1.01382
$231$	0	0
$232$	−41.7777	−2.74284
$233$	24.0536	1.57580	0.787900	−	0.615803i	$-0.211168\pi$
$233$	24.0536	1.57580	0.787900	+	0.615803i	$0.211168\pi$
$234$	0	0
$235$	3.52148	0.229716
$236$	−44.2220	−2.87861
$237$	0	0
$238$	17.4138	1.12877
$239$	−5.01391	−0.324323	−0.162162	−	0.986764i	$-0.551847\pi$
$239$	−5.01391	−0.324323	−0.162162	+	0.986764i	$0.551847\pi$
$240$	0	0
$241$	−24.9890	−1.60968	−0.804841	−	0.593491i	$-0.797749\pi$
$241$	−24.9890	−1.60968	−0.804841	+	0.593491i	$0.797749\pi$
$242$	−2.46793	−0.158645
$243$	0	0
$244$	−20.5678	−1.31672
$245$	0.190565	0.0121748
$246$	0	0
$247$	6.69175	0.425786
$248$	32.7179	2.07759
$249$	0	0
$250$	18.0016	1.13852
$251$	−18.0406	−1.13871	−0.569356	−	0.822091i	$-0.692807\pi$
$251$	−18.0406	−1.13871	−0.569356	+	0.822091i	$0.692807\pi$
$252$	0	0
$253$	8.02655	0.504625
$254$	−49.2813	−3.09219
$255$	0	0
$256$	−32.5206	−2.03254
$257$	21.1298	1.31804	0.659019	−	0.752126i	$-0.270972\pi$
$257$	21.1298	1.31804	0.659019	+	0.752126i	$0.270972\pi$
$258$	0	0
$259$	−1.07385	−0.0667257
$260$	3.17512	0.196913
$261$	0	0
$262$	12.1505	0.750660
$263$	−11.8592	−0.731271	−0.365635	−	0.930758i	$-0.619148\pi$
$263$	−11.8592	−0.731271	−0.365635	+	0.930758i	$0.619148\pi$
$264$	0	0
$265$	2.13798	0.131335
$266$	−44.4537	−2.72563
$267$	0	0
$268$	−20.2799	−1.23879
$269$	0.322165	0.0196427	0.00982136	−	0.999952i	$-0.496874\pi$
$269$	0.322165	0.0196427	0.00982136	+	0.999952i	$0.496874\pi$
$270$	0	0
$271$	26.2944	1.59727	0.798636	−	0.601814i	$-0.205555\pi$
$271$	26.2944	1.59727	0.798636	+	0.601814i	$0.205555\pi$
$272$	−11.9334	−0.723567
$273$	0	0
$274$	−32.5462	−1.96619
$275$	−4.39754	−0.265182
$276$	0	0
$277$	−8.89247	−0.534297	−0.267148	−	0.963655i	$-0.586081\pi$
$277$	−8.89247	−0.534297	−0.267148	+	0.963655i	$0.586081\pi$
$278$	−27.1974	−1.63119
$279$	0	0
$280$	−10.7801	−0.644232
$281$	−23.5833	−1.40686	−0.703432	−	0.710763i	$-0.748350\pi$
$281$	−23.5833	−1.40686	−0.703432	+	0.710763i	$0.748350\pi$
$282$	0	0
$283$	8.13754	0.483727	0.241863	−	0.970310i	$-0.422241\pi$
$283$	8.13754	0.483727	0.241863	+	0.970310i	$0.422241\pi$
$284$	24.3961	1.44764
$285$	0	0
$286$	−2.46793	−0.145932
$287$	−18.5007	−1.09206
$288$	0	0
$289$	−10.1285	−0.595793
$290$	−15.5102	−0.910793
$291$	0	0
$292$	15.5302	0.908838
$293$	−19.8649	−1.16052	−0.580260	−	0.814431i	$-0.697049\pi$
$293$	−19.8649	−1.16052	−0.580260	+	0.814431i	$0.697049\pi$
$294$	0	0
$295$	−8.39084	−0.488534
$296$	2.05841	0.119642
$297$	0	0
$298$	20.3768	1.18040
$299$	8.02655	0.464188
$300$	0	0
$301$	19.4825	1.12295
$302$	8.92182	0.513393
$303$	0	0
$304$	30.4633	1.74719
$305$	−3.90261	−0.223463
$306$	0	0
$307$	−23.3305	−1.33154	−0.665770	−	0.746157i	$-0.731897\pi$
$307$	−23.3305	−1.33154	−0.665770	+	0.746157i	$0.731897\pi$
$308$	11.0111	0.627416
$309$	0	0
$310$	12.1467	0.689888
$311$	4.43359	0.251406	0.125703	−	0.992068i	$-0.459881\pi$
$311$	4.43359	0.251406	0.125703	+	0.992068i	$0.459881\pi$
$312$	0	0
$313$	−17.4129	−0.984233	−0.492116	−	0.870529i	$-0.663777\pi$
$313$	−17.4129	−0.984233	−0.492116	+	0.870529i	$0.663777\pi$
$314$	−49.5244	−2.79482
$315$	0	0
$316$	8.77715	0.493753
$317$	30.8752	1.73412	0.867062	−	0.498201i	$-0.166006\pi$
$317$	30.8752	1.73412	0.867062	+	0.498201i	$0.166006\pi$
$318$	0	0
$319$	8.09695	0.453342
$320$	−5.31311	−0.297012
$321$	0	0
$322$	−53.3209	−2.97145
$323$	−17.5415	−0.976033
$324$	0	0
$325$	−4.39754	−0.243932
$326$	−44.3801	−2.45799
$327$	0	0
$328$	35.4632	1.95813
$329$	12.2123	0.673284
$330$	0	0
$331$	6.72749	0.369776	0.184888	−	0.982760i	$-0.440808\pi$
$331$	6.72749	0.369776	0.184888	+	0.982760i	$0.440808\pi$
$332$	−15.2736	−0.838248
$333$	0	0
$334$	24.0531	1.31613
$335$	−3.84798	−0.210237
