LMFDB - Embedded newform 405.3.d.a.404.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(404,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.404"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{18} - 19 x^{16} + 66 x^{14} + 109 x^{12} - 813 x^{10} + 981 x^{8} + 5346 x^{6} + \cdots + 59049 $ x^20 - 3x^18 - 19x^16 + 66x^14 + 109x^12 - 813x^10 + 981x^8 + 5346x^6 - 13851x^4 - 19683*x^2 + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.5
Root			$1.72886 - 0.105167i$ of defining polynomial
Character	$\chi$	$=$	405.404
Dual form			405.3.d.a.404.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.29340 q^{2} +1.25967 q^{4} +(-4.62307 - 1.90452i) q^{5} +6.98084i q^{7} +6.28466 q^{8} +(10.6025 + 4.36783i) q^{10} -18.2517i q^{11} -4.23723i q^{13} -16.0098i q^{14} -19.4519 q^{16} -17.3066 q^{17} -3.96601 q^{19} +(-5.82355 - 2.39907i) q^{20} +41.8584i q^{22} -0.575351 q^{23} +(17.7456 + 17.6095i) q^{25} +9.71765i q^{26} +8.79356i q^{28} +20.9881i q^{29} +33.4653 q^{31} +19.4723 q^{32} +39.6910 q^{34} +(13.2952 - 32.2729i) q^{35} +21.4222i q^{37} +9.09563 q^{38} +(-29.0544 - 11.9693i) q^{40} +50.9227i q^{41} +8.26205i q^{43} -22.9911i q^{44} +1.31951 q^{46} +15.1558 q^{47} +0.267815 q^{49} +(-40.6976 - 40.3856i) q^{50} -5.33751i q^{52} +24.5806 q^{53} +(-34.7608 + 84.3789i) q^{55} +43.8723i q^{56} -48.1340i q^{58} +49.8356i q^{59} +62.9347 q^{61} -76.7492 q^{62} +33.1499 q^{64} +(-8.06991 + 19.5890i) q^{65} +119.564i q^{67} -21.8006 q^{68} +(-30.4911 + 74.0147i) q^{70} +66.8256i q^{71} -48.9419i q^{73} -49.1296i q^{74} -4.99586 q^{76} +127.412 q^{77} -117.932 q^{79} +(89.9276 + 37.0466i) q^{80} -116.786i q^{82} -7.32127 q^{83} +(80.0098 + 32.9609i) q^{85} -18.9482i q^{86} -114.706i q^{88} +100.624i q^{89} +29.5794 q^{91} -0.724753 q^{92} -34.7582 q^{94} +(18.3351 + 7.55336i) q^{95} +4.14812i q^{97} -0.614205 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) $ 20 * q + 36 * q^4 + 4 * q^10 + 52 * q^16 - 8 * q^19 - 4 * q^25 - 56 * q^31 + 8 * q^34 + 68 * q^40 + 116 * q^46 + 80 * q^49 + 36 * q^55 + 100 * q^61 + 140 * q^64 + 108 * q^70 + 192 * q^76 + 256 * q^79 + 148 * q^85 - 288 * q^91 - 436 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−2.29340			−1.14670			−0.573349	−	0.819311i	$-0.694356\pi$
$2$	−2.29340			−1.14670			−0.573349	+	0.819311i	$0.694356\pi$
$3$		0			0
$3$		0			0
$4$	1.25967			0.314918
$5$	−4.62307	−	1.90452i	−0.924614	−	0.380905i
$5$	−4.62307	−	1.90452i	−0.924614	−	0.380905i
$6$		0			0
$7$			6.98084i			0.997263i	0.866814	+	0.498632i	$0.166164\pi$
$7$			6.98084i			0.997263i	−0.866814	+	0.498632i	$0.833836\pi$
$8$	6.28466			0.785583
$9$		0			0
$10$	10.6025	+	4.36783i	1.06025	+	0.436783i
$11$		−	18.2517i		−	1.65924i	−0.558325	−	0.829622i	$-0.688556\pi$
$11$		−	18.2517i		−	1.65924i	0.558325	−	0.829622i	$-0.311444\pi$
$12$		0			0
$13$		−	4.23723i		−	0.325941i	−0.986631	−	0.162970i	$-0.947892\pi$
$13$		−	4.23723i		−	0.325941i	0.986631	−	0.162970i	$-0.0521075\pi$
$14$		−	16.0098i		−	1.14356i
$15$		0			0
$16$	−19.4519			−1.21574
$17$	−17.3066			−1.01804			−0.509019	−	0.860756i	$-0.669992\pi$
$17$	−17.3066			−1.01804			−0.509019	+	0.860756i	$0.669992\pi$
$18$		0			0
$19$	−3.96601			−0.208737			−0.104369	−	0.994539i	$-0.533282\pi$
$19$	−3.96601			−0.208737			−0.104369	+	0.994539i	$0.533282\pi$
$20$	−5.82355	−	2.39907i	−0.291177	−	0.119954i
$21$		0			0
$22$			41.8584i			1.90265i
$23$	−0.575351			−0.0250153			−0.0125076	−	0.999922i	$-0.503981\pi$
$23$	−0.575351			−0.0250153			−0.0125076	+	0.999922i	$0.503981\pi$
$24$		0			0
$25$	17.7456	+	17.6095i	0.709823	+	0.704380i
$26$			9.71765i			0.373756i
$27$		0			0
$28$			8.79356i			0.314056i
$29$			20.9881i			0.723727i	0.932231	+	0.361863i	$0.117859\pi$
$29$			20.9881i			0.723727i	−0.932231	+	0.361863i	$0.882141\pi$
$30$		0			0
$31$	33.4653			1.07953			0.539763	−	0.841817i	$-0.318514\pi$
$31$	33.4653			1.07953			0.539763	+	0.841817i	$0.318514\pi$
$32$	19.4723			0.608509
$33$		0			0
$34$	39.6910			1.16738
$35$	13.2952	−	32.2729i	0.379863	−	0.922084i
$36$		0			0
$37$			21.4222i			0.578978i	0.957181	+	0.289489i	$0.0934854\pi$
$37$			21.4222i			0.578978i	−0.957181	+	0.289489i	$0.906515\pi$
$38$	9.09563			0.239359
$39$		0			0
$40$	−29.0544	−	11.9693i	−0.726361	−	0.299232i
$41$			50.9227i			1.24202i	0.783804	+	0.621008i	$0.213277\pi$
$41$			50.9227i			1.24202i	−0.783804	+	0.621008i	$0.786723\pi$
$42$		0			0
$43$			8.26205i			0.192141i	0.995375	+	0.0960703i	$0.0306274\pi$
$43$			8.26205i			0.192141i	−0.995375	+	0.0960703i	$0.969373\pi$
$44$		−	22.9911i		−	0.522525i
$45$		0			0
$46$	1.31951			0.0286850
$47$	15.1558			0.322463			0.161232	−	0.986917i	$-0.448453\pi$
$47$	15.1558			0.322463			0.161232	+	0.986917i	$0.448453\pi$
$48$		0			0
$49$	0.267815			0.00546560
$50$	−40.6976	−	40.3856i	−0.813953	−	0.807712i
$51$		0			0
$52$		−	5.33751i		−	0.102644i
$53$	24.5806			0.463784			0.231892	−	0.972741i	$-0.425508\pi$
$53$	24.5806			0.463784			0.231892	+	0.972741i	$0.425508\pi$
$54$		0			0
$55$	−34.7608	+	84.3789i	−0.632014	+	1.53416i
$56$			43.8723i			0.783433i
$57$		0			0
$58$		−	48.1340i		−	0.829897i
$59$			49.8356i			0.844671i	0.906440	+	0.422335i	$0.138790\pi$
$59$			49.8356i			0.844671i	−0.906440	+	0.422335i	$0.861210\pi$
$60$		0			0
$61$	62.9347			1.03172			0.515858	−	0.856674i	$-0.327473\pi$
$61$	62.9347			1.03172			0.515858	+	0.856674i	$0.327473\pi$
$62$	−76.7492			−1.23789
$63$		0			0
$64$	33.1499			0.517968
$65$	−8.06991	+	19.5890i	−0.124152	+	0.301370i
$66$		0			0
$67$			119.564i			1.78453i	0.451511	+	0.892266i	$0.350885\pi$
$67$			119.564i			1.78453i	−0.451511	+	0.892266i	$0.649115\pi$
$68$	−21.8006			−0.320598
$69$		0			0
$70$	−30.4911	+	74.0147i	−0.435588	+	1.05735i
$71$			66.8256i			0.941205i	0.882345	+	0.470603i	$0.155963\pi$
$71$			66.8256i			0.941205i	−0.882345	+	0.470603i	$0.844037\pi$
$72$		0			0
$73$		−	48.9419i		−	0.670437i	−0.942140	−	0.335218i	$-0.891190\pi$
$73$		−	48.9419i		−	0.670437i	0.942140	−	0.335218i	$-0.108810\pi$
$74$		−	49.1296i		−	0.663913i
$75$		0			0
