LMFDB - Embedded newform 6019.2.a.b.1.19

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:	$101$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q-2.03639 q^{2} -1.20777 q^{3} +2.14690 q^{4} +3.16451 q^{5} +2.45949 q^{6} +3.67207 q^{7} -0.299141 q^{8} -1.54130 q^{9} +O(q^{10})$	\(q-2.03639 q^{2} -1.20777 q^{3} +2.14690 q^{4} +3.16451 q^{5} +2.45949 q^{6} +3.67207 q^{7} -0.299141 q^{8} -1.54130 q^{9} -6.44419 q^{10} +0.325684 q^{11} -2.59296 q^{12} +1.00000 q^{13} -7.47779 q^{14} -3.82200 q^{15} -3.68463 q^{16} -5.38625 q^{17} +3.13868 q^{18} +2.98579 q^{19} +6.79389 q^{20} -4.43502 q^{21} -0.663222 q^{22} -4.80284 q^{23} +0.361293 q^{24} +5.01415 q^{25} -2.03639 q^{26} +5.48483 q^{27} +7.88357 q^{28} -8.51496 q^{29} +7.78310 q^{30} +3.30196 q^{31} +8.10163 q^{32} -0.393351 q^{33} +10.9685 q^{34} +11.6203 q^{35} -3.30900 q^{36} +6.47891 q^{37} -6.08025 q^{38} -1.20777 q^{39} -0.946636 q^{40} +1.14589 q^{41} +9.03144 q^{42} -3.87638 q^{43} +0.699211 q^{44} -4.87745 q^{45} +9.78047 q^{46} -7.55277 q^{47} +4.45018 q^{48} +6.48413 q^{49} -10.2108 q^{50} +6.50535 q^{51} +2.14690 q^{52} -1.66269 q^{53} -11.1693 q^{54} +1.03063 q^{55} -1.09847 q^{56} -3.60615 q^{57} +17.3398 q^{58} -7.23942 q^{59} -8.20544 q^{60} +7.34531 q^{61} -6.72410 q^{62} -5.65975 q^{63} -9.12885 q^{64} +3.16451 q^{65} +0.801018 q^{66} -10.4604 q^{67} -11.5637 q^{68} +5.80072 q^{69} -23.6636 q^{70} -9.22106 q^{71} +0.461064 q^{72} -3.96706 q^{73} -13.1936 q^{74} -6.05593 q^{75} +6.41019 q^{76} +1.19594 q^{77} +2.45949 q^{78} -14.9799 q^{79} -11.6601 q^{80} -2.00052 q^{81} -2.33349 q^{82} -4.17697 q^{83} -9.52152 q^{84} -17.0449 q^{85} +7.89383 q^{86} +10.2841 q^{87} -0.0974256 q^{88} -0.223120 q^{89} +9.93240 q^{90} +3.67207 q^{91} -10.3112 q^{92} -3.98801 q^{93} +15.3804 q^{94} +9.44858 q^{95} -9.78489 q^{96} -8.90776 q^{97} -13.2042 q^{98} -0.501976 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 101 q - 8 q^{2} - 13 q^{3} + 86 q^{4} - 43 q^{5} - 10 q^{6} - q^{7} - 24 q^{8} + 52 q^{9}+O(q^{10}) $ 101 * q - 8 * q^2 - 13 * q^3 + 86 * q^4 - 43 * q^5 - 10 * q^6 - q^7 - 24 * q^8 + 52 * q^9	\( 101 q - 8 q^{2} - 13 q^{3} + 86 q^{4} - 43 q^{5} - 10 q^{6} - q^{7} - 24 q^{8} + 52 q^{9} - 19 q^{10} - 42 q^{11} - 28 q^{12} + 101 q^{13} - 45 q^{14} - 15 q^{15} + 48 q^{16} - 83 q^{17} - 4 q^{18} - 18 q^{19} - 51 q^{20} - 50 q^{21} - 20 q^{22} - 64 q^{23} - 23 q^{24} + 46 q^{25} - 8 q^{26} - 37 q^{27} - 11 q^{28} - 117 q^{29} - 28 q^{30} - 10 q^{31} - 36 q^{32} - 20 q^{33} - 10 q^{34} - 53 q^{35} - 16 q^{36} - 27 q^{37} - 68 q^{38} - 13 q^{39} - 42 q^{40} - 60 q^{41} - 31 q^{42} - 16 q^{43} - 89 q^{44} - 56 q^{45} + 5 q^{46} - 23 q^{47} - 37 q^{48} + 48 q^{49} - 30 q^{50} - 68 q^{51} + 86 q^{52} - 189 q^{53} - 23 q^{54} + 3 q^{55} - 106 q^{56} - 25 q^{57} - 82 q^{59} + 6 q^{60} - 68 q^{61} - 57 q^{62} + 3 q^{63} - 2 q^{64} - 43 q^{65} - 40 q^{66} - 13 q^{67} - 138 q^{68} - 92 q^{69} + 18 q^{70} - 39 q^{71} - 20 q^{72} + 19 q^{73} - 88 q^{74} - 21 q^{75} - 53 q^{76} - 147 q^{77} - 10 q^{78} - 19 q^{79} - 104 q^{80} - 55 q^{81} + 27 q^{82} - 49 q^{83} - 59 q^{84} - 27 q^{85} - 99 q^{86} - 33 q^{87} - 41 q^{88} - 70 q^{89} - 49 q^{90} - q^{91} - 111 q^{92} - 84 q^{93} + 4 q^{94} - 82 q^{95} - 7 q^{96} + 25 q^{97} - 37 q^{98} - 41 q^{99}+O(q^{100}) \) 101 * q - 8 * q^2 - 13 * q^3 + 86 * q^4 - 43 * q^5 - 10 * q^6 - q^7 - 24 * q^8 + 52 * q^9 - 19 * q^10 - 42 * q^11 - 28 * q^12 + 101 * q^13 - 45 * q^14 - 15 * q^15 + 48 * q^16 - 83 * q^17 - 4 * q^18 - 18 * q^19 - 51 * q^20 - 50 * q^21 - 20 * q^22 - 64 * q^23 - 23 * q^24 + 46 * q^25 - 8 * q^26 - 37 * q^27 - 11 * q^28 - 117 * q^29 - 28 * q^30 - 10 * q^31 - 36 * q^32 - 20 * q^33 - 10 * q^34 - 53 * q^35 - 16 * q^36 - 27 * q^37 - 68 * q^38 - 13 * q^39 - 42 * q^40 - 60 * q^41 - 31 * q^42 - 16 * q^43 - 89 * q^44 - 56 * q^45 + 5 * q^46 - 23 * q^47 - 37 * q^48 + 48 * q^49 - 30 * q^50 - 68 * q^51 + 86 * q^52 - 189 * q^53 - 23 * q^54 + 3 * q^55 - 106 * q^56 - 25 * q^57 - 82 * q^59 + 6 * q^60 - 68 * q^61 - 57 * q^62 + 3 * q^63 - 2 * q^64 - 43 * q^65 - 40 * q^66 - 13 * q^67 - 138 * q^68 - 92 * q^69 + 18 * q^70 - 39 * q^71 - 20 * q^72 + 19 * q^73 - 88 * q^74 - 21 * q^75 - 53 * q^76 - 147 * q^77 - 10 * q^78 - 19 * q^79 - 104 * q^80 - 55 * q^81 + 27 * q^82 - 49 * q^83 - 59 * q^84 - 27 * q^85 - 99 * q^86 - 33 * q^87 - 41 * q^88 - 70 * q^89 - 49 * q^90 - q^91 - 111 * q^92 - 84 * q^93 + 4 * q^94 - 82 * q^95 - 7 * q^96 + 25 * q^97 - 37 * q^98 - 41 * 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 $

$2$	−2.03639	−1.43995	−0.719974	−	0.694001i	$-0.755846\pi$
$2$	−2.03639	−1.43995	−0.719974	+	0.694001i	$0.755846\pi$
$3$	−1.20777	−0.697306	−0.348653	−	0.937252i	$-0.613361\pi$
$3$	−1.20777	−0.697306	−0.348653	+	0.937252i	$0.613361\pi$
$4$	2.14690	1.07345
$5$	3.16451	1.41521	0.707607	−	0.706606i	$-0.249775\pi$
$5$	3.16451	1.41521	0.707607	+	0.706606i	$0.249775\pi$
$6$	2.45949	1.00408
$7$	3.67207	1.38791	0.693957	−	0.720017i	$-0.255866\pi$
$7$	3.67207	1.38791	0.693957	+	0.720017i	$0.255866\pi$
$8$	−0.299141	−0.105762
$9$	−1.54130	−0.513765
$10$	−6.44419	−2.03783
$11$	0.325684	0.0981976	0.0490988	−	0.998794i	$-0.484365\pi$
$11$	0.325684	0.0981976	0.0490988	+	0.998794i	$0.484365\pi$
$12$	−2.59296	−0.748522
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−7.47779	−1.99852
$15$	−3.82200	−0.986836
$16$	−3.68463	−0.921157
$17$	−5.38625	−1.30636	−0.653179	−	0.757203i	$-0.726565\pi$
$17$	−5.38625	−1.30636	−0.653179	+	0.757203i	$0.726565\pi$
$18$	3.13868	0.739795
$19$	2.98579	0.684988	0.342494	−	0.939520i	$-0.388728\pi$
$19$	2.98579	0.684988	0.342494	+	0.939520i	$0.388728\pi$
$20$	6.79389	1.51916
$21$	−4.43502	−0.967800
$22$	−0.663222	−0.141399
$23$	−4.80284	−1.00146	−0.500730	−	0.865603i	$-0.666935\pi$
$23$	−4.80284	−1.00146	−0.500730	+	0.865603i	$0.666935\pi$
$24$	0.361293	0.0737486
$25$	5.01415	1.00283
$26$	−2.03639	−0.399370
$27$	5.48483	1.05556
$28$	7.88357	1.48985
$29$	−8.51496	−1.58119	−0.790594	−	0.612340i	$-0.790228\pi$
$29$	−8.51496	−1.58119	−0.790594	+	0.612340i	$0.790228\pi$
$30$	7.78310	1.42099
$31$	3.30196	0.593050	0.296525	−	0.955025i	$-0.404172\pi$
