LMFDB - Embedded newform 2280.2.r.b.1481.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2280,2,Mod(1481,2280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2280.1481");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2280.r (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:	$18.2058916609$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1481.12
Character	$\chi$	$=$	2280.1481
Dual form			2280.2.r.b.1481.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.13508 + 1.30828i) q^{3} +1.00000i q^{5} -2.58104 q^{7} +(-0.423200 - 2.97000i) q^{9} +O(q^{10})$	\(q+(-1.13508 + 1.30828i) q^{3} +1.00000i q^{5} -2.58104 q^{7} +(-0.423200 - 2.97000i) q^{9} -0.296245i q^{11} -2.55568i q^{13} +(-1.30828 - 1.13508i) q^{15} -4.65753i q^{17} +(1.78732 + 3.97561i) q^{19} +(2.92968 - 3.37673i) q^{21} +0.600591i q^{23} -1.00000 q^{25} +(4.36596 + 2.81752i) q^{27} +7.17443 q^{29} -0.557927i q^{31} +(0.387572 + 0.336261i) q^{33} -2.58104i q^{35} +4.66470i q^{37} +(3.34355 + 2.90090i) q^{39} +2.09586 q^{41} +3.40408 q^{43} +(2.97000 - 0.423200i) q^{45} +2.33842i q^{47} -0.338215 q^{49} +(6.09336 + 5.28666i) q^{51} +3.90829 q^{53} +0.296245 q^{55} +(-7.22997 - 2.17430i) q^{57} +2.05020 q^{59} -5.02480 q^{61} +(1.09230 + 7.66570i) q^{63} +2.55568 q^{65} +11.6045i q^{67} +(-0.785742 - 0.681717i) q^{69} +10.1067 q^{71} -12.5132 q^{73} +(1.13508 - 1.30828i) q^{75} +0.764621i q^{77} +12.6416i q^{79} +(-8.64180 + 2.51381i) q^{81} -14.3967i q^{83} +4.65753 q^{85} +(-8.14354 + 9.38618i) q^{87} +6.91423 q^{89} +6.59633i q^{91} +(0.729925 + 0.633290i) q^{93} +(-3.97561 + 1.78732i) q^{95} +1.58325i q^{97} +(-0.879847 + 0.125371i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 2 q^{3} + 4 q^{7} - 6 q^{9}+O(q^{10}) $ 40 * q + 2 * q^3 + 4 * q^7 - 6 * q^9	$ 40 q + 2 q^{3} + 4 q^{7} - 6 q^{9} - 4 q^{19} + 6 q^{21} - 40 q^{25} + 20 q^{27} + 12 q^{29} + 8 q^{33} - 6 q^{39} + 8 q^{41} - 40 q^{43} + 28 q^{49} - 22 q^{51} + 28 q^{53} + 2 q^{57} + 20 q^{59} + 8 q^{61} + 6 q^{63} + 8 q^{65} + 30 q^{69} - 40 q^{71} + 36 q^{73} - 2 q^{75} - 14 q^{81} - 6 q^{87} - 48 q^{89} - 4 q^{95} - 16 q^{99}+O(q^{100}) $ 40 * q + 2 * q^3 + 4 * q^7 - 6 * q^9 - 4 * q^19 + 6 * q^21 - 40 * q^25 + 20 * q^27 + 12 * q^29 + 8 * q^33 - 6 * q^39 + 8 * q^41 - 40 * q^43 + 28 * q^49 - 22 * q^51 + 28 * q^53 + 2 * q^57 + 20 * q^59 + 8 * q^61 + 6 * q^63 + 8 * q^65 + 30 * q^69 - 40 * q^71 + 36 * q^73 - 2 * q^75 - 14 * q^81 - 6 * q^87 - 48 * q^89 - 4 * q^95 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times$.

$n$	$457$	$761$	$1141$	$1711$	$1921$
$\chi(n)$	$1$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.13508	+	1.30828i	−0.655337	+	0.755337i
$3$	−1.13508	+	1.30828i	−0.655337	+	0.755337i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−2.58104			−0.975543			−0.487771	−	0.872971i	$-0.662190\pi$
$7$	−2.58104			−0.975543			−0.487771	+	0.872971i	$0.662190\pi$
$8$		0			0
$9$	−0.423200	−	2.97000i	−0.141067	−	0.990000i
$10$		0			0
$11$		−	0.296245i		−	0.0893212i	−0.999002	−	0.0446606i	$-0.985779\pi$
$11$		−	0.296245i		−	0.0893212i	0.999002	−	0.0446606i	$-0.0142206\pi$
$12$		0			0
$13$		−	2.55568i		−	0.708819i	−0.935090	−	0.354410i	$-0.884682\pi$
$13$		−	2.55568i		−	0.708819i	0.935090	−	0.354410i	$-0.115318\pi$
$14$		0			0
$15$	−1.30828	−	1.13508i	−0.337797	−	0.293076i
$16$		0			0
$17$		−	4.65753i		−	1.12962i	−0.825222	−	0.564808i	$-0.808950\pi$
$17$		−	4.65753i		−	1.12962i	0.825222	−	0.564808i	$-0.191050\pi$
$18$		0			0
$19$	1.78732	+	3.97561i	0.410040	+	0.912067i
$19$	1.78732	+	3.97561i	0.410040	+	0.912067i
$20$		0			0
$21$	2.92968	−	3.37673i	0.639309	−	0.736863i
$22$		0			0
$23$			0.600591i			0.125232i	0.998038	+	0.0626159i	$0.0199443\pi$
$23$			0.600591i			0.125232i	−0.998038	+	0.0626159i	$0.980056\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	4.36596	+	2.81752i	0.840229	+	0.542231i
$28$		0			0
$29$	7.17443			1.33226			0.666129	−	0.745836i	$-0.267950\pi$
$29$	7.17443			1.33226			0.666129	+	0.745836i	$0.267950\pi$
$30$		0			0
$31$		−	0.557927i		−	0.100207i	−0.998744	−	0.0501033i	$-0.984045\pi$
$31$		−	0.557927i		−	0.100207i	0.998744	−	0.0501033i	$-0.0159551\pi$
$32$		0			0
$33$	0.387572	+	0.336261i	0.0674676	+	0.0585355i
$34$		0			0
$35$		−	2.58104i		−	0.436276i
$36$		0			0
$37$			4.66470i			0.766872i	0.923567	+	0.383436i	$0.125259\pi$
$37$			4.66470i			0.766872i	−0.923567	+	0.383436i	$0.874741\pi$
$38$		0			0
$39$	3.34355	+	2.90090i	0.535397	+	0.464516i
$40$		0			0
$41$	2.09586			0.327318			0.163659	−	0.986517i	$-0.447670\pi$
$41$	2.09586			0.327318			0.163659	+	0.986517i	$0.447670\pi$
$42$		0			0
$43$	3.40408			0.519117			0.259559	−	0.965727i	$-0.416423\pi$
$43$	3.40408			0.519117			0.259559	+	0.965727i	$0.416423\pi$
$44$		0			0
$45$	2.97000	−	0.423200i	0.442742	−	0.0630869i
$46$		0			0
$47$			2.33842i			0.341094i	0.985350	+	0.170547i	$0.0545535\pi$
$47$			2.33842i			0.341094i	−0.985350	+	0.170547i	$0.945447\pi$
$48$		0			0
$49$	−0.338215			−0.0483164
$50$		0			0
$51$	6.09336	+	5.28666i	0.853241	+	0.740280i
$52$		0			0
$53$	3.90829			0.536845			0.268423	−	0.963301i	$-0.413498\pi$
$53$	3.90829			0.536845			0.268423	+	0.963301i	$0.413498\pi$
$54$		0			0
$55$	0.296245			0.0399456
$56$		0			0
$57$	−7.22997	−	2.17430i	−0.957632	−	0.287993i
$58$		0			0
$59$	2.05020			0.266913			0.133456	−	0.991055i	$-0.457392\pi$
$59$	2.05020			0.266913			0.133456	+	0.991055i	$0.457392\pi$
$60$		0			0
$61$	−5.02480			−0.643359			−0.321680	−	0.946849i	$-0.604247\pi$
$61$	−5.02480			−0.643359			−0.321680	+	0.946849i	$0.604247\pi$
$62$		0			0
$63$	1.09230	+	7.66570i	0.137616	+	0.965787i
$64$		0			0
$65$	2.55568			0.316994
$66$		0			0
$67$			11.6045i			1.41772i	0.705351	+	0.708858i	$0.250789\pi$
$67$			11.6045i			1.41772i	−0.705351	+	0.708858i	$0.749211\pi$
$68$		0			0
