LMFDB - Embedded newform 4007.2.a.a.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4007,2,Mod(1,4007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4007, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4007 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4007.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:	$31.9960560899$
Analytic rank:	$1$
Dimension:	$139$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	4007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24444 q^{2} -1.81943 q^{3} +3.03751 q^{4} +1.76577 q^{5} +4.08359 q^{6} +0.689140 q^{7} -2.32862 q^{8} +0.310309 q^{9} +O(q^{10})$	\(q-2.24444 q^{2} -1.81943 q^{3} +3.03751 q^{4} +1.76577 q^{5} +4.08359 q^{6} +0.689140 q^{7} -2.32862 q^{8} +0.310309 q^{9} -3.96317 q^{10} -4.87681 q^{11} -5.52652 q^{12} +2.16250 q^{13} -1.54673 q^{14} -3.21270 q^{15} -0.848559 q^{16} -4.16658 q^{17} -0.696471 q^{18} +3.29423 q^{19} +5.36356 q^{20} -1.25384 q^{21} +10.9457 q^{22} +5.32922 q^{23} +4.23676 q^{24} -1.88204 q^{25} -4.85359 q^{26} +4.89369 q^{27} +2.09327 q^{28} +0.0325181 q^{29} +7.21070 q^{30} -1.94624 q^{31} +6.56179 q^{32} +8.87300 q^{33} +9.35163 q^{34} +1.21687 q^{35} +0.942567 q^{36} +3.57382 q^{37} -7.39371 q^{38} -3.93450 q^{39} -4.11183 q^{40} +5.78254 q^{41} +2.81417 q^{42} -11.4231 q^{43} -14.8134 q^{44} +0.547936 q^{45} -11.9611 q^{46} +5.99308 q^{47} +1.54389 q^{48} -6.52509 q^{49} +4.22412 q^{50} +7.58078 q^{51} +6.56860 q^{52} +5.62083 q^{53} -10.9836 q^{54} -8.61135 q^{55} -1.60475 q^{56} -5.99361 q^{57} -0.0729848 q^{58} +14.7725 q^{59} -9.75859 q^{60} -14.2665 q^{61} +4.36821 q^{62} +0.213847 q^{63} -13.0304 q^{64} +3.81848 q^{65} -19.9149 q^{66} -5.18380 q^{67} -12.6560 q^{68} -9.69611 q^{69} -2.73118 q^{70} +6.16509 q^{71} -0.722594 q^{72} -6.45176 q^{73} -8.02123 q^{74} +3.42423 q^{75} +10.0063 q^{76} -3.36081 q^{77} +8.83075 q^{78} +4.45613 q^{79} -1.49836 q^{80} -9.83464 q^{81} -12.9786 q^{82} -15.2792 q^{83} -3.80855 q^{84} -7.35724 q^{85} +25.6385 q^{86} -0.0591642 q^{87} +11.3563 q^{88} -10.6436 q^{89} -1.22981 q^{90} +1.49026 q^{91} +16.1875 q^{92} +3.54104 q^{93} -13.4511 q^{94} +5.81687 q^{95} -11.9387 q^{96} +16.3375 q^{97} +14.6452 q^{98} -1.51332 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9}+O(q^{10}) $ 139 * q - 13 * q^2 - 22 * q^3 + 113 * q^4 - 16 * q^5 - 15 * q^6 - 44 * q^7 - 36 * q^8 + 87 * q^9	\( 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9} - 40 q^{10} - 17 q^{11} - 59 q^{12} - 89 q^{13} - 15 q^{14} - 29 q^{15} + 73 q^{16} - 58 q^{17} - 51 q^{18} - 37 q^{19} - 24 q^{20} - 37 q^{21} - 99 q^{22} - 42 q^{23} - 27 q^{24} - 11 q^{25} + 2 q^{26} - 73 q^{27} - 113 q^{28} - 57 q^{29} - 29 q^{30} - 51 q^{31} - 80 q^{32} - 78 q^{33} - 28 q^{34} - 34 q^{35} + 28 q^{36} - 117 q^{37} - 31 q^{38} - 36 q^{39} - 107 q^{40} - 60 q^{41} - 41 q^{42} - 109 q^{43} - 21 q^{44} - 62 q^{45} - 92 q^{46} - 26 q^{47} - 90 q^{48} - 7 q^{49} - 22 q^{50} - 47 q^{51} - 182 q^{52} - 83 q^{53} - 19 q^{54} - 53 q^{55} - 23 q^{56} - 201 q^{57} - 112 q^{58} + 14 q^{59} - 64 q^{60} - 73 q^{61} - 21 q^{62} - 94 q^{63} + 14 q^{64} - 123 q^{65} - 10 q^{66} - 135 q^{67} - 84 q^{68} - 50 q^{69} - 35 q^{70} - 29 q^{71} - 143 q^{72} - 266 q^{73} - 53 q^{74} - 32 q^{75} - 66 q^{76} - 69 q^{77} - 59 q^{78} - 124 q^{79} - 20 q^{80} - 33 q^{81} - 93 q^{82} - 28 q^{83} - 4 q^{84} - 179 q^{85} + 6 q^{86} - 40 q^{87} - 259 q^{88} - 41 q^{89} + 2 q^{90} - 50 q^{91} - 77 q^{92} - 60 q^{93} - 48 q^{94} - 37 q^{95} + 3 q^{96} - 220 q^{97} - 9 q^{98} - 35 q^{99}+O(q^{100}) \) 139 * q - 13 * q^2 - 22 * q^3 + 113 * q^4 - 16 * q^5 - 15 * q^6 - 44 * q^7 - 36 * q^8 + 87 * q^9 - 40 * q^10 - 17 * q^11 - 59 * q^12 - 89 * q^13 - 15 * q^14 - 29 * q^15 + 73 * q^16 - 58 * q^17 - 51 * q^18 - 37 * q^19 - 24 * q^20 - 37 * q^21 - 99 * q^22 - 42 * q^23 - 27 * q^24 - 11 * q^25 + 2 * q^26 - 73 * q^27 - 113 * q^28 - 57 * q^29 - 29 * q^30 - 51 * q^31 - 80 * q^32 - 78 * q^33 - 28 * q^34 - 34 * q^35 + 28 * q^36 - 117 * q^37 - 31 * q^38 - 36 * q^39 - 107 * q^40 - 60 * q^41 - 41 * q^42 - 109 * q^43 - 21 * q^44 - 62 * q^45 - 92 * q^46 - 26 * q^47 - 90 * q^48 - 7 * q^49 - 22 * q^50 - 47 * q^51 - 182 * q^52 - 83 * q^53 - 19 * q^54 - 53 * q^55 - 23 * q^56 - 201 * q^57 - 112 * q^58 + 14 * q^59 - 64 * q^60 - 73 * q^61 - 21 * q^62 - 94 * q^63 + 14 * q^64 - 123 * q^65 - 10 * q^66 - 135 * q^67 - 84 * q^68 - 50 * q^69 - 35 * q^70 - 29 * q^71 - 143 * q^72 - 266 * q^73 - 53 * q^74 - 32 * q^75 - 66 * q^76 - 69 * q^77 - 59 * q^78 - 124 * q^79 - 20 * q^80 - 33 * q^81 - 93 * q^82 - 28 * q^83 - 4 * q^84 - 179 * q^85 + 6 * q^86 - 40 * q^87 - 259 * q^88 - 41 * q^89 + 2 * q^90 - 50 * q^91 - 77 * q^92 - 60 * q^93 - 48 * q^94 - 37 * q^95 + 3 * q^96 - 220 * q^97 - 9 * q^98 - 35 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.24444	−1.58706	−0.793529	−	0.608532i	$-0.791759\pi$
$2$	−2.24444	−1.58706	−0.793529	+	0.608532i	$0.791759\pi$
$3$	−1.81943	−1.05045	−0.525223	−	0.850965i	$-0.676018\pi$
$3$	−1.81943	−1.05045	−0.525223	+	0.850965i	$0.676018\pi$
$4$	3.03751	1.51875
$5$	1.76577	0.789679	0.394839	−	0.918750i	$-0.370800\pi$
$5$	1.76577	0.789679	0.394839	+	0.918750i	$0.370800\pi$
$6$	4.08359	1.66712
$7$	0.689140	0.260470	0.130235	−	0.991483i	$-0.458427\pi$
$7$	0.689140	0.260470	0.130235	+	0.991483i	$0.458427\pi$
$8$	−2.32862	−0.823293
$9$	0.310309	0.103436
$10$	−3.96317	−1.25327
$11$	−4.87681	−1.47041	−0.735207	−	0.677842i	$-0.762915\pi$
$11$	−4.87681	−1.47041	−0.735207	+	0.677842i	$0.762915\pi$
$12$	−5.52652	−1.59537
$13$	2.16250	0.599768	0.299884	−	0.953976i	$-0.403052\pi$
$13$	2.16250	0.599768	0.299884	+	0.953976i	$0.403052\pi$
$14$	−1.54673	−0.413382
$15$	−3.21270	−0.829515
$16$	−0.848559	−0.212140
$17$	−4.16658	−1.01054	−0.505272	−	0.862960i	$-0.668608\pi$
$17$	−4.16658	−1.01054	−0.505272	+	0.862960i	$0.668608\pi$
$18$	−0.696471	−0.164160
$19$	3.29423	0.755749	0.377874	−	0.925857i	$-0.376655\pi$
$19$	3.29423	0.755749	0.377874	+	0.925857i	$0.376655\pi$
$20$	5.36356	1.19933
$21$	−1.25384	−0.273610
$22$	10.9457	2.33363
$23$	5.32922	1.11122	0.555609	−	0.831444i	$-0.312485\pi$
$23$	5.32922	1.11122	0.555609	+	0.831444i	$0.312485\pi$
$24$	4.23676	0.864825
$25$	−1.88204	−0.376408
