LMFDB - Embedded newform 399.2.a.g.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [399,2,Mod(1,399)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(399, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("399.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,3,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.368464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.78948$ of defining polynomial
Character	$\chi$	$=$	399.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.78948 q^{2} -1.00000 q^{3} +5.78120 q^{4} -0.430991 q^{5} -2.78948 q^{6} +1.00000 q^{7} +10.5476 q^{8} +1.00000 q^{9} -1.20224 q^{10} -4.67080 q^{11} -5.78120 q^{12} -4.61014 q^{13} +2.78948 q^{14} +0.430991 q^{15} +17.8599 q^{16} +5.52283 q^{17} +2.78948 q^{18} -1.00000 q^{19} -2.49165 q^{20} -1.00000 q^{21} -13.0291 q^{22} -1.98344 q^{23} -10.5476 q^{24} -4.81425 q^{25} -12.8599 q^{26} -1.00000 q^{27} +5.78120 q^{28} +6.41443 q^{29} +1.20224 q^{30} -6.95226 q^{31} +28.7247 q^{32} +4.67080 q^{33} +15.4058 q^{34} -0.430991 q^{35} +5.78120 q^{36} -3.95382 q^{37} -2.78948 q^{38} +4.61014 q^{39} -4.54592 q^{40} -0.487127 q^{41} -2.78948 q^{42} -2.25184 q^{43} -27.0028 q^{44} -0.430991 q^{45} -5.53278 q^{46} +7.10179 q^{47} -17.8599 q^{48} +1.00000 q^{49} -13.4292 q^{50} -5.52283 q^{51} -26.6522 q^{52} -7.50627 q^{53} -2.78948 q^{54} +2.01307 q^{55} +10.5476 q^{56} +1.00000 q^{57} +17.8929 q^{58} -2.68735 q^{59} +2.49165 q^{60} -2.22985 q^{61} -19.3932 q^{62} +1.00000 q^{63} +44.4071 q^{64} +1.98693 q^{65} +13.0291 q^{66} +0.358299 q^{67} +31.9286 q^{68} +1.98344 q^{69} -1.20224 q^{70} -13.0852 q^{71} +10.5476 q^{72} -1.77919 q^{73} -11.0291 q^{74} +4.81425 q^{75} -5.78120 q^{76} -4.67080 q^{77} +12.8599 q^{78} +13.1117 q^{79} -7.69746 q^{80} +1.00000 q^{81} -1.35883 q^{82} +2.40136 q^{83} -5.78120 q^{84} -2.38029 q^{85} -6.28147 q^{86} -6.41443 q^{87} -49.2656 q^{88} +16.4540 q^{89} -1.20224 q^{90} -4.61014 q^{91} -11.4667 q^{92} +6.95226 q^{93} +19.8103 q^{94} +0.430991 q^{95} -28.7247 q^{96} +3.22028 q^{97} +2.78948 q^{98} -4.67080 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} - 5 q^{3} + 7 q^{4} + 4 q^{5} - 3 q^{6} + 5 q^{7} + 9 q^{8} + 5 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 6 q^{13} + 3 q^{14} - 4 q^{15} + 19 q^{16} + 12 q^{17} + 3 q^{18} - 5 q^{19} + 8 q^{20}+ \cdots + 8 q^{99}+O(q^{100}) $ 5 * q + 3 * q^2 - 5 * q^3 + 7 * q^4 + 4 * q^5 - 3 * q^6 + 5 * q^7 + 9 * q^8 + 5 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 6 * q^13 + 3 * q^14 - 4 * q^15 + 19 * q^16 + 12 * q^17 + 3 * q^18 - 5 * q^19 + 8 * q^20 - 5 * q^21 - 12 * q^22 + 12 * q^23 - 9 * q^24 + 15 * q^25 + 6 * q^26 - 5 * q^27 + 7 * q^28 + 4 * q^29 + 6 * q^30 - 8 * q^31 + 31 * q^32 - 8 * q^33 + 2 * q^34 + 4 * q^35 + 7 * q^36 + 2 * q^37 - 3 * q^38 + 6 * q^39 - 18 * q^40 + 10 * q^41 - 3 * q^42 - 16 * q^43 - 4 * q^44 + 4 * q^45 + 16 * q^46 - 2 * q^47 - 19 * q^48 + 5 * q^49 - 39 * q^50 - 12 * q^51 - 22 * q^52 - 3 * q^54 - 12 * q^55 + 9 * q^56 + 5 * q^57 + 2 * q^58 - 4 * q^59 - 8 * q^60 - 14 * q^61 - 36 * q^62 + 5 * q^63 + 11 * q^64 + 32 * q^65 + 12 * q^66 - 20 * q^67 + 8 * q^68 - 12 * q^69 - 6 * q^70 - 6 * q^71 + 9 * q^72 + 10 * q^73 - 2 * q^74 - 15 * q^75 - 7 * q^76 + 8 * q^77 - 6 * q^78 - 20 * q^80 + 5 * q^81 - 38 * q^82 + 6 * q^83 - 7 * q^84 + 8 * q^85 - 36 * q^86 - 4 * q^87 - 40 * q^88 + 26 * q^89 - 6 * q^90 - 6 * q^91 - 48 * q^92 + 8 * q^93 + 24 * q^94 - 4 * q^95 - 31 * q^96 - 18 * q^97 + 3 * q^98 + 8 * 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 $

$2$	2.78948	1.97246	0.986230	−	0.165377i	$-0.0528842\pi$
$2$	2.78948	1.97246	0.986230	+	0.165377i	$0.0528842\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.78120	2.89060
$5$	−0.430991	−0.192745	−0.0963725	−	0.995345i	$-0.530724\pi$
$5$	−0.430991	−0.192745	−0.0963725	+	0.995345i	$0.530724\pi$
$6$	−2.78948	−1.13880
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	10.5476	3.72914
$9$	1.00000	0.333333
$10$	−1.20224	−0.380182
$11$	−4.67080	−1.40830	−0.704149	−	0.710052i	$-0.748671\pi$
$11$	−4.67080	−1.40830	−0.704149	+	0.710052i	$0.748671\pi$
$12$	−5.78120	−1.66889
$13$	−4.61014	−1.27862	−0.639311	−	0.768948i	$-0.720781\pi$
$13$	−4.61014	−1.27862	−0.639311	+	0.768948i	$0.720781\pi$
$14$	2.78948	0.745520
$15$	0.430991	0.111281
$16$	17.8599	4.46497
$17$	5.52283	1.33948	0.669741	−	0.742595i	$-0.266405\pi$
$17$	5.52283	1.33948	0.669741	+	0.742595i	$0.266405\pi$
$18$	2.78948	0.657487
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.49165	−0.557149
$21$	−1.00000	−0.218218
$22$	−13.0291	−2.77781
$23$	−1.98344	−0.413577	−0.206788	−	0.978386i	$-0.566301\pi$
$23$	−1.98344	−0.413577	−0.206788	+	0.978386i	$0.566301\pi$
$24$	−10.5476	−2.15302
$25$	−4.81425	−0.962849
$26$	−12.8599	−2.52203
$27$	−1.00000	−0.192450
$28$	5.78120	1.09254
$29$	6.41443	1.19113	0.595565	−	0.803307i	$-0.296928\pi$
$29$	6.41443	1.19113	0.595565	+	0.803307i	$0.296928\pi$
$30$	1.20224	0.219498
$31$	−6.95226	−1.24866	−0.624332	−	0.781159i	$-0.714629\pi$
$31$	−6.95226	−1.24866	−0.624332	+	0.781159i	$0.714629\pi$
$32$	28.7247	5.07785
$33$	4.67080	0.813081
$34$	15.4058	2.64208
$35$	−0.430991	−0.0728508
$36$	5.78120	0.963534
$37$	−3.95382	−0.650003	−0.325002	−	0.945713i	$-0.605365\pi$
$37$	−3.95382	−0.650003	−0.325002	+	0.945713i	$0.605365\pi$
$38$	−2.78948	−0.452514
$39$	4.61014	0.738213
$40$	−4.54592	−0.718773
$41$	−0.487127	−0.0760765	−0.0380382	−	0.999276i	$-0.512111\pi$
$41$	−0.487127	−0.0760765	−0.0380382	+	0.999276i	$0.512111\pi$
$42$	−2.78948	−0.430426
$43$	−2.25184	−0.343403	−0.171701	−	0.985149i	$-0.554926\pi$
$43$	−2.25184	−0.343403	−0.171701	+	0.985149i	$0.554926\pi$
$44$	−27.0028	−4.07083
$45$	−0.430991	−0.0642483
$46$	−5.53278	−0.815764
$47$	7.10179	1.03590	0.517951	−	0.855410i	$-0.326695\pi$
$47$	7.10179	1.03590	0.517951	+	0.855410i	$0.326695\pi$
$48$	−17.8599	−2.57785
$49$	1.00000	0.142857
$50$	−13.4292	−1.89918
