LMFDB - Embedded newform 1120.2.h.a.111.4

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			111.4
Root			$0.244064 + 1.39299i$ of defining polynomial
Character	$\chi$	$=$	1120.111
Dual form			1120.2.h.a.111.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68420i q^{3} -1.00000 q^{5} +(0.695780 - 2.55262i) q^{7} +0.163484 q^{9} +O(q^{10})$	$q-1.68420i q^{3} -1.00000 q^{5} +(0.695780 - 2.55262i) q^{7} +0.163484 q^{9} -1.45385 q^{11} -5.12034 q^{13} +1.68420i q^{15} -0.313877i q^{17} +0.250100i q^{19} +(-4.29912 - 1.17183i) q^{21} -4.27001i q^{23} +1.00000 q^{25} -5.32793i q^{27} -1.63961i q^{29} -8.96308 q^{31} +2.44857i q^{33} +(-0.695780 + 2.55262i) q^{35} +3.47842i q^{37} +8.62365i q^{39} +9.88374i q^{41} +8.65164 q^{43} -0.163484 q^{45} -7.77853 q^{47} +(-6.03178 - 3.55213i) q^{49} -0.528630 q^{51} +1.90687i q^{53} +1.45385 q^{55} +0.421218 q^{57} -7.73295i q^{59} -0.415877 q^{61} +(0.113749 - 0.417313i) q^{63} +5.12034 q^{65} -15.2878 q^{67} -7.19153 q^{69} +1.50413i q^{71} -10.7072i q^{73} -1.68420i q^{75} +(-1.01156 + 3.71114i) q^{77} +9.36521i q^{79} -8.48282 q^{81} -3.45276i q^{83} +0.313877i q^{85} -2.76142 q^{87} +9.12705i q^{89} +(-3.56263 + 13.0703i) q^{91} +15.0956i q^{93} -0.250100i q^{95} -16.5442i q^{97} -0.237682 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.68420i		−	0.972371i	−0.873856	−	0.486185i	$-0.838388\pi$
$3$		−	1.68420i		−	0.972371i	0.873856	−	0.486185i	$-0.161612\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	0.695780	−	2.55262i	0.262980	−	0.964801i
$7$	0.695780	−	2.55262i	0.262980	−	0.964801i
$8$		0			0
$9$	0.163484			0.0544947
$10$		0			0
$11$	−1.45385			−0.438353			−0.219177	−	0.975685i	$-0.570337\pi$
$11$	−1.45385			−0.438353			−0.219177	+	0.975685i	$0.570337\pi$
$12$		0			0
$13$	−5.12034			−1.42013			−0.710063	−	0.704138i	$-0.751334\pi$
$13$	−5.12034			−1.42013			−0.710063	+	0.704138i	$0.751334\pi$
$14$		0			0
$15$			1.68420i			0.434858i
$16$		0			0
$17$		−	0.313877i		−	0.0761263i	−0.999275	−	0.0380631i	$-0.987881\pi$
$17$		−	0.313877i		−	0.0761263i	0.999275	−	0.0380631i	$-0.0121188\pi$
$18$		0			0
$19$			0.250100i			0.0573769i	0.999588	+	0.0286884i	$0.00913307\pi$
$19$			0.250100i			0.0573769i	−0.999588	+	0.0286884i	$0.990867\pi$
$20$		0			0
$21$	−4.29912	−	1.17183i	−0.938145	−	0.255714i
$22$		0			0
$23$		−	4.27001i		−	0.890358i	−0.895442	−	0.445179i	$-0.853140\pi$
$23$		−	4.27001i		−	0.890358i	0.895442	−	0.445179i	$-0.146860\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	5.32793i		−	1.02536i
$28$		0			0
$29$		−	1.63961i		−	0.304468i	−0.988345	−	0.152234i	$-0.951353\pi$
$29$		−	1.63961i		−	0.304468i	0.988345	−	0.152234i	$-0.0486467\pi$
$30$		0			0
$31$	−8.96308			−1.60982			−0.804909	−	0.593399i	$-0.797786\pi$
$31$	−8.96308			−1.60982			−0.804909	+	0.593399i	$0.797786\pi$
$32$		0			0
$33$			2.44857i			0.426242i
$34$		0			0
$35$	−0.695780	+	2.55262i	−0.117608	+	0.431472i
$36$		0			0
$37$			3.47842i			0.571848i	0.958252	+	0.285924i	$0.0923005\pi$
$37$			3.47842i			0.571848i	−0.958252	+	0.285924i	$0.907699\pi$
$38$		0			0
$39$			8.62365i			1.38089i
$40$		0			0
$41$			9.88374i			1.54358i	0.635877	+	0.771790i	$0.280638\pi$
$41$			9.88374i			1.54358i	−0.635877	+	0.771790i	$0.719362\pi$
$42$		0			0
$43$	8.65164			1.31936			0.659681	−	0.751545i	$-0.270691\pi$
$43$	8.65164			1.31936			0.659681	+	0.751545i	$0.270691\pi$
$44$		0			0
$45$	−0.163484			−0.0243708
$46$		0			0
$47$	−7.77853			−1.13461			−0.567307	−	0.823506i	$-0.692015\pi$
$47$	−7.77853			−1.13461			−0.567307	+	0.823506i	$0.692015\pi$
$48$		0			0
$49$	−6.03178	−	3.55213i	−0.861683	−	0.507447i
$50$		0			0
$51$	−0.528630			−0.0740230
$52$		0			0
$53$			1.90687i			0.261929i	0.991387	+	0.130965i	$0.0418075\pi$
$53$			1.90687i			0.261929i	−0.991387	+	0.130965i	$0.958193\pi$
$54$		0			0
$55$	1.45385			0.196038
$56$		0			0
$57$	0.421218			0.0557916
$58$		0			0
$59$		−	7.73295i		−	1.00674i	−0.864070	−	0.503372i	$-0.832093\pi$
$59$		−	7.73295i		−	1.00674i	0.864070	−	0.503372i	$-0.167907\pi$
$60$		0			0
$61$	−0.415877			−0.0532475			−0.0266238	−	0.999646i	$-0.508476\pi$
$61$	−0.415877			−0.0532475			−0.0266238	+	0.999646i	$0.508476\pi$
$62$		0			0
$63$	0.113749	−	0.417313i	0.0143310	−	0.0525766i
$64$		0			0
$65$	5.12034			0.635100
$66$		0			0
$67$	−15.2878			−1.86770			−0.933848	−	0.357670i	$-0.883571\pi$
$67$	−15.2878			−1.86770			−0.933848	+	0.357670i	$0.883571\pi$
$68$		0			0
$69$	−7.19153			−0.865759
$70$		0			0
$71$			1.50413i			0.178507i	0.996009	+	0.0892537i	$0.0284482\pi$
$71$			1.50413i			0.178507i	−0.996009	+	0.0892537i	$0.971552\pi$
$72$		0			0
$73$		−	10.7072i		−	1.25318i	−0.779349	−	0.626590i	$-0.784450\pi$
$73$		−	10.7072i		−	1.25318i	0.779349	−	0.626590i	$-0.215550\pi$
$74$		0			0
$75$		−	1.68420i		−	0.194474i
$76$		0			0
$77$	−1.01156	+	3.71114i	−0.115278	+	0.422924i
$78$		0			0
$79$			9.36521i			1.05367i	0.849968	+	0.526834i	$0.176621\pi$
$79$			9.36521i			1.05367i	−0.849968	+	0.526834i	$0.823379\pi$
$80$		0			0
$81$	−8.48282			−0.942536
$82$		0			0
$83$		−	3.45276i		−	0.378989i	−0.981882	−	0.189495i	$-0.939315\pi$
$83$		−	3.45276i		−	0.378989i	0.981882	−	0.189495i	$-0.0606850\pi$
$84$		0			0
$85$			0.313877i			0.0340447i
$86$		0			0
$87$	−2.76142			−0.296055
$88$		0			0
$89$			9.12705i			0.967466i	0.875216	+	0.483733i	$0.160719\pi$
