LMFDB - Embedded newform 735.3.h.a.391.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,3,Mod(391,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("735.391");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	735.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$20.0272994305$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.523596960000.16
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100 $ x^8 - 2x^7 + 13x^6 - 2x^5 + 91x^4 - 50x^3 + 190x^2 + 100*x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			391.5
Root			$0.836732 + 1.44926i$ of defining polynomial
Character	$\chi$	$=$	735.391
Dual form			735.3.h.a.391.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.673464 q^{2} -1.73205i q^{3} -3.54645 q^{4} -2.23607i q^{5} -1.16647i q^{6} -5.08226 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+0.673464 q^{2} -1.73205i q^{3} -3.54645 q^{4} -2.23607i q^{5} -1.16647i q^{6} -5.08226 q^{8} -3.00000 q^{9} -1.50591i q^{10} -0.0447599 q^{11} +6.14263i q^{12} +23.0010i q^{13} -3.87298 q^{15} +10.7631 q^{16} -9.42573i q^{17} -2.02039 q^{18} -1.14437i q^{19} +7.93010i q^{20} -0.0301442 q^{22} -44.2404 q^{23} +8.80273i q^{24} -5.00000 q^{25} +15.4904i q^{26} +5.19615i q^{27} +53.0004 q^{29} -2.60831 q^{30} +22.5963i q^{31} +27.5776 q^{32} +0.0775264i q^{33} -6.34788i q^{34} +10.6393 q^{36} +42.2836 q^{37} -0.770689i q^{38} +39.8390 q^{39} +11.3643i q^{40} +38.2787i q^{41} +76.5222 q^{43} +0.158739 q^{44} +6.70820i q^{45} -29.7943 q^{46} +27.1436i q^{47} -18.6422i q^{48} -3.36732 q^{50} -16.3258 q^{51} -81.5719i q^{52} +18.9878 q^{53} +3.49942i q^{54} +0.100086i q^{55} -1.98210 q^{57} +35.6938 q^{58} -4.86973i q^{59} +13.7353 q^{60} +38.8479i q^{61} +15.2178i q^{62} -24.4798 q^{64} +51.4319 q^{65} +0.0522112i q^{66} -7.00878 q^{67} +33.4278i q^{68} +76.6266i q^{69} -46.8735 q^{71} +15.2468 q^{72} +83.5952i q^{73} +28.4764 q^{74} +8.66025i q^{75} +4.05843i q^{76} +26.8301 q^{78} +20.4794 q^{79} -24.0670i q^{80} +9.00000 q^{81} +25.7793i q^{82} +125.683i q^{83} -21.0766 q^{85} +51.5349 q^{86} -91.7993i q^{87} +0.227481 q^{88} -46.7024i q^{89} +4.51773i q^{90} +156.896 q^{92} +39.1380 q^{93} +18.2802i q^{94} -2.55888 q^{95} -47.7657i q^{96} +3.11494i q^{97} +0.134280 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 12 q^{4} - 32 q^{8} - 24 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 + 12 * q^4 - 32 * q^8 - 24 * q^9	$ 8 q - 4 q^{2} + 12 q^{4} - 32 q^{8} - 24 q^{9} - 40 q^{11} + 4 q^{16} + 12 q^{18} - 16 q^{22} - 124 q^{23} - 40 q^{25} - 100 q^{29} - 72 q^{32} - 36 q^{36} + 160 q^{37} + 24 q^{39} + 352 q^{43} + 36 q^{44} + 164 q^{46} + 20 q^{50} - 36 q^{51} + 152 q^{53} + 80 q^{58} + 120 q^{60} - 4 q^{64} + 120 q^{65} - 368 q^{67} + 164 q^{71} + 96 q^{72} + 280 q^{74} - 240 q^{78} + 412 q^{79} + 72 q^{81} - 60 q^{85} - 356 q^{86} - 248 q^{88} - 288 q^{92} - 252 q^{93} + 240 q^{95} + 120 q^{99}+O(q^{100}) $ 8 * q - 4 * q^2 + 12 * q^4 - 32 * q^8 - 24 * q^9 - 40 * q^11 + 4 * q^16 + 12 * q^18 - 16 * q^22 - 124 * q^23 - 40 * q^25 - 100 * q^29 - 72 * q^32 - 36 * q^36 + 160 * q^37 + 24 * q^39 + 352 * q^43 + 36 * q^44 + 164 * q^46 + 20 * q^50 - 36 * q^51 + 152 * q^53 + 80 * q^58 + 120 * q^60 - 4 * q^64 + 120 * q^65 - 368 * q^67 + 164 * q^71 + 96 * q^72 + 280 * q^74 - 240 * q^78 + 412 * q^79 + 72 * q^81 - 60 * q^85 - 356 * q^86 - 248 * q^88 - 288 * q^92 - 252 * q^93 + 240 * q^95 + 120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$.

$n$	$346$	$442$	$491$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0.673464	0.336732	0.168366	−	0.985725i	$-0.446151\pi$
$2$	0.673464	0.336732	0.168366	+	0.985725i	$0.446151\pi$
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	−3.54645	−0.886612
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	− 1.16647i	− 0.194412i
$7$	0	0
$7$	0	0
$8$	−5.08226	−0.635282
$9$	−3.00000	−0.333333
$10$	− 1.50591i	− 0.150591i
$11$	−0.0447599	−0.00406908	−0.00203454	−	0.999998i	$-0.500648\pi$
$11$	−0.0447599	−0.00406908	−0.00203454	+	0.999998i	$0.500648\pi$
$12$	6.14263i	0.511886i
$13$	23.0010i	1.76931i	0.466246	+	0.884655i	$0.345606\pi$
$13$	23.0010i	1.76931i	−0.466246	+	0.884655i	$0.654394\pi$
$14$	0	0
$15$	−3.87298	−0.258199
$16$	10.7631	0.672692
$17$	− 9.42573i	− 0.554455i	−0.960804	−	0.277227i	$-0.910584\pi$
$17$	− 9.42573i	− 0.554455i	0.960804	−	0.277227i	$-0.0894155\pi$
$18$	−2.02039	−0.112244
$19$	− 1.14437i	− 0.0602298i	−0.999546	−	0.0301149i	$-0.990413\pi$
$19$	− 1.14437i	− 0.0602298i	0.999546	−	0.0301149i	$-0.00958732\pi$
$20$	7.93010i	0.396505i
$21$	0	0
$22$	−0.0301442	−0.00137019
$23$	−44.2404	−1.92350	−0.961748	−	0.273937i	$-0.911674\pi$
$23$	−44.2404	−1.92350	−0.961748	+	0.273937i	$0.911674\pi$
$24$	8.80273i	0.366780i
$25$	−5.00000	−0.200000
$26$	15.4904i	0.595783i
$27$	5.19615i	0.192450i
$28$	0	0
$29$	53.0004	1.82760	0.913799	−	0.406166i	$-0.133134\pi$
$29$	53.0004	1.82760	0.913799	+	0.406166i	$0.133134\pi$
$30$	−2.60831	−0.0869438
$31$	22.5963i	0.728914i	0.931220	+	0.364457i	$0.118745\pi$
$31$	22.5963i	0.728914i	−0.931220	+	0.364457i	$0.881255\pi$
$32$	27.5776	0.861799
$33$	0.0775264i	0.00234929i
$34$	− 6.34788i	− 0.186702i
$35$	0	0
$36$	10.6393	0.295537
$37$	42.2836	1.14280	0.571400	−	0.820672i	$-0.306401\pi$
$37$	42.2836	1.14280	0.571400	+	0.820672i	$0.306401\pi$
$38$	− 0.770689i	− 0.0202813i
$39$	39.8390	1.02151
$40$	11.3643i	0.284107i
$41$	38.2787i	0.933628i	0.884356	+	0.466814i	$0.154598\pi$
$41$	38.2787i	0.933628i	−0.884356	+	0.466814i	$0.845402\pi$
$42$	0	0
$43$	76.5222	1.77959	0.889793	−	0.456365i	$-0.150849\pi$
$43$	76.5222	1.77959	0.889793	+	0.456365i	$0.150849\pi$
$44$	0.158739	0.00360770
$45$	6.70820i	0.149071i
$46$	−29.7943	−0.647702
$47$	27.1436i	0.577522i	0.957401	+	0.288761i	$0.0932434\pi$
$47$	27.1436i	0.577522i	−0.957401	+	0.288761i	$0.906757\pi$
$48$	− 18.6422i	− 0.388379i
$49$	0	0
$50$	−3.36732	−0.0673464
$51$	−16.3258	−0.320115
$52$	− 81.5719i	− 1.56869i
