LMFDB - Embedded newform 900.3.c.n.451.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(451,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("900.451");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.c (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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4069419264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{6} + 50x^{4} - 84x^{3} + 55x^{2} - 12x + 1 $ x^8 - 7x^6 + 50x^4 - 84x^3 + 55x^2 - 12*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			451.8
Root			$0.151747 + 0.0876113i$ of defining polynomial
Character	$\chi$	$=$	900.451
Dual form			900.3.c.n.451.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.33290 + 1.49110i) q^{2} +(-0.446749 + 3.97497i) q^{4} -6.56834i q^{7} +(-6.52255 + 4.63210i) q^{8} +O(q^{10})$	\(q+(1.33290 + 1.49110i) q^{2} +(-0.446749 + 3.97497i) q^{4} -6.56834i q^{7} +(-6.52255 + 4.63210i) q^{8} +2.26696i q^{11} +14.8772 q^{13} +(9.79404 - 8.75495i) q^{14} +(-15.6008 - 3.55163i) q^{16} +26.8250 q^{17} -10.8680i q^{19} +(-3.38027 + 3.02164i) q^{22} +36.4610i q^{23} +(19.8298 + 22.1834i) q^{26} +(26.1090 + 2.93440i) q^{28} +35.2510 q^{29} +23.8330i q^{31} +(-15.4985 - 27.9963i) q^{32} +(35.7550 + 39.9987i) q^{34} -54.7495 q^{37} +(16.2053 - 14.4860i) q^{38} +23.8298 q^{41} +56.2515i q^{43} +(-9.01112 - 1.01276i) q^{44} +(-54.3670 + 48.5989i) q^{46} +51.4177i q^{47} +5.85689 q^{49} +(-6.64636 + 59.1364i) q^{52} +30.6465 q^{53} +(30.4252 + 42.8423i) q^{56} +(46.9861 + 52.5627i) q^{58} -6.92483i q^{59} +107.426 q^{61} +(-35.5374 + 31.7671i) q^{62} +(21.0873 - 60.4262i) q^{64} -111.444i q^{67} +(-11.9840 + 106.629i) q^{68} -31.3190i q^{71} -110.909 q^{73} +(-72.9757 - 81.6369i) q^{74} +(43.2002 + 4.85528i) q^{76} +14.8902 q^{77} -59.0065i q^{79} +(31.7628 + 35.5326i) q^{82} +142.416i q^{83} +(-83.8765 + 74.9776i) q^{86} +(-10.5008 - 14.7864i) q^{88} -7.14798 q^{89} -97.7185i q^{91} +(-144.932 - 16.2889i) q^{92} +(-76.6689 + 68.5347i) q^{94} +126.308 q^{97} +(7.80665 + 8.73319i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 8 q^{4} - 20 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 - 8 * q^4 - 20 * q^8	$ 8 q - 2 q^{2} - 8 q^{4} - 20 q^{8} + 8 q^{13} - 22 q^{14} + 40 q^{16} + 4 q^{22} + 66 q^{26} + 104 q^{28} + 32 q^{29} - 112 q^{32} + 124 q^{34} - 176 q^{37} + 170 q^{38} + 16 q^{41} - 40 q^{44} - 76 q^{46} + 16 q^{49} + 56 q^{52} + 304 q^{53} + 172 q^{56} - 12 q^{58} + 136 q^{61} + 238 q^{62} + 16 q^{64} - 88 q^{68} + 240 q^{73} + 108 q^{74} + 120 q^{76} + 384 q^{77} + 320 q^{82} - 214 q^{86} - 200 q^{88} - 128 q^{89} - 312 q^{92} + 12 q^{94} + 216 q^{97} - 60 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 - 8 * q^4 - 20 * q^8 + 8 * q^13 - 22 * q^14 + 40 * q^16 + 4 * q^22 + 66 * q^26 + 104 * q^28 + 32 * q^29 - 112 * q^32 + 124 * q^34 - 176 * q^37 + 170 * q^38 + 16 * q^41 - 40 * q^44 - 76 * q^46 + 16 * q^49 + 56 * q^52 + 304 * q^53 + 172 * q^56 - 12 * q^58 + 136 * q^61 + 238 * q^62 + 16 * q^64 - 88 * q^68 + 240 * q^73 + 108 * q^74 + 120 * q^76 + 384 * q^77 + 320 * q^82 - 214 * q^86 - 200 * q^88 - 128 * q^89 - 312 * q^92 + 12 * q^94 + 216 * q^97 - 60 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.33290	+	1.49110i	0.666451	+	0.745549i
$2$	1.33290	+	1.49110i	0.666451	+	0.745549i
$3$		0			0
$3$		0			0
$4$	−0.446749	+	3.97497i	−0.111687	+	0.993743i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	6.56834i		−	0.938335i	−0.883109	−	0.469167i	$-0.844554\pi$
$7$		−	6.56834i		−	0.938335i	0.883109	−	0.469167i	$-0.155446\pi$
$8$	−6.52255	+	4.63210i	−0.815319	+	0.579013i
$9$		0			0
$10$		0			0
$11$			2.26696i			0.206088i	0.994677	+	0.103044i	$0.0328582\pi$
$11$			2.26696i			0.206088i	−0.994677	+	0.103044i	$0.967142\pi$
$12$		0			0
$13$	14.8772			1.14440			0.572200	−	0.820114i	$-0.306090\pi$
$13$	14.8772			1.14440			0.572200	+	0.820114i	$0.306090\pi$
$14$	9.79404	−	8.75495i	0.699575	−	0.625354i
$15$		0			0
$16$	−15.6008	−	3.55163i	−0.975052	−	0.221977i
$17$	26.8250			1.57794			0.788969	−	0.614432i	$-0.210615\pi$
$17$	26.8250			1.57794			0.788969	+	0.614432i	$0.210615\pi$
$18$		0			0
$19$		−	10.8680i		−	0.572002i	−0.958229	−	0.286001i	$-0.907674\pi$
$19$		−	10.8680i		−	0.572002i	0.958229	−	0.286001i	$-0.0923260\pi$
$20$		0			0
$21$		0			0
$22$	−3.38027	+	3.02164i	−0.153648	+	0.137347i
$23$			36.4610i			1.58526i	0.609702	+	0.792631i	$0.291289\pi$
$23$			36.4610i			1.58526i	−0.609702	+	0.792631i	$0.708711\pi$
$24$		0			0
$25$		0			0
$26$	19.8298	+	22.1834i	0.762685	+	0.853206i
$27$		0			0
$28$	26.1090	+	2.93440i	0.932464	+	0.104800i
$29$	35.2510			1.21555			0.607775	−	0.794109i	$-0.292062\pi$
$29$	35.2510			1.21555			0.607775	+	0.794109i	$0.292062\pi$
$30$		0			0
$31$			23.8330i			0.768808i	0.923165	+	0.384404i	$0.125593\pi$
$31$			23.8330i			0.768808i	−0.923165	+	0.384404i	$0.874407\pi$
$32$	−15.4985	−	27.9963i	−0.484329	−	0.874886i
$33$		0			0
$34$	35.7550	+	39.9987i	1.05162	+	1.17643i
$35$		0			0
$36$		0			0
$37$	−54.7495			−1.47972			−0.739858	−	0.672763i	$-0.765107\pi$
$37$	−54.7495			−1.47972			−0.739858	+	0.672763i	$0.765107\pi$
$38$	16.2053	−	14.4860i	0.426455	−	0.381211i
$39$		0			0
$40$		0			0
$41$	23.8298			0.581215			0.290608	−	0.956842i	$-0.406143\pi$
$41$	23.8298			0.581215			0.290608	+	0.956842i	$0.406143\pi$
$42$		0			0
$43$			56.2515i			1.30817i	0.756420	+	0.654087i	$0.226947\pi$
$43$			56.2515i			1.30817i	−0.756420	+	0.654087i	$0.773053\pi$
$44$	−9.01112	−	1.01276i	−0.204798	−	0.0230173i
$45$		0			0
$46$	−54.3670	+	48.5989i	−1.18189	+	1.05650i
$47$			51.4177i			1.09399i	0.837135	+	0.546997i	$0.184229\pi$
$47$			51.4177i			1.09399i	−0.837135	+	0.546997i	$0.815771\pi$
$48$		0			0
$49$	5.85689			0.119528
$50$		0			0
$51$		0			0
$52$	−6.64636	+	59.1364i	−0.127815	+	1.13724i
$53$	30.6465			0.578236			0.289118	−	0.957293i	$-0.406638\pi$
$53$	30.6465			0.578236			0.289118	+	0.957293i	$0.406638\pi$
$54$		0			0
$55$		0			0
$56$	30.4252	+	42.8423i	0.543308	+	0.765042i
$57$		0			0
$58$	46.9861	+	52.5627i	0.810105	+	0.906253i
$59$		−	6.92483i		−	0.117370i	−0.998277	−	0.0586850i	$-0.981309\pi$
$59$		−	6.92483i		−	0.117370i	0.998277	−	0.0586850i	$-0.0186908\pi$
$60$		0			0
$61$	107.426			1.76107			0.880537	−	0.473977i	$-0.157182\pi$
$61$	107.426			1.76107			0.880537	+	0.473977i	$0.157182\pi$
