LMFDB - Embedded newform 588.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,8,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("588.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	588.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:	$183.682394985$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21961}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5490 $ x^2 - x - 5490
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$74.5962$ of defining polynomial
Character	$\chi$	$=$	588.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} -344.385 q^{5} +729.000 q^{9} +7108.54 q^{11} +2499.24 q^{13} +9298.39 q^{15} -7854.38 q^{17} +23470.0 q^{19} -7657.02 q^{23} +40475.9 q^{25} -19683.0 q^{27} +246789. q^{29} -24007.8 q^{31} -191931. q^{33} -5112.02 q^{37} -67479.4 q^{39} -199345. q^{41} +440325. q^{43} -251057. q^{45} -697260. q^{47} +212068. q^{51} -488537. q^{53} -2.44807e6 q^{55} -633691. q^{57} -271749. q^{59} -3.29982e6 q^{61} -860700. q^{65} +228888. q^{67} +206740. q^{69} -2.34949e6 q^{71} +6.08175e6 q^{73} -1.09285e6 q^{75} -7.78437e6 q^{79} +531441. q^{81} +2.18086e6 q^{83} +2.70493e6 q^{85} -6.66330e6 q^{87} +4.94045e6 q^{89} +648210. q^{93} -8.08272e6 q^{95} -3.68264e6 q^{97} +5.18213e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 54 q^{3} - 96 q^{5} + 1458 q^{9} + 4140 q^{11} - 9228 q^{13} + 2592 q^{15} - 30528 q^{17} + 704 q^{19} + 38628 q^{23} + 24046 q^{25} - 39366 q^{27} + 131988 q^{29} - 165384 q^{31} - 111780 q^{33}+ \cdots + 3018060 q^{99}+O(q^{100}) $ 2 * q - 54 * q^3 - 96 * q^5 + 1458 * q^9 + 4140 * q^11 - 9228 * q^13 + 2592 * q^15 - 30528 * q^17 + 704 * q^19 + 38628 * q^23 + 24046 * q^25 - 39366 * q^27 + 131988 * q^29 - 165384 * q^31 - 111780 * q^33 + 381004 * q^37 + 249156 * q^39 - 379128 * q^41 + 916216 * q^43 - 69984 * q^45 - 914376 * q^47 + 824256 * q^51 + 1498332 * q^53 - 3185416 * q^55 - 19008 * q^57 - 893232 * q^59 - 478692 * q^61 - 3773568 * q^65 + 112784 * q^67 - 1042956 * q^69 - 6384228 * q^71 + 1433188 * q^73 - 649242 * q^75 - 5649336 * q^79 + 1062882 * q^81 + 8086680 * q^83 - 2926856 * q^85 - 3563676 * q^87 + 10113864 * q^89 + 4465368 * q^93 - 13737456 * q^95 - 7390172 * q^97 + 3018060 * 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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−344.385	−1.23211	−0.616054	−	0.787704i	$-0.711270\pi$
$5$	−344.385	−1.23211	−0.616054	+	0.787704i	$0.711270\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	7108.54	1.61030	0.805149	−	0.593073i	$-0.202085\pi$
$11$	7108.54	1.61030	0.805149	+	0.593073i	$0.202085\pi$
$12$	0	0
$13$	2499.24	0.315505	0.157752	−	0.987479i	$-0.449575\pi$
$13$	2499.24	0.315505	0.157752	+	0.987479i	$0.449575\pi$
$14$	0	0
$15$	9298.39	0.711358
$16$	0	0
$17$	−7854.38	−0.387740	−0.193870	−	0.981027i	$-0.562104\pi$
$17$	−7854.38	−0.387740	−0.193870	+	0.981027i	$0.562104\pi$
$18$	0	0
$19$	23470.0	0.785011	0.392506	−	0.919750i	$-0.371608\pi$
$19$	23470.0	0.785011	0.392506	+	0.919750i	$0.371608\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7657.02	−0.131224	−0.0656119	−	0.997845i	$-0.520900\pi$
$23$	−7657.02	−0.131224	−0.0656119	+	0.997845i	$0.520900\pi$
$24$	0	0
$25$	40475.9	0.518092
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	246789.	1.87902	0.939512	−	0.342515i	$-0.111279\pi$
$29$	246789.	1.87902	0.939512	+	0.342515i	$0.111279\pi$
$30$	0	0
$31$	−24007.8	−0.144739	−0.0723696	−	0.997378i	$-0.523056\pi$
$31$	−24007.8	−0.144739	−0.0723696	+	0.997378i	$0.523056\pi$
$32$	0	0
$33$	−191931.	−0.929706
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5112.02	−0.0165915	−0.00829577	−	0.999966i	$-0.502641\pi$
$37$	−5112.02	−0.0165915	−0.00829577	+	0.999966i	$0.502641\pi$
$38$	0	0
$39$	−67479.4	−0.182157
$40$	0	0
$41$	−199345.	−0.451712	−0.225856	−	0.974161i	$-0.572518\pi$
$41$	−199345.	−0.451712	−0.225856	+	0.974161i	$0.572518\pi$
$42$	0	0
$43$	440325.	0.844566	0.422283	−	0.906464i	$-0.361229\pi$
$43$	440325.	0.844566	0.422283	+	0.906464i	$0.361229\pi$
$44$	0	0
$45$	−251057.	−0.410703
$46$	0	0
$47$	−697260.	−0.979607	−0.489804	−	0.871833i	$-0.662932\pi$
$47$	−697260.	−0.979607	−0.489804	+	0.871833i	$0.662932\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	212068.	0.223862
$52$	0	0
$53$	−488537.	−0.450746	−0.225373	−	0.974273i	$-0.572360\pi$
$53$	−488537.	−0.450746	−0.225373	+	0.974273i	$0.572360\pi$
$54$	0	0
$55$	−2.44807e6	−1.98406
$56$	0	0
$57$	−633691.	−0.453226
$58$	0	0
$59$	−271749.	−0.172261	−0.0861303	−	0.996284i	$-0.527450\pi$
$59$	−271749.	−0.172261	−0.0861303	+	0.996284i	$0.527450\pi$
$60$	0	0
$61$	−3.29982e6	−1.86138	−0.930691	−	0.365807i	$-0.880793\pi$
$61$	−3.29982e6	−1.86138	−0.930691	+	0.365807i	$0.880793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−860700.	−0.388736
$66$	0	0
$67$	228888.	0.0929739	0.0464870	−	0.998919i	$-0.485197\pi$
