LMFDB - Embedded newform 6014.2.a.i.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6014,2,Mod(1,6014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6014, 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("6014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6014 = 2 \cdot 31 \cdot 97 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6014.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:	$48.0220317756$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.370415 q^{3} +1.00000 q^{4} +2.36567 q^{5} -0.370415 q^{6} +1.74646 q^{7} +1.00000 q^{8} -2.86279 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.370415 q^{3} +1.00000 q^{4} +2.36567 q^{5} -0.370415 q^{6} +1.74646 q^{7} +1.00000 q^{8} -2.86279 q^{9} +2.36567 q^{10} -0.565188 q^{11} -0.370415 q^{12} -1.80284 q^{13} +1.74646 q^{14} -0.876280 q^{15} +1.00000 q^{16} +4.37249 q^{17} -2.86279 q^{18} +2.88258 q^{19} +2.36567 q^{20} -0.646916 q^{21} -0.565188 q^{22} -0.747632 q^{23} -0.370415 q^{24} +0.596390 q^{25} -1.80284 q^{26} +2.17167 q^{27} +1.74646 q^{28} +7.97470 q^{29} -0.876280 q^{30} -1.00000 q^{31} +1.00000 q^{32} +0.209354 q^{33} +4.37249 q^{34} +4.13155 q^{35} -2.86279 q^{36} +4.19701 q^{37} +2.88258 q^{38} +0.667800 q^{39} +2.36567 q^{40} -3.11848 q^{41} -0.646916 q^{42} +3.55904 q^{43} -0.565188 q^{44} -6.77242 q^{45} -0.747632 q^{46} -0.928054 q^{47} -0.370415 q^{48} -3.94988 q^{49} +0.596390 q^{50} -1.61964 q^{51} -1.80284 q^{52} -5.73725 q^{53} +2.17167 q^{54} -1.33705 q^{55} +1.74646 q^{56} -1.06775 q^{57} +7.97470 q^{58} +9.35201 q^{59} -0.876280 q^{60} +4.49636 q^{61} -1.00000 q^{62} -4.99975 q^{63} +1.00000 q^{64} -4.26493 q^{65} +0.209354 q^{66} -3.08762 q^{67} +4.37249 q^{68} +0.276934 q^{69} +4.13155 q^{70} +10.3840 q^{71} -2.86279 q^{72} +0.723606 q^{73} +4.19701 q^{74} -0.220912 q^{75} +2.88258 q^{76} -0.987079 q^{77} +0.667800 q^{78} -3.87552 q^{79} +2.36567 q^{80} +7.78396 q^{81} -3.11848 q^{82} -1.70487 q^{83} -0.646916 q^{84} +10.3439 q^{85} +3.55904 q^{86} -2.95395 q^{87} -0.565188 q^{88} +12.3457 q^{89} -6.77242 q^{90} -3.14859 q^{91} -0.747632 q^{92} +0.370415 q^{93} -0.928054 q^{94} +6.81922 q^{95} -0.370415 q^{96} -1.00000 q^{97} -3.94988 q^{98} +1.61802 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 28 q^{2} + 12 q^{3} + 28 q^{4} + 10 q^{5} + 12 q^{6} + 13 q^{7} + 28 q^{8} + 38 q^{9}+O(q^{10}) $ 28 * q + 28 * q^2 + 12 * q^3 + 28 * q^4 + 10 * q^5 + 12 * q^6 + 13 * q^7 + 28 * q^8 + 38 * q^9	\( 28 q + 28 q^{2} + 12 q^{3} + 28 q^{4} + 10 q^{5} + 12 q^{6} + 13 q^{7} + 28 q^{8} + 38 q^{9} + 10 q^{10} + 12 q^{11} + 12 q^{12} + 20 q^{13} + 13 q^{14} + 19 q^{15} + 28 q^{16} - 4 q^{17} + 38 q^{18} + 35 q^{19} + 10 q^{20} + 30 q^{21} + 12 q^{22} + 20 q^{23} + 12 q^{24} + 46 q^{25} + 20 q^{26} + 39 q^{27} + 13 q^{28} + 5 q^{29} + 19 q^{30} - 28 q^{31} + 28 q^{32} + 12 q^{33} - 4 q^{34} + 36 q^{35} + 38 q^{36} + 11 q^{37} + 35 q^{38} - 4 q^{39} + 10 q^{40} - 5 q^{41} + 30 q^{42} + 43 q^{43} + 12 q^{44} + 11 q^{45} + 20 q^{46} + 18 q^{47} + 12 q^{48} + 99 q^{49} + 46 q^{50} - 43 q^{51} + 20 q^{52} + 11 q^{53} + 39 q^{54} + 66 q^{55} + 13 q^{56} - 15 q^{57} + 5 q^{58} + 34 q^{59} + 19 q^{60} + 66 q^{61} - 28 q^{62} + 65 q^{63} + 28 q^{64} - 16 q^{65} + 12 q^{66} + 5 q^{67} - 4 q^{68} - 33 q^{69} + 36 q^{70} + 25 q^{71} + 38 q^{72} - 9 q^{73} + 11 q^{74} + 92 q^{75} + 35 q^{76} + 4 q^{77} - 4 q^{78} + 15 q^{79} + 10 q^{80} - 5 q^{82} - 12 q^{83} + 30 q^{84} + 88 q^{85} + 43 q^{86} + 31 q^{87} + 12 q^{88} + 8 q^{89} + 11 q^{90} + 34 q^{91} + 20 q^{92} - 12 q^{93} + 18 q^{94} + 32 q^{95} + 12 q^{96} - 28 q^{97} + 99 q^{98} + 51 q^{99}+O(q^{100}) \) 28 * q + 28 * q^2 + 12 * q^3 + 28 * q^4 + 10 * q^5 + 12 * q^6 + 13 * q^7 + 28 * q^8 + 38 * q^9 + 10 * q^10 + 12 * q^11 + 12 * q^12 + 20 * q^13 + 13 * q^14 + 19 * q^15 + 28 * q^16 - 4 * q^17 + 38 * q^18 + 35 * q^19 + 10 * q^20 + 30 * q^21 + 12 * q^22 + 20 * q^23 + 12 * q^24 + 46 * q^25 + 20 * q^26 + 39 * q^27 + 13 * q^28 + 5 * q^29 + 19 * q^30 - 28 * q^31 + 28 * q^32 + 12 * q^33 - 4 * q^34 + 36 * q^35 + 38 * q^36 + 11 * q^37 + 35 * q^38 - 4 * q^39 + 10 * q^40 - 5 * q^41 + 30 * q^42 + 43 * q^43 + 12 * q^44 + 11 * q^45 + 20 * q^46 + 18 * q^47 + 12 * q^48 + 99 * q^49 + 46 * q^50 - 43 * q^51 + 20 * q^52 + 11 * q^53 + 39 * q^54 + 66 * q^55 + 13 * q^56 - 15 * q^57 + 5 * q^58 + 34 * q^59 + 19 * q^60 + 66 * q^61 - 28 * q^62 + 65 * q^63 + 28 * q^64 - 16 * q^65 + 12 * q^66 + 5 * q^67 - 4 * q^68 - 33 * q^69 + 36 * q^70 + 25 * q^71 + 38 * q^72 - 9 * q^73 + 11 * q^74 + 92 * q^75 + 35 * q^76 + 4 * q^77 - 4 * q^78 + 15 * q^79 + 10 * q^80 - 5 * q^82 - 12 * q^83 + 30 * q^84 + 88 * q^85 + 43 * q^86 + 31 * q^87 + 12 * q^88 + 8 * q^89 + 11 * q^90 + 34 * q^91 + 20 * q^92 - 12 * q^93 + 18 * q^94 + 32 * q^95 + 12 * q^96 - 28 * q^97 + 99 * q^98 + 51 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.370415	−0.213859	−0.106930	−	0.994267i	$-0.534102\pi$
$3$	−0.370415	−0.213859	−0.106930	+	0.994267i	$0.534102\pi$
$4$	1.00000	0.500000
$5$	2.36567	1.05796	0.528980	−	0.848634i	$-0.322575\pi$
$5$	2.36567	1.05796	0.528980	+	0.848634i	$0.322575\pi$
$6$	−0.370415	−0.151221
$7$	1.74646	0.660100	0.330050	−	0.943963i	$-0.392934\pi$
$7$	1.74646	0.660100	0.330050	+	0.943963i	$0.392934\pi$
$8$	1.00000	0.353553
$9$	−2.86279	−0.954264
$10$	2.36567	0.748090
$11$	−0.565188	−0.170411	−0.0852053	−	0.996363i	$-0.527155\pi$
$11$	−0.565188	−0.170411	−0.0852053	+	0.996363i	$0.527155\pi$
$12$	−0.370415	−0.106930
$13$	−1.80284	−0.500019	−0.250009	−	0.968243i	$-0.580434\pi$
$13$	−1.80284	−0.500019	−0.250009	+	0.968243i	$0.580434\pi$
$14$	1.74646	0.466761
$15$	−0.876280	−0.226255
$16$	1.00000	0.250000
$17$	4.37249	1.06048	0.530242	−	0.847846i	$-0.322101\pi$
$17$	4.37249	1.06048	0.530242	+	0.847846i	$0.322101\pi$
$18$	−2.86279	−0.674767
$19$	2.88258	0.661308	0.330654	−	0.943752i	$-0.392731\pi$
$19$	2.88258	0.661308	0.330654	+	0.943752i	$0.392731\pi$
$20$	2.36567	0.528980
$21$	−0.646916	−0.141169
$22$	−0.565188	−0.120499
$23$	−0.747632	−0.155892	−0.0779461	−	0.996958i	$-0.524836\pi$
$23$	−0.747632	−0.155892	−0.0779461	+	0.996958i	$0.524836\pi$
$24$	−0.370415	−0.0756107
$25$	0.596390	0.119278
$26$	−1.80284	−0.353567
$27$	2.17167	0.417938
$28$	1.74646	0.330050