$53$	18.9878	0.358260	0.179130	−	0.983825i	$-0.442672\pi$
$53$	18.9878	0.358260	0.179130	+	0.983825i	$0.442672\pi$
$54$	3.49942i	0.0648041i
$55$	0.100086i	0.00181975i
$56$	0	0
$57$	−1.98210	−0.0347737
$58$	35.6938	0.615411
$59$	− 4.86973i	− 0.0825377i	−0.999148	−	0.0412689i	$-0.986860\pi$
$59$	− 4.86973i	− 0.0825377i	0.999148	−	0.0412689i	$-0.0131400\pi$
$60$	13.7353	0.228922
$61$	38.8479i	0.636850i	0.947948	+	0.318425i	$0.103154\pi$
$61$	38.8479i	0.636850i	−0.947948	+	0.318425i	$0.896846\pi$
$62$	15.2178i	0.245449i
$63$	0	0
$64$	−24.4798	−0.382497
$65$	51.4319	0.791259
$66$	0.0522112i	0 0.000791079i
$67$	−7.00878	−0.104609	−0.0523043	−	0.998631i	$-0.516657\pi$
$67$	−7.00878	−0.104609	−0.0523043	+	0.998631i	$0.516657\pi$
$68$	33.4278i	0.491586i
$69$	76.6266i	1.11053i
$70$	0	0
$71$	−46.8735	−0.660190	−0.330095	−	0.943948i	$-0.607081\pi$
$71$	−46.8735	−0.660190	−0.330095	+	0.943948i	$0.607081\pi$
$72$	15.2468	0.211761
$73$	83.5952i	1.14514i	0.819856	+	0.572570i	$0.194053\pi$
$73$	83.5952i	1.14514i	−0.819856	+	0.572570i	$0.805947\pi$
$74$	28.4764	0.384817
$75$	8.66025i	0.115470i
$76$	4.05843i	0.0534004i
$77$	0	0
$78$	26.8301	0.343975
$79$	20.4794	0.259233	0.129617	−	0.991564i	$-0.458625\pi$
$79$	20.4794	0.259233	0.129617	+	0.991564i	$0.458625\pi$
$80$	− 24.0670i	− 0.300837i
$81$	9.00000	0.111111
$82$	25.7793i	0.314382i
$83$	125.683i	1.51425i	0.653271	+	0.757124i	$0.273396\pi$
$83$	125.683i	1.51425i	−0.653271	+	0.757124i	$0.726604\pi$
$84$	0	0
$85$	−21.0766	−0.247960
$86$	51.5349	0.599243
$87$	− 91.7993i	− 1.05516i
$88$	0.227481	0.00258502
$89$	− 46.7024i	− 0.524746i	−0.964966	−	0.262373i	$-0.915495\pi$
$89$	− 46.7024i	− 0.524746i	0.964966	−	0.262373i	$-0.0845051\pi$
$90$	4.51773i	0.0501970i
$91$	0	0
$92$	156.896	1.70539
$93$	39.1380	0.420839
$94$	18.2802i	0.194470i
$95$	−2.55888	−0.0269356
$96$	− 47.7657i	− 0.497560i
$97$	3.11494i	0.0321128i	0.999871	+	0.0160564i	$0.00511112\pi$
$97$	3.11494i	0.0321128i	−0.999871	+	0.0160564i	$0.994889\pi$
$98$	0	0
$99$	0.134280	0.00135636
$100$	17.7322	0.177322
$101$	− 89.4379i	− 0.885523i	−0.896639	−	0.442762i	$-0.853999\pi$
$101$	− 89.4379i	− 0.885523i	0.896639	−	0.442762i	$-0.146001\pi$
$102$	−10.9949	−0.107793
$103$	91.3813i	0.887197i	0.896226	+	0.443598i	$0.146298\pi$
$103$	91.3813i	0.887197i	−0.896226	+	0.443598i	$0.853702\pi$
$104$	− 116.897i	− 1.12401i
$105$	0	0
$106$	12.7876	0.120637
$107$	−105.103	−0.982271	−0.491136	−	0.871083i	$-0.663418\pi$
$107$	−105.103	−0.982271	−0.491136	+	0.871083i	$0.663418\pi$
$108$	− 18.4279i	− 0.170629i
$109$	55.7014	0.511022	0.255511	−	0.966806i	$-0.417756\pi$
$109$	55.7014	0.511022	0.255511	+	0.966806i	$0.417756\pi$
$110$	0.0674044i	0 0.000612767i
$111$	− 73.2373i	− 0.659795i
$112$	0	0
$113$	−5.25425	−0.0464978	−0.0232489	−	0.999730i	$-0.507401\pi$
$113$	−5.25425	−0.0464978	−0.0232489	+	0.999730i	$0.507401\pi$
$114$	−1.33487	−0.0117094
$115$	98.9245i	0.860213i
$116$	−187.963	−1.62037
$117$	− 69.0031i	− 0.589770i
$118$	− 3.27958i	− 0.0277931i
$119$	0	0
$120$	19.6835	0.164029
$121$	−120.998	−0.999983
$122$	26.1626i	0.214448i
$123$	66.3007	0.539030
$124$	− 80.1367i	− 0.646264i
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	−5.54989	−0.0436999	−0.0218500	−	0.999761i	$-0.506956\pi$
$127$	−5.54989	−0.0436999	−0.0218500	+	0.999761i	$0.506956\pi$
$128$	−126.797	−0.990598
$129$	− 132.540i	− 1.02744i
$130$	34.6375	0.266442
$131$	166.521i	1.27115i	0.772037	+	0.635577i	$0.219238\pi$
$131$	166.521i	1.27115i	−0.772037	+	0.635577i	$0.780762\pi$
$132$	− 0.274943i	− 0.00208290i
$133$	0	0
$134$	−4.72016	−0.0352251
$135$	11.6190	0.0860663
$136$	47.9040i	0.352235i
$137$	72.9463	0.532455	0.266227	−	0.963910i	$-0.414223\pi$
$137$	72.9463	0.532455	0.266227	+	0.963910i	$0.414223\pi$
$138$	51.6052i	0.373951i
$139$	114.994i	0.827292i	0.910438	+	0.413646i	$0.135745\pi$
$139$	114.994i	0.827292i	−0.910438	+	0.413646i	$0.864255\pi$
$140$	0	0
$141$	47.0140	0.333433
$142$	−31.5676	−0.222307
$143$	− 1.02952i	− 0.00719947i
$144$	−32.2892	−0.224231
$145$	− 118.512i	− 0.817327i
$146$	56.2983i	0.385605i
$147$	0	0
$148$	−149.956	−1.01322
$149$	72.7457	0.488226	0.244113	−	0.969747i	$-0.421503\pi$
$149$	72.7457	0.488226	0.244113	+	0.969747i	$0.421503\pi$
$150$	5.83237i	0.0388824i
$151$	−127.129	−0.841911	−0.420956	−	0.907081i	$-0.638305\pi$
$151$	−127.129	−0.841911	−0.420956	+	0.907081i	$0.638305\pi$
$152$	5.81596i	0.0382629i
$153$	28.2772i	0.184818i
$154$	0	0
$155$	50.5270	0.325980
$156$	−141.287	−0.905684
$157$	151.065i	0.962200i	0.876666	+	0.481100i	$0.159763\pi$
$157$	151.065i	0.962200i	−0.876666	+	0.481100i	$0.840237\pi$
$158$	13.7921	0.0872920
$159$	− 32.8878i	− 0.206841i
$160$	− 61.6653i	− 0.385408i
$161$	0	0
$162$	6.06117	0.0374146
$163$	−59.9278	−0.367655	−0.183828	−	0.982959i	$-0.558849\pi$
$163$	−59.9278	−0.367655	−0.183828	+	0.982959i	$0.558849\pi$
$164$	− 135.754i	− 0.827766i
$165$	0.173354	0.00105063
$166$	84.6426i	0.509895i
$167$	− 224.089i	− 1.34185i	−0.741526	−	0.670924i	$-0.765898\pi$
$167$	− 224.089i	− 1.34185i	0.741526	−	0.670924i	$-0.234102\pi$
$168$	0	0
$169$	−360.047	−2.13046
$170$	−14.1943	−0.0834959
$171$	3.43310i	0.0200766i
$172$	−271.382	−1.57780
$173$	− 190.618i	− 1.10184i	−0.834558	−	0.550920i	$-0.814277\pi$
$173$	− 190.618i	− 1.10184i	0.834558	−	0.550920i	$-0.185723\pi$
$174$	− 61.8235i	− 0.355307i
$175$	0	0
$176$	−0.481754	−0.00273724
$177$	−8.43461	−0.0476532
$178$	− 31.4524i	− 0.176699i
$179$	217.862	1.21711	0.608553	−	0.793513i	$-0.291750\pi$
$179$	217.862	1.21711	0.608553	+	0.793513i	$0.291750\pi$
$180$	− 23.7903i	− 0.132168i
$181$	39.0804i	0.215914i	0.994156	+	0.107957i	$0.0344309\pi$
$181$	39.0804i	0.215914i	−0.994156	+	0.107957i	$0.965569\pi$
$182$	0	0
$183$	67.2865	0.367686
$184$	224.841	1.22196
$185$	− 94.5489i	− 0.511075i
$186$	26.3580	0.141710
$187$	0.421895i	0.00225612i
$188$	− 96.2632i	− 0.512038i
$189$	0	0
$190$	−1.72331	−0.00907006
$191$	189.517	0.992236	0.496118	−	0.868255i	$-0.334758\pi$