$62$	−35.5374	+	31.7671i	−0.573184	+	0.512372i
$63$		0			0
$64$	21.0873	−	60.4262i	0.329489	−	0.944160i
$65$		0			0
$66$		0			0
$67$		−	111.444i		−	1.66334i	−0.555271	−	0.831670i	$-0.687385\pi$
$67$		−	111.444i		−	1.66334i	0.555271	−	0.831670i	$-0.312615\pi$
$68$	−11.9840	+	106.629i	−0.176235	+	1.56807i
$69$		0			0
$70$		0			0
$71$		−	31.3190i		−	0.441113i	−0.975374	−	0.220556i	$-0.929213\pi$
$71$		−	31.3190i		−	0.441113i	0.975374	−	0.220556i	$-0.0707873\pi$
$72$		0			0
$73$	−110.909			−1.51930			−0.759652	−	0.650330i	$-0.774631\pi$
$73$	−110.909			−1.51930			−0.759652	+	0.650330i	$0.774631\pi$
$74$	−72.9757	−	81.6369i	−0.986158	−	1.10320i
$75$		0			0
$76$	43.2002	+	4.85528i	0.568423	+	0.0638853i
$77$	14.8902			0.193379
$78$		0			0
$79$		−	59.0065i		−	0.746917i	−0.927647	−	0.373459i	$-0.878172\pi$
$79$		−	59.0065i		−	0.746917i	0.927647	−	0.373459i	$-0.121828\pi$
$80$		0			0
$81$		0			0
$82$	31.7628	+	35.5326i	0.387351	+	0.433325i
$83$			142.416i			1.71586i	0.513767	+	0.857930i	$0.328249\pi$
$83$			142.416i			1.71586i	−0.513767	+	0.857930i	$0.671751\pi$
$84$		0			0
$85$		0			0
$86$	−83.8765	+	74.9776i	−0.975308	+	0.871833i
$87$		0			0
$88$	−10.5008	−	14.7864i	−0.119327	−	0.168027i
$89$	−7.14798			−0.0803144			−0.0401572	−	0.999193i	$-0.512786\pi$
$89$	−7.14798			−0.0803144			−0.0401572	+	0.999193i	$0.512786\pi$
$90$		0			0
$91$		−	97.7185i		−	1.07383i
$92$	−144.932	−	16.2889i	−1.57534	−	0.177053i
$93$		0			0
$94$	−76.6689	+	68.5347i	−0.815626	+	0.729093i
$95$		0			0
$96$		0			0
$97$	126.308			1.30214			0.651070	−	0.759017i	$-0.274320\pi$
$97$	126.308			1.30214			0.651070	+	0.759017i	$0.274320\pi$
$98$	7.80665	+	8.73319i	0.0796597	+	0.0891142i
$99$		0			0
$100$		0			0
$101$	−86.7133			−0.858547			−0.429274	−	0.903174i	$-0.641230\pi$
$101$	−86.7133			−0.858547			−0.429274	+	0.903174i	$0.641230\pi$
$102$		0			0
$103$			21.9281i			0.212895i	0.994318	+	0.106447i	$0.0339475\pi$
$103$			21.9281i			0.212895i	−0.994318	+	0.106447i	$0.966052\pi$
$104$	−97.0372	+	68.9126i	−0.933050	+	0.662622i
$105$		0			0
$106$	40.8488	+	45.6970i	0.385366	+	0.431104i
$107$			7.17725i			0.0670771i	0.999437	+	0.0335385i	$0.0106777\pi$
$107$			7.17725i			0.0670771i	−0.999437	+	0.0335385i	$0.989322\pi$
$108$		0			0
$109$	25.4256			0.233262			0.116631	−	0.993175i	$-0.462790\pi$
$109$	25.4256			0.233262			0.116631	+	0.993175i	$0.462790\pi$
$110$		0			0
$111$		0			0
$112$	−23.3283	+	102.472i	−0.208288	+	0.914925i
$113$	−78.3588			−0.693441			−0.346720	−	0.937968i	$-0.612705\pi$
$113$	−78.3588			−0.693441			−0.346720	+	0.937968i	$0.612705\pi$
$114$		0			0
$115$		0			0
$116$	−15.7483	+	140.122i	−0.135761	+	1.20795i
$117$		0			0
$118$	10.3256	−	9.23012i	0.0875052	−	0.0782214i
$119$		−	176.196i		−	1.48063i
$120$		0			0
$121$	115.861			0.957528
$122$	143.188	+	160.182i	1.17367	+	1.31297i
$123$		0			0
$124$	−94.7357	−	10.6474i	−0.763997	−	0.0858659i
$125$		0			0
$126$		0			0
$127$			71.6077i			0.563840i	0.959438	+	0.281920i	$0.0909713\pi$
$127$			71.6077i			0.563840i	−0.959438	+	0.281920i	$0.909029\pi$
$128$	118.209	−	49.0990i	0.923505	−	0.383586i
$129$		0			0
$130$		0			0
$131$			103.978i			0.793728i	0.917877	+	0.396864i	$0.129902\pi$
$131$			103.978i			0.793728i	−0.917877	+	0.396864i	$0.870098\pi$
$132$		0			0
$133$	−71.3850			−0.536729
$134$	166.174	−	148.543i	1.24010	−	1.10853i
$135$		0			0
$136$	−174.967	+	124.256i	−1.28652	+	0.913647i
$137$	7.16645			0.0523099			0.0261549	−	0.999658i	$-0.491674\pi$
$137$	7.16645			0.0523099			0.0261549	+	0.999658i	$0.491674\pi$
$138$		0			0
$139$		−	146.909i		−	1.05690i	−0.848965	−	0.528449i	$-0.822774\pi$
$139$		−	146.909i		−	1.05690i	0.848965	−	0.528449i	$-0.177226\pi$
$140$		0			0
$141$		0			0
$142$	46.6997	−	41.7451i	0.328871	−	0.293980i
$143$			33.7261i			0.235847i
$144$		0			0
$145$		0			0
$146$	−147.831	−	165.376i	−1.01254	−	1.13272i
$147$		0			0
$148$	24.4593	−	217.628i	0.165265	−	1.47046i
$149$	−79.6054			−0.534265			−0.267132	−	0.963660i	$-0.586076\pi$
$149$	−79.6054			−0.534265			−0.267132	+	0.963660i	$0.586076\pi$
$150$		0			0
$151$		−	182.722i		−	1.21008i	−0.796196	−	0.605039i	$-0.793158\pi$
$151$		−	182.722i		−	1.21008i	0.796196	−	0.605039i	$-0.206842\pi$
$152$	50.3418	+	70.8873i	0.331196	+	0.466364i
$153$		0			0
$154$	19.8472	+	22.2027i	0.128878	+	0.144174i
$155$		0			0
$156$		0			0
$157$	212.182			1.35148			0.675739	−	0.737141i	$-0.263825\pi$
$157$	212.182			1.35148			0.675739	+	0.737141i	$0.263825\pi$
$158$	87.9844	−	78.6498i	0.556864	−	0.497783i
$159$		0			0
$160$		0			0
$161$	239.488			1.48751
$162$		0			0
$163$			243.400i			1.49325i	0.665244	+	0.746626i	$0.268327\pi$
$163$			243.400i			1.49325i	−0.665244	+	0.746626i	$0.731673\pi$
$164$	−10.6459	+	94.7229i	−0.0649143	+	0.577579i
$165$		0			0
$166$	−212.357	+	189.827i	−1.27926	+	1.14354i
$167$		−	211.395i		−	1.26584i	−0.774218	−	0.632919i	$-0.781857\pi$
$167$		−	211.395i		−	1.26584i	0.774218	−	0.632919i	$-0.218143\pi$
$168$		0			0
$169$	52.3307			0.309649
$170$		0			0
$171$		0			0
$172$	−223.598	−	25.1303i	−1.29999	−	0.146106i
$173$	22.3138			0.128982			0.0644909	−	0.997918i	$-0.479458\pi$
$173$	22.3138			0.128982			0.0644909	+	0.997918i	$0.479458\pi$
$174$		0			0
$175$		0			0
$176$	8.05141	−	35.3665i	0.0457467	−	0.200946i
$177$		0			0
$178$	−9.52755	−	10.6583i	−0.0535255	−	0.0598783i
$179$			94.5219i			0.528055i	0.964515	+	0.264028i	$0.0850510\pi$
$179$			94.5219i			0.528055i	−0.964515	+	0.264028i	$0.914949\pi$
$180$		0			0
$181$	−80.6179			−0.445403			−0.222702	−	0.974887i	$-0.571488\pi$
$181$	−80.6179			−0.445403			−0.222702	+	0.974887i	$0.571488\pi$
$182$	145.708	−	130.249i	0.800592	−	0.715654i
$183$		0			0
$184$	−168.891	−	237.819i	−0.917887	−	1.29249i
$185$		0			0
$186$		0			0
$187$			60.8112i			0.325194i
$188$	−204.384	−	22.9708i	−1.08715	−	0.122185i
$189$		0			0
$190$		0			0
$191$		−	330.540i		−	1.73058i	−0.501275	−	0.865288i	$-0.667135\pi$
$191$		−	330.540i		−	1.73058i	0.501275	−	0.865288i	$-0.332865\pi$
$192$		0			0
$193$	103.609			0.536836			0.268418	−	0.963303i	$-0.413499\pi$
$193$	103.609			0.536836			0.268418	+	0.963303i	$0.413499\pi$
$194$	168.356	+	188.337i	0.867812	+	0.970810i
$195$		0			0
$196$	−2.61656	+	23.2810i	−0.0133498	+	0.118780i
$197$	160.633			0.815394			0.407697	−	0.913117i	$-0.366332\pi$
