LMFDB - Embedded newform 224.2.x.a.139.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(27,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 1, 4]))

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

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

chi := DirichletCharacter("224.27");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	224.x (of order $8$, degree $4$, 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:	$1.78864900528$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{8}]$

Embedding invariants

Embedding label			139.2
Root			$1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	224.139
Dual form			224.2.x.a.195.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.28897 - 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(1.87083 - 1.87083i) q^{7} +(0.832353 - 2.70318i) q^{8} +(-2.12132 + 2.12132i) q^{9} +O(q^{10})$	\(q+(1.28897 - 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(1.87083 - 1.87083i) q^{7} +(0.832353 - 2.70318i) q^{8} +(-2.12132 + 2.12132i) q^{9} +(-0.639291 + 1.54338i) q^{11} +(1.32288 - 3.50000i) q^{14} +(-0.500000 - 3.96863i) q^{16} +(-1.50000 + 3.96863i) q^{18} +(0.0740100 + 2.36135i) q^{22} +(6.64575 + 6.64575i) q^{23} +(-3.53553 - 3.53553i) q^{25} +(-0.331369 - 5.28112i) q^{28} +(-9.74027 + 4.03455i) q^{29} +(-2.95367 - 4.82450i) q^{32} +(0.375737 + 5.98822i) q^{36} +(-3.67117 + 8.86298i) q^{37} +(0.113469 - 0.273939i) q^{43} +(1.46937 + 3.00064i) q^{44} +(12.4331 + 4.69926i) q^{46} -7.00000i q^{49} +(-6.61438 - 2.50000i) q^{50} +(4.79388 + 1.98569i) q^{53} +(-3.50000 - 6.61438i) q^{56} +(-10.2073 + 10.8679i) q^{58} +7.93725i q^{63} +(-6.61438 - 4.50000i) q^{64} +(-5.02670 - 12.1355i) q^{67} +(11.3137 - 11.3137i) q^{71} +(3.96863 + 7.50000i) q^{72} +(0.425007 + 13.5602i) q^{74} +(1.69140 + 4.08341i) q^{77} +5.56812 q^{79} -9.00000i q^{81} +(-0.0131362 - 0.419122i) q^{86} +(3.63993 + 3.01276i) q^{88} +(18.7601 - 1.17712i) q^{92} +(-4.07303 - 9.02277i) q^{98} +(-1.91787 - 4.63015i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 4 q^{16} - 12 q^{18} - 36 q^{22} + 32 q^{23} - 48 q^{43} + 52 q^{44} + 40 q^{53} - 28 q^{56} + 16 q^{67} + 44 q^{74} - 56 q^{77} + 76 q^{92}+O(q^{100}) $ 8 * q - 4 * q^16 - 12 * q^18 - 36 * q^22 + 32 * q^23 - 48 * q^43 + 52 * q^44 + 40 * q^53 - 28 * q^56 + 16 * q^67 + 44 * q^74 - 56 * q^77 + 76 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{8}\right)$

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 $

$2$	1.28897	−	0.581861i	0.911438	−	0.411438i
$2$	1.28897	−	0.581861i	0.911438	−	0.411438i
$3$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$3$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$4$	1.32288	−	1.50000i	0.661438	−	0.750000i
$5$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$5$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$6$		0			0
$7$	1.87083	−	1.87083i	0.707107	−	0.707107i
$7$	1.87083	−	1.87083i	0.707107	−	0.707107i
$8$	0.832353	−	2.70318i	0.294281	−	0.955719i
$9$	−2.12132	+	2.12132i	−0.707107	+	0.707107i
$10$		0			0
$11$	−0.639291	+	1.54338i	−0.192753	+	0.465348i	−0.990478	−	0.137675i	$-0.956037\pi$
$11$	−0.639291	+	1.54338i	−0.192753	+	0.465348i	0.797724	+	0.603023i	$0.206037\pi$
$12$		0			0
$13$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$13$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$14$	1.32288	−	3.50000i	0.353553	−	0.935414i
$15$		0			0
$16$	−0.500000	−	3.96863i	−0.125000	−	0.992157i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−1.50000	+	3.96863i	−0.353553	+	0.935414i
$19$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$19$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$20$		0			0
$21$		0			0
$22$	0.0740100	+	2.36135i	0.0157790	+	0.503442i
$23$	6.64575	+	6.64575i	1.38573	+	1.38573i	0.834058	+	0.551677i	$0.186012\pi$
$23$	6.64575	+	6.64575i	1.38573	+	1.38573i	0.551677	+	0.834058i	$0.313988\pi$
$24$		0			0
$25$	−3.53553	−	3.53553i	−0.707107	−	0.707107i
$26$		0			0
$27$		0			0
$28$	−0.331369	−	5.28112i	−0.0626229	−	0.998037i
$29$	−9.74027	+	4.03455i	−1.80872	+	0.749197i	−0.826115	+	0.563502i	$0.809454\pi$
$29$	−9.74027	+	4.03455i	−1.80872	+	0.749197i	−0.982607	+	0.185695i	$0.940546\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−2.95367	−	4.82450i	−0.522141	−	0.852859i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.375737	+	5.98822i	0.0626229	+	0.998037i
$37$	−3.67117	+	8.86298i	−0.603536	+	1.45706i	0.266382	+	0.963868i	$0.414172\pi$
$37$	−3.67117	+	8.86298i	−0.603536	+	1.45706i	−0.869918	+	0.493197i	$0.835828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$	0.113469	−	0.273939i	0.0173039	−	0.0417754i	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	0.113469	−	0.273939i	0.0173039	−	0.0417754i	0.932295	+	0.361698i	$0.117803\pi$
$44$	1.46937	+	3.00064i	0.221517	+	0.452364i
$45$		0			0
$46$	12.4331	+	4.69926i	1.83316	+	0.692867i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$	−6.61438	−	2.50000i	−0.935414	−	0.353553i
$51$		0			0
$52$		0			0
$53$	4.79388	+	1.98569i	0.658490	+	0.272755i	0.686803	−	0.726844i	$-0.259014\pi$
$53$	4.79388	+	1.98569i	0.658490	+	0.272755i	−0.0283132	+	0.999599i	$0.509014\pi$