$67$	228888.	0.0929739	0.0464870	+	0.998919i	$0.485197\pi$
$68$	0	0
$69$	206740.	0.0757621
$70$	0	0
$71$	−2.34949e6	−0.779058	−0.389529	−	0.921014i	$-0.627362\pi$
$71$	−2.34949e6	−0.779058	−0.389529	+	0.921014i	$0.627362\pi$
$72$	0	0
$73$	6.08175e6	1.82978	0.914890	−	0.403704i	$-0.132277\pi$
$73$	6.08175e6	1.82978	0.914890	+	0.403704i	$0.132277\pi$
$74$	0	0
$75$	−1.09285e6	−0.299121
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.78437e6	−1.77635	−0.888176	−	0.459504i	$-0.848027\pi$
$79$	−7.78437e6	−1.77635	−0.888176	+	0.459504i	$0.848027\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	2.18086e6	0.418653	0.209326	−	0.977846i	$-0.432873\pi$
$83$	2.18086e6	0.418653	0.209326	+	0.977846i	$0.432873\pi$
$84$	0	0
$85$	2.70493e6	0.477738
$86$	0	0
$87$	−6.66330e6	−1.08486
$88$	0	0
$89$	4.94045e6	0.742851	0.371426	−	0.928463i	$-0.378869\pi$
$89$	4.94045e6	0.742851	0.371426	+	0.928463i	$0.378869\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	648210.	0.0835652
$94$	0	0
$95$	−8.08272e6	−0.967219
$96$	0	0
$97$	−3.68264e6	−0.409692	−0.204846	−	0.978794i	$-0.565669\pi$
$97$	−3.68264e6	−0.409692	−0.204846	+	0.978794i	$0.565669\pi$
$98$	0	0
$99$	5.18213e6	0.536766
$100$	0	0
$101$	1.08159e6	0.104457	0.0522285	−	0.998635i	$-0.483368\pi$
$101$	1.08159e6	0.104457	0.0522285	+	0.998635i	$0.483368\pi$
$102$	0	0
$103$	1.53446e7	1.38364	0.691822	−	0.722068i	$-0.256808\pi$
$103$	1.53446e7	1.38364	0.691822	+	0.722068i	$0.256808\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.29274e7	−1.02016	−0.510080	−	0.860127i	$-0.670384\pi$
$107$	−1.29274e7	−1.02016	−0.510080	+	0.860127i	$0.670384\pi$
$108$	0	0
$109$	2.22964e7	1.64908	0.824540	−	0.565803i	$-0.191434\pi$
$109$	2.22964e7	1.64908	0.824540	+	0.565803i	$0.191434\pi$
$110$	0	0
$111$	138025.	0.00957913
$112$	0	0
$113$	−1.84044e7	−1.19991	−0.599954	−	0.800035i	$-0.704814\pi$
$113$	−1.84044e7	−1.19991	−0.599954	+	0.800035i	$0.704814\pi$
$114$	0	0
$115$	2.63696e6	0.161682
$116$	0	0
$117$	1.82194e6	0.105168
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.10442e7	1.59306
$122$	0	0
$123$	5.38231e6	0.260796
$124$	0	0
$125$	1.29658e7	0.593763
$126$	0	0
$127$	−4.15136e7	−1.79836	−0.899181	−	0.437576i	$-0.855837\pi$
$127$	−4.15136e7	−1.79836	−0.899181	+	0.437576i	$0.855837\pi$
$128$	0	0
$129$	−1.18888e7	−0.487610
$130$	0	0
$131$	3.67570e7	1.42854	0.714268	−	0.699873i	$-0.246760\pi$
$131$	3.67570e7	1.42854	0.714268	+	0.699873i	$0.246760\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.77853e6	0.237119
$136$	0	0
$137$	3.45324e7	1.14737	0.573686	−	0.819075i	$-0.305513\pi$
$137$	3.45324e7	1.14737	0.573686	+	0.819075i	$0.305513\pi$
$138$	0	0
$139$	−4.35370e7	−1.37501	−0.687507	−	0.726178i	$-0.741295\pi$
$139$	−4.35370e7	−1.37501	−0.687507	+	0.726178i	$0.741295\pi$
$140$	0	0
$141$	1.88260e7	0.565577
$142$	0	0
$143$	1.77659e7	0.508056
$144$	0	0
$145$	−8.49903e7	−2.31516
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.02156e6	0.173893	0.0869464	−	0.996213i	$-0.472289\pi$
$149$	7.02156e6	0.173893	0.0869464	+	0.996213i	$0.472289\pi$
$150$	0	0
$151$	1.55352e7	0.367195	0.183598	−	0.983001i	$-0.441226\pi$
$151$	1.55352e7	0.367195	0.183598	+	0.983001i	$0.441226\pi$
$152$	0	0
$153$	−5.72584e6	−0.129247
$154$	0	0
$155$	8.26792e6	0.178335
$156$	0	0
$157$	−1.96373e7	−0.404979	−0.202489	−	0.979284i	$-0.564903\pi$
$157$	−1.96373e7	−0.404979	−0.202489	+	0.979284i	$0.564903\pi$
$158$	0	0
$159$	1.31905e7	0.260238
$160$	0	0
$161$	0	0
$162$	0	0
$163$	5.09898e7	0.922204	0.461102	−	0.887347i	$-0.347454\pi$
$163$	5.09898e7	0.922204	0.461102	+	0.887347i	$0.347454\pi$
$164$	0	0
$165$	6.60980e7	1.14550
$166$	0	0
$167$	6.83363e7	1.13539	0.567693	−	0.823240i	$-0.307836\pi$
$167$	6.83363e7	1.13539	0.567693	+	0.823240i	$0.307836\pi$
$168$	0	0
$169$	−5.65023e7	−0.900457
$170$	0	0
$171$	1.71096e7	0.261670
$172$	0	0
$173$	1.28289e7	0.188376	0.0941882	−	0.995554i	$-0.469974\pi$
$173$	1.28289e7	0.188376	0.0941882	+	0.995554i	$0.469974\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.33722e6	0.0994547
$178$	0	0
$179$	6.79517e7	0.885553	0.442776	−	0.896632i	$-0.353994\pi$
$179$	6.79517e7	0.885553	0.442776	+	0.896632i	$0.353994\pi$
$180$	0	0
$181$	6.19724e7	0.776825	0.388413	−	0.921486i	$-0.373024\pi$
$181$	6.19724e7	0.776825	0.388413	+	0.921486i	$0.373024\pi$
$182$	0	0
$183$	8.90950e7	1.07467
$184$	0	0
$185$	1.76050e6	0.0204426
$186$	0	0
$187$	−5.58332e7	−0.624377
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.29061e8	−1.34023	−0.670113	−	0.742259i	$-0.733755\pi$
$191$	−1.29061e8	−1.34023	−0.670113	+	0.742259i	$0.733755\pi$
$192$	0	0
$193$	−1.03318e8	−1.03449	−0.517243	−	0.855839i	$-0.673041\pi$
$193$	−1.03318e8	−1.03449	−0.517243	+	0.855839i	$0.673041\pi$
$194$	0	0
$195$	2.32389e7	0.224437
$196$	0	0
$197$	1.72525e8	1.60776	0.803879	−	0.594793i	$-0.202766\pi$