$197$	160.633			0.815394			0.407697	+	0.913117i	$0.366332\pi$
$198$		0			0
$199$			27.5518i			0.138451i	0.997601	+	0.0692255i	$0.0220528\pi$
$199$			27.5518i			0.138451i	−0.997601	+	0.0692255i	$0.977947\pi$
$200$		0			0
$201$		0			0
$202$	−115.580	−	129.298i	−0.572179	−	0.640089i
$203$		−	231.540i		−	1.14059i
$204$		0			0
$205$		0			0
$206$	−32.6970	+	29.2280i	−0.158723	+	0.141884i
$207$		0			0
$208$	−232.097	−	52.8382i	−1.11585	−	0.254030i
$209$	24.6374			0.117883
$210$		0			0
$211$		−	269.808i		−	1.27871i	−0.768911	−	0.639355i	$-0.779201\pi$
$211$		−	269.808i		−	1.27871i	0.768911	−	0.639355i	$-0.220799\pi$
$212$	−13.6913	+	121.819i	−0.0645816	+	0.574618i
$213$		0			0
$214$	−10.7020	+	9.56656i	−0.0500093	+	0.0447036i
$215$		0			0
$216$		0			0
$217$	156.544			0.721399
$218$	33.8898	+	37.9120i	0.155458	+	0.173908i
$219$		0			0
$220$		0			0
$221$	399.080			1.80579
$222$		0			0
$223$		−	41.3345i		−	0.185356i	−0.995696	−	0.0926782i	$-0.970457\pi$
$223$		−	41.3345i		−	0.185356i	0.995696	−	0.0926782i	$-0.0295428\pi$
$224$	−183.890	+	101.800i	−0.820935	+	0.454463i
$225$		0			0
$226$	−104.445	−	116.841i	−0.462144	−	0.516994i
$227$		−	149.837i		−	0.660076i	−0.943968	−	0.330038i	$-0.892939\pi$
$227$		−	149.837i		−	0.660076i	0.943968	−	0.330038i	$-0.107061\pi$
$228$		0			0
$229$	−61.6770			−0.269332			−0.134666	−	0.990891i	$-0.542996\pi$
$229$	−61.6770			−0.269332			−0.134666	+	0.990891i	$0.542996\pi$
$230$		0			0
$231$		0			0
$232$	−229.926	+	163.286i	−0.991061	+	0.703819i
$233$	−405.585			−1.74071			−0.870355	−	0.492425i	$-0.836110\pi$
$233$	−405.585			−1.74071			−0.870355	+	0.492425i	$0.836110\pi$
$234$		0			0
$235$		0			0
$236$	27.5260	+	3.09366i	0.116636	+	0.0131087i
$237$		0			0
$238$	262.725	−	234.851i	1.10389	−	0.986770i
$239$		−	267.769i		−	1.12037i	−0.828367	−	0.560185i	$-0.810730\pi$
$239$		−	267.769i		−	1.12037i	0.828367	−	0.560185i	$-0.189270\pi$
$240$		0			0
$241$	−89.5377			−0.371526			−0.185763	−	0.982595i	$-0.559476\pi$
$241$	−89.5377			−0.371526			−0.185763	+	0.982595i	$0.559476\pi$
$242$	154.431	+	172.760i	0.638145	+	0.713884i
$243$		0			0
$244$	−47.9922	+	427.014i	−0.196689	+	1.75006i
$245$		0			0
$246$		0			0
$247$		−	161.686i		−	0.654598i
$248$	−110.397	−	155.452i	−0.445149	−	0.626823i
$249$		0			0
$250$		0			0
$251$			227.844i			0.907745i	0.891067	+	0.453873i	$0.149958\pi$
$251$			227.844i			0.907745i	−0.891067	+	0.453873i	$0.850042\pi$
$252$		0			0
$253$	−82.6559			−0.326703
$254$	−106.774	+	95.4460i	−0.420371	+	0.375772i
$255$		0			0
$256$	230.772	+	110.817i	0.901453	+	0.432878i
$257$	−442.129			−1.72035			−0.860173	−	0.510003i	$-0.829644\pi$
$257$	−442.129			−1.72035			−0.860173	+	0.510003i	$0.829644\pi$
$258$		0			0
$259$			359.614i			1.38847i
$260$		0			0
$261$		0			0
$262$	−155.042	+	138.593i	−0.591763	+	0.528981i
$263$		−	34.1556i		−	0.129869i	−0.997890	−	0.0649346i	$-0.979316\pi$
$263$		−	34.1556i		−	0.129869i	0.997890	−	0.0649346i	$-0.0206839\pi$
$264$		0			0
$265$		0			0
$266$	−95.1491	−	106.442i	−0.357703	−	0.400158i
$267$		0			0
$268$	442.986	+	49.7873i	1.65293	+	0.185774i
$269$	9.96085			0.0370292			0.0185146	−	0.999829i	$-0.494106\pi$
$269$	9.96085			0.0370292			0.0185146	+	0.999829i	$0.494106\pi$
$270$		0			0
$271$			56.5791i			0.208779i	0.994536	+	0.104390i	$0.0332889\pi$
$271$			56.5791i			0.208779i	−0.994536	+	0.104390i	$0.966711\pi$
$272$	−418.492	−	95.2723i	−1.53857	−	0.350266i
$273$		0			0
$274$	9.55218	+	10.6859i	0.0348620	+	0.0389996i
$275$		0			0
$276$		0			0
$277$	−103.794			−0.374708			−0.187354	−	0.982292i	$-0.559991\pi$
$277$	−103.794			−0.374708			−0.187354	+	0.982292i	$0.559991\pi$
$278$	219.056	−	195.815i	0.787969	−	0.704370i
$279$		0			0
$280$		0			0
$281$	−393.069			−1.39882			−0.699411	−	0.714720i	$-0.746554\pi$
$281$	−393.069			−1.39882			−0.699411	+	0.714720i	$0.746554\pi$
$282$		0			0
$283$		−	114.027i		−	0.402923i	−0.979496	−	0.201462i	$-0.935431\pi$
$283$		−	114.027i		−	0.402923i	0.979496	−	0.201462i	$-0.0645692\pi$
$284$	124.492	+	13.9917i	0.438353	+	0.0492666i
$285$		0			0
$286$	−50.2889	+	44.9535i	−0.175835	+	0.157180i
$287$		−	156.522i		−	0.545374i
$288$		0			0
$289$	430.579			1.48989
$290$		0			0
$291$		0			0
$292$	49.5485	−	440.861i	0.169687	−	1.50980i
$293$	126.796			0.432750			0.216375	−	0.976310i	$-0.430577\pi$
$293$	126.796			0.432750			0.216375	+	0.976310i	$0.430577\pi$
$294$		0			0
$295$		0			0
$296$	357.106	−	253.605i	1.20644	−	0.856775i
$297$		0			0
$298$	−106.106	−	118.700i	−0.356061	−	0.398321i
$299$			542.438i			1.81417i
$300$		0			0
$301$	369.479			1.22750
$302$	272.456	−	243.550i	0.902172	−	0.806457i
$303$		0			0
$304$	−38.5992	+	169.550i	−0.126971	+	0.557732i
$305$		0			0
$306$		0			0
$307$		−	408.420i		−	1.33036i	−0.746683	−	0.665180i	$-0.768355\pi$
$307$		−	408.420i		−	1.33036i	0.746683	−	0.665180i	$-0.231645\pi$
$308$	−6.65217	+	59.1881i	−0.0215980	+	0.192169i
$309$		0			0
$310$		0			0
$311$		−	472.495i		−	1.51928i	−0.650345	−	0.759639i	$-0.725376\pi$
$311$		−	472.495i		−	1.51928i	0.650345	−	0.759639i	$-0.274624\pi$
$312$		0			0
$313$	−54.6519			−0.174607			−0.0873033	−	0.996182i	$-0.527825\pi$
$313$	−54.6519			−0.174607			−0.0873033	+	0.996182i	$0.527825\pi$
$314$	282.818	+	316.384i	0.900693	+	1.00759i
$315$		0			0
$316$	234.549	+	26.3611i	0.742244	+	0.0834211i
$317$	−63.3734			−0.199916			−0.0999581	−	0.994992i	$-0.531871\pi$
$317$	−63.3734			−0.199916			−0.0999581	+	0.994992i	$0.531871\pi$
$318$		0			0
$319$			79.9127i			0.250510i
$320$		0			0
$321$		0			0
$322$	319.215	+	357.101i	0.991349	+	1.10901i
$323$		−	291.535i		−	0.902584i
$324$		0			0
$325$		0			0
$326$	−362.933	+	324.428i	−1.11329	+	0.995179i
$327$		0			0
$328$	−155.431	+	110.382i	−0.473876	+	0.336531i
$329$	337.729			1.02653
$330$		0			0
$331$		−	431.595i		−	1.30391i	−0.758257	−	0.651955i	$-0.773949\pi$
$331$		−	431.595i		−	1.30391i	0.758257	−	0.651955i	$-0.226051\pi$
$332$	−566.101	−	63.6243i	−1.70512	−	0.191639i
$333$		0			0
$334$	315.211	−	281.768i	0.943744	−	0.843618i
$335$		0			0
$336$		0			0
$337$	−486.091			−1.44241			−0.721203	−	0.692723i	$-0.756411\pi$
$337$	−486.091			−1.44241			−0.721203	+	0.692723i	$0.756411\pi$
$338$	69.7517	+	78.0303i	0.206366	+	0.230859i
$339$		0			0
$340$		0			0
$341$	−54.0286			−0.158442
$342$		0			0