$54$		0			0
$55$		0			0
$56$	−3.50000	−	6.61438i	−0.467707	−	0.883883i
$57$		0			0
$58$	−10.2073	+	10.8679i	−1.34029	+	1.42702i
$59$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$59$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$60$		0			0
$61$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$61$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$62$		0			0
$63$			7.93725i			1.00000i
$64$	−6.61438	−	4.50000i	−0.826797	−	0.562500i
$65$		0			0
$66$		0			0
$67$	−5.02670	−	12.1355i	−0.614109	−	1.48259i	−0.858448	−	0.512901i	$-0.828571\pi$
$67$	−5.02670	−	12.1355i	−0.614109	−	1.48259i	0.244339	−	0.969690i	$-0.421429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.3137	−	11.3137i	1.34269	−	1.34269i	0.449319	−	0.893372i	$-0.351667\pi$
$71$	11.3137	−	11.3137i	1.34269	−	1.34269i	0.893372	−	0.449319i	$-0.148333\pi$
$72$	3.96863	+	7.50000i	0.467707	+	0.883883i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$	0.425007	+	13.5602i	0.0494060	+	1.57634i
$75$		0			0
$76$		0			0
$77$	1.69140	+	4.08341i	0.192753	+	0.465348i
$78$		0			0
$79$	5.56812			0.626462			0.313231	−	0.949677i	$-0.398589\pi$
$79$	5.56812			0.626462			0.313231	+	0.949677i	$0.398589\pi$
$80$		0			0
$81$		−	9.00000i		−	1.00000i
$82$		0			0
$83$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$83$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$84$		0			0
$85$		0			0
$86$	−0.0131362	−	0.419122i	−0.00141652	−	0.0451951i
$87$		0			0
$88$	3.63993	+	3.01276i	0.388018	+	0.321161i
$89$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$89$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$90$		0			0
$91$		0			0
$92$	18.7601	−	1.17712i	1.95588	−	0.122724i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−4.07303	−	9.02277i	−0.411438	−	0.911438i
$99$	−1.91787	−	4.63015i	−0.192753	−	0.465348i
$100$	−9.98037	+	0.626229i	−0.998037	+	0.0626229i
$101$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$101$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$	7.33455	−	0.229881i	0.712394	−	0.0223280i
$107$	−4.79976	+	11.5876i	−0.464010	+	1.12022i	0.502726	+	0.864446i	$0.332330\pi$
$107$	−4.79976	+	11.5876i	−0.464010	+	1.12022i	−0.966736	+	0.255774i	$0.917670\pi$
$108$		0			0
$109$	−6.37770	−	15.3971i	−0.610873	−	1.47478i	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−6.37770	−	15.3971i	−0.610873	−	1.47478i	0.251171	−	0.967943i	$-0.419184\pi$
$110$		0			0
$111$		0			0
$112$	−8.36004	−	6.48921i	−0.789949	−	0.613172i
$113$		−	13.5524i		−	1.27490i	−0.770490	−	0.637452i	$-0.779988\pi$
$113$		−	13.5524i		−	1.27490i	0.770490	−	0.637452i	$-0.220012\pi$
$114$		0			0
$115$		0			0
$116$	−6.83334	+	19.9476i	−0.634459	+	1.85209i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.80483	+	5.80483i	0.527712	+	0.527712i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	4.61838	+	10.2309i	0.411438	+	0.911438i
$127$		−	16.0000i		−	1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$		−	16.0000i		−	1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	−11.1441	−	1.95171i	−0.985008	−	0.172508i
$129$		0			0
$130$		0			0
$131$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$131$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$132$		0			0
$133$		0			0
$134$	−13.5404	−	12.7175i	−1.16972	−	1.09862i
$135$		0			0
$136$		0			0
$137$	5.58301	−	5.58301i	0.476988	−	0.476988i	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	5.58301	−	5.58301i	0.476988	−	0.476988i	0.904167	+	0.427179i	$0.140493\pi$
$138$		0			0
$139$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$139$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$140$		0			0
$141$		0			0
$142$	8.00000	−	21.1660i	0.671345	−	1.77621i
$143$		0			0
$144$	9.47939	+	7.35807i	0.789949	+	0.613172i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	8.43797	+	17.2314i	0.693597	+	1.41641i
$149$	22.5198	+	9.32802i	1.84490	+	0.764181i	0.943741	+	0.330684i	$0.107280\pi$
$149$	22.5198	+	9.32802i	1.84490	+	0.764181i	0.901155	+	0.433497i	$0.142720\pi$
$150$		0			0
$151$	9.35425	+	9.35425i	0.761238	+	0.761238i	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	9.35425	+	9.35425i	0.761238	+	0.761238i	−0.215308	+	0.976546i	$0.569076\pi$
$152$		0			0
$153$		0			0
$154$	4.55614	+	4.27922i	0.367145	+	0.344830i
$155$		0			0
$156$		0			0
$157$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$157$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$158$	7.17712	−	3.23987i	0.570981	−	0.257750i
$159$		0			0
$160$		0			0
$161$	24.8661			1.95973
$162$	−5.23675	−	11.6007i	−0.411438	−	0.911438i
$163$	−9.39584	−	22.6836i	−0.735939	−	1.77671i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	−9.39584	−	22.6836i	−0.735939	−	1.77671i	−0.114245	−	0.993453i	$-0.536445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$	−9.19239	+	9.19239i	−0.707107	+	0.707107i
$170$		0			0
$171$		0			0
$172$	−0.260803	−	0.532592i	−0.0198860	−	0.0406097i
$173$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$173$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$174$		0			0
$175$	−13.2288			−1.00000
$176$	6.44476	+	1.76541i	0.485792	+	0.133073i
$177$		0			0
