LMFDB - Embedded newform 6019.2.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6019,2,Mod(1,6019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6019, 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("6019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6019 = 13 \cdot 463 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6019.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:	$48.0619569766$
Analytic rank:	$0$
Dimension:	$123$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11210 q^{2} -0.313775 q^{3} +2.46095 q^{4} -4.01398 q^{5} +0.662722 q^{6} +2.82593 q^{7} -0.973578 q^{8} -2.90155 q^{9} +O(q^{10})$	\(q-2.11210 q^{2} -0.313775 q^{3} +2.46095 q^{4} -4.01398 q^{5} +0.662722 q^{6} +2.82593 q^{7} -0.973578 q^{8} -2.90155 q^{9} +8.47792 q^{10} +3.25267 q^{11} -0.772185 q^{12} -1.00000 q^{13} -5.96864 q^{14} +1.25949 q^{15} -2.86562 q^{16} +0.901185 q^{17} +6.12835 q^{18} -2.07796 q^{19} -9.87822 q^{20} -0.886706 q^{21} -6.86996 q^{22} +4.85703 q^{23} +0.305484 q^{24} +11.1120 q^{25} +2.11210 q^{26} +1.85176 q^{27} +6.95449 q^{28} +6.20287 q^{29} -2.66016 q^{30} -7.96160 q^{31} +7.99961 q^{32} -1.02061 q^{33} -1.90339 q^{34} -11.3432 q^{35} -7.14057 q^{36} +8.89311 q^{37} +4.38886 q^{38} +0.313775 q^{39} +3.90792 q^{40} +6.17079 q^{41} +1.87281 q^{42} -6.94512 q^{43} +8.00467 q^{44} +11.6467 q^{45} -10.2585 q^{46} -5.49251 q^{47} +0.899157 q^{48} +0.985897 q^{49} -23.4697 q^{50} -0.282769 q^{51} -2.46095 q^{52} +2.94960 q^{53} -3.91109 q^{54} -13.0562 q^{55} -2.75127 q^{56} +0.652013 q^{57} -13.1011 q^{58} -5.17988 q^{59} +3.09954 q^{60} -1.20977 q^{61} +16.8157 q^{62} -8.19957 q^{63} -11.1647 q^{64} +4.01398 q^{65} +2.15562 q^{66} +13.9615 q^{67} +2.21777 q^{68} -1.52401 q^{69} +23.9580 q^{70} -0.0312980 q^{71} +2.82488 q^{72} -4.81619 q^{73} -18.7831 q^{74} -3.48668 q^{75} -5.11377 q^{76} +9.19183 q^{77} -0.662722 q^{78} -11.4355 q^{79} +11.5025 q^{80} +8.12360 q^{81} -13.0333 q^{82} -1.89790 q^{83} -2.18214 q^{84} -3.61734 q^{85} +14.6688 q^{86} -1.94630 q^{87} -3.16673 q^{88} +14.8237 q^{89} -24.5991 q^{90} -2.82593 q^{91} +11.9529 q^{92} +2.49815 q^{93} +11.6007 q^{94} +8.34091 q^{95} -2.51008 q^{96} +1.21112 q^{97} -2.08231 q^{98} -9.43777 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 123 q + 10 q^{2} + q^{3} + 136 q^{4} + 46 q^{5} + 16 q^{6} + 12 q^{7} + 30 q^{8} + 154 q^{9}+O(q^{10}) $ 123 * q + 10 * q^2 + q^3 + 136 * q^4 + 46 * q^5 + 16 * q^6 + 12 * q^7 + 30 * q^8 + 154 * q^9	\( 123 q + 10 q^{2} + q^{3} + 136 q^{4} + 46 q^{5} + 16 q^{6} + 12 q^{7} + 30 q^{8} + 154 q^{9} + 5 q^{10} + 53 q^{11} - 6 q^{12} - 123 q^{13} + 21 q^{14} + 29 q^{15} + 166 q^{16} - 35 q^{17} + 28 q^{18} + 23 q^{19} + 93 q^{20} + 72 q^{21} + 8 q^{22} + 42 q^{23} + 55 q^{24} + 153 q^{25} - 10 q^{26} + 7 q^{27} + 39 q^{28} + 86 q^{29} + 44 q^{30} + 16 q^{31} + 70 q^{32} + 40 q^{33} + 10 q^{34} + 6 q^{35} + 222 q^{36} + 52 q^{37} + 12 q^{38} - q^{39} + 14 q^{40} + 80 q^{41} + 29 q^{42} + 2 q^{43} + 143 q^{44} + 137 q^{45} + 39 q^{46} + 45 q^{47} - 27 q^{48} + 163 q^{49} + 102 q^{50} + 48 q^{51} - 136 q^{52} + 117 q^{53} + 75 q^{54} + 20 q^{55} + 88 q^{56} + 67 q^{57} + 56 q^{58} + 88 q^{59} + 96 q^{60} + 57 q^{61} - 13 q^{62} + 48 q^{63} + 228 q^{64} - 46 q^{65} + 28 q^{66} + 43 q^{67} - 56 q^{68} + 92 q^{69} + 14 q^{70} + 90 q^{71} + 98 q^{72} + 25 q^{73} + 80 q^{74} + 21 q^{75} + 75 q^{76} + 112 q^{77} - 16 q^{78} + 36 q^{79} + 208 q^{80} + 231 q^{81} - 27 q^{82} + 93 q^{83} + 175 q^{84} + 77 q^{85} + 199 q^{86} + 15 q^{87} + 43 q^{88} + 140 q^{89} + 11 q^{90} - 12 q^{91} + 93 q^{92} + 140 q^{93} + 4 q^{94} + 23 q^{95} + 105 q^{96} + 43 q^{97} + 67 q^{98} + 140 q^{99}+O(q^{100}) \) 123 * q + 10 * q^2 + q^3 + 136 * q^4 + 46 * q^5 + 16 * q^6 + 12 * q^7 + 30 * q^8 + 154 * q^9 + 5 * q^10 + 53 * q^11 - 6 * q^12 - 123 * q^13 + 21 * q^14 + 29 * q^15 + 166 * q^16 - 35 * q^17 + 28 * q^18 + 23 * q^19 + 93 * q^20 + 72 * q^21 + 8 * q^22 + 42 * q^23 + 55 * q^24 + 153 * q^25 - 10 * q^26 + 7 * q^27 + 39 * q^28 + 86 * q^29 + 44 * q^30 + 16 * q^31 + 70 * q^32 + 40 * q^33 + 10 * q^34 + 6 * q^35 + 222 * q^36 + 52 * q^37 + 12 * q^38 - q^39 + 14 * q^40 + 80 * q^41 + 29 * q^42 + 2 * q^43 + 143 * q^44 + 137 * q^45 + 39 * q^46 + 45 * q^47 - 27 * q^48 + 163 * q^49 + 102 * q^50 + 48 * q^51 - 136 * q^52 + 117 * q^53 + 75 * q^54 + 20 * q^55 + 88 * q^56 + 67 * q^57 + 56 * q^58 + 88 * q^59 + 96 * q^60 + 57 * q^61 - 13 * q^62 + 48 * q^63 + 228 * q^64 - 46 * q^65 + 28 * q^66 + 43 * q^67 - 56 * q^68 + 92 * q^69 + 14 * q^70 + 90 * q^71 + 98 * q^72 + 25 * q^73 + 80 * q^74 + 21 * q^75 + 75 * q^76 + 112 * q^77 - 16 * q^78 + 36 * q^79 + 208 * q^80 + 231 * q^81 - 27 * q^82 + 93 * q^83 + 175 * q^84 + 77 * q^85 + 199 * q^86 + 15 * q^87 + 43 * q^88 + 140 * q^89 + 11 * q^90 - 12 * q^91 + 93 * q^92 + 140 * q^93 + 4 * q^94 + 23 * q^95 + 105 * q^96 + 43 * q^97 + 67 * q^98 + 140 * 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 $

$2$	−2.11210	−1.49348	−0.746739	−	0.665117i	$-0.768382\pi$
$2$	−2.11210	−1.49348	−0.746739	+	0.665117i	$0.768382\pi$
$3$	−0.313775	−0.181158	−0.0905789	−	0.995889i	$-0.528872\pi$
$3$	−0.313775	−0.181158	−0.0905789	+	0.995889i	$0.528872\pi$
$4$	2.46095	1.23048
$5$	−4.01398	−1.79511	−0.897554	−	0.440906i	$-0.854657\pi$
$5$	−4.01398	−1.79511	−0.897554	+	0.440906i	$0.854657\pi$
$6$	0.662722	0.270555
$7$	2.82593	1.06810	0.534051	−	0.845452i	$-0.320669\pi$
$7$	2.82593	1.06810	0.534051	+	0.845452i	$0.320669\pi$
$8$	−0.973578	−0.344212
$9$	−2.90155	−0.967182
$10$	8.47792	2.68095
$11$	3.25267	0.980717	0.490359	−	0.871521i	$-0.336866\pi$
$11$	3.25267	0.980717	0.490359	+	0.871521i	$0.336866\pi$
$12$	−0.772185	−0.222911
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−5.96864	−1.59519
$15$	1.25949	0.325198
$16$	−2.86562	−0.716404
$17$	0.901185	0.218569	0.109285	−	0.994010i	$-0.465144\pi$
$17$	0.901185	0.218569	0.109285	+	0.994010i	$0.465144\pi$
$18$	6.12835	1.44446
$19$	−2.07796	−0.476718	−0.238359	−	0.971177i	$-0.576609\pi$
$19$	−2.07796	−0.476718	−0.238359	+	0.971177i	$0.576609\pi$
$20$	−9.87822	−2.20884
$21$	−0.886706	−0.193495
$22$	−6.86996	−1.46468
$23$	4.85703	1.01276	0.506381	−	0.862310i	$-0.330983\pi$
$23$	4.85703	1.01276	0.506381	+	0.862310i	$0.330983\pi$