$89$			9.12705i			0.967466i	−0.875216	+	0.483733i	$0.839281\pi$
$90$		0			0
$91$	−3.56263	+	13.0703i	−0.373465	+	1.37014i
$92$		0			0
$93$			15.0956i			1.56534i
$94$		0			0
$95$		−	0.250100i		−	0.0256597i
$96$		0			0
$97$		−	16.5442i		−	1.67981i	−0.542732	−	0.839906i	$-0.682610\pi$
$97$		−	16.5442i		−	1.67981i	0.542732	−	0.839906i	$-0.317390\pi$
$98$		0			0
$99$	−0.237682			−0.0238879
$100$		0			0
$101$	11.1440			1.10887			0.554437	−	0.832226i	$-0.312934\pi$
$101$	11.1440			1.10887			0.554437	+	0.832226i	$0.312934\pi$
$102$		0			0
$103$	−11.5479			−1.13785			−0.568923	−	0.822391i	$-0.692640\pi$
$103$	−11.5479			−1.13785			−0.568923	+	0.822391i	$0.692640\pi$
$104$		0			0
$105$	4.29912	+	1.17183i	0.419551	+	0.114359i
$106$		0			0
$107$	3.37404			0.326181			0.163091	−	0.986611i	$-0.447854\pi$
$107$	3.37404			0.326181			0.163091	+	0.986611i	$0.447854\pi$
$108$		0			0
$109$		−	12.6125i		−	1.20805i	−0.796964	−	0.604027i	$-0.793562\pi$
$109$		−	12.6125i		−	1.20805i	0.796964	−	0.604027i	$-0.206438\pi$
$110$		0			0
$111$	5.85834			0.556049
$112$		0			0
$113$	13.5643			1.27602			0.638009	−	0.770029i	$-0.279758\pi$
$113$	13.5643			1.27602			0.638009	+	0.770029i	$0.279758\pi$
$114$		0			0
$115$			4.27001i			0.398180i
$116$		0			0
$117$	−0.837094			−0.0773894
$118$		0			0
$119$	−0.801209	−	0.218389i	−0.0734467	−	0.0200197i
$120$		0			0
$121$	−8.88631			−0.807846
$122$		0			0
$123$	16.6461			1.50093
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	2.17011i		−	0.192566i	−0.995354	−	0.0962832i	$-0.969305\pi$
$127$		−	2.17011i		−	0.192566i	0.995354	−	0.0962832i	$-0.0306954\pi$
$128$		0			0
$129$		−	14.5711i		−	1.28291i
$130$		0			0
$131$			8.82257i			0.770832i	0.922743	+	0.385416i	$0.125942\pi$
$131$			8.82257i			0.770832i	−0.922743	+	0.385416i	$0.874058\pi$
$132$		0			0
$133$	0.638411	+	0.174015i	0.0553573	+	0.0150890i
$134$		0			0
$135$			5.32793i			0.458555i
$136$		0			0
$137$	0.361903			0.0309195			0.0154597	−	0.999880i	$-0.495079\pi$
$137$	0.361903			0.0309195			0.0154597	+	0.999880i	$0.495079\pi$
$138$		0			0
$139$			8.49740i			0.720740i	0.932809	+	0.360370i	$0.117350\pi$
$139$			8.49740i			0.720740i	−0.932809	+	0.360370i	$0.882650\pi$
$140$		0			0
$141$			13.1006i			1.10327i
$142$		0			0
$143$	7.44422			0.622517
$144$		0			0
$145$			1.63961i			0.136162i
$146$		0			0
$147$	−5.98248	+	10.1587i	−0.493427	+	0.837875i
$148$		0			0
$149$		−	14.7688i		−	1.20991i	−0.796260	−	0.604955i	$-0.793191\pi$
$149$		−	14.7688i		−	1.20991i	0.796260	−	0.604955i	$-0.206809\pi$
$150$		0			0
$151$		−	17.6833i		−	1.43905i	−0.694468	−	0.719523i	$-0.744360\pi$
$151$		−	17.6833i		−	1.43905i	0.694468	−	0.719523i	$-0.255640\pi$
$152$		0			0
$153$		−	0.0513138i		−	0.00414848i
$154$		0			0
$155$	8.96308			0.719932
$156$		0			0
$157$	−0.628228			−0.0501380			−0.0250690	−	0.999686i	$-0.507981\pi$
$157$	−0.628228			−0.0501380			−0.0250690	+	0.999686i	$0.507981\pi$
$158$		0			0
$159$	3.21155			0.254693
$160$		0			0
$161$	−10.8997	−	2.97099i	−0.859019	−	0.234147i
$162$		0			0
$163$	15.7162			1.23099			0.615493	−	0.788142i	$-0.288957\pi$
$163$	15.7162			1.23099			0.615493	+	0.788142i	$0.288957\pi$
$164$		0			0
$165$		−	2.44857i		−	0.190621i
$166$		0			0
$167$	18.6731			1.44497			0.722485	−	0.691386i	$-0.243000\pi$
$167$	18.6731			1.44497			0.722485	+	0.691386i	$0.243000\pi$
$168$		0			0
$169$	13.2179			1.01676
$170$		0			0
$171$			0.0408874i			0.00312674i
$172$		0			0
$173$	24.7373			1.88074			0.940372	−	0.340147i	$-0.110477\pi$
$173$	24.7373			1.88074			0.940372	+	0.340147i	$0.110477\pi$
$174$		0			0
$175$	0.695780	−	2.55262i	0.0525960	−	0.192960i
$176$		0			0
$177$	−13.0238			−0.978929
$178$		0			0
$179$	−6.86464			−0.513088			−0.256544	−	0.966533i	$-0.582584\pi$
$179$	−6.86464			−0.513088			−0.256544	+	0.966533i	$0.582584\pi$
$180$		0			0
$181$	−13.8072			−1.02628			−0.513141	−	0.858304i	$-0.671518\pi$
$181$	−13.8072			−1.02628			−0.513141	+	0.858304i	$0.671518\pi$
$182$		0			0
$183$			0.700418i			0.0517764i
$184$		0			0
$185$		−	3.47842i		−	0.255738i
$186$		0			0
$187$			0.456331i			0.0333702i
$188$		0			0
$189$	−13.6002	−	3.70707i	−0.989269	−	0.269649i
$190$		0			0
$191$		−	6.79373i		−	0.491577i	−0.969324	−	0.245788i	$-0.920953\pi$
$191$		−	6.79373i		−	0.491577i	0.969324	−	0.245788i	$-0.0790468\pi$
$192$		0			0
$193$	−6.94798			−0.500127			−0.250063	−	0.968229i	$-0.580451\pi$
$193$	−6.94798			−0.500127			−0.250063	+	0.968229i	$0.580451\pi$
$194$		0			0
$195$		−	8.62365i		−	0.617553i
$196$		0			0
$197$		−	25.0020i		−	1.78132i	−0.454670	−	0.890660i	$-0.650243\pi$
$197$		−	25.0020i		−	1.78132i	0.454670	−	0.890660i	$-0.349757\pi$
$198$		0			0
$199$	8.86022			0.628084			0.314042	−	0.949409i	$-0.398317\pi$
$199$	8.86022			0.628084			0.314042	+	0.949409i	$0.398317\pi$
$200$		0			0
$201$			25.7476i			1.81609i
$202$		0			0
$203$	−4.18530	−	1.14081i	−0.293751	−	0.0800689i
$204$		0			0
$205$		−	9.88374i		−	0.690310i
$206$		0			0
$207$		−	0.698079i		−	0.0485198i
$208$		0			0
$209$		−	0.363609i		−	0.0251514i
$210$		0			0
$211$	−13.6470			−0.939501			−0.469751	−	0.882799i	$-0.655656\pi$
$211$	−13.6470			−0.939501			−0.469751	+	0.882799i	$0.655656\pi$
$212$		0			0
$213$	2.53325			0.173576
$214$		0			0
$215$	−8.65164			−0.590037
$216$		0			0
