LMFDB - Embedded newform 2842.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2842,2,Mod(1,2842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2842, 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("2842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2842 = 2 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2842.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:	$22.6934842544$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.52756992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 13x^{4} + 18x^{3} + 22x^{2} - 20x - 7 $ x^6 - 2x^5 - 13x^4 + 18x^3 + 22x^2 - 20*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.87472$ of defining polynomial
Character	$\chi$	$=$	2842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +4.31913 q^{5} +1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +4.31913 q^{5} +1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -4.31913 q^{10} +1.18716 q^{11} -1.41421 q^{12} -4.31913 q^{13} -6.10817 q^{15} +1.00000 q^{16} +4.05444 q^{17} +1.00000 q^{18} -5.54515 q^{19} +4.31913 q^{20} -1.18716 q^{22} -1.07899 q^{23} +1.41421 q^{24} +13.6549 q^{25} +4.31913 q^{26} +5.65685 q^{27} -1.00000 q^{29} +6.10817 q^{30} -9.59959 q^{31} -1.00000 q^{32} -1.67890 q^{33} -4.05444 q^{34} -1.00000 q^{36} -11.2953 q^{37} +5.54515 q^{38} +6.10817 q^{39} -4.31913 q^{40} -0.0764866 q^{41} -11.8420 q^{43} +1.18716 q^{44} -4.31913 q^{45} +1.07899 q^{46} -1.86709 q^{47} -1.41421 q^{48} -13.6549 q^{50} -5.73385 q^{51} -4.31913 q^{52} -3.84202 q^{53} -5.65685 q^{54} +5.12749 q^{55} +7.84202 q^{57} +1.00000 q^{58} -1.22602 q^{59} -6.10817 q^{60} -9.78779 q^{61} +9.59959 q^{62} +1.00000 q^{64} -18.6549 q^{65} +1.67890 q^{66} -4.00000 q^{67} +4.05444 q^{68} +1.52592 q^{69} +8.76303 q^{71} +1.00000 q^{72} +5.35695 q^{73} +11.2953 q^{74} -19.3109 q^{75} -5.54515 q^{76} -6.10817 q^{78} -3.73385 q^{79} +4.31913 q^{80} -5.00000 q^{81} +0.0764866 q^{82} +0.452878 q^{83} +17.5117 q^{85} +11.8420 q^{86} +1.41421 q^{87} -1.18716 q^{88} +14.3716 q^{89} +4.31913 q^{90} -1.07899 q^{92} +13.5759 q^{93} +1.86709 q^{94} -23.9502 q^{95} +1.41421 q^{96} +9.71130 q^{97} -1.18716 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 - 6 * q^9	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9} - 4 q^{11} + 6 q^{16} + 6 q^{18} + 4 q^{22} - 32 q^{23} + 42 q^{25} - 6 q^{29} - 6 q^{32} - 6 q^{36} - 20 q^{37} - 20 q^{43} - 4 q^{44} + 32 q^{46} - 42 q^{50} - 20 q^{51} + 28 q^{53} - 4 q^{57} + 6 q^{58} + 6 q^{64} - 72 q^{65} - 24 q^{67} - 24 q^{71} + 6 q^{72} + 20 q^{74} - 8 q^{79} - 30 q^{81} - 16 q^{85} + 20 q^{86} + 4 q^{88} - 32 q^{92} + 16 q^{93} - 56 q^{95} + 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 - 6 * q^9 - 4 * q^11 + 6 * q^16 + 6 * q^18 + 4 * q^22 - 32 * q^23 + 42 * q^25 - 6 * q^29 - 6 * q^32 - 6 * q^36 - 20 * q^37 - 20 * q^43 - 4 * q^44 + 32 * q^46 - 42 * q^50 - 20 * q^51 + 28 * q^53 - 4 * q^57 + 6 * q^58 + 6 * q^64 - 72 * q^65 - 24 * q^67 - 24 * q^71 + 6 * q^72 + 20 * q^74 - 8 * q^79 - 30 * q^81 - 16 * q^85 + 20 * q^86 + 4 * q^88 - 32 * q^92 + 16 * q^93 - 56 * q^95 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	1.00000	0.500000
$5$	4.31913	1.93157	0.965786	−	0.259339i	$-0.0835048\pi$
$5$	4.31913	1.93157	0.965786	+	0.259339i	$0.0835048\pi$
$6$	1.41421	0.577350
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	−4.31913	−1.36583
$11$	1.18716	0.357942	0.178971	−	0.983854i	$-0.442723\pi$
$11$	1.18716	0.357942	0.178971	+	0.983854i	$0.442723\pi$
$12$	−1.41421	−0.408248
$13$	−4.31913	−1.19791	−0.598955	−	0.800783i	$-0.704417\pi$
$13$	−4.31913	−1.19791	−0.598955	+	0.800783i	$0.704417\pi$
$14$	0	0
$15$	−6.10817	−1.57712
$16$	1.00000	0.250000
$17$	4.05444	0.983347	0.491674	−	0.870780i	$-0.336385\pi$
$17$	4.05444	0.983347	0.491674	+	0.870780i	$0.336385\pi$
$18$	1.00000	0.235702
$19$	−5.54515	−1.27214	−0.636072	−	0.771630i	$-0.719442\pi$
$19$	−5.54515	−1.27214	−0.636072	+	0.771630i	$0.719442\pi$
$20$	4.31913	0.965786
$21$	0	0
$22$	−1.18716	−0.253103
$23$	−1.07899	−0.224985	−0.112493	−	0.993653i	$-0.535883\pi$
$23$	−1.07899	−0.224985	−0.112493	+	0.993653i	$0.535883\pi$
$24$	1.41421	0.288675
$25$	13.6549	2.73097
$26$	4.31913	0.847051
$27$	5.65685	1.08866
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	6.10817	1.11519
$31$	−9.59959	−1.72414	−0.862069	−	0.506792i	$-0.830831\pi$
$31$	−9.59959	−1.72414	−0.862069	+	0.506792i	$0.830831\pi$
$32$	−1.00000	−0.176777
$33$	−1.67890	−0.292258
$34$	−4.05444	−0.695332
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−11.2953	−1.85694	−0.928470	−	0.371407i	$-0.878875\pi$
$37$	−11.2953	−1.85694	−0.928470	+	0.371407i	$0.878875\pi$
$38$	5.54515	0.899541
$39$	6.10817	0.978090
$40$	−4.31913	−0.682914
$41$	−0.0764866	−0.0119452	−0.00597260	−	0.999982i	$-0.501901\pi$
$41$	−0.0764866	−0.0119452	−0.00597260	+	0.999982i	$0.501901\pi$
$42$	0	0
$43$	−11.8420	−1.80589	−0.902946	−	0.429755i	$-0.858600\pi$
$43$	−11.8420	−1.80589	−0.902946	+	0.429755i	$0.858600\pi$
$44$	1.18716	0.178971
$45$	−4.31913	−0.643857
$46$	1.07899	0.159088
$47$	−1.86709	−0.272343	−0.136172	−	0.990685i	$-0.543480\pi$
$47$	−1.86709	−0.272343	−0.136172	+	0.990685i	$0.543480\pi$
$48$	−1.41421	−0.204124
$49$	0	0
$50$	−13.6549	−1.93109
$51$	−5.73385	−0.802900
$52$	−4.31913	−0.598955
$53$	−3.84202	−0.527742	−0.263871	−	0.964558i	$-0.584999\pi$
$53$	−3.84202	−0.527742	−0.263871	+	0.964558i	$0.584999\pi$
$54$	−5.65685	−0.769800
$55$	5.12749	0.691391
$56$	0	0
