LMFDB - Embedded newform 5040.2.f.f.881.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(881,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, 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("5040.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.7442857984.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 $ x^8 + 26x^6 + 205x^4 + 540*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$0.916813i$ of defining polynomial
Character	$\chi$	$=$	5040.881
Dual form			5040.2.f.f.881.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +(2.56510 - 0.648285i) q^{7} +O(q^{10})$	$q-1.00000 q^{5} +(2.56510 - 0.648285i) q^{7} +1.29657i q^{11} +3.13020i q^{13} +5.53921 q^{17} -7.37284i q^{19} -1.83363i q^{23} +1.00000 q^{25} +1.83363i q^{29} -10.4268i q^{31} +(-2.56510 + 0.648285i) q^{35} -10.6694 q^{37} +3.13020 q^{41} -3.53921 q^{43} +10.7797 q^{47} +(6.15945 - 3.32583i) q^{49} +4.42677i q^{53} -1.29657i q^{55} -7.18871 q^{59} -4.88755i q^{61} -3.13020i q^{65} +9.79960 q^{67} +7.37284i q^{71} +3.40686i q^{73} +(0.840546 + 3.32583i) q^{77} -9.01990 q^{79} +6.26039 q^{83} -5.53921 q^{85} -7.94822 q^{89} +(2.02926 + 8.02926i) q^{91} +7.37284i q^{95} -8.09402i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 4 q^{7}+O(q^{10}) $ 8 * q - 8 * q^5 + 4 * q^7	$ 8 q - 8 q^{5} + 4 q^{7} + 8 q^{25} - 4 q^{35} - 8 q^{37} - 8 q^{41} + 16 q^{43} + 40 q^{47} + 4 q^{49} - 32 q^{67} + 52 q^{77} - 8 q^{79} - 16 q^{83} - 8 q^{89} + 4 q^{91}+O(q^{100}) $ 8 * q - 8 * q^5 + 4 * q^7 + 8 * q^25 - 4 * q^35 - 8 * q^37 - 8 * q^41 + 16 * q^43 + 40 * q^47 + 4 * q^49 - 32 * q^67 + 52 * q^77 - 8 * q^79 - 16 * q^83 - 8 * q^89 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$1$	$-1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	2.56510	−	0.648285i	0.969516	−	0.245029i
$7$	2.56510	−	0.648285i	0.969516	−	0.245029i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.29657i			0.390930i	0.980711	+	0.195465i	$0.0626217\pi$
$11$			1.29657i			0.390930i	−0.980711	+	0.195465i	$0.937378\pi$
$12$		0			0
$13$			3.13020i			0.868160i	0.900874	+	0.434080i	$0.142926\pi$
$13$			3.13020i			0.868160i	−0.900874	+	0.434080i	$0.857074\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.53921			1.34346			0.671728	−	0.740798i	$-0.265552\pi$
$17$	5.53921			1.34346			0.671728	+	0.740798i	$0.265552\pi$
$18$		0			0
$19$		−	7.37284i		−	1.69144i	−0.533623	−	0.845722i	$-0.679170\pi$
$19$		−	7.37284i		−	1.69144i	0.533623	−	0.845722i	$-0.320830\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.83363i		−	0.382337i	−0.981557	−	0.191169i	$-0.938772\pi$
$23$		−	1.83363i		−	0.382337i	0.981557	−	0.191169i	$-0.0612278\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.83363i			0.340496i	0.985401	+	0.170248i	$0.0544569\pi$
$29$			1.83363i			0.340496i	−0.985401	+	0.170248i	$0.945543\pi$
$30$		0			0
$31$		−	10.4268i		−	1.87270i	−0.351065	−	0.936351i	$-0.614180\pi$
$31$		−	10.4268i		−	1.87270i	0.351065	−	0.936351i	$-0.385820\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.56510	+	0.648285i	−0.433581	+	0.109580i
$36$		0			0
$37$	−10.6694			−1.75404			−0.877020	−	0.480454i	$-0.840472\pi$
$37$	−10.6694			−1.75404			−0.877020	+	0.480454i	$0.840472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.13020			0.488854			0.244427	−	0.969668i	$-0.421400\pi$
$41$	3.13020			0.488854			0.244427	+	0.969668i	$0.421400\pi$
$42$		0			0
$43$	−3.53921			−0.539724			−0.269862	−	0.962899i	$-0.586978\pi$
$43$	−3.53921			−0.539724			−0.269862	+	0.962899i	$0.586978\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.7797			1.57238			0.786190	−	0.617985i	$-0.212051\pi$
$47$	10.7797			1.57238			0.786190	+	0.617985i	$0.212051\pi$
$48$		0			0
$49$	6.15945	−	3.32583i	0.879922	−	0.475118i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.42677i			0.608063i	0.952662	+	0.304031i	$0.0983328\pi$
$53$			4.42677i			0.608063i	−0.952662	+	0.304031i	$0.901667\pi$
$54$		0			0
$55$		−	1.29657i		−	0.174829i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.18871			−0.935891			−0.467945	−	0.883757i	$-0.655006\pi$
$59$	−7.18871			−0.935891			−0.467945	+	0.883757i	$0.655006\pi$
$60$		0			0
$61$		−	4.88755i		−	0.625787i	−0.949788	−	0.312894i	$-0.898702\pi$
$61$		−	4.88755i		−	0.625787i	0.949788	−	0.312894i	$-0.101298\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	3.13020i		−	0.388253i
$66$		0			0
$67$	9.79960			1.19721			0.598606	−	0.801044i	$-0.295722\pi$
$67$	9.79960			1.19721			0.598606	+	0.801044i	$0.295722\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.37284i			0.874995i	0.899220	+	0.437497i	$0.144135\pi$
$71$			7.37284i			0.874995i	−0.899220	+	0.437497i	$0.855865\pi$
$72$		0			0
$73$			3.40686i			0.398743i	0.979924	+	0.199371i	$0.0638900\pi$
$73$			3.40686i			0.398743i	−0.979924	+	0.199371i	$0.936110\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.840546	+	3.32583i	0.0957891	+	0.379013i
$78$		0			0
$79$	−9.01990			−1.01482			−0.507409	−	0.861705i	$-0.669397\pi$
$79$	−9.01990			−1.01482			−0.507409	+	0.861705i	$0.669397\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.26039			0.687167			0.343584	−	0.939122i	$-0.388359\pi$