$197$	1.72525e8	1.60776	0.803879	+	0.594793i	$0.202766\pi$
$198$	0	0
$199$	6.44856e7	0.580065	0.290033	−	0.957017i	$-0.406334\pi$
$199$	6.44856e7	0.580065	0.290033	+	0.957017i	$0.406334\pi$
$200$	0	0
$201$	−6.17998e6	−0.0536785
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.86513e7	0.556558
$206$	0	0
$207$	−5.58197e6	−0.0437413
$208$	0	0
$209$	1.66838e8	1.26410
$210$	0	0
$211$	−2.21550e8	−1.62361	−0.811807	−	0.583926i	$-0.801516\pi$
$211$	−2.21550e8	−1.62361	−0.811807	+	0.583926i	$0.801516\pi$
$212$	0	0
$213$	6.34363e7	0.449789
$214$	0	0
$215$	−1.51641e8	−1.04060
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.64207e8	−1.05642
$220$	0	0
$221$	−1.96300e7	−0.122334
$222$	0	0
$223$	1.05439e8	0.636697	0.318349	−	0.947974i	$-0.396872\pi$
$223$	1.05439e8	0.636697	0.318349	+	0.947974i	$0.396872\pi$
$224$	0	0
$225$	2.95070e7	0.172697
$226$	0	0
$227$	3.44669e8	1.95574	0.977871	−	0.209207i	$-0.0670882\pi$
$227$	3.44669e8	1.95574	0.977871	+	0.209207i	$0.0670882\pi$
$228$	0	0
$229$	−3.22018e8	−1.77197	−0.885985	−	0.463713i	$-0.846517\pi$
$229$	−3.22018e8	−1.77197	−0.885985	+	0.463713i	$0.846517\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.87923e8	1.49118	0.745592	−	0.666403i	$-0.232167\pi$
$233$	2.87923e8	1.49118	0.745592	+	0.666403i	$0.232167\pi$
$234$	0	0
$235$	2.40126e8	1.20698
$236$	0	0
$237$	2.10178e8	1.02558
$238$	0	0
$239$	−2.46950e8	−1.17008	−0.585040	−	0.811004i	$-0.698921\pi$
$239$	−2.46950e8	−1.17008	−0.585040	+	0.811004i	$0.698921\pi$
$240$	0	0
$241$	3.63210e8	1.67147	0.835735	−	0.549133i	$-0.185042\pi$
$241$	3.63210e8	1.67147	0.835735	+	0.549133i	$0.185042\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.86572e7	0.247675
$248$	0	0
$249$	−5.88831e7	−0.241709
$250$	0	0
$251$	−5.98237e7	−0.238790	−0.119395	−	0.992847i	$-0.538095\pi$
$251$	−5.98237e7	−0.238790	−0.119395	+	0.992847i	$0.538095\pi$
$252$	0	0
$253$	−5.44303e7	−0.211309
$254$	0	0
$255$	−7.30331e7	−0.275822
$256$	0	0
$257$	−1.99156e8	−0.731858	−0.365929	−	0.930643i	$-0.619249\pi$
$257$	−1.99156e8	−0.731858	−0.365929	+	0.930643i	$0.619249\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.79909e8	0.626341
$262$	0	0
$263$	5.42320e8	1.83828	0.919138	−	0.393936i	$-0.128887\pi$
$263$	5.42320e8	1.83828	0.919138	+	0.393936i	$0.128887\pi$
$264$	0	0
$265$	1.68245e8	0.555368
$266$	0	0
$267$	−1.33392e8	−0.428885
$268$	0	0
$269$	−3.89160e8	−1.21898	−0.609488	−	0.792795i	$-0.708625\pi$
$269$	−3.89160e8	−1.21898	−0.609488	+	0.792795i	$0.708625\pi$
$270$	0	0
$271$	8.73813e7	0.266702	0.133351	−	0.991069i	$-0.457426\pi$
$271$	8.73813e7	0.266702	0.133351	+	0.991069i	$0.457426\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.87725e8	0.834283
$276$	0	0
$277$	2.37346e8	0.670969	0.335485	−	0.942046i	$-0.391100\pi$
$277$	2.37346e8	0.670969	0.335485	+	0.942046i	$0.391100\pi$
$278$	0	0
$279$	−1.75017e7	−0.0482464
$280$	0	0
$281$	1.61007e8	0.432884	0.216442	−	0.976295i	$-0.430555\pi$
$281$	1.61007e8	0.432884	0.216442	+	0.976295i	$0.430555\pi$
$282$	0	0
$283$	5.03269e8	1.31992	0.659961	−	0.751300i	$-0.270573\pi$
$283$	5.03269e8	1.31992	0.659961	+	0.751300i	$0.270573\pi$
$284$	0	0
$285$	2.18233e8	0.558424
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.48647e8	−0.849658
$290$	0	0
$291$	9.94312e7	0.236536
$292$	0	0
$293$	3.58565e8	0.832781	0.416391	−	0.909186i	$-0.363295\pi$
$293$	3.58565e8	0.832781	0.416391	+	0.909186i	$0.363295\pi$
$294$	0	0
$295$	9.35862e7	0.212244
$296$	0	0
$297$	−1.39917e8	−0.309902
$298$	0	0
$299$	−1.91367e7	−0.0414017
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.92029e7	−0.0603083
$304$	0	0
$305$	1.13641e9	2.29342
$306$	0	0
$307$	2.31844e8	0.457311	0.228655	−	0.973507i	$-0.426567\pi$
$307$	2.31844e8	0.457311	0.228655	+	0.973507i	$0.426567\pi$
$308$	0	0
$309$	−4.14304e8	−0.798848
$310$	0	0
$311$	−5.96200e8	−1.12391	−0.561954	−	0.827169i	$-0.689950\pi$
$311$	−5.96200e8	−1.12391	−0.561954	+	0.827169i	$0.689950\pi$
$312$	0	0
$313$	8.70791e7	0.160512	0.0802562	−	0.996774i	$-0.474426\pi$
$313$	8.70791e7	0.160512	0.0802562	+	0.996774i	$0.474426\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.67049e8	1.35243	0.676217	−	0.736703i	$-0.263618\pi$
$317$	7.67049e8	1.35243	0.676217	+	0.736703i	$0.263618\pi$
$318$	0	0
$319$	1.75431e9	3.02579
$320$	0	0
$321$	3.49040e8	0.588990
$322$	0	0
$323$	−1.84342e8	−0.304380
$324$	0	0
$325$	1.01159e8	0.163460
$326$	0	0
$327$	−6.02003e8	−0.952097
$328$	0	0
$329$	0	0
$330$	0	0
$331$	3.99231e8	0.605099	0.302550	−	0.953134i	$-0.402162\pi$
$331$	3.99231e8	0.605099	0.302550	+	0.953134i	$0.402162\pi$
$332$	0	0
$333$	−3.72667e6	−0.00553052
$334$	0	0
$335$	−7.88256e7	−0.114554
$336$	0	0
$337$	−2.11383e6	−0.00300861	−0.00150431	−	0.999999i	$-0.500479\pi$
$337$	−2.11383e6	−0.00300861	−0.00150431	+	0.999999i	$0.500479\pi$
$338$	0	0
$339$	4.96919e8	0.692767
$340$	0	0
$341$	−1.70660e8	−0.233073