$217$	−6.23634	+	22.8794i	−0.423350	+	1.55315i
$218$		0			0
$219$	−18.0330			−1.21856
$220$		0			0
$221$			1.60716i			0.108109i
$222$		0			0
$223$	7.67172			0.513737			0.256868	−	0.966446i	$-0.417309\pi$
$223$	7.67172			0.513737			0.256868	+	0.966446i	$0.417309\pi$
$224$		0			0
$225$	0.163484			0.0108989
$226$		0			0
$227$		−	18.4879i		−	1.22708i	−0.789662	−	0.613542i	$-0.789744\pi$
$227$		−	18.4879i		−	1.22708i	0.789662	−	0.613542i	$-0.210256\pi$
$228$		0			0
$229$	22.6715			1.49817			0.749087	−	0.662472i	$-0.230493\pi$
$229$	22.6715			1.49817			0.749087	+	0.662472i	$0.230493\pi$
$230$		0			0
$231$	6.25029	+	1.70367i	0.411239	+	0.112093i
$232$		0			0
$233$	25.4680			1.66847			0.834234	−	0.551411i	$-0.185910\pi$
$233$	25.4680			1.66847			0.834234	+	0.551411i	$0.185910\pi$
$234$		0			0
$235$	7.77853			0.507415
$236$		0			0
$237$	15.7728			1.02456
$238$		0			0
$239$		−	17.5044i		−	1.13227i	−0.824314	−	0.566134i	$-0.808439\pi$
$239$		−	17.5044i		−	1.13227i	0.824314	−	0.566134i	$-0.191561\pi$
$240$		0			0
$241$		−	20.7733i		−	1.33813i	−0.743206	−	0.669063i	$-0.766696\pi$
$241$		−	20.7733i		−	1.33813i	0.743206	−	0.669063i	$-0.233304\pi$
$242$		0			0
$243$		−	1.69705i		−	0.108866i
$244$		0			0
$245$	6.03178	+	3.55213i	0.385356	+	0.226937i
$246$		0			0
$247$		−	1.28060i		−	0.0814825i
$248$		0			0
$249$	−5.81512			−0.368518
$250$		0			0
$251$		−	18.9279i		−	1.19472i	−0.801973	−	0.597361i	$-0.796216\pi$
$251$		−	18.9279i		−	1.19472i	0.801973	−	0.597361i	$-0.203784\pi$
$252$		0			0
$253$			6.20797i			0.390292i
$254$		0			0
$255$	0.528630			0.0331041
$256$		0			0
$257$			4.31841i			0.269375i	0.990888	+	0.134688i	$0.0430031\pi$
$257$			4.31841i			0.269375i	−0.990888	+	0.134688i	$0.956997\pi$
$258$		0			0
$259$	8.87909	+	2.42021i	0.551720	+	0.150385i
$260$		0			0
$261$		−	0.268050i		−	0.0165919i
$262$		0			0
$263$			1.43331i			0.0883814i	0.999023	+	0.0441907i	$0.0140709\pi$
$263$			1.43331i			0.0883814i	−0.999023	+	0.0441907i	$0.985929\pi$
$264$		0			0
$265$		−	1.90687i		−	0.117138i
$266$		0			0
$267$	15.3717			0.940736
$268$		0			0
$269$	12.1457			0.740537			0.370268	−	0.928925i	$-0.379266\pi$
$269$	12.1457			0.740537			0.370268	+	0.928925i	$0.379266\pi$
$270$		0			0
$271$	−6.48716			−0.394067			−0.197034	−	0.980397i	$-0.563131\pi$
$271$	−6.48716			−0.394067			−0.197034	+	0.980397i	$0.563131\pi$
$272$		0			0
$273$	22.0129	+	6.00017i	1.33228	+	0.363147i
$274$		0			0
$275$	−1.45385			−0.0876707
$276$		0			0
$277$			23.8861i			1.43518i	0.696467	+	0.717589i	$0.254754\pi$
$277$			23.8861i			1.43518i	−0.696467	+	0.717589i	$0.745246\pi$
$278$		0			0
$279$	−1.46532			−0.0877265
$280$		0			0
$281$	−14.4015			−0.859120			−0.429560	−	0.903038i	$-0.641331\pi$
$281$	−14.4015			−0.859120			−0.429560	+	0.903038i	$0.641331\pi$
$282$		0			0
$283$		−	24.0556i		−	1.42996i	−0.699146	−	0.714979i	$-0.746436\pi$
$283$		−	24.0556i		−	1.42996i	0.699146	−	0.714979i	$-0.253564\pi$
$284$		0			0
$285$	−0.421218			−0.0249508
$286$		0			0
$287$	25.2295	+	6.87691i	1.48925	+	0.405931i
$288$		0			0
$289$	16.9015			0.994205
$290$		0			0
$291$	−27.8637			−1.63340
$292$		0			0
$293$	4.78803			0.279720			0.139860	−	0.990171i	$-0.455335\pi$
$293$	4.78803			0.279720			0.139860	+	0.990171i	$0.455335\pi$
$294$		0			0
$295$			7.73295i			0.450230i
$296$		0			0
$297$			7.74603i			0.449470i
$298$		0			0
$299$			21.8639i			1.26442i
$300$		0			0
$301$	6.01964	−	22.0844i	0.346966	−	1.27292i
$302$		0			0
$303$		−	18.7688i		−	1.07824i
$304$		0			0
$305$	0.415877			0.0238130
$306$		0			0
$307$			23.6121i			1.34762i	0.738907	+	0.673808i	$0.235342\pi$
$307$			23.6121i			1.34762i	−0.738907	+	0.673808i	$0.764658\pi$
$308$		0			0
$309$			19.4489i			1.10641i
$310$		0			0
$311$	23.6203			1.33939			0.669693	−	0.742638i	$-0.266426\pi$
$311$	23.6203			1.33939			0.669693	+	0.742638i	$0.266426\pi$
$312$		0			0
$313$			20.6514i			1.16728i	0.812011	+	0.583642i	$0.198373\pi$
$313$			20.6514i			1.16728i	−0.812011	+	0.583642i	$0.801627\pi$
$314$		0			0
$315$	−0.113749	+	0.417313i	−0.00640903	+	0.0235130i
$316$		0			0
$317$		−	13.1857i		−	0.740580i	−0.928916	−	0.370290i	$-0.879258\pi$
$317$		−	13.1857i		−	0.740580i	0.928916	−	0.370290i	$-0.120742\pi$
$318$		0			0
$319$			2.38375i			0.133464i
$320$		0			0
$321$		−	5.68255i		−	0.317169i
$322$		0			0
$323$	0.0785006			0.00436789
$324$		0			0
$325$	−5.12034			−0.284025
$326$		0			0
$327$	−21.2419			−1.17468
$328$		0			0
$329$	−5.41214	+	19.8557i	−0.298381	+	1.09468i
$330$		0			0
$331$	−4.29166			−0.235891			−0.117945	−	0.993020i	$-0.537631\pi$
$331$	−4.29166			−0.235891			−0.117945	+	0.993020i	$0.537631\pi$
$332$		0			0
$333$			0.568666i			0.0311627i
$334$		0			0
$335$	15.2878			0.835259
$336$		0			0
$337$	−23.8194			−1.29752			−0.648762	−	0.760991i	$-0.724713\pi$
$337$	−23.8194			−1.29752			−0.648762	+	0.760991i	$0.724713\pi$
$338$		0			0
$339$		−	22.8449i		−	1.24076i
$340$		0			0
$341$	13.0310			0.705669
$342$		0			0
$343$	−13.2640	+	12.9254i	−0.716191	+	0.697904i
$344$		0			0
$345$	7.19153			0.387179
$346$		0			0
$347$	1.81968			0.0976858			0.0488429	−	0.998806i	$-0.484447\pi$
$347$	1.81968			0.0976858			0.0488429	+	0.998806i	$0.484447\pi$
$348$		0			0
$349$	−8.94236			−0.478673			−0.239337	−	0.970937i	$-0.576930\pi$
$349$	−8.94236			−0.478673			−0.239337	+	0.970937i	$0.576930\pi$
