LMFDB - Embedded newform 735.3.h.a.391.1

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			391.1
Root			$-1.26021 - 2.18275i$ of defining polynomial
Character	$\chi$	$=$	735.391
Dual form			735.3.h.a.391.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.52043 q^{2} -1.73205i q^{3} +8.39341 q^{4} +2.23607i q^{5} +6.09756i q^{6} -15.4667 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-3.52043 q^{2} -1.73205i q^{3} +8.39341 q^{4} +2.23607i q^{5} +6.09756i q^{6} -15.4667 q^{8} -3.00000 q^{9} -7.87192i q^{10} +2.59370 q^{11} -14.5378i q^{12} +11.5763i q^{13} +3.87298 q^{15} +20.8757 q^{16} +23.2045i q^{17} +10.5613 q^{18} -29.9689i q^{19} +18.7682i q^{20} -9.13094 q^{22} -35.1084 q^{23} +26.7891i q^{24} -5.00000 q^{25} -40.7535i q^{26} +5.19615i q^{27} -24.4905 q^{29} -13.6346 q^{30} -37.4533i q^{31} -11.6247 q^{32} -4.49242i q^{33} -81.6898i q^{34} -25.1802 q^{36} +25.7486 q^{37} +105.503i q^{38} +20.0507 q^{39} -34.5846i q^{40} +3.71113i q^{41} +74.2225 q^{43} +21.7700 q^{44} -6.70820i q^{45} +123.597 q^{46} -3.37919i q^{47} -36.1578i q^{48} +17.6021 q^{50} +40.1914 q^{51} +97.1647i q^{52} -40.0385 q^{53} -18.2927i q^{54} +5.79969i q^{55} -51.9076 q^{57} +86.2170 q^{58} -49.3264i q^{59} +32.5075 q^{60} +0.883454i q^{61} +131.852i q^{62} -42.5790 q^{64} -25.8854 q^{65} +15.8153i q^{66} -65.0544 q^{67} +194.765i q^{68} +60.8096i q^{69} +86.0786 q^{71} +46.4001 q^{72} -61.5772i q^{73} -90.6462 q^{74} +8.66025i q^{75} -251.541i q^{76} -70.5872 q^{78} +27.5435 q^{79} +46.6795i q^{80} +9.00000 q^{81} -13.0648i q^{82} -131.445i q^{83} -51.8869 q^{85} -261.295 q^{86} +42.4187i q^{87} -40.1160 q^{88} +65.2531i q^{89} +23.6157i q^{90} -294.679 q^{92} -64.8710 q^{93} +11.8962i q^{94} +67.0124 q^{95} +20.1345i q^{96} +42.2375i q^{97} -7.78111 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.52043	−1.76021	−0.880107	−	0.474776i	$-0.842529\pi$
$2$	−3.52043	−1.76021	−0.880107	+	0.474776i	$0.842529\pi$
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	8.39341	2.09835
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	6.09756i	1.01626i
$7$	0	0
$7$	0	0
$8$	−15.4667	−1.93334
$9$	−3.00000	−0.333333
$10$	− 7.87192i	− 0.787192i
$11$	2.59370	0.235791	0.117896	−	0.993026i	$-0.462385\pi$
$11$	2.59370	0.235791	0.117896	+	0.993026i	$0.462385\pi$
$12$	− 14.5378i	− 1.21148i
$13$	11.5763i	0.890485i	0.895410	+	0.445242i	$0.146883\pi$
$13$	11.5763i	0.890485i	−0.895410	+	0.445242i	$0.853117\pi$
$14$	0	0
$15$	3.87298	0.258199
$16$	20.8757	1.30473
$17$	23.2045i	1.36497i	0.730899	+	0.682486i	$0.239101\pi$
$17$	23.2045i	1.36497i	−0.730899	+	0.682486i	$0.760899\pi$
$18$	10.5613	0.586738
$19$	− 29.9689i	− 1.57731i	−0.614837	−	0.788654i	$-0.710778\pi$
$19$	− 29.9689i	− 1.57731i	0.614837	−	0.788654i	$-0.289222\pi$
$20$	18.7682i	0.938412i
$21$	0	0
$22$	−9.13094	−0.415043
$23$	−35.1084	−1.52645	−0.763226	−	0.646131i	$-0.776386\pi$
$23$	−35.1084	−1.52645	−0.763226	+	0.646131i	$0.776386\pi$
$24$	26.7891i	1.11621i
$25$	−5.00000	−0.200000
$26$	− 40.7535i	− 1.56744i
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−24.4905	−0.844499	−0.422250	−	0.906480i	$-0.638759\pi$
$29$	−24.4905	−0.844499	−0.422250	+	0.906480i	$0.638759\pi$
$30$	−13.6346	−0.454485
$31$	− 37.4533i	− 1.20817i	−0.796920	−	0.604085i	$-0.793539\pi$
$31$	− 37.4533i	− 1.20817i	0.796920	−	0.604085i	$-0.206461\pi$
$32$	−11.6247	−0.363271
$33$	− 4.49242i	− 0.136134i
$34$	− 81.6898i	− 2.40264i
$35$	0	0
$36$	−25.1802	−0.699451
$37$	25.7486	0.695909	0.347954	−	0.937511i	$-0.386876\pi$
$37$	25.7486	0.695909	0.347954	+	0.937511i	$0.386876\pi$
$38$	105.503i	2.77640i
$39$	20.0507	0.514122
$40$	− 34.5846i	− 0.864614i
$41$	3.71113i	0.0905155i	0.998975	+	0.0452577i	$0.0144109\pi$
$41$	3.71113i	0.0905155i	−0.998975	+	0.0452577i	$0.985589\pi$
$42$	0	0
$43$	74.2225	1.72611	0.863053	−	0.505114i	$-0.168549\pi$
$43$	74.2225	1.72611	0.863053	+	0.505114i	$0.168549\pi$
$44$	21.7700	0.494773
$45$	− 6.70820i	− 0.149071i
$46$	123.597	2.68688
$47$	− 3.37919i	− 0.0718976i	−0.999354	−	0.0359488i	$-0.988555\pi$
$47$	− 3.37919i	− 0.0718976i	0.999354	−	0.0359488i	$-0.0114453\pi$
$48$	− 36.1578i	− 0.753287i
$49$	0	0
$50$	17.6021	0.352043
$51$	40.1914	0.788066
$52$	97.1647i	1.86855i
$53$	−40.0385	−0.755444	−0.377722	−	0.925919i	$-0.623293\pi$
$53$	−40.0385	−0.755444	−0.377722	+	0.925919i	$0.623293\pi$
$54$	− 18.2927i	− 0.338753i
$55$	5.79969i	0.105449i
$56$	0	0
$57$	−51.9076	−0.910660
$58$	86.2170	1.48650
$59$	− 49.3264i	− 0.836041i	−0.908438	−	0.418021i	$-0.862724\pi$
$59$	− 49.3264i	− 0.836041i	0.908438	−	0.418021i	$-0.137276\pi$
$60$	32.5075	0.541792
$61$	0.883454i	0.0144829i	0.999974	+	0.00724143i	$0.00230504\pi$
$61$	0.883454i	0.0144829i	−0.999974	+	0.00724143i	$0.997695\pi$
$62$	131.852i	2.12664i
$63$	0	0
$64$	−42.5790	−0.665297
$65$	−25.8854	−0.398237
$66$	15.8153i	0.239625i
$67$	−65.0544	−0.970961	−0.485481	−	0.874247i	$-0.661355\pi$
$67$	−65.0544	−0.970961	−0.485481	+	0.874247i	$0.661355\pi$
$68$	194.765i	2.86419i
$69$	60.8096i	0.881298i
$70$	0	0
$71$	86.0786	1.21237	0.606187	−	0.795322i	$-0.292698\pi$
$71$	86.0786	1.21237	0.606187	+	0.795322i	$0.292698\pi$
$72$	46.4001	0.644445
$73$	− 61.5772i	− 0.843523i	−0.906707	−	0.421761i	$-0.861412\pi$
$73$	− 61.5772i	− 0.843523i	0.906707	−	0.421761i	$-0.138588\pi$
$74$	−90.6462	−1.22495
$75$	8.66025i	0.115470i
$76$	− 251.541i	− 3.30975i
$77$	0	0
$78$	−70.5872	−0.904964
$79$	27.5435	0.348652	0.174326	−	0.984688i	$-0.444225\pi$
$79$	27.5435	0.348652	0.174326	+	0.984688i	$0.444225\pi$
$80$	46.6795i	0.583494i
