LMFDB - Embedded newform 560.2.k.a.111.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,2,Mod(111,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("560.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	560.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.47162251319$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.6
Root			$0.396143 + 1.68614i$ of defining polynomial
Character	$\chi$	$=$	560.111
Dual form			560.2.k.a.111.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.792287 q^{3} +1.00000i q^{5} +(0.792287 + 2.52434i) q^{7} -2.37228 q^{9} +O(q^{10})$	$q+0.792287 q^{3} +1.00000i q^{5} +(0.792287 + 2.52434i) q^{7} -2.37228 q^{9} -0.792287i q^{11} +5.37228i q^{13} +0.792287i q^{15} +3.37228i q^{17} +3.46410 q^{19} +(0.627719 + 2.00000i) q^{21} -1.87953i q^{23} -1.00000 q^{25} -4.25639 q^{27} -5.37228 q^{29} +8.51278 q^{31} -0.627719i q^{33} +(-2.52434 + 0.792287i) q^{35} -0.744563 q^{37} +4.25639i q^{39} +2.74456i q^{41} -3.46410i q^{43} -2.37228i q^{45} +11.1846 q^{47} +(-5.74456 + 4.00000i) q^{49} +2.67181i q^{51} +11.4891 q^{53} +0.792287 q^{55} +2.74456 q^{57} -6.63325 q^{59} +0.744563i q^{61} +(-1.87953 - 5.98844i) q^{63} -5.37228 q^{65} -6.63325i q^{67} -1.48913i q^{69} -6.63325i q^{71} +2.74456i q^{73} -0.792287 q^{75} +(2.00000 - 0.627719i) q^{77} +14.0588i q^{79} +3.74456 q^{81} -10.3923 q^{83} -3.37228 q^{85} -4.25639 q^{87} -17.4891i q^{89} +(-13.5615 + 4.25639i) q^{91} +6.74456 q^{93} +3.46410i q^{95} -2.11684i q^{97} +1.87953i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} + 28 q^{21} - 8 q^{25} - 20 q^{29} + 40 q^{37} - 24 q^{57} - 20 q^{65} + 16 q^{77} - 16 q^{81} - 4 q^{85} + 8 q^{93}+O(q^{100}) $ 8 * q + 4 * q^9 + 28 * q^21 - 8 * q^25 - 20 * q^29 + 40 * q^37 - 24 * q^57 - 20 * q^65 + 16 * q^77 - 16 * q^81 - 4 * q^85 + 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0.792287			0.457427			0.228714	−	0.973494i	$-0.426548\pi$
$3$	0.792287			0.457427			0.228714	+	0.973494i	$0.426548\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	0.792287	+	2.52434i	0.299456	+	0.954110i
$7$	0.792287	+	2.52434i	0.299456	+	0.954110i
$8$		0			0
$9$	−2.37228			−0.790760
$10$		0			0
$11$		−	0.792287i		−	0.238884i	−0.992841	−	0.119442i	$-0.961890\pi$
$11$		−	0.792287i		−	0.238884i	0.992841	−	0.119442i	$-0.0381105\pi$
$12$		0			0
$13$			5.37228i			1.49000i	0.667063	+	0.745001i	$0.267551\pi$
$13$			5.37228i			1.49000i	−0.667063	+	0.745001i	$0.732449\pi$
$14$		0			0
$15$			0.792287i			0.204568i
$16$		0			0
$17$			3.37228i			0.817898i	0.912557	+	0.408949i	$0.134105\pi$
$17$			3.37228i			0.817898i	−0.912557	+	0.408949i	$0.865895\pi$
$18$		0			0
$19$	3.46410			0.794719			0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410			0.794719			0.397360	+	0.917663i	$0.369927\pi$
$20$		0			0
$21$	0.627719	+	2.00000i	0.136979	+	0.436436i
$22$		0			0
$23$		−	1.87953i		−	0.391909i	−0.980613	−	0.195954i	$-0.937220\pi$
$23$		−	1.87953i		−	0.391909i	0.980613	−	0.195954i	$-0.0627804\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−4.25639			−0.819142
$28$		0			0
$29$	−5.37228			−0.997608			−0.498804	−	0.866715i	$-0.666227\pi$
$29$	−5.37228			−0.997608			−0.498804	+	0.866715i	$0.666227\pi$
$30$		0			0
$31$	8.51278			1.52894			0.764470	−	0.644659i	$-0.223001\pi$
$31$	8.51278			1.52894			0.764470	+	0.644659i	$0.223001\pi$
$32$		0			0
$33$		−	0.627719i		−	0.109272i
$34$		0			0
$35$	−2.52434	+	0.792287i	−0.426691	+	0.133921i
$36$		0			0
$37$	−0.744563			−0.122405			−0.0612027	−	0.998125i	$-0.519494\pi$
$37$	−0.744563			−0.122405			−0.0612027	+	0.998125i	$0.519494\pi$
$38$		0			0
$39$			4.25639i			0.681568i
$40$		0			0
$41$			2.74456i			0.428629i	0.976765	+	0.214314i	$0.0687517\pi$
$41$			2.74456i			0.428629i	−0.976765	+	0.214314i	$0.931248\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$		0			0
$45$		−	2.37228i		−	0.353639i
$46$		0			0
$47$	11.1846			1.63144			0.815720	−	0.578447i	$-0.196341\pi$
$47$	11.1846			1.63144			0.815720	+	0.578447i	$0.196341\pi$
$48$		0			0
$49$	−5.74456	+	4.00000i	−0.820652	+	0.571429i
$50$		0			0
$51$			2.67181i			0.374129i
$52$		0			0
$53$	11.4891			1.57815			0.789076	−	0.614295i	$-0.210560\pi$
$53$	11.4891			1.57815			0.789076	+	0.614295i	$0.210560\pi$
$54$		0			0
$55$	0.792287			0.106832
$56$		0			0
$57$	2.74456			0.363526
$58$		0			0
$59$	−6.63325			−0.863576			−0.431788	−	0.901975i	$-0.642117\pi$
$59$	−6.63325			−0.863576			−0.431788	+	0.901975i	$0.642117\pi$
$60$		0			0
$61$			0.744563i			0.0953315i	0.998863	+	0.0476657i	$0.0151782\pi$
$61$			0.744563i			0.0953315i	−0.998863	+	0.0476657i	$0.984822\pi$
$62$		0			0
$63$	−1.87953	−	5.98844i	−0.236798	−	0.754472i
$64$		0			0
$65$	−5.37228			−0.666349
$66$		0			0
$67$		−	6.63325i		−	0.810380i	−0.914232	−	0.405190i	$-0.867205\pi$
$67$		−	6.63325i		−	0.810380i	0.914232	−	0.405190i	$-0.132795\pi$
$68$		0			0
$69$		−	1.48913i		−	0.179270i
$70$		0			0
$71$		−	6.63325i		−	0.787222i	−0.919277	−	0.393611i	$-0.871226\pi$
$71$		−	6.63325i		−	0.787222i	0.919277	−	0.393611i	$-0.128774\pi$
$72$		0			0
$73$			2.74456i			0.321227i	0.987017	+	0.160613i	$0.0513472\pi$
$73$			2.74456i			0.321227i	−0.987017	+	0.160613i	$0.948653\pi$
$74$		0			0
$75$	−0.792287			−0.0914854
$76$		0			0
$77$	2.00000	−	0.627719i	0.227921	−	0.0715352i
$78$		0			0
$79$			14.0588i			1.58174i	0.611986	+	0.790869i	$0.290371\pi$
$79$			14.0588i			1.58174i	−0.611986	+	0.790869i	$0.709629\pi$
$80$		0			0
$81$	3.74456			0.416063