$31$	3.30196	0.593050	0.296525	+	0.955025i	$0.404172\pi$
$32$	8.10163	1.43218
$33$	−0.393351	−0.0684737
$34$	10.9685	1.88109
$35$	11.6203	1.96419
$36$	−3.30900	−0.551500
$37$	6.47891	1.06513	0.532564	−	0.846390i	$-0.321229\pi$
$37$	6.47891	1.06513	0.532564	+	0.846390i	$0.321229\pi$
$38$	−6.08025	−0.986346
$39$	−1.20777	−0.193398
$40$	−0.946636	−0.149676
$41$	1.14589	0.178958	0.0894791	−	0.995989i	$-0.471480\pi$
$41$	1.14589	0.178958	0.0894791	+	0.995989i	$0.471480\pi$
$42$	9.03144	1.39358
$43$	−3.87638	−0.591142	−0.295571	−	0.955321i	$-0.595510\pi$
$43$	−3.87638	−0.591142	−0.295571	+	0.955321i	$0.595510\pi$
$44$	0.699211	0.105410
$45$	−4.87745	−0.727087
$46$	9.78047	1.44205
$47$	−7.55277	−1.10169	−0.550843	−	0.834609i	$-0.685694\pi$
$47$	−7.55277	−1.10169	−0.550843	+	0.834609i	$0.685694\pi$
$48$	4.45018	0.642328
$49$	6.48413	0.926304
$50$	−10.2108	−1.44402
$51$	6.50535	0.910931
$52$	2.14690	0.297721
$53$	−1.66269	−0.228388	−0.114194	−	0.993458i	$-0.536429\pi$
$53$	−1.66269	−0.228388	−0.114194	+	0.993458i	$0.536429\pi$
$54$	−11.1693	−1.51995
$55$	1.03063	0.138971
$56$	−1.09847	−0.146789
$57$	−3.60615	−0.477646
$58$	17.3398	2.27683
$59$	−7.23942	−0.942493	−0.471246	−	0.882002i	$-0.656196\pi$
$59$	−7.23942	−0.942493	−0.471246	+	0.882002i	$0.656196\pi$
$60$	−8.20544	−1.05932
$61$	7.34531	0.940470	0.470235	−	0.882541i	$-0.344169\pi$
$61$	7.34531	0.940470	0.470235	+	0.882541i	$0.344169\pi$
$62$	−6.72410	−0.853961
$63$	−5.65975	−0.713061
$64$	−9.12885	−1.14111
$65$	3.16451	0.392510
$66$	0.801018	0.0985985
$67$	−10.4604	−1.27794	−0.638972	−	0.769230i	$-0.720640\pi$
$67$	−10.4604	−1.27794	−0.638972	+	0.769230i	$0.720640\pi$
$68$	−11.5637	−1.40231
$69$	5.80072	0.698324
$70$	−23.6636	−2.82834
$71$	−9.22106	−1.09434	−0.547169	−	0.837022i	$-0.684295\pi$
$71$	−9.22106	−1.09434	−0.547169	+	0.837022i	$0.684295\pi$
$72$	0.461064	0.0543370
$73$	−3.96706	−0.464309	−0.232154	−	0.972679i	$-0.574577\pi$
$73$	−3.96706	−0.464309	−0.232154	+	0.972679i	$0.574577\pi$
$74$	−13.1936	−1.53373
$75$	−6.05593	−0.699278
$76$	6.41019	0.735299
$77$	1.19594	0.136290
$78$	2.45949	0.278483
$79$	−14.9799	−1.68537	−0.842684	−	0.538408i	$-0.819026\pi$
$79$	−14.9799	−1.68537	−0.842684	+	0.538408i	$0.819026\pi$
$80$	−11.6601	−1.30363
$81$	−2.00052	−0.222280
$82$	−2.33349	−0.257690
$83$	−4.17697	−0.458482	−0.229241	−	0.973370i	$-0.573624\pi$
$83$	−4.17697	−0.458482	−0.229241	+	0.973370i	$0.573624\pi$
$84$	−9.52152	−1.03888
$85$	−17.0449	−1.84878
$86$	7.89383	0.851213
$87$	10.2841	1.10257
$88$	−0.0974256	−0.0103856
$89$	−0.223120	−0.0236507	−0.0118254	−	0.999930i	$-0.503764\pi$
$89$	−0.223120	−0.0236507	−0.0118254	+	0.999930i	$0.503764\pi$
$90$	9.93240	1.04697
$91$	3.67207	0.384938
$92$	−10.3112	−1.07502
$93$	−3.98801	−0.413537
$94$	15.3804	1.58637
$95$	9.44858	0.969404
$96$	−9.78489	−0.998667
$97$	−8.90776	−0.904446	−0.452223	−	0.891905i	$-0.649369\pi$
$97$	−8.90776	−0.904446	−0.452223	+	0.891905i	$0.649369\pi$
$98$	−13.2042	−1.33383
$99$	−0.501976	−0.0504505
$100$	10.7649	1.07649
$101$	2.18936	0.217849	0.108925	−	0.994050i	$-0.465259\pi$
$101$	2.18936	0.217849	0.108925	+	0.994050i	$0.465259\pi$
$102$	−13.2474	−1.31169
$103$	9.61468	0.947362	0.473681	−	0.880696i	$-0.342925\pi$
$103$	9.61468	0.947362	0.473681	+	0.880696i	$0.342925\pi$
$104$	−0.299141	−0.0293332
$105$	−14.0347	−1.36964
$106$	3.38589	0.328867
$107$	−1.10814	−0.107128	−0.0535639	−	0.998564i	$-0.517058\pi$
$107$	−1.10814	−0.107128	−0.0535639	+	0.998564i	$0.517058\pi$
$108$	11.7754	1.13309
$109$	−0.760592	−0.0728515	−0.0364258	−	0.999336i	$-0.511597\pi$
$109$	−0.760592	−0.0728515	−0.0364258	+	0.999336i	$0.511597\pi$
$110$	−2.09877	−0.200110
$111$	−7.82503	−0.742719
$112$	−13.5302	−1.27849
$113$	−0.375624	−0.0353358	−0.0176679	−	0.999844i	$-0.505624\pi$
$113$	−0.375624	−0.0353358	−0.0176679	+	0.999844i	$0.505624\pi$
$114$	7.34353	0.687785
$115$	−15.1986	−1.41728
$116$	−18.2807	−1.69732
$117$	−1.54130	−0.142493
$118$	14.7423	1.35714
$119$	−19.7787	−1.81311
$120$	1.14332	0.104370
$121$	−10.8939	−0.990357
$122$	−14.9579	−1.35423
$123$	−1.38397	−0.124788
$124$	7.08898	0.636609
$125$	0.0447683	0.00400419
$126$	11.5255	1.02677
$127$	−21.4785	−1.90591	−0.952956	−	0.303107i	$-0.901976\pi$
$127$	−21.4785	−1.90591	−0.952956	+	0.303107i	$0.901976\pi$
$128$	2.38667	0.210954
$129$	4.68176	0.412206
$130$	−6.44419	−0.565193
$131$	−9.49902	−0.829934	−0.414967	−	0.909837i	$-0.636207\pi$
$131$	−9.49902	−0.829934	−0.414967	+	0.909837i	$0.636207\pi$
$132$	−0.844485	−0.0735030
$133$	10.9640	0.950704
$134$	21.3015	1.84017
$135$	17.3568	1.49384
$136$	1.61125	0.138163
$137$	9.39725	0.802862	0.401431	−	0.915889i	$-0.368513\pi$
$137$	9.39725	0.802862	0.401431	+	0.915889i	$0.368513\pi$
$138$	−11.8125	−1.00555
$139$	17.3835	1.47445	0.737226	−	0.675646i	$-0.236135\pi$
$139$	17.3835	1.47445	0.737226	+	0.675646i	$0.236135\pi$
$140$	24.9477	2.10846
$141$	9.12200	0.768211
$142$	18.7777	1.57579
$143$	0.325684	0.0272351
$144$	5.67910	0.473258
$145$	−26.9457	−2.23772
$146$	8.07849	0.668580
$147$	−7.83133	−0.645917
$148$	13.9096	1.14336
$149$	2.72607	0.223329	0.111664	−	0.993746i	$-0.464382\pi$
$149$	2.72607	0.223329	0.111664	+	0.993746i	$0.464382\pi$
$150$	12.3323	1.00692
$151$	10.7122	0.871743	0.435871	−	0.900009i	$-0.356440\pi$
$151$	10.7122	0.871743	0.435871	+	0.900009i	$0.356440\pi$
$152$	−0.893173	−0.0724459
$153$	8.30180	0.671161
$154$	−2.43540	−0.196250
$155$	10.4491	0.839292
$156$	−2.59296	−0.207603
$157$	−16.4675	−1.31425	−0.657126	−	0.753781i	$-0.728228\pi$
$157$	−16.4675	−1.31425	−0.657126	+	0.753781i	$0.728228\pi$
$158$	30.5049	2.42684
$159$	2.00814	0.159256
$160$	25.6377	2.02684
$161$	−17.6364	−1.38994
$162$	4.07385	0.320072
$163$	21.6294	1.69414	0.847072	−	0.531478i	$-0.178363\pi$
$163$	21.6294	1.69414	0.847072	+	0.531478i	$0.178363\pi$
$164$	2.46011	0.192102
$165$	−1.24477	−0.0969049
$166$	8.50595	0.660190
$167$	−23.8419	−1.84495	−0.922473	−	0.386062i	$-0.873835\pi$
$167$	−23.8419	−1.84495	−0.922473	+	0.386062i	$0.873835\pi$
$168$	1.32669	0.102357
$169$	1.00000	0.0769231
$170$	34.7101	2.66214
$171$	−4.60199	−0.351923
$172$	−8.32218	−0.634560
$173$	8.88297	0.675360	0.337680	−	0.941261i	$-0.390358\pi$