$26$	−4.85359	−0.951867
$27$	4.89369	0.941791
$28$	2.09327	0.395591
$29$	0.0325181	0.00603845	0.00301923	−	0.999995i	$-0.499039\pi$
$29$	0.0325181	0.00603845	0.00301923	+	0.999995i	$0.499039\pi$
$30$	7.21070	1.31649
$31$	−1.94624	−0.349555	−0.174777	−	0.984608i	$-0.555921\pi$
$31$	−1.94624	−0.349555	−0.174777	+	0.984608i	$0.555921\pi$
$32$	6.56179	1.15997
$33$	8.87300	1.54459
$34$	9.35163	1.60379
$35$	1.21687	0.205688
$36$	0.942567	0.157095
$37$	3.57382	0.587533	0.293766	−	0.955877i	$-0.405091\pi$
$37$	3.57382	0.587533	0.293766	+	0.955877i	$0.405091\pi$
$38$	−7.39371	−1.19942
$39$	−3.93450	−0.630024
$40$	−4.11183	−0.650137
$41$	5.78254	0.903081	0.451541	−	0.892251i	$-0.350875\pi$
$41$	5.78254	0.903081	0.451541	+	0.892251i	$0.350875\pi$
$42$	2.81417	0.434235
$43$	−11.4231	−1.74201	−0.871006	−	0.491272i	$-0.836532\pi$
$43$	−11.4231	−1.74201	−0.871006	+	0.491272i	$0.836532\pi$
$44$	−14.8134	−2.23320
$45$	0.547936	0.0816815
$46$	−11.9611	−1.76357
$47$	5.99308	0.874181	0.437091	−	0.899417i	$-0.356009\pi$
$47$	5.99308	0.874181	0.437091	+	0.899417i	$0.356009\pi$
$48$	1.54389	0.222841
$49$	−6.52509	−0.932155
$50$	4.22412	0.597381
$51$	7.58078	1.06152
$52$	6.56860	0.910901
$53$	5.62083	0.772081	0.386040	−	0.922482i	$-0.373843\pi$
$53$	5.62083	0.772081	0.386040	+	0.922482i	$0.373843\pi$
$54$	−10.9836	−1.49468
$55$	−8.61135	−1.16115
$56$	−1.60475	−0.214444
$57$	−5.99361	−0.793873
$58$	−0.0729848	−0.00958338
$59$	14.7725	1.92322	0.961608	−	0.274428i	$-0.0884886\pi$
$59$	14.7725	1.92322	0.961608	+	0.274428i	$0.0884886\pi$
$60$	−9.75859	−1.25983
$61$	−14.2665	−1.82663	−0.913316	−	0.407251i	$-0.866487\pi$
$61$	−14.2665	−1.82663	−0.913316	+	0.407251i	$0.866487\pi$
$62$	4.36821	0.554764
$63$	0.213847	0.0269421
$64$	−13.0304	−1.62880
$65$	3.81848	0.473624
$66$	−19.9149	−2.45136
$67$	−5.18380	−0.633302	−0.316651	−	0.948542i	$-0.602558\pi$
$67$	−5.18380	−0.633302	−0.316651	+	0.948542i	$0.602558\pi$
$68$	−12.6560	−1.53477
$69$	−9.69611	−1.16727
$70$	−2.73118	−0.326439
$71$	6.16509	0.731662	0.365831	−	0.930681i	$-0.380785\pi$
$71$	6.16509	0.731662	0.365831	+	0.930681i	$0.380785\pi$
$72$	−0.722594	−0.0851585
$73$	−6.45176	−0.755121	−0.377561	−	0.925985i	$-0.623237\pi$
$73$	−6.45176	−0.755121	−0.377561	+	0.925985i	$0.623237\pi$
$74$	−8.02123	−0.932449
$75$	3.42423	0.395396
$76$	10.0063	1.14780
$77$	−3.36081	−0.383000
$78$	8.83075	0.999885
$79$	4.45613	0.501353	0.250677	−	0.968071i	$-0.419347\pi$
$79$	4.45613	0.501353	0.250677	+	0.968071i	$0.419347\pi$
$80$	−1.49836	−0.167522
$81$	−9.83464	−1.09274
$82$	−12.9786	−1.43324
$83$	−15.2792	−1.67711	−0.838555	−	0.544817i	$-0.816599\pi$
$83$	−15.2792	−1.67711	−0.838555	+	0.544817i	$0.816599\pi$
$84$	−3.80855	−0.415547
$85$	−7.35724	−0.798004
$86$	25.6385	2.76467
$87$	−0.0591642	−0.00634307
$88$	11.3563	1.21058
$89$	−10.6436	−1.12822	−0.564112	−	0.825698i	$-0.690781\pi$
$89$	−10.6436	−1.12822	−0.564112	+	0.825698i	$0.690781\pi$
$90$	−1.22981	−0.129633
$91$	1.49026	0.156222
$92$	16.1875	1.68767
$93$	3.54104	0.367188
$94$	−13.4511	−1.38738
$95$	5.81687	0.596799
$96$	−11.9387	−1.21849
$97$	16.3375	1.65883	0.829413	−	0.558637i	$-0.188675\pi$
$97$	16.3375	1.65883	0.829413	+	0.558637i	$0.188675\pi$
$98$	14.6452	1.47938
$99$	−1.51332	−0.152094
$100$	−5.71671	−0.571671
$101$	8.89516	0.885102	0.442551	−	0.896743i	$-0.354074\pi$
$101$	8.89516	0.885102	0.442551	+	0.896743i	$0.354074\pi$
$102$	−17.0146	−1.68470
$103$	2.92915	0.288618	0.144309	−	0.989533i	$-0.453904\pi$
$103$	2.92915	0.288618	0.144309	+	0.989533i	$0.453904\pi$
$104$	−5.03564	−0.493785
$105$	−2.21400	−0.216064
$106$	−12.6156	−1.22534
$107$	−14.1200	−1.36503	−0.682517	−	0.730870i	$-0.739115\pi$
$107$	−14.1200	−1.36503	−0.682517	+	0.730870i	$0.739115\pi$
$108$	14.8646	1.43035
$109$	5.35808	0.513211	0.256606	−	0.966516i	$-0.417396\pi$
$109$	5.35808	0.513211	0.256606	+	0.966516i	$0.417396\pi$
$110$	19.3277	1.84282
$111$	−6.50230	−0.617171
$112$	−0.584776	−0.0552562
$113$	2.68044	0.252155	0.126077	−	0.992020i	$-0.459761\pi$
$113$	2.68044	0.252155	0.126077	+	0.992020i	$0.459761\pi$
$114$	13.4523	1.25992
$115$	9.41020	0.877505
$116$	0.0987739	0.00917093
$117$	0.671043	0.0620379
$118$	−33.1560	−3.05225
$119$	−2.87136	−0.263217
$120$	7.48116	0.682934
$121$	12.7833	1.16212
$122$	32.0202	2.89897
$123$	−10.5209	−0.948638
$124$	−5.91172	−0.530888
$125$	−12.1521	−1.08692
$126$	−0.479966	−0.0427588
$127$	−1.49675	−0.132815	−0.0664077	−	0.997793i	$-0.521154\pi$
$127$	−1.49675	−0.132815	−0.0664077	+	0.997793i	$0.521154\pi$
$128$	16.1224	1.42503
$129$	20.7835	1.82989
$130$	−8.57035	−0.751669
$131$	18.7621	1.63925	0.819627	−	0.572898i	$-0.194181\pi$
$131$	18.7621	1.63925	0.819627	+	0.572898i	$0.194181\pi$
$132$	26.9518	2.34585
$133$	2.27019	0.196850
$134$	11.6347	1.00509
$135$	8.64116	0.743713
$136$	9.70239	0.831973
$137$	−10.1776	−0.869533	−0.434767	−	0.900543i	$-0.643169\pi$
$137$	−10.1776	−0.869533	−0.434767	+	0.900543i	$0.643169\pi$
$138$	21.7623	1.85253
$139$	−16.3738	−1.38881	−0.694403	−	0.719586i	$-0.744332\pi$
$139$	−16.3738	−1.38881	−0.694403	+	0.719586i	$0.744332\pi$
$140$	3.69624	0.312389
$141$	−10.9040	−0.918280
$142$	−13.8372	−1.16119
$143$	−10.5461	−0.881908
$144$	−0.263316	−0.0219430
$145$	0.0574196	0.00476844
$146$	14.4806	1.19842
$147$	11.8719	0.979178
$148$	10.8555	0.892318
$149$	12.7565	1.04505	0.522527	−	0.852623i	$-0.324989\pi$
$149$	12.7565	1.04505	0.522527	+	0.852623i	$0.324989\pi$
$150$	−7.68548	−0.627517
$151$	2.19587	0.178697	0.0893486	−	0.996000i	$-0.471521\pi$
$151$	2.19587	0.178697	0.0893486	+	0.996000i	$0.471521\pi$
$152$	−7.67103	−0.622203
$153$	−1.29293	−0.104527
$154$	7.54313	0.607843
$155$	−3.43662	−0.276036
$156$	−11.9511	−0.956852
$157$	6.66988	0.532315	0.266157	−	0.963930i	$-0.414246\pi$
$157$	6.66988	0.532315	0.266157	+	0.963930i	$0.414246\pi$
$158$	−10.0015	−0.795677
$159$	−10.2267	−0.811029
$160$	11.5866	0.916005
$161$	3.67258	0.289440
$162$	22.0732	1.73424
$163$	11.0890	0.868560	0.434280	−	0.900778i	$-0.357003\pi$
$163$	11.0890	0.868560	0.434280	+	0.900778i	$0.357003\pi$
$164$	17.5645	1.37156
$165$	15.6677	1.21973
$166$	34.2932	2.66167
$167$	6.66713	0.515918	0.257959	−	0.966156i	$-0.416950\pi$
$167$	6.66713	0.515918	0.257959	+	0.966156i	$0.416950\pi$
