LMFDB - Embedded newform 1120.2.h.b.111.9

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			111.9
Root			$-0.470943 + 1.33350i$ of defining polynomial
Character	$\chi$	$=$	1120.111
Dual form			1120.2.h.b.111.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.528177i q^{3} +1.00000 q^{5} +(-2.17448 - 1.50719i) q^{7} +2.72103 q^{9} +O(q^{10})$	$q+0.528177i q^{3} +1.00000 q^{5} +(-2.17448 - 1.50719i) q^{7} +2.72103 q^{9} -3.04781 q^{11} -4.75069 q^{13} +0.528177i q^{15} -6.78894i q^{17} -0.584285i q^{19} +(0.796062 - 1.14851i) q^{21} -5.80481i q^{23} +1.00000 q^{25} +3.02172i q^{27} +0.185682i q^{29} -3.12589 q^{31} -1.60978i q^{33} +(-2.17448 - 1.50719i) q^{35} -7.04672i q^{37} -2.50920i q^{39} +3.83583i q^{41} +1.43554 q^{43} +2.72103 q^{45} -2.95871 q^{47} +(2.45676 + 6.55472i) q^{49} +3.58576 q^{51} +0.535513i q^{53} -3.04781 q^{55} +0.308606 q^{57} -1.52116i q^{59} -13.2354 q^{61} +(-5.91683 - 4.10111i) q^{63} -4.75069 q^{65} +9.13242 q^{67} +3.06597 q^{69} -9.68233i q^{71} -12.4295i q^{73} +0.528177i q^{75} +(6.62742 + 4.59363i) q^{77} -2.81383i q^{79} +6.56709 q^{81} -13.4535i q^{83} -6.78894i q^{85} -0.0980728 q^{87} +3.10041i q^{89} +(10.3303 + 7.16019i) q^{91} -1.65102i q^{93} -0.584285i q^{95} +13.5027i q^{97} -8.29319 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			0.528177i			0.304943i	0.988308	+	0.152472i	$0.0487232\pi$
$3$			0.528177i			0.304943i	−0.988308	+	0.152472i	$0.951277\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−2.17448	−	1.50719i	−0.821878	−	0.569664i
$7$	−2.17448	−	1.50719i	−0.821878	−	0.569664i
$8$		0			0
$9$	2.72103			0.907010
$10$		0			0
$11$	−3.04781			−0.918950			−0.459475	−	0.888191i	$-0.651962\pi$
$11$	−3.04781			−0.918950			−0.459475	+	0.888191i	$0.651962\pi$
$12$		0			0
$13$	−4.75069			−1.31760			−0.658802	−	0.752317i	$-0.728936\pi$
$13$	−4.75069			−1.31760			−0.658802	+	0.752317i	$0.728936\pi$
$14$		0			0
$15$			0.528177i			0.136375i
$16$		0			0
$17$		−	6.78894i		−	1.64656i	−0.567635	−	0.823280i	$-0.692142\pi$
$17$		−	6.78894i		−	1.64656i	0.567635	−	0.823280i	$-0.307858\pi$
$18$		0			0
$19$		−	0.584285i		−	0.134044i	−0.997751	−	0.0670221i	$-0.978650\pi$
$19$		−	0.584285i		−	0.134044i	0.997751	−	0.0670221i	$-0.0213498\pi$
$20$		0			0
$21$	0.796062	−	1.14851i	0.173715	−	0.250626i
$22$		0			0
$23$		−	5.80481i		−	1.21039i	−0.796079	−	0.605193i	$-0.793096\pi$
$23$		−	5.80481i		−	1.21039i	0.796079	−	0.605193i	$-0.206904\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			3.02172i			0.581529i
$28$		0			0
$29$			0.185682i			0.0344802i	0.999851	+	0.0172401i	$0.00548797\pi$
$29$			0.185682i			0.0344802i	−0.999851	+	0.0172401i	$0.994512\pi$
$30$		0			0
$31$	−3.12589			−0.561427			−0.280714	−	0.959792i	$-0.590571\pi$
$31$	−3.12589			−0.561427			−0.280714	+	0.959792i	$0.590571\pi$
$32$		0			0
$33$		−	1.60978i		−	0.280227i
$34$		0			0
$35$	−2.17448	−	1.50719i	−0.367555	−	0.254761i
$36$		0			0
$37$		−	7.04672i		−	1.15847i	−0.815159	−	0.579237i	$-0.803351\pi$
$37$		−	7.04672i		−	1.15847i	0.815159	−	0.579237i	$-0.196649\pi$
$38$		0			0
$39$		−	2.50920i		−	0.401794i
$40$		0			0
$41$			3.83583i			0.599056i	0.954087	+	0.299528i	$0.0968292\pi$
$41$			3.83583i			0.599056i	−0.954087	+	0.299528i	$0.903171\pi$
$42$		0			0
$43$	1.43554			0.218917			0.109459	−	0.993991i	$-0.465088\pi$
$43$	1.43554			0.218917			0.109459	+	0.993991i	$0.465088\pi$
$44$		0			0
$45$	2.72103			0.405627
$46$		0			0
$47$	−2.95871			−0.431572			−0.215786	−	0.976441i	$-0.569231\pi$
$47$	−2.95871			−0.431572			−0.215786	+	0.976441i	$0.569231\pi$
$48$		0			0
$49$	2.45676	+	6.55472i	0.350966	+	0.936388i
$50$		0			0
$51$	3.58576			0.502107
$52$		0			0
$53$			0.535513i			0.0735584i	0.999323	+	0.0367792i	$0.0117098\pi$
$53$			0.535513i			0.0735584i	−0.999323	+	0.0367792i	$0.988290\pi$
$54$		0			0
$55$	−3.04781			−0.410967
$56$		0			0
$57$	0.308606			0.0408759
$58$		0			0
$59$		−	1.52116i		−	0.198038i	−0.995086	−	0.0990188i	$-0.968430\pi$
$59$		−	1.52116i		−	0.198038i	0.995086	−	0.0990188i	$-0.0315704\pi$
$60$		0			0
$61$	−13.2354			−1.69461			−0.847307	−	0.531103i	$-0.821778\pi$
$61$	−13.2354			−1.69461			−0.847307	+	0.531103i	$0.821778\pi$
$62$		0			0
$63$	−5.91683	−	4.10111i	−0.745451	−	0.516691i
$64$		0			0
$65$	−4.75069			−0.589250
$66$		0			0
$67$	9.13242			1.11570			0.557851	−	0.829941i	$-0.311626\pi$
$67$	9.13242			1.11570			0.557851	+	0.829941i	$0.311626\pi$
$68$		0			0
$69$	3.06597			0.369099
$70$		0			0
$71$		−	9.68233i		−	1.14908i	−0.818476	−	0.574540i	$-0.805181\pi$
$71$		−	9.68233i		−	1.14908i	0.818476	−	0.574540i	$-0.194819\pi$
$72$		0			0
$73$		−	12.4295i		−	1.45476i	−0.686235	−	0.727380i	$-0.740738\pi$
$73$		−	12.4295i		−	1.45476i	0.686235	−	0.727380i	$-0.259262\pi$
$74$		0			0
$75$			0.528177i			0.0609886i
$76$		0			0
$77$	6.62742	+	4.59363i	0.755264	+	0.523493i
$78$		0			0
$79$		−	2.81383i		−	0.316580i	−0.987393	−	0.158290i	$-0.949402\pi$
$79$		−	2.81383i		−	0.316580i	0.987393	−	0.158290i	$-0.0505982\pi$
$80$		0			0
$81$	6.56709			0.729676
$82$		0			0
$83$		−	13.4535i		−	1.47671i	−0.674412	−	0.738355i	$-0.735603\pi$
$83$		−	13.4535i		−	1.47671i	0.674412	−	0.738355i	$-0.264397\pi$
$84$		0			0
$85$		−	6.78894i		−	0.736364i
$86$		0			0
$87$	−0.0980728			−0.0105145
$88$		0			0