$24$	0.305484	0.0623567
$25$	11.1120	2.22241
$26$	2.11210	0.414216
$27$	1.85176	0.356370
$28$	6.95449	1.31427
$29$	6.20287	1.15184	0.575922	−	0.817505i	$-0.304643\pi$
$29$	6.20287	1.15184	0.575922	+	0.817505i	$0.304643\pi$
$30$	−2.66016	−0.485676
$31$	−7.96160	−1.42994	−0.714972	−	0.699153i	$-0.753561\pi$
$31$	−7.96160	−1.42994	−0.714972	+	0.699153i	$0.753561\pi$
$32$	7.99961	1.41415
$33$	−1.02061	−0.177665
$34$	−1.90339	−0.326429
$35$	−11.3432	−1.91736
$36$	−7.14057	−1.19009
$37$	8.89311	1.46202	0.731009	−	0.682368i	$-0.239050\pi$
$37$	8.89311	1.46202	0.731009	+	0.682368i	$0.239050\pi$
$38$	4.38886	0.711968
$39$	0.313775	0.0502442
$40$	3.90792	0.617897
$41$	6.17079	0.963716	0.481858	−	0.876249i	$-0.339962\pi$
$41$	6.17079	0.963716	0.481858	+	0.876249i	$0.339962\pi$
$42$	1.87281	0.288981
$43$	−6.94512	−1.05912	−0.529561	−	0.848272i	$-0.677643\pi$
$43$	−6.94512	−1.05912	−0.529561	+	0.848272i	$0.677643\pi$
$44$	8.00467	1.20675
$45$	11.6467	1.73619
$46$	−10.2585	−1.51254
$47$	−5.49251	−0.801165	−0.400583	−	0.916261i	$-0.631192\pi$
$47$	−5.49251	−0.801165	−0.400583	+	0.916261i	$0.631192\pi$
$48$	0.899157	0.129782
$49$	0.985897	0.140842
$50$	−23.4697	−3.31912
$51$	−0.282769	−0.0395956
$52$	−2.46095	−0.341273
$53$	2.94960	0.405159	0.202579	−	0.979266i	$-0.435068\pi$
$53$	2.94960	0.405159	0.202579	+	0.979266i	$0.435068\pi$
$54$	−3.91109	−0.532231
$55$	−13.0562	−1.76049
$56$	−2.75127	−0.367653
$57$	0.652013	0.0863612
$58$	−13.1011	−1.72025
$59$	−5.17988	−0.674363	−0.337181	−	0.941440i	$-0.609474\pi$
$59$	−5.17988	−0.674363	−0.337181	+	0.941440i	$0.609474\pi$
$60$	3.09954	0.400148
$61$	−1.20977	−0.154895	−0.0774474	−	0.996996i	$-0.524677\pi$
$61$	−1.20977	−0.154895	−0.0774474	+	0.996996i	$0.524677\pi$
$62$	16.8157	2.13559
$63$	−8.19957	−1.03305
$64$	−11.1647	−1.39559
$65$	4.01398	0.497873
$66$	2.15562	0.265338
$67$	13.9615	1.70567	0.852835	−	0.522180i	$-0.174881\pi$
$67$	13.9615	1.70567	0.852835	+	0.522180i	$0.174881\pi$
$68$	2.21777	0.268945
$69$	−1.52401	−0.183470
$70$	23.9580	2.86353
$71$	−0.0312980	−0.00371439	−0.00185719	−	0.999998i	$-0.500591\pi$
$71$	−0.0312980	−0.00371439	−0.00185719	+	0.999998i	$0.500591\pi$
$72$	2.82488	0.332915
$73$	−4.81619	−0.563692	−0.281846	−	0.959460i	$-0.590947\pi$
$73$	−4.81619	−0.563692	−0.281846	+	0.959460i	$0.590947\pi$
$74$	−18.7831	−2.18349
$75$	−3.48668	−0.402607
$76$	−5.11377	−0.586590
$77$	9.19183	1.04751
$78$	−0.662722	−0.0750385
$79$	−11.4355	−1.28659	−0.643297	−	0.765617i	$-0.722434\pi$
$79$	−11.4355	−1.28659	−0.643297	+	0.765617i	$0.722434\pi$
$80$	11.5025	1.28602
$81$	8.12360	0.902623
$82$	−13.0333	−1.43929
$83$	−1.89790	−0.208322	−0.104161	−	0.994560i	$-0.533216\pi$
$83$	−1.89790	−0.208322	−0.104161	+	0.994560i	$0.533216\pi$
$84$	−2.18214	−0.238091
$85$	−3.61734	−0.392356
$86$	14.6688	1.58177
$87$	−1.94630	−0.208666
$88$	−3.16673	−0.337574
$89$	14.8237	1.57131	0.785657	−	0.618663i	$-0.212325\pi$
$89$	14.8237	1.57131	0.785657	+	0.618663i	$0.212325\pi$
$90$	−24.5991	−2.59297
$91$	−2.82593	−0.296238
$92$	11.9529	1.24618
$93$	2.49815	0.259046
$94$	11.6007	1.19652
$95$	8.34091	0.855760
$96$	−2.51008	−0.256184
$97$	1.21112	0.122971	0.0614855	−	0.998108i	$-0.480416\pi$
$97$	1.21112	0.122971	0.0614855	+	0.998108i	$0.480416\pi$
$98$	−2.08231	−0.210345
$99$	−9.43777	−0.948532
$100$	27.3462	2.73462
$101$	17.9462	1.78571	0.892856	−	0.450343i	$-0.148698\pi$
$101$	17.9462	1.78571	0.892856	+	0.450343i	$0.148698\pi$
$102$	0.597235	0.0591351
$103$	−2.11827	−0.208720	−0.104360	−	0.994540i	$-0.533279\pi$
$103$	−2.11827	−0.208720	−0.104360	+	0.994540i	$0.533279\pi$
$104$	0.973578	0.0954672
$105$	3.55922	0.347344
$106$	−6.22984	−0.605096
$107$	−6.60203	−0.638242	−0.319121	−	0.947714i	$-0.603388\pi$
$107$	−6.60203	−0.638242	−0.319121	+	0.947714i	$0.603388\pi$
$108$	4.55708	0.438506
$109$	−6.14299	−0.588392	−0.294196	−	0.955745i	$-0.595052\pi$
$109$	−6.14299	−0.588392	−0.294196	+	0.955745i	$0.595052\pi$
$110$	27.5759	2.62926
$111$	−2.79043	−0.264856
$112$	−8.09804	−0.765193
$113$	4.68741	0.440955	0.220477	−	0.975392i	$-0.429238\pi$
$113$	4.68741	0.440955	0.220477	+	0.975392i	$0.429238\pi$
$114$	−1.37711	−0.128979
$115$	−19.4960	−1.81802
$116$	15.2650	1.41732
$117$	2.90155	0.268248
$118$	10.9404	1.00715
$119$	2.54669	0.233454
$120$	−1.22621	−0.111937
$121$	−0.420132	−0.0381938
$122$	2.55514	0.231332
$123$	−1.93624	−0.174585
$124$	−19.5931	−1.75951
$125$	−24.5336	−2.19436
$126$	17.3183	1.54284
$127$	9.01623	0.800061	0.400031	−	0.916502i	$-0.369000\pi$
$127$	9.01623	0.800061	0.400031	+	0.916502i	$0.369000\pi$
$128$	7.58176	0.670139
$129$	2.17920	0.191868
$130$	−8.47792	−0.743563
$131$	−20.8104	−1.81821	−0.909106	−	0.416564i	$-0.863234\pi$
$131$	−20.8104	−1.81821	−0.909106	+	0.416564i	$0.863234\pi$
$132$	−2.51166	−0.218612
$133$	−5.87219	−0.509183
$134$	−29.4881	−2.54738
$135$	−7.43291	−0.639723
$136$	−0.877374	−0.0752342
$137$	12.9540	1.10674	0.553369	−	0.832936i	$-0.313342\pi$
$137$	12.9540	1.10674	0.553369	+	0.832936i	$0.313342\pi$
$138$	3.21887	0.274008
$139$	−16.8445	−1.42873	−0.714366	−	0.699773i	$-0.753285\pi$
$139$	−16.8445	−1.42873	−0.714366	+	0.699773i	$0.753285\pi$
$140$	−27.9152	−2.35926
$141$	1.72341	0.145137
$142$	0.0661043	0.00554735
$143$	−3.25267	−0.272002
$144$	8.31471	0.692893
$145$	−24.8982	−2.06768
$146$	10.1723	0.841861
$147$	−0.309349	−0.0255147
$148$	21.8855	1.79898
$149$	−17.7816	−1.45672	−0.728361	−	0.685194i	$-0.759718\pi$
$149$	−17.7816	−1.45672	−0.728361	+	0.685194i	$0.759718\pi$
$150$	7.36420	0.601285
$151$	−13.2555	−1.07872	−0.539358	−	0.842076i	$-0.681333\pi$
$151$	−13.2555	−1.07872	−0.539358	+	0.842076i	$0.681333\pi$
$152$	2.02306	0.164092
$153$	−2.61483	−0.211396
$154$	−19.4140	−1.56443
$155$	31.9577	2.56690
$156$	0.772185	0.0618243
$157$	23.0933	1.84305	0.921524	−	0.388322i	$-0.126945\pi$
$157$	23.0933	1.84305	0.921524	+	0.388322i	$0.126945\pi$
$158$	24.1529	1.92150
$159$	−0.925510	−0.0733977
$160$	−32.1103	−2.53854
$161$	13.7257	1.08173
$162$	−17.1578	−1.34805
$163$	−5.99560	−0.469611	−0.234806	−	0.972042i	$-0.575445\pi$
$163$	−5.99560	−0.469611	−0.234806	+	0.972042i	$0.575445\pi$
$164$	15.1860	1.18583
$165$	4.09669	0.318927
$166$	4.00855	0.311124