$81$	9.00000	0.111111
$82$	− 13.0648i	− 0.159327i
$83$	− 131.445i	− 1.58367i	−0.610732	−	0.791837i	$-0.709125\pi$
$83$	− 131.445i	− 1.58367i	0.610732	−	0.791837i	$-0.290875\pi$
$84$	0	0
$85$	−51.8869	−0.610434
$86$	−261.295	−3.03832
$87$	42.4187i	0.487572i
$88$	−40.1160	−0.455863
$89$	65.2531i	0.733181i	0.930382	+	0.366590i	$0.119475\pi$
$89$	65.2531i	0.733181i	−0.930382	+	0.366590i	$0.880525\pi$
$90$	23.6157i	0.262397i
$91$	0	0
$92$	−294.679	−3.20304
$93$	−64.8710	−0.697538
$94$	11.8962i	0.126555i
$95$	67.0124	0.705394
$96$	20.1345i	0.209734i
$97$	42.2375i	0.435438i	0.976011	+	0.217719i	$0.0698616\pi$
$97$	42.2375i	0.435438i	−0.976011	+	0.217719i	$0.930138\pi$
$98$	0	0
$99$	−7.78111	−0.0785970
$100$	−41.9671	−0.419671
$101$	− 149.966i	− 1.48481i	−0.669950	−	0.742406i	$-0.733684\pi$
$101$	− 149.966i	− 1.48481i	0.669950	−	0.742406i	$-0.266316\pi$
$102$	−141.491	−1.38717
$103$	− 139.678i	− 1.35609i	−0.735019	−	0.678047i	$-0.762827\pi$
$103$	− 139.678i	− 1.35609i	0.735019	−	0.678047i	$-0.237173\pi$
$104$	− 179.047i	− 1.72161i
$105$	0	0
$106$	140.953	1.32974
$107$	−181.240	−1.69383	−0.846914	−	0.531730i	$-0.821542\pi$
$107$	−181.240	−1.69383	−0.846914	+	0.531730i	$0.821542\pi$
$108$	43.6134i	0.403828i
$109$	−73.8098	−0.677154	−0.338577	−	0.940939i	$-0.609946\pi$
$109$	−73.8098	−0.677154	−0.338577	+	0.940939i	$0.609946\pi$
$110$	− 20.4174i	− 0.185613i
$111$	− 44.5979i	− 0.401783i
$112$	0	0
$113$	7.38562	0.0653595	0.0326797	−	0.999466i	$-0.489596\pi$
$113$	7.38562	0.0653595	0.0326797	+	0.999466i	$0.489596\pi$
$114$	182.737	1.60296
$115$	− 78.5048i	− 0.682650i
$116$	−205.559	−1.77206
$117$	− 34.7289i	− 0.296828i
$118$	173.650i	1.47161i
$119$	0	0
$120$	−59.9022	−0.499185
$121$	−114.273	−0.944403
$122$	− 3.11014i	− 0.0254929i
$123$	6.42787	0.0522591
$124$	− 314.361i	− 2.53517i
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−208.640	−1.64283	−0.821416	−	0.570329i	$-0.806816\pi$
$127$	−208.640	−1.64283	−0.821416	+	0.570329i	$0.806816\pi$
$128$	196.395	1.53434
$129$	− 128.557i	− 0.996568i
$130$	91.1277	0.700982
$131$	− 109.581i	− 0.836494i	−0.908333	−	0.418247i	$-0.862645\pi$
$131$	− 109.581i	− 0.836494i	0.908333	−	0.418247i	$-0.137355\pi$
$132$	− 37.7068i	− 0.285657i
$133$	0	0
$134$	229.019	1.70910
$135$	−11.6190	−0.0860663
$136$	− 358.897i	− 2.63895i
$137$	−177.452	−1.29527	−0.647637	−	0.761949i	$-0.724242\pi$
$137$	−177.452	−1.29527	−0.647637	+	0.761949i	$0.724242\pi$
$138$	− 214.076i	− 1.55127i
$139$	− 169.894i	− 1.22226i	−0.791532	−	0.611128i	$-0.790716\pi$
$139$	− 169.894i	− 1.22226i	0.791532	−	0.611128i	$-0.209284\pi$
$140$	0	0
$141$	−5.85293	−0.0415101
$142$	−303.033	−2.13404
$143$	30.0255i	0.209968i
$144$	−62.6271	−0.434911
$145$	− 54.7624i	− 0.377672i
$146$	216.778i	1.48478i
$147$	0	0
$148$	216.119	1.46026
$149$	85.3856	0.573058	0.286529	−	0.958072i	$-0.407499\pi$
$149$	85.3856	0.573058	0.286529	+	0.958072i	$0.407499\pi$
$150$	− 30.4878i	− 0.203252i
$151$	−137.995	−0.913876	−0.456938	−	0.889498i	$-0.651054\pi$
$151$	−137.995	−0.913876	−0.456938	+	0.889498i	$0.651054\pi$
$152$	463.519i	3.04947i
$153$	− 69.6135i	− 0.454990i
$154$	0	0
$155$	83.7481	0.540310
$156$	168.294	1.07881
$157$	6.86671i	0.0437370i	0.999761	+	0.0218685i	$0.00696151\pi$
$157$	6.86671i	0.0437370i	−0.999761	+	0.0218685i	$0.993038\pi$
$158$	−96.9650	−0.613703
$159$	69.3488i	0.436156i
$160$	− 25.9935i	− 0.162460i
$161$	0	0
$162$	−31.6838	−0.195579
$163$	−276.726	−1.69771	−0.848854	−	0.528627i	$-0.822707\pi$
$163$	−276.726	−1.69771	−0.848854	+	0.528627i	$0.822707\pi$
$164$	31.1491i	0.189933i
$165$	10.0454	0.0608810
$166$	462.743i	2.78761i
$167$	− 42.3799i	− 0.253772i	−0.991917	−	0.126886i	$-0.959502\pi$
$167$	− 42.3799i	− 0.253772i	0.991917	−	0.126886i	$-0.0404982\pi$
$168$	0	0
$169$	34.9892	0.207036
$170$	182.664	1.07449
$171$	89.9066i	0.525770i
$172$	622.980	3.62198
$173$	− 29.5602i	− 0.170868i	−0.996344	−	0.0854342i	$-0.972772\pi$
$173$	− 29.5602i	− 0.170868i	0.996344	−	0.0854342i	$-0.0272277\pi$
$174$	− 149.332i	− 0.858231i
$175$	0	0
$176$	54.1454	0.307644
$177$	−85.4359	−0.482689
$178$	− 229.719i	− 1.29055i
$179$	−148.682	−0.830624	−0.415312	−	0.909679i	$-0.636328\pi$
$179$	−148.682	−0.830624	−0.415312	+	0.909679i	$0.636328\pi$
$180$	− 56.3047i	− 0.312804i
$181$	257.302i	1.42156i	0.703414	+	0.710780i	$0.251658\pi$
$181$	257.302i	1.42156i	−0.703414	+	0.710780i	$0.748342\pi$
$182$	0	0
$183$	1.53019	0.00836168
$184$	543.011	2.95115
$185$	57.5757i	0.311220i
$186$	228.374	1.22782
$187$	60.1856i	0.321848i
$188$	− 28.3629i	− 0.150867i
$189$	0	0
$190$	−235.912	−1.24164
$191$	121.604	0.636671	0.318336	−	0.947978i	$-0.396876\pi$
$191$	121.604	0.636671	0.318336	+	0.947978i	$0.396876\pi$
$192$	73.7490i	0.384110i
$193$	242.533	1.25665	0.628323	−	0.777953i	$-0.283742\pi$
$193$	242.533	1.25665	0.628323	+	0.777953i	$0.283742\pi$
$194$	− 148.694i	− 0.766464i
$195$	44.8348i	0.229922i
$196$	0	0
$197$	−98.9929	−0.502502	−0.251251	−	0.967922i	$-0.580842\pi$
$197$	−98.9929	−0.502502	−0.251251	+	0.967922i	$0.580842\pi$
$198$	27.3928	0.138348
$199$	78.7639i	0.395798i	0.980222	+	0.197899i	$0.0634119\pi$
$199$	78.7639i	0.395798i	−0.980222	+	0.197899i	$0.936588\pi$
$200$	77.3334	0.386667
$201$	112.678i	0.560585i
$202$	527.945i	2.61359i
$203$	0	0
$204$	337.343	1.65364
$205$	−8.29835	−0.0404797
$206$	491.725i	2.38702i
$207$	105.325	0.508818
$208$	241.664i	1.16184i
$209$	− 77.7303i	− 0.371915i
$210$	0	0
$211$	−107.144	−0.507790	−0.253895	−	0.967232i	$-0.581712\pi$
$211$	−107.144	−0.507790	−0.253895	+	0.967232i	$0.581712\pi$
