LMFDB - Embedded newform 2842.2.a.bb.1.5

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.401917952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 14x^{4} + 49x^{2} - 18 $ x^6 - 14x^4 + 49x^2 - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.26409$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.26409 q^{3} +1.00000 q^{4} +3.67831 q^{5} +2.26409 q^{6} +1.00000 q^{8} +2.12612 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.26409 q^{3} +1.00000 q^{4} +3.67831 q^{5} +2.26409 q^{6} +1.00000 q^{8} +2.12612 q^{9} +3.67831 q^{10} -2.32803 q^{11} +2.26409 q^{12} +5.27088 q^{13} +8.32803 q^{15} +1.00000 q^{16} -4.70655 q^{17} +2.12612 q^{18} +5.94240 q^{19} +3.67831 q^{20} -2.32803 q^{22} -8.65606 q^{23} +2.26409 q^{24} +8.52994 q^{25} +5.27088 q^{26} -1.97855 q^{27} -1.00000 q^{29} +8.32803 q^{30} -1.97855 q^{31} +1.00000 q^{32} -5.27088 q^{33} -4.70655 q^{34} +2.12612 q^{36} -2.00000 q^{37} +5.94240 q^{38} +11.9338 q^{39} +3.67831 q^{40} +10.3634 q^{41} -12.1860 q^{43} -2.32803 q^{44} +7.82052 q^{45} -8.65606 q^{46} -1.97855 q^{47} +2.26409 q^{48} +8.52994 q^{50} -10.6561 q^{51} +5.27088 q^{52} +6.32803 q^{53} -1.97855 q^{54} -8.56321 q^{55} +13.4542 q^{57} -1.00000 q^{58} -7.17825 q^{59} +8.32803 q^{60} -11.1129 q^{61} -1.97855 q^{62} +1.00000 q^{64} +19.3879 q^{65} -5.27088 q^{66} -2.25224 q^{67} -4.70655 q^{68} -19.5981 q^{69} +8.00000 q^{71} +2.12612 q^{72} -7.17825 q^{73} -2.00000 q^{74} +19.3126 q^{75} +5.94240 q^{76} +11.9338 q^{78} +11.1261 q^{79} +3.67831 q^{80} -10.8580 q^{81} +10.3634 q^{82} -10.3634 q^{83} -17.3121 q^{85} -12.1860 q^{86} -2.26409 q^{87} -2.32803 q^{88} +5.83521 q^{89} +7.82052 q^{90} -8.65606 q^{92} -4.47962 q^{93} -1.97855 q^{94} +21.8580 q^{95} +2.26409 q^{96} +4.13545 q^{97} -4.94967 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9} + 8 q^{11} + 28 q^{15} + 6 q^{16} + 10 q^{18} + 8 q^{22} - 8 q^{23} + 10 q^{25} - 6 q^{29} + 28 q^{30} + 6 q^{32} + 10 q^{36} - 12 q^{37} - 8 q^{39} + 12 q^{43} + 8 q^{44} - 8 q^{46} + 10 q^{50} - 20 q^{51} + 16 q^{53} + 56 q^{57} - 6 q^{58} + 28 q^{60} + 6 q^{64} + 12 q^{65} - 8 q^{67} + 48 q^{71} + 10 q^{72} - 12 q^{74} - 8 q^{78} + 64 q^{79} - 2 q^{81} - 16 q^{85} + 12 q^{86} + 8 q^{88} - 8 q^{92} + 28 q^{93} + 68 q^{95} - 16 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9 + 8 * q^11 + 28 * q^15 + 6 * q^16 + 10 * q^18 + 8 * q^22 - 8 * q^23 + 10 * q^25 - 6 * q^29 + 28 * q^30 + 6 * q^32 + 10 * q^36 - 12 * q^37 - 8 * q^39 + 12 * q^43 + 8 * q^44 - 8 * q^46 + 10 * q^50 - 20 * q^51 + 16 * q^53 + 56 * q^57 - 6 * q^58 + 28 * q^60 + 6 * q^64 + 12 * q^65 - 8 * q^67 + 48 * q^71 + 10 * q^72 - 12 * q^74 - 8 * q^78 + 64 * q^79 - 2 * q^81 - 16 * q^85 + 12 * q^86 + 8 * q^88 - 8 * q^92 + 28 * q^93 + 68 * q^95 - 16 * 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
$3$	2.26409	1.30718	0.653588	−	0.756851i	$-0.273263\pi$
$3$	2.26409	1.30718	0.653588	+	0.756851i	$0.273263\pi$
$4$	1.00000	0.500000
$5$	3.67831	1.64499	0.822494	−	0.568773i	$-0.192582\pi$
$5$	3.67831	1.64499	0.822494	+	0.568773i	$0.192582\pi$
$6$	2.26409	0.924312
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	2.12612	0.708707
$10$	3.67831	1.16318
$11$	−2.32803	−0.701928	−0.350964	−	0.936389i	$-0.614146\pi$
$11$	−2.32803	−0.701928	−0.350964	+	0.936389i	$0.614146\pi$
$12$	2.26409	0.653588
$13$	5.27088	1.46188	0.730940	−	0.682442i	$-0.239082\pi$
$13$	5.27088	1.46188	0.730940	+	0.682442i	$0.239082\pi$
$14$	0	0
$15$	8.32803	2.15029
$16$	1.00000	0.250000
$17$	−4.70655	−1.14151	−0.570753	−	0.821122i	$-0.693348\pi$
$17$	−4.70655	−1.14151	−0.570753	+	0.821122i	$0.693348\pi$
$18$	2.12612	0.501131
$19$	5.94240	1.36328	0.681640	−	0.731688i	$-0.261267\pi$
$19$	5.94240	1.36328	0.681640	+	0.731688i	$0.261267\pi$
$20$	3.67831	0.822494
$21$	0	0
$22$	−2.32803	−0.496338
$23$	−8.65606	−1.80491	−0.902457	−	0.430780i	$-0.858238\pi$
$23$	−8.65606	−1.80491	−0.902457	+	0.430780i	$0.858238\pi$
$24$	2.26409	0.462156
$25$	8.52994	1.70599
$26$	5.27088	1.03370
$27$	−1.97855	−0.380772
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	8.32803	1.52048
$31$	−1.97855	−0.355358	−0.177679	−	0.984089i	$-0.556859\pi$
$31$	−1.97855	−0.355358	−0.177679	+	0.984089i	$0.556859\pi$
$32$	1.00000	0.176777
$33$	−5.27088	−0.917543
$34$	−4.70655	−0.807166
$35$	0	0
$36$	2.12612	0.354353
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	5.94240	0.963985
$39$	11.9338	1.91093
$40$	3.67831	0.581591
$41$	10.3634	1.61849	0.809246	−	0.587470i	$-0.199876\pi$
$41$	10.3634	1.61849	0.809246	+	0.587470i	$0.199876\pi$
$42$	0	0
$43$	−12.1860	−1.85835	−0.929174	−	0.369642i	$-0.879480\pi$
$43$	−12.1860	−1.85835	−0.929174	+	0.369642i	$0.879480\pi$
$44$	−2.32803	−0.350964
$45$	7.82052	1.16581
$46$	−8.65606	−1.27627
$47$	−1.97855	−0.288601	−0.144300	−	0.989534i	$-0.546093\pi$
$47$	−1.97855	−0.288601	−0.144300	+	0.989534i	$0.546093\pi$
$48$	2.26409	0.326794
$49$	0	0
$50$	8.52994	1.20632
$51$	−10.6561	−1.49215
$52$	5.27088	0.730940
$53$	6.32803	0.869222	0.434611	−	0.900618i	$-0.356886\pi$
$53$	6.32803	0.869222	0.434611	+	0.900618i	$0.356886\pi$
$54$	−1.97855	−0.269246
$55$	−8.56321	−1.15466
$56$	0	0
$57$	13.4542	1.78205
$58$	−1.00000	−0.131306
$59$	−7.17825	−0.934529	−0.467265	−	0.884118i	$-0.654760\pi$
$59$	−7.17825	−0.934529	−0.467265	+	0.884118i	$0.654760\pi$
$60$	8.32803	1.07514
$61$	−11.1129	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$61$	−11.1129	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$62$	−1.97855	−0.251276
