LMFDB - Embedded newform 1519.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1519,2,Mod(1,1519)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1519, 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("1519.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1519 = 7^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1519.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:	$12.1292760670$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 9 x^{11} + 76 x^{10} - 17 x^{9} - 387 x^{8} + 332 x^{7} + 758 x^{6} - 875 x^{5} + \cdots + 21 $ x^13 - 5x^12 - 9x^11 + 76x^10 - 17x^9 - 387x^8 + 332x^7 + 758x^6 - 875x^5 - 475x^4 + 705x^3 - 48x^2 - 108x + 21
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 217)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.47057$ of defining polynomial
Character	$\chi$	$=$	1519.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47057 q^{2} -1.55945 q^{3} +0.162576 q^{4} +4.37105 q^{5} +2.29329 q^{6} +2.70206 q^{8} -0.568105 q^{9} +O(q^{10})$	\(q-1.47057 q^{2} -1.55945 q^{3} +0.162576 q^{4} +4.37105 q^{5} +2.29329 q^{6} +2.70206 q^{8} -0.568105 q^{9} -6.42794 q^{10} +0.334425 q^{11} -0.253529 q^{12} +5.80839 q^{13} -6.81646 q^{15} -4.29872 q^{16} +2.86861 q^{17} +0.835438 q^{18} +0.421184 q^{19} +0.710628 q^{20} -0.491795 q^{22} +2.00865 q^{23} -4.21374 q^{24} +14.1061 q^{25} -8.54164 q^{26} +5.56429 q^{27} +4.99544 q^{29} +10.0241 q^{30} +1.00000 q^{31} +0.917448 q^{32} -0.521520 q^{33} -4.21849 q^{34} -0.0923602 q^{36} -4.42560 q^{37} -0.619380 q^{38} -9.05791 q^{39} +11.8109 q^{40} -10.1180 q^{41} -4.91910 q^{43} +0.0543694 q^{44} -2.48322 q^{45} -2.95386 q^{46} -1.20240 q^{47} +6.70365 q^{48} -20.7440 q^{50} -4.47346 q^{51} +0.944304 q^{52} -5.10760 q^{53} -8.18268 q^{54} +1.46179 q^{55} -0.656816 q^{57} -7.34614 q^{58} -7.94464 q^{59} -1.10819 q^{60} -1.02514 q^{61} -1.47057 q^{62} +7.24827 q^{64} +25.3888 q^{65} +0.766932 q^{66} +7.16201 q^{67} +0.466367 q^{68} -3.13240 q^{69} +1.51205 q^{71} -1.53505 q^{72} +10.3665 q^{73} +6.50815 q^{74} -21.9978 q^{75} +0.0684743 q^{76} +13.3203 q^{78} +7.74342 q^{79} -18.7899 q^{80} -6.97294 q^{81} +14.8792 q^{82} -12.5211 q^{83} +12.5388 q^{85} +7.23387 q^{86} -7.79015 q^{87} +0.903636 q^{88} +2.43545 q^{89} +3.65175 q^{90} +0.326558 q^{92} -1.55945 q^{93} +1.76821 q^{94} +1.84102 q^{95} -1.43072 q^{96} +4.26302 q^{97} -0.189988 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 5 q^{2} + 17 q^{4} + q^{5} - 2 q^{6} + 12 q^{8} + 25 q^{9}+O(q^{10}) $ 13 * q + 5 * q^2 + 17 * q^4 + q^5 - 2 * q^6 + 12 * q^8 + 25 * q^9	$ 13 q + 5 q^{2} + 17 q^{4} + q^{5} - 2 q^{6} + 12 q^{8} + 25 q^{9} - 7 q^{10} + 15 q^{11} - 5 q^{12} + 4 q^{13} + 4 q^{15} + 29 q^{16} - 4 q^{17} + 16 q^{18} + 2 q^{19} + 26 q^{20} - 10 q^{22} + 14 q^{23} - 28 q^{24} + 24 q^{25} - 7 q^{26} + 12 q^{27} + 22 q^{29} + 6 q^{30} + 13 q^{31} + 19 q^{32} - 5 q^{33} + 20 q^{34} + 11 q^{36} + 12 q^{37} - 11 q^{38} + 11 q^{39} - 6 q^{40} + 4 q^{41} - 3 q^{43} + 52 q^{44} - 12 q^{45} - 3 q^{46} - 14 q^{47} + 48 q^{48} + 15 q^{50} + 16 q^{51} + 4 q^{52} + 19 q^{53} - 25 q^{54} + 18 q^{55} + 13 q^{57} + 24 q^{58} + 19 q^{59} + 6 q^{60} - 11 q^{61} + 5 q^{62} + 10 q^{64} + 68 q^{65} - 52 q^{66} - 25 q^{67} - 26 q^{68} + 52 q^{69} + 28 q^{71} + 52 q^{72} - 29 q^{73} + 54 q^{74} - 71 q^{75} + 37 q^{76} - 71 q^{78} + 30 q^{79} + 3 q^{80} + 25 q^{81} + 5 q^{82} - 10 q^{83} - q^{85} + 10 q^{86} + 50 q^{87} + 18 q^{88} - 11 q^{89} + 81 q^{90} + 35 q^{92} - 36 q^{94} - 20 q^{95} - 12 q^{96} - 3 q^{97} + 42 q^{99}+O(q^{100}) $ 13 * q + 5 * q^2 + 17 * q^4 + q^5 - 2 * q^6 + 12 * q^8 + 25 * q^9 - 7 * q^10 + 15 * q^11 - 5 * q^12 + 4 * q^13 + 4 * q^15 + 29 * q^16 - 4 * q^17 + 16 * q^18 + 2 * q^19 + 26 * q^20 - 10 * q^22 + 14 * q^23 - 28 * q^24 + 24 * q^25 - 7 * q^26 + 12 * q^27 + 22 * q^29 + 6 * q^30 + 13 * q^31 + 19 * q^32 - 5 * q^33 + 20 * q^34 + 11 * q^36 + 12 * q^37 - 11 * q^38 + 11 * q^39 - 6 * q^40 + 4 * q^41 - 3 * q^43 + 52 * q^44 - 12 * q^45 - 3 * q^46 - 14 * q^47 + 48 * q^48 + 15 * q^50 + 16 * q^51 + 4 * q^52 + 19 * q^53 - 25 * q^54 + 18 * q^55 + 13 * q^57 + 24 * q^58 + 19 * q^59 + 6 * q^60 - 11 * q^61 + 5 * q^62 + 10 * q^64 + 68 * q^65 - 52 * q^66 - 25 * q^67 - 26 * q^68 + 52 * q^69 + 28 * q^71 + 52 * q^72 - 29 * q^73 + 54 * q^74 - 71 * q^75 + 37 * q^76 - 71 * q^78 + 30 * q^79 + 3 * q^80 + 25 * q^81 + 5 * q^82 - 10 * q^83 - q^85 + 10 * q^86 + 50 * q^87 + 18 * q^88 - 11 * q^89 + 81 * q^90 + 35 * q^92 - 36 * q^94 - 20 * q^95 - 12 * q^96 - 3 * q^97 + 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 $

$2$	−1.47057	−1.03985	−0.519925	−	0.854212i	$-0.674040\pi$
$2$	−1.47057	−1.03985	−0.519925	+	0.854212i	$0.674040\pi$
$3$	−1.55945	−0.900351	−0.450175	−	0.892940i	$-0.648639\pi$
$3$	−1.55945	−0.900351	−0.450175	+	0.892940i	$0.648639\pi$
$4$	0.162576	0.0812879
$5$	4.37105	1.95479	0.977397	−	0.211410i	$-0.0678055\pi$
$5$	4.37105	1.95479	0.977397	+	0.211410i	$0.0678055\pi$
$6$	2.29329	0.936230
$7$	0	0
$7$	0	0
$8$	2.70206	0.955323
$9$	−0.568105	−0.189368
$10$	−6.42794	−2.03269
$11$	0.334425	0.100833	0.0504164	−	0.998728i	$-0.483945\pi$
$11$	0.334425	0.100833	0.0504164	+	0.998728i	$0.483945\pi$
$12$	−0.253529	−0.0731877
$13$	5.80839	1.61096	0.805479	−	0.592625i	$-0.201908\pi$
$13$	5.80839	1.61096	0.805479	+	0.592625i	$0.201908\pi$
$14$	0	0
$15$	−6.81646	−1.76000
$16$	−4.29872	−1.07468
$17$	2.86861	0.695740	0.347870	−	0.937543i	$-0.386905\pi$
$17$	2.86861	0.695740	0.347870	+	0.937543i	$0.386905\pi$
$18$	0.835438	0.196915
$19$	0.421184	0.0966262	0.0483131	−	0.998832i	$-0.484615\pi$
$19$	0.421184	0.0966262	0.0483131	+	0.998832i	$0.484615\pi$
$20$	0.710628	0.158901
$21$	0	0
$22$	−0.491795	−0.104851
$23$	2.00865	0.418833	0.209416	−	0.977827i	$-0.432844\pi$
$23$	2.00865	0.418833	0.209416	+	0.977827i	$0.432844\pi$
$24$	−4.21374	−0.860126
$25$	14.1061	2.82122
$26$	−8.54164	−1.67515