$191$	189.517	0.992236	0.496118	+	0.868255i	$0.334758\pi$
$192$	42.4003i	0.220835i
$193$	−273.141	−1.41524	−0.707619	−	0.706594i	$-0.750231\pi$
$193$	−273.141	−1.41524	−0.707619	+	0.706594i	$0.750231\pi$
$194$	2.09780i	0.0108134i
$195$	− 89.0826i	− 0.456834i
$196$	0	0
$197$	198.898	1.00963	0.504817	−	0.863226i	$-0.331560\pi$
$197$	198.898	1.00963	0.504817	+	0.863226i	$0.331560\pi$
$198$	0.0904325	0.000456730 0
$199$	38.3746i	0.192837i	0.995341	+	0.0964185i	$0.0307387\pi$
$199$	38.3746i	0.192837i	−0.995341	+	0.0964185i	$0.969261\pi$
$200$	25.4113	0.127056
$201$	12.1396i	0.0603958i
$202$	− 60.2331i	− 0.298184i
$203$	0	0
$204$	57.8987	0.283817
$205$	85.5939	0.417531
$206$	61.5420i	0.298747i
$207$	132.721	0.641165
$208$	247.562i	1.19020i
$209$	0.0512217i	0 0.000245080i
$210$	0	0
$211$	−127.283	−0.603238	−0.301619	−	0.953429i	$-0.597527\pi$
$211$	−127.283	−0.603238	−0.301619	+	0.953429i	$0.597527\pi$
$212$	−67.3391	−0.317637
$213$	81.1872i	0.381161i
$214$	−70.7830	−0.330762
$215$	− 171.109i	− 0.795855i
$216$	− 26.4082i	− 0.122260i
$217$	0	0
$218$	37.5129	0.172077
$219$	144.791	0.661147
$220$	− 0.354950i	− 0.00161341i
$221$	216.801	0.981002
$222$	− 49.3226i	− 0.222174i
$223$	293.558i	1.31641i	0.752841	+	0.658203i	$0.228683\pi$
$223$	293.558i	1.31641i	−0.752841	+	0.658203i	$0.771317\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	−3.53855	−0.0156573
$227$	215.480i	0.949252i	0.880187	+	0.474626i	$0.157417\pi$
$227$	215.480i	0.949252i	−0.880187	+	0.474626i	$0.842583\pi$
$228$	7.02941	0.0308308
$229$	− 144.266i	− 0.629984i	−0.949094	−	0.314992i	$-0.897998\pi$
$229$	− 144.266i	− 0.629984i	0.949094	−	0.314992i	$-0.102002\pi$
$230$	66.6221i	0.289661i
$231$	0	0
$232$	−269.361	−1.16104
$233$	−286.433	−1.22932	−0.614662	−	0.788790i	$-0.710708\pi$
$233$	−286.433	−1.22932	−0.614662	+	0.788790i	$0.710708\pi$
$234$	− 46.4711i	− 0.198594i
$235$	60.6948	0.258276
$236$	17.2702i	0.0731789i
$237$	− 35.4714i	− 0.149668i
$238$	0	0
$239$	−413.420	−1.72979	−0.864895	−	0.501954i	$-0.832615\pi$
$239$	−413.420	−1.72979	−0.864895	+	0.501954i	$0.832615\pi$
$240$	−41.6852	−0.173688
$241$	295.895i	1.22778i	0.789392	+	0.613890i	$0.210396\pi$
$241$	295.895i	1.22778i	−0.789392	+	0.613890i	$0.789604\pi$
$242$	−81.4877	−0.336726
$243$	− 15.5885i	− 0.0641500i
$244$	− 137.772i	− 0.564639i
$245$	0	0
$246$	44.6511	0.181509
$247$	26.3216	0.106565
$248$	− 114.840i	− 0.463066i
$249$	217.689	0.874251
$250$	7.52955i	0.0301182i
$251$	− 311.712i	− 1.24188i	−0.783858	−	0.620940i	$-0.786751\pi$
$251$	− 311.712i	− 1.24188i	0.783858	−	0.620940i	$-0.213249\pi$
$252$	0	0
$253$	1.98020	0.00782686
$254$	−3.73765	−0.0147152
$255$	36.5057i	0.143160i
$256$	12.5264	0.0489313
$257$	144.724i	0.563128i	0.959542	+	0.281564i	$0.0908531\pi$
$257$	144.724i	0.563128i	−0.959542	+	0.281564i	$0.909147\pi$
$258$	− 89.2610i	− 0.345973i
$259$	0	0
$260$	−182.400	−0.701540
$261$	−159.001	−0.609200
$262$	112.146i	0.428038i
$263$	−229.665	−0.873251	−0.436626	−	0.899643i	$-0.643827\pi$
$263$	−229.665	−0.873251	−0.436626	+	0.899643i	$0.643827\pi$
$264$	− 0.394009i	− 0.00149246i
$265$	− 42.4579i	− 0.160219i
$266$	0	0
$267$	−80.8910	−0.302962
$268$	24.8563	0.0927473
$269$	424.361i	1.57755i	0.614681	+	0.788776i	$0.289285\pi$
$269$	424.361i	1.57755i	−0.614681	+	0.788776i	$0.710715\pi$
$270$	7.82494	0.0289813
$271$	− 291.804i	− 1.07677i	−0.842699	−	0.538385i	$-0.819035\pi$
$271$	− 291.804i	− 1.07677i	0.842699	−	0.538385i	$-0.180965\pi$
$272$	− 101.450i	− 0.372977i
$273$	0	0
$274$	49.1267	0.179294
$275$	0.223800	0.000813817 0
$276$	− 271.752i	− 0.984609i
$277$	−202.901	−0.732494	−0.366247	−	0.930518i	$-0.619357\pi$
$277$	−202.901	−0.732494	−0.366247	+	0.930518i	$0.619357\pi$
$278$	77.4439i	0.278575i
$279$	− 67.7890i	− 0.242971i
$280$	0	0
$281$	254.325	0.905071	0.452536	−	0.891746i	$-0.350520\pi$
$281$	254.325	0.905071	0.452536	+	0.891746i	$0.350520\pi$
$282$	31.6622	0.112277
$283$	443.704i	1.56786i	0.620849	+	0.783930i	$0.286788\pi$
$283$	443.704i	1.56786i	−0.620849	+	0.783930i	$0.713212\pi$
$284$	166.234	0.585332
$285$	4.43211i	0.0155513i
$286$	− 0.693347i	− 0.00242429i
$287$	0	0
$288$	−82.7327	−0.287266
$289$	200.156	0.692580
$290$	− 79.8138i	− 0.275220i
$291$	5.39523	0.0185403
$292$	− 296.466i	− 1.01529i
$293$	− 223.513i	− 0.762845i	−0.924401	−	0.381422i	$-0.875434\pi$
$293$	− 223.513i	− 0.762845i	0.924401	−	0.381422i	$-0.124566\pi$
$294$	0	0
$295$	−10.8890	−0.0369120
$296$	−214.896	−0.726000
$297$	− 0.232579i	0 0.000783095i
$298$	48.9916	0.164401
$299$	− 1017.57i	− 3.40326i
$300$	− 30.7131i	− 0.102377i
$301$	0	0
$302$	−85.6165	−0.283498
$303$	−154.911	−0.511257
$304$	− 12.3169i	− 0.0405161i
$305$	86.8664	0.284808
$306$	19.0437i	0.0622342i
$307$	− 47.3887i	− 0.154361i	−0.997017	−	0.0771803i	$-0.975408\pi$
$307$	− 47.3887i	− 0.154361i	0.997017	−	0.0771803i	$-0.0245917\pi$
$308$	0	0
$309$	158.277	0.512223
$310$	34.0281	0.109768
$311$	− 362.108i	− 1.16433i	−0.813069	−	0.582167i	$-0.802205\pi$
$311$	− 362.108i	− 1.16433i	0.813069	−	0.582167i	$-0.197795\pi$
$312$	−202.472	−0.648948
$313$	393.614i	1.25755i	0.777586	+	0.628777i	$0.216444\pi$
$313$	393.614i	1.25755i	−0.777586	+	0.628777i	$0.783556\pi$
$314$	101.737i	0.324003i
$315$	0	0
$316$	−72.6291	−0.229839
$317$	577.577	1.82201	0.911004	−	0.412397i	$-0.135308\pi$
$317$	577.577	1.82201	0.911004	+	0.412397i	$0.135308\pi$
$318$	− 22.1487i	− 0.0696500i
$319$	−2.37229	−0.00743665
$320$	54.7385i	0.171058i
$321$	182.044i	0.567114i
$322$	0	0
$323$	−10.7865	−0.0333947
$324$	−31.9180	−0.0985124
$325$	− 115.005i	− 0.353862i
$326$	−40.3592	−0.123801
$327$	− 96.4776i	− 0.295039i
$328$	− 194.542i	− 0.593117i
$329$	0	0
$330$	0.116748	0.000353781 0
$331$	−183.595	−0.554667	−0.277333	−	0.960774i	$-0.589451\pi$
$331$	−183.595	−0.554667	−0.277333	+	0.960774i	$0.589451\pi$
$332$	− 445.727i	− 1.34255i
$333$	−126.851	−0.380933
$334$	− 150.916i	− 0.451843i
$335$	15.6721i	0.0467824i
