LMFDB - Embedded newform 5239.2.a.u.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5239,2,Mod(1,5239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5239, 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("5239.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.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:	$41.8336256189$
Analytic rank:	$0$
Dimension:	$54$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	5239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54225 q^{2} +0.532354 q^{3} +4.46305 q^{4} -3.72054 q^{5} -1.35338 q^{6} -4.87084 q^{7} -6.26171 q^{8} -2.71660 q^{9} +O(q^{10})$	\(q-2.54225 q^{2} +0.532354 q^{3} +4.46305 q^{4} -3.72054 q^{5} -1.35338 q^{6} -4.87084 q^{7} -6.26171 q^{8} -2.71660 q^{9} +9.45857 q^{10} +4.55621 q^{11} +2.37593 q^{12} +12.3829 q^{14} -1.98065 q^{15} +6.99275 q^{16} +1.64964 q^{17} +6.90628 q^{18} +3.55793 q^{19} -16.6050 q^{20} -2.59301 q^{21} -11.5831 q^{22} -4.85665 q^{23} -3.33345 q^{24} +8.84245 q^{25} -3.04326 q^{27} -21.7388 q^{28} +0.682716 q^{29} +5.03531 q^{30} -1.00000 q^{31} -5.25392 q^{32} +2.42552 q^{33} -4.19382 q^{34} +18.1222 q^{35} -12.1243 q^{36} -8.45676 q^{37} -9.04515 q^{38} +23.2970 q^{40} +1.77602 q^{41} +6.59210 q^{42} -12.2856 q^{43} +20.3346 q^{44} +10.1072 q^{45} +12.3468 q^{46} -9.80027 q^{47} +3.72262 q^{48} +16.7251 q^{49} -22.4798 q^{50} +0.878195 q^{51} +1.80433 q^{53} +7.73673 q^{54} -16.9516 q^{55} +30.4998 q^{56} +1.89408 q^{57} -1.73564 q^{58} -3.23388 q^{59} -8.83974 q^{60} -2.95677 q^{61} +2.54225 q^{62} +13.2321 q^{63} -0.628702 q^{64} -6.16629 q^{66} -0.394392 q^{67} +7.36245 q^{68} -2.58546 q^{69} -46.0712 q^{70} -1.02705 q^{71} +17.0106 q^{72} -13.9671 q^{73} +21.4992 q^{74} +4.70732 q^{75} +15.8792 q^{76} -22.1926 q^{77} -16.0274 q^{79} -26.0168 q^{80} +6.52971 q^{81} -4.51509 q^{82} -6.69429 q^{83} -11.5728 q^{84} -6.13758 q^{85} +31.2331 q^{86} +0.363447 q^{87} -28.5297 q^{88} -3.60657 q^{89} -25.6951 q^{90} -21.6755 q^{92} -0.532354 q^{93} +24.9148 q^{94} -13.2374 q^{95} -2.79694 q^{96} -3.84782 q^{97} -42.5195 q^{98} -12.3774 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 2 q^{2} + 7 q^{3} + 64 q^{4} - 5 q^{5} + 3 q^{6} - 5 q^{7} - 6 q^{8} + 95 q^{9}+O(q^{10}) $ 54 * q - 2 * q^2 + 7 * q^3 + 64 * q^4 - 5 * q^5 + 3 * q^6 - 5 * q^7 - 6 * q^8 + 95 * q^9	\( 54 q - 2 q^{2} + 7 q^{3} + 64 q^{4} - 5 q^{5} + 3 q^{6} - 5 q^{7} - 6 q^{8} + 95 q^{9} - 6 q^{10} + 7 q^{11} + 5 q^{12} + 38 q^{14} - 4 q^{15} + 76 q^{16} + 62 q^{17} + 9 q^{18} - 8 q^{19} - 16 q^{20} + 6 q^{21} + 15 q^{22} + 38 q^{23} + 99 q^{24} + 87 q^{25} + 25 q^{27} - 19 q^{28} + 95 q^{29} + 41 q^{30} - 54 q^{31} - 9 q^{32} - 12 q^{33} - 7 q^{34} + 53 q^{35} + 97 q^{36} + 24 q^{37} - 16 q^{38} - 28 q^{40} - 22 q^{41} + 11 q^{42} + 11 q^{43} + 24 q^{44} - 8 q^{45} - 9 q^{46} - 45 q^{47} + 2 q^{48} + 105 q^{49} - 6 q^{50} + 58 q^{51} + 56 q^{53} - 50 q^{54} + q^{55} + 91 q^{56} + 51 q^{57} - 25 q^{58} - 36 q^{59} - 100 q^{60} + 48 q^{61} + 2 q^{62} + 56 q^{63} + 90 q^{64} - 24 q^{66} - 26 q^{67} + 140 q^{68} + 47 q^{69} + 24 q^{70} - 40 q^{71} - 7 q^{72} - 9 q^{73} + 114 q^{74} + 18 q^{75} + 67 q^{76} + 65 q^{77} + 33 q^{79} - 53 q^{80} + 210 q^{81} - 6 q^{82} + 41 q^{83} + 37 q^{84} - 37 q^{85} + 42 q^{86} - 16 q^{87} - 22 q^{88} + 24 q^{89} - 40 q^{90} + 87 q^{92} - 7 q^{93} - 4 q^{94} + 61 q^{95} + 200 q^{96} - 28 q^{97} - 68 q^{98} - 39 q^{99}+O(q^{100}) \) 54 * q - 2 * q^2 + 7 * q^3 + 64 * q^4 - 5 * q^5 + 3 * q^6 - 5 * q^7 - 6 * q^8 + 95 * q^9 - 6 * q^10 + 7 * q^11 + 5 * q^12 + 38 * q^14 - 4 * q^15 + 76 * q^16 + 62 * q^17 + 9 * q^18 - 8 * q^19 - 16 * q^20 + 6 * q^21 + 15 * q^22 + 38 * q^23 + 99 * q^24 + 87 * q^25 + 25 * q^27 - 19 * q^28 + 95 * q^29 + 41 * q^30 - 54 * q^31 - 9 * q^32 - 12 * q^33 - 7 * q^34 + 53 * q^35 + 97 * q^36 + 24 * q^37 - 16 * q^38 - 28 * q^40 - 22 * q^41 + 11 * q^42 + 11 * q^43 + 24 * q^44 - 8 * q^45 - 9 * q^46 - 45 * q^47 + 2 * q^48 + 105 * q^49 - 6 * q^50 + 58 * q^51 + 56 * q^53 - 50 * q^54 + q^55 + 91 * q^56 + 51 * q^57 - 25 * q^58 - 36 * q^59 - 100 * q^60 + 48 * q^61 + 2 * q^62 + 56 * q^63 + 90 * q^64 - 24 * q^66 - 26 * q^67 + 140 * q^68 + 47 * q^69 + 24 * q^70 - 40 * q^71 - 7 * q^72 - 9 * q^73 + 114 * q^74 + 18 * q^75 + 67 * q^76 + 65 * q^77 + 33 * q^79 - 53 * q^80 + 210 * q^81 - 6 * q^82 + 41 * q^83 + 37 * q^84 - 37 * q^85 + 42 * q^86 - 16 * q^87 - 22 * q^88 + 24 * q^89 - 40 * q^90 + 87 * q^92 - 7 * q^93 - 4 * q^94 + 61 * q^95 + 200 * q^96 - 28 * q^97 - 68 * q^98 - 39 * 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 $

$2$	−2.54225	−1.79764	−0.898822	−	0.438313i	$-0.855576\pi$
$2$	−2.54225	−1.79764	−0.898822	+	0.438313i	$0.855576\pi$
$3$	0.532354	0.307355	0.153677	−	0.988121i	$-0.450888\pi$
$3$	0.532354	0.307355	0.153677	+	0.988121i	$0.450888\pi$
$4$	4.46305	2.23153
$5$	−3.72054	−1.66388	−0.831939	−	0.554867i	$-0.812769\pi$
$5$	−3.72054	−1.66388	−0.831939	+	0.554867i	$0.812769\pi$
$6$	−1.35338	−0.552515
$7$	−4.87084	−1.84101	−0.920503	−	0.390736i	$-0.872220\pi$
$7$	−4.87084	−1.84101	−0.920503	+	0.390736i	$0.872220\pi$
$8$	−6.26171	−2.21385
$9$	−2.71660	−0.905533
$10$	9.45857	2.99106
$11$	4.55621	1.37375	0.686875	−	0.726776i	$-0.258982\pi$
$11$	4.55621	1.37375	0.686875	+	0.726776i	$0.258982\pi$
$12$	2.37593	0.685871
$13$	0	0
$13$	0	0
$14$	12.3829	3.30947
$15$	−1.98065	−0.511401
$16$	6.99275	1.74819
$17$	1.64964	0.400098	0.200049	−	0.979786i	$-0.435890\pi$
$17$	1.64964	0.400098	0.200049	+	0.979786i	$0.435890\pi$
$18$	6.90628	1.62783
$19$	3.55793	0.816244	0.408122	−	0.912927i	$-0.366184\pi$
$19$	3.55793	0.816244	0.408122	+	0.912927i	$0.366184\pi$
$20$	−16.6050	−3.71299
$21$	−2.59301	−0.565842
$22$	−11.5831	−2.46951
$23$	−4.85665	−1.01268	−0.506341	−	0.862334i	$-0.669002\pi$
$23$	−4.85665	−1.01268	−0.506341	+	0.862334i	$0.669002\pi$
$24$	−3.33345	−0.680437
$25$	8.84245	1.76849
$26$	0	0
$27$	−3.04326	−0.585675
$28$	−21.7388	−4.10825
$29$	0.682716	0.126777	0.0633886	−	0.997989i	$-0.479809\pi$