$27$	5.56429	1.07085
$28$	0	0
$29$	4.99544	0.927629	0.463814	−	0.885932i	$-0.346480\pi$
$29$	4.99544	0.927629	0.463814	+	0.885932i	$0.346480\pi$
$30$	10.0241	1.83014
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0.917448	0.162183
$33$	−0.521520	−0.0907850
$34$	−4.21849	−0.723465
$35$	0	0
$36$	−0.0923602	−0.0153934
$37$	−4.42560	−0.727564	−0.363782	−	0.931484i	$-0.618515\pi$
$37$	−4.42560	−0.727564	−0.363782	+	0.931484i	$0.618515\pi$
$38$	−0.619380	−0.100477
$39$	−9.05791	−1.45043
$40$	11.8109	1.86746
$41$	−10.1180	−1.58016	−0.790081	−	0.613002i	$-0.789962\pi$
$41$	−10.1180	−1.58016	−0.790081	+	0.613002i	$0.789962\pi$
$42$	0	0
$43$	−4.91910	−0.750155	−0.375078	−	0.926993i	$-0.622384\pi$
$43$	−4.91910	−0.750155	−0.375078	+	0.926993i	$0.622384\pi$
$44$	0.0543694	0.00819650
$45$	−2.48322	−0.370176
$46$	−2.95386	−0.435523
$47$	−1.20240	−0.175388	−0.0876940	−	0.996147i	$-0.527950\pi$
$47$	−1.20240	−0.175388	−0.0876940	+	0.996147i	$0.527950\pi$
$48$	6.70365	0.967589
$49$	0	0
$50$	−20.7440	−2.93365
$51$	−4.47346	−0.626410
$52$	0.944304	0.130951
$53$	−5.10760	−0.701583	−0.350792	−	0.936454i	$-0.614087\pi$
$53$	−5.10760	−0.701583	−0.350792	+	0.936454i	$0.614087\pi$
$54$	−8.18268	−1.11352
$55$	1.46179	0.197108
$56$	0	0
$57$	−0.656816	−0.0869974
$58$	−7.34614	−0.964595
$59$	−7.94464	−1.03430	−0.517152	−	0.855893i	$-0.673008\pi$
$59$	−7.94464	−1.03430	−0.517152	+	0.855893i	$0.673008\pi$
$60$	−1.10819	−0.143067
$61$	−1.02514	−0.131256	−0.0656280	−	0.997844i	$-0.520905\pi$
$61$	−1.02514	−0.131256	−0.0656280	+	0.997844i	$0.520905\pi$
$62$	−1.47057	−0.186763
$63$	0	0
$64$	7.24827	0.906034
$65$	25.3888	3.14909
$66$	0.766932	0.0944028
$67$	7.16201	0.874979	0.437490	−	0.899224i	$-0.355868\pi$
$67$	7.16201	0.874979	0.437490	+	0.899224i	$0.355868\pi$
$68$	0.466367	0.0565553
$69$	−3.13240	−0.377096
$70$	0	0
$71$	1.51205	0.179447	0.0897237	−	0.995967i	$-0.471402\pi$
$71$	1.51205	0.179447	0.0897237	+	0.995967i	$0.471402\pi$
$72$	−1.53505	−0.180908
$73$	10.3665	1.21331	0.606653	−	0.794967i	$-0.292512\pi$
$73$	10.3665	1.21331	0.606653	+	0.794967i	$0.292512\pi$
$74$	6.50815	0.756557
$75$	−21.9978	−2.54009
$76$	0.0684743	0.00785454
$77$	0	0
$78$	13.3203	1.50823
$79$	7.74342	0.871203	0.435601	−	0.900140i	$-0.356536\pi$
$79$	7.74342	0.871203	0.435601	+	0.900140i	$0.356536\pi$
$80$	−18.7899	−2.10078
$81$	−6.97294	−0.774771
$82$	14.8792	1.64313
$83$	−12.5211	−1.37437	−0.687184	−	0.726483i	$-0.741153\pi$
$83$	−12.5211	−1.37437	−0.687184	+	0.726483i	$0.741153\pi$
$84$	0	0
$85$	12.5388	1.36003
$86$	7.23387	0.780049
$87$	−7.79015	−0.835192
$88$	0.903636	0.0963279
$89$	2.43545	0.258157	0.129078	−	0.991634i	$-0.458798\pi$
$89$	2.43545	0.258157	0.129078	+	0.991634i	$0.458798\pi$
$90$	3.65175	0.384928
$91$	0	0
$92$	0.326558	0.0340461
$93$	−1.55945	−0.161708
$94$	1.76821	0.182377
$95$	1.84102	0.188884
$96$	−1.43072	−0.146022
$97$	4.26302	0.432844	0.216422	−	0.976300i	$-0.430561\pi$
$97$	4.26302	0.432844	0.216422	+	0.976300i	$0.430561\pi$
$98$	0	0
$99$	−0.189988	−0.0190946
$100$	2.29331	0.229331
$101$	1.21018	0.120417	0.0602086	−	0.998186i	$-0.480823\pi$
$101$	1.21018	0.120417	0.0602086	+	0.998186i	$0.480823\pi$
$102$	6.57854	0.651372
$103$	−16.5647	−1.63217	−0.816086	−	0.577931i	$-0.803860\pi$
$103$	−16.5647	−1.63217	−0.816086	+	0.577931i	$0.803860\pi$
$104$	15.6946	1.53898
$105$	0	0
$106$	7.51109	0.729541
$107$	15.9724	1.54411	0.772053	−	0.635558i	$-0.219230\pi$
$107$	15.9724	1.54411	0.772053	+	0.635558i	$0.219230\pi$
$108$	0.904620	0.0870471
$109$	5.22794	0.500746	0.250373	−	0.968149i	$-0.419447\pi$
$109$	5.22794	0.500746	0.250373	+	0.968149i	$0.419447\pi$
$110$	−2.14966	−0.204962
$111$	6.90151	0.655063
$112$	0	0
$113$	9.16180	0.861870	0.430935	−	0.902383i	$-0.358184\pi$
$113$	9.16180	0.861870	0.430935	+	0.902383i	$0.358184\pi$
$114$	0.965894	0.0904643
$115$	8.77993	0.818732
$116$	0.812137	0.0754050
$117$	−3.29978	−0.305064
$118$	11.6832	1.07552
$119$	0	0
$120$	−18.4185	−1.68137
$121$	−10.8882	−0.989833
$122$	1.50754	0.136487
$123$	15.7785	1.42270
$124$	0.162576	0.0145997
$125$	39.8033	3.56012
$126$	0	0
$127$	−3.84670	−0.341339	−0.170670	−	0.985328i	$-0.554593\pi$
$127$	−3.84670	−0.341339	−0.170670	+	0.985328i	$0.554593\pi$
$128$	−12.4940	−1.10432
$129$	7.67110	0.675403
$130$	−37.3360	−3.27458
$131$	2.97936	0.260308	0.130154	−	0.991494i	$-0.458453\pi$
$131$	2.97936	0.260308	0.130154	+	0.991494i	$0.458453\pi$
$132$	−0.0847866	−0.00737972
$133$	0	0
$134$	−10.5322	−0.909847
$135$	24.3218	2.09329
$136$	7.75116	0.664656
$137$	12.6454	1.08037	0.540184	−	0.841547i	$-0.318355\pi$
$137$	12.6454	1.08037	0.540184	+	0.841547i	$0.318355\pi$
$138$	4.60641	0.392124
$139$	−9.18944	−0.779438	−0.389719	−	0.920934i	$-0.627428\pi$
$139$	−9.18944	−0.779438	−0.389719	+	0.920934i	$0.627428\pi$
$140$	0	0
$141$	1.87509	0.157911
$142$	−2.22358	−0.186598
$143$	1.94247	0.162438
$144$	2.44213	0.203510
$145$	21.8353	1.81332
$146$	−15.2446	−1.26166
$147$	0	0
$148$	−0.719495	−0.0591421
$149$	8.23664	0.674772	0.337386	−	0.941366i	$-0.390457\pi$
$149$	8.23664	0.674772	0.337386	+	0.941366i	$0.390457\pi$
$150$	32.3493	2.64131
$151$	−6.49472	−0.528533	−0.264266	−	0.964450i	$-0.585130\pi$
$151$	−6.49472	−0.528533	−0.264266	+	0.964450i	$0.585130\pi$
$152$	1.13806	0.0923092
$153$	−1.62967	−0.131751
$154$	0	0
$155$	4.37105	0.351092
$156$	−1.47260	−0.117902
$157$	17.0392	1.35988	0.679938	−	0.733270i	$-0.262007\pi$
$157$	17.0392	1.35988	0.679938	+	0.733270i	$0.262007\pi$
$158$	−11.3872	−0.905920
$159$	7.96507	0.631671
$160$	4.01022	0.317035
$161$	0	0
$162$	10.2542	0.805646
$163$	3.54835	0.277928	0.138964	−	0.990297i	$-0.455623\pi$
$163$	3.54835	0.277928	0.138964	+	0.990297i	$0.455623\pi$
$164$	−1.64494	−0.128448
$165$	−2.27959	−0.177466
$166$	18.4131	1.42914
$167$	−24.6131	−1.90462	−0.952310	−	0.305133i	$-0.901299\pi$
$167$	−24.6131	−1.90462	−0.952310	+	0.305133i	$0.901299\pi$
$168$	0	0
