LMFDB - Embedded newform 1120.2.h.a.111.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(111,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1120.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 2x^{12} + 6x^{11} - 12x^{9} + 8x^{8} - 24x^{7} + 48x^{5} - 32x^{4} - 128x + 256 $ x^16 - x^15 - 2x^12 + 6x^11 - 12x^9 + 8x^8 - 24x^7 + 48x^5 - 32x^4 - 128x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.12
Root			$-1.24098 - 0.678208i$ of defining polynomial
Character	$\chi$	$=$	1120.111
Dual form			1120.2.h.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)

$f(q)$	$=$	$q+1.61069i q^{3} -1.00000 q^{5} +(2.13463 - 1.56312i) q^{7} +0.405694 q^{9} +O(q^{10})$	$q+1.61069i q^{3} -1.00000 q^{5} +(2.13463 - 1.56312i) q^{7} +0.405694 q^{9} +6.01679 q^{11} -4.25411 q^{13} -1.61069i q^{15} -5.42124i q^{17} -4.53588i q^{19} +(2.51769 + 3.43822i) q^{21} -5.19890i q^{23} +1.00000 q^{25} +5.48550i q^{27} +0.376818i q^{29} +2.93636 q^{31} +9.69116i q^{33} +(-2.13463 + 1.56312i) q^{35} -0.372265i q^{37} -6.85203i q^{39} +5.75560i q^{41} +4.96515 q^{43} -0.405694 q^{45} +3.86303 q^{47} +(2.11332 - 6.67337i) q^{49} +8.73190 q^{51} -10.2224i q^{53} -6.01679 q^{55} +7.30587 q^{57} +10.5835i q^{59} +5.58001 q^{61} +(0.866007 - 0.634147i) q^{63} +4.25411 q^{65} +0.782596 q^{67} +8.37378 q^{69} +15.3803i q^{71} +8.77164i q^{73} +1.61069i q^{75} +(12.8437 - 9.40496i) q^{77} +8.74609i q^{79} -7.61833 q^{81} -8.42742i q^{83} +5.42124i q^{85} -0.606935 q^{87} +1.94765i q^{89} +(-9.08097 + 6.64968i) q^{91} +4.72954i q^{93} +4.53588i q^{95} +3.14377i q^{97} +2.44098 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^5 - 16 * q^9	$ 16 q - 16 q^{5} - 16 q^{9} + 4 q^{11} + 4 q^{21} + 16 q^{25} - 16 q^{31} + 4 q^{43} + 16 q^{45} - 8 q^{49} + 40 q^{51} - 4 q^{55} - 16 q^{57} + 8 q^{61} + 28 q^{63} - 20 q^{67} + 40 q^{69} + 4 q^{77} + 24 q^{81} + 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) $ 16 * q - 16 * q^5 - 16 * q^9 + 4 * q^11 + 4 * q^21 + 16 * q^25 - 16 * q^31 + 4 * q^43 + 16 * q^45 - 8 * q^49 + 40 * q^51 - 4 * q^55 - 16 * q^57 + 8 * q^61 + 28 * q^63 - 20 * q^67 + 40 * q^69 + 4 * q^77 + 24 * q^81 + 72 * q^87 + 32 * q^91 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$			1.61069i			0.929929i	0.885329	+	0.464965i	$0.153933\pi$
$3$			1.61069i			0.929929i	−0.885329	+	0.464965i	$0.846067\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	2.13463	−	1.56312i	0.806816	−	0.590803i
$7$	2.13463	−	1.56312i	0.806816	−	0.590803i
$8$		0			0
$9$	0.405694			0.135231
$10$		0			0
$11$	6.01679			1.81413			0.907066	−	0.420989i	$-0.138317\pi$
$11$	6.01679			1.81413			0.907066	+	0.420989i	$0.138317\pi$
$12$		0			0
$13$	−4.25411			−1.17988			−0.589939	−	0.807448i	$-0.700848\pi$
$13$	−4.25411			−1.17988			−0.589939	+	0.807448i	$0.700848\pi$
$14$		0			0
$15$		−	1.61069i		−	0.415877i
$16$		0			0
$17$		−	5.42124i		−	1.31484i	−0.753523	−	0.657421i	$-0.771647\pi$
$17$		−	5.42124i		−	1.31484i	0.753523	−	0.657421i	$-0.228353\pi$
$18$		0			0
$19$		−	4.53588i		−	1.04060i	−0.853983	−	0.520301i	$-0.825820\pi$
$19$		−	4.53588i		−	1.04060i	0.853983	−	0.520301i	$-0.174180\pi$
$20$		0			0
$21$	2.51769	+	3.43822i	0.549405	+	0.750282i
$22$		0			0
$23$		−	5.19890i		−	1.08404i	−0.840364	−	0.542022i	$-0.817659\pi$
$23$		−	5.19890i		−	1.08404i	0.840364	−	0.542022i	$-0.182341\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			5.48550i			1.05568i
$28$		0			0
$29$			0.376818i			0.0699733i	0.999388	+	0.0349866i	$0.0111389\pi$
$29$			0.376818i			0.0699733i	−0.999388	+	0.0349866i	$0.988861\pi$
$30$		0			0
$31$	2.93636			0.527385			0.263693	−	0.964607i	$-0.415060\pi$
$31$	2.93636			0.527385			0.263693	+	0.964607i	$0.415060\pi$
$32$		0			0
$33$			9.69116i			1.68701i
$34$		0			0
$35$	−2.13463	+	1.56312i	−0.360819	+	0.264215i
$36$		0			0
$37$		−	0.372265i		−	0.0612000i	−0.999532	−	0.0306000i	$-0.990258\pi$
$37$		−	0.372265i		−	0.0612000i	0.999532	−	0.0306000i	$-0.00974181\pi$
$38$		0			0
$39$		−	6.85203i		−	1.09720i
$40$		0			0
$41$			5.75560i			0.898874i	0.893312	+	0.449437i	$0.148375\pi$
$41$			5.75560i			0.898874i	−0.893312	+	0.449437i	$0.851625\pi$
$42$		0			0
$43$	4.96515			0.757178			0.378589	−	0.925565i	$-0.376409\pi$
$43$	4.96515			0.757178			0.378589	+	0.925565i	$0.376409\pi$
$44$		0			0
$45$	−0.405694			−0.0604772
$46$		0			0
$47$	3.86303			0.563481			0.281741	−	0.959491i	$-0.409088\pi$
$47$	3.86303			0.563481			0.281741	+	0.959491i	$0.409088\pi$
$48$		0			0
$49$	2.11332	−	6.67337i	0.301903	−	0.953339i
$50$		0			0
$51$	8.73190			1.22271
$52$		0			0
$53$		−	10.2224i		−	1.40416i	−0.712099	−	0.702079i	$-0.752255\pi$
$53$		−	10.2224i		−	1.40416i	0.712099	−	0.702079i	$-0.247745\pi$
$54$		0			0
$55$	−6.01679			−0.811304
$56$		0			0
$57$	7.30587			0.967686
$58$		0			0
$59$			10.5835i			1.37786i	0.724828	+	0.688930i	$0.241919\pi$
$59$			10.5835i			1.37786i	−0.724828	+	0.688930i	$0.758081\pi$
$60$		0			0
$61$	5.58001			0.714447			0.357223	−	0.934019i	$-0.383723\pi$
$61$	5.58001			0.714447			0.357223	+	0.934019i	$0.383723\pi$
$62$		0			0
$63$	0.866007	−	0.634147i	0.109107	−	0.0798950i
$64$		0			0
$65$	4.25411			0.527657
$66$		0			0
$67$	0.782596			0.0956093			0.0478047	−	0.998857i	$-0.484777\pi$
$67$	0.782596			0.0956093			0.0478047	+	0.998857i	$0.484777\pi$
$68$		0			0
$69$	8.37378			1.00809
$70$		0			0
$71$			15.3803i			1.82531i	0.408735	+	0.912653i	$0.365970\pi$
$71$			15.3803i			1.82531i	−0.408735	+	0.912653i	$0.634030\pi$
$72$		0			0