$29$	0.682716	0.126777	0.0633886	+	0.997989i	$0.479809\pi$
$30$	5.03531	0.919317
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.25392	−0.928770
$33$	2.42552	0.422229
$34$	−4.19382	−0.719233
$35$	18.1222	3.06321
$36$	−12.1243	−2.02072
$37$	−8.45676	−1.39028	−0.695142	−	0.718873i	$-0.744658\pi$
$37$	−8.45676	−1.39028	−0.695142	+	0.718873i	$0.744658\pi$
$38$	−9.04515	−1.46732
$39$	0	0
$40$	23.2970	3.68357
$41$	1.77602	0.277367	0.138684	−	0.990337i	$-0.455713\pi$
$41$	1.77602	0.277367	0.138684	+	0.990337i	$0.455713\pi$
$42$	6.59210	1.01718
$43$	−12.2856	−1.87353	−0.936766	−	0.349955i	$-0.886197\pi$
$43$	−12.2856	−1.87353	−0.936766	+	0.349955i	$0.886197\pi$
$44$	20.3346	3.06556
$45$	10.1072	1.50670
$46$	12.3468	1.82044
$47$	−9.80027	−1.42952	−0.714758	−	0.699371i	$-0.753463\pi$
$47$	−9.80027	−1.42952	−0.714758	+	0.699371i	$0.753463\pi$
$48$	3.72262	0.537314
$49$	16.7251	2.38930
$50$	−22.4798	−3.17912
$51$	0.878195	0.122972
$52$	0	0
$53$	1.80433	0.247844	0.123922	−	0.992292i	$-0.460453\pi$
$53$	1.80433	0.247844	0.123922	+	0.992292i	$0.460453\pi$
$54$	7.73673	1.05284
$55$	−16.9516	−2.28575
$56$	30.4998	4.07571
$57$	1.89408	0.250877
$58$	−1.73564	−0.227900
$59$	−3.23388	−0.421016	−0.210508	−	0.977592i	$-0.567512\pi$
$59$	−3.23388	−0.421016	−0.210508	+	0.977592i	$0.567512\pi$
$60$	−8.83974	−1.14121
$61$	−2.95677	−0.378576	−0.189288	−	0.981922i	$-0.560618\pi$
$61$	−2.95677	−0.378576	−0.189288	+	0.981922i	$0.560618\pi$
$62$	2.54225	0.322867
$63$	13.2321	1.66709
$64$	−0.628702	−0.0785878
$65$	0	0
$66$	−6.16629	−0.759017
$67$	−0.394392	−0.0481826	−0.0240913	−	0.999710i	$-0.507669\pi$
$67$	−0.394392	−0.0481826	−0.0240913	+	0.999710i	$0.507669\pi$
$68$	7.36245	0.892829
$69$	−2.58546	−0.311252
$70$	−46.0712	−5.50656
$71$	−1.02705	−0.121889	−0.0609444	−	0.998141i	$-0.519411\pi$
$71$	−1.02705	−0.121889	−0.0609444	+	0.998141i	$0.519411\pi$
$72$	17.0106	2.00471
$73$	−13.9671	−1.63473	−0.817365	−	0.576120i	$-0.804566\pi$
$73$	−13.9671	−1.63473	−0.817365	+	0.576120i	$0.804566\pi$
$74$	21.4992	2.49924
$75$	4.70732	0.543554
$76$	15.8792	1.82147
$77$	−22.1926	−2.52908
$78$	0	0
$79$	−16.0274	−1.80322	−0.901612	−	0.432545i	$-0.857616\pi$
$79$	−16.0274	−1.80322	−0.901612	+	0.432545i	$0.857616\pi$
$80$	−26.0168	−2.90877
$81$	6.52971	0.725523
$82$	−4.51509	−0.498608
$83$	−6.69429	−0.734794	−0.367397	−	0.930064i	$-0.619751\pi$
$83$	−6.69429	−0.734794	−0.367397	+	0.930064i	$0.619751\pi$
$84$	−11.5728	−1.26269
$85$	−6.13758	−0.665714
$86$	31.2331	3.36795
$87$	0.363447	0.0389656
$88$	−28.5297	−3.04127
$89$	−3.60657	−0.382296	−0.191148	−	0.981561i	$-0.561221\pi$
$89$	−3.60657	−0.382296	−0.191148	+	0.981561i	$0.561221\pi$
$90$	−25.6951	−2.70851
$91$	0	0
$92$	−21.6755	−2.25983
$93$	−0.532354	−0.0552026
$94$	24.9148	2.56976
$95$	−13.2374	−1.35813
$96$	−2.79694	−0.285462
$97$	−3.84782	−0.390687	−0.195344	−	0.980735i	$-0.562582\pi$
$97$	−3.84782	−0.390687	−0.195344	+	0.980735i	$0.562582\pi$
$98$	−42.5195	−4.29511
$99$	−12.3774	−1.24398
$100$	39.4644	3.94644
$101$	−10.6880	−1.06349	−0.531746	−	0.846904i	$-0.678464\pi$
$101$	−10.6880	−1.06349	−0.531746	+	0.846904i	$0.678464\pi$
$102$	−2.23260	−0.221060
$103$	−2.13180	−0.210053	−0.105026	−	0.994469i	$-0.533493\pi$
$103$	−2.13180	−0.210053	−0.105026	+	0.994469i	$0.533493\pi$
$104$	0	0
$105$	9.64742	0.941492
$106$	−4.58707	−0.445536
$107$	−12.8198	−1.23934	−0.619670	−	0.784863i	$-0.712733\pi$
$107$	−12.8198	−1.23934	−0.619670	+	0.784863i	$0.712733\pi$
$108$	−13.5822	−1.30695
$109$	−7.27370	−0.696694	−0.348347	−	0.937366i	$-0.613257\pi$
$109$	−7.27370	−0.696694	−0.348347	+	0.937366i	$0.613257\pi$
$110$	43.0953	4.10897
$111$	−4.50199	−0.427310
$112$	−34.0606	−3.21842
$113$	7.52224	0.707632	0.353816	−	0.935315i	$-0.384884\pi$
$113$	7.52224	0.707632	0.353816	+	0.935315i	$0.384884\pi$
$114$	−4.81522	−0.450987
$115$	18.0694	1.68498
$116$	3.04700	0.282907
$117$	0	0
$118$	8.22135	0.756837
$119$	−8.03516	−0.736582
$120$	12.4022	1.13216
$121$	9.75908	0.887189
$122$	7.51686	0.680545
$123$	0.945471	0.0852502
$124$	−4.46305	−0.400794
$125$	−14.2960	−1.27868
$126$	−33.6394	−2.99684
$127$	−15.3834	−1.36505	−0.682527	−	0.730860i	$-0.739119\pi$
$127$	−15.3834	−1.36505	−0.682527	+	0.730860i	$0.739119\pi$
$128$	12.1062	1.07004
$129$	−6.54028	−0.575839
$130$	0	0
$131$	3.22938	0.282152	0.141076	−	0.989999i	$-0.454944\pi$
$131$	3.22938	0.282152	0.141076	+	0.989999i	$0.454944\pi$
$132$	10.8252	0.942215
$133$	−17.3301	−1.50271
$134$	1.00264	0.0866152
$135$	11.3226	0.974492
$136$	−10.3296	−0.885756
$137$	7.27484	0.621531	0.310766	−	0.950487i	$-0.399415\pi$
$137$	7.27484	0.621531	0.310766	+	0.950487i	$0.399415\pi$
$138$	6.57289	0.559521
$139$	3.92390	0.332820	0.166410	−	0.986057i	$-0.446782\pi$
$139$	3.92390	0.332820	0.166410	+	0.986057i	$0.446782\pi$
$140$	80.8803	6.83563
$141$	−5.21722	−0.439369
$142$	2.61103	0.219113
$143$	0	0
$144$	−18.9965	−1.58304
$145$	−2.54008	−0.210942
$146$	35.5080	2.93867
$147$	8.90368	0.734363
$148$	−37.7430	−3.10246
$149$	22.4361	1.83804	0.919020	−	0.394211i	$-0.128982\pi$
$149$	22.4361	1.83804	0.919020	+	0.394211i	$0.128982\pi$
$150$	−11.9672	−0.977117
$151$	4.76677	0.387914	0.193957	−	0.981010i	$-0.437868\pi$
$151$	4.76677	0.387914	0.193957	+	0.981010i	$0.437868\pi$
$152$	−22.2787	−1.80704
$153$	−4.48142	−0.362302
$154$	56.4192	4.54639
$155$	3.72054	0.298841
$156$	0	0
$157$	−7.36440	−0.587743	−0.293872	−	0.955845i	$-0.594944\pi$
$157$	−7.36440	−0.587743	−0.293872	+	0.955845i	$0.594944\pi$
$158$	40.7457	3.24156
$159$	0.960545	0.0761761
$160$	19.5474	1.54536
$161$	23.6560	1.86435
$162$	−16.6002	−1.30423
$163$	11.9933	0.939390	0.469695	−	0.882829i	$-0.344364\pi$
$163$	11.9933	0.939390	0.469695	+	0.882829i	$0.344364\pi$
$164$	7.92647	0.618953
$165$	−9.02425	−0.702537
$166$	17.0186	1.32090
$167$	18.0038	1.39318	0.696589	−	0.717470i	$-0.254700\pi$
$167$	18.0038	1.39318	0.696589	+	0.717470i	$0.254700\pi$
$168$	16.2367	1.25269
$169$	0	0
$170$	15.6033	1.19672