$169$	20.7374	1.59518
$170$	−18.4393	−1.41423
$171$	−0.239277	−0.0182979
$172$	−0.799726	−0.0609786
$173$	15.4342	1.17344	0.586721	−	0.809789i	$-0.300418\pi$
$173$	15.4342	1.17344	0.586721	+	0.809789i	$0.300418\pi$
$174$	11.4560	0.868474
$175$	0	0
$176$	−1.43760	−0.108363
$177$	12.3893	0.931237
$178$	−3.58149	−0.268444
$179$	19.6717	1.47033	0.735165	−	0.677889i	$-0.237105\pi$
$179$	19.6717	1.47033	0.735165	+	0.677889i	$0.237105\pi$
$180$	−0.403711	−0.0300909
$181$	12.4376	0.924477	0.462239	−	0.886756i	$-0.347046\pi$
$181$	12.4376	0.924477	0.462239	+	0.886756i	$0.347046\pi$
$182$	0	0
$183$	1.59866	0.118176
$184$	5.42750	0.400121
$185$	−19.3445	−1.42224
$186$	2.29329	0.168152
$187$	0.959334	0.0701535
$188$	−0.195481	−0.0142569
$189$	0	0
$190$	−2.70734	−0.196411
$191$	−4.72288	−0.341736	−0.170868	−	0.985294i	$-0.554657\pi$
$191$	−4.72288	−0.341736	−0.170868	+	0.985294i	$0.554657\pi$
$192$	−11.3033	−0.815748
$193$	−2.10268	−0.151354	−0.0756771	−	0.997132i	$-0.524112\pi$
$193$	−2.10268	−0.151354	−0.0756771	+	0.997132i	$0.524112\pi$
$194$	−6.26907	−0.450093
$195$	−39.5926	−2.83529
$196$	0	0
$197$	−4.31625	−0.307520	−0.153760	−	0.988108i	$-0.549138\pi$
$197$	−4.31625	−0.307520	−0.153760	+	0.988108i	$0.549138\pi$
$198$	0.279391	0.0198555
$199$	−6.25485	−0.443394	−0.221697	−	0.975116i	$-0.571160\pi$
$199$	−6.25485	−0.443394	−0.221697	+	0.975116i	$0.571160\pi$
$200$	38.1156	2.69518
$201$	−11.1688	−0.787788
$202$	−1.77965	−0.125216
$203$	0	0
$204$	−0.727277	−0.0509196
$205$	−44.2262	−3.08889
$206$	24.3596	1.69721
$207$	−1.14113	−0.0793137
$208$	−24.9686	−1.73126
$209$	0.140854	0.00974309
$210$	0	0
$211$	1.12522	0.0774631	0.0387316	−	0.999250i	$-0.487668\pi$
$211$	1.12522	0.0774631	0.0387316	+	0.999250i	$0.487668\pi$
$212$	−0.830373	−0.0570302
$213$	−2.35797	−0.161566
$214$	−23.4885	−1.60564
$215$	−21.5016	−1.46640
$216$	15.0351	1.02301
$217$	0	0
$218$	−7.68805	−0.520700
$219$	−16.1661	−1.09240
$220$	0.237652	0.0160225
$221$	16.6620	1.12081
$222$	−10.1492	−0.681167
$223$	−26.0719	−1.74590	−0.872952	−	0.487806i	$-0.837797\pi$
$223$	−26.0719	−1.74590	−0.872952	+	0.487806i	$0.837797\pi$
$224$	0	0
$225$	−8.01376	−0.534250
$226$	−13.4731	−0.896215
$227$	15.5167	1.02988	0.514941	−	0.857226i	$-0.327814\pi$
$227$	15.5167	1.02988	0.514941	+	0.857226i	$0.327814\pi$
$228$	−0.106782	−0.00707184
$229$	17.7363	1.17205	0.586025	−	0.810293i	$-0.300692\pi$
$229$	17.7363	1.17205	0.586025	+	0.810293i	$0.300692\pi$
$230$	−12.9115	−0.851359
$231$	0	0
$232$	13.4980	0.886185
$233$	−0.155388	−0.0101798	−0.00508991	−	0.999987i	$-0.501620\pi$
$233$	−0.155388	−0.0101798	−0.00508991	+	0.999987i	$0.501620\pi$
$234$	4.85255	0.317221
$235$	−5.25576	−0.342848
$236$	−1.29161	−0.0840765
$237$	−12.0755	−0.784388
$238$	0	0
$239$	18.2663	1.18155	0.590773	−	0.806838i	$-0.298823\pi$
$239$	18.2663	1.18155	0.590773	+	0.806838i	$0.298823\pi$
$240$	29.3020	1.89144
$241$	−18.5803	−1.19686	−0.598431	−	0.801175i	$-0.704209\pi$
$241$	−18.5803	−1.19686	−0.598431	+	0.801175i	$0.704209\pi$
$242$	16.0118	1.02928
$243$	−5.81890	−0.373283
$244$	−0.166663	−0.0106695
$245$	0	0
$246$	−23.2034	−1.47940
$247$	2.44640	0.155661
$248$	2.70206	0.171581
$249$	19.5261	1.23741
$250$	−58.5336	−3.70199
$251$	17.8295	1.12539	0.562693	−	0.826666i	$-0.309765\pi$
$251$	17.8295	1.12539	0.562693	+	0.826666i	$0.309765\pi$
$252$	0	0
$253$	0.671743	0.0422321
$254$	5.65684	0.354942
$255$	−19.5537	−1.22450
$256$	3.87674	0.242296
$257$	9.25708	0.577441	0.288720	−	0.957413i	$-0.406770\pi$
$257$	9.25708	0.577441	0.288720	+	0.957413i	$0.406770\pi$
$258$	−11.2809	−0.702318
$259$	0	0
$260$	4.12760	0.255983
$261$	−2.83793	−0.175664
$262$	−4.38136	−0.270681
$263$	−18.8778	−1.16405	−0.582027	−	0.813169i	$-0.697740\pi$
$263$	−18.8778	−1.16405	−0.582027	+	0.813169i	$0.697740\pi$
$264$	−1.40918	−0.0867289
$265$	−22.3256	−1.37145
$266$	0	0
$267$	−3.79797	−0.232432
$268$	1.16437	0.0711252
$269$	−18.4723	−1.12628	−0.563138	−	0.826363i	$-0.690406\pi$
$269$	−18.4723	−1.12628	−0.563138	+	0.826363i	$0.690406\pi$
$270$	−35.7670	−2.17671
$271$	30.5777	1.85746	0.928731	−	0.370753i	$-0.120901\pi$
$271$	30.5777	1.85746	0.928731	+	0.370753i	$0.120901\pi$
$272$	−12.3314	−0.747698
$273$	0	0
$274$	−18.5959	−1.12342
$275$	4.71744	0.284472
$276$	−0.509252	−0.0306534
$277$	−27.2746	−1.63877	−0.819387	−	0.573240i	$-0.805686\pi$
$277$	−27.2746	−1.63877	−0.819387	+	0.573240i	$0.805686\pi$
$278$	13.5137	0.810499
$279$	−0.568105	−0.0340116
$280$	0	0
$281$	15.9872	0.953716	0.476858	−	0.878980i	$-0.341776\pi$
$281$	15.9872	0.953716	0.476858	+	0.878980i	$0.341776\pi$
$282$	−2.75745	−0.164204
$283$	−11.7377	−0.697733	−0.348867	−	0.937172i	$-0.613433\pi$
$283$	−11.7377	−0.697733	−0.348867	+	0.937172i	$0.613433\pi$
$284$	0.245823	0.0145869
$285$	−2.87098	−0.170062
$286$	−2.85654	−0.168911
$287$	0	0
$288$	−0.521207	−0.0307124
$289$	−8.77108	−0.515946
$290$	−32.1104	−1.88559
$291$	−6.64798	−0.389712
$292$	1.68534	0.0986271
$293$	4.91819	0.287324	0.143662	−	0.989627i	$-0.454112\pi$
$293$	4.91819	0.287324	0.143662	+	0.989627i	$0.454112\pi$
$294$	0	0
$295$	−34.7265	−2.02185
$296$	−11.9582	−0.695058
$297$	1.86084	0.107977
$298$	−12.1126	−0.701662
$299$	11.6670	0.674722
$300$	−3.57632	−0.206479
$301$	0	0
$302$	9.55094	0.549595
$303$	−1.88722	−0.108418
$304$	−1.81055	−0.103842
$305$	−4.48095	−0.256579
$306$	2.39655	0.137001
$307$	17.5101	0.999355	0.499677	−	0.866212i	$-0.333452\pi$
$307$	17.5101	0.999355	0.499677	+	0.866212i	$0.333452\pi$
$308$	0	0
$309$	25.8319	1.46953
$310$	−6.42794	−0.365083
$311$	11.2651	0.638786	0.319393	−	0.947622i	$-0.396521\pi$
$311$	11.2651	0.638786	0.319393	+	0.947622i	$0.396521\pi$
$312$	−24.4750	−1.38563
$313$	−24.7326	−1.39797	−0.698983	−	0.715138i	$-0.746364\pi$
$313$	−24.7326	−1.39797	−0.698983	+	0.715138i	$0.746364\pi$
$314$	−25.0573	−1.41407
$315$	0	0
$316$	1.25889	0.0708183
$317$	−27.2078	−1.52814	−0.764071	−	0.645132i	$-0.776802\pi$
