LMFDB - Embedded newform 224.2.x.a.139.1

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			139.1
Root			$-0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	224.139
Dual form			224.2.x.a.195.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} +(-1.87083 + 1.87083i) q^{7} +(2.70318 - 0.832353i) q^{8} +(-2.12132 + 2.12132i) q^{9} +O(q^{10})$	\(q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} +(-1.87083 + 1.87083i) q^{7} +(2.70318 - 0.832353i) q^{8} +(-2.12132 + 2.12132i) q^{9} +(-2.18914 + 5.28504i) q^{11} +(-1.32288 - 3.50000i) q^{14} +(-0.500000 + 3.96863i) q^{16} +(-1.50000 - 3.96863i) q^{18} +(-5.53848 - 5.89689i) q^{22} +(1.35425 + 1.35425i) q^{23} +(-3.53553 - 3.53553i) q^{25} +(5.28112 + 0.331369i) q^{28} +(8.32605 - 3.44876i) q^{29} +(-4.82450 - 2.95367i) q^{32} +(5.98822 + 0.375737i) q^{36} +(-0.571475 + 1.37966i) q^{37} +(-3.62819 + 8.75922i) q^{43} +(10.8235 - 3.70775i) q^{44} +(-2.53357 + 0.957598i) q^{46} -7.00000i q^{49} +(6.61438 - 2.50000i) q^{50} +(12.2772 + 5.08538i) q^{53} +(-3.50000 + 6.61438i) q^{56} +(-0.399260 + 12.7387i) q^{58} -7.93725i q^{63} +(6.61438 - 4.50000i) q^{64} +(6.19827 + 14.9640i) q^{67} +(11.3137 - 11.3137i) q^{71} +(-3.96863 + 7.50000i) q^{72} +(-1.44582 - 1.53938i) q^{74} +(-5.79191 - 13.9829i) q^{77} -16.8818 q^{79} -9.00000i q^{81} +(-9.17925 - 9.77327i) q^{86} +(-1.51861 + 16.1086i) q^{88} +(0.239870 - 3.82288i) q^{92} +(9.02277 + 4.07303i) q^{98} +(-6.56741 - 15.8551i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.581861	+	1.28897i	−0.411438	+	0.911438i
$2$	−0.581861	+	1.28897i	−0.411438	+	0.911438i
$3$		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
$8$	2.70318	−	0.832353i	0.955719	−	0.294281i
$9$	−2.12132	+	2.12132i	−0.707107	+	0.707107i
$10$		0			0
$11$	−2.18914	+	5.28504i	−0.660049	+	1.59350i	0.137675	+	0.990478i	$0.456037\pi$
$11$	−2.18914	+	5.28504i	−0.660049	+	1.59350i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		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
$22$	−5.53848	−	5.89689i	−1.18081	−	1.25722i
$23$	1.35425	+	1.35425i	0.282380	+	0.282380i	0.834058	−	0.551677i	$-0.186012\pi$
$23$	1.35425	+	1.35425i	0.282380	+	0.282380i	−0.551677	+	0.834058i	$0.686012\pi$
$24$		0			0
$25$	−3.53553	−	3.53553i	−0.707107	−	0.707107i
$26$		0			0
$27$		0			0
$28$	5.28112	+	0.331369i	0.998037	+	0.0626229i
$29$	8.32605	−	3.44876i	1.54611	−	0.640419i	0.563502	−	0.826115i	$-0.309454\pi$
$29$	8.32605	−	3.44876i	1.54611	−	0.640419i	0.982607	+	0.185695i	$0.0594537\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−4.82450	−	2.95367i	−0.852859	−	0.522141i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	5.98822	+	0.375737i	0.998037	+	0.0626229i
$37$	−0.571475	+	1.37966i	−0.0939499	+	0.226815i	−0.963868	−	0.266382i	$-0.914172\pi$
$37$	−0.571475	+	1.37966i	−0.0939499	+	0.226815i	0.869918	+	0.493197i	$0.164172\pi$
$38$		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
$43$	−3.62819	+	8.75922i	−0.553293	+	1.33577i	0.361698	+	0.932295i	$0.382197\pi$
$43$	−3.62819	+	8.75922i	−0.553293	+	1.33577i	−0.914991	+	0.403473i	$0.867803\pi$
$44$	10.8235	−	3.70775i	1.63171	−	0.558964i
$45$		0			0
$46$	−2.53357	+	0.957598i	−0.373554	+	0.141190i
$47$		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
$53$	12.2772	+	5.08538i	1.68640	+	0.698531i	0.999599	−	0.0283132i	$-0.00901359\pi$
$53$	12.2772	+	5.08538i	1.68640	+	0.698531i	0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$		0			0
$56$	−3.50000	+	6.61438i	−0.467707	+	0.883883i
$57$		0			0
$58$	−0.399260	+	12.7387i	−0.0524253	+	1.67268i
$59$		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
$67$	6.19827	+	14.9640i	0.757240	+	1.82814i	0.512901	+	0.858448i	$0.328571\pi$
$67$	6.19827	+	14.9640i	0.757240	+	1.82814i	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	$0.750000\pi$
$74$	−1.44582	−	1.53938i	−0.168073	−	0.178950i
$75$		0			0
$76$		0			0
$77$	−5.79191	−	13.9829i	−0.660049	−	1.59350i
$78$		0			0
$79$	−16.8818			−1.89935			−0.949677	−	0.313231i	$-0.898589\pi$
$79$	−16.8818			−1.89935			−0.949677	+	0.313231i	$0.898589\pi$
$80$		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
$86$	−9.17925	−	9.77327i	−0.989824	−	1.05388i
