LMFDB - Embedded newform 6013.2.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6013,2,Mod(1,6013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6013, 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("6013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6013 = 7 \cdot 859 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6013.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:	$48.0140467354$
Analytic rank:	$0$
Dimension:	$109$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74589 q^{2} -1.96009 q^{3} +1.04812 q^{4} -3.28540 q^{5} +3.42209 q^{6} +1.00000 q^{7} +1.66188 q^{8} +0.841938 q^{9} +O(q^{10})$	\(q-1.74589 q^{2} -1.96009 q^{3} +1.04812 q^{4} -3.28540 q^{5} +3.42209 q^{6} +1.00000 q^{7} +1.66188 q^{8} +0.841938 q^{9} +5.73592 q^{10} +1.91069 q^{11} -2.05440 q^{12} -3.31629 q^{13} -1.74589 q^{14} +6.43966 q^{15} -4.99768 q^{16} -1.86606 q^{17} -1.46993 q^{18} +8.48604 q^{19} -3.44348 q^{20} -1.96009 q^{21} -3.33585 q^{22} +2.57229 q^{23} -3.25743 q^{24} +5.79382 q^{25} +5.78986 q^{26} +4.22999 q^{27} +1.04812 q^{28} +0.829641 q^{29} -11.2429 q^{30} -2.16770 q^{31} +5.40163 q^{32} -3.74512 q^{33} +3.25793 q^{34} -3.28540 q^{35} +0.882450 q^{36} +2.77673 q^{37} -14.8157 q^{38} +6.50021 q^{39} -5.45993 q^{40} +10.8038 q^{41} +3.42209 q^{42} -2.93374 q^{43} +2.00263 q^{44} -2.76610 q^{45} -4.49092 q^{46} +11.9434 q^{47} +9.79589 q^{48} +1.00000 q^{49} -10.1154 q^{50} +3.65764 q^{51} -3.47586 q^{52} +2.58333 q^{53} -7.38507 q^{54} -6.27737 q^{55} +1.66188 q^{56} -16.6334 q^{57} -1.44846 q^{58} -8.63134 q^{59} +6.74952 q^{60} +3.85505 q^{61} +3.78455 q^{62} +0.841938 q^{63} +0.564743 q^{64} +10.8953 q^{65} +6.53855 q^{66} +2.23311 q^{67} -1.95585 q^{68} -5.04190 q^{69} +5.73592 q^{70} -8.59384 q^{71} +1.39920 q^{72} +2.15469 q^{73} -4.84785 q^{74} -11.3564 q^{75} +8.89436 q^{76} +1.91069 q^{77} -11.3486 q^{78} -6.24433 q^{79} +16.4194 q^{80} -10.8170 q^{81} -18.8621 q^{82} +9.36943 q^{83} -2.05440 q^{84} +6.13076 q^{85} +5.12198 q^{86} -1.62617 q^{87} +3.17534 q^{88} +9.54575 q^{89} +4.82929 q^{90} -3.31629 q^{91} +2.69606 q^{92} +4.24888 q^{93} -20.8518 q^{94} -27.8800 q^{95} -10.5877 q^{96} +16.4209 q^{97} -1.74589 q^{98} +1.60868 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) $ 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9 + 15 * q^10 + 48 * q^11 + 72 * q^12 + 29 * q^13 + 19 * q^14 + 29 * q^15 + 115 * q^16 + 72 * q^17 + 33 * q^18 + 58 * q^19 + 88 * q^20 + 38 * q^21 + 4 * q^22 + 65 * q^23 + 46 * q^24 + 124 * q^25 + 49 * q^26 + 131 * q^27 + 111 * q^28 + 25 * q^29 + 2 * q^30 + 41 * q^31 + 75 * q^32 + 54 * q^33 + 23 * q^34 + 43 * q^35 + 111 * q^36 + 25 * q^37 + 54 * q^38 + 27 * q^39 + 30 * q^40 + 109 * q^41 + 14 * q^42 + 38 * q^43 + 68 * q^44 + 84 * q^45 - 9 * q^46 + 121 * q^47 + 106 * q^48 + 109 * q^49 + 14 * q^50 + 36 * q^51 + 38 * q^52 + 61 * q^53 + 31 * q^54 + 50 * q^55 + 48 * q^56 + 5 * q^57 - 20 * q^58 + 181 * q^59 + 25 * q^60 + 34 * q^61 + 75 * q^62 + 119 * q^63 + 96 * q^64 + 12 * q^65 + 19 * q^66 + 87 * q^67 + 150 * q^68 + 89 * q^69 + 15 * q^70 + 83 * q^71 + 65 * q^72 + 32 * q^73 - 19 * q^74 + 112 * q^75 + 84 * q^76 + 48 * q^77 - 34 * q^78 - 9 * q^79 + 137 * q^80 + 109 * q^81 - 19 * q^82 + 136 * q^83 + 72 * q^84 - 32 * q^85 - 24 * q^86 + 28 * q^87 - 24 * q^88 + 142 * q^89 + 19 * q^90 + 29 * q^91 + 96 * q^92 + 29 * q^93 + 9 * q^94 + 52 * q^95 + 88 * q^96 + 75 * q^97 + 19 * q^98 + 84 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.74589	−1.23453	−0.617264	−	0.786756i	$-0.711759\pi$
$2$	−1.74589	−1.23453	−0.617264	+	0.786756i	$0.711759\pi$
$3$	−1.96009	−1.13166	−0.565828	−	0.824523i	$-0.691443\pi$
$3$	−1.96009	−1.13166	−0.565828	+	0.824523i	$0.691443\pi$
$4$	1.04812	0.524059
$5$	−3.28540	−1.46927	−0.734637	−	0.678461i	$-0.762647\pi$
$5$	−3.28540	−1.46927	−0.734637	+	0.678461i	$0.762647\pi$
$6$	3.42209	1.39706
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.66188	0.587563
$9$	0.841938	0.280646
$10$	5.73592	1.81386
$11$	1.91069	0.576095	0.288047	−	0.957616i	$-0.406994\pi$
$11$	1.91069	0.576095	0.288047	+	0.957616i	$0.406994\pi$
$12$	−2.05440	−0.593054
$13$	−3.31629	−0.919772	−0.459886	−	0.887978i	$-0.652110\pi$
$13$	−3.31629	−0.919772	−0.459886	+	0.887978i	$0.652110\pi$
$14$	−1.74589	−0.466608
$15$	6.43966	1.66271
$16$	−4.99768	−1.24942
$17$	−1.86606	−0.452587	−0.226293	−	0.974059i	$-0.572661\pi$
$17$	−1.86606	−0.452587	−0.226293	+	0.974059i	$0.572661\pi$
$18$	−1.46993	−0.346465
$19$	8.48604	1.94683	0.973416	−	0.229046i	$-0.0735607\pi$
$19$	8.48604	1.94683	0.973416	+	0.229046i	$0.0735607\pi$
$20$	−3.44348	−0.769985
$21$	−1.96009	−0.427726
$22$	−3.33585	−0.711205
$23$	2.57229	0.536359	0.268179	−	0.963369i	$-0.413578\pi$
$23$	2.57229	0.536359	0.268179	+	0.963369i	$0.413578\pi$
$24$	−3.25743	−0.664919
$25$	5.79382	1.15876
$26$	5.78986	1.13548
$27$	4.22999	0.814061
$28$	1.04812	0.198076
$29$	0.829641	0.154060	0.0770302	−	0.997029i	$-0.475456\pi$
$29$	0.829641	0.154060	0.0770302	+	0.997029i	$0.475456\pi$
$30$	−11.2429	−2.05266
$31$	−2.16770	−0.389330	−0.194665	−	0.980870i	$-0.562362\pi$
$31$	−2.16770	−0.389330	−0.194665	+	0.980870i	$0.562362\pi$
$32$	5.40163	0.954882
$33$	−3.74512	−0.651941
$34$	3.25793	0.558731
$35$	−3.28540	−0.555333
$36$	0.882450	0.147075
$37$	2.77673	0.456491	0.228246	−	0.973604i	$-0.426701\pi$
$37$	2.77673	0.456491	0.228246	+	0.973604i	$0.426701\pi$
$38$	−14.8157	−2.40342
$39$	6.50021	1.04087
$40$	−5.45993	−0.863291
$41$	10.8038	1.68726	0.843632	−	0.536922i	$-0.180413\pi$
$41$	10.8038	1.68726	0.843632	+	0.536922i	$0.180413\pi$
$42$	3.42209	0.528039
$43$	−2.93374	−0.447391	−0.223696	−	0.974659i	$-0.571812\pi$
$43$	−2.93374	−0.447391	−0.223696	+	0.974659i	$0.571812\pi$
$44$	2.00263	0.301907
$45$	−2.76610	−0.412346
$46$	−4.49092	−0.662149
$47$	11.9434	1.74212	0.871062	−	0.491173i	$-0.163432\pi$
