LMFDB - Embedded newform 2760.2.k.e.2209.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(2209,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2760.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.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:	$22.0387109579$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 141 x^{12} - 298 x^{11} + 314 x^{10} + 144 x^{9} + 4788 x^{8} + \cdots + 20736 $ x^16 - 2x^15 + 2x^14 + 141x^12 - 298x^11 + 314x^10 + 144x^9 + 4788x^8 - 10568x^7 + 11688x^6 + 8928x^5 + 8320x^4 - 18624x^3 + 28800x^2 + 34560x + 20736
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.9
Root			$2.09040 - 2.09040i$ of defining polynomial
Character	$\chi$	$=$	2760.2209
Dual form			2760.2.k.e.2209.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.09040 - 0.793869i) q^{5} +4.15840i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-2.09040 - 0.793869i) q^{5} +4.15840i q^{7} -1.00000 q^{9} -2.25773 q^{11} +0.730564i q^{13} +(0.793869 - 2.09040i) q^{15} +4.88896i q^{17} -4.91136 q^{19} -4.15840 q^{21} -1.00000i q^{23} +(3.73954 + 3.31901i) q^{25} -1.00000i q^{27} +7.68506 q^{29} -4.81203 q^{31} -2.25773i q^{33} +(3.30122 - 8.69271i) q^{35} +1.42733i q^{37} -0.730564 q^{39} -5.48840 q^{41} -2.69970i q^{43} +(2.09040 + 0.793869i) q^{45} -13.4124i q^{47} -10.2923 q^{49} -4.88896 q^{51} +6.11730i q^{53} +(4.71957 + 1.79235i) q^{55} -4.91136i q^{57} +13.5406 q^{59} -2.27771 q^{61} -4.15840i q^{63} +(0.579972 - 1.52717i) q^{65} -7.22429i q^{67} +1.00000 q^{69} -12.0946 q^{71} +10.5827i q^{73} +(-3.31901 + 3.73954i) q^{75} -9.38856i q^{77} -14.8635 q^{79} +1.00000 q^{81} -15.4631i q^{83} +(3.88119 - 10.2199i) q^{85} +7.68506i q^{87} +4.89024 q^{89} -3.03797 q^{91} -4.81203i q^{93} +(10.2667 + 3.89898i) q^{95} +2.40242i q^{97} +2.25773 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^5 - 16 * q^9	$ 16 q - 2 q^{5} - 16 q^{9} - 2 q^{15} - 8 q^{19} + 6 q^{21} + 22 q^{29} + 30 q^{31} + 2 q^{35} - 4 q^{39} - 22 q^{41} + 2 q^{45} - 38 q^{49} + 2 q^{51} + 12 q^{55} - 6 q^{59} - 4 q^{61} + 20 q^{65} + 16 q^{69} - 10 q^{71} - 4 q^{75} - 84 q^{79} + 16 q^{81} + 22 q^{85} + 16 q^{89} + 36 q^{91} + 76 q^{95}+O(q^{100}) $ 16 * q - 2 * q^5 - 16 * q^9 - 2 * q^15 - 8 * q^19 + 6 * q^21 + 22 * q^29 + 30 * q^31 + 2 * q^35 - 4 * q^39 - 22 * q^41 + 2 * q^45 - 38 * q^49 + 2 * q^51 + 12 * q^55 - 6 * q^59 - 4 * q^61 + 20 * q^65 + 16 * q^69 - 10 * q^71 - 4 * q^75 - 84 * q^79 + 16 * q^81 + 22 * q^85 + 16 * q^89 + 36 * q^91 + 76 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1657$	$1841$	$2071$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−2.09040	−	0.793869i	−0.934855	−	0.355029i
$5$	−2.09040	−	0.793869i	−0.934855	−	0.355029i
