LMFDB - Embedded newform 6019.2.a.c.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:	$1$
Dimension:	$108$
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)

$f(q)$	$=$	$q-2.12654 q^{2} -0.167601 q^{3} +2.52216 q^{4} -1.92531 q^{5} +0.356410 q^{6} +0.0410631 q^{7} -1.11038 q^{8} -2.97191 q^{9} +O(q^{10})$	\(q-2.12654 q^{2} -0.167601 q^{3} +2.52216 q^{4} -1.92531 q^{5} +0.356410 q^{6} +0.0410631 q^{7} -1.11038 q^{8} -2.97191 q^{9} +4.09424 q^{10} +3.07732 q^{11} -0.422716 q^{12} -1.00000 q^{13} -0.0873222 q^{14} +0.322684 q^{15} -2.68304 q^{16} -4.74077 q^{17} +6.31987 q^{18} -1.80299 q^{19} -4.85593 q^{20} -0.00688222 q^{21} -6.54404 q^{22} -4.58005 q^{23} +0.186102 q^{24} -1.29318 q^{25} +2.12654 q^{26} +1.00090 q^{27} +0.103568 q^{28} +2.69979 q^{29} -0.686199 q^{30} +8.90052 q^{31} +7.92635 q^{32} -0.515762 q^{33} +10.0814 q^{34} -0.0790592 q^{35} -7.49562 q^{36} +7.81755 q^{37} +3.83412 q^{38} +0.167601 q^{39} +2.13783 q^{40} +9.02010 q^{41} +0.0146353 q^{42} +5.06430 q^{43} +7.76149 q^{44} +5.72185 q^{45} +9.73965 q^{46} +12.4793 q^{47} +0.449680 q^{48} -6.99831 q^{49} +2.75000 q^{50} +0.794558 q^{51} -2.52216 q^{52} -1.95071 q^{53} -2.12845 q^{54} -5.92480 q^{55} -0.0455959 q^{56} +0.302183 q^{57} -5.74121 q^{58} -14.0622 q^{59} +0.813859 q^{60} -0.519818 q^{61} -18.9273 q^{62} -0.122036 q^{63} -11.4896 q^{64} +1.92531 q^{65} +1.09679 q^{66} -14.4946 q^{67} -11.9570 q^{68} +0.767621 q^{69} +0.168122 q^{70} -2.74485 q^{71} +3.29996 q^{72} -3.91130 q^{73} -16.6243 q^{74} +0.216739 q^{75} -4.54742 q^{76} +0.126364 q^{77} -0.356410 q^{78} +12.3492 q^{79} +5.16568 q^{80} +8.74798 q^{81} -19.1816 q^{82} -2.28778 q^{83} -0.0173580 q^{84} +9.12745 q^{85} -10.7694 q^{86} -0.452488 q^{87} -3.41701 q^{88} -6.45962 q^{89} -12.1677 q^{90} -0.0410631 q^{91} -11.5516 q^{92} -1.49174 q^{93} -26.5376 q^{94} +3.47131 q^{95} -1.32846 q^{96} +5.74742 q^{97} +14.8822 q^{98} -9.14552 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 108 q - 11 q^{2} + q^{3} + 95 q^{4} - 40 q^{5} - 10 q^{6} - 8 q^{7} - 33 q^{8} + 79 q^{9}+O(q^{10}) $ 108 * q - 11 * q^2 + q^3 + 95 * q^4 - 40 * q^5 - 10 * q^6 - 8 * q^7 - 33 * q^8 + 79 * q^9	\( 108 q - 11 q^{2} + q^{3} + 95 q^{4} - 40 q^{5} - 10 q^{6} - 8 q^{7} - 33 q^{8} + 79 q^{9} - q^{10} - 45 q^{11} - 6 q^{12} - 108 q^{13} - 31 q^{14} - 39 q^{15} + 73 q^{16} + 21 q^{17} - 35 q^{18} - 19 q^{19} - 79 q^{20} - 72 q^{21} - 26 q^{23} - 23 q^{24} + 92 q^{25} + 11 q^{26} + 7 q^{27} - 21 q^{28} - 94 q^{29} - 24 q^{30} - 36 q^{31} - 77 q^{32} - 32 q^{33} - 58 q^{34} - 10 q^{35} + 17 q^{36} - 54 q^{37} - 12 q^{38} - q^{39} - 4 q^{40} - 68 q^{41} - 11 q^{42} - 32 q^{43} - 151 q^{44} - 121 q^{45} - 33 q^{46} - 51 q^{47} - 27 q^{48} + 72 q^{49} - 45 q^{50} - 24 q^{51} - 95 q^{52} - 81 q^{53} - 29 q^{54} + 4 q^{55} - 68 q^{56} - 45 q^{57} - 30 q^{58} - 94 q^{59} - 108 q^{60} - 39 q^{61} - 9 q^{62} - 52 q^{63} + 31 q^{64} + 40 q^{65} - 40 q^{66} - 47 q^{67} + 24 q^{68} - 60 q^{69} - 66 q^{70} - 86 q^{71} - 91 q^{72} - 51 q^{73} - 110 q^{74} - 7 q^{75} - 51 q^{76} - 96 q^{77} + 10 q^{78} - 18 q^{79} - 136 q^{80} - 24 q^{81} - 33 q^{82} - 77 q^{83} - 113 q^{84} - 95 q^{85} - 137 q^{86} + 23 q^{87} + 19 q^{88} - 112 q^{89} - 19 q^{90} + 8 q^{91} - 111 q^{92} - 124 q^{93} - 20 q^{94} - 73 q^{95} - 77 q^{96} - 41 q^{97} - 80 q^{98} - 154 q^{99}+O(q^{100}) \) 108 * q - 11 * q^2 + q^3 + 95 * q^4 - 40 * q^5 - 10 * q^6 - 8 * q^7 - 33 * q^8 + 79 * q^9 - q^10 - 45 * q^11 - 6 * q^12 - 108 * q^13 - 31 * q^14 - 39 * q^15 + 73 * q^16 + 21 * q^17 - 35 * q^18 - 19 * q^19 - 79 * q^20 - 72 * q^21 - 26 * q^23 - 23 * q^24 + 92 * q^25 + 11 * q^26 + 7 * q^27 - 21 * q^28 - 94 * q^29 - 24 * q^30 - 36 * q^31 - 77 * q^32 - 32 * q^33 - 58 * q^34 - 10 * q^35 + 17 * q^36 - 54 * q^37 - 12 * q^38 - q^39 - 4 * q^40 - 68 * q^41 - 11 * q^42 - 32 * q^43 - 151 * q^44 - 121 * q^45 - 33 * q^46 - 51 * q^47 - 27 * q^48 + 72 * q^49 - 45 * q^50 - 24 * q^51 - 95 * q^52 - 81 * q^53 - 29 * q^54 + 4 * q^55 - 68 * q^56 - 45 * q^57 - 30 * q^58 - 94 * q^59 - 108 * q^60 - 39 * q^61 - 9 * q^62 - 52 * q^63 + 31 * q^64 + 40 * q^65 - 40 * q^66 - 47 * q^67 + 24 * q^68 - 60 * q^69 - 66 * q^70 - 86 * q^71 - 91 * q^72 - 51 * q^73 - 110 * q^74 - 7 * q^75 - 51 * q^76 - 96 * q^77 + 10 * q^78 - 18 * q^79 - 136 * q^80 - 24 * q^81 - 33 * q^82 - 77 * q^83 - 113 * q^84 - 95 * q^85 - 137 * q^86 + 23 * q^87 + 19 * q^88 - 112 * q^89 - 19 * q^90 + 8 * q^91 - 111 * q^92 - 124 * q^93 - 20 * q^94 - 73 * q^95 - 77 * q^96 - 41 * q^97 - 80 * q^98 - 154 * 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 $

$2$	−2.12654	−1.50369	−0.751844	−	0.659341i	$-0.770835\pi$
$2$	−2.12654	−1.50369	−0.751844	+	0.659341i	$0.770835\pi$
$3$	−0.167601	−0.0967645	−0.0483822	−	0.998829i	$-0.515407\pi$
$3$	−0.167601	−0.0967645	−0.0483822	+	0.998829i	$0.515407\pi$
$4$	2.52216	1.26108
$5$	−1.92531	−0.861024	−0.430512	−	0.902585i	$-0.641667\pi$
$5$	−1.92531	−0.861024	−0.430512	+	0.902585i	$0.641667\pi$
$6$	0.356410	0.145504
$7$	0.0410631	0.0155204	0.00776020	−	0.999970i	$-0.497530\pi$
$7$	0.0410631	0.0155204	0.00776020	+	0.999970i	$0.497530\pi$
$8$	−1.11038	−0.392580
$9$	−2.97191	−0.990637
$10$	4.09424	1.29471
$11$	3.07732	0.927847	0.463924	−	0.885875i	$-0.346441\pi$
$11$	3.07732	0.927847	0.463924	+	0.885875i	$0.346441\pi$
$12$	−0.422716	−0.122028
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.0873222	−0.0233379
$15$	0.322684	0.0833166
$16$	−2.68304	−0.670760
$17$	−4.74077	−1.14981	−0.574903	−	0.818222i	$-0.694960\pi$
$17$	−4.74077	−1.14981	−0.574903	+	0.818222i	$0.694960\pi$
$18$	6.31987	1.48961
$19$	−1.80299	−0.413634	−0.206817	−	0.978380i	$-0.566310\pi$
$19$	−1.80299	−0.413634	−0.206817	+	0.978380i	$0.566310\pi$
$20$	−4.85593	−1.08582
$21$	−0.00688222	−0.00150182
$22$	−6.54404	−1.39519
$23$	−4.58005	−0.955007	−0.477503	−	0.878630i	$-0.658458\pi$
$23$	−4.58005	−0.955007	−0.477503	+	0.878630i	$0.658458\pi$
$24$	0.186102	0.0379878
$25$	−1.29318	−0.258637
$26$	2.12654	0.417048
$27$	1.00090	0.192623
$28$	0.103568	0.0195724
$29$	2.69979	0.501339	0.250669	−	0.968073i	$-0.419349\pi$
$29$	2.69979	0.501339	0.250669	+	0.968073i	$0.419349\pi$
$30$	−0.686199	−0.125282
$31$	8.90052	1.59858	0.799290	−	0.600945i	$-0.205209\pi$