$212$	−336.060	−1.58519
$213$	− 149.092i	− 0.699965i
$214$	638.041	2.98150
$215$	165.967i	0.771938i
$216$	− 80.3673i	− 0.372071i
$217$	0	0
$218$	259.842	1.19194
$219$	−106.655	−0.487008
$220$	48.6792i	0.221269i
$221$	−268.622	−1.21549
$222$	157.004i	0.707224i
$223$	− 8.72021i	− 0.0391041i	−0.999809	−	0.0195520i	$-0.993776\pi$
$223$	− 8.72021i	− 0.0391041i	0.999809	−	0.0195520i	$-0.00622400\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	−26.0005	−0.115047
$227$	− 253.021i	− 1.11463i	−0.830301	−	0.557316i	$-0.811831\pi$
$227$	− 253.021i	− 1.11463i	0.830301	−	0.557316i	$-0.188169\pi$
$228$	−435.682	−1.91089
$229$	− 145.479i	− 0.635278i	−0.948212	−	0.317639i	$-0.897110\pi$
$229$	− 145.479i	− 0.635278i	0.948212	−	0.317639i	$-0.102890\pi$
$230$	276.370i	1.20161i
$231$	0	0
$232$	378.787	1.63270
$233$	257.516	1.10522	0.552609	−	0.833441i	$-0.313632\pi$
$233$	257.516	1.10522	0.552609	+	0.833441i	$0.313632\pi$
$234$	122.261i	0.522481i
$235$	7.55609	0.0321536
$236$	− 414.017i	− 1.75431i
$237$	− 47.7068i	− 0.201295i
$238$	0	0
$239$	−128.682	−0.538418	−0.269209	−	0.963082i	$-0.586762\pi$
$239$	−128.682	−0.538418	−0.269209	+	0.963082i	$0.586762\pi$
$240$	80.8513	0.336880
$241$	− 9.26332i	− 0.0384370i	−0.999815	−	0.0192185i	$-0.993882\pi$
$241$	− 9.26332i	− 0.0384370i	0.999815	−	0.0192185i	$-0.00611782\pi$
$242$	402.289	1.66235
$243$	− 15.5885i	− 0.0641500i
$244$	7.41519i	0.0303901i
$245$	0	0
$246$	−22.6289	−0.0919872
$247$	346.929	1.40457
$248$	579.278i	2.33580i
$249$	−227.669	−0.914335
$250$	39.3596i	0.157438i
$251$	− 29.9212i	− 0.119208i	−0.998222	−	0.0596040i	$-0.981016\pi$
$251$	− 29.9212i	− 0.119208i	0.998222	−	0.0596040i	$-0.0189838\pi$
$252$	0	0
$253$	−91.0608	−0.359924
$254$	734.501	2.89174
$255$	89.8707i	0.352434i
$256$	−521.078	−2.03546
$257$	− 96.6875i	− 0.376216i	−0.982148	−	0.188108i	$-0.939765\pi$
$257$	− 96.6875i	− 0.376216i	0.982148	−	0.188108i	$-0.0602355\pi$
$258$	452.576i	1.75417i
$259$	0	0
$260$	−217.267	−0.835642
$261$	73.4714	0.281500
$262$	385.771i	1.47241i
$263$	64.2462	0.244282	0.122141	−	0.992513i	$-0.461024\pi$
$263$	64.2462	0.244282	0.122141	+	0.992513i	$0.461024\pi$
$264$	69.4829i	0.263193i
$265$	− 89.5289i	− 0.337845i
$266$	0	0
$267$	113.022	0.423302
$268$	−546.028	−2.03742
$269$	− 197.565i	− 0.734441i	−0.930134	−	0.367220i	$-0.880310\pi$
$269$	− 197.565i	− 0.734441i	0.930134	−	0.367220i	$-0.119690\pi$
$270$	40.9037	0.151495
$271$	154.133i	0.568758i	0.958712	+	0.284379i	$0.0917874\pi$
$271$	154.133i	0.568758i	−0.958712	+	0.284379i	$0.908213\pi$
$272$	484.410i	1.78092i
$273$	0	0
$274$	624.708	2.27996
$275$	−12.9685	−0.0471582
$276$	510.400i	1.84927i
$277$	339.203	1.22456	0.612280	−	0.790641i	$-0.290253\pi$
$277$	339.203	1.22456	0.612280	+	0.790641i	$0.290253\pi$
$278$	598.098i	2.15143i
$279$	112.360i	0.402724i
$280$	0	0
$281$	−111.976	−0.398492	−0.199246	−	0.979949i	$-0.563849\pi$
$281$	−111.976	−0.398492	−0.199246	+	0.979949i	$0.563849\pi$
$282$	20.6048	0.0730667
$283$	− 65.4084i	− 0.231125i	−0.993300	−	0.115563i	$-0.963133\pi$
$283$	− 65.4084i	− 0.231125i	0.993300	−	0.115563i	$-0.0368671\pi$
$284$	722.493	2.54399
$285$	− 116.069i	− 0.407259i
$286$	− 105.703i	− 0.369589i
$287$	0	0
$288$	34.8740	0.121090
$289$	−249.449	−0.863146
$290$	192.787i	0.664783i
$291$	73.1575	0.251400
$292$	− 516.842i	− 1.77001i
$293$	− 100.992i	− 0.344681i	−0.985037	−	0.172341i	$-0.944867\pi$
$293$	− 100.992i	− 0.344681i	0.985037	−	0.172341i	$-0.0551330\pi$
$294$	0	0
$295$	110.297	0.373889
$296$	−398.246	−1.34543
$297$	13.4773i	0.0453780i
$298$	−300.594	−1.00870
$299$	− 406.426i	− 1.35928i
$300$	72.6891i	0.242297i
$301$	0	0
$302$	485.802	1.60862
$303$	−259.749	−0.857257
$304$	− 625.621i	− 2.05796i
$305$	−1.97546	−0.00647693
$306$	245.069i	0.800880i
$307$	400.388i	1.30420i	0.758135	+	0.652098i	$0.226111\pi$
$307$	400.388i	1.30420i	−0.758135	+	0.652098i	$0.773889\pi$
$308$	0	0
$309$	−241.929	−0.782941
$310$	−294.829	−0.951062
$311$	− 35.9987i	− 0.115751i	−0.998324	−	0.0578757i	$-0.981567\pi$
$311$	− 35.9987i	− 0.115751i	0.998324	−	0.0578757i	$-0.0184327\pi$
$312$	−310.119	−0.993970
$313$	114.495i	0.365799i	0.983132	+	0.182900i	$0.0585484\pi$
$313$	114.495i	0.365799i	−0.983132	+	0.182900i	$0.941452\pi$
$314$	− 24.1737i	− 0.0769864i
$315$	0	0
$316$	231.184	0.731596
$317$	4.03216	0.0127197	0.00635987	−	0.999980i	$-0.497976\pi$
$317$	4.03216	0.0127197	0.00635987	+	0.999980i	$0.497976\pi$
$318$	− 244.137i	− 0.767728i
$319$	−63.5210	−0.199125
$320$	− 95.2096i	− 0.297530i
$321$	313.916i	0.977932i
$322$	0	0
$323$	695.413	2.15298
$324$	75.5407	0.233150
$325$	− 57.8815i	− 0.178097i
$326$	974.196	2.98833
$327$	127.842i	0.390955i
$328$	− 57.3989i	− 0.174997i
$329$	0	0
$330$	−35.3640	−0.107164
$331$	507.383	1.53288	0.766439	−	0.642317i	$-0.222027\pi$
$331$	507.383	1.53288	0.766439	+	0.642317i	$0.222027\pi$
$332$	− 1103.27i	− 3.32311i
$333$	−77.2459	−0.231970
$334$	149.195i	0.446692i
$335$	− 145.466i	− 0.434227i
$336$	0	0
$337$	−264.279	−0.784210	−0.392105	−	0.919921i	$-0.628253\pi$
$337$	−264.279	−0.784210	−0.392105	+	0.919921i	$0.628253\pi$
$338$	−123.177	−0.364428
$339$	− 12.7923i	− 0.0377353i
$340$	−435.508	−1.28091
$341$	− 97.1427i	− 0.284876i
$342$	− 316.510i	− 0.925467i
$343$	0	0
$344$	−1147.98	−3.33714
$345$	−135.974	−0.394128
$346$	104.065i	0.300765i
$347$	−234.467	−0.675699	−0.337849	−	0.941200i	$-0.609699\pi$
$347$	−234.467	−0.675699	−0.337849	+	0.941200i	$0.609699\pi$
$348$	356.038i	1.02310i
$349$	54.2133i	0.155339i	0.996979	+	0.0776695i	$0.0247479\pi$