$343$		−	360.319i		−	1.05049i
$344$	−260.562	−	366.903i	−0.757449	−	1.06658i
$345$		0			0
$346$	29.7422	+	33.2721i	0.0859600	+	0.0961623i
$347$			294.297i			0.848119i	0.905634	+	0.424060i	$0.139395\pi$
$347$			294.297i			0.848119i	−0.905634	+	0.424060i	$0.860605\pi$
$348$		0			0
$349$	83.0428			0.237945			0.118972	−	0.992898i	$-0.462040\pi$
$349$	83.0428			0.237945			0.118972	+	0.992898i	$0.462040\pi$
$350$		0			0
$351$		0			0
$352$	63.4667	−	35.1346i	0.180303	−	0.0998143i
$353$	570.733			1.61681			0.808404	−	0.588628i	$-0.200332\pi$
$353$	570.733			1.61681			0.808404	+	0.588628i	$0.200332\pi$
$354$		0			0
$355$		0			0
$356$	3.19335	−	28.4130i	0.00897008	−	0.0798119i
$357$		0			0
$358$	−140.941	+	125.988i	−0.393691	+	0.351923i
$359$			558.265i			1.55506i	0.628848	+	0.777528i	$0.283527\pi$
$359$			558.265i			1.55506i	−0.628848	+	0.777528i	$0.716473\pi$
$360$		0			0
$361$	242.886			0.672814
$362$	−107.456	−	120.209i	−0.296839	−	0.332070i
$363$		0			0
$364$	388.428	+	43.6556i	1.06711	+	0.119933i
$365$		0			0
$366$		0			0
$367$			446.467i			1.21653i	0.793734	+	0.608265i	$0.208134\pi$
$367$			446.467i			1.21653i	−0.793734	+	0.608265i	$0.791866\pi$
$368$	129.496	−	568.822i	0.351891	−	1.54571i
$369$		0			0
$370$		0			0
$371$		−	201.297i		−	0.542579i
$372$		0			0
$373$	−112.924			−0.302744			−0.151372	−	0.988477i	$-0.548369\pi$
$373$	−112.924			−0.302744			−0.151372	+	0.988477i	$0.548369\pi$
$374$	−90.6755	+	81.0554i	−0.242448	+	0.216726i
$375$		0			0
$376$	−238.172	−	335.375i	−0.633436	−	0.891954i
$377$	524.435			1.39108
$378$		0			0
$379$		−	321.457i		−	0.848173i	−0.905622	−	0.424086i	$-0.860595\pi$
$379$		−	321.457i		−	0.848173i	0.905622	−	0.424086i	$-0.139405\pi$
$380$		0			0
$381$		0			0
$382$	492.868	−	440.577i	1.29023	−	1.15334i
$383$		−	89.2269i		−	0.232968i	−0.993193	−	0.116484i	$-0.962838\pi$
$383$		−	89.2269i		−	0.232968i	0.993193	−	0.116484i	$-0.0371624\pi$
$384$		0			0
$385$		0			0
$386$	138.101	+	154.492i	0.357775	+	0.400238i
$387$		0			0
$388$	−56.4278	+	502.069i	−0.145432	+	1.29399i
$389$	−260.714			−0.670217			−0.335108	−	0.942180i	$-0.608773\pi$
$389$	−260.714			−0.670217			−0.335108	+	0.942180i	$0.608773\pi$
$390$		0			0
$391$			978.066i			2.50145i
$392$	−38.2018	+	27.1297i	−0.0974536	+	0.0692084i
$393$		0			0
$394$	214.107	+	239.519i	0.543420	+	0.607916i
$395$		0			0
$396$		0			0
$397$	112.607			0.283644			0.141822	−	0.989892i	$-0.454704\pi$
$397$	112.607			0.283644			0.141822	+	0.989892i	$0.454704\pi$
$398$	−41.0824	+	36.7238i	−0.103222	+	0.0922708i
$399$		0			0
$400$		0			0
$401$	−577.513			−1.44018			−0.720091	−	0.693880i	$-0.755900\pi$
$401$	−577.513			−1.44018			−0.720091	+	0.693880i	$0.755900\pi$
$402$		0			0
$403$			354.569i			0.879823i
$404$	38.7390	−	344.683i	0.0958887	−	0.853176i
$405$		0			0
$406$	345.250	−	308.621i	0.850368	−	0.760149i
$407$		−	124.115i		−	0.304951i
$408$		0			0
$409$	−276.255			−0.675441			−0.337721	−	0.941246i	$-0.609656\pi$
$409$	−276.255			−0.675441			−0.337721	+	0.941246i	$0.609656\pi$
$410$		0			0
$411$		0			0
$412$	−87.1638	−	9.79636i	−0.211563	−	0.0237776i
$413$	−45.4847			−0.110132
$414$		0			0
$415$		0			0
$416$	−230.575	−	416.507i	−0.554266	−	1.00122i
$417$		0			0
$418$	32.8393	+	36.7369i	0.0785629	+	0.0878872i
$419$		−	247.520i		−	0.590739i	−0.955383	−	0.295370i	$-0.904557\pi$
$419$		−	247.520i		−	0.590739i	0.955383	−	0.295370i	$-0.0954428\pi$
$420$		0			0
$421$	−77.7303			−0.184632			−0.0923162	−	0.995730i	$-0.529427\pi$
$421$	−77.7303			−0.184632			−0.0923162	+	0.995730i	$0.529427\pi$
$422$	402.310	−	359.627i	0.953342	−	0.852197i
$423$		0			0
$424$	−199.893	+	141.958i	−0.471447	+	0.334806i
$425$		0			0
$426$		0			0
$427$		−	705.608i		−	1.65248i
$428$	−28.5294	−	3.20643i	−0.0666574	−	0.00749165i
$429$		0			0
$430$		0			0
$431$			317.184i			0.735926i	0.929840	+	0.367963i	$0.119945\pi$
$431$			317.184i			0.735926i	−0.929840	+	0.367963i	$0.880055\pi$
$432$		0			0
$433$	−82.9688			−0.191614			−0.0958069	−	0.995400i	$-0.530543\pi$
$433$	−82.9688			−0.191614			−0.0958069	+	0.995400i	$0.530543\pi$
$434$	208.657	+	233.422i	0.480777	+	0.537838i
$435$		0			0
$436$	−11.3588	+	101.066i	−0.0260524	+	0.231803i
$437$	396.260			0.906773
$438$		0			0
$439$			117.621i			0.267930i	0.990986	+	0.133965i	$0.0427709\pi$
$439$			117.621i			0.267930i	−0.990986	+	0.133965i	$0.957229\pi$
$440$		0			0
$441$		0			0
$442$	531.934	+	595.068i	1.20347	+	1.34631i
$443$		−	35.1780i		−	0.0794086i	−0.999211	−	0.0397043i	$-0.987358\pi$
$443$		−	35.1780i		−	0.0794086i	0.999211	−	0.0397043i	$-0.0126416\pi$
$444$		0			0
$445$		0			0
$446$	61.6337	−	55.0948i	0.138192	−	0.123531i
$447$		0			0
$448$	−396.900	−	138.508i	−0.885938	−	0.309171i
$449$	67.4253			0.150168			0.0750838	−	0.997177i	$-0.476078\pi$
$449$	67.4253			0.150168			0.0750838	+	0.997177i	$0.476078\pi$
$450$		0			0
$451$			54.0214i			0.119781i
$452$	35.0067	−	311.474i	0.0774484	−	0.689102i
$453$		0			0
$454$	223.422	−	199.718i	0.492119	−	0.439908i
$455$		0			0
$456$		0			0
$457$	−204.153			−0.446724			−0.223362	−	0.974736i	$-0.571703\pi$
$457$	−204.153			−0.446724			−0.223362	+	0.974736i	$0.571703\pi$
$458$	−82.2094	−	91.9665i	−0.179496	−	0.200800i
$459$		0			0
$460$		0			0
$461$	−125.762			−0.272802			−0.136401	−	0.990654i	$-0.543554\pi$
$461$	−125.762			−0.272802			−0.136401	+	0.990654i	$0.543554\pi$
$462$		0			0
$463$		−	553.629i		−	1.19574i	−0.801592	−	0.597871i	$-0.796013\pi$
$463$		−	553.629i		−	1.19574i	0.801592	−	0.597871i	$-0.203987\pi$
$464$	−549.944	−	125.198i	−1.18523	−	0.269824i
$465$		0			0
$466$	−540.605	−	604.768i	−1.16010	−	1.29778i
$467$			625.772i			1.33998i	0.742369	+	0.669991i	$0.233702\pi$
$467$			625.772i			1.33998i	−0.742369	+	0.669991i	$0.766298\pi$
$468$		0			0
$469$	−732.000			−1.56077
$470$		0			0
$471$		0			0
$472$	32.0765	+	45.1676i	0.0679588	+	0.0956940i
$473$	−127.520			−0.269598
$474$		0			0
$475$		0			0
$476$	700.373	+	78.7151i	1.47137	+	0.165368i
$477$		0			0
$478$	399.269	−	356.909i	0.835291	−	0.746672i
$479$		−	488.207i		−	1.01922i	−0.860405	−	0.509610i	$-0.829790\pi$
$479$		−	488.207i		−	1.01922i	0.860405	−	0.509610i	$-0.170210\pi$
$480$		0			0
$481$	−814.519			−1.69339
$482$	−119.345	−	133.509i	−0.247604	−	0.276991i
$483$		0			0
$484$	−51.7607	+	460.544i	−0.106944	+	0.951537i