$29$	7.97470	1.48087	0.740433	−	0.672130i	$-0.234621\pi$
$29$	7.97470	1.48087	0.740433	+	0.672130i	$0.234621\pi$
$30$	−0.876280	−0.159986
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	0.209354	0.0364439
$34$	4.37249	0.749876
$35$	4.13155	0.698359
$36$	−2.86279	−0.477132
$37$	4.19701	0.689983	0.344992	−	0.938606i	$-0.387882\pi$
$37$	4.19701	0.689983	0.344992	+	0.938606i	$0.387882\pi$
$38$	2.88258	0.467616
$39$	0.667800	0.106934
$40$	2.36567	0.374045
$41$	−3.11848	−0.487025	−0.243512	−	0.969898i	$-0.578300\pi$
$41$	−3.11848	−0.487025	−0.243512	+	0.969898i	$0.578300\pi$
$42$	−0.646916	−0.0998213
$43$	3.55904	0.542748	0.271374	−	0.962474i	$-0.412522\pi$
$43$	3.55904	0.542748	0.271374	+	0.962474i	$0.412522\pi$
$44$	−0.565188	−0.0852053
$45$	−6.77242	−1.00957
$46$	−0.747632	−0.110232
$47$	−0.928054	−0.135371	−0.0676853	−	0.997707i	$-0.521561\pi$
$47$	−0.928054	−0.135371	−0.0676853	+	0.997707i	$0.521561\pi$
$48$	−0.370415	−0.0534648
$49$	−3.94988	−0.564268
$50$	0.596390	0.0843423
$51$	−1.61964	−0.226795
$52$	−1.80284	−0.250009
$53$	−5.73725	−0.788072	−0.394036	−	0.919095i	$-0.628921\pi$
$53$	−5.73725	−0.788072	−0.394036	+	0.919095i	$0.628921\pi$
$54$	2.17167	0.295527
$55$	−1.33705	−0.180288
$56$	1.74646	0.233381
$57$	−1.06775	−0.141427
$58$	7.97470	1.04713
$59$	9.35201	1.21753	0.608764	−	0.793351i	$-0.291666\pi$
$59$	9.35201	1.21753	0.608764	+	0.793351i	$0.291666\pi$
$60$	−0.876280	−0.113127
$61$	4.49636	0.575700	0.287850	−	0.957676i	$-0.407060\pi$
$61$	4.49636	0.575700	0.287850	+	0.957676i	$0.407060\pi$
$62$	−1.00000	−0.127000
$63$	−4.99975	−0.629910
$64$	1.00000	0.125000
$65$	−4.26493	−0.528999
$66$	0.209354	0.0257697
$67$	−3.08762	−0.377213	−0.188607	−	0.982053i	$-0.560397\pi$
$67$	−3.08762	−0.377213	−0.188607	+	0.982053i	$0.560397\pi$
$68$	4.37249	0.530242
$69$	0.276934	0.0333390
$70$	4.13155	0.493814
$71$	10.3840	1.23236	0.616178	−	0.787607i	$-0.288680\pi$
$71$	10.3840	1.23236	0.616178	+	0.787607i	$0.288680\pi$
$72$	−2.86279	−0.337383
$73$	0.723606	0.0846917	0.0423459	−	0.999103i	$-0.486517\pi$
$73$	0.723606	0.0846917	0.0423459	+	0.999103i	$0.486517\pi$
$74$	4.19701	0.487892
$75$	−0.220912	−0.0255087
$76$	2.88258	0.330654
$77$	−0.987079	−0.112488
$78$	0.667800	0.0756135
$79$	−3.87552	−0.436030	−0.218015	−	0.975945i	$-0.569958\pi$
$79$	−3.87552	−0.436030	−0.218015	+	0.975945i	$0.569958\pi$
$80$	2.36567	0.264490
$81$	7.78396	0.864884
$82$	−3.11848	−0.344378
$83$	−1.70487	−0.187134	−0.0935672	−	0.995613i	$-0.529827\pi$
$83$	−1.70487	−0.187134	−0.0935672	+	0.995613i	$0.529827\pi$
$84$	−0.646916	−0.0705843
$85$	10.3439	1.12195
$86$	3.55904	0.383781
$87$	−2.95395	−0.316697
$88$	−0.565188	−0.0602493
$89$	12.3457	1.30864	0.654320	−	0.756218i	$-0.272955\pi$
$89$	12.3457	1.30864	0.654320	+	0.756218i	$0.272955\pi$
$90$	−6.77242	−0.713876
$91$	−3.14859	−0.330062
$92$	−0.747632	−0.0779461
$93$	0.370415	0.0384103
$94$	−0.928054	−0.0957215
$95$	6.81922	0.699637
$96$	−0.370415	−0.0378054
$97$	−1.00000	−0.101535
$97$	−1.00000	−0.101535
$98$	−3.94988	−0.398998
$99$	1.61802	0.162617
$100$	0.596390	0.0596390
$101$	10.4494	1.03976	0.519879	−	0.854240i	$-0.325977\pi$
$101$	10.4494	1.03976	0.519879	+	0.854240i	$0.325977\pi$
$102$	−1.61964	−0.160368
$103$	16.8461	1.65990	0.829949	−	0.557840i	$-0.188370\pi$
$103$	16.8461	1.65990	0.829949	+	0.557840i	$0.188370\pi$
$104$	−1.80284	−0.176783
$105$	−1.53039	−0.149351
$106$	−5.73725	−0.557251
$107$	−7.58813	−0.733572	−0.366786	−	0.930305i	$-0.619542\pi$
$107$	−7.58813	−0.733572	−0.366786	+	0.930305i	$0.619542\pi$
$108$	2.17167	0.208969
$109$	14.3982	1.37910	0.689551	−	0.724238i	$-0.257808\pi$
$109$	14.3982	1.37910	0.689551	+	0.724238i	$0.257808\pi$
$110$	−1.33705	−0.127483
$111$	−1.55463	−0.147559
$112$	1.74646	0.165025
$113$	12.8125	1.20530	0.602651	−	0.798005i	$-0.294111\pi$
$113$	12.8125	1.20530	0.602651	+	0.798005i	$0.294111\pi$
$114$	−1.06775	−0.100004
$115$	−1.76865	−0.164928
$116$	7.97470	0.740433
$117$	5.16116	0.477150
$118$	9.35201	0.860923
$119$	7.63638	0.700026
$120$	−0.876280	−0.0799931
$121$	−10.6806	−0.970960
$122$	4.49636	0.407081
$123$	1.15513	0.104155
$124$	−1.00000	−0.0898027
$125$	−10.4175	−0.931768
$126$	−4.99975	−0.445414
$127$	−11.0218	−0.978025	−0.489013	−	0.872277i	$-0.662643\pi$
$127$	−11.0218	−0.978025	−0.489013	+	0.872277i	$0.662643\pi$
$128$	1.00000	0.0883883
$129$	−1.31832	−0.116072
$130$	−4.26493	−0.374059
$131$	3.08963	0.269942	0.134971	−	0.990850i	$-0.456906\pi$
$131$	3.08963	0.269942	0.134971	+	0.990850i	$0.456906\pi$
$132$	0.209354	0.0182220
$133$	5.03431	0.436530
$134$	−3.08762	−0.266730
$135$	5.13745	0.442161
$136$	4.37249	0.374938
$137$	19.1912	1.63962	0.819808	−	0.572639i	$-0.194080\pi$
$137$	19.1912	1.63962	0.819808	+	0.572639i	$0.194080\pi$
$138$	0.276934	0.0235742
$139$	1.60664	0.136273	0.0681367	−	0.997676i	$-0.478295\pi$
$139$	1.60664	0.136273	0.0681367	+	0.997676i	$0.478295\pi$
$140$	4.13155	0.349180
$141$	0.343765	0.0289503
$142$	10.3840	0.871407
$143$	1.01895	0.0852085
$144$	−2.86279	−0.238566
$145$	18.8655	1.56670
$146$	0.723606	0.0598861
$147$	1.46309	0.120674
$148$	4.19701	0.344992
$149$	−19.8491	−1.62610	−0.813050	−	0.582193i	$-0.802195\pi$
$149$	−19.8491	−1.62610	−0.813050	+	0.582193i	$0.802195\pi$
$150$	−0.220912	−0.0180374
$151$	5.70601	0.464349	0.232174	−	0.972674i	$-0.425416\pi$
$151$	5.70601	0.464349	0.232174	+	0.972674i	$0.425416\pi$
$152$	2.88258	0.233808
$153$	−12.5175	−1.01198
$154$	−0.987079	−0.0795411
$155$	−2.36567	−0.190015
$156$	0.667800	0.0534668
$157$	−3.55525	−0.283740	−0.141870	−	0.989885i	$-0.545311\pi$
$157$	−3.55525	−0.283740	−0.141870	+	0.989885i	$0.545311\pi$
$158$	−3.87552	−0.308320
$159$	2.12517	0.168537
$160$	2.36567	0.187023
$161$	−1.30571	−0.102904
$162$	7.78396	0.611566
$163$	10.6049	0.830642	0.415321	−	0.909675i	$-0.363669\pi$
$163$	10.6049	0.830642	0.415321	+	0.909675i	$0.363669\pi$
$164$	−3.11848	−0.243512
$165$	0.495263	0.0385562
$166$	−1.70487	−0.132324
$167$	−23.2359	−1.79805	−0.899023	−	0.437901i	$-0.855722\pi$
$167$	−23.2359	−1.79805	−0.899023	+	0.437901i	$0.855722\pi$
$168$	−0.646916	−0.0499106
$169$	−9.74976	−0.749981
$170$	10.3439	0.793338
$171$	−8.25222	−0.631063
$172$	3.55904	0.271374
$173$	−8.07784	−0.614147	−0.307074	−	0.951686i	$-0.599350\pi$