$178$		0			0
$179$	−23.9971	+	9.93993i	−1.79363	+	0.742945i	−0.804865	+	0.593458i	$0.797762\pi$
$179$	−23.9971	+	9.93993i	−1.79363	+	0.742945i	−0.988764	+	0.149487i	$0.952238\pi$
$180$		0			0
$181$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$181$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$182$		0			0
$183$		0			0
$184$	23.4963	−	12.4331i	1.73217	−	0.916578i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			24.3651i			1.76300i	0.472184	+	0.881500i	$0.343466\pi$
$191$			24.3651i			1.76300i	−0.472184	+	0.881500i	$0.656534\pi$
$192$		0			0
$193$	−2.23871			−0.161146			−0.0805728	−	0.996749i	$-0.525675\pi$
$193$	−2.23871			−0.161146			−0.0805728	+	0.996749i	$0.525675\pi$
$194$		0			0
$195$		0			0
$196$	−10.5000	−	9.26013i	−0.750000	−	0.661438i
$197$	−7.64254	+	18.4507i	−0.544509	+	1.31456i	0.377004	+	0.926212i	$0.376954\pi$
$197$	−7.64254	+	18.4507i	−0.544509	+	1.31456i	−0.921513	+	0.388348i	$0.873046\pi$
$198$	−5.16618	−	4.85218i	−0.367145	−	0.344830i
$199$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$199$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$200$	−12.5000	+	6.61438i	−0.883883	+	0.467707i
$201$		0			0
$202$		0			0
$203$	−10.6744	+	25.7703i	−0.749197	+	1.80872i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−28.1955			−1.95973
$208$		0			0
$209$		0			0
$210$		0			0
$211$	19.5968	−	8.11725i	1.34910	−	0.558815i	0.413057	−	0.910705i	$-0.364461\pi$
$211$	19.5968	−	8.11725i	1.34910	−	0.558815i	0.936041	+	0.351890i	$0.114461\pi$
$212$	9.32024	−	4.56400i	0.640116	−	0.313457i
$213$		0			0
$214$	0.555663	+	17.7289i	0.0379843	+	1.21192i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−17.1796	−	16.1355i	−1.16355	−	1.09283i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−14.5516	−	3.50000i	−0.972272	−	0.233854i
$225$	15.0000			1.00000
$226$	−7.88562	−	17.4686i	−0.524544	−	1.16200i
$227$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$227$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$228$		0			0
$229$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$229$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$230$		0			0
$231$		0			0
$232$	2.79879	+	29.6879i	0.183749	+	1.94910i
$233$	−21.5830	+	21.5830i	−1.41395	+	1.41395i	−0.693316	+	0.720634i	$0.743851\pi$
$233$	−21.5830	+	21.5830i	−1.41395	+	1.41395i	−0.720634	+	0.693316i	$0.756149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	10.8598	+	4.10464i	0.698097	+	0.263856i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$251$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$252$	11.9059	+	10.5000i	0.750000	+	0.661438i
$253$	−14.5055	+	6.00838i	−0.911954	+	0.377744i
$254$	−9.30978	−	20.6235i	−0.584147	−	1.29403i
$255$		0			0
$256$	−15.5000	+	3.96863i	−0.968750	+	0.248039i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	9.71299	+	23.4492i	0.603536	+	1.45706i
$260$		0			0
$261$	12.1037	−	29.2208i	0.749197	−	1.80872i
$262$		0			0
$263$	−3.74166	+	3.74166i	−0.230720	+	0.230720i	−0.812993	−	0.582273i	$-0.802164\pi$
$263$	−3.74166	+	3.74166i	−0.230720	+	0.230720i	0.582273	+	0.812993i	$0.302164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−24.8530	−	8.51374i	−1.51814	−	0.520060i
$269$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$269$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	3.94778	−	10.4448i	0.238494	−	0.630996i
$275$	7.71692	−	3.19645i	0.465348	−	0.192753i
$276$		0			0
$277$	23.5639	+	9.76051i	1.41582	+	0.586452i	0.953807	−	0.300421i	$-0.0971271\pi$
$277$	23.5639	+	9.76051i	1.41582	+	0.586452i	0.462014	+	0.886873i	$0.347127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.41699	+	2.41699i	0.144186	+	0.144186i	0.775515	−	0.631329i	$-0.217490\pi$
$281$	2.41699	+	2.41699i	0.144186	+	0.144186i	−0.631329	+	0.775515i	$0.717490\pi$
$282$		0			0
$283$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$283$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$284$	−2.00393	−	31.9372i	−0.118912	−	1.89512i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	16.5000	+	3.96863i	0.972272	+	0.233854i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$293$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$294$		0			0
$295$		0			0
$296$	20.9025	+	17.3009i	1.21493	+	1.00560i
$297$		0			0
$298$	34.4549	−	1.07989i	1.99592	−	0.0625566i
$299$		0			0
$300$		0			0
$301$	−0.300212	−	0.724775i	−0.0173039	−	0.0417754i
$302$	17.5002	+	6.61445i	1.00702	+	0.380619i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$307$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$308$	8.36264	+	2.86474i	0.476505	+	0.163234i
$309$		0			0
$310$		0			0
$311$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$311$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$	7.36593	−	8.35218i	0.414366	−	0.469847i
$317$	21.0540	−	8.72084i	1.18251	−	0.489811i	0.297200	−	0.954815i	$-0.403947\pi$
$317$	21.0540	−	8.72084i	1.18251	−	0.489811i	0.885309	+	0.465004i	$0.153947\pi$
$318$		0			0
$319$		−	17.6122i		−	0.986096i
$320$		0			0
$321$		0			0
$322$	32.0516	−	14.4686i	1.78617	−	0.806305i
$323$		0			0
$324$	−13.5000	−	11.9059i	−0.750000	−	0.661438i
$325$		0			0