$73$			8.77164i			1.02664i	0.858196	+	0.513321i	$0.171585\pi$
$73$			8.77164i			1.02664i	−0.858196	+	0.513321i	$0.828415\pi$
$74$		0			0
$75$			1.61069i			0.185986i
$76$		0			0
$77$	12.8437	−	9.40496i	1.46367	−	1.07179i
$78$		0			0
$79$			8.74609i			0.984012i	0.870592	+	0.492006i	$0.163736\pi$
$79$			8.74609i			0.984012i	−0.870592	+	0.492006i	$0.836264\pi$
$80$		0			0
$81$	−7.61833			−0.846481
$82$		0			0
$83$		−	8.42742i		−	0.925029i	−0.886612	−	0.462515i	$-0.846947\pi$
$83$		−	8.42742i		−	0.925029i	0.886612	−	0.462515i	$-0.153053\pi$
$84$		0			0
$85$			5.42124i			0.588016i
$86$		0			0
$87$	−0.606935			−0.0650702
$88$		0			0
$89$			1.94765i			0.206451i	0.994658	+	0.103225i	$0.0329163\pi$
$89$			1.94765i			0.206451i	−0.994658	+	0.103225i	$0.967084\pi$
$90$		0			0
$91$	−9.08097	+	6.64968i	−0.951944	+	0.697076i
$92$		0			0
$93$			4.72954i			0.490431i
$94$		0			0
$95$			4.53588i			0.465371i
$96$		0			0
$97$			3.14377i			0.319201i	0.987182	+	0.159601i	$0.0510206\pi$
$97$			3.14377i			0.319201i	−0.987182	+	0.159601i	$0.948979\pi$
$98$		0			0
$99$	2.44098			0.245327
$100$		0			0
$101$	−16.3016			−1.62207			−0.811037	−	0.584995i	$-0.801096\pi$
$101$	−16.3016			−1.62207			−0.811037	+	0.584995i	$0.801096\pi$
$102$		0			0
$103$	11.8940			1.17195			0.585974	−	0.810330i	$-0.300712\pi$
$103$	11.8940			1.17195			0.585974	+	0.810330i	$0.300712\pi$
$104$		0			0
$105$	−2.51769	−	3.43822i	−0.245702	−	0.335536i
$106$		0			0
$107$	−10.4794			−1.01308			−0.506542	−	0.862215i	$-0.669077\pi$
$107$	−10.4794			−1.01308			−0.506542	+	0.862215i	$0.669077\pi$
$108$		0			0
$109$		−	0.676794i		−	0.0648251i	−0.999475	−	0.0324126i	$-0.989681\pi$
$109$		−	0.676794i		−	0.0648251i	0.999475	−	0.0324126i	$-0.0103190\pi$
$110$		0			0
$111$	0.599602			0.0569117
$112$		0			0
$113$	−9.10537			−0.856562			−0.428281	−	0.903646i	$-0.640881\pi$
$113$	−9.10537			−0.856562			−0.428281	+	0.903646i	$0.640881\pi$
$114$		0			0
$115$			5.19890i			0.484800i
$116$		0			0
$117$	−1.72587			−0.159556
$118$		0			0
$119$	−8.47403	−	11.5724i	−0.776813	−	1.06084i
$120$		0			0
$121$	25.2018			2.29107
$122$		0			0
$123$	−9.27046			−0.835889
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	3.74123i		−	0.331980i	−0.986127	−	0.165990i	$-0.946918\pi$
$127$		−	3.74123i		−	0.331980i	0.986127	−	0.165990i	$-0.0530820\pi$
$128$		0			0
$129$			7.99729i			0.704122i
$130$		0			0
$131$		−	17.5774i		−	1.53575i	−0.640601	−	0.767874i	$-0.721315\pi$
$131$		−	17.5774i		−	1.53575i	0.640601	−	0.767874i	$-0.278685\pi$
$132$		0			0
$133$	−7.09012	−	9.68244i	−0.614791	−	0.839574i
$134$		0			0
$135$		−	5.48550i		−	0.472117i
$136$		0			0
$137$	−0.767338			−0.0655581			−0.0327791	−	0.999463i	$-0.510436\pi$
$137$	−0.767338			−0.0655581			−0.0327791	+	0.999463i	$0.510436\pi$
$138$		0			0
$139$			2.01854i			0.171210i	0.996329	+	0.0856052i	$0.0272824\pi$
$139$			2.01854i			0.171210i	−0.996329	+	0.0856052i	$0.972718\pi$
$140$		0			0
$141$			6.22213i			0.523998i
$142$		0			0
$143$	−25.5961			−2.14045
$144$		0			0
$145$		−	0.376818i		−	0.0312930i
$146$		0			0
$147$	10.7487	+	3.40390i	0.886538	+	0.280749i
$148$		0			0
$149$		−	4.23267i		−	0.346754i	−0.984856	−	0.173377i	$-0.944532\pi$
$149$		−	4.23267i		−	0.346754i	0.984856	−	0.173377i	$-0.0554679\pi$
$150$		0			0
$151$		−	0.625768i		−	0.0509243i	−0.999676	−	0.0254622i	$-0.991894\pi$
$151$		−	0.625768i		−	0.0509243i	0.999676	−	0.0254622i	$-0.00810573\pi$
$152$		0			0
$153$		−	2.19936i		−	0.177808i
$154$		0			0
$155$	−2.93636			−0.235854
$156$		0			0
$157$	2.91713			0.232812			0.116406	−	0.993202i	$-0.462863\pi$
$157$	2.91713			0.232812			0.116406	+	0.993202i	$0.462863\pi$
$158$		0			0
$159$	16.4651			1.30577
$160$		0			0
$161$	−8.12649	−	11.0977i	−0.640457	−	0.874624i
$162$		0			0
$163$	7.14397			0.559559			0.279779	−	0.960064i	$-0.409739\pi$
$163$	7.14397			0.559559			0.279779	+	0.960064i	$0.409739\pi$
$164$		0			0
$165$		−	9.69116i		−	0.754456i
$166$		0			0
$167$	6.26796			0.485029			0.242515	−	0.970148i	$-0.422028\pi$
$167$	6.26796			0.485029			0.242515	+	0.970148i	$0.422028\pi$
$168$		0			0
$169$	5.09746			0.392112
$170$		0			0
$171$		−	1.84018i		−	0.140722i
$172$		0			0
$173$	−9.11366			−0.692898			−0.346449	−	0.938069i	$-0.612613\pi$
$173$	−9.11366			−0.692898			−0.346449	+	0.938069i	$0.612613\pi$
$174$		0			0
$175$	2.13463	−	1.56312i	0.161363	−	0.118161i
$176$		0			0
$177$	−17.0468			−1.28131
$178$		0			0
$179$	−10.9518			−0.818579			−0.409290	−	0.912405i	$-0.634223\pi$
$179$	−10.9518			−0.818579			−0.409290	+	0.912405i	$0.634223\pi$
$180$		0			0
$181$	18.7975			1.39721			0.698603	−	0.715509i	$-0.253805\pi$
$181$	18.7975			1.39721			0.698603	+	0.715509i	$0.253805\pi$
$182$		0			0
$183$			8.98763i			0.664385i
$184$		0			0
$185$			0.372265i			0.0273695i
$186$		0			0
$187$		−	32.6185i		−	2.38530i
$188$		0			0
$189$	8.57449	+	11.7095i	0.623702	+	0.851743i
$190$		0			0
$191$		−	1.13019i		−	0.0817778i	−0.999164	−	0.0408889i	$-0.986981\pi$
$191$		−	1.13019i		−	0.0817778i	0.999164	−	0.0408889i	$-0.0130190\pi$
$192$		0			0
$193$	−3.27357			−0.235636			−0.117818	−	0.993035i	$-0.537590\pi$
$193$	−3.27357			−0.235636			−0.117818	+	0.993035i	$0.537590\pi$
$194$		0			0
$195$			6.85203i			0.490684i
$196$		0			0
$197$			21.2252i			1.51223i	0.654437	+	0.756116i	$0.272906\pi$
$197$			21.2252i			1.51223i	−0.654437	+	0.756116i	$0.727094\pi$
$198$		0			0
$199$	−17.9614			−1.27325			−0.636626	−	0.771172i	$-0.719671\pi$