$31$	8.90052	1.59858	0.799290	+	0.600945i	$0.205209\pi$
$32$	7.92635	1.40119
$33$	−0.515762	−0.0897827
$34$	10.0814	1.72895
$35$	−0.0790592	−0.0133634
$36$	−7.49562	−1.24927
$37$	7.81755	1.28520	0.642598	−	0.766203i	$-0.277856\pi$
$37$	7.81755	1.28520	0.642598	+	0.766203i	$0.277856\pi$
$38$	3.83412	0.621976
$39$	0.167601	0.0268376
$40$	2.13783	0.338021
$41$	9.02010	1.40870	0.704351	−	0.709852i	$-0.251238\pi$
$41$	9.02010	1.40870	0.704351	+	0.709852i	$0.251238\pi$
$42$	0.0146353	0.00225828
$43$	5.06430	0.772299	0.386149	−	0.922436i	$-0.373805\pi$
$43$	5.06430	0.772299	0.386149	+	0.922436i	$0.373805\pi$
$44$	7.76149	1.17009
$45$	5.72185	0.852962
$46$	9.73965	1.43603
$47$	12.4793	1.82029	0.910145	−	0.414290i	$-0.135970\pi$
$47$	12.4793	1.82029	0.910145	+	0.414290i	$0.135970\pi$
$48$	0.449680	0.0649057
$49$	−6.99831	−0.999759
$50$	2.75000	0.388909
$51$	0.794558	0.111260
$52$	−2.52216	−0.349760
$53$	−1.95071	−0.267951	−0.133976	−	0.990985i	$-0.542774\pi$
$53$	−1.95071	−0.267951	−0.133976	+	0.990985i	$0.542774\pi$
$54$	−2.12845	−0.289645
$55$	−5.92480	−0.798899
$56$	−0.0455959	−0.00609301
$57$	0.302183	0.0400251
$58$	−5.74121	−0.753857
$59$	−14.0622	−1.83074	−0.915370	−	0.402613i	$-0.868102\pi$
$59$	−14.0622	−1.83074	−0.915370	+	0.402613i	$0.868102\pi$
$60$	0.813859	0.105069
$61$	−0.519818	−0.0665558	−0.0332779	−	0.999446i	$-0.510595\pi$
$61$	−0.519818	−0.0665558	−0.0332779	+	0.999446i	$0.510595\pi$
$62$	−18.9273	−2.40377
$63$	−0.122036	−0.0153751
$64$	−11.4896	−1.43620
$65$	1.92531	0.238805
$66$	1.09679	0.135005
$67$	−14.4946	−1.77080	−0.885401	−	0.464828i	$-0.846116\pi$
$67$	−14.4946	−1.77080	−0.885401	+	0.464828i	$0.846116\pi$
$68$	−11.9570	−1.45000
$69$	0.767621	0.0924107
$70$	0.168122	0.0200945
$71$	−2.74485	−0.325753	−0.162877	−	0.986646i	$-0.552077\pi$
$71$	−2.74485	−0.325753	−0.162877	+	0.986646i	$0.552077\pi$
$72$	3.29996	0.388904
$73$	−3.91130	−0.457783	−0.228891	−	0.973452i	$-0.573510\pi$
$73$	−3.91130	−0.457783	−0.228891	+	0.973452i	$0.573510\pi$
$74$	−16.6243	−1.93253
$75$	0.216739	0.0250269
$76$	−4.54742	−0.521624
$77$	0.126364	0.0144006
$78$	−0.356410	−0.0403554
$79$	12.3492	1.38939	0.694697	−	0.719303i	$-0.255539\pi$
$79$	12.3492	1.38939	0.694697	+	0.719303i	$0.255539\pi$
$80$	5.16568	0.577541
$81$	8.74798	0.971998
$82$	−19.1816	−2.11825
$83$	−2.28778	−0.251116	−0.125558	−	0.992086i	$-0.540072\pi$
$83$	−2.28778	−0.251116	−0.125558	+	0.992086i	$0.540072\pi$
$84$	−0.0173580	−0.00189392
$85$	9.12745	0.990011
$86$	−10.7694	−1.16130
$87$	−0.452488	−0.0485118
$88$	−3.41701	−0.364255
$89$	−6.45962	−0.684718	−0.342359	−	0.939569i	$-0.611226\pi$
$89$	−6.45962	−0.684718	−0.342359	+	0.939569i	$0.611226\pi$
$90$	−12.1677	−1.28259
$91$	−0.0410631	−0.00430459
$92$	−11.5516	−1.20434
$93$	−1.49174	−0.154686
$94$	−26.5376	−2.73715
$95$	3.47131	0.356149
$96$	−1.32846	−0.135586
$97$	5.74742	0.583562	0.291781	−	0.956485i	$-0.405752\pi$
$97$	5.74742	0.583562	0.291781	+	0.956485i	$0.405752\pi$
$98$	14.8822	1.50333
$99$	−9.14552	−0.919160
$100$	−3.26161	−0.326161
$101$	−13.4576	−1.33908	−0.669539	−	0.742777i	$-0.733508\pi$
$101$	−13.4576	−1.33908	−0.669539	+	0.742777i	$0.733508\pi$
$102$	−1.68966	−0.167301
$103$	15.2889	1.50646	0.753230	−	0.657757i	$-0.228494\pi$
$103$	15.2889	1.50646	0.753230	+	0.657757i	$0.228494\pi$
$104$	1.11038	0.108882
$105$	0.0132504	0.00129311
$106$	4.14826	0.402915
$107$	13.1298	1.26931	0.634655	−	0.772796i	$-0.281142\pi$
$107$	13.1298	1.26931	0.634655	+	0.772796i	$0.281142\pi$
$108$	2.52442	0.242913
$109$	4.07881	0.390679	0.195339	−	0.980736i	$-0.437419\pi$
$109$	4.07881	0.390679	0.195339	+	0.980736i	$0.437419\pi$
$110$	12.5993	1.20130
$111$	−1.31023	−0.124361
$112$	−0.110174	−0.0104105
$113$	5.48600	0.516079	0.258040	−	0.966134i	$-0.416923\pi$
$113$	5.48600	0.516079	0.258040	+	0.966134i	$0.416923\pi$
$114$	−0.642602	−0.0601852
$115$	8.81801	0.822284
$116$	6.80930	0.632228
$117$	2.97191	0.274753
$118$	29.9038	2.75286
$119$	−0.194671	−0.0178455
$120$	−0.358303	−0.0327085
$121$	−1.53009	−0.139099
$122$	1.10541	0.100079
$123$	−1.51178	−0.136312
$124$	22.4485	2.01594
$125$	12.1163	1.08372
$126$	0.259514	0.0231193
$127$	15.4543	1.37135	0.685674	−	0.727909i	$-0.259508\pi$
$127$	15.4543	1.37135	0.685674	+	0.727909i	$0.259508\pi$
$128$	8.58034	0.758402
$129$	−0.848782	−0.0747311
$130$	−4.09424	−0.359089
$131$	−12.8990	−1.12699	−0.563495	−	0.826119i	$-0.690544\pi$
$131$	−12.8990	−1.12699	−0.563495	+	0.826119i	$0.690544\pi$
$132$	−1.30083	−0.113223
$133$	−0.0740363	−0.00641976
$134$	30.8234	2.66273
$135$	−1.92704	−0.165853
$136$	5.26408	0.451391
$137$	−17.4507	−1.49092	−0.745458	−	0.666553i	$-0.767769\pi$
$137$	−17.4507	−1.49092	−0.745458	+	0.666553i	$0.767769\pi$
$138$	−1.63237	−0.138957
$139$	−18.0474	−1.53076	−0.765378	−	0.643580i	$-0.777448\pi$
$139$	−18.0474	−1.53076	−0.765378	+	0.643580i	$0.777448\pi$
$140$	−0.199400	−0.0168524
$141$	−2.09154	−0.176139
$142$	5.83701	0.489831
$143$	−3.07732	−0.257339
$144$	7.97375	0.664479
$145$	−5.19794	−0.431665
$146$	8.31751	0.688362
$147$	1.17292	0.0967412
$148$	19.7171	1.62073
$149$	20.8768	1.71030	0.855148	−	0.518385i	$-0.173466\pi$
$149$	20.8768	1.71030	0.855148	+	0.518385i	$0.173466\pi$
$150$	−0.460904	−0.0376326
$151$	−4.34170	−0.353323	−0.176661	−	0.984272i	$-0.556530\pi$
$151$	−4.34170	−0.353323	−0.176661	+	0.984272i	$0.556530\pi$
$152$	2.00201	0.162384
$153$	14.0891	1.13904
$154$	−0.268719	−0.0216540
$155$	−17.1363	−1.37642
$156$	0.422716	0.0338444
$157$	14.5110	1.15810	0.579052	−	0.815291i	$-0.303423\pi$
$157$	14.5110	1.15810	0.579052	+	0.815291i	$0.303423\pi$
$158$	−26.2610	−2.08921
$159$	0.326942	0.0259282
$160$	−15.2607	−1.20646
$161$	−0.188071	−0.0148221
$162$	−18.6029	−1.46158
$163$	23.9646	1.87705	0.938527	−	0.345206i	$-0.112191\pi$
$163$	23.9646	1.87705	0.938527	+	0.345206i	$0.112191\pi$
$164$	22.7501	1.77648
$165$	0.993002	0.0773051
$166$	4.86504	0.377601
$167$	−20.5263	−1.58837	−0.794185	−	0.607676i	$-0.792102\pi$
$167$	−20.5263	−1.58837	−0.794185	+	0.607676i	$0.792102\pi$
$168$	0.00764192	0.000589587 0
$169$	1.00000	0.0769231
$170$	−19.4099	−1.48867
$171$	5.35832	0.409761
$172$	12.7730	0.973929
$173$	1.79921	0.136791	0.0683956	−	0.997658i	$-0.478212\pi$
$173$	1.79921	0.136791	0.0683956	+	0.997658i	$0.478212\pi$