$485$		0			0
$486$		0			0
$487$			609.476i			1.25149i	0.780027	+	0.625746i	$0.215205\pi$
$487$			609.476i			1.25149i	−0.780027	+	0.625746i	$0.784795\pi$
$488$	−700.689	+	497.606i	−1.43584	+	1.01968i
$489$		0			0
$490$		0			0
$491$			689.074i			1.40341i	0.712468	+	0.701705i	$0.247578\pi$
$491$			689.074i			1.40341i	−0.712468	+	0.701705i	$0.752422\pi$
$492$		0			0
$493$	945.606			1.91806
$494$	241.089	−	215.511i	0.488035	−	0.436257i
$495$		0			0
$496$	84.6461	−	371.815i	0.170657	−	0.749627i
$497$	−205.714			−0.413911
$498$		0			0
$499$		−	700.401i		−	1.40361i	−0.712370	−	0.701804i	$-0.752378\pi$
$499$		−	700.401i		−	1.40361i	0.712370	−	0.701804i	$-0.247622\pi$
$500$		0			0
$501$		0			0
$502$	−339.738	+	303.694i	−0.676769	+	0.604967i
$503$			943.945i			1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$503$			943.945i			1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$504$		0			0
$505$		0			0
$506$	−110.172	−	123.248i	−0.217731	−	0.243573i
$507$		0			0
$508$	−284.639	−	31.9906i	−0.560313	−	0.0629737i
$509$	357.147			0.701665			0.350832	−	0.936438i	$-0.385899\pi$
$509$	357.147			0.701665			0.350832	+	0.936438i	$0.385899\pi$
$510$		0			0
$511$			728.489i			1.42562i
$512$	142.358	+	491.811i	0.278042	+	0.960569i
$513$		0			0
$514$	−589.314	−	659.258i	−1.14653	−	1.28260i
$515$		0			0
$516$		0			0
$517$	−116.562			−0.225459
$518$	−536.219	+	479.329i	−1.03517	+	0.925346i
$519$		0			0
$520$		0			0
$521$	88.8415			0.170521			0.0852605	−	0.996359i	$-0.472828\pi$
$521$	88.8415			0.170521			0.0852605	+	0.996359i	$0.472828\pi$
$522$		0			0
$523$		−	220.427i		−	0.421467i	−0.977544	−	0.210734i	$-0.932415\pi$
$523$		−	220.427i		−	0.421467i	0.977544	−	0.210734i	$-0.0675852\pi$
$524$	−413.311	−	46.4522i	−0.788762	−	0.0886492i
$525$		0			0
$526$	50.9294	−	45.5261i	0.0968239	−	0.0865514i
$527$			639.320i			1.21313i
$528$		0			0
$529$	−800.407			−1.51306
$530$		0			0
$531$		0			0
$532$	31.8911	−	283.753i	0.0599457	−	0.533371i
$533$	354.521			0.665142
$534$		0			0
$535$		0			0
$536$	516.219	+	726.897i	0.963094	+	1.35615i
$537$		0			0
$538$	13.2768	+	14.8526i	0.0246781	+	0.0276071i
$539$			13.2774i			0.0246333i
$540$		0			0
$541$	−411.560			−0.760740			−0.380370	−	0.924834i	$-0.624203\pi$
$541$	−411.560			−0.760740			−0.380370	+	0.924834i	$0.624203\pi$
$542$	−84.3651	+	75.4144i	−0.155655	+	0.139141i
$543$		0			0
$544$	−415.748	−	751.001i	−0.764242	−	1.38052i
$545$		0			0
$546$		0			0
$547$			851.537i			1.55674i	0.627806	+	0.778370i	$0.283953\pi$
$547$			851.537i			1.55674i	−0.627806	+	0.778370i	$0.716047\pi$
$548$	−3.20160	+	28.4865i	−0.00584234	+	0.0519826i
$549$		0			0
$550$		0			0
$551$		−	383.109i		−	0.695297i
$552$		0			0
$553$	−387.575			−0.700858
$554$	−138.347	−	154.767i	−0.249724	−	0.279363i
$555$		0			0
$556$	583.959	+	65.6313i	1.05029	+	0.118042i
$557$	−211.553			−0.379808			−0.189904	−	0.981803i	$-0.560818\pi$
$557$	−211.553			−0.379808			−0.189904	+	0.981803i	$0.560818\pi$
$558$		0			0
$559$			836.864i			1.49707i
$560$		0			0
$561$		0			0
$562$	−523.922	−	586.104i	−0.932246	−	1.04289i
$563$		−	404.044i		−	0.717663i	−0.933402	−	0.358832i	$-0.883175\pi$
$563$		−	404.044i		−	0.717663i	0.933402	−	0.358832i	$-0.116825\pi$
$564$		0			0
$565$		0			0
$566$	170.026	−	151.987i	0.300399	−	0.268529i
$567$		0			0
$568$	145.073	+	204.280i	0.255410	+	0.359647i
$569$	−230.465			−0.405036			−0.202518	−	0.979279i	$-0.564912\pi$
$569$	−230.465			−0.405036			−0.202518	+	0.979279i	$0.564912\pi$
$570$		0			0
$571$			351.234i			0.615121i	0.951529	+	0.307560i	$0.0995126\pi$
$571$			351.234i			0.615121i	−0.951529	+	0.307560i	$0.900487\pi$
$572$	−134.060	−	15.0671i	−0.234371	−	0.0263410i
$573$		0			0
$574$	233.390	−	208.629i	0.406603	−	0.363465i
$575$		0			0
$576$		0			0
$577$	−638.575			−1.10672			−0.553358	−	0.832944i	$-0.686654\pi$
$577$	−638.575			−1.10672			−0.553358	+	0.832944i	$0.686654\pi$
$578$	573.919	+	642.035i	0.992939	+	1.11079i
$579$		0			0
$580$		0			0
$581$	935.439			1.61005
$582$		0			0
$583$			69.4746i			0.119167i
$584$	723.410	−	513.742i	1.23872	−	0.879696i
$585$		0			0
$586$	169.006	+	189.065i	0.288407	+	0.322637i
$587$			105.047i			0.178956i	0.995989	+	0.0894779i	$0.0285198\pi$
$587$			105.047i			0.178956i	−0.995989	+	0.0894779i	$0.971480\pi$
$588$		0			0
$589$	259.018			0.439759
$590$		0			0
$591$		0			0
$592$	854.138	+	194.450i	1.44280	+	0.328463i
$593$	−990.175			−1.66977			−0.834886	−	0.550422i	$-0.814467\pi$
$593$	−990.175			−1.66977			−0.834886	+	0.550422i	$0.814467\pi$
$594$		0			0
$595$		0			0
$596$	35.5636	−	316.429i	0.0596705	−	0.530922i
$597$		0			0
$598$	−808.828	+	723.016i	−1.35255	+	1.20906i
$599$		−	78.0745i		−	0.130341i	−0.997874	−	0.0651707i	$-0.979241\pi$
$599$		−	78.0745i		−	0.130341i	0.997874	−	0.0651707i	$-0.0207592\pi$
$600$		0			0
$601$	−616.498			−1.02579			−0.512893	−	0.858452i	$-0.671426\pi$
$601$	−616.498			−1.02579			−0.512893	+	0.858452i	$0.671426\pi$
$602$	492.479	+	550.929i	0.818071	+	0.915165i
$603$		0			0
$604$	726.314	+	81.6306i	1.20251	+	0.135150i
$605$		0			0
$606$		0			0
$607$		−	226.520i		−	0.373179i	−0.982438	−	0.186590i	$-0.940257\pi$
$607$		−	226.520i		−	0.373179i	0.982438	−	0.186590i	$-0.0597435\pi$
$608$	−304.265	+	168.439i	−0.500436	+	0.277037i
$609$		0			0
$610$		0			0
$611$			764.951i			1.25197i
$612$		0			0
$613$	732.519			1.19497			0.597487	−	0.801879i	$-0.296166\pi$
$613$	732.519			1.19497			0.597487	+	0.801879i	$0.296166\pi$
$614$	608.995	−	544.384i	0.991848	−	0.886619i
$615$		0			0
$616$	−97.1220	+	68.9729i	−0.157666	+	0.111969i
$617$	−350.585			−0.568209			−0.284105	−	0.958793i	$-0.591696\pi$
$617$	−350.585			−0.568209			−0.284105	+	0.958793i	$0.591696\pi$
$618$		0			0
$619$			237.923i			0.384367i	0.981359	+	0.192184i	$0.0615569\pi$
$619$			237.923i			0.384367i	−0.981359	+	0.192184i	$0.938443\pi$
$620$		0			0
$621$		0			0
$622$	704.537	−	629.790i	1.13270	−	1.01252i
$623$			46.9504i			0.0753617i
$624$		0			0
$625$		0			0
$626$	−72.8455	−	81.4913i	−0.116367	−	0.130178i
$627$		0			0
$628$	−94.7920	+	843.417i	−0.150943	+	1.34302i
$629$	−1468.65			−2.33490
$630$		0			0
$631$		−	200.923i		−	0.318419i	−0.987245	−	0.159210i	$-0.949105\pi$
$631$		−	200.923i		−	0.318419i	0.987245	−	0.159210i	$-0.0508946\pi$
$632$	273.324	+	384.873i	0.432475	+	0.608976i
