LMFDB - Embedded newform 560.2.k.a.111.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,2,Mod(111,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("560.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	560.k (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:	$4.47162251319$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.2
Root			$-1.26217 - 1.18614i$ of defining polynomial
Character	$\chi$	$=$	560.111
Dual form			560.2.k.a.111.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52434 q^{3} +1.00000i q^{5} +(-2.52434 - 0.792287i) q^{7} +3.37228 q^{9} +O(q^{10})$	$q-2.52434 q^{3} +1.00000i q^{5} +(-2.52434 - 0.792287i) q^{7} +3.37228 q^{9} +2.52434i q^{11} -0.372281i q^{13} -2.52434i q^{15} -2.37228i q^{17} +3.46410 q^{19} +(6.37228 + 2.00000i) q^{21} -8.51278i q^{23} -1.00000 q^{25} -0.939764 q^{27} +0.372281 q^{29} +1.87953 q^{31} -6.37228i q^{33} +(0.792287 - 2.52434i) q^{35} +10.7446 q^{37} +0.939764i q^{39} -8.74456i q^{41} -3.46410i q^{43} +3.37228i q^{45} +7.86797 q^{47} +(5.74456 + 4.00000i) q^{49} +5.98844i q^{51} -11.4891 q^{53} -2.52434 q^{55} -8.74456 q^{57} +6.63325 q^{59} -10.7446i q^{61} +(-8.51278 - 2.67181i) q^{63} +0.372281 q^{65} +6.63325i q^{67} +21.4891i q^{69} +6.63325i q^{71} -8.74456i q^{73} +2.52434 q^{75} +(2.00000 - 6.37228i) q^{77} -15.7908i q^{79} -7.74456 q^{81} -10.3923 q^{83} +2.37228 q^{85} -0.939764 q^{87} +5.48913i q^{89} +(-0.294954 + 0.939764i) q^{91} -4.74456 q^{93} +3.46410i q^{95} +15.1168i q^{97} +8.51278i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} + 28 q^{21} - 8 q^{25} - 20 q^{29} + 40 q^{37} - 24 q^{57} - 20 q^{65} + 16 q^{77} - 16 q^{81} - 4 q^{85} + 8 q^{93}+O(q^{100}) $ 8 * q + 4 * q^9 + 28 * q^21 - 8 * q^25 - 20 * q^29 + 40 * q^37 - 24 * q^57 - 20 * q^65 + 16 * q^77 - 16 * q^81 - 4 * q^85 + 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−2.52434			−1.45743			−0.728714	−	0.684819i	$-0.759881\pi$
$3$	−2.52434			−1.45743			−0.728714	+	0.684819i	$0.759881\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−2.52434	−	0.792287i	−0.954110	−	0.299456i
$7$	−2.52434	−	0.792287i	−0.954110	−	0.299456i
$8$		0			0
$9$	3.37228			1.12409
$10$		0			0
$11$			2.52434i			0.761116i	0.924757	+	0.380558i	$0.124268\pi$
$11$			2.52434i			0.761116i	−0.924757	+	0.380558i	$0.875732\pi$
$12$		0			0
$13$		−	0.372281i		−	0.103252i	−0.998666	−	0.0516261i	$-0.983560\pi$
$13$		−	0.372281i		−	0.103252i	0.998666	−	0.0516261i	$-0.0164404\pi$
$14$		0			0
$15$		−	2.52434i		−	0.651781i
$16$		0			0
$17$		−	2.37228i		−	0.575363i	−0.957726	−	0.287681i	$-0.907116\pi$
$17$		−	2.37228i		−	0.575363i	0.957726	−	0.287681i	$-0.0928844\pi$
$18$		0			0
$19$	3.46410			0.794719			0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410			0.794719			0.397360	+	0.917663i	$0.369927\pi$
$20$		0			0
$21$	6.37228	+	2.00000i	1.39055	+	0.436436i
$22$		0			0
$23$		−	8.51278i		−	1.77504i	−0.460772	−	0.887518i	$-0.652428\pi$
$23$		−	8.51278i		−	1.77504i	0.460772	−	0.887518i	$-0.347572\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−0.939764			−0.180858
$28$		0			0
$29$	0.372281			0.0691309			0.0345655	−	0.999402i	$-0.488995\pi$
$29$	0.372281			0.0691309			0.0345655	+	0.999402i	$0.488995\pi$
$30$		0			0
$31$	1.87953			0.337573			0.168787	−	0.985653i	$-0.446015\pi$
$31$	1.87953			0.337573			0.168787	+	0.985653i	$0.446015\pi$
$32$		0			0
$33$		−	6.37228i		−	1.10927i
$34$		0			0
$35$	0.792287	−	2.52434i	0.133921	−	0.426691i
$36$		0			0
$37$	10.7446			1.76640			0.883198	−	0.469001i	$-0.155386\pi$
$37$	10.7446			1.76640			0.883198	+	0.469001i	$0.155386\pi$
$38$		0			0
$39$			0.939764i			0.150483i
$40$		0			0
$41$		−	8.74456i		−	1.36567i	−0.730572	−	0.682836i	$-0.760747\pi$
$41$		−	8.74456i		−	1.36567i	0.730572	−	0.682836i	$-0.239253\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$		0			0
$45$			3.37228i			0.502710i
$46$		0			0
$47$	7.86797			1.14766			0.573830	−	0.818974i	$-0.305457\pi$
$47$	7.86797			1.14766			0.573830	+	0.818974i	$0.305457\pi$
$48$		0			0
$49$	5.74456	+	4.00000i	0.820652	+	0.571429i
$50$		0			0
$51$			5.98844i			0.838549i
$52$		0			0
$53$	−11.4891			−1.57815			−0.789076	−	0.614295i	$-0.789440\pi$
$53$	−11.4891			−1.57815			−0.789076	+	0.614295i	$0.789440\pi$
$54$		0			0
$55$	−2.52434			−0.340382
$56$		0			0
$57$	−8.74456			−1.15825
$58$		0			0
$59$	6.63325			0.863576			0.431788	−	0.901975i	$-0.357883\pi$
$59$	6.63325			0.863576			0.431788	+	0.901975i	$0.357883\pi$
$60$		0			0
$61$		−	10.7446i		−	1.37570i	−0.725853	−	0.687850i	$-0.758555\pi$
$61$		−	10.7446i		−	1.37570i	0.725853	−	0.687850i	$-0.241445\pi$
$62$		0			0
$63$	−8.51278	−	2.67181i	−1.07251	−	0.336617i
$64$		0			0
$65$	0.372281			0.0461758
$66$		0			0
$67$			6.63325i			0.810380i	0.914232	+	0.405190i	$0.132795\pi$
$67$			6.63325i			0.810380i	−0.914232	+	0.405190i	$0.867205\pi$
$68$		0			0
$69$			21.4891i			2.58699i
$70$		0			0
$71$			6.63325i			0.787222i	0.919277	+	0.393611i	$0.128774\pi$
$71$			6.63325i			0.787222i	−0.919277	+	0.393611i	$0.871226\pi$
$72$		0			0
$73$		−	8.74456i		−	1.02347i	−0.859142	−	0.511737i	$-0.829002\pi$
$73$		−	8.74456i		−	1.02347i	0.859142	−	0.511737i	$-0.170998\pi$
$74$		0			0
$75$	2.52434			0.291485
$76$		0			0
$77$	2.00000	−	6.37228i	0.227921	−	0.726189i
$78$		0			0
$79$		−	15.7908i		−	1.77661i	−0.459256	−	0.888304i	$-0.651884\pi$
$79$		−	15.7908i		−	1.77661i	0.459256	−	0.888304i	$-0.348116\pi$
$80$		0			0
$81$	−7.74456			−0.860507