$63$	0	0
$64$	1.00000	0.125000
$65$	19.3879	2.40478
$66$	−5.27088	−0.648801
$67$	−2.25224	−0.275155	−0.137577	−	0.990491i	$-0.543932\pi$
$67$	−2.25224	−0.275155	−0.137577	+	0.990491i	$0.543932\pi$
$68$	−4.70655	−0.570753
$69$	−19.5981	−2.35934
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	2.12612	0.250566
$73$	−7.17825	−0.840151	−0.420076	−	0.907489i	$-0.637996\pi$
$73$	−7.17825	−0.840151	−0.420076	+	0.907489i	$0.637996\pi$
$74$	−2.00000	−0.232495
$75$	19.3126	2.23003
$76$	5.94240	0.681640
$77$	0	0
$78$	11.9338	1.35123
$79$	11.1261	1.25179	0.625893	−	0.779909i	$-0.284735\pi$
$79$	11.1261	1.25179	0.625893	+	0.779909i	$0.284735\pi$
$80$	3.67831	0.411247
$81$	−10.8580	−1.20644
$82$	10.3634	1.14445
$83$	−10.3634	−1.13753	−0.568766	−	0.822500i	$-0.692579\pi$
$83$	−10.3634	−1.13753	−0.568766	+	0.822500i	$0.692579\pi$
$84$	0	0
$85$	−17.3121	−1.87776
$86$	−12.1860	−1.31405
$87$	−2.26409	−0.242736
$88$	−2.32803	−0.248169
$89$	5.83521	0.618531	0.309266	−	0.950976i	$-0.399917\pi$
$89$	5.83521	0.618531	0.309266	+	0.950976i	$0.399917\pi$
$90$	7.82052	0.824355
$91$	0	0
$92$	−8.65606	−0.902457
$93$	−4.47962	−0.464515
$94$	−1.97855	−0.204072
$95$	21.8580	2.24258
$96$	2.26409	0.231078
$97$	4.13545	0.419892	0.209946	−	0.977713i	$-0.432671\pi$
$97$	4.13545	0.419892	0.209946	+	0.977713i	$0.432671\pi$
$98$	0	0
$99$	−4.94967	−0.497461
$100$	8.52994	0.852994
$101$	3.75624	0.373760	0.186880	−	0.982383i	$-0.440162\pi$
$101$	3.75624	0.373760	0.186880	+	0.982383i	$0.440162\pi$
$102$	−10.6561	−1.05511
$103$	8.48528	0.836080	0.418040	−	0.908429i	$-0.362717\pi$
$103$	8.48528	0.836080	0.418040	+	0.908429i	$0.362717\pi$
$104$	5.27088	0.516852
$105$	0	0
$106$	6.32803	0.614633
$107$	4.65606	0.450119	0.225059	−	0.974345i	$-0.427742\pi$
$107$	4.65606	0.450119	0.225059	+	0.974345i	$0.427742\pi$
$108$	−1.97855	−0.190386
$109$	−10.9841	−1.05209	−0.526043	−	0.850458i	$-0.676325\pi$
$109$	−10.9841	−1.05209	−0.526043	+	0.850458i	$0.676325\pi$
$110$	−8.56321	−0.816470
$111$	−4.52819	−0.429796
$112$	0	0
$113$	20.8076	1.95742	0.978709	−	0.205251i	$-0.0658010\pi$
$113$	20.8076	1.95742	0.978709	+	0.205251i	$0.0658010\pi$
$114$	13.4542	1.26010
$115$	−31.8397	−2.96906
$116$	−1.00000	−0.0928477
$117$	11.2065	1.03604
$118$	−7.17825	−0.660812
$119$	0	0
$120$	8.32803	0.760242
$121$	−5.58027	−0.507297
$122$	−11.1129	−1.00611
$123$	23.4637	2.11565
$124$	−1.97855	−0.177679
$125$	12.9842	1.16134
$126$	0	0
$127$	2.25224	0.199854	0.0999269	−	0.994995i	$-0.468139\pi$
$127$	2.25224	0.199854	0.0999269	+	0.994995i	$0.468139\pi$
$128$	1.00000	0.0883883
$129$	−27.5903	−2.42919
$130$	19.3879	1.70043
$131$	9.48427	0.828644	0.414322	−	0.910130i	$-0.364019\pi$
$131$	9.48427	0.828644	0.414322	+	0.910130i	$0.364019\pi$
$132$	−5.27088	−0.458771
$133$	0	0
$134$	−2.25224	−0.194564
$135$	−7.27770	−0.626365
$136$	−4.70655	−0.403583
$137$	13.1605	1.12438	0.562190	−	0.827008i	$-0.309959\pi$
$137$	13.1605	1.12438	0.562190	+	0.827008i	$0.309959\pi$
$138$	−19.5981	−1.66830
$139$	−8.66364	−0.734840	−0.367420	−	0.930055i	$-0.619759\pi$
$139$	−8.66364	−0.734840	−0.367420	+	0.930055i	$0.619759\pi$
$140$	0	0
$141$	−4.47962	−0.377252
$142$	8.00000	0.671345
$143$	−12.2708	−1.02613
$144$	2.12612	0.177177
$145$	−3.67831	−0.305467
$146$	−7.17825	−0.594077
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−5.12612	−0.419948	−0.209974	−	0.977707i	$-0.567338\pi$
$149$	−5.12612	−0.419948	−0.209974	+	0.977707i	$0.567338\pi$
$150$	19.3126	1.57687
$151$	−15.0599	−1.22556	−0.612778	−	0.790255i	$-0.709948\pi$
$151$	−15.0599	−1.22556	−0.612778	+	0.790255i	$0.709948\pi$
$152$	5.94240	0.481992
$153$	−10.0067	−0.808992
$154$	0	0
$155$	−7.27770	−0.584559
$156$	11.9338	0.955466
$157$	−8.12856	−0.648730	−0.324365	−	0.945932i	$-0.605151\pi$
$157$	−8.12856	−0.648730	−0.324365	+	0.945932i	$0.605151\pi$
$158$	11.1261	0.885147
$159$	14.3273	1.13623
$160$	3.67831	0.290796
$161$	0	0
$162$	−10.8580	−0.853083
$163$	5.82355	0.456136	0.228068	−	0.973645i	$-0.426759\pi$
$163$	5.82355	0.456136	0.228068	+	0.973645i	$0.426759\pi$
$164$	10.3634	0.809246
$165$	−19.3879	−1.50935
$166$	−10.3634	−0.804356
$167$	−6.22795	−0.481933	−0.240967	−	0.970533i	$-0.577464\pi$
$167$	−6.22795	−0.481933	−0.240967	+	0.970533i	$0.577464\pi$
$168$	0	0
$169$	14.7822	1.13709
$170$	−17.3121	−1.32778
$171$	12.6343	0.966166
$172$	−12.1860	−0.929174
$173$	2.36452	0.179771	0.0898856	−	0.995952i	$-0.471350\pi$
$173$	2.36452	0.179771	0.0898856	+	0.995952i	$0.471350\pi$
$174$	−2.26409	−0.171640
$175$	0	0
$176$	−2.32803	−0.175482
$177$	−16.2522	−1.22159
$178$	5.83521	0.437368
$179$	−8.50448	−0.635655	−0.317827	−	0.948149i	$-0.602953\pi$
$179$	−8.50448	−0.635655	−0.317827	+	0.948149i	$0.602953\pi$
$180$	7.82052	0.582907
$181$	−7.99212	−0.594050	−0.297025	−	0.954870i	$-0.595994\pi$
$181$	−7.99212	−0.594050	−0.297025	+	0.954870i	$0.595994\pi$
$182$	0	0
$183$	−25.1605	−1.85992
$184$	−8.65606	−0.638133
$185$	−7.35661	−0.540869
$186$	−4.47962	−0.328461
$187$	10.9570	0.801254
$188$	−1.97855	−0.144300
$189$	0	0
$190$	21.8580	1.58574
$191$	1.74776	0.126464	0.0632318	−	0.997999i	$-0.479859\pi$
$191$	1.74776	0.126464	0.0632318	+	0.997999i	$0.479859\pi$
$192$	2.26409	0.163397
$193$	−7.49552	−0.539539	−0.269770	−	0.962925i	$-0.586948\pi$
$193$	−7.49552	−0.539539	−0.269770	+	0.962925i	$0.586948\pi$
$194$	4.13545	0.296908
$195$	43.8961	3.14346
