LMFDB - Embedded newform 1120.2.h.a.111.10

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			111.10
Root			$1.14218 + 0.833926i$ of defining polynomial
Character	$\chi$	$=$	1120.111
Dual form			1120.2.h.a.111.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.586834i q^{3} -1.00000 q^{5} +(-2.52442 - 0.792014i) q^{7} +2.65563 q^{9} +O(q^{10})$	$q+0.586834i q^{3} -1.00000 q^{5} +(-2.52442 - 0.792014i) q^{7} +2.65563 q^{9} +0.580189 q^{11} -1.14766 q^{13} -0.586834i q^{15} -1.82880i q^{17} -4.72462i q^{19} +(0.464781 - 1.48142i) q^{21} +2.79961i q^{23} +1.00000 q^{25} +3.31891i q^{27} -7.40518i q^{29} +4.73007 q^{31} +0.340475i q^{33} +(2.52442 + 0.792014i) q^{35} -4.35175i q^{37} -0.673485i q^{39} -8.46264i q^{41} +4.32314 q^{43} -2.65563 q^{45} +3.56830 q^{47} +(5.74543 + 3.99876i) q^{49} +1.07320 q^{51} -5.48977i q^{53} -0.580189 q^{55} +2.77257 q^{57} -13.2957i q^{59} -0.275184 q^{61} +(-6.70392 - 2.10329i) q^{63} +1.14766 q^{65} +7.71818 q^{67} -1.64291 q^{69} -1.13454i q^{71} +1.78121i q^{73} +0.586834i q^{75} +(-1.46464 - 0.459518i) q^{77} -10.0108i q^{79} +6.01923 q^{81} +15.6230i q^{83} +1.82880i q^{85} +4.34561 q^{87} +15.3393i q^{89} +(2.89717 + 0.908961i) q^{91} +2.77577i q^{93} +4.72462i q^{95} -14.9759i q^{97} +1.54077 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			0.586834i			0.338809i	0.985547	+	0.169404i	$0.0541844\pi$
$3$			0.586834i			0.338809i	−0.985547	+	0.169404i	$0.945816\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−2.52442	−	0.792014i	−0.954142	−	0.299353i
$7$	−2.52442	−	0.792014i	−0.954142	−	0.299353i
$8$		0			0
$9$	2.65563			0.885209
$10$		0			0
$11$	0.580189			0.174934			0.0874668	−	0.996167i	$-0.472123\pi$
$11$	0.580189			0.174934			0.0874668	+	0.996167i	$0.472123\pi$
$12$		0			0
$13$	−1.14766			−0.318303			−0.159152	−	0.987254i	$-0.550876\pi$
$13$	−1.14766			−0.318303			−0.159152	+	0.987254i	$0.550876\pi$
$14$		0			0
$15$		−	0.586834i		−	0.151520i
$16$		0			0
$17$		−	1.82880i		−	0.443550i	−0.975098	−	0.221775i	$-0.928815\pi$
$17$		−	1.82880i		−	0.443550i	0.975098	−	0.221775i	$-0.0711851\pi$
$18$		0			0
$19$		−	4.72462i		−	1.08390i	−0.840410	−	0.541951i	$-0.817686\pi$
$19$		−	4.72462i		−	1.08390i	0.840410	−	0.541951i	$-0.182314\pi$
$20$		0			0
$21$	0.464781	−	1.48142i	0.101423	−	0.323272i
$22$		0			0
$23$			2.79961i			0.583759i	0.956455	+	0.291879i	$0.0942806\pi$
$23$			2.79961i			0.583759i	−0.956455	+	0.291879i	$0.905719\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			3.31891i			0.638725i
$28$		0			0
$29$		−	7.40518i		−	1.37511i	−0.726133	−	0.687554i	$-0.758684\pi$
$29$		−	7.40518i		−	1.37511i	0.726133	−	0.687554i	$-0.241316\pi$
$30$		0			0
$31$	4.73007			0.849546			0.424773	−	0.905300i	$-0.360354\pi$
$31$	4.73007			0.849546			0.424773	+	0.905300i	$0.360354\pi$
$32$		0			0
$33$			0.340475i			0.0592691i
$34$		0			0
$35$	2.52442	+	0.792014i	0.426705	+	0.133875i
$36$		0			0
$37$		−	4.35175i		−	0.715423i	−0.933832	−	0.357712i	$-0.883557\pi$
$37$		−	4.35175i		−	0.715423i	0.933832	−	0.357712i	$-0.116443\pi$
$38$		0			0
$39$		−	0.673485i		−	0.107844i
$40$		0			0
$41$		−	8.46264i		−	1.32164i	−0.750544	−	0.660821i	$-0.770208\pi$
$41$		−	8.46264i		−	1.32164i	0.750544	−	0.660821i	$-0.229792\pi$
$42$		0			0
$43$	4.32314			0.659273			0.329637	−	0.944108i	$-0.393074\pi$
$43$	4.32314			0.659273			0.329637	+	0.944108i	$0.393074\pi$
$44$		0			0
$45$	−2.65563			−0.395877
$46$		0			0
$47$	3.56830			0.520490			0.260245	−	0.965543i	$-0.416197\pi$
$47$	3.56830			0.520490			0.260245	+	0.965543i	$0.416197\pi$
$48$		0			0
$49$	5.74543	+	3.99876i	0.820775	+	0.571251i
$50$		0			0
$51$	1.07320			0.150279
$52$		0			0
$53$		−	5.48977i		−	0.754079i	−0.926197	−	0.377039i	$-0.876942\pi$
$53$		−	5.48977i		−	0.754079i	0.926197	−	0.377039i	$-0.123058\pi$
$54$		0			0
$55$	−0.580189			−0.0782327
$56$		0			0
$57$	2.77257			0.367235
$58$		0			0
$59$		−	13.2957i		−	1.73095i	−0.500948	−	0.865477i	$-0.667015\pi$
$59$		−	13.2957i		−	1.73095i	0.500948	−	0.865477i	$-0.332985\pi$
$60$		0			0
$61$	−0.275184			−0.0352337			−0.0176168	−	0.999845i	$-0.505608\pi$
$61$	−0.275184			−0.0352337			−0.0176168	+	0.999845i	$0.505608\pi$
$62$		0			0
$63$	−6.70392	−	2.10329i	−0.844615	−	0.264990i
$64$		0			0
$65$	1.14766			0.142349
$66$		0			0
$67$	7.71818			0.942925			0.471463	−	0.881886i	$-0.343726\pi$
$67$	7.71818			0.942925			0.471463	+	0.881886i	$0.343726\pi$
$68$		0			0
$69$	−1.64291			−0.197783
$70$		0			0
$71$		−	1.13454i		−	0.134645i	−0.997731	−	0.0673224i	$-0.978554\pi$
$71$		−	1.13454i		−	0.134645i	0.997731	−	0.0673224i	$-0.0214456\pi$
$72$		0			0
$73$			1.78121i			0.208474i	0.994552	+	0.104237i	$0.0332401\pi$
$73$			1.78121i			0.208474i	−0.994552	+	0.104237i	$0.966760\pi$
$74$		0			0
$75$			0.586834i			0.0677617i
$76$		0			0
$77$	−1.46464	−	0.459518i	−0.166912	−	0.0523669i
$78$		0			0
$79$		−	10.0108i		−	1.12630i	−0.826355	−	0.563149i	$-0.809590\pi$
$79$		−	10.0108i		−	1.12630i	0.826355	−	0.563149i	$-0.190410\pi$
$80$		0			0
$81$	6.01923			0.668803
$82$		0			0
$83$			15.6230i			1.71485i	0.514607	+	0.857426i	$0.327938\pi$
$83$			15.6230i			1.71485i	−0.514607	+	0.857426i	$0.672062\pi$
$84$		0			0
$85$			1.82880i			0.198362i
$86$		0			0
$87$	4.34561			0.465899
$88$		0			0
$89$			15.3393i			1.62596i	0.582293	+	0.812979i	$0.302156\pi$