$82$		0			0
$83$	−10.3923			−1.14070			−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923			−1.14070			−0.570352	+	0.821401i	$0.693193\pi$
$84$		0			0
$85$	2.37228			0.257310
$86$		0			0
$87$	−0.939764			−0.100753
$88$		0			0
$89$			5.48913i			0.581846i	0.956746	+	0.290923i	$0.0939624\pi$
$89$			5.48913i			0.581846i	−0.956746	+	0.290923i	$0.906038\pi$
$90$		0			0
$91$	−0.294954	+	0.939764i	−0.0309195	+	0.0985140i
$92$		0			0
$93$	−4.74456			−0.491988
$94$		0			0
$95$			3.46410i			0.355409i
$96$		0			0
$97$			15.1168i			1.53488i	0.641119	+	0.767441i	$0.278470\pi$
$97$			15.1168i			1.53488i	−0.641119	+	0.767441i	$0.721530\pi$
$98$		0			0
$99$			8.51278i			0.855566i
$100$		0			0
$101$			7.48913i			0.745196i	0.927993	+	0.372598i	$0.121533\pi$
$101$			7.48913i			0.745196i	−0.927993	+	0.372598i	$0.878467\pi$
$102$		0			0
$103$	2.81929			0.277793			0.138897	−	0.990307i	$-0.455644\pi$
$103$	2.81929			0.277793			0.138897	+	0.990307i	$0.455644\pi$
$104$		0			0
$105$	−2.00000	+	6.37228i	−0.195180	+	0.621871i
$106$		0			0
$107$			16.7306i			1.61741i	0.588216	+	0.808704i	$0.299831\pi$
$107$			16.7306i			1.61741i	−0.588216	+	0.808704i	$0.700169\pi$
$108$		0			0
$109$	−7.62772			−0.730603			−0.365301	−	0.930889i	$-0.619034\pi$
$109$	−7.62772			−0.730603			−0.365301	+	0.930889i	$0.619034\pi$
$110$		0			0
$111$	−27.1229			−2.57439
$112$		0			0
$113$	1.25544			0.118102			0.0590508	−	0.998255i	$-0.481193\pi$
$113$	1.25544			0.118102			0.0590508	+	0.998255i	$0.481193\pi$
$114$		0			0
$115$	8.51278			0.793821
$116$		0			0
$117$		−	1.25544i		−	0.116065i
$118$		0			0
$119$	−1.87953	+	5.98844i	−0.172296	+	0.548959i
$120$		0			0
$121$	4.62772			0.420702
$122$		0			0
$123$			22.0742i			1.99037i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	6.63325i		−	0.588606i	−0.955712	−	0.294303i	$-0.904913\pi$
$127$		−	6.63325i		−	0.588606i	0.955712	−	0.294303i	$-0.0950874\pi$
$128$		0			0
$129$			8.74456i			0.769916i
$130$		0			0
$131$	15.4410			1.34908			0.674542	−	0.738236i	$-0.264341\pi$
$131$	15.4410			1.34908			0.674542	+	0.738236i	$0.264341\pi$
$132$		0			0
$133$	−8.74456	−	2.74456i	−0.758250	−	0.237984i
$134$		0			0
$135$		−	0.939764i		−	0.0808820i
$136$		0			0
$137$	−10.7446			−0.917970			−0.458985	−	0.888444i	$-0.651787\pi$
$137$	−10.7446			−0.917970			−0.458985	+	0.888444i	$0.651787\pi$
$138$		0			0
$139$	4.75372			0.403205			0.201603	−	0.979467i	$-0.435385\pi$
$139$	4.75372			0.403205			0.201603	+	0.979467i	$0.435385\pi$
$140$		0			0
$141$	−19.8614			−1.67263
$142$		0			0
$143$	0.939764			0.0785870
$144$		0			0
$145$			0.372281i			0.0309163i
$146$		0			0
$147$	−14.5012	−	10.0974i	−1.19604	−	0.832815i
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		−	18.2603i		−	1.48600i	−0.669291	−	0.743000i	$-0.733402\pi$
$151$		−	18.2603i		−	1.48600i	0.669291	−	0.743000i	$-0.266598\pi$
$152$		0			0
$153$		−	8.00000i		−	0.646762i
$154$		0			0
$155$			1.87953i			0.150967i
$156$		0			0
$157$		−	23.4891i		−	1.87464i	−0.348475	−	0.937318i	$-0.613300\pi$
$157$		−	23.4891i		−	1.87464i	0.348475	−	0.937318i	$-0.386700\pi$
$158$		0			0
$159$	29.0024			2.30004
$160$		0			0
$161$	−6.74456	+	21.4891i	−0.531546	+	1.69358i
$162$		0			0
$163$			4.75372i			0.372340i	0.982518	+	0.186170i	$0.0596076\pi$
$163$			4.75372i			0.372340i	−0.982518	+	0.186170i	$0.940392\pi$
$164$		0			0
$165$	6.37228			0.496081
$166$		0			0
$167$	7.27806			0.563193			0.281597	−	0.959533i	$-0.409136\pi$
$167$	7.27806			0.563193			0.281597	+	0.959533i	$0.409136\pi$
$168$		0			0
$169$	12.8614			0.989339
$170$		0			0
$171$	11.6819			0.893339
$172$		0			0
$173$		−	16.3723i		−	1.24476i	−0.782715	−	0.622381i	$-0.786166\pi$
$173$		−	16.3723i		−	1.24476i	0.782715	−	0.622381i	$-0.213834\pi$
$174$		0			0
$175$	2.52434	+	0.792287i	0.190822	+	0.0598913i
$176$		0			0
$177$	−16.7446			−1.25860
$178$		0			0
$179$		−	13.5615i		−	1.01363i	−0.862055	−	0.506815i	$-0.830823\pi$
$179$		−	13.5615i		−	1.01363i	0.862055	−	0.506815i	$-0.169177\pi$
$180$		0			0
$181$		−	20.9783i		−	1.55930i	−0.626215	−	0.779651i	$-0.715397\pi$
$181$		−	20.9783i		−	1.55930i	0.626215	−	0.779651i	$-0.284603\pi$
$182$		0			0
$183$			27.1229i			2.00498i
$184$		0			0
$185$			10.7446i			0.789956i
$186$		0			0
$187$	5.98844			0.437918
$188$		0			0
$189$	2.37228	+	0.744563i	0.172558	+	0.0541590i
$190$		0			0
$191$		−	7.57301i		−	0.547964i	−0.961735	−	0.273982i	$-0.911659\pi$
$191$		−	7.57301i		−	0.547964i	0.961735	−	0.273982i	$-0.0883409\pi$
$192$		0			0
$193$	26.7446			1.92512			0.962558	−	0.271076i	$-0.0873796\pi$
$193$	26.7446			1.92512			0.962558	+	0.271076i	$0.0873796\pi$
$194$		0			0
$195$	−0.939764			−0.0672979
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−1.28962			−0.0914188			−0.0457094	−	0.998955i	$-0.514555\pi$
$199$	−1.28962			−0.0914188			−0.0457094	+	0.998955i	$0.514555\pi$
$200$		0			0
$201$		−	16.7446i		−	1.18107i
$202$		0			0
$203$	−0.939764	−	0.294954i	−0.0659585	−	0.0207017i
$204$		0			0
$205$	8.74456			0.610747
$206$		0			0
$207$		−	28.7075i		−	1.99531i
$208$		0			0
$209$			8.74456i			0.604874i
$210$		0			0
$211$		−	17.6704i		−	1.21648i	−0.793754	−	0.608239i	$-0.791876\pi$
$211$		−	17.6704i		−	1.21648i	0.793754	−	0.608239i	$-0.208124\pi$
$212$		0			0
$213$		−	16.7446i		−	1.14732i
$214$		0			0
$215$	3.46410			0.236250
$216$		0			0
$217$	−4.74456	−	1.48913i	−0.322082	−	0.101088i
$218$		0			0
$219$			22.0742i			1.49164i
$220$		0			0
$221$	−0.883156			−0.0594075