$51$	−5.52283	−0.773350
$52$	−26.6522	−3.69599
$53$	−7.50627	−1.03107	−0.515533	−	0.856870i	$-0.672406\pi$
$53$	−7.50627	−1.03107	−0.515533	+	0.856870i	$0.672406\pi$
$54$	−2.78948	−0.379600
$55$	2.01307	0.271442
$56$	10.5476	1.40948
$57$	1.00000	0.132453
$58$	17.8929	2.34946
$59$	−2.68735	−0.349863	−0.174932	−	0.984581i	$-0.555970\pi$
$59$	−2.68735	−0.349863	−0.174932	+	0.984581i	$0.555970\pi$
$60$	2.49165	0.321670
$61$	−2.22985	−0.285503	−0.142752	−	0.989759i	$-0.545595\pi$
$61$	−2.22985	−0.285503	−0.142752	+	0.989759i	$0.545595\pi$
$62$	−19.3932	−2.46294
$63$	1.00000	0.125988
$64$	44.4071	5.55088
$65$	1.98693	0.246448
$66$	13.0291	1.60377
$67$	0.358299	0.0437731	0.0218866	−	0.999760i	$-0.493033\pi$
$67$	0.358299	0.0437731	0.0218866	+	0.999760i	$0.493033\pi$
$68$	31.9286	3.87191
$69$	1.98344	0.238779
$70$	−1.20224	−0.143695
$71$	−13.0852	−1.55293	−0.776466	−	0.630160i	$-0.782989\pi$
$71$	−13.0852	−1.55293	−0.776466	+	0.630160i	$0.782989\pi$
$72$	10.5476	1.24305
$73$	−1.77919	−0.208238	−0.104119	−	0.994565i	$-0.533202\pi$
$73$	−1.77919	−0.208238	−0.104119	+	0.994565i	$0.533202\pi$
$74$	−11.0291	−1.28211
$75$	4.81425	0.555901
$76$	−5.78120	−0.663149
$77$	−4.67080	−0.532287
$78$	12.8599	1.45610
$79$	13.1117	1.47519	0.737593	−	0.675246i	$-0.235962\pi$
$79$	13.1117	1.47519	0.737593	+	0.675246i	$0.235962\pi$
$80$	−7.69746	−0.860602
$81$	1.00000	0.111111
$82$	−1.35883	−0.150058
$83$	2.40136	0.263584	0.131792	−	0.991277i	$-0.457927\pi$
$83$	2.40136	0.263584	0.131792	+	0.991277i	$0.457927\pi$
$84$	−5.78120	−0.630781
$85$	−2.38029	−0.258178
$86$	−6.28147	−0.677348
$87$	−6.41443	−0.687700
$88$	−49.2656	−5.25174
$89$	16.4540	1.74412	0.872061	−	0.489397i	$-0.162783\pi$
$89$	16.4540	1.74412	0.872061	+	0.489397i	$0.162783\pi$
$90$	−1.20224	−0.126727
$91$	−4.61014	−0.483274
$92$	−11.4667	−1.19548
$93$	6.95226	0.720916
$94$	19.8103	2.04328
$95$	0.430991	0.0442187
$96$	−28.7247	−2.93170
$97$	3.22028	0.326970	0.163485	−	0.986546i	$-0.447726\pi$
$97$	3.22028	0.326970	0.163485	+	0.986546i	$0.447726\pi$
$98$	2.78948	0.281780
$99$	−4.67080	−0.469433
$100$	−27.8321	−2.78321
$101$	19.8188	1.97204	0.986020	−	0.166624i	$-0.0532867\pi$
$101$	19.8188	1.97204	0.986020	+	0.166624i	$0.0532867\pi$
$102$	−15.4058	−1.52540
$103$	9.34159	0.920454	0.460227	−	0.887801i	$-0.347768\pi$
$103$	9.34159	0.920454	0.460227	+	0.887801i	$0.347768\pi$
$104$	−48.6259	−4.76816
$105$	0.430991	0.0420604
$106$	−20.9386	−2.03374
$107$	1.22688	0.118607	0.0593037	−	0.998240i	$-0.481112\pi$
$107$	1.22688	0.118607	0.0593037	+	0.998240i	$0.481112\pi$
$108$	−5.78120	−0.556296
$109$	−2.11227	−0.202319	−0.101159	−	0.994870i	$-0.532255\pi$
$109$	−2.11227	−0.202319	−0.101159	+	0.994870i	$0.532255\pi$
$110$	5.61542	0.535410
$111$	3.95382	0.375280
$112$	17.8599	1.68760
$113$	10.4144	0.979708	0.489854	−	0.871805i	$-0.337050\pi$
$113$	10.4144	0.979708	0.489854	+	0.871805i	$0.337050\pi$
$114$	2.78948	0.261259
$115$	0.854846	0.0797148
$116$	37.0831	3.44308
$117$	−4.61014	−0.426208
$118$	−7.49632	−0.690092
$119$	5.52283	0.506277
$120$	4.54592	0.414984
$121$	10.8163	0.983303
$122$	−6.22013	−0.563144
$123$	0.487127	0.0439228
$124$	−40.1925	−3.60939
$125$	4.22985	0.378329
$126$	2.78948	0.248507
$127$	−2.17463	−0.192967	−0.0964836	−	0.995335i	$-0.530760\pi$
$127$	−2.17463	−0.192967	−0.0964836	+	0.995335i	$0.530760\pi$
$128$	66.4234	5.87105
$129$	2.25184	0.198264
$130$	5.54250	0.486109
$131$	0.789672	0.0689939	0.0344970	−	0.999405i	$-0.489017\pi$
$131$	0.789672	0.0689939	0.0344970	+	0.999405i	$0.489017\pi$
$132$	27.0028	2.35029
$133$	−1.00000	−0.0867110
$134$	0.999467	0.0863408
$135$	0.430991	0.0370938
$136$	58.2525	4.99511
$137$	16.3158	1.39396	0.696978	−	0.717092i	$-0.254527\pi$
$137$	16.3158	1.39396	0.696978	+	0.717092i	$0.254527\pi$
$138$	5.53278	0.470981
$139$	−18.9040	−1.60342	−0.801708	−	0.597716i	$-0.796075\pi$
$139$	−18.9040	−1.60342	−0.801708	+	0.597716i	$0.796075\pi$
$140$	−2.49165	−0.210583
$141$	−7.10179	−0.598078
$142$	−36.5010	−3.06310
$143$	21.5330	1.80068
$144$	17.8599	1.48832
$145$	−2.76456	−0.229585
$146$	−4.96301	−0.410741
$147$	−1.00000	−0.0824786
$148$	−22.8578	−1.87890
$149$	2.45067	0.200766	0.100383	−	0.994949i	$-0.467993\pi$
$149$	2.45067	0.200766	0.100383	+	0.994949i	$0.467993\pi$
$150$	13.4292	1.09649
$151$	−17.9737	−1.46268	−0.731340	−	0.682013i	$-0.761105\pi$
$151$	−17.9737	−1.46268	−0.731340	+	0.682013i	$0.761105\pi$
$152$	−10.5476	−0.855523
$153$	5.52283	0.446494
$154$	−13.0291	−1.04991
$155$	2.99636	0.240674
$156$	26.6522	2.13388
$157$	6.64117	0.530023	0.265011	−	0.964245i	$-0.414624\pi$
$157$	6.64117	0.530023	0.265011	+	0.964245i	$0.414624\pi$
$158$	36.5749	2.90975
$159$	7.50627	0.595286
$160$	−12.3801	−0.978730
$161$	−1.98344	−0.156317
$162$	2.78948	0.219162
$163$	18.1264	1.41977	0.709883	−	0.704320i	$-0.248748\pi$
$163$	18.1264	1.41977	0.709883	+	0.704320i	$0.248748\pi$
$164$	−2.81618	−0.219907
$165$	−2.01307	−0.156717
$166$	6.69856	0.519909
$167$	−8.35830	−0.646785	−0.323392	−	0.946265i	$-0.604823\pi$
$167$	−8.35830	−0.646785	−0.323392	+	0.946265i	$0.604823\pi$
$168$	−10.5476	−0.813764
$169$	8.25339	0.634876
$170$	−6.63977	−0.509247
$171$	−1.00000	−0.0764719
$172$	−13.0184	−0.992640
$173$	−10.1671	−0.772991	−0.386496	−	0.922291i	$-0.626315\pi$
$173$	−10.1671	−0.772991	−0.386496	+	0.922291i	$0.626315\pi$
$174$	−17.8929	−1.35646
$175$	−4.81425	−0.363923
$176$	−83.4199	−6.28801
$177$	2.68735	0.201994
$178$	45.8982	3.44021
$179$	−1.91113	−0.142845	−0.0714224	−	0.997446i	$-0.522754\pi$
$179$	−1.91113	−0.142845	−0.0714224	+	0.997446i	$0.522754\pi$
$180$	−2.49165	−0.185716
$181$	13.0656	0.971155	0.485578	−	0.874194i	$-0.338609\pi$
$181$	13.0656	0.971155	0.485578	+	0.874194i	$0.338609\pi$
$182$	−12.8599	−0.953239
$183$	2.22985	0.164835
$184$	−20.9206	−1.54228
$185$	1.70406	0.125285
$186$	19.3932	1.42198
$187$	−25.7960	−1.88639
$188$	41.0569	2.99438
$189$	−1.00000	−0.0727393