$89$			15.3393i			1.62596i	−0.582293	+	0.812979i	$0.697844\pi$
$90$		0			0
$91$	2.89717	+	0.908961i	0.303706	+	0.0952850i
$92$		0			0
$93$			2.77577i			0.287833i
$94$		0			0
$95$			4.72462i			0.484736i
$96$		0			0
$97$		−	14.9759i		−	1.52058i	−0.649585	−	0.760289i	$-0.725057\pi$
$97$		−	14.9759i		−	1.52058i	0.649585	−	0.760289i	$-0.274943\pi$
$98$		0			0
$99$	1.54077			0.154853
$100$		0			0
$101$	12.7803			1.27169			0.635845	−	0.771817i	$-0.280652\pi$
$101$	12.7803			1.27169			0.635845	+	0.771817i	$0.280652\pi$
$102$		0			0
$103$	−15.1439			−1.49217			−0.746085	−	0.665851i	$-0.768069\pi$
$103$	−15.1439			−1.49217			−0.746085	+	0.665851i	$0.768069\pi$
$104$		0			0
$105$	−0.464781	+	1.48142i	−0.0453579	+	0.144572i
$106$		0			0
$107$	−6.70224			−0.647930			−0.323965	−	0.946069i	$-0.605016\pi$
$107$	−6.70224			−0.647930			−0.323965	+	0.946069i	$0.605016\pi$
$108$		0			0
$109$			11.2152i			1.07422i	0.843511	+	0.537112i	$0.180485\pi$
$109$			11.2152i			1.07422i	−0.843511	+	0.537112i	$0.819515\pi$
$110$		0			0
$111$	2.55376			0.242392
$112$		0			0
$113$	1.41012			0.132653			0.0663265	−	0.997798i	$-0.478872\pi$
$113$	1.41012			0.132653			0.0663265	+	0.997798i	$0.478872\pi$
$114$		0			0
$115$		−	2.79961i		−	0.261065i
$116$		0			0
$117$	−3.04775			−0.281765
$118$		0			0
$119$	−1.44844	+	4.61668i	−0.132778	+	0.423210i
$120$		0			0
$121$	−10.6634			−0.969398
$122$		0			0
$123$	4.96616			0.447784
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	15.7103i		−	1.39406i	−0.717041	−	0.697031i	$-0.754504\pi$
$127$		−	15.7103i		−	1.39406i	0.717041	−	0.697031i	$-0.245496\pi$
$128$		0			0
$129$			2.53697i			0.223368i
$130$		0			0
$131$		−	5.01749i		−	0.438380i	−0.975682	−	0.219190i	$-0.929658\pi$
$131$		−	5.01749i		−	0.438380i	0.975682	−	0.219190i	$-0.0703415\pi$
$132$		0			0
$133$	−3.74196	+	11.9269i	−0.324469	+	1.03420i
$134$		0			0
$135$		−	3.31891i		−	0.285647i
$136$		0			0
$137$	−14.8703			−1.27045			−0.635226	−	0.772326i	$-0.719093\pi$
$137$	−14.8703			−1.27045			−0.635226	+	0.772326i	$0.719093\pi$
$138$		0			0
$139$		−	0.148572i		−	0.0126017i	−0.999980	−	0.00630086i	$-0.997994\pi$
$139$		−	0.148572i		−	0.0126017i	0.999980	−	0.00630086i	$-0.00200564\pi$
$140$		0			0
$141$			2.09400i			0.176346i
$142$		0			0
$143$	−0.665859			−0.0556819
$144$		0			0
$145$			7.40518i			0.614967i
$146$		0			0
$147$	−2.34661	+	3.37161i	−0.193545	+	0.278086i
$148$		0			0
$149$			8.85493i			0.725424i	0.931901	+	0.362712i	$0.118149\pi$
$149$			8.85493i			0.725424i	−0.931901	+	0.362712i	$0.881851\pi$
$150$		0			0
$151$		−	12.3932i		−	1.00855i	−0.863544	−	0.504273i	$-0.831761\pi$
$151$		−	12.3932i		−	1.00855i	0.863544	−	0.504273i	$-0.168239\pi$
$152$		0			0
$153$		−	4.85662i		−	0.392635i
$154$		0			0
$155$	−4.73007			−0.379928
$156$		0			0
$157$	15.6090			1.24574			0.622869	−	0.782326i	$-0.285967\pi$
$157$	15.6090			1.24574			0.622869	+	0.782326i	$0.285967\pi$
$158$		0			0
$159$	3.22159			0.255488
$160$		0			0
$161$	2.21733	−	7.06740i	0.174750	−	0.556989i
$162$		0			0
$163$	5.60110			0.438712			0.219356	−	0.975645i	$-0.429604\pi$
$163$	5.60110			0.438712			0.219356	+	0.975645i	$0.429604\pi$
$164$		0			0
$165$		−	0.340475i		−	0.0265059i
$166$		0			0
$167$	9.34952			0.723488			0.361744	−	0.932278i	$-0.382181\pi$
$167$	9.34952			0.723488			0.361744	+	0.932278i	$0.382181\pi$
$168$		0			0
$169$	−11.6829			−0.898683
$170$		0			0
$171$		−	12.5468i		−	0.959479i
$172$		0			0
$173$	−23.5706			−1.79204			−0.896018	−	0.444018i	$-0.853553\pi$
$173$	−23.5706			−1.79204			−0.896018	+	0.444018i	$0.853553\pi$
$174$		0			0
$175$	−2.52442	−	0.792014i	−0.190828	−	0.0598706i
$176$		0			0
$177$	7.80238			0.586463
$178$		0			0
$179$	−19.7086			−1.47309			−0.736546	−	0.676387i	$-0.763545\pi$
$179$	−19.7086			−1.47309			−0.736546	+	0.676387i	$0.763545\pi$
$180$		0			0
$181$	21.0102			1.56168			0.780839	−	0.624732i	$-0.214792\pi$
$181$	21.0102			1.56168			0.780839	+	0.624732i	$0.214792\pi$
$182$		0			0
$183$		−	0.161487i		−	0.0119375i
$184$		0			0
$185$			4.35175i			0.319947i
$186$		0			0
$187$		−	1.06105i		−	0.0775919i
$188$		0			0
$189$	2.62862	−	8.37834i	0.191204	−	0.609435i
$190$		0			0
$191$			16.9750i			1.22827i	0.789201	+	0.614135i	$0.210495\pi$
$191$			16.9750i			1.22827i	−0.789201	+	0.614135i	$0.789505\pi$
$192$		0			0
$193$	14.8445			1.06853			0.534266	−	0.845317i	$-0.320588\pi$
$193$	14.8445			1.06853			0.534266	+	0.845317i	$0.320588\pi$
$194$		0			0
$195$			0.673485i			0.0482292i
$196$		0			0
$197$			6.30785i			0.449416i	0.974426	+	0.224708i	$0.0721428\pi$
$197$			6.30785i			0.449416i	−0.974426	+	0.224708i	$0.927857\pi$
$198$		0			0
$199$	11.9770			0.849030			0.424515	−	0.905421i	$-0.360445\pi$
$199$	11.9770			0.849030			0.424515	+	0.905421i	$0.360445\pi$
$200$		0			0
$201$			4.52929i			0.319471i
$202$		0			0
$203$	−5.86501	+	18.6938i	−0.411643	+	1.31205i
$204$		0			0
$205$			8.46264i			0.591056i
$206$		0			0
$207$			7.43471i			0.516748i
$208$		0			0
$209$		−	2.74117i		−	0.189611i
$210$		0			0
$211$	−10.8525			−0.747120			−0.373560	−	0.927606i	$-0.621863\pi$
$211$	−10.8525			−0.747120			−0.373560	+	0.927606i	$0.621863\pi$
$212$		0			0
$213$	0.665785			0.0456189
$214$		0			0
$215$	−4.32314			−0.294836
$216$		0			0
$217$	−11.9407	−	3.74628i	−0.810588	−	0.254314i