$57$	7.84202	1.03870
$58$	1.00000	0.131306
$59$	−1.22602	−0.159614	−0.0798070	−	0.996810i	$-0.525430\pi$
$59$	−1.22602	−0.159614	−0.0798070	+	0.996810i	$0.525430\pi$
$60$	−6.10817	−0.788561
$61$	−9.78779	−1.25320	−0.626599	−	0.779342i	$-0.715554\pi$
$61$	−9.78779	−1.25320	−0.626599	+	0.779342i	$0.715554\pi$
$62$	9.59959	1.21915
$63$	0	0
$64$	1.00000	0.125000
$65$	−18.6549	−2.31385
$66$	1.67890	0.206658
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.05444	0.491674
$69$	1.52592	0.183700
$70$	0	0
$71$	8.76303	1.03998	0.519990	−	0.854172i	$-0.325936\pi$
$71$	8.76303	1.03998	0.519990	+	0.854172i	$0.325936\pi$
$72$	1.00000	0.117851
$73$	5.35695	0.626984	0.313492	−	0.949591i	$-0.398501\pi$
$73$	5.35695	0.626984	0.313492	+	0.949591i	$0.398501\pi$
$74$	11.2953	1.31306
$75$	−19.3109	−2.22983
$76$	−5.54515	−0.636072
$77$	0	0
$78$	−6.10817	−0.691614
$79$	−3.73385	−0.420091	−0.210046	−	0.977692i	$-0.567361\pi$
$79$	−3.73385	−0.420091	−0.210046	+	0.977692i	$0.567361\pi$
$80$	4.31913	0.482893
$81$	−5.00000	−0.555556
$82$	0.0764866	0.00844654
$83$	0.452878	0.0497098	0.0248549	−	0.999691i	$-0.492088\pi$
$83$	0.452878	0.0497098	0.0248549	+	0.999691i	$0.492088\pi$
$84$	0	0
$85$	17.5117	1.89941
$86$	11.8420	1.27696
$87$	1.41421	0.151620
$88$	−1.18716	−0.126552
$89$	14.3716	1.52339	0.761693	−	0.647938i	$-0.224368\pi$
$89$	14.3716	1.52339	0.761693	+	0.647938i	$0.224368\pi$
$90$	4.31913	0.455276
$91$	0	0
$92$	−1.07899	−0.112493
$93$	13.5759	1.40775
$94$	1.86709	0.192576
$95$	−23.9502	−2.45724
$96$	1.41421	0.144338
$97$	9.71130	0.986033	0.493017	−	0.870020i	$-0.335894\pi$
$97$	9.71130	0.986033	0.493017	+	0.870020i	$0.335894\pi$
$98$	0	0
$99$	−1.18716	−0.119314
$100$	13.6549	1.36549
$101$	15.4446	1.53680	0.768400	−	0.639970i	$-0.221053\pi$
$101$	15.4446	1.53680	0.768400	+	0.639970i	$0.221053\pi$
$102$	5.73385	0.567736
$103$	2.45204	0.241606	0.120803	−	0.992676i	$-0.461453\pi$
$103$	2.45204	0.241606	0.120803	+	0.992676i	$0.461453\pi$
$104$	4.31913	0.423525
$105$	0	0
$106$	3.84202	0.373170
$107$	−6.37432	−0.616229	−0.308114	−	0.951349i	$-0.599698\pi$
$107$	−6.37432	−0.616229	−0.308114	+	0.951349i	$0.599698\pi$
$108$	5.65685	0.544331
$109$	−3.84202	−0.367999	−0.183999	−	0.982926i	$-0.558904\pi$
$109$	−3.84202	−0.367999	−0.183999	+	0.982926i	$0.558904\pi$
$110$	−5.12749	−0.488887
$111$	15.9740	1.51619
$112$	0	0
$113$	−1.89183	−0.177969	−0.0889843	−	0.996033i	$-0.528362\pi$
$113$	−1.89183	−0.177969	−0.0889843	+	0.996033i	$0.528362\pi$
$114$	−7.84202	−0.734472
$115$	−4.66030	−0.434575
$116$	−1.00000	−0.0928477
$117$	4.31913	0.399303
$118$	1.22602	0.112864
$119$	0	0
$120$	6.10817	0.557597
$121$	−9.59065	−0.871878
$122$	9.78779	0.886144
$123$	0.108168	0.00975322
$124$	−9.59959	−0.862069
$125$	37.3814	3.34350
$126$	0	0
$127$	−9.57587	−0.849721	−0.424861	−	0.905259i	$-0.639677\pi$
$127$	−9.57587	−0.849721	−0.424861	+	0.905259i	$0.639677\pi$
$128$	−1.00000	−0.0883883
$129$	16.7471	1.47450
$130$	18.6549	1.63614
$131$	−5.76856	−0.504002	−0.252001	−	0.967727i	$-0.581089\pi$
$131$	−5.76856	−0.504002	−0.252001	+	0.967727i	$0.581089\pi$
$132$	−1.67890	−0.146129
$133$	0	0
$134$	4.00000	0.345547
$135$	24.4327	2.10283
$136$	−4.05444	−0.347666
$137$	−6.64047	−0.567333	−0.283667	−	0.958923i	$-0.591551\pi$
$137$	−6.64047	−0.567333	−0.283667	+	0.958923i	$0.591551\pi$
$138$	−1.52592	−0.129895
$139$	−17.2000	−1.45889	−0.729443	−	0.684041i	$-0.760221\pi$
$139$	−17.2000	−1.45889	−0.729443	+	0.684041i	$0.760221\pi$
$140$	0	0
$141$	2.64047	0.222367
$142$	−8.76303	−0.735377
$143$	−5.12749	−0.428782
$144$	−1.00000	−0.0833333
$145$	−4.31913	−0.358684
$146$	−5.35695	−0.443344
$147$	0	0
$148$	−11.2953	−0.928470
$149$	−19.5759	−1.60372	−0.801859	−	0.597513i	$-0.796155\pi$
$149$	−19.5759	−1.60372	−0.801859	+	0.597513i	$0.796155\pi$
$150$	19.3109	1.57673
$151$	4.54669	0.370005	0.185002	−	0.982738i	$-0.440771\pi$
$151$	4.54669	0.370005	0.185002	+	0.982738i	$0.440771\pi$
$152$	5.54515	0.449771
$153$	−4.05444	−0.327782
$154$	0	0
$155$	−41.4619	−3.33030
$156$	6.10817	0.489045
$157$	14.8245	1.18312	0.591561	−	0.806260i	$-0.298512\pi$
$157$	14.8245	1.18312	0.591561	+	0.806260i	$0.298512\pi$
$158$	3.73385	0.297049
$159$	5.43344	0.430900
$160$	−4.31913	−0.341457
$161$	0	0
$162$	5.00000	0.392837
$163$	−23.0292	−1.80378	−0.901892	−	0.431961i	$-0.857822\pi$
$163$	−23.0292	−1.80378	−0.901892	+	0.431961i	$0.857822\pi$
$164$	−0.0764866	−0.00597260
$165$	−7.25137	−0.564518
$166$	−0.452878	−0.0351502
$167$	4.35435	0.336950	0.168475	−	0.985706i	$-0.446116\pi$
$167$	4.35435	0.336950	0.168475	+	0.985706i	$0.446116\pi$
$168$	0	0
$169$	5.65486	0.434989
$170$	−17.5117	−1.34308
$171$	5.54515	0.424048
$172$	−11.8420	−0.902946
$173$	−12.5810	−0.956515	−0.478257	−	0.878220i	$-0.658731\pi$
$173$	−12.5810	−0.956515	−0.478257	+	0.878220i	$0.658731\pi$
$174$	−1.41421	−0.107211
$175$	0	0
$176$	1.18716	0.0894855
$177$	1.73385	0.130324
$178$	−14.3716	−1.07720
$179$	23.6840	1.77023	0.885114	−	0.465374i	$-0.154080\pi$
$179$	23.6840	1.77023	0.885114	+	0.465374i	$0.154080\pi$
$180$	−4.31913	−0.321929
$181$	−1.71412	−0.127409	−0.0637047	−	0.997969i	$-0.520292\pi$
$181$	−1.71412	−0.127409	−0.0637047	+	0.997969i	$0.520292\pi$
$182$	0	0
$183$	13.8420	1.02323
$184$	1.07899	0.0795442
$185$	−48.7860	−3.58681
$186$	−13.5759	−0.995431
$187$	4.81327	0.351981
$188$	−1.86709	−0.136172
$189$	0	0
$190$	23.9502	1.73753
$191$	11.2016	0.810516	0.405258	−	0.914202i	$-0.367182\pi$
$191$	11.2016	0.810516	0.405258	+	0.914202i	$0.367182\pi$