$326$	−25.3096	−	23.7713i	−1.40177	−	1.31657i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	11.9530	−	28.8571i	0.656996	−	1.58613i	−0.145424	−	0.989369i	$-0.546455\pi$
$331$	11.9530	−	28.8571i	0.656996	−	1.58613i	0.802420	−	0.596760i	$-0.203545\pi$
$332$		0			0
$333$	−11.0135	−	26.5889i	−0.603536	−	1.45706i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			36.1798i			1.97084i	0.170136	+	0.985421i	$0.445579\pi$
$337$			36.1798i			1.97084i	−0.170136	+	0.985421i	$0.554421\pi$
$338$	−6.50000	+	17.1974i	−0.353553	+	0.935414i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−13.0958	−	13.0958i	−0.707107	−	0.707107i
$344$	−0.646061	−	0.534742i	−0.0348333	−	0.0288314i
$345$		0			0
$346$		0			0
$347$	16.5100	+	6.83867i	0.886304	+	0.367119i	0.778938	−	0.627100i	$-0.215758\pi$
$347$	16.5100	+	6.83867i	0.886304	+	0.367119i	0.107366	+	0.994220i	$0.465758\pi$
$348$		0			0
$349$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$349$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$350$	−17.0514	+	7.69730i	−0.911438	+	0.411438i
$351$		0			0
$352$	9.33432	−	1.47439i	0.497521	−	0.0785855i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−25.1478	+	26.7752i	−1.32911	+	1.41512i
$359$	14.5203	−	14.5203i	0.766350	−	0.766350i	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	14.5203	−	14.5203i	0.766350	−	0.766350i	0.977462	+	0.211112i	$0.0677085\pi$
$360$		0			0
$361$	13.4350	−	13.4350i	0.707107	−	0.707107i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	23.0516	−	29.6974i	1.20165	−	1.54808i
$369$		0			0
$370$		0			0
$371$	12.6834	−	5.25364i	0.658490	−	0.272755i
$372$		0			0
$373$	−34.8777	−	14.4468i	−1.80590	−	0.748027i	−0.983946	−	0.178468i	$-0.942886\pi$
$373$	−34.8777	−	14.4468i	−1.80590	−	0.748027i	−0.821951	−	0.569558i	$-0.807114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	35.8587	+	14.8532i	1.84194	+	0.762956i	0.951322	+	0.308199i	$0.0997264\pi$
$379$	35.8587	+	14.8532i	1.84194	+	0.762956i	0.890616	+	0.454756i	$0.150274\pi$
$380$		0			0
$381$		0			0
$382$	14.1771	+	31.4059i	0.725365	+	1.60686i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	−2.88562	+	1.30262i	−0.146874	+	0.0663014i
$387$	0.340408	+	0.821818i	0.0173039	+	0.0417754i
$388$		0			0
$389$	14.9849	−	36.1767i	0.759763	−	1.83423i	0.268290	−	0.963338i	$-0.413542\pi$
$389$	14.9849	−	36.1767i	0.759763	−	1.83423i	0.491473	−	0.870893i	$-0.336458\pi$
$390$		0			0
$391$		0			0
$392$	−18.9223	−	5.82647i	−0.955719	−	0.294281i
$393$		0			0
$394$	0.884769	+	28.2293i	0.0445740	+	1.42217i
$395$		0			0
$396$	−9.48234	−	3.24831i	−0.476505	−	0.163234i
$397$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$397$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$398$		0			0
$399$		0			0
$400$	−12.2634	+	15.7990i	−0.613172	+	0.789949i
$401$		−	9.07500i		−	0.453184i	−0.973990	−	0.226592i	$-0.927242\pi$
$401$		−	9.07500i		−	0.453184i	0.973990	−	0.226592i	$-0.0727584\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	1.23577	+	39.4281i	0.0613300	+	1.95679i
$407$	−11.3320	−	11.3320i	−0.561709	−	0.561709i
$408$		0			0
$409$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$409$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−36.3431	+	16.4059i	−1.78617	+	0.806305i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$419$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$420$		0			0
$421$	−7.41736	+	17.9071i	−0.361500	+	0.872738i	0.633581	+	0.773676i	$0.281584\pi$
$421$	−7.41736	+	17.9071i	−0.361500	+	0.872738i	−0.995081	+	0.0990621i	$0.968416\pi$
$422$	20.5365	−	21.8655i	0.999702	−	1.06440i
$423$		0			0
$424$	9.35787	−	11.3059i	0.454458	−	0.549064i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	11.0320	+	22.5287i	0.533251	+	1.08896i
$429$		0			0
$430$		0			0
$431$	26.4575			1.27441			0.637207	−	0.770693i	$-0.280090\pi$
$431$	26.4575			1.27441			0.637207	+	0.770693i	$0.280090\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−31.5326	−	10.8019i	−1.51014	−	0.517319i
$437$		0			0
$438$		0			0
$439$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$439$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$440$		0			0
$441$	14.8492	+	14.8492i	0.707107	+	0.707107i
$442$		0			0
$443$	−24.5450	−	10.1669i	−1.16617	−	0.483043i	−0.286244	−	0.958157i	$-0.592407\pi$
$443$	−24.5450	−	10.1669i	−1.16617	−	0.483043i	−0.879924	+	0.475114i	$0.842407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−20.7931	+	3.95564i	−0.982382	+	0.186886i
$449$	−42.3320			−1.99777			−0.998886	−	0.0471929i	$-0.984972\pi$
$449$	−42.3320			−1.99777			−0.998886	+	0.0471929i	$0.984972\pi$
$450$	19.3345	−	8.72791i	0.911438	−	0.411438i
$451$		0			0
$452$	−20.3286	−	17.9282i	−0.956178	−	0.843270i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−29.9333	+	29.9333i	−1.40022	+	1.40022i	−0.600878	+	0.799341i	$0.705182\pi$
$457$	−29.9333	+	29.9333i	−1.40022	+	1.40022i	−0.799341	+	0.600878i	$0.794818\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$461$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$462$		0			0
$463$	17.0593			0.792813			0.396406	−	0.918075i	$-0.370257\pi$
$463$	17.0593			0.792813			0.396406	+	0.918075i	$0.370257\pi$
