LMFDB - Embedded newform 363.8.a.n.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,8,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$113.395764251$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 1139x^{10} + 474732x^{8} - 87245660x^{6} + 6439013152x^{4} - 100879962816x^{2} + 359539347456 $ x^12 - 1139x^10 + 474732x^8 - 87245660x^6 + 6439013152x^4 - 100879962816*x^2 + 359539347456
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3\cdot 11^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.26509$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.26509 q^{2} -27.0000 q^{3} -122.869 q^{4} -77.8195 q^{5} +61.1574 q^{6} -799.987 q^{7} +568.242 q^{8} +729.000 q^{9} +176.268 q^{10} +3317.47 q^{12} -3.56354 q^{13} +1812.04 q^{14} +2101.13 q^{15} +14440.2 q^{16} +7190.73 q^{17} -1651.25 q^{18} -28280.5 q^{19} +9561.64 q^{20} +21599.7 q^{21} -69227.9 q^{23} -15342.5 q^{24} -72069.1 q^{25} +8.07175 q^{26} -19683.0 q^{27} +98293.9 q^{28} -117962. q^{29} -4759.24 q^{30} -102566. q^{31} -105443. q^{32} -16287.6 q^{34} +62254.6 q^{35} -89571.8 q^{36} -326829. q^{37} +64057.8 q^{38} +96.2157 q^{39} -44220.3 q^{40} -401254. q^{41} -48925.1 q^{42} +319823. q^{43} -56730.5 q^{45} +156807. q^{46} +131421. q^{47} -389884. q^{48} -183563. q^{49} +163243. q^{50} -194150. q^{51} +437.850 q^{52} -2.06067e6 q^{53} +44583.8 q^{54} -454586. q^{56} +763573. q^{57} +267194. q^{58} -374567. q^{59} -258164. q^{60} -25265.2 q^{61} +232322. q^{62} -583191. q^{63} -1.60950e6 q^{64} +277.313 q^{65} +675543. q^{67} -883521. q^{68} +1.86915e6 q^{69} -141012. q^{70} +1.04454e6 q^{71} +414248. q^{72} -2.73896e6 q^{73} +740297. q^{74} +1.94587e6 q^{75} +3.47481e6 q^{76} -217.937 q^{78} +808349. q^{79} -1.12373e6 q^{80} +531441. q^{81} +908877. q^{82} -4.64558e6 q^{83} -2.65394e6 q^{84} -559579. q^{85} -724427. q^{86} +3.18497e6 q^{87} -4.27067e6 q^{89} +128500. q^{90} +2850.79 q^{91} +8.50599e6 q^{92} +2.76930e6 q^{93} -297681. q^{94} +2.20078e6 q^{95} +2.84697e6 q^{96} +8.25067e6 q^{97} +415788. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 324 q^{3} + 742 q^{4} + 54 q^{5} + 8748 q^{9} - 20034 q^{12} + 25512 q^{14} - 1458 q^{15} + 17570 q^{16} - 14730 q^{20} - 108720 q^{23} + 395498 q^{25} + 498174 q^{26} - 236196 q^{27} - 261362 q^{31}+ \cdots - 51696396 q^{97}+O(q^{100}) $ 12 * q - 324 * q^3 + 742 * q^4 + 54 * q^5 + 8748 * q^9 - 20034 * q^12 + 25512 * q^14 - 1458 * q^15 + 17570 * q^16 - 14730 * q^20 - 108720 * q^23 + 395498 * q^25 + 498174 * q^26 - 236196 * q^27 - 261362 * q^31 - 225430 * q^34 + 540918 * q^36 + 102548 * q^37 + 87324 * q^38 - 688824 * q^42 + 39366 * q^45 - 51252 * q^47 - 474390 * q^48 + 60838 * q^49 + 3925254 * q^53 - 8454996 * q^56 + 5992958 * q^58 - 3013200 * q^59 + 397710 * q^60 - 12033562 * q^64 - 6969974 * q^67 + 2935440 * q^69 + 2942924 * q^70 + 8330556 * q^71 - 10678446 * q^75 - 13450698 * q^78 + 34706754 * q^80 + 6377292 * q^81 + 19876174 * q^82 - 17505540 * q^86 + 45480366 * q^89 - 2614500 * q^91 - 49064676 * q^92 + 7056774 * q^93 - 51696396 * q^97

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 $

$2$	−2.26509	−0.200208	−0.100104	−	0.994977i	$-0.531917\pi$
$2$	−2.26509	−0.200208	−0.100104	+	0.994977i	$0.531917\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−122.869	−0.959917
$5$	−77.8195	−0.278416	−0.139208	−	0.990263i	$-0.544456\pi$
$5$	−77.8195	−0.278416	−0.139208	+	0.990263i	$0.544456\pi$
$6$	61.1574	0.115590
$7$	−799.987	−0.881536	−0.440768	−	0.897621i	$-0.645294\pi$
$7$	−799.987	−0.881536	−0.440768	+	0.897621i	$0.645294\pi$
$8$	568.242	0.392390
$9$	729.000	0.333333
$10$	176.268	0.0557409
$11$	0	0
$11$	0	0
$12$	3317.47	0.554208
$13$	−3.56354	−0.000449863 0	−0.000224932	−	1.00000i	$-0.500072\pi$
$13$	−3.56354	−0.000449863 0	−0.000224932		1.00000i	$0.500072\pi$
$14$	1812.04	0.176490
$15$	2101.13	0.160743
$16$	14440.2	0.881358
$17$	7190.73	0.354978	0.177489	−	0.984123i	$-0.443203\pi$
$17$	7190.73	0.354978	0.177489	+	0.984123i	$0.443203\pi$
$18$	−1651.25	−0.0667358
$19$	−28280.5	−0.945909	−0.472955	−	0.881087i	$-0.656813\pi$
$19$	−28280.5	−0.945909	−0.472955	+	0.881087i	$0.656813\pi$
$20$	9561.64	0.267256
$21$	21599.7	0.508955
$22$	0	0
$23$	−69227.9	−1.18641	−0.593204	−	0.805052i	$-0.702137\pi$
$23$	−69227.9	−1.18641	−0.593204	+	0.805052i	$0.702137\pi$
$24$	−15342.5	−0.226547
$25$	−72069.1	−0.922485
$26$	8.07175	9.00660e−5 0
$27$	−19683.0	−0.192450
$28$	98293.9	0.846201
$29$	−117962.	−0.898148	−0.449074	−	0.893495i	$-0.648246\pi$
$29$	−117962.	−0.898148	−0.449074	+	0.893495i	$0.648246\pi$
$30$	−4759.24	−0.0321820
$31$	−102566.	−0.618357	−0.309179	−	0.951004i	$-0.600054\pi$
$31$	−102566.	−0.618357	−0.309179	+	0.951004i	$0.600054\pi$
$32$	−105443.	−0.568844
$33$	0	0
$34$	−16287.6	−0.0710693
$35$	62254.6	0.245433
$36$	−89571.8	−0.319972
$37$	−326829.	−1.06075	−0.530377	−	0.847762i	$-0.677950\pi$
$37$	−326829.	−1.06075	−0.530377	+	0.847762i	$0.677950\pi$
$38$	64057.8	0.189378
$39$	96.2157	0.000259729 0
$40$	−44220.3	−0.109248
$41$	−401254.	−0.909235	−0.454618	−	0.890687i	$-0.650224\pi$
$41$	−401254.	−0.909235	−0.454618	+	0.890687i	$0.650224\pi$
$42$	−48925.1	−0.101897
$43$	319823.	0.613436	0.306718	−	0.951800i	$-0.400769\pi$
$43$	319823.	0.613436	0.306718	+	0.951800i	$0.400769\pi$
$44$	0	0
$45$	−56730.5	−0.0928052
$46$	156807.	0.237528
$47$	131421.	0.184639	0.0923195	−	0.995729i	$-0.470572\pi$
$47$	131421.	0.184639	0.0923195	+	0.995729i	$0.470572\pi$
$48$	−389884.	−0.508852
$49$	−183563.	−0.222895
$50$	163243.	0.184688
$51$	−194150.	−0.204947
$52$	437.850	0.000431831 0
$53$	−2.06067e6	−1.90126	−0.950631	−	0.310324i	$-0.899562\pi$
$53$	−2.06067e6	−1.90126	−0.950631	+	0.310324i	$0.899562\pi$
$54$	44583.8	0.0385300
$55$	0	0
$56$	−454586.	−0.345906
$57$	763573.	0.546121
$58$	267194.	0.179816
$59$	−374567.	−0.237436	−0.118718	−	0.992928i	$-0.537879\pi$
$59$	−374567.	−0.237436	−0.118718	+	0.992928i	$0.537879\pi$
