LMFDB - Embedded newform 1441.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1441,2,Mod(1,1441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1441, 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("1441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1441 = 11 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1441.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:	$11.5064429313$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	1441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33619 q^{2} +2.29063 q^{3} +3.45776 q^{4} -0.314509 q^{5} -5.35135 q^{6} +0.543204 q^{7} -3.40560 q^{8} +2.24701 q^{9} +O(q^{10})$	\(q-2.33619 q^{2} +2.29063 q^{3} +3.45776 q^{4} -0.314509 q^{5} -5.35135 q^{6} +0.543204 q^{7} -3.40560 q^{8} +2.24701 q^{9} +0.734752 q^{10} +1.00000 q^{11} +7.92047 q^{12} +6.37371 q^{13} -1.26903 q^{14} -0.720426 q^{15} +1.04060 q^{16} +7.80692 q^{17} -5.24942 q^{18} -4.51268 q^{19} -1.08750 q^{20} +1.24428 q^{21} -2.33619 q^{22} +3.63556 q^{23} -7.80099 q^{24} -4.90108 q^{25} -14.8902 q^{26} -1.72483 q^{27} +1.87827 q^{28} +1.45649 q^{29} +1.68305 q^{30} -3.82273 q^{31} +4.38018 q^{32} +2.29063 q^{33} -18.2384 q^{34} -0.170843 q^{35} +7.76961 q^{36} -1.07120 q^{37} +10.5425 q^{38} +14.5998 q^{39} +1.07109 q^{40} +4.72274 q^{41} -2.90687 q^{42} +0.536888 q^{43} +3.45776 q^{44} -0.706705 q^{45} -8.49333 q^{46} +2.22765 q^{47} +2.38362 q^{48} -6.70493 q^{49} +11.4498 q^{50} +17.8828 q^{51} +22.0388 q^{52} -3.76717 q^{53} +4.02953 q^{54} -0.314509 q^{55} -1.84994 q^{56} -10.3369 q^{57} -3.40263 q^{58} +9.78696 q^{59} -2.49106 q^{60} -4.79737 q^{61} +8.93061 q^{62} +1.22058 q^{63} -12.3141 q^{64} -2.00459 q^{65} -5.35135 q^{66} +15.1411 q^{67} +26.9945 q^{68} +8.32773 q^{69} +0.399121 q^{70} +8.17140 q^{71} -7.65241 q^{72} +10.2037 q^{73} +2.50252 q^{74} -11.2266 q^{75} -15.6038 q^{76} +0.543204 q^{77} -34.1079 q^{78} -14.4673 q^{79} -0.327277 q^{80} -10.6920 q^{81} -11.0332 q^{82} +2.12709 q^{83} +4.30243 q^{84} -2.45535 q^{85} -1.25427 q^{86} +3.33629 q^{87} -3.40560 q^{88} -7.45804 q^{89} +1.65099 q^{90} +3.46223 q^{91} +12.5709 q^{92} -8.75648 q^{93} -5.20420 q^{94} +1.41928 q^{95} +10.0334 q^{96} -0.288063 q^{97} +15.6640 q^{98} +2.24701 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 7 q^{2} - 2 q^{3} + 27 q^{4} + 3 q^{5} - q^{6} + 7 q^{7} + 21 q^{8} + 28 q^{9}+O(q^{10}) $ 28 * q + 7 * q^2 - 2 * q^3 + 27 * q^4 + 3 * q^5 - q^6 + 7 * q^7 + 21 * q^8 + 28 * q^9	\( 28 q + 7 q^{2} - 2 q^{3} + 27 q^{4} + 3 q^{5} - q^{6} + 7 q^{7} + 21 q^{8} + 28 q^{9} + 14 q^{10} + 28 q^{11} - 6 q^{12} + 3 q^{13} - q^{14} + 19 q^{15} + 29 q^{16} + 9 q^{17} - 2 q^{18} + 20 q^{19} + 6 q^{20} + 6 q^{21} + 7 q^{22} + 24 q^{23} + 20 q^{24} + 23 q^{25} + 10 q^{26} - 20 q^{27} - 3 q^{28} + 43 q^{29} + 11 q^{30} + 3 q^{31} + 44 q^{32} - 2 q^{33} - 28 q^{34} + 32 q^{35} + 24 q^{36} - 4 q^{37} + 24 q^{38} + 37 q^{39} + 22 q^{40} + 32 q^{41} - 27 q^{42} + 25 q^{43} + 27 q^{44} - 36 q^{45} + 10 q^{46} + 19 q^{47} + 42 q^{48} + 17 q^{49} + q^{50} + 39 q^{51} - 19 q^{52} + 5 q^{53} + 6 q^{54} + 3 q^{55} + 8 q^{56} + 2 q^{57} + 21 q^{58} + 44 q^{59} + 65 q^{60} + 28 q^{61} + 60 q^{62} - 8 q^{63} + 5 q^{64} + 33 q^{65} - q^{66} + 7 q^{67} + 13 q^{68} - 22 q^{69} + 9 q^{70} + 117 q^{71} - 17 q^{72} + 7 q^{73} + 41 q^{74} - 40 q^{75} + 34 q^{76} + 7 q^{77} - 97 q^{78} + 48 q^{79} + 41 q^{80} + 40 q^{81} + 2 q^{82} + 22 q^{83} + 27 q^{84} + 30 q^{85} + 24 q^{86} + 37 q^{87} + 21 q^{88} - 6 q^{89} + 4 q^{90} - 33 q^{91} + 18 q^{92} + 5 q^{93} - 43 q^{94} + 64 q^{95} + 55 q^{96} - 50 q^{97} + 97 q^{98} + 28 q^{99}+O(q^{100}) \) 28 * q + 7 * q^2 - 2 * q^3 + 27 * q^4 + 3 * q^5 - q^6 + 7 * q^7 + 21 * q^8 + 28 * q^9 + 14 * q^10 + 28 * q^11 - 6 * q^12 + 3 * q^13 - q^14 + 19 * q^15 + 29 * q^16 + 9 * q^17 - 2 * q^18 + 20 * q^19 + 6 * q^20 + 6 * q^21 + 7 * q^22 + 24 * q^23 + 20 * q^24 + 23 * q^25 + 10 * q^26 - 20 * q^27 - 3 * q^28 + 43 * q^29 + 11 * q^30 + 3 * q^31 + 44 * q^32 - 2 * q^33 - 28 * q^34 + 32 * q^35 + 24 * q^36 - 4 * q^37 + 24 * q^38 + 37 * q^39 + 22 * q^40 + 32 * q^41 - 27 * q^42 + 25 * q^43 + 27 * q^44 - 36 * q^45 + 10 * q^46 + 19 * q^47 + 42 * q^48 + 17 * q^49 + q^50 + 39 * q^51 - 19 * q^52 + 5 * q^53 + 6 * q^54 + 3 * q^55 + 8 * q^56 + 2 * q^57 + 21 * q^58 + 44 * q^59 + 65 * q^60 + 28 * q^61 + 60 * q^62 - 8 * q^63 + 5 * q^64 + 33 * q^65 - q^66 + 7 * q^67 + 13 * q^68 - 22 * q^69 + 9 * q^70 + 117 * q^71 - 17 * q^72 + 7 * q^73 + 41 * q^74 - 40 * q^75 + 34 * q^76 + 7 * q^77 - 97 * q^78 + 48 * q^79 + 41 * q^80 + 40 * q^81 + 2 * q^82 + 22 * q^83 + 27 * q^84 + 30 * q^85 + 24 * q^86 + 37 * q^87 + 21 * q^88 - 6 * q^89 + 4 * q^90 - 33 * q^91 + 18 * q^92 + 5 * q^93 - 43 * q^94 + 64 * q^95 + 55 * q^96 - 50 * q^97 + 97 * q^98 + 28 * 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 $

$2$	−2.33619	−1.65193	−0.825966	−	0.563720i	$-0.809370\pi$
$2$	−2.33619	−1.65193	−0.825966	+	0.563720i	$0.809370\pi$
$3$	2.29063	1.32250	0.661249	−	0.750166i	$-0.270027\pi$
$3$	2.29063	1.32250	0.661249	+	0.750166i	$0.270027\pi$
$4$	3.45776	1.72888
$5$	−0.314509	−0.140653	−0.0703265	−	0.997524i	$-0.522404\pi$
$5$	−0.314509	−0.140653	−0.0703265	+	0.997524i	$0.522404\pi$
$6$	−5.35135	−2.18468
$7$	0.543204	0.205312	0.102656	−	0.994717i	$-0.467266\pi$
$7$	0.543204	0.205312	0.102656	+	0.994717i	$0.467266\pi$
$8$	−3.40560	−1.20406
$9$	2.24701	0.749002
$10$	0.734752	0.232349
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	7.92047	2.28644
$13$	6.37371	1.76775	0.883875	−	0.467723i	$-0.154926\pi$
$13$	6.37371	1.76775	0.883875	+	0.467723i	$0.154926\pi$
$14$	−1.26903	−0.339161
$15$	−0.720426	−0.186013
$16$	1.04060	0.260149
$17$	7.80692	1.89346	0.946728	−	0.322035i	$-0.104367\pi$
$17$	7.80692	1.89346	0.946728	+	0.322035i	$0.104367\pi$
$18$	−5.24942	−1.23730
$19$	−4.51268	−1.03528	−0.517640	−	0.855599i	$-0.673189\pi$
$19$	−4.51268	−1.03528	−0.517640	+	0.855599i	$0.673189\pi$
$20$	−1.08750	−0.243172
$21$	1.24428	0.271525
$22$	−2.33619	−0.498076
$23$	3.63556	0.758066	0.379033	−	0.925383i	$-0.376257\pi$
$23$	3.63556	0.758066	0.379033	+	0.925383i	$0.376257\pi$
$24$	−7.80099	−1.59237
$25$	−4.90108	−0.980217
$26$	−14.8902	−2.92020
$27$	−1.72483	−0.331944