$76$	−4.99586			−0.0657350
$77$	127.412			1.65470
$78$		0			0
$79$	−117.932			−1.49281			−0.746407	−	0.665490i	$-0.768223\pi$
$79$	−117.932			−1.49281			−0.746407	+	0.665490i	$0.768223\pi$
$80$	89.9276	+	37.0466i	1.12409	+	0.463083i
$81$		0			0
$82$		−	116.786i		−	1.42422i
$83$	−7.32127			−0.0882080			−0.0441040	−	0.999027i	$-0.514043\pi$
$83$	−7.32127			−0.0882080			−0.0441040	+	0.999027i	$0.514043\pi$
$84$		0			0
$85$	80.0098	+	32.9609i	0.941292	+	0.387775i
$86$		−	18.9482i		−	0.220327i
$87$		0			0
$88$		−	114.706i		−	1.30347i
$89$			100.624i			1.13060i	0.824884	+	0.565302i	$0.191241\pi$
$89$			100.624i			1.13060i	−0.824884	+	0.565302i	$0.808759\pi$
$90$		0			0
$91$	29.5794			0.325049
$92$	−0.724753			−0.00787774
$93$		0			0
$94$	−34.7582			−0.369768
$95$	18.3351	+	7.55336i	0.193001	+	0.0795091i
$96$		0			0
$97$			4.14812i			0.0427642i	0.999771	+	0.0213821i	$0.00680665\pi$
$97$			4.14812i			0.0427642i	−0.999771	+	0.0213821i	$0.993193\pi$
$98$	−0.614205			−0.00626740
$99$		0			0
$100$	22.3536	+	22.1822i	0.223536	+	0.221822i
$101$			161.965i			1.60362i	0.597582	+	0.801808i	$0.296128\pi$
$101$			161.965i			1.60362i	−0.597582	+	0.801808i	$0.703872\pi$
$102$		0			0
$103$		−	164.143i		−	1.59363i	−0.604226	−	0.796813i	$-0.706518\pi$
$103$		−	164.143i		−	1.59363i	0.604226	−	0.796813i	$-0.293482\pi$
$104$		−	26.6296i		−	0.256054i
$105$		0			0
$106$	−56.3730			−0.531821
$107$	48.4086			0.452416			0.226208	−	0.974079i	$-0.427367\pi$
$107$	48.4086			0.452416			0.226208	+	0.974079i	$0.427367\pi$
$108$		0			0
$109$	23.7458			0.217851			0.108925	−	0.994050i	$-0.465259\pi$
$109$	23.7458			0.217851			0.108925	+	0.994050i	$0.465259\pi$
$110$	79.7203	−	193.514i	0.724730	−	1.75922i
$111$		0			0
$112$		−	135.791i		−	1.21242i
$113$	−112.794			−0.998180			−0.499090	−	0.866550i	$-0.666332\pi$
$113$	−112.794			−0.998180			−0.499090	+	0.866550i	$0.666332\pi$
$114$		0			0
$115$	2.65989	+	1.09577i	0.0231295	+	0.00952843i
$116$			26.4381i			0.227914i
$117$		0			0
$118$		−	114.293i		−	0.968582i
$119$		−	120.815i		−	1.01525i
$120$		0			0
$121$	−212.124			−1.75309
$122$	−144.334			−1.18307
$123$		0			0
$124$	42.1552			0.339962
$125$	−48.5013	−	115.207i	−0.388011	−	0.921655i
$126$		0			0
$127$			97.5047i			0.767754i	0.923384	+	0.383877i	$0.125411\pi$
$127$			97.5047i			0.767754i	−0.923384	+	0.383877i	$0.874589\pi$
$128$	−153.915			−1.20246
$129$		0			0
$130$	18.5075	−	44.9254i	0.142365	−	0.345580i
$131$			20.5419i			0.156808i	0.996922	+	0.0784041i	$0.0249825\pi$
$131$			20.5419i			0.156808i	−0.996922	+	0.0784041i	$0.975018\pi$
$132$		0			0
$133$		−	27.6861i		−	0.208166i
$134$		−	274.207i		−	2.04632i
$135$		0			0
$136$	−108.766			−0.799753
$137$	19.6714			0.143587			0.0717935	−	0.997420i	$-0.477128\pi$
$137$	19.6714			0.143587			0.0717935	+	0.997420i	$0.477128\pi$
$138$		0			0
$139$	108.043			0.777287			0.388644	−	0.921388i	$-0.372944\pi$
$139$	108.043			0.777287			0.388644	+	0.921388i	$0.372944\pi$
$140$	16.7476	−	40.6533i	0.119625	−	0.290380i
$141$		0			0
$142$		−	153.258i		−	1.07928i
$143$	−77.3366			−0.540816
$144$		0			0
$145$	39.9723	−	97.0294i	0.275671	−	0.669168i
$146$			112.243i			0.768789i
$147$		0			0
$148$			26.9849i			0.182330i
$149$		−	226.960i		−	1.52322i	−0.648034	−	0.761612i	$-0.724408\pi$
$149$		−	226.960i		−	1.52322i	0.648034	−	0.761612i	$-0.275592\pi$
$150$		0			0
$151$	55.8282			0.369723			0.184862	−	0.982765i	$-0.440816\pi$
$151$	55.8282			0.369723			0.184862	+	0.982765i	$0.440816\pi$
$152$	−24.9250			−0.163980
$153$		0			0
$154$	−292.207			−1.89745
$155$	−154.712	−	63.7355i	−0.998145	−	0.411197i
$156$		0			0
$157$			68.3997i			0.435667i	0.975986	+	0.217833i	$0.0698990\pi$
$157$			68.3997i			0.435667i	−0.975986	+	0.217833i	$0.930101\pi$
$158$	270.466			1.71181
$159$		0			0
$160$	−90.0218	−	37.0855i	−0.562636	−	0.231784i
$161$		−	4.01644i		−	0.0249468i
$162$		0			0
$163$		−	80.5043i		−	0.493892i	−0.969029	−	0.246946i	$-0.920573\pi$
$163$		−	80.5043i		−	0.493892i	0.969029	−	0.246946i	$-0.0794269\pi$
$164$			64.1458i			0.391133i
$165$		0			0
$166$	16.7906			0.101148
$167$	−74.8228			−0.448041			−0.224020	−	0.974584i	$-0.571918\pi$
$167$	−74.8228			−0.448041			−0.224020	+	0.974584i	$0.571918\pi$
$168$		0			0
$169$	151.046			0.893763
$170$	−183.494	−	75.5924i	−1.07938	−	0.444661i
$171$		0			0
$172$			10.4075i			0.0605085i
$173$	318.424			1.84060			0.920301	−	0.391212i	$-0.127944\pi$
$173$	318.424			1.84060			0.920301	+	0.391212i	$0.127944\pi$
$174$		0			0
$175$	−122.929	+	123.879i	−0.702453	+	0.707880i
$176$			355.030i			2.01722i
$177$		0			0
$178$		−	230.770i		−	1.29646i
$179$		−	3.14738i		−	0.0175831i	−0.999961	−	0.00879155i	$-0.997202\pi$
$179$		−	3.14738i		−	0.0175831i	0.999961	−	0.00879155i	$-0.00279847\pi$
$180$		0			0
$181$	−0.833264			−0.00460367			−0.00230183	−	0.999997i	$-0.500733\pi$
$181$	−0.833264			−0.00460367			−0.00230183	+	0.999997i	$0.500733\pi$
$182$	−67.8374			−0.372733
$183$		0			0
$184$	−3.61589			−0.0196516
$185$	40.7991	−	99.0363i	0.220536	−	0.535331i
$186$		0			0
$187$			315.875i			1.68917i
$188$	19.0913			0.101549
$189$		0			0
$190$	−42.0498	−	17.3229i	−0.221315	−	0.0911729i
$191$		−	57.5187i		−	0.301145i	−0.988599	−	0.150573i	$-0.951888\pi$
$191$		−	57.5187i		−	0.301145i	0.988599	−	0.150573i	$-0.0481117\pi$
$192$		0			0
$193$			36.3611i			0.188400i	0.995553	+	0.0941999i	$0.0300293\pi$
$193$			36.3611i			0.188400i	−0.995553	+	0.0941999i	$0.969971\pi$
$194$		−	9.51330i		−	0.0490376i
$195$		0			0
$196$	0.337358			0.00172121
$197$	151.285			0.767943			0.383972	−	0.923345i	$-0.374556\pi$
$197$	151.285			0.767943			0.383972	+	0.923345i	$0.374556\pi$
$198$		0			0
$199$	268.742			1.35046			0.675232	−	0.737605i	$-0.264043\pi$
$199$	268.742			1.35046			0.675232	+	0.737605i	$0.264043\pi$
$200$	111.525	+	110.670i	0.557625	+	0.553349i
$201$		0			0
$202$		−	371.451i		−	1.83886i
$203$	−146.515			−0.721746
$204$		0			0
$205$	96.9834	−	235.419i	0.473090	−	1.14839i
$206$			376.446i			1.82741i
$207$		0			0
$208$			82.4222i			0.396261i