$173$	−8.07784	−0.614147	−0.307074	+	0.951686i	$0.599350\pi$
$174$	−2.95395	−0.223939
$175$	1.04157	0.0787355
$176$	−0.565188	−0.0426027
$177$	−3.46413	−0.260380
$178$	12.3457	0.925348
$179$	−5.43083	−0.405919	−0.202960	−	0.979187i	$-0.565056\pi$
$179$	−5.43083	−0.405919	−0.202960	+	0.979187i	$0.565056\pi$
$180$	−6.77242	−0.504786
$181$	−21.8417	−1.62348	−0.811739	−	0.584020i	$-0.801479\pi$
$181$	−21.8417	−1.62348	−0.811739	+	0.584020i	$0.801479\pi$
$182$	−3.14859	−0.233389
$183$	−1.66552	−0.123119
$184$	−0.747632	−0.0551162
$185$	9.92873	0.729974
$186$	0.370415	0.0271602
$187$	−2.47128	−0.180718
$188$	−0.928054	−0.0676853
$189$	3.79273	0.275881
$190$	6.81922	0.494718
$191$	11.3198	0.819070	0.409535	−	0.912294i	$-0.365691\pi$
$191$	11.3198	0.819070	0.409535	+	0.912294i	$0.365691\pi$
$192$	−0.370415	−0.0267324
$193$	−12.7210	−0.915675	−0.457837	−	0.889036i	$-0.651376\pi$
$193$	−12.7210	−0.915675	−0.457837	+	0.889036i	$0.651376\pi$
$194$	−1.00000	−0.0717958
$195$	1.57979	0.113131
$196$	−3.94988	−0.282134
$197$	−8.74955	−0.623380	−0.311690	−	0.950184i	$-0.600895\pi$
$197$	−8.74955	−0.623380	−0.311690	+	0.950184i	$0.600895\pi$
$198$	1.61802	0.114987
$199$	6.11035	0.433151	0.216576	−	0.976266i	$-0.430511\pi$
$199$	6.11035	0.433151	0.216576	+	0.976266i	$0.430511\pi$
$200$	0.596390	0.0421712
$201$	1.14370	0.0806706
$202$	10.4494	0.735220
$203$	13.9275	0.977519
$204$	−1.61964	−0.113397
$205$	−7.37729	−0.515252
$206$	16.8461	1.17372
$207$	2.14032	0.148762
$208$	−1.80284	−0.125005
$209$	−1.62920	−0.112694
$210$	−1.53039	−0.105607
$211$	−8.57087	−0.590043	−0.295022	−	0.955491i	$-0.595327\pi$
$211$	−8.57087	−0.590043	−0.295022	+	0.955491i	$0.595327\pi$
$212$	−5.73725	−0.394036
$213$	−3.84640	−0.263551
$214$	−7.58813	−0.518714
$215$	8.41950	0.574205
$216$	2.17167	0.147763
$217$	−1.74646	−0.118557
$218$	14.3982	0.975172
$219$	−0.268035	−0.0181121
$220$	−1.33705	−0.0901438
$221$	−7.88291	−0.530262
$222$	−1.55463	−0.104340
$223$	13.3600	0.894651	0.447326	−	0.894371i	$-0.352377\pi$
$223$	13.3600	0.894651	0.447326	+	0.894371i	$0.352377\pi$
$224$	1.74646	0.116690
$225$	−1.70734	−0.113823
$226$	12.8125	0.852277
$227$	−28.7817	−1.91031	−0.955155	−	0.296107i	$-0.904311\pi$
$227$	−28.7817	−1.91031	−0.955155	+	0.296107i	$0.904311\pi$
$228$	−1.06775	−0.0707135
$229$	−9.76681	−0.645409	−0.322705	−	0.946500i	$-0.604592\pi$
$229$	−9.76681	−0.645409	−0.322705	+	0.946500i	$0.604592\pi$
$230$	−1.76865	−0.116621
$231$	0.365629	0.0240566
$232$	7.97470	0.523565
$233$	10.7189	0.702216	0.351108	−	0.936335i	$-0.385805\pi$
$233$	10.7189	0.702216	0.351108	+	0.936335i	$0.385805\pi$
$234$	5.16116	0.337396
$235$	−2.19547	−0.143217
$236$	9.35201	0.608764
$237$	1.43555	0.0932490
$238$	7.63638	0.494993
$239$	27.0901	1.75231	0.876156	−	0.482027i	$-0.160099\pi$
$239$	27.0901	1.75231	0.876156	+	0.482027i	$0.160099\pi$
$240$	−0.876280	−0.0565636
$241$	12.9919	0.836885	0.418442	−	0.908243i	$-0.362576\pi$
$241$	12.9919	0.836885	0.418442	+	0.908243i	$0.362576\pi$
$242$	−10.6806	−0.686573
$243$	−9.39830	−0.602901
$244$	4.49636	0.287850
$245$	−9.34410	−0.596973
$246$	1.15513	0.0736485
$247$	−5.19683	−0.330667
$248$	−1.00000	−0.0635001
$249$	0.631512	0.0400204
$250$	−10.4175	−0.658859
$251$	19.8677	1.25404	0.627020	−	0.779003i	$-0.284275\pi$
$251$	19.8677	1.25404	0.627020	+	0.779003i	$0.284275\pi$
$252$	−4.99975	−0.314955
$253$	0.422553	0.0265657
$254$	−11.0218	−0.691568
$255$	−3.83153	−0.239939
$256$	1.00000	0.0625000
$257$	−18.7126	−1.16726	−0.583630	−	0.812020i	$-0.698368\pi$
$257$	−18.7126	−1.16726	−0.583630	+	0.812020i	$0.698368\pi$
$258$	−1.31832	−0.0820751
$259$	7.32990	0.455458
$260$	−4.26493	−0.264500
$261$	−22.8299	−1.41314
$262$	3.08963	0.190878
$263$	10.9353	0.674302	0.337151	−	0.941451i	$-0.390537\pi$
$263$	10.9353	0.674302	0.337151	+	0.941451i	$0.390537\pi$
$264$	0.209354	0.0128849
$265$	−13.5724	−0.833748
$266$	5.03431	0.308673
$267$	−4.57303	−0.279865
$268$	−3.08762	−0.188607
$269$	−17.7952	−1.08499	−0.542497	−	0.840058i	$-0.682521\pi$
$269$	−17.7952	−1.08499	−0.542497	+	0.840058i	$0.682521\pi$
$270$	5.13745	0.312655
$271$	7.13709	0.433547	0.216774	−	0.976222i	$-0.430447\pi$
$271$	7.13709	0.433547	0.216774	+	0.976222i	$0.430447\pi$
$272$	4.37249	0.265121
$273$	1.16629	0.0705869
$274$	19.1912	1.15938
$275$	−0.337073	−0.0203263
$276$	0.276934	0.0166695
$277$	15.7550	0.946624	0.473312	−	0.880895i	$-0.343058\pi$
$277$	15.7550	0.946624	0.473312	+	0.880895i	$0.343058\pi$
$278$	1.60664	0.0963599
$279$	2.86279	0.171391
$280$	4.13155	0.246907
$281$	19.6606	1.17285	0.586425	−	0.810003i	$-0.300535\pi$
$281$	19.6606	1.17285	0.586425	+	0.810003i	$0.300535\pi$
$282$	0.343765	0.0204709
$283$	−2.94422	−0.175016	−0.0875079	−	0.996164i	$-0.527890\pi$
$283$	−2.94422	−0.175016	−0.0875079	+	0.996164i	$0.527890\pi$
$284$	10.3840	0.616178
$285$	−2.52594	−0.149624
$286$	1.01895	0.0602515
$287$	−5.44630	−0.321485
$288$	−2.86279	−0.168692
$289$	2.11867	0.124628
$290$	18.8655	1.10782
$291$	0.370415	0.0217141
$292$	0.723606	0.0423459
$293$	30.7681	1.79749	0.898747	−	0.438467i	$-0.144479\pi$
$293$	30.7681	1.79749	0.898747	+	0.438467i	$0.144479\pi$
$294$	1.46309	0.0853294
$295$	22.1238	1.28810
$296$	4.19701	0.243946
$297$	−1.22740	−0.0712210
$298$	−19.8491	−1.14983
$299$	1.34786	0.0779490
$300$	−0.220912	−0.0127544
$301$	6.21572	0.358268
$302$	5.70601	0.328344
$303$	−3.87063	−0.222362
$304$	2.88258	0.165327
$305$	10.6369	0.609067
$306$	−12.5175	−0.715580
$307$	−25.3299	−1.44565	−0.722827	−	0.691029i	$-0.757158\pi$
$307$	−25.3299	−1.44565	−0.722827	+	0.691029i	$0.757158\pi$
$308$	−0.987079	−0.0562440
$309$	−6.24006	−0.354985
$310$	−2.36567	−0.134361
$311$	9.74925	0.552829	0.276415	−	0.961039i	$-0.410854\pi$
$311$	9.74925	0.552829	0.276415	+	0.961039i	$0.410854\pi$
$312$	0.667800	0.0378068
$313$	−27.4992	−1.55434	−0.777172	−	0.629288i	$-0.783347\pi$
$313$	−27.4992	−1.55434	−0.777172	+	0.629288i	$0.783347\pi$
$314$	−3.55525	−0.200634
$315$	−11.8278	−0.666419
$316$	−3.87552	−0.218015
$317$	−2.63516	−0.148005	−0.0740026	−	0.997258i	$-0.523577\pi$
$317$	−2.63516	−0.148005	−0.0740026	+	0.997258i	$0.523577\pi$
$318$	2.12517	0.119173
$319$	−4.50721	−0.252355
$320$	2.36567	0.132245
$321$	2.81076	0.156881
$322$	−1.30571	−0.0727644