$218$		0			0
$219$	−1.04527			−0.0706329
$220$		0			0
$221$			2.09884i			0.141183i
$222$		0			0
$223$	−14.0737			−0.942445			−0.471223	−	0.882014i	$-0.656187\pi$
$223$	−14.0737			−0.942445			−0.471223	+	0.882014i	$0.656187\pi$
$224$		0			0
$225$	2.65563			0.177042
$226$		0			0
$227$		−	14.9683i		−	0.993478i	−0.867900	−	0.496739i	$-0.834531\pi$
$227$		−	14.9683i		−	0.993478i	0.867900	−	0.496739i	$-0.165469\pi$
$228$		0			0
$229$	−21.5684			−1.42528			−0.712641	−	0.701529i	$-0.752501\pi$
$229$	−21.5684			−1.42528			−0.712641	+	0.701529i	$0.752501\pi$
$230$		0			0
$231$	0.269661	−	0.859503i	0.0177424	−	0.0565511i
$232$		0			0
$233$	−19.5888			−1.28331			−0.641653	−	0.766995i	$-0.721751\pi$
$233$	−19.5888			−1.28331			−0.641653	+	0.766995i	$0.721751\pi$
$234$		0			0
$235$	−3.56830			−0.232770
$236$		0			0
$237$	5.87465			0.381600
$238$		0			0
$239$			23.2562i			1.50432i	0.658981	+	0.752160i	$0.270988\pi$
$239$			23.2562i			1.50432i	−0.658981	+	0.752160i	$0.729012\pi$
$240$		0			0
$241$		−	20.9056i		−	1.34665i	−0.739347	−	0.673324i	$-0.764866\pi$
$241$		−	20.9056i		−	1.34665i	0.739347	−	0.673324i	$-0.235134\pi$
$242$		0			0
$243$			13.4890i			0.865321i
$244$		0			0
$245$	−5.74543	−	3.99876i	−0.367062	−	0.255471i
$246$		0			0
$247$			5.42225i			0.345009i
$248$		0			0
$249$	−9.16813			−0.581007
$250$		0			0
$251$			3.33668i			0.210610i	0.994440	+	0.105305i	$0.0335818\pi$
$251$			3.33668i			0.210610i	−0.994440	+	0.105305i	$0.966418\pi$
$252$		0			0
$253$			1.62430i			0.102119i
$254$		0			0
$255$	−1.07320			−0.0672067
$256$		0			0
$257$			24.4200i			1.52328i	0.648002	+	0.761639i	$0.275605\pi$
$257$			24.4200i			1.52328i	−0.648002	+	0.761639i	$0.724395\pi$
$258$		0			0
$259$	−3.44665	+	10.9857i	−0.214164	+	0.682616i
$260$		0			0
$261$		−	19.6654i		−	1.21726i
$262$		0			0
$263$			27.1113i			1.67176i	0.548915	+	0.835878i	$0.315041\pi$
$263$			27.1113i			1.67176i	−0.548915	+	0.835878i	$0.684959\pi$
$264$		0			0
$265$			5.48977i			0.337234i
$266$		0			0
$267$	−9.00160			−0.550889
$268$		0			0
$269$	18.5701			1.13224			0.566121	−	0.824322i	$-0.308444\pi$
$269$	18.5701			1.13224			0.566121	+	0.824322i	$0.308444\pi$
$270$		0			0
$271$	12.9119			0.784344			0.392172	−	0.919892i	$-0.371724\pi$
$271$	12.9119			0.784344			0.392172	+	0.919892i	$0.371724\pi$
$272$		0			0
$273$	−0.533409	+	1.70016i	−0.0322834	+	0.102898i
$274$		0			0
$275$	0.580189			0.0349867
$276$		0			0
$277$			0.557017i			0.0334679i	0.999860	+	0.0167340i	$0.00532683\pi$
$277$			0.557017i			0.0334679i	−0.999860	+	0.0167340i	$0.994673\pi$
$278$		0			0
$279$	12.5613			0.752025
$280$		0			0
$281$	−27.3862			−1.63372			−0.816860	−	0.576835i	$-0.804287\pi$
$281$	−27.3862			−1.63372			−0.816860	+	0.576835i	$0.804287\pi$
$282$		0			0
$283$			5.24149i			0.311574i	0.987791	+	0.155787i	$0.0497914\pi$
$283$			5.24149i			0.311574i	−0.987791	+	0.155787i	$0.950209\pi$
$284$		0			0
$285$	−2.77257			−0.164233
$286$		0			0
$287$	−6.70253	+	21.3633i	−0.395638	+	1.26103i
$288$		0			0
$289$	13.6555			0.803263
$290$		0			0
$291$	8.78840			0.515185
$292$		0			0
$293$	−6.66098			−0.389138			−0.194569	−	0.980889i	$-0.562331\pi$
$293$	−6.66098			−0.389138			−0.194569	+	0.980889i	$0.562331\pi$
$294$		0			0
$295$			13.2957i			0.774107i
$296$		0			0
$297$			1.92560i			0.111735i
$298$		0			0
$299$		−	3.21299i		−	0.185812i
$300$		0			0
$301$	−10.9134	−	3.42399i	−0.629041	−	0.197355i
$302$		0			0
$303$			7.49993i			0.430860i
$304$		0			0
$305$	0.275184			0.0157570
$306$		0			0
$307$		−	17.0826i		−	0.974957i	−0.873135	−	0.487479i	$-0.837917\pi$
$307$		−	17.0826i		−	0.974957i	0.873135	−	0.487479i	$-0.162083\pi$
$308$		0			0
$309$		−	8.88694i		−	0.505560i
$310$		0			0
$311$	29.4721			1.67121			0.835604	−	0.549332i	$-0.185118\pi$
$311$	29.4721			1.67121			0.835604	+	0.549332i	$0.185118\pi$
$312$		0			0
$313$			26.1624i			1.47879i	0.673275	+	0.739393i	$0.264887\pi$
$313$			26.1624i			1.47879i	−0.673275	+	0.739393i	$0.735113\pi$
$314$		0			0
$315$	6.70392	+	2.10329i	0.377723	+	0.118507i
$316$		0			0
$317$		−	5.72675i		−	0.321646i	−0.986983	−	0.160823i	$-0.948585\pi$
$317$		−	5.72675i		−	0.321646i	0.986983	−	0.160823i	$-0.0514149\pi$
$318$		0			0
$319$		−	4.29641i		−	0.240553i
$320$		0			0
$321$		−	3.93310i		−	0.219524i
$322$		0			0
$323$	−8.64040			−0.480765
$324$		0			0
$325$	−1.14766			−0.0636606
$326$		0			0
$327$	−6.58148			−0.363957
$328$		0			0
$329$	−9.00789	−	2.82614i	−0.496621	−	0.155810i
$330$		0			0
$331$	−17.6047			−0.967641			−0.483820	−	0.875167i	$-0.660751\pi$
$331$	−17.6047			−0.967641			−0.483820	+	0.875167i	$0.660751\pi$
$332$		0			0
$333$		−	11.5566i		−	0.633299i
$334$		0			0
$335$	−7.71818			−0.421689
$336$		0			0
$337$	24.3443			1.32612			0.663059	−	0.748567i	$-0.269258\pi$
$337$	24.3443			1.32612			0.663059	+	0.748567i	$0.269258\pi$
$338$		0			0
$339$			0.827507i			0.0449440i
$340$		0			0
$341$	2.74434			0.148614
$342$		0			0
$343$	−11.3368	−	14.6450i	−0.612131	−	0.790756i
$344$		0			0
$345$	1.64291			0.0884510
$346$		0			0
$347$	−16.7400			−0.898650			−0.449325	−	0.893368i	$-0.648336\pi$
$347$	−16.7400			−0.898650			−0.449325	+	0.893368i	$0.648336\pi$
$348$		0			0
$349$	14.2317			0.761807			0.380903	−	0.924615i	$-0.375613\pi$
$349$	14.2317			0.761807			0.380903	+	0.924615i	$0.375613\pi$
