LMFDB - Embedded newform 2005.2.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50483 q^{2} +0.906859 q^{3} +0.264506 q^{4} +1.00000 q^{5} +1.36467 q^{6} -0.905011 q^{7} -2.61162 q^{8} -2.17761 q^{9} +O(q^{10})$	\(q+1.50483 q^{2} +0.906859 q^{3} +0.264506 q^{4} +1.00000 q^{5} +1.36467 q^{6} -0.905011 q^{7} -2.61162 q^{8} -2.17761 q^{9} +1.50483 q^{10} -5.12534 q^{11} +0.239869 q^{12} -2.87981 q^{13} -1.36188 q^{14} +0.906859 q^{15} -4.45905 q^{16} +7.86878 q^{17} -3.27692 q^{18} +1.79978 q^{19} +0.264506 q^{20} -0.820717 q^{21} -7.71274 q^{22} -1.04970 q^{23} -2.36837 q^{24} +1.00000 q^{25} -4.33362 q^{26} -4.69536 q^{27} -0.239380 q^{28} -6.45900 q^{29} +1.36467 q^{30} -9.04334 q^{31} -1.48686 q^{32} -4.64796 q^{33} +11.8412 q^{34} -0.905011 q^{35} -0.575989 q^{36} -1.31037 q^{37} +2.70835 q^{38} -2.61158 q^{39} -2.61162 q^{40} +0.565794 q^{41} -1.23504 q^{42} -7.14995 q^{43} -1.35568 q^{44} -2.17761 q^{45} -1.57962 q^{46} +8.14264 q^{47} -4.04373 q^{48} -6.18096 q^{49} +1.50483 q^{50} +7.13588 q^{51} -0.761726 q^{52} +5.60696 q^{53} -7.06571 q^{54} -5.12534 q^{55} +2.36354 q^{56} +1.63215 q^{57} -9.71969 q^{58} -2.43162 q^{59} +0.239869 q^{60} -10.7929 q^{61} -13.6087 q^{62} +1.97076 q^{63} +6.68063 q^{64} -2.87981 q^{65} -6.99438 q^{66} -6.59936 q^{67} +2.08134 q^{68} -0.951930 q^{69} -1.36188 q^{70} -9.32040 q^{71} +5.68708 q^{72} +13.5074 q^{73} -1.97188 q^{74} +0.906859 q^{75} +0.476051 q^{76} +4.63848 q^{77} -3.92998 q^{78} +1.69847 q^{79} -4.45905 q^{80} +2.27478 q^{81} +0.851423 q^{82} +0.788704 q^{83} -0.217084 q^{84} +7.86878 q^{85} -10.7594 q^{86} -5.85741 q^{87} +13.3854 q^{88} -3.10466 q^{89} -3.27692 q^{90} +2.60626 q^{91} -0.277651 q^{92} -8.20104 q^{93} +12.2533 q^{94} +1.79978 q^{95} -1.34837 q^{96} -15.4091 q^{97} -9.30127 q^{98} +11.1610 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9}+O(q^{10}) $ 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9} - 5 q^{10} - 30 q^{11} - 29 q^{12} - 18 q^{13} + 6 q^{14} - 10 q^{15} + 21 q^{16} - 18 q^{17} - 30 q^{18} - 17 q^{19} + 25 q^{20} + 6 q^{21} - 2 q^{22} - 44 q^{23} + 11 q^{24} + 25 q^{25} - 14 q^{26} - 25 q^{27} - 50 q^{28} - 9 q^{29} + 2 q^{30} - 13 q^{31} - 45 q^{32} - 21 q^{33} - 21 q^{34} - 31 q^{35} + 5 q^{36} - 28 q^{37} - 32 q^{38} + 9 q^{39} - 30 q^{40} + 28 q^{41} - 67 q^{42} - 61 q^{43} - 49 q^{44} + 17 q^{45} + 18 q^{46} - 53 q^{47} - 44 q^{48} + 28 q^{49} - 5 q^{50} - 30 q^{51} - 3 q^{52} - 36 q^{53} + 17 q^{54} - 30 q^{55} - 3 q^{56} - 13 q^{57} + 2 q^{58} - 39 q^{59} - 29 q^{60} + 10 q^{61} - 30 q^{62} - 44 q^{63} - 4 q^{64} - 18 q^{65} + 33 q^{66} - 10 q^{67} - 18 q^{68} - 6 q^{69} + 6 q^{70} - 7 q^{71} - q^{72} - 26 q^{73} - 3 q^{74} - 10 q^{75} + 12 q^{76} + 29 q^{77} - 5 q^{78} - 6 q^{79} + 21 q^{80} + 13 q^{81} - 30 q^{82} - 35 q^{83} + 117 q^{84} - 18 q^{85} + 14 q^{86} - 104 q^{87} + 53 q^{88} + 7 q^{89} - 30 q^{90} - 25 q^{91} - 31 q^{92} + 2 q^{93} + 68 q^{94} - 17 q^{95} + 92 q^{96} + 6 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9 - 5 * q^10 - 30 * q^11 - 29 * q^12 - 18 * q^13 + 6 * q^14 - 10 * q^15 + 21 * q^16 - 18 * q^17 - 30 * q^18 - 17 * q^19 + 25 * q^20 + 6 * q^21 - 2 * q^22 - 44 * q^23 + 11 * q^24 + 25 * q^25 - 14 * q^26 - 25 * q^27 - 50 * q^28 - 9 * q^29 + 2 * q^30 - 13 * q^31 - 45 * q^32 - 21 * q^33 - 21 * q^34 - 31 * q^35 + 5 * q^36 - 28 * q^37 - 32 * q^38 + 9 * q^39 - 30 * q^40 + 28 * q^41 - 67 * q^42 - 61 * q^43 - 49 * q^44 + 17 * q^45 + 18 * q^46 - 53 * q^47 - 44 * q^48 + 28 * q^49 - 5 * q^50 - 30 * q^51 - 3 * q^52 - 36 * q^53 + 17 * q^54 - 30 * q^55 - 3 * q^56 - 13 * q^57 + 2 * q^58 - 39 * q^59 - 29 * q^60 + 10 * q^61 - 30 * q^62 - 44 * q^63 - 4 * q^64 - 18 * q^65 + 33 * q^66 - 10 * q^67 - 18 * q^68 - 6 * q^69 + 6 * q^70 - 7 * q^71 - q^72 - 26 * q^73 - 3 * q^74 - 10 * q^75 + 12 * q^76 + 29 * q^77 - 5 * q^78 - 6 * q^79 + 21 * q^80 + 13 * q^81 - 30 * q^82 - 35 * q^83 + 117 * q^84 - 18 * q^85 + 14 * q^86 - 104 * q^87 + 53 * q^88 + 7 * q^89 - 30 * q^90 - 25 * q^91 - 31 * q^92 + 2 * q^93 + 68 * q^94 - 17 * q^95 + 92 * q^96 + 6 * q^97 + 15 * q^98 - 36 * 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 $

$2$	1.50483	1.06407	0.532037	−	0.846721i	$-0.321427\pi$
$2$	1.50483	1.06407	0.532037	+	0.846721i	$0.321427\pi$
$3$	0.906859	0.523576	0.261788	−	0.965125i	$-0.415688\pi$
$3$	0.906859	0.523576	0.261788	+	0.965125i	$0.415688\pi$
$4$	0.264506	0.132253
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.36467	0.557123
$7$	−0.905011	−0.342062	−0.171031	−	0.985266i	$-0.554710\pi$
$7$	−0.905011	−0.342062	−0.171031	+	0.985266i	$0.554710\pi$
$8$	−2.61162	−0.923347
$9$	−2.17761	−0.725869
$10$	1.50483	0.475868
$11$	−5.12534	−1.54535	−0.772673	−	0.634804i	$-0.781081\pi$
$11$	−5.12534	−1.54535	−0.772673	+	0.634804i	$0.781081\pi$
$12$	0.239869	0.0692443
$13$	−2.87981	−0.798716	−0.399358	−	0.916795i	$-0.630767\pi$
$13$	−2.87981	−0.798716	−0.399358	+	0.916795i	$0.630767\pi$
$14$	−1.36188	−0.363979
$15$	0.906859	0.234150
$16$	−4.45905	−1.11476
$17$	7.86878	1.90846	0.954230	−	0.299074i	$-0.0966776\pi$
$17$	7.86878	1.90846	0.954230	+	0.299074i	$0.0966776\pi$
$18$	−3.27692	−0.772378
$19$	1.79978	0.412897	0.206449	−	0.978457i	$-0.433809\pi$
$19$	1.79978	0.412897	0.206449	+	0.978457i	$0.433809\pi$
$20$	0.264506	0.0591452
$21$	−0.820717	−0.179095
$22$	−7.71274	−1.64436
$23$	−1.04970	−0.218877	−0.109439	−	0.993994i	$-0.534905\pi$
$23$	−1.04970	−0.218877	−0.109439	+	0.993994i	$0.534905\pi$
$24$	−2.36837	−0.483442
$25$	1.00000	0.200000
$26$	−4.33362	−0.849893
$27$	−4.69536	−0.903623
$28$	−0.239380	−0.0452386
$29$	−6.45900	−1.19941	−0.599704	−	0.800222i	$-0.704715\pi$
$29$	−6.45900	−1.19941	−0.599704	+	0.800222i	$0.704715\pi$
$30$	1.36467	0.249153
$31$	−9.04334	−1.62423	−0.812116	−	0.583496i	$-0.801684\pi$
$31$	−9.04334	−1.62423	−0.812116	+	0.583496i	$0.801684\pi$
$32$	−1.48686	−0.262842
$33$	−4.64796	−0.809106
$34$	11.8412	2.03074
$35$	−0.905011	−0.152975
$36$	−0.575989	−0.0959981
$37$	−1.31037	−0.215424	−0.107712	−	0.994182i	$-0.534352\pi$
$37$	−1.31037	−0.215424	−0.107712	+	0.994182i	$0.534352\pi$
$38$	2.70835	0.439353
$39$	−2.61158	−0.418188