$350$		0			0
$351$			27.2808i			1.45614i
$352$		0			0
$353$		−	16.0890i		−	0.856331i	−0.903700	−	0.428165i	$-0.859160\pi$
$353$		−	16.0890i		−	0.856331i	0.903700	−	0.428165i	$-0.140840\pi$
$354$		0			0
$355$		−	1.50413i		−	0.0798310i
$356$		0			0
$357$	−0.367810	+	1.34939i	−0.0194666	+	0.0714175i
$358$		0			0
$359$			10.1838i			0.537481i	0.963213	+	0.268740i	$0.0866073\pi$
$359$			10.1838i			0.537481i	−0.963213	+	0.268740i	$0.913393\pi$
$360$		0			0
$361$	18.9374			0.996708
$362$		0			0
$363$			14.9663i			0.785526i
$364$		0			0
$365$			10.7072i			0.560439i
$366$		0			0
$367$	−3.78238			−0.197438			−0.0987192	−	0.995115i	$-0.531475\pi$
$367$	−3.78238			−0.197438			−0.0987192	+	0.995115i	$0.531475\pi$
$368$		0			0
$369$			1.61583i			0.0841169i
$370$		0			0
$371$	4.86753	+	1.32677i	0.252710	+	0.0688822i
$372$		0			0
$373$			2.84339i			0.147225i	0.997287	+	0.0736127i	$0.0234529\pi$
$373$			2.84339i			0.147225i	−0.997287	+	0.0736127i	$0.976547\pi$
$374$		0			0
$375$			1.68420i			0.0869715i
$376$		0			0
$377$			8.39535i			0.432383i
$378$		0			0
$379$	−5.22176			−0.268224			−0.134112	−	0.990966i	$-0.542818\pi$
$379$	−5.22176			−0.268224			−0.134112	+	0.990966i	$0.542818\pi$
$380$		0			0
$381$	−3.65490			−0.187246
$382$		0			0
$383$	25.8702			1.32191			0.660953	−	0.750428i	$-0.270152\pi$
$383$	25.8702			1.32191			0.660953	+	0.750428i	$0.270152\pi$
$384$		0			0
$385$	1.01156	−	3.71114i	0.0515540	−	0.189137i
$386$		0			0
$387$	1.41441			0.0718983
$388$		0			0
$389$			26.1792i			1.32734i	0.748025	+	0.663670i	$0.231002\pi$
$389$			26.1792i			1.32734i	−0.748025	+	0.663670i	$0.768998\pi$
$390$		0			0
$391$	−1.34026			−0.0677797
$392$		0			0
$393$	14.8589			0.749534
$394$		0			0
$395$		−	9.36521i		−	0.471215i
$396$		0			0
$397$	−30.6192			−1.53674			−0.768368	−	0.640009i	$-0.778931\pi$
$397$	−30.6192			−1.53674			−0.768368	+	0.640009i	$0.778931\pi$
$398$		0			0
$399$	0.293075	−	1.07521i	0.0146721	−	0.0538278i
$400$		0			0
$401$	−32.6992			−1.63292			−0.816461	−	0.577401i	$-0.804067\pi$
$401$	−32.6992			−1.63292			−0.816461	+	0.577401i	$0.804067\pi$
$402$		0			0
$403$	45.8940			2.28614
$404$		0			0
$405$	8.48282			0.421515
$406$		0			0
$407$		−	5.05711i		−	0.250672i
$408$		0			0
$409$			15.2212i			0.752642i	0.926489	+	0.376321i	$0.122811\pi$
$409$			15.2212i			0.752642i	−0.926489	+	0.376321i	$0.877189\pi$
$410$		0			0
$411$		−	0.609516i		−	0.0300652i
$412$		0			0
$413$	−19.7393	−	5.38043i	−0.971308	−	0.264754i
$414$		0			0
$415$			3.45276i			0.169489i
$416$		0			0
$417$	14.3113			0.700827
$418$		0			0
$419$			1.51292i			0.0739111i	0.999317	+	0.0369555i	$0.0117660\pi$
$419$			1.51292i			0.0739111i	−0.999317	+	0.0369555i	$0.988234\pi$
$420$		0			0
$421$		−	4.42587i		−	0.215704i	−0.994167	−	0.107852i	$-0.965603\pi$
$421$		−	4.42587i		−	0.215704i	0.994167	−	0.107852i	$-0.0343973\pi$
$422$		0			0
$423$	−1.27167			−0.0618305
$424$		0			0
$425$		−	0.313877i		−	0.0152253i
$426$		0			0
$427$	−0.289359	+	1.06158i	−0.0140030	+	0.0513733i
$428$		0			0
$429$		−	12.5375i		−	0.605318i
$430$		0			0
$431$		−	15.7807i		−	0.760128i	−0.924960	−	0.380064i	$-0.875902\pi$
$431$		−	15.7807i		−	0.760128i	0.924960	−	0.380064i	$-0.124098\pi$
$432$		0			0
$433$		−	38.5048i		−	1.85042i	−0.379452	−	0.925211i	$-0.623888\pi$
$433$		−	38.5048i		−	1.85042i	0.379452	−	0.925211i	$-0.376112\pi$
$434$		0			0
$435$	2.76142			0.132400
$436$		0			0
$437$	1.06793			0.0510860
$438$		0			0
$439$	8.16215			0.389558			0.194779	−	0.980847i	$-0.437601\pi$
$439$	8.16215			0.389558			0.194779	+	0.980847i	$0.437601\pi$
$440$		0			0
$441$	−0.986100	−	0.580717i	−0.0469572	−	0.0276532i
$442$		0			0
$443$	−14.4962			−0.688735			−0.344368	−	0.938835i	$-0.611907\pi$
$443$	−14.4962			−0.688735			−0.344368	+	0.938835i	$0.611907\pi$
$444$		0			0
$445$		−	9.12705i		−	0.432664i
$446$		0			0
$447$	−24.8736			−1.17648
$448$		0			0
$449$	−26.2203			−1.23741			−0.618705	−	0.785623i	$-0.712342\pi$
$449$	−26.2203			−1.23741			−0.618705	+	0.785623i	$0.712342\pi$
$450$		0			0
$451$		−	14.3695i		−	0.676634i
$452$		0			0
$453$	−29.7821			−1.39929
$454$		0			0
$455$	3.56263	−	13.0703i	0.167019	−	0.612745i
$456$		0			0
$457$	−10.8518			−0.507624			−0.253812	−	0.967254i	$-0.581684\pi$
$457$	−10.8518			−0.507624			−0.253812	+	0.967254i	$0.581684\pi$
$458$		0			0
$459$	−1.67231			−0.0780568
$460$		0			0
$461$	10.9727			0.511049			0.255525	−	0.966803i	$-0.417752\pi$
$461$	10.9727			0.511049			0.255525	+	0.966803i	$0.417752\pi$
$462$		0			0
$463$			12.8266i			0.596102i	0.954550	+	0.298051i	$0.0963365\pi$
$463$			12.8266i			0.596102i	−0.954550	+	0.298051i	$0.903663\pi$
$464$		0			0
$465$		−	15.0956i		−	0.700041i
$466$		0			0
$467$			27.2475i			1.26087i	0.776244	+	0.630433i	$0.217122\pi$
$467$			27.2475i			1.26087i	−0.776244	+	0.630433i	$0.782878\pi$
$468$		0			0
$469$	−10.6369	+	39.0239i	−0.491167	+	1.80196i
$470$		0			0
$471$			1.05806i			0.0487527i
$472$		0			0
$473$	−12.5782			−0.578347
$474$		0			0
$475$			0.250100i			0.0114754i
$476$		0			0
$477$			0.311744i			0.0142738i
$478$		0			0
$479$	−5.93280			−0.271076			−0.135538	−	0.990772i	$-0.543276\pi$
$479$	−5.93280			−0.271076			−0.135538	+	0.990772i	$0.543276\pi$
$480$		0			0
$481$		−	17.8107i		−	0.812097i
$482$		0			0
$483$	−5.00372	+	18.3573i	−0.227677	+	0.835285i
