LMFDB - Embedded newform 4015.2.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.39893 q^{2} -0.528085 q^{3} -0.0429944 q^{4} -1.00000 q^{5} -0.738755 q^{6} -3.55790 q^{7} -2.85801 q^{8} -2.72113 q^{9} +O(q^{10})$	\(q+1.39893 q^{2} -0.528085 q^{3} -0.0429944 q^{4} -1.00000 q^{5} -0.738755 q^{6} -3.55790 q^{7} -2.85801 q^{8} -2.72113 q^{9} -1.39893 q^{10} +1.00000 q^{11} +0.0227047 q^{12} -3.96432 q^{13} -4.97726 q^{14} +0.528085 q^{15} -3.91216 q^{16} +3.39299 q^{17} -3.80667 q^{18} -2.71428 q^{19} +0.0429944 q^{20} +1.87888 q^{21} +1.39893 q^{22} +3.30605 q^{23} +1.50927 q^{24} +1.00000 q^{25} -5.54580 q^{26} +3.02124 q^{27} +0.152970 q^{28} -4.80001 q^{29} +0.738755 q^{30} -6.72751 q^{31} +0.243171 q^{32} -0.528085 q^{33} +4.74655 q^{34} +3.55790 q^{35} +0.116993 q^{36} -5.54301 q^{37} -3.79708 q^{38} +2.09350 q^{39} +2.85801 q^{40} +7.29794 q^{41} +2.62842 q^{42} +1.66847 q^{43} -0.0429944 q^{44} +2.72113 q^{45} +4.62493 q^{46} +3.93485 q^{47} +2.06596 q^{48} +5.65866 q^{49} +1.39893 q^{50} -1.79179 q^{51} +0.170443 q^{52} -9.88807 q^{53} +4.22651 q^{54} -1.00000 q^{55} +10.1685 q^{56} +1.43337 q^{57} -6.71488 q^{58} +4.81052 q^{59} -0.0227047 q^{60} -0.353343 q^{61} -9.41132 q^{62} +9.68150 q^{63} +8.16450 q^{64} +3.96432 q^{65} -0.738755 q^{66} +9.25191 q^{67} -0.145879 q^{68} -1.74587 q^{69} +4.97726 q^{70} -0.985408 q^{71} +7.77700 q^{72} -1.00000 q^{73} -7.75429 q^{74} -0.528085 q^{75} +0.116699 q^{76} -3.55790 q^{77} +2.92866 q^{78} +12.0385 q^{79} +3.91216 q^{80} +6.56790 q^{81} +10.2093 q^{82} -11.8606 q^{83} -0.0807811 q^{84} -3.39299 q^{85} +2.33408 q^{86} +2.53482 q^{87} -2.85801 q^{88} -0.167608 q^{89} +3.80667 q^{90} +14.1046 q^{91} -0.142141 q^{92} +3.55270 q^{93} +5.50459 q^{94} +2.71428 q^{95} -0.128415 q^{96} +10.0914 q^{97} +7.91607 q^{98} -2.72113 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9}+O(q^{10}) $ 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9} - 6 q^{10} + 27 q^{11} + 28 q^{12} + 21 q^{13} + 11 q^{14} - 5 q^{15} + 12 q^{16} + 30 q^{17} + 10 q^{18} + 20 q^{19} - 22 q^{20} - 14 q^{21} + 6 q^{22} + 16 q^{23} + 23 q^{24} + 27 q^{25} + 13 q^{26} + 17 q^{27} + 6 q^{28} - 2 q^{29} - 7 q^{30} - 6 q^{31} + 41 q^{32} + 5 q^{33} + 8 q^{34} - 10 q^{35} + 13 q^{36} + 20 q^{37} + 11 q^{38} - 10 q^{39} - 15 q^{40} + 38 q^{41} - 33 q^{42} + 29 q^{43} + 22 q^{44} - 24 q^{45} + 23 q^{46} + 17 q^{47} + 37 q^{48} + 21 q^{49} + 6 q^{50} + 13 q^{51} + 29 q^{52} + 4 q^{53} + q^{54} - 27 q^{55} + 28 q^{56} + 51 q^{57} - 6 q^{58} + 35 q^{59} - 28 q^{60} - 13 q^{61} + 28 q^{62} + 41 q^{63} - 5 q^{64} - 21 q^{65} + 7 q^{66} + 10 q^{67} + 65 q^{68} - 12 q^{69} - 11 q^{70} + 22 q^{71} - 12 q^{72} - 27 q^{73} - 5 q^{74} + 5 q^{75} + 13 q^{76} + 10 q^{77} - 9 q^{78} - 18 q^{79} - 12 q^{80} + 39 q^{81} + 20 q^{82} + 54 q^{83} - 50 q^{84} - 30 q^{85} + 3 q^{86} + 15 q^{87} + 15 q^{88} + 89 q^{89} - 10 q^{90} - 18 q^{91} + 57 q^{92} + 20 q^{93} - 29 q^{94} - 20 q^{95} + 105 q^{96} + 35 q^{97} + 10 q^{98} + 24 q^{99}+O(q^{100}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9 - 6 * q^10 + 27 * q^11 + 28 * q^12 + 21 * q^13 + 11 * q^14 - 5 * q^15 + 12 * q^16 + 30 * q^17 + 10 * q^18 + 20 * q^19 - 22 * q^20 - 14 * q^21 + 6 * q^22 + 16 * q^23 + 23 * q^24 + 27 * q^25 + 13 * q^26 + 17 * q^27 + 6 * q^28 - 2 * q^29 - 7 * q^30 - 6 * q^31 + 41 * q^32 + 5 * q^33 + 8 * q^34 - 10 * q^35 + 13 * q^36 + 20 * q^37 + 11 * q^38 - 10 * q^39 - 15 * q^40 + 38 * q^41 - 33 * q^42 + 29 * q^43 + 22 * q^44 - 24 * q^45 + 23 * q^46 + 17 * q^47 + 37 * q^48 + 21 * q^49 + 6 * q^50 + 13 * q^51 + 29 * q^52 + 4 * q^53 + q^54 - 27 * q^55 + 28 * q^56 + 51 * q^57 - 6 * q^58 + 35 * q^59 - 28 * q^60 - 13 * q^61 + 28 * q^62 + 41 * q^63 - 5 * q^64 - 21 * q^65 + 7 * q^66 + 10 * q^67 + 65 * q^68 - 12 * q^69 - 11 * q^70 + 22 * q^71 - 12 * q^72 - 27 * q^73 - 5 * q^74 + 5 * q^75 + 13 * q^76 + 10 * q^77 - 9 * q^78 - 18 * q^79 - 12 * q^80 + 39 * q^81 + 20 * q^82 + 54 * q^83 - 50 * q^84 - 30 * q^85 + 3 * q^86 + 15 * q^87 + 15 * q^88 + 89 * q^89 - 10 * q^90 - 18 * q^91 + 57 * q^92 + 20 * q^93 - 29 * q^94 - 20 * q^95 + 105 * q^96 + 35 * q^97 + 10 * q^98 + 24 * 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 $

$2$	1.39893	0.989193	0.494597	−	0.869123i	$-0.335316\pi$
$2$	1.39893	0.989193	0.494597	+	0.869123i	$0.335316\pi$
$3$	−0.528085	−0.304890	−0.152445	−	0.988312i	$-0.548715\pi$
$3$	−0.528085	−0.304890	−0.152445	+	0.988312i	$0.548715\pi$
$4$	−0.0429944	−0.0214972
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.738755	−0.301595
$7$	−3.55790	−1.34476	−0.672380	−	0.740206i	$-0.734728\pi$
$7$	−3.55790	−1.34476	−0.672380	+	0.740206i	$0.734728\pi$
$8$	−2.85801	−1.01046
$9$	−2.72113	−0.907042
$10$	−1.39893	−0.442381
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.0227047	0.00655429
$13$	−3.96432	−1.09950	−0.549752	−	0.835328i	$-0.685278\pi$
$13$	−3.96432	−1.09950	−0.549752	+	0.835328i	$0.685278\pi$
$14$	−4.97726	−1.33023
$15$	0.528085	0.136351
$16$	−3.91216	−0.978041
$17$	3.39299	0.822921	0.411460	−	0.911428i	$-0.365019\pi$
$17$	3.39299	0.822921	0.411460	+	0.911428i	$0.365019\pi$
$18$	−3.80667	−0.897240
$19$	−2.71428	−0.622697	−0.311349	−	0.950296i	$-0.600781\pi$
$19$	−2.71428	−0.622697	−0.311349	+	0.950296i	$0.600781\pi$
$20$	0.0429944	0.00961384
$21$	1.87888	0.410004
$22$	1.39893	0.298253
$23$	3.30605	0.689358	0.344679	−	0.938721i	$-0.387988\pi$
$23$	3.30605	0.689358	0.344679	+	0.938721i	$0.387988\pi$
$24$	1.50927	0.308079
$25$	1.00000	0.200000
$26$	−5.54580	−1.08762
$27$	3.02124	0.581438
$28$	0.152970	0.0289086
$29$	−4.80001	−0.891340	−0.445670	−	0.895197i	$-0.647034\pi$
$29$	−4.80001	−0.891340	−0.445670	+	0.895197i	$0.647034\pi$
$30$	0.738755	0.134878
$31$	−6.72751	−1.20830	−0.604148	−	0.796872i	$-0.706487\pi$
$31$	−6.72751	−1.20830	−0.604148	+	0.796872i	$0.706487\pi$
$32$	0.243171	0.0429869
$33$	−0.528085	−0.0919279
$34$	4.74655	0.814027
$35$	3.55790	0.601395
$36$	0.116993	0.0194989
$37$	−5.54301	−0.911266	−0.455633	−	0.890168i	$-0.650587\pi$
$37$	−5.54301	−0.911266	−0.455633	+	0.890168i	$0.650587\pi$
$38$	−3.79708	−0.615968
$39$	2.09350	0.335228
$40$	2.85801	0.451891
