LMFDB - Embedded newform 2240.2.n.a.1119.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1119,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))

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

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

chi := DirichletCharacter("2240.1119");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.n (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1731891456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
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			1119.8
Root			$1.35234 - 0.780776i$ of defining polynomial
Character	$\chi$	$=$	2240.1119
Dual form			2240.2.n.a.1119.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +(-2.00000 + 1.00000i) q^{5} +(2.00000 + 1.73205i) q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} +(-2.00000 + 1.00000i) q^{5} +(2.00000 + 1.73205i) q^{7} -0.561553 q^{9} +6.16879 q^{11} +3.56155i q^{13} +(-3.12311 + 1.56155i) q^{15} -2.70469 q^{17} -7.12311i q^{19} +(3.12311 + 2.70469i) q^{21} +7.12311 q^{23} +(3.00000 - 4.00000i) q^{25} -5.56155 q^{27} +2.70469i q^{29} +5.40938 q^{31} +9.63289 q^{33} +(-5.73205 - 1.46410i) q^{35} -5.40938 q^{37} +5.56155i q^{39} +5.40938i q^{41} +12.3376i q^{43} +(1.12311 - 0.561553i) q^{45} +6.16879i q^{47} +(1.00000 + 6.92820i) q^{49} -4.22351 q^{51} +(-12.3376 + 6.16879i) q^{55} -11.1231i q^{57} -4.00000i q^{59} -7.12311 q^{61} +(-1.12311 - 0.972638i) q^{63} +(-3.56155 - 7.12311i) q^{65} +6.92820i q^{67} +11.1231 q^{69} -2.24621i q^{71} +6.92820 q^{73} +(4.68466 - 6.24621i) q^{75} +(12.3376 + 10.6847i) q^{77} +4.68466i q^{79} -7.00000 q^{81} +4.00000 q^{83} +(5.40938 - 2.70469i) q^{85} +4.22351i q^{87} -5.40938i q^{89} +(-6.16879 + 7.12311i) q^{91} +8.44703 q^{93} +(7.12311 + 14.2462i) q^{95} +16.5611 q^{97} -3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 16 q^{5} + 16 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 - 16 * q^5 + 16 * q^7 + 12 * q^9	$ 8 q - 4 q^{3} - 16 q^{5} + 16 q^{7} + 12 q^{9} + 8 q^{15} - 8 q^{21} + 24 q^{23} + 24 q^{25} - 28 q^{27} - 32 q^{35} - 24 q^{45} + 8 q^{49} - 24 q^{61} + 24 q^{63} - 12 q^{65} + 56 q^{69} - 12 q^{75} - 56 q^{81} + 32 q^{83} + 24 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 - 16 * q^5 + 16 * q^7 + 12 * q^9 + 8 * q^15 - 8 * q^21 + 24 * q^23 + 24 * q^25 - 28 * q^27 - 32 * q^35 - 24 * q^45 + 8 * q^49 - 24 * q^61 + 24 * q^63 - 12 * q^65 + 56 * q^69 - 12 * q^75 - 56 * q^81 + 32 * q^83 + 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$3$	1.56155			0.901563			0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155			0.901563			0.450781	+	0.892634i	$0.351145\pi$
$4$		0			0
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$	−0.561553			−0.187184
$10$		0			0
$11$	6.16879			1.85996			0.929980	−	0.367610i	$-0.119824\pi$
$11$	6.16879			1.85996			0.929980	+	0.367610i	$0.119824\pi$
$12$		0			0
$13$			3.56155i			0.987797i	0.869520	+	0.493899i	$0.164429\pi$
$13$			3.56155i			0.987797i	−0.869520	+	0.493899i	$0.835571\pi$
$14$		0			0
$15$	−3.12311	+	1.56155i	−0.806382	+	0.403191i
$16$		0			0
$17$	−2.70469			−0.655983			−0.327992	−	0.944681i	$-0.606372\pi$
$17$	−2.70469			−0.655983			−0.327992	+	0.944681i	$0.606372\pi$
$18$		0			0
$19$		−	7.12311i		−	1.63415i	−0.576530	−	0.817076i	$-0.695593\pi$
$19$		−	7.12311i		−	1.63415i	0.576530	−	0.817076i	$-0.304407\pi$
$20$		0			0
$21$	3.12311	+	2.70469i	0.681518	+	0.590211i
$22$		0			0
$23$	7.12311			1.48527			0.742635	−	0.669696i	$-0.233576\pi$
$23$	7.12311			1.48527			0.742635	+	0.669696i	$0.233576\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$	−5.56155			−1.07032
$28$		0			0
$29$			2.70469i			0.502248i	0.967955	+	0.251124i	$0.0808002\pi$
$29$			2.70469i			0.502248i	−0.967955	+	0.251124i	$0.919200\pi$
$30$		0			0
$31$	5.40938			0.971553			0.485776	−	0.874083i	$-0.338537\pi$
$31$	5.40938			0.971553			0.485776	+	0.874083i	$0.338537\pi$
$32$		0			0
$33$	9.63289			1.67687
$34$		0			0
$35$	−5.73205	−	1.46410i	−0.968893	−	0.247478i
$36$		0			0
$37$	−5.40938			−0.889296			−0.444648	−	0.895705i	$-0.646671\pi$
$37$	−5.40938			−0.889296			−0.444648	+	0.895705i	$0.646671\pi$
$38$		0			0
$39$			5.56155i			0.890561i
$40$		0			0
$41$			5.40938i			0.844803i	0.906409	+	0.422401i	$0.138813\pi$
$41$			5.40938i			0.844803i	−0.906409	+	0.422401i	$0.861187\pi$
$42$		0			0
$43$			12.3376i			1.88146i	0.339152	+	0.940732i	$0.389860\pi$
$43$			12.3376i			1.88146i	−0.339152	+	0.940732i	$0.610140\pi$
$44$		0			0
$45$	1.12311	−	0.561553i	0.167423	−	0.0837114i
$46$		0			0
$47$			6.16879i			0.899811i	0.893076	+	0.449905i	$0.148542\pi$
$47$			6.16879i			0.899811i	−0.893076	+	0.449905i	$0.851458\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$	−4.22351			−0.591410
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−12.3376	+	6.16879i	−1.66360	+	0.831800i
$56$		0			0
$57$		−	11.1231i		−	1.47329i
$58$		0			0
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	−7.12311			−0.912020			−0.456010	−	0.889975i	$-0.650722\pi$
$61$	−7.12311			−0.912020			−0.456010	+	0.889975i	$0.650722\pi$
$62$		0			0
$63$	−1.12311	−	0.972638i	−0.141498	−	0.122541i
$64$		0			0
$65$	−3.56155	−	7.12311i	−0.441756	−	0.883513i
$66$		0			0
$67$			6.92820i			0.846415i	0.906033	+	0.423207i	$0.139096\pi$
$67$			6.92820i			0.846415i	−0.906033	+	0.423207i	$0.860904\pi$
$68$		0			0
$69$	11.1231			1.33906
$70$		0			0
$71$		−	2.24621i		−	0.266576i	−0.991077	−	0.133288i	$-0.957446\pi$
$71$		−	2.24621i		−	0.266576i	0.991077	−	0.133288i	$-0.0425536\pi$
$72$		0			0
$73$	6.92820			0.810885			0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820			0.810885			0.405442	+	0.914121i	$0.367117\pi$
$74$		0			0