$484$		0			0
$485$			16.5442i			0.751235i
$486$		0			0
$487$		−	0.799878i		−	0.0362459i	−0.999836	−	0.0181230i	$-0.994231\pi$
$487$		−	0.799878i		−	0.0362459i	0.999836	−	0.0181230i	$-0.00576903\pi$
$488$		0			0
$489$		−	26.4691i		−	1.19698i
$490$		0			0
$491$	−35.4782			−1.60111			−0.800555	−	0.599259i	$-0.795462\pi$
$491$	−35.4782			−1.60111			−0.800555	+	0.599259i	$0.795462\pi$
$492$		0			0
$493$	−0.514635			−0.0231780
$494$		0			0
$495$	0.237682			0.0106830
$496$		0			0
$497$	3.83948	+	1.04654i	0.172224	+	0.0469439i
$498$		0			0
$499$	13.8343			0.619306			0.309653	−	0.950850i	$-0.399787\pi$
$499$	13.8343			0.619306			0.309653	+	0.950850i	$0.399787\pi$
$500$		0			0
$501$		−	31.4492i		−	1.40505i
$502$		0			0
$503$	−24.9750			−1.11358			−0.556790	−	0.830653i	$-0.687967\pi$
$503$	−24.9750			−1.11358			−0.556790	+	0.830653i	$0.687967\pi$
$504$		0			0
$505$	−11.1440			−0.495903
$506$		0			0
$507$		−	22.2615i		−	0.988668i
$508$		0			0
$509$	6.97858			0.309320			0.154660	−	0.987968i	$-0.450572\pi$
$509$	6.97858			0.309320			0.154660	+	0.987968i	$0.450572\pi$
$510$		0			0
$511$	−27.3314	−	7.44983i	−1.20907	−	0.329561i
$512$		0			0
$513$	1.33251			0.0588320
$514$		0			0
$515$	11.5479			0.508860
$516$		0			0
$517$	11.3088			0.497362
$518$		0			0
$519$		−	41.6625i		−	1.82878i
$520$		0			0
$521$		−	16.4401i		−	0.720253i	−0.932904	−	0.360126i	$-0.882734\pi$
$521$		−	16.4401i		−	0.720253i	0.932904	−	0.360126i	$-0.117266\pi$
$522$		0			0
$523$			23.0349i			1.00725i	0.863923	+	0.503624i	$0.168000\pi$
$523$			23.0349i			1.00725i	−0.863923	+	0.503624i	$0.832000\pi$
$524$		0			0
$525$	−4.29912	−	1.17183i	−0.187629	−	0.0511429i
$526$		0			0
$527$			2.81330i			0.122549i
$528$		0			0
$529$	4.76703			0.207262
$530$		0			0
$531$		−	1.26421i		−	0.0548622i
$532$		0			0
$533$		−	50.6081i		−	2.19208i
$534$		0			0
$535$	−3.37404			−0.145873
$536$		0			0
$537$			11.5614i			0.498911i
$538$		0			0
$539$	8.76933	+	5.16428i	0.377722	+	0.222441i
$540$		0			0
$541$			12.6920i			0.545671i	0.962061	+	0.272835i	$0.0879615\pi$
$541$			12.6920i			0.545671i	−0.962061	+	0.272835i	$0.912039\pi$
$542$		0			0
$543$			23.2540i			0.997926i
$544$		0			0
$545$			12.6125i			0.540258i
$546$		0			0
$547$	−21.0334			−0.899321			−0.449661	−	0.893199i	$-0.648455\pi$
$547$	−21.0334			−0.899321			−0.449661	+	0.893199i	$0.648455\pi$
$548$		0			0
$549$	−0.0679892			−0.00290171
$550$		0			0
$551$	0.410066			0.0174694
$552$		0			0
$553$	23.9059	+	6.51613i	1.01658	+	0.277094i
$554$		0			0
$555$	−5.85834			−0.248672
$556$		0			0
$557$		−	0.724648i		−	0.0307043i	−0.999882	−	0.0153522i	$-0.995113\pi$
$557$		−	0.724648i		−	0.0307043i	0.999882	−	0.0153522i	$-0.00488694\pi$
$558$		0			0
$559$	−44.2993			−1.87366
$560$		0			0
$561$	0.768550			0.0324482
$562$		0			0
$563$		−	8.62465i		−	0.363486i	−0.983346	−	0.181743i	$-0.941826\pi$
$563$		−	8.62465i		−	0.363486i	0.983346	−	0.181743i	$-0.0581739\pi$
$564$		0			0
$565$	−13.5643			−0.570653
$566$		0			0
$567$	−5.90218	+	21.6535i	−0.247868	+	0.909360i
$568$		0			0
$569$	2.20005			0.0922311			0.0461155	−	0.998936i	$-0.485316\pi$
$569$	2.20005			0.0922311			0.0461155	+	0.998936i	$0.485316\pi$
$570$		0			0
$571$	−22.8120			−0.954654			−0.477327	−	0.878726i	$-0.658394\pi$
$571$	−22.8120			−0.954654			−0.477327	+	0.878726i	$0.658394\pi$
$572$		0			0
$573$	−11.4420			−0.477995
$574$		0			0
$575$		−	4.27001i		−	0.178072i
$576$		0			0
$577$		−	19.6510i		−	0.818081i	−0.912516	−	0.409040i	$-0.865864\pi$
$577$		−	19.6510i		−	0.818081i	0.912516	−	0.409040i	$-0.134136\pi$
$578$		0			0
$579$			11.7018i			0.486309i
$580$		0			0
$581$	−8.81359	−	2.40236i	−0.365649	−	0.0996667i
$582$		0			0
$583$		−	2.77232i		−	0.114818i
$584$		0			0
$585$	0.837094			0.0346096
$586$		0			0
$587$			24.2904i			1.00257i	0.865281	+	0.501287i	$0.167140\pi$
$587$			24.2904i			1.00257i	−0.865281	+	0.501287i	$0.832860\pi$
$588$		0			0
$589$		−	2.24167i		−	0.0923663i
$590$		0			0
$591$	−42.1083			−1.73210
$592$		0			0
$593$		−	25.4337i		−	1.04444i	−0.852811	−	0.522219i	$-0.825104\pi$
$593$		−	25.4337i		−	1.04444i	0.852811	−	0.522219i	$-0.174896\pi$
$594$		0			0
$595$	0.801209	+	0.218389i	0.0328464	+	0.00895308i
$596$		0			0
$597$		−	14.9223i		−	0.610731i
$598$		0			0
$599$			4.24448i			0.173425i	0.996233	+	0.0867124i	$0.0276361\pi$
$599$			4.24448i			0.173425i	−0.996233	+	0.0867124i	$0.972364\pi$
$600$		0			0
$601$		−	1.00464i		−	0.0409800i	−0.999790	−	0.0204900i	$-0.993477\pi$
$601$		−	1.00464i		−	0.0409800i	0.999790	−	0.0204900i	$-0.00652263\pi$
$602$		0			0
$603$	−2.49930			−0.101780
$604$		0			0
$605$	8.88631			0.361280
$606$		0			0
$607$	−31.5181			−1.27928			−0.639640	−	0.768675i	$-0.720917\pi$
$607$	−31.5181			−1.27928			−0.639640	+	0.768675i	$0.720917\pi$
$608$		0			0
$609$	−1.92134	+	7.04887i	−0.0778567	+	0.285635i
$610$		0			0
$611$	39.8287			1.61130
$612$		0			0
$613$		−	22.3851i		−	0.904126i	−0.891986	−	0.452063i	$-0.850688\pi$
$613$		−	22.3851i		−	0.904126i	0.891986	−	0.452063i	$-0.149312\pi$
$614$		0			0
$615$	−16.6461			−0.671237
$616$		0			0
$617$	27.1721			1.09391			0.546954	−	0.837163i	$-0.315787\pi$
$617$	27.1721			1.09391			0.546954	+	0.837163i	$0.315787\pi$
$618$		0			0
$619$			16.7190i			0.671993i	0.941863	+	0.335997i	$0.109073\pi$
$619$			16.7190i			0.671993i	−0.941863	+	0.335997i	$0.890927\pi$