$28$	1.87827	0.354960
$29$	1.45649	0.270464	0.135232	−	0.990814i	$-0.456822\pi$
$29$	1.45649	0.270464	0.135232	+	0.990814i	$0.456822\pi$
$30$	1.68305	0.307281
$31$	−3.82273	−0.686583	−0.343291	−	0.939229i	$-0.611542\pi$
$31$	−3.82273	−0.686583	−0.343291	+	0.939229i	$0.611542\pi$
$32$	4.38018	0.774314
$33$	2.29063	0.398748
$34$	−18.2384	−3.12786
$35$	−0.170843	−0.0288777
$36$	7.76961	1.29494
$37$	−1.07120	−0.176104	−0.0880520	−	0.996116i	$-0.528064\pi$
$37$	−1.07120	−0.176104	−0.0880520	+	0.996116i	$0.528064\pi$
$38$	10.5425	1.71021
$39$	14.5998	2.33785
$40$	1.07109	0.169355
$41$	4.72274	0.737568	0.368784	−	0.929515i	$-0.379774\pi$
$41$	4.72274	0.737568	0.368784	+	0.929515i	$0.379774\pi$
$42$	−2.90687	−0.448540
$43$	0.536888	0.0818747	0.0409373	−	0.999162i	$-0.486966\pi$
$43$	0.536888	0.0818747	0.0409373	+	0.999162i	$0.486966\pi$
$44$	3.45776	0.521277
$45$	−0.706705	−0.105349
$46$	−8.49333	−1.25227
$47$	2.22765	0.324936	0.162468	−	0.986714i	$-0.448055\pi$
$47$	2.22765	0.324936	0.162468	+	0.986714i	$0.448055\pi$
$48$	2.38362	0.344046
$49$	−6.70493	−0.957847
$50$	11.4498	1.61925
$51$	17.8828	2.50409
$52$	22.0388	3.05623
$53$	−3.76717	−0.517461	−0.258731	−	0.965950i	$-0.583304\pi$
$53$	−3.76717	−0.517461	−0.258731	+	0.965950i	$0.583304\pi$
$54$	4.02953	0.548349
$55$	−0.314509	−0.0424085
$56$	−1.84994	−0.247208
$57$	−10.3369	−1.36916
$58$	−3.40263	−0.446788
$59$	9.78696	1.27415	0.637077	−	0.770800i	$-0.280143\pi$
$59$	9.78696	1.27415	0.637077	+	0.770800i	$0.280143\pi$
$60$	−2.49106	−0.321595
$61$	−4.79737	−0.614240	−0.307120	−	0.951671i	$-0.599365\pi$
$61$	−4.79737	−0.614240	−0.307120	+	0.951671i	$0.599365\pi$
$62$	8.93061	1.13419
$63$	1.22058	0.153779
$64$	−12.3141	−1.53926
$65$	−2.00459	−0.248639
$66$	−5.35135	−0.658705
$67$	15.1411	1.84978	0.924888	−	0.380240i	$-0.124159\pi$
$67$	15.1411	1.84978	0.924888	+	0.380240i	$0.124159\pi$
$68$	26.9945	3.27356
$69$	8.32773	1.00254
$70$	0.399121	0.0477040
$71$	8.17140	0.969767	0.484883	−	0.874579i	$-0.338862\pi$
$71$	8.17140	0.969767	0.484883	+	0.874579i	$0.338862\pi$
$72$	−7.65241	−0.901845
$73$	10.2037	1.19425	0.597126	−	0.802147i	$-0.296309\pi$
$73$	10.2037	1.19425	0.597126	+	0.802147i	$0.296309\pi$
$74$	2.50252	0.290912
$75$	−11.2266	−1.29634
$76$	−15.6038	−1.78988
$77$	0.543204	0.0619039
$78$	−34.1079	−3.86196
$79$	−14.4673	−1.62769	−0.813847	−	0.581079i	$-0.802631\pi$
$79$	−14.4673	−1.62769	−0.813847	+	0.581079i	$0.802631\pi$
$80$	−0.327277	−0.0365907
$81$	−10.6920	−1.18800
$82$	−11.0332	−1.21841
$83$	2.12709	0.233479	0.116739	−	0.993163i	$-0.462756\pi$
$83$	2.12709	0.233479	0.116739	+	0.993163i	$0.462756\pi$
$84$	4.30243	0.469434
$85$	−2.45535	−0.266320
$86$	−1.25427	−0.135251
$87$	3.33629	0.357688
$88$	−3.40560	−0.363038
$89$	−7.45804	−0.790551	−0.395275	−	0.918563i	$-0.629351\pi$
$89$	−7.45804	−0.790551	−0.395275	+	0.918563i	$0.629351\pi$
$90$	1.65099	0.174030
$91$	3.46223	0.362940
$92$	12.5709	1.31061
$93$	−8.75648	−0.908004
$94$	−5.20420	−0.536772
$95$	1.41928	0.145615
$96$	10.0334	1.02403
$97$	−0.288063	−0.0292484	−0.0146242	−	0.999893i	$-0.504655\pi$
$97$	−0.288063	−0.0292484	−0.0146242	+	0.999893i	$0.504655\pi$
$98$	15.6640	1.58230
$99$	2.24701	0.225833
$100$	−16.9468	−1.69468
$101$	−5.67036	−0.564222	−0.282111	−	0.959382i	$-0.591035\pi$
$101$	−5.67036	−0.564222	−0.282111	+	0.959382i	$0.591035\pi$
$102$	−41.7775	−4.13659
$103$	0.0917267	0.00903810	0.00451905	−	0.999990i	$-0.498562\pi$
$103$	0.0917267	0.00903810	0.00451905	+	0.999990i	$0.498562\pi$
$104$	−21.7063	−2.12848
$105$	−0.391339	−0.0381907
$106$	8.80082	0.854811
$107$	6.80587	0.657949	0.328974	−	0.944339i	$-0.393297\pi$
$107$	6.80587	0.657949	0.328974	+	0.944339i	$0.393297\pi$
$108$	−5.96406	−0.573892
$109$	5.44692	0.521721	0.260860	−	0.965377i	$-0.415994\pi$
$109$	5.44692	0.521721	0.260860	+	0.965377i	$0.415994\pi$
$110$	0.734752	0.0700559
$111$	−2.45372	−0.232897
$112$	0.565256	0.0534116
$113$	−17.5873	−1.65448	−0.827238	−	0.561851i	$-0.810089\pi$
$113$	−17.5873	−1.65448	−0.827238	+	0.561851i	$0.810089\pi$
$114$	24.1489	2.26175
$115$	−1.14342	−0.106624
$116$	5.03620	0.467599
$117$	14.3218	1.32405
$118$	−22.8641	−2.10482
$119$	4.24075	0.388749
$120$	2.45349	0.223972
$121$	1.00000	0.0909091
$122$	11.2075	1.01468
$123$	10.8181	0.975432
$124$	−13.2181	−1.18702
$125$	3.11398	0.278523
$126$	−2.85151	−0.254033
$127$	11.8787	1.05406	0.527030	−	0.849846i	$-0.323305\pi$
$127$	11.8787	1.05406	0.527030	+	0.849846i	$0.323305\pi$
$128$	20.0077	1.76845
$129$	1.22981	0.108279
$130$	4.68310	0.410735
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	7.92047	0.689388
$133$	−2.45131	−0.212555
$134$	−35.3723	−3.05570
$135$	0.542476	0.0466889
$136$	−26.5873	−2.27984
$137$	9.56395	0.817103	0.408552	−	0.912735i	$-0.366034\pi$
$137$	9.56395	0.817103	0.408552	+	0.912735i	$0.366034\pi$
$138$	−19.4551	−1.65613
$139$	16.4826	1.39804	0.699019	−	0.715103i	$-0.253620\pi$
$139$	16.4826	1.39804	0.699019	+	0.715103i	$0.253620\pi$
$140$	−0.590734	−0.0499261
$141$	5.10273	0.429727
$142$	−19.0899	−1.60199
$143$	6.37371	0.532997
$144$	2.33822	0.194852
$145$	−0.458080	−0.0380415
$146$	−23.8377	−1.97283
$147$	−15.3585	−1.26675
$148$	−3.70395	−0.304463
$149$	4.34525	0.355977	0.177988	−	0.984033i	$-0.443041\pi$
$149$	4.34525	0.355977	0.177988	+	0.984033i	$0.443041\pi$
$150$	26.2274	2.14146
$151$	−1.55343	−0.126416	−0.0632082	−	0.998000i	$-0.520133\pi$
$151$	−1.55343	−0.126416	−0.0632082	+	0.998000i	$0.520133\pi$
$152$	15.3684	1.24654
$153$	17.5422	1.41820
$154$	−1.26903	−0.102261
$155$	1.20228	0.0965698
$156$	50.4828	4.04186
$157$	3.81663	0.304600	0.152300	−	0.988334i	$-0.451332\pi$
$157$	3.81663	0.304600	0.152300	+	0.988334i	$0.451332\pi$
$158$	33.7982	2.68884
$159$	−8.62922	−0.684342
$160$	−1.37761	−0.108910
$161$	1.97485	0.155640
$162$	24.9784	1.96249
$163$	−20.0385	−1.56953	−0.784766	−	0.619792i	$-0.787217\pi$
$163$	−20.0385	−1.56953	−0.784766	+	0.619792i	$0.787217\pi$
$164$	16.3301	1.27517
$165$	−0.720426	−0.0560851
$166$	−4.96928	−0.385691
$167$	25.4400	1.96861	0.984304	−	0.176483i	$-0.0564722\pi$
$167$	25.4400	1.96861	0.984304	+	0.176483i	$0.0564722\pi$
$168$	−4.23753	−0.326933
$169$	27.6242	2.12494
$170$	5.73615	0.439943