$323$	12.6040	0.701307
$324$	7.78396	0.432442
$325$	−1.07520	−0.0596413
$326$	10.6049	0.587353
$327$	−5.33333	−0.294934
$328$	−3.11848	−0.172189
$329$	−1.62081	−0.0893582
$330$	0.495263	0.0272633
$331$	7.68855	0.422601	0.211300	−	0.977421i	$-0.432230\pi$
$331$	7.68855	0.422601	0.211300	+	0.977421i	$0.432230\pi$
$332$	−1.70487	−0.0935672
$333$	−12.0152	−0.658426
$334$	−23.2359	−1.27141
$335$	−7.30429	−0.399076
$336$	−0.646916	−0.0352921
$337$	29.1649	1.58871	0.794356	−	0.607452i	$-0.207808\pi$
$337$	29.1649	1.58871	0.794356	+	0.607452i	$0.207808\pi$
$338$	−9.74976	−0.530317
$339$	−4.74596	−0.257765
$340$	10.3439	0.560975
$341$	0.565188	0.0306067
$342$	−8.25222	−0.446229
$343$	−19.1235	−1.03257
$344$	3.55904	0.191890
$345$	0.655135	0.0352713
$346$	−8.07784	−0.434268
$347$	1.95085	0.104727	0.0523636	−	0.998628i	$-0.483325\pi$
$347$	1.95085	0.104727	0.0523636	+	0.998628i	$0.483325\pi$
$348$	−2.95395	−0.158348
$349$	−3.57699	−0.191472	−0.0957359	−	0.995407i	$-0.530520\pi$
$349$	−3.57699	−0.191472	−0.0957359	+	0.995407i	$0.530520\pi$
$350$	1.04157	0.0556744
$351$	−3.91518	−0.208977
$352$	−0.565188	−0.0301246
$353$	4.27889	0.227742	0.113871	−	0.993496i	$-0.463675\pi$
$353$	4.27889	0.227742	0.113871	+	0.993496i	$0.463675\pi$
$354$	−3.46413	−0.184116
$355$	24.5652	1.30378
$356$	12.3457	0.654320
$357$	−2.82863	−0.149707
$358$	−5.43083	−0.287028
$359$	−24.7341	−1.30542	−0.652708	−	0.757609i	$-0.726367\pi$
$359$	−24.7341	−1.30542	−0.652708	+	0.757609i	$0.726367\pi$
$360$	−6.77242	−0.356938
$361$	−10.6908	−0.562671
$362$	−21.8417	−1.14797
$363$	3.95624	0.207649
$364$	−3.14859	−0.165031
$365$	1.71181	0.0896004
$366$	−1.66552	−0.0870581
$367$	−10.6574	−0.556312	−0.278156	−	0.960536i	$-0.589723\pi$
$367$	−10.6574	−0.556312	−0.278156	+	0.960536i	$0.589723\pi$
$368$	−0.747632	−0.0389730
$369$	8.92756	0.464750
$370$	9.92873	0.516170
$371$	−10.0199	−0.520206
$372$	0.370415	0.0192051
$373$	17.7286	0.917952	0.458976	−	0.888449i	$-0.348216\pi$
$373$	17.7286	0.917952	0.458976	+	0.888449i	$0.348216\pi$
$374$	−2.47128	−0.127787
$375$	3.85879	0.199267
$376$	−0.928054	−0.0478607
$377$	−14.3771	−0.740460
$378$	3.79273	0.195077
$379$	−4.96321	−0.254943	−0.127472	−	0.991842i	$-0.540686\pi$
$379$	−4.96321	−0.254943	−0.127472	+	0.991842i	$0.540686\pi$
$380$	6.81922	0.349819
$381$	4.08264	0.209160
$382$	11.3198	0.579170
$383$	15.6728	0.800845	0.400422	−	0.916331i	$-0.368863\pi$
$383$	15.6728	0.800845	0.400422	+	0.916331i	$0.368863\pi$
$384$	−0.370415	−0.0189027
$385$	−2.33510	−0.119008
$386$	−12.7210	−0.647480
$387$	−10.1888	−0.517925
$388$	−1.00000	−0.0507673
$389$	−26.2582	−1.33134	−0.665671	−	0.746246i	$-0.731854\pi$
$389$	−26.2582	−1.33134	−0.665671	+	0.746246i	$0.731854\pi$
$390$	1.57979	0.0799960
$391$	−3.26901	−0.165321
$392$	−3.94988	−0.199499
$393$	−1.14445	−0.0577296
$394$	−8.74955	−0.440796
$395$	−9.16819	−0.461302
$396$	1.61802	0.0813084
$397$	33.9878	1.70580	0.852899	−	0.522076i	$-0.174842\pi$
$397$	33.9878	1.70580	0.852899	+	0.522076i	$0.174842\pi$
$398$	6.11035	0.306284
$399$	−1.86478	−0.0933560
$400$	0.596390	0.0298195
$401$	6.73082	0.336121	0.168061	−	0.985777i	$-0.446250\pi$
$401$	6.73082	0.336121	0.168061	+	0.985777i	$0.446250\pi$
$402$	1.14370	0.0570427
$403$	1.80284	0.0898060
$404$	10.4494	0.519879
$405$	18.4143	0.915012
$406$	13.9275	0.691211
$407$	−2.37210	−0.117581
$408$	−1.61964	−0.0801840
$409$	−39.4666	−1.95150	−0.975749	−	0.218891i	$-0.929756\pi$
$409$	−39.4666	−1.95150	−0.975749	+	0.218891i	$0.929756\pi$
$410$	−7.37729	−0.364338
$411$	−7.10872	−0.350647
$412$	16.8461	0.829949
$413$	16.3329	0.803691
$414$	2.14032	0.105191
$415$	−4.03317	−0.197980
$416$	−1.80284	−0.0883916
$417$	−0.595124	−0.0291434
$418$	−1.62920	−0.0796867
$419$	12.7138	0.621108	0.310554	−	0.950556i	$-0.399485\pi$
$419$	12.7138	0.621108	0.310554	+	0.950556i	$0.399485\pi$
$420$	−1.53039	−0.0746753
$421$	−33.1149	−1.61392	−0.806961	−	0.590605i	$-0.798889\pi$
$421$	−33.1149	−1.61392	−0.806961	+	0.590605i	$0.798889\pi$
$422$	−8.57087	−0.417224
$423$	2.65683	0.129179
$424$	−5.73725	−0.278626
$425$	2.60771	0.126493
$426$	−3.84640	−0.186359
$427$	7.85271	0.380019
$428$	−7.58813	−0.366786
$429$	−0.377433	−0.0182226
$430$	8.41950	0.406025
$431$	−16.7178	−0.805269	−0.402634	−	0.915361i	$-0.631905\pi$
$431$	−16.7178	−0.805269	−0.402634	+	0.915361i	$0.631905\pi$
$432$	2.17167	0.104484
$433$	20.7091	0.995216	0.497608	−	0.867402i	$-0.334212\pi$
$433$	20.7091	0.995216	0.497608	+	0.867402i	$0.334212\pi$
$434$	−1.74646	−0.0838328
$435$	−6.98807	−0.335053
$436$	14.3982	0.689551
$437$	−2.15511	−0.103093
$438$	−0.268035	−0.0128072
$439$	30.1985	1.44130	0.720649	−	0.693300i	$-0.243844\pi$
$439$	30.1985	1.44130	0.720649	+	0.693300i	$0.243844\pi$
$440$	−1.33705	−0.0637413
$441$	11.3077	0.538461
$442$	−7.88291	−0.374952
$443$	31.8659	1.51399	0.756996	−	0.653419i	$-0.226666\pi$
$443$	31.8659	1.51399	0.756996	+	0.653419i	$0.226666\pi$
$444$	−1.55463	−0.0737797
$445$	29.2058	1.38449
$446$	13.3600	0.632614
$447$	7.35241	0.347757
$448$	1.74646	0.0825125
$449$	−3.32746	−0.157033	−0.0785163	−	0.996913i	$-0.525018\pi$
$449$	−3.32746	−0.157033	−0.0785163	+	0.996913i	$0.525018\pi$
$450$	−1.70734	−0.0804849
$451$	1.76253	0.0829942
$452$	12.8125	0.602651
$453$	−2.11359	−0.0993053
$454$	−28.7817	−1.35079
$455$	−7.44853	−0.349193
$456$	−1.06775	−0.0500020
$457$	−15.6941	−0.734138	−0.367069	−	0.930194i	$-0.619639\pi$
$457$	−15.6941	−0.734138	−0.367069	+	0.930194i	$0.619639\pi$
$458$	−9.76681	−0.456373
$459$	9.49560	0.443216
$460$	−1.76865	−0.0824638
$461$	−20.1840	−0.940065	−0.470032	−	0.882649i	$-0.655758\pi$
$461$	−20.1840	−0.940065	−0.470032	+	0.882649i	$0.655758\pi$
$462$	0.365629	0.0170106
$463$	22.1096	1.02752	0.513761	−	0.857933i	$-0.328252\pi$
$463$	22.1096	1.02752	0.513761	+	0.857933i	$0.328252\pi$
$464$	7.97470	0.370216
$465$	0.876280	0.0406365
$466$	10.7189	0.496541
$467$	2.40009	0.111063	0.0555314	−	0.998457i	$-0.482315\pi$
$467$	2.40009	0.111063	0.0555314	+	0.998457i	$0.482315\pi$
$468$	5.16116	0.238575
$469$	−5.39241	−0.248998
$470$	−2.19547	−0.101269
$471$	1.31692	0.0606804
$472$	9.35201	0.430461
$473$	−2.01153	−0.0924900
$474$	1.43555	0.0659370
$475$	1.71914	0.0788796
$476$	7.63638	0.350013
$477$	16.4246	0.752029
