LMFDB - Embedded newform 1339.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$10.6919688306$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50015 q^{2} -2.63774 q^{3} +4.25075 q^{4} -1.17684 q^{5} +6.59474 q^{6} +4.82219 q^{7} -5.62720 q^{8} +3.95767 q^{9} +O(q^{10})$	\(q-2.50015 q^{2} -2.63774 q^{3} +4.25075 q^{4} -1.17684 q^{5} +6.59474 q^{6} +4.82219 q^{7} -5.62720 q^{8} +3.95767 q^{9} +2.94227 q^{10} -2.58003 q^{11} -11.2124 q^{12} -1.00000 q^{13} -12.0562 q^{14} +3.10419 q^{15} +5.56735 q^{16} +1.15915 q^{17} -9.89477 q^{18} -0.129120 q^{19} -5.00244 q^{20} -12.7197 q^{21} +6.45045 q^{22} -4.19248 q^{23} +14.8431 q^{24} -3.61505 q^{25} +2.50015 q^{26} -2.52609 q^{27} +20.4979 q^{28} +1.74484 q^{29} -7.76095 q^{30} -4.73087 q^{31} -2.66480 q^{32} +6.80544 q^{33} -2.89804 q^{34} -5.67494 q^{35} +16.8231 q^{36} +2.82874 q^{37} +0.322819 q^{38} +2.63774 q^{39} +6.62231 q^{40} -3.67998 q^{41} +31.8011 q^{42} +6.93571 q^{43} -10.9670 q^{44} -4.65754 q^{45} +10.4818 q^{46} +8.13246 q^{47} -14.6852 q^{48} +16.2535 q^{49} +9.03817 q^{50} -3.05753 q^{51} -4.25075 q^{52} +0.536946 q^{53} +6.31561 q^{54} +3.03627 q^{55} -27.1354 q^{56} +0.340585 q^{57} -4.36235 q^{58} -7.91490 q^{59} +13.1951 q^{60} -5.98348 q^{61} +11.8279 q^{62} +19.0846 q^{63} -4.47230 q^{64} +1.17684 q^{65} -17.0146 q^{66} -5.06585 q^{67} +4.92724 q^{68} +11.0587 q^{69} +14.1882 q^{70} +5.24961 q^{71} -22.2706 q^{72} +9.44481 q^{73} -7.07226 q^{74} +9.53557 q^{75} -0.548857 q^{76} -12.4414 q^{77} -6.59474 q^{78} +9.71315 q^{79} -6.55187 q^{80} -5.20984 q^{81} +9.20049 q^{82} -6.03263 q^{83} -54.0681 q^{84} -1.36413 q^{85} -17.3403 q^{86} -4.60242 q^{87} +14.5183 q^{88} -3.08020 q^{89} +11.6445 q^{90} -4.82219 q^{91} -17.8212 q^{92} +12.4788 q^{93} -20.3324 q^{94} +0.151953 q^{95} +7.02905 q^{96} +7.01383 q^{97} -40.6362 q^{98} -10.2109 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) $ 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.50015	−1.76787	−0.883936	−	0.467607i	$-0.845116\pi$
$2$	−2.50015	−1.76787	−0.883936	+	0.467607i	$0.845116\pi$
$3$	−2.63774	−1.52290	−0.761450	−	0.648224i	$-0.775512\pi$
$3$	−2.63774	−1.52290	−0.761450	+	0.648224i	$0.775512\pi$
$4$	4.25075	2.12537
$5$	−1.17684	−0.526298	−0.263149	−	0.964755i	$-0.584761\pi$
$5$	−1.17684	−0.526298	−0.263149	+	0.964755i	$0.584761\pi$
$6$	6.59474	2.69229
$7$	4.82219	1.82262	0.911308	−	0.411726i	$-0.135074\pi$
$7$	4.82219	1.82262	0.911308	+	0.411726i	$0.135074\pi$
$8$	−5.62720	−1.98952
$9$	3.95767	1.31922
$10$	2.94227	0.930428
$11$	−2.58003	−0.777907	−0.388954	−	0.921257i	$-0.627163\pi$
$11$	−2.58003	−0.777907	−0.388954	+	0.921257i	$0.627163\pi$
$12$	−11.2124	−3.23673
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−12.0562	−3.22215
$15$	3.10419	0.801499
$16$	5.56735	1.39184
$17$	1.15915	0.281134	0.140567	−	0.990071i	$-0.455107\pi$
$17$	1.15915	0.281134	0.140567	+	0.990071i	$0.455107\pi$
$18$	−9.89477	−2.33222
$19$	−0.129120	−0.0296222	−0.0148111	−	0.999890i	$-0.504715\pi$
$19$	−0.129120	−0.0296222	−0.0148111	+	0.999890i	$0.504715\pi$
$20$	−5.00244	−1.11858
$21$	−12.7197	−2.77566
$22$	6.45045	1.37524
$23$	−4.19248	−0.874192	−0.437096	−	0.899415i	$-0.643993\pi$
$23$	−4.19248	−0.874192	−0.437096	+	0.899415i	$0.643993\pi$
$24$	14.8431	3.02983
$25$	−3.61505	−0.723010
$26$	2.50015	0.490320
$27$	−2.52609	−0.486147
$28$	20.4979	3.87374
$29$	1.74484	0.324008	0.162004	−	0.986790i	$-0.448204\pi$
$29$	1.74484	0.324008	0.162004	+	0.986790i	$0.448204\pi$
$30$	−7.76095	−1.41695
$31$	−4.73087	−0.849690	−0.424845	−	0.905266i	$-0.639671\pi$
$31$	−4.73087	−0.849690	−0.424845	+	0.905266i	$0.639671\pi$
$32$	−2.66480	−0.471075
$33$	6.80544	1.18468
$34$	−2.89804	−0.497010
$35$	−5.67494	−0.959239
$36$	16.8231	2.80384
$37$	2.82874	0.465041	0.232521	−	0.972591i	$-0.425303\pi$
$37$	2.82874	0.465041	0.232521	+	0.972591i	$0.425303\pi$
$38$	0.322819	0.0523682
$39$	2.63774	0.422376
$40$	6.62231	1.04708
$41$	−3.67998	−0.574716	−0.287358	−	0.957823i	$-0.592777\pi$
$41$	−3.67998	−0.574716	−0.287358	+	0.957823i	$0.592777\pi$
$42$	31.8011	4.90702
$43$	6.93571	1.05769	0.528843	−	0.848720i	$-0.322626\pi$
$43$	6.93571	1.05769	0.528843	+	0.848720i	$0.322626\pi$
$44$	−10.9670	−1.65334
$45$	−4.65754	−0.694305
$46$	10.4818	1.54546
$47$	8.13246	1.18624	0.593121	−	0.805113i	$-0.297896\pi$
$47$	8.13246	1.18624	0.593121	+	0.805113i	$0.297896\pi$
$48$	−14.6852	−2.11963
$49$	16.2535	2.32193
$50$	9.03817	1.27819
$51$	−3.05753	−0.428139
$52$	−4.25075	−0.589472
$53$	0.536946	0.0737552	0.0368776	−	0.999320i	$-0.488259\pi$
$53$	0.536946	0.0737552	0.0368776	+	0.999320i	$0.488259\pi$
$54$	6.31561	0.859446
$55$	3.03627	0.409411
$56$	−27.1354	−3.62612
$57$	0.340585	0.0451116
$58$	−4.36235	−0.572805
$59$	−7.91490	−1.03043	−0.515216	−	0.857060i	$-0.672288\pi$
$59$	−7.91490	−1.03043	−0.515216	+	0.857060i	$0.672288\pi$
$60$	13.1951	1.70349
$61$	−5.98348	−0.766106	−0.383053	−	0.923726i	$-0.625127\pi$
$61$	−5.98348	−0.766106	−0.383053	+	0.923726i	$0.625127\pi$
$62$	11.8279	1.50214
$63$	19.0846	2.40444
$64$	−4.47230	−0.559037
$65$	1.17684	0.145969
$66$	−17.0146	−2.09435
$67$	−5.06585	−0.618893	−0.309446	−	0.950917i	$-0.600144\pi$
$67$	−5.06585	−0.618893	−0.309446	+	0.950917i	$0.600144\pi$
$68$	4.92724	0.597515
$69$	11.0587	1.33131
$70$	14.1882	1.69581
$71$	5.24961	0.623014	0.311507	−	0.950244i	$-0.399166\pi$
$71$	5.24961	0.623014	0.311507	+	0.950244i	$0.399166\pi$
$72$	−22.2706	−2.62462
$73$	9.44481	1.10543	0.552716	−	0.833370i	$-0.313592\pi$