$82$		0			0
$83$	−10.3923			−1.14070			−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923			−1.14070			−0.570352	+	0.821401i	$0.693193\pi$
$84$		0			0
$85$	−3.37228			−0.365775
$86$		0			0
$87$	−4.25639			−0.456333
$88$		0			0
$89$		−	17.4891i		−	1.85384i	−0.375255	−	0.926922i	$-0.622445\pi$
$89$		−	17.4891i		−	1.85384i	0.375255	−	0.926922i	$-0.377555\pi$
$90$		0			0
$91$	−13.5615	+	4.25639i	−1.42163	+	0.446191i
$92$		0			0
$93$	6.74456			0.699379
$94$		0			0
$95$			3.46410i			0.355409i
$96$		0			0
$97$		−	2.11684i		−	0.214933i	−0.994209	−	0.107466i	$-0.965726\pi$
$97$		−	2.11684i		−	0.214933i	0.994209	−	0.107466i	$-0.0342738\pi$
$98$		0			0
$99$			1.87953i			0.188900i
$100$		0			0
$101$		−	15.4891i		−	1.54123i	−0.637304	−	0.770613i	$-0.719950\pi$
$101$		−	15.4891i		−	1.54123i	0.637304	−	0.770613i	$-0.280050\pi$
$102$		0			0
$103$	12.7692			1.25818			0.629092	−	0.777331i	$-0.283427\pi$
$103$	12.7692			1.25818			0.629092	+	0.777331i	$0.283427\pi$
$104$		0			0
$105$	−2.00000	+	0.627719i	−0.195180	+	0.0612591i
$106$		0			0
$107$		−	9.80240i		−	0.947634i	−0.880623	−	0.473817i	$-0.842876\pi$
$107$		−	9.80240i		−	0.947634i	0.880623	−	0.473817i	$-0.157124\pi$
$108$		0			0
$109$	−13.3723			−1.28083			−0.640416	−	0.768028i	$-0.721238\pi$
$109$	−13.3723			−1.28083			−0.640416	+	0.768028i	$0.721238\pi$
$110$		0			0
$111$	−0.589907			−0.0559915
$112$		0			0
$113$	12.7446			1.19891			0.599454	−	0.800409i	$-0.295385\pi$
$113$	12.7446			1.19891			0.599454	+	0.800409i	$0.295385\pi$
$114$		0			0
$115$	1.87953			0.175267
$116$		0			0
$117$		−	12.7446i		−	1.17824i
$118$		0			0
$119$	−8.51278	+	2.67181i	−0.780365	+	0.244925i
$120$		0			0
$121$	10.3723			0.942935
$122$		0			0
$123$			2.17448i			0.196066i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			6.63325i			0.588606i	0.955712	+	0.294303i	$0.0950874\pi$
$127$			6.63325i			0.588606i	−0.955712	+	0.294303i	$0.904913\pi$
$128$		0			0
$129$		−	2.74456i		−	0.241645i
$130$		0			0
$131$	8.80773			0.769535			0.384768	−	0.923014i	$-0.374282\pi$
$131$	8.80773			0.769535			0.384768	+	0.923014i	$0.374282\pi$
$132$		0			0
$133$	2.74456	+	8.74456i	0.237984	+	0.758250i
$134$		0			0
$135$		−	4.25639i		−	0.366332i
$136$		0			0
$137$	0.744563			0.0636123			0.0318061	−	0.999494i	$-0.489874\pi$
$137$	0.744563			0.0636123			0.0318061	+	0.999494i	$0.489874\pi$
$138$		0			0
$139$	−15.1460			−1.28467			−0.642335	−	0.766424i	$-0.722034\pi$
$139$	−15.1460			−1.28467			−0.642335	+	0.766424i	$0.722034\pi$
$140$		0			0
$141$	8.86141			0.746265
$142$		0			0
$143$	4.25639			0.355937
$144$		0			0
$145$		−	5.37228i		−	0.446144i
$146$		0			0
$147$	−4.55134	+	3.16915i	−0.375388	+	0.261387i
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		−	21.5769i		−	1.75590i	−0.478750	−	0.877951i	$-0.658910\pi$
$151$		−	21.5769i		−	1.75590i	0.478750	−	0.877951i	$-0.341090\pi$
$152$		0			0
$153$		−	8.00000i		−	0.646762i
$154$		0			0
$155$			8.51278i			0.683763i
$156$		0			0
$157$		−	0.510875i		−	0.0407722i	−0.999792	−	0.0203861i	$-0.993510\pi$
$157$		−	0.510875i		−	0.0407722i	0.999792	−	0.0203861i	$-0.00648955\pi$
$158$		0			0
$159$	9.10268			0.721890
$160$		0			0
$161$	4.74456	−	1.48913i	0.373924	−	0.117360i
$162$		0			0
$163$		−	15.1460i		−	1.18633i	−0.805082	−	0.593164i	$-0.797878\pi$
$163$		−	15.1460i		−	1.18633i	0.805082	−	0.593164i	$-0.202122\pi$
$164$		0			0
$165$	0.627719			0.0488678
$166$		0			0
$167$	−15.9383			−1.23334			−0.616672	−	0.787220i	$-0.711519\pi$
$167$	−15.9383			−1.23334			−0.616672	+	0.787220i	$0.711519\pi$
$168$		0			0
$169$	−15.8614			−1.22011
$170$		0			0
$171$	−8.21782			−0.628433
$172$		0			0
$173$		−	10.6277i		−	0.808010i	−0.914757	−	0.404005i	$-0.867618\pi$
$173$		−	10.6277i		−	0.808010i	0.914757	−	0.404005i	$-0.132382\pi$
$174$		0			0
$175$	−0.792287	−	2.52434i	−0.0598913	−	0.190822i
$176$		0			0
$177$	−5.25544			−0.395023
$178$		0			0
$179$		−	0.294954i		−	0.0220459i	−0.999939	−	0.0110229i	$-0.996491\pi$
$179$		−	0.294954i		−	0.0220459i	0.999939	−	0.0110229i	$-0.00350878\pi$
$180$		0			0
$181$			24.9783i			1.85662i	0.371809	+	0.928309i	$0.378738\pi$
$181$			24.9783i			1.85662i	−0.371809	+	0.928309i	$0.621262\pi$
$182$		0			0
$183$			0.589907i			0.0436072i
$184$		0			0
$185$		−	0.744563i		−	0.0547413i
$186$		0			0
$187$	2.67181			0.195382
$188$		0			0
$189$	−3.37228	−	10.7446i	−0.245297	−	0.781552i
$190$		0			0
$191$			2.37686i			0.171984i	0.996296	+	0.0859918i	$0.0274059\pi$
$191$			2.37686i			0.171984i	−0.996296	+	0.0859918i	$0.972594\pi$
$192$		0			0
$193$	15.2554			1.09811			0.549055	−	0.835786i	$-0.314988\pi$
$193$	15.2554			1.09811			0.549055	+	0.835786i	$0.314988\pi$
$194$		0			0
$195$	−4.25639			−0.304806
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	18.6101			1.31924			0.659619	−	0.751601i	$-0.270718\pi$
$199$	18.6101			1.31924			0.659619	+	0.751601i	$0.270718\pi$
$200$		0			0
$201$		−	5.25544i		−	0.370690i
$202$		0			0
$203$	−4.25639	−	13.5615i	−0.298740	−	0.951827i
$204$		0			0
$205$	−2.74456			−0.191689
$206$		0			0
$207$			4.45877i			0.309906i
$208$		0			0
$209$		−	2.74456i		−	0.189845i
$210$		0			0
$211$			5.54601i			0.381803i	0.981609	+	0.190902i	$0.0611411\pi$
$211$			5.54601i			0.381803i	−0.981609	+	0.190902i	$0.938859\pi$
$212$		0			0
$213$		−	5.25544i		−	0.360097i
$214$		0			0
$215$	3.46410			0.236250
$216$		0			0
$217$	6.74456	+	21.4891i	0.457851	+	1.45878i
$218$		0			0
$219$			2.17448i			0.146938i
$220$		0			0