$190$	1.20224	0.0872197
$191$	19.0790	1.38051	0.690254	−	0.723567i	$-0.257499\pi$
$191$	19.0790	1.38051	0.690254	+	0.723567i	$0.257499\pi$
$192$	−44.4071	−3.20480
$193$	−4.91720	−0.353948	−0.176974	−	0.984216i	$-0.556631\pi$
$193$	−4.91720	−0.353948	−0.176974	+	0.984216i	$0.556631\pi$
$194$	8.98291	0.644935
$195$	−1.98693	−0.142287
$196$	5.78120	0.412943
$197$	−10.8819	−0.775302	−0.387651	−	0.921806i	$-0.626713\pi$
$197$	−10.8819	−0.775302	−0.387651	+	0.921806i	$0.626713\pi$
$198$	−13.0291	−0.925937
$199$	−14.2036	−1.00686	−0.503432	−	0.864035i	$-0.667930\pi$
$199$	−14.2036	−1.00686	−0.503432	+	0.864035i	$0.667930\pi$
$200$	−50.7787	−3.59060
$201$	−0.358299	−0.0252724
$202$	55.2841	3.88977
$203$	6.41443	0.450205
$204$	−31.9286	−2.23545
$205$	0.209947	0.0146634
$206$	26.0582	1.81556
$207$	−1.98344	−0.137859
$208$	−82.3366	−5.70902
$209$	4.67080	0.323086
$210$	1.20224	0.0829625
$211$	−11.4963	−0.791439	−0.395720	−	0.918371i	$-0.629505\pi$
$211$	−11.4963	−0.791439	−0.395720	+	0.918371i	$0.629505\pi$
$212$	−43.3953	−2.98040
$213$	13.0852	0.896585
$214$	3.42237	0.233948
$215$	0.970524	0.0661892
$216$	−10.5476	−0.717673
$217$	−6.95226	−0.471950
$218$	−5.89214	−0.399066
$219$	1.77919	0.120226
$220$	11.6380	0.784632
$221$	−25.4610	−1.71269
$222$	11.0291	0.740224
$223$	−16.9223	−1.13320	−0.566599	−	0.823994i	$-0.691741\pi$
$223$	−16.9223	−1.13320	−0.566599	+	0.823994i	$0.691741\pi$
$224$	28.7247	1.91925
$225$	−4.81425	−0.320950
$226$	29.0509	1.93243
$227$	−6.29594	−0.417876	−0.208938	−	0.977929i	$-0.567001\pi$
$227$	−6.29594	−0.417876	−0.208938	+	0.977929i	$0.567001\pi$
$228$	5.78120	0.382869
$229$	8.76651	0.579307	0.289654	−	0.957132i	$-0.406460\pi$
$229$	8.76651	0.579307	0.289654	+	0.957132i	$0.406460\pi$
$230$	2.38458	0.157234
$231$	4.67080	0.307316
$232$	67.6568	4.44189
$233$	−6.76961	−0.443492	−0.221746	−	0.975104i	$-0.571176\pi$
$233$	−6.76961	−0.443492	−0.221746	+	0.975104i	$0.571176\pi$
$234$	−12.8599	−0.840678
$235$	−3.06081	−0.199665
$236$	−15.5361	−1.01132
$237$	−13.1117	−0.851699
$238$	15.4058	0.998611
$239$	1.87117	0.121036	0.0605180	−	0.998167i	$-0.480725\pi$
$239$	1.87117	0.121036	0.0605180	+	0.998167i	$0.480725\pi$
$240$	7.69746	0.496869
$241$	13.1579	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$241$	13.1579	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$242$	30.1719	1.93953
$243$	−1.00000	−0.0641500
$244$	−12.8912	−0.825276
$245$	−0.430991	−0.0275350
$246$	1.35883	0.0866359
$247$	4.61014	0.293336
$248$	−73.3296	−4.65644
$249$	−2.40136	−0.152180
$250$	11.7991	0.746240
$251$	−10.2067	−0.644241	−0.322120	−	0.946699i	$-0.604396\pi$
$251$	−10.2067	−0.644241	−0.322120	+	0.946699i	$0.604396\pi$
$252$	5.78120	0.364182
$253$	9.26426	0.582439
$254$	−6.06609	−0.380620
$255$	2.38029	0.149059
$256$	96.4726	6.02954
$257$	8.96674	0.559330	0.279665	−	0.960098i	$-0.409777\pi$
$257$	8.96674	0.559330	0.279665	+	0.960098i	$0.409777\pi$
$258$	6.28147	0.391067
$259$	−3.95382	−0.245678
$260$	11.4868	0.712384
$261$	6.41443	0.397044
$262$	2.20277	0.136088
$263$	−11.3491	−0.699816	−0.349908	−	0.936784i	$-0.613787\pi$
$263$	−11.3491	−0.699816	−0.349908	+	0.936784i	$0.613787\pi$
$264$	49.2656	3.03209
$265$	3.23513	0.198733
$266$	−2.78948	−0.171034
$267$	−16.4540	−1.00697
$268$	2.07140	0.126531
$269$	−1.39750	−0.0852069	−0.0426035	−	0.999092i	$-0.513565\pi$
$269$	−1.39750	−0.0852069	−0.0426035	+	0.999092i	$0.513565\pi$
$270$	1.20224	0.0731661
$271$	29.5492	1.79499	0.897493	−	0.441029i	$-0.145386\pi$
$271$	29.5492	1.79499	0.897493	+	0.441029i	$0.145386\pi$
$272$	98.6371	5.98075
$273$	4.61014	0.279018
$274$	45.5127	2.74952
$275$	22.4864	1.35598
$276$	11.4667	0.690214
$277$	−4.09028	−0.245761	−0.122881	−	0.992421i	$-0.539213\pi$
$277$	−4.09028	−0.245761	−0.122881	+	0.992421i	$0.539213\pi$
$278$	−52.7323	−3.16268
$279$	−6.95226	−0.416221
$280$	−4.54592	−0.271671
$281$	17.2302	1.02787	0.513935	−	0.857829i	$-0.328187\pi$
$281$	17.2302	1.02787	0.513935	+	0.857829i	$0.328187\pi$
$282$	−19.8103	−1.17969
$283$	−29.6999	−1.76548	−0.882738	−	0.469866i	$-0.844302\pi$
$283$	−29.6999	−1.76548	−0.882738	+	0.469866i	$0.844302\pi$
$284$	−75.6484	−4.48890
$285$	−0.430991	−0.0255297
$286$	60.0660	3.55177
$287$	−0.487127	−0.0287542
$288$	28.7247	1.69262
$289$	13.5016	0.794212
$290$	−7.71170	−0.452846
$291$	−3.22028	−0.188776
$292$	−10.2858	−0.601933
$293$	−27.0490	−1.58022	−0.790110	−	0.612965i	$-0.789976\pi$
$293$	−27.0490	−1.58022	−0.790110	+	0.612965i	$0.789976\pi$
$294$	−2.78948	−0.162686
$295$	1.15822	0.0674344
$296$	−41.7032	−2.42395
$297$	4.67080	0.271027
$298$	6.83609	0.396004
$299$	9.14395	0.528808
$300$	27.8321	1.60689
$301$	−2.25184	−0.129794
$302$	−50.1373	−2.88508
$303$	−19.8188	−1.13856
$304$	−17.8599	−1.02434
$305$	0.961046	0.0550293
$306$	15.4058	0.880692
$307$	−22.3893	−1.27783	−0.638913	−	0.769279i	$-0.720616\pi$
$307$	−22.3893	−1.27783	−0.638913	+	0.769279i	$0.720616\pi$
$308$	−27.0028	−1.53863
$309$	−9.34159	−0.531425
$310$	8.35830	0.474719
$311$	18.0389	1.02289	0.511446	−	0.859315i	$-0.329110\pi$
$311$	18.0389	1.02289	0.511446	+	0.859315i	$0.329110\pi$
$312$	48.6259	2.75290
$313$	−10.6673	−0.602952	−0.301476	−	0.953474i	$-0.597479\pi$
$313$	−10.6673	−0.602952	−0.301476	+	0.953474i	$0.597479\pi$
$314$	18.5254	1.04545
$315$	−0.430991	−0.0242836
$316$	75.8016	4.26417
$317$	15.7560	0.884947	0.442473	−	0.896782i	$-0.354101\pi$
$317$	15.7560	0.884947	0.442473	+	0.896782i	$0.354101\pi$
$318$	20.9386	1.17418
$319$	−29.9605	−1.67747
$320$	−19.1391	−1.06991
$321$	−1.22688	−0.0684780
$322$	−5.53278	−0.308330
$323$	−5.52283	−0.307298
$324$	5.78120	0.321178
$325$	22.1944	1.23112
$326$	50.5631	2.80043
$327$	2.11227	0.116809
$328$	−5.13802	−0.283700
$329$	7.10179	0.391534
$330$	−5.61542	−0.309119
$331$	−22.2790	−1.22456	−0.612282	−	0.790639i	$-0.709748\pi$
$331$	−22.2790	−1.22456	−0.612282	+	0.790639i	$0.709748\pi$
$332$	13.8828	0.761916
$333$	−3.95382	−0.216668