$47$	11.9434	1.74212	0.871062	+	0.491173i	$0.163432\pi$
$48$	9.79589	1.41392
$49$	1.00000	0.142857
$50$	−10.1154	−1.43053
$51$	3.65764	0.512173
$52$	−3.47586	−0.482015
$53$	2.58333	0.354847	0.177424	−	0.984135i	$-0.443224\pi$
$53$	2.58333	0.354847	0.177424	+	0.984135i	$0.443224\pi$
$54$	−7.38507	−1.00498
$55$	−6.27737	−0.846441
$56$	1.66188	0.222078
$57$	−16.6334	−2.20314
$58$	−1.44846	−0.190192
$59$	−8.63134	−1.12370	−0.561852	−	0.827238i	$-0.689911\pi$
$59$	−8.63134	−1.12370	−0.561852	+	0.827238i	$0.689911\pi$
$60$	6.74952	0.871359
$61$	3.85505	0.493588	0.246794	−	0.969068i	$-0.420623\pi$
$61$	3.85505	0.493588	0.246794	+	0.969068i	$0.420623\pi$
$62$	3.78455	0.480639
$63$	0.841938	0.106074
$64$	0.564743	0.0705928
$65$	10.8953	1.35140
$66$	6.53855	0.804840
$67$	2.23311	0.272818	0.136409	−	0.990653i	$-0.456444\pi$
$67$	2.23311	0.272818	0.136409	+	0.990653i	$0.456444\pi$
$68$	−1.95585	−0.237182
$69$	−5.04190	−0.606974
$70$	5.73592	0.685574
$71$	−8.59384	−1.01990	−0.509950	−	0.860204i	$-0.670336\pi$
$71$	−8.59384	−1.01990	−0.509950	+	0.860204i	$0.670336\pi$
$72$	1.39920	0.164897
$73$	2.15469	0.252187	0.126093	−	0.992018i	$-0.459756\pi$
$73$	2.15469	0.252187	0.126093	+	0.992018i	$0.459756\pi$
$74$	−4.84785	−0.563551
$75$	−11.3564	−1.31132
$76$	8.89436	1.02025
$77$	1.91069	0.217743
$78$	−11.3486	−1.28498
$79$	−6.24433	−0.702542	−0.351271	−	0.936274i	$-0.614250\pi$
$79$	−6.24433	−0.702542	−0.351271	+	0.936274i	$0.614250\pi$
$80$	16.4194	1.83574
$81$	−10.8170	−1.20188
$82$	−18.8621	−2.08297
$83$	9.36943	1.02843	0.514214	−	0.857662i	$-0.328084\pi$
$83$	9.36943	1.02843	0.514214	+	0.857662i	$0.328084\pi$
$84$	−2.05440	−0.224153
$85$	6.13076	0.664974
$86$	5.12198	0.552317
$87$	−1.62617	−0.174343
$88$	3.17534	0.338492
$89$	9.54575	1.01185	0.505923	−	0.862578i	$-0.331152\pi$
$89$	9.54575	1.01185	0.505923	+	0.862578i	$0.331152\pi$
$90$	4.82929	0.509052
$91$	−3.31629	−0.347641
$92$	2.69606	0.281083
$93$	4.24888	0.440588
$94$	−20.8518	−2.15070
$95$	−27.8800	−2.86043
$96$	−10.5877	−1.08060
$97$	16.4209	1.66729	0.833646	−	0.552300i	$-0.186250\pi$
$97$	16.4209	1.66729	0.833646	+	0.552300i	$0.186250\pi$
$98$	−1.74589	−0.176361
$99$	1.60868	0.161679
$100$	6.07260	0.607260
$101$	1.09604	0.109060	0.0545302	−	0.998512i	$-0.482634\pi$
$101$	1.09604	0.109060	0.0545302	+	0.998512i	$0.482634\pi$
$102$	−6.38583	−0.632291
$103$	−13.3210	−1.31256	−0.656280	−	0.754518i	$-0.727871\pi$
$103$	−13.3210	−1.31256	−0.656280	+	0.754518i	$0.727871\pi$
$104$	−5.51127	−0.540424
$105$	6.43966	0.628446
$106$	−4.51019	−0.438068
$107$	14.0155	1.35493	0.677464	−	0.735556i	$-0.263079\pi$
$107$	14.0155	1.35493	0.677464	+	0.735556i	$0.263079\pi$
$108$	4.43352	0.426616
$109$	−13.3420	−1.27793	−0.638967	−	0.769234i	$-0.720638\pi$
$109$	−13.3420	−1.27793	−0.638967	+	0.769234i	$0.720638\pi$
$110$	10.9596	1.04495
$111$	−5.44263	−0.516591
$112$	−4.99768	−0.472237
$113$	10.1519	0.955007	0.477503	−	0.878630i	$-0.341542\pi$
$113$	10.1519	0.955007	0.477503	+	0.878630i	$0.341542\pi$
$114$	29.0400	2.71984
$115$	−8.45097	−0.788057
$116$	0.869561	0.0807367
$117$	−2.79211	−0.258131
$118$	15.0693	1.38724
$119$	−1.86606	−0.171062
$120$	10.7019	0.976948
$121$	−7.34926	−0.668115
$122$	−6.73047	−0.609348
$123$	−21.1763	−1.90940
$124$	−2.27200	−0.204032
$125$	−2.60802	−0.233268
$126$	−1.46993	−0.130952
$127$	−5.41375	−0.480393	−0.240196	−	0.970724i	$-0.577212\pi$
$127$	−5.41375	−0.480393	−0.240196	+	0.970724i	$0.577212\pi$
$128$	−11.7892	−1.04203
$129$	5.75039	0.506293
$130$	−19.0220	−1.66834
$131$	0.315403	0.0275569	0.0137784	−	0.999905i	$-0.495614\pi$
$131$	0.315403	0.0275569	0.0137784	+	0.999905i	$0.495614\pi$
$132$	−3.92532	−0.341655
$133$	8.48604	0.735833
$134$	−3.89876	−0.336801
$135$	−13.8972	−1.19608
$136$	−3.10117	−0.265923
$137$	−13.4607	−1.15002	−0.575011	−	0.818146i	$-0.695002\pi$
$137$	−13.4607	−1.15002	−0.575011	+	0.818146i	$0.695002\pi$
$138$	8.80258	0.749326
$139$	12.5108	1.06115	0.530577	−	0.847637i	$-0.321975\pi$
$139$	12.5108	1.06115	0.530577	+	0.847637i	$0.321975\pi$
$140$	−3.44348	−0.291027
$141$	−23.4101	−1.97149
$142$	15.0039	1.25910
$143$	−6.33640	−0.529876
$144$	−4.20774	−0.350645
$145$	−2.72570	−0.226357
$146$	−3.76183	−0.311332
$147$	−1.96009	−0.161665
$148$	2.91034	0.239228
$149$	−19.8234	−1.62400	−0.811999	−	0.583658i	$-0.801621\pi$
$149$	−19.8234	−1.62400	−0.811999	+	0.583658i	$0.801621\pi$
$150$	19.8270	1.61886
$151$	−9.24647	−0.752467	−0.376233	−	0.926525i	$-0.622781\pi$
$151$	−9.24647	−0.752467	−0.376233	+	0.926525i	$0.622781\pi$
$152$	14.1028	1.14389
$153$	−1.57111	−0.127017
$154$	−3.33585	−0.268810
$155$	7.12175	0.572033
$156$	6.81298	0.545475
$157$	7.62036	0.608171	0.304086	−	0.952645i	$-0.401649\pi$
$157$	7.62036	0.608171	0.304086	+	0.952645i	$0.401649\pi$
$158$	10.9019	0.867307
$159$	−5.06354	−0.401565
$160$	−17.7465	−1.40298
$161$	2.57229	0.202724
$162$	18.8852	1.48376
$163$	4.42505	0.346597	0.173299	−	0.984869i	$-0.444557\pi$
$163$	4.42505	0.346597	0.173299	+	0.984869i	$0.444557\pi$
$164$	11.3236	0.884225
$165$	12.3042	0.957880
$166$	−16.3580	−1.26962
$167$	−8.29704	−0.642044	−0.321022	−	0.947072i	$-0.604026\pi$
$167$	−8.29704	−0.642044	−0.321022	+	0.947072i	$0.604026\pi$
$168$	−3.25743	−0.251316
$169$	−2.00224	−0.154019
$170$	−10.7036	−0.820928
$171$	7.14472	0.546371
$172$	−3.07490	−0.234459
$173$	−9.53468	−0.724908	−0.362454	−	0.932002i	$-0.618061\pi$
$173$	−9.53468	−0.724908	−0.362454	+	0.932002i	$0.618061\pi$
$174$	2.83910	0.215232
$175$	5.79382	0.437972
$176$	−9.54903	−0.719785
$177$	16.9182	1.27165
$178$	−16.6658	−1.24915
$179$	7.01788	0.524541	0.262271	−	0.964994i	$-0.415529\pi$
$179$	7.01788	0.524541	0.262271	+	0.964994i	$0.415529\pi$
$180$	−2.89920	−0.216093
$181$	−22.2976	−1.65736	−0.828682	−	0.559720i	$-0.810909\pi$
$181$	−22.2976	−1.65736	−0.828682	+	0.559720i	$0.810909\pi$
$182$	5.78986	0.429173