$464$	20.8818	+	36.6382i	0.969411	+	1.70089i
$465$		0			0
$466$	−15.2615	+	40.3781i	−0.706975	+	1.87048i
$467$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$467$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$468$		0			0
$469$	−32.1076	−	13.2994i	−1.48259	−	0.614109i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.350254	+	0.350254i	0.0161047	+	0.0161047i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−14.3816	+	5.95707i	−0.658490	+	0.272755i
$478$	−20.6235	+	9.30978i	−0.943296	+	0.425819i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	16.3863	−	1.02818i	0.744833	−	0.0467353i
$485$		0			0
$486$		0			0
$487$	−30.5203	+	30.5203i	−1.38300	+	1.38300i	−0.543772	+	0.839233i	$0.683004\pi$
$487$	−30.5203	+	30.5203i	−1.38300	+	1.38300i	−0.839233	+	0.543772i	$0.816996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.57282	−	11.0398i	0.206369	−	0.498218i	−0.786478	−	0.617619i	$-0.788097\pi$
$491$	4.57282	−	11.0398i	0.206369	−	0.498218i	0.992846	+	0.119401i	$0.0380974\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	42.3320i		−	1.89885i
$498$		0			0
$499$	35.3108	−	14.6262i	1.58073	−	0.654760i	0.592200	−	0.805791i	$-0.298259\pi$
$499$	35.3108	−	14.6262i	1.58073	−	0.654760i	0.988529	+	0.151031i	$0.0482594\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$	21.4558	+	6.60659i	0.955719	+	0.294281i
$505$		0			0
$506$	−15.2011	+	16.1848i	−0.675772	+	0.719502i
$507$		0			0
$508$	−24.0000	−	21.1660i	−1.06483	−	0.939090i
$509$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$509$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$510$		0			0
$511$		0			0
$512$	−17.6698	+	14.1343i	−0.780903	+	0.624653i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	26.1639	+	24.5737i	1.14958	+	1.07971i
$519$		0			0
$520$		0			0
$521$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$521$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$522$	−1.40123	−	44.7073i	−0.0613300	−	1.95679i
$523$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$523$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$524$		0			0
$525$		0			0
$526$	−2.64575	+	7.00000i	−0.115360	+	0.305215i
$527$		0			0
$528$		0			0
$529$			65.3320i			2.84052i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−36.9885	+	3.48704i	−1.59766	+	0.150617i
$537$		0			0
$538$		0			0
$539$	10.8037	+	4.47504i	0.465348	+	0.192753i
$540$		0			0
$541$	40.2458	−	16.6704i	1.73030	−	0.716714i	0.730887	−	0.682498i	$-0.239107\pi$
$541$	40.2458	−	16.6704i	1.73030	−	0.716714i	0.999414	−	0.0342160i	$-0.0108934\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13.2316	+	31.9439i	0.565742	+	1.36582i	0.905114	+	0.425169i	$0.139785\pi$
$547$	13.2316	+	31.9439i	0.565742	+	1.36582i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	−0.988886	−	15.7601i	−0.0422431	−	0.673239i
$549$		0			0
$550$	8.08697	−	8.61030i	0.344830	−	0.367145i
$551$		0			0
$552$		0			0
$553$	10.4170	−	10.4170i	0.442976	−	0.442976i
$554$	36.0524	−	1.12996i	1.53172	−	0.0480075i
$555$		0			0
$556$		0			0
$557$	2.99489	+	7.23030i	0.126897	+	0.306357i	0.974541	−	0.224208i	$-0.0719796\pi$
$557$	2.99489	+	7.23030i	0.126897	+	0.306357i	−0.847644	+	0.530566i	$0.821980\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	4.52178	+	1.70907i	0.190740	+	0.0720929i
$563$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$563$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−16.8375	−	16.8375i	−0.707107	−	0.707107i
$568$	−21.1660	−	40.0000i	−0.888106	−	1.67836i
$569$	−29.9333	−	29.9333i	−1.25487	−	1.25487i	−0.953512	−	0.301356i	$-0.902561\pi$
$569$	−29.9333	−	29.9333i	−1.25487	−	1.25487i	−0.301356	−	0.953512i	$-0.597439\pi$
$570$		0			0
$571$	13.4232	+	5.56009i	0.561745	+	0.232683i	0.645443	−	0.763809i	$-0.276673\pi$
$571$	13.4232	+	5.56009i	0.561745	+	0.232683i	−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	46.9926i		−	1.95973i
$576$	23.5772	−	4.48527i	0.982382	−	0.186886i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−21.9125	+	9.89164i	−0.911438	+	0.411438i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.12936	+	6.12936i	−0.253852	+	0.253852i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	37.0094	+	10.1380i	1.52108	+	0.416669i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	43.7830	−	21.4399i	1.79342	−	0.878214i
$597$		0			0
$598$		0			0
$599$	26.1916	+	26.1916i	1.07016	+	1.07016i	0.997346	+	0.0728143i	$0.0231980\pi$
$599$	26.1916	+	26.1916i	1.07016	+	1.07016i	0.0728143	+	0.997346i	$0.476802\pi$
$600$		0			0
$601$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$601$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$602$	−0.808682	−	0.759530i	−0.0329594	−	0.0309562i
$603$	36.4066	+	15.0801i	1.48259	+	0.614109i
$604$	26.4059	−	1.65687i	1.07444	−	0.0674169i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−16.7899	+	40.5345i	−0.678139	+	1.63717i	0.0892633	+	0.996008i	$0.471549\pi$
$613$	−16.7899	+	40.5345i	−0.678139	+	1.63717i	−0.767403	+	0.641165i	$0.778451\pi$
$614$		0			0
$615$		0			0
$616$	12.4460	−	1.17333i	0.501466	−	0.0472750i
$617$	29.9333	−	29.9333i	1.20507	−	1.20507i	0.232462	−	0.972605i	$-0.425322\pi$