$87$		0			0
$88$	−1.51861	+	16.1086i	−0.161885	+	1.71718i
$89$		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
$92$	0.239870	−	3.82288i	0.0250082	−	0.398562i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	9.02277	+	4.07303i	0.911438	+	0.411438i
$99$	−6.56741	−	15.8551i	−0.660049	−	1.59350i
$100$	−0.626229	+	9.98037i	−0.0626229	+	0.998037i
$101$		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
$106$	−13.6985	+	12.8659i	−1.33052	+	1.24965i
$107$	−1.05810	+	2.55449i	−0.102291	+	0.246952i	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−1.05810	+	2.55449i	−0.102291	+	0.246952i	0.864446	+	0.502726i	$0.167670\pi$
$108$		0			0
$109$	1.10562	+	2.66920i	0.105899	+	0.255663i	0.967943	−	0.251171i	$-0.0808157\pi$
$109$	1.10562	+	2.66920i	0.105899	+	0.255663i	−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$		0			0
$112$	−6.48921	−	8.36004i	−0.613172	−	0.789949i
$113$			16.3808i			1.54098i	0.637452	+	0.770490i	$0.279988\pi$
$113$			16.3808i			1.54098i	−0.637452	+	0.770490i	$0.720012\pi$
$114$		0			0
$115$		0			0
$116$	−16.1875	−	7.92679i	−1.50297	−	0.735984i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−15.3612	−	15.3612i	−1.39647	−	1.39647i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	10.2309	+	4.61838i	0.911438	+	0.411438i
$127$		−	16.0000i		−	1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$		−	16.0000i		−	1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	1.95171	+	11.1441i	0.172508	+	0.985008i
$129$		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
$134$	−22.8946	−	0.717567i	−1.97779	−	0.0619884i
$135$		0			0
$136$		0			0
$137$	−15.5830	+	15.5830i	−1.33135	+	1.33135i	−0.427179	+	0.904167i	$0.640493\pi$
$137$	−15.5830	+	15.5830i	−1.33135	+	1.33135i	−0.904167	+	0.427179i	$0.859507\pi$
$138$		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
$144$	−7.35807	−	9.47939i	−0.613172	−	0.789949i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	2.82548	−	0.967910i	0.232253	−	0.0795617i
$149$	15.0365	+	6.22833i	1.23184	+	0.510245i	0.901155	−	0.433497i	$-0.142720\pi$
$149$	15.0365	+	6.22833i	1.23184	+	0.510245i	0.330684	+	0.943741i	$0.392720\pi$
$150$		0			0
$151$	14.6458	+	14.6458i	1.19185	+	1.19185i	0.976546	+	0.215308i	$0.0690756\pi$
$151$	14.6458	+	14.6458i	1.19185	+	1.19185i	0.215308	+	0.976546i	$0.430924\pi$
$152$		0			0
$153$		0			0
$154$	21.3936	+	0.670523i	1.72395	+	0.0540323i
$155$		0			0
$156$		0			0
$157$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$157$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$158$	9.82288	−	21.7601i	0.781466	−	1.73114i
$159$		0			0
$160$		0			0
$161$	−5.06713			−0.399346
$162$	11.6007	+	5.23675i	0.911438	+	0.411438i
$163$	−4.74630	−	11.4586i	−0.371759	−	0.897505i	−0.993453	−	0.114245i	$-0.963555\pi$
$163$	−4.74630	−	11.4586i	−0.371759	−	0.897505i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		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
$172$	17.9385	−	6.14508i	1.36780	−	0.468558i
$173$		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
$176$	−19.8798	−	11.3304i	−1.49850	−	0.854060i
$177$		0			0
$178$		0			0
$179$	21.1687	−	8.76836i	1.58222	−	0.655378i	0.593458	−	0.804865i	$-0.297762\pi$
$179$	21.1687	−	8.76836i	1.58222	−	0.655378i	0.988764	+	0.149487i	$0.0477622\pi$
$180$		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
$184$	4.78799	+	2.53357i	0.352975	+	0.186777i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.0514i		−	0.944369i	−0.881500	−	0.472184i	$-0.843466\pi$
$191$		−	13.0514i		−	0.944369i	0.881500	−	0.472184i	$-0.156534\pi$
$192$		0			0
$193$	27.6946			1.99350			0.996749	−	0.0805728i	$-0.0256750\pi$
$193$	27.6946			1.99350			0.996749	+	0.0805728i	$0.0256750\pi$
$194$		0			0
$195$		0			0
$196$	−10.5000	+	9.26013i	−0.750000	+	0.661438i
$197$	−10.7422	+	25.9340i	−0.765352	+	1.84772i	−0.377004	+	0.926212i	$0.623046\pi$
$197$	−10.7422	+	25.9340i	−0.765352	+	1.84772i	−0.388348	+	0.921513i	$0.626954\pi$
$198$	24.2581	+	0.760301i	1.72395	+	0.0540323i
$199$		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
$203$	−9.12457	+	22.0287i	−0.640419	+	1.54611i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.74559			−0.399346
