LMFDB - Embedded newform 588.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,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, 6, names="a")

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

chi := DirichletCharacter("588.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$94.3056860500$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7081}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1770 $ x^2 - x - 1770
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-41.5743$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +18.5743 q^{5} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +18.5743 q^{5} +81.0000 q^{9} -161.426 q^{11} -14.1283 q^{13} -167.169 q^{15} +765.703 q^{17} -1414.29 q^{19} +4185.43 q^{23} -2779.99 q^{25} -729.000 q^{27} -4202.60 q^{29} +2387.35 q^{31} +1452.83 q^{33} -672.939 q^{37} +127.155 q^{39} +4173.45 q^{41} -5430.94 q^{43} +1504.52 q^{45} -6302.68 q^{47} -6891.32 q^{51} +16416.7 q^{53} -2998.37 q^{55} +12728.6 q^{57} -3966.49 q^{59} +50338.7 q^{61} -262.424 q^{65} +13645.3 q^{67} -37668.9 q^{69} -83957.2 q^{71} +28578.3 q^{73} +25019.9 q^{75} -59955.5 q^{79} +6561.00 q^{81} +61583.0 q^{83} +14222.4 q^{85} +37823.4 q^{87} -42298.3 q^{89} -21486.2 q^{93} -26269.5 q^{95} -44638.1 q^{97} -13075.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} - 47 q^{5} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 - 47 * q^5 + 162 * q^9	$ 2 q - 18 q^{3} - 47 q^{5} + 162 q^{9} - 407 q^{11} - 449 q^{13} + 423 q^{15} + 1868 q^{17} + 1463 q^{19} - 44 q^{23} - 1605 q^{25} - 1458 q^{27} + 767 q^{29} + 11170 q^{31} + 3663 q^{33} - 3113 q^{37} + 4041 q^{39} + 7842 q^{41} - 12629 q^{43} - 3807 q^{45} - 9576 q^{47} - 16812 q^{51} + 13395 q^{53} + 13105 q^{55} - 13167 q^{57} + 47521 q^{59} + 63652 q^{61} + 28254 q^{65} + 44541 q^{67} + 396 q^{69} - 125840 q^{71} - 6039 q^{73} + 14445 q^{75} + 17588 q^{79} + 13122 q^{81} - 39325 q^{83} - 58060 q^{85} - 6903 q^{87} - 83082 q^{89} - 100530 q^{93} - 214946 q^{95} - 184785 q^{97} - 32967 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 - 47 * q^5 + 162 * q^9 - 407 * q^11 - 449 * q^13 + 423 * q^15 + 1868 * q^17 + 1463 * q^19 - 44 * q^23 - 1605 * q^25 - 1458 * q^27 + 767 * q^29 + 11170 * q^31 + 3663 * q^33 - 3113 * q^37 + 4041 * q^39 + 7842 * q^41 - 12629 * q^43 - 3807 * q^45 - 9576 * q^47 - 16812 * q^51 + 13395 * q^53 + 13105 * q^55 - 13167 * q^57 + 47521 * q^59 + 63652 * q^61 + 28254 * q^65 + 44541 * q^67 + 396 * q^69 - 125840 * q^71 - 6039 * q^73 + 14445 * q^75 + 17588 * q^79 + 13122 * q^81 - 39325 * q^83 - 58060 * q^85 - 6903 * q^87 - 83082 * q^89 - 100530 * q^93 - 214946 * q^95 - 184785 * q^97 - 32967 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	18.5743	0.332268	0.166134	−	0.986103i	$-0.446872\pi$
$5$	18.5743	0.332268	0.166134	+	0.986103i	$0.446872\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−161.426	−0.402245	−0.201123	−	0.979566i	$-0.564459\pi$
$11$	−161.426	−0.402245	−0.201123	+	0.979566i	$0.564459\pi$
$12$	0	0
$13$	−14.1283	−0.0231863	−0.0115932	−	0.999933i	$-0.503690\pi$
$13$	−14.1283	−0.0231863	−0.0115932	+	0.999933i	$0.503690\pi$
$14$	0	0
$15$	−167.169	−0.191835
$16$	0	0
$17$	765.703	0.642596	0.321298	−	0.946978i	$-0.395881\pi$
$17$	765.703	0.642596	0.321298	+	0.946978i	$0.395881\pi$
$18$	0	0
$19$	−1414.29	−0.898783	−0.449392	−	0.893335i	$-0.648359\pi$
$19$	−1414.29	−0.898783	−0.449392	+	0.893335i	$0.648359\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4185.43	1.64976	0.824880	−	0.565308i	$-0.191242\pi$
$23$	4185.43	1.64976	0.824880	+	0.565308i	$0.191242\pi$
$24$	0	0
$25$	−2779.99	−0.889598
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−4202.60	−0.927947	−0.463974	−	0.885849i	$-0.653577\pi$
$29$	−4202.60	−0.927947	−0.463974	+	0.885849i	$0.653577\pi$
$30$	0	0
$31$	2387.35	0.446182	0.223091	−	0.974798i	$-0.428385\pi$
$31$	2387.35	0.446182	0.223091	+	0.974798i	$0.428385\pi$
$32$	0	0
$33$	1452.83	0.232236
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−672.939	−0.0808112	−0.0404056	−	0.999183i	$-0.512865\pi$
$37$	−672.939	−0.0808112	−0.0404056	+	0.999183i	$0.512865\pi$
$38$	0	0
$39$	127.155	0.0133866
$40$	0	0
$41$	4173.45	0.387735	0.193868	−	0.981028i	$-0.437897\pi$
$41$	4173.45	0.387735	0.193868	+	0.981028i	$0.437897\pi$
$42$	0	0
$43$	−5430.94	−0.447923	−0.223962	−	0.974598i	$-0.571899\pi$
$43$	−5430.94	−0.447923	−0.223962	+	0.974598i	$0.571899\pi$
$44$	0	0
$45$	1504.52	0.110756
$46$	0	0
$47$	−6302.68	−0.416179	−0.208090	−	0.978110i	$-0.566725\pi$
$47$	−6302.68	−0.416179	−0.208090	+	0.978110i	$0.566725\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6891.32	−0.371003
$52$	0	0
$53$	16416.7	0.802778	0.401389	−	0.915908i	$-0.368528\pi$
$53$	16416.7	0.802778	0.401389	+	0.915908i	$0.368528\pi$
$54$	0	0
$55$	−2998.37	−0.133653
$56$	0	0
$57$	12728.6	0.518913
$58$	0	0
$59$	−3966.49	−0.148346	−0.0741731	−	0.997245i	$-0.523632\pi$