$334$	−23.3153	−1.27576
$335$	−0.154423	−0.00843705
$336$	−17.8599	−0.974337
$337$	−12.6452	−0.688828	−0.344414	−	0.938818i	$-0.611922\pi$
$337$	−12.6452	−0.688828	−0.344414	+	0.938818i	$0.611922\pi$
$338$	23.0227	1.25227
$339$	−10.4144	−0.565634
$340$	−13.7609	−0.746291
$341$	32.4726	1.75849
$342$	−2.78948	−0.150838
$343$	1.00000	0.0539949
$344$	−23.7515	−1.28060
$345$	−0.854846	−0.0460234
$346$	−28.3610	−1.52469
$347$	5.87117	0.315181	0.157590	−	0.987505i	$-0.449627\pi$
$347$	5.87117	0.315181	0.157590	+	0.987505i	$0.449627\pi$
$348$	−37.0831	−1.98787
$349$	13.6285	0.729517	0.364758	−	0.931102i	$-0.381152\pi$
$349$	13.6285	0.729517	0.364758	+	0.931102i	$0.381152\pi$
$350$	−13.4292	−0.717824
$351$	4.61014	0.246071
$352$	−134.167	−7.15112
$353$	−9.75268	−0.519083	−0.259541	−	0.965732i	$-0.583571\pi$
$353$	−9.75268	−0.519083	−0.259541	+	0.965732i	$0.583571\pi$
$354$	7.49632	0.398425
$355$	5.63962	0.299320
$356$	95.1240	5.04156
$357$	−5.52283	−0.292299
$358$	−5.33107	−0.281756
$359$	−8.73005	−0.460754	−0.230377	−	0.973101i	$-0.573996\pi$
$359$	−8.73005	−0.460754	−0.230377	+	0.973101i	$0.573996\pi$
$360$	−4.54592	−0.239591
$361$	1.00000	0.0526316
$362$	36.4461	1.91557
$363$	−10.8163	−0.567710
$364$	−26.6522	−1.39695
$365$	0.766813	0.0401368
$366$	6.22013	0.325131
$367$	13.7923	0.719950	0.359975	−	0.932962i	$-0.382785\pi$
$367$	13.7923	0.719950	0.359975	+	0.932962i	$0.382785\pi$
$368$	−35.4241	−1.84661
$369$	−0.487127	−0.0253588
$370$	4.75344	0.247120
$371$	−7.50627	−0.389706
$372$	40.1925	2.08388
$373$	−18.2569	−0.945306	−0.472653	−	0.881249i	$-0.656704\pi$
$373$	−18.2569	−0.945306	−0.472653	+	0.881249i	$0.656704\pi$
$374$	−71.9574	−3.72083
$375$	−4.22985	−0.218429
$376$	74.9067	3.86302
$377$	−29.5714	−1.52301
$378$	−2.78948	−0.143475
$379$	22.9149	1.17706	0.588529	−	0.808476i	$-0.299707\pi$
$379$	22.9149	1.17706	0.588529	+	0.808476i	$0.299707\pi$
$380$	2.49165	0.127819
$381$	2.17463	0.111410
$382$	53.2205	2.72300
$383$	−38.6741	−1.97616	−0.988078	−	0.153953i	$-0.950800\pi$
$383$	−38.6741	−1.97616	−0.988078	+	0.153953i	$0.950800\pi$
$384$	−66.4234	−3.38965
$385$	2.01307	0.102596
$386$	−13.7164	−0.698149
$387$	−2.25184	−0.114468
$388$	18.6171	0.945140
$389$	8.76651	0.444480	0.222240	−	0.974992i	$-0.428663\pi$
$389$	8.76651	0.444480	0.222240	+	0.974992i	$0.428663\pi$
$390$	−5.54250	−0.280655
$391$	−10.9542	−0.553978
$392$	10.5476	0.532734
$393$	−0.789672	−0.0398337
$394$	−30.3548	−1.52925
$395$	−5.65104	−0.284335
$396$	−27.0028	−1.35694
$397$	−23.8561	−1.19731	−0.598653	−	0.801009i	$-0.704297\pi$
$397$	−23.8561	−1.19731	−0.598653	+	0.801009i	$0.704297\pi$
$398$	−39.6206	−1.98600
$399$	1.00000	0.0500626
$400$	−85.9820	−4.29910
$401$	−35.5461	−1.77509	−0.887543	−	0.460724i	$-0.847590\pi$
$401$	−35.5461	−1.77509	−0.887543	+	0.460724i	$0.847590\pi$
$402$	−0.999467	−0.0498489
$403$	32.0509	1.59657
$404$	114.576	5.70038
$405$	−0.430991	−0.0214161
$406$	17.8929	0.888012
$407$	18.4675	0.915398
$408$	−58.2525	−2.88393
$409$	−12.1947	−0.602987	−0.301493	−	0.953468i	$-0.597485\pi$
$409$	−12.1947	−0.602987	−0.301493	+	0.953468i	$0.597485\pi$
$410$	0.585644	0.0289229
$411$	−16.3158	−0.804801
$412$	54.0056	2.66067
$413$	−2.68735	−0.132236
$414$	−5.53278	−0.271921
$415$	−1.03497	−0.0508045
$416$	−132.425	−6.49265
$417$	18.9040	0.925733
$418$	13.0291	0.637274
$419$	16.4827	0.805234	0.402617	−	0.915369i	$-0.368101\pi$
$419$	16.4827	0.805234	0.402617	+	0.915369i	$0.368101\pi$
$420$	2.49165	0.121580
$421$	10.6213	0.517649	0.258824	−	0.965924i	$-0.416665\pi$
$421$	10.6213	0.517649	0.258824	+	0.965924i	$0.416665\pi$
$422$	−32.0688	−1.56108
$423$	7.10179	0.345301
$424$	−79.1731	−3.84498
$425$	−26.5882	−1.28972
$426$	36.5010	1.76848
$427$	−2.22985	−0.107910
$428$	7.09287	0.342847
$429$	−21.5330	−1.03962
$430$	2.70726	0.130556
$431$	5.66851	0.273043	0.136521	−	0.990637i	$-0.456408\pi$
$431$	5.66851	0.273043	0.136521	+	0.990637i	$0.456408\pi$
$432$	−17.8599	−0.859285
$433$	39.1499	1.88142	0.940712	−	0.339207i	$-0.110159\pi$
$433$	39.1499	1.88142	0.940712	+	0.339207i	$0.110159\pi$
$434$	−19.3932	−0.930904
$435$	2.76456	0.132551
$436$	−12.2115	−0.584824
$437$	1.98344	0.0948810
$438$	4.96301	0.237142
$439$	−17.6065	−0.840313	−0.420156	−	0.907452i	$-0.638025\pi$
$439$	−17.6065	−0.840313	−0.420156	+	0.907452i	$0.638025\pi$
$440$	21.2330	1.01225
$441$	1.00000	0.0476190
$442$	−71.0230	−3.37822
$443$	23.4373	1.11354	0.556770	−	0.830667i	$-0.312040\pi$
$443$	23.4373	1.11354	0.556770	+	0.830667i	$0.312040\pi$
$444$	22.8578	1.08478
$445$	−7.09153	−0.336171
$446$	−47.2043	−2.23519
$447$	−2.45067	−0.115913
$448$	44.4071	2.09804
$449$	−36.3882	−1.71726	−0.858632	−	0.512593i	$-0.828685\pi$
$449$	−36.3882	−1.71726	−0.858632	+	0.512593i	$0.828685\pi$
$450$	−13.4292	−0.633061
$451$	2.27527	0.107138
$452$	60.2080	2.83194
$453$	17.9737	0.844479
$454$	−17.5624	−0.824245
$455$	1.98693	0.0931487
$456$	10.5476	0.493936
$457$	16.4395	0.769009	0.384505	−	0.923123i	$-0.374372\pi$
$457$	16.4395	0.769009	0.384505	+	0.923123i	$0.374372\pi$
$458$	24.4540	1.14266
$459$	−5.52283	−0.257783
$460$	4.94204	0.230424
$461$	−19.6734	−0.916281	−0.458140	−	0.888880i	$-0.651484\pi$
$461$	−19.6734	−0.916281	−0.458140	+	0.888880i	$0.651484\pi$
$462$	13.0291	0.606168
$463$	−11.1248	−0.517014	−0.258507	−	0.966009i	$-0.583230\pi$
$463$	−11.1248	−0.517014	−0.258507	+	0.966009i	$0.583230\pi$
$464$	114.561	5.31837
$465$	−2.99636	−0.138953
$466$	−18.8837	−0.874771
$467$	−24.5917	−1.13797	−0.568985	−	0.822348i	$-0.692664\pi$
$467$	−24.5917	−1.13797	−0.568985	+	0.822348i	$0.692664\pi$
$468$	−26.6522	−1.23200
$469$	0.358299	0.0165447
$470$	−8.53806	−0.393831
$471$	−6.64117	−0.306009
$472$	−28.3451	−1.30469
$473$	10.5179	0.483613
$474$	−36.5749	−1.67994
$475$	4.81425	0.220893
$476$	31.9286	1.46344
$477$	−7.50627	−0.343688
$478$	5.21960	0.238739
$479$	−16.0429	−0.733020	−0.366510	−	0.930414i	$-0.619447\pi$