$336$	0	0
$337$	−205.885	−0.610934	−0.305467	−	0.952203i	$-0.598813\pi$
$337$	−205.885	−0.610934	−0.305467	+	0.952203i	$0.598813\pi$
$338$	−242.479	−0.717393
$339$	9.10063i	0.0268455i
$340$	74.7469	0.219844
$341$	− 1.01141i	− 0.00296601i
$342$	2.31207i	0.00676043i
$343$	0	0
$344$	−388.905	−1.13054
$345$	171.342	0.496644
$346$	− 128.374i	− 0.371025i
$347$	199.451	0.574787	0.287394	−	0.957813i	$-0.407211\pi$
$347$	199.451	0.574787	0.287394	+	0.957813i	$0.407211\pi$
$348$	325.561i	0.935521i
$349$	− 391.231i	− 1.12101i	−0.828152	−	0.560503i	$-0.810608\pi$
$349$	− 391.231i	− 1.12101i	0.828152	−	0.560503i	$-0.189392\pi$
$350$	0	0
$351$	−119.517	−0.340504
$352$	−1.23437	−0.00350673
$353$	− 93.6486i	− 0.265293i	−0.991163	−	0.132647i	$-0.957652\pi$
$353$	− 93.6486i	− 0.265293i	0.991163	−	0.132647i	$-0.0423476\pi$
$354$	−5.68040	−0.0160463
$355$	104.812i	0.295246i
$356$	165.628i	0.465246i
$357$	0	0
$358$	146.722	0.409838
$359$	147.752	0.411565	0.205782	−	0.978598i	$-0.434026\pi$
$359$	147.752	0.411565	0.205782	+	0.978598i	$0.434026\pi$
$360$	− 34.0928i	− 0.0947023i
$361$	359.690	0.996372
$362$	26.3192i	0.0727051i
$363$	209.575i	0.577341i
$364$	0	0
$365$	186.925	0.512122
$366$	45.3150	0.123811
$367$	− 82.6171i	− 0.225115i	−0.993645	−	0.112557i	$-0.964096\pi$
$367$	− 82.6171i	− 0.225115i	0.993645	−	0.112557i	$-0.0359042\pi$
$368$	−476.163	−1.29392
$369$	− 114.836i	− 0.311209i
$370$	− 63.6753i	− 0.172095i
$371$	0	0
$372$	−138.801	−0.373121
$373$	−342.651	−0.918635	−0.459318	−	0.888272i	$-0.651906\pi$
$373$	−342.651	−0.918635	−0.459318	+	0.888272i	$0.651906\pi$
$374$	0.284131i	0 0.000759708i
$375$	19.3649	0.0516398
$376$	− 137.950i	− 0.366890i
$377$	1219.06i	3.23359i
$378$	0	0
$379$	355.679	0.938467	0.469233	−	0.883074i	$-0.344530\pi$
$379$	355.679	0.938467	0.469233	+	0.883074i	$0.344530\pi$
$380$	9.07493	0.0238814
$381$	9.61269i	0.0252302i
$382$	127.633	0.334118
$383$	166.988i	0.435999i	0.975949	+	0.218000i	$0.0699532\pi$
$383$	166.988i	0.435999i	−0.975949	+	0.218000i	$0.930047\pi$
$384$	219.618i	0.571922i
$385$	0	0
$386$	−183.950	−0.476556
$387$	−229.567	−0.593195
$388$	− 11.0470i	− 0.0284715i
$389$	159.290	0.409487	0.204744	−	0.978816i	$-0.434364\pi$
$389$	159.290	0.409487	0.204744	+	0.978816i	$0.434364\pi$
$390$	− 59.9939i	− 0.153830i
$391$	416.998i	1.06649i
$392$	0	0
$393$	288.423	0.733901
$394$	133.951	0.339976
$395$	− 45.7934i	− 0.115933i
$396$	−0.476216	−0.00120257
$397$	589.303i	1.48439i	0.670183	+	0.742196i	$0.266216\pi$
$397$	589.303i	1.48439i	−0.670183	+	0.742196i	$0.733784\pi$
$398$	25.8439i	0.0649344i
$399$	0	0
$400$	−53.8154	−0.134538
$401$	−166.307	−0.414731	−0.207365	−	0.978264i	$-0.566489\pi$
$401$	−166.307	−0.414731	−0.207365	+	0.978264i	$0.566489\pi$
$402$	8.17555i	0.0203372i
$403$	−519.739	−1.28968
$404$	317.187i	0.785115i
$405$	− 20.1246i	− 0.0496904i
$406$	0	0
$407$	−1.89261	−0.00465014
$408$	82.9721	0.203363
$409$	− 219.095i	− 0.535685i	−0.963463	−	0.267843i	$-0.913689\pi$
$409$	− 219.095i	− 0.535685i	0.963463	−	0.267843i	$-0.0863108\pi$
$410$	57.6444	0.140596
$411$	− 126.347i	− 0.307413i
$412$	− 324.079i	− 0.786599i
$413$	0	0
$414$	89.3829	0.215901
$415$	281.035	0.677192
$416$	634.312i	1.52479i
$417$	199.175	0.477637
$418$	0.0344960i	0 8.25262e-5i
$419$	− 554.704i	− 1.32388i	−0.749558	−	0.661938i	$-0.769734\pi$
$419$	− 554.704i	− 1.32388i	0.749558	−	0.661938i	$-0.230266\pi$
$420$	0	0
$421$	642.342	1.52575	0.762876	−	0.646545i	$-0.223787\pi$
$421$	642.342	1.52575	0.762876	+	0.646545i	$0.223787\pi$
$422$	−85.7206	−0.203129
$423$	− 81.4307i	− 0.192507i
$424$	−96.5007	−0.227596
$425$	47.1286i	0.110891i
$426$	54.6767i	0.128349i
$427$	0	0
$428$	372.742	0.870893
$429$	−1.78319	−0.00415661
$430$	− 115.236i	− 0.267990i
$431$	75.3321	0.174785	0.0873923	−	0.996174i	$-0.472147\pi$
$431$	75.3321	0.174785	0.0873923	+	0.996174i	$0.472147\pi$
$432$	55.9266i	0.129460i
$433$	− 353.064i	− 0.815391i	−0.913118	−	0.407695i	$-0.866333\pi$
$433$	− 353.064i	− 0.815391i	0.913118	−	0.407695i	$-0.133667\pi$
$434$	0	0
$435$	−205.270	−0.471884
$436$	−197.542	−0.453078
$437$	50.6272i	0.115852i
$438$	97.5115	0.222629
$439$	271.946i	0.619466i	0.950824	+	0.309733i	$0.100240\pi$
$439$	271.946i	0.619466i	−0.950824	+	0.309733i	$0.899760\pi$
$440$	− 0.508664i	− 0.00115605i
$441$	0	0
$442$	146.008	0.330335
$443$	−110.241	−0.248851	−0.124425	−	0.992229i	$-0.539709\pi$
$443$	−110.241	−0.248851	−0.124425	+	0.992229i	$0.539709\pi$
$444$	259.732i	0.584982i
$445$	−104.430	−0.234674
$446$	197.701i	0.443275i
$447$	− 125.999i	− 0.281878i
$448$	0	0
$449$	59.1007	0.131627	0.0658137	−	0.997832i	$-0.479036\pi$
$449$	59.1007	0.131627	0.0658137	+	0.997832i	$0.479036\pi$
$450$	10.1020	0.0224488
$451$	− 1.71335i	− 0.00379901i
$452$	18.6339	0.0412255
$453$	220.193i	0.486078i
$454$	145.118i	0.319643i
$455$	0	0
$456$	10.0735	0.0220911
$457$	−205.277	−0.449184	−0.224592	−	0.974453i	$-0.572105\pi$
$457$	−205.277	−0.449184	−0.224592	+	0.974453i	$0.572105\pi$
$458$	− 97.1581i	− 0.212136i
$459$	48.9775	0.106705
$460$	− 350.831i	− 0.762675i
$461$	− 466.172i	− 1.01122i	−0.862762	−	0.505610i	$-0.831268\pi$
$461$	− 466.172i	− 1.01122i	0.862762	−	0.505610i	$-0.168732\pi$
$462$	0	0
$463$	191.705	0.414051	0.207025	−	0.978336i	$-0.433622\pi$
$463$	191.705	0.414051	0.207025	+	0.978336i	$0.433622\pi$
$464$	570.447	1.22941
$465$	− 87.5153i	− 0.188205i
$466$	−192.902	−0.413953
$467$	843.233i	1.80564i	0.430020	+	0.902819i	$0.358507\pi$
$467$	843.233i	1.80564i	−0.430020	+	0.902819i	$0.641493\pi$
$468$	244.716i	0.522897i
$469$	0	0
$470$	40.8758	0.0869697
$471$	261.653	0.555526
$472$	24.7492i	0.0524347i
$473$	−3.42513	−0.00724128
$474$	− 23.8887i	− 0.0503981i
$475$	5.72183i	0.0120460i
$476$	0	0
$477$	−56.9633	−0.119420
$478$	−278.423	−0.582475
$479$	− 284.680i	− 0.594322i	−0.954827	−	0.297161i	$-0.903960\pi$
$479$	− 284.680i	− 0.594322i	0.954827	−	0.297161i	$-0.0960399\pi$
$480$	−106.807	−0.222516