$173$	8.88297	0.675360	0.337680	+	0.941261i	$0.390358\pi$
$174$	−20.9425	−1.58764
$175$	18.4123	1.39184
$176$	−1.20003	−0.0904553
$177$	8.74355	0.657205
$178$	0.454361	0.0340558
$179$	14.2881	1.06794	0.533972	−	0.845502i	$-0.320699\pi$
$179$	14.2881	1.06794	0.533972	+	0.845502i	$0.320699\pi$
$180$	−10.4714	−0.780491
$181$	−19.5901	−1.45612	−0.728059	−	0.685515i	$-0.759577\pi$
$181$	−19.5901	−1.45612	−0.728059	+	0.685515i	$0.759577\pi$
$182$	−7.47779	−0.554290
$183$	−8.87143	−0.655795
$184$	1.43673	0.105917
$185$	20.5026	1.50738
$186$	8.12115	0.595472
$187$	−1.75422	−0.128281
$188$	−16.2150	−1.18260
$189$	20.1407	1.46502
$190$	−19.2410	−1.39589
$191$	−19.4316	−1.40602	−0.703011	−	0.711179i	$-0.748162\pi$
$191$	−19.4316	−1.40602	−0.703011	+	0.711179i	$0.748162\pi$
$192$	11.0255	0.795700
$193$	−23.9115	−1.72119	−0.860593	−	0.509293i	$-0.829907\pi$
$193$	−23.9115	−1.72119	−0.860593	+	0.509293i	$0.829907\pi$
$194$	18.1397	1.30235
$195$	−3.82200	−0.273699
$196$	13.9208	0.994340
$197$	−7.53785	−0.537050	−0.268525	−	0.963273i	$-0.586536\pi$
$197$	−7.53785	−0.537050	−0.268525	+	0.963273i	$0.586536\pi$
$198$	1.02222	0.0726460
$199$	8.15967	0.578424	0.289212	−	0.957265i	$-0.406607\pi$
$199$	8.15967	0.578424	0.289212	+	0.957265i	$0.406607\pi$
$200$	−1.49994	−0.106062
$201$	12.6338	0.891118
$202$	−4.45840	−0.313692
$203$	−31.2676	−2.19455
$204$	13.9663	0.977837
$205$	3.62619	0.253264
$206$	−19.5793	−1.36415
$207$	7.40259	0.514516
$208$	−3.68463	−0.255483
$209$	0.972426	0.0672641
$210$	28.5801	1.97221
$211$	1.82738	0.125802	0.0629009	−	0.998020i	$-0.479965\pi$
$211$	1.82738	0.125802	0.0629009	+	0.998020i	$0.479965\pi$
$212$	−3.56962	−0.245163
$213$	11.1369	0.763088
$214$	2.25661	0.154258
$215$	−12.2668	−0.836592
$216$	−1.64074	−0.111638
$217$	12.1251	0.823102
$218$	1.54887	0.104902
$219$	4.79129	0.323765
$220$	2.21266	0.149178
$221$	−5.38625	−0.362319
$222$	15.9348	1.06948
$223$	20.6252	1.38116	0.690582	−	0.723254i	$-0.257354\pi$
$223$	20.6252	1.38116	0.690582	+	0.723254i	$0.257354\pi$
$224$	29.7498	1.98774
$225$	−7.72828	−0.515219
$226$	0.764919	0.0508816
$227$	18.5219	1.22934	0.614671	−	0.788784i	$-0.289289\pi$
$227$	18.5219	1.22934	0.614671	+	0.788784i	$0.289289\pi$
$228$	−7.74202	−0.512728
$229$	−16.1829	−1.06940	−0.534698	−	0.845043i	$-0.679574\pi$
$229$	−16.1829	−1.06940	−0.534698	+	0.845043i	$0.679574\pi$
$230$	30.9504	2.04081
$231$	−1.44442	−0.0950356
$232$	2.54717	0.167230
$233$	14.2178	0.931437	0.465718	−	0.884933i	$-0.345796\pi$
$233$	14.2178	0.931437	0.465718	+	0.884933i	$0.345796\pi$
$234$	3.13868	0.205182
$235$	−23.9009	−1.55912
$236$	−15.5423	−1.01172
$237$	18.0922	1.17522
$238$	40.2773	2.61079
$239$	−11.6325	−0.752444	−0.376222	−	0.926529i	$-0.622777\pi$
$239$	−11.6325	−0.752444	−0.376222	+	0.926529i	$0.622777\pi$
$240$	14.0826	0.909031
$241$	6.93142	0.446492	0.223246	−	0.974762i	$-0.428335\pi$
$241$	6.93142	0.446492	0.223246	+	0.974762i	$0.428335\pi$
$242$	22.1843	1.42606
$243$	−14.0383	−0.900559
$244$	15.7696	1.00955
$245$	20.5191	1.31092
$246$	2.81831	0.179689
$247$	2.98579	0.189981
$248$	−0.987752	−0.0627223
$249$	5.04481	0.319702
$250$	−0.0911658	−0.00576583
$251$	−2.80608	−0.177118	−0.0885591	−	0.996071i	$-0.528226\pi$
$251$	−2.80608	−0.177118	−0.0885591	+	0.996071i	$0.528226\pi$
$252$	−12.1509	−0.765435
$253$	−1.56421	−0.0983410
$254$	43.7388	2.74441
$255$	20.5863	1.28916
$256$	13.3975	0.837344
$257$	−17.7374	−1.10643	−0.553215	−	0.833039i	$-0.686599\pi$
$257$	−17.7374	−1.10643	−0.553215	+	0.833039i	$0.686599\pi$
$258$	−9.53391	−0.593556
$259$	23.7911	1.47830
$260$	6.79389	0.421339
$261$	13.1241	0.812359
$262$	19.3437	1.19506
$263$	4.00485	0.246950	0.123475	−	0.992348i	$-0.460596\pi$
$263$	4.00485	0.246950	0.123475	+	0.992348i	$0.460596\pi$
$264$	0.117668	0.00724194
$265$	−5.26160	−0.323218
$266$	−22.3271	−1.36896
$267$	0.269478	0.0164918
$268$	−22.4575	−1.37181
$269$	4.66055	0.284159	0.142079	−	0.989855i	$-0.454621\pi$
$269$	4.66055	0.284159	0.142079	+	0.989855i	$0.454621\pi$
$270$	−35.3453	−2.15105
$271$	−1.74512	−0.106008	−0.0530041	−	0.998594i	$-0.516880\pi$
$271$	−1.74512	−0.106008	−0.0530041	+	0.998594i	$0.516880\pi$
$272$	19.8463	1.20336
$273$	−4.43502	−0.268419
$274$	−19.1365	−1.15608
$275$	1.63303	0.0984754
$276$	12.4535	0.749615
$277$	−12.3319	−0.740951	−0.370476	−	0.928842i	$-0.620805\pi$
$277$	−12.3319	−0.740951	−0.370476	+	0.928842i	$0.620805\pi$
$278$	−35.3997	−2.12313
$279$	−5.08930	−0.304688
$280$	−3.47612	−0.207738
$281$	20.4775	1.22159	0.610793	−	0.791790i	$-0.290851\pi$
$281$	20.4775	1.22159	0.610793	+	0.791790i	$0.290851\pi$
$282$	−18.5760	−1.10618
$283$	1.37089	0.0814911	0.0407456	−	0.999170i	$-0.487027\pi$
$283$	1.37089	0.0814911	0.0407456	+	0.999170i	$0.487027\pi$
$284$	−19.7967	−1.17472
$285$	−11.4117	−0.675971
$286$	−0.663222	−0.0392171
$287$	4.20780	0.248378
$288$	−12.4870	−0.735804
$289$	12.0117	0.706572
$290$	54.8721	3.22220
$291$	10.7585	0.630675
$292$	−8.51686	−0.498412
$293$	12.4523	0.727472	0.363736	−	0.931502i	$-0.381501\pi$
$293$	12.4523	0.727472	0.363736	+	0.931502i	$0.381501\pi$
$294$	15.9477	0.930087
$295$	−22.9093	−1.33383
$296$	−1.93811	−0.112650
$297$	1.78633	0.103653
$298$	−5.55136	−0.321581
$299$	−4.80284	−0.277755
$300$	−13.0015	−0.750640
$301$	−14.2343	−0.820454
$302$	−21.8142	−1.25526
$303$	−2.64424	−0.151908
$304$	−11.0015	−0.630981
$305$	23.2443	1.33097
$306$	−16.9057	−0.966437
$307$	18.2623	1.04228	0.521142	−	0.853470i	$-0.325506\pi$
$307$	18.2623	1.04228	0.521142	+	0.853470i	$0.325506\pi$
$308$	2.56756	0.146300
$309$	−11.6123	−0.660601
$310$	−21.2785	−1.20854
$311$	16.1164	0.913878	0.456939	−	0.889498i	$-0.348946\pi$
$311$	16.1164	0.913878	0.456939	+	0.889498i	$0.348946\pi$
$312$	0.361293	0.0204542
$313$	24.7079	1.39657	0.698287	−	0.715818i	$-0.253946\pi$
$313$	24.7079	1.39657	0.698287	+	0.715818i	$0.253946\pi$
$314$	33.5343	1.89245
$315$	−17.9104	−1.00913
$316$	−32.1603	−1.80916
$317$	−7.78108	−0.437029	−0.218514	−	0.975834i	$-0.570121\pi$
$317$	−7.78108	−0.437029	−0.218514	+	0.975834i	$0.570121\pi$
$318$	−4.08937	−0.229321
$319$	−2.77319	−0.155269
$320$	−28.8884	−1.61491
$321$	1.33838	0.0747008
$322$	35.9146	2.00144