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.11980e7	−0.0933471
$346$	0	0
$347$	4.05619e8	0.521153	0.260576	−	0.965453i	$-0.416087\pi$
$347$	4.05619e8	0.521153	0.260576	+	0.965453i	$0.416087\pi$
$348$	0	0
$349$	6.06769e8	0.764072	0.382036	−	0.924147i	$-0.375223\pi$
$349$	6.06769e8	0.764072	0.382036	+	0.924147i	$0.375223\pi$
$350$	0	0
$351$	−4.91925e7	−0.0607189
$352$	0	0
$353$	8.47805e8	1.02585	0.512926	−	0.858433i	$-0.328562\pi$
$353$	8.47805e8	1.02585	0.512926	+	0.858433i	$0.328562\pi$
$354$	0	0
$355$	8.09129e8	0.959884
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.43048e8	−0.733522	−0.366761	−	0.930315i	$-0.619533\pi$
$359$	−6.43048e8	−0.733522	−0.366761	+	0.930315i	$0.619533\pi$
$360$	0	0
$361$	−3.43030e8	−0.383757
$362$	0	0
$363$	−8.38194e8	−0.919753
$364$	0	0
$365$	−2.09446e9	−2.25449
$366$	0	0
$367$	−2.58748e8	−0.273241	−0.136621	−	0.990623i	$-0.543624\pi$
$367$	−2.58748e8	−0.273241	−0.136621	+	0.990623i	$0.543624\pi$
$368$	0	0
$369$	−1.45322e8	−0.150571
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.78964e8	−0.278335	−0.139167	−	0.990269i	$-0.544443\pi$
$373$	−2.78964e8	−0.278335	−0.139167	+	0.990269i	$0.544443\pi$
$374$	0	0
$375$	−3.50076e8	−0.342809
$376$	0	0
$377$	6.16784e8	0.592841
$378$	0	0
$379$	−1.19101e8	−0.112377	−0.0561885	−	0.998420i	$-0.517895\pi$
$379$	−1.19101e8	−0.112377	−0.0561885	+	0.998420i	$0.517895\pi$
$380$	0	0
$381$	1.12087e9	1.03829
$382$	0	0
$383$	−2.53193e8	−0.230280	−0.115140	−	0.993349i	$-0.536732\pi$
$383$	−2.53193e8	−0.230280	−0.115140	+	0.993349i	$0.536732\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.20997e8	0.281522
$388$	0	0
$389$	7.13924e8	0.614934	0.307467	−	0.951559i	$-0.400519\pi$
$389$	7.13924e8	0.614934	0.307467	+	0.951559i	$0.400519\pi$
$390$	0	0
$391$	6.01412e7	0.0508807
$392$	0	0
$393$	−9.92440e8	−0.824765
$394$	0	0
$395$	2.68082e9	2.18866
$396$	0	0
$397$	−1.53079e9	−1.22786	−0.613931	−	0.789360i	$-0.710413\pi$
$397$	−1.53079e9	−1.22786	−0.613931	+	0.789360i	$0.710413\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.26388e8	−0.485108	−0.242554	−	0.970138i	$-0.577985\pi$
$401$	−6.26388e8	−0.485108	−0.242554	+	0.970138i	$0.577985\pi$
$402$	0	0
$403$	−6.00012e7	−0.0456659
$404$	0	0
$405$	−1.83020e8	−0.136901
$406$	0	0
$407$	−3.63390e7	−0.0267173
$408$	0	0
$409$	−5.70438e8	−0.412266	−0.206133	−	0.978524i	$-0.566088\pi$
$409$	−5.70438e8	−0.412266	−0.206133	+	0.978524i	$0.566088\pi$
$410$	0	0
$411$	−9.32374e8	−0.662436
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7.51054e8	−0.515826
$416$	0	0
$417$	1.17550e9	0.793865
$418$	0	0
$419$	−9.98546e8	−0.663161	−0.331581	−	0.943427i	$-0.607582\pi$
$419$	−9.98546e8	−0.663161	−0.331581	+	0.943427i	$0.607582\pi$
$420$	0	0
$421$	1.03711e9	0.677389	0.338694	−	0.940896i	$-0.390015\pi$
$421$	1.03711e9	0.677389	0.338694	+	0.940896i	$0.390015\pi$
$422$	0	0
$423$	−5.08302e8	−0.326536
$424$	0	0
$425$	−3.17913e8	−0.200885
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.79680e8	−0.293327
$430$	0	0
$431$	−1.31918e9	−0.793656	−0.396828	−	0.917893i	$-0.629889\pi$
$431$	−1.31918e9	−0.793656	−0.396828	+	0.917893i	$0.629889\pi$
$432$	0	0
$433$	1.73484e9	1.02695	0.513477	−	0.858103i	$-0.328357\pi$
$433$	1.73484e9	1.02695	0.513477	+	0.858103i	$0.328357\pi$
$434$	0	0
$435$	2.29474e9	1.33666
$436$	0	0
$437$	−1.79711e8	−0.103012
$438$	0	0
$439$	−1.63687e9	−0.923396	−0.461698	−	0.887037i	$-0.652760\pi$
$439$	−1.63687e9	−0.923396	−0.461698	+	0.887037i	$0.652760\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.43477e8	0.406307	0.203154	−	0.979147i	$-0.434881\pi$
$443$	7.43477e8	0.406307	0.203154	+	0.979147i	$0.434881\pi$
$444$	0	0
$445$	−1.70142e9	−0.915273
$446$	0	0
$447$	−1.89582e8	−0.100397
$448$	0	0
$449$	−7.27404e8	−0.379240	−0.189620	−	0.981858i	$-0.560726\pi$
$449$	−7.27404e8	−0.379240	−0.189620	+	0.981858i	$0.560726\pi$
$450$	0	0
$451$	−1.41705e9	−0.727390
$452$	0	0
$453$	−4.19450e8	−0.212000
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.06345e9	−0.521206	−0.260603	−	0.965446i	$-0.583921\pi$
$457$	−1.06345e9	−0.521206	−0.260603	+	0.965446i	$0.583921\pi$
$458$	0	0
$459$	1.54598e8	0.0746206
$460$	0	0
$461$	1.70184e9	0.809029	0.404515	−	0.914532i	$-0.367440\pi$
$461$	1.70184e9	0.809029	0.404515	+	0.914532i	$0.367440\pi$
$462$	0	0
$463$	3.48810e9	1.63326	0.816630	−	0.577162i	$-0.195840\pi$
$463$	3.48810e9	1.63326	0.816630	+	0.577162i	$0.195840\pi$
$464$	0	0
$465$	−2.23234e8	−0.102961
$466$	0	0
$467$	2.52499e9	1.14723	0.573615	−	0.819125i	$-0.305540\pi$
$467$	2.52499e9	1.14723	0.573615	+	0.819125i	$0.305540\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.30206e8	0.233815
$472$	0	0
$473$	3.13007e9	1.36000
$474$	0	0
$475$	9.49971e8	0.406708
$476$	0	0
$477$	−3.56144e8	−0.150249
$478$	0	0
$479$	4.34770e8	0.180753	0.0903765	−	0.995908i	$-0.471193\pi$
$479$	4.34770e8	0.180753	0.0903765	+	0.995908i	$0.471193\pi$
