LMFDB - Embedded newform 560.6.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,6,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("560.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.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:	$89.8149390953$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 232x^{2} + 60x + 5808 $ x^4 - 2x^3 - 232x^2 + 60*x + 5808
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$15.2911$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.08191 q^{3} +25.0000 q^{5} -49.0000 q^{7} -206.010 q^{9} +O(q^{10})$	$q-6.08191 q^{3} +25.0000 q^{5} -49.0000 q^{7} -206.010 q^{9} -407.671 q^{11} -1075.68 q^{13} -152.048 q^{15} -1775.88 q^{17} -2156.68 q^{19} +298.014 q^{21} +4540.75 q^{23} +625.000 q^{25} +2730.84 q^{27} +6114.50 q^{29} +2856.57 q^{31} +2479.42 q^{33} -1225.00 q^{35} -7036.59 q^{37} +6542.17 q^{39} -11206.3 q^{41} -10684.8 q^{43} -5150.26 q^{45} +7637.19 q^{47} +2401.00 q^{49} +10800.7 q^{51} +19261.2 q^{53} -10191.8 q^{55} +13116.8 q^{57} +9188.24 q^{59} -33709.8 q^{61} +10094.5 q^{63} -26891.9 q^{65} +29762.1 q^{67} -27616.4 q^{69} +29860.8 q^{71} -40725.0 q^{73} -3801.19 q^{75} +19975.9 q^{77} +39180.1 q^{79} +33451.8 q^{81} -34423.7 q^{83} -44396.9 q^{85} -37187.8 q^{87} -7995.49 q^{89} +52708.2 q^{91} -17373.4 q^{93} -53917.1 q^{95} -58573.9 q^{97} +83984.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9}+O(q^{10}) $ 4 * q + 13 * q^3 + 100 * q^5 - 196 * q^7 - 193 * q^9	$ 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9} + 595 q^{11} - 969 q^{13} + 325 q^{15} - 1315 q^{17} + 1090 q^{19} - 637 q^{21} + 1534 q^{23} + 2500 q^{25} - 173 q^{27} + 4099 q^{29} + 4820 q^{31} + 4149 q^{33} - 4900 q^{35} + 7692 q^{37} + 6371 q^{39} - 9722 q^{41} + 20610 q^{43} - 4825 q^{45} + 1661 q^{47} + 9604 q^{49} + 73361 q^{51} - 28898 q^{53} + 14875 q^{55} - 21246 q^{57} + 101872 q^{59} - 24742 q^{61} + 9457 q^{63} - 24225 q^{65} + 82060 q^{67} + 16914 q^{69} + 102784 q^{71} - 80652 q^{73} + 8125 q^{75} - 29155 q^{77} + 117801 q^{79} - 141052 q^{81} + 155440 q^{83} - 32875 q^{85} + 82519 q^{87} + 56426 q^{89} + 47481 q^{91} - 17332 q^{93} + 27250 q^{95} - 261031 q^{97} + 61686 q^{99}+O(q^{100}) $ 4 * q + 13 * q^3 + 100 * q^5 - 196 * q^7 - 193 * q^9 + 595 * q^11 - 969 * q^13 + 325 * q^15 - 1315 * q^17 + 1090 * q^19 - 637 * q^21 + 1534 * q^23 + 2500 * q^25 - 173 * q^27 + 4099 * q^29 + 4820 * q^31 + 4149 * q^33 - 4900 * q^35 + 7692 * q^37 + 6371 * q^39 - 9722 * q^41 + 20610 * q^43 - 4825 * q^45 + 1661 * q^47 + 9604 * q^49 + 73361 * q^51 - 28898 * q^53 + 14875 * q^55 - 21246 * q^57 + 101872 * q^59 - 24742 * q^61 + 9457 * q^63 - 24225 * q^65 + 82060 * q^67 + 16914 * q^69 + 102784 * q^71 - 80652 * q^73 + 8125 * q^75 - 29155 * q^77 + 117801 * q^79 - 141052 * q^81 + 155440 * q^83 - 32875 * q^85 + 82519 * q^87 + 56426 * q^89 + 47481 * q^91 - 17332 * q^93 + 27250 * q^95 - 261031 * q^97 + 61686 * 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 $

$2$	0	0
$2$	0	0
$3$	−6.08191	−0.390155	−0.195077	−	0.980788i	$-0.562496\pi$
$3$	−6.08191	−0.390155	−0.195077	+	0.980788i	$0.562496\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−206.010	−0.847779
$10$	0	0
$11$	−407.671	−1.01585	−0.507924	−	0.861402i	$-0.669587\pi$
$11$	−407.671	−1.01585	−0.507924	+	0.861402i	$0.669587\pi$
$12$	0	0
$13$	−1075.68	−1.76532	−0.882661	−	0.470011i	$-0.844250\pi$
$13$	−1075.68	−1.76532	−0.882661	+	0.470011i	$0.844250\pi$
$14$	0	0
$15$	−152.048	−0.174482
$16$	0	0
$17$	−1775.88	−1.49036	−0.745178	−	0.666865i	$-0.767636\pi$
$17$	−1775.88	−1.49036	−0.745178	+	0.666865i	$0.767636\pi$
$18$	0	0
$19$	−2156.68	−1.37057	−0.685287	−	0.728273i	$-0.740323\pi$
$19$	−2156.68	−1.37057	−0.685287	+	0.728273i	$0.740323\pi$
$20$	0	0
$21$	298.014	0.147465
$22$	0	0
$23$	4540.75	1.78981	0.894907	−	0.446253i	$-0.147242\pi$
$23$	4540.75	1.78981	0.894907	+	0.446253i	$0.147242\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	2730.84	0.720920
$28$	0	0
$29$	6114.50	1.35010	0.675050	−	0.737772i	$-0.264122\pi$
$29$	6114.50	1.35010	0.675050	+	0.737772i	$0.264122\pi$
$30$	0	0
$31$	2856.57	0.533876	0.266938	−	0.963714i	$-0.413988\pi$
$31$	2856.57	0.533876	0.266938	+	0.963714i	$0.413988\pi$
$32$	0	0
$33$	2479.42	0.396337
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−7036.59	−0.845002	−0.422501	−	0.906362i	$-0.638848\pi$
$37$	−7036.59	−0.845002	−0.422501	+	0.906362i	$0.638848\pi$
$38$	0	0
$39$	6542.17	0.688748
$40$	0	0
$41$	−11206.3	−1.04113	−0.520563	−	0.853824i	$-0.674278\pi$
$41$	−11206.3	−1.04113	−0.520563	+	0.853824i	$0.674278\pi$
$42$	0	0
$43$	−10684.8	−0.881241	−0.440620	−	0.897694i	$-0.645242\pi$
$43$	−10684.8	−0.881241	−0.440620	+	0.897694i	$0.645242\pi$
$44$	0	0
$45$	−5150.26	−0.379138
$46$	0	0
$47$	7637.19	0.504300	0.252150	−	0.967688i	$-0.418862\pi$
$47$	7637.19	0.504300	0.252150	+	0.967688i	$0.418862\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	10800.7	0.581469
$52$	0	0