$40$	−2.61162	−0.412933
$41$	0.565794	0.0883622	0.0441811	−	0.999024i	$-0.485932\pi$
$41$	0.565794	0.0883622	0.0441811	+	0.999024i	$0.485932\pi$
$42$	−1.23504	−0.190571
$43$	−7.14995	−1.09036	−0.545179	−	0.838320i	$-0.683538\pi$
$43$	−7.14995	−1.09036	−0.545179	+	0.838320i	$0.683538\pi$
$44$	−1.35568	−0.204376
$45$	−2.17761	−0.324618
$46$	−1.57962	−0.232902
$47$	8.14264	1.18773	0.593863	−	0.804566i	$-0.297602\pi$
$47$	8.14264	1.18773	0.593863	+	0.804566i	$0.297602\pi$
$48$	−4.04373	−0.583662
$49$	−6.18096	−0.882994
$50$	1.50483	0.212815
$51$	7.13588	0.999223
$52$	−0.761726	−0.105632
$53$	5.60696	0.770175	0.385088	−	0.922880i	$-0.374171\pi$
$53$	5.60696	0.770175	0.385088	+	0.922880i	$0.374171\pi$
$54$	−7.06571	−0.961521
$55$	−5.12534	−0.691100
$56$	2.36354	0.315842
$57$	1.63215	0.216183
$58$	−9.71969	−1.27626
$59$	−2.43162	−0.316570	−0.158285	−	0.987393i	$-0.550596\pi$
$59$	−2.43162	−0.316570	−0.158285	+	0.987393i	$0.550596\pi$
$60$	0.239869	0.0309670
$61$	−10.7929	−1.38189	−0.690946	−	0.722906i	$-0.742806\pi$
$61$	−10.7929	−1.38189	−0.690946	+	0.722906i	$0.742806\pi$
$62$	−13.6087	−1.72830
$63$	1.97076	0.248292
$64$	6.68063	0.835079
$65$	−2.87981	−0.357197
$66$	−6.99438	−0.860948
$67$	−6.59936	−0.806240	−0.403120	−	0.915147i	$-0.632074\pi$
$67$	−6.59936	−0.806240	−0.403120	+	0.915147i	$0.632074\pi$
$68$	2.08134	0.252399
$69$	−0.951930	−0.114599
$70$	−1.36188	−0.162776
$71$	−9.32040	−1.10613	−0.553064	−	0.833139i	$-0.686541\pi$
$71$	−9.32040	−1.10613	−0.553064	+	0.833139i	$0.686541\pi$
$72$	5.68708	0.670229
$73$	13.5074	1.58092	0.790459	−	0.612515i	$-0.209842\pi$
$73$	13.5074	1.58092	0.790459	+	0.612515i	$0.209842\pi$
$74$	−1.97188	−0.229227
$75$	0.906859	0.104715
$76$	0.476051	0.0546068
$77$	4.63848	0.528604
$78$	−3.92998	−0.444983
$79$	1.69847	0.191093	0.0955464	−	0.995425i	$-0.469540\pi$
$79$	1.69847	0.191093	0.0955464	+	0.995425i	$0.469540\pi$
$80$	−4.45905	−0.498537
$81$	2.27478	0.252754
$82$	0.851423	0.0940239
$83$	0.788704	0.0865716	0.0432858	−	0.999063i	$-0.486217\pi$
$83$	0.788704	0.0865716	0.0432858	+	0.999063i	$0.486217\pi$
$84$	−0.217084	−0.0236858
$85$	7.86878	0.853489
$86$	−10.7594	−1.16022
$87$	−5.85741	−0.627980
$88$	13.3854	1.42689
$89$	−3.10466	−0.329094	−0.164547	−	0.986369i	$-0.552616\pi$
$89$	−3.10466	−0.329094	−0.164547	+	0.986369i	$0.552616\pi$
$90$	−3.27692	−0.345418
$91$	2.60626	0.273210
$92$	−0.277651	−0.0289472
$93$	−8.20104	−0.850408
$94$	12.2533	1.26383
$95$	1.79978	0.184653
$96$	−1.34837	−0.137618
$97$	−15.4091	−1.56455	−0.782276	−	0.622931i	$-0.785942\pi$
$97$	−15.4091	−1.56455	−0.782276	+	0.622931i	$0.785942\pi$
$98$	−9.30127	−0.939570
$99$	11.1610	1.12172
$100$	0.264506	0.0264506
$101$	5.65405	0.562599	0.281299	−	0.959620i	$-0.409235\pi$
$101$	5.65405	0.562599	0.281299	+	0.959620i	$0.409235\pi$
$102$	10.7383	1.06325
$103$	17.4559	1.71998	0.859991	−	0.510309i	$-0.170469\pi$
$103$	17.4559	1.71998	0.859991	+	0.510309i	$0.170469\pi$
$104$	7.52097	0.737492
$105$	−0.820717	−0.0800938
$106$	8.43751	0.819523
$107$	−5.01731	−0.485041	−0.242521	−	0.970146i	$-0.577974\pi$
$107$	−5.01731	−0.485041	−0.242521	+	0.970146i	$0.577974\pi$
$108$	−1.24195	−0.119507
$109$	19.2276	1.84167	0.920833	−	0.389956i	$-0.127510\pi$
$109$	19.2276	1.84167	0.920833	+	0.389956i	$0.127510\pi$
$110$	−7.71274	−0.735381
$111$	−1.18832	−0.112791
$112$	4.03549	0.381318
$113$	16.1258	1.51699	0.758493	−	0.651681i	$-0.225936\pi$
$113$	16.1258	1.51699	0.758493	+	0.651681i	$0.225936\pi$
$114$	2.45610	0.230035
$115$	−1.04970	−0.0978850
$116$	−1.70844	−0.158625
$117$	6.27109	0.579763
$118$	−3.65916	−0.336853
$119$	−7.12133	−0.652811
$120$	−2.36837	−0.216202
$121$	15.2691	1.38810
$122$	−16.2415	−1.47044
$123$	0.513096	0.0462643
$124$	−2.39201	−0.214809
$125$	1.00000	0.0894427
$126$	2.96565	0.264201
$127$	5.19649	0.461114	0.230557	−	0.973059i	$-0.425945\pi$
$127$	5.19649	0.461114	0.230557	+	0.973059i	$0.425945\pi$
$128$	13.0269	1.15143
$129$	−6.48400	−0.570884
$130$	−4.33362	−0.380084
$131$	−9.09290	−0.794450	−0.397225	−	0.917721i	$-0.630027\pi$
$131$	−9.09290	−0.794450	−0.397225	+	0.917721i	$0.630027\pi$
$132$	−1.22941	−0.107006
$133$	−1.62882	−0.141236
$134$	−9.93090	−0.857899
$135$	−4.69536	−0.404112
$136$	−20.5503	−1.76217
$137$	5.16187	0.441009	0.220504	−	0.975386i	$-0.429230\pi$
$137$	5.16187	0.441009	0.220504	+	0.975386i	$0.429230\pi$
$138$	−1.43249	−0.121942
$139$	−3.71058	−0.314728	−0.157364	−	0.987541i	$-0.550300\pi$
$139$	−3.71058	−0.314728	−0.157364	+	0.987541i	$0.550300\pi$
$140$	−0.239380	−0.0202313
$141$	7.38423	0.621865
$142$	−14.0256	−1.17700
$143$	14.7600	1.23429
$144$	9.71005	0.809171
$145$	−6.45900	−0.536391
$146$	20.3263	1.68221
$147$	−5.60526	−0.462314
$148$	−0.346600	−0.0284904
$149$	14.6078	1.19672	0.598359	−	0.801228i	$-0.295820\pi$
$149$	14.6078	1.19672	0.598359	+	0.801228i	$0.295820\pi$
$150$	1.36467	0.111425
$151$	−1.30171	−0.105932	−0.0529658	−	0.998596i	$-0.516867\pi$
$151$	−1.30171	−0.105932	−0.0529658	+	0.998596i	$0.516867\pi$
$152$	−4.70034	−0.381248
$153$	−17.1351	−1.38529
$154$	6.98011	0.562474
$155$	−9.04334	−0.726379
$156$	−0.690778	−0.0553065
$157$	−7.17547	−0.572665	−0.286333	−	0.958130i	$-0.592436\pi$
$157$	−7.17547	−0.572665	−0.286333	+	0.958130i	$0.592436\pi$
$158$	2.55591	0.203337
$159$	5.08473	0.403245
$160$	−1.48686	−0.117546
$161$	0.949989	0.0748696
$162$	3.42316	0.268949
$163$	−1.93356	−0.151448	−0.0757241	−	0.997129i	$-0.524127\pi$
$163$	−1.93356	−0.151448	−0.0757241	+	0.997129i	$0.524127\pi$
$164$	0.149656	0.0116861
$165$	−4.64796	−0.361843
$166$	1.18686	0.0921185
$167$	18.2932	1.41557	0.707783	−	0.706430i	$-0.249695\pi$
$167$	18.2932	1.41557	0.707783	+	0.706430i	$0.249695\pi$
$168$	2.14340	0.165367
$169$	−4.70669	−0.362053
$170$	11.8412	0.908176
$171$	−3.91921	−0.299709
$172$	−1.89120	−0.144203
$173$	−23.2939	−1.77100	−0.885502	−	0.464636i	$-0.846185\pi$
$173$	−23.2939	−1.77100	−0.885502	+	0.464636i	$0.846185\pi$
$174$	−8.81439	−0.668217
$175$	−0.905011	−0.0684124
$176$	22.8541	1.72269
$177$	−2.20514	−0.165748
$178$	−4.67198	−0.350180
$179$	−13.0764	−0.977377	−0.488688	−	0.872458i	$-0.662524\pi$
