LMFDB - Embedded newform 1470.3.f.d.391.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,3,Mod(391,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, 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("1470.391");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1470.f (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:	$40.0545988610$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + \cdots + 101626561 $ x^16 + 92x^14 - 112x^13 + 5846x^12 - 7728x^11 + 197216x^10 - 298200x^9 + 4836403x^8 - 6808704x^7 + 64376800x^6 - 91953512x^5 + 595763862x^4 - 630430976x^3 + 1087013404x^2 + 294123256x + 101626561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			391.16
Root			$1.92573 - 3.33546i$ of defining polynomial
Character	$\chi$	$=$	1470.391
Dual form			1470.3.f.d.391.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} +1.15679 q^{11} +3.46410i q^{12} +14.8176i q^{13} -3.87298 q^{15} +4.00000 q^{16} -12.6176i q^{17} -4.24264 q^{18} +19.3022i q^{19} +4.47214i q^{20} +1.63595 q^{22} -24.3007 q^{23} +4.89898i q^{24} -5.00000 q^{25} +20.9552i q^{26} -5.19615i q^{27} +49.0382 q^{29} -5.47723 q^{30} +28.8191i q^{31} +5.65685 q^{32} +2.00362i q^{33} -17.8439i q^{34} -6.00000 q^{36} -53.3158 q^{37} +27.2974i q^{38} -25.6648 q^{39} +6.32456i q^{40} +38.0398i q^{41} -63.5774 q^{43} +2.31358 q^{44} -6.70820i q^{45} -34.3664 q^{46} +25.1927i q^{47} +6.92820i q^{48} -7.07107 q^{50} +21.8543 q^{51} +29.6351i q^{52} +20.8320 q^{53} -7.34847i q^{54} +2.58666i q^{55} -33.4323 q^{57} +69.3505 q^{58} -24.4125i q^{59} -7.74597 q^{60} +6.38984i q^{61} +40.7564i q^{62} +8.00000 q^{64} -33.1331 q^{65} +2.83354i q^{66} +124.490 q^{67} -25.2351i q^{68} -42.0901i q^{69} -118.973 q^{71} -8.48528 q^{72} +39.5295i q^{73} -75.3999 q^{74} -8.66025i q^{75} +38.6043i q^{76} -36.2955 q^{78} -92.8711 q^{79} +8.94427i q^{80} +9.00000 q^{81} +53.7964i q^{82} +5.79665i q^{83} +28.2137 q^{85} -89.9120 q^{86} +84.9366i q^{87} +3.27189 q^{88} +151.984i q^{89} -9.48683i q^{90} -48.6014 q^{92} -49.9162 q^{93} +35.6279i q^{94} -43.1609 q^{95} +9.79796i q^{96} -144.310i q^{97} -3.47036 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) $ 16 * q + 32 * q^4 - 48 * q^9	$ 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) $ 16 * q + 32 * q^4 - 48 * q^9 + 8 * q^11 + 64 * q^16 - 48 * q^22 + 24 * q^23 - 80 * q^25 + 72 * q^29 - 96 * q^36 - 88 * q^37 - 72 * q^39 - 56 * q^43 + 16 * q^44 - 16 * q^46 + 24 * q^51 - 64 * q^53 + 144 * q^57 + 176 * q^58 + 128 * q^64 - 40 * q^65 + 328 * q^67 - 136 * q^71 + 224 * q^74 - 560 * q^79 + 144 * q^81 + 120 * q^85 + 176 * q^86 - 96 * q^88 + 48 * q^92 + 240 * q^93 - 400 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$.

$n$	$491$	$1081$	$1177$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	2.00000	0.500000
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	2.44949i	0.408248i
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	−3.00000	−0.333333
$10$	3.16228i	0.316228i
$11$	1.15679	0.105163	0.0525813	−	0.998617i	$-0.483255\pi$
$11$	1.15679	0.105163	0.0525813	+	0.998617i	$0.483255\pi$
$12$	3.46410i	0.288675i
$13$	14.8176i	1.13981i	0.821710	+	0.569906i	$0.193021\pi$
$13$	14.8176i	1.13981i	−0.821710	+	0.569906i	$0.806979\pi$
$14$	0	0
$15$	−3.87298	−0.258199
$16$	4.00000	0.250000
$17$	− 12.6176i	− 0.742210i	−0.928591	−	0.371105i	$-0.878979\pi$
$17$	− 12.6176i	− 0.742210i	0.928591	−	0.371105i	$-0.121021\pi$
$18$	−4.24264	−0.235702
$19$	19.3022i	1.01590i	0.861386	+	0.507951i	$0.169597\pi$
$19$	19.3022i	1.01590i	−0.861386	+	0.507951i	$0.830403\pi$
$20$	4.47214i	0.223607i
$21$	0	0
$22$	1.63595	0.0743612
$23$	−24.3007	−1.05655	−0.528277	−	0.849072i	$-0.677162\pi$
$23$	−24.3007	−1.05655	−0.528277	+	0.849072i	$0.677162\pi$
$24$	4.89898i	0.204124i
$25$	−5.00000	−0.200000
$26$	20.9552i	0.805969i
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	49.0382	1.69097	0.845486	−	0.533998i	$-0.179311\pi$
$29$	49.0382	1.69097	0.845486	+	0.533998i	$0.179311\pi$
$30$	−5.47723	−0.182574
$31$	28.8191i	0.929649i	0.885403	+	0.464825i	$0.153883\pi$
$31$	28.8191i	0.929649i	−0.885403	+	0.464825i	$0.846117\pi$
$32$	5.65685	0.176777
$33$	2.00362i	0.0607156i
$34$	− 17.8439i	− 0.524822i
$35$	0	0
$36$	−6.00000	−0.166667
$37$	−53.3158	−1.44097	−0.720484	−	0.693472i	$-0.756080\pi$
$37$	−53.3158	−1.44097	−0.720484	+	0.693472i	$0.756080\pi$
$38$	27.2974i	0.718352i
$39$	−25.6648	−0.658071
$40$	6.32456i	0.158114i
$41$	38.0398i	0.927800i	0.885888	+	0.463900i	$0.153550\pi$
$41$	38.0398i	0.927800i	−0.885888	+	0.463900i	$0.846450\pi$
$42$	0	0
$43$	−63.5774	−1.47854	−0.739272	−	0.673407i	$-0.764830\pi$
$43$	−63.5774	−1.47854	−0.739272	+	0.673407i	$0.764830\pi$
$44$	2.31358	0.0525813
$45$	− 6.70820i	− 0.149071i
$46$	−34.3664	−0.747096
$47$	25.1927i	0.536015i	0.963417	+	0.268008i	$0.0863652\pi$
$47$	25.1927i	0.536015i	−0.963417	+	0.268008i	$0.913635\pi$
$48$	6.92820i	0.144338i
$49$	0	0
$50$	−7.07107	−0.141421
$51$	21.8543	0.428515
$52$	29.6351i	0.569906i
$53$	20.8320	0.393057	0.196529	−	0.980498i	$-0.437033\pi$
$53$	20.8320	0.393057	0.196529	+	0.980498i	$0.437033\pi$
$54$	− 7.34847i	− 0.136083i
$55$	2.58666i	0.0470301i
$56$	0	0
$57$	−33.4323	−0.586532
$58$	69.3505	1.19570
$59$	− 24.4125i	− 0.413772i	−0.978365	−	0.206886i	$-0.933667\pi$
$59$	− 24.4125i	− 0.413772i	0.978365	−	0.206886i	$-0.0663329\pi$
$60$	−7.74597	−0.129099
$61$	6.38984i	0.104751i	0.998627	+	0.0523757i	$0.0166793\pi$
$61$	6.38984i	0.104751i	−0.998627	+	0.0523757i	$0.983321\pi$
$62$	40.7564i	0.657361i
$63$	0	0
$64$	8.00000	0.125000
$65$	−33.1331	−0.509740
$66$	2.83354i	0.0429324i
$67$	124.490	1.85806	0.929031	−	0.370002i	$-0.120643\pi$