$41$	7.29794	1.13975	0.569873	−	0.821733i	$-0.306992\pi$
$41$	7.29794	1.13975	0.569873	+	0.821733i	$0.306992\pi$
$42$	2.62842	0.405573
$43$	1.66847	0.254440	0.127220	−	0.991875i	$-0.459395\pi$
$43$	1.66847	0.254440	0.127220	+	0.991875i	$0.459395\pi$
$44$	−0.0429944	−0.00648165
$45$	2.72113	0.405641
$46$	4.62493	0.681908
$47$	3.93485	0.573957	0.286979	−	0.957937i	$-0.407349\pi$
$47$	3.93485	0.573957	0.286979	+	0.957937i	$0.407349\pi$
$48$	2.06596	0.298195
$49$	5.65866	0.808380
$50$	1.39893	0.197839
$51$	−1.79179	−0.250900
$52$	0.170443	0.0236362
$53$	−9.88807	−1.35823	−0.679115	−	0.734032i	$-0.737636\pi$
$53$	−9.88807	−1.35823	−0.679115	+	0.734032i	$0.737636\pi$
$54$	4.22651	0.575155
$55$	−1.00000	−0.134840
$56$	10.1685	1.35882
$57$	1.43337	0.189854
$58$	−6.71488	−0.881707
$59$	4.81052	0.626276	0.313138	−	0.949708i	$-0.398620\pi$
$59$	4.81052	0.626276	0.313138	+	0.949708i	$0.398620\pi$
$60$	−0.0227047	−0.00293117
$61$	−0.353343	−0.0452409	−0.0226204	−	0.999744i	$-0.507201\pi$
$61$	−0.353343	−0.0452409	−0.0226204	+	0.999744i	$0.507201\pi$
$62$	−9.41132	−1.19524
$63$	9.68150	1.21975
$64$	8.16450	1.02056
$65$	3.96432	0.491713
$66$	−0.738755	−0.0909344
$67$	9.25191	1.13030	0.565150	−	0.824988i	$-0.308818\pi$
$67$	9.25191	1.13030	0.565150	+	0.824988i	$0.308818\pi$
$68$	−0.145879	−0.0176905
$69$	−1.74587	−0.210179
$70$	4.97726	0.594896
$71$	−0.985408	−0.116946	−0.0584732	−	0.998289i	$-0.518623\pi$
$71$	−0.985408	−0.116946	−0.0584732	+	0.998289i	$0.518623\pi$
$72$	7.77700	0.916528
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	−7.75429	−0.901418
$75$	−0.528085	−0.0609780
$76$	0.116699	0.0133863
$77$	−3.55790	−0.405460
$78$	2.92866	0.331605
$79$	12.0385	1.35444	0.677218	−	0.735782i	$-0.263185\pi$
$79$	12.0385	1.35444	0.677218	+	0.735782i	$0.263185\pi$
$80$	3.91216	0.437393
$81$	6.56790	0.729767
$82$	10.2093	1.12743
$83$	−11.8606	−1.30187	−0.650935	−	0.759134i	$-0.725623\pi$
$83$	−11.8606	−1.30187	−0.650935	+	0.759134i	$0.725623\pi$
$84$	−0.0807811	−0.00881394
$85$	−3.39299	−0.368021
$86$	2.33408	0.251690
$87$	2.53482	0.271761
$88$	−2.85801	−0.304665
$89$	−0.167608	−0.0177664	−0.00888319	−	0.999961i	$-0.502828\pi$
$89$	−0.167608	−0.0177664	−0.00888319	+	0.999961i	$0.502828\pi$
$90$	3.80667	0.401258
$91$	14.1046	1.47857
$92$	−0.142141	−0.0148193
$93$	3.55270	0.368398
$94$	5.50459	0.567755
$95$	2.71428	0.278479
$96$	−0.128415	−0.0131063
$97$	10.0914	1.02463	0.512315	−	0.858798i	$-0.328788\pi$
$97$	10.0914	1.02463	0.512315	+	0.858798i	$0.328788\pi$
$98$	7.91607	0.799644
$99$	−2.72113	−0.273483
$100$	−0.0429944	−0.00429944
$101$	−5.64797	−0.561994	−0.280997	−	0.959709i	$-0.590665\pi$
$101$	−5.64797	−0.561994	−0.280997	+	0.959709i	$0.590665\pi$
$102$	−2.50659	−0.248189
$103$	7.18825	0.708279	0.354139	−	0.935193i	$-0.384774\pi$
$103$	7.18825	0.708279	0.354139	+	0.935193i	$0.384774\pi$
$104$	11.3300	1.11100
$105$	−1.87888	−0.183359
$106$	−13.8327	−1.34355
$107$	−2.71687	−0.262650	−0.131325	−	0.991339i	$-0.541923\pi$
$107$	−2.71687	−0.262650	−0.131325	+	0.991339i	$0.541923\pi$
$108$	−0.129897	−0.0124993
$109$	10.4072	0.996833	0.498416	−	0.866938i	$-0.333915\pi$
$109$	10.4072	0.996833	0.498416	+	0.866938i	$0.333915\pi$
$110$	−1.39893	−0.133383
$111$	2.92718	0.277836
$112$	13.9191	1.31523
$113$	−1.75208	−0.164822	−0.0824109	−	0.996598i	$-0.526262\pi$
$113$	−1.75208	−0.164822	−0.0824109	+	0.996598i	$0.526262\pi$
$114$	2.00518	0.187803
$115$	−3.30605	−0.308290
$116$	0.206374	0.0191613
$117$	10.7874	0.997296
$118$	6.72958	0.619508
$119$	−12.0719	−1.10663
$120$	−1.50927	−0.137777
$121$	1.00000	0.0909091
$122$	−0.494302	−0.0447520
$123$	−3.85393	−0.347498
$124$	0.289245	0.0259750
$125$	−1.00000	−0.0894427
$126$	13.5437	1.20657
$127$	7.56881	0.671623	0.335812	−	0.941929i	$-0.390989\pi$
$127$	7.56881	0.671623	0.335812	+	0.941929i	$0.390989\pi$
$128$	10.9352	0.966547
$129$	−0.881096	−0.0775762
$130$	5.54580	0.486399
$131$	0.467978	0.0408874	0.0204437	−	0.999791i	$-0.493492\pi$
$131$	0.467978	0.0408874	0.0204437	+	0.999791i	$0.493492\pi$
$132$	0.0227047	0.00197619
$133$	9.65712	0.837379
$134$	12.9428	1.11809
$135$	−3.02124	−0.260027
$136$	−9.69718	−0.831527
$137$	5.39424	0.460861	0.230431	−	0.973089i	$-0.425986\pi$
$137$	5.39424	0.460861	0.230431	+	0.973089i	$0.425986\pi$
$138$	−2.44236	−0.207907
$139$	0.476231	0.0403934	0.0201967	−	0.999796i	$-0.493571\pi$
$139$	0.476231	0.0403934	0.0201967	+	0.999796i	$0.493571\pi$
$140$	−0.152970	−0.0129283
$141$	−2.07794	−0.174994
$142$	−1.37852	−0.115683
$143$	−3.96432	−0.331513
$144$	10.6455	0.887124
$145$	4.80001	0.398619
$146$	−1.39893	−0.115776
$147$	−2.98826	−0.246467
$148$	0.238319	0.0195897
$149$	−4.27993	−0.350625	−0.175313	−	0.984513i	$-0.556094\pi$
$149$	−4.27993	−0.350625	−0.175313	+	0.984513i	$0.556094\pi$
$150$	−0.738755	−0.0603191
$151$	6.97241	0.567407	0.283703	−	0.958912i	$-0.408437\pi$
$151$	6.97241	0.567407	0.283703	+	0.958912i	$0.408437\pi$
$152$	7.75742	0.629210
$153$	−9.23275	−0.746423
$154$	−4.97726	−0.401079
$155$	6.72751	0.540367
$156$	−0.0900087	−0.00720646
$157$	14.5194	1.15878	0.579388	−	0.815052i	$-0.303292\pi$
$157$	14.5194	1.15878	0.579388	+	0.815052i	$0.303292\pi$
$158$	16.8410	1.33980
$159$	5.22174	0.414111
$160$	−0.243171	−0.0192243
$161$	−11.7626	−0.927022
$162$	9.18804	0.721880
$163$	−12.6471	−0.990599	−0.495300	−	0.868722i	$-0.664942\pi$
$163$	−12.6471	−0.990599	−0.495300	+	0.868722i	$0.664942\pi$
$164$	−0.313770	−0.0245014
$165$	0.528085	0.0411114
$166$	−16.5921	−1.28780
$167$	−4.26160	−0.329773	−0.164886	−	0.986313i	$-0.552726\pi$
$167$	−4.26160	−0.329773	−0.164886	+	0.986313i	$0.552726\pi$
$168$	−5.36984	−0.414292
$169$	2.71580	0.208908
$170$	−4.74655	−0.364044
$171$	7.38588	0.564813
$172$	−0.0717350	−0.00546974
$173$	−0.746370	−0.0567455	−0.0283727	−	0.999597i	$-0.509033\pi$
$173$	−0.746370	−0.0567455	−0.0283727	+	0.999597i	$0.509033\pi$
$174$	3.54603	0.268824
$175$	−3.55790	−0.268952
$176$	−3.91216	−0.294890
$177$	−2.54037	−0.190946
$178$	−0.234472	−0.0175744
$179$	−13.2661	−0.991558	−0.495779	−	0.868449i	$-0.665117\pi$
$179$	−13.2661	−0.991558	−0.495779	+	0.868449i	$0.665117\pi$
$180$	−0.116993	−0.00872016