$317$	−27.2078	−1.52814	−0.764071	+	0.645132i	$0.776802\pi$
$318$	−11.7132	−0.656843
$319$	1.67060	0.0935355
$320$	31.6826	1.77111
$321$	−24.9081	−1.39024
$322$	0	0
$323$	1.20821	0.0672267
$324$	−1.13363	−0.0629795
$325$	81.9338	4.54487
$326$	−5.21810	−0.289004
$327$	−8.15273	−0.450847
$328$	−27.3394	−1.50957
$329$	0	0
$330$	3.35230	0.184538
$331$	26.2818	1.44458	0.722289	−	0.691591i	$-0.243090\pi$
$331$	26.2818	1.44458	0.722289	+	0.691591i	$0.243090\pi$
$332$	−2.03563	−0.111720
$333$	2.51420	0.137778
$334$	36.1953	1.98052
$335$	31.3055	1.71040
$336$	0	0
$337$	−12.5855	−0.685576	−0.342788	−	0.939413i	$-0.611371\pi$
$337$	−12.5855	−0.685576	−0.342788	+	0.939413i	$0.611371\pi$
$338$	−30.4958	−1.65875
$339$	−14.2874	−0.775985
$340$	2.03851	0.110554
$341$	0.334425	0.0181101
$342$	0.351873	0.0190271
$343$	0	0
$344$	−13.2917	−0.716640
$345$	−13.6919	−0.737146
$346$	−22.6971	−1.22020
$347$	10.1617	0.545509	0.272755	−	0.962084i	$-0.412065\pi$
$347$	10.1617	0.545509	0.272755	+	0.962084i	$0.412065\pi$
$348$	−1.26649	−0.0678910
$349$	2.58840	0.138554	0.0692769	−	0.997597i	$-0.477931\pi$
$349$	2.58840	0.138554	0.0692769	+	0.997597i	$0.477931\pi$
$350$	0	0
$351$	32.3196	1.72509
$352$	0.306817	0.0163534
$353$	−2.76436	−0.147132	−0.0735659	−	0.997290i	$-0.523438\pi$
$353$	−2.76436	−0.147132	−0.0735659	+	0.997290i	$0.523438\pi$
$354$	−18.2193	−0.968347
$355$	6.60925	0.350783
$356$	0.395945	0.0209850
$357$	0	0
$358$	−28.9286	−1.52892
$359$	22.1626	1.16970	0.584848	−	0.811143i	$-0.301154\pi$
$359$	22.1626	1.16970	0.584848	+	0.811143i	$0.301154\pi$
$360$	−6.70981	−0.353638
$361$	−18.8226	−0.990663
$362$	−18.2903	−0.961318
$363$	16.9796	0.891197
$364$	0	0
$365$	45.3125	2.37176
$366$	−2.35094	−0.122886
$367$	6.89463	0.359897	0.179948	−	0.983676i	$-0.442407\pi$
$367$	6.89463	0.359897	0.179948	+	0.983676i	$0.442407\pi$
$368$	−8.63463	−0.450111
$369$	5.74808	0.299233
$370$	28.4475	1.47891
$371$	0	0
$372$	−0.253529	−0.0131449
$373$	−10.1121	−0.523585	−0.261792	−	0.965124i	$-0.584314\pi$
$373$	−10.1121	−0.523585	−0.261792	+	0.965124i	$0.584314\pi$
$374$	−1.41077	−0.0729491
$375$	−62.0714	−3.20536
$376$	−3.24896	−0.167552
$377$	29.0154	1.49437
$378$	0	0
$379$	−0.654720	−0.0336307	−0.0168154	−	0.999859i	$-0.505353\pi$
$379$	−0.654720	−0.0336307	−0.0168154	+	0.999859i	$0.505353\pi$
$380$	0.299305	0.0153540
$381$	5.99875	0.307325
$382$	6.94533	0.355354
$383$	−17.3588	−0.886995	−0.443498	−	0.896276i	$-0.646263\pi$
$383$	−17.3588	−0.886995	−0.443498	+	0.896276i	$0.646263\pi$
$384$	19.4838	0.994278
$385$	0	0
$386$	3.09214	0.157386
$387$	2.79456	0.142056
$388$	0.693064	0.0351850
$389$	17.9331	0.909247	0.454623	−	0.890684i	$-0.349774\pi$
$389$	17.9331	0.909247	0.454623	+	0.890684i	$0.349774\pi$
$390$	58.2237	2.94827
$391$	5.76204	0.291399
$392$	0	0
$393$	−4.64618	−0.234369
$394$	6.34734	0.319775
$395$	33.8469	1.70302
$396$	−0.0308875	−0.00155216
$397$	2.34293	0.117588	0.0587942	−	0.998270i	$-0.481274\pi$
$397$	2.34293	0.117588	0.0587942	+	0.998270i	$0.481274\pi$
$398$	9.19819	0.461064
$399$	0	0
$400$	−60.6383	−3.03191
$401$	−2.67963	−0.133815	−0.0669073	−	0.997759i	$-0.521313\pi$
$401$	−2.67963	−0.133815	−0.0669073	+	0.997759i	$0.521313\pi$
$402$	16.4245	0.819181
$403$	5.80839	0.289337
$404$	0.196746	0.00978847
$405$	−30.4791	−1.51452
$406$	0	0
$407$	−1.48003	−0.0733623
$408$	−12.0876	−0.598424
$409$	−25.7060	−1.27108	−0.635539	−	0.772069i	$-0.719222\pi$
$409$	−25.7060	−1.27108	−0.635539	+	0.772069i	$0.719222\pi$
$410$	65.0378	3.21199
$411$	−19.7199	−0.972710
$412$	−2.69303	−0.132676
$413$	0	0
$414$	1.67810	0.0824743
$415$	−54.7304	−2.68661
$416$	5.32890	0.261271
$417$	14.3305	0.701768
$418$	−0.207136	−0.0101314
$419$	−10.5209	−0.513979	−0.256989	−	0.966414i	$-0.582731\pi$
$419$	−10.5209	−0.513979	−0.256989	+	0.966414i	$0.582731\pi$
$420$	0	0
$421$	8.38713	0.408764	0.204382	−	0.978891i	$-0.434482\pi$
$421$	8.38713	0.408764	0.204382	+	0.978891i	$0.434482\pi$
$422$	−1.65471	−0.0805500
$423$	0.683090	0.0332129
$424$	−13.8011	−0.670238
$425$	40.4649	1.96284
$426$	3.46756	0.168004
$427$	0	0
$428$	2.59672	0.125517
$429$	−3.02919	−0.146251
$430$	31.6197	1.52484
$431$	−8.68680	−0.418429	−0.209214	−	0.977870i	$-0.567091\pi$
$431$	−8.68680	−0.418429	−0.209214	+	0.977870i	$0.567091\pi$
$432$	−23.9193	−1.15082
$433$	−31.1680	−1.49784	−0.748918	−	0.662663i	$-0.769426\pi$
$433$	−31.1680	−1.49784	−0.748918	+	0.662663i	$0.769426\pi$
$434$	0	0
$435$	−34.0512	−1.63263
$436$	0.849937	0.0407046
$437$	0.846011	0.0404702
$438$	23.7733	1.13593
$439$	−5.38928	−0.257216	−0.128608	−	0.991695i	$-0.541051\pi$
$439$	−5.38928	−0.257216	−0.128608	+	0.991695i	$0.541051\pi$
$440$	3.94984	0.188301
$441$	0	0
$442$	−24.5026	−1.16547
$443$	−13.6858	−0.650231	−0.325115	−	0.945674i	$-0.605403\pi$
$443$	−13.6858	−0.650231	−0.325115	+	0.945674i	$0.605403\pi$
$444$	1.12202	0.0532487
$445$	10.6455	0.504644
$446$	38.3406	1.81548
$447$	−12.8447	−0.607532
$448$	0	0
$449$	−19.7774	−0.933352	−0.466676	−	0.884428i	$-0.654549\pi$
$449$	−19.7774	−0.933352	−0.466676	+	0.884428i	$0.654549\pi$
$450$	11.7848	0.555540
$451$	−3.38370	−0.159332
$452$	1.48949	0.0700596
$453$	10.1282	0.475865
$454$	−22.8185	−1.07092
$455$	0	0
$456$	−1.77476	−0.0831106
$457$	−1.60745	−0.0751933	−0.0375967	−	0.999293i	$-0.511970\pi$
$457$	−1.60745	−0.0751933	−0.0375967	+	0.999293i	$0.511970\pi$
$458$	−26.0825	−1.21876
$459$	15.9618	0.745032
$460$	1.42740	0.0665531
$461$	0.957604	0.0446001	0.0223000	−	0.999751i	$-0.492901\pi$
$461$	0.957604	0.0446001	0.0223000	+	0.999751i	$0.492901\pi$
$462$	0	0
$463$	−14.3693	−0.667798	−0.333899	−	0.942609i	$-0.608365\pi$
$463$	−14.3693	−0.667798	−0.333899	+	0.942609i	$0.608365\pi$
$464$	−21.4740	−0.996905
$465$	−6.81646	−0.316106
$466$	0.228509	0.0105855
$467$	−12.8116	−0.592850	−0.296425	−	0.955056i	$-0.595794\pi$
$467$	−12.8116	−0.592850	−0.296425	+	0.955056i	$0.595794\pi$