$67$	124.490	1.85806	0.929031	+	0.370002i	$0.120643\pi$
$68$	− 25.2351i	− 0.371105i
$69$	− 42.0901i	− 0.610001i
$70$	0	0
$71$	−118.973	−1.67567	−0.837835	−	0.545924i	$-0.816179\pi$
$71$	−118.973	−1.67567	−0.837835	+	0.545924i	$0.816179\pi$
$72$	−8.48528	−0.117851
$73$	39.5295i	0.541500i	0.962650	+	0.270750i	$0.0872717\pi$
$73$	39.5295i	0.541500i	−0.962650	+	0.270750i	$0.912728\pi$
$74$	−75.3999	−1.01892
$75$	− 8.66025i	− 0.115470i
$76$	38.6043i	0.507951i
$77$	0	0
$78$	−36.2955	−0.465327
$79$	−92.8711	−1.17558	−0.587792	−	0.809012i	$-0.700003\pi$
$79$	−92.8711	−1.17558	−0.587792	+	0.809012i	$0.700003\pi$
$80$	8.94427i	0.111803i
$81$	9.00000	0.111111
$82$	53.7964i	0.656054i
$83$	5.79665i	0.0698392i	0.999390	+	0.0349196i	$0.0111175\pi$
$83$	5.79665i	0.0698392i	−0.999390	+	0.0349196i	$0.988882\pi$
$84$	0	0
$85$	28.2137	0.331926
$86$	−89.9120	−1.04549
$87$	84.9366i	0.976283i
$88$	3.27189	0.0371806
$89$	151.984i	1.70768i	0.520533	+	0.853842i	$0.325733\pi$
$89$	151.984i	1.70768i	−0.520533	+	0.853842i	$0.674267\pi$
$90$	− 9.48683i	− 0.105409i
$91$	0	0
$92$	−48.6014	−0.528277
$93$	−49.9162	−0.536733
$94$	35.6279i	0.379020i
$95$	−43.1609	−0.454326
$96$	9.79796i	0.102062i
$97$	− 144.310i	− 1.48773i	−0.668331	−	0.743864i	$-0.732991\pi$
$97$	− 144.310i	− 1.48773i	0.668331	−	0.743864i	$-0.267009\pi$
$98$	0	0
$99$	−3.47036	−0.0350542
$100$	−10.0000	−0.100000
$101$	39.0843i	0.386973i	0.981103	+	0.193487i	$0.0619796\pi$
$101$	39.0843i	0.386973i	−0.981103	+	0.193487i	$0.938020\pi$
$102$	30.9066	0.303006
$103$	37.8430i	0.367408i	0.982982	+	0.183704i	$0.0588088\pi$
$103$	37.8430i	0.367408i	−0.982982	+	0.183704i	$0.941191\pi$
$104$	41.9104i	0.402985i
$105$	0	0
$106$	29.4609	0.277933
$107$	83.0321	0.776001	0.388000	−	0.921659i	$-0.373166\pi$
$107$	83.0321	0.776001	0.388000	+	0.921659i	$0.373166\pi$
$108$	− 10.3923i	− 0.0962250i
$109$	−47.7924	−0.438462	−0.219231	−	0.975673i	$-0.570355\pi$
$109$	−47.7924	−0.438462	−0.219231	+	0.975673i	$0.570355\pi$
$110$	3.65809i	0.0332553i
$111$	− 92.3457i	− 0.831943i
$112$	0	0
$113$	−16.2283	−0.143613	−0.0718064	−	0.997419i	$-0.522876\pi$
$113$	−16.2283	−0.143613	−0.0718064	+	0.997419i	$0.522876\pi$
$114$	−47.2804	−0.414741
$115$	− 54.3381i	− 0.472505i
$116$	98.0764	0.845486
$117$	− 44.4527i	− 0.379938i
$118$	− 34.5246i	− 0.292581i
$119$	0	0
$120$	−10.9545	−0.0912871
$121$	−119.662	−0.988941
$122$	9.03659i	0.0740704i
$123$	−65.8869	−0.535666
$124$	57.6383i	0.464825i
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	80.5643	0.634365	0.317182	−	0.948365i	$-0.397263\pi$
$127$	80.5643	0.634365	0.317182	+	0.948365i	$0.397263\pi$
$128$	11.3137	0.0883883
$129$	− 110.119i	− 0.853638i
$130$	−46.8573	−0.360440
$131$	124.686i	0.951800i	0.879499	+	0.475900i	$0.157878\pi$
$131$	124.686i	0.951800i	−0.879499	+	0.475900i	$0.842122\pi$
$132$	4.00723i	0.0303578i
$133$	0	0
$134$	176.056	1.31385
$135$	11.6190	0.0860663
$136$	− 35.6879i	− 0.262411i
$137$	−76.0659	−0.555226	−0.277613	−	0.960693i	$-0.589543\pi$
$137$	−76.0659	−0.555226	−0.277613	+	0.960693i	$0.589543\pi$
$138$	− 59.5244i	− 0.431336i
$139$	− 91.7680i	− 0.660201i	−0.943946	−	0.330101i	$-0.892917\pi$
$139$	− 91.7680i	− 0.660201i	0.943946	−	0.330101i	$-0.107083\pi$
$140$	0	0
$141$	−43.6350	−0.309468
$142$	−168.253	−1.18488
$143$	17.1408i	0.119866i
$144$	−12.0000	−0.0833333
$145$	109.653i	0.756226i
$146$	55.9032i	0.382898i
$147$	0	0
$148$	−106.632	−0.720484
$149$	−98.9159	−0.663865	−0.331933	−	0.943303i	$-0.607701\pi$
$149$	−98.9159	−0.663865	−0.331933	+	0.943303i	$0.607701\pi$
$150$	− 12.2474i	− 0.0816497i
$151$	97.7900	0.647616	0.323808	−	0.946123i	$-0.395037\pi$
$151$	97.7900	0.647616	0.323808	+	0.946123i	$0.395037\pi$
$152$	54.5947i	0.359176i
$153$	37.8527i	0.247403i
$154$	0	0
$155$	−64.4415	−0.415752
$156$	−51.3296	−0.329036
$157$	− 133.403i	− 0.849698i	−0.905264	−	0.424849i	$-0.860327\pi$
$157$	− 133.403i	− 0.849698i	0.905264	−	0.424849i	$-0.139673\pi$
$158$	−131.340	−0.831264
$159$	36.0821i	0.226932i
$160$	12.6491i	0.0790569i
$161$	0	0
$162$	12.7279	0.0785674
$163$	−32.7492	−0.200915	−0.100458	−	0.994941i	$-0.532031\pi$
$163$	−32.7492	−0.200915	−0.100458	+	0.994941i	$0.532031\pi$
$164$	76.0796i	0.463900i
$165$	−4.48022	−0.0271529
$166$	8.19770i	0.0493838i
$167$	− 171.659i	− 1.02790i	−0.857821	−	0.513948i	$-0.828182\pi$
$167$	− 171.659i	− 1.02790i	0.857821	−	0.513948i	$-0.171818\pi$
$168$	0	0
$169$	−50.5603	−0.299173
$170$	39.9002	0.234707
$171$	− 57.9065i	− 0.338634i
$172$	−127.155	−0.739272
$173$	202.460i	1.17029i	0.810928	+	0.585145i	$0.198963\pi$
$173$	202.460i	1.17029i	−0.810928	+	0.585145i	$0.801037\pi$
$174$	120.119i	0.690336i
$175$	0	0
$176$	4.62715	0.0262906
$177$	42.2838	0.238891
$178$	214.938i	1.20751i
$179$	−78.6918	−0.439619	−0.219810	−	0.975543i	$-0.570544\pi$
$179$	−78.6918	−0.439619	−0.219810	+	0.975543i	$0.570544\pi$
$180$	− 13.4164i	− 0.0745356i
$181$	− 58.1509i	− 0.321276i	−0.987013	−	0.160638i	$-0.948645\pi$
$181$	− 58.1509i	− 0.321276i	0.987013	−	0.160638i	$-0.0513551\pi$
$182$	0	0
$183$	−11.0675	−0.0604783
$184$	−68.7328	−0.373548
$185$	− 119.218i	− 0.644420i
$186$	−70.5922	−0.379528
$187$	− 14.5959i	− 0.0780527i
$188$	50.3854i	0.268008i
$189$	0	0
$190$	−61.0388	−0.321257
$191$	368.408	1.92884	0.964419	−	0.264379i	$-0.0851669\pi$
$191$	368.408	1.92884	0.964419	+	0.264379i	$0.0851669\pi$
$192$	13.8564i	0.0721688i
$193$	280.819	1.45502	0.727510	−	0.686097i	$-0.240677\pi$
$193$	280.819	1.45502	0.727510	+	0.686097i	$0.240677\pi$
$194$	− 204.085i	− 1.05198i
$195$	− 57.3882i	− 0.294298i
$196$	0	0
$197$	7.61779	0.0386690	0.0193345	−	0.999813i	$-0.493845\pi$
$197$	7.61779	0.0386690	0.0193345	+	0.999813i	$0.493845\pi$
$198$	−4.90784	−0.0247871