$60$	−258164.	−0.154300
$61$	−25265.2	−0.0142517	−0.00712587	−	0.999975i	$-0.502268\pi$
$61$	−25265.2	−0.0142517	−0.00712587	+	0.999975i	$0.502268\pi$
$62$	232322.	0.123800
$63$	−583191.	−0.293845
$64$	−1.60950e6	−0.767471
$65$	277.313	0.000125249 0
$66$	0	0
$67$	675543.	0.274405	0.137202	−	0.990543i	$-0.456189\pi$
$67$	675543.	0.274405	0.137202	+	0.990543i	$0.456189\pi$
$68$	−883521.	−0.340750
$69$	1.86915e6	0.684973
$70$	−141012.	−0.0491376
$71$	1.04454e6	0.346356	0.173178	−	0.984891i	$-0.444596\pi$
$71$	1.04454e6	0.346356	0.173178	+	0.984891i	$0.444596\pi$
$72$	414248.	0.130797
$73$	−2.73896e6	−0.824053	−0.412026	−	0.911172i	$-0.635179\pi$
$73$	−2.73896e6	−0.824053	−0.412026	+	0.911172i	$0.635179\pi$
$74$	740297.	0.212371
$75$	1.94587e6	0.532597
$76$	3.47481e6	0.907994
$77$	0	0
$78$	−217.937	−5.19996e−5 0
$79$	808349.	0.184461	0.0922304	−	0.995738i	$-0.470600\pi$
$79$	808349.	0.184461	0.0922304	+	0.995738i	$0.470600\pi$
$80$	−1.12373e6	−0.245384
$81$	531441.	0.111111
$82$	908877.	0.182036
$83$	−4.64558e6	−0.891798	−0.445899	−	0.895083i	$-0.647116\pi$
$83$	−4.64558e6	−0.891798	−0.445899	+	0.895083i	$0.647116\pi$
$84$	−2.65394e6	−0.488554
$85$	−559579.	−0.0988315
$86$	−724427.	−0.122815
$87$	3.18497e6	0.518546
$88$	0	0
$89$	−4.27067e6	−0.642142	−0.321071	−	0.947055i	$-0.604043\pi$
$89$	−4.27067e6	−0.642142	−0.321071	+	0.947055i	$0.604043\pi$
$90$	128500.	0.0185803
$91$	2850.79	0.000396570 0
$92$	8.50599e6	1.13885
$93$	2.76930e6	0.357009
$94$	−297681.	−0.0369661
$95$	2.20078e6	0.263356
$96$	2.84697e6	0.328423
$97$	8.25067e6	0.917884	0.458942	−	0.888466i	$-0.348229\pi$
$97$	8.25067e6	0.917884	0.458942	+	0.888466i	$0.348229\pi$
$98$	415788.	0.0446252
$99$	0	0
$100$	8.85509e6	0.885509
$101$	−1.98620e7	−1.91822	−0.959108	−	0.283039i	$-0.908657\pi$
$101$	−1.98620e7	−1.91822	−0.959108	+	0.283039i	$0.908657\pi$
$102$	439766.	0.0410319
$103$	−1.24682e6	−0.112428	−0.0562138	−	0.998419i	$-0.517903\pi$
$103$	−1.24682e6	−0.112428	−0.0562138	+	0.998419i	$0.517903\pi$
$104$	−2024.95	−0.000176522 0
$105$	−1.68088e6	−0.141701
$106$	4.66759e6	0.380647
$107$	−1.59947e7	−1.26222	−0.631109	−	0.775694i	$-0.717400\pi$
$107$	−1.59947e7	−1.26222	−0.631109	+	0.775694i	$0.717400\pi$
$108$	2.41844e6	0.184736
$109$	1.27763e7	0.944961	0.472480	−	0.881341i	$-0.343359\pi$
$109$	1.27763e7	0.944961	0.472480	+	0.881341i	$0.343359\pi$
$110$	0	0
$111$	8.82438e6	0.612426
$112$	−1.15519e7	−0.776948
$113$	1.95390e7	1.27388	0.636939	−	0.770914i	$-0.280200\pi$
$113$	1.95390e7	1.27388	0.636939	+	0.770914i	$0.280200\pi$
$114$	−1.72956e6	−0.109337
$115$	5.38729e6	0.330314
$116$	1.44939e7	0.862148
$117$	−2597.82	−0.000149954 0
$118$	848427.	0.0475366
$119$	−5.75249e6	−0.312926
$120$	1.19395e6	0.0630741
$121$	0	0
$122$	57227.9	0.00285331
$123$	1.08339e7	0.524947
$124$	1.26023e7	0.593572
$125$	1.16880e7	0.535250
$126$	1.32098e6	0.0588300
$127$	−1.88123e7	−0.814946	−0.407473	−	0.913217i	$-0.633590\pi$
$127$	−1.88123e7	−0.814946	−0.407473	+	0.913217i	$0.633590\pi$
$128$	1.71424e7	0.722498
$129$	−8.63521e6	−0.354168
$130$	−628.140	−2.50758e−5 0
$131$	6.79288e6	0.264000	0.132000	−	0.991250i	$-0.457860\pi$
$131$	6.79288e6	0.264000	0.132000	+	0.991250i	$0.457860\pi$
$132$	0	0
$133$	2.26240e7	0.833853
$134$	−1.53017e6	−0.0549379
$135$	1.53172e6	0.0535811
$136$	4.08607e6	0.139290
$137$	−2.32183e6	−0.0771451	−0.0385726	−	0.999256i	$-0.512281\pi$
$137$	−2.32183e6	−0.0771451	−0.0385726	+	0.999256i	$0.512281\pi$
$138$	−4.23380e6	−0.137137
$139$	2.48978e7	0.786337	0.393168	−	0.919467i	$-0.371379\pi$
$139$	2.48978e7	0.786337	0.393168	+	0.919467i	$0.371379\pi$
$140$	−7.64919e6	−0.235596
$141$	−3.54838e6	−0.106601
$142$	−2.36599e6	−0.0693431
$143$	0	0
$144$	1.05269e7	0.293786
$145$	9.17973e6	0.250059
$146$	6.20398e6	0.164982
$147$	4.95621e6	0.128688
$148$	4.01573e7	1.01824
$149$	7.41339e7	1.83597	0.917984	−	0.396618i	$-0.129816\pi$
$149$	7.41339e7	1.83597	0.917984	+	0.396618i	$0.129816\pi$
$150$	−4.40756e6	−0.106630
$151$	−8.36707e7	−1.97767	−0.988836	−	0.149007i	$-0.952392\pi$
$151$	−8.36707e7	−1.97767	−0.988836	+	0.149007i	$0.952392\pi$
$152$	−1.60702e7	−0.371165
$153$	5.24204e6	0.118326
$154$	0	0
$155$	7.98168e6	0.172160
$156$	−11822.0	−0.000249318 0
$157$	4.71338e7	0.972040	0.486020	−	0.873948i	$-0.338448\pi$
$157$	4.71338e7	0.972040	0.486020	+	0.873948i	$0.338448\pi$
$158$	−1.83098e6	−0.0369304
$159$	5.56380e7	1.09769
$160$	8.20554e6	0.158375
$161$	5.53815e7	1.04586
$162$	−1.20376e6	−0.0222453
$163$	−4.54460e7	−0.821937	−0.410969	−	0.911649i	$-0.634809\pi$
$163$	−4.54460e7	−0.821937	−0.410969	+	0.911649i	$0.634809\pi$
$164$	4.93019e7	0.872790
$165$	0	0
$166$	1.05226e7	0.178545
$167$	−7.33883e7	−1.21932	−0.609662	−	0.792661i	$-0.708695\pi$
$167$	−7.33883e7	−1.21932	−0.609662	+	0.792661i	$0.708695\pi$
$168$	1.22738e7	0.199709
$169$	−6.27485e7	−1.00000
$170$	1.26750e6	0.0197868
$171$	−2.06165e7	−0.315303
$172$	−3.92964e7	−0.588848
$173$	−1.06183e8	−1.55917	−0.779583	−	0.626299i	$-0.784569\pi$
$173$	−1.06183e8	−1.55917	−0.779583	+	0.626299i	$0.784569\pi$
$174$	−7.21423e6	−0.103817
$175$	5.76544e7	0.813203
$176$	0	0
$177$	1.01133e7	0.137084
$178$	9.67345e6	0.128562
$179$	1.85019e7	0.241119	0.120559	−	0.992706i	$-0.461531\pi$
$179$	1.85019e7	0.241119	0.120559	+	0.992706i	$0.461531\pi$
$180$	6.97043e6	0.0890853
$181$	1.31714e8	1.65103	0.825516	−	0.564378i	$-0.190884\pi$
$181$	1.31714e8	1.65103	0.825516	+	0.564378i	$0.190884\pi$
$182$	−6457.29	−7.93964e−5 0
$183$	682160.	0.00822825
$184$	−3.93382e7	−0.465535
$185$	2.54337e7	0.295330
$186$	−6.27270e6	−0.0714758
$187$	0	0
$188$	−1.61477e7	−0.177238
$189$	1.57461e7	0.169652
$190$	−4.98495e6	−0.0527258
$191$	−7.57338e7	−0.786453	−0.393226	−	0.919442i	$-0.628641\pi$
$191$	−7.57338e7	−0.786453	−0.393226	+	0.919442i	$0.628641\pi$
$192$	4.34566e7	0.443099
$193$	2.24931e7	0.225216	0.112608	−	0.993640i	$-0.464080\pi$
$193$	2.24931e7	0.225216	0.112608	+	0.993640i	$0.464080\pi$