$192$	−1.41421	−0.102062
$193$	−7.73385	−0.556695	−0.278347	−	0.960480i	$-0.589787\pi$
$193$	−7.73385	−0.556695	−0.278347	+	0.960480i	$0.589787\pi$
$194$	−9.71130	−0.697231
$195$	26.3820	1.88925
$196$	0	0
$197$	−1.73385	−0.123532	−0.0617659	−	0.998091i	$-0.519673\pi$
$197$	−1.73385	−0.123532	−0.0617659	+	0.998091i	$0.519673\pi$
$198$	1.18716	0.0843677
$199$	26.1585	1.85433	0.927165	−	0.374654i	$-0.122238\pi$
$199$	26.1585	1.85433	0.927165	+	0.374654i	$0.122238\pi$
$200$	−13.6549	−0.965544
$201$	5.65685	0.399004
$202$	−15.4446	−1.08668
$203$	0	0
$204$	−5.73385	−0.401450
$205$	−0.330355	−0.0230730
$206$	−2.45204	−0.170841
$207$	1.07899	0.0749950
$208$	−4.31913	−0.299478
$209$	−6.58297	−0.455353
$210$	0	0
$211$	−6.15798	−0.423933	−0.211966	−	0.977277i	$-0.567987\pi$
$211$	−6.15798	−0.423933	−0.211966	+	0.977277i	$0.567987\pi$
$212$	−3.84202	−0.263871
$213$	−12.3928	−0.849140
$214$	6.37432	0.435739
$215$	−51.1472	−3.48821
$216$	−5.65685	−0.384900
$217$	0	0
$218$	3.84202	0.260214
$219$	−7.57587	−0.511930
$220$	5.12749	0.345695
$221$	−17.5117	−1.17796
$222$	−15.9740	−1.07211
$223$	−24.0829	−1.61271	−0.806355	−	0.591432i	$-0.798563\pi$
$223$	−24.0829	−1.61271	−0.806355	+	0.591432i	$0.798563\pi$
$224$	0	0
$225$	−13.6549	−0.910324
$226$	1.89183	0.125843
$227$	2.52852	0.167824	0.0839120	−	0.996473i	$-0.473259\pi$
$227$	2.52852	0.167824	0.0839120	+	0.996473i	$0.473259\pi$
$228$	7.84202	0.519350
$229$	9.94076	0.656904	0.328452	−	0.944521i	$-0.393473\pi$
$229$	9.94076	0.656904	0.328452	+	0.944521i	$0.393473\pi$
$230$	4.66030	0.307291
$231$	0	0
$232$	1.00000	0.0656532
$233$	12.2163	0.800319	0.400159	−	0.916446i	$-0.368955\pi$
$233$	12.2163	0.800319	0.400159	+	0.916446i	$0.368955\pi$
$234$	−4.31913	−0.282350
$235$	−8.06421	−0.526051
$236$	−1.22602	−0.0798070
$237$	5.28046	0.343003
$238$	0	0
$239$	24.4327	1.58042	0.790209	−	0.612837i	$-0.209972\pi$
$239$	24.4327	1.58042	0.790209	+	0.612837i	$0.209972\pi$
$240$	−6.10817	−0.394281
$241$	17.5412	1.12993	0.564964	−	0.825116i	$-0.308890\pi$
$241$	17.5412	1.12993	0.564964	+	0.825116i	$0.308890\pi$
$242$	9.59065	0.616511
$243$	−9.89949	−0.635053
$244$	−9.78779	−0.626599
$245$	0	0
$246$	−0.108168	−0.00689657
$247$	23.9502	1.52391
$248$	9.59959	0.609575
$249$	−0.640466	−0.0405879
$250$	−37.3814	−2.36421
$251$	−10.8053	−0.682021	−0.341011	−	0.940059i	$-0.610769\pi$
$251$	−10.8053	−0.682021	−0.341011	+	0.940059i	$0.610769\pi$
$252$	0	0
$253$	−1.28093	−0.0805315
$254$	9.57587	0.600844
$255$	−24.7652	−1.55086
$256$	1.00000	0.0625000
$257$	21.3662	1.33279	0.666393	−	0.745601i	$-0.267837\pi$
$257$	21.3662	1.33279	0.666393	+	0.745601i	$0.267837\pi$
$258$	−16.7471	−1.04263
$259$	0	0
$260$	−18.6549	−1.15693
$261$	1.00000	0.0618984
$262$	5.76856	0.356383
$263$	−4.04981	−0.249722	−0.124861	−	0.992174i	$-0.539849\pi$
$263$	−4.04981	−0.249722	−0.124861	+	0.992174i	$0.539849\pi$
$264$	1.67890	0.103329
$265$	−16.5942	−1.01937
$266$	0	0
$267$	−20.3245	−1.24384
$268$	−4.00000	−0.244339
$269$	2.82843	0.172452	0.0862261	−	0.996276i	$-0.472519\pi$
$269$	2.82843	0.172452	0.0862261	+	0.996276i	$0.472519\pi$
$270$	−24.4327	−1.48693
$271$	6.99458	0.424891	0.212445	−	0.977173i	$-0.431857\pi$
$271$	6.99458	0.424891	0.212445	+	0.977173i	$0.431857\pi$
$272$	4.05444	0.245837
$273$	0	0
$274$	6.64047	0.401165
$275$	16.2105	0.977529
$276$	1.52592	0.0918498
$277$	5.95019	0.357512	0.178756	−	0.983893i	$-0.442793\pi$
$277$	5.95019	0.357512	0.178756	+	0.983893i	$0.442793\pi$
$278$	17.2000	1.03159
$279$	9.59959	0.574712
$280$	0	0
$281$	−21.7778	−1.29916	−0.649578	−	0.760295i	$-0.725054\pi$
$281$	−21.7778	−1.29916	−0.649578	+	0.760295i	$0.725054\pi$
$282$	−2.64047	−0.157237
$283$	−12.5397	−0.745409	−0.372705	−	0.927950i	$-0.621570\pi$
$283$	−12.5397	−0.745409	−0.372705	+	0.927950i	$0.621570\pi$
$284$	8.76303	0.519990
$285$	33.8707	2.00633
$286$	5.12749	0.303195
$287$	0	0
$288$	1.00000	0.0589256
$289$	−0.561476	−0.0330280
$290$	4.31913	0.253628
$291$	−13.7339	−0.805093
$292$	5.35695	0.313492
$293$	−17.3470	−1.01342	−0.506710	−	0.862117i	$-0.669138\pi$
$293$	−17.3470	−1.01342	−0.506710	+	0.862117i	$0.669138\pi$
$294$	0	0
$295$	−5.29533	−0.308306
$296$	11.2953	0.656528
$297$	6.71558	0.389678
$298$	19.5759	1.13400
$299$	4.66030	0.269512
$300$	−19.3109	−1.11491
$301$	0	0
$302$	−4.54669	−0.261633
$303$	−21.8420	−1.25479
$304$	−5.54515	−0.318036
$305$	−42.2747	−2.42064
$306$	4.05444	0.231777
$307$	−27.5728	−1.57366	−0.786830	−	0.617170i	$-0.788279\pi$
$307$	−27.5728	−1.57366	−0.786830	+	0.617170i	$0.788279\pi$
$308$	0	0
$309$	−3.46770	−0.197271
$310$	41.4619	2.35487
$311$	−33.2857	−1.88746	−0.943730	−	0.330716i	$-0.892710\pi$
$311$	−33.2857	−1.88746	−0.943730	+	0.330716i	$0.892710\pi$
$312$	−6.10817	−0.345807
$313$	−11.3550	−0.641821	−0.320911	−	0.947110i	$-0.603989\pi$
$313$	−11.3550	−0.641821	−0.320911	+	0.947110i	$0.603989\pi$
$314$	−14.8245	−0.836593
$315$	0	0
$316$	−3.73385	−0.210046
$317$	2.21634	0.124482	0.0622409	−	0.998061i	$-0.480175\pi$
$317$	2.21634	0.124482	0.0622409	+	0.998061i	$0.480175\pi$
$318$	−5.43344	−0.304692
$319$	−1.18716	−0.0664681
$320$	4.31913	0.241447
$321$	9.01465	0.503148
$322$	0	0
$323$	−22.4825	−1.25096
$324$	−5.00000	−0.277778
$325$	−58.9771	−3.27146
$326$	23.0292	1.27547
$327$	5.43344	0.300470
$328$	0.0764866	0.00422327
$329$	0	0
$330$	7.25137	0.399175
$331$	−29.9004	−1.64347	−0.821737	−	0.569867i	$-0.806995\pi$
$331$	−29.9004	−1.64347	−0.821737	+	0.569867i	$0.806995\pi$