$174$	0.962232	0.0729466
$175$	−0.0531022	−0.00401415
$176$	−8.25658	−0.622363
$177$	2.35684	0.177151
$178$	13.7366	1.02960
$179$	−7.36235	−0.550288	−0.275144	−	0.961403i	$-0.588725\pi$
$179$	−7.36235	−0.550288	−0.275144	+	0.961403i	$0.588725\pi$
$180$	14.4314	1.07565
$181$	6.61714	0.491848	0.245924	−	0.969289i	$-0.420909\pi$
$181$	6.61714	0.491848	0.245924	+	0.969289i	$0.420909\pi$
$182$	0.0873222	0.00647276
$183$	0.0871220	0.00644024
$184$	5.08562	0.374917
$185$	−15.0512	−1.10659
$186$	3.17223	0.232599
$187$	−14.5889	−1.06684
$188$	31.4747	2.29553
$189$	0.0411000	0.00298959
$190$	−7.38186	−0.535537
$191$	−13.6371	−0.986743	−0.493372	−	0.869819i	$-0.664236\pi$
$191$	−13.6371	−0.986743	−0.493372	+	0.869819i	$0.664236\pi$
$192$	1.92567	0.138973
$193$	1.01066	0.0727488	0.0363744	−	0.999338i	$-0.488419\pi$
$193$	1.01066	0.0727488	0.0363744	+	0.999338i	$0.488419\pi$
$194$	−12.2221	−0.877496
$195$	−0.322684	−0.0231079
$196$	−17.6508	−1.26077
$197$	−0.540203	−0.0384878	−0.0192439	−	0.999815i	$-0.506126\pi$
$197$	−0.540203	−0.0384878	−0.0192439	+	0.999815i	$0.506126\pi$
$198$	19.4483	1.38213
$199$	22.3714	1.58587	0.792933	−	0.609309i	$-0.208553\pi$
$199$	22.3714	1.58587	0.792933	+	0.609309i	$0.208553\pi$
$200$	1.43593	0.101536
$201$	2.42932	0.171351
$202$	28.6180	2.01355
$203$	0.110862	0.00778098
$204$	2.00400	0.140308
$205$	−17.3665	−1.21293
$206$	−32.5124	−2.26525
$207$	13.6115	0.946065
$208$	2.68304	0.186035
$209$	−5.54837	−0.383789
$210$	−0.0281775	−0.00194443
$211$	2.76956	0.190664	0.0953321	−	0.995446i	$-0.469609\pi$
$211$	2.76956	0.190664	0.0953321	+	0.995446i	$0.469609\pi$
$212$	−4.92001	−0.337907
$213$	0.460039	0.0315214
$214$	−27.9211	−1.90865
$215$	−9.75035	−0.664968
$216$	−1.11138	−0.0756200
$217$	0.365483	0.0248106
$218$	−8.67373	−0.587459
$219$	0.655537	0.0442971
$220$	−14.9433	−1.00747
$221$	4.74077	0.318899
$222$	2.78625	0.187001
$223$	−5.90104	−0.395163	−0.197581	−	0.980286i	$-0.563309\pi$
$223$	−5.90104	−0.395163	−0.197581	+	0.980286i	$0.563309\pi$
$224$	0.325481	0.0217471
$225$	3.84323	0.256215
$226$	−11.6662	−0.776022
$227$	−18.6060	−1.23492	−0.617461	−	0.786601i	$-0.711839\pi$
$227$	−18.6060	−1.23492	−0.617461	+	0.786601i	$0.711839\pi$
$228$	0.762152	0.0504747
$229$	12.8389	0.848420	0.424210	−	0.905564i	$-0.360552\pi$
$229$	12.8389	0.848420	0.424210	+	0.905564i	$0.360552\pi$
$230$	−18.7518	−1.23646
$231$	−0.0211788	−0.00139346
$232$	−2.99781	−0.196816
$233$	−8.40261	−0.550473	−0.275237	−	0.961377i	$-0.588756\pi$
$233$	−8.40261	−0.550473	−0.275237	+	0.961377i	$0.588756\pi$
$234$	−6.31987	−0.413143
$235$	−24.0265	−1.56731
$236$	−35.4670	−2.30871
$237$	−2.06974	−0.134444
$238$	0.413975	0.0268340
$239$	−20.7790	−1.34408	−0.672041	−	0.740514i	$-0.734582\pi$
$239$	−20.7790	−1.34408	−0.672041	+	0.740514i	$0.734582\pi$
$240$	−0.865773	−0.0558854
$241$	−3.84725	−0.247823	−0.123912	−	0.992293i	$-0.539544\pi$
$241$	−3.84725	−0.247823	−0.123912	+	0.992293i	$0.539544\pi$
$242$	3.25379	0.209162
$243$	−4.46886	−0.286678
$244$	−1.31106	−0.0839321
$245$	13.4739	0.860817
$246$	3.21485	0.204971
$247$	1.80299	0.114721
$248$	−9.88300	−0.627571
$249$	0.383434	0.0242991
$250$	−25.7658	−1.62957
$251$	24.5615	1.55031	0.775155	−	0.631771i	$-0.217672\pi$
$251$	24.5615	1.55031	0.775155	+	0.631771i	$0.217672\pi$
$252$	−0.307794	−0.0193892
$253$	−14.0943	−0.886101
$254$	−32.8641	−2.06208
$255$	−1.52977	−0.0957979
$256$	4.73279	0.295799
$257$	2.71235	0.169192	0.0845959	−	0.996415i	$-0.473040\pi$
$257$	2.71235	0.169192	0.0845959	+	0.996415i	$0.473040\pi$
$258$	1.80497	0.112372
$259$	0.321013	0.0199468
$260$	4.85593	0.301152
$261$	−8.02354	−0.496645
$262$	27.4302	1.69464
$263$	1.75020	0.107922	0.0539611	−	0.998543i	$-0.482815\pi$
$263$	1.75020	0.107922	0.0539611	+	0.998543i	$0.482815\pi$
$264$	0.572695	0.0352469
$265$	3.75573	0.230712
$266$	0.157441	0.00965332
$267$	1.08264	0.0662564
$268$	−36.5577	−2.23312
$269$	4.66312	0.284315	0.142158	−	0.989844i	$-0.454596\pi$
$269$	4.66312	0.284315	0.142158	+	0.989844i	$0.454596\pi$
$270$	4.09792	0.249391
$271$	−1.11372	−0.0676538	−0.0338269	−	0.999428i	$-0.510769\pi$
$271$	−1.11372	−0.0676538	−0.0338269	+	0.999428i	$0.510769\pi$
$272$	12.7197	0.771244
$273$	0.00688222	0.000416531 0
$274$	37.1096	2.24187
$275$	−3.97955	−0.239976
$276$	1.93606	0.116537
$277$	−3.57329	−0.214698	−0.107349	−	0.994221i	$-0.534236\pi$
$277$	−3.57329	−0.214698	−0.107349	+	0.994221i	$0.534236\pi$
$278$	38.3784	2.30178
$279$	−26.4515	−1.58361
$280$	0.0877862	0.00524623
$281$	−13.2494	−0.790393	−0.395196	−	0.918597i	$-0.629323\pi$
$281$	−13.2494	−0.790393	−0.395196	+	0.918597i	$0.629323\pi$
$282$	4.44774	0.264859
$283$	13.8599	0.823884	0.411942	−	0.911210i	$-0.364851\pi$
$283$	13.8599	0.823884	0.411942	+	0.911210i	$0.364851\pi$
$284$	−6.92293	−0.410800
$285$	−0.581795	−0.0344625
$286$	6.54404	0.386957
$287$	0.370393	0.0218636
$288$	−23.5564	−1.38807
$289$	5.47491	0.322054
$290$	11.0536	0.649090
$291$	−0.963274	−0.0564681
$292$	−9.86490	−0.577300
$293$	31.8458	1.86045	0.930226	−	0.366987i	$-0.119611\pi$
$293$	31.8458	1.86045	0.930226	+	0.366987i	$0.119611\pi$
$294$	−2.49427	−0.145469
$295$	27.0741	1.57631
$296$	−8.68048	−0.504543
$297$	3.08009	0.178725
$298$	−44.3953	−2.57175
$299$	4.58005	0.264871
$300$	0.546650	0.0315608
$301$	0.207956	0.0119864
$302$	9.23279	0.531287
$303$	2.25550	0.129575
$304$	4.83749	0.277449
$305$	1.00081	0.0573062
$306$	−29.9611	−1.71276
$307$	−0.113410	−0.00647268	−0.00323634	−	0.999995i	$-0.501030\pi$
$307$	−0.113410	−0.00647268	−0.00323634	+	0.999995i	$0.501030\pi$
$308$	0.318711	0.0181602
$309$	−2.56244	−0.145772
$310$	36.4409	2.06970
$311$	18.4376	1.04550	0.522750	−	0.852486i	$-0.324906\pi$
$311$	18.4376	1.04550	0.522750	+	0.852486i	$0.324906\pi$
$312$	−0.186102	−0.0105359
$313$	−9.47042	−0.535300	−0.267650	−	0.963516i	$-0.586247\pi$
$313$	−9.47042	−0.535300	−0.267650	+	0.963516i	$0.586247\pi$
$314$	−30.8581	−1.74143
$315$	0.234957	0.0132383
$316$	31.1466	1.75213
$317$	−0.327446	−0.0183912	−0.00919560	−	0.999958i	$-0.502927\pi$
$317$	−0.327446	−0.0183912	−0.00919560	+	0.999958i	$0.502927\pi$
$318$	−0.695253	−0.0389879
$319$	8.30813	0.465166
$320$	22.1210	1.23660
$321$	−2.20057	−0.122824
$322$	0.399940	0.0222878