$171$	−10.1400	−0.775427
$172$	1.85643	0.141552
$173$	−24.6773	−1.87618	−0.938088	−	0.346396i	$-0.887405\pi$
$173$	−24.6773	−1.87618	−0.938088	+	0.346396i	$0.887405\pi$
$174$	−7.79419	−0.590876
$175$	−2.66229	−0.201250
$176$	1.04060	0.0784378
$177$	22.4183	1.68507
$178$	17.4234	1.30594
$179$	−10.3609	−0.774411	−0.387205	−	0.921994i	$-0.626560\pi$
$179$	−10.3609	−0.774411	−0.387205	+	0.921994i	$0.626560\pi$
$180$	−2.44362	−0.182137
$181$	−11.3881	−0.846474	−0.423237	−	0.906019i	$-0.639106\pi$
$181$	−11.3881	−0.846474	−0.423237	+	0.906019i	$0.639106\pi$
$182$	−8.08840	−0.599552
$183$	−10.9890	−0.812332
$184$	−12.3813	−0.912759
$185$	0.336902	0.0247695
$186$	20.4568	1.49996
$187$	7.80692	0.570898
$188$	7.70268	0.561776
$189$	−0.936936	−0.0681521
$190$	−3.31570	−0.240546
$191$	3.38638	0.245030	0.122515	−	0.992467i	$-0.460904\pi$
$191$	3.38638	0.245030	0.122515	+	0.992467i	$0.460904\pi$
$192$	−28.2071	−2.03567
$193$	−4.21318	−0.303271	−0.151636	−	0.988436i	$-0.548454\pi$
$193$	−4.21318	−0.303271	−0.151636	+	0.988436i	$0.548454\pi$
$194$	0.672969	0.0483163
$195$	−4.59179	−0.328825
$196$	−23.1841	−1.65600
$197$	23.3051	1.66042	0.830210	−	0.557451i	$-0.188220\pi$
$197$	23.3051	1.66042	0.830210	+	0.557451i	$0.188220\pi$
$198$	−5.24942	−0.373060
$199$	17.4522	1.23715	0.618576	−	0.785725i	$-0.287710\pi$
$199$	17.4522	1.23715	0.618576	+	0.785725i	$0.287710\pi$
$200$	16.6911	1.18024
$201$	34.6826	2.44633
$202$	13.2470	0.932057
$203$	0.791172	0.0555294
$204$	61.8344	4.32928
$205$	−1.48535	−0.103741
$206$	−0.214291	−0.0149303
$207$	8.16912	0.567793
$208$	6.63245	0.459878
$209$	−4.51268	−0.312149
$210$	0.914239	0.0630885
$211$	10.3257	0.710849	0.355424	−	0.934705i	$-0.384336\pi$
$211$	10.3257	0.710849	0.355424	+	0.934705i	$0.384336\pi$
$212$	−13.0260	−0.894629
$213$	18.7177	1.28252
$214$	−15.8998	−1.08689
$215$	−0.168856	−0.0115159
$216$	5.87409	0.399681
$217$	−2.07652	−0.140964
$218$	−12.7250	−0.861847
$219$	23.3730	1.57940
$220$	−1.08750	−0.0733192
$221$	49.7590	3.34716
$222$	5.73236	0.384731
$223$	20.5939	1.37907	0.689533	−	0.724254i	$-0.257816\pi$
$223$	20.5939	1.37907	0.689533	+	0.724254i	$0.257816\pi$
$224$	2.37933	0.158976
$225$	−11.0128	−0.734185
$226$	41.0873	2.73308
$227$	−18.6562	−1.23826	−0.619128	−	0.785290i	$-0.712514\pi$
$227$	−18.6562	−1.23826	−0.619128	+	0.785290i	$0.712514\pi$
$228$	−35.7425	−2.36711
$229$	3.48451	0.230263	0.115131	−	0.993350i	$-0.463271\pi$
$229$	3.48451	0.230263	0.115131	+	0.993350i	$0.463271\pi$
$230$	2.67123	0.176136
$231$	1.24428	0.0818678
$232$	−4.96023	−0.325655
$233$	−21.7891	−1.42745	−0.713727	−	0.700424i	$-0.752994\pi$
$233$	−21.7891	−1.42745	−0.713727	+	0.700424i	$0.752994\pi$
$234$	−33.4583	−2.18724
$235$	−0.700617	−0.0457032
$236$	33.8410	2.20286
$237$	−33.1392	−2.15262
$238$	−9.90718	−0.642187
$239$	−3.75514	−0.242900	−0.121450	−	0.992598i	$-0.538754\pi$
$239$	−3.75514	−0.242900	−0.121450	+	0.992598i	$0.538754\pi$
$240$	−0.749672	−0.0483911
$241$	−1.40733	−0.0906542	−0.0453271	−	0.998972i	$-0.514433\pi$
$241$	−1.40733	−0.0906542	−0.0453271	+	0.998972i	$0.514433\pi$
$242$	−2.33619	−0.150176
$243$	−19.3169	−1.23918
$244$	−16.5882	−1.06195
$245$	2.10876	0.134724
$246$	−25.2730	−1.61135
$247$	−28.7625	−1.83012
$248$	13.0187	0.826688
$249$	4.87239	0.308775
$250$	−7.27485	−0.460102
$251$	11.9983	0.757325	0.378663	−	0.925535i	$-0.376384\pi$
$251$	11.9983	0.757325	0.378663	+	0.925535i	$0.376384\pi$
$252$	4.22049	0.265866
$253$	3.63556	0.228565
$254$	−27.7508	−1.74124
$255$	−5.62431	−0.352208
$256$	−22.1134	−1.38209
$257$	11.9080	0.742800	0.371400	−	0.928473i	$-0.378878\pi$
$257$	11.9080	0.742800	0.371400	+	0.928473i	$0.378878\pi$
$258$	−2.87307	−0.178870
$259$	−0.581880	−0.0361562
$260$	−6.93141	−0.429868
$261$	3.27274	0.202578
$262$	2.33619	0.144330
$263$	−22.2094	−1.36949	−0.684746	−	0.728782i	$-0.740087\pi$
$263$	−22.2094	−1.36949	−0.684746	+	0.728782i	$0.740087\pi$
$264$	−7.80099	−0.480118
$265$	1.18481	0.0727824
$266$	5.72671	0.351127
$267$	−17.0836	−1.04550
$268$	52.3542	3.19804
$269$	−12.4304	−0.757898	−0.378949	−	0.925418i	$-0.623714\pi$
$269$	−12.4304	−0.757898	−0.378949	+	0.925418i	$0.623714\pi$
$270$	−1.26732	−0.0771269
$271$	−18.2743	−1.11009	−0.555043	−	0.831822i	$-0.687298\pi$
$271$	−18.2743	−1.11009	−0.555043	+	0.831822i	$0.687298\pi$
$272$	8.12384	0.492580
$273$	7.93070	0.479988
$274$	−22.3432	−1.34980
$275$	−4.90108	−0.295546
$276$	28.7953	1.73327
$277$	21.3833	1.28480	0.642400	−	0.766370i	$-0.277939\pi$
$277$	21.3833	1.28480	0.642400	+	0.766370i	$0.277939\pi$
$278$	−38.5065	−2.30946
$279$	−8.58970	−0.514252
$280$	0.581823	0.0347706
$281$	1.85893	0.110894	0.0554471	−	0.998462i	$-0.482342\pi$
$281$	1.85893	0.110894	0.0554471	+	0.998462i	$0.482342\pi$
$282$	−11.9209	−0.709880
$283$	−25.1027	−1.49220	−0.746102	−	0.665832i	$-0.768077\pi$
$283$	−25.1027	−1.49220	−0.746102	+	0.665832i	$0.768077\pi$
$284$	28.2548	1.67661
$285$	3.25105	0.192576
$286$	−14.8902	−0.880474
$287$	2.56541	0.151431
$288$	9.84230	0.579963
$289$	43.9480	2.58517
$290$	1.07016	0.0628420
$291$	−0.659847	−0.0386809
$292$	35.2820	2.06472
$293$	−12.5964	−0.735891	−0.367946	−	0.929847i	$-0.619939\pi$
$293$	−12.5964	−0.735891	−0.367946	+	0.929847i	$0.619939\pi$
$294$	35.8804	2.09259
$295$	−3.07809	−0.179213
$296$	3.64808	0.212040
$297$	−1.72483	−0.100085
$298$	−10.1513	−0.588049
$299$	23.1720	1.34007
$300$	−38.8189	−2.24121
$301$	0.291640	0.0168098
$302$	3.62910	0.208831
$303$	−12.9887	−0.746183
$304$	−4.69587	−0.269327
$305$	1.50882	0.0863947
$306$	−40.9818	−2.34277
$307$	−10.6504	−0.607852	−0.303926	−	0.952696i	$-0.598298\pi$
$307$	−10.6504	−0.607852	−0.303926	+	0.952696i	$0.598298\pi$
$308$	1.87827	0.107024
$309$	0.210112	0.0119529
$310$	−2.80876	−0.159527
$311$	−17.1880	−0.974643	−0.487322	−	0.873223i	$-0.662026\pi$
$311$	−17.1880	−0.974643	−0.487322	+	0.873223i	$0.662026\pi$
$312$	−49.7213	−2.81491
$313$	−24.9637	−1.41103	−0.705516	−	0.708694i	$-0.749285\pi$
$313$	−24.9637	−1.41103	−0.705516	+	0.708694i	$0.749285\pi$
$314$	−8.91635	−0.503179
$315$	−0.383885	−0.0216295
$316$	−50.0243	−2.81409
$317$	−1.60050	−0.0898933	−0.0449466	−	0.998989i	$-0.514312\pi$
$317$	−1.60050	−0.0898933	−0.0449466	+	0.998989i	$0.514312\pi$