$179$	−13.0764	−0.977377	−0.488688	+	0.872458i	$0.662524\pi$
$180$	−0.575989	−0.0429317
$181$	11.9609	0.889048	0.444524	−	0.895767i	$-0.353373\pi$
$181$	11.9609	0.889048	0.444524	+	0.895767i	$0.353373\pi$
$182$	3.92197	0.290716
$183$	−9.78767	−0.723525
$184$	2.74142	0.202100
$185$	−1.31037	−0.0963404
$186$	−12.3412	−0.904897
$187$	−40.3301	−2.94923
$188$	2.15377	0.157080
$189$	4.24935	0.309095
$190$	2.70835	0.196485
$191$	−12.5275	−0.906458	−0.453229	−	0.891394i	$-0.649728\pi$
$191$	−12.5275	−0.906458	−0.453229	+	0.891394i	$0.649728\pi$
$192$	6.05839	0.437227
$193$	−0.201854	−0.0145298	−0.00726490	−	0.999974i	$-0.502313\pi$
$193$	−0.201854	−0.0145298	−0.00726490	+	0.999974i	$0.502313\pi$
$194$	−23.1880	−1.66480
$195$	−2.61158	−0.187019
$196$	−1.63490	−0.116778
$197$	−3.61465	−0.257533	−0.128766	−	0.991675i	$-0.541102\pi$
$197$	−3.61465	−0.257533	−0.128766	+	0.991675i	$0.541102\pi$
$198$	16.7953	1.19359
$199$	3.57850	0.253673	0.126837	−	0.991924i	$-0.459518\pi$
$199$	3.57850	0.253673	0.126837	+	0.991924i	$0.459518\pi$
$200$	−2.61162	−0.184669
$201$	−5.98469	−0.422128
$202$	8.50837	0.598647
$203$	5.84547	0.410271
$204$	1.88748	0.132150
$205$	0.565794	0.0395168
$206$	26.2681	1.83019
$207$	2.28583	0.158876
$208$	12.8412	0.890378
$209$	−9.22446	−0.638070
$210$	−1.23504	−0.0852257
$211$	−12.6124	−0.868273	−0.434136	−	0.900847i	$-0.642946\pi$
$211$	−12.6124	−0.868273	−0.434136	+	0.900847i	$0.642946\pi$
$212$	1.48307	0.101858
$213$	−8.45229	−0.579141
$214$	−7.55018	−0.516120
$215$	−7.14995	−0.487623
$216$	12.2625	0.834357
$217$	8.18432	0.555588
$218$	28.9342	1.95967
$219$	12.2493	0.827730
$220$	−1.35568	−0.0913999
$221$	−22.6606	−1.52432
$222$	−1.78822	−0.120017
$223$	−11.3614	−0.760814	−0.380407	−	0.924819i	$-0.624216\pi$
$223$	−11.3614	−0.760814	−0.380407	+	0.924819i	$0.624216\pi$
$224$	1.34562	0.0899082
$225$	−2.17761	−0.145174
$226$	24.2665	1.61418
$227$	15.3506	1.01886	0.509429	−	0.860513i	$-0.329857\pi$
$227$	15.3506	1.01886	0.509429	+	0.860513i	$0.329857\pi$
$228$	0.431712	0.0285908
$229$	−0.554889	−0.0366681	−0.0183340	−	0.999832i	$-0.505836\pi$
$229$	−0.554889	−0.0366681	−0.0183340	+	0.999832i	$0.505836\pi$
$230$	−1.57962	−0.104157
$231$	4.20645	0.276764
$232$	16.8685	1.10747
$233$	19.9855	1.30930	0.654648	−	0.755934i	$-0.272817\pi$
$233$	19.9855	1.30930	0.654648	+	0.755934i	$0.272817\pi$
$234$	9.43691	0.616910
$235$	8.14264	0.531167
$236$	−0.643176	−0.0418672
$237$	1.54027	0.100052
$238$	−10.7164	−0.694639
$239$	−4.57434	−0.295889	−0.147945	−	0.988996i	$-0.547266\pi$
$239$	−4.57434	−0.295889	−0.147945	+	0.988996i	$0.547266\pi$
$240$	−4.04373	−0.261022
$241$	−6.08836	−0.392186	−0.196093	−	0.980585i	$-0.562825\pi$
$241$	−6.08836	−0.392186	−0.196093	+	0.980585i	$0.562825\pi$
$242$	22.9773	1.47704
$243$	16.1490	1.03596
$244$	−2.85479	−0.182759
$245$	−6.18096	−0.394887
$246$	0.772121	0.0492286
$247$	−5.18302	−0.329788
$248$	23.6178	1.49973
$249$	0.715244	0.0453268
$250$	1.50483	0.0951736
$251$	−15.8410	−0.999877	−0.499938	−	0.866061i	$-0.666644\pi$
$251$	−15.8410	−0.999877	−0.499938	+	0.866061i	$0.666644\pi$
$252$	0.521276	0.0328373
$253$	5.38006	0.338242
$254$	7.81982	0.490659
$255$	7.13588	0.446866
$256$	6.24199	0.390125
$257$	−21.2335	−1.32451	−0.662253	−	0.749280i	$-0.730400\pi$
$257$	−21.2335	−1.32451	−0.662253	+	0.749280i	$0.730400\pi$
$258$	−9.75730	−0.607463
$259$	1.18590	0.0736882
$260$	−0.761726	−0.0472402
$261$	14.0652	0.870612
$262$	−13.6832	−0.845354
$263$	4.36049	0.268880	0.134440	−	0.990922i	$-0.457077\pi$
$263$	4.36049	0.268880	0.134440	+	0.990922i	$0.457077\pi$
$264$	12.1387	0.747085
$265$	5.60696	0.344433
$266$	−2.45109	−0.150286
$267$	−2.81549	−0.172305
$268$	−1.74557	−0.106627
$269$	−9.04795	−0.551663	−0.275832	−	0.961206i	$-0.588953\pi$
$269$	−9.04795	−0.551663	−0.275832	+	0.961206i	$0.588953\pi$
$270$	−7.06571	−0.430005
$271$	−14.3679	−0.872788	−0.436394	−	0.899756i	$-0.643745\pi$
$271$	−14.3679	−0.872788	−0.436394	+	0.899756i	$0.643745\pi$
$272$	−35.0873	−2.12748
$273$	2.36351	0.143046
$274$	7.76773	0.469266
$275$	−5.12534	−0.309069
$276$	−0.251791	−0.0151560
$277$	1.11122	0.0667667	0.0333834	−	0.999443i	$-0.489372\pi$
$277$	1.11122	0.0667667	0.0333834	+	0.999443i	$0.489372\pi$
$278$	−5.58379	−0.334893
$279$	19.6928	1.17898
$280$	2.36354	0.141249
$281$	−23.6825	−1.41278	−0.706388	−	0.707824i	$-0.749677\pi$
$281$	−23.6825	−1.41278	−0.706388	+	0.707824i	$0.749677\pi$
$282$	11.1120	0.661710
$283$	−5.68888	−0.338169	−0.169084	−	0.985602i	$-0.554081\pi$
$283$	−5.68888	−0.338169	−0.169084	+	0.985602i	$0.554081\pi$
$284$	−2.46530	−0.146288
$285$	1.63215	0.0966800
$286$	22.2112	1.31338
$287$	−0.512050	−0.0302253
$288$	3.23779	0.190789
$289$	44.9177	2.64222
$290$	−9.71969	−0.570760
$291$	−13.9739	−0.819162
$292$	3.57277	0.209081
$293$	−26.2391	−1.53291	−0.766453	−	0.642300i	$-0.777980\pi$
$293$	−26.2391	−1.53291	−0.766453	+	0.642300i	$0.777980\pi$
$294$	−8.43495	−0.491936
$295$	−2.43162	−0.141574
$296$	3.42219	0.198911
$297$	24.0653	1.39641
$298$	21.9822	1.27340
$299$	3.02294	0.174821
$300$	0.239869	0.0138489
$301$	6.47078	0.372970
$302$	−1.95885	−0.112719
$303$	5.12743	0.294563
$304$	−8.02530	−0.460282
$305$	−10.7929	−0.618001
$306$	−25.7854	−1.47405
$307$	1.31234	0.0748992	0.0374496	−	0.999299i	$-0.488077\pi$
$307$	1.31234	0.0748992	0.0374496	+	0.999299i	$0.488077\pi$
$308$	1.22690	0.0699094
$309$	15.8301	0.900541
$310$	−13.6087	−0.772920
$311$	−2.91302	−0.165182	−0.0825911	−	0.996584i	$-0.526320\pi$
$311$	−2.91302	−0.165182	−0.0825911	+	0.996584i	$0.526320\pi$
$312$	6.82046	0.386133
$313$	−29.1746	−1.64904	−0.824522	−	0.565830i	$-0.808556\pi$
$313$	−29.1746	−1.64904	−0.824522	+	0.565830i	$0.808556\pi$
$314$	−10.7979	−0.609358
$315$	1.97076	0.111040
$316$	0.449255	0.0252726
$317$	−5.94835	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$317$	−5.94835	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$318$	7.65164	0.429082
$319$	33.1046	1.85350
$320$	6.68063	0.373459
$321$	−4.54999	−0.253956
$322$	1.42957	0.0796668
$323$	14.1621	0.787998
$324$	0.601693	0.0334274
$325$	−2.87981	−0.159743
$326$	−2.90968	−0.161152