$633$		0			0
$634$	−84.4705	−	94.4960i	−0.133234	−	0.149047i
$635$		0			0
$636$		0			0
$637$	87.1340			0.136788
$638$	−119.158	+	106.516i	−0.186768	+	0.166953i
$639$		0			0
$640$		0			0
$641$	−216.861			−0.338316			−0.169158	−	0.985589i	$-0.554105\pi$
$641$	−216.861			−0.338316			−0.169158	+	0.985589i	$0.554105\pi$
$642$		0			0
$643$		−	37.5349i		−	0.0583746i	−0.999574	−	0.0291873i	$-0.990708\pi$
$643$		−	37.5349i		−	0.0583746i	0.999574	−	0.0291873i	$-0.00929193\pi$
$644$	−106.991	+	951.960i	−0.166135	+	1.47820i
$645$		0			0
$646$	434.707	−	388.587i	0.672921	−	0.601528i
$647$		−	1192.56i		−	1.84321i	−0.388125	−	0.921607i	$-0.626877\pi$
$647$		−	1192.56i		−	1.84321i	0.388125	−	0.921607i	$-0.373123\pi$
$648$		0			0
$649$	15.6984			0.0241885
$650$		0			0
$651$		0			0
$652$	−967.509	−	108.739i	−1.48391	−	0.166777i
$653$	1087.78			1.66582			0.832908	−	0.553412i	$-0.186675\pi$
$653$	1087.78			1.66582			0.832908	+	0.553412i	$0.186675\pi$
$654$		0			0
$655$		0			0
$656$	−371.765	−	84.6347i	−0.566715	−	0.129016i
$657$		0			0
$658$	450.160	+	503.587i	0.684133	+	0.765330i
$659$		−	852.957i		−	1.29432i	−0.762354	−	0.647160i	$-0.775956\pi$
$659$		−	852.957i		−	1.29432i	0.762354	−	0.647160i	$-0.224044\pi$
$660$		0			0
$661$	−504.933			−0.763892			−0.381946	−	0.924185i	$-0.624746\pi$
$661$	−504.933			−0.763892			−0.381946	+	0.924185i	$0.624746\pi$
$662$	643.550	−	575.273i	0.972130	−	0.868992i
$663$		0			0
$664$	−659.687	−	928.917i	−0.993504	−	1.39897i
$665$		0			0
$666$		0			0
$667$			1285.29i			1.92697i
$668$	840.289	+	94.4403i	1.25792	+	0.141378i
$669$		0			0
$670$		0			0
$671$			243.530i			0.362936i
$672$		0			0
$673$	−902.689			−1.34129			−0.670646	−	0.741778i	$-0.733983\pi$
$673$	−902.689			−1.34129			−0.670646	+	0.741778i	$0.733983\pi$
$674$	−647.911	−	724.810i	−0.961293	−	1.07539i
$675$		0			0
$676$	−23.3787	+	208.013i	−0.0345838	+	0.307712i
$677$	−930.750			−1.37482			−0.687408	−	0.726272i	$-0.741252\pi$
$677$	−930.750			−1.37482			−0.687408	+	0.726272i	$0.741252\pi$
$678$		0			0
$679$		−	829.632i		−	1.22184i
$680$		0			0
$681$		0			0
$682$	−72.0148	−	80.5620i	−0.105594	−	0.118126i
$683$			64.9023i			0.0950253i	0.998871	+	0.0475127i	$0.0151294\pi$
$683$			64.9023i			0.0950253i	−0.998871	+	0.0475127i	$0.984871\pi$
$684$		0			0
$685$		0			0
$686$	537.271	−	480.269i	0.783193	−	0.700101i
$687$		0			0
$688$	199.784	−	877.570i	0.290384	−	1.27554i
$689$	455.934			0.661733
$690$		0			0
$691$		−	348.329i		−	0.504094i	−0.967715	−	0.252047i	$-0.918896\pi$
$691$		−	348.329i		−	0.504094i	0.967715	−	0.252047i	$-0.0811038\pi$
$692$	−9.96868	+	88.6970i	−0.0144056	+	0.128175i
$693$		0			0
$694$	−438.826	+	392.269i	−0.632315	+	0.565230i
$695$		0			0
$696$		0			0
$697$	639.234			0.917122
$698$	110.688	+	123.825i	0.158579	+	0.177400i
$699$		0			0
$700$		0			0
$701$	−815.159			−1.16285			−0.581426	−	0.813600i	$-0.697505\pi$
$701$	−815.159			−1.16285			−0.581426	+	0.813600i	$0.697505\pi$
$702$		0			0
$703$			595.020i			0.846401i
$704$	136.984	+	47.8041i	0.194580	+	0.0679036i
$705$		0			0
$706$	760.731	+	851.020i	1.07752	+	1.20541i
$707$			569.562i			0.805605i
$708$		0			0
$709$	1300.08			1.83368			0.916839	−	0.399257i	$-0.130732\pi$
$709$	1300.08			1.83368			0.916839	+	0.399257i	$0.130732\pi$
$710$		0			0
$711$		0			0
$712$	46.6230	−	33.1102i	0.0654818	−	0.0465030i
$713$	−868.977			−1.21876
$714$		0			0
$715$		0			0
$716$	−375.722	−	42.2275i	−0.524751	−	0.0589770i
$717$		0			0
$718$	−832.428	+	744.112i	−1.15937	+	1.03637i
$719$			782.612i			1.08847i	0.838932	+	0.544237i	$0.183181\pi$
$719$			782.612i			1.08847i	−0.838932	+	0.544237i	$0.816819\pi$
$720$		0			0
$721$	144.031			0.199766
$722$	323.743	+	362.167i	0.448397	+	0.501616i
$723$		0			0
$724$	36.0160	−	320.454i	0.0497458	−	0.442616i
$725$		0			0
$726$		0			0
$727$			850.638i			1.17007i	0.811009	+	0.585033i	$0.198919\pi$
$727$			850.638i			1.17007i	−0.811009	+	0.585033i	$0.801081\pi$
$728$	452.642	+	637.373i	0.621761	+	0.875513i
$729$		0			0
$730$		0			0
$731$			1508.94i			2.06422i
$732$		0			0
$733$	365.781			0.499019			0.249510	−	0.968372i	$-0.419731\pi$
$733$	365.781			0.499019			0.249510	+	0.968372i	$0.419731\pi$
$734$	−665.726	+	595.096i	−0.906984	+	0.810758i
$735$		0			0
$736$	1020.78	−	565.093i	1.38692	−	0.767789i
$737$	252.639			0.342794
$738$		0			0
$739$			727.328i			0.984205i	0.870537	+	0.492103i	$0.163772\pi$
$739$			727.328i			0.984205i	−0.870537	+	0.492103i	$0.836228\pi$
$740$		0			0
$741$		0			0
$742$	300.153	−	268.309i	0.404519	−	0.361602i
$743$			481.526i			0.648083i	0.946043	+	0.324041i	$0.105042\pi$
$743$			481.526i			0.648083i	−0.946043	+	0.324041i	$0.894958\pi$
$744$		0			0
$745$		0			0
$746$	−150.516	−	168.380i	−0.201764	−	0.225711i
$747$		0			0
$748$	−241.723	−	27.1673i	−0.323159	−	0.0363200i
$749$	47.1426			0.0629408
$750$		0			0
$751$			1316.30i			1.75273i	0.481647	+	0.876365i	$0.340039\pi$
$751$			1316.30i			1.75273i	−0.481647	+	0.876365i	$0.659961\pi$
$752$	182.617	−	802.159i	0.242841	−	1.06670i
$753$		0			0
$754$	699.021	+	781.985i	0.927083	+	1.03711i
$755$		0			0
$756$		0			0
$757$	483.813			0.639118			0.319559	−	0.947566i	$-0.396465\pi$
$757$	483.813			0.639118			0.319559	+	0.947566i	$0.396465\pi$
$758$	479.325	−	428.471i	0.632354	−	0.565265i
$759$		0			0
$760$		0			0
$761$	−1027.03			−1.34958			−0.674789	−	0.738011i	$-0.735765\pi$
$761$	−1027.03			−1.34958			−0.674789	+	0.738011i	$0.735765\pi$
$762$		0			0
$763$		−	167.004i		−	0.218878i
$764$	1313.89	+	147.668i	1.71975	+	0.193283i
$765$		0			0
$766$	133.046	−	118.931i	0.173689	−	0.155262i
$767$		−	103.022i		−	0.134318i
$768$		0			0
$769$	1024.79			1.33263			0.666314	−	0.745671i	$-0.267871\pi$
$769$	1024.79			1.33263			0.666314	+	0.745671i	$0.267871\pi$
$770$		0			0
$771$		0			0
$772$	−46.2873	+	411.844i	−0.0599577	+	0.533477i
$773$	1092.74			1.41364			0.706819	−	0.707395i	$-0.250130\pi$
$773$	1092.74			1.41364			0.706819	+	0.707395i	$0.250130\pi$
$774$		0			0
$775$		0			0
$776$	−823.848	+	585.070i	−1.06166	+	0.753956i
$777$		0			0
$778$	−347.506	−	388.751i	−0.446666	−	0.499679i
$779$		−	258.983i		−	0.332456i
$780$		0			0
$781$	70.9991			0.0909079
$782$	−1458.39	+	1303.66i	−1.86495	+	1.66709i
$783$		0			0
$784$	−91.3723	−	20.8015i	−0.116546	−	0.0265325i
$785$		0			0
$786$		0			0