$620$		0			0
$621$	−22.7503			−0.912938
$622$		0			0
$623$	23.2979	+	6.35042i	0.933412	+	0.254424i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−0.612389			−0.0244564
$628$		0			0
$629$	1.09179			0.0435327
$630$		0			0
$631$			25.9280i			1.03218i	0.856535	+	0.516090i	$0.172613\pi$
$631$			25.9280i			1.03218i	−0.856535	+	0.516090i	$0.827387\pi$
$632$		0			0
$633$			22.9843i			0.913544i
$634$		0			0
$635$			2.17011i			0.0861183i
$636$		0			0
$637$	30.8848	+	18.1881i	1.22370	+	0.720639i
$638$		0			0
$639$			0.245901i			0.00972771i
$640$		0			0
$641$	40.1639			1.58638			0.793189	−	0.608975i	$-0.208419\pi$
$641$	40.1639			1.58638			0.793189	+	0.608975i	$0.208419\pi$
$642$		0			0
$643$		−	40.5245i		−	1.59813i	−0.601244	−	0.799065i	$-0.705328\pi$
$643$		−	40.5245i		−	1.59813i	0.601244	−	0.799065i	$-0.294672\pi$
$644$		0			0
$645$			14.5711i			0.573735i
$646$		0			0
$647$	14.3559			0.564387			0.282193	−	0.959358i	$-0.408938\pi$
$647$	14.3559			0.564387			0.282193	+	0.959358i	$0.408938\pi$
$648$		0			0
$649$			11.2426i			0.441310i
$650$		0			0
$651$	38.5334	+	10.5032i	1.51024	+	0.411653i
$652$		0			0
$653$			34.5999i			1.35400i	0.735984	+	0.676999i	$0.236720\pi$
$653$			34.5999i			1.35400i	−0.735984	+	0.676999i	$0.763280\pi$
$654$		0			0
$655$		−	8.82257i		−	0.344726i
$656$		0			0
$657$		−	1.75045i		−	0.0682916i
$658$		0			0
$659$	10.6325			0.414185			0.207092	−	0.978321i	$-0.433600\pi$
$659$	10.6325			0.414185			0.207092	+	0.978321i	$0.433600\pi$
$660$		0			0
$661$	−16.0278			−0.623410			−0.311705	−	0.950179i	$-0.600900\pi$
$661$	−16.0278			−0.623410			−0.311705	+	0.950179i	$0.600900\pi$
$662$		0			0
$663$	2.70676			0.105122
$664$		0			0
$665$	−0.638411	−	0.174015i	−0.0247565	−	0.00674800i
$666$		0			0
$667$	−7.00114			−0.271085
$668$		0			0
$669$		−	12.9207i		−	0.499542i
$670$		0			0
$671$	0.604624			0.0233412
$672$		0			0
$673$	−24.0119			−0.925591			−0.462796	−	0.886465i	$-0.653154\pi$
$673$	−24.0119			−0.925591			−0.462796	+	0.886465i	$0.653154\pi$
$674$		0			0
$675$		−	5.32793i		−	0.205072i
$676$		0			0
$677$	−32.1429			−1.23535			−0.617676	−	0.786432i	$-0.711926\pi$
$677$	−32.1429			−1.23535			−0.617676	+	0.786432i	$0.711926\pi$
$678$		0			0
$679$	−42.2312	−	11.5111i	−1.62068	−	0.441757i
$680$		0			0
$681$	−31.1372			−1.19318
$682$		0			0
$683$	2.97604			0.113875			0.0569374	−	0.998378i	$-0.481866\pi$
$683$	2.97604			0.113875			0.0569374	+	0.998378i	$0.481866\pi$
$684$		0			0
$685$	−0.361903			−0.0138276
$686$		0			0
$687$		−	38.1832i		−	1.45678i
$688$		0			0
$689$		−	9.76385i		−	0.371973i
$690$		0			0
$691$		−	13.7448i		−	0.522877i	−0.965220	−	0.261439i	$-0.915803\pi$
$691$		−	13.7448i		−	0.522877i	0.965220	−	0.261439i	$-0.0841969\pi$
$692$		0			0
$693$	−0.165374	+	0.606713i	−0.00628205	+	0.0230471i
$694$		0			0
$695$		−	8.49740i		−	0.322325i
$696$		0			0
$697$	3.10227			0.117507
$698$		0			0
$699$		−	42.8932i		−	1.62237i
$700$		0			0
$701$			49.8152i			1.88149i	0.339111	+	0.940746i	$0.389874\pi$
$701$			49.8152i			1.88149i	−0.339111	+	0.940746i	$0.610126\pi$
$702$		0			0
$703$	−0.869952			−0.0328109
$704$		0			0
$705$		−	13.1006i		−	0.493396i
$706$		0			0
$707$	7.75380	−	28.4466i	0.291612	−	1.06984i
$708$		0			0
$709$			5.24482i			0.196973i	0.995138	+	0.0984867i	$0.0314002\pi$
$709$			5.24482i			0.196973i	−0.995138	+	0.0984867i	$0.968600\pi$
$710$		0			0
$711$			1.53106i			0.0574193i
$712$		0			0
$713$			38.2724i			1.43331i
$714$		0			0
$715$	−7.44422			−0.278398
$716$		0			0
$717$	−29.4809			−1.10098
$718$		0			0
$719$	35.6884			1.33095			0.665476	−	0.746419i	$-0.268229\pi$
$719$	35.6884			1.33095			0.665476	+	0.746419i	$0.268229\pi$
$720$		0			0
$721$	−8.03479	+	29.4774i	−0.299231	+	1.09780i
$722$		0			0
$723$	−34.9863			−1.30116
$724$		0			0
$725$		−	1.63961i		−	0.0608935i
$726$		0			0
$727$	49.5564			1.83794			0.918972	−	0.394322i	$-0.129020\pi$
$727$	49.5564			1.83794			0.918972	+	0.394322i	$0.129020\pi$
$728$		0			0
$729$	−28.3066			−1.04839
$730$		0			0
$731$		−	2.71555i		−	0.100438i
$732$		0			0
$733$	9.76884			0.360820			0.180410	−	0.983591i	$-0.442257\pi$
$733$	9.76884			0.360820			0.180410	+	0.983591i	$0.442257\pi$
$734$		0			0
$735$	5.98248	−	10.1587i	0.220667	−	0.374709i
$736$		0			0
$737$	22.2262			0.818711
$738$		0			0
$739$	28.7691			1.05829			0.529145	−	0.848531i	$-0.322513\pi$
$739$	28.7691			1.05829			0.529145	+	0.848531i	$0.322513\pi$
$740$		0			0
$741$	−2.15678			−0.0792312
$742$		0			0
$743$			34.3244i			1.25924i	0.776903	+	0.629620i	$0.216789\pi$
$743$			34.3244i			1.25924i	−0.776903	+	0.629620i	$0.783211\pi$
$744$		0			0
$745$			14.7688i			0.541088i
$746$		0			0
$747$		−	0.564471i		−	0.0206529i
$748$		0			0
$749$	2.34759	−	8.61267i	0.0857792	−	0.314700i
$750$		0			0
$751$			40.8689i			1.49133i	0.666323	+	0.745663i	$0.267867\pi$
$751$			40.8689i			1.49133i	−0.666323	+	0.745663i	$0.732133\pi$
$752$		0			0
$753$	−31.8784			−1.16171
$754$		0			0
$755$			17.6833i			0.643561i
$756$		0			0
$757$		−	35.1958i		−	1.27921i	−0.768703	−	0.639606i	$-0.779098\pi$
$757$		−	35.1958i		−	1.27921i	0.768703	−	0.639606i	$-0.220902\pi$
$758$		0			0
$759$	10.4554			0.379508
$760$		0			0
$761$			9.03383i			0.327476i	0.986504	+	0.163738i	$0.0523552\pi$
$761$			9.03383i			0.327476i	−0.986504	+	0.163738i	$0.947645\pi$