$75$	4.68466	−	6.24621i	0.540938	−	0.721250i
$76$		0			0
$77$	12.3376	+	10.6847i	1.40600	+	1.21763i
$78$		0			0
$79$			4.68466i			0.527065i	0.964650	+	0.263533i	$0.0848877\pi$
$79$			4.68466i			0.527065i	−0.964650	+	0.263533i	$0.915112\pi$
$80$		0			0
$81$	−7.00000			−0.777778
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$	5.40938	−	2.70469i	0.586729	−	0.293365i
$86$		0			0
$87$			4.22351i			0.452808i
$88$		0			0
$89$		−	5.40938i		−	0.573393i	−0.958021	−	0.286696i	$-0.907443\pi$
$89$		−	5.40938i		−	0.573393i	0.958021	−	0.286696i	$-0.0925571\pi$
$90$		0			0
$91$	−6.16879	+	7.12311i	−0.646665	+	0.746704i
$92$		0			0
$93$	8.44703			0.875916
$94$		0			0
$95$	7.12311	+	14.2462i	0.730815	+	1.46163i
$96$		0			0
$97$	16.5611			1.68152			0.840762	−	0.541405i	$-0.182107\pi$
$97$	16.5611			1.68152			0.840762	+	0.541405i	$0.182107\pi$
$98$		0			0
$99$	−3.46410			−0.348155
$100$		0			0
$101$	−13.3693			−1.33030			−0.665148	−	0.746711i	$-0.731632\pi$
$101$	−13.3693			−1.33030			−0.665148	+	0.746711i	$0.731632\pi$
$102$		0			0
$103$			7.68762i			0.757483i	0.925502	+	0.378742i	$0.123643\pi$
$103$			7.68762i			0.757483i	−0.925502	+	0.378742i	$0.876357\pi$
$104$		0			0
$105$	−8.95090	−	2.28627i	−0.873518	−	0.223117i
$106$		0			0
$107$		−	17.7470i		−	1.71566i	−0.513931	−	0.857832i	$-0.671811\pi$
$107$		−	17.7470i		−	1.71566i	0.513931	−	0.857832i	$-0.328189\pi$
$108$		0			0
$109$			11.1517i			1.06814i	0.845440	+	0.534070i	$0.179338\pi$
$109$			11.1517i			1.06814i	−0.845440	+	0.534070i	$0.820662\pi$
$110$		0			0
$111$	−8.44703			−0.801756
$112$		0			0
$113$			9.36932i			0.881391i	0.897657	+	0.440696i	$0.145268\pi$
$113$			9.36932i			0.881391i	−0.897657	+	0.440696i	$0.854732\pi$
$114$		0			0
$115$	−14.2462	+	7.12311i	−1.32847	+	0.664233i
$116$		0			0
$117$		−	2.00000i		−	0.184900i
$118$		0			0
$119$	−5.40938	−	4.68466i	−0.495877	−	0.429442i
$120$		0			0
$121$	27.0540			2.45945
$122$		0			0
$123$			8.44703i			0.761643i
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$			19.2658i			1.69626i
$130$		0			0
$131$			5.36932i			0.469119i	0.972102	+	0.234560i	$0.0753648\pi$
$131$			5.36932i			0.469119i	−0.972102	+	0.234560i	$0.924635\pi$
$132$		0			0
$133$	12.3376	−	14.2462i	1.06980	−	1.23530i
$134$		0			0
$135$	11.1231	−	5.56155i	0.957325	−	0.478662i
$136$		0			0
$137$		−	4.87689i		−	0.416661i	−0.978058	−	0.208331i	$-0.933197\pi$
$137$		−	4.87689i		−	0.416661i	0.978058	−	0.208331i	$-0.0668030\pi$
$138$		0			0
$139$			2.24621i			0.190521i	0.995452	+	0.0952606i	$0.0303684\pi$
$139$			2.24621i			0.190521i	−0.995452	+	0.0952606i	$0.969632\pi$
$140$		0			0
$141$			9.63289i			0.811236i
$142$		0			0
$143$			21.9705i			1.83726i
$144$		0			0
$145$	−2.70469	−	5.40938i	−0.224612	−	0.449224i
$146$		0			0
$147$	1.56155	+	10.8188i	0.128795	+	0.892316i
$148$		0			0
$149$		−	17.7470i		−	1.45389i	−0.686697	−	0.726944i	$-0.740940\pi$
$149$		−	17.7470i		−	1.45389i	0.686697	−	0.726944i	$-0.259060\pi$
$150$		0			0
$151$		−	22.0540i		−	1.79473i	−0.441292	−	0.897364i	$-0.645480\pi$
$151$		−	22.0540i		−	1.79473i	0.441292	−	0.897364i	$-0.354520\pi$
$152$		0			0
$153$	1.51883			0.122790
$154$		0			0
$155$	−10.8188	+	5.40938i	−0.868983	+	0.434492i
$156$		0			0
$157$			3.75379i			0.299585i	0.988717	+	0.149792i	$0.0478606\pi$
$157$			3.75379i			0.299585i	−0.988717	+	0.149792i	$0.952139\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	14.2462	+	12.3376i	1.12276	+	0.972338i
$162$		0			0
$163$		−	3.89055i		−	0.304732i	−0.988324	−	0.152366i	$-0.951311\pi$
$163$		−	3.89055i		−	0.304732i	0.988324	−	0.152366i	$-0.0486892\pi$
$164$		0			0
$165$	−19.2658	+	9.63289i	−1.49984	+	0.749920i
$166$		0			0
$167$		−	18.5064i		−	1.43207i	−0.698066	−	0.716033i	$-0.745956\pi$
$167$		−	18.5064i		−	1.43207i	0.698066	−	0.716033i	$-0.254044\pi$
$168$		0			0
$169$	0.315342			0.0242570
$170$		0			0
$171$			4.00000i			0.305888i
$172$		0			0
$173$		−	2.68466i		−	0.204111i	−0.994779	−	0.102055i	$-0.967458\pi$
$173$		−	2.68466i		−	0.204111i	0.994779	−	0.102055i	$-0.0325419\pi$
$174$		0			0
$175$	12.9282	−	2.80385i	0.977280	−	0.211951i
$176$		0			0
$177$		−	6.24621i		−	0.469494i
$178$		0			0
$179$	−7.35465			−0.549713			−0.274856	−	0.961485i	$-0.588630\pi$
$179$	−7.35465			−0.549713			−0.274856	+	0.961485i	$0.588630\pi$
$180$		0			0
$181$	−2.24621			−0.166960			−0.0834798	−	0.996509i	$-0.526603\pi$
$181$	−2.24621			−0.166960			−0.0834798	+	0.996509i	$0.526603\pi$
$182$		0			0
$183$	−11.1231			−0.822244
$184$		0			0
$185$	10.8188	−	5.40938i	0.795411	−	0.397705i
$186$		0			0
$187$	−16.6847			−1.22010
$188$		0			0
$189$	−11.1231	−	9.63289i	−0.809087	−	0.700690i
$190$		0			0
$191$			9.56155i			0.691850i	0.938262	+	0.345925i	$0.112435\pi$
$191$			9.56155i			0.691850i	−0.938262	+	0.345925i	$0.887565\pi$
$192$		0			0
$193$		−	17.3693i		−	1.25027i	−0.780516	−	0.625135i	$-0.785044\pi$
$193$		−	17.3693i		−	1.25027i	0.780516	−	0.625135i	$-0.214956\pi$
$194$		0			0
$195$	−5.56155	−	11.1231i	−0.398271	−	0.796542i
$196$		0			0
$197$	−8.44703			−0.601826			−0.300913	−	0.953652i	$-0.597291\pi$
$197$	−8.44703			−0.601826			−0.300913	+	0.953652i	$0.597291\pi$
$198$		0			0
$199$	−10.8188			−0.766921			−0.383461	−	0.923557i	$-0.625268\pi$
$199$	−10.8188			−0.766921			−0.383461	+	0.923557i	$0.625268\pi$
$200$		0			0
$201$			10.8188i			0.763096i
$202$		0			0
$203$	−4.68466	+	5.40938i	−0.328799	+	0.379664i
$204$		0			0
$205$	−5.40938	−	10.8188i	−0.377807	−	0.755615i
$206$		0			0
$207$	−4.00000			−0.278019
$208$		0			0
$209$		−	43.9409i		−	3.03946i
$210$		0			0
$211$	7.68762			0.529237			0.264619	−	0.964353i	$-0.414754\pi$
$211$	7.68762			0.529237			0.264619	+	0.964353i	$0.414754\pi$
$212$		0			0
$213$		−	3.50758i		−	0.240335i