$221$	−18.1168			−1.21867
$222$		0			0
$223$	5.84096			0.391140			0.195570	−	0.980690i	$-0.437344\pi$
$223$	5.84096			0.391140			0.195570	+	0.980690i	$0.437344\pi$
$224$		0			0
$225$	2.37228			0.158152
$226$		0			0
$227$	15.6434			1.03829			0.519143	−	0.854687i	$-0.326251\pi$
$227$	15.6434			1.03829			0.519143	+	0.854687i	$0.326251\pi$
$228$		0			0
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$		0			0
$231$	1.58457	−	0.497333i	0.104257	−	0.0327221i
$232$		0			0
$233$	0.744563			0.0487779			0.0243890	−	0.999703i	$-0.492236\pi$
$233$	0.744563			0.0487779			0.0243890	+	0.999703i	$0.492236\pi$
$234$		0			0
$235$			11.1846i			0.729602i
$236$		0			0
$237$			11.1386i			0.723529i
$238$		0			0
$239$			3.96143i			0.256244i	0.991758	+	0.128122i	$0.0408949\pi$
$239$			3.96143i			0.256244i	−0.991758	+	0.128122i	$0.959105\pi$
$240$		0			0
$241$			16.0000i			1.03065i	0.856995	+	0.515325i	$0.172329\pi$
$241$			16.0000i			1.03065i	−0.856995	+	0.515325i	$0.827671\pi$
$242$		0			0
$243$	15.7359			1.00946
$244$		0			0
$245$	−4.00000	−	5.74456i	−0.255551	−	0.367007i
$246$		0			0
$247$			18.6101i			1.18413i
$248$		0			0
$249$	−8.23369			−0.521789
$250$		0			0
$251$	−8.80773			−0.555939			−0.277970	−	0.960590i	$-0.589661\pi$
$251$	−8.80773			−0.555939			−0.277970	+	0.960590i	$0.589661\pi$
$252$		0			0
$253$	−1.48913			−0.0936205
$254$		0			0
$255$	−2.67181			−0.167316
$256$		0			0
$257$		−	1.25544i		−	0.0783120i	−0.999233	−	0.0391560i	$-0.987533\pi$
$257$		−	1.25544i		−	0.0783120i	0.999233	−	0.0391560i	$-0.0124669\pi$
$258$		0			0
$259$	−0.589907	−	1.87953i	−0.0366551	−	0.116788i
$260$		0			0
$261$	12.7446			0.788869
$262$		0			0
$263$			27.4179i			1.69066i	0.534246	+	0.845329i	$0.320595\pi$
$263$			27.4179i			1.69066i	−0.534246	+	0.845329i	$0.679405\pi$
$264$		0			0
$265$			11.4891i			0.705771i
$266$		0			0
$267$		−	13.8564i		−	0.847998i
$268$		0			0
$269$			0.744563i			0.0453968i	0.999742	+	0.0226984i	$0.00722574\pi$
$269$			0.744563i			0.0453968i	−0.999742	+	0.0226984i	$0.992774\pi$
$270$		0			0
$271$	26.5330			1.61176			0.805882	−	0.592076i	$-0.201691\pi$
$271$	26.5330			1.61176			0.805882	+	0.592076i	$0.201691\pi$
$272$		0			0
$273$	−10.7446	+	3.37228i	−0.650291	+	0.204100i
$274$		0			0
$275$			0.792287i			0.0477767i
$276$		0			0
$277$	−14.2337			−0.855219			−0.427610	−	0.903963i	$-0.640644\pi$
$277$	−14.2337			−0.855219			−0.427610	+	0.903963i	$0.640644\pi$
$278$		0			0
$279$	−20.1947			−1.20903
$280$		0			0
$281$	−21.6060			−1.28890			−0.644452	−	0.764645i	$-0.722914\pi$
$281$	−21.6060			−1.28890			−0.644452	+	0.764645i	$0.722914\pi$
$282$		0			0
$283$	10.8896			0.647322			0.323661	−	0.946173i	$-0.395086\pi$
$283$	10.8896			0.647322			0.323661	+	0.946173i	$0.395086\pi$
$284$		0			0
$285$			2.74456i			0.162574i
$286$		0			0
$287$	−6.92820	+	2.17448i	−0.408959	+	0.128356i
$288$		0			0
$289$	5.62772			0.331042
$290$		0			0
$291$		−	1.67715i		−	0.0983162i
$292$		0			0
$293$			25.3723i			1.48226i	0.671359	+	0.741132i	$0.265711\pi$
$293$			25.3723i			1.48226i	−0.671359	+	0.741132i	$0.734289\pi$
$294$		0			0
$295$		−	6.63325i		−	0.386203i
$296$		0			0
$297$			3.37228i			0.195680i
$298$		0			0
$299$	10.0974			0.583945
$300$		0			0
$301$	8.74456	−	2.74456i	0.504028	−	0.158194i
$302$		0			0
$303$		−	12.2718i		−	0.704998i
$304$		0			0
$305$	−0.744563			−0.0426335
$306$		0			0
$307$	9.30506			0.531068			0.265534	−	0.964101i	$-0.414452\pi$
$307$	9.30506			0.531068			0.265534	+	0.964101i	$0.414452\pi$
$308$		0			0
$309$	10.1168			0.575527
$310$		0			0
$311$	−21.7793			−1.23499			−0.617495	−	0.786575i	$-0.711852\pi$
$311$	−21.7793			−1.23499			−0.617495	+	0.786575i	$0.711852\pi$
$312$		0			0
$313$			14.1168i			0.797931i	0.916966	+	0.398966i	$0.130631\pi$
$313$			14.1168i			0.797931i	−0.916966	+	0.398966i	$0.869369\pi$
$314$		0			0
$315$	5.98844	−	1.87953i	0.337410	−	0.105899i
$316$		0			0
$317$	27.4891			1.54394			0.771972	−	0.635657i	$-0.219271\pi$
$317$	27.4891			1.54394			0.771972	+	0.635657i	$0.219271\pi$
$318$		0			0
$319$			4.25639i			0.238312i
$320$		0			0
$321$		−	7.76631i		−	0.433473i
$322$		0			0
$323$			11.6819i			0.650000i
$324$		0			0
$325$		−	5.37228i		−	0.298001i
$326$		0			0
$327$	−10.5947			−0.585887
$328$		0			0
$329$	8.86141	+	28.2337i	0.488545	+	1.55657i
$330$		0			0
$331$			24.2487i			1.33283i	0.745581	+	0.666415i	$0.232172\pi$
$331$			24.2487i			1.33283i	−0.745581	+	0.666415i	$0.767828\pi$
$332$		0			0
$333$	1.76631			0.0967933
$334$		0			0
$335$	6.63325			0.362413
$336$		0			0
$337$	−14.2337			−0.775358			−0.387679	−	0.921794i	$-0.626723\pi$
$337$	−14.2337			−0.775358			−0.387679	+	0.921794i	$0.626723\pi$
$338$		0			0
$339$	10.0974			0.548413
$340$		0			0
$341$		−	6.74456i		−	0.365239i
$342$		0			0
$343$	−14.6487	−	11.3321i	−0.790955	−	0.611874i
$344$		0			0
$345$	1.48913			0.0801718
$346$		0			0
$347$		−	32.1716i		−	1.72706i	−0.504297	−	0.863530i	$-0.668248\pi$
$347$		−	32.1716i		−	1.72706i	0.504297	−	0.863530i	$-0.331752\pi$
$348$		0			0
$349$			10.0000i			0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$			10.0000i			0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$		−	22.8665i		−	1.22052i
$352$		0			0
$353$			12.6277i			0.672106i	0.941843	+	0.336053i	$0.109092\pi$
$353$			12.6277i			0.672106i	−0.941843	+	0.336053i	$0.890908\pi$
$354$		0			0
$355$	6.63325			0.352056
$356$		0			0
$357$	−6.74456	+	2.11684i	−0.356960	+	0.112035i
$358$		0			0
$359$		−	13.5615i		−	0.715746i	−0.933770	−	0.357873i	$-0.883502\pi$