$199$	−17.9614			−1.27325			−0.636626	+	0.771172i	$0.719671\pi$
$200$		0			0
$201$			1.26052i			0.0889099i
$202$		0			0
$203$	0.589011	+	0.804368i	0.0413404	+	0.0564556i
$204$		0			0
$205$		−	5.75560i		−	0.401989i
$206$		0			0
$207$		−	2.10916i		−	0.146597i
$208$		0			0
$209$		−	27.2915i		−	1.88779i
$210$		0			0
$211$	7.43642			0.511944			0.255972	−	0.966684i	$-0.417604\pi$
$211$	7.43642			0.511944			0.255972	+	0.966684i	$0.417604\pi$
$212$		0			0
$213$	−24.7728			−1.69741
$214$		0			0
$215$	−4.96515			−0.338620
$216$		0			0
$217$	6.26804	−	4.58987i	0.425503	−	0.311581i
$218$		0			0
$219$	−14.1283			−0.954705
$220$		0			0
$221$			23.0625i			1.55135i
$222$		0			0
$223$	−23.8641			−1.59806			−0.799029	−	0.601293i	$-0.794653\pi$
$223$	−23.8641			−1.59806			−0.799029	+	0.601293i	$0.794653\pi$
$224$		0			0
$225$	0.405694			0.0270462
$226$		0			0
$227$		−	12.1282i		−	0.804974i	−0.915426	−	0.402487i	$-0.868146\pi$
$227$		−	12.1282i		−	0.804974i	0.915426	−	0.402487i	$-0.131854\pi$
$228$		0			0
$229$	16.4001			1.08375			0.541875	−	0.840459i	$-0.317715\pi$
$229$	16.4001			1.08375			0.541875	+	0.840459i	$0.317715\pi$
$230$		0			0
$231$	15.1484	+	20.6871i	0.996694	+	1.36111i
$232$		0			0
$233$	−26.4108			−1.73023			−0.865114	−	0.501575i	$-0.832754\pi$
$233$	−26.4108			−1.73023			−0.865114	+	0.501575i	$0.832754\pi$
$234$		0			0
$235$	−3.86303			−0.251997
$236$		0			0
$237$	−14.0872			−0.915062
$238$		0			0
$239$			1.68909i			0.109258i	0.998507	+	0.0546292i	$0.0173977\pi$
$239$			1.68909i			0.109258i	−0.998507	+	0.0546292i	$0.982602\pi$
$240$		0			0
$241$		−	5.25118i		−	0.338258i	−0.985594	−	0.169129i	$-0.945905\pi$
$241$		−	5.25118i		−	0.338258i	0.985594	−	0.169129i	$-0.0540955\pi$
$242$		0			0
$243$			4.18577i			0.268517i
$244$		0			0
$245$	−2.11332	+	6.67337i	−0.135015	+	0.426346i
$246$		0			0
$247$			19.2961i			1.22778i
$248$		0			0
$249$	13.5739			0.860212
$250$		0			0
$251$		−	13.9692i		−	0.881730i	−0.897574	−	0.440865i	$-0.854672\pi$
$251$		−	13.9692i		−	0.881730i	0.897574	−	0.440865i	$-0.145328\pi$
$252$		0			0
$253$		−	31.2807i		−	1.96660i
$254$		0			0
$255$	−8.73190			−0.546813
$256$		0			0
$257$		−	21.6938i		−	1.35322i	−0.736341	−	0.676610i	$-0.763448\pi$
$257$		−	21.6938i		−	1.35322i	0.736341	−	0.676610i	$-0.236552\pi$
$258$		0			0
$259$	−0.581895	−	0.794650i	−0.0361572	−	0.0493771i
$260$		0			0
$261$			0.152873i			0.00946257i
$262$		0			0
$263$		−	5.88968i		−	0.363173i	−0.983375	−	0.181587i	$-0.941877\pi$
$263$		−	5.88968i		−	0.363173i	0.983375	−	0.181587i	$-0.0581232\pi$
$264$		0			0
$265$			10.2224i			0.627958i
$266$		0			0
$267$	−3.13706			−0.191985
$268$		0			0
$269$	−13.3475			−0.813810			−0.406905	−	0.913470i	$-0.633392\pi$
$269$	−13.3475			−0.813810			−0.406905	+	0.913470i	$0.633392\pi$
$270$		0			0
$271$	24.7908			1.50593			0.752966	−	0.658059i	$-0.228622\pi$
$271$	24.7908			1.50593			0.752966	+	0.658059i	$0.228622\pi$
$272$		0			0
$273$	−10.7105	−	14.6266i	−0.648231	−	0.885241i
$274$		0			0
$275$	6.01679			0.362826
$276$		0			0
$277$		−	2.09066i		−	0.125616i	−0.998026	−	0.0628078i	$-0.979994\pi$
$277$		−	2.09066i		−	0.125616i	0.998026	−	0.0628078i	$-0.0200055\pi$
$278$		0			0
$279$	1.19126			0.0713189
$280$		0			0
$281$	−4.09181			−0.244097			−0.122048	−	0.992524i	$-0.538946\pi$
$281$	−4.09181			−0.244097			−0.122048	+	0.992524i	$0.538946\pi$
$282$		0			0
$283$			5.92294i			0.352082i	0.984383	+	0.176041i	$0.0563291\pi$
$283$			5.92294i			0.352082i	−0.984383	+	0.176041i	$0.943671\pi$
$284$		0			0
$285$	−7.30587			−0.432763
$286$		0			0
$287$	8.99669	+	12.2861i	0.531058	+	0.725226i
$288$		0			0
$289$	−12.3898			−0.728812
$290$		0			0
$291$	−5.06362			−0.296835
$292$		0			0
$293$	−9.44316			−0.551675			−0.275838	−	0.961204i	$-0.588955\pi$
$293$	−9.44316			−0.551675			−0.275838	+	0.961204i	$0.588955\pi$
$294$		0			0
$295$		−	10.5835i		−	0.616198i
$296$		0			0
$297$			33.0051i			1.91515i
$298$		0			0
$299$			22.1167i			1.27904i
$300$		0			0
$301$	10.5988	−	7.76111i	0.610903	−	0.447343i
$302$		0			0
$303$		−	26.2568i		−	1.50841i
$304$		0			0
$305$	−5.58001			−0.319510
$306$		0			0
$307$			24.1204i			1.37663i	0.725414	+	0.688313i	$0.241649\pi$
$307$			24.1204i			1.37663i	−0.725414	+	0.688313i	$0.758351\pi$
$308$		0			0
$309$			19.1574i			1.08983i
$310$		0			0
$311$	16.5947			0.941001			0.470500	−	0.882400i	$-0.344073\pi$
$311$	16.5947			0.941001			0.470500	+	0.882400i	$0.344073\pi$
$312$		0			0
$313$			11.3871i			0.643637i	0.946801	+	0.321819i	$0.104294\pi$
$313$			11.3871i			0.643637i	−0.946801	+	0.321819i	$0.895706\pi$
$314$		0			0
$315$	−0.866007	+	0.634147i	−0.0487940	+	0.0357301i
$316$		0			0
$317$			26.7084i			1.50009i	0.661386	+	0.750046i	$0.269969\pi$
$317$			26.7084i			1.50009i	−0.661386	+	0.750046i	$0.730031\pi$
$318$		0			0
$319$			2.26723i			0.126941i
$320$		0			0
$321$		−	16.8791i		−	0.942097i
$322$		0			0
$323$	−24.5901			−1.36823
$324$		0			0
$325$	−4.25411			−0.235976
$326$		0			0
$327$	1.09010			0.0602828
$328$		0			0
$329$	8.24616	−	6.03838i	0.454626	−	0.332907i
$330$		0			0
$331$	3.25364			0.178836			0.0894182	−	0.995994i	$-0.471499\pi$
$331$	3.25364			0.178836			0.0894182	+	0.995994i	$0.471499\pi$
$332$		0			0
$333$		−	0.151026i		−	0.00827615i
$334$		0			0
$335$	−0.782596			−0.0427578
$336$		0			0
$337$	−12.0157			−0.654539			−0.327270	−	0.944931i	$-0.606129\pi$
$337$	−12.0157			−0.654539			−0.327270	+	0.944931i	$0.606129\pi$
$338$		0			0
$339$		−	14.6659i		−	0.796542i
$340$		0			0
$341$	17.6675			0.956746