$69$	−0.785742	−	0.681717i	−0.0945922	−	0.0820691i
$70$		0			0
$71$	10.1067			1.19944			0.599721	−	0.800209i	$-0.295278\pi$
$71$	10.1067			1.19944			0.599721	+	0.800209i	$0.295278\pi$
$72$		0			0
$73$	−12.5132			−1.46456			−0.732278	−	0.681006i	$-0.761543\pi$
$73$	−12.5132			−1.46456			−0.732278	+	0.681006i	$0.761543\pi$
$74$		0			0
$75$	1.13508	−	1.30828i	0.131067	−	0.151067i
$76$		0			0
$77$			0.764621i			0.0871366i
$78$		0			0
$79$			12.6416i			1.42229i	0.703043	+	0.711147i	$0.251824\pi$
$79$			12.6416i			1.42229i	−0.703043	+	0.711147i	$0.748176\pi$
$80$		0			0
$81$	−8.64180	+	2.51381i	−0.960200	+	0.279312i
$82$		0			0
$83$		−	14.3967i		−	1.58024i	−0.612953	−	0.790119i	$-0.710019\pi$
$83$		−	14.3967i		−	1.58024i	0.612953	−	0.790119i	$-0.289981\pi$
$84$		0			0
$85$	4.65753			0.505180
$86$		0			0
$87$	−8.14354	+	9.38618i	−0.873079	+	1.00630i
$88$		0			0
$89$	6.91423			0.732907			0.366454	−	0.930436i	$-0.380572\pi$
$89$	6.91423			0.732907			0.366454	+	0.930436i	$0.380572\pi$
$90$		0			0
$91$			6.59633i			0.691484i
$92$		0			0
$93$	0.729925	+	0.633290i	0.0756897	+	0.0656691i
$94$		0			0
$95$	−3.97561	+	1.78732i	−0.407889	+	0.183376i
$96$		0			0
$97$			1.58325i			0.160754i	0.996765	+	0.0803771i	$0.0256125\pi$
$97$			1.58325i			0.160754i	−0.996765	+	0.0803771i	$0.974388\pi$
$98$		0			0
$99$	−0.879847	+	0.125371i	−0.0884280	+	0.0126002i
$100$		0			0
$101$			8.41249i			0.837074i	0.908200	+	0.418537i	$0.137457\pi$
$101$			8.41249i			0.837074i	−0.908200	+	0.418537i	$0.862543\pi$
$102$		0			0
$103$		−	0.438171i		−	0.0431742i	−0.999767	−	0.0215871i	$-0.993128\pi$
$103$		−	0.438171i		−	0.0431742i	0.999767	−	0.0215871i	$-0.00687193\pi$
$104$		0			0
$105$	3.37673	+	2.92968i	0.329535	+	0.285908i
$106$		0			0
$107$	−13.9263			−1.34631			−0.673155	−	0.739502i	$-0.735061\pi$
$107$	−13.9263			−1.34631			−0.673155	+	0.739502i	$0.735061\pi$
$108$		0			0
$109$		−	4.17791i		−	0.400172i	−0.979778	−	0.200086i	$-0.935878\pi$
$109$		−	4.17791i		−	0.400172i	0.979778	−	0.200086i	$-0.0641221\pi$
$110$		0			0
$111$	−6.10274	−	5.29480i	−0.579247	−	0.502560i
$112$		0			0
$113$	7.22552			0.679720			0.339860	−	0.940476i	$-0.389620\pi$
$113$	7.22552			0.679720			0.339860	+	0.940476i	$0.389620\pi$
$114$		0			0
$115$	−0.600591			−0.0560054
$116$		0			0
$117$	−7.59038	+	1.08156i	−0.701731	+	0.0999907i
$118$		0			0
$119$			12.0213i			1.10199i
$120$		0			0
$121$	10.9122			0.992022
$122$		0			0
$123$	−2.37896	+	2.74197i	−0.214504	+	0.247235i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			11.1174i			0.986507i	0.869886	+	0.493253i	$0.164192\pi$
$127$			11.1174i			0.986507i	−0.869886	+	0.493253i	$0.835808\pi$
$128$		0			0
$129$	−3.86389	+	4.45349i	−0.340197	+	0.392108i
$130$		0			0
$131$			14.5122i			1.26793i	0.773360	+	0.633967i	$0.218574\pi$
$131$			14.5122i			1.26793i	−0.773360	+	0.633967i	$0.781426\pi$
$132$		0			0
$133$	−4.61316	−	10.2612i	−0.400012	−	0.889761i
$134$		0			0
$135$	−2.81752	+	4.36596i	−0.242493	+	0.375762i
$136$		0			0
$137$		−	2.27625i		−	0.194473i	−0.995261	−	0.0972367i	$-0.969000\pi$
$137$		−	2.27625i		−	0.194473i	0.995261	−	0.0972367i	$-0.0310004\pi$
$138$		0			0
$139$	16.7561			1.42123			0.710616	−	0.703580i	$-0.248416\pi$
$139$	16.7561			1.42123			0.710616	+	0.703580i	$0.248416\pi$
$140$		0			0
$141$	−3.05931	−	2.65429i	−0.257641	−	0.223532i
$142$		0			0
$143$	−0.757108			−0.0633126
$144$		0			0
$145$			7.17443i			0.595804i
$146$		0			0
$147$	0.383900	−	0.442480i	0.0316635	−	0.0364952i
$148$		0			0
$149$			3.64387i			0.298518i	0.988798	+	0.149259i	$0.0476888\pi$
$149$			3.64387i			0.298518i	−0.988798	+	0.149259i	$0.952311\pi$
$150$		0			0
$151$		−	1.69605i		−	0.138022i	−0.997616	−	0.0690112i	$-0.978016\pi$
$151$		−	1.69605i		−	0.138022i	0.997616	−	0.0690112i	$-0.0219844\pi$
$152$		0			0
$153$	−13.8329	+	1.97107i	−1.11832	+	0.159351i
$154$		0			0
$155$	0.557927			0.0448137
$156$		0			0
$157$	8.37926			0.668738			0.334369	−	0.942442i	$-0.391477\pi$
$157$	8.37926			0.668738			0.334369	+	0.942442i	$0.391477\pi$
$158$		0			0
$159$	−4.43621	+	5.11314i	−0.351814	+	0.405499i
$160$		0			0
$161$		−	1.55015i		−	0.122169i
$162$		0			0
$163$	−8.85512			−0.693586			−0.346793	−	0.937942i	$-0.612729\pi$
$163$	−8.85512			−0.693586			−0.346793	+	0.937942i	$0.612729\pi$
$164$		0			0
$165$	−0.336261	+	0.387572i	−0.0261779	+	0.0301724i
$166$		0			0
$167$	−13.9029			−1.07584			−0.537920	−	0.842996i	$-0.680790\pi$
$167$	−13.9029			−1.07584			−0.537920	+	0.842996i	$0.680790\pi$
$168$		0			0
$169$	6.46848			0.497575
$170$		0			0
$171$	11.0512	−	6.99083i	0.845104	−	0.534602i
$172$		0			0
$173$	20.6427			1.56944			0.784719	−	0.619851i	$-0.212807\pi$
$173$	20.6427			1.56944			0.784719	+	0.619851i	$0.212807\pi$
$174$		0			0
$175$	2.58104			0.195109
$176$		0			0
$177$	−2.32713	+	2.68223i	−0.174918	+	0.201609i
$178$		0			0
$179$	3.14327			0.234939			0.117470	−	0.993076i	$-0.462522\pi$
$179$	3.14327			0.234939			0.117470	+	0.993076i	$0.462522\pi$
$180$		0			0
$181$			14.1400i			1.05102i	0.850788	+	0.525509i	$0.176125\pi$
$181$			14.1400i			1.05102i	−0.850788	+	0.525509i	$0.823875\pi$
$182$		0			0
$183$	5.70353	−	6.57385i	0.421617	−	0.485953i
$184$		0			0
$185$	−4.66470			−0.342956
$186$		0			0
$187$	−1.37977			−0.100899
$188$		0			0
$189$	−11.2687	−	7.27213i	−0.819680	−	0.528970i
$190$		0			0
$191$			13.7639i			0.995918i	0.867201	+	0.497959i	$0.165917\pi$
$191$			13.7639i			0.995918i	−0.867201	+	0.497959i	$0.834083\pi$
$192$		0			0
$193$		−	0.516714i		−	0.0371939i	−0.999827	−	0.0185970i	$-0.994080\pi$
$193$		−	0.516714i		−	0.0371939i	0.999827	−	0.0185970i	$-0.00591994\pi$
$194$		0			0
$195$	−2.90090	+	3.34355i	−0.207738	+	0.239437i
$196$		0			0
$197$		−	16.9942i		−	1.21079i	−0.795927	−	0.605393i	$-0.793016\pi$
$197$		−	16.9942i		−	1.21079i	0.795927	−	0.605393i	$-0.206984\pi$
$198$		0			0