$349$	54.2133i	0.155339i	−0.996979	+	0.0776695i	$0.975252\pi$
$350$	0	0
$351$	−60.1522	−0.171374
$352$	−30.1509	−0.0856560
$353$	− 521.825i	− 1.47826i	−0.673564	−	0.739129i	$-0.735237\pi$
$353$	− 521.825i	− 1.47826i	0.673564	−	0.739129i	$-0.264763\pi$
$354$	300.771	0.849635
$355$	192.478i	0.542190i
$356$	547.696i	1.53847i
$357$	0	0
$358$	523.423	1.46208
$359$	−467.946	−1.30347	−0.651735	−	0.758447i	$-0.725959\pi$
$359$	−467.946	−1.30347	−0.651735	+	0.758447i	$0.725959\pi$
$360$	103.754i	0.288205i
$361$	−537.133	−1.48790
$362$	− 905.814i	− 2.50225i
$363$	197.926i	0.545251i
$364$	0	0
$365$	137.691	0.377235
$366$	−5.38691	−0.0147183
$367$	− 172.225i	− 0.469277i	−0.972083	−	0.234638i	$-0.924609\pi$
$367$	− 172.225i	− 0.469277i	0.972083	−	0.234638i	$-0.0753906\pi$
$368$	−732.913	−1.99161
$369$	− 11.1334i	− 0.0301718i
$370$	− 202.691i	− 0.547814i
$371$	0	0
$372$	−544.489	−1.46368
$373$	−460.971	−1.23585	−0.617924	−	0.786238i	$-0.712026\pi$
$373$	−460.971	−1.23585	−0.617924	+	0.786238i	$0.712026\pi$
$374$	− 211.879i	− 0.566521i
$375$	−19.3649	−0.0516398
$376$	52.2648i	0.139002i
$377$	− 283.509i	− 0.752014i
$378$	0	0
$379$	444.638	1.17319	0.586594	−	0.809881i	$-0.300468\pi$
$379$	444.638	1.17319	0.586594	+	0.809881i	$0.300468\pi$
$380$	562.463	1.48017
$381$	361.375i	0.948490i
$382$	−428.099	−1.12068
$383$	529.463i	1.38241i	0.722659	+	0.691205i	$0.242920\pi$
$383$	529.463i	1.38241i	−0.722659	+	0.691205i	$0.757080\pi$
$384$	− 340.166i	− 0.885849i
$385$	0	0
$386$	−853.818	−2.21196
$387$	−222.668	−0.575369
$388$	354.517i	0.913703i
$389$	195.511	0.502598	0.251299	−	0.967909i	$-0.419142\pi$
$389$	195.511	0.502598	0.251299	+	0.967909i	$0.419142\pi$
$390$	− 157.838i	− 0.404712i
$391$	− 814.674i	− 2.08356i
$392$	0	0
$393$	−189.799	−0.482950
$394$	348.497	0.884511
$395$	61.5892i	0.155922i
$396$	−65.3100	−0.164924
$397$	− 29.2552i	− 0.0736908i	−0.999321	−	0.0368454i	$-0.988269\pi$
$397$	− 29.2552i	− 0.0736908i	0.999321	−	0.0368454i	$-0.0117309\pi$
$398$	− 277.283i	− 0.696690i
$399$	0	0
$400$	−104.379	−0.260946
$401$	210.792	0.525667	0.262833	−	0.964841i	$-0.415343\pi$
$401$	210.792	0.525667	0.262833	+	0.964841i	$0.415343\pi$
$402$	− 396.673i	− 0.986749i
$403$	433.571	1.07586
$404$	− 1258.73i	− 3.11566i
$405$	20.1246i	0.0496904i
$406$	0	0
$407$	66.7843	0.164089
$408$	−621.628	−1.52360
$409$	324.487i	0.793368i	0.917955	+	0.396684i	$0.129839\pi$
$409$	324.487i	0.793368i	−0.917955	+	0.396684i	$0.870161\pi$
$410$	29.2137	0.0712530
$411$	307.357i	0.747826i
$412$	− 1172.37i	− 2.84556i
$413$	0	0
$414$	−370.790	−0.895628
$415$	293.920	0.708241
$416$	− 134.571i	− 0.323487i
$417$	−294.264	−0.705670
$418$	273.644i	0.654651i
$419$	− 693.958i	− 1.65622i	−0.560563	−	0.828112i	$-0.689415\pi$
$419$	− 693.958i	− 1.65622i	0.560563	−	0.828112i	$-0.310585\pi$
$420$	0	0
$421$	−341.554	−0.811292	−0.405646	−	0.914030i	$-0.632953\pi$
$421$	−341.554	−0.811292	−0.405646	+	0.914030i	$0.632953\pi$
$422$	377.191	0.893819
$423$	10.1376i	0.0239659i
$424$	619.264	1.46053
$425$	− 116.023i	− 0.272994i
$426$	524.869i	1.23209i
$427$	0	0
$428$	−1521.22	−3.55425
$429$	52.0057	0.121225
$430$	− 584.274i	− 1.35878i
$431$	−411.452	−0.954646	−0.477323	−	0.878728i	$-0.658393\pi$
$431$	−411.452	−0.954646	−0.477323	+	0.878728i	$0.658393\pi$
$432$	108.473i	0.251096i
$433$	− 443.458i	− 1.02415i	−0.858940	−	0.512077i	$-0.828876\pi$
$433$	− 443.458i	− 1.02415i	0.858940	−	0.512077i	$-0.171124\pi$
$434$	0	0
$435$	−94.8512	−0.218049
$436$	−619.516	−1.42091
$437$	1052.16i	2.40769i
$438$	375.470	0.857238
$439$	309.462i	0.704926i	0.935826	+	0.352463i	$0.114656\pi$
$439$	309.462i	0.704926i	−0.935826	+	0.352463i	$0.885344\pi$
$440$	− 89.7021i	− 0.203868i
$441$	0	0
$442$	945.666	2.13952
$443$	818.661	1.84799	0.923996	−	0.382401i	$-0.124903\pi$
$443$	818.661	1.84799	0.923996	+	0.382401i	$0.124903\pi$
$444$	− 374.329i	− 0.843083i
$445$	−145.910	−0.327888
$446$	30.6989i	0.0688315i
$447$	− 147.892i	− 0.330855i
$448$	0	0
$449$	315.756	0.703243	0.351621	−	0.936142i	$-0.385630\pi$
$449$	315.756	0.703243	0.351621	+	0.936142i	$0.385630\pi$
$450$	−52.8064	−0.117348
$451$	9.62558i	0.0213427i
$452$	61.9906	0.137147
$453$	239.015i	0.527627i
$454$	890.743i	1.96199i
$455$	0	0
$456$	802.839	1.76061
$457$	−187.307	−0.409861	−0.204931	−	0.978776i	$-0.565697\pi$
$457$	−187.307	−0.409861	−0.204931	+	0.978776i	$0.565697\pi$
$458$	512.148i	1.11823i
$459$	−120.574	−0.262689
$460$	− 658.923i	− 1.43244i
$461$	− 8.27599i	− 0.0179523i	−0.999960	−	0.00897613i	$-0.997143\pi$
$461$	− 8.27599i	− 0.0179523i	0.999960	−	0.00897613i	$-0.00285723\pi$
$462$	0	0
$463$	−472.925	−1.02144	−0.510718	−	0.859748i	$-0.670620\pi$
$463$	−472.925	−1.02144	−0.510718	+	0.859748i	$0.670620\pi$
$464$	−511.256	−1.10184
$465$	− 145.056i	− 0.311948i
$466$	−906.565	−1.94542
$467$	716.913i	1.53515i	0.640962	+	0.767573i	$0.278536\pi$
$467$	716.913i	1.53515i	−0.640962	+	0.767573i	$0.721464\pi$
$468$	− 291.494i	− 0.622851i
$469$	0	0
$470$	−26.6007	−0.0565972
$471$	11.8935	0.0252516
$472$	762.917i	1.61635i
$473$	192.511	0.407000
$474$	167.948i	0.354321i
$475$	149.844i	0.315462i
$476$	0	0
$477$	120.116	0.251815
$478$	453.015	0.947731
$479$	205.906i	0.429867i	0.976629	+	0.214933i	$0.0689534\pi$
$479$	205.906i	0.429867i	−0.976629	+	0.214933i	$0.931047\pi$
$480$	−45.0221	−0.0937961
$481$	298.074i	0.619696i
$482$	32.6108i	0.0676573i
$483$	0	0
$484$	−959.138	−1.98169
$485$	−94.4459	−0.194734
$486$	54.8780i	0.112918i
$487$	322.879	0.662996	0.331498	−	0.943456i	$-0.392446\pi$
$487$	322.879	0.662996	0.331498	+	0.943456i	$0.392446\pi$