$59$	−3966.49	−0.148346	−0.0741731	+	0.997245i	$0.523632\pi$
$60$	0	0
$61$	50338.7	1.73212	0.866059	−	0.499942i	$-0.166645\pi$
$61$	50338.7	1.73212	0.866059	+	0.499942i	$0.166645\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−262.424	−0.00770407
$66$	0	0
$67$	13645.3	0.371360	0.185680	−	0.982610i	$-0.440551\pi$
$67$	13645.3	0.371360	0.185680	+	0.982610i	$0.440551\pi$
$68$	0	0
$69$	−37668.9	−0.952490
$70$	0	0
$71$	−83957.2	−1.97657	−0.988284	−	0.152624i	$-0.951228\pi$
$71$	−83957.2	−1.97657	−0.988284	+	0.152624i	$0.951228\pi$
$72$	0	0
$73$	28578.3	0.627668	0.313834	−	0.949478i	$-0.398387\pi$
$73$	28578.3	0.627668	0.313834	+	0.949478i	$0.398387\pi$
$74$	0	0
$75$	25019.9	0.513610
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−59955.5	−1.08084	−0.540420	−	0.841396i	$-0.681734\pi$
$79$	−59955.5	−1.08084	−0.540420	+	0.841396i	$0.681734\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	61583.0	0.981219	0.490610	−	0.871380i	$-0.336774\pi$
$83$	61583.0	0.981219	0.490610	+	0.871380i	$0.336774\pi$
$84$	0	0
$85$	14222.4	0.213514
$86$	0	0
$87$	37823.4	0.535751
$88$	0	0
$89$	−42298.3	−0.566042	−0.283021	−	0.959114i	$-0.591337\pi$
$89$	−42298.3	−0.566042	−0.283021	+	0.959114i	$0.591337\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−21486.2	−0.257603
$94$	0	0
$95$	−26269.5	−0.298637
$96$	0	0
$97$	−44638.1	−0.481700	−0.240850	−	0.970562i	$-0.577426\pi$
$97$	−44638.1	−0.481700	−0.240850	+	0.970562i	$0.577426\pi$
$98$	0	0
$99$	−13075.5	−0.134082
$100$	0	0
$101$	111924.	1.09175	0.545873	−	0.837868i	$-0.316198\pi$
$101$	111924.	1.09175	0.545873	+	0.837868i	$0.316198\pi$
$102$	0	0
$103$	−166949.	−1.55057	−0.775285	−	0.631612i	$-0.782394\pi$
$103$	−166949.	−1.55057	−0.775285	+	0.631612i	$0.782394\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	102986.	0.869595	0.434798	−	0.900528i	$-0.356820\pi$
$107$	102986.	0.869595	0.434798	+	0.900528i	$0.356820\pi$
$108$	0	0
$109$	−137821.	−1.11109	−0.555546	−	0.831486i	$-0.687491\pi$
$109$	−137821.	−1.11109	−0.555546	+	0.831486i	$0.687491\pi$
$110$	0	0
$111$	6056.45	0.0466563
$112$	0	0
$113$	−129967.	−0.957495	−0.478748	−	0.877953i	$-0.658909\pi$
$113$	−129967.	−0.957495	−0.478748	+	0.877953i	$0.658909\pi$
$114$	0	0
$115$	77741.7	0.548162
$116$	0	0
$117$	−1144.39	−0.00772877
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−134993.	−0.838199
$122$	0	0
$123$	−37561.0	−0.223859
$124$	0	0
$125$	−109681.	−0.627853
$126$	0	0
$127$	−243265.	−1.33835	−0.669177	−	0.743103i	$-0.733353\pi$
$127$	−243265.	−1.33835	−0.669177	+	0.743103i	$0.733353\pi$
$128$	0	0
$129$	48878.4	0.258609
$130$	0	0
$131$	−194336.	−0.989405	−0.494703	−	0.869062i	$-0.664723\pi$
$131$	−194336.	−0.989405	−0.494703	+	0.869062i	$0.664723\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−13540.7	−0.0639450
$136$	0	0
$137$	−3093.70	−0.0140824	−0.00704121	−	0.999975i	$-0.502241\pi$
$137$	−3093.70	−0.0140824	−0.00704121	+	0.999975i	$0.502241\pi$
$138$	0	0
$139$	−22600.4	−0.0992155	−0.0496078	−	0.998769i	$-0.515797\pi$
$139$	−22600.4	−0.0992155	−0.0496078	+	0.998769i	$0.515797\pi$
$140$	0	0
$141$	56724.1	0.240281
$142$	0	0
$143$	2280.67	0.00932658
$144$	0	0
$145$	−78060.6	−0.308327
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−352794.	−1.30183	−0.650917	−	0.759149i	$-0.725616\pi$
$149$	−352794.	−1.30183	−0.650917	+	0.759149i	$0.725616\pi$
$150$	0	0
$151$	145096.	0.517863	0.258931	−	0.965896i	$-0.416630\pi$
$151$	145096.	0.517863	0.258931	+	0.965896i	$0.416630\pi$
$152$	0	0
$153$	62021.9	0.214199
$154$	0	0
$155$	44343.5	0.148252
$156$	0	0
$157$	217895.	0.705503	0.352751	−	0.935717i	$-0.385246\pi$
$157$	217895.	0.705503	0.352751	+	0.935717i	$0.385246\pi$
$158$	0	0
$159$	−147750.	−0.463484
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−322111.	−0.949591	−0.474795	−	0.880096i	$-0.657478\pi$
$163$	−322111.	−0.949591	−0.474795	+	0.880096i	$0.657478\pi$
$164$	0	0
$165$	26985.4	0.0771647
$166$	0	0
$167$	−707392.	−1.96277	−0.981384	−	0.192056i	$-0.938485\pi$
$167$	−707392.	−1.96277	−0.981384	+	0.192056i	$0.938485\pi$
$168$	0	0
$169$	−371093.	−0.999462
$170$	0	0
$171$	−114558.	−0.299594
$172$	0	0
$173$	170507.	0.433140	0.216570	−	0.976267i	$-0.430513\pi$
$173$	170507.	0.433140	0.216570	+	0.976267i	$0.430513\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	35698.4	0.0856477
$178$	0	0
$179$	−415406.	−0.969037	−0.484519	−	0.874781i	$-0.661005\pi$
$179$	−415406.	−0.969037	−0.484519	+	0.874781i	$0.661005\pi$
$180$	0	0
$181$	1162.38	0.00263726	0.00131863	−	0.999999i	$-0.499580\pi$
$181$	1162.38	0.00263726	0.00131863	+	0.999999i	$0.499580\pi$
$182$	0	0
$183$	−453048.	−1.00004
$184$	0	0
$185$	−12499.4	−0.0268510
$186$	0	0
$187$	−123604.	−0.258481
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−745614.	−1.47887	−0.739436	−	0.673227i	$-0.764908\pi$
$191$	−745614.	−1.47887	−0.739436	+	0.673227i	$0.764908\pi$
$192$	0	0
$193$	−198373.	−0.383344	−0.191672	−	0.981459i	$-0.561391\pi$