$323$	8.54755	0.475598
$324$	22.0638	1.22577
$325$	1.29318	0.0717330
$326$	−50.9616	−2.82250
$327$	−0.683612	−0.0378039
$328$	−10.0158	−0.553029
$329$	0.512438	0.0282516
$330$	−2.11165	−0.116243
$331$	−31.2009	−1.71496	−0.857479	−	0.514519i	$-0.827971\pi$
$331$	−31.2009	−1.71496	−0.857479	+	0.514519i	$0.827971\pi$
$332$	−5.77014	−0.316677
$333$	−23.2330	−1.27316
$334$	43.6499	2.38841
$335$	27.9067	1.52470
$336$	0.0184653	0.00100736
$337$	2.42583	0.132143	0.0660717	−	0.997815i	$-0.478953\pi$
$337$	2.42583	0.132143	0.0660717	+	0.997815i	$0.478953\pi$
$338$	−2.12654	−0.115668
$339$	−0.919459	−0.0499381
$340$	23.0209	1.24848
$341$	27.3898	1.48324
$342$	−11.3947	−0.616152
$343$	−0.574815	−0.0310371
$344$	−5.62332	−0.303189
$345$	−1.47791	−0.0795679
$346$	−3.82608	−0.205691
$347$	−26.7225	−1.43454	−0.717269	−	0.696796i	$-0.754608\pi$
$347$	−26.7225	−1.43454	−0.717269	+	0.696796i	$0.754608\pi$
$348$	−1.14125	−0.0611772
$349$	13.9159	0.744903	0.372451	−	0.928052i	$-0.378517\pi$
$349$	13.9159	0.744903	0.372451	+	0.928052i	$0.378517\pi$
$350$	0.112924	0.00603603
$351$	−1.00090	−0.0534240
$352$	24.3919	1.30009
$353$	−0.453367	−0.0241303	−0.0120651	−	0.999927i	$-0.503841\pi$
$353$	−0.453367	−0.0241303	−0.0120651	+	0.999927i	$0.503841\pi$
$354$	−5.01190	−0.266379
$355$	5.28468	0.280482
$356$	−16.2922	−0.863484
$357$	0.0326270	0.00172681
$358$	15.6563	0.827461
$359$	−29.9920	−1.58292	−0.791460	−	0.611221i	$-0.790678\pi$
$359$	−29.9920	−1.58292	−0.791460	+	0.611221i	$0.790678\pi$
$360$	−6.35345	−0.334856
$361$	−15.7492	−0.828907
$362$	−14.0716	−0.739586
$363$	0.256445	0.0134599
$364$	−0.103568	−0.00542842
$365$	7.53045	0.394162
$366$	−0.185268	−0.00968411
$367$	−23.8742	−1.24622	−0.623110	−	0.782134i	$-0.714131\pi$
$367$	−23.8742	−1.24622	−0.623110	+	0.782134i	$0.714131\pi$
$368$	12.2885	0.640580
$369$	−26.8069	−1.39551
$370$	32.0069	1.66396
$371$	−0.0801024	−0.00415871
$372$	−3.76239	−0.195071
$373$	−20.1275	−1.04216	−0.521081	−	0.853507i	$-0.674471\pi$
$373$	−20.1275	−1.04216	−0.521081	+	0.853507i	$0.674471\pi$
$374$	31.0238	1.60420
$375$	−2.03071	−0.104865
$376$	−13.8568	−0.714610
$377$	−2.69979	−0.139046
$378$	−0.0874007	−0.00449541
$379$	−30.0630	−1.54423	−0.772116	−	0.635481i	$-0.780802\pi$
$379$	−30.0630	−1.54423	−0.772116	+	0.635481i	$0.780802\pi$
$380$	8.75518	0.449131
$381$	−2.59016	−0.132698
$382$	28.9997	1.48375
$383$	−23.4974	−1.20066	−0.600332	−	0.799751i	$-0.704965\pi$
$383$	−23.4974	−1.20066	−0.600332	+	0.799751i	$0.704965\pi$
$384$	−1.43807	−0.0733863
$385$	−0.243291	−0.0123992
$386$	−2.14920	−0.109391
$387$	−15.0506	−0.765067
$388$	14.4959	0.735918
$389$	−20.9710	−1.06327	−0.531637	−	0.846973i	$-0.678423\pi$
$389$	−20.9710	−1.06327	−0.531637	+	0.846973i	$0.678423\pi$
$390$	0.686199	0.0347470
$391$	21.7130	1.09807
$392$	7.77082	0.392486
$393$	2.16189	0.109053
$394$	1.14876	0.0578737
$395$	−23.7760	−1.19630
$396$	−23.0664	−1.15913
$397$	−30.4229	−1.52688	−0.763442	−	0.645876i	$-0.776492\pi$
$397$	−30.4229	−1.52688	−0.763442	+	0.645876i	$0.776492\pi$
$398$	−47.5736	−2.38465
$399$	0.0124086	0.000621205 0
$400$	3.46967	0.173483
$401$	24.0363	1.20032	0.600159	−	0.799881i	$-0.295104\pi$
$401$	24.0363	1.20032	0.600159	+	0.799881i	$0.295104\pi$
$402$	−5.16603	−0.257658
$403$	−8.90052	−0.443367
$404$	−33.9421	−1.68868
$405$	−16.8426	−0.836914
$406$	−0.235752	−0.0117002
$407$	24.0571	1.19247
$408$	−0.882265	−0.0436786
$409$	−14.1843	−0.701366	−0.350683	−	0.936494i	$-0.614051\pi$
$409$	−14.1843	−0.701366	−0.350683	+	0.936494i	$0.614051\pi$
$410$	36.9304	1.82386
$411$	2.92476	0.144268
$412$	38.5610	1.89977
$413$	−0.577437	−0.0284138
$414$	−28.9453	−1.42259
$415$	4.40468	0.216217
$416$	−7.92635	−0.388621
$417$	3.02476	0.148123
$418$	11.7988	0.577099
$419$	−19.3169	−0.943693	−0.471846	−	0.881681i	$-0.656412\pi$
$419$	−19.3169	−0.943693	−0.471846	+	0.881681i	$0.656412\pi$
$420$	0.0334196	0.00163071
$421$	9.36974	0.456653	0.228327	−	0.973585i	$-0.426675\pi$
$421$	9.36974	0.456653	0.228327	+	0.973585i	$0.426675\pi$
$422$	−5.88957	−0.286700
$423$	−37.0873	−1.80325
$424$	2.16604	0.105192
$425$	6.13069	0.297382
$426$	−0.978290	−0.0473983
$427$	−0.0213453	−0.00103297
$428$	33.1155	1.60070
$429$	0.515762	0.0249012
$430$	20.7345	0.999905
$431$	−11.3595	−0.547166	−0.273583	−	0.961848i	$-0.588209\pi$
$431$	−11.3595	−0.547166	−0.273583	+	0.961848i	$0.588209\pi$
$432$	−2.68545	−0.129204
$433$	34.1724	1.64222	0.821110	−	0.570769i	$-0.193355\pi$
$433$	34.1724	1.64222	0.821110	+	0.570769i	$0.193355\pi$
$434$	−0.777213	−0.0373074
$435$	0.871179	0.0417698
$436$	10.2874	0.492677
$437$	8.25778	0.395023
$438$	−1.39402	−0.0666090
$439$	−22.6638	−1.08168	−0.540841	−	0.841125i	$-0.681894\pi$
$439$	−22.6638	−1.08168	−0.540841	+	0.841125i	$0.681894\pi$
$440$	6.57880	0.313632
$441$	20.7984	0.990398
$442$	−10.0814	−0.479524
$443$	−18.2731	−0.868179	−0.434090	−	0.900870i	$-0.642930\pi$
$443$	−18.2731	−0.868179	−0.434090	+	0.900870i	$0.642930\pi$
$444$	−3.30460	−0.156829
$445$	12.4368	0.589559
$446$	12.5488	0.594202
$447$	−3.49897	−0.165496
$448$	−0.471799	−0.0222904
$449$	−5.98117	−0.282269	−0.141134	−	0.989990i	$-0.545075\pi$
$449$	−5.98117	−0.282269	−0.141134	+	0.989990i	$0.545075\pi$
$450$	−8.17277	−0.385268
$451$	27.7577	1.30706
$452$	13.8365	0.650816
$453$	0.727674	0.0341891
$454$	39.5663	1.85694
$455$	0.0790592	0.00370635
$456$	−0.335539	−0.0157130
$457$	29.1975	1.36580	0.682901	−	0.730511i	$-0.260718\pi$
$457$	29.1975	1.36580	0.682901	+	0.730511i	$0.260718\pi$
$458$	−27.3024	−1.27576
$459$	−4.74503	−0.221479
$460$	22.2404	1.03696
$461$	−18.4840	−0.860885	−0.430442	−	0.902618i	$-0.641642\pi$
$461$	−18.4840	−0.860885	−0.430442	+	0.902618i	$0.641642\pi$
$462$	0.0450375	0.00209533
$463$	−1.00000	−0.0464739
$463$	−1.00000	−0.0464739
$464$	−7.24365	−0.336278
$465$	2.87205	0.133188
$466$	17.8685	0.827740
$467$	−24.2985	−1.12440	−0.562201	−	0.827001i	$-0.690045\pi$
$467$	−24.2985	−1.12440	−0.562201	+	0.827001i	$0.690045\pi$
$468$	7.49562	0.346485
$469$	−0.595195	−0.0274836
$470$	51.0932	2.35675
$471$	−2.43206	−0.112063
$472$	15.6144	0.718713
$473$	15.5845	0.716575
$474$	4.40137	0.202162
$475$	2.33160	0.106981
$476$	−0.490991	−0.0225045
$477$	5.79735	0.265442
$478$	44.1873	2.02108