$480$	0	0
$481$	−1.27762e7	−0.00523471
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.26824e9	0.504786
$486$	0	0
$487$	4.99726e9	1.96056	0.980280	−	0.197611i	$-0.0633184\pi$
$487$	4.99726e9	1.96056	0.980280	+	0.197611i	$0.0633184\pi$
$488$	0	0
$489$	−1.37673e9	−0.532435
$490$	0	0
$491$	−5.79958e8	−0.221111	−0.110556	−	0.993870i	$-0.535263\pi$
$491$	−5.79958e8	−0.221111	−0.110556	+	0.993870i	$0.535263\pi$
$492$	0	0
$493$	−1.93837e9	−0.728573
$494$	0	0
$495$	−1.78465e9	−0.661354
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.97783e9	1.79345	0.896723	−	0.442591i	$-0.145941\pi$
$499$	4.97783e9	1.79345	0.896723	+	0.442591i	$0.145941\pi$
$500$	0	0
$501$	−1.84508e9	−0.655516
$502$	0	0
$503$	4.35283e9	1.52505	0.762524	−	0.646960i	$-0.223960\pi$
$503$	4.35283e9	1.52505	0.762524	+	0.646960i	$0.223960\pi$
$504$	0	0
$505$	−3.72483e8	−0.128702
$506$	0	0
$507$	1.52556e9	0.519879
$508$	0	0
$509$	2.22837e9	0.748987	0.374493	−	0.927230i	$-0.377817\pi$
$509$	2.22837e9	0.748987	0.374493	+	0.927230i	$0.377817\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.61960e8	−0.151075
$514$	0	0
$515$	−5.28444e9	−1.70480
$516$	0	0
$517$	−4.95650e9	−1.57746
$518$	0	0
$519$	−3.46379e8	−0.108759
$520$	0	0
$521$	−4.77969e9	−1.48070	−0.740352	−	0.672220i	$-0.765341\pi$
$521$	−4.77969e9	−1.48070	−0.740352	+	0.672220i	$0.765341\pi$
$522$	0	0
$523$	−9.69432e8	−0.296320	−0.148160	−	0.988963i	$-0.547335\pi$
$523$	−9.69432e8	−0.296320	−0.148160	+	0.988963i	$0.547335\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.88566e8	0.0561212
$528$	0	0
$529$	−3.34620e9	−0.982780
$530$	0	0
$531$	−1.98105e8	−0.0574202
$532$	0	0
$533$	−4.98210e8	−0.142517
$534$	0	0
$535$	4.45201e9	1.25695
$536$	0	0
$537$	−1.83470e9	−0.511274
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.04192e9	−0.554431	−0.277216	−	0.960808i	$-0.589412\pi$
$541$	−2.04192e9	−0.554431	−0.277216	+	0.960808i	$0.589412\pi$
$542$	0	0
$543$	−1.67325e9	−0.448500
$544$	0	0
$545$	−7.67854e9	−2.03185
$546$	0	0
$547$	−2.15208e9	−0.562215	−0.281108	−	0.959676i	$-0.590702\pi$
$547$	−2.15208e9	−0.562215	−0.281108	+	0.959676i	$0.590702\pi$
$548$	0	0
$549$	−2.40557e9	−0.620460
$550$	0	0
$551$	5.79214e9	1.47506
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.75336e7	−0.0118025
$556$	0	0
$557$	8.55391e8	0.209735	0.104868	−	0.994486i	$-0.466558\pi$
$557$	8.55391e8	0.209735	0.104868	+	0.994486i	$0.466558\pi$
$558$	0	0
$559$	1.10048e9	0.266465
$560$	0	0
$561$	1.50750e9	0.360484
$562$	0	0
$563$	1.97565e8	0.0466584	0.0233292	−	0.999728i	$-0.492573\pi$
$563$	1.97565e8	0.0466584	0.0233292	+	0.999728i	$0.492573\pi$
$564$	0	0
$565$	6.33820e9	1.47842
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.22011e9	1.41548	0.707742	−	0.706471i	$-0.249714\pi$
$569$	6.22011e9	1.41548	0.707742	+	0.706471i	$0.249714\pi$
$570$	0	0
$571$	−5.38583e9	−1.21067	−0.605335	−	0.795971i	$-0.706961\pi$
$571$	−5.38583e9	−1.21067	−0.605335	+	0.795971i	$0.706961\pi$
$572$	0	0
$573$	3.48465e9	0.773780
$574$	0	0
$575$	−3.09925e8	−0.0679860
$576$	0	0
$577$	2.97524e9	0.644773	0.322387	−	0.946608i	$-0.395515\pi$
$577$	2.97524e9	0.644773	0.322387	+	0.946608i	$0.395515\pi$
$578$	0	0
$579$	2.78958e9	0.597260
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.47279e9	−0.725836
$584$	0	0
$585$	−6.27450e8	−0.129579
$586$	0	0
$587$	9.49841e9	1.93828	0.969142	−	0.246502i	$-0.0792813\pi$
$587$	9.49841e9	1.93828	0.969142	+	0.246502i	$0.0792813\pi$
$588$	0	0
$589$	−5.63463e8	−0.113622
$590$	0	0
$591$	−4.65818e9	−0.928239
$592$	0	0
$593$	3.85157e9	0.758483	0.379242	−	0.925298i	$-0.376185\pi$
$593$	3.85157e9	0.758483	0.379242	+	0.925298i	$0.376185\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.74111e9	−0.334901
$598$	0	0
$599$	4.36355e9	0.829557	0.414778	−	0.909922i	$-0.363859\pi$
$599$	4.36355e9	0.829557	0.414778	+	0.909922i	$0.363859\pi$
$600$	0	0
$601$	−6.33773e9	−1.19089	−0.595447	−	0.803395i	$-0.703025\pi$
$601$	−6.33773e9	−1.19089	−0.595447	+	0.803395i	$0.703025\pi$
$602$	0	0
$603$	1.66859e8	0.0309913
$604$	0	0
$605$	−1.06912e10	−1.96282
$606$	0	0
$607$	−6.89059e9	−1.25053	−0.625267	−	0.780411i	$-0.715010\pi$
$607$	−6.89059e9	−1.25053	−0.625267	+	0.780411i	$0.715010\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.74262e9	−0.309071
$612$	0	0
$613$	−9.32798e9	−1.63560	−0.817798	−	0.575506i	$-0.804805\pi$
$613$	−9.32798e9	−1.63560	−0.817798	+	0.575506i	$0.804805\pi$
$614$	0	0
$615$	−1.85359e9	−0.321329
$616$	0	0
$617$	2.67877e9	0.459132	0.229566	−	0.973293i	$-0.426269\pi$
$617$	2.67877e9	0.459132	0.229566	+	0.973293i	$0.426269\pi$
$618$	0	0
$619$	9.17841e9	1.55543	0.777715	−	0.628618i	$-0.216379\pi$
$619$	9.17841e9	1.55543	0.777715	+	0.628618i	$0.216379\pi$
$620$	0	0
$621$	1.50713e8	0.0252540
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.62740e9	−1.24967
$626$	0	0