$214$		0			0
$215$	−12.3376	−	24.6752i	−0.841416	−	1.68283i
$216$		0			0
$217$	10.8188	+	9.36932i	0.734425	+	0.636031i
$218$		0			0
$219$	10.8188			0.731064
$220$		0			0
$221$		−	9.63289i		−	0.647978i
$222$		0			0
$223$			6.16879i			0.413093i	0.978437	+	0.206546i	$0.0662224\pi$
$223$			6.16879i			0.413093i	−0.978437	+	0.206546i	$0.933778\pi$
$224$		0			0
$225$	−1.68466	+	2.24621i	−0.112311	+	0.149747i
$226$		0			0
$227$	−3.31534			−0.220047			−0.110023	−	0.993929i	$-0.535093\pi$
$227$	−3.31534			−0.220047			−0.110023	+	0.993929i	$0.535093\pi$
$228$		0			0
$229$	7.12311			0.470708			0.235354	−	0.971910i	$-0.424375\pi$
$229$	7.12311			0.470708			0.235354	+	0.971910i	$0.424375\pi$
$230$		0			0
$231$	19.2658	+	16.6847i	1.26760	+	1.09777i
$232$		0			0
$233$		−	4.49242i		−	0.294308i	−0.989114	−	0.147154i	$-0.952989\pi$
$233$		−	4.49242i		−	0.294308i	0.989114	−	0.147154i	$-0.0470114\pi$
$234$		0			0
$235$	−6.16879	−	12.3376i	−0.402408	−	0.804815i
$236$		0			0
$237$			7.31534i			0.475182i
$238$		0			0
$239$			4.68466i			0.303025i	0.988455	+	0.151513i	$0.0484144\pi$
$239$			4.68466i			0.303025i	−0.988455	+	0.151513i	$0.951586\pi$
$240$		0			0
$241$			19.2658i			1.24102i	0.784199	+	0.620509i	$0.213074\pi$
$241$			19.2658i			1.24102i	−0.784199	+	0.620509i	$0.786926\pi$
$242$		0			0
$243$	5.75379			0.369106
$244$		0			0
$245$	−8.92820	−	12.8564i	−0.570402	−	0.821366i
$246$		0			0
$247$	25.3693			1.61421
$248$		0			0
$249$	6.24621			0.395838
$250$		0			0
$251$		−	20.0000i		−	1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$		−	20.0000i		−	1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$		0			0
$253$	43.9409			2.76254
$254$		0			0
$255$	8.44703	−	4.22351i	0.528973	−	0.264487i
$256$		0			0
$257$	−6.92820			−0.432169			−0.216085	−	0.976375i	$-0.569329\pi$
$257$	−6.92820			−0.432169			−0.216085	+	0.976375i	$0.569329\pi$
$258$		0			0
$259$	−10.8188	−	9.36932i	−0.672245	−	0.582181i
$260$		0			0
$261$		−	1.51883i		−	0.0940129i
$262$		0			0
$263$	−12.0000			−0.739952			−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000			−0.739952			−0.369976	+	0.929041i	$0.620634\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	8.44703i		−	0.516950i
$268$		0			0
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$	−8.44703			−0.513120			−0.256560	−	0.966528i	$-0.582589\pi$
$271$	−8.44703			−0.513120			−0.256560	+	0.966528i	$0.582589\pi$
$272$		0			0
$273$	−9.63289	+	11.1231i	−0.583009	+	0.673201i
$274$		0			0
$275$	18.5064	−	24.6752i	1.11598	−	1.48797i
$276$		0			0
$277$	−13.8564			−0.832551			−0.416275	−	0.909239i	$-0.636665\pi$
$277$	−13.8564			−0.832551			−0.416275	+	0.909239i	$0.636665\pi$
$278$		0			0
$279$	−3.03765			−0.181859
$280$		0			0
$281$	−13.3153			−0.794327			−0.397163	−	0.917748i	$-0.630005\pi$
$281$	−13.3153			−0.794327			−0.397163	+	0.917748i	$0.630005\pi$
$282$		0			0
$283$	9.56155			0.568375			0.284188	−	0.958769i	$-0.408276\pi$
$283$	9.56155			0.568375			0.284188	+	0.958769i	$0.408276\pi$
$284$		0			0
$285$	11.1231	+	22.2462i	0.658876	+	1.31775i
$286$		0			0
$287$	−9.36932	+	10.8188i	−0.553053	+	0.638611i
$288$		0			0
$289$	−9.68466			−0.569686
$290$		0			0
$291$	25.8610			1.51600
$292$		0			0
$293$		−	26.6847i		−	1.55893i	−0.626443	−	0.779467i	$-0.715490\pi$
$293$		−	26.6847i		−	1.55893i	0.626443	−	0.779467i	$-0.284510\pi$
$294$		0			0
$295$	4.00000	+	8.00000i	0.232889	+	0.465778i
$296$		0			0
$297$	−34.3081			−1.99076
$298$		0			0
$299$			25.3693i			1.46715i
$300$		0			0
$301$	−21.3693	+	24.6752i	−1.23171	+	1.42225i
$302$		0			0
$303$	−20.8769			−1.19935
$304$		0			0
$305$	14.2462	−	7.12311i	0.815736	−	0.407868i
$306$		0			0
$307$	18.9309			1.08044			0.540221	−	0.841523i	$-0.318341\pi$
$307$	18.9309			1.08044			0.540221	+	0.841523i	$0.318341\pi$
$308$		0			0
$309$			12.0046i			0.682919i
$310$		0			0
$311$	33.1222			1.87819			0.939094	−	0.343662i	$-0.111667\pi$
$311$	33.1222			1.87819			0.939094	+	0.343662i	$0.111667\pi$
$312$		0			0
$313$	11.1517			0.630332			0.315166	−	0.949037i	$-0.397940\pi$
$313$	11.1517			0.630332			0.315166	+	0.949037i	$0.397940\pi$
$314$		0			0
$315$	3.21885	+	0.822170i	0.181362	+	0.0463241i
$316$		0			0
$317$	−10.8188			−0.607642			−0.303821	−	0.952729i	$-0.598262\pi$
$317$	−10.8188			−0.607642			−0.303821	+	0.952729i	$0.598262\pi$
$318$		0			0
$319$			16.6847i			0.934162i
$320$		0			0
$321$		−	27.7128i		−	1.54678i
$322$		0			0
$323$			19.2658i			1.07198i
$324$		0			0
$325$	14.2462	+	10.6847i	0.790238	+	0.592678i
$326$		0			0
$327$			17.4140i			0.962996i
$328$		0			0
$329$	−10.6847	+	12.3376i	−0.589064	+	0.680193i
$330$		0			0
$331$	−17.3205			−0.952021			−0.476011	−	0.879440i	$-0.657918\pi$
$331$	−17.3205			−0.952021			−0.476011	+	0.879440i	$0.657918\pi$
$332$		0			0
$333$	3.03765			0.166462
$334$		0			0
$335$	−6.92820	−	13.8564i	−0.378528	−	0.757056i
$336$		0			0
$337$			26.7386i			1.45655i	0.685287	+	0.728273i	$0.259677\pi$
$337$			26.7386i			1.45655i	−0.685287	+	0.728273i	$0.740323\pi$
$338$		0			0
$339$			14.6307i			0.794630i
$340$		0			0
$341$	33.3693			1.80705
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$	−22.2462	+	11.1231i	−1.19770	+	0.598848i
$346$		0			0
$347$		−	1.51883i		−	0.0815348i	−0.999169	−	0.0407674i	$-0.987020\pi$
$347$		−	1.51883i		−	0.0815348i	0.999169	−	0.0407674i	$-0.0129803\pi$
$348$		0			0
$349$	7.12311			0.381291			0.190646	−	0.981659i	$-0.438942\pi$
$349$	7.12311			0.381291			0.190646	+	0.981659i	$0.438942\pi$
$350$		0			0
$351$		−	19.8078i		−	1.05726i
$352$		0			0
$353$	−11.1517			−0.593546			−0.296773	−	0.954948i	$-0.595910\pi$
$353$	−11.1517			−0.593546			−0.296773	+	0.954948i	$0.595910\pi$
$354$		0			0