$342$		0			0
$343$	−5.92010	−	17.5486i	−0.319655	−	0.947534i
$344$		0			0
$345$	−8.37378			−0.450829
$346$		0			0
$347$	15.5740			0.836057			0.418028	−	0.908434i	$-0.362721\pi$
$347$	15.5740			0.836057			0.418028	+	0.908434i	$0.362721\pi$
$348$		0			0
$349$	−34.7705			−1.86122			−0.930610	−	0.366012i	$-0.880723\pi$
$349$	−34.7705			−1.86122			−0.930610	+	0.366012i	$0.880723\pi$
$350$		0			0
$351$		−	23.3359i		−	1.24558i
$352$		0			0
$353$			12.5154i			0.666126i	0.942905	+	0.333063i	$0.108082\pi$
$353$			12.5154i			0.666126i	−0.942905	+	0.333063i	$0.891918\pi$
$354$		0			0
$355$		−	15.3803i		−	0.816302i
$356$		0			0
$357$	18.6394	−	13.6490i	0.986502	−	0.722382i
$358$		0			0
$359$			31.0752i			1.64009i	0.572301	+	0.820044i	$0.306051\pi$
$359$			31.0752i			1.64009i	−0.572301	+	0.820044i	$0.693949\pi$
$360$		0			0
$361$	−1.57420			−0.0828526
$362$		0			0
$363$			40.5922i			2.13054i
$364$		0			0
$365$		−	8.77164i		−	0.459129i
$366$		0			0
$367$	−14.0542			−0.733624			−0.366812	−	0.930295i	$-0.619551\pi$
$367$	−14.0542			−0.733624			−0.366812	+	0.930295i	$0.619551\pi$
$368$		0			0
$369$			2.33501i			0.121556i
$370$		0			0
$371$	−15.9789	−	21.8211i	−0.829581	−	1.13290i
$372$		0			0
$373$			19.7598i			1.02312i	0.859247	+	0.511561i	$0.170933\pi$
$373$			19.7598i			1.02312i	−0.859247	+	0.511561i	$0.829067\pi$
$374$		0			0
$375$		−	1.61069i		−	0.0831754i
$376$		0			0
$377$		−	1.60302i		−	0.0825600i
$378$		0			0
$379$	0.483016			0.0248108			0.0124054	−	0.999923i	$-0.496051\pi$
$379$	0.483016			0.0248108			0.0124054	+	0.999923i	$0.496051\pi$
$380$		0			0
$381$	6.02594			0.308718
$382$		0			0
$383$	−1.09585			−0.0559953			−0.0279977	−	0.999608i	$-0.508913\pi$
$383$	−1.09585			−0.0559953			−0.0279977	+	0.999608i	$0.508913\pi$
$384$		0			0
$385$	−12.8437	+	9.40496i	−0.654573	+	0.479321i
$386$		0			0
$387$	2.01433			0.102394
$388$		0			0
$389$			18.3250i			0.929114i	0.885543	+	0.464557i	$0.153786\pi$
$389$			18.3250i			0.929114i	−0.885543	+	0.464557i	$0.846214\pi$
$390$		0			0
$391$	−28.1844			−1.42535
$392$		0			0
$393$	28.3117			1.42814
$394$		0			0
$395$		−	8.74609i		−	0.440064i
$396$		0			0
$397$	−0.717950			−0.0360329			−0.0180164	−	0.999838i	$-0.505735\pi$
$397$	−0.717950			−0.0360329			−0.0180164	+	0.999838i	$0.505735\pi$
$398$		0			0
$399$	15.5954	−	11.4199i	0.780745	−	0.571712i
$400$		0			0
$401$	−21.1139			−1.05438			−0.527189	−	0.849748i	$-0.676754\pi$
$401$	−21.1139			−1.05438			−0.527189	+	0.849748i	$0.676754\pi$
$402$		0			0
$403$	−12.4916			−0.622250
$404$		0			0
$405$	7.61833			0.378558
$406$		0			0
$407$		−	2.23984i		−	0.111025i
$408$		0			0
$409$		−	1.08609i		−	0.0537038i	−0.999639	−	0.0268519i	$-0.991452\pi$
$409$		−	1.08609i		−	0.0537038i	0.999639	−	0.0268519i	$-0.00854825\pi$
$410$		0			0
$411$		−	1.23594i		−	0.0609644i
$412$		0			0
$413$	16.5433	+	22.5920i	0.814044	+	1.11168i
$414$		0			0
$415$			8.42742i			0.413686i
$416$		0			0
$417$	−3.25123			−0.159214
$418$		0			0
$419$		−	0.208107i		−	0.0101667i	−0.999987	−	0.00508334i	$-0.998382\pi$
$419$		−	0.208107i		−	0.0101667i	0.999987	−	0.00508334i	$-0.00161808\pi$
$420$		0			0
$421$		−	9.48511i		−	0.462276i	−0.972921	−	0.231138i	$-0.925755\pi$
$421$		−	9.48511i		−	0.462276i	0.972921	−	0.231138i	$-0.0742449\pi$
$422$		0			0
$423$	1.56721			0.0762003
$424$		0			0
$425$		−	5.42124i		−	0.262969i
$426$		0			0
$427$	11.9113	−	8.72221i	0.576427	−	0.422097i
$428$		0			0
$429$		−	41.2273i		−	1.99047i
$430$		0			0
$431$		−	13.6900i		−	0.659423i	−0.944082	−	0.329711i	$-0.893049\pi$
$431$		−	13.6900i		−	0.659423i	0.944082	−	0.329711i	$-0.106951\pi$
$432$		0			0
$433$			6.48406i			0.311604i	0.987788	+	0.155802i	$0.0497962\pi$
$433$			6.48406i			0.311604i	−0.987788	+	0.155802i	$0.950204\pi$
$434$		0			0
$435$	0.606935			0.0291003
$436$		0			0
$437$	−23.5816			−1.12806
$438$		0			0
$439$	9.04411			0.431652			0.215826	−	0.976432i	$-0.430756\pi$
$439$	9.04411			0.431652			0.215826	+	0.976432i	$0.430756\pi$
$440$		0			0
$441$	0.857361	−	2.70734i	0.0408267	−	0.128921i
$442$		0			0
$443$	−8.66264			−0.411574			−0.205787	−	0.978597i	$-0.565975\pi$
$443$	−8.66264			−0.411574			−0.205787	+	0.978597i	$0.565975\pi$
$444$		0			0
$445$		−	1.94765i		−	0.0923277i
$446$		0			0
$447$	6.81750			0.322457
$448$		0			0
$449$	−26.5808			−1.25443			−0.627213	−	0.778848i	$-0.715804\pi$
$449$	−26.5808			−1.25443			−0.627213	+	0.778848i	$0.715804\pi$
$450$		0			0
$451$			34.6303i			1.63068i
$452$		0			0
$453$	1.00792			0.0473560
$454$		0			0
$455$	9.08097	−	6.64968i	0.425722	−	0.311742i
$456$		0			0
$457$	33.4034			1.56254			0.781272	−	0.624191i	$-0.214571\pi$
$457$	33.4034			1.56254			0.781272	+	0.624191i	$0.214571\pi$
$458$		0			0
$459$	29.7382			1.38806
$460$		0			0
$461$	9.70385			0.451954			0.225977	−	0.974133i	$-0.427443\pi$
$461$	9.70385			0.451954			0.225977	+	0.974133i	$0.427443\pi$
$462$		0			0
$463$			28.6011i			1.32921i	0.747196	+	0.664604i	$0.231400\pi$
$463$			28.6011i			1.32921i	−0.747196	+	0.664604i	$0.768600\pi$
$464$		0			0
$465$		−	4.72954i		−	0.219327i
$466$		0			0
$467$			8.61107i			0.398473i	0.979951	+	0.199236i	$0.0638462\pi$
$467$			8.61107i			0.398473i	−0.979951	+	0.199236i	$0.936154\pi$
$468$		0			0
$469$	1.67056	−	1.22329i	0.0771391	−	0.0564863i
$470$		0			0
$471$			4.69857i			0.216499i
$472$		0			0
$473$	29.8743			1.37362
$474$		0			0
$475$		−	4.53588i		−	0.208120i
$476$		0			0
$477$		−	4.14717i		−	0.189886i
$478$		0			0