$199$	19.2810			1.36679			0.683397	−	0.730047i	$-0.260502\pi$
$199$	19.2810			1.36679			0.683397	+	0.730047i	$0.260502\pi$
$200$		0			0
$201$	−15.1820	−	13.1720i	−1.07085	−	0.929082i
$202$		0			0
$203$	−18.5175			−1.29968
$204$		0			0
$205$			2.09586i			0.146381i
$206$		0			0
$207$	1.78376	−	0.254170i	0.123980	−	0.0176660i
$208$		0			0
$209$	1.17775	−	0.529485i	0.0814670	−	0.0366253i
$210$		0			0
$211$		−	3.99690i		−	0.275158i	−0.990491	−	0.137579i	$-0.956068\pi$
$211$		−	3.99690i		−	0.275158i	0.990491	−	0.137579i	$-0.0439321\pi$
$212$		0			0
$213$	−11.4719	+	13.2224i	−0.786039	+	0.905982i
$214$		0			0
$215$			3.40408i			0.232156i
$216$		0			0
$217$			1.44003i			0.0977558i
$218$		0			0
$219$	14.2034	−	16.3707i	0.959778	−	1.10623i
$220$		0			0
$221$	−11.9032			−0.800694
$222$		0			0
$223$			16.6565i			1.11540i	0.830041	+	0.557702i	$0.188317\pi$
$223$			16.6565i			1.11540i	−0.830041	+	0.557702i	$0.811683\pi$
$224$		0			0
$225$	0.423200	+	2.97000i	0.0282133	+	0.198000i
$226$		0			0
$227$	0.345921			0.0229596			0.0114798	−	0.999934i	$-0.496346\pi$
$227$	0.345921			0.0229596			0.0114798	+	0.999934i	$0.496346\pi$
$228$		0			0
$229$	26.9597			1.78155			0.890775	−	0.454445i	$-0.150162\pi$
$229$	26.9597			1.78155			0.890775	+	0.454445i	$0.150162\pi$
$230$		0			0
$231$	−1.00034	−	0.867904i	−0.0658175	−	0.0571039i
$232$		0			0
$233$			12.8711i			0.843211i	0.906779	+	0.421605i	$0.138533\pi$
$233$			12.8711i			0.843211i	−0.906779	+	0.421605i	$0.861467\pi$
$234$		0			0
$235$	−2.33842			−0.152542
$236$		0			0
$237$	−16.5388	−	14.3492i	−1.07431	−	0.932082i
$238$		0			0
$239$			24.9047i			1.61095i	0.592627	+	0.805477i	$0.298091\pi$
$239$			24.9047i			1.61095i	−0.592627	+	0.805477i	$0.701909\pi$
$240$		0			0
$241$		−	9.01536i		−	0.580730i	−0.956916	−	0.290365i	$-0.906223\pi$
$241$		−	9.01536i		−	0.580730i	0.956916	−	0.290365i	$-0.0937768\pi$
$242$		0			0
$243$	6.52035	−	14.1593i	0.418281	−	0.908318i
$244$		0			0
$245$		−	0.338215i		−	0.0216078i
$246$		0			0
$247$	10.1604	−	4.56783i	0.646491	−	0.290644i
$248$		0			0
$249$	18.8349	+	16.3413i	1.19361	+	1.03559i
$250$		0			0
$251$			7.89531i			0.498348i	0.968459	+	0.249174i	$0.0801591\pi$
$251$			7.89531i			0.498348i	−0.968459	+	0.249174i	$0.919841\pi$
$252$		0			0
$253$	0.177922			0.0111859
$254$		0			0
$255$	−5.28666	+	6.09336i	−0.331063	+	0.381581i
$256$		0			0
$257$	16.6375			1.03782			0.518910	−	0.854829i	$-0.326338\pi$
$257$	16.6375			1.03782			0.518910	+	0.854829i	$0.326338\pi$
$258$		0			0
$259$		−	12.0398i		−	0.748117i
$260$		0			0
$261$	−3.03622	−	21.3081i	−0.187937	−	1.31894i
$262$		0			0
$263$		−	0.227845i		−	0.0140495i	−0.999975	−	0.00702476i	$-0.997764\pi$
$263$		−	0.227845i		−	0.0140495i	0.999975	−	0.00702476i	$-0.00223607\pi$
$264$		0			0
$265$			3.90829i			0.240084i
$266$		0			0
$267$	−7.84819	+	9.04576i	−0.480301	+	0.553592i
$268$		0			0
$269$	22.1176			1.34854			0.674268	−	0.738486i	$-0.264459\pi$
$269$	22.1176			1.34854			0.674268	+	0.738486i	$0.264459\pi$
$270$		0			0
$271$	11.1137			0.675111			0.337555	−	0.941306i	$-0.390400\pi$
$271$	11.1137			0.675111			0.337555	+	0.941306i	$0.390400\pi$
$272$		0			0
$273$	−8.62986	−	7.48735i	−0.522303	−	0.453155i
$274$		0			0
$275$			0.296245i			0.0178642i
$276$		0			0
$277$	−20.4210			−1.22698			−0.613488	−	0.789704i	$-0.710234\pi$
$277$	−20.4210			−1.22698			−0.613488	+	0.789704i	$0.710234\pi$
$278$		0			0
$279$	−1.65704	+	0.236114i	−0.0992045	+	0.0141358i
$280$		0			0
$281$	−15.4182			−0.919774			−0.459887	−	0.887977i	$-0.652110\pi$
$281$	−15.4182			−0.919774			−0.459887	+	0.887977i	$0.652110\pi$
$282$		0			0
$283$	16.3954			0.974606			0.487303	−	0.873233i	$-0.337981\pi$
$283$	16.3954			0.974606			0.487303	+	0.873233i	$0.337981\pi$
$284$		0			0
$285$	2.17430	−	7.22997i	0.128795	−	0.428266i
$286$		0			0
$287$	−5.40950			−0.319313
$288$		0			0
$289$	−4.69258			−0.276034
$290$		0			0
$291$	−2.07133	−	1.79711i	−0.121424	−	0.105348i
$292$		0			0
$293$	17.3810			1.01541			0.507706	−	0.861530i	$-0.330494\pi$
$293$	17.3810			1.01541			0.507706	+	0.861530i	$0.330494\pi$
$294$		0			0
$295$			2.05020i			0.119367i
$296$		0			0
$297$	0.834674	−	1.29339i	0.0484327	−	0.0750503i
$298$		0			0
$299$	1.53492			0.0887668
$300$		0			0
$301$	−8.78608			−0.506421
$302$		0			0
$303$	−11.0059	−	9.54883i	−0.632273	−	0.548566i
$304$		0			0
$305$		−	5.02480i		−	0.287719i
$306$		0			0
$307$			4.66161i			0.266052i	0.991113	+	0.133026i	$0.0424694\pi$
$307$			4.66161i			0.266052i	−0.991113	+	0.133026i	$0.957531\pi$
$308$		0			0
$309$	0.573251	+	0.497358i	0.0326111	+	0.0282937i
$310$		0			0
$311$		−	18.2618i		−	1.03553i	−0.855522	−	0.517766i	$-0.826764\pi$
$311$		−	18.2618i		−	1.03553i	0.855522	−	0.517766i	$-0.173236\pi$
$312$		0			0
$313$	2.23029			0.126063			0.0630316	−	0.998012i	$-0.479923\pi$
$313$	2.23029			0.126063			0.0630316	+	0.998012i	$0.479923\pi$
$314$		0			0
$315$	−7.66570	+	1.09230i	−0.431913	+	0.0615440i
$316$		0			0
$317$	−20.7067			−1.16301			−0.581503	−	0.813544i	$-0.697535\pi$
$317$	−20.7067			−1.16301			−0.581503	+	0.813544i	$0.697535\pi$
$318$		0			0
$319$		−	2.12539i		−	0.118999i
$320$		0			0
$321$	15.8075	−	18.2196i	0.882287	−	1.01692i
$322$		0			0
$323$	18.5165	−	8.32451i	1.03029	−	0.463188i
$324$		0			0
$325$			2.55568i			0.141764i
$326$		0			0
$327$	5.46589	+	4.74226i	0.302264	+	0.262247i
$328$		0			0
$329$		−	6.03557i		−	0.332752i
$330$		0			0
$331$		−	14.1936i		−	0.780153i	−0.920783	−	0.390076i	$-0.872449\pi$
$331$		−	14.1936i		−	0.780153i	0.920783	−	0.390076i	$-0.127551\pi$
$332$		0			0
$333$	13.8542	−	1.97410i	0.759204	−	0.108180i
$334$		0			0
$335$	−11.6045			−0.634022
$336$		0			0
$337$			26.2582i			1.43038i	0.698931	+	0.715189i	$0.253659\pi$
$337$			26.2582i			1.43038i	−0.698931	+	0.715189i	$0.746341\pi$
$338$		0			0
$339$	−8.20152	+	9.45301i	−0.445446	+	0.513417i