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.888497	−	0.368028i	0.0611667	−	0.0253361i	−0.351890	−	0.936041i	$-0.614461\pi$
$211$	0.888497	−	0.368028i	0.0611667	−	0.0253361i	0.413057	+	0.910705i	$0.364461\pi$
$212$	−8.61313	−	25.1431i	−0.591552	−	1.72684i
$213$		0			0
$214$	−2.67698	−	2.85022i	−0.182995	−	0.194837i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−4.08383	−	0.127996i	−0.276592	−	0.00866900i
$219$		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
$226$	−21.1144	−	9.53137i	−1.40451	−	0.634018i
$227$		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
$232$	19.6362	−	16.2528i	1.28918	−	1.06705i
$233$	−0.416995	+	0.416995i	−0.0273182	+	0.0273182i	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−0.416995	+	0.416995i	−0.0273182	+	0.0273182i	0.693316	+	0.720634i	$0.256149\pi$
$234$		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	$0.5\pi$
$242$	28.7381	−	10.8620i	1.84736	−	0.698235i
$243$		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
$253$	−10.1219	+	4.19263i	−0.636358	+	0.263588i
$254$	20.6235	+	9.30978i	1.29403	+	0.584147i
$255$		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
$259$	−1.51198	−	3.65025i	−0.0939499	−	0.226815i
$260$		0			0
$261$	−10.3463	+	24.9782i	−0.640419	+	1.54611i
$262$		0			0
$263$	3.74166	−	3.74166i	0.230720	−	0.230720i	−0.582273	−	0.812993i	$-0.697836\pi$
$263$	3.74166	−	3.74166i	0.230720	−	0.230720i	0.812993	+	0.582273i	$0.197836\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	14.2464	−	29.0929i	0.870237	−	1.77713i
$269$		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
$274$	−11.0188	−	29.1531i	−0.665673	−	1.76121i
$275$	26.4252	−	10.9457i	1.59350	−	0.660049i
$276$		0			0
$277$	−30.6350	−	12.6894i	−1.84068	−	0.762434i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	−30.6350	−	12.6894i	−1.84068	−	0.762434i	−0.886873	−	0.462014i	$-0.847127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	23.5830	+	23.5830i	1.40684	+	1.40684i	0.775515	+	0.631329i	$0.217490\pi$
$281$	23.5830	+	23.5830i	1.40684	+	1.40684i	0.631329	+	0.775515i	$0.282510\pi$
$282$		0			0
$283$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$283$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$284$	−31.9372	−	2.00393i	−1.89512	−	0.118912i
$285$		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
$296$	−0.396435	+	4.20515i	−0.0230423	+	0.244419i
$297$		0			0
$298$	−16.7773	+	15.7576i	−0.971882	+	0.912811i
$299$		0			0
$300$		0			0
$301$	−9.59928	−	23.1747i	−0.553293	−	1.33577i
$302$	−27.3997	+	10.3561i	−1.57668	+	0.595927i
$303$		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	$0.875000\pi$
$308$	−13.3124	+	27.1855i	−0.758544	+	1.54904i
$309$		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
$316$	22.3326	+	25.3227i	1.25630	+	1.42452i
$317$	2.98766	−	1.23753i	0.167803	−	0.0695065i	−0.297200	−	0.954815i	$-0.596053\pi$
$317$	2.98766	−	1.23753i	0.167803	−	0.0695065i	0.465004	+	0.885309i	$0.346053\pi$
$318$		0			0
$319$			51.5534i			2.88643i
$320$		0			0
$321$		0			0
$322$	2.94837	−	6.53137i	0.164306	−	0.363979i
$323$		0			0
$324$	−13.5000	+	11.9059i	−0.750000	+	0.661438i
$325$		0			0
$326$	17.5314	+	0.549474i	0.970976	+	0.0304325i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.5028	−	32.5988i	0.742183	−	1.79179i	0.145424	−	0.989369i	$-0.453545\pi$
$331$	13.5028	−	32.5988i	0.742183	−	1.79179i	0.596760	−	0.802420i	$-0.296455\pi$
$332$		0			0
$333$	−1.71443	−	4.13899i	−0.0939499	−	0.226815i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			6.24657i			0.340273i	0.985421	+	0.170136i	$0.0544208\pi$
$337$			6.24657i			0.340273i	−0.985421	+	0.170136i	$0.945579\pi$
$338$	−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
$344$	−2.51689	+	26.6977i	−0.135702	+	1.43944i
$345$		0			0
$346$		0			0
$347$	−9.68159	−	4.01024i	−0.519735	−	0.215281i	0.107366	−	0.994220i	$-0.465758\pi$
$347$	−9.68159	−	4.01024i	−0.519735	−	0.215281i	−0.627100	+	0.778938i	$0.715758\pi$
$348$		0			0
$349$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$349$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$350$	−7.69730	+	17.0514i	−0.411438	+	0.911438i
$351$		0			0