$479$	−6.75741			−0.308754			−0.154377	−	0.988012i	$-0.549337\pi$
$479$	−6.75741			−0.308754			−0.154377	+	0.988012i	$0.549337\pi$
$480$		0			0
$481$			1.58366i			0.0722086i
$482$		0			0
$483$	17.8750	−	13.0892i	0.813339	−	0.595580i
$484$		0			0
$485$		−	3.14377i		−	0.142751i
$486$		0			0
$487$			7.72534i			0.350069i	0.984562	+	0.175034i	$0.0560036\pi$
$487$			7.72534i			0.350069i	−0.984562	+	0.175034i	$0.943996\pi$
$488$		0			0
$489$			11.5067i			0.520350i
$490$		0			0
$491$	33.1648			1.49671			0.748354	−	0.663300i	$-0.230844\pi$
$491$	33.1648			1.49671			0.748354	+	0.663300i	$0.230844\pi$
$492$		0			0
$493$	2.04282			0.0920039
$494$		0			0
$495$	−2.44098			−0.109714
$496$		0			0
$497$	24.0412	+	32.8313i	1.07840	+	1.47269i
$498$		0			0
$499$	9.13507			0.408942			0.204471	−	0.978873i	$-0.434453\pi$
$499$	9.13507			0.408942			0.204471	+	0.978873i	$0.434453\pi$
$500$		0			0
$501$			10.0957i			0.451043i
$502$		0			0
$503$	13.9338			0.621277			0.310639	−	0.950528i	$-0.399457\pi$
$503$	13.9338			0.621277			0.310639	+	0.950528i	$0.399457\pi$
$504$		0			0
$505$	16.3016			0.725413
$506$		0			0
$507$			8.21040i			0.364637i
$508$		0			0
$509$	11.4366			0.506918			0.253459	−	0.967346i	$-0.418432\pi$
$509$	11.4366			0.506918			0.253459	+	0.967346i	$0.418432\pi$
$510$		0			0
$511$	13.7111	+	18.7242i	0.606544	+	0.828311i
$512$		0			0
$513$	24.8816			1.09855
$514$		0			0
$515$	−11.8940			−0.524111
$516$		0			0
$517$	23.2431			1.02223
$518$		0			0
$519$		−	14.6792i		−	0.644347i
$520$		0			0
$521$			13.5526i			0.593751i	0.954916	+	0.296875i	$0.0959446\pi$
$521$			13.5526i			0.593751i	−0.954916	+	0.296875i	$0.904055\pi$
$522$		0			0
$523$		−	16.3634i		−	0.715522i	−0.933813	−	0.357761i	$-0.883540\pi$
$523$		−	16.3634i		−	0.715522i	0.933813	−	0.357761i	$-0.116460\pi$
$524$		0			0
$525$	2.51769	+	3.43822i	0.109881	+	0.150056i
$526$		0			0
$527$		−	15.9187i		−	0.693429i
$528$		0			0
$529$	−4.02852			−0.175153
$530$		0			0
$531$			4.29368i			0.186330i
$532$		0			0
$533$		−	24.4850i		−	1.06056i
$534$		0			0
$535$	10.4794			0.453065
$536$		0			0
$537$		−	17.6400i		−	0.761221i
$538$		0			0
$539$	12.7154	−	40.1523i	0.547692	−	1.72948i
$540$		0			0
$541$		−	23.4050i		−	1.00626i	−0.864211	−	0.503129i	$-0.832182\pi$
$541$		−	23.4050i		−	1.00626i	0.864211	−	0.503129i	$-0.167818\pi$
$542$		0			0
$543$			30.2768i			1.29930i
$544$		0			0
$545$			0.676794i			0.0289907i
$546$		0			0
$547$	−21.2928			−0.910416			−0.455208	−	0.890385i	$-0.650435\pi$
$547$	−21.2928			−0.910416			−0.455208	+	0.890385i	$0.650435\pi$
$548$		0			0
$549$	2.26377			0.0966155
$550$		0			0
$551$	1.70920			0.0728144
$552$		0			0
$553$	13.6712	+	18.6697i	0.581358	+	0.793917i
$554$		0			0
$555$	−0.599602			−0.0254517
$556$		0			0
$557$		−	2.06871i		−	0.0876541i	−0.999039	−	0.0438270i	$-0.986045\pi$
$557$		−	2.06871i		−	0.0876541i	0.999039	−	0.0438270i	$-0.0139551\pi$
$558$		0			0
$559$	−21.1223			−0.893378
$560$		0			0
$561$	52.5381			2.21816
$562$		0			0
$563$			25.5312i			1.07601i	0.842941	+	0.538006i	$0.180822\pi$
$563$			25.5312i			1.07601i	−0.842941	+	0.538006i	$0.819178\pi$
$564$		0			0
$565$	9.10537			0.383066
$566$		0			0
$567$	−16.2623	+	11.9084i	−0.682954	+	0.500104i
$568$		0			0
$569$	26.3859			1.10616			0.553078	−	0.833130i	$-0.313453\pi$
$569$	26.3859			1.10616			0.553078	+	0.833130i	$0.313453\pi$
$570$		0			0
$571$	−17.6202			−0.737384			−0.368692	−	0.929552i	$-0.620194\pi$
$571$	−17.6202			−0.737384			−0.368692	+	0.929552i	$0.620194\pi$
$572$		0			0
$573$	1.82038			0.0760476
$574$		0			0
$575$		−	5.19890i		−	0.216809i
$576$		0			0
$577$		−	29.5306i		−	1.22938i	−0.788770	−	0.614688i	$-0.789282\pi$
$577$		−	29.5306i		−	1.22938i	0.788770	−	0.614688i	$-0.210718\pi$
$578$		0			0
$579$		−	5.27268i		−	0.219125i
$580$		0			0
$581$	−13.1731	−	17.9895i	−0.546510	−	0.746328i
$582$		0			0
$583$		−	61.5062i		−	2.54733i
$584$		0			0
$585$	1.72587			0.0713558
$586$		0			0
$587$			5.32153i			0.219643i	0.993951	+	0.109822i	$0.0350279\pi$
$587$			5.32153i			0.219643i	−0.993951	+	0.109822i	$0.964972\pi$
$588$		0			0
$589$		−	13.3190i		−	0.548798i
$590$		0			0
$591$	−34.1871			−1.40627
$592$		0			0
$593$		−	45.6864i		−	1.87611i	−0.346481	−	0.938057i	$-0.612623\pi$
$593$		−	45.6864i		−	1.87611i	0.346481	−	0.938057i	$-0.387377\pi$
$594$		0			0
$595$	8.47403	+	11.5724i	0.347401	+	0.474420i
$596$		0			0
$597$		−	28.9302i		−	1.18404i
$598$		0			0
$599$			22.5049i			0.919525i	0.888042	+	0.459762i	$0.152065\pi$
$599$			22.5049i			0.919525i	−0.888042	+	0.459762i	$0.847935\pi$
$600$		0			0
$601$		−	4.49699i		−	0.183436i	−0.995785	−	0.0917181i	$-0.970764\pi$
$601$		−	4.49699i		−	0.183436i	0.995785	−	0.0917181i	$-0.0292359\pi$
$602$		0			0
$603$	0.317494			0.0129294
$604$		0			0
$605$	−25.2018			−1.02460
$606$		0			0
$607$	10.8910			0.442051			0.221025	−	0.975268i	$-0.429060\pi$
$607$	10.8910			0.442051			0.221025	+	0.975268i	$0.429060\pi$
$608$		0			0
$609$	−1.29558	+	0.948711i	−0.0524997	+	0.0384437i
$610$		0			0
$611$	−16.4338			−0.664839
$612$		0			0
$613$		−	30.5045i		−	1.23206i	−0.787721	−	0.616032i	$-0.788739\pi$
$613$		−	30.5045i		−	1.23206i	0.787721	−	0.616032i	$-0.211261\pi$
$614$		0			0
$615$	9.27046			0.373821
$616$		0			0
$617$	−39.6859			−1.59769			−0.798847	−	0.601534i	$-0.794557\pi$
$617$	−39.6859			−1.59769			−0.798847	+	0.601534i	$0.794557\pi$
$618$		0			0
$619$			13.7231i			0.551579i	0.961218	+	0.275789i	$0.0889392\pi$
$619$			13.7231i			0.551579i	−0.961218	+	0.275789i	$0.911061\pi$