$340$		0			0
$341$	−0.165283			−0.00895057
$342$		0			0
$343$	18.9403			1.02268
$344$		0			0
$345$	0.681717	−	0.785742i	0.0367024	−	0.0423029i
$346$		0			0
$347$			7.47142i			0.401087i	0.979685	+	0.200543i	$0.0642707\pi$
$347$			7.47142i			0.401087i	−0.979685	+	0.200543i	$0.935729\pi$
$348$		0			0
$349$	15.7087			0.840865			0.420433	−	0.907324i	$-0.361878\pi$
$349$	15.7087			0.840865			0.420433	+	0.907324i	$0.361878\pi$
$350$		0			0
$351$	7.20068	−	11.1580i	0.384344	−	0.595571i
$352$		0			0
$353$		−	0.971351i		−	0.0516998i	−0.999666	−	0.0258499i	$-0.991771\pi$
$353$		−	0.971351i		−	0.0516998i	0.999666	−	0.0258499i	$-0.00822919\pi$
$354$		0			0
$355$			10.1067i			0.536407i
$356$		0			0
$357$	−15.7272	−	13.6451i	−0.832373	−	0.722175i
$358$		0			0
$359$		−	6.75347i		−	0.356434i	−0.983991	−	0.178217i	$-0.942967\pi$
$359$		−	6.75347i		−	0.356434i	0.983991	−	0.178217i	$-0.0570330\pi$
$360$		0			0
$361$	−12.6110	+	14.2114i	−0.663734	+	0.747969i
$362$		0			0
$363$	−12.3862	+	14.2763i	−0.650109	+	0.749310i
$364$		0			0
$365$		−	12.5132i		−	0.654969i
$366$		0			0
$367$	−0.864670			−0.0451354			−0.0225677	−	0.999745i	$-0.507184\pi$
$367$	−0.864670			−0.0451354			−0.0225677	+	0.999745i	$0.507184\pi$
$368$		0			0
$369$	−0.886966	−	6.22470i	−0.0461736	−	0.324045i
$370$		0			0
$371$	−10.0875			−0.523715
$372$		0			0
$373$			21.8812i			1.13297i	0.824074	+	0.566483i	$0.191696\pi$
$373$			21.8812i			1.13297i	−0.824074	+	0.566483i	$0.808304\pi$
$374$		0			0
$375$	1.30828	+	1.13508i	0.0675594	+	0.0586151i
$376$		0			0
$377$		−	18.3356i		−	0.944331i
$378$		0			0
$379$		−	12.6169i		−	0.648086i	−0.946042	−	0.324043i	$-0.894958\pi$
$379$		−	12.6169i		−	0.648086i	0.946042	−	0.324043i	$-0.105042\pi$
$380$		0			0
$381$	−14.5446	−	12.6191i	−0.745145	−	0.646494i
$382$		0			0
$383$	6.59344			0.336909			0.168455	−	0.985709i	$-0.446122\pi$
$383$	6.59344			0.336909			0.168455	+	0.985709i	$0.446122\pi$
$384$		0			0
$385$	−0.764621			−0.0389687
$386$		0			0
$387$	−1.44061	−	10.1101i	−0.0732301	−	0.513926i
$388$		0			0
$389$		−	21.5975i		−	1.09504i	−0.836793	−	0.547519i	$-0.815572\pi$
$389$		−	21.5975i		−	1.09504i	0.836793	−	0.547519i	$-0.184428\pi$
$390$		0			0
$391$	2.79727			0.141464
$392$		0			0
$393$	−18.9860	−	16.4724i	−0.957717	−	0.830924i
$394$		0			0
$395$	−12.6416			−0.636069
$396$		0			0
$397$	12.1802			0.611305			0.305652	−	0.952143i	$-0.401125\pi$
$397$	12.1802			0.611305			0.305652	+	0.952143i	$0.401125\pi$
$398$		0			0
$399$	18.6609	+	5.61197i	0.934211	+	0.280950i
$400$		0			0
$401$	0.483332			0.0241364			0.0120682	−	0.999927i	$-0.496158\pi$
$401$	0.483332			0.0241364			0.0120682	+	0.999927i	$0.496158\pi$
$402$		0			0
$403$	−1.42588			−0.0710284
$404$		0			0
$405$	−2.51381	−	8.64180i	−0.124912	−	0.429415i
$406$		0			0
$407$	1.38189			0.0684980
$408$		0			0
$409$		−	8.45256i		−	0.417952i	−0.977921	−	0.208976i	$-0.932987\pi$
$409$		−	8.45256i		−	0.417952i	0.977921	−	0.208976i	$-0.0670131\pi$
$410$		0			0
$411$	2.97798	+	2.58372i	0.146893	+	0.127446i
$412$		0			0
$413$	−5.29165			−0.260385
$414$		0			0
$415$	14.3967			0.706704
$416$		0			0
$417$	−19.0194	+	21.9217i	−0.931386	+	1.07351i
$418$		0			0
$419$		−	33.5946i		−	1.64121i	−0.571499	−	0.820603i	$-0.693638\pi$
$419$		−	33.5946i		−	1.64121i	0.571499	−	0.820603i	$-0.306362\pi$
$420$		0			0
$421$		−	22.0347i		−	1.07390i	−0.843613	−	0.536952i	$-0.819576\pi$
$421$		−	22.0347i		−	1.07390i	0.843613	−	0.536952i	$-0.180424\pi$
$422$		0			0
$423$	6.94512	−	0.989620i	0.337683	−	0.0481170i
$424$		0			0
$425$			4.65753i			0.225923i
$426$		0			0
$427$	12.9692			0.627624
$428$		0			0
$429$	0.859376	−	0.990511i	0.0414911	−	0.0478223i
$430$		0			0
$431$	−22.1837			−1.06855			−0.534277	−	0.845310i	$-0.679416\pi$
$431$	−22.1837			−1.06855			−0.534277	+	0.845310i	$0.679416\pi$
$432$		0			0
$433$		−	0.730580i		−	0.0351095i	−0.999846	−	0.0175547i	$-0.994412\pi$
$433$		−	0.730580i		−	0.0351095i	0.999846	−	0.0175547i	$-0.00558813\pi$
$434$		0			0
$435$	−9.38618	−	8.14354i	−0.450033	−	0.390453i
$436$		0			0
$437$	−2.38772	+	1.07345i	−0.114220	+	0.0513501i
$438$		0			0
$439$		−	11.9541i		−	0.570538i	−0.958447	−	0.285269i	$-0.907917\pi$
$439$		−	11.9541i		−	0.570538i	0.958447	−	0.285269i	$-0.0920830\pi$
$440$		0			0
$441$	0.143132	+	1.00450i	0.00681583	+	0.0478333i
$442$		0			0
$443$		−	8.30874i		−	0.394760i	−0.980327	−	0.197380i	$-0.936757\pi$
$443$		−	8.30874i		−	0.394760i	0.980327	−	0.197380i	$-0.0632433\pi$
$444$		0			0
$445$			6.91423i			0.327766i
$446$		0			0
$447$	−4.76721	−	4.13608i	−0.225481	−	0.195630i
$448$		0			0
$449$	−17.7130			−0.835928			−0.417964	−	0.908464i	$-0.637256\pi$
$449$	−17.7130			−0.835928			−0.417964	+	0.908464i	$0.637256\pi$
$450$		0			0
$451$		−	0.620887i		−	0.0292364i
$452$		0			0
$453$	2.21891	+	1.92514i	0.104253	+	0.0904512i
$454$		0			0
$455$	−6.59633			−0.309241
$456$		0			0
$457$	−28.0818			−1.31361			−0.656807	−	0.754059i	$-0.728093\pi$
$457$	−28.0818			−1.31361			−0.656807	+	0.754059i	$0.728093\pi$
$458$		0			0
$459$	13.1227	−	20.3346i	0.612513	−	0.949137i
$460$		0			0
$461$			7.16544i			0.333728i	0.985980	+	0.166864i	$0.0533641\pi$
$461$			7.16544i			0.333728i	−0.985980	+	0.166864i	$0.946636\pi$
$462$		0			0
$463$	31.8939			1.48224			0.741118	−	0.671375i	$-0.234296\pi$
$463$	31.8939			1.48224			0.741118	+	0.671375i	$0.234296\pi$
$464$		0			0
$465$	−0.633290	+	0.729925i	−0.0293681	+	0.0338495i
$466$		0			0
$467$		−	33.6876i		−	1.55888i	−0.626480	−	0.779438i	$-0.715505\pi$
$467$		−	33.6876i		−	1.55888i	0.626480	−	0.779438i	$-0.284495\pi$
$468$		0			0
$469$		−	29.9517i		−	1.38304i
$470$		0			0
$471$	−9.51110	+	10.9624i	−0.438249	+	0.505122i
$472$		0			0
$473$		−	1.00844i		−	0.0463682i
$474$		0			0
$475$	−1.78732	−	3.97561i	−0.0820080	−	0.182413i
$476$		0			0