$209$			72.3864i			0.346346i
$210$		0			0
$211$	357.665			1.69510			0.847548	−	0.530718i	$-0.178078\pi$
$211$	357.665			1.69510			0.847548	+	0.530718i	$0.178078\pi$
$212$	30.9634			0.146054
$213$		0			0
$214$	−111.020			−0.518785
$215$	15.7353	−	38.1960i	0.0731873	−	0.177656i
$216$		0			0
$217$			233.616i			1.07657i
$218$	−54.4584			−0.249809
$219$		0			0
$220$	−43.7871	+	106.290i	−0.199032	+	0.483134i
$221$			73.3322i			0.331820i
$222$		0			0
$223$			357.273i			1.60212i	0.598582	+	0.801062i	$0.295731\pi$
$223$			357.273i			1.60212i	−0.598582	+	0.801062i	$0.704269\pi$
$224$			135.933i			0.606844i
$225$		0			0
$226$	258.682			1.14461
$227$	−128.800			−0.567403			−0.283701	−	0.958913i	$-0.591562\pi$
$227$	−128.800			−0.567403			−0.283701	+	0.958913i	$0.591562\pi$
$228$		0			0
$229$	−253.105			−1.10526			−0.552631	−	0.833426i	$-0.686376\pi$
$229$	−253.105			−1.10526			−0.552631	+	0.833426i	$0.686376\pi$
$230$	−6.10018	−	2.51304i	−0.0265225	−	0.0109262i
$231$		0			0
$232$			131.903i			0.568548i
$233$	96.3566			0.413548			0.206774	−	0.978389i	$-0.433704\pi$
$233$	96.3566			0.413548			0.206774	+	0.978389i	$0.433704\pi$
$234$		0			0
$235$	−70.0663	−	28.8646i	−0.298154	−	0.122828i
$236$			62.7764i			0.266002i
$237$		0			0
$238$			277.077i			1.16419i
$239$			109.392i			0.457706i	0.973461	+	0.228853i	$0.0734976\pi$
$239$			109.392i			0.457706i	−0.973461	+	0.228853i	$0.926502\pi$
$240$		0			0
$241$	−313.624			−1.30134			−0.650672	−	0.759359i	$-0.725513\pi$
$241$	−313.624			−1.30134			−0.650672	+	0.759359i	$0.725513\pi$
$242$	486.485			2.01027
$243$		0			0
$244$	79.2770			0.324906
$245$	−1.23813	−	0.510059i	−0.00505357	−	0.00208187i
$246$		0			0
$247$			16.8049i			0.0680360i
$248$	210.318			0.848057
$249$		0			0
$250$	111.233	+	264.215i	0.444931	+	1.05686i
$251$			192.888i			0.768478i	0.923234	+	0.384239i	$0.125536\pi$
$251$			192.888i			0.768478i	−0.923234	+	0.384239i	$0.874464\pi$
$252$		0			0
$253$			10.5011i			0.0415064i
$254$		−	223.617i		−	0.880382i
$255$		0			0
$256$	220.389			0.860894
$257$	−373.514			−1.45336			−0.726680	−	0.686976i	$-0.758938\pi$
$257$	−373.514			−1.45336			−0.726680	+	0.686976i	$0.758938\pi$
$258$		0			0
$259$	−149.545			−0.577394
$260$	−10.1654	+	24.6757i	−0.0390978	+	0.0949066i
$261$		0			0
$262$		−	47.1107i		−	0.179812i
$263$	−201.579			−0.766460			−0.383230	−	0.923653i	$-0.625188\pi$
$263$	−201.579			−0.766460			−0.383230	+	0.923653i	$0.625188\pi$
$264$		0			0
$265$	−113.638	−	46.8143i	−0.428822	−	0.176658i
$266$			63.4952i			0.238704i
$267$		0			0
$268$			150.611i			0.561980i
$269$		−	7.31695i		−	0.0272006i	−0.999908	−	0.0136003i	$-0.995671\pi$
$269$		−	7.31695i		−	0.0272006i	0.999908	−	0.0136003i	$-0.00432924\pi$
$270$		0			0
$271$	−93.8451			−0.346292			−0.173146	−	0.984896i	$-0.555393\pi$
$271$	−93.8451			−0.346292			−0.173146	+	0.984896i	$0.555393\pi$
$272$	336.647			1.23767
$273$		0			0
$274$	−45.1144			−0.164651
$275$	321.403	−	323.887i	1.16874	−	1.17777i
$276$		0			0
$277$		−	266.869i		−	0.963428i	−0.876329	−	0.481714i	$-0.840014\pi$
$277$		−	266.869i		−	0.963428i	0.876329	−	0.481714i	$-0.159986\pi$
$278$	−247.785			−0.891314
$279$		0			0
$280$	83.5558	−	202.825i	0.298414	−	0.724374i
$281$		−	6.31405i		−	0.0224699i	−0.999937	−	0.0112350i	$-0.996424\pi$
$281$		−	6.31405i		−	0.0224699i	0.999937	−	0.0112350i	$-0.00357628\pi$
$282$		0			0
$283$			346.480i			1.22431i	0.790738	+	0.612155i	$0.209697\pi$
$283$			346.480i			1.22431i	−0.790738	+	0.612155i	$0.790303\pi$
$284$			84.1782i			0.296402i
$285$		0			0
$286$	177.364			0.620152
$287$	−355.483			−1.23862
$288$		0			0
$289$	10.5195			0.0363996
$290$	−91.6724	+	222.527i	−0.316112	+	0.767334i
$291$		0			0
$292$		−	61.6506i		−	0.211132i
$293$	106.692			0.364137			0.182068	−	0.983286i	$-0.441721\pi$
$293$	106.692			0.364137			0.182068	+	0.983286i	$0.441721\pi$
$294$		0			0
$295$	94.9130	−	230.393i	0.321739	−	0.780994i
$296$			134.631i			0.454835i
$297$		0			0
$298$			520.510i			1.74668i
$299$			2.43789i			0.00815349i
$300$		0			0
$301$	−57.6761			−0.191615
$302$	−128.036			−0.423961
$303$		0			0
$304$	77.1465			0.253771
$305$	−290.952	−	119.861i	−0.953940	−	0.392986i
$306$		0			0
$307$			161.083i			0.524702i	0.964973	+	0.262351i	$0.0844978\pi$
$307$			161.083i			0.524702i	−0.964973	+	0.262351i	$0.915502\pi$
$308$	160.497			0.521095
$309$		0			0
$310$	354.817	+	146.171i	1.14457	+	0.471519i
$311$			17.3745i			0.0558667i	0.999610	+	0.0279333i	$0.00889261\pi$
$311$			17.3745i			0.0558667i	−0.999610	+	0.0279333i	$0.991107\pi$
$312$		0			0
$313$			348.475i			1.11334i	0.830734	+	0.556670i	$0.187921\pi$
$313$			348.475i			1.11334i	−0.830734	+	0.556670i	$0.812079\pi$
$314$		−	156.868i		−	0.499579i
$315$		0			0
$316$	−148.556			−0.470113
$317$	−296.850			−0.936437			−0.468218	−	0.883613i	$-0.655104\pi$
$317$	−296.850			−0.936437			−0.468218	+	0.883613i	$0.655104\pi$
$318$		0			0
$319$	383.068			1.20084
$320$	−153.254	−	63.1348i	−0.478920	−	0.197296i
$321$		0			0
$322$			9.21128i			0.0286065i
$323$	68.6383			0.212502
$324$		0			0
$325$	74.6155	−	75.1921i	0.229586	−	0.231360i
$326$			184.628i			0.566345i
$327$		0			0
$328$			320.032i			0.975707i
$329$			105.800i			0.321581i
$330$		0			0
$331$	202.694			0.612367			0.306184	−	0.951972i	$-0.400948\pi$
$331$	202.694			0.612367			0.306184	+	0.951972i	$0.400948\pi$
$332$	−9.22238			−0.0277783
$333$		0			0
$334$	171.598			0.513768
$335$	227.712	−	552.751i	0.679737	−	1.65000i
$336$		0			0
$337$		−	32.7607i		−	0.0972129i	−0.998818	−	0.0486064i	$-0.984522\pi$
$337$		−	32.7607i		−	0.0972129i	0.998818	−	0.0486064i	$-0.0154780\pi$
$338$	−346.408			−1.02488
$339$		0			0
$340$	100.786	+	41.5199i	0.296429	+	0.122117i
$341$		−	610.798i		−	1.79120i
$342$		0			0
$343$			343.931i			1.00271i
$344$			51.9242i			0.150942i
$345$		0			0
$346$	−730.273			−2.11061
$347$	669.391			1.92908			0.964540	−	0.263935i	$-0.0850205\pi$
$347$	669.391			1.92908			0.964540	+	0.263935i	$0.0850205\pi$
$348$		0			0
$349$	548.274			1.57098			0.785492	−	0.618871i	$-0.212410\pi$