$350$		0			0
$351$		−	3.80898i		−	0.203308i
$352$		0			0
$353$		−	6.08004i		−	0.323608i	−0.986823	−	0.161804i	$-0.948269\pi$
$353$		−	6.08004i		−	0.323608i	0.986823	−	0.161804i	$-0.0517312\pi$
$354$		0			0
$355$			1.13454i			0.0602150i
$356$		0			0
$357$	−2.70922	−	0.849993i	−0.143387	−	0.0449864i
$358$		0			0
$359$		−	16.3854i		−	0.864787i	−0.901685	−	0.432394i	$-0.857669\pi$
$359$		−	16.3854i		−	0.864787i	0.901685	−	0.432394i	$-0.142331\pi$
$360$		0			0
$361$	−3.32202			−0.174843
$362$		0			0
$363$		−	6.25763i		−	0.328441i
$364$		0			0
$365$		−	1.78121i		−	0.0932326i
$366$		0			0
$367$	13.1245			0.685092			0.342546	−	0.939501i	$-0.388711\pi$
$367$	13.1245			0.685092			0.342546	+	0.939501i	$0.388711\pi$
$368$		0			0
$369$		−	22.4736i		−	1.16993i
$370$		0			0
$371$	−4.34798	+	13.8585i	−0.225736	+	0.719498i
$372$		0			0
$373$			0.552362i			0.0286002i	0.999898	+	0.0143001i	$0.00455202\pi$
$373$			0.552362i			0.0286002i	−0.999898	+	0.0143001i	$0.995448\pi$
$374$		0			0
$375$		−	0.586834i		−	0.0303040i
$376$		0			0
$377$			8.49862i			0.437701i
$378$		0			0
$379$	29.1600			1.49785			0.748925	−	0.662655i	$-0.230570\pi$
$379$	29.1600			1.49785			0.748925	+	0.662655i	$0.230570\pi$
$380$		0			0
$381$	9.21933			0.472320
$382$		0			0
$383$	−5.89451			−0.301195			−0.150598	−	0.988595i	$-0.548120\pi$
$383$	−5.89451			−0.301195			−0.150598	+	0.988595i	$0.548120\pi$
$384$		0			0
$385$	1.46464	+	0.459518i	0.0746451	+	0.0234192i
$386$		0			0
$387$	11.4807			0.583594
$388$		0			0
$389$			13.5376i			0.686385i	0.939265	+	0.343192i	$0.111508\pi$
$389$			13.5376i			0.686385i	−0.939265	+	0.343192i	$0.888492\pi$
$390$		0			0
$391$	5.11994			0.258926
$392$		0			0
$393$	2.94444			0.148527
$394$		0			0
$395$			10.0108i			0.503696i
$396$		0			0
$397$	4.22501			0.212047			0.106024	−	0.994364i	$-0.466188\pi$
$397$	4.22501			0.212047			0.106024	+	0.994364i	$0.466188\pi$
$398$		0			0
$399$	−6.99913	−	2.19591i	−0.350395	−	0.109933i
$400$		0			0
$401$	8.09225			0.404108			0.202054	−	0.979374i	$-0.435238\pi$
$401$	8.09225			0.404108			0.202054	+	0.979374i	$0.435238\pi$
$402$		0			0
$403$	−5.42850			−0.270413
$404$		0			0
$405$	−6.01923			−0.299098
$406$		0			0
$407$		−	2.52484i		−	0.125152i
$408$		0			0
$409$		−	27.6608i		−	1.36774i	−0.729605	−	0.683868i	$-0.760296\pi$
$409$		−	27.6608i		−	1.36774i	0.729605	−	0.683868i	$-0.239704\pi$
$410$		0			0
$411$		−	8.72637i		−	0.430440i
$412$		0			0
$413$	−10.5304	+	33.5640i	−0.518167	+	1.65158i
$414$		0			0
$415$		−	15.6230i		−	0.766905i
$416$		0			0
$417$	0.0871872			0.00426957
$418$		0			0
$419$			10.5982i			0.517755i	0.965910	+	0.258877i	$0.0833526\pi$
$419$			10.5982i			0.517755i	−0.965910	+	0.258877i	$0.916647\pi$
$420$		0			0
$421$			25.5951i			1.24743i	0.781652	+	0.623715i	$0.214377\pi$
$421$			25.5951i			1.24743i	−0.781652	+	0.623715i	$0.785623\pi$
$422$		0			0
$423$	9.47606			0.460742
$424$		0			0
$425$		−	1.82880i		−	0.0887101i
$426$		0			0
$427$	0.694680	+	0.217949i	0.0336179	+	0.0105473i
$428$		0			0
$429$		−	0.390749i		−	0.0188655i
$430$		0			0
$431$		−	32.7406i		−	1.57706i	−0.614997	−	0.788529i	$-0.710843\pi$
$431$		−	32.7406i		−	1.57706i	0.614997	−	0.788529i	$-0.289157\pi$
$432$		0			0
$433$			16.8933i			0.811840i	0.913909	+	0.405920i	$0.133049\pi$
$433$			16.8933i			0.811840i	−0.913909	+	0.405920i	$0.866951\pi$
$434$		0			0
$435$	−4.34561			−0.208356
$436$		0			0
$437$	13.2271			0.632737
$438$		0			0
$439$	−31.6749			−1.51176			−0.755881	−	0.654709i	$-0.772791\pi$
$439$	−31.6749			−1.51176			−0.755881	+	0.654709i	$0.772791\pi$
$440$		0			0
$441$	15.2577	+	10.6192i	0.726558	+	0.505676i
$442$		0			0
$443$	12.8303			0.609587			0.304794	−	0.952418i	$-0.401413\pi$
$443$	12.8303			0.609587			0.304794	+	0.952418i	$0.401413\pi$
$444$		0			0
$445$		−	15.3393i		−	0.727151i
$446$		0			0
$447$	−5.19637			−0.245780
$448$		0			0
$449$	9.10148			0.429525			0.214763	−	0.976666i	$-0.431102\pi$
$449$	9.10148			0.429525			0.214763	+	0.976666i	$0.431102\pi$
$450$		0			0
$451$		−	4.90993i		−	0.231200i
$452$		0			0
$453$	7.27276			0.341704
$454$		0			0
$455$	−2.89717	−	0.908961i	−0.135822	−	0.0426127i
$456$		0			0
$457$	−5.29969			−0.247909			−0.123954	−	0.992288i	$-0.539558\pi$
$457$	−5.29969			−0.247909			−0.123954	+	0.992288i	$0.539558\pi$
$458$		0			0
$459$	6.06964			0.283307
$460$		0			0
$461$	−0.755725			−0.0351976			−0.0175988	−	0.999845i	$-0.505602\pi$
$461$	−0.755725			−0.0351976			−0.0175988	+	0.999845i	$0.505602\pi$
$462$		0			0
$463$		−	3.99447i		−	0.185639i	−0.995683	−	0.0928193i	$-0.970412\pi$
$463$		−	3.99447i		−	0.185639i	0.995683	−	0.0928193i	$-0.0295879\pi$
$464$		0			0
$465$		−	2.77577i		−	0.128723i
$466$		0			0
$467$			3.90589i			0.180743i	0.995908	+	0.0903716i	$0.0288055\pi$
$467$			3.90589i			0.180743i	−0.995908	+	0.0903716i	$0.971195\pi$
$468$		0			0
$469$	−19.4839	−	6.11290i	−0.899685	−	0.282268i
$470$		0			0
$471$			9.15992i			0.422067i
$472$		0			0
$473$	2.50824			0.115329
$474$		0			0
$475$		−	4.72462i		−	0.216780i
$476$		0			0
$477$		−	14.5788i		−	0.667517i
$478$		0			0
$479$	9.21055			0.420841			0.210420	−	0.977611i	$-0.432517\pi$
$479$	9.21055			0.420841			0.210420	+	0.977611i	$0.432517\pi$
$480$		0			0
$481$			4.99432i			0.227721i
$482$		0			0
$483$	4.14739	+	1.30120i	0.188713	+	0.0592068i
$484$		0			0