$359$		−	13.5615i		−	0.715746i	0.933770	−	0.357873i	$-0.116498\pi$
$360$		0			0
$361$	−7.00000			−0.368421
$362$		0			0
$363$	8.21782			0.431324
$364$		0			0
$365$	−2.74456			−0.143657
$366$		0			0
$367$	25.0410			1.30713			0.653565	−	0.756871i	$-0.273273\pi$
$367$	25.0410			1.30713			0.653565	+	0.756871i	$0.273273\pi$
$368$		0			0
$369$		−	6.51087i		−	0.338943i
$370$		0			0
$371$	9.10268	+	29.0024i	0.472588	+	1.50573i
$372$		0			0
$373$	−32.7446			−1.69545			−0.847725	−	0.530437i	$-0.822028\pi$
$373$	−32.7446			−1.69545			−0.847725	+	0.530437i	$0.822028\pi$
$374$		0			0
$375$		−	0.792287i		−	0.0409135i
$376$		0			0
$377$		−	28.8614i		−	1.48644i
$378$		0			0
$379$			2.87419i			0.147637i	0.997272	+	0.0738187i	$0.0235186\pi$
$379$			2.87419i			0.147637i	−0.997272	+	0.0738187i	$0.976481\pi$
$380$		0			0
$381$			5.25544i			0.269244i
$382$		0			0
$383$	−5.34363			−0.273047			−0.136523	−	0.990637i	$-0.543593\pi$
$383$	−5.34363			−0.273047			−0.136523	+	0.990637i	$0.543593\pi$
$384$		0			0
$385$	0.627719	+	2.00000i	0.0319915	+	0.101929i
$386$		0			0
$387$			8.21782i			0.417735i
$388$		0			0
$389$	16.1168			0.817156			0.408578	−	0.912723i	$-0.366025\pi$
$389$	16.1168			0.817156			0.408578	+	0.912723i	$0.366025\pi$
$390$		0			0
$391$	6.33830			0.320541
$392$		0			0
$393$	6.97825			0.352006
$394$		0			0
$395$	−14.0588			−0.707374
$396$		0			0
$397$			6.62772i			0.332636i	0.986072	+	0.166318i	$0.0531878\pi$
$397$			6.62772i			0.332636i	−0.986072	+	0.166318i	$0.946812\pi$
$398$		0			0
$399$	2.17448	+	6.92820i	0.108860	+	0.346844i
$400$		0			0
$401$	−12.1168			−0.605086			−0.302543	−	0.953136i	$-0.597836\pi$
$401$	−12.1168			−0.605086			−0.302543	+	0.953136i	$0.597836\pi$
$402$		0			0
$403$			45.7330i			2.27812i
$404$		0			0
$405$			3.74456i			0.186069i
$406$		0			0
$407$			0.589907i			0.0292406i
$408$		0			0
$409$		−	5.25544i		−	0.259865i	−0.991523	−	0.129932i	$-0.958524\pi$
$409$		−	5.25544i		−	0.259865i	0.991523	−	0.129932i	$-0.0414760\pi$
$410$		0			0
$411$	0.589907			0.0290980
$412$		0			0
$413$	−5.25544	−	16.7446i	−0.258603	−	0.823946i
$414$		0			0
$415$		−	10.3923i		−	0.510138i
$416$		0			0
$417$	−12.0000			−0.587643
$418$		0			0
$419$	26.8280			1.31063			0.655316	−	0.755355i	$-0.272536\pi$
$419$	26.8280			1.31063			0.655316	+	0.755355i	$0.272536\pi$
$420$		0			0
$421$	3.88316			0.189253			0.0946267	−	0.995513i	$-0.469834\pi$
$421$	3.88316			0.189253			0.0946267	+	0.995513i	$0.469834\pi$
$422$		0			0
$423$	−26.5330			−1.29008
$424$		0			0
$425$		−	3.37228i		−	0.163580i
$426$		0			0
$427$	−1.87953	+	0.589907i	−0.0909567	+	0.0285476i
$428$		0			0
$429$	3.37228			0.162815
$430$		0			0
$431$		−	31.0843i		−	1.49728i	−0.662977	−	0.748640i	$-0.730707\pi$
$431$		−	31.0843i		−	1.49728i	0.662977	−	0.748640i	$-0.269293\pi$
$432$		0			0
$433$		−	28.2337i		−	1.35682i	−0.734681	−	0.678412i	$-0.762668\pi$
$433$		−	28.2337i		−	1.35682i	0.734681	−	0.678412i	$-0.237332\pi$
$434$		0			0
$435$		−	4.25639i		−	0.204078i
$436$		0			0
$437$		−	6.51087i		−	0.311457i
$438$		0			0
$439$	−27.1229			−1.29451			−0.647253	−	0.762275i	$-0.724082\pi$
$439$	−27.1229			−1.29451			−0.647253	+	0.762275i	$0.724082\pi$
$440$		0			0
$441$	13.6277	−	9.48913i	0.648939	−	0.451863i
$442$		0			0
$443$		−	34.3461i		−	1.63183i	−0.578171	−	0.815915i	$-0.696234\pi$
$443$		−	34.3461i		−	1.63183i	0.578171	−	0.815915i	$-0.303766\pi$
$444$		0			0
$445$	17.4891			0.829064
$446$		0			0
$447$	−7.92287			−0.374739
$448$		0			0
$449$	17.6060			0.830877			0.415439	−	0.909621i	$-0.363628\pi$
$449$	17.6060			0.830877			0.415439	+	0.909621i	$0.363628\pi$
$450$		0			0
$451$	2.17448			0.102392
$452$		0			0
$453$		−	17.0951i		−	0.803198i
$454$		0			0
$455$	−4.25639	−	13.5615i	−0.199543	−	0.635771i
$456$		0			0
$457$	−7.48913			−0.350327			−0.175163	−	0.984539i	$-0.556045\pi$
$457$	−7.48913			−0.350327			−0.175163	+	0.984539i	$0.556045\pi$
$458$		0			0
$459$		−	14.3537i		−	0.669975i
$460$		0			0
$461$			11.2554i			0.524218i	0.965038	+	0.262109i	$0.0844180\pi$
$461$			11.2554i			0.524218i	−0.965038	+	0.262109i	$0.915582\pi$
$462$		0			0
$463$			11.3870i			0.529197i	0.964359	+	0.264599i	$0.0852395\pi$
$463$			11.3870i			0.529197i	−0.964359	+	0.264599i	$0.914760\pi$
$464$		0			0
$465$			6.74456i			0.312772i
$466$		0			0
$467$	12.4742			0.577238			0.288619	−	0.957444i	$-0.406804\pi$
$467$	12.4742			0.577238			0.288619	+	0.957444i	$0.406804\pi$
$468$		0			0
$469$	16.7446	−	5.25544i	0.773192	−	0.242674i
$470$		0			0
$471$		−	0.404759i		−	0.0186503i
$472$		0			0
$473$	−2.74456			−0.126195
$474$		0			0
$475$	−3.46410			−0.158944
$476$		0			0
$477$	−27.2554			−1.24794
$478$		0			0
$479$	−24.5437			−1.12143			−0.560714	−	0.828009i	$-0.689473\pi$
$479$	−24.5437			−1.12143			−0.560714	+	0.828009i	$0.689473\pi$
$480$		0			0
$481$		−	4.00000i		−	0.182384i
$482$		0			0
$483$	3.75906	−	1.17981i	0.171043	−	0.0536834i
$484$		0			0
$485$	2.11684			0.0961209
$486$		0			0
$487$			15.1460i			0.686332i	0.939275	+	0.343166i	$0.111499\pi$
$487$			15.1460i			0.686332i	−0.939275	+	0.343166i	$0.888501\pi$
$488$		0			0
$489$		−	12.0000i		−	0.542659i
$490$		0			0
$491$			25.7407i			1.16166i	0.814024	+	0.580831i	$0.197272\pi$
$491$			25.7407i			1.16166i	−0.814024	+	0.580831i	$0.802728\pi$
$492$		0			0
$493$		−	18.1168i		−	0.815942i
$494$		0			0
$495$	−1.87953			−0.0844785
$496$		0			0
$497$	16.7446	−	5.25544i	0.751096	−	0.235739i
$498$		0			0