$83$	6.26039			0.687167			0.343584	+	0.939122i	$0.388359\pi$
$84$		0			0
$85$	−5.53921			−0.600812
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.94822			−0.842510			−0.421255	−	0.906942i	$-0.638410\pi$
$89$	−7.94822			−0.842510			−0.421255	+	0.906942i	$0.638410\pi$
$90$		0			0
$91$	2.02926	+	8.02926i	0.212724	+	0.841695i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			7.37284i			0.756437i
$96$		0			0
$97$		−	8.09402i		−	0.821823i	−0.911675	−	0.410911i	$-0.865211\pi$
$97$		−	8.09402i		−	0.821823i	0.911675	−	0.410911i	$-0.134789\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.05852			0.403837			0.201919	−	0.979402i	$-0.435282\pi$
$101$	4.05852			0.403837			0.201919	+	0.979402i	$0.435282\pi$
$102$		0			0
$103$		−	5.61548i		−	0.553309i	−0.960969	−	0.276655i	$-0.910774\pi$
$103$		−	5.61548i		−	0.553309i	0.960969	−	0.276655i	$-0.0892258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.651655i		−	0.0629979i	−0.999504	−	0.0314989i	$-0.989972\pi$
$107$		−	0.651655i		−	0.0629979i	0.999504	−	0.0314989i	$-0.0100281\pi$
$108$		0			0
$109$	15.3388			1.46919			0.734596	−	0.678505i	$-0.237372\pi$
$109$	15.3388			1.46919			0.734596	+	0.678505i	$0.237372\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.0784i			1.60660i	0.595573	+	0.803301i	$0.296925\pi$
$113$			17.0784i			1.60660i	−0.595573	+	0.803301i	$0.703075\pi$
$114$		0			0
$115$			1.83363i			0.170987i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.2086	−	3.59099i	1.30250	−	0.329185i
$120$		0			0
$121$	9.31891			0.847173
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	7.39059			0.655809			0.327904	−	0.944711i	$-0.393658\pi$
$127$	7.39059			0.655809			0.327904	+	0.944711i	$0.393658\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	20.2086			1.76563			0.882817	−	0.469716i	$-0.155644\pi$
$131$	20.2086			1.76563			0.882817	+	0.469716i	$0.155644\pi$
$132$		0			0
$133$	−4.77970	−	18.9120i	−0.414452	−	1.63988i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	20.7457i		−	1.77242i	−0.463281	−	0.886211i	$-0.653328\pi$
$137$		−	20.7457i		−	1.77242i	0.463281	−	0.886211i	$-0.346672\pi$
$138$		0			0
$139$		−	9.20646i		−	0.780882i	−0.920628	−	0.390441i	$-0.872323\pi$
$139$		−	9.20646i		−	0.780882i	0.920628	−	0.390441i	$-0.127677\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.05852			−0.339390
$144$		0			0
$145$		−	1.83363i		−	0.152274i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	10.3189i		−	0.845358i	−0.906279	−	0.422679i	$-0.861090\pi$
$149$		−	10.3189i		−	0.845358i	0.906279	−	0.422679i	$-0.138910\pi$
$150$		0			0
$151$	1.24049			0.100949			0.0504747	−	0.998725i	$-0.483927\pi$
$151$	1.24049			0.100949			0.0504747	+	0.998725i	$0.483927\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			10.4268i			0.837498i
$156$		0			0
$157$			11.4629i			0.914842i	0.889250	+	0.457421i	$0.151227\pi$
$157$			11.4629i			0.914842i	−0.889250	+	0.457421i	$0.848773\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.18871	−	4.70343i	−0.0936836	−	0.370682i
$162$		0			0
$163$	−1.27882			−0.100165			−0.0500824	−	0.998745i	$-0.515948\pi$
$163$	−1.27882			−0.100165			−0.0500824	+	0.998745i	$0.515948\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.72118			0.520101			0.260050	−	0.965595i	$-0.416261\pi$
$167$	6.72118			0.520101			0.260050	+	0.965595i	$0.416261\pi$
$168$		0			0
$169$	3.20188			0.246298
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.3189			−1.24070			−0.620352	−	0.784324i	$-0.713010\pi$
$173$	−16.3189			−1.24070			−0.620352	+	0.784324i	$0.713010\pi$
$174$		0			0
$175$	2.56510	−	0.648285i	0.193903	−	0.0490057i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	19.1887i		−	1.43423i	−0.696954	−	0.717116i	$-0.745462\pi$
$179$		−	19.1887i		−	1.43423i	0.696954	−	0.717116i	$-0.254538\pi$
$180$		0			0
$181$			10.7797i			0.801249i	0.916242	+	0.400624i	$0.131207\pi$
$181$			10.7797i			0.801249i	−0.916242	+	0.400624i	$0.868793\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.6694			0.784430
$186$		0			0
$187$			7.18197i			0.525198i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	19.3728i		−	1.40177i	−0.713275	−	0.700885i	$-0.752789\pi$
$191$		−	19.3728i		−	1.40177i	0.713275	−	0.700885i	$-0.247211\pi$
$192$		0			0
$193$	−6.81803			−0.490772			−0.245386	−	0.969425i	$-0.578915\pi$
$193$	−6.81803			−0.490772			−0.245386	+	0.969425i	$0.578915\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.7457i		−	1.05059i	−0.850922	−	0.525293i	$-0.823956\pi$
$197$		−	14.7457i		−	1.05059i	0.850922	−	0.525293i	$-0.176044\pi$
$198$		0			0
$199$		−	1.94148i		−	0.137628i	−0.997630	−	0.0688141i	$-0.978078\pi$
$199$		−	1.94148i		−	0.137628i	0.997630	−	0.0688141i	$-0.0219215\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.18871	+	4.70343i	0.0834312	+	0.330116i
$204$		0			0
$205$	−3.13020			−0.218622
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.55939			0.661237
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.53921			0.241372
$216$		0			0
$217$	−6.75951	−	26.7457i	−0.458866	−	1.81561i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			17.3388i			1.16633i