$488$	− 13.6641i	− 0.0280002i
$489$	479.304i	0.980172i
$490$	0	0
$491$	272.380	0.554745	0.277372	−	0.960763i	$-0.410536\pi$
$491$	272.380	0.554745	0.277372	+	0.960763i	$0.410536\pi$
$492$	53.9518	0.109658
$493$	− 568.290i	− 1.15272i
$494$	−1221.34	−2.47234
$495$	− 17.3991i	− 0.0351497i
$496$	− 781.864i	− 1.57634i
$497$	0	0
$498$	801.494	1.60943
$499$	529.195	1.06051	0.530255	−	0.847838i	$-0.322096\pi$
$499$	529.195	1.06051	0.530255	+	0.847838i	$0.322096\pi$
$500$	− 93.8412i	− 0.187682i
$501$	−73.4041	−0.146515
$502$	105.335i	0.209831i
$503$	− 204.695i	− 0.406948i	−0.979080	−	0.203474i	$-0.934777\pi$
$503$	− 204.695i	− 0.406948i	0.979080	−	0.203474i	$-0.0652232\pi$
$504$	0	0
$505$	335.334	0.664028
$506$	320.573	0.633543
$507$	− 60.6030i	− 0.119533i
$508$	−1751.20	−3.44724
$509$	547.204i	1.07506i	0.843246	+	0.537528i	$0.180642\pi$
$509$	547.204i	1.07506i	−0.843246	+	0.537528i	$0.819358\pi$
$510$	− 316.383i	− 0.620359i
$511$	0	0
$512$	1048.84	2.04851
$513$	155.723	0.303553
$514$	340.381i	0.662221i
$515$	312.329	0.606464
$516$	− 1079.03i	− 2.09115i
$517$	− 8.76461i	− 0.0169528i
$518$	0	0
$519$	−51.1998	−0.0986509
$520$	400.361	0.769926
$521$	34.8121i	0.0668179i	0.999442	+	0.0334090i	$0.0106364\pi$
$521$	34.8121i	0.0668179i	−0.999442	+	0.0334090i	$0.989364\pi$
$522$	−258.651	−0.495500
$523$	− 831.719i	− 1.59029i	−0.606422	−	0.795143i	$-0.707396\pi$
$523$	− 831.719i	− 1.59029i	0.606422	−	0.795143i	$-0.292604\pi$
$524$	− 919.756i	− 1.75526i
$525$	0	0
$526$	−226.174	−0.429989
$527$	869.085	1.64912
$528$	− 93.7825i	− 0.177618i
$529$	703.601	1.33006
$530$	315.180i	0.594679i
$531$	147.979i	0.278680i
$532$	0	0
$533$	−42.9612	−0.0806027
$534$	−397.885	−0.745102
$535$	− 405.264i	− 0.757503i
$536$	1006.18	1.87719
$537$	257.524i	0.479561i
$538$	695.512i	1.29277i
$539$	0	0
$540$	−97.5226	−0.180597
$541$	−224.354	−0.414702	−0.207351	−	0.978267i	$-0.566484\pi$
$541$	−224.354	−0.414702	−0.207351	+	0.978267i	$0.566484\pi$
$542$	− 542.615i	− 1.00114i
$543$	445.661	0.820738
$544$	− 269.745i	− 0.495854i
$545$	− 165.044i	− 0.302833i
$546$	0	0
$547$	−456.739	−0.834989	−0.417495	−	0.908679i	$-0.637092\pi$
$547$	−456.739	−0.834989	−0.417495	+	0.908679i	$0.637092\pi$
$548$	−1489.43	−2.71794
$549$	− 2.65036i	− 0.00482762i
$550$	45.6547	0.0830086
$551$	733.952i	1.33204i
$552$	− 940.522i	− 1.70384i
$553$	0	0
$554$	−1194.14	−2.15549
$555$	99.7240	0.179683
$556$	− 1425.99i	− 2.56473i
$557$	−777.050	−1.39506	−0.697532	−	0.716554i	$-0.745718\pi$
$557$	−777.050	−1.39506	−0.697532	+	0.716554i	$0.745718\pi$
$558$	− 395.555i	− 0.708880i
$559$	859.223i	1.53707i
$560$	0	0
$561$	104.245	0.185819
$562$	394.204	0.701431
$563$	− 83.4188i	− 0.148168i	−0.997252	−	0.0740842i	$-0.976397\pi$
$563$	− 83.4188i	− 0.148168i	0.997252	−	0.0740842i	$-0.0236033\pi$
$564$	−49.1260	−0.0871028
$565$	16.5148i	0.0292296i
$566$	230.265i	0.406829i
$567$	0	0
$568$	−1331.35	−2.34393
$569$	−223.343	−0.392518	−0.196259	−	0.980552i	$-0.562879\pi$
$569$	−223.343	−0.392518	−0.196259	+	0.980552i	$0.562879\pi$
$570$	408.612i	0.716864i
$571$	176.726	0.309503	0.154751	−	0.987953i	$-0.450542\pi$
$571$	176.726	0.309503	0.154751	+	0.987953i	$0.450542\pi$
$572$	252.016i	0.440588i
$573$	− 210.625i	− 0.367582i
$574$	0	0
$575$	175.542	0.305291
$576$	127.737	0.221766
$577$	901.604i	1.56257i	0.624173	+	0.781286i	$0.285436\pi$
$577$	901.604i	1.56257i	−0.624173	+	0.781286i	$0.714564\pi$
$578$	878.168	1.51932
$579$	− 420.079i	− 0.725525i
$580$	− 459.643i	− 0.792488i
$581$	0	0
$582$	−257.546	−0.442518
$583$	−103.848	−0.178127
$584$	952.395i	1.63081i
$585$	77.6562	0.132746
$586$	355.534i	0.606713i
$587$	− 163.544i	− 0.278610i	−0.990250	−	0.139305i	$-0.955513\pi$
$587$	− 163.544i	− 0.278610i	0.990250	−	0.139305i	$-0.0444868\pi$
$588$	0	0
$589$	−1122.43	−1.90566
$590$	−388.294	−0.658125
$591$	171.461i	0.290120i
$592$	537.521	0.907974
$593$	− 1159.86i	− 1.95593i	−0.208775	−	0.977964i	$-0.566948\pi$
$593$	− 1159.86i	− 1.95593i	0.208775	−	0.977964i	$-0.433052\pi$
$594$	− 47.4458i	− 0.0798750i
$595$	0	0
$596$	716.677	1.20248
$597$	136.423	0.228514
$598$	1430.79i	2.39263i
$599$	32.7980	0.0547545	0.0273773	−	0.999625i	$-0.491284\pi$
$599$	32.7980	0.0547545	0.0273773	+	0.999625i	$0.491284\pi$
$600$	− 133.945i	− 0.223242i
$601$	− 796.834i	− 1.32585i	−0.748687	−	0.662924i	$-0.769315\pi$
$601$	− 796.834i	− 1.32585i	0.748687	−	0.662924i	$-0.230685\pi$
$602$	0	0
$603$	195.163	0.323654
$604$	−1158.25	−1.91763
$605$	− 255.522i	− 0.422350i
$606$	914.427	1.50896
$607$	524.093i	0.863414i	0.902014	+	0.431707i	$0.142089\pi$
$607$	524.093i	0.863414i	−0.902014	+	0.431707i	$0.857911\pi$
$608$	348.378i	0.572990i
$609$	0	0
$610$	6.95448	0.0114008
$611$	39.1185	0.0640237
$612$	− 584.295i	− 0.954730i
$613$	−253.731	−0.413917	−0.206959	−	0.978350i	$-0.566357\pi$
$613$	−253.731	−0.413917	−0.206959	+	0.978350i	$0.566357\pi$
$614$	− 1409.54i	− 2.29566i
$615$	14.3732i	0.0233710i
$616$	0	0
$617$	620.813	1.00618	0.503090	−	0.864234i	$-0.332196\pi$
$617$	620.813	1.00618	0.503090	+	0.864234i	$0.332196\pi$
$618$	851.693	1.37814
$619$	641.602i	1.03651i	0.855225	+	0.518257i	$0.173419\pi$
$619$	641.602i	1.03651i	−0.855225	+	0.518257i	$0.826581\pi$
$620$	702.932	1.13376
$621$	− 182.429i	− 0.293766i
$622$	126.731i	0.203747i
$623$	0	0
$624$	418.573	0.670791
$625$	25.0000	0.0400000
$626$	− 403.072i	− 0.643885i
$627$	−134.633	−0.214725
$628$	57.6351i	0.0917756i
$629$	597.484i	0.949896i
$630$	0	0
$631$	−166.338	−0.263610	−0.131805	−	0.991276i	$-0.542077\pi$
$631$	−166.338	−0.263610	−0.131805	+	0.991276i	$0.542077\pi$