$196$	0	0
$197$	22.6561	1.61418	0.807089	−	0.590430i	$-0.201042\pi$
$197$	22.6561	1.61418	0.807089	+	0.590430i	$0.201042\pi$
$198$	−4.94967	−0.351758
$199$	−17.5417	−1.24349	−0.621747	−	0.783218i	$-0.713577\pi$
$199$	−17.5417	−1.24349	−0.621747	+	0.783218i	$0.713577\pi$
$200$	8.52994	0.603158
$201$	−5.09928	−0.359675
$202$	3.75624	0.264288
$203$	0	0
$204$	−10.6561	−0.746074
$205$	38.1198	2.66240
$206$	8.48528	0.591198
$207$	−18.4038	−1.27915
$208$	5.27088	0.365470
$209$	−13.8341	−0.956924
$210$	0	0
$211$	−16.4796	−1.13450	−0.567252	−	0.823544i	$-0.691993\pi$
$211$	−16.4796	−1.13450	−0.567252	+	0.823544i	$0.691993\pi$
$212$	6.32803	0.434611
$213$	18.1127	1.24107
$214$	4.65606	0.318282
$215$	−44.8239	−3.05696
$216$	−1.97855	−0.134623
$217$	0	0
$218$	−10.9841	−0.743937
$219$	−16.2522	−1.09822
$220$	−8.56321	−0.577332
$221$	−24.8076	−1.66874
$222$	−4.52819	−0.303912
$223$	−21.0835	−1.41186	−0.705929	−	0.708283i	$-0.749470\pi$
$223$	−21.0835	−1.41186	−0.705929	+	0.708283i	$0.749470\pi$
$224$	0	0
$225$	18.1357	1.20905
$226$	20.8076	1.38410
$227$	24.3047	1.61316	0.806579	−	0.591126i	$-0.201316\pi$
$227$	24.3047	1.61316	0.806579	+	0.591126i	$0.201316\pi$
$228$	13.4542	0.891023
$229$	12.0857	0.798643	0.399321	−	0.916811i	$-0.369246\pi$
$229$	12.0857	0.798643	0.399321	+	0.916811i	$0.369246\pi$
$230$	−31.8397	−2.09944
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	7.63060	0.499897	0.249949	−	0.968259i	$-0.419586\pi$
$233$	7.63060	0.499897	0.249949	+	0.968259i	$0.419586\pi$
$234$	11.2065	0.732593
$235$	−7.27770	−0.474745
$236$	−7.17825	−0.467265
$237$	25.1906	1.63630
$238$	0	0
$239$	0.504478	0.0326320	0.0163160	−	0.999867i	$-0.494806\pi$
$239$	0.504478	0.0326320	0.0163160	+	0.999867i	$0.494806\pi$
$240$	8.32803	0.537572
$241$	−21.8622	−1.40827	−0.704135	−	0.710066i	$-0.748665\pi$
$241$	−21.8622	−1.40827	−0.704135	+	0.710066i	$0.748665\pi$
$242$	−5.58027	−0.358713
$243$	−18.6478	−1.19626
$244$	−11.1129	−0.711428
$245$	0	0
$246$	23.4637	1.49599
$247$	31.3217	1.99295
$248$	−1.97855	−0.125638
$249$	−23.4637	−1.48695
$250$	12.9842	0.821194
$251$	9.87024	0.623004	0.311502	−	0.950245i	$-0.399168\pi$
$251$	9.87024	0.623004	0.311502	+	0.950245i	$0.399168\pi$
$252$	0	0
$253$	20.1516	1.26692
$254$	2.25224	0.141318
$255$	−39.1963	−2.45457
$256$	1.00000	0.0625000
$257$	11.6772	0.728403	0.364201	−	0.931320i	$-0.381342\pi$
$257$	11.6772	0.728403	0.364201	+	0.931320i	$0.381342\pi$
$258$	−27.5903	−1.71769
$259$	0	0
$260$	19.3879	1.20239
$261$	−2.12612	−0.131603
$262$	9.48427	0.585940
$263$	−17.6815	−1.09029	−0.545145	−	0.838342i	$-0.683525\pi$
$263$	−17.6815	−1.09029	−0.545145	+	0.838342i	$0.683525\pi$
$264$	−5.27088	−0.324400
$265$	23.2764	1.42986
$266$	0	0
$267$	13.2115	0.808529
$268$	−2.25224	−0.137577
$269$	−4.32733	−0.263842	−0.131921	−	0.991260i	$-0.542115\pi$
$269$	−4.32733	−0.263842	−0.131921	+	0.991260i	$0.542115\pi$
$270$	−7.27770	−0.442907
$271$	24.8690	1.51069	0.755343	−	0.655330i	$-0.227470\pi$
$271$	24.8690	1.51069	0.755343	+	0.655330i	$0.227470\pi$
$272$	−4.70655	−0.285376
$273$	0	0
$274$	13.1605	0.795057
$275$	−19.8580	−1.19748
$276$	−19.5981	−1.17967
$277$	−5.64711	−0.339302	−0.169651	−	0.985504i	$-0.554264\pi$
$277$	−5.64711	−0.339302	−0.169651	+	0.985504i	$0.554264\pi$
$278$	−8.66364	−0.519611
$279$	−4.20663	−0.251844
$280$	0	0
$281$	14.2274	0.848734	0.424367	−	0.905490i	$-0.360497\pi$
$281$	14.2274	0.848734	0.424367	+	0.905490i	$0.360497\pi$
$282$	−4.47962	−0.266757
$283$	18.2911	1.08729	0.543647	−	0.839314i	$-0.317043\pi$
$283$	18.2911	1.08729	0.543647	+	0.839314i	$0.317043\pi$
$284$	8.00000	0.474713
$285$	49.4885	2.93145
$286$	−12.2708	−0.725586
$287$	0	0
$288$	2.12612	0.125283
$289$	5.15158	0.303034
$290$	−3.67831	−0.215998
$291$	9.36306	0.548872
$292$	−7.17825	−0.420076
$293$	−7.71333	−0.450618	−0.225309	−	0.974287i	$-0.572339\pi$
$293$	−7.71333	−0.450618	−0.225309	+	0.974287i	$0.572339\pi$
$294$	0	0
$295$	−26.4038	−1.53729
$296$	−2.00000	−0.116248
$297$	4.60612	0.267274
$298$	−5.12612	−0.296948
$299$	−45.6251	−2.63857
$300$	19.3126	1.11501
$301$	0	0
$302$	−15.0599	−0.866599
$303$	8.50448	0.488569
$304$	5.94240	0.340820
$305$	−40.8765	−2.34058
$306$	−10.0067	−0.572044
$307$	30.5034	1.74092	0.870460	−	0.492240i	$-0.163822\pi$
$307$	30.5034	1.74092	0.870460	+	0.492240i	$0.163822\pi$
$308$	0	0
$309$	19.2115	1.09290
$310$	−7.27770	−0.413346
$311$	6.32162	0.358466	0.179233	−	0.983807i	$-0.442638\pi$
$311$	6.32162	0.358466	0.179233	+	0.983807i	$0.442638\pi$
$312$	11.9338	0.675616
$313$	30.5970	1.72945	0.864724	−	0.502248i	$-0.167494\pi$
$313$	30.5970	1.72945	0.864724	+	0.502248i	$0.167494\pi$
$314$	−8.12856	−0.458721
$315$	0	0
$316$	11.1261	0.625893
$317$	33.2115	1.86534	0.932671	−	0.360728i	$-0.117472\pi$
$317$	33.2115	1.86534	0.932671	+	0.360728i	$0.117472\pi$
$318$	14.3273	0.803433
$319$	2.32803	0.130345
$320$	3.67831	0.205624
$321$	10.5418	0.588384
$322$	0	0
$323$	−27.9682	−1.55619
$324$	−10.8580	−0.603221
$325$	44.9603	2.49395
$326$	5.82355	0.322537
$327$	−24.8690	−1.37526
$328$	10.3634	0.572223
$329$	0	0
$330$	−19.3879	−1.06727
$331$	11.2459	0.618130	0.309065	−	0.951041i	$-0.399984\pi$
$331$	11.2459	0.618130	0.309065	+	0.951041i	$0.399984\pi$
$332$	−10.3634	−0.568766
$333$	−4.25224	−0.233021
$334$	−6.22795	−0.340778
$335$	−8.28443	−0.452627
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	14.7822	0.804045
$339$	47.1105	2.55869
$340$	−17.3121	−0.938882