$194$	−1.86885e7	−0.183767
$195$	−7487.46	−7.23125e−5 0
$196$	2.25543e7	0.213960
$197$	2.01700e8	1.87964	0.939818	−	0.341674i	$-0.110994\pi$
$197$	2.01700e8	1.87964	0.939818	+	0.341674i	$0.110994\pi$
$198$	0	0
$199$	8.69613e7	0.782240	0.391120	−	0.920340i	$-0.372088\pi$
$199$	8.69613e7	0.782240	0.391120	+	0.920340i	$0.372088\pi$
$200$	−4.09527e7	−0.361974
$201$	−1.82397e7	−0.158428
$202$	4.49892e7	0.384041
$203$	9.43678e7	0.791750
$204$	2.38551e7	0.196732
$205$	3.12254e7	0.253145
$206$	2.82416e6	0.0225089
$207$	−5.04672e7	−0.395469
$208$	−51458.2	−0.000396490 0
$209$	0	0
$210$	3.80733e6	0.0283696
$211$	−2.00011e8	−1.46577	−0.732883	−	0.680355i	$-0.761826\pi$
$211$	−2.00011e8	−1.46577	−0.732883	+	0.680355i	$0.761826\pi$
$212$	2.53193e8	1.82505
$213$	−2.82027e7	−0.199969
$214$	3.62295e7	0.252705
$215$	−2.48884e7	−0.170790
$216$	−1.11847e7	−0.0755155
$217$	8.20519e7	0.545104
$218$	−2.89396e7	−0.189188
$219$	7.39518e7	0.475767
$220$	0	0
$221$	−25624.5	−0.000159692 0
$222$	−1.99880e7	−0.122612
$223$	2.27414e8	1.37325	0.686624	−	0.727012i	$-0.259092\pi$
$223$	2.27414e8	1.37325	0.686624	+	0.727012i	$0.259092\pi$
$224$	8.43532e7	0.501457
$225$	−5.25384e7	−0.307495
$226$	−4.42576e7	−0.255040
$227$	8.42594e7	0.478110	0.239055	−	0.971006i	$-0.423162\pi$
$227$	8.42594e7	0.478110	0.239055	+	0.971006i	$0.423162\pi$
$228$	−9.38198e7	−0.524231
$229$	−1.31617e8	−0.724246	−0.362123	−	0.932130i	$-0.617948\pi$
$229$	−1.31617e8	−0.724246	−0.362123	+	0.932130i	$0.617948\pi$
$230$	−1.22027e7	−0.0661314
$231$	0	0
$232$	−6.70307e7	−0.352424
$233$	−1.16265e8	−0.602148	−0.301074	−	0.953601i	$-0.597345\pi$
$233$	−1.16265e8	−0.602148	−0.301074	+	0.953601i	$0.597345\pi$
$234$	5884.30	3.00220e−5 0
$235$	−1.02272e7	−0.0514064
$236$	4.60228e7	0.227919
$237$	−2.18254e7	−0.106498
$238$	1.30299e7	0.0626501
$239$	1.01515e8	0.480991	0.240496	−	0.970650i	$-0.422690\pi$
$239$	1.01515e8	0.480991	0.240496	+	0.970650i	$0.422690\pi$
$240$	3.03406e7	0.141672
$241$	2.69100e8	1.23838	0.619189	−	0.785242i	$-0.287461\pi$
$241$	2.69100e8	1.23838	0.619189	+	0.785242i	$0.287461\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	3.10432e6	0.0136805
$245$	1.42848e7	0.0620574
$246$	−2.45397e7	−0.105098
$247$	100779.	0.000425530 0
$248$	−5.82825e7	−0.242637
$249$	1.25431e8	0.514880
$250$	−2.64745e7	−0.107161
$251$	1.29877e8	0.518411	0.259205	−	0.965822i	$-0.416539\pi$
$251$	1.29877e8	0.518411	0.259205	+	0.965822i	$0.416539\pi$
$252$	7.16563e7	0.282067
$253$	0	0
$254$	4.26115e7	0.163158
$255$	1.51086e7	0.0570604
$256$	1.67187e8	0.622821
$257$	3.65091e8	1.34164	0.670820	−	0.741621i	$-0.265942\pi$
$257$	3.65091e8	1.34164	0.670820	+	0.741621i	$0.265942\pi$
$258$	1.95595e7	0.0709070
$259$	2.61459e8	0.935092
$260$	−34073.3	−0.000120229 0
$261$	−8.59941e7	−0.299383
$262$	−1.53865e7	−0.0528549
$263$	1.75897e8	0.596229	0.298114	−	0.954530i	$-0.403642\pi$
$263$	1.75897e8	0.596229	0.298114	+	0.954530i	$0.403642\pi$
$264$	0	0
$265$	1.60360e8	0.529341
$266$	−5.12455e7	−0.166944
$267$	1.15308e8	0.370741
$268$	−8.30036e7	−0.263406
$269$	2.96594e8	0.929031	0.464515	−	0.885565i	$-0.346229\pi$
$269$	2.96594e8	0.929031	0.464515	+	0.885565i	$0.346229\pi$
$270$	−3.46949e6	−0.0107273
$271$	1.24423e8	0.379760	0.189880	−	0.981807i	$-0.439190\pi$
$271$	1.24423e8	0.379760	0.189880	+	0.981807i	$0.439190\pi$
$272$	1.03835e8	0.312863
$273$	−76971.3	−0.000228960 0
$274$	5.25915e6	0.0154450
$275$	0	0
$276$	−2.29662e8	−0.657517
$277$	−6.39650e7	−0.180827	−0.0904134	−	0.995904i	$-0.528819\pi$
$277$	−6.39650e7	−0.180827	−0.0904134	+	0.995904i	$0.528819\pi$
$278$	−5.63956e7	−0.157430
$279$	−7.47710e7	−0.206119
$280$	3.53757e7	0.0963056
$281$	−6.17644e8	−1.66061	−0.830303	−	0.557313i	$-0.811832\pi$
$281$	−6.17644e8	−1.66061	−0.830303	+	0.557313i	$0.811832\pi$
$282$	8.03739e6	0.0213424
$283$	−4.76817e8	−1.25055	−0.625273	−	0.780406i	$-0.715012\pi$
$283$	−4.76817e8	−1.25055	−0.625273	+	0.780406i	$0.715012\pi$
$284$	−1.28343e8	−0.332473
$285$	−5.94209e7	−0.152049
$286$	0	0
$287$	3.20998e8	0.801523
$288$	−7.68681e7	−0.189615
$289$	−3.58632e8	−0.873990
$290$	−2.07929e7	−0.0500636
$291$	−2.22768e8	−0.529941
$292$	3.36534e8	0.791022
$293$	−1.02390e8	−0.237805	−0.118902	−	0.992906i	$-0.537938\pi$
$293$	−1.02390e8	−0.237805	−0.118902	+	0.992906i	$0.537938\pi$
$294$	−1.12263e7	−0.0257644
$295$	2.91486e7	0.0661060
$296$	−1.85718e8	−0.416229
$297$	0	0
$298$	−1.67920e8	−0.367574
$299$	246697.	0.000533721 0
$300$	−2.39087e8	−0.511249
$301$	−2.55854e8	−0.540766
$302$	1.89522e8	0.395945
$303$	5.36273e8	1.10748
$304$	−4.08375e8	−0.833684
$305$	1.96612e6	0.00396791
$306$	−1.18737e7	−0.0236898
$307$	−8.82787e8	−1.74129	−0.870646	−	0.491910i	$-0.836299\pi$
$307$	−8.82787e8	−1.74129	−0.870646	+	0.491910i	$0.836299\pi$
$308$	0	0
$309$	3.36641e7	0.0649101
$310$	−1.80792e7	−0.0344678
$311$	−3.83982e8	−0.723852	−0.361926	−	0.932207i	$-0.617881\pi$
$311$	−3.83982e8	−0.723852	−0.361926	+	0.932207i	$0.617881\pi$
$312$	54673.8	0.000101915 0
$313$	−6.24076e8	−1.15036	−0.575178	−	0.818028i	$-0.695067\pi$
$313$	−6.24076e8	−1.15036	−0.575178	+	0.818028i	$0.695067\pi$
$314$	−1.06762e8	−0.194610
$315$	4.53836e7	0.0818111
$316$	−9.93213e7	−0.177067
$317$	7.79021e8	1.37354	0.686771	−	0.726874i	$-0.259028\pi$
$317$	7.79021e8	1.37354	0.686771	+	0.726874i	$0.259028\pi$
$318$	−1.26025e8	−0.219767
$319$	0	0
$320$	1.25251e8	0.213676
$321$	4.31858e8	0.728742
$322$	−1.25444e8	−0.209389
$323$	−2.03357e8	−0.335777
$324$	−6.52978e7	−0.106657
$325$	256821.	0.000414992 0
$326$	1.02939e8	0.164558
$327$	−3.44961e8	−0.545573
$328$	−2.28009e8	−0.356775
$329$	−1.05135e8	−0.162766
$330$	0	0
$331$	2.14211e8	0.324671	0.162335	−	0.986736i	$-0.448097\pi$
$331$	2.14211e8	0.324671	0.162335	+	0.986736i	$0.448097\pi$
$332$	5.70799e8	0.856052
$333$	−2.38258e8	−0.353585
$334$	1.66231e8	0.244118
$335$	−5.25705e7	−0.0763986
$336$	3.11903e8	0.448571