$171$	−9.66546	−0.739136
$172$	−54.8312	−4.18084
$173$	6.50330	0.494437	0.247218	−	0.968960i	$-0.420484\pi$
$173$	6.50330	0.494437	0.247218	+	0.968960i	$0.420484\pi$
$174$	−0.923974	−0.0700463
$175$	−43.0702	−3.25580
$176$	31.8604	2.40157
$177$	−1.72157	−0.129401
$178$	9.16882	0.687232
$179$	−22.1561	−1.65602	−0.828010	−	0.560713i	$-0.810527\pi$
$179$	−22.1561	−1.65602	−0.828010	+	0.560713i	$0.810527\pi$
$180$	45.1091	3.36223
$181$	−19.0939	−1.41924	−0.709618	−	0.704586i	$-0.751133\pi$
$181$	−19.0939	−1.41924	−0.709618	+	0.704586i	$0.751133\pi$
$182$	0	0
$183$	−1.57405	−0.116357
$184$	30.4109	2.24192
$185$	31.4638	2.31326
$186$	1.35338	0.0992346
$187$	7.51613	0.549634
$188$	−43.7392	−3.19001
$189$	14.8232	1.07823
$190$	33.6529	2.44144
$191$	1.88040	0.136061	0.0680303	−	0.997683i	$-0.478329\pi$
$191$	1.88040	0.136061	0.0680303	+	0.997683i	$0.478329\pi$
$192$	−0.334692	−0.0241543
$193$	18.1627	1.30738	0.653688	−	0.756764i	$-0.273221\pi$
$193$	18.1627	1.30738	0.653688	+	0.756764i	$0.273221\pi$
$194$	9.78214	0.702317
$195$	0	0
$196$	74.6450	5.33179
$197$	4.64788	0.331148	0.165574	−	0.986197i	$-0.447052\pi$
$197$	4.64788	0.331148	0.165574	+	0.986197i	$0.447052\pi$
$198$	31.4665	2.23623
$199$	−7.63159	−0.540989	−0.270495	−	0.962721i	$-0.587187\pi$
$199$	−7.63159	−0.540989	−0.270495	+	0.962721i	$0.587187\pi$
$200$	−55.3689	−3.91517
$201$	−0.209956	−0.0148092
$202$	27.1715	1.91178
$203$	−3.32540	−0.233398
$204$	3.91943	0.274415
$205$	−6.60776	−0.461506
$206$	5.41958	0.377600
$207$	13.1936	0.917016
$208$	0	0
$209$	16.2107	1.12132
$210$	−24.5262	−1.69247
$211$	−7.37363	−0.507621	−0.253811	−	0.967254i	$-0.581684\pi$
$211$	−7.37363	−0.507621	−0.253811	+	0.967254i	$0.581684\pi$
$212$	8.05284	0.553071
$213$	−0.546756	−0.0374631
$214$	32.5912	2.22789
$215$	45.7090	3.11733
$216$	19.0560	1.29660
$217$	4.87084	0.330654
$218$	18.4916	1.25241
$219$	−7.43547	−0.502442
$220$	−75.6559	−5.10072
$221$	0	0
$222$	11.4452	0.768152
$223$	23.9961	1.60690	0.803448	−	0.595375i	$-0.202996\pi$
$223$	23.9961	1.60690	0.803448	+	0.595375i	$0.202996\pi$
$224$	25.5910	1.70987
$225$	−24.0214	−1.60143
$226$	−19.1234	−1.27207
$227$	9.70273	0.643993	0.321996	−	0.946741i	$-0.395646\pi$
$227$	9.70273	0.643993	0.321996	+	0.946741i	$0.395646\pi$
$228$	8.45337	0.559838
$229$	4.03720	0.266786	0.133393	−	0.991063i	$-0.457413\pi$
$229$	4.03720	0.266786	0.133393	+	0.991063i	$0.457413\pi$
$230$	−45.9369	−3.02899
$231$	−11.8143	−0.777325
$232$	−4.27497	−0.280666
$233$	22.8715	1.49836	0.749182	−	0.662365i	$-0.230447\pi$
$233$	22.8715	1.49836	0.749182	+	0.662365i	$0.230447\pi$
$234$	0	0
$235$	36.4624	2.37854
$236$	−14.4330	−0.939508
$237$	−8.53226	−0.554230
$238$	20.4274	1.32411
$239$	−27.6314	−1.78733	−0.893663	−	0.448738i	$-0.851874\pi$
$239$	−27.6314	−1.78733	−0.893663	+	0.448738i	$0.851874\pi$
$240$	−13.8502	−0.894024
$241$	−17.2929	−1.11393	−0.556966	−	0.830535i	$-0.688035\pi$
$241$	−17.2929	−1.11393	−0.556966	+	0.830535i	$0.688035\pi$
$242$	−24.8101	−1.59485
$243$	12.6059	0.808668
$244$	−13.1962	−0.844802
$245$	−62.2265	−3.97550
$246$	−2.40363	−0.153250
$247$	0	0
$248$	6.26171	0.397619
$249$	−3.56373	−0.225842
$250$	36.3441	2.29860
$251$	30.7008	1.93782	0.968908	−	0.247420i	$-0.0795826\pi$
$251$	30.7008	1.93782	0.968908	+	0.247420i	$0.0795826\pi$
$252$	59.0557	3.72016
$253$	−22.1279	−1.39117
$254$	39.1084	2.45388
$255$	−3.26736	−0.204610
$256$	−29.5195	−1.84497
$257$	−8.01686	−0.500078	−0.250039	−	0.968236i	$-0.580443\pi$
$257$	−8.01686	−0.500078	−0.250039	+	0.968236i	$0.580443\pi$
$258$	16.6270	1.03515
$259$	41.1916	2.55952
$260$	0	0
$261$	−1.85467	−0.114801
$262$	−8.20989	−0.507209
$263$	15.7406	0.970606	0.485303	−	0.874346i	$-0.338709\pi$
$263$	15.7406	0.970606	0.485303	+	0.874346i	$0.338709\pi$
$264$	−15.1879	−0.934751
$265$	−6.71310	−0.412383
$266$	44.0575	2.70134
$267$	−1.91997	−0.117500
$268$	−1.76019	−0.107521
$269$	19.8905	1.21275	0.606374	−	0.795180i	$-0.292624\pi$
$269$	19.8905	1.21275	0.606374	+	0.795180i	$0.292624\pi$
$270$	−28.7848	−1.75179
$271$	−22.9325	−1.39305	−0.696524	−	0.717533i	$-0.745271\pi$
$271$	−22.9325	−1.39305	−0.696524	+	0.717533i	$0.745271\pi$
$272$	11.5355	0.699445
$273$	0	0
$274$	−18.4945	−1.11729
$275$	40.2881	2.42946
$276$	−11.5390	−0.694568
$277$	−3.56590	−0.214254	−0.107127	−	0.994245i	$-0.534165\pi$
$277$	−3.56590	−0.214254	−0.107127	+	0.994245i	$0.534165\pi$
$278$	−9.97554	−0.598293
$279$	2.71660	0.162639
$280$	−113.476	−6.78148
$281$	−0.721187	−0.0430224	−0.0215112	−	0.999769i	$-0.506848\pi$
$281$	−0.721187	−0.0430224	−0.0215112	+	0.999769i	$0.506848\pi$
$282$	13.2635	0.789829
$283$	1.17477	0.0698329	0.0349165	−	0.999390i	$-0.488883\pi$
$283$	1.17477	0.0698329	0.0349165	+	0.999390i	$0.488883\pi$
$284$	−4.58380	−0.271998
$285$	−7.04700	−0.417428
$286$	0	0
$287$	−8.65071	−0.510635
$288$	14.2728	0.841032
$289$	−14.2787	−0.839922
$290$	6.45752	0.379199
$291$	−2.04841	−0.120080
$292$	−62.3361	−3.64795
$293$	−11.6330	−0.679605	−0.339803	−	0.940497i	$-0.610360\pi$
$293$	−11.6330	−0.679605	−0.339803	+	0.940497i	$0.610360\pi$
$294$	−22.6354	−1.32012
$295$	12.0318	0.700519
$296$	52.9538	3.07788
$297$	−13.8657	−0.804571
$298$	−57.0383	−3.30414
$299$	0	0
$300$	21.0090	1.21296
$301$	59.8411	3.44918
$302$	−12.1183	−0.697332
$303$	−5.68978	−0.326869
$304$	24.8797	1.42695
$305$	11.0008	0.629904
$306$	11.3929	0.651290
$307$	−26.6286	−1.51977	−0.759887	−	0.650055i	$-0.774746\pi$
$307$	−26.6286	−1.51977	−0.759887	+	0.650055i	$0.774746\pi$
$308$	−99.0468	−5.64371
$309$	−1.13487	−0.0645607
$310$	−9.45857	−0.537211
$311$	−14.5879	−0.827202	−0.413601	−	0.910458i	$-0.635729\pi$
$311$	−14.5879	−0.827202	−0.413601	+	0.910458i	$0.635729\pi$
$312$	0	0
$313$	13.0380	0.736952	0.368476	−	0.929637i	$-0.379880\pi$
$313$	13.0380	0.736952	0.368476	+	0.929637i	$0.379880\pi$
$314$	18.7222	1.05655
$315$	−49.2307	−2.77384
$316$	−71.5312	−4.02394
$317$	2.57656	0.144714	0.0723569	−	0.997379i	$-0.476948\pi$