$53$	19261.2	0.941875	0.470937	−	0.882167i	$-0.343916\pi$
$53$	19261.2	0.941875	0.470937	+	0.882167i	$0.343916\pi$
$54$	0	0
$55$	−10191.8	−0.454301
$56$	0	0
$57$	13116.8	0.534736
$58$	0	0
$59$	9188.24	0.343639	0.171819	−	0.985128i	$-0.445035\pi$
$59$	9188.24	0.343639	0.171819	+	0.985128i	$0.445035\pi$
$60$	0	0
$61$	−33709.8	−1.15993	−0.579964	−	0.814642i	$-0.696933\pi$
$61$	−33709.8	−1.15993	−0.579964	+	0.814642i	$0.696933\pi$
$62$	0	0
$63$	10094.5	0.320430
$64$	0	0
$65$	−26891.9	−0.789476
$66$	0	0
$67$	29762.1	0.809985	0.404993	−	0.914320i	$-0.367274\pi$
$67$	29762.1	0.809985	0.404993	+	0.914320i	$0.367274\pi$
$68$	0	0
$69$	−27616.4	−0.698304
$70$	0	0
$71$	29860.8	0.703001	0.351500	−	0.936188i	$-0.385672\pi$
$71$	29860.8	0.703001	0.351500	+	0.936188i	$0.385672\pi$
$72$	0	0
$73$	−40725.0	−0.894445	−0.447222	−	0.894423i	$-0.647587\pi$
$73$	−40725.0	−0.894445	−0.447222	+	0.894423i	$0.647587\pi$
$74$	0	0
$75$	−3801.19	−0.0780309
$76$	0	0
$77$	19975.9	0.383954
$78$	0	0
$79$	39180.1	0.706313	0.353157	−	0.935564i	$-0.385108\pi$
$79$	39180.1	0.706313	0.353157	+	0.935564i	$0.385108\pi$
$80$	0	0
$81$	33451.8	0.566509
$82$	0	0
$83$	−34423.7	−0.548483	−0.274241	−	0.961661i	$-0.588427\pi$
$83$	−34423.7	−0.548483	−0.274241	+	0.961661i	$0.588427\pi$
$84$	0	0
$85$	−44396.9	−0.666508
$86$	0	0
$87$	−37187.8	−0.526748
$88$	0	0
$89$	−7995.49	−0.106997	−0.0534983	−	0.998568i	$-0.517037\pi$
$89$	−7995.49	−0.106997	−0.0534983	+	0.998568i	$0.517037\pi$
$90$	0	0
$91$	52708.2	0.667229
$92$	0	0
$93$	−17373.4	−0.208294
$94$	0	0
$95$	−53917.1	−0.612940
$96$	0	0
$97$	−58573.9	−0.632084	−0.316042	−	0.948745i	$-0.602354\pi$
$97$	−58573.9	−0.632084	−0.316042	+	0.948745i	$0.602354\pi$
$98$	0	0
$99$	83984.5	0.861214
$100$	0	0
$101$	159709.	1.55785	0.778925	−	0.627117i	$-0.215765\pi$
$101$	159709.	1.55785	0.778925	+	0.627117i	$0.215765\pi$
$102$	0	0
$103$	69774.5	0.648043	0.324021	−	0.946050i	$-0.394965\pi$
$103$	69774.5	0.648043	0.324021	+	0.946050i	$0.394965\pi$
$104$	0	0
$105$	7450.34	0.0659482
$106$	0	0
$107$	136651.	1.15386	0.576929	−	0.816794i	$-0.304251\pi$
$107$	136651.	1.15386	0.576929	+	0.816794i	$0.304251\pi$
$108$	0	0
$109$	−164476.	−1.32598	−0.662990	−	0.748628i	$-0.730713\pi$
$109$	−164476.	−1.32598	−0.662990	+	0.748628i	$0.730713\pi$
$110$	0	0
$111$	42795.9	0.329682
$112$	0	0
$113$	−57417.3	−0.423006	−0.211503	−	0.977377i	$-0.567836\pi$
$113$	−57417.3	−0.423006	−0.211503	+	0.977377i	$0.567836\pi$
$114$	0	0
$115$	113519.	0.800429
$116$	0	0
$117$	221601.	1.49660
$118$	0	0
$119$	87017.9	0.563302
$120$	0	0
$121$	5144.87	0.0319456
$122$	0	0
$123$	68155.7	0.406200
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	132464.	0.728766	0.364383	−	0.931249i	$-0.381280\pi$
$127$	132464.	0.728766	0.364383	+	0.931249i	$0.381280\pi$
$128$	0	0
$129$	64983.9	0.343820
$130$	0	0
$131$	−68982.8	−0.351206	−0.175603	−	0.984461i	$-0.556188\pi$
$131$	−68982.8	−0.351206	−0.175603	+	0.984461i	$0.556188\pi$
$132$	0	0
$133$	105678.	0.518028
$134$	0	0
$135$	68271.0	0.322405
$136$	0	0
$137$	−379201.	−1.72611	−0.863055	−	0.505110i	$-0.831452\pi$
$137$	−379201.	−1.72611	−0.863055	+	0.505110i	$0.831452\pi$
$138$	0	0
$139$	−87581.5	−0.384481	−0.192241	−	0.981348i	$-0.561575\pi$
$139$	−87581.5	−0.384481	−0.192241	+	0.981348i	$0.561575\pi$
$140$	0	0
$141$	−46448.7	−0.196755
$142$	0	0
$143$	438523.	1.79330
$144$	0	0
$145$	152863.	0.603783
$146$	0	0
$147$	−14602.7	−0.0557364
$148$	0	0
$149$	278024.	1.02593	0.512963	−	0.858411i	$-0.328548\pi$
$149$	278024.	1.02593	0.512963	+	0.858411i	$0.328548\pi$
$150$	0	0
$151$	−214016.	−0.763843	−0.381921	−	0.924195i	$-0.624737\pi$
$151$	−214016.	−0.763843	−0.381921	+	0.924195i	$0.624737\pi$
$152$	0	0
$153$	365849.	1.26349
$154$	0	0
$155$	71414.2	0.238757
$156$	0	0
$157$	372088.	1.20475	0.602374	−	0.798214i	$-0.294222\pi$
$157$	372088.	1.20475	0.602374	+	0.798214i	$0.294222\pi$
$158$	0	0
$159$	−117145.	−0.367477
$160$	0	0
$161$	−222497.	−0.676486
$162$	0	0
$163$	−505449.	−1.49008	−0.745038	−	0.667022i	$-0.767568\pi$
$163$	−505449.	−1.49008	−0.745038	+	0.667022i	$0.767568\pi$
$164$	0	0
$165$	61985.5	0.177248
$166$	0	0
$167$	210214.	0.583271	0.291635	−	0.956530i	$-0.405801\pi$
$167$	210214.	0.583271	0.291635	+	0.956530i	$0.405801\pi$
$168$	0	0
$169$	785790.	2.11636
$170$	0	0
$171$	444299.	1.16194
$172$	0	0
$173$	126094.	0.320316	0.160158	−	0.987091i	$-0.448800\pi$
$173$	126094.	0.320316	0.160158	+	0.987091i	$0.448800\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−55882.0	−0.134072
$178$	0	0
$179$	133191.	0.310701	0.155350	−	0.987859i	$-0.450349\pi$
$179$	133191.	0.310701	0.155350	+	0.987859i	$0.450349\pi$
$180$	0	0
$181$	137432.	0.311810	0.155905	−	0.987772i	$-0.450171\pi$
$181$	137432.	0.311810	0.155905	+	0.987772i	$0.450171\pi$
$182$	0	0
$183$	205020.	0.452551
$184$	0	0
$185$	−175915.	−0.377897
$186$	0	0
$187$	723973.	1.51397
$188$	0	0
$189$	−133811.	−0.272482
$190$	0	0
$191$	−223165.	−0.442631	−0.221316	−	0.975202i	$-0.571035\pi$