$183$	−7.55622	−0.558572
$184$	4.27483	0.315144
$185$	−9.12265	−0.670711
$186$	−7.41805	−0.543918
$187$	−3.56547	−0.260733
$188$	12.5181	0.912975
$189$	4.22999	0.307686
$190$	48.6753	3.53128
$191$	−6.16315	−0.445950	−0.222975	−	0.974824i	$-0.571577\pi$
$191$	−6.16315	−0.445950	−0.222975	+	0.974824i	$0.571577\pi$
$192$	−1.10694	−0.0798868
$193$	−10.2000	−0.734214	−0.367107	−	0.930179i	$-0.619652\pi$
$193$	−10.2000	−0.734214	−0.367107	+	0.930179i	$0.619652\pi$
$194$	−28.6690	−2.05832
$195$	−21.3558	−1.52932
$196$	1.04812	0.0748655
$197$	−1.74104	−0.124044	−0.0620219	−	0.998075i	$-0.519755\pi$
$197$	−1.74104	−0.124044	−0.0620219	+	0.998075i	$0.519755\pi$
$198$	−2.80858	−0.199597
$199$	−10.6091	−0.752062	−0.376031	−	0.926607i	$-0.622711\pi$
$199$	−10.6091	−0.752062	−0.376031	+	0.926607i	$0.622711\pi$
$200$	9.62863	0.680847
$201$	−4.37709	−0.308736
$202$	−1.91357	−0.134638
$203$	0.829641	0.0582294
$204$	3.83364	0.268408
$205$	−35.4946	−2.47905
$206$	23.2570	1.62039
$207$	2.16571	0.150527
$208$	16.5738	1.14918
$209$	16.2142	1.12156
$210$	−11.2429	−0.775834
$211$	−6.15523	−0.423743	−0.211872	−	0.977297i	$-0.567956\pi$
$211$	−6.15523	−0.423743	−0.211872	+	0.977297i	$0.567956\pi$
$212$	2.70763	0.185961
$213$	16.8447	1.15418
$214$	−24.4694	−1.67270
$215$	9.63850	0.657340
$216$	7.02973	0.478312
$217$	−2.16770	−0.147153
$218$	23.2936	1.57764
$219$	−4.22337	−0.285389
$220$	−6.57942	−0.443585
$221$	6.18840	0.416277
$222$	9.50221	0.637746
$223$	−10.8906	−0.729287	−0.364644	−	0.931147i	$-0.618809\pi$
$223$	−10.8906	−0.729287	−0.364644	+	0.931147i	$0.618809\pi$
$224$	5.40163	0.360912
$225$	4.87804	0.325203
$226$	−17.7240	−1.17898
$227$	6.65267	0.441553	0.220776	−	0.975324i	$-0.429141\pi$
$227$	6.65267	0.441553	0.220776	+	0.975324i	$0.429141\pi$
$228$	−17.4337	−1.15458
$229$	5.25229	0.347081	0.173541	−	0.984827i	$-0.444479\pi$
$229$	5.25229	0.347081	0.173541	+	0.984827i	$0.444479\pi$
$230$	14.7544	0.972879
$231$	−3.74512	−0.246411
$232$	1.37876	0.0905202
$233$	27.3980	1.79491	0.897453	−	0.441111i	$-0.145416\pi$
$233$	27.3980	1.79491	0.897453	+	0.441111i	$0.145416\pi$
$234$	4.87470	0.318669
$235$	−39.2388	−2.55966
$236$	−9.04665	−0.588887
$237$	12.2394	0.795036
$238$	3.25793	0.211180
$239$	−15.3051	−0.990005	−0.495003	−	0.868891i	$-0.664833\pi$
$239$	−15.3051	−0.990005	−0.495003	+	0.868891i	$0.664833\pi$
$240$	−32.1834	−2.07743
$241$	−5.67145	−0.365330	−0.182665	−	0.983175i	$-0.558472\pi$
$241$	−5.67145	−0.365330	−0.182665	+	0.983175i	$0.558472\pi$
$242$	12.8310	0.824806
$243$	8.51220	0.546058
$244$	4.04054	0.258669
$245$	−3.28540	−0.209896
$246$	36.9714	2.35721
$247$	−28.1421	−1.79064
$248$	−3.60245	−0.228756
$249$	−18.3649	−1.16383
$250$	4.55331	0.287976
$251$	12.8211	0.809259	0.404629	−	0.914481i	$-0.367401\pi$
$251$	12.8211	0.809259	0.404629	+	0.914481i	$0.367401\pi$
$252$	0.882450	0.0555891
$253$	4.91484	0.308993
$254$	9.45179	0.593058
$255$	−12.0168	−0.752522
$256$	19.4532	1.21582
$257$	21.7714	1.35806	0.679032	−	0.734109i	$-0.262400\pi$
$257$	21.7714	1.35806	0.679032	+	0.734109i	$0.262400\pi$
$258$	−10.0395	−0.625033
$259$	2.77673	0.172538
$260$	11.4196	0.708211
$261$	0.698506	0.0432365
$262$	−0.550657	−0.0340197
$263$	−15.1009	−0.931161	−0.465581	−	0.885006i	$-0.654154\pi$
$263$	−15.1009	−0.931161	−0.465581	+	0.885006i	$0.654154\pi$
$264$	−6.22393	−0.383057
$265$	−8.48725	−0.521367
$266$	−14.8157	−0.908406
$267$	−18.7105	−1.14506
$268$	2.34056	0.142973
$269$	26.2716	1.60181	0.800903	−	0.598794i	$-0.204353\pi$
$269$	26.2716	1.60181	0.800903	+	0.598794i	$0.204353\pi$
$270$	24.2629	1.47659
$271$	−2.05618	−0.124904	−0.0624520	−	0.998048i	$-0.519892\pi$
$271$	−2.05618	−0.124904	−0.0624520	+	0.998048i	$0.519892\pi$
$272$	9.32600	0.565472
$273$	6.50021	0.393410
$274$	23.5008	1.41973
$275$	11.0702	0.667558
$276$	−5.28450	−0.318090
$277$	−5.60176	−0.336577	−0.168289	−	0.985738i	$-0.553824\pi$
$277$	−5.60176	−0.336577	−0.168289	+	0.985738i	$0.553824\pi$
$278$	−21.8425	−1.31002
$279$	−1.82507	−0.109264
$280$	−5.45993	−0.326293
$281$	29.4980	1.75970	0.879851	−	0.475250i	$-0.157642\pi$
$281$	29.4980	1.75970	0.879851	+	0.475250i	$0.157642\pi$
$282$	40.8714	2.43385
$283$	−11.6430	−0.692107	−0.346053	−	0.938215i	$-0.612478\pi$
$283$	−11.6430	−0.692107	−0.346053	+	0.938215i	$0.612478\pi$
$284$	−9.00735	−0.534488
$285$	54.6472	3.23702
$286$	11.0626	0.654147
$287$	10.8038	0.637726
$288$	4.54784	0.267984
$289$	−13.5178	−0.795165
$290$	4.75876	0.279444
$291$	−32.1864	−1.88680
$292$	2.25836	0.132161
$293$	−6.47962	−0.378543	−0.189272	−	0.981925i	$-0.560613\pi$
$293$	−6.47962	−0.378543	−0.189272	+	0.981925i	$0.560613\pi$
$294$	3.42209	0.199580
$295$	28.3574	1.65103
$296$	4.61459	0.268217
$297$	8.08220	0.468977
$298$	34.6094	2.00487
$299$	−8.53043	−0.493328
$300$	−11.9028	−0.687210
$301$	−2.93374	−0.169098
$302$	16.1433	0.928941
$303$	−2.14834	−0.123419
$304$	−42.4106	−2.43241
$305$	−12.6654	−0.725216
$306$	2.74298	0.156806
$307$	25.6854	1.46595	0.732973	−	0.680258i	$-0.238132\pi$
$307$	25.6854	1.46595	0.732973	+	0.680258i	$0.238132\pi$
$308$	2.00263	0.114110
$309$	26.1104	1.48537
$310$	−12.4338	−0.706190
$311$	−28.0808	−1.59232	−0.796158	−	0.605089i	$-0.793138\pi$
$311$	−28.0808	−1.59232	−0.796158	+	0.605089i	$0.793138\pi$
$312$	10.8026	0.611574
$313$	22.6216	1.27865	0.639325	−	0.768936i	$-0.279214\pi$
$313$	22.6216	1.27865	0.639325	+	0.768936i	$0.279214\pi$
$314$	−13.3043	−0.750804
$315$	−2.76610	−0.155852
$316$	−6.54479	−0.368173
$317$	16.2021	0.909999	0.454999	−	0.890492i	$-0.349639\pi$
$317$	16.2021	0.909999	0.454999	+	0.890492i	$0.349639\pi$
$318$	8.84036	0.495743
$319$	1.58519	0.0887534
$320$	−1.85540	−0.103720
$321$	−27.4716	−1.53331
$322$	−4.49092	−0.250269
$323$	−15.8355	−0.881110
$324$	−11.3374	−0.629857
$325$	−19.2140	−1.06580
$326$	−7.72564	−0.427884
$327$	26.1515	1.44618
$328$	17.9545	0.991374