$479$	−16.0429	−0.733020	−0.366510	+	0.930414i	$0.619447\pi$
$480$	12.3801	0.565070
$481$	18.2276	0.831109
$482$	36.7038	1.67181
$483$	1.98344	0.0902498
$484$	62.5314	2.84234
$485$	−1.38791	−0.0630218
$486$	−2.78948	−0.126533
$487$	−13.2823	−0.601880	−0.300940	−	0.953643i	$-0.597300\pi$
$487$	−13.2823	−0.601880	−0.300940	+	0.953643i	$0.597300\pi$
$488$	−23.5196	−1.06468
$489$	−18.1264	−0.819702
$490$	−1.20224	−0.0543117
$491$	30.6276	1.38220	0.691102	−	0.722758i	$-0.257126\pi$
$491$	30.6276	1.38220	0.691102	+	0.722758i	$0.257126\pi$
$492$	2.81618	0.126963
$493$	35.4258	1.59550
$494$	12.8599	0.578594
$495$	2.01307	0.0904808
$496$	−124.167	−5.57525
$497$	−13.0852	−0.586953
$498$	−6.69856	−0.300170
$499$	37.5985	1.68314	0.841570	−	0.540149i	$-0.181632\pi$
$499$	37.5985	1.68314	0.841570	+	0.540149i	$0.181632\pi$
$500$	24.4536	1.09360
$501$	8.35830	0.373421
$502$	−28.4714	−1.27074
$503$	−19.9344	−0.888830	−0.444415	−	0.895821i	$-0.646588\pi$
$503$	−19.9344	−0.888830	−0.444415	+	0.895821i	$0.646588\pi$
$504$	10.5476	0.469827
$505$	−8.54171	−0.380101
$506$	25.8425	1.14884
$507$	−8.25339	−0.366546
$508$	−12.5720	−0.557791
$509$	10.6949	0.474042	0.237021	−	0.971505i	$-0.423829\pi$
$509$	10.6949	0.474042	0.237021	+	0.971505i	$0.423829\pi$
$510$	6.63977	0.294014
$511$	−1.77919	−0.0787066
$512$	136.262	6.02197
$513$	1.00000	0.0441511
$514$	25.0125	1.10326
$515$	−4.02614	−0.177413
$516$	13.0184	0.573101
$517$	−33.1710	−1.45886
$518$	−11.0291	−0.484591
$519$	10.1671	0.446287
$520$	20.9573	0.919039
$521$	24.5070	1.07367	0.536836	−	0.843686i	$-0.319619\pi$
$521$	24.5070	1.07367	0.536836	+	0.843686i	$0.319619\pi$
$522$	17.8929	0.783153
$523$	8.32697	0.364113	0.182056	−	0.983288i	$-0.441725\pi$
$523$	8.32697	0.364113	0.182056	+	0.983288i	$0.441725\pi$
$524$	4.56525	0.199434
$525$	4.81425	0.210111
$526$	−31.6581	−1.38036
$527$	−38.3961	−1.67256
$528$	83.4199	3.63039
$529$	−19.0660	−0.828954
$530$	9.02435	0.391993
$531$	−2.68735	−0.116621
$532$	−5.78120	−0.250647
$533$	2.24572	0.0972731
$534$	−45.8982	−1.98621
$535$	−0.528776	−0.0228610
$536$	3.77919	0.163236
$537$	1.91113	0.0824715
$538$	−3.89829	−0.168067
$539$	−4.67080	−0.201185
$540$	2.49165	0.107223
$541$	45.0614	1.93734	0.968670	−	0.248351i	$-0.0798886\pi$
$541$	45.0614	1.93734	0.968670	+	0.248351i	$0.0798886\pi$
$542$	82.4269	3.54054
$543$	−13.0656	−0.560697
$544$	158.641	6.80169
$545$	0.910370	0.0389960
$546$	12.8599	0.550353
$547$	−14.6050	−0.624463	−0.312231	−	0.950006i	$-0.601076\pi$
$547$	−14.6050	−0.624463	−0.312231	+	0.950006i	$0.601076\pi$
$548$	94.3252	4.02937
$549$	−2.22985	−0.0951678
$550$	62.7253	2.67461
$551$	−6.41443	−0.273264
$552$	20.9206	0.890438
$553$	13.1117	0.557568
$554$	−11.4098	−0.484754
$555$	−1.70406	−0.0723333
$556$	−109.288	−4.63484
$557$	−34.6441	−1.46792	−0.733960	−	0.679193i	$-0.762330\pi$
$557$	−34.6441	−1.46792	−0.733960	+	0.679193i	$0.762330\pi$
$558$	−19.3932	−0.820980
$559$	10.3813	0.439082
$560$	−7.69746	−0.325277
$561$	25.7960	1.08911
$562$	48.0634	2.02743
$563$	33.4497	1.40974	0.704868	−	0.709338i	$-0.251006\pi$
$563$	33.4497	1.40974	0.704868	+	0.709338i	$0.251006\pi$
$564$	−41.0569	−1.72881
$565$	−4.48853	−0.188834
$566$	−82.8473	−3.48233
$567$	1.00000	0.0419961
$568$	−138.018	−5.79109
$569$	−8.24277	−0.345555	−0.172777	−	0.984961i	$-0.555274\pi$
$569$	−8.24277	−0.345555	−0.172777	+	0.984961i	$0.555274\pi$
$570$	−1.20224	−0.0503563
$571$	15.7498	0.659109	0.329554	−	0.944137i	$-0.393101\pi$
$571$	15.7498	0.659109	0.329554	+	0.944137i	$0.393101\pi$
$572$	124.487	5.20505
$573$	−19.0790	−0.797037
$574$	−1.35883	−0.0567165
$575$	9.54879	0.398212
$576$	44.4071	1.85029
$577$	−20.4094	−0.849652	−0.424826	−	0.905275i	$-0.639665\pi$
$577$	−20.4094	−0.849652	−0.424826	+	0.905275i	$0.639665\pi$
$578$	37.6624	1.56655
$579$	4.91720	0.204352
$580$	−15.9825	−0.663637
$581$	2.40136	0.0996254
$582$	−8.98291	−0.372354
$583$	35.0602	1.45205
$584$	−18.7661	−0.776548
$585$	1.98693	0.0821494
$586$	−75.4527	−3.11692
$587$	20.0429	0.827260	0.413630	−	0.910445i	$-0.364261\pi$
$587$	20.0429	0.827260	0.413630	+	0.910445i	$0.364261\pi$
$588$	−5.78120	−0.238413
$589$	6.95226	0.286463
$590$	3.23085	0.133012
$591$	10.8819	0.447621
$592$	−70.6148	−2.90225
$593$	−11.7074	−0.480766	−0.240383	−	0.970678i	$-0.577273\pi$
$593$	−11.7074	−0.480766	−0.240383	+	0.970678i	$0.577273\pi$
$594$	13.0291	0.534590
$595$	−2.38029	−0.0975823
$596$	14.1678	0.580336
$597$	14.2036	0.581314
$598$	25.5069	1.04305
$599$	−2.41822	−0.0988059	−0.0494029	−	0.998779i	$-0.515732\pi$
$599$	−2.41822	−0.0988059	−0.0494029	+	0.998779i	$0.515732\pi$
$600$	50.7787	2.07303
$601$	−9.78632	−0.399192	−0.199596	−	0.979878i	$-0.563963\pi$
$601$	−9.78632	−0.399192	−0.199596	+	0.979878i	$0.563963\pi$
$602$	−6.28147	−0.256014
$603$	0.358299	0.0145910
$604$	−103.910	−4.22803
$605$	−4.66174	−0.189527
$606$	−55.2841	−2.24576
$607$	−23.6574	−0.960226	−0.480113	−	0.877207i	$-0.659404\pi$
$607$	−23.6574	−0.960226	−0.480113	+	0.877207i	$0.659404\pi$
$608$	−28.7247	−1.16494
$609$	−6.41443	−0.259926
$610$	2.68082	0.108543
$611$	−32.7402	−1.32453
$612$	31.9286	1.29064
$613$	−27.9055	−1.12709	−0.563547	−	0.826084i	$-0.690564\pi$
$613$	−27.9055	−1.12709	−0.563547	+	0.826084i	$0.690564\pi$
$614$	−62.4546	−2.52046
$615$	−0.209947	−0.00846590
$616$	−49.2656	−1.98497
$617$	1.12898	0.0454510	0.0227255	−	0.999742i	$-0.492766\pi$
$617$	1.12898	0.0454510	0.0227255	+	0.999742i	$0.492766\pi$
$618$	−26.0582	−1.04821
$619$	−48.2125	−1.93782	−0.968911	−	0.247408i	$-0.920421\pi$
$619$	−48.2125	−1.93782	−0.968911	+	0.247408i	$0.920421\pi$
$620$	17.3226	0.695692
$621$	1.98344	0.0795929
$622$	50.3192	2.01761
$623$	16.4540	0.659216
$624$	82.3366	3.29610
$625$	22.2482	0.889928
$626$	−29.7563	−1.18930
$627$	−4.67080	−0.186534
$628$	38.3939	1.53208
$629$	−21.8362	−0.870668
$630$	−1.20224	−0.0478984
$631$	−29.4819	−1.17366	−0.586828	−	0.809711i	$-0.699624\pi$