$222$		0			0
$223$	−4.10891			−0.275153			−0.137577	−	0.990491i	$-0.543931\pi$
$223$	−4.10891			−0.275153			−0.137577	+	0.990491i	$0.543931\pi$
$224$		0			0
$225$	−3.37228			−0.224819
$226$		0			0
$227$	−20.8395			−1.38317			−0.691584	−	0.722297i	$-0.743087\pi$
$227$	−20.8395			−1.38317			−0.691584	+	0.722297i	$0.743087\pi$
$228$		0			0
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$		0			0
$231$	−5.04868	+	16.0858i	−0.332178	+	1.05837i
$232$		0			0
$233$	−10.7446			−0.703900			−0.351950	−	0.936019i	$-0.614481\pi$
$233$	−10.7446			−0.703900			−0.351950	+	0.936019i	$0.614481\pi$
$234$		0			0
$235$			7.86797i			0.513250i
$236$		0			0
$237$			39.8614i			2.58928i
$238$		0			0
$239$		−	12.6217i		−	0.816429i	−0.912886	−	0.408215i	$-0.866152\pi$
$239$		−	12.6217i		−	0.816429i	0.912886	−	0.408215i	$-0.133848\pi$
$240$		0			0
$241$			16.0000i			1.03065i	0.856995	+	0.515325i	$0.172329\pi$
$241$			16.0000i			1.03065i	−0.856995	+	0.515325i	$0.827671\pi$
$242$		0			0
$243$	22.3692			1.43498
$244$		0			0
$245$	−4.00000	+	5.74456i	−0.255551	+	0.367007i
$246$		0			0
$247$		−	1.28962i		−	0.0820566i
$248$		0			0
$249$	26.2337			1.66249
$250$		0			0
$251$	−15.4410			−0.974626			−0.487313	−	0.873227i	$-0.662023\pi$
$251$	−15.4410			−0.974626			−0.487313	+	0.873227i	$0.662023\pi$
$252$		0			0
$253$	21.4891			1.35101
$254$		0			0
$255$	−5.98844			−0.375011
$256$		0			0
$257$		−	12.7446i		−	0.794984i	−0.917606	−	0.397492i	$-0.869881\pi$
$257$		−	12.7446i		−	0.794984i	0.917606	−	0.397492i	$-0.130119\pi$
$258$		0			0
$259$	−27.1229	−	8.51278i	−1.68534	−	0.528958i
$260$		0			0
$261$	1.25544			0.0777096
$262$		0			0
$263$			14.1514i			0.872610i	0.899799	+	0.436305i	$0.143713\pi$
$263$			14.1514i			0.872610i	−0.899799	+	0.436305i	$0.856287\pi$
$264$		0			0
$265$		−	11.4891i		−	0.705771i
$266$		0			0
$267$		−	13.8564i		−	0.847998i
$268$		0			0
$269$		−	10.7446i		−	0.655108i	−0.944833	−	0.327554i	$-0.893776\pi$
$269$		−	10.7446i		−	0.655108i	0.944833	−	0.327554i	$-0.106224\pi$
$270$		0			0
$271$	−26.5330			−1.61176			−0.805882	−	0.592076i	$-0.798309\pi$
$271$	−26.5330			−1.61176			−0.805882	+	0.592076i	$0.798309\pi$
$272$		0			0
$273$	0.744563	−	2.37228i	0.0450630	−	0.143577i
$274$		0			0
$275$		−	2.52434i		−	0.152223i
$276$		0			0
$277$	20.2337			1.21572			0.607862	−	0.794043i	$-0.292027\pi$
$277$	20.2337			1.21572			0.607862	+	0.794043i	$0.292027\pi$
$278$		0			0
$279$	6.33830			0.379464
$280$		0			0
$281$	18.6060			1.10994			0.554970	−	0.831871i	$-0.312730\pi$
$281$	18.6060			1.10994			0.554970	+	0.831871i	$0.312730\pi$
$282$		0			0
$283$	−5.69349			−0.338443			−0.169221	−	0.985578i	$-0.554125\pi$
$283$	−5.69349			−0.338443			−0.169221	+	0.985578i	$0.554125\pi$
$284$		0			0
$285$		−	8.74456i		−	0.517983i
$286$		0			0
$287$	−6.92820	+	22.0742i	−0.408959	+	1.30300i
$288$		0			0
$289$	11.3723			0.668958
$290$		0			0
$291$		−	38.1600i		−	2.23698i
$292$		0			0
$293$			19.6277i			1.14666i	0.819323	+	0.573332i	$0.194349\pi$
$293$			19.6277i			1.14666i	−0.819323	+	0.573332i	$0.805651\pi$
$294$		0			0
$295$			6.63325i			0.386203i
$296$		0			0
$297$		−	2.37228i		−	0.137654i
$298$		0			0
$299$	−3.16915			−0.183277
$300$		0			0
$301$	−2.74456	+	8.74456i	−0.158194	+	0.504028i
$302$		0			0
$303$		−	18.9051i		−	1.08607i
$304$		0			0
$305$	10.7446			0.615232
$306$		0			0
$307$	−0.644810			−0.0368013			−0.0184006	−	0.999831i	$-0.505857\pi$
$307$	−0.644810			−0.0368013			−0.0184006	+	0.999831i	$0.505857\pi$
$308$		0			0
$309$	−7.11684			−0.404863
$310$		0			0
$311$	11.3870			0.645696			0.322848	−	0.946451i	$-0.395360\pi$
$311$	11.3870			0.645696			0.322848	+	0.946451i	$0.395360\pi$
$312$		0			0
$313$		−	3.11684i		−	0.176174i	−0.996113	−	0.0880872i	$-0.971925\pi$
$313$		−	3.11684i		−	0.176174i	0.996113	−	0.0880872i	$-0.0280754\pi$
$314$		0			0
$315$	2.67181	−	8.51278i	0.150540	−	0.479641i
$316$		0			0
$317$	4.51087			0.253356			0.126678	−	0.991944i	$-0.459569\pi$
$317$	4.51087			0.253356			0.126678	+	0.991944i	$0.459569\pi$
$318$		0			0
$319$			0.939764i			0.0526167i
$320$		0			0
$321$		−	42.2337i		−	2.35725i
$322$		0			0
$323$		−	8.21782i		−	0.457252i
$324$		0			0
$325$			0.372281i			0.0206505i
$326$		0			0
$327$	19.2549			1.06480
$328$		0			0
$329$	−19.8614	−	6.23369i	−1.09499	−	0.343674i
$330$		0			0
$331$			24.2487i			1.33283i	0.745581	+	0.666415i	$0.232172\pi$
$331$			24.2487i			1.33283i	−0.745581	+	0.666415i	$0.767828\pi$
$332$		0			0
$333$	36.2337			1.98559
$334$		0			0
$335$	−6.63325			−0.362413
$336$		0			0
$337$	20.2337			1.10220			0.551100	−	0.834439i	$-0.314208\pi$
$337$	20.2337			1.10220			0.551100	+	0.834439i	$0.314208\pi$
$338$		0			0
$339$	−3.16915			−0.172124
$340$		0			0
$341$			4.74456i			0.256932i
$342$		0			0
$343$	−11.3321	−	14.6487i	−0.611874	−	0.790955i
$344$		0			0
$345$	−21.4891			−1.15694
$346$		0			0
$347$			0.994667i			0.0533965i	0.999644	+	0.0266983i	$0.00849933\pi$
$347$			0.994667i			0.0533965i	−0.999644	+	0.0266983i	$0.991501\pi$
$348$		0			0
$349$			10.0000i			0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$			10.0000i			0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$			0.349857i			0.0186740i
$352$		0			0
$353$			18.3723i			0.977858i	0.872324	+	0.488929i	$0.162612\pi$
$353$			18.3723i			0.977858i	−0.872324	+	0.488929i	$0.837388\pi$
$354$		0			0
$355$	−6.63325			−0.352056
$356$		0			0
$357$	4.74456	−	15.1168i	0.251109	−	0.800068i
$358$		0			0