$167$	−24.1298	−1.86722	−0.933609	−	0.358293i	$-0.883359\pi$
$167$	−24.1298	−1.86722	−0.933609	+	0.358293i	$0.883359\pi$
$168$	0.863278	0.0666033
$169$	1.00000	0.0769231
$170$	7.64017	0.585974
$171$	6.02931	0.461073
$172$	−17.0916	−1.30322
$173$	−15.4454	−1.17429	−0.587146	−	0.809481i	$-0.699749\pi$
$173$	−15.4454	−1.17429	−0.587146	+	0.809481i	$0.699749\pi$
$174$	4.11078	0.311638
$175$	31.4019	2.37376
$176$	−9.32090	−0.702590
$177$	1.62531	0.122166
$178$	−31.3092	−2.34672
$179$	11.5823	0.865704	0.432852	−	0.901465i	$-0.357507\pi$
$179$	11.5823	0.865704	0.432852	+	0.901465i	$0.357507\pi$
$180$	28.6621	2.13635
$181$	−24.0359	−1.78658	−0.893288	−	0.449484i	$-0.851608\pi$
$181$	−24.0359	−1.78658	−0.893288	+	0.449484i	$0.851608\pi$
$182$	5.96864	0.442425
$183$	0.379594	0.0280604
$184$	−4.72870	−0.348605
$185$	−35.6968	−2.62448
$186$	−5.27633	−0.386879
$187$	2.93126	0.214355
$188$	−13.5168	−0.985815
$189$	5.23294	0.380640
$190$	−17.6168	−1.27806
$191$	11.8022	0.853977	0.426988	−	0.904257i	$-0.359574\pi$
$191$	11.8022	0.853977	0.426988	+	0.904257i	$0.359574\pi$
$192$	3.50321	0.252822
$193$	15.2787	1.09979	0.549894	−	0.835235i	$-0.314668\pi$
$193$	15.2787	1.09979	0.549894	+	0.835235i	$0.314668\pi$
$194$	−2.55801	−0.183654
$195$	−1.25949	−0.0901936
$196$	2.42625	0.173303
$197$	10.7229	0.763973	0.381986	−	0.924168i	$-0.375240\pi$
$197$	10.7229	0.763973	0.381986	+	0.924168i	$0.375240\pi$
$198$	19.9335	1.41661
$199$	3.56954	0.253038	0.126519	−	0.991964i	$-0.459620\pi$
$199$	3.56954	0.253038	0.126519	+	0.991964i	$0.459620\pi$
$200$	−10.8184	−0.764980
$201$	−4.38077	−0.308996
$202$	−37.9041	−2.66692
$203$	17.5289	1.23029
$204$	−0.695881	−0.0487214
$205$	−24.7695	−1.72997
$206$	4.47400	0.311718
$207$	−14.0929	−0.979525
$208$	2.86562	0.198695
$209$	−6.75894	−0.467525
$210$	−7.51742	−0.518751
$211$	13.1795	0.907316	0.453658	−	0.891176i	$-0.350119\pi$
$211$	13.1795	0.907316	0.453658	+	0.891176i	$0.350119\pi$
$212$	7.25883	0.498539
$213$	0.00982051	0.000672890 0
$214$	13.9441	0.953201
$215$	27.8776	1.90124
$216$	−1.80283	−0.122667
$217$	−22.4989	−1.52733
$218$	12.9746	0.878751
$219$	1.51120	0.102117
$220$	−32.1306	−2.16624
$221$	−0.901185	−0.0606203
$222$	5.89366	0.395557
$223$	−7.81156	−0.523101	−0.261551	−	0.965190i	$-0.584234\pi$
$223$	−7.81156	−0.523101	−0.261551	+	0.965190i	$0.584234\pi$
$224$	22.6064	1.51045
$225$	−32.2421	−2.14947
$226$	−9.90027	−0.658556
$227$	8.77589	0.582476	0.291238	−	0.956651i	$-0.405933\pi$
$227$	8.77589	0.582476	0.291238	+	0.956651i	$0.405933\pi$
$228$	1.60457	0.106265
$229$	17.1609	1.13402	0.567011	−	0.823710i	$-0.308100\pi$
$229$	17.1609	1.13402	0.567011	+	0.823710i	$0.308100\pi$
$230$	41.1775	2.71517
$231$	−2.88416	−0.189764
$232$	−6.03898	−0.396478
$233$	−19.4588	−1.27479	−0.637394	−	0.770538i	$-0.719988\pi$
$233$	−19.4588	−1.27479	−0.637394	+	0.770538i	$0.719988\pi$
$234$	−6.12835	−0.400622
$235$	22.0468	1.43818
$236$	−12.7474	−0.829788
$237$	3.58817	0.233077
$238$	−5.37885	−0.348659
$239$	23.5776	1.52511	0.762554	−	0.646925i	$-0.223945\pi$
$239$	23.5776	1.52511	0.762554	+	0.646925i	$0.223945\pi$
$240$	−3.60920	−0.232973
$241$	−21.4372	−1.38089	−0.690445	−	0.723385i	$-0.742585\pi$
$241$	−21.4372	−1.38089	−0.690445	+	0.723385i	$0.742585\pi$
$242$	0.887359	0.0570416
$243$	−8.10425	−0.519888
$244$	−2.97718	−0.190594
$245$	−3.95737	−0.252827
$246$	4.08952	0.260739
$247$	2.07796	0.132218
$248$	7.75124	0.492204
$249$	0.595513	0.0377391
$250$	51.8174	3.27722
$251$	−30.8492	−1.94718	−0.973591	−	0.228300i	$-0.926683\pi$
$251$	−30.8492	−1.94718	−0.973591	+	0.228300i	$0.926683\pi$
$252$	−20.1788	−1.27114
$253$	15.7983	0.993233
$254$	−19.0432	−1.19487
$255$	1.13503	0.0710783
$256$	6.31604	0.394753
$257$	8.48232	0.529112	0.264556	−	0.964370i	$-0.414775\pi$
$257$	8.48232	0.529112	0.264556	+	0.964370i	$0.414775\pi$
$258$	−4.60269	−0.286551
$259$	25.1313	1.56159
$260$	9.87822	0.612621
$261$	−17.9979	−1.11404
$262$	43.9536	2.71546
$263$	−10.0190	−0.617797	−0.308899	−	0.951095i	$-0.599960\pi$
$263$	−10.0190	−0.617797	−0.308899	+	0.951095i	$0.599960\pi$
$264$	0.993639	0.0611543
$265$	−11.8396	−0.727303
$266$	12.4026	0.760454
$267$	−4.65131	−0.284656
$268$	34.3586	2.09879
$269$	9.74945	0.594435	0.297217	−	0.954810i	$-0.403941\pi$
$269$	9.74945	0.594435	0.297217	+	0.954810i	$0.403941\pi$
$270$	15.6990	0.955412
$271$	−28.1922	−1.71255	−0.856277	−	0.516516i	$-0.827229\pi$
$271$	−28.1922	−1.71255	−0.856277	+	0.516516i	$0.827229\pi$
$272$	−2.58245	−0.156584
$273$	0.886706	0.0536659
$274$	−27.3602	−1.65289
$275$	36.1438	2.17955
$276$	−3.75053	−0.225755
$277$	4.06804	0.244425	0.122212	−	0.992504i	$-0.461001\pi$
$277$	4.06804	0.244425	0.122212	+	0.992504i	$0.461001\pi$
$278$	35.5772	2.13378
$279$	23.1009	1.38302
$280$	11.0435	0.659977
$281$	12.3246	0.735221	0.367611	−	0.929980i	$-0.380176\pi$
$281$	12.3246	0.735221	0.367611	+	0.929980i	$0.380176\pi$
$282$	−3.64001	−0.216759
$283$	−4.18501	−0.248773	−0.124386	−	0.992234i	$-0.539696\pi$
$283$	−4.18501	−0.248773	−0.124386	+	0.992234i	$0.539696\pi$
$284$	−0.0770228	−0.00457046
$285$	−2.61717	−0.155028
$286$	6.86996	0.406229
$287$	17.4383	1.02935
$288$	−23.2112	−1.36774
$289$	−16.1879	−0.952227
$290$	52.5874	3.08804
$291$	−0.380020	−0.0222772
$292$	−11.8524	−0.693610
$293$	29.8615	1.74453	0.872264	−	0.489036i	$-0.162651\pi$
$293$	29.8615	1.74453	0.872264	+	0.489036i	$0.162651\pi$
$294$	0.653376	0.0381057
$295$	20.7919	1.21055
$296$	−8.65814	−0.503244
$297$	6.02315	0.349499
$298$	37.5564	2.17558
$299$	−4.85703	−0.280890
$300$	−8.58055	−0.495398
$301$	−19.6265	−1.13125
$302$	27.9969	1.61104
$303$	−5.63106	−0.323496
$304$	5.95465	0.341522
$305$	4.85598	0.278053
$306$	5.52277	0.315716
$307$	−4.08929	−0.233388	−0.116694	−	0.993168i	$-0.537230\pi$
$307$	−4.08929	−0.233388	−0.116694	+	0.993168i	$0.537230\pi$
$308$	22.6207	1.28893
$309$	0.664661	0.0378112
$310$	−67.4978	−3.83361
$311$	18.6701	1.05868	0.529342	−	0.848409i	$-0.322439\pi$
$311$	18.6701	1.05868	0.529342	+	0.848409i	$0.322439\pi$
$312$	−0.305484	−0.0172946
$313$	1.45737	0.0823756	0.0411878	−	0.999151i	$-0.486886\pi$
$313$	1.45737	0.0823756	0.0411878	+	0.999151i	$0.486886\pi$
$314$	−48.7753	−2.75255
$315$	32.9129	1.85443
$316$	−28.1422	−1.58312