$193$	−198373.	−0.383344	−0.191672	+	0.981459i	$0.561391\pi$
$194$	0	0
$195$	2361.82	0.00444795
$196$	0	0
$197$	−469368.	−0.861684	−0.430842	−	0.902427i	$-0.641783\pi$
$197$	−469368.	−0.861684	−0.430842	+	0.902427i	$0.641783\pi$
$198$	0	0
$199$	193593.	0.346544	0.173272	−	0.984874i	$-0.444566\pi$
$199$	193593.	0.346544	0.173272	+	0.984874i	$0.444566\pi$
$200$	0	0
$201$	−122807.	−0.214405
$202$	0	0
$203$	0	0
$204$	0	0
$205$	77519.0	0.128832
$206$	0	0
$207$	339020.	0.549920
$208$	0	0
$209$	228303.	0.361531
$210$	0	0
$211$	−298066.	−0.460900	−0.230450	−	0.973084i	$-0.574020\pi$
$211$	−298066.	−0.460900	−0.230450	+	0.973084i	$0.574020\pi$
$212$	0	0
$213$	755615.	1.14117
$214$	0	0
$215$	−100876.	−0.148831
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−257205.	−0.362384
$220$	0	0
$221$	−10818.1	−0.0148994
$222$	0	0
$223$	187215.	0.252103	0.126051	−	0.992024i	$-0.459770\pi$
$223$	187215.	0.252103	0.126051	+	0.992024i	$0.459770\pi$
$224$	0	0
$225$	−225180.	−0.296533
$226$	0	0
$227$	1.33896e6	1.72466	0.862332	−	0.506344i	$-0.169003\pi$
$227$	1.33896e6	1.72466	0.862332	+	0.506344i	$0.169003\pi$
$228$	0	0
$229$	−951656.	−1.19920	−0.599600	−	0.800300i	$-0.704674\pi$
$229$	−951656.	−1.19920	−0.599600	+	0.800300i	$0.704674\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	743419.	0.897106	0.448553	−	0.893756i	$-0.351940\pi$
$233$	743419.	0.897106	0.448553	+	0.893756i	$0.351940\pi$
$234$	0	0
$235$	−117068.	−0.138283
$236$	0	0
$237$	539599.	0.624023
$238$	0	0
$239$	−625847.	−0.708718	−0.354359	−	0.935110i	$-0.615301\pi$
$239$	−625847.	−0.708718	−0.354359	+	0.935110i	$0.615301\pi$
$240$	0	0
$241$	1.33282e6	1.47819	0.739094	−	0.673602i	$-0.235254\pi$
$241$	1.33282e6	1.47819	0.739094	+	0.673602i	$0.235254\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19981.5	0.0208395
$248$	0	0
$249$	−554247.	−0.566507
$250$	0	0
$251$	−1.78809e6	−1.79145	−0.895726	−	0.444606i	$-0.853344\pi$
$251$	−1.78809e6	−1.79145	−0.895726	+	0.444606i	$0.853344\pi$
$252$	0	0
$253$	−675636.	−0.663608
$254$	0	0
$255$	−128002.	−0.123272
$256$	0	0
$257$	−508924.	−0.480640	−0.240320	−	0.970694i	$-0.577252\pi$
$257$	−508924.	−0.480640	−0.240320	+	0.970694i	$0.577252\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−340411.	−0.309316
$262$	0	0
$263$	−336493.	−0.299976	−0.149988	−	0.988688i	$-0.547924\pi$
$263$	−336493.	−0.299976	−0.149988	+	0.988688i	$0.547924\pi$
$264$	0	0
$265$	304929.	0.266737
$266$	0	0
$267$	380685.	0.326804
$268$	0	0
$269$	−1.49553e6	−1.26013	−0.630064	−	0.776544i	$-0.716971\pi$
$269$	−1.49553e6	−1.26013	−0.630064	+	0.776544i	$0.716971\pi$
$270$	0	0
$271$	1.77697e6	1.46979	0.734897	−	0.678179i	$-0.237230\pi$
$271$	1.77697e6	1.46979	0.734897	+	0.678179i	$0.237230\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	448762.	0.357837
$276$	0	0
$277$	881265.	0.690092	0.345046	−	0.938586i	$-0.387863\pi$
$277$	881265.	0.690092	0.345046	+	0.938586i	$0.387863\pi$
$278$	0	0
$279$	193375.	0.148727
$280$	0	0
$281$	−1.13932e6	−0.860756	−0.430378	−	0.902649i	$-0.641620\pi$
$281$	−1.13932e6	−0.860756	−0.430378	+	0.902649i	$0.641620\pi$
$282$	0	0
$283$	897466.	0.666119	0.333059	−	0.942906i	$-0.391919\pi$
$283$	897466.	0.666119	0.333059	+	0.942906i	$0.391919\pi$
$284$	0	0
$285$	236426.	0.172418
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−833556.	−0.587071
$290$	0	0
$291$	401743.	0.278110
$292$	0	0
$293$	1.84614e6	1.25631	0.628153	−	0.778090i	$-0.283811\pi$
$293$	1.84614e6	1.25631	0.628153	+	0.778090i	$0.283811\pi$
$294$	0	0
$295$	−73674.9	−0.0492907
$296$	0	0
$297$	117679.	0.0774121
$298$	0	0
$299$	−59133.1	−0.0382519
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.00732e6	−0.630320
$304$	0	0
$305$	935008.	0.575527
$306$	0	0
$307$	−1.38565e6	−0.839089	−0.419544	−	0.907735i	$-0.637810\pi$
$307$	−1.38565e6	−0.839089	−0.419544	+	0.907735i	$0.637810\pi$
$308$	0	0
$309$	1.50254e6	0.895221
$310$	0	0
$311$	−2.31248e6	−1.35574	−0.677871	−	0.735181i	$-0.737097\pi$
$311$	−2.31248e6	−1.35574	−0.677871	+	0.735181i	$0.737097\pi$
$312$	0	0
$313$	1.16880e6	0.674341	0.337171	−	0.941444i	$-0.390530\pi$
$313$	1.16880e6	0.674341	0.337171	+	0.941444i	$0.390530\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.43433e6	0.801681	0.400840	−	0.916148i	$-0.368718\pi$
$317$	1.43433e6	0.801681	0.400840	+	0.916148i	$0.368718\pi$
$318$	0	0
$319$	678408.	0.373262
$320$	0	0
$321$	−926871.	−0.502061
$322$	0	0
$323$	−1.08293e6	−0.577554
$324$	0	0
$325$	39276.6	0.0206265
$326$	0	0
$327$	1.24039e6	0.641489
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.23418e6	−1.12085	−0.560426	−	0.828205i	$-0.689362\pi$
$331$	−2.23418e6	−1.12085	−0.560426	+	0.828205i	$0.689362\pi$
$332$	0	0
$333$	−54508.0	−0.0269371
$334$	0	0
$335$	253452.	0.123391
$336$	0	0
$337$	3.08787e6	1.48110	0.740549	−	0.672002i	$-0.234565\pi$
$337$	3.08787e6	1.48110	0.740549	+	0.672002i	$0.234565\pi$
$338$	0	0
$339$	1.16970e6	0.552810
$340$	0	0
$341$	−385380.	−0.179475