$222$		0			0
$223$		−	8.76195i		−	0.586743i	−0.955998	−	0.293372i	$-0.905223\pi$
$223$		−	8.76195i		−	0.586743i	0.955998	−	0.293372i	$-0.0947773\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−23.0784			−1.53177			−0.765884	−	0.642978i	$-0.777698\pi$
$227$	−23.0784			−1.53177			−0.765884	+	0.642978i	$0.777698\pi$
$228$		0			0
$229$		−	9.70558i		−	0.641363i	−0.947187	−	0.320682i	$-0.896088\pi$
$229$		−	9.70558i		−	0.641363i	0.947187	−	0.320682i	$-0.103912\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6.36825i			0.417198i	0.978001	+	0.208599i	$0.0668903\pi$
$233$			6.36825i			0.417198i	−0.978001	+	0.208599i	$0.933110\pi$
$234$		0			0
$235$	−10.7797			−0.703190
$236$		0			0
$237$		0			0
$238$		0			0
$239$			19.3728i			1.25312i	0.779371	+	0.626562i	$0.215539\pi$
$239$			19.3728i			1.25312i	−0.779371	+	0.626562i	$0.784461\pi$
$240$		0			0
$241$		−	24.3587i		−	1.56908i	−0.620076	−	0.784541i	$-0.712898\pi$
$241$		−	24.3587i		−	1.56908i	0.620076	−	0.784541i	$-0.287102\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.15945	+	3.32583i	−0.393513	+	0.212479i
$246$		0			0
$247$	23.0784			1.46844
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.94822			−0.122971			−0.0614854	−	0.998108i	$-0.519584\pi$
$251$	−1.94822			−0.122971			−0.0614854	+	0.998108i	$0.519584\pi$
$252$		0			0
$253$	2.37742			0.149467
$254$		0			0
$255$		0			0
$256$		0			0
$257$	25.2420			1.57455			0.787275	−	0.616602i	$-0.211491\pi$
$257$	25.2420			1.57455			0.787275	+	0.616602i	$0.211491\pi$
$258$		0			0
$259$	−27.3681	+	6.91681i	−1.70057	+	0.429790i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	0.912047i		−	0.0562392i	−0.999605	−	0.0281196i	$-0.991048\pi$
$263$		−	0.912047i		−	0.0562392i	0.999605	−	0.0281196i	$-0.00895193\pi$
$264$		0			0
$265$		−	4.42677i		−	0.271934i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.5793			1.01086			0.505429	−	0.862868i	$-0.331334\pi$
$269$	16.5793			1.01086			0.505429	+	0.862868i	$0.331334\pi$
$270$		0			0
$271$		−	7.83363i		−	0.475859i	−0.971282	−	0.237929i	$-0.923531\pi$
$271$		−	7.83363i		−	0.475859i	0.971282	−	0.237929i	$-0.0764687\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.29657i			0.0781861i
$276$		0			0
$277$	11.5910			0.696435			0.348217	−	0.937414i	$-0.386787\pi$
$277$	11.5910			0.696435			0.348217	+	0.937414i	$0.386787\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			19.3566i			1.15472i	0.816491	+	0.577358i	$0.195916\pi$
$281$			19.3566i			1.15472i	−0.816491	+	0.577358i	$0.804084\pi$
$282$		0			0
$283$		−	1.28983i		−	0.0766724i	−0.999265	−	0.0383362i	$-0.987794\pi$
$283$		−	1.28983i		−	0.0766724i	0.999265	−	0.0383362i	$-0.0122058\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.02926	−	2.02926i	0.473952	−	0.119783i
$288$		0			0
$289$	13.6828			0.804873
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.5992			1.02816			0.514078	−	0.857743i	$-0.328134\pi$
$293$	17.5992			1.02816			0.514078	+	0.857743i	$0.328134\pi$
$294$		0			0
$295$	7.18871			0.418543
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.73961			0.331930
$300$		0			0
$301$	−9.07842	+	2.29442i	−0.523271	+	0.132248i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			4.88755i			0.279861i
$306$		0			0
$307$		−	26.7457i		−	1.52646i	−0.646129	−	0.763228i	$-0.723613\pi$
$307$		−	26.7457i		−	1.52646i	0.646129	−	0.763228i	$-0.276387\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−29.3388			−1.66365			−0.831826	−	0.555037i	$-0.812704\pi$
$311$	−29.3388			−1.66365			−0.831826	+	0.555037i	$0.812704\pi$
$312$		0			0
$313$		−	34.4172i		−	1.94538i	−0.232114	−	0.972688i	$-0.574564\pi$
$313$		−	34.4172i		−	1.94538i	0.232114	−	0.972688i	$-0.425436\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.53462i		−	0.254690i	−0.991858	−	0.127345i	$-0.959354\pi$
$317$		−	4.53462i		−	0.254690i	0.991858	−	0.127345i	$-0.0406455\pi$
$318$		0			0
$319$	−2.37742			−0.133110
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	40.8397i		−	2.27238i
$324$		0			0
$325$			3.13020i			0.173632i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	27.6510	−	6.98831i	1.52445	−	0.385278i
$330$		0			0
$331$	3.94148			0.216644			0.108322	−	0.994116i	$-0.465452\pi$
$331$	3.94148			0.216644			0.108322	+	0.994116i	$0.465452\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.79960			−0.535409
$336$		0			0
$337$	25.8198			1.40649			0.703247	−	0.710946i	$-0.251733\pi$
$337$	25.8198			1.40649			0.703247	+	0.710946i	$0.251733\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.5190			0.732096
$342$		0			0
$343$	13.6435	−	12.5242i	0.736681	−	0.676241i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			10.7102i			0.574952i	0.957788	+	0.287476i	$0.0928162\pi$
$347$			10.7102i			0.574952i	−0.957788	+	0.287476i	$0.907184\pi$
$348$		0			0
$349$			14.5236i			0.777431i	0.921358	+	0.388716i	$0.127081\pi$
$349$			14.5236i			0.777431i	−0.921358	+	0.388716i	$0.872919\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.6176			1.52316			0.761581	−	0.648069i	$-0.224423\pi$