$355$	2.24621	+	4.49242i	0.119217	+	0.238433i
$356$		0			0
$357$	−8.44703	−	7.31534i	−0.447064	−	0.387169i
$358$		0			0
$359$		−	12.0000i		−	0.633336i	−0.948536	−	0.316668i	$-0.897436\pi$
$359$		−	12.0000i		−	0.633336i	0.948536	−	0.316668i	$-0.102564\pi$
$360$		0			0
$361$	−31.7386			−1.67045
$362$		0			0
$363$	42.2462			2.21735
$364$		0			0
$365$	−13.8564	+	6.92820i	−0.725277	+	0.362639i
$366$		0			0
$367$		−	27.8063i		−	1.45148i	−0.687971	−	0.725739i	$-0.741498\pi$
$367$		−	27.8063i		−	1.45148i	0.687971	−	0.725739i	$-0.258502\pi$
$368$		0			0
$369$		−	3.03765i		−	0.158134i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−16.2281			−0.840261			−0.420130	−	0.907464i	$-0.638016\pi$
$373$	−16.2281			−0.840261			−0.420130	+	0.907464i	$0.638016\pi$
$374$		0			0
$375$	−3.12311	+	17.1771i	−0.161276	+	0.887021i
$376$		0			0
$377$	−9.63289			−0.496119
$378$		0			0
$379$	0.426450			0.0219053			0.0109526	−	0.999940i	$-0.496514\pi$
$379$	0.426450			0.0219053			0.0109526	+	0.999940i	$0.496514\pi$
$380$		0			0
$381$	18.7386			0.960009
$382$		0			0
$383$		−	28.1393i		−	1.43785i	−0.695088	−	0.718925i	$-0.744635\pi$
$383$		−	28.1393i		−	1.43785i	0.695088	−	0.718925i	$-0.255365\pi$
$384$		0			0
$385$	−35.3598	−	9.03174i	−1.80210	−	0.460300i
$386$		0			0
$387$		−	6.92820i		−	0.352180i
$388$		0			0
$389$		−	21.9705i		−	1.11395i	−0.830530	−	0.556974i	$-0.811962\pi$
$389$		−	21.9705i		−	1.11395i	0.830530	−	0.556974i	$-0.188038\pi$
$390$		0			0
$391$	−19.2658			−0.974313
$392$		0			0
$393$			8.38447i			0.422941i
$394$		0			0
$395$	−4.68466	−	9.36932i	−0.235711	−	0.471421i
$396$		0			0
$397$			24.9309i			1.25124i	0.780126	+	0.625622i	$0.215155\pi$
$397$			24.9309i			1.25124i	−0.780126	+	0.625622i	$0.784845\pi$
$398$		0			0
$399$	19.2658	−	22.2462i	0.964496	−	1.11370i
$400$		0			0
$401$	17.8078			0.889277			0.444639	−	0.895710i	$-0.353332\pi$
$401$	17.8078			0.889277			0.444639	+	0.895710i	$0.353332\pi$
$402$		0			0
$403$			19.2658i			0.959697i
$404$		0			0
$405$	14.0000	−	7.00000i	0.695666	−	0.347833i
$406$		0			0
$407$	−33.3693			−1.65406
$408$		0			0
$409$			3.03765i			0.150202i	0.997176	+	0.0751011i	$0.0239279\pi$
$409$			3.03765i			0.150202i	−0.997176	+	0.0751011i	$0.976072\pi$
$410$		0			0
$411$		−	7.61553i		−	0.375646i
$412$		0			0
$413$	6.92820	−	8.00000i	0.340915	−	0.393654i
$414$		0			0
$415$	−8.00000	+	4.00000i	−0.392705	+	0.196352i
$416$		0			0
$417$			3.50758i			0.171767i
$418$		0			0
$419$		−	22.7386i		−	1.11085i	−0.831565	−	0.555427i	$-0.812555\pi$
$419$		−	22.7386i		−	1.11085i	0.831565	−	0.555427i	$-0.187445\pi$
$420$		0			0
$421$		−	2.70469i		−	0.131818i	−0.997826	−	0.0659092i	$-0.979005\pi$
$421$		−	2.70469i		−	0.131818i	0.997826	−	0.0659092i	$-0.0209948\pi$
$422$		0			0
$423$		−	3.46410i		−	0.168430i
$424$		0			0
$425$	−8.11407	+	10.8188i	−0.393590	+	0.524787i
$426$		0			0
$427$	−14.2462	−	12.3376i	−0.689422	−	0.597057i
$428$		0			0
$429$			34.3081i			1.65641i
$430$		0			0
$431$		−	28.6847i		−	1.38169i	−0.723002	−	0.690846i	$-0.757238\pi$
$431$		−	28.6847i		−	1.38169i	0.723002	−	0.690846i	$-0.242762\pi$
$432$		0			0
$433$	31.6034			1.51876			0.759380	−	0.650647i	$-0.225502\pi$
$433$	31.6034			1.51876			0.759380	+	0.650647i	$0.225502\pi$
$434$		0			0
$435$	−4.22351	−	8.44703i	−0.202502	−	0.405004i
$436$		0			0
$437$		−	50.7386i		−	2.42716i
$438$		0			0
$439$	−13.8564			−0.661330			−0.330665	−	0.943748i	$-0.607273\pi$
$439$	−13.8564			−0.661330			−0.330665	+	0.943748i	$0.607273\pi$
$440$		0			0
$441$	−0.561553	−	3.89055i	−0.0267406	−	0.185264i
$442$		0			0
$443$			15.3752i			0.730499i	0.930910	+	0.365250i	$0.119016\pi$
$443$			15.3752i			0.730499i	−0.930910	+	0.365250i	$0.880984\pi$
$444$		0			0
$445$	5.40938	+	10.8188i	0.256429	+	0.512858i
$446$		0			0
$447$		−	27.7128i		−	1.31077i
$448$		0			0
$449$	−8.05398			−0.380091			−0.190045	−	0.981775i	$-0.560863\pi$
$449$	−8.05398			−0.380091			−0.190045	+	0.981775i	$0.560863\pi$
$450$		0			0
$451$			33.3693i			1.57130i
$452$		0			0
$453$		−	34.4384i		−	1.61806i
$454$		0			0
$455$	5.21448	−	20.4150i	0.244458	−	0.957070i
$456$		0			0
$457$		−	25.3693i		−	1.18673i	−0.804935	−	0.593363i	$-0.797800\pi$
$457$		−	25.3693i		−	1.18673i	0.804935	−	0.593363i	$-0.202200\pi$
$458$		0			0
$459$	15.0423			0.702113
$460$		0			0
$461$	20.0000			0.931493			0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000			0.931493			0.465746	+	0.884918i	$0.345786\pi$
$462$		0			0
$463$	−13.3693			−0.621325			−0.310662	−	0.950520i	$-0.600551\pi$
$463$	−13.3693			−0.621325			−0.310662	+	0.950520i	$0.600551\pi$
$464$		0			0
$465$	−16.8941	+	8.44703i	−0.783443	+	0.391722i
$466$		0			0
$467$	−6.05398			−0.280145			−0.140072	−	0.990141i	$-0.544733\pi$
$467$	−6.05398			−0.280145			−0.140072	+	0.990141i	$0.544733\pi$
$468$		0			0
$469$	−12.0000	+	13.8564i	−0.554109	+	0.639829i
$470$		0			0
$471$			5.86174i			0.270095i
$472$		0			0
$473$			76.1080i			3.49945i
$474$		0			0
$475$	−28.4924	−	21.3693i	−1.30732	−	0.980492i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	33.1222			1.51339			0.756696	−	0.653767i	$-0.226812\pi$
$479$	33.1222			1.51339			0.756696	+	0.653767i	$0.226812\pi$
$480$		0			0
$481$		−	19.2658i		−	0.878444i
$482$		0			0
$483$	22.2462	+	19.2658i	1.01224	+	0.876624i
$484$		0			0
$485$	−33.1222	+	16.5611i	−1.50400	+	0.752001i
$486$		0			0
$487$	12.0000			0.543772			0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000			0.543772			0.271886	+	0.962329i	$0.412353\pi$
$488$		0			0
$489$		−	6.07530i		−	0.274735i
$490$		0			0
$491$	21.5440			0.972268			0.486134	−	0.873884i	$-0.338407\pi$
$491$	21.5440			0.972268			0.486134	+	0.873884i	$0.338407\pi$
$492$		0			0
$493$		−	7.31534i		−	0.329466i
$494$		0			0