$499$		−	32.6689i		−	1.46246i	−0.682130	−	0.731231i	$-0.738946\pi$
$499$		−	32.6689i		−	1.46246i	0.682130	−	0.731231i	$-0.261054\pi$
$500$		0			0
$501$	−12.6277			−0.564165
$502$		0			0
$503$	−33.9585			−1.51414			−0.757068	−	0.653336i	$-0.773369\pi$
$503$	−33.9585			−1.51414			−0.757068	+	0.653336i	$0.773369\pi$
$504$		0			0
$505$	15.4891			0.689257
$506$		0			0
$507$	−12.5668			−0.558111
$508$		0			0
$509$			0.744563i			0.0330022i	0.999864	+	0.0165011i	$0.00525270\pi$
$509$			0.744563i			0.0330022i	−0.999864	+	0.0165011i	$0.994747\pi$
$510$		0			0
$511$	−6.92820	+	2.17448i	−0.306486	+	0.0961934i
$512$		0			0
$513$	−14.7446			−0.650988
$514$		0			0
$515$			12.7692i			0.562677i
$516$		0			0
$517$		−	8.86141i		−	0.389724i
$518$		0			0
$519$		−	8.42020i		−	0.369606i
$520$		0			0
$521$		−	21.7228i		−	0.951694i	−0.879528	−	0.475847i	$-0.842142\pi$
$521$		−	21.7228i		−	0.951694i	0.879528	−	0.475847i	$-0.157858\pi$
$522$		0			0
$523$	−37.5152			−1.64043			−0.820213	−	0.572059i	$-0.806145\pi$
$523$	−37.5152			−1.64043			−0.820213	+	0.572059i	$0.806145\pi$
$524$		0			0
$525$	−0.627719	−	2.00000i	−0.0273959	−	0.0872872i
$526$		0			0
$527$			28.7075i			1.25052i
$528$		0			0
$529$	19.4674			0.846408
$530$		0			0
$531$	15.7359			0.682881
$532$		0			0
$533$	−14.7446			−0.638658
$534$		0			0
$535$	9.80240			0.423795
$536$		0			0
$537$		−	0.233688i		−	0.0100844i
$538$		0			0
$539$	3.16915	+	4.55134i	0.136505	+	0.196040i
$540$		0			0
$541$	9.37228			0.402946			0.201473	−	0.979494i	$-0.435427\pi$
$541$	9.37228			0.402946			0.201473	+	0.979494i	$0.435427\pi$
$542$		0			0
$543$			19.7899i			0.849268i
$544$		0			0
$545$		−	13.3723i		−	0.572806i
$546$		0			0
$547$		−	21.4843i		−	0.918603i	−0.888280	−	0.459302i	$-0.848100\pi$
$547$		−	21.4843i		−	0.918603i	0.888280	−	0.459302i	$-0.151900\pi$
$548$		0			0
$549$		−	1.76631i		−	0.0753844i
$550$		0			0
$551$	−18.6101			−0.792818
$552$		0			0
$553$	−35.4891	+	11.1386i	−1.50915	+	0.473661i
$554$		0			0
$555$		−	0.589907i		−	0.0250402i
$556$		0			0
$557$	23.7228			1.00517			0.502584	−	0.864528i	$-0.332383\pi$
$557$	23.7228			1.00517			0.502584	+	0.864528i	$0.332383\pi$
$558$		0			0
$559$	18.6101			0.787125
$560$		0			0
$561$	2.11684			0.0893732
$562$		0			0
$563$	16.1407			0.680249			0.340125	−	0.940380i	$-0.389531\pi$
$563$	16.1407			0.680249			0.340125	+	0.940380i	$0.389531\pi$
$564$		0			0
$565$			12.7446i			0.536168i
$566$		0			0
$567$	2.96677	+	9.45254i	0.124593	+	0.396969i
$568$		0			0
$569$	35.4891			1.48778			0.743891	−	0.668301i	$-0.232978\pi$
$569$	35.4891			1.48778			0.743891	+	0.668301i	$0.232978\pi$
$570$		0			0
$571$		−	23.6588i		−	0.990090i	−0.868867	−	0.495045i	$-0.835152\pi$
$571$		−	23.6588i		−	0.990090i	0.868867	−	0.495045i	$-0.164848\pi$
$572$		0			0
$573$			1.88316i			0.0786700i
$574$		0			0
$575$			1.87953i			0.0783817i
$576$		0			0
$577$			5.88316i			0.244919i	0.992474	+	0.122459i	$0.0390782\pi$
$577$			5.88316i			0.244919i	−0.992474	+	0.122459i	$0.960922\pi$
$578$		0			0
$579$	12.0867			0.502305
$580$		0			0
$581$	−8.23369	−	26.2337i	−0.341591	−	1.08836i
$582$		0			0
$583$		−	9.10268i		−	0.376995i
$584$		0			0
$585$	12.7446			0.526923
$586$		0			0
$587$	−3.46410			−0.142979			−0.0714894	−	0.997441i	$-0.522775\pi$
$587$	−3.46410			−0.142979			−0.0714894	+	0.997441i	$0.522775\pi$
$588$		0			0
$589$	29.4891			1.21508
$590$		0			0
$591$	4.75372			0.195542
$592$		0			0
$593$		−	11.3723i		−	0.467004i	−0.972356	−	0.233502i	$-0.924982\pi$
$593$		−	11.3723i		−	0.467004i	0.972356	−	0.233502i	$-0.0750185\pi$
$594$		0			0
$595$	−2.67181	−	8.51278i	−0.109534	−	0.348990i
$596$		0			0
$597$	14.7446			0.603455
$598$		0			0
$599$		−	32.6689i		−	1.33482i	−0.744692	−	0.667408i	$-0.767404\pi$
$599$		−	32.6689i		−	1.33482i	0.744692	−	0.667408i	$-0.232596\pi$
$600$		0			0
$601$		−	13.2554i		−	0.540701i	−0.962762	−	0.270350i	$-0.912860\pi$
$601$		−	13.2554i		−	0.540701i	0.962762	−	0.270350i	$-0.0871395\pi$
$602$		0			0
$603$			15.7359i			0.640817i
$604$		0			0
$605$			10.3723i			0.421693i
$606$		0			0
$607$	−39.8921			−1.61917			−0.809585	−	0.587003i	$-0.800308\pi$
$607$	−39.8921			−1.61917			−0.809585	+	0.587003i	$0.800308\pi$
$608$		0			0
$609$	−3.37228	−	10.7446i	−0.136652	−	0.435392i
$610$		0			0
$611$			60.0868i			2.43085i
$612$		0			0
$613$	24.9783			1.00886			0.504431	−	0.863452i	$-0.331702\pi$
$613$	24.9783			1.00886			0.504431	+	0.863452i	$0.331702\pi$
$614$		0			0
$615$	−2.17448			−0.0876835
$616$		0			0
$617$	31.2554			1.25830			0.629148	−	0.777285i	$-0.283404\pi$
$617$	31.2554			1.25830			0.629148	+	0.777285i	$0.283404\pi$
$618$		0			0
$619$	9.39764			0.377723			0.188861	−	0.982004i	$-0.439520\pi$
$619$	9.39764			0.377723			0.188861	+	0.982004i	$0.439520\pi$
$620$		0			0
$621$			8.00000i			0.321029i
$622$		0			0
$623$	44.1485	−	13.8564i	1.76877	−	0.555145i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	2.17448i		−	0.0868404i
$628$		0			0
$629$		−	2.51087i		−	0.100115i
$630$		0			0
$631$		−	20.9870i		−	0.835479i	−0.908567	−	0.417739i	$-0.862822\pi$
$631$		−	20.9870i		−	0.835479i	0.908567	−	0.417739i	$-0.137178\pi$
$632$		0			0
$633$			4.39403i			0.174647i
$634$		0			0
$635$	−6.63325			−0.263232
$636$		0			0
$637$	−21.4891	−	30.8614i	−0.851430	−	1.22277i
$638$		0			0
$639$			15.7359i			0.622504i
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	8.31040			0.327730			0.163865	−	0.986483i	$-0.447604\pi$
$643$	8.31040			0.327730			0.163865	+	0.986483i	$0.447604\pi$