$631$	−29.4819	−1.17366	−0.586828	+	0.809711i	$0.699624\pi$
$632$	138.297	5.50117
$633$	11.4963	0.456938
$634$	43.9511	1.74552
$635$	0.937246	0.0371935
$636$	43.3953	1.72073
$637$	−4.61014	−0.182660
$638$	−83.5743	−3.30874
$639$	−13.0852	−0.517644
$640$	−28.6279	−1.13162
$641$	−4.03313	−0.159299	−0.0796495	−	0.996823i	$-0.525380\pi$
$641$	−4.03313	−0.159299	−0.0796495	+	0.996823i	$0.525380\pi$
$642$	−3.42237	−0.135070
$643$	29.0555	1.14584	0.572918	−	0.819613i	$-0.305812\pi$
$643$	29.0555	1.14584	0.572918	+	0.819613i	$0.305812\pi$
$644$	−11.4667	−0.451851
$645$	−0.970524	−0.0382143
$646$	−15.4058	−0.606134
$647$	9.75943	0.383683	0.191841	−	0.981426i	$-0.438554\pi$
$647$	9.75943	0.383683	0.191841	+	0.981426i	$0.438554\pi$
$648$	10.5476	0.414349
$649$	12.5521	0.492712
$650$	61.9107	2.42834
$651$	6.95226	0.272481
$652$	104.792	4.10398
$653$	−20.1174	−0.787256	−0.393628	−	0.919270i	$-0.628780\pi$
$653$	−20.1174	−0.787256	−0.393628	+	0.919270i	$0.628780\pi$
$654$	5.89214	0.230401
$655$	−0.340341	−0.0132982
$656$	−8.70004	−0.339680
$657$	−1.77919	−0.0694127
$658$	19.8103	0.772286
$659$	17.4676	0.680441	0.340221	−	0.940346i	$-0.389498\pi$
$659$	17.4676	0.680441	0.340221	+	0.940346i	$0.389498\pi$
$660$	−11.6380	−0.453007
$661$	−16.2717	−0.632898	−0.316449	−	0.948610i	$-0.602491\pi$
$661$	−16.2717	−0.632898	−0.316449	+	0.948610i	$0.602491\pi$
$662$	−62.1468	−2.41541
$663$	25.4610	0.988823
$664$	25.3286	0.982940
$665$	0.430991	0.0167131
$666$	−11.0291	−0.427369
$667$	−12.7227	−0.492624
$668$	−48.3210	−1.86960
$669$	16.9223	0.654252
$670$	−0.430761	−0.0166418
$671$	10.4152	0.402074
$672$	−28.7247	−1.10808
$673$	41.0755	1.58335	0.791673	−	0.610946i	$-0.209211\pi$
$673$	41.0755	1.58335	0.791673	+	0.610946i	$0.209211\pi$
$674$	−35.2735	−1.35869
$675$	4.81425	0.185300
$676$	47.7145	1.83517
$677$	2.81542	0.108205	0.0541026	−	0.998535i	$-0.482770\pi$
$677$	2.81542	0.108205	0.0541026	+	0.998535i	$0.482770\pi$
$678$	−29.0509	−1.11569
$679$	3.22028	0.123583
$680$	−25.1063	−0.962783
$681$	6.29594	0.241261
$682$	90.5817	3.46855
$683$	−28.9174	−1.10649	−0.553247	−	0.833017i	$-0.686612\pi$
$683$	−28.9174	−1.10649	−0.553247	+	0.833017i	$0.686612\pi$
$684$	−5.78120	−0.221050
$685$	−7.03198	−0.268678
$686$	2.78948	0.106503
$687$	−8.76651	−0.334463
$688$	−40.2177	−1.53328
$689$	34.6050	1.31834
$690$	−2.38458	−0.0907793
$691$	35.4760	1.34957	0.674785	−	0.738015i	$-0.264236\pi$
$691$	35.4760	1.34957	0.674785	+	0.738015i	$0.264236\pi$
$692$	−58.7781	−2.23441
$693$	−4.67080	−0.177429
$694$	16.3775	0.621682
$695$	8.14745	0.309050
$696$	−67.6568	−2.56453
$697$	−2.69032	−0.101903
$698$	38.0164	1.43894
$699$	6.76961	0.256050
$700$	−27.8321	−1.05196
$701$	30.5104	1.15236	0.576181	−	0.817322i	$-0.304542\pi$
$701$	30.5104	1.15236	0.576181	+	0.817322i	$0.304542\pi$
$702$	12.8599	0.485366
$703$	3.95382	0.149121
$704$	−207.416	−7.81730
$705$	3.06081	0.115277
$706$	−27.2049	−1.02387
$707$	19.8188	0.745361
$708$	15.5361	0.583883
$709$	43.7808	1.64422	0.822112	−	0.569326i	$-0.192796\pi$
$709$	43.7808	1.64422	0.822112	+	0.569326i	$0.192796\pi$
$710$	15.7316	0.590396
$711$	13.1117	0.491729
$712$	173.550	6.50407
$713$	13.7894	0.516418
$714$	−15.4058	−0.576548
$715$	−9.28054	−0.347072
$716$	−11.0486	−0.412907
$717$	−1.87117	−0.0698802
$718$	−24.3523	−0.908820
$719$	14.5597	0.542985	0.271492	−	0.962441i	$-0.412483\pi$
$719$	14.5597	0.542985	0.271492	+	0.962441i	$0.412483\pi$
$720$	−7.69746	−0.286867
$721$	9.34159	0.347899
$722$	2.78948	0.103814
$723$	−13.1579	−0.489348
$724$	75.5346	2.80722
$725$	−30.8807	−1.14688
$726$	−30.1719	−1.11979
$727$	35.4537	1.31491	0.657453	−	0.753496i	$-0.271634\pi$
$727$	35.4537	1.31491	0.657453	+	0.753496i	$0.271634\pi$
$728$	−48.6259	−1.80219
$729$	1.00000	0.0370370
$730$	2.13901	0.0791683
$731$	−12.4365	−0.459982
$732$	12.8912	0.476473
$733$	2.00373	0.0740095	0.0370047	−	0.999315i	$-0.488218\pi$
$733$	2.00373	0.0740095	0.0370047	+	0.999315i	$0.488218\pi$
$734$	38.4732	1.42007
$735$	0.430991	0.0158973
$736$	−56.9737	−2.10008
$737$	−1.67354	−0.0616456
$738$	−1.35883	−0.0500193
$739$	33.6847	1.23911	0.619556	−	0.784952i	$-0.287312\pi$
$739$	33.6847	1.23911	0.619556	+	0.784952i	$0.287312\pi$
$740$	9.85151	0.362149
$741$	−4.61014	−0.169358
$742$	−20.9386	−0.768680
$743$	18.0757	0.663132	0.331566	−	0.943432i	$-0.392423\pi$
$743$	18.0757	0.663132	0.331566	+	0.943432i	$0.392423\pi$
$744$	73.3296	2.68840
$745$	−1.05622	−0.0386967
$746$	−50.9272	−1.86458
$747$	2.40136	0.0878613
$748$	−149.132	−5.45280
$749$	1.22688	0.0448294
$750$	−11.7991	−0.430842
$751$	6.55814	0.239310	0.119655	−	0.992816i	$-0.461821\pi$
$751$	6.55814	0.239310	0.119655	+	0.992816i	$0.461821\pi$
$752$	126.837	4.62528
$753$	10.2067	0.371953
$754$	−82.4890	−3.00407
$755$	7.74651	0.281924
$756$	−5.78120	−0.210260
$757$	−44.0279	−1.60022	−0.800110	−	0.599853i	$-0.795226\pi$
$757$	−44.0279	−1.60022	−0.800110	+	0.599853i	$0.795226\pi$
$758$	63.9206	2.32170
$759$	−9.26426	−0.336271
$760$	4.54592	0.164898
$761$	−8.08173	−0.292963	−0.146481	−	0.989213i	$-0.546795\pi$
$761$	−8.08173	−0.292963	−0.146481	+	0.989213i	$0.546795\pi$
$762$	6.06609	0.219751
$763$	−2.11227	−0.0764694
$764$	110.300	3.99050
$765$	−2.38029	−0.0860595
$766$	−107.881	−3.89789
$767$	12.3891	0.447343
$768$	−96.4726	−3.48115
$769$	−12.9570	−0.467242	−0.233621	−	0.972328i	$-0.575057\pi$
$769$	−12.9570	−0.467242	−0.233621	+	0.972328i	$0.575057\pi$
$770$	5.61542	0.202366
$771$	−8.96674	−0.322929
$772$	−28.4274	−1.02312
$773$	−6.70111	−0.241022	−0.120511	−	0.992712i	$-0.538453\pi$
$773$	−6.70111	−0.241022	−0.120511	+	0.992712i	$0.538453\pi$
$774$	−6.28147	−0.225783
$775$	33.4699	1.20227
$776$	33.9662	1.21932
$777$	3.95382	0.141842
$778$	24.4540	0.876719
$779$	0.487127	0.0174531
$780$	−11.4868	−0.411295
$781$	61.1184	2.18699
$782$	−30.5566	−1.09270
$783$	−6.41443	−0.229233
$784$	17.8599	0.637853
$785$	−2.86228	−0.102159
$786$	−2.20277	−0.0785704