$73$	9.44481	1.10543	0.552716	+	0.833370i	$0.313592\pi$
$74$	−7.07226	−0.822133
$75$	9.53557	1.10107
$76$	−0.548857	−0.0629582
$77$	−12.4414	−1.41783
$78$	−6.59474	−0.746708
$79$	9.71315	1.09281	0.546407	−	0.837519i	$-0.315995\pi$
$79$	9.71315	1.09281	0.546407	+	0.837519i	$0.315995\pi$
$80$	−6.55187	−0.732521
$81$	−5.20984	−0.578871
$82$	9.20049	1.01602
$83$	−6.03263	−0.662167	−0.331083	−	0.943601i	$-0.607414\pi$
$83$	−6.03263	−0.662167	−0.331083	+	0.943601i	$0.607414\pi$
$84$	−54.0681	−5.89932
$85$	−1.36413	−0.147960
$86$	−17.3403	−1.86985
$87$	−4.60242	−0.493432
$88$	14.5183	1.54766
$89$	−3.08020	−0.326501	−0.163250	−	0.986585i	$-0.552198\pi$
$89$	−3.08020	−0.326501	−0.163250	+	0.986585i	$0.552198\pi$
$90$	11.6445	1.22744
$91$	−4.82219	−0.505503
$92$	−17.8212	−1.85798
$93$	12.4788	1.29399
$94$	−20.3324	−2.09712
$95$	0.151953	0.0155901
$96$	7.02905	0.717400
$97$	7.01383	0.712147	0.356073	−	0.934458i	$-0.384115\pi$
$97$	7.01383	0.712147	0.356073	+	0.934458i	$0.384115\pi$
$98$	−40.6362	−4.10487
$99$	−10.2109	−1.02623
$100$	−15.3667	−1.53667
$101$	−10.5961	−1.05435	−0.527175	−	0.849757i	$-0.676749\pi$
$101$	−10.5961	−1.05435	−0.527175	+	0.849757i	$0.676749\pi$
$102$	7.64427	0.756896
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	5.62720	0.551792
$105$	14.9690	1.46083
$106$	−1.34245	−0.130390
$107$	−4.76407	−0.460560	−0.230280	−	0.973124i	$-0.573964\pi$
$107$	−4.76407	−0.460560	−0.230280	+	0.973124i	$0.573964\pi$
$108$	−10.7378	−1.03324
$109$	−15.7507	−1.50864	−0.754320	−	0.656507i	$-0.772033\pi$
$109$	−15.7507	−1.50864	−0.754320	+	0.656507i	$0.772033\pi$
$110$	−7.59114	−0.723787
$111$	−7.46147	−0.708211
$112$	26.8468	2.53678
$113$	−17.9203	−1.68580	−0.842899	−	0.538072i	$-0.819153\pi$
$113$	−17.9203	−1.68580	−0.842899	+	0.538072i	$0.819153\pi$
$114$	−0.851514	−0.0797516
$115$	4.93387	0.460085
$116$	7.41685	0.688638
$117$	−3.95767	−0.365887
$118$	19.7884	1.82167
$119$	5.58962	0.512400
$120$	−17.4679	−1.59460
$121$	−4.34346	−0.394860
$122$	14.9596	1.35438
$123$	9.70682	0.875235
$124$	−20.1097	−1.80591
$125$	10.1385	0.906817
$126$	−47.7145	−4.25074
$127$	−0.327407	−0.0290526	−0.0145263	−	0.999894i	$-0.504624\pi$
$127$	−0.327407	−0.0290526	−0.0145263	+	0.999894i	$0.504624\pi$
$128$	16.5110	1.45938
$129$	−18.2946	−1.61075
$130$	−2.94227	−0.258054
$131$	12.0556	1.05330	0.526652	−	0.850081i	$-0.323447\pi$
$131$	12.0556	1.05330	0.526652	+	0.850081i	$0.323447\pi$
$132$	28.9282	2.51788
$133$	−0.622641	−0.0539898
$134$	12.6654	1.09412
$135$	2.97280	0.255858
$136$	−6.52275	−0.559321
$137$	22.9927	1.96440	0.982198	−	0.187850i	$-0.0601519\pi$
$137$	22.9927	1.96440	0.982198	+	0.187850i	$0.0601519\pi$
$138$	−27.6483	−2.35358
$139$	−14.1803	−1.20275	−0.601377	−	0.798966i	$-0.705381\pi$
$139$	−14.1803	−1.20275	−0.601377	+	0.798966i	$0.705381\pi$
$140$	−24.1227	−2.03874
$141$	−21.4513	−1.80653
$142$	−13.1248	−1.10141
$143$	2.58003	0.215753
$144$	22.0337	1.83615
$145$	−2.05339	−0.170525
$146$	−23.6134	−1.95426
$147$	−42.8725	−3.53606
$148$	12.0242	0.988386
$149$	21.7133	1.77882	0.889410	−	0.457111i	$-0.151116\pi$
$149$	21.7133	1.77882	0.889410	+	0.457111i	$0.151116\pi$
$150$	−23.8403	−1.94656
$151$	−21.6274	−1.76002	−0.880008	−	0.474958i	$-0.842463\pi$
$151$	−21.6274	−1.76002	−0.880008	+	0.474958i	$0.842463\pi$
$152$	0.726585	0.0589338
$153$	4.58752	0.370879
$154$	31.1053	2.50654
$155$	5.56747	0.447190
$156$	11.2124	0.897708
$157$	−12.8226	−1.02335	−0.511676	−	0.859178i	$-0.670975\pi$
$157$	−12.8226	−1.02335	−0.511676	+	0.859178i	$0.670975\pi$
$158$	−24.2843	−1.93196
$159$	−1.41632	−0.112322
$160$	3.13604	0.247926
$161$	−20.2169	−1.59332
$162$	13.0254	1.02337
$163$	3.23710	0.253549	0.126775	−	0.991932i	$-0.459538\pi$
$163$	3.23710	0.253549	0.126775	+	0.991932i	$0.459538\pi$
$164$	−15.6426	−1.22149
$165$	−8.00890	−0.623492
$166$	15.0825	1.17063
$167$	−3.13468	−0.242569	−0.121285	−	0.992618i	$-0.538701\pi$
$167$	−3.13468	−0.242569	−0.121285	+	0.992618i	$0.538701\pi$
$168$	71.5762	5.52222
$169$	1.00000	0.0769231
$170$	3.41052	0.261575
$171$	−0.511015	−0.0390783
$172$	29.4819	2.24798
$173$	−16.3537	−1.24335	−0.621674	−	0.783276i	$-0.713547\pi$
$173$	−16.3537	−1.24335	−0.621674	+	0.783276i	$0.713547\pi$
$174$	11.5067	0.872324
$175$	−17.4325	−1.31777
$176$	−14.3639	−1.08272
$177$	20.8774	1.56925
$178$	7.70097	0.577212
$179$	4.50744	0.336902	0.168451	−	0.985710i	$-0.446124\pi$
$179$	4.50744	0.336902	0.168451	+	0.985710i	$0.446124\pi$
$180$	−19.7980	−1.47566
$181$	−12.2360	−0.909493	−0.454746	−	0.890621i	$-0.650270\pi$
$181$	−12.2360	−0.909493	−0.454746	+	0.890621i	$0.650270\pi$
$182$	12.0562	0.893664
$183$	15.7829	1.16670
$184$	23.5919	1.73922
$185$	−3.32896	−0.244750
$186$	−31.1989	−2.28761
$187$	−2.99063	−0.218696
$188$	34.5690	2.52121
$189$	−12.1813	−0.886059
$190$	−0.379906	−0.0275613
$191$	−4.20165	−0.304021	−0.152010	−	0.988379i	$-0.548575\pi$
$191$	−4.20165	−0.304021	−0.152010	+	0.988379i	$0.548575\pi$
$192$	11.7968	0.851357
$193$	−4.89956	−0.352678	−0.176339	−	0.984330i	$-0.556425\pi$
$193$	−4.89956	−0.352678	−0.176339	+	0.984330i	$0.556425\pi$
$194$	−17.5356	−1.25898
$195$	−3.10419	−0.222296
$196$	69.0895	4.93496
$197$	−21.1011	−1.50339	−0.751697	−	0.659509i	$-0.770764\pi$
$197$	−21.1011	−1.50339	−0.751697	+	0.659509i	$0.770764\pi$
$198$	25.5288	1.81425
$199$	25.9041	1.83629	0.918145	−	0.396244i	$-0.129686\pi$
$199$	25.9041	1.83629	0.918145	+	0.396244i	$0.129686\pi$
$200$	20.3426	1.43844
$201$	13.3624	0.942512