$627$	−4.50462e9	−0.729830
$628$	0	0
$629$	4.01518e7	0.00643321
$630$	0	0
$631$	3.35341e9	0.531354	0.265677	−	0.964062i	$-0.414404\pi$
$631$	3.35341e9	0.531354	0.265677	+	0.964062i	$0.414404\pi$
$632$	0	0
$633$	5.98184e9	0.937394
$634$	0	0
$635$	1.42967e10	2.21578
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.71278e9	−0.259686
$640$	0	0
$641$	−1.55893e9	−0.233788	−0.116894	−	0.993144i	$-0.537294\pi$
$641$	−1.55893e9	−0.233788	−0.116894	+	0.993144i	$0.537294\pi$
$642$	0	0
$643$	−3.52069e9	−0.522263	−0.261131	−	0.965303i	$-0.584096\pi$
$643$	−3.52069e9	−0.522263	−0.261131	+	0.965303i	$0.584096\pi$
$644$	0	0
$645$	4.09431e9	0.600789
$646$	0	0
$647$	−8.36160e9	−1.21374	−0.606869	−	0.794802i	$-0.707575\pi$
$647$	−8.36160e9	−1.21374	−0.606869	+	0.794802i	$0.707575\pi$
$648$	0	0
$649$	−1.93174e9	−0.277391
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.84949e9	−1.24372	−0.621859	−	0.783129i	$-0.713622\pi$
$653$	−8.84949e9	−1.24372	−0.621859	+	0.783129i	$0.713622\pi$
$654$	0	0
$655$	−1.26586e10	−1.76011
$656$	0	0
$657$	4.43360e9	0.609927
$658$	0	0
$659$	3.78573e9	0.515289	0.257644	−	0.966240i	$-0.417054\pi$
$659$	3.78573e9	0.515289	0.257644	+	0.966240i	$0.417054\pi$
$660$	0	0
$661$	−1.47274e10	−1.98344	−0.991722	−	0.128406i	$-0.959014\pi$
$661$	−1.47274e10	−1.98344	−0.991722	+	0.128406i	$0.959014\pi$
$662$	0	0
$663$	5.30009e8	0.0706294
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.88967e9	−0.246573
$668$	0	0
$669$	−2.84684e9	−0.367597
$670$	0	0
$671$	−2.34569e10	−2.99738
$672$	0	0
$673$	−1.79335e9	−0.226783	−0.113392	−	0.993550i	$-0.536171\pi$
$673$	−1.79335e9	−0.226783	−0.113392	+	0.993550i	$0.536171\pi$
$674$	0	0
$675$	−7.96688e8	−0.0997069
$676$	0	0
$677$	−1.05751e10	−1.30985	−0.654926	−	0.755693i	$-0.727300\pi$
$677$	−1.05751e10	−1.30985	−0.654926	+	0.755693i	$0.727300\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−9.30606e9	−1.12915
$682$	0	0
$683$	−5.18773e9	−0.623024	−0.311512	−	0.950242i	$-0.600835\pi$
$683$	−5.18773e9	−0.623024	−0.311512	+	0.950242i	$0.600835\pi$
$684$	0	0
$685$	−1.18924e10	−1.41369
$686$	0	0
$687$	8.69450e9	1.02305
$688$	0	0
$689$	−1.22097e9	−0.142213
$690$	0	0
$691$	−7.34485e9	−0.846856	−0.423428	−	0.905930i	$-0.639173\pi$
$691$	−7.34485e9	−0.846856	−0.423428	+	0.905930i	$0.639173\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.49935e10	1.69417
$696$	0	0
$697$	1.56573e9	0.175147
$698$	0	0
$699$	−7.77393e9	−0.860935
$700$	0	0
$701$	7.16925e9	0.786069	0.393034	−	0.919524i	$-0.371425\pi$
$701$	7.16925e9	0.786069	0.393034	+	0.919524i	$0.371425\pi$
$702$	0	0
$703$	−1.19979e8	−0.0130246
$704$	0	0
$705$	−6.48339e9	−0.696852
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−2.01023e9	−0.211829	−0.105914	−	0.994375i	$-0.533777\pi$
$709$	−2.01023e9	−0.211829	−0.105914	+	0.994375i	$0.533777\pi$
$710$	0	0
$711$	−5.67481e9	−0.592117
$712$	0	0
$713$	1.83828e8	0.0189932
$714$	0	0
$715$	−6.11832e9	−0.625981
$716$	0	0
$717$	6.66764e9	0.675546
$718$	0	0
$719$	7.55427e9	0.757952	0.378976	−	0.925407i	$-0.376276\pi$
$719$	7.55427e9	0.757952	0.378976	+	0.925407i	$0.376276\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.80668e9	−0.965024
$724$	0	0
$725$	9.98901e9	0.973508
$726$	0	0
$727$	−1.09036e10	−1.05244	−0.526221	−	0.850348i	$-0.676392\pi$
$727$	−1.09036e10	−1.05244	−0.526221	+	0.850348i	$0.676392\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−3.45848e9	−0.327472
$732$	0	0
$733$	−7.10538e9	−0.666382	−0.333191	−	0.942859i	$-0.608125\pi$
$733$	−7.10538e9	−0.666382	−0.333191	+	0.942859i	$0.608125\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.62706e9	0.149716
$738$	0	0
$739$	−1.28604e10	−1.17219	−0.586096	−	0.810241i	$-0.699336\pi$
$739$	−1.28604e10	−1.17219	−0.586096	+	0.810241i	$0.699336\pi$
$740$	0	0
$741$	−1.58374e9	−0.142995
$742$	0	0
$743$	−1.03498e10	−0.925701	−0.462850	−	0.886436i	$-0.653173\pi$
$743$	−1.03498e10	−0.925701	−0.462850	+	0.886436i	$0.653173\pi$
$744$	0	0
$745$	−2.41812e9	−0.214255
$746$	0	0
$747$	1.58985e9	0.139551
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.33169e10	1.14726	0.573630	−	0.819114i	$-0.305535\pi$
$751$	1.33169e10	1.14726	0.573630	+	0.819114i	$0.305535\pi$
$752$	0	0
$753$	1.61524e9	0.137865
$754$	0	0
$755$	−5.35008e9	−0.452425
$756$	0	0
$757$	−9.86095e9	−0.826196	−0.413098	−	0.910687i	$-0.635553\pi$
$757$	−9.86095e9	−0.826196	−0.413098	+	0.910687i	$0.635553\pi$
$758$	0	0
$759$	1.46962e9	0.122000
$760$	0	0
$761$	1.97462e10	1.62419	0.812094	−	0.583527i	$-0.198328\pi$
$761$	1.97462e10	1.62419	0.812094	+	0.583527i	$0.198328\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.97189e9	0.159246
$766$	0	0
$767$	−6.79165e8	−0.0543490
$768$	0	0
$769$	1.24911e10	0.990509	0.495254	−	0.868748i	$-0.335075\pi$
$769$	1.24911e10	0.990509	0.495254	+	0.868748i	$0.335075\pi$
$770$	0	0
$771$	5.37720e9	0.422538
$772$	0	0