$329$	11.9434	0.658461
$330$	−21.4817	−1.18253
$331$	4.70979	0.258873	0.129437	−	0.991588i	$-0.458683\pi$
$331$	4.70979	0.258873	0.129437	+	0.991588i	$0.458683\pi$
$332$	9.82026	0.538957
$333$	2.33783	0.128113
$334$	14.4857	0.792621
$335$	−7.33666	−0.400844
$336$	9.79589	0.534410
$337$	−11.9069	−0.648609	−0.324304	−	0.945953i	$-0.605130\pi$
$337$	−11.9069	−0.648609	−0.324304	+	0.945953i	$0.605130\pi$
$338$	3.49569	0.190141
$339$	−19.8985	−1.08074
$340$	6.42575	0.348485
$341$	−4.14180	−0.224291
$342$	−12.4739	−0.674510
$343$	1.00000	0.0539949
$344$	−4.87552	−0.262871
$345$	16.5646	0.891810
$346$	16.6465	0.894919
$347$	−27.0007	−1.44947	−0.724737	−	0.689026i	$-0.758039\pi$
$347$	−27.0007	−1.44947	−0.724737	+	0.689026i	$0.758039\pi$
$348$	−1.70441	−0.0913662
$349$	26.8913	1.43946	0.719730	−	0.694254i	$-0.244265\pi$
$349$	26.8913	1.43946	0.719730	+	0.694254i	$0.244265\pi$
$350$	−10.1154	−0.540688
$351$	−14.0278	−0.748751
$352$	10.3208	0.550103
$353$	−6.88105	−0.366241	−0.183121	−	0.983090i	$-0.558620\pi$
$353$	−6.88105	−0.366241	−0.183121	+	0.983090i	$0.558620\pi$
$354$	−29.5372	−1.56988
$355$	28.2342	1.49851
$356$	10.0051	0.530267
$357$	3.65764	0.193583
$358$	−12.2524	−0.647561
$359$	16.8498	0.889298	0.444649	−	0.895705i	$-0.353328\pi$
$359$	16.8498	0.889298	0.444649	+	0.895705i	$0.353328\pi$
$360$	−4.59692	−0.242279
$361$	53.0129	2.79015
$362$	38.9290	2.04606
$363$	14.4052	0.756076
$364$	−3.47586	−0.182184
$365$	−7.07899	−0.370531
$366$	13.1923	0.689573
$367$	8.12816	0.424286	0.212143	−	0.977239i	$-0.431956\pi$
$367$	8.12816	0.424286	0.212143	+	0.977239i	$0.431956\pi$
$368$	−12.8555	−0.670138
$369$	9.09610	0.473524
$370$	15.9271	0.828011
$371$	2.58333	0.134120
$372$	4.45332	0.230894
$373$	9.87937	0.511534	0.255767	−	0.966738i	$-0.417672\pi$
$373$	9.87937	0.511534	0.255767	+	0.966738i	$0.417672\pi$
$374$	6.22490	0.321882
$375$	5.11195	0.263980
$376$	19.8485	1.02361
$377$	−2.75133	−0.141701
$378$	−7.38507	−0.379847
$379$	9.70783	0.498657	0.249329	−	0.968419i	$-0.419790\pi$
$379$	9.70783	0.498657	0.249329	+	0.968419i	$0.419790\pi$
$380$	−29.2215	−1.49903
$381$	10.6114	0.543639
$382$	10.7602	0.550537
$383$	9.86406	0.504030	0.252015	−	0.967723i	$-0.418907\pi$
$383$	9.86406	0.504030	0.252015	+	0.967723i	$0.418907\pi$
$384$	23.1079	1.17922
$385$	−6.27737	−0.319925
$386$	17.8081	0.906407
$387$	−2.47003	−0.125559
$388$	17.2110	0.873758
$389$	27.3709	1.38776	0.693879	−	0.720091i	$-0.255900\pi$
$389$	27.3709	1.38776	0.693879	+	0.720091i	$0.255900\pi$
$390$	37.2847	1.88798
$391$	−4.80005	−0.242749
$392$	1.66188	0.0839376
$393$	−0.618216	−0.0311849
$394$	3.03965	0.153136
$395$	20.5151	1.03223
$396$	1.68609	0.0847291
$397$	−5.39048	−0.270540	−0.135270	−	0.990809i	$-0.543190\pi$
$397$	−5.39048	−0.270540	−0.135270	+	0.990809i	$0.543190\pi$
$398$	18.5223	0.928441
$399$	−16.6334	−0.832710
$400$	−28.9557	−1.44778
$401$	28.7668	1.43655	0.718273	−	0.695761i	$-0.244933\pi$
$401$	28.7668	1.43655	0.718273	+	0.695761i	$0.244933\pi$
$402$	7.64190	0.381143
$403$	7.18871	0.358095
$404$	1.14878	0.0571540
$405$	35.5380	1.76590
$406$	−1.44846	−0.0718858
$407$	5.30547	0.262982
$408$	6.07856	0.300934
$409$	−13.2751	−0.656411	−0.328206	−	0.944606i	$-0.606444\pi$
$409$	−13.2751	−0.656411	−0.328206	+	0.944606i	$0.606444\pi$
$410$	61.9696	3.06046
$411$	26.3840	1.30143
$412$	−13.9620	−0.687858
$413$	−8.63134	−0.424720
$414$	−3.78107	−0.185830
$415$	−30.7823	−1.51104
$416$	−17.9133	−0.878274
$417$	−24.5223	−1.20086
$418$	−28.3081	−1.38460
$419$	29.6675	1.44935	0.724676	−	0.689089i	$-0.241989\pi$
$419$	29.6675	1.44935	0.724676	+	0.689089i	$0.241989\pi$
$420$	6.74952	0.329343
$421$	28.2124	1.37499	0.687494	−	0.726190i	$-0.258711\pi$
$421$	28.2124	1.37499	0.687494	+	0.726190i	$0.258711\pi$
$422$	10.7463	0.523123
$423$	10.0556	0.488920
$424$	4.29317	0.208495
$425$	−10.8116	−0.524442
$426$	−29.4089	−1.42486
$427$	3.85505	0.186559
$428$	14.6899	0.710061
$429$	12.4199	0.599638
$430$	−16.8277	−0.811505
$431$	21.7657	1.04842	0.524208	−	0.851590i	$-0.324362\pi$
$431$	21.7657	1.04842	0.524208	+	0.851590i	$0.324362\pi$
$432$	−21.1401	−1.01711
$433$	17.0310	0.818459	0.409230	−	0.912431i	$-0.365797\pi$
$433$	17.0310	0.818459	0.409230	+	0.912431i	$0.365797\pi$
$434$	3.78455	0.181664
$435$	5.34260	0.256158
$436$	−13.9840	−0.669712
$437$	21.8285	1.04420
$438$	7.37352	0.352320
$439$	8.11277	0.387202	0.193601	−	0.981080i	$-0.437983\pi$
$439$	8.11277	0.387202	0.193601	+	0.981080i	$0.437983\pi$
$440$	−10.4322	−0.497337
$441$	0.841938	0.0400923
$442$	−10.8042	−0.513905
$443$	6.76886	0.321598	0.160799	−	0.986987i	$-0.448593\pi$
$443$	6.76886	0.321598	0.160799	+	0.986987i	$0.448593\pi$
$444$	−5.70451	−0.270724
$445$	−31.3615	−1.48668
$446$	19.0137	0.900325
$447$	38.8556	1.83781
$448$	0.564743	0.0266816
$449$	−34.2148	−1.61470	−0.807348	−	0.590075i	$-0.799098\pi$
$449$	−34.2148	−1.61470	−0.807348	+	0.590075i	$0.799098\pi$
$450$	−8.51650	−0.401472
$451$	20.6426	0.972024
$452$	10.6403	0.500479
$453$	18.1239	0.851534
$454$	−11.6148	−0.545109
$455$	10.8953	0.510780
$456$	−27.6426	−1.29449
$457$	7.17176	0.335481	0.167740	−	0.985831i	$-0.446353\pi$
$457$	7.17176	0.335481	0.167740	+	0.985831i	$0.446353\pi$
$458$	−9.16990	−0.428481
$459$	−7.89342	−0.368433
$460$	−8.85761	−0.412988
$461$	4.96680	0.231327	0.115664	−	0.993288i	$-0.463101\pi$
$461$	4.96680	0.231327	0.115664	+	0.993288i	$0.463101\pi$
$462$	6.53855	0.304201
$463$	25.9335	1.20523	0.602615	−	0.798032i	$-0.294125\pi$
$463$	25.9335	1.20523	0.602615	+	0.798032i	$0.294125\pi$
$464$	−4.14628	−0.192486
$465$	−13.9592	−0.647344
$466$	−47.8338	−2.21586
$467$	−21.8989	−1.01336	−0.506679	−	0.862135i	$-0.669127\pi$
$467$	−21.8989	−1.01336	−0.506679	+	0.862135i	$0.669127\pi$
$468$	−2.92646	−0.135276
$469$	2.23311	0.103116
$470$	68.5065	3.15997
$471$	−14.9366	−0.688241
$472$	−14.3442	−0.660247