$323$	−16.0822	−0.894839
$324$	−4.29492	−0.238607
$325$	5.01415	0.278135
$326$	−44.0459	−2.43948
$327$	0.918620	0.0507998
$328$	−0.342783	−0.0189270
$329$	−27.7343	−1.52904
$330$	2.53483	0.139538
$331$	−6.47813	−0.356070	−0.178035	−	0.984024i	$-0.556974\pi$
$331$	−6.47813	−0.356070	−0.178035	+	0.984024i	$0.556974\pi$
$332$	−8.96752	−0.492157
$333$	−9.98592	−0.547225
$334$	48.5516	2.65662
$335$	−33.1022	−1.80856
$336$	16.3414	0.891495
$337$	6.73009	0.366611	0.183306	−	0.983056i	$-0.441320\pi$
$337$	6.73009	0.366611	0.183306	+	0.983056i	$0.441320\pi$
$338$	−2.03639	−0.110765
$339$	0.453667	0.0246398
$340$	−36.5936	−1.98457
$341$	1.07540	0.0582361
$342$	9.37145	0.506750
$343$	−1.89431	−0.102283
$344$	1.15958	0.0625205
$345$	18.3564	0.988278
$346$	−18.0892	−0.972482
$347$	31.7021	1.70186	0.850930	−	0.525279i	$-0.176039\pi$
$347$	31.7021	1.70186	0.850930	+	0.525279i	$0.176039\pi$
$348$	22.0789	1.18355
$349$	−7.86184	−0.420835	−0.210417	−	0.977612i	$-0.567482\pi$
$349$	−7.86184	−0.420835	−0.210417	+	0.977612i	$0.567482\pi$
$350$	−37.4947	−2.00418
$351$	5.48483	0.292759
$352$	2.63858	0.140637
$353$	−9.13306	−0.486104	−0.243052	−	0.970013i	$-0.578149\pi$
$353$	−9.13306	−0.486104	−0.243052	+	0.970013i	$0.578149\pi$
$354$	−17.8053	−0.946341
$355$	−29.1802	−1.54872
$356$	−0.479017	−0.0253878
$357$	23.8881	1.26429
$358$	−29.0962	−1.53778
$359$	−19.0364	−1.00470	−0.502352	−	0.864663i	$-0.667532\pi$
$359$	−19.0364	−1.00470	−0.502352	+	0.864663i	$0.667532\pi$
$360$	1.45904	0.0768984
$361$	−10.0850	−0.530792
$362$	39.8931	2.09673
$363$	13.1573	0.690582
$364$	7.88357	0.413211
$365$	−12.5538	−0.657096
$366$	18.0657	0.944311
$367$	−34.4748	−1.79957	−0.899786	−	0.436332i	$-0.856277\pi$
$367$	−34.4748	−1.79957	−0.899786	+	0.436332i	$0.856277\pi$
$368$	17.6967	0.922502
$369$	−1.76616	−0.0919424
$370$	−41.7514	−2.17055
$371$	−6.10552	−0.316983
$372$	−8.56184	−0.443911
$373$	20.8863	1.08145	0.540725	−	0.841199i	$-0.318150\pi$
$373$	20.8863	1.08145	0.540725	+	0.841199i	$0.318150\pi$
$374$	3.57228	0.184718
$375$	−0.0540697	−0.00279215
$376$	2.25934	0.116517
$377$	−8.51496	−0.438543
$378$	−41.0144	−2.10955
$379$	27.5913	1.41727	0.708635	−	0.705576i	$-0.249312\pi$
$379$	27.5913	1.41727	0.708635	+	0.705576i	$0.249312\pi$
$380$	20.2851	1.04061
$381$	25.9411	1.32900
$382$	39.5704	2.02460
$383$	1.49392	0.0763359	0.0381680	−	0.999271i	$-0.487848\pi$
$383$	1.49392	0.0763359	0.0381680	+	0.999271i	$0.487848\pi$
$384$	−2.88255	−0.147099
$385$	3.78456	0.192879
$386$	48.6932	2.47842
$387$	5.97464	0.303708
$388$	−19.1240	−0.970876
$389$	23.5649	1.19479	0.597393	−	0.801948i	$-0.296203\pi$
$389$	23.5649	1.19479	0.597393	+	0.801948i	$0.296203\pi$
$390$	7.78310	0.394112
$391$	25.8693	1.30827
$392$	−1.93967	−0.0979681
$393$	11.4726	0.578717
$394$	15.3500	0.773324
$395$	−47.4040	−2.38516
$396$	−1.07769	−0.0541560
$397$	−18.8816	−0.947638	−0.473819	−	0.880622i	$-0.657125\pi$
$397$	−18.8816	−0.947638	−0.473819	+	0.880622i	$0.657125\pi$
$398$	−16.6163	−0.832900
$399$	−13.2420	−0.662931
$400$	−18.4753	−0.923763
$401$	13.6578	0.682038	0.341019	−	0.940056i	$-0.389228\pi$
$401$	13.6578	0.682038	0.341019	+	0.940056i	$0.389228\pi$
$402$	−25.7273	−1.28316
$403$	3.30196	0.164482
$404$	4.70033	0.233850
$405$	−6.33069	−0.314574
$406$	63.6731	3.16004
$407$	2.11008	0.104593
$408$	−1.94602	−0.0963421
$409$	4.14536	0.204975	0.102487	−	0.994734i	$-0.467320\pi$
$409$	4.14536	0.204975	0.102487	+	0.994734i	$0.467320\pi$
$410$	−7.38435	−0.364687
$411$	−11.3497	−0.559840
$412$	20.6417	1.01694
$413$	−26.5837	−1.30810
$414$	−15.0746	−0.740875
$415$	−13.2181	−0.648850
$416$	8.10163	0.397215
$417$	−20.9953	−1.02814
$418$	−1.98024	−0.0968568
$419$	−3.77620	−0.184480	−0.0922398	−	0.995737i	$-0.529403\pi$
$419$	−3.77620	−0.184480	−0.0922398	+	0.995737i	$0.529403\pi$
$420$	−30.1310	−1.47024
$421$	12.9866	0.632930	0.316465	−	0.948604i	$-0.397504\pi$
$421$	12.9866	0.632930	0.316465	+	0.948604i	$0.397504\pi$
$422$	−3.72125	−0.181148
$423$	11.6411	0.566007
$424$	0.497379	0.0241548
$425$	−27.0075	−1.31005
$426$	−22.6791	−1.09881
$427$	26.9725	1.30529
$428$	−2.37906	−0.114996
$429$	−0.393351	−0.0189912
$430$	24.9801	1.20465
$431$	−23.6577	−1.13955	−0.569776	−	0.821800i	$-0.692970\pi$
$431$	−23.6577	−1.13955	−0.569776	+	0.821800i	$0.692970\pi$
$432$	−20.2096	−0.972333
$433$	1.49669	0.0719265	0.0359633	−	0.999353i	$-0.488550\pi$
$433$	1.49669	0.0719265	0.0359633	+	0.999353i	$0.488550\pi$
$434$	−24.6914	−1.18522
$435$	32.5442	1.56037
$436$	−1.63291	−0.0782024
$437$	−14.3403	−0.685988
$438$	−9.75694	−0.466205
$439$	−30.6552	−1.46310	−0.731548	−	0.681790i	$-0.761202\pi$
$439$	−30.6552	−1.46310	−0.731548	+	0.681790i	$0.761202\pi$
$440$	−0.308305	−0.0146978
$441$	−9.99396	−0.475903
$442$	10.9685	0.521720
$443$	6.73875	0.320168	0.160084	−	0.987103i	$-0.448824\pi$
$443$	6.73875	0.320168	0.160084	+	0.987103i	$0.448824\pi$
$444$	−16.7995	−0.797271
$445$	−0.706067	−0.0334708
$446$	−42.0010	−1.98880
$447$	−3.29247	−0.155728
$448$	−33.5218	−1.58376
$449$	−16.9851	−0.801578	−0.400789	−	0.916170i	$-0.631264\pi$
$449$	−16.9851	−0.801578	−0.400789	+	0.916170i	$0.631264\pi$
$450$	15.7378	0.741888
$451$	0.373199	0.0175733
$452$	−0.806427	−0.0379311
$453$	−12.9378	−0.607871
$454$	−37.7179	−1.77019
$455$	11.6203	0.544769
$456$	1.07875	0.0505169
$457$	7.20357	0.336969	0.168484	−	0.985704i	$-0.446113\pi$
$457$	7.20357	0.336969	0.168484	+	0.985704i	$0.446113\pi$
$458$	32.9547	1.53987
$459$	−29.5427	−1.37894
$460$	−32.6299	−1.52138
$461$	11.3545	0.528830	0.264415	−	0.964409i	$-0.414821\pi$
$461$	11.3545	0.528830	0.264415	+	0.964409i	$0.414821\pi$
$462$	2.94140	0.136846
$463$	1.00000	0.0464739
$463$	1.00000	0.0464739
$464$	31.3744	1.45652
$465$	−12.6201	−0.585243
$466$	−28.9530	−1.34122
$467$	10.4290	0.482598	0.241299	−	0.970451i	$-0.422427\pi$
$467$	10.4290	0.482598	0.241299	+	0.970451i	$0.422427\pi$
$468$	−3.30900	−0.152959
$469$	−38.4115	−1.77368
$470$	48.6715	2.24505
$471$	19.8890	0.916434
$472$	2.16561	0.0996802
$473$	−1.26248	−0.0580487
$474$	−36.8429	−1.69225
$475$	14.9712	0.686926
$476$	−42.4629	−1.94628
$477$	2.56270	0.117338
$478$	23.6884	1.08348