$317$	2.57656	0.144714	0.0723569	+	0.997379i	$0.476948\pi$
$318$	−2.44195	−0.136938
$319$	3.11060	0.174160
$320$	2.33911	0.130760
$321$	−6.82469	−0.380917
$322$	−60.1395	−3.35144
$323$	5.86931	0.326577
$324$	29.1424	1.61902
$325$	0	0
$326$	−30.4901	−1.68869
$327$	−3.87218	−0.214132
$328$	−11.1209	−0.614050
$329$	47.7356	2.63175
$330$	22.9419	1.26291
$331$	−3.08219	−0.169413	−0.0847063	−	0.996406i	$-0.526995\pi$
$331$	−3.08219	−0.169413	−0.0847063	+	0.996406i	$0.526995\pi$
$332$	−29.8770	−1.63971
$333$	22.9736	1.25895
$334$	−45.7703	−2.50444
$335$	1.46735	0.0801700
$336$	−18.1323	−0.989197
$337$	−5.00846	−0.272828	−0.136414	−	0.990652i	$-0.543558\pi$
$337$	−5.00846	−0.272828	−0.136414	+	0.990652i	$0.543558\pi$
$338$	0	0
$339$	4.00449	0.217494
$340$	−27.3923	−1.48556
$341$	−4.55621	−0.246733
$342$	24.5720	1.32870
$343$	−47.3694	−2.55771
$344$	76.9287	4.14772
$345$	9.61931	0.517886
$346$	−16.5330	−0.888822
$347$	−5.63394	−0.302446	−0.151223	−	0.988500i	$-0.548321\pi$
$347$	−5.63394	−0.302446	−0.151223	+	0.988500i	$0.548321\pi$
$348$	1.62208	0.0869528
$349$	−15.9925	−0.856061	−0.428030	−	0.903764i	$-0.640792\pi$
$349$	−15.9925	−0.856061	−0.428030	+	0.903764i	$0.640792\pi$
$350$	109.495	5.85277
$351$	0	0
$352$	−23.9380	−1.27590
$353$	−12.4485	−0.662566	−0.331283	−	0.943531i	$-0.607481\pi$
$353$	−12.4485	−0.662566	−0.331283	+	0.943531i	$0.607481\pi$
$354$	4.37667	0.232617
$355$	3.82120	0.202808
$356$	−16.0963	−0.853103
$357$	−4.27755	−0.226392
$358$	56.3263	2.97694
$359$	−12.0581	−0.636404	−0.318202	−	0.948023i	$-0.603079\pi$
$359$	−12.0581	−0.636404	−0.318202	+	0.948023i	$0.603079\pi$
$360$	−63.2885	−3.33560
$361$	−6.34117	−0.333746
$362$	48.5415	2.55128
$363$	5.19529	0.272682
$364$	0	0
$365$	51.9654	2.71999
$366$	4.00163	0.209169
$367$	23.5877	1.23127	0.615635	−	0.788032i	$-0.288900\pi$
$367$	23.5877	1.23127	0.615635	+	0.788032i	$0.288900\pi$
$368$	−33.9613	−1.77036
$369$	−4.82473	−0.251165
$370$	−79.9889	−4.15842
$371$	−8.78862	−0.456283
$372$	−2.37593	−0.123186
$373$	12.0044	0.621564	0.310782	−	0.950481i	$-0.399409\pi$
$373$	12.0044	0.621564	0.310782	+	0.950481i	$0.399409\pi$
$374$	−19.1079	−0.988047
$375$	−7.61055	−0.393007
$376$	61.3665	3.16473
$377$	0	0
$378$	−37.6844	−1.93828
$379$	13.2346	0.679818	0.339909	−	0.940458i	$-0.389604\pi$
$379$	13.2346	0.679818	0.339909	+	0.940458i	$0.389604\pi$
$380$	−59.0793	−3.03071
$381$	−8.18941	−0.419556
$382$	−4.78044	−0.244589
$383$	−2.36202	−0.120694	−0.0603468	−	0.998177i	$-0.519221\pi$
$383$	−2.36202	−0.120694	−0.0603468	+	0.998177i	$0.519221\pi$
$384$	6.44476	0.328883
$385$	82.5685	4.20808
$386$	−46.1741	−2.35020
$387$	33.3750	1.69655
$388$	−17.1730	−0.871829
$389$	−9.84725	−0.499275	−0.249638	−	0.968339i	$-0.580312\pi$
$389$	−9.84725	−0.499275	−0.249638	+	0.968339i	$0.580312\pi$
$390$	0	0
$391$	−8.01174	−0.405171
$392$	−104.728	−5.28955
$393$	1.71917	0.0867207
$394$	−11.8161	−0.595286
$395$	59.6307	3.00035
$396$	−55.2410	−2.77597
$397$	−22.5727	−1.13289	−0.566446	−	0.824099i	$-0.691682\pi$
$397$	−22.5727	−1.13289	−0.566446	+	0.824099i	$0.691682\pi$
$398$	19.4014	0.972507
$399$	−9.22575	−0.461865
$400$	61.8330	3.09165
$401$	−20.3280	−1.01513	−0.507566	−	0.861613i	$-0.669455\pi$
$401$	−20.3280	−1.01513	−0.507566	+	0.861613i	$0.669455\pi$
$402$	0.533761	0.0266216
$403$	0	0
$404$	−47.7009	−2.37321
$405$	−24.2941	−1.20718
$406$	8.45402	0.419566
$407$	−38.5308	−1.90990
$408$	−5.49900	−0.272241
$409$	−30.6268	−1.51440	−0.757199	−	0.653184i	$-0.773433\pi$
$409$	−30.6268	−1.51440	−0.757199	+	0.653184i	$0.773433\pi$
$410$	16.7986	0.829623
$411$	3.87279	0.191031
$412$	−9.51435	−0.468738
$413$	15.7517	0.775092
$414$	−33.5414	−1.64847
$415$	24.9064	1.22261
$416$	0	0
$417$	2.08890	0.102294
$418$	−41.2116	−2.01573
$419$	4.07101	0.198882	0.0994409	−	0.995043i	$-0.468295\pi$
$419$	4.07101	0.198882	0.0994409	+	0.995043i	$0.468295\pi$
$420$	43.0570	2.10097
$421$	31.4933	1.53489	0.767445	−	0.641115i	$-0.221528\pi$
$421$	31.4933	1.53489	0.767445	+	0.641115i	$0.221528\pi$
$422$	18.7456	0.912523
$423$	26.6234	1.29447
$424$	−11.2982	−0.548690
$425$	14.5869	0.707569
$426$	1.38999	0.0673454
$427$	14.4020	0.696960
$428$	−57.2156	−2.76562
$429$	0	0
$430$	−116.204	−5.60385
$431$	26.6890	1.28556	0.642781	−	0.766050i	$-0.277780\pi$
$431$	26.6890	1.28556	0.642781	+	0.766050i	$0.277780\pi$
$432$	−21.2807	−1.02387
$433$	−10.4852	−0.503888	−0.251944	−	0.967742i	$-0.581070\pi$
$433$	−10.4852	−0.503888	−0.251944	+	0.967742i	$0.581070\pi$
$434$	−12.3829	−0.594399
$435$	−1.35222	−0.0648340
$436$	−32.4629	−1.55469
$437$	−17.2796	−0.826595
$438$	18.9029	0.903213
$439$	34.7054	1.65640	0.828198	−	0.560435i	$-0.189366\pi$
$439$	34.7054	1.65640	0.828198	+	0.560435i	$0.189366\pi$
$440$	106.146	5.06031
$441$	−45.4354	−2.16359
$442$	0	0
$443$	−8.77932	−0.417118	−0.208559	−	0.978010i	$-0.566877\pi$
$443$	−8.77932	−0.417118	−0.208559	+	0.978010i	$0.566877\pi$
$444$	−20.0926	−0.953555
$445$	13.4184	0.636094
$446$	−61.0041	−2.88863
$447$	11.9440	0.564930
$448$	3.06231	0.144680
$449$	−26.3922	−1.24553	−0.622763	−	0.782411i	$-0.713990\pi$
$449$	−26.3922	−1.24553	−0.622763	+	0.782411i	$0.713990\pi$
$450$	61.0685	2.87880
$451$	8.09192	0.381034
$452$	33.5721	1.57910
$453$	2.53761	0.119227
$454$	−24.6668	−1.15767
$455$	0	0
$456$	−11.8602	−0.555403
$457$	−5.05064	−0.236259	−0.118129	−	0.992998i	$-0.537690\pi$
$457$	−5.05064	−0.236259	−0.118129	+	0.992998i	$0.537690\pi$
$458$	−10.2636	−0.479586
$459$	−5.02029	−0.234327
$460$	80.6446	3.76008
$461$	29.0005	1.35069	0.675344	−	0.737503i	$-0.263995\pi$
$461$	29.0005	1.35069	0.675344	+	0.737503i	$0.263995\pi$
$462$	30.0350	1.39735
$463$	6.55577	0.304672	0.152336	−	0.988329i	$-0.451320\pi$
$463$	6.55577	0.304672	0.152336	+	0.988329i	$0.451320\pi$
$464$	4.77406	0.221630
$465$	1.98065	0.0918503
$466$	−58.1452	−2.69353
$467$	13.7234	0.635042	0.317521	−	0.948251i	$-0.397150\pi$
$467$	13.7234	0.635042	0.317521	+	0.948251i	$0.397150\pi$
$468$	0	0
$469$	1.92102	0.0887044
$470$	−92.6966	−4.27577