$468$	−0.536464	−0.0247981
$469$	0	0
$470$	7.72896	0.356510
$471$	−26.5718	−1.22437
$472$	−21.4669	−0.988095
$473$	−1.64507	−0.0756403
$474$	17.7579	0.815646
$475$	5.94127	0.272604
$476$	0	0
$477$	2.90166	0.132858
$478$	−26.8618	−1.22863
$479$	−12.1914	−0.557039	−0.278519	−	0.960431i	$-0.589844\pi$
$479$	−12.1914	−0.557039	−0.278519	+	0.960431i	$0.589844\pi$
$480$	−6.25374	−0.285443
$481$	−25.7056	−1.17207
$482$	27.3236	1.24456
$483$	0	0
$484$	−1.77015	−0.0804615
$485$	18.6339	0.846122
$486$	8.55711	0.388158
$487$	39.5308	1.79131	0.895656	−	0.444747i	$-0.146706\pi$
$487$	39.5308	1.79131	0.895656	+	0.444747i	$0.146706\pi$
$488$	−2.77000	−0.125392
$489$	−5.53349	−0.250233
$490$	0	0
$491$	−5.77830	−0.260771	−0.130386	−	0.991463i	$-0.541622\pi$
$491$	−5.77830	−0.260771	−0.130386	+	0.991463i	$0.541622\pi$
$492$	2.56521	0.115648
$493$	14.3300	0.645389
$494$	−3.59760	−0.161864
$495$	−0.830450	−0.0373260
$496$	−4.29872	−0.193018
$497$	0	0
$498$	−28.7144	−1.28672
$499$	5.57172	0.249424	0.124712	−	0.992193i	$-0.460199\pi$
$499$	5.57172	0.249424	0.124712	+	0.992193i	$0.460199\pi$
$500$	6.47106	0.289395
$501$	38.3830	1.71483
$502$	−26.2195	−1.17023
$503$	8.93145	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$503$	8.93145	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$504$	0	0
$505$	5.28976	0.235391
$506$	−0.987845	−0.0439151
$507$	−32.3390	−1.43623
$508$	−0.625381	−0.0277468
$509$	30.4587	1.35006	0.675030	−	0.737790i	$-0.264131\pi$
$509$	30.4587	1.35006	0.675030	+	0.737790i	$0.264131\pi$
$510$	28.7552	1.27330
$511$	0	0
$512$	19.2870	0.852371
$513$	2.34359	0.103472
$514$	−13.6132	−0.600452
$515$	−72.4054	−3.19056
$516$	1.24714	0.0549021
$517$	−0.402112	−0.0176849
$518$	0	0
$519$	−24.0690	−1.05651
$520$	68.6021	3.00840
$521$	40.5530	1.77666	0.888331	−	0.459204i	$-0.151865\pi$
$521$	40.5530	1.77666	0.888331	+	0.459204i	$0.151865\pi$
$522$	4.17338	0.182664
$523$	−31.5736	−1.38062	−0.690308	−	0.723515i	$-0.742525\pi$
$523$	−31.5736	−1.38062	−0.690308	+	0.723515i	$0.742525\pi$
$524$	0.484372	0.0211599
$525$	0	0
$526$	27.7611	1.21044
$527$	2.86861	0.124959
$528$	2.24187	0.0975648
$529$	−18.9653	−0.824579
$530$	32.8314	1.42610
$531$	4.51339	0.195865
$532$	0	0
$533$	−58.7692	−2.54558
$534$	5.58517	0.241694
$535$	69.8160	3.01841
$536$	19.3522	0.835887
$537$	−30.6770	−1.32381
$538$	27.1648	1.17116
$539$	0	0
$540$	3.95414	0.170159
$541$	1.68274	0.0723466	0.0361733	−	0.999346i	$-0.488483\pi$
$541$	1.68274	0.0723466	0.0361733	+	0.999346i	$0.488483\pi$
$542$	−44.9666	−1.93148
$543$	−19.3958	−0.832354
$544$	2.63180	0.112838
$545$	22.8516	0.978855
$546$	0	0
$547$	−0.227486	−0.00972661	−0.00486331	−	0.999988i	$-0.501548\pi$
$547$	−0.227486	−0.00972661	−0.00486331	+	0.999988i	$0.501548\pi$
$548$	2.05583	0.0878209
$549$	0.582388	0.0248557
$550$	−6.93732	−0.295808
$551$	2.10400	0.0896332
$552$	−8.46393	−0.360249
$553$	0	0
$554$	40.1093	1.70408
$555$	30.1669	1.28051
$556$	−1.49398	−0.0633589
$557$	−7.88211	−0.333976	−0.166988	−	0.985959i	$-0.553404\pi$
$557$	−7.88211	−0.333976	−0.166988	+	0.985959i	$0.553404\pi$
$558$	0.835438	0.0353669
$559$	−28.5720	−1.20847
$560$	0	0
$561$	−1.49604	−0.0631627
$562$	−23.5103	−0.991721
$563$	20.5424	0.865760	0.432880	−	0.901451i	$-0.357497\pi$
$563$	20.5424	0.865760	0.432880	+	0.901451i	$0.357497\pi$
$564$	0.304844	0.0128362
$565$	40.0467	1.68478
$566$	17.2611	0.725538
$567$	0	0
$568$	4.08565	0.171430
$569$	26.0926	1.09386	0.546930	−	0.837178i	$-0.315796\pi$
$569$	26.0926	1.09386	0.546930	+	0.837178i	$0.315796\pi$
$570$	4.22198	0.176839
$571$	−12.6705	−0.530243	−0.265122	−	0.964215i	$-0.585412\pi$
$571$	−12.6705	−0.530243	−0.265122	+	0.964215i	$0.585412\pi$
$572$	0.315799	0.0132042
$573$	7.36511	0.307682
$574$	0	0
$575$	28.3343	1.18162
$576$	−4.11778	−0.171574
$577$	0.893748	0.0372072	0.0186036	−	0.999827i	$-0.494078\pi$
$577$	0.893748	0.0372072	0.0186036	+	0.999827i	$0.494078\pi$
$578$	12.8985	0.536506
$579$	3.27903	0.136272
$580$	3.54990	0.147401
$581$	0	0
$582$	9.77632	0.405242
$583$	−1.70811	−0.0707427
$584$	28.0109	1.15910
$585$	−14.4235	−0.596338
$586$	−7.23254	−0.298773
$587$	−24.3556	−1.00526	−0.502631	−	0.864501i	$-0.667635\pi$
$587$	−24.3556	−1.00526	−0.502631	+	0.864501i	$0.667635\pi$
$588$	0	0
$589$	0.421184	0.0173546
$590$	51.0677	2.10242
$591$	6.73099	0.276876
$592$	19.0244	0.781898
$593$	−13.3120	−0.546660	−0.273330	−	0.961920i	$-0.588125\pi$
$593$	−13.3120	−0.546660	−0.273330	+	0.961920i	$0.588125\pi$
$594$	−2.73649	−0.112280
$595$	0	0
$596$	1.33908	0.0548508
$597$	9.75414	0.399211
$598$	−17.1572	−0.701610
$599$	8.92277	0.364574	0.182287	−	0.983245i	$-0.441650\pi$
$599$	8.92277	0.364574	0.182287	+	0.983245i	$0.441650\pi$
$600$	−59.4395	−2.42661
$601$	23.3119	0.950911	0.475455	−	0.879740i	$-0.342283\pi$
$601$	23.3119	0.950911	0.475455	+	0.879740i	$0.342283\pi$
$602$	0	0
$603$	−4.06878	−0.165693
$604$	−1.05588	−0.0429633
$605$	−47.5927	−1.93492
$606$	2.77528	0.112738
$607$	−14.7724	−0.599592	−0.299796	−	0.954003i	$-0.596919\pi$
$607$	−14.7724	−0.599592	−0.299796	+	0.954003i	$0.596919\pi$
$608$	0.386414	0.0156712
$609$	0	0
$610$	6.58955	0.266803
$611$	−6.98401	−0.282543
$612$	−0.264945	−0.0107098
$613$	−11.0055	−0.444507	−0.222254	−	0.974989i	$-0.571341\pi$
$613$	−11.0055	−0.444507	−0.222254	+	0.974989i	$0.571341\pi$
$614$	−25.7498	−1.03918
$615$	68.9688	2.78109
$616$	0	0
$617$	25.9673	1.04540	0.522701	−	0.852516i	$-0.324924\pi$
$617$	25.9673	1.04540	0.522701	+	0.852516i	$0.324924\pi$
$618$	−37.9877	−1.52809
$619$	−21.9804	−0.883468	−0.441734	−	0.897146i	$-0.645637\pi$
$619$	−21.9804	−0.883468	−0.441734	+	0.897146i	$0.645637\pi$
$620$	0.710628	0.0285395
$621$	11.1767	0.448507
$622$	−16.5661	−0.664242
$623$	0	0
$624$	38.9374	1.55875
$625$	103.452	4.13808
$626$	36.3709	1.45368
$627$	−0.219656	−0.00877220
$628$	2.77016	0.110541
$629$	−12.6953	−0.506195
$630$	0	0