$773$	−6.46836e9	−0.503693	−0.251846	−	0.967767i	$-0.581038\pi$
$773$	−6.46836e9	−0.503693	−0.251846	+	0.967767i	$0.581038\pi$
$774$	0	0
$775$	−9.71738e8	−0.0749883
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.67862e9	−0.354599
$780$	0	0
$781$	−1.67015e10	−1.25452
$782$	0	0
$783$	−4.85754e9	−0.361618
$784$	0	0
$785$	6.76278e9	0.498978
$786$	0	0
$787$	−4.41920e9	−0.323171	−0.161585	−	0.986859i	$-0.551661\pi$
$787$	−4.41920e9	−0.323171	−0.161585	+	0.986859i	$0.551661\pi$
$788$	0	0
$789$	−1.46426e10	−1.06133
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.24702e9	−0.587274
$794$	0	0
$795$	−4.54261e9	−0.320642
$796$	0	0
$797$	1.57455e10	1.10167	0.550835	−	0.834614i	$-0.314309\pi$
$797$	1.57455e10	1.10167	0.550835	+	0.834614i	$0.314309\pi$
$798$	0	0
$799$	5.47654e9	0.379833
$800$	0	0
$801$	3.60159e9	0.247617
$802$	0	0
$803$	4.32324e10	2.94649
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.05073e10	0.703776
$808$	0	0
$809$	2.16144e10	1.43523	0.717617	−	0.696438i	$-0.245233\pi$
$809$	2.16144e10	1.43523	0.717617	+	0.696438i	$0.245233\pi$
$810$	0	0
$811$	−2.00798e9	−0.132186	−0.0660931	−	0.997813i	$-0.521053\pi$
$811$	−2.00798e9	−0.132186	−0.0660931	+	0.997813i	$0.521053\pi$
$812$	0	0
$813$	−2.35930e9	−0.153980
$814$	0	0
$815$	−1.75601e10	−1.13626
$816$	0	0
$817$	1.03344e10	0.662994
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.78890e10	1.12820	0.564099	−	0.825707i	$-0.309224\pi$
$821$	1.78890e10	1.12820	0.564099	+	0.825707i	$0.309224\pi$
$822$	0	0
$823$	1.04292e10	0.652158	0.326079	−	0.945342i	$-0.394272\pi$
$823$	1.04292e10	0.652158	0.326079	+	0.945342i	$0.394272\pi$
$824$	0	0
$825$	−7.76858e9	−0.481673
$826$	0	0
$827$	−1.17153e10	−0.720252	−0.360126	−	0.932904i	$-0.617266\pi$
$827$	−1.17153e10	−0.720252	−0.360126	+	0.932904i	$0.617266\pi$
$828$	0	0
$829$	−3.75378e9	−0.228838	−0.114419	−	0.993433i	$-0.536501\pi$
$829$	−3.75378e9	−0.228838	−0.114419	+	0.993433i	$0.536501\pi$
$830$	0	0
$831$	−6.40834e9	−0.387384
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.35340e10	−1.39892
$836$	0	0
$837$	4.72545e8	0.0278551
$838$	0	0
$839$	1.70900e10	0.999021	0.499510	−	0.866308i	$-0.333513\pi$
$839$	1.70900e10	0.999021	0.499510	+	0.866308i	$0.333513\pi$
$840$	0	0
$841$	4.36548e10	2.53073
$842$	0	0
$843$	−4.34718e9	−0.249926
$844$	0	0
$845$	1.94585e10	1.10946
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.35883e10	−0.762057
$850$	0	0
$851$	3.91429e7	0.00217721
$852$	0	0
$853$	2.48124e10	1.36882	0.684411	−	0.729097i	$-0.260060\pi$
$853$	2.48124e10	1.36882	0.684411	+	0.729097i	$0.260060\pi$
$854$	0	0
$855$	−5.89230e9	−0.322406
$856$	0	0
$857$	3.68069e10	1.99755	0.998773	−	0.0495262i	$-0.0157711\pi$
$857$	3.68069e10	1.99755	0.998773	+	0.0495262i	$0.0157711\pi$
$858$	0	0
$859$	−1.50948e10	−0.812554	−0.406277	−	0.913750i	$-0.633173\pi$
$859$	−1.50948e10	−0.812554	−0.406277	+	0.913750i	$0.633173\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.13151e10	1.12888	0.564442	−	0.825473i	$-0.309091\pi$
$863$	2.13151e10	1.12888	0.564442	+	0.825473i	$0.309091\pi$
$864$	0	0
$865$	−4.41806e9	−0.232100
$866$	0	0
$867$	9.41348e9	0.490550
$868$	0	0
$869$	−5.53355e10	−2.86045
$870$	0	0
$871$	5.72045e8	0.0293337
$872$	0	0
$873$	−2.68464e9	−0.136564
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.84440e9	−0.442761	−0.221381	−	0.975187i	$-0.571056\pi$
$877$	−8.84440e9	−0.442761	−0.221381	+	0.975187i	$0.571056\pi$
$878$	0	0
$879$	−9.68125e9	−0.480807
$880$	0	0
$881$	2.69301e10	1.32685	0.663425	−	0.748243i	$-0.269102\pi$
$881$	2.69301e10	1.32685	0.663425	+	0.748243i	$0.269102\pi$
$882$	0	0
$883$	1.67021e10	0.816411	0.408205	−	0.912890i	$-0.366155\pi$
$883$	1.67021e10	0.816411	0.408205	+	0.912890i	$0.366155\pi$
$884$	0	0
$885$	−2.52683e9	−0.122539
$886$	0	0
$887$	1.57597e10	0.758253	0.379127	−	0.925345i	$-0.376224\pi$
$887$	1.57597e10	0.758253	0.379127	+	0.925345i	$0.376224\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.77777e9	0.178922
$892$	0	0
$893$	−1.63647e10	−0.769003
$894$	0	0
$895$	−2.34015e10	−1.09110
$896$	0	0
$897$	5.16691e8	0.0239033
$898$	0	0
$899$	−5.92485e9	−0.271969
$900$	0	0
$901$	3.83716e9	0.174772
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.13424e10	−0.957133
$906$	0	0
$907$	3.00786e10	1.33854	0.669272	−	0.743018i	$-0.266606\pi$
$907$	3.00786e10	1.33854	0.669272	+	0.743018i	$0.266606\pi$
$908$	0	0
$909$	7.88478e8	0.0348190
$910$	0	0
$911$	−8.21925e9	−0.360179	−0.180089	−	0.983650i	$-0.557639\pi$
$911$	−8.21925e9	−0.360179	−0.180089	+	0.983650i	$0.557639\pi$
$912$	0	0
$913$	1.55027e10	0.674156
$914$	0	0
$915$	−3.06830e10	−1.32411
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.36844e10	1.43161	0.715805	−	0.698300i	$-0.246060\pi$
$919$	3.36844e10	1.43161	0.715805	+	0.698300i	$0.246060\pi$
$920$	0	0
$921$	−6.25979e9	−0.264029
$922$	0	0
$923$	−5.87194e9	−0.245796