$620$		0			0
$621$	28.5185			1.14441
$622$		0			0
$623$	3.04441	+	4.15753i	0.121972	+	0.166568i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	43.9579			1.75551
$628$		0			0
$629$	−2.01814			−0.0804684
$630$		0			0
$631$		−	36.7721i		−	1.46387i	−0.681373	−	0.731936i	$-0.738617\pi$
$631$		−	36.7721i		−	1.46387i	0.681373	−	0.731936i	$-0.261383\pi$
$632$		0			0
$633$			11.9777i			0.476072i
$634$		0			0
$635$			3.74123i			0.148466i
$636$		0			0
$637$	−8.99031	+	28.3893i	−0.356209	+	1.12482i
$638$		0			0
$639$			6.23969i			0.246838i
$640$		0			0
$641$	−40.2483			−1.58971			−0.794855	−	0.606799i	$-0.792453\pi$
$641$	−40.2483			−1.58971			−0.794855	+	0.606799i	$0.792453\pi$
$642$		0			0
$643$		−	12.3361i		−	0.486487i	−0.969965	−	0.243243i	$-0.921789\pi$
$643$		−	12.3361i		−	0.486487i	0.969965	−	0.243243i	$-0.0782114\pi$
$644$		0			0
$645$		−	7.99729i		−	0.314893i
$646$		0			0
$647$	−47.9622			−1.88559			−0.942794	−	0.333376i	$-0.891812\pi$
$647$	−47.9622			−1.88559			−0.942794	+	0.333376i	$0.891812\pi$
$648$		0			0
$649$			63.6790i			2.49962i
$650$		0			0
$651$	7.39284	+	10.0958i	0.289748	+	0.395687i
$652$		0			0
$653$			36.7504i			1.43815i	0.694930	+	0.719077i	$0.255435\pi$
$653$			36.7504i			1.43815i	−0.694930	+	0.719077i	$0.744565\pi$
$654$		0			0
$655$			17.5774i			0.686808i
$656$		0			0
$657$			3.55860i			0.138834i
$658$		0			0
$659$	−15.4039			−0.600050			−0.300025	−	0.953931i	$-0.596995\pi$
$659$	−15.4039			−0.600050			−0.300025	+	0.953931i	$0.596995\pi$
$660$		0			0
$661$	6.20639			0.241400			0.120700	−	0.992689i	$-0.461486\pi$
$661$	6.20639			0.241400			0.120700	+	0.992689i	$0.461486\pi$
$662$		0			0
$663$	−37.1465			−1.44265
$664$		0			0
$665$	7.09012	+	9.68244i	0.274943	+	0.375469i
$666$		0			0
$667$	1.95904			0.0758542
$668$		0			0
$669$		−	38.4375i		−	1.48608i
$670$		0			0
$671$	33.5738			1.29610
$672$		0			0
$673$	−46.2742			−1.78374			−0.891870	−	0.452291i	$-0.850607\pi$
$673$	−46.2742			−1.78374			−0.891870	+	0.452291i	$0.850607\pi$
$674$		0			0
$675$			5.48550i			0.211137i
$676$		0			0
$677$	0.112071			0.00430726			0.00215363	−	0.999998i	$-0.499314\pi$
$677$	0.112071			0.00430726			0.00215363	+	0.999998i	$0.499314\pi$
$678$		0			0
$679$	4.91408	+	6.71079i	0.188585	+	0.257537i
$680$		0			0
$681$	19.5346			0.748569
$682$		0			0
$683$	−0.561153			−0.0214719			−0.0107360	−	0.999942i	$-0.503417\pi$
$683$	−0.561153			−0.0214719			−0.0107360	+	0.999942i	$0.503417\pi$
$684$		0			0
$685$	0.767338			0.0293185
$686$		0			0
$687$			26.4154i			1.00781i
$688$		0			0
$689$			43.4873i			1.65673i
$690$		0			0
$691$			39.1762i			1.49033i	0.666879	+	0.745166i	$0.267630\pi$
$691$			39.1762i			1.49033i	−0.666879	+	0.745166i	$0.732370\pi$
$692$		0			0
$693$	5.21059	−	3.81553i	0.197934	−	0.144940i
$694$		0			0
$695$		−	2.01854i		−	0.0765676i
$696$		0			0
$697$	31.2025			1.18188
$698$		0			0
$699$		−	42.5395i		−	1.60899i
$700$		0			0
$701$		−	49.6478i		−	1.87517i	−0.347753	−	0.937586i	$-0.613055\pi$
$701$		−	49.6478i		−	1.87517i	0.347753	−	0.937586i	$-0.386945\pi$
$702$		0			0
$703$	−1.68855			−0.0636849
$704$		0			0
$705$		−	6.22213i		−	0.234339i
$706$		0			0
$707$	−34.7980	+	25.4814i	−1.30871	+	0.958326i
$708$		0			0
$709$			13.0064i			0.488465i	0.969717	+	0.244233i	$0.0785360\pi$
$709$			13.0064i			0.488465i	−0.969717	+	0.244233i	$0.921464\pi$
$710$		0			0
$711$			3.54823i			0.133069i
$712$		0			0
$713$		−	15.2658i		−	0.571709i
$714$		0			0
$715$	25.5961			0.957240
$716$		0			0
$717$	−2.72060			−0.101603
$718$		0			0
$719$	12.4976			0.466081			0.233041	−	0.972467i	$-0.425132\pi$
$719$	12.4976			0.466081			0.233041	+	0.972467i	$0.425132\pi$
$720$		0			0
$721$	25.3893	−	18.5917i	0.945546	−	0.692391i
$722$		0			0
$723$	8.45800			0.314556
$724$		0			0
$725$			0.376818i			0.0139947i
$726$		0			0
$727$	49.8167			1.84760			0.923800	−	0.382875i	$-0.125066\pi$
$727$	49.8167			1.84760			0.923800	+	0.382875i	$0.125066\pi$
$728$		0			0
$729$	−29.5969			−1.09618
$730$		0			0
$731$		−	26.9172i		−	0.995570i
$732$		0			0
$733$	48.8875			1.80570			0.902850	−	0.429955i	$-0.141471\pi$
$733$	48.8875			1.80570			0.902850	+	0.429955i	$0.141471\pi$
$734$		0			0
$735$	−10.7487	−	3.40390i	−0.396472	−	0.125555i
$736$		0			0
$737$	4.70872			0.173448
$738$		0			0
$739$	39.3743			1.44841			0.724204	−	0.689585i	$-0.242207\pi$
$739$	39.3743			1.44841			0.724204	+	0.689585i	$0.242207\pi$
$740$		0			0
$741$	−31.0800			−1.14175
$742$		0			0
$743$		−	35.6179i		−	1.30669i	−0.757059	−	0.653346i	$-0.773365\pi$
$743$		−	35.6179i		−	1.30669i	0.757059	−	0.653346i	$-0.226635\pi$
$744$		0			0
$745$			4.23267i			0.155073i
$746$		0			0
$747$		−	3.41895i		−	0.125093i
$748$		0			0
$749$	−22.3697	+	16.3806i	−0.817373	+	0.598534i
$750$		0			0
$751$			13.2839i			0.484738i	0.970184	+	0.242369i	$0.0779245\pi$
$751$			13.2839i			0.484738i	−0.970184	+	0.242369i	$0.922076\pi$
$752$		0			0
$753$	22.5000			0.819946
$754$		0			0
$755$			0.625768i			0.0227740i
$756$		0			0
$757$			20.0483i			0.728666i	0.931269	+	0.364333i	$0.118703\pi$
$757$			20.0483i			0.728666i	−0.931269	+	0.364333i	$0.881297\pi$
$758$		0			0
$759$	50.3833			1.82880
$760$		0			0
$761$		−	28.5860i		−	1.03624i	−0.855307	−	0.518121i	$-0.826632\pi$
$761$		−	28.5860i		−	1.03624i	0.855307	−	0.518121i	$-0.173368\pi$
$762$		0			0
$763$	−1.05791	−	1.44471i	−0.0382989	−	0.0523019i
$764$		0			0
$765$			2.19936i			0.0795181i
$766$		0			0
$767$		−	45.0236i		−	1.62571i
$768$		0			0