$191$	−223165.	−0.442631	−0.221316	+	0.975202i	$0.571035\pi$
$192$	0	0
$193$	849159.	1.64095	0.820476	−	0.571681i	$-0.193708\pi$
$193$	849159.	1.64095	0.820476	+	0.571681i	$0.193708\pi$
$194$	0	0
$195$	163554.	0.308018
$196$	0	0
$197$	−198502.	−0.364418	−0.182209	−	0.983260i	$-0.558325\pi$
$197$	−198502.	−0.364418	−0.182209	+	0.983260i	$0.558325\pi$
$198$	0	0
$199$	485537.	0.869139	0.434570	−	0.900638i	$-0.356900\pi$
$199$	485537.	0.869139	0.434570	+	0.900638i	$0.356900\pi$
$200$	0	0
$201$	−181011.	−0.316020
$202$	0	0
$203$	−299611.	−0.510290
$204$	0	0
$205$	−280158.	−0.465605
$206$	0	0
$207$	−935441.	−1.51737
$208$	0	0
$209$	879218.	1.39229
$210$	0	0
$211$	−1.03186e6	−1.59557	−0.797785	−	0.602943i	$-0.793995\pi$
$211$	−1.03186e6	−1.59557	−0.797785	+	0.602943i	$0.793995\pi$
$212$	0	0
$213$	−181611.	−0.274279
$214$	0	0
$215$	−267120.	−0.394103
$216$	0	0
$217$	−139972.	−0.201786
$218$	0	0
$219$	247685.	0.348972
$220$	0	0
$221$	1.91027e6	2.63096
$222$	0	0
$223$	−1.24200e6	−1.67247	−0.836235	−	0.548372i	$-0.815248\pi$
$223$	−1.24200e6	−1.67247	−0.836235	+	0.548372i	$0.815248\pi$
$224$	0	0
$225$	−128756.	−0.169556
$226$	0	0
$227$	−1.45644e6	−1.87597	−0.937987	−	0.346670i	$-0.887312\pi$
$227$	−1.45644e6	−1.87597	−0.937987	+	0.346670i	$0.887312\pi$
$228$	0	0
$229$	−602646.	−0.759406	−0.379703	−	0.925109i	$-0.623974\pi$
$229$	−602646.	−0.759406	−0.379703	+	0.925109i	$0.623974\pi$
$230$	0	0
$231$	−121492.	−0.149801
$232$	0	0
$233$	754395.	0.910351	0.455175	−	0.890402i	$-0.349577\pi$
$233$	754395.	0.910351	0.455175	+	0.890402i	$0.349577\pi$
$234$	0	0
$235$	190930.	0.225530
$236$	0	0
$237$	−238290.	−0.275571
$238$	0	0
$239$	43290.4	0.0490226	0.0245113	−	0.999700i	$-0.492197\pi$
$239$	43290.4	0.0490226	0.0245113	+	0.999700i	$0.492197\pi$
$240$	0	0
$241$	696883.	0.772890	0.386445	−	0.922313i	$-0.373703\pi$
$241$	696883.	0.772890	0.386445	+	0.922313i	$0.373703\pi$
$242$	0	0
$243$	−867045.	−0.941946
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	2.31990e6	2.41950
$248$	0	0
$249$	209362.	0.213993
$250$	0	0
$251$	1.98194e6	1.98566	0.992831	−	0.119529i	$-0.0381384\pi$
$251$	1.98194e6	1.98566	0.992831	+	0.119529i	$0.0381384\pi$
$252$	0	0
$253$	−1.85113e6	−1.81818
$254$	0	0
$255$	270018.	0.260041
$256$	0	0
$257$	160003.	0.151110	0.0755552	−	0.997142i	$-0.475927\pi$
$257$	160003.	0.151110	0.0755552	+	0.997142i	$0.475927\pi$
$258$	0	0
$259$	344793.	0.319381
$260$	0	0
$261$	−1.25965e6	−1.14459
$262$	0	0
$263$	913315.	0.814200	0.407100	−	0.913384i	$-0.366540\pi$
$263$	913315.	0.814200	0.407100	+	0.913384i	$0.366540\pi$
$264$	0	0
$265$	481529.	0.421219
$266$	0	0
$267$	48627.8	0.0417452
$268$	0	0
$269$	842402.	0.709805	0.354902	−	0.934903i	$-0.384514\pi$
$269$	842402.	0.709805	0.354902	+	0.934903i	$0.384514\pi$
$270$	0	0
$271$	1.25079e6	1.03457	0.517285	−	0.855813i	$-0.326943\pi$
$271$	1.25079e6	1.03457	0.517285	+	0.855813i	$0.326943\pi$
$272$	0	0
$273$	−320567.	−0.260322
$274$	0	0
$275$	−254795.	−0.203169
$276$	0	0
$277$	−370062.	−0.289784	−0.144892	−	0.989447i	$-0.546284\pi$
$277$	−370062.	−0.289784	−0.144892	+	0.989447i	$0.546284\pi$
$278$	0	0
$279$	−588483.	−0.452609
$280$	0	0
$281$	1.08198e6	0.817435	0.408717	−	0.912661i	$-0.365976\pi$
$281$	1.08198e6	0.817435	0.408717	+	0.912661i	$0.365976\pi$
$282$	0	0
$283$	−2.17327e6	−1.61305	−0.806525	−	0.591200i	$-0.798654\pi$
$283$	−2.17327e6	−1.61305	−0.806525	+	0.591200i	$0.798654\pi$
$284$	0	0
$285$	327919.	0.239141
$286$	0	0
$287$	549109.	0.393508
$288$	0	0
$289$	1.73388e6	1.22116
$290$	0	0
$291$	356241.	0.246611
$292$	0	0
$293$	−2.06185e6	−1.40310	−0.701550	−	0.712620i	$-0.747508\pi$
$293$	−2.06185e6	−1.40310	−0.701550	+	0.712620i	$0.747508\pi$
$294$	0	0
$295$	229706.	0.153680
$296$	0	0
$297$	−1.11329e6	−0.732344
$298$	0	0
$299$	−4.88438e6	−3.15960
$300$	0	0
$301$	523554.	0.333078
$302$	0	0
$303$	−971335.	−0.607803
$304$	0	0
$305$	−842744.	−0.518736
$306$	0	0
$307$	1.45256e6	0.879603	0.439802	−	0.898095i	$-0.355049\pi$
$307$	1.45256e6	0.879603	0.439802	+	0.898095i	$0.355049\pi$
$308$	0	0
$309$	−424362.	−0.252837
$310$	0	0
$311$	2.66514e6	1.56250	0.781248	−	0.624221i	$-0.214583\pi$
$311$	2.66514e6	1.56250	0.781248	+	0.624221i	$0.214583\pi$
$312$	0	0
$313$	−1.21556e6	−0.701317	−0.350658	−	0.936503i	$-0.614042\pi$
$313$	−1.21556e6	−0.701317	−0.350658	+	0.936503i	$0.614042\pi$
$314$	0	0
$315$	252363.	0.143301
$316$	0	0
$317$	2.41341e6	1.34891	0.674455	−	0.738316i	$-0.264379\pi$
$317$	2.41341e6	1.34891	0.674455	+	0.738316i	$0.264379\pi$
$318$	0	0
$319$	−2.49271e6	−1.37150
$320$	0	0
$321$	−831097.	−0.450183
$322$	0	0
$323$	3.83000e6	2.04264
$324$	0	0
$325$	−672299.	−0.353064
$326$	0	0
$327$	1.00033e6	0.517338
$328$	0	0
$329$	−374223.	−0.190608
$330$	0	0
$331$	−1.47548e6	−0.740226	−0.370113	−	0.928987i	$-0.620681\pi$
$331$	−1.47548e6	−0.740226	−0.370113	+	0.928987i	$0.620681\pi$
$332$	0	0
$333$	1.44961e6	0.716376
$334$	0	0
$335$	744054.	0.362236
$336$	0	0