$479$	19.5729	0.894309	0.447155	−	0.894457i	$-0.352437\pi$
$479$	19.5729	0.894309	0.447155	+	0.894457i	$0.352437\pi$
$480$	2.55770	0.116743
$481$	−7.81755	−0.356449
$482$	8.18132	0.372649
$483$	0.0315209	0.00143425
$484$	−3.85913	−0.175415
$485$	−11.0656	−0.502461
$486$	9.50320	0.431074
$487$	−23.1114	−1.04728	−0.523639	−	0.851941i	$-0.675426\pi$
$487$	−23.1114	−1.04728	−0.523639	+	0.851941i	$0.675426\pi$
$488$	0.577198	0.0261285
$489$	−4.01649	−0.181632
$490$	−28.6528	−1.29440
$491$	27.6965	1.24992	0.624962	−	0.780655i	$-0.285114\pi$
$491$	27.6965	1.24992	0.624962	+	0.780655i	$0.285114\pi$
$492$	−3.81294	−0.171901
$493$	−12.7991	−0.576442
$494$	−3.83412	−0.172505
$495$	17.6080	0.791419
$496$	−23.8804	−1.07226
$497$	−0.112712	−0.00505582
$498$	−0.815386	−0.0365383
$499$	0.616476	0.0275973	0.0137986	−	0.999905i	$-0.495608\pi$
$499$	0.616476	0.0275973	0.0137986	+	0.999905i	$0.495608\pi$
$500$	30.5593	1.36665
$501$	3.44022	0.153698
$502$	−52.2310	−2.33118
$503$	15.1551	0.675731	0.337866	−	0.941194i	$-0.390295\pi$
$503$	15.1551	0.675731	0.337866	+	0.941194i	$0.390295\pi$
$504$	0.135507	0.00603595
$505$	25.9100	1.15298
$506$	29.9720	1.33242
$507$	−0.167601	−0.00744342
$508$	38.9782	1.72938
$509$	−36.2041	−1.60472	−0.802360	−	0.596840i	$-0.796423\pi$
$509$	−36.2041	−1.60472	−0.802360	+	0.596840i	$0.796423\pi$
$510$	3.25311	0.144050
$511$	−0.160610	−0.00710497
$512$	−27.2251	−1.20319
$513$	−1.80461	−0.0796753
$514$	−5.76791	−0.254412
$515$	−29.4359	−1.29710
$516$	−2.14076	−0.0942418
$517$	38.4028	1.68895
$518$	−0.682646	−0.0299937
$519$	−0.301549	−0.0132365
$520$	−2.13783	−0.0937502
$521$	−7.94924	−0.348263	−0.174131	−	0.984722i	$-0.555712\pi$
$521$	−7.94924	−0.348263	−0.174131	+	0.984722i	$0.555712\pi$
$522$	17.0624	0.746799
$523$	19.5301	0.853992	0.426996	−	0.904254i	$-0.359572\pi$
$523$	19.5301	0.853992	0.426996	+	0.904254i	$0.359572\pi$
$524$	−32.5333	−1.42122
$525$	0.00889999	0.000388427 0
$526$	−3.72187	−0.162281
$527$	−42.1953	−1.83806
$528$	1.38381	0.0602226
$529$	−2.02313	−0.0879621
$530$	−7.98669	−0.346920
$531$	41.7916	1.81360
$532$	−0.186731	−0.00809582
$533$	−9.02010	−0.390704
$534$	−2.30227	−0.0996290
$535$	−25.2790	−1.09291
$536$	16.0946	0.695182
$537$	1.23394	0.0532483
$538$	−9.91629	−0.427521
$539$	−21.5361	−0.927624
$540$	−4.86029	−0.209154
$541$	32.1231	1.38108	0.690541	−	0.723294i	$-0.257373\pi$
$541$	32.1231	1.38108	0.690541	+	0.723294i	$0.257373\pi$
$542$	2.36837	0.101730
$543$	−1.10904	−0.0475934
$544$	−37.5770	−1.61110
$545$	−7.85297	−0.336384
$546$	−0.0146353	−0.000626333 0
$547$	−42.0649	−1.79856	−0.899282	−	0.437370i	$-0.855910\pi$
$547$	−42.0649	−1.79856	−0.899282	+	0.437370i	$0.855910\pi$
$548$	−44.0134	−1.88016
$549$	1.54485	0.0659326
$550$	8.46265	0.360849
$551$	−4.86769	−0.207371
$552$	−0.852355	−0.0362786
$553$	0.507097	0.0215640
$554$	7.59874	0.322840
$555$	2.52259	0.107078
$556$	−45.5183	−1.93040
$557$	−19.7273	−0.835873	−0.417936	−	0.908476i	$-0.637246\pi$
$557$	−19.7273	−0.835873	−0.417936	+	0.908476i	$0.637246\pi$
$558$	56.2502	2.38126
$559$	−5.06430	−0.214197
$560$	0.212119	0.00896366
$561$	2.44511	0.103233
$562$	28.1753	1.18850
$563$	−23.6207	−0.995496	−0.497748	−	0.867322i	$-0.665839\pi$
$563$	−23.6207	−0.995496	−0.497748	+	0.867322i	$0.665839\pi$
$564$	−5.27519	−0.222126
$565$	−10.5622	−0.444357
$566$	−29.4735	−1.23887
$567$	0.359219	0.0150858
$568$	3.04784	0.127884
$569$	−12.4247	−0.520871	−0.260435	−	0.965491i	$-0.583866\pi$
$569$	−12.4247	−0.520871	−0.260435	+	0.965491i	$0.583866\pi$
$570$	1.23721	0.0518209
$571$	−4.18465	−0.175122	−0.0875612	−	0.996159i	$-0.527907\pi$
$571$	−4.18465	−0.175122	−0.0875612	+	0.996159i	$0.527907\pi$
$572$	−7.76149	−0.324524
$573$	2.28559	0.0954817
$574$	−0.787655	−0.0328761
$575$	5.92285	0.247000
$576$	34.1460	1.42275
$577$	13.7380	0.571922	0.285961	−	0.958241i	$-0.407687\pi$
$577$	13.7380	0.571922	0.285961	+	0.958241i	$0.407687\pi$
$578$	−11.6426	−0.484268
$579$	−0.169387	−0.00703950
$580$	−13.1100	−0.544363
$581$	−0.0939434	−0.00389743
$582$	2.04844	0.0849104
$583$	−6.00297	−0.248618
$584$	4.34304	0.179716
$585$	−5.72185	−0.236569
$586$	−67.7213	−2.79754
$587$	20.0531	0.827678	0.413839	−	0.910350i	$-0.364188\pi$
$587$	20.0531	0.827678	0.413839	+	0.910350i	$0.364188\pi$
$588$	2.95830	0.121998
$589$	−16.0475	−0.661227
$590$	−57.5740	−2.37028
$591$	0.0905385	0.00372426
$592$	−20.9748	−0.862058
$593$	−19.4705	−0.799557	−0.399779	−	0.916612i	$-0.630913\pi$
$593$	−19.4705	−0.799557	−0.399779	+	0.916612i	$0.630913\pi$
$594$	−6.54991	−0.268746
$595$	0.374802	0.0153654
$596$	52.6546	2.15682
$597$	−3.74947	−0.153456
$598$	−9.73965	−0.398284
$599$	−28.6342	−1.16996	−0.584980	−	0.811048i	$-0.698898\pi$
$599$	−28.6342	−1.16996	−0.584980	+	0.811048i	$0.698898\pi$
$600$	−0.240664	−0.00982506
$601$	−13.7836	−0.562244	−0.281122	−	0.959672i	$-0.590706\pi$
$601$	−13.7836	−0.562244	−0.281122	+	0.959672i	$0.590706\pi$
$602$	−0.442226	−0.0180238
$603$	43.0768	1.75422
$604$	−10.9505	−0.445568
$605$	2.94590	0.119768
$606$	−4.79640	−0.194841
$607$	−37.5822	−1.52542	−0.762708	−	0.646743i	$-0.776130\pi$
$607$	−37.5822	−1.52542	−0.762708	+	0.646743i	$0.776130\pi$
$608$	−14.2911	−0.579581
$609$	−0.0185806	−0.000752923 0
$610$	−2.12826	−0.0861706
$611$	−12.4793	−0.504858
$612$	35.5350	1.43642
$613$	0.394258	0.0159239	0.00796196	−	0.999968i	$-0.497466\pi$
$613$	0.394258	0.0159239	0.00796196	+	0.999968i	$0.497466\pi$
$614$	0.241171	0.00973289
$615$	2.91064	0.117368
$616$	−0.140313	−0.00565338
$617$	48.2584	1.94281	0.971404	−	0.237431i	$-0.0763055\pi$
$617$	48.2584	1.94281	0.971404	+	0.237431i	$0.0763055\pi$
$618$	5.44911	0.219196
$619$	35.7294	1.43608	0.718042	−	0.696000i	$-0.245039\pi$
$619$	35.7294	1.43608	0.718042	+	0.696000i	$0.245039\pi$
$620$	−43.2203	−1.73577
$621$	−4.58416	−0.183956
$622$	−39.2082	−1.57211
$623$	−0.265252	−0.0106271
$624$	−0.449680	−0.0180016
$625$	−16.8617	−0.674470
$626$	20.1392	0.804924
$627$	0.929913	0.0371371
$628$	36.5990	1.46046
$629$	−37.0612	−1.47773
$630$	−0.499644	−0.0199063
$631$	−6.56552	−0.261369	−0.130685	−	0.991424i	$-0.541718\pi$
$631$	−6.56552	−0.261369	−0.130685	+	0.991424i	$0.541718\pi$
$632$	−13.7124	−0.545449
$633$	−0.464181	−0.0184495
$634$	0.696326	0.0276546
$635$	−29.7543	−1.18076