$181$	−10.4078	−0.773606	−0.386803	−	0.922162i	$-0.626421\pi$
$181$	−10.4078	−0.773606	−0.386803	+	0.922162i	$0.626421\pi$
$182$	19.7314	1.46259
$183$	0.186595	0.0137935
$184$	−9.44870	−0.696568
$185$	5.54301	0.407530
$186$	4.96998	0.364417
$187$	3.39299	0.248120
$188$	−0.169177	−0.0123385
$189$	−10.7493	−0.781895
$190$	3.79708	0.275469
$191$	−7.83772	−0.567118	−0.283559	−	0.958955i	$-0.591515\pi$
$191$	−7.83772	−0.567118	−0.283559	+	0.958955i	$0.591515\pi$
$192$	−4.31156	−0.311160
$193$	3.76343	0.270897	0.135449	−	0.990784i	$-0.456752\pi$
$193$	3.76343	0.270897	0.135449	+	0.990784i	$0.456752\pi$
$194$	14.1172	1.01356
$195$	−2.09350	−0.149918
$196$	−0.243291	−0.0173779
$197$	−0.650921	−0.0463762	−0.0231881	−	0.999731i	$-0.507382\pi$
$197$	−0.650921	−0.0463762	−0.0231881	+	0.999731i	$0.507382\pi$
$198$	−3.80667	−0.270528
$199$	−15.7777	−1.11845	−0.559227	−	0.829014i	$-0.688902\pi$
$199$	−15.7777	−1.11845	−0.559227	+	0.829014i	$0.688902\pi$
$200$	−2.85801	−0.202092
$201$	−4.88580	−0.344618
$202$	−7.90112	−0.555921
$203$	17.0780	1.19864
$204$	0.0770368	0.00539366
$205$	−7.29794	−0.509710
$206$	10.0559	0.700625
$207$	−8.99617	−0.625277
$208$	15.5090	1.07536
$209$	−2.71428	−0.187750
$210$	−2.62842	−0.181378
$211$	7.59063	0.522561	0.261280	−	0.965263i	$-0.415855\pi$
$211$	7.59063	0.522561	0.261280	+	0.965263i	$0.415855\pi$
$212$	0.425131	0.0291981
$213$	0.520380	0.0356558
$214$	−3.80071	−0.259811
$215$	−1.66847	−0.113789
$216$	−8.63473	−0.587519
$217$	23.9358	1.62487
$218$	14.5590	0.986060
$219$	0.528085	0.0356847
$220$	0.0429944	0.00289868
$221$	−13.4509	−0.904804
$222$	4.09493	0.274833
$223$	9.94939	0.666260	0.333130	−	0.942881i	$-0.391895\pi$
$223$	9.94939	0.666260	0.333130	+	0.942881i	$0.391895\pi$
$224$	−0.865177	−0.0578071
$225$	−2.72113	−0.181408
$226$	−2.45104	−0.163041
$227$	−7.28846	−0.483752	−0.241876	−	0.970307i	$-0.577763\pi$
$227$	−7.28846	−0.483752	−0.241876	+	0.970307i	$0.577763\pi$
$228$	−0.0616268	−0.00408134
$229$	−17.8177	−1.17742	−0.588712	−	0.808343i	$-0.700365\pi$
$229$	−17.8177	−1.17742	−0.588712	+	0.808343i	$0.700365\pi$
$230$	−4.62493	−0.304959
$231$	1.87888	0.123621
$232$	13.7185	0.900661
$233$	9.94568	0.651563	0.325782	−	0.945445i	$-0.394373\pi$
$233$	9.94568	0.651563	0.325782	+	0.945445i	$0.394373\pi$
$234$	15.0908	0.986518
$235$	−3.93485	−0.256682
$236$	−0.206825	−0.0134632
$237$	−6.35735	−0.412955
$238$	−16.8878	−1.09467
$239$	−20.3841	−1.31854	−0.659269	−	0.751908i	$-0.729134\pi$
$239$	−20.3841	−1.31854	−0.659269	+	0.751908i	$0.729134\pi$
$240$	−2.06596	−0.133357
$241$	−1.69131	−0.108947	−0.0544733	−	0.998515i	$-0.517348\pi$
$241$	−1.69131	−0.108947	−0.0544733	+	0.998515i	$0.517348\pi$
$242$	1.39893	0.0899266
$243$	−12.5321	−0.803937
$244$	0.0151918	0.000972552 0
$245$	−5.65866	−0.361519
$246$	−5.39138	−0.343742
$247$	10.7602	0.684658
$248$	19.2273	1.22093
$249$	6.26341	0.396927
$250$	−1.39893	−0.0884761
$251$	−7.64134	−0.482317	−0.241159	−	0.970486i	$-0.577527\pi$
$251$	−7.64134	−0.482317	−0.241159	+	0.970486i	$0.577527\pi$
$252$	−0.416250	−0.0262213
$253$	3.30605	0.207849
$254$	10.5882	0.664365
$255$	1.79179	0.112206
$256$	−1.03139	−0.0644616
$257$	4.84086	0.301964	0.150982	−	0.988536i	$-0.451756\pi$
$257$	4.84086	0.301964	0.150982	+	0.988536i	$0.451756\pi$
$258$	−1.23259	−0.0767378
$259$	19.7215	1.22543
$260$	−0.170443	−0.0105704
$261$	13.0614	0.808483
$262$	0.654669	0.0404456
$263$	2.01674	0.124357	0.0621786	−	0.998065i	$-0.480195\pi$
$263$	2.01674	0.124357	0.0621786	+	0.998065i	$0.480195\pi$
$264$	1.50927	0.0928892
$265$	9.88807	0.607419
$266$	13.5096	0.828329
$267$	0.0885112	0.00541680
$268$	−0.397780	−0.0242983
$269$	−3.95023	−0.240850	−0.120425	−	0.992722i	$-0.538426\pi$
$269$	−3.95023	−0.240850	−0.120425	+	0.992722i	$0.538426\pi$
$270$	−4.22651	−0.257217
$271$	−12.2146	−0.741985	−0.370992	−	0.928636i	$-0.620982\pi$
$271$	−12.2146	−0.741985	−0.370992	+	0.928636i	$0.620982\pi$
$272$	−13.2739	−0.804850
$273$	−7.44846	−0.450801
$274$	7.54617	0.455881
$275$	1.00000	0.0603023
$276$	0.0750628	0.00451825
$277$	0.441192	0.0265087	0.0132543	−	0.999912i	$-0.495781\pi$
$277$	0.441192	0.0265087	0.0132543	+	0.999912i	$0.495781\pi$
$278$	0.666213	0.0399568
$279$	18.3064	1.09598
$280$	−10.1685	−0.607684
$281$	33.1087	1.97510	0.987550	−	0.157305i	$-0.0502805\pi$
$281$	33.1087	1.97510	0.987550	+	0.157305i	$0.0502805\pi$
$282$	−2.90689	−0.173103
$283$	−6.76496	−0.402135	−0.201067	−	0.979577i	$-0.564441\pi$
$283$	−6.76496	−0.402135	−0.201067	+	0.979577i	$0.564441\pi$
$284$	0.0423670	0.00251402
$285$	−1.43337	−0.0849055
$286$	−5.54580	−0.327930
$287$	−25.9653	−1.53269
$288$	−0.661698	−0.0389909
$289$	−5.48763	−0.322802
$290$	6.71488	0.394311
$291$	−5.32914	−0.312400
$292$	0.0429944	0.00251606
$293$	9.13696	0.533787	0.266893	−	0.963726i	$-0.414003\pi$
$293$	9.13696	0.533787	0.266893	+	0.963726i	$0.414003\pi$
$294$	−4.18036	−0.243804
$295$	−4.81052	−0.280079
$296$	15.8420	0.920796
$297$	3.02124	0.175310
$298$	−5.98732	−0.346836
$299$	−13.1062	−0.757952
$300$	0.0227047	0.00131086
$301$	−5.93626	−0.342160
$302$	9.75392	0.561275
$303$	2.98261	0.171347
$304$	10.6187	0.609023
$305$	0.353343	0.0202323
$306$	−12.9160	−0.738357
$307$	2.23686	0.127664	0.0638322	−	0.997961i	$-0.479668\pi$
$307$	2.23686	0.127664	0.0638322	+	0.997961i	$0.479668\pi$
$308$	0.152970	0.00871626
$309$	−3.79601	−0.215947
$310$	9.41132	0.534527
$311$	19.0064	1.07775	0.538877	−	0.842384i	$-0.318849\pi$
$311$	19.0064	1.07775	0.538877	+	0.842384i	$0.318849\pi$
$312$	−5.98323	−0.338734
$313$	13.9596	0.789045	0.394523	−	0.918886i	$-0.370910\pi$
$313$	13.9596	0.789045	0.394523	+	0.918886i	$0.370910\pi$
$314$	20.3117	1.14625
$315$	−9.68150	−0.545491
$316$	−0.517588	−0.0291166
$317$	−6.88367	−0.386626	−0.193313	−	0.981137i	$-0.561923\pi$
$317$	−6.88367	−0.386626	−0.193313	+	0.981137i	$0.561923\pi$
$318$	7.30485	0.409636
$319$	−4.80001	−0.268749
$320$	−8.16450	−0.456410
$321$	1.43474	0.0800794
$322$	−16.4550	−0.917003
$323$	−9.20950	−0.512431
$324$	−0.282383	−0.0156879
$325$	−3.96432	−0.219901
$326$	−17.6924	−0.979894
$327$	−5.49591	−0.303925
$328$	−20.8576	−1.15167
$329$	−13.9998	−0.771835
$330$	0.738755	0.0406671