$478$	27.0901	1.23907
$479$	5.89438	0.269321	0.134661	−	0.990892i	$-0.457006\pi$
$479$	5.89438	0.269321	0.134661	+	0.990892i	$0.457006\pi$
$480$	−0.876280	−0.0399965
$481$	−7.56654	−0.345005
$482$	12.9919	0.591767
$483$	0.483655	0.0220071
$484$	−10.6806	−0.485480
$485$	−2.36567	−0.107420
$486$	−9.39830	−0.426316
$487$	−12.9026	−0.584670	−0.292335	−	0.956316i	$-0.594432\pi$
$487$	−12.9026	−0.584670	−0.292335	+	0.956316i	$0.594432\pi$
$488$	4.49636	0.203541
$489$	−3.92823	−0.177641
$490$	−9.34410	−0.422123
$491$	−30.1741	−1.36174	−0.680869	−	0.732405i	$-0.738398\pi$
$491$	−30.1741	−1.36174	−0.680869	+	0.732405i	$0.738398\pi$
$492$	1.15513	0.0520774
$493$	34.8693	1.57044
$494$	−5.19683	−0.233817
$495$	3.82769	0.172042
$496$	−1.00000	−0.0449013
$497$	18.1353	0.813478
$498$	0.631512	0.0282987
$499$	−24.9132	−1.11527	−0.557633	−	0.830087i	$-0.688290\pi$
$499$	−24.9132	−1.11527	−0.557633	+	0.830087i	$0.688290\pi$
$500$	−10.4175	−0.465884
$501$	8.60692	0.384529
$502$	19.8677	0.886740
$503$	−23.6338	−1.05378	−0.526890	−	0.849934i	$-0.676642\pi$
$503$	−23.6338	−1.05378	−0.526890	+	0.849934i	$0.676642\pi$
$504$	−4.99975	−0.222707
$505$	24.7199	1.10002
$506$	0.422553	0.0187848
$507$	3.61146	0.160391
$508$	−11.0218	−0.489013
$509$	−0.722200	−0.0320109	−0.0160055	−	0.999872i	$-0.505095\pi$
$509$	−0.722200	−0.0320109	−0.0160055	+	0.999872i	$0.505095\pi$
$510$	−3.83153	−0.169663
$511$	1.26375	0.0559050
$512$	1.00000	0.0441942
$513$	6.26000	0.276386
$514$	−18.7126	−0.825378
$515$	39.8523	1.75610
$516$	−1.31832	−0.0580359
$517$	0.524525	0.0230686
$518$	7.32990	0.322058
$519$	2.99216	0.131341
$520$	−4.26493	−0.187030
$521$	0.990380	0.0433894	0.0216947	−	0.999765i	$-0.493094\pi$
$521$	0.990380	0.0433894	0.0216947	+	0.999765i	$0.493094\pi$
$522$	−22.8299	−0.999239
$523$	23.9397	1.04681	0.523405	−	0.852084i	$-0.324661\pi$
$523$	23.9397	1.04681	0.523405	+	0.852084i	$0.324661\pi$
$524$	3.08963	0.134971
$525$	−0.385814	−0.0168383
$526$	10.9353	0.476803
$527$	−4.37249	−0.190469
$528$	0.209354	0.00911098
$529$	−22.4410	−0.975698
$530$	−13.5724	−0.589549
$531$	−26.7729	−1.16184
$532$	5.03431	0.218265
$533$	5.62213	0.243521
$534$	−4.57303	−0.197894
$535$	−17.9510	−0.776089
$536$	−3.08762	−0.133365
$537$	2.01166	0.0868097
$538$	−17.7952	−0.767207
$539$	2.23242	0.0961572
$540$	5.13745	0.221081
$541$	20.5309	0.882693	0.441347	−	0.897337i	$-0.354501\pi$
$541$	20.5309	0.882693	0.441347	+	0.897337i	$0.354501\pi$
$542$	7.13709	0.306564
$543$	8.09049	0.347196
$544$	4.37249	0.187469
$545$	34.0615	1.45903
$546$	1.16629	0.0499125
$547$	8.88704	0.379983	0.189991	−	0.981786i	$-0.439154\pi$
$547$	8.88704	0.379983	0.189991	+	0.981786i	$0.439154\pi$
$548$	19.1912	0.819808
$549$	−12.8721	−0.549370
$550$	−0.337073	−0.0143728
$551$	22.9877	0.979309
$552$	0.276934	0.0117871
$553$	−6.76844	−0.287823
$554$	15.7550	0.669364
$555$	−3.67775	−0.156112
$556$	1.60664	0.0681367
$557$	3.48839	0.147808	0.0739038	−	0.997265i	$-0.476454\pi$
$557$	3.48839	0.147808	0.0739038	+	0.997265i	$0.476454\pi$
$558$	2.86279	0.121192
$559$	−6.41638	−0.271384
$560$	4.13155	0.174590
$561$	0.915400	0.0386482
$562$	19.6606	0.829330
$563$	−12.5159	−0.527481	−0.263741	−	0.964594i	$-0.584956\pi$
$563$	−12.5159	−0.527481	−0.263741	+	0.964594i	$0.584956\pi$
$564$	0.343765	0.0144751
$565$	30.3102	1.27516
$566$	−2.94422	−0.123755
$567$	13.5944	0.570910
$568$	10.3840	0.435704
$569$	22.9462	0.961955	0.480977	−	0.876733i	$-0.340282\pi$
$569$	22.9462	0.961955	0.480977	+	0.876733i	$0.340282\pi$
$570$	−2.52594	−0.105800
$571$	17.5678	0.735189	0.367594	−	0.929986i	$-0.380182\pi$
$571$	17.5678	0.735189	0.367594	+	0.929986i	$0.380182\pi$
$572$	1.01895	0.0426042
$573$	−4.19302	−0.175166
$574$	−5.44630	−0.227324
$575$	−0.445881	−0.0185945
$576$	−2.86279	−0.119283
$577$	−42.7969	−1.78166	−0.890830	−	0.454338i	$-0.849876\pi$
$577$	−42.7969	−1.78166	−0.890830	+	0.454338i	$0.849876\pi$
$578$	2.11867	0.0881250
$579$	4.71204	0.195826
$580$	18.8655	0.783348
$581$	−2.97750	−0.123527
$582$	0.370415	0.0153542
$583$	3.24263	0.134296
$584$	0.723606	0.0299430
$585$	12.2096	0.504805
$586$	30.7681	1.27102
$587$	9.67681	0.399405	0.199702	−	0.979857i	$-0.436002\pi$
$587$	9.67681	0.399405	0.199702	+	0.979857i	$0.436002\pi$
$588$	1.46309	0.0603370
$589$	−2.88258	−0.118775
$590$	22.1238	0.910821
$591$	3.24097	0.133316
$592$	4.19701	0.172496
$593$	−31.6128	−1.29818	−0.649091	−	0.760711i	$-0.724851\pi$
$593$	−31.6128	−1.29818	−0.649091	+	0.760711i	$0.724851\pi$
$594$	−1.22740	−0.0503609
$595$	18.0652	0.740599
$596$	−19.8491	−0.813050
$597$	−2.26337	−0.0926334
$598$	1.34786	0.0551182
$599$	−35.5553	−1.45275	−0.726375	−	0.687298i	$-0.758796\pi$
$599$	−35.5553	−1.45275	−0.726375	+	0.687298i	$0.758796\pi$
$600$	−0.220912	−0.00901870
$601$	32.9786	1.34523	0.672613	−	0.739994i	$-0.265172\pi$
$601$	32.9786	1.34523	0.672613	+	0.739994i	$0.265172\pi$
$602$	6.21572	0.253334
$603$	8.83922	0.359961
$604$	5.70601	0.232174
$605$	−25.2667	−1.02724
$606$	−3.87063	−0.157234
$607$	−13.2064	−0.536033	−0.268016	−	0.963414i	$-0.586368\pi$
$607$	−13.2064	−0.536033	−0.268016	+	0.963414i	$0.586368\pi$
$608$	2.88258	0.116904
$609$	−5.15896	−0.209052
$610$	10.6369	0.430675
$611$	1.67314	0.0676878
$612$	−12.5175	−0.505991
$613$	−11.8787	−0.479775	−0.239887	−	0.970801i	$-0.577111\pi$
$613$	−11.8787	−0.479775	−0.239887	+	0.970801i	$0.577111\pi$
$614$	−25.3299	−1.02223
$615$	2.73266	0.110192
$616$	−0.987079	−0.0397705
$617$	−29.9464	−1.20560	−0.602798	−	0.797894i	$-0.705947\pi$
$617$	−29.9464	−1.20560	−0.602798	+	0.797894i	$0.705947\pi$
$618$	−6.24006	−0.251012
$619$	−39.0310	−1.56879	−0.784393	−	0.620264i	$-0.787026\pi$
$619$	−39.0310	−1.56879	−0.784393	+	0.620264i	$0.787026\pi$
$620$	−2.36567	−0.0950076
$621$	−1.62361	−0.0651532
$622$	9.74925	0.390909
$623$	21.5612	0.863833
$624$	0.667800	0.0267334
$625$	−27.6263	−1.10505
$626$	−27.4992	−1.09909
$627$	0.603480	0.0241007
$628$	−3.55525	−0.141870
$629$	18.3514	0.731717
$630$	−11.8278	−0.471229
$631$	43.8433	1.74537	0.872687	−	0.488280i	$-0.162376\pi$
$631$	43.8433	1.74537	0.872687	+	0.488280i	$0.162376\pi$
$632$	−3.87552	−0.154160
$633$	3.17478	0.126186
$634$	−2.63516	−0.104655
$635$	−26.0739	−1.03471
$636$	2.12517	0.0842683
$637$	7.12100	0.282144