$349$	548.274			1.57098			0.785492	+	0.618871i	$0.212410\pi$
$350$	281.925	−	284.104i	0.805501	−	0.811726i
$351$		0			0
$352$		−	355.402i		−	1.00967i
$353$	20.0111			0.0566888			0.0283444	−	0.999598i	$-0.490976\pi$
$353$	20.0111			0.0566888			0.0283444	+	0.999598i	$0.490976\pi$
$354$		0			0
$355$	127.271	−	308.939i	0.358510	−	0.870252i
$356$			126.753i			0.356047i
$357$		0			0
$358$			7.21818i			0.0201625i
$359$		−	224.934i		−	0.626556i	−0.949661	−	0.313278i	$-0.898573\pi$
$359$		−	224.934i		−	0.626556i	0.949661	−	0.313278i	$-0.101427\pi$
$360$		0			0
$361$	−345.271			−0.956429
$362$	1.91100			0.00527902
$363$		0			0
$364$	37.2604			0.102364
$365$	−93.2110	+	226.262i	−0.255373	+	0.619895i
$366$		0			0
$367$			50.5726i			0.137800i	0.997624	+	0.0689000i	$0.0219490\pi$
$367$			50.5726i			0.137800i	−0.997624	+	0.0689000i	$0.978051\pi$
$368$	11.1917			0.0304122
$369$		0			0
$370$	−93.5685	+	227.130i	−0.252888	+	0.613864i
$371$			171.593i			0.462515i
$372$		0			0
$373$		−	282.741i		−	0.758019i	−0.925393	−	0.379010i	$-0.876265\pi$
$373$		−	282.741i		−	0.758019i	0.925393	−	0.379010i	$-0.123735\pi$
$374$		−	724.427i		−	1.93697i
$375$		0			0
$376$	95.2490			0.253322
$377$	88.9313			0.235892
$378$		0			0
$379$	−435.602			−1.14935			−0.574673	−	0.818383i	$-0.694871\pi$
$379$	−435.602			−1.14935			−0.574673	+	0.818383i	$0.694871\pi$
$380$	23.0962	+	9.51474i	0.0607796	+	0.0250388i
$381$		0			0
$382$			131.913i			0.345323i
$383$	−312.449			−0.815795			−0.407897	−	0.913028i	$-0.633738\pi$
$383$	−312.449			−0.815795			−0.407897	+	0.913028i	$0.633738\pi$
$384$		0			0
$385$	−589.036	−	242.660i	−1.52996	−	0.630285i
$386$		−	83.3905i		−	0.216038i
$387$		0			0
$388$			5.22527i			0.0134672i
$389$			251.407i			0.646290i	0.946349	+	0.323145i	$0.104740\pi$
$389$			251.407i			0.646290i	−0.946349	+	0.323145i	$0.895260\pi$
$390$		0			0
$391$	9.95739			0.0254665
$392$	1.68312			0.00429368
$393$		0			0
$394$	−346.956			−0.880600
$395$	545.209	+	224.605i	1.38028	+	0.568620i
$396$		0			0
$397$		−	237.163i		−	0.597388i	−0.954349	−	0.298694i	$-0.903449\pi$
$397$		−	237.163i		−	0.597388i	0.954349	−	0.298694i	$-0.0965510\pi$
$398$	−616.333			−1.54858
$399$		0			0
$400$	−345.185	−	342.539i	−0.862963	−	0.856346i
$401$		−	536.897i		−	1.33890i	−0.742859	−	0.669448i	$-0.766531\pi$
$401$		−	536.897i		−	1.33890i	0.742859	−	0.669448i	$-0.233469\pi$
$402$		0			0
$403$		−	141.800i		−	0.351862i
$404$			204.023i			0.505007i
$405$		0			0
$406$	336.016			0.827626
$407$	390.991			0.960666
$408$		0			0
$409$	−41.5034			−0.101475			−0.0507376	−	0.998712i	$-0.516157\pi$
$409$	−41.5034			−0.101475			−0.0507376	+	0.998712i	$0.516157\pi$
$410$	−222.422	+	539.909i	−0.542492	+	1.31685i
$411$		0			0
$412$		−	206.767i		−	0.501861i
$413$	−347.894			−0.842359
$414$		0			0
$415$	33.8467	+	13.9435i	0.0815584	+	0.0335989i
$416$		−	82.5086i		−	0.198338i
$417$		0			0
$418$		−	166.011i		−	0.397155i
$419$			734.867i			1.75386i	0.480619	+	0.876929i	$0.340412\pi$
$419$			734.867i			1.75386i	−0.480619	+	0.876929i	$0.659588\pi$
$420$		0			0
$421$	−15.3176			−0.0363839			−0.0181920	−	0.999835i	$-0.505791\pi$
$421$	−15.3176			−0.0363839			−0.0181920	+	0.999835i	$0.505791\pi$
$422$	−820.269			−1.94377
$423$		0			0
$424$	154.481			0.364341
$425$	−307.116	−	304.761i	−0.722626	−	0.717085i
$426$		0			0
$427$			439.338i			1.02889i
$428$	60.9788			0.142474
$429$		0			0
$430$	−36.0872	+	87.5987i	−0.0839238	+	0.203718i
$431$		−	98.8622i		−	0.229379i	−0.993401	−	0.114689i	$-0.963413\pi$
$431$		−	98.8622i		−	0.229379i	0.993401	−	0.114689i	$-0.0365872\pi$
$432$		0			0
$433$		−	573.821i		−	1.32522i	−0.748964	−	0.662610i	$-0.769449\pi$
$433$		−	573.821i		−	1.32522i	0.748964	−	0.662610i	$-0.230551\pi$
$434$		−	535.774i		−	1.23450i
$435$		0			0
$436$	29.9118			0.0686051
$437$	2.28185			0.00522162
$438$		0			0
$439$	−239.854			−0.546364			−0.273182	−	0.961962i	$-0.588076\pi$
$439$	−239.854			−0.546364			−0.273182	+	0.961962i	$0.588076\pi$
$440$	−218.460	+	530.293i	−0.496500	+	1.20521i
$441$		0			0
$442$		−	168.180i		−	0.380497i
$443$	366.515			0.827347			0.413673	−	0.910425i	$-0.364246\pi$
$443$	366.515			0.827347			0.413673	+	0.910425i	$0.364246\pi$
$444$		0			0
$445$	191.641	−	465.191i	0.430653	−	1.04537i
$446$		−	819.370i		−	1.83715i
$447$		0			0
$448$			231.414i			0.516550i
$449$			623.682i			1.38905i	0.719470	+	0.694523i	$0.244385\pi$
$449$			623.682i			1.38905i	−0.719470	+	0.694523i	$0.755615\pi$
$450$		0			0
$451$	929.424			2.06081
$452$	−142.084			−0.314344
$453$		0			0
$454$	295.391			0.650640
$455$	−136.748	−	56.3348i	−0.300545	−	0.123813i
$456$		0			0
$457$		−	388.452i		−	0.850005i	−0.905192	−	0.425002i	$-0.860273\pi$
$457$		−	388.452i		−	0.850005i	0.905192	−	0.425002i	$-0.139727\pi$
$458$	580.470			1.26740
$459$		0			0
$460$	3.35058	+	1.38031i	0.00728387	+	0.00300067i
$461$		−	379.855i		−	0.823980i	−0.911188	−	0.411990i	$-0.864834\pi$
$461$		−	379.855i		−	0.823980i	0.911188	−	0.411990i	$-0.135166\pi$
$462$		0			0
$463$		−	382.252i		−	0.825599i	−0.910822	−	0.412800i	$-0.864551\pi$
$463$		−	382.252i		−	0.825599i	0.910822	−	0.412800i	$-0.135449\pi$
$464$		−	408.258i		−	0.879867i
$465$		0			0
$466$	−220.984			−0.474215
$467$	35.3515			0.0756992			0.0378496	−	0.999283i	$-0.487949\pi$
$467$	35.3515			0.0756992			0.0378496	+	0.999283i	$0.487949\pi$
$468$		0			0
$469$	−834.655			−1.77965
$470$	160.690	+	66.1979i	0.341893	+	0.140847i
$471$		0			0
$472$			313.200i			0.663559i
$473$	150.796			0.318808
$474$		0			0
$475$	−70.3791	−	69.8394i	−0.148167	−	0.147030i
$476$		−	152.187i		−	0.319720i
$477$		0			0
$478$		−	250.879i		−	0.524851i
$479$			578.766i			1.20828i	0.796879	+	0.604139i	$0.206483\pi$
$479$			578.766i			1.20828i	−0.796879	+	0.604139i	$0.793517\pi$
$480$		0			0
$481$	90.7708			0.188713
$482$	719.264			1.49225
$483$		0			0
$484$	−267.207			−0.552080
$485$	7.90020	−	19.1771i	0.0162891	−	0.0395404i
$486$		0			0
$487$		−	424.778i		−	0.872235i	−0.899890	−	0.436117i	$-0.856353\pi$