$202$	26.4918	1.86396
$203$	8.41393	0.590542
$204$	−12.9968	−0.909956
$205$	4.33074	0.302472
$206$	2.50015	0.174194
$207$	−16.5925	−1.15326
$208$	−5.56735	−0.386026
$209$	0.333133	0.0230433
$210$	−37.4247	−2.58255
$211$	−19.3022	−1.32881	−0.664407	−	0.747371i	$-0.731316\pi$
$211$	−19.3022	−1.32881	−0.664407	+	0.747371i	$0.731316\pi$
$212$	2.28242	0.156757
$213$	−13.8471	−0.948788
$214$	11.9109	0.814211
$215$	−8.16221	−0.556658
$216$	14.2148	0.967197
$217$	−22.8132	−1.54866
$218$	39.3790	2.66708
$219$	−24.9130	−1.68346
$220$	12.9064	0.870152
$221$	−1.15915	−0.0779726
$222$	18.6548	1.25203
$223$	18.2865	1.22455	0.612276	−	0.790644i	$-0.290254\pi$
$223$	18.2865	1.22455	0.612276	+	0.790644i	$0.290254\pi$
$224$	−12.8502	−0.858588
$225$	−14.3072	−0.953813
$226$	44.8034	2.98028
$227$	−5.44606	−0.361468	−0.180734	−	0.983532i	$-0.557847\pi$
$227$	−5.44606	−0.361468	−0.180734	+	0.983532i	$0.557847\pi$
$228$	1.44774	0.0958790
$229$	−18.8536	−1.24588	−0.622941	−	0.782269i	$-0.714062\pi$
$229$	−18.8536	−1.24588	−0.622941	+	0.782269i	$0.714062\pi$
$230$	−12.3354	−0.813372
$231$	32.8171	2.15921
$232$	−9.81854	−0.644619
$233$	23.8420	1.56194	0.780972	−	0.624566i	$-0.214724\pi$
$233$	23.8420	1.56194	0.780972	+	0.624566i	$0.214724\pi$
$234$	9.89477	0.646842
$235$	−9.57059	−0.624317
$236$	−33.6442	−2.19005
$237$	−25.6208	−1.66425
$238$	−13.9749	−0.905857
$239$	10.1793	0.658447	0.329223	−	0.944252i	$-0.393213\pi$
$239$	10.1793	0.658447	0.329223	+	0.944252i	$0.393213\pi$
$240$	17.2821	1.11556
$241$	−23.0861	−1.48711	−0.743553	−	0.668677i	$-0.766861\pi$
$241$	−23.0861	−1.48711	−0.743553	+	0.668677i	$0.766861\pi$
$242$	10.8593	0.698062
$243$	21.3205	1.36771
$244$	−25.4342	−1.62826
$245$	−19.1277	−1.22203
$246$	−24.2685	−1.54730
$247$	0.129120	0.00821571
$248$	26.6216	1.69047
$249$	15.9125	1.00841
$250$	−25.3478	−1.60314
$251$	−12.2968	−0.776167	−0.388083	−	0.921624i	$-0.626863\pi$
$251$	−12.2968	−0.776167	−0.388083	+	0.921624i	$0.626863\pi$
$252$	81.1240	5.11033
$253$	10.8167	0.680040
$254$	0.818565	0.0513614
$255$	3.59822	0.225329
$256$	−32.3354	−2.02096
$257$	−20.2943	−1.26592	−0.632962	−	0.774183i	$-0.718161\pi$
$257$	−20.2943	−1.26592	−0.632962	+	0.774183i	$0.718161\pi$
$258$	45.7392	2.84760
$259$	13.6407	0.847591
$260$	5.00244	0.310238
$261$	6.90549	0.427439
$262$	−30.1408	−1.86211
$263$	−30.2337	−1.86429	−0.932145	−	0.362086i	$-0.882065\pi$
$263$	−30.2337	−1.86429	−0.932145	+	0.362086i	$0.882065\pi$
$264$	−38.2956	−2.35693
$265$	−0.631899	−0.0388172
$266$	1.55670	0.0954472
$267$	8.12478	0.497228
$268$	−21.5337	−1.31538
$269$	−22.1638	−1.35135	−0.675674	−	0.737200i	$-0.736147\pi$
$269$	−22.1638	−1.35135	−0.675674	+	0.737200i	$0.736147\pi$
$270$	−7.43245	−0.452325
$271$	−18.5325	−1.12577	−0.562883	−	0.826536i	$-0.690308\pi$
$271$	−18.5325	−1.12577	−0.562883	+	0.826536i	$0.690308\pi$
$272$	6.45337	0.391293
$273$	12.7197	0.769830
$274$	−57.4851	−3.47280
$275$	9.32693	0.562435
$276$	47.0076	2.82952
$277$	−1.80255	−0.108305	−0.0541524	−	0.998533i	$-0.517246\pi$
$277$	−1.80255	−0.108305	−0.0541524	+	0.998533i	$0.517246\pi$
$278$	35.4528	2.12631
$279$	−18.7233	−1.12093
$280$	31.9340	1.90842
$281$	8.62480	0.514512	0.257256	−	0.966343i	$-0.417182\pi$
$281$	8.62480	0.514512	0.257256	+	0.966343i	$0.417182\pi$
$282$	53.6315	3.19371
$283$	25.3858	1.50903	0.754515	−	0.656283i	$-0.227872\pi$
$283$	25.3858	1.50903	0.754515	+	0.656283i	$0.227872\pi$
$284$	22.3147	1.32414
$285$	−0.400814	−0.0237422
$286$	−6.45045	−0.381423
$287$	−17.7455	−1.04749
$288$	−10.5464	−0.621453
$289$	−15.6564	−0.920964
$290$	5.13378	0.301466
$291$	−18.5007	−1.08453
$292$	40.1475	2.34945
$293$	−1.99265	−0.116412	−0.0582058	−	0.998305i	$-0.518538\pi$
$293$	−1.99265	−0.116412	−0.0582058	+	0.998305i	$0.518538\pi$
$294$	107.188	6.25131
$295$	9.31456	0.542315
$296$	−15.9179	−0.925207
$297$	6.51739	0.378177
$298$	−54.2864	−3.14473
$299$	4.19248	0.242457
$300$	40.5333	2.34019
$301$	33.4453	1.92775
$302$	54.0718	3.11148
$303$	27.9497	1.60567
$304$	−0.718856	−0.0412292
$305$	7.04159	0.403200
$306$	−11.4695	−0.655667
$307$	10.1784	0.580913	0.290456	−	0.956888i	$-0.406193\pi$
$307$	10.1784	0.580913	0.290456	+	0.956888i	$0.406193\pi$
$308$	−52.8851	−3.01341
$309$	2.63774	0.150056
$310$	−13.9195	−0.790575
$311$	−1.30324	−0.0738998	−0.0369499	−	0.999317i	$-0.511764\pi$
$311$	−1.30324	−0.0738998	−0.0369499	+	0.999317i	$0.511764\pi$
$312$	−14.8431	−0.840325
$313$	−29.7755	−1.68301	−0.841505	−	0.540249i	$-0.818330\pi$
$313$	−29.7755	−1.68301	−0.841505	+	0.540249i	$0.818330\pi$
$314$	32.0583	1.80916
$315$	−22.4595	−1.26545
$316$	41.2881	2.32264
$317$	12.7389	0.715487	0.357744	−	0.933820i	$-0.383546\pi$
$317$	12.7389	0.715487	0.357744	+	0.933820i	$0.383546\pi$
$318$	3.54102	0.198571
$319$	−4.50172	−0.252048
$320$	5.26317	0.294220
$321$	12.5664	0.701386
$322$	50.5453	2.81678
$323$	−0.149669	−0.00832781
$324$	−22.1457	−1.23032
$325$	3.61505	0.200527
$326$	−8.09323	−0.448243
$327$	41.5461	2.29751
$328$	20.7080	1.14341
$329$	39.2163	2.16206
$330$	20.0235	1.10225
$331$	18.7518	1.03069	0.515347	−	0.856982i	$-0.327663\pi$
$331$	18.7518	1.03069	0.515347	+	0.856982i	$0.327663\pi$
$332$	−25.6432	−1.40735
$333$	11.1952	0.613494
$334$	7.83718	0.428831
$335$	5.96169	0.325722
$336$	−70.8149	−3.86327
$337$	11.2071	0.610489	0.305245	−	0.952274i	$-0.401262\pi$
$337$	11.2071	0.610489	0.305245	+	0.952274i	$0.401262\pi$
$338$	−2.50015	−0.135990
$339$	47.2690	2.56730
$340$	−5.79856	−0.314471
$341$	12.2058	0.660980