$327$	17.4367	0.964252
$328$	−1.47764	−0.0815890
$329$	−7.36918	−0.406276
$330$	−6.99438	−0.385028
$331$	−28.1863	−1.54926	−0.774630	−	0.632414i	$-0.782064\pi$
$331$	−28.1863	−1.54926	−0.774630	+	0.632414i	$0.782064\pi$
$332$	0.208617	0.0114493
$333$	2.85347	0.156369
$334$	27.5280	1.50627
$335$	−6.59936	−0.360562
$336$	3.65962	0.199649
$337$	34.6584	1.88796	0.943981	−	0.330001i	$-0.107049\pi$
$337$	34.6584	1.88796	0.943981	+	0.330001i	$0.107049\pi$
$338$	−7.08275	−0.385251
$339$	14.6238	0.794257
$340$	2.08134	0.112876
$341$	46.3502	2.51000
$342$	−5.89773	−0.318913
$343$	11.9289	0.644100
$344$	18.6730	1.00678
$345$	−0.951930	−0.0512502
$346$	−35.0533	−1.88448
$347$	−21.7805	−1.16924	−0.584619	−	0.811308i	$-0.698756\pi$
$347$	−21.7805	−1.16924	−0.584619	+	0.811308i	$0.698756\pi$
$348$	−1.54932	−0.0830521
$349$	−15.3681	−0.822635	−0.411318	−	0.911492i	$-0.634931\pi$
$349$	−15.3681	−0.822635	−0.411318	+	0.911492i	$0.634931\pi$
$350$	−1.36188	−0.0727958
$351$	13.5218	0.721738
$352$	7.62065	0.406182
$353$	22.6991	1.20815	0.604076	−	0.796927i	$-0.293542\pi$
$353$	22.6991	1.20815	0.604076	+	0.796927i	$0.293542\pi$
$354$	−3.31835	−0.176368
$355$	−9.32040	−0.494675
$356$	−0.821201	−0.0435236
$357$	−6.45805	−0.341796
$358$	−19.6777	−1.04000
$359$	30.0149	1.58413	0.792063	−	0.610440i	$-0.209007\pi$
$359$	30.0149	1.58413	0.792063	+	0.610440i	$0.209007\pi$
$360$	5.68708	0.299735
$361$	−15.7608	−0.829516
$362$	17.9991	0.946012
$363$	13.8469	0.726773
$364$	0.689370	0.0361328
$365$	13.5074	0.707008
$366$	−14.7288	−0.769884
$367$	17.2878	0.902416	0.451208	−	0.892419i	$-0.350993\pi$
$367$	17.2878	0.902416	0.451208	+	0.892419i	$0.350993\pi$
$368$	4.68066	0.243996
$369$	−1.23208	−0.0641394
$370$	−1.97188	−0.102513
$371$	−5.07436	−0.263448
$372$	−2.16922	−0.112469
$373$	15.3509	0.794840	0.397420	−	0.917637i	$-0.369906\pi$
$373$	15.3509	0.794840	0.397420	+	0.917637i	$0.369906\pi$
$374$	−60.6899	−3.13820
$375$	0.906859	0.0468300
$376$	−21.2655	−1.09668
$377$	18.6007	0.957986
$378$	6.39454	0.328900
$379$	1.18743	0.0609944	0.0304972	−	0.999535i	$-0.490291\pi$
$379$	1.18743	0.0609944	0.0304972	+	0.999535i	$0.490291\pi$
$380$	0.476051	0.0244209
$381$	4.71249	0.241428
$382$	−18.8517	−0.964539
$383$	−9.27092	−0.473722	−0.236861	−	0.971544i	$-0.576119\pi$
$383$	−9.27092	−0.473722	−0.236861	+	0.971544i	$0.576119\pi$
$384$	11.8136	0.602859
$385$	4.63848	0.236399
$386$	−0.303756	−0.0154608
$387$	15.5698	0.791456
$388$	−4.07578	−0.206916
$389$	−8.58642	−0.435349	−0.217674	−	0.976021i	$-0.569847\pi$
$389$	−8.58642	−0.435349	−0.217674	+	0.976021i	$0.569847\pi$
$390$	−3.92998	−0.199002
$391$	−8.25986	−0.417719
$392$	16.1423	0.815310
$393$	−8.24598	−0.415955
$394$	−5.43942	−0.274034
$395$	1.69847	0.0854593
$396$	2.95214	0.148350
$397$	−4.17447	−0.209511	−0.104755	−	0.994498i	$-0.533406\pi$
$397$	−4.17447	−0.209511	−0.104755	+	0.994498i	$0.533406\pi$
$398$	5.38503	0.269927
$399$	−1.47711	−0.0739479
$400$	−4.45905	−0.222952
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−9.00593	−0.449175
$403$	26.0431	1.29730
$404$	1.49553	0.0744053
$405$	2.27478	0.113035
$406$	8.79642	0.436559
$407$	6.71609	0.332904
$408$	−18.6362	−0.922630
$409$	−21.7873	−1.07731	−0.538656	−	0.842526i	$-0.681068\pi$
$409$	−21.7873	−1.07731	−0.538656	+	0.842526i	$0.681068\pi$
$410$	0.851423	0.0420488
$411$	4.68109	0.230901
$412$	4.61719	0.227472
$413$	2.20064	0.108286
$414$	3.43978	0.169056
$415$	0.788704	0.0387160
$416$	4.28187	0.209936
$417$	−3.36498	−0.164784
$418$	−13.8812	−0.678953
$419$	30.2829	1.47942	0.739708	−	0.672928i	$-0.234964\pi$
$419$	30.2829	1.47942	0.739708	+	0.672928i	$0.234964\pi$
$420$	−0.217084	−0.0105926
$421$	5.26387	0.256545	0.128273	−	0.991739i	$-0.459057\pi$
$421$	5.26387	0.256545	0.128273	+	0.991739i	$0.459057\pi$
$422$	−18.9795	−0.923906
$423$	−17.7315	−0.862133
$424$	−14.6433	−0.711139
$425$	7.86878	0.381692
$426$	−12.7192	−0.616249
$427$	9.76771	0.472693
$428$	−1.32711	−0.0641481
$429$	13.3852	0.646246
$430$	−10.7594	−0.518866
$431$	1.93311	0.0931149	0.0465574	−	0.998916i	$-0.485175\pi$
$431$	1.93311	0.0931149	0.0465574	+	0.998916i	$0.485175\pi$
$432$	20.9368	1.00732
$433$	6.61302	0.317801	0.158901	−	0.987295i	$-0.449205\pi$
$433$	6.61302	0.317801	0.158901	+	0.987295i	$0.449205\pi$
$434$	12.3160	0.591186
$435$	−5.85741	−0.280841
$436$	5.08580	0.243566
$437$	−1.88923	−0.0903739
$438$	18.4331	0.880766
$439$	−3.60781	−0.172191	−0.0860956	−	0.996287i	$-0.527439\pi$
$439$	−3.60781	−0.172191	−0.0860956	+	0.996287i	$0.527439\pi$
$440$	13.3854	0.638125
$441$	13.4597	0.640937
$442$	−34.1003	−1.62199
$443$	−18.2064	−0.865014	−0.432507	−	0.901631i	$-0.642371\pi$
$443$	−18.2064	−0.865014	−0.432507	+	0.901631i	$0.642371\pi$
$444$	−0.314318	−0.0149169
$445$	−3.10466	−0.147175
$446$	−17.0969	−0.809562
$447$	13.2472	0.626572
$448$	−6.04604	−0.285649
$449$	−10.1417	−0.478615	−0.239307	−	0.970944i	$-0.576920\pi$
$449$	−10.1417	−0.478615	−0.239307	+	0.970944i	$0.576920\pi$
$450$	−3.27692	−0.154476
$451$	−2.89988	−0.136550
$452$	4.26536	0.200626
$453$	−1.18047	−0.0554632
$454$	23.1001	1.08414
$455$	2.60626	0.122183
$456$	−4.26254	−0.199612
$457$	39.7766	1.86067	0.930335	−	0.366711i	$-0.119516\pi$
$457$	39.7766	1.86067	0.930335	+	0.366711i	$0.119516\pi$
$458$	−0.835012	−0.0390175
$459$	−36.9468	−1.72453
$460$	−0.277651	−0.0129456
$461$	−20.4495	−0.952430	−0.476215	−	0.879329i	$-0.657992\pi$
$461$	−20.4495	−0.952430	−0.476215	+	0.879329i	$0.657992\pi$
$462$	6.32998	0.294497
$463$	−33.7457	−1.56830	−0.784149	−	0.620573i	$-0.786900\pi$
$463$	−33.7457	−1.56830	−0.784149	+	0.620573i	$0.786900\pi$
$464$	28.8010	1.33705
$465$	−8.20104	−0.380314
$466$	30.0748	1.39319
$467$	−4.32924	−0.200334	−0.100167	−	0.994971i	$-0.531938\pi$
$467$	−4.32924	−0.200334	−0.100167	+	0.994971i	$0.531938\pi$
$468$	1.65874	0.0766752
$469$	5.97249	0.275784
$470$	12.2533	0.565201
$471$	−6.50715	−0.299834
$472$	6.35046	0.292304
$473$	36.6459	1.68498
$474$	2.31785	0.106462
$475$	1.79978	0.0825795
$476$	−1.88363	−0.0863361
$477$	−12.2098	−0.559046
$478$	−6.88359	−0.314848
$479$	14.3940	0.657679	0.328840	−	0.944386i	$-0.393342\pi$