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−699675.	−0.316482
$346$	0	0
$347$	−3.11754e6	−1.38992	−0.694959	−	0.719050i	$-0.744577\pi$
$347$	−3.11754e6	−1.38992	−0.694959	+	0.719050i	$0.744577\pi$
$348$	0	0
$349$	−613026.	−0.269411	−0.134706	−	0.990886i	$-0.543009\pi$
$349$	−613026.	−0.269411	−0.134706	+	0.990886i	$0.543009\pi$
$350$	0	0
$351$	10299.5	0.00446221
$352$	0	0
$353$	−3.78370e6	−1.61614	−0.808071	−	0.589085i	$-0.799488\pi$
$353$	−3.78370e6	−1.61614	−0.808071	+	0.589085i	$0.799488\pi$
$354$	0	0
$355$	−1.55945e6	−0.656750
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.72213e6	1.52425	0.762125	−	0.647430i	$-0.224156\pi$
$359$	3.72213e6	1.52425	0.762125	+	0.647430i	$0.224156\pi$
$360$	0	0
$361$	−475879.	−0.192189
$362$	0	0
$363$	1.21493e6	0.483934
$364$	0	0
$365$	530824.	0.208554
$366$	0	0
$367$	−4.13795e6	−1.60369	−0.801845	−	0.597532i	$-0.796148\pi$
$367$	−4.13795e6	−1.60369	−0.801845	+	0.597532i	$0.796148\pi$
$368$	0	0
$369$	338049.	0.129245
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.94418e6	1.09570	0.547851	−	0.836576i	$-0.315446\pi$
$373$	2.94418e6	1.09570	0.547851	+	0.836576i	$0.315446\pi$
$374$	0	0
$375$	987132.	0.362491
$376$	0	0
$377$	59375.7	0.0215157
$378$	0	0
$379$	2.97504e6	1.06388	0.531942	−	0.846781i	$-0.321462\pi$
$379$	2.97504e6	1.06388	0.531942	+	0.846781i	$0.321462\pi$
$380$	0	0
$381$	2.18939e6	0.772699
$382$	0	0
$383$	−2.06104e6	−0.717942	−0.358971	−	0.933349i	$-0.616872\pi$
$383$	−2.06104e6	−0.717942	−0.358971	+	0.933349i	$0.616872\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−439906.	−0.149308
$388$	0	0
$389$	2.09084e6	0.700562	0.350281	−	0.936645i	$-0.386086\pi$
$389$	2.09084e6	0.700562	0.350281	+	0.936645i	$0.386086\pi$
$390$	0	0
$391$	3.20480e6	1.06013
$392$	0	0
$393$	1.74902e6	0.571233
$394$	0	0
$395$	−1.11363e6	−0.359128
$396$	0	0
$397$	−1.17129e6	−0.372984	−0.186492	−	0.982457i	$-0.559712\pi$
$397$	−1.17129e6	−0.372984	−0.186492	+	0.982457i	$0.559712\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.09149e6	0.960078	0.480039	−	0.877247i	$-0.340623\pi$
$401$	3.09149e6	0.960078	0.480039	+	0.877247i	$0.340623\pi$
$402$	0	0
$403$	−33729.2	−0.0103453
$404$	0	0
$405$	121866.	0.0369187
$406$	0	0
$407$	108630.	0.0325059
$408$	0	0
$409$	4.34853e6	1.28539	0.642693	−	0.766124i	$-0.277817\pi$
$409$	4.34853e6	1.28539	0.642693	+	0.766124i	$0.277817\pi$
$410$	0	0
$411$	27843.3	0.00813048
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.14386e6	0.326028
$416$	0	0
$417$	203404.	0.0572821
$418$	0	0
$419$	−3.13660e6	−0.872818	−0.436409	−	0.899748i	$-0.643750\pi$
$419$	−3.13660e6	−0.872818	−0.436409	+	0.899748i	$0.643750\pi$
$420$	0	0
$421$	6.02560e6	1.65690	0.828448	−	0.560066i	$-0.189224\pi$
$421$	6.02560e6	1.65690	0.828448	+	0.560066i	$0.189224\pi$
$422$	0	0
$423$	−510517.	−0.138726
$424$	0	0
$425$	−2.12865e6	−0.571652
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−20526.0	−0.00538471
$430$	0	0
$431$	−2.29863e6	−0.596040	−0.298020	−	0.954560i	$-0.596326\pi$
$431$	−2.29863e6	−0.596040	−0.298020	+	0.954560i	$0.596326\pi$
$432$	0	0
$433$	5.62982e6	1.44303	0.721515	−	0.692399i	$-0.243446\pi$
$433$	5.62982e6	1.44303	0.721515	+	0.692399i	$0.243446\pi$
$434$	0	0
$435$	702545.	0.178013
$436$	0	0
$437$	−5.91942e6	−1.48278
$438$	0	0
$439$	5.44963e6	1.34960	0.674801	−	0.737999i	$-0.264229\pi$
$439$	5.44963e6	1.34960	0.674801	+	0.737999i	$0.264229\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.70933e6	1.38222	0.691108	−	0.722751i	$-0.257123\pi$
$443$	5.70933e6	1.38222	0.691108	+	0.722751i	$0.257123\pi$
$444$	0	0
$445$	−785664.	−0.188077
$446$	0	0
$447$	3.17515e6	0.751615
$448$	0	0
$449$	5.38234e6	1.25996	0.629978	−	0.776613i	$-0.283064\pi$
$449$	5.38234e6	1.25996	0.629978	+	0.776613i	$0.283064\pi$
$450$	0	0
$451$	−673701.	−0.155965
$452$	0	0
$453$	−1.30587e6	−0.298988
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.39051e6	−1.43135	−0.715673	−	0.698435i	$-0.753880\pi$
$457$	−6.39051e6	−1.43135	−0.715673	+	0.698435i	$0.753880\pi$
$458$	0	0
$459$	−558197.	−0.123668
$460$	0	0
$461$	−7.34511e6	−1.60970	−0.804851	−	0.593476i	$-0.797755\pi$
$461$	−7.34511e6	−1.60970	−0.804851	+	0.593476i	$0.797755\pi$
$462$	0	0
$463$	−4.63416e6	−1.00466	−0.502329	−	0.864677i	$-0.667523\pi$
$463$	−4.63416e6	−1.00466	−0.502329	+	0.864677i	$0.667523\pi$
$464$	0	0
$465$	−399091.	−0.0855933
$466$	0	0
$467$	7.63195e6	1.61936	0.809680	−	0.586872i	$-0.199641\pi$
$467$	7.63195e6	1.61936	0.809680	+	0.586872i	$0.199641\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.96106e6	−0.407322
$472$	0	0
$473$	876693.	0.180175
$474$	0	0
$475$	3.93172e6	0.799556
$476$	0	0
$477$	1.32975e6	0.267593
$478$	0	0
$479$	−9.61262e6	−1.91427	−0.957135	−	0.289644i	$-0.906463\pi$
$479$	−9.61262e6	−1.91427	−0.957135	+	0.289644i	$0.906463\pi$
$480$	0	0
$481$	9507.49	0.00187371
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−829124.	−0.160053
$486$	0	0