$336$	0	0
$337$	−5.06133	−0.275708	−0.137854	−	0.990453i	$-0.544021\pi$
$337$	−5.06133	−0.275708	−0.137854	+	0.990453i	$0.544021\pi$
$338$	−2.46793	−0.134238
$339$	0	0
$340$	−8.32313	−0.451385
$341$	−6.34106	−0.343388
$342$	0	0
$343$	−18.1814	−0.981702
$344$	−37.3450	−2.01351
$345$	0	0
$346$	−36.9798	−1.98805
$347$	4.34303	0.233146	0.116573	−	0.993182i	$-0.462809\pi$
$347$	4.34303	0.233146	0.116573	+	0.993182i	$0.462809\pi$
$348$	0	0
$349$	20.2779	1.08545	0.542725	−	0.839910i	$-0.317393\pi$
$349$	20.2779	1.08545	0.542725	+	0.839910i	$0.317393\pi$
$350$	29.2131	1.56151
$351$	0	0
$352$	−0.915567	−0.0487999
$353$	36.5630	1.94605	0.973027	−	0.230691i	$-0.0740986\pi$
$353$	36.5630	1.94605	0.973027	+	0.230691i	$0.0740986\pi$
$354$	0	0
$355$	4.62901	0.245683
$356$	−9.72456	−0.515401
$357$	0	0
$358$	−35.8524	−1.89486
$359$	27.8621	1.47051	0.735253	−	0.677793i	$-0.237063\pi$
$359$	27.8621	1.47051	0.735253	+	0.677793i	$0.237063\pi$
$360$	0	0
$361$	25.7795	1.35682
$362$	−2.24412	−0.117948
$363$	0	0
$364$	11.0111	0.577139
$365$	2.94676	0.154241
$366$	0	0
$367$	10.4476	0.545362	0.272681	−	0.962104i	$-0.412090\pi$
$367$	10.4476	0.545362	0.272681	+	0.962104i	$0.412090\pi$
$368$	36.5398	1.90477
$369$	0	0
$370$	0.764197	0.0397287
$371$	7.41438	0.384936
$372$	0	0
$373$	−17.8842	−0.926011	−0.463006	−	0.886355i	$-0.653229\pi$
$373$	−17.8842	−0.926011	−0.463006	+	0.886355i	$0.653229\pi$
$374$	6.46933	0.334521
$375$	0	0
$376$	−23.4091	−1.20723
$377$	8.09695	0.417014
$378$	0	0
$379$	−13.6200	−0.699610	−0.349805	−	0.936822i	$-0.613752\pi$
$379$	−13.6200	−0.699610	−0.349805	+	0.936822i	$0.613752\pi$
$380$	21.2471	1.08995
$381$	0	0
$382$	30.7130	1.57141
$383$	−4.62901	−0.236532	−0.118266	−	0.992982i	$-0.537733\pi$
$383$	−4.62901	−0.236532	−0.118266	+	0.992982i	$0.537733\pi$
$384$	0	0
$385$	2.08929	0.106480
$386$	−45.2634	−2.30385
$387$	0	0
$388$	−72.9597	−3.70397
$389$	−5.34290	−0.270896	−0.135448	−	0.990784i	$-0.543247\pi$
$389$	−5.34290	−0.270896	−0.135448	+	0.990784i	$0.543247\pi$
$390$	0	0
$391$	−21.0405	−1.06406
$392$	−1.26678	−0.0639822
$393$	0	0
$394$	−19.6579	−0.990353
$395$	1.66541	0.0837958
$396$	0	0
$397$	28.0700	1.40879	0.704397	−	0.709806i	$-0.251217\pi$
$397$	28.0700	1.40879	0.704397	+	0.709806i	$0.251217\pi$
$398$	36.1343	1.81125
$399$	0	0
$400$	−20.0192	−1.00096
$401$	−32.3942	−1.61769	−0.808844	−	0.588023i	$-0.799906\pi$
$401$	−32.3942	−1.61769	−0.808844	+	0.588023i	$0.799906\pi$
$402$	0	0
$403$	−6.34106	−0.315871
$404$	45.4459	2.26102
$405$	0	0
$406$	−53.7885	−2.66948
$407$	−0.398941	−0.0197748
$408$	0	0
$409$	−22.4380	−1.10949	−0.554744	−	0.832021i	$-0.687184\pi$
$409$	−22.4380	−1.10949	−0.554744	+	0.832021i	$0.687184\pi$
$410$	13.1659	0.650219
$411$	0	0
$412$	41.6489	2.05189
$413$	−29.0989	−1.43186
$414$	0	0
$415$	−2.89807	−0.142261
$416$	−0.915567	−0.0448894
$417$	0	0
$418$	−16.5148	−0.807765
$419$	−40.6113	−1.98399	−0.991996	−	0.126272i	$-0.959699\pi$
$419$	−40.6113	−1.98399	−0.991996	+	0.126272i	$0.959699\pi$
$420$	0	0
$421$	−2.67241	−0.130245	−0.0651227	−	0.997877i	$-0.520744\pi$
$421$	−2.67241	−0.130245	−0.0651227	+	0.997877i	$0.520744\pi$
$422$	−36.9489	−1.79864
$423$	0	0
$424$	−14.2123	−0.690209
$425$	11.5275	0.559167
$426$	0	0
$427$	−13.5340	−0.654956
$428$	−12.5230	−0.605321
$429$	0	0
$430$	−13.8646	−0.668610
$431$	25.9972	1.25224	0.626121	−	0.779726i	$-0.284642\pi$
$431$	25.9972	1.25224	0.626121	+	0.779726i	$0.284642\pi$
$432$	0	0
$433$	18.9765	0.911951	0.455975	−	0.889992i	$-0.349291\pi$
$433$	18.9765	0.911951	0.455975	+	0.889992i	$0.349291\pi$
$434$	42.1240	2.02202
$435$	0	0
$436$	53.9152	2.58207
$437$	53.7117	2.56938
$438$	0	0
$439$	16.8586	0.804619	0.402310	−	0.915504i	$-0.368208\pi$