$471$	−3.92047	−0.180646
$472$	20.2496	0.932065
$473$	−55.9757	−2.57377
$474$	21.6912	0.996308
$475$	31.4608	1.44352
$476$	−35.8614	−1.64370
$477$	−4.90165	−0.224431
$478$	70.2460	3.21298
$479$	−33.3601	−1.52426	−0.762130	−	0.647424i	$-0.775846\pi$
$479$	−33.3601	−1.52426	−0.762130	+	0.647424i	$0.775846\pi$
$480$	10.4062	0.474974
$481$	0	0
$482$	43.9629	2.00246
$483$	12.5934	0.573017
$484$	43.5553	1.97979
$485$	14.3160	0.650056
$486$	−32.0474	−1.45370
$487$	19.8142	0.897868	0.448934	−	0.893565i	$-0.351804\pi$
$487$	19.8142	0.897868	0.448934	+	0.893565i	$0.351804\pi$
$488$	18.5144	0.838109
$489$	6.38470	0.288726
$490$	158.196	7.14655
$491$	9.33530	0.421296	0.210648	−	0.977562i	$-0.432443\pi$
$491$	9.33530	0.421296	0.210648	+	0.977562i	$0.432443\pi$
$492$	4.21969	0.190238
$493$	1.12624	0.0507233
$494$	0	0
$495$	46.0507	2.06982
$496$	−6.99275	−0.313984
$497$	5.00262	0.224398
$498$	9.05991	0.405984
$499$	13.5785	0.607856	0.303928	−	0.952695i	$-0.401702\pi$
$499$	13.5785	0.607856	0.303928	+	0.952695i	$0.401702\pi$
$500$	−63.8039	−2.85340
$501$	9.58442	0.428200
$502$	−78.0492	−3.48351
$503$	27.3533	1.21962	0.609812	−	0.792546i	$-0.291245\pi$
$503$	27.3533	1.21962	0.609812	+	0.792546i	$0.291245\pi$
$504$	−82.8557	−3.69069
$505$	39.7650	1.76952
$506$	56.2548	2.50083
$507$	0	0
$508$	−68.6568	−3.04616
$509$	3.87392	0.171708	0.0858542	−	0.996308i	$-0.472638\pi$
$509$	3.87392	0.171708	0.0858542	+	0.996308i	$0.472638\pi$
$510$	8.30647	0.367817
$511$	68.0318	3.00955
$512$	50.8338	2.24656
$513$	−10.8277	−0.478054
$514$	20.3809	0.898963
$515$	7.93146	0.349502
$516$	−29.1896	−1.28500
$517$	−44.6521	−1.96380
$518$	−104.719	−4.60111
$519$	3.46206	0.151967
$520$	0	0
$521$	1.53769	0.0673676	0.0336838	−	0.999433i	$-0.489276\pi$
$521$	1.53769	0.0673676	0.0336838	+	0.999433i	$0.489276\pi$
$522$	4.71503	0.206371
$523$	24.0096	1.04987	0.524934	−	0.851143i	$-0.324090\pi$
$523$	24.0096	1.04987	0.524934	+	0.851143i	$0.324090\pi$
$524$	14.4129	0.629630
$525$	−22.9286	−1.00069
$526$	−40.0166	−1.74481
$527$	−1.64964	−0.0718596
$528$	16.9610	0.738135
$529$	0.587036	0.0255233
$530$	17.0664	0.741318
$531$	8.78516	0.381244
$532$	−77.3451	−3.35334
$533$	0	0
$534$	4.88106	0.211224
$535$	47.6967	2.06211
$536$	2.46957	0.106669
$537$	−11.7949	−0.508986
$538$	−50.5668	−2.18009
$539$	76.2031	3.28230
$540$	50.5332	2.17460
$541$	12.7086	0.546387	0.273194	−	0.961959i	$-0.411920\pi$
$541$	12.7086	0.546387	0.273194	+	0.961959i	$0.411920\pi$
$542$	58.3002	2.50421
$543$	−10.1647	−0.436209
$544$	−8.66710	−0.371599
$545$	27.0621	1.15921
$546$	0	0
$547$	−2.49676	−0.106754	−0.0533769	−	0.998574i	$-0.516998\pi$
$547$	−2.49676	−0.106754	−0.0533769	+	0.998574i	$0.516998\pi$
$548$	32.4680	1.38696
$549$	8.03236	0.342813
$550$	−102.423	−4.36731
$551$	2.42905	0.103481
$552$	16.1894	0.689066
$553$	78.0670	3.31975
$554$	9.06541	0.385152
$555$	16.7499	0.710992
$556$	17.5126	0.742698
$557$	−4.65197	−0.197110	−0.0985552	−	0.995132i	$-0.531422\pi$
$557$	−4.65197	−0.197110	−0.0985552	+	0.995132i	$0.531422\pi$
$558$	−6.90628	−0.292366
$559$	0	0
$560$	126.724	5.35506
$561$	4.00125	0.168933
$562$	1.83344	0.0773391
$563$	27.0281	1.13910	0.569549	−	0.821957i	$-0.307118\pi$
$563$	27.0281	1.13910	0.569549	+	0.821957i	$0.307118\pi$
$564$	−23.2847	−0.980464
$565$	−27.9868	−1.17741
$566$	−2.98657	−0.125535
$567$	−31.8052	−1.33569
$568$	6.43111	0.269843
$569$	30.4997	1.27861	0.639306	−	0.768952i	$-0.279222\pi$
$569$	30.4997	1.27861	0.639306	+	0.768952i	$0.279222\pi$
$570$	17.9153	0.750387
$571$	−9.55983	−0.400067	−0.200033	−	0.979789i	$-0.564105\pi$
$571$	−9.55983	−0.400067	−0.200033	+	0.979789i	$0.564105\pi$
$572$	0	0
$573$	1.00104	0.0418189
$574$	21.9923	0.917940
$575$	−42.9447	−1.79092
$576$	1.70793	0.0711638
$577$	17.9940	0.749099	0.374550	−	0.927207i	$-0.377797\pi$
$577$	17.9940	0.749099	0.374550	+	0.927207i	$0.377797\pi$
$578$	36.3000	1.50988
$579$	9.66897	0.401828
$580$	−11.3365	−0.470723
$581$	32.6068	1.35276
$582$	5.20757	0.215861
$583$	8.22093	0.340476
$584$	87.4582	3.61905
$585$	0	0
$586$	29.5740	1.22169
$587$	−14.0201	−0.578669	−0.289335	−	0.957228i	$-0.593434\pi$
$587$	−14.0201	−0.578669	−0.289335	+	0.957228i	$0.593434\pi$
$588$	39.7376	1.63875
$589$	−3.55793	−0.146602
$590$	−30.5879	−1.25928
$591$	2.47432	0.101780
$592$	−59.1360	−2.43048
$593$	−5.14173	−0.211145	−0.105573	−	0.994412i	$-0.533668\pi$
$593$	−5.14173	−0.211145	−0.105573	+	0.994412i	$0.533668\pi$
$594$	35.2502	1.44633
$595$	29.8952	1.22558
$596$	100.134	4.10164
$597$	−4.06271	−0.166276
$598$	0	0
$599$	−40.2281	−1.64367	−0.821837	−	0.569723i	$-0.807051\pi$
$599$	−40.2281	−1.64367	−0.821837	+	0.569723i	$0.807051\pi$
$600$	−29.4759	−1.20335
$601$	−4.52012	−0.184379	−0.0921897	−	0.995741i	$-0.529387\pi$
$601$	−4.52012	−0.184379	−0.0921897	+	0.995741i	$0.529387\pi$
$602$	−152.131	−6.20041
$603$	1.07140	0.0436309
$604$	21.2743	0.865641
$605$	−36.3091	−1.47617
$606$	14.4649	0.587595
$607$	−29.4974	−1.19726	−0.598631	−	0.801025i	$-0.704288\pi$
$607$	−29.4974	−1.19726	−0.598631	+	0.801025i	$0.704288\pi$
$608$	−18.6930	−0.758103
$609$	−1.77029	−0.0717359
$610$	−27.9668	−1.13234
$611$	0	0
$612$	−20.0008	−0.808486
$613$	−18.6232	−0.752182	−0.376091	−	0.926583i	$-0.622732\pi$
$613$	−18.6232	−0.752182	−0.376091	+	0.926583i	$0.622732\pi$
$614$	67.6967	2.73201
$615$	−3.51767	−0.141846
$616$	138.964	5.59900
$617$	−38.1771	−1.53695	−0.768476	−	0.639879i	$-0.778985\pi$
$617$	−38.1771	−1.53695	−0.768476	+	0.639879i	$0.778985\pi$
$618$	2.88514	0.116057
$619$	−17.8402	−0.717057	−0.358528	−	0.933519i	$-0.616721\pi$
$619$	−17.8402	−0.717057	−0.358528	+	0.933519i	$0.616721\pi$
$620$	16.6050	0.666873
$621$	14.7800	0.593102
$622$	37.0861	1.48702
$623$	17.5670	0.703809
$624$	0	0
$625$	8.97673	0.359069
$626$	−33.1459	−1.32478
$627$	8.62982	0.344642
$628$	−32.8677	−1.31156
$629$	−13.9507	−0.556249
$630$	125.157	4.98637
$631$	−7.06024	−0.281064	−0.140532	−	0.990076i	$-0.544881\pi$