$473$	−5.60547	−0.257740
$474$	−21.3686	−0.981494
$475$	49.1666	2.25592
$476$	−1.95585	−0.0896464
$477$	2.17500	0.0995864
$478$	26.7210	1.22219
$479$	−10.2278	−0.467319	−0.233659	−	0.972319i	$-0.575070\pi$
$479$	−10.2278	−0.467319	−0.233659	+	0.972319i	$0.575070\pi$
$480$	34.7846	1.58769
$481$	−9.20843	−0.419868
$482$	9.90170	0.451010
$483$	−5.04190	−0.229414
$484$	−7.70289	−0.350131
$485$	−53.9492	−2.44971
$486$	−14.8613	−0.674124
$487$	−40.0561	−1.81511	−0.907557	−	0.419930i	$-0.862055\pi$
$487$	−40.0561	−1.81511	−0.907557	+	0.419930i	$0.862055\pi$
$488$	6.40662	0.290014
$489$	−8.67349	−0.392229
$490$	5.73592	0.259123
$491$	−18.8272	−0.849658	−0.424829	−	0.905274i	$-0.639666\pi$
$491$	−18.8272	−0.849658	−0.424829	+	0.905274i	$0.639666\pi$
$492$	−22.1952	−1.00064
$493$	−1.54816	−0.0697257
$494$	49.1330	2.21060
$495$	−5.28516	−0.237550
$496$	10.8335	0.486437
$497$	−8.59384	−0.385486
$498$	32.0630	1.43678
$499$	−40.7116	−1.82250	−0.911251	−	0.411851i	$-0.864882\pi$
$499$	−40.7116	−1.82250	−0.911251	+	0.411851i	$0.864882\pi$
$500$	−2.73351	−0.122246
$501$	16.2629	0.726573
$502$	−22.3841	−0.999052
$503$	4.82188	0.214997	0.107499	−	0.994205i	$-0.465716\pi$
$503$	4.82188	0.214997	0.107499	+	0.994205i	$0.465716\pi$
$504$	1.39920	0.0623253
$505$	−3.60094	−0.160240
$506$	−8.58075	−0.381461
$507$	3.92457	0.174296
$508$	−5.67424	−0.251754
$509$	−21.9411	−0.972524	−0.486262	−	0.873813i	$-0.661640\pi$
$509$	−21.9411	−0.972524	−0.486262	+	0.873813i	$0.661640\pi$
$510$	20.9800	0.929009
$511$	2.15469	0.0953177
$512$	−10.3845	−0.458936
$513$	35.8958	1.58484
$514$	−38.0104	−1.67657
$515$	43.7648	1.92851
$516$	6.02708	0.265327
$517$	22.8201	1.00363
$518$	−4.84785	−0.213002
$519$	18.6888	0.820347
$520$	18.1067	0.794031
$521$	−17.4090	−0.762702	−0.381351	−	0.924430i	$-0.624541\pi$
$521$	−17.4090	−0.762702	−0.381351	+	0.924430i	$0.624541\pi$
$522$	−1.21951	−0.0533766
$523$	−5.44139	−0.237935	−0.118968	−	0.992898i	$-0.537958\pi$
$523$	−5.44139	−0.237935	−0.118968	+	0.992898i	$0.537958\pi$
$524$	0.330579	0.0144414
$525$	−11.3564	−0.495634
$526$	26.3644	1.14954
$527$	4.04506	0.176206
$528$	18.7169	0.814549
$529$	−16.3833	−0.712320
$530$	14.8178	0.643642
$531$	−7.26705	−0.315363
$532$	8.89436	0.385620
$533$	−35.8284	−1.55190
$534$	32.6664	1.41361
$535$	−46.0464	−1.99076
$536$	3.71116	0.160298
$537$	−13.7557	−0.593600
$538$	−45.8672	−1.97747
$539$	1.91069	0.0822993
$540$	−14.5659	−0.626815
$541$	−5.90035	−0.253676	−0.126838	−	0.991923i	$-0.540483\pi$
$541$	−5.90035	−0.253676	−0.126838	+	0.991923i	$0.540483\pi$
$542$	3.58986	0.154198
$543$	43.7051	1.87557
$544$	−10.0798	−0.432167
$545$	43.8338	1.87763
$546$	−11.3486	−0.485676
$547$	−15.3011	−0.654227	−0.327113	−	0.944985i	$-0.606076\pi$
$547$	−15.3011	−0.654227	−0.327113	+	0.944985i	$0.606076\pi$
$548$	−14.1083	−0.602679
$549$	3.24571	0.138524
$550$	−19.3273	−0.824119
$551$	7.04037	0.299930
$552$	−8.37903	−0.356635
$553$	−6.24433	−0.265536
$554$	9.78003	0.415514
$555$	17.8812	0.759014
$556$	13.1128	0.556107
$557$	1.46309	0.0619931	0.0309966	−	0.999519i	$-0.490132\pi$
$557$	1.46309	0.0619931	0.0309966	+	0.999519i	$0.490132\pi$
$558$	3.18636	0.134889
$559$	9.72913	0.411498
$560$	16.4194	0.693845
$561$	6.98863	0.295060
$562$	−51.5001	−2.17240
$563$	31.4951	1.32736	0.663679	−	0.748017i	$-0.268994\pi$
$563$	31.4951	1.32736	0.663679	+	0.748017i	$0.268994\pi$
$564$	−24.5365	−1.03317
$565$	−33.3529	−1.40317
$566$	20.3274	0.854425
$567$	−10.8170	−0.454269
$568$	−14.2819	−0.599256
$569$	−6.50207	−0.272581	−0.136290	−	0.990669i	$-0.543518\pi$
$569$	−6.50207	−0.272581	−0.136290	+	0.990669i	$0.543518\pi$
$570$	−95.4078	−3.99619
$571$	16.3126	0.682663	0.341331	−	0.939943i	$-0.389122\pi$
$571$	16.3126	0.682663	0.341331	+	0.939943i	$0.389122\pi$
$572$	−6.64129	−0.277686
$573$	12.0803	0.504662
$574$	−18.8621	−0.787290
$575$	14.9034	0.621513
$576$	0.475479	0.0198116
$577$	10.6963	0.445295	0.222647	−	0.974899i	$-0.428530\pi$
$577$	10.6963	0.445295	0.222647	+	0.974899i	$0.428530\pi$
$578$	23.6005	0.981653
$579$	19.9929	0.830878
$580$	−2.85685	−0.118624
$581$	9.36943	0.388710
$582$	56.1938	2.32931
$583$	4.93594	0.204426
$584$	3.58083	0.148176
$585$	9.17318	0.379264
$586$	11.3127	0.467322
$587$	−24.0346	−0.992012	−0.496006	−	0.868319i	$-0.665201\pi$
$587$	−24.0346	−0.992012	−0.496006	+	0.868319i	$0.665201\pi$
$588$	−2.05440	−0.0847220
$589$	−18.3952	−0.757960
$590$	−49.5087	−2.03824
$591$	3.41259	0.140375
$592$	−13.8772	−0.570350
$593$	−44.4349	−1.82472	−0.912360	−	0.409388i	$-0.865742\pi$
$593$	−44.4349	−1.82472	−0.912360	+	0.409388i	$0.865742\pi$
$594$	−14.1106	−0.578965
$595$	6.13076	0.251336
$596$	−20.7773	−0.851070
$597$	20.7948	0.851076
$598$	14.8932	0.609027
$599$	47.5175	1.94151	0.970756	−	0.240068i	$-0.0771697\pi$
$599$	47.5175	1.94151	0.970756	+	0.240068i	$0.0771697\pi$
$600$	−18.8729	−0.770485
$601$	−7.71627	−0.314753	−0.157377	−	0.987539i	$-0.550304\pi$
$601$	−7.71627	−0.314753	−0.157377	+	0.987539i	$0.550304\pi$
$602$	5.12198	0.208756
$603$	1.88014	0.0765653
$604$	−9.69138	−0.394337
$605$	24.1452	0.981643
$606$	3.75076	0.152364
$607$	−16.3509	−0.663663	−0.331832	−	0.943339i	$-0.607667\pi$
$607$	−16.3509	−0.663663	−0.331832	+	0.943339i	$0.607667\pi$
$608$	45.8384	1.85899
$609$	−1.62617	−0.0658956
$610$	22.1123	0.895299
$611$	−39.6077	−1.60236
$612$	−1.64671	−0.0665642
$613$	−9.78766	−0.395320	−0.197660	−	0.980271i	$-0.563334\pi$
$613$	−9.78766	−0.395320	−0.197660	+	0.980271i	$0.563334\pi$
$614$	−44.8438	−1.80975
$615$	69.5725	2.80544
$616$	3.17534	0.127938
$617$	41.2809	1.66191	0.830953	−	0.556342i	$-0.187796\pi$
$617$	41.2809	1.66191	0.830953	+	0.556342i	$0.187796\pi$
$618$	−45.5857	−1.83373
$619$	−28.1587	−1.13180	−0.565898	−	0.824476i	$-0.691470\pi$
$619$	−28.1587	−1.13180	−0.565898	+	0.824476i	$0.691470\pi$
$620$	7.46443	0.299779
$621$	10.8807	0.436629