$331$	9.60301	0.527829	0.263914	−	0.964546i	$-0.414986\pi$
$331$	9.60301	0.527829	0.263914	+	0.964546i	$0.414986\pi$
$332$	0.509939	0.0279866
$333$	15.0832	0.826556
$334$	−5.96168	−0.326209
$335$	−9.25191	−0.505486
$336$	−7.35047	−0.401001
$337$	21.8908	1.19247	0.596234	−	0.802811i	$-0.296663\pi$
$337$	21.8908	1.19247	0.596234	+	0.802811i	$0.296663\pi$
$338$	3.79922	0.206650
$339$	0.925248	0.0502525
$340$	0.145879	0.00791143
$341$	−6.72751	−0.364315
$342$	10.3323	0.558709
$343$	4.77235	0.257683
$344$	−4.76851	−0.257101
$345$	1.74587	0.0939947
$346$	−1.04412	−0.0561322
$347$	−22.8298	−1.22557	−0.612784	−	0.790250i	$-0.709950\pi$
$347$	−22.8298	−1.22557	−0.612784	+	0.790250i	$0.709950\pi$
$348$	−0.108983	−0.00584210
$349$	21.2792	1.13905	0.569525	−	0.821974i	$-0.307127\pi$
$349$	21.2792	1.13905	0.569525	+	0.821974i	$0.307127\pi$
$350$	−4.97726	−0.266045
$351$	−11.9772	−0.639294
$352$	0.243171	0.0129610
$353$	−3.96538	−0.211056	−0.105528	−	0.994416i	$-0.533653\pi$
$353$	−3.96538	−0.211056	−0.105528	+	0.994416i	$0.533653\pi$
$354$	−3.55379	−0.188882
$355$	0.985408	0.0523000
$356$	0.00720619	0.000381928 0
$357$	6.37500	0.337401
$358$	−18.5584	−0.980842
$359$	−14.0832	−0.743281	−0.371640	−	0.928377i	$-0.621205\pi$
$359$	−14.0832	−0.743281	−0.371640	+	0.928377i	$0.621205\pi$
$360$	−7.77700	−0.409884
$361$	−11.6327	−0.612248
$362$	−14.5598	−0.765246
$363$	−0.528085	−0.0277173
$364$	−0.606421	−0.0317851
$365$	1.00000	0.0523424
$366$	0.261033	0.0136444
$367$	12.5523	0.655224	0.327612	−	0.944812i	$-0.393756\pi$
$367$	12.5523	0.655224	0.327612	+	0.944812i	$0.393756\pi$
$368$	−12.9338	−0.674220
$369$	−19.8586	−1.03380
$370$	7.75429	0.403126
$371$	35.1808	1.82649
$372$	−0.152746	−0.00791952
$373$	−21.8183	−1.12971	−0.564853	−	0.825191i	$-0.691067\pi$
$373$	−21.8183	−1.12971	−0.564853	+	0.825191i	$0.691067\pi$
$374$	4.74655	0.245438
$375$	0.528085	0.0272702
$376$	−11.2458	−0.579960
$377$	19.0288	0.980031
$378$	−15.0375	−0.773445
$379$	−24.6189	−1.26459	−0.632294	−	0.774728i	$-0.717887\pi$
$379$	−24.6189	−1.26459	−0.632294	+	0.774728i	$0.717887\pi$
$380$	−0.116699	−0.00598651
$381$	−3.99698	−0.204771
$382$	−10.9644	−0.560989
$383$	−20.1531	−1.02977	−0.514887	−	0.857258i	$-0.672166\pi$
$383$	−20.1531	−1.02977	−0.514887	+	0.857258i	$0.672166\pi$
$384$	−5.77473	−0.294691
$385$	3.55790	0.181327
$386$	5.26477	0.267970
$387$	−4.54012	−0.230787
$388$	−0.433875	−0.0220267
$389$	−26.9820	−1.36804	−0.684022	−	0.729461i	$-0.739771\pi$
$389$	−26.9820	−1.36804	−0.684022	+	0.729461i	$0.739771\pi$
$390$	−2.92866	−0.148298
$391$	11.2174	0.567287
$392$	−16.1725	−0.816834
$393$	−0.247132	−0.0124662
$394$	−0.910593	−0.0458750
$395$	−12.0385	−0.605722
$396$	0.116993	0.00587913
$397$	5.70320	0.286236	0.143118	−	0.989706i	$-0.454287\pi$
$397$	5.70320	0.286236	0.143118	+	0.989706i	$0.454287\pi$
$398$	−22.0720	−1.10637
$399$	−5.09979	−0.255309
$400$	−3.91216	−0.195608
$401$	−11.9915	−0.598829	−0.299415	−	0.954123i	$-0.596791\pi$
$401$	−11.9915	−0.598829	−0.299415	+	0.954123i	$0.596791\pi$
$402$	−6.83489	−0.340893
$403$	26.6700	1.32853
$404$	0.242831	0.0120813
$405$	−6.56790	−0.326362
$406$	23.8909	1.18568
$407$	−5.54301	−0.274757
$408$	5.12094	0.253524
$409$	14.2684	0.705528	0.352764	−	0.935712i	$-0.385242\pi$
$409$	14.2684	0.705528	0.352764	+	0.935712i	$0.385242\pi$
$410$	−10.2093	−0.504202
$411$	−2.84862	−0.140512
$412$	−0.309054	−0.0152260
$413$	−17.1154	−0.842191
$414$	−12.5850	−0.618520
$415$	11.8606	0.582214
$416$	−0.964005	−0.0472642
$417$	−0.251490	−0.0123155
$418$	−3.79708	−0.185721
$419$	30.9212	1.51060	0.755299	−	0.655380i	$-0.227492\pi$
$419$	30.9212	1.51060	0.755299	+	0.655380i	$0.227492\pi$
$420$	0.0807811	0.00394172
$421$	−15.1545	−0.738586	−0.369293	−	0.929313i	$-0.620400\pi$
$421$	−15.1545	−0.738586	−0.369293	+	0.929313i	$0.620400\pi$
$422$	10.6188	0.516913
$423$	−10.7072	−0.520603
$424$	28.2602	1.37243
$425$	3.39299	0.164584
$426$	0.727975	0.0352705
$427$	1.25716	0.0608381
$428$	0.116810	0.00564624
$429$	2.09350	0.101075
$430$	−2.33408	−0.112559
$431$	39.5141	1.90333	0.951663	−	0.307145i	$-0.0993738\pi$
$431$	39.5141	1.90333	0.951663	+	0.307145i	$0.0993738\pi$
$432$	−11.8196	−0.568670
$433$	22.8149	1.09641	0.548207	−	0.836343i	$-0.315311\pi$
$433$	22.8149	1.09641	0.548207	+	0.836343i	$0.315311\pi$
$434$	33.4845	1.60731
$435$	−2.53482	−0.121535
$436$	−0.447453	−0.0214291
$437$	−8.97352	−0.429262
$438$	0.738755	0.0352991
$439$	−33.3653	−1.59244	−0.796220	−	0.605007i	$-0.793170\pi$
$439$	−33.3653	−1.59244	−0.796220	+	0.605007i	$0.793170\pi$
$440$	2.85801	0.136250
$441$	−15.3979	−0.733235
$442$	−18.8168	−0.895026
$443$	15.0696	0.715978	0.357989	−	0.933726i	$-0.383463\pi$
$443$	15.0696	0.715978	0.357989	+	0.933726i	$0.383463\pi$
$444$	−0.125853	−0.00597270
$445$	0.167608	0.00794537
$446$	13.9185	0.659060
$447$	2.26017	0.106902
$448$	−29.0485	−1.37241
$449$	33.3846	1.57552	0.787758	−	0.615985i	$-0.211242\pi$
$449$	33.3846	1.57552	0.787758	+	0.615985i	$0.211242\pi$
$450$	−3.80667	−0.179448
$451$	7.29794	0.343646
$452$	0.0753296	0.00354321
$453$	−3.68203	−0.172997
$454$	−10.1960	−0.478524
$455$	−14.1046	−0.661236
$456$	−4.09658	−0.191840
$457$	18.2641	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$457$	18.2641	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$458$	−24.9257	−1.16470
$459$	10.2510	0.478478
$460$	0.142141	0.00662738
$461$	−3.36156	−0.156563	−0.0782817	−	0.996931i	$-0.524943\pi$
$461$	−3.36156	−0.156563	−0.0782817	+	0.996931i	$0.524943\pi$
$462$	2.62842	0.122285
$463$	26.1593	1.21572	0.607862	−	0.794043i	$-0.292027\pi$
$463$	26.1593	1.21572	0.607862	+	0.794043i	$0.292027\pi$
$464$	18.7784	0.871767
$465$	−3.55270	−0.164753
$466$	13.9133	0.644522
$467$	24.4205	1.13005	0.565023	−	0.825075i	$-0.308867\pi$
$467$	24.4205	1.13005	0.565023	+	0.825075i	$0.308867\pi$
$468$	−0.463798	−0.0214391
$469$	−32.9174	−1.51998
$470$	−5.50459	−0.253908
$471$	−7.66749	−0.353299
$472$	−13.7485	−0.632826
$473$	1.66847	0.0767165
$474$	−8.89349	−0.408492
$475$	−2.71428	−0.124539
$476$	0.519025	0.0237895
$477$	26.9067	1.23197
$478$	−28.5159	−1.30429
$479$	22.4049	1.02371	0.511854	−	0.859072i	$-0.328959\pi$