$6$		0			0
$7$			4.15840i			1.57173i	0.618400	+	0.785863i	$0.287781\pi$
$7$			4.15840i			1.57173i	−0.618400	+	0.785863i	$0.712219\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.25773			−0.680732			−0.340366	−	0.940293i	$-0.610551\pi$
$11$	−2.25773			−0.680732			−0.340366	+	0.940293i	$0.610551\pi$
$12$		0			0
$13$			0.730564i			0.202622i	0.994855	+	0.101311i	$0.0323037\pi$
$13$			0.730564i			0.202622i	−0.994855	+	0.101311i	$0.967696\pi$
$14$		0			0
$15$	0.793869	−	2.09040i	0.204976	−	0.539739i
$16$		0			0
$17$			4.88896i			1.18575i	0.805296	+	0.592874i	$0.202007\pi$
$17$			4.88896i			1.18575i	−0.805296	+	0.592874i	$0.797993\pi$
$18$		0			0
$19$	−4.91136			−1.12674			−0.563372	−	0.826203i	$-0.690496\pi$
$19$	−4.91136			−1.12674			−0.563372	+	0.826203i	$0.690496\pi$
$20$		0			0
$21$	−4.15840			−0.907437
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	3.73954	+	3.31901i	0.747909	+	0.663802i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	7.68506			1.42708			0.713540	−	0.700615i	$-0.247091\pi$
$29$	7.68506			1.42708			0.713540	+	0.700615i	$0.247091\pi$
$30$		0			0
$31$	−4.81203			−0.864265			−0.432133	−	0.901810i	$-0.642239\pi$
$31$	−4.81203			−0.864265			−0.432133	+	0.901810i	$0.642239\pi$
$32$		0			0
$33$		−	2.25773i		−	0.393021i
$34$		0			0
$35$	3.30122	−	8.69271i	0.558008	−	1.46934i
$36$		0			0
$37$			1.42733i			0.234651i	0.993094	+	0.117325i	$0.0374320\pi$
$37$			1.42733i			0.234651i	−0.993094	+	0.117325i	$0.962568\pi$
$38$		0			0
$39$	−0.730564			−0.116984
$40$		0			0
$41$	−5.48840			−0.857144			−0.428572	−	0.903508i	$-0.640983\pi$
$41$	−5.48840			−0.857144			−0.428572	+	0.903508i	$0.640983\pi$
$42$		0			0
$43$		−	2.69970i		−	0.411700i	−0.978584	−	0.205850i	$-0.934004\pi$
$43$		−	2.69970i		−	0.411700i	0.978584	−	0.205850i	$-0.0659959\pi$
$44$		0			0
$45$	2.09040	+	0.793869i	0.311618	+	0.118343i
$46$		0			0
$47$		−	13.4124i		−	1.95640i	−0.207671	−	0.978199i	$-0.566588\pi$
$47$		−	13.4124i		−	1.95640i	0.207671	−	0.978199i	$-0.433412\pi$
$48$		0			0
$49$	−10.2923			−1.47032
$50$		0			0
$51$	−4.88896			−0.684591
$52$		0			0
$53$			6.11730i			0.840275i	0.907460	+	0.420138i	$0.138018\pi$
$53$			6.11730i			0.840275i	−0.907460	+	0.420138i	$0.861982\pi$
$54$		0			0
$55$	4.71957	+	1.79235i	0.636386	+	0.241680i
$56$		0			0
$57$		−	4.91136i		−	0.650526i
$58$		0			0
$59$	13.5406			1.76283			0.881416	−	0.472340i	$-0.156591\pi$
$59$	13.5406			1.76283			0.881416	+	0.472340i	$0.156591\pi$
$60$		0			0
$61$	−2.27771			−0.291631			−0.145815	−	0.989312i	$-0.546581\pi$
$61$	−2.27771			−0.291631			−0.145815	+	0.989312i	$0.546581\pi$
$62$		0			0
$63$		−	4.15840i		−	0.523909i
$64$		0			0
$65$	0.579972	−	1.52717i	0.0719367	−	0.189422i