$342$	1.27761	0.0690854
$343$	44.6221	2.40937
$344$	−39.0286	−2.10428
$345$	−13.0143	−0.700664
$346$	40.8867	2.19808
$347$	−19.7763	−1.06165	−0.530823	−	0.847483i	$-0.678117\pi$
$347$	−19.7763	−1.06165	−0.530823	+	0.847483i	$0.678117\pi$
$348$	−19.5637	−1.04873
$349$	−23.9380	−1.28137	−0.640687	−	0.767802i	$-0.721350\pi$
$349$	−23.9380	−1.28137	−0.640687	+	0.767802i	$0.721350\pi$
$350$	43.5837	2.32965
$351$	2.52609	0.134833
$352$	6.87526	0.366453
$353$	−7.18444	−0.382389	−0.191194	−	0.981552i	$-0.561236\pi$
$353$	−7.18444	−0.382389	−0.191194	+	0.981552i	$0.561236\pi$
$354$	−52.1967	−2.77423
$355$	−6.17794	−0.327891
$356$	−13.0932	−0.693936
$357$	−14.7440	−0.780334
$358$	−11.2693	−0.595599
$359$	29.8840	1.57722	0.788608	−	0.614896i	$-0.210802\pi$
$359$	29.8840	1.57722	0.788608	+	0.614896i	$0.210802\pi$
$360$	26.2089	1.38133
$361$	−18.9833	−0.999123
$362$	30.5918	1.60787
$363$	11.4569	0.601332
$364$	−20.4979	−1.07438
$365$	−11.1150	−0.581786
$366$	−39.4595	−2.06258
$367$	23.7836	1.24150	0.620748	−	0.784010i	$-0.286829\pi$
$367$	23.7836	1.24150	0.620748	+	0.784010i	$0.286829\pi$
$368$	−23.3410	−1.21673
$369$	−14.5641	−0.758179
$370$	8.32291	0.432687
$371$	2.58926	0.134427
$372$	53.0443	2.75022
$373$	−22.5736	−1.16881	−0.584407	−	0.811460i	$-0.698673\pi$
$373$	−22.5736	−1.16881	−0.584407	+	0.811460i	$0.698673\pi$
$374$	7.47702	0.386627
$375$	−26.7428	−1.38099
$376$	−45.7630	−2.36005
$377$	−1.74484	−0.0898636
$378$	30.4551	1.56644
$379$	−8.31755	−0.427244	−0.213622	−	0.976916i	$-0.568526\pi$
$379$	−8.31755	−0.427244	−0.213622	+	0.976916i	$0.568526\pi$
$380$	0.645916	0.0331348
$381$	0.863614	0.0442443
$382$	10.5047	0.537469
$383$	13.3184	0.680539	0.340269	−	0.940328i	$-0.389482\pi$
$383$	13.3184	0.680539	0.340269	+	0.940328i	$0.389482\pi$
$384$	−43.5518	−2.22249
$385$	14.6415	0.746199
$386$	12.2496	0.623489
$387$	27.4493	1.39532
$388$	29.8140	1.51358
$389$	−32.4648	−1.64603	−0.823014	−	0.568021i	$-0.807709\pi$
$389$	−32.4648	−1.64603	−0.823014	+	0.568021i	$0.807709\pi$
$390$	7.76095	0.392991
$391$	−4.85969	−0.245765
$392$	−91.4617	−4.61951
$393$	−31.7996	−1.60408
$394$	52.7560	2.65781
$395$	−11.4308	−0.575146
$396$	−43.4040	−2.18113
$397$	−6.55244	−0.328857	−0.164429	−	0.986389i	$-0.552578\pi$
$397$	−6.55244	−0.328857	−0.164429	+	0.986389i	$0.552578\pi$
$398$	−64.7640	−3.24633
$399$	1.64237	0.0822211
$400$	−20.1263	−1.00631
$401$	2.12447	0.106091	0.0530456	−	0.998592i	$-0.483107\pi$
$401$	2.12447	0.106091	0.0530456	+	0.998592i	$0.483107\pi$
$402$	−33.4080	−1.66624
$403$	4.73087	0.235662
$404$	−45.0412	−2.24089
$405$	6.13114	0.304659
$406$	−21.0361	−1.04400
$407$	−7.29821	−0.361759
$408$	17.2053	0.851790
$409$	1.12837	0.0557945	0.0278972	−	0.999611i	$-0.491119\pi$
$409$	1.12837	0.0557945	0.0278972	+	0.999611i	$0.491119\pi$
$410$	−10.8275	−0.534732
$411$	−60.6487	−2.99158
$412$	−4.25075	−0.209419
$413$	−38.1671	−1.87808
$414$	41.4836	2.03881
$415$	7.09943	0.348497
$416$	2.66480	0.130653
$417$	37.4038	1.83167
$418$	−0.832883	−0.0407376
$419$	19.6665	0.960771	0.480385	−	0.877058i	$-0.340497\pi$
$419$	19.6665	0.960771	0.480385	+	0.877058i	$0.340497\pi$
$420$	63.6294	3.10480
$421$	19.5367	0.952162	0.476081	−	0.879401i	$-0.342057\pi$
$421$	19.5367	0.952162	0.476081	+	0.879401i	$0.342057\pi$
$422$	48.2583	2.34918
$423$	32.1856	1.56492
$424$	−3.02150	−0.146737
$425$	−4.19037	−0.203263
$426$	34.6198	1.67734
$427$	−28.8535	−1.39632
$428$	−20.2508	−0.978861
$429$	−6.80544	−0.328570
$430$	20.4067	0.984100
$431$	−6.55915	−0.315943	−0.157972	−	0.987444i	$-0.550495\pi$
$431$	−6.55915	−0.315943	−0.157972	+	0.987444i	$0.550495\pi$
$432$	−14.0636	−0.676637
$433$	−37.3803	−1.79638	−0.898191	−	0.439606i	$-0.855118\pi$
$433$	−37.3803	−1.79638	−0.898191	+	0.439606i	$0.855118\pi$
$434$	57.0363	2.73783
$435$	5.41631	0.259692
$436$	−66.9521	−3.20642
$437$	0.541333	0.0258955
$438$	62.2861	2.97615
$439$	5.06754	0.241860	0.120930	−	0.992661i	$-0.461412\pi$
$439$	5.06754	0.241860	0.120930	+	0.992661i	$0.461412\pi$
$440$	−17.0857	−0.814530
$441$	64.3260	3.06314
$442$	2.89804	0.137846
$443$	−1.25634	−0.0596905	−0.0298452	−	0.999555i	$-0.509501\pi$
$443$	−1.25634	−0.0596905	−0.0298452	+	0.999555i	$0.509501\pi$
$444$	−31.7168	−1.50521
$445$	3.62490	0.171837
$446$	−45.7189	−2.16485
$447$	−57.2739	−2.70896
$448$	−21.5663	−1.01891
$449$	7.46156	0.352133	0.176066	−	0.984378i	$-0.443663\pi$
$449$	7.46156	0.352133	0.176066	+	0.984378i	$0.443663\pi$
$450$	35.7701	1.68622
$451$	9.49444	0.447076
$452$	−76.1745	−3.58295
$453$	57.0476	2.68033
$454$	13.6160	0.639029
$455$	5.67494	0.266045
$456$	−1.91654	−0.0897503
$457$	−35.9580	−1.68205	−0.841023	−	0.540999i	$-0.818046\pi$
$457$	−35.9580	−1.68205	−0.841023	+	0.540999i	$0.818046\pi$
$458$	47.1368	2.20256
$459$	−2.92811	−0.136673
$460$	20.9726	0.977853
$461$	17.5630	0.817992	0.408996	−	0.912536i	$-0.365879\pi$
$461$	17.5630	0.817992	0.408996	+	0.912536i	$0.365879\pi$
$462$	−82.0477	−3.81720
$463$	−11.6790	−0.542768	−0.271384	−	0.962471i	$-0.587481\pi$
$463$	−11.6790	−0.542768	−0.271384	+	0.962471i	$0.587481\pi$
$464$	9.71411	0.450966
$465$	−14.6856	−0.681026
$466$	−59.6086	−2.76132
$467$	−37.6846	−1.74384	−0.871918	−	0.489653i	$-0.837124\pi$
$467$	−37.6846	−1.74384	−0.871918	+	0.489653i	$0.837124\pi$
$468$	−16.8231	−0.777646
$469$	−24.4285	−1.12800
$470$	23.9279	1.10371
$471$	33.8226	1.55846
$472$	44.5387	2.05006
$473$	−17.8943	−0.822781
$474$	64.0557	2.94218
$475$	0.466776	0.0214171
$476$	23.7601	1.08904
$477$	2.12506	0.0972997