$332$	0.452878	0.0248549
$333$	11.2953	0.618980
$334$	−4.35435	−0.238259
$335$	−17.2765	−0.943917
$336$	0	0
$337$	−24.0584	−1.31054	−0.655271	−	0.755394i	$-0.727446\pi$
$337$	−24.0584	−1.31054	−0.655271	+	0.755394i	$0.727446\pi$
$338$	−5.65486	−0.307584
$339$	2.67545	0.145311
$340$	17.5117	0.949703
$341$	−11.3962	−0.617141
$342$	−5.54515	−0.299847
$343$	0	0
$344$	11.8420	0.638479
$345$	6.59065	0.354829
$346$	12.5810	0.676358
$347$	−6.26615	−0.336385	−0.168192	−	0.985754i	$-0.553793\pi$
$347$	−6.26615	−0.336385	−0.168192	+	0.985754i	$0.553793\pi$
$348$	1.41421	0.0758098
$349$	32.0036	1.71311	0.856556	−	0.516053i	$-0.172599\pi$
$349$	32.0036	1.71311	0.856556	+	0.516053i	$0.172599\pi$
$350$	0	0
$351$	−24.4327	−1.30412
$352$	−1.18716	−0.0632758
$353$	5.69812	0.303280	0.151640	−	0.988436i	$-0.451545\pi$
$353$	5.69812	0.303280	0.151640	+	0.988436i	$0.451545\pi$
$354$	−1.73385	−0.0921531
$355$	37.8486	2.00880
$356$	14.3716	0.761693
$357$	0	0
$358$	−23.6840	−1.25174
$359$	10.1082	0.533489	0.266744	−	0.963767i	$-0.414052\pi$
$359$	10.1082	0.533489	0.266744	+	0.963767i	$0.414052\pi$
$360$	4.31913	0.227638
$361$	11.7486	0.618349
$362$	1.71412	0.0900921
$363$	13.5632	0.711885
$364$	0	0
$365$	23.1373	1.21106
$366$	−13.8420	−0.723534
$367$	−9.97598	−0.520742	−0.260371	−	0.965509i	$-0.583845\pi$
$367$	−9.97598	−0.520742	−0.260371	+	0.965509i	$0.583845\pi$
$368$	−1.07899	−0.0562463
$369$	0.0764866	0.00398173
$370$	48.7860	2.53626
$371$	0	0
$372$	13.5759	0.703876
$373$	−6.48249	−0.335650	−0.167825	−	0.985817i	$-0.553674\pi$
$373$	−6.48249	−0.335650	−0.167825	+	0.985817i	$0.553674\pi$
$374$	−4.81327	−0.248888
$375$	−52.8653	−2.72995
$376$	1.86709	0.0962879
$377$	4.31913	0.222446
$378$	0	0
$379$	−3.09338	−0.158896	−0.0794482	−	0.996839i	$-0.525316\pi$
$379$	−3.09338	−0.158896	−0.0794482	+	0.996839i	$0.525316\pi$
$380$	−23.9502	−1.22862
$381$	13.5423	0.693794
$382$	−11.2016	−0.573121
$383$	−1.30250	−0.0665549	−0.0332774	−	0.999446i	$-0.510594\pi$
$383$	−1.30250	−0.0665549	−0.0332774	+	0.999446i	$0.510594\pi$
$384$	1.41421	0.0721688
$385$	0	0
$386$	7.73385	0.393643
$387$	11.8420	0.601964
$388$	9.71130	0.493017
$389$	3.07899	0.156111	0.0780555	−	0.996949i	$-0.475129\pi$
$389$	3.07899	0.156111	0.0780555	+	0.996949i	$0.475129\pi$
$390$	−26.3820	−1.33590
$391$	−4.37471	−0.221238
$392$	0	0
$393$	8.15798	0.411516
$394$	1.73385	0.0873502
$395$	−16.1270	−0.811436
$396$	−1.18716	−0.0596570
$397$	4.54255	0.227984	0.113992	−	0.993482i	$-0.463636\pi$
$397$	4.54255	0.227984	0.113992	+	0.993482i	$0.463636\pi$
$398$	−26.1585	−1.31121
$399$	0	0
$400$	13.6549	0.682743
$401$	−14.2163	−0.709930	−0.354965	−	0.934880i	$-0.615507\pi$
$401$	−14.2163	−0.709930	−0.354965	+	0.934880i	$0.615507\pi$
$402$	−5.65685	−0.282138
$403$	41.4619	2.06536
$404$	15.4446	0.768400
$405$	−21.5956	−1.07310
$406$	0	0
$407$	−13.4093	−0.664677
$408$	5.73385	0.283868
$409$	29.5928	1.46327	0.731635	−	0.681696i	$-0.238757\pi$
$409$	29.5928	1.46327	0.731635	+	0.681696i	$0.238757\pi$
$410$	0.330355	0.0163151
$411$	9.39104	0.463226
$412$	2.45204	0.120803
$413$	0	0
$414$	−1.07899	−0.0530295
$415$	1.95604	0.0960181
$416$	4.31913	0.211763
$417$	24.3245	1.19118
$418$	6.58297	0.321983
$419$	35.4731	1.73297	0.866487	−	0.499200i	$-0.166373\pi$
$419$	35.4731	1.73297	0.866487	+	0.499200i	$0.166373\pi$
$420$	0	0
$421$	21.6696	1.05611	0.528057	−	0.849209i	$-0.322921\pi$
$421$	21.6696	1.05611	0.528057	+	0.849209i	$0.322921\pi$
$422$	6.15798	0.299766
$423$	1.86709	0.0907811
$424$	3.84202	0.186585
$425$	55.3629	2.68549
$426$	12.3928	0.600433
$427$	0	0
$428$	−6.37432	−0.308114
$429$	7.25137	0.350099
$430$	51.1472	2.46654
$431$	−17.8420	−0.859420	−0.429710	−	0.902967i	$-0.641384\pi$
$431$	−17.8420	−0.859420	−0.429710	+	0.902967i	$0.641384\pi$
$432$	5.65685	0.272166
$433$	−14.5950	−0.701392	−0.350696	−	0.936489i	$-0.614055\pi$
$433$	−14.5950	−0.701392	−0.350696	+	0.936489i	$0.614055\pi$
$434$	0	0
$435$	6.10817	0.292864
$436$	−3.84202	−0.183999
$437$	5.98316	0.286213
$438$	7.57587	0.361989
$439$	−12.3928	−0.591476	−0.295738	−	0.955269i	$-0.595566\pi$
$439$	−12.3928	−0.591476	−0.295738	+	0.955269i	$0.595566\pi$
$440$	−5.12749	−0.244443
$441$	0	0
$442$	17.5117	0.832945
$443$	−17.7778	−0.844649	−0.422325	−	0.906445i	$-0.638786\pi$
$443$	−17.7778	−0.844649	−0.422325	+	0.906445i	$0.638786\pi$
$444$	15.9740	0.758093
$445$	62.0728	2.94253
$446$	24.0829	1.14036
$447$	27.6845	1.30943
$448$	0	0
$449$	−38.4327	−1.81375	−0.906875	−	0.421400i	$-0.861539\pi$
$449$	−38.4327	−1.81375	−0.906875	+	0.421400i	$0.861539\pi$
$450$	13.6549	0.643696
$451$	−0.0908018	−0.00427569
$452$	−1.89183	−0.0889843
$453$	−6.42999	−0.302107
$454$	−2.52852	−0.118669
$455$	0	0
$456$	−7.84202	−0.367236
$457$	−5.24552	−0.245375	−0.122687	−	0.992445i	$-0.539151\pi$
$457$	−5.24552	−0.245375	−0.122687	+	0.992445i	$0.539151\pi$
$458$	−9.94076	−0.464501
$459$	22.9354	1.07053
$460$	−4.66030	−0.217287
$461$	−11.0903	−0.516526	−0.258263	−	0.966075i	$-0.583150\pi$
$461$	−11.0903	−0.516526	−0.258263	+	0.966075i	$0.583150\pi$
$462$	0	0
$463$	27.1373	1.26118	0.630590	−	0.776116i	$-0.282813\pi$
$463$	27.1373	1.26118	0.630590	+	0.776116i	$0.282813\pi$
$464$	−1.00000	−0.0464238
$465$	58.6359	2.71918
$466$	−12.2163	−0.565911
$467$	−9.65572	−0.446813	−0.223407	−	0.974725i	$-0.571718\pi$
$467$	−9.65572	−0.446813	−0.223407	+	0.974725i	$0.571718\pi$
$468$	4.31913	0.199652
$469$	0	0
$470$	8.06421	0.371974
$471$	−20.9650	−0.966015