$631$	−7.06024	−0.281064	−0.140532	+	0.990076i	$0.544881\pi$
$632$	100.359	3.99207
$633$	−3.92538	−0.156020
$634$	−6.55026	−0.260144
$635$	57.2345	2.27128
$636$	4.28696	0.169989
$637$	0	0
$638$	−7.90794	−0.313078
$639$	2.79009	0.110374
$640$	−45.0415	−1.78042
$641$	−7.79245	−0.307783	−0.153892	−	0.988088i	$-0.549181\pi$
$641$	−7.79245	−0.307783	−0.153892	+	0.988088i	$0.549181\pi$
$642$	17.3501	0.684753
$643$	−17.5665	−0.692754	−0.346377	−	0.938095i	$-0.612588\pi$
$643$	−17.5665	−0.692754	−0.346377	+	0.938095i	$0.612588\pi$
$644$	105.578	4.16035
$645$	24.3334	0.958127
$646$	−14.9213	−0.587070
$647$	−7.37114	−0.289789	−0.144895	−	0.989447i	$-0.546284\pi$
$647$	−7.37114	−0.289789	−0.144895	+	0.989447i	$0.546284\pi$
$648$	−40.8871	−1.60620
$649$	−14.7343	−0.578370
$650$	0	0
$651$	2.59301	0.101628
$652$	53.5269	2.09628
$653$	15.2035	0.594958	0.297479	−	0.954728i	$-0.403854\pi$
$653$	15.2035	0.594958	0.297479	+	0.954728i	$0.403854\pi$
$654$	9.84408	0.384934
$655$	−12.0150	−0.469466
$656$	12.4192	0.484890
$657$	37.9431	1.48030
$658$	−121.356	−4.73095
$659$	11.9928	0.467175	0.233587	−	0.972336i	$-0.424953\pi$
$659$	11.9928	0.467175	0.233587	+	0.972336i	$0.424953\pi$
$660$	−40.2757	−1.56773
$661$	−9.12059	−0.354750	−0.177375	−	0.984143i	$-0.556761\pi$
$661$	−9.12059	−0.354750	−0.177375	+	0.984143i	$0.556761\pi$
$662$	7.83571	0.304544
$663$	0	0
$664$	41.9177	1.62672
$665$	64.4774	2.50033
$666$	−58.4048	−2.26314
$667$	−3.31571	−0.128385
$668$	80.3521	3.10892
$669$	12.7744	0.493887
$670$	−3.73038	−0.144117
$671$	−13.4717	−0.520068
$672$	13.6235	0.525537
$673$	−6.01646	−0.231917	−0.115959	−	0.993254i	$-0.536994\pi$
$673$	−6.01646	−0.231917	−0.115959	+	0.993254i	$0.536994\pi$
$674$	12.7328	0.490448
$675$	−26.9098	−1.03576
$676$	0	0
$677$	9.86659	0.379204	0.189602	−	0.981861i	$-0.439280\pi$
$677$	9.86659	0.379204	0.189602	+	0.981861i	$0.439280\pi$
$678$	−10.1804	−0.390977
$679$	18.7421	0.719257
$680$	38.4317	1.47379
$681$	5.16529	0.197934
$682$	11.5831	0.443538
$683$	−46.1492	−1.76585	−0.882926	−	0.469513i	$-0.844429\pi$
$683$	−46.1492	−1.76585	−0.882926	+	0.469513i	$0.844429\pi$
$684$	−43.1375	−1.64940
$685$	−27.0664	−1.03415
$686$	120.425	4.59785
$687$	2.14922	0.0819979
$688$	−85.9099	−3.27529
$689$	0	0
$690$	−24.4547	−0.930976
$691$	32.7622	1.24633	0.623166	−	0.782089i	$-0.285846\pi$
$691$	32.7622	1.24633	0.623166	+	0.782089i	$0.285846\pi$
$692$	29.0246	1.10335
$693$	60.2884	2.29017
$694$	14.3229	0.543690
$695$	−14.5990	−0.553773
$696$	−2.27580	−0.0862639
$697$	2.92980	0.110974
$698$	40.6571	1.53889
$699$	12.1758	0.460529
$700$	−192.225	−7.26541
$701$	24.9900	0.943859	0.471930	−	0.881636i	$-0.343558\pi$
$701$	24.9900	0.943859	0.471930	+	0.881636i	$0.343558\pi$
$702$	0	0
$703$	−30.0885	−1.13481
$704$	−2.86450	−0.107960
$705$	19.4109	0.731056
$706$	31.6472	1.19106
$707$	52.0594	1.95789
$708$	−7.68347	−0.288762
$709$	38.9073	1.46119	0.730597	−	0.682809i	$-0.239242\pi$
$709$	38.9073	1.46119	0.730597	+	0.682809i	$0.239242\pi$
$710$	−9.71446	−0.364577
$711$	43.5400	1.63288
$712$	22.5833	0.846345
$713$	4.85665	0.181883
$714$	10.8746	0.406972
$715$	0	0
$716$	−98.8837	−3.69546
$717$	−14.7097	−0.549343
$718$	30.6548	1.14403
$719$	−3.94640	−0.147176	−0.0735879	−	0.997289i	$-0.523445\pi$
$719$	−3.94640	−0.147176	−0.0735879	+	0.997289i	$0.523445\pi$
$720$	70.6773	2.63399
$721$	10.3837	0.386708
$722$	16.1209	0.599956
$723$	−9.20594	−0.342373
$724$	−85.2170	−3.16707
$725$	6.03689	0.224204
$726$	−13.2077	−0.490185
$727$	−35.9019	−1.33153	−0.665764	−	0.746162i	$-0.731894\pi$
$727$	−35.9019	−1.33153	−0.665764	+	0.746162i	$0.731894\pi$
$728$	0	0
$729$	−12.8783	−0.476975
$730$	−132.109	−4.88958
$731$	−20.2668	−0.749596
$732$	−7.02507	−0.259654
$733$	36.9093	1.36328	0.681639	−	0.731689i	$-0.261268\pi$
$733$	36.9093	1.36328	0.681639	+	0.731689i	$0.261268\pi$
$734$	−59.9660	−2.21339
$735$	−33.1265	−1.22189
$736$	25.5164	0.940548
$737$	−1.79693	−0.0661908
$738$	12.2657	0.451506
$739$	−22.9910	−0.845737	−0.422869	−	0.906191i	$-0.638977\pi$
$739$	−22.9910	−0.845737	−0.422869	+	0.906191i	$0.638977\pi$
$740$	140.425	5.16211
$741$	0	0
$742$	22.3429	0.820234
$743$	−36.8191	−1.35076	−0.675381	−	0.737469i	$-0.736021\pi$
$743$	−36.8191	−1.35076	−0.675381	+	0.737469i	$0.736021\pi$
$744$	3.33345	0.122210
$745$	−83.4746	−3.05827
$746$	−30.5182	−1.11735
$747$	18.1857	0.665380
$748$	33.5449	1.22652
$749$	62.4433	2.28163
$750$	19.3479	0.706487
$751$	−33.9315	−1.23818	−0.619089	−	0.785321i	$-0.712498\pi$
$751$	−33.9315	−1.23818	−0.619089	+	0.785321i	$0.712498\pi$
$752$	−68.5308	−2.49906
$753$	16.3437	0.595597
$754$	0	0
$755$	−17.7350	−0.645442
$756$	66.1568	2.40610
$757$	−10.8159	−0.393110	−0.196555	−	0.980493i	$-0.562975\pi$
$757$	−10.8159	−0.393110	−0.196555	+	0.980493i	$0.562975\pi$
$758$	−33.6458	−1.22207
$759$	−11.7799	−0.427583
$760$	82.8889	3.00670
$761$	3.19400	0.115783	0.0578913	−	0.998323i	$-0.481562\pi$
$761$	3.19400	0.115783	0.0578913	+	0.998323i	$0.481562\pi$
$762$	20.8195	0.754213
$763$	35.4290	1.28262
$764$	8.39231	0.303623
$765$	16.6733	0.602826
$766$	6.00485	0.216964
$767$	0	0
$768$	−15.7148	−0.567060
$769$	−13.4268	−0.484184	−0.242092	−	0.970253i	$-0.577834\pi$
$769$	−13.4268	−0.484184	−0.242092	+	0.970253i	$0.577834\pi$
$770$	−209.910	−7.56464
$771$	−4.26781	−0.153701
$772$	81.0609	2.91745
$773$	27.3330	0.983100	0.491550	−	0.870849i	$-0.336431\pi$
$773$	27.3330	0.983100	0.491550	+	0.870849i	$0.336431\pi$
$774$	−84.8477	−3.04979
$775$	−8.84245	−0.317630
$776$	24.0940	0.864923
$777$	21.9285	0.786681
$778$	25.0342	0.897520
$779$	6.31894	0.226400
$780$	0	0
$781$	−4.67948	−0.167445
$782$	20.3679	0.728354
$783$	−2.07768	−0.0742502
$784$	116.954	4.17694
$785$	27.3996	0.977933
$786$	−4.37057	−0.155893
$787$	32.0659	1.14303	0.571513	−	0.820593i	$-0.306357\pi$
$787$	32.0659	1.14303	0.571513	+	0.820593i	$0.306357\pi$
$788$	20.7438	0.738966
$789$	8.37957	0.298321
$790$	−151.596	−5.39356
$791$	−36.6396	−1.30275
$792$	77.5037	2.75397
$793$	0	0