$631$	42.6583	1.69820	0.849100	−	0.528232i	$-0.177145\pi$
$631$	42.6583	1.69820	0.849100	+	0.528232i	$0.177145\pi$
$632$	20.9232	0.832280
$633$	−1.75472	−0.0697440
$634$	40.0110	1.58904
$635$	−16.8141	−0.667249
$636$	1.29493	0.0513472
$637$	0	0
$638$	−2.45673	−0.0972629
$639$	−0.859004	−0.0339817
$640$	−54.6119	−2.15872
$641$	−3.95319	−0.156142	−0.0780709	−	0.996948i	$-0.524876\pi$
$641$	−3.95319	−0.156142	−0.0780709	+	0.996948i	$0.524876\pi$
$642$	36.6292	1.44564
$643$	7.43895	0.293364	0.146682	−	0.989184i	$-0.453141\pi$
$643$	7.43895	0.293364	0.146682	+	0.989184i	$0.453141\pi$
$644$	0	0
$645$	33.5308	1.32027
$646$	−1.77676	−0.0699057
$647$	11.3252	0.445241	0.222621	−	0.974905i	$-0.428539\pi$
$647$	11.3252	0.445241	0.222621	+	0.974905i	$0.428539\pi$
$648$	−18.8413	−0.740157
$649$	−2.65689	−0.104292
$650$	−120.489	−4.72598
$651$	0	0
$652$	0.576876	0.0225922
$653$	40.3302	1.57824	0.789122	−	0.614237i	$-0.210536\pi$
$653$	40.3302	1.57824	0.789122	+	0.614237i	$0.210536\pi$
$654$	11.9892	0.468813
$655$	13.0230	0.508849
$656$	43.4944	1.69817
$657$	−5.88926	−0.229762
$658$	0	0
$659$	−1.42326	−0.0554424	−0.0277212	−	0.999616i	$-0.508825\pi$
$659$	−1.42326	−0.0554424	−0.0277212	+	0.999616i	$0.508825\pi$
$660$	−0.370607	−0.0144258
$661$	−36.1402	−1.40569	−0.702845	−	0.711343i	$-0.748087\pi$
$661$	−36.1402	−1.40569	−0.702845	+	0.711343i	$0.748087\pi$
$662$	−38.6492	−1.50214
$663$	−25.9836	−1.00912
$664$	−33.8327	−1.31296
$665$	0	0
$666$	−3.69731	−0.143268
$667$	10.0341	0.388521
$668$	−4.00150	−0.154823
$669$	40.6579	1.57193
$670$	−46.0370	−1.77856
$671$	−0.342833	−0.0132349
$672$	0	0
$673$	−27.3356	−1.05371	−0.526855	−	0.849955i	$-0.676629\pi$
$673$	−27.3356	−1.05371	−0.526855	+	0.849955i	$0.676629\pi$
$674$	18.5079	0.712896
$675$	78.4906	3.02110
$676$	3.37140	0.129669
$677$	−38.4725	−1.47862	−0.739309	−	0.673366i	$-0.764848\pi$
$677$	−38.4725	−1.47862	−0.739309	+	0.673366i	$0.764848\pi$
$678$	21.0106	0.806908
$679$	0	0
$680$	33.8807	1.29927
$681$	−24.1976	−0.927255
$682$	−0.491795	−0.0188318
$683$	−23.8648	−0.913161	−0.456580	−	0.889682i	$-0.650926\pi$
$683$	−23.8648	−0.913161	−0.456580	+	0.889682i	$0.650926\pi$
$684$	−0.0389006	−0.00148740
$685$	55.2736	2.11190
$686$	0	0
$687$	−27.6590	−1.05526
$688$	21.1458	0.806177
$689$	−29.6670	−1.13022
$690$	20.1349	0.766522
$691$	−15.1546	−0.576508	−0.288254	−	0.957554i	$-0.593075\pi$
$691$	−15.1546	−0.576508	−0.288254	+	0.957554i	$0.593075\pi$
$692$	2.50923	0.0953867
$693$	0	0
$694$	−14.9435	−0.567248
$695$	−40.1675	−1.52364
$696$	−21.0495	−0.797877
$697$	−29.0245	−1.09938
$698$	−3.80642	−0.144075
$699$	0.242321	0.00916541
$700$	0	0
$701$	−24.5410	−0.926900	−0.463450	−	0.886123i	$-0.653389\pi$
$701$	−24.5410	−0.926900	−0.463450	+	0.886123i	$0.653389\pi$
$702$	−47.5282	−1.79384
$703$	−1.86399	−0.0703017
$704$	2.42400	0.0913580
$705$	8.19611	0.308683
$706$	4.06518	0.152995
$707$	0	0
$708$	2.01420	0.0756983
$709$	−16.7029	−0.627290	−0.313645	−	0.949540i	$-0.601550\pi$
$709$	−16.7029	−0.627290	−0.313645	+	0.949540i	$0.601550\pi$
$710$	−9.71937	−0.364762
$711$	−4.39908	−0.164978
$712$	6.58072	0.246623
$713$	2.00865	0.0752246
$714$	0	0
$715$	8.49064	0.317532
$716$	3.19814	0.119520
$717$	−28.4854	−1.06381
$718$	−32.5916	−1.21631
$719$	−0.150659	−0.00561864	−0.00280932	−	0.999996i	$-0.500894\pi$
$719$	−0.150659	−0.00561864	−0.00280932	+	0.999996i	$0.500894\pi$
$720$	10.6747	0.397821
$721$	0	0
$722$	27.6800	1.03014
$723$	28.9751	1.07760
$724$	2.02205	0.0751489
$725$	70.4662	2.61705
$726$	−24.9697	−0.926711
$727$	−8.32486	−0.308752	−0.154376	−	0.988012i	$-0.549337\pi$
$727$	−8.32486	−0.308752	−0.154376	+	0.988012i	$0.549337\pi$
$728$	0	0
$729$	29.9931	1.11086
$730$	−66.6352	−2.46628
$731$	−14.1110	−0.521913
$732$	0.259904	0.00960632
$733$	21.8943	0.808685	0.404343	−	0.914608i	$-0.367500\pi$
$733$	21.8943	0.808685	0.404343	+	0.914608i	$0.367500\pi$
$734$	−10.1390	−0.374239
$735$	0	0
$736$	1.84283	0.0679278
$737$	2.39515	0.0882267
$738$	−8.45295	−0.311157
$739$	−1.64886	−0.0606542	−0.0303271	−	0.999540i	$-0.509655\pi$
$739$	−1.64886	−0.0606542	−0.0303271	+	0.999540i	$0.509655\pi$
$740$	−3.14495	−0.115611
$741$	−3.81505	−0.140149
$742$	0	0
$743$	5.14880	0.188891	0.0944455	−	0.995530i	$-0.469892\pi$
$743$	5.14880	0.188891	0.0944455	+	0.995530i	$0.469892\pi$
$744$	−4.21374	−0.154483
$745$	36.0028	1.31904
$746$	14.8706	0.544450
$747$	7.11329	0.260262
$748$	0.155965	0.00570263
$749$	0	0
$750$	91.2804	3.33309
$751$	34.8988	1.27347	0.636737	−	0.771081i	$-0.280284\pi$
$751$	34.8988	1.27347	0.636737	+	0.771081i	$0.280284\pi$
$752$	5.16878	0.188486
$753$	−27.8042	−1.01324
$754$	−42.6692	−1.55392
$755$	−28.3888	−1.03317
$756$	0	0
$757$	−24.3537	−0.885150	−0.442575	−	0.896732i	$-0.645935\pi$
$757$	−24.3537	−0.885150	−0.442575	+	0.896732i	$0.645935\pi$
$758$	0.962812	0.0349709
$759$	−1.04755	−0.0380237
$760$	4.97454	0.180445
$761$	−32.9130	−1.19310	−0.596548	−	0.802577i	$-0.703461\pi$
$761$	−32.9130	−1.19310	−0.596548	+	0.802577i	$0.703461\pi$
$762$	−8.82158	−0.319572
$763$	0	0
$764$	−0.767827	−0.0277790
$765$	−7.12338	−0.257546
$766$	25.5274	0.922342
$767$	−46.1456	−1.66622
$768$	−6.04559	−0.218151
$769$	−29.1096	−1.04972	−0.524859	−	0.851189i	$-0.675882\pi$
$769$	−29.1096	−1.04972	−0.524859	+	0.851189i	$0.675882\pi$
$770$	0	0
$771$	−14.4360	−0.519899
$772$	−0.341845	−0.0123033
$773$	−15.1138	−0.543605	−0.271803	−	0.962353i	$-0.587620\pi$
$773$	−15.1138	−0.543605	−0.271803	+	0.962353i	$0.587620\pi$
$774$	−4.10960	−0.147717
$775$	14.1061	0.506707
$776$	11.5189	0.413506
$777$	0	0
$778$	−26.3719	−0.945480
$779$	−4.26153	−0.152685
$780$	−6.43681	−0.230475
$781$	0.505667	0.0180942
$782$	−8.47348	−0.303011
$783$	27.7961	0.993350
$784$	0	0
$785$	74.4792	2.65828
$786$	6.83253	0.243708
$787$	−43.2670	−1.54230	−0.771151	−	0.636653i	$-0.780319\pi$
$787$	−43.2670	−1.54230	−0.771151	+	0.636653i	$0.780319\pi$
$788$	−0.701718	−0.0249977