$481$	972.566i	2.02197i
$482$	199.274i	0.413432i
$483$	0	0
$484$	429.113	0.886597
$485$	6.96521	0.0143613
$486$	− 10.4983i	− 0.0216014i
$487$	−195.646	−0.401736	−0.200868	−	0.979618i	$-0.564376\pi$
$487$	−195.646	−0.401736	−0.200868	+	0.979618i	$0.564376\pi$
$488$	− 197.435i	− 0.404579i
$489$	103.798i	0.212266i
$490$	0	0
$491$	745.464	1.51826	0.759128	−	0.650941i	$-0.225625\pi$
$491$	745.464	1.51826	0.759128	+	0.650941i	$0.225625\pi$
$492$	−235.132	−0.477911
$493$	− 499.567i	− 1.01332i
$494$	17.7266	0.0358839
$495$	− 0.300259i	0 0.000606583i
$496$	243.206i	0.490335i
$497$	0	0
$498$	146.605	0.294388
$499$	91.9495	0.184267	0.0921337	−	0.995747i	$-0.470631\pi$
$499$	91.9495	0.184267	0.0921337	+	0.995747i	$0.470631\pi$
$500$	− 39.6505i	− 0.0793010i
$501$	−388.133	−0.774717
$502$	− 209.927i	− 0.418181i
$503$	672.220i	1.33642i	0.743972	+	0.668211i	$0.232940\pi$
$503$	672.220i	1.33642i	−0.743972	+	0.668211i	$0.767060\pi$
$504$	0	0
$505$	−199.989	−0.396018
$506$	1.33359	0.00263555
$507$	623.620i	1.23002i
$508$	19.6824	0.0387449
$509$	− 325.901i	− 0.640276i	−0.947371	−	0.320138i	$-0.896271\pi$
$509$	− 325.901i	− 0.640276i	0.947371	−	0.320138i	$-0.103729\pi$
$510$	24.5853i	0.0482064i
$511$	0	0
$512$	515.622	1.00707
$513$	5.94630	0.0115912
$514$	97.4662i	0.189623i
$515$	204.335	0.396767
$516$	470.047i	0.910944i
$517$	− 1.21494i	− 0.00234999i
$518$	0	0
$519$	−330.161	−0.636148
$520$	−261.390	−0.502673
$521$	595.189i	1.14240i	0.820812	+	0.571199i	$0.193522\pi$
$521$	595.189i	1.14240i	−0.820812	+	0.571199i	$0.806478\pi$
$522$	−107.081	−0.205137
$523$	50.4015i	0.0963700i	0.998838	+	0.0481850i	$0.0153437\pi$
$523$	50.4015i	0.0963700i	−0.998838	+	0.0481850i	$0.984656\pi$
$524$	− 590.559i	− 1.12702i
$525$	0	0
$526$	−154.671	−0.294051
$527$	212.987	0.404150
$528$	0.834423i	0.00158035i
$529$	1428.21	2.69983
$530$	− 28.5939i	− 0.0539507i
$531$	14.6092i	0.0275126i
$532$	0	0
$533$	−880.451	−1.65188
$534$	−54.4771	−0.102017
$535$	235.017i	0.439285i
$536$	35.6204	0.0664560
$537$	− 377.348i	− 0.702696i
$538$	285.792i	0.531212i
$539$	0	0
$540$	−41.2060	−0.0763074
$541$	936.762	1.73154	0.865769	−	0.500444i	$-0.166830\pi$
$541$	936.762	1.73154	0.865769	+	0.500444i	$0.166830\pi$
$542$	− 196.520i	− 0.362582i
$543$	67.6893	0.124658
$544$	− 259.939i	− 0.477828i
$545$	− 124.552i	− 0.228536i
$546$	0	0
$547$	−3.89041	−0.00711227	−0.00355613	−	0.999994i	$-0.501132\pi$
$547$	−3.89041	−0.00711227	−0.00355613	+	0.999994i	$0.501132\pi$
$548$	−258.700	−0.472081
$549$	− 116.544i	− 0.212283i
$550$	0.150721	0.000274038 0
$551$	− 60.6518i	− 0.110076i
$552$	− 389.436i	− 0.705500i
$553$	0	0
$554$	−136.646	−0.246654
$555$	−163.764	−0.295070
$556$	− 407.818i	− 0.733486i
$557$	386.761	0.694365	0.347183	−	0.937798i	$-0.387138\pi$
$557$	386.761	0.694365	0.347183	+	0.937798i	$0.387138\pi$
$558$	− 45.6534i	− 0.0818162i
$559$	1760.09i	3.14864i
$560$	0	0
$561$	0.730743	0.00130257
$562$	171.279	0.304766
$563$	− 121.245i	− 0.215356i	−0.994186	−	0.107678i	$-0.965659\pi$
$563$	− 121.245i	− 0.215356i	0.994186	−	0.107678i	$-0.0343415\pi$
$564$	−166.733	−0.295625
$565$	11.7489i	0.0207944i
$566$	298.819i	0.527948i
$567$	0	0
$568$	238.223	0.419407
$569$	409.910	0.720405	0.360202	−	0.932874i	$-0.382708\pi$
$569$	409.910	0.720405	0.360202	+	0.932874i	$0.382708\pi$
$570$	2.98486i	0.00523660i
$571$	−575.723	−1.00827	−0.504136	−	0.863625i	$-0.668189\pi$
$571$	−575.723	−1.00827	−0.504136	+	0.863625i	$0.668189\pi$
$572$	3.65115i	0.00638313i
$573$	− 328.253i	− 0.572868i
$574$	0	0
$575$	221.202	0.384699
$576$	73.4394	0.127499
$577$	− 233.543i	− 0.404754i	−0.979308	−	0.202377i	$-0.935133\pi$
$577$	− 233.543i	− 0.404754i	0.979308	−	0.202377i	$-0.0648666\pi$
$578$	134.798	0.233214
$579$	473.094i	0.817088i
$580$	420.298i	0.724652i
$581$	0	0
$582$	3.63349	0.00624311
$583$	−0.849890	−0.00145779
$584$	− 424.852i	− 0.727487i
$585$	−154.296	−0.263753
$586$	− 150.528i	− 0.256874i
$587$	606.882i	1.03387i	0.856024	+	0.516935i	$0.172927\pi$
$587$	606.882i	1.03387i	−0.856024	+	0.516935i	$0.827073\pi$
$588$	0	0
$589$	25.8585	0.0439024
$590$	−7.33337	−0.0124294
$591$	− 344.502i	− 0.582913i
$592$	455.101	0.768752
$593$	810.597i	1.36694i	0.729977	+	0.683472i	$0.239531\pi$
$593$	810.597i	1.36694i	−0.729977	+	0.683472i	$0.760469\pi$
$594$	− 0.156634i	0 0.000263693i
$595$	0	0
$596$	−257.989	−0.432867
$597$	66.4667	0.111335
$598$	− 685.299i	− 1.14599i
$599$	−1022.78	−1.70748	−0.853738	−	0.520703i	$-0.825670\pi$
$599$	−1022.78	−1.70748	−0.853738	+	0.520703i	$0.825670\pi$
$600$	− 44.0136i	− 0.0733561i
$601$	147.884i	0.246063i	0.992403	+	0.123032i	$0.0392617\pi$
$601$	147.884i	0.246063i	−0.992403	+	0.123032i	$0.960738\pi$
$602$	0	0
$603$	21.0263	0.0348696
$604$	450.855	0.746448
$605$	270.560i	0.447206i
$606$	−104.327	−0.172157
$607$	− 941.647i	− 1.55131i	−0.631156	−	0.775656i	$-0.717419\pi$
$607$	− 941.647i	− 1.55131i	0.631156	−	0.775656i	$-0.282581\pi$
$608$	− 31.5588i	− 0.0519060i
$609$	0	0
$610$	58.5014	0.0959039
$611$	−624.330	−1.02182
$612$	− 100.284i	− 0.163862i
$613$	360.135	0.587496	0.293748	−	0.955883i	$-0.405097\pi$
$613$	360.135	0.587496	0.293748	+	0.955883i	$0.405097\pi$
$614$	− 31.9146i	− 0.0519781i
$615$	− 148.253i	− 0.241062i
$616$	0	0
$617$	769.687	1.24747	0.623734	−	0.781637i	$-0.285615\pi$
$617$	769.687	1.24747	0.623734	+	0.781637i	$0.285615\pi$
$618$	106.594	0.172482
$619$	− 985.586i	− 1.59222i	−0.605150	−	0.796111i	$-0.706887\pi$
$619$	− 985.586i	− 1.59222i	0.605150	−	0.796111i	$-0.293113\pi$
$620$	−179.191	−0.289018
$621$	− 229.880i	− 0.370177i
$622$	− 243.866i	− 0.392068i
$623$	0	0
$624$	428.790	0.687163
$625$	25.0000	0.0400000
$626$	265.085i	0.423458i
$627$	0.0887186	0.000141497 0
$628$	− 535.745i	− 0.853098i
$629$	− 398.554i	− 0.633630i
$630$	0	0
$631$	−89.7688	−0.142264	−0.0711322	−	0.997467i	$-0.522661\pi$
$631$	−89.7688	−0.142264	−0.0711322	+	0.997467i	$0.522661\pi$
$632$	−104.082	−0.164686
$633$	220.461i	0.348279i
$634$	388.977	0.613528