$318$	20.1595	1.13049
$319$	1.45649	0.0815478
$320$	3.87290	0.216502
$321$	15.5898	0.870136
$322$	−4.61361	−0.257107
$323$	−35.2301	−1.96026
$324$	−36.9703	−2.05391
$325$	−31.2381	−1.73278
$326$	46.8135	2.59276
$327$	12.4769	0.689975
$328$	−16.0838	−0.888078
$329$	1.21007	0.0667132
$330$	1.68305	0.0926488
$331$	−19.3992	−1.06628	−0.533138	−	0.846028i	$-0.678987\pi$
$331$	−19.3992	−1.06628	−0.533138	+	0.846028i	$0.678987\pi$
$332$	7.35498	0.403657
$333$	−2.40699	−0.131902
$334$	−59.4326	−3.25201
$335$	−4.76201	−0.260176
$336$	1.29479	0.0706368
$337$	−26.6789	−1.45329	−0.726647	−	0.687011i	$-0.758923\pi$
$337$	−26.6789	−1.45329	−0.726647	+	0.687011i	$0.758923\pi$
$338$	−64.5353	−3.51026
$339$	−40.2861	−2.18804
$340$	−8.49001	−0.460436
$341$	−3.82273	−0.207012
$342$	23.6890	1.28095
$343$	−7.44457	−0.401969
$344$	−1.82843	−0.0985822
$345$	−2.61915	−0.141010
$346$	57.6506	3.09932
$347$	22.1118	1.18703	0.593513	−	0.804825i	$-0.297741\pi$
$347$	22.1118	1.18703	0.593513	+	0.804825i	$0.297741\pi$
$348$	11.5361	0.618399
$349$	−8.51031	−0.455546	−0.227773	−	0.973714i	$-0.573144\pi$
$349$	−8.51031	−0.455546	−0.227773	+	0.973714i	$0.573144\pi$
$350$	6.21960	0.332452
$351$	−10.9936	−0.586794
$352$	4.38018	0.233465
$353$	28.5988	1.52216	0.761080	−	0.648658i	$-0.224670\pi$
$353$	28.5988	1.52216	0.761080	+	0.648658i	$0.224670\pi$
$354$	−52.3734	−2.78362
$355$	−2.56998	−0.136401
$356$	−25.7881	−1.36677
$357$	9.71401	0.514120
$358$	24.2050	1.27927
$359$	12.0731	0.637196	0.318598	−	0.947890i	$-0.396788\pi$
$359$	12.0731	0.637196	0.318598	+	0.947890i	$0.396788\pi$
$360$	2.40676	0.126847
$361$	1.36428	0.0718042
$362$	26.6048	1.39832
$363$	2.29063	0.120227
$364$	11.9716	0.627480
$365$	−3.20916	−0.167975
$366$	25.6724	1.34192
$367$	27.2886	1.42445	0.712227	−	0.701949i	$-0.247687\pi$
$367$	27.2886	1.42445	0.712227	+	0.701949i	$0.247687\pi$
$368$	3.78314	0.197210
$369$	10.6120	0.552440
$370$	−0.787066	−0.0409176
$371$	−2.04634	−0.106241
$372$	−30.2778	−1.56983
$373$	−20.0361	−1.03743	−0.518716	−	0.854947i	$-0.673590\pi$
$373$	−20.0361	−1.03743	−0.518716	+	0.854947i	$0.673590\pi$
$374$	−18.2384	−0.943085
$375$	7.13300	0.368347
$376$	−7.58649	−0.391243
$377$	9.28325	0.478112
$378$	2.18886	0.112583
$379$	−4.54617	−0.233521	−0.116761	−	0.993160i	$-0.537251\pi$
$379$	−4.54617	−0.233521	−0.116761	+	0.993160i	$0.537251\pi$
$380$	4.90753	0.251751
$381$	27.2097	1.39399
$382$	−7.91121	−0.404773
$383$	2.20834	0.112841	0.0564204	−	0.998407i	$-0.482031\pi$
$383$	2.20834	0.112841	0.0564204	+	0.998407i	$0.482031\pi$
$384$	45.8303	2.33877
$385$	−0.170843	−0.00870696
$386$	9.84277	0.500984
$387$	1.20639	0.0613243
$388$	−0.996054	−0.0505670
$389$	−20.1882	−1.02358	−0.511791	−	0.859110i	$-0.671018\pi$
$389$	−20.1882	−1.02358	−0.511791	+	0.859110i	$0.671018\pi$
$390$	10.7273	0.543197
$391$	28.3825	1.43536
$392$	22.8343	1.15331
$393$	−2.29063	−0.115547
$394$	−54.4451	−2.74290
$395$	4.55009	0.228940
$396$	7.76961	0.390438
$397$	−30.8583	−1.54873	−0.774367	−	0.632737i	$-0.781931\pi$
$397$	−30.8583	−1.54873	−0.774367	+	0.632737i	$0.781931\pi$
$398$	−40.7715	−2.04369
$399$	−5.61505	−0.281104
$400$	−5.10004	−0.255002
$401$	−17.0035	−0.849113	−0.424556	−	0.905401i	$-0.639570\pi$
$401$	−17.0035	−0.849113	−0.424556	+	0.905401i	$0.639570\pi$
$402$	−81.0251	−4.04116
$403$	−24.3650	−1.21371
$404$	−19.6068	−0.975473
$405$	3.36273	0.167095
$406$	−1.84832	−0.0917308
$407$	−1.07120	−0.0530974
$408$	−60.9017	−3.01508
$409$	−3.98834	−0.197211	−0.0986053	−	0.995127i	$-0.531438\pi$
$409$	−3.98834	−0.197211	−0.0986053	+	0.995127i	$0.531438\pi$
$410$	3.47004	0.171373
$411$	21.9075	1.08062
$412$	0.317169	0.0156258
$413$	5.31632	0.261599
$414$	−19.0846	−0.937956
$415$	−0.668991	−0.0328395
$416$	27.9180	1.36879
$417$	37.7557	1.84890
$418$	10.5425	0.515648
$419$	−17.9681	−0.877801	−0.438900	−	0.898536i	$-0.644632\pi$
$419$	−17.9681	−0.877801	−0.438900	+	0.898536i	$0.644632\pi$
$420$	−1.35316	−0.0660272
$421$	−17.6538	−0.860392	−0.430196	−	0.902736i	$-0.641556\pi$
$421$	−17.6538	−0.860392	−0.430196	+	0.902736i	$0.641556\pi$
$422$	−24.1227	−1.17427
$423$	5.00554	0.243378
$424$	12.8295	0.623056
$425$	−38.2624	−1.85600
$426$	−43.7280	−2.11863
$427$	−2.60595	−0.126111
$428$	23.5331	1.13751
$429$	14.5998	0.704887
$430$	0.394480	0.0190235
$431$	29.4012	1.41621	0.708104	−	0.706109i	$-0.249551\pi$
$431$	29.4012	1.41621	0.708104	+	0.706109i	$0.249551\pi$
$432$	−1.79485	−0.0863549
$433$	15.5782	0.748639	0.374320	−	0.927300i	$-0.377876\pi$
$433$	15.5782	0.748639	0.374320	+	0.927300i	$0.377876\pi$
$434$	4.85114	0.232862
$435$	−1.04929	−0.0503098
$436$	18.8342	0.901993
$437$	−16.4061	−0.784810
$438$	−54.6035	−2.60906
$439$	33.0192	1.57592	0.787961	−	0.615725i	$-0.211137\pi$
$439$	33.0192	1.57592	0.787961	+	0.615725i	$0.211137\pi$
$440$	1.07109	0.0510624
$441$	−15.0660	−0.717430
$442$	−116.246	−5.52927
$443$	−3.91943	−0.186218	−0.0931088	−	0.995656i	$-0.529680\pi$
$443$	−3.91943	−0.186218	−0.0931088	+	0.995656i	$0.529680\pi$
$444$	−8.48440	−0.402652
$445$	2.34562	0.111193
$446$	−48.1111	−2.27812
$447$	9.95338	0.470779
$448$	−6.68907	−0.316029
$449$	−38.1005	−1.79807	−0.899036	−	0.437874i	$-0.855732\pi$
$449$	−38.1005	−1.79807	−0.899036	+	0.437874i	$0.855732\pi$
$450$	25.7279	1.21282
$451$	4.72274	0.222385
$452$	−60.8128	−2.86039
$453$	−3.55834	−0.167186
$454$	43.5844	2.04552
$455$	−1.08890	−0.0510486
$456$	35.2034	1.64855
$457$	−24.5233	−1.14715	−0.573576	−	0.819152i	$-0.694444\pi$
$457$	−24.5233	−1.14715	−0.573576	+	0.819152i	$0.694444\pi$
$458$	−8.14045	−0.380378
$459$	−13.4656	−0.628521
$460$	−3.95366	−0.184341
$461$	14.3001	0.666024	0.333012	−	0.942923i	$-0.391935\pi$
$461$	14.3001	0.666024	0.333012	+	0.942923i	$0.391935\pi$
$462$	−2.90687	−0.135240
$463$	−28.1884	−1.31003	−0.655013	−	0.755618i	$-0.727337\pi$
$463$	−28.1884	−1.31003	−0.655013	+	0.755618i	$0.727337\pi$
$464$	1.51562	0.0703608
$465$	2.75400	0.127713
$466$	50.9035	2.35806
$467$	31.4165	1.45378	0.726891	−	0.686753i	$-0.240964\pi$
$467$	31.4165	1.45378	0.726891	+	0.686753i	$0.240964\pi$
$468$	49.5213	2.28912
$469$	8.22469	0.379781
$470$	1.63677	0.0754986
$471$	8.74250	0.402833
$472$	−33.3305	−1.53416
$473$	0.536888	0.0246861
$474$	77.4193	3.55599