$439$	16.8586	0.804619	0.402310	+	0.915504i	$0.368208\pi$
$440$	−4.00486	−0.190924
$441$	0	0
$442$	6.46933	0.307715
$443$	−34.5105	−1.63964	−0.819821	−	0.572620i	$-0.805927\pi$
$443$	−34.5105	−1.63964	−0.819821	+	0.572620i	$0.805927\pi$
$444$	0	0
$445$	−1.84517	−0.0874697
$446$	64.6605	3.06176
$447$	0	0
$448$	−18.4255	−0.870523
$449$	−5.12363	−0.241799	−0.120899	−	0.992665i	$-0.538578\pi$
$449$	−5.12363	−0.241799	−0.120899	+	0.992665i	$0.538578\pi$
$450$	0	0
$451$	−6.87313	−0.323643
$452$	−18.4111	−0.865988
$453$	0	0
$454$	59.8328	2.80809
$455$	2.08929	0.0979474
$456$	0	0
$457$	−19.9248	−0.932045	−0.466022	−	0.884773i	$-0.654313\pi$
$457$	−19.9248	−0.932045	−0.466022	+	0.884773i	$0.654313\pi$
$458$	15.8923	0.742600
$459$	0	0
$460$	25.4853	1.18826
$461$	10.4661	0.487454	0.243727	−	0.969844i	$-0.421630\pi$
$461$	10.4661	0.487454	0.243727	+	0.969844i	$0.421630\pi$
$462$	0	0
$463$	−36.5558	−1.69889	−0.849447	−	0.527675i	$-0.823064\pi$
$463$	−36.5558	−1.69889	−0.849447	+	0.527675i	$0.823064\pi$
$464$	36.8603	1.71119
$465$	0	0
$466$	−59.3625	−2.74992
$467$	2.43079	0.112484	0.0562418	−	0.998417i	$-0.482088\pi$
$467$	2.43079	0.112484	0.0562418	+	0.998417i	$0.482088\pi$
$468$	0	0
$469$	−13.3445	−0.616193
$470$	−8.69078	−0.400876
$471$	0	0
$472$	55.7782	2.56740
$473$	7.23786	0.332797
$474$	0	0
$475$	−29.4272	−1.35021
$476$	−28.8641	−1.32298
$477$	0	0
$478$	12.3740	0.565974
$479$	1.96535	0.0897990	0.0448995	−	0.998992i	$-0.485703\pi$
$479$	1.96535	0.0897990	0.0448995	+	0.998992i	$0.485703\pi$
$480$	0	0
$481$	−0.398941	−0.0181901
$482$	61.6711	2.80904
$483$	0	0
$484$	4.09069	0.185940
$485$	−13.8436	−0.628608
$486$	0	0
$487$	22.4863	1.01895	0.509475	−	0.860486i	$-0.329840\pi$
$487$	22.4863	1.01895	0.509475	+	0.860486i	$0.329840\pi$
$488$	25.9426	1.17437
$489$	0	0
$490$	−0.470301	−0.0212461
$491$	−17.2914	−0.780350	−0.390175	−	0.920741i	$-0.627586\pi$
$491$	−17.2914	−0.780350	−0.390175	+	0.920741i	$0.627586\pi$
$492$	0	0
$493$	−21.2250	−0.955926
$494$	−16.5148	−0.743035
$495$	0	0
$496$	−28.8668	−1.29616
$497$	16.0531	0.720080
$498$	0	0
$499$	3.71485	0.166299	0.0831497	−	0.996537i	$-0.473502\pi$
$499$	3.71485	0.166299	0.0831497	+	0.996537i	$0.473502\pi$
$500$	−29.8384	−1.33441
$501$	0	0
$502$	44.5230	1.98716
$503$	7.68064	0.342463	0.171231	−	0.985231i	$-0.445225\pi$
$503$	7.68064	0.342463	0.171231	+	0.985231i	$0.445225\pi$
$504$	0	0
$505$	8.62307	0.383722
$506$	−19.8090	−0.880617
$507$	0	0
$508$	81.6857	3.62422
$509$	−29.0035	−1.28556	−0.642778	−	0.766053i	$-0.722218\pi$
$509$	−29.0035	−1.28556	−0.642778	+	0.766053i	$0.722218\pi$
$510$	0	0
$511$	10.2192	0.452070
$512$	42.8095	1.89193
$513$	0	0
$514$	−52.1468	−2.30010
$515$	7.90261	0.348231
$516$	0	0
$517$	4.53692	0.199534
$518$	2.65018	0.116442
$519$	0	0
$520$	−4.00486	−0.175625
$521$	−32.5973	−1.42811	−0.714056	−	0.700089i	$-0.753144\pi$
$521$	−32.5973	−1.42811	−0.714056	+	0.700089i	$0.753144\pi$
$522$	0	0
$523$	−10.7304	−0.469207	−0.234603	−	0.972091i	$-0.575379\pi$
$523$	−10.7304	−0.469207	−0.234603	+	0.972091i	$0.575379\pi$
$524$	−20.1399	−0.879816
$525$	0	0
$526$	29.2677	1.27613
$527$	16.6222	0.724074
$528$	0	0
$529$	41.4256	1.80111
$530$	−5.27640	−0.229192
$531$	0	0
$532$	73.6836	3.19459
$533$	−6.87313	−0.297708
$534$	0	0
$535$	−2.37616	−0.102730
$536$	25.5794	1.10486
$537$	0	0
$538$	−0.795080	−0.0342784
$539$	0.245516	0.0105751
$540$	0	0
$541$	12.4910	0.537031	0.268516	−	0.963275i	$-0.413467\pi$
$541$	12.4910	0.537031	0.268516	+	0.963275i	$0.413467\pi$
$542$	−64.8929	−2.78739
$543$	0	0
$544$	2.40003	0.102900
$545$	10.2301	0.438208
$546$	0	0
$547$	14.2377	0.608761	0.304381	−	0.952550i	$-0.401550\pi$
$547$	14.2377	0.608761	0.304381	+	0.952550i	$0.401550\pi$