$359$		−	0.294954i		−	0.0155671i	−0.999970	−	0.00778353i	$-0.997522\pi$
$359$		−	0.294954i		−	0.0155671i	0.999970	−	0.00778353i	$-0.00247760\pi$
$360$		0			0
$361$	−7.00000			−0.368421
$362$		0			0
$363$	−11.6819			−0.613142
$364$		0			0
$365$	8.74456			0.457711
$366$		0			0
$367$	21.7244			1.13400			0.567002	−	0.823717i	$-0.308103\pi$
$367$	21.7244			1.13400			0.567002	+	0.823717i	$0.308103\pi$
$368$		0			0
$369$		−	29.4891i		−	1.53514i
$370$		0			0
$371$	29.0024	+	9.10268i	1.50573	+	0.472588i
$372$		0			0
$373$	−21.2554			−1.10056			−0.550282	−	0.834979i	$-0.685480\pi$
$373$	−21.2554			−1.10056			−0.550282	+	0.834979i	$0.685480\pi$
$374$		0			0
$375$			2.52434i			0.130356i
$376$		0			0
$377$		−	0.138593i		−	0.00713792i
$378$		0			0
$379$		−	23.6588i		−	1.21527i	−0.794216	−	0.607636i	$-0.792118\pi$
$379$		−	23.6588i		−	1.21527i	0.794216	−	0.607636i	$-0.207882\pi$
$380$		0			0
$381$			16.7446i			0.857850i
$382$		0			0
$383$	−11.9769			−0.611990			−0.305995	−	0.952033i	$-0.598989\pi$
$383$	−11.9769			−0.611990			−0.305995	+	0.952033i	$0.598989\pi$
$384$		0			0
$385$	6.37228	+	2.00000i	0.324762	+	0.101929i
$386$		0			0
$387$		−	11.6819i		−	0.593826i
$388$		0			0
$389$	−1.11684			−0.0566262			−0.0283131	−	0.999599i	$-0.509014\pi$
$389$	−1.11684			−0.0566262			−0.0283131	+	0.999599i	$0.509014\pi$
$390$		0			0
$391$	−20.1947			−1.02129
$392$		0			0
$393$	−38.9783			−1.96619
$394$		0			0
$395$	15.7908			0.794523
$396$		0			0
$397$			12.3723i			0.620947i	0.950582	+	0.310473i	$0.100488\pi$
$397$			12.3723i			0.620947i	−0.950582	+	0.310473i	$0.899512\pi$
$398$		0			0
$399$	22.0742	+	6.92820i	1.10509	+	0.346844i
$400$		0			0
$401$	5.11684			0.255523			0.127761	−	0.991805i	$-0.459221\pi$
$401$	5.11684			0.255523			0.127761	+	0.991805i	$0.459221\pi$
$402$		0			0
$403$		−	0.699713i		−	0.0348552i
$404$		0			0
$405$		−	7.74456i		−	0.384830i
$406$		0			0
$407$			27.1229i			1.34443i
$408$		0			0
$409$		−	16.7446i		−	0.827965i	−0.910285	−	0.413983i	$-0.864137\pi$
$409$		−	16.7446i		−	0.827965i	0.910285	−	0.413983i	$-0.135863\pi$
$410$		0			0
$411$	27.1229			1.33787
$412$		0			0
$413$	−16.7446	−	5.25544i	−0.823946	−	0.258603i
$414$		0			0
$415$		−	10.3923i		−	0.510138i
$416$		0			0
$417$	−12.0000			−0.587643
$418$		0			0
$419$	−12.9715			−0.633701			−0.316851	−	0.948475i	$-0.602625\pi$
$419$	−12.9715			−0.633701			−0.316851	+	0.948475i	$0.602625\pi$
$420$		0			0
$421$	21.1168			1.02917			0.514586	−	0.857439i	$-0.327946\pi$
$421$	21.1168			1.02917			0.514586	+	0.857439i	$0.327946\pi$
$422$		0			0
$423$	26.5330			1.29008
$424$		0			0
$425$			2.37228i			0.115073i
$426$		0			0
$427$	−8.51278	+	27.1229i	−0.411962	+	1.31257i
$428$		0			0
$429$	−2.37228			−0.114535
$430$		0			0
$431$			12.0318i			0.579551i	0.957095	+	0.289775i	$0.0935806\pi$
$431$			12.0318i			0.579551i	−0.957095	+	0.289775i	$0.906419\pi$
$432$		0			0
$433$			6.23369i			0.299572i	0.988718	+	0.149786i	$0.0478585\pi$
$433$			6.23369i			0.299572i	−0.988718	+	0.149786i	$0.952142\pi$
$434$		0			0
$435$		−	0.939764i		−	0.0450582i
$436$		0			0
$437$		−	29.4891i		−	1.41066i
$438$		0			0
$439$	−0.589907			−0.0281547			−0.0140774	−	0.999901i	$-0.504481\pi$
$439$	−0.589907			−0.0281547			−0.0140774	+	0.999901i	$0.504481\pi$
$440$		0			0
$441$	19.3723	+	13.4891i	0.922490	+	0.642339i
$442$		0			0
$443$		−	21.0796i		−	1.00152i	−0.865586	−	0.500760i	$-0.833054\pi$
$443$		−	21.0796i		−	1.00152i	0.865586	−	0.500760i	$-0.166946\pi$
$444$		0			0
$445$	−5.48913			−0.260209
$446$		0			0
$447$	25.2434			1.19397
$448$		0			0
$449$	−22.6060			−1.06684			−0.533421	−	0.845850i	$-0.679094\pi$
$449$	−22.6060			−1.06684			−0.533421	+	0.845850i	$0.679094\pi$
$450$		0			0
$451$	22.0742			1.03943
$452$		0			0
$453$			46.0951i			2.16574i
$454$		0			0
$455$	−0.939764	−	0.294954i	−0.0440568	−	0.0138276i
$456$		0			0
$457$	15.4891			0.724551			0.362275	−	0.932071i	$-0.382000\pi$
$457$	15.4891			0.724551			0.362275	+	0.932071i	$0.382000\pi$
$458$		0			0
$459$			2.22938i			0.104059i
$460$		0			0
$461$			22.7446i			1.05932i	0.848210	+	0.529660i	$0.177680\pi$
$461$			22.7446i			1.05932i	−0.848210	+	0.529660i	$0.822320\pi$
$462$		0			0
$463$		−	21.7793i		−	1.01217i	−0.862484	−	0.506084i	$-0.831092\pi$
$463$		−	21.7793i		−	1.01217i	0.862484	−	0.506084i	$-0.168908\pi$
$464$		0			0
$465$		−	4.74456i		−	0.220024i
$466$		0			0
$467$	−10.7422			−0.497088			−0.248544	−	0.968621i	$-0.579952\pi$
$467$	−10.7422			−0.497088			−0.248544	+	0.968621i	$0.579952\pi$
$468$		0			0
$469$	5.25544	−	16.7446i	0.242674	−	0.773192i
$470$		0			0
$471$			59.2945i			2.73215i
$472$		0			0
$473$	8.74456			0.402075
$474$		0			0
$475$	−3.46410			−0.158944
$476$		0			0
$477$	−38.7446			−1.77399
$478$		0			0
$479$	−37.8102			−1.72759			−0.863795	−	0.503843i	$-0.831919\pi$
$479$	−37.8102			−1.72759			−0.863795	+	0.503843i	$0.831919\pi$
$480$		0			0
$481$		−	4.00000i		−	0.182384i
$482$		0			0
$483$	17.0256	−	54.2458i	0.774690	−	2.46827i
$484$		0			0
$485$	−15.1168			−0.686421
$486$		0			0
$487$		−	4.75372i		−	0.215412i	−0.994183	−	0.107706i	$-0.965650\pi$
$487$		−	4.75372i		−	0.215412i	0.994183	−	0.107706i	$-0.0343505\pi$
$488$		0			0
$489$		−	12.0000i		−	0.542659i
$490$		0			0
$491$		−	24.0087i		−	1.08350i	−0.840541	−	0.541748i	$-0.817763\pi$
$491$		−	24.0087i		−	1.08350i	0.840541	−	0.541748i	$-0.182237\pi$
$492$		0			0
$493$		−	0.883156i		−	0.0397753i
$494$		0			0
$495$	−8.51278			−0.382621
$496$		0			0