$341$	4.60612	0.249435
$342$	12.6343	0.683182
$343$	0	0
$344$	−12.1860	−0.657025
$345$	−72.0880	−3.88109
$346$	2.36452	0.127117
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−2.26409	−0.121368
$349$	−28.3622	−1.51819	−0.759097	−	0.650978i	$-0.774359\pi$
$349$	−28.3622	−1.51819	−0.759097	+	0.650978i	$0.774359\pi$
$350$	0	0
$351$	−10.4287	−0.556642
$352$	−2.32803	−0.124084
$353$	−5.74155	−0.305592	−0.152796	−	0.988258i	$-0.548828\pi$
$353$	−5.74155	−0.305592	−0.152796	+	0.988258i	$0.548828\pi$
$354$	−16.2522	−0.863797
$355$	29.4265	1.56179
$356$	5.83521	0.309266
$357$	0	0
$358$	−8.50448	−0.449476
$359$	16.4382	0.867577	0.433789	−	0.901015i	$-0.357176\pi$
$359$	16.4382	0.867577	0.433789	+	0.901015i	$0.357176\pi$
$360$	7.82052	0.412178
$361$	16.3121	0.858533
$362$	−7.99212	−0.420057
$363$	−12.6343	−0.663126
$364$	0	0
$365$	−26.4038	−1.38204
$366$	−25.1605	−1.31516
$367$	14.0485	0.733324	0.366662	−	0.930354i	$-0.380501\pi$
$367$	14.0485	0.733324	0.366662	+	0.930354i	$0.380501\pi$
$368$	−8.65606	−0.451228
$369$	22.0338	1.14704
$370$	−7.35661	−0.382452
$371$	0	0
$372$	−4.47962	−0.232257
$373$	−17.4981	−0.906019	−0.453009	−	0.891506i	$-0.649650\pi$
$373$	−17.4981	−0.906019	−0.453009	+	0.891506i	$0.649650\pi$
$374$	10.9570	0.566572
$375$	29.3975	1.51808
$376$	−1.97855	−0.102036
$377$	−5.27088	−0.271464
$378$	0	0
$379$	−7.89935	−0.405762	−0.202881	−	0.979203i	$-0.565030\pi$
$379$	−7.89935	−0.405762	−0.202881	+	0.979203i	$0.565030\pi$
$380$	21.8580	1.12129
$381$	5.09928	0.261244
$382$	1.74776	0.0894232
$383$	−11.5146	−0.588367	−0.294183	−	0.955749i	$-0.595048\pi$
$383$	−11.5146	−0.588367	−0.294183	+	0.955749i	$0.595048\pi$
$384$	2.26409	0.115539
$385$	0	0
$386$	−7.49552	−0.381512
$387$	−25.9089	−1.31702
$388$	4.13545	0.209946
$389$	0.756717	0.0383671	0.0191835	−	0.999816i	$-0.493893\pi$
$389$	0.756717	0.0383671	0.0191835	+	0.999816i	$0.493893\pi$
$390$	43.8961	2.22276
$391$	40.7402	2.06032
$392$	0	0
$393$	21.4733	1.08318
$394$	22.6561	1.14140
$395$	40.9253	2.05917
$396$	−4.94967	−0.248730
$397$	−27.4966	−1.38001	−0.690007	−	0.723803i	$-0.742393\pi$
$397$	−27.4966	−1.38001	−0.690007	+	0.723803i	$0.742393\pi$
$398$	−17.5417	−0.879284
$399$	0	0
$400$	8.52994	0.426497
$401$	11.9242	0.595467	0.297733	−	0.954649i	$-0.403769\pi$
$401$	11.9242	0.595467	0.297733	+	0.954649i	$0.403769\pi$
$402$	−5.09928	−0.254329
$403$	−10.4287	−0.519490
$404$	3.75624	0.186880
$405$	−39.9390	−1.98458
$406$	0	0
$407$	4.65606	0.230792
$408$	−10.6561	−0.527554
$409$	36.7471	1.81703	0.908513	−	0.417858i	$-0.137219\pi$
$409$	36.7471	1.81703	0.908513	+	0.417858i	$0.137219\pi$
$410$	38.1198	1.88260
$411$	29.7967	1.46976
$412$	8.48528	0.418040
$413$	0	0
$414$	−18.4038	−0.904499
$415$	−38.1198	−1.87123
$416$	5.27088	0.258426
$417$	−19.6153	−0.960565
$418$	−13.8341	−0.676648
$419$	−1.16468	−0.0568983	−0.0284492	−	0.999595i	$-0.509057\pi$
$419$	−1.16468	−0.0568983	−0.0284492	+	0.999595i	$0.509057\pi$
$420$	0	0
$421$	−25.4128	−1.23854	−0.619272	−	0.785177i	$-0.712572\pi$
$421$	−25.4128	−1.23854	−0.619272	+	0.785177i	$0.712572\pi$
$422$	−16.4796	−0.802215
$423$	−4.20663	−0.204533
$424$	6.32803	0.307316
$425$	−40.1466	−1.94740
$426$	18.1127	0.877566
$427$	0	0
$428$	4.65606	0.225059
$429$	−27.7822	−1.34134
$430$	−44.8239	−2.16160
$431$	30.4038	1.46450	0.732250	−	0.681036i	$-0.238470\pi$
$431$	30.4038	1.46450	0.732250	+	0.681036i	$0.238470\pi$
$432$	−1.97855	−0.0951929
$433$	−14.5349	−0.698501	−0.349251	−	0.937029i	$-0.613564\pi$
$433$	−14.5349	−0.698501	−0.349251	+	0.937029i	$0.613564\pi$
$434$	0	0
$435$	−8.32803	−0.399299
$436$	−10.9841	−0.526043
$437$	−51.4378	−2.46060
$438$	−16.2522	−0.776562
$439$	1.91413	0.0913566	0.0456783	−	0.998956i	$-0.485455\pi$
$439$	1.91413	0.0913566	0.0456783	+	0.998956i	$0.485455\pi$
$440$	−8.56321	−0.408235
$441$	0	0
$442$	−24.8076	−1.17998
$443$	−37.1797	−1.76646	−0.883229	−	0.468941i	$-0.844636\pi$
$443$	−37.1797	−1.76646	−0.883229	+	0.468941i	$0.844636\pi$
$444$	−4.52819	−0.214898
$445$	21.4637	1.01748
$446$	−21.0835	−0.998334
$447$	−11.6060	−0.548946
$448$	0	0
$449$	−36.2714	−1.71175	−0.855876	−	0.517182i	$-0.826981\pi$
$449$	−36.2714	−1.71175	−0.855876	+	0.517182i	$0.826981\pi$
$450$	18.1357	0.854924
$451$	−24.1263	−1.13606
$452$	20.8076	0.978709
$453$	−34.0970	−1.60202
$454$	24.3047	1.14068
$455$	0	0
$456$	13.4542	0.630048
$457$	−16.9592	−0.793319	−0.396660	−	0.917966i	$-0.629831\pi$
$457$	−16.9592	−0.793319	−0.396660	+	0.917966i	$0.629831\pi$
$458$	12.0857	0.564726
$459$	9.31213	0.434653
$460$	−31.8397	−1.48453
$461$	24.6974	1.15027	0.575137	−	0.818057i	$-0.304949\pi$
$461$	24.6974	1.15027	0.575137	+	0.818057i	$0.304949\pi$
$462$	0	0
$463$	−0.807647	−0.0375345	−0.0187673	−	0.999824i	$-0.505974\pi$
$463$	−0.807647	−0.0375345	−0.0187673	+	0.999824i	$0.505974\pi$
$464$	−1.00000	−0.0464238
$465$	−16.4774	−0.764121
$466$	7.63060	0.353481
$467$	27.4254	1.26910	0.634548	−	0.772883i	$-0.281186\pi$
$467$	27.4254	1.26910	0.634548	+	0.772883i	$0.281186\pi$
$468$	11.2065	0.518022
$469$	0	0
$470$	−7.27770	−0.335695
$471$	−18.4038	−0.848003
$472$	−7.17825	−0.330406
$473$	28.3694	1.30443
$474$	25.1906	1.15704
$475$	50.6883	2.32574
$476$	0	0
$477$	13.4542	0.616023
$478$	0.504478	0.0230743
$479$	−18.1420	−0.828929	−0.414465	−	0.910065i	$-0.636031\pi$
$479$	−18.1420	−0.828929	−0.414465	+	0.910065i	$0.636031\pi$
$480$	8.32803	0.380121
$481$	−10.5418	−0.480663
$482$	−21.8622	−0.995797
$483$	0	0