$475$	22.1170	1.01480
$476$	14.6635	0.672101
$477$	−8.46487	−0.387580
$478$	8.77271	0.401255
$479$	−41.4746	−1.89502	−0.947512	−	0.319720i	$-0.896411\pi$
$479$	−41.4746	−1.89502	−0.947512	+	0.319720i	$0.896411\pi$
$480$	−3.15560	−0.144033
$481$	−6.82751	−0.311308
$482$	3.28779	0.149755
$483$	4.52366	0.205834
$484$	3.45776	0.157171
$485$	0.0905986	0.00411387
$486$	45.1279	2.04704
$487$	−16.2605	−0.736832	−0.368416	−	0.929661i	$-0.620100\pi$
$487$	−16.2605	−0.736832	−0.368416	+	0.929661i	$0.620100\pi$
$488$	16.3379	0.739583
$489$	−45.9008	−2.07570
$490$	−4.92646	−0.222555
$491$	−20.6771	−0.933144	−0.466572	−	0.884483i	$-0.654511\pi$
$491$	−20.6771	−0.933144	−0.466572	+	0.884483i	$0.654511\pi$
$492$	37.4063	1.68641
$493$	11.3707	0.512111
$494$	67.1946	3.02323
$495$	−0.706705	−0.0317640
$496$	−3.97791	−0.178614
$497$	4.43874	0.199105
$498$	−11.3828	−0.510076
$499$	−0.843697	−0.0377691	−0.0188845	−	0.999822i	$-0.506011\pi$
$499$	−0.843697	−0.0377691	−0.0188845	+	0.999822i	$0.506011\pi$
$500$	10.7674	0.481534
$501$	58.2738	2.60348
$502$	−28.0302	−1.25105
$503$	25.3863	1.13192	0.565959	−	0.824433i	$-0.308506\pi$
$503$	25.3863	1.13192	0.565959	+	0.824433i	$0.308506\pi$
$504$	−4.15682	−0.185160
$505$	1.78338	0.0793595
$506$	−8.49333	−0.377575
$507$	63.2770	2.81023
$508$	41.0736	1.82235
$509$	36.9272	1.63677	0.818384	−	0.574672i	$-0.194870\pi$
$509$	36.9272	1.63677	0.818384	+	0.574672i	$0.194870\pi$
$510$	13.1394	0.581824
$511$	5.54269	0.245194
$512$	11.6457	0.514672
$513$	7.78362	0.343655
$514$	−27.8193	−1.22706
$515$	−0.0288489	−0.00127124
$516$	4.25241	0.187202
$517$	2.22765	0.0979719
$518$	1.35938	0.0597277
$519$	−56.5266	−2.48124
$520$	6.82685	0.299377
$521$	−29.8608	−1.30822	−0.654112	−	0.756398i	$-0.726958\pi$
$521$	−29.8608	−1.30822	−0.654112	+	0.756398i	$0.726958\pi$
$522$	−7.64574	−0.334645
$523$	14.2719	0.624066	0.312033	−	0.950071i	$-0.398990\pi$
$523$	14.2719	0.624066	0.312033	+	0.950071i	$0.398990\pi$
$524$	−3.45776	−0.151053
$525$	−6.09833	−0.266153
$526$	51.8853	2.26231
$527$	−29.8437	−1.30001
$528$	2.38362	0.103734
$529$	−9.78273	−0.425336
$530$	−2.76794	−0.120232
$531$	21.9914	0.954344
$532$	−8.47603	−0.367483
$533$	30.1014	1.30384
$534$	39.9106	1.72710
$535$	−2.14051	−0.0925424
$536$	−51.5645	−2.22725
$537$	−23.7331	−1.02416
$538$	29.0398	1.25200
$539$	−6.70493	−0.288802
$540$	1.87575	0.0807196
$541$	27.5187	1.18312	0.591560	−	0.806261i	$-0.298512\pi$
$541$	27.5187	1.18312	0.591560	+	0.806261i	$0.298512\pi$
$542$	42.6922	1.83379
$543$	−26.0861	−1.11946
$544$	34.1957	1.46613
$545$	−1.71311	−0.0733815
$546$	−18.5276	−0.792907
$547$	−42.6872	−1.82517	−0.912586	−	0.408885i	$-0.865918\pi$
$547$	−42.6872	−1.82517	−0.912586	+	0.408885i	$0.865918\pi$
$548$	33.0699	1.41267
$549$	−10.7797	−0.460067
$550$	11.4498	0.488223
$551$	−6.57268	−0.280005
$552$	−28.3609	−1.20712
$553$	−7.85867	−0.334185
$554$	−49.9554	−2.12240
$555$	0.771720	0.0327577
$556$	56.9930	2.41704
$557$	22.0904	0.936002	0.468001	−	0.883728i	$-0.344974\pi$
$557$	22.0904	0.936002	0.468001	+	0.883728i	$0.344974\pi$
$558$	20.0671	0.849509
$559$	3.42197	0.144734
$560$	−0.177778	−0.00751250
$561$	17.8828	0.755012
$562$	−4.34279	−0.183190
$563$	27.6699	1.16615	0.583074	−	0.812419i	$-0.301850\pi$
$563$	27.6699	1.16615	0.583074	+	0.812419i	$0.301850\pi$
$564$	17.6440	0.742947
$565$	5.53138	0.232707
$566$	58.6447	2.46502
$567$	−5.80793	−0.243910
$568$	−27.8285	−1.16766
$569$	−20.7074	−0.868098	−0.434049	−	0.900889i	$-0.642916\pi$
$569$	−20.7074	−0.868098	−0.434049	+	0.900889i	$0.642916\pi$
$570$	−7.59506	−0.318122
$571$	38.7837	1.62305	0.811523	−	0.584320i	$-0.198639\pi$
$571$	38.7837	1.62305	0.811523	+	0.584320i	$0.198639\pi$
$572$	22.0388	0.921488
$573$	7.75695	0.324051
$574$	−5.99328	−0.250155
$575$	−17.8182	−0.743069
$576$	−27.6699	−1.15291
$577$	−25.7724	−1.07292	−0.536460	−	0.843926i	$-0.680239\pi$
$577$	−25.7724	−1.07292	−0.536460	+	0.843926i	$0.680239\pi$
$578$	−102.671	−4.27053
$579$	−9.65086	−0.401076
$580$	−1.58393	−0.0657692
$581$	1.15545	0.0479359
$582$	1.54153	0.0638983
$583$	−3.76717	−0.156020
$584$	−34.7498	−1.43796
$585$	−4.50433	−0.186231
$586$	29.4276	1.21564
$587$	−1.73455	−0.0715925	−0.0357963	−	0.999359i	$-0.511397\pi$
$587$	−1.73455	−0.0715925	−0.0357963	+	0.999359i	$0.511397\pi$
$588$	−53.1062	−2.19006
$589$	17.2508	0.710805
$590$	7.19099	0.296048
$591$	53.3835	2.19590
$592$	−1.11468	−0.0458132
$593$	33.8954	1.39192	0.695958	−	0.718082i	$-0.254980\pi$
$593$	33.8954	1.39192	0.695958	+	0.718082i	$0.254980\pi$
$594$	4.02953	0.165334
$595$	−1.33376	−0.0546787
$596$	15.0248	0.615441
$597$	39.9766	1.63613
$598$	−54.1341	−2.21371
$599$	10.0090	0.408958	0.204479	−	0.978871i	$-0.434450\pi$
$599$	10.0090	0.408958	0.204479	+	0.978871i	$0.434450\pi$
$600$	38.2333	1.56087
$601$	−16.6990	−0.681167	−0.340583	−	0.940214i	$-0.610625\pi$
$601$	−16.6990	−0.681167	−0.340583	+	0.940214i	$0.610625\pi$
$602$	−0.681325	−0.0277687
$603$	34.0221	1.38549
$604$	−5.37140	−0.218559
$605$	−0.314509	−0.0127866
$606$	30.3441	1.23264
$607$	19.0334	0.772540	0.386270	−	0.922386i	$-0.373763\pi$
$607$	19.0334	0.772540	0.386270	+	0.922386i	$0.373763\pi$
$608$	−19.7664	−0.801632
$609$	1.81229	0.0734375
$610$	−3.52488	−0.142718
$611$	14.1984	0.574405
$612$	60.6567	2.45190
$613$	−35.1293	−1.41886	−0.709429	−	0.704777i	$-0.751047\pi$
$613$	−35.1293	−1.41886	−0.709429	+	0.704777i	$0.751047\pi$
$614$	24.8814	1.00413
$615$	−3.40239	−0.137197
$616$	−1.84994	−0.0745361
$617$	−19.0192	−0.765684	−0.382842	−	0.923814i	$-0.625055\pi$
$617$	−19.0192	−0.765684	−0.382842	+	0.923814i	$0.625055\pi$
$618$	−0.490862	−0.0197454
$619$	−26.6743	−1.07213	−0.536065	−	0.844177i	$-0.680090\pi$
$619$	−26.6743	−1.07213	−0.536065	+	0.844177i	$0.680090\pi$
$620$	4.15721	0.166958
$621$	−6.27073	−0.251636
$622$	40.1544	1.61004
$623$	−4.05124	−0.162309
$624$	15.1925	0.608188
$625$	23.5260	0.941042
$626$	58.3199	2.33093
$627$	−10.3369	−0.412816
$628$	13.1970	0.526617
$629$	−8.36276	−0.333445
$630$	0.896827	0.0357304
$631$	−47.6453	−1.89673	−0.948365	−	0.317181i	$-0.897264\pi$
$631$	−47.6453	−1.89673	−0.948365	+	0.317181i	$0.897264\pi$
$632$	49.2697	1.95984
$633$	23.6523	0.940096
$634$	3.73907	0.148498