$497$	5.25544	−	16.7446i	0.235739	−	0.751096i
$498$		0			0
$499$			17.0805i			0.764626i	0.924033	+	0.382313i	$0.124872\pi$
$499$			17.0805i			0.764626i	−0.924033	+	0.382313i	$0.875128\pi$
$500$		0			0
$501$	−18.3723			−0.820813
$502$		0			0
$503$	35.6906			1.59136			0.795682	−	0.605714i	$-0.207113\pi$
$503$	35.6906			1.59136			0.795682	+	0.605714i	$0.207113\pi$
$504$		0			0
$505$	−7.48913			−0.333262
$506$		0			0
$507$	−32.4665			−1.44189
$508$		0			0
$509$		−	10.7446i		−	0.476244i	−0.971235	−	0.238122i	$-0.923468\pi$
$509$		−	10.7446i		−	0.476244i	0.971235	−	0.238122i	$-0.0765319\pi$
$510$		0			0
$511$	−6.92820	+	22.0742i	−0.306486	+	0.976506i
$512$		0			0
$513$	−3.25544			−0.143731
$514$		0			0
$515$			2.81929i			0.124233i
$516$		0			0
$517$			19.8614i			0.873504i
$518$		0			0
$519$			41.3292i			1.81415i
$520$		0			0
$521$			35.7228i			1.56504i	0.622622	+	0.782522i	$0.286067\pi$
$521$			35.7228i			1.56504i	−0.622622	+	0.782522i	$0.713933\pi$
$522$		0			0
$523$	−10.9822			−0.480219			−0.240109	−	0.970746i	$-0.577183\pi$
$523$	−10.9822			−0.480219			−0.240109	+	0.970746i	$0.577183\pi$
$524$		0			0
$525$	−6.37228	−	2.00000i	−0.278109	−	0.0872872i
$526$		0			0
$527$		−	4.45877i		−	0.194227i
$528$		0			0
$529$	−49.4674			−2.15076
$530$		0			0
$531$	22.3692			0.970740
$532$		0			0
$533$	−3.25544			−0.141009
$534$		0			0
$535$	−16.7306			−0.723327
$536$		0			0
$537$			34.2337i			1.47729i
$538$		0			0
$539$	−10.0974	+	14.5012i	−0.434924	+	0.624612i
$540$		0			0
$541$	3.62772			0.155968			0.0779839	−	0.996955i	$-0.475152\pi$
$541$	3.62772			0.155968			0.0779839	+	0.996955i	$0.475152\pi$
$542$		0			0
$543$			52.9562i			2.27257i
$544$		0			0
$545$		−	7.62772i		−	0.326736i
$546$		0			0
$547$			24.9484i			1.06672i	0.845889	+	0.533359i	$0.179070\pi$
$547$			24.9484i			1.06672i	−0.845889	+	0.533359i	$0.820930\pi$
$548$		0			0
$549$		−	36.2337i		−	1.54642i
$550$		0			0
$551$	1.28962			0.0549397
$552$		0			0
$553$	−12.5109	+	39.8614i	−0.532017	+	1.69508i
$554$		0			0
$555$		−	27.1229i		−	1.15130i
$556$		0			0
$557$	−33.7228			−1.42888			−0.714441	−	0.699696i	$-0.753319\pi$
$557$	−33.7228			−1.42888			−0.714441	+	0.699696i	$0.753319\pi$
$558$		0			0
$559$	−1.28962			−0.0545451
$560$		0			0
$561$	−15.1168			−0.638234
$562$		0			0
$563$	−36.9253			−1.55622			−0.778108	−	0.628131i	$-0.783820\pi$
$563$	−36.9253			−1.55622			−0.778108	+	0.628131i	$0.783820\pi$
$564$		0			0
$565$			1.25544i			0.0528166i
$566$		0			0
$567$	19.5499	+	6.13592i	0.821018	+	0.257684i
$568$		0			0
$569$	12.5109			0.524483			0.262242	−	0.965002i	$-0.415538\pi$
$569$	12.5109			0.524483			0.262242	+	0.965002i	$0.415538\pi$
$570$		0			0
$571$			2.87419i			0.120281i	0.998190	+	0.0601406i	$0.0191549\pi$
$571$			2.87419i			0.120281i	−0.998190	+	0.0601406i	$0.980845\pi$
$572$		0			0
$573$			19.1168i			0.798618i
$574$		0			0
$575$			8.51278i			0.355007i
$576$		0			0
$577$			23.1168i			0.962367i	0.876620	+	0.481183i	$0.159793\pi$
$577$			23.1168i			0.962367i	−0.876620	+	0.481183i	$0.840207\pi$
$578$		0			0
$579$	−67.5123			−2.80572
$580$		0			0
$581$	26.2337	+	8.23369i	1.08836	+	0.341591i
$582$		0			0
$583$		−	29.0024i		−	1.20116i
$584$		0			0
$585$	1.25544			0.0519059
$586$		0			0
$587$	−3.46410			−0.142979			−0.0714894	−	0.997441i	$-0.522775\pi$
$587$	−3.46410			−0.142979			−0.0714894	+	0.997441i	$0.522775\pi$
$588$		0			0
$589$	6.51087			0.268276
$590$		0			0
$591$	−15.1460			−0.623024
$592$		0			0
$593$		−	5.62772i		−	0.231103i	−0.993302	−	0.115551i	$-0.963137\pi$
$593$		−	5.62772i		−	0.231103i	0.993302	−	0.115551i	$-0.0368635\pi$
$594$		0			0
$595$	−5.98844	−	1.87953i	−0.245502	−	0.0770531i
$596$		0			0
$597$	3.25544			0.133236
$598$		0			0
$599$			17.0805i			0.697889i	0.937143	+	0.348944i	$0.113460\pi$
$599$			17.0805i			0.697889i	−0.937143	+	0.348944i	$0.886540\pi$
$600$		0			0
$601$		−	24.7446i		−	1.00935i	−0.863309	−	0.504676i	$-0.831612\pi$
$601$		−	24.7446i		−	1.00935i	0.863309	−	0.504676i	$-0.168388\pi$
$602$		0			0
$603$			22.3692i			0.910944i
$604$		0			0
$605$			4.62772i			0.188144i
$606$		0			0
$607$	−3.40920			−0.138375			−0.0691876	−	0.997604i	$-0.522041\pi$
$607$	−3.40920			−0.138375			−0.0691876	+	0.997604i	$0.522041\pi$
$608$		0			0
$609$	2.37228	+	0.744563i	0.0961297	+	0.0301712i
$610$		0			0
$611$		−	2.92910i		−	0.118499i
$612$		0			0
$613$	−20.9783			−0.847304			−0.423652	−	0.905825i	$-0.639252\pi$
$613$	−20.9783			−0.847304			−0.423652	+	0.905825i	$0.639252\pi$
$614$		0			0
$615$	−22.0742			−0.890119
$616$		0			0
$617$	42.7446			1.72083			0.860416	−	0.509593i	$-0.170204\pi$
$617$	42.7446			1.72083			0.860416	+	0.509593i	$0.170204\pi$
$618$		0			0
$619$	42.5639			1.71079			0.855394	−	0.517979i	$-0.173315\pi$
$619$	42.5639			1.71079			0.855394	+	0.517979i	$0.173315\pi$
$620$		0			0
$621$			8.00000i			0.321029i
$622$		0			0
$623$	4.34896	−	13.8564i	0.174238	−	0.555145i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	22.0742i		−	0.881560i
$628$		0			0
$629$		−	25.4891i		−	1.01632i
$630$		0			0
$631$			8.86263i			0.352816i	0.984317	+	0.176408i	$0.0564478\pi$
$631$			8.86263i			0.352816i	−0.984317	+	0.176408i	$0.943552\pi$
$632$		0			0
$633$			44.6060i			1.77293i
$634$		0			0
$635$	6.63325			0.263232
$636$		0			0
$637$	1.48913	−	2.13859i	0.0590013	−	0.0847342i
$638$		0			0
$639$			22.3692i			0.884911i
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	31.5268			1.24329			0.621647	−	0.783297i	$-0.286464\pi$