$485$			14.9759i			0.680023i
$486$		0			0
$487$		−	21.4403i		−	0.971552i	−0.874083	−	0.485776i	$-0.838537\pi$
$487$		−	21.4403i		−	0.971552i	0.874083	−	0.485776i	$-0.161463\pi$
$488$		0			0
$489$			3.28692i			0.148640i
$490$		0			0
$491$	−19.6672			−0.887570			−0.443785	−	0.896133i	$-0.646365\pi$
$491$	−19.6672			−0.887570			−0.443785	+	0.896133i	$0.646365\pi$
$492$		0			0
$493$	−13.5426			−0.609930
$494$		0			0
$495$	−1.54077			−0.0692523
$496$		0			0
$497$	−0.898570	+	2.86405i	−0.0403064	+	0.128470i
$498$		0			0
$499$	12.2344			0.547686			0.273843	−	0.961774i	$-0.411705\pi$
$499$	12.2344			0.547686			0.273843	+	0.961774i	$0.411705\pi$
$500$		0			0
$501$			5.48662i			0.245124i
$502$		0			0
$503$	5.42742			0.241997			0.120998	−	0.992653i	$-0.461390\pi$
$503$	5.42742			0.241997			0.120998	+	0.992653i	$0.461390\pi$
$504$		0			0
$505$	−12.7803			−0.568717
$506$		0			0
$507$		−	6.85591i		−	0.304482i
$508$		0			0
$509$	17.3838			0.770522			0.385261	−	0.922808i	$-0.374111\pi$
$509$	17.3838			0.770522			0.385261	+	0.922808i	$0.374111\pi$
$510$		0			0
$511$	1.41074	−	4.49652i	0.0624074	−	0.198914i
$512$		0			0
$513$	15.6806			0.692315
$514$		0			0
$515$	15.1439			0.667319
$516$		0			0
$517$	2.07029			0.0910511
$518$		0			0
$519$		−	13.8320i		−	0.607157i
$520$		0			0
$521$		−	12.8139i		−	0.561388i	−0.959797	−	0.280694i	$-0.909435\pi$
$521$		−	12.8139i		−	0.561388i	0.959797	−	0.280694i	$-0.0905646\pi$
$522$		0			0
$523$		−	8.05393i		−	0.352174i	−0.984375	−	0.176087i	$-0.943656\pi$
$523$		−	8.05393i		−	0.352174i	0.984375	−	0.176087i	$-0.0563439\pi$
$524$		0			0
$525$	0.464781	−	1.48142i	0.0202847	−	0.0646544i
$526$		0			0
$527$		−	8.65037i		−	0.376816i
$528$		0			0
$529$	15.1622			0.659226
$530$		0			0
$531$		−	35.3085i		−	1.53226i
$532$		0			0
$533$			9.71221i			0.420683i
$534$		0			0
$535$	6.70224			0.289763
$536$		0			0
$537$		−	11.5657i		−	0.499096i
$538$		0			0
$539$	3.33344	+	2.32004i	0.143581	+	0.0999310i
$540$		0			0
$541$		−	15.9099i		−	0.684018i	−0.939697	−	0.342009i	$-0.888893\pi$
$541$		−	15.9099i		−	0.684018i	0.939697	−	0.342009i	$-0.111107\pi$
$542$		0			0
$543$			12.3295i			0.529110i
$544$		0			0
$545$		−	11.2152i		−	0.480408i
$546$		0			0
$547$	28.6893			1.22667			0.613333	−	0.789825i	$-0.289828\pi$
$547$	28.6893			1.22667			0.613333	+	0.789825i	$0.289828\pi$
$548$		0			0
$549$	−0.730785			−0.0311891
$550$		0			0
$551$	−34.9867			−1.49048
$552$		0			0
$553$	−7.92866	+	25.2714i	−0.337161	+	1.07465i
$554$		0			0
$555$	−2.55376			−0.108401
$556$		0			0
$557$			29.6910i			1.25805i	0.777386	+	0.629024i	$0.216545\pi$
$557$			29.6910i			1.25805i	−0.777386	+	0.629024i	$0.783455\pi$
$558$		0			0
$559$	−4.96149			−0.209849
$560$		0			0
$561$	0.622662			0.0262888
$562$		0			0
$563$		−	25.7978i		−	1.08725i	−0.839329	−	0.543624i	$-0.817052\pi$
$563$		−	25.7978i		−	1.08725i	0.839329	−	0.543624i	$-0.182948\pi$
$564$		0			0
$565$	−1.41012			−0.0593242
$566$		0			0
$567$	−15.1951	−	4.76731i	−0.638133	−	0.200208i
$568$		0			0
$569$	−6.17604			−0.258913			−0.129457	−	0.991585i	$-0.541323\pi$
$569$	−6.17604			−0.258913			−0.129457	+	0.991585i	$0.541323\pi$
$570$		0			0
$571$	44.7626			1.87326			0.936628	−	0.350325i	$-0.113929\pi$
$571$	44.7626			1.87326			0.936628	+	0.350325i	$0.113929\pi$
$572$		0			0
$573$	−9.96153			−0.416149
$574$		0			0
$575$			2.79961i			0.116752i
$576$		0			0
$577$		−	45.0609i		−	1.87591i	−0.346760	−	0.937954i	$-0.612718\pi$
$577$		−	45.0609i		−	1.87591i	0.346760	−	0.937954i	$-0.387282\pi$
$578$		0			0
$579$			8.71126i			0.362028i
$580$		0			0
$581$	12.3737	−	39.4392i	0.513346	−	1.63621i
$582$		0			0
$583$		−	3.18511i		−	0.131914i
$584$		0			0
$585$	3.04775			0.126009
$586$		0			0
$587$		−	0.738708i		−	0.0304897i	−0.999884	−	0.0152449i	$-0.995147\pi$
$587$		−	0.738708i		−	0.0304897i	0.999884	−	0.0152449i	$-0.00485278\pi$
$588$		0			0
$589$		−	22.3478i		−	0.920824i
$590$		0			0
$591$	−3.70166			−0.152266
$592$		0			0
$593$			5.51961i			0.226663i	0.993557	+	0.113332i	$0.0361522\pi$
$593$			5.51961i			0.226663i	−0.993557	+	0.113332i	$0.963848\pi$
$594$		0			0
$595$	1.44844	−	4.61668i	0.0593802	−	0.189265i
$596$		0			0
$597$			7.02853i			0.287659i
$598$		0			0
$599$		−	0.145815i		−	0.00595785i	−0.999996	−	0.00297893i	$-0.999052\pi$
$599$		−	0.145815i		−	0.00595785i	0.999996	−	0.00297893i	$-0.000948223\pi$
$600$		0			0
$601$			21.7269i			0.886257i	0.896458	+	0.443128i	$0.146131\pi$
$601$			21.7269i			0.886257i	−0.896458	+	0.443128i	$0.853869\pi$
$602$		0			0
$603$	20.4966			0.834686
$604$		0			0
$605$	10.6634			0.433528
$606$		0			0
$607$	18.6456			0.756804			0.378402	−	0.925641i	$-0.376474\pi$
$607$	18.6456			0.756804			0.378402	+	0.925641i	$0.376474\pi$
$608$		0			0
$609$	−10.9702	−	3.44179i	−0.444534	−	0.139468i
$610$		0			0
$611$	−4.09518			−0.165673
$612$		0			0
$613$		−	47.1735i		−	1.90532i	−0.304041	−	0.952659i	$-0.598336\pi$
$613$		−	47.1735i		−	1.90532i	0.304041	−	0.952659i	$-0.401664\pi$
$614$		0			0
$615$	−4.96616			−0.200255
$616$		0			0
$617$	16.2076			0.652493			0.326246	−	0.945285i	$-0.394216\pi$
$617$	16.2076			0.652493			0.326246	+	0.945285i	$0.394216\pi$
$618$		0			0
$619$			30.2319i			1.21512i	0.794273	+	0.607561i	$0.207852\pi$
$619$			30.2319i			1.21512i	−0.794273	+	0.607561i	$0.792148\pi$
$620$		0			0