$478$	−25.4499	−1.16405
$479$	−28.9027	−1.32060	−0.660299	−	0.751003i	$-0.729570\pi$
$479$	−28.9027	−1.32060	−0.660299	+	0.751003i	$0.729570\pi$
$480$	−8.27206	−0.377566
$481$	−2.82874	−0.128979
$482$	57.7187	2.62901
$483$	53.3269	2.42646
$484$	−18.4629	−0.839225
$485$	−8.25415	−0.374802
$486$	−53.3044	−2.41794
$487$	−11.6120	−0.526191	−0.263095	−	0.964770i	$-0.584743\pi$
$487$	−11.6120	−0.526191	−0.263095	+	0.964770i	$0.584743\pi$
$488$	33.6702	1.52418
$489$	−8.53863	−0.386130
$490$	47.8222	2.16039
$491$	7.45983	0.336657	0.168329	−	0.985731i	$-0.446163\pi$
$491$	7.45983	0.336657	0.168329	+	0.985731i	$0.446163\pi$
$492$	41.2612	1.86020
$493$	2.02252	0.0910897
$494$	−0.322819	−0.0145243
$495$	12.0166	0.540105
$496$	−26.3384	−1.18263
$497$	25.3146	1.13551
$498$	−39.7836	−1.78275
$499$	20.9951	0.939872	0.469936	−	0.882701i	$-0.344277\pi$
$499$	20.9951	0.939872	0.469936	+	0.882701i	$0.344277\pi$
$500$	43.0963	1.92732
$501$	8.26848	0.369409
$502$	30.7438	1.37216
$503$	−21.6741	−0.966402	−0.483201	−	0.875509i	$-0.660526\pi$
$503$	−21.6741	−0.966402	−0.483201	+	0.875509i	$0.660526\pi$
$504$	−107.393	−4.78367
$505$	12.4699	0.554902
$506$	−27.0434	−1.20222
$507$	−2.63774	−0.117146
$508$	−1.39172	−0.0617477
$509$	−32.6790	−1.44847	−0.724235	−	0.689553i	$-0.757807\pi$
$509$	−32.6790	−1.44847	−0.724235	+	0.689553i	$0.757807\pi$
$510$	−8.99607	−0.398353
$511$	45.5446	2.01478
$512$	47.8213	2.11342
$513$	0.326169	0.0144007
$514$	50.7388	2.23799
$515$	1.17684	0.0518577
$516$	−77.7657	−3.42344
$517$	−20.9820	−0.922786
$518$	−34.1038	−1.49843
$519$	43.1368	1.89350
$520$	−6.62231	−0.290407
$521$	−40.7143	−1.78373	−0.891864	−	0.452304i	$-0.850602\pi$
$521$	−40.7143	−1.78373	−0.891864	+	0.452304i	$0.850602\pi$
$522$	−17.2648	−0.755658
$523$	−22.9532	−1.00367	−0.501836	−	0.864963i	$-0.667342\pi$
$523$	−22.9532	−1.00367	−0.501836	+	0.864963i	$0.667342\pi$
$524$	51.2454	2.23866
$525$	45.9823	2.00683
$526$	75.5887	3.29583
$527$	−5.48378	−0.238877
$528$	37.8883	1.64888
$529$	−5.42314	−0.235789
$530$	1.57984	0.0686239
$531$	−31.3246	−1.35937
$532$	−2.64669	−0.114749
$533$	3.67998	0.159397
$534$	−20.3132	−0.879036
$535$	5.60653	0.242392
$536$	28.5066	1.23130
$537$	−11.8894	−0.513067
$538$	55.4127	2.38901
$539$	−41.9345	−1.80624
$540$	12.6366	0.543794
$541$	8.83756	0.379956	0.189978	−	0.981788i	$-0.439158\pi$
$541$	8.83756	0.379956	0.189978	+	0.981788i	$0.439158\pi$
$542$	46.3339	1.99021
$543$	32.2753	1.38507
$544$	−3.08890	−0.132435
$545$	18.5360	0.793994
$546$	−31.8011	−1.36096
$547$	−20.6480	−0.882843	−0.441421	−	0.897300i	$-0.645526\pi$
$547$	−20.6480	−0.882843	−0.441421	+	0.897300i	$0.645526\pi$
$548$	97.7359	4.17507
$549$	−23.6807	−1.01067
$550$	−23.3187	−0.994313
$551$	−0.225293	−0.00959782
$552$	−62.2293	−2.64866
$553$	46.8386	1.99178
$554$	4.50665	0.191469
$555$	8.78094	0.372730
$556$	−60.2767	−2.55630
$557$	4.75539	0.201492	0.100746	−	0.994912i	$-0.467877\pi$
$557$	4.75539	0.201492	0.100746	+	0.994912i	$0.467877\pi$
$558$	46.8109	1.98166
$559$	−6.93571	−0.293349
$560$	−31.5943	−1.33510
$561$	7.88850	0.333053
$562$	−21.5633	−0.909592
$563$	−44.8101	−1.88852	−0.944261	−	0.329198i	$-0.893222\pi$
$563$	−44.8101	−1.88852	−0.944261	+	0.329198i	$0.893222\pi$
$564$	−91.1841	−3.83954
$565$	21.0893	0.887232
$566$	−63.4683	−2.66777
$567$	−25.1228	−1.05506
$568$	−29.5406	−1.23950
$569$	−12.5797	−0.527368	−0.263684	−	0.964609i	$-0.584938\pi$
$569$	−12.5797	−0.527368	−0.263684	+	0.964609i	$0.584938\pi$
$570$	1.00209	0.0419731
$571$	−25.5224	−1.06808	−0.534041	−	0.845459i	$-0.679327\pi$
$571$	−25.5224	−1.06808	−0.534041	+	0.845459i	$0.679327\pi$
$572$	10.9670	0.458555
$573$	11.0829	0.462993
$574$	44.3665	1.85182
$575$	15.1560	0.632050
$576$	−17.6999	−0.737495
$577$	−21.6466	−0.901159	−0.450580	−	0.892736i	$-0.648783\pi$
$577$	−21.6466	−0.901159	−0.450580	+	0.892736i	$0.648783\pi$
$578$	39.1433	1.62815
$579$	12.9238	0.537093
$580$	−8.72844	−0.362429
$581$	−29.0905	−1.20688
$582$	46.2544	1.91731
$583$	−1.38534	−0.0573747
$584$	−53.1478	−2.19927
$585$	4.65754	0.192566
$586$	4.98191	0.205801
$587$	20.6346	0.851682	0.425841	−	0.904798i	$-0.359978\pi$
$587$	20.6346	0.851682	0.425841	+	0.904798i	$0.359978\pi$
$588$	−182.240	−7.51545
$589$	0.610851	0.0251697
$590$	−23.2878	−0.958743
$591$	55.6593	2.28952
$592$	15.7486	0.647262
$593$	8.82209	0.362280	0.181140	−	0.983457i	$-0.442021\pi$
$593$	8.82209	0.362280	0.181140	+	0.983457i	$0.442021\pi$
$594$	−16.2944	−0.668569
$595$	−6.57808	−0.269675
$596$	92.2975	3.78065
$597$	−68.3282	−2.79649
$598$	−10.4818	−0.428633
$599$	−1.58475	−0.0647510	−0.0323755	−	0.999476i	$-0.510307\pi$
$599$	−1.58475	−0.0647510	−0.0323755	+	0.999476i	$0.510307\pi$
$600$	−53.6585	−2.19060
$601$	−0.733816	−0.0299330	−0.0149665	−	0.999888i	$-0.504764\pi$
$601$	−0.733816	−0.0299330	−0.0149665	+	0.999888i	$0.504764\pi$
$602$	−83.6182	−3.40802
$603$	−20.0490	−0.816458
$604$	−91.9328	−3.74069
$605$	5.11155	0.207814
$606$	−69.8784	−2.83862
$607$	18.8963	0.766976	0.383488	−	0.923546i	$-0.374723\pi$
$607$	18.8963	0.766976	0.383488	+	0.923546i	$0.374723\pi$
$608$	0.344079	0.0139543
$609$	−22.1938	−0.899336
$610$	−17.6050	−0.712806
$611$	−8.13246	−0.329004
$612$	19.5004	0.788257
$613$	3.04907	0.123151	0.0615753	−	0.998102i	$-0.480388\pi$
$613$	3.04907	0.123151	0.0615753	+	0.998102i	$0.480388\pi$
$614$	−25.4476	−1.02698
$615$	−11.4234	−0.460634
$616$	70.0101	2.82079
$617$	20.0009	0.805205	0.402603	−	0.915375i	$-0.368106\pi$