$643$	31.5268			1.24329			0.621647	+	0.783297i	$0.286464\pi$
$644$		0			0
$645$	−8.74456			−0.344317
$646$		0			0
$647$	−2.46943			−0.0970835			−0.0485418	−	0.998821i	$-0.515457\pi$
$647$	−2.46943			−0.0970835			−0.0485418	+	0.998821i	$0.515457\pi$
$648$		0			0
$649$			16.7446i			0.657282i
$650$		0			0
$651$	11.9769	+	3.75906i	0.469411	+	0.147329i
$652$		0			0
$653$	18.4674			0.722684			0.361342	−	0.932433i	$-0.382319\pi$
$653$	18.4674			0.722684			0.361342	+	0.932433i	$0.382319\pi$
$654$		0			0
$655$			15.4410i			0.603329i
$656$		0			0
$657$		−	29.4891i		−	1.15048i
$658$		0			0
$659$		−	16.9707i		−	0.661083i	−0.943792	−	0.330541i	$-0.892769\pi$
$659$		−	16.9707i		−	0.661083i	0.943792	−	0.330541i	$-0.107231\pi$
$660$		0			0
$661$		−	37.2554i		−	1.44907i	−0.689239	−	0.724534i	$-0.742055\pi$
$661$		−	37.2554i		−	1.44907i	0.689239	−	0.724534i	$-0.257945\pi$
$662$		0			0
$663$	2.22938			0.0865821
$664$		0			0
$665$	2.74456	−	8.74456i	0.106430	−	0.339100i
$666$		0			0
$667$		−	3.16915i		−	0.122710i
$668$		0			0
$669$	10.3723			0.401016
$670$		0			0
$671$	27.1229			1.04707
$672$		0			0
$673$	16.2337			0.625763			0.312881	−	0.949792i	$-0.398706\pi$
$673$	16.2337			0.625763			0.312881	+	0.949792i	$0.398706\pi$
$674$		0			0
$675$	0.939764			0.0361715
$676$		0			0
$677$			6.88316i			0.264541i	0.991214	+	0.132271i	$0.0422268\pi$
$677$			6.88316i			0.264541i	−0.991214	+	0.132271i	$0.957773\pi$
$678$		0			0
$679$	11.9769	−	38.1600i	0.459630	−	1.46445i
$680$		0			0
$681$	52.6060			2.01587
$682$		0			0
$683$			36.2256i			1.38613i	0.720873	+	0.693067i	$0.243741\pi$
$683$			36.2256i			1.38613i	−0.720873	+	0.693067i	$0.756259\pi$
$684$		0			0
$685$		−	10.7446i		−	0.410529i
$686$		0			0
$687$		−	35.3407i		−	1.34833i
$688$		0			0
$689$			4.27719i			0.162948i
$690$		0			0
$691$	−29.8873			−1.13697			−0.568483	−	0.822695i	$-0.692470\pi$
$691$	−29.8873			−1.13697			−0.568483	+	0.822695i	$0.692470\pi$
$692$		0			0
$693$	6.74456	−	21.4891i	0.256205	−	0.816304i
$694$		0			0
$695$			4.75372i			0.180319i
$696$		0			0
$697$	−20.7446			−0.785756
$698$		0			0
$699$	27.1229			1.02588
$700$		0			0
$701$	−18.8832			−0.713207			−0.356603	−	0.934256i	$-0.616065\pi$
$701$	−18.8832			−0.713207			−0.356603	+	0.934256i	$0.616065\pi$
$702$		0			0
$703$	37.2203			1.40379
$704$		0			0
$705$		−	19.8614i		−	0.748024i
$706$		0			0
$707$	5.93354	−	18.9051i	0.223154	−	0.710999i
$708$		0			0
$709$	0.372281			0.0139813			0.00699066	−	0.999976i	$-0.497775\pi$
$709$	0.372281			0.0139813			0.00699066	+	0.999976i	$0.497775\pi$
$710$		0			0
$711$		−	53.2511i		−	1.99707i
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$			0.939764i			0.0351452i
$716$		0			0
$717$			31.8614i			1.18989i
$718$		0			0
$719$	20.1947			0.753135			0.376568	−	0.926389i	$-0.377104\pi$
$719$	20.1947			0.753135			0.376568	+	0.926389i	$0.377104\pi$
$720$		0			0
$721$	−7.11684	−	2.23369i	−0.265045	−	0.0831869i
$722$		0			0
$723$		−	40.3894i		−	1.50210i
$724$		0			0
$725$	−0.372281			−0.0138262
$726$		0			0
$727$	42.8588			1.58955			0.794773	−	0.606907i	$-0.207590\pi$
$727$	42.8588			1.58955			0.794773	+	0.606907i	$0.207590\pi$
$728$		0			0
$729$	−33.2337			−1.23088
$730$		0			0
$731$	−8.21782			−0.303947
$732$		0			0
$733$			6.13859i			0.226734i	0.993553	+	0.113367i	$0.0361636\pi$
$733$			6.13859i			0.226734i	−0.993553	+	0.113367i	$0.963836\pi$
$734$		0			0
$735$	10.0974	−	14.5012i	0.372446	−	0.534885i
$736$		0			0
$737$	−16.7446			−0.616794
$738$		0			0
$739$			17.0805i			0.628315i	0.949371	+	0.314157i	$0.101722\pi$
$739$			17.0805i			0.628315i	−0.949371	+	0.314157i	$0.898278\pi$
$740$		0			0
$741$			3.25544i			0.119591i
$742$		0			0
$743$			7.22316i			0.264992i	0.991184	+	0.132496i	$0.0422992\pi$
$743$			7.22316i			0.264992i	−0.991184	+	0.132496i	$0.957701\pi$
$744$		0			0
$745$		−	10.0000i		−	0.366372i
$746$		0			0
$747$	−35.0458			−1.28226
$748$		0			0
$749$	13.2554	−	42.2337i	0.484343	−	1.54319i
$750$		0			0
$751$		−	24.5986i		−	0.897614i	−0.893629	−	0.448807i	$-0.851849\pi$
$751$		−	24.5986i		−	0.897614i	0.893629	−	0.448807i	$-0.148151\pi$
$752$		0			0
$753$	38.9783			1.42045
$754$		0			0
$755$	18.2603			0.664559
$756$		0			0
$757$	−22.7446			−0.826665			−0.413333	−	0.910580i	$-0.635635\pi$
$757$	−22.7446			−0.826665			−0.413333	+	0.910580i	$0.635635\pi$
$758$		0			0
$759$	−54.2458			−1.96900
$760$		0			0
$761$			36.0000i			1.30500i	0.757789	+	0.652499i	$0.226280\pi$
$761$			36.0000i			1.30500i	−0.757789	+	0.652499i	$0.773720\pi$
$762$		0			0
$763$	19.2549	+	6.04334i	0.697076	+	0.218784i
$764$		0			0
$765$	8.00000			0.289241
$766$		0			0
$767$		−	2.46943i		−	0.0891661i
$768$		0			0
$769$		−	38.2337i		−	1.37874i	−0.724408	−	0.689371i	$-0.757887\pi$
$769$		−	38.2337i		−	1.37874i	0.724408	−	0.689371i	$-0.242113\pi$
$770$		0			0
$771$			32.1716i			1.15863i
$772$		0			0
$773$			5.11684i			0.184040i	0.995757	+	0.0920200i	$0.0293324\pi$
$773$			5.11684i			0.184040i	−0.995757	+	0.0920200i	$0.970668\pi$
$774$		0			0
$775$	−1.87953			−0.0675146
$776$		0			0
$777$	68.4674	+	21.4891i	2.45625	+	0.770918i
$778$		0			0
$779$		−	30.2921i		−	1.08533i
$780$		0			0
$781$	−16.7446			−0.599168
$782$		0			0
$783$	−0.349857			−0.0125029
$784$		0			0
$785$	23.4891			0.838363
$786$		0			0
$787$	29.6472			1.05681			0.528405	−	0.848992i	$-0.322790\pi$
$787$	29.6472			1.05681			0.528405	+	0.848992i	$0.322790\pi$
$788$		0			0
$789$		−	35.7228i		−	1.27177i
$790$		0			0