$632$	−426.007	−0.674062
$633$	185.578i	0.293173i
$634$	−14.1949	−0.0223895
$635$	− 466.533i	− 0.734697i
$636$	582.073i	0.915209i
$637$	0	0
$638$	223.621	0.350503
$639$	−258.236	−0.404125
$640$	439.153i	0.686176i
$641$	137.060	0.213823	0.106911	−	0.994269i	$-0.465904\pi$
$641$	137.060	0.213823	0.106911	+	0.994269i	$0.465904\pi$
$642$	− 1105.12i	− 1.72137i
$643$	812.010i	1.26285i	0.775438	+	0.631423i	$0.217529\pi$
$643$	812.010i	1.26285i	−0.775438	+	0.631423i	$0.782471\pi$
$644$	0	0
$645$	287.463	0.445679
$646$	−2448.15	−3.78971
$647$	− 228.295i	− 0.352852i	−0.984314	−	0.176426i	$-0.943546\pi$
$647$	− 228.295i	− 0.352852i	0.984314	−	0.176426i	$-0.0564536\pi$
$648$	−139.200	−0.214815
$649$	− 127.938i	− 0.197131i
$650$	203.768i	0.313489i
$651$	0	0
$652$	−2322.68	−3.56239
$653$	−871.113	−1.33402	−0.667009	−	0.745050i	$-0.732426\pi$
$653$	−871.113	−1.33402	−0.667009	+	0.745050i	$0.732426\pi$
$654$	− 450.060i	− 0.688165i
$655$	245.030	0.374092
$656$	77.4725i	0.118098i
$657$	184.731i	0.281174i
$658$	0	0
$659$	−677.945	−1.02875	−0.514374	−	0.857566i	$-0.671976\pi$
$659$	−677.945	−1.02875	−0.514374	+	0.857566i	$0.671976\pi$
$660$	84.3149	0.127750
$661$	702.844i	1.06330i	0.846963	+	0.531652i	$0.178429\pi$
$661$	702.844i	1.06330i	−0.846963	+	0.531652i	$0.821571\pi$
$662$	−1786.20	−2.69819
$663$	465.268i	0.701761i
$664$	2033.02i	3.06177i
$665$	0	0
$666$	271.939	0.408316
$667$	859.822	1.28909
$668$	− 355.712i	− 0.532502i
$669$	−15.1038	−0.0225768
$670$	512.103i	0.764333i
$671$	2.29142i	0.00341493i
$672$	0	0
$673$	612.283	0.909782	0.454891	−	0.890547i	$-0.349678\pi$
$673$	612.283	0.909782	0.454891	+	0.890547i	$0.349678\pi$
$674$	930.374	1.38038
$675$	− 25.9808i	− 0.0384900i
$676$	293.678	0.434436
$677$	5.78817i	0.00854973i	0.999991	+	0.00427487i	$0.00136074\pi$
$677$	5.78817i	0.00854973i	−0.999991	+	0.00427487i	$0.998639\pi$
$678$	45.0343i	0.0664222i
$679$	0	0
$680$	802.518	1.18017
$681$	−438.246	−0.643533
$682$	341.984i	0.501443i
$683$	−116.766	−0.170961	−0.0854806	−	0.996340i	$-0.527243\pi$
$683$	−116.766	−0.170961	−0.0854806	+	0.996340i	$0.527243\pi$
$684$	754.623i	1.10325i
$685$	− 396.796i	− 0.579264i
$686$	0	0
$687$	−251.977	−0.366778
$688$	1549.45	2.25210
$689$	− 463.498i	− 0.672712i
$690$	478.688	0.693750
$691$	− 947.479i	− 1.37117i	−0.727992	−	0.685585i	$-0.759546\pi$
$691$	− 947.479i	− 1.37117i	0.727992	−	0.685585i	$-0.240454\pi$
$692$	− 248.111i	− 0.358542i
$693$	0	0
$694$	825.425	1.18937
$695$	379.894	0.546610
$696$	− 656.077i	− 0.942640i
$697$	−86.1151	−0.123551
$698$	− 190.854i	− 0.273430i
$699$	− 446.030i	− 0.638097i
$700$	0	0
$701$	1168.56	1.66700	0.833498	−	0.552523i	$-0.186335\pi$
$701$	1168.56	1.66700	0.833498	+	0.552523i	$0.186335\pi$
$702$	211.762	0.301655
$703$	− 771.657i	− 1.09766i
$704$	−110.437	−0.156871
$705$	− 13.0875i	− 0.0185639i
$706$	1837.05i	2.60205i
$707$	0	0
$708$	−717.099	−1.01285
$709$	669.561	0.944374	0.472187	−	0.881498i	$-0.343465\pi$
$709$	669.561	0.944374	0.472187	+	0.881498i	$0.343465\pi$
$710$	− 677.603i	− 0.954371i
$711$	−82.6306	−0.116217
$712$	− 1009.25i	− 1.41748i
$713$	1314.93i	1.84422i
$714$	0	0
$715$	−67.1390	−0.0939008
$716$	−1247.95	−1.74294
$717$	222.884i	0.310856i
$718$	1647.37	2.29439
$719$	930.668i	1.29439i	0.762323	+	0.647196i	$0.224059\pi$
$719$	930.668i	1.29439i	−0.762323	+	0.647196i	$0.775941\pi$
$720$	− 140.038i	− 0.194498i
$721$	0	0
$722$	1890.94	2.61903
$723$	−16.0445	−0.0221916
$724$	2159.64i	2.98293i
$725$	122.452	0.168900
$726$	− 696.785i	− 0.959758i
$727$	− 290.932i	− 0.400182i	−0.979777	−	0.200091i	$-0.935876\pi$
$727$	− 290.932i	− 0.400182i	0.979777	−	0.200091i	$-0.0641238\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	−484.730	−0.664014
$731$	1722.30i	2.35608i
$732$	12.8435	0.0175458
$733$	0.965424i	0.00131709i	1.00000		0.000658543i	$0.000209621\pi$
$733$	0.965424i	0.00131709i	−1.00000		0.000658543i	$0.999790\pi$
$734$	606.304i	0.826027i
$735$	0	0
$736$	408.124	0.554516
$737$	−168.732	−0.228944
$738$	39.1943i	0.0531089i
$739$	−99.8730	−0.135146	−0.0675731	−	0.997714i	$-0.521526\pi$
$739$	−99.8730	−0.135146	−0.0675731	+	0.997714i	$0.521526\pi$
$740$	483.256i	0.653049i
$741$	− 600.898i	− 0.810929i
$742$	0	0
$743$	890.635	1.19870	0.599351	−	0.800486i	$-0.295425\pi$
$743$	890.635	1.19870	0.599351	+	0.800486i	$0.295425\pi$
$744$	1003.34	1.34857
$745$	190.928i	0.256279i
$746$	1622.82	2.17536
$747$	394.335i	0.527892i
$748$	505.162i	0.675351i
$749$	0	0
$750$	68.1728	0.0908971
$751$	621.075	0.826997	0.413499	−	0.910505i	$-0.364307\pi$
$751$	621.075	0.826997	0.413499	+	0.910505i	$0.364307\pi$
$752$	− 70.5429i	− 0.0938071i
$753$	−51.8250	−0.0688247
$754$	998.074i	1.32371i
$755$	− 308.567i	− 0.408698i
$756$	0	0
$757$	675.637	0.892519	0.446259	−	0.894904i	$-0.352756\pi$
$757$	675.637	0.892519	0.446259	+	0.894904i	$0.352756\pi$
$758$	−1565.32	−2.06506
$759$	157.722i	0.207802i
$760$	−1036.46	−1.36376
$761$	− 634.318i	− 0.833533i	−0.909014	−	0.416766i	$-0.863163\pi$
$761$	− 634.318i	− 0.833533i	0.909014	−	0.416766i	$-0.136837\pi$
$762$	− 1272.19i	− 1.66954i
$763$	0	0
$764$	1020.67	1.33596
$765$	155.661	0.203478
$766$	− 1863.93i	− 2.43334i
$767$	571.018	0.744482
$768$	902.534i	1.17517i
$769$	17.8434i	0.0232033i	0.999933	+	0.0116017i	$0.00369301\pi$
$769$	17.8434i	0.0232033i	−0.999933	+	0.0116017i	$0.996307\pi$
$770$	0	0
$771$	−167.468	−0.217208
$772$	2035.68	2.63689
$773$	− 258.578i	− 0.334513i	−0.985913	−	0.167256i	$-0.946509\pi$
$773$	− 258.578i	− 0.334513i	0.985913	−	0.167256i	$-0.0534908\pi$