$635$	−3.73595	−0.148257
$636$	−29.8378	−1.18315
$637$	−42.7353	−1.69323
$638$	−3.40263	−0.134712
$639$	18.3612	0.726357
$640$	−6.29260	−0.248737
$641$	−1.18002	−0.0466082	−0.0233041	−	0.999728i	$-0.507419\pi$
$641$	−1.18002	−0.0466082	−0.0233041	+	0.999728i	$0.507419\pi$
$642$	−36.4206	−1.43741
$643$	−18.6281	−0.734620	−0.367310	−	0.930099i	$-0.619721\pi$
$643$	−18.6281	−0.734620	−0.367310	+	0.930099i	$0.619721\pi$
$644$	6.82856	0.269083
$645$	−0.386788	−0.0152298
$646$	82.3041	3.23821
$647$	−1.04305	−0.0410065	−0.0205032	−	0.999790i	$-0.506527\pi$
$647$	−1.04305	−0.0410065	−0.0205032	+	0.999790i	$0.506527\pi$
$648$	36.4126	1.43042
$649$	9.78696	0.384172
$650$	72.9780	2.86243
$651$	−4.75655	−0.186424
$652$	−69.2882	−2.71354
$653$	−17.3841	−0.680292	−0.340146	−	0.940373i	$-0.610476\pi$
$653$	−17.3841	−0.680292	−0.340146	+	0.940373i	$0.610476\pi$
$654$	−29.1484	−1.13979
$655$	0.314509	0.0122889
$656$	4.91446	0.191877
$657$	22.9278	0.894498
$658$	−2.82694	−0.110206
$659$	−3.23750	−0.126115	−0.0630575	−	0.998010i	$-0.520085\pi$
$659$	−3.23750	−0.126115	−0.0630575	+	0.998010i	$0.520085\pi$
$660$	−2.49106	−0.0969645
$661$	15.3210	0.595920	0.297960	−	0.954578i	$-0.403694\pi$
$661$	15.3210	0.595920	0.297960	+	0.954578i	$0.403694\pi$
$662$	45.3201	1.76142
$663$	113.980	4.42661
$664$	−7.24403	−0.281123
$665$	0.770959	0.0298965
$666$	5.62318	0.217894
$667$	5.29516	0.205029
$668$	87.9655	3.40349
$669$	47.1730	1.82381
$670$	11.1249	0.429794
$671$	−4.79737	−0.185200
$672$	5.45018	0.210245
$673$	−26.4810	−1.02077	−0.510384	−	0.859946i	$-0.670497\pi$
$673$	−26.4810	−1.02077	−0.510384	+	0.859946i	$0.670497\pi$
$674$	62.3270	2.40074
$675$	8.45355	0.325377
$676$	95.5179	3.67377
$677$	−17.9841	−0.691186	−0.345593	−	0.938385i	$-0.612322\pi$
$677$	−17.9841	−0.691186	−0.345593	+	0.938385i	$0.612322\pi$
$678$	94.1159	3.61450
$679$	−0.156477	−0.00600504
$680$	8.36194	0.320666
$681$	−42.7346	−1.63759
$682$	8.93061	0.341971
$683$	−7.47119	−0.285877	−0.142939	−	0.989732i	$-0.545655\pi$
$683$	−7.47119	−0.285877	−0.142939	+	0.989732i	$0.545655\pi$
$684$	−35.0618	−1.34062
$685$	−3.00795	−0.114928
$686$	17.3919	0.664026
$687$	7.98173	0.304522
$688$	0.558683	0.0212996
$689$	−24.0109	−0.914742
$690$	6.11882	0.232940
$691$	−17.2576	−0.656511	−0.328255	−	0.944589i	$-0.606461\pi$
$691$	−17.2576	−0.656511	−0.328255	+	0.944589i	$0.606461\pi$
$692$	−85.3281	−3.24369
$693$	1.22058	0.0463661
$694$	−51.6573	−1.96089
$695$	−5.18394	−0.196638
$696$	−11.3621	−0.430678
$697$	36.8700	1.39655
$698$	19.8817	0.752532
$699$	−49.9110	−1.88781
$700$	−9.20556	−0.347938
$701$	15.4973	0.585324	0.292662	−	0.956216i	$-0.405459\pi$
$701$	15.4973	0.585324	0.292662	+	0.956216i	$0.405459\pi$
$702$	25.6831	0.969344
$703$	4.83398	0.182317
$704$	−12.3141	−0.464105
$705$	−1.60486	−0.0604424
$706$	−66.8120	−2.51450
$707$	−3.08016	−0.115841
$708$	77.5173	2.91328
$709$	19.0070	0.713823	0.356911	−	0.934138i	$-0.383830\pi$
$709$	19.0070	0.713823	0.356911	+	0.934138i	$0.383830\pi$
$710$	6.00396	0.225324
$711$	−32.5080	−1.21915
$712$	25.3991	0.951872
$713$	−13.8977	−0.520475
$714$	−22.6937	−0.849291
$715$	−2.00459	−0.0749675
$716$	−35.8256	−1.33886
$717$	−8.60166	−0.321235
$718$	−28.2051	−1.05261
$719$	−26.8539	−1.00148	−0.500740	−	0.865597i	$-0.666939\pi$
$719$	−26.8539	−1.00148	−0.500740	+	0.865597i	$0.666939\pi$
$720$	−0.735394	−0.0274065
$721$	0.0498263	0.00185563
$722$	−3.18721	−0.118616
$723$	−3.22368	−0.119890
$724$	−39.3775	−1.46345
$725$	−7.13838	−0.265113
$726$	−5.35135	−0.198607
$727$	−7.95940	−0.295198	−0.147599	−	0.989047i	$-0.547154\pi$
$727$	−7.95940	−0.295198	−0.147599	+	0.989047i	$0.547154\pi$
$728$	−11.7910	−0.437002
$729$	−12.1721	−0.450817
$730$	7.49720	0.277484
$731$	4.19144	0.155026
$732$	−37.9974	−1.40442
$733$	−23.0074	−0.849797	−0.424898	−	0.905241i	$-0.639690\pi$
$733$	−23.0074	−0.849797	−0.424898	+	0.905241i	$0.639690\pi$
$734$	−63.7513	−2.35310
$735$	4.83041	0.178172
$736$	15.9244	0.586981
$737$	15.1411	0.557728
$738$	−24.7917	−0.912594
$739$	−13.9978	−0.514917	−0.257459	−	0.966289i	$-0.582885\pi$
$739$	−13.9978	−0.514917	−0.257459	+	0.966289i	$0.582885\pi$
$740$	1.16493	0.0428236
$741$	−65.8844	−2.42032
$742$	4.78064	0.175503
$743$	−4.46936	−0.163965	−0.0819825	−	0.996634i	$-0.526125\pi$
$743$	−4.46936	−0.163965	−0.0819825	+	0.996634i	$0.526125\pi$
$744$	29.8211	1.09329
$745$	−1.36662	−0.0500692
$746$	46.8081	1.71377
$747$	4.77959	0.174876
$748$	26.9945	0.987015
$749$	3.69698	0.135085
$750$	−16.6640	−0.608484
$751$	26.2771	0.958864	0.479432	−	0.877579i	$-0.340843\pi$
$751$	26.2771	0.958864	0.479432	+	0.877579i	$0.340843\pi$
$752$	2.31808	0.0845317
$753$	27.4837	1.00156
$754$	−21.6874	−0.789809
$755$	0.488569	0.0177808
$756$	−3.23970	−0.117827
$757$	−8.29482	−0.301480	−0.150740	−	0.988573i	$-0.548166\pi$
$757$	−8.29482	−0.301480	−0.150740	+	0.988573i	$0.548166\pi$
$758$	10.6207	0.385761
$759$	8.32773	0.302278
$760$	−4.83351	−0.175330
$761$	−10.6435	−0.385828	−0.192914	−	0.981216i	$-0.561794\pi$
$761$	−10.6435	−0.385828	−0.192914	+	0.981216i	$0.561794\pi$
$762$	−63.5668	−2.30278
$763$	2.95879	0.107115
$764$	11.7093	0.423627
$765$	−5.51719	−0.199474
$766$	−5.15909	−0.186405
$767$	62.3793	2.25238
$768$	−50.6538	−1.82781
$769$	−28.8981	−1.04209	−0.521046	−	0.853528i	$-0.674458\pi$
$769$	−28.8981	−1.04209	−0.521046	+	0.853528i	$0.674458\pi$
$770$	0.399121	0.0143833
$771$	27.2769	0.982352
$772$	−14.5682	−0.524320
$773$	19.4491	0.699534	0.349767	−	0.936837i	$-0.386261\pi$
$773$	19.4491	0.699534	0.349767	+	0.936837i	$0.386261\pi$
$774$	−2.81835	−0.101304
$775$	18.7355	0.673000
$776$	0.981029	0.0352169
$777$	−1.33287	−0.0478166
$778$	47.1634	1.69089
$779$	−21.3122	−0.763589
$780$	−15.8773	−0.568499
$781$	8.17140	0.292396
$782$	−66.3068	−2.37112
$783$	−2.51220	−0.0897788
$784$	−6.97712	−0.249183
$785$	−1.20037	−0.0428429
$786$	5.35135	0.190876
$787$	−33.7652	−1.20360	−0.601800	−	0.798647i	$-0.705549\pi$
$787$	−33.7652	−1.20360	−0.601800	+	0.798647i	$0.705549\pi$
$788$	80.5835	2.87067
$789$	−50.8737	−1.81115
$790$	−10.6299	−0.378193
$791$	−9.55351	−0.339684
$792$	−7.65241	−0.271917
$793$	−30.5770	−1.08582
$794$	72.0907	2.55840
$795$	2.71397	0.0962547
$796$	60.3455	2.13889
