LMFDB - Embedded newform 1120.2.g.b.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(449,1120)]

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([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1120.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 $ x^10 + 13x^8 + 56x^6 + 97x^4 + 61x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$2.52064i$ of defining polynomial
Character	$\chi$	$=$	1120.449
Dual form			1120.2.g.b.449.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83297i q^{3} +(1.86302 - 1.23660i) q^{5} -1.00000i q^{7} -0.359777 q^{9} +O(q^{10})$	$q-1.83297i q^{3} +(1.86302 - 1.23660i) q^{5} -1.00000i q^{7} -0.359777 q^{9} -4.40105 q^{11} -3.20830i q^{13} +(-2.26664 - 3.41485i) q^{15} -1.14821i q^{17} -1.72603 q^{19} -1.83297 q^{21} +3.25284i q^{23} +(1.94166 - 4.60760i) q^{25} -4.83945i q^{27} -4.18948 q^{29} -1.36952 q^{31} +8.06699i q^{33} +(-1.23660 - 1.86302i) q^{35} -4.19923i q^{37} -5.88072 q^{39} +11.7485 q^{41} -2.64997i q^{43} +(-0.670270 + 0.444899i) q^{45} -0.106937i q^{47} -1.00000 q^{49} -2.10463 q^{51} +7.86516i q^{53} +(-8.19923 + 5.44232i) q^{55} +3.16376i q^{57} +13.3029 q^{59} -10.0224 q^{61} +0.359777i q^{63} +(-3.96737 - 5.97712i) q^{65} -1.28045i q^{67} +5.96236 q^{69} -14.1616 q^{71} +11.8652i q^{73} +(-8.44559 - 3.55900i) q^{75} +4.40105i q^{77} +5.48227 q^{79} -9.94989 q^{81} -17.1586i q^{83} +(-1.41987 - 2.13913i) q^{85} +7.67919i q^{87} -4.31591 q^{89} -3.20830 q^{91} +2.51029i q^{93} +(-3.21563 + 2.13441i) q^{95} -7.65389i q^{97} +1.58340 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 14 q^{9}+O(q^{10}) $ 10 * q - 2 * q^5 - 14 * q^9	$ 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65} - 8 q^{69} - 8 q^{71} - 64 q^{75} + 64 q^{79} + 2 q^{81} - 12 q^{85} - 4 q^{89} - 60 q^{95} + 80 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * 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
$3$		−	1.83297i		−	1.05827i	−0.848539	−	0.529133i	$-0.822517\pi$
$3$		−	1.83297i		−	1.05827i	0.848539	−	0.529133i	$-0.177483\pi$
$4$		0			0
$5$	1.86302	−	1.23660i	0.833166	−	0.553023i
$5$	1.86302	−	1.23660i	0.833166	−	0.553023i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−0.359777			−0.119926
$10$		0			0
$11$	−4.40105			−1.32697			−0.663483	−	0.748191i	$-0.730923\pi$
$11$	−4.40105			−1.32697			−0.663483	+	0.748191i	$0.730923\pi$
$12$		0			0
$13$		−	3.20830i		−	0.889823i	−0.895575	−	0.444911i	$-0.853235\pi$
$13$		−	3.20830i		−	0.889823i	0.895575	−	0.444911i	$-0.146765\pi$
$14$		0			0
$15$	−2.26664	−	3.41485i	−0.585245	−	0.881711i
$16$		0			0
$17$		−	1.14821i		−	0.278482i	−0.990259	−	0.139241i	$-0.955534\pi$
$17$		−	1.14821i		−	0.278482i	0.990259	−	0.139241i	$-0.0444662\pi$
$18$		0			0
$19$	−1.72603			−0.395979			−0.197990	−	0.980204i	$-0.563441\pi$
$19$	−1.72603			−0.395979			−0.197990	+	0.980204i	$0.563441\pi$
$20$		0			0
$21$	−1.83297			−0.399987
$22$		0			0
$23$			3.25284i			0.678264i	0.940739	+	0.339132i	$0.110133\pi$
$23$			3.25284i			0.678264i	−0.940739	+	0.339132i	$0.889867\pi$
$24$		0			0
$25$	1.94166	−	4.60760i	0.388332	−	0.921520i
$26$		0			0
$27$		−	4.83945i		−	0.931352i
$28$		0			0
$29$	−4.18948			−0.777967			−0.388983	−	0.921245i	$-0.627174\pi$
$29$	−4.18948			−0.777967			−0.388983	+	0.921245i	$0.627174\pi$
$30$		0			0
$31$	−1.36952			−0.245973			−0.122987	−	0.992408i	$-0.539247\pi$
$31$	−1.36952			−0.245973			−0.122987	+	0.992408i	$0.539247\pi$
$32$		0			0
$33$			8.06699i			1.40428i
$34$		0			0
$35$	−1.23660	−	1.86302i	−0.209023	−	0.314907i
$36$		0			0
$37$		−	4.19923i		−	0.690348i	−0.938539	−	0.345174i	$-0.887820\pi$
$37$		−	4.19923i		−	0.690348i	0.938539	−	0.345174i	$-0.112180\pi$
$38$		0			0
$39$	−5.88072			−0.941669
$40$		0			0
$41$	11.7485			1.83480			0.917402	−	0.397961i	$-0.130282\pi$
$41$	11.7485			1.83480			0.917402	+	0.397961i	$0.130282\pi$
$42$		0			0
$43$		−	2.64997i		−	0.404116i	−0.979374	−	0.202058i	$-0.935237\pi$
$43$		−	2.64997i		−	0.404116i	0.979374	−	0.202058i	$-0.0647630\pi$
$44$		0			0
$45$	−0.670270	+	0.444899i	−0.0999180	+	0.0663216i
$46$		0			0
$47$		−	0.106937i		−	0.0155983i	−0.999970	−	0.00779917i	$-0.997517\pi$
$47$		−	0.106937i		−	0.0155983i	0.999970	−	0.00779917i	$-0.00248258\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−2.10463			−0.294707
$52$		0			0
$53$			7.86516i			1.08036i	0.841548	+	0.540182i	$0.181644\pi$
$53$			7.86516i			1.08036i	−0.841548	+	0.540182i	$0.818356\pi$
$54$		0			0
$55$	−8.19923	+	5.44232i	−1.10558	+	0.733842i
$56$		0			0
$57$			3.16376i			0.419051i
$58$		0			0
$59$	13.3029			1.73189			0.865945	−	0.500140i	$-0.166718\pi$
$59$	13.3029			1.73189			0.865945	+	0.500140i	$0.166718\pi$
$60$		0			0
$61$	−10.0224			−1.28324			−0.641622	−	0.767021i	$-0.721738\pi$
$61$	−10.0224			−1.28324			−0.641622	+	0.767021i	$0.721738\pi$
$62$		0			0
$63$			0.359777i			0.0453276i
$64$		0			0
$65$	−3.96737	−	5.97712i	−0.492092	−	0.741370i
$66$		0			0
$67$		−	1.28045i		−	0.156431i	−0.996936	−	0.0782157i	$-0.975078\pi$
$67$		−	1.28045i		−	0.156431i	0.996936	−	0.0782157i	$-0.0249223\pi$
$68$		0			0
$69$	5.96236			0.717783
$70$		0			0
$71$	−14.1616			−1.68067			−0.840335	−	0.542067i	$-0.817642\pi$
$71$	−14.1616			−1.68067			−0.840335	+	0.542067i	$0.817642\pi$
$72$		0			0
$73$			11.8652i			1.38871i	0.719631	+	0.694356i	$0.244311\pi$
$73$			11.8652i			1.38871i	−0.719631	+	0.694356i	$0.755689\pi$
$74$		0			0
$75$	−8.44559	−	3.55900i	−0.975212	−	0.410958i
$76$		0			0
$77$			4.40105i			0.501546i
$78$		0			0