$787$		−	1385.63i		−	1.76064i	−0.474376	−	0.880322i	$-0.657326\pi$
$787$		−	1385.63i		−	1.76064i	0.474376	−	0.880322i	$-0.342674\pi$
$788$	−71.7624	+	638.511i	−0.0910691	+	0.810293i
$789$		0			0
$790$		0			0
$791$			514.688i			0.650680i
$792$		0			0
$793$	1598.19			2.01537
$794$	150.093	+	167.907i	0.189034	+	0.211470i
$795$		0			0
$796$	−109.518	−	12.3087i	−0.137585	−	0.0154632i
$797$	98.5251			0.123620			0.0618100	−	0.998088i	$-0.480313\pi$
$797$	98.5251			0.123620			0.0618100	+	0.998088i	$0.480313\pi$
$798$		0			0
$799$			1379.28i			1.72626i
$800$		0			0
$801$		0			0
$802$	−769.767	−	861.128i	−0.959810	−	1.07373i
$803$		−	251.427i		−	0.313110i
$804$		0			0
$805$		0			0
$806$	−528.697	+	472.605i	−0.655951	+	0.586358i
$807$		0			0
$808$	565.592	−	401.665i	0.699990	−	0.497110i
$809$	212.100			0.262176			0.131088	−	0.991371i	$-0.458153\pi$
$809$	212.100			0.262176			0.131088	+	0.991371i	$0.458153\pi$
$810$		0			0
$811$			485.409i			0.598531i	0.954170	+	0.299266i	$0.0967417\pi$
$811$			485.409i			0.598531i	−0.954170	+	0.299266i	$0.903258\pi$
$812$	920.367	+	103.440i	1.13346	+	0.127390i
$813$		0			0
$814$	185.068	−	165.433i	0.227356	−	0.203235i
$815$		0			0
$816$		0			0
$817$	611.343			0.748278
$818$	−368.221	−	411.924i	−0.450148	−	0.503575i
$819$		0			0
$820$		0			0
$821$	862.231			1.05022			0.525110	−	0.851034i	$-0.324024\pi$
$821$	862.231			1.05022			0.525110	+	0.851034i	$0.324024\pi$
$822$		0			0
$823$		−	485.042i		−	0.589358i	−0.955596	−	0.294679i	$-0.904787\pi$
$823$		−	485.042i		−	0.589358i	0.955596	−	0.294679i	$-0.0952127\pi$
$824$	−101.573	−	143.027i	−0.123269	−	0.173577i
$825$		0			0
$826$	−60.6266	−	67.8221i	−0.0733978	−	0.0821091i
$827$		−	590.547i		−	0.714084i	−0.934088	−	0.357042i	$-0.883785\pi$
$827$		−	590.547i		−	0.714084i	0.934088	−	0.357042i	$-0.116215\pi$
$828$		0			0
$829$	−1071.30			−1.29228			−0.646139	−	0.763220i	$-0.723617\pi$
$829$	−1071.30			−1.29228			−0.646139	+	0.763220i	$0.723617\pi$
$830$		0			0
$831$		0			0
$832$	313.719	−	898.972i	0.377067	−	1.08050i
$833$	157.111			0.188608
$834$		0			0
$835$		0			0
$836$	−11.0067	+	97.9332i	−0.0131660	+	0.117145i
$837$		0			0
$838$	369.076	−	329.919i	0.440425	−	0.393698i
$839$		−	783.581i		−	0.933946i	−0.884272	−	0.466973i	$-0.845345\pi$
$839$		−	783.581i		−	0.933946i	0.884272	−	0.466973i	$-0.154655\pi$
$840$		0			0
$841$	401.631			0.477564
$842$	−103.607	−	115.903i	−0.123048	−	0.137653i
$843$		0			0
$844$	1072.48	+	120.536i	1.27071	+	0.142816i
$845$		0			0
$846$		0			0
$847$		−	761.014i		−	0.898481i
$848$	−478.111	−	108.845i	−0.563810	−	0.128355i
$849$		0			0
$850$		0			0
$851$		−	1996.22i		−	2.34574i
$852$		0			0
$853$	−104.807			−0.122868			−0.0614341	−	0.998111i	$-0.519567\pi$
$853$	−104.807			−0.122868			−0.0614341	+	0.998111i	$0.519567\pi$
$854$	1052.13	−	940.506i	1.23200	−	1.10129i
$855$		0			0
$856$	−33.2457	−	46.8140i	−0.0388385	−	0.0546892i
$857$	−1163.30			−1.35741			−0.678705	−	0.734411i	$-0.737458\pi$
$857$	−1163.30			−1.35741			−0.678705	+	0.734411i	$0.737458\pi$
$858$		0			0
$859$		−	337.818i		−	0.393269i	−0.980477	−	0.196634i	$-0.936999\pi$
$859$		−	337.818i		−	0.393269i	0.980477	−	0.196634i	$-0.0630012\pi$
$860$		0			0
$861$		0			0
$862$	−472.953	+	422.775i	−0.548669	+	0.490458i
$863$		−	199.699i		−	0.231400i	−0.993284	−	0.115700i	$-0.963089\pi$
$863$		−	199.699i		−	0.231400i	0.993284	−	0.115700i	$-0.0369112\pi$
$864$		0			0
$865$		0			0
$866$	−110.589	−	123.715i	−0.127701	−	0.142858i
$867$		0			0
$868$	−69.9356	+	622.256i	−0.0805710	+	0.716885i
$869$	133.766			0.153930
$870$		0			0
$871$		−	1657.97i		−	1.90352i
$872$	−165.840	+	117.774i	−0.190183	+	0.135062i
$873$		0			0
$874$	528.175	+	590.862i	0.604319	+	0.676044i
$875$		0			0
$876$		0			0
$877$	−1032.82			−1.17767			−0.588836	−	0.808252i	$-0.700414\pi$
$877$	−1032.82			−1.17767			−0.588836	+	0.808252i	$0.700414\pi$
$878$	−175.385	+	156.777i	−0.199755	+	0.178562i
$879$		0			0
$880$		0			0
$881$	585.412			0.664486			0.332243	−	0.943194i	$-0.392195\pi$
$881$	585.412			0.664486			0.332243	+	0.943194i	$0.392195\pi$
$882$		0			0
$883$			536.231i			0.607283i	0.952786	+	0.303641i	$0.0982025\pi$
$883$			536.231i			0.607283i	−0.952786	+	0.303641i	$0.901797\pi$
$884$	−178.288	+	1586.33i	−0.201684	+	1.79449i
$885$		0			0
$886$	52.4539	−	46.8888i	0.0592030	−	0.0529219i
$887$			173.466i			0.195564i	0.995208	+	0.0977822i	$0.0311749\pi$
$887$			173.466i			0.195564i	−0.995208	+	0.0977822i	$0.968825\pi$
$888$		0			0
$889$	470.344			0.529071
$890$		0			0
$891$		0			0
$892$	164.303	+	18.4661i	0.184197	+	0.0207019i
$893$	558.810			0.625767
$894$		0			0
$895$		0			0
$896$	−322.499	−	776.435i	−0.359932	−	0.866557i
$897$		0			0
$898$	89.8712	+	100.538i	0.100079	+	0.111957i
$899$			840.138i			0.934525i
$900$		0			0
$901$	822.092			0.912421
$902$	−80.5512	+	72.0051i	−0.0893028	+	0.0798283i
$903$		0			0
$904$	511.099	−	362.966i	0.565375	−	0.401511i
$905$		0			0
$906$		0			0
$907$			817.237i			0.901033i	0.892768	+	0.450516i	$0.148760\pi$
$907$			817.237i			0.901033i	−0.892768	+	0.450516i	$0.851240\pi$
$908$	595.599	+	66.9395i	0.655946	+	0.0737220i
$909$		0			0
$910$		0			0
$911$			227.911i			0.250177i	0.992146	+	0.125088i	$0.0399215\pi$
$911$			227.911i			0.250177i	−0.992146	+	0.125088i	$0.960079\pi$
$912$		0			0
$913$	−322.853			−0.353617
$914$	−272.115	−	304.412i	−0.297719	−	0.333055i
$915$		0			0
$916$	27.5541	−	245.165i	0.0300809	−	0.267647i
$917$	682.966			0.744783
$918$		0			0
$919$		−	669.088i		−	0.728061i	−0.931387	−	0.364031i	$-0.881400\pi$
$919$		−	669.088i		−	0.728061i	0.931387	−	0.364031i	$-0.118600\pi$
$920$		0			0
$921$		0			0
$922$	−167.628	−	187.523i	−0.181809	−	0.203387i
$923$		−	465.939i		−	0.504809i
$924$		0			0
$925$		0			0
$926$	825.515	−	737.933i	0.891485	−	0.796904i
$927$		0			0
$928$	−546.339	−	986.898i	−0.588727	−	1.06347i
$929$	1192.84			1.28400			0.642000	−	0.766705i	$-0.278105\pi$
$929$	1192.84			1.28400			0.642000	+	0.766705i	$0.278105\pi$
$930$		0			0
$931$		−	63.6528i		−	0.0683704i
$932$	181.195	−	1612.19i	0.194415	−	1.72982i
$933$		0			0
$934$	−933.088	+	834.092i	−0.999023	+	0.893032i
$935$		0			0
$936$		0			0
$937$	−530.621			−0.566298			−0.283149	−	0.959076i	$-0.591379\pi$
$937$	−530.621			−0.566298			−0.283149	+	0.959076i	$0.591379\pi$