$89$			3.10041i			0.328642i	0.986407	+	0.164321i	$0.0525434\pi$
$89$			3.10041i			0.328642i	−0.986407	+	0.164321i	$0.947457\pi$
$90$		0			0
$91$	10.3303	+	7.16019i	1.08291	+	0.750591i
$92$		0			0
$93$		−	1.65102i		−	0.171203i
$94$		0			0
$95$		−	0.584285i		−	0.0599464i
$96$		0			0
$97$			13.5027i			1.37099i	0.728077	+	0.685495i	$0.240414\pi$
$97$			13.5027i			1.37099i	−0.728077	+	0.685495i	$0.759586\pi$
$98$		0			0
$99$	−8.29319			−0.833497
$100$		0			0
$101$	−1.29842			−0.129198			−0.0645988	−	0.997911i	$-0.520577\pi$
$101$	−1.29842			−0.129198			−0.0645988	+	0.997911i	$0.520577\pi$
$102$		0			0
$103$	19.0846			1.88046			0.940232	−	0.340535i	$-0.110608\pi$
$103$	19.0846			1.88046			0.940232	+	0.340535i	$0.110608\pi$
$104$		0			0
$105$	0.796062	−	1.14851i	0.0776877	−	0.112083i
$106$		0			0
$107$	3.81102			0.368425			0.184213	−	0.982886i	$-0.441027\pi$
$107$	3.81102			0.368425			0.184213	+	0.982886i	$0.441027\pi$
$108$		0			0
$109$		−	15.1487i		−	1.45098i	−0.688233	−	0.725490i	$-0.741613\pi$
$109$		−	15.1487i		−	1.45098i	0.688233	−	0.725490i	$-0.258387\pi$
$110$		0			0
$111$	3.72192			0.353269
$112$		0			0
$113$	−16.6411			−1.56546			−0.782732	−	0.622359i	$-0.786175\pi$
$113$	−16.6411			−1.56546			−0.782732	+	0.622359i	$0.786175\pi$
$114$		0			0
$115$		−	5.80481i		−	0.541301i
$116$		0			0
$117$	−12.9268			−1.19508
$118$		0			0
$119$	−10.2322	+	14.7624i	−0.937986	+	1.35327i
$120$		0			0
$121$	−1.71084			−0.155531
$122$		0			0
$123$	−2.02600			−0.182678
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			15.9073i			1.41154i	0.708440	+	0.705771i	$0.249399\pi$
$127$			15.9073i			1.41154i	−0.708440	+	0.705771i	$0.750601\pi$
$128$		0			0
$129$			0.758217i			0.0667573i
$130$		0			0
$131$			20.2439i			1.76872i	0.466806	+	0.884360i	$0.345405\pi$
$131$			20.2439i			1.76872i	−0.466806	+	0.884360i	$0.654595\pi$
$132$		0			0
$133$	−0.880629	+	1.27052i	−0.0763602	+	0.110168i
$134$		0			0
$135$			3.02172i			0.260068i
$136$		0			0
$137$	6.38933			0.545877			0.272939	−	0.962031i	$-0.412004\pi$
$137$	6.38933			0.545877			0.272939	+	0.962031i	$0.412004\pi$
$138$		0			0
$139$		−	21.8128i		−	1.85014i	−0.379801	−	0.925068i	$-0.624008\pi$
$139$		−	21.8128i		−	1.85014i	0.379801	−	0.925068i	$-0.375992\pi$
$140$		0			0
$141$		−	1.56272i		−	0.131605i
$142$		0			0
$143$	14.4792			1.21081
$144$		0			0
$145$			0.185682i			0.0154200i
$146$		0			0
$147$	−3.46205	+	1.29760i	−0.285545	+	0.107025i
$148$		0			0
$149$			11.5156i			0.943396i	0.881760	+	0.471698i	$0.156359\pi$
$149$			11.5156i			0.943396i	−0.881760	+	0.471698i	$0.843641\pi$
$150$		0			0
$151$			7.70970i			0.627407i	0.949521	+	0.313703i	$0.101570\pi$
$151$			7.70970i			0.627407i	−0.949521	+	0.313703i	$0.898430\pi$
$152$		0			0
$153$		−	18.4729i		−	1.49345i
$154$		0			0
$155$	−3.12589			−0.251078
$156$		0			0
$157$	4.42111			0.352843			0.176422	−	0.984315i	$-0.443548\pi$
$157$	4.42111			0.352843			0.176422	+	0.984315i	$0.443548\pi$
$158$		0			0
$159$	−0.282846			−0.0224311
$160$		0			0
$161$	−8.74895	+	12.6225i	−0.689514	+	0.994790i
$162$		0			0
$163$	−17.5475			−1.37443			−0.687213	−	0.726456i	$-0.741166\pi$
$163$	−17.5475			−1.37443			−0.687213	+	0.726456i	$0.741166\pi$
$164$		0			0
$165$		−	1.60978i		−	0.125321i
$166$		0			0
$167$	1.57597			0.121952			0.0609760	−	0.998139i	$-0.480579\pi$
$167$	1.57597			0.121952			0.0609760	+	0.998139i	$0.480579\pi$
$168$		0			0
$169$	9.56903			0.736079
$170$		0			0
$171$		−	1.58986i		−	0.121579i
$172$		0			0
$173$	0.208914			0.0158835			0.00794173	−	0.999968i	$-0.497472\pi$
$173$	0.208914			0.0158835			0.00794173	+	0.999968i	$0.497472\pi$
$174$		0			0
$175$	−2.17448	−	1.50719i	−0.164376	−	0.113933i
$176$		0			0
$177$	0.803439			0.0603902
$178$		0			0
$179$	5.84685			0.437014			0.218507	−	0.975835i	$-0.429881\pi$
$179$	5.84685			0.437014			0.218507	+	0.975835i	$0.429881\pi$
$180$		0			0
$181$	8.66680			0.644198			0.322099	−	0.946706i	$-0.395612\pi$
$181$	8.66680			0.644198			0.322099	+	0.946706i	$0.395612\pi$
$182$		0			0
$183$		−	6.99061i		−	0.516761i
$184$		0			0
$185$		−	7.04672i		−	0.518085i
$186$		0			0
$187$			20.6914i			1.51311i
$188$		0			0
$189$	4.55430	−	6.57067i	0.331276	−	0.477946i
$190$		0			0
$191$			12.9741i			0.938770i	0.882993	+	0.469385i	$0.155524\pi$
$191$			12.9741i			0.938770i	−0.882993	+	0.469385i	$0.844476\pi$
$192$		0			0
$193$	−24.5869			−1.76980			−0.884901	−	0.465780i	$-0.845774\pi$
$193$	−24.5869			−1.76980			−0.884901	+	0.465780i	$0.845774\pi$
$194$		0			0
$195$		−	2.50920i		−	0.179688i
$196$		0			0
$197$			7.21194i			0.513829i	0.966434	+	0.256915i	$0.0827059\pi$
$197$			7.21194i			0.513829i	−0.966434	+	0.256915i	$0.917294\pi$
$198$		0			0
$199$	−6.32808			−0.448585			−0.224293	−	0.974522i	$-0.572007\pi$
$199$	−6.32808			−0.448585			−0.224293	+	0.974522i	$0.572007\pi$
$200$		0			0
$201$			4.82353i			0.340226i
$202$		0			0
$203$	0.279858	−	0.403762i	0.0196422	−	0.0283385i
$204$		0			0
$205$			3.83583i			0.267906i
$206$		0			0
$207$		−	15.7951i		−	1.09783i
$208$		0			0
$209$			1.78079i			0.123180i
$210$		0			0
$211$	10.2211			0.703650			0.351825	−	0.936066i	$-0.385561\pi$
$211$	10.2211			0.703650			0.351825	+	0.936066i	$0.385561\pi$
$212$		0			0
$213$	5.11398			0.350404
$214$		0			0
$215$	1.43554			0.0979028
$216$		0			0
$217$	6.79721	+	4.71131i	0.461424	+	0.319825i