$636$	0.824598	0.0326974
$637$	6.99831	0.277283
$638$	−17.6675	−0.699465
$639$	8.15744	0.322703
$640$	−16.5198	−0.653002
$641$	9.86437	0.389619	0.194810	−	0.980841i	$-0.437591\pi$
$641$	9.86437	0.389619	0.194810	+	0.980841i	$0.437591\pi$
$642$	4.67960	0.184689
$643$	−32.7342	−1.29091	−0.645455	−	0.763798i	$-0.723332\pi$
$643$	−32.7342	−1.29091	−0.645455	+	0.763798i	$0.723332\pi$
$644$	−0.474345	−0.0186918
$645$	1.63417	0.0643453
$646$	−18.1767	−0.715152
$647$	26.3856	1.03732	0.518662	−	0.854979i	$-0.326430\pi$
$647$	26.3856	1.03732	0.518662	+	0.854979i	$0.326430\pi$
$648$	−9.71362	−0.381587
$649$	−43.2739	−1.69865
$650$	−2.75000	−0.107864
$651$	−0.0612554	−0.00240079
$652$	60.4425	2.36711
$653$	−47.1434	−1.84486	−0.922432	−	0.386159i	$-0.873802\pi$
$653$	−47.1434	−1.84486	−0.922432	+	0.386159i	$0.873802\pi$
$654$	1.45373	0.0568452
$655$	24.8346	0.970366
$656$	−24.2013	−0.944901
$657$	11.6240	0.453496
$658$	−1.08972	−0.0424817
$659$	7.54754	0.294010	0.147005	−	0.989136i	$-0.453037\pi$
$659$	7.54754	0.294010	0.147005	+	0.989136i	$0.453037\pi$
$660$	2.50451	0.0974878
$661$	39.1731	1.52366	0.761829	−	0.647778i	$-0.224302\pi$
$661$	39.1731	1.52366	0.761829	+	0.647778i	$0.224302\pi$
$662$	66.3499	2.57876
$663$	−0.794558	−0.0308581
$664$	2.54031	0.0985833
$665$	0.142543	0.00552757
$666$	49.4059	1.91444
$667$	−12.3652	−0.478782
$668$	−51.7705	−2.00306
$669$	0.989020	0.0382377
$670$	−59.3445	−2.29268
$671$	−1.59965	−0.0617536
$672$	−0.0545509	−0.00210435
$673$	23.9209	0.922083	0.461041	−	0.887379i	$-0.347476\pi$
$673$	23.9209	0.922083	0.461041	+	0.887379i	$0.347476\pi$
$674$	−5.15862	−0.198702
$675$	−1.29435	−0.0498194
$676$	2.52216	0.0970060
$677$	29.2001	1.12225	0.561125	−	0.827731i	$-0.310369\pi$
$677$	29.2001	1.12225	0.561125	+	0.827731i	$0.310369\pi$
$678$	1.95526	0.0750914
$679$	0.236007	0.00905712
$680$	−10.1350	−0.388659
$681$	3.11838	0.119497
$682$	−58.2453	−2.23033
$683$	−33.8080	−1.29363	−0.646813	−	0.762649i	$-0.723898\pi$
$683$	−33.8080	−1.29363	−0.646813	+	0.762649i	$0.723898\pi$
$684$	13.5145	0.516740
$685$	33.5980	1.28371
$686$	1.22236	0.0466701
$687$	−2.15182	−0.0820969
$688$	−13.5877	−0.518027
$689$	1.95071	0.0743163
$690$	3.14283	0.119645
$691$	−6.36915	−0.242294	−0.121147	−	0.992635i	$-0.538657\pi$
$691$	−6.36915	−0.242294	−0.121147	+	0.992635i	$0.538657\pi$
$692$	4.53788	0.172504
$693$	−0.375544	−0.0142657
$694$	56.8263	2.15710
$695$	34.7467	1.31802
$696$	0.502436	0.0190448
$697$	−42.7622	−1.61973
$698$	−29.5927	−1.12010
$699$	1.40829	0.0532663
$700$	−0.133932	−0.00506216
$701$	9.74768	0.368165	0.184082	−	0.982911i	$-0.441069\pi$
$701$	9.74768	0.368165	0.184082	+	0.982911i	$0.441069\pi$
$702$	2.12845	0.0803330
$703$	−14.0949	−0.531601
$704$	−35.3572	−1.33257
$705$	4.02686	0.151660
$706$	0.964102	0.0362844
$707$	−0.552610	−0.0207830
$708$	5.94431	0.223401
$709$	−12.1235	−0.455306	−0.227653	−	0.973742i	$-0.573105\pi$
$709$	−12.1235	−0.455306	−0.227653	+	0.973742i	$0.573105\pi$
$710$	−11.2381	−0.421757
$711$	−36.7007	−1.37638
$712$	7.17267	0.268807
$713$	−40.7648	−1.52666
$714$	−0.0693826	−0.00259658
$715$	5.92480	0.221575
$716$	−18.5690	−0.693956
$717$	3.48258	0.130059
$718$	63.7792	2.38022
$719$	17.1380	0.639140	0.319570	−	0.947563i	$-0.396461\pi$
$719$	17.1380	0.639140	0.319570	+	0.947563i	$0.396461\pi$
$720$	−15.3519	−0.572133
$721$	0.627810	0.0233809
$722$	33.4913	1.24642
$723$	0.644804	0.0239805
$724$	16.6895	0.620259
$725$	−3.49133	−0.129665
$726$	−0.545339	−0.0202394
$727$	13.1194	0.486572	0.243286	−	0.969955i	$-0.421775\pi$
$727$	13.1194	0.486572	0.243286	+	0.969955i	$0.421775\pi$
$728$	0.0455959	0.00168990
$729$	−25.4949	−0.944257
$730$	−16.0138	−0.592697
$731$	−24.0087	−0.887994
$732$	0.219735	0.00812165
$733$	45.3647	1.67558	0.837792	−	0.545989i	$-0.183846\pi$
$733$	45.3647	1.67558	0.837792	+	0.545989i	$0.183846\pi$
$734$	50.7692	1.87393
$735$	−2.25824	−0.0832965
$736$	−36.3031	−1.33815
$737$	−44.6047	−1.64303
$738$	57.0059	2.09842
$739$	−10.7344	−0.394871	−0.197435	−	0.980316i	$-0.563261\pi$
$739$	−10.7344	−0.394871	−0.197435	+	0.980316i	$0.563261\pi$
$740$	−37.9615	−1.39549
$741$	−0.302183	−0.0111010
$742$	0.170341	0.00625340
$743$	−4.77880	−0.175317	−0.0876585	−	0.996151i	$-0.527938\pi$
$743$	−4.77880	−0.175317	−0.0876585	+	0.996151i	$0.527938\pi$
$744$	1.65640	0.0607266
$745$	−40.1943	−1.47261
$746$	42.8019	1.56709
$747$	6.79907	0.248765
$748$	−36.7954	−1.34537
$749$	0.539152	0.0197002
$750$	4.31838	0.157685
$751$	−9.45753	−0.345110	−0.172555	−	0.985000i	$-0.555202\pi$
$751$	−9.45753	−0.345110	−0.172555	+	0.985000i	$0.555202\pi$
$752$	−33.4824	−1.22098
$753$	−4.11654	−0.150015
$754$	5.74121	0.209082
$755$	8.35912	0.304219
$756$	0.103661	0.00377010
$757$	21.9693	0.798487	0.399243	−	0.916845i	$-0.369273\pi$
$757$	21.9693	0.798487	0.399243	+	0.916845i	$0.369273\pi$
$758$	63.9300	2.32204
$759$	2.36222	0.0857431
$760$	−3.85449	−0.139817
$761$	−53.0060	−1.92147	−0.960733	−	0.277476i	$-0.910502\pi$
$761$	−53.0060	−1.92147	−0.960733	+	0.277476i	$0.910502\pi$
$762$	5.50806	0.199536
$763$	0.167489	0.00606350
$764$	−34.3948	−1.24436
$765$	−27.1260	−0.980741
$766$	49.9682	1.80542
$767$	14.0622	0.507756
$768$	−0.793221	−0.0286229
$769$	−29.9712	−1.08079	−0.540395	−	0.841412i	$-0.681725\pi$
$769$	−29.9712	−1.08079	−0.540395	+	0.841412i	$0.681725\pi$
$770$	0.517366	0.0186446
$771$	−0.454593	−0.0163718
$772$	2.54904	0.0917419
$773$	−42.8860	−1.54250	−0.771252	−	0.636530i	$-0.780369\pi$
$773$	−42.8860	−1.54250	−0.771252	+	0.636530i	$0.780369\pi$
$774$	32.0057	1.15042
$775$	−11.5100	−0.413452
$776$	−6.38185	−0.229095
$777$	−0.0538021	−0.00193014
$778$	44.5956	1.59883
$779$	−16.2631	−0.582687
$780$	−0.813859	−0.0291408
$781$	−8.44677	−0.302249
$782$	−46.1734	−1.65116
$783$	2.70222	0.0965694
$784$	18.7768	0.670598
$785$	−27.9381	−0.997155
$786$	−4.59733	−0.163981
$787$	41.1824	1.46799	0.733996	−	0.679153i	$-0.237653\pi$
$787$	41.1824	1.46799	0.733996	+	0.679153i	$0.237653\pi$
$788$	−1.36248	−0.0485362
$789$	−0.293336	−0.0104430
$790$	50.5606	1.79886
$791$	0.225272	0.00800976
$792$	10.1551	0.360844
$793$	0.519818	0.0184593
$794$	64.6955	2.29596
$795$	−0.629464	−0.0223248
$796$	56.4242	1.99990
$797$	26.5727	0.941255	0.470627	−	0.882332i	$-0.344028\pi$