$352$	26.1718	−	19.0317i	1.39496	−	1.01439i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.01510	+	32.3877i	−0.0536499	+	1.71174i
$359$	−22.5203	+	22.5203i	−1.18857	+	1.18857i	−0.211112	+	0.977462i	$0.567708\pi$
$359$	−22.5203	+	22.5203i	−1.18857	+	1.18857i	−0.977462	+	0.211112i	$0.932292\pi$
$360$		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			$\pi$
$368$	−6.05163	+	4.69738i	−0.315463	+	0.244868i
$369$		0			0
$370$		0			0
$371$	−32.4824	+	13.4546i	−1.68640	+	0.698531i
$372$		0			0
$373$	19.3213	+	8.00315i	1.00042	+	0.414387i	0.821951	−	0.569558i	$-0.192886\pi$
$373$	19.3213	+	8.00315i	1.00042	+	0.414387i	0.178468	+	0.983946i	$0.442886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−27.3734	−	11.3384i	−1.40608	−	0.582417i	−0.454756	−	0.890616i	$-0.650274\pi$
$379$	−27.3734	−	11.3384i	−1.40608	−	0.582417i	−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$		0			0
$382$	16.8229	+	7.59412i	0.860733	+	0.388549i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	−16.1144	+	35.6974i	−0.820200	+	1.81695i
$387$	−10.8846	−	26.2777i	−0.553293	−	1.33577i
$388$		0			0
$389$	11.8852	−	28.6934i	0.602603	−	1.45481i	−0.268290	−	0.963338i	$-0.586458\pi$
$389$	11.8852	−	28.6934i	0.602603	−	1.45481i	0.870893	−	0.491473i	$-0.163542\pi$
$390$		0			0
$391$		0			0
$392$	−5.82647	−	18.9223i	−0.294281	−	0.955719i
$393$		0			0
$394$	−27.1777	−	28.9364i	−1.36919	−	1.45779i
$395$		0			0
$396$	−15.0948	+	30.8255i	−0.758544	+	1.54904i
$397$		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
$400$	15.7990	−	12.2634i	0.789949	−	0.613172i
$401$		−	39.0083i		−	1.94798i	−0.226592	−	0.973990i	$-0.572758\pi$
$401$		−	39.0083i		−	1.94798i	0.226592	−	0.973990i	$-0.427242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−23.0850	−	24.5789i	−1.14569	−	1.21983i
$407$	−6.04054	−	6.04054i	−0.299418	−	0.299418i
$408$		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
$414$	3.34313	−	7.40588i	0.164306	−	0.363979i
$415$		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
$421$	15.0326	−	36.2919i	0.732643	−	1.76876i	0.0990621	−	0.995081i	$-0.468416\pi$
$421$	15.0326	−	36.2919i	0.732643	−	1.76876i	0.633581	−	0.773676i	$-0.281584\pi$
$422$	−0.0426061	+	1.35939i	−0.00207403	+	0.0661738i
$423$		0			0
$424$	37.4203	+	3.52775i	1.81729	+	0.171323i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	5.23147	−	1.79211i	0.252873	−	0.0866251i
$429$		0			0
$430$		0			0
$431$	−26.4575			−1.27441			−0.637207	−	0.770693i	$-0.719910\pi$
$431$	−26.4575			−1.27441			−0.637207	+	0.770693i	$0.719910\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	2.54120	−	5.18945i	0.121702	−	0.248529i
$437$		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
$443$	38.6871	+	16.0247i	1.83808	+	0.761358i	0.958157	+	0.286244i	$0.0924067\pi$
$443$	38.6871	+	16.0247i	1.83808	+	0.761358i	0.879924	+	0.475114i	$0.157593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−3.95564	+	20.7931i	−0.186886	+	0.982382i
$449$	42.3320			1.99777			0.998886	−	0.0471929i	$-0.0150276\pi$
$449$	42.3320			1.99777			0.998886	+	0.0471929i	$0.0150276\pi$
$450$	−8.72791	+	19.3345i	−0.411438	+	0.911438i
$451$		0			0
$452$	24.5713	−	21.6698i	1.15574	−	1.01926i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	29.9333	−	29.9333i	1.40022	−	1.40022i	0.600878	−	0.799341i	$-0.294818\pi$
$457$	29.9333	−	29.9333i	1.40022	−	1.40022i	0.799341	−	0.600878i	$-0.205182\pi$
$458$		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
$463$	39.5092			1.83615			0.918075	−	0.396406i	$-0.129743\pi$
$463$	39.5092			1.83615			0.918075	+	0.396406i	$0.129743\pi$
$464$	9.52383	+	34.7674i	0.442133	+	1.61404i
$465$		0			0
$466$	−0.294860	−	0.780126i	−0.0136591	−	0.0361386i
$467$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$467$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$468$		0			0
$469$	−39.5909	−	16.3991i	−1.82814	−	0.757240i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−38.3503	−	38.3503i	−1.76335	−	1.76335i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−36.8316	+	15.2561i	−1.68640	+	0.698531i
$478$	9.30978	−	20.6235i	0.425819	−	0.943296i
$479$		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
$484$	−2.72084	+	43.3627i	−0.123674	+	1.97103i
$485$		0			0
$486$		0			0