$762$		0			0
$763$	−32.1949	−	8.77550i	−1.16553	−	0.317694i
$764$		0			0
$765$			0.0513138i			0.00185526i
$766$		0			0
$767$			39.5953i			1.42970i
$768$		0			0
$769$		−	16.4068i		−	0.591645i	−0.955243	−	0.295823i	$-0.904406\pi$
$769$		−	16.4068i		−	0.591645i	0.955243	−	0.295823i	$-0.0955937\pi$
$770$		0			0
$771$	7.27305			0.261933
$772$		0			0
$773$	9.49352			0.341458			0.170729	−	0.985318i	$-0.445388\pi$
$773$	9.49352			0.341458			0.170729	+	0.985318i	$0.445388\pi$
$774$		0			0
$775$	−8.96308			−0.321963
$776$		0			0
$777$	4.07611	−	14.9541i	0.146230	−	0.536476i
$778$		0			0
$779$	−2.47192			−0.0885658
$780$		0			0
$781$		−	2.18679i		−	0.0782494i
$782$		0			0
$783$	−8.73571			−0.312189
$784$		0			0
$785$	0.628228			0.0224224
$786$		0			0
$787$			2.43467i			0.0867866i	0.999058	+	0.0433933i	$0.0138169\pi$
$787$			2.43467i			0.0867866i	−0.999058	+	0.0433933i	$0.986183\pi$
$788$		0			0
$789$	2.41397			0.0859395
$790$		0			0
$791$	9.43774	−	34.6245i	0.335568	−	1.23110i
$792$		0			0
$793$	2.12943			0.0756182
$794$		0			0
$795$	−3.21155			−0.113902
$796$		0			0
$797$	18.3689			0.650660			0.325330	−	0.945601i	$-0.394525\pi$
$797$	18.3689			0.650660			0.325330	+	0.945601i	$0.394525\pi$
$798$		0			0
$799$			2.44150i			0.0863740i
$800$		0			0
$801$			1.49213i			0.0527218i
$802$		0			0
$803$			15.5667i			0.549335i
$804$		0			0
$805$	10.8997	+	2.97099i	0.384165	+	0.104714i
$806$		0			0
$807$		−	20.4557i		−	0.720076i
$808$		0			0
$809$	18.8508			0.662759			0.331379	−	0.943498i	$-0.392486\pi$
$809$	18.8508			0.662759			0.331379	+	0.943498i	$0.392486\pi$
$810$		0			0
$811$			7.40501i			0.260025i	0.991512	+	0.130013i	$0.0415018\pi$
$811$			7.40501i			0.260025i	−0.991512	+	0.130013i	$0.958498\pi$
$812$		0			0
$813$			10.9257i			0.383179i
$814$		0			0
$815$	−15.7162			−0.550514
$816$		0			0
$817$			2.16378i			0.0757009i
$818$		0			0
$819$	−0.582433	+	2.13679i	−0.0203519	+	0.0746654i
$820$		0			0
$821$			16.2192i			0.566054i	0.959112	+	0.283027i	$0.0913385\pi$
$821$			16.2192i			0.566054i	−0.959112	+	0.283027i	$0.908661\pi$
$822$		0			0
$823$			31.7375i			1.10630i	0.833081	+	0.553150i	$0.186575\pi$
$823$			31.7375i			1.10630i	−0.833081	+	0.553150i	$0.813425\pi$
$824$		0			0
$825$			2.44857i			0.0852484i
$826$		0			0
$827$	−35.1357			−1.22179			−0.610894	−	0.791713i	$-0.709190\pi$
$827$	−35.1357			−1.22179			−0.610894	+	0.791713i	$0.709190\pi$
$828$		0			0
$829$	−30.5400			−1.06070			−0.530350	−	0.847779i	$-0.677939\pi$
$829$	−30.5400			−1.06070			−0.530350	+	0.847779i	$0.677939\pi$
$830$		0			0
$831$	40.2289			1.39553
$832$		0			0
$833$	−1.11493	+	1.89324i	−0.0386301	+	0.0655967i
$834$		0			0
$835$	−18.6731			−0.646211
$836$		0			0
$837$			47.7547i			1.65064i
$838$		0			0
$839$	12.5696			0.433950			0.216975	−	0.976177i	$-0.430381\pi$
$839$	12.5696			0.433950			0.216975	+	0.976177i	$0.430381\pi$
$840$		0			0
$841$	26.3117			0.907299
$842$		0			0
$843$			24.2549i			0.835384i
$844$		0			0
$845$	−13.2179			−0.454709
$846$		0			0
$847$	−6.18292	+	22.6834i	−0.212448	+	0.779411i
$848$		0			0
$849$	−40.5143			−1.39045
$850$		0			0
$851$	14.8529			0.509150
$852$		0			0
$853$	−2.56221			−0.0877284			−0.0438642	−	0.999038i	$-0.513967\pi$
$853$	−2.56221			−0.0877284			−0.0438642	+	0.999038i	$0.513967\pi$
$854$		0			0
$855$		−	0.0408874i		−	0.00139832i
$856$		0			0
$857$		−	50.0346i		−	1.70915i	−0.519329	−	0.854574i	$-0.673818\pi$
$857$		−	50.0346i		−	1.70915i	0.519329	−	0.854574i	$-0.326182\pi$
$858$		0			0
$859$		−	38.9502i		−	1.32896i	−0.747304	−	0.664482i	$-0.768652\pi$
$859$		−	38.9502i		−	1.32896i	0.747304	−	0.664482i	$-0.231348\pi$
$860$		0			0
$861$	11.5821	−	42.4914i	0.394716	−	1.44810i
$862$		0			0
$863$		−	18.6149i		−	0.633657i	−0.948483	−	0.316828i	$-0.897382\pi$
$863$		−	18.6149i		−	0.633657i	0.948483	−	0.316828i	$-0.102618\pi$
$864$		0			0
$865$	−24.7373			−0.841095
$866$		0			0
$867$		−	28.4654i		−	0.966736i
$868$		0			0
$869$		−	13.6156i		−	0.461879i
$870$		0			0
$871$	78.2785			2.65237
$872$		0			0
$873$		−	2.70472i		−	0.0915408i
$874$		0			0
$875$	−0.695780	+	2.55262i	−0.0235217	+	0.0862944i
$876$		0			0
$877$			26.4153i			0.891982i	0.895037	+	0.445991i	$0.147149\pi$
$877$			26.4153i			0.891982i	−0.895037	+	0.445991i	$0.852851\pi$
$878$		0			0
$879$		−	8.06398i		−	0.271991i
$880$		0			0
$881$			27.3582i			0.921722i	0.887472	+	0.460861i	$0.152459\pi$
$881$			27.3582i			0.921722i	−0.887472	+	0.460861i	$0.847541\pi$
$882$		0			0
$883$	−15.3050			−0.515053			−0.257526	−	0.966271i	$-0.582907\pi$
$883$	−15.3050			−0.515053			−0.257526	+	0.966271i	$0.582907\pi$
$884$		0			0
$885$	13.0238			0.437790
$886$		0			0
$887$	4.33168			0.145444			0.0727218	−	0.997352i	$-0.476831\pi$
$887$	4.33168			0.145444			0.0727218	+	0.997352i	$0.476831\pi$
$888$		0			0
$889$	−5.53948	−	1.50992i	−0.185788	−	0.0506411i
$890$		0			0
$891$	12.3328			0.413164
$892$		0			0
$893$		−	1.94541i		−	0.0651007i
$894$		0			0
$895$	6.86464			0.229460
$896$		0			0
$897$	36.8231			1.22949
$898$		0			0
$899$			14.6959i			0.490137i
$900$		0			0
$901$	0.598523			0.0199397
$902$		0			0
$903$	−37.1944	−	10.1383i	−1.23775	−	0.337380i
$904$		0			0
$905$	13.8072			0.458967
$906$		0			0
$907$	−38.3842			−1.27453			−0.637263	−	0.770646i	$-0.719934\pi$