$479$	14.3940	0.657679	0.328840	+	0.944386i	$0.393342\pi$
$480$	−1.34837	−0.0615445
$481$	3.77362	0.172062
$482$	−9.16194	−0.417315
$483$	0.861507	0.0391999
$484$	4.03875	0.183580
$485$	−15.4091	−0.699689
$486$	24.3014	1.10234
$487$	40.9172	1.85413	0.927067	−	0.374896i	$-0.122322\pi$
$487$	40.9172	1.85413	0.927067	+	0.374896i	$0.122322\pi$
$488$	28.1870	1.27597
$489$	−1.75347	−0.0792946
$490$	−9.30127	−0.420189
$491$	9.88539	0.446121	0.223061	−	0.974805i	$-0.428395\pi$
$491$	9.88539	0.446121	0.223061	+	0.974805i	$0.428395\pi$
$492$	0.135717	0.00611858
$493$	−50.8245	−2.28902
$494$	−7.79955	−0.350918
$495$	11.1610	0.501648
$496$	40.3247	1.81063
$497$	8.43506	0.378364
$498$	1.07632	0.0482310
$499$	20.5388	0.919443	0.459721	−	0.888063i	$-0.347949\pi$
$499$	20.5388	0.919443	0.459721	+	0.888063i	$0.347949\pi$
$500$	0.264506	0.0118290
$501$	16.5893	0.741156
$502$	−23.8380	−1.06394
$503$	−11.5783	−0.516251	−0.258125	−	0.966111i	$-0.583105\pi$
$503$	−11.5783	−0.516251	−0.258125	+	0.966111i	$0.583105\pi$
$504$	−5.14687	−0.229260
$505$	5.65405	0.251602
$506$	8.09606	0.359914
$507$	−4.26830	−0.189562
$508$	1.37450	0.0609836
$509$	−22.7558	−1.00863	−0.504316	−	0.863519i	$-0.668255\pi$
$509$	−22.7558	−1.00863	−0.504316	+	0.863519i	$0.668255\pi$
$510$	10.7383	0.475499
$511$	−12.2243	−0.540772
$512$	−16.6607	−0.736306
$513$	−8.45061	−0.373103
$514$	−31.9527	−1.40937
$515$	17.4559	0.769200
$516$	−1.71505	−0.0755011
$517$	−41.7338	−1.83545
$518$	1.78457	0.0784097
$519$	−21.1243	−0.927254
$520$	7.52097	0.329816
$521$	20.4726	0.896920	0.448460	−	0.893803i	$-0.351973\pi$
$521$	20.4726	0.896920	0.448460	+	0.893803i	$0.351973\pi$
$522$	21.1656	0.926395
$523$	−37.5310	−1.64112	−0.820558	−	0.571563i	$-0.806337\pi$
$523$	−37.5310	−1.64112	−0.820558	+	0.571563i	$0.806337\pi$
$524$	−2.40512	−0.105068
$525$	−0.820717	−0.0358190
$526$	6.56179	0.286108
$527$	−71.1601	−3.09978
$528$	20.7255	0.901960
$529$	−21.8981	−0.952093
$530$	8.43751	0.366502
$531$	5.29510	0.229788
$532$	−0.430831	−0.0186789
$533$	−1.62938	−0.0705763
$534$	−4.23683	−0.183346
$535$	−5.01731	−0.216917
$536$	17.2350	0.744439
$537$	−11.8585	−0.511731
$538$	−13.6156	−0.587010
$539$	31.6795	1.36453
$540$	−1.24195	−0.0534450
$541$	−30.0067	−1.29009	−0.645044	−	0.764146i	$-0.723161\pi$
$541$	−30.0067	−1.29009	−0.645044	+	0.764146i	$0.723161\pi$
$542$	−21.6212	−0.928710
$543$	10.8469	0.465484
$544$	−11.6998	−0.501623
$545$	19.2276	0.823618
$546$	3.55668	0.152212
$547$	−43.5301	−1.86121	−0.930607	−	0.366019i	$-0.880721\pi$
$547$	−43.5301	−1.86121	−0.930607	+	0.366019i	$0.880721\pi$
$548$	1.36534	0.0583246
$549$	23.5027	1.00307
$550$	−7.71274	−0.328873
$551$	−11.6248	−0.495232
$552$	2.48608	0.105815
$553$	−1.53713	−0.0653656
$554$	1.67219	0.0710447
$555$	−1.18832	−0.0504415
$556$	−0.981470	−0.0416236
$557$	43.3176	1.83543	0.917713	−	0.397243i	$-0.130033\pi$
$557$	43.3176	1.83543	0.917713	+	0.397243i	$0.130033\pi$
$558$	29.6343	1.25452
$559$	20.5905	0.870886
$560$	4.03549	0.170530
$561$	−36.5738	−1.54415
$562$	−35.6380	−1.50330
$563$	27.3474	1.15256	0.576278	−	0.817254i	$-0.304505\pi$
$563$	27.3474	1.15256	0.576278	+	0.817254i	$0.304505\pi$
$564$	1.95317	0.0822433
$565$	16.1258	0.678417
$566$	−8.56078	−0.359836
$567$	−2.05870	−0.0864575
$568$	24.3413	1.02134
$569$	−11.3766	−0.476932	−0.238466	−	0.971151i	$-0.576645\pi$
$569$	−11.3766	−0.476932	−0.238466	+	0.971151i	$0.576645\pi$
$570$	2.45610	0.102875
$571$	−23.1540	−0.968965	−0.484482	−	0.874801i	$-0.660992\pi$
$571$	−23.1540	−0.968965	−0.484482	+	0.874801i	$0.660992\pi$
$572$	3.90410	0.163239
$573$	−11.3607	−0.474600
$574$	−0.770546	−0.0321620
$575$	−1.04970	−0.0437755
$576$	−14.5478	−0.606158
$577$	2.34491	0.0976200	0.0488100	−	0.998808i	$-0.484457\pi$
$577$	2.34491	0.0976200	0.0488100	+	0.998808i	$0.484457\pi$
$578$	67.5934	2.81152
$579$	−0.183054	−0.00760745
$580$	−1.70844	−0.0709392
$581$	−0.713786	−0.0296128
$582$	−21.0282	−0.871648
$583$	−28.7376	−1.19019
$584$	−35.2761	−1.45974
$585$	6.27109	0.259278
$586$	−39.4854	−1.63113
$587$	−37.1551	−1.53355	−0.766777	−	0.641913i	$-0.778141\pi$
$587$	−37.1551	−1.53355	−0.766777	+	0.641913i	$0.778141\pi$
$588$	−1.48262	−0.0611423
$589$	−16.2760	−0.670641
$590$	−3.65916	−0.150645
$591$	−3.27798	−0.134838
$592$	5.84301	0.240146
$593$	−20.3806	−0.836932	−0.418466	−	0.908232i	$-0.637432\pi$
$593$	−20.3806	−0.836932	−0.418466	+	0.908232i	$0.637432\pi$
$594$	36.2141	1.48588
$595$	−7.12133	−0.291946
$596$	3.86384	0.158269
$597$	3.24520	0.132817
$598$	4.54900	0.186022
$599$	3.27243	0.133708	0.0668540	−	0.997763i	$-0.478704\pi$
$599$	3.27243	0.133708	0.0668540	+	0.997763i	$0.478704\pi$
$600$	−2.36837	−0.0966884
$601$	20.8695	0.851287	0.425643	−	0.904891i	$-0.360048\pi$
$601$	20.8695	0.851287	0.425643	+	0.904891i	$0.360048\pi$
$602$	9.73741	0.396867
$603$	14.3708	0.585224
$604$	−0.344309	−0.0140097
$605$	15.2691	0.620776
$606$	7.71589	0.313437
$607$	−5.84964	−0.237429	−0.118715	−	0.992928i	$-0.537877\pi$
$607$	−5.84964	−0.237429	−0.118715	+	0.992928i	$0.537877\pi$
$608$	−2.67601	−0.108527
$609$	5.30102	0.214808
$610$	−16.2415	−0.657599
$611$	−23.4493	−0.948656
$612$	−4.53233	−0.183209
$613$	−23.7121	−0.957721	−0.478860	−	0.877891i	$-0.658950\pi$
$613$	−23.7121	−0.957721	−0.478860	+	0.877891i	$0.658950\pi$
$614$	1.97485	0.0796983
$615$	0.513096	0.0206900
$616$	−12.1140	−0.488085
$617$	−30.4953	−1.22770	−0.613848	−	0.789425i	$-0.710379\pi$
$617$	−30.4953	−1.22770	−0.613848	+	0.789425i	$0.710379\pi$
$618$	23.8215	0.958242
$619$	26.4911	1.06477	0.532385	−	0.846503i	$-0.321296\pi$
$619$	26.4911	1.06477	0.532385	+	0.846503i	$0.321296\pi$
$620$	−2.39201	−0.0960656
$621$	4.92872	0.197783
$622$	−4.38359	−0.175766
$623$	2.80975	0.112570
$624$	11.6452	0.466180
$625$	1.00000	0.0400000
$626$	−43.9027	−1.75470
$627$	−8.36529	−0.334078
$628$	−1.89795	−0.0757366
$629$	−10.3110	−0.411127
$630$	2.96565	0.118154
$631$	−22.6558	−0.901912	−0.450956	−	0.892546i	$-0.648917\pi$
$631$	−22.6558	−0.901912	−0.450956	+	0.892546i	$0.648917\pi$
$632$	−4.43576	−0.176445
$633$	−11.4377	−0.454606
$634$	−8.95124	−0.355499
$635$	5.19649	0.206216