$487$	6.52026	−	6.52026i	0.295461	−	0.295461i	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	6.52026	−	6.52026i	0.295461	−	0.295461i	0.839233	+	0.543772i	$0.183004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.31448	−	20.0729i	0.375227	−	0.905879i	−0.617619	−	0.786478i	$-0.711903\pi$
$491$	8.31448	−	20.0729i	0.375227	−	0.905879i	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
$499$	−9.85498	+	4.08207i	−0.441169	+	0.182738i	−0.592200	−	0.805791i	$-0.701741\pi$
$499$	−9.85498	+	4.08207i	−0.441169	+	0.182738i	0.151031	+	0.988529i	$0.451741\pi$
$500$		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	$0.250000\pi$
$504$	−6.60659	−	21.4558i	−0.294281	−	0.955719i
$505$		0			0
$506$	0.485376	−	15.4863i	0.0215776	−	0.688451i
$507$		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
$512$	14.1343	−	17.6698i	0.624653	−	0.780903i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	5.58481	+	0.175040i	0.245383	+	0.00769083i
$519$		0			0
$520$		0			0
$521$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$521$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$522$	−26.1759	−	27.8699i	−1.14569	−	1.21983i
$523$		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
$529$		−	19.3320i		−	0.840523i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	29.2103	+	35.2911i	1.26169	+	1.52434i
$537$		0			0
$538$		0			0
$539$	36.9953	+	15.3240i	1.59350	+	0.660049i
$540$		0			0
$541$	17.7958	−	7.37128i	0.765103	−	0.316916i	0.0342160	−	0.999414i	$-0.489107\pi$
$541$	17.7958	−	7.37128i	0.765103	−	0.316916i	0.730887	+	0.682498i	$0.239107\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	17.8811	+	43.1688i	0.764541	+	1.84577i	0.425169	+	0.905114i	$0.360215\pi$
$547$	17.8811	+	43.1688i	0.764541	+	1.84577i	0.339372	+	0.940652i	$0.389785\pi$
$548$	43.9889	+	2.76013i	1.87911	+	0.117907i
$549$		0			0
$550$	−1.26717	+	40.4301i	−0.0540323	+	1.72395i
$551$		0			0
$552$		0			0
$553$	31.5830	−	31.5830i	1.34305	−	1.34305i
$554$	34.1816	−	32.1041i	1.45224	−	1.36397i
$555$		0			0
$556$		0			0
$557$	10.4782	+	25.2966i	0.443976	+	1.07185i	0.974541	+	0.224208i	$0.0719796\pi$
$557$	10.4782	+	25.2966i	0.443976	+	1.07185i	−0.530566	+	0.847644i	$0.678020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−44.1198	+	16.6757i	−1.86108	+	0.703422i
$563$		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	$0.0974392\pi$
$569$	29.9333	+	29.9333i	1.25487	+	1.25487i	0.301356	+	0.953512i	$0.402561\pi$
$570$		0			0
$571$	−20.2517	−	8.38852i	−0.847506	−	0.351049i	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−20.2517	−	8.38852i	−0.847506	−	0.351049i	−0.763809	+	0.645443i	$0.776673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	9.57598i		−	0.399346i
$576$	−4.48527	+	23.5772i	−0.186886	+	0.982382i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	9.89164	−	21.9125i	0.411438	−	0.911438i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−53.7529	+	53.7529i	−2.22622	+	2.22622i
$584$		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
$592$	−5.18963	−	2.95780i	−0.213292	−	0.121565i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−10.5489	−	30.7941i	−0.432102	−	1.26137i
$597$		0			0
$598$		0			0
$599$	−26.1916	−	26.1916i	−1.07016	−	1.07016i	−0.997346	−	0.0728143i	$-0.976802\pi$
$599$	−26.1916	−	26.1916i	−1.07016	−	1.07016i	−0.0728143	−	0.997346i	$-0.523198\pi$
$600$		0			0
$601$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$601$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$602$	35.4569	+	1.11130i	1.44512	+	0.0452931i
$603$	−44.8919	−	18.5948i	−1.82814	−	0.757240i
$604$	2.59412	−	41.3431i	0.105553	−	1.68223i
$605$		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
$613$	5.66000	−	13.6645i	0.228605	−	0.551902i	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	5.66000	−	13.6645i	0.228605	−	0.551902i	0.996008	+	0.0892633i	$0.0284513\pi$
$614$		0			0
$615$		0			0
$616$	−27.2953	−	32.9774i	−1.09976	−	1.32870i
$617$	−29.9333	+	29.9333i	−1.20507	+	1.20507i	−0.232462	+	0.972605i	$0.574678\pi$
$617$	−29.9333	+	29.9333i	−1.20507	+	1.20507i	−0.972605	+	0.232462i	$0.925322\pi$
$618$		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	$0.853171\pi$