$218$		0			0
$219$	6.56496			0.443619
$220$		0			0
$221$			32.2521i			2.16951i
$222$		0			0
$223$	2.64553			0.177158			0.0885790	−	0.996069i	$-0.471767\pi$
$223$	2.64553			0.177158			0.0885790	+	0.996069i	$0.471767\pi$
$224$		0			0
$225$	2.72103			0.181402
$226$		0			0
$227$		−	16.2979i		−	1.08173i	−0.841109	−	0.540866i	$-0.818097\pi$
$227$		−	16.2979i		−	1.08173i	0.841109	−	0.540866i	$-0.181903\pi$
$228$		0			0
$229$	−5.51840			−0.364666			−0.182333	−	0.983237i	$-0.558365\pi$
$229$	−5.51840			−0.364666			−0.182333	+	0.983237i	$0.558365\pi$
$230$		0			0
$231$	−2.42625	+	3.50045i	−0.159635	+	0.230313i
$232$		0			0
$233$	18.6314			1.22058			0.610291	−	0.792177i	$-0.291053\pi$
$233$	18.6314			1.22058			0.610291	+	0.792177i	$0.291053\pi$
$234$		0			0
$235$	−2.95871			−0.193005
$236$		0			0
$237$	1.48620			0.0965390
$238$		0			0
$239$			5.77674i			0.373666i	0.982392	+	0.186833i	$0.0598223\pi$
$239$			5.77674i			0.373666i	−0.982392	+	0.186833i	$0.940178\pi$
$240$		0			0
$241$			1.42853i			0.0920200i	0.998941	+	0.0460100i	$0.0146506\pi$
$241$			1.42853i			0.0920200i	−0.998941	+	0.0460100i	$0.985349\pi$
$242$		0			0
$243$			12.5337i			0.804039i
$244$		0			0
$245$	2.45676	+	6.55472i	0.156957	+	0.418766i
$246$		0			0
$247$			2.77576i			0.176617i
$248$		0			0
$249$	7.10581			0.450312
$250$		0			0
$251$			11.6354i			0.734419i	0.930138	+	0.367210i	$0.119687\pi$
$251$			11.6354i			0.734419i	−0.930138	+	0.367210i	$0.880313\pi$
$252$		0			0
$253$			17.6920i			1.11228i
$254$		0			0
$255$	3.58576			0.224549
$256$		0			0
$257$			12.0011i			0.748611i	0.927305	+	0.374306i	$0.122119\pi$
$257$			12.0011i			0.748611i	−0.927305	+	0.374306i	$0.877881\pi$
$258$		0			0
$259$	−10.6207	+	15.3230i	−0.659941	+	0.952124i
$260$		0			0
$261$			0.505246i			0.0312739i
$262$		0			0
$263$			26.7775i			1.65117i	0.564277	+	0.825585i	$0.309155\pi$
$263$			26.7775i			1.65117i	−0.564277	+	0.825585i	$0.690845\pi$
$264$		0			0
$265$			0.535513i			0.0328963i
$266$		0			0
$267$	−1.63756			−0.100217
$268$		0			0
$269$	19.0365			1.16067			0.580337	−	0.814376i	$-0.302921\pi$
$269$	19.0365			1.16067			0.580337	+	0.814376i	$0.302921\pi$
$270$		0			0
$271$	−2.47835			−0.150549			−0.0752745	−	0.997163i	$-0.523983\pi$
$271$	−2.47835			−0.150549			−0.0752745	+	0.997163i	$0.523983\pi$
$272$		0			0
$273$	−3.78184	+	5.45622i	−0.228888	+	0.330226i
$274$		0			0
$275$	−3.04781			−0.183790
$276$		0			0
$277$			16.0967i			0.967159i	0.875301	+	0.483579i	$0.160664\pi$
$277$			16.0967i			0.967159i	−0.875301	+	0.483579i	$0.839336\pi$
$278$		0			0
$279$	−8.50565			−0.509220
$280$		0			0
$281$	17.4111			1.03866			0.519328	−	0.854575i	$-0.326182\pi$
$281$	17.4111			1.03866			0.519328	+	0.854575i	$0.326182\pi$
$282$		0			0
$283$		−	29.4211i		−	1.74890i	−0.485114	−	0.874451i	$-0.661222\pi$
$283$		−	29.4211i		−	1.74890i	0.485114	−	0.874451i	$-0.338778\pi$
$284$		0			0
$285$	0.308606			0.0182802
$286$		0			0
$287$	5.78132	−	8.34095i	0.341261	−	0.492351i
$288$		0			0
$289$	−29.0897			−1.71116
$290$		0			0
$291$	−7.13181			−0.418074
$292$		0			0
$293$	23.7367			1.38671			0.693357	−	0.720594i	$-0.256131\pi$
$293$	23.7367			1.38671			0.693357	+	0.720594i	$0.256131\pi$
$294$		0			0
$295$		−	1.52116i		−	0.0885651i
$296$		0			0
$297$		−	9.20962i		−	0.534396i
$298$		0			0
$299$			27.5768i			1.59481i
$300$		0			0
$301$	−3.12155	−	2.16362i	−0.179923	−	0.124709i
$302$		0			0
$303$		−	0.685795i		−	0.0393979i
$304$		0			0
$305$	−13.2354			−0.757855
$306$		0			0
$307$		−	3.27590i		−	0.186965i	−0.995621	−	0.0934827i	$-0.970200\pi$
$307$		−	3.27590i		−	0.186965i	0.995621	−	0.0934827i	$-0.0298000\pi$
$308$		0			0
$309$			10.0801i			0.573434i
$310$		0			0
$311$	20.9116			1.18579			0.592893	−	0.805282i	$-0.297986\pi$
$311$	20.9116			1.18579			0.592893	+	0.805282i	$0.297986\pi$
$312$		0			0
$313$		−	13.8819i		−	0.784650i	−0.919827	−	0.392325i	$-0.871671\pi$
$313$		−	13.8819i		−	0.784650i	0.919827	−	0.392325i	$-0.128329\pi$
$314$		0			0
$315$	−5.91683	−	4.10111i	−0.333376	−	0.231071i
$316$		0			0
$317$			5.93227i			0.333190i	0.986025	+	0.166595i	$0.0532772\pi$
$317$			5.93227i			0.333190i	−0.986025	+	0.166595i	$0.946723\pi$
$318$		0			0
$319$		−	0.565923i		−	0.0316856i
$320$		0			0
$321$			2.01289i			0.112349i
$322$		0			0
$323$	−3.96668			−0.220712
$324$		0			0
$325$	−4.75069			−0.263521
$326$		0			0
$327$	8.00118			0.442466
$328$		0			0
$329$	6.43367	+	4.45934i	0.354700	+	0.245851i
$330$		0			0
$331$	7.33151			0.402976			0.201488	−	0.979491i	$-0.435422\pi$
$331$	7.33151			0.402976			0.201488	+	0.979491i	$0.435422\pi$
$332$		0			0
$333$		−	19.1743i		−	1.05075i
$334$		0			0
$335$	9.13242			0.498957
$336$		0			0
$337$	−16.5861			−0.903505			−0.451752	−	0.892143i	$-0.649201\pi$
$337$	−16.5861			−0.903505			−0.451752	+	0.892143i	$0.649201\pi$
$338$		0			0
$339$		−	8.78945i		−	0.477378i
$340$		0			0
$341$	9.52714			0.515923
$342$		0			0
$343$	4.53701	−	17.9559i	0.244976	−	0.969529i
$344$		0			0
$345$	3.06597			0.165066
$346$		0			0
$347$	−28.9802			−1.55574			−0.777870	−	0.628425i	$-0.783700\pi$
$347$	−28.9802			−1.55574			−0.777870	+	0.628425i	$0.783700\pi$
$348$		0			0
$349$	10.2706			0.549772			0.274886	−	0.961477i	$-0.411360\pi$
$349$	10.2706			0.549772			0.274886	+	0.961477i	$0.411360\pi$
$350$		0			0