$769$			26.3204i			0.949137i	0.880219	+	0.474569i	$0.157396\pi$
$769$			26.3204i			0.949137i	−0.880219	+	0.474569i	$0.842604\pi$
$770$		0			0
$771$	34.9418			1.25840
$772$		0			0
$773$	−0.586223			−0.0210850			−0.0105425	−	0.999944i	$-0.503356\pi$
$773$	−0.586223			−0.0210850			−0.0105425	+	0.999944i	$0.503356\pi$
$774$		0			0
$775$	2.93636			0.105477
$776$		0			0
$777$	1.27993	−	0.937249i	0.0459173	−	0.0336236i
$778$		0			0
$779$	26.1067			0.935370
$780$		0			0
$781$			92.5401i			3.31135i
$782$		0			0
$783$	−2.06703			−0.0738698
$784$		0			0
$785$	−2.91713			−0.104117
$786$		0			0
$787$			12.7886i			0.455864i	0.973677	+	0.227932i	$0.0731964\pi$
$787$			12.7886i			0.455864i	−0.973677	+	0.227932i	$0.926804\pi$
$788$		0			0
$789$	9.48642			0.337725
$790$		0			0
$791$	−19.4366	+	14.2328i	−0.691087	+	0.506059i
$792$		0			0
$793$	−23.7380			−0.842960
$794$		0			0
$795$	−16.4651			−0.583957
$796$		0			0
$797$	27.5991			0.977609			0.488805	−	0.872393i	$-0.337433\pi$
$797$	27.5991			0.977609			0.488805	+	0.872393i	$0.337433\pi$
$798$		0			0
$799$		−	20.9424i		−	0.740890i
$800$		0			0
$801$			0.790151i			0.0279186i
$802$		0			0
$803$			52.7771i			1.86247i
$804$		0			0
$805$	8.12649	+	11.0977i	0.286421	+	0.391144i
$806$		0			0
$807$		−	21.4986i		−	0.756786i
$808$		0			0
$809$	−19.2283			−0.676030			−0.338015	−	0.941141i	$-0.609755\pi$
$809$	−19.2283			−0.676030			−0.338015	+	0.941141i	$0.609755\pi$
$810$		0			0
$811$			7.93942i			0.278791i	0.990237	+	0.139395i	$0.0445159\pi$
$811$			7.93942i			0.278791i	−0.990237	+	0.139395i	$0.955484\pi$
$812$		0			0
$813$			39.9301i			1.40041i
$814$		0			0
$815$	−7.14397			−0.250242
$816$		0			0
$817$		−	22.5213i		−	0.787921i
$818$		0			0
$819$	−3.68409	+	2.69773i	−0.128733	+	0.0942664i
$820$		0			0
$821$			26.3775i			0.920582i	0.887768	+	0.460291i	$0.152255\pi$
$821$			26.3775i			0.920582i	−0.887768	+	0.460291i	$0.847745\pi$
$822$		0			0
$823$			21.7419i			0.757875i	0.925422	+	0.378938i	$0.123711\pi$
$823$			21.7419i			0.757875i	−0.925422	+	0.378938i	$0.876289\pi$
$824$		0			0
$825$			9.69116i			0.337403i
$826$		0			0
$827$	−57.4711			−1.99846			−0.999232	−	0.0391762i	$-0.987527\pi$
$827$	−57.4711			−1.99846			−0.999232	+	0.0391762i	$0.987527\pi$
$828$		0			0
$829$	−7.16668			−0.248909			−0.124455	−	0.992225i	$-0.539718\pi$
$829$	−7.16668			−0.248909			−0.124455	+	0.992225i	$0.539718\pi$
$830$		0			0
$831$	3.36740			0.116814
$832$		0			0
$833$	−36.1779	−	11.4568i	−1.25349	−	0.396955i
$834$		0			0
$835$	−6.26796			−0.216912
$836$		0			0
$837$			16.1074i			0.556753i
$838$		0			0
$839$	12.0764			0.416922			0.208461	−	0.978031i	$-0.433155\pi$
$839$	12.0764			0.416922			0.208461	+	0.978031i	$0.433155\pi$
$840$		0			0
$841$	28.8580			0.995104
$842$		0			0
$843$		−	6.59061i		−	0.226993i
$844$		0			0
$845$	−5.09746			−0.175358
$846$		0			0
$847$	53.7966	−	39.3934i	1.84847	−	1.35357i
$848$		0			0
$849$	−9.53998			−0.327411
$850$		0			0
$851$	−1.93537			−0.0663436
$852$		0			0
$853$	−40.3302			−1.38088			−0.690440	−	0.723390i	$-0.742583\pi$
$853$	−40.3302			−1.38088			−0.690440	+	0.723390i	$0.742583\pi$
$854$		0			0
$855$			1.84018i			0.0629327i
$856$		0			0
$857$		−	13.5485i		−	0.462810i	−0.972858	−	0.231405i	$-0.925668\pi$
$857$		−	13.5485i		−	0.462810i	0.972858	−	0.231405i	$-0.0743322\pi$
$858$		0			0
$859$			24.4181i			0.833136i	0.909105	+	0.416568i	$0.136767\pi$
$859$			24.4181i			0.833136i	−0.909105	+	0.416568i	$0.863233\pi$
$860$		0			0
$861$	−19.7890	+	14.4908i	−0.674409	+	0.493846i
$862$		0			0
$863$			33.4276i			1.13789i	0.822376	+	0.568945i	$0.192648\pi$
$863$			33.4276i			1.13789i	−0.822376	+	0.568945i	$0.807352\pi$
$864$		0			0
$865$	9.11366			0.309874
$866$		0			0
$867$		−	19.9561i		−	0.677743i
$868$		0			0
$869$			52.6234i			1.78513i
$870$		0			0
$871$	−3.32925			−0.112807
$872$		0			0
$873$			1.27541i			0.0431660i
$874$		0			0
$875$	−2.13463	+	1.56312i	−0.0721638	+	0.0528430i
$876$		0			0
$877$		−	33.1528i		−	1.11949i	−0.828664	−	0.559746i	$-0.810899\pi$
$877$		−	33.1528i		−	1.11949i	0.828664	−	0.559746i	$-0.189101\pi$
$878$		0			0
$879$		−	15.2100i		−	0.513019i
$880$		0			0
$881$			52.3710i			1.76442i	0.470853	+	0.882212i	$0.343946\pi$
$881$			52.3710i			1.76442i	−0.470853	+	0.882212i	$0.656054\pi$
$882$		0			0
$883$	−3.17165			−0.106735			−0.0533673	−	0.998575i	$-0.516995\pi$
$883$	−3.17165			−0.106735			−0.0533673	+	0.998575i	$0.516995\pi$
$884$		0			0
$885$	17.0468			0.573020
$886$		0			0
$887$	−36.3723			−1.22126			−0.610632	−	0.791915i	$-0.709084\pi$
$887$	−36.3723			−1.22126			−0.610632	+	0.791915i	$0.709084\pi$
$888$		0			0
$889$	−5.84798	−	7.98615i	−0.196135	−	0.267847i
$890$		0			0
$891$	−45.8379			−1.53563
$892$		0			0
$893$		−	17.5223i		−	0.586360i
$894$		0			0
$895$	10.9518			0.366080
$896$		0			0
$897$	−35.6230			−1.18942
$898$		0			0
$899$			1.10647i			0.0369029i
$900$		0			0
$901$	−55.4182			−1.84625
$902$		0			0
$903$	12.5007	+	17.0713i	0.415998	+	0.568097i
$904$		0			0
$905$	−18.7975			−0.624850
$906$		0			0
$907$	13.5038			0.448388			0.224194	−	0.974545i	$-0.428025\pi$
$907$	13.5038			0.448388			0.224194	+	0.974545i	$0.428025\pi$
$908$		0			0
$909$	−6.61347			−0.219355
$910$		0			0
$911$		−	6.93039i		−	0.229614i	−0.993388	−	0.114807i	$-0.963375\pi$
$911$		−	6.93039i		−	0.229614i	0.993388	−	0.114807i	$-0.0366250\pi$
$912$		0			0
$913$		−	50.7060i		−	1.67813i
$914$		0			0
$915$		−	8.98763i		−	0.297122i
$916$		0			0
$917$	−27.4756	−	37.5214i	−0.907325	−	1.23907i