$477$	−1.65399	−	11.6076i	−0.0757309	−	0.531477i
$478$		0			0
$479$			6.39218i			0.292066i	0.989280	+	0.146033i	$0.0466506\pi$
$479$			6.39218i			0.292066i	−0.989280	+	0.146033i	$0.953349\pi$
$480$		0			0
$481$	11.9215			0.543574
$482$		0			0
$483$	2.02803	+	1.75954i	0.0922787	+	0.0800619i
$484$		0			0
$485$	−1.58325			−0.0718915
$486$		0			0
$487$		−	27.9921i		−	1.26844i	−0.773152	−	0.634221i	$-0.781321\pi$
$487$		−	27.9921i		−	1.26844i	0.773152	−	0.634221i	$-0.218679\pi$
$488$		0			0
$489$	10.0512	−	11.5850i	0.454533	−	0.523891i
$490$		0			0
$491$			2.99385i			0.135111i	0.997716	+	0.0675553i	$0.0215199\pi$
$491$			2.99385i			0.135111i	−0.997716	+	0.0675553i	$0.978480\pi$
$492$		0			0
$493$		−	33.4151i		−	1.50494i
$494$		0			0
$495$	−0.125371	−	0.879847i	−0.00563500	−	0.0395462i
$496$		0			0
$497$	−26.0858			−1.17011
$498$		0			0
$499$	−40.3591			−1.80672			−0.903361	−	0.428880i	$-0.858908\pi$
$499$	−40.3591			−1.80672			−0.903361	+	0.428880i	$0.858908\pi$
$500$		0			0
$501$	15.7809	−	18.1889i	0.705038	−	0.812621i
$502$		0			0
$503$		−	9.04952i		−	0.403498i	−0.979437	−	0.201749i	$-0.935337\pi$
$503$		−	9.04952i		−	0.403498i	0.979437	−	0.201749i	$-0.0646626\pi$
$504$		0			0
$505$	−8.41249			−0.374351
$506$		0			0
$507$	−7.34222	+	8.46259i	−0.326079	+	0.375837i
$508$		0			0
$509$	42.9662			1.90445			0.952223	−	0.305405i	$-0.0987917\pi$
$509$	42.9662			1.90445			0.952223	+	0.305405i	$0.0987917\pi$
$510$		0			0
$511$	32.2970			1.42874
$512$		0			0
$513$	−3.39796	+	22.3932i	−0.150024	+	0.988682i
$514$		0			0
$515$	0.438171			0.0193081
$516$		0			0
$517$	0.692746			0.0304669
$518$		0			0
$519$	−23.4311	+	27.0065i	−1.02851	+	1.18545i
$520$		0			0
$521$	−27.8288			−1.21920			−0.609600	−	0.792709i	$-0.708670\pi$
$521$	−27.8288			−1.21920			−0.609600	+	0.792709i	$0.708670\pi$
$522$		0			0
$523$		−	33.5951i		−	1.46901i	−0.678602	−	0.734506i	$-0.737414\pi$
$523$		−	33.5951i		−	1.46901i	0.678602	−	0.734506i	$-0.262586\pi$
$524$		0			0
$525$	−2.92968	+	3.37673i	−0.127862	+	0.147373i
$526$		0			0
$527$	−2.59856			−0.113195
$528$		0			0
$529$	22.6393			0.984317
$530$		0			0
$531$	−0.867643	−	6.08909i	−0.0376525	−	0.264244i
$532$		0			0
$533$		−	5.35635i		−	0.232009i
$534$		0			0
$535$		−	13.9263i		−	0.602088i
$536$		0			0
$537$	−3.56786	+	4.11229i	−0.153964	+	0.177458i
$538$		0			0
$539$			0.100194i			0.00431568i
$540$		0			0
$541$	−28.4376			−1.22263			−0.611314	−	0.791388i	$-0.709359\pi$
$541$	−28.4376			−1.22263			−0.611314	+	0.791388i	$0.709359\pi$
$542$		0			0
$543$	−18.4991	−	16.0500i	−0.793872	−	0.688771i
$544$		0			0
$545$	4.17791			0.178962
$546$		0			0
$547$			6.62876i			0.283425i	0.989908	+	0.141713i	$0.0452609\pi$
$547$			6.62876i			0.283425i	−0.989908	+	0.141713i	$0.954739\pi$
$548$		0			0
$549$	2.12649	+	14.9236i	0.0907565	+	0.636926i
$550$		0			0
$551$	12.8230	+	28.5228i	0.546280	+	1.21511i
$552$		0			0
$553$		−	32.6286i		−	1.38751i
$554$		0			0
$555$	5.29480	−	6.10274i	0.224752	−	0.259047i
$556$		0			0
$557$			26.2068i			1.11042i	0.831711	+	0.555208i	$0.187362\pi$
$557$			26.2068i			1.11042i	−0.831711	+	0.555208i	$0.812638\pi$
$558$		0			0
$559$		−	8.69975i		−	0.367960i
$560$		0			0
$561$	1.56614	−	1.80513i	0.0661227	−	0.0762125i
$562$		0			0
$563$	−24.1562			−1.01806			−0.509031	−	0.860748i	$-0.669996\pi$
$563$	−24.1562			−1.01806			−0.509031	+	0.860748i	$0.669996\pi$
$564$		0			0
$565$			7.22552i			0.303980i
$566$		0			0
$567$	22.3049	−	6.48824i	0.936717	−	0.272481i
$568$		0			0
$569$	−23.9538			−1.00420			−0.502099	−	0.864810i	$-0.667439\pi$
$569$	−23.9538			−1.00420			−0.502099	+	0.864810i	$0.667439\pi$
$570$		0			0
$571$	32.8293			1.37386			0.686932	−	0.726721i	$-0.258957\pi$
$571$	32.8293			1.37386			0.686932	+	0.726721i	$0.258957\pi$
$572$		0			0
$573$	−18.0070	−	15.6230i	−0.752253	−	0.652662i
$574$		0			0
$575$		−	0.600591i		−	0.0250464i
$576$		0			0
$577$	34.0925			1.41929			0.709644	−	0.704561i	$-0.248856\pi$
$577$	34.0925			1.41929			0.709644	+	0.704561i	$0.248856\pi$
$578$		0			0
$579$	0.676008	+	0.586511i	0.0280939	+	0.0243746i
$580$		0			0
$581$			37.1584i			1.54159i
$582$		0			0
$583$		−	1.15781i		−	0.0479516i
$584$		0			0
$585$	−1.08156	−	7.59038i	−0.0447172	−	0.313824i
$586$		0			0
$587$		−	36.4828i		−	1.50580i	−0.658132	−	0.752902i	$-0.728653\pi$
$587$		−	36.4828i		−	1.50580i	0.658132	−	0.752902i	$-0.271347\pi$
$588$		0			0
$589$	2.21810	−	0.997195i	0.0913952	−	0.0410887i
$590$		0			0
$591$	22.2332	+	19.2897i	0.914550	+	0.793472i
$592$		0			0
$593$			40.1355i			1.64817i	0.566468	+	0.824084i	$0.308310\pi$
$593$			40.1355i			1.64817i	−0.566468	+	0.824084i	$0.691690\pi$
$594$		0			0
$595$	−12.0213			−0.492825
$596$		0			0
$597$	−21.8854	+	25.2250i	−0.895711	+	1.03239i
$598$		0			0
$599$	7.65347			0.312712			0.156356	−	0.987701i	$-0.450025\pi$
$599$	7.65347			0.312712			0.156356	+	0.987701i	$0.450025\pi$
$600$		0			0
$601$			13.6959i			0.558666i	0.960194	+	0.279333i	$0.0901133\pi$
$601$			13.6959i			0.558666i	−0.960194	+	0.279333i	$0.909887\pi$
$602$		0			0
$603$	34.4654	−	4.91102i	1.40354	−	0.199992i
$604$		0			0
$605$			10.9122i			0.443646i
$606$		0			0
$607$			41.1225i			1.66911i	0.550923	+	0.834556i	$0.314276\pi$
$607$			41.1225i			1.66911i	−0.550923	+	0.834556i	$0.685724\pi$
$608$		0			0
$609$	21.0188	−	24.2261i	0.851726	−	0.981692i
$610$		0			0
$611$	5.97627			0.241774
$612$		0			0
$613$	44.3286			1.79041			0.895207	−	0.445651i	$-0.147028\pi$
$613$	44.3286			1.79041			0.895207	+	0.445651i	$0.147028\pi$
$614$		0			0
$615$	−2.74197	−	2.37896i	−0.110567	−	0.0959289i
$616$		0			0
$617$		−	4.81100i		−	0.193683i	−0.995300	−	0.0968417i	$-0.969126\pi$
$617$		−	4.81100i		−	0.193683i	0.995300	−	0.0968417i	$-0.0308741\pi$
$618$		0			0
$619$	−22.5391			−0.905923			−0.452961	−	0.891530i	$-0.649632\pi$
$619$	−22.5391			−0.905923			−0.452961	+	0.891530i	$0.649632\pi$