$636$	1.34494	0.0533303
$637$	17.8000	0.705261
$638$	49.8167	1.97226
$639$	20.2962	0.802903
$640$	13.0269	0.514934
$641$	1.34095	0.0529642	0.0264821	−	0.999649i	$-0.491570\pi$
$641$	1.34095	0.0529642	0.0264821	+	0.999649i	$0.491570\pi$
$642$	−6.84695	−0.270228
$643$	1.14682	0.0452261	0.0226131	−	0.999744i	$-0.492801\pi$
$643$	1.14682	0.0452261	0.0226131	+	0.999744i	$0.492801\pi$
$644$	0.251277	0.00990172
$645$	−6.48400	−0.255307
$646$	21.3115	0.838488
$647$	−27.7096	−1.08938	−0.544688	−	0.838639i	$-0.683352\pi$
$647$	−27.7096	−1.08938	−0.544688	+	0.838639i	$0.683352\pi$
$648$	−5.94087	−0.233380
$649$	12.4629	0.489210
$650$	−4.33362	−0.169979
$651$	7.42203	0.290892
$652$	−0.511438	−0.0200295
$653$	28.1999	1.10355	0.551774	−	0.833994i	$-0.313951\pi$
$653$	28.1999	1.10355	0.551774	+	0.833994i	$0.313951\pi$
$654$	26.2392	1.02603
$655$	−9.09290	−0.355289
$656$	−2.52290	−0.0985028
$657$	−29.4137	−1.14754
$658$	−11.0893	−0.432307
$659$	37.9255	1.47737	0.738684	−	0.674052i	$-0.235448\pi$
$659$	37.9255	1.47737	0.738684	+	0.674052i	$0.235448\pi$
$660$	−1.22941	−0.0478548
$661$	30.1805	1.17389	0.586943	−	0.809628i	$-0.300331\pi$
$661$	30.1805	1.17389	0.586943	+	0.809628i	$0.300331\pi$
$662$	−42.4156	−1.64853
$663$	−20.5500	−0.798095
$664$	−2.05980	−0.0799356
$665$	−1.62882	−0.0631628
$666$	4.29398	0.166388
$667$	6.78001	0.262523
$668$	4.83864	0.187213
$669$	−10.3032	−0.398344
$670$	−9.93090	−0.383664
$671$	55.3174	2.13550
$672$	1.22029	0.0470737
$673$	14.4410	0.556659	0.278329	−	0.960486i	$-0.410219\pi$
$673$	14.4410	0.556659	0.278329	+	0.960486i	$0.410219\pi$
$674$	52.1549	2.00893
$675$	−4.69536	−0.180725
$676$	−1.24495	−0.0478825
$677$	−22.4039	−0.861051	−0.430526	−	0.902578i	$-0.641672\pi$
$677$	−22.4039	−0.861051	−0.430526	+	0.902578i	$0.641672\pi$
$678$	22.0063	0.845148
$679$	13.9454	0.535174
$680$	−20.5503	−0.788067
$681$	13.9209	0.533449
$682$	69.7490	2.67083
$683$	−45.0731	−1.72467	−0.862336	−	0.506336i	$-0.831000\pi$
$683$	−45.0731	−1.72467	−0.862336	+	0.506336i	$0.831000\pi$
$684$	−1.03665	−0.0396374
$685$	5.16187	0.197225
$686$	17.9509	0.685370
$687$	−0.503206	−0.0191985
$688$	31.8820	1.21549
$689$	−16.1470	−0.615151
$690$	−1.43249	−0.0545340
$691$	18.9265	0.719997	0.359998	−	0.932953i	$-0.382777\pi$
$691$	18.9265	0.719997	0.359998	+	0.932953i	$0.382777\pi$
$692$	−6.16137	−0.234220
$693$	−10.1008	−0.383697
$694$	−32.7759	−1.24416
$695$	−3.71058	−0.140750
$696$	15.2973	0.579844
$697$	4.45211	0.168636
$698$	−23.1263	−0.875345
$699$	18.1241	0.685515
$700$	−0.239380	−0.00904772
$701$	35.8251	1.35310	0.676548	−	0.736399i	$-0.263475\pi$
$701$	35.8251	1.35310	0.676548	+	0.736399i	$0.263475\pi$
$702$	20.3479	0.767982
$703$	−2.35838	−0.0889478
$704$	−34.2405	−1.29049
$705$	7.38423	0.278106
$706$	34.1582	1.28556
$707$	−5.11697	−0.192444
$708$	−0.583271	−0.0219206
$709$	7.29265	0.273881	0.136941	−	0.990579i	$-0.456273\pi$
$709$	7.29265	0.273881	0.136941	+	0.990579i	$0.456273\pi$
$710$	−14.0256	−0.526371
$711$	−3.69860	−0.138708
$712$	8.10820	0.303868
$713$	9.49279	0.355508
$714$	−9.71825	−0.363696
$715$	14.7600	0.551993
$716$	−3.45878	−0.129261
$717$	−4.14828	−0.154920
$718$	45.1672	1.68563
$719$	−27.2772	−1.01727	−0.508634	−	0.860983i	$-0.669849\pi$
$719$	−27.2772	−1.01727	−0.508634	+	0.860983i	$0.669849\pi$
$720$	9.71005	0.361872
$721$	−15.7978	−0.588340
$722$	−23.7173	−0.882666
$723$	−5.52129	−0.205339
$724$	3.16373	0.117579
$725$	−6.45900	−0.239881
$726$	20.8372	0.773340
$727$	−41.9999	−1.55769	−0.778845	−	0.627217i	$-0.784194\pi$
$727$	−41.9999	−1.55769	−0.778845	+	0.627217i	$0.784194\pi$
$728$	−6.80656	−0.252268
$729$	7.82051	0.289649
$730$	20.3263	0.752309
$731$	−56.2614	−2.08090
$732$	−2.58889	−0.0956882
$733$	−24.5520	−0.906849	−0.453424	−	0.891295i	$-0.649798\pi$
$733$	−24.5520	−0.906849	−0.453424	+	0.891295i	$0.649798\pi$
$734$	26.0152	0.960237
$735$	−5.60526	−0.206753
$736$	1.56075	0.0575302
$737$	33.8239	1.24592
$738$	−1.85406	−0.0682490
$739$	−9.45163	−0.347684	−0.173842	−	0.984774i	$-0.555618\pi$
$739$	−9.45163	−0.347684	−0.173842	+	0.984774i	$0.555618\pi$
$740$	−0.346600	−0.0127413
$741$	−4.70027	−0.172669
$742$	−7.63604	−0.280328
$743$	−40.0308	−1.46859	−0.734294	−	0.678832i	$-0.762487\pi$
$743$	−40.0308	−1.46859	−0.734294	+	0.678832i	$0.762487\pi$
$744$	21.4180	0.785222
$745$	14.6078	0.535188
$746$	23.1005	0.845769
$747$	−1.71749	−0.0628396
$748$	−10.6675	−0.390044
$749$	4.54072	0.165914
$750$	1.36467	0.0498306
$751$	21.5337	0.785777	0.392889	−	0.919586i	$-0.371476\pi$
$751$	21.5337	0.785777	0.392889	+	0.919586i	$0.371476\pi$
$752$	−36.3084	−1.32403
$753$	−14.3656	−0.523511
$754$	27.9909	1.01937
$755$	−1.30171	−0.0473741
$756$	1.12398	0.0408786
$757$	0.137593	0.00500090	0.00250045	−	0.999997i	$-0.499204\pi$
$757$	0.137593	0.00500090	0.00250045	+	0.999997i	$0.499204\pi$
$758$	1.78688	0.0649025
$759$	4.87896	0.177095
$760$	−4.70034	−0.170499
$761$	25.2632	0.915790	0.457895	−	0.889006i	$-0.348604\pi$
$761$	25.2632	0.915790	0.457895	+	0.889006i	$0.348604\pi$
$762$	7.09148	0.256897
$763$	−17.4011	−0.629964
$764$	−3.31359	−0.119882
$765$	−17.1351	−0.619521
$766$	−13.9511	−0.504075
$767$	7.00260	0.252849
$768$	5.66061	0.204260
$769$	−24.4549	−0.881867	−0.440933	−	0.897540i	$-0.645352\pi$
$769$	−24.4549	−0.881867	−0.440933	+	0.897540i	$0.645352\pi$
$770$	6.98011	0.251546
$771$	−19.2558	−0.693479
$772$	−0.0533916	−0.00192161
$773$	8.77066	0.315459	0.157729	−	0.987482i	$-0.449583\pi$
$773$	8.77066	0.315459	0.157729	+	0.987482i	$0.449583\pi$
$774$	23.4298	0.842168
$775$	−9.04334	−0.324846
$776$	40.2426	1.44463
$777$	1.07544	0.0385813
$778$	−12.9211	−0.463243
$779$	1.01830	0.0364845
$780$	−0.690778	−0.0247338
$781$	47.7702	1.70935
$782$	−12.4297	−0.444484
$783$	30.3274	1.08381
$784$	27.5612	0.984328
$785$	−7.17547	−0.256104
$786$	−12.4088	−0.442606
$787$	−15.9221	−0.567562	−0.283781	−	0.958889i	$-0.591589\pi$
$787$	−15.9221	−0.567562	−0.283781	+	0.958889i	$0.591589\pi$
$788$	−0.956094	−0.0340594
$789$	3.95436	0.140779
$790$	2.55591	0.0909350
$791$	−14.5940	−0.518903
$792$	−29.1482	−1.03574
$793$	31.0816	1.10374
$794$	−6.28186	−0.222935