$479$	22.4049	1.02371	0.511854	+	0.859072i	$0.328959\pi$
$480$	0.128415	0.00586131
$481$	21.9743	1.00194
$482$	−2.36602	−0.107769
$483$	6.21165	0.282640
$484$	−0.0429944	−0.00195429
$485$	−10.0914	−0.458229
$486$	−17.5316	−0.795249
$487$	−6.86445	−0.311058	−0.155529	−	0.987831i	$-0.549708\pi$
$487$	−6.86445	−0.311058	−0.155529	+	0.987831i	$0.549708\pi$
$488$	1.00986	0.0457140
$489$	6.67876	0.302024
$490$	−7.91607	−0.357612
$491$	3.62773	0.163717	0.0818585	−	0.996644i	$-0.473914\pi$
$491$	3.62773	0.163717	0.0818585	+	0.996644i	$0.473914\pi$
$492$	0.165698	0.00747022
$493$	−16.2864	−0.733502
$494$	15.0528	0.677259
$495$	2.72113	0.122306
$496$	26.3191	1.18176
$497$	3.50598	0.157265
$498$	8.76207	0.392638
$499$	4.35479	0.194947	0.0974736	−	0.995238i	$-0.468924\pi$
$499$	4.35479	0.194947	0.0974736	+	0.995238i	$0.468924\pi$
$500$	0.0429944	0.00192277
$501$	2.25049	0.100544
$502$	−10.6897	−0.477105
$503$	16.8272	0.750288	0.375144	−	0.926966i	$-0.377593\pi$
$503$	16.8272	0.750288	0.375144	+	0.926966i	$0.377593\pi$
$504$	−27.6698	−1.23251
$505$	5.64797	0.251331
$506$	4.62493	0.205603
$507$	−1.43417	−0.0636939
$508$	−0.325416	−0.0144380
$509$	29.6356	1.31358	0.656788	−	0.754075i	$-0.271915\pi$
$509$	29.6356	1.31358	0.656788	+	0.754075i	$0.271915\pi$
$510$	2.50659	0.110993
$511$	3.55790	0.157392
$512$	−23.3133	−1.03031
$513$	−8.20049	−0.362060
$514$	6.77202	0.298701
$515$	−7.18825	−0.316752
$516$	0.0378822	0.00166767
$517$	3.93485	0.173055
$518$	27.5890	1.21219
$519$	0.394147	0.0173011
$520$	−11.3300	−0.496855
$521$	−2.77028	−0.121368	−0.0606842	−	0.998157i	$-0.519328\pi$
$521$	−2.77028	−0.121368	−0.0606842	+	0.998157i	$0.519328\pi$
$522$	18.2720	0.799745
$523$	14.4594	0.632264	0.316132	−	0.948715i	$-0.397616\pi$
$523$	14.4594	0.632264	0.316132	+	0.948715i	$0.397616\pi$
$524$	−0.0201204	−0.000878966 0
$525$	1.87888	0.0820009
$526$	2.82127	0.123013
$527$	−22.8264	−0.994332
$528$	2.06596	0.0899092
$529$	−12.0701	−0.524785
$530$	13.8327	0.600855
$531$	−13.0900	−0.568059
$532$	−0.415202	−0.0180013
$533$	−28.9313	−1.25316
$534$	0.123821	0.00535826
$535$	2.71687	0.117461
$536$	−26.4420	−1.14212
$537$	7.00565	0.302316
$538$	−5.52610	−0.238247
$539$	5.65866	0.243736
$540$	0.129897	0.00558986
$541$	−29.0143	−1.24742	−0.623710	−	0.781655i	$-0.714376\pi$
$541$	−29.0143	−1.24742	−0.623710	+	0.781655i	$0.714376\pi$
$542$	−17.0874	−0.733966
$543$	5.49621	0.235865
$544$	0.825075	0.0353748
$545$	−10.4072	−0.445797
$546$	−10.4199	−0.445929
$547$	31.9684	1.36687	0.683434	−	0.730012i	$-0.260486\pi$
$547$	31.9684	1.36687	0.683434	+	0.730012i	$0.260486\pi$
$548$	−0.231922	−0.00990723
$549$	0.961490	0.0410354
$550$	1.39893	0.0596506
$551$	13.0286	0.555035
$552$	4.98972	0.212377
$553$	−42.8318	−1.82139
$554$	0.617197	0.0262222
$555$	−2.92718	−0.124252
$556$	−0.0204752	−0.000868344 0
$557$	33.4712	1.41822	0.709109	−	0.705098i	$-0.249097\pi$
$557$	33.4712	1.41822	0.709109	+	0.705098i	$0.249097\pi$
$558$	25.6094	1.08413
$559$	−6.61435	−0.279757
$560$	−13.9191	−0.588189
$561$	−1.79179	−0.0756493
$562$	46.3168	1.95376
$563$	−40.4929	−1.70657	−0.853286	−	0.521444i	$-0.825394\pi$
$563$	−40.4929	−1.70657	−0.853286	+	0.521444i	$0.825394\pi$
$564$	0.0893397	0.00376188
$565$	1.75208	0.0737105
$566$	−9.46370	−0.397789
$567$	−23.3680	−0.981362
$568$	2.81630	0.118169
$569$	26.4783	1.11003	0.555013	−	0.831842i	$-0.312713\pi$
$569$	26.4783	1.11003	0.555013	+	0.831842i	$0.312713\pi$
$570$	−2.00518	−0.0839879
$571$	18.8438	0.788588	0.394294	−	0.918984i	$-0.370989\pi$
$571$	18.8438	0.788588	0.394294	+	0.918984i	$0.370989\pi$
$572$	0.170443	0.00712660
$573$	4.13899	0.172909
$574$	−36.3237	−1.51612
$575$	3.30605	0.137872
$576$	−22.2166	−0.925693
$577$	16.8817	0.702796	0.351398	−	0.936226i	$-0.385706\pi$
$577$	16.8817	0.702796	0.351398	+	0.936226i	$0.385706\pi$
$578$	−7.67681	−0.319313
$579$	−1.98741	−0.0825939
$580$	−0.206374	−0.00856920
$581$	42.1988	1.75070
$582$	−7.45510	−0.309024
$583$	−9.88807	−0.409522
$584$	2.85801	0.118265
$585$	−10.7874	−0.446004
$586$	12.7820	0.528018
$587$	26.6375	1.09945	0.549723	−	0.835347i	$-0.314733\pi$
$587$	26.6375	1.09945	0.549723	+	0.835347i	$0.314733\pi$
$588$	0.128478	0.00529835
$589$	18.2603	0.752403
$590$	−6.72958	−0.277052
$591$	0.343742	0.0141397
$592$	21.6852	0.891255
$593$	11.6564	0.478669	0.239334	−	0.970937i	$-0.423071\pi$
$593$	11.6564	0.478669	0.239334	+	0.970937i	$0.423071\pi$
$594$	4.22651	0.173416
$595$	12.0719	0.494900
$596$	0.184013	0.00753746
$597$	8.33200	0.341006
$598$	−18.3347	−0.749761
$599$	−41.5128	−1.69617	−0.848084	−	0.529861i	$-0.822244\pi$
$599$	−41.5128	−1.69617	−0.848084	+	0.529861i	$0.822244\pi$
$600$	1.50927	0.0616157
$601$	20.5066	0.836482	0.418241	−	0.908336i	$-0.362647\pi$
$601$	20.5066	0.836482	0.418241	+	0.908336i	$0.362647\pi$
$602$	−8.30441	−0.338463
$603$	−25.1756	−1.02523
$604$	−0.299775	−0.0121977
$605$	−1.00000	−0.0406558
$606$	4.17247	0.169495
$607$	−19.2467	−0.781199	−0.390599	−	0.920561i	$-0.627732\pi$
$607$	−19.2467	−0.781199	−0.390599	+	0.920561i	$0.627732\pi$
$608$	−0.660032	−0.0267678
$609$	−9.01863	−0.365453
$610$	0.494302	0.0200137
$611$	−15.5990	−0.631068
$612$	0.396956	0.0160460
$613$	8.28660	0.334693	0.167346	−	0.985898i	$-0.446480\pi$
$613$	8.28660	0.334693	0.167346	+	0.985898i	$0.446480\pi$
$614$	3.12921	0.126285
$615$	3.85393	0.155406
$616$	10.1685	0.409701
$617$	40.6610	1.63695	0.818475	−	0.574541i	$-0.194820\pi$
$617$	40.6610	1.63695	0.818475	+	0.574541i	$0.194820\pi$
$618$	−5.31035	−0.213614
$619$	14.8071	0.595149	0.297574	−	0.954699i	$-0.403822\pi$
$619$	14.8071	0.595149	0.297574	+	0.954699i	$0.403822\pi$
$620$	−0.289245	−0.0116164
$621$	9.98837	0.400819
$622$	26.5886	1.06611
$623$	0.596332	0.0238915
$624$	−8.19010	−0.327866
$625$	1.00000	0.0400000
$626$	19.5286	0.780518
$627$	1.43337	0.0572432
$628$	−0.624254	−0.0249104
$629$	−18.8074	−0.749899
$630$	−13.5437	−0.539595
$631$	3.92848	0.156390	0.0781952	−	0.996938i	$-0.475084\pi$
$631$	3.92848	0.156390	0.0781952	+	0.996938i	$0.475084\pi$
$632$	−34.4061	−1.36860
$633$	−4.00850	−0.159324
$634$	−9.62978	−0.382447
$635$	−7.56881	−0.300359
$636$	−0.224506	−0.00890223
$637$	−22.4327	−0.888817