$797$	29.9480	1.06081	0.530406	−	0.847744i	$-0.322040\pi$
$797$	29.9480	1.06081	0.530406	+	0.847744i	$0.322040\pi$
$798$	13.1178	0.464365
$799$	17.3911	0.615252
$800$	−21.4676	−0.758996
$801$	−16.7583	−0.592124
$802$	39.7233	1.40268
$803$	10.2037	0.360081
$804$	119.924	4.22941
$805$	−0.621109	−0.0218912
$806$	56.9211	2.00496
$807$	−28.4736	−1.00232
$808$	19.3110	0.679358
$809$	−13.7240	−0.482510	−0.241255	−	0.970462i	$-0.577559\pi$
$809$	−13.7240	−0.482510	−0.241255	+	0.970462i	$0.577559\pi$
$810$	−7.85596	−0.276030
$811$	−55.3907	−1.94503	−0.972516	−	0.232837i	$-0.925199\pi$
$811$	−55.3907	−1.94503	−0.972516	+	0.232837i	$0.925199\pi$
$812$	2.73568	0.0960037
$813$	−41.8598	−1.46809
$814$	2.50252	0.0877132
$815$	6.30228	0.220759
$816$	18.6087	0.651436
$817$	−2.42280	−0.0847632
$818$	9.31749	0.325779
$819$	7.77965	0.271843
$820$	−5.13597	−0.179356
$821$	6.34584	0.221471	0.110736	−	0.993850i	$-0.464679\pi$
$821$	6.34584	0.221471	0.110736	+	0.993850i	$0.464679\pi$
$822$	−51.1800	−1.78511
$823$	−48.8338	−1.70224	−0.851120	−	0.524972i	$-0.824076\pi$
$823$	−48.8338	−1.70224	−0.851120	+	0.524972i	$0.824076\pi$
$824$	−0.312385	−0.0108824
$825$	−11.2266	−0.390860
$826$	−12.4199	−0.432144
$827$	−40.3974	−1.40476	−0.702378	−	0.711804i	$-0.747878\pi$
$827$	−40.3974	−1.40476	−0.702378	+	0.711804i	$0.747878\pi$
$828$	28.2469	0.981647
$829$	8.82269	0.306425	0.153212	−	0.988193i	$-0.451038\pi$
$829$	8.82269	0.306425	0.153212	+	0.988193i	$0.451038\pi$
$830$	1.56289	0.0542486
$831$	48.9814	1.69915
$832$	−78.4866	−2.72103
$833$	−52.3448	−1.81364
$834$	−88.2042	−3.05426
$835$	−8.00113	−0.276890
$836$	−15.6038	−0.539668
$837$	6.59357	0.227907
$838$	41.9769	1.45007
$839$	43.1858	1.49094	0.745470	−	0.666539i	$-0.232225\pi$
$839$	43.1858	1.49094	0.745470	+	0.666539i	$0.232225\pi$
$840$	1.33274	0.0459840
$841$	−26.8786	−0.926849
$842$	41.2425	1.42131
$843$	4.25812	0.146657
$844$	35.7037	1.22897
$845$	−8.68808	−0.298879
$846$	−11.6939	−0.402044
$847$	0.543204	0.0186647
$848$	−3.92010	−0.134617
$849$	−57.5012	−1.97344
$850$	89.3880	3.06598
$851$	−3.89440	−0.133498
$852$	64.7213	2.21732
$853$	36.8173	1.26060	0.630301	−	0.776351i	$-0.282932\pi$
$853$	36.8173	1.26060	0.630301	+	0.776351i	$0.282932\pi$
$854$	6.08798	0.208326
$855$	3.18913	0.109066
$856$	−23.1781	−0.792211
$857$	−44.1996	−1.50983	−0.754915	−	0.655822i	$-0.772322\pi$
$857$	−44.1996	−1.50983	−0.754915	+	0.655822i	$0.772322\pi$
$858$	−34.1079	−1.16443
$859$	−17.6542	−0.602352	−0.301176	−	0.953569i	$-0.597379\pi$
$859$	−17.6542	−0.602352	−0.301176	+	0.953569i	$0.597379\pi$
$860$	−0.583865	−0.0199096
$861$	5.87642	0.200268
$862$	−68.6867	−2.33948
$863$	−8.93498	−0.304150	−0.152075	−	0.988369i	$-0.548596\pi$
$863$	−8.93498	−0.304150	−0.152075	+	0.988369i	$0.548596\pi$
$864$	−7.55508	−0.257029
$865$	7.76123	0.263890
$866$	−36.3935	−1.23670
$867$	100.669	3.41889
$868$	−7.18012	−0.243709
$869$	−14.4673	−0.490768
$870$	2.45135	0.0831084
$871$	96.5048	3.26994
$872$	−18.5501	−0.628184
$873$	−0.647280	−0.0219071
$874$	38.3277	1.29645
$875$	1.69153	0.0571841
$876$	80.8181	2.73059
$877$	11.8641	0.400621	0.200310	−	0.979732i	$-0.435805\pi$
$877$	11.8641	0.400621	0.200310	+	0.979732i	$0.435805\pi$
$878$	−77.1390	−2.60332
$879$	−28.8538	−0.973215
$880$	−0.327277	−0.0110325
$881$	−7.19783	−0.242501	−0.121251	−	0.992622i	$-0.538690\pi$
$881$	−7.19783	−0.242501	−0.121251	+	0.992622i	$0.538690\pi$
$882$	35.1970	1.18515
$883$	−13.1306	−0.441878	−0.220939	−	0.975288i	$-0.570912\pi$
$883$	−13.1306	−0.441878	−0.220939	+	0.975288i	$0.570912\pi$
$884$	172.055	5.78683
$885$	−7.05078	−0.237009
$886$	9.15651	0.307619
$887$	24.9610	0.838110	0.419055	−	0.907961i	$-0.362362\pi$
$887$	24.9610	0.838110	0.419055	+	0.907961i	$0.362362\pi$
$888$	8.35641	0.280423
$889$	6.45254	0.216411
$890$	−5.47981	−0.183684
$891$	−10.6920	−0.358195
$892$	71.2087	2.38424
$893$	−10.0527	−0.336400
$894$	−23.2529	−0.777695
$895$	3.25860	0.108923
$896$	10.8683	0.363083
$897$	53.0786	1.77224
$898$	89.0098	2.97029
$899$	−5.56777	−0.185696
$900$	−38.0795	−1.26932
$901$	−29.4100	−0.979790
$902$	−11.0332	−0.367365
$903$	0.668040	0.0222310
$904$	59.8955	1.99209
$905$	3.58168	0.119059
$906$	8.31295	0.276179
$907$	31.1653	1.03483	0.517414	−	0.855735i	$-0.326895\pi$
$907$	31.1653	1.03483	0.517414	+	0.855735i	$0.326895\pi$
$908$	−64.5087	−2.14080
$909$	−12.7413	−0.422604
$910$	2.54388	0.0843288
$911$	−7.55230	−0.250219	−0.125109	−	0.992143i	$-0.539928\pi$
$911$	−7.55230	−0.250219	−0.125109	+	0.992143i	$0.539928\pi$
$912$	−10.7565	−0.356184
$913$	2.12709	0.0703965
$914$	57.2910	1.89502
$915$	3.45615	0.114257
$916$	12.0486	0.398097
$917$	−0.543204	−0.0179382
$918$	31.4582	1.03828
$919$	20.0776	0.662300	0.331150	−	0.943578i	$-0.392563\pi$
$919$	20.0776	0.662300	0.331150	+	0.943578i	$0.392563\pi$
$920$	3.89402	0.128382
$921$	−24.3962	−0.803884
$922$	−33.4078	−1.10023
$923$	52.0821	1.71430
$924$	4.30243	0.141540
$925$	5.25004	0.172620
$926$	65.8533	2.16407
$927$	0.206111	0.00676956
$928$	6.37970	0.209424
$929$	−36.4399	−1.19555	−0.597777	−	0.801662i	$-0.703949\pi$
$929$	−36.4399	−1.19555	−0.597777	+	0.801662i	$0.703949\pi$
$930$	−6.43384	−0.210974
$931$	30.2572	0.991640
$932$	−75.3417	−2.46790
$933$	−39.3715	−1.28896
$934$	−73.3948	−2.40155
$935$	−2.45535	−0.0802985
$936$	−48.7743	−1.59424
$937$	−22.7359	−0.742750	−0.371375	−	0.928483i	$-0.621114\pi$
$937$	−22.7359	−0.742750	−0.371375	+	0.928483i	$0.621114\pi$
$938$	−19.2144	−0.627372
$939$	−57.1828	−1.86609
$940$	−2.42257	−0.0790154
$941$	−1.59355	−0.0519483	−0.0259741	−	0.999663i	$-0.508269\pi$
$941$	−1.59355	−0.0519483	−0.0259741	+	0.999663i	$0.508269\pi$
$942$	−20.4241	−0.665453
$943$	17.1698	0.559125
$944$	10.1843	0.331469
$945$	0.294675	0.00958579
$946$	−1.25427	−0.0407798
$947$	45.0572	1.46416	0.732081	−	0.681218i	$-0.238549\pi$
$947$	45.0572	1.46416	0.732081	+	0.681218i	$0.238549\pi$
$948$	−114.587	−3.72163
$949$	65.0355	2.11114
$950$	−51.6695	−1.67638
$951$	−3.66617	−0.118884
$952$	−14.4423	−0.468078
$953$	11.6770	0.378256	0.189128	−	0.981952i	$-0.439434\pi$
$953$	11.6770	0.378256	0.189128	+	0.981952i	$0.439434\pi$
$954$	19.7755	0.640255
$955$	−1.06505	−0.0344641
$956$	−12.9844	−0.419945