$487$		−	424.778i		−	0.872235i	0.899890	−	0.436117i	$-0.143647\pi$
$488$	395.524			0.810499
$489$		0			0
$490$	2.83951	+	1.16977i	0.00579493	+	0.00238728i
$491$			390.005i			0.794307i	0.917752	+	0.397154i	$0.130002\pi$
$491$			390.005i			0.794307i	−0.917752	+	0.397154i	$0.869998\pi$
$492$		0			0
$493$		−	363.233i		−	0.736781i
$494$		−	38.5403i		−	0.0780168i
$495$		0			0
$496$	−650.964			−1.31243
$497$	−466.499			−0.938630
$498$		0			0
$499$	−617.553			−1.23758			−0.618790	−	0.785556i	$-0.712377\pi$
$499$	−617.553			−1.23758			−0.618790	+	0.785556i	$0.712377\pi$
$500$	−61.0957	−	145.123i	−0.122191	−	0.290245i
$501$		0			0
$502$		−	442.368i		−	0.881212i
$503$	551.752			1.09692			0.548461	−	0.836176i	$-0.315214\pi$
$503$	551.752			1.09692			0.548461	+	0.836176i	$0.315214\pi$
$504$		0			0
$505$	308.467	−	748.777i	0.610825	−	1.48273i
$506$		−	24.0833i		−	0.0475954i
$507$		0			0
$508$			122.824i			0.241779i
$509$			443.153i			0.870635i	0.900277	+	0.435317i	$0.143364\pi$
$509$			443.153i			0.870635i	−0.900277	+	0.435317i	$0.856636\pi$
$510$		0			0
$511$	341.656			0.668602
$512$	110.221			0.215276
$513$		0			0
$514$	856.615			1.66657
$515$	−312.615	+	758.847i	−0.607020	+	1.47349i
$516$		0			0
$517$		−	276.619i		−	0.535046i
$518$	342.966			0.662097
$519$		0			0
$520$	−50.7167	+	123.110i	−0.0975321	+	0.236751i
$521$			716.733i			1.37569i	0.725859	+	0.687843i	$0.241442\pi$
$521$			716.733i			1.37569i	−0.725859	+	0.687843i	$0.758558\pi$
$522$		0			0
$523$			417.591i			0.798453i	0.916852	+	0.399227i	$0.130721\pi$
$523$			417.591i			0.798453i	−0.916852	+	0.399227i	$0.869279\pi$
$524$			25.8760i			0.0493817i
$525$		0			0
$526$	462.301			0.878899
$527$	−579.172			−1.09900
$528$		0			0
$529$	−528.669			−0.999374
$530$	260.616	+	107.364i	0.491729	+	0.202573i
$531$		0			0
$532$		−	34.8753i		−	0.0655552i
$533$	215.771			0.404824
$534$		0			0
$535$	−223.796	−	92.1953i	−0.418311	−	0.172328i
$536$			751.417i			1.40190i
$537$		0			0
$538$			16.7807i			0.0311908i
$539$		−	4.88807i		−	0.00906877i
$540$		0			0
$541$	−365.297			−0.675225			−0.337612	−	0.941285i	$-0.609619\pi$
$541$	−365.297			−0.675225			−0.337612	+	0.941285i	$0.609619\pi$
$542$	215.224			0.397092
$543$		0			0
$544$	−337.000			−0.619485
$545$	−109.778	−	45.2244i	−0.201428	−	0.0829805i
$546$		0			0
$547$			611.588i			1.11808i	0.829142	+	0.559039i	$0.188830\pi$
$547$			611.588i			1.11808i	−0.829142	+	0.559039i	$0.811170\pi$
$548$	24.7795			0.0452181
$549$		0			0
$550$	−737.105	+	742.801i	−1.34019	+	1.35055i
$551$		−	83.2389i		−	0.151069i
$552$		0			0
$553$		−	823.267i		−	1.48873i
$554$			612.038i			1.10476i
$555$		0			0
$556$	136.098			0.244781
$557$	−510.888			−0.917214			−0.458607	−	0.888639i	$-0.651651\pi$
$557$	−510.888			−0.917214			−0.458607	+	0.888639i	$0.651651\pi$
$558$		0			0
$559$	35.0082			0.0626265
$560$	−258.617	+	627.770i	−0.461816	+	1.12102i
$561$		0			0
$562$			14.4806i			0.0257662i
$563$	−157.752			−0.280199			−0.140099	−	0.990137i	$-0.544742\pi$
$563$	−157.752			−0.280199			−0.140099	+	0.990137i	$0.544742\pi$
$564$		0			0
$565$	521.456	+	214.820i	0.922931	+	0.380212i
$566$		−	794.615i		−	1.40391i
$567$		0			0
$568$			419.976i			0.739395i
$569$			175.875i			0.309095i	0.987985	+	0.154547i	$0.0493919\pi$
$569$			175.875i			0.309095i	−0.987985	+	0.154547i	$0.950608\pi$
$570$		0			0
$571$	−588.516			−1.03068			−0.515338	−	0.856987i	$-0.672334\pi$
$571$	−588.516			−1.03068			−0.515338	+	0.856987i	$0.672334\pi$
$572$	−97.4186			−0.170312
$573$		0			0
$574$	815.264			1.42032
$575$	−10.2099	−	10.1316i	−0.0177564	−	0.0176203i
$576$		0			0
$577$			323.853i			0.561270i	0.959815	+	0.280635i	$0.0905451\pi$
$577$			323.853i			0.561270i	−0.959815	+	0.280635i	$0.909455\pi$
$578$	−24.1254			−0.0417394
$579$		0			0
$580$	50.3519	−	122.225i	0.0868137	−	0.210733i
$581$		−	51.1086i		−	0.0879667i
$582$		0			0
$583$		−	448.637i		−	0.769532i
$584$		−	307.583i		−	0.526684i
$585$		0			0
$586$	−244.687			−0.417555
$587$	1031.07			1.75652			0.878258	−	0.478187i	$-0.158706\pi$
$587$	1031.07			1.75652			0.878258	+	0.478187i	$0.158706\pi$
$588$		0			0
$589$	−132.724			−0.225337
$590$	−217.673	+	528.383i	−0.368938	+	0.895565i
$591$		0			0
$592$		−	416.703i		−	0.703890i
$593$	193.131			0.325685			0.162842	−	0.986652i	$-0.447934\pi$
$593$	193.131			0.325685			0.162842	+	0.986652i	$0.447934\pi$
$594$		0			0
$595$	−230.095	+	558.536i	−0.386714	+	0.938716i
$596$		−	285.895i		−	0.479690i
$597$		0			0
$598$		−	5.59106i		−	0.00934960i
$599$		−	409.269i		−	0.683254i	−0.939836	−	0.341627i	$-0.889022\pi$
$599$		−	409.269i		−	0.683254i	0.939836	−	0.341627i	$-0.110978\pi$
$600$		0			0
$601$	−1040.86			−1.73189			−0.865943	−	0.500143i	$-0.833281\pi$
$601$	−1040.86			−1.73189			−0.865943	+	0.500143i	$0.833281\pi$
$602$	132.274			0.219724
$603$		0			0
$604$	70.3251			0.116432
$605$	980.665	+	403.996i	1.62093	+	0.667761i
$606$		0			0
$607$			680.854i			1.12167i	0.827927	+	0.560836i	$0.189520\pi$
$607$			680.854i			1.12167i	−0.827927	+	0.560836i	$0.810480\pi$
$608$	−77.2273			−0.127019
$609$		0			0
$610$	667.268	+	274.888i	1.09388	+	0.450637i
$611$		−	64.2185i		−	0.105104i
$612$		0			0
$613$		−	1024.19i		−	1.67079i	−0.549651	−	0.835395i	$-0.685239\pi$
$613$		−	1024.19i		−	1.67079i	0.549651	−	0.835395i	$-0.314761\pi$
$614$		−	369.428i		−	0.601675i
$615$		0			0
$616$	800.743			1.29991
$617$	1111.27			1.80109			0.900544	−	0.434765i	$-0.143169\pi$
$617$	1111.27			1.80109			0.900544	+	0.434765i	$0.143169\pi$
$618$		0			0
$619$	801.879			1.29544			0.647722	−	0.761877i	$-0.275722\pi$
$619$	801.879			1.29544			0.647722	+	0.761877i	$0.275722\pi$
$620$	−194.887	−	80.2857i	−0.314333	−	0.129493i
$621$		0			0
$622$		−	39.8467i		−	0.0640622i
$623$	−702.439			−1.12751
$624$		0			0
$625$	4.81074	+	624.981i	0.00769719	+	0.999970i
$626$		−	799.192i		−	1.27666i
$627$		0			0
$628$			86.1611i			0.137199i
$629$		−	370.746i		−	0.589421i
$630$		0			0
$631$	−564.192			−0.894124			−0.447062	−	0.894503i	$-0.647530\pi$
$631$	−564.192			−0.894124			−0.447062	+	0.894503i	$0.647530\pi$