$168$	2.91972	0.225261
$169$	−8.32361	−0.640278
$170$	16.5129	1.26648
$171$	1.02223	0.0781720
$172$	−34.6979	−2.64569
$173$	2.97162	0.225928	0.112964	−	0.993599i	$-0.463965\pi$
$173$	2.97162	0.225928	0.112964	+	0.993599i	$0.463965\pi$
$174$	0.132790	0.0100668
$175$	−1.29699	−0.0980431
$176$	4.13827	0.311934
$177$	−26.8775	−2.02023
$178$	23.8890	1.79056
$179$	9.54067	0.713103	0.356552	−	0.934276i	$-0.383952\pi$
$179$	9.54067	0.713103	0.356552	+	0.934276i	$0.383952\pi$
$180$	1.66436	0.124054
$181$	0.535499	0.0398034	0.0199017	−	0.999802i	$-0.493665\pi$
$181$	0.535499	0.0398034	0.0199017	+	0.999802i	$0.493665\pi$
$182$	−3.34480	−0.247933
$183$	25.9568	1.91878
$184$	−12.4097	−0.914859
$185$	6.31057	0.463962
$186$	−7.94764	−0.582749
$187$	20.3196	1.48592
$188$	18.2040	1.32767
$189$	3.37244	0.245309
$190$	−13.0556	−0.947154
$191$	18.8929	1.36704	0.683521	−	0.729931i	$-0.260448\pi$
$191$	18.8929	1.36704	0.683521	+	0.729931i	$0.260448\pi$
$192$	23.7079	1.71097
$193$	−21.5864	−1.55382	−0.776911	−	0.629611i	$-0.783214\pi$
$193$	−21.5864	−1.55382	−0.776911	+	0.629611i	$0.783214\pi$
$194$	−36.6686	−2.63265
$195$	−6.94744	−0.497517
$196$	−19.8200	−1.41571
$197$	−9.04408	−0.644364	−0.322182	−	0.946678i	$-0.604416\pi$
$197$	−9.04408	−0.644364	−0.322182	+	0.946678i	$0.604416\pi$
$198$	3.39656	0.241383
$199$	−2.40483	−0.170474	−0.0852369	−	0.996361i	$-0.527165\pi$
$199$	−2.40483	−0.170474	−0.0852369	+	0.996361i	$0.527165\pi$
$200$	4.38256	0.309894
$201$	9.43154	0.665250
$202$	−19.9647	−1.40471
$203$	0.0224095	0.00157284
$204$	23.0267	1.61219
$205$	10.2107	0.713144
$206$	−6.57430	−0.458053
$207$	1.65371	0.114940
$208$	−1.83501	−0.127235
$209$	−16.0654	−1.11126
$210$	4.96918	0.342906
$211$	22.5325	1.55120	0.775601	−	0.631223i	$-0.217447\pi$
$211$	22.5325	1.55120	0.775601	+	0.631223i	$0.217447\pi$
$212$	17.0733	1.17260
$213$	−11.2169	−0.768571
$214$	31.6915	2.16639
$215$	−20.1707	−1.37563
$216$	−11.3956	−0.775370
$217$	−1.34123	−0.0910487
$218$	−12.0259	−0.814496
$219$	11.7385	0.793214
$220$	−26.1571	−1.76351
$221$	−9.01020	−0.606092
$222$	14.5940	0.979487
$223$	−26.8387	−1.79725	−0.898625	−	0.438718i	$-0.855433\pi$
$223$	−26.8387	−1.79725	−0.898625	+	0.438718i	$0.855433\pi$
$224$	4.52199	0.302138
$225$	−0.584014	−0.0389343
$226$	−6.01609	−0.400184
$227$	−21.4035	−1.42060	−0.710300	−	0.703899i	$-0.751441\pi$
$227$	−21.4035	−1.42060	−0.710300	+	0.703899i	$0.751441\pi$
$228$	−18.2056	−1.20570
$229$	−9.93861	−0.656762	−0.328381	−	0.944545i	$-0.606503\pi$
$229$	−9.93861	−0.656762	−0.328381	+	0.944545i	$0.606503\pi$
$230$	−21.1206	−1.39265
$231$	6.11474	0.402320
$232$	−0.0757224	−0.00497142
$233$	−11.6866	−0.765615	−0.382807	−	0.923828i	$-0.625043\pi$
$233$	−11.6866	−0.765615	−0.382807	+	0.923828i	$0.625043\pi$
$234$	−1.50611	−0.0984578
$235$	10.5824	0.690322
$236$	44.8716	2.92089
$237$	−8.10759	−0.526645
$238$	6.44458	0.417740
$239$	−27.3874	−1.77155	−0.885773	−	0.464118i	$-0.846371\pi$
$239$	−27.3874	−1.77155	−0.885773	+	0.464118i	$0.846371\pi$
$240$	2.72616	0.175973
$241$	−13.3238	−0.858261	−0.429131	−	0.903242i	$-0.641180\pi$
$241$	−13.3238	−0.858261	−0.429131	+	0.903242i	$0.641180\pi$
$242$	−28.6914	−1.84435
$243$	3.21231	0.206070
$244$	−43.3345	−2.77421
$245$	−11.5218	−0.736103
$246$	23.6135	1.50554
$247$	7.12376	0.453274
$248$	4.53206	0.287786
$249$	27.7994	1.76171
$250$	27.2747	1.72500
$251$	9.18469	0.579732	0.289866	−	0.957067i	$-0.406389\pi$
$251$	9.18469	0.579732	0.289866	+	0.957067i	$0.406389\pi$
$252$	0.649561	0.0409185
$253$	−25.9896	−1.63395
$254$	3.35937	0.210786
$255$	13.3859	0.838260
$256$	−10.1249	−0.632808
$257$	−2.22164	−0.138582	−0.0692909	−	0.997596i	$-0.522074\pi$
$257$	−2.22164	−0.138582	−0.0692909	+	0.997596i	$0.522074\pi$
$258$	−46.6474	−2.90414
$259$	2.46286	0.153035
$260$	11.5987	0.719319
$261$	0.0100907	0.000624596 0
$262$	−42.1104	−2.60159
$263$	26.2518	1.61875	0.809377	−	0.587289i	$-0.199805\pi$
$263$	26.2518	1.61875	0.809377	+	0.587289i	$0.199805\pi$
$264$	−20.6619	−1.27165
$265$	9.92513	0.609696
$266$	−5.09530	−0.312413
$267$	19.3653	1.18514
$268$	−15.7458	−0.961831
$269$	3.63985	0.221926	0.110963	−	0.993825i	$-0.464607\pi$
$269$	3.63985	0.221926	0.110963	+	0.993825i	$0.464607\pi$
$270$	−19.3946	−1.18032
$271$	15.8456	0.962552	0.481276	−	0.876569i	$-0.340173\pi$
$271$	15.8456	0.962552	0.481276	+	0.876569i	$0.340173\pi$
$272$	3.53559	0.214376
$273$	−2.71142	−0.164103
$274$	22.8431	1.38000
$275$	9.17835	0.553475
$276$	−29.4520	−1.77280
$277$	−19.7956	−1.18940	−0.594702	−	0.803947i	$-0.702730\pi$
$277$	−19.7956	−1.18940	−0.594702	+	0.803947i	$0.702730\pi$
$278$	36.7500	2.20412
$279$	−0.603936	−0.0361567
$280$	−2.83363	−0.169342
$281$	7.81806	0.466387	0.233193	−	0.972430i	$-0.425083\pi$
$281$	7.81806	0.466387	0.233193	+	0.972430i	$0.425083\pi$
$282$	24.4733	1.45736
$283$	−17.0846	−1.01558	−0.507788	−	0.861482i	$-0.669537\pi$
$283$	−17.0846	−1.01558	−0.507788	+	0.861482i	$0.669537\pi$
$284$	18.7265	1.11121
$285$	−10.5834	−0.626905
$286$	23.6701	1.39964
$287$	3.98498	0.235226
$288$	2.03618	0.119983
$289$	0.360360	0.0211976
$290$	−0.128875	−0.00756779
$291$	−29.7249	−1.74251
$292$	−19.5973	−1.14684
$293$	19.1615	1.11943	0.559714	−	0.828686i	$-0.310911\pi$
$293$	19.1615	1.11943	0.559714	+	0.828686i	$0.310911\pi$
$294$	−26.6458	−1.55401
$295$	26.0849	1.51872
$296$	−8.32209	−0.483712
$297$	−23.8656	−1.38482
$298$	−28.6312	−1.65856
$299$	11.5244	0.666474
$300$	10.4011	0.600509
$301$	−7.87214	−0.453743
$302$	−4.92849	−0.283603
$303$	−16.1841	−0.929751
$304$	−2.79535	−0.160324
$305$	−25.1914	−1.44245
$306$	2.90190	0.165890
$307$	10.9482	0.624848	0.312424	−	0.949943i	$-0.398859\pi$
$307$	10.9482	0.624848	0.312424	+	0.949943i	$0.398859\pi$
$308$	−10.2085	−0.581682
$309$	−5.32937	−0.303177
$310$	7.71328	0.438085
$311$	−11.0720	−0.627837	−0.313919	−	0.949450i	$-0.601642\pi$
$311$	−11.0720	−0.627837	−0.313919	+	0.949450i	$0.601642\pi$
$312$	9.16197	0.518695
$313$	25.9248	1.46535	0.732677	−	0.680576i	$-0.238270\pi$
$313$	25.9248	1.46535	0.732677	+	0.680576i	$0.238270\pi$
$314$	−14.9701	−0.844814
$315$	0.377605	0.0212756
$316$	13.5355	0.761433
$317$	−31.5639	−1.77281	−0.886404	−	0.462912i	$-0.846804\pi$