$79$	5.48227			0.616804			0.308402	−	0.951256i	$-0.400206\pi$
$79$	5.48227			0.616804			0.308402	+	0.951256i	$0.400206\pi$
$80$		0			0
$81$	−9.94989			−1.10554
$82$		0			0
$83$		−	17.1586i		−	1.88340i	−0.336451	−	0.941701i	$-0.609227\pi$
$83$		−	17.1586i		−	1.88340i	0.336451	−	0.941701i	$-0.390773\pi$
$84$		0			0
$85$	−1.41987	−	2.13913i	−0.154007	−	0.232021i
$86$		0			0
$87$			7.67919i			0.823296i
$88$		0			0
$89$	−4.31591			−0.457485			−0.228743	−	0.973487i	$-0.573461\pi$
$89$	−4.31591			−0.457485			−0.228743	+	0.973487i	$0.573461\pi$
$90$		0			0
$91$	−3.20830			−0.336321
$92$		0			0
$93$			2.51029i			0.260305i
$94$		0			0
$95$	−3.21563	+	2.13441i	−0.329916	+	0.218985i
$96$		0			0
$97$		−	7.65389i		−	0.777135i	−0.921420	−	0.388567i	$-0.872970\pi$
$97$		−	7.65389i		−	0.777135i	0.921420	−	0.388567i	$-0.127030\pi$
$98$		0			0
$99$	1.58340			0.159137
$100$		0			0
$101$	−2.25581			−0.224462			−0.112231	−	0.993682i	$-0.535800\pi$
$101$	−2.25581			−0.224462			−0.112231	+	0.993682i	$0.535800\pi$
$102$		0			0
$103$			6.61262i			0.651560i	0.945446	+	0.325780i	$0.105627\pi$
$103$			6.61262i			0.651560i	−0.945446	+	0.325780i	$0.894373\pi$
$104$		0			0
$105$	−3.41485	+	2.26664i	−0.333255	+	0.221202i
$106$		0			0
$107$		−	5.83756i		−	0.564338i	−0.959365	−	0.282169i	$-0.908946\pi$
$107$		−	5.83756i		−	0.564338i	0.959365	−	0.282169i	$-0.0910539\pi$
$108$		0			0
$109$	10.6952			1.02441			0.512205	−	0.858863i	$-0.328829\pi$
$109$	10.6952			1.02441			0.512205	+	0.858863i	$0.328829\pi$
$110$		0			0
$111$	−7.69705			−0.730572
$112$		0			0
$113$		−	0.867348i		−	0.0815932i	−0.999167	−	0.0407966i	$-0.987010\pi$
$113$		−	0.867348i		−	0.0815932i	0.999167	−	0.0407966i	$-0.0129896\pi$
$114$		0			0
$115$	4.02245	+	6.06009i	0.375095	+	0.565107i
$116$		0			0
$117$			1.15427i			0.106713i
$118$		0			0
$119$	−1.14821			−0.105256
$120$		0			0
$121$	8.36923			0.760839
$122$		0			0
$123$		−	21.5346i		−	1.94171i
$124$		0			0
$125$	−2.08040	−	10.9851i	−0.186076	−	0.982535i
$126$		0			0
$127$		−	14.6158i		−	1.29695i	−0.761238	−	0.648473i	$-0.775408\pi$
$127$		−	14.6158i		−	1.29695i	0.761238	−	0.648473i	$-0.224592\pi$
$128$		0			0
$129$	−4.85731			−0.427662
$130$		0			0
$131$	17.8600			1.56044			0.780218	−	0.625508i	$-0.215108\pi$
$131$	17.8600			1.56044			0.780218	+	0.625508i	$0.215108\pi$
$132$		0			0
$133$			1.72603i			0.149666i
$134$		0			0
$135$	−5.98444	−	9.01597i	−0.515059	−	0.775971i
$136$		0			0
$137$			5.22523i			0.446422i	0.974770	+	0.223211i	$0.0716539\pi$
$137$			5.22523i			0.446422i	−0.974770	+	0.223211i	$0.928346\pi$
$138$		0			0
$139$	5.25800			0.445977			0.222989	−	0.974821i	$-0.428419\pi$
$139$	5.25800			0.445977			0.222989	+	0.974821i	$0.428419\pi$
$140$		0			0
$141$	−0.196012			−0.0165072
$142$		0			0
$143$			14.1199i			1.18076i
$144$		0			0
$145$	−7.80507	+	5.18070i	−0.648176	+	0.430233i
$146$		0			0
$147$			1.83297i			0.151181i
$148$		0			0
$149$	11.8928			0.974294			0.487147	−	0.873320i	$-0.338038\pi$
$149$	11.8928			0.974294			0.487147	+	0.873320i	$0.338038\pi$
$150$		0			0
$151$	−8.54534			−0.695410			−0.347705	−	0.937604i	$-0.613039\pi$
$151$	−8.54534			−0.695410			−0.347705	+	0.937604i	$0.613039\pi$
$152$		0			0
$153$			0.413099i			0.0333971i
$154$		0			0
$155$	−2.55144	+	1.69355i	−0.204937	+	0.136029i
$156$		0			0
$157$			3.36436i			0.268506i	0.990947	+	0.134253i	$0.0428634\pi$
$157$			3.36436i			0.268506i	−0.990947	+	0.134253i	$0.957137\pi$
$158$		0			0
$159$	14.4166			1.14331
$160$		0			0
$161$	3.25284			0.256360
$162$		0			0
$163$			15.8310i			1.23998i	0.784609	+	0.619991i	$0.212864\pi$
$163$			15.8310i			1.23998i	−0.784609	+	0.619991i	$0.787136\pi$
$164$		0			0
$165$	9.97561	+	15.0289i	0.776600	+	1.17000i
$166$		0			0
$167$			20.8843i			1.61608i	0.589128	+	0.808040i	$0.299471\pi$
$167$			20.8843i			1.61608i	−0.589128	+	0.808040i	$0.700529\pi$
$168$		0			0
$169$	2.70680			0.208215
$170$		0			0
$171$	0.620986			0.0474880
$172$		0			0
$173$			5.94735i			0.452168i	0.974108	+	0.226084i	$0.0725924\pi$
$173$			5.94735i			0.452168i	−0.974108	+	0.226084i	$0.927408\pi$
$174$		0			0
$175$	−4.60760	−	1.94166i	−0.348302	−	0.146776i
$176$		0			0
$177$		−	24.3838i		−	1.83280i
$178$		0			0
$179$	22.9041			1.71194			0.855968	−	0.517030i	$-0.172962\pi$
$179$	22.9041			1.71194			0.855968	+	0.517030i	$0.172962\pi$
$180$		0			0
$181$	−1.77746			−0.132118			−0.0660589	−	0.997816i	$-0.521043\pi$
$181$	−1.77746			−0.132118			−0.0660589	+	0.997816i	$0.521043\pi$
$182$		0			0
$183$			18.3708i			1.35801i
$184$		0			0
$185$	−5.19275	−	7.82322i	−0.381778	−	0.575175i
$186$		0			0
$187$			5.05332i			0.369536i
$188$		0			0
$189$	−4.83945			−0.352018
$190$		0			0
$191$	19.7183			1.42676			0.713382	−	0.700775i	$-0.247162\pi$
$191$	19.7183			1.42676			0.713382	+	0.700775i	$0.247162\pi$
$192$		0			0
$193$		−	18.4956i		−	1.33135i	−0.746244	−	0.665673i	$-0.768145\pi$
$193$		−	18.4956i		−	1.33135i	0.746244	−	0.665673i	$-0.231855\pi$
$194$		0			0
$195$	−10.9559	+	7.27208i	−0.784567	+	0.520764i
$196$		0			0
$197$		−	19.4179i		−	1.38347i	−0.722151	−	0.691735i	$-0.756847\pi$
$197$		−	19.4179i		−	1.38347i	0.722151	−	0.691735i	$-0.243153\pi$
$198$		0			0
$199$	21.7485			1.54171			0.770855	−	0.637011i	$-0.219830\pi$
$199$	21.7485			1.54171			0.770855	+	0.637011i	$0.219830\pi$
$200$		0			0
$201$	−2.34702			−0.165546
$202$		0			0
$203$			4.18948i			0.294044i
$204$		0			0
$205$	21.8876	−	14.5281i	1.52870	−	1.01469i
$206$		0			0
$207$		−	1.17030i		−	0.0813412i
$208$		0			0