$487$	2.80223e6	0.535404	0.267702	−	0.963502i	$-0.413736\pi$
$487$	2.80223e6	0.535404	0.267702	+	0.963502i	$0.413736\pi$
$488$	0	0
$489$	2.89900e6	0.548246
$490$	0	0
$491$	4.82008e6	0.902299	0.451149	−	0.892448i	$-0.351014\pi$
$491$	4.82008e6	0.902299	0.451149	+	0.892448i	$0.351014\pi$
$492$	0	0
$493$	−3.21794e6	−0.596295
$494$	0	0
$495$	−242868.	−0.0445511
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.59658e6	0.826387	0.413194	−	0.910643i	$-0.364413\pi$
$499$	4.59658e6	0.826387	0.413194	+	0.910643i	$0.364413\pi$
$500$	0	0
$501$	6.36653e6	1.13320
$502$	0	0
$503$	−1.80055e6	−0.317311	−0.158656	−	0.987334i	$-0.550716\pi$
$503$	−1.80055e6	−0.317311	−0.158656	+	0.987334i	$0.550716\pi$
$504$	0	0
$505$	2.07892e6	0.362752
$506$	0	0
$507$	3.33984e6	0.577040
$508$	0	0
$509$	4.83205e6	0.826678	0.413339	−	0.910577i	$-0.364362\pi$
$509$	4.83205e6	0.826678	0.413339	+	0.910577i	$0.364362\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.03102e6	0.172971
$514$	0	0
$515$	−3.10097e6	−0.515204
$516$	0	0
$517$	1.01741e6	0.167406
$518$	0	0
$519$	−1.53457e6	−0.250073
$520$	0	0
$521$	−2.81125e6	−0.453738	−0.226869	−	0.973925i	$-0.572849\pi$
$521$	−2.81125e6	−0.453738	−0.226869	+	0.973925i	$0.572849\pi$
$522$	0	0
$523$	2.45429e6	0.392349	0.196174	−	0.980569i	$-0.437148\pi$
$523$	2.45429e6	0.392349	0.196174	+	0.980569i	$0.437148\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.82800e6	0.286715
$528$	0	0
$529$	1.10815e7	1.72171
$530$	0	0
$531$	−321286.	−0.0494487
$532$	0	0
$533$	−58963.7	−0.00899015
$534$	0	0
$535$	1.91289e6	0.288939
$536$	0	0
$537$	3.73866e6	0.559474
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.15846e6	−0.170172	−0.0850862	−	0.996374i	$-0.527117\pi$
$541$	−1.15846e6	−0.170172	−0.0850862	+	0.996374i	$0.527117\pi$
$542$	0	0
$543$	−10461.4	−0.00152262
$544$	0	0
$545$	−2.55994e6	−0.369180
$546$	0	0
$547$	4.63638e6	0.662538	0.331269	−	0.943536i	$-0.392523\pi$
$547$	4.63638e6	0.662538	0.331269	+	0.943536i	$0.392523\pi$
$548$	0	0
$549$	4.07744e6	0.577373
$550$	0	0
$551$	5.94370e6	0.834023
$552$	0	0
$553$	0	0
$554$	0	0
$555$	112495.	0.0155024
$556$	0	0
$557$	271832.	0.0371247	0.0185624	−	0.999828i	$-0.494091\pi$
$557$	271832.	0.0371247	0.0185624	+	0.999828i	$0.494091\pi$
$558$	0	0
$559$	76730.0	0.0103857
$560$	0	0
$561$	1.11244e6	0.149234
$562$	0	0
$563$	−3.38244e6	−0.449738	−0.224869	−	0.974389i	$-0.572195\pi$
$563$	−3.38244e6	−0.449738	−0.224869	+	0.974389i	$0.572195\pi$
$564$	0	0
$565$	−2.41405e6	−0.318145
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.31226e7	1.69918	0.849590	−	0.527443i	$-0.176849\pi$
$569$	1.31226e7	1.69918	0.849590	+	0.527443i	$0.176849\pi$
$570$	0	0
$571$	−1.47162e7	−1.88889	−0.944443	−	0.328674i	$-0.893398\pi$
$571$	−1.47162e7	−1.88889	−0.944443	+	0.328674i	$0.893398\pi$
$572$	0	0
$573$	6.71053e6	0.853828
$574$	0	0
$575$	−1.16355e7	−1.46762
$576$	0	0
$577$	3.88295e6	0.485537	0.242769	−	0.970084i	$-0.421944\pi$
$577$	3.88295e6	0.485537	0.242769	+	0.970084i	$0.421944\pi$
$578$	0	0
$579$	1.78536e6	0.221324
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.65007e6	−0.322914
$584$	0	0
$585$	−21256.3	−0.00256802
$586$	0	0
$587$	−4.97913e6	−0.596428	−0.298214	−	0.954499i	$-0.596391\pi$
$587$	−4.97913e6	−0.596428	−0.298214	+	0.954499i	$0.596391\pi$
$588$	0	0
$589$	−3.37641e6	−0.401021
$590$	0	0
$591$	4.22431e6	0.497493
$592$	0	0
$593$	−1.53269e7	−1.78985	−0.894926	−	0.446215i	$-0.852772\pi$
$593$	−1.53269e7	−1.78985	−0.894926	+	0.446215i	$0.852772\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.74234e6	−0.200077
$598$	0	0
$599$	−6.62961e6	−0.754955	−0.377477	−	0.926019i	$-0.623208\pi$
$599$	−6.62961e6	−0.754955	−0.377477	+	0.926019i	$0.623208\pi$
$600$	0	0
$601$	1.45010e6	0.163762	0.0818808	−	0.996642i	$-0.473907\pi$
$601$	1.45010e6	0.163762	0.0818808	+	0.996642i	$0.473907\pi$
$602$	0	0
$603$	1.10527e6	0.123787
$604$	0	0
$605$	−2.50740e6	−0.278507
$606$	0	0
$607$	−3.91110e6	−0.430851	−0.215425	−	0.976520i	$-0.569114\pi$
$607$	−3.91110e6	−0.430851	−0.215425	+	0.976520i	$0.569114\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	89046.1	0.00964966
$612$	0	0
$613$	1.17753e7	1.26567	0.632835	−	0.774287i	$-0.281891\pi$
$613$	1.17753e7	1.26567	0.632835	+	0.774287i	$0.281891\pi$
$614$	0	0
$615$	−697671.	−0.0743812
$616$	0	0
$617$	−4.61462e6	−0.488004	−0.244002	−	0.969775i	$-0.578460\pi$
$617$	−4.61462e6	−0.488004	−0.244002	+	0.969775i	$0.578460\pi$
$618$	0	0
$619$	5.67466e6	0.595269	0.297634	−	0.954680i	$-0.403802\pi$
$619$	5.67466e6	0.595269	0.297634	+	0.954680i	$0.403802\pi$
$620$	0	0
$621$	−3.05118e6	−0.317497
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.65022e6	0.680983
$626$	0	0
$627$	−2.05473e6	−0.208730
$628$	0	0
$629$	−515271.	−0.0519289
$630$	0	0
$631$	5.67894e6	0.567798	0.283899	−	0.958854i	$-0.408372\pi$
$631$	5.67894e6	0.567798	0.283899	+	0.958854i	$0.408372\pi$
$632$	0	0
$633$	2.68260e6	0.266101
$634$	0	0
$635$	−4.51849e6	−0.444692