$638$	−6.71488	−0.265845
$639$	2.68142	0.106075
$640$	−10.9352	−0.432253
$641$	−39.4307	−1.55742	−0.778709	−	0.627385i	$-0.784125\pi$
$641$	−39.4307	−1.55742	−0.778709	+	0.627385i	$0.784125\pi$
$642$	2.00710	0.0792140
$643$	−22.9131	−0.903605	−0.451803	−	0.892118i	$-0.649219\pi$
$643$	−22.9131	−0.903605	−0.451803	+	0.892118i	$0.649219\pi$
$644$	0.505725	0.0199284
$645$	0.881096	0.0346931
$646$	−12.8835	−0.506893
$647$	38.3029	1.50584	0.752921	−	0.658111i	$-0.228644\pi$
$647$	38.3029	1.50584	0.752921	+	0.658111i	$0.228644\pi$
$648$	−18.7711	−0.737399
$649$	4.81052	0.188829
$650$	−5.54580	−0.217524
$651$	−12.6402	−0.495407
$652$	0.543755	0.0212951
$653$	19.4844	0.762485	0.381242	−	0.924475i	$-0.375496\pi$
$653$	19.4844	0.762485	0.381242	+	0.924475i	$0.375496\pi$
$654$	−7.68840	−0.300640
$655$	−0.467978	−0.0182854
$656$	−28.5507	−1.11472
$657$	2.72113	0.106161
$658$	−19.5848	−0.763494
$659$	14.4986	0.564784	0.282392	−	0.959299i	$-0.408872\pi$
$659$	14.4986	0.564784	0.282392	+	0.959299i	$0.408872\pi$
$660$	−0.0227047	−0.000883780 0
$661$	−39.9167	−1.55258	−0.776290	−	0.630376i	$-0.782901\pi$
$661$	−39.9167	−1.55258	−0.776290	+	0.630376i	$0.782901\pi$
$662$	13.4339	0.522125
$663$	7.10321	0.275866
$664$	33.8977	1.31548
$665$	−9.65712	−0.374487
$666$	21.1004	0.817624
$667$	−15.8691	−0.614453
$668$	0.183225	0.00708919
$669$	−5.25413	−0.203136
$670$	−12.9428	−0.500023
$671$	−0.353343	−0.0136406
$672$	0.456887	0.0176248
$673$	−18.3060	−0.705643	−0.352821	−	0.935691i	$-0.614778\pi$
$673$	−18.3060	−0.705643	−0.352821	+	0.935691i	$0.614778\pi$
$674$	30.6237	1.17958
$675$	3.02124	0.116288
$676$	−0.116764	−0.00449093
$677$	−5.72497	−0.220029	−0.110014	−	0.993930i	$-0.535090\pi$
$677$	−5.72497	−0.220029	−0.110014	+	0.993930i	$0.535090\pi$
$678$	1.29436	0.0497095
$679$	−35.9043	−1.37788
$680$	9.69718	0.371870
$681$	3.84893	0.147491
$682$	−9.41132	−0.360378
$683$	−0.706315	−0.0270264	−0.0135132	−	0.999909i	$-0.504302\pi$
$683$	−0.706315	−0.0270264	−0.0135132	+	0.999909i	$0.504302\pi$
$684$	−0.317552	−0.0121419
$685$	−5.39424	−0.206103
$686$	6.67619	0.254898
$687$	9.40925	0.358985
$688$	−6.52734	−0.248852
$689$	39.1994	1.49338
$690$	2.44236	0.0929789
$691$	−10.7298	−0.408182	−0.204091	−	0.978952i	$-0.565424\pi$
$691$	−10.7298	−0.408182	−0.204091	+	0.978952i	$0.565424\pi$
$692$	0.0320897	0.00121987
$693$	9.68150	0.367770
$694$	−31.9373	−1.21232
$695$	−0.476231	−0.0180645
$696$	−7.24452	−0.274603
$697$	24.7618	0.937921
$698$	29.7681	1.12674
$699$	−5.25217	−0.198655
$700$	0.152970	0.00578172
$701$	0.599648	0.0226484	0.0113242	−	0.999936i	$-0.496395\pi$
$701$	0.599648	0.0226484	0.0113242	+	0.999936i	$0.496395\pi$
$702$	−16.7552	−0.632385
$703$	15.0453	0.567443
$704$	8.16450	0.307711
$705$	2.07794	0.0782597
$706$	−5.54730	−0.208775
$707$	20.0949	0.755747
$708$	0.109221	0.00410479
$709$	−24.4734	−0.919119	−0.459559	−	0.888147i	$-0.651993\pi$
$709$	−24.4734	−0.919119	−0.459559	+	0.888147i	$0.651993\pi$
$710$	1.37852	0.0517348
$711$	−32.7583	−1.22853
$712$	0.479024	0.0179522
$713$	−22.2415	−0.832949
$714$	8.91818	0.333755
$715$	3.96432	0.148257
$716$	0.570370	0.0213157
$717$	10.7645	0.402009
$718$	−19.7014	−0.735248
$719$	−7.01754	−0.261710	−0.130855	−	0.991402i	$-0.541772\pi$
$719$	−7.01754	−0.261710	−0.130855	+	0.991402i	$0.541772\pi$
$720$	−10.6455	−0.396734
$721$	−25.5751	−0.952465
$722$	−16.2733	−0.605631
$723$	0.893155	0.0332168
$724$	0.447477	0.0166304
$725$	−4.80001	−0.178268
$726$	−0.738755	−0.0274178
$727$	−16.7741	−0.622117	−0.311058	−	0.950391i	$-0.600683\pi$
$727$	−16.7741	−0.622117	−0.311058	+	0.950391i	$0.600683\pi$
$728$	−40.3112	−1.49403
$729$	−13.0857	−0.484654
$730$	1.39893	0.0517767
$731$	5.66111	0.209384
$732$	−0.00802254	−0.000296522 0
$733$	16.1579	0.596806	0.298403	−	0.954440i	$-0.403546\pi$
$733$	16.1579	0.596806	0.298403	+	0.954440i	$0.403546\pi$
$734$	17.5598	0.648143
$735$	2.98826	0.110223
$736$	0.803933	0.0296334
$737$	9.25191	0.340799
$738$	−27.7808	−1.02263
$739$	13.4021	0.493004	0.246502	−	0.969142i	$-0.420719\pi$
$739$	13.4021	0.493004	0.246502	+	0.969142i	$0.420719\pi$
$740$	−0.238319	−0.00876076
$741$	−5.68233	−0.208746
$742$	49.2154	1.80676
$743$	7.63447	0.280082	0.140041	−	0.990146i	$-0.455277\pi$
$743$	7.63447	0.280082	0.140041	+	0.990146i	$0.455277\pi$
$744$	−10.1536	−0.372251
$745$	4.27993	0.156804
$746$	−30.5222	−1.11750
$747$	32.2742	1.18085
$748$	−0.145879	−0.00533388
$749$	9.66636	0.353201
$750$	0.738755	0.0269755
$751$	−47.9553	−1.74992	−0.874958	−	0.484199i	$-0.839111\pi$
$751$	−47.9553	−1.74992	−0.874958	+	0.484199i	$0.839111\pi$
$752$	−15.3938	−0.561354
$753$	4.03528	0.147054
$754$	26.6199	0.969440
$755$	−6.97241	−0.253752
$756$	0.462159	0.0168086
$757$	20.1402	0.732007	0.366004	−	0.930613i	$-0.380726\pi$
$757$	20.1402	0.732007	0.366004	+	0.930613i	$0.380726\pi$
$758$	−34.4401	−1.25092
$759$	−1.74587	−0.0633712
$760$	−7.75742	−0.281391
$761$	−47.0852	−1.70684	−0.853419	−	0.521225i	$-0.825475\pi$
$761$	−47.0852	−1.70684	−0.853419	+	0.521225i	$0.825475\pi$
$762$	−5.59149	−0.202558
$763$	−37.0279	−1.34050
$764$	0.336978	0.0121914
$765$	9.23275	0.333811
$766$	−28.1927	−1.01865
$767$	−19.0704	−0.688593
$768$	0.544659	0.0196537
$769$	13.1735	0.475050	0.237525	−	0.971381i	$-0.423664\pi$
$769$	13.1735	0.475050	0.237525	+	0.971381i	$0.423664\pi$
$770$	4.97726	0.179368
$771$	−2.55639	−0.0920660
$772$	−0.161806	−0.00582353
$773$	−8.92826	−0.321127	−0.160564	−	0.987026i	$-0.551331\pi$
$773$	−8.92826	−0.321127	−0.160564	+	0.987026i	$0.551331\pi$
$774$	−6.35132	−0.228293
$775$	−6.72751	−0.241659
$776$	−28.8414	−1.03535
$777$	−10.4146	−0.373623
$778$	−37.7460	−1.35326
$779$	−19.8086	−0.709717
$780$	0.0900087	0.00322283
$781$	−0.985408	−0.0352607
$782$	15.6923	0.561156
$783$	−14.5020	−0.518259
$784$	−22.1376	−0.790629
$785$	−14.5194	−0.518220
$786$	−0.345721	−0.0123315
$787$	7.95271	0.283484	0.141742	−	0.989904i	$-0.454730\pi$
$787$	7.95271	0.283484	0.141742	+	0.989904i	$0.454730\pi$
$788$	0.0279860	0.000996958 0
$789$	−1.06501	−0.0379153
$790$	−16.8410	−0.599176
$791$	6.23373	0.221646
$792$	7.77700	0.276343
$793$	1.40076	0.0497425
$794$	7.97838	0.283142