$209$	7.59635			0.525451
$210$		0			0
$211$	−23.2091			−1.59778			−0.798890	−	0.601477i	$-0.794579\pi$
$211$	−23.2091			−1.59778			−0.798890	+	0.601477i	$0.794579\pi$
$212$		0			0
$213$			25.9577i			1.77860i
$214$		0			0
$215$	−3.27694	−	4.93693i	−0.223485	−	0.336696i
$216$		0			0
$217$			1.36952i			0.0929692i
$218$		0			0
$219$	21.7485			1.46963
$220$		0			0
$221$	−3.68380			−0.247799
$222$		0			0
$223$		−	0.784235i		−	0.0525163i	−0.999655	−	0.0262581i	$-0.991641\pi$
$223$		−	0.784235i		−	0.0525163i	0.999655	−	0.0262581i	$-0.00835919\pi$
$224$		0			0
$225$	−0.698564	+	1.65771i	−0.0465709	+	0.110514i
$226$		0			0
$227$			11.7257i			0.778265i	0.921182	+	0.389132i	$0.127225\pi$
$227$			11.7257i			0.778265i	−0.921182	+	0.389132i	$0.872775\pi$
$228$		0			0
$229$	1.09555			0.0723962			0.0361981	−	0.999345i	$-0.488475\pi$
$229$	1.09555			0.0723962			0.0361981	+	0.999345i	$0.488475\pi$
$230$		0			0
$231$	8.06699			0.530769
$232$		0			0
$233$		−	25.3345i		−	1.65972i	−0.557972	−	0.829860i	$-0.688420\pi$
$233$		−	25.3345i		−	1.65972i	0.557972	−	0.829860i	$-0.311580\pi$
$234$		0			0
$235$	−0.132238	−	0.199225i	−0.00862624	−	0.0129960i
$236$		0			0
$237$		−	10.0488i		−	0.652742i
$238$		0			0
$239$	28.8886			1.86865			0.934323	−	0.356427i	$-0.116005\pi$
$239$	28.8886			1.86865			0.934323	+	0.356427i	$0.116005\pi$
$240$		0			0
$241$	10.2542			0.660529			0.330264	−	0.943888i	$-0.392862\pi$
$241$	10.2542			0.660529			0.330264	+	0.943888i	$0.392862\pi$
$242$		0			0
$243$			3.71950i			0.238606i
$244$		0			0
$245$	−1.86302	+	1.23660i	−0.119024	+	0.0790032i
$246$		0			0
$247$			5.53763i			0.352351i
$248$		0			0
$249$	−31.4512			−1.99314
$250$		0			0
$251$	27.5389			1.73824			0.869120	−	0.494601i	$-0.164686\pi$
$251$	27.5389			1.73824			0.869120	+	0.494601i	$0.164686\pi$
$252$		0			0
$253$		−	14.3159i		−	0.900033i
$254$		0			0
$255$	−3.92096	+	2.60258i	−0.245540	+	0.162980i
$256$		0			0
$257$			12.2636i			0.764983i	0.923959	+	0.382492i	$0.124934\pi$
$257$			12.2636i			0.764983i	−0.923959	+	0.382492i	$0.875066\pi$
$258$		0			0
$259$	−4.19923			−0.260927
$260$		0			0
$261$	1.50728			0.0932982
$262$		0			0
$263$			17.8928i			1.10332i	0.834071	+	0.551658i	$0.186005\pi$
$263$			17.8928i			1.10332i	−0.834071	+	0.551658i	$0.813995\pi$
$264$		0			0
$265$	9.72603	+	14.6529i	0.597465	+	0.900122i
$266$		0			0
$267$			7.91092i			0.484141i
$268$		0			0
$269$	−25.2295			−1.53827			−0.769136	−	0.639085i	$-0.779313\pi$
$269$	−25.2295			−1.53827			−0.769136	+	0.639085i	$0.779313\pi$
$270$		0			0
$271$	−9.65999			−0.586803			−0.293401	−	0.955989i	$-0.594787\pi$
$271$	−9.65999			−0.586803			−0.293401	+	0.955989i	$0.594787\pi$
$272$		0			0
$273$			5.88072i			0.355917i
$274$		0			0
$275$	−8.54534	+	20.2783i	−0.515303	+	1.22283i
$276$		0			0
$277$			23.1970i			1.39378i	0.717180	+	0.696888i	$0.245432\pi$
$277$			23.1970i			1.39378i	−0.717180	+	0.696888i	$0.754568\pi$
$278$		0			0
$279$	0.492722			0.0294985
$280$		0			0
$281$	26.9734			1.60910			0.804550	−	0.593885i	$-0.202407\pi$
$281$	26.9734			1.60910			0.804550	+	0.593885i	$0.202407\pi$
$282$		0			0
$283$		−	22.2119i		−	1.32036i	−0.751106	−	0.660181i	$-0.770480\pi$
$283$		−	22.2119i		−	1.32036i	0.751106	−	0.660181i	$-0.229520\pi$
$284$		0			0
$285$	3.91230	+	5.89415i	0.231745	+	0.349139i
$286$		0			0
$287$		−	11.7485i		−	0.693491i
$288$		0			0
$289$	15.6816			0.922448
$290$		0			0
$291$	−14.0293			−0.822415
$292$		0			0
$293$		−	20.8756i		−	1.21956i	−0.792569	−	0.609782i	$-0.791257\pi$
$293$		−	20.8756i		−	1.21956i	0.792569	−	0.609782i	$-0.208743\pi$
$294$		0			0
$295$	24.7835	−	16.4503i	1.44295	−	0.957774i
$296$		0			0
$297$			21.2986i			1.23587i
$298$		0			0
$299$	10.4361			0.603535
$300$		0			0
$301$	−2.64997			−0.152742
$302$		0			0
$303$			4.13484i			0.237540i
$304$		0			0
$305$	−18.6720	+	12.3937i	−1.06915	+	0.709663i
$306$		0			0
$307$		−	2.97374i		−	0.169720i	−0.996393	−	0.0848601i	$-0.972956\pi$
$307$		−	2.97374i		−	0.169720i	0.996393	−	0.0848601i	$-0.0270443\pi$
$308$		0			0
$309$	12.1207			0.689524
$310$		0			0
$311$	−1.04007			−0.0589770			−0.0294885	−	0.999565i	$-0.509388\pi$
$311$	−1.04007			−0.0589770			−0.0294885	+	0.999565i	$0.509388\pi$
$312$		0			0
$313$			9.12411i			0.515725i	0.966182	+	0.257863i	$0.0830182\pi$
$313$			9.12411i			0.515725i	−0.966182	+	0.257863i	$0.916982\pi$
$314$		0			0
$315$	0.444899	+	0.670270i	0.0250672	+	0.0377654i
$316$		0			0
$317$		−	4.74255i		−	0.266368i	−0.991091	−	0.133184i	$-0.957480\pi$
$317$		−	4.74255i		−	0.266368i	0.991091	−	0.133184i	$-0.0425201\pi$
$318$		0			0
$319$	18.4381			1.03234
$320$		0			0
$321$	−10.7001			−0.597219
$322$		0			0
$323$			1.98185i			0.110273i
$324$		0			0
$325$	−14.7826	−	6.22943i	−0.819989	−	0.345547i
$326$		0			0
$327$		−	19.6039i		−	1.08410i
$328$		0			0
$329$	−0.106937			−0.00589562
$330$		0			0
$331$	−9.38114			−0.515634			−0.257817	−	0.966194i	$-0.583003\pi$
$331$	−9.38114			−0.515634			−0.257817	+	0.966194i	$0.583003\pi$
$332$		0			0
$333$			1.51078i			0.0827904i
$334$		0			0
$335$	−1.58340	−	2.38549i	−0.0865101	−	0.130333i
$336$		0			0
$337$			17.4797i			0.952178i	0.879397	+	0.476089i	$0.157946\pi$
$337$			17.4797i			0.952178i	−0.879397	+	0.476089i	$0.842054\pi$
$338$		0			0
$339$	−1.58982			−0.0863472
$340$		0			0
$341$	6.02733			0.326398
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	11.1080	−	7.37303i	0.598033	−	0.396950i
$346$		0			0
$347$			19.2623i			1.03405i	0.855969	+	0.517027i	$0.172961\pi$