$918$		0			0
$919$		−	18.9826i		−	0.626178i	−0.949724	−	0.313089i	$-0.898636\pi$
$919$		−	18.9826i		−	0.626178i	0.949724	−	0.313089i	$-0.101364\pi$
$920$		0			0
$921$	−38.8504			−1.28017
$922$		0			0
$923$		−	65.4295i		−	2.15364i
$924$		0			0
$925$		−	0.372265i		−	0.0122400i
$926$		0			0
$927$	4.82531			0.158484
$928$		0			0
$929$		−	55.6476i		−	1.82574i	−0.408250	−	0.912870i	$-0.633861\pi$
$929$		−	55.6476i		−	1.82574i	0.408250	−	0.912870i	$-0.366139\pi$
$930$		0			0
$931$	−30.2696	−	9.58577i	−0.992046	−	0.314161i
$932$		0			0
$933$			26.7289i			0.875064i
$934$		0			0
$935$			32.6185i			1.06674i
$936$		0			0
$937$			30.8690i			1.00845i	0.863574	+	0.504223i	$0.168221\pi$
$937$			30.8690i			1.00845i	−0.863574	+	0.504223i	$0.831779\pi$
$938$		0			0
$939$	−18.3410			−0.598537
$940$		0			0
$941$	−14.1430			−0.461048			−0.230524	−	0.973067i	$-0.574044\pi$
$941$	−14.1430			−0.461048			−0.230524	+	0.973067i	$0.574044\pi$
$942$		0			0
$943$	29.9228			0.974420
$944$		0			0
$945$	−8.57449	−	11.7095i	−0.278928	−	0.380911i
$946$		0			0
$947$	−10.5002			−0.341212			−0.170606	−	0.985339i	$-0.554572\pi$
$947$	−10.5002			−0.341212			−0.170606	+	0.985339i	$0.554572\pi$
$948$		0			0
$949$		−	37.3155i		−	1.21131i
$950$		0			0
$951$	−43.0188			−1.39498
$952$		0			0
$953$	36.8415			1.19341			0.596706	−	0.802460i	$-0.296476\pi$
$953$	36.8415			1.19341			0.596706	+	0.802460i	$0.296476\pi$
$954$		0			0
$955$			1.13019i			0.0365721i
$956$		0			0
$957$	−3.65180			−0.118046
$958$		0			0
$959$	−1.63799	+	1.19944i	−0.0528933	+	0.0387319i
$960$		0			0
$961$	−22.3778			−0.721865
$962$		0			0
$963$	−4.25144			−0.137001
$964$		0			0
$965$	3.27357			0.105380
$966$		0			0
$967$			48.0941i			1.54660i	0.634040	+	0.773301i	$0.281396\pi$
$967$			48.0941i			1.54660i	−0.634040	+	0.773301i	$0.718604\pi$
$968$		0			0
$969$		−	39.6069i		−	1.27236i
$970$		0			0
$971$			13.7904i			0.442555i	0.975211	+	0.221277i	$0.0710226\pi$
$971$			13.7904i			0.442555i	−0.975211	+	0.221277i	$0.928977\pi$
$972$		0			0
$973$	3.15522	+	4.30884i	0.101152	+	0.138135i
$974$		0			0
$975$		−	6.85203i		−	0.219441i
$976$		0			0
$977$	19.9730			0.638992			0.319496	−	0.947588i	$-0.396486\pi$
$977$	19.9730			0.638992			0.319496	+	0.947588i	$0.396486\pi$
$978$		0			0
$979$			11.7186i			0.374529i
$980$		0			0
$981$		−	0.274571i		−	0.00876638i
$982$		0			0
$983$	50.8845			1.62296			0.811482	−	0.584377i	$-0.198661\pi$
$983$	50.8845			1.62296			0.811482	+	0.584377i	$0.198661\pi$
$984$		0			0
$985$		−	21.2252i		−	0.676291i
$986$		0			0
$987$	9.72593	+	13.2820i	0.309580	+	0.422770i
$988$		0			0
$989$		−	25.8133i		−	0.820815i
$990$		0			0
$991$		−	41.0233i		−	1.30315i	−0.758585	−	0.651574i	$-0.774109\pi$
$991$		−	41.0233i		−	1.30315i	0.758585	−	0.651574i	$-0.225891\pi$
$992$		0			0
$993$			5.24060i			0.166305i
$994$		0			0
$995$	17.9614			0.569416
$996$		0			0
$997$	29.5605			0.936189			0.468095	−	0.883678i	$-0.344941\pi$
$997$	29.5605			0.936189			0.468095	+	0.883678i	$0.344941\pi$
$998$		0			0
$999$	2.04206			0.0646079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.h.a.111.12		16
4.3	odd	2		280.2.h.a.251.3	✓	16
7.6	odd	2		1120.2.h.b.111.5		16
8.3	odd	2		1120.2.h.b.111.12		16
8.5	even	2		280.2.h.b.251.4	yes	16
28.27	even	2		280.2.h.b.251.3	yes	16
56.13	odd	2		280.2.h.a.251.4	yes	16
56.27	even	2	inner	1120.2.h.a.111.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.h.a.251.3	✓	16	4.3	odd	2
280.2.h.a.251.4	yes	16	56.13	odd	2
280.2.h.b.251.3	yes	16	28.27	even	2
280.2.h.b.251.4	yes	16	8.5	even	2
1120.2.h.a.111.5		16	56.27	even	2	inner
1120.2.h.a.111.12		16	1.1	even	1	trivial
1120.2.h.b.111.5		16	7.6	odd	2
1120.2.h.b.111.12		16	8.3	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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.61069i q^{3} -1.00000 q^{5} +(2.13463 - 1.56312i) q^{7} +0.405694 q^{9} +O(q^{10})\)	\(q+1.61069i q^{3} -1.00000 q^{5} +(2.13463 - 1.56312i) q^{7} +0.405694 q^{9} +6.01679 q^{11} -4.25411 q^{13} -1.61069i q^{15} -5.42124i q^{17} -4.53588i q^{19} +(2.51769 + 3.43822i) q^{21} -5.19890i q^{23} +1.00000 q^{25} +5.48550i q^{27} +0.376818i q^{29} +2.93636 q^{31} +9.69116i q^{33} +(-2.13463 + 1.56312i) q^{35} -0.372265i q^{37} -6.85203i q^{39} +5.75560i q^{41} +4.96515 q^{43} -0.405694 q^{45} +3.86303 q^{47} +(2.11332 - 6.67337i) q^{49} +8.73190 q^{51} -10.2224i q^{53} -6.01679 q^{55} +7.30587 q^{57} +10.5835i q^{59} +5.58001 q^{61} +(0.866007 - 0.634147i) q^{63} +4.25411 q^{65} +0.782596 q^{67} +8.37378 q^{69} +15.3803i q^{71} +8.77164i q^{73} +1.61069i q^{75} +(12.8437 - 9.40496i) q^{77} +8.74609i q^{79} -7.61833 q^{81} -8.42742i q^{83} +5.42124i q^{85} -0.606935 q^{87} +1.94765i q^{89} +(-9.08097 + 6.64968i) q^{91} +4.72954i q^{93} +4.53588i q^{95} +3.14377i q^{97} +2.44098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^5 - 16 * q^9	\( 16 q - 16 q^{5} - 16 q^{9} + 4 q^{11} + 4 q^{21} + 16 q^{25} - 16 q^{31} + 4 q^{43} + 16 q^{45} - 8 q^{49} + 40 q^{51} - 4 q^{55} - 16 q^{57} + 8 q^{61} + 28 q^{63} - 20 q^{67} + 40 q^{69} + 4 q^{77} + 24 q^{81} + 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) \) 16 * q - 16 * q^5 - 16 * q^9 + 4 * q^11 + 4 * q^21 + 16 * q^25 - 16 * q^31 + 4 * q^43 + 16 * q^45 - 8 * q^49 + 40 * q^51 - 4 * q^55 - 16 * q^57 + 8 * q^61 + 28 * q^63 - 20 * q^67 + 40 * q^69 + 4 * q^77 + 24 * q^81 + 72 * q^87 + 32 * q^91 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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