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.80053e6	−0.658856
$640$	0	0
$641$	−2.10737e6	−0.202580	−0.101290	−	0.994857i	$-0.532297\pi$
$641$	−2.10737e6	−0.202580	−0.101290	+	0.994857i	$0.532297\pi$
$642$	0	0
$643$	2.30987e6	0.220323	0.110162	−	0.993914i	$-0.464863\pi$
$643$	2.30987e6	0.220323	0.110162	+	0.993914i	$0.464863\pi$
$644$	0	0
$645$	907885.	0.0859274
$646$	0	0
$647$	−9.86066e6	−0.926073	−0.463036	−	0.886339i	$-0.653240\pi$
$647$	−9.86066e6	−0.926073	−0.463036	+	0.886339i	$0.653240\pi$
$648$	0	0
$649$	640293.	0.0596715
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.00309e7	0.920570	0.460285	−	0.887771i	$-0.347747\pi$
$653$	1.00309e7	0.920570	0.460285	+	0.887771i	$0.347747\pi$
$654$	0	0
$655$	−3.60966e6	−0.328748
$656$	0	0
$657$	2.31484e6	0.209223
$658$	0	0
$659$	−1.42828e7	−1.28115	−0.640575	−	0.767895i	$-0.721304\pi$
$659$	−1.42828e7	−1.28115	−0.640575	+	0.767895i	$0.721304\pi$
$660$	0	0
$661$	1.96419e7	1.74856	0.874280	−	0.485421i	$-0.161334\pi$
$661$	1.96419e7	1.74856	0.874280	+	0.485421i	$0.161334\pi$
$662$	0	0
$663$	97362.7	0.00860219
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.75897e7	−1.53089
$668$	0	0
$669$	−1.68493e6	−0.145552
$670$	0	0
$671$	−8.12596e6	−0.696736
$672$	0	0
$673$	−9.00150e6	−0.766086	−0.383043	−	0.923731i	$-0.625124\pi$
$673$	−9.00150e6	−0.766086	−0.383043	+	0.923731i	$0.625124\pi$
$674$	0	0
$675$	2.02662e6	0.171203
$676$	0	0
$677$	−1.34236e7	−1.12564	−0.562818	−	0.826581i	$-0.690283\pi$
$677$	−1.34236e7	−1.12564	−0.562818	+	0.826581i	$0.690283\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.20507e7	−0.995735
$682$	0	0
$683$	−8.01681e6	−0.657582	−0.328791	−	0.944403i	$-0.606641\pi$
$683$	−8.01681e6	−0.657582	−0.328791	+	0.944403i	$0.606641\pi$
$684$	0	0
$685$	−57463.5	−0.00467913
$686$	0	0
$687$	8.56491e6	0.692358
$688$	0	0
$689$	−231940.	−0.0186135
$690$	0	0
$691$	2.41973e7	1.92785	0.963923	−	0.266181i	$-0.0857618\pi$
$691$	2.41973e7	1.92785	0.963923	+	0.266181i	$0.0857618\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−419788.	−0.0329661
$696$	0	0
$697$	3.19562e6	0.249157
$698$	0	0
$699$	−6.69077e6	−0.517944
$700$	0	0
$701$	−1.56268e6	−0.120109	−0.0600543	−	0.998195i	$-0.519127\pi$
$701$	−1.56268e6	−0.120109	−0.0600543	+	0.998195i	$0.519127\pi$
$702$	0	0
$703$	951732.	0.0726317
$704$	0	0
$705$	1.05361e6	0.0798377
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.95390e7	−1.45978	−0.729889	−	0.683566i	$-0.760428\pi$
$709$	−1.95390e7	−1.45978	−0.729889	+	0.683566i	$0.760428\pi$
$710$	0	0
$711$	−4.85639e6	−0.360280
$712$	0	0
$713$	9.99210e6	0.736093
$714$	0	0
$715$	42362.0	0.00309892
$716$	0	0
$717$	5.63262e6	0.409178
$718$	0	0
$719$	8.10190e6	0.584473	0.292237	−	0.956346i	$-0.405601\pi$
$719$	8.10190e6	0.584473	0.292237	+	0.956346i	$0.405601\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.19954e7	−0.853432
$724$	0	0
$725$	1.16832e7	0.825500
$726$	0	0
$727$	−2.52759e7	−1.77366	−0.886829	−	0.462098i	$-0.847097\pi$
$727$	−2.52759e7	−1.77366	−0.886829	+	0.462098i	$0.847097\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−4.15848e6	−0.287834
$732$	0	0
$733$	−2.74414e7	−1.88646	−0.943228	−	0.332145i	$-0.892228\pi$
$733$	−2.74414e7	−1.88646	−0.943228	+	0.332145i	$0.892228\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.20270e6	−0.149378
$738$	0	0
$739$	−6.53569e6	−0.440231	−0.220115	−	0.975474i	$-0.570643\pi$
$739$	−6.53569e6	−0.440231	−0.220115	+	0.975474i	$0.570643\pi$
$740$	0	0
$741$	−179834.	−0.0120317
$742$	0	0
$743$	−2.46784e7	−1.64000	−0.820001	−	0.572362i	$-0.806027\pi$
$743$	−2.46784e7	−1.64000	−0.820001	+	0.572362i	$0.806027\pi$
$744$	0	0
$745$	−6.55292e6	−0.432558
$746$	0	0
$747$	4.98823e6	0.327073
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3.27822e6	−0.212099	−0.106049	−	0.994361i	$-0.533820\pi$
$751$	−3.27822e6	−0.212099	−0.106049	+	0.994361i	$0.533820\pi$
$752$	0	0
$753$	1.60928e7	1.03430
$754$	0	0
$755$	2.69507e6	0.172069
$756$	0	0
$757$	3.68090e6	0.233461	0.116731	−	0.993164i	$-0.462759\pi$
$757$	3.68090e6	0.233461	0.116731	+	0.993164i	$0.462759\pi$
$758$	0	0
$759$	6.08073e6	0.383134
$760$	0	0
$761$	−2.10198e7	−1.31573	−0.657866	−	0.753135i	$-0.728541\pi$
$761$	−2.10198e7	−1.31573	−0.657866	+	0.753135i	$0.728541\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.15202e6	0.0711713
$766$	0	0
$767$	56039.8	0.00343960
$768$	0	0
$769$	4.15418e6	0.253320	0.126660	−	0.991946i	$-0.459574\pi$
$769$	4.15418e6	0.253320	0.126660	+	0.991946i	$0.459574\pi$
$770$	0	0
$771$	4.58032e6	0.277498
$772$	0	0
$773$	−1.46348e6	−0.0880924	−0.0440462	−	0.999029i	$-0.514025\pi$
$773$	−1.46348e6	−0.0880924	−0.0440462	+	0.999029i	$0.514025\pi$
$774$	0	0
$775$	−6.63682e6	−0.396923
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.90247e6	−0.348490
$780$	0	0
$781$	1.35528e7	0.795065
$782$	0	0
$783$	3.06370e6	0.178584
$784$	0	0
$785$	4.04726e6	0.234416
$786$	0	0