$479$	−5.63936	−0.257669	−0.128835	−	0.991666i	$-0.541124\pi$
$479$	−5.63936	−0.257669	−0.128835	+	0.991666i	$0.541124\pi$
$480$	−30.9644	−1.41333
$481$	6.47891	0.295413
$482$	−14.1151	−0.642925
$483$	21.3007	0.969214
$484$	−23.3881	−1.06310
$485$	−28.1887	−1.27998
$486$	28.5876	1.29676
$487$	−32.3977	−1.46808	−0.734040	−	0.679107i	$-0.762367\pi$
$487$	−32.3977	−1.46808	−0.734040	+	0.679107i	$0.762367\pi$
$488$	−2.19728	−0.0994663
$489$	−26.1233	−1.18134
$490$	−41.7850	−1.88765
$491$	−6.11571	−0.275998	−0.137999	−	0.990432i	$-0.544067\pi$
$491$	−6.11571	−0.275998	−0.137999	+	0.990432i	$0.544067\pi$
$492$	−2.97124	−0.133954
$493$	45.8637	2.06560
$494$	−6.08025	−0.273563
$495$	−1.58851	−0.0713982
$496$	−12.1665	−0.546292
$497$	−33.8604	−1.51885
$498$	−10.2732	−0.460354
$499$	21.7925	0.975568	0.487784	−	0.872964i	$-0.337805\pi$
$499$	21.7925	0.975568	0.487784	+	0.872964i	$0.337805\pi$
$500$	0.0961128	0.00429830
$501$	28.7956	1.28649
$502$	5.71429	0.255041
$503$	5.60934	0.250108	0.125054	−	0.992150i	$-0.460090\pi$
$503$	5.60934	0.250108	0.125054	+	0.992150i	$0.460090\pi$
$504$	1.69306	0.0754150
$505$	6.92826	0.308304
$506$	3.18535	0.141606
$507$	−1.20777	−0.0536389
$508$	−46.1122	−2.04590
$509$	−42.5667	−1.88674	−0.943368	−	0.331749i	$-0.892361\pi$
$509$	−42.5667	−1.88674	−0.943368	+	0.331749i	$0.892361\pi$
$510$	−41.9217	−1.85632
$511$	−14.5673	−0.644420
$512$	−32.0559	−1.41669
$513$	16.3766	0.723043
$514$	36.1203	1.59320
$515$	30.4258	1.34072
$516$	10.0513	0.442482
$517$	−2.45982	−0.108183
$518$	−48.4479	−2.12868
$519$	−10.7286	−0.470932
$520$	−0.946636	−0.0415127
$521$	10.7733	0.471988	0.235994	−	0.971754i	$-0.424165\pi$
$521$	10.7733	0.471988	0.235994	+	0.971754i	$0.424165\pi$
$522$	−26.7258	−1.16975
$523$	−12.9543	−0.566450	−0.283225	−	0.959053i	$-0.591404\pi$
$523$	−12.9543	−0.566450	−0.283225	+	0.959053i	$0.591404\pi$
$524$	−20.3934	−0.890891
$525$	−22.2378	−0.970538
$526$	−8.15545	−0.355594
$527$	−17.7852	−0.774736
$528$	1.44935	0.0630750
$529$	0.0672478	0.00292382
$530$	10.7147	0.465417
$531$	11.1581	0.484220
$532$	23.5387	1.02053
$533$	1.14589	0.0496341
$534$	−0.548763	−0.0237473
$535$	−3.50672	−0.151609
$536$	3.12914	0.135158
$537$	−17.2567	−0.744683
$538$	−9.49072	−0.409174
$539$	2.11178	0.0909608
$540$	37.2633	1.60356
$541$	−5.89003	−0.253232	−0.126616	−	0.991952i	$-0.540412\pi$
$541$	−5.89003	−0.253232	−0.126616	+	0.991952i	$0.540412\pi$
$542$	3.55374	0.152646
$543$	23.6603	1.01536
$544$	−43.6374	−1.87094
$545$	−2.40691	−0.103100
$546$	9.03144	0.386510
$547$	18.2027	0.778293	0.389146	−	0.921176i	$-0.372770\pi$
$547$	18.2027	0.778293	0.389146	+	0.921176i	$0.372770\pi$
$548$	20.1749	0.861831
$549$	−11.3213	−0.483181
$550$	−3.32549	−0.141799
$551$	−25.4239	−1.08309
$552$	−1.73523	−0.0738564
$553$	−55.0072	−2.33915
$554$	25.1126	1.06693
$555$	−24.7624	−1.05111
$556$	37.3207	1.58275
$557$	29.2708	1.24024	0.620121	−	0.784506i	$-0.287083\pi$
$557$	29.2708	1.24024	0.620121	+	0.784506i	$0.287083\pi$
$558$	10.3638	0.438735
$559$	−3.87638	−0.163953
$560$	−42.8166	−1.80933
$561$	2.11869	0.0894512
$562$	−41.7003	−1.75902
$563$	−22.9899	−0.968910	−0.484455	−	0.874816i	$-0.660982\pi$
$563$	−22.9899	−0.968910	−0.484455	+	0.874816i	$0.660982\pi$
$564$	19.5840	0.824635
$565$	−1.18867	−0.0500076
$566$	−2.79168	−0.117343
$567$	−7.34607	−0.308506
$568$	2.75840	0.115740
$569$	−15.3968	−0.645466	−0.322733	−	0.946490i	$-0.604602\pi$
$569$	−15.3968	−0.645466	−0.322733	+	0.946490i	$0.604602\pi$
$570$	23.2387	0.973362
$571$	1.66828	0.0698154	0.0349077	−	0.999391i	$-0.488886\pi$
$571$	1.66828	0.0698154	0.0349077	+	0.999391i	$0.488886\pi$
$572$	0.699211	0.0292355
$573$	23.4689	0.980427
$574$	−8.56873	−0.357652
$575$	−24.0821	−1.00429
$576$	14.0703	0.586261
$577$	10.8345	0.451047	0.225523	−	0.974238i	$-0.427591\pi$
$577$	10.8345	0.451047	0.225523	+	0.974238i	$0.427591\pi$
$578$	−24.4606	−1.01743
$579$	28.8795	1.20019
$580$	−57.8497	−2.40208
$581$	−15.3381	−0.636333
$582$	−21.9086	−0.908139
$583$	−0.541512	−0.0224271
$584$	1.18671	0.0491064
$585$	−4.87745	−0.201658
$586$	−25.3578	−1.04752
$587$	−35.2224	−1.45378	−0.726892	−	0.686751i	$-0.759036\pi$
$587$	−35.2224	−1.45378	−0.726892	+	0.686751i	$0.759036\pi$
$588$	−16.8131	−0.693359
$589$	9.85897	0.406232
$590$	46.6523	1.92064
$591$	9.10398	0.374488
$592$	−23.8724	−0.981149
$593$	−33.5346	−1.37710	−0.688551	−	0.725188i	$-0.741753\pi$
$593$	−33.5346	−1.37710	−0.688551	+	0.725188i	$0.741753\pi$
$594$	−3.63766	−0.149255
$595$	−62.5900	−2.56594
$596$	5.85260	0.239732
$597$	−9.85500	−0.403338
$598$	9.78047	0.399953
$599$	23.3052	0.952223	0.476112	−	0.879385i	$-0.342046\pi$
$599$	23.3052	0.952223	0.476112	+	0.879385i	$0.342046\pi$
$600$	1.81158	0.0739573
$601$	34.3096	1.39952	0.699759	−	0.714379i	$-0.253291\pi$
$601$	34.3096	1.39952	0.699759	+	0.714379i	$0.253291\pi$
$602$	28.9867	1.18141
$603$	16.1226	0.656563
$604$	22.9979	0.935771
$605$	−34.4740	−1.40157
$606$	5.38471	0.218739
$607$	3.25199	0.131994	0.0659972	−	0.997820i	$-0.478977\pi$
$607$	3.25199	0.131994	0.0659972	+	0.997820i	$0.478977\pi$
$608$	24.1898	0.981025
$609$	37.7640	1.53027
$610$	−47.3346	−1.91652
$611$	−7.55277	−0.305552
$612$	17.8231	0.720457
$613$	40.9545	1.65414	0.827069	−	0.562101i	$-0.190007\pi$
$613$	40.9545	1.65414	0.827069	+	0.562101i	$0.190007\pi$
$614$	−37.1892	−1.50083
$615$	−4.37960	−0.176602
$616$	−0.357754	−0.0144143
$617$	2.51153	0.101110	0.0505552	−	0.998721i	$-0.483901\pi$
$617$	2.51153	0.101110	0.0505552	+	0.998721i	$0.483901\pi$
$618$	23.6472	0.951231
$619$	44.6593	1.79501	0.897505	−	0.441005i	$-0.145378\pi$
$619$	44.6593	1.79501	0.897505	+	0.441005i	$0.145378\pi$
$620$	22.4332	0.900937
$621$	−26.3428	−1.05710
$622$	−32.8194	−1.31594
$623$	−0.819315	−0.0328251
$624$	4.45018	0.178150
$625$	−24.9291	−0.997163
$626$	−50.3150	−2.01099
$627$	−1.17447	−0.0469036
$628$	−35.3541	−1.41078
$629$	−34.8971	−1.39144
$630$	36.4725	1.45310
$631$	0.155159	0.00617678	0.00308839	−	0.999995i	$-0.499017\pi$
$631$	0.155159	0.00617678	0.00308839	+	0.999995i	$0.499017\pi$
$632$	4.48110	0.178248
$633$	−2.20705	−0.0877222
$634$	15.8453	0.629299
$635$	−67.9692	−2.69727
$636$	4.31128	0.170953
$637$	6.48413	0.256911