$632$	−741.165			−1.17273
$633$		0			0
$634$	680.796			1.07381
$635$	185.700	−	450.771i	0.292441	−	0.709876i
$636$		0			0
$637$		−	1.13479i		−	0.00178146i
$638$	−878.527			−1.37700
$639$		0			0
$640$	711.561	+	293.135i	1.11181	+	0.458024i
$641$		−	1205.50i		−	1.88065i	−0.340273	−	0.940327i	$-0.610519\pi$
$641$		−	1205.50i		−	1.88065i	0.340273	−	0.940327i	$-0.389481\pi$
$642$		0			0
$643$			464.316i			0.722109i	0.932545	+	0.361055i	$0.117583\pi$
$643$			464.316i			0.722109i	−0.932545	+	0.361055i	$0.882417\pi$
$644$		−	5.05938i		−	0.00785619i
$645$		0			0
$646$	−157.415			−0.243676
$647$	−372.702			−0.576046			−0.288023	−	0.957624i	$-0.592998\pi$
$647$	−372.702			−0.576046			−0.288023	+	0.957624i	$0.592998\pi$
$648$		0			0
$649$	909.583			1.40151
$650$	−171.123	+	172.445i	−0.263266	+	0.265300i
$651$		0			0
$652$		−	101.409i		−	0.155535i
$653$	221.410			0.339066			0.169533	−	0.985525i	$-0.445774\pi$
$653$	221.410			0.339066			0.169533	+	0.985525i	$0.445774\pi$
$654$		0			0
$655$	39.1225	−	94.9666i	0.0597290	−	0.144987i
$656$		−	990.543i		−	1.50997i
$657$		0			0
$658$		−	242.642i		−	0.368757i
$659$			624.807i			0.948113i	0.880494	+	0.474057i	$0.157211\pi$
$659$			624.807i			0.948113i	−0.880494	+	0.474057i	$0.842789\pi$
$660$		0			0
$661$	334.515			0.506074			0.253037	−	0.967457i	$-0.418571\pi$
$661$	334.515			0.506074			0.253037	+	0.967457i	$0.418571\pi$
$662$	−464.857			−0.702201
$663$		0			0
$664$	−46.0117			−0.0692947
$665$	−52.7288	+	127.995i	−0.0792915	+	0.192473i
$666$		0			0
$667$		−	12.0755i		−	0.0181042i
$668$	−94.2521			−0.141096
$669$		0			0
$670$	−522.234	+	1267.68i	−0.779453	+	1.89206i
$671$		−	1148.67i		−	1.71187i
$672$		0			0
$673$		−	947.251i		−	1.40751i	−0.710445	−	0.703753i	$-0.751506\pi$
$673$		−	947.251i		−	1.40751i	0.710445	−	0.703753i	$-0.248494\pi$
$674$			75.1334i			0.111474i
$675$		0			0
$676$	190.268			0.281462
$677$	−1009.24			−1.49075			−0.745377	−	0.666643i	$-0.767731\pi$
$677$	−1009.24			−1.49075			−0.745377	+	0.666643i	$0.767731\pi$
$678$		0			0
$679$	−28.9574			−0.0426471
$680$	502.835	+	207.148i	0.739463	+	0.304630i
$681$		0			0
$682$			1400.80i			2.05396i
$683$	−697.555			−1.02131			−0.510655	−	0.859786i	$-0.670597\pi$
$683$	−697.555			−1.02131			−0.510655	+	0.859786i	$0.670597\pi$
$684$		0			0
$685$	−90.9424	−	37.4647i	−0.132763	−	0.0546930i
$686$		−	788.770i		−	1.14981i
$687$		0			0
$688$		−	160.713i		−	0.233594i
$689$		−	104.154i		−	0.151166i
$690$		0			0
$691$	−179.186			−0.259314			−0.129657	−	0.991559i	$-0.541388\pi$
$691$	−179.186			−0.259314			−0.129657	+	0.991559i	$0.541388\pi$
$692$	401.109			0.579638
$693$		0			0
$694$	−1535.18			−2.21207
$695$	−499.490	−	205.770i	−0.718691	−	0.296072i
$696$		0			0
$697$		−	881.300i		−	1.26442i
$698$	−1257.41			−1.80145
$699$		0			0
$700$	−154.850	+	156.047i	−0.221215	+	0.222924i
$701$		−	211.499i		−	0.301710i	−0.988556	−	0.150855i	$-0.951797\pi$
$701$		−	211.499i		−	0.301710i	0.988556	−	0.150855i	$-0.0482026\pi$
$702$		0			0
$703$		−	84.9606i		−	0.120854i
$704$		−	605.042i		−	0.859435i
$705$		0			0
$706$	−45.8935			−0.0650050
$707$	−1130.65			−1.59923
$708$		0			0
$709$	−433.249			−0.611071			−0.305536	−	0.952181i	$-0.598835\pi$
$709$	−433.249			−0.611071			−0.305536	+	0.952181i	$0.598835\pi$
$710$	−291.883	+	708.521i	−0.411103	+	0.997916i
$711$		0			0
$712$			632.387i			0.888184i
$713$	−19.2543			−0.0270046
$714$		0			0
$715$	357.533	+	147.289i	0.500046	+	0.205999i
$716$		−	3.96466i		−	0.00553723i
$717$		0			0
$718$			515.862i			0.718471i
$719$			588.734i			0.818824i	0.912350	+	0.409412i	$0.134266\pi$
$719$			588.734i			0.818824i	−0.912350	+	0.409412i	$0.865734\pi$
$720$		0			0
$721$	1145.86			1.58927
$722$	791.843			1.09674
$723$		0			0
$724$	−1.04964			−0.00144978
$725$	−369.590	+	372.445i	−0.509779	+	0.513718i
$726$		0			0
$727$			252.040i			0.346685i	0.984862	+	0.173343i	$0.0554568\pi$
$727$			252.040i			0.346685i	−0.984862	+	0.173343i	$0.944543\pi$
$728$	185.897			0.255353
$729$		0			0
$730$	213.770	−	518.908i	0.292835	−	0.710833i
$731$		−	142.988i		−	0.195606i
$732$		0			0
$733$			192.473i			0.262583i	0.991344	+	0.131291i	$0.0419124\pi$
$733$			192.473i			0.262583i	−0.991344	+	0.131291i	$0.958088\pi$
$734$		−	115.983i		−	0.158015i
$735$		0			0
$736$	−11.2034			−0.0152220
$737$	2182.24			2.96097
$738$		0			0
$739$	811.337			1.09788			0.548942	−	0.835860i	$-0.315031\pi$
$739$	811.337			1.09788			0.548942	+	0.835860i	$0.315031\pi$
$740$	51.3934	−	124.753i	0.0694505	−	0.168585i
$741$		0			0
$742$		−	393.531i		−	0.530365i
$743$	1021.43			1.37474			0.687370	−	0.726307i	$-0.258765\pi$
$743$	1021.43			1.37474			0.687370	+	0.726307i	$0.258765\pi$
$744$		0			0
$745$	−432.251	+	1049.25i	−0.580203	+	1.40839i
$746$			648.438i			0.869219i
$747$		0			0
$748$			397.899i			0.531950i
$749$			337.933i			0.451178i
$750$		0			0
$751$	−954.377			−1.27081			−0.635404	−	0.772180i	$-0.719167\pi$
$751$	−954.377			−1.27081			−0.635404	+	0.772180i	$0.719167\pi$
$752$	−294.809			−0.392033
$753$		0			0
$754$	−203.955			−0.270497
$755$	−258.098	−	106.326i	−0.341851	−	0.140829i
$756$		0			0
$757$			1462.32i			1.93173i	0.259040	+	0.965866i	$0.416594\pi$
$757$			1462.32i			1.93173i	−0.259040	+	0.965866i	$0.583406\pi$
$758$	999.008			1.31795
$759$		0			0
$760$	115.230	+	47.4703i	0.151619	+	0.0624610i
$761$		−	38.1592i		−	0.0501435i	−0.999686	−	0.0250718i	$-0.992019\pi$
$761$		−	38.1592i		−	0.0501435i	0.999686	−	0.0250718i	$-0.00798143\pi$
$762$		0			0
$763$			165.765i			0.217255i
$764$		−	72.4546i		−	0.0948359i
$765$		0			0
$766$	716.570			0.935470
$767$	211.165			0.275313
$768$		0			0
$769$	602.497			0.783481			0.391741	−	0.920076i	$-0.371873\pi$
$769$	602.497			0.783481			0.391741	+	0.920076i	$0.371873\pi$
$770$	1350.89	+	556.515i	1.75441	+	0.722747i
$771$		0			0
$772$			45.8031i			0.0593304i
$773$	394.816			0.510758			0.255379	−	0.966841i	$-0.417800\pi$
$773$	394.816			0.510758			0.255379	+	0.966841i	$0.417800\pi$
$774$		0			0
$775$	593.861	+	589.307i	0.766272	+	0.760397i