$787$	2.01711e7	1.16089	0.580446	−	0.814299i	$-0.302878\pi$
$787$	2.01711e7	1.16089	0.580446	+	0.814299i	$0.302878\pi$
$788$	0	0
$789$	3.02844e6	0.173191
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−711201.	−0.0401614
$794$	0	0
$795$	−2.74436e6	−0.154001
$796$	0	0
$797$	−2.19496e7	−1.22400	−0.612000	−	0.790858i	$-0.709635\pi$
$797$	−2.19496e7	−1.22400	−0.612000	+	0.790858i	$0.709635\pi$
$798$	0	0
$799$	−4.82598e6	−0.267435
$800$	0	0
$801$	−3.42617e6	−0.188681
$802$	0	0
$803$	−4.61328e6	−0.252476
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.34598e7	0.727535
$808$	0	0
$809$	−2.36276e7	−1.26926	−0.634628	−	0.772818i	$-0.718847\pi$
$809$	−2.36276e7	−1.26926	−0.634628	+	0.772818i	$0.718847\pi$
$810$	0	0
$811$	2.85032e6	0.152174	0.0760872	−	0.997101i	$-0.475757\pi$
$811$	2.85032e6	0.152174	0.0760872	+	0.997101i	$0.475757\pi$
$812$	0	0
$813$	−1.59927e7	−0.848586
$814$	0	0
$815$	−5.98300e6	−0.315518
$816$	0	0
$817$	7.68093e6	0.402586
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.41313e7	−0.731685	−0.365843	−	0.930677i	$-0.619219\pi$
$821$	−1.41313e7	−0.731685	−0.365843	+	0.930677i	$0.619219\pi$
$822$	0	0
$823$	−3.23188e7	−1.66324	−0.831620	−	0.555345i	$-0.812586\pi$
$823$	−3.23188e7	−1.66324	−0.831620	+	0.555345i	$0.812586\pi$
$824$	0	0
$825$	−4.03886e6	−0.206597
$826$	0	0
$827$	−3.35291e7	−1.70474	−0.852369	−	0.522941i	$-0.824835\pi$
$827$	−3.35291e7	−1.70474	−0.852369	+	0.522941i	$0.824835\pi$
$828$	0	0
$829$	1.99517e7	1.00831	0.504154	−	0.863614i	$-0.331805\pi$
$829$	1.99517e7	1.00831	0.504154	+	0.863614i	$0.331805\pi$
$830$	0	0
$831$	−7.93138e6	−0.398425
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.31393e7	−0.652165
$836$	0	0
$837$	−1.74038e6	−0.0858677
$838$	0	0
$839$	2.34067e6	0.114798	0.0573990	−	0.998351i	$-0.481719\pi$
$839$	2.34067e6	0.114798	0.0573990	+	0.998351i	$0.481719\pi$
$840$	0	0
$841$	−2.84928e6	−0.138914
$842$	0	0
$843$	1.02539e7	0.496958
$844$	0	0
$845$	−6.89281e6	−0.332089
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.07719e6	−0.384584
$850$	0	0
$851$	−2.81654e6	−0.133319
$852$	0	0
$853$	1.79465e7	0.844514	0.422257	−	0.906476i	$-0.361238\pi$
$853$	1.79465e7	0.844514	0.422257	+	0.906476i	$0.361238\pi$
$854$	0	0
$855$	−2.12783e6	−0.0995456
$856$	0	0
$857$	3.68442e7	1.71363	0.856815	−	0.515624i	$-0.172440\pi$
$857$	3.68442e7	1.71363	0.856815	+	0.515624i	$0.172440\pi$
$858$	0	0
$859$	−2.40612e7	−1.11259	−0.556293	−	0.830986i	$-0.687777\pi$
$859$	−2.40612e7	−1.11259	−0.556293	+	0.830986i	$0.687777\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.64770e7	0.753098	0.376549	−	0.926397i	$-0.377111\pi$
$863$	1.64770e7	0.753098	0.376549	+	0.926397i	$0.377111\pi$
$864$	0	0
$865$	3.16706e6	0.143918
$866$	0	0
$867$	7.50201e6	0.338945
$868$	0	0
$869$	9.67835e6	0.434762
$870$	0	0
$871$	−192784.	−0.00861046
$872$	0	0
$873$	−3.61569e6	−0.160567
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.04983e7	1.33899	0.669494	−	0.742817i	$-0.266511\pi$
$877$	3.04983e7	1.33899	0.669494	+	0.742817i	$0.266511\pi$
$878$	0	0
$879$	−1.66153e7	−0.725329
$880$	0	0
$881$	−2.35008e7	−1.02010	−0.510051	−	0.860144i	$-0.670373\pi$
$881$	−2.35008e7	−1.02010	−0.510051	+	0.860144i	$0.670373\pi$
$882$	0	0
$883$	−4.42812e7	−1.91125	−0.955625	−	0.294587i	$-0.904818\pi$
$883$	−4.42812e7	−1.91125	−0.955625	+	0.294587i	$0.904818\pi$
$884$	0	0
$885$	663074.	0.0284580
$886$	0	0
$887$	1.52662e7	0.651510	0.325755	−	0.945454i	$-0.394382\pi$
$887$	1.52662e7	0.651510	0.325755	+	0.945454i	$0.394382\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.05911e6	−0.0446939
$892$	0	0
$893$	8.91382e6	0.374055
$894$	0	0
$895$	−7.71590e6	−0.321980
$896$	0	0
$897$	532198.	0.0220847
$898$	0	0
$899$	−1.00331e7	−0.414033
$900$	0	0
$901$	1.25703e7	0.515862
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21590.5	0.000876275 0
$906$	0	0
$907$	2.67694e7	1.08049	0.540245	−	0.841508i	$-0.318332\pi$
$907$	2.67694e7	1.08049	0.540245	+	0.841508i	$0.318332\pi$
$908$	0	0
$909$	9.06588e6	0.363915
$910$	0	0
$911$	1.60537e7	0.640883	0.320442	−	0.947268i	$-0.396169\pi$
$911$	1.60537e7	0.640883	0.320442	+	0.947268i	$0.396169\pi$
$912$	0	0
$913$	−9.94108e6	−0.394691
$914$	0	0
$915$	−8.41507e6	−0.332281
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.08072e7	0.422107	0.211053	−	0.977475i	$-0.432311\pi$
$919$	1.08072e7	0.422107	0.211053	+	0.977475i	$0.432311\pi$
$920$	0	0
$921$	1.24709e7	0.484448
$922$	0	0
$923$	1.18617e6	0.0458293
$924$	0	0
$925$	1.87077e6	0.0718894
$926$	0	0
$927$	−1.35229e7	−0.516856
$928$	0	0
$929$	2.16937e6	0.0824695	0.0412348	−	0.999149i	$-0.486871\pi$
$929$	2.16937e6	0.0824695	0.0412348	+	0.999149i	$0.486871\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.08123e7	0.782738
$934$	0	0
$935$	−2.29586e6	−0.0858849
$936$	0	0
$937$	3.51305e6	0.130718	0.0653589	−	0.997862i	$-0.479181\pi$
$937$	3.51305e6	0.130718	0.0653589	+	0.997862i	$0.479181\pi$
$938$	0	0
$939$	−1.05192e7	−0.389331
$940$	0	0