$484$	−5.58027	−0.253649
$485$	15.2115	0.690717
$486$	−18.6478	−0.845883
$487$	−17.8993	−0.811097	−0.405548	−	0.914074i	$-0.632919\pi$
$487$	−17.8993	−0.811097	−0.405548	+	0.914074i	$0.632919\pi$
$488$	−11.1129	−0.503055
$489$	13.1851	0.596249
$490$	0	0
$491$	10.1446	0.457821	0.228910	−	0.973447i	$-0.426484\pi$
$491$	10.1446	0.457821	0.228910	+	0.973447i	$0.426484\pi$
$492$	23.4637	1.05783
$493$	4.70655	0.211972
$494$	31.3217	1.40923
$495$	−18.2064	−0.818318
$496$	−1.97855	−0.0888394
$497$	0	0
$498$	−23.4637	−1.05143
$499$	−24.1516	−1.08117	−0.540587	−	0.841288i	$-0.681798\pi$
$499$	−24.1516	−1.08117	−0.540587	+	0.841288i	$0.681798\pi$
$500$	12.9842	0.580672
$501$	−14.1007	−0.629971
$502$	9.87024	0.440530
$503$	33.9254	1.51266	0.756329	−	0.654191i	$-0.226991\pi$
$503$	33.9254	1.51266	0.756329	+	0.654191i	$0.226991\pi$
$504$	0	0
$505$	13.8166	0.614831
$506$	20.1516	0.895847
$507$	33.4682	1.48638
$508$	2.25224	0.0999269
$509$	−39.1183	−1.73389	−0.866945	−	0.498404i	$-0.833920\pi$
$509$	−39.1183	−1.73389	−0.866945	+	0.498404i	$0.833920\pi$
$510$	−39.1963	−1.73564
$511$	0	0
$512$	1.00000	0.0441942
$513$	−11.7573	−0.519098
$514$	11.6772	0.515059
$515$	31.2115	1.37534
$516$	−27.5903	−1.21459
$517$	4.60612	0.202577
$518$	0	0
$519$	5.35350	0.234992
$520$	19.3879	0.850216
$521$	14.7200	0.644894	0.322447	−	0.946587i	$-0.395495\pi$
$521$	14.7200	0.644894	0.322447	+	0.946587i	$0.395495\pi$
$522$	−2.12612	−0.0930577
$523$	38.8035	1.69676	0.848380	−	0.529388i	$-0.177578\pi$
$523$	38.8035	1.69676	0.848380	+	0.529388i	$0.177578\pi$
$524$	9.48427	0.414322
$525$	0	0
$526$	−17.6815	−0.770951
$527$	9.31213	0.405643
$528$	−5.27088	−0.229386
$529$	51.9274	2.25771
$530$	23.2764	1.01106
$531$	−15.2618	−0.662307
$532$	0	0
$533$	54.6243	2.36604
$534$	13.2115	0.571716
$535$	17.1264	0.740440
$536$	−2.25224	−0.0972819
$537$	−19.2549	−0.830912
$538$	−4.32733	−0.186565
$539$	0	0
$540$	−7.27770	−0.313183
$541$	16.7758	0.721249	0.360625	−	0.932711i	$-0.382564\pi$
$541$	16.7758	0.721249	0.360625	+	0.932711i	$0.382564\pi$
$542$	24.8690	1.06822
$543$	−18.0949	−0.776527
$544$	−4.70655	−0.201792
$545$	−40.4029	−1.73067
$546$	0	0
$547$	−19.8993	−0.850835	−0.425417	−	0.904997i	$-0.639873\pi$
$547$	−19.8993	−0.850835	−0.425417	+	0.904997i	$0.639873\pi$
$548$	13.1605	0.562190
$549$	−23.6273	−1.00839
$550$	−19.8580	−0.846747
$551$	−5.94240	−0.253155
$552$	−19.5981	−0.834152
$553$	0	0
$554$	−5.64711	−0.239923
$555$	−16.6561	−0.707010
$556$	−8.66364	−0.367420
$557$	40.1198	1.69993	0.849965	−	0.526840i	$-0.176623\pi$
$557$	40.1198	1.69993	0.849965	+	0.526840i	$0.176623\pi$
$558$	−4.20663	−0.178081
$559$	−64.2310	−2.71668
$560$	0	0
$561$	24.8076	1.04738
$562$	14.2274	0.600146
$563$	−39.1895	−1.65164	−0.825820	−	0.563933i	$-0.809288\pi$
$563$	−39.1895	−1.65164	−0.825820	+	0.563933i	$0.809288\pi$
$564$	−4.47962	−0.188626
$565$	76.5369	3.21993
$566$	18.2911	0.768833
$567$	0	0
$568$	8.00000	0.335673
$569$	41.2115	1.72767	0.863837	−	0.503771i	$-0.168054\pi$
$569$	41.2115	1.72767	0.863837	+	0.503771i	$0.168054\pi$
$570$	49.4885	2.07285
$571$	−33.4128	−1.39828	−0.699141	−	0.714984i	$-0.746434\pi$
$571$	−33.4128	−1.39828	−0.699141	+	0.714984i	$0.746434\pi$
$572$	−12.2708	−0.513067
$573$	3.95709	0.165310
$574$	0	0
$575$	−73.8357	−3.07916
$576$	2.12612	0.0885883
$577$	−32.0765	−1.33536	−0.667682	−	0.744447i	$-0.732713\pi$
$577$	−32.0765	−1.33536	−0.667682	+	0.744447i	$0.732713\pi$
$578$	5.15158	0.214278
$579$	−16.9706	−0.705273
$580$	−3.67831	−0.152733
$581$	0	0
$582$	9.36306	0.388111
$583$	−14.7319	−0.610131
$584$	−7.17825	−0.297038
$585$	41.2210	1.70428
$586$	−7.71333	−0.318635
$587$	−28.2033	−1.16407	−0.582037	−	0.813162i	$-0.697744\pi$
$587$	−28.2033	−1.16407	−0.582037	+	0.813162i	$0.697744\pi$
$588$	0	0
$589$	−11.7573	−0.484452
$590$	−26.4038	−1.08703
$591$	51.2954	2.11001
$592$	−2.00000	−0.0821995
$593$	−42.5890	−1.74892	−0.874461	−	0.485096i	$-0.838785\pi$
$593$	−42.5890	−1.74892	−0.874461	+	0.485096i	$0.838785\pi$
$594$	4.60612	0.188991
$595$	0	0
$596$	−5.12612	−0.209974
$597$	−39.7159	−1.62547
$598$	−45.6251	−1.86575
$599$	−10.3185	−0.421601	−0.210801	−	0.977529i	$-0.567607\pi$
$599$	−10.3185	−0.421601	−0.210801	+	0.977529i	$0.567607\pi$
$600$	19.3126	0.788433
$601$	−34.2754	−1.39812	−0.699060	−	0.715063i	$-0.746398\pi$
$601$	−34.2754	−1.39812	−0.699060	+	0.715063i	$0.746398\pi$
$602$	0	0
$603$	−4.78853	−0.195004
$604$	−15.0599	−0.612778
$605$	−20.5259	−0.834498
$606$	8.50448	0.345471
$607$	−11.6997	−0.474875	−0.237438	−	0.971403i	$-0.576308\pi$
$607$	−11.6997	−0.474875	−0.237438	+	0.971403i	$0.576308\pi$
$608$	5.94240	0.240996
$609$	0	0
$610$	−40.8765	−1.65504
$611$	−10.4287	−0.421899
$612$	−10.0067	−0.404496
$613$	−14.1351	−0.570910	−0.285455	−	0.958392i	$-0.592145\pi$
$613$	−14.1351	−0.570910	−0.285455	+	0.958392i	$0.592145\pi$
$614$	30.5034	1.23102
$615$	86.3067	3.48022
$616$	0	0
$617$	−21.0599	−0.847839	−0.423920	−	0.905700i	$-0.639346\pi$
$617$	−21.0599	−0.847839	−0.423920	+	0.905700i	$0.639346\pi$
$618$	19.2115	0.772799
$619$	−10.0261	−0.402983	−0.201492	−	0.979490i	$-0.564579\pi$
$619$	−10.0261	−0.402983	−0.201492	+	0.979490i	$0.564579\pi$
$620$	−7.27770	−0.292280
$621$	17.1264	0.687260
$622$	6.32162	0.253474
$623$	0	0
$624$	11.9338	0.477733
$625$	5.11021	0.204409
$626$	30.5970	1.22290
$627$	−31.3217	−1.25087
$628$	−8.12856	−0.324365
$629$	9.41309	0.375325
$630$	0	0