$620$		0			0
$621$	−1.69217	+	2.62216i	−0.0679046	+	0.105223i
$622$		0			0
$623$	−17.8459			−0.714982
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−0.644126	+	2.14184i	−0.0257239	+	0.0855369i
$628$		0			0
$629$	21.7260			0.866272
$630$		0			0
$631$	3.26283			0.129891			0.0649455	−	0.997889i	$-0.479313\pi$
$631$	3.26283			0.129891			0.0649455	+	0.997889i	$0.479313\pi$
$632$		0			0
$633$	5.22907	+	4.53679i	0.207837	+	0.180321i
$634$		0			0
$635$	−11.1174			−0.441179
$636$		0			0
$637$			0.864371i			0.0342476i
$638$		0			0
$639$	−4.27714	−	30.0168i	−0.169201	−	1.18745i
$640$		0			0
$641$	32.7643			1.29411			0.647057	−	0.762442i	$-0.276000\pi$
$641$	32.7643			1.29411			0.647057	+	0.762442i	$0.276000\pi$
$642$		0			0
$643$	8.15790			0.321716			0.160858	−	0.986978i	$-0.448574\pi$
$643$	8.15790			0.321716			0.160858	+	0.986978i	$0.448574\pi$
$644$		0			0
$645$	−4.45349	−	3.86389i	−0.175356	−	0.152141i
$646$		0			0
$647$		−	22.6009i		−	0.888531i	−0.895895	−	0.444266i	$-0.853465\pi$
$647$		−	22.6009i		−	0.888531i	0.895895	−	0.444266i	$-0.146535\pi$
$648$		0			0
$649$		−	0.607360i		−	0.0238410i
$650$		0			0
$651$	−1.88397	−	1.63455i	−0.0738385	−	0.0640630i
$652$		0			0
$653$		−	5.42121i		−	0.212148i	−0.994358	−	0.106074i	$-0.966172\pi$
$653$		−	5.42121i		−	0.212148i	0.994358	−	0.106074i	$-0.0338281\pi$
$654$		0			0
$655$	−14.5122			−0.567037
$656$		0			0
$657$	5.29557	+	37.1641i	0.206600	+	1.44991i
$658$		0			0
$659$	−28.2739			−1.10139			−0.550697	−	0.834705i	$-0.685638\pi$
$659$	−28.2739			−1.10139			−0.550697	+	0.834705i	$0.685638\pi$
$660$		0			0
$661$		−	1.91456i		−	0.0744679i	−0.999307	−	0.0372340i	$-0.988145\pi$
$661$		−	1.91456i		−	0.0744679i	0.999307	−	0.0372340i	$-0.0118547\pi$
$662$		0			0
$663$	13.5110	−	15.5727i	0.524725	−	0.604794i
$664$		0			0
$665$	10.2612	−	4.61316i	0.397913	−	0.178891i
$666$		0			0
$667$			4.30890i			0.166841i
$668$		0			0
$669$	−21.7914	−	18.9064i	−0.842505	−	0.730965i
$670$		0			0
$671$			1.48857i			0.0574656i
$672$		0			0
$673$			7.11183i			0.274141i	0.990561	+	0.137070i	$0.0437687\pi$
$673$			7.11183i			0.274141i	−0.990561	+	0.137070i	$0.956231\pi$
$674$		0			0
$675$	−4.36596	−	2.81752i	−0.168046	−	0.108446i
$676$		0			0
$677$	−28.2124			−1.08429			−0.542144	−	0.840285i	$-0.682387\pi$
$677$	−28.2124			−1.08429			−0.542144	+	0.840285i	$0.682387\pi$
$678$		0			0
$679$		−	4.08643i		−	0.156823i
$680$		0			0
$681$	−0.392647	+	0.452563i	−0.0150463	+	0.0173422i
$682$		0			0
$683$	14.4263			0.552006			0.276003	−	0.961157i	$-0.410990\pi$
$683$	14.4263			0.552006			0.276003	+	0.961157i	$0.410990\pi$
$684$		0			0
$685$	2.27625			0.0869712
$686$		0			0
$687$	−30.6014	+	35.2709i	−1.16752	+	1.34567i
$688$		0			0
$689$		−	9.98836i		−	0.380526i
$690$		0			0
$691$	−40.4611			−1.53921			−0.769606	−	0.638518i	$-0.779548\pi$
$691$	−40.4611			−1.53921			−0.769606	+	0.638518i	$0.779548\pi$
$692$		0			0
$693$	2.27092	−	0.323587i	0.0862653	−	0.0122921i
$694$		0			0
$695$			16.7561i			0.635594i
$696$		0			0
$697$		−	9.76152i		−	0.369744i
$698$		0			0
$699$	−16.8390	−	14.6096i	−0.636908	−	0.552587i
$700$		0			0
$701$		−	5.92053i		−	0.223615i	−0.993730	−	0.111808i	$-0.964336\pi$
$701$		−	5.92053i		−	0.223615i	0.993730	−	0.111808i	$-0.0356641\pi$
$702$		0			0
$703$	−18.5450	+	8.33733i	−0.699439	+	0.314448i
$704$		0			0
$705$	2.65429	−	3.05931i	0.0999664	−	0.115220i
$706$		0			0
$707$		−	21.7130i		−	0.816602i
$708$		0			0
$709$	−0.0635385			−0.00238624			−0.00119312	−	0.999999i	$-0.500380\pi$
$709$	−0.0635385			−0.00238624			−0.00119312	+	0.999999i	$0.500380\pi$
$710$		0			0
$711$	37.5456	−	5.34993i	1.40807	−	0.200638i
$712$		0			0
$713$	0.335086			0.0125491
$714$		0			0
$715$		−	0.757108i		−	0.0283142i
$716$		0			0
$717$	−32.5824	−	28.2688i	−1.21681	−	1.05572i
$718$		0			0
$719$			45.1098i			1.68231i	0.540793	+	0.841155i	$0.318124\pi$
$719$			45.1098i			1.68231i	−0.540793	+	0.841155i	$0.681876\pi$
$720$		0			0
$721$			1.13094i			0.0421183i
$722$		0			0
$723$	11.7946	+	10.2331i	0.438647	+	0.380574i
$724$		0			0
$725$	−7.17443			−0.266452
$726$		0			0
$727$	−41.3537			−1.53372			−0.766862	−	0.641811i	$-0.778183\pi$
$727$	−41.3537			−1.53372			−0.766862	+	0.641811i	$0.778183\pi$
$728$		0			0
$729$	11.1232	+	24.6023i	0.411971	+	0.911197i
$730$		0			0
$731$		−	15.8546i		−	0.586404i
$732$		0			0
$733$	−0.632469			−0.0233608			−0.0116804	−	0.999932i	$-0.503718\pi$
$733$	−0.632469			−0.0233608			−0.0116804	+	0.999932i	$0.503718\pi$
$734$		0			0
$735$	0.442480	+	0.383900i	0.0163211	+	0.0141604i
$736$		0			0
$737$	3.43778			0.126632
$738$		0			0
$739$	−36.0083			−1.32458			−0.662292	−	0.749245i	$-0.730416\pi$
$739$	−36.0083			−1.32458			−0.662292	+	0.749245i	$0.730416\pi$
$740$		0			0
$741$	−5.55683	+	18.4775i	−0.204135	+	0.678788i
$742$		0			0
$743$	−13.9034			−0.510065			−0.255032	−	0.966933i	$-0.582086\pi$
$743$	−13.9034			−0.510065			−0.255032	+	0.966933i	$0.582086\pi$
$744$		0			0
$745$	−3.64387			−0.133501
$746$		0			0
$747$	−42.7581	+	6.09266i	−1.56444	+	0.222919i
$748$		0			0
$749$	35.9445			1.31338
$750$		0			0
$751$			45.5362i			1.66164i	0.556541	+	0.830820i	$0.312128\pi$
$751$			45.5362i			1.66164i	−0.556541	+	0.830820i	$0.687872\pi$
$752$		0			0
$753$	−10.3293	−	8.96179i	−0.376420	−	0.326586i
$754$		0			0
$755$	1.69605			0.0617255
$756$		0			0
$757$	−1.32425			−0.0481307			−0.0240653	−	0.999710i	$-0.507661\pi$
$757$	−1.32425			−0.0481307			−0.0240653	+	0.999710i	$0.507661\pi$
$758$		0			0
$759$	−0.201955	+	0.232772i	−0.00733051	+	0.00844909i
$760$		0			0
$761$			10.8656i			0.393876i	0.980416	+	0.196938i	$0.0630998\pi$
$761$			10.8656i			0.393876i	−0.980416	+	0.196938i	$0.936900\pi$
$762$		0			0
$763$			10.7834i			0.390385i
$764$		0			0
$765$	−1.97107	−	13.8329i	−0.0712640	−	0.500128i
$766$		0			0