$787$	−5.64379	−0.201179	−0.100590	−	0.994928i	$-0.532073\pi$
$787$	−5.64379	−0.201179	−0.100590	+	0.994928i	$0.532073\pi$
$788$	−62.9104	−2.24109
$789$	11.3491	0.404039
$790$	−15.7635	−0.560839
$791$	10.4144	0.370295
$792$	−49.2656	−1.75058
$793$	10.2799	0.365051
$794$	−66.5462	−2.36164
$795$	−3.23513	−0.114738
$796$	−82.1137	−2.91044
$797$	−34.0525	−1.20620	−0.603101	−	0.797665i	$-0.706068\pi$
$797$	−34.0525	−1.20620	−0.603101	+	0.797665i	$0.706068\pi$
$798$	2.78948	0.0987465
$799$	39.2219	1.38757
$800$	−138.288	−4.88920
$801$	16.4540	0.581374
$802$	−99.1551	−3.50129
$803$	8.31022	0.293261
$804$	−2.07140	−0.0730525
$805$	0.854846	0.0301294
$806$	89.4054	3.14917
$807$	1.39750	0.0491942
$808$	209.040	7.35401
$809$	3.90376	0.137249	0.0686245	−	0.997643i	$-0.478139\pi$
$809$	3.90376	0.137249	0.0686245	+	0.997643i	$0.478139\pi$
$810$	−1.20224	−0.0422424
$811$	24.6511	0.865618	0.432809	−	0.901486i	$-0.357522\pi$
$811$	24.6511	0.865618	0.432809	+	0.901486i	$0.357522\pi$
$812$	37.0831	1.30136
$813$	−29.5492	−1.03634
$814$	51.5146	1.80559
$815$	−7.81230	−0.273653
$816$	−98.6371	−3.45299
$817$	2.25184	0.0787820
$818$	−34.0168	−1.18937
$819$	−4.61014	−0.161091
$820$	1.21375	0.0423859
$821$	17.9234	0.625530	0.312765	−	0.949831i	$-0.398745\pi$
$821$	17.9234	0.625530	0.312765	+	0.949831i	$0.398745\pi$
$822$	−45.5127	−1.58744
$823$	41.8653	1.45933	0.729666	−	0.683803i	$-0.239675\pi$
$823$	41.8653	1.45933	0.729666	+	0.683803i	$0.239675\pi$
$824$	98.5313	3.43250
$825$	−22.4864	−0.782875
$826$	−7.49632	−0.260830
$827$	40.8151	1.41928	0.709641	−	0.704564i	$-0.248857\pi$
$827$	40.8151	1.41928	0.709641	+	0.704564i	$0.248857\pi$
$828$	−11.4667	−0.398495
$829$	−50.9370	−1.76911	−0.884557	−	0.466432i	$-0.845539\pi$
$829$	−50.9370	−1.76911	−0.884557	+	0.466432i	$0.845539\pi$
$830$	−2.88702	−0.100210
$831$	4.09028	0.141890
$832$	−204.723	−7.09749
$833$	5.52283	0.191355
$834$	52.7323	1.82597
$835$	3.60235	0.124665
$836$	27.0028	0.933912
$837$	6.95226	0.240305
$838$	45.9783	1.58829
$839$	12.2527	0.423011	0.211505	−	0.977377i	$-0.432163\pi$
$839$	12.2527	0.423011	0.211505	+	0.977377i	$0.432163\pi$
$840$	4.54592	0.156849
$841$	12.1450	0.418792
$842$	29.6278	1.02104
$843$	−17.2302	−0.593441
$844$	−66.4625	−2.28773
$845$	−3.55714	−0.122369
$846$	19.8103	0.681092
$847$	10.8163	0.371654
$848$	−134.061	−4.60368
$849$	29.6999	1.01930
$850$	−74.1674	−2.54392
$851$	7.84217	0.268826
$852$	75.6484	2.59167
$853$	−31.7198	−1.08607	−0.543033	−	0.839712i	$-0.682724\pi$
$853$	−31.7198	−1.08607	−0.543033	+	0.839712i	$0.682724\pi$
$854$	−6.22013	−0.212848
$855$	0.430991	0.0147396
$856$	12.9407	0.442303
$857$	35.4906	1.21234	0.606168	−	0.795336i	$-0.292706\pi$
$857$	35.4906	1.21234	0.606168	+	0.795336i	$0.292706\pi$
$858$	−60.0660	−2.05062
$859$	24.2527	0.827492	0.413746	−	0.910392i	$-0.364220\pi$
$859$	24.2527	0.827492	0.413746	+	0.910392i	$0.364220\pi$
$860$	5.61079	0.191326
$861$	0.487127	0.0166012
$862$	15.8122	0.538566
$863$	−22.2259	−0.756579	−0.378289	−	0.925687i	$-0.623488\pi$
$863$	−22.2259	−0.756579	−0.378289	+	0.925687i	$0.623488\pi$
$864$	−28.7247	−0.977233
$865$	4.38193	0.148990
$866$	109.208	3.71103
$867$	−13.5016	−0.458538
$868$	−40.1925	−1.36422
$869$	−61.2423	−2.07750
$870$	7.71170	0.261451
$871$	−1.65181	−0.0559693
$872$	−22.2794	−0.754475
$873$	3.22028	0.108990
$874$	5.53278	0.187149
$875$	4.22985	0.142995
$876$	10.2858	0.347526
$877$	25.3600	0.856345	0.428173	−	0.903697i	$-0.359158\pi$
$877$	25.3600	0.856345	0.428173	+	0.903697i	$0.359158\pi$
$878$	−49.1130	−1.65748
$879$	27.0490	0.912340
$880$	35.9532	1.21198
$881$	7.42362	0.250108	0.125054	−	0.992150i	$-0.460090\pi$
$881$	7.42362	0.250108	0.125054	+	0.992150i	$0.460090\pi$
$882$	2.78948	0.0939267
$883$	−14.7424	−0.496123	−0.248061	−	0.968744i	$-0.579793\pi$
$883$	−14.7424	−0.496123	−0.248061	+	0.968744i	$0.579793\pi$
$884$	−147.195	−4.95071
$885$	−1.15822	−0.0389333
$886$	65.3779	2.19641
$887$	−40.5456	−1.36139	−0.680693	−	0.732568i	$-0.738321\pi$
$887$	−40.5456	−1.36139	−0.680693	+	0.732568i	$0.738321\pi$
$888$	41.7032	1.39947
$889$	−2.17463	−0.0729348
$890$	−19.7817	−0.663084
$891$	−4.67080	−0.156478
$892$	−97.8310	−3.27562
$893$	−7.10179	−0.237652
$894$	−6.83609	−0.228633
$895$	0.823681	0.0275326
$896$	66.4234	2.21905
$897$	−9.14395	−0.305308
$898$	−101.504	−3.38723
$899$	−44.5948	−1.48732
$900$	−27.8321	−0.927738
$901$	−41.4558	−1.38109
$902$	6.34683	0.211326
$903$	2.25184	0.0749366
$904$	109.847	3.65346
$905$	−5.63114	−0.187185
$906$	50.1373	1.66570
$907$	−45.5330	−1.51190	−0.755950	−	0.654630i	$-0.772825\pi$
$907$	−45.5330	−1.51190	−0.755950	+	0.654630i	$0.772825\pi$
$908$	−36.3981	−1.20791
$909$	19.8188	0.657347
$910$	5.54250	0.183732
$911$	−33.8615	−1.12188	−0.560941	−	0.827855i	$-0.689561\pi$
$911$	−33.8615	−1.12188	−0.560941	+	0.827855i	$0.689561\pi$
$912$	17.8599	0.591400
$913$	−11.2163	−0.371205
$914$	45.8578	1.51684
$915$	−0.961046	−0.0317712
$916$	50.6810	1.67455
$917$	0.789672	0.0260773
$918$	−15.4058	−0.508468
$919$	9.24999	0.305129	0.152564	−	0.988294i	$-0.451247\pi$
$919$	9.24999	0.305129	0.152564	+	0.988294i	$0.451247\pi$
$920$	9.01657	0.297268
$921$	22.3893	0.737753
$922$	−54.8785	−1.80733
$923$	60.3247	1.98561
$924$	27.0028	0.888327
$925$	19.0346	0.625855
$926$	−31.0324	−1.01979
$927$	9.34159	0.306818
$928$	184.252	6.04838
$929$	48.7467	1.59933	0.799664	−	0.600448i	$-0.205011\pi$
$929$	48.7467	1.59933	0.799664	+	0.600448i	$0.205011\pi$
$930$	−8.35830	−0.274079
$931$	−1.00000	−0.0327737
$932$	−39.1365	−1.28196
$933$	−18.0389	−0.590567
$934$	−68.5982	−2.24460
$935$	11.1178	0.363592
$936$	−48.6259	−1.58939
$937$	33.4921	1.09414	0.547070	−	0.837087i	$-0.315743\pi$
$937$	33.4921	1.09414	0.547070	+	0.837087i	$0.315743\pi$
$938$	0.999467	0.0326337
$939$	10.6673	0.348115
$940$	−17.6951	−0.577152
$941$	−23.3742	−0.761977	−0.380988	−	0.924580i	$-0.624416\pi$
$941$	−23.3742	−0.761977	−0.380988	+	0.924580i	$0.624416\pi$