$617$	20.0009	0.805205	0.402603	+	0.915375i	$0.368106\pi$
$618$	−6.59474	−0.265280
$619$	−40.0787	−1.61090	−0.805450	−	0.592664i	$-0.798076\pi$
$619$	−40.0787	−1.61090	−0.805450	+	0.592664i	$0.798076\pi$
$620$	23.6659	0.950446
$621$	10.5906	0.424986
$622$	3.25829	0.130645
$623$	−14.8533	−0.595086
$624$	14.6852	0.587879
$625$	6.14385	0.245754
$626$	74.4432	2.97535
$627$	−0.878719	−0.0350927
$628$	−54.5055	−2.17501
$629$	3.27892	0.130739
$630$	56.1522	2.23716
$631$	15.5540	0.619196	0.309598	−	0.950868i	$-0.399806\pi$
$631$	15.5540	0.619196	0.309598	+	0.950868i	$0.399806\pi$
$632$	−54.6579	−2.17417
$633$	50.9141	2.02365
$634$	−31.8491	−1.26489
$635$	0.385305	0.0152904
$636$	−6.02044	−0.238726
$637$	−16.2535	−0.643987
$638$	11.2550	0.445589
$639$	20.7762	0.821895
$640$	−19.4308	−0.768069
$641$	42.8849	1.69385	0.846926	−	0.531711i	$-0.178451\pi$
$641$	42.8849	1.69385	0.846926	+	0.531711i	$0.178451\pi$
$642$	−31.4178	−1.23996
$643$	−10.8806	−0.429089	−0.214544	−	0.976714i	$-0.568827\pi$
$643$	−10.8806	−0.429089	−0.214544	+	0.976714i	$0.568827\pi$
$644$	−85.9369	−3.38639
$645$	21.5298	0.847734
$646$	0.374195	0.0147225
$647$	46.5230	1.82901	0.914503	−	0.404578i	$-0.132582\pi$
$647$	46.5230	1.82901	0.914503	+	0.404578i	$0.132582\pi$
$648$	29.3168	1.15167
$649$	20.4207	0.801581
$650$	−9.03817	−0.354506
$651$	60.1752	2.35845
$652$	13.7601	0.538887
$653$	−10.8693	−0.425349	−0.212674	−	0.977123i	$-0.568217\pi$
$653$	−10.8693	−0.425349	−0.212674	+	0.977123i	$0.568217\pi$
$654$	−103.872	−4.06170
$655$	−14.1875	−0.554352
$656$	−20.4877	−0.799911
$657$	37.3795	1.45831
$658$	−98.0465	−3.82225
$659$	−7.65806	−0.298316	−0.149158	−	0.988813i	$-0.547656\pi$
$659$	−7.65806	−0.298316	−0.149158	+	0.988813i	$0.547656\pi$
$660$	−34.0438	−1.32515
$661$	27.6720	1.07632	0.538158	−	0.842844i	$-0.319120\pi$
$661$	27.6720	1.07632	0.538158	+	0.842844i	$0.319120\pi$
$662$	−46.8824	−1.82214
$663$	3.05753	0.118745
$664$	33.9468	1.31739
$665$	0.732748	0.0284148
$666$	−27.9897	−1.08458
$667$	−7.31518	−0.283245
$668$	−13.3247	−0.515550
$669$	−48.2349	−1.86487
$670$	−14.9051	−0.575835
$671$	15.4375	0.595959
$672$	33.8954	1.30754
$673$	−26.7900	−1.03268	−0.516340	−	0.856384i	$-0.672706\pi$
$673$	−26.7900	−1.03268	−0.516340	+	0.856384i	$0.672706\pi$
$674$	−28.0194	−1.07927
$675$	9.13196	0.351489
$676$	4.25075	0.163490
$677$	−1.15836	−0.0445196	−0.0222598	−	0.999752i	$-0.507086\pi$
$677$	−1.15836	−0.0445196	−0.0222598	+	0.999752i	$0.507086\pi$
$678$	−118.180	−4.53866
$679$	33.8220	1.29797
$680$	7.67622	0.294370
$681$	14.3653	0.550479
$682$	−30.5163	−1.16853
$683$	19.7643	0.756260	0.378130	−	0.925752i	$-0.376567\pi$
$683$	19.7643	0.756260	0.378130	+	0.925752i	$0.376567\pi$
$684$	−2.17220	−0.0830560
$685$	−27.0586	−1.03386
$686$	−111.562	−4.25945
$687$	49.7309	1.89735
$688$	38.6135	1.47213
$689$	−0.536946	−0.0204560
$690$	32.5376	1.23868
$691$	48.1110	1.83023	0.915114	−	0.403195i	$-0.132100\pi$
$691$	48.1110	1.83023	0.915114	+	0.403195i	$0.132100\pi$
$692$	−69.5154	−2.64258
$693$	−49.2389	−1.87043
$694$	49.4437	1.87686
$695$	16.6879	0.633007
$696$	25.8988	0.981690
$697$	−4.26563	−0.161572
$698$	59.8486	2.26530
$699$	−62.8891	−2.37868
$700$	−74.1009	−2.80075
$701$	13.6992	0.517412	0.258706	−	0.965956i	$-0.416704\pi$
$701$	13.6992	0.517412	0.258706	+	0.965956i	$0.416704\pi$
$702$	−6.31561	−0.238367
$703$	−0.365246	−0.0137755
$704$	11.5386	0.434879
$705$	25.2447	0.950772
$706$	17.9622	0.676015
$707$	−51.0963	−1.92167
$708$	88.7447	3.33523
$709$	44.2748	1.66278	0.831388	−	0.555692i	$-0.187547\pi$
$709$	44.2748	1.66278	0.831388	+	0.555692i	$0.187547\pi$
$710$	15.4458	0.579670
$711$	38.4415	1.44167
$712$	17.3329	0.649579
$713$	19.8341	0.742792
$714$	36.8621	1.37953
$715$	−3.03627	−0.113550
$716$	19.1600	0.716042
$717$	−26.8504	−1.00275
$718$	−74.7144	−2.78832
$719$	10.0250	0.373871	0.186936	−	0.982372i	$-0.440144\pi$
$719$	10.0250	0.373871	0.186936	+	0.982372i	$0.440144\pi$
$720$	−25.9302	−0.966360
$721$	−4.82219	−0.179588
$722$	47.4612	1.76632
$723$	60.8951	2.26471
$724$	−52.0120	−1.93301
$725$	−6.30767	−0.234261
$726$	−28.6440	−1.06308
$727$	45.1921	1.67608	0.838042	−	0.545606i	$-0.183700\pi$
$727$	45.1921	1.67608	0.838042	+	0.545606i	$0.183700\pi$
$728$	27.1354	1.00571
$729$	−40.6084	−1.50401
$730$	27.7892	1.02852
$731$	8.03950	0.297352
$732$	67.0889	2.47968
$733$	−0.815037	−0.0301041	−0.0150520	−	0.999887i	$-0.504791\pi$
$733$	−0.815037	−0.0301041	−0.0150520	+	0.999887i	$0.504791\pi$
$734$	−59.4626	−2.19481
$735$	50.4540	1.86102
$736$	11.1721	0.411810
$737$	13.0700	0.481441
$738$	36.4125	1.34036
$739$	22.7545	0.837036	0.418518	−	0.908209i	$-0.362550\pi$
$739$	22.7545	0.837036	0.418518	+	0.908209i	$0.362550\pi$
$740$	−14.1506	−0.520186
$741$	−0.340585	−0.0125117
$742$	−6.47352	−0.237651
$743$	21.3031	0.781536	0.390768	−	0.920489i	$-0.372210\pi$
$743$	21.3031	0.781536	0.390768	+	0.920489i	$0.372210\pi$
$744$	−70.2208	−2.57442
$745$	−25.5530	−0.936189
$746$	56.4373	2.06632
$747$	−23.8752	−0.873547
$748$	−12.7124	−0.464812
$749$	−22.9732	−0.839423
$750$	66.8610	2.44142
$751$	50.2339	1.83306	0.916530	−	0.399966i	$-0.130978\pi$
$751$	50.2339	1.83306	0.916530	+	0.399966i	$0.130978\pi$
$752$	45.2763	1.65106
$753$	32.4357	1.18202
$754$	4.36235	0.158867
$755$	25.4520	0.926293
$756$	−51.7796	−1.88321
$757$	19.4975	0.708649	0.354324	−	0.935123i	$-0.384711\pi$
$757$	19.4975	0.708649	0.354324	+	0.935123i	$0.384711\pi$
$758$	20.7951	0.755312
$759$	−28.5316	−1.03563
$760$	−0.855073	−0.0310167