$774$	783.885	1.01277
$775$	187.266i	0.241634i
$776$	− 653.274i	− 0.841848i
$777$	0	0
$778$	−688.281	−0.884680
$779$	111.218	0.142771
$780$	376.317i	0.482458i
$781$	223.262	0.285867
$782$	2868.00i	3.66752i
$783$	− 127.256i	− 0.162524i
$784$	0	0
$785$	−15.3544	−0.0195598
$786$	668.175	0.850096
$787$	1081.37i	1.37404i	0.726638	+	0.687020i	$0.241082\pi$
$787$	1081.37i	1.37404i	−0.726638	+	0.687020i	$0.758918\pi$
$788$	−830.888	−1.05443
$789$	− 111.278i	− 0.141036i
$790$	− 216.820i	− 0.274456i
$791$	0	0
$792$	120.348	0.151954
$793$	−10.2271	−0.0128968
$794$	102.991i	0.129712i
$795$	−155.069	−0.195055
$796$	661.098i	0.830525i
$797$	145.723i	0.182839i	0.995812	+	0.0914195i	$0.0291404\pi$
$797$	145.723i	0.182839i	−0.995812	+	0.0914195i	$0.970860\pi$
$798$	0	0
$799$	78.4124	0.0981382
$800$	58.1233	0.0726542
$801$	− 195.759i	− 0.244394i
$802$	−742.079	−0.925286
$803$	− 159.713i	− 0.198895i
$804$	945.749i	1.17630i
$805$	0	0
$806$	−1526.35	−1.89374
$807$	−342.192	−0.424029
$808$	2319.48i	2.87064i
$809$	−216.849	−0.268046	−0.134023	−	0.990978i	$-0.542790\pi$
$809$	−216.849	−0.268046	−0.134023	+	0.990978i	$0.542790\pi$
$810$	− 70.8472i	− 0.0874657i
$811$	− 8.20233i	− 0.0101139i	−0.999987	−	0.00505693i	$-0.998390\pi$
$811$	− 8.20233i	− 0.0101139i	0.999987	−	0.00505693i	$-0.00160968\pi$
$812$	0	0
$813$	266.967	0.328372
$814$	−235.109	−0.288832
$815$	− 618.779i	− 0.759238i
$816$	839.024	1.02822
$817$	− 2224.37i	− 2.72260i
$818$	− 1142.33i	− 1.39650i
$819$	0	0
$820$	−69.6514	−0.0849408
$821$	1572.95	1.91589	0.957946	−	0.286948i	$-0.0926407\pi$
$821$	1572.95	1.91589	0.957946	+	0.286948i	$0.0926407\pi$
$822$	− 1082.03i	− 1.31633i
$823$	998.679	1.21346	0.606731	−	0.794907i	$-0.292481\pi$
$823$	998.679	1.21346	0.606731	+	0.794907i	$0.292481\pi$
$824$	2160.35i	2.62178i
$825$	22.4621i	0.0272268i
$826$	0	0
$827$	−1344.24	−1.62544	−0.812718	−	0.582658i	$-0.802013\pi$
$827$	−1344.24	−1.62544	−0.812718	+	0.582658i	$0.802013\pi$
$828$	884.038	1.06768
$829$	− 1515.22i	− 1.82777i	−0.405972	−	0.913885i	$-0.633067\pi$
$829$	− 1515.22i	− 1.82777i	0.405972	−	0.913885i	$-0.366933\pi$
$830$	−1034.72	−1.24666
$831$	− 587.517i	− 0.707000i
$832$	− 492.908i	− 0.592437i
$833$	0	0
$834$	1035.94	1.24213
$835$	94.7642	0.113490
$836$	− 652.423i	− 0.780410i
$837$	194.613	0.232513
$838$	2443.03i	2.91531i
$839$	439.769i	0.524159i	0.965046	+	0.262079i	$0.0844082\pi$
$839$	439.769i	0.524159i	−0.965046	+	0.262079i	$0.915592\pi$
$840$	0	0
$841$	−241.217	−0.286821
$842$	1202.42	1.42805
$843$	193.949i	0.230070i
$844$	−899.301	−1.06552
$845$	78.2382i	0.0925895i
$846$	− 35.6886i	− 0.0421851i
$847$	0	0
$848$	−835.833	−0.985652
$849$	−113.291	−0.133440
$850$	408.449i	0.480528i
$851$	−903.994	−1.06227
$852$	− 1251.39i	− 1.46877i
$853$	222.026i	0.260288i	0.991495	+	0.130144i	$0.0415439\pi$
$853$	222.026i	0.260288i	−0.991495	+	0.130144i	$0.958456\pi$
$854$	0	0
$855$	−201.037	−0.235131
$856$	2803.18	3.27474
$857$	− 1042.35i	− 1.21627i	−0.793832	−	0.608137i	$-0.791917\pi$
$857$	− 1042.35i	− 1.21627i	0.793832	−	0.608137i	$-0.208083\pi$
$858$	−183.082	−0.213383
$859$	− 153.464i	− 0.178654i	−0.996002	−	0.0893272i	$-0.971528\pi$
$859$	− 153.464i	− 0.178654i	0.996002	−	0.0893272i	$-0.0284717\pi$
$860$	1393.03i	1.61980i
$861$	0	0
$862$	1448.49	1.68038
$863$	−802.985	−0.930458	−0.465229	−	0.885190i	$-0.654028\pi$
$863$	−802.985	−0.930458	−0.465229	+	0.885190i	$0.654028\pi$
$864$	− 60.4035i	− 0.0699115i
$865$	66.0987	0.0764147
$866$	1561.16i	1.80273i
$867$	432.059i	0.498338i
$868$	0	0
$869$	71.4397	0.0822091
$870$	333.917	0.383812
$871$	− 753.090i	− 0.864627i
$872$	1141.59	1.30917
$873$	− 126.712i	− 0.145146i
$874$	− 3704.05i	− 4.23804i
$875$	0	0
$876$	−895.197	−1.02191
$877$	−1344.66	−1.53324	−0.766622	−	0.642099i	$-0.778064\pi$
$877$	−1344.66	−1.53324	−0.766622	+	0.642099i	$0.778064\pi$
$878$	− 1089.44i	− 1.24082i
$879$	−174.923	−0.199002
$880$	121.073i	0.137583i
$881$	− 1052.91i	− 1.19513i	−0.801822	−	0.597563i	$-0.796136\pi$
$881$	− 1052.91i	− 1.19513i	0.801822	−	0.597563i	$-0.203864\pi$
$882$	0	0
$883$	1372.84	1.55475	0.777375	−	0.629037i	$-0.216551\pi$
$883$	1372.84	1.55475	0.777375	+	0.629037i	$0.216551\pi$
$884$	−2254.66	−2.55052
$885$	− 191.040i	− 0.215865i
$886$	−2882.04	−3.25286
$887$	629.109i	0.709255i	0.935008	+	0.354628i	$0.115392\pi$
$887$	629.109i	0.709255i	−0.935008	+	0.354628i	$0.884608\pi$
$888$	689.782i	0.776782i
$889$	0	0
$890$	513.667	0.577154
$891$	23.3433	0.0261990
$892$	− 73.1923i	− 0.0820542i
$893$	−101.270	−0.113405
$894$	520.644i	0.582376i
$895$	− 332.462i	− 0.371466i
$896$	0	0
$897$	−703.950	−0.784783
$898$	−1111.60	−1.23786
$899$	917.249i	1.02030i
$900$	125.901	0.139890
$901$	− 929.075i	− 1.03116i
$902$	− 33.8862i	− 0.0375678i
$903$	0	0
$904$	−114.231	−0.126362
$905$	−575.346	−0.635741
$906$	− 841.435i	− 0.928736i
$907$	−59.7908	−0.0659215	−0.0329608	−	0.999457i	$-0.510494\pi$
$907$	−59.7908	−0.0659215	−0.0329608	+	0.999457i	$0.510494\pi$
$908$	− 2123.71i	− 2.33889i
$909$	449.898i	0.494938i
$910$	0	0
$911$	−850.964	−0.934099	−0.467050	−	0.884231i	$-0.654683\pi$
$911$	−850.964	−0.934099	−0.467050	+	0.884231i	$0.654683\pi$
$912$	−1083.61	−1.18817
$913$	− 340.929i	− 0.373416i
$914$	659.399	0.721444
$915$	3.42160i	0.00373946i
$916$	− 1221.06i	− 1.33304i
$917$	0	0
$918$	424.473	0.462389
$919$	661.011	0.719272	0.359636	−	0.933093i	$-0.382901\pi$
$919$	661.011	0.719272	0.359636	+	0.933093i	$0.382901\pi$
$920$	1214.21i	1.31979i
$921$	693.492	0.752978
$922$	29.1350i	0.0315998i
$923$	996.472i	1.07960i
$924$	0	0