$795$	−5.22174	−0.185196
$796$	0.678355	0.0240436
$797$	29.3609	1.04002	0.520009	−	0.854161i	$-0.325929\pi$
$797$	29.3609	1.04002	0.520009	+	0.854161i	$0.325929\pi$
$798$	−7.13424	−0.252550
$799$	13.3509	0.472321
$800$	0.243171	0.00859738
$801$	0.456082	0.0161149
$802$	−16.7753	−0.592358
$803$	−1.00000	−0.0352892
$804$	0.210062	0.00740832
$805$	11.7626	0.414577
$806$	37.3094	1.31417
$807$	2.08606	0.0734327
$808$	16.1419	0.567871
$809$	50.7147	1.78303	0.891517	−	0.452988i	$-0.149642\pi$
$809$	50.7147	1.78303	0.891517	+	0.452988i	$0.149642\pi$
$810$	−9.18804	−0.322835
$811$	48.7802	1.71290	0.856451	−	0.516228i	$-0.172664\pi$
$811$	48.7802	1.71290	0.856451	+	0.516228i	$0.172664\pi$
$812$	−0.734257	−0.0257674
$813$	6.45036	0.226224
$814$	−7.75429	−0.271788
$815$	12.6471	0.443009
$816$	7.00976	0.245391
$817$	−4.52869	−0.158439
$818$	19.9605	0.697904
$819$	−38.3805	−1.34112
$820$	0.313770	0.0109573
$821$	31.2438	1.09042	0.545209	−	0.838300i	$-0.316450\pi$
$821$	31.2438	1.09042	0.545209	+	0.838300i	$0.316450\pi$
$822$	−3.98502	−0.138994
$823$	−15.9541	−0.556126	−0.278063	−	0.960563i	$-0.589692\pi$
$823$	−15.9541	−0.556126	−0.278063	+	0.960563i	$0.589692\pi$
$824$	−20.5441	−0.715686
$825$	−0.528085	−0.0183856
$826$	−23.9432	−0.833090
$827$	−29.6576	−1.03130	−0.515648	−	0.856801i	$-0.672449\pi$
$827$	−29.6576	−1.03130	−0.515648	+	0.856801i	$0.672449\pi$
$828$	0.386785	0.0134417
$829$	−0.867789	−0.0301396	−0.0150698	−	0.999886i	$-0.504797\pi$
$829$	−0.867789	−0.0301396	−0.0150698	+	0.999886i	$0.504797\pi$
$830$	16.5921	0.575922
$831$	−0.232987	−0.00808223
$832$	−32.3667	−1.12211
$833$	19.1998	0.665233
$834$	−0.351818	−0.0121824
$835$	4.26160	0.147479
$836$	0.116699	0.00403611
$837$	−20.3254	−0.702550
$838$	43.2566	1.49427
$839$	−35.2149	−1.21575	−0.607877	−	0.794031i	$-0.707979\pi$
$839$	−35.2149	−1.21575	−0.607877	+	0.794031i	$0.707979\pi$
$840$	5.36984	0.185277
$841$	−5.95988	−0.205513
$842$	−21.2001	−0.730604
$843$	−17.4842	−0.602189
$844$	−0.326355	−0.0112336
$845$	−2.71580	−0.0934264
$846$	−14.9787	−0.514977
$847$	−3.55790	−0.122251
$848$	38.6837	1.32840
$849$	3.57247	0.122607
$850$	4.74655	0.162805
$851$	−18.3255	−0.628189
$852$	−0.0223734	−0.000766500 0
$853$	−53.5786	−1.83450	−0.917249	−	0.398315i	$-0.869595\pi$
$853$	−53.5786	−1.83450	−0.917249	+	0.398315i	$0.869595\pi$
$854$	1.75868	0.0601807
$855$	−7.38588	−0.252592
$856$	7.76483	0.265397
$857$	24.5313	0.837973	0.418987	−	0.907992i	$-0.362385\pi$
$857$	24.5313	0.837973	0.418987	+	0.907992i	$0.362385\pi$
$858$	2.92866	0.0999827
$859$	35.3774	1.20706	0.603530	−	0.797340i	$-0.293760\pi$
$859$	35.3774	1.20706	0.603530	+	0.797340i	$0.293760\pi$
$860$	0.0717350	0.00244614
$861$	13.7119	0.467301
$862$	55.2774	1.88276
$863$	9.20917	0.313484	0.156742	−	0.987640i	$-0.449901\pi$
$863$	9.20917	0.313484	0.156742	+	0.987640i	$0.449901\pi$
$864$	0.734677	0.0249942
$865$	0.746370	0.0253774
$866$	31.9165	1.08457
$867$	2.89794	0.0984191
$868$	−1.02911	−0.0349301
$869$	12.0385	0.408378
$870$	−3.54603	−0.120222
$871$	−36.6775	−1.24277
$872$	−29.7440	−1.00726
$873$	−27.4601	−0.929383
$874$	−12.5533	−0.424623
$875$	3.55790	0.120279
$876$	−0.0227047	−0.000767121 0
$877$	51.2584	1.73087	0.865437	−	0.501018i	$-0.167041\pi$
$877$	51.2584	1.73087	0.865437	+	0.501018i	$0.167041\pi$
$878$	−46.6758	−1.57523
$879$	−4.82509	−0.162746
$880$	3.91216	0.131879
$881$	57.6523	1.94236	0.971178	−	0.238354i	$-0.0766077\pi$
$881$	57.6523	1.94236	0.971178	+	0.238354i	$0.0766077\pi$
$882$	−21.5406	−0.725311
$883$	5.52476	0.185923	0.0929614	−	0.995670i	$-0.470367\pi$
$883$	5.52476	0.185923	0.0929614	+	0.995670i	$0.470367\pi$
$884$	0.578312	0.0194508
$885$	2.54037	0.0853934
$886$	21.0813	0.708240
$887$	−28.7047	−0.963810	−0.481905	−	0.876223i	$-0.660055\pi$
$887$	−28.7047	−0.963810	−0.481905	+	0.876223i	$0.660055\pi$
$888$	−8.36591	−0.280742
$889$	−26.9291	−0.903172
$890$	0.234472	0.00785950
$891$	6.56790	0.220033
$892$	−0.427768	−0.0143227
$893$	−10.6803	−0.357402
$894$	3.16181	0.105747
$895$	13.2661	0.443438
$896$	−38.9065	−1.29977
$897$	6.92120	0.231092
$898$	46.7027	1.55849
$899$	32.2921	1.07700
$900$	0.116993	0.00389977
$901$	−33.5501	−1.11772
$902$	10.2093	0.339933
$903$	3.13485	0.104321
$904$	5.00745	0.166545
$905$	10.4078	0.345967
$906$	−5.15090	−0.171127
$907$	−24.0936	−0.800015	−0.400008	−	0.916512i	$-0.630993\pi$
$907$	−24.0936	−0.800015	−0.400008	+	0.916512i	$0.630993\pi$
$908$	0.313363	0.0103993
$909$	15.3688	0.509752
$910$	−19.7314	−0.654090
$911$	−7.03023	−0.232922	−0.116461	−	0.993195i	$-0.537155\pi$
$911$	−7.03023	−0.232922	−0.116461	+	0.993195i	$0.537155\pi$
$912$	−5.60757	−0.185685
$913$	−11.8606	−0.392529
$914$	25.5502	0.845124
$915$	−0.186595	−0.00616864
$916$	0.766060	0.0253113
$917$	−1.66502	−0.0549838
$918$	14.3405	0.473307
$919$	31.6155	1.04290	0.521449	−	0.853282i	$-0.325392\pi$
$919$	31.6155	1.04290	0.521449	+	0.853282i	$0.325392\pi$
$920$	9.44870	0.311514
$921$	−1.18125	−0.0389236
$922$	−4.70259	−0.154871
$923$	3.90647	0.128583
$924$	−0.0807811	−0.00265750
$925$	−5.54301	−0.182253
$926$	36.5950	1.20259
$927$	−19.5601	−0.642439
$928$	−1.16722	−0.0383159
$929$	−57.7395	−1.89437	−0.947186	−	0.320684i	$-0.896087\pi$
$929$	−57.7395	−1.89437	−0.947186	+	0.320684i	$0.896087\pi$
$930$	−4.96998	−0.162972
$931$	−15.3592	−0.503376
$932$	−0.427608	−0.0140068
$933$	−10.0370	−0.328597
$934$	34.1626	1.11783
$935$	−3.39299	−0.110963
$936$	−30.8305	−1.00773
$937$	−1.20473	−0.0393567	−0.0196784	−	0.999806i	$-0.506264\pi$
$937$	−1.20473	−0.0393567	−0.0196784	+	0.999806i	$0.506264\pi$
$938$	−46.0491	−1.50356
$939$	−7.37188	−0.240572
$940$	0.169177	0.00551794
$941$	−1.05878	−0.0345151	−0.0172575	−	0.999851i	$-0.505494\pi$
$941$	−1.05878	−0.0345151	−0.0172575	+	0.999851i	$0.505494\pi$
$942$	−10.7263	−0.349481
$943$	24.1273	0.785694
$944$	−18.8195	−0.612524
$945$	10.7493	0.349674
$946$	2.33408	0.0758874
$947$	15.8726	0.515791	0.257896	−	0.966173i	$-0.416971\pi$
$947$	15.8726	0.515791	0.257896	+	0.966173i	$0.416971\pi$
$948$	0.273331	0.00887737
$949$	3.96432	0.128687
$950$	−3.79708	−0.123194
$951$	3.63517	0.117878
$952$	34.5016	1.11820