$353$	28.6176			1.52316			0.761581	+	0.648069i	$0.224423\pi$
$354$		0			0
$355$		−	7.37284i		−	0.391310i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			29.3771i			1.55047i	0.631675	+	0.775233i	$0.282368\pi$
$359$			29.3771i			1.55047i	−0.631675	+	0.775233i	$0.717632\pi$
$360$		0			0
$361$	−35.3587			−1.86099
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	3.40686i		−	0.178323i
$366$		0			0
$367$			18.2671i			0.953537i	0.879029	+	0.476768i	$0.158192\pi$
$367$			18.2671i			0.953537i	−0.879029	+	0.476768i	$0.841808\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.86980	+	11.3551i	0.148993	+	0.589527i
$372$		0			0
$373$	22.6694			1.17378			0.586889	−	0.809668i	$-0.300353\pi$
$373$	22.6694			1.17378			0.586889	+	0.809668i	$0.300353\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.73961			−0.295605
$378$		0			0
$379$	15.1955			0.780538			0.390269	−	0.920701i	$-0.372382\pi$
$379$	15.1955			0.780538			0.390269	+	0.920701i	$0.372382\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.22030			−0.0623546			−0.0311773	−	0.999514i	$-0.509926\pi$
$383$	−1.22030			−0.0623546			−0.0311773	+	0.999514i	$0.509926\pi$
$384$		0			0
$385$	−0.840546	−	3.32583i	−0.0428382	−	0.169500i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			13.4654i			0.682722i	0.939932	+	0.341361i	$0.110888\pi$
$389$			13.4654i			0.682722i	−0.939932	+	0.341361i	$0.889112\pi$
$390$		0			0
$391$		−	10.1568i		−	0.513654i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.01990			0.453841
$396$		0			0
$397$			12.9053i			0.647699i	0.946109	+	0.323849i	$0.104977\pi$
$397$			12.9053i			0.647699i	−0.946109	+	0.323849i	$0.895023\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			26.7677i			1.33672i	0.743839	+	0.668358i	$0.233003\pi$
$401$			26.7677i			1.33672i	−0.743839	+	0.668358i	$0.766997\pi$
$402$		0			0
$403$	32.6378			1.62581
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	13.8336i		−	0.685707i
$408$		0			0
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.4397	+	4.66033i	−0.907361	+	0.229320i
$414$		0			0
$415$	−6.26039			−0.307311
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.9283			0.631590			0.315795	−	0.948827i	$-0.397729\pi$
$419$	12.9283			0.631590			0.315795	+	0.948827i	$0.397729\pi$
$420$		0			0
$421$	−11.4424			−0.557667			−0.278833	−	0.960340i	$-0.589948\pi$
$421$	−11.4424			−0.557667			−0.278833	+	0.960340i	$0.589948\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.53921			0.268691
$426$		0			0
$427$	−3.16853	−	12.5371i	−0.153336	−	0.606711i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			19.7411i			0.950895i	0.879744	+	0.475447i	$0.157714\pi$
$431$			19.7411i			0.950895i	−0.879744	+	0.475447i	$0.842286\pi$
$432$		0			0
$433$			9.95066i			0.478198i	0.970995	+	0.239099i	$0.0768521\pi$
$433$			9.95066i			0.478198i	−0.970995	+	0.239099i	$0.923148\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−13.5190			−0.646703
$438$		0			0
$439$			2.64735i			0.126351i	0.998002	+	0.0631755i	$0.0201228\pi$
$439$			2.64735i			0.126351i	−0.998002	+	0.0631755i	$0.979877\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			26.8982i			1.27797i	0.769218	+	0.638986i	$0.220646\pi$
$443$			26.8982i			1.27797i	−0.769218	+	0.638986i	$0.779354\pi$
$444$		0			0
$445$	7.94822			0.376782
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	41.1317i		−	1.94112i	−0.240852	−	0.970562i	$-0.577427\pi$
$449$		−	41.1317i		−	1.94112i	0.240852	−	0.970562i	$-0.422573\pi$
$450$		0			0
$451$			4.05852i			0.191108i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.02926	−	8.02926i	−0.0951331	−	0.376417i
$456$		0			0
$457$	12.0398			0.563198			0.281599	−	0.959532i	$-0.409135\pi$
$457$	12.0398			0.563198			0.281599	+	0.959532i	$0.409135\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−34.8397			−1.62265			−0.811323	−	0.584598i	$-0.801252\pi$
$461$	−34.8397			−1.62265			−0.811323	+	0.584598i	$0.801252\pi$
$462$		0			0
$463$	−24.3256			−1.13051			−0.565254	−	0.824917i	$-0.691222\pi$
$463$	−24.3256			−1.13051			−0.565254	+	0.824917i	$0.691222\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.7812			0.869089			0.434545	−	0.900650i	$-0.356909\pi$
$467$	18.7812			0.869089			0.434545	+	0.900650i	$0.356909\pi$
$468$		0			0
$469$	25.1369	−	6.35293i	1.16072	−	0.293351i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4.58883i		−	0.210995i
$474$		0			0
$475$		−	7.37284i		−	0.338289i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.66119			−0.304357			−0.152179	−	0.988353i	$-0.548629\pi$
$479$	−6.66119			−0.304357			−0.152179	+	0.988353i	$0.548629\pi$
$480$		0			0
$481$		−	33.3973i		−	1.52279i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			8.09402i			0.367530i
$486$		0			0
$487$	−20.9096			−0.947505			−0.473752	−	0.880658i	$-0.657101\pi$
$487$	−20.9096			−0.947505			−0.473752	+	0.880658i	$0.657101\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			0.374990i			0.0169231i	0.999964	+	0.00846153i	$0.00269342\pi$
$491$			0.374990i			0.0169231i	−0.999964	+	0.00846153i	$0.997307\pi$
$492$		0			0
