LMFDB - Embedded newform 520.6.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,6,Mod(1,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, 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("520.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	520.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:	$83.3995863027$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 1344x^{6} - 288x^{5} + 542568x^{4} - 84480x^{3} - 68942848x^{2} + 12558720x + 2674054800 $ x^8 - 1344x^6 - 288x^5 + 542568x^4 - 84480x^3 - 68942848x^2 + 12558720x + 2674054800
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$9.12057$ of defining polynomial
Character	$\chi$	$=$	520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.12057 q^{3} -25.0000 q^{5} -144.052 q^{7} -159.815 q^{9} +23.7578 q^{11} +169.000 q^{13} +228.014 q^{15} +2132.02 q^{17} +1778.03 q^{19} +1313.84 q^{21} -1674.13 q^{23} +625.000 q^{25} +3673.90 q^{27} +4178.58 q^{29} +16.6679 q^{31} -216.685 q^{33} +3601.31 q^{35} -1720.13 q^{37} -1541.38 q^{39} -16050.6 q^{41} -4407.92 q^{43} +3995.38 q^{45} +1552.64 q^{47} +3944.04 q^{49} -19445.3 q^{51} +9246.04 q^{53} -593.945 q^{55} -16216.6 q^{57} -6141.23 q^{59} +43277.4 q^{61} +23021.7 q^{63} -4225.00 q^{65} -22908.3 q^{67} +15269.0 q^{69} +19984.4 q^{71} -44846.4 q^{73} -5700.36 q^{75} -3422.36 q^{77} -82083.4 q^{79} +5327.01 q^{81} +6507.80 q^{83} -53300.6 q^{85} -38111.0 q^{87} +42668.5 q^{89} -24344.8 q^{91} -152.021 q^{93} -44450.7 q^{95} +38025.5 q^{97} -3796.86 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 200 q^{5} - 184 q^{7} + 744 q^{9} - 720 q^{11} + 1352 q^{13} - 248 q^{17} - 2544 q^{19} + 4520 q^{21} - 88 q^{23} + 5000 q^{25} - 864 q^{27} + 2224 q^{29} + 5368 q^{31} + 9720 q^{33} + 4600 q^{35}+ \cdots - 395808 q^{99}+O(q^{100}) $ 8 * q - 200 * q^5 - 184 * q^7 + 744 * q^9 - 720 * q^11 + 1352 * q^13 - 248 * q^17 - 2544 * q^19 + 4520 * q^21 - 88 * q^23 + 5000 * q^25 - 864 * q^27 + 2224 * q^29 + 5368 * q^31 + 9720 * q^33 + 4600 * q^35 + 7096 * q^37 + 6232 * q^41 + 10528 * q^43 - 18600 * q^45 - 16424 * q^47 - 7192 * q^49 - 7904 * q^51 - 20856 * q^53 + 18000 * q^55 - 26408 * q^57 - 44864 * q^59 - 39712 * q^61 - 112696 * q^63 - 33800 * q^65 + 10824 * q^67 - 221200 * q^69 + 32552 * q^71 - 120104 * q^73 - 76120 * q^77 - 113472 * q^79 - 44760 * q^81 - 173416 * q^83 + 6200 * q^85 - 71056 * q^87 - 214368 * q^89 - 31096 * q^91 - 171032 * q^93 + 63600 * q^95 - 95440 * q^97 - 395808 * q^99

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
$3$	−9.12057	−0.585085	−0.292542	−	0.956253i	$-0.594501\pi$
$3$	−9.12057	−0.585085	−0.292542	+	0.956253i	$0.594501\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−144.052	−1.11116	−0.555578	−	0.831465i	$-0.687503\pi$
$7$	−144.052	−1.11116	−0.555578	+	0.831465i	$0.687503\pi$
$8$	0	0
$9$	−159.815	−0.657676
$10$	0	0
$11$	23.7578	0.0592004	0.0296002	−	0.999562i	$-0.490577\pi$
$11$	23.7578	0.0592004	0.0296002	+	0.999562i	$0.490577\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	228.014	0.261658
$16$	0	0
$17$	2132.02	1.78924	0.894622	−	0.446824i	$-0.147445\pi$
$17$	2132.02	1.78924	0.894622	+	0.446824i	$0.147445\pi$
$18$	0	0
$19$	1778.03	1.12994	0.564969	−	0.825112i	$-0.308888\pi$
$19$	1778.03	1.12994	0.564969	+	0.825112i	$0.308888\pi$
$20$	0	0
$21$	1313.84	0.650120
$22$	0	0
$23$	−1674.13	−0.659888	−0.329944	−	0.944001i	$-0.607030\pi$
$23$	−1674.13	−0.659888	−0.329944	+	0.944001i	$0.607030\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	3673.90	0.969881
$28$	0	0
$29$	4178.58	0.922643	0.461321	−	0.887233i	$-0.347376\pi$
$29$	4178.58	0.922643	0.461321	+	0.887233i	$0.347376\pi$
$30$	0	0
$31$	16.6679	0.00311513	0.00155757	−	0.999999i	$-0.499504\pi$
$31$	16.6679	0.00311513	0.00155757	+	0.999999i	$0.499504\pi$
$32$	0	0
$33$	−216.685	−0.0346372
$34$	0	0
$35$	3601.31	0.496924
$36$	0	0
$37$	−1720.13	−0.206565	−0.103283	−	0.994652i	$-0.532935\pi$
$37$	−1720.13	−0.206565	−0.103283	+	0.994652i	$0.532935\pi$
$38$	0	0
$39$	−1541.38	−0.162273
$40$	0	0
$41$	−16050.6	−1.49118	−0.745591	−	0.666404i	$-0.767833\pi$
$41$	−16050.6	−1.49118	−0.745591	+	0.666404i	$0.767833\pi$
$42$	0	0
$43$	−4407.92	−0.363549	−0.181774	−	0.983340i	$-0.558184\pi$
$43$	−4407.92	−0.363549	−0.181774	+	0.983340i	$0.558184\pi$
$44$	0	0
$45$	3995.38	0.294122
$46$	0	0
$47$	1552.64	0.102524	0.0512622	−	0.998685i	$-0.483676\pi$
$47$	1552.64	0.102524	0.0512622	+	0.998685i	$0.483676\pi$
$48$	0	0
$49$	3944.04	0.234666
$50$	0	0
$51$	−19445.3	−1.04686
$52$	0	0
$53$	9246.04	0.452133	0.226067	−	0.974112i	$-0.427413\pi$
$53$	9246.04	0.452133	0.226067	+	0.974112i	$0.427413\pi$
$54$	0	0
$55$	−593.945	−0.0264752
$56$	0	0
$57$	−16216.6	−0.661110
$58$	0	0
$59$	−6141.23	−0.229681	−0.114841	−	0.993384i	$-0.536636\pi$
$59$	−6141.23	−0.229681	−0.114841	+	0.993384i	$0.536636\pi$