$632$	−45.6346	+	14.0516i	−1.81525	+	0.558944i
$633$		0			0
$634$	−0.143267	+	4.57106i	−0.00568987	+	0.181540i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−66.4506	−	29.9969i	−2.63081	−	1.18759i
$639$			48.0000i			1.89885i
$640$		0			0
$641$	−17.5603			−0.693589			−0.346795	−	0.937941i	$-0.612730\pi$
$641$	−17.5603			−0.693589			−0.346795	+	0.937941i	$0.612730\pi$
$642$		0			0
$643$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$643$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$644$	6.70319	+	7.60070i	0.264143	+	0.299510i
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	−7.49117	−	24.3286i	−0.294281	−	0.955719i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−10.9091	+	22.2777i	−0.427234	+	0.872463i
$653$	3.58067	+	8.64451i	0.140123	+	0.338286i	0.978326	−	0.207072i	$-0.0663936\pi$
$653$	3.58067	+	8.64451i	0.140123	+	0.338286i	−0.838203	+	0.545358i	$0.816394\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−28.2022	+	11.6817i	−1.09860	+	0.455056i	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−28.2022	+	11.6817i	−1.09860	+	0.455056i	−0.241604	+	0.970375i	$0.577673\pi$
$660$		0			0
$661$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$661$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$662$	34.1620	+	36.3727i	1.32774	+	1.41366i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	6.33258	+	0.198477i	0.245383	+	0.00769083i
$667$	15.9460	+	6.60506i	0.617433	+	0.255749i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−42.3320			−1.63178			−0.815890	−	0.578208i	$-0.803752\pi$
$673$	−42.3320			−1.63178			−0.815890	+	0.578208i	$0.803752\pi$
$674$	−8.05163	−	3.63464i	−0.310137	−	0.140001i
$675$		0			0
$676$	25.9490	+	1.62820i	0.998037	+	0.0626229i
$677$		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
$683$	5.74439	−	13.8682i	0.219803	−	0.530652i	−0.775059	−	0.631889i	$-0.782280\pi$
$683$	5.74439	−	13.8682i	0.219803	−	0.530652i	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
$688$	−32.9480	−	18.7785i	−1.25613	−	0.715925i
$689$		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
$693$	41.9487	+	17.3757i	1.59350	+	0.660049i
$694$	10.8024	−	10.1458i	0.410054	−	0.385131i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−17.5000	−	19.8431i	−0.661438	−	0.750000i
$701$	20.4154	−	8.45633i	0.771079	−	0.319391i	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	20.4154	−	8.45633i	0.771079	−	0.319391i	0.733309	+	0.679895i	$0.237975\pi$
$702$		0			0
$703$		0			0
$704$	9.30292	+	44.8084i	0.350617	+	1.68878i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−19.5870	+	47.2871i	−0.735604	+	1.77591i	−0.112667	+	0.993633i	$0.535939\pi$
$709$	−19.5870	+	47.2871i	−0.735604	+	1.77591i	−0.622937	+	0.782272i	$0.714061\pi$
$710$		0			0
$711$	35.8118	−	35.8118i	1.34305	−	1.34305i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−41.1561	−	20.1536i	−1.53807	−	0.753175i
$717$		0			0
$718$	−15.9242	−	42.1315i	−0.594287	−	1.57234i
$719$		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
$725$	−41.6303	−	17.2438i	−1.54611	−	0.640419i
$726$		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
$736$	−2.53357	−	10.5336i	−0.0933885	−	0.388273i
$737$	−92.6540			−3.41295
$738$		0			0
$739$	16.0600	+	38.7723i	0.590777	+	1.42626i	0.882754	+	0.469836i	$0.155687\pi$
$739$	16.0600	+	38.7723i	0.590777	+	1.42626i	−0.291977	+	0.956425i	$0.594313\pi$
$740$		0			0
$741$		0			0
$742$	1.55763	−	49.6975i	0.0571824	−	1.82445i
$743$	38.5203	−	38.5203i	1.41317	−	1.41317i	0.679442	−	0.733729i	$-0.262222\pi$
$743$	38.5203	−	38.5203i	1.41317	−	1.41317i	0.733729	−	0.679442i	$-0.237778\pi$
$744$		0			0
$745$		0			0
$746$	−21.5581	+	20.2478i	−0.789299	+	0.741325i
$747$		0			0
$748$		0			0
$749$	−2.79948	−	6.75854i	−0.102291	−	0.246952i
$750$		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
$757$	−42.3502	−	17.5420i	−1.53925	−	0.637576i	−0.557913	−	0.829899i	$-0.688398\pi$
$757$	−42.3502	−	17.5420i	−1.53925	−	0.637576i	−0.981332	+	0.192323i	$0.938398\pi$
$758$	30.5424	−	28.6861i	1.10935	−	1.04192i
$759$		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
$763$	−7.06204	−	2.92519i	−0.255663	−	0.105899i
$764$	−19.5771	+	17.2654i	−0.708276	+	0.624641i