$794$	57.3855	2.03654
$795$	−3.57375	−0.126748
$796$	−34.0602	−1.20723
$797$	−11.5689	−0.409791	−0.204896	−	0.978784i	$-0.565685\pi$
$797$	−11.5689	−0.409791	−0.204896	+	0.978784i	$0.565685\pi$
$798$	23.4542	0.830269
$799$	−16.1670	−0.571946
$800$	−46.4575	−1.64252
$801$	9.79761	0.346181
$802$	51.6789	1.82485
$803$	−63.6373	−2.24571
$804$	−0.937045	−0.0330470
$805$	−88.0131	−3.10205
$806$	0	0
$807$	10.5888	0.372744
$808$	66.9249	2.35441
$809$	46.5170	1.63545	0.817725	−	0.575609i	$-0.195235\pi$
$809$	46.5170	1.63545	0.817725	+	0.575609i	$0.195235\pi$
$810$	61.7617	2.17008
$811$	−32.2098	−1.13104	−0.565520	−	0.824735i	$-0.691324\pi$
$811$	−32.2098	−1.13104	−0.565520	+	0.824735i	$0.691324\pi$
$812$	−14.8415	−0.520833
$813$	−12.2082	−0.428160
$814$	97.9551	3.43333
$815$	−44.6217	−1.56303
$816$	6.14100	0.214978
$817$	−43.7112	−1.52926
$818$	77.8611	2.72235
$819$	0	0
$820$	−29.4908	−1.02986
$821$	41.6850	1.45481	0.727407	−	0.686206i	$-0.240725\pi$
$821$	41.6850	1.45481	0.727407	+	0.686206i	$0.240725\pi$
$822$	−9.84561	−0.343405
$823$	27.6179	0.962701	0.481350	−	0.876528i	$-0.340147\pi$
$823$	27.6179	0.962701	0.481350	+	0.876528i	$0.340147\pi$
$824$	13.3487	0.465025
$825$	21.4475	0.746708
$826$	−40.0449	−1.39334
$827$	−44.4377	−1.54525	−0.772626	−	0.634862i	$-0.781057\pi$
$827$	−44.4377	−1.54525	−0.772626	+	0.634862i	$0.781057\pi$
$828$	58.8836	2.04635
$829$	10.3017	0.357792	0.178896	−	0.983868i	$-0.442748\pi$
$829$	10.3017	0.357792	0.178896	+	0.983868i	$0.442748\pi$
$830$	−63.3184	−2.19781
$831$	−1.89832	−0.0658520
$832$	0	0
$833$	27.5905	0.955953
$834$	−5.31052	−0.183888
$835$	−66.9841	−2.31808
$836$	72.3491	2.50225
$837$	3.04326	0.105190
$838$	−10.3495	−0.357519
$839$	−32.6316	−1.12657	−0.563283	−	0.826264i	$-0.690462\pi$
$839$	−32.6316	−1.12657	−0.563283	+	0.826264i	$0.690462\pi$
$840$	−60.4094	−2.08432
$841$	−28.5339	−0.983928
$842$	−80.0640	−2.75919
$843$	−0.383927	−0.0132232
$844$	−32.9089	−1.13277
$845$	0	0
$846$	−67.6835	−2.32701
$847$	−47.5349	−1.63332
$848$	12.6172	0.433278
$849$	0.625395	0.0214635
$850$	−37.0836	−1.27196
$851$	41.0715	1.40791
$852$	−2.44020	−0.0836000
$853$	−43.2379	−1.48044	−0.740220	−	0.672365i	$-0.765278\pi$
$853$	−43.2379	−1.48044	−0.740220	+	0.672365i	$0.765278\pi$
$854$	−36.6134	−1.25289
$855$	35.9608	1.22983
$856$	80.2740	2.74371
$857$	−7.00810	−0.239392	−0.119696	−	0.992811i	$-0.538192\pi$
$857$	−7.00810	−0.239392	−0.119696	+	0.992811i	$0.538192\pi$
$858$	0	0
$859$	18.8484	0.643100	0.321550	−	0.946893i	$-0.395796\pi$
$859$	18.8484	0.643100	0.321550	+	0.946893i	$0.395796\pi$
$860$	204.002	6.95641
$861$	−4.60524	−0.156946
$862$	−67.8501	−2.31098
$863$	−31.2186	−1.06269	−0.531347	−	0.847154i	$-0.678314\pi$
$863$	−31.2186	−1.06269	−0.531347	+	0.847154i	$0.678314\pi$
$864$	15.9890	0.543957
$865$	−24.1958	−0.822682
$866$	26.6561	0.905812
$867$	−7.60131	−0.258154
$868$	21.7388	0.737864
$869$	−73.0243	−2.47718
$870$	3.43769	0.116549
$871$	0	0
$872$	45.5458	1.54238
$873$	10.4530	0.353780
$874$	43.9291	1.48592
$875$	69.6337	2.35405
$876$	−33.1849	−1.12121
$877$	12.9192	0.436249	0.218125	−	0.975921i	$-0.430006\pi$
$877$	12.9192	0.436249	0.218125	+	0.975921i	$0.430006\pi$
$878$	−88.2299	−2.97761
$879$	−6.19286	−0.208880
$880$	−118.538	−3.99592
$881$	15.6346	0.526743	0.263371	−	0.964695i	$-0.415166\pi$
$881$	15.6346	0.526743	0.263371	+	0.964695i	$0.415166\pi$
$882$	115.508	3.88937
$883$	10.6582	0.358676	0.179338	−	0.983787i	$-0.442604\pi$
$883$	10.6582	0.358676	0.179338	+	0.983787i	$0.442604\pi$
$884$	0	0
$885$	6.40518	0.215308
$886$	22.3193	0.749830
$887$	34.2337	1.14946	0.574728	−	0.818345i	$-0.305108\pi$
$887$	34.2337	1.14946	0.574728	+	0.818345i	$0.305108\pi$
$888$	28.1902	0.946001
$889$	74.9300	2.51307
$890$	−34.1130	−1.14347
$891$	29.7507	0.996687
$892$	107.096	3.58583
$893$	−34.8686	−1.16683
$894$	−30.3646	−1.01554
$895$	82.4326	2.75542
$896$	−58.9672	−1.96995
$897$	0	0
$898$	67.0957	2.23901
$899$	−0.682716	−0.0227699
$900$	−107.209	−3.57363
$901$	2.97651	0.0991619
$902$	−20.5717	−0.684963
$903$	31.8567	1.06012
$904$	−47.1021	−1.56659
$905$	71.0396	2.36144
$906$	−6.45125	−0.214328
$907$	43.4512	1.44277	0.721387	−	0.692532i	$-0.243505\pi$
$907$	43.4512	1.44277	0.721387	+	0.692532i	$0.243505\pi$
$908$	43.3038	1.43709
$909$	29.0349	0.963027
$910$	0	0
$911$	42.3190	1.40209	0.701045	−	0.713117i	$-0.252717\pi$
$911$	42.3190	1.40209	0.701045	+	0.713117i	$0.252717\pi$
$912$	13.2448	0.438579
$913$	−30.5006	−1.00942
$914$	12.8400	0.424710
$915$	5.85632	0.193604
$916$	18.0182	0.595340
$917$	−15.7298	−0.519443
$918$	12.7629	0.421237
$919$	−16.5210	−0.544978	−0.272489	−	0.962159i	$-0.587847\pi$
$919$	−16.5210	−0.544978	−0.272489	+	0.962159i	$0.587847\pi$
$920$	−113.145	−3.73029
$921$	−14.1758	−0.467110
$922$	−73.7266	−2.42806
$923$	0	0
$924$	−52.7280	−1.73462
$925$	−74.7786	−2.45870
$926$	−16.6664	−0.547693
$927$	5.79125	0.190210
$928$	−3.58693	−0.117747
$929$	28.1231	0.922688	0.461344	−	0.887221i	$-0.347367\pi$
$929$	28.1231	0.922688	0.461344	+	0.887221i	$0.347367\pi$
$930$	−5.03531	−0.165114
$931$	59.5067	1.95025
$932$	102.077	3.34364
$933$	−7.76592	−0.254245
$934$	−34.8883	−1.14158
$935$	−27.9641	−0.914524
$936$	0	0
$937$	36.0347	1.17720	0.588602	−	0.808423i	$-0.299679\pi$
$937$	36.0347	1.17720	0.588602	+	0.808423i	$0.299679\pi$
$938$	−4.88372	−0.159459
$939$	6.94084	0.226506
$940$	162.733	5.30778
$941$	−28.7168	−0.936140	−0.468070	−	0.883691i	$-0.655050\pi$
$941$	−28.7168	−0.936140	−0.468070	+	0.883691i	$0.655050\pi$
$942$	9.96683	0.324737
$943$	−8.62550	−0.280885
$944$	−22.6137	−0.736014
$945$	−55.1504	−1.79404
$946$	142.304	4.62672
$947$	28.2144	0.916844	0.458422	−	0.888735i	$-0.348415\pi$
$947$	28.2144	0.916844	0.458422	+	0.888735i	$0.348415\pi$
$948$	−38.0799	−1.23678
$949$	0	0
$950$	−79.9813	−2.59494
$951$	1.37164	0.0444785
$952$	50.3138	1.63068
$953$	−15.3567	−0.497451	−0.248725	−	0.968574i	$-0.580012\pi$
$953$	−15.3567	−0.497451	−0.248725	+	0.968574i	$0.580012\pi$