$472$	1.22602	0.0564320
$473$	−14.0584	−0.646404
$474$	−5.28046	−0.242540
$475$	−75.7182	−3.47419
$476$	0	0
$477$	3.84202	0.175914
$478$	−24.4327	−1.11752
$479$	−37.8134	−1.72774	−0.863870	−	0.503716i	$-0.831966\pi$
$479$	−37.8134	−1.72774	−0.863870	+	0.503716i	$0.831966\pi$
$480$	6.10817	0.278798
$481$	48.7860	2.22445
$482$	−17.5412	−0.798979
$483$	0	0
$484$	−9.59065	−0.435939
$485$	41.9443	1.90459
$486$	9.89949	0.449050
$487$	−17.3097	−0.784378	−0.392189	−	0.919885i	$-0.628282\pi$
$487$	−17.3097	−0.784378	−0.392189	+	0.919885i	$0.628282\pi$
$488$	9.78779	0.443072
$489$	32.5682	1.47278
$490$	0	0
$491$	40.1167	1.81044	0.905221	−	0.424941i	$-0.139705\pi$
$491$	40.1167	1.81044	0.905221	+	0.424941i	$0.139705\pi$
$492$	0.108168	0.00487661
$493$	−4.05444	−0.182603
$494$	−23.9502	−1.07757
$495$	−5.12749	−0.230464
$496$	−9.59959	−0.431034
$497$	0	0
$498$	0.640466	0.0287000
$499$	−15.4677	−0.692429	−0.346215	−	0.938155i	$-0.612533\pi$
$499$	−15.4677	−0.692429	−0.346215	+	0.938155i	$0.612533\pi$
$500$	37.3814	1.67175
$501$	−6.15798	−0.275118
$502$	10.8053	0.482262
$503$	−7.67692	−0.342297	−0.171148	−	0.985245i	$-0.554748\pi$
$503$	−7.67692	−0.342297	−0.171148	+	0.985245i	$0.554748\pi$
$504$	0	0
$505$	66.7074	2.96844
$506$	1.28093	0.0569444
$507$	−7.99718	−0.355167
$508$	−9.57587	−0.424861
$509$	5.07191	0.224808	0.112404	−	0.993663i	$-0.464145\pi$
$509$	5.07191	0.224808	0.112404	+	0.993663i	$0.464145\pi$
$510$	24.7652	1.09662
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−31.3681	−1.38493
$514$	−21.3662	−0.942422
$515$	10.5907	0.466680
$516$	16.7471	0.737252
$517$	−2.21653	−0.0974831
$518$	0	0
$519$	17.7922	0.780991
$520$	18.6549	0.818070
$521$	−17.9176	−0.784984	−0.392492	−	0.919756i	$-0.628387\pi$
$521$	−17.9176	−0.784984	−0.392492	+	0.919756i	$0.628387\pi$
$522$	−1.00000	−0.0437688
$523$	19.8755	0.869094	0.434547	−	0.900649i	$-0.356909\pi$
$523$	19.8755	0.869094	0.434547	+	0.900649i	$0.356909\pi$
$524$	−5.76856	−0.252001
$525$	0	0
$526$	4.04981	0.176580
$527$	−38.9210	−1.69543
$528$	−1.67890	−0.0730646
$529$	−21.8358	−0.949382
$530$	16.5942	0.720805
$531$	1.22602	0.0532046
$532$	0	0
$533$	0.330355	0.0143093
$534$	20.3245	0.879527
$535$	−27.5315	−1.19029
$536$	4.00000	0.172774
$537$	−33.4943	−1.44539
$538$	−2.82843	−0.121942
$539$	0	0
$540$	24.4327	1.05141
$541$	5.12295	0.220253	0.110126	−	0.993918i	$-0.464874\pi$
$541$	5.12295	0.220253	0.110126	+	0.993918i	$0.464874\pi$
$542$	−6.99458	−0.300443
$543$	2.42413	0.104029
$544$	−4.05444	−0.173833
$545$	−16.5942	−0.710816
$546$	0	0
$547$	9.20155	0.393430	0.196715	−	0.980461i	$-0.436973\pi$
$547$	9.20155	0.393430	0.196715	+	0.980461i	$0.436973\pi$
$548$	−6.64047	−0.283667
$549$	9.78779	0.417733
$550$	−16.2105	−0.691217
$551$	5.54515	0.236231
$552$	−1.52592	−0.0649476
$553$	0	0
$554$	−5.95019	−0.252799
$555$	68.9938	2.92862
$556$	−17.2000	−0.729443
$557$	20.9066	0.885842	0.442921	−	0.896561i	$-0.353942\pi$
$557$	20.9066	0.885842	0.442921	+	0.896561i	$0.353942\pi$
$558$	−9.59959	−0.406383
$559$	51.1472	2.16330
$560$	0	0
$561$	−6.80699	−0.287391
$562$	21.7778	0.918642
$563$	26.4029	1.11275	0.556374	−	0.830932i	$-0.312192\pi$
$563$	26.4029	1.11275	0.556374	+	0.830932i	$0.312192\pi$
$564$	2.64047	0.111184
$565$	−8.17106	−0.343759
$566$	12.5397	0.527084
$567$	0	0
$568$	−8.76303	−0.367689
$569$	19.5261	0.818575	0.409287	−	0.912405i	$-0.365777\pi$
$569$	19.5261	0.818575	0.409287	+	0.912405i	$0.365777\pi$
$570$	−33.8707	−1.41869
$571$	12.0085	0.502542	0.251271	−	0.967917i	$-0.419151\pi$
$571$	12.0085	0.502542	0.251271	+	0.967917i	$0.419151\pi$
$572$	−5.12749	−0.214391
$573$	−15.8414	−0.661783
$574$	0	0
$575$	−14.7335	−0.614428
$576$	−1.00000	−0.0416667
$577$	−38.8512	−1.61740	−0.808699	−	0.588223i	$-0.799828\pi$
$577$	−38.8512	−1.61740	−0.808699	+	0.588223i	$0.799828\pi$
$578$	0.561476	0.0233543
$579$	10.9373	0.454539
$580$	−4.31913	−0.179342
$581$	0	0
$582$	13.7339	0.569286
$583$	−4.56109	−0.188901
$584$	−5.35695	−0.221672
$585$	18.6549	0.771284
$586$	17.3470	0.716596
$587$	21.7778	0.898866	0.449433	−	0.893314i	$-0.351626\pi$
$587$	21.7778	0.898866	0.449433	+	0.893314i	$0.351626\pi$
$588$	0	0
$589$	53.2311	2.19335
$590$	5.29533	0.218005
$591$	2.45204	0.100863
$592$	−11.2953	−0.464235
$593$	9.14671	0.375610	0.187805	−	0.982206i	$-0.439863\pi$
$593$	9.14671	0.375610	0.187805	+	0.982206i	$0.439863\pi$
$594$	−6.71558	−0.275544
$595$	0	0
$596$	−19.5759	−0.801859
$597$	−36.9938	−1.51405
$598$	−4.66030	−0.190574
$599$	−18.3245	−0.748719	−0.374359	−	0.927284i	$-0.622137\pi$
$599$	−18.3245	−0.748719	−0.374359	+	0.927284i	$0.622137\pi$
$600$	19.3109	0.788364
$601$	9.55833	0.389892	0.194946	−	0.980814i	$-0.437547\pi$
$601$	9.55833	0.389892	0.194946	+	0.980814i	$0.437547\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	4.54669	0.185002
$605$	−41.4233	−1.68409
$606$	21.8420	0.887271
$607$	−7.07711	−0.287251	−0.143626	−	0.989632i	$-0.545876\pi$
$607$	−7.07711	−0.287251	−0.143626	+	0.989632i	$0.545876\pi$
$608$	5.54515	0.224885
$609$	0	0
$610$	42.2747	1.71165
$611$	8.06421	0.326243
$612$	−4.05444	−0.163891
$613$	27.8918	1.12654	0.563270	−	0.826273i	$-0.309543\pi$
$613$	27.8918	1.12654	0.563270	+	0.826273i	$0.309543\pi$
$614$	27.5728	1.11275
$615$	0.467193	0.0188390
$616$	0	0
$617$	−19.0934	−0.768671	−0.384335	−	0.923194i	$-0.625569\pi$
$617$	−19.0934	−0.768671	−0.384335	+	0.923194i	$0.625569\pi$
$618$	3.46770	0.139491
$619$	13.5011	0.542653	0.271327	−	0.962487i	$-0.412538\pi$