$351$		−	14.3552i		−	0.766225i
$352$		0			0
$353$		−	24.6097i		−	1.30984i	−0.755697	−	0.654921i	$-0.772702\pi$
$353$		−	24.6097i		−	1.30984i	0.755697	−	0.654921i	$-0.227298\pi$
$354$		0			0
$355$		−	9.68233i		−	0.513885i
$356$		0			0
$357$	−7.79718	−	5.40442i	−0.412671	−	0.286032i
$358$		0			0
$359$			5.06081i			0.267099i	0.991042	+	0.133550i	$0.0426375\pi$
$359$			5.06081i			0.267099i	−0.991042	+	0.133550i	$0.957362\pi$
$360$		0			0
$361$	18.6586			0.982032
$362$		0			0
$363$		−	0.903627i		−	0.0474281i
$364$		0			0
$365$		−	12.4295i		−	0.650589i
$366$		0			0
$367$	12.8106			0.668708			0.334354	−	0.942447i	$-0.391482\pi$
$367$	12.8106			0.668708			0.334354	+	0.942447i	$0.391482\pi$
$368$		0			0
$369$			10.4374i			0.543350i
$370$		0			0
$371$	0.807120	−	1.16447i	0.0419036	−	0.0604560i
$372$		0			0
$373$		−	12.4155i		−	0.642849i	−0.946935	−	0.321425i	$-0.895838\pi$
$373$		−	12.4155i		−	0.642849i	0.946935	−	0.321425i	$-0.104162\pi$
$374$		0			0
$375$			0.528177i			0.0272749i
$376$		0			0
$377$		−	0.882116i		−	0.0454313i
$378$		0			0
$379$	−9.41586			−0.483660			−0.241830	−	0.970319i	$-0.577748\pi$
$379$	−9.41586			−0.483660			−0.241830	+	0.970319i	$0.577748\pi$
$380$		0			0
$381$	−8.40185			−0.430440
$382$		0			0
$383$	−28.8384			−1.47358			−0.736788	−	0.676124i	$-0.763658\pi$
$383$	−28.8384			−1.47358			−0.736788	+	0.676124i	$0.763658\pi$
$384$		0			0
$385$	6.62742	+	4.59363i	0.337765	+	0.234113i
$386$		0			0
$387$	3.90614			0.198560
$388$		0			0
$389$			4.35382i			0.220747i	0.993890	+	0.110374i	$0.0352048\pi$
$389$			4.35382i			0.220747i	−0.993890	+	0.110374i	$0.964795\pi$
$390$		0			0
$391$	−39.4085			−1.99297
$392$		0			0
$393$	−10.6924			−0.539359
$394$		0			0
$395$		−	2.81383i		−	0.141579i
$396$		0			0
$397$	−16.9988			−0.853148			−0.426574	−	0.904453i	$-0.640280\pi$
$397$	−16.9988			−0.853148			−0.426574	+	0.904453i	$0.640280\pi$
$398$		0			0
$399$	−0.671059	−	0.465128i	−0.0335950	−	0.0232855i
$400$		0			0
$401$	10.4337			0.521035			0.260518	−	0.965469i	$-0.416107\pi$
$401$	10.4337			0.521035			0.260518	+	0.965469i	$0.416107\pi$
$402$		0			0
$403$	14.8501			0.739738
$404$		0			0
$405$	6.56709			0.326321
$406$		0			0
$407$			21.4771i			1.06458i
$408$		0			0
$409$		−	27.7527i		−	1.37228i	−0.727469	−	0.686141i	$-0.759303\pi$
$409$		−	27.7527i		−	1.37228i	0.727469	−	0.686141i	$-0.240697\pi$
$410$		0			0
$411$			3.37470i			0.166461i
$412$		0			0
$413$	−2.29267	+	3.30773i	−0.112815	+	0.162763i
$414$		0			0
$415$		−	13.4535i		−	0.660405i
$416$		0			0
$417$	11.5210			0.564186
$418$		0			0
$419$			30.1125i			1.47109i	0.677476	+	0.735545i	$0.263074\pi$
$419$			30.1125i			1.47109i	−0.677476	+	0.735545i	$0.736926\pi$
$420$		0			0
$421$		−	27.1321i		−	1.32234i	−0.750236	−	0.661170i	$-0.770060\pi$
$421$		−	27.1321i		−	1.32234i	0.750236	−	0.661170i	$-0.229940\pi$
$422$		0			0
$423$	−8.05074			−0.391440
$424$		0			0
$425$		−	6.78894i		−	0.329312i
$426$		0			0
$427$	28.7801	+	19.9482i	1.39277	+	0.965361i
$428$		0			0
$429$			7.64758i			0.369229i
$430$		0			0
$431$			3.80266i			0.183167i	0.995797	+	0.0915837i	$0.0291929\pi$
$431$			3.80266i			0.183167i	−0.995797	+	0.0915837i	$0.970807\pi$
$432$		0			0
$433$			6.04283i			0.290400i	0.989402	+	0.145200i	$0.0463825\pi$
$433$			6.04283i			0.290400i	−0.989402	+	0.145200i	$0.953617\pi$
$434$		0			0
$435$	−0.0980728			−0.00470223
$436$		0			0
$437$	−3.39167			−0.162245
$438$		0			0
$439$	20.7511			0.990398			0.495199	−	0.868780i	$-0.335095\pi$
$439$	20.7511			0.990398			0.495199	+	0.868780i	$0.335095\pi$
$440$		0			0
$441$	6.68492	+	17.8356i	0.318329	+	0.849313i
$442$		0			0
$443$	−40.9174			−1.94404			−0.972022	−	0.234891i	$-0.924527\pi$
$443$	−40.9174			−1.94404			−0.972022	+	0.234891i	$0.924527\pi$
$444$		0			0
$445$			3.10041i			0.146973i
$446$		0			0
$447$	−6.08228			−0.287682
$448$		0			0
$449$	9.64737			0.455288			0.227644	−	0.973744i	$-0.426898\pi$
$449$	9.64737			0.455288			0.227644	+	0.973744i	$0.426898\pi$
$450$		0			0
$451$		−	11.6909i		−	0.550503i
$452$		0			0
$453$	−4.07209			−0.191323
$454$		0			0
$455$	10.3303	+	7.16019i	0.484292	+	0.335675i
$456$		0			0
$457$	8.31667			0.389037			0.194519	−	0.980899i	$-0.437686\pi$
$457$	8.31667			0.389037			0.194519	+	0.980899i	$0.437686\pi$
$458$		0			0
$459$	20.5142			0.957523
$460$		0			0
$461$	−14.8017			−0.689385			−0.344693	−	0.938716i	$-0.612017\pi$
$461$	−14.8017			−0.689385			−0.344693	+	0.938716i	$0.612017\pi$
$462$		0			0
$463$		−	10.2328i		−	0.475556i	−0.971319	−	0.237778i	$-0.923581\pi$
$463$		−	10.2328i		−	0.475556i	0.971319	−	0.237778i	$-0.0764191\pi$
$464$		0			0
$465$		−	1.65102i		−	0.0765644i
$466$		0			0
$467$		−	1.49149i		−	0.0690179i	−0.999404	−	0.0345090i	$-0.989013\pi$
$467$		−	1.49149i		−	0.0690179i	0.999404	−	0.0345090i	$-0.0109867\pi$
$468$		0			0
$469$	−19.8583	−	13.7643i	−0.916971	−	0.635575i
$470$		0			0
$471$			2.33513i			0.107597i
$472$		0			0
$473$	−4.37524			−0.201174
$474$		0			0
$475$		−	0.584285i		−	0.0268089i
$476$		0			0
$477$			1.45715i			0.0667182i
$478$		0			0
$479$	33.0698			1.51100			0.755499	−	0.655150i	$-0.227395\pi$
$479$	33.0698			1.51100			0.755499	+	0.655150i	$0.227395\pi$
$480$		0			0
$481$			33.4768i			1.52641i
$482$		0			0
$483$	−6.66689	−	4.62099i	−0.303354	−	0.210262i
$484$		0			0