$617$	29.9333	−	29.9333i	1.20507	−	1.20507i	0.972605	−	0.232462i	$-0.0746782\pi$
$618$		0			0
$619$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$619$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$			25.0000i			1.00000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−11.3137	−	11.3137i	−0.450392	−	0.450392i	0.445093	−	0.895484i	$-0.353171\pi$
$631$	−11.3137	−	11.3137i	−0.450392	−	0.450392i	−0.895484	+	0.445093i	$0.853171\pi$
$632$	4.63464	−	15.0516i	0.184356	−	0.598722i
$633$		0			0
$634$	22.0636	−	23.4914i	0.876257	−	0.932961i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−10.2479	−	22.7016i	−0.405717	−	0.898765i
$639$			48.0000i			1.89885i
$640$		0			0
$641$	−47.4935			−1.87588			−0.937941	−	0.346795i	$-0.887270\pi$
$641$	−47.4935			−1.87588			−0.937941	+	0.346795i	$0.887270\pi$
$642$		0			0
$643$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$643$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$644$	32.8948	−	37.2992i	1.29624	−	1.46979i
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	−24.3286	−	7.49117i	−0.955719	−	0.294281i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−46.4549	−	15.9138i	−1.81931	−	0.623231i
$653$	11.0640	+	26.7108i	0.432967	+	1.04528i	0.978326	+	0.207072i	$0.0663936\pi$
$653$	11.0640	+	26.7108i	0.432967	+	1.04528i	−0.545358	+	0.838203i	$0.683606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−46.9105	+	19.4310i	−1.82737	+	0.756923i	−0.856998	+	0.515319i	$0.827673\pi$
$659$	−46.9105	+	19.4310i	−1.82737	+	0.756923i	−0.970375	+	0.241604i	$0.922327\pi$
$660$		0			0
$661$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$661$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$662$	−1.38379	−	44.1508i	−0.0537823	−	1.71597i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−29.6671	−	27.8640i	−1.14958	−	1.07971i
$667$	−91.5440	−	37.9188i	−3.54460	−	1.46822i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	42.3320			1.63178			0.815890	−	0.578208i	$-0.196248\pi$
$673$	42.3320			1.63178			0.815890	+	0.578208i	$0.196248\pi$
$674$	21.0516	+	46.6346i	0.810879	+	1.79630i
$675$		0			0
$676$	1.62820	+	25.9490i	0.0626229	+	0.998037i
$677$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$677$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.48605	−	22.9014i	0.362974	−	0.876296i	−0.631889	−	0.775059i	$-0.717720\pi$
$683$	9.48605	−	22.9014i	0.362974	−	0.876296i	0.994862	−	0.101237i	$-0.0322800\pi$
$684$		0			0
$685$		0			0
$686$	−24.5000	−	9.26013i	−0.935414	−	0.353553i
$687$		0			0
$688$	−1.14390	−	0.313348i	−0.0436107	−	0.0119463i
$689$		0			0
$690$		0			0
$691$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$691$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$692$		0			0
$693$	−12.2502	−	5.07421i	−0.465348	−	0.192753i
$694$	25.2600	−	0.791705i	0.958858	−	0.0300527i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−17.5000	+	19.8431i	−0.661438	+	0.750000i
$701$	−17.0012	+	7.04212i	−0.642126	+	0.265977i	−0.679895	−	0.733309i	$-0.737975\pi$
$701$	−17.0012	+	7.04212i	−0.642126	+	0.265977i	0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$		0			0
$704$	11.1737	−	7.33172i	0.421126	−	0.276325i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	17.8296	−	43.0445i	0.669605	−	1.61657i	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	17.8296	−	43.0445i	0.669605	−	1.61657i	0.782272	−	0.622937i	$-0.214061\pi$
$710$		0			0
$711$	−11.8118	+	11.8118i	−0.442976	+	0.442976i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−16.8353	+	49.1450i	−0.629165	+	1.83663i
$717$		0			0
$718$	10.2674	−	27.1649i	0.383175	−	1.01379i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	9.50000	−	25.1346i	0.353553	−	0.935414i
$723$		0			0
$724$		0			0
$725$	48.7013	+	20.1728i	1.80872	+	0.749197i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	19.0919	+	19.0919i	0.707107	+	0.707107i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$733$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$734$		0			0
$735$		0			0
$736$	12.4331	−	51.6918i	0.458289	−	1.90539i
$737$	21.9433			0.808292
$738$		0			0
$739$	20.7095	+	49.9973i	0.761813	+	1.83918i	0.469836	+	0.882754i	$0.344313\pi$
$739$	20.7095	+	49.9973i	0.761813	+	1.83918i	0.291977	+	0.956425i	$0.405687\pi$
$740$		0			0
$741$		0			0
$742$	13.2916	−	14.1517i	0.487950	−	0.519527i
$743$	1.47974	−	1.47974i	0.0542864	−	0.0542864i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	1.47974	−	1.47974i	0.0542864	−	0.0542864i	0.733729	+	0.679442i	$0.237778\pi$
$744$		0			0
$745$		0			0
$746$	−53.3622	+	1.67249i	−1.95373	+	0.0612342i
$747$		0			0
$748$		0			0
$749$	12.6990	+	30.6580i	0.464010	+	1.12022i
$750$		0			0
$751$	48.0000			1.75154			0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000			1.75154			0.875772	+	0.482724i	$0.160353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−49.8335	−	20.6417i	−1.81123	−	0.750236i	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−49.8335	−	20.6417i	−1.81123	−	0.750236i	−0.829899	−	0.557913i	$-0.811602\pi$
$758$	54.8632	−	1.71953i	1.99272	−	0.0624563i
$759$		0			0
$760$		0			0