$621$	−9.29166			−0.372861
$622$		0			0
$623$	12.1489	−	38.7228i	0.486736	−	1.55140i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	1.60861			0.0642418
$628$		0			0
$629$	−7.95850			−0.317326
$630$		0			0
$631$			6.27504i			0.249806i	0.992169	+	0.124903i	$0.0398619\pi$
$631$			6.27504i			0.249806i	−0.992169	+	0.124903i	$0.960138\pi$
$632$		0			0
$633$		−	6.36864i		−	0.253131i
$634$		0			0
$635$			15.7103i			0.623443i
$636$		0			0
$637$	−6.59379	−	4.58920i	−0.261255	−	0.181831i
$638$		0			0
$639$		−	3.01291i		−	0.119189i
$640$		0			0
$641$	−0.954180			−0.0376878			−0.0188439	−	0.999822i	$-0.505999\pi$
$641$	−0.954180			−0.0376878			−0.0188439	+	0.999822i	$0.505999\pi$
$642$		0			0
$643$		−	29.7150i		−	1.17184i	−0.810367	−	0.585922i	$-0.800732\pi$
$643$		−	29.7150i		−	1.17184i	0.810367	−	0.585922i	$-0.199268\pi$
$644$		0			0
$645$		−	2.53697i		−	0.0998930i
$646$		0			0
$647$	22.3760			0.879693			0.439846	−	0.898073i	$-0.355033\pi$
$647$	22.3760			0.879693			0.439846	+	0.898073i	$0.355033\pi$
$648$		0			0
$649$		−	7.71403i		−	0.302802i
$650$		0			0
$651$	2.19844	−	7.00721i	0.0861638	−	0.274634i
$652$		0			0
$653$		−	3.18538i		−	0.124653i	−0.998056	−	0.0623267i	$-0.980148\pi$
$653$		−	3.18538i		−	0.124653i	0.998056	−	0.0623267i	$-0.0198521\pi$
$654$		0			0
$655$			5.01749i			0.196050i
$656$		0			0
$657$			4.73022i			0.184543i
$658$		0			0
$659$	1.50751			0.0587242			0.0293621	−	0.999569i	$-0.490652\pi$
$659$	1.50751			0.0587242			0.0293621	+	0.999569i	$0.490652\pi$
$660$		0			0
$661$	−11.0607			−0.430210			−0.215105	−	0.976591i	$-0.569009\pi$
$661$	−11.0607			−0.430210			−0.215105	+	0.976591i	$0.569009\pi$
$662$		0			0
$663$	−1.23167			−0.0478342
$664$		0			0
$665$	3.74196	−	11.9269i	0.145107	−	0.462507i
$666$		0			0
$667$	20.7316			0.802731
$668$		0			0
$669$		−	8.25893i		−	0.319309i
$670$		0			0
$671$	−0.159659			−0.00616355
$672$		0			0
$673$	31.5781			1.21725			0.608624	−	0.793459i	$-0.291722\pi$
$673$	31.5781			1.21725			0.608624	+	0.793459i	$0.291722\pi$
$674$		0			0
$675$			3.31891i			0.127745i
$676$		0			0
$677$	−5.96254			−0.229159			−0.114580	−	0.993414i	$-0.536552\pi$
$677$	−5.96254			−0.229159			−0.114580	+	0.993414i	$0.536552\pi$
$678$		0			0
$679$	−11.8612	+	37.8056i	−0.455189	+	1.45085i
$680$		0			0
$681$	8.78388			0.336599
$682$		0			0
$683$	−32.0466			−1.22623			−0.613114	−	0.789994i	$-0.710084\pi$
$683$	−32.0466			−1.22623			−0.613114	+	0.789994i	$0.710084\pi$
$684$		0			0
$685$	14.8703			0.568163
$686$		0			0
$687$		−	12.6571i		−	0.482898i
$688$		0			0
$689$			6.30038i			0.240026i
$690$		0			0
$691$			17.0740i			0.649526i	0.945795	+	0.324763i	$0.105285\pi$
$691$			17.0740i			0.649526i	−0.945795	+	0.324763i	$0.894715\pi$
$692$		0			0
$693$	−3.88955	−	1.22031i	−0.147752	−	0.0463557i
$694$		0			0
$695$			0.148572i			0.00563566i
$696$		0			0
$697$	−15.4765			−0.586215
$698$		0			0
$699$		−	11.4954i		−	0.434796i
$700$		0			0
$701$			32.7289i			1.23615i	0.786118	+	0.618076i	$0.212088\pi$
$701$			32.7289i			1.23615i	−0.786118	+	0.618076i	$0.787912\pi$
$702$		0			0
$703$	−20.5604			−0.775449
$704$		0			0
$705$		−	2.09400i		−	0.0788645i
$706$		0			0
$707$	−32.2629	−	10.1222i	−1.21337	−	0.380684i
$708$		0			0
$709$			20.9447i			0.786594i	0.919412	+	0.393297i	$0.128666\pi$
$709$			20.9447i			0.786594i	−0.919412	+	0.393297i	$0.871334\pi$
$710$		0			0
$711$		−	26.5848i		−	0.997009i
$712$		0			0
$713$			13.2423i			0.495930i
$714$		0			0
$715$	0.665859			0.0249017
$716$		0			0
$717$	−13.6475			−0.509677
$718$		0			0
$719$	−0.417030			−0.0155526			−0.00777630	−	0.999970i	$-0.502475\pi$
$719$	−0.417030			−0.0155526			−0.00777630	+	0.999970i	$0.502475\pi$
$720$		0			0
$721$	38.2295	+	11.9942i	1.42374	+	0.446686i
$722$		0			0
$723$	12.2681			0.456256
$724$		0			0
$725$		−	7.40518i		−	0.275022i
$726$		0			0
$727$	4.48987			0.166520			0.0832601	−	0.996528i	$-0.473467\pi$
$727$	4.48987			0.166520			0.0832601	+	0.996528i	$0.473467\pi$
$728$		0			0
$729$	10.1419			0.375624
$730$		0			0
$731$		−	7.90619i		−	0.292421i
$732$		0			0
$733$	−22.8948			−0.845640			−0.422820	−	0.906214i	$-0.638960\pi$
$733$	−22.8948			−0.845640			−0.422820	+	0.906214i	$0.638960\pi$
$734$		0			0
$735$	2.34661	−	3.37161i	0.0865559	−	0.124364i
$736$		0			0
$737$	4.47800			0.164949
$738$		0			0
$739$	0.354985			0.0130583			0.00652917	−	0.999979i	$-0.497922\pi$
$739$	0.354985			0.0130583			0.00652917	+	0.999979i	$0.497922\pi$
$740$		0			0
$741$	−3.18196			−0.116892
$742$		0			0
$743$		−	6.75171i		−	0.247696i	−0.992301	−	0.123848i	$-0.960476\pi$
$743$		−	6.75171i		−	0.247696i	0.992301	−	0.123848i	$-0.0395235\pi$
$744$		0			0
$745$		−	8.85493i		−	0.324419i
$746$		0			0
$747$			41.4890i			1.51800i
$748$		0			0
$749$	16.9193	+	5.30827i	0.618218	+	0.193960i
$750$		0			0
$751$			27.4120i			1.00028i	0.865945	+	0.500139i	$0.166718\pi$
$751$			27.4120i			1.00028i	−0.865945	+	0.500139i	$0.833282\pi$
$752$		0			0
$753$	−1.95808			−0.0713564
$754$		0			0
$755$			12.3932i			0.451035i
$756$		0			0
$757$		−	27.6550i		−	1.00514i	−0.864537	−	0.502570i	$-0.832388\pi$
$757$		−	27.6550i		−	1.00514i	0.864537	−	0.502570i	$-0.167612\pi$
$758$		0			0
$759$	−0.953196			−0.0345988
$760$		0			0
$761$		−	14.1083i		−	0.511424i	−0.966753	−	0.255712i	$-0.917690\pi$