$619$	13.5011	0.542653	0.271327	+	0.962487i	$0.412538\pi$
$620$	−41.4619	−1.66515
$621$	−6.10369	−0.244933
$622$	33.2857	1.33464
$623$	0	0
$624$	6.10817	0.244522
$625$	93.1809	3.72724
$626$	11.3550	0.453836
$627$	9.30972	0.371794
$628$	14.8245	0.591561
$629$	−45.7963	−1.82602
$630$	0	0
$631$	36.1167	1.43778	0.718892	−	0.695122i	$-0.244650\pi$
$631$	36.1167	1.43778	0.718892	+	0.695122i	$0.244650\pi$
$632$	3.73385	0.148525
$633$	8.70870	0.346140
$634$	−2.21634	−0.0880220
$635$	−41.3594	−1.64130
$636$	5.43344	0.215450
$637$	0	0
$638$	1.18716	0.0470001
$639$	−8.76303	−0.346660
$640$	−4.31913	−0.170728
$641$	31.7424	1.25375	0.626875	−	0.779120i	$-0.284334\pi$
$641$	31.7424	1.25375	0.626875	+	0.779120i	$0.284334\pi$
$642$	−9.01465	−0.355780
$643$	−39.5215	−1.55858	−0.779288	−	0.626666i	$-0.784419\pi$
$643$	−39.5215	−1.55858	−0.779288	+	0.626666i	$0.784419\pi$
$644$	0	0
$645$	72.3330	2.84811
$646$	22.4825	0.884562
$647$	34.4204	1.35321	0.676603	−	0.736348i	$-0.263451\pi$
$647$	34.4204	1.35321	0.676603	+	0.736348i	$0.263451\pi$
$648$	5.00000	0.196419
$649$	−1.45548	−0.0571325
$650$	58.9771	2.31327
$651$	0	0
$652$	−23.0292	−0.901892
$653$	−11.8420	−0.463414	−0.231707	−	0.972786i	$-0.574431\pi$
$653$	−11.8420	−0.463414	−0.231707	+	0.972786i	$0.574431\pi$
$654$	−5.43344	−0.212464
$655$	−24.9152	−0.973516
$656$	−0.0764866	−0.00298630
$657$	−5.35695	−0.208995
$658$	0	0
$659$	−18.9296	−0.737391	−0.368695	−	0.929550i	$-0.620195\pi$
$659$	−18.9296	−0.737391	−0.368695	+	0.929550i	$0.620195\pi$
$660$	−7.25137	−0.282259
$661$	43.0939	1.67616	0.838079	−	0.545549i	$-0.183679\pi$
$661$	43.0939	1.67616	0.838079	+	0.545549i	$0.183679\pi$
$662$	29.9004	1.16211
$663$	24.7652	0.961802
$664$	−0.452878	−0.0175751
$665$	0	0
$666$	−11.2953	−0.437685
$667$	1.07899	0.0417787
$668$	4.35435	0.168475
$669$	34.0584	1.31677
$670$	17.2765	0.667450
$671$	−11.6197	−0.448572
$672$	0	0
$673$	−10.8712	−0.419054	−0.209527	−	0.977803i	$-0.567192\pi$
$673$	−10.8712	−0.419054	−0.209527	+	0.977803i	$0.567192\pi$
$674$	24.0584	0.926693
$675$	77.2436	2.97311
$676$	5.65486	0.217495
$677$	33.6269	1.29239	0.646193	−	0.763174i	$-0.276360\pi$
$677$	33.6269	1.29239	0.646193	+	0.763174i	$0.276360\pi$
$678$	−2.67545	−0.102750
$679$	0	0
$680$	−17.5117	−0.671542
$681$	−3.57587	−0.137028
$682$	11.3962	0.436384
$683$	−46.2747	−1.77065	−0.885326	−	0.464971i	$-0.846065\pi$
$683$	−46.2747	−1.77065	−0.885326	+	0.464971i	$0.846065\pi$
$684$	5.54515	0.212024
$685$	−28.6810	−1.09585
$686$	0	0
$687$	−14.0584	−0.536360
$688$	−11.8420	−0.451473
$689$	16.5942	0.632188
$690$	−6.59065	−0.250902
$691$	18.8789	0.718188	0.359094	−	0.933301i	$-0.383086\pi$
$691$	18.8789	0.718188	0.359094	+	0.933301i	$0.383086\pi$
$692$	−12.5810	−0.478257
$693$	0	0
$694$	6.26615	0.237860
$695$	−74.2891	−2.81795
$696$	−1.41421	−0.0536056
$697$	−0.310111	−0.0117463
$698$	−32.0036	−1.21135
$699$	−17.2765	−0.653458
$700$	0	0
$701$	13.7339	0.518720	0.259360	−	0.965781i	$-0.416488\pi$
$701$	13.7339	0.518720	0.259360	+	0.965781i	$0.416488\pi$
$702$	24.4327	0.922152
$703$	62.6342	2.36229
$704$	1.18716	0.0447427
$705$	11.4045	0.429519
$706$	−5.69812	−0.214452
$707$	0	0
$708$	1.73385	0.0651621
$709$	3.09338	0.116175	0.0580873	−	0.998312i	$-0.481500\pi$
$709$	3.09338	0.116175	0.0580873	+	0.998312i	$0.481500\pi$
$710$	−37.8486	−1.42043
$711$	3.73385	0.140030
$712$	−14.3716	−0.538598
$713$	10.3579	0.387905
$714$	0	0
$715$	−22.1463	−0.828224
$716$	23.6840	0.885114
$717$	−34.5530	−1.29041
$718$	−10.1082	−0.377233
$719$	−34.0941	−1.27150	−0.635748	−	0.771897i	$-0.719308\pi$
$719$	−34.0941	−1.27150	−0.635748	+	0.771897i	$0.719308\pi$
$720$	−4.31913	−0.160964
$721$	0	0
$722$	−11.7486	−0.437239
$723$	−24.8070	−0.922582
$724$	−1.71412	−0.0637047
$725$	−13.6549	−0.507129
$726$	−13.5632	−0.503379
$727$	−9.29364	−0.344682	−0.172341	−	0.985037i	$-0.555133\pi$
$727$	−9.29364	−0.344682	−0.172341	+	0.985037i	$0.555133\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	−23.1373	−0.856352
$731$	−48.0128	−1.77582
$732$	13.8420	0.511616
$733$	27.8171	1.02745	0.513724	−	0.857956i	$-0.328266\pi$
$733$	27.8171	1.02745	0.513724	+	0.857956i	$0.328266\pi$
$734$	9.97598	0.368220
$735$	0	0
$736$	1.07899	0.0397721
$737$	−4.74863	−0.174918
$738$	−0.0764866	−0.00281551
$739$	13.6553	0.502316	0.251158	−	0.967946i	$-0.419189\pi$
$739$	13.6553	0.502316	0.251158	+	0.967946i	$0.419189\pi$
$740$	−48.7860	−1.79341
$741$	−33.8707	−1.24427
$742$	0	0
$743$	−24.9650	−0.915876	−0.457938	−	0.888984i	$-0.651412\pi$
$743$	−24.9650	−0.915876	−0.457938	+	0.888984i	$0.651412\pi$
$744$	−13.5759	−0.497716
$745$	−84.5507	−3.09770
$746$	6.48249	0.237341
$747$	−0.452878	−0.0165699
$748$	4.81327	0.175991
$749$	0	0
$750$	52.8653	1.93037
$751$	42.6194	1.55521	0.777603	−	0.628756i	$-0.216436\pi$
$751$	42.6194	1.55521	0.777603	+	0.628756i	$0.216436\pi$
$752$	−1.86709	−0.0680858
$753$	15.2809	0.556868
$754$	−4.31913	−0.157293
$755$	19.6377	0.714691
$756$	0	0
$757$	−12.9210	−0.469622	−0.234811	−	0.972041i	$-0.575447\pi$
$757$	−12.9210	−0.469622	−0.234811	+	0.972041i	$0.575447\pi$
$758$	3.09338	0.112357
$759$	1.81151	0.0657537
$760$	23.9502	0.868765
$761$	−14.5801	−0.528530	−0.264265	−	0.964450i	$-0.585129\pi$
$761$	−14.5801	−0.528530	−0.264265	+	0.964450i	$0.585129\pi$
$762$	−13.5423	−0.490587
$763$	0	0
$764$	11.2016	0.405258
$765$	−17.5117	−0.633136
$766$	1.30250	0.0470614
$767$	5.29533	0.191203
$768$	−1.41421	−0.0510310
$769$	23.7830	0.857637	0.428818	−	0.903391i	$-0.358930\pi$