$347$			19.2623i			1.03405i	−0.855969	+	0.517027i	$0.827039\pi$
$348$		0			0
$349$	−28.5730			−1.52948			−0.764740	−	0.644339i	$-0.777132\pi$
$349$	−28.5730			−1.52948			−0.764740	+	0.644339i	$0.777132\pi$
$350$		0			0
$351$	−15.5264			−0.828739
$352$		0			0
$353$			23.8293i			1.26831i	0.773208	+	0.634153i	$0.218651\pi$
$353$			23.8293i			1.26831i	−0.773208	+	0.634153i	$0.781349\pi$
$354$		0			0
$355$	−26.3833	+	17.5122i	−1.40028	+	0.929449i
$356$		0			0
$357$			2.10463i			0.111389i
$358$		0			0
$359$	−2.03194			−0.107242			−0.0536209	−	0.998561i	$-0.517076\pi$
$359$	−2.03194			−0.107242			−0.0536209	+	0.998561i	$0.517076\pi$
$360$		0			0
$361$	−16.0208			−0.843201
$362$		0			0
$363$		−	15.3405i		−	0.805170i
$364$		0			0
$365$	14.6724	+	22.1050i	0.767989	+	1.15703i
$366$		0			0
$367$		−	22.6965i		−	1.18475i	−0.805663	−	0.592374i	$-0.798191\pi$
$367$		−	22.6965i		−	1.18475i	0.805663	−	0.592374i	$-0.201809\pi$
$368$		0			0
$369$	−4.22683			−0.220040
$370$		0			0
$371$	7.86516			0.408339
$372$		0			0
$373$		−	30.2909i		−	1.56841i	−0.620505	−	0.784203i	$-0.713072\pi$
$373$		−	30.2909i		−	1.56841i	0.620505	−	0.784203i	$-0.286928\pi$
$374$		0			0
$375$	−20.1353	+	3.81330i	−1.03978	+	0.196918i
$376$		0			0
$377$			13.4411i			0.692253i
$378$		0			0
$379$	14.9783			0.769385			0.384692	−	0.923045i	$-0.374307\pi$
$379$	14.9783			0.769385			0.384692	+	0.923045i	$0.374307\pi$
$380$		0			0
$381$	−26.7904			−1.37251
$382$		0			0
$383$			5.37413i			0.274605i	0.990529	+	0.137303i	$0.0438433\pi$
$383$			5.37413i			0.274605i	−0.990529	+	0.137303i	$0.956157\pi$
$384$		0			0
$385$	5.44232	+	8.19923i	0.277366	+	0.417871i
$386$		0			0
$387$			0.953397i			0.0484639i
$388$		0			0
$389$	2.87331			0.145683			0.0728413	−	0.997344i	$-0.476793\pi$
$389$	2.87331			0.145683			0.0728413	+	0.997344i	$0.476793\pi$
$390$		0			0
$391$	3.73494			0.188884
$392$		0			0
$393$		−	32.7368i		−	1.65136i
$394$		0			0
$395$	10.2136	−	6.77935i	0.513900	−	0.341106i
$396$		0			0
$397$			24.2887i			1.21901i	0.792781	+	0.609507i	$0.208632\pi$
$397$			24.2887i			1.21901i	−0.792781	+	0.609507i	$0.791368\pi$
$398$		0			0
$399$	3.16376			0.158386
$400$		0			0
$401$	12.7274			0.635575			0.317787	−	0.948162i	$-0.397060\pi$
$401$	12.7274			0.635575			0.317787	+	0.948162i	$0.397060\pi$
$402$		0			0
$403$			4.39384i			0.218873i
$404$		0			0
$405$	−18.5368	+	12.3040i	−0.921101	+	0.611391i
$406$		0			0
$407$			18.4810i			0.916069i
$408$		0			0
$409$	10.0455			0.496717			0.248359	−	0.968668i	$-0.420109\pi$
$409$	10.0455			0.496717			0.248359	+	0.968668i	$0.420109\pi$
$410$		0			0
$411$	9.57769			0.472433
$412$		0			0
$413$		−	13.3029i		−	0.654593i
$414$		0			0
$415$	−21.2183	−	31.9668i	−1.04156	−	1.56919i
$416$		0			0
$417$		−	9.63775i		−	0.471962i
$418$		0			0
$419$	16.2433			0.793539			0.396769	−	0.917918i	$-0.370131\pi$
$419$	16.2433			0.793539			0.396769	+	0.917918i	$0.370131\pi$
$420$		0			0
$421$	−31.8066			−1.55016			−0.775080	−	0.631863i	$-0.782291\pi$
$421$	−31.8066			−1.55016			−0.775080	+	0.631863i	$0.782291\pi$
$422$		0			0
$423$			0.0384734i			0.00187064i
$424$		0			0
$425$	−5.29048	−	2.22943i	−0.256626	−	0.108143i
$426$		0			0
$427$			10.0224i			0.485020i
$428$		0			0
$429$	25.8813			1.24956
$430$		0			0
$431$	18.8177			0.906414			0.453207	−	0.891405i	$-0.350280\pi$
$431$	18.8177			0.906414			0.453207	+	0.891405i	$0.350280\pi$
$432$		0			0
$433$		−	21.5825i		−	1.03719i	−0.855020	−	0.518595i	$-0.826455\pi$
$433$		−	21.5825i		−	1.03719i	0.855020	−	0.518595i	$-0.173545\pi$
$434$		0			0
$435$	9.49606	+	14.3065i	0.455301	+	0.685942i
$436$		0			0
$437$		−	5.61451i		−	0.268578i
$438$		0			0
$439$	−2.69487			−0.128619			−0.0643095	−	0.997930i	$-0.520484\pi$
$439$	−2.69487			−0.128619			−0.0643095	+	0.997930i	$0.520484\pi$
$440$		0			0
$441$	0.359777			0.0171322
$442$		0			0
$443$		−	18.5368i		−	0.880710i	−0.897824	−	0.440355i	$-0.854853\pi$
$443$		−	18.5368i		−	0.880710i	0.897824	−	0.440355i	$-0.145147\pi$
$444$		0			0
$445$	−8.04060	+	5.33703i	−0.381161	+	0.253000i
$446$		0			0
$447$		−	21.7991i		−	1.03106i
$448$		0			0
$449$	12.6816			0.598483			0.299241	−	0.954177i	$-0.403266\pi$
$449$	12.6816			0.598483			0.299241	+	0.954177i	$0.403266\pi$
$450$		0			0
$451$	−51.7056			−2.43472
$452$		0			0
$453$			15.6633i			0.735928i
$454$		0			0
$455$	−5.97712	+	3.96737i	−0.280212	+	0.185993i
$456$		0			0
$457$		−	19.5381i		−	0.913954i	−0.889478	−	0.456977i	$-0.848932\pi$
$457$		−	19.5381i		−	0.913954i	0.889478	−	0.456977i	$-0.151068\pi$
$458$		0			0
$459$	−5.55670			−0.259364
$460$		0			0
$461$	14.9929			0.698291			0.349145	−	0.937069i	$-0.386472\pi$
$461$	14.9929			0.698291			0.349145	+	0.937069i	$0.386472\pi$
$462$		0			0
$463$		−	24.9309i		−	1.15864i	−0.815102	−	0.579318i	$-0.803319\pi$
$463$		−	24.9309i		−	1.15864i	0.815102	−	0.579318i	$-0.196681\pi$
$464$		0			0
$465$	3.10422	+	4.67671i	0.143955	+	0.216877i
$466$		0			0
$467$			30.4350i			1.40836i	0.710020	+	0.704181i	$0.248686\pi$
$467$			30.4350i			1.40836i	−0.710020	+	0.704181i	$0.751314\pi$
$468$		0			0
$469$	−1.28045			−0.0591255
$470$		0			0
$471$	6.16678			0.284150
$472$		0			0
$473$			11.6626i			0.536249i
$474$		0			0
$475$	−3.35137	+	7.95286i	−0.153771	+	0.364902i
$476$		0			0
$477$		−	2.82970i		−	0.129563i
$478$		0			0
$479$	−17.9694			−0.821041			−0.410521	−	0.911851i	$-0.634653\pi$
$479$	−17.9694			−0.821041			−0.410521	+	0.911851i	$0.634653\pi$
$480$		0			0
$481$	−13.4724			−0.614288
$482$		0			0
$483$		−	5.96236i		−	0.271297i
$484$		0			0