$199$	201.836i	1.01425i	0.861873	+	0.507125i	$0.169292\pi$
$199$	201.836i	1.01425i	−0.861873	+	0.507125i	$0.830708\pi$
$200$	−14.1421	−0.0707107
$201$	215.623i	1.07275i
$202$	55.2736i	0.273632i
$203$	0	0
$204$	43.7085	0.214257
$205$	−85.0596	−0.414925
$206$	53.5181i	0.259797i
$207$	72.9022	0.352184
$208$	59.2703i	0.284953i
$209$	22.3285i	0.106835i
$210$	0	0
$211$	30.3818	0.143989	0.0719947	−	0.997405i	$-0.477064\pi$
$211$	30.3818	0.143989	0.0719947	+	0.997405i	$0.477064\pi$
$212$	41.6641	0.196529
$213$	− 206.067i	− 0.967448i
$214$	117.425	0.548715
$215$	− 142.163i	− 0.661225i
$216$	− 14.6969i	− 0.0680414i
$217$	0	0
$218$	−67.5886	−0.310039
$219$	−68.4671	−0.312635
$220$	5.17331i	0.0235151i
$221$	186.962	0.845980
$222$	− 130.597i	− 0.588273i
$223$	16.7377i	0.0750569i	0.999296	+	0.0375284i	$0.0119485\pi$
$223$	16.7377i	0.0750569i	−0.999296	+	0.0375284i	$0.988052\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	−22.9502	−0.101550
$227$	423.472i	1.86552i	0.360503	+	0.932758i	$0.382605\pi$
$227$	423.472i	1.86552i	−0.360503	+	0.932758i	$0.617395\pi$
$228$	−66.8646	−0.293266
$229$	− 404.834i	− 1.76783i	−0.467645	−	0.883916i	$-0.654897\pi$
$229$	− 404.834i	− 1.76783i	0.467645	−	0.883916i	$-0.345103\pi$
$230$	− 76.8456i	− 0.334111i
$231$	0	0
$232$	138.701	0.597849
$233$	277.298	1.19012	0.595061	−	0.803681i	$-0.297128\pi$
$233$	277.298	1.19012	0.595061	+	0.803681i	$0.297128\pi$
$234$	− 62.8656i	− 0.268656i
$235$	−56.3326	−0.239713
$236$	− 48.8251i	− 0.206886i
$237$	− 160.858i	− 0.678724i
$238$	0	0
$239$	290.247	1.21442	0.607211	−	0.794541i	$-0.292288\pi$
$239$	290.247	1.21442	0.607211	+	0.794541i	$0.292288\pi$
$240$	−15.4919	−0.0645497
$241$	404.808i	1.67970i	0.542819	+	0.839850i	$0.317357\pi$
$241$	404.808i	1.67970i	−0.542819	+	0.839850i	$0.682643\pi$
$242$	−169.227	−0.699287
$243$	15.5885i	0.0641500i
$244$	12.7797i	0.0523757i
$245$	0	0
$246$	−93.1781	−0.378773
$247$	−286.011	−1.15794
$248$	81.5128i	0.328681i
$249$	−10.0401	−0.0403217
$250$	− 15.8114i	− 0.0632456i
$251$	155.805i	0.620739i	0.950616	+	0.310369i	$0.100453\pi$
$251$	155.805i	0.620739i	−0.950616	+	0.310369i	$0.899547\pi$
$252$	0	0
$253$	−28.1108	−0.111110
$254$	113.935	0.448563
$255$	48.8676i	0.191638i
$256$	16.0000	0.0625000
$257$	− 25.9017i	− 0.100785i	−0.998729	−	0.0503923i	$-0.983953\pi$
$257$	− 25.9017i	− 0.100785i	0.998729	−	0.0503923i	$-0.0160472\pi$
$258$	− 155.732i	− 0.603613i
$259$	0	0
$260$	−66.2662	−0.254870
$261$	−147.115	−0.563657
$262$	176.332i	0.673024i
$263$	−315.805	−1.20078	−0.600389	−	0.799708i	$-0.704988\pi$
$263$	−315.805	−1.20078	−0.600389	+	0.799708i	$0.704988\pi$
$264$	5.66708i	0.0214662i
$265$	46.5818i	0.175781i
$266$	0	0
$267$	−263.244	−0.985931
$268$	248.980	0.929031
$269$	47.8228i	0.177780i	0.996041	+	0.0888899i	$0.0283319\pi$
$269$	47.8228i	0.177780i	−0.996041	+	0.0888899i	$0.971668\pi$
$270$	16.4317	0.0608581
$271$	340.152i	1.25517i	0.778547	+	0.627587i	$0.215957\pi$
$271$	340.152i	1.25517i	−0.778547	+	0.627587i	$0.784043\pi$
$272$	− 50.4703i	− 0.185552i
$273$	0	0
$274$	−107.573	−0.392604
$275$	−5.78394	−0.0210325
$276$	− 84.1802i	− 0.305001i
$277$	210.587	0.760241	0.380121	−	0.924937i	$-0.375882\pi$
$277$	210.587	0.760241	0.380121	+	0.924937i	$0.375882\pi$
$278$	− 129.780i	− 0.466833i
$279$	− 86.4574i	− 0.309883i
$280$	0	0
$281$	471.785	1.67895	0.839476	−	0.543397i	$-0.182862\pi$
$281$	471.785	1.67895	0.839476	+	0.543397i	$0.182862\pi$
$282$	−61.7093	−0.218827
$283$	− 468.270i	− 1.65467i	−0.561712	−	0.827333i	$-0.689857\pi$
$283$	− 468.270i	− 1.65467i	0.561712	−	0.827333i	$-0.310143\pi$
$284$	−237.945	−0.837835
$285$	− 74.7569i	− 0.262305i
$286$	24.2407i	0.0847578i
$287$	0	0
$288$	−16.9706	−0.0589256
$289$	129.797	0.449125
$290$	155.072i	0.534732i
$291$	249.952	0.858940
$292$	79.0590i	0.270750i
$293$	− 63.5067i	− 0.216746i	−0.994110	−	0.108373i	$-0.965436\pi$
$293$	− 63.5067i	− 0.216746i	0.994110	−	0.108373i	$-0.0345642\pi$
$294$	0	0
$295$	54.5881	0.185044
$296$	−150.800	−0.509459
$297$	− 6.01085i	− 0.0202385i
$298$	−139.888	−0.469423
$299$	− 360.078i	− 1.20427i
$300$	− 17.3205i	− 0.0577350i
$301$	0	0
$302$	138.296	0.457934
$303$	−67.6960	−0.223419
$304$	77.2086i	0.253976i
$305$	−14.2881	−0.0468463
$306$	53.5318i	0.174941i
$307$	− 211.610i	− 0.689283i	−0.938734	−	0.344642i	$-0.888000\pi$
$307$	− 211.610i	− 0.689283i	0.938734	−	0.344642i	$-0.112000\pi$
$308$	0	0
$309$	−65.5460	−0.212123
$310$	−91.1341	−0.293981
$311$	− 67.3294i	− 0.216493i	−0.994124	−	0.108247i	$-0.965476\pi$
$311$	− 67.3294i	− 0.216493i	0.994124	−	0.108247i	$-0.0345236\pi$
$312$	−72.5910	−0.232663
$313$	− 138.234i	− 0.441641i	−0.975314	−	0.220821i	$-0.929126\pi$
$313$	− 138.234i	− 0.441641i	0.975314	−	0.220821i	$-0.0708735\pi$
$314$	− 188.660i	− 0.600827i
$315$	0	0
$316$	−185.742	−0.587792
$317$	19.8243	0.0625373	0.0312686	−	0.999511i	$-0.490045\pi$
$317$	19.8243	0.0625373	0.0312686	+	0.999511i	$0.490045\pi$
$318$	51.0278i	0.160465i
$319$	56.7268	0.177827
$320$	17.8885i	0.0559017i
$321$	143.816i	0.448024i
$322$	0	0
$323$	243.546	0.754013
$324$	18.0000	0.0555556
$325$	− 74.0878i	− 0.227963i
$326$	−46.3143	−0.142068
$327$	− 82.7788i	− 0.253146i
$328$	107.593i	0.328027i
$329$	0	0
$330$	−6.33599	−0.0192000
$331$	−262.277	−0.792379	−0.396189	−	0.918169i	$-0.629668\pi$
$331$	−262.277	−0.792379	−0.396189	+	0.918169i	$0.629668\pi$
$332$	11.5933i	0.0349196i
$333$	159.947	0.480323
$334$	− 242.762i	− 0.726832i
$335$	278.368i	0.830951i
$336$	0	0
$337$	578.125	1.71550	0.857752	−	0.514063i	$-0.171860\pi$
$337$	578.125	1.71550	0.857752	+	0.514063i	$0.171860\pi$
$338$	−71.5031	−0.211548
$339$	− 28.1082i	− 0.0829149i
$340$	56.4275	0.165963
$341$	33.3376i	0.0977643i