$789$	29.4390	1.04806
$790$	−49.7742	−1.77089
$791$	0	0
$792$	−0.513360	−0.0182415
$793$	−5.95443	−0.211448
$794$	−3.44545	−0.122274
$795$	34.8157	1.23479
$796$	−1.01689	−0.0360426
$797$	−9.61790	−0.340683	−0.170342	−	0.985385i	$-0.554487\pi$
$797$	−9.61790	−0.340683	−0.170342	+	0.985385i	$0.554487\pi$
$798$	0	0
$799$	−3.44922	−0.122024
$800$	12.9416	0.457556
$801$	−1.38359	−0.0488867
$802$	3.94059	0.139147
$803$	3.46681	0.122341
$804$	−1.81578	−0.0640377
$805$	0	0
$806$	−8.54164	−0.300867
$807$	28.8067	1.01404
$808$	3.26998	0.115037
$809$	18.0374	0.634161	0.317080	−	0.948399i	$-0.397297\pi$
$809$	18.0374	0.634161	0.317080	+	0.948399i	$0.397297\pi$
$810$	44.8217	1.57487
$811$	−28.6423	−1.00577	−0.502884	−	0.864354i	$-0.667728\pi$
$811$	−28.6423	−1.00577	−0.502884	+	0.864354i	$0.667728\pi$
$812$	0	0
$813$	−47.6845	−1.67237
$814$	2.17649	0.0762858
$815$	15.5100	0.543293
$816$	19.2302	0.673191
$817$	−2.07184	−0.0724846
$818$	37.8024	1.32173
$819$	0	0
$820$	−7.19012	−0.251090
$821$	−14.6425	−0.511025	−0.255513	−	0.966806i	$-0.582244\pi$
$821$	−14.6425	−0.511025	−0.255513	+	0.966806i	$0.582244\pi$
$822$	28.9995	1.01147
$823$	−21.6422	−0.754400	−0.377200	−	0.926132i	$-0.623113\pi$
$823$	−21.6422	−0.754400	−0.377200	+	0.926132i	$0.623113\pi$
$824$	−44.7589	−1.55925
$825$	−7.35662	−0.256125
$826$	0	0
$827$	5.94455	0.206712	0.103356	−	0.994644i	$-0.467042\pi$
$827$	5.94455	0.206712	0.103356	+	0.994644i	$0.467042\pi$
$828$	−0.185519	−0.00644725
$829$	10.0735	0.349868	0.174934	−	0.984580i	$-0.444029\pi$
$829$	10.0735	0.349868	0.174934	+	0.984580i	$0.444029\pi$
$830$	80.4848	2.79367
$831$	42.5335	1.47547
$832$	42.1008	1.45958
$833$	0	0
$834$	−21.0740	−0.729733
$835$	−107.585	−3.72314
$836$	0.0228995	0.000791996 0
$837$	5.56429	0.192330
$838$	15.4717	0.534461
$839$	14.2644	0.492461	0.246231	−	0.969211i	$-0.420808\pi$
$839$	14.2644	0.492461	0.246231	+	0.969211i	$0.420808\pi$
$840$	0	0
$841$	−4.04563	−0.139504
$842$	−12.3339	−0.425053
$843$	−24.9313	−0.858679
$844$	0.182933	0.00629682
$845$	90.6443	3.11826
$846$	−1.00453	−0.0345365
$847$	0	0
$848$	21.9562	0.753978
$849$	18.3044	0.628205
$850$	−59.5065	−2.04106
$851$	−8.88948	−0.304728
$852$	−0.383349	−0.0131333
$853$	3.41266	0.116847	0.0584236	−	0.998292i	$-0.481393\pi$
$853$	3.41266	0.116847	0.0584236	+	0.998292i	$0.481393\pi$
$854$	0	0
$855$	−1.04589	−0.0357687
$856$	43.1583	1.47512
$857$	−0.980031	−0.0334772	−0.0167386	−	0.999860i	$-0.505328\pi$
$857$	−0.980031	−0.0334772	−0.0167386	+	0.999860i	$0.505328\pi$
$858$	4.45464	0.152079
$859$	−44.5634	−1.52048	−0.760242	−	0.649640i	$-0.774920\pi$
$859$	−44.5634	−1.52048	−0.760242	+	0.649640i	$0.774920\pi$
$860$	−3.49565	−0.119201
$861$	0	0
$862$	12.7746	0.435103
$863$	44.8459	1.52657	0.763286	−	0.646061i	$-0.223585\pi$
$863$	44.8459	1.52657	0.763286	+	0.646061i	$0.223585\pi$
$864$	5.10495	0.173674
$865$	67.4638	2.29384
$866$	45.8347	1.55752
$867$	13.6781	0.464532
$868$	0	0
$869$	2.58959	0.0878459
$870$	50.0746	1.69769
$871$	41.5998	1.40955
$872$	14.1262	0.478374
$873$	−2.42184	−0.0819670
$874$	−1.24412	−0.0420829
$875$	0	0
$876$	−2.62821	−0.0887990
$877$	17.4548	0.589407	0.294704	−	0.955589i	$-0.404779\pi$
$877$	17.4548	0.589407	0.294704	+	0.955589i	$0.404779\pi$
$878$	7.92531	0.267466
$879$	−7.66968	−0.258692
$880$	−6.28382	−0.211828
$881$	−6.27922	−0.211552	−0.105776	−	0.994390i	$-0.533733\pi$
$881$	−6.27922	−0.211552	−0.105776	+	0.994390i	$0.533733\pi$
$882$	0	0
$883$	8.89048	0.299189	0.149594	−	0.988747i	$-0.452203\pi$
$883$	8.89048	0.299189	0.149594	+	0.988747i	$0.452203\pi$
$884$	2.70884	0.0911081
$885$	54.1543	1.82038
$886$	20.1259	0.676143
$887$	−21.7374	−0.729870	−0.364935	−	0.931033i	$-0.618909\pi$
$887$	−21.7374	−0.729870	−0.364935	+	0.931033i	$0.618909\pi$
$888$	18.6483	0.625796
$889$	0	0
$890$	−15.6549	−0.524754
$891$	−2.33192	−0.0781224
$892$	−4.23866	−0.141921
$893$	−0.506431	−0.0169471
$894$	18.8890	0.631742
$895$	85.9859	2.87419
$896$	0	0
$897$	−18.1942	−0.607486
$898$	29.0840	0.970546
$899$	4.99544	0.166607
$900$	−1.30284	−0.0434281
$901$	−14.6517	−0.488119
$902$	4.97597	0.165682
$903$	0	0
$904$	24.7557	0.823364
$905$	54.3653	1.80716
$906$	−14.8942	−0.494828
$907$	7.17151	0.238126	0.119063	−	0.992887i	$-0.462011\pi$
$907$	7.17151	0.238126	0.119063	+	0.992887i	$0.462011\pi$
$908$	2.52265	0.0837170
$909$	−0.687509	−0.0228032
$910$	0	0
$911$	34.7540	1.15145	0.575726	−	0.817643i	$-0.304720\pi$
$911$	34.7540	1.15145	0.575726	+	0.817643i	$0.304720\pi$
$912$	2.82347	0.0934944
$913$	−4.18736	−0.138581
$914$	2.36387	0.0781898
$915$	6.98784	0.231011
$916$	2.88350	0.0952736
$917$	0	0
$918$	−23.4729	−0.774722
$919$	51.0930	1.68540	0.842700	−	0.538383i	$-0.180964\pi$
$919$	51.0930	1.68540	0.842700	+	0.538383i	$0.180964\pi$
$920$	23.7239	0.782154
$921$	−27.3062	−0.899770
$922$	−1.40822	−0.0463774
$923$	8.78258	0.289082
$924$	0	0
$925$	−62.4280	−2.05262
$926$	21.1311	0.694410
$927$	9.41051	0.309082
$928$	4.58305	0.150446
$929$	−28.3627	−0.930550	−0.465275	−	0.885166i	$-0.654045\pi$
$929$	−28.3627	−0.930550	−0.465275	+	0.885166i	$0.654045\pi$
$930$	10.0241	0.328702
$931$	0	0
$932$	−0.0252624	−0.000827497 0
$933$	−17.5674	−0.575132
$934$	18.8403	0.616475
$935$	4.19330	0.137136
$936$	−8.91620	−0.291435
$937$	43.8732	1.43328	0.716638	−	0.697446i	$-0.245680\pi$
$937$	43.8732	1.43328	0.716638	+	0.697446i	$0.245680\pi$
$938$	0	0
$939$	38.5693	1.25866
$940$	−0.854459	−0.0278694
$941$	15.9227	0.519066	0.259533	−	0.965734i	$-0.416431\pi$
$941$	15.9227	0.519066	0.259533	+	0.965734i	$0.416431\pi$
$942$	39.0757	1.27316
$943$	−20.3235	−0.661824
$944$	34.1518	1.11155
$945$	0	0
$946$	2.41919	0.0786546
$947$	11.6994	0.380181	0.190090	−	0.981767i	$-0.439122\pi$
$947$	11.6994	0.380181	0.190090	+	0.981767i	$0.439122\pi$
$948$	−1.96319	−0.0637613
$949$	60.2126	1.95458
$950$	−8.73705	−0.283467
$951$	42.4293	1.37586
$952$	0	0