$485$	−9.46477	−	14.2593i	−0.429773	−	0.647482i
$486$		0			0
$487$		−	9.51295i		−	0.431073i	−0.976496	−	0.215536i	$-0.930850\pi$
$487$		−	9.51295i		−	0.431073i	0.976496	−	0.215536i	$-0.0691500\pi$
$488$		0			0
$489$	29.0178			1.31223
$490$		0			0
$491$	−21.0466			−0.949822			−0.474911	−	0.880034i	$-0.657520\pi$
$491$	−21.0466			−0.949822			−0.474911	+	0.880034i	$0.657520\pi$
$492$		0			0
$493$			4.81040i			0.216649i
$494$		0			0
$495$	2.94989	−	1.95802i	0.132588	−	0.0880065i
$496$		0			0
$497$			14.1616i			0.635234i
$498$		0			0
$499$	−36.4408			−1.63131			−0.815656	−	0.578537i	$-0.803624\pi$
$499$	−36.4408			−1.63131			−0.815656	+	0.578537i	$0.803624\pi$
$500$		0			0
$501$	38.2804			1.71024
$502$		0			0
$503$			32.1783i			1.43476i	0.696681	+	0.717381i	$0.254659\pi$
$503$			32.1783i			1.43476i	−0.696681	+	0.717381i	$0.745341\pi$
$504$		0			0
$505$	−4.20262	+	2.78953i	−0.187014	+	0.124132i
$506$		0			0
$507$		−	4.96147i		−	0.220347i
$508$		0			0
$509$	−2.06604			−0.0915756			−0.0457878	−	0.998951i	$-0.514580\pi$
$509$	−2.06604			−0.0915756			−0.0457878	+	0.998951i	$0.514580\pi$
$510$		0			0
$511$	11.8652			0.524884
$512$		0			0
$513$			8.35305i			0.368796i
$514$		0			0
$515$	8.17714	+	12.3194i	0.360328	+	0.542858i
$516$		0			0
$517$			0.470634i			0.0206985i
$518$		0			0
$519$	10.9013			0.478514
$520$		0			0
$521$	−23.0609			−1.01032			−0.505158	−	0.863027i	$-0.668566\pi$
$521$	−23.0609			−1.01032			−0.505158	+	0.863027i	$0.668566\pi$
$522$		0			0
$523$		−	26.1602i		−	1.14391i	−0.820286	−	0.571953i	$-0.806186\pi$
$523$		−	26.1602i		−	1.14391i	0.820286	−	0.571953i	$-0.193814\pi$
$524$		0			0
$525$	−3.55900	+	8.44559i	−0.155328	+	0.368596i
$526$		0			0
$527$			1.57250i			0.0684990i
$528$		0			0
$529$	12.4190			0.539958
$530$		0			0
$531$	−4.78607			−0.207698
$532$		0			0
$533$		−	37.6927i		−	1.63265i
$534$		0			0
$535$	−7.21870	−	10.8755i	−0.312092	−	0.470187i
$536$		0			0
$537$		−	41.9826i		−	1.81168i
$538$		0			0
$539$	4.40105			0.189567
$540$		0			0
$541$	41.2094			1.77173			0.885866	−	0.463941i	$-0.153565\pi$
$541$	41.2094			1.77173			0.885866	+	0.463941i	$0.153565\pi$
$542$		0			0
$543$			3.25804i			0.139816i
$544$		0			0
$545$	19.9253	−	13.2256i	0.853504	−	0.566522i
$546$		0			0
$547$		−	7.67055i		−	0.327969i	−0.986463	−	0.163985i	$-0.947565\pi$
$547$		−	7.67055i		−	0.327969i	0.986463	−	0.163985i	$-0.0524347\pi$
$548$		0			0
$549$	3.60585			0.153894
$550$		0			0
$551$	7.23118			0.308059
$552$		0			0
$553$		−	5.48227i		−	0.233130i
$554$		0			0
$555$	−14.3397	+	9.51814i	−0.608688	+	0.404023i
$556$		0			0
$557$			31.8600i			1.34995i	0.737841	+	0.674975i	$0.235846\pi$
$557$			31.8600i			1.34995i	−0.737841	+	0.674975i	$0.764154\pi$
$558$		0			0
$559$	−8.50190			−0.359592
$560$		0			0
$561$	9.26258			0.391067
$562$		0			0
$563$			9.20351i			0.387882i	0.981013	+	0.193941i	$0.0621270\pi$
$563$			9.20351i			0.387882i	−0.981013	+	0.193941i	$0.937873\pi$
$564$		0			0
$565$	−1.07256	−	1.61588i	−0.0451229	−	0.0679807i
$566$		0			0
$567$			9.94989i			0.417856i
$568$		0			0
$569$	−8.47615			−0.355339			−0.177669	−	0.984090i	$-0.556856\pi$
$569$	−8.47615			−0.355339			−0.177669	+	0.984090i	$0.556856\pi$
$570$		0			0
$571$	−13.1245			−0.549245			−0.274622	−	0.961552i	$-0.588553\pi$
$571$	−13.1245			−0.549245			−0.274622	+	0.961552i	$0.588553\pi$
$572$		0			0
$573$		−	36.1430i		−	1.50990i
$574$		0			0
$575$	14.9878	+	6.31591i	0.625034	+	0.263392i
$576$		0			0
$577$			16.5640i			0.689567i	0.938682	+	0.344783i	$0.112048\pi$
$577$			16.5640i			0.689567i	−0.938682	+	0.344783i	$0.887952\pi$
$578$		0			0
$579$	−33.9019			−1.40892
$580$		0			0
$581$	−17.1586			−0.711859
$582$		0			0
$583$		−	34.6150i		−	1.43361i
$584$		0			0
$585$	1.42737	+	2.15043i	0.0590145	+	0.0889093i
$586$		0			0
$587$			0.890659i			0.0367614i	0.999831	+	0.0183807i	$0.00585109\pi$
$587$			0.890659i			0.0367614i	−0.999831	+	0.0183807i	$0.994149\pi$
$588$		0			0
$589$	2.36384			0.0974003
$590$		0			0
$591$	−35.5925			−1.46408
$592$		0			0
$593$		−	8.12654i		−	0.333717i	−0.985981	−	0.166858i	$-0.946638\pi$
$593$		−	8.12654i		−	0.333717i	0.985981	−	0.166858i	$-0.0533623\pi$
$594$		0			0
$595$	−2.13913	+	1.41987i	−0.0876958	+	0.0582090i
$596$		0			0
$597$		−	39.8643i		−	1.63154i
$598$		0			0
$599$	9.38288			0.383374			0.191687	−	0.981456i	$-0.438604\pi$
$599$	9.38288			0.383374			0.191687	+	0.981456i	$0.438604\pi$
$600$		0			0
$601$	−5.44289			−0.222020			−0.111010	−	0.993819i	$-0.535409\pi$
$601$	−5.44289			−0.222020			−0.111010	+	0.993819i	$0.535409\pi$
$602$		0			0
$603$			0.460675i			0.0187601i
$604$		0			0
$605$	15.5920	−	10.3494i	0.633905	−	0.420761i
$606$		0			0
$607$		−	36.7479i		−	1.49155i	−0.666197	−	0.745776i	$-0.732079\pi$
$607$		−	36.7479i		−	1.49155i	0.666197	−	0.745776i	$-0.267921\pi$
$608$		0			0
$609$	7.67919			0.311176
$610$		0			0
$611$	−0.343086			−0.0138798
$612$		0			0
$613$		−	16.8238i		−	0.679505i	−0.940515	−	0.339753i	$-0.889657\pi$
$613$		−	16.8238i		−	0.679505i	0.940515	−	0.339753i	$-0.110343\pi$
$614$		0			0
$615$	−26.6296	−	40.1193i	−1.07381	−	1.61777i
$616$		0			0
$617$			5.67406i			0.228429i	0.993456	+	0.114214i	$0.0364351\pi$
$617$			5.67406i			0.228429i	−0.993456	+	0.114214i	$0.963565\pi$
$618$		0			0
$619$	−4.33702			−0.174320			−0.0871598	−	0.996194i	$-0.527779\pi$
$619$	−4.33702			−0.174320			−0.0871598	+	0.996194i	$0.527779\pi$
$620$		0			0
$621$	15.7420			0.631703
$622$		0			0
$623$			4.31591i			0.172913i
$624$		0			0