$769$	23.7830	0.857637	0.428818	+	0.903391i	$0.358930\pi$
$770$	0	0
$771$	−30.2163	−1.08821
$772$	−7.73385	−0.278347
$773$	−26.3820	−0.948893	−0.474447	−	0.880284i	$-0.657352\pi$
$773$	−26.3820	−0.948893	−0.474447	+	0.880284i	$0.657352\pi$
$774$	−11.8420	−0.425653
$775$	−131.081	−4.70857
$776$	−9.71130	−0.348615
$777$	0	0
$778$	−3.07899	−0.110387
$779$	0.424129	0.0151960
$780$	26.3820	0.944626
$781$	10.4031	0.372252
$782$	4.37471	0.156439
$783$	−5.65685	−0.202159
$784$	0	0
$785$	64.0288	2.28529
$786$	−8.15798	−0.290986
$787$	18.7964	0.670019	0.335009	−	0.942215i	$-0.391260\pi$
$787$	18.7964	0.670019	0.335009	+	0.942215i	$0.391260\pi$
$788$	−1.73385	−0.0617659
$789$	5.72730	0.203897
$790$	16.1270	0.573772
$791$	0	0
$792$	1.18716	0.0421838
$793$	42.2747	1.50122
$794$	−4.54255	−0.161209
$795$	23.4677	0.832314
$796$	26.1585	0.927165
$797$	41.2972	1.46282	0.731412	−	0.681936i	$-0.238862\pi$
$797$	41.2972	1.46282	0.731412	+	0.681936i	$0.238862\pi$
$798$	0	0
$799$	−7.57002	−0.267808
$800$	−13.6549	−0.482772
$801$	−14.3716	−0.507795
$802$	14.2163	0.501996
$803$	6.35955	0.224424
$804$	5.65685	0.199502
$805$	0	0
$806$	−41.4619	−1.46043
$807$	−4.00000	−0.140807
$808$	−15.4446	−0.543341
$809$	4.16652	0.146487	0.0732436	−	0.997314i	$-0.476665\pi$
$809$	4.16652	0.146487	0.0732436	+	0.997314i	$0.476665\pi$
$810$	21.5956	0.758793
$811$	10.0877	0.354227	0.177113	−	0.984190i	$-0.443324\pi$
$811$	10.0877	0.354227	0.177113	+	0.984190i	$0.443324\pi$
$812$	0	0
$813$	−9.89183	−0.346922
$814$	13.4093	0.469997
$815$	−99.4660	−3.48414
$816$	−5.73385	−0.200725
$817$	65.6657	2.29735
$818$	−29.5928	−1.03469
$819$	0	0
$820$	−0.330355	−0.0115365
$821$	9.78366	0.341452	0.170726	−	0.985319i	$-0.445389\pi$
$821$	9.78366	0.341452	0.170726	+	0.985319i	$0.445389\pi$
$822$	−9.39104	−0.327550
$823$	37.9587	1.32316	0.661579	−	0.749875i	$-0.269887\pi$
$823$	37.9587	1.32316	0.661579	+	0.749875i	$0.269887\pi$
$824$	−2.45204	−0.0854207
$825$	−22.9251	−0.798149
$826$	0	0
$827$	25.2455	0.877873	0.438936	−	0.898518i	$-0.355355\pi$
$827$	25.2455	0.877873	0.438936	+	0.898518i	$0.355355\pi$
$828$	1.07899	0.0374975
$829$	−35.0202	−1.21630	−0.608151	−	0.793821i	$-0.708089\pi$
$829$	−35.0202	−1.21630	−0.608151	+	0.793821i	$0.708089\pi$
$830$	−1.95604	−0.0678951
$831$	−8.41484	−0.291907
$832$	−4.31913	−0.149739
$833$	0	0
$834$	−24.3245	−0.842289
$835$	18.8070	0.650843
$836$	−6.58297	−0.227677
$837$	−54.3035	−1.87700
$838$	−35.4731	−1.22540
$839$	35.5848	1.22852	0.614262	−	0.789102i	$-0.289454\pi$
$839$	35.5848	1.22852	0.614262	+	0.789102i	$0.289454\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−21.6696	−0.746785
$843$	30.7985	1.06076
$844$	−6.15798	−0.211966
$845$	24.4241	0.840213
$846$	−1.86709	−0.0641919
$847$	0	0
$848$	−3.84202	−0.131935
$849$	17.7339	0.608624
$850$	−55.3629	−1.89893
$851$	12.1875	0.417784
$852$	−12.3928	−0.424570
$853$	−46.8633	−1.60457	−0.802284	−	0.596942i	$-0.796382\pi$
$853$	−46.8633	−1.60457	−0.802284	+	0.596942i	$0.796382\pi$
$854$	0	0
$855$	23.9502	0.819079
$856$	6.37432	0.217870
$857$	−40.6980	−1.39022	−0.695108	−	0.718905i	$-0.744644\pi$
$857$	−40.6980	−1.39022	−0.695108	+	0.718905i	$0.744644\pi$
$858$	−7.25137	−0.247558
$859$	8.83249	0.301361	0.150680	−	0.988583i	$-0.451854\pi$
$859$	8.83249	0.301361	0.150680	+	0.988583i	$0.451854\pi$
$860$	−51.1472	−1.74410
$861$	0	0
$862$	17.8420	0.607702
$863$	7.13735	0.242958	0.121479	−	0.992594i	$-0.461236\pi$
$863$	7.13735	0.242958	0.121479	+	0.992594i	$0.461236\pi$
$864$	−5.65685	−0.192450
$865$	−54.3389	−1.84758
$866$	14.5950	0.495959
$867$	0.794047	0.0269673
$868$	0	0
$869$	−4.43267	−0.150368
$870$	−6.10817	−0.207086
$871$	17.2765	0.585392
$872$	3.84202	0.130107
$873$	−9.71130	−0.328678
$874$	−5.98316	−0.202383
$875$	0	0
$876$	−7.57587	−0.255965
$877$	−39.0436	−1.31841	−0.659204	−	0.751964i	$-0.729107\pi$
$877$	−39.0436	−1.31841	−0.659204	+	0.751964i	$0.729107\pi$
$878$	12.3928	0.418237
$879$	24.5323	0.827454
$880$	5.12749	0.172848
$881$	32.4212	1.09230	0.546150	−	0.837688i	$-0.316093\pi$
$881$	32.4212	1.09230	0.546150	+	0.837688i	$0.316093\pi$
$882$	0	0
$883$	−24.5611	−0.826546	−0.413273	−	0.910607i	$-0.635615\pi$
$883$	−24.5611	−0.826546	−0.413273	+	0.910607i	$0.635615\pi$
$884$	−17.5117	−0.588981
$885$	7.48872	0.251731
$886$	17.7778	0.597257
$887$	−37.9664	−1.27479	−0.637393	−	0.770539i	$-0.719987\pi$
$887$	−37.9664	−1.27479	−0.637393	+	0.770539i	$0.719987\pi$
$888$	−15.9740	−0.536053
$889$	0	0
$890$	−62.0728	−2.08068
$891$	−5.93579	−0.198857
$892$	−24.0829	−0.806355
$893$	10.3533	0.346460
$894$	−27.6845	−0.925907
$895$	102.294	3.41932
$896$	0	0
$897$	−6.59065	−0.220056
$898$	38.4327	1.28251
$899$	9.59959	0.320164
$900$	−13.6549	−0.455162
$901$	−15.5773	−0.518954
$902$	0.0908018	0.00302337
$903$	0	0
$904$	1.89183	0.0629214
$905$	−7.40350	−0.246101
$906$	6.42999	0.213622
$907$	−6.09377	−0.202340	−0.101170	−	0.994869i	$-0.532259\pi$
$907$	−6.09377	−0.202340	−0.101170	+	0.994869i	$0.532259\pi$
$908$	2.52852	0.0839120
$909$	−15.4446	−0.512266
$910$	0	0
$911$	1.54708	0.0512571	0.0256286	−	0.999672i	$-0.491841\pi$
$911$	1.54708	0.0512571	0.0256286	+	0.999672i	$0.491841\pi$
$912$	7.84202	0.259675
$913$	0.537638	0.0177932
$914$	5.24552	0.173506
$915$	59.7854	1.97645
$916$	9.94076	0.328452
$917$	0	0
$918$	−22.9354	−0.756981
$919$	34.0440	1.12301	0.561503	−	0.827474i	$-0.310223\pi$
$919$	34.0440	1.12301	0.561503	+	0.827474i	$0.310223\pi$
$920$	4.66030	0.153645