$337$	−1.98021e6	−0.949812	−0.474906	−	0.880037i	$-0.657518\pi$
$337$	−1.98021e6	−0.949812	−0.474906	+	0.880037i	$0.657518\pi$
$338$	0	0
$339$	349207.	0.165038
$340$	0	0
$341$	−1.16454e6	−0.542337
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−690410.	−0.312291
$346$	0	0
$347$	212122.	0.0945720	0.0472860	−	0.998881i	$-0.484943\pi$
$347$	212122.	0.0945720	0.0472860	+	0.998881i	$0.484943\pi$
$348$	0	0
$349$	−852633.	−0.374713	−0.187356	−	0.982292i	$-0.559992\pi$
$349$	−852633.	−0.374713	−0.187356	+	0.982292i	$0.559992\pi$
$350$	0	0
$351$	−2.93750e6	−1.27265
$352$	0	0
$353$	−987395.	−0.421749	−0.210875	−	0.977513i	$-0.567631\pi$
$353$	−987395.	−0.421749	−0.210875	+	0.977513i	$0.567631\pi$
$354$	0	0
$355$	746520.	0.314391
$356$	0	0
$357$	−529235.	−0.219775
$358$	0	0
$359$	940935.	0.385322	0.192661	−	0.981265i	$-0.438288\pi$
$359$	940935.	0.385322	0.192661	+	0.981265i	$0.438288\pi$
$360$	0	0
$361$	2.17519e6	0.878474
$362$	0	0
$363$	−31290.6	−0.0124637
$364$	0	0
$365$	−1.01812e6	−0.400008
$366$	0	0
$367$	4.12708e6	1.59948	0.799739	−	0.600348i	$-0.204971\pi$
$367$	4.12708e6	1.59948	0.799739	+	0.600348i	$0.204971\pi$
$368$	0	0
$369$	2.30862e6	0.882644
$370$	0	0
$371$	−943798.	−0.355995
$372$	0	0
$373$	2.26644e6	0.843476	0.421738	−	0.906718i	$-0.361420\pi$
$373$	2.26644e6	0.843476	0.421738	+	0.906718i	$0.361420\pi$
$374$	0	0
$375$	−95029.8	−0.0348965
$376$	0	0
$377$	−6.57723e6	−2.38336
$378$	0	0
$379$	2.24173e6	0.801653	0.400826	−	0.916154i	$-0.368723\pi$
$379$	2.24173e6	0.801653	0.400826	+	0.916154i	$0.368723\pi$
$380$	0	0
$381$	−805633.	−0.284331
$382$	0	0
$383$	−1.03804e6	−0.361591	−0.180795	−	0.983521i	$-0.557867\pi$
$383$	−1.03804e6	−0.361591	−0.180795	+	0.983521i	$0.557867\pi$
$384$	0	0
$385$	499397.	0.171710
$386$	0	0
$387$	2.20118e6	0.747098
$388$	0	0
$389$	−259366.	−0.0869040	−0.0434520	−	0.999056i	$-0.513836\pi$
$389$	−259366.	−0.0869040	−0.0434520	+	0.999056i	$0.513836\pi$
$390$	0	0
$391$	−8.06380e6	−2.66746
$392$	0	0
$393$	419547.	0.137025
$394$	0	0
$395$	979502.	0.315873
$396$	0	0
$397$	−2.75320e6	−0.876720	−0.438360	−	0.898799i	$-0.644441\pi$
$397$	−2.75320e6	−0.876720	−0.438360	+	0.898799i	$0.644441\pi$
$398$	0	0
$399$	−642721.	−0.202111
$400$	0	0
$401$	−5.92473e6	−1.83996	−0.919979	−	0.391968i	$-0.871794\pi$
$401$	−5.92473e6	−1.83996	−0.919979	+	0.391968i	$0.871794\pi$
$402$	0	0
$403$	−3.07275e6	−0.942463
$404$	0	0
$405$	836295.	0.253351
$406$	0	0
$407$	2.86862e6	0.858393
$408$	0	0
$409$	3.69799e6	1.09309	0.546547	−	0.837428i	$-0.315942\pi$
$409$	3.69799e6	1.09309	0.546547	+	0.837428i	$0.315942\pi$
$410$	0	0
$411$	2.30627e6	0.673450
$412$	0	0
$413$	−450224.	−0.129883
$414$	0	0
$415$	−860594.	−0.245289
$416$	0	0
$417$	532662.	0.150007
$418$	0	0
$419$	4.15994e6	1.15758	0.578791	−	0.815476i	$-0.303525\pi$
$419$	4.15994e6	1.15758	0.578791	+	0.815476i	$0.303525\pi$
$420$	0	0
$421$	−5.20770e6	−1.43199	−0.715996	−	0.698104i	$-0.754027\pi$
$421$	−5.20770e6	−1.43199	−0.715996	+	0.698104i	$0.754027\pi$
$422$	0	0
$423$	−1.57334e6	−0.427535
$424$	0	0
$425$	−1.10992e6	−0.298071
$426$	0	0
$427$	1.65178e6	0.438412
$428$	0	0
$429$	−2.66706e6	−0.699663
$430$	0	0
$431$	−1.68636e6	−0.437276	−0.218638	−	0.975806i	$-0.570161\pi$
$431$	−1.68636e6	−0.437276	−0.218638	+	0.975806i	$0.570161\pi$
$432$	0	0
$433$	−3.10555e6	−0.796010	−0.398005	−	0.917383i	$-0.630297\pi$
$433$	−3.10555e6	−0.796010	−0.398005	+	0.917383i	$0.630297\pi$
$434$	0	0
$435$	−929696.	−0.235569
$436$	0	0
$437$	−9.79296e6	−2.45307
$438$	0	0
$439$	−1.22048e6	−0.302252	−0.151126	−	0.988515i	$-0.548290\pi$
$439$	−1.22048e6	−0.302252	−0.151126	+	0.988515i	$0.548290\pi$
$440$	0	0
$441$	−494631.	−0.121111
$442$	0	0
$443$	−2.25319e6	−0.545493	−0.272747	−	0.962086i	$-0.587932\pi$
$443$	−2.25319e6	−0.545493	−0.272747	+	0.962086i	$0.587932\pi$
$444$	0	0
$445$	−199887.	−0.0478503
$446$	0	0
$447$	−1.69091e6	−0.400270
$448$	0	0
$449$	7.67885e6	1.79755	0.898774	−	0.438412i	$-0.144459\pi$
$449$	7.67885e6	1.79755	0.898774	+	0.438412i	$0.144459\pi$
$450$	0	0
$451$	4.56849e6	1.05762
$452$	0	0
$453$	1.30163e6	0.298017
$454$	0	0
$455$	1.31771e6	0.298394
$456$	0	0
$457$	7.98138e6	1.78767	0.893835	−	0.448396i	$-0.148005\pi$
$457$	7.98138e6	1.78767	0.893835	+	0.448396i	$0.148005\pi$
$458$	0	0
$459$	−4.84963e6	−1.07443
$460$	0	0
$461$	−6.10635e6	−1.33823	−0.669113	−	0.743161i	$-0.733326\pi$
$461$	−6.10635e6	−1.33823	−0.669113	+	0.743161i	$0.733326\pi$
$462$	0	0
$463$	−7.69353e6	−1.66791	−0.833957	−	0.551830i	$-0.813930\pi$
$463$	−7.69353e6	−1.66791	−0.833957	+	0.551830i	$0.813930\pi$
$464$	0	0
$465$	−434335.	−0.0931520
$466$	0	0
$467$	−4.41567e6	−0.936924	−0.468462	−	0.883484i	$-0.655192\pi$
$467$	−4.41567e6	−0.936924	−0.468462	+	0.883484i	$0.655192\pi$
$468$	0	0
$469$	−1.45835e6	−0.306146
$470$	0	0
$471$	−2.26300e6	−0.470038
$472$	0	0
$473$	4.35588e6	0.895206
$474$	0	0
$475$	−1.34793e6	−0.274115
$476$	0	0
$477$	−3.96800e6	−0.798502
$478$	0	0