$60$	0	0
$61$	43277.4	1.48914	0.744571	−	0.667543i	$-0.232654\pi$
$61$	43277.4	1.48914	0.744571	+	0.667543i	$0.232654\pi$
$62$	0	0
$63$	23021.7	0.730780
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	−22908.3	−0.623455	−0.311728	−	0.950172i	$-0.600908\pi$
$67$	−22908.3	−0.623455	−0.311728	+	0.950172i	$0.600908\pi$
$68$	0	0
$69$	15269.0	0.386090
$70$	0	0
$71$	19984.4	0.470485	0.235243	−	0.971937i	$-0.424412\pi$
$71$	19984.4	0.470485	0.235243	+	0.971937i	$0.424412\pi$
$72$	0	0
$73$	−44846.4	−0.984965	−0.492482	−	0.870322i	$-0.663910\pi$
$73$	−44846.4	−0.984965	−0.492482	+	0.870322i	$0.663910\pi$
$74$	0	0
$75$	−5700.36	−0.117017
$76$	0	0
$77$	−3422.36	−0.0657808
$78$	0	0
$79$	−82083.4	−1.47975	−0.739873	−	0.672746i	$-0.765115\pi$
$79$	−82083.4	−1.47975	−0.739873	+	0.672746i	$0.765115\pi$
$80$	0	0
$81$	5327.01	0.0902134
$82$	0	0
$83$	6507.80	0.103690	0.0518452	−	0.998655i	$-0.483490\pi$
$83$	6507.80	0.103690	0.0518452	+	0.998655i	$0.483490\pi$
$84$	0	0
$85$	−53300.6	−0.800174
$86$	0	0
$87$	−38111.0	−0.539824
$88$	0	0
$89$	42668.5	0.570995	0.285498	−	0.958379i	$-0.407841\pi$
$89$	42668.5	0.570995	0.285498	+	0.958379i	$0.407841\pi$
$90$	0	0
$91$	−24344.8	−0.308179
$92$	0	0
$93$	−152.021	−0.00182262
$94$	0	0
$95$	−44450.7	−0.505324
$96$	0	0
$97$	38025.5	0.410341	0.205171	−	0.978726i	$-0.434225\pi$
$97$	38025.5	0.410341	0.205171	+	0.978726i	$0.434225\pi$
$98$	0	0
$99$	−3796.86	−0.0389347
$100$	0	0
$101$	56768.0	0.553733	0.276867	−	0.960908i	$-0.410704\pi$
$101$	56768.0	0.553733	0.276867	+	0.960908i	$0.410704\pi$
$102$	0	0
$103$	162419.	1.50850	0.754248	−	0.656590i	$-0.228002\pi$
$103$	162419.	1.50850	0.754248	+	0.656590i	$0.228002\pi$
$104$	0	0
$105$	−32845.9	−0.290743
$106$	0	0
$107$	−173227.	−1.46270	−0.731349	−	0.682003i	$-0.761109\pi$
$107$	−173227.	−1.46270	−0.731349	+	0.682003i	$0.761109\pi$
$108$	0	0
$109$	−111738.	−0.900815	−0.450408	−	0.892823i	$-0.648721\pi$
$109$	−111738.	−0.900815	−0.450408	+	0.892823i	$0.648721\pi$
$110$	0	0
$111$	15688.6	0.120858
$112$	0	0
$113$	59496.9	0.438327	0.219164	−	0.975688i	$-0.429667\pi$
$113$	59496.9	0.438327	0.219164	+	0.975688i	$0.429667\pi$
$114$	0	0
$115$	41853.3	0.295111
$116$	0	0
$117$	−27008.8	−0.182406
$118$	0	0
$119$	−307122.	−1.98813
$120$	0	0
$121$	−160487.	−0.996495
$122$	0	0
$123$	146390.	0.872468
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	50779.4	0.279369	0.139685	−	0.990196i	$-0.455391\pi$
$127$	50779.4	0.279369	0.139685	+	0.990196i	$0.455391\pi$
$128$	0	0
$129$	40202.7	0.212707
$130$	0	0
$131$	−172865.	−0.880094	−0.440047	−	0.897975i	$-0.645038\pi$
$131$	−172865.	−0.880094	−0.440047	+	0.897975i	$0.645038\pi$
$132$	0	0
$133$	−256129.	−1.25554
$134$	0	0
$135$	−91847.6	−0.433744
$136$	0	0
$137$	−149678.	−0.681329	−0.340664	−	0.940185i	$-0.610652\pi$
$137$	−149678.	−0.681329	−0.340664	+	0.940185i	$0.610652\pi$
$138$	0	0
$139$	−331638.	−1.45589	−0.727943	−	0.685638i	$-0.759523\pi$
$139$	−331638.	−1.45589	−0.727943	+	0.685638i	$0.759523\pi$
$140$	0	0
$141$	−14161.0	−0.0599854
$142$	0	0
$143$	4015.07	0.0164192
$144$	0	0
$145$	−104464.	−0.412618
$146$	0	0
$147$	−35971.9	−0.137300
$148$	0	0
$149$	97173.1	0.358575	0.179288	−	0.983797i	$-0.442621\pi$
$149$	97173.1	0.358575	0.179288	+	0.983797i	$0.442621\pi$
$150$	0	0
$151$	−451968.	−1.61311	−0.806557	−	0.591156i	$-0.798672\pi$
$151$	−451968.	−1.61311	−0.806557	+	0.591156i	$0.798672\pi$
$152$	0	0
$153$	−340730.	−1.17674
$154$	0	0
$155$	−416.697	−0.00139313
$156$	0	0
$157$	274661.	0.889299	0.444649	−	0.895705i	$-0.353328\pi$
$157$	274661.	0.889299	0.444649	+	0.895705i	$0.353328\pi$
$158$	0	0
$159$	−84329.2	−0.264536
$160$	0	0
$161$	241162.	0.733238
$162$	0	0
$163$	27177.3	0.0801193	0.0400596	−	0.999197i	$-0.487245\pi$
$163$	27177.3	0.0801193	0.0400596	+	0.999197i	$0.487245\pi$
$164$	0	0
$165$	5417.12	0.0154902
$166$	0	0
$167$	−1208.42	−0.00335294	−0.00167647	−	0.999999i	$-0.500534\pi$
$167$	−1208.42	−0.00335294	−0.00167647	+	0.999999i	$0.500534\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−284156.	−0.743133
$172$	0	0
$173$	447636.	1.13713	0.568565	−	0.822639i	$-0.307499\pi$
$173$	447636.	1.13713	0.568565	+	0.822639i	$0.307499\pi$
$174$	0	0
$175$	−90032.6	−0.222231
$176$	0	0
$177$	56011.5	0.134383
$178$	0	0
$179$	398599.	0.929831	0.464915	−	0.885355i	$-0.346085\pi$
$179$	398599.	0.929831	0.464915	+	0.885355i	$0.346085\pi$
$180$	0	0
$181$	−179681.	−0.407667	−0.203834	−	0.979006i	$-0.565340\pi$
$181$	−179681.	−0.407667	−0.203834	+	0.979006i	$0.565340\pi$
$182$	0	0
$183$	−394714.	−0.871275
$184$	0	0
$185$	43003.3	0.0923788
$186$	0	0
$187$	50652.2	0.105924
$188$	0	0
$189$	−529234.	−1.07769
$190$	0	0
$191$	−717340.	−1.42279	−0.711397	−	0.702791i	$-0.751937\pi$
$191$	−717340.	−1.42279	−0.711397	+	0.702791i	$0.751937\pi$
$192$	0	0
$193$	12670.9	0.0244859	0.0122429	−	0.999925i	$-0.496103\pi$