$317$	−31.5639	−1.77281	−0.886404	+	0.462912i	$0.846804\pi$
$318$	22.9532	1.28715
$319$	−0.158585	−0.00887903
$320$	−23.0088	−1.28623
$321$	25.6903	1.43389
$322$	−8.24288	−0.459358
$323$	−13.7257	−0.763717
$324$	−29.8728	−1.65960
$325$	−4.06990	−0.225757
$326$	−24.8887	−1.37846
$327$	−9.74863	−0.539101
$328$	−13.4654	−0.743501
$329$	4.13008	0.227698
$330$	−35.1652	−1.93578
$331$	−5.40000	−0.296811	−0.148405	−	0.988927i	$-0.547414\pi$
$331$	−5.40000	−0.296811	−0.148405	+	0.988927i	$0.547414\pi$
$332$	−46.4107	−2.54712
$333$	1.10899	0.0607723
$334$	−14.9640	−0.818792
$335$	−9.15343	−0.500105
$336$	1.06396	0.0580436
$337$	−23.5894	−1.28500	−0.642499	−	0.766286i	$-0.722103\pi$
$337$	−23.5894	−1.28500	−0.642499	+	0.766286i	$0.722103\pi$
$338$	18.6818	1.01616
$339$	−4.87686	−0.264875
$340$	−22.3477	−1.21197
$341$	9.49144	0.513990
$342$	−2.29434	−0.124063
$343$	−9.32068	−0.503269
$344$	26.6002	1.43419
$345$	−17.1212	−0.921772
$346$	−6.66962	−0.358561
$347$	−21.7027	−1.16506	−0.582532	−	0.812808i	$-0.697938\pi$
$347$	−21.7027	−1.16506	−0.582532	+	0.812808i	$0.697938\pi$
$348$	−0.179712	−0.00963356
$349$	−3.69121	−0.197586	−0.0987928	−	0.995108i	$-0.531498\pi$
$349$	−3.69121	−0.197586	−0.0987928	+	0.995108i	$0.531498\pi$
$350$	2.91101	0.155600
$351$	10.5826	0.564857
$352$	−32.0006	−1.70564
$353$	−29.7196	−1.58182	−0.790908	−	0.611935i	$-0.790391\pi$
$353$	−29.7196	−1.58182	−0.790908	+	0.611935i	$0.790391\pi$
$354$	60.3248	3.20623
$355$	10.8862	0.577778
$356$	−32.3301	−1.71349
$357$	5.22422	0.276495
$358$	−21.4135	−1.13174
$359$	−8.83604	−0.466348	−0.233174	−	0.972435i	$-0.574911\pi$
$359$	−8.83604	−0.466348	−0.233174	+	0.972435i	$0.574911\pi$
$360$	−1.27594	−0.0672479
$361$	−8.14803	−0.428844
$362$	−1.20190	−0.0631702
$363$	−23.2583	−1.22074
$364$	4.52668	0.237263
$365$	−11.3924	−0.596303
$366$	−58.2584	−3.04521
$367$	20.2735	1.05827	0.529135	−	0.848538i	$-0.322517\pi$
$367$	20.2735	1.05827	0.529135	+	0.848538i	$0.322517\pi$
$368$	−4.52216	−0.235734
$369$	1.79438	0.0934115
$370$	−14.1637	−0.736335
$371$	3.87354	0.201104
$372$	10.7559	0.557669
$373$	29.4283	1.52374	0.761871	−	0.647729i	$-0.224281\pi$
$373$	29.4283	1.52374	0.761871	+	0.647729i	$0.224281\pi$
$374$	−45.6061	−2.35824
$375$	22.1099	1.14175
$376$	−13.9556	−0.719707
$377$	0.0703202	0.00362167
$378$	−7.56924	−0.389320
$379$	10.3666	0.532495	0.266248	−	0.963905i	$-0.414216\pi$
$379$	10.3666	0.532495	0.266248	+	0.963905i	$0.414216\pi$
$380$	17.6688	0.906390
$381$	2.72323	0.139515
$382$	−42.4039	−2.16957
$383$	7.38647	0.377431	0.188715	−	0.982032i	$-0.439568\pi$
$383$	7.38647	0.377431	0.188715	+	0.982032i	$0.439568\pi$
$384$	−29.3335	−1.49692
$385$	−5.93443	−0.302447
$386$	48.4493	2.46601
$387$	−3.54471	−0.180188
$388$	49.6254	2.51935
$389$	−16.8459	−0.854123	−0.427061	−	0.904223i	$-0.640451\pi$
$389$	−16.8459	−0.854123	−0.427061	+	0.904223i	$0.640451\pi$
$390$	15.5931	0.789588
$391$	−22.2046	−1.12293
$392$	15.1945	0.767437
$393$	−34.1363	−1.72195
$394$	20.2989	1.02264
$395$	7.86852	0.395908
$396$	−4.59672	−0.230994
$397$	−31.5316	−1.58253	−0.791264	−	0.611475i	$-0.790577\pi$
$397$	−31.5316	−1.58253	−0.791264	+	0.611475i	$0.790577\pi$
$398$	5.39749	0.270552
$399$	−4.13044	−0.206781
$400$	1.59702	0.0798511
$401$	−8.49152	−0.424046	−0.212023	−	0.977265i	$-0.568005\pi$
$401$	−8.49152	−0.424046	−0.212023	+	0.977265i	$0.568005\pi$
$402$	−21.1685	−1.05579
$403$	−4.20873	−0.209652
$404$	27.0191	1.34425
$405$	−17.3658	−0.862911
$406$	−0.0502968	−0.00249619
$407$	−17.4289	−0.863917
$408$	−17.6528	−0.873943
$409$	−30.7885	−1.52239	−0.761196	−	0.648522i	$-0.775388\pi$
$409$	−30.7885	−1.52239	−0.761196	+	0.648522i	$0.775388\pi$
$410$	−22.9172	−1.13180
$411$	18.5174	0.913398
$412$	8.89732	0.438340
$413$	10.1803	0.500941
$414$	−3.71164	−0.182417
$415$	−26.9796	−1.32438
$416$	14.1898	0.695714
$417$	29.7909	1.45887
$418$	36.0577	1.76364
$419$	−9.62286	−0.470107	−0.235054	−	0.971982i	$-0.575527\pi$
$419$	−9.62286	−0.470107	−0.235054	+	0.971982i	$0.575527\pi$
$420$	−6.72504	−0.328148
$421$	13.5259	0.659214	0.329607	−	0.944118i	$-0.393084\pi$
$421$	13.5259	0.659214	0.329607	+	0.944118i	$0.393084\pi$
$422$	−50.5729	−2.46185
$423$	1.85971	0.0904222
$424$	−13.0888	−0.635649
$425$	7.84166	0.380376
$426$	25.1757	1.21977
$427$	−9.83159	−0.475784
$428$	−42.8897	−2.07315
$429$	19.1878	0.926397
$430$	45.2719	2.18320
$431$	−25.3968	−1.22332	−0.611660	−	0.791121i	$-0.709498\pi$
$431$	−25.3968	−1.22332	−0.611660	+	0.791121i	$0.709498\pi$
$432$	−4.15259	−0.199791
$433$	−22.2083	−1.06726	−0.533632	−	0.845717i	$-0.679173\pi$
$433$	−22.2083	−1.06726	−0.533632	+	0.845717i	$0.679173\pi$
$434$	3.01031	0.144500
$435$	−0.104471	−0.00500898
$436$	16.2752	0.779442
$437$	17.5557	0.839802
$438$	−26.3463	−1.25888
$439$	10.0092	0.477711	0.238856	−	0.971055i	$-0.423228\pi$
$439$	10.0092	0.477711	0.238856	+	0.971055i	$0.423228\pi$
$440$	20.0526	0.955971
$441$	−2.02480	−0.0964188
$442$	20.2229	0.961903
$443$	−17.2941	−0.821668	−0.410834	−	0.911710i	$-0.634762\pi$
$443$	−17.2941	−0.821668	−0.410834	+	0.911710i	$0.634762\pi$
$444$	−19.7508	−0.937332
$445$	−18.7943	−0.890934
$446$	60.2378	2.85234
$447$	−23.2095	−1.09777
$448$	−8.97979	−0.424255
$449$	10.3553	0.488698	0.244349	−	0.969687i	$-0.421426\pi$
$449$	10.3553	0.488698	0.244349	+	0.969687i	$0.421426\pi$
$450$	1.31078	0.0617910
$451$	−28.2004	−1.32790
$452$	8.14186	0.382961
$453$	−3.99522	−0.187712
$454$	48.0389	2.25458
$455$	2.63147	0.123365
$456$	13.9569	0.653590
$457$	−12.2513	−0.573090	−0.286545	−	0.958067i	$-0.592507\pi$
$457$	−12.2513	−0.573090	−0.286545	+	0.958067i	$0.592507\pi$
$458$	22.3066	1.04232
$459$	−20.3899	−0.951721
$460$	28.5836	1.33271
$461$	−26.3347	−1.22653	−0.613263	−	0.789878i	$-0.710144\pi$
$461$	−26.3347	−1.22653	−0.613263	+	0.789878i	$0.710144\pi$
$462$	−13.7242	−0.638506
$463$	0.865706	0.0402328	0.0201164	−	0.999798i	$-0.493596\pi$
$463$	0.865706	0.0402328	0.0201164	+	0.999798i	$0.493596\pi$
$464$	−0.0275935	−0.00128100
$465$	6.25267	0.289961
$466$	26.2299	1.21507
$467$	4.36606	0.202037	0.101019	−	0.994885i	$-0.467790\pi$
$467$	4.36606	0.202037	0.101019	+	0.994885i	$0.467790\pi$
$468$	2.03830	0.0942203
$469$	−3.57237	−0.164957
$470$	−23.7516	−1.09558