$797$	26.5727	0.941255	0.470627	+	0.882332i	$0.344028\pi$
$798$	−0.0263873	−0.000934099 0
$799$	−59.1614	−2.09298
$800$	−10.2502	−0.362401
$801$	19.1974	0.678307
$802$	−51.1142	−1.80490
$803$	−12.0363	−0.424752
$804$	6.12711	0.216087
$805$	0.362095	0.0127622
$806$	18.9273	0.666685
$807$	−0.781543	−0.0275116
$808$	14.9431	0.525695
$809$	−23.9655	−0.842582	−0.421291	−	0.906926i	$-0.638423\pi$
$809$	−23.9655	−0.842582	−0.421291	+	0.906926i	$0.638423\pi$
$810$	35.8163	1.25846
$811$	11.3020	0.396868	0.198434	−	0.980114i	$-0.436414\pi$
$811$	11.3020	0.396868	0.198434	+	0.980114i	$0.436414\pi$
$812$	0.279611	0.00981243
$813$	0.186661	0.00654649
$814$	−51.1583	−1.79310
$815$	−46.1393	−1.61619
$816$	−2.13183	−0.0746290
$817$	−9.13087	−0.319449
$818$	30.1633	1.05464
$819$	0.122036	0.00426428
$820$	−43.8010	−1.52960
$821$	−22.8285	−0.796720	−0.398360	−	0.917229i	$-0.630420\pi$
$821$	−22.8285	−0.796720	−0.398360	+	0.917229i	$0.630420\pi$
$822$	−6.21960	−0.216934
$823$	−7.37589	−0.257107	−0.128554	−	0.991703i	$-0.541033\pi$
$823$	−7.37589	−0.257107	−0.128554	+	0.991703i	$0.541033\pi$
$824$	−16.9766	−0.591407
$825$	0.666976	0.0232211
$826$	1.22794	0.0427256
$827$	27.5616	0.958412	0.479206	−	0.877702i	$-0.340925\pi$
$827$	27.5616	0.958412	0.479206	+	0.877702i	$0.340925\pi$
$828$	34.3303	1.19306
$829$	−21.5148	−0.747240	−0.373620	−	0.927582i	$-0.621884\pi$
$829$	−21.5148	−0.747240	−0.373620	+	0.927582i	$0.621884\pi$
$830$	−9.36671	−0.325123
$831$	0.598888	0.0207752
$832$	11.4896	0.398330
$833$	33.1774	1.14953
$834$	−6.43225	−0.222731
$835$	39.5194	1.36763
$836$	−13.9939	−0.483988
$837$	8.90851	0.307923
$838$	41.0781	1.41902
$839$	3.95055	0.136388	0.0681941	−	0.997672i	$-0.478276\pi$
$839$	3.95055	0.136388	0.0681941	+	0.997672i	$0.478276\pi$
$840$	−0.0147130	−0.000507648 0
$841$	−21.7111	−0.748659
$842$	−19.9251	−0.686664
$843$	2.22061	0.0764819
$844$	6.98526	0.240443
$845$	−1.92531	−0.0662326
$846$	78.8675	2.71152
$847$	−0.0628303	−0.00215887
$848$	5.23384	0.179731
$849$	−2.32293	−0.0797227
$850$	−13.0371	−0.447170
$851$	−35.8048	−1.22737
$852$	1.16029	0.0397509
$853$	33.9708	1.16314	0.581569	−	0.813497i	$-0.302439\pi$
$853$	33.9708	1.16314	0.581569	+	0.813497i	$0.302439\pi$
$854$	0.0453916	0.00155327
$855$	−10.3164	−0.352814
$856$	−14.5792	−0.498306
$857$	−34.4912	−1.17820	−0.589099	−	0.808061i	$-0.700517\pi$
$857$	−34.4912	−1.17820	−0.589099	+	0.808061i	$0.700517\pi$
$858$	−1.09679	−0.0374437
$859$	−45.2599	−1.54425	−0.772124	−	0.635471i	$-0.780806\pi$
$859$	−45.2599	−1.54425	−0.772124	+	0.635471i	$0.780806\pi$
$860$	−24.5919	−0.838577
$861$	−0.0620783	−0.00211562
$862$	24.1563	0.822767
$863$	30.5418	1.03966	0.519828	−	0.854271i	$-0.325996\pi$
$863$	30.5418	1.03966	0.519828	+	0.854271i	$0.325996\pi$
$864$	7.93347	0.269902
$865$	−3.46403	−0.117780
$866$	−72.6689	−2.46939
$867$	−0.917601	−0.0311634
$868$	0.921806	0.0312881
$869$	38.0025	1.28915
$870$	−1.85259	−0.0628088
$871$	14.4946	0.491132
$872$	−4.52905	−0.153373
$873$	−17.0808	−0.578098
$874$	−17.5605	−0.593991
$875$	0.497534	0.0168197
$876$	1.65337	0.0558621
$877$	−37.6125	−1.27008	−0.635042	−	0.772477i	$-0.719017\pi$
$877$	−37.6125	−1.27008	−0.635042	+	0.772477i	$0.719017\pi$
$878$	48.1953	1.62651
$879$	−5.33739	−0.180026
$880$	15.8965	0.535870
$881$	14.0753	0.474208	0.237104	−	0.971484i	$-0.423802\pi$
$881$	14.0753	0.474208	0.237104	+	0.971484i	$0.423802\pi$
$882$	−44.2285	−1.48925
$883$	37.1421	1.24993	0.624966	−	0.780652i	$-0.285113\pi$
$883$	37.1421	1.24993	0.624966	+	0.780652i	$0.285113\pi$
$884$	11.9570	0.402156
$885$	−4.53764	−0.152531
$886$	38.8583	1.30547
$887$	−47.6824	−1.60102	−0.800510	−	0.599320i	$-0.795438\pi$
$887$	−47.6824	−1.60102	−0.800510	+	0.599320i	$0.795438\pi$
$888$	1.45486	0.0488218
$889$	0.634602	0.0212839
$890$	−26.4472	−0.886513
$891$	26.9203	0.901865
$892$	−14.8833	−0.498331
$893$	−22.5000	−0.752933
$894$	7.44070	0.248854
$895$	14.1748	0.473811
$896$	0.352335	0.0117707
$897$	−0.767621	−0.0256301
$898$	12.7192	0.424444
$899$	24.0296	0.801431
$900$	9.69322	0.323107
$901$	9.24789	0.308092
$902$	−59.0278	−1.96541
$903$	−0.0348536	−0.00115986
$904$	−6.09157	−0.202603
$905$	−12.7400	−0.423493
$906$	−1.54742	−0.0514097
$907$	21.0593	0.699263	0.349631	−	0.936887i	$-0.386307\pi$
$907$	21.0593	0.699263	0.349631	+	0.936887i	$0.386307\pi$
$908$	−46.9272	−1.55733
$909$	39.9947	1.32654
$910$	−0.168122	−0.00557320
$911$	9.82605	0.325552	0.162776	−	0.986663i	$-0.447955\pi$
$911$	9.82605	0.325552	0.162776	+	0.986663i	$0.447955\pi$
$912$	−0.810768	−0.0268472
$913$	−7.04023	−0.232998
$914$	−62.0896	−2.05374
$915$	−0.167737	−0.00554520
$916$	32.3818	1.06992
$917$	−0.529673	−0.0174914
$918$	10.0905	0.333035
$919$	−20.2081	−0.666603	−0.333301	−	0.942820i	$-0.608163\pi$
$919$	−20.2081	−0.666603	−0.333301	+	0.942820i	$0.608163\pi$
$920$	−9.79139	−0.322813
$921$	0.0190077	0.000626325 0
$922$	39.3069	1.29450
$923$	2.74485	0.0903477
$924$	−0.0534163	−0.00175727
$925$	−10.1095	−0.332399
$926$	2.12654	0.0698823
$927$	−45.4373	−1.49236
$928$	21.3995	0.702473
$929$	33.4399	1.09713	0.548565	−	0.836108i	$-0.315174\pi$
$929$	33.4399	1.09713	0.548565	+	0.836108i	$0.315174\pi$
$930$	−6.10753	−0.200274
$931$	12.6179	0.413534
$932$	−21.1927	−0.694190
$933$	−3.09016	−0.101167
$934$	51.6717	1.69075
$935$	28.0881	0.918579
$936$	−3.29996	−0.107863
$937$	11.2836	0.368618	0.184309	−	0.982868i	$-0.440995\pi$
$937$	11.2836	0.368618	0.184309	+	0.982868i	$0.440995\pi$
$938$	1.26570	0.0413267
$939$	1.58725	0.0517980
$940$	−60.5985	−1.97651
$941$	−20.3569	−0.663616	−0.331808	−	0.943347i	$-0.607659\pi$
$941$	−20.3569	−0.663616	−0.331808	+	0.943347i	$0.607659\pi$
$942$	5.17186	0.168508
$943$	−41.3125	−1.34532
$944$	37.7294	1.22799
$945$	−0.0791302	−0.00257411
$946$	−33.1410	−1.07751
$947$	18.2561	0.593243	0.296621	−	0.954995i	$-0.404140\pi$
$947$	18.2561	0.593243	0.296621	+	0.954995i	$0.404140\pi$
$948$	−5.22020	−0.169544
$949$	3.91130	0.126966
$950$	−4.95822	−0.160866
$951$	0.0548803	0.00177961
$952$	0.216160	0.00700577
$953$	0.106277	0.00344265	0.00172132	−	0.999999i	$-0.499452\pi$
$953$	0.106277	0.00344265	0.00172132	+	0.999999i	$0.499452\pi$
$954$	−12.3283	−0.399142
$955$	26.2556	0.849610
$956$	−52.4079	−1.69499
$957$	−1.39245	−0.0450116