$761$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$761$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$762$		0			0
$763$	−40.7370	−	16.8738i	−1.47478	−	0.610873i
$764$	36.5477	+	32.2321i	1.32225	+	1.16611i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−2.96153	+	3.35806i	−0.106588	+	0.120859i
$773$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$773$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$774$	0.916959	+	0.861227i	0.0329594	+	0.0309562i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.73478	−	55.3497i	−0.0621949	−	1.98438i
$779$		0			0
$780$		0			0
$781$	10.2287	+	24.6942i	0.366010	+	0.883626i
$782$		0			0
$783$		0			0
$784$	−27.7804	+	3.50000i	−0.992157	+	0.125000i
$785$		0			0
$786$		0			0
$787$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$787$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$788$	17.5660	+	35.8718i	0.625761	+	1.27788i
$789$		0			0
$790$		0			0
$791$	−25.3542	−	25.3542i	−0.901493	−	0.901493i
$792$	−14.1125	+	1.33044i	−0.501466	+	0.0472750i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$797$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$798$		0			0
$799$		0			0
$800$	−6.61438	+	27.5000i	−0.233854	+	0.972272i
$801$		0			0
$802$	−5.28039	−	11.6974i	−0.186457	−	0.413049i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	29.9333	−	29.9333i	1.05240	−	1.05240i	0.0538482	−	0.998549i	$-0.482851\pi$
$809$	29.9333	−	29.9333i	1.05240	−	1.05240i	0.998549	−	0.0538482i	$-0.0171487\pi$
$810$		0			0
$811$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$811$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$812$	24.5346	+	50.1026i	0.860994	+	1.75826i
$813$		0			0
$814$	−21.2003	−	8.01296i	−0.743071	−	0.280854i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−37.3876	−	15.4865i	−1.30484	−	0.540481i	−0.381464	−	0.924384i	$-0.624580\pi$
$821$	−37.3876	−	15.4865i	−1.30484	−	0.540481i	−0.923374	+	0.383903i	$0.874580\pi$
$822$		0			0
$823$	33.6749	+	33.6749i	1.17383	+	1.17383i	0.981288	+	0.192546i	$0.0616744\pi$
$823$	33.6749	+	33.6749i	1.17383	+	1.17383i	0.192546	+	0.981288i	$0.438326\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−47.1724	−	19.5395i	−1.64035	−	0.679453i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$	−47.1724	−	19.5395i	−1.64035	−	0.679453i	−0.996333	+	0.0855616i	$0.972732\pi$
$828$	−37.2992	+	42.2933i	−1.29624	+	1.46979i
$829$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$829$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	58.0891	−	58.0891i	2.00307	−	2.00307i
$842$	0.858699	+	27.3975i	0.0295927	+	0.944181i
$843$		0			0
$844$	13.7482	−	40.1333i	0.473234	−	1.38144i
$845$		0			0
$846$		0			0
$847$	21.7197			0.746297
$848$	5.48352	−	20.0180i	0.188305	−	0.687419i
$849$		0			0
$850$		0			0
$851$	−83.2988	+	34.5035i	−2.85545	+	1.18276i
$852$		0			0
$853$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$853$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$854$		0			0
$855$		0			0
$856$	27.3284	+	22.6196i	0.934066	+	0.773123i
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$859$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$860$		0			0
$861$		0			0
$862$	34.1029	−	15.3946i	1.16155	−	0.524342i
$863$			35.5014i			1.20848i	0.796802	+	0.604240i	$0.206523\pi$
$863$			35.5014i			1.20848i	−0.796802	+	0.604240i	$0.793477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.55965	+	8.59375i	−0.120753	+	0.291523i
$870$		0			0
$871$		0			0
$872$	−46.9297	+	4.42424i	−1.58924	+	0.149824i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	22.3272	+	53.9026i	0.753936	+	1.82016i	0.536044	+	0.844190i	$0.319918\pi$
$877$	22.3272	+	53.9026i	0.753936	+	1.82016i	0.217892	+	0.975973i	$0.430082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	27.7804	+	10.5000i	0.935414	+	0.353553i
$883$	10.3365	−	4.28151i	0.347850	−	0.144084i	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	10.3365	−	4.28151i	0.347850	−	0.144084i	0.549766	+	0.835319i	$0.314717\pi$
$884$		0			0
$885$		0			0
$886$	−37.5534	+	1.17701i	−1.26163	+	0.0395423i
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$	−29.9333	−	29.9333i	−1.00393	−	1.00393i
$890$		0			0
$891$	13.8905	+	5.75362i	0.465348	+	0.192753i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−24.5000	+	17.1974i	−0.818488	+	0.574524i
$897$		0			0
$898$	−54.5646	+	24.6314i	−1.82084	+	0.821959i
$899$		0			0
$900$	19.8431	−	22.5000i	0.661438	−	0.750000i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−36.6346	−	11.2804i	−1.21845	−	0.375180i
$905$		0			0
$906$		0			0
$907$	21.9881	−	53.0840i	0.730104	−	1.76263i	0.0878507	−	0.996134i	$-0.472000\pi$
$907$	21.9881	−	53.0840i	0.730104	−	1.76263i	0.642253	−	0.766493i	$-0.278000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.0000			0.530104			0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000			0.530104			0.265052	+	0.964234i	$0.414611\pi$
$912$		0			0
$913$		0			0
$914$	−21.1660	+	56.0000i	−0.700109	+	1.85232i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−33.9411	−	33.9411i	−1.11961	−	1.11961i	−0.991798	−	0.127817i	$-0.959203\pi$
$919$	−33.9411	−	33.9411i	−1.11961	−	1.11961i	−0.127817	−	0.991798i	$-0.540797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	44.3149	−	18.3558i	1.45706	−	0.603536i