$493$			10.1568i			0.457441i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.77970	+	18.9120i	0.214399	+	0.848321i
$498$		0			0
$499$	36.1785			1.61957			0.809786	−	0.586725i	$-0.199583\pi$
$499$	36.1785			1.61957			0.809786	+	0.586725i	$0.199583\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	36.8195			1.64170			0.820850	−	0.571143i	$-0.193500\pi$
$503$	36.8195			1.64170			0.820850	+	0.571143i	$0.193500\pi$
$504$		0			0
$505$	−4.05852			−0.180602
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.7396			0.520349			0.260175	−	0.965562i	$-0.416220\pi$
$509$	11.7396			0.520349			0.260175	+	0.965562i	$0.416220\pi$
$510$		0			0
$511$	2.20862	+	8.73893i	0.0977034	+	0.386588i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			5.61548i			0.247447i
$516$		0			0
$517$			13.9766i			0.614691i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−30.9500			−1.35594			−0.677972	−	0.735088i	$-0.737141\pi$
$521$	−30.9500			−1.35594			−0.677972	+	0.735088i	$0.737141\pi$
$522$		0			0
$523$		−	20.4853i		−	0.895759i	−0.894094	−	0.447879i	$-0.852179\pi$
$523$		−	20.4853i		−	0.895759i	0.894094	−	0.447879i	$-0.147821\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	57.7560i		−	2.51589i
$528$		0			0
$529$	19.6378			0.853818
$530$		0			0
$531$		0			0
$532$		0			0
$533$			9.79812i			0.424404i
$534$		0			0
$535$			0.651655i			0.0281735i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.31217	+	7.98616i	0.185738	+	0.343988i
$540$		0			0
$541$	−14.7414			−0.633781			−0.316890	−	0.948462i	$-0.602639\pi$
$541$	−14.7414			−0.633781			−0.316890	+	0.948462i	$0.602639\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.3388			−0.657043
$546$		0			0
$547$	−6.61763			−0.282949			−0.141475	−	0.989942i	$-0.545184\pi$
$547$	−6.61763			−0.282949			−0.141475	+	0.989942i	$0.545184\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.5190			0.575930
$552$		0			0
$553$	−23.1369	+	5.84747i	−0.983883	+	0.248660i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	16.1568i		−	0.684587i	−0.939593	−	0.342294i	$-0.888796\pi$
$557$		−	16.1568i		−	0.684587i	0.939593	−	0.342294i	$-0.111204\pi$
$558$		0			0
$559$		−	11.0784i		−	0.468567i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.55939			0.402880			0.201440	−	0.979501i	$-0.435438\pi$
$563$	9.55939			0.402880			0.201440	+	0.979501i	$0.435438\pi$
$564$		0			0
$565$		−	17.0784i		−	0.718495i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	30.9883i		−	1.29910i	−0.760320	−	0.649549i	$-0.774958\pi$
$569$		−	30.9883i		−	1.29910i	0.760320	−	0.649549i	$-0.225042\pi$
$570$		0			0
$571$	−16.4623			−0.688924			−0.344462	−	0.938800i	$-0.611939\pi$
$571$	−16.4623			−0.688924			−0.344462	+	0.938800i	$0.611939\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1.83363i		−	0.0764675i
$576$		0			0
$577$		−	23.1863i		−	0.965257i	−0.875825	−	0.482629i	$-0.839682\pi$
$577$		−	23.1863i		−	0.965257i	0.875825	−	0.482629i	$-0.160318\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.0585	−	4.05852i	0.666220	−	0.168376i
$582$		0			0
$583$	−5.73961			−0.237710
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.18197			−0.296432			−0.148216	−	0.988955i	$-0.547353\pi$
$587$	−7.18197			−0.296432			−0.148216	+	0.988955i	$0.547353\pi$
$588$		0			0
$589$	−76.8748			−3.16757
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.75979			0.400787			0.200393	−	0.979716i	$-0.435778\pi$
$593$	9.75979			0.400787			0.200393	+	0.979716i	$0.435778\pi$
$594$		0			0
$595$	−14.2086	+	3.59099i	−0.582496	+	0.147216i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	14.4158i		−	0.589012i	−0.955650	−	0.294506i	$-0.904845\pi$
$599$		−	14.4158i		−	0.589012i	0.955650	−	0.294506i	$-0.0951551\pi$
$600$		0			0
$601$			17.1326i			0.698855i	0.936963	+	0.349427i	$0.113624\pi$
$601$			17.1326i			0.698855i	−0.936963	+	0.349427i	$0.886376\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.31891			−0.378867
$606$		0			0
$607$			19.1345i			0.776646i	0.921523	+	0.388323i	$0.126945\pi$
$607$			19.1345i			0.776646i	−0.921523	+	0.388323i	$0.873055\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			33.7426i			1.36508i
$612$		0			0
$613$	−33.5676			−1.35578			−0.677892	−	0.735162i	$-0.737106\pi$
$613$	−33.5676			−1.35578			−0.677892	+	0.735162i	$0.737106\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			0.476107i			0.0191673i	0.999954	+	0.00958366i	$0.00305062\pi$
$617$			0.476107i			0.0191673i	−0.999954	+	0.00958366i	$0.996949\pi$
$618$		0			0
$619$			35.9521i			1.44504i	0.691351	+	0.722519i	$0.257016\pi$
$619$			35.9521i			1.44504i	−0.691351	+	0.722519i	$0.742984\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−20.3880	+	5.15271i	−0.816827	+	0.206439i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−59.1001			−2.35647
$630$		0			0
$631$	−46.8982			−1.86699			−0.933494	−	0.358593i	$-0.883257\pi$
$631$	−46.8982			−1.86699			−0.933494	+	0.358593i	$0.883257\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.39059			−0.293286
$636$		0			0
$637$	10.4105	+	19.2803i	0.412479	+	0.763913i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			15.6893i			0.619690i	0.950787	+	0.309845i	$0.100277\pi$