$638$	−4.50721	−0.178442
$639$	−29.7273	−1.17599
$640$	2.36567	0.0935113
$641$	−17.7415	−0.700748	−0.350374	−	0.936610i	$-0.613946\pi$
$641$	−17.7415	−0.700748	−0.350374	+	0.936610i	$0.613946\pi$
$642$	2.81076	0.110932
$643$	21.2817	0.839268	0.419634	−	0.907693i	$-0.362158\pi$
$643$	21.2817	0.839268	0.419634	+	0.907693i	$0.362158\pi$
$644$	−1.30571	−0.0514522
$645$	−3.11871	−0.122799
$646$	12.6040	0.495899
$647$	11.2475	0.442184	0.221092	−	0.975253i	$-0.429038\pi$
$647$	11.2475	0.442184	0.221092	+	0.975253i	$0.429038\pi$
$648$	7.78396	0.305783
$649$	−5.28565	−0.207480
$650$	−1.07520	−0.0421727
$651$	0.646916	0.0253546
$652$	10.6049	0.415321
$653$	−32.4553	−1.27007	−0.635036	−	0.772482i	$-0.719015\pi$
$653$	−32.4553	−1.27007	−0.635036	+	0.772482i	$0.719015\pi$
$654$	−5.33333	−0.208550
$655$	7.30904	0.285588
$656$	−3.11848	−0.121756
$657$	−2.07154	−0.0808183
$658$	−1.62081	−0.0631858
$659$	−36.2790	−1.41323	−0.706615	−	0.707598i	$-0.749779\pi$
$659$	−36.2790	−1.41323	−0.706615	+	0.707598i	$0.749779\pi$
$660$	0.495263	0.0192781
$661$	−44.8660	−1.74508	−0.872542	−	0.488539i	$-0.837530\pi$
$661$	−44.8660	−1.74508	−0.872542	+	0.488539i	$0.837530\pi$
$662$	7.68855	0.298824
$663$	2.91995	0.113401
$664$	−1.70487	−0.0661620
$665$	11.9095	0.461831
$666$	−12.0152	−0.465578
$667$	−5.96215	−0.230855
$668$	−23.2359	−0.899023
$669$	−4.94874	−0.191330
$670$	−7.30429	−0.282190
$671$	−2.54129	−0.0981053
$672$	−0.646916	−0.0249553
$673$	7.00879	0.270169	0.135084	−	0.990834i	$-0.456869\pi$
$673$	7.00879	0.270169	0.135084	+	0.990834i	$0.456869\pi$
$674$	29.1649	1.12339
$675$	1.29516	0.0498508
$676$	−9.74976	−0.374991
$677$	−28.3094	−1.08802	−0.544008	−	0.839080i	$-0.683094\pi$
$677$	−28.3094	−1.08802	−0.544008	+	0.839080i	$0.683094\pi$
$678$	−4.74596	−0.182267
$679$	−1.74646	−0.0670230
$680$	10.3439	0.396669
$681$	10.6612	0.408538
$682$	0.565188	0.0216422
$683$	−1.17699	−0.0450364	−0.0225182	−	0.999746i	$-0.507168\pi$
$683$	−1.17699	−0.0450364	−0.0225182	+	0.999746i	$0.507168\pi$
$684$	−8.25222	−0.315531
$685$	45.4001	1.73465
$686$	−19.1235	−0.730140
$687$	3.61778	0.138027
$688$	3.55904	0.135687
$689$	10.3434	0.394051
$690$	0.655135	0.0249406
$691$	−16.6648	−0.633959	−0.316979	−	0.948432i	$-0.602669\pi$
$691$	−16.6648	−0.633959	−0.316979	+	0.948432i	$0.602669\pi$
$692$	−8.07784	−0.307074
$693$	2.82580	0.107343
$694$	1.95085	0.0740533
$695$	3.80078	0.144172
$696$	−2.95395	−0.111969
$697$	−13.6355	−0.516482
$698$	−3.57699	−0.135391
$699$	−3.97043	−0.150175
$700$	1.04157	0.0393677
$701$	11.9774	0.452382	0.226191	−	0.974083i	$-0.427373\pi$
$701$	11.9774	0.452382	0.226191	+	0.974083i	$0.427373\pi$
$702$	−3.91518	−0.147769
$703$	12.0982	0.456292
$704$	−0.565188	−0.0213013
$705$	0.813235	0.0306282
$706$	4.27889	0.161038
$707$	18.2495	0.686344
$708$	−3.46413	−0.130190
$709$	19.1610	0.719605	0.359802	−	0.933029i	$-0.382844\pi$
$709$	19.1610	0.719605	0.359802	+	0.933029i	$0.382844\pi$
$710$	24.5652	0.921914
$711$	11.0948	0.416088
$712$	12.3457	0.462674
$713$	0.747632	0.0279991
$714$	−2.82863	−0.105859
$715$	2.41049	0.0901471
$716$	−5.43083	−0.202960
$717$	−10.0346	−0.374748
$718$	−24.7341	−0.923069
$719$	49.4608	1.84458	0.922288	−	0.386503i	$-0.126317\pi$
$719$	49.4608	1.84458	0.922288	+	0.386503i	$0.126317\pi$
$720$	−6.77242	−0.252393
$721$	29.4211	1.09570
$722$	−10.6908	−0.397869
$723$	−4.81241	−0.178976
$724$	−21.8417	−0.811739
$725$	4.75604	0.176635
$726$	3.95624	0.146830
$727$	−11.7595	−0.436136	−0.218068	−	0.975934i	$-0.569975\pi$
$727$	−11.7595	−0.436136	−0.218068	+	0.975934i	$0.569975\pi$
$728$	−3.14859	−0.116695
$729$	−19.8706	−0.735948
$730$	1.71181	0.0633571
$731$	15.5619	0.575576
$732$	−1.66552	−0.0615594
$733$	−43.3394	−1.60078	−0.800389	−	0.599480i	$-0.795374\pi$
$733$	−43.3394	−1.60078	−0.800389	+	0.599480i	$0.795374\pi$
$734$	−10.6574	−0.393372
$735$	3.46120	0.127668
$736$	−0.747632	−0.0275581
$737$	1.74509	0.0642811
$738$	8.92756	0.328628
$739$	36.1643	1.33033	0.665163	−	0.746698i	$-0.268362\pi$
$739$	36.1643	1.33033	0.665163	+	0.746698i	$0.268362\pi$
$740$	9.92873	0.364987
$741$	1.92499	0.0707161
$742$	−10.0199	−0.367841
$743$	−26.4823	−0.971542	−0.485771	−	0.874086i	$-0.661461\pi$
$743$	−26.4823	−0.971542	−0.485771	+	0.874086i	$0.661461\pi$
$744$	0.370415	0.0135801
$745$	−46.9564	−1.72035
$746$	17.7286	0.649090
$747$	4.88070	0.178576
$748$	−2.47128	−0.0903589
$749$	−13.2524	−0.484231
$750$	3.85879	0.140903
$751$	−4.79469	−0.174961	−0.0874803	−	0.996166i	$-0.527881\pi$
$751$	−4.79469	−0.174961	−0.0874803	+	0.996166i	$0.527881\pi$
$752$	−0.928054	−0.0338427
$753$	−7.35931	−0.268188
$754$	−14.3771	−0.523585
$755$	13.4985	0.491262
$756$	3.79273	0.137940
$757$	−51.5627	−1.87408	−0.937039	−	0.349225i	$-0.886445\pi$
$757$	−51.5627	−1.87408	−0.937039	+	0.349225i	$0.886445\pi$
$758$	−4.96321	−0.180272
$759$	−0.156520	−0.00568132
$760$	6.81922	0.247359
$761$	−14.9630	−0.542407	−0.271203	−	0.962522i	$-0.587422\pi$
$761$	−14.9630	−0.542407	−0.271203	+	0.962522i	$0.587422\pi$
$762$	4.08264	0.147898
$763$	25.1460	0.910345
$764$	11.3198	0.409535
$765$	−29.6123	−1.07064
$766$	15.6728	0.566283
$767$	−16.8602	−0.608787
$768$	−0.370415	−0.0133662
$769$	13.0920	0.472109	0.236055	−	0.971740i	$-0.424146\pi$
$769$	13.0920	0.472109	0.236055	+	0.971740i	$0.424146\pi$
$770$	−2.33510	−0.0841512
$771$	6.93143	0.249630
$772$	−12.7210	−0.457837
$773$	−1.30166	−0.0468173	−0.0234087	−	0.999726i	$-0.507452\pi$
$773$	−1.30166	−0.0468173	−0.0234087	+	0.999726i	$0.507452\pi$
$774$	−10.1888	−0.366228
$775$	−0.596390	−0.0214230
$776$	−1.00000	−0.0358979
$777$	−2.71511	−0.0974040
$778$	−26.2582	−0.941401
$779$	−8.98925	−0.322073
$780$	1.57979	0.0565657
$781$	−5.86892	−0.210007
$782$	−3.26901	−0.116900
$783$	17.3184	0.618910
$784$	−3.94988	−0.141067
$785$	−8.41054	−0.300185
$786$	−1.14445	−0.0408210
$787$	45.4195	1.61903	0.809516	−	0.587098i	$-0.199730\pi$
$787$	45.4195	1.61903	0.809516	+	0.587098i	$0.199730\pi$
$788$	−8.74955	−0.311690
$789$	−4.05062	−0.144206
$790$	−9.16819	−0.326190
$791$	22.3766	0.795620
$792$	1.61802	0.0574937
$793$	−8.10623	−0.287861
$794$	33.9878	1.20618
$795$	5.02744	0.178305
$796$	6.11035	0.216576
$797$	53.0886	1.88050	0.940248	−	0.340490i	$-0.110593\pi$
$797$	53.0886	1.88050	0.940248	+	0.340490i	$0.110593\pi$