$957$	3.33629	0.107847
$958$	96.8924	3.13045
$959$	5.19518	0.167761
$960$	8.87141	0.286323
$961$	−16.3867	−0.528604
$962$	15.9503	0.514259
$963$	15.2928	0.492805
$964$	−4.86622	−0.156730
$965$	1.32509	0.0426560
$966$	−10.5681	−0.340023
$967$	26.3853	0.848495	0.424247	−	0.905546i	$-0.360539\pi$
$967$	26.3853	0.848495	0.424247	+	0.905546i	$0.360539\pi$
$968$	−3.40560	−0.109460
$969$	−80.6993	−2.59244
$970$	−0.211655	−0.00679584
$971$	−32.2931	−1.03633	−0.518167	−	0.855279i	$-0.673386\pi$
$971$	−32.2931	−1.03633	−0.518167	+	0.855279i	$0.673386\pi$
$972$	−66.7933	−2.14240
$973$	8.95343	0.287034
$974$	37.9875	1.21720
$975$	−71.5551	−2.29160
$976$	−4.99212	−0.159794
$977$	−19.9381	−0.637875	−0.318938	−	0.947776i	$-0.603326\pi$
$977$	−19.9381	−0.637875	−0.318938	+	0.947776i	$0.603326\pi$
$978$	107.233	3.42892
$979$	−7.45804	−0.238360
$980$	7.29160	0.232922
$981$	12.2393	0.390770
$982$	48.3055	1.54149
$983$	14.6538	0.467382	0.233691	−	0.972311i	$-0.424920\pi$
$983$	14.6538	0.467382	0.233691	+	0.972311i	$0.424920\pi$
$984$	−36.8420	−1.17448
$985$	−7.32968	−0.233543
$986$	−26.5641	−0.845972
$987$	2.77182	0.0882281
$988$	−99.4540	−3.16405
$989$	1.95189	0.0620664
$990$	1.65099	0.0524720
$991$	29.2410	0.928871	0.464435	−	0.885607i	$-0.346257\pi$
$991$	29.2410	0.928871	0.464435	+	0.885607i	$0.346257\pi$
$992$	−16.7443	−0.531631
$993$	−44.4365	−1.41015
$994$	−10.3697	−0.328907
$995$	−5.48888	−0.174009
$996$	16.8476	0.533836
$997$	40.5825	1.28526	0.642631	−	0.766176i	$-0.277843\pi$
$997$	40.5825	1.28526	0.642631	+	0.766176i	$0.277843\pi$
$998$	1.97103	0.0623919
$999$	1.84764	0.0584567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1441.2.a.e.1.3	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1441.2.a.e.1.3	✓	28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1441 = 11 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1441.a (trivial)

\(f(q)\)	\(=\)	\(q-2.33619 q^{2} +2.29063 q^{3} +3.45776 q^{4} -0.314509 q^{5} -5.35135 q^{6} +0.543204 q^{7} -3.40560 q^{8} +2.24701 q^{9} +O(q^{10})\)	\(q-2.33619 q^{2} +2.29063 q^{3} +3.45776 q^{4} -0.314509 q^{5} -5.35135 q^{6} +0.543204 q^{7} -3.40560 q^{8} +2.24701 q^{9} +0.734752 q^{10} +1.00000 q^{11} +7.92047 q^{12} +6.37371 q^{13} -1.26903 q^{14} -0.720426 q^{15} +1.04060 q^{16} +7.80692 q^{17} -5.24942 q^{18} -4.51268 q^{19} -1.08750 q^{20} +1.24428 q^{21} -2.33619 q^{22} +3.63556 q^{23} -7.80099 q^{24} -4.90108 q^{25} -14.8902 q^{26} -1.72483 q^{27} +1.87827 q^{28} +1.45649 q^{29} +1.68305 q^{30} -3.82273 q^{31} +4.38018 q^{32} +2.29063 q^{33} -18.2384 q^{34} -0.170843 q^{35} +7.76961 q^{36} -1.07120 q^{37} +10.5425 q^{38} +14.5998 q^{39} +1.07109 q^{40} +4.72274 q^{41} -2.90687 q^{42} +0.536888 q^{43} +3.45776 q^{44} -0.706705 q^{45} -8.49333 q^{46} +2.22765 q^{47} +2.38362 q^{48} -6.70493 q^{49} +11.4498 q^{50} +17.8828 q^{51} +22.0388 q^{52} -3.76717 q^{53} +4.02953 q^{54} -0.314509 q^{55} -1.84994 q^{56} -10.3369 q^{57} -3.40263 q^{58} +9.78696 q^{59} -2.49106 q^{60} -4.79737 q^{61} +8.93061 q^{62} +1.22058 q^{63} -12.3141 q^{64} -2.00459 q^{65} -5.35135 q^{66} +15.1411 q^{67} +26.9945 q^{68} +8.32773 q^{69} +0.399121 q^{70} +8.17140 q^{71} -7.65241 q^{72} +10.2037 q^{73} +2.50252 q^{74} -11.2266 q^{75} -15.6038 q^{76} +0.543204 q^{77} -34.1079 q^{78} -14.4673 q^{79} -0.327277 q^{80} -10.6920 q^{81} -11.0332 q^{82} +2.12709 q^{83} +4.30243 q^{84} -2.45535 q^{85} -1.25427 q^{86} +3.33629 q^{87} -3.40560 q^{88} -7.45804 q^{89} +1.65099 q^{90} +3.46223 q^{91} +12.5709 q^{92} -8.75648 q^{93} -5.20420 q^{94} +1.41928 q^{95} +10.0334 q^{96} -0.288063 q^{97} +15.6640 q^{98} +2.24701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 7 q^{2} - 2 q^{3} + 27 q^{4} + 3 q^{5} - q^{6} + 7 q^{7} + 21 q^{8} + 28 q^{9}+O(q^{10}) \) 28 * q + 7 * q^2 - 2 * q^3 + 27 * q^4 + 3 * q^5 - q^6 + 7 * q^7 + 21 * q^8 + 28 * q^9	\( 28 q + 7 q^{2} - 2 q^{3} + 27 q^{4} + 3 q^{5} - q^{6} + 7 q^{7} + 21 q^{8} + 28 q^{9} + 14 q^{10} + 28 q^{11} - 6 q^{12} + 3 q^{13} - q^{14} + 19 q^{15} + 29 q^{16} + 9 q^{17} - 2 q^{18} + 20 q^{19} + 6 q^{20} + 6 q^{21} + 7 q^{22} + 24 q^{23} + 20 q^{24} + 23 q^{25} + 10 q^{26} - 20 q^{27} - 3 q^{28} + 43 q^{29} + 11 q^{30} + 3 q^{31} + 44 q^{32} - 2 q^{33} - 28 q^{34} + 32 q^{35} + 24 q^{36} - 4 q^{37} + 24 q^{38} + 37 q^{39} + 22 q^{40} + 32 q^{41} - 27 q^{42} + 25 q^{43} + 27 q^{44} - 36 q^{45} + 10 q^{46} + 19 q^{47} + 42 q^{48} + 17 q^{49} + q^{50} + 39 q^{51} - 19 q^{52} + 5 q^{53} + 6 q^{54} + 3 q^{55} + 8 q^{56} + 2 q^{57} + 21 q^{58} + 44 q^{59} + 65 q^{60} + 28 q^{61} + 60 q^{62} - 8 q^{63} + 5 q^{64} + 33 q^{65} - q^{66} + 7 q^{67} + 13 q^{68} - 22 q^{69} + 9 q^{70} + 117 q^{71} - 17 q^{72} + 7 q^{73} + 41 q^{74} - 40 q^{75} + 34 q^{76} + 7 q^{77} - 97 q^{78} + 48 q^{79} + 41 q^{80} + 40 q^{81} + 2 q^{82} + 22 q^{83} + 27 q^{84} + 30 q^{85} + 24 q^{86} + 37 q^{87} + 21 q^{88} - 6 q^{89} + 4 q^{90} - 33 q^{91} + 18 q^{92} + 5 q^{93} - 43 q^{94} + 64 q^{95} + 55 q^{96} - 50 q^{97} + 97 q^{98} + 28 q^{99}+O(q^{100}) \) 28 * q + 7 * q^2 - 2 * q^3 + 27 * q^4 + 3 * q^5 - q^6 + 7 * q^7 + 21 * q^8 + 28 * q^9 + 14 * q^10 + 28 * q^11 - 6 * q^12 + 3 * q^13 - q^14 + 19 * q^15 + 29 * q^16 + 9 * q^17 - 2 * q^18 + 20 * q^19 + 6 * q^20 + 6 * q^21 + 7 * q^22 + 24 * q^23 + 20 * q^24 + 23 * q^25 + 10 * q^26 - 20 * q^27 - 3 * q^28 + 43 * q^29 + 11 * q^30 + 3 * q^31 + 44 * q^32 - 2 * q^33 - 28 * q^34 + 32 * q^35 + 24 * q^36 - 4 * q^37 + 24 * q^38 + 37 * q^39 + 22 * q^40 + 32 * q^41 - 27 * q^42 + 25 * q^43 + 27 * q^44 - 36 * q^45 + 10 * q^46 + 19 * q^47 + 42 * q^48 + 17 * q^49 + q^50 + 39 * q^51 - 19 * q^52 + 5 * q^53 + 6 * q^54 + 3 * q^55 + 8 * q^56 + 2 * q^57 + 21 * q^58 + 44 * q^59 + 65 * q^60 + 28 * q^61 + 60 * q^62 - 8 * q^63 + 5 * q^64 + 33 * q^65 - q^66 + 7 * q^67 + 13 * q^68 - 22 * q^69 + 9 * q^70 + 117 * q^71 - 17 * q^72 + 7 * q^73 + 41 * q^74 - 40 * q^75 + 34 * q^76 + 7 * q^77 - 97 * q^78 + 48 * q^79 + 41 * q^80 + 40 * q^81 + 2 * q^82 + 22 * q^83 + 27 * q^84 + 30 * q^85 + 24 * q^86 + 37 * q^87 + 21 * q^88 - 6 * q^89 + 4 * q^90 - 33 * q^91 + 18 * q^92 + 5 * q^93 - 43 * q^94 + 64 * q^95 + 55 * q^96 - 50 * q^97 + 97 * q^98 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more