$644$		0			0
$645$	2.74456			0.108067
$646$		0			0
$647$	−35.6357			−1.40098			−0.700492	−	0.713661i	$-0.747036\pi$
$647$	−35.6357			−1.40098			−0.700492	+	0.713661i	$0.747036\pi$
$648$		0			0
$649$			5.25544i			0.206294i
$650$		0			0
$651$	5.34363	+	17.0256i	0.209433	+	0.667284i
$652$		0			0
$653$	−50.4674			−1.97494			−0.987471	−	0.157804i	$-0.949559\pi$
$653$	−50.4674			−1.97494			−0.987471	+	0.157804i	$0.949559\pi$
$654$		0			0
$655$			8.80773i			0.344147i
$656$		0			0
$657$		−	6.51087i		−	0.254013i
$658$		0			0
$659$		−	40.1870i		−	1.56546i	−0.622359	−	0.782732i	$-0.713825\pi$
$659$		−	40.1870i		−	1.56546i	0.622359	−	0.782732i	$-0.286175\pi$
$660$		0			0
$661$		−	48.7446i		−	1.89594i	−0.318354	−	0.947972i	$-0.603130\pi$
$661$		−	48.7446i		−	1.89594i	0.318354	−	0.947972i	$-0.396870\pi$
$662$		0			0
$663$	−14.3537			−0.557453
$664$		0			0
$665$	−8.74456	+	2.74456i	−0.339100	+	0.106430i
$666$		0			0
$667$			10.0974i			0.390971i
$668$		0			0
$669$	4.62772			0.178918
$670$		0			0
$671$	0.589907			0.0227731
$672$		0			0
$673$	−18.2337			−0.702857			−0.351429	−	0.936215i	$-0.614304\pi$
$673$	−18.2337			−0.702857			−0.351429	+	0.936215i	$0.614304\pi$
$674$		0			0
$675$	4.25639			0.163828
$676$		0			0
$677$			24.1168i			0.926886i	0.886127	+	0.463443i	$0.153386\pi$
$677$			24.1168i			0.926886i	−0.886127	+	0.463443i	$0.846614\pi$
$678$		0			0
$679$	5.34363	−	1.67715i	0.205070	−	0.0643630i
$680$		0			0
$681$	12.3940			0.474940
$682$		0			0
$683$			29.5923i			1.13232i	0.824296	+	0.566160i	$0.191571\pi$
$683$			29.5923i			1.13232i	−0.824296	+	0.566160i	$0.808429\pi$
$684$		0			0
$685$			0.744563i			0.0284483i
$686$		0			0
$687$			11.0920i			0.423187i
$688$		0			0
$689$			61.7228i			2.35145i
$690$		0			0
$691$	−49.7870			−1.89399			−0.946994	−	0.321251i	$-0.895897\pi$
$691$	−49.7870			−1.89399			−0.946994	+	0.321251i	$0.895897\pi$
$692$		0			0
$693$	−4.74456	+	1.48913i	−0.180231	+	0.0565672i
$694$		0			0
$695$		−	15.1460i		−	0.574522i
$696$		0			0
$697$	−9.25544			−0.350575
$698$		0			0
$699$	0.589907			0.0223123
$700$		0			0
$701$	−36.1168			−1.36411			−0.682057	−	0.731299i	$-0.738914\pi$
$701$	−36.1168			−1.36411			−0.682057	+	0.731299i	$0.738914\pi$
$702$		0			0
$703$	−2.57924			−0.0972779
$704$		0			0
$705$			8.86141i			0.333740i
$706$		0			0
$707$	39.0998	−	12.2718i	1.47050	−	0.461530i
$708$		0			0
$709$	−5.37228			−0.201760			−0.100880	−	0.994899i	$-0.532166\pi$
$709$	−5.37228			−0.201760			−0.100880	+	0.994899i	$0.532166\pi$
$710$		0			0
$711$		−	33.3514i		−	1.25078i
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$			4.25639i			0.159180i
$716$		0			0
$717$			3.13859i			0.117213i
$718$		0			0
$719$	−6.33830			−0.236379			−0.118189	−	0.992991i	$-0.537709\pi$
$719$	−6.33830			−0.236379			−0.118189	+	0.992991i	$0.537709\pi$
$720$		0			0
$721$	10.1168	+	32.2337i	0.376771	+	1.20045i
$722$		0			0
$723$			12.6766i			0.471448i
$724$		0			0
$725$	5.37228			0.199522
$726$		0			0
$727$	22.9591			0.851506			0.425753	−	0.904840i	$-0.360009\pi$
$727$	22.9591			0.851506			0.425753	+	0.904840i	$0.360009\pi$
$728$		0			0
$729$	1.23369			0.0456921
$730$		0			0
$731$	11.6819			0.432072
$732$		0			0
$733$			34.8614i			1.28764i	0.765179	+	0.643818i	$0.222650\pi$
$733$			34.8614i			1.28764i	−0.765179	+	0.643818i	$0.777350\pi$
$734$		0			0
$735$	−3.16915	−	4.55134i	−0.116896	−	0.167879i
$736$		0			0
$737$	−5.25544			−0.193587
$738$		0			0
$739$		−	32.6689i		−	1.20175i	−0.799345	−	0.600873i	$-0.794820\pi$
$739$		−	32.6689i		−	1.20175i	0.799345	−	0.600873i	$-0.205180\pi$
$740$		0			0
$741$			14.7446i			0.541655i
$742$		0			0
$743$			20.4897i			0.751693i	0.926682	+	0.375846i	$0.122648\pi$
$743$			20.4897i			0.751693i	−0.926682	+	0.375846i	$0.877352\pi$
$744$		0			0
$745$		−	10.0000i		−	0.366372i
$746$		0			0
$747$	24.6535			0.902023
$748$		0			0
$749$	24.7446	−	7.76631i	0.904147	−	0.283775i
$750$		0			0
$751$		−	1.38219i		−	0.0504370i	−0.999682	−	0.0252185i	$-0.991972\pi$
$751$		−	1.38219i		−	0.0504370i	0.999682	−	0.0252185i	$-0.00802815\pi$
$752$		0			0
$753$	−6.97825			−0.254302
$754$		0			0
$755$	21.5769			0.785264
$756$		0			0
$757$	−11.2554			−0.409086			−0.204543	−	0.978858i	$-0.565571\pi$
$757$	−11.2554			−0.409086			−0.204543	+	0.978858i	$0.565571\pi$
$758$		0			0
$759$	−1.17981			−0.0428246
$760$		0			0
$761$			36.0000i			1.30500i	0.757789	+	0.652499i	$0.226280\pi$
$761$			36.0000i			1.30500i	−0.757789	+	0.652499i	$0.773720\pi$
$762$		0			0
$763$	−10.5947	−	33.7562i	−0.383553	−	1.22205i
$764$		0			0
$765$	8.00000			0.289241
$766$		0			0
$767$		−	35.6357i		−	1.28673i
$768$		0			0
$769$		−	3.76631i		−	0.135817i	−0.997692	−	0.0679083i	$-0.978367\pi$
$769$		−	3.76631i		−	0.135817i	0.997692	−	0.0679083i	$-0.0216325\pi$
$770$		0			0
$771$		−	0.994667i		−	0.0358220i
$772$		0			0
$773$		−	12.1168i		−	0.435813i	−0.975970	−	0.217906i	$-0.930077\pi$
$773$		−	12.1168i		−	0.435813i	0.975970	−	0.217906i	$-0.0699227\pi$
$774$		0			0
$775$	−8.51278			−0.305788
$776$		0			0
$777$	−0.467376	−	1.48913i	−0.0167670	−	0.0534221i
$778$		0			0
$779$			9.50744i			0.340640i
$780$		0			0
$781$	−5.25544			−0.188054
$782$		0			0
$783$	22.8665			0.817183
$784$		0			0
$785$	0.510875			0.0182339
$786$		0			0
$787$	−0.202380			−0.00721406			−0.00360703	−	0.999993i	$-0.501148\pi$
$787$	−0.202380			−0.00721406			−0.00360703	+	0.999993i	$0.501148\pi$
$788$		0			0
$789$			21.7228i			0.773353i
$790$		0			0