$638$	5.64731	0.223579
$639$	14.2124	0.562233
$640$	7.55266	0.298545
$641$	−20.0369	−0.791410	−0.395705	−	0.918378i	$-0.629500\pi$
$641$	−20.0369	−0.791410	−0.395705	+	0.918378i	$0.629500\pi$
$642$	−2.72546	−0.107565
$643$	34.1066	1.34503	0.672516	−	0.740082i	$-0.265213\pi$
$643$	34.1066	1.34503	0.672516	+	0.740082i	$0.265213\pi$
$644$	−37.8635	−1.49203
$645$	14.8155	0.583360
$646$	32.7497	1.28852
$647$	−1.07534	−0.0422760	−0.0211380	−	0.999777i	$-0.506729\pi$
$647$	−1.07534	−0.0422760	−0.0211380	+	0.999777i	$0.506729\pi$
$648$	0.598439	0.0235089
$649$	−2.35777	−0.0925505
$650$	−10.2108	−0.400500
$651$	−14.6443	−0.573954
$652$	46.4361	1.81858
$653$	−39.5979	−1.54959	−0.774793	−	0.632215i	$-0.782146\pi$
$653$	−39.5979	−1.54959	−0.774793	+	0.632215i	$0.782146\pi$
$654$	−1.87067	−0.0731490
$655$	−30.0598	−1.17453
$656$	−4.22218	−0.164848
$657$	6.11440	0.238546
$658$	56.4780	2.20174
$659$	−8.20392	−0.319579	−0.159790	−	0.987151i	$-0.551082\pi$
$659$	−8.20392	−0.319579	−0.159790	+	0.987151i	$0.551082\pi$
$660$	−2.67239	−0.104022
$661$	−45.5675	−1.77237	−0.886186	−	0.463330i	$-0.846655\pi$
$661$	−45.5675	−1.77237	−0.886186	+	0.463330i	$0.846655\pi$
$662$	13.1920	0.512722
$663$	6.50535	0.252647
$664$	1.24950	0.0484901
$665$	34.6959	1.34545
$666$	20.3353	0.787975
$667$	40.8960	1.58350
$668$	−51.1862	−1.98045
$669$	−24.9104	−0.963093
$670$	67.4090	2.60424
$671$	2.39225	0.0923519
$672$	−35.9309	−1.38606
$673$	4.58451	0.176720	0.0883600	−	0.996089i	$-0.471837\pi$
$673$	4.58451	0.176720	0.0883600	+	0.996089i	$0.471837\pi$
$674$	−13.7051	−0.527901
$675$	27.5018	1.05854
$676$	2.14690	0.0825730
$677$	−23.2884	−0.895048	−0.447524	−	0.894272i	$-0.647694\pi$
$677$	−23.2884	−0.895048	−0.447524	+	0.894272i	$0.647694\pi$
$678$	−0.923845	−0.0354800
$679$	−32.7099	−1.25529
$680$	5.09882	0.195531
$681$	−22.3702	−0.857227
$682$	−2.18993	−0.0838569
$683$	24.8311	0.950134	0.475067	−	0.879950i	$-0.342424\pi$
$683$	24.8311	0.950134	0.475067	+	0.879950i	$0.342424\pi$
$684$	−9.87999	−0.377771
$685$	29.7377	1.13622
$686$	3.85756	0.147282
$687$	19.5452	0.745695
$688$	14.2830	0.544534
$689$	−1.66269	−0.0633434
$690$	−37.3809	−1.42307
$691$	14.6146	0.555967	0.277984	−	0.960586i	$-0.410334\pi$
$691$	14.6146	0.555967	0.277984	+	0.960586i	$0.410334\pi$
$692$	19.0708	0.724964
$693$	−1.84329	−0.0700209
$694$	−64.5580	−2.45059
$695$	55.0104	2.08666
$696$	−3.07640	−0.116610
$697$	−6.17206	−0.233783
$698$	16.0098	0.605980
$699$	−17.1718	−0.649496
$700$	39.5294	1.49407
$701$	−50.0742	−1.89127	−0.945637	−	0.325223i	$-0.894561\pi$
$701$	−50.0742	−1.89127	−0.945637	+	0.325223i	$0.894561\pi$
$702$	−11.1693	−0.421557
$703$	19.3447	0.729599
$704$	−2.97313	−0.112054
$705$	28.8667	1.08718
$706$	18.5985	0.699964
$707$	8.03949	0.302356
$708$	18.7715	0.705476
$709$	−18.1534	−0.681766	−0.340883	−	0.940106i	$-0.610726\pi$
$709$	−18.1534	−0.681766	−0.340883	+	0.940106i	$0.610726\pi$
$710$	59.4223	2.23008
$711$	23.0884	0.865883
$712$	0.0667444	0.00250135
$713$	−15.8588	−0.593916
$714$	−48.6456	−1.82052
$715$	1.03063	0.0385435
$716$	30.6751	1.14638
$717$	14.0494	0.524684
$718$	38.7656	1.44672
$719$	−29.9414	−1.11663	−0.558313	−	0.829631i	$-0.688551\pi$
$719$	−29.9414	−1.11663	−0.558313	+	0.829631i	$0.688551\pi$
$720$	17.9716	0.669761
$721$	35.3058	1.31486
$722$	20.5371	0.764313
$723$	−8.37155	−0.311341
$724$	−42.0578	−1.56307
$725$	−42.6953	−1.58566
$726$	−26.7935	−0.994401
$727$	18.1382	0.672710	0.336355	−	0.941735i	$-0.390806\pi$
$727$	18.1382	0.672710	0.336355	+	0.941735i	$0.390806\pi$
$728$	−1.09847	−0.0407119
$729$	22.9566	0.850245
$730$	25.5645	0.946184
$731$	20.8791	0.772243
$732$	−19.0461	−0.703962
$733$	−20.5141	−0.757705	−0.378852	−	0.925457i	$-0.623681\pi$
$733$	−20.5141	−0.757705	−0.378852	+	0.925457i	$0.623681\pi$
$734$	70.2043	2.59129
$735$	−24.7823	−0.914111
$736$	−38.9108	−1.43427
$737$	−3.40680	−0.125491
$738$	3.59659	0.132392
$739$	−29.4397	−1.08296	−0.541479	−	0.840715i	$-0.682135\pi$
$739$	−29.4397	−1.08296	−0.541479	+	0.840715i	$0.682135\pi$
$740$	44.0170	1.61810
$741$	−3.60615	−0.132475
$742$	12.4332	0.456439
$743$	10.9248	0.400791	0.200396	−	0.979715i	$-0.435777\pi$
$743$	10.9248	0.400791	0.200396	+	0.979715i	$0.435777\pi$
$744$	1.19298	0.0437366
$745$	8.62670	0.316058
$746$	−42.5327	−1.55723
$747$	6.43794	0.235552
$748$	−3.76613	−0.137703
$749$	−4.06917	−0.148684
$750$	0.110107	0.00402054
$751$	47.3938	1.72942	0.864712	−	0.502268i	$-0.167501\pi$
$751$	47.3938	1.72942	0.864712	+	0.502268i	$0.167501\pi$
$752$	27.8291	1.01482
$753$	3.38910	0.123506
$754$	17.3398	0.631479
$755$	33.8988	1.23370
$756$	43.2401	1.57263
$757$	20.6881	0.751923	0.375961	−	0.926635i	$-0.377312\pi$
$757$	20.6881	0.751923	0.375961	+	0.926635i	$0.377312\pi$
$758$	−56.1867	−2.04079
$759$	1.88920	0.0685737
$760$	−2.82646	−0.102526
$761$	3.15824	0.114486	0.0572431	−	0.998360i	$-0.481769\pi$
$761$	3.15824	0.114486	0.0572431	+	0.998360i	$0.481769\pi$
$762$	−52.8263	−1.91370
$763$	−2.79295	−0.101112
$764$	−41.7177	−1.50929
$765$	26.2712	0.949836
$766$	−3.04222	−0.109920
$767$	−7.23942	−0.261400
$768$	−16.1811	−0.583884
$769$	6.44023	0.232241	0.116120	−	0.993235i	$-0.462954\pi$
$769$	6.44023	0.232241	0.116120	+	0.993235i	$0.462954\pi$
$770$	−7.70685	−0.277736
$771$	21.4227	0.771519
$772$	−51.3355	−1.84761
$773$	3.65404	0.131427	0.0657133	−	0.997839i	$-0.479068\pi$
$773$	3.65404	0.131427	0.0657133	+	0.997839i	$0.479068\pi$
$774$	−12.1667	−0.437324
$775$	16.5565	0.594728
$776$	2.66467	0.0956562
$777$	−28.7341	−1.03083
$778$	−47.9874	−1.72043
$779$	3.42139	0.122584
$780$	−8.20544	−0.293802
$781$	−3.00316	−0.107461
$782$	−52.6801	−1.88384
$783$	−46.7031	−1.66903
$784$	−23.8916	−0.853271
$785$	−52.1117	−1.85995
$786$	−23.3628	−0.833323
$787$	22.6759	0.808308	0.404154	−	0.914691i	$-0.367566\pi$
$787$	22.6759	0.808308	0.404154	+	0.914691i	$0.367566\pi$
$788$	−16.1830	−0.576495
$789$	−4.83693	−0.172199
$790$	96.5333	3.43450
$791$	−1.37932	−0.0490430
$792$	0.150162	0.00533576
$793$	7.34531	0.260840
$794$	38.4503	1.36455
$795$	6.35480	0.225382
$796$	17.5180	0.620908
$797$	25.2459	0.894256	0.447128	−	0.894470i	$-0.352447\pi$