$548$	53.9465	2.30448
$549$	0	0
$550$	10.8528	0.462766
$551$	54.1827	2.30826
$552$	0	0
$553$	5.77553	0.245600
$554$	21.9460	0.932397
$555$	0	0
$556$	45.0806	1.91184
$557$	28.4633	1.20603	0.603014	−	0.797730i	$-0.293966\pi$
$557$	28.4633	1.20603	0.603014	+	0.797730i	$0.293966\pi$
$558$	0	0
$559$	7.23786	0.306129
$560$	9.51121	0.401922
$561$	0	0
$562$	58.2021	2.45511
$563$	29.8998	1.26013	0.630063	−	0.776544i	$-0.283029\pi$
$563$	29.8998	1.26013	0.630063	+	0.776544i	$0.283029\pi$
$564$	0	0
$565$	−3.49340	−0.146968
$566$	−20.0829	−0.844147
$567$	0	0
$568$	−30.7714	−1.29114
$569$	−21.5572	−0.903726	−0.451863	−	0.892087i	$-0.649240\pi$
$569$	−21.5572	−0.903726	−0.451863	+	0.892087i	$0.649240\pi$
$570$	0	0
$571$	−38.0830	−1.59372	−0.796862	−	0.604162i	$-0.793508\pi$
$571$	−38.0830	−1.59372	−0.796862	+	0.604162i	$0.793508\pi$
$572$	4.09069	0.171040
$573$	0	0
$574$	45.6586	1.90575
$575$	−35.2971	−1.47199
$576$	0	0
$577$	19.7463	0.822048	0.411024	−	0.911625i	$-0.365171\pi$
$577$	19.7463	0.822048	0.411024	+	0.911625i	$0.365171\pi$
$578$	24.9964	1.03971
$579$	0	0
$580$	25.7088	1.06750
$581$	−10.0503	−0.416957
$582$	0	0
$583$	2.75448	0.114079
$584$	−19.5886	−0.810583
$585$	0	0
$586$	49.0252	2.02521
$587$	5.48001	0.226184	0.113092	−	0.993585i	$-0.463925\pi$
$587$	5.48001	0.226184	0.113092	+	0.993585i	$0.463925\pi$
$588$	0	0
$589$	−42.4328	−1.74841
$590$	20.7080	0.852536
$591$	0	0
$592$	−1.81612	−0.0746422
$593$	−35.0670	−1.44003	−0.720015	−	0.693958i	$-0.755865\pi$
$593$	−35.0670	−1.44003	−0.720015	+	0.693958i	$0.755865\pi$
$594$	0	0
$595$	−5.47678	−0.224526
$596$	−33.7753	−1.38349
$597$	0	0
$598$	−19.8090	−0.810050
$599$	−17.3401	−0.708497	−0.354249	−	0.935151i	$-0.615263\pi$
$599$	−17.3401	−0.708497	−0.354249	+	0.935151i	$0.615263\pi$
$600$	0	0
$601$	16.0253	0.653685	0.326842	−	0.945079i	$-0.394015\pi$
$601$	16.0253	0.653685	0.326842	+	0.945079i	$0.394015\pi$
$602$	−48.0815	−1.95966
$603$	0	0
$604$	−14.7883	−0.601726
$605$	0.776183	0.0315563
$606$	0	0
$607$	−27.2673	−1.10675	−0.553373	−	0.832934i	$-0.686660\pi$
$607$	−27.2673	−1.10675	−0.553373	+	0.832934i	$0.686660\pi$
$608$	−6.12674	−0.248472
$609$	0	0
$610$	9.63138	0.389963
$611$	4.53692	0.183544
$612$	0	0
$613$	−7.92195	−0.319965	−0.159982	−	0.987120i	$-0.551144\pi$
$613$	−7.92195	−0.319965	−0.159982	+	0.987120i	$0.551144\pi$
$614$	57.5780	2.32366
$615$	0	0
$616$	−13.8886	−0.559586
$617$	28.8293	1.16062	0.580312	−	0.814394i	$-0.302931\pi$
$617$	28.8293	1.16062	0.580312	+	0.814394i	$0.302931\pi$
$618$	0	0
$619$	3.75794	0.151044	0.0755222	−	0.997144i	$-0.475938\pi$
$619$	3.75794	0.151044	0.0755222	+	0.997144i	$0.475938\pi$
$620$	−20.1337	−0.808587
$621$	0	0
$622$	−10.9418	−0.438727
$623$	−6.39894	−0.256368
$624$	0	0
$625$	16.3261	0.653042
$626$	42.9737	1.71758
$627$	0	0
$628$	82.0885	3.27569
$629$	1.04577	0.0416974
$630$	0	0
$631$	14.7525	0.587288	0.293644	−	0.955915i	$-0.405132\pi$
$631$	14.7525	0.587288	0.293644	+	0.955915i	$0.405132\pi$
$632$	−11.0708	−0.440374
$633$	0	0
$634$	−76.1979	−3.02620
$635$	15.4993	0.615073
$636$	0	0
$637$	0.245516	0.00972768
$638$	−19.9827	−0.791123
$639$	0	0
$640$	14.5337	0.574494
$641$	26.5133	1.04721	0.523605	−	0.851961i	$-0.324587\pi$
$641$	26.5133	1.04721	0.523605	+	0.851961i	$0.324587\pi$
$642$	0	0
$643$	−17.7371	−0.699482	−0.349741	−	0.936846i	$-0.613730\pi$
$643$	−17.7371	−0.699482	−0.349741	+	0.936846i	$0.613730\pi$
$644$	88.3813	3.48271
$645$	0	0
$646$	43.2912	1.70327
$647$	32.2751	1.26886	0.634432	−	0.772978i	$-0.281234\pi$
$647$	32.2751	1.26886	0.634432	+	0.772978i	$0.281234\pi$
$648$	0	0
$649$	−10.8104	−0.424345
$650$	10.8528	0.425683
$651$	0	0
$652$	73.5617	2.88090