$471$	−12.1354	−0.559168
$472$	−34.3996	−1.58337
$473$	55.7085	2.56148
$474$	18.1970	0.835816
$475$	−6.19987	−0.284470
$476$	−8.72177	−0.399761
$477$	1.74420	0.0798613
$478$	61.4695	2.81155
$479$	22.3629	1.02179	0.510894	−	0.859643i	$-0.329314\pi$
$479$	22.3629	1.02179	0.510894	+	0.859643i	$0.329314\pi$
$480$	−21.0810	−0.962213
$481$	7.72838	0.352384
$482$	29.9045	1.36211
$483$	−6.68198	−0.304041
$484$	38.8294	1.76497
$485$	28.8484	1.30994
$486$	−7.20984	−0.327045
$487$	19.4913	0.883234	0.441617	−	0.897204i	$-0.354405\pi$
$487$	19.4913	0.883234	0.441617	+	0.897204i	$0.354405\pi$
$488$	33.2212	1.50385
$489$	−20.1757	−0.912375
$490$	25.8601	1.16824
$491$	24.5511	1.10797	0.553987	−	0.832525i	$-0.313106\pi$
$491$	24.5511	1.10797	0.553987	+	0.832525i	$0.313106\pi$
$492$	−31.9573	−1.44075
$493$	−0.135489	−0.00610212
$494$	−15.9889	−0.719373
$495$	−2.67218	−0.120106
$496$	1.65150	0.0741545
$497$	4.24861	0.190576
$498$	−62.3940	−2.79594
$499$	30.9492	1.38548	0.692738	−	0.721189i	$-0.256404\pi$
$499$	30.9492	1.38548	0.692738	+	0.721189i	$0.256404\pi$
$500$	−36.9122	−1.65076
$501$	−12.1303	−0.541944
$502$	−20.6145	−0.920069
$503$	−37.9160	−1.69059	−0.845295	−	0.534300i	$-0.820575\pi$
$503$	−37.9160	−1.69059	−0.845295	+	0.534300i	$0.820575\pi$
$504$	−0.497969	−0.0221813
$505$	15.7069	0.698946
$506$	58.3321	2.59318
$507$	15.1442	0.672577
$508$	−4.54640	−0.201714
$509$	−21.3591	−0.946725	−0.473362	−	0.880868i	$-0.656960\pi$
$509$	−21.3591	−0.946725	−0.473362	+	0.880868i	$0.656960\pi$
$510$	−30.0439	−1.33037
$511$	−4.44616	−0.196687
$512$	−9.52002	−0.420729
$513$	16.1210	0.711758
$514$	4.98633	0.219937
$515$	5.17222	0.227915
$516$	63.1302	2.77915
$517$	−29.2272	−1.28541
$518$	−5.52775	−0.242875
$519$	−5.40664	−0.237325
$520$	−8.89181	−0.389932
$521$	−20.7828	−0.910510	−0.455255	−	0.890361i	$-0.650452\pi$
$521$	−20.7828	−0.910510	−0.455255	+	0.890361i	$0.650452\pi$
$522$	−0.0226479	−0.000991271 0
$523$	−31.3440	−1.37058	−0.685290	−	0.728271i	$-0.740324\pi$
$523$	−31.3440	−1.37058	−0.685290	+	0.728271i	$0.740324\pi$
$524$	56.9901	2.48962
$525$	2.35977	0.102989
$526$	−58.9205	−2.56906
$527$	8.10915	0.353240
$528$	−7.52927	−0.327669
$529$	5.40055	0.234806
$530$	−22.2763	−0.967623
$531$	4.58404	0.198931
$532$	6.89571	0.298967
$533$	12.5047	0.541640
$534$	−43.4643	−1.88088
$535$	−24.9328	−1.07794
$536$	12.0711	0.521394
$537$	−17.3585	−0.749076
$538$	−8.16942	−0.352209
$539$	31.8216	1.37065
$540$	26.2476	1.12952
$541$	30.7282	1.32111	0.660554	−	0.750779i	$-0.270322\pi$
$541$	30.7282	1.32111	0.660554	+	0.750779i	$0.270322\pi$
$542$	−35.5645	−1.52763
$543$	−0.974301	−0.0418113
$544$	−27.3402	−1.17220
$545$	9.46117	0.405272
$546$	6.08562	0.260441
$547$	−21.4699	−0.917986	−0.458993	−	0.888440i	$-0.651790\pi$
$547$	−21.4699	−0.917986	−0.458993	+	0.888440i	$0.651790\pi$
$548$	−30.9146	−1.32061
$549$	−4.42702	−0.188940
$550$	−20.6003	−0.878398
$551$	0.107122	0.00456355
$552$	22.5786	0.961009
$553$	3.07090	0.130588
$554$	44.4301	1.88765
$555$	−11.4816	−0.487367
$556$	−49.7355	−2.10926
$557$	−22.5803	−0.956758	−0.478379	−	0.878153i	$-0.658776\pi$
$557$	−22.5803	−0.956758	−0.478379	+	0.878153i	$0.658776\pi$
$558$	1.35550	0.0573828
$559$	−24.7025	−1.04480
$560$	−1.03258	−0.0436346
$561$	−36.9700	−1.56088
$562$	−17.5472	−0.740183
$563$	42.9912	1.81186	0.905932	−	0.423422i	$-0.139171\pi$
$563$	42.9912	1.81186	0.905932	+	0.423422i	$0.139171\pi$
$564$	−33.1209	−1.39464
$565$	4.73305	0.199121
$566$	38.3454	1.61178
$567$	−6.77744	−0.284626
$568$	−14.3562	−0.602372
$569$	34.6777	1.45376	0.726882	−	0.686763i	$-0.240969\pi$
$569$	34.6777	1.45376	0.726882	+	0.686763i	$0.240969\pi$
$570$	23.7537	0.994934
$571$	−25.7783	−1.07879	−0.539395	−	0.842053i	$-0.681347\pi$
$571$	−25.7783	−1.07879	−0.539395	+	0.842053i	$0.681347\pi$
$572$	−32.0338	−1.33940
$573$	−34.3742	−1.43600
$574$	−8.94405	−0.373317
$575$	−10.0298	−0.418271
$576$	−4.04346	−0.168478
$577$	1.39986	0.0582769	0.0291385	−	0.999575i	$-0.490724\pi$
$577$	1.39986	0.0582769	0.0291385	+	0.999575i	$0.490724\pi$
$578$	−0.808806	−0.0336419
$579$	39.2748	1.63221
$580$	0.174412	0.00724208
$581$	−10.5295	−0.436838
$582$	66.7158	2.76546
$583$	−27.4118	−1.13528
$584$	15.0237	0.621686
$585$	1.18491	0.0489900
$586$	−43.0068	−1.77660
$587$	−36.1562	−1.49233	−0.746164	−	0.665762i	$-0.768106\pi$
$587$	−36.1562	−1.49233	−0.746164	+	0.665762i	$0.768106\pi$
$588$	36.0610	1.48713
$589$	−6.41136	−0.264176
$590$	−58.5460	−2.41030
$591$	16.4550	0.676870
$592$	−3.03260	−0.124639
$593$	21.7012	0.891160	0.445580	−	0.895242i	$-0.352998\pi$
$593$	21.7012	0.891160	0.445580	+	0.895242i	$0.352998\pi$
$594$	53.5649	2.19780
$595$	−5.07017	−0.207857
$596$	38.7480	1.58718
$597$	4.37541	0.179073
$598$	−25.8658	−1.05773
$599$	23.0646	0.942393	0.471197	−	0.882028i	$-0.343822\pi$
$599$	23.0646	0.942393	0.471197	+	0.882028i	$0.343822\pi$
$600$	−7.97375	−0.325527
$601$	−35.7625	−1.45878	−0.729392	−	0.684096i	$-0.760197\pi$
$601$	−35.7625	−1.45878	−0.729392	+	0.684096i	$0.760197\pi$
$602$	17.6685	0.720116
$603$	−1.60858	−0.0655065
$604$	6.66997	0.271397
$605$	22.5724	0.917700
$606$	36.3242	1.47557
$607$	34.1147	1.38467	0.692337	−	0.721574i	$-0.256581\pi$
$607$	34.1147	1.38467	0.692337	+	0.721574i	$0.256581\pi$
$608$	21.6161	0.876647
$609$	−0.0407724	−0.00165218
$610$	56.5405	2.28926
$611$	12.9600	0.524306
$612$	−3.92728	−0.158751
$613$	−39.7638	−1.60605	−0.803023	−	0.595949i	$-0.796776\pi$
$613$	−39.7638	−1.60605	−0.803023	+	0.595949i	$0.796776\pi$
$614$	−24.5726	−0.991671
$615$	−18.5775	−0.749119
$616$	7.82606	0.315321
$617$	−7.58312	−0.305285	−0.152642	−	0.988281i	$-0.548778\pi$
$617$	−7.58312	−0.305285	−0.152642	+	0.988281i	$0.548778\pi$
$618$	11.9615	0.481160
$619$	−22.3138	−0.896867	−0.448433	−	0.893816i	$-0.648018\pi$
$619$	−22.3138	−0.896867	−0.448433	+	0.893816i	$0.648018\pi$
$620$	−10.4388	−0.419231
$621$	26.0795	1.04654
$622$	24.8505	0.996415
$623$	−7.33496	−0.293869
$624$	3.33866	0.133653
$625$	−12.0477	−0.481909
$626$	−58.1866	−2.32560
$627$	29.2297	1.16732
$628$	20.2598	0.808455
$629$	−14.8906	−0.593727
$630$	−0.847512	−0.0337657
$631$	12.0481	0.479629	0.239814	−	0.970819i	$-0.422913\pi$
$631$	12.0481	0.479629	0.239814	+	0.970819i	$0.422913\pi$