$193$	12670.9	0.0244859	0.0122429	+	0.999925i	$0.496103\pi$
$194$	0	0
$195$	38534.4	0.0725708
$196$	0	0
$197$	−22270.9	−0.0408859	−0.0204429	−	0.999791i	$-0.506508\pi$
$197$	−22270.9	−0.0408859	−0.0204429	+	0.999791i	$0.506508\pi$
$198$	0	0
$199$	−585539.	−1.04815	−0.524074	−	0.851673i	$-0.675589\pi$
$199$	−585539.	−1.04815	−0.524074	+	0.851673i	$0.675589\pi$
$200$	0	0
$201$	208936.	0.364774
$202$	0	0
$203$	−601933.	−1.02520
$204$	0	0
$205$	401264.	0.666877
$206$	0	0
$207$	267552.	0.433992
$208$	0	0
$209$	42242.1	0.0668928
$210$	0	0
$211$	−410386.	−0.634581	−0.317290	−	0.948328i	$-0.602773\pi$
$211$	−410386.	−0.634581	−0.317290	+	0.948328i	$0.602773\pi$
$212$	0	0
$213$	−182269.	−0.275274
$214$	0	0
$215$	110198.	0.162584
$216$	0	0
$217$	−2401.05	−0.00346139
$218$	0	0
$219$	409025.	0.576288
$220$	0	0
$221$	360312.	0.496247
$222$	0	0
$223$	360489.	0.485434	0.242717	−	0.970097i	$-0.421961\pi$
$223$	360489.	0.485434	0.242717	+	0.970097i	$0.421961\pi$
$224$	0	0
$225$	−99884.5	−0.131535
$226$	0	0
$227$	−942901.	−1.21451	−0.607256	−	0.794506i	$-0.707730\pi$
$227$	−942901.	−1.21451	−0.607256	+	0.794506i	$0.707730\pi$
$228$	0	0
$229$	1.17645e6	1.48247	0.741233	−	0.671248i	$-0.234241\pi$
$229$	1.17645e6	1.48247	0.741233	+	0.671248i	$0.234241\pi$
$230$	0	0
$231$	31213.9	0.0384874
$232$	0	0
$233$	−30735.0	−0.0370889	−0.0185445	−	0.999828i	$-0.505903\pi$
$233$	−30735.0	−0.0370889	−0.0185445	+	0.999828i	$0.505903\pi$
$234$	0	0
$235$	−38816.1	−0.0458503
$236$	0	0
$237$	748647.	0.865777
$238$	0	0
$239$	−36995.5	−0.0418942	−0.0209471	−	0.999781i	$-0.506668\pi$
$239$	−36995.5	−0.0418942	−0.0209471	+	0.999781i	$0.506668\pi$
$240$	0	0
$241$	−1.24246e6	−1.37797	−0.688985	−	0.724776i	$-0.741943\pi$
$241$	−1.24246e6	−1.37797	−0.688985	+	0.724776i	$0.741943\pi$
$242$	0	0
$243$	−941344.	−1.02266
$244$	0	0
$245$	−98601.0	−0.104946
$246$	0	0
$247$	300487.	0.313389
$248$	0	0
$249$	−59354.8	−0.0606677
$250$	0	0
$251$	974724.	0.976556	0.488278	−	0.872688i	$-0.337625\pi$
$251$	974724.	0.976556	0.488278	+	0.872688i	$0.337625\pi$
$252$	0	0
$253$	−39773.7	−0.0390656
$254$	0	0
$255$	486131.	0.468170
$256$	0	0
$257$	435882.	0.411658	0.205829	−	0.978588i	$-0.434011\pi$
$257$	435882.	0.411658	0.205829	+	0.978588i	$0.434011\pi$
$258$	0	0
$259$	247789.	0.229526
$260$	0	0
$261$	−667801.	−0.606800
$262$	0	0
$263$	−80454.7	−0.0717236	−0.0358618	−	0.999357i	$-0.511418\pi$
$263$	−80454.7	−0.0717236	−0.0358618	+	0.999357i	$0.511418\pi$
$264$	0	0
$265$	−231151.	−0.202200
$266$	0	0
$267$	−389161.	−0.334081
$268$	0	0
$269$	−235559.	−0.198481	−0.0992406	−	0.995063i	$-0.531641\pi$
$269$	−235559.	−0.198481	−0.0992406	+	0.995063i	$0.531641\pi$
$270$	0	0
$271$	−311240.	−0.257438	−0.128719	−	0.991681i	$-0.541086\pi$
$271$	−311240.	−0.257438	−0.128719	+	0.991681i	$0.541086\pi$
$272$	0	0
$273$	222039.	0.180311
$274$	0	0
$275$	14848.6	0.0118401
$276$	0	0
$277$	−286468.	−0.224325	−0.112162	−	0.993690i	$-0.535778\pi$
$277$	−286468.	−0.224325	−0.112162	+	0.993690i	$0.535778\pi$
$278$	0	0
$279$	−2663.78	−0.00204875
$280$	0	0
$281$	−1.26444e6	−0.955280	−0.477640	−	0.878556i	$-0.658508\pi$
$281$	−1.26444e6	−0.955280	−0.477640	+	0.878556i	$0.658508\pi$
$282$	0	0
$283$	−162953.	−0.120947	−0.0604737	−	0.998170i	$-0.519261\pi$
$283$	−162953.	−0.120947	−0.0604737	+	0.998170i	$0.519261\pi$
$284$	0	0
$285$	405416.	0.295657
$286$	0	0
$287$	2.31212e6	1.65693
$288$	0	0
$289$	3.12566e6	2.20139
$290$	0	0
$291$	−346814.	−0.240084
$292$	0	0
$293$	446869.	0.304096	0.152048	−	0.988373i	$-0.451413\pi$
$293$	446869.	0.304096	0.152048	+	0.988373i	$0.451413\pi$
$294$	0	0
$295$	153531.	0.102717
$296$	0	0
$297$	87283.9	0.0574173
$298$	0	0
$299$	−282928.	−0.183020
$300$	0	0
$301$	634971.	0.403959
$302$	0	0
$303$	−517757.	−0.323981
$304$	0	0
$305$	−1.08193e6	−0.665965
$306$	0	0
$307$	−1.57280e6	−0.952417	−0.476209	−	0.879332i	$-0.657989\pi$
$307$	−1.57280e6	−0.952417	−0.476209	+	0.879332i	$0.657989\pi$
$308$	0	0
$309$	−1.48136e6	−0.882598
$310$	0	0
$311$	−326273.	−0.191285	−0.0956423	−	0.995416i	$-0.530490\pi$
$311$	−326273.	−0.191285	−0.0956423	+	0.995416i	$0.530490\pi$
$312$	0	0
$313$	1.65961e6	0.957514	0.478757	−	0.877948i	$-0.341088\pi$
$313$	1.65961e6	0.957514	0.478757	+	0.877948i	$0.341088\pi$
$314$	0	0
$315$	−575543.	−0.326815
$316$	0	0
$317$	−2.30895e6	−1.29052	−0.645262	−	0.763962i	$-0.723252\pi$
$317$	−2.30895e6	−1.29052	−0.645262	+	0.763962i	$0.723252\pi$
$318$	0	0
$319$	99273.9	0.0546208
$320$	0	0
$321$	1.57992e6	0.855803
$322$	0	0
$323$	3.79080e6	2.02174
$324$	0	0
$325$	105625.	0.0554700
$326$	0	0
$327$	1.01912e6	0.527053
$328$	0	0
$329$	−223662.	−0.113920
$330$	0	0
$331$	−3.59194e6	−1.80202	−0.901008	−	0.433802i	$-0.857172\pi$
$331$	−3.59194e6	−1.80202	−0.901008	+	0.433802i	$0.857172\pi$
$332$	0	0
$333$	274903.	0.135853
$334$	0	0