$342$	− 81.8921i	− 0.239451i
$343$	0	0
$344$	−179.824	−0.522744
$345$	94.1163	0.272801
$346$	286.322i	0.827521i
$347$	459.722	1.32485	0.662423	−	0.749130i	$-0.269528\pi$
$347$	459.722	1.32485	0.662423	+	0.749130i	$0.269528\pi$
$348$	169.873i	0.488142i
$349$	− 389.147i	− 1.11504i	−0.830165	−	0.557518i	$-0.811754\pi$
$349$	− 389.147i	− 1.11504i	0.830165	−	0.557518i	$-0.188246\pi$
$350$	0	0
$351$	76.9943	0.219357
$352$	6.54378	0.0185903
$353$	− 645.957i	− 1.82991i	−0.403560	−	0.914953i	$-0.632227\pi$
$353$	− 645.957i	− 1.82991i	0.403560	−	0.914953i	$-0.367773\pi$
$354$	59.7983	0.168922
$355$	− 266.031i	− 0.749382i
$356$	303.968i	0.853842i
$357$	0	0
$358$	−111.287	−0.310858
$359$	−66.0412	−0.183959	−0.0919794	−	0.995761i	$-0.529319\pi$
$359$	−66.0412	−0.183959	−0.0919794	+	0.995761i	$0.529319\pi$
$360$	− 18.9737i	− 0.0527046i
$361$	−11.5732	−0.0320588
$362$	− 82.2378i	− 0.227176i
$363$	− 207.260i	− 0.570965i
$364$	0	0
$365$	−88.3907	−0.242166
$366$	−15.6518	−0.0427646
$367$	− 599.932i	− 1.63469i	−0.576148	−	0.817346i	$-0.695445\pi$
$367$	− 599.932i	− 1.63469i	0.576148	−	0.817346i	$-0.304555\pi$
$368$	−97.2029	−0.264138
$369$	− 114.119i	− 0.309267i
$370$	− 168.599i	− 0.455674i
$371$	0	0
$372$	−99.8324	−0.268367
$373$	404.753	1.08513	0.542564	−	0.840015i	$-0.317454\pi$
$373$	404.753	1.08513	0.542564	+	0.840015i	$0.317454\pi$
$374$	− 20.6417i	− 0.0551916i
$375$	19.3649	0.0516398
$376$	71.2557i	0.189510i
$377$	726.627i	1.92739i
$378$	0	0
$379$	−239.675	−0.632388	−0.316194	−	0.948695i	$-0.602405\pi$
$379$	−239.675	−0.632388	−0.316194	+	0.948695i	$0.602405\pi$
$380$	−86.3219	−0.227163
$381$	139.541i	0.366251i
$382$	521.008	1.36389
$383$	639.110i	1.66869i	0.551239	+	0.834347i	$0.314155\pi$
$383$	639.110i	1.66869i	−0.551239	+	0.834347i	$0.685845\pi$
$384$	19.5959i	0.0510310i
$385$	0	0
$386$	397.138	1.02885
$387$	190.732	0.492848
$388$	− 288.619i	− 0.743864i
$389$	521.594	1.34086	0.670429	−	0.741973i	$-0.266110\pi$
$389$	521.594	1.34086	0.670429	+	0.741973i	$0.266110\pi$
$390$	− 81.1592i	− 0.208100i
$391$	306.616i	0.784184i
$392$	0	0
$393$	−215.962	−0.549522
$394$	10.7732	0.0273431
$395$	− 207.666i	− 0.525737i
$396$	−6.94073	−0.0175271
$397$	145.160i	0.365642i	0.983146	+	0.182821i	$0.0585230\pi$
$397$	145.160i	0.365642i	−0.983146	+	0.182821i	$0.941477\pi$
$398$	285.439i	0.717182i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	214.043	0.533772	0.266886	−	0.963728i	$-0.414005\pi$
$401$	214.043	0.533772	0.266886	+	0.963728i	$0.414005\pi$
$402$	304.937i	0.758551i
$403$	−427.029	−1.05963
$404$	78.1686i	0.193487i
$405$	20.1246i	0.0496904i
$406$	0	0
$407$	−61.6751	−0.151536
$408$	61.8132	0.151503
$409$	553.972i	1.35446i	0.735773	+	0.677228i	$0.236819\pi$
$409$	553.972i	1.35446i	−0.735773	+	0.677228i	$0.763181\pi$
$410$	−120.292	−0.293396
$411$	− 131.750i	− 0.320560i
$412$	75.6861i	0.183704i
$413$	0	0
$414$	103.099	0.249032
$415$	−12.9617	−0.0312330
$416$	83.8208i	0.201492i
$417$	158.947	0.381167
$418$	31.5773i	0.0755437i
$419$	268.003i	0.639626i	0.947481	+	0.319813i	$0.103620\pi$
$419$	268.003i	0.639626i	−0.947481	+	0.319813i	$0.896380\pi$
$420$	0	0
$421$	−9.12915	−0.0216844	−0.0108422	−	0.999941i	$-0.503451\pi$
$421$	−9.12915	−0.0216844	−0.0108422	+	0.999941i	$0.503451\pi$
$422$	42.9663	0.101816
$423$	− 75.5781i	− 0.178672i
$424$	58.9219	0.138967
$425$	63.0878i	0.148442i
$426$	− 291.422i	− 0.684089i
$427$	0	0
$428$	166.064	0.388000
$429$	−29.6887	−0.0692045
$430$	− 201.049i	− 0.467557i
$431$	−268.443	−0.622837	−0.311419	−	0.950273i	$-0.600804\pi$
$431$	−268.443	−0.622837	−0.311419	+	0.950273i	$0.600804\pi$
$432$	− 20.7846i	− 0.0481125i
$433$	− 472.254i	− 1.09066i	−0.838223	−	0.545328i	$-0.816405\pi$
$433$	− 472.254i	− 1.09066i	0.838223	−	0.545328i	$-0.183595\pi$
$434$	0	0
$435$	−189.924	−0.436607
$436$	−95.5847	−0.219231
$437$	− 469.056i	− 1.07336i
$438$	−96.8271	−0.221067
$439$	549.392i	1.25146i	0.780039	+	0.625731i	$0.215199\pi$
$439$	549.392i	1.25146i	−0.780039	+	0.625731i	$0.784801\pi$
$440$	7.31617i	0.0166277i
$441$	0	0
$442$	264.404	0.598198
$443$	470.810	1.06278	0.531388	−	0.847128i	$-0.321670\pi$
$443$	470.810	1.06278	0.531388	+	0.847128i	$0.321670\pi$
$444$	− 184.691i	− 0.415972i
$445$	−339.846	−0.763699
$446$	23.6707i	0.0530732i
$447$	− 171.327i	− 0.383283i
$448$	0	0
$449$	559.525	1.24616	0.623079	−	0.782159i	$-0.285882\pi$
$449$	559.525	1.24616	0.623079	+	0.782159i	$0.285882\pi$
$450$	21.2132	0.0471405
$451$	44.0040i	0.0975699i
$452$	−32.4565	−0.0718064
$453$	169.377i	0.373901i
$454$	598.880i	1.31912i
$455$	0	0
$456$	−94.5609	−0.207370
$457$	627.378	1.37282	0.686409	−	0.727216i	$-0.259186\pi$
$457$	627.378	1.37282	0.686409	+	0.727216i	$0.259186\pi$
$458$	− 572.521i	− 1.25005i
$459$	−65.5628	−0.142838
$460$	− 108.676i	− 0.236252i
$461$	223.659i	0.485160i	0.970131	+	0.242580i	$0.0779936\pi$
$461$	223.659i	0.485160i	−0.970131	+	0.242580i	$0.922006\pi$
$462$	0	0
$463$	−397.204	−0.857893	−0.428946	−	0.903330i	$-0.641115\pi$
$463$	−397.204	−0.857893	−0.428946	+	0.903330i	$0.641115\pi$
$464$	196.153	0.422743
$465$	− 111.616i	− 0.240034i
$466$	392.159	0.841543
$467$	− 333.725i	− 0.714614i	−0.933987	−	0.357307i	$-0.883695\pi$
$467$	− 333.725i	− 0.714614i	0.933987	−	0.357307i	$-0.116305\pi$
$468$	− 88.9054i	− 0.189969i
$469$	0	0
$470$	−79.6663	−0.169503
$471$	231.060	0.490573
$472$	− 69.0491i	− 0.146290i
$473$	−73.5456	−0.155488
$474$	− 227.487i	− 0.479930i
$475$	− 96.5108i	− 0.203181i
$476$	0	0
$477$	−62.4961	−0.131019
$478$	410.471	0.858726
$479$	− 608.833i	− 1.27105i	−0.772080	−	0.635526i	$-0.780783\pi$
$479$	− 608.833i	− 1.27105i	0.772080	−	0.635526i	$-0.219217\pi$
$480$	−21.9089	−0.0456435
$481$	− 790.011i	− 1.64243i