$941$	1.29297e7	0.476008	0.238004	−	0.971264i	$-0.423507\pi$
$941$	1.29297e7	0.476008	0.238004	+	0.971264i	$0.423507\pi$
$942$	0	0
$943$	1.74677e7	0.639670
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.48662e7	−0.901019	−0.450510	−	0.892772i	$-0.648758\pi$
$947$	−2.48662e7	−0.901019	−0.450510	+	0.892772i	$0.648758\pi$
$948$	0	0
$949$	−403763.	−0.0145533
$950$	0	0
$951$	−1.29090e7	−0.462850
$952$	0	0
$953$	3.33465e7	1.18937	0.594686	−	0.803958i	$-0.297276\pi$
$953$	3.33465e7	1.18937	0.594686	+	0.803958i	$0.297276\pi$
$954$	0	0
$955$	−1.38493e7	−0.491382
$956$	0	0
$957$	−6.10567e6	−0.215503
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.29297e7	−0.800922
$962$	0	0
$963$	8.34184e6	0.289865
$964$	0	0
$965$	−3.68465e6	−0.127373
$966$	0	0
$967$	−2.48706e7	−0.855305	−0.427652	−	0.903943i	$-0.640659\pi$
$967$	−2.48706e7	−0.855305	−0.427652	+	0.903943i	$0.640659\pi$
$968$	0	0
$969$	9.74634e6	0.333451
$970$	0	0
$971$	4.79491e7	1.63205	0.816024	−	0.578018i	$-0.196174\pi$
$971$	4.79491e7	1.63205	0.816024	+	0.578018i	$0.196174\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−353489.	−0.0119087
$976$	0	0
$977$	−7.85191e6	−0.263171	−0.131586	−	0.991305i	$-0.542007\pi$
$977$	−7.85191e6	−0.263171	−0.131586	+	0.991305i	$0.542007\pi$
$978$	0	0
$979$	6.82804e6	0.227687
$980$	0	0
$981$	−1.11635e7	−0.370364
$982$	0	0
$983$	2.84868e7	0.940284	0.470142	−	0.882591i	$-0.344203\pi$
$983$	2.84868e7	0.940284	0.470142	+	0.882591i	$0.344203\pi$
$984$	0	0
$985$	−8.71820e6	−0.286310
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.27308e7	−0.738966
$990$	0	0
$991$	−1.03795e7	−0.335733	−0.167867	−	0.985810i	$-0.553688\pi$
$991$	−1.03795e7	−0.335733	−0.167867	+	0.985810i	$0.553688\pi$
$992$	0	0
$993$	2.01076e7	0.647124
$994$	0	0
$995$	3.59587e6	0.115145
$996$	0	0
$997$	−1.68161e7	−0.535781	−0.267890	−	0.963449i	$-0.586326\pi$
$997$	−1.68161e7	−0.535781	−0.267890	+	0.963449i	$0.586326\pi$
$998$	0	0
$999$	490572.	0.0155521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.a.h.1.2		2
7.2	even	3		588.6.i.m.361.1		4
7.3	odd	6		84.6.i.b.37.2	yes	4
7.4	even	3		588.6.i.m.373.1		4
7.5	odd	6		84.6.i.b.25.2	✓	4
7.6	odd	2		588.6.a.l.1.1		2
21.5	even	6		252.6.k.e.109.1		4
21.17	even	6		252.6.k.e.37.1		4
28.3	even	6		336.6.q.g.289.2		4
28.19	even	6		336.6.q.g.193.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.b.25.2	✓	4	7.5	odd	6
84.6.i.b.37.2	yes	4	7.3	odd	6
252.6.k.e.37.1		4	21.17	even	6
252.6.k.e.109.1		4	21.5	even	6
336.6.q.g.193.2		4	28.19	even	6
336.6.q.g.289.2		4	28.3	even	6
588.6.a.h.1.2		2	1.1	even	1	trivial
588.6.a.l.1.1		2	7.6	odd	2
588.6.i.m.361.1		4	7.2	even	3
588.6.i.m.373.1		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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +18.5743 q^{5} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +18.5743 q^{5} +81.0000 q^{9} -161.426 q^{11} -14.1283 q^{13} -167.169 q^{15} +765.703 q^{17} -1414.29 q^{19} +4185.43 q^{23} -2779.99 q^{25} -729.000 q^{27} -4202.60 q^{29} +2387.35 q^{31} +1452.83 q^{33} -672.939 q^{37} +127.155 q^{39} +4173.45 q^{41} -5430.94 q^{43} +1504.52 q^{45} -6302.68 q^{47} -6891.32 q^{51} +16416.7 q^{53} -2998.37 q^{55} +12728.6 q^{57} -3966.49 q^{59} +50338.7 q^{61} -262.424 q^{65} +13645.3 q^{67} -37668.9 q^{69} -83957.2 q^{71} +28578.3 q^{73} +25019.9 q^{75} -59955.5 q^{79} +6561.00 q^{81} +61583.0 q^{83} +14222.4 q^{85} +37823.4 q^{87} -42298.3 q^{89} -21486.2 q^{93} -26269.5 q^{95} -44638.1 q^{97} -13075.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} - 47 q^{5} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 - 47 * q^5 + 162 * q^9	\( 2 q - 18 q^{3} - 47 q^{5} + 162 q^{9} - 407 q^{11} - 449 q^{13} + 423 q^{15} + 1868 q^{17} + 1463 q^{19} - 44 q^{23} - 1605 q^{25} - 1458 q^{27} + 767 q^{29} + 11170 q^{31} + 3663 q^{33} - 3113 q^{37} + 4041 q^{39} + 7842 q^{41} - 12629 q^{43} - 3807 q^{45} - 9576 q^{47} - 16812 q^{51} + 13395 q^{53} + 13105 q^{55} - 13167 q^{57} + 47521 q^{59} + 63652 q^{61} + 28254 q^{65} + 44541 q^{67} + 396 q^{69} - 125840 q^{71} - 6039 q^{73} + 14445 q^{75} + 17588 q^{79} + 13122 q^{81} - 39325 q^{83} - 58060 q^{85} - 6903 q^{87} - 83082 q^{89} - 100530 q^{93} - 214946 q^{95} - 184785 q^{97} - 32967 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 - 47 * q^5 + 162 * q^9 - 407 * q^11 - 449 * q^13 + 423 * q^15 + 1868 * q^17 + 1463 * q^19 - 44 * q^23 - 1605 * q^25 - 1458 * q^27 + 767 * q^29 + 11170 * q^31 + 3663 * q^33 - 3113 * q^37 + 4041 * q^39 + 7842 * q^41 - 12629 * q^43 - 3807 * q^45 - 9576 * q^47 - 16812 * q^51 + 13395 * q^53 + 13105 * q^55 - 13167 * q^57 + 47521 * q^59 + 63652 * q^61 + 28254 * q^65 + 44541 * q^67 + 396 * q^69 - 125840 * q^71 - 6039 * q^73 + 14445 * q^75 + 17588 * q^79 + 13122 * q^81 - 39325 * q^83 - 58060 * q^85 - 6903 * q^87 - 83082 * q^89 - 100530 * q^93 - 214946 * q^95 - 184785 * q^97 - 32967 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more