$635$	12.4099i	0.0195432i
$636$	116.635i	0.183388i
$637$	0	0
$638$	−1.59765	−0.00250416
$639$	140.620	0.220063
$640$	283.526i	0.443009i
$641$	−428.333	−0.668226	−0.334113	−	0.942533i	$-0.608437\pi$
$641$	−428.333	−0.668226	−0.334113	+	0.942533i	$0.608437\pi$
$642$	122.600i	0.190965i
$643$	111.498i	0.173403i	0.996234	+	0.0867015i	$0.0276326\pi$
$643$	111.498i	0.173403i	−0.996234	+	0.0867015i	$0.972367\pi$
$644$	0	0
$645$	−296.369	−0.459487
$646$	−7.26430	−0.0112451
$647$	− 288.714i	− 0.446234i	−0.974792	−	0.223117i	$-0.928377\pi$
$647$	− 288.714i	− 0.446234i	0.974792	−	0.223117i	$-0.0716233\pi$
$648$	−45.7403	−0.0705869
$649$	0.217968i	0 0.000335853i
$650$	− 77.4518i	− 0.119157i
$651$	0	0
$652$	212.531	0.325967
$653$	853.695	1.30734	0.653672	−	0.756778i	$-0.273228\pi$
$653$	853.695	1.30734	0.653672	+	0.756778i	$0.273228\pi$
$654$	− 64.9742i	− 0.0993489i
$655$	372.353	0.568477
$656$	411.997i	0.628044i
$657$	− 250.786i	− 0.381713i
$658$	0	0
$659$	−288.693	−0.438077	−0.219039	−	0.975716i	$-0.570292\pi$
$659$	−288.693	−0.438077	−0.219039	+	0.975716i	$0.570292\pi$
$660$	−0.614792	−0.000931503 0
$661$	210.579i	0.318576i	0.987232	+	0.159288i	$0.0509199\pi$
$661$	210.579i	0.318576i	−0.987232	+	0.159288i	$0.949080\pi$
$662$	−123.644	−0.186774
$663$	− 375.511i	− 0.566382i
$664$	− 638.751i	− 0.961974i
$665$	0	0
$666$	−85.4293	−0.128272
$667$	−2344.76	−3.51538
$668$	794.719i	1.18970i
$669$	508.458	0.760027
$670$	10.5546i	0.0157531i
$671$	− 1.73883i	− 0.00259140i
$672$	0	0
$673$	760.139	1.12948	0.564739	−	0.825269i	$-0.308977\pi$
$673$	760.139	1.12948	0.564739	+	0.825269i	$0.308977\pi$
$674$	−138.656	−0.205721
$675$	− 25.9808i	− 0.0384900i
$676$	1276.89	1.88889
$677$	− 188.520i	− 0.278464i	−0.990260	−	0.139232i	$-0.955537\pi$
$677$	− 188.520i	− 0.278464i	0.990260	−	0.139232i	$-0.0444633\pi$
$678$	6.12894i	0.00903974i
$679$	0	0
$680$	107.117	0.157524
$681$	373.223	0.548051
$682$	− 0.681148i	0 0.000998751i
$683$	520.486	0.762058	0.381029	−	0.924563i	$-0.375570\pi$
$683$	520.486	0.762058	0.381029	+	0.924563i	$0.375570\pi$
$684$	− 12.1753i	− 0.0178001i
$685$	− 163.113i	− 0.238121i
$686$	0	0
$687$	−249.877	−0.363721
$688$	823.614	1.19711
$689$	436.738i	0.633872i
$690$	115.393	0.167236
$691$	− 681.983i	− 0.986951i	−0.869760	−	0.493475i	$-0.835726\pi$
$691$	− 681.983i	− 0.986951i	0.869760	−	0.493475i	$-0.164274\pi$
$692$	676.018i	0.976904i
$693$	0	0
$694$	134.323	0.193549
$695$	257.133	0.369976
$696$	466.548i	0.670327i
$697$	360.805	0.517654
$698$	− 263.480i	− 0.377478i
$699$	496.116i	0.709751i
$700$	0	0
$701$	−946.473	−1.35018	−0.675088	−	0.737737i	$-0.735894\pi$
$701$	−946.473	−1.35018	−0.675088	+	0.737737i	$0.735894\pi$
$702$	−80.4902	−0.114658
$703$	− 48.3879i	− 0.0688306i
$704$	1.09571	0.00155641
$705$	− 105.127i	− 0.149116i
$706$	− 63.0689i	− 0.0893327i
$707$	0	0
$708$	29.9129	0.0422499
$709$	−1009.57	−1.42393	−0.711967	−	0.702213i	$-0.752196\pi$
$709$	−1009.57	−1.42393	−0.711967	+	0.702213i	$0.752196\pi$
$710$	70.5873i	0.0994187i
$711$	−61.4382	−0.0864110
$712$	237.354i	0.333362i
$713$	− 999.671i	− 1.40206i
$714$	0	0
$715$	−2.30209	−0.00321970
$716$	−772.636	−1.07910
$717$	716.064i	0.998694i
$718$	99.5054	0.138587
$719$	− 904.572i	− 1.25810i	−0.777366	−	0.629049i	$-0.783445\pi$
$719$	− 904.572i	− 1.25810i	0.777366	−	0.629049i	$-0.216555\pi$
$720$	72.2009i	0.100279i
$721$	0	0
$722$	242.238	0.335510
$723$	512.505	0.708859
$724$	− 138.597i	− 0.191432i
$725$	−265.002	−0.365520
$726$	141.141i	0.194409i
$727$	− 535.515i	− 0.736609i	−0.929705	−	0.368304i	$-0.879938\pi$
$727$	− 535.515i	− 0.736609i	0.929705	−	0.368304i	$-0.120062\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	125.887	0.172448
$731$	− 721.277i	− 0.986699i
$732$	−238.628	−0.325994
$733$	753.537i	1.02802i	0.857785	+	0.514009i	$0.171840\pi$
$733$	753.537i	1.02802i	−0.857785	+	0.514009i	$0.828160\pi$
$734$	− 55.6396i	− 0.0758033i
$735$	0	0
$736$	−1220.04	−1.65767
$737$	0.313712	0.000425661 0
$738$	− 77.3380i	− 0.104794i
$739$	−1092.31	−1.47810	−0.739049	−	0.673652i	$-0.764725\pi$
$739$	−1092.31	−1.47810	−0.739049	+	0.673652i	$0.764725\pi$
$740$	335.313i	0.453125i
$741$	− 45.5903i	− 0.0615254i
$742$	0	0
$743$	−362.303	−0.487622	−0.243811	−	0.969823i	$-0.578398\pi$
$743$	−362.303	−0.487622	−0.243811	+	0.969823i	$0.578398\pi$
$744$	−198.909	−0.267351
$745$	− 162.664i	− 0.218342i
$746$	−230.763	−0.309334
$747$	− 377.048i	− 0.504749i
$748$	− 1.49623i	− 0.00200030i
$749$	0	0
$750$	13.0416	0.0173888
$751$	−672.540	−0.895526	−0.447763	−	0.894152i	$-0.647779\pi$
$751$	−672.540	−0.895526	−0.447763	+	0.894152i	$0.647779\pi$
$752$	292.148i	0.388495i
$753$	−539.901	−0.717000
$754$	820.994i	1.08885i
$755$	284.268i	0.376514i
$756$	0	0
$757$	−368.166	−0.486349	−0.243174	−	0.969983i	$-0.578189\pi$
$757$	−368.166	−0.486349	−0.243174	+	0.969983i	$0.578189\pi$
$758$	239.537	0.316012
$759$	− 3.42980i	− 0.00451884i
$760$	13.0049	0.0171117
$761$	− 653.168i	− 0.858302i	−0.903233	−	0.429151i	$-0.858813\pi$
$761$	− 653.168i	− 0.858302i	0.903233	−	0.429151i	$-0.141187\pi$
$762$	6.47380i	0.00849580i
$763$	0	0
$764$	−672.113	−0.879728
$765$	63.2297	0.0826532
$766$	112.460i	0.146815i
$767$	112.009	0.146035
$768$	− 21.6964i	− 0.0282505i
$769$	− 1393.19i	− 1.81170i	−0.423602	−	0.905848i	$-0.639235\pi$
$769$	− 1393.19i	− 1.81170i	0.423602	−	0.905848i	$-0.360765\pi$
$770$	0	0
$771$	250.669	0.325122
$772$	968.680	1.25477
$773$	1304.60i	1.68772i	0.536567	+	0.843858i	$0.319721\pi$
$773$	1304.60i	1.68772i	−0.536567	+	0.843858i	$0.680279\pi$
$774$	−154.605	−0.199748
$775$	− 112.982i	− 0.145783i
$776$	− 15.8309i	− 0.0204007i
$777$	0	0
$778$	107.276	0.137887
$779$	43.8049	0.0562322
$780$	315.927i	0.405034i
$781$	2.09805	0.00268637
$782$	280.833i	0.359121i
$783$	275.398i	0.351722i
$784$	0	0
$785$	337.793	0.430309
$786$	194.243	0.247128
$787$	210.473i	0.267437i	0.991019	+	0.133719i	$0.0426918\pi$
$787$	210.473i	0.267437i	−0.991019	+	0.133719i	$0.957308\pi$