$317$	−6.49600	−0.364852	−0.182426	−	0.983220i	$-0.558395\pi$
$317$	−6.49600	−0.364852	−0.182426	+	0.983220i	$0.558395\pi$
$318$	1.95477	0.109618
$319$	20.1759	1.12963
$320$	44.8150	2.50524
$321$	2.07155	0.115623
$322$	−28.9899	−1.61554
$323$	−1.87263	−0.104196
$324$	19.9918	1.11066
$325$	−11.1120	−0.616385
$326$	12.6633	0.701354
$327$	1.92752	0.106592
$328$	−6.00775	−0.331723
$329$	−15.5215	−0.855726
$330$	−8.65261	−0.476311
$331$	4.19667	0.230670	0.115335	−	0.993327i	$-0.463206\pi$
$331$	4.19667	0.230670	0.115335	+	0.993327i	$0.463206\pi$
$332$	−4.67064	−0.256335
$333$	−25.8038	−1.41404
$334$	50.9644	2.78865
$335$	−56.0413	−3.06186
$336$	2.54096	0.138621
$337$	−21.9617	−1.19633	−0.598165	−	0.801373i	$-0.704103\pi$
$337$	−21.9617	−1.19633	−0.598165	+	0.801373i	$0.704103\pi$
$338$	−2.11210	−0.114883
$339$	−1.47079	−0.0798824
$340$	−8.90210	−0.482784
$341$	−25.8965	−1.40237
$342$	−12.7345	−0.688602
$343$	−16.9955	−0.917668
$344$	6.76162	0.364562
$345$	6.11736	0.329348
$346$	32.6222	1.75378
$347$	−5.98428	−0.321253	−0.160626	−	0.987015i	$-0.551351\pi$
$347$	−5.98428	−0.321253	−0.160626	+	0.987015i	$0.551351\pi$
$348$	−4.78976	−0.256758
$349$	23.9038	1.27954	0.639769	−	0.768567i	$-0.279030\pi$
$349$	23.9038	1.27954	0.639769	+	0.768567i	$0.279030\pi$
$350$	−66.3239	−3.54516
$351$	−1.85176	−0.0988394
$352$	26.0201	1.38688
$353$	17.9891	0.957463	0.478732	−	0.877961i	$-0.341097\pi$
$353$	17.9891	0.957463	0.478732	+	0.877961i	$0.341097\pi$
$354$	−3.43282	−0.182452
$355$	0.125629	0.00666772
$356$	36.4805	1.93346
$357$	−0.799086	−0.0422921
$358$	−24.4630	−1.29291
$359$	12.7298	0.671853	0.335927	−	0.941888i	$-0.390951\pi$
$359$	12.7298	0.671853	0.335927	+	0.941888i	$0.390951\pi$
$360$	−11.3390	−0.597619
$361$	−14.6821	−0.772740
$362$	50.7662	2.66821
$363$	0.131827	0.00691911
$364$	−6.95449	−0.364514
$365$	19.3321	1.01189
$366$	−0.801740	−0.0419076
$367$	−15.6070	−0.814676	−0.407338	−	0.913277i	$-0.633543\pi$
$367$	−15.6070	−0.814676	−0.407338	+	0.913277i	$0.633543\pi$
$368$	−13.9184	−0.725546
$369$	−17.9048	−0.932089
$370$	75.3951	3.91960
$371$	8.33537	0.432751
$372$	6.14782	0.318750
$373$	28.7256	1.48736	0.743679	−	0.668537i	$-0.233079\pi$
$373$	28.7256	1.48736	0.743679	+	0.668537i	$0.233079\pi$
$374$	−6.19110	−0.320134
$375$	7.69803	0.397525
$376$	5.34739	0.275771
$377$	−6.20287	−0.319464
$378$	−11.0525	−0.568478
$379$	35.1620	1.80615	0.903077	−	0.429480i	$-0.141303\pi$
$379$	35.1620	1.80615	0.903077	+	0.429480i	$0.141303\pi$
$380$	20.5266	1.05299
$381$	−2.82906	−0.144937
$382$	−24.9274	−1.27540
$383$	30.0651	1.53626	0.768128	−	0.640297i	$-0.221189\pi$
$383$	30.0651	1.53626	0.768128	+	0.640297i	$0.221189\pi$
$384$	−2.37896	−0.121401
$385$	−36.8958	−1.88039
$386$	−32.2702	−1.64251
$387$	20.1516	1.02436
$388$	2.98052	0.151313
$389$	22.5325	1.14244	0.571221	−	0.820796i	$-0.306470\pi$
$389$	22.5325	1.14244	0.571221	+	0.820796i	$0.306470\pi$
$390$	2.66016	0.134702
$391$	4.37708	0.221359
$392$	−0.959848	−0.0484796
$393$	6.52977	0.329383
$394$	−22.6477	−1.14098
$395$	45.9019	2.30957
$396$	−23.2259	−1.16715
$397$	13.3709	0.671066	0.335533	−	0.942028i	$-0.391084\pi$
$397$	13.3709	0.671066	0.335533	+	0.942028i	$0.391084\pi$
$398$	−7.53921	−0.377906
$399$	1.84254	0.0922426
$400$	−31.8428	−1.59214
$401$	−12.5431	−0.626373	−0.313186	−	0.949692i	$-0.601396\pi$
$401$	−12.5431	−0.626373	−0.313186	+	0.949692i	$0.601396\pi$
$402$	9.25261	0.461478
$403$	7.96160	0.396595
$404$	44.1647	2.19728
$405$	−32.6080	−1.62030
$406$	−37.0227	−1.83741
$407$	28.9264	1.43383
$408$	0.275298	0.0136293
$409$	34.5918	1.71045	0.855227	−	0.518253i	$-0.173418\pi$
$409$	34.5918	1.71045	0.855227	+	0.518253i	$0.173418\pi$
$410$	52.3155	2.58368
$411$	−4.06465	−0.200494
$412$	−5.21297	−0.256825
$413$	−14.6380	−0.720289
$414$	29.7656	1.46290
$415$	7.61814	0.373960
$416$	−7.99961	−0.392213
$417$	5.28537	0.258826
$418$	14.2755	0.698239
$419$	5.48822	0.268117	0.134059	−	0.990973i	$-0.457199\pi$
$419$	5.48822	0.268117	0.134059	+	0.990973i	$0.457199\pi$
$420$	8.75908	0.427399
$421$	31.4473	1.53265	0.766324	−	0.642454i	$-0.222084\pi$
$421$	31.4473	1.53265	0.766324	+	0.642454i	$0.222084\pi$
$422$	−27.8364	−1.35506
$423$	15.9368	0.774872
$424$	−2.87167	−0.139460
$425$	10.0140	0.485751
$426$	−0.0207419	−0.00100495
$427$	−3.41872	−0.165443
$428$	−16.2473	−0.785342
$429$	1.02061	0.0492753
$430$	−58.8802	−2.83946
$431$	8.81324	0.424519	0.212259	−	0.977213i	$-0.431918\pi$
$431$	8.81324	0.424519	0.212259	+	0.977213i	$0.431918\pi$
$432$	−5.30642	−0.255305
$433$	27.7220	1.33223	0.666117	−	0.745847i	$-0.267955\pi$
$433$	27.7220	1.33223	0.666117	+	0.745847i	$0.267955\pi$
$434$	47.5199	2.28103
$435$	7.81243	0.374577
$436$	−15.1176	−0.724003
$437$	−10.0927	−0.482801
$438$	−3.19179	−0.152510
$439$	−17.9332	−0.855904	−0.427952	−	0.903801i	$-0.640765\pi$
$439$	−17.9332	−0.855904	−0.427952	+	0.903801i	$0.640765\pi$
$440$	12.7112	0.605982
$441$	−2.86062	−0.136220
$442$	1.90339	0.0905350
$443$	25.6954	1.22083	0.610413	−	0.792084i	$-0.291004\pi$
$443$	25.6954	1.22083	0.610413	+	0.792084i	$0.291004\pi$
$444$	−6.86713	−0.325899
$445$	−59.5022	−2.82068
$446$	16.4988	0.781240
$447$	5.57940	0.263897
$448$	−31.5508	−1.49063
$449$	20.5335	0.969035	0.484517	−	0.874782i	$-0.338995\pi$
$449$	20.5335	0.969035	0.484517	+	0.874782i	$0.338995\pi$
$450$	68.0985	3.21019
$451$	20.0716	0.945133
$452$	11.5355	0.542584
$453$	4.15924	0.195418
$454$	−18.5355	−0.869915
$455$	11.3432	0.531779
$456$	−0.634785	−0.0297265
$457$	32.3504	1.51329	0.756644	−	0.653827i	$-0.226837\pi$
$457$	32.3504	1.51329	0.756644	+	0.653827i	$0.226837\pi$
$458$	−36.2454	−1.69364
$459$	1.66877	0.0778917
$460$	−47.9788	−2.23703
$461$	0.123140	0.00573521	0.00286760	−	0.999996i	$-0.499087\pi$
$461$	0.123140	0.00573521	0.00286760	+	0.999996i	$0.499087\pi$
$462$	6.09163	0.283408
$463$	1.00000	0.0464739
$463$	1.00000	0.0464739
$464$	−17.7750	−0.825186
$465$	−10.0275	−0.465015
$466$	41.0989	1.90387
$467$	−2.09087	−0.0967539	−0.0483770	−	0.998829i	$-0.515405\pi$
$467$	−2.09087	−0.0967539	−0.0483770	+	0.998829i	$0.515405\pi$
$468$	7.14057	0.330073
$469$	39.4543	1.82183
$470$	−46.5651	−2.14789
$471$	−7.24610	−0.333883
$472$	5.04302	0.232124
$473$	−22.5902	−1.03870
$474$	−7.57856	−0.348095