$337$	2.46893e8	0.351402	0.175701	−	0.984444i	$-0.443781\pi$
$337$	2.46893e8	0.351402	0.175701	+	0.984444i	$0.443781\pi$
$338$	1.42131e8	0.200207
$339$	−5.27553e8	−0.735473
$340$	6.87552e7	0.0948701
$341$	0	0
$342$	4.66982e7	0.0631260
$343$	8.05672e8	1.07803
$344$	1.81736e8	0.240706
$345$	−1.45457e8	−0.190707
$346$	2.40513e8	0.312157
$347$	−4.19266e8	−0.538687	−0.269343	−	0.963044i	$-0.586807\pi$
$347$	−4.19266e8	−0.538687	−0.269343	+	0.963044i	$0.586807\pi$
$348$	−3.91335e8	−0.497761
$349$	2.24540e8	0.282752	0.141376	−	0.989956i	$-0.454847\pi$
$349$	2.24540e8	0.282752	0.141376	+	0.989956i	$0.454847\pi$
$350$	−1.30592e8	−0.162809
$351$	70141.2	8.65762e−5 0
$352$	0	0
$353$	1.83753e7	0.0222343	0.0111172	−	0.999938i	$-0.496461\pi$
$353$	1.83753e7	0.0222343	0.0111172	+	0.999938i	$0.496461\pi$
$354$	−2.29075e7	−0.0274452
$355$	−8.12860e7	−0.0964309
$356$	5.24735e8	0.616403
$357$	1.55317e8	0.180668
$358$	−4.19084e7	−0.0482737
$359$	−5.15913e8	−0.588499	−0.294250	−	0.955729i	$-0.595070\pi$
$359$	−5.15913e8	−0.588499	−0.294250	+	0.955729i	$0.595070\pi$
$360$	−3.22366e7	−0.0364159
$361$	−9.40854e7	−0.105256
$362$	−2.98343e8	−0.330549
$363$	0	0
$364$	−350275.	−0.000380675 0
$365$	2.13144e8	0.229429
$366$	−1.54515e6	−0.00164736
$367$	−6.45634e8	−0.681797	−0.340899	−	0.940100i	$-0.610731\pi$
$367$	−6.45634e8	−0.681797	−0.340899	+	0.940100i	$0.610731\pi$
$368$	−9.99662e8	−1.04565
$369$	−2.92514e8	−0.303078
$370$	−5.76096e7	−0.0591274
$371$	1.64851e9	1.67603
$372$	−3.40262e8	−0.342699
$373$	−9.47464e8	−0.945326	−0.472663	−	0.881243i	$-0.656707\pi$
$373$	−9.47464e8	−0.945326	−0.472663	+	0.881243i	$0.656707\pi$
$374$	0	0
$375$	−3.15577e8	−0.309027
$376$	7.46791e7	0.0724505
$377$	420362.	0.000404044 0
$378$	−3.56664e7	−0.0339655
$379$	−1.44544e9	−1.36384	−0.681919	−	0.731427i	$-0.738854\pi$
$379$	−1.44544e9	−1.36384	−0.681919	+	0.731427i	$0.738854\pi$
$380$	−2.70408e8	−0.252800
$381$	5.07932e8	0.470509
$382$	1.71544e8	0.157454
$383$	1.80823e9	1.64459	0.822294	−	0.569064i	$-0.192694\pi$
$383$	1.80823e9	1.64459	0.822294	+	0.569064i	$0.192694\pi$
$384$	−4.62845e8	−0.417134
$385$	0	0
$386$	−5.09489e7	−0.0450899
$387$	2.33151e8	0.204479
$388$	−1.01375e9	−0.881093
$389$	1.85934e9	1.60153	0.800765	−	0.598979i	$-0.204427\pi$
$389$	1.85934e9	1.60153	0.800765	+	0.598979i	$0.204427\pi$
$390$	16959.8	1.44775e−5 0
$391$	−4.97799e8	−0.421149
$392$	−1.04308e8	−0.0874617
$393$	−1.83408e8	−0.152421
$394$	−4.56868e8	−0.376317
$395$	−6.29053e7	−0.0513568
$396$	0	0
$397$	−1.90505e9	−1.52806	−0.764030	−	0.645181i	$-0.776782\pi$
$397$	−1.90505e9	−1.52806	−0.764030	+	0.645181i	$0.776782\pi$
$398$	−1.96975e8	−0.156610
$399$	−6.10849e8	−0.481425
$400$	−1.04069e9	−0.813039
$401$	−1.97671e9	−1.53087	−0.765435	−	0.643513i	$-0.777476\pi$
$401$	−1.97671e9	−1.53087	−0.765435	+	0.643513i	$0.777476\pi$
$402$	4.13145e7	0.0317184
$403$	365500.	0.000278176 0
$404$	2.44043e9	1.84133
$405$	−4.13565e7	−0.0309351
$406$	−2.13752e8	−0.158514
$407$	0	0
$408$	−1.10324e8	−0.0804191
$409$	5.98142e8	0.432287	0.216144	−	0.976362i	$-0.430652\pi$
$409$	5.98142e8	0.432287	0.216144	+	0.976362i	$0.430652\pi$
$410$	−7.07284e7	−0.0506816
$411$	6.26894e7	0.0445397
$412$	1.53196e8	0.107921
$413$	2.99649e8	0.209309
$414$	1.14313e8	0.0791759
$415$	3.61517e8	0.248291
$416$	375751.	0.000255902 0
$417$	−6.72239e8	−0.453992
$418$	0	0
$419$	−9.31473e8	−0.618616	−0.309308	−	0.950962i	$-0.600097\pi$
$419$	−9.31473e8	−0.618616	−0.309308	+	0.950962i	$0.600097\pi$
$420$	2.06528e8	0.136021
$421$	−1.57225e9	−1.02692	−0.513459	−	0.858114i	$-0.671636\pi$
$421$	−1.57225e9	−1.02692	−0.513459	+	0.858114i	$0.671636\pi$
$422$	4.53042e8	0.293457
$423$	9.58062e7	0.0615463
$424$	−1.17096e9	−0.746036
$425$	−5.18230e8	−0.327462
$426$	6.38816e7	0.0400352
$427$	2.02118e7	0.0125634
$428$	1.96526e9	1.21162
$429$	0	0
$430$	5.63746e7	0.0341935
$431$	−2.71393e9	−1.63278	−0.816391	−	0.577499i	$-0.804029\pi$
$431$	−2.71393e9	−1.63278	−0.816391	+	0.577499i	$0.804029\pi$
$432$	−2.84226e8	−0.169617
$433$	3.30747e8	0.195789	0.0978944	−	0.995197i	$-0.468789\pi$
$433$	3.30747e8	0.195789	0.0978944	+	0.995197i	$0.468789\pi$
$434$	−1.85855e8	−0.109134
$435$	−2.47853e8	−0.144371
$436$	−1.56982e9	−0.907084
$437$	1.95780e9	1.12223
$438$	−1.67507e8	−0.0952521
$439$	−1.52144e9	−0.858281	−0.429140	−	0.903238i	$-0.641183\pi$
$439$	−1.52144e9	−0.858281	−0.429140	+	0.903238i	$0.641183\pi$
$440$	0	0
$441$	−1.33818e8	−0.0742982
$442$	58041.8	3.19715e−5 0
$443$	1.26944e8	0.0693744	0.0346872	−	0.999398i	$-0.488957\pi$
$443$	1.26944e8	0.0693744	0.0346872	+	0.999398i	$0.488957\pi$
$444$	−1.08425e9	−0.587879
$445$	3.32342e8	0.178782
$446$	−5.15112e8	−0.274935
$447$	−2.00162e9	−1.06000
$448$	1.28758e9	0.676553
$449$	3.33352e9	1.73796	0.868982	−	0.494844i	$-0.164775\pi$
$449$	3.33352e9	1.73796	0.868982	+	0.494844i	$0.164775\pi$
$450$	1.19004e8	0.0615628
$451$	0	0
$452$	−2.40074e9	−1.22282
$453$	2.25911e9	1.14181
$454$	−1.90855e8	−0.0957212
$455$	−221847.	−0.000110411 0
$456$	4.33894e8	0.214292
$457$	9.91262e8	0.485827	0.242913	−	0.970048i	$-0.421897\pi$
$457$	9.91262e8	0.485827	0.242913	+	0.970048i	$0.421897\pi$
$458$	2.98123e8	0.145000
$459$	−1.41535e8	−0.0683156
$460$	−6.61932e8	−0.317074
$461$	−2.44787e9	−1.16368	−0.581842	−	0.813302i	$-0.697668\pi$
$461$	−2.44787e9	−1.16368	−0.581842	+	0.813302i	$0.697668\pi$
$462$	0	0
$463$	2.06525e9	0.967027	0.483514	−	0.875337i	$-0.339360\pi$
$463$	2.06525e9	0.967027	0.483514	+	0.875337i	$0.339360\pi$
$464$	−1.70339e9	−0.791590
$465$	−2.15505e8	−0.0993968
$466$	2.63350e8	0.120554
$467$	−2.10797e9	−0.957756	−0.478878	−	0.877881i	$-0.658956\pi$
$467$	−2.10797e9	−0.957756	−0.478878	+	0.877881i	$0.658956\pi$
$468$	319193.	0.000143944 0
$469$	−5.40426e8	−0.241898
$470$	2.31654e7	0.0102919
$471$	−1.27261e9	−0.561207
$472$	−2.12844e8	−0.0931677
$473$	0	0
$474$	4.94365e7	0.0213218
$475$	2.03815e9	0.872587
$476$	7.06805e8	0.300383
$477$	−1.50222e9	−0.633754