$479$	2.46498e6	0.490878	0.245439	−	0.969412i	$-0.421068\pi$
$479$	2.46498e6	0.490878	0.245439	+	0.969412i	$0.421068\pi$
$480$	0	0
$481$	7.56910e6	1.49170
$482$	0	0
$483$	1.35320e6	0.263934
$484$	0	0
$485$	−1.46435e6	−0.282677
$486$	0	0
$487$	−29208.5	−0.00558068	−0.00279034	−	0.999996i	$-0.500888\pi$
$487$	−29208.5	−0.00558068	−0.00279034	+	0.999996i	$0.500888\pi$
$488$	0	0
$489$	3.07409e6	0.581360
$490$	0	0
$491$	1.41942e6	0.265710	0.132855	−	0.991135i	$-0.457586\pi$
$491$	1.41942e6	0.265710	0.132855	+	0.991135i	$0.457586\pi$
$492$	0	0
$493$	−1.08586e7	−2.01213
$494$	0	0
$495$	2.09961e6	0.385147
$496$	0	0
$497$	−1.46318e6	−0.265709
$498$	0	0
$499$	−6.68747e6	−1.20229	−0.601147	−	0.799138i	$-0.705289\pi$
$499$	−6.68747e6	−1.20229	−0.601147	+	0.799138i	$0.705289\pi$
$500$	0	0
$501$	−1.27850e6	−0.227566
$502$	0	0
$503$	5.12274e6	0.902782	0.451391	−	0.892326i	$-0.350928\pi$
$503$	5.12274e6	0.902782	0.451391	+	0.892326i	$0.350928\pi$
$504$	0	0
$505$	3.99272e6	0.696692
$506$	0	0
$507$	−4.77910e6	−0.825708
$508$	0	0
$509$	2.78484e6	0.476438	0.238219	−	0.971211i	$-0.423436\pi$
$509$	2.78484e6	0.476438	0.238219	+	0.971211i	$0.423436\pi$
$510$	0	0
$511$	1.99552e6	0.338068
$512$	0	0
$513$	−5.88956e6	−0.988074
$514$	0	0
$515$	1.74436e6	0.289814
$516$	0	0
$517$	−3.11346e6	−0.512292
$518$	0	0
$519$	−766891.	−0.124973
$520$	0	0
$521$	6.73984e6	1.08781	0.543907	−	0.839145i	$-0.316944\pi$
$521$	6.73984e6	1.08781	0.543907	+	0.839145i	$0.316944\pi$
$522$	0	0
$523$	−6.94281e6	−1.10989	−0.554947	−	0.831886i	$-0.687261\pi$
$523$	−6.94281e6	−1.10989	−0.554947	+	0.831886i	$0.687261\pi$
$524$	0	0
$525$	186258.	0.0294929
$526$	0	0
$527$	−5.07291e6	−0.795666
$528$	0	0
$529$	1.41820e7	2.20343
$530$	0	0
$531$	−1.89287e6	−0.291330
$532$	0	0
$533$	1.20544e7	1.83792
$534$	0	0
$535$	3.41627e6	0.516021
$536$	0	0
$537$	−810055.	−0.121221
$538$	0	0
$539$	−978819.	−0.145121
$540$	0	0
$541$	5.25286e6	0.771619	0.385810	−	0.922578i	$-0.373922\pi$
$541$	5.25286e6	0.771619	0.385810	+	0.922578i	$0.373922\pi$
$542$	0	0
$543$	−835847.	−0.121654
$544$	0	0
$545$	−4.11191e6	−0.592997
$546$	0	0
$547$	8.43518e6	1.20539	0.602693	−	0.797974i	$-0.294095\pi$
$547$	8.43518e6	1.20539	0.602693	+	0.797974i	$0.294095\pi$
$548$	0	0
$549$	6.94456e6	0.983363
$550$	0	0
$551$	−1.31871e7	−1.85041
$552$	0	0
$553$	−1.91982e6	−0.266961
$554$	0	0
$555$	1.06990e6	0.147438
$556$	0	0
$557$	−8.14320e6	−1.11213	−0.556067	−	0.831137i	$-0.687690\pi$
$557$	−8.14320e6	−1.11213	−0.556067	+	0.831137i	$0.687690\pi$
$558$	0	0
$559$	1.14934e7	1.55567
$560$	0	0
$561$	−4.40314e6	−0.590684
$562$	0	0
$563$	−5.32372e6	−0.707854	−0.353927	−	0.935273i	$-0.615154\pi$
$563$	−5.32372e6	−0.707854	−0.353927	+	0.935273i	$0.615154\pi$
$564$	0	0
$565$	−1.43543e6	−0.189174
$566$	0	0
$567$	−1.63914e6	−0.214120
$568$	0	0
$569$	594659.	0.0769994	0.0384997	−	0.999259i	$-0.487742\pi$
$569$	594659.	0.0769994	0.0384997	+	0.999259i	$0.487742\pi$
$570$	0	0
$571$	−5.08376e6	−0.652521	−0.326260	−	0.945280i	$-0.605789\pi$
$571$	−5.08376e6	−0.652521	−0.326260	+	0.945280i	$0.605789\pi$
$572$	0	0
$573$	1.35727e6	0.172695
$574$	0	0
$575$	2.83797e6	0.357963
$576$	0	0
$577$	6.83943e6	0.855225	0.427612	−	0.903962i	$-0.359355\pi$
$577$	6.83943e6	0.855225	0.427612	+	0.903962i	$0.359355\pi$
$578$	0	0
$579$	−5.16451e6	−0.640225
$580$	0	0
$581$	1.68676e6	0.207307
$582$	0	0
$583$	−7.85223e6	−0.956801
$584$	0	0
$585$	5.54002e6	0.669301
$586$	0	0
$587$	1.24602e7	1.49256	0.746278	−	0.665634i	$-0.231839\pi$
$587$	1.24602e7	1.49256	0.746278	+	0.665634i	$0.231839\pi$
$588$	0	0
$589$	−6.16072e6	−0.731717
$590$	0	0
$591$	1.20727e6	0.142179
$592$	0	0
$593$	−1.04774e7	−1.22353	−0.611765	−	0.791039i	$-0.709540\pi$
$593$	−1.04774e7	−1.22353	−0.611765	+	0.791039i	$0.709540\pi$
$594$	0	0
$595$	2.17545e6	0.251916
$596$	0	0
$597$	−2.95299e6	−0.339099
$598$	0	0
$599$	−2.01252e6	−0.229179	−0.114589	−	0.993413i	$-0.536555\pi$
$599$	−2.01252e6	−0.229179	−0.114589	+	0.993413i	$0.536555\pi$
$600$	0	0
$601$	−9.45258e6	−1.06749	−0.533746	−	0.845645i	$-0.679216\pi$
$601$	−9.45258e6	−1.06749	−0.533746	+	0.845645i	$0.679216\pi$
$602$	0	0
$603$	−6.13131e6	−0.686689
$604$	0	0
$605$	128622.	0.0142865
$606$	0	0
$607$	−1.58379e7	−1.74473	−0.872363	−	0.488859i	$-0.837413\pi$
$607$	−1.58379e7	−1.74473	−0.872363	+	0.488859i	$0.837413\pi$
$608$	0	0
$609$	1.82220e6	0.199092
$610$	0	0
$611$	−8.21516e6	−0.890252
$612$	0	0
$613$	1.15645e7	1.24301	0.621505	−	0.783410i	$-0.286521\pi$
$613$	1.15645e7	1.24301	0.621505	+	0.783410i	$0.286521\pi$
$614$	0	0
$615$	1.70389e6	0.181658
$616$	0	0
$617$	−1.24967e6	−0.132154	−0.0660771	−	0.997815i	$-0.521048\pi$
$617$	−1.24967e6	−0.132154	−0.0660771	+	0.997815i	$0.521048\pi$
$618$	0	0
$619$	1.11632e6	0.117101	0.0585505	−	0.998284i	$-0.481352\pi$
$619$	1.11632e6	0.117101	0.0585505	+	0.998284i	$0.481352\pi$
$620$	0	0
$621$	1.24001e7	1.29031
$622$	0	0
$623$	391779.	0.0404409
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−5.34733e6	−0.543210