$495$	6.92820	−	3.46410i	0.311400	−	0.155700i
$496$		0			0
$497$	3.89055	−	4.49242i	0.174515	−	0.201513i
$498$		0			0
$499$	7.68762			0.344145			0.172072	−	0.985084i	$-0.444954\pi$
$499$	7.68762			0.344145			0.172072	+	0.985084i	$0.444954\pi$
$500$		0			0
$501$		−	28.8987i		−	1.29110i
$502$		0			0
$503$		−	9.20644i		−	0.410495i	−0.978710	−	0.205247i	$-0.934200\pi$
$503$		−	9.20644i		−	0.410495i	0.978710	−	0.205247i	$-0.0657999\pi$
$504$		0			0
$505$	26.7386	−	13.3693i	1.18985	−	0.594927i
$506$		0			0
$507$	0.492423			0.0218693
$508$		0			0
$509$	4.00000			0.177297			0.0886484	−	0.996063i	$-0.471745\pi$
$509$	4.00000			0.177297			0.0886484	+	0.996063i	$0.471745\pi$
$510$		0			0
$511$	13.8564	+	12.0000i	0.612971	+	0.530849i
$512$		0			0
$513$			39.6155i			1.74907i
$514$		0			0
$515$	−7.68762	−	15.3752i	−0.338757	−	0.677514i
$516$		0			0
$517$			38.0540i			1.67361i
$518$		0			0
$519$		−	4.19224i		−	0.184019i
$520$		0			0
$521$			8.44703i			0.370071i	0.982732	+	0.185036i	$0.0592400\pi$
$521$			8.44703i			0.370071i	−0.982732	+	0.185036i	$0.940760\pi$
$522$		0			0
$523$	40.4924			1.77061			0.885305	−	0.465011i	$-0.153950\pi$
$523$	40.4924			1.77061			0.885305	+	0.465011i	$0.153950\pi$
$524$		0			0
$525$	20.1881	−	4.37836i	0.881080	−	0.191087i
$526$		0			0
$527$	−14.6307			−0.637323
$528$		0			0
$529$	27.7386			1.20603
$530$		0			0
$531$			2.24621i			0.0974773i
$532$		0			0
$533$	−19.2658			−0.834494
$534$		0			0
$535$	17.7470	+	35.4939i	0.767268	+	1.53454i
$536$		0			0
$537$	−11.4847			−0.495601
$538$		0			0
$539$	6.16879	+	42.7386i	0.265709	+	1.84088i
$540$		0			0
$541$		−	27.3799i		−	1.17715i	−0.808442	−	0.588576i	$-0.799689\pi$
$541$		−	27.3799i		−	1.17715i	0.808442	−	0.588576i	$-0.200311\pi$
$542$		0			0
$543$	−3.50758			−0.150525
$544$		0			0
$545$	−11.1517	−	22.3034i	−0.477687	−	0.955374i
$546$		0			0
$547$		−	1.51883i		−	0.0649403i	−0.999473	−	0.0324701i	$-0.989663\pi$
$547$		−	1.51883i		−	0.0649403i	0.999473	−	0.0324701i	$-0.0103374\pi$
$548$		0			0
$549$	4.00000			0.170716
$550$		0			0
$551$	19.2658			0.820750
$552$		0			0
$553$	−8.11407	+	9.36932i	−0.345045	+	0.398424i
$554$		0			0
$555$	16.8941	−	8.44703i	0.717113	−	0.358556i
$556$		0			0
$557$	13.8564			0.587115			0.293557	−	0.955941i	$-0.405161\pi$
$557$	13.8564			0.587115			0.293557	+	0.955941i	$0.405161\pi$
$558$		0			0
$559$	−43.9409			−1.85850
$560$		0			0
$561$	−26.0540			−1.10000
$562$		0			0
$563$	20.0000			0.842900			0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000			0.842900			0.421450	+	0.906852i	$0.361521\pi$
$564$		0			0
$565$	−9.36932	−	18.7386i	−0.394170	−	0.788340i
$566$		0			0
$567$	−14.0000	−	12.1244i	−0.587945	−	0.509175i
$568$		0			0
$569$	−15.7538			−0.660433			−0.330217	−	0.943905i	$-0.607122\pi$
$569$	−15.7538			−0.660433			−0.330217	+	0.943905i	$0.607122\pi$
$570$		0			0
$571$	−21.2111			−0.887655			−0.443828	−	0.896112i	$-0.646380\pi$
$571$	−21.2111			−0.887655			−0.443828	+	0.896112i	$0.646380\pi$
$572$		0			0
$573$			14.9309i			0.623746i
$574$		0			0
$575$	21.3693	−	28.4924i	0.891162	−	1.18822i
$576$		0			0
$577$	−24.3422			−1.01338			−0.506690	−	0.862129i	$-0.669131\pi$
$577$	−24.3422			−1.01338			−0.506690	+	0.862129i	$0.669131\pi$
$578$		0			0
$579$		−	27.1231i		−	1.12720i
$580$		0			0
$581$	8.00000	+	6.92820i	0.331896	+	0.287430i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	2.00000	+	4.00000i	0.0826898	+	0.165380i
$586$		0			0
$587$	28.0000			1.15568			0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000			1.15568			0.577842	+	0.816149i	$0.303895\pi$
$588$		0			0
$589$		−	38.5316i		−	1.58767i
$590$		0			0
$591$	−13.1905			−0.542584
$592$		0			0
$593$	−21.9705			−0.902219			−0.451110	−	0.892469i	$-0.648972\pi$
$593$	−21.9705			−0.902219			−0.451110	+	0.892469i	$0.648972\pi$
$594$		0			0
$595$	15.5034	+	3.95994i	0.635578	+	0.162342i
$596$		0			0
$597$	−16.8941			−0.691428
$598$		0			0
$599$		−	9.56155i		−	0.390674i	−0.980736	−	0.195337i	$-0.937420\pi$
$599$		−	9.56155i		−	0.390674i	0.980736	−	0.195337i	$-0.0625801\pi$
$600$		0			0
$601$		−	38.5316i		−	1.57174i	−0.618395	−	0.785868i	$-0.712217\pi$
$601$		−	38.5316i		−	1.57174i	0.618395	−	0.785868i	$-0.287783\pi$
$602$		0			0
$603$		−	3.89055i		−	0.158436i
$604$		0			0
$605$	−54.1080	+	27.0540i	−2.19980	+	1.09990i
$606$		0			0
$607$			4.64996i			0.188736i	0.995537	+	0.0943681i	$0.0300831\pi$
$607$			4.64996i			0.188736i	−0.995537	+	0.0943681i	$0.969917\pi$
$608$		0			0
$609$	−7.31534	+	8.44703i	−0.296433	+	0.342291i
$610$		0			0
$611$	−21.9705			−0.888830
$612$		0			0
$613$	38.5316			1.55628			0.778138	−	0.628094i	$-0.216165\pi$
$613$	38.5316			1.55628			0.778138	+	0.628094i	$0.216165\pi$
$614$		0			0
$615$	−8.44703	−	16.8941i	−0.340617	−	0.681234i
$616$		0			0
$617$		−	18.7386i		−	0.754389i	−0.926134	−	0.377194i	$-0.876889\pi$
$617$		−	18.7386i		−	0.754389i	0.926134	−	0.377194i	$-0.123111\pi$
$618$		0			0
$619$			11.6155i			0.466867i	0.972373	+	0.233434i	$0.0749962\pi$
$619$			11.6155i			0.466867i	−0.972373	+	0.233434i	$0.925004\pi$
$620$		0			0
$621$	−39.6155			−1.58972
$622$		0			0
$623$	9.36932	−	10.8188i	0.375374	−	0.433444i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		−	68.6161i		−	2.74026i
$628$		0			0
$629$	14.6307			0.583364
$630$		0			0
$631$			28.6847i			1.14192i	0.820978	+	0.570959i	$0.193429\pi$
$631$			28.6847i			1.14192i	−0.820978	+	0.570959i	$0.806571\pi$
$632$		0			0
$633$	12.0046			0.477141
$634$		0			0
$635$	−24.0000	+	12.0000i	−0.952411	+	0.476205i
$636$		0			0
$637$	−24.6752	+	3.56155i	−0.977666	+	0.141114i
$638$		0			0
$639$			1.26137i			0.0498989i
$640$		0			0
$641$	−8.24621			−0.325706			−0.162853	−	0.986650i	$-0.552070\pi$