$478$	−2.29940e8	−0.0962981
$479$	−6.05048e8	−0.251545	−0.125772	−	0.992059i	$-0.540141\pi$
$479$	−6.05048e8	−0.251545	−0.125772	+	0.992059i	$0.540141\pi$
$480$	−2.21550e8	−0.0914380
$481$	1.16467e6	0.000477194 0
$482$	−6.09535e8	−0.247933
$483$	−1.49530e9	−0.603828
$484$	0	0
$485$	−6.42063e8	−0.255553
$486$	3.25016e7	0.0128433
$487$	−9.16610e8	−0.359611	−0.179806	−	0.983702i	$-0.557547\pi$
$487$	−9.16610e8	−0.359611	−0.179806	+	0.983702i	$0.557547\pi$
$488$	−1.43567e7	−0.00559224
$489$	1.22704e9	0.474546
$490$	−3.23564e7	−0.0124244
$491$	3.56549e9	1.35936	0.679680	−	0.733509i	$-0.262119\pi$
$491$	3.56549e9	1.35936	0.679680	+	0.733509i	$0.262119\pi$
$492$	−1.33115e9	−0.503906
$493$	−8.48231e8	−0.318823
$494$	−228273.	−8.51942e−5 0
$495$	0	0
$496$	−1.48108e9	−0.544994
$497$	−8.35622e8	−0.305325
$498$	−2.84112e8	−0.103083
$499$	−1.99010e9	−0.717007	−0.358504	−	0.933528i	$-0.616713\pi$
$499$	−1.99010e9	−0.717007	−0.358504	+	0.933528i	$0.616713\pi$
$500$	−1.43610e9	−0.513795
$501$	1.98148e9	0.703977
$502$	−2.94183e8	−0.103790
$503$	1.49394e9	0.523413	0.261706	−	0.965148i	$-0.415715\pi$
$503$	1.49394e9	0.523413	0.261706	+	0.965148i	$0.415715\pi$
$504$	−3.31393e8	−0.115302
$505$	1.54565e9	0.534062
$506$	0	0
$507$	1.69421e9	0.577350
$508$	2.31145e9	0.782280
$509$	2.30865e7	0.00775973	0.00387986	−	0.999992i	$-0.498765\pi$
$509$	2.30865e7	0.00775973	0.00387986	+	0.999992i	$0.498765\pi$
$510$	−3.42224e7	−0.0114239
$511$	2.19113e9	0.726432
$512$	−2.57292e9	−0.847191
$513$	5.56645e8	0.182040
$514$	−8.26965e8	−0.268606
$515$	9.70269e7	0.0313016
$516$	1.06100e9	0.339972
$517$	0	0
$518$	−5.92228e8	−0.187213
$519$	2.86693e9	0.900185
$520$	157581.	4.91464e−5 0
$521$	5.37086e9	1.66384	0.831921	−	0.554894i	$-0.187241\pi$
$521$	5.37086e9	1.66384	0.831921	+	0.554894i	$0.187241\pi$
$522$	1.94784e8	0.0599387
$523$	5.89444e9	1.80172	0.900859	−	0.434112i	$-0.142938\pi$
$523$	5.89444e9	1.80172	0.900859	+	0.434112i	$0.142938\pi$
$524$	−8.34637e8	−0.253419
$525$	−1.55667e9	−0.469503
$526$	−3.98422e8	−0.119369
$527$	−7.37528e8	−0.219503
$528$	0	0
$529$	1.38768e9	0.407563
$530$	−3.63230e8	−0.105978
$531$	−2.73059e8	−0.0791455
$532$	−2.77980e9	−0.800429
$533$	1.42989e6	0.000409031 0
$534$	−2.61183e8	−0.0742251
$535$	1.24470e9	0.351421
$536$	3.83872e8	0.107674
$537$	−4.99551e8	−0.139210
$538$	−6.71813e8	−0.185999
$539$	0	0
$540$	−1.88202e8	−0.0514334
$541$	6.69416e9	1.81763	0.908815	−	0.417199i	$-0.136988\pi$
$541$	6.69416e9	1.81763	0.908815	+	0.417199i	$0.136988\pi$
$542$	−2.81830e8	−0.0760308
$543$	−3.55627e9	−0.953224
$544$	−7.58213e8	−0.201927
$545$	−9.94249e8	−0.263092
$546$	174347.	4.58395e−5 0
$547$	−6.03028e9	−1.57537	−0.787684	−	0.616079i	$-0.788720\pi$
$547$	−6.03028e9	−1.57537	−0.787684	+	0.616079i	$0.788720\pi$
$548$	2.85282e8	0.0740529
$549$	−1.84183e7	−0.00475058
$550$	0	0
$551$	3.33601e9	0.849566
$552$	1.06213e9	0.268777
$553$	−6.46669e8	−0.162609
$554$	1.44886e8	0.0362029
$555$	−6.86710e8	−0.170509
$556$	−3.05917e9	−0.754818
$557$	3.54097e9	0.868218	0.434109	−	0.900860i	$-0.357063\pi$
$557$	3.54097e9	0.868218	0.434109	+	0.900860i	$0.357063\pi$
$558$	1.69363e8	0.0412666
$559$	−1.13970e6	−0.000275962 0
$560$	8.98967e8	0.216315
$561$	0	0
$562$	1.39902e9	0.332466
$563$	−7.36921e9	−1.74037	−0.870185	−	0.492725i	$-0.836001\pi$
$563$	−7.36921e9	−1.74037	−0.870185	+	0.492725i	$0.836001\pi$
$564$	4.35987e8	0.102328
$565$	−1.52052e9	−0.354667
$566$	1.08003e9	0.250369
$567$	−4.25146e8	−0.0979484
$568$	5.93553e8	0.135907
$569$	2.18893e9	0.498126	0.249063	−	0.968487i	$-0.419877\pi$
$569$	2.18893e9	0.498126	0.249063	+	0.968487i	$0.419877\pi$
$570$	1.34594e8	0.0304413
$571$	−4.60805e9	−1.03584	−0.517918	−	0.855430i	$-0.673293\pi$
$571$	−4.60805e9	−1.03584	−0.517918	+	0.855430i	$0.673293\pi$
$572$	0	0
$573$	2.04481e9	0.454059
$574$	−7.27090e8	−0.160471
$575$	4.98920e9	1.09444
$576$	−1.17333e9	−0.255824
$577$	5.22645e9	1.13264	0.566319	−	0.824186i	$-0.308367\pi$
$577$	5.22645e9	1.13264	0.566319	+	0.824186i	$0.308367\pi$
$578$	8.12334e8	0.174979
$579$	−6.07313e8	−0.130028
$580$	−1.12791e9	−0.240035
$581$	3.71640e9	0.786152
$582$	5.04589e8	0.106098
$583$	0	0
$584$	−1.55639e9	−0.323350
$585$	202161.	4.17497e−5 0
$586$	2.31922e8	0.0476103
$587$	9.51178e9	1.94101	0.970506	−	0.241075i	$-0.0775000\pi$
$587$	9.51178e9	1.94101	0.970506	+	0.241075i	$0.0775000\pi$
$588$	−6.08967e8	−0.123530
$589$	2.90063e9	0.584910
$590$	−6.60242e7	−0.0132349
$591$	−5.44590e9	−1.08521
$592$	−4.71946e9	−0.934903
$593$	4.83658e9	0.952460	0.476230	−	0.879321i	$-0.342003\pi$
$593$	4.83658e9	0.952460	0.476230	+	0.879321i	$0.342003\pi$
$594$	0	0
$595$	4.47656e8	0.0871235
$596$	−9.10879e9	−1.76238
$597$	−2.34795e9	−0.451627
$598$	−558790.	−0.000106855 0
$599$	5.91584e9	1.12466	0.562332	−	0.826912i	$-0.309904\pi$
$599$	5.91584e9	1.12466	0.562332	+	0.826912i	$0.309904\pi$
$600$	1.10572e9	0.208986
$601$	−9.30263e8	−0.174802	−0.0874008	−	0.996173i	$-0.527856\pi$
$601$	−9.30263e8	−0.174802	−0.0874008	+	0.996173i	$0.527856\pi$
$602$	5.79532e8	0.108265
$603$	4.92471e8	0.0914682
$604$	1.02806e10	1.89840
$605$	0	0
$606$	−1.21471e9	−0.221726
$607$	−6.20128e9	−1.12544	−0.562718	−	0.826649i	$-0.690244\pi$
$607$	−6.20128e9	−1.12544	−0.562718	+	0.826649i	$0.690244\pi$
$608$	2.98198e9	0.538075
$609$	−2.54793e9	−0.457117
$610$	−4.45345e6	−0.000794405 0
$611$	−468326.	−8.30623e−5 0
$612$	−6.44086e8	−0.113583
$613$	−7.83969e9	−1.37464	−0.687318	−	0.726357i	$-0.741212\pi$
$613$	−7.83969e9	−1.37464	−0.687318	+	0.726357i	$0.741212\pi$
$614$	1.99959e9	0.348620
$615$	−8.43087e8	−0.146154
$616$	0	0
$617$	6.98558e9	1.19730	0.598652	−	0.801009i	$-0.295703\pi$
$617$	6.98558e9	1.19730	0.598652	+	0.801009i	$0.295703\pi$
$618$	−7.62522e7	−0.0129955
$619$	3.68405e9	0.624321	0.312161	−	0.950029i	$-0.398947\pi$
$619$	3.68405e9	0.624321	0.312161	+	0.950029i	$0.398947\pi$
$620$	−9.80704e8	−0.165260
$621$	1.36261e9	0.228324
$622$	8.69754e8	0.144921