$628$	0	0
$629$	1.24961e7	1.25935
$630$	0	0
$631$	1.52038e7	1.52012	0.760062	−	0.649851i	$-0.225169\pi$
$631$	1.52038e7	1.52012	0.760062	+	0.649851i	$0.225169\pi$
$632$	0	0
$633$	6.27569e6	0.622519
$634$	0	0
$635$	3.31160e6	0.325914
$636$	0	0
$637$	−2.58270e6	−0.252189
$638$	0	0
$639$	−6.15164e6	−0.595989
$640$	0	0
$641$	1.02239e7	0.982814	0.491407	−	0.870930i	$-0.336483\pi$
$641$	1.02239e7	0.982814	0.491407	+	0.870930i	$0.336483\pi$
$642$	0	0
$643$	1.70419e6	0.162551	0.0812757	−	0.996692i	$-0.474101\pi$
$643$	1.70419e6	0.162551	0.0812757	+	0.996692i	$0.474101\pi$
$644$	0	0
$645$	1.62460e6	0.153761
$646$	0	0
$647$	1.34708e7	1.26512	0.632560	−	0.774512i	$-0.282004\pi$
$647$	1.34708e7	1.26512	0.632560	+	0.774512i	$0.282004\pi$
$648$	0	0
$649$	−3.74578e6	−0.349085
$650$	0	0
$651$	851296.	0.0787278
$652$	0	0
$653$	−1.55928e7	−1.43100	−0.715500	−	0.698612i	$-0.753801\pi$
$653$	−1.55928e7	−1.43100	−0.715500	+	0.698612i	$0.753801\pi$
$654$	0	0
$655$	−1.72457e6	−0.157064
$656$	0	0
$657$	8.38976e6	0.758292
$658$	0	0
$659$	2.02390e7	1.81542	0.907708	−	0.419603i	$-0.137831\pi$
$659$	2.02390e7	1.81542	0.907708	+	0.419603i	$0.137831\pi$
$660$	0	0
$661$	−5.28966e6	−0.470895	−0.235448	−	0.971887i	$-0.575656\pi$
$661$	−5.28966e6	−0.470895	−0.235448	+	0.971887i	$0.575656\pi$
$662$	0	0
$663$	−1.16181e7	−1.02648
$664$	0	0
$665$	2.64194e6	0.231669
$666$	0	0
$667$	2.77644e7	2.41643
$668$	0	0
$669$	7.55371e6	0.652522
$670$	0	0
$671$	1.37425e7	1.17831
$672$	0	0
$673$	7.68769e6	0.654271	0.327136	−	0.944977i	$-0.393917\pi$
$673$	7.68769e6	0.654271	0.327136	+	0.944977i	$0.393917\pi$
$674$	0	0
$675$	1.70678e6	0.144184
$676$	0	0
$677$	−1.13880e7	−0.954939	−0.477469	−	0.878648i	$-0.658446\pi$
$677$	−1.13880e7	−0.954939	−0.477469	+	0.878648i	$0.658446\pi$
$678$	0	0
$679$	2.87012e6	0.238905
$680$	0	0
$681$	8.85791e6	0.731920
$682$	0	0
$683$	−1.15735e7	−0.949320	−0.474660	−	0.880169i	$-0.657429\pi$
$683$	−1.15735e7	−0.949320	−0.474660	+	0.880169i	$0.657429\pi$
$684$	0	0
$685$	−9.48003e6	−0.771940
$686$	0	0
$687$	3.66524e6	0.296286
$688$	0	0
$689$	−2.07188e7	−1.66271
$690$	0	0
$691$	−1.33398e7	−1.06281	−0.531404	−	0.847119i	$-0.678335\pi$
$691$	−1.33398e7	−1.06281	−0.531404	+	0.847119i	$0.678335\pi$
$692$	0	0
$693$	−4.11524e6	−0.325508
$694$	0	0
$695$	−2.18954e6	−0.171945
$696$	0	0
$697$	1.99010e7	1.55165
$698$	0	0
$699$	−4.58816e6	−0.355178
$700$	0	0
$701$	−4.67569e6	−0.359377	−0.179689	−	0.983724i	$-0.557509\pi$
$701$	−4.67569e6	−0.359377	−0.179689	+	0.983724i	$0.557509\pi$
$702$	0	0
$703$	1.51757e7	1.15814
$704$	0	0
$705$	−1.16122e6	−0.0879916
$706$	0	0
$707$	−7.82574e6	−0.588812
$708$	0	0
$709$	2.08491e7	1.55766	0.778829	−	0.627236i	$-0.215814\pi$
$709$	2.08491e7	1.55766	0.778829	+	0.627236i	$0.215814\pi$
$710$	0	0
$711$	−8.07150e6	−0.598798
$712$	0	0
$713$	1.29710e7	0.955539
$714$	0	0
$715$	1.09631e7	0.801987
$716$	0	0
$717$	−263288.	−0.0191264
$718$	0	0
$719$	1.97250e7	1.42297	0.711485	−	0.702702i	$-0.248023\pi$
$719$	1.97250e7	1.42297	0.711485	+	0.702702i	$0.248023\pi$
$720$	0	0
$721$	−3.41895e6	−0.244937
$722$	0	0
$723$	−4.23838e6	−0.301547
$724$	0	0
$725$	3.82156e6	0.270020
$726$	0	0
$727$	−1.84776e7	−1.29661	−0.648306	−	0.761380i	$-0.724522\pi$
$727$	−1.84776e7	−1.29661	−0.648306	+	0.761380i	$0.724522\pi$
$728$	0	0
$729$	−2.85550e6	−0.199005
$730$	0	0
$731$	1.89748e7	1.31336
$732$	0	0
$733$	1.95578e6	0.134449	0.0672247	−	0.997738i	$-0.478586\pi$
$733$	1.95578e6	0.134449	0.0672247	+	0.997738i	$0.478586\pi$
$734$	0	0
$735$	−365067.	−0.0249261
$736$	0	0
$737$	−1.21332e7	−0.822821
$738$	0	0
$739$	1.55560e7	1.04782	0.523911	−	0.851773i	$-0.324472\pi$
$739$	1.55560e7	1.04782	0.523911	+	0.851773i	$0.324472\pi$
$740$	0	0
$741$	−1.41094e7	−0.943981
$742$	0	0
$743$	1.21540e7	0.807694	0.403847	−	0.914827i	$-0.367673\pi$
$743$	1.21540e7	0.807694	0.403847	+	0.914827i	$0.367673\pi$
$744$	0	0
$745$	6.95059e6	0.458808
$746$	0	0
$747$	7.09165e6	0.464992
$748$	0	0
$749$	−6.69588e6	−0.436117
$750$	0	0
$751$	7.81290e6	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$751$	7.81290e6	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$752$	0	0
$753$	−1.20540e7	−0.774715
$754$	0	0
$755$	−5.35040e6	−0.341601
$756$	0	0
$757$	2.04844e7	1.29922	0.649610	−	0.760268i	$-0.274932\pi$
$757$	2.04844e7	1.29922	0.649610	+	0.760268i	$0.274932\pi$
$758$	0	0
$759$	1.12584e7	0.709370
$760$	0	0
$761$	−8.17274e6	−0.511571	−0.255786	−	0.966734i	$-0.582334\pi$
$761$	−8.17274e6	−0.511571	−0.255786	+	0.966734i	$0.582334\pi$
$762$	0	0
$763$	8.05934e6	0.501174
$764$	0	0
$765$	9.14622e6	0.565051
$766$	0	0
$767$	−9.88359e6	−0.606633
$768$	0	0
$769$	6.09119e6	0.371438	0.185719	−	0.982603i	$-0.440539\pi$
$769$	6.09119e6	0.371438	0.185719	+	0.982603i	$0.440539\pi$
$770$	0	0
$771$	−973122.	−0.0589564
$772$	0	0
$773$	−2.91960e7	−1.75741	−0.878707	−	0.477361i	$-0.841593\pi$
$773$	−2.91960e7	−1.75741	−0.878707	+	0.477361i	$0.841593\pi$
$774$	0	0