$926$	21.9889	−	9.92614i	0.722600	−	0.326193i
$927$		0			0
$928$	48.2343	+	35.0752i	1.58337	+	1.15140i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	3.82288	+	60.9261i	0.125222	+	1.99570i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$	−49.1240	+	1.53966i	−1.60396	+	0.0502716i
$939$		0			0
$940$		0			0
$941$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$941$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.655265	+	0.247667i	0.0213045	+	0.00805234i
$947$	−37.6502	+	15.5952i	−1.22347	+	0.506776i	−0.898510	−	0.438953i	$-0.855350\pi$
$947$	−37.6502	+	15.5952i	−1.22347	+	0.506776i	−0.324956	+	0.945729i	$0.605350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−18.4170	−	18.4170i	−0.596585	−	0.596585i	0.342817	−	0.939402i	$-0.388619\pi$
$953$	−18.4170	−	18.4170i	−0.596585	−	0.596585i	−0.939402	+	0.342817i	$0.888619\pi$
$954$	−15.0713	+	16.0466i	−0.487950	+	0.519527i
$955$		0			0
$956$	−21.1660	+	24.0000i	−0.684558	+	0.776215i
$957$		0			0
$958$		0			0
$959$		−	20.8897i		−	0.674563i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	−14.3993	−	34.7629i	−0.464010	−	1.12022i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−43.8118	+	43.8118i	−1.40889	+	1.40889i	−0.643157	+	0.765735i	$0.722376\pi$
$967$	−43.8118	+	43.8118i	−1.40889	+	1.40889i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	20.5232	−	10.8598i	0.659640	−	0.349049i
$969$		0			0
$970$		0			0
$971$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$971$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$972$		0			0
$973$		0			0
$974$	−21.5811	+	57.0982i	−0.691502	+	1.82954i
$975$		0			0
$976$		0			0
$977$			42.3320i			1.35432i	0.735835	+	0.677161i	$0.236790\pi$
$977$			42.3320i			1.35432i	−0.735835	+	0.677161i	$0.763210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	46.1914	+	19.1331i	1.47478	+	0.610873i
$982$	−0.529390	−	16.8907i	−0.0168935	−	0.539002i
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.57462	−	1.06644i	0.0818682	−	0.0339109i
$990$		0			0
$991$		−	58.1288i		−	1.84652i	−0.384173	−	0.923261i	$-0.625513\pi$
$991$		−	58.1288i		−	1.84652i	0.384173	−	0.923261i	$-0.374487\pi$
$992$		0			0
$993$		0			0
$994$	−24.6314	−	54.5646i	−0.781259	−	1.73068i
$995$		0			0
$996$		0			0
$997$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$997$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$998$	37.0041	−	39.3987i	1.17134	−	1.24714i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.x.a.139.2	✓	8
4.3	odd	2		896.2.x.a.335.1		8
7.6	odd	2	CM	224.2.x.a.139.2	✓	8
28.27	even	2		896.2.x.a.335.1		8
32.3	odd	8	inner	224.2.x.a.195.2	yes	8
32.29	even	8		896.2.x.a.559.1		8
224.125	odd	8		896.2.x.a.559.1		8
224.195	even	8	inner	224.2.x.a.195.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.x.a.139.2	✓	8	1.1	even	1	trivial
224.2.x.a.139.2	✓	8	7.6	odd	2	CM
224.2.x.a.195.2	yes	8	32.3	odd	8	inner
224.2.x.a.195.2	yes	8	224.195	even	8	inner
896.2.x.a.335.1		8	4.3	odd	2
896.2.x.a.335.1		8	28.27	even	2
896.2.x.a.559.1		8	32.29	even	8
896.2.x.a.559.1		8	224.125	odd	8

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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.x (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.28897 - 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(1.87083 - 1.87083i) q^{7} +(0.832353 - 2.70318i) q^{8} +(-2.12132 + 2.12132i) q^{9} +O(q^{10})\)	\(q+(1.28897 - 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(1.87083 - 1.87083i) q^{7} +(0.832353 - 2.70318i) q^{8} +(-2.12132 + 2.12132i) q^{9} +(-0.639291 + 1.54338i) q^{11} +(1.32288 - 3.50000i) q^{14} +(-0.500000 - 3.96863i) q^{16} +(-1.50000 + 3.96863i) q^{18} +(0.0740100 + 2.36135i) q^{22} +(6.64575 + 6.64575i) q^{23} +(-3.53553 - 3.53553i) q^{25} +(-0.331369 - 5.28112i) q^{28} +(-9.74027 + 4.03455i) q^{29} +(-2.95367 - 4.82450i) q^{32} +(0.375737 + 5.98822i) q^{36} +(-3.67117 + 8.86298i) q^{37} +(0.113469 - 0.273939i) q^{43} +(1.46937 + 3.00064i) q^{44} +(12.4331 + 4.69926i) q^{46} -7.00000i q^{49} +(-6.61438 - 2.50000i) q^{50} +(4.79388 + 1.98569i) q^{53} +(-3.50000 - 6.61438i) q^{56} +(-10.2073 + 10.8679i) q^{58} +7.93725i q^{63} +(-6.61438 - 4.50000i) q^{64} +(-5.02670 - 12.1355i) q^{67} +(11.3137 - 11.3137i) q^{71} +(3.96863 + 7.50000i) q^{72} +(0.425007 + 13.5602i) q^{74} +(1.69140 + 4.08341i) q^{77} +5.56812 q^{79} -9.00000i q^{81} +(-0.0131362 - 0.419122i) q^{86} +(3.63993 + 3.01276i) q^{88} +(18.7601 - 1.17712i) q^{92} +(-4.07303 - 9.02277i) q^{98} +(-1.91787 - 4.63015i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 4 q^{16} - 12 q^{18} - 36 q^{22} + 32 q^{23} - 48 q^{43} + 52 q^{44} + 40 q^{53} - 28 q^{56} + 16 q^{67} + 44 q^{74} - 56 q^{77} + 76 q^{92}+O(q^{100}) \) 8 * q - 4 * q^16 - 12 * q^18 - 36 * q^22 + 32 * q^23 - 48 * q^43 + 52 * q^44 + 40 * q^53 - 28 * q^56 + 16 * q^67 + 44 * q^74 - 56 * q^77 + 76 * q^92

Introduction

L-functions

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Varieties

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Database

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