$625$	−17.4599	−	17.8928i	−0.698397	−	0.715711i
$626$		0			0
$627$		−	13.9239i		−	0.556066i
$628$		0			0
$629$	−4.82159			−0.192249
$630$		0			0
$631$	−41.9784			−1.67113			−0.835567	−	0.549389i	$-0.814860\pi$
$631$	−41.9784			−1.67113			−0.835567	+	0.549389i	$0.814860\pi$
$632$		0			0
$633$			42.5416i			1.69088i
$634$		0			0
$635$	−18.0739	−	27.2295i	−0.717240	−	1.08057i
$636$		0			0
$637$			3.20830i			0.127118i
$638$		0			0
$639$	5.09501			0.201555
$640$		0			0
$641$	−40.4035			−1.59584			−0.797922	−	0.602761i	$-0.794067\pi$
$641$	−40.4035			−1.59584			−0.797922	+	0.602761i	$0.794067\pi$
$642$		0			0
$643$			28.7826i			1.13507i	0.823348	+	0.567537i	$0.192104\pi$
$643$			28.7826i			1.13507i	−0.823348	+	0.567537i	$0.807896\pi$
$644$		0			0
$645$	−9.04925	+	6.00653i	−0.356314	+	0.236507i
$646$		0			0
$647$		−	50.0475i		−	1.96757i	−0.179346	−	0.983786i	$-0.557398\pi$
$647$		−	50.0475i		−	1.96757i	0.179346	−	0.983786i	$-0.442602\pi$
$648$		0			0
$649$	−58.5467			−2.29816
$650$		0			0
$651$	2.51029			0.0983861
$652$		0			0
$653$			39.5622i			1.54819i	0.633070	+	0.774095i	$0.281795\pi$
$653$			39.5622i			1.54819i	−0.633070	+	0.774095i	$0.718205\pi$
$654$		0			0
$655$	33.2735	−	22.0856i	1.30010	−	0.862957i
$656$		0			0
$657$		−	4.26881i		−	0.166542i
$658$		0			0
$659$	10.0994			0.393418			0.196709	−	0.980462i	$-0.436975\pi$
$659$	10.0994			0.393418			0.196709	+	0.980462i	$0.436975\pi$
$660$		0			0
$661$	−4.51434			−0.175588			−0.0877938	−	0.996139i	$-0.527982\pi$
$661$	−4.51434			−0.175588			−0.0877938	+	0.996139i	$0.527982\pi$
$662$		0			0
$663$			6.75229i			0.262237i
$664$		0			0
$665$	2.13441	+	3.21563i	0.0827687	+	0.124697i
$666$		0			0
$667$		−	13.6277i		−	0.527667i
$668$		0			0
$669$	−1.43748			−0.0555762
$670$		0			0
$671$	44.1093			1.70282
$672$		0			0
$673$		−	3.62104i		−	0.139581i	−0.997562	−	0.0697904i	$-0.977767\pi$
$673$		−	3.62104i		−	0.139581i	0.997562	−	0.0697904i	$-0.0222330\pi$
$674$		0			0
$675$	−22.2982	−	9.39656i	−0.858259	−	0.361674i
$676$		0			0
$677$			35.1617i			1.35137i	0.737189	+	0.675687i	$0.236153\pi$
$677$			35.1617i			1.35137i	−0.737189	+	0.675687i	$0.763847\pi$
$678$		0			0
$679$	−7.65389			−0.293729
$680$		0			0
$681$	21.4929			0.823611
$682$		0			0
$683$			4.39058i			0.168001i	0.996466	+	0.0840005i	$0.0267697\pi$
$683$			4.39058i			0.168001i	−0.996466	+	0.0840005i	$0.973230\pi$
$684$		0			0
$685$	6.46150	+	9.73470i	0.246881	+	0.371944i
$686$		0			0
$687$		−	2.00812i		−	0.0766144i
$688$		0			0
$689$	25.2338			0.961332
$690$		0			0
$691$	6.28314			0.239022			0.119511	−	0.992833i	$-0.461867\pi$
$691$	6.28314			0.239022			0.119511	+	0.992833i	$0.461867\pi$
$692$		0			0
$693$		−	1.58340i		−	0.0601482i
$694$		0			0
$695$	9.79573	−	6.50202i	0.371573	−	0.246636i
$696$		0			0
$697$		−	13.4897i		−	0.510959i
$698$		0			0
$699$	−46.4374			−1.75642
$700$		0			0
$701$	5.41413			0.204489			0.102244	−	0.994759i	$-0.467398\pi$
$701$	5.41413			0.204489			0.102244	+	0.994759i	$0.467398\pi$
$702$		0			0
$703$			7.24800i			0.273363i
$704$		0			0
$705$	−0.365174	+	0.242388i	−0.0137532	+	0.00912885i
$706$		0			0
$707$			2.25581i			0.0848386i
$708$		0			0
$709$	−15.5284			−0.583180			−0.291590	−	0.956543i	$-0.594184\pi$
$709$	−15.5284			−0.583180			−0.291590	+	0.956543i	$0.594184\pi$
$710$		0			0
$711$	−1.97239			−0.0739705
$712$		0			0
$713$		−	4.45483i		−	0.166835i
$714$		0			0
$715$	17.4606	+	26.3056i	0.652990	+	0.983773i
$716$		0			0
$717$		−	52.9519i		−	1.97752i
$718$		0			0
$719$	−19.9304			−0.743279			−0.371640	−	0.928377i	$-0.621204\pi$
$719$	−19.9304			−0.743279			−0.371640	+	0.928377i	$0.621204\pi$
$720$		0			0
$721$	6.61262			0.246267
$722$		0			0
$723$		−	18.7956i		−	0.699015i
$724$		0			0
$725$	−8.13454	+	19.3034i	−0.302109	+	0.716912i
$726$		0			0
$727$			24.0839i			0.893222i	0.894728	+	0.446611i	$0.147369\pi$
$727$			24.0839i			0.893222i	−0.894728	+	0.446611i	$0.852631\pi$
$728$		0			0
$729$	−23.0319			−0.853035
$730$		0			0
$731$	−3.04272			−0.112539
$732$		0			0
$733$			35.8771i			1.32515i	0.748994	+	0.662576i	$0.230537\pi$
$733$			35.8771i			1.32515i	−0.748994	+	0.662576i	$0.769463\pi$
$734$		0			0
$735$	2.26664	+	3.41485i	0.0836064	+	0.125959i
$736$		0			0
$737$			5.63531i			0.207579i
$738$		0			0
$739$	−1.29734			−0.0477233			−0.0238617	−	0.999715i	$-0.507596\pi$
$739$	−1.29734			−0.0477233			−0.0238617	+	0.999715i	$0.507596\pi$
$740$		0			0
$741$	10.1503			0.372881
$742$		0			0
$743$			8.53790i			0.313225i	0.987660	+	0.156613i	$0.0500574\pi$
$743$			8.53790i			0.313225i	−0.987660	+	0.156613i	$0.949943\pi$
$744$		0			0
$745$	22.1564	−	14.7066i	0.811749	−	0.538807i
$746$		0			0
$747$			6.17327i			0.225868i
$748$		0			0
$749$	−5.83756			−0.213300
$750$		0			0
$751$	−17.5930			−0.641978			−0.320989	−	0.947083i	$-0.604015\pi$
$751$	−17.5930			−0.641978			−0.320989	+	0.947083i	$0.604015\pi$
$752$		0			0
$753$		−	50.4780i		−	1.83952i
$754$		0			0
$755$	−15.9201	+	10.5671i	−0.579392	+	0.384577i
$756$		0			0
$757$			33.5195i			1.21829i	0.793061	+	0.609143i	$0.208486\pi$
$757$			33.5195i			1.21829i	−0.793061	+	0.609143i	$0.791514\pi$
$758$		0			0
$759$	−26.2406			−0.952474
$760$		0			0
$761$	−25.8063			−0.935479			−0.467740	−	0.883866i	$-0.654931\pi$
$761$	−25.8063			−0.935479			−0.467740	+	0.883866i	$0.654931\pi$
$762$		0			0
$763$		−	10.6952i		−	0.387191i
$764$		0			0
$765$	0.510837	+	0.769610i	0.0184693	+	0.0278253i
$766$		0			0
$767$		−	42.6797i		−	1.54108i
$768$		0			0
$769$	27.6616			0.997502			0.498751	−	0.866745i	$-0.333792\pi$