$953$	−53.1045	−1.72022	−0.860112	−	0.510105i	$-0.829606\pi$
$953$	−53.1045	−1.72022	−0.860112	+	0.510105i	$0.829606\pi$
$954$	−4.26709	−0.138152
$955$	−20.6440	−0.668023
$956$	2.96965	0.0960455
$957$	−2.60522	−0.0842148
$958$	17.9283	0.579237
$959$	0	0
$960$	−49.4075	−1.59462
$961$	1.00000	0.0322581
$962$	37.8019	1.21878
$963$	−9.07398	−0.292405
$964$	−3.02071	−0.0972904
$965$	−9.19093	−0.295866
$966$	0	0
$967$	25.5960	0.823112	0.411556	−	0.911385i	$-0.364985\pi$
$967$	25.5960	0.823112	0.411556	+	0.911385i	$0.364985\pi$
$968$	−29.4205	−0.945610
$969$	−1.88415	−0.0605276
$970$	−27.4024	−0.879839
$971$	−54.0563	−1.73475	−0.867375	−	0.497655i	$-0.834194\pi$
$971$	−54.0563	−1.73475	−0.867375	+	0.497655i	$0.834194\pi$
$972$	−0.946013	−0.0303434
$973$	0	0
$974$	−58.1328	−1.86270
$975$	−127.772	−4.09198
$976$	4.40680	0.141058
$977$	−17.3425	−0.554834	−0.277417	−	0.960750i	$-0.589478\pi$
$977$	−17.3425	−0.554834	−0.277417	+	0.960750i	$0.589478\pi$
$978$	8.13738	0.260205
$979$	0.814474	0.0260307
$980$	0	0
$981$	−2.97002	−0.0948254
$982$	8.49740	0.271163
$983$	−46.7270	−1.49036	−0.745181	−	0.666863i	$-0.767637\pi$
$983$	−46.7270	−1.49036	−0.745181	+	0.666863i	$0.767637\pi$
$984$	42.6345	1.35914
$985$	−18.8666	−0.601138
$986$	−21.0732	−0.671107
$987$	0	0
$988$	0.397725	0.0126533
$989$	−9.88075	−0.314190
$990$	1.22123	0.0388134
$991$	−57.0083	−1.81093	−0.905464	−	0.424423i	$-0.860477\pi$
$991$	−57.0083	−1.81093	−0.905464	+	0.424423i	$0.860477\pi$
$992$	0.917448	0.0291290
$993$	−40.9852	−1.30063
$994$	0	0
$995$	−27.3403	−0.866745
$996$	3.17446	0.100587
$997$	30.6155	0.969601	0.484801	−	0.874625i	$-0.338892\pi$
$997$	30.6155	0.969601	0.484801	+	0.874625i	$0.338892\pi$
$998$	−8.19360	−0.259364
$999$	−24.6253	−0.779111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.2.a.k.1.3		13
7.2	even	3		217.2.f.b.32.11	✓	26
7.4	even	3		217.2.f.b.156.11	yes	26
7.6	odd	2		1519.2.a.j.1.3		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
217.2.f.b.32.11	✓	26	7.2	even	3
217.2.f.b.156.11	yes	26	7.4	even	3
1519.2.a.j.1.3		13	7.6	odd	2
1519.2.a.k.1.3		13	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:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1519.a (trivial)

\(f(q)\)	\(=\)	\(q-1.47057 q^{2} -1.55945 q^{3} +0.162576 q^{4} +4.37105 q^{5} +2.29329 q^{6} +2.70206 q^{8} -0.568105 q^{9} +O(q^{10})\)	\(q-1.47057 q^{2} -1.55945 q^{3} +0.162576 q^{4} +4.37105 q^{5} +2.29329 q^{6} +2.70206 q^{8} -0.568105 q^{9} -6.42794 q^{10} +0.334425 q^{11} -0.253529 q^{12} +5.80839 q^{13} -6.81646 q^{15} -4.29872 q^{16} +2.86861 q^{17} +0.835438 q^{18} +0.421184 q^{19} +0.710628 q^{20} -0.491795 q^{22} +2.00865 q^{23} -4.21374 q^{24} +14.1061 q^{25} -8.54164 q^{26} +5.56429 q^{27} +4.99544 q^{29} +10.0241 q^{30} +1.00000 q^{31} +0.917448 q^{32} -0.521520 q^{33} -4.21849 q^{34} -0.0923602 q^{36} -4.42560 q^{37} -0.619380 q^{38} -9.05791 q^{39} +11.8109 q^{40} -10.1180 q^{41} -4.91910 q^{43} +0.0543694 q^{44} -2.48322 q^{45} -2.95386 q^{46} -1.20240 q^{47} +6.70365 q^{48} -20.7440 q^{50} -4.47346 q^{51} +0.944304 q^{52} -5.10760 q^{53} -8.18268 q^{54} +1.46179 q^{55} -0.656816 q^{57} -7.34614 q^{58} -7.94464 q^{59} -1.10819 q^{60} -1.02514 q^{61} -1.47057 q^{62} +7.24827 q^{64} +25.3888 q^{65} +0.766932 q^{66} +7.16201 q^{67} +0.466367 q^{68} -3.13240 q^{69} +1.51205 q^{71} -1.53505 q^{72} +10.3665 q^{73} +6.50815 q^{74} -21.9978 q^{75} +0.0684743 q^{76} +13.3203 q^{78} +7.74342 q^{79} -18.7899 q^{80} -6.97294 q^{81} +14.8792 q^{82} -12.5211 q^{83} +12.5388 q^{85} +7.23387 q^{86} -7.79015 q^{87} +0.903636 q^{88} +2.43545 q^{89} +3.65175 q^{90} +0.326558 q^{92} -1.55945 q^{93} +1.76821 q^{94} +1.84102 q^{95} -1.43072 q^{96} +4.26302 q^{97} -0.189988 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 5 q^{2} + 17 q^{4} + q^{5} - 2 q^{6} + 12 q^{8} + 25 q^{9}+O(q^{10}) \) 13 * q + 5 * q^2 + 17 * q^4 + q^5 - 2 * q^6 + 12 * q^8 + 25 * q^9	\( 13 q + 5 q^{2} + 17 q^{4} + q^{5} - 2 q^{6} + 12 q^{8} + 25 q^{9} - 7 q^{10} + 15 q^{11} - 5 q^{12} + 4 q^{13} + 4 q^{15} + 29 q^{16} - 4 q^{17} + 16 q^{18} + 2 q^{19} + 26 q^{20} - 10 q^{22} + 14 q^{23} - 28 q^{24} + 24 q^{25} - 7 q^{26} + 12 q^{27} + 22 q^{29} + 6 q^{30} + 13 q^{31} + 19 q^{32} - 5 q^{33} + 20 q^{34} + 11 q^{36} + 12 q^{37} - 11 q^{38} + 11 q^{39} - 6 q^{40} + 4 q^{41} - 3 q^{43} + 52 q^{44} - 12 q^{45} - 3 q^{46} - 14 q^{47} + 48 q^{48} + 15 q^{50} + 16 q^{51} + 4 q^{52} + 19 q^{53} - 25 q^{54} + 18 q^{55} + 13 q^{57} + 24 q^{58} + 19 q^{59} + 6 q^{60} - 11 q^{61} + 5 q^{62} + 10 q^{64} + 68 q^{65} - 52 q^{66} - 25 q^{67} - 26 q^{68} + 52 q^{69} + 28 q^{71} + 52 q^{72} - 29 q^{73} + 54 q^{74} - 71 q^{75} + 37 q^{76} - 71 q^{78} + 30 q^{79} + 3 q^{80} + 25 q^{81} + 5 q^{82} - 10 q^{83} - q^{85} + 10 q^{86} + 50 q^{87} + 18 q^{88} - 11 q^{89} + 81 q^{90} + 35 q^{92} - 36 q^{94} - 20 q^{95} - 12 q^{96} - 3 q^{97} + 42 q^{99}+O(q^{100}) \) 13 * q + 5 * q^2 + 17 * q^4 + q^5 - 2 * q^6 + 12 * q^8 + 25 * q^9 - 7 * q^10 + 15 * q^11 - 5 * q^12 + 4 * q^13 + 4 * q^15 + 29 * q^16 - 4 * q^17 + 16 * q^18 + 2 * q^19 + 26 * q^20 - 10 * q^22 + 14 * q^23 - 28 * q^24 + 24 * q^25 - 7 * q^26 + 12 * q^27 + 22 * q^29 + 6 * q^30 + 13 * q^31 + 19 * q^32 - 5 * q^33 + 20 * q^34 + 11 * q^36 + 12 * q^37 - 11 * q^38 + 11 * q^39 - 6 * q^40 + 4 * q^41 - 3 * q^43 + 52 * q^44 - 12 * q^45 - 3 * q^46 - 14 * q^47 + 48 * q^48 + 15 * q^50 + 16 * q^51 + 4 * q^52 + 19 * q^53 - 25 * q^54 + 18 * q^55 + 13 * q^57 + 24 * q^58 + 19 * q^59 + 6 * q^60 - 11 * q^61 + 5 * q^62 + 10 * q^64 + 68 * q^65 - 52 * q^66 - 25 * q^67 - 26 * q^68 + 52 * q^69 + 28 * q^71 + 52 * q^72 - 29 * q^73 + 54 * q^74 - 71 * q^75 + 37 * q^76 - 71 * q^78 + 30 * q^79 + 3 * q^80 + 25 * q^81 + 5 * q^82 - 10 * q^83 - q^85 + 10 * q^86 + 50 * q^87 + 18 * q^88 - 11 * q^89 + 81 * q^90 + 35 * q^92 - 36 * q^94 - 20 * q^95 - 12 * q^96 - 3 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more