$653$	3.25803	0.127497	0.0637483	−	0.997966i	$-0.479695\pi$
$653$	3.25803	0.127497	0.0637483	+	0.997966i	$0.479695\pi$
$654$	0	0
$655$	−3.82142	−0.149315
$656$	−31.2890	−1.22163
$657$	0	0
$658$	−30.1390	−1.17494
$659$	−39.0368	−1.52066	−0.760329	−	0.649539i	$-0.774962\pi$
$659$	−39.0368	−1.52066	−0.760329	+	0.649539i	$0.774962\pi$
$660$	0	0
$661$	−38.9194	−1.51379	−0.756895	−	0.653537i	$-0.773285\pi$
$661$	−38.9194	−1.51379	−0.756895	+	0.653537i	$0.773285\pi$
$662$	−16.6030	−0.645293
$663$	0	0
$664$	19.2649	0.747625
$665$	13.9810	0.542160
$666$	0	0
$667$	64.9906	2.51645
$668$	−39.8689	−1.54258
$669$	0	0
$670$	9.49654	0.366883
$671$	−5.02796	−0.194102
$672$	0	0
$673$	34.5466	1.33167	0.665837	−	0.746097i	$-0.268074\pi$
$673$	34.5466	1.33167	0.665837	+	0.746097i	$0.268074\pi$
$674$	12.4910	0.481137
$675$	0	0
$676$	4.09069	0.157334
$677$	−25.4362	−0.977591	−0.488796	−	0.872398i	$-0.662564\pi$
$677$	−25.4362	−0.977591	−0.488796	+	0.872398i	$0.662564\pi$
$678$	0	0
$679$	−48.0088	−1.84241
$680$	10.4982	0.402586
$681$	0	0
$682$	15.6493	0.599243
$683$	−9.81027	−0.375379	−0.187690	−	0.982228i	$-0.560100\pi$
$683$	−9.81027	−0.375379	−0.187690	+	0.982228i	$0.560100\pi$
$684$	0	0
$685$	10.2360	0.391098
$686$	44.8704	1.71316
$687$	0	0
$688$	32.9494	1.25618
$689$	2.75448	0.104937
$690$	0	0
$691$	−6.32133	−0.240475	−0.120237	−	0.992745i	$-0.538366\pi$
$691$	−6.32133	−0.240475	−0.120237	+	0.992745i	$0.538366\pi$
$692$	61.2954	2.33010
$693$	0	0
$694$	−10.7183	−0.406861
$695$	8.55377	0.324463
$696$	0	0
$697$	18.0169	0.682440
$698$	−50.0445	−1.89421
$699$	0	0
$700$	−48.4218	−1.83017
$701$	−30.4082	−1.14850	−0.574251	−	0.818679i	$-0.694707\pi$
$701$	−30.4082	−1.14850	−0.574251	+	0.818679i	$0.694707\pi$
$702$	0	0
$703$	−2.66961	−0.100686
$704$	−6.84517	−0.257987
$705$	0	0
$706$	−90.2351	−3.39604
$707$	29.9042	1.12466
$708$	0	0
$709$	−38.0014	−1.42717	−0.713586	−	0.700568i	$-0.752930\pi$
$709$	−38.0014	−1.42717	−0.713586	+	0.700568i	$0.752930\pi$
$710$	−11.4241	−0.428739
$711$	0	0
$712$	12.2658	0.459681
$713$	−50.8969	−1.90610
$714$	0	0
$715$	0.776183	0.0290276
$716$	59.4267	2.22088
$717$	0	0
$718$	−68.7618	−2.56617
$719$	−34.7670	−1.29659	−0.648295	−	0.761389i	$-0.724518\pi$
$719$	−34.7670	−1.29659	−0.648295	+	0.761389i	$0.724518\pi$
$720$	0	0
$721$	27.4057	1.02064
$722$	−63.6221	−2.36777
$723$	0	0
$724$	3.71970	0.138242
$725$	−35.6067	−1.32240
$726$	0	0
$727$	27.5746	1.02269	0.511343	−	0.859377i	$-0.329148\pi$
$727$	27.5746	1.02269	0.511343	+	0.859377i	$0.329148\pi$
$728$	−13.8886	−0.514745
$729$	0	0
$730$	−7.27241	−0.269164
$731$	−18.9730	−0.701742
$732$	0	0
$733$	20.5076	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$733$	20.5076	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$734$	−25.7841	−0.951707
$735$	0	0
$736$	−7.34885	−0.270882
$737$	−4.95756	−0.182614
$738$	0	0
$739$	−29.7241	−1.09342	−0.546710	−	0.837322i	$-0.684120\pi$
$739$	−29.7241	−1.09342	−0.546710	+	0.837322i	$0.684120\pi$
$740$	−1.26669	−0.0465643
$741$	0	0
$742$	−18.2982	−0.671748
$743$	−5.01986	−0.184161	−0.0920804	−	0.995752i	$-0.529352\pi$
$743$	−5.01986	−0.184161	−0.0920804	+	0.995752i	$0.529352\pi$
$744$	0	0
$745$	−6.40865	−0.234795
$746$	44.1371	1.61597
$747$	0	0
$748$	−10.7232	−0.392078
$749$	−8.24035	−0.301096
$750$	0	0
$751$	−17.7309	−0.647011	−0.323506	−	0.946226i	$-0.604861\pi$
$751$	−17.7309	−0.647011	−0.323506	+	0.946226i	$0.604861\pi$
$752$	20.6537	0.753164
$753$	0	0
$754$	−19.9827	−0.727728
$755$	−2.80598	−0.102120
$756$	0	0
$757$	−32.8369	−1.19348	−0.596740	−	0.802435i	$-0.703537\pi$
$757$	−32.8369	−1.19348	−0.596740	+	0.802435i	$0.703537\pi$
$758$	33.6131	1.22088
$759$	0	0
$760$	−26.7995	−0.972120