$623$	3.41648e9	0.566071
$624$	1.38937e6	0.000228914 0
$625$	4.72084e9	0.773463
$626$	1.41359e9	0.230310
$627$	0	0
$628$	−5.79130e9	−0.933077
$629$	−2.35014e9	−0.376545
$630$	−1.02798e8	−0.0163792
$631$	7.28179e9	1.15381	0.576906	−	0.816810i	$-0.304260\pi$
$631$	7.28179e9	1.15381	0.576906	+	0.816810i	$0.304260\pi$
$632$	4.59337e8	0.0723806
$633$	5.40029e9	0.846260
$634$	−1.76455e9	−0.274993
$635$	1.46396e9	0.226894
$636$	−6.83620e9	−1.05370
$637$	654136.	0.000100272 0
$638$	0	0
$639$	7.61473e8	0.115452
$640$	−1.33401e9	−0.201155
$641$	6.73350e9	1.00981	0.504903	−	0.863176i	$-0.331528\pi$
$641$	6.73350e9	1.00981	0.504903	+	0.863176i	$0.331528\pi$
$642$	−9.78197e8	−0.145900
$643$	−2.62177e9	−0.388917	−0.194458	−	0.980911i	$-0.562295\pi$
$643$	−2.62177e9	−0.388917	−0.194458	+	0.980911i	$0.562295\pi$
$644$	−6.80469e9	−1.00394
$645$	6.71988e8	0.0986058
$646$	4.60623e8	0.0672251
$647$	8.24161e9	1.19632	0.598159	−	0.801377i	$-0.295899\pi$
$647$	8.24161e9	1.19632	0.598159	+	0.801377i	$0.295899\pi$
$648$	3.01987e8	0.0435989
$649$	0	0
$650$	−581724.	−8.30845e−5 0
$651$	−2.21540e9	−0.314716
$652$	5.58392e9	0.788992
$653$	4.66174e9	0.655167	0.327583	−	0.944822i	$-0.393766\pi$
$653$	4.66174e9	0.655167	0.327583	+	0.944822i	$0.393766\pi$
$654$	7.81368e8	0.109228
$655$	−5.28619e8	−0.0735019
$656$	−5.79418e9	−0.801361
$657$	−1.99670e9	−0.274684
$658$	2.38141e8	0.0325870
$659$	7.22025e9	0.982773	0.491386	−	0.870942i	$-0.336490\pi$
$659$	7.22025e9	0.982773	0.491386	+	0.870942i	$0.336490\pi$
$660$	0	0
$661$	5.14171e9	0.692473	0.346236	−	0.938147i	$-0.387460\pi$
$661$	5.14171e9	0.692473	0.346236	+	0.938147i	$0.387460\pi$
$662$	−4.85206e8	−0.0650015
$663$	691861.	9.21980e−5 0
$664$	−2.63981e9	−0.349933
$665$	−1.76059e9	−0.232158
$666$	5.39676e8	0.0707903
$667$	8.16624e9	1.06557
$668$	9.01718e9	1.17045
$669$	−6.14017e9	−0.792846
$670$	1.19077e8	0.0152956
$671$	0	0
$672$	−2.27754e9	−0.289516
$673$	−2.15401e9	−0.272392	−0.136196	−	0.990682i	$-0.543488\pi$
$673$	−2.15401e9	−0.272392	−0.136196	+	0.990682i	$0.543488\pi$
$674$	−5.59235e8	−0.0703534
$675$	1.41854e9	0.177532
$676$	7.70987e9	0.959917
$677$	−1.44000e10	−1.78362	−0.891812	−	0.452406i	$-0.850566\pi$
$677$	−1.44000e10	−1.78362	−0.891812	+	0.452406i	$0.850566\pi$
$678$	1.19495e9	0.147247
$679$	−6.60043e9	−0.809148
$680$	−3.17976e8	−0.0387805
$681$	−2.27500e9	−0.276037
$682$	0	0
$683$	5.24344e9	0.629715	0.314857	−	0.949139i	$-0.398043\pi$
$683$	5.24344e9	0.629715	0.314857	+	0.949139i	$0.398043\pi$
$684$	2.53313e9	0.302665
$685$	1.80684e8	0.0214784
$686$	−1.82492e9	−0.215829
$687$	3.55365e9	0.418144
$688$	4.61829e9	0.540657
$689$	7.34327e6	0.000855308 0
$690$	3.29472e8	0.0381810
$691$	−1.15792e10	−1.33507	−0.667537	−	0.744577i	$-0.732651\pi$
$691$	−1.15792e10	−1.33507	−0.667537	+	0.744577i	$0.732651\pi$
$692$	1.30466e10	1.49667
$693$	0	0
$694$	9.49674e8	0.107849
$695$	−1.93753e9	−0.218928
$696$	1.80983e9	0.203472
$697$	−2.88531e9	−0.322759
$698$	−5.08604e8	−0.0566091
$699$	3.13915e9	0.347650
$700$	−7.08396e9	−0.780608
$701$	−2.05637e9	−0.225469	−0.112735	−	0.993625i	$-0.535961\pi$
$701$	−2.05637e9	−0.225469	−0.112735	+	0.993625i	$0.535961\pi$
$702$	−158876.	−1.73332e−5 0
$703$	9.24289e9	1.00338
$704$	0	0
$705$	2.76133e8	0.0296795
$706$	−4.16218e7	−0.00445148
$707$	1.58893e10	1.69098
$708$	−1.24262e9	−0.131589
$709$	6.02846e9	0.635250	0.317625	−	0.948216i	$-0.397115\pi$
$709$	6.02846e9	0.635250	0.317625	+	0.948216i	$0.397115\pi$
$710$	1.84120e8	0.0193062
$711$	5.89286e8	0.0614869
$712$	−2.42677e9	−0.251970
$713$	7.10047e9	0.733624
$714$	−3.51808e8	−0.0361711
$715$	0	0
$716$	−2.27332e9	−0.231454
$717$	−2.74090e9	−0.277701
$718$	1.16859e9	0.117822
$719$	9.77890e8	0.0981158	0.0490579	−	0.998796i	$-0.484378\pi$
$719$	9.77890e8	0.0981158	0.0490579	+	0.998796i	$0.484378\pi$
$720$	−8.19197e8	−0.0817946
$721$	9.97439e8	0.0991090
$722$	2.13112e8	0.0210731
$723$	−7.26569e9	−0.714978
$724$	−1.61836e10	−1.58485
$725$	8.50139e9	0.828528
$726$	0	0
$727$	−7.88188e9	−0.760780	−0.380390	−	0.924826i	$-0.624210\pi$
$727$	−7.88188e9	−0.760780	−0.380390	+	0.924826i	$0.624210\pi$
$728$	1.61994e6	0.000155610 0
$729$	3.87420e8	0.0370370
$730$	−4.82791e8	−0.0459334
$731$	2.29976e9	0.217757
$732$	−8.38166e7	−0.00789843
$733$	9.93763e9	0.932006	0.466003	−	0.884783i	$-0.345694\pi$
$733$	9.93763e9	0.932006	0.466003	+	0.884783i	$0.345694\pi$
$734$	1.46242e9	0.136501
$735$	−3.85690e8	−0.0358289
$736$	7.29961e9	0.674881
$737$	0	0
$738$	6.62571e8	0.0606786
$739$	8.69246e9	0.792295	0.396148	−	0.918187i	$-0.370347\pi$
$739$	8.69246e9	0.792295	0.396148	+	0.918187i	$0.370347\pi$
$740$	−3.12502e9	−0.283493
$741$	−2.72103e6	−0.000245680 0
$742$	−3.73401e9	−0.335554
$743$	7.30730e9	0.653576	0.326788	−	0.945098i	$-0.394034\pi$
$743$	7.30730e9	0.653576	0.326788	+	0.945098i	$0.394034\pi$
$744$	1.57363e9	0.140087
$745$	−5.76907e9	−0.511162
$746$	2.14609e9	0.189261
$747$	−3.38663e9	−0.297266
$748$	0	0
$749$	1.27956e10	1.11269
$750$	7.14810e8	0.0618695
$751$	−8.72580e9	−0.751737	−0.375868	−	0.926673i	$-0.622656\pi$
$751$	−8.72580e9	−0.751737	−0.375868	+	0.926673i	$0.622656\pi$
$752$	1.89775e9	0.162733
$753$	−3.50668e9	−0.299305
$754$	−952157.	−8.08926e−5 0
$755$	6.51122e9	0.550615
$756$	−1.93472e9	−0.162851
$757$	−1.41514e9	−0.118567	−0.0592836	−	0.998241i	$-0.518882\pi$
$757$	−1.41514e9	−0.118567	−0.0592836	+	0.998241i	$0.518882\pi$
$758$	3.27405e9	0.273051
$759$	0	0
$760$	1.25057e9	0.103338
$761$	1.70403e9	0.140162	0.0700809	−	0.997541i	$-0.477674\pi$
$761$	1.70403e9	0.140162	0.0700809	+	0.997541i	$0.477674\pi$
$762$	−1.15051e9	−0.0941995
$763$	−1.02209e10	−0.833017
$764$	9.30536e9	0.754929
$765$	−4.07933e8	−0.0329438
$766$	−4.09579e9	−0.329259
$767$	1.33479e6	0.000106814 0
$768$	−4.51406e9	−0.359586
$769$	−2.42942e10	−1.92646	−0.963231	−	0.268676i	$-0.913414\pi$
$769$	−2.42942e10	−1.92646	−0.963231	+	0.268676i	$0.913414\pi$
$770$	0	0
$771$	−9.85747e9	−0.774596