$761$		−	14.1083i		−	0.511424i	0.966753	−	0.255712i	$-0.0823099\pi$
$762$		0			0
$763$	8.88262	−	28.3120i	0.321572	−	1.02496i
$764$		0			0
$765$			4.85662i			0.175592i
$766$		0			0
$767$			15.2589i			0.550968i
$768$		0			0
$769$		−	2.82610i		−	0.101912i	−0.998701	−	0.0509558i	$-0.983773\pi$
$769$		−	2.82610i		−	0.101912i	0.998701	−	0.0509558i	$-0.0162268\pi$
$770$		0			0
$771$	−14.3305			−0.516100
$772$		0			0
$773$	44.5587			1.60266			0.801332	−	0.598220i	$-0.204125\pi$
$773$	44.5587			1.60266			0.801332	+	0.598220i	$0.204125\pi$
$774$		0			0
$775$	4.73007			0.169909
$776$		0			0
$777$	−6.44676	−	2.02261i	−0.231276	−	0.0725607i
$778$		0			0
$779$	−39.9827			−1.43253
$780$		0			0
$781$		−	0.658247i		−	0.0235539i
$782$		0			0
$783$	24.5772			0.878316
$784$		0			0
$785$	−15.6090			−0.557111
$786$		0			0
$787$			30.3744i			1.08273i	0.840788	+	0.541365i	$0.182092\pi$
$787$			30.3744i			1.08273i	−0.840788	+	0.541365i	$0.817908\pi$
$788$		0			0
$789$	−15.9098			−0.566406
$790$		0			0
$791$	−3.55974	−	1.11684i	−0.126570	−	0.0397101i
$792$		0			0
$793$	0.315817			0.0112150
$794$		0			0
$795$	−3.22159			−0.114258
$796$		0			0
$797$	−43.5064			−1.54108			−0.770538	−	0.637394i	$-0.780012\pi$
$797$	−43.5064			−1.54108			−0.770538	+	0.637394i	$0.780012\pi$
$798$		0			0
$799$		−	6.52572i		−	0.230863i
$800$		0			0
$801$			40.7353i			1.43931i
$802$		0			0
$803$			1.03344i			0.0364692i
$804$		0			0
$805$	−2.21733	+	7.06740i	−0.0781506	+	0.249093i
$806$		0			0
$807$			10.8976i			0.383613i
$808$		0			0
$809$	−17.7818			−0.625176			−0.312588	−	0.949889i	$-0.601196\pi$
$809$	−17.7818			−0.625176			−0.312588	+	0.949889i	$0.601196\pi$
$810$		0			0
$811$			7.03219i			0.246934i	0.992349	+	0.123467i	$0.0394012\pi$
$811$			7.03219i			0.246934i	−0.992349	+	0.123467i	$0.960599\pi$
$812$		0			0
$813$			7.57716i			0.265743i
$814$		0			0
$815$	−5.60110			−0.196198
$816$		0			0
$817$		−	20.4252i		−	0.714588i
$818$		0			0
$819$	7.69381	+	2.41386i	0.268844	+	0.0843471i
$820$		0			0
$821$		−	7.24453i		−	0.252836i	−0.991977	−	0.126418i	$-0.959652\pi$
$821$		−	7.24453i		−	0.252836i	0.991977	−	0.126418i	$-0.0403481\pi$
$822$		0			0
$823$			46.4665i			1.61972i	0.586624	+	0.809860i	$0.300457\pi$
$823$			46.4665i			1.61972i	−0.586624	+	0.809860i	$0.699543\pi$
$824$		0			0
$825$			0.340475i			0.0118538i
$826$		0			0
$827$	51.0385			1.77478			0.887392	−	0.461016i	$-0.152515\pi$
$827$	51.0385			1.77478			0.887392	+	0.461016i	$0.152515\pi$
$828$		0			0
$829$	14.8970			0.517393			0.258696	−	0.965959i	$-0.416707\pi$
$829$	14.8970			0.517393			0.258696	+	0.965959i	$0.416707\pi$
$830$		0			0
$831$	−0.326876			−0.0113392
$832$		0			0
$833$	7.31294	−	10.5073i	0.253379	−	0.364055i
$834$		0			0
$835$	−9.34952			−0.323554
$836$		0			0
$837$			15.6987i			0.542626i
$838$		0			0
$839$	4.53355			0.156516			0.0782578	−	0.996933i	$-0.475064\pi$
$839$	4.53355			0.156516			0.0782578	+	0.996933i	$0.475064\pi$
$840$		0			0
$841$	−25.8367			−0.890922
$842$		0			0
$843$		−	16.0711i		−	0.553519i
$844$		0			0
$845$	11.6829			0.401903
$846$		0			0
$847$	26.9189	+	8.44554i	0.924944	+	0.290192i
$848$		0			0
$849$	−3.07588			−0.105564
$850$		0			0
$851$	12.1832			0.417635
$852$		0			0
$853$	39.0584			1.33734			0.668668	−	0.743561i	$-0.266865\pi$
$853$	39.0584			1.33734			0.668668	+	0.743561i	$0.266865\pi$
$854$		0			0
$855$			12.5468i			0.429092i
$856$		0			0
$857$			52.9271i			1.80795i	0.427580	+	0.903977i	$0.359366\pi$
$857$			52.9271i			1.80795i	−0.427580	+	0.903977i	$0.640634\pi$
$858$		0			0
$859$			25.8556i			0.882182i	0.897462	+	0.441091i	$0.145408\pi$
$859$			25.8556i			0.882182i	−0.897462	+	0.441091i	$0.854592\pi$
$860$		0			0
$861$	−12.5367	−	3.93327i	−0.427250	−	0.134045i
$862$		0			0
$863$		−	31.9510i		−	1.08762i	−0.839207	−	0.543812i	$-0.816980\pi$
$863$		−	31.9510i		−	1.08762i	0.839207	−	0.543812i	$-0.183020\pi$
$864$		0			0
$865$	23.5706			0.801423
$866$		0			0
$867$			8.01350i			0.272153i
$868$		0			0
$869$		−	5.80813i		−	0.197027i
$870$		0			0
$871$	−8.85783			−0.300136
$872$		0			0
$873$		−	39.7705i		−	1.34603i
$874$		0			0
$875$	2.52442	+	0.792014i	0.0853411	+	0.0267750i
$876$		0			0
$877$		−	40.3185i		−	1.36146i	−0.732534	−	0.680730i	$-0.761663\pi$
$877$		−	40.3185i		−	1.36146i	0.732534	−	0.680730i	$-0.238337\pi$
$878$		0			0
$879$		−	3.90889i		−	0.131843i
$880$		0			0
$881$		−	1.03841i		−	0.0349848i	−0.999847	−	0.0174924i	$-0.994432\pi$
$881$		−	1.03841i		−	0.0349848i	0.999847	−	0.0174924i	$-0.00556829\pi$
$882$		0			0
$883$	−31.1810			−1.04932			−0.524662	−	0.851310i	$-0.675808\pi$
$883$	−31.1810			−1.04932			−0.524662	+	0.851310i	$0.675808\pi$
$884$		0			0
$885$	−7.80238			−0.262274
$886$		0			0
$887$	−32.4273			−1.08880			−0.544402	−	0.838825i	$-0.683243\pi$
$887$	−32.4273			−1.08880			−0.544402	+	0.838825i	$0.683243\pi$
$888$		0			0
$889$	−12.4428	+	39.6594i	−0.417317	+	1.33013i
$890$		0			0
$891$	3.49229			0.116996
$892$		0			0
$893$		−	16.8588i		−	0.564160i
$894$		0			0
$895$	19.7086			0.658787
$896$		0			0
$897$	1.88549			0.0629548
$898$		0			0
$899$		−	35.0270i		−	1.16822i
$900$		0			0
$901$	−10.0397			−0.334472
$902$		0			0
$903$	2.00931	−	6.40438i	0.0668658	−	0.213124i
$904$		0			0
$905$	−21.0102			−0.698404
$906$		0			0
$907$	−48.3550			−1.60560			−0.802801	−	0.596248i	$-0.796658\pi$