$954$	12.4612	0.403448
$955$	−6.99610	−0.226388
$956$	−123.320	−3.98847
$957$	1.65594	0.0535290
$958$	84.8097	2.74008
$959$	−35.4346	−1.14424
$960$	1.24524	0.0401899
$961$	1.00000	0.0322581
$962$	0	0
$963$	34.8263	1.12226
$964$	−77.1791	−2.48577
$965$	−67.5750	−2.17532
$966$	−32.0155	−1.03008
$967$	34.5484	1.11100	0.555501	−	0.831516i	$-0.312526\pi$
$967$	34.5484	1.11100	0.555501	+	0.831516i	$0.312526\pi$
$968$	−61.1085	−1.96410
$969$	3.12455	0.100375
$970$	−36.3949	−1.16857
$971$	0.200447	0.00643266	0.00321633	−	0.999995i	$-0.498976\pi$
$971$	0.200447	0.00643266	0.00321633	+	0.999995i	$0.498976\pi$
$972$	56.2607	1.80456
$973$	−19.1127	−0.612724
$974$	−50.3728	−1.61405
$975$	0	0
$976$	−20.6759	−0.661821
$977$	33.4915	1.07149	0.535744	−	0.844381i	$-0.320031\pi$
$977$	33.4915	1.07149	0.535744	+	0.844381i	$0.320031\pi$
$978$	−16.2315	−0.519027
$979$	−16.4323	−0.525179
$980$	−277.720	−8.87145
$981$	19.7597	0.630879
$982$	−23.7327	−0.757341
$983$	38.7894	1.23719	0.618595	−	0.785710i	$-0.287702\pi$
$983$	38.7894	1.23719	0.618595	+	0.785710i	$0.287702\pi$
$984$	−5.92026	−0.188731
$985$	−17.2927	−0.550990
$986$	−2.86319	−0.0911824
$987$	25.4122	0.808881
$988$	0	0
$989$	59.6667	1.89729
$990$	−117.073	−3.72081
$991$	11.2686	0.357958	0.178979	−	0.983853i	$-0.442721\pi$
$991$	11.2686	0.357958	0.178979	+	0.983853i	$0.442721\pi$
$992$	5.25392	0.166812
$993$	−1.64082	−0.0520698
$994$	−12.7179	−0.403388
$995$	28.3937	0.900140
$996$	−15.9051	−0.503973
$997$	0.971709	0.0307743	0.0153872	−	0.999882i	$-0.495102\pi$
$997$	0.971709	0.0307743	0.0153872	+	0.999882i	$0.495102\pi$
$998$	−34.5199	−1.09271
$999$	25.7361	0.814254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.u.1.5	✓	54
13.12	even	2		5239.2.a.v.1.50	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5239.2.a.u.1.5	✓	54	1.1	even	1	trivial
5239.2.a.v.1.50	yes	54	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54225 q^{2} +0.532354 q^{3} +4.46305 q^{4} -3.72054 q^{5} -1.35338 q^{6} -4.87084 q^{7} -6.26171 q^{8} -2.71660 q^{9} +O(q^{10})\)	\(q-2.54225 q^{2} +0.532354 q^{3} +4.46305 q^{4} -3.72054 q^{5} -1.35338 q^{6} -4.87084 q^{7} -6.26171 q^{8} -2.71660 q^{9} +9.45857 q^{10} +4.55621 q^{11} +2.37593 q^{12} +12.3829 q^{14} -1.98065 q^{15} +6.99275 q^{16} +1.64964 q^{17} +6.90628 q^{18} +3.55793 q^{19} -16.6050 q^{20} -2.59301 q^{21} -11.5831 q^{22} -4.85665 q^{23} -3.33345 q^{24} +8.84245 q^{25} -3.04326 q^{27} -21.7388 q^{28} +0.682716 q^{29} +5.03531 q^{30} -1.00000 q^{31} -5.25392 q^{32} +2.42552 q^{33} -4.19382 q^{34} +18.1222 q^{35} -12.1243 q^{36} -8.45676 q^{37} -9.04515 q^{38} +23.2970 q^{40} +1.77602 q^{41} +6.59210 q^{42} -12.2856 q^{43} +20.3346 q^{44} +10.1072 q^{45} +12.3468 q^{46} -9.80027 q^{47} +3.72262 q^{48} +16.7251 q^{49} -22.4798 q^{50} +0.878195 q^{51} +1.80433 q^{53} +7.73673 q^{54} -16.9516 q^{55} +30.4998 q^{56} +1.89408 q^{57} -1.73564 q^{58} -3.23388 q^{59} -8.83974 q^{60} -2.95677 q^{61} +2.54225 q^{62} +13.2321 q^{63} -0.628702 q^{64} -6.16629 q^{66} -0.394392 q^{67} +7.36245 q^{68} -2.58546 q^{69} -46.0712 q^{70} -1.02705 q^{71} +17.0106 q^{72} -13.9671 q^{73} +21.4992 q^{74} +4.70732 q^{75} +15.8792 q^{76} -22.1926 q^{77} -16.0274 q^{79} -26.0168 q^{80} +6.52971 q^{81} -4.51509 q^{82} -6.69429 q^{83} -11.5728 q^{84} -6.13758 q^{85} +31.2331 q^{86} +0.363447 q^{87} -28.5297 q^{88} -3.60657 q^{89} -25.6951 q^{90} -21.6755 q^{92} -0.532354 q^{93} +24.9148 q^{94} -13.2374 q^{95} -2.79694 q^{96} -3.84782 q^{97} -42.5195 q^{98} -12.3774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} + 7 q^{3} + 64 q^{4} - 5 q^{5} + 3 q^{6} - 5 q^{7} - 6 q^{8} + 95 q^{9}+O(q^{10}) \) 54 * q - 2 * q^2 + 7 * q^3 + 64 * q^4 - 5 * q^5 + 3 * q^6 - 5 * q^7 - 6 * q^8 + 95 * q^9	\( 54 q - 2 q^{2} + 7 q^{3} + 64 q^{4} - 5 q^{5} + 3 q^{6} - 5 q^{7} - 6 q^{8} + 95 q^{9} - 6 q^{10} + 7 q^{11} + 5 q^{12} + 38 q^{14} - 4 q^{15} + 76 q^{16} + 62 q^{17} + 9 q^{18} - 8 q^{19} - 16 q^{20} + 6 q^{21} + 15 q^{22} + 38 q^{23} + 99 q^{24} + 87 q^{25} + 25 q^{27} - 19 q^{28} + 95 q^{29} + 41 q^{30} - 54 q^{31} - 9 q^{32} - 12 q^{33} - 7 q^{34} + 53 q^{35} + 97 q^{36} + 24 q^{37} - 16 q^{38} - 28 q^{40} - 22 q^{41} + 11 q^{42} + 11 q^{43} + 24 q^{44} - 8 q^{45} - 9 q^{46} - 45 q^{47} + 2 q^{48} + 105 q^{49} - 6 q^{50} + 58 q^{51} + 56 q^{53} - 50 q^{54} + q^{55} + 91 q^{56} + 51 q^{57} - 25 q^{58} - 36 q^{59} - 100 q^{60} + 48 q^{61} + 2 q^{62} + 56 q^{63} + 90 q^{64} - 24 q^{66} - 26 q^{67} + 140 q^{68} + 47 q^{69} + 24 q^{70} - 40 q^{71} - 7 q^{72} - 9 q^{73} + 114 q^{74} + 18 q^{75} + 67 q^{76} + 65 q^{77} + 33 q^{79} - 53 q^{80} + 210 q^{81} - 6 q^{82} + 41 q^{83} + 37 q^{84} - 37 q^{85} + 42 q^{86} - 16 q^{87} - 22 q^{88} + 24 q^{89} - 40 q^{90} + 87 q^{92} - 7 q^{93} - 4 q^{94} + 61 q^{95} + 200 q^{96} - 28 q^{97} - 68 q^{98} - 39 q^{99}+O(q^{100}) \) 54 * q - 2 * q^2 + 7 * q^3 + 64 * q^4 - 5 * q^5 + 3 * q^6 - 5 * q^7 - 6 * q^8 + 95 * q^9 - 6 * q^10 + 7 * q^11 + 5 * q^12 + 38 * q^14 - 4 * q^15 + 76 * q^16 + 62 * q^17 + 9 * q^18 - 8 * q^19 - 16 * q^20 + 6 * q^21 + 15 * q^22 + 38 * q^23 + 99 * q^24 + 87 * q^25 + 25 * q^27 - 19 * q^28 + 95 * q^29 + 41 * q^30 - 54 * q^31 - 9 * q^32 - 12 * q^33 - 7 * q^34 + 53 * q^35 + 97 * q^36 + 24 * q^37 - 16 * q^38 - 28 * q^40 - 22 * q^41 + 11 * q^42 + 11 * q^43 + 24 * q^44 - 8 * q^45 - 9 * q^46 - 45 * q^47 + 2 * q^48 + 105 * q^49 - 6 * q^50 + 58 * q^51 + 56 * q^53 - 50 * q^54 + q^55 + 91 * q^56 + 51 * q^57 - 25 * q^58 - 36 * q^59 - 100 * q^60 + 48 * q^61 + 2 * q^62 + 56 * q^63 + 90 * q^64 - 24 * q^66 - 26 * q^67 + 140 * q^68 + 47 * q^69 + 24 * q^70 - 40 * q^71 - 7 * q^72 - 9 * q^73 + 114 * q^74 + 18 * q^75 + 67 * q^76 + 65 * q^77 + 33 * q^79 - 53 * q^80 + 210 * q^81 - 6 * q^82 + 41 * q^83 + 37 * q^84 - 37 * q^85 + 42 * q^86 - 16 * q^87 - 22 * q^88 + 24 * q^89 - 40 * q^90 + 87 * q^92 - 7 * q^93 - 4 * q^94 + 61 * q^95 + 200 * q^96 - 28 * q^97 - 68 * q^98 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more