$791$	10.0974	+	32.1716i	0.359020	+	1.14389i
$792$		0			0
$793$	−4.00000			−0.142044
$794$		0			0
$795$			9.10268i			0.322839i
$796$		0			0
$797$			47.0951i			1.66819i	0.551618	+	0.834097i	$0.314011\pi$
$797$			47.0951i			1.66819i	−0.551618	+	0.834097i	$0.685989\pi$
$798$		0			0
$799$			37.7176i			1.33435i
$800$		0			0
$801$			41.4891i			1.46595i
$802$		0			0
$803$	2.17448			0.0767358
$804$		0			0
$805$	1.48913	+	4.74456i	0.0524848	+	0.167224i
$806$		0			0
$807$			0.589907i			0.0207657i
$808$		0			0
$809$	21.6060			0.759625			0.379813	−	0.925063i	$-0.375988\pi$
$809$	21.6060			0.759625			0.379813	+	0.925063i	$0.375988\pi$
$810$		0			0
$811$	15.7359			0.552563			0.276282	−	0.961077i	$-0.410898\pi$
$811$	15.7359			0.552563			0.276282	+	0.961077i	$0.410898\pi$
$812$		0			0
$813$	21.0217			0.737265
$814$		0			0
$815$	15.1460			0.530542
$816$		0			0
$817$		−	12.0000i		−	0.419827i
$818$		0			0
$819$	32.1716	−	10.0974i	1.12417	−	0.352830i
$820$		0			0
$821$	−15.0951			−0.526822			−0.263411	−	0.964684i	$-0.584848\pi$
$821$	−15.0951			−0.526822			−0.263411	+	0.964684i	$0.584848\pi$
$822$		0			0
$823$			55.1307i			1.92173i	0.277009	+	0.960867i	$0.410657\pi$
$823$			55.1307i			1.92173i	−0.277009	+	0.960867i	$0.589343\pi$
$824$		0			0
$825$			0.627719i			0.0218544i
$826$		0			0
$827$			5.04868i			0.175560i	0.996140	+	0.0877798i	$0.0279772\pi$
$827$			5.04868i			0.175560i	−0.996140	+	0.0877798i	$0.972023\pi$
$828$		0			0
$829$		−	19.4891i		−	0.676885i	−0.940987	−	0.338443i	$-0.890100\pi$
$829$		−	19.4891i		−	0.676885i	0.940987	−	0.338443i	$-0.109900\pi$
$830$		0			0
$831$	−11.2772			−0.391201
$832$		0			0
$833$	−13.4891	−	19.3723i	−0.467370	−	0.671210i
$834$		0			0
$835$		−	15.9383i		−	0.551568i
$836$		0			0
$837$	−36.2337			−1.25242
$838$		0			0
$839$	36.6303			1.26462			0.632310	−	0.774715i	$-0.282107\pi$
$839$	36.6303			1.26462			0.632310	+	0.774715i	$0.282107\pi$
$840$		0			0
$841$	−0.138593			−0.00477908
$842$		0			0
$843$	−17.1181			−0.589580
$844$		0			0
$845$		−	15.8614i		−	0.545649i
$846$		0			0
$847$	8.21782	+	26.1831i	0.282368	+	0.899663i
$848$		0			0
$849$	8.62772			0.296103
$850$		0			0
$851$			1.39943i			0.0479717i
$852$		0			0
$853$		−	7.48913i		−	0.256423i	−0.991747	−	0.128211i	$-0.959076\pi$
$853$		−	7.48913i		−	0.256423i	0.991747	−	0.128211i	$-0.0409236\pi$
$854$		0			0
$855$		−	8.21782i		−	0.281044i
$856$		0			0
$857$			43.2119i			1.47609i	0.674751	+	0.738046i	$0.264251\pi$
$857$			43.2119i			1.47609i	−0.674751	+	0.738046i	$0.735749\pi$
$858$		0			0
$859$	−36.5205			−1.24606			−0.623032	−	0.782196i	$-0.714100\pi$
$859$	−36.5205			−1.24606			−0.623032	+	0.782196i	$0.714100\pi$
$860$		0			0
$861$	−5.48913	+	1.72281i	−0.187069	+	0.0587133i
$862$		0			0
$863$		−	40.6844i		−	1.38491i	−0.721460	−	0.692456i	$-0.756529\pi$
$863$		−	40.6844i		−	1.38491i	0.721460	−	0.692456i	$-0.243471\pi$
$864$		0			0
$865$	10.6277			0.361353
$866$		0			0
$867$	4.45877			0.151428
$868$		0			0
$869$	11.1386			0.377851
$870$		0			0
$871$	35.6357			1.20747
$872$		0			0
$873$			5.02175i			0.169960i
$874$		0			0
$875$	2.52434	−	0.792287i	0.0853382	−	0.0267842i
$876$		0			0
$877$	−12.9783			−0.438244			−0.219122	−	0.975697i	$-0.570319\pi$
$877$	−12.9783			−0.438244			−0.219122	+	0.975697i	$0.570319\pi$
$878$		0			0
$879$			20.1021i			0.678028i
$880$		0			0
$881$		−	45.9565i		−	1.54831i	−0.632994	−	0.774157i	$-0.718174\pi$
$881$		−	45.9565i		−	1.54831i	0.632994	−	0.774157i	$-0.281826\pi$
$882$		0			0
$883$			26.4232i			0.889211i	0.895726	+	0.444606i	$0.146656\pi$
$883$			26.4232i			0.889211i	−0.895726	+	0.444606i	$0.853344\pi$
$884$		0			0
$885$		−	5.25544i		−	0.176660i
$886$		0			0
$887$	7.92287			0.266024			0.133012	−	0.991114i	$-0.457535\pi$
$887$	7.92287			0.266024			0.133012	+	0.991114i	$0.457535\pi$
$888$		0			0
$889$	−16.7446	+	5.25544i	−0.561595	+	0.176262i
$890$		0			0
$891$		−	2.96677i		−	0.0993905i
$892$		0			0
$893$	38.7446			1.29654
$894$		0			0
$895$	0.294954			0.00985921
$896$		0			0
$897$	8.00000			0.267112
$898$		0			0
$899$	−45.7330			−1.52528
$900$		0			0
$901$			38.7446i			1.29077i
$902$		0			0
$903$	6.92820	−	2.17448i	0.230556	−	0.0723622i
$904$		0			0
$905$	−24.9783			−0.830305
$906$		0			0
$907$		−	29.0024i		−	0.963010i	−0.876443	−	0.481505i	$-0.840090\pi$
$907$		−	29.0024i		−	0.963010i	0.876443	−	0.481505i	$-0.159910\pi$
$908$		0			0
$909$			36.7446i			1.21874i
$910$		0			0
$911$			2.28429i			0.0756818i	0.999284	+	0.0378409i	$0.0120480\pi$
$911$			2.28429i			0.0756818i	−0.999284	+	0.0378409i	$0.987952\pi$
$912$		0			0
$913$			8.23369i			0.272495i
$914$		0			0
$915$	−0.589907			−0.0195017
$916$		0			0
$917$	6.97825	+	22.2337i	0.230442	+	0.734221i
$918$		0			0
$919$			1.38219i			0.0455944i	0.999740	+	0.0227972i	$0.00725720\pi$
$919$			1.38219i			0.0455944i	−0.999740	+	0.0227972i	$0.992743\pi$
$920$		0			0
$921$	7.37228			0.242925
$922$		0			0
$923$	35.6357			1.17296
$924$		0			0
$925$	0.744563			0.0244811
$926$		0			0
$927$	−30.2921			−0.994922
$928$		0			0
$929$			30.7446i			1.00870i	0.863500	+	0.504348i	$0.168267\pi$
$929$			30.7446i			1.00870i	−0.863500	+	0.504348i	$0.831733\pi$
$930$		0			0
$931$	−19.8997	+	13.8564i	−0.652188	+	0.454125i
$932$		0			0
$933$	−17.2554			−0.564918
$934$		0			0
$935$			2.67181i			0.0873777i
$936$		0			0
$937$		−	38.1168i		−	1.24522i	−0.782531	−	0.622612i	$-0.786072\pi$
$937$		−	38.1168i		−	1.24522i	0.782531	−	0.622612i	$-0.213928\pi$
$938$		0			0
$939$			11.1846i			0.364995i