$938$	−975.684	−	1091.48i	−1.04018	−	1.16363i
$939$		0			0
$940$		0			0
$941$	507.433			0.539248			0.269624	−	0.962966i	$-0.413101\pi$
$941$	507.433			0.539248			0.269624	+	0.962966i	$0.413101\pi$
$942$		0			0
$943$			868.860i			0.921378i
$944$	−24.5944	+	108.033i	−0.0260534	+	0.114442i
$945$		0			0
$946$	−169.972	−	190.145i	−0.179674	−	0.200999i
$947$			453.872i			0.479274i	0.970863	+	0.239637i	$0.0770284\pi$
$947$			453.872i			0.479274i	−0.970863	+	0.239637i	$0.922972\pi$
$948$		0			0
$949$	−1650.02			−1.73869
$950$		0			0
$951$		0			0
$952$	816.155	+	1149.24i	0.857306	+	1.20719i
$953$	−1220.52			−1.28071			−0.640356	−	0.768078i	$-0.721213\pi$
$953$	−1220.52			−1.28071			−0.640356	+	0.768078i	$0.721213\pi$
$954$		0			0
$955$		0			0
$956$	1064.37	+	119.625i	1.11336	+	0.125131i
$957$		0			0
$958$	727.964	−	650.731i	0.759879	−	0.679260i
$959$		−	47.0717i		−	0.0490842i
$960$		0			0
$961$	392.986			0.408935
$962$	−1085.67	−	1214.53i	−1.12856	−	1.26250i
$963$		0			0
$964$	40.0008	−	355.910i	0.0414946	−	0.369201i
$965$		0			0
$966$		0			0
$967$		−	292.088i		−	0.302056i	−0.988530	−	0.151028i	$-0.951742\pi$
$967$		−	292.088i		−	0.302056i	0.988530	−	0.151028i	$-0.0482583\pi$
$968$	−755.708	+	536.679i	−0.780690	+	0.554421i
$969$		0			0
$970$		0			0
$971$		−	1097.69i		−	1.13047i	−0.824928	−	0.565237i	$-0.808785\pi$
$971$		−	1097.69i		−	1.13047i	0.824928	−	0.565237i	$-0.191215\pi$
$972$		0			0
$973$	−964.947			−0.991724
$974$	−908.789	+	812.372i	−0.933048	+	0.834057i
$975$		0			0
$976$	−1675.93	−	381.536i	−1.71714	−	0.390918i
$977$	−338.550			−0.346520			−0.173260	−	0.984876i	$-0.555430\pi$
$977$	−338.550			−0.346520			−0.173260	+	0.984876i	$0.555430\pi$
$978$		0			0
$979$		−	16.2042i		−	0.0165518i
$980$		0			0
$981$		0			0
$982$	−1027.48	+	918.468i	−1.04631	+	0.935304i
$983$		−	346.346i		−	0.352336i	−0.984360	−	0.176168i	$-0.943630\pi$
$983$		−	346.346i		−	0.352336i	0.984360	−	0.176168i	$-0.0563702\pi$
$984$		0			0
$985$		0			0
$986$	1260.40	+	1409.99i	1.27830	+	1.43001i
$987$		0			0
$988$	642.697	+	72.2329i	0.650503	+	0.0731102i
$989$	−2050.99			−2.07380
$990$		0			0
$991$		−	242.030i		−	0.244228i	−0.992516	−	0.122114i	$-0.961033\pi$
$991$		−	242.030i		−	0.244228i	0.992516	−	0.122114i	$-0.0389673\pi$
$992$	667.238	−	369.377i	0.672619	−	0.372356i
$993$		0			0
$994$	−274.196	−	306.740i	−0.275851	−	0.308591i
$995$		0			0
$996$		0			0
$997$	−178.452			−0.178989			−0.0894946	−	0.995987i	$-0.528525\pi$
$997$	−178.452			−0.178989			−0.0894946	+	0.995987i	$0.528525\pi$
$998$	1044.37	−	933.565i	1.04646	−	0.935436i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.c.n.451.8		8
3.2	odd	2		300.3.c.g.151.1	yes	8
4.3	odd	2	inner	900.3.c.n.451.7		8
5.2	odd	4		900.3.f.h.199.6		16
5.3	odd	4		900.3.f.h.199.11		16
5.4	even	2		900.3.c.t.451.1		8
12.11	even	2		300.3.c.g.151.2	yes	8
15.2	even	4		300.3.f.c.199.11		16
15.8	even	4		300.3.f.c.199.6		16
15.14	odd	2		300.3.c.e.151.8	yes	8
20.3	even	4		900.3.f.h.199.5		16
20.7	even	4		900.3.f.h.199.12		16
20.19	odd	2		900.3.c.t.451.2		8
60.23	odd	4		300.3.f.c.199.12		16
60.47	odd	4		300.3.f.c.199.5		16
60.59	even	2		300.3.c.e.151.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.3.c.e.151.7	✓	8	60.59	even	2
300.3.c.e.151.8	yes	8	15.14	odd	2
300.3.c.g.151.1	yes	8	3.2	odd	2
300.3.c.g.151.2	yes	8	12.11	even	2
300.3.f.c.199.5		16	60.47	odd	4
300.3.f.c.199.6		16	15.8	even	4
300.3.f.c.199.11		16	15.2	even	4
300.3.f.c.199.12		16	60.23	odd	4
900.3.c.n.451.7		8	4.3	odd	2	inner
900.3.c.n.451.8		8	1.1	even	1	trivial
900.3.c.t.451.1		8	5.4	even	2
900.3.c.t.451.2		8	20.19	odd	2
900.3.f.h.199.5		16	20.3	even	4
900.3.f.h.199.6		16	5.2	odd	4
900.3.f.h.199.11		16	5.3	odd	4
900.3.f.h.199.12		16	20.7	even	4

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.33290 + 1.49110i) q^{2} +(-0.446749 + 3.97497i) q^{4} -6.56834i q^{7} +(-6.52255 + 4.63210i) q^{8} +O(q^{10})\)	\(q+(1.33290 + 1.49110i) q^{2} +(-0.446749 + 3.97497i) q^{4} -6.56834i q^{7} +(-6.52255 + 4.63210i) q^{8} +2.26696i q^{11} +14.8772 q^{13} +(9.79404 - 8.75495i) q^{14} +(-15.6008 - 3.55163i) q^{16} +26.8250 q^{17} -10.8680i q^{19} +(-3.38027 + 3.02164i) q^{22} +36.4610i q^{23} +(19.8298 + 22.1834i) q^{26} +(26.1090 + 2.93440i) q^{28} +35.2510 q^{29} +23.8330i q^{31} +(-15.4985 - 27.9963i) q^{32} +(35.7550 + 39.9987i) q^{34} -54.7495 q^{37} +(16.2053 - 14.4860i) q^{38} +23.8298 q^{41} +56.2515i q^{43} +(-9.01112 - 1.01276i) q^{44} +(-54.3670 + 48.5989i) q^{46} +51.4177i q^{47} +5.85689 q^{49} +(-6.64636 + 59.1364i) q^{52} +30.6465 q^{53} +(30.4252 + 42.8423i) q^{56} +(46.9861 + 52.5627i) q^{58} -6.92483i q^{59} +107.426 q^{61} +(-35.5374 + 31.7671i) q^{62} +(21.0873 - 60.4262i) q^{64} -111.444i q^{67} +(-11.9840 + 106.629i) q^{68} -31.3190i q^{71} -110.909 q^{73} +(-72.9757 - 81.6369i) q^{74} +(43.2002 + 4.85528i) q^{76} +14.8902 q^{77} -59.0065i q^{79} +(31.7628 + 35.5326i) q^{82} +142.416i q^{83} +(-83.8765 + 74.9776i) q^{86} +(-10.5008 - 14.7864i) q^{88} -7.14798 q^{89} -97.7185i q^{91} +(-144.932 - 16.2889i) q^{92} +(-76.6689 + 68.5347i) q^{94} +126.308 q^{97} +(7.80665 + 8.73319i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{4} - 20 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 - 8 * q^4 - 20 * q^8	\( 8 q - 2 q^{2} - 8 q^{4} - 20 q^{8} + 8 q^{13} - 22 q^{14} + 40 q^{16} + 4 q^{22} + 66 q^{26} + 104 q^{28} + 32 q^{29} - 112 q^{32} + 124 q^{34} - 176 q^{37} + 170 q^{38} + 16 q^{41} - 40 q^{44} - 76 q^{46} + 16 q^{49} + 56 q^{52} + 304 q^{53} + 172 q^{56} - 12 q^{58} + 136 q^{61} + 238 q^{62} + 16 q^{64} - 88 q^{68} + 240 q^{73} + 108 q^{74} + 120 q^{76} + 384 q^{77} + 320 q^{82} - 214 q^{86} - 200 q^{88} - 128 q^{89} - 312 q^{92} + 12 q^{94} + 216 q^{97} - 60 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 - 8 * q^4 - 20 * q^8 + 8 * q^13 - 22 * q^14 + 40 * q^16 + 4 * q^22 + 66 * q^26 + 104 * q^28 + 32 * q^29 - 112 * q^32 + 124 * q^34 - 176 * q^37 + 170 * q^38 + 16 * q^41 - 40 * q^44 - 76 * q^46 + 16 * q^49 + 56 * q^52 + 304 * q^53 + 172 * q^56 - 12 * q^58 + 136 * q^61 + 238 * q^62 + 16 * q^64 - 88 * q^68 + 240 * q^73 + 108 * q^74 + 120 * q^76 + 384 * q^77 + 320 * q^82 - 214 * q^86 - 200 * q^88 - 128 * q^89 - 312 * q^92 + 12 * q^94 + 216 * q^97 - 60 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more