$788$	−705.381	−0.895154
$789$	397.792i	0.504172i
$790$	− 30.8402i	− 0.0390382i
$791$	0	0
$792$	−0.682444	−0.000861672 0
$793$	−893.541	−1.12679
$794$	396.874i	0.499842i
$795$	−73.5393	−0.0925022
$796$	− 136.093i	− 0.170972i
$797$	254.794i	0.319691i	0.987142	+	0.159845i	$0.0510996\pi$
$797$	254.794i	0.319691i	−0.987142	+	0.159845i	$0.948900\pi$
$798$	0	0
$799$	255.848	0.320210
$800$	−137.888	−0.172360
$801$	140.107i	0.174915i
$802$	−112.002	−0.139653
$803$	− 3.74171i	− 0.00465967i
$804$	− 43.0523i	− 0.0535477i
$805$	0	0
$806$	−350.025	−0.434275
$807$	735.016	0.910800
$808$	454.546i	0.562557i
$809$	−229.366	−0.283518	−0.141759	−	0.989901i	$-0.545276\pi$
$809$	−229.366	−0.283518	−0.141759	+	0.989901i	$0.545276\pi$
$810$	− 13.5532i	− 0.0167323i
$811$	− 1108.59i	− 1.36694i	−0.729978	−	0.683470i	$-0.760470\pi$
$811$	− 1108.59i	− 1.36694i	0.729978	−	0.683470i	$-0.239530\pi$
$812$	0	0
$813$	−505.420	−0.621673
$814$	−1.27460	−0.00156585
$815$	134.003i	0.164420i
$816$	−175.716	−0.215339
$817$	− 87.5694i	− 0.107184i
$818$	− 147.553i	− 0.180382i
$819$	0	0
$820$	−303.554	−0.370188
$821$	867.524	1.05667	0.528333	−	0.849037i	$-0.322817\pi$
$821$	867.524	1.05667	0.528333	+	0.849037i	$0.322817\pi$
$822$	− 85.0899i	− 0.103516i
$823$	984.880	1.19670	0.598348	−	0.801237i	$-0.295824\pi$
$823$	984.880	1.19670	0.598348	+	0.801237i	$0.295824\pi$
$824$	− 464.423i	− 0.563620i
$825$	− 0.387632i	0 0.000469857i
$826$	0	0
$827$	−767.641	−0.928224	−0.464112	−	0.885777i	$-0.653626\pi$
$827$	−767.641	−0.928224	−0.464112	+	0.885777i	$0.653626\pi$
$828$	−470.689	−0.568464
$829$	− 177.337i	− 0.213916i	−0.994264	−	0.106958i	$-0.965889\pi$
$829$	− 177.337i	− 0.213916i	0.994264	−	0.106958i	$-0.0341111\pi$
$830$	189.267	0.228032
$831$	351.434i	0.422905i
$832$	− 563.061i	− 0.676756i
$833$	0	0
$834$	134.137	0.160836
$835$	−501.078	−0.600093
$836$	− 0.181655i	0 0.000217291i
$837$	−117.414	−0.140280
$838$	− 373.573i	− 0.445791i
$839$	− 15.9176i	− 0.0189721i	−0.999955	−	0.00948606i	$-0.996980\pi$
$839$	− 15.9176i	− 0.0189721i	0.999955	−	0.00948606i	$-0.00301955\pi$
$840$	0	0
$841$	1968.04	2.34012
$842$	432.594	0.513769
$843$	− 440.504i	− 0.522543i
$844$	451.403	0.534838
$845$	805.090i	0.952770i
$846$	− 54.8406i	− 0.0648234i
$847$	0	0
$848$	204.367	0.240998
$849$	768.518	0.905204
$850$	31.7394i	0.0373405i
$851$	−1870.64	−2.19817
$852$	− 287.926i	− 0.337942i
$853$	694.629i	0.814336i	0.913353	+	0.407168i	$0.133484\pi$
$853$	694.629i	0.814336i	−0.913353	+	0.407168i	$0.866516\pi$
$854$	0	0
$855$	7.67664	0.00897853
$856$	534.160	0.624019
$857$	− 1502.46i	− 1.75317i	−0.481251	−	0.876583i	$-0.659817\pi$
$857$	− 1502.46i	− 1.75317i	0.481251	−	0.876583i	$-0.340183\pi$
$858$	−1.20091	−0.00139966
$859$	293.797i	0.342022i	0.985269	+	0.171011i	$0.0547034\pi$
$859$	293.797i	0.342022i	−0.985269	+	0.171011i	$0.945297\pi$
$860$	606.828i	0.705614i
$861$	0	0
$862$	50.7334	0.0588555
$863$	−258.118	−0.299094	−0.149547	−	0.988755i	$-0.547782\pi$
$863$	−258.118	−0.299094	−0.149547	+	0.988755i	$0.547782\pi$
$864$	143.297i	0.165853i
$865$	−426.236	−0.492758
$866$	− 237.776i	− 0.274568i
$867$	− 346.680i	− 0.399861i
$868$	0	0
$869$	−0.916657	−0.00105484
$870$	−138.242	−0.158898
$871$	− 161.209i	− 0.185085i
$872$	−283.089	−0.324643
$873$	− 9.34481i	− 0.0107043i
$874$	34.0956i	0.0390109i
$875$	0	0
$876$	−513.494	−0.586180
$877$	1337.49	1.52508	0.762539	−	0.646942i	$-0.223952\pi$
$877$	1337.49	1.52508	0.762539	+	0.646942i	$0.223952\pi$
$878$	183.145i	0.208594i
$879$	−387.137	−0.440428
$880$	1.07724i	0.00122413i
$881$	− 606.188i	− 0.688069i	−0.938957	−	0.344034i	$-0.888206\pi$
$881$	− 606.188i	− 0.688069i	0.938957	−	0.344034i	$-0.111794\pi$
$882$	0	0
$883$	−862.650	−0.976953	−0.488477	−	0.872577i	$-0.662447\pi$
$883$	−862.650	−0.976953	−0.488477	+	0.872577i	$0.662447\pi$
$884$	−768.875	−0.869768
$885$	18.8604i	0.0213111i
$886$	−74.2432	−0.0837960
$887$	923.371i	1.04100i	0.853860	+	0.520502i	$0.174255\pi$
$887$	923.371i	1.04100i	−0.853860	+	0.520502i	$0.825745\pi$
$888$	372.211i	0.419156i
$889$	0	0
$890$	−70.3297	−0.0790221
$891$	−0.402839	−0.000452120 0
$892$	− 1041.09i	− 1.16714i
$893$	31.0622	0.0347840
$894$	− 84.8560i	− 0.0949172i
$895$	− 487.154i	− 0.544306i
$896$	0	0
$897$	−1762.49	−1.96487
$898$	39.8022	0.0443231
$899$	1197.61i	1.33216i
$900$	−53.1967	−0.0591074
$901$	− 178.973i	− 0.198639i
$902$	− 1.15388i	− 0.00127925i
$903$	0	0
$904$	26.7035	0.0295392
$905$	87.3865	0.0965596
$906$	148.292i	0.163678i
$907$	−94.7134	−0.104425	−0.0522125	−	0.998636i	$-0.516627\pi$
$907$	−94.7134	−0.104425	−0.0522125	+	0.998636i	$0.516627\pi$
$908$	− 764.189i	− 0.841618i
$909$	268.314i	0.295174i
$910$	0	0
$911$	−556.948	−0.611359	−0.305679	−	0.952134i	$-0.598884\pi$
$911$	−556.948	−0.611359	−0.305679	+	0.952134i	$0.598884\pi$
$912$	−21.3335	−0.0233920
$913$	− 5.62554i	− 0.00616160i
$914$	−138.247	−0.151254
$915$	− 150.457i	− 0.164434i
$916$	511.633i	0.558551i
$917$	0	0
$918$	32.9846	0.0359309
$919$	−957.161	−1.04152	−0.520762	−	0.853702i	$-0.674352\pi$
$919$	−957.161	−1.04152	−0.520762	+	0.853702i	$0.674352\pi$
$920$	− 502.760i	− 0.546478i
$921$	−82.0797	−0.0891202
$922$	− 313.950i	− 0.340510i
$923$	− 1078.14i	− 1.16808i
$924$	0	0
$925$	−211.418	−0.228560
$926$	129.107	0.139424
$927$	− 274.144i	− 0.295732i
$928$	1461.62	1.57502
$929$	1143.63i	1.23103i	0.788123	+	0.615517i	$0.211053\pi$
$929$	1143.63i	1.23103i	−0.788123	+	0.615517i	$0.788947\pi$
$930$	− 58.9383i	− 0.0633746i
$931$	0	0
$932$	1015.82	1.08993
$933$	−627.189	−0.672228
$934$	567.887i	0.608016i
$935$	0.943385	0.00100897
$936$	350.691i	0.374670i
$937$	− 578.660i	− 0.617567i	−0.951132	−	0.308783i	$-0.900078\pi$
$937$	− 578.660i	− 0.617567i	0.951132	−	0.308783i	$-0.0999218\pi$
$938$	0	0
$939$	681.760	0.726049
$940$	−215.251	−0.228990
$941$	− 1254.22i	− 1.33285i	−0.745571	−	0.666427i	$-0.767823\pi$
$941$	− 1254.22i	− 1.33285i	0.745571	−	0.666427i	$-0.232177\pi$