$791$	−3.16915	−	0.994667i	−0.112682	−	0.0353663i
$792$		0			0
$793$	−4.00000			−0.142044
$794$		0			0
$795$			29.0024i			1.02861i
$796$		0			0
$797$		−	16.0951i		−	0.570117i	−0.958510	−	0.285059i	$-0.907987\pi$
$797$		−	16.0951i		−	0.570117i	0.958510	−	0.285059i	$-0.0920131\pi$
$798$		0			0
$799$		−	18.6650i		−	0.660321i
$800$		0			0
$801$			18.5109i			0.654050i
$802$		0			0
$803$	22.0742			0.778983
$804$		0			0
$805$	−21.4891	−	6.74456i	−0.757392	−	0.237715i
$806$		0			0
$807$			27.1229i			0.954771i
$808$		0			0
$809$	−18.6060			−0.654151			−0.327076	−	0.944998i	$-0.606063\pi$
$809$	−18.6060			−0.654151			−0.327076	+	0.944998i	$0.606063\pi$
$810$		0			0
$811$	22.3692			0.785488			0.392744	−	0.919648i	$-0.371526\pi$
$811$	22.3692			0.785488			0.392744	+	0.919648i	$0.371526\pi$
$812$		0			0
$813$	66.9783			2.34903
$814$		0			0
$815$	−4.75372			−0.166516
$816$		0			0
$817$		−	12.0000i		−	0.419827i
$818$		0			0
$819$	−0.994667	+	3.16915i	−0.0347565	+	0.110739i
$820$		0			0
$821$	48.0951			1.67853			0.839265	−	0.543722i	$-0.182986\pi$
$821$	48.0951			1.67853			0.839265	+	0.543722i	$0.182986\pi$
$822$		0			0
$823$			41.8642i			1.45929i	0.683824	+	0.729647i	$0.260315\pi$
$823$			41.8642i			1.45929i	−0.683824	+	0.729647i	$0.739685\pi$
$824$		0			0
$825$			6.37228i			0.221854i
$826$		0			0
$827$		−	1.58457i		−	0.0551010i	−0.999620	−	0.0275505i	$-0.991229\pi$
$827$		−	1.58457i		−	0.0551010i	0.999620	−	0.0275505i	$-0.00877071\pi$
$828$		0			0
$829$			3.48913i			0.121182i	0.998163	+	0.0605912i	$0.0192986\pi$
$829$			3.48913i			0.121182i	−0.998163	+	0.0605912i	$0.980701\pi$
$830$		0			0
$831$	−51.0767			−1.77183
$832$		0			0
$833$	9.48913	−	13.6277i	0.328779	−	0.472172i
$834$		0			0
$835$			7.27806i			0.251868i
$836$		0			0
$837$	−1.76631			−0.0610527
$838$		0			0
$839$	−29.7021			−1.02543			−0.512716	−	0.858558i	$-0.671361\pi$
$839$	−29.7021			−1.02543			−0.512716	+	0.858558i	$0.671361\pi$
$840$		0			0
$841$	−28.8614			−0.995221
$842$		0			0
$843$	−46.9678			−1.61766
$844$		0			0
$845$			12.8614i			0.442446i
$846$		0			0
$847$	−11.6819	−	3.66648i	−0.401396	−	0.125982i
$848$		0			0
$849$	14.3723			0.493255
$850$		0			0
$851$		−	91.4661i		−	3.13542i
$852$		0			0
$853$			15.4891i			0.530338i	0.964202	+	0.265169i	$0.0854277\pi$
$853$			15.4891i			0.530338i	−0.964202	+	0.265169i	$0.914572\pi$
$854$		0			0
$855$			11.6819i			0.399513i
$856$		0			0
$857$		−	37.2119i		−	1.27114i	−0.772045	−	0.635568i	$-0.780766\pi$
$857$		−	37.2119i		−	1.27114i	0.772045	−	0.635568i	$-0.219234\pi$
$858$		0			0
$859$	−43.1538			−1.47239			−0.736194	−	0.676770i	$-0.763379\pi$
$859$	−43.1538			−1.47239			−0.736194	+	0.676770i	$0.763379\pi$
$860$		0			0
$861$	17.4891	−	55.7228i	0.596028	−	1.89903i
$862$		0			0
$863$		−	0.884861i		−	0.0301210i	−0.999887	−	0.0150605i	$-0.995206\pi$
$863$		−	0.884861i		−	0.0301210i	0.999887	−	0.0150605i	$-0.00479409\pi$
$864$		0			0
$865$	16.3723			0.556674
$866$		0			0
$867$	−28.7075			−0.974957
$868$		0			0
$869$	39.8614			1.35221
$870$		0			0
$871$	2.46943			0.0836736
$872$		0			0
$873$			50.9783i			1.72535i
$874$		0			0
$875$	−0.792287	+	2.52434i	−0.0267842	+	0.0853382i
$876$		0			0
$877$	32.9783			1.11360			0.556798	−	0.830648i	$-0.312030\pi$
$877$	32.9783			1.11360			0.556798	+	0.830648i	$0.312030\pi$
$878$		0			0
$879$		−	49.5470i		−	1.67118i
$880$		0			0
$881$			45.9565i			1.54831i	0.632994	+	0.774157i	$0.281826\pi$
$881$			45.9565i			1.54831i	−0.632994	+	0.774157i	$0.718174\pi$
$882$		0			0
$883$			46.3229i			1.55889i	0.626470	+	0.779446i	$0.284499\pi$
$883$			46.3229i			1.55889i	−0.626470	+	0.779446i	$0.715501\pi$
$884$		0			0
$885$		−	16.7446i		−	0.562862i
$886$		0			0
$887$	−25.2434			−0.847590			−0.423795	−	0.905758i	$-0.639302\pi$
$887$	−25.2434			−0.847590			−0.423795	+	0.905758i	$0.639302\pi$
$888$		0			0
$889$	−5.25544	+	16.7446i	−0.176262	+	0.561595i
$890$		0			0
$891$		−	19.5499i		−	0.654946i
$892$		0			0
$893$	27.2554			0.912068
$894$		0			0
$895$	13.5615			0.453309
$896$		0			0
$897$	8.00000			0.267112
$898$		0			0
$899$	0.699713			0.0233367
$900$		0			0
$901$			27.2554i			0.908010i
$902$		0			0
$903$	6.92820	−	22.0742i	0.230556	−	0.734584i
$904$		0			0
$905$	20.9783			0.697341
$906$		0			0
$907$		−	9.10268i		−	0.302250i	−0.988515	−	0.151125i	$-0.951710\pi$
$907$		−	9.10268i		−	0.302250i	0.988515	−	0.151125i	$-0.0482895\pi$
$908$		0			0
$909$			25.2554i			0.837670i
$910$		0			0
$911$		−	50.7817i		−	1.68247i	−0.540667	−	0.841237i	$-0.681828\pi$
$911$		−	50.7817i		−	1.68247i	0.540667	−	0.841237i	$-0.318172\pi$
$912$		0			0
$913$		−	26.2337i		−	0.868208i
$914$		0			0
$915$	−27.1229			−0.896656
$916$		0			0
$917$	−38.9783	−	12.2337i	−1.28718	−	0.403992i
$918$		0			0
$919$			24.5986i			0.811432i	0.913999	+	0.405716i	$0.132978\pi$
$919$			24.5986i			0.811432i	−0.913999	+	0.405716i	$0.867022\pi$
$920$		0			0
$921$	1.62772			0.0536352
$922$		0			0
$923$	2.46943			0.0812824
$924$		0			0
$925$	−10.7446			−0.353279
$926$		0			0
$927$	9.50744			0.312265
$928$		0			0
$929$			19.2554i			0.631750i	0.948801	+	0.315875i	$0.102298\pi$
$929$			19.2554i			0.631750i	−0.948801	+	0.315875i	$0.897702\pi$
$930$		0			0
$931$	19.8997	+	13.8564i	0.652188	+	0.454125i
$932$		0			0
$933$	−28.7446			−0.941055
$934$		0			0
$935$			5.98844i			0.195843i
$936$		0			0
$937$		−	20.8832i		−	0.682223i	−0.940023	−	0.341111i	$-0.889197\pi$
$937$		−	20.8832i		−	0.682223i	0.940023	−	0.341111i	$-0.110803\pi$
$938$		0			0