$772$	−2.76371e9	−0.216188
$773$	5.99115e9	0.466533	0.233266	−	0.972413i	$-0.425059\pi$
$773$	5.99115e9	0.466533	0.233266	+	0.972413i	$0.425059\pi$
$774$	−5.28107e8	−0.0409382
$775$	7.39188e9	0.570425
$776$	4.68837e9	0.360169
$777$	−7.05939e9	−0.539876
$778$	−4.21157e9	−0.320638
$779$	1.13477e10	0.860054
$780$	919980.	6.94140e−5 0
$781$	0	0
$782$	1.12756e9	0.0843172
$783$	2.32184e9	0.172849
$784$	−2.65069e9	−0.196450
$785$	−3.66793e9	−0.270631
$786$	4.15435e8	0.0305158
$787$	6.36260e9	0.465289	0.232645	−	0.972562i	$-0.425262\pi$
$787$	6.36260e9	0.465289	0.232645	+	0.972562i	$0.425262\pi$
$788$	−2.47827e10	−1.80430
$789$	−4.74921e9	−0.344233
$790$	1.42486e8	0.0102820
$791$	−1.56309e10	−1.12297
$792$	0	0
$793$	90033.6	6.41133e−6 0
$794$	4.31512e9	0.305929
$795$	−4.32972e9	−0.305615
$796$	−1.06849e10	−0.750886
$797$	1.01302e10	0.708782	0.354391	−	0.935097i	$-0.384688\pi$
$797$	1.01302e10	0.708782	0.354391	+	0.935097i	$0.384688\pi$
$798$	1.38363e9	0.0963849
$799$	9.45016e8	0.0655428
$800$	7.59920e9	0.524750
$801$	−3.11332e9	−0.214047
$802$	4.47743e9	0.306492
$803$	0	0
$804$	2.24110e9	0.152077
$805$	−4.30976e9	−0.291184
$806$	−827891.	−5.56930e−5 0
$807$	−8.00805e9	−0.536376
$808$	−1.12864e10	−0.752689
$809$	−6.02620e9	−0.400151	−0.200075	−	0.979781i	$-0.564119\pi$
$809$	−6.02620e9	−0.400151	−0.200075	+	0.979781i	$0.564119\pi$
$810$	9.36762e7	0.00619343
$811$	−6.27695e9	−0.413214	−0.206607	−	0.978424i	$-0.566242\pi$
$811$	−6.27695e9	−0.413214	−0.206607	+	0.978424i	$0.566242\pi$
$812$	−1.15949e10	−0.760014
$813$	−3.35943e9	−0.219254
$814$	0	0
$815$	3.53658e9	0.228840
$816$	−2.80355e9	−0.180631
$817$	−9.04474e9	−0.580255
$818$	−1.35484e9	−0.0865472
$819$	2.07823e6	0.000132190 0
$820$	−3.83665e9	−0.242998
$821$	1.09386e10	0.689860	0.344930	−	0.938628i	$-0.387903\pi$
$821$	1.09386e10	0.689860	0.344930	+	0.938628i	$0.387903\pi$
$822$	−1.41997e8	−0.00891719
$823$	2.37581e10	1.48564	0.742819	−	0.669493i	$-0.233488\pi$
$823$	2.37581e10	1.48564	0.742819	+	0.669493i	$0.233488\pi$
$824$	−7.08494e8	−0.0441155
$825$	0	0
$826$	−6.78731e8	−0.0419052
$827$	−4.55312e9	−0.279924	−0.139962	−	0.990157i	$-0.544698\pi$
$827$	−4.55312e9	−0.279924	−0.139962	+	0.990157i	$0.544698\pi$
$828$	6.20087e9	0.379618
$829$	8.33322e9	0.508009	0.254005	−	0.967203i	$-0.418252\pi$
$829$	8.33322e9	0.508009	0.254005	+	0.967203i	$0.418252\pi$
$830$	−8.18868e8	−0.0497096
$831$	1.72705e9	0.104400
$832$	5.73553e6	0.000345257 0
$833$	−1.31996e9	−0.0791228
$834$	1.52268e9	0.0908925
$835$	5.71104e9	0.339479
$836$	0	0
$837$	2.01882e9	0.119003
$838$	2.10987e9	0.123852
$839$	−2.45148e10	−1.43305	−0.716524	−	0.697562i	$-0.754268\pi$
$839$	−2.45148e10	−1.43305	−0.716524	+	0.697562i	$0.754268\pi$
$840$	−9.55143e8	−0.0556021
$841$	−3.33492e9	−0.193330
$842$	3.56130e9	0.205597
$843$	1.66764e10	0.958751
$844$	2.45752e10	1.40701
$845$	4.88306e9	0.278416
$846$	−2.17010e8	−0.0123220
$847$	0	0
$848$	−2.97563e10	−1.67569
$849$	1.28741e10	0.722003
$850$	1.17384e9	0.0655604
$851$	2.26257e10	1.25849
$852$	3.46525e9	0.191953
$853$	7.97219e9	0.439801	0.219900	−	0.975522i	$-0.429427\pi$
$853$	7.97219e9	0.439801	0.219900	+	0.975522i	$0.429427\pi$
$854$	−4.57816e7	−0.00251529
$855$	1.60437e9	0.0877853
$856$	−9.08888e9	−0.495282
$857$	−2.96733e10	−1.61040	−0.805199	−	0.593004i	$-0.797942\pi$
$857$	−2.96733e10	−1.61040	−0.805199	+	0.593004i	$0.797942\pi$
$858$	0	0
$859$	2.21657e10	1.19318	0.596589	−	0.802547i	$-0.296522\pi$
$859$	2.21657e10	1.19318	0.596589	+	0.802547i	$0.296522\pi$
$860$	3.05803e9	0.163944
$861$	−8.66695e9	−0.462760
$862$	6.14730e9	0.326895
$863$	−2.95417e10	−1.56458	−0.782289	−	0.622916i	$-0.785948\pi$
$863$	−2.95417e10	−1.56458	−0.782289	+	0.622916i	$0.785948\pi$
$864$	2.07544e9	0.109474
$865$	8.26309e9	0.434096
$866$	−7.49171e8	−0.0391984
$867$	9.68307e9	0.504599
$868$	−1.00817e10	−0.523255
$869$	0	0
$870$	5.61408e8	0.0289042
$871$	−2.40733e6	−0.000123445 0
$872$	7.26005e9	0.370793
$873$	6.01473e9	0.305961
$874$	−4.43459e9	−0.224680
$875$	−9.35028e9	−0.471842
$876$	−9.08641e9	−0.456697
$877$	7.93157e9	0.397064	0.198532	−	0.980094i	$-0.436383\pi$
$877$	7.93157e9	0.397064	0.198532	+	0.980094i	$0.436383\pi$
$878$	3.44620e9	0.171834
$879$	2.76453e9	0.137297
$880$	0	0
$881$	−6.17779e9	−0.304381	−0.152190	−	0.988351i	$-0.548633\pi$
$881$	−6.17779e9	−0.304381	−0.152190	+	0.988351i	$0.548633\pi$
$882$	3.03109e8	0.0148751
$883$	−1.01620e10	−0.496725	−0.248362	−	0.968667i	$-0.579892\pi$
$883$	−1.01620e10	−0.496725	−0.248362	+	0.968667i	$0.579892\pi$
$884$	3.14846e6	0.000153291 0
$885$	−7.87013e8	−0.0381663
$886$	−2.87539e8	−0.0138893
$887$	3.06982e10	1.47700	0.738500	−	0.674254i	$-0.235535\pi$
$887$	3.06982e10	1.47700	0.738500	+	0.674254i	$0.235535\pi$
$888$	5.01438e9	0.240310
$889$	1.50496e10	0.718404
$890$	−7.52783e8	−0.0357936
$891$	0	0
$892$	−2.79422e10	−1.31820
$893$	−3.71666e9	−0.174652
$894$	4.53384e9	0.212219
$895$	−1.43981e9	−0.0671312
$896$	−1.37137e10	−0.636908
$897$	−6.66081e6	−0.000308144 0
$898$	−7.55072e9	−0.347953
$899$	1.20989e10	0.555377
$900$	6.45536e9	0.295170
$901$	−1.48177e10	−0.674907
$902$	0	0
$903$	6.90806e9	0.312211
$904$	1.11029e10	0.499857
$905$	−1.02499e10	−0.459673
$906$	−5.11709e9	−0.228599
$907$	−1.56760e10	−0.697605	−0.348803	−	0.937196i	$-0.613412\pi$
$907$	−1.56760e10	−0.697605	−0.348803	+	0.937196i	$0.613412\pi$
$908$	−1.03529e10	−0.458946
$909$	−1.44794e10	−0.639406
$910$	502504.	2.21052e−5 0
$911$	−5.69823e9	−0.249704	−0.124852	−	0.992175i	$-0.539846\pi$
$911$	−5.69823e9	−0.249704	−0.124852	+	0.992175i	$0.539846\pi$
$912$	1.10261e10	0.481328
$913$	0	0
$914$	−2.24530e9	−0.0972662
$915$	−5.30854e7	−0.00229087
$916$	1.61716e10	0.695216
$917$	−5.43422e9	−0.232726
$918$	3.20590e8	0.0136773
$919$	−2.08793e10	−0.887383	−0.443692	−	0.896180i	$-0.646331\pi$
$919$	−2.08793e10	−0.887383	−0.443692	+	0.896180i	$0.646331\pi$
$920$	3.06128e9	0.129612
$921$	2.38352e10	1.00534