$622$	49.0259	1.96576
$623$	9.54575	0.382442
$624$	−32.4860	−1.30048
$625$	−20.4007	−0.816029
$626$	−39.4948	−1.57853
$627$	−31.7812	−1.26922
$628$	7.98703	0.318717
$629$	−5.18155	−0.206602
$630$	4.82929	0.192404
$631$	−31.8845	−1.26930	−0.634650	−	0.772799i	$-0.718856\pi$
$631$	−31.8845	−1.26930	−0.634650	+	0.772799i	$0.718856\pi$
$632$	−10.3773	−0.412787
$633$	12.0648	0.479532
$634$	−28.2870	−1.12342
$635$	17.7863	0.705828
$636$	−5.30718	−0.210444
$637$	−3.31629	−0.131396
$638$	−2.76756	−0.109569
$639$	−7.23548	−0.286231
$640$	38.7323	1.53103
$641$	−27.6404	−1.09173	−0.545864	−	0.837874i	$-0.683799\pi$
$641$	−27.6404	−1.09173	−0.545864	+	0.837874i	$0.683799\pi$
$642$	47.9622	1.89292
$643$	39.8607	1.57195	0.785976	−	0.618257i	$-0.212161\pi$
$643$	39.8607	1.57195	0.785976	+	0.618257i	$0.212161\pi$
$644$	2.69606	0.106239
$645$	−18.8923	−0.743883
$646$	27.6470	1.08775
$647$	30.4658	1.19773	0.598867	−	0.800849i	$-0.295618\pi$
$647$	30.4658	1.19773	0.598867	+	0.800849i	$0.295618\pi$
$648$	−17.9765	−0.706182
$649$	−16.4918	−0.647360
$650$	33.5454	1.31576
$651$	4.24888	0.166527
$652$	4.63797	0.181637
$653$	−19.0497	−0.745473	−0.372736	−	0.927937i	$-0.621580\pi$
$653$	−19.0497	−0.745473	−0.372736	+	0.927937i	$0.621580\pi$
$654$	−45.6575	−1.78535
$655$	−1.03622	−0.0404886
$656$	−53.9938	−2.10810
$657$	1.81411	0.0707753
$658$	−20.8518	−0.812888
$659$	11.9209	0.464371	0.232186	−	0.972671i	$-0.425412\pi$
$659$	11.9209	0.464371	0.232186	+	0.972671i	$0.425412\pi$
$660$	12.8962	0.501985
$661$	22.9270	0.891755	0.445878	−	0.895094i	$-0.352892\pi$
$661$	22.9270	0.891755	0.445878	+	0.895094i	$0.352892\pi$
$662$	−8.22276	−0.319586
$663$	−12.1298	−0.471082
$664$	15.5709	0.604267
$665$	−27.8800	−1.08114
$666$	−4.08159	−0.158158
$667$	2.13407	0.0826316
$668$	−8.69627	−0.336469
$669$	21.3465	0.825302
$670$	12.8090	0.494853
$671$	7.36580	0.284354
$672$	−10.5877	−0.408428
$673$	−21.8310	−0.841524	−0.420762	−	0.907171i	$-0.638237\pi$
$673$	−21.8310	−0.841524	−0.420762	+	0.907171i	$0.638237\pi$
$674$	20.7880	0.800726
$675$	24.5078	0.943305
$676$	−2.09859	−0.0807149
$677$	30.4099	1.16875	0.584373	−	0.811485i	$-0.301341\pi$
$677$	30.4099	1.16875	0.584373	+	0.811485i	$0.301341\pi$
$678$	34.7405	1.33420
$679$	16.4209	0.630177
$680$	10.1886	0.390714
$681$	−13.0398	−0.499686
$682$	7.23111	0.276894
$683$	51.8359	1.98345	0.991723	−	0.128400i	$-0.0409840\pi$
$683$	51.8359	1.98345	0.991723	+	0.128400i	$0.0409840\pi$
$684$	7.48851	0.286330
$685$	44.2236	1.68970
$686$	−1.74589	−0.0666582
$687$	−10.2949	−0.392777
$688$	14.6619	0.558980
$689$	−8.56705	−0.326378
$690$	−28.9200	−1.10096
$691$	22.8195	0.868095	0.434047	−	0.900890i	$-0.357085\pi$
$691$	22.8195	0.868095	0.434047	+	0.900890i	$0.357085\pi$
$692$	−9.99346	−0.379894
$693$	1.60868	0.0611088
$694$	47.1401	1.78941
$695$	−41.1030	−1.55913
$696$	−2.70249	−0.102438
$697$	−20.1605	−0.763633
$698$	−46.9492	−1.77705
$699$	−53.7025	−2.03122
$700$	6.07260	0.229523
$701$	41.7773	1.57791	0.788953	−	0.614454i	$-0.210624\pi$
$701$	41.7773	1.57791	0.788953	+	0.614454i	$0.210624\pi$
$702$	24.4910	0.924354
$703$	23.5634	0.888712
$704$	1.07905	0.0406682
$705$	76.9114	2.89665
$706$	12.0135	0.452135
$707$	1.09604	0.0412210
$708$	17.7322	0.666418
$709$	−13.6129	−0.511242	−0.255621	−	0.966777i	$-0.582280\pi$
$709$	−13.6129	−0.511242	−0.255621	+	0.966777i	$0.582280\pi$
$710$	−49.2936	−1.84996
$711$	−5.25734	−0.197166
$712$	15.8639	0.594524
$713$	−5.57594	−0.208821
$714$	−6.38583	−0.238984
$715$	20.8176	0.778533
$716$	7.35556	0.274890
$717$	29.9993	1.12035
$718$	−29.4178	−1.09786
$719$	33.1382	1.23584	0.617922	−	0.786239i	$-0.287975\pi$
$719$	33.1382	1.23584	0.617922	+	0.786239i	$0.287975\pi$
$720$	13.8241	0.515194
$721$	−13.3210	−0.496101
$722$	−92.5544	−3.44452
$723$	11.1165	0.413428
$724$	−23.3704	−0.868556
$725$	4.80679	0.178520
$726$	−25.1498	−0.933397
$727$	33.4495	1.24057	0.620287	−	0.784375i	$-0.287016\pi$
$727$	33.4495	1.24057	0.620287	+	0.784375i	$0.287016\pi$
$728$	−5.51127	−0.204261
$729$	15.7662	0.583934
$730$	12.3591	0.457431
$731$	5.47455	0.202483
$732$	−7.91981	−0.292724
$733$	1.13408	0.0418884	0.0209442	−	0.999781i	$-0.493333\pi$
$733$	1.13408	0.0418884	0.0209442	+	0.999781i	$0.493333\pi$
$734$	−14.1908	−0.523793
$735$	6.43966	0.237530
$736$	13.8945	0.512159
$737$	4.26679	0.157169
$738$	−15.8808	−0.584579
$739$	−1.24976	−0.0459733	−0.0229867	−	0.999736i	$-0.507318\pi$
$739$	−1.24976	−0.0459733	−0.0229867	+	0.999736i	$0.507318\pi$
$740$	−9.56161	−0.351492
$741$	55.1610	2.02639
$742$	−4.51019	−0.165574
$743$	28.9431	1.06182	0.530910	−	0.847428i	$-0.321850\pi$
$743$	28.9431	1.06182	0.530910	+	0.847428i	$0.321850\pi$
$744$	7.06112	0.258873
$745$	65.1278	2.38610
$746$	−17.2482	−0.631503
$747$	7.88848	0.288625
$748$	−3.73703	−0.136639
$749$	14.0155	0.512115
$750$	−8.92487	−0.325890
$751$	−15.4679	−0.564431	−0.282215	−	0.959351i	$-0.591069\pi$
$751$	−15.4679	−0.564431	−0.282215	+	0.959351i	$0.591069\pi$
$752$	−59.6894	−2.17665
$753$	−25.1304	−0.915803
$754$	4.80350	0.174933
$755$	30.3783	1.10558
$756$	4.43352	0.161246
$757$	44.1814	1.60580	0.802899	−	0.596114i	$-0.203290\pi$
$757$	44.1814	1.60580	0.802899	+	0.596114i	$0.203290\pi$
$758$	−16.9488	−0.615606
$759$	−9.63351	−0.349674
$760$	−46.3332	−1.68068
$761$	10.4630	0.379283	0.189642	−	0.981853i	$-0.439267\pi$
$761$	10.4630	0.379283	0.189642	+	0.981853i	$0.439267\pi$
$762$	−18.5263	−0.671138
$763$	−13.3420	−0.483014
$764$	−6.45970	−0.233704
$765$	5.16172	0.186622
$766$	−17.2215	−0.622239
$767$	28.6240	1.03355
$768$	−38.1299	−1.37589
$769$	−4.42904	−0.159715	−0.0798577	−	0.996806i	$-0.525447\pi$
$769$	−4.42904	−0.159715	−0.0798577	+	0.996806i	$0.525447\pi$
$770$	10.9596	0.394956
$771$	−42.6738	−1.53686
$772$	−10.6908	−0.384771
$773$	0.340986	0.0122644	0.00613221	−	0.999981i	$-0.498048\pi$