$775$	1.78536e6	0.106775
$776$	0	0
$777$	−2.09700e6	−0.124608
$778$	0	0
$779$	2.41685e7	1.42694
$780$	0	0
$781$	−1.21734e7	−0.714141
$782$	0	0
$783$	1.66977e7	0.973314
$784$	0	0
$785$	9.30219e6	0.538780
$786$	0	0
$787$	9.27644e6	0.533881	0.266940	−	0.963713i	$-0.413987\pi$
$787$	9.27644e6	0.533881	0.266940	+	0.963713i	$0.413987\pi$
$788$	0	0
$789$	−5.55470e6	−0.317664
$790$	0	0
$791$	2.81345e6	0.159881
$792$	0	0
$793$	3.62608e7	2.04765
$794$	0	0
$795$	−2.92862e6	−0.164341
$796$	0	0
$797$	−3.37629e7	−1.88275	−0.941377	−	0.337356i	$-0.890467\pi$
$797$	−3.37629e7	−1.88275	−0.941377	+	0.337356i	$0.890467\pi$
$798$	0	0
$799$	−1.35627e7	−0.751587
$800$	0	0
$801$	1.64715e6	0.0907095
$802$	0	0
$803$	1.66024e7	0.908619
$804$	0	0
$805$	−5.56242e6	−0.302534
$806$	0	0
$807$	−5.12341e6	−0.276934
$808$	0	0
$809$	2.68549e7	1.44262	0.721312	−	0.692611i	$-0.243540\pi$
$809$	2.68549e7	1.44262	0.721312	+	0.692611i	$0.243540\pi$
$810$	0	0
$811$	−8.99843e6	−0.480413	−0.240206	−	0.970722i	$-0.577215\pi$
$811$	−8.99843e6	−0.480413	−0.240206	+	0.970722i	$0.577215\pi$
$812$	0	0
$813$	−7.60717e6	−0.403642
$814$	0	0
$815$	−1.26362e7	−0.666382
$816$	0	0
$817$	2.30437e7	1.20781
$818$	0	0
$819$	−1.08584e7	−0.565663
$820$	0	0
$821$	−2.53478e7	−1.31245	−0.656225	−	0.754565i	$-0.727848\pi$
$821$	−2.53478e7	−1.31245	−0.656225	+	0.754565i	$0.727848\pi$
$822$	0	0
$823$	2.80232e7	1.44217	0.721087	−	0.692844i	$-0.243643\pi$
$823$	2.80232e7	1.44217	0.721087	+	0.692844i	$0.243643\pi$
$824$	0	0
$825$	1.54964e6	0.0792675
$826$	0	0
$827$	1.08632e7	0.552325	0.276162	−	0.961111i	$-0.410937\pi$
$827$	1.08632e7	0.552325	0.276162	+	0.961111i	$0.410937\pi$
$828$	0	0
$829$	2.61760e7	1.32287	0.661435	−	0.750002i	$-0.269948\pi$
$829$	2.61760e7	1.32287	0.661435	+	0.750002i	$0.269948\pi$
$830$	0	0
$831$	2.25068e6	0.113061
$832$	0	0
$833$	−4.26388e6	−0.212908
$834$	0	0
$835$	5.25535e6	0.260847
$836$	0	0
$837$	7.80083e6	0.384882
$838$	0	0
$839$	5.33086e6	0.261452	0.130726	−	0.991419i	$-0.458269\pi$
$839$	5.33086e6	0.261452	0.130726	+	0.991419i	$0.458269\pi$
$840$	0	0
$841$	1.68760e7	0.822771
$842$	0	0
$843$	−6.58050e6	−0.318926
$844$	0	0
$845$	1.96447e7	0.946465
$846$	0	0
$847$	−252098.	−0.0120743
$848$	0	0
$849$	1.32176e7	0.629339
$850$	0	0
$851$	−3.19514e7	−1.51240
$852$	0	0
$853$	−3.52267e7	−1.65768	−0.828838	−	0.559488i	$-0.810998\pi$
$853$	−3.52267e7	−1.65768	−0.828838	+	0.559488i	$0.810998\pi$
$854$	0	0
$855$	1.11075e7	0.519638
$856$	0	0
$857$	−1.93997e6	−0.0902283	−0.0451141	−	0.998982i	$-0.514365\pi$
$857$	−1.93997e6	−0.0902283	−0.0451141	+	0.998982i	$0.514365\pi$
$858$	0	0
$859$	2.90322e7	1.34245	0.671223	−	0.741255i	$-0.265769\pi$
$859$	2.90322e7	1.34245	0.671223	+	0.741255i	$0.265769\pi$
$860$	0	0
$861$	−3.33963e6	−0.153529
$862$	0	0
$863$	−3.76000e7	−1.71854	−0.859272	−	0.511518i	$-0.829083\pi$
$863$	−3.76000e7	−1.71854	−0.859272	+	0.511518i	$0.829083\pi$
$864$	0	0
$865$	3.15235e6	0.143250
$866$	0	0
$867$	−1.05453e7	−0.476442
$868$	0	0
$869$	−1.59726e7	−0.717506
$870$	0	0
$871$	−3.20145e7	−1.42988
$872$	0	0
$873$	1.20668e7	0.535868
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	−2.05117e7	−0.900539	−0.450269	−	0.892893i	$-0.648672\pi$
$877$	−2.05117e7	−0.900539	−0.450269	+	0.892893i	$0.648672\pi$
$878$	0	0
$879$	1.25400e7	0.547426
$880$	0	0
$881$	−1.36918e7	−0.594322	−0.297161	−	0.954827i	$-0.596040\pi$
$881$	−1.36918e7	−0.594322	−0.297161	+	0.954827i	$0.596040\pi$
$882$	0	0
$883$	2.73738e7	1.18150	0.590750	−	0.806855i	$-0.298832\pi$
$883$	2.73738e7	1.18150	0.590750	+	0.806855i	$0.298832\pi$
$884$	0	0
$885$	−1.39705e6	−0.0599590
$886$	0	0
$887$	1.44173e7	0.615284	0.307642	−	0.951502i	$-0.400460\pi$
$887$	1.44173e7	0.615284	0.307642	+	0.951502i	$0.400460\pi$
$888$	0	0
$889$	−6.49073e6	−0.275448
$890$	0	0
$891$	−1.36373e7	−0.575487
$892$	0	0
$893$	−1.64710e7	−0.691181
$894$	0	0
$895$	3.32977e6	0.138950
$896$	0	0
$897$	2.97064e7	1.23273
$898$	0	0
$899$	1.74665e7	0.720786
$900$	0	0
$901$	−3.42054e7	−1.40373
$902$	0	0
$903$	−3.18421e6	−0.129952
$904$	0	0
$905$	3.43579e6	0.139446
$906$	0	0
$907$	−2.74482e7	−1.10789	−0.553944	−	0.832554i	$-0.686878\pi$
$907$	−2.74482e7	−1.10789	−0.553944	+	0.832554i	$0.686878\pi$
$908$	0	0
$909$	−3.29017e7	−1.32071
$910$	0	0
$911$	3.10414e7	1.23921	0.619607	−	0.784913i	$-0.287292\pi$
$911$	3.10414e7	1.23921	0.619607	+	0.784913i	$0.287292\pi$
$912$	0	0
$913$	1.40336e7	0.557175
$914$	0	0
$915$	5.12549e6	0.202387
$916$	0	0
$917$	3.38016e6	0.132744
$918$	0	0
$919$	1.13810e7	0.444519	0.222260	−	0.974988i	$-0.428657\pi$
$919$	1.13810e7	0.444519	0.222260	+	0.974988i	$0.428657\pi$
$920$	0	0
$921$	−8.83431e6	−0.343181
$922$	0	0
$923$	−3.21206e7	−1.24102
$924$	0	0
$925$	−4.39787e6	−0.169000
$926$	0	0
$927$	−1.43743e7	−0.549397
$928$	0	0
$929$	1.77569e7	0.675038	0.337519	−	0.941319i	$-0.390412\pi$
$929$	1.77569e7	0.675038	0.337519	+	0.941319i	$0.390412\pi$
$930$	0	0
$931$	−5.17820e6	−0.195796
$932$	0	0
$933$	−1.62091e7	−0.609615