$335$	572707.	0.278818
$336$	0	0
$337$	1.19237e6	0.571920	0.285960	−	0.958242i	$-0.407688\pi$
$337$	1.19237e6	0.571920	0.285960	+	0.958242i	$0.407688\pi$
$338$	0	0
$339$	−542646.	−0.256458
$340$	0	0
$341$	395.992	0.000184417 0
$342$	0	0
$343$	1.85294e6	0.850405
$344$	0	0
$345$	−381726.	−0.172665
$346$	0	0
$347$	−2.93534e6	−1.30868	−0.654341	−	0.756200i	$-0.727054\pi$
$347$	−2.93534e6	−1.30868	−0.654341	+	0.756200i	$0.727054\pi$
$348$	0	0
$349$	3.71742e6	1.63372	0.816861	−	0.576835i	$-0.195712\pi$
$349$	3.71742e6	1.63372	0.816861	+	0.576835i	$0.195712\pi$
$350$	0	0
$351$	620890.	0.268997
$352$	0	0
$353$	66123.6	0.0282436	0.0141218	−	0.999900i	$-0.495505\pi$
$353$	66123.6	0.0282436	0.0141218	+	0.999900i	$0.495505\pi$
$354$	0	0
$355$	−499611.	−0.210407
$356$	0	0
$357$	2.80113e6	1.16322
$358$	0	0
$359$	−526246.	−0.215503	−0.107751	−	0.994178i	$-0.534365\pi$
$359$	−526246.	−0.215503	−0.107751	+	0.994178i	$0.534365\pi$
$360$	0	0
$361$	685289.	0.276762
$362$	0	0
$363$	1.46373e6	0.583034
$364$	0	0
$365$	1.12116e6	0.440490
$366$	0	0
$367$	2.00018e6	0.775181	0.387591	−	0.921832i	$-0.373307\pi$
$367$	2.00018e6	0.775181	0.387591	+	0.921832i	$0.373307\pi$
$368$	0	0
$369$	2.56512e6	0.980714
$370$	0	0
$371$	−1.33191e6	−0.502390
$372$	0	0
$373$	−2.00969e6	−0.747923	−0.373962	−	0.927444i	$-0.622001\pi$
$373$	−2.00969e6	−0.747923	−0.373962	+	0.927444i	$0.622001\pi$
$374$	0	0
$375$	142509.	0.0523316
$376$	0	0
$377$	706180.	0.255895
$378$	0	0
$379$	−2.66112e6	−0.951627	−0.475814	−	0.879546i	$-0.657846\pi$
$379$	−2.66112e6	−0.951627	−0.475814	+	0.879546i	$0.657846\pi$
$380$	0	0
$381$	−463137.	−0.163455
$382$	0	0
$383$	620509.	0.216148	0.108074	−	0.994143i	$-0.465532\pi$
$383$	620509.	0.216148	0.108074	+	0.994143i	$0.465532\pi$
$384$	0	0
$385$	85559.1	0.0294181
$386$	0	0
$387$	704453.	0.239097
$388$	0	0
$389$	−2.57567e6	−0.863010	−0.431505	−	0.902111i	$-0.642017\pi$
$389$	−2.57567e6	−0.863010	−0.431505	+	0.902111i	$0.642017\pi$
$390$	0	0
$391$	−3.56929e6	−1.18070
$392$	0	0
$393$	1.57663e6	0.514929
$394$	0	0
$395$	2.05208e6	0.661763
$396$	0	0
$397$	1.35006e6	0.429909	0.214954	−	0.976624i	$-0.431040\pi$
$397$	1.35006e6	0.429909	0.214954	+	0.976624i	$0.431040\pi$
$398$	0	0
$399$	2.33604e6	0.734596
$400$	0	0
$401$	4.11854e6	1.27903	0.639517	−	0.768777i	$-0.279134\pi$
$401$	4.11854e6	1.27903	0.639517	+	0.768777i	$0.279134\pi$
$402$	0	0
$403$	2816.87	0.000863982 0
$404$	0	0
$405$	−133175.	−0.0403447
$406$	0	0
$407$	−40866.5	−0.0122287
$408$	0	0
$409$	4.11782e6	1.21719	0.608597	−	0.793480i	$-0.291733\pi$
$409$	4.11782e6	1.21719	0.608597	+	0.793480i	$0.291733\pi$
$410$	0	0
$411$	1.36515e6	0.398635
$412$	0	0
$413$	884658.	0.255212
$414$	0	0
$415$	−162695.	−0.0463718
$416$	0	0
$417$	3.02473e6	0.851816
$418$	0	0
$419$	−2.18976e6	−0.609343	−0.304672	−	0.952457i	$-0.598547\pi$
$419$	−2.18976e6	−0.609343	−0.304672	+	0.952457i	$0.598547\pi$
$420$	0	0
$421$	−3.46745e6	−0.953465	−0.476732	−	0.879048i	$-0.658179\pi$
$421$	−3.46745e6	−0.953465	−0.476732	+	0.879048i	$0.658179\pi$
$422$	0	0
$423$	−248136.	−0.0674278
$424$	0	0
$425$	1.33251e6	0.357849
$426$	0	0
$427$	−6.23420e6	−1.65467
$428$	0	0
$429$	−36619.7	−0.00960664
$430$	0	0
$431$	−1.63303e6	−0.423449	−0.211724	−	0.977329i	$-0.567908\pi$
$431$	−1.63303e6	−0.423449	−0.211724	+	0.977329i	$0.567908\pi$
$432$	0	0
$433$	6.86624e6	1.75994	0.879972	−	0.475025i	$-0.157561\pi$
$433$	6.86624e6	1.75994	0.879972	+	0.475025i	$0.157561\pi$
$434$	0	0
$435$	952775.	0.241417
$436$	0	0
$437$	−2.97666e6	−0.745633
$438$	0	0
$439$	−4.93122e6	−1.22122	−0.610608	−	0.791933i	$-0.709075\pi$
$439$	−4.93122e6	−1.22122	−0.610608	+	0.791933i	$0.709075\pi$
$440$	0	0
$441$	−630317.	−0.154334
$442$	0	0
$443$	−3.50949e6	−0.849640	−0.424820	−	0.905278i	$-0.639663\pi$
$443$	−3.50949e6	−0.849640	−0.424820	+	0.905278i	$0.639663\pi$
$444$	0	0
$445$	−1.06671e6	−0.255357
$446$	0	0
$447$	−886274.	−0.209797
$448$	0	0
$449$	3.39090e6	0.793777	0.396889	−	0.917867i	$-0.370090\pi$
$449$	3.39090e6	0.793777	0.396889	+	0.917867i	$0.370090\pi$
$450$	0	0
$451$	−381326.	−0.0882785
$452$	0	0
$453$	4.12220e6	0.943809
$454$	0	0
$455$	608621.	0.137822
$456$	0	0
$457$	4.51109e6	1.01039	0.505197	−	0.863004i	$-0.331420\pi$
$457$	4.51109e6	1.01039	0.505197	+	0.863004i	$0.331420\pi$
$458$	0	0
$459$	7.83284e6	1.73535
$460$	0	0
$461$	6.59378e6	1.44505	0.722524	−	0.691346i	$-0.242982\pi$
$461$	6.59378e6	1.44505	0.722524	+	0.691346i	$0.242982\pi$
$462$	0	0
$463$	−2.93685e6	−0.636692	−0.318346	−	0.947975i	$-0.603127\pi$
$463$	−2.93685e6	−0.636692	−0.318346	+	0.947975i	$0.603127\pi$
$464$	0	0
$465$	3800.51	0.000815098 0
$466$	0	0
$467$	2.25729e6	0.478956	0.239478	−	0.970902i	$-0.423024\pi$
$467$	2.25729e6	0.478956	0.239478	+	0.970902i	$0.423024\pi$
$468$	0	0
$469$	3.29999e6	0.692756
$470$	0	0
$471$	−2.50506e6	−0.520315
$472$	0	0