$485$			13.5027i			0.613126i
$486$		0			0
$487$		−	30.3143i		−	1.37367i	−0.726811	−	0.686837i	$-0.758999\pi$
$487$		−	30.3143i		−	1.37367i	0.726811	−	0.686837i	$-0.241001\pi$
$488$		0			0
$489$		−	9.26819i		−	0.419122i
$490$		0			0
$491$	24.0080			1.08347			0.541733	−	0.840551i	$-0.317768\pi$
$491$	24.0080			1.08347			0.541733	+	0.840551i	$0.317768\pi$
$492$		0			0
$493$	1.26058			0.0567738
$494$		0			0
$495$	−8.29319			−0.372751
$496$		0			0
$497$	−14.5931	+	21.0541i	−0.654590	+	0.944404i
$498$		0			0
$499$	36.6238			1.63951			0.819754	−	0.572716i	$-0.194110\pi$
$499$	36.6238			1.63951			0.819754	+	0.572716i	$0.194110\pi$
$500$		0			0
$501$			0.832390i			0.0371884i
$502$		0			0
$503$	0.225539			0.0100563			0.00502815	−	0.999987i	$-0.498399\pi$
$503$	0.225539			0.0100563			0.00502815	+	0.999987i	$0.498399\pi$
$504$		0			0
$505$	−1.29842			−0.0577789
$506$		0			0
$507$			5.05414i			0.224462i
$508$		0			0
$509$	11.4130			0.505871			0.252935	−	0.967483i	$-0.418604\pi$
$509$	11.4130			0.505871			0.252935	+	0.967483i	$0.418604\pi$
$510$		0			0
$511$	−18.7336	+	27.0277i	−0.828725	+	1.19564i
$512$		0			0
$513$	1.76554			0.0779507
$514$		0			0
$515$	19.0846			0.840969
$516$		0			0
$517$	9.01759			0.396593
$518$		0			0
$519$			0.110344i			0.00484355i
$520$		0			0
$521$		−	18.6069i		−	0.815183i	−0.913164	−	0.407591i	$-0.866369\pi$
$521$		−	18.6069i		−	0.815183i	0.913164	−	0.407591i	$-0.133631\pi$
$522$		0			0
$523$		−	35.5417i		−	1.55413i	−0.629420	−	0.777065i	$-0.716708\pi$
$523$		−	35.5417i		−	1.55413i	0.629420	−	0.777065i	$-0.283292\pi$
$524$		0			0
$525$	0.796062	−	1.14851i	0.0347430	−	0.0501252i
$526$		0			0
$527$			21.2215i			0.924424i
$528$		0			0
$529$	−10.6958			−0.465035
$530$		0			0
$531$		−	4.13911i		−	0.179622i
$532$		0			0
$533$		−	18.2228i		−	0.789319i
$534$		0			0
$535$	3.81102			0.164765
$536$		0			0
$537$			3.08817i			0.133265i
$538$		0			0
$539$	−7.48775	−	19.9775i	−0.322520	−	0.860494i
$540$		0			0
$541$		−	40.2046i		−	1.72853i	−0.503035	−	0.864266i	$-0.667784\pi$
$541$		−	40.2046i		−	1.72853i	0.503035	−	0.864266i	$-0.332216\pi$
$542$		0			0
$543$			4.57760i			0.196444i
$544$		0			0
$545$		−	15.1487i		−	0.648898i
$546$		0			0
$547$	6.38283			0.272910			0.136455	−	0.990646i	$-0.456429\pi$
$547$	6.38283			0.272910			0.136455	+	0.990646i	$0.456429\pi$
$548$		0			0
$549$	−36.0138			−1.53703
$550$		0			0
$551$	0.108491			0.00462188
$552$		0			0
$553$	−4.24097	+	6.11863i	−0.180344	+	0.260190i
$554$		0			0
$555$	3.72192			0.157987
$556$		0			0
$557$		−	33.9096i		−	1.43680i	−0.695631	−	0.718399i	$-0.744875\pi$
$557$		−	33.9096i		−	1.43680i	0.695631	−	0.718399i	$-0.255125\pi$
$558$		0			0
$559$	−6.81978			−0.288446
$560$		0			0
$561$	−10.9287			−0.461411
$562$		0			0
$563$		−	13.9112i		−	0.586288i	−0.956068	−	0.293144i	$-0.905298\pi$
$563$		−	13.9112i		−	0.586288i	0.956068	−	0.293144i	$-0.0947016\pi$
$564$		0			0
$565$	−16.6411			−0.700097
$566$		0			0
$567$	−14.2800	−	9.89784i	−0.599705	−	0.415670i
$568$		0			0
$569$	−23.6948			−0.993336			−0.496668	−	0.867940i	$-0.665443\pi$
$569$	−23.6948			−0.993336			−0.496668	+	0.867940i	$0.665443\pi$
$570$		0			0
$571$	−20.9646			−0.877340			−0.438670	−	0.898648i	$-0.644550\pi$
$571$	−20.9646			−0.877340			−0.438670	+	0.898648i	$0.644550\pi$
$572$		0			0
$573$	−6.85260			−0.286271
$574$		0			0
$575$		−	5.80481i		−	0.242077i
$576$		0			0
$577$			11.5447i			0.480614i	0.970697	+	0.240307i	$0.0772481\pi$
$577$			11.5447i			0.480614i	−0.970697	+	0.240307i	$0.922752\pi$
$578$		0			0
$579$		−	12.9862i		−	0.539689i
$580$		0			0
$581$	−20.2769	+	29.2544i	−0.841229	+	1.21368i
$582$		0			0
$583$		−	1.63214i		−	0.0675965i
$584$		0			0
$585$	−12.9268			−0.534456
$586$		0			0
$587$		−	8.38593i		−	0.346124i	−0.984911	−	0.173062i	$-0.944634\pi$
$587$		−	8.38593i		−	0.346124i	0.984911	−	0.173062i	$-0.0553661\pi$
$588$		0			0
$589$			1.82641i			0.0752561i
$590$		0			0
$591$	−3.80918			−0.156689
$592$		0			0
$593$			26.8713i			1.10347i	0.834018	+	0.551737i	$0.186035\pi$
$593$			26.8713i			1.10347i	−0.834018	+	0.551737i	$0.813965\pi$
$594$		0			0
$595$	−10.2322	+	14.7624i	−0.419480	+	0.605201i
$596$		0			0
$597$		−	3.34234i		−	0.136793i
$598$		0			0
$599$		−	34.0473i		−	1.39113i	−0.718462	−	0.695566i	$-0.755153\pi$
$599$		−	34.0473i		−	1.39113i	0.718462	−	0.695566i	$-0.244847\pi$
$600$		0			0
$601$		−	24.5067i		−	0.999649i	−0.866127	−	0.499825i	$-0.833398\pi$
$601$		−	24.5067i		−	0.999649i	0.866127	−	0.499825i	$-0.166602\pi$
$602$		0			0
$603$	24.8496			1.01195
$604$		0			0
$605$	−1.71084			−0.0695556
$606$		0			0
$607$	−36.7082			−1.48994			−0.744971	−	0.667097i	$-0.767537\pi$
$607$	−36.7082			−1.48994			−0.744971	+	0.667097i	$0.767537\pi$
$608$		0			0
$609$	0.213258	+	0.147814i	0.00864164	+	0.00598974i
$610$		0			0
$611$	14.0559			0.568641
$612$		0			0
$613$		−	41.6702i		−	1.68304i	−0.540224	−	0.841521i	$-0.681661\pi$
$613$		−	41.6702i		−	1.68304i	0.540224	−	0.841521i	$-0.318339\pi$
$614$		0			0
$615$	−2.02600			−0.0816961
$616$		0			0
$617$	18.6715			0.751688			0.375844	−	0.926683i	$-0.377353\pi$
$617$	18.6715			0.751688			0.375844	+	0.926683i	$0.377353\pi$
$618$		0			0
$619$		−	19.0302i		−	0.764888i	−0.923979	−	0.382444i	$-0.875083\pi$
$619$		−	19.0302i		−	0.764888i	0.923979	−	0.382444i	$-0.124917\pi$
$620$		0			0