$767$		−	5.23966i		−	0.189193i
$768$		0			0
$769$	40.3466			1.45493			0.727467	−	0.686142i	$-0.240697\pi$
$769$	40.3466			1.45493			0.727467	+	0.686142i	$0.240697\pi$
$770$		0			0
$771$	−18.8849	+	21.7666i	−0.680122	+	0.783904i
$772$		0			0
$773$	17.3033			0.622356			0.311178	−	0.950352i	$-0.399276\pi$
$773$	17.3033			0.622356			0.311178	+	0.950352i	$0.399276\pi$
$774$		0			0
$775$			0.557927i			0.0200413i
$776$		0			0
$777$	15.7514	+	13.6661i	0.565080	+	0.490269i
$778$		0			0
$779$	3.74597	+	8.33231i	0.134213	+	0.298536i
$780$		0			0
$781$		−	2.99405i		−	0.107136i
$782$		0			0
$783$	31.3233	+	20.2141i	1.11940	+	0.722392i
$784$		0			0
$785$			8.37926i			0.299069i
$786$		0			0
$787$		−	6.15164i		−	0.219282i	−0.993971	−	0.109641i	$-0.965030\pi$
$787$		−	6.15164i		−	0.219282i	0.993971	−	0.109641i	$-0.0349702\pi$
$788$		0			0
$789$	0.298085	+	0.258621i	0.0106121	+	0.00920717i
$790$		0			0
$791$	−18.6494			−0.663096
$792$		0			0
$793$			12.8418i			0.456025i
$794$		0			0
$795$	−5.11314	−	4.43621i	−0.181345	−	0.157336i
$796$		0			0
$797$	−9.71606			−0.344161			−0.172080	−	0.985083i	$-0.555049\pi$
$797$	−9.71606			−0.344161			−0.172080	+	0.985083i	$0.555049\pi$
$798$		0			0
$799$	10.8913			0.385306
$800$		0			0
$801$	−2.92610	−	20.5353i	−0.103389	−	0.725578i
$802$		0			0
$803$			3.70696i			0.130816i
$804$		0			0
$805$	1.55015			0.0546357
$806$		0			0
$807$	−25.1052	+	28.9361i	−0.883746	+	1.01860i
$808$		0			0
$809$			9.52466i			0.334869i	0.985883	+	0.167435i	$0.0535483\pi$
$809$			9.52466i			0.334869i	−0.985883	+	0.167435i	$0.946452\pi$
$810$		0			0
$811$		−	38.1220i		−	1.33864i	−0.742972	−	0.669322i	$-0.766585\pi$
$811$		−	38.1220i		−	1.33864i	0.742972	−	0.669322i	$-0.233415\pi$
$812$		0			0
$813$	−12.6149	+	14.5399i	−0.442425	+	0.509936i
$814$		0			0
$815$		−	8.85512i		−	0.310181i
$816$		0			0
$817$	6.08419	+	13.5333i	0.212859	+	0.473470i
$818$		0			0
$819$	19.5911	−	2.79157i	0.684569	−	0.0975452i
$820$		0			0
$821$			37.0981i			1.29473i	0.762179	+	0.647367i	$0.224130\pi$
$821$			37.0981i			1.29473i	−0.762179	+	0.647367i	$0.775870\pi$
$822$		0			0
$823$	7.96920			0.277789			0.138895	−	0.990307i	$-0.455645\pi$
$823$	7.96920			0.277789			0.138895	+	0.990307i	$0.455645\pi$
$824$		0			0
$825$	−0.387572	−	0.336261i	−0.0134935	−	0.0117071i
$826$		0			0
$827$	29.2051			1.01556			0.507780	−	0.861487i	$-0.330466\pi$
$827$	29.2051			1.01556			0.507780	+	0.861487i	$0.330466\pi$
$828$		0			0
$829$			14.5551i			0.505519i	0.967529	+	0.252760i	$0.0813382\pi$
$829$			14.5551i			0.505519i	−0.967529	+	0.252760i	$0.918662\pi$
$830$		0			0
$831$	23.1794	−	26.7164i	0.804083	−	0.926780i
$832$		0			0
$833$			1.57525i			0.0545790i
$834$		0			0
$835$		−	13.9029i		−	0.481130i
$836$		0			0
$837$	1.57197	−	2.43589i	0.0543351	−	0.0841965i
$838$		0			0
$839$	1.34488			0.0464304			0.0232152	−	0.999730i	$-0.492610\pi$
$839$	1.34488			0.0464304			0.0232152	+	0.999730i	$0.492610\pi$
$840$		0			0
$841$	22.4725			0.774914
$842$		0			0
$843$	17.5009	−	20.1714i	0.602762	−	0.694739i
$844$		0			0
$845$			6.46848i			0.222522i
$846$		0			0
$847$	−28.1650			−0.967760
$848$		0			0
$849$	−18.6101	+	21.4498i	−0.638696	+	0.736156i
$850$		0			0
$851$	−2.80158			−0.0960369
$852$		0			0
$853$	14.2882			0.489218			0.244609	−	0.969622i	$-0.421340\pi$
$853$	14.2882			0.489218			0.244609	+	0.969622i	$0.421340\pi$
$854$		0			0
$855$	6.99083	+	11.0512i	0.239081	+	0.377942i
$856$		0			0
$857$	19.0155			0.649556			0.324778	−	0.945790i	$-0.394710\pi$
$857$	19.0155			0.649556			0.324778	+	0.945790i	$0.394710\pi$
$858$		0			0
$859$	−49.4412			−1.68691			−0.843457	−	0.537197i	$-0.819483\pi$
$859$	−49.4412			−1.68691			−0.843457	+	0.537197i	$0.819483\pi$
$860$		0			0
$861$	6.14020	−	7.07715i	0.209257	−	0.241188i
$862$		0			0
$863$	−16.7918			−0.571600			−0.285800	−	0.958289i	$-0.592259\pi$
$863$	−16.7918			−0.571600			−0.285800	+	0.958289i	$0.592259\pi$
$864$		0			0
$865$			20.6427i			0.701874i
$866$		0			0
$867$	5.32645	−	6.13922i	0.180896	−	0.208499i
$868$		0			0
$869$	3.74502			0.127041
$870$		0			0
$871$	29.6575			1.00490
$872$		0			0
$873$	4.70224	−	0.670029i	0.159147	−	0.0226771i
$874$		0			0
$875$			2.58104i			0.0872552i
$876$		0			0
$877$			26.6135i			0.898673i	0.893363	+	0.449337i	$0.148340\pi$
$877$			26.6135i			0.898673i	−0.893363	+	0.449337i	$0.851660\pi$
$878$		0			0
$879$	−19.7288	+	22.7393i	−0.665437	+	0.766978i
$880$		0			0
$881$			7.39901i			0.249279i	0.992202	+	0.124639i	$0.0397774\pi$
$881$			7.39901i			0.249279i	−0.992202	+	0.124639i	$0.960223\pi$
$882$		0			0
$883$	20.2432			0.681240			0.340620	−	0.940201i	$-0.389363\pi$
$883$	20.2432			0.681240			0.340620	+	0.940201i	$0.389363\pi$
$884$		0			0
$885$	−2.68223	−	2.32713i	−0.0901623	−	0.0782257i
$886$		0			0
$887$	−43.6432			−1.46540			−0.732698	−	0.680554i	$-0.761739\pi$
$887$	−43.6432			−1.46540			−0.732698	+	0.680554i	$0.761739\pi$
$888$		0			0
$889$		−	28.6944i		−	0.962379i
$890$		0			0
$891$	0.744702	+	2.56009i	0.0249485	+	0.0857662i
$892$		0			0
$893$	−9.29666	+	4.17952i	−0.311101	+	0.139862i
$894$		0			0
$895$			3.14327i			0.105068i
$896$		0			0
$897$	−1.74225	+	2.00811i	−0.0581722	+	0.0670488i
$898$		0			0
$899$		−	4.00281i		−	0.133501i
$900$		0			0
$901$		−	18.2030i		−	0.606429i
$902$		0			0
$903$	9.97287	−	11.4947i	0.331876	−	0.382518i
$904$		0			0
$905$	−14.1400			−0.470029
$906$		0			0
$907$			49.5193i			1.64426i	0.569298	+	0.822131i	$0.307215\pi$
$907$			49.5193i			1.64426i	−0.569298	+	0.822131i	$0.692785\pi$
$908$		0			0
$909$	24.9851	−	3.56016i	0.828704	−	0.118083i
$910$		0			0
$911$	13.9623			0.462592			0.231296	−	0.972883i	$-0.425703\pi$
$911$	13.9623			0.462592			0.231296	+	0.972883i	$0.425703\pi$
$912$		0			0
$913$	−4.26494			−0.141149
$914$		0			0
$915$	6.57385	+	5.70353i	0.217325	+	0.188553i
$916$		0			0