$473$	−104722.	−0.0215222
$474$	0	0
$475$	1.11127e6	0.225988
$476$	0	0
$477$	−1.47766e6	−0.297357
$478$	0	0
$479$	−1.84731e6	−0.367875	−0.183937	−	0.982938i	$-0.558884\pi$
$479$	−1.84731e6	−0.367875	−0.183937	+	0.982938i	$0.558884\pi$
$480$	0	0
$481$	−290702.	−0.0572909
$482$	0	0
$483$	−2.19954e6	−0.429006
$484$	0	0
$485$	−950637.	−0.183510
$486$	0	0
$487$	816228.	0.155951	0.0779757	−	0.996955i	$-0.475154\pi$
$487$	816228.	0.155951	0.0779757	+	0.996955i	$0.475154\pi$
$488$	0	0
$489$	−247872.	−0.0468766
$490$	0	0
$491$	2.50572e6	0.469060	0.234530	−	0.972109i	$-0.424645\pi$
$491$	2.50572e6	0.469060	0.234530	+	0.972109i	$0.424645\pi$
$492$	0	0
$493$	8.90882e6	1.65083
$494$	0	0
$495$	94921.5	0.0174121
$496$	0	0
$497$	−2.87880e6	−0.522782
$498$	0	0
$499$	2.27095e6	0.408279	0.204139	−	0.978942i	$-0.434560\pi$
$499$	2.27095e6	0.408279	0.204139	+	0.978942i	$0.434560\pi$
$500$	0	0
$501$	11021.4	0.00196175
$502$	0	0
$503$	2.03375e6	0.358407	0.179204	−	0.983812i	$-0.442648\pi$
$503$	2.03375e6	0.358407	0.179204	+	0.983812i	$0.442648\pi$
$504$	0	0
$505$	−1.41920e6	−0.247637
$506$	0	0
$507$	−260493.	−0.0450065
$508$	0	0
$509$	−7.83309e6	−1.34011	−0.670053	−	0.742314i	$-0.733728\pi$
$509$	−7.83309e6	−1.34011	−0.670053	+	0.742314i	$0.733728\pi$
$510$	0	0
$511$	6.46023e6	1.09445
$512$	0	0
$513$	6.53231e6	1.09591
$514$	0	0
$515$	−4.06048e6	−0.674620
$516$	0	0
$517$	36887.4	0.00606948
$518$	0	0
$519$	−4.08270e6	−0.665317
$520$	0	0
$521$	−4.79078e6	−0.773236	−0.386618	−	0.922240i	$-0.626357\pi$
$521$	−4.79078e6	−0.773236	−0.386618	+	0.922240i	$0.626357\pi$
$522$	0	0
$523$	9.54776e6	1.52633	0.763163	−	0.646206i	$-0.223645\pi$
$523$	9.54776e6	1.52633	0.763163	+	0.646206i	$0.223645\pi$
$524$	0	0
$525$	821149.	0.130024
$526$	0	0
$527$	35536.3	0.00557373
$528$	0	0
$529$	−3.63362e6	−0.564548
$530$	0	0
$531$	981462.	0.151056
$532$	0	0
$533$	−2.71254e6	−0.413579
$534$	0	0
$535$	4.33066e6	0.654139
$536$	0	0
$537$	−3.63545e6	−0.544030
$538$	0	0
$539$	93701.7	0.0138923
$540$	0	0
$541$	−8.13405e6	−1.19485	−0.597425	−	0.801925i	$-0.703809\pi$
$541$	−8.13405e6	−1.19485	−0.597425	+	0.801925i	$0.703809\pi$
$542$	0	0
$543$	1.63879e6	0.238520
$544$	0	0
$545$	2.79346e6	0.402857
$546$	0	0
$547$	1.23827e7	1.76948	0.884741	−	0.466084i	$-0.154335\pi$
$547$	1.23827e7	1.76948	0.884741	+	0.466084i	$0.154335\pi$
$548$	0	0
$549$	−6.91638e6	−0.979373
$550$	0	0
$551$	7.42964e6	1.04253
$552$	0	0
$553$	1.18243e7	1.64423
$554$	0	0
$555$	−392214.	−0.0540494
$556$	0	0
$557$	−1.28798e7	−1.75902	−0.879512	−	0.475876i	$-0.842131\pi$
$557$	−1.28798e7	−1.75902	−0.879512	+	0.475876i	$0.842131\pi$
$558$	0	0
$559$	−744938.	−0.100830
$560$	0	0
$561$	−461976.	−0.0619745
$562$	0	0
$563$	81173.2	0.0107930	0.00539649	−	0.999985i	$-0.498282\pi$
$563$	81173.2	0.0107930	0.00539649	+	0.999985i	$0.498282\pi$
$564$	0	0
$565$	−1.48742e6	−0.196026
$566$	0	0
$567$	−767368.	−0.100241
$568$	0	0
$569$	−1.30067e7	−1.68418	−0.842089	−	0.539339i	$-0.818674\pi$
$569$	−1.30067e7	−1.68418	−0.842089	+	0.539339i	$0.818674\pi$
$570$	0	0
$571$	−4.60477e6	−0.591041	−0.295521	−	0.955336i	$-0.595493\pi$
$571$	−4.60477e6	−0.591041	−0.295521	+	0.955336i	$0.595493\pi$
$572$	0	0
$573$	6.54255e6	0.832455
$574$	0	0
$575$	−1.04633e6	−0.131978
$576$	0	0
$577$	−1.11277e7	−1.39144	−0.695722	−	0.718312i	$-0.744915\pi$
$577$	−1.11277e7	−1.39144	−0.695722	+	0.718312i	$0.744915\pi$
$578$	0	0
$579$	−115566.	−0.0143263
$580$	0	0
$581$	−937463.	−0.115216
$582$	0	0
$583$	219666.	0.0267665
$584$	0	0
$585$	675219.	0.0815747
$586$	0	0
$587$	−1.20002e7	−1.43745	−0.718727	−	0.695292i	$-0.755275\pi$
$587$	−1.20002e7	−1.43745	−0.718727	+	0.695292i	$0.755275\pi$
$588$	0	0
$589$	29636.0	0.00351991
$590$	0	0
$591$	203124.	0.0239217
$592$	0	0
$593$	6.68928e6	0.781165	0.390582	−	0.920568i	$-0.372274\pi$
$593$	6.68928e6	0.781165	0.390582	+	0.920568i	$0.372274\pi$
$594$	0	0
$595$	7.67806e6	0.889118
$596$	0	0
$597$	5.34044e6	0.613256
$598$	0	0
$599$	−2.55395e6	−0.290834	−0.145417	−	0.989370i	$-0.546452\pi$
$599$	−2.55395e6	−0.290834	−0.145417	+	0.989370i	$0.546452\pi$
$600$	0	0
$601$	−1.22145e7	−1.37939	−0.689697	−	0.724098i	$-0.742256\pi$
$601$	−1.22145e7	−1.37939	−0.689697	+	0.724098i	$0.742256\pi$
$602$	0	0
$603$	3.66109e6	0.410031
$604$	0	0
$605$	4.01216e6	0.445646
$606$	0	0
$607$	7.84309e6	0.864004	0.432002	−	0.901873i	$-0.357807\pi$
$607$	7.84309e6	0.864004	0.432002	+	0.901873i	$0.357807\pi$
$608$	0	0
$609$	5.48998e6	0.599829
$610$	0	0
$611$	262397.	0.0284351
$612$	0	0
$613$	6.92510e6	0.744347	0.372173	−	0.928163i	$-0.378613\pi$
$613$	6.92510e6	0.744347	0.372173	+	0.928163i	$0.378613\pi$
$614$	0	0
$615$	−3.65976e6	−0.390179
$616$	0	0
$617$	−1.34307e7	−1.42032	−0.710159	−	0.704042i	$-0.751377\pi$
$617$	−1.34307e7	−1.42032	−0.710159	+	0.704042i	$0.751377\pi$
$618$	0	0