$641$			15.6893i			0.619690i	−0.950787	+	0.309845i	$0.899723\pi$
$642$		0			0
$643$			39.8198i			1.57034i	0.619281	+	0.785170i	$0.287424\pi$
$643$			39.8198i			1.57034i	−0.619281	+	0.785170i	$0.712576\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.3005			0.444268			0.222134	−	0.975016i	$-0.428698\pi$
$647$	11.3005			0.444268			0.222134	+	0.975016i	$0.428698\pi$
$648$		0			0
$649$		−	9.32066i		−	0.365868i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	42.1430i		−	1.64918i	−0.565729	−	0.824592i	$-0.691405\pi$
$653$		−	42.1430i		−	1.64918i	0.565729	−	0.824592i	$-0.308595\pi$
$654$		0			0
$655$	−20.2086			−0.789616
$656$		0			0
$657$		0			0
$658$		0			0
$659$			17.5172i			0.682371i	0.939996	+	0.341186i	$0.110828\pi$
$659$			17.5172i			0.682371i	−0.939996	+	0.341186i	$0.889172\pi$
$660$		0			0
$661$			32.2307i			1.25363i	0.779169	+	0.626814i	$0.215641\pi$
$661$			32.2307i			1.25363i	−0.779169	+	0.626814i	$0.784359\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.77970	+	18.9120i	0.185349	+	0.733378i
$666$		0			0
$667$	3.36218			0.130184
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.33705			0.244639
$672$		0			0
$673$	10.9351			0.421516			0.210758	−	0.977538i	$-0.432407\pi$
$673$	10.9351			0.421516			0.210758	+	0.977538i	$0.432407\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−44.5606			−1.71260			−0.856301	−	0.516477i	$-0.827243\pi$
$677$	−44.5606			−1.71260			−0.856301	+	0.516477i	$0.827243\pi$
$678$		0			0
$679$	−5.24723	−	20.7619i	−0.201370	−	0.796770i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	7.98616i		−	0.305582i	−0.988259	−	0.152791i	$-0.951174\pi$
$683$		−	7.98616i		−	0.305582i	0.988259	−	0.152791i	$-0.0488261\pi$
$684$		0			0
$685$			20.7457i			0.792651i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−13.8566			−0.527896
$690$		0			0
$691$		−	45.1969i		−	1.71937i	−0.510823	−	0.859686i	$-0.670659\pi$
$691$		−	45.1969i		−	1.71937i	0.510823	−	0.859686i	$-0.329341\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			9.20646i			0.349221i
$696$		0			0
$697$	17.3388			0.656754
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.31284i			0.0495854i	0.999693	+	0.0247927i	$0.00789257\pi$
$701$			1.31284i			0.0495854i	−0.999693	+	0.0247927i	$0.992107\pi$
$702$		0			0
$703$			78.6638i			2.96686i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10.4105	−	2.63107i	0.391527	−	0.0989517i
$708$		0			0
$709$	−13.0784			−0.491170			−0.245585	−	0.969375i	$-0.578980\pi$
$709$	−13.0784			−0.491170			−0.245585	+	0.969375i	$0.578980\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−19.1188			−0.716004
$714$		0			0
$715$	4.05852			0.151780
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−34.1568			−1.27384			−0.636918	−	0.770932i	$-0.719791\pi$
$719$	−34.1568			−1.27384			−0.636918	+	0.770932i	$0.719791\pi$
$720$		0			0
$721$	−3.64043	−	14.4042i	−0.135577	−	0.536442i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			1.83363i			0.0680992i
$726$		0			0
$727$		−	16.0965i		−	0.596984i	−0.954412	−	0.298492i	$-0.903516\pi$
$727$		−	16.0965i		−	0.596984i	0.954412	−	0.298492i	$-0.0964837\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−19.6044			−0.725096
$732$		0			0
$733$			40.2484i			1.48661i	0.668953	+	0.743305i	$0.266743\pi$
$733$			40.2484i			1.48661i	−0.668953	+	0.743305i	$0.733257\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.7059i			0.468027i
$738$		0			0
$739$	13.3838			0.492333			0.246166	−	0.969228i	$-0.420829\pi$
$739$	13.3838			0.492333			0.246166	+	0.969228i	$0.420829\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4.18803i			0.153644i	0.997045	+	0.0768221i	$0.0244773\pi$
$743$			4.18803i			0.153644i	−0.997045	+	0.0768221i	$0.975523\pi$
$744$		0			0
$745$			10.3189i			0.378056i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.422458	−	1.67156i	−0.0154363	−	0.0610774i
$750$		0			0
$751$	−22.7812			−0.831297			−0.415648	−	0.909525i	$-0.636445\pi$
$751$	−22.7812			−0.831297			−0.415648	+	0.909525i	$0.636445\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.24049			−0.0451460
$756$		0			0
$757$	44.7092			1.62498			0.812492	−	0.582972i	$-0.198110\pi$
$757$	44.7092			1.62498			0.812492	+	0.582972i	$0.198110\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	43.5474			1.57859			0.789297	−	0.614012i	$-0.210445\pi$
$761$	43.5474			1.57859			0.789297	+	0.614012i	$0.210445\pi$
$762$		0			0
$763$	39.3456	−	9.94392i	1.42440	−	0.359994i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	22.5021i		−	0.812503i
$768$		0			0
$769$		−	37.2803i		−	1.34436i	−0.740387	−	0.672181i	$-0.765358\pi$
$769$		−	37.2803i		−	1.34436i	0.740387	−	0.672181i	$-0.234642\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−39.2353			−1.41119			−0.705597	−	0.708613i	$-0.749321\pi$
$773$	−39.2353			−1.41119			−0.705597	+	0.708613i	$0.749321\pi$
$774$		0			0
$775$		−	10.4268i		−	0.374540i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	23.0784i		−	0.826870i
$780$		0			0
$781$	−9.55939			−0.342062
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	11.4629i		−	0.409130i