$773$	0.340986	0.0122644	0.00613221	+	0.999981i	$0.498048\pi$
$774$	4.31239	0.155006
$775$	−12.5593	−0.451142
$776$	27.2896	0.979639
$777$	−5.44263	−0.195253
$778$	−47.7864	−1.71323
$779$	91.6812	3.28482
$780$	−22.3833	−0.801452
$781$	−16.4202	−0.587560
$782$	8.38033	0.299680
$783$	3.50937	0.125415
$784$	−4.99768	−0.178489
$785$	−25.0359	−0.893570
$786$	1.07934	0.0384986
$787$	−31.0067	−1.10527	−0.552635	−	0.833424i	$-0.686377\pi$
$787$	−31.0067	−1.10527	−0.552635	+	0.833424i	$0.686377\pi$
$788$	−1.82481	−0.0650062
$789$	29.5991	1.05375
$790$	−35.8170	−1.27431
$791$	10.1519	0.360959
$792$	2.67344	0.0949965
$793$	−12.7844	−0.453989
$794$	9.41116	0.333989
$795$	16.6357	0.590009
$796$	−11.1196	−0.394125
$797$	30.2850	1.07275	0.536374	−	0.843980i	$-0.319793\pi$
$797$	30.2850	1.07275	0.536374	+	0.843980i	$0.319793\pi$
$798$	29.0400	1.02800
$799$	−22.2871	−0.788462
$800$	31.2961	1.10648
$801$	8.03693	0.283971
$802$	−50.2236	−1.77346
$803$	4.11694	0.145284
$804$	−4.58771	−0.161796
$805$	−8.45097	−0.297858
$806$	−12.5507	−0.442078
$807$	−51.4945	−1.81269
$808$	1.82149	0.0640798
$809$	−47.4570	−1.66850	−0.834250	−	0.551386i	$-0.814099\pi$
$809$	−47.4570	−1.66850	−0.834250	+	0.551386i	$0.814099\pi$
$810$	−62.0452	−2.18005
$811$	−33.1888	−1.16542	−0.582709	−	0.812681i	$-0.698007\pi$
$811$	−33.1888	−1.16542	−0.582709	+	0.812681i	$0.698007\pi$
$812$	0.869561	0.0305156
$813$	4.03029	0.141348
$814$	−9.26274	−0.324659
$815$	−14.5381	−0.509246
$816$	−18.2798	−0.639919
$817$	−24.8959	−0.870996
$818$	23.1768	0.810358
$819$	−2.79211	−0.0975642
$820$	−37.2025	−1.29917
$821$	14.9224	0.520795	0.260398	−	0.965501i	$-0.416146\pi$
$821$	14.9224	0.520795	0.260398	+	0.965501i	$0.416146\pi$
$822$	−46.0635	−1.60665
$823$	31.7130	1.10545	0.552723	−	0.833365i	$-0.313589\pi$
$823$	31.7130	1.10545	0.552723	+	0.833365i	$0.313589\pi$
$824$	−22.1379	−0.771211
$825$	−21.6986	−0.755447
$826$	15.0693	0.524329
$827$	−48.7849	−1.69642	−0.848209	−	0.529662i	$-0.822319\pi$
$827$	−48.7849	−1.69642	−0.848209	+	0.529662i	$0.822319\pi$
$828$	2.26991	0.0788849
$829$	27.6527	0.960419	0.480210	−	0.877154i	$-0.340561\pi$
$829$	27.6527	0.960419	0.480210	+	0.877154i	$0.340561\pi$
$830$	53.7424	1.86542
$831$	10.9799	0.380890
$832$	−1.87285	−0.0649293
$833$	−1.86606	−0.0646553
$834$	42.8131	1.48250
$835$	27.2591	0.943339
$836$	16.9944	0.587763
$837$	−9.16934	−0.316939
$838$	−51.7961	−1.78927
$839$	2.95740	0.102101	0.0510504	−	0.998696i	$-0.483743\pi$
$839$	2.95740	0.102101	0.0510504	+	0.998696i	$0.483743\pi$
$840$	10.7019	0.369252
$841$	−28.3117	−0.976265
$842$	−49.2556	−1.69746
$843$	−57.8186	−1.99138
$844$	−6.45140	−0.222066
$845$	6.57817	0.226296
$846$	−17.5559	−0.603586
$847$	−7.34926	−0.252524
$848$	−12.9106	−0.443353
$849$	22.8214	0.783227
$850$	18.8759	0.647438
$851$	7.14254	0.244843
$852$	17.6552	0.604856
$853$	−52.0891	−1.78350	−0.891748	−	0.452532i	$-0.850521\pi$
$853$	−52.0891	−1.78350	−0.891748	+	0.452532i	$0.850521\pi$
$854$	−6.73047	−0.230312
$855$	−23.4732	−0.802768
$856$	23.2920	0.796105
$857$	48.1988	1.64644	0.823219	−	0.567723i	$-0.192176\pi$
$857$	48.1988	1.64644	0.823219	+	0.567723i	$0.192176\pi$
$858$	−21.6837	−0.740269
$859$	−1.00000	−0.0341196
$859$	−1.00000	−0.0341196
$860$	10.1023	0.344485
$861$	−21.1763	−0.721687
$862$	−38.0004	−1.29430
$863$	28.3074	0.963596	0.481798	−	0.876282i	$-0.339984\pi$
$863$	28.3074	0.963596	0.481798	+	0.876282i	$0.339984\pi$
$864$	22.8488	0.777333
$865$	31.3252	1.06509
$866$	−29.7342	−1.01041
$867$	26.4961	0.899854
$868$	−2.27200	−0.0771168
$869$	−11.9310	−0.404731
$870$	−9.32758	−0.316234
$871$	−7.40564	−0.250931
$872$	−22.1728	−0.750866
$873$	13.8254	0.467919
$874$	−38.1101	−1.28909
$875$	−2.60802	−0.0881672
$876$	−4.42659	−0.149560
$877$	−13.7382	−0.463907	−0.231953	−	0.972727i	$-0.574512\pi$
$877$	−13.7382	−0.463907	−0.231953	+	0.972727i	$0.574512\pi$
$878$	−14.1640	−0.478011
$879$	12.7006	0.428381
$880$	31.3723	1.05756
$881$	−34.2910	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$881$	−34.2910	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$882$	−1.46993	−0.0494951
$883$	46.4387	1.56279	0.781394	−	0.624038i	$-0.214509\pi$
$883$	46.4387	1.56279	0.781394	+	0.624038i	$0.214509\pi$
$884$	6.48617	0.218153
$885$	−55.5829	−1.86840
$886$	−11.8177	−0.397022
$887$	−43.5715	−1.46299	−0.731494	−	0.681848i	$-0.761177\pi$
$887$	−43.5715	−1.46299	−0.731494	+	0.681848i	$0.761177\pi$
$888$	−9.04499	−0.303530
$889$	−5.41375	−0.181571
$890$	54.7537	1.83535
$891$	−20.6679	−0.692399
$892$	−11.4146	−0.382189
$893$	101.352	3.39162
$894$	−67.8375	−2.26883
$895$	−23.0565	−0.770694
$896$	−11.7892	−0.393851
$897$	16.7204	0.558277
$898$	59.7352	1.99339
$899$	−1.79841	−0.0599804
$900$	5.11276	0.170425
$901$	−4.82065	−0.160599
$902$	−36.0397	−1.19999
$903$	5.75039	0.191361
$904$	16.8712	0.561126
$905$	73.2563	2.43512
$906$	−31.6422	−1.05124
$907$	8.11369	0.269411	0.134705	−	0.990886i	$-0.456991\pi$
$907$	8.11369	0.269411	0.134705	+	0.990886i	$0.456991\pi$
$908$	6.97277	0.231400
$909$	0.922801	0.0306074
$910$	−19.0220	−0.630572
$911$	28.4699	0.943249	0.471625	−	0.881799i	$-0.343668\pi$
$911$	28.4699	0.943249	0.471625	+	0.881799i	$0.343668\pi$
$912$	83.1283	2.75265
$913$	17.9021	0.592473
$914$	−12.5211	−0.414160
$915$	24.8252	0.820695
$916$	5.50502	0.181891
$917$	0.315403	0.0104155
$918$	13.7810	0.454841
$919$	42.8095	1.41215	0.706077	−	0.708135i	$-0.250463\pi$
$919$	42.8095	1.41215	0.706077	+	0.708135i	$0.250463\pi$
$920$	−14.0445	−0.463033
$921$	−50.3457	−1.65895
$922$	−8.67147	−0.285580
$923$	28.4996	0.938077
$924$	−3.92532	−0.129134
$925$	16.0879	0.528966
$926$	−45.2769	−1.48789
$927$	−11.2155	−0.368365
$928$	4.48141	0.147110
$929$	−28.9205	−0.948850	−0.474425	−	0.880296i	$-0.657344\pi$
$929$	−28.9205	−0.948850	−0.474425	+	0.880296i	$0.657344\pi$
$930$	24.3712	0.799164
$931$	8.48604	0.278119
$932$	28.7163	0.940635