$940$		0			0
$941$			46.2337i			1.50718i	0.657348	+	0.753588i	$0.271678\pi$
$941$			46.2337i			1.50718i	−0.657348	+	0.753588i	$0.728322\pi$
$942$		0			0
$943$	5.15848			0.167983
$944$		0			0
$945$	10.7446	−	3.37228i	0.349521	−	0.109700i
$946$		0			0
$947$		−	12.9715i		−	0.421519i	−0.977538	−	0.210759i	$-0.932406\pi$
$947$		−	12.9715i		−	0.421519i	0.977538	−	0.210759i	$-0.0675936\pi$
$948$		0			0
$949$	−14.7446			−0.478629
$950$		0			0
$951$	21.7793			0.706241
$952$		0			0
$953$	−16.9783			−0.549979			−0.274990	−	0.961447i	$-0.588674\pi$
$953$	−16.9783			−0.549979			−0.274990	+	0.961447i	$0.588674\pi$
$954$		0			0
$955$	−2.37686			−0.0769134
$956$		0			0
$957$			3.37228i			0.109010i
$958$		0			0
$959$	0.589907	+	1.87953i	0.0190491	+	0.0606931i
$960$		0			0
$961$	41.4674			1.33766
$962$		0			0
$963$			23.2540i			0.749351i
$964$		0			0
$965$			15.2554i			0.491090i
$966$		0			0
$967$		−	0.884861i		−	0.0284552i	−0.999899	−	0.0142276i	$-0.995471\pi$
$967$		−	0.884861i		−	0.0284552i	0.999899	−	0.0142276i	$-0.00452894\pi$
$968$		0			0
$969$			9.25544i			0.297327i
$970$		0			0
$971$	−33.1662			−1.06436			−0.532178	−	0.846633i	$-0.678626\pi$
$971$	−33.1662			−1.06436			−0.532178	+	0.846633i	$0.678626\pi$
$972$		0			0
$973$	−12.0000	−	38.2337i	−0.384702	−	1.22572i
$974$		0			0
$975$		−	4.25639i		−	0.136314i
$976$		0			0
$977$	−24.9783			−0.799125			−0.399563	−	0.916706i	$-0.630838\pi$
$977$	−24.9783			−0.799125			−0.399563	+	0.916706i	$0.630838\pi$
$978$		0			0
$979$	−13.8564			−0.442853
$980$		0			0
$981$	31.7228			1.01283
$982$		0			0
$983$	−3.07657			−0.0981275			−0.0490637	−	0.998796i	$-0.515624\pi$
$983$	−3.07657			−0.0981275			−0.0490637	+	0.998796i	$0.515624\pi$
$984$		0			0
$985$			6.00000i			0.191176i
$986$		0			0
$987$	7.02078	+	22.3692i	0.223474	+	0.712019i
$988$		0			0
$989$	−6.51087			−0.207034
$990$		0			0
$991$		−	57.7099i		−	1.83322i	−0.399787	−	0.916608i	$-0.630916\pi$
$991$		−	57.7099i		−	1.83322i	0.399787	−	0.916608i	$-0.369084\pi$
$992$		0			0
$993$			19.2119i			0.609672i
$994$		0			0
$995$			18.6101i			0.589981i
$996$		0			0
$997$		−	32.3505i		−	1.02455i	−0.858821	−	0.512276i	$-0.828803\pi$
$997$		−	32.3505i		−	1.02455i	0.858821	−	0.512276i	$-0.171197\pi$
$998$		0			0
$999$	3.16915			0.100267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.k.a.111.6	yes	8
3.2	odd	2		5040.2.d.e.4591.3		8
4.3	odd	2	inner	560.2.k.a.111.4	yes	8
5.2	odd	4		2800.2.e.j.2799.3		8
5.3	odd	4		2800.2.e.i.2799.6		8
5.4	even	2		2800.2.k.l.2351.3		8
7.6	odd	2	inner	560.2.k.a.111.3	✓	8
8.3	odd	2		2240.2.k.c.1791.5		8
8.5	even	2		2240.2.k.c.1791.3		8
12.11	even	2		5040.2.d.e.4591.2		8
20.3	even	4		2800.2.e.i.2799.3		8
20.7	even	4		2800.2.e.j.2799.6		8
20.19	odd	2		2800.2.k.l.2351.6		8
21.20	even	2		5040.2.d.e.4591.6		8
28.27	even	2	inner	560.2.k.a.111.5	yes	8
35.13	even	4		2800.2.e.j.2799.4		8
35.27	even	4		2800.2.e.i.2799.5		8
35.34	odd	2		2800.2.k.l.2351.5		8
56.13	odd	2		2240.2.k.c.1791.6		8
56.27	even	2		2240.2.k.c.1791.4		8
84.83	odd	2		5040.2.d.e.4591.7		8
140.27	odd	4		2800.2.e.i.2799.4		8
140.83	odd	4		2800.2.e.j.2799.5		8
140.139	even	2		2800.2.k.l.2351.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.k.a.111.3	✓	8	7.6	odd	2	inner
560.2.k.a.111.4	yes	8	4.3	odd	2	inner
560.2.k.a.111.5	yes	8	28.27	even	2	inner
560.2.k.a.111.6	yes	8	1.1	even	1	trivial
2240.2.k.c.1791.3		8	8.5	even	2
2240.2.k.c.1791.4		8	56.27	even	2
2240.2.k.c.1791.5		8	8.3	odd	2
2240.2.k.c.1791.6		8	56.13	odd	2
2800.2.e.i.2799.3		8	20.3	even	4
2800.2.e.i.2799.4		8	140.27	odd	4
2800.2.e.i.2799.5		8	35.27	even	4
2800.2.e.i.2799.6		8	5.3	odd	4
2800.2.e.j.2799.3		8	5.2	odd	4
2800.2.e.j.2799.4		8	35.13	even	4
2800.2.e.j.2799.5		8	140.83	odd	4
2800.2.e.j.2799.6		8	20.7	even	4
2800.2.k.l.2351.3		8	5.4	even	2
2800.2.k.l.2351.4		8	140.139	even	2
2800.2.k.l.2351.5		8	35.34	odd	2
2800.2.k.l.2351.6		8	20.19	odd	2
5040.2.d.e.4591.2		8	12.11	even	2
5040.2.d.e.4591.3		8	3.2	odd	2
5040.2.d.e.4591.6		8	21.20	even	2
5040.2.d.e.4591.7		8	84.83	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.792287 q^{3} +1.00000i q^{5} +(0.792287 + 2.52434i) q^{7} -2.37228 q^{9} +O(q^{10})\)	\(q+0.792287 q^{3} +1.00000i q^{5} +(0.792287 + 2.52434i) q^{7} -2.37228 q^{9} -0.792287i q^{11} +5.37228i q^{13} +0.792287i q^{15} +3.37228i q^{17} +3.46410 q^{19} +(0.627719 + 2.00000i) q^{21} -1.87953i q^{23} -1.00000 q^{25} -4.25639 q^{27} -5.37228 q^{29} +8.51278 q^{31} -0.627719i q^{33} +(-2.52434 + 0.792287i) q^{35} -0.744563 q^{37} +4.25639i q^{39} +2.74456i q^{41} -3.46410i q^{43} -2.37228i q^{45} +11.1846 q^{47} +(-5.74456 + 4.00000i) q^{49} +2.67181i q^{51} +11.4891 q^{53} +0.792287 q^{55} +2.74456 q^{57} -6.63325 q^{59} +0.744563i q^{61} +(-1.87953 - 5.98844i) q^{63} -5.37228 q^{65} -6.63325i q^{67} -1.48913i q^{69} -6.63325i q^{71} +2.74456i q^{73} -0.792287 q^{75} +(2.00000 - 0.627719i) q^{77} +14.0588i q^{79} +3.74456 q^{81} -10.3923 q^{83} -3.37228 q^{85} -4.25639 q^{87} -17.4891i q^{89} +(-13.5615 + 4.25639i) q^{91} +6.74456 q^{93} +3.46410i q^{95} -2.11684i q^{97} +1.87953i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} + 28 q^{21} - 8 q^{25} - 20 q^{29} + 40 q^{37} - 24 q^{57} - 20 q^{65} + 16 q^{77} - 16 q^{81} - 4 q^{85} + 8 q^{93}+O(q^{100}) \) 8 * q + 4 * q^9 + 28 * q^21 - 8 * q^25 - 20 * q^29 + 40 * q^37 - 24 * q^57 - 20 * q^65 + 16 * q^77 - 16 * q^81 - 4 * q^85 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more