$632$	−10.3766	−0.412761
$633$	−40.9962	−1.62945
$634$	70.8434	2.81355
$635$	−2.64293	−0.104881
$636$	−31.0637	−1.23175
$637$	−14.1105	−0.559077
$638$	0.355933	0.0140915
$639$	1.91309	0.0756805
$640$	28.4686	1.12532
$641$	47.1270	1.86141	0.930703	−	0.365775i	$-0.119196\pi$
$641$	47.1270	1.86141	0.930703	+	0.365775i	$0.119196\pi$
$642$	−57.6604	−2.27567
$643$	−26.1567	−1.03152	−0.515759	−	0.856734i	$-0.672490\pi$
$643$	−26.1567	−1.03152	−0.515759	+	0.856734i	$0.672490\pi$
$644$	11.1555	0.439588
$645$	36.6991	1.44502
$646$	30.8064	1.21206
$647$	4.05073	0.159251	0.0796253	−	0.996825i	$-0.474628\pi$
$647$	4.05073	0.159251	0.0796253	+	0.996825i	$0.474628\pi$
$648$	22.9012	0.899643
$649$	−72.0427	−2.82792
$650$	9.13465	0.358290
$651$	2.44027	0.0956417
$652$	33.6830	1.31913
$653$	−3.14117	−0.122924	−0.0614618	−	0.998109i	$-0.519576\pi$
$653$	−3.14117	−0.122924	−0.0614618	+	0.998109i	$0.519576\pi$
$654$	21.8802	0.855584
$655$	33.1297	1.29448
$656$	−4.90683	−0.191580
$657$	−2.00204	−0.0781070
$658$	−9.26970	−0.361371
$659$	12.5053	0.487136	0.243568	−	0.969884i	$-0.421682\pi$
$659$	12.5053	0.487136	0.243568	+	0.969884i	$0.421682\pi$
$660$	47.5908	1.85247
$661$	−34.6722	−1.34859	−0.674297	−	0.738460i	$-0.735553\pi$
$661$	−34.6722	−1.34859	−0.674297	+	0.738460i	$0.735553\pi$
$662$	12.1200	0.471056
$663$	16.3934	0.636667
$664$	35.5795	1.38075
$665$	4.00864	0.155448
$666$	−2.48906	−0.0964492
$667$	0.173296	0.00671004
$668$	20.2515	0.783553
$669$	48.8309	1.88791
$670$	20.5443	0.793696
$671$	69.5748	2.68591
$672$	−8.22743	−0.317380
$673$	−17.7713	−0.685032	−0.342516	−	0.939512i	$-0.611279\pi$
$673$	−17.7713	−0.685032	−0.342516	+	0.939512i	$0.611279\pi$
$674$	52.9451	2.03937
$675$	−9.21012	−0.354498
$676$	−25.2830	−0.972425
$677$	−37.8510	−1.45473	−0.727367	−	0.686249i	$-0.759256\pi$
$677$	−37.8510	−1.45473	−0.727367	+	0.686249i	$0.759256\pi$
$678$	10.9458	0.420372
$679$	11.2588	0.432075
$680$	17.1322	0.656992
$681$	38.9421	1.49226
$682$	−21.3030	−0.815733
$683$	5.70060	0.218127	0.109064	−	0.994035i	$-0.465215\pi$
$683$	5.70060	0.218127	0.109064	+	0.994035i	$0.465215\pi$
$684$	3.10504	0.118724
$685$	−17.9714	−0.686652
$686$	20.9197	0.798718
$687$	18.0826	0.689893
$688$	9.69321	0.369550
$689$	12.1550	0.463070
$690$	38.4274	1.46291
$691$	−17.4628	−0.664318	−0.332159	−	0.943223i	$-0.607777\pi$
$691$	−17.4628	−0.664318	−0.332159	+	0.943223i	$0.607777\pi$
$692$	9.02632	0.343129
$693$	−1.04289	−0.0396161
$694$	48.7105	1.84902
$695$	−28.9124	−1.09671
$696$	0.137771	0.00522220
$697$	−24.0934	−0.912603
$698$	8.28469	0.313580
$699$	21.2629	0.804237
$700$	−3.93961	−0.148903
$701$	−48.8852	−1.84637	−0.923184	−	0.384358i	$-0.874423\pi$
$701$	−48.8852	−1.84637	−0.923184	+	0.384358i	$0.874423\pi$
$702$	−23.7520	−0.896461
$703$	11.7730	0.444027
$704$	63.5469	2.39501
$705$	−19.2540	−0.725146
$706$	66.7039	2.51043
$707$	6.13001	0.230543
$708$	−81.6405	−3.06824
$709$	1.29179	0.0485142	0.0242571	−	0.999706i	$-0.492278\pi$
$709$	1.29179	0.0485142	0.0242571	+	0.999706i	$0.492278\pi$
$710$	−24.4333	−0.916967
$711$	1.38278	0.0518582
$712$	24.7850	0.928859
$713$	−10.3719	−0.388432
$714$	−11.7254	−0.438814
$715$	−18.6220	−0.696424
$716$	28.9799	1.08303
$717$	49.8294	1.86091
$718$	19.8320	0.740122
$719$	37.1738	1.38635	0.693175	−	0.720769i	$-0.256211\pi$
$719$	37.1738	1.38635	0.693175	+	0.720769i	$0.256211\pi$
$720$	−0.464957	−0.0173279
$721$	2.01860	0.0751764
$722$	18.2878	0.680600
$723$	24.2417	0.901557
$724$	1.62658	0.0604515
$725$	−0.0612003	−0.00227292
$726$	52.2018	1.93739
$727$	1.14188	0.0423498	0.0211749	−	0.999776i	$-0.493259\pi$
$727$	1.14188	0.0423498	0.0211749	+	0.999776i	$0.493259\pi$
$728$	−3.47026	−0.128616
$729$	23.6593	0.876272
$730$	25.5694	0.946367
$731$	47.5954	1.76038
$732$	78.8439	2.91415
$733$	−35.7883	−1.32187	−0.660936	−	0.750442i	$-0.729841\pi$
$733$	−35.7883	−1.32187	−0.660936	+	0.750442i	$0.729841\pi$
$734$	−45.5027	−1.67953
$735$	20.9631	0.773236
$736$	34.9692	1.28898
$737$	25.2804	0.931217
$738$	−4.02737	−0.148250
$739$	−45.1010	−1.65907	−0.829533	−	0.558458i	$-0.811393\pi$
$739$	−45.1010	−1.65907	−0.829533	+	0.558458i	$0.811393\pi$
$740$	19.1684	0.704644
$741$	−12.9612	−0.476140
$742$	−8.69393	−0.319164
$743$	1.85005	0.0678717	0.0339359	−	0.999424i	$-0.489196\pi$
$743$	1.85005	0.0678717	0.0339359	+	0.999424i	$0.489196\pi$
$744$	−8.24575	−0.302304
$745$	22.5251	0.825256
$746$	−66.0501	−2.41827
$747$	−4.74128	−0.173474
$748$	61.7210	2.25674
$749$	−9.73067	−0.355551
$750$	−49.6243	−1.81202
$751$	−24.7302	−0.902418	−0.451209	−	0.892418i	$-0.649007\pi$
$751$	−24.7302	−0.902418	−0.451209	+	0.892418i	$0.649007\pi$
$752$	−5.08549	−0.185449
$753$	−16.7109	−0.608977
$754$	−0.157829	−0.00574781
$755$	3.87741	0.141113
$756$	10.2438	0.372564
$757$	1.40694	0.0511361	0.0255681	−	0.999673i	$-0.491861\pi$
$757$	1.40694	0.0511361	0.0255681	+	0.999673i	$0.491861\pi$
$758$	−23.2672	−0.845101
$759$	47.2861	1.71638
$760$	−13.5453	−0.491340
$761$	16.3755	0.593611	0.296806	−	0.954938i	$-0.404079\pi$
$761$	16.3755	0.593611	0.296806	+	0.954938i	$0.404079\pi$
$762$	−6.11213	−0.221419
$763$	3.69247	0.133676
$764$	57.3873	2.07620
$765$	−2.28302	−0.0825427
$766$	−16.5785	−0.599005
$767$	31.9455	1.15348
$768$	18.4216	0.664731
$769$	−20.2918	−0.731740	−0.365870	−	0.930666i	$-0.619229\pi$
$769$	−20.2918	−0.731740	−0.365870	+	0.930666i	$0.619229\pi$
$770$	13.3195	0.480000
$771$	4.04210	0.145573
$772$	−65.5688	−2.35987
$773$	18.2914	0.657894	0.328947	−	0.944348i	$-0.393306\pi$
$773$	18.2914	0.657894	0.328947	+	0.944348i	$0.393306\pi$
$774$	7.95588	0.285968
$775$	3.66290	0.131575
$776$	−38.0440	−1.36570
$777$	−4.48100	−0.160755
$778$	37.8097	1.35554
$779$	19.0490	0.682503
$780$	−21.1029	−0.755605
$781$	−30.0660	−1.07585
$782$	49.8369	1.78216
$783$	0.159133	0.00568696
$784$	5.53692	0.197747
$785$	11.7775	0.420357
$786$	76.6168	2.73283
$787$	1.29494	0.0461597	0.0230798	−	0.999734i	$-0.492653\pi$
$787$	1.29494	0.0461597	0.0230798	+	0.999734i	$0.492653\pi$
$788$	−27.4715	−0.978631
$789$	−47.7632	−1.70041
$790$	−17.6604	−0.628329
$791$	1.84720	0.0656788
$792$	3.52396	0.125218
$793$	−30.8512	−1.09556
$794$	70.7709	2.51156
$795$	−18.0580	−0.640452
$796$	−7.30469	−0.258908