$769$	27.6616			0.997502			0.498751	+	0.866745i	$0.333792\pi$
$770$		0			0
$771$	22.4788			0.809555
$772$		0			0
$773$			49.4294i			1.77785i	0.458050	+	0.888927i	$0.348548\pi$
$773$			49.4294i			1.77785i	−0.458050	+	0.888927i	$0.651452\pi$
$774$		0			0
$775$	−2.65914	+	6.31020i	−0.0955193	+	0.226669i
$776$		0			0
$777$			7.69705i			0.276130i
$778$		0			0
$779$	−20.2783			−0.726544
$780$		0			0
$781$	62.3258			2.23019
$782$		0			0
$783$			20.2748i			0.724561i
$784$		0			0
$785$	4.16036	+	6.26787i	0.148490	+	0.223710i
$786$		0			0
$787$			44.2790i			1.57837i	0.614153	+	0.789187i	$0.289498\pi$
$787$			44.2790i			1.57837i	−0.614153	+	0.789187i	$0.710502\pi$
$788$		0			0
$789$	32.7969			1.16760
$790$		0			0
$791$	−0.867348			−0.0308393
$792$		0			0
$793$			32.1550i			1.14186i
$794$		0			0
$795$	26.8584	−	17.8275i	0.952568	−	0.632277i
$796$		0			0
$797$		−	17.7589i		−	0.629052i	−0.949249	−	0.314526i	$-0.898154\pi$
$797$		−	17.7589i		−	0.629052i	0.949249	−	0.314526i	$-0.101846\pi$
$798$		0			0
$799$	−0.122786			−0.00434385
$800$		0			0
$801$	1.55276			0.0548642
$802$		0			0
$803$		−	52.2192i		−	1.84277i
$804$		0			0
$805$	6.06009	−	4.02245i	0.213590	−	0.141773i
$806$		0			0
$807$			46.2450i			1.62790i
$808$		0			0
$809$	−31.2427			−1.09844			−0.549218	−	0.835679i	$-0.685074\pi$
$809$	−31.2427			−1.09844			−0.549218	+	0.835679i	$0.685074\pi$
$810$		0			0
$811$	6.66672			0.234100			0.117050	−	0.993126i	$-0.462656\pi$
$811$	6.66672			0.234100			0.117050	+	0.993126i	$0.462656\pi$
$812$		0			0
$813$			17.7065i			0.620993i
$814$		0			0
$815$	19.5766	+	29.4935i	0.685738	+	1.03311i
$816$		0			0
$817$			4.57393i			0.160022i
$818$		0			0
$819$	1.15427			0.0403336
$820$		0			0
$821$	−43.2138			−1.50817			−0.754086	−	0.656776i	$-0.771920\pi$
$821$	−43.2138			−1.50817			−0.754086	+	0.656776i	$0.771920\pi$
$822$		0			0
$823$			17.8129i			0.620918i	0.950587	+	0.310459i	$0.100483\pi$
$823$			17.8129i			0.620918i	−0.950587	+	0.310459i	$0.899517\pi$
$824$		0			0
$825$	37.1694	+	15.6633i	1.29407	+	0.545328i
$826$		0			0
$827$			18.9139i			0.657702i	0.944382	+	0.328851i	$0.106661\pi$
$827$			18.9139i			0.657702i	−0.944382	+	0.328851i	$0.893339\pi$
$828$		0			0
$829$	−37.5969			−1.30579			−0.652897	−	0.757447i	$-0.726446\pi$
$829$	−37.5969			−1.30579			−0.652897	+	0.757447i	$0.726446\pi$
$830$		0			0
$831$	42.5195			1.47498
$832$		0			0
$833$			1.14821i			0.0397831i
$834$		0			0
$835$	25.8255	+	38.9079i	0.893729	+	1.34646i
$836$		0			0
$837$			6.62773i			0.229088i
$838$		0			0
$839$	32.2054			1.11185			0.555927	−	0.831231i	$-0.312364\pi$
$839$	32.2054			1.11185			0.555927	+	0.831231i	$0.312364\pi$
$840$		0			0
$841$	−11.4483			−0.394767
$842$		0			0
$843$		−	49.4415i		−	1.70285i
$844$		0			0
$845$	5.04280	−	3.34721i	0.173478	−	0.115148i
$846$		0			0
$847$		−	8.36923i		−	0.287570i
$848$		0			0
$849$	−40.7138			−1.39729
$850$		0			0
$851$	13.6594			0.468238
$852$		0			0
$853$			20.6540i			0.707179i	0.935401	+	0.353590i	$0.115039\pi$
$853$			20.6540i			0.707179i	−0.935401	+	0.353590i	$0.884961\pi$
$854$		0			0
$855$	1.15691	−	0.767910i	0.0395654	−	0.0262620i
$856$		0			0
$857$		−	42.6561i		−	1.45711i	−0.684990	−	0.728553i	$-0.740193\pi$
$857$		−	42.6561i		−	1.45711i	0.684990	−	0.728553i	$-0.259807\pi$
$858$		0			0
$859$	−7.95286			−0.271348			−0.135674	−	0.990754i	$-0.543320\pi$
$859$	−7.95286			−0.271348			−0.135674	+	0.990754i	$0.543320\pi$
$860$		0			0
$861$	−21.5346			−0.733897
$862$		0			0
$863$			29.0419i			0.988597i	0.869292	+	0.494299i	$0.164575\pi$
$863$			29.0419i			0.988597i	−0.869292	+	0.494299i	$0.835425\pi$
$864$		0			0
$865$	7.35447	+	11.0800i	0.250059	+	0.376731i
$866$		0			0
$867$		−	28.7439i		−	0.976195i
$868$		0			0
$869$	−24.1277			−0.818477
$870$		0			0
$871$	−4.10806			−0.139196
$872$		0			0
$873$			2.75369i			0.0931983i
$874$		0			0
$875$	−10.9851	+	2.08040i	−0.371363	+	0.0703302i
$876$		0			0
$877$			15.1827i			0.512682i	0.966586	+	0.256341i	$0.0825170\pi$
$877$			15.1827i			0.512682i	−0.966586	+	0.256341i	$0.917483\pi$
$878$		0			0
$879$	−38.2643			−1.29062
$880$		0			0
$881$	−17.3877			−0.585806			−0.292903	−	0.956142i	$-0.594621\pi$
$881$	−17.3877			−0.585806			−0.292903	+	0.956142i	$0.594621\pi$
$882$		0			0
$883$		−	8.46587i		−	0.284899i	−0.989802	−	0.142450i	$-0.954502\pi$
$883$		−	8.46587i		−	0.284899i	0.989802	−	0.142450i	$-0.0454979\pi$
$884$		0			0
$885$	−30.1529	−	45.4274i	−1.01358	−	1.52703i
$886$		0			0
$887$			32.8229i			1.10208i	0.834477	+	0.551042i	$0.185770\pi$
$887$			32.8229i			1.10208i	−0.834477	+	0.551042i	$0.814230\pi$
$888$		0			0
$889$	−14.6158			−0.490199
$890$		0			0
$891$	43.7900			1.46702
$892$		0			0
$893$			0.184577i			0.00617662i
$894$		0			0
$895$	42.6708	−	28.3232i	1.42633	−	0.946739i
$896$		0			0
$897$		−	19.1290i		−	0.638700i
$898$		0			0
$899$	5.73758			0.191359
$900$		0			0
$901$	9.03085			0.300861
$902$		0			0
$903$			4.85731i			0.161641i
$904$		0			0
$905$	−3.31144	+	2.19801i	−0.110076	+	0.0730642i
$906$		0			0
$907$			54.6260i			1.81383i	0.421317	+	0.906913i	$0.361568\pi$
$907$			54.6260i			1.81383i	−0.421317	+	0.906913i	$0.638432\pi$
$908$		0			0
$909$	0.811589			0.0269187
$910$		0			0
$911$	−4.21329			−0.139592			−0.0697962	−	0.997561i	$-0.522235\pi$
$911$	−4.21329			−0.139592			−0.0697962	+	0.997561i	$0.522235\pi$
$912$		0			0
$913$			75.5159i			2.49921i
$914$		0			0
$915$	22.7173	+	34.2252i	0.751011	+	1.13145i
$916$		0			0
$917$		−	17.8600i		−	0.589789i