$475$	−23.0904	−1.05946
$476$	6.26728	0.287260
$477$	−8.55840	−0.391862
$478$	−49.7982	−2.27771
$479$	−18.1265	−0.828222	−0.414111	−	0.910226i	$-0.635907\pi$
$479$	−18.1265	−0.828222	−0.414111	+	0.910226i	$0.635907\pi$
$480$	10.0754	0.459877
$481$	−8.89311	−0.405491
$482$	45.2774	2.06233
$483$	−4.30676	−0.195964
$484$	−1.03393	−0.0469966
$485$	−4.86143	−0.220746
$486$	17.1170	0.776441
$487$	−6.67688	−0.302558	−0.151279	−	0.988491i	$-0.548339\pi$
$487$	−6.67688	−0.302558	−0.151279	+	0.988491i	$0.548339\pi$
$488$	1.17780	0.0533166
$489$	1.88127	0.0850738
$490$	8.35835	0.377592
$491$	−8.82036	−0.398057	−0.199029	−	0.979994i	$-0.563779\pi$
$491$	−8.82036	−0.398057	−0.199029	+	0.979994i	$0.563779\pi$
$492$	−4.76499	−0.214822
$493$	5.58993	0.251758
$494$	−4.38886	−0.197464
$495$	37.8830	1.70272
$496$	22.8149	1.02442
$497$	−0.0884460	−0.00396734
$498$	−1.25778	−0.0563625
$499$	−28.8622	−1.29205	−0.646025	−	0.763316i	$-0.723570\pi$
$499$	−28.8622	−1.29205	−0.646025	+	0.763316i	$0.723570\pi$
$500$	−60.3761	−2.70010
$501$	7.57131	0.338261
$502$	65.1564	2.90807
$503$	−33.6793	−1.50169	−0.750843	−	0.660481i	$-0.770353\pi$
$503$	−33.6793	−1.50169	−0.750843	+	0.660481i	$0.770353\pi$
$504$	7.98293	0.355588
$505$	−72.0356	−3.20554
$506$	−33.3676	−1.48337
$507$	−0.313775	−0.0139352
$508$	22.1885	0.984457
$509$	−0.944625	−0.0418698	−0.0209349	−	0.999781i	$-0.506664\pi$
$509$	−0.944625	−0.0418698	−0.0209349	+	0.999781i	$0.506664\pi$
$510$	−2.39729	−0.106154
$511$	−13.6102	−0.602081
$512$	−28.5036	−1.25969
$513$	−3.84788	−0.169888
$514$	−17.9155	−0.790218
$515$	8.50271	0.374674
$516$	5.36292	0.236089
$517$	−17.8653	−0.785716
$518$	−53.0798	−2.33219
$519$	4.84638	0.212732
$520$	−3.90792	−0.171374
$521$	35.4586	1.55347	0.776734	−	0.629828i	$-0.216875\pi$
$521$	35.4586	1.55347	0.776734	+	0.629828i	$0.216875\pi$
$522$	38.0133	1.66380
$523$	−12.3437	−0.539751	−0.269875	−	0.962895i	$-0.586982\pi$
$523$	−12.3437	−0.539751	−0.269875	+	0.962895i	$0.586982\pi$
$524$	−51.2134	−2.23727
$525$	−9.85312	−0.430025
$526$	21.1611	0.922667
$527$	−7.17487	−0.312542
$528$	2.92466	0.127280
$529$	0.590775	0.0256859
$530$	25.0065	1.08621
$531$	15.0297	0.652232
$532$	−14.4512	−0.626538
$533$	−6.17079	−0.267287
$534$	9.82402	0.425127
$535$	26.5004	1.14571
$536$	−13.5926	−0.587112
$537$	−3.63424	−0.156829
$538$	−20.5918	−0.887775
$539$	3.20680	0.138127
$540$	−18.2920	−0.787164
$541$	−15.6907	−0.674595	−0.337298	−	0.941398i	$-0.609513\pi$
$541$	−15.6907	−0.674595	−0.337298	+	0.941398i	$0.609513\pi$
$542$	59.5447	2.55766
$543$	7.54186	0.323652
$544$	7.20913	0.309089
$545$	24.6579	1.05623
$546$	−1.87281	−0.0801488
$547$	−12.0139	−0.513679	−0.256840	−	0.966454i	$-0.582681\pi$
$547$	−12.0139	−0.513679	−0.256840	+	0.966454i	$0.582681\pi$
$548$	31.8793	1.36181
$549$	3.51019	0.149811
$550$	−76.3393	−3.25512
$551$	−12.8893	−0.549105
$552$	1.48375	0.0631525
$553$	−32.3159	−1.37421
$554$	−8.59210	−0.365043
$555$	11.2007	0.475445
$556$	−41.4535	−1.75802
$557$	−17.7173	−0.750707	−0.375354	−	0.926882i	$-0.622479\pi$
$557$	−17.7173	−0.750707	−0.375354	+	0.926882i	$0.622479\pi$
$558$	−48.7914	−2.06551
$559$	6.94512	0.293747
$560$	32.5054	1.37360
$561$	−0.919754	−0.0388321
$562$	−26.0307	−1.09804
$563$	−13.3403	−0.562228	−0.281114	−	0.959674i	$-0.590704\pi$
$563$	−13.3403	−0.562228	−0.281114	+	0.959674i	$0.590704\pi$
$564$	4.24123	0.178588
$565$	−18.8152	−0.791561
$566$	8.83914	0.371537
$567$	22.9568	0.964093
$568$	0.0304710	0.00127854
$569$	20.2350	0.848295	0.424147	−	0.905593i	$-0.360574\pi$
$569$	20.2350	0.848295	0.424147	+	0.905593i	$0.360574\pi$
$570$	5.52771	0.231530
$571$	14.0717	0.588883	0.294442	−	0.955669i	$-0.404866\pi$
$571$	14.0717	0.588883	0.294442	+	0.955669i	$0.404866\pi$
$572$	−8.00467	−0.334692
$573$	−3.70323	−0.154705
$574$	−36.8313	−1.53731
$575$	53.9716	2.25077
$576$	32.3950	1.34979
$577$	−41.3107	−1.71979	−0.859894	−	0.510473i	$-0.829470\pi$
$577$	−41.3107	−1.71979	−0.859894	+	0.510473i	$0.829470\pi$
$578$	34.1903	1.42213
$579$	−4.79408	−0.199235
$580$	−61.2733	−2.54424
$581$	−5.36334	−0.222509
$582$	0.802639	0.0332705
$583$	9.59408	0.397346
$584$	4.68893	0.194029
$585$	−11.6467	−0.481534
$586$	−63.0704	−2.60541
$587$	23.0352	0.950765	0.475382	−	0.879779i	$-0.342310\pi$
$587$	23.0352	0.950765	0.475382	+	0.879779i	$0.342310\pi$
$588$	−0.761295	−0.0313953
$589$	16.5439	0.681680
$590$	−43.9146	−1.80794
$591$	−3.36457	−0.138400
$592$	−25.4842	−1.04740
$593$	15.5805	0.639816	0.319908	−	0.947449i	$-0.396348\pi$
$593$	15.5805	0.639816	0.319908	+	0.947449i	$0.396348\pi$
$594$	−12.7215	−0.521969
$595$	−10.2224	−0.419076
$596$	−43.7596	−1.79246
$597$	−1.12003	−0.0458398
$598$	10.2585	0.419502
$599$	44.8216	1.83136	0.915681	−	0.401906i	$-0.131652\pi$
$599$	44.8216	1.83136	0.915681	+	0.401906i	$0.131652\pi$
$600$	3.39455	0.138582
$601$	20.8627	0.851008	0.425504	−	0.904957i	$-0.360097\pi$
$601$	20.8627	0.851008	0.425504	+	0.904957i	$0.360097\pi$
$602$	41.4530	1.68950
$603$	−40.5100	−1.64969
$604$	−32.6212	−1.32734
$605$	1.68640	0.0685620
$606$	11.8933	0.483134
$607$	26.4350	1.07296	0.536482	−	0.843912i	$-0.319753\pi$
$607$	26.4350	1.07296	0.536482	+	0.843912i	$0.319753\pi$
$608$	−16.6229	−0.674148
$609$	−5.50012	−0.222876
$610$	−10.2563	−0.415266
$611$	5.49251	0.222203
$612$	−6.43497	−0.260118
$613$	−2.80387	−0.113247	−0.0566236	−	0.998396i	$-0.518033\pi$
$613$	−2.80387	−0.113247	−0.0566236	+	0.998396i	$0.518033\pi$
$614$	8.63699	0.348560
$615$	7.77203	0.313398
$616$	−8.94896	−0.360564
$617$	34.6458	1.39479	0.697394	−	0.716688i	$-0.254343\pi$
$617$	34.6458	1.39479	0.697394	+	0.716688i	$0.254343\pi$
$618$	−1.40383	−0.0564702
$619$	21.1064	0.848337	0.424169	−	0.905583i	$-0.360566\pi$
$619$	21.1064	0.848337	0.424169	+	0.905583i	$0.360566\pi$
$620$	78.6464	3.15852
$621$	8.99404	0.360918
$622$	−39.4330	−1.58112
$623$	41.8909	1.67832
$624$	−0.899157	−0.0359951
$625$	42.9173	1.71669
$626$	−3.07811	−0.123026
$627$	2.12078	0.0846959
$628$	56.8316	2.26783
$629$	8.01434	0.319553
$630$	−69.5153	−2.76956
$631$	−2.55593	−0.101750	−0.0508749	−	0.998705i	$-0.516201\pi$
$631$	−2.55593	−0.101750	−0.0508749	+	0.998705i	$0.516201\pi$
$632$	11.1334	0.442861
$633$	−4.13540	−0.164367
$634$	13.7202	0.544898