$795$	5.08473	0.180337
$796$	0.946533	0.0335490
$797$	39.3041	1.39222	0.696111	−	0.717934i	$-0.254912\pi$
$797$	39.3041	1.39222	0.696111	+	0.717934i	$0.254912\pi$
$798$	−2.22279	−0.0786861
$799$	64.0727	2.26673
$800$	−1.48686	−0.0525684
$801$	6.76074	0.238879
$802$	1.50483	0.0531373
$803$	−69.2298	−2.44307
$804$	−1.58298	−0.0558275
$805$	0.949989	0.0334827
$806$	39.1904	1.38042
$807$	−8.20522	−0.288837
$808$	−14.7662	−0.519474
$809$	−30.9555	−1.08834	−0.544169	−	0.838976i	$-0.683155\pi$
$809$	−30.9555	−1.08834	−0.544169	+	0.838976i	$0.683155\pi$
$810$	3.42316	0.120278
$811$	−6.86714	−0.241138	−0.120569	−	0.992705i	$-0.538472\pi$
$811$	−6.86714	−0.241138	−0.120569	+	0.992705i	$0.538472\pi$
$812$	1.54616	0.0542595
$813$	−13.0297	−0.456970
$814$	10.1066	0.354235
$815$	−1.93356	−0.0677297
$816$	−31.8192	−1.11390
$817$	−12.8683	−0.450206
$818$	−32.7861	−1.14634
$819$	−5.67541	−0.198315
$820$	0.149656	0.00522620
$821$	−17.6705	−0.616705	−0.308353	−	0.951272i	$-0.599778\pi$
$821$	−17.6705	−0.616705	−0.308353	+	0.951272i	$0.599778\pi$
$822$	7.04424	0.245696
$823$	9.76372	0.340342	0.170171	−	0.985415i	$-0.445568\pi$
$823$	9.76372	0.340342	0.170171	+	0.985415i	$0.445568\pi$
$824$	−45.5882	−1.58814
$825$	−4.64796	−0.161821
$826$	3.31158	0.115225
$827$	−29.3508	−1.02063	−0.510314	−	0.859988i	$-0.670471\pi$
$827$	−29.3508	−1.02063	−0.510314	+	0.859988i	$0.670471\pi$
$828$	0.604615	0.0210118
$829$	48.6451	1.68951	0.844757	−	0.535150i	$-0.179745\pi$
$829$	48.6451	1.68951	0.844757	+	0.535150i	$0.179745\pi$
$830$	1.18686	0.0411966
$831$	1.00772	0.0349574
$832$	−19.2390	−0.666991
$833$	−48.6366	−1.68516
$834$	−5.06371	−0.175342
$835$	18.2932	0.633061
$836$	−2.43992	−0.0843865
$837$	42.4618	1.46769
$838$	45.5705	1.57421
$839$	−12.7931	−0.441665	−0.220833	−	0.975312i	$-0.570877\pi$
$839$	−12.7931	−0.441665	−0.220833	+	0.975312i	$0.570877\pi$
$840$	2.14340	0.0739544
$841$	12.7187	0.438577
$842$	7.92122	0.272983
$843$	−21.4767	−0.739695
$844$	−3.33605	−0.114831
$845$	−4.70669	−0.161915
$846$	−26.6828	−0.917373
$847$	−13.8187	−0.474815
$848$	−25.0017	−0.858562
$849$	−5.15901	−0.177057
$850$	11.8412	0.406148
$851$	1.37550	0.0471514
$852$	−2.23568	−0.0765931
$853$	15.1573	0.518975	0.259488	−	0.965746i	$-0.416446\pi$
$853$	15.1573	0.518975	0.259488	+	0.965746i	$0.416446\pi$
$854$	14.6987	0.502980
$855$	−3.91921	−0.134034
$856$	13.1033	0.447861
$857$	−48.5199	−1.65741	−0.828704	−	0.559687i	$-0.810921\pi$
$857$	−48.5199	−1.65741	−0.828704	+	0.559687i	$0.810921\pi$
$858$	20.1425	0.687653
$859$	21.4379	0.731451	0.365725	−	0.930723i	$-0.380821\pi$
$859$	21.4379	0.731451	0.365725	+	0.930723i	$0.380821\pi$
$860$	−1.89120	−0.0644894
$861$	−0.464357	−0.0158252
$862$	2.90900	0.0990811
$863$	22.4526	0.764296	0.382148	−	0.924101i	$-0.375184\pi$
$863$	22.4526	0.764296	0.382148	+	0.924101i	$0.375184\pi$
$864$	6.98134	0.237510
$865$	−23.2939	−0.792017
$866$	9.95145	0.338164
$867$	40.7341	1.38340
$868$	2.16480	0.0734780
$869$	−8.70523	−0.295305
$870$	−8.81439	−0.298836
$871$	19.0049	0.643957
$872$	−50.2151	−1.70050
$873$	33.5549	1.13566
$874$	−2.84296	−0.0961645
$875$	−0.905011	−0.0305949
$876$	3.24000	0.109470
$877$	−14.2417	−0.480908	−0.240454	−	0.970661i	$-0.577296\pi$
$877$	−14.2417	−0.480908	−0.240454	+	0.970661i	$0.577296\pi$
$878$	−5.42913	−0.183224
$879$	−23.7952	−0.802592
$880$	22.8541	0.770412
$881$	−39.8143	−1.34138	−0.670689	−	0.741739i	$-0.734001\pi$
$881$	−39.8143	−1.34138	−0.670689	+	0.741739i	$0.734001\pi$
$882$	20.2545	0.682005
$883$	−5.88988	−0.198210	−0.0991051	−	0.995077i	$-0.531598\pi$
$883$	−5.88988	−0.198210	−0.0991051	+	0.995077i	$0.531598\pi$
$884$	−5.99386	−0.201595
$885$	−2.20514	−0.0741248
$886$	−27.3976	−0.920439
$887$	18.3695	0.616788	0.308394	−	0.951259i	$-0.400208\pi$
$887$	18.3695	0.616788	0.308394	+	0.951259i	$0.400208\pi$
$888$	3.10345	0.104145
$889$	−4.70288	−0.157730
$890$	−4.67198	−0.156605
$891$	−11.6590	−0.390592
$892$	−3.00515	−0.100620
$893$	14.6549	0.490409
$894$	19.9348	0.666719
$895$	−13.0764	−0.437096
$896$	−11.7895	−0.393859
$897$	2.74138	0.0915320
$898$	−15.2614	−0.509281
$899$	58.4110	1.94812
$900$	−0.575989	−0.0191996
$901$	44.1200	1.46985
$902$	−4.36383	−0.145300
$903$	5.86809	0.195278
$904$	−42.1144	−1.40070
$905$	11.9609	0.397594
$906$	−1.77640	−0.0590169
$907$	37.4080	1.24211	0.621056	−	0.783766i	$-0.286704\pi$
$907$	37.4080	1.24211	0.621056	+	0.783766i	$0.286704\pi$
$908$	4.06033	0.134747
$909$	−12.3123	−0.408373
$910$	3.92197	0.130012
$911$	42.8953	1.42118	0.710592	−	0.703604i	$-0.248427\pi$
$911$	42.8953	1.42118	0.710592	+	0.703604i	$0.248427\pi$
$912$	−7.27782	−0.240993
$913$	−4.04237	−0.133783
$914$	59.8569	1.97989
$915$	−9.78767	−0.323570
$916$	−0.146771	−0.00484945
$917$	8.22917	0.271751
$918$	−55.5985	−1.83502
$919$	−8.60970	−0.284008	−0.142004	−	0.989866i	$-0.545355\pi$
$919$	−8.60970	−0.284008	−0.142004	+	0.989866i	$0.545355\pi$
$920$	2.74142	0.0903818
$921$	1.19011	0.0392154
$922$	−30.7730	−1.01346
$923$	26.8410	0.883482
$924$	1.11263	0.0366028
$925$	−1.31037	−0.0430847
$926$	−50.7815	−1.66878
$927$	−38.0121	−1.24848
$928$	9.60362	0.315254
$929$	−1.50311	−0.0493154	−0.0246577	−	0.999696i	$-0.507850\pi$
$929$	−1.50311	−0.0493154	−0.0246577	+	0.999696i	$0.507850\pi$
$930$	−12.3412	−0.404682
$931$	−11.1243	−0.364586
$932$	5.28628	0.173158
$933$	−2.64170	−0.0864853
$934$	−6.51477	−0.213170
$935$	−40.3301	−1.31894
$936$	−16.3777	−0.535322
$937$	38.2056	1.24812	0.624061	−	0.781376i	$-0.285482\pi$
$937$	38.2056	1.24812	0.624061	+	0.781376i	$0.285482\pi$
$938$	8.98757	0.293454
$939$	−26.4572	−0.863399
$940$	2.15377	0.0702484
$941$	−0.563825	−0.0183802	−0.00919009	−	0.999958i	$-0.502925\pi$
$941$	−0.563825	−0.0183802	−0.00919009	+	0.999958i	$0.502925\pi$
$942$	−9.79213	−0.319045
$943$	−0.593914	−0.0193405
$944$	10.8427	0.352900
$945$	4.24935	0.138231
$946$	55.1457	1.79294
$947$	−2.22751	−0.0723844	−0.0361922	−	0.999345i	$-0.511523\pi$
$947$	−2.22751	−0.0723844	−0.0361922	+	0.999345i	$0.511523\pi$
$948$	0.407411	0.0132321
$949$	−38.8987	−1.26270
$950$	2.70835	0.0878706
$951$	−5.39431	−0.174923
$952$	18.5982	0.602771