$765$		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
$772$	−36.6364	−	41.5418i	−1.31857	−	1.49512i
$773$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$773$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$774$	40.2044	+	1.26009i	1.44512	+	0.0452931i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	30.0693	+	32.0152i	1.07804	+	1.14780i
$779$		0			0
$780$		0			0
$781$	35.0262	+	84.5607i	1.25334	+	3.02582i
$782$		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	$0.875000\pi$
$788$	53.1117	−	18.1942i	1.89203	−	0.648140i
$789$		0			0
$790$		0			0
$791$	−30.6458	−	30.6458i	−1.08964	−	1.08964i
$792$	−30.9500	−	37.3929i	−1.09976	−	1.32870i
$793$		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
$802$	50.2804	+	22.6974i	1.77546	+	0.801472i
$803$		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	$0.517149\pi$
$809$	−29.9333	+	29.9333i	−1.05240	+	1.05240i	−0.998549	+	0.0538482i	$0.982851\pi$
$810$		0			0
$811$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$811$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$812$	45.1137	−	15.4543i	1.58318	−	0.542341i
$813$		0			0
$814$	11.3008	−	4.27131i	0.396093	−	0.149709i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	52.9440	+	21.9301i	1.84776	+	0.765366i	0.924384	+	0.381464i	$0.124580\pi$
$821$	52.9440	+	21.9301i	1.84776	+	0.765366i	0.923374	+	0.383903i	$0.125420\pi$
$822$		0			0
$823$	−33.6749	−	33.6749i	−1.17383	−	1.17383i	−0.981288	−	0.192546i	$-0.938326\pi$
$823$	−33.6749	−	33.6749i	−1.17383	−	1.17383i	−0.192546	−	0.981288i	$-0.561674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.0597	+	6.65215i	0.558451	+	0.231318i	0.644013	−	0.765015i	$-0.277268\pi$
$827$	16.0597	+	6.65215i	0.558451	+	0.231318i	−0.0855616	+	0.996333i	$0.527268\pi$
$828$	7.60070	+	8.61839i	0.264143	+	0.299510i
$829$		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
$841$	36.9231	−	36.9231i	1.27321	−	1.27321i
$842$	38.0322	+	40.4933i	1.31068	+	1.39549i
$843$		0			0
$844$	−1.72741	−	0.845891i	−0.0594600	−	0.0291168i
$845$		0			0
$846$		0			0
$847$	57.4763			1.97491
$848$	−26.3206	+	46.1809i	−0.903852	+	1.58586i
$849$		0			0
$850$		0			0
$851$	−2.64233	+	1.09449i	−0.0905778	+	0.0375185i
$852$		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
$856$	−0.734011	+	7.78596i	−0.0250880	+	0.266119i
$857$		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
$862$	15.3946	−	34.1029i	0.524342	−	1.16155i
$863$		−	46.8151i		−	1.59360i	−0.604240	−	0.796802i	$-0.706523\pi$
$863$		−	46.8151i		−	1.59360i	0.604240	−	0.796802i	$-0.293477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	36.9566	−	89.2212i	1.25367	−	3.02662i
$870$		0			0
$871$		0			0
$872$	5.21040	+	6.29506i	0.176446	+	0.213178i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	13.0281	+	31.4527i	0.439929	+	1.06208i	0.975973	+	0.217892i	$0.0699181\pi$
$877$	13.0281	+	31.4527i	0.439929	+	1.06208i	−0.536044	+	0.844190i	$0.680082\pi$
$878$		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
$883$	−30.8218	+	12.7668i	−1.03723	+	0.429637i	−0.835319	−	0.549766i	$-0.814717\pi$
$883$	−30.8218	+	12.7668i	−1.03723	+	0.429637i	−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$		0			0
$886$	−43.1659	+	40.5423i	−1.45019	+	1.36204i
$887$		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
$891$	47.5654	+	19.7022i	1.59350	+	0.660049i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−24.5000	−	17.1974i	−0.818488	−	0.574524i
$897$		0			0
$898$	−24.6314	+	54.5646i	−0.821959	+	1.82084i
$899$		0			0
$900$	−19.8431	−	22.5000i	−0.661438	−	0.750000i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	13.6346	+	44.2804i	0.453481	+	1.47274i
$905$		0			0
$906$		0			0
$907$	20.4383	−	49.3424i	0.678642	−	1.63839i	−0.0878507	−	0.996134i	$-0.528000\pi$
$907$	20.4383	−	49.3424i	0.678642	−	1.63839i	0.766493	−	0.642253i	$-0.222000\pi$
$908$		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
$925$	6.89831	−	2.85738i	0.226815	−	0.0939499i
$926$	−22.9889	+	50.9261i	−0.755462	+	1.67354i
$927$		0			0
$928$	−50.3556	−	7.95387i	−1.65300	−	0.261099i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	1.17712	+	0.0738599i	0.0385580	+	0.00241936i