$798$	−1.86478	−0.0660126
$799$	−4.05791	−0.143558
$800$	0.596390	0.0210856
$801$	−35.3431	−1.24879
$802$	6.73082	0.237674
$803$	−0.408974	−0.0144324
$804$	1.14370	0.0403353
$805$	−3.08888	−0.108869
$806$	1.80284	0.0635024
$807$	6.59163	0.232036
$808$	10.4494	0.367610
$809$	22.1606	0.779125	0.389562	−	0.921000i	$-0.372626\pi$
$809$	22.1606	0.779125	0.389562	+	0.921000i	$0.372626\pi$
$810$	18.4143	0.647012
$811$	−8.25673	−0.289933	−0.144967	−	0.989437i	$-0.546307\pi$
$811$	−8.25673	−0.289933	−0.144967	+	0.989437i	$0.546307\pi$
$812$	13.9275	0.488760
$813$	−2.64369	−0.0927181
$814$	−2.37210	−0.0831420
$815$	25.0877	0.878785
$816$	−1.61964	−0.0566986
$817$	10.2592	0.358924
$818$	−39.4666	−1.37992
$819$	9.01377	0.314967
$820$	−7.37729	−0.257626
$821$	22.9544	0.801115	0.400558	−	0.916272i	$-0.368816\pi$
$821$	22.9544	0.801115	0.400558	+	0.916272i	$0.368816\pi$
$822$	−7.10872	−0.247945
$823$	−2.73683	−0.0954000	−0.0477000	−	0.998862i	$-0.515189\pi$
$823$	−2.73683	−0.0954000	−0.0477000	+	0.998862i	$0.515189\pi$
$824$	16.8461	0.586862
$825$	0.124857	0.00434696
$826$	16.3329	0.568295
$827$	5.21597	0.181377	0.0906885	−	0.995879i	$-0.471093\pi$
$827$	5.21597	0.181377	0.0906885	+	0.995879i	$0.471093\pi$
$828$	2.14032	0.0743811
$829$	−27.0320	−0.938859	−0.469430	−	0.882970i	$-0.655540\pi$
$829$	−27.0320	−0.938859	−0.469430	+	0.882970i	$0.655540\pi$
$830$	−4.03317	−0.139993
$831$	−5.83588	−0.202444
$832$	−1.80284	−0.0625023
$833$	−17.2708	−0.598397
$834$	−0.595124	−0.0206075
$835$	−54.9684	−1.90226
$836$	−1.62920	−0.0563470
$837$	−2.17167	−0.0750638
$838$	12.7138	0.439190
$839$	−40.4865	−1.39775	−0.698875	−	0.715244i	$-0.746316\pi$
$839$	−40.4865	−1.39775	−0.698875	+	0.715244i	$0.746316\pi$
$840$	−1.53039	−0.0528034
$841$	34.5959	1.19296
$842$	−33.1149	−1.14121
$843$	−7.28257	−0.250825
$844$	−8.57087	−0.295022
$845$	−23.0647	−0.793450
$846$	2.65683	0.0913436
$847$	−18.6532	−0.640931
$848$	−5.73725	−0.197018
$849$	1.09058	0.0374288
$850$	2.60771	0.0894438
$851$	−3.13782	−0.107563
$852$	−3.84640	−0.131775
$853$	−11.1156	−0.380592	−0.190296	−	0.981727i	$-0.560945\pi$
$853$	−11.1156	−0.380592	−0.190296	+	0.981727i	$0.560945\pi$
$854$	7.85271	0.268714
$855$	−19.5220	−0.667639
$856$	−7.58813	−0.259357
$857$	−4.24955	−0.145162	−0.0725810	−	0.997363i	$-0.523124\pi$
$857$	−4.24955	−0.145162	−0.0725810	+	0.997363i	$0.523124\pi$
$858$	−0.377433	−0.0128853
$859$	39.7122	1.35496	0.677482	−	0.735540i	$-0.263071\pi$
$859$	39.7122	1.35496	0.677482	+	0.735540i	$0.263071\pi$
$860$	8.41950	0.287103
$861$	2.01739	0.0687526
$862$	−16.7178	−0.569411
$863$	−23.3265	−0.794043	−0.397021	−	0.917809i	$-0.629956\pi$
$863$	−23.3265	−0.794043	−0.397021	+	0.917809i	$0.629956\pi$
$864$	2.17167	0.0738816
$865$	−19.1095	−0.649743
$866$	20.7091	0.703724
$867$	−0.784787	−0.0266528
$868$	−1.74646	−0.0592787
$869$	2.19040	0.0743041
$870$	−6.98807	−0.236918
$871$	5.56650	0.188614
$872$	14.3982	0.487586
$873$	2.86279	0.0968908
$874$	−2.15511	−0.0728976
$875$	−18.1937	−0.615060
$876$	−0.268035	−0.00905606
$877$	−26.7052	−0.901771	−0.450886	−	0.892582i	$-0.648892\pi$
$877$	−26.7052	−0.901771	−0.450886	+	0.892582i	$0.648892\pi$
$878$	30.1985	1.01915
$879$	−11.3970	−0.384411
$880$	−1.33705	−0.0450719
$881$	−48.1788	−1.62318	−0.811592	−	0.584225i	$-0.801399\pi$
$881$	−48.1788	−1.62318	−0.811592	+	0.584225i	$0.801399\pi$
$882$	11.3077	0.380749
$883$	−43.1203	−1.45111	−0.725557	−	0.688162i	$-0.758418\pi$
$883$	−43.1203	−1.45111	−0.725557	+	0.688162i	$0.758418\pi$
$884$	−7.88291	−0.265131
$885$	−8.19498	−0.275471
$886$	31.8659	1.07055
$887$	−20.0637	−0.673675	−0.336837	−	0.941563i	$-0.609357\pi$
$887$	−20.0637	−0.673675	−0.336837	+	0.941563i	$0.609357\pi$
$888$	−1.55463	−0.0521701
$889$	−19.2491	−0.645595
$890$	29.2058	0.978980
$891$	−4.39940	−0.147385
$892$	13.3600	0.447326
$893$	−2.67519	−0.0895217
$894$	7.35241	0.245901
$895$	−12.8476	−0.429446
$896$	1.74646	0.0583452
$897$	−0.499269	−0.0166701
$898$	−3.32746	−0.111039
$899$	−7.97470	−0.265971
$900$	−1.70734	−0.0569114
$901$	−25.0861	−0.835738
$902$	1.76253	0.0586857
$903$	−2.30240	−0.0766190
$904$	12.8125	0.426139
$905$	−51.6701	−1.71757
$906$	−2.11359	−0.0702195
$907$	50.9835	1.69288	0.846439	−	0.532485i	$-0.178742\pi$
$907$	50.9835	1.69288	0.846439	+	0.532485i	$0.178742\pi$
$908$	−28.7817	−0.955155
$909$	−29.9146	−0.992204
$910$	−7.44853	−0.246916
$911$	11.0629	0.366530	0.183265	−	0.983064i	$-0.441333\pi$
$911$	11.0629	0.366530	0.183265	+	0.983064i	$0.441333\pi$
$912$	−1.06775	−0.0353568
$913$	0.963575	0.0318897
$914$	−15.6941	−0.519114
$915$	−3.94007	−0.130255
$916$	−9.76681	−0.322705
$917$	5.39591	0.178189
$918$	9.49560	0.313401
$919$	8.26554	0.272655	0.136327	−	0.990664i	$-0.456470\pi$
$919$	8.26554	0.272655	0.136327	+	0.990664i	$0.456470\pi$
$920$	−1.76865	−0.0583107
$921$	9.38259	0.309167
$922$	−20.1840	−0.664726
$923$	−18.7208	−0.616201
$924$	0.365629	0.0120283
$925$	2.50305	0.0822999
$926$	22.1096	0.726568
$927$	−48.2269	−1.58398
$928$	7.97470	0.261783
$929$	19.1331	0.627737	0.313868	−	0.949466i	$-0.398375\pi$
$929$	19.1331	0.627737	0.313868	+	0.949466i	$0.398375\pi$
$930$	0.876280	0.0287344
$931$	−11.3858	−0.373155
$932$	10.7189	0.351108
$933$	−3.61127	−0.118228
$934$	2.40009	0.0785333
$935$	−5.84623	−0.191192
$936$	5.16116	0.168698
$937$	−19.8042	−0.646976	−0.323488	−	0.946232i	$-0.604856\pi$
$937$	−19.8042	−0.646976	−0.323488	+	0.946232i	$0.604856\pi$
$938$	−5.39241	−0.176068
$939$	10.1861	0.332411
$940$	−2.19547	−0.0716083
$941$	47.3539	1.54369	0.771847	−	0.635808i	$-0.219333\pi$
$941$	47.3539	1.54369	0.771847	+	0.635808i	$0.219333\pi$
$942$	1.31692	0.0429075
$943$	2.33148	0.0759233
$944$	9.35201	0.304382
$945$	8.97235	0.291871
$946$	−2.01153	−0.0654003
$947$	45.6894	1.48471	0.742353	−	0.670009i	$-0.233710\pi$
$947$	45.6894	1.48471	0.742353	+	0.670009i	$0.233710\pi$
$948$	1.43555	0.0466245
$949$	−1.30455	−0.0423474
$950$	1.71914	0.0557763
$951$	0.976103	0.0316523
$952$	7.63638	0.247497
$953$	−49.0426	−1.58865	−0.794323	−	0.607495i	$-0.792174\pi$
$953$	−49.0426	−1.58865	−0.794323	+	0.607495i	$0.792174\pi$
$954$	16.4246	0.531765
$955$	26.7788	0.866543
$956$	27.0901	0.876156
$957$	1.66954	0.0539685
$958$	5.89438	0.190439