$958$	−41.6225	−1.34476
$959$	−0.716581	−0.0231396
$960$	−3.70750	−0.119659
$961$	48.2193	1.55546
$962$	16.6243	0.535989
$963$	−39.0207	−1.25742
$964$	−9.70338	−0.312525
$965$	−1.94583	−0.0626385
$966$	−0.0670304	−0.00215667
$967$	−37.9888	−1.22164	−0.610819	−	0.791770i	$-0.709160\pi$
$967$	−37.9888	−1.22164	−0.610819	+	0.791770i	$0.709160\pi$
$968$	1.69899	0.0546076
$969$	−1.43258	−0.0460210
$970$	23.5313	0.755545
$971$	−22.5744	−0.724448	−0.362224	−	0.932091i	$-0.617983\pi$
$971$	−22.5744	−0.724448	−0.362224	+	0.932091i	$0.617983\pi$
$972$	−11.2712	−0.361523
$973$	−0.741081	−0.0237580
$974$	49.1472	1.57478
$975$	−0.216739	−0.00694121
$976$	1.39469	0.0446430
$977$	3.94105	0.126085	0.0630427	−	0.998011i	$-0.479920\pi$
$977$	3.94105	0.126085	0.0630427	+	0.998011i	$0.479920\pi$
$978$	8.54122	0.273118
$979$	−19.8783	−0.635314
$980$	33.9833	1.08556
$981$	−12.1219	−0.387021
$982$	−58.8975	−1.87950
$983$	7.53961	0.240476	0.120238	−	0.992745i	$-0.461634\pi$
$983$	7.53961	0.240476	0.120238	+	0.992745i	$0.461634\pi$
$984$	1.67865	0.0535136
$985$	1.04006	0.0331390
$986$	27.2178	0.866790
$987$	−0.0858852	−0.00273376
$988$	4.54742	0.144673
$989$	−23.1948	−0.737550
$990$	−37.4440	−1.19005
$991$	22.6447	0.719332	0.359666	−	0.933081i	$-0.382891\pi$
$991$	22.6447	0.719332	0.359666	+	0.933081i	$0.382891\pi$
$992$	70.5486	2.23992
$993$	5.22931	0.165947
$994$	0.239686	0.00760238
$995$	−43.0718	−1.36547
$996$	0.967081	0.0306431
$997$	1.86014	0.0589113	0.0294556	−	0.999566i	$-0.490623\pi$
$997$	1.86014	0.0589113	0.0294556	+	0.999566i	$0.490623\pi$
$998$	−1.31096	−0.0414977
$999$	7.82457	0.247558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6019.2.a.c.1.20	✓	108

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6019.2.a.c.1.20	✓	108	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)

\(f(q)\)	\(=\)	\(q-2.12654 q^{2} -0.167601 q^{3} +2.52216 q^{4} -1.92531 q^{5} +0.356410 q^{6} +0.0410631 q^{7} -1.11038 q^{8} -2.97191 q^{9} +O(q^{10})\)	\(q-2.12654 q^{2} -0.167601 q^{3} +2.52216 q^{4} -1.92531 q^{5} +0.356410 q^{6} +0.0410631 q^{7} -1.11038 q^{8} -2.97191 q^{9} +4.09424 q^{10} +3.07732 q^{11} -0.422716 q^{12} -1.00000 q^{13} -0.0873222 q^{14} +0.322684 q^{15} -2.68304 q^{16} -4.74077 q^{17} +6.31987 q^{18} -1.80299 q^{19} -4.85593 q^{20} -0.00688222 q^{21} -6.54404 q^{22} -4.58005 q^{23} +0.186102 q^{24} -1.29318 q^{25} +2.12654 q^{26} +1.00090 q^{27} +0.103568 q^{28} +2.69979 q^{29} -0.686199 q^{30} +8.90052 q^{31} +7.92635 q^{32} -0.515762 q^{33} +10.0814 q^{34} -0.0790592 q^{35} -7.49562 q^{36} +7.81755 q^{37} +3.83412 q^{38} +0.167601 q^{39} +2.13783 q^{40} +9.02010 q^{41} +0.0146353 q^{42} +5.06430 q^{43} +7.76149 q^{44} +5.72185 q^{45} +9.73965 q^{46} +12.4793 q^{47} +0.449680 q^{48} -6.99831 q^{49} +2.75000 q^{50} +0.794558 q^{51} -2.52216 q^{52} -1.95071 q^{53} -2.12845 q^{54} -5.92480 q^{55} -0.0455959 q^{56} +0.302183 q^{57} -5.74121 q^{58} -14.0622 q^{59} +0.813859 q^{60} -0.519818 q^{61} -18.9273 q^{62} -0.122036 q^{63} -11.4896 q^{64} +1.92531 q^{65} +1.09679 q^{66} -14.4946 q^{67} -11.9570 q^{68} +0.767621 q^{69} +0.168122 q^{70} -2.74485 q^{71} +3.29996 q^{72} -3.91130 q^{73} -16.6243 q^{74} +0.216739 q^{75} -4.54742 q^{76} +0.126364 q^{77} -0.356410 q^{78} +12.3492 q^{79} +5.16568 q^{80} +8.74798 q^{81} -19.1816 q^{82} -2.28778 q^{83} -0.0173580 q^{84} +9.12745 q^{85} -10.7694 q^{86} -0.452488 q^{87} -3.41701 q^{88} -6.45962 q^{89} -12.1677 q^{90} -0.0410631 q^{91} -11.5516 q^{92} -1.49174 q^{93} -26.5376 q^{94} +3.47131 q^{95} -1.32846 q^{96} +5.74742 q^{97} +14.8822 q^{98} -9.14552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 11 q^{2} + q^{3} + 95 q^{4} - 40 q^{5} - 10 q^{6} - 8 q^{7} - 33 q^{8} + 79 q^{9}+O(q^{10}) \) 108 * q - 11 * q^2 + q^3 + 95 * q^4 - 40 * q^5 - 10 * q^6 - 8 * q^7 - 33 * q^8 + 79 * q^9	\( 108 q - 11 q^{2} + q^{3} + 95 q^{4} - 40 q^{5} - 10 q^{6} - 8 q^{7} - 33 q^{8} + 79 q^{9} - q^{10} - 45 q^{11} - 6 q^{12} - 108 q^{13} - 31 q^{14} - 39 q^{15} + 73 q^{16} + 21 q^{17} - 35 q^{18} - 19 q^{19} - 79 q^{20} - 72 q^{21} - 26 q^{23} - 23 q^{24} + 92 q^{25} + 11 q^{26} + 7 q^{27} - 21 q^{28} - 94 q^{29} - 24 q^{30} - 36 q^{31} - 77 q^{32} - 32 q^{33} - 58 q^{34} - 10 q^{35} + 17 q^{36} - 54 q^{37} - 12 q^{38} - q^{39} - 4 q^{40} - 68 q^{41} - 11 q^{42} - 32 q^{43} - 151 q^{44} - 121 q^{45} - 33 q^{46} - 51 q^{47} - 27 q^{48} + 72 q^{49} - 45 q^{50} - 24 q^{51} - 95 q^{52} - 81 q^{53} - 29 q^{54} + 4 q^{55} - 68 q^{56} - 45 q^{57} - 30 q^{58} - 94 q^{59} - 108 q^{60} - 39 q^{61} - 9 q^{62} - 52 q^{63} + 31 q^{64} + 40 q^{65} - 40 q^{66} - 47 q^{67} + 24 q^{68} - 60 q^{69} - 66 q^{70} - 86 q^{71} - 91 q^{72} - 51 q^{73} - 110 q^{74} - 7 q^{75} - 51 q^{76} - 96 q^{77} + 10 q^{78} - 18 q^{79} - 136 q^{80} - 24 q^{81} - 33 q^{82} - 77 q^{83} - 113 q^{84} - 95 q^{85} - 137 q^{86} + 23 q^{87} + 19 q^{88} - 112 q^{89} - 19 q^{90} + 8 q^{91} - 111 q^{92} - 124 q^{93} - 20 q^{94} - 73 q^{95} - 77 q^{96} - 41 q^{97} - 80 q^{98} - 154 q^{99}+O(q^{100}) \) 108 * q - 11 * q^2 + q^3 + 95 * q^4 - 40 * q^5 - 10 * q^6 - 8 * q^7 - 33 * q^8 + 79 * q^9 - q^10 - 45 * q^11 - 6 * q^12 - 108 * q^13 - 31 * q^14 - 39 * q^15 + 73 * q^16 + 21 * q^17 - 35 * q^18 - 19 * q^19 - 79 * q^20 - 72 * q^21 - 26 * q^23 - 23 * q^24 + 92 * q^25 + 11 * q^26 + 7 * q^27 - 21 * q^28 - 94 * q^29 - 24 * q^30 - 36 * q^31 - 77 * q^32 - 32 * q^33 - 58 * q^34 - 10 * q^35 + 17 * q^36 - 54 * q^37 - 12 * q^38 - q^39 - 4 * q^40 - 68 * q^41 - 11 * q^42 - 32 * q^43 - 151 * q^44 - 121 * q^45 - 33 * q^46 - 51 * q^47 - 27 * q^48 + 72 * q^49 - 45 * q^50 - 24 * q^51 - 95 * q^52 - 81 * q^53 - 29 * q^54 + 4 * q^55 - 68 * q^56 - 45 * q^57 - 30 * q^58 - 94 * q^59 - 108 * q^60 - 39 * q^61 - 9 * q^62 - 52 * q^63 + 31 * q^64 + 40 * q^65 - 40 * q^66 - 47 * q^67 + 24 * q^68 - 60 * q^69 - 66 * q^70 - 86 * q^71 - 91 * q^72 - 51 * q^73 - 110 * q^74 - 7 * q^75 - 51 * q^76 - 96 * q^77 + 10 * q^78 - 18 * q^79 - 136 * q^80 - 24 * q^81 - 33 * q^82 - 77 * q^83 - 113 * q^84 - 95 * q^85 - 137 * q^86 + 23 * q^87 + 19 * q^88 - 112 * q^89 - 19 * q^90 + 8 * q^91 - 111 * q^92 - 124 * q^93 - 20 * q^94 - 73 * q^95 - 77 * q^96 - 41 * q^97 - 80 * q^98 - 154 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more