$631$	−14.4038	−0.573407	−0.286704	−	0.958019i	$-0.592559\pi$
$631$	−14.4038	−0.573407	−0.286704	+	0.958019i	$0.592559\pi$
$632$	11.1261	0.442573
$633$	−37.3114	−1.48299
$634$	33.2115	1.31900
$635$	8.28443	0.328757
$636$	14.3273	0.568113
$637$	0	0
$638$	2.32803	0.0921676
$639$	17.0090	0.672864
$640$	3.67831	0.145398
$641$	−40.2714	−1.59062	−0.795311	−	0.606201i	$-0.792693\pi$
$641$	−40.2714	−1.59062	−0.795311	+	0.606201i	$0.792693\pi$
$642$	10.5418	0.416050
$643$	−2.23484	−0.0881335	−0.0440667	−	0.999029i	$-0.514031\pi$
$643$	−2.23484	−0.0881335	−0.0440667	+	0.999029i	$0.514031\pi$
$644$	0	0
$645$	−101.485	−3.99599
$646$	−27.9682	−1.10039
$647$	−11.1714	−0.439192	−0.219596	−	0.975591i	$-0.570474\pi$
$647$	−11.1714	−0.439192	−0.219596	+	0.975591i	$0.570474\pi$
$648$	−10.8580	−0.426542
$649$	16.7112	0.655972
$650$	44.9603	1.76349
$651$	0	0
$652$	5.82355	0.228068
$653$	26.1516	1.02339	0.511695	−	0.859167i	$-0.329018\pi$
$653$	26.1516	1.02339	0.511695	+	0.859167i	$0.329018\pi$
$654$	−24.8690	−0.972455
$655$	34.8860	1.36311
$656$	10.3634	0.404623
$657$	−15.2618	−0.595421
$658$	0	0
$659$	21.9751	0.856030	0.428015	−	0.903772i	$-0.359213\pi$
$659$	21.9751	0.856030	0.428015	+	0.903772i	$0.359213\pi$
$660$	−19.3879	−0.754674
$661$	16.1049	0.626410	0.313205	−	0.949686i	$-0.398597\pi$
$661$	16.1049	0.626410	0.313205	+	0.949686i	$0.398597\pi$
$662$	11.2459	0.437084
$663$	−56.1668	−2.18134
$664$	−10.3634	−0.402178
$665$	0	0
$666$	−4.25224	−0.164771
$667$	8.65606	0.335164
$668$	−6.22795	−0.240967
$669$	−47.7351	−1.84554
$670$	−8.28443	−0.320055
$671$	25.8711	0.998742
$672$	0	0
$673$	43.5395	1.67832	0.839162	−	0.543881i	$-0.183046\pi$
$673$	43.5395	1.67832	0.839162	+	0.543881i	$0.183046\pi$
$674$	10.0000	0.385186
$675$	−16.8769	−0.649592
$676$	14.7822	0.568545
$677$	−12.0271	−0.462241	−0.231120	−	0.972925i	$-0.574239\pi$
$677$	−12.0271	−0.462241	−0.231120	+	0.972925i	$0.574239\pi$
$678$	47.1105	1.80927
$679$	0	0
$680$	−17.3121	−0.663890
$681$	55.0281	2.10868
$682$	4.60612	0.176377
$683$	−11.2115	−0.428995	−0.214498	−	0.976725i	$-0.568811\pi$
$683$	−11.2115	−0.428995	−0.214498	+	0.976725i	$0.568811\pi$
$684$	12.6343	0.483083
$685$	48.4085	1.84959
$686$	0	0
$687$	27.3631	1.04397
$688$	−12.1860	−0.464587
$689$	33.3543	1.27070
$690$	−72.0880	−2.74434
$691$	−17.0904	−0.650150	−0.325075	−	0.945688i	$-0.605390\pi$
$691$	−17.0904	−0.650150	−0.325075	+	0.945688i	$0.605390\pi$
$692$	2.36452	0.0898856
$693$	0	0
$694$	6.00000	0.227757
$695$	−31.8675	−1.20880
$696$	−2.26409	−0.0858202
$697$	−48.7758	−1.84752
$698$	−28.3622	−1.07353
$699$	17.2764	0.653453
$700$	0	0
$701$	−36.4892	−1.37818	−0.689089	−	0.724677i	$-0.741989\pi$
$701$	−36.4892	−1.37818	−0.689089	+	0.724677i	$0.741989\pi$
$702$	−10.4287	−0.393605
$703$	−11.8848	−0.448244
$704$	−2.32803	−0.0877410
$705$	−16.4774	−0.620575
$706$	−5.74155	−0.216086
$707$	0	0
$708$	−16.2522	−0.610797
$709$	19.9433	0.748987	0.374494	−	0.927229i	$-0.377817\pi$
$709$	19.9433	0.748987	0.374494	+	0.927229i	$0.377817\pi$
$710$	29.4265	1.10436
$711$	23.6555	0.887149
$712$	5.83521	0.218684
$713$	17.1264	0.641390
$714$	0	0
$715$	−45.1357	−1.68798
$716$	−8.50448	−0.317827
$717$	1.14219	0.0426557
$718$	16.4382	0.613470
$719$	−33.2412	−1.23969	−0.619844	−	0.784725i	$-0.712804\pi$
$719$	−33.2412	−1.23969	−0.619844	+	0.784725i	$0.712804\pi$
$720$	7.82052	0.291454
$721$	0	0
$722$	16.3121	0.607074
$723$	−49.4981	−1.84086
$724$	−7.99212	−0.297025
$725$	−8.52994	−0.316794
$726$	−12.6343	−0.468901
$727$	36.3192	1.34700	0.673502	−	0.739186i	$-0.264789\pi$
$727$	36.3192	1.34700	0.673502	+	0.739186i	$0.264789\pi$
$728$	0	0
$729$	−9.64650	−0.357278
$730$	−26.4038	−0.977249
$731$	57.3540	2.12131
$732$	−25.1605	−0.929960
$733$	13.1558	0.485921	0.242961	−	0.970036i	$-0.421881\pi$
$733$	13.1558	0.485921	0.242961	+	0.970036i	$0.421881\pi$
$734$	14.0485	0.518538
$735$	0	0
$736$	−8.65606	−0.319067
$737$	5.24328	0.193139
$738$	22.0338	0.811076
$739$	14.7414	0.542272	0.271136	−	0.962541i	$-0.412601\pi$
$739$	14.7414	0.542272	0.271136	+	0.962541i	$0.412601\pi$
$740$	−7.35661	−0.270435
$741$	70.9152	2.60514
$742$	0	0
$743$	50.9274	1.86835	0.934173	−	0.356821i	$-0.116139\pi$
$743$	50.9274	1.86835	0.934173	+	0.356821i	$0.116139\pi$
$744$	−4.47962	−0.164231
$745$	−18.8554	−0.690810
$746$	−17.4981	−0.640652
$747$	−22.0338	−0.806176
$748$	10.9570	0.400627
$749$	0	0
$750$	29.3975	1.07344
$751$	10.1198	0.369276	0.184638	−	0.982807i	$-0.440889\pi$
$751$	10.1198	0.369276	0.184638	+	0.982807i	$0.440889\pi$
$752$	−1.97855	−0.0721502
$753$	22.3471	0.814375
$754$	−5.27088	−0.191954
$755$	−55.3949	−2.01603
$756$	0	0
$757$	41.3631	1.50337	0.751683	−	0.659525i	$-0.229242\pi$
$757$	41.3631	1.50337	0.751683	+	0.659525i	$0.229242\pi$
$758$	−7.89935	−0.286917
$759$	45.6251	1.65609
$760$	21.8580	0.792872
$761$	24.2110	0.877649	0.438824	−	0.898573i	$-0.355395\pi$
$761$	24.2110	0.877649	0.438824	+	0.898573i	$0.355395\pi$
$762$	5.09928	0.184727
$763$	0	0
$764$	1.74776	0.0632318
$765$	−36.8076	−1.33078
$766$	−11.5146	−0.416038
$767$	−37.8357	−1.36617
$768$	2.26409	0.0816984
$769$	3.99311	0.143995	0.0719976	−	0.997405i	$-0.477063\pi$
$769$	3.99311	0.143995	0.0719976	+	0.997405i	$0.477063\pi$
$770$	0	0
$771$	26.4382	0.952150
$772$	−7.49552	−0.269770
$773$	−2.42672	−0.0872830	−0.0436415	−	0.999047i	$-0.513896\pi$
$773$	−2.42672	−0.0872830	−0.0436415	+	0.999047i	$0.513896\pi$
$774$	−25.9089	−0.931276
$775$	−16.8769	−0.606236
$776$	4.13545	0.148454