$619$	−1.53000e7	−1.60497	−0.802483	−	0.596675i	$-0.796488\pi$
$619$	−1.53000e7	−1.60497	−0.802483	+	0.596675i	$0.796488\pi$
$620$	0	0
$621$	−6.15060e6	−0.640013
$622$	0	0
$623$	−6.14649e6	−0.634464
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−385272.	−0.0391380
$628$	0	0
$629$	−3.66736e6	−0.369595
$630$	0	0
$631$	−8.91520e6	−0.891369	−0.445685	−	0.895190i	$-0.647040\pi$
$631$	−8.91520e6	−0.891369	−0.445685	+	0.895190i	$0.647040\pi$
$632$	0	0
$633$	3.74296e6	0.371283
$634$	0	0
$635$	−1.26949e6	−0.124938
$636$	0	0
$637$	666543.	0.0650848
$638$	0	0
$639$	−3.19382e6	−0.309427
$640$	0	0
$641$	−1.09269e7	−1.05039	−0.525196	−	0.850981i	$-0.676008\pi$
$641$	−1.09269e7	−1.05039	−0.525196	+	0.850981i	$0.676008\pi$
$642$	0	0
$643$	−1.40738e7	−1.34241	−0.671203	−	0.741274i	$-0.734222\pi$
$643$	−1.40738e7	−1.34241	−0.671203	+	0.741274i	$0.734222\pi$
$644$	0	0
$645$	−1.00507e6	−0.0951254
$646$	0	0
$647$	1.05543e7	0.991214	0.495607	−	0.868547i	$-0.334946\pi$
$647$	1.05543e7	0.991214	0.495607	+	0.868547i	$0.334946\pi$
$648$	0	0
$649$	−145902.	−0.0135972
$650$	0	0
$651$	21898.9	0.00202521
$652$	0	0
$653$	−1.91597e7	−1.75835	−0.879174	−	0.476501i	$-0.841905\pi$
$653$	−1.91597e7	−1.75835	−0.879174	+	0.476501i	$0.841905\pi$
$654$	0	0
$655$	4.32163e6	0.393590
$656$	0	0
$657$	7.16714e6	0.647787
$658$	0	0
$659$	−1.85170e7	−1.66095	−0.830475	−	0.557056i	$-0.811931\pi$
$659$	−1.85170e7	−1.66095	−0.830475	+	0.557056i	$0.811931\pi$
$660$	0	0
$661$	−1.14698e7	−1.02106	−0.510530	−	0.859860i	$-0.670551\pi$
$661$	−1.14698e7	−1.02106	−0.510530	+	0.859860i	$0.670551\pi$
$662$	0	0
$663$	−3.28625e6	−0.290346
$664$	0	0
$665$	6.40323e6	0.561493
$666$	0	0
$667$	−6.99549e6	−0.608841
$668$	0	0
$669$	−3.28787e6	−0.284020
$670$	0	0
$671$	1.02818e6	0.0881578
$672$	0	0
$673$	−1.29759e7	−1.10433	−0.552167	−	0.833734i	$-0.686199\pi$
$673$	−1.29759e7	−1.10433	−0.552167	+	0.833734i	$0.686199\pi$
$674$	0	0
$675$	2.29619e6	0.193976
$676$	0	0
$677$	−3.51672e6	−0.294894	−0.147447	−	0.989070i	$-0.547106\pi$
$677$	−3.51672e6	−0.294894	−0.147447	+	0.989070i	$0.547106\pi$
$678$	0	0
$679$	−5.47765e6	−0.455953
$680$	0	0
$681$	8.59980e6	0.710592
$682$	0	0
$683$	−6.51964e6	−0.534776	−0.267388	−	0.963589i	$-0.586161\pi$
$683$	−6.51964e6	−0.534776	−0.267388	+	0.963589i	$0.586161\pi$
$684$	0	0
$685$	3.74195e6	0.304700
$686$	0	0
$687$	−1.07299e7	−0.867368
$688$	0	0
$689$	1.56258e6	0.125399
$690$	0	0
$691$	1.36775e7	1.08971	0.544855	−	0.838530i	$-0.316585\pi$
$691$	1.36775e7	1.08971	0.544855	+	0.838530i	$0.316585\pi$
$692$	0	0
$693$	546946.	0.0432625
$694$	0	0
$695$	8.29095e6	0.651092
$696$	0	0
$697$	−3.42202e7	−2.66809
$698$	0	0
$699$	280321.	0.0217002
$700$	0	0
$701$	−2.01729e7	−1.55051	−0.775253	−	0.631651i	$-0.782377\pi$
$701$	−2.01729e7	−1.55051	−0.775253	+	0.631651i	$0.782377\pi$
$702$	0	0
$703$	−3.05844e6	−0.233406
$704$	0	0
$705$	354025.	0.0268263
$706$	0	0
$707$	−8.17756e6	−0.615284
$708$	0	0
$709$	6.80504e6	0.508411	0.254206	−	0.967150i	$-0.418186\pi$
$709$	6.80504e6	0.508411	0.254206	+	0.967150i	$0.418186\pi$
$710$	0	0
$711$	1.31182e7	0.973194
$712$	0	0
$713$	−27904.2	−0.00205564
$714$	0	0
$715$	−100377.	−0.00734290
$716$	0	0
$717$	337420.	0.0245117
$718$	0	0
$719$	2.66952e7	1.92580	0.962899	−	0.269861i	$-0.0869776\pi$
$719$	2.66952e7	1.92580	0.962899	+	0.269861i	$0.0869776\pi$
$720$	0	0
$721$	−2.33968e7	−1.67617
$722$	0	0
$723$	1.13319e7	0.806229
$724$	0	0
$725$	2.61161e6	0.184529
$726$	0	0
$727$	−1.89543e7	−1.33006	−0.665029	−	0.746818i	$-0.731581\pi$
$727$	−1.89543e7	−1.33006	−0.665029	+	0.746818i	$0.731581\pi$
$728$	0	0
$729$	7.29113e6	0.508131
$730$	0	0
$731$	−9.39778e6	−0.650477
$732$	0	0
$733$	−1.46646e7	−1.00811	−0.504056	−	0.863671i	$-0.668160\pi$
$733$	−1.46646e7	−1.00811	−0.504056	+	0.863671i	$0.668160\pi$
$734$	0	0
$735$	899297.	0.0614023
$736$	0	0
$737$	−544250.	−0.0369088
$738$	0	0
$739$	2.43251e7	1.63849	0.819244	−	0.573445i	$-0.194393\pi$
$739$	2.43251e7	1.63849	0.819244	+	0.573445i	$0.194393\pi$
$740$	0	0
$741$	−2.74061e6	−0.183359
$742$	0	0
$743$	2.73841e7	1.81981	0.909907	−	0.414812i	$-0.136153\pi$
$743$	2.73841e7	1.81981	0.909907	+	0.414812i	$0.136153\pi$
$744$	0	0
$745$	−2.42933e6	−0.160360
$746$	0	0
$747$	−1.04005e6	−0.0681947
$748$	0	0
$749$	2.49537e7	1.62529
$750$	0	0
$751$	−1.15493e7	−0.747235	−0.373617	−	0.927583i	$-0.621883\pi$
$751$	−1.15493e7	−0.747235	−0.373617	+	0.927583i	$0.621883\pi$
$752$	0	0
$753$	−8.89003e6	−0.571368
$754$	0	0
$755$	1.12992e7	0.721407
$756$	0	0
$757$	2.30747e7	1.46351	0.731755	−	0.681568i	$-0.238702\pi$
$757$	2.30747e7	1.46351	0.731755	+	0.681568i	$0.238702\pi$
$758$	0	0
$759$	362759.	0.0228567
$760$	0	0
$761$	−7.63715e6	−0.478046	−0.239023	−	0.971014i	$-0.576827\pi$
$761$	−7.63715e6	−0.478046	−0.239023	+	0.971014i	$0.576827\pi$
$762$	0	0
$763$	1.60961e7	1.00095
$764$	0	0
$765$	8.51824e6	0.526255