$776$			26.0696i			0.0335948i
$777$		0			0
$778$		−	576.576i		−	0.741100i
$779$		−	201.960i		−	0.259255i
$780$		0			0
$781$	1219.68			1.56169
$782$	−22.8362			−0.0292024
$783$		0			0
$784$	−5.20950			−0.00664478
$785$	130.269	−	316.217i	0.165948	−	0.402824i
$786$		0			0
$787$			823.709i			1.04664i	0.852135	+	0.523322i	$0.175308\pi$
$787$			823.709i			1.04664i	−0.852135	+	0.523322i	$0.824692\pi$
$788$	190.569			0.241839
$789$		0			0
$790$	−1250.38	−	515.108i	−1.58276	−	0.652036i
$791$		−	787.400i		−	0.995448i
$792$		0			0
$793$		−	266.669i		−	0.336279i
$794$			543.909i			0.685024i
$795$		0			0
$796$	338.527			0.425285
$797$	−919.499			−1.15370			−0.576850	−	0.816850i	$-0.695718\pi$
$797$	−919.499			−1.15370			−0.576850	+	0.816850i	$0.695718\pi$
$798$		0			0
$799$	−262.296			−0.328280
$800$	345.547	+	342.898i	0.431934	+	0.428622i
$801$		0			0
$802$			1231.32i			1.53531i
$803$	−893.272			−1.11242
$804$		0			0
$805$	−7.64940	+	18.5683i	−0.00950236	+	0.0230662i
$806$			325.204i			0.403479i
$807$		0			0
$808$			1017.90i			1.25977i
$809$			1432.18i			1.77031i	0.465300	+	0.885153i	$0.345946\pi$
$809$			1432.18i			1.77031i	−0.465300	+	0.885153i	$0.654054\pi$
$810$		0			0
$811$	−422.921			−0.521481			−0.260740	−	0.965409i	$-0.583967\pi$
$811$	−422.921			−0.521481			−0.260740	+	0.965409i	$0.583967\pi$
$812$	−184.560			−0.227291
$813$		0			0
$814$	−896.698			−1.10159
$815$	−153.322	+	372.177i	−0.188126	+	0.456659i
$816$		0			0
$817$		−	32.7674i		−	0.0401069i
$818$	95.1837			0.116362
$819$		0			0
$820$	122.167	−	296.550i	0.148984	−	0.361647i
$821$			174.385i			0.212406i	0.994344	+	0.106203i	$0.0338693\pi$
$821$			174.385i			0.212406i	−0.994344	+	0.106203i	$0.966131\pi$
$822$		0			0
$823$			897.360i			1.09035i	0.838322	+	0.545176i	$0.183537\pi$
$823$			897.360i			1.09035i	−0.838322	+	0.545176i	$0.816463\pi$
$824$		−	1031.59i		−	1.25193i
$825$		0			0
$826$	797.860			0.965932
$827$	−233.440			−0.282273			−0.141137	−	0.989990i	$-0.545076\pi$
$827$	−233.440			−0.282273			−0.141137	+	0.989990i	$0.545076\pi$
$828$		0			0
$829$	−990.934			−1.19534			−0.597668	−	0.801744i	$-0.703906\pi$
$829$	−990.934			−1.19534			−0.597668	+	0.801744i	$0.703906\pi$
$830$	−77.6240	−	31.9781i	−0.0935229	−	0.0385278i
$831$		0			0
$832$		−	140.464i		−	0.168827i
$833$	−4.63497			−0.00556419
$834$		0			0
$835$	345.911	+	142.502i	0.414265	+	0.170661i
$836$			91.1829i			0.109071i
$837$		0			0
$838$		−	1685.34i		−	2.01115i
$839$			347.594i			0.414295i	0.978310	+	0.207148i	$0.0664181\pi$
$839$			347.594i			0.414295i	−0.978310	+	0.207148i	$0.933582\pi$
$840$		0			0
$841$	400.501			0.476219
$842$	35.1294			0.0417214
$843$		0			0
$844$	450.541			0.533816
$845$	−698.296	−	287.671i	−0.826386	−	0.340439i
$846$		0			0
$847$		−	1480.81i		−	1.74829i
$848$	−478.139			−0.563843
$849$		0			0
$850$	704.339	+	698.938i	0.828634	+	0.822281i
$851$		−	12.3253i		−	0.0144833i
$852$		0			0
$853$		−	1236.38i		−	1.44945i	−0.689036	−	0.724727i	$-0.741966\pi$
$853$		−	1236.38i		−	1.44945i	0.689036	−	0.724727i	$-0.258034\pi$
$854$		−	1007.58i		−	1.17983i
$855$		0			0
$856$	304.231			0.355411
$857$	914.705			1.06733			0.533667	−	0.845695i	$-0.320814\pi$
$857$	914.705			1.06733			0.533667	+	0.845695i	$0.320814\pi$
$858$		0			0
$859$	1143.02			1.33064			0.665320	−	0.746559i	$-0.268295\pi$
$859$	1143.02			1.33064			0.665320	+	0.746559i	$0.268295\pi$
$860$	19.8213	−	48.1144i	0.0230480	−	0.0559470i
$861$		0			0
$862$			226.730i			0.263028i
$863$	−1228.14			−1.42311			−0.711554	−	0.702632i	$-0.752008\pi$
$863$	−1228.14			−1.42311			−0.711554	+	0.702632i	$0.752008\pi$
$864$		0			0
$865$	−1472.10	−	606.446i	−1.70185	−	0.701094i
$866$			1316.00i			1.51963i
$867$		0			0
$868$			294.279i			0.339031i
$869$			2152.46i			2.47694i
$870$		0			0
$871$	506.619			0.581652
$872$	149.234			0.171140
$873$		0			0
$874$	−5.23318			−0.00598762
$875$	804.241	−	338.580i	0.919133	−	0.386949i
$876$		0			0
$877$		−	1377.73i		−	1.57096i	−0.618889	−	0.785478i	$-0.712417\pi$
$877$		−	1377.73i		−	1.57096i	0.618889	−	0.785478i	$-0.287583\pi$
$878$	550.080			0.626515
$879$		0			0
$880$	676.164	−	1641.33i	0.768368	−	1.86515i
$881$		−	1087.45i		−	1.23434i	−0.786831	−	0.617168i	$-0.788280\pi$
$881$		−	1087.45i		−	1.23434i	0.786831	−	0.617168i	$-0.211720\pi$
$882$		0			0
$883$			922.151i			1.04434i	0.852842	+	0.522170i	$0.174877\pi$
$883$			922.151i			1.04434i	−0.852842	+	0.522170i	$0.825123\pi$
$884$			92.3744i			0.104496i
$885$		0			0
$886$	−840.564			−0.948717
$887$	1060.99			1.19616			0.598079	−	0.801437i	$-0.295931\pi$
$887$	1060.99			1.19616			0.598079	+	0.801437i	$0.295931\pi$
$888$		0			0
$889$	−680.665			−0.765653
$890$	−439.508	+	1066.87i	−0.493829	+	1.19873i
$891$		0			0
$892$			450.047i			0.504537i
$893$	−60.1080			−0.0673102
$894$		0			0
$895$	−5.99425	+	14.5505i	−0.00669749	+	0.0162576i
$896$		−	1074.46i		−	1.19917i
$897$		0			0
$898$		−	1430.35i		−	1.59282i
$899$			702.372i			0.781282i
$900$		0			0
$901$	−425.407			−0.472150
$902$	−2131.54			−2.36313
$903$		0			0
$904$	−708.875			−0.784153
$905$	3.85224	+	1.58697i	0.00425661	+	0.00175356i
$906$		0			0
$907$			1395.98i			1.53912i	0.638576	+	0.769559i	$0.279524\pi$
$907$			1395.98i			1.53912i	−0.638576	+	0.769559i	$0.720476\pi$
$908$	−162.246			−0.178685
$909$		0			0
$910$	313.617	+	129.198i	0.344634	+	0.141976i
$911$			1270.25i			1.39434i	0.716905	+	0.697171i	$0.245558\pi$
$911$			1270.25i			1.39434i	−0.716905	+	0.697171i	$0.754442\pi$
$912$		0			0
$913$			133.626i			0.146359i
$914$			890.875i			0.974699i
$915$		0			0
$916$	−318.829			−0.348066
$917$	−143.400			−0.156379
$918$		0			0
$919$	269.489			0.293242			0.146621	−	0.989193i	$-0.453160\pi$
$919$	269.489			0.293242			0.146621	+	0.989193i	$0.453160\pi$
$920$	16.7165	+	6.88655i	0.0181701	+	0.00748538i
$921$		0			0
$922$			871.158i			0.944857i
$923$	283.155			0.306777
$924$		0			0
$925$	−377.234	+	380.149i	−0.407821	+	0.410972i
$926$			876.657i			0.946713i
$927$		0			0
$928$			408.686i			0.440395i
$929$		−	1106.34i		−	1.19089i	−0.803394	−	0.595447i	$-0.796975\pi$