$907$	−48.3550			−1.60560			−0.802801	+	0.596248i	$0.796658\pi$
$908$		0			0
$909$	33.9398			1.12571
$910$		0			0
$911$			48.2821i			1.59966i	0.600229	+	0.799828i	$0.295076\pi$
$911$			48.2821i			1.59966i	−0.600229	+	0.799828i	$0.704924\pi$
$912$		0			0
$913$			9.06432i			0.299985i
$914$		0			0
$915$			0.161487i			0.00533860i
$916$		0			0
$917$	−3.97392	+	12.6663i	−0.131231	+	0.418277i
$918$		0			0
$919$			6.72784i			0.221931i	0.993824	+	0.110965i	$0.0353943\pi$
$919$			6.72784i			0.221931i	−0.993824	+	0.110965i	$0.964606\pi$
$920$		0			0
$921$	10.0247			0.330324
$922$		0			0
$923$			1.30206i			0.0428579i
$924$		0			0
$925$		−	4.35175i		−	0.143085i
$926$		0			0
$927$	−40.2165			−1.32088
$928$		0			0
$929$		−	15.7311i		−	0.516120i	−0.966129	−	0.258060i	$-0.916917\pi$
$929$		−	15.7311i		−	0.516120i	0.966129	−	0.258060i	$-0.0830832\pi$
$930$		0			0
$931$	18.8926	−	27.1450i	0.619180	−	0.889640i
$932$		0			0
$933$			17.2952i			0.566220i
$934$		0			0
$935$			1.06105i			0.0347001i
$936$		0			0
$937$		−	32.8060i		−	1.07172i	−0.844306	−	0.535862i	$-0.819987\pi$
$937$		−	32.8060i		−	1.07172i	0.844306	−	0.535862i	$-0.180013\pi$
$938$		0			0
$939$	−15.3530			−0.501025
$940$		0			0
$941$	−10.5674			−0.344488			−0.172244	−	0.985054i	$-0.555102\pi$
$941$	−10.5674			−0.344488			−0.172244	+	0.985054i	$0.555102\pi$
$942$		0			0
$943$	23.6921			0.771520
$944$		0			0
$945$	−2.62862	+	8.37834i	−0.0855092	+	0.272548i
$946$		0			0
$947$	−34.2414			−1.11269			−0.556347	−	0.830950i	$-0.687798\pi$
$947$	−34.2414			−1.11269			−0.556347	+	0.830950i	$0.687798\pi$
$948$		0			0
$949$		−	2.04421i		−	0.0663580i
$950$		0			0
$951$	3.36065			0.108977
$952$		0			0
$953$	−3.54692			−0.114896			−0.0574480	−	0.998348i	$-0.518296\pi$
$953$	−3.54692			−0.114896			−0.0574480	+	0.998348i	$0.518296\pi$
$954$		0			0
$955$		−	16.9750i		−	0.549299i
$956$		0			0
$957$	2.52128			0.0815013
$958$		0			0
$959$	37.5388	+	11.7775i	1.21219	+	0.380314i
$960$		0			0
$961$	−8.62644			−0.278272
$962$		0			0
$963$	−17.7986			−0.573553
$964$		0			0
$965$	−14.8445			−0.477862
$966$		0			0
$967$		−	13.7899i		−	0.443454i	−0.975109	−	0.221727i	$-0.928831\pi$
$967$		−	13.7899i		−	0.443454i	0.975109	−	0.221727i	$-0.0711693\pi$
$968$		0			0
$969$		−	5.07048i		−	0.162887i
$970$		0			0
$971$			39.0911i			1.25449i	0.778821	+	0.627246i	$0.215818\pi$
$971$			39.0911i			1.25449i	−0.778821	+	0.627246i	$0.784182\pi$
$972$		0			0
$973$	−0.117671	+	0.375059i	−0.00377236	+	0.0120238i
$974$		0			0
$975$		−	0.673485i		−	0.0215688i
$976$		0			0
$977$	−18.1910			−0.581982			−0.290991	−	0.956726i	$-0.593985\pi$
$977$	−18.1910			−0.581982			−0.290991	+	0.956726i	$0.593985\pi$
$978$		0			0
$979$			8.89968i			0.284435i
$980$		0			0
$981$			29.7835i			0.950913i
$982$		0			0
$983$	44.1968			1.40966			0.704830	−	0.709376i	$-0.251023\pi$
$983$	44.1968			1.40966			0.704830	+	0.709376i	$0.251023\pi$
$984$		0			0
$985$		−	6.30785i		−	0.200985i
$986$		0			0
$987$	1.65847	−	5.28614i	0.0527898	−	0.168260i
$988$		0			0
$989$			12.1031i			0.384857i
$990$		0			0
$991$		−	15.9016i		−	0.505132i	−0.967580	−	0.252566i	$-0.918726\pi$
$991$		−	15.9016i		−	0.505132i	0.967580	−	0.252566i	$-0.0812745\pi$
$992$		0			0
$993$		−	10.3310i		−	0.327845i
$994$		0			0
$995$	−11.9770			−0.379698
$996$		0			0
$997$	13.5139			0.427988			0.213994	−	0.976835i	$-0.431353\pi$
$997$	13.5139			0.427988			0.213994	+	0.976835i	$0.431353\pi$
$998$		0			0
$999$	14.4431			0.456959

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.h.a.111.10		16
4.3	odd	2		280.2.h.a.251.14	yes	16
7.6	odd	2		1120.2.h.b.111.7		16
8.3	odd	2		1120.2.h.b.111.10		16
8.5	even	2		280.2.h.b.251.13	yes	16
28.27	even	2		280.2.h.b.251.14	yes	16
56.13	odd	2		280.2.h.a.251.13	✓	16
56.27	even	2	inner	1120.2.h.a.111.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.h.a.251.13	✓	16	56.13	odd	2
280.2.h.a.251.14	yes	16	4.3	odd	2
280.2.h.b.251.13	yes	16	8.5	even	2
280.2.h.b.251.14	yes	16	28.27	even	2
1120.2.h.a.111.7		16	56.27	even	2	inner
1120.2.h.a.111.10		16	1.1	even	1	trivial
1120.2.h.b.111.7		16	7.6	odd	2
1120.2.h.b.111.10		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)

\(f(q)\)	\(=\)	\(q+0.586834i q^{3} -1.00000 q^{5} +(-2.52442 - 0.792014i) q^{7} +2.65563 q^{9} +O(q^{10})\)	\(q+0.586834i q^{3} -1.00000 q^{5} +(-2.52442 - 0.792014i) q^{7} +2.65563 q^{9} +0.580189 q^{11} -1.14766 q^{13} -0.586834i q^{15} -1.82880i q^{17} -4.72462i q^{19} +(0.464781 - 1.48142i) q^{21} +2.79961i q^{23} +1.00000 q^{25} +3.31891i q^{27} -7.40518i q^{29} +4.73007 q^{31} +0.340475i q^{33} +(2.52442 + 0.792014i) q^{35} -4.35175i q^{37} -0.673485i q^{39} -8.46264i q^{41} +4.32314 q^{43} -2.65563 q^{45} +3.56830 q^{47} +(5.74543 + 3.99876i) q^{49} +1.07320 q^{51} -5.48977i q^{53} -0.580189 q^{55} +2.77257 q^{57} -13.2957i q^{59} -0.275184 q^{61} +(-6.70392 - 2.10329i) q^{63} +1.14766 q^{65} +7.71818 q^{67} -1.64291 q^{69} -1.13454i q^{71} +1.78121i q^{73} +0.586834i q^{75} +(-1.46464 - 0.459518i) q^{77} -10.0108i q^{79} +6.01923 q^{81} +15.6230i q^{83} +1.82880i q^{85} +4.34561 q^{87} +15.3393i q^{89} +(2.89717 + 0.908961i) q^{91} +2.77577i q^{93} +4.72462i q^{95} -14.9759i q^{97} +1.54077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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