$933$		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	$0.750000\pi$
$938$	44.1743	−	41.4894i	1.44234	−	1.35468i
$939$		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
$946$	71.7468	−	27.1177i	2.33269	−	0.881673i
$947$	3.50805	−	1.45308i	0.113996	−	0.0472188i	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	3.50805	−	1.45308i	0.113996	−	0.0472188i	0.438953	+	0.898510i	$0.355350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−39.5830	−	39.5830i	−1.28222	−	1.28222i	−0.939402	−	0.342817i	$-0.888619\pi$
$953$	−39.5830	−	39.5830i	−1.28222	−	1.28222i	−0.342817	−	0.939402i	$-0.611381\pi$
$954$	1.76619	−	56.3517i	0.0571824	−	1.82445i
$955$		0			0
$956$	21.1660	+	24.0000i	0.684558	+	0.776215i
$957$		0			0
$958$		0			0
$959$		−	58.3063i		−	1.88281i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	−3.17431	−	7.66346i	−0.102291	−	0.246952i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	3.81176	−	3.81176i	0.122578	−	0.122578i	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	3.81176	−	3.81176i	0.122578	−	0.122578i	0.765735	+	0.643157i	$0.222376\pi$
$968$	−54.3100	−	28.7381i	−1.74559	−	0.923679i
$969$		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
$974$	4.61052	+	12.1983i	0.147731	+	0.390858i
$975$		0			0
$976$		0			0
$977$		−	42.3320i		−	1.35432i	−0.735835	−	0.677161i	$-0.763210\pi$
$977$		−	42.3320i		−	1.35432i	0.735835	−	0.677161i	$-0.236790\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−8.00760	−	3.31686i	−0.255663	−	0.105899i
$982$	21.0355	+	22.3967i	0.671269	+	0.714709i
$983$		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
$989$	−16.7756	+	6.94869i	−0.533434	+	0.220956i
$990$		0			0
$991$			24.1877i			0.768347i	0.923261	+	0.384173i	$0.125513\pi$
$991$			24.1877i			0.768347i	−0.923261	+	0.384173i	$0.874487\pi$
$992$		0			0
$993$		0			0
$994$	−54.5646	−	24.6314i	−1.73068	−	0.781259i
$995$		0			0
$996$		0			0
$997$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$997$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$998$	0.472576	−	15.0779i	0.0149591	−	0.477284i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.x.a.139.1	✓	8
4.3	odd	2		896.2.x.a.335.2		8
7.6	odd	2	CM	224.2.x.a.139.1	✓	8
28.27	even	2		896.2.x.a.335.2		8
32.3	odd	8	inner	224.2.x.a.195.1	yes	8
32.29	even	8		896.2.x.a.559.2		8
224.125	odd	8		896.2.x.a.559.2		8
224.195	even	8	inner	224.2.x.a.195.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.x.a.139.1	✓	8	1.1	even	1	trivial
224.2.x.a.139.1	✓	8	7.6	odd	2	CM
224.2.x.a.195.1	yes	8	32.3	odd	8	inner
224.2.x.a.195.1	yes	8	224.195	even	8	inner
896.2.x.a.335.2		8	4.3	odd	2
896.2.x.a.335.2		8	28.27	even	2
896.2.x.a.559.2		8	32.29	even	8
896.2.x.a.559.2		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)

\(f(q)\)	\(=\)	\(q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} +(-1.87083 + 1.87083i) q^{7} +(2.70318 - 0.832353i) q^{8} +(-2.12132 + 2.12132i) q^{9} +O(q^{10})\)	\(q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} +(-1.87083 + 1.87083i) q^{7} +(2.70318 - 0.832353i) q^{8} +(-2.12132 + 2.12132i) q^{9} +(-2.18914 + 5.28504i) q^{11} +(-1.32288 - 3.50000i) q^{14} +(-0.500000 + 3.96863i) q^{16} +(-1.50000 - 3.96863i) q^{18} +(-5.53848 - 5.89689i) q^{22} +(1.35425 + 1.35425i) q^{23} +(-3.53553 - 3.53553i) q^{25} +(5.28112 + 0.331369i) q^{28} +(8.32605 - 3.44876i) q^{29} +(-4.82450 - 2.95367i) q^{32} +(5.98822 + 0.375737i) q^{36} +(-0.571475 + 1.37966i) q^{37} +(-3.62819 + 8.75922i) q^{43} +(10.8235 - 3.70775i) q^{44} +(-2.53357 + 0.957598i) q^{46} -7.00000i q^{49} +(6.61438 - 2.50000i) q^{50} +(12.2772 + 5.08538i) q^{53} +(-3.50000 + 6.61438i) q^{56} +(-0.399260 + 12.7387i) q^{58} -7.93725i q^{63} +(6.61438 - 4.50000i) q^{64} +(6.19827 + 14.9640i) q^{67} +(11.3137 - 11.3137i) q^{71} +(-3.96863 + 7.50000i) q^{72} +(-1.44582 - 1.53938i) q^{74} +(-5.79191 - 13.9829i) q^{77} -16.8818 q^{79} -9.00000i q^{81} +(-9.17925 - 9.77327i) q^{86} +(-1.51861 + 16.1086i) q^{88} +(0.239870 - 3.82288i) q^{92} +(9.02277 + 4.07303i) q^{98} +(-6.56741 - 15.8551i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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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