$641$	−8.24621			−0.325706			−0.162853	+	0.986650i	$0.552070\pi$
$642$		0			0
$643$	−28.3002			−1.11605			−0.558025	−	0.829824i	$-0.688441\pi$
$643$	−28.3002			−1.11605			−0.558025	+	0.829824i	$0.688441\pi$
$644$		0			0
$645$	−19.2658	−	38.5316i	−0.758590	−	1.51718i
$646$		0			0
$647$		−	25.1016i		−	0.986846i	−0.869789	−	0.493423i	$-0.835745\pi$
$647$		−	25.1016i		−	0.986846i	0.869789	−	0.493423i	$-0.164255\pi$
$648$		0			0
$649$		−	24.6752i		−	0.968585i
$650$		0			0
$651$	16.8941	+	14.6307i	0.662130	+	0.573422i
$652$		0			0
$653$	30.0845			1.17730			0.588649	−	0.808388i	$-0.299660\pi$
$653$	30.0845			1.17730			0.588649	+	0.808388i	$0.299660\pi$
$654$		0			0
$655$	−5.36932	−	10.7386i	−0.209797	−	0.419593i
$656$		0			0
$657$	−3.89055			−0.151785
$658$		0			0
$659$	−18.5064			−0.720906			−0.360453	−	0.932777i	$-0.617378\pi$
$659$	−18.5064			−0.720906			−0.360453	+	0.932777i	$0.617378\pi$
$660$		0			0
$661$	−21.7538			−0.846124			−0.423062	−	0.906101i	$-0.639045\pi$
$661$	−21.7538			−0.846124			−0.423062	+	0.906101i	$0.639045\pi$
$662$		0			0
$663$		−	15.0423i		−	0.584193i
$664$		0			0
$665$	−10.4290	+	40.8300i	−0.404417	+	1.58332i
$666$		0			0
$667$			19.2658i			0.745974i
$668$		0			0
$669$			9.63289i			0.372429i
$670$		0			0
$671$	−43.9409			−1.69632
$672$		0			0
$673$			9.36932i			0.361161i	0.983560	+	0.180580i	$0.0577976\pi$
$673$			9.36932i			0.361161i	−0.983560	+	0.180580i	$0.942202\pi$
$674$		0			0
$675$	−16.6847	+	22.2462i	−0.642193	+	0.856257i
$676$		0			0
$677$		−	5.31534i		−	0.204285i	−0.994770	−	0.102143i	$-0.967430\pi$
$677$		−	5.31534i		−	0.204285i	0.994770	−	0.102143i	$-0.0325698\pi$
$678$		0			0
$679$	33.1222	+	28.6847i	1.27111	+	1.10082i
$680$		0			0
$681$	−5.17708			−0.198386
$682$		0			0
$683$		−	15.3752i		−	0.588317i	−0.955757	−	0.294158i	$-0.904961\pi$
$683$		−	15.3752i		−	0.588317i	0.955757	−	0.294158i	$-0.0950393\pi$
$684$		0			0
$685$	4.87689	+	9.75379i	0.186337	+	0.372673i
$686$		0			0
$687$	11.1231			0.424373
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	40.4924i		−	1.54040i	−0.637800	−	0.770202i	$-0.720155\pi$
$691$		−	40.4924i		−	1.54040i	0.637800	−	0.770202i	$-0.279845\pi$
$692$		0			0
$693$	−6.92820	−	6.00000i	−0.263181	−	0.227921i
$694$		0			0
$695$	−2.24621	−	4.49242i	−0.0852036	−	0.170407i
$696$		0			0
$697$		−	14.6307i		−	0.554177i
$698$		0			0
$699$		−	7.01515i		−	0.265338i
$700$		0			0
$701$			19.5987i			0.740234i	0.928985	+	0.370117i	$0.120682\pi$
$701$			19.5987i			0.740234i	−0.928985	+	0.370117i	$0.879318\pi$
$702$		0			0
$703$			38.5316i			1.45325i
$704$		0			0
$705$	−9.63289	−	19.2658i	−0.362796	−	0.725591i
$706$		0			0
$707$	−26.7386	−	23.1563i	−1.00561	−	0.870884i
$708$		0			0
$709$		−	21.9705i		−	0.825118i	−0.910931	−	0.412559i	$-0.864635\pi$
$709$		−	21.9705i		−	0.825118i	0.910931	−	0.412559i	$-0.135365\pi$
$710$		0			0
$711$		−	2.63068i		−	0.0986583i
$712$		0			0
$713$	38.5316			1.44302
$714$		0			0
$715$	−21.9705	−	43.9409i	−0.821649	−	1.64330i
$716$		0			0
$717$			7.31534i			0.273196i
$718$		0			0
$719$	−5.40938			−0.201736			−0.100868	−	0.994900i	$-0.532162\pi$
$719$	−5.40938			−0.201736			−0.100868	+	0.994900i	$0.532162\pi$
$720$		0			0
$721$	−13.3153	+	15.3752i	−0.495889	+	0.572604i
$722$		0			0
$723$			30.0845i			1.11886i
$724$		0			0
$725$	10.8188	+	8.11407i	0.401798	+	0.301349i
$726$		0			0
$727$		−	14.2829i		−	0.529722i	−0.964287	−	0.264861i	$-0.914674\pi$
$727$		−	14.2829i		−	0.529722i	0.964287	−	0.264861i	$-0.0853260\pi$
$728$		0			0
$729$	29.9848			1.11055
$730$		0			0
$731$		−	33.3693i		−	1.23421i
$732$		0			0
$733$		−	22.3002i		−	0.823676i	−0.911257	−	0.411838i	$-0.864887\pi$
$733$		−	22.3002i		−	0.823676i	0.911257	−	0.411838i	$-0.135113\pi$
$734$		0			0
$735$	−13.9419	−	20.0760i	−0.514253	−	0.740513i
$736$		0			0
$737$			42.7386i			1.57430i
$738$		0			0
$739$	−29.3251			−1.07874			−0.539371	−	0.842068i	$-0.681338\pi$
$739$	−29.3251			−1.07874			−0.539371	+	0.842068i	$0.681338\pi$
$740$		0			0
$741$	39.6155			1.45531
$742$		0			0
$743$	−21.7538			−0.798069			−0.399035	−	0.916936i	$-0.630655\pi$
$743$	−21.7538			−0.798069			−0.399035	+	0.916936i	$0.630655\pi$
$744$		0			0
$745$	17.7470	+	35.4939i	0.650198	+	1.30040i
$746$		0			0
$747$	−2.24621			−0.0821846
$748$		0			0
$749$	30.7386	−	35.4939i	1.12317	−	1.29692i
$750$		0			0
$751$			3.31534i			0.120979i	0.998169	+	0.0604893i	$0.0192661\pi$
$751$			3.31534i			0.120979i	−0.998169	+	0.0604893i	$0.980734\pi$
$752$		0			0
$753$		−	31.2311i		−	1.13812i
$754$		0			0
$755$	22.0540	+	44.1080i	0.802626	+	1.60525i
$756$		0			0
$757$	13.8564			0.503620			0.251810	−	0.967777i	$-0.418974\pi$
$757$	13.8564			0.503620			0.251810	+	0.967777i	$0.418974\pi$
$758$		0			0
$759$	68.6161			2.49061
$760$		0			0
$761$		−	40.9033i		−	1.48274i	−0.671095	−	0.741372i	$-0.734176\pi$
$761$		−	40.9033i		−	1.48274i	0.671095	−	0.741372i	$-0.265824\pi$
$762$		0			0
$763$	−19.3153	+	22.3034i	−0.699262	+	0.807439i
$764$		0			0
$765$	−3.03765	+	1.51883i	−0.109827	+	0.0549133i
$766$		0			0
$767$	14.2462			0.514401
$768$		0			0
$769$			24.6752i			0.889809i	0.895578	+	0.444905i	$0.146762\pi$
$769$			24.6752i			0.889809i	−0.895578	+	0.444905i	$0.853238\pi$
$770$		0			0
$771$	−10.8188			−0.389628
$772$		0			0
$773$		−	40.0540i		−	1.44064i	−0.693641	−	0.720321i	$-0.743995\pi$
$773$		−	40.0540i		−	1.44064i	0.693641	−	0.720321i	$-0.256005\pi$
$774$		0			0
$775$	16.2281	−	21.6375i	0.582932	−	0.777242i
$776$		0			0
$777$	−16.8941	−	14.6307i	−0.606071	−	0.524873i
$778$		0			0
$779$	38.5316			1.38054
$780$		0			0
$781$		−	13.8564i		−	0.495821i
$782$		0			0
$783$		−	15.0423i		−	0.537567i
$784$		0			0
$785$	−3.75379	−	7.50758i	−0.133978	−	0.267957i