$942$	176.214	0.187063
$943$	− 1693.47i	− 1.79583i
$944$	− 52.4132i	− 0.0555225i
$945$	0	0
$946$	−2.30670	−0.00243837
$947$	−1756.02	−1.85430	−0.927151	−	0.374688i	$-0.877750\pi$
$947$	−1756.02	−1.85430	−0.927151	+	0.374688i	$0.877750\pi$
$948$	125.797i	0.132698i
$949$	−1922.78	−2.02611
$950$	3.85344i	0.00405626i
$951$	− 1000.39i	− 1.05194i
$952$	0	0
$953$	−1048.32	−1.10002	−0.550011	−	0.835157i	$-0.685376\pi$
$953$	−1048.32	−1.10002	−0.550011	+	0.835157i	$0.685376\pi$
$954$	−38.3627	−0.0402125
$955$	− 423.773i	− 0.443742i
$956$	1466.17	1.53365
$957$	4.10893i	0.00429355i
$958$	− 191.722i	− 0.200127i
$959$	0	0
$960$	94.8099	0.0987603
$961$	450.405	0.468684
$962$	654.988i	0.680860i
$963$	315.309	0.327424
$964$	− 1049.38i	− 1.08856i
$965$	610.762i	0.632914i
$966$	0	0
$967$	−1770.86	−1.83130	−0.915648	−	0.401982i	$-0.868321\pi$
$967$	−1770.86	−1.83130	−0.915648	+	0.401982i	$0.868321\pi$
$968$	614.943	0.635272
$969$	18.6827i	0.0192804i
$970$	4.69082	0.00483589
$971$	− 963.020i	− 0.991781i	−0.868385	−	0.495891i	$-0.834842\pi$
$971$	− 963.020i	− 0.991781i	0.868385	−	0.495891i	$-0.165158\pi$
$972$	55.2836i	0.0568762i
$973$	0	0
$974$	−131.760	−0.135277
$975$	−199.195	−0.204302
$976$	418.122i	0.428404i
$977$	−538.301	−0.550973	−0.275487	−	0.961305i	$-0.588839\pi$
$977$	−538.301	−0.550973	−0.275487	+	0.961305i	$0.588839\pi$
$978$	69.9041i	0.0714766i
$979$	2.09040i	0.00213524i
$980$	0	0
$981$	−167.104	−0.170341
$982$	502.043	0.511245
$983$	− 676.924i	− 0.688630i	−0.938854	−	0.344315i	$-0.888111\pi$
$983$	− 676.924i	− 0.688630i	0.938854	−	0.344315i	$-0.111889\pi$
$984$	−336.957	−0.342436
$985$	− 444.750i	− 0.451522i
$986$	− 336.440i	− 0.341217i
$987$	0	0
$988$	−93.3481	−0.0944819
$989$	−3385.37	−3.42302
$990$	− 0.202213i	0 0.000204256i
$991$	1169.80	1.18042	0.590212	−	0.807248i	$-0.299044\pi$
$991$	1169.80	1.18042	0.590212	+	0.807248i	$0.299044\pi$
$992$	623.152i	0.628178i
$993$	317.995i	0.320237i
$994$	0	0
$995$	85.8082	0.0862394
$996$	−772.021	−0.775122
$997$	603.927i	0.605744i	0.953031	+	0.302872i	$0.0979454\pi$
$997$	603.927i	0.605744i	−0.953031	+	0.302872i	$0.902055\pi$
$998$	61.9246	0.0620487
$999$	219.712i	0.219932i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.3.h.a.391.5		8
7.4	even	3		105.3.n.a.61.2	yes	8
7.5	odd	6		105.3.n.a.31.2	✓	8
7.6	odd	2	inner	735.3.h.a.391.6		8
21.5	even	6		315.3.w.a.136.3		8
21.11	odd	6		315.3.w.a.271.3		8
35.4	even	6		525.3.o.l.376.3		8
35.12	even	12		525.3.s.h.199.5		16
35.18	odd	12		525.3.s.h.124.5		16
35.19	odd	6		525.3.o.l.451.3		8
35.32	odd	12		525.3.s.h.124.4		16
35.33	even	12		525.3.s.h.199.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.n.a.31.2	✓	8	7.5	odd	6
105.3.n.a.61.2	yes	8	7.4	even	3
315.3.w.a.136.3		8	21.5	even	6
315.3.w.a.271.3		8	21.11	odd	6
525.3.o.l.376.3		8	35.4	even	6
525.3.o.l.451.3		8	35.19	odd	6
525.3.s.h.124.4		16	35.32	odd	12
525.3.s.h.124.5		16	35.18	odd	12
525.3.s.h.199.4		16	35.33	even	12
525.3.s.h.199.5		16	35.12	even	12
735.3.h.a.391.5		8	1.1	even	1	trivial
735.3.h.a.391.6		8	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	735.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.673464 q^{2} -1.73205i q^{3} -3.54645 q^{4} -2.23607i q^{5} -1.16647i q^{6} -5.08226 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+0.673464 q^{2} -1.73205i q^{3} -3.54645 q^{4} -2.23607i q^{5} -1.16647i q^{6} -5.08226 q^{8} -3.00000 q^{9} -1.50591i q^{10} -0.0447599 q^{11} +6.14263i q^{12} +23.0010i q^{13} -3.87298 q^{15} +10.7631 q^{16} -9.42573i q^{17} -2.02039 q^{18} -1.14437i q^{19} +7.93010i q^{20} -0.0301442 q^{22} -44.2404 q^{23} +8.80273i q^{24} -5.00000 q^{25} +15.4904i q^{26} +5.19615i q^{27} +53.0004 q^{29} -2.60831 q^{30} +22.5963i q^{31} +27.5776 q^{32} +0.0775264i q^{33} -6.34788i q^{34} +10.6393 q^{36} +42.2836 q^{37} -0.770689i q^{38} +39.8390 q^{39} +11.3643i q^{40} +38.2787i q^{41} +76.5222 q^{43} +0.158739 q^{44} +6.70820i q^{45} -29.7943 q^{46} +27.1436i q^{47} -18.6422i q^{48} -3.36732 q^{50} -16.3258 q^{51} -81.5719i q^{52} +18.9878 q^{53} +3.49942i q^{54} +0.100086i q^{55} -1.98210 q^{57} +35.6938 q^{58} -4.86973i q^{59} +13.7353 q^{60} +38.8479i q^{61} +15.2178i q^{62} -24.4798 q^{64} +51.4319 q^{65} +0.0522112i q^{66} -7.00878 q^{67} +33.4278i q^{68} +76.6266i q^{69} -46.8735 q^{71} +15.2468 q^{72} +83.5952i q^{73} +28.4764 q^{74} +8.66025i q^{75} +4.05843i q^{76} +26.8301 q^{78} +20.4794 q^{79} -24.0670i q^{80} +9.00000 q^{81} +25.7793i q^{82} +125.683i q^{83} -21.0766 q^{85} +51.5349 q^{86} -91.7993i q^{87} +0.227481 q^{88} -46.7024i q^{89} +4.51773i q^{90} +156.896 q^{92} +39.1380 q^{93} +18.2802i q^{94} -2.55888 q^{95} -47.7657i q^{96} +3.11494i q^{97} +0.134280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 12 q^{4} - 32 q^{8} - 24 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 + 12 * q^4 - 32 * q^8 - 24 * q^9	\( 8 q - 4 q^{2} + 12 q^{4} - 32 q^{8} - 24 q^{9} - 40 q^{11} + 4 q^{16} + 12 q^{18} - 16 q^{22} - 124 q^{23} - 40 q^{25} - 100 q^{29} - 72 q^{32} - 36 q^{36} + 160 q^{37} + 24 q^{39} + 352 q^{43} + 36 q^{44} + 164 q^{46} + 20 q^{50} - 36 q^{51} + 152 q^{53} + 80 q^{58} + 120 q^{60} - 4 q^{64} + 120 q^{65} - 368 q^{67} + 164 q^{71} + 96 q^{72} + 280 q^{74} - 240 q^{78} + 412 q^{79} + 72 q^{81} - 60 q^{85} - 356 q^{86} - 248 q^{88} - 288 q^{92} - 252 q^{93} + 240 q^{95} + 120 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 12 * q^4 - 32 * q^8 - 24 * q^9 - 40 * q^11 + 4 * q^16 + 12 * q^18 - 16 * q^22 - 124 * q^23 - 40 * q^25 - 100 * q^29 - 72 * q^32 - 36 * q^36 + 160 * q^37 + 24 * q^39 + 352 * q^43 + 36 * q^44 + 164 * q^46 + 20 * q^50 - 36 * q^51 + 152 * q^53 + 80 * q^58 + 120 * q^60 - 4 * q^64 + 120 * q^65 - 368 * q^67 + 164 * q^71 + 96 * q^72 + 280 * q^74 - 240 * q^78 + 412 * q^79 + 72 * q^81 - 60 * q^85 - 356 * q^86 - 248 * q^88 - 288 * q^92 - 252 * q^93 + 240 * q^95 + 120 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more