$934$	0	0
$935$	1.80993e7	0.677070
$936$	0	0
$937$	−1.93662e7	−0.720602	−0.360301	−	0.932836i	$-0.617326\pi$
$937$	−1.93662e7	−0.720602	−0.360301	+	0.932836i	$0.617326\pi$
$938$	0	0
$939$	7.39290e6	0.273622
$940$	0	0
$941$	2.75371e7	1.01378	0.506891	−	0.862010i	$-0.330795\pi$
$941$	2.75371e7	1.01378	0.506891	+	0.862010i	$0.330795\pi$
$942$	0	0
$943$	−5.08850e7	−1.86342
$944$	0	0
$945$	−3.34528e6	−0.121858
$946$	0	0
$947$	−4.71647e7	−1.70900	−0.854501	−	0.519450i	$-0.826137\pi$
$947$	−4.71647e7	−1.70900	−0.854501	+	0.519450i	$0.826137\pi$
$948$	0	0
$949$	4.38069e7	1.57898
$950$	0	0
$951$	−1.46781e7	−0.526283
$952$	0	0
$953$	−3.72865e7	−1.32990	−0.664951	−	0.746887i	$-0.731547\pi$
$953$	−3.72865e7	−1.32990	−0.664951	+	0.746887i	$0.731547\pi$
$954$	0	0
$955$	−5.57912e6	−0.197951
$956$	0	0
$957$	1.51604e7	0.535095
$958$	0	0
$959$	1.85809e7	0.652408
$960$	0	0
$961$	−2.04692e7	−0.714976
$962$	0	0
$963$	−2.81515e7	−0.978217
$964$	0	0
$965$	2.12290e7	0.733856
$966$	0	0
$967$	2.46193e7	0.846662	0.423331	−	0.905975i	$-0.360861\pi$
$967$	2.46193e7	0.846662	0.423331	+	0.905975i	$0.360861\pi$
$968$	0	0
$969$	−2.32937e7	−0.796947
$970$	0	0
$971$	2.00471e6	0.0682344	0.0341172	−	0.999418i	$-0.489138\pi$
$971$	2.00471e6	0.0682344	0.0341172	+	0.999418i	$0.489138\pi$
$972$	0	0
$973$	4.29149e6	0.145320
$974$	0	0
$975$	4.08886e6	0.137750
$976$	0	0
$977$	2.77180e7	0.929022	0.464511	−	0.885567i	$-0.346230\pi$
$977$	2.77180e7	0.929022	0.464511	+	0.885567i	$0.346230\pi$
$978$	0	0
$979$	3.25953e6	0.108692
$980$	0	0
$981$	3.38838e7	1.12414
$982$	0	0
$983$	1.79222e7	0.591573	0.295787	−	0.955254i	$-0.404418\pi$
$983$	1.79222e7	0.591573	0.295787	+	0.955254i	$0.404418\pi$
$984$	0	0
$985$	−4.96255e6	−0.162973
$986$	0	0
$987$	2.27599e6	0.0743664
$988$	0	0
$989$	−4.85169e7	−1.57726
$990$	0	0
$991$	1.38359e7	0.447532	0.223766	−	0.974643i	$-0.428165\pi$
$991$	1.38359e7	0.447532	0.223766	+	0.974643i	$0.428165\pi$
$992$	0	0
$993$	8.97375e6	0.288803
$994$	0	0
$995$	1.21384e7	0.388691
$996$	0	0
$997$	3.19988e7	1.01952	0.509761	−	0.860316i	$-0.329734\pi$
$997$	3.19988e7	1.01952	0.509761	+	0.860316i	$0.329734\pi$
$998$	0	0
$999$	−1.92158e7	−0.609179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.x.1.2		4
4.3	odd	2		280.6.a.g.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.g.1.3	✓	4	4.3	odd	2
560.6.a.x.1.2		4	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-6.08191 q^{3} +25.0000 q^{5} -49.0000 q^{7} -206.010 q^{9} +O(q^{10})\)	\(q-6.08191 q^{3} +25.0000 q^{5} -49.0000 q^{7} -206.010 q^{9} -407.671 q^{11} -1075.68 q^{13} -152.048 q^{15} -1775.88 q^{17} -2156.68 q^{19} +298.014 q^{21} +4540.75 q^{23} +625.000 q^{25} +2730.84 q^{27} +6114.50 q^{29} +2856.57 q^{31} +2479.42 q^{33} -1225.00 q^{35} -7036.59 q^{37} +6542.17 q^{39} -11206.3 q^{41} -10684.8 q^{43} -5150.26 q^{45} +7637.19 q^{47} +2401.00 q^{49} +10800.7 q^{51} +19261.2 q^{53} -10191.8 q^{55} +13116.8 q^{57} +9188.24 q^{59} -33709.8 q^{61} +10094.5 q^{63} -26891.9 q^{65} +29762.1 q^{67} -27616.4 q^{69} +29860.8 q^{71} -40725.0 q^{73} -3801.19 q^{75} +19975.9 q^{77} +39180.1 q^{79} +33451.8 q^{81} -34423.7 q^{83} -44396.9 q^{85} -37187.8 q^{87} -7995.49 q^{89} +52708.2 q^{91} -17373.4 q^{93} -53917.1 q^{95} -58573.9 q^{97} +83984.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9}+O(q^{10}) \) 4 * q + 13 * q^3 + 100 * q^5 - 196 * q^7 - 193 * q^9	\( 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9} + 595 q^{11} - 969 q^{13} + 325 q^{15} - 1315 q^{17} + 1090 q^{19} - 637 q^{21} + 1534 q^{23} + 2500 q^{25} - 173 q^{27} + 4099 q^{29} + 4820 q^{31} + 4149 q^{33} - 4900 q^{35} + 7692 q^{37} + 6371 q^{39} - 9722 q^{41} + 20610 q^{43} - 4825 q^{45} + 1661 q^{47} + 9604 q^{49} + 73361 q^{51} - 28898 q^{53} + 14875 q^{55} - 21246 q^{57} + 101872 q^{59} - 24742 q^{61} + 9457 q^{63} - 24225 q^{65} + 82060 q^{67} + 16914 q^{69} + 102784 q^{71} - 80652 q^{73} + 8125 q^{75} - 29155 q^{77} + 117801 q^{79} - 141052 q^{81} + 155440 q^{83} - 32875 q^{85} + 82519 q^{87} + 56426 q^{89} + 47481 q^{91} - 17332 q^{93} + 27250 q^{95} - 261031 q^{97} + 61686 q^{99}+O(q^{100}) \) 4 * q + 13 * q^3 + 100 * q^5 - 196 * q^7 - 193 * q^9 + 595 * q^11 - 969 * q^13 + 325 * q^15 - 1315 * q^17 + 1090 * q^19 - 637 * q^21 + 1534 * q^23 + 2500 * q^25 - 173 * q^27 + 4099 * q^29 + 4820 * q^31 + 4149 * q^33 - 4900 * q^35 + 7692 * q^37 + 6371 * q^39 - 9722 * q^41 + 20610 * q^43 - 4825 * q^45 + 1661 * q^47 + 9604 * q^49 + 73361 * q^51 - 28898 * q^53 + 14875 * q^55 - 21246 * q^57 + 101872 * q^59 - 24742 * q^61 + 9457 * q^63 - 24225 * q^65 + 82060 * q^67 + 16914 * q^69 + 102784 * q^71 - 80652 * q^73 + 8125 * q^75 - 29155 * q^77 + 117801 * q^79 - 141052 * q^81 + 155440 * q^83 - 32875 * q^85 + 82519 * q^87 + 56426 * q^89 + 47481 * q^91 - 17332 * q^93 + 27250 * q^95 - 261031 * q^97 + 61686 * q^99

Introduction

L-functions

Modular forms

Varieties

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