$766$	0	0
$767$	−1.03787e6	−0.0637021
$768$	0	0
$769$	1.43630e7	0.875850	0.437925	−	0.899012i	$-0.355714\pi$
$769$	1.43630e7	0.875850	0.437925	+	0.899012i	$0.355714\pi$
$770$	0	0
$771$	−3.97549e6	−0.240855
$772$	0	0
$773$	1.23053e7	0.740704	0.370352	−	0.928892i	$-0.379237\pi$
$773$	1.23053e7	0.740704	0.370352	+	0.928892i	$0.379237\pi$
$774$	0	0
$775$	10417.4	0.000623026 0
$776$	0	0
$777$	−2.25997e6	−0.134292
$778$	0	0
$779$	−2.85384e7	−1.68494
$780$	0	0
$781$	474786.	0.0278529
$782$	0	0
$783$	1.53517e7	0.894854
$784$	0	0
$785$	−6.86652e6	−0.397706
$786$	0	0
$787$	−1.03667e6	−0.0596629	−0.0298315	−	0.999555i	$-0.509497\pi$
$787$	−1.03667e6	−0.0596629	−0.0298315	+	0.999555i	$0.509497\pi$
$788$	0	0
$789$	733792.	0.0419644
$790$	0	0
$791$	−8.57066e6	−0.487050
$792$	0	0
$793$	7.31388e6	0.413014
$794$	0	0
$795$	2.10823e6	0.118304
$796$	0	0
$797$	−1.78851e7	−0.997347	−0.498673	−	0.866790i	$-0.666179\pi$
$797$	−1.78851e7	−0.997347	−0.498673	+	0.866790i	$0.666179\pi$
$798$	0	0
$799$	3.31027e6	0.183441
$800$	0	0
$801$	−6.81908e6	−0.375530
$802$	0	0
$803$	−1.06545e6	−0.0583103
$804$	0	0
$805$	−6.02906e6	−0.327914
$806$	0	0
$807$	2.14843e6	0.116128
$808$	0	0
$809$	7.95290e6	0.427222	0.213611	−	0.976919i	$-0.431477\pi$
$809$	7.95290e6	0.427222	0.213611	+	0.976919i	$0.431477\pi$
$810$	0	0
$811$	−2.30520e6	−0.123071	−0.0615357	−	0.998105i	$-0.519600\pi$
$811$	−2.30520e6	−0.123071	−0.0615357	+	0.998105i	$0.519600\pi$
$812$	0	0
$813$	2.83869e6	0.150623
$814$	0	0
$815$	−679432.	−0.0358304
$816$	0	0
$817$	−7.83741e6	−0.410788
$818$	0	0
$819$	3.89067e6	0.202682
$820$	0	0
$821$	8.72052e6	0.451528	0.225764	−	0.974182i	$-0.427512\pi$
$821$	8.72052e6	0.451528	0.225764	+	0.974182i	$0.427512\pi$
$822$	0	0
$823$	2.09138e7	1.07630	0.538149	−	0.842850i	$-0.319124\pi$
$823$	2.09138e7	1.07630	0.538149	+	0.842850i	$0.319124\pi$
$824$	0	0
$825$	−135428.	−0.00692745
$826$	0	0
$827$	−2.94201e7	−1.49582	−0.747912	−	0.663798i	$-0.768944\pi$
$827$	−2.94201e7	−1.49582	−0.747912	+	0.663798i	$0.768944\pi$
$828$	0	0
$829$	−2.20194e7	−1.11281	−0.556403	−	0.830913i	$-0.687819\pi$
$829$	−2.20194e7	−1.11281	−0.556403	+	0.830913i	$0.687819\pi$
$830$	0	0
$831$	2.61275e6	0.131249
$832$	0	0
$833$	8.40878e6	0.419875
$834$	0	0
$835$	30210.4	0.00149948
$836$	0	0
$837$	61236.2	0.00302131
$838$	0	0
$839$	3.18362e7	1.56141	0.780703	−	0.624903i	$-0.214861\pi$
$839$	3.18362e7	1.56141	0.780703	+	0.624903i	$0.214861\pi$
$840$	0	0
$841$	−3.05063e6	−0.148730
$842$	0	0
$843$	1.15324e7	0.558920
$844$	0	0
$845$	−714025.	−0.0344010
$846$	0	0
$847$	2.31184e7	1.10726
$848$	0	0
$849$	1.48622e6	0.0707645
$850$	0	0
$851$	2.87973e6	0.136310
$852$	0	0
$853$	−1.61744e7	−0.761124	−0.380562	−	0.924756i	$-0.624269\pi$
$853$	−1.61744e7	−0.761124	−0.380562	+	0.924756i	$0.624269\pi$
$854$	0	0
$855$	7.10390e6	0.332339
$856$	0	0
$857$	−1.71014e6	−0.0795390	−0.0397695	−	0.999209i	$-0.512662\pi$
$857$	−1.71014e6	−0.0795390	−0.0397695	+	0.999209i	$0.512662\pi$
$858$	0	0
$859$	−3.10226e7	−1.43448	−0.717242	−	0.696825i	$-0.754596\pi$
$859$	−3.10226e7	−1.43448	−0.717242	+	0.696825i	$0.754596\pi$
$860$	0	0
$861$	−2.10878e7	−0.969447
$862$	0	0
$863$	3.39307e7	1.55084	0.775418	−	0.631449i	$-0.217539\pi$
$863$	3.39307e7	1.55084	0.775418	+	0.631449i	$0.217539\pi$
$864$	0	0
$865$	−1.11909e7	−0.508540
$866$	0	0
$867$	−2.85078e7	−1.28800
$868$	0	0
$869$	−1.95012e6	−0.0876016
$870$	0	0
$871$	−3.87150e6	−0.172915
$872$	0	0
$873$	−6.07705e6	−0.269872
$874$	0	0
$875$	2.25082e6	0.0993848
$876$	0	0
$877$	2.23180e7	0.979844	0.489922	−	0.871766i	$-0.337025\pi$
$877$	2.23180e7	0.979844	0.489922	+	0.871766i	$0.337025\pi$
$878$	0	0
$879$	−4.07570e6	−0.177922
$880$	0	0
$881$	−2.25314e6	−0.0978021	−0.0489011	−	0.998804i	$-0.515572\pi$
$881$	−2.25314e6	−0.0978021	−0.0489011	+	0.998804i	$0.515572\pi$
$882$	0	0
$883$	1.27130e7	0.548713	0.274357	−	0.961628i	$-0.411535\pi$
$883$	1.27130e7	0.548713	0.274357	+	0.961628i	$0.411535\pi$
$884$	0	0
$885$	−1.40029e6	−0.0600979
$886$	0	0
$887$	−2.09571e7	−0.894382	−0.447191	−	0.894439i	$-0.647576\pi$
$887$	−2.09571e7	−0.894382	−0.447191	+	0.894439i	$0.647576\pi$
$888$	0	0
$889$	−7.31489e6	−0.310422
$890$	0	0
$891$	126558.	0.00534067
$892$	0	0
$893$	2.76064e6	0.115846
$894$	0	0
$895$	−9.96498e6	−0.415833
$896$	0	0
$897$	2.58047e6	0.107082
$898$	0	0
$899$	69648.1	0.00287415
$900$	0	0
$901$	1.97128e7	0.808976
$902$	0	0
$903$	−5.79129e6	−0.236350
$904$	0	0
$905$	4.49203e6	0.182314
$906$	0	0
$907$	1.30726e7	0.527648	0.263824	−	0.964571i	$-0.415016\pi$
$907$	1.30726e7	0.527648	0.263824	+	0.964571i	$0.415016\pi$
$908$	0	0
$909$	−9.07240e6	−0.364177
$910$	0	0
$911$	−7.01062e6	−0.279873	−0.139936	−	0.990161i	$-0.544690\pi$
$911$	−7.01062e6	−0.279873	−0.139936	+	0.990161i	$0.544690\pi$
$912$	0	0
$913$	154611.	0.00613852
$914$	0	0
$915$	9.86786e6	0.389646