$924$	0	0
$925$	−2.06914e8	−0.00859595
$926$	0	0
$927$	1.11862e10	0.461215
$928$	0	0
$929$	−1.02153e10	−0.418019	−0.209009	−	0.977914i	$-0.567024\pi$
$929$	−1.02153e10	−0.418019	−0.209009	+	0.977914i	$0.567024\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.60974e10	0.648888
$934$	0	0
$935$	1.92281e10	0.769300
$936$	0	0
$937$	1.16234e10	0.461576	0.230788	−	0.973004i	$-0.425870\pi$
$937$	1.16234e10	0.461576	0.230788	+	0.973004i	$0.425870\pi$
$938$	0	0
$939$	−2.35113e9	−0.0926719
$940$	0	0
$941$	−2.82759e10	−1.10625	−0.553123	−	0.833099i	$-0.686564\pi$
$941$	−2.82759e10	−1.10625	−0.553123	+	0.833099i	$0.686564\pi$
$942$	0	0
$943$	1.52639e9	0.0592753
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.08655e10	1.94625	0.973125	−	0.230279i	$-0.0739640\pi$
$947$	5.08655e10	1.94625	0.973125	+	0.230279i	$0.0739640\pi$
$948$	0	0
$949$	1.51997e10	0.577304
$950$	0	0
$951$	−2.07103e10	−0.780828
$952$	0	0
$953$	−3.76585e10	−1.40941	−0.704705	−	0.709500i	$-0.748921\pi$
$953$	−3.76585e10	−1.40941	−0.704705	+	0.709500i	$0.748921\pi$
$954$	0	0
$955$	4.44467e10	1.65131
$956$	0	0
$957$	−4.73663e10	−1.74694
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.69362e10	−0.979051
$962$	0	0
$963$	−9.42409e9	−0.340053
$964$	0	0
$965$	3.55811e10	1.27460
$966$	0	0
$967$	3.35567e10	1.19340	0.596701	−	0.802464i	$-0.296478\pi$
$967$	3.35567e10	1.19340	0.596701	+	0.802464i	$0.296478\pi$
$968$	0	0
$969$	4.97725e9	0.175734
$970$	0	0
$971$	−4.77955e10	−1.67540	−0.837702	−	0.546128i	$-0.816101\pi$
$971$	−4.77955e10	−1.67540	−0.837702	+	0.546128i	$0.816101\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.73129e9	−0.0943739
$976$	0	0
$977$	4.18966e10	1.43730	0.718651	−	0.695371i	$-0.244760\pi$
$977$	4.18966e10	1.43730	0.718651	+	0.695371i	$0.244760\pi$
$978$	0	0
$979$	3.51194e10	1.19621
$980$	0	0
$981$	1.62541e10	0.549694
$982$	0	0
$983$	3.44206e10	1.15580	0.577899	−	0.816109i	$-0.303873\pi$
$983$	3.44206e10	1.15580	0.577899	+	0.816109i	$0.303873\pi$
$984$	0	0
$985$	−5.94150e10	−1.98093
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.37158e9	−0.110827
$990$	0	0
$991$	2.63968e10	0.861576	0.430788	−	0.902453i	$-0.358236\pi$
$991$	2.63968e10	0.861576	0.430788	+	0.902453i	$0.358236\pi$
$992$	0	0
$993$	−1.07792e10	−0.349354
$994$	0	0
$995$	−2.22079e10	−0.714704
$996$	0	0
$997$	2.56904e9	0.0820990	0.0410495	−	0.999157i	$-0.486930\pi$
$997$	2.56904e9	0.0820990	0.0410495	+	0.999157i	$0.486930\pi$
$998$	0	0
$999$	1.00620e8	0.00319304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.8.a.e.1.1		2
7.2	even	3		588.8.i.l.361.2		4
7.3	odd	6		588.8.i.i.373.1		4
7.4	even	3		588.8.i.l.373.2		4
7.5	odd	6		588.8.i.i.361.1		4
7.6	odd	2		84.8.a.d.1.2	✓	2
21.20	even	2		252.8.a.d.1.1		2
28.27	even	2		336.8.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.8.a.d.1.2	✓	2	7.6	odd	2
252.8.a.d.1.1		2	21.20	even	2
336.8.a.k.1.2		2	28.27	even	2
588.8.a.e.1.1		2	1.1	even	1	trivial
588.8.i.i.361.1		4	7.5	odd	6
588.8.i.i.373.1		4	7.3	odd	6
588.8.i.l.361.2		4	7.2	even	3
588.8.i.l.373.2		4	7.4	even	3

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} -344.385 q^{5} +729.000 q^{9} +7108.54 q^{11} +2499.24 q^{13} +9298.39 q^{15} -7854.38 q^{17} +23470.0 q^{19} -7657.02 q^{23} +40475.9 q^{25} -19683.0 q^{27} +246789. q^{29} -24007.8 q^{31} -191931. q^{33} -5112.02 q^{37} -67479.4 q^{39} -199345. q^{41} +440325. q^{43} -251057. q^{45} -697260. q^{47} +212068. q^{51} -488537. q^{53} -2.44807e6 q^{55} -633691. q^{57} -271749. q^{59} -3.29982e6 q^{61} -860700. q^{65} +228888. q^{67} +206740. q^{69} -2.34949e6 q^{71} +6.08175e6 q^{73} -1.09285e6 q^{75} -7.78437e6 q^{79} +531441. q^{81} +2.18086e6 q^{83} +2.70493e6 q^{85} -6.66330e6 q^{87} +4.94045e6 q^{89} +648210. q^{93} -8.08272e6 q^{95} -3.68264e6 q^{97} +5.18213e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} - 96 q^{5} + 1458 q^{9} + 4140 q^{11} - 9228 q^{13} + 2592 q^{15} - 30528 q^{17} + 704 q^{19} + 38628 q^{23} + 24046 q^{25} - 39366 q^{27} + 131988 q^{29} - 165384 q^{31} - 111780 q^{33}+ \cdots + 3018060 q^{99}+O(q^{100}) \) 2 * q - 54 * q^3 - 96 * q^5 + 1458 * q^9 + 4140 * q^11 - 9228 * q^13 + 2592 * q^15 - 30528 * q^17 + 704 * q^19 + 38628 * q^23 + 24046 * q^25 - 39366 * q^27 + 131988 * q^29 - 165384 * q^31 - 111780 * q^33 + 381004 * q^37 + 249156 * q^39 - 379128 * q^41 + 916216 * q^43 - 69984 * q^45 - 914376 * q^47 + 824256 * q^51 + 1498332 * q^53 - 3185416 * q^55 - 19008 * q^57 - 893232 * q^59 - 478692 * q^61 - 3773568 * q^65 + 112784 * q^67 - 1042956 * q^69 - 6384228 * q^71 + 1433188 * q^73 - 649242 * q^75 - 5649336 * q^79 + 1062882 * q^81 + 8086680 * q^83 - 2926856 * q^85 - 3563676 * q^87 + 10113864 * q^89 + 4465368 * q^93 - 13737456 * q^95 - 7390172 * q^97 + 3018060 * q^99

Introduction

L-functions

Modular forms

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