$925$	−128.743	−0.139182
$926$	1664.90	1.79794
$927$	419.033i	0.452031i
$928$	284.694	0.306782
$929$	267.472i	0.287914i	0.989584	+	0.143957i	$0.0459827\pi$
$929$	267.472i	0.287914i	−0.989584	+	0.143957i	$0.954017\pi$
$930$	510.659i	0.549096i
$931$	0	0
$932$	2161.43	2.31914
$933$	−62.3515	−0.0668291
$934$	− 2523.84i	− 2.70218i
$935$	−134.579	−0.143935
$936$	537.141i	0.573869i
$937$	1625.83i	1.73515i	0.497309	+	0.867573i	$0.334321\pi$
$937$	1625.83i	1.73515i	−0.497309	+	0.867573i	$0.665679\pi$
$938$	0	0
$939$	198.312	0.211194
$940$	63.4214	0.0674696
$941$	− 297.904i	− 0.316583i	−0.987392	−	0.158291i	$-0.949402\pi$
$941$	− 297.904i	− 0.316583i	0.987392	−	0.158291i	$-0.0505985\pi$
$942$	−41.8701	−0.0444481
$943$	− 130.292i	− 0.138168i
$944$	− 1029.72i	− 1.09081i
$945$	0	0
$946$	−677.722	−0.716408
$947$	−913.753	−0.964892	−0.482446	−	0.875926i	$-0.660252\pi$
$947$	−913.753	−0.964892	−0.482446	+	0.875926i	$0.660252\pi$
$948$	− 400.423i	− 0.422387i
$949$	712.836	0.751144
$950$	− 527.516i	− 0.555280i
$951$	− 6.98391i	− 0.00734375i
$952$	0	0
$953$	−1451.89	−1.52349	−0.761746	−	0.647875i	$-0.775658\pi$
$953$	−1451.89	−1.52349	−0.761746	+	0.647875i	$0.775658\pi$
$954$	−422.858	−0.443248
$955$	271.915i	0.284728i
$956$	−1080.08	−1.12979
$957$	110.022i	0.114965i
$958$	− 724.878i	− 0.756657i
$959$	0	0
$960$	−164.908	−0.171779
$961$	−441.749	−0.459676
$962$	− 1049.35i	− 1.09080i
$963$	543.719	0.564609
$964$	− 77.7508i	− 0.0806544i
$965$	542.319i	0.561989i
$966$	0	0
$967$	235.985	0.244039	0.122019	−	0.992528i	$-0.461063\pi$
$967$	235.985	0.244039	0.122019	+	0.992528i	$0.461063\pi$
$968$	1767.42	1.82585
$969$	− 1204.49i	− 1.24302i
$970$	332.490	0.342773
$971$	641.865i	0.661035i	0.943800	+	0.330518i	$0.107223\pi$
$971$	641.865i	0.661035i	−0.943800	+	0.330518i	$0.892777\pi$
$972$	− 130.840i	− 0.134609i
$973$	0	0
$974$	−1136.67	−1.16701
$975$	−100.254	−0.102824
$976$	18.4427i	0.0188962i
$977$	640.847	0.655933	0.327967	−	0.944689i	$-0.393637\pi$
$977$	640.847	0.655933	0.327967	+	0.944689i	$0.393637\pi$
$978$	− 1687.36i	− 1.72531i
$979$	169.247i	0.172878i
$980$	0	0
$981$	221.429	0.225718
$982$	−958.893	−0.976469
$983$	− 425.729i	− 0.433092i	−0.976272	−	0.216546i	$-0.930521\pi$
$983$	− 425.729i	− 0.433092i	0.976272	−	0.216546i	$-0.0694791\pi$
$984$	−99.4179	−0.101034
$985$	− 221.355i	− 0.224726i
$986$	2000.62i	2.02903i
$987$	0	0
$988$	2911.92	2.94728
$989$	−2605.84	−2.63482
$990$	61.2522i	0.0618709i
$991$	−1097.25	−1.10722	−0.553608	−	0.832777i	$-0.686749\pi$
$991$	−1097.25	−1.10722	−0.553608	+	0.832777i	$0.686749\pi$
$992$	435.382i	0.438893i
$993$	− 878.813i	− 0.885008i
$994$	0	0
$995$	−176.121	−0.177006
$996$	−1910.92	−1.91860
$997$	− 1274.97i	− 1.27881i	−0.768870	−	0.639405i	$-0.779181\pi$
$997$	− 1274.97i	− 1.27881i	0.768870	−	0.639405i	$-0.220819\pi$
$998$	−1862.99	−1.86672
$999$	133.794i	0.133928i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.3.h.a.391.1		8
7.4	even	3		105.3.n.a.61.4	yes	8
7.5	odd	6		105.3.n.a.31.4	✓	8
7.6	odd	2	inner	735.3.h.a.391.2		8
21.5	even	6		315.3.w.a.136.1		8
21.11	odd	6		315.3.w.a.271.1		8
35.4	even	6		525.3.o.l.376.1		8
35.12	even	12		525.3.s.h.199.1		16
35.18	odd	12		525.3.s.h.124.1		16
35.19	odd	6		525.3.o.l.451.1		8
35.32	odd	12		525.3.s.h.124.8		16
35.33	even	12		525.3.s.h.199.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.n.a.31.4	✓	8	7.5	odd	6
105.3.n.a.61.4	yes	8	7.4	even	3
315.3.w.a.136.1		8	21.5	even	6
315.3.w.a.271.1		8	21.11	odd	6
525.3.o.l.376.1		8	35.4	even	6
525.3.o.l.451.1		8	35.19	odd	6
525.3.s.h.124.1		16	35.18	odd	12
525.3.s.h.124.8		16	35.32	odd	12
525.3.s.h.199.1		16	35.12	even	12
525.3.s.h.199.8		16	35.33	even	12
735.3.h.a.391.1		8	1.1	even	1	trivial
735.3.h.a.391.2		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)

\(f(q)\)	\(=\)	\(q-3.52043 q^{2} -1.73205i q^{3} +8.39341 q^{4} +2.23607i q^{5} +6.09756i q^{6} -15.4667 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-3.52043 q^{2} -1.73205i q^{3} +8.39341 q^{4} +2.23607i q^{5} +6.09756i q^{6} -15.4667 q^{8} -3.00000 q^{9} -7.87192i q^{10} +2.59370 q^{11} -14.5378i q^{12} +11.5763i q^{13} +3.87298 q^{15} +20.8757 q^{16} +23.2045i q^{17} +10.5613 q^{18} -29.9689i q^{19} +18.7682i q^{20} -9.13094 q^{22} -35.1084 q^{23} +26.7891i q^{24} -5.00000 q^{25} -40.7535i q^{26} +5.19615i q^{27} -24.4905 q^{29} -13.6346 q^{30} -37.4533i q^{31} -11.6247 q^{32} -4.49242i q^{33} -81.6898i q^{34} -25.1802 q^{36} +25.7486 q^{37} +105.503i q^{38} +20.0507 q^{39} -34.5846i q^{40} +3.71113i q^{41} +74.2225 q^{43} +21.7700 q^{44} -6.70820i q^{45} +123.597 q^{46} -3.37919i q^{47} -36.1578i q^{48} +17.6021 q^{50} +40.1914 q^{51} +97.1647i q^{52} -40.0385 q^{53} -18.2927i q^{54} +5.79969i q^{55} -51.9076 q^{57} +86.2170 q^{58} -49.3264i q^{59} +32.5075 q^{60} +0.883454i q^{61} +131.852i q^{62} -42.5790 q^{64} -25.8854 q^{65} +15.8153i q^{66} -65.0544 q^{67} +194.765i q^{68} +60.8096i q^{69} +86.0786 q^{71} +46.4001 q^{72} -61.5772i q^{73} -90.6462 q^{74} +8.66025i q^{75} -251.541i q^{76} -70.5872 q^{78} +27.5435 q^{79} +46.6795i q^{80} +9.00000 q^{81} -13.0648i q^{82} -131.445i q^{83} -51.8869 q^{85} -261.295 q^{86} +42.4187i q^{87} -40.1160 q^{88} +65.2531i q^{89} +23.6157i q^{90} -294.679 q^{92} -64.8710 q^{93} +11.8962i q^{94} +67.0124 q^{95} +20.1345i q^{96} +42.2375i q^{97} -7.78111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 12 q^{4} - 32 q^{8} - 24 q^{9}+O(q^{10}) \) 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

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