$635$	−36.1910	−1.43620
$636$	−2.27764	−0.0903142
$637$	−0.985897	−0.0390627
$638$	−42.6134	−1.68708
$639$	0.0908125	0.00359249
$640$	−30.4330	−1.20297
$641$	−8.33107	−0.329057	−0.164529	−	0.986372i	$-0.552610\pi$
$641$	−8.33107	−0.329057	−0.164529	+	0.986372i	$0.552610\pi$
$642$	−4.37531	−0.172680
$643$	29.5206	1.16418	0.582089	−	0.813125i	$-0.302235\pi$
$643$	29.5206	1.16418	0.582089	+	0.813125i	$0.302235\pi$
$644$	33.7782	1.33105
$645$	−8.74728	−0.344424
$646$	3.95518	0.155614
$647$	−27.4619	−1.07964	−0.539818	−	0.841782i	$-0.681507\pi$
$647$	−27.4619	−1.07964	−0.539818	+	0.841782i	$0.681507\pi$
$648$	−7.90896	−0.310693
$649$	−16.8484	−0.661359
$650$	23.4697	0.920558
$651$	7.05960	0.276687
$652$	−14.7549	−0.577846
$653$	12.1705	0.476267	0.238134	−	0.971232i	$-0.423464\pi$
$653$	12.1705	0.476267	0.238134	+	0.971232i	$0.423464\pi$
$654$	−4.07110	−0.159193
$655$	83.5325	3.26389
$656$	−17.6831	−0.690410
$657$	13.9744	0.545193
$658$	32.7828	1.27801
$659$	−19.8789	−0.774370	−0.387185	−	0.922002i	$-0.626553\pi$
$659$	−19.8789	−0.774370	−0.387185	+	0.922002i	$0.626553\pi$
$660$	10.0818	0.392432
$661$	42.9874	1.67202	0.836008	−	0.548717i	$-0.184884\pi$
$661$	42.9874	1.67202	0.836008	+	0.548717i	$0.184884\pi$
$662$	−8.86376	−0.344500
$663$	0.282769	0.0109818
$664$	1.84775	0.0717068
$665$	23.5709	0.914039
$666$	54.5001	2.11183
$667$	30.1275	1.16654
$668$	−59.3823	−2.29757
$669$	2.45107	0.0947639
$670$	118.365	4.57282
$671$	−3.93497	−0.151908
$672$	−7.09331	−0.273630
$673$	−49.9818	−1.92666	−0.963328	−	0.268325i	$-0.913530\pi$
$673$	−49.9818	−1.92666	−0.963328	+	0.268325i	$0.913530\pi$
$674$	46.3852	1.78669
$675$	20.5768	0.792001
$676$	2.46095	0.0946521
$677$	32.2907	1.24103	0.620516	−	0.784194i	$-0.286923\pi$
$677$	32.2907	1.24103	0.620516	+	0.784194i	$0.286923\pi$
$678$	3.10645	0.119303
$679$	3.42256	0.131346
$680$	3.52176	0.135053
$681$	−2.75365	−0.105520
$682$	54.6958	2.09441
$683$	23.8134	0.911195	0.455598	−	0.890186i	$-0.349426\pi$
$683$	23.8134	0.911195	0.455598	+	0.890186i	$0.349426\pi$
$684$	14.8378	0.567339
$685$	−51.9972	−1.98671
$686$	35.8960	1.37052
$687$	−5.38464	−0.205437
$688$	19.9021	0.758759
$689$	−2.94960	−0.112371
$690$	−12.9205	−0.491874
$691$	40.4397	1.53840	0.769199	−	0.639009i	$-0.220655\pi$
$691$	40.4397	1.53840	0.769199	+	0.639009i	$0.220655\pi$
$692$	−38.0104	−1.44494
$693$	−26.6705	−1.01313
$694$	12.6394	0.479784
$695$	67.6135	2.56473
$696$	1.89488	0.0718252
$697$	5.56103	0.210639
$698$	−50.4870	−1.91096
$699$	6.10568	0.230938
$700$	77.2786	2.92086
$701$	−36.1846	−1.36668	−0.683338	−	0.730103i	$-0.739472\pi$
$701$	−36.1846	−1.36668	−0.683338	+	0.730103i	$0.739472\pi$
$702$	3.91109	0.147614
$703$	−18.4796	−0.696970
$704$	−36.3152	−1.36868
$705$	−6.91774	−0.260537
$706$	−37.9947	−1.42995
$707$	50.7147	1.90732
$708$	3.99982	0.150323
$709$	−14.3410	−0.538588	−0.269294	−	0.963058i	$-0.586790\pi$
$709$	−14.3410	−0.538588	−0.269294	+	0.963058i	$0.586790\pi$
$710$	−0.265342	−0.00995809
$711$	33.1806	1.24437
$712$	−14.4321	−0.540865
$713$	−38.6697	−1.44819
$714$	1.68775	0.0631624
$715$	13.0562	0.488273
$716$	28.5036	1.06523
$717$	−7.39805	−0.276285
$718$	−26.8866	−1.00340
$719$	42.1490	1.57189	0.785946	−	0.618296i	$-0.212177\pi$
$719$	42.1490	1.57189	0.785946	+	0.618296i	$0.212177\pi$
$720$	−33.3751	−1.24382
$721$	−5.98610	−0.222934
$722$	31.0099	1.15407
$723$	6.72644	0.250159
$724$	−59.1513	−2.19834
$725$	68.9266	2.55987
$726$	−0.278431	−0.0103335
$727$	17.0085	0.630810	0.315405	−	0.948957i	$-0.397860\pi$
$727$	17.0085	0.630810	0.315405	+	0.948957i	$0.397860\pi$
$728$	2.75127	0.101969
$729$	−21.8279	−0.808441
$730$	−40.8312	−1.51123
$731$	−6.25884	−0.231492
$732$	0.934164	0.0345277
$733$	11.1541	0.411985	0.205993	−	0.978554i	$-0.433958\pi$
$733$	11.1541	0.411985	0.205993	+	0.978554i	$0.433958\pi$
$734$	32.9634	1.21670
$735$	1.24172	0.0458016
$736$	38.8544	1.43219
$737$	45.4122	1.67278
$738$	37.8168	1.39205
$739$	29.9129	1.10036	0.550182	−	0.835045i	$-0.314558\pi$
$739$	29.9129	1.10036	0.550182	+	0.835045i	$0.314558\pi$
$740$	−87.8481	−3.22936
$741$	−0.652013	−0.0239523
$742$	−17.6051	−0.646304
$743$	28.8763	1.05937	0.529685	−	0.848194i	$-0.322310\pi$
$743$	28.8763	1.05937	0.529685	+	0.848194i	$0.322310\pi$
$744$	−2.43214	−0.0891666
$745$	71.3748	2.61497
$746$	−60.6713	−2.22134
$747$	5.50684	0.201485
$748$	7.21369	0.263759
$749$	−18.6569	−0.681708
$750$	−16.2590	−0.593695
$751$	33.4202	1.21952	0.609760	−	0.792586i	$-0.291266\pi$
$751$	33.4202	1.21952	0.609760	+	0.792586i	$0.291266\pi$
$752$	15.7394	0.573958
$753$	9.67968	0.352747
$754$	13.1011	0.477113
$755$	53.2073	1.93641
$756$	12.8780	0.468369
$757$	−11.6540	−0.423573	−0.211786	−	0.977316i	$-0.567928\pi$
$757$	−11.6540	−0.423573	−0.211786	+	0.977316i	$0.567928\pi$
$758$	−74.2657	−2.69745
$759$	−4.95712	−0.179932
$760$	−8.12053	−0.294563
$761$	26.8598	0.973667	0.486833	−	0.873495i	$-0.338152\pi$
$761$	26.8598	0.973667	0.486833	+	0.873495i	$0.338152\pi$
$762$	5.97526	0.216461
$763$	−17.3597	−0.628463
$764$	29.0447	1.05080
$765$	10.4959	0.379479
$766$	−63.5004	−2.29436
$767$	5.17988	0.187035
$768$	−1.98181	−0.0715125
$769$	29.1265	1.05033	0.525164	−	0.851001i	$-0.324004\pi$
$769$	29.1265	1.05033	0.525164	+	0.851001i	$0.324004\pi$
$770$	77.9276	2.80831
$771$	−2.66154	−0.0958529
$772$	37.6003	1.35326
$773$	−21.2427	−0.764045	−0.382023	−	0.924153i	$-0.624772\pi$
$773$	−21.2427	−0.764045	−0.382023	+	0.924153i	$0.624772\pi$
$774$	−42.5621	−1.52986
$775$	−88.4696	−3.17792
$776$	−1.17912	−0.0423281
$777$	−7.88558	−0.282893
$778$	−47.5908	−1.70621
$779$	−12.8227	−0.459421
$780$	−3.09954	−0.110981
$781$	−0.101802	−0.00364276
$782$	−9.24483	−0.330594
$783$	11.4862	0.410483
$784$	−2.82520	−0.100900
$785$	−92.6962	−3.30847
$786$	−13.7915	−0.491927
$787$	−42.1121	−1.50114	−0.750568	−	0.660793i	$-0.770220\pi$
$787$	−42.1121	−1.50114	−0.750568	+	0.660793i	$0.770220\pi$
$788$	26.3885	0.940051
$789$	3.14370	0.111919
$790$	−96.9492	−3.44930
$791$	13.2463	0.470985
$792$	9.18841	0.326496
$793$	1.20977	0.0429601
$794$	−28.2406	−1.00222
$795$	3.71498	0.131757
$796$	8.78446	0.311357
$797$	28.3321	1.00358	0.501788	−	0.864991i	$-0.332676\pi$
$797$	28.3321	1.00358	0.501788	+	0.864991i	$0.332676\pi$