$761$	−6.38849	−0.231583	−0.115791	−	0.993274i	$-0.536940\pi$
$761$	−6.38849	−0.231583	−0.115791	+	0.993274i	$0.536940\pi$
$762$	−2.15916	−0.0782182
$763$	−75.9526	−2.74967
$764$	−17.8601	−0.646157
$765$	−5.39877	−0.195193
$766$	−33.2980	−1.20311
$767$	7.91490	0.285790
$768$	85.2924	3.07772
$769$	13.2904	0.479264	0.239632	−	0.970864i	$-0.422973\pi$
$769$	13.2904	0.479264	0.239632	+	0.970864i	$0.422973\pi$
$770$	−36.6059	−1.31919
$771$	53.5311	1.92788
$772$	−20.8268	−0.749572
$773$	−34.0591	−1.22502	−0.612510	−	0.790463i	$-0.709840\pi$
$773$	−34.0591	−1.22502	−0.612510	+	0.790463i	$0.709840\pi$
$774$	−68.6272	−2.46676
$775$	17.1024	0.614335
$776$	−39.4682	−1.41683
$777$	−35.9806	−1.29080
$778$	81.1667	2.90997
$779$	0.475159	0.0170243
$780$	−13.1951	−0.472462
$781$	−13.5441	−0.484647
$782$	12.1500	0.434482
$783$	−4.40762	−0.157515
$784$	90.4889	3.23175
$785$	15.0901	0.538589
$786$	79.5037	2.83580
$787$	3.32075	0.118372	0.0591860	−	0.998247i	$-0.481149\pi$
$787$	3.32075	0.118372	0.0591860	+	0.998247i	$0.481149\pi$
$788$	−89.6955	−3.19527
$789$	79.7486	2.83913
$790$	28.5787	1.01679
$791$	−86.4149	−3.07256
$792$	57.4588	2.04171
$793$	5.98348	0.212480
$794$	16.3821	0.581378
$795$	1.66679	0.0591148
$796$	110.112	3.90280
$797$	48.4010	1.71445	0.857226	−	0.514940i	$-0.172186\pi$
$797$	48.4010	1.71445	0.857226	+	0.514940i	$0.172186\pi$
$798$	−4.10616	−0.145356
$799$	9.42671	0.333493
$800$	9.63340	0.340592
$801$	−12.1904	−0.430728
$802$	−5.31150	−0.187556
$803$	−24.3679	−0.859923
$804$	56.8002	2.00319
$805$	23.7920	0.838559
$806$	−11.8279	−0.416620
$807$	58.4623	2.05797
$808$	59.6263	2.09764
$809$	8.71397	0.306367	0.153183	−	0.988198i	$-0.451047\pi$
$809$	8.71397	0.306367	0.153183	+	0.988198i	$0.451047\pi$
$810$	−15.3288	−0.538598
$811$	−42.5055	−1.49257	−0.746285	−	0.665627i	$-0.768164\pi$
$811$	−42.5055	−1.49257	−0.746285	+	0.665627i	$0.768164\pi$
$812$	35.7655	1.25512
$813$	48.8838	1.71443
$814$	18.2466	0.639544
$815$	−3.80954	−0.133442
$816$	−17.0223	−0.595900
$817$	−0.895539	−0.0313309
$818$	−2.82110	−0.0986375
$819$	−19.0846	−0.666871
$820$	18.4089	0.642866
$821$	26.5527	0.926695	0.463347	−	0.886177i	$-0.346648\pi$
$821$	26.5527	0.926695	0.463347	+	0.886177i	$0.346648\pi$
$822$	151.631	5.28873
$823$	19.8002	0.690192	0.345096	−	0.938567i	$-0.387846\pi$
$823$	19.8002	0.690192	0.345096	+	0.938567i	$0.387846\pi$
$824$	5.62720	0.196033
$825$	−24.6020	−0.856532
$826$	95.4235	3.32021
$827$	9.51769	0.330963	0.165481	−	0.986213i	$-0.447082\pi$
$827$	9.51769	0.330963	0.165481	+	0.986213i	$0.447082\pi$
$828$	−70.5303	−2.45110
$829$	−33.8044	−1.17408	−0.587038	−	0.809559i	$-0.699706\pi$
$829$	−33.8044	−1.17408	−0.587038	+	0.809559i	$0.699706\pi$
$830$	−17.7496	−0.616099
$831$	4.75466	0.164937
$832$	4.47230	0.155049
$833$	18.8402	0.652774
$834$	−93.5151	−3.23816
$835$	3.68902	0.127664
$836$	1.41606	0.0489756
$837$	11.9506	0.413074
$838$	−49.1691	−1.69852
$839$	12.6195	0.435673	0.217837	−	0.975985i	$-0.430100\pi$
$839$	12.6195	0.435673	0.217837	+	0.975985i	$0.430100\pi$
$840$	−84.2336	−2.90634
$841$	−25.9555	−0.895019
$842$	−48.8448	−1.68330
$843$	−22.7500	−0.783551
$844$	−82.0485	−2.82423
$845$	−1.17684	−0.0404845
$846$	−80.4689	−2.76658
$847$	−20.9450	−0.719678
$848$	2.98937	0.102655
$849$	−66.9612	−2.29810
$850$	10.4766	0.359343
$851$	−11.8594	−0.406535
$852$	−58.8605	−2.01653
$853$	−31.5223	−1.07930	−0.539652	−	0.841888i	$-0.681444\pi$
$853$	−31.5223	−1.07930	−0.539652	+	0.841888i	$0.681444\pi$
$854$	72.1379	2.46851
$855$	0.601382	0.0205668
$856$	26.8084	0.916291
$857$	−23.7259	−0.810460	−0.405230	−	0.914215i	$-0.632809\pi$
$857$	−23.7259	−0.810460	−0.405230	+	0.914215i	$0.632809\pi$
$858$	17.0146	0.580869
$859$	−5.26941	−0.179790	−0.0898949	−	0.995951i	$-0.528653\pi$
$859$	−5.26941	−0.179790	−0.0898949	+	0.995951i	$0.528653\pi$
$860$	−34.6955	−1.18311
$861$	46.8081	1.59522
$862$	16.3989	0.558547
$863$	20.0654	0.683034	0.341517	−	0.939876i	$-0.389059\pi$
$863$	20.0654	0.683034	0.341517	+	0.939876i	$0.389059\pi$
$864$	6.73154	0.229012
$865$	19.2457	0.654372
$866$	93.4563	3.17577
$867$	41.2975	1.40254
$868$	−96.9730	−3.29148
$869$	−25.0602	−0.850109
$870$	−13.5416	−0.459103
$871$	5.06585	0.171650
$872$	88.6321	3.00146
$873$	27.7585	0.939482
$874$	−1.35341	−0.0457799
$875$	48.8899	1.65278
$876$	−105.899	−3.57798
$877$	−26.4821	−0.894238	−0.447119	−	0.894475i	$-0.647550\pi$
$877$	−26.4821	−0.894238	−0.447119	+	0.894475i	$0.647550\pi$
$878$	−12.6696	−0.427578
$879$	5.25608	0.177283
$880$	16.9040	0.569834
$881$	4.16565	0.140344	0.0701721	−	0.997535i	$-0.477645\pi$
$881$	4.16565	0.140344	0.0701721	+	0.997535i	$0.477645\pi$
$882$	−160.825	−5.41525
$883$	33.6606	1.13277	0.566385	−	0.824141i	$-0.308342\pi$
$883$	33.6606	1.13277	0.566385	+	0.824141i	$0.308342\pi$
$884$	−4.92724	−0.165721
$885$	−24.5694	−0.825891
$886$	3.14103	0.105525
$887$	−33.3886	−1.12108	−0.560540	−	0.828127i	$-0.689407\pi$
$887$	−33.3886	−1.12108	−0.560540	+	0.828127i	$0.689407\pi$
$888$	41.9872	1.40900
$889$	−1.57882	−0.0529518
$890$	−9.06279	−0.303786
$891$	13.4415	0.450308
$892$	77.7311	2.60263
$893$	−1.05006	−0.0351391
$894$	143.193	4.78910
$895$	−5.30452	−0.177311
$896$	79.6192	2.65989
$897$	−11.0587	−0.369238
$898$	−18.6550	−0.622526
$899$	−8.25460	−0.275306
$900$	−60.8162	−2.02721
$901$	0.622399	0.0207351
$902$	−23.7375	−0.790373
$903$	−88.2200	−2.93578
$904$	100.841	3.35392
$905$	14.3998	0.478664
$906$	−142.627	−4.73848
$907$	30.0120	0.996531	0.498266	−	0.867024i	$-0.333970\pi$