$621$	17.5405			0.703875
$622$		0			0
$623$	4.67290	−	6.74178i	0.187216	−	0.270104i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−0.940573			−0.0375629
$628$		0			0
$629$	−47.8398			−1.90750
$630$		0			0
$631$		−	28.1199i		−	1.11944i	−0.828683	−	0.559718i	$-0.810910\pi$
$631$		−	28.1199i		−	1.11944i	0.828683	−	0.559718i	$-0.189090\pi$
$632$		0			0
$633$			5.39855i			0.214573i
$634$		0			0
$635$			15.9073i			0.631261i
$636$		0			0
$637$	−11.6713	−	31.1394i	−0.462434	−	1.23379i
$638$		0			0
$639$		−	26.3459i		−	1.04223i
$640$		0			0
$641$	−7.40297			−0.292400			−0.146200	−	0.989255i	$-0.546704\pi$
$641$	−7.40297			−0.292400			−0.146200	+	0.989255i	$0.546704\pi$
$642$		0			0
$643$		−	10.9752i		−	0.432819i	−0.976303	−	0.216409i	$-0.930565\pi$
$643$		−	10.9752i		−	0.432819i	0.976303	−	0.216409i	$-0.0694346\pi$
$644$		0			0
$645$			0.758217i			0.0298548i
$646$		0			0
$647$	−37.3313			−1.46765			−0.733823	−	0.679340i	$-0.762266\pi$
$647$	−37.3313			−1.46765			−0.733823	+	0.679340i	$0.762266\pi$
$648$		0			0
$649$			4.63620i			0.181987i
$650$		0			0
$651$	−2.48841	+	3.59013i	−0.0975283	+	0.140708i
$652$		0			0
$653$			27.0398i			1.05815i	0.848575	+	0.529074i	$0.177461\pi$
$653$			27.0398i			1.05815i	−0.848575	+	0.529074i	$0.822539\pi$
$654$		0			0
$655$			20.2439i			0.790995i
$656$		0			0
$657$		−	33.8210i		−	1.31948i
$658$		0			0
$659$	−16.0036			−0.623410			−0.311705	−	0.950179i	$-0.600900\pi$
$659$	−16.0036			−0.623410			−0.311705	+	0.950179i	$0.600900\pi$
$660$		0			0
$661$	28.7776			1.11932			0.559659	−	0.828723i	$-0.310932\pi$
$661$	28.7776			1.11932			0.559659	+	0.828723i	$0.310932\pi$
$662$		0			0
$663$	−17.0348			−0.661578
$664$		0			0
$665$	−0.880629	+	1.27052i	−0.0341493	+	0.0492686i
$666$		0			0
$667$	1.07785			0.0417344
$668$		0			0
$669$			1.39731i			0.0540231i
$670$		0			0
$671$	40.3389			1.55727
$672$		0			0
$673$	19.6820			0.758685			0.379342	−	0.925256i	$-0.376150\pi$
$673$	19.6820			0.758685			0.379342	+	0.925256i	$0.376150\pi$
$674$		0			0
$675$			3.02172i			0.116306i
$676$		0			0
$677$	41.7128			1.60315			0.801576	−	0.597893i	$-0.203995\pi$
$677$	41.7128			1.60315			0.801576	+	0.597893i	$0.203995\pi$
$678$		0			0
$679$	20.3511	−	29.3614i	0.781004	−	1.12679i
$680$		0			0
$681$	8.60819			0.329866
$682$		0			0
$683$	22.1942			0.849237			0.424618	−	0.905372i	$-0.360408\pi$
$683$	22.1942			0.849237			0.424618	+	0.905372i	$0.360408\pi$
$684$		0			0
$685$	6.38933			0.244124
$686$		0			0
$687$		−	2.91469i		−	0.111202i
$688$		0			0
$689$		−	2.54406i		−	0.0969208i
$690$		0			0
$691$			7.35566i			0.279822i	0.990164	+	0.139911i	$0.0446817\pi$
$691$			7.35566i			0.279822i	−0.990164	+	0.139911i	$0.955318\pi$
$692$		0			0
$693$	18.0334	+	12.4994i	0.685032	+	0.474813i
$694$		0			0
$695$		−	21.8128i		−	0.827406i
$696$		0			0
$697$	26.0412			0.986382
$698$		0			0
$699$			9.84066i			0.372208i
$700$		0			0
$701$			10.2820i			0.388346i	0.980967	+	0.194173i	$0.0622024\pi$
$701$			10.2820i			0.388346i	−0.980967	+	0.194173i	$0.937798\pi$
$702$		0			0
$703$	−4.11730			−0.155287
$704$		0			0
$705$		−	1.56272i		−	0.0588555i
$706$		0			0
$707$	2.82339	+	1.95696i	0.106185	+	0.0735992i
$708$		0			0
$709$			10.8180i			0.406277i	0.979150	+	0.203139i	$0.0651142\pi$
$709$			10.8180i			0.406277i	−0.979150	+	0.203139i	$0.934886\pi$
$710$		0			0
$711$		−	7.65651i		−	0.287142i
$712$		0			0
$713$			18.1452i			0.679544i
$714$		0			0
$715$	14.4792			0.541491
$716$		0			0
$717$	−3.05114			−0.113947
$718$		0			0
$719$	−29.4274			−1.09746			−0.548729	−	0.836000i	$-0.684888\pi$
$719$	−29.4274			−1.09746			−0.548729	+	0.836000i	$0.684888\pi$
$720$		0			0
$721$	−41.4992	−	28.7641i	−1.54551	−	1.07123i
$722$		0			0
$723$	−0.754518			−0.0280608
$724$		0			0
$725$			0.185682i			0.00689605i
$726$		0			0
$727$	31.0932			1.15318			0.576591	−	0.817033i	$-0.304383\pi$
$727$	31.0932			1.15318			0.576591	+	0.817033i	$0.304383\pi$
$728$		0			0
$729$	13.0812			0.484490
$730$		0			0
$731$		−	9.74577i		−	0.360460i
$732$		0			0
$733$	−10.0076			−0.369639			−0.184820	−	0.982772i	$-0.559170\pi$
$733$	−10.0076			−0.369639			−0.184820	+	0.982772i	$0.559170\pi$
$734$		0			0
$735$	−3.46205	+	1.29760i	−0.127700	+	0.0478629i
$736$		0			0
$737$	−27.8339			−1.02527
$738$		0			0
$739$	−40.1592			−1.47728			−0.738641	−	0.674099i	$-0.764532\pi$
$739$	−40.1592			−1.47728			−0.738641	+	0.674099i	$0.764532\pi$
$740$		0			0
$741$	−1.46609			−0.0538582
$742$		0			0
$743$			41.3554i			1.51718i	0.651567	+	0.758591i	$0.274112\pi$
$743$			41.3554i			1.51718i	−0.651567	+	0.758591i	$0.725888\pi$
$744$		0			0
$745$			11.5156i			0.421899i
$746$		0			0
$747$		−	36.6073i		−	1.33939i
$748$		0			0
$749$	−8.28700	−	5.74393i	−0.302800	−	0.209878i
$750$		0			0
$751$		−	10.4171i		−	0.380126i	−0.981772	−	0.190063i	$-0.939131\pi$
$751$		−	10.4171i		−	0.380126i	0.981772	−	0.190063i	$-0.0608693\pi$
$752$		0			0
$753$	−6.14554			−0.223956
$754$		0			0
$755$			7.70970i			0.280585i
$756$		0			0
$757$			31.7145i			1.15268i	0.817209	+	0.576341i	$0.195520\pi$
$757$			31.7145i			1.15268i	−0.817209	+	0.576341i	$0.804480\pi$
$758$		0			0
$759$	−9.34449			−0.339183
$760$		0			0
$761$		−	11.9199i		−	0.432095i	−0.976383	−	0.216047i	$-0.930683\pi$
$761$		−	11.9199i		−	0.432095i	0.976383	−	0.216047i	$-0.0693166\pi$