$916$	0	0
$917$	2.49016e7	0.977921
$918$	0	0
$919$	6.65846e6	0.260067	0.130033	−	0.991510i	$-0.458492\pi$
$919$	6.65846e6	0.260067	0.130033	+	0.991510i	$0.458492\pi$
$920$	0	0
$921$	1.43448e7	0.557245
$922$	0	0
$923$	3.37737e6	0.130489
$924$	0	0
$925$	−1.07508e6	−0.0413130
$926$	0	0
$927$	−2.59571e7	−0.992102
$928$	0	0
$929$	−4.66408e7	−1.77307	−0.886537	−	0.462659i	$-0.846896\pi$
$929$	−4.66408e7	−1.77307	−0.886537	+	0.462659i	$0.846896\pi$
$930$	0	0
$931$	7.01262e6	0.265159
$932$	0	0
$933$	2.97579e6	0.111918
$934$	0	0
$935$	−1.26630e6	−0.0473706
$936$	0	0
$937$	−2.64464e7	−0.984049	−0.492025	−	0.870581i	$-0.663743\pi$
$937$	−2.64464e7	−0.984049	−0.492025	+	0.870581i	$0.663743\pi$
$938$	0	0
$939$	−1.51366e7	−0.560227
$940$	0	0
$941$	−3.76271e7	−1.38525	−0.692623	−	0.721299i	$-0.743545\pi$
$941$	−3.76271e7	−1.38525	−0.692623	+	0.721299i	$0.743545\pi$
$942$	0	0
$943$	2.68708e7	0.984013
$944$	0	0
$945$	1.32308e7	0.481957
$946$	0	0
$947$	4.11437e7	1.49083	0.745415	−	0.666600i	$-0.232251\pi$
$947$	4.11437e7	1.49083	0.745415	+	0.666600i	$0.232251\pi$
$948$	0	0
$949$	−7.57904e6	−0.273180
$950$	0	0
$951$	2.10589e7	0.755065
$952$	0	0
$953$	3.51213e7	1.25268	0.626338	−	0.779552i	$-0.284553\pi$
$953$	3.51213e7	1.25268	0.626338	+	0.779552i	$0.284553\pi$
$954$	0	0
$955$	1.79335e7	0.636292
$956$	0	0
$957$	−905434.	−0.0319578
$958$	0	0
$959$	2.15615e7	0.757062
$960$	0	0
$961$	−2.86289e7	−0.999990
$962$	0	0
$963$	2.76842e7	0.961982
$964$	0	0
$965$	−316773.	−0.0109504
$966$	0	0
$967$	3.36172e7	1.15610	0.578049	−	0.816002i	$-0.303814\pi$
$967$	3.36172e7	1.15610	0.578049	+	0.816002i	$0.303814\pi$
$968$	0	0
$969$	−3.45742e7	−1.18289
$970$	0	0
$971$	−1.80332e7	−0.613796	−0.306898	−	0.951742i	$-0.599291\pi$
$971$	−1.80332e7	−0.613796	−0.306898	+	0.951742i	$0.599291\pi$
$972$	0	0
$973$	4.77732e7	1.61772
$974$	0	0
$975$	−963360.	−0.0324547
$976$	0	0
$977$	−3.45101e7	−1.15667	−0.578336	−	0.815798i	$-0.696298\pi$
$977$	−3.45101e7	−1.15667	−0.578336	+	0.815798i	$0.696298\pi$
$978$	0	0
$979$	1.01371e6	0.0338031
$980$	0	0
$981$	1.78575e7	0.592444
$982$	0	0
$983$	1.98276e7	0.654466	0.327233	−	0.944944i	$-0.393884\pi$
$983$	1.98276e7	0.654466	0.327233	+	0.944944i	$0.393884\pi$
$984$	0	0
$985$	556774.	0.0182847
$986$	0	0
$987$	2.03992e6	0.0666531
$988$	0	0
$989$	7.37944e6	0.239901
$990$	0	0
$991$	−2.34575e7	−0.758747	−0.379373	−	0.925244i	$-0.623860\pi$
$991$	−2.34575e7	−0.758747	−0.379373	+	0.925244i	$0.623860\pi$
$992$	0	0
$993$	3.27605e7	1.05433
$994$	0	0
$995$	1.46385e7	0.468746
$996$	0	0
$997$	3.22271e7	1.02680	0.513398	−	0.858151i	$-0.328387\pi$
$997$	3.22271e7	1.02680	0.513398	+	0.858151i	$0.328387\pi$
$998$	0	0
$999$	−6.31960e6	−0.200344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.6.a.f.1.4	✓	8
4.3	odd	2		1040.6.a.y.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.6.a.f.1.4	✓	8	1.1	even	1	trivial
1040.6.a.y.1.5		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	520.a (trivial)

\(f(q)\)	\(=\)	\(q-9.12057 q^{3} -25.0000 q^{5} -144.052 q^{7} -159.815 q^{9} +23.7578 q^{11} +169.000 q^{13} +228.014 q^{15} +2132.02 q^{17} +1778.03 q^{19} +1313.84 q^{21} -1674.13 q^{23} +625.000 q^{25} +3673.90 q^{27} +4178.58 q^{29} +16.6679 q^{31} -216.685 q^{33} +3601.31 q^{35} -1720.13 q^{37} -1541.38 q^{39} -16050.6 q^{41} -4407.92 q^{43} +3995.38 q^{45} +1552.64 q^{47} +3944.04 q^{49} -19445.3 q^{51} +9246.04 q^{53} -593.945 q^{55} -16216.6 q^{57} -6141.23 q^{59} +43277.4 q^{61} +23021.7 q^{63} -4225.00 q^{65} -22908.3 q^{67} +15269.0 q^{69} +19984.4 q^{71} -44846.4 q^{73} -5700.36 q^{75} -3422.36 q^{77} -82083.4 q^{79} +5327.01 q^{81} +6507.80 q^{83} -53300.6 q^{85} -38111.0 q^{87} +42668.5 q^{89} -24344.8 q^{91} -152.021 q^{93} -44450.7 q^{95} +38025.5 q^{97} -3796.86 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 200 q^{5} - 184 q^{7} + 744 q^{9} - 720 q^{11} + 1352 q^{13} - 248 q^{17} - 2544 q^{19} + 4520 q^{21} - 88 q^{23} + 5000 q^{25} - 864 q^{27} + 2224 q^{29} + 5368 q^{31} + 9720 q^{33} + 4600 q^{35}+ \cdots - 395808 q^{99}+O(q^{100}) \) 8 * q - 200 * q^5 - 184 * q^7 + 744 * q^9 - 720 * q^11 + 1352 * q^13 - 248 * q^17 - 2544 * q^19 + 4520 * q^21 - 88 * q^23 + 5000 * q^25 - 864 * q^27 + 2224 * q^29 + 5368 * q^31 + 9720 * q^33 + 4600 * q^35 + 7096 * q^37 + 6232 * q^41 + 10528 * q^43 - 18600 * q^45 - 16424 * q^47 - 7192 * q^49 - 7904 * q^51 - 20856 * q^53 + 18000 * q^55 - 26408 * q^57 - 44864 * q^59 - 39712 * q^61 - 112696 * q^63 - 33800 * q^65 + 10824 * q^67 - 221200 * q^69 + 32552 * q^71 - 120104 * q^73 - 76120 * q^77 - 113472 * q^79 - 44760 * q^81 - 173416 * q^83 + 6200 * q^85 - 71056 * q^87 - 214368 * q^89 - 31096 * q^91 - 171032 * q^93 + 63600 * q^95 - 95440 * q^97 - 395808 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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