$786$		0			0
$787$			1.15078i			0.0410208i	0.999790	+	0.0205104i	$0.00652911\pi$
$787$			1.15078i			0.0410208i	−0.999790	+	0.0205104i	$0.993471\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.0717	+	43.8078i	0.393664	+	1.55763i
$792$		0			0
$793$	15.2990			0.543284
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16.3189			0.578045			0.289023	−	0.957322i	$-0.406670\pi$
$797$	16.3189			0.578045			0.289023	+	0.957322i	$0.406670\pi$
$798$		0			0
$799$	59.7110			2.11242
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.41723			−0.155881
$804$		0			0
$805$	1.18871	+	4.70343i	0.0418966	+	0.165774i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			13.4970i			0.474528i	0.971445	+	0.237264i	$0.0762507\pi$
$809$			13.4970i			0.474528i	−0.971445	+	0.237264i	$0.923749\pi$
$810$		0			0
$811$			31.2866i			1.09862i	0.835618	+	0.549311i	$0.185110\pi$
$811$			31.2866i			1.09862i	−0.835618	+	0.549311i	$0.814890\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.27882			0.0447951
$816$		0			0
$817$			26.0940i			0.912914i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			53.8059i			1.87784i	0.344135	+	0.938920i	$0.388172\pi$
$821$			53.8059i			1.87784i	−0.344135	+	0.938920i	$0.611828\pi$
$822$		0			0
$823$	26.5860			0.926731			0.463366	−	0.886167i	$-0.346642\pi$
$823$	26.5860			0.926731			0.463366	+	0.886167i	$0.346642\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	23.7517i		−	0.825929i	−0.910747	−	0.412964i	$-0.864493\pi$
$827$		−	23.7517i		−	0.825929i	0.910747	−	0.412964i	$-0.135507\pi$
$828$		0			0
$829$			30.3109i			1.05274i	0.850256	+	0.526370i	$0.176447\pi$
$829$			30.3109i			1.05274i	−0.850256	+	0.526370i	$0.823553\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34.1185	−	18.4225i	1.18214	−	0.638300i
$834$		0			0
$835$	−6.72118			−0.232596
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−37.1182			−1.28146			−0.640732	−	0.767765i	$-0.721369\pi$
$839$	−37.1182			−1.28146			−0.640732	+	0.767765i	$0.721369\pi$
$840$		0			0
$841$	25.6378			0.884063
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.20188			−0.110148
$846$		0			0
$847$	23.9039	−	6.04131i	0.821348	−	0.207582i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			19.5637i			0.670635i
$852$		0			0
$853$			52.7692i			1.80678i	0.428816	+	0.903392i	$0.358931\pi$
$853$			52.7692i			1.80678i	−0.428816	+	0.903392i	$0.641069\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−27.6961			−0.946079			−0.473040	−	0.881041i	$-0.656843\pi$
$857$	−27.6961			−0.946079			−0.473040	+	0.881041i	$0.656843\pi$
$858$		0			0
$859$		−	8.46996i		−	0.288991i	−0.989505	−	0.144496i	$-0.953844\pi$
$859$		−	8.46996i		−	0.288991i	0.989505	−	0.144496i	$-0.0461560\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	30.2374i		−	1.02929i	−0.857403	−	0.514646i	$-0.827923\pi$
$863$		−	30.2374i		−	1.02929i	0.857403	−	0.514646i	$-0.172077\pi$
$864$		0			0
$865$	16.3189			0.554860
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	11.6949i		−	0.396723i
$870$		0			0
$871$			30.6747i			1.03937i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.56510	+	0.648285i	−0.0867161	+	0.0219160i
$876$		0			0
$877$	8.30546			0.280456			0.140228	−	0.990119i	$-0.455217\pi$
$877$	8.30546			0.280456			0.140228	+	0.990119i	$0.455217\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.98803			0.336505			0.168253	−	0.985744i	$-0.446188\pi$
$881$	9.98803			0.336505			0.168253	+	0.985744i	$0.446188\pi$
$882$		0			0
$883$	6.08337			0.204722			0.102361	−	0.994747i	$-0.467360\pi$
$883$	6.08337			0.204722			0.102361	+	0.994747i	$0.467360\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−37.5158			−1.25966			−0.629829	−	0.776734i	$-0.716875\pi$
$887$	−37.5158			−1.25966			−0.629829	+	0.776734i	$0.716875\pi$
$888$		0			0
$889$	18.9576	−	4.79120i	0.635817	−	0.160692i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	79.4769i		−	2.65959i
$894$		0			0
$895$			19.1887i			0.641408i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	19.1188			0.637647
$900$		0			0
$901$			24.5208i			0.816906i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	10.7797i		−	0.358329i
$906$		0			0
$907$	−35.5796			−1.18140			−0.590700	−	0.806891i	$-0.701148\pi$
$907$	−35.5796			−1.18140			−0.590700	+	0.806891i	$0.701148\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			45.3495i			1.50249i	0.660021	+	0.751247i	$0.270547\pi$
$911$			45.3495i			1.50249i	−0.660021	+	0.751247i	$0.729453\pi$
$912$		0			0
$913$			8.11703i			0.268635i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	51.8371	−	13.1009i	1.71181	−	0.432631i
$918$		0			0
$919$	0.741366			0.0244554			0.0122277	−	0.999925i	$-0.496108\pi$
$919$	0.741366			0.0244554			0.0122277	+	0.999925i	$0.496108\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−23.0784			−0.759635
$924$		0			0
$925$	−10.6694			−0.350808
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.49238			−0.213008			−0.106504	−	0.994312i	$-0.533966\pi$
$929$	−6.49238			−0.213008			−0.106504	+	0.994312i	$0.533966\pi$
$930$		0			0
$931$	−24.5208	−	45.4126i	−0.803636	−	1.48834i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	7.18197i		−	0.234876i