$798$	−3.89163	−0.137762
$799$	−4.94977	−0.175110
$800$	88.8921	3.14281
$801$	−43.0117	−1.51975
$802$	26.4922	0.935474
$803$	−15.6655	−0.552822
$804$	−10.7809	−0.380212
$805$	−55.0945	−1.94183
$806$	−16.8157	−0.592306
$807$	−3.05913	−0.107687
$808$	−17.4720	−0.614663
$809$	−20.3924	−0.716960	−0.358480	−	0.933537i	$-0.616705\pi$
$809$	−20.3924	−0.716960	−0.358480	+	0.933537i	$0.616705\pi$
$810$	68.8712	2.41989
$811$	6.45512	0.226670	0.113335	−	0.993557i	$-0.463847\pi$
$811$	6.45512	0.226670	0.113335	+	0.993557i	$0.463847\pi$
$812$	43.1378	1.51384
$813$	8.84600	0.310243
$814$	−61.0953	−2.14139
$815$	24.0662	0.843003
$816$	0.810307	0.0283664
$817$	14.4317	0.504902
$818$	−73.0612	−2.55453
$819$	8.19957	0.286516
$820$	−60.9565	−2.12869
$821$	26.4004	0.921380	0.460690	−	0.887561i	$-0.347602\pi$
$821$	26.4004	0.921380	0.460690	+	0.887561i	$0.347602\pi$
$822$	8.58493	0.299434
$823$	−7.97822	−0.278103	−0.139052	−	0.990285i	$-0.544405\pi$
$823$	−7.97822	−0.278103	−0.139052	+	0.990285i	$0.544405\pi$
$824$	2.06231	0.0718438
$825$	−11.3410	−0.394844
$826$	30.9169	1.07574
$827$	0.885759	0.0308009	0.0154004	−	0.999881i	$-0.495098\pi$
$827$	0.885759	0.0308009	0.0154004	+	0.999881i	$0.495098\pi$
$828$	−34.6820	−1.20528
$829$	4.95555	0.172113	0.0860566	−	0.996290i	$-0.472573\pi$
$829$	4.95555	0.172113	0.0860566	+	0.996290i	$0.472573\pi$
$830$	−16.0902	−0.558500
$831$	−1.27645	−0.0442795
$832$	11.1647	0.387067
$833$	0.888475	0.0307838
$834$	−11.1632	−0.386551
$835$	96.8565	3.35186
$836$	−16.6334	−0.575279
$837$	−14.7429	−0.509590
$838$	−11.5917	−0.400427
$839$	10.3787	0.358311	0.179155	−	0.983821i	$-0.442663\pi$
$839$	10.3787	0.358311	0.179155	+	0.983821i	$0.442663\pi$
$840$	−3.46518	−0.119560
$841$	9.47560	0.326745
$842$	−66.4198	−2.28898
$843$	−3.86713	−0.133191
$844$	32.4342	1.11643
$845$	−4.01398	−0.138085
$846$	−33.6600	−1.15725
$847$	−1.18726	−0.0407949
$848$	−8.45242	−0.290257
$849$	1.31315	0.0450671
$850$	−21.1506	−0.725458
$851$	43.1941	1.48068
$852$	0.0241678	0.000827976 0
$853$	40.1655	1.37524	0.687621	−	0.726070i	$-0.258655\pi$
$853$	40.1655	1.37524	0.687621	+	0.726070i	$0.258655\pi$
$854$	7.22067	0.247086
$855$	−24.2015	−0.827675
$856$	6.42759	0.219691
$857$	11.9560	0.408411	0.204205	−	0.978928i	$-0.434539\pi$
$857$	11.9560	0.408411	0.204205	+	0.978928i	$0.434539\pi$
$858$	−2.15562	−0.0735916
$859$	24.4694	0.834885	0.417443	−	0.908703i	$-0.362926\pi$
$859$	24.4694	0.834885	0.417443	+	0.908703i	$0.362926\pi$
$860$	68.6055	2.33943
$861$	−5.47168	−0.186474
$862$	−18.6144	−0.634010
$863$	12.8486	0.437372	0.218686	−	0.975795i	$-0.429823\pi$
$863$	12.8486	0.437372	0.218686	+	0.975795i	$0.429823\pi$
$864$	14.8133	0.503960
$865$	61.9976	2.10798
$866$	−58.5516	−1.98966
$867$	5.07934	0.172503
$868$	−55.3688	−1.87934
$869$	−37.1959	−1.26178
$870$	−16.5006	−0.559423
$871$	−13.9615	−0.473068
$872$	5.98068	0.202532
$873$	−3.51413	−0.118935
$874$	21.3169	0.721053
$875$	−69.3304	−2.34380
$876$	3.71899	0.125653
$877$	30.2178	1.02038	0.510191	−	0.860061i	$-0.329575\pi$
$877$	30.2178	1.02038	0.510191	+	0.860061i	$0.329575\pi$
$878$	37.8766	1.27827
$879$	−9.36978	−0.316035
$880$	37.4139	1.26122
$881$	20.7672	0.699666	0.349833	−	0.936812i	$-0.386238\pi$
$881$	20.7672	0.699666	0.349833	+	0.936812i	$0.386238\pi$
$882$	6.04192	0.203442
$883$	13.6598	0.459688	0.229844	−	0.973227i	$-0.426178\pi$
$883$	13.6598	0.459688	0.229844	+	0.973227i	$0.426178\pi$
$884$	−2.21777	−0.0745918
$885$	−6.52398	−0.219301
$886$	−54.2712	−1.82328
$887$	−28.2895	−0.949868	−0.474934	−	0.880021i	$-0.657528\pi$
$887$	−28.2895	−0.949868	−0.474934	+	0.880021i	$0.657528\pi$
$888$	2.71670	0.0911666
$889$	25.4793	0.854547
$890$	125.674	4.21262
$891$	26.4234	0.885217
$892$	−19.2239	−0.643664
$893$	11.4132	0.381930
$894$	−11.7842	−0.394124
$895$	−46.4913	−1.55403
$896$	21.4255	0.715777
$897$	1.52401	0.0508853
$898$	−43.3687	−1.44723
$899$	−49.3848	−1.64707
$900$	−79.3463	−2.64488
$901$	2.65814	0.0885553
$902$	−42.3931	−1.41154
$903$	6.15828	0.204935
$904$	−4.56356	−0.151782
$905$	96.4798	3.20710
$906$	−8.78471	−0.291853
$907$	32.8934	1.09221	0.546103	−	0.837718i	$-0.316111\pi$
$907$	32.8934	1.09221	0.546103	+	0.837718i	$0.316111\pi$
$908$	21.5970	0.716723
$909$	−52.0717	−1.72711
$910$	−23.9580	−0.794201
$911$	−32.7960	−1.08658	−0.543291	−	0.839545i	$-0.682822\pi$
$911$	−32.7960	−1.08658	−0.543291	+	0.839545i	$0.682822\pi$
$912$	−1.86842	−0.0618695
$913$	−6.17324	−0.204305
$914$	−68.3272	−2.26006
$915$	−1.52368	−0.0503714
$916$	42.2321	1.39539
$917$	−58.8088	−1.94204
$918$	−3.52461	−0.116330
$919$	−22.9862	−0.758246	−0.379123	−	0.925346i	$-0.623774\pi$
$919$	−22.9862	−0.758246	−0.379123	+	0.925346i	$0.623774\pi$
$920$	18.9809	0.625782
$921$	1.28312	0.0422801
$922$	−0.260084	−0.00856541
$923$	0.0312980	0.00103019
$924$	−7.09779	−0.233500
$925$	98.8207	3.24920
$926$	−2.11210	−0.0694078
$927$	6.14627	0.201870
$928$	49.6206	1.62887
$929$	−40.1498	−1.31727	−0.658635	−	0.752462i	$-0.728866\pi$
$929$	−40.1498	−1.31727	−0.658635	+	0.752462i	$0.728866\pi$
$930$	21.1791	0.694490
$931$	−2.04866	−0.0671421
$932$	−47.8872	−1.56860
$933$	−5.85820	−0.191789
$934$	4.41612	0.144500
$935$	−11.7660	−0.384790
$936$	−2.82488	−0.0923341
$937$	−31.4645	−1.02790	−0.513950	−	0.857820i	$-0.671818\pi$
$937$	−31.4645	−1.02790	−0.513950	+	0.857820i	$0.671818\pi$
$938$	−83.3313	−2.72086
$939$	−0.457287	−0.0149230
$940$	54.2562	1.76964
$941$	39.3419	1.28251	0.641255	−	0.767328i	$-0.278414\pi$
$941$	39.3419	1.28251	0.641255	+	0.767328i	$0.278414\pi$
$942$	15.3045	0.498646
$943$	29.9718	0.976015
$944$	14.8435	0.483116
$945$	−21.0049	−0.683290
$946$	47.7127	1.55127
$947$	11.2906	0.366894	0.183447	−	0.983030i	$-0.441274\pi$
$947$	11.2906	0.366894	0.183447	+	0.983030i	$0.441274\pi$
$948$	8.83032	0.286795
$949$	4.81619	0.156340
$950$	48.7693	1.58228
$951$	2.03828	0.0660957
$952$	−2.47940	−0.0803578
$953$	−5.37654	−0.174163	−0.0870816	−	0.996201i	$-0.527754\pi$
$953$	−5.37654	−0.174163	−0.0870816	+	0.996201i	$0.527754\pi$
$954$	18.0762	0.585238
$955$	−47.3738	−1.53298
$956$	58.0233	1.87661
$957$	−6.33068	−0.204642
$958$	38.2850	1.23693
$959$	36.6072	1.18211
$960$	−14.0618	−0.453843
$961$	32.3870	1.04474
$962$	18.7831	0.605592