$953$	−43.5635	−1.41116	−0.705580	−	0.708630i	$-0.749313\pi$
$953$	−43.5635	−1.41116	−0.705580	+	0.708630i	$0.749313\pi$
$954$	−18.3736	−0.594866
$955$	−12.5275	−0.405381
$956$	−1.20994	−0.0391322
$957$	30.0212	0.970447
$958$	21.6605	0.699819
$959$	−4.67155	−0.150852
$960$	6.05839	0.195534
$961$	50.7820	1.63813
$962$	5.67865	0.183087
$963$	10.9257	0.352076
$964$	−1.61041	−0.0518677
$965$	−0.201854	−0.00649792
$966$	1.29642	0.0417116
$967$	23.5932	0.758705	0.379352	−	0.925252i	$-0.376147\pi$
$967$	23.5932	0.758705	0.379352	+	0.925252i	$0.376147\pi$
$968$	−39.8770	−1.28169
$969$	12.8430	0.412577
$970$	−23.1880	−0.744521
$971$	−29.9607	−0.961486	−0.480743	−	0.876862i	$-0.659633\pi$
$971$	−29.9607	−0.961486	−0.480743	+	0.876862i	$0.659633\pi$
$972$	4.27150	0.137008
$973$	3.35812	0.107656
$974$	61.5733	1.97293
$975$	−2.61158	−0.0836376
$976$	48.1262	1.54048
$977$	−41.4513	−1.32615	−0.663073	−	0.748555i	$-0.730748\pi$
$977$	−41.4513	−1.32615	−0.663073	+	0.748555i	$0.730748\pi$
$978$	−2.63867	−0.0843753
$979$	15.9124	0.508564
$980$	−1.63490	−0.0522249
$981$	−41.8701	−1.33681
$982$	14.8758	0.474706
$983$	49.3974	1.57553	0.787767	−	0.615974i	$-0.211237\pi$
$983$	49.3974	1.57553	0.787767	+	0.615974i	$0.211237\pi$
$984$	−1.34001	−0.0427180
$985$	−3.61465	−0.115172
$986$	−76.4821	−2.43569
$987$	−6.68281	−0.212716
$988$	−1.37094	−0.0436153
$989$	7.50530	0.238655
$990$	16.7953	0.533790
$991$	−9.39113	−0.298319	−0.149160	−	0.988813i	$-0.547657\pi$
$991$	−9.39113	−0.298319	−0.149160	+	0.988813i	$0.547657\pi$
$992$	13.4462	0.426916
$993$	−25.5610	−0.811155
$994$	12.6933	0.402607
$995$	3.57850	0.113446
$996$	0.189186	0.00599459
$997$	−52.6711	−1.66811	−0.834055	−	0.551681i	$-0.813987\pi$
$997$	−52.6711	−1.66811	−0.834055	+	0.551681i	$0.813987\pi$
$998$	30.9073	0.978355
$999$	6.15266	0.194662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.d.1.20	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.d.1.20	✓	25	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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q+1.50483 q^{2} +0.906859 q^{3} +0.264506 q^{4} +1.00000 q^{5} +1.36467 q^{6} -0.905011 q^{7} -2.61162 q^{8} -2.17761 q^{9} +O(q^{10})\)	\(q+1.50483 q^{2} +0.906859 q^{3} +0.264506 q^{4} +1.00000 q^{5} +1.36467 q^{6} -0.905011 q^{7} -2.61162 q^{8} -2.17761 q^{9} +1.50483 q^{10} -5.12534 q^{11} +0.239869 q^{12} -2.87981 q^{13} -1.36188 q^{14} +0.906859 q^{15} -4.45905 q^{16} +7.86878 q^{17} -3.27692 q^{18} +1.79978 q^{19} +0.264506 q^{20} -0.820717 q^{21} -7.71274 q^{22} -1.04970 q^{23} -2.36837 q^{24} +1.00000 q^{25} -4.33362 q^{26} -4.69536 q^{27} -0.239380 q^{28} -6.45900 q^{29} +1.36467 q^{30} -9.04334 q^{31} -1.48686 q^{32} -4.64796 q^{33} +11.8412 q^{34} -0.905011 q^{35} -0.575989 q^{36} -1.31037 q^{37} +2.70835 q^{38} -2.61158 q^{39} -2.61162 q^{40} +0.565794 q^{41} -1.23504 q^{42} -7.14995 q^{43} -1.35568 q^{44} -2.17761 q^{45} -1.57962 q^{46} +8.14264 q^{47} -4.04373 q^{48} -6.18096 q^{49} +1.50483 q^{50} +7.13588 q^{51} -0.761726 q^{52} +5.60696 q^{53} -7.06571 q^{54} -5.12534 q^{55} +2.36354 q^{56} +1.63215 q^{57} -9.71969 q^{58} -2.43162 q^{59} +0.239869 q^{60} -10.7929 q^{61} -13.6087 q^{62} +1.97076 q^{63} +6.68063 q^{64} -2.87981 q^{65} -6.99438 q^{66} -6.59936 q^{67} +2.08134 q^{68} -0.951930 q^{69} -1.36188 q^{70} -9.32040 q^{71} +5.68708 q^{72} +13.5074 q^{73} -1.97188 q^{74} +0.906859 q^{75} +0.476051 q^{76} +4.63848 q^{77} -3.92998 q^{78} +1.69847 q^{79} -4.45905 q^{80} +2.27478 q^{81} +0.851423 q^{82} +0.788704 q^{83} -0.217084 q^{84} +7.86878 q^{85} -10.7594 q^{86} -5.85741 q^{87} +13.3854 q^{88} -3.10466 q^{89} -3.27692 q^{90} +2.60626 q^{91} -0.277651 q^{92} -8.20104 q^{93} +12.2533 q^{94} +1.79978 q^{95} -1.34837 q^{96} -15.4091 q^{97} -9.30127 q^{98} +11.1610 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9}+O(q^{10}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9} - 5 q^{10} - 30 q^{11} - 29 q^{12} - 18 q^{13} + 6 q^{14} - 10 q^{15} + 21 q^{16} - 18 q^{17} - 30 q^{18} - 17 q^{19} + 25 q^{20} + 6 q^{21} - 2 q^{22} - 44 q^{23} + 11 q^{24} + 25 q^{25} - 14 q^{26} - 25 q^{27} - 50 q^{28} - 9 q^{29} + 2 q^{30} - 13 q^{31} - 45 q^{32} - 21 q^{33} - 21 q^{34} - 31 q^{35} + 5 q^{36} - 28 q^{37} - 32 q^{38} + 9 q^{39} - 30 q^{40} + 28 q^{41} - 67 q^{42} - 61 q^{43} - 49 q^{44} + 17 q^{45} + 18 q^{46} - 53 q^{47} - 44 q^{48} + 28 q^{49} - 5 q^{50} - 30 q^{51} - 3 q^{52} - 36 q^{53} + 17 q^{54} - 30 q^{55} - 3 q^{56} - 13 q^{57} + 2 q^{58} - 39 q^{59} - 29 q^{60} + 10 q^{61} - 30 q^{62} - 44 q^{63} - 4 q^{64} - 18 q^{65} + 33 q^{66} - 10 q^{67} - 18 q^{68} - 6 q^{69} + 6 q^{70} - 7 q^{71} - q^{72} - 26 q^{73} - 3 q^{74} - 10 q^{75} + 12 q^{76} + 29 q^{77} - 5 q^{78} - 6 q^{79} + 21 q^{80} + 13 q^{81} - 30 q^{82} - 35 q^{83} + 117 q^{84} - 18 q^{85} + 14 q^{86} - 104 q^{87} + 53 q^{88} + 7 q^{89} - 30 q^{90} - 25 q^{91} - 31 q^{92} + 2 q^{93} + 68 q^{94} - 17 q^{95} + 92 q^{96} + 6 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9 - 5 * q^10 - 30 * q^11 - 29 * q^12 - 18 * q^13 + 6 * q^14 - 10 * q^15 + 21 * q^16 - 18 * q^17 - 30 * q^18 - 17 * q^19 + 25 * q^20 + 6 * q^21 - 2 * q^22 - 44 * q^23 + 11 * q^24 + 25 * q^25 - 14 * q^26 - 25 * q^27 - 50 * q^28 - 9 * q^29 + 2 * q^30 - 13 * q^31 - 45 * q^32 - 21 * q^33 - 21 * q^34 - 31 * q^35 + 5 * q^36 - 28 * q^37 - 32 * q^38 + 9 * q^39 - 30 * q^40 + 28 * q^41 - 67 * q^42 - 61 * q^43 - 49 * q^44 + 17 * q^45 + 18 * q^46 - 53 * q^47 - 44 * q^48 + 28 * q^49 - 5 * q^50 - 30 * q^51 - 3 * q^52 - 36 * q^53 + 17 * q^54 - 30 * q^55 - 3 * q^56 - 13 * q^57 + 2 * q^58 - 39 * q^59 - 29 * q^60 + 10 * q^61 - 30 * q^62 - 44 * q^63 - 4 * q^64 - 18 * q^65 + 33 * q^66 - 10 * q^67 - 18 * q^68 - 6 * q^69 + 6 * q^70 - 7 * q^71 - q^72 - 26 * q^73 - 3 * q^74 - 10 * q^75 + 12 * q^76 + 29 * q^77 - 5 * q^78 - 6 * q^79 + 21 * q^80 + 13 * q^81 - 30 * q^82 - 35 * q^83 + 117 * q^84 - 18 * q^85 + 14 * q^86 - 104 * q^87 + 53 * q^88 + 7 * q^89 - 30 * q^90 - 25 * q^91 - 31 * q^92 + 2 * q^93 + 68 * q^94 - 17 * q^95 + 92 * q^96 + 6 * q^97 + 15 * q^98 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more