$939$			7.86797i			0.256761i
$940$		0			0
$941$			11.7663i			0.383571i	0.981437	+	0.191785i	$0.0614278\pi$
$941$			11.7663i			0.383571i	−0.981437	+	0.191785i	$0.938572\pi$
$942$		0			0
$943$	−74.4405			−2.42412
$944$		0			0
$945$	−0.744563	+	2.37228i	−0.0242206	+	0.0771703i
$946$		0			0
$947$			26.8280i			0.871791i	0.899997	+	0.435896i	$0.143568\pi$
$947$			26.8280i			0.871791i	−0.899997	+	0.435896i	$0.856432\pi$
$948$		0			0
$949$	−3.25544			−0.105676
$950$		0			0
$951$	−11.3870			−0.369248
$952$		0			0
$953$	28.9783			0.938698			0.469349	−	0.883013i	$-0.344489\pi$
$953$	28.9783			0.938698			0.469349	+	0.883013i	$0.344489\pi$
$954$		0			0
$955$	7.57301			0.245057
$956$		0			0
$957$		−	2.37228i		−	0.0766850i
$958$		0			0
$959$	27.1229	+	8.51278i	0.875844	+	0.274892i
$960$		0			0
$961$	−27.4674			−0.886044
$962$		0			0
$963$			56.4203i			1.81812i
$964$		0			0
$965$			26.7446i			0.860938i
$966$		0			0
$967$		−	40.6844i		−	1.30832i	−0.756356	−	0.654160i	$-0.773022\pi$
$967$		−	40.6844i		−	1.30832i	0.756356	−	0.654160i	$-0.226978\pi$
$968$		0			0
$969$			20.7446i			0.666411i
$970$		0			0
$971$	33.1662			1.06436			0.532178	−	0.846633i	$-0.321374\pi$
$971$	33.1662			1.06436			0.532178	+	0.846633i	$0.321374\pi$
$972$		0			0
$973$	−12.0000	−	3.76631i	−0.384702	−	0.120742i
$974$		0			0
$975$		−	0.939764i		−	0.0300965i
$976$		0			0
$977$	20.9783			0.671154			0.335577	−	0.942013i	$-0.391069\pi$
$977$	20.9783			0.671154			0.335577	+	0.942013i	$0.391069\pi$
$978$		0			0
$979$	−13.8564			−0.442853
$980$		0			0
$981$	−25.7228			−0.821266
$982$		0			0
$983$	53.3060			1.70020			0.850099	−	0.526622i	$-0.176542\pi$
$983$	53.3060			1.70020			0.850099	+	0.526622i	$0.176542\pi$
$984$		0			0
$985$			6.00000i			0.191176i
$986$		0			0
$987$	50.1369	+	15.7359i	1.59588	+	0.500880i
$988$		0			0
$989$	−29.4891			−0.937700
$990$		0			0
$991$		−	4.64392i		−	0.147519i	−0.997276	−	0.0737594i	$-0.976500\pi$
$991$		−	4.64392i		−	0.147519i	0.997276	−	0.0737594i	$-0.0234997\pi$
$992$		0			0
$993$		−	61.2119i		−	1.94250i
$994$		0			0
$995$		−	1.28962i		−	0.0408837i
$996$		0			0
$997$			19.3505i			0.612837i	0.951897	+	0.306419i	$0.0991308\pi$
$997$			19.3505i			0.612837i	−0.951897	+	0.306419i	$0.900869\pi$
$998$		0			0
$999$	−10.0974			−0.319466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.k.a.111.2	yes	8
3.2	odd	2		5040.2.d.e.4591.1		8
4.3	odd	2	inner	560.2.k.a.111.8	yes	8
5.2	odd	4		2800.2.e.j.2799.8		8
5.3	odd	4		2800.2.e.i.2799.1		8
5.4	even	2		2800.2.k.l.2351.8		8
7.6	odd	2	inner	560.2.k.a.111.7	yes	8
8.3	odd	2		2240.2.k.c.1791.1		8
8.5	even	2		2240.2.k.c.1791.7		8
12.11	even	2		5040.2.d.e.4591.4		8
20.3	even	4		2800.2.e.i.2799.8		8
20.7	even	4		2800.2.e.j.2799.1		8
20.19	odd	2		2800.2.k.l.2351.1		8
21.20	even	2		5040.2.d.e.4591.8		8
28.27	even	2	inner	560.2.k.a.111.1	✓	8
35.13	even	4		2800.2.e.j.2799.7		8
35.27	even	4		2800.2.e.i.2799.2		8
35.34	odd	2		2800.2.k.l.2351.2		8
56.13	odd	2		2240.2.k.c.1791.2		8
56.27	even	2		2240.2.k.c.1791.8		8
84.83	odd	2		5040.2.d.e.4591.5		8
140.27	odd	4		2800.2.e.i.2799.7		8
140.83	odd	4		2800.2.e.j.2799.2		8
140.139	even	2		2800.2.k.l.2351.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.k.a.111.1	✓	8	28.27	even	2	inner
560.2.k.a.111.2	yes	8	1.1	even	1	trivial
560.2.k.a.111.7	yes	8	7.6	odd	2	inner
560.2.k.a.111.8	yes	8	4.3	odd	2	inner
2240.2.k.c.1791.1		8	8.3	odd	2
2240.2.k.c.1791.2		8	56.13	odd	2
2240.2.k.c.1791.7		8	8.5	even	2
2240.2.k.c.1791.8		8	56.27	even	2
2800.2.e.i.2799.1		8	5.3	odd	4
2800.2.e.i.2799.2		8	35.27	even	4
2800.2.e.i.2799.7		8	140.27	odd	4
2800.2.e.i.2799.8		8	20.3	even	4
2800.2.e.j.2799.1		8	20.7	even	4
2800.2.e.j.2799.2		8	140.83	odd	4
2800.2.e.j.2799.7		8	35.13	even	4
2800.2.e.j.2799.8		8	5.2	odd	4
2800.2.k.l.2351.1		8	20.19	odd	2
2800.2.k.l.2351.2		8	35.34	odd	2
2800.2.k.l.2351.7		8	140.139	even	2
2800.2.k.l.2351.8		8	5.4	even	2
5040.2.d.e.4591.1		8	3.2	odd	2
5040.2.d.e.4591.4		8	12.11	even	2
5040.2.d.e.4591.5		8	84.83	odd	2
5040.2.d.e.4591.8		8	21.20	even	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.52434 q^{3} +1.00000i q^{5} +(-2.52434 - 0.792287i) q^{7} +3.37228 q^{9} +O(q^{10})\)	\(q-2.52434 q^{3} +1.00000i q^{5} +(-2.52434 - 0.792287i) q^{7} +3.37228 q^{9} +2.52434i q^{11} -0.372281i q^{13} -2.52434i q^{15} -2.37228i q^{17} +3.46410 q^{19} +(6.37228 + 2.00000i) q^{21} -8.51278i q^{23} -1.00000 q^{25} -0.939764 q^{27} +0.372281 q^{29} +1.87953 q^{31} -6.37228i q^{33} +(0.792287 - 2.52434i) q^{35} +10.7446 q^{37} +0.939764i q^{39} -8.74456i q^{41} -3.46410i q^{43} +3.37228i q^{45} +7.86797 q^{47} +(5.74456 + 4.00000i) q^{49} +5.98844i q^{51} -11.4891 q^{53} -2.52434 q^{55} -8.74456 q^{57} +6.63325 q^{59} -10.7446i q^{61} +(-8.51278 - 2.67181i) q^{63} +0.372281 q^{65} +6.63325i q^{67} +21.4891i q^{69} +6.63325i q^{71} -8.74456i q^{73} +2.52434 q^{75} +(2.00000 - 6.37228i) q^{77} -15.7908i q^{79} -7.74456 q^{81} -10.3923 q^{83} +2.37228 q^{85} -0.939764 q^{87} +5.48913i q^{89} +(-0.294954 + 0.939764i) q^{91} -4.74456 q^{93} +3.46410i q^{95} +15.1168i q^{97} +8.51278i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} + 28 q^{21} - 8 q^{25} - 20 q^{29} + 40 q^{37} - 24 q^{57} - 20 q^{65} + 16 q^{77} - 16 q^{81} - 4 q^{85} + 8 q^{93}+O(q^{100}) \) 8 * q + 4 * q^9 + 28 * q^21 - 8 * q^25 - 20 * q^29 + 40 * q^37 - 24 * q^57 - 20 * q^65 + 16 * q^77 - 16 * q^81 - 4 * q^85 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more