$797$	25.2459	0.894256	0.447128	+	0.894470i	$0.352447\pi$
$798$	26.9660	0.954586
$799$	40.6811	1.43920
$800$	40.6228	1.43623
$801$	0.343894	0.0121509
$802$	−27.8126	−0.982099
$803$	−1.29201	−0.0455940
$804$	27.1234	0.956569
$805$	−55.8106	−1.96706
$806$	−6.72410	−0.236846
$807$	−5.62887	−0.198146
$808$	−0.654927	−0.0230403
$809$	−16.1100	−0.566396	−0.283198	−	0.959061i	$-0.591395\pi$
$809$	−16.1100	−0.566396	−0.283198	+	0.959061i	$0.591395\pi$
$810$	12.8918	0.452971
$811$	27.7160	0.973239	0.486620	−	0.873614i	$-0.338230\pi$
$811$	27.7160	0.973239	0.486620	+	0.873614i	$0.338230\pi$
$812$	−67.1283	−2.35574
$813$	2.10770	0.0739202
$814$	−4.29696	−0.150608
$815$	68.4465	2.39758
$816$	−23.9698	−0.839110
$817$	−11.5741	−0.404925
$818$	−8.44159	−0.295153
$819$	−5.65975	−0.197768
$820$	7.78505	0.271866
$821$	−47.9281	−1.67270	−0.836352	−	0.548193i	$-0.815316\pi$
$821$	−47.9281	−1.67270	−0.836352	+	0.548193i	$0.815316\pi$
$822$	23.1125	0.806140
$823$	8.70562	0.303459	0.151729	−	0.988422i	$-0.451516\pi$
$823$	8.70562	0.303459	0.151729	+	0.988422i	$0.451516\pi$
$824$	−2.87614	−0.100195
$825$	−1.97232	−0.0686674
$826$	54.1349	1.88359
$827$	28.2245	0.981462	0.490731	−	0.871311i	$-0.336730\pi$
$827$	28.2245	0.981462	0.490731	+	0.871311i	$0.336730\pi$
$828$	15.8926	0.552306
$829$	−20.4121	−0.708943	−0.354472	−	0.935067i	$-0.615339\pi$
$829$	−20.4121	−0.708943	−0.354472	+	0.935067i	$0.615339\pi$
$830$	26.9172	0.934310
$831$	14.8941	0.516669
$832$	−9.12885	−0.316486
$833$	−34.9252	−1.21009
$834$	42.7547	1.48047
$835$	−75.4482	−2.61099
$836$	2.08770	0.0722046
$837$	18.1107	0.625998
$838$	7.68984	0.265641
$839$	0.349416	0.0120632	0.00603158	−	0.999982i	$-0.498080\pi$
$839$	0.349416	0.0120632	0.00603158	+	0.999982i	$0.498080\pi$
$840$	4.19834	0.144857
$841$	43.5045	1.50016
$842$	−26.4459	−0.911386
$843$	−24.7321	−0.851819
$844$	3.92319	0.135042
$845$	3.16451	0.108863
$846$	−23.7058	−0.815021
$847$	−40.0033	−1.37453
$848$	6.12639	0.210381
$849$	−1.65572	−0.0568242
$850$	54.9978	1.88641
$851$	−31.1172	−1.06668
$852$	23.9098	0.819136
$853$	−32.8259	−1.12394	−0.561968	−	0.827159i	$-0.689956\pi$
$853$	−32.8259	−1.12394	−0.561968	+	0.827159i	$0.689956\pi$
$854$	−54.9267	−1.87955
$855$	−14.5630	−0.498046
$856$	0.331490	0.0113301
$857$	34.6302	1.18295	0.591473	−	0.806325i	$-0.298547\pi$
$857$	34.6302	1.18295	0.591473	+	0.806325i	$0.298547\pi$
$858$	0.801018	0.0273463
$859$	48.3834	1.65082	0.825410	−	0.564534i	$-0.190944\pi$
$859$	48.3834	1.65082	0.825410	+	0.564534i	$0.190944\pi$
$860$	−26.3357	−0.898038
$861$	−5.08205	−0.173196
$862$	48.1764	1.64089
$863$	27.8819	0.949110	0.474555	−	0.880226i	$-0.342609\pi$
$863$	27.8819	0.949110	0.474555	+	0.880226i	$0.342609\pi$
$864$	44.4361	1.51175
$865$	28.1103	0.955778
$866$	−3.04786	−0.103570
$867$	−14.5074	−0.492696
$868$	26.0312	0.883558
$869$	−4.87872	−0.165499
$870$	−66.2727	−2.24686
$871$	−10.4604	−0.354438
$872$	0.227524	0.00770495
$873$	13.7295	0.464672
$874$	29.2024	0.987787
$875$	0.164392	0.00555748
$876$	10.2864	0.347545
$877$	−16.8052	−0.567471	−0.283736	−	0.958903i	$-0.591574\pi$
$877$	−16.8052	−0.567471	−0.283736	+	0.958903i	$0.591574\pi$
$878$	62.4261	2.10678
$879$	−15.0395	−0.507270
$880$	−3.79750	−0.128014
$881$	6.95008	0.234154	0.117077	−	0.993123i	$-0.462648\pi$
$881$	6.95008	0.234154	0.117077	+	0.993123i	$0.462648\pi$
$882$	20.3516	0.685275
$883$	28.0030	0.942376	0.471188	−	0.882033i	$-0.343825\pi$
$883$	28.0030	0.942376	0.471188	+	0.882033i	$0.343825\pi$
$884$	−11.5637	−0.388930
$885$	27.6691	0.930086
$886$	−13.7227	−0.461024
$887$	50.2416	1.68695	0.843475	−	0.537169i	$-0.180506\pi$
$887$	50.2416	1.68695	0.843475	+	0.537169i	$0.180506\pi$
$888$	2.34079	0.0785517
$889$	−78.8708	−2.64524
$890$	1.43783	0.0481962
$891$	−0.651540	−0.0218274
$892$	44.2801	1.48261
$893$	−22.5510	−0.754641
$894$	6.70475	0.224241
$895$	45.2149	1.51137
$896$	8.76404	0.292786
$897$	5.80072	0.193680
$898$	34.5884	1.15423
$899$	−28.1161	−0.937724
$900$	−16.5918	−0.553061
$901$	8.95567	0.298357
$902$	−0.759980	−0.0253046
$903$	17.1918	0.572107
$904$	0.112365	0.00373719
$905$	−61.9930	−2.06072
$906$	26.3465	0.875302
$907$	52.1191	1.73059	0.865293	−	0.501267i	$-0.167133\pi$
$907$	52.1191	1.73059	0.865293	+	0.501267i	$0.167133\pi$
$908$	39.7646	1.31964
$909$	−3.37445	−0.111923
$910$	−23.6636	−0.784439
$911$	−48.6531	−1.61195	−0.805974	−	0.591951i	$-0.798358\pi$
$911$	−48.6531	−1.61195	−0.805974	+	0.591951i	$0.798358\pi$
$912$	13.2873	0.439986
$913$	−1.36037	−0.0450218
$914$	−14.6693	−0.485217
$915$	−28.0738	−0.928090
$916$	−34.7430	−1.14794
$917$	−34.8811	−1.15188
$918$	60.1606	1.98559
$919$	−16.4346	−0.542128	−0.271064	−	0.962561i	$-0.587376\pi$
$919$	−16.4346	−0.542128	−0.271064	+	0.962561i	$0.587376\pi$
$920$	4.54654	0.149895
$921$	−22.0566	−0.726791
$922$	−23.1222	−0.761488
$923$	−9.22106	−0.303515
$924$	−3.10101	−0.102016
$925$	32.4862	1.06814
$926$	−2.03639	−0.0669200
$927$	−14.8191	−0.486722
$928$	−68.9851	−2.26455
$929$	−8.18395	−0.268507	−0.134253	−	0.990947i	$-0.542864\pi$
$929$	−8.18395	−0.268507	−0.134253	+	0.990947i	$0.542864\pi$
$930$	25.6995	0.842720
$931$	19.3603	0.634507
$932$	30.5241	0.999850
$933$	−19.4649	−0.637252
$934$	−21.2376	−0.694915
$935$	−5.55125	−0.181545
$936$	0.461064	0.0150704
$937$	−40.7435	−1.33103	−0.665516	−	0.746384i	$-0.731788\pi$
$937$	−40.7435	−1.33103	−0.665516	+	0.746384i	$0.731788\pi$
$938$	78.2208	2.55400
$939$	−29.8414	−0.973838
$940$	−51.3127	−1.67363
$941$	−20.8235	−0.678826	−0.339413	−	0.940637i	$-0.610228\pi$
$941$	−20.8235	−0.678826	−0.339413	+	0.940637i	$0.610228\pi$
$942$	−40.5017	−1.31962
$943$	−5.50353	−0.179220
$944$	26.6746	0.868183
$945$	63.7356	2.07332
$946$	2.57090	0.0835871
$947$	−4.38750	−0.142574	−0.0712872	−	0.997456i	$-0.522711\pi$
$947$	−4.38750	−0.142574	−0.0712872	+	0.997456i	$0.522711\pi$
$948$	38.8422	1.26153
$949$	−3.96706	−0.128776
$950$	−30.4872	−0.989137
$951$	9.39774	0.304743
$952$	5.91663	0.191759
$953$	−7.57524	−0.245386	−0.122693	−	0.992445i	$-0.539153\pi$
$953$	−7.57524	−0.245386	−0.122693	+	0.992445i	$0.539153\pi$
$954$	−5.21866	−0.168960
$955$	−61.4916	−1.98982
$956$	−24.9738	−0.807710
$957$	3.34937	0.108270
$958$	11.4840	0.371030