$959$	33.5167	1.08231
$960$	−0.876280	−0.0282818
$961$	1.00000	0.0322581
$962$	−7.56654	−0.243955
$963$	21.7232	0.700022
$964$	12.9919	0.418442
$965$	−30.0936	−0.968747
$966$	0.483655	0.0155613
$967$	58.3282	1.87571	0.937855	−	0.347029i	$-0.112809\pi$
$967$	58.3282	1.87571	0.937855	+	0.347029i	$0.112809\pi$
$968$	−10.6806	−0.343286
$969$	−4.66873	−0.149981
$970$	−2.36567	−0.0759571
$971$	−56.8796	−1.82535	−0.912676	−	0.408684i	$-0.865988\pi$
$971$	−56.8796	−1.82535	−0.912676	+	0.408684i	$0.865988\pi$
$972$	−9.39830	−0.301451
$973$	2.80593	0.0899541
$974$	−12.9026	−0.413424
$975$	0.398270	0.0127548
$976$	4.49636	0.143925
$977$	7.60771	0.243392	0.121696	−	0.992567i	$-0.461167\pi$
$977$	7.60771	0.243392	0.121696	+	0.992567i	$0.461167\pi$
$978$	−3.92823	−0.125611
$979$	−6.97763	−0.223006
$980$	−9.34410	−0.298486
$981$	−41.2192	−1.31603
$982$	−30.1741	−0.962895
$983$	43.7675	1.39597	0.697983	−	0.716114i	$-0.254081\pi$
$983$	43.7675	1.39597	0.697983	+	0.716114i	$0.254081\pi$
$984$	1.15513	0.0368243
$985$	−20.6985	−0.659511
$986$	34.8693	1.11047
$987$	0.600373	0.0191101
$988$	−5.19683	−0.165333
$989$	−2.66085	−0.0846101
$990$	3.82769	0.121652
$991$	−3.67290	−0.116673	−0.0583367	−	0.998297i	$-0.518580\pi$
$991$	−3.67290	−0.116673	−0.0583367	+	0.998297i	$0.518580\pi$
$992$	−1.00000	−0.0317500
$993$	−2.84796	−0.0903771
$994$	18.1353	0.575216
$995$	14.4551	0.458256
$996$	0.631512	0.0200102
$997$	−14.9414	−0.473199	−0.236600	−	0.971607i	$-0.576033\pi$
$997$	−14.9414	−0.473199	−0.236600	+	0.971607i	$0.576033\pi$
$998$	−24.9132	−0.788613
$999$	9.11450	0.288370

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6014.2.a.i.1.12	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6014.2.a.i.1.12	✓	28	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:	\( N \)	\(=\)	\( 6014 = 2 \cdot 31 \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.370415 q^{3} +1.00000 q^{4} +2.36567 q^{5} -0.370415 q^{6} +1.74646 q^{7} +1.00000 q^{8} -2.86279 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.370415 q^{3} +1.00000 q^{4} +2.36567 q^{5} -0.370415 q^{6} +1.74646 q^{7} +1.00000 q^{8} -2.86279 q^{9} +2.36567 q^{10} -0.565188 q^{11} -0.370415 q^{12} -1.80284 q^{13} +1.74646 q^{14} -0.876280 q^{15} +1.00000 q^{16} +4.37249 q^{17} -2.86279 q^{18} +2.88258 q^{19} +2.36567 q^{20} -0.646916 q^{21} -0.565188 q^{22} -0.747632 q^{23} -0.370415 q^{24} +0.596390 q^{25} -1.80284 q^{26} +2.17167 q^{27} +1.74646 q^{28} +7.97470 q^{29} -0.876280 q^{30} -1.00000 q^{31} +1.00000 q^{32} +0.209354 q^{33} +4.37249 q^{34} +4.13155 q^{35} -2.86279 q^{36} +4.19701 q^{37} +2.88258 q^{38} +0.667800 q^{39} +2.36567 q^{40} -3.11848 q^{41} -0.646916 q^{42} +3.55904 q^{43} -0.565188 q^{44} -6.77242 q^{45} -0.747632 q^{46} -0.928054 q^{47} -0.370415 q^{48} -3.94988 q^{49} +0.596390 q^{50} -1.61964 q^{51} -1.80284 q^{52} -5.73725 q^{53} +2.17167 q^{54} -1.33705 q^{55} +1.74646 q^{56} -1.06775 q^{57} +7.97470 q^{58} +9.35201 q^{59} -0.876280 q^{60} +4.49636 q^{61} -1.00000 q^{62} -4.99975 q^{63} +1.00000 q^{64} -4.26493 q^{65} +0.209354 q^{66} -3.08762 q^{67} +4.37249 q^{68} +0.276934 q^{69} +4.13155 q^{70} +10.3840 q^{71} -2.86279 q^{72} +0.723606 q^{73} +4.19701 q^{74} -0.220912 q^{75} +2.88258 q^{76} -0.987079 q^{77} +0.667800 q^{78} -3.87552 q^{79} +2.36567 q^{80} +7.78396 q^{81} -3.11848 q^{82} -1.70487 q^{83} -0.646916 q^{84} +10.3439 q^{85} +3.55904 q^{86} -2.95395 q^{87} -0.565188 q^{88} +12.3457 q^{89} -6.77242 q^{90} -3.14859 q^{91} -0.747632 q^{92} +0.370415 q^{93} -0.928054 q^{94} +6.81922 q^{95} -0.370415 q^{96} -1.00000 q^{97} -3.94988 q^{98} +1.61802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} + 12 q^{3} + 28 q^{4} + 10 q^{5} + 12 q^{6} + 13 q^{7} + 28 q^{8} + 38 q^{9}+O(q^{10}) \) 28 * q + 28 * q^2 + 12 * q^3 + 28 * q^4 + 10 * q^5 + 12 * q^6 + 13 * q^7 + 28 * q^8 + 38 * q^9	\( 28 q + 28 q^{2} + 12 q^{3} + 28 q^{4} + 10 q^{5} + 12 q^{6} + 13 q^{7} + 28 q^{8} + 38 q^{9} + 10 q^{10} + 12 q^{11} + 12 q^{12} + 20 q^{13} + 13 q^{14} + 19 q^{15} + 28 q^{16} - 4 q^{17} + 38 q^{18} + 35 q^{19} + 10 q^{20} + 30 q^{21} + 12 q^{22} + 20 q^{23} + 12 q^{24} + 46 q^{25} + 20 q^{26} + 39 q^{27} + 13 q^{28} + 5 q^{29} + 19 q^{30} - 28 q^{31} + 28 q^{32} + 12 q^{33} - 4 q^{34} + 36 q^{35} + 38 q^{36} + 11 q^{37} + 35 q^{38} - 4 q^{39} + 10 q^{40} - 5 q^{41} + 30 q^{42} + 43 q^{43} + 12 q^{44} + 11 q^{45} + 20 q^{46} + 18 q^{47} + 12 q^{48} + 99 q^{49} + 46 q^{50} - 43 q^{51} + 20 q^{52} + 11 q^{53} + 39 q^{54} + 66 q^{55} + 13 q^{56} - 15 q^{57} + 5 q^{58} + 34 q^{59} + 19 q^{60} + 66 q^{61} - 28 q^{62} + 65 q^{63} + 28 q^{64} - 16 q^{65} + 12 q^{66} + 5 q^{67} - 4 q^{68} - 33 q^{69} + 36 q^{70} + 25 q^{71} + 38 q^{72} - 9 q^{73} + 11 q^{74} + 92 q^{75} + 35 q^{76} + 4 q^{77} - 4 q^{78} + 15 q^{79} + 10 q^{80} - 5 q^{82} - 12 q^{83} + 30 q^{84} + 88 q^{85} + 43 q^{86} + 31 q^{87} + 12 q^{88} + 8 q^{89} + 11 q^{90} + 34 q^{91} + 20 q^{92} - 12 q^{93} + 18 q^{94} + 32 q^{95} + 12 q^{96} - 28 q^{97} + 99 q^{98} + 51 q^{99}+O(q^{100}) \) 28 * q + 28 * q^2 + 12 * q^3 + 28 * q^4 + 10 * q^5 + 12 * q^6 + 13 * q^7 + 28 * q^8 + 38 * q^9 + 10 * q^10 + 12 * q^11 + 12 * q^12 + 20 * q^13 + 13 * q^14 + 19 * q^15 + 28 * q^16 - 4 * q^17 + 38 * q^18 + 35 * q^19 + 10 * q^20 + 30 * q^21 + 12 * q^22 + 20 * q^23 + 12 * q^24 + 46 * q^25 + 20 * q^26 + 39 * q^27 + 13 * q^28 + 5 * q^29 + 19 * q^30 - 28 * q^31 + 28 * q^32 + 12 * q^33 - 4 * q^34 + 36 * q^35 + 38 * q^36 + 11 * q^37 + 35 * q^38 - 4 * q^39 + 10 * q^40 - 5 * q^41 + 30 * q^42 + 43 * q^43 + 12 * q^44 + 11 * q^45 + 20 * q^46 + 18 * q^47 + 12 * q^48 + 99 * q^49 + 46 * q^50 - 43 * q^51 + 20 * q^52 + 11 * q^53 + 39 * q^54 + 66 * q^55 + 13 * q^56 - 15 * q^57 + 5 * q^58 + 34 * q^59 + 19 * q^60 + 66 * q^61 - 28 * q^62 + 65 * q^63 + 28 * q^64 - 16 * q^65 + 12 * q^66 + 5 * q^67 - 4 * q^68 - 33 * q^69 + 36 * q^70 + 25 * q^71 + 38 * q^72 - 9 * q^73 + 11 * q^74 + 92 * q^75 + 35 * q^76 + 4 * q^77 - 4 * q^78 + 15 * q^79 + 10 * q^80 - 5 * q^82 - 12 * q^83 + 30 * q^84 + 88 * q^85 + 43 * q^86 + 31 * q^87 + 12 * q^88 + 8 * q^89 + 11 * q^90 + 34 * q^91 + 20 * q^92 - 12 * q^93 + 18 * q^94 + 32 * q^95 + 12 * q^96 - 28 * q^97 + 99 * q^98 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more