$786$		0			0
$787$	−24.1922			−0.862360			−0.431180	−	0.902266i	$-0.641903\pi$
$787$	−24.1922			−0.862360			−0.431180	+	0.902266i	$0.641903\pi$
$788$		0			0
$789$	−18.7386			−0.667113
$790$		0			0
$791$	−16.2281	+	18.7386i	−0.577006	+	0.666269i
$792$		0			0
$793$		−	25.3693i		−	0.900891i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	16.0540i		−	0.568661i	−0.958726	−	0.284330i	$-0.908229\pi$
$797$		−	16.0540i		−	0.568661i	0.958726	−	0.284330i	$-0.0917713\pi$
$798$		0			0
$799$		−	16.6847i		−	0.590261i
$800$		0			0
$801$			3.03765i			0.107330i
$802$		0			0
$803$	42.7386			1.50821
$804$		0			0
$805$	−40.8300	−	10.4290i	−1.43907	−	0.367572i
$806$		0			0
$807$	−31.2311			−1.09939
$808$		0			0
$809$	−22.3002			−0.784033			−0.392016	−	0.919958i	$-0.628222\pi$
$809$	−22.3002			−0.784033			−0.392016	+	0.919958i	$0.628222\pi$
$810$		0			0
$811$		−	2.24621i		−	0.0788751i	−0.999222	−	0.0394376i	$-0.987443\pi$
$811$		−	2.24621i		−	0.0788751i	0.999222	−	0.0394376i	$-0.0125566\pi$
$812$		0			0
$813$	−13.1905			−0.462610
$814$		0			0
$815$	3.89055	+	7.78110i	0.136280	+	0.272560i
$816$		0			0
$817$	87.8819			3.07460
$818$		0			0
$819$	3.46410	−	4.00000i	0.121046	−	0.139771i
$820$		0			0
$821$		−	11.1517i		−	0.389198i	−0.980883	−	0.194599i	$-0.937660\pi$
$821$		−	11.1517i		−	0.389198i	0.980883	−	0.194599i	$-0.0623405\pi$
$822$		0			0
$823$	10.6307			0.370562			0.185281	−	0.982686i	$-0.440680\pi$
$823$	10.6307			0.370562			0.185281	+	0.982686i	$0.440680\pi$
$824$		0			0
$825$	28.8987	−	38.5316i	1.00612	−	1.34150i
$826$		0			0
$827$			34.6410i			1.20459i	0.798275	+	0.602293i	$0.205746\pi$
$827$			34.6410i			1.20459i	−0.798275	+	0.602293i	$0.794254\pi$
$828$		0			0
$829$	−11.6155			−0.403424			−0.201712	−	0.979445i	$-0.564650\pi$
$829$	−11.6155			−0.403424			−0.201712	+	0.979445i	$0.564650\pi$
$830$		0			0
$831$	−21.6375			−0.750597
$832$		0			0
$833$	−2.70469	−	18.7386i	−0.0937119	−	0.649255i
$834$		0			0
$835$	18.5064	+	37.0127i	0.640439	+	1.28088i
$836$		0			0
$837$	−30.0845			−1.03987
$838$		0			0
$839$	−46.9786			−1.62188			−0.810941	−	0.585128i	$-0.801044\pi$
$839$	−46.9786			−1.62188			−0.810941	+	0.585128i	$0.801044\pi$
$840$		0			0
$841$	21.6847			0.747747
$842$		0			0
$843$	−20.7926			−0.716135
$844$		0			0
$845$	−0.630683	+	0.315342i	−0.0216962	+	0.0108481i
$846$		0			0
$847$	54.1080	+	46.8589i	1.85917	+	1.61009i
$848$		0			0
$849$	14.9309			0.512426
$850$		0			0
$851$	−38.5316			−1.32085
$852$		0			0
$853$		−	10.4924i		−	0.359254i	−0.983735	−	0.179627i	$-0.942511\pi$
$853$		−	10.4924i		−	0.359254i	0.983735	−	0.179627i	$-0.0574890\pi$
$854$		0			0
$855$	−4.00000	−	8.00000i	−0.136797	−	0.273594i
$856$		0			0
$857$	−53.2409			−1.81867			−0.909337	−	0.416061i	$-0.863410\pi$
$857$	−53.2409			−1.81867			−0.909337	+	0.416061i	$0.863410\pi$
$858$		0			0
$859$		−	26.2462i		−	0.895509i	−0.894156	−	0.447755i	$-0.852224\pi$
$859$		−	26.2462i		−	0.895509i	0.894156	−	0.447755i	$-0.147776\pi$
$860$		0			0
$861$	−14.6307	+	16.8941i	−0.498612	+	0.575748i
$862$		0			0
$863$	−21.3693			−0.727420			−0.363710	−	0.931512i	$-0.618490\pi$
$863$	−21.3693			−0.727420			−0.363710	+	0.931512i	$0.618490\pi$
$864$		0			0
$865$	2.68466	+	5.36932i	0.0912811	+	0.182562i
$866$		0			0
$867$	−15.1231			−0.513608
$868$		0			0
$869$			28.8987i			0.980320i
$870$		0			0
$871$	−24.6752			−0.836086
$872$		0			0
$873$	−9.29993			−0.314755
$874$		0			0
$875$	−23.0526	+	18.5359i	−0.779319	+	0.626628i
$876$		0			0
$877$	16.2281			0.547985			0.273993	−	0.961732i	$-0.411656\pi$
$877$	16.2281			0.547985			0.273993	+	0.961732i	$0.411656\pi$
$878$		0			0
$879$		−	41.6695i		−	1.40548i
$880$		0			0
$881$		−	13.8564i		−	0.466834i	−0.972377	−	0.233417i	$-0.925009\pi$
$881$		−	13.8564i		−	0.466834i	0.972377	−	0.233417i	$-0.0749907\pi$
$882$		0			0
$883$			1.51883i			0.0511126i	0.999673	+	0.0255563i	$0.00813570\pi$
$883$			1.51883i			0.0511126i	−0.999673	+	0.0255563i	$0.991864\pi$
$884$		0			0
$885$	6.24621	+	12.4924i	0.209964	+	0.419928i
$886$		0			0
$887$			25.1016i			0.842830i	0.906868	+	0.421415i	$0.138466\pi$
$887$			25.1016i			0.842830i	−0.906868	+	0.421415i	$0.861534\pi$
$888$		0			0
$889$	24.0000	+	20.7846i	0.804934	+	0.697093i
$890$		0			0
$891$	−43.1815			−1.44664
$892$		0			0
$893$	43.9409			1.47043
$894$		0			0
$895$	14.7093	−	7.35465i	0.491678	−	0.245839i
$896$		0			0
$897$			39.6155i			1.32272i
$898$		0			0
$899$			14.6307i			0.487961i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−33.3693	+	38.5316i	−1.11046	+	1.28225i
$904$		0			0
$905$	4.49242	−	2.24621i	0.149333	−	0.0746666i
$906$		0			0
$907$			9.29993i			0.308799i	0.988008	+	0.154400i	$0.0493443\pi$
$907$			9.29993i			0.308799i	−0.988008	+	0.154400i	$0.950656\pi$
$908$		0			0
$909$	7.50758			0.249011
$910$		0			0
$911$			30.7386i			1.01842i	0.860643	+	0.509208i	$0.170062\pi$
$911$			30.7386i			1.01842i	−0.860643	+	0.509208i	$0.829938\pi$
$912$		0			0
$913$	24.6752			0.816629
$914$		0			0
$915$	22.2462	−	11.1231i	0.735437	−	0.367719i
$916$		0			0
$917$	−9.29993	+	10.7386i	−0.307111	+	0.354621i
$918$		0			0
$919$		−	4.68466i		−	0.154533i	−0.997010	−	0.0772663i	$-0.975381\pi$
$919$		−	4.68466i		−	0.154533i	0.997010	−	0.0772663i	$-0.0246192\pi$
$920$		0			0
$921$	29.5616			0.974086
$922$		0			0
$923$	8.00000			0.263323
$924$		0			0
$925$	−16.2281	+	21.6375i	−0.533578	+	0.711437i
$926$		0			0
$927$		−	4.31700i		−	0.141789i
$928$		0			0
$929$			38.5316i			1.26418i	0.774895	+	0.632090i	$0.217803\pi$
$929$			38.5316i			1.26418i	−0.774895	+	0.632090i	$0.782197\pi$
$930$		0			0
$931$	49.3503	−	7.12311i	1.61739	−	0.233450i
$932$		0			0
$933$	51.7220			1.69330
$934$		0			0