$918$		0			0
$919$	−25.0033			−0.824784			−0.412392	−	0.911007i	$-0.635307\pi$
$919$	−25.0033			−0.824784			−0.412392	+	0.911007i	$0.635307\pi$
$920$		0			0
$921$	−5.45077			−0.179609
$922$		0			0
$923$			45.4346i			1.49550i
$924$		0			0
$925$	−19.3483	−	8.15346i	−0.636170	−	0.268084i
$926$		0			0
$927$		−	2.37907i		−	0.0781388i
$928$		0			0
$929$	34.5462			1.13342			0.566712	−	0.823916i	$-0.308215\pi$
$929$	34.5462			1.13342			0.566712	+	0.823916i	$0.308215\pi$
$930$		0			0
$931$	1.72603			0.0565684
$932$		0			0
$933$			1.90642i			0.0624134i
$934$		0			0
$935$	6.24892	+	9.41442i	0.204362	+	0.307884i
$936$		0			0
$937$			19.7689i			0.645821i	0.946429	+	0.322911i	$0.104661\pi$
$937$			19.7689i			0.645821i	−0.946429	+	0.322911i	$0.895339\pi$
$938$		0			0
$939$	16.7242			0.545774
$940$		0			0
$941$	23.0820			0.752452			0.376226	−	0.926528i	$-0.377222\pi$
$941$	23.0820			0.752452			0.376226	+	0.926528i	$0.377222\pi$
$942$		0			0
$943$			38.2159i			1.24448i
$944$		0			0
$945$	−9.01597	+	5.98444i	−0.293290	+	0.194674i
$946$		0			0
$947$		−	60.0386i		−	1.95099i	−0.220019	−	0.975496i	$-0.570612\pi$
$947$		−	60.0386i		−	1.95099i	0.220019	−	0.975496i	$-0.429388\pi$
$948$		0			0
$949$	38.0670			1.23571
$950$		0			0
$951$	−8.69295			−0.281888
$952$		0			0
$953$		−	11.4188i		−	0.369891i	−0.982749	−	0.184945i	$-0.940789\pi$
$953$		−	11.4188i		−	0.369891i	0.982749	−	0.184945i	$-0.0592108\pi$
$954$		0			0
$955$	36.7355	−	24.3835i	1.18873	−	0.789033i
$956$		0			0
$957$		−	33.7965i		−	1.09249i
$958$		0			0
$959$	5.22523			0.168732
$960$		0			0
$961$	−29.1244			−0.939497
$962$		0			0
$963$			2.10022i			0.0676786i
$964$		0			0
$965$	−22.8716	−	34.4577i	−0.736264	−	1.10923i
$966$		0			0
$967$			28.9169i			0.929903i	0.885336	+	0.464952i	$0.153928\pi$
$967$			28.9169i			0.929903i	−0.885336	+	0.464952i	$0.846072\pi$
$968$		0			0
$969$	3.63266			0.116698
$970$		0			0
$971$	43.1410			1.38446			0.692230	−	0.721677i	$-0.256629\pi$
$971$	43.1410			1.38446			0.692230	+	0.721677i	$0.256629\pi$
$972$		0			0
$973$		−	5.25800i		−	0.168564i
$974$		0			0
$975$	−11.4184	+	27.0960i	−0.365680	+	0.867766i
$976$		0			0
$977$			38.0221i			1.21644i	0.793770	+	0.608218i	$0.208115\pi$
$977$			38.0221i			1.21644i	−0.793770	+	0.608218i	$0.791885\pi$
$978$		0			0
$979$	18.9945			0.607067
$980$		0			0
$981$	−3.84787			−0.122853
$982$		0			0
$983$			12.7030i			0.405164i	0.979265	+	0.202582i	$0.0649332\pi$
$983$			12.7030i			0.405164i	−0.979265	+	0.202582i	$0.935067\pi$
$984$		0			0
$985$	−24.0121	−	36.1759i	−0.765090	−	1.15266i
$986$		0			0
$987$			0.196012i			0.00623913i
$988$		0			0
$989$	8.61992			0.274098
$990$		0			0
$991$	−25.4382			−0.808073			−0.404036	−	0.914743i	$-0.632393\pi$
$991$	−25.4382			−0.808073			−0.404036	+	0.914743i	$0.632393\pi$
$992$		0			0
$993$			17.1954i			0.545678i
$994$		0			0
$995$	40.5178	−	26.8941i	1.28450	−	0.852600i
$996$		0			0
$997$		−	13.5489i		−	0.429098i	−0.976713	−	0.214549i	$-0.931172\pi$
$997$		−	13.5489i		−	0.429098i	0.976713	−	0.214549i	$-0.0688281\pi$
$998$		0			0
$999$	−20.3219			−0.642958

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.g.b.449.3	✓	10
4.3	odd	2		1120.2.g.c.449.8	yes	10
5.2	odd	4		5600.2.a.bu.1.2		5
5.3	odd	4		5600.2.a.bw.1.4		5
5.4	even	2	inner	1120.2.g.b.449.8	yes	10
8.3	odd	2		2240.2.g.n.449.3		10
8.5	even	2		2240.2.g.o.449.8		10
20.3	even	4		5600.2.a.bv.1.2		5
20.7	even	4		5600.2.a.bx.1.4		5
20.19	odd	2		1120.2.g.c.449.3	yes	10
40.19	odd	2		2240.2.g.n.449.8		10
40.29	even	2		2240.2.g.o.449.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.b.449.3	✓	10	1.1	even	1	trivial
1120.2.g.b.449.8	yes	10	5.4	even	2	inner
1120.2.g.c.449.3	yes	10	20.19	odd	2
1120.2.g.c.449.8	yes	10	4.3	odd	2
2240.2.g.n.449.3		10	8.3	odd	2
2240.2.g.n.449.8		10	40.19	odd	2
2240.2.g.o.449.3		10	40.29	even	2
2240.2.g.o.449.8		10	8.5	even	2
5600.2.a.bu.1.2		5	5.2	odd	4
5600.2.a.bv.1.2		5	20.3	even	4
5600.2.a.bw.1.4		5	5.3	odd	4
5600.2.a.bx.1.4		5	20.7	even	4

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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.83297i q^{3} +(1.86302 - 1.23660i) q^{5} -1.00000i q^{7} -0.359777 q^{9} +O(q^{10})\)	\(q-1.83297i q^{3} +(1.86302 - 1.23660i) q^{5} -1.00000i q^{7} -0.359777 q^{9} -4.40105 q^{11} -3.20830i q^{13} +(-2.26664 - 3.41485i) q^{15} -1.14821i q^{17} -1.72603 q^{19} -1.83297 q^{21} +3.25284i q^{23} +(1.94166 - 4.60760i) q^{25} -4.83945i q^{27} -4.18948 q^{29} -1.36952 q^{31} +8.06699i q^{33} +(-1.23660 - 1.86302i) q^{35} -4.19923i q^{37} -5.88072 q^{39} +11.7485 q^{41} -2.64997i q^{43} +(-0.670270 + 0.444899i) q^{45} -0.106937i q^{47} -1.00000 q^{49} -2.10463 q^{51} +7.86516i q^{53} +(-8.19923 + 5.44232i) q^{55} +3.16376i q^{57} +13.3029 q^{59} -10.0224 q^{61} +0.359777i q^{63} +(-3.96737 - 5.97712i) q^{65} -1.28045i q^{67} +5.96236 q^{69} -14.1616 q^{71} +11.8652i q^{73} +(-8.44559 - 3.55900i) q^{75} +4.40105i q^{77} +5.48227 q^{79} -9.94989 q^{81} -17.1586i q^{83} +(-1.41987 - 2.13913i) q^{85} +7.67919i q^{87} -4.31591 q^{89} -3.20830 q^{91} +2.51029i q^{93} +(-3.21563 + 2.13441i) q^{95} -7.65389i q^{97} +1.58340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 14 q^{9}+O(q^{10}) \) 10 * q - 2 * q^5 - 14 * q^9	\( 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65} - 8 q^{69} - 8 q^{71} - 64 q^{75} + 64 q^{79} + 2 q^{81} - 12 q^{85} - 4 q^{89} - 60 q^{95} + 80 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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