$797$	−28.2805	−1.00175	−0.500874	−	0.865520i	$-0.666988\pi$
$797$	−28.2805	−1.00175	−0.500874	+	0.865520i	$0.666988\pi$
$798$	9.27052	0.328173
$799$	−24.9706	−0.883398
$800$	−12.3495	−0.436622
$801$	−3.30282	−0.116699
$802$	19.0587	0.672986
$803$	31.4640	1.11034
$804$	28.6484	1.01035
$805$	6.48494	0.228564
$806$	9.44625	0.332730
$807$	−6.62244	−0.233121
$808$	−20.7135	−0.728698
$809$	25.6139	0.900537	0.450268	−	0.892893i	$-0.351328\pi$
$809$	25.6139	0.900537	0.450268	+	0.892893i	$0.351328\pi$
$810$	38.9764	1.36949
$811$	−33.0647	−1.16106	−0.580529	−	0.814239i	$-0.697154\pi$
$811$	−33.0647	−1.16106	−0.580529	+	0.814239i	$0.697154\pi$
$812$	0.0680691	0.00238876
$813$	−28.8299	−1.01111
$814$	39.1180	1.37109
$815$	19.5807	0.685883
$816$	−6.43274	−0.225191
$817$	−37.6305	−1.31652
$818$	69.1029	2.41613
$819$	0.462442	0.0161590
$820$	31.0150	1.08309
$821$	3.91290	0.136561	0.0682805	−	0.997666i	$-0.478249\pi$
$821$	3.91290	0.136561	0.0682805	+	0.997666i	$0.478249\pi$
$822$	−41.5613	−1.44962
$823$	2.05213	0.0715327	0.0357663	−	0.999360i	$-0.488613\pi$
$823$	2.05213	0.0715327	0.0357663	+	0.999360i	$0.488613\pi$
$824$	−6.82090	−0.237617
$825$	−16.6993	−0.581396
$826$	−22.8491	−0.795022
$827$	2.78004	0.0966716	0.0483358	−	0.998831i	$-0.484608\pi$
$827$	2.78004	0.0966716	0.0483358	+	0.998831i	$0.484608\pi$
$828$	5.02315	0.174566
$829$	−23.9494	−0.831799	−0.415899	−	0.909411i	$-0.636533\pi$
$829$	−23.9494	−0.831799	−0.415899	+	0.909411i	$0.636533\pi$
$830$	60.5541	2.10186
$831$	36.0166	1.24940
$832$	−28.1782	−0.976904
$833$	27.1873	0.941983
$834$	−66.8638	−2.31531
$835$	11.7727	0.407409
$836$	−48.7987	−1.68774
$837$	−9.52429	−0.329208
$838$	21.5979	0.746088
$839$	−39.5757	−1.36631	−0.683153	−	0.730275i	$-0.739392\pi$
$839$	−39.5757	−1.36631	−0.683153	+	0.730275i	$0.739392\pi$
$840$	5.15557	0.177884
$841$	−28.9989	−0.999964
$842$	−30.3582	−1.04621
$843$	−14.2244	−0.489914
$844$	68.4427	2.35590
$845$	−14.6976	−0.505614
$846$	−4.17401	−0.143505
$847$	8.80949	0.302698
$848$	−4.76961	−0.163789
$849$	31.0842	1.06681
$850$	−17.6001	−0.603679
$851$	19.0457	0.652877
$852$	−34.0715	−1.16727
$853$	−23.4118	−0.801605	−0.400802	−	0.916164i	$-0.631269\pi$
$853$	−23.4118	−0.801605	−0.400802	+	0.916164i	$0.631269\pi$
$854$	22.0664	0.755097
$855$	1.80503	0.0617307
$856$	32.8802	1.12382
$857$	−11.3057	−0.386196	−0.193098	−	0.981179i	$-0.561854\pi$
$857$	−11.3057	−0.386196	−0.193098	+	0.981179i	$0.561854\pi$
$858$	−43.0659	−1.47025
$859$	37.6589	1.28491	0.642453	−	0.766325i	$-0.277917\pi$
$859$	37.6589	1.28491	0.642453	+	0.766325i	$0.277917\pi$
$860$	−61.2686	−2.08924
$861$	−7.25038	−0.247092
$862$	57.0015	1.94148
$863$	−16.6002	−0.565079	−0.282539	−	0.959256i	$-0.591177\pi$
$863$	−16.6002	−0.565079	−0.282539	+	0.959256i	$0.591177\pi$
$864$	32.1114	1.09245
$865$	5.24721	0.178411
$866$	49.8452	1.69381
$867$	−0.655648	−0.0222670
$868$	−4.07400	−0.138281
$869$	−21.7317	−0.737197
$870$	0.234478	0.00794955
$871$	−11.2100	−0.379835
$872$	−12.4770	−0.422523
$873$	5.06969	0.171583
$874$	−39.4027	−1.33281
$875$	−8.37452	−0.283111
$876$	35.6558	1.20470
$877$	27.1173	0.915688	0.457844	−	0.889033i	$-0.348622\pi$
$877$	27.1173	0.915688	0.457844	+	0.889033i	$0.348622\pi$
$878$	−22.4650	−0.758156
$879$	−34.8629	−1.17590
$880$	7.30725	0.246327
$881$	−43.4326	−1.46328	−0.731641	−	0.681690i	$-0.761245\pi$
$881$	−43.4326	−1.46328	−0.731641	+	0.681690i	$0.761245\pi$
$882$	4.54453	0.153022
$883$	20.0862	0.675954	0.337977	−	0.941154i	$-0.390257\pi$
$883$	20.0862	0.675954	0.337977	+	0.941154i	$0.390257\pi$
$884$	−27.3686	−0.920505
$885$	−47.4595	−1.59533
$886$	38.8156	1.30404
$887$	−38.7275	−1.30034	−0.650171	−	0.759788i	$-0.725303\pi$
$887$	−38.7275	−1.30034	−0.650171	+	0.759788i	$0.725303\pi$
$888$	15.1414	0.508113
$889$	−1.03147	−0.0345945
$890$	42.1826	1.41396
$891$	47.9617	1.60678
$892$	−81.5227	−2.72958
$893$	19.7426	0.660661
$894$	52.0923	1.74223
$895$	16.8467	0.563122
$896$	11.1106	0.371179
$897$	−20.9678	−0.700094
$898$	−23.2419	−0.775592
$899$	−0.0632879	−0.00211077
$900$	−1.77395	−0.0591316
$901$	−23.4196	−0.780221
$902$	63.2940	2.10746
$903$	14.3228	0.476632
$904$	−6.24174	−0.207597
$905$	0.945571	0.0314319
$906$	8.96703	0.297909
$907$	31.0455	1.03085	0.515424	−	0.856935i	$-0.327635\pi$
$907$	31.0455	1.03085	0.515424	+	0.856935i	$0.327635\pi$
$908$	−65.0133	−2.15754
$909$	2.76025	0.0915518
$910$	−5.90617	−0.195788
$911$	−6.57923	−0.217980	−0.108990	−	0.994043i	$-0.534762\pi$
$911$	−6.57923	−0.217980	−0.108990	+	0.994043i	$0.534762\pi$
$912$	5.08593	0.168412
$913$	74.5138	2.46605
$914$	27.4972	0.909528
$915$	45.8338	1.51522
$916$	−30.1886	−0.997460
$917$	12.9297	0.426977
$918$	45.7640	1.51044
$919$	37.4006	1.23373	0.616867	−	0.787068i	$-0.288402\pi$
$919$	37.4006	1.23373	0.616867	+	0.787068i	$0.288402\pi$
$920$	−21.9128	−0.722444
$921$	−19.9195	−0.656369
$922$	59.1065	1.94657
$923$	13.3320	0.438828
$924$	18.5736	0.611026
$925$	−6.72607	−0.221152
$926$	−1.94302	−0.0638518
$927$	0.908943	0.0298536
$928$	0.213377	0.00700443
$929$	−21.5175	−0.705966	−0.352983	−	0.935630i	$-0.614833\pi$
$929$	−21.5175	−0.705966	−0.352983	+	0.935630i	$0.614833\pi$
$930$	−14.0337	−0.460185
$931$	−21.4951	−0.704475
$932$	−35.4981	−1.16278
$933$	20.1447	0.659509
$934$	−9.79937	−0.320645
$935$	35.8799	1.17340
$936$	−1.56261	−0.0510754
$937$	3.20778	0.104794	0.0523968	−	0.998626i	$-0.483314\pi$
$937$	3.20778	0.104794	0.0523968	+	0.998626i	$0.483314\pi$
$938$	8.01796	0.261796
$939$	−47.1682	−1.53928
$940$	32.1442	1.04843
$941$	35.5483	1.15884	0.579421	−	0.815028i	$-0.303279\pi$
$941$	35.5483	1.15884	0.579421	+	0.815028i	$0.303279\pi$
$942$	27.2371	0.887432
$943$	30.8164	1.00352
$944$	−12.5353	−0.407991
$945$	5.95497	0.193715
$946$	−125.034	−4.06522
$947$	−17.8849	−0.581183	−0.290591	−	0.956847i	$-0.593852\pi$
$947$	−17.8849	−0.581183	−0.290591	+	0.956847i	$0.593852\pi$
$948$	−24.6269	−0.799844
$949$	−13.9519	−0.452898
$950$	13.9152	0.451470
$951$	57.4283	1.86224
$952$	6.68631	0.216705
$953$	−22.4325	−0.726659	−0.363329	−	0.931661i	$-0.618360\pi$
$953$	−22.4325	−0.726659	−0.363329	+	0.931661i	$0.618360\pi$
$954$	−3.91475	−0.126745
$955$	33.3606	1.07952
$956$	−83.1896	−2.69054