$935$	33.3693	−	16.6847i	1.09129	−	0.545647i
$936$		0			0
$937$	−18.9328			−0.618508			−0.309254	−	0.950979i	$-0.600079\pi$
$937$	−18.9328			−0.618508			−0.309254	+	0.950979i	$0.600079\pi$
$938$		0			0
$939$	17.4140			0.568284
$940$		0			0
$941$	−5.36932			−0.175035			−0.0875174	−	0.996163i	$-0.527893\pi$
$941$	−5.36932			−0.175035			−0.0875174	+	0.996163i	$0.527893\pi$
$942$		0			0
$943$			38.5316i			1.25476i
$944$		0			0
$945$	31.8791	+	8.14268i	1.03703	+	0.264881i
$946$		0			0
$947$			23.1563i			0.752480i	0.926522	+	0.376240i	$0.122783\pi$
$947$			23.1563i			0.752480i	−0.926522	+	0.376240i	$0.877217\pi$
$948$		0			0
$949$			24.6752i			0.800990i
$950$		0			0
$951$	−16.8941			−0.547827
$952$		0			0
$953$		−	23.6155i		−	0.764982i	−0.923959	−	0.382491i	$-0.875066\pi$
$953$		−	23.6155i		−	0.764982i	0.923959	−	0.382491i	$-0.124934\pi$
$954$		0			0
$955$	−9.56155	−	19.1231i	−0.309405	−	0.618809i
$956$		0			0
$957$			26.0540i			0.842205i
$958$		0			0
$959$	8.44703	−	9.75379i	0.272769	−	0.314966i
$960$		0			0
$961$	−1.73863			−0.0560850
$962$		0			0
$963$			9.96585i			0.321145i
$964$		0			0
$965$	17.3693	+	34.7386i	0.559138	+	1.11828i
$966$		0			0
$967$	54.7386			1.76028			0.880138	−	0.474718i	$-0.157450\pi$
$967$	54.7386			1.76028			0.880138	+	0.474718i	$0.157450\pi$
$968$		0			0
$969$			30.0845i			0.966455i
$970$		0			0
$971$			37.3693i			1.19924i	0.800285	+	0.599619i	$0.204681\pi$
$971$			37.3693i			1.19924i	−0.800285	+	0.599619i	$0.795319\pi$
$972$		0			0
$973$	−3.89055	+	4.49242i	−0.124725	+	0.144020i
$974$		0			0
$975$	22.2462	+	16.6847i	0.712449	+	0.534337i
$976$		0			0
$977$			28.8769i			0.923854i	0.886918	+	0.461927i	$0.152842\pi$
$977$			28.8769i			0.923854i	−0.886918	+	0.461927i	$0.847158\pi$
$978$		0			0
$979$		−	33.3693i		−	1.06649i
$980$		0			0
$981$		−	6.26228i		−	0.199939i
$982$		0			0
$983$			30.8440i			0.983769i	0.870660	+	0.491885i	$0.163692\pi$
$983$			30.8440i			0.983769i	−0.870660	+	0.491885i	$0.836308\pi$
$984$		0			0
$985$	16.8941	−	8.44703i	0.538289	−	0.269145i
$986$		0			0
$987$	−16.6847	+	19.2658i	−0.531079	+	0.613237i
$988$		0			0
$989$			87.8819i			2.79448i
$990$		0			0
$991$			49.4773i			1.57170i	0.618419	+	0.785849i	$0.287774\pi$
$991$			49.4773i			1.57170i	−0.618419	+	0.785849i	$0.712226\pi$
$992$		0			0
$993$	−27.0469			−0.858307
$994$		0			0
$995$	21.6375	−	10.8188i	0.685955	−	0.342978i
$996$		0			0
$997$			8.82292i			0.279425i	0.990192	+	0.139712i	$0.0446178\pi$
$997$			8.82292i			0.279425i	−0.990192	+	0.139712i	$0.955382\pi$
$998$		0			0
$999$	30.0845			0.951833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.n.a.1119.8	yes	8
4.3	odd	2		2240.2.n.g.1119.3	yes	8
5.4	even	2		2240.2.n.g.1119.1	yes	8
7.6	odd	2		2240.2.n.h.1119.1	yes	8
8.3	odd	2		2240.2.n.b.1119.5	yes	8
8.5	even	2		2240.2.n.h.1119.2	yes	8
20.19	odd	2	inner	2240.2.n.a.1119.6	yes	8
28.27	even	2		2240.2.n.b.1119.6	yes	8
35.34	odd	2		2240.2.n.b.1119.8	yes	8
40.19	odd	2		2240.2.n.h.1119.4	yes	8
40.29	even	2		2240.2.n.b.1119.7	yes	8
56.13	odd	2	inner	2240.2.n.a.1119.7	yes	8
56.27	even	2		2240.2.n.g.1119.4	yes	8
140.139	even	2		2240.2.n.h.1119.3	yes	8
280.69	odd	2		2240.2.n.g.1119.2	yes	8
280.139	even	2	inner	2240.2.n.a.1119.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.n.a.1119.5	✓	8	280.139	even	2	inner
2240.2.n.a.1119.6	yes	8	20.19	odd	2	inner
2240.2.n.a.1119.7	yes	8	56.13	odd	2	inner
2240.2.n.a.1119.8	yes	8	1.1	even	1	trivial
2240.2.n.b.1119.5	yes	8	8.3	odd	2
2240.2.n.b.1119.6	yes	8	28.27	even	2
2240.2.n.b.1119.7	yes	8	40.29	even	2
2240.2.n.b.1119.8	yes	8	35.34	odd	2
2240.2.n.g.1119.1	yes	8	5.4	even	2
2240.2.n.g.1119.2	yes	8	280.69	odd	2
2240.2.n.g.1119.3	yes	8	4.3	odd	2
2240.2.n.g.1119.4	yes	8	56.27	even	2
2240.2.n.h.1119.1	yes	8	7.6	odd	2
2240.2.n.h.1119.2	yes	8	8.5	even	2
2240.2.n.h.1119.3	yes	8	140.139	even	2
2240.2.n.h.1119.4	yes	8	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +(-2.00000 + 1.00000i) q^{5} +(2.00000 + 1.73205i) q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} +(-2.00000 + 1.00000i) q^{5} +(2.00000 + 1.73205i) q^{7} -0.561553 q^{9} +6.16879 q^{11} +3.56155i q^{13} +(-3.12311 + 1.56155i) q^{15} -2.70469 q^{17} -7.12311i q^{19} +(3.12311 + 2.70469i) q^{21} +7.12311 q^{23} +(3.00000 - 4.00000i) q^{25} -5.56155 q^{27} +2.70469i q^{29} +5.40938 q^{31} +9.63289 q^{33} +(-5.73205 - 1.46410i) q^{35} -5.40938 q^{37} +5.56155i q^{39} +5.40938i q^{41} +12.3376i q^{43} +(1.12311 - 0.561553i) q^{45} +6.16879i q^{47} +(1.00000 + 6.92820i) q^{49} -4.22351 q^{51} +(-12.3376 + 6.16879i) q^{55} -11.1231i q^{57} -4.00000i q^{59} -7.12311 q^{61} +(-1.12311 - 0.972638i) q^{63} +(-3.56155 - 7.12311i) q^{65} +6.92820i q^{67} +11.1231 q^{69} -2.24621i q^{71} +6.92820 q^{73} +(4.68466 - 6.24621i) q^{75} +(12.3376 + 10.6847i) q^{77} +4.68466i q^{79} -7.00000 q^{81} +4.00000 q^{83} +(5.40938 - 2.70469i) q^{85} +4.22351i q^{87} -5.40938i q^{89} +(-6.16879 + 7.12311i) q^{91} +8.44703 q^{93} +(7.12311 + 14.2462i) q^{95} +16.5611 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 16 q^{5} + 16 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 - 16 * q^5 + 16 * q^7 + 12 * q^9	\( 8 q - 4 q^{3} - 16 q^{5} + 16 q^{7} + 12 q^{9} + 8 q^{15} - 8 q^{21} + 24 q^{23} + 24 q^{25} - 28 q^{27} - 32 q^{35} - 24 q^{45} + 8 q^{49} - 24 q^{61} + 24 q^{63} - 12 q^{65} + 56 q^{69} - 12 q^{75} - 56 q^{81} + 32 q^{83} + 24 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 - 16 * q^5 + 16 * q^7 + 12 * q^9 + 8 * q^15 - 8 * q^21 + 24 * q^23 + 24 * q^25 - 28 * q^27 - 32 * q^35 - 24 * q^45 + 8 * q^49 - 24 * q^61 + 24 * q^63 - 12 * q^65 + 56 * q^69 - 12 * q^75 - 56 * q^81 + 32 * q^83 + 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more