$917$		−	37.4565i		−	1.23692i
$918$		0			0
$919$	−17.8091			−0.587469			−0.293734	−	0.955887i	$-0.594898\pi$
$919$	−17.8091			−0.587469			−0.293734	+	0.955887i	$0.594898\pi$
$920$		0			0
$921$	−6.09870	−	5.29129i	−0.200959	−	0.174354i
$922$		0			0
$923$		−	25.8295i		−	0.850188i
$924$		0			0
$925$		−	4.66470i		−	0.153374i
$926$		0			0
$927$	−1.30137	+	0.185434i	−0.0427425	+	0.00609044i
$928$		0			0
$929$		−	50.3730i		−	1.65268i	−0.563168	−	0.826342i	$-0.690418\pi$
$929$		−	50.3730i		−	1.65268i	0.563168	−	0.826342i	$-0.309582\pi$
$930$		0			0
$931$	−0.604499	−	1.34461i	−0.0198117	−	0.0440678i
$932$		0			0
$933$	23.8916	+	20.7286i	0.782176	+	0.678623i
$934$		0			0
$935$		−	1.37977i		−	0.0451233i
$936$		0			0
$937$	−45.0075			−1.47033			−0.735165	−	0.677888i	$-0.762895\pi$
$937$	−45.0075			−1.47033			−0.735165	+	0.677888i	$0.762895\pi$
$938$		0			0
$939$	−2.53155	+	2.91784i	−0.0826139	+	0.0952202i
$940$		0			0
$941$	39.1021			1.27469			0.637346	−	0.770577i	$-0.280032\pi$
$941$	39.1021			1.27469			0.637346	+	0.770577i	$0.280032\pi$
$942$		0			0
$943$			1.25875i			0.0409906i
$944$		0			0
$945$	7.27213	−	11.2687i	0.236562	−	0.366572i
$946$		0			0
$947$		−	44.3344i		−	1.44068i	−0.693623	−	0.720338i	$-0.743987\pi$
$947$		−	44.3344i		−	1.44068i	0.693623	−	0.720338i	$-0.256013\pi$
$948$		0			0
$949$			31.9797i			1.03811i
$950$		0			0
$951$	23.5037	−	27.0902i	0.762161	−	0.878461i
$952$		0			0
$953$	−15.2786			−0.494921			−0.247461	−	0.968898i	$-0.579596\pi$
$953$	−15.2786			−0.494921			−0.247461	+	0.968898i	$0.579596\pi$
$954$		0			0
$955$	−13.7639			−0.445388
$956$		0			0
$957$	2.78061	+	2.41248i	0.0898843	+	0.0779844i
$958$		0			0
$959$			5.87511i			0.189717i
$960$		0			0
$961$	30.6887			0.989959
$962$		0			0
$963$	5.89362	+	41.3612i	0.189919	+	1.33285i
$964$		0			0
$965$	0.516714			0.0166336
$966$		0			0
$967$	53.4117			1.71760			0.858802	−	0.512308i	$-0.171209\pi$
$967$	53.4117			1.71760			0.858802	+	0.512308i	$0.171209\pi$
$968$		0			0
$969$	−10.1269	+	33.6738i	−0.325322	+	1.08176i
$970$		0			0
$971$	−8.90569			−0.285797			−0.142899	−	0.989737i	$-0.545642\pi$
$971$	−8.90569			−0.285797			−0.142899	+	0.989737i	$0.545642\pi$
$972$		0			0
$973$	−43.2482			−1.38647
$974$		0			0
$975$	−3.34355	−	2.90090i	−0.107079	−	0.0929031i
$976$		0			0
$977$	10.9003			0.348733			0.174366	−	0.984681i	$-0.444212\pi$
$977$	10.9003			0.348733			0.174366	+	0.984681i	$0.444212\pi$
$978$		0			0
$979$		−	2.04831i		−	0.0654641i
$980$		0			0
$981$	−12.4084	+	1.76809i	−0.396170	+	0.0564508i
$982$		0			0
$983$	34.1384			1.08885			0.544423	−	0.838811i	$-0.316749\pi$
$983$	34.1384			1.08885			0.544423	+	0.838811i	$0.316749\pi$
$984$		0			0
$985$	16.9942			0.541480
$986$		0			0
$987$	7.89622	+	6.85084i	0.251340	+	0.218065i
$988$		0			0
$989$			2.04446i			0.0650100i
$990$		0			0
$991$		−	5.04875i		−	0.160379i	−0.996780	−	0.0801894i	$-0.974447\pi$
$991$		−	5.04875i		−	0.160379i	0.996780	−	0.0801894i	$-0.0255525\pi$
$992$		0			0
$993$	18.5693	+	16.1109i	0.589278	+	0.511263i
$994$		0			0
$995$			19.2810i			0.611249i
$996$		0			0
$997$	−38.6060			−1.22266			−0.611332	−	0.791374i	$-0.709366\pi$
$997$	−38.6060			−1.22266			−0.611332	+	0.791374i	$0.709366\pi$
$998$		0			0
$999$	−13.1429	+	20.3659i	−0.415822	+	0.644349i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.2.r.b.1481.12	yes	40
3.2	odd	2		2280.2.r.a.1481.30	yes	40
19.18	odd	2		2280.2.r.a.1481.29	✓	40
57.56	even	2	inner	2280.2.r.b.1481.11	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.r.a.1481.29	✓	40	19.18	odd	2
2280.2.r.a.1481.30	yes	40	3.2	odd	2
2280.2.r.b.1481.11	yes	40	57.56	even	2	inner
2280.2.r.b.1481.12	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2280.r (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.13508 + 1.30828i) q^{3} +1.00000i q^{5} -2.58104 q^{7} +(-0.423200 - 2.97000i) q^{9} +O(q^{10})\)	\(q+(-1.13508 + 1.30828i) q^{3} +1.00000i q^{5} -2.58104 q^{7} +(-0.423200 - 2.97000i) q^{9} -0.296245i q^{11} -2.55568i q^{13} +(-1.30828 - 1.13508i) q^{15} -4.65753i q^{17} +(1.78732 + 3.97561i) q^{19} +(2.92968 - 3.37673i) q^{21} +0.600591i q^{23} -1.00000 q^{25} +(4.36596 + 2.81752i) q^{27} +7.17443 q^{29} -0.557927i q^{31} +(0.387572 + 0.336261i) q^{33} -2.58104i q^{35} +4.66470i q^{37} +(3.34355 + 2.90090i) q^{39} +2.09586 q^{41} +3.40408 q^{43} +(2.97000 - 0.423200i) q^{45} +2.33842i q^{47} -0.338215 q^{49} +(6.09336 + 5.28666i) q^{51} +3.90829 q^{53} +0.296245 q^{55} +(-7.22997 - 2.17430i) q^{57} +2.05020 q^{59} -5.02480 q^{61} +(1.09230 + 7.66570i) q^{63} +2.55568 q^{65} +11.6045i q^{67} +(-0.785742 - 0.681717i) q^{69} +10.1067 q^{71} -12.5132 q^{73} +(1.13508 - 1.30828i) q^{75} +0.764621i q^{77} +12.6416i q^{79} +(-8.64180 + 2.51381i) q^{81} -14.3967i q^{83} +4.65753 q^{85} +(-8.14354 + 9.38618i) q^{87} +6.91423 q^{89} +6.59633i q^{91} +(0.729925 + 0.633290i) q^{93} +(-3.97561 + 1.78732i) q^{95} +1.58325i q^{97} +(-0.879847 + 0.125371i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 2 q^{3} + 4 q^{7} - 6 q^{9}+O(q^{10}) \) 40 * q + 2 * q^3 + 4 * q^7 - 6 * q^9	\( 40 q + 2 q^{3} + 4 q^{7} - 6 q^{9} - 4 q^{19} + 6 q^{21} - 40 q^{25} + 20 q^{27} + 12 q^{29} + 8 q^{33} - 6 q^{39} + 8 q^{41} - 40 q^{43} + 28 q^{49} - 22 q^{51} + 28 q^{53} + 2 q^{57} + 20 q^{59} + 8 q^{61} + 6 q^{63} + 8 q^{65} + 30 q^{69} - 40 q^{71} + 36 q^{73} - 2 q^{75} - 14 q^{81} - 6 q^{87} - 48 q^{89} - 4 q^{95} - 16 q^{99}+O(q^{100}) \) 40 * q + 2 * q^3 + 4 * q^7 - 6 * q^9 - 4 * q^19 + 6 * q^21 - 40 * q^25 + 20 * q^27 + 12 * q^29 + 8 * q^33 - 6 * q^39 + 8 * q^41 - 40 * q^43 + 28 * q^49 - 22 * q^51 + 28 * q^53 + 2 * q^57 + 20 * q^59 + 8 * q^61 + 6 * q^63 + 8 * q^65 + 30 * q^69 - 40 * q^71 + 36 * q^73 - 2 * q^75 - 14 * q^81 - 6 * q^87 - 48 * q^89 - 4 * q^95 - 16 * q^99

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