$761$	−40.7015	−1.47543	−0.737713	−	0.675114i	$-0.764094\pi$
$761$	−40.7015	−1.47543	−0.737713	+	0.675114i	$0.764094\pi$
$762$	0	0
$763$	35.4772	1.28436
$764$	−50.9080	−1.84179
$765$	0	0
$766$	11.4241	0.412769
$767$	−10.8104	−0.390341
$768$	0	0
$769$	−31.1712	−1.12406	−0.562032	−	0.827116i	$-0.689980\pi$
$769$	−31.1712	−1.12406	−0.562032	+	0.827116i	$0.689980\pi$
$770$	−5.15623	−0.185817
$771$	0	0
$772$	75.0258	2.70024
$773$	−9.63553	−0.346566	−0.173283	−	0.984872i	$-0.555437\pi$
$773$	−9.63553	−0.346566	−0.173283	+	0.984872i	$0.555437\pi$
$774$	0	0
$775$	27.8851	1.00166
$776$	92.0258	3.30353
$777$	0	0
$778$	13.1859	0.472738
$779$	−45.9933	−1.64788
$780$	0	0
$781$	5.96382	0.213402
$782$	51.9265	1.85689
$783$	0	0
$784$	1.11768	0.0399170
$785$	15.5758	0.555923
$786$	0	0
$787$	−29.7241	−1.05955	−0.529775	−	0.848138i	$-0.677724\pi$
$787$	−29.7241	−1.05955	−0.529775	+	0.848138i	$0.677724\pi$
$788$	32.5838	1.16075
$789$	0	0
$790$	−4.11012	−0.146231
$791$	−12.1149	−0.430755
$792$	0	0
$793$	−5.02796	−0.178548
$794$	−69.2750	−2.45848
$795$	0	0
$796$	−59.8940	−2.12288
$797$	−4.82571	−0.170935	−0.0854677	−	0.996341i	$-0.527238\pi$
$797$	−4.82571	−0.170935	−0.0854677	+	0.996341i	$0.527238\pi$
$798$	0	0
$799$	−11.8929	−0.420741
$800$	4.02624	0.142349
$801$	0	0
$802$	79.9466	2.82301
$803$	3.79648	0.133975
$804$	0	0
$805$	16.7698	0.591058
$806$	15.6493	0.551224
$807$	0	0
$808$	−57.3219	−2.01658
$809$	31.6284	1.11200	0.555998	−	0.831183i	$-0.312336\pi$
$809$	31.6284	1.11200	0.555998	+	0.831183i	$0.312336\pi$
$810$	0	0
$811$	−30.2655	−1.06276	−0.531382	−	0.847132i	$-0.678327\pi$
$811$	−30.2655	−1.06276	−0.531382	+	0.847132i	$0.678327\pi$
$812$	89.1564	3.12878
$813$	0	0
$814$	0.984558	0.0345088
$815$	13.9579	0.488923
$816$	0	0
$817$	48.4339	1.69449
$818$	55.3755	1.93616
$819$	0	0
$820$	−21.8230	−0.762093
$821$	18.9833	0.662521	0.331261	−	0.943539i	$-0.392526\pi$
$821$	18.9833	0.662521	0.331261	+	0.943539i	$0.392526\pi$
$822$	0	0
$823$	−24.6234	−0.858318	−0.429159	−	0.903229i	$-0.641190\pi$
$823$	−24.6234	−0.858318	−0.429159	+	0.903229i	$0.641190\pi$
$824$	−52.5327	−1.83006
$825$	0	0
$826$	71.8141	2.49873
$827$	−23.7784	−0.826855	−0.413427	−	0.910537i	$-0.635668\pi$
$827$	−23.7784	−0.826855	−0.413427	+	0.910537i	$0.635668\pi$
$828$	0	0
$829$	44.5973	1.54893	0.774463	−	0.632619i	$-0.218020\pi$
$829$	44.5973	1.54893	0.774463	+	0.632619i	$0.218020\pi$
$830$	7.15224	0.248258
$831$	0	0
$832$	−6.84517	−0.237314
$833$	−0.643584	−0.0222989
$834$	0	0
$835$	−7.56488	−0.261793
$836$	27.3739	0.946745
$837$	0	0
$838$	100.226	3.46225
$839$	22.1090	0.763288	0.381644	−	0.924309i	$-0.375358\pi$
$839$	22.1090	0.763288	0.381644	+	0.924309i	$0.375358\pi$
$840$	0	0
$841$	36.5606	1.26071
$842$	6.59533	0.227290
$843$	0	0
$844$	61.2442	2.10811
$845$	0.776183	0.0267015
$846$	0	0
$847$	2.69175	0.0924896
$848$	12.5394	0.430605
$849$	0	0
$850$	−28.4492	−0.975798
$851$	−3.20212	−0.109767
$852$	0	0
$853$	−17.8689	−0.611820	−0.305910	−	0.952060i	$-0.598961\pi$
$853$	−17.8689	−0.611820	−0.305910	+	0.952060i	$0.598961\pi$
$854$	33.4010	1.14296
$855$	0	0
$856$	15.7955	0.539879
$857$	−11.6521	−0.398029	−0.199014	−	0.979997i	$-0.563774\pi$
$857$	−11.6521	−0.398029	−0.199014	+	0.979997i	$0.563774\pi$
$858$	0	0
$859$	52.8576	1.80348	0.901740	−	0.432279i	$-0.142290\pi$
$859$	52.8576	1.80348	0.901740	+	0.432279i	$0.142290\pi$
$860$	22.9811	0.783649
$861$	0	0
$862$	−64.1593	−2.18528
$863$	−51.1483	−1.74111	−0.870554	−	0.492072i	$-0.836240\pi$
$863$	−51.1483	−1.74111	−0.870554	+	0.492072i	$0.836240\pi$
$864$	0	0
$865$	11.6304	0.395446
$866$	−46.8326	−1.59144
$867$	0	0
$868$	−69.8222	−2.36992
$869$	2.14564	0.0727859
$870$	0	0
$871$	−4.95756	−0.167981
$872$	−68.0045	−2.30292