$963$	19.1561	0.617296
$964$	−52.7559	−1.69915
$965$	−61.3286	−1.97424
$966$	9.09630	0.292669
$967$	−3.27078	−0.105181	−0.0525906	−	0.998616i	$-0.516748\pi$
$967$	−3.27078	−0.105181	−0.0525906	+	0.998616i	$0.516748\pi$
$968$	0.409031	0.0131468
$969$	0.587584	0.0188759
$970$	10.2678	0.329679
$971$	8.81074	0.282750	0.141375	−	0.989956i	$-0.454848\pi$
$971$	8.81074	0.282750	0.141375	+	0.989956i	$0.454848\pi$
$972$	−19.9442	−0.639710
$973$	−47.6014	−1.52603
$974$	14.1022	0.451864
$975$	3.48668	0.111663
$976$	3.46673	0.110967
$977$	−20.5121	−0.656239	−0.328119	−	0.944636i	$-0.606415\pi$
$977$	−20.5121	−0.656239	−0.328119	+	0.944636i	$0.606415\pi$
$978$	−3.97342	−0.127056
$979$	48.2167	1.54101
$980$	−9.73891	−0.311098
$981$	17.8242	0.569082
$982$	18.6295	0.594490
$983$	−39.2372	−1.25147	−0.625737	−	0.780034i	$-0.715202\pi$
$983$	−39.2372	−1.25147	−0.625737	+	0.780034i	$0.715202\pi$
$984$	1.88508	0.0600941
$985$	−43.0414	−1.37141
$986$	−11.8065	−0.375995
$987$	4.87024	0.155022
$988$	5.11377	0.162691
$989$	−33.7327	−1.07264
$990$	−80.0127	−2.54297
$991$	−34.2644	−1.08844	−0.544222	−	0.838941i	$-0.683175\pi$
$991$	−34.2644	−1.08844	−0.544222	+	0.838941i	$0.683175\pi$
$992$	−63.6897	−2.02215
$993$	−1.31681	−0.0417876
$994$	0.186806	0.00592514
$995$	−14.3280	−0.454230
$996$	1.46553	0.0464371
$997$	−52.8573	−1.67401	−0.837004	−	0.547196i	$-0.815695\pi$
$997$	−52.8573	−1.67401	−0.837004	+	0.547196i	$0.815695\pi$
$998$	60.9598	1.92965
$999$	16.4679	0.521020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6019.2.a.d.1.20	✓	123

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6019.2.a.d.1.20	✓	123	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:	\( N \)	\(=\)	\( 6019 = 13 \cdot 463 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11210 q^{2} -0.313775 q^{3} +2.46095 q^{4} -4.01398 q^{5} +0.662722 q^{6} +2.82593 q^{7} -0.973578 q^{8} -2.90155 q^{9} +O(q^{10})\)	\(q-2.11210 q^{2} -0.313775 q^{3} +2.46095 q^{4} -4.01398 q^{5} +0.662722 q^{6} +2.82593 q^{7} -0.973578 q^{8} -2.90155 q^{9} +8.47792 q^{10} +3.25267 q^{11} -0.772185 q^{12} -1.00000 q^{13} -5.96864 q^{14} +1.25949 q^{15} -2.86562 q^{16} +0.901185 q^{17} +6.12835 q^{18} -2.07796 q^{19} -9.87822 q^{20} -0.886706 q^{21} -6.86996 q^{22} +4.85703 q^{23} +0.305484 q^{24} +11.1120 q^{25} +2.11210 q^{26} +1.85176 q^{27} +6.95449 q^{28} +6.20287 q^{29} -2.66016 q^{30} -7.96160 q^{31} +7.99961 q^{32} -1.02061 q^{33} -1.90339 q^{34} -11.3432 q^{35} -7.14057 q^{36} +8.89311 q^{37} +4.38886 q^{38} +0.313775 q^{39} +3.90792 q^{40} +6.17079 q^{41} +1.87281 q^{42} -6.94512 q^{43} +8.00467 q^{44} +11.6467 q^{45} -10.2585 q^{46} -5.49251 q^{47} +0.899157 q^{48} +0.985897 q^{49} -23.4697 q^{50} -0.282769 q^{51} -2.46095 q^{52} +2.94960 q^{53} -3.91109 q^{54} -13.0562 q^{55} -2.75127 q^{56} +0.652013 q^{57} -13.1011 q^{58} -5.17988 q^{59} +3.09954 q^{60} -1.20977 q^{61} +16.8157 q^{62} -8.19957 q^{63} -11.1647 q^{64} +4.01398 q^{65} +2.15562 q^{66} +13.9615 q^{67} +2.21777 q^{68} -1.52401 q^{69} +23.9580 q^{70} -0.0312980 q^{71} +2.82488 q^{72} -4.81619 q^{73} -18.7831 q^{74} -3.48668 q^{75} -5.11377 q^{76} +9.19183 q^{77} -0.662722 q^{78} -11.4355 q^{79} +11.5025 q^{80} +8.12360 q^{81} -13.0333 q^{82} -1.89790 q^{83} -2.18214 q^{84} -3.61734 q^{85} +14.6688 q^{86} -1.94630 q^{87} -3.16673 q^{88} +14.8237 q^{89} -24.5991 q^{90} -2.82593 q^{91} +11.9529 q^{92} +2.49815 q^{93} +11.6007 q^{94} +8.34091 q^{95} -2.51008 q^{96} +1.21112 q^{97} -2.08231 q^{98} -9.43777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 123 q + 10 q^{2} + q^{3} + 136 q^{4} + 46 q^{5} + 16 q^{6} + 12 q^{7} + 30 q^{8} + 154 q^{9}+O(q^{10}) \) 123 * q + 10 * q^2 + q^3 + 136 * q^4 + 46 * q^5 + 16 * q^6 + 12 * q^7 + 30 * q^8 + 154 * q^9	\( 123 q + 10 q^{2} + q^{3} + 136 q^{4} + 46 q^{5} + 16 q^{6} + 12 q^{7} + 30 q^{8} + 154 q^{9} + 5 q^{10} + 53 q^{11} - 6 q^{12} - 123 q^{13} + 21 q^{14} + 29 q^{15} + 166 q^{16} - 35 q^{17} + 28 q^{18} + 23 q^{19} + 93 q^{20} + 72 q^{21} + 8 q^{22} + 42 q^{23} + 55 q^{24} + 153 q^{25} - 10 q^{26} + 7 q^{27} + 39 q^{28} + 86 q^{29} + 44 q^{30} + 16 q^{31} + 70 q^{32} + 40 q^{33} + 10 q^{34} + 6 q^{35} + 222 q^{36} + 52 q^{37} + 12 q^{38} - q^{39} + 14 q^{40} + 80 q^{41} + 29 q^{42} + 2 q^{43} + 143 q^{44} + 137 q^{45} + 39 q^{46} + 45 q^{47} - 27 q^{48} + 163 q^{49} + 102 q^{50} + 48 q^{51} - 136 q^{52} + 117 q^{53} + 75 q^{54} + 20 q^{55} + 88 q^{56} + 67 q^{57} + 56 q^{58} + 88 q^{59} + 96 q^{60} + 57 q^{61} - 13 q^{62} + 48 q^{63} + 228 q^{64} - 46 q^{65} + 28 q^{66} + 43 q^{67} - 56 q^{68} + 92 q^{69} + 14 q^{70} + 90 q^{71} + 98 q^{72} + 25 q^{73} + 80 q^{74} + 21 q^{75} + 75 q^{76} + 112 q^{77} - 16 q^{78} + 36 q^{79} + 208 q^{80} + 231 q^{81} - 27 q^{82} + 93 q^{83} + 175 q^{84} + 77 q^{85} + 199 q^{86} + 15 q^{87} + 43 q^{88} + 140 q^{89} + 11 q^{90} - 12 q^{91} + 93 q^{92} + 140 q^{93} + 4 q^{94} + 23 q^{95} + 105 q^{96} + 43 q^{97} + 67 q^{98} + 140 q^{99}+O(q^{100}) \) 123 * q + 10 * q^2 + q^3 + 136 * q^4 + 46 * q^5 + 16 * q^6 + 12 * q^7 + 30 * q^8 + 154 * q^9 + 5 * q^10 + 53 * q^11 - 6 * q^12 - 123 * q^13 + 21 * q^14 + 29 * q^15 + 166 * q^16 - 35 * q^17 + 28 * q^18 + 23 * q^19 + 93 * q^20 + 72 * q^21 + 8 * q^22 + 42 * q^23 + 55 * q^24 + 153 * q^25 - 10 * q^26 + 7 * q^27 + 39 * q^28 + 86 * q^29 + 44 * q^30 + 16 * q^31 + 70 * q^32 + 40 * q^33 + 10 * q^34 + 6 * q^35 + 222 * q^36 + 52 * q^37 + 12 * q^38 - q^39 + 14 * q^40 + 80 * q^41 + 29 * q^42 + 2 * q^43 + 143 * q^44 + 137 * q^45 + 39 * q^46 + 45 * q^47 - 27 * q^48 + 163 * q^49 + 102 * q^50 + 48 * q^51 - 136 * q^52 + 117 * q^53 + 75 * q^54 + 20 * q^55 + 88 * q^56 + 67 * q^57 + 56 * q^58 + 88 * q^59 + 96 * q^60 + 57 * q^61 - 13 * q^62 + 48 * q^63 + 228 * q^64 - 46 * q^65 + 28 * q^66 + 43 * q^67 - 56 * q^68 + 92 * q^69 + 14 * q^70 + 90 * q^71 + 98 * q^72 + 25 * q^73 + 80 * q^74 + 21 * q^75 + 75 * q^76 + 112 * q^77 - 16 * q^78 + 36 * q^79 + 208 * q^80 + 231 * q^81 - 27 * q^82 + 93 * q^83 + 175 * q^84 + 77 * q^85 + 199 * q^86 + 15 * q^87 + 43 * q^88 + 140 * q^89 + 11 * q^90 - 12 * q^91 + 93 * q^92 + 140 * q^93 + 4 * q^94 + 23 * q^95 + 105 * q^96 + 43 * q^97 + 67 * q^98 + 140 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more