$482$	572.484i	1.18773i
$483$	0	0
$484$	−239.324	−0.494470
$485$	322.686	0.665332
$486$	22.0454i	0.0453609i
$487$	−318.477	−0.653957	−0.326978	−	0.945032i	$-0.606030\pi$
$487$	−318.477	−0.653957	−0.326978	+	0.945032i	$0.606030\pi$
$488$	18.0732i	0.0370352i
$489$	− 56.7232i	− 0.115998i
$490$	0	0
$491$	523.303	1.06579	0.532895	−	0.846181i	$-0.321104\pi$
$491$	523.303	1.06579	0.532895	+	0.846181i	$0.321104\pi$
$492$	−131.774	−0.267833
$493$	− 618.742i	− 1.25506i
$494$	−404.481	−0.818787
$495$	− 7.75997i	− 0.0156767i
$496$	115.277i	0.232412i
$497$	0	0
$498$	−14.1988	−0.0285117
$499$	−783.817	−1.57078	−0.785388	−	0.619004i	$-0.787536\pi$
$499$	−783.817	−1.57078	−0.785388	+	0.619004i	$0.787536\pi$
$500$	− 22.3607i	− 0.0447214i
$501$	297.322	0.593456
$502$	220.342i	0.438929i
$503$	− 58.0772i	− 0.115462i	−0.998332	−	0.0577308i	$-0.981613\pi$
$503$	− 58.0772i	− 0.115462i	0.998332	−	0.0577308i	$-0.0183865\pi$
$504$	0	0
$505$	−87.3952	−0.173060
$506$	−39.7547	−0.0785665
$507$	− 87.5730i	− 0.172728i
$508$	161.129	0.317182
$509$	− 936.589i	− 1.84006i	−0.391851	−	0.920029i	$-0.628165\pi$
$509$	− 936.589i	− 1.84006i	0.391851	−	0.920029i	$-0.371835\pi$
$510$	69.1092i	0.135508i
$511$	0	0
$512$	22.6274	0.0441942
$513$	100.297	0.195511
$514$	− 36.6305i	− 0.0712655i
$515$	−84.6196	−0.164310
$516$	− 220.239i	− 0.426819i
$517$	29.1426i	0.0563687i
$518$	0	0
$519$	−350.672	−0.675668
$520$	−93.7145	−0.180220
$521$	701.057i	1.34560i	0.739825	+	0.672799i	$0.234908\pi$
$521$	701.057i	1.34560i	−0.739825	+	0.672799i	$0.765092\pi$
$522$	−208.051	−0.398566
$523$	− 686.428i	− 1.31248i	−0.754551	−	0.656241i	$-0.772145\pi$
$523$	− 686.428i	− 1.31248i	0.754551	−	0.656241i	$-0.227855\pi$
$524$	249.372i	0.475900i
$525$	0	0
$526$	−446.615	−0.849078
$527$	363.627	0.689995
$528$	8.01446i	0.0151789i
$529$	61.5252	0.116305
$530$	65.8767i	0.124296i
$531$	73.2376i	0.137924i
$532$	0	0
$533$	−563.657	−1.05752
$534$	−372.283	−0.697159
$535$	185.665i	0.347038i
$536$	352.111	0.656924
$537$	− 136.298i	− 0.253814i
$538$	67.6316i	0.125709i
$539$	0	0
$540$	23.2379	0.0430331
$541$	796.499	1.47227	0.736136	−	0.676833i	$-0.236648\pi$
$541$	796.499	1.47227	0.736136	+	0.676833i	$0.236648\pi$
$542$	481.048i	0.887542i
$543$	100.720	0.185489
$544$	− 71.3757i	− 0.131205i
$545$	− 106.867i	− 0.196086i
$546$	0	0
$547$	−395.055	−0.722221	−0.361111	−	0.932523i	$-0.617602\pi$
$547$	−395.055	−0.722221	−0.361111	+	0.932523i	$0.617602\pi$
$548$	−152.132	−0.277613
$549$	− 19.1695i	− 0.0349171i
$550$	−8.17973	−0.0148722
$551$	946.543i	1.71786i
$552$	− 119.049i	− 0.215668i
$553$	0	0
$554$	297.815	0.537572
$555$	206.491	0.372056
$556$	− 183.536i	− 0.330101i
$557$	−41.6886	−0.0748449	−0.0374224	−	0.999300i	$-0.511915\pi$
$557$	−41.6886	−0.0748449	−0.0374224	+	0.999300i	$0.511915\pi$
$558$	− 122.269i	− 0.219120i
$559$	− 942.063i	− 1.68526i
$560$	0	0
$561$	25.2808	0.0450637
$562$	667.205	1.18720
$563$	739.750i	1.31394i	0.753915	+	0.656972i	$0.228163\pi$
$563$	739.750i	1.31394i	−0.753915	+	0.656972i	$0.771837\pi$
$564$	−87.2701	−0.154734
$565$	− 36.2875i	− 0.0642256i
$566$	− 662.234i	− 1.17003i
$567$	0	0
$568$	−336.505	−0.592439
$569$	4.76061	0.00836662	0.00418331	−	0.999991i	$-0.498668\pi$
$569$	4.76061	0.00836662	0.00418331	+	0.999991i	$0.498668\pi$
$570$	− 105.722i	− 0.185478i
$571$	799.696	1.40052	0.700260	−	0.713888i	$-0.253068\pi$
$571$	799.696	1.40052	0.700260	+	0.713888i	$0.253068\pi$
$572$	34.2816i	0.0599328i
$573$	638.101i	1.11362i
$574$	0	0
$575$	121.504	0.211311
$576$	−24.0000	−0.0416667
$577$	171.902i	0.297924i	0.988843	+	0.148962i	$0.0475931\pi$
$577$	171.902i	0.297924i	−0.988843	+	0.148962i	$0.952407\pi$
$578$	183.561	0.317579
$579$	486.393i	0.840056i
$580$	219.305i	0.378113i
$581$	0	0
$582$	353.485	0.607362
$583$	24.0982	0.0413349
$584$	111.806i	0.191449i
$585$	99.3993	0.169913
$586$	− 89.8121i	− 0.153263i
$587$	724.352i	1.23399i	0.786967	+	0.616994i	$0.211650\pi$
$587$	724.352i	1.23399i	−0.786967	+	0.616994i	$0.788350\pi$
$588$	0	0
$589$	−556.271	−0.944433
$590$	77.1992	0.130846
$591$	13.1944i	0.0223256i
$592$	−213.263	−0.360242
$593$	− 1032.64i	− 1.74138i	−0.491829	−	0.870692i	$-0.663672\pi$
$593$	− 1032.64i	− 1.74138i	0.491829	−	0.870692i	$-0.336328\pi$
$594$	− 8.50062i	− 0.0143108i
$595$	0	0
$596$	−197.832	−0.331933
$597$	−349.589	−0.585577
$598$	− 509.227i	− 0.851550i
$599$	−424.871	−0.709301	−0.354650	−	0.934999i	$-0.615400\pi$
$599$	−424.871	−0.709301	−0.354650	+	0.934999i	$0.615400\pi$
$600$	− 24.4949i	− 0.0408248i
$601$	749.418i	1.24695i	0.781843	+	0.623476i	$0.214280\pi$
$601$	749.418i	1.24695i	−0.781843	+	0.623476i	$0.785720\pi$
$602$	0	0
$603$	−373.471	−0.619354
$604$	195.580	0.323808
$605$	− 267.572i	− 0.442268i
$606$	−95.7366	−0.157981
$607$	− 236.867i	− 0.390226i	−0.980781	−	0.195113i	$-0.937493\pi$
$607$	− 236.867i	− 0.390226i	0.980781	−	0.195113i	$-0.0625073\pi$
$608$	109.189i	0.179588i
$609$	0	0
$610$	−20.2064	−0.0331253
$611$	−373.295	−0.610957
$612$	75.7054i	0.123702i
$613$	−938.378	−1.53080	−0.765398	−	0.643557i	$-0.777458\pi$
$613$	−938.378	−1.53080	−0.765398	+	0.643557i	$0.777458\pi$
$614$	− 299.262i	− 0.487397i
$615$	− 147.328i	− 0.239557i
$616$	0	0
$617$	225.176	0.364952	0.182476	−	0.983210i	$-0.441589\pi$
$617$	225.176	0.364952	0.182476	+	0.983210i	$0.441589\pi$
$618$	−92.6961	−0.149994
$619$	− 1057.84i	− 1.70895i	−0.519495	−	0.854473i	$-0.673880\pi$
$619$	− 1057.84i	− 1.70895i	0.519495	−	0.854473i	$-0.326120\pi$
$620$	−128.883	−0.207876
$621$	126.270i	0.203334i
$622$	− 95.2182i	− 0.153084i
$623$	0	0
$624$	−102.659	−0.164518
$625$	25.0000	0.0400000
$626$	− 195.492i	− 0.312287i
$627$	−38.6741	−0.0616812
$628$	− 266.805i	− 0.424849i