$873$	0	0
$874$	−132.557	−4.48380
$875$	−19.6342	−0.663757
$876$	0	0
$877$	−28.0125	−0.945915	−0.472958	−	0.881085i	$-0.656814\pi$
$877$	−28.0125	−0.945915	−0.472958	+	0.881085i	$0.656814\pi$
$878$	−41.6060	−1.40413
$879$	0	0
$880$	3.53347	0.119113
$881$	−16.2095	−0.546111	−0.273055	−	0.961998i	$-0.588034\pi$
$881$	−16.2095	−0.546111	−0.273055	+	0.961998i	$0.588034\pi$
$882$	0	0
$883$	−50.4100	−1.69643	−0.848216	−	0.529650i	$-0.822323\pi$
$883$	−50.4100	−1.69643	−0.848216	+	0.529650i	$0.822323\pi$
$884$	−10.7232	−0.360659
$885$	0	0
$886$	85.1695	2.86132
$887$	−49.4717	−1.66110	−0.830548	−	0.556947i	$-0.811973\pi$
$887$	−49.4717	−1.66110	−0.830548	+	0.556947i	$0.811973\pi$
$888$	0	0
$889$	53.7507	1.80274
$890$	4.55377	0.152643
$891$	0	0
$892$	−107.177	−3.58856
$893$	30.3600	1.01596
$894$	0	0
$895$	11.2758	0.376910
$896$	50.4018	1.68381
$897$	0	0
$898$	12.6448	0.421961
$899$	−51.3432	−1.71239
$900$	0	0
$901$	−7.22049	−0.240549
$902$	16.9624	0.564787
$903$	0	0
$904$	23.2224	0.772366
$905$	0.705791	0.0234613
$906$	0	0
$907$	−59.3623	−1.97109	−0.985547	−	0.169403i	$-0.945816\pi$
$907$	−59.3623	−1.97109	−0.985547	+	0.169403i	$0.945816\pi$
$908$	−99.1752	−3.29124
$909$	0	0
$910$	−5.15623	−0.170927
$911$	23.7337	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$911$	23.7337	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$912$	0	0
$913$	−3.73375	−0.123569
$914$	49.1732	1.62650
$915$	0	0
$916$	−26.3421	−0.870369
$917$	−13.2524	−0.437634
$918$	0	0
$919$	37.1293	1.22478	0.612392	−	0.790555i	$-0.290208\pi$
$919$	37.1293	1.22478	0.612392	+	0.790555i	$0.290208\pi$
$920$	−32.1452	−1.05980
$921$	0	0
$922$	−25.8296	−0.850653
$923$	5.96382	0.196302
$924$	0	0
$925$	1.75436	0.0576829
$926$	90.2173	2.96472
$927$	0	0
$928$	−7.41330	−0.243353
$929$	3.10364	0.101827	0.0509136	−	0.998703i	$-0.483787\pi$
$929$	3.10364	0.101827	0.0509136	+	0.998703i	$0.483787\pi$
$930$	0	0
$931$	1.64293	0.0538448
$932$	98.3956	3.22306
$933$	0	0
$934$	−5.99903	−0.196294
$935$	−2.03465	−0.0665403
$936$	0	0
$937$	23.1072	0.754880	0.377440	−	0.926034i	$-0.376804\pi$
$937$	23.1072	0.754880	0.377440	+	0.926034i	$0.376804\pi$
$938$	32.9334	1.07531
$939$	0	0
$940$	14.4053	0.469849
$941$	−1.36416	−0.0444704	−0.0222352	−	0.999753i	$-0.507078\pi$
$941$	−1.36416	−0.0444704	−0.0222352	+	0.999753i	$0.507078\pi$
$942$	0	0
$943$	−55.1676	−1.79650
$944$	−49.2129	−1.60174
$945$	0	0
$946$	−17.8625	−0.580761
$947$	18.6209	0.605098	0.302549	−	0.953134i	$-0.402162\pi$
$947$	18.6209	0.605098	0.302549	+	0.953134i	$0.402162\pi$
$948$	0	0
$949$	3.79648	0.123239
$950$	72.6244	2.35625
$951$	0	0
$952$	36.4069	1.17995
$953$	−19.4237	−0.629194	−0.314597	−	0.949225i	$-0.601869\pi$
$953$	−19.4237	−0.629194	−0.314597	+	0.949225i	$0.601869\pi$
$954$	0	0
$955$	−9.65947	−0.312573
$956$	−20.5104	−0.663353
$957$	0	0
$958$	−4.85034	−0.156708
$959$	35.4978	1.14628
$960$	0	0
$961$	9.20907	0.297067
$962$	0.984558	0.0317434
$963$	0	0
$964$	−102.222	−3.29235
$965$	14.2357	0.458263
$966$	0	0
$967$	−10.2413	−0.329338	−0.164669	−	0.986349i	$-0.552656\pi$
$967$	−10.2413	−0.329338	−0.164669	+	0.986349i	$0.552656\pi$
$968$	−5.15968	−0.165838
$969$	0	0
$970$	34.1652	1.09698
$971$	−8.71822	−0.279781	−0.139890	−	0.990167i	$-0.544675\pi$
$971$	−8.71822	−0.279781	−0.139890	+	0.990167i	$0.544675\pi$
$972$	0	0
$973$	29.6639	0.950980
$974$	−55.4946	−1.77816
$975$	0	0
$976$	−22.8891	−0.732662
$977$	56.9127	1.82080	0.910400	−	0.413730i	$-0.135774\pi$
$977$	56.9127	1.82080	0.910400	+	0.413730i	$0.135774\pi$
$978$	0	0
$979$	−2.37724	−0.0759770
$980$	0.779542	0.0249016
$981$	0	0
$982$	42.6740	1.36178
$983$	−25.1270	−0.801425	−0.400713	−	0.916204i	$-0.631237\pi$