$762$		0			0
$763$	−22.8319	+	32.9406i	−0.826571	+	1.19253i
$764$		0			0
$765$		−	18.4729i		−	0.667890i
$766$		0			0
$767$			7.22653i			0.260935i
$768$		0			0
$769$		−	9.56458i		−	0.344908i	−0.985018	−	0.172454i	$-0.944830\pi$
$769$		−	9.56458i		−	0.344908i	0.985018	−	0.172454i	$-0.0551695\pi$
$770$		0			0
$771$	−6.33873			−0.228284
$772$		0			0
$773$	30.6270			1.10158			0.550788	−	0.834645i	$-0.314327\pi$
$773$	30.6270			1.10158			0.550788	+	0.834645i	$0.314327\pi$
$774$		0			0
$775$	−3.12589			−0.112285
$776$		0			0
$777$	−8.09325	−	5.60963i	−0.290344	−	0.201244i
$778$		0			0
$779$	2.24122			0.0803001
$780$		0			0
$781$			29.5099i			1.05595i
$782$		0			0
$783$	−0.561078			−0.0200513
$784$		0			0
$785$	4.42111			0.157796
$786$		0			0
$787$			26.7295i			0.952804i	0.879228	+	0.476402i	$0.158059\pi$
$787$			26.7295i			0.952804i	−0.879228	+	0.476402i	$0.841941\pi$
$788$		0			0
$789$	−14.1432			−0.503513
$790$		0			0
$791$	36.1858	+	25.0813i	1.28662	+	0.891789i
$792$		0			0
$793$	62.8771			2.23283
$794$		0			0
$795$	−0.282846			−0.0100315
$796$		0			0
$797$	1.38276			0.0489799			0.0244899	−	0.999700i	$-0.492204\pi$
$797$	1.38276			0.0489799			0.0244899	+	0.999700i	$0.492204\pi$
$798$		0			0
$799$			20.0865i			0.710610i
$800$		0			0
$801$			8.43630i			0.298082i
$802$		0			0
$803$			37.8827i			1.33685i
$804$		0			0
$805$	−8.74895	+	12.6225i	−0.308360	+	0.444883i
$806$		0			0
$807$			10.0546i			0.353939i
$808$		0			0
$809$	3.98813			0.140215			0.0701075	−	0.997539i	$-0.477666\pi$
$809$	3.98813			0.140215			0.0701075	+	0.997539i	$0.477666\pi$
$810$		0			0
$811$		−	19.2038i		−	0.674338i	−0.941444	−	0.337169i	$-0.890531\pi$
$811$		−	19.2038i		−	0.674338i	0.941444	−	0.337169i	$-0.109469\pi$
$812$		0			0
$813$		−	1.30901i		−	0.0459089i
$814$		0			0
$815$	−17.5475			−0.614662
$816$		0			0
$817$		−	0.838763i		−	0.0293446i
$818$		0			0
$819$	28.1090	+	19.4831i	0.982209	+	0.680794i
$820$		0			0
$821$		−	39.4413i		−	1.37651i	−0.725468	−	0.688256i	$-0.758377\pi$
$821$		−	39.4413i		−	1.37651i	0.725468	−	0.688256i	$-0.241623\pi$
$822$		0			0
$823$			0.408196i			0.0142288i	0.999975	+	0.00711440i	$0.00226460\pi$
$823$			0.408196i			0.0142288i	−0.999975	+	0.00711440i	$0.997735\pi$
$824$		0			0
$825$		−	1.60978i		−	0.0560455i
$826$		0			0
$827$	−33.9219			−1.17958			−0.589790	−	0.807557i	$-0.700789\pi$
$827$	−33.9219			−1.17958			−0.589790	+	0.807557i	$0.700789\pi$
$828$		0			0
$829$	16.5466			0.574686			0.287343	−	0.957828i	$-0.407228\pi$
$829$	16.5466			0.574686			0.287343	+	0.957828i	$0.407228\pi$
$830$		0			0
$831$	−8.50192			−0.294928
$832$		0			0
$833$	44.4996	−	16.6788i	1.54182	−	0.577887i
$834$		0			0
$835$	1.57597			0.0545386
$836$		0			0
$837$		−	9.44556i		−	0.326486i
$838$		0			0
$839$	33.4148			1.15361			0.576804	−	0.816883i	$-0.304300\pi$
$839$	33.4148			1.15361			0.576804	+	0.816883i	$0.304300\pi$
$840$		0			0
$841$	28.9655			0.998811
$842$		0			0
$843$			9.19612i			0.316731i
$844$		0			0
$845$	9.56903			0.329185
$846$		0			0
$847$	3.72020	+	2.57856i	0.127828	+	0.0886005i
$848$		0			0
$849$	15.5395			0.533315
$850$		0			0
$851$	−40.9049			−1.40220
$852$		0			0
$853$	−12.1851			−0.417211			−0.208606	−	0.978000i	$-0.566893\pi$
$853$	−12.1851			−0.417211			−0.208606	+	0.978000i	$0.566893\pi$
$854$		0			0
$855$		−	1.58986i		−	0.0543720i
$856$		0			0
$857$		−	16.0726i		−	0.549031i	−0.961583	−	0.274515i	$-0.911483\pi$
$857$		−	16.0726i		−	0.549031i	0.961583	−	0.274515i	$-0.0885174\pi$
$858$		0			0
$859$			21.5250i			0.734425i	0.930137	+	0.367212i	$0.119688\pi$
$859$			21.5250i			0.734425i	−0.930137	+	0.367212i	$0.880312\pi$
$860$		0			0
$861$	4.40550	+	3.05356i	0.150139	+	0.104065i
$862$		0			0
$863$		−	3.22755i		−	0.109867i	−0.998490	−	0.0549336i	$-0.982505\pi$
$863$		−	3.22755i		−	0.109867i	0.998490	−	0.0549336i	$-0.0174947\pi$
$864$		0			0
$865$	0.208914			0.00710330
$866$		0			0
$867$		−	15.3645i		−	0.521807i
$868$		0			0
$869$			8.57602i			0.290922i
$870$		0			0
$871$	−43.3853			−1.47005
$872$		0			0
$873$			36.7412i			1.24350i
$874$		0			0
$875$	−2.17448	−	1.50719i	−0.0735110	−	0.0509523i
$876$		0			0
$877$			27.5777i			0.931233i	0.884987	+	0.465616i	$0.154167\pi$
$877$			27.5777i			0.931233i	−0.884987	+	0.465616i	$0.845833\pi$
$878$		0			0
$879$			12.5372i			0.422869i
$880$		0			0
$881$			20.1792i			0.679856i	0.940452	+	0.339928i	$0.110403\pi$
$881$			20.1792i			0.679856i	−0.940452	+	0.339928i	$0.889597\pi$
$882$		0			0
$883$	−4.97245			−0.167336			−0.0836681	−	0.996494i	$-0.526664\pi$
$883$	−4.97245			−0.167336			−0.0836681	+	0.996494i	$0.526664\pi$
$884$		0			0
$885$	0.803439			0.0270073
$886$		0			0
$887$	−5.11354			−0.171696			−0.0858480	−	0.996308i	$-0.527360\pi$
$887$	−5.11354			−0.171696			−0.0858480	+	0.996308i	$0.527360\pi$
$888$		0			0
$889$	23.9753	−	34.5901i	0.804105	−	1.16012i
$890$		0			0
$891$	−20.0152			−0.670536
$892$		0			0
$893$			1.72873i			0.0578498i
$894$		0			0
$895$	5.84685			0.195439
$896$		0			0
$897$	−14.5654			−0.486326
$898$		0			0
$899$		−	0.580422i		−	0.0193581i
$900$		0			0
$901$	3.63557			0.121118
$902$		0			0
$903$	1.14278	−	1.64873i	0.0380292	−	0.0548663i
$904$		0			0
$905$	8.66680			0.288094
$906$		0			0
$907$	26.8798			0.892530			0.446265	−	0.894901i	$-0.352754\pi$