$929$		−	1106.34i		−	1.19089i	0.803394	−	0.595447i	$-0.203025\pi$
$930$		0			0
$931$	−1.06215			−0.00114088
$932$	121.378			0.130233
$933$		0			0
$934$	−81.0751			−0.0868042
$935$	601.592	−	1460.31i	0.643414	−	1.56183i
$936$		0			0
$937$		−	130.956i		−	0.139761i	−0.997555	−	0.0698805i	$-0.977738\pi$
$937$		−	130.956i		−	0.139761i	0.997555	−	0.0698805i	$-0.0222618\pi$
$938$	1914.19			2.04072
$939$		0			0
$940$	−88.2604	−	36.3598i	−0.0938940	−	0.0386807i
$941$		−	355.742i		−	0.378047i	−0.981973	−	0.189023i	$-0.939468\pi$
$941$		−	355.742i		−	0.378047i	0.981973	−	0.189023i	$-0.0605322\pi$
$942$		0			0
$943$		−	29.2984i		−	0.0310694i
$944$		−	969.397i		−	1.02690i
$945$		0			0
$946$	−345.836			−0.365577
$947$	1022.67			1.07991			0.539953	−	0.841695i	$-0.318442\pi$
$947$	1022.67			1.07991			0.539953	+	0.841695i	$0.318442\pi$
$948$		0			0
$949$	−207.378			−0.218523
$950$	161.407	+	160.170i	0.169902	+	0.168600i
$951$		0			0
$952$		−	759.281i		−	0.797564i
$953$	475.336			0.498778			0.249389	−	0.968403i	$-0.419770\pi$
$953$	475.336			0.498778			0.249389	+	0.968403i	$0.419770\pi$
$954$		0			0
$955$	−109.546	+	265.913i	−0.114708	+	0.278443i
$956$			137.798i			0.144140i
$957$		0			0
$958$		−	1327.34i		−	1.38553i
$959$			137.323i			0.143194i
$960$		0			0
$961$	158.926			0.165376
$962$	−208.173			−0.216397
$963$		0			0
$964$	−395.063			−0.409816
$965$	69.2507	−	168.100i	0.0717624	−	0.174197i
$966$		0			0
$967$		−	1252.37i		−	1.29511i	−0.762020	−	0.647554i	$-0.775792\pi$
$967$		−	1252.37i		−	1.29511i	0.762020	−	0.647554i	$-0.224208\pi$
$968$	−1333.13			−1.37720
$969$		0			0
$970$	−18.1183	+	43.9806i	−0.0186787	+	0.0453409i
$971$		−	280.624i		−	0.289005i	−0.989504	−	0.144502i	$-0.953842\pi$
$971$		−	280.624i		−	0.289005i	0.989504	−	0.144502i	$-0.0461581\pi$
$972$		0			0
$973$			754.231i			0.775160i
$974$			974.185i			1.00019i
$975$		0			0
$976$	−1224.20			−1.25430
$977$	−331.594			−0.339400			−0.169700	−	0.985496i	$-0.554280\pi$
$977$	−331.594			−0.339400			−0.169700	+	0.985496i	$0.554280\pi$
$978$		0			0
$979$	1836.56			1.87595
$980$	−1.55963	−	0.642507i	−0.00159146	−	0.000655619i
$981$		0			0
$982$		−	894.436i		−	0.910831i
$983$	−12.9829			−0.0132074			−0.00660372	−	0.999978i	$-0.502102\pi$
$983$	−12.9829			−0.0132074			−0.00660372	+	0.999978i	$0.502102\pi$
$984$		0			0
$985$	−699.401	−	288.126i	−0.710051	−	0.292513i
$986$			833.037i			0.844866i
$987$		0			0
$988$			21.1686i			0.0214257i
$989$		−	4.75358i		−	0.00480645i
$990$		0			0
$991$	−1366.83			−1.37924			−0.689619	−	0.724172i	$-0.742222\pi$
$991$	−1366.83			−1.37924			−0.689619	+	0.724172i	$0.742222\pi$
$992$	651.646			0.656902
$993$		0			0
$994$	1069.87			1.07633
$995$	−1242.42	−	511.826i	−1.24866	−	0.514398i
$996$		0			0
$997$		−	1027.81i		−	1.03090i	−0.856918	−	0.515452i	$-0.827624\pi$
$997$		−	1027.81i		−	1.03090i	0.856918	−	0.515452i	$-0.172376\pi$
$998$	1416.29			1.41913
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.a.404.5		20
3.2	odd	2	inner	405.3.d.a.404.16		20
5.4	even	2	inner	405.3.d.a.404.15		20
9.2	odd	6		45.3.h.a.14.3	✓	20
9.4	even	3		45.3.h.a.29.8	yes	20
9.5	odd	6		135.3.h.a.89.3		20
9.7	even	3		135.3.h.a.44.8		20
15.14	odd	2	inner	405.3.d.a.404.6		20
45.2	even	12		225.3.j.e.176.8		20
45.4	even	6		45.3.h.a.29.3	yes	20
45.7	odd	12		675.3.j.e.476.3		20
45.13	odd	12		225.3.j.e.101.3		20
45.14	odd	6		135.3.h.a.89.8		20
45.22	odd	12		225.3.j.e.101.8		20
45.23	even	12		675.3.j.e.251.8		20
45.29	odd	6		45.3.h.a.14.8	yes	20
45.32	even	12		675.3.j.e.251.3		20
45.34	even	6		135.3.h.a.44.3		20
45.38	even	12		225.3.j.e.176.3		20
45.43	odd	12		675.3.j.e.476.8		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.3	✓	20	9.2	odd	6
45.3.h.a.14.8	yes	20	45.29	odd	6
45.3.h.a.29.3	yes	20	45.4	even	6
45.3.h.a.29.8	yes	20	9.4	even	3
135.3.h.a.44.3		20	45.34	even	6
135.3.h.a.44.8		20	9.7	even	3
135.3.h.a.89.3		20	9.5	odd	6
135.3.h.a.89.8		20	45.14	odd	6
225.3.j.e.101.3		20	45.13	odd	12
225.3.j.e.101.8		20	45.22	odd	12
225.3.j.e.176.3		20	45.38	even	12
225.3.j.e.176.8		20	45.2	even	12
405.3.d.a.404.5		20	1.1	even	1	trivial
405.3.d.a.404.6		20	15.14	odd	2	inner
405.3.d.a.404.15		20	5.4	even	2	inner
405.3.d.a.404.16		20	3.2	odd	2	inner
675.3.j.e.251.3		20	45.32	even	12
675.3.j.e.251.8		20	45.23	even	12
675.3.j.e.476.3		20	45.7	odd	12
675.3.j.e.476.8		20	45.43	odd	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.29340 q^{2} +1.25967 q^{4} +(-4.62307 - 1.90452i) q^{5} +6.98084i q^{7} +6.28466 q^{8} +(10.6025 + 4.36783i) q^{10} -18.2517i q^{11} -4.23723i q^{13} -16.0098i q^{14} -19.4519 q^{16} -17.3066 q^{17} -3.96601 q^{19} +(-5.82355 - 2.39907i) q^{20} +41.8584i q^{22} -0.575351 q^{23} +(17.7456 + 17.6095i) q^{25} +9.71765i q^{26} +8.79356i q^{28} +20.9881i q^{29} +33.4653 q^{31} +19.4723 q^{32} +39.6910 q^{34} +(13.2952 - 32.2729i) q^{35} +21.4222i q^{37} +9.09563 q^{38} +(-29.0544 - 11.9693i) q^{40} +50.9227i q^{41} +8.26205i q^{43} -22.9911i q^{44} +1.31951 q^{46} +15.1558 q^{47} +0.267815 q^{49} +(-40.6976 - 40.3856i) q^{50} -5.33751i q^{52} +24.5806 q^{53} +(-34.7608 + 84.3789i) q^{55} +43.8723i q^{56} -48.1340i q^{58} +49.8356i q^{59} +62.9347 q^{61} -76.7492 q^{62} +33.1499 q^{64} +(-8.06991 + 19.5890i) q^{65} +119.564i q^{67} -21.8006 q^{68} +(-30.4911 + 74.0147i) q^{70} +66.8256i q^{71} -48.9419i q^{73} -49.1296i q^{74} -4.99586 q^{76} +127.412 q^{77} -117.932 q^{79} +(89.9276 + 37.0466i) q^{80} -116.786i q^{82} -7.32127 q^{83} +(80.0098 + 32.9609i) q^{85} -18.9482i q^{86} -114.706i q^{88} +100.624i q^{89} +29.5794 q^{91} -0.724753 q^{92} -34.7582 q^{94} +(18.3351 + 7.55336i) q^{95} +4.14812i q^{97} -0.614205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) \) 20 * q + 36 * q^4 + 4 * q^10 + 52 * q^16 - 8 * q^19 - 4 * q^25 - 56 * q^31 + 8 * q^34 + 68 * q^40 + 116 * q^46 + 80 * q^49 + 36 * q^55 + 100 * q^61 + 140 * q^64 + 108 * q^70 + 192 * q^76 + 256 * q^79 + 148 * q^85 - 288 * q^91 - 436 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more