$959$	34.5074	1.11430
$960$	34.8905	1.12609
$961$	−20.0970	−0.648292
$962$	−13.1936	−0.425379
$963$	1.70797	0.0550385
$964$	14.8810	0.479286
$965$	−75.6682	−2.43585
$966$	−43.3765	−1.39562
$967$	−27.7049	−0.890931	−0.445466	−	0.895299i	$-0.646962\pi$
$967$	−27.7049	−0.890931	−0.445466	+	0.895299i	$0.646962\pi$
$968$	3.25882	0.104742
$969$	19.4236	0.623976
$970$	57.4033	1.84311
$971$	7.79820	0.250256	0.125128	−	0.992141i	$-0.460066\pi$
$971$	7.79820	0.250256	0.125128	+	0.992141i	$0.460066\pi$
$972$	−30.1389	−0.966704
$973$	63.8336	2.04641
$974$	65.9744	2.11396
$975$	−6.05593	−0.193945
$976$	−27.0647	−0.866320
$977$	42.1784	1.34941	0.674703	−	0.738089i	$-0.264272\pi$
$977$	42.1784	1.34941	0.674703	+	0.738089i	$0.264272\pi$
$978$	53.1973	1.70106
$979$	−0.0726668	−0.00232244
$980$	44.0524	1.40720
$981$	1.17230	0.0374286
$982$	12.4540	0.397423
$983$	44.9360	1.43323	0.716617	−	0.697467i	$-0.245689\pi$
$983$	44.9360	1.43323	0.716617	+	0.697467i	$0.245689\pi$
$984$	0.414003	0.0131979
$985$	−23.8536	−0.760040
$986$	−93.3966	−2.97435
$987$	33.4967	1.06621
$988$	6.41019	0.203935
$989$	18.6176	0.592005
$990$	3.23483	0.102810
$991$	−36.1289	−1.14767	−0.573836	−	0.818970i	$-0.694545\pi$
$991$	−36.1289	−1.14767	−0.573836	+	0.818970i	$0.694545\pi$
$992$	26.7513	0.849354
$993$	7.82408	0.248290
$994$	68.9531	2.18706
$995$	25.8214	0.818593
$996$	10.8307	0.343184
$997$	7.54780	0.239041	0.119521	−	0.992832i	$-0.461864\pi$
$997$	7.54780	0.239041	0.119521	+	0.992832i	$0.461864\pi$
$998$	−44.3782	−1.40477
$999$	35.5358	1.12430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6019.2.a.b.1.19	✓	101

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6019.2.a.b.1.19	✓	101	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)

\(f(q)\)	\(=\)	\(q-2.03639 q^{2} -1.20777 q^{3} +2.14690 q^{4} +3.16451 q^{5} +2.45949 q^{6} +3.67207 q^{7} -0.299141 q^{8} -1.54130 q^{9} +O(q^{10})\)	\(q-2.03639 q^{2} -1.20777 q^{3} +2.14690 q^{4} +3.16451 q^{5} +2.45949 q^{6} +3.67207 q^{7} -0.299141 q^{8} -1.54130 q^{9} -6.44419 q^{10} +0.325684 q^{11} -2.59296 q^{12} +1.00000 q^{13} -7.47779 q^{14} -3.82200 q^{15} -3.68463 q^{16} -5.38625 q^{17} +3.13868 q^{18} +2.98579 q^{19} +6.79389 q^{20} -4.43502 q^{21} -0.663222 q^{22} -4.80284 q^{23} +0.361293 q^{24} +5.01415 q^{25} -2.03639 q^{26} +5.48483 q^{27} +7.88357 q^{28} -8.51496 q^{29} +7.78310 q^{30} +3.30196 q^{31} +8.10163 q^{32} -0.393351 q^{33} +10.9685 q^{34} +11.6203 q^{35} -3.30900 q^{36} +6.47891 q^{37} -6.08025 q^{38} -1.20777 q^{39} -0.946636 q^{40} +1.14589 q^{41} +9.03144 q^{42} -3.87638 q^{43} +0.699211 q^{44} -4.87745 q^{45} +9.78047 q^{46} -7.55277 q^{47} +4.45018 q^{48} +6.48413 q^{49} -10.2108 q^{50} +6.50535 q^{51} +2.14690 q^{52} -1.66269 q^{53} -11.1693 q^{54} +1.03063 q^{55} -1.09847 q^{56} -3.60615 q^{57} +17.3398 q^{58} -7.23942 q^{59} -8.20544 q^{60} +7.34531 q^{61} -6.72410 q^{62} -5.65975 q^{63} -9.12885 q^{64} +3.16451 q^{65} +0.801018 q^{66} -10.4604 q^{67} -11.5637 q^{68} +5.80072 q^{69} -23.6636 q^{70} -9.22106 q^{71} +0.461064 q^{72} -3.96706 q^{73} -13.1936 q^{74} -6.05593 q^{75} +6.41019 q^{76} +1.19594 q^{77} +2.45949 q^{78} -14.9799 q^{79} -11.6601 q^{80} -2.00052 q^{81} -2.33349 q^{82} -4.17697 q^{83} -9.52152 q^{84} -17.0449 q^{85} +7.89383 q^{86} +10.2841 q^{87} -0.0974256 q^{88} -0.223120 q^{89} +9.93240 q^{90} +3.67207 q^{91} -10.3112 q^{92} -3.98801 q^{93} +15.3804 q^{94} +9.44858 q^{95} -9.78489 q^{96} -8.90776 q^{97} -13.2042 q^{98} -0.501976 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 101 q - 8 q^{2} - 13 q^{3} + 86 q^{4} - 43 q^{5} - 10 q^{6} - q^{7} - 24 q^{8} + 52 q^{9}+O(q^{10}) \) 101 * q - 8 * q^2 - 13 * q^3 + 86 * q^4 - 43 * q^5 - 10 * q^6 - q^7 - 24 * q^8 + 52 * q^9	\( 101 q - 8 q^{2} - 13 q^{3} + 86 q^{4} - 43 q^{5} - 10 q^{6} - q^{7} - 24 q^{8} + 52 q^{9} - 19 q^{10} - 42 q^{11} - 28 q^{12} + 101 q^{13} - 45 q^{14} - 15 q^{15} + 48 q^{16} - 83 q^{17} - 4 q^{18} - 18 q^{19} - 51 q^{20} - 50 q^{21} - 20 q^{22} - 64 q^{23} - 23 q^{24} + 46 q^{25} - 8 q^{26} - 37 q^{27} - 11 q^{28} - 117 q^{29} - 28 q^{30} - 10 q^{31} - 36 q^{32} - 20 q^{33} - 10 q^{34} - 53 q^{35} - 16 q^{36} - 27 q^{37} - 68 q^{38} - 13 q^{39} - 42 q^{40} - 60 q^{41} - 31 q^{42} - 16 q^{43} - 89 q^{44} - 56 q^{45} + 5 q^{46} - 23 q^{47} - 37 q^{48} + 48 q^{49} - 30 q^{50} - 68 q^{51} + 86 q^{52} - 189 q^{53} - 23 q^{54} + 3 q^{55} - 106 q^{56} - 25 q^{57} - 82 q^{59} + 6 q^{60} - 68 q^{61} - 57 q^{62} + 3 q^{63} - 2 q^{64} - 43 q^{65} - 40 q^{66} - 13 q^{67} - 138 q^{68} - 92 q^{69} + 18 q^{70} - 39 q^{71} - 20 q^{72} + 19 q^{73} - 88 q^{74} - 21 q^{75} - 53 q^{76} - 147 q^{77} - 10 q^{78} - 19 q^{79} - 104 q^{80} - 55 q^{81} + 27 q^{82} - 49 q^{83} - 59 q^{84} - 27 q^{85} - 99 q^{86} - 33 q^{87} - 41 q^{88} - 70 q^{89} - 49 q^{90} - q^{91} - 111 q^{92} - 84 q^{93} + 4 q^{94} - 82 q^{95} - 7 q^{96} + 25 q^{97} - 37 q^{98} - 41 q^{99}+O(q^{100}) \) 101 * q - 8 * q^2 - 13 * q^3 + 86 * q^4 - 43 * q^5 - 10 * q^6 - q^7 - 24 * q^8 + 52 * q^9 - 19 * q^10 - 42 * q^11 - 28 * q^12 + 101 * q^13 - 45 * q^14 - 15 * q^15 + 48 * q^16 - 83 * q^17 - 4 * q^18 - 18 * q^19 - 51 * q^20 - 50 * q^21 - 20 * q^22 - 64 * q^23 - 23 * q^24 + 46 * q^25 - 8 * q^26 - 37 * q^27 - 11 * q^28 - 117 * q^29 - 28 * q^30 - 10 * q^31 - 36 * q^32 - 20 * q^33 - 10 * q^34 - 53 * q^35 - 16 * q^36 - 27 * q^37 - 68 * q^38 - 13 * q^39 - 42 * q^40 - 60 * q^41 - 31 * q^42 - 16 * q^43 - 89 * q^44 - 56 * q^45 + 5 * q^46 - 23 * q^47 - 37 * q^48 + 48 * q^49 - 30 * q^50 - 68 * q^51 + 86 * q^52 - 189 * q^53 - 23 * q^54 + 3 * q^55 - 106 * q^56 - 25 * q^57 - 82 * q^59 + 6 * q^60 - 68 * q^61 - 57 * q^62 + 3 * q^63 - 2 * q^64 - 43 * q^65 - 40 * q^66 - 13 * q^67 - 138 * q^68 - 92 * q^69 + 18 * q^70 - 39 * q^71 - 20 * q^72 + 19 * q^73 - 88 * q^74 - 21 * q^75 - 53 * q^76 - 147 * q^77 - 10 * q^78 - 19 * q^79 - 104 * q^80 - 55 * q^81 + 27 * q^82 - 49 * q^83 - 59 * q^84 - 27 * q^85 - 99 * q^86 - 33 * q^87 - 41 * q^88 - 70 * q^89 - 49 * q^90 - q^91 - 111 * q^92 - 84 * q^93 + 4 * q^94 - 82 * q^95 - 7 * q^96 + 25 * q^97 - 37 * q^98 - 41 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more