$933$	55.0408	1.80195
$934$	38.2329	1.25102
$935$	11.7140	0.383088
$936$	−4.64015	−0.151668
$937$	5.43546	0.177569	0.0887844	−	0.996051i	$-0.471702\pi$
$937$	5.43546	0.177569	0.0887844	+	0.996051i	$0.471702\pi$
$938$	−3.89876	−0.127299
$939$	−44.3404	−1.44699
$940$	−41.1269	−1.34141
$941$	−38.5710	−1.25738	−0.628689	−	0.777657i	$-0.716408\pi$
$941$	−38.5710	−1.25738	−0.628689	+	0.777657i	$0.716408\pi$
$942$	26.0775	0.849652
$943$	27.7904	0.904979
$944$	43.1367	1.40398
$945$	−13.8972	−0.452075
$946$	9.78652	0.318187
$947$	56.9289	1.84994	0.924970	−	0.380040i	$-0.124090\pi$
$947$	56.9289	1.84994	0.924970	+	0.380040i	$0.124090\pi$
$948$	12.8283	0.416645
$949$	−7.14555	−0.231955
$950$	−85.8393	−2.78499
$951$	−31.7575	−1.02981
$952$	−3.10117	−0.100510
$953$	−35.1039	−1.13713	−0.568563	−	0.822640i	$-0.692500\pi$
$953$	−35.1039	−1.13713	−0.568563	+	0.822640i	$0.692500\pi$
$954$	−3.79730	−0.122942
$955$	20.2484	0.655222
$956$	−16.0415	−0.518821
$957$	−3.10710	−0.100438
$958$	17.8565	0.576918
$959$	−13.4607	−0.434667
$960$	3.63675	0.117376
$961$	−26.3011	−0.848422
$962$	16.0769	0.518339
$963$	11.8002	0.380255
$964$	−5.94434	−0.191454
$965$	33.5111	1.07876
$966$	8.80258	0.283218
$967$	−17.1347	−0.551016	−0.275508	−	0.961299i	$-0.588846\pi$
$967$	−17.1347	−0.551016	−0.275508	+	0.961299i	$0.588846\pi$
$968$	−12.2136	−0.392559
$969$	31.0389	0.997114
$970$	94.1891	3.02423
$971$	−24.3608	−0.781774	−0.390887	−	0.920439i	$-0.627832\pi$
$971$	−24.3608	−0.781774	−0.390887	+	0.920439i	$0.627832\pi$
$972$	8.92179	0.286166
$973$	12.5108	0.401079
$974$	69.9333	2.24081
$975$	37.6610	1.20612
$976$	−19.2663	−0.616699
$977$	32.1297	1.02792	0.513960	−	0.857814i	$-0.328178\pi$
$977$	32.1297	1.02792	0.513960	+	0.857814i	$0.328178\pi$
$978$	15.1429	0.484217
$979$	18.2390	0.582920
$980$	−3.44348	−0.109998
$981$	−11.2332	−0.358647
$982$	32.8701	1.04893
$983$	−38.8126	−1.23793	−0.618965	−	0.785419i	$-0.712448\pi$
$983$	−38.8126	−1.23793	−0.618965	+	0.785419i	$0.712448\pi$
$984$	−35.1925	−1.12189
$985$	5.72000	0.182254
$986$	2.70291	0.0860783
$987$	−23.4101	−0.745152
$988$	−29.4963	−0.938401
$989$	−7.54642	−0.239962
$990$	9.22729	0.293262
$991$	−56.8249	−1.80510	−0.902551	−	0.430584i	$-0.858308\pi$
$991$	−56.8249	−1.80510	−0.902551	+	0.430584i	$0.858308\pi$
$992$	−11.7091	−0.371764
$993$	−9.23160	−0.292956
$994$	15.0039	0.475893
$995$	34.8552	1.10498
$996$	−19.2486	−0.609914
$997$	52.3753	1.65874	0.829371	−	0.558698i	$-0.188699\pi$
$997$	52.3753	1.65874	0.829371	+	0.558698i	$0.188699\pi$
$998$	71.0778	2.24993
$999$	11.7455	0.371612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6013.2.a.e.1.20	✓	109

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6013.2.a.e.1.20	✓	109	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:	\( N \)	\(=\)	\( 6013 = 7 \cdot 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6013.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74589 q^{2} -1.96009 q^{3} +1.04812 q^{4} -3.28540 q^{5} +3.42209 q^{6} +1.00000 q^{7} +1.66188 q^{8} +0.841938 q^{9} +O(q^{10})\)	\(q-1.74589 q^{2} -1.96009 q^{3} +1.04812 q^{4} -3.28540 q^{5} +3.42209 q^{6} +1.00000 q^{7} +1.66188 q^{8} +0.841938 q^{9} +5.73592 q^{10} +1.91069 q^{11} -2.05440 q^{12} -3.31629 q^{13} -1.74589 q^{14} +6.43966 q^{15} -4.99768 q^{16} -1.86606 q^{17} -1.46993 q^{18} +8.48604 q^{19} -3.44348 q^{20} -1.96009 q^{21} -3.33585 q^{22} +2.57229 q^{23} -3.25743 q^{24} +5.79382 q^{25} +5.78986 q^{26} +4.22999 q^{27} +1.04812 q^{28} +0.829641 q^{29} -11.2429 q^{30} -2.16770 q^{31} +5.40163 q^{32} -3.74512 q^{33} +3.25793 q^{34} -3.28540 q^{35} +0.882450 q^{36} +2.77673 q^{37} -14.8157 q^{38} +6.50021 q^{39} -5.45993 q^{40} +10.8038 q^{41} +3.42209 q^{42} -2.93374 q^{43} +2.00263 q^{44} -2.76610 q^{45} -4.49092 q^{46} +11.9434 q^{47} +9.79589 q^{48} +1.00000 q^{49} -10.1154 q^{50} +3.65764 q^{51} -3.47586 q^{52} +2.58333 q^{53} -7.38507 q^{54} -6.27737 q^{55} +1.66188 q^{56} -16.6334 q^{57} -1.44846 q^{58} -8.63134 q^{59} +6.74952 q^{60} +3.85505 q^{61} +3.78455 q^{62} +0.841938 q^{63} +0.564743 q^{64} +10.8953 q^{65} +6.53855 q^{66} +2.23311 q^{67} -1.95585 q^{68} -5.04190 q^{69} +5.73592 q^{70} -8.59384 q^{71} +1.39920 q^{72} +2.15469 q^{73} -4.84785 q^{74} -11.3564 q^{75} +8.89436 q^{76} +1.91069 q^{77} -11.3486 q^{78} -6.24433 q^{79} +16.4194 q^{80} -10.8170 q^{81} -18.8621 q^{82} +9.36943 q^{83} -2.05440 q^{84} +6.13076 q^{85} +5.12198 q^{86} -1.62617 q^{87} +3.17534 q^{88} +9.54575 q^{89} +4.82929 q^{90} -3.31629 q^{91} +2.69606 q^{92} +4.24888 q^{93} -20.8518 q^{94} -27.8800 q^{95} -10.5877 q^{96} +16.4209 q^{97} -1.74589 q^{98} +1.60868 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9 + 15 * q^10 + 48 * q^11 + 72 * q^12 + 29 * q^13 + 19 * q^14 + 29 * q^15 + 115 * q^16 + 72 * q^17 + 33 * q^18 + 58 * q^19 + 88 * q^20 + 38 * q^21 + 4 * q^22 + 65 * q^23 + 46 * q^24 + 124 * q^25 + 49 * q^26 + 131 * q^27 + 111 * q^28 + 25 * q^29 + 2 * q^30 + 41 * q^31 + 75 * q^32 + 54 * q^33 + 23 * q^34 + 43 * q^35 + 111 * q^36 + 25 * q^37 + 54 * q^38 + 27 * q^39 + 30 * q^40 + 109 * q^41 + 14 * q^42 + 38 * q^43 + 68 * q^44 + 84 * q^45 - 9 * q^46 + 121 * q^47 + 106 * q^48 + 109 * q^49 + 14 * q^50 + 36 * q^51 + 38 * q^52 + 61 * q^53 + 31 * q^54 + 50 * q^55 + 48 * q^56 + 5 * q^57 - 20 * q^58 + 181 * q^59 + 25 * q^60 + 34 * q^61 + 75 * q^62 + 119 * q^63 + 96 * q^64 + 12 * q^65 + 19 * q^66 + 87 * q^67 + 150 * q^68 + 89 * q^69 + 15 * q^70 + 83 * q^71 + 65 * q^72 + 32 * q^73 - 19 * q^74 + 112 * q^75 + 84 * q^76 + 48 * q^77 - 34 * q^78 - 9 * q^79 + 137 * q^80 + 109 * q^81 - 19 * q^82 + 136 * q^83 + 72 * q^84 - 32 * q^85 - 24 * q^86 + 28 * q^87 - 24 * q^88 + 142 * q^89 + 19 * q^90 + 29 * q^91 + 96 * q^92 + 29 * q^93 + 9 * q^94 + 52 * q^95 + 88 * q^96 + 75 * q^97 + 19 * q^98 + 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more