$957$	0.288533	0.00932694
$958$	−50.1923	−1.62164
$959$	−7.01381	−0.226488
$960$	41.8628	1.35112
$961$	−27.2122	−0.877811
$962$	−17.3459	−0.559253
$963$	−4.38157	−0.141194
$964$	−40.4712	−1.30349
$965$	−38.1167	−1.22702
$966$	14.9973	0.482530
$967$	−49.0565	−1.57755	−0.788775	−	0.614682i	$-0.789284\pi$
$967$	−49.0565	−1.57755	−0.788775	+	0.614682i	$0.789284\pi$
$968$	−29.7675	−0.956764
$969$	24.9728	0.802243
$970$	−64.7485	−2.07895
$971$	9.24914	0.296819	0.148410	−	0.988926i	$-0.452585\pi$
$971$	9.24914	0.296819	0.148410	+	0.988926i	$0.452585\pi$
$972$	9.75743	0.312970
$973$	−11.2838	−0.361743
$974$	−43.7470	−1.40174
$975$	7.40488	0.237146
$976$	12.1059	0.387502
$977$	−3.84617	−0.123050	−0.0615249	−	0.998106i	$-0.519596\pi$
$977$	−3.84617	−0.123050	−0.0615249	+	0.998106i	$0.519596\pi$
$978$	45.2831	1.44799
$979$	51.9070	1.65896
$980$	−34.9977	−1.11796
$981$	1.66266	0.0530848
$982$	−55.1034	−1.75842
$983$	−59.7633	−1.90615	−0.953076	−	0.302730i	$-0.902102\pi$
$983$	−59.7633	−1.90615	−0.953076	+	0.302730i	$0.902102\pi$
$984$	24.4992	0.781007
$985$	−15.9698	−0.508841
$986$	0.304097	0.00968442
$987$	−7.51436	−0.239185
$988$	21.6385	0.688412
$989$	−60.8764	−1.93576
$990$	5.99755	0.190615
$991$	−49.7825	−1.58139	−0.790697	−	0.612208i	$-0.790282\pi$
$991$	−49.7825	−1.58139	−0.790697	+	0.612208i	$0.790282\pi$
$992$	−12.7708	−0.405474
$993$	9.82489	0.311784
$994$	−9.53575	−0.302456
$995$	−4.24639	−0.134619
$996$	84.4408	2.67561
$997$	51.4978	1.63095	0.815475	−	0.578792i	$-0.196476\pi$
$997$	51.4978	1.63095	0.815475	+	0.578792i	$0.196476\pi$
$998$	−69.4636	−2.19883
$999$	17.4892	0.553333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4007.2.a.a.1.17	✓	139

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4007.2.a.a.1.17	✓	139	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:	\( N \)	\(=\)	\( 4007 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24444 q^{2} -1.81943 q^{3} +3.03751 q^{4} +1.76577 q^{5} +4.08359 q^{6} +0.689140 q^{7} -2.32862 q^{8} +0.310309 q^{9} +O(q^{10})\)	\(q-2.24444 q^{2} -1.81943 q^{3} +3.03751 q^{4} +1.76577 q^{5} +4.08359 q^{6} +0.689140 q^{7} -2.32862 q^{8} +0.310309 q^{9} -3.96317 q^{10} -4.87681 q^{11} -5.52652 q^{12} +2.16250 q^{13} -1.54673 q^{14} -3.21270 q^{15} -0.848559 q^{16} -4.16658 q^{17} -0.696471 q^{18} +3.29423 q^{19} +5.36356 q^{20} -1.25384 q^{21} +10.9457 q^{22} +5.32922 q^{23} +4.23676 q^{24} -1.88204 q^{25} -4.85359 q^{26} +4.89369 q^{27} +2.09327 q^{28} +0.0325181 q^{29} +7.21070 q^{30} -1.94624 q^{31} +6.56179 q^{32} +8.87300 q^{33} +9.35163 q^{34} +1.21687 q^{35} +0.942567 q^{36} +3.57382 q^{37} -7.39371 q^{38} -3.93450 q^{39} -4.11183 q^{40} +5.78254 q^{41} +2.81417 q^{42} -11.4231 q^{43} -14.8134 q^{44} +0.547936 q^{45} -11.9611 q^{46} +5.99308 q^{47} +1.54389 q^{48} -6.52509 q^{49} +4.22412 q^{50} +7.58078 q^{51} +6.56860 q^{52} +5.62083 q^{53} -10.9836 q^{54} -8.61135 q^{55} -1.60475 q^{56} -5.99361 q^{57} -0.0729848 q^{58} +14.7725 q^{59} -9.75859 q^{60} -14.2665 q^{61} +4.36821 q^{62} +0.213847 q^{63} -13.0304 q^{64} +3.81848 q^{65} -19.9149 q^{66} -5.18380 q^{67} -12.6560 q^{68} -9.69611 q^{69} -2.73118 q^{70} +6.16509 q^{71} -0.722594 q^{72} -6.45176 q^{73} -8.02123 q^{74} +3.42423 q^{75} +10.0063 q^{76} -3.36081 q^{77} +8.83075 q^{78} +4.45613 q^{79} -1.49836 q^{80} -9.83464 q^{81} -12.9786 q^{82} -15.2792 q^{83} -3.80855 q^{84} -7.35724 q^{85} +25.6385 q^{86} -0.0591642 q^{87} +11.3563 q^{88} -10.6436 q^{89} -1.22981 q^{90} +1.49026 q^{91} +16.1875 q^{92} +3.54104 q^{93} -13.4511 q^{94} +5.81687 q^{95} -11.9387 q^{96} +16.3375 q^{97} +14.6452 q^{98} -1.51332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9}+O(q^{10}) \) 139 * q - 13 * q^2 - 22 * q^3 + 113 * q^4 - 16 * q^5 - 15 * q^6 - 44 * q^7 - 36 * q^8 + 87 * q^9	\( 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9} - 40 q^{10} - 17 q^{11} - 59 q^{12} - 89 q^{13} - 15 q^{14} - 29 q^{15} + 73 q^{16} - 58 q^{17} - 51 q^{18} - 37 q^{19} - 24 q^{20} - 37 q^{21} - 99 q^{22} - 42 q^{23} - 27 q^{24} - 11 q^{25} + 2 q^{26} - 73 q^{27} - 113 q^{28} - 57 q^{29} - 29 q^{30} - 51 q^{31} - 80 q^{32} - 78 q^{33} - 28 q^{34} - 34 q^{35} + 28 q^{36} - 117 q^{37} - 31 q^{38} - 36 q^{39} - 107 q^{40} - 60 q^{41} - 41 q^{42} - 109 q^{43} - 21 q^{44} - 62 q^{45} - 92 q^{46} - 26 q^{47} - 90 q^{48} - 7 q^{49} - 22 q^{50} - 47 q^{51} - 182 q^{52} - 83 q^{53} - 19 q^{54} - 53 q^{55} - 23 q^{56} - 201 q^{57} - 112 q^{58} + 14 q^{59} - 64 q^{60} - 73 q^{61} - 21 q^{62} - 94 q^{63} + 14 q^{64} - 123 q^{65} - 10 q^{66} - 135 q^{67} - 84 q^{68} - 50 q^{69} - 35 q^{70} - 29 q^{71} - 143 q^{72} - 266 q^{73} - 53 q^{74} - 32 q^{75} - 66 q^{76} - 69 q^{77} - 59 q^{78} - 124 q^{79} - 20 q^{80} - 33 q^{81} - 93 q^{82} - 28 q^{83} - 4 q^{84} - 179 q^{85} + 6 q^{86} - 40 q^{87} - 259 q^{88} - 41 q^{89} + 2 q^{90} - 50 q^{91} - 77 q^{92} - 60 q^{93} - 48 q^{94} - 37 q^{95} + 3 q^{96} - 220 q^{97} - 9 q^{98} - 35 q^{99}+O(q^{100}) \) 139 * q - 13 * q^2 - 22 * q^3 + 113 * q^4 - 16 * q^5 - 15 * q^6 - 44 * q^7 - 36 * q^8 + 87 * q^9 - 40 * q^10 - 17 * q^11 - 59 * q^12 - 89 * q^13 - 15 * q^14 - 29 * q^15 + 73 * q^16 - 58 * q^17 - 51 * q^18 - 37 * q^19 - 24 * q^20 - 37 * q^21 - 99 * q^22 - 42 * q^23 - 27 * q^24 - 11 * q^25 + 2 * q^26 - 73 * q^27 - 113 * q^28 - 57 * q^29 - 29 * q^30 - 51 * q^31 - 80 * q^32 - 78 * q^33 - 28 * q^34 - 34 * q^35 + 28 * q^36 - 117 * q^37 - 31 * q^38 - 36 * q^39 - 107 * q^40 - 60 * q^41 - 41 * q^42 - 109 * q^43 - 21 * q^44 - 62 * q^45 - 92 * q^46 - 26 * q^47 - 90 * q^48 - 7 * q^49 - 22 * q^50 - 47 * q^51 - 182 * q^52 - 83 * q^53 - 19 * q^54 - 53 * q^55 - 23 * q^56 - 201 * q^57 - 112 * q^58 + 14 * q^59 - 64 * q^60 - 73 * q^61 - 21 * q^62 - 94 * q^63 + 14 * q^64 - 123 * q^65 - 10 * q^66 - 135 * q^67 - 84 * q^68 - 50 * q^69 - 35 * q^70 - 29 * q^71 - 143 * q^72 - 266 * q^73 - 53 * q^74 - 32 * q^75 - 66 * q^76 - 69 * q^77 - 59 * q^78 - 124 * q^79 - 20 * q^80 - 33 * q^81 - 93 * q^82 - 28 * q^83 - 4 * q^84 - 179 * q^85 + 6 * q^86 - 40 * q^87 - 259 * q^88 - 41 * q^89 + 2 * q^90 - 50 * q^91 - 77 * q^92 - 60 * q^93 - 48 * q^94 - 37 * q^95 + 3 * q^96 - 220 * q^97 - 9 * q^98 - 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more