$629$	672.716i	1.06950i
$630$	0	0
$631$	877.283	1.39031	0.695153	−	0.718862i	$-0.255336\pi$
$631$	877.283	1.39031	0.695153	+	0.718862i	$0.255336\pi$
$632$	−262.679	−0.415632
$633$	52.6228i	0.0831323i
$634$	28.0358	0.0442205
$635$	180.147i	0.283696i
$636$	72.1643i	0.113466i
$637$	0	0
$638$	80.2238	0.125743
$639$	356.918	0.558557
$640$	25.2982i	0.0395285i
$641$	53.7367	0.0838327	0.0419163	−	0.999121i	$-0.486654\pi$
$641$	53.7367	0.0838327	0.0419163	+	0.999121i	$0.486654\pi$
$642$	203.386i	0.316801i
$643$	99.7799i	0.155179i	0.996985	+	0.0775893i	$0.0247223\pi$
$643$	99.7799i	0.155179i	−0.996985	+	0.0775893i	$0.975278\pi$
$644$	0	0
$645$	246.234	0.381759
$646$	344.426	0.533168
$647$	759.683i	1.17416i	0.809528	+	0.587081i	$0.199723\pi$
$647$	759.683i	1.17416i	−0.809528	+	0.587081i	$0.800277\pi$
$648$	25.4558	0.0392837
$649$	− 28.2401i	− 0.0435133i
$650$	− 104.776i	− 0.161194i
$651$	0	0
$652$	−65.4983	−0.100458
$653$	1149.79	1.76078	0.880392	−	0.474247i	$-0.157280\pi$
$653$	1149.79	1.76078	0.880392	+	0.474247i	$0.157280\pi$
$654$	− 117.067i	− 0.179001i
$655$	−278.806	−0.425658
$656$	152.159i	0.231950i
$657$	− 118.589i	− 0.180500i
$658$	0	0
$659$	213.700	0.324280	0.162140	−	0.986768i	$-0.448160\pi$
$659$	213.700	0.324280	0.162140	+	0.986768i	$0.448160\pi$
$660$	−8.96044	−0.0135764
$661$	− 768.957i	− 1.16332i	−0.813431	−	0.581662i	$-0.802403\pi$
$661$	− 768.957i	− 1.16332i	0.813431	−	0.581662i	$-0.197597\pi$
$662$	−370.916	−0.560296
$663$	323.827i	0.488427i
$664$	16.3954i	0.0246919i
$665$	0	0
$666$	226.200	0.339639
$667$	−1191.66	−1.78660
$668$	− 343.317i	− 0.513948i
$669$	−28.9905	−0.0433341
$670$	393.672i	0.587571i
$671$	7.39169i	0.0110159i
$672$	0	0
$673$	−1299.78	−1.93133	−0.965664	−	0.259796i	$-0.916345\pi$
$673$	−1299.78	−1.93133	−0.965664	+	0.259796i	$0.916345\pi$
$674$	817.592	1.21305
$675$	25.9808i	0.0384900i
$676$	−101.121	−0.149587
$677$	1239.68i	1.83114i	0.402160	+	0.915569i	$0.368259\pi$
$677$	1239.68i	1.83114i	−0.402160	+	0.915569i	$0.631741\pi$
$678$	− 39.7509i	− 0.0586297i
$679$	0	0
$680$	79.8005	0.117354
$681$	−733.475	−1.07706
$682$	47.1465i	0.0691298i
$683$	−466.086	−0.682410	−0.341205	−	0.939989i	$-0.610835\pi$
$683$	−466.086	−0.682410	−0.341205	+	0.939989i	$0.610835\pi$
$684$	− 115.813i	− 0.169317i
$685$	− 170.089i	− 0.248304i
$686$	0	0
$687$	701.193	1.02066
$688$	−254.310	−0.369636
$689$	308.680i	0.448012i
$690$	133.101	0.192899
$691$	− 108.458i	− 0.156958i	−0.996916	−	0.0784788i	$-0.974994\pi$
$691$	− 108.458i	− 0.156958i	0.996916	−	0.0784788i	$-0.0250063\pi$
$692$	404.921i	0.585145i
$693$	0	0
$694$	650.145	0.936808
$695$	205.199	0.295251
$696$	240.237i	0.345168i
$697$	479.970	0.688622
$698$	− 550.337i	− 0.788449i
$699$	480.295i	0.687117i
$700$	0	0
$701$	528.400	0.753780	0.376890	−	0.926258i	$-0.376993\pi$
$701$	528.400	0.753780	0.376890	+	0.926258i	$0.376993\pi$
$702$	108.886	0.155109
$703$	− 1029.11i	− 1.46388i
$704$	9.25431	0.0131453
$705$	− 97.5709i	− 0.138398i
$706$	− 913.521i	− 1.29394i
$707$	0	0
$708$	84.5675	0.119446
$709$	−933.557	−1.31672	−0.658362	−	0.752701i	$-0.728750\pi$
$709$	−933.557	−1.31672	−0.658362	+	0.752701i	$0.728750\pi$
$710$	− 376.224i	− 0.529893i
$711$	278.613	0.391861
$712$	429.875i	0.603757i
$713$	− 700.326i	− 0.982224i
$714$	0	0
$715$	−38.3280	−0.0536055
$716$	−157.384	−0.219810
$717$	502.722i	0.701147i
$718$	−93.3964	−0.130078
$719$	643.690i	0.895258i	0.894219	+	0.447629i	$0.147731\pi$
$719$	643.690i	0.895258i	−0.894219	+	0.447629i	$0.852269\pi$
$720$	− 26.8328i	− 0.0372678i
$721$	0	0
$722$	−16.3670	−0.0226690
$723$	−701.147	−0.969775
$724$	− 116.302i	− 0.160638i
$725$	−245.191	−0.338194
$726$	− 293.110i	− 0.403733i
$727$	− 317.353i	− 0.436524i	−0.975890	−	0.218262i	$-0.929961\pi$
$727$	− 317.353i	− 0.436524i	0.975890	−	0.218262i	$-0.0700388\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	−125.003	−0.171237
$731$	802.192i	1.09739i
$732$	−22.1350	−0.0302391
$733$	− 839.494i	− 1.14529i	−0.819805	−	0.572643i	$-0.805918\pi$
$733$	− 839.494i	− 1.14529i	0.819805	−	0.572643i	$-0.194082\pi$
$734$	− 848.431i	− 1.15590i
$735$	0	0
$736$	−137.466	−0.186774
$737$	144.009	0.195399
$738$	− 161.389i	− 0.218685i
$739$	−918.805	−1.24331	−0.621654	−	0.783292i	$-0.713539\pi$
$739$	−918.805	−1.24331	−0.621654	+	0.783292i	$0.713539\pi$
$740$	− 238.436i	− 0.322210i
$741$	− 495.386i	− 0.668537i
$742$	0	0
$743$	−1034.18	−1.39190	−0.695952	−	0.718088i	$-0.745017\pi$
$743$	−1034.18	−1.39190	−0.695952	+	0.718088i	$0.745017\pi$
$744$	−141.184	−0.189764
$745$	− 221.183i	− 0.296889i
$746$	572.407	0.767301
$747$	− 17.3900i	− 0.0232797i
$748$	− 29.1917i	− 0.0390263i
$749$	0	0
$750$	27.3861	0.0365148
$751$	−681.896	−0.907985	−0.453992	−	0.891006i	$-0.650001\pi$
$751$	−681.896	−0.907985	−0.453992	+	0.891006i	$0.650001\pi$
$752$	100.771i	0.134004i
$753$	−269.863	−0.358384
$754$	1027.61i	1.36287i
$755$	218.665i	0.289623i
$756$	0	0
$757$	183.172	0.241971	0.120985	−	0.992654i	$-0.461395\pi$
$757$	183.172	0.241971	0.120985	+	0.992654i	$0.461395\pi$
$758$	−338.952	−0.447166
$759$	− 48.6893i	− 0.0641493i
$760$	−122.078	−0.160628
$761$	− 777.971i	− 1.02230i	−0.859491	−	0.511150i	$-0.829220\pi$
$761$	− 777.971i	− 1.02230i	0.859491	−	0.511150i	$-0.170780\pi$
$762$	197.341i	0.258978i
$763$	0	0
$764$	736.816	0.964419
$765$	−84.6412	−0.110642
$766$	903.838i	1.17995i
$767$	361.735	0.471623
$768$	27.7128i	0.0360844i
$769$	302.546i	0.393428i	0.980461	+	0.196714i	$0.0630271\pi$
$769$	302.546i	0.393428i	−0.980461	+	0.196714i	$0.936973\pi$
$770$	0	0
$771$	44.8630	0.0581881
$772$	561.638	0.727510
$773$	128.167i	0.165804i	0.996558	+	0.0829020i	$0.0264189\pi$
$773$	128.167i	0.165804i	−0.996558	+	0.0829020i	$0.973581\pi$