$983$	−25.1270	−0.801425	−0.400713	+	0.916204i	$0.631237\pi$
$984$	0	0
$985$	6.18257	0.196993
$986$	52.3818	1.66818
$987$	0	0
$988$	27.3739	0.870879
$989$	58.0951	1.84732
$990$	0	0
$991$	−12.8268	−0.407458	−0.203729	−	0.979027i	$-0.565306\pi$
$991$	−12.8268	−0.407458	−0.203729	+	0.979027i	$0.565306\pi$
$992$	5.80567	0.184330
$993$	0	0
$994$	−39.6180	−1.25661
$995$	−11.3645	−0.360279
$996$	0	0
$997$	2.62596	0.0831650	0.0415825	−	0.999135i	$-0.486760\pi$
$997$	2.62596	0.0831650	0.0415825	+	0.999135i	$0.486760\pi$
$998$	−9.16799	−0.290208
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.m.1.1		4
3.2	odd	2		429.2.a.h.1.4	✓	4
12.11	even	2		6864.2.a.bz.1.2		4
33.32	even	2		4719.2.a.z.1.1		4
39.38	odd	2		5577.2.a.m.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.h.1.4	✓	4	3.2	odd	2
1287.2.a.m.1.1		4	1.1	even	1	trivial
4719.2.a.z.1.1		4	33.32	even	2
5577.2.a.m.1.1		4	39.38	odd	2
6864.2.a.bz.1.2		4	12.11	even	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46793 q^{2} +4.09069 q^{4} +0.776183 q^{5} +2.69175 q^{7} -5.15968 q^{8} +O(q^{10})\)	\(q-2.46793 q^{2} +4.09069 q^{4} +0.776183 q^{5} +2.69175 q^{7} -5.15968 q^{8} -1.91557 q^{10} +1.00000 q^{11} +1.00000 q^{13} -6.64306 q^{14} +4.55237 q^{16} -2.62136 q^{17} +6.69175 q^{19} +3.17512 q^{20} -2.46793 q^{22} +8.02655 q^{23} -4.39754 q^{25} -2.46793 q^{26} +11.0111 q^{28} +8.09695 q^{29} -6.34106 q^{31} -0.915567 q^{32} +6.46933 q^{34} +2.08929 q^{35} -0.398941 q^{37} -16.5148 q^{38} -4.00486 q^{40} -6.87313 q^{41} +7.23786 q^{43} +4.09069 q^{44} -19.8090 q^{46} +4.53692 q^{47} +0.245516 q^{49} +10.8528 q^{50} +4.09069 q^{52} +2.75448 q^{53} +0.776183 q^{55} -13.8886 q^{56} -19.9827 q^{58} -10.8104 q^{59} -5.02796 q^{61} +15.6493 q^{62} -6.84517 q^{64} +0.776183 q^{65} -4.95756 q^{67} -10.7232 q^{68} -5.15623 q^{70} +5.96382 q^{71} +3.79648 q^{73} +0.984558 q^{74} +27.3739 q^{76} +2.69175 q^{77} +2.14564 q^{79} +3.53347 q^{80} +16.9624 q^{82} -3.73375 q^{83} -2.03465 q^{85} -17.8625 q^{86} -5.15968 q^{88} -2.37724 q^{89} +2.69175 q^{91} +32.8341 q^{92} -11.1968 q^{94} +5.19402 q^{95} -17.8355 q^{97} -0.605916 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7	\( 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7} - 2 q^{10} + 4 q^{11} + 4 q^{13} - 12 q^{14} + 12 q^{16} + 8 q^{17} + 18 q^{19} + 10 q^{20} + 2 q^{22} + 4 q^{25} + 2 q^{26} - 6 q^{28} + 10 q^{29} + 12 q^{31} + 2 q^{32} + 36 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} + 20 q^{40} - 2 q^{41} + 28 q^{43} + 8 q^{44} - 30 q^{46} - 6 q^{47} + 8 q^{49} + 36 q^{50} + 8 q^{52} + 4 q^{53} - 48 q^{56} - 6 q^{58} - 16 q^{59} - 10 q^{61} + 34 q^{62} - 12 q^{64} + 34 q^{68} - 58 q^{70} - 10 q^{71} - 6 q^{73} - 14 q^{74} + 26 q^{76} + 2 q^{77} - 8 q^{79} + 48 q^{80} + 12 q^{82} + 8 q^{83} - 18 q^{85} + 8 q^{86} - 6 q^{89} + 2 q^{91} + 28 q^{92} - 46 q^{94} - 22 q^{95} + 10 q^{97} + 34 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7 - 2 * q^10 + 4 * q^11 + 4 * q^13 - 12 * q^14 + 12 * q^16 + 8 * q^17 + 18 * q^19 + 10 * q^20 + 2 * q^22 + 4 * q^25 + 2 * q^26 - 6 * q^28 + 10 * q^29 + 12 * q^31 + 2 * q^32 + 36 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 + 20 * q^40 - 2 * q^41 + 28 * q^43 + 8 * q^44 - 30 * q^46 - 6 * q^47 + 8 * q^49 + 36 * q^50 + 8 * q^52 + 4 * q^53 - 48 * q^56 - 6 * q^58 - 16 * q^59 - 10 * q^61 + 34 * q^62 - 12 * q^64 + 34 * q^68 - 58 * q^70 - 10 * q^71 - 6 * q^73 - 14 * q^74 + 26 * q^76 + 2 * q^77 - 8 * q^79 + 48 * q^80 + 12 * q^82 + 8 * q^83 - 18 * q^85 + 8 * q^86 - 6 * q^89 + 2 * q^91 + 28 * q^92 - 46 * q^94 - 22 * q^95 + 10 * q^97 + 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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