$942$	−18.5254	−0.603590
$943$	0.966189	0.0314634
$944$	−47.9958	−1.56213
$945$	0.430991	0.0140201
$946$	29.3395	0.953908
$947$	−23.9643	−0.778735	−0.389368	−	0.921082i	$-0.627306\pi$
$947$	−23.9643	−0.778735	−0.389368	+	0.921082i	$0.627306\pi$
$948$	−75.8016	−2.46192
$949$	8.20230	0.266258
$950$	13.4292	0.435702
$951$	−15.7560	−0.510924
$952$	58.2525	1.88797
$953$	36.5266	1.18321	0.591606	−	0.806227i	$-0.298494\pi$
$953$	36.5266	1.18321	0.591606	+	0.806227i	$0.298494\pi$
$954$	−20.9386	−0.677912
$955$	−8.22288	−0.266086
$956$	10.8176	0.349867
$957$	29.9605	0.968486
$958$	−44.7514	−1.44585
$959$	16.3158	0.526866
$960$	19.1391	0.617710
$961$	17.3340	0.559161
$962$	50.8457	1.63933
$963$	1.22688	0.0395358
$964$	76.0686	2.45001
$965$	2.11927	0.0682217
$966$	5.53278	0.178014
$967$	−15.6636	−0.503706	−0.251853	−	0.967766i	$-0.581040\pi$
$967$	−15.6636	−0.503706	−0.251853	+	0.967766i	$0.581040\pi$
$968$	114.086	3.66687
$969$	5.52283	0.177419
$970$	−3.87155	−0.124308
$971$	−5.25339	−0.168589	−0.0842947	−	0.996441i	$-0.526864\pi$
$971$	−5.25339	−0.168589	−0.0842947	+	0.996441i	$0.526864\pi$
$972$	−5.78120	−0.185432
$973$	−18.9040	−0.606034
$974$	−37.0508	−1.18718
$975$	−22.1944	−0.710788
$976$	−39.8249	−1.27476
$977$	21.9525	0.702322	0.351161	−	0.936315i	$-0.385787\pi$
$977$	21.9525	0.702322	0.351161	+	0.936315i	$0.385787\pi$
$978$	−50.5631	−1.61683
$979$	−76.8533	−2.45624
$980$	−2.49165	−0.0795927
$981$	−2.11227	−0.0674397
$982$	85.4350	2.72634
$983$	−18.7244	−0.597214	−0.298607	−	0.954376i	$-0.596522\pi$
$983$	−18.7244	−0.597214	−0.298607	+	0.954376i	$0.596522\pi$
$984$	5.13802	0.163794
$985$	4.69000	0.149436
$986$	98.8196	3.14706
$987$	−7.10179	−0.226052
$988$	26.6522	0.847918
$989$	4.46640	0.142023
$990$	5.61542	0.178470
$991$	20.1060	0.638689	0.319345	−	0.947639i	$-0.396537\pi$
$991$	20.1060	0.638689	0.319345	+	0.947639i	$0.396537\pi$
$992$	−199.701	−6.34053
$993$	22.2790	0.707003
$994$	−36.5010	−1.15774
$995$	6.12161	0.194068
$996$	−13.8828	−0.439892
$997$	29.5676	0.936415	0.468207	−	0.883619i	$-0.344900\pi$
$997$	29.5676	0.936415	0.468207	+	0.883619i	$0.344900\pi$
$998$	104.880	3.31993
$999$	3.95382	0.125093

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.a.g.1.5	✓	5
3.2	odd	2		1197.2.a.o.1.1		5
4.3	odd	2		6384.2.a.cf.1.2		5
5.4	even	2		9975.2.a.bp.1.1		5
7.6	odd	2		2793.2.a.bg.1.5		5
19.18	odd	2		7581.2.a.w.1.1		5
21.20	even	2		8379.2.a.cb.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.g.1.5	✓	5	1.1	even	1	trivial
1197.2.a.o.1.1		5	3.2	odd	2
2793.2.a.bg.1.5		5	7.6	odd	2
6384.2.a.cf.1.2		5	4.3	odd	2
7581.2.a.w.1.1		5	19.18	odd	2
8379.2.a.cb.1.1		5	21.20	even	2
9975.2.a.bp.1.1		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.a (trivial)

\(f(q)\)	\(=\)	\(q+2.78948 q^{2} -1.00000 q^{3} +5.78120 q^{4} -0.430991 q^{5} -2.78948 q^{6} +1.00000 q^{7} +10.5476 q^{8} +1.00000 q^{9} -1.20224 q^{10} -4.67080 q^{11} -5.78120 q^{12} -4.61014 q^{13} +2.78948 q^{14} +0.430991 q^{15} +17.8599 q^{16} +5.52283 q^{17} +2.78948 q^{18} -1.00000 q^{19} -2.49165 q^{20} -1.00000 q^{21} -13.0291 q^{22} -1.98344 q^{23} -10.5476 q^{24} -4.81425 q^{25} -12.8599 q^{26} -1.00000 q^{27} +5.78120 q^{28} +6.41443 q^{29} +1.20224 q^{30} -6.95226 q^{31} +28.7247 q^{32} +4.67080 q^{33} +15.4058 q^{34} -0.430991 q^{35} +5.78120 q^{36} -3.95382 q^{37} -2.78948 q^{38} +4.61014 q^{39} -4.54592 q^{40} -0.487127 q^{41} -2.78948 q^{42} -2.25184 q^{43} -27.0028 q^{44} -0.430991 q^{45} -5.53278 q^{46} +7.10179 q^{47} -17.8599 q^{48} +1.00000 q^{49} -13.4292 q^{50} -5.52283 q^{51} -26.6522 q^{52} -7.50627 q^{53} -2.78948 q^{54} +2.01307 q^{55} +10.5476 q^{56} +1.00000 q^{57} +17.8929 q^{58} -2.68735 q^{59} +2.49165 q^{60} -2.22985 q^{61} -19.3932 q^{62} +1.00000 q^{63} +44.4071 q^{64} +1.98693 q^{65} +13.0291 q^{66} +0.358299 q^{67} +31.9286 q^{68} +1.98344 q^{69} -1.20224 q^{70} -13.0852 q^{71} +10.5476 q^{72} -1.77919 q^{73} -11.0291 q^{74} +4.81425 q^{75} -5.78120 q^{76} -4.67080 q^{77} +12.8599 q^{78} +13.1117 q^{79} -7.69746 q^{80} +1.00000 q^{81} -1.35883 q^{82} +2.40136 q^{83} -5.78120 q^{84} -2.38029 q^{85} -6.28147 q^{86} -6.41443 q^{87} -49.2656 q^{88} +16.4540 q^{89} -1.20224 q^{90} -4.61014 q^{91} -11.4667 q^{92} +6.95226 q^{93} +19.8103 q^{94} +0.430991 q^{95} -28.7247 q^{96} +3.22028 q^{97} +2.78948 q^{98} -4.67080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} - 5 q^{3} + 7 q^{4} + 4 q^{5} - 3 q^{6} + 5 q^{7} + 9 q^{8} + 5 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 6 q^{13} + 3 q^{14} - 4 q^{15} + 19 q^{16} + 12 q^{17} + 3 q^{18} - 5 q^{19} + 8 q^{20}+ \cdots + 8 q^{99}+O(q^{100}) \) 5 * q + 3 * q^2 - 5 * q^3 + 7 * q^4 + 4 * q^5 - 3 * q^6 + 5 * q^7 + 9 * q^8 + 5 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 6 * q^13 + 3 * q^14 - 4 * q^15 + 19 * q^16 + 12 * q^17 + 3 * q^18 - 5 * q^19 + 8 * q^20 - 5 * q^21 - 12 * q^22 + 12 * q^23 - 9 * q^24 + 15 * q^25 + 6 * q^26 - 5 * q^27 + 7 * q^28 + 4 * q^29 + 6 * q^30 - 8 * q^31 + 31 * q^32 - 8 * q^33 + 2 * q^34 + 4 * q^35 + 7 * q^36 + 2 * q^37 - 3 * q^38 + 6 * q^39 - 18 * q^40 + 10 * q^41 - 3 * q^42 - 16 * q^43 - 4 * q^44 + 4 * q^45 + 16 * q^46 - 2 * q^47 - 19 * q^48 + 5 * q^49 - 39 * q^50 - 12 * q^51 - 22 * q^52 - 3 * q^54 - 12 * q^55 + 9 * q^56 + 5 * q^57 + 2 * q^58 - 4 * q^59 - 8 * q^60 - 14 * q^61 - 36 * q^62 + 5 * q^63 + 11 * q^64 + 32 * q^65 + 12 * q^66 - 20 * q^67 + 8 * q^68 - 12 * q^69 - 6 * q^70 - 6 * q^71 + 9 * q^72 + 10 * q^73 - 2 * q^74 - 15 * q^75 - 7 * q^76 + 8 * q^77 - 6 * q^78 - 20 * q^80 + 5 * q^81 - 38 * q^82 + 6 * q^83 - 7 * q^84 + 8 * q^85 - 36 * q^86 - 4 * q^87 - 40 * q^88 + 26 * q^89 - 6 * q^90 - 6 * q^91 - 48 * q^92 + 8 * q^93 + 24 * q^94 - 4 * q^95 - 31 * q^96 - 18 * q^97 + 3 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

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Database

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