$953$	−11.9368	−0.386670	−0.193335	−	0.981133i	$-0.561930\pi$
$953$	−11.9368	−0.386670	−0.193335	+	0.981133i	$0.561930\pi$
$954$	37.6406	1.21866
$955$	7.83772	0.253623
$956$	0.876402	0.0283449
$957$	2.53482	0.0819390
$958$	31.3430	1.01264
$959$	−19.1922	−0.619748
$960$	4.31156	0.139155
$961$	14.2594	0.459981
$962$	30.7405	0.991112
$963$	7.39295	0.238234
$964$	0.0727168	0.00234205
$965$	−3.76343	−0.121149
$966$	8.68966	0.279585
$967$	−42.3805	−1.36286	−0.681432	−	0.731881i	$-0.738643\pi$
$967$	−42.3805	−1.36286	−0.681432	+	0.731881i	$0.738643\pi$
$968$	−2.85801	−0.0918598
$969$	4.86340	0.156235
$970$	−14.1172	−0.453277
$971$	−7.02327	−0.225387	−0.112694	−	0.993630i	$-0.535948\pi$
$971$	−7.02327	−0.225387	−0.112694	+	0.993630i	$0.535948\pi$
$972$	0.538812	0.0172824
$973$	−1.69438	−0.0543194
$974$	−9.60288	−0.307696
$975$	2.09350	0.0670456
$976$	1.38233	0.0442474
$977$	−32.1174	−1.02753	−0.513763	−	0.857932i	$-0.671749\pi$
$977$	−32.1174	−1.02753	−0.513763	+	0.857932i	$0.671749\pi$
$978$	9.34312	0.298760
$979$	−0.167608	−0.00535677
$980$	0.243291	0.00777164
$981$	−28.3194	−0.904169
$982$	5.07494	0.161948
$983$	22.3416	0.712585	0.356293	−	0.934374i	$-0.384041\pi$
$983$	22.3416	0.712585	0.356293	+	0.934374i	$0.384041\pi$
$984$	11.0146	0.351132
$985$	0.650921	0.0207401
$986$	−22.7835	−0.725575
$987$	7.39310	0.235325
$988$	−0.462630	−0.0147182
$989$	5.51605	0.175400
$990$	3.80667	0.120984
$991$	25.0573	0.795971	0.397985	−	0.917392i	$-0.369710\pi$
$991$	25.0573	0.795971	0.397985	+	0.917392i	$0.369710\pi$
$992$	−1.63593	−0.0519409
$993$	−5.07121	−0.160930
$994$	4.90463	0.155565
$995$	15.7777	0.500188
$996$	−0.269291	−0.00853283
$997$	25.6278	0.811640	0.405820	−	0.913953i	$-0.366986\pi$
$997$	25.6278	0.811640	0.405820	+	0.913953i	$0.366986\pi$
$998$	6.09205	0.192840
$999$	−16.7468	−0.529845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.e.1.20	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.e.1.20	✓	27	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:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q+1.39893 q^{2} -0.528085 q^{3} -0.0429944 q^{4} -1.00000 q^{5} -0.738755 q^{6} -3.55790 q^{7} -2.85801 q^{8} -2.72113 q^{9} +O(q^{10})\)	\(q+1.39893 q^{2} -0.528085 q^{3} -0.0429944 q^{4} -1.00000 q^{5} -0.738755 q^{6} -3.55790 q^{7} -2.85801 q^{8} -2.72113 q^{9} -1.39893 q^{10} +1.00000 q^{11} +0.0227047 q^{12} -3.96432 q^{13} -4.97726 q^{14} +0.528085 q^{15} -3.91216 q^{16} +3.39299 q^{17} -3.80667 q^{18} -2.71428 q^{19} +0.0429944 q^{20} +1.87888 q^{21} +1.39893 q^{22} +3.30605 q^{23} +1.50927 q^{24} +1.00000 q^{25} -5.54580 q^{26} +3.02124 q^{27} +0.152970 q^{28} -4.80001 q^{29} +0.738755 q^{30} -6.72751 q^{31} +0.243171 q^{32} -0.528085 q^{33} +4.74655 q^{34} +3.55790 q^{35} +0.116993 q^{36} -5.54301 q^{37} -3.79708 q^{38} +2.09350 q^{39} +2.85801 q^{40} +7.29794 q^{41} +2.62842 q^{42} +1.66847 q^{43} -0.0429944 q^{44} +2.72113 q^{45} +4.62493 q^{46} +3.93485 q^{47} +2.06596 q^{48} +5.65866 q^{49} +1.39893 q^{50} -1.79179 q^{51} +0.170443 q^{52} -9.88807 q^{53} +4.22651 q^{54} -1.00000 q^{55} +10.1685 q^{56} +1.43337 q^{57} -6.71488 q^{58} +4.81052 q^{59} -0.0227047 q^{60} -0.353343 q^{61} -9.41132 q^{62} +9.68150 q^{63} +8.16450 q^{64} +3.96432 q^{65} -0.738755 q^{66} +9.25191 q^{67} -0.145879 q^{68} -1.74587 q^{69} +4.97726 q^{70} -0.985408 q^{71} +7.77700 q^{72} -1.00000 q^{73} -7.75429 q^{74} -0.528085 q^{75} +0.116699 q^{76} -3.55790 q^{77} +2.92866 q^{78} +12.0385 q^{79} +3.91216 q^{80} +6.56790 q^{81} +10.2093 q^{82} -11.8606 q^{83} -0.0807811 q^{84} -3.39299 q^{85} +2.33408 q^{86} +2.53482 q^{87} -2.85801 q^{88} -0.167608 q^{89} +3.80667 q^{90} +14.1046 q^{91} -0.142141 q^{92} +3.55270 q^{93} +5.50459 q^{94} +2.71428 q^{95} -0.128415 q^{96} +10.0914 q^{97} +7.91607 q^{98} -2.72113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9}+O(q^{10}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9} - 6 q^{10} + 27 q^{11} + 28 q^{12} + 21 q^{13} + 11 q^{14} - 5 q^{15} + 12 q^{16} + 30 q^{17} + 10 q^{18} + 20 q^{19} - 22 q^{20} - 14 q^{21} + 6 q^{22} + 16 q^{23} + 23 q^{24} + 27 q^{25} + 13 q^{26} + 17 q^{27} + 6 q^{28} - 2 q^{29} - 7 q^{30} - 6 q^{31} + 41 q^{32} + 5 q^{33} + 8 q^{34} - 10 q^{35} + 13 q^{36} + 20 q^{37} + 11 q^{38} - 10 q^{39} - 15 q^{40} + 38 q^{41} - 33 q^{42} + 29 q^{43} + 22 q^{44} - 24 q^{45} + 23 q^{46} + 17 q^{47} + 37 q^{48} + 21 q^{49} + 6 q^{50} + 13 q^{51} + 29 q^{52} + 4 q^{53} + q^{54} - 27 q^{55} + 28 q^{56} + 51 q^{57} - 6 q^{58} + 35 q^{59} - 28 q^{60} - 13 q^{61} + 28 q^{62} + 41 q^{63} - 5 q^{64} - 21 q^{65} + 7 q^{66} + 10 q^{67} + 65 q^{68} - 12 q^{69} - 11 q^{70} + 22 q^{71} - 12 q^{72} - 27 q^{73} - 5 q^{74} + 5 q^{75} + 13 q^{76} + 10 q^{77} - 9 q^{78} - 18 q^{79} - 12 q^{80} + 39 q^{81} + 20 q^{82} + 54 q^{83} - 50 q^{84} - 30 q^{85} + 3 q^{86} + 15 q^{87} + 15 q^{88} + 89 q^{89} - 10 q^{90} - 18 q^{91} + 57 q^{92} + 20 q^{93} - 29 q^{94} - 20 q^{95} + 105 q^{96} + 35 q^{97} + 10 q^{98} + 24 q^{99}+O(q^{100}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9 - 6 * q^10 + 27 * q^11 + 28 * q^12 + 21 * q^13 + 11 * q^14 - 5 * q^15 + 12 * q^16 + 30 * q^17 + 10 * q^18 + 20 * q^19 - 22 * q^20 - 14 * q^21 + 6 * q^22 + 16 * q^23 + 23 * q^24 + 27 * q^25 + 13 * q^26 + 17 * q^27 + 6 * q^28 - 2 * q^29 - 7 * q^30 - 6 * q^31 + 41 * q^32 + 5 * q^33 + 8 * q^34 - 10 * q^35 + 13 * q^36 + 20 * q^37 + 11 * q^38 - 10 * q^39 - 15 * q^40 + 38 * q^41 - 33 * q^42 + 29 * q^43 + 22 * q^44 - 24 * q^45 + 23 * q^46 + 17 * q^47 + 37 * q^48 + 21 * q^49 + 6 * q^50 + 13 * q^51 + 29 * q^52 + 4 * q^53 + q^54 - 27 * q^55 + 28 * q^56 + 51 * q^57 - 6 * q^58 + 35 * q^59 - 28 * q^60 - 13 * q^61 + 28 * q^62 + 41 * q^63 - 5 * q^64 - 21 * q^65 + 7 * q^66 + 10 * q^67 + 65 * q^68 - 12 * q^69 - 11 * q^70 + 22 * q^71 - 12 * q^72 - 27 * q^73 - 5 * q^74 + 5 * q^75 + 13 * q^76 + 10 * q^77 - 9 * q^78 - 18 * q^79 - 12 * q^80 + 39 * q^81 + 20 * q^82 + 54 * q^83 - 50 * q^84 - 30 * q^85 + 3 * q^86 + 15 * q^87 + 15 * q^88 + 89 * q^89 - 10 * q^90 - 18 * q^91 + 57 * q^92 + 20 * q^93 - 29 * q^94 - 20 * q^95 + 105 * q^96 + 35 * q^97 + 10 * q^98 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more