$777$	0	0
$778$	0.756717	0.0271296
$779$	61.5835	2.20646
$780$	43.8961	1.57173
$781$	−18.6243	−0.666428
$782$	40.7402	1.45687
$783$	1.97855	0.0707075
$784$	0	0
$785$	−29.8993	−1.06715
$786$	21.4733	0.765926
$787$	−50.2731	−1.79204	−0.896021	−	0.444011i	$-0.853555\pi$
$787$	−50.2731	−1.79204	−0.896021	+	0.444011i	$0.853555\pi$
$788$	22.6561	0.807089
$789$	−40.0326	−1.42520
$790$	40.9253	1.45606
$791$	0	0
$792$	−4.94967	−0.175879
$793$	−58.5745	−2.08004
$794$	−27.4966	−0.975817
$795$	52.7000	1.86908
$796$	−17.5417	−0.621747
$797$	−1.64125	−0.0581361	−0.0290681	−	0.999577i	$-0.509254\pi$
$797$	−1.64125	−0.0581361	−0.0290681	+	0.999577i	$0.509254\pi$
$798$	0	0
$799$	9.31213	0.329439
$800$	8.52994	0.301579
$801$	12.4064	0.438357
$802$	11.9242	0.421058
$803$	16.7112	0.589725
$804$	−5.09928	−0.179838
$805$	0	0
$806$	−10.4287	−0.367335
$807$	−9.79749	−0.344888
$808$	3.75624	0.132144
$809$	−3.34394	−0.117567	−0.0587833	−	0.998271i	$-0.518722\pi$
$809$	−3.34394	−0.117567	−0.0587833	+	0.998271i	$0.518722\pi$
$810$	−39.9390	−1.40331
$811$	−3.00679	−0.105583	−0.0527913	−	0.998606i	$-0.516812\pi$
$811$	−3.00679	−0.105583	−0.0527913	+	0.998606i	$0.516812\pi$
$812$	0	0
$813$	56.3058	1.97473
$814$	4.65606	0.163195
$815$	21.4208	0.750338
$816$	−10.6561	−0.373037
$817$	−72.4141	−2.53345
$818$	36.7471	1.28483
$819$	0	0
$820$	38.1198	1.33120
$821$	39.9433	1.39403	0.697016	−	0.717056i	$-0.254511\pi$
$821$	39.9433	1.39403	0.697016	+	0.717056i	$0.254511\pi$
$822$	29.7967	1.03928
$823$	17.0090	0.592895	0.296447	−	0.955049i	$-0.404198\pi$
$823$	17.0090	0.592895	0.296447	+	0.955049i	$0.404198\pi$
$824$	8.48528	0.295599
$825$	−44.9603	−1.56532
$826$	0	0
$827$	33.0943	1.15080	0.575401	−	0.817872i	$-0.304846\pi$
$827$	33.0943	1.15080	0.575401	+	0.817872i	$0.304846\pi$
$828$	−18.4038	−0.639577
$829$	−7.15576	−0.248530	−0.124265	−	0.992249i	$-0.539657\pi$
$829$	−7.15576	−0.248530	−0.124265	+	0.992249i	$0.539657\pi$
$830$	−38.1198	−1.32316
$831$	−12.7856	−0.443527
$832$	5.27088	0.182735
$833$	0	0
$834$	−19.6153	−0.679222
$835$	−22.9083	−0.792774
$836$	−13.8341	−0.478462
$837$	3.91465	0.135310
$838$	−1.16468	−0.0402332
$839$	0.684186	0.0236207	0.0118104	−	0.999930i	$-0.496241\pi$
$839$	0.684186	0.0236207	0.0118104	+	0.999930i	$0.496241\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−25.4128	−0.875782
$843$	32.2121	1.10944
$844$	−16.4796	−0.567252
$845$	54.3734	1.87050
$846$	−4.20663	−0.144627
$847$	0	0
$848$	6.32803	0.217305
$849$	41.4128	1.42128
$850$	−40.1466	−1.37702
$851$	17.3121	0.593452
$852$	18.1127	0.620533
$853$	−32.6701	−1.11860	−0.559302	−	0.828964i	$-0.688931\pi$
$853$	−32.6701	−1.11860	−0.559302	+	0.828964i	$0.688931\pi$
$854$	0	0
$855$	46.4727	1.58933
$856$	4.65606	0.159141
$857$	18.0475	0.616490	0.308245	−	0.951307i	$-0.400258\pi$
$857$	18.0475	0.616490	0.308245	+	0.951307i	$0.400258\pi$
$858$	−27.7822	−0.948468
$859$	32.4490	1.10714	0.553572	−	0.832801i	$-0.313264\pi$
$859$	32.4490	1.10714	0.553572	+	0.832801i	$0.313264\pi$
$860$	−44.8239	−1.52848
$861$	0	0
$862$	30.4038	1.03556
$863$	0.132467	0.00450922	0.00225461	−	0.999997i	$-0.499282\pi$
$863$	0.132467	0.00450922	0.00225461	+	0.999997i	$0.499282\pi$
$864$	−1.97855	−0.0673115
$865$	8.69743	0.295722
$866$	−14.5349	−0.493915
$867$	11.6637	0.396119
$868$	0	0
$869$	−25.9020	−0.878664
$870$	−8.32803	−0.282347
$871$	−11.8713	−0.402243
$872$	−10.9841	−0.371968
$873$	8.79247	0.297580
$874$	−51.4378	−1.73991
$875$	0	0
$876$	−16.2522	−0.549112
$877$	1.56176	0.0527367	0.0263684	−	0.999652i	$-0.491606\pi$
$877$	1.56176	0.0527367	0.0263684	+	0.999652i	$0.491606\pi$
$878$	1.91413	0.0645989
$879$	−17.4637	−0.589036
$880$	−8.56321	−0.288666
$881$	10.0202	0.337589	0.168795	−	0.985651i	$-0.446013\pi$
$881$	10.0202	0.337589	0.168795	+	0.985651i	$0.446013\pi$
$882$	0	0
$883$	−5.96819	−0.200846	−0.100423	−	0.994945i	$-0.532020\pi$
$883$	−5.96819	−0.200846	−0.100423	+	0.994945i	$0.532020\pi$
$884$	−24.8076	−0.834371
$885$	−59.7807	−2.00951
$886$	−37.1797	−1.24908
$887$	13.4481	0.451544	0.225772	−	0.974180i	$-0.427510\pi$
$887$	13.4481	0.451544	0.225772	+	0.974180i	$0.427510\pi$
$888$	−4.52819	−0.151956
$889$	0	0
$890$	21.4637	0.719465
$891$	25.2777	0.846835
$892$	−21.0835	−0.705929
$893$	−11.7573	−0.393444
$894$	−11.6060	−0.388163
$895$	−31.2821	−1.04564
$896$	0	0
$897$	−103.299	−3.44907
$898$	−36.2714	−1.21039
$899$	1.97855	0.0659882
$900$	18.1357	0.604523
$901$	−29.7832	−0.992221
$902$	−24.1263	−0.803319
$903$	0	0
$904$	20.8076	0.692052
$905$	−29.3975	−0.977205
$906$	−34.0970	−1.13280
$907$	33.6841	1.11846	0.559232	−	0.829011i	$-0.311096\pi$
$907$	33.6841	1.11846	0.559232	+	0.829011i	$0.311096\pi$
$908$	24.3047	0.806579
$909$	7.98621	0.264886
$910$	0	0
$911$	−22.5580	−0.747380	−0.373690	−	0.927554i	$-0.621908\pi$
$911$	−22.5580	−0.747380	−0.373690	+	0.927554i	$0.621908\pi$
$912$	13.4542	0.445511
$913$	24.1263	0.798465
$914$	−16.9592	−0.560962
$915$	−92.5482	−3.05955
$916$	12.0857	0.399321
$917$	0	0
$918$	9.31213	0.307346
$919$	−24.0688	−0.793958	−0.396979	−	0.917828i	$-0.629941\pi$
$919$	−24.0688	−0.793958	−0.396979	+	0.917828i	$0.629941\pi$
$920$	−31.8397	−1.04972
$921$	69.0625	2.27569
$922$	24.6974	0.813366
$923$	42.1670	1.38795
$924$	0	0
$925$	−17.0599	−0.560926
$926$	−0.807647	−0.0265409
$927$	18.0407	0.592535
$928$	−1.00000	−0.0328266
$929$	−39.9944	−1.31217	−0.656087	−	0.754685i	$-0.727790\pi$