$922$	5.54465e9	0.232978
$923$	−3.72228e6	−0.000155813 0
$924$	0	0
$925$	2.35543e10	0.978529
$926$	−4.67797e9	−0.193606
$927$	−9.08931e8	−0.0374759
$928$	1.24383e10	0.510907
$929$	−3.55975e9	−0.145668	−0.0728341	−	0.997344i	$-0.523204\pi$
$929$	−3.55975e9	−0.145668	−0.0728341	+	0.997344i	$0.523204\pi$
$930$	4.88139e8	0.0199000
$931$	5.19126e9	0.210838
$932$	1.42854e10	0.578012
$933$	1.03675e10	0.417916
$934$	4.77474e9	0.191750
$935$	0	0
$936$	−1.47619e6	−5.88406e−5 0
$937$	2.36199e10	0.937972	0.468986	−	0.883206i	$-0.344619\pi$
$937$	2.36199e10	0.937972	0.468986	+	0.883206i	$0.344619\pi$
$938$	1.22411e9	0.0484297
$939$	1.68501e10	0.664159
$940$	1.25660e9	0.0493459
$941$	1.92156e9	0.0751780	0.0375890	−	0.999293i	$-0.488032\pi$
$941$	1.92156e9	0.0751780	0.0375890	+	0.999293i	$0.488032\pi$
$942$	2.88258e9	0.112358
$943$	2.77780e10	1.07872
$944$	−5.40881e9	−0.209266
$945$	−1.22536e9	−0.0472337
$946$	0	0
$947$	−3.92895e9	−0.150332	−0.0751661	−	0.997171i	$-0.523949\pi$
$947$	−3.92895e9	−0.150332	−0.0751661	+	0.997171i	$0.523949\pi$
$948$	2.68168e9	0.102230
$949$	9.76039e6	0.000370711 0
$950$	−4.61659e9	−0.174698
$951$	−2.10336e10	−0.793014
$952$	−3.26881e9	−0.122789
$953$	−5.71086e9	−0.213735	−0.106868	−	0.994273i	$-0.534082\pi$
$953$	−5.71086e9	−0.213735	−0.106868	+	0.994273i	$0.534082\pi$
$954$	3.40267e9	0.126882
$955$	5.89357e9	0.218961
$956$	−1.24731e10	−0.461712
$957$	0	0
$958$	1.37049e9	0.0503612
$959$	1.85743e9	0.0680062
$960$	−3.38177e9	−0.123366
$961$	−1.69927e10	−0.617634
$962$	−2.63808e6	−9.55378e−5 0
$963$	−1.16602e10	−0.420739
$964$	−3.30641e10	−1.18874
$965$	−1.75040e9	−0.0627036
$966$	3.38699e9	0.120891
$967$	−3.57036e10	−1.26975	−0.634877	−	0.772613i	$-0.718949\pi$
$967$	−3.57036e10	−1.26975	−0.634877	+	0.772613i	$0.718949\pi$
$968$	0	0
$969$	5.49065e9	0.193861
$970$	1.45433e9	0.0511637
$971$	−4.92075e10	−1.72490	−0.862450	−	0.506143i	$-0.831071\pi$
$971$	−4.92075e10	−1.72490	−0.862450	+	0.506143i	$0.831071\pi$
$972$	1.76304e9	0.0615787
$973$	−1.99179e10	−0.693184
$974$	2.07620e9	0.0719969
$975$	−6.93418e6	−0.000239596 0
$976$	−3.64833e8	−0.0125609
$977$	−1.58498e10	−0.543741	−0.271871	−	0.962334i	$-0.587642\pi$
$977$	−1.58498e10	−0.543741	−0.271871	+	0.962334i	$0.587642\pi$
$978$	−2.77936e9	−0.0950076
$979$	0	0
$980$	−1.75517e9	−0.0595699
$981$	9.31395e9	0.314987
$982$	−8.07616e9	−0.272154
$983$	9.11932e9	0.306214	0.153107	−	0.988210i	$-0.451072\pi$
$983$	9.11932e9	0.306214	0.153107	+	0.988210i	$0.451072\pi$
$984$	6.15625e9	0.205984
$985$	−1.56962e10	−0.523320
$986$	1.92132e9	0.0638308
$987$	2.83866e9	0.0939729
$988$	−1.23826e7	−0.000408473 0
$989$	−2.21407e10	−0.727786
$990$	0	0
$991$	−4.27508e10	−1.39536	−0.697680	−	0.716410i	$-0.745784\pi$
$991$	−4.27508e10	−1.39536	−0.697680	+	0.716410i	$0.745784\pi$
$992$	1.08149e10	0.351749
$993$	−5.78369e9	−0.187449
$994$	1.89276e9	0.0611284
$995$	−6.76729e9	−0.217788
$996$	−1.54116e10	−0.494242
$997$	−4.17291e10	−1.33354	−0.666770	−	0.745264i	$-0.732324\pi$
$997$	−4.17291e10	−1.33354	−0.666770	+	0.745264i	$0.732324\pi$
$998$	4.50776e9	0.143550
$999$	6.43298e9	0.204142

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.8.a.n.1.6	✓	12
11.10	odd	2	inner	363.8.a.n.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.8.a.n.1.6	✓	12	1.1	even	1	trivial
363.8.a.n.1.7	yes	12	11.10	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26509 q^{2} -27.0000 q^{3} -122.869 q^{4} -77.8195 q^{5} +61.1574 q^{6} -799.987 q^{7} +568.242 q^{8} +729.000 q^{9} +176.268 q^{10} +3317.47 q^{12} -3.56354 q^{13} +1812.04 q^{14} +2101.13 q^{15} +14440.2 q^{16} +7190.73 q^{17} -1651.25 q^{18} -28280.5 q^{19} +9561.64 q^{20} +21599.7 q^{21} -69227.9 q^{23} -15342.5 q^{24} -72069.1 q^{25} +8.07175 q^{26} -19683.0 q^{27} +98293.9 q^{28} -117962. q^{29} -4759.24 q^{30} -102566. q^{31} -105443. q^{32} -16287.6 q^{34} +62254.6 q^{35} -89571.8 q^{36} -326829. q^{37} +64057.8 q^{38} +96.2157 q^{39} -44220.3 q^{40} -401254. q^{41} -48925.1 q^{42} +319823. q^{43} -56730.5 q^{45} +156807. q^{46} +131421. q^{47} -389884. q^{48} -183563. q^{49} +163243. q^{50} -194150. q^{51} +437.850 q^{52} -2.06067e6 q^{53} +44583.8 q^{54} -454586. q^{56} +763573. q^{57} +267194. q^{58} -374567. q^{59} -258164. q^{60} -25265.2 q^{61} +232322. q^{62} -583191. q^{63} -1.60950e6 q^{64} +277.313 q^{65} +675543. q^{67} -883521. q^{68} +1.86915e6 q^{69} -141012. q^{70} +1.04454e6 q^{71} +414248. q^{72} -2.73896e6 q^{73} +740297. q^{74} +1.94587e6 q^{75} +3.47481e6 q^{76} -217.937 q^{78} +808349. q^{79} -1.12373e6 q^{80} +531441. q^{81} +908877. q^{82} -4.64558e6 q^{83} -2.65394e6 q^{84} -559579. q^{85} -724427. q^{86} +3.18497e6 q^{87} -4.27067e6 q^{89} +128500. q^{90} +2850.79 q^{91} +8.50599e6 q^{92} +2.76930e6 q^{93} -297681. q^{94} +2.20078e6 q^{95} +2.84697e6 q^{96} +8.25067e6 q^{97} +415788. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 324 q^{3} + 742 q^{4} + 54 q^{5} + 8748 q^{9} - 20034 q^{12} + 25512 q^{14} - 1458 q^{15} + 17570 q^{16} - 14730 q^{20} - 108720 q^{23} + 395498 q^{25} + 498174 q^{26} - 236196 q^{27} - 261362 q^{31}+ \cdots - 51696396 q^{97}+O(q^{100}) \) 12 * q - 324 * q^3 + 742 * q^4 + 54 * q^5 + 8748 * q^9 - 20034 * q^12 + 25512 * q^14 - 1458 * q^15 + 17570 * q^16 - 14730 * q^20 - 108720 * q^23 + 395498 * q^25 + 498174 * q^26 - 236196 * q^27 - 261362 * q^31 - 225430 * q^34 + 540918 * q^36 + 102548 * q^37 + 87324 * q^38 - 688824 * q^42 + 39366 * q^45 - 51252 * q^47 - 474390 * q^48 + 60838 * q^49 + 3925254 * q^53 - 8454996 * q^56 + 5992958 * q^58 - 3013200 * q^59 + 397710 * q^60 - 12033562 * q^64 - 6969974 * q^67 + 2935440 * q^69 + 2942924 * q^70 + 8330556 * q^71 - 10678446 * q^75 - 13450698 * q^78 + 34706754 * q^80 + 6377292 * q^81 + 19876174 * q^82 - 17505540 * q^86 + 45480366 * q^89 - 2614500 * q^91 - 49064676 * q^92 + 7056774 * q^93 - 51696396 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more