$936$		0			0
$937$			11.4562i			0.374258i	0.982335	+	0.187129i	$0.0599182\pi$
$937$			11.4562i			0.374258i	−0.982335	+	0.187129i	$0.940082\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−46.3189			−1.50995			−0.754977	−	0.655752i	$-0.772352\pi$
$941$	−46.3189			−1.50995			−0.754977	+	0.655752i	$0.772352\pi$
$942$		0			0
$943$		−	5.73961i		−	0.186907i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	45.2569i		−	1.47065i	−0.677713	−	0.735326i	$-0.737029\pi$
$947$		−	45.2569i		−	1.47065i	0.677713	−	0.735326i	$-0.262971\pi$
$948$		0			0
$949$	−10.6641			−0.346173
$950$		0			0
$951$		0			0
$952$		0			0
$953$			57.9723i			1.87791i	0.344043	+	0.938954i	$0.388203\pi$
$953$			57.9723i			1.87791i	−0.344043	+	0.938954i	$0.611797\pi$
$954$		0			0
$955$			19.3728i			0.626890i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13.4491	−	53.2147i	−0.434294	−	1.71839i
$960$		0			0
$961$	−77.7174			−2.50701
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.81803			0.219480
$966$		0			0
$967$	17.8716			0.574711			0.287355	−	0.957824i	$-0.407224\pi$
$967$	17.8716			0.574711			0.287355	+	0.957824i	$0.407224\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−26.9898			−0.866144			−0.433072	−	0.901359i	$-0.642570\pi$
$971$	−26.9898			−0.866144			−0.433072	+	0.901359i	$0.642570\pi$
$972$		0			0
$973$	−5.96841	−	23.6155i	−0.191338	−	0.757077i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			14.4853i			0.463425i	0.972784	+	0.231713i	$0.0744329\pi$
$977$			14.4853i			0.463425i	−0.972784	+	0.231713i	$0.925567\pi$
$978$		0			0
$979$		−	10.3054i		−	0.329363i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.6241			−0.562120			−0.281060	−	0.959690i	$-0.590686\pi$
$983$	−17.6241			−0.562120			−0.281060	+	0.959690i	$0.590686\pi$
$984$		0			0
$985$			14.7457i			0.469836i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			6.48959i			0.206357i
$990$		0			0
$991$	19.1551			0.608481			0.304241	−	0.952595i	$-0.401597\pi$
$991$	19.1551			0.608481			0.304241	+	0.952595i	$0.401597\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			1.94148i			0.0615492i
$996$		0			0
$997$			36.8862i			1.16820i	0.811682	+	0.584099i	$0.198552\pi$
$997$			36.8862i			1.16820i	−0.811682	+	0.584099i	$0.801448\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.f.f.881.7		8
3.2	odd	2		5040.2.f.i.881.7		8
4.3	odd	2		630.2.b.a.251.1	✓	8
7.6	odd	2		5040.2.f.i.881.8		8
12.11	even	2		630.2.b.b.251.5	yes	8
20.3	even	4		3150.2.d.c.3149.6		8
20.7	even	4		3150.2.d.d.3149.3		8
20.19	odd	2		3150.2.b.e.251.8		8
21.20	even	2	inner	5040.2.f.f.881.8		8
28.27	even	2		630.2.b.b.251.1	yes	8
60.23	odd	4		3150.2.d.f.3149.6		8
60.47	odd	4		3150.2.d.a.3149.3		8
60.59	even	2		3150.2.b.f.251.4		8
84.83	odd	2		630.2.b.a.251.5	yes	8
140.27	odd	4		3150.2.d.f.3149.5		8
140.83	odd	4		3150.2.d.a.3149.4		8
140.139	even	2		3150.2.b.f.251.8		8
420.83	even	4		3150.2.d.d.3149.4		8
420.167	even	4		3150.2.d.c.3149.5		8
420.419	odd	2		3150.2.b.e.251.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.1	✓	8	4.3	odd	2
630.2.b.a.251.5	yes	8	84.83	odd	2
630.2.b.b.251.1	yes	8	28.27	even	2
630.2.b.b.251.5	yes	8	12.11	even	2
3150.2.b.e.251.4		8	420.419	odd	2
3150.2.b.e.251.8		8	20.19	odd	2
3150.2.b.f.251.4		8	60.59	even	2
3150.2.b.f.251.8		8	140.139	even	2
3150.2.d.a.3149.3		8	60.47	odd	4
3150.2.d.a.3149.4		8	140.83	odd	4
3150.2.d.c.3149.5		8	420.167	even	4
3150.2.d.c.3149.6		8	20.3	even	4
3150.2.d.d.3149.3		8	20.7	even	4
3150.2.d.d.3149.4		8	420.83	even	4
3150.2.d.f.3149.5		8	140.27	odd	4
3150.2.d.f.3149.6		8	60.23	odd	4
5040.2.f.f.881.7		8	1.1	even	1	trivial
5040.2.f.f.881.8		8	21.20	even	2	inner
5040.2.f.i.881.7		8	3.2	odd	2
5040.2.f.i.881.8		8	7.6	odd	2

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:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +(2.56510 - 0.648285i) q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +(2.56510 - 0.648285i) q^{7} +1.29657i q^{11} +3.13020i q^{13} +5.53921 q^{17} -7.37284i q^{19} -1.83363i q^{23} +1.00000 q^{25} +1.83363i q^{29} -10.4268i q^{31} +(-2.56510 + 0.648285i) q^{35} -10.6694 q^{37} +3.13020 q^{41} -3.53921 q^{43} +10.7797 q^{47} +(6.15945 - 3.32583i) q^{49} +4.42677i q^{53} -1.29657i q^{55} -7.18871 q^{59} -4.88755i q^{61} -3.13020i q^{65} +9.79960 q^{67} +7.37284i q^{71} +3.40686i q^{73} +(0.840546 + 3.32583i) q^{77} -9.01990 q^{79} +6.26039 q^{83} -5.53921 q^{85} -7.94822 q^{89} +(2.02926 + 8.02926i) q^{91} +7.37284i q^{95} -8.09402i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 4 q^{7}+O(q^{10}) \) 8 * q - 8 * q^5 + 4 * q^7	\( 8 q - 8 q^{5} + 4 q^{7} + 8 q^{25} - 4 q^{35} - 8 q^{37} - 8 q^{41} + 16 q^{43} + 40 q^{47} + 4 q^{49} - 32 q^{67} + 52 q^{77} - 8 q^{79} - 16 q^{83} - 8 q^{89} + 4 q^{91}+O(q^{100}) \) 8 * q - 8 * q^5 + 4 * q^7 + 8 * q^25 - 4 * q^35 - 8 * q^37 - 8 * q^41 + 16 * q^43 + 40 * q^47 + 4 * q^49 - 32 * q^67 + 52 * q^77 - 8 * q^79 - 16 * q^83 - 8 * q^89 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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