$907$	26.8798			0.892530			0.446265	+	0.894901i	$0.352754\pi$
$908$		0			0
$909$	−3.53304			−0.117183
$910$		0			0
$911$		−	43.0775i		−	1.42722i	−0.700543	−	0.713610i	$-0.747059\pi$
$911$		−	43.0775i		−	1.42722i	0.700543	−	0.713610i	$-0.252941\pi$
$912$		0			0
$913$			41.0036i			1.35702i
$914$		0			0
$915$		−	6.99061i		−	0.231103i
$916$		0			0
$917$	30.5114	−	44.0201i	1.00758	−	1.45367i
$918$		0			0
$919$		−	6.42394i		−	0.211906i	−0.994371	−	0.105953i	$-0.966211\pi$
$919$		−	6.42394i		−	0.211906i	0.994371	−	0.105953i	$-0.0337893\pi$
$920$		0			0
$921$	1.73025			0.0570138
$922$		0			0
$923$			45.9977i			1.51403i
$924$		0			0
$925$		−	7.04672i		−	0.231695i
$926$		0			0
$927$	51.9298			1.70560
$928$		0			0
$929$			8.92552i			0.292837i	0.989223	+	0.146418i	$0.0467746\pi$
$929$			8.92552i			0.292837i	−0.989223	+	0.146418i	$0.953225\pi$
$930$		0			0
$931$	3.82983	−	1.43545i	0.125517	−	0.0470450i
$932$		0			0
$933$			11.0450i			0.361597i
$934$		0			0
$935$			20.6914i			0.676682i
$936$		0			0
$937$		−	23.5181i		−	0.768304i	−0.923270	−	0.384152i	$-0.874494\pi$
$937$		−	23.5181i		−	0.768304i	0.923270	−	0.384152i	$-0.125506\pi$
$938$		0			0
$939$	7.33209			0.239274
$940$		0			0
$941$	−0.932766			−0.0304073			−0.0152037	−	0.999884i	$-0.504840\pi$
$941$	−0.932766			−0.0304073			−0.0152037	+	0.999884i	$0.504840\pi$
$942$		0			0
$943$	22.2663			0.725089
$944$		0			0
$945$	4.55430	−	6.57067i	0.148151	−	0.213744i
$946$		0			0
$947$	−18.3452			−0.596140			−0.298070	−	0.954544i	$-0.596343\pi$
$947$	−18.3452			−0.596140			−0.298070	+	0.954544i	$0.596343\pi$
$948$		0			0
$949$			59.0486i			1.91680i
$950$		0			0
$951$	−3.13329			−0.101604
$952$		0			0
$953$	−47.3388			−1.53345			−0.766727	−	0.641973i	$-0.778116\pi$
$953$	−47.3388			−1.53345			−0.766727	+	0.641973i	$0.778116\pi$
$954$		0			0
$955$			12.9741i			0.419831i
$956$		0			0
$957$	0.298908			0.00966231
$958$		0			0
$959$	−13.8935	−	9.62993i	−0.448644	−	0.310967i
$960$		0			0
$961$	−21.2288			−0.684800
$962$		0			0
$963$	10.3699			0.334165
$964$		0			0
$965$	−24.5869			−0.791479
$966$		0			0
$967$			6.55903i			0.210924i	0.994423	+	0.105462i	$0.0336322\pi$
$967$			6.55903i			0.210924i	−0.994423	+	0.105462i	$0.966368\pi$
$968$		0			0
$969$		−	2.09511i		−	0.0673046i
$970$		0			0
$971$		−	40.6590i		−	1.30481i	−0.757871	−	0.652404i	$-0.773760\pi$
$971$		−	40.6590i		−	1.30481i	0.757871	−	0.652404i	$-0.226240\pi$
$972$		0			0
$973$	−32.8760	+	47.4316i	−1.05396	+	1.52059i
$974$		0			0
$975$		−	2.50920i		−	0.0803588i
$976$		0			0
$977$	5.28576			0.169106			0.0845532	−	0.996419i	$-0.473054\pi$
$977$	5.28576			0.169106			0.0845532	+	0.996419i	$0.473054\pi$
$978$		0			0
$979$		−	9.44946i		−	0.302006i
$980$		0			0
$981$		−	41.2200i		−	1.31605i
$982$		0			0
$983$	−11.2577			−0.359066			−0.179533	−	0.983752i	$-0.557459\pi$
$983$	−11.2577			−0.359066			−0.179533	+	0.983752i	$0.557459\pi$
$984$		0			0
$985$			7.21194i			0.229791i
$986$		0			0
$987$	−2.35532	+	3.39811i	−0.0749706	+	0.108163i
$988$		0			0
$989$		−	8.33301i		−	0.264974i
$990$		0			0
$991$			14.3170i			0.454796i	0.973802	+	0.227398i	$0.0730218\pi$
$991$			14.3170i			0.454796i	−0.973802	+	0.227398i	$0.926978\pi$
$992$		0			0
$993$			3.87233i			0.122885i
$994$		0			0
$995$	−6.32808			−0.200614
$996$		0			0
$997$	18.4076			0.582974			0.291487	−	0.956575i	$-0.405850\pi$
$997$	18.4076			0.582974			0.291487	+	0.956575i	$0.405850\pi$
$998$		0			0
$999$	21.2932			0.673687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.h.b.111.9		16
4.3	odd	2		280.2.h.b.251.5	yes	16
7.6	odd	2		1120.2.h.a.111.8		16
8.3	odd	2		1120.2.h.a.111.9		16
8.5	even	2		280.2.h.a.251.6	yes	16
28.27	even	2		280.2.h.a.251.5	✓	16
56.13	odd	2		280.2.h.b.251.6	yes	16
56.27	even	2	inner	1120.2.h.b.111.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.h.a.251.5	✓	16	28.27	even	2
280.2.h.a.251.6	yes	16	8.5	even	2
280.2.h.b.251.5	yes	16	4.3	odd	2
280.2.h.b.251.6	yes	16	56.13	odd	2
1120.2.h.a.111.8		16	7.6	odd	2
1120.2.h.a.111.9		16	8.3	odd	2
1120.2.h.b.111.8		16	56.27	even	2	inner
1120.2.h.b.111.9		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.528177i q^{3} +1.00000 q^{5} +(-2.17448 - 1.50719i) q^{7} +2.72103 q^{9} +O(q^{10})\)	\(q+0.528177i q^{3} +1.00000 q^{5} +(-2.17448 - 1.50719i) q^{7} +2.72103 q^{9} -3.04781 q^{11} -4.75069 q^{13} +0.528177i q^{15} -6.78894i q^{17} -0.584285i q^{19} +(0.796062 - 1.14851i) q^{21} -5.80481i q^{23} +1.00000 q^{25} +3.02172i q^{27} +0.185682i q^{29} -3.12589 q^{31} -1.60978i q^{33} +(-2.17448 - 1.50719i) q^{35} -7.04672i q^{37} -2.50920i q^{39} +3.83583i q^{41} +1.43554 q^{43} +2.72103 q^{45} -2.95871 q^{47} +(2.45676 + 6.55472i) q^{49} +3.58576 q^{51} +0.535513i q^{53} -3.04781 q^{55} +0.308606 q^{57} -1.52116i q^{59} -13.2354 q^{61} +(-5.91683 - 4.10111i) q^{63} -4.75069 q^{65} +9.13242 q^{67} +3.06597 q^{69} -9.68233i q^{71} -12.4295i q^{73} +0.528177i q^{75} +(6.62742 + 4.59363i) q^{77} -2.81383i q^{79} +6.56709 q^{81} -13.4535i q^{83} -6.78894i q^{85} -0.0980728 q^{87} +3.10041i q^{89} +(10.3303 + 7.16019i) q^{91} -1.65102i q^{93} -0.584285i q^{95} +13.5027i q^{97} -8.29319 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} - 16 q^{9}+O(q^{10}) \) 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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