$774$	269.736	0.348496
$775$	− 144.096i	− 0.185930i
$776$	− 408.169i	− 0.525991i
$777$	0	0
$778$	737.645	0.948130
$779$	−734.250	−0.942555
$780$	− 114.776i	− 0.147149i
$781$	−137.626	−0.176218
$782$	433.620i	0.554502i
$783$	− 254.810i	− 0.325428i
$784$	0	0
$785$	298.297	0.379996
$786$	−305.417	−0.388571
$787$	538.450i	0.684180i	0.939667	+	0.342090i	$0.111135\pi$
$787$	538.450i	0.684180i	−0.939667	+	0.342090i	$0.888865\pi$
$788$	15.2356	0.0193345
$789$	− 546.990i	− 0.693270i
$790$	− 293.684i	− 0.371752i
$791$	0	0
$792$	−9.81567	−0.0123935
$793$	−94.6818	−0.119397
$794$	205.287i	0.258548i
$795$	−80.6821	−0.101487
$796$	403.671i	0.507125i
$797$	− 207.481i	− 0.260328i	−0.991492	−	0.130164i	$-0.958450\pi$
$797$	− 207.481i	− 0.260328i	0.991492	−	0.130164i	$-0.0415504\pi$
$798$	0	0
$799$	317.871	0.397836
$800$	−28.2843	−0.0353553
$801$	− 455.951i	− 0.569228i
$802$	302.702	0.377434
$803$	45.7273i	0.0569455i
$804$	431.247i	0.536376i
$805$	0	0
$806$	−603.911	−0.749269
$807$	−82.8315	−0.102641
$808$	110.547i	0.136816i
$809$	1445.72	1.78704	0.893520	−	0.449022i	$-0.148228\pi$
$809$	1445.72	1.78704	0.893520	+	0.449022i	$0.148228\pi$
$810$	28.4605i	0.0351364i
$811$	207.868i	0.256310i	0.991754	+	0.128155i	$0.0409055\pi$
$811$	207.868i	0.256310i	−0.991754	+	0.128155i	$0.959094\pi$
$812$	0	0
$813$	−589.161	−0.724675
$814$	−87.2218	−0.107152
$815$	− 73.2294i	− 0.0898520i
$816$	87.4171	0.107129
$817$	− 1227.18i	− 1.50206i
$818$	783.435i	0.957745i
$819$	0	0
$820$	−170.119	−0.207462
$821$	615.284	0.749432	0.374716	−	0.927140i	$-0.377740\pi$
$821$	615.284	0.749432	0.374716	+	0.927140i	$0.377740\pi$
$822$	− 186.323i	− 0.226670i
$823$	−335.608	−0.407786	−0.203893	−	0.978993i	$-0.565359\pi$
$823$	−335.608	−0.407786	−0.203893	+	0.978993i	$0.565359\pi$
$824$	107.036i	0.129898i
$825$	− 10.0181i	− 0.0121431i
$826$	0	0
$827$	−1453.38	−1.75741	−0.878707	−	0.477362i	$-0.841593\pi$
$827$	−1453.38	−1.75741	−0.878707	+	0.477362i	$0.841593\pi$
$828$	145.804	0.176092
$829$	12.9867i	0.0156655i	0.999969	+	0.00783273i	$0.00249326\pi$
$829$	12.9867i	0.0156655i	−0.999969	+	0.00783273i	$0.997507\pi$
$830$	−18.3306	−0.0220851
$831$	364.747i	0.438925i
$832$	118.541i	0.142477i
$833$	0	0
$834$	224.785	0.269526
$835$	383.840	0.459689
$836$	44.6570i	0.0534175i
$837$	149.749	0.178911
$838$	379.014i	0.452284i
$839$	− 940.714i	− 1.12123i	−0.828076	−	0.560616i	$-0.810564\pi$
$839$	− 940.714i	− 1.12123i	0.828076	−	0.560616i	$-0.189436\pi$
$840$	0	0
$841$	1563.74	1.85939
$842$	−12.9106	−0.0153332
$843$	817.156i	0.969343i
$844$	60.7635	0.0719947
$845$	− 113.056i	− 0.133794i
$846$	− 106.884i	− 0.126340i
$847$	0	0
$848$	83.3281	0.0982643
$849$	811.068	0.955322
$850$	89.2197i	0.104964i
$851$	1295.61	1.52246
$852$	− 412.133i	− 0.483724i
$853$	1176.97i	1.37980i	0.723903	+	0.689902i	$0.242346\pi$
$853$	1176.97i	1.37980i	−0.723903	+	0.689902i	$0.757654\pi$
$854$	0	0
$855$	129.483	0.151442
$856$	234.850	0.274358
$857$	− 479.805i	− 0.559865i	−0.960020	−	0.279933i	$-0.909688\pi$
$857$	− 479.805i	− 0.559865i	0.960020	−	0.279933i	$-0.0903121\pi$
$858$	−41.9862	−0.0489349
$859$	1368.94i	1.59364i	0.604215	+	0.796821i	$0.293487\pi$
$859$	1368.94i	1.59364i	−0.604215	+	0.796821i	$0.706513\pi$
$860$	− 284.327i	− 0.330613i
$861$	0	0
$862$	−379.635	−0.440412
$863$	85.5353	0.0991139	0.0495570	−	0.998771i	$-0.484219\pi$
$863$	85.5353	0.0991139	0.0495570	+	0.998771i	$0.484219\pi$
$864$	− 29.3939i	− 0.0340207i
$865$	−452.715	−0.523370
$866$	− 667.868i	− 0.771210i
$867$	224.815i	0.259302i
$868$	0	0
$869$	−107.432	−0.123627
$870$	−268.593	−0.308728
$871$	1844.64i	2.11784i
$872$	−135.177	−0.155020
$873$	432.929i	0.495909i
$874$	− 663.346i	− 0.758977i
$875$	0	0
$876$	−136.934	−0.156318
$877$	−473.269	−0.539646	−0.269823	−	0.962910i	$-0.586965\pi$
$877$	−473.269	−0.539646	−0.269823	+	0.962910i	$0.586965\pi$
$878$	776.958i	0.884918i
$879$	109.997	0.125139
$880$	10.3466i	0.0117575i
$881$	442.658i	0.502449i	0.967929	+	0.251225i	$0.0808333\pi$
$881$	442.658i	0.502449i	−0.967929	+	0.251225i	$0.919167\pi$
$882$	0	0
$883$	432.227	0.489498	0.244749	−	0.969586i	$-0.421294\pi$
$883$	432.227	0.489498	0.244749	+	0.969586i	$0.421294\pi$
$884$	373.923	0.422990
$885$	94.5494i	0.106835i
$886$	665.826	0.751497
$887$	521.477i	0.587911i	0.955819	+	0.293956i	$0.0949718\pi$
$887$	521.477i	0.587911i	−0.955819	+	0.293956i	$0.905028\pi$
$888$	− 261.193i	− 0.294136i
$889$	0	0
$890$	−480.615	−0.540017
$891$	10.4111	0.0116847
$892$	33.4754i	0.0375284i
$893$	−486.274	−0.544539
$894$	− 242.293i	− 0.271022i
$895$	− 175.960i	− 0.196604i
$896$	0	0
$897$	623.673	0.695287
$898$	791.288	0.881167
$899$	1413.24i	1.57201i
$900$	30.0000	0.0333333
$901$	− 262.849i	− 0.291731i
$902$	62.2311i	0.0689923i
$903$	0	0
$904$	−45.9004	−0.0507748
$905$	130.029	0.143679
$906$	239.536i	0.264388i
$907$	1267.74	1.39773	0.698865	−	0.715253i	$-0.253689\pi$
$907$	1267.74	1.39773	0.698865	+	0.715253i	$0.253689\pi$
$908$	846.944i	0.932758i
$909$	− 117.253i	− 0.128991i
$910$	0	0
$911$	789.834	0.866997	0.433498	−	0.901154i	$-0.357279\pi$
$911$	789.834	0.866997	0.433498	+	0.901154i	$0.357279\pi$
$912$	−133.729	−0.146633
$913$	6.70550i	0.00734447i
$914$	887.246	0.970729
$915$	− 24.7477i	− 0.0270467i
$916$	− 809.668i	− 0.883916i
$917$	0	0
$918$	−92.7198	−0.101002
$919$	−1013.63	−1.10297	−0.551486	−	0.834184i	$-0.685939\pi$
$919$	−1013.63	−1.10297	−0.551486	+	0.834184i	$0.685939\pi$
$920$	− 153.691i	− 0.167056i
$921$	366.519	0.397958
$922$	316.301i	0.343060i
$923$	− 1762.88i	− 1.90995i
$924$	0	0
$925$	266.579	0.288194
$926$	−561.732	−0.606622
$927$	− 113.529i	− 0.122469i
$928$	277.402	0.298924
$929$	631.122i	0.679357i	0.940542	+	0.339678i	$0.110318\pi$