$907$	−38.3842			−1.27453			−0.637263	+	0.770646i	$0.719934\pi$
$908$		0			0
$909$	1.82187			0.0604277
$910$		0			0
$911$			21.1601i			0.701065i	0.936551	+	0.350533i	$0.113999\pi$
$911$			21.1601i			0.701065i	−0.936551	+	0.350533i	$0.886001\pi$
$912$		0			0
$913$			5.01980i			0.166131i
$914$		0			0
$915$		−	0.700418i		−	0.0231551i
$916$		0			0
$917$	22.5207	+	6.13857i	0.743699	+	0.202713i
$918$		0			0
$919$		−	11.5795i		−	0.381971i	−0.981593	−	0.190986i	$-0.938832\pi$
$919$		−	11.5795i		−	0.381971i	0.981593	−	0.190986i	$-0.0611684\pi$
$920$		0			0
$921$	39.7674			1.31038
$922$		0			0
$923$		−	7.70166i		−	0.253503i
$924$		0			0
$925$			3.47842i			0.114370i
$926$		0			0
$927$	−1.88789			−0.0620066
$928$		0			0
$929$			16.9537i			0.556232i	0.960547	+	0.278116i	$0.0897100\pi$
$929$			16.9537i			0.556232i	−0.960547	+	0.278116i	$0.910290\pi$
$930$		0			0
$931$	0.888388	−	1.50855i	0.0291157	−	0.0494407i
$932$		0			0
$933$		−	39.7813i		−	1.30238i
$934$		0			0
$935$		−	0.456331i		−	0.0149236i
$936$		0			0
$937$			36.4798i			1.19174i	0.803079	+	0.595872i	$0.203193\pi$
$937$			36.4798i			1.19174i	−0.803079	+	0.595872i	$0.796807\pi$
$938$		0			0
$939$	34.7810			1.13503
$940$		0			0
$941$	9.71945			0.316845			0.158423	−	0.987371i	$-0.449359\pi$
$941$	9.71945			0.316845			0.158423	+	0.987371i	$0.449359\pi$
$942$		0			0
$943$	42.2036			1.37434
$944$		0			0
$945$	13.6002	+	3.70707i	0.442414	+	0.120591i
$946$		0			0
$947$	32.8605			1.06782			0.533911	−	0.845541i	$-0.320722\pi$
$947$	32.8605			1.06782			0.533911	+	0.845541i	$0.320722\pi$
$948$		0			0
$949$			54.8243i			1.77967i
$950$		0			0
$951$	−22.2072			−0.720118
$952$		0			0
$953$	−52.0838			−1.68716			−0.843580	−	0.537003i	$-0.819556\pi$
$953$	−52.0838			−1.68716			−0.843580	+	0.537003i	$0.819556\pi$
$954$		0			0
$955$			6.79373i			0.219840i
$956$		0			0
$957$	4.01470			0.129777
$958$		0			0
$959$	0.251805	−	0.923803i	0.00813121	−	0.0298311i
$960$		0			0
$961$	49.3369			1.59151
$962$		0			0
$963$	0.551603			0.0177751
$964$		0			0
$965$	6.94798			0.223663
$966$		0			0
$967$		−	53.8786i		−	1.73262i	−0.499509	−	0.866309i	$-0.666486\pi$
$967$		−	53.8786i		−	1.73262i	0.499509	−	0.866309i	$-0.333514\pi$
$968$		0			0
$969$		−	0.132210i		−	0.00424721i
$970$		0			0
$971$			41.7861i			1.34098i	0.741919	+	0.670490i	$0.233916\pi$
$971$			41.7861i			1.34098i	−0.741919	+	0.670490i	$0.766084\pi$
$972$		0			0
$973$	21.6907	+	5.91232i	0.695371	+	0.189540i
$974$		0			0
$975$			8.62365i			0.276178i
$976$		0			0
$977$	−11.1568			−0.356939			−0.178469	−	0.983945i	$-0.557115\pi$
$977$	−11.1568			−0.356939			−0.178469	+	0.983945i	$0.557115\pi$
$978$		0			0
$979$		−	13.2694i		−	0.424092i
$980$		0			0
$981$		−	2.06194i		−	0.0658326i
$982$		0			0
$983$	−6.51094			−0.207667			−0.103833	−	0.994595i	$-0.533111\pi$
$983$	−6.51094			−0.207667			−0.103833	+	0.994595i	$0.533111\pi$
$984$		0			0
$985$			25.0020i			0.796631i
$986$		0			0
$987$	33.4408	+	9.11511i	1.06443	+	0.290137i
$988$		0			0
$989$		−	36.9426i		−	1.17471i
$990$		0			0
$991$		−	41.7400i		−	1.32591i	−0.748658	−	0.662957i	$-0.769301\pi$
$991$		−	41.7400i		−	1.32591i	0.748658	−	0.662957i	$-0.230699\pi$
$992$		0			0
$993$			7.22799i			0.229373i
$994$		0			0
$995$	−8.86022			−0.280888
$996$		0			0
$997$	−13.1237			−0.415632			−0.207816	−	0.978168i	$-0.566635\pi$
$997$	−13.1237			−0.415632			−0.207816	+	0.978168i	$0.566635\pi$
$998$		0			0
$999$	18.5327			0.586350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.h.a.111.4		16
4.3	odd	2		280.2.h.a.251.10	yes	16
7.6	odd	2		1120.2.h.b.111.13		16
8.3	odd	2		1120.2.h.b.111.4		16
8.5	even	2		280.2.h.b.251.9	yes	16
28.27	even	2		280.2.h.b.251.10	yes	16
56.13	odd	2		280.2.h.a.251.9	✓	16
56.27	even	2	inner	1120.2.h.a.111.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.h.a.251.9	✓	16	56.13	odd	2
280.2.h.a.251.10	yes	16	4.3	odd	2
280.2.h.b.251.9	yes	16	8.5	even	2
280.2.h.b.251.10	yes	16	28.27	even	2
1120.2.h.a.111.4		16	1.1	even	1	trivial
1120.2.h.a.111.13		16	56.27	even	2	inner
1120.2.h.b.111.4		16	8.3	odd	2
1120.2.h.b.111.13		16	7.6	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)

\(f(q)\)	\(=\)	\(q-1.68420i q^{3} -1.00000 q^{5} +(0.695780 - 2.55262i) q^{7} +0.163484 q^{9} +O(q^{10})\)	\(q-1.68420i q^{3} -1.00000 q^{5} +(0.695780 - 2.55262i) q^{7} +0.163484 q^{9} -1.45385 q^{11} -5.12034 q^{13} +1.68420i q^{15} -0.313877i q^{17} +0.250100i q^{19} +(-4.29912 - 1.17183i) q^{21} -4.27001i q^{23} +1.00000 q^{25} -5.32793i q^{27} -1.63961i q^{29} -8.96308 q^{31} +2.44857i q^{33} +(-0.695780 + 2.55262i) q^{35} +3.47842i q^{37} +8.62365i q^{39} +9.88374i q^{41} +8.65164 q^{43} -0.163484 q^{45} -7.77853 q^{47} +(-6.03178 - 3.55213i) q^{49} -0.528630 q^{51} +1.90687i q^{53} +1.45385 q^{55} +0.421218 q^{57} -7.73295i q^{59} -0.415877 q^{61} +(0.113749 - 0.417313i) q^{63} +5.12034 q^{65} -15.2878 q^{67} -7.19153 q^{69} +1.50413i q^{71} -10.7072i q^{73} -1.68420i q^{75} +(-1.01156 + 3.71114i) q^{77} +9.36521i q^{79} -8.48282 q^{81} -3.45276i q^{83} +0.313877i q^{85} -2.76142 q^{87} +9.12705i q^{89} +(-3.56263 + 13.0703i) q^{91} +15.0956i q^{93} -0.250100i q^{95} -16.5442i q^{97} -0.237682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) \) 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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