$907$	30.0120	0.996531	0.498266	+	0.867024i	$0.333970\pi$
$908$	−23.1498	−0.768254
$909$	−41.9358	−1.39092
$910$	−14.1882	−0.470334
$911$	13.3046	0.440802	0.220401	−	0.975409i	$-0.429263\pi$
$911$	13.3046	0.440802	0.220401	+	0.975409i	$0.429263\pi$
$912$	1.89616	0.0627880
$913$	15.5643	0.515105
$914$	89.9004	2.97364
$915$	−18.5739	−0.614033
$916$	−80.1419	−2.64796
$917$	58.1344	1.91977
$918$	7.32072	0.241620
$919$	−23.7347	−0.782936	−0.391468	−	0.920192i	$-0.628033\pi$
$919$	−23.7347	−0.782936	−0.391468	+	0.920192i	$0.628033\pi$
$920$	−27.7639	−0.915347
$921$	−26.8480	−0.884672
$922$	−43.9102	−1.44611
$923$	−5.24961	−0.172793
$924$	139.497	4.58912
$925$	−10.2260	−0.336230
$926$	29.1992	0.959545
$927$	−3.95767	−0.129987
$928$	−4.64964	−0.152632
$929$	0.529200	0.0173625	0.00868124	−	0.999962i	$-0.497237\pi$
$929$	0.529200	0.0173625	0.00868124	+	0.999962i	$0.497237\pi$
$930$	36.7161	1.20397
$931$	−2.09865	−0.0687806
$932$	101.346	3.31971
$933$	3.43760	0.112542
$934$	94.2171	3.08288
$935$	3.51949	0.115100
$936$	22.2706	0.727938
$937$	45.0820	1.47276	0.736382	−	0.676566i	$-0.236532\pi$
$937$	45.0820	1.47276	0.736382	+	0.676566i	$0.236532\pi$
$938$	61.0749	1.99417
$939$	78.5400	2.56306
$940$	−40.6822	−1.32691
$941$	0.536000	0.0174731	0.00873654	−	0.999962i	$-0.497219\pi$
$941$	0.536000	0.0174731	0.00873654	+	0.999962i	$0.497219\pi$
$942$	−84.5616	−2.75517
$943$	15.4282	0.502412
$944$	−44.0650	−1.43419
$945$	14.3354	0.466331
$946$	44.7384	1.45457
$947$	−55.3387	−1.79827	−0.899133	−	0.437675i	$-0.855802\pi$
$947$	−55.3387	−1.79827	−0.899133	+	0.437675i	$0.855802\pi$
$948$	−108.907	−3.53715
$949$	−9.44481	−0.306592
$950$	−1.16701	−0.0378628
$951$	−33.6019	−1.08962
$952$	−31.4539	−1.01943
$953$	23.3564	0.756587	0.378294	−	0.925686i	$-0.376511\pi$
$953$	23.3564	0.756587	0.378294	+	0.925686i	$0.376511\pi$
$954$	−5.31296	−0.172013
$955$	4.94466	0.160005
$956$	43.2698	1.39944
$957$	11.8744	0.383844
$958$	72.2610	2.33465
$959$	110.875	3.58034
$960$	−13.8829	−0.448068
$961$	−8.61883	−0.278027
$962$	7.07226	0.228019
$963$	−18.8546	−0.607581
$964$	−98.1332	−3.16066
$965$	5.76599	0.185614
$966$	−133.325	−4.28967
$967$	−3.11634	−0.100215	−0.0501073	−	0.998744i	$-0.515956\pi$
$967$	−3.11634	−0.100215	−0.0501073	+	0.998744i	$0.515956\pi$
$968$	24.4415	0.785580
$969$	0.394788	0.0126824
$970$	20.6366	0.662601
$971$	48.0381	1.54161	0.770807	−	0.637068i	$-0.219853\pi$
$971$	48.0381	1.54161	0.770807	+	0.637068i	$0.219853\pi$
$972$	90.6280	2.90689
$973$	−68.3798	−2.19216
$974$	29.0318	0.930238
$975$	−9.53557	−0.305383
$976$	−33.3121	−1.06629
$977$	−47.9140	−1.53291	−0.766453	−	0.642301i	$-0.777980\pi$
$977$	−47.9140	−1.53291	−0.766453	+	0.642301i	$0.777980\pi$
$978$	21.3478	0.682629
$979$	7.94701	0.253987
$980$	−81.3071	−2.59726
$981$	−62.3360	−1.99023
$982$	−18.6507	−0.595167
$983$	18.5859	0.592800	0.296400	−	0.955064i	$-0.404214\pi$
$983$	18.5859	0.592800	0.296400	+	0.955064i	$0.404214\pi$
$984$	−54.6222	−1.74129
$985$	24.8326	0.791233
$986$	−5.05660	−0.161035
$987$	−103.442	−3.29260
$988$	0.548857	0.0174615
$989$	−29.0778	−0.924620
$990$	−30.0433	−0.954837
$991$	−17.8446	−0.566853	−0.283427	−	0.958994i	$-0.591471\pi$
$991$	−17.8446	−0.566853	−0.283427	+	0.958994i	$0.591471\pi$
$992$	12.6068	0.400268
$993$	−49.4625	−1.56964
$994$	−63.2903	−2.00745
$995$	−30.4849	−0.966436
$996$	67.6400	2.14326
$997$	58.5966	1.85577	0.927886	−	0.372865i	$-0.121624\pi$
$997$	58.5966	1.85577	0.927886	+	0.372865i	$0.121624\pi$
$998$	−52.4910	−1.66157
$999$	−7.14565	−0.226078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.e.1.1	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.e.1.1	✓	21	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50015 q^{2} -2.63774 q^{3} +4.25075 q^{4} -1.17684 q^{5} +6.59474 q^{6} +4.82219 q^{7} -5.62720 q^{8} +3.95767 q^{9} +O(q^{10})\)	\(q-2.50015 q^{2} -2.63774 q^{3} +4.25075 q^{4} -1.17684 q^{5} +6.59474 q^{6} +4.82219 q^{7} -5.62720 q^{8} +3.95767 q^{9} +2.94227 q^{10} -2.58003 q^{11} -11.2124 q^{12} -1.00000 q^{13} -12.0562 q^{14} +3.10419 q^{15} +5.56735 q^{16} +1.15915 q^{17} -9.89477 q^{18} -0.129120 q^{19} -5.00244 q^{20} -12.7197 q^{21} +6.45045 q^{22} -4.19248 q^{23} +14.8431 q^{24} -3.61505 q^{25} +2.50015 q^{26} -2.52609 q^{27} +20.4979 q^{28} +1.74484 q^{29} -7.76095 q^{30} -4.73087 q^{31} -2.66480 q^{32} +6.80544 q^{33} -2.89804 q^{34} -5.67494 q^{35} +16.8231 q^{36} +2.82874 q^{37} +0.322819 q^{38} +2.63774 q^{39} +6.62231 q^{40} -3.67998 q^{41} +31.8011 q^{42} +6.93571 q^{43} -10.9670 q^{44} -4.65754 q^{45} +10.4818 q^{46} +8.13246 q^{47} -14.6852 q^{48} +16.2535 q^{49} +9.03817 q^{50} -3.05753 q^{51} -4.25075 q^{52} +0.536946 q^{53} +6.31561 q^{54} +3.03627 q^{55} -27.1354 q^{56} +0.340585 q^{57} -4.36235 q^{58} -7.91490 q^{59} +13.1951 q^{60} -5.98348 q^{61} +11.8279 q^{62} +19.0846 q^{63} -4.47230 q^{64} +1.17684 q^{65} -17.0146 q^{66} -5.06585 q^{67} +4.92724 q^{68} +11.0587 q^{69} +14.1882 q^{70} +5.24961 q^{71} -22.2706 q^{72} +9.44481 q^{73} -7.07226 q^{74} +9.53557 q^{75} -0.548857 q^{76} -12.4414 q^{77} -6.59474 q^{78} +9.71315 q^{79} -6.55187 q^{80} -5.20984 q^{81} +9.20049 q^{82} -6.03263 q^{83} -54.0681 q^{84} -1.36413 q^{85} -17.3403 q^{86} -4.60242 q^{87} +14.5183 q^{88} -3.08020 q^{89} +11.6445 q^{90} -4.82219 q^{91} -17.8212 q^{92} +12.4788 q^{93} -20.3324 q^{94} +0.151953 q^{95} +7.02905 q^{96} +7.01383 q^{97} -40.6362 q^{98} -10.2109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more