$929$	−39.9944	−1.31217	−0.656087	+	0.754685i	$0.727790\pi$
$930$	−16.4774	−0.540315
$931$	0	0
$932$	7.63060	0.249949
$933$	14.3127	0.468578
$934$	27.4254	0.897387
$935$	40.3032	1.31805
$936$	11.2065	0.366297
$937$	−49.8812	−1.62955	−0.814774	−	0.579778i	$-0.803139\pi$
$937$	−49.8812	−1.62955	−0.814774	+	0.579778i	$0.803139\pi$
$938$	0	0
$939$	69.2746	2.26069
$940$	−7.27770	−0.237373
$941$	13.1401	0.428354	0.214177	−	0.976795i	$-0.431293\pi$
$941$	13.1401	0.428354	0.214177	+	0.976795i	$0.431293\pi$
$942$	−18.4038	−0.599629
$943$	−89.7062	−2.92124
$944$	−7.17825	−0.233632
$945$	0	0
$946$	28.3694	0.922369
$947$	−27.2459	−0.885373	−0.442686	−	0.896677i	$-0.645974\pi$
$947$	−27.2459	−0.885373	−0.442686	+	0.896677i	$0.645974\pi$
$948$	25.1906	0.818152
$949$	−37.8357	−1.22820
$950$	50.6883	1.64455
$951$	75.1939	2.43833
$952$	0	0
$953$	40.3376	1.30666	0.653331	−	0.757072i	$-0.273371\pi$
$953$	40.3376	1.30666	0.653331	+	0.757072i	$0.273371\pi$
$954$	13.4542	0.435594
$955$	6.42880	0.208031
$956$	0.504478	0.0163160
$957$	5.27088	0.170383
$958$	−18.1420	−0.586141
$959$	0	0
$960$	8.32803	0.268786
$961$	−27.0854	−0.873721
$962$	−10.5418	−0.339880
$963$	9.89935	0.319002
$964$	−21.8622	−0.704135
$965$	−27.5708	−0.887536
$966$	0	0
$967$	47.3879	1.52389	0.761946	−	0.647640i	$-0.224244\pi$
$967$	47.3879	1.52389	0.761946	+	0.647640i	$0.224244\pi$
$968$	−5.58027	−0.179357
$969$	−63.3226	−2.03421
$970$	15.2115	0.488411
$971$	9.55629	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$971$	9.55629	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$972$	−18.6478	−0.598129
$973$	0	0
$974$	−17.8993	−0.573532
$975$	101.794	3.26003
$976$	−11.1129	−0.355714
$977$	12.5485	0.401461	0.200730	−	0.979647i	$-0.435668\pi$
$977$	12.5485	0.401461	0.200730	+	0.979647i	$0.435668\pi$
$978$	13.1851	0.421612
$979$	−13.5846	−0.434164
$980$	0	0
$981$	−23.3535	−0.745620
$982$	10.1446	0.323728
$983$	1.51464	0.0483095	0.0241548	−	0.999708i	$-0.492311\pi$
$983$	1.51464	0.0483095	0.0241548	+	0.999708i	$0.492311\pi$
$984$	23.4637	0.747996
$985$	83.3360	2.65530
$986$	4.70655	0.149887
$987$	0	0
$988$	31.3217	0.996476
$989$	105.483	3.35416
$990$	−18.2064	−0.578638
$991$	−2.53629	−0.0805679	−0.0402840	−	0.999188i	$-0.512826\pi$
$991$	−2.53629	−0.0805679	−0.0402840	+	0.999188i	$0.512826\pi$
$992$	−1.97855	−0.0628189
$993$	25.4618	0.808004
$994$	0	0
$995$	−64.5236	−2.04554
$996$	−23.4637	−0.743476
$997$	−29.0112	−0.918795	−0.459397	−	0.888231i	$-0.651935\pi$
$997$	−29.0112	−0.918795	−0.459397	+	0.888231i	$0.651935\pi$
$998$	−24.1516	−0.764505
$999$	3.95709	0.125197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.bb.1.5	yes	6
7.6	odd	2	inner	2842.2.a.bb.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.bb.1.2	✓	6	7.6	odd	2	inner
2842.2.a.bb.1.5	yes	6	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:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.26409 q^{3} +1.00000 q^{4} +3.67831 q^{5} +2.26409 q^{6} +1.00000 q^{8} +2.12612 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.26409 q^{3} +1.00000 q^{4} +3.67831 q^{5} +2.26409 q^{6} +1.00000 q^{8} +2.12612 q^{9} +3.67831 q^{10} -2.32803 q^{11} +2.26409 q^{12} +5.27088 q^{13} +8.32803 q^{15} +1.00000 q^{16} -4.70655 q^{17} +2.12612 q^{18} +5.94240 q^{19} +3.67831 q^{20} -2.32803 q^{22} -8.65606 q^{23} +2.26409 q^{24} +8.52994 q^{25} +5.27088 q^{26} -1.97855 q^{27} -1.00000 q^{29} +8.32803 q^{30} -1.97855 q^{31} +1.00000 q^{32} -5.27088 q^{33} -4.70655 q^{34} +2.12612 q^{36} -2.00000 q^{37} +5.94240 q^{38} +11.9338 q^{39} +3.67831 q^{40} +10.3634 q^{41} -12.1860 q^{43} -2.32803 q^{44} +7.82052 q^{45} -8.65606 q^{46} -1.97855 q^{47} +2.26409 q^{48} +8.52994 q^{50} -10.6561 q^{51} +5.27088 q^{52} +6.32803 q^{53} -1.97855 q^{54} -8.56321 q^{55} +13.4542 q^{57} -1.00000 q^{58} -7.17825 q^{59} +8.32803 q^{60} -11.1129 q^{61} -1.97855 q^{62} +1.00000 q^{64} +19.3879 q^{65} -5.27088 q^{66} -2.25224 q^{67} -4.70655 q^{68} -19.5981 q^{69} +8.00000 q^{71} +2.12612 q^{72} -7.17825 q^{73} -2.00000 q^{74} +19.3126 q^{75} +5.94240 q^{76} +11.9338 q^{78} +11.1261 q^{79} +3.67831 q^{80} -10.8580 q^{81} +10.3634 q^{82} -10.3634 q^{83} -17.3121 q^{85} -12.1860 q^{86} -2.26409 q^{87} -2.32803 q^{88} +5.83521 q^{89} +7.82052 q^{90} -8.65606 q^{92} -4.47962 q^{93} -1.97855 q^{94} +21.8580 q^{95} +2.26409 q^{96} +4.13545 q^{97} -4.94967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9} + 8 q^{11} + 28 q^{15} + 6 q^{16} + 10 q^{18} + 8 q^{22} - 8 q^{23} + 10 q^{25} - 6 q^{29} + 28 q^{30} + 6 q^{32} + 10 q^{36} - 12 q^{37} - 8 q^{39} + 12 q^{43} + 8 q^{44} - 8 q^{46} + 10 q^{50} - 20 q^{51} + 16 q^{53} + 56 q^{57} - 6 q^{58} + 28 q^{60} + 6 q^{64} + 12 q^{65} - 8 q^{67} + 48 q^{71} + 10 q^{72} - 12 q^{74} - 8 q^{78} + 64 q^{79} - 2 q^{81} - 16 q^{85} + 12 q^{86} + 8 q^{88} - 8 q^{92} + 28 q^{93} + 68 q^{95} - 16 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9 + 8 * q^11 + 28 * q^15 + 6 * q^16 + 10 * q^18 + 8 * q^22 - 8 * q^23 + 10 * q^25 - 6 * q^29 + 28 * q^30 + 6 * q^32 + 10 * q^36 - 12 * q^37 - 8 * q^39 + 12 * q^43 + 8 * q^44 - 8 * q^46 + 10 * q^50 - 20 * q^51 + 16 * q^53 + 56 * q^57 - 6 * q^58 + 28 * q^60 + 6 * q^64 + 12 * q^65 - 8 * q^67 + 48 * q^71 + 10 * q^72 - 12 * q^74 - 8 * q^78 + 64 * q^79 - 2 * q^81 - 16 * q^85 + 12 * q^86 + 8 * q^88 - 8 * q^92 + 28 * q^93 + 68 * q^95 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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