$929$	631.122i	0.679357i	−0.940542	+	0.339678i	$0.889682\pi$
$930$	− 157.849i	− 0.169730i
$931$	0	0
$932$	554.596	0.595061
$933$	116.618	0.124993
$934$	− 471.958i	− 0.505308i
$935$	32.6373	0.0349062
$936$	− 125.731i	− 0.134328i
$937$	− 867.113i	− 0.925414i	−0.886511	−	0.462707i	$-0.846878\pi$
$937$	− 867.113i	− 0.925414i	0.886511	−	0.462707i	$-0.153122\pi$
$938$	0	0
$939$	239.428	0.254982
$940$	−112.665	−0.119857
$941$	1329.38i	1.41273i	0.707846	+	0.706367i	$0.249667\pi$
$941$	1329.38i	1.41273i	−0.707846	+	0.706367i	$0.750333\pi$
$942$	326.768	0.346888
$943$	− 924.395i	− 0.980270i
$944$	− 97.6502i	− 0.103443i
$945$	0	0
$946$	−104.009	−0.109946
$947$	−1541.48	−1.62775	−0.813877	−	0.581037i	$-0.802647\pi$
$947$	−1541.48	−1.62775	−0.813877	+	0.581037i	$0.802647\pi$
$948$	− 321.715i	− 0.339362i
$949$	−585.731	−0.617209
$950$	− 136.487i	− 0.143670i
$951$	34.3367i	0.0361059i
$952$	0	0
$953$	−114.779	−0.120439	−0.0602196	−	0.998185i	$-0.519180\pi$
$953$	−114.779	−0.120439	−0.0602196	+	0.998185i	$0.519180\pi$
$954$	−88.3828	−0.0926445
$955$	823.785i	0.862603i
$956$	580.493	0.607211
$957$	98.2537i	0.102668i
$958$	− 861.020i	− 0.898769i
$959$	0	0
$960$	−30.9839	−0.0322749
$961$	130.458	0.135752
$962$	− 1117.24i	− 1.16138i
$963$	−249.096	−0.258667
$964$	809.615i	0.839850i
$965$	627.930i	0.650705i
$966$	0	0
$967$	−881.904	−0.912000	−0.456000	−	0.889980i	$-0.650718\pi$
$967$	−881.904	−0.912000	−0.456000	+	0.889980i	$0.650718\pi$
$968$	−338.455	−0.349643
$969$	421.834i	0.435330i
$970$	456.347	0.470461
$971$	470.081i	0.484121i	0.970261	+	0.242060i	$0.0778232\pi$
$971$	470.081i	0.484121i	−0.970261	+	0.242060i	$0.922177\pi$
$972$	31.1769i	0.0320750i
$973$	0	0
$974$	−450.394	−0.462417
$975$	128.324	0.131614
$976$	25.5593i	0.0261879i
$977$	−1055.94	−1.08080	−0.540400	−	0.841408i	$-0.681727\pi$
$977$	−1055.94	−1.08080	−0.540400	+	0.841408i	$0.681727\pi$
$978$	− 80.2187i	− 0.0820233i
$979$	175.813i	0.179584i
$980$	0	0
$981$	143.377	0.146154
$982$	740.063	0.753628
$983$	147.960i	0.150519i	0.997164	+	0.0752593i	$0.0239784\pi$
$983$	147.960i	0.150519i	−0.997164	+	0.0752593i	$0.976022\pi$
$984$	−186.356	−0.189386
$985$	17.0339i	0.0172933i
$986$	− 875.034i	− 0.887458i
$987$	0	0
$988$	−572.022	−0.578970
$989$	1544.98	1.56216
$990$	− 10.9743i	− 0.0110851i
$991$	−1306.44	−1.31831	−0.659153	−	0.752009i	$-0.729085\pi$
$991$	−1306.44	−1.31831	−0.659153	+	0.752009i	$0.729085\pi$
$992$	163.026i	0.164340i
$993$	− 454.278i	− 0.457480i
$994$	0	0
$995$	−451.318	−0.453586
$996$	−20.0802	−0.0201608
$997$	353.596i	0.354660i	0.984151	+	0.177330i	$0.0567460\pi$
$997$	353.596i	0.354660i	−0.984151	+	0.177330i	$0.943254\pi$
$998$	−1108.48	−1.11071
$999$	277.037i	0.277314i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.3.f.d.391.16		16
7.2	even	3		210.3.o.b.31.3	✓	16
7.3	odd	6		210.3.o.b.61.3	yes	16
7.6	odd	2	inner	1470.3.f.d.391.10		16
21.2	odd	6		630.3.v.c.451.5		16
21.17	even	6		630.3.v.c.271.5		16
35.2	odd	12		1050.3.q.e.199.13		32
35.3	even	12		1050.3.q.e.649.14		32
35.9	even	6		1050.3.p.i.451.8		16
35.17	even	12		1050.3.q.e.649.6		32
35.23	odd	12		1050.3.q.e.199.6		32
35.24	odd	6		1050.3.p.i.901.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.o.b.31.3	✓	16	7.2	even	3
210.3.o.b.61.3	yes	16	7.3	odd	6
630.3.v.c.271.5		16	21.17	even	6
630.3.v.c.451.5		16	21.2	odd	6
1050.3.p.i.451.8		16	35.9	even	6
1050.3.p.i.901.8		16	35.24	odd	6
1050.3.q.e.199.6		32	35.23	odd	12
1050.3.q.e.199.13		32	35.2	odd	12
1050.3.q.e.649.6		32	35.17	even	12
1050.3.q.e.649.14		32	35.3	even	12
1470.3.f.d.391.10		16	7.6	odd	2	inner
1470.3.f.d.391.16		16	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:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1470.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} +1.15679 q^{11} +3.46410i q^{12} +14.8176i q^{13} -3.87298 q^{15} +4.00000 q^{16} -12.6176i q^{17} -4.24264 q^{18} +19.3022i q^{19} +4.47214i q^{20} +1.63595 q^{22} -24.3007 q^{23} +4.89898i q^{24} -5.00000 q^{25} +20.9552i q^{26} -5.19615i q^{27} +49.0382 q^{29} -5.47723 q^{30} +28.8191i q^{31} +5.65685 q^{32} +2.00362i q^{33} -17.8439i q^{34} -6.00000 q^{36} -53.3158 q^{37} +27.2974i q^{38} -25.6648 q^{39} +6.32456i q^{40} +38.0398i q^{41} -63.5774 q^{43} +2.31358 q^{44} -6.70820i q^{45} -34.3664 q^{46} +25.1927i q^{47} +6.92820i q^{48} -7.07107 q^{50} +21.8543 q^{51} +29.6351i q^{52} +20.8320 q^{53} -7.34847i q^{54} +2.58666i q^{55} -33.4323 q^{57} +69.3505 q^{58} -24.4125i q^{59} -7.74597 q^{60} +6.38984i q^{61} +40.7564i q^{62} +8.00000 q^{64} -33.1331 q^{65} +2.83354i q^{66} +124.490 q^{67} -25.2351i q^{68} -42.0901i q^{69} -118.973 q^{71} -8.48528 q^{72} +39.5295i q^{73} -75.3999 q^{74} -8.66025i q^{75} +38.6043i q^{76} -36.2955 q^{78} -92.8711 q^{79} +8.94427i q^{80} +9.00000 q^{81} +53.7964i q^{82} +5.79665i q^{83} +28.2137 q^{85} -89.9120 q^{86} +84.9366i q^{87} +3.27189 q^{88} +151.984i q^{89} -9.48683i q^{90} -48.6014 q^{92} -49.9162 q^{93} +35.6279i q^{94} -43.1609 q^{95} +9.79796i q^{96} -144.310i q^{97} -3.47036 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) 16 * q + 32 * q^4 - 48 * q^9	\( 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) \) 16 * q + 32 * q^4 - 48 * q^9 + 8 * q^11 + 64 * q^16 - 48 * q^22 + 24 * q^23 - 80 * q^25 + 72 * q^29 - 96 * q^36 - 88 * q^37 - 72 * q^39 - 56 * q^43 + 16 * q^44 - 16 * q^46 + 24 * q^51 - 64 * q^53 + 144 * q^57 + 176 * q^58 + 128 * q^64 - 40 * q^65 + 328 * q^67 - 136 * q^71 + 224 * q^74 - 560 * q^79 + 144 * q^81 + 120 * q^85 + 176 * q^86 - 96 * q^88 + 48 * q^92 + 240 * q^93 - 400 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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