$921$	38.9938	1.28489
$922$	11.0903	0.365239
$923$	−37.8486	−1.24580
$924$	0	0
$925$	−154.236	−5.07125
$926$	−27.1373	−0.891789
$927$	−2.45204	−0.0805354
$928$	1.00000	0.0328266
$929$	22.2218	0.729075	0.364538	−	0.931189i	$-0.381227\pi$
$929$	22.2218	0.729075	0.364538	+	0.931189i	$0.381227\pi$
$930$	−58.6359	−1.92275
$931$	0	0
$932$	12.2163	0.400159
$933$	47.0731	1.54111
$934$	9.65572	0.315945
$935$	20.7891	0.679877
$936$	−4.31913	−0.141175
$937$	−50.2827	−1.64266	−0.821332	−	0.570450i	$-0.806769\pi$
$937$	−50.2827	−1.64266	−0.821332	+	0.570450i	$0.806769\pi$
$938$	0	0
$939$	16.0584	0.524045
$940$	−8.06421	−0.263025
$941$	−44.0701	−1.43664	−0.718322	−	0.695711i	$-0.755089\pi$
$941$	−44.0701	−1.43664	−0.718322	+	0.695711i	$0.755089\pi$
$942$	20.9650	0.683076
$943$	0.0825283	0.00268749
$944$	−1.22602	−0.0399035
$945$	0	0
$946$	14.0584	0.457077
$947$	23.5261	0.764494	0.382247	−	0.924060i	$-0.375150\pi$
$947$	23.5261	0.764494	0.382247	+	0.924060i	$0.375150\pi$
$948$	5.28046	0.171501
$949$	−23.1373	−0.751070
$950$	75.7182	2.45662
$951$	−3.13437	−0.101639
$952$	0	0
$953$	49.6198	1.60734	0.803672	−	0.595073i	$-0.202877\pi$
$953$	49.6198	1.60734	0.803672	+	0.595073i	$0.202877\pi$
$954$	−3.84202	−0.124390
$955$	48.3809	1.56557
$956$	24.4327	0.790209
$957$	1.67890	0.0542710
$958$	37.8134	1.22170
$959$	0	0
$960$	−6.10817	−0.197140
$961$	61.1521	1.97265
$962$	−48.7860	−1.57292
$963$	6.37432	0.205410
$964$	17.5412	0.564964
$965$	−33.4035	−1.07530
$966$	0	0
$967$	8.77742	0.282263	0.141131	−	0.989991i	$-0.454926\pi$
$967$	8.77742	0.282263	0.141131	+	0.989991i	$0.454926\pi$
$968$	9.59065	0.308255
$969$	31.7950	1.02140
$970$	−41.9443	−1.34675
$971$	2.62592	0.0842697	0.0421348	−	0.999112i	$-0.486584\pi$
$971$	2.62592	0.0842697	0.0421348	+	0.999112i	$0.486584\pi$
$972$	−9.89949	−0.317526
$973$	0	0
$974$	17.3097	0.554639
$975$	83.4062	2.67114
$976$	−9.78779	−0.313299
$977$	20.9296	0.669596	0.334798	−	0.942290i	$-0.391332\pi$
$977$	20.9296	0.669596	0.334798	+	0.942290i	$0.391332\pi$
$978$	−32.5682	−1.04142
$979$	17.0614	0.545284
$980$	0	0
$981$	3.84202	0.122666
$982$	−40.1167	−1.28018
$983$	46.1457	1.47182	0.735910	−	0.677079i	$-0.236755\pi$
$983$	46.1457	1.47182	0.735910	+	0.677079i	$0.236755\pi$
$984$	−0.108168	−0.00344828
$985$	−7.48872	−0.238611
$986$	4.05444	0.129120
$987$	0	0
$988$	23.9502	0.761957
$989$	12.7774	0.406298
$990$	5.12749	0.162962
$991$	−46.2747	−1.46996	−0.734982	−	0.678087i	$-0.762809\pi$
$991$	−46.2747	−1.46996	−0.734982	+	0.678087i	$0.762809\pi$
$992$	9.59959	0.304787
$993$	42.2855	1.34189
$994$	0	0
$995$	112.982	3.58177
$996$	−0.640466	−0.0202940
$997$	−30.4017	−0.962832	−0.481416	−	0.876492i	$-0.659877\pi$
$997$	−30.4017	−0.962832	−0.481416	+	0.876492i	$0.659877\pi$
$998$	15.4677	0.489622
$999$	−63.8960	−2.02158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.ba.1.3	✓	6
7.6	odd	2	inner	2842.2.a.ba.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.ba.1.3	✓	6	1.1	even	1	trivial
2842.2.a.ba.1.4	yes	6	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +4.31913 q^{5} +1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +4.31913 q^{5} +1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -4.31913 q^{10} +1.18716 q^{11} -1.41421 q^{12} -4.31913 q^{13} -6.10817 q^{15} +1.00000 q^{16} +4.05444 q^{17} +1.00000 q^{18} -5.54515 q^{19} +4.31913 q^{20} -1.18716 q^{22} -1.07899 q^{23} +1.41421 q^{24} +13.6549 q^{25} +4.31913 q^{26} +5.65685 q^{27} -1.00000 q^{29} +6.10817 q^{30} -9.59959 q^{31} -1.00000 q^{32} -1.67890 q^{33} -4.05444 q^{34} -1.00000 q^{36} -11.2953 q^{37} +5.54515 q^{38} +6.10817 q^{39} -4.31913 q^{40} -0.0764866 q^{41} -11.8420 q^{43} +1.18716 q^{44} -4.31913 q^{45} +1.07899 q^{46} -1.86709 q^{47} -1.41421 q^{48} -13.6549 q^{50} -5.73385 q^{51} -4.31913 q^{52} -3.84202 q^{53} -5.65685 q^{54} +5.12749 q^{55} +7.84202 q^{57} +1.00000 q^{58} -1.22602 q^{59} -6.10817 q^{60} -9.78779 q^{61} +9.59959 q^{62} +1.00000 q^{64} -18.6549 q^{65} +1.67890 q^{66} -4.00000 q^{67} +4.05444 q^{68} +1.52592 q^{69} +8.76303 q^{71} +1.00000 q^{72} +5.35695 q^{73} +11.2953 q^{74} -19.3109 q^{75} -5.54515 q^{76} -6.10817 q^{78} -3.73385 q^{79} +4.31913 q^{80} -5.00000 q^{81} +0.0764866 q^{82} +0.452878 q^{83} +17.5117 q^{85} +11.8420 q^{86} +1.41421 q^{87} -1.18716 q^{88} +14.3716 q^{89} +4.31913 q^{90} -1.07899 q^{92} +13.5759 q^{93} +1.86709 q^{94} -23.9502 q^{95} +1.41421 q^{96} +9.71130 q^{97} -1.18716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 - 6 * q^9	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9} - 4 q^{11} + 6 q^{16} + 6 q^{18} + 4 q^{22} - 32 q^{23} + 42 q^{25} - 6 q^{29} - 6 q^{32} - 6 q^{36} - 20 q^{37} - 20 q^{43} - 4 q^{44} + 32 q^{46} - 42 q^{50} - 20 q^{51} + 28 q^{53} - 4 q^{57} + 6 q^{58} + 6 q^{64} - 72 q^{65} - 24 q^{67} - 24 q^{71} + 6 q^{72} + 20 q^{74} - 8 q^{79} - 30 q^{81} - 16 q^{85} + 20 q^{86} + 4 q^{88} - 32 q^{92} + 16 q^{93} - 56 q^{95} + 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 - 6 * q^9 - 4 * q^11 + 6 * q^16 + 6 * q^18 + 4 * q^22 - 32 * q^23 + 42 * q^25 - 6 * q^29 - 6 * q^32 - 6 * q^36 - 20 * q^37 - 20 * q^43 - 4 * q^44 + 32 * q^46 - 42 * q^50 - 20 * q^51 + 28 * q^53 - 4 * q^57 + 6 * q^58 + 6 * q^64 - 72 * q^65 - 24 * q^67 - 24 * q^71 + 6 * q^72 + 20 * q^74 - 8 * q^79 - 30 * q^81 - 16 * q^85 + 20 * q^86 + 4 * q^88 - 32 * q^92 + 16 * q^93 - 56 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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