$66$		0			0
$67$		−	7.22429i		−	0.882587i	−0.897363	−	0.441294i	$-0.854520\pi$
$67$		−	7.22429i		−	0.882587i	0.897363	−	0.441294i	$-0.145480\pi$
$68$		0			0
$69$	1.00000			0.120386
$70$		0			0
$71$	−12.0946			−1.43537			−0.717685	−	0.696368i	$-0.754798\pi$
$71$	−12.0946			−1.43537			−0.717685	+	0.696368i	$0.754798\pi$
$72$		0			0
$73$			10.5827i			1.23861i	0.785149	+	0.619306i	$0.212586\pi$
$73$			10.5827i			1.23861i	−0.785149	+	0.619306i	$0.787414\pi$
$74$		0			0
$75$	−3.31901	+	3.73954i	−0.383246	+	0.431805i
$76$		0			0
$77$		−	9.38856i		−	1.06993i
$78$		0			0
$79$	−14.8635			−1.67227			−0.836136	−	0.548523i	$-0.815190\pi$
$79$	−14.8635			−1.67227			−0.836136	+	0.548523i	$0.815190\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	15.4631i		−	1.69730i	−0.528955	−	0.848650i	$-0.677416\pi$
$83$		−	15.4631i		−	1.69730i	0.528955	−	0.848650i	$-0.322584\pi$
$84$		0			0
$85$	3.88119	−	10.2199i	0.420975	−	1.10850i
$86$		0			0
$87$			7.68506i			0.823925i
$88$		0			0
$89$	4.89024			0.518364			0.259182	−	0.965828i	$-0.416547\pi$
$89$	4.89024			0.518364			0.259182	+	0.965828i	$0.416547\pi$
$90$		0			0
$91$	−3.03797			−0.318466
$92$		0			0
$93$		−	4.81203i		−	0.498984i
$94$		0			0
$95$	10.2667	+	3.89898i	1.05334	+	0.400027i
$96$		0			0
$97$			2.40242i			0.243929i	0.992534	+	0.121964i	$0.0389194\pi$
$97$			2.40242i			0.243929i	−0.992534	+	0.121964i	$0.961081\pi$
$98$		0			0
$99$	2.25773			0.226911
$100$		0			0
$101$	13.1031			1.30380			0.651902	−	0.758303i	$-0.273971\pi$
$101$	13.1031			1.30380			0.651902	+	0.758303i	$0.273971\pi$
$102$		0			0
$103$			9.15986i			0.902548i	0.892385	+	0.451274i	$0.149030\pi$
$103$			9.15986i			0.902548i	−0.892385	+	0.451274i	$0.850970\pi$
$104$		0			0
$105$	8.69271	+	3.30122i	0.848322	+	0.322166i
$106$		0			0
$107$		−	17.0677i		−	1.65000i	−0.565132	−	0.825001i	$-0.691175\pi$
$107$		−	17.0677i		−	1.65000i	0.565132	−	0.825001i	$-0.308825\pi$
$108$		0			0
$109$	6.03908			0.578439			0.289220	−	0.957263i	$-0.406604\pi$
$109$	6.03908			0.578439			0.289220	+	0.957263i	$0.406604\pi$
$110$		0			0
$111$	−1.42733			−0.135476
$112$		0			0
$113$		−	2.65951i		−	0.250186i	−0.992145	−	0.125093i	$-0.960077\pi$
$113$		−	2.65951i		−	0.250186i	0.992145	−	0.125093i	$-0.0399229\pi$
$114$		0			0
$115$	−0.793869	+	2.09040i	−0.0740287	+	0.194931i
$116$		0			0
$117$		−	0.730564i		−	0.0675406i
$118$		0			0
$119$	−20.3302			−1.86367
$120$		0			0
$121$	−5.90264			−0.536603
$122$		0			0
$123$		−	5.48840i		−	0.494872i
$124$		0			0
$125$	−5.18228	−	9.90676i	−0.463518	−	0.886088i
$126$		0			0
$127$		−	8.41075i		−	0.746334i	−0.927764	−	0.373167i	$-0.878272\pi$
$127$		−	8.41075i		−	0.746334i	0.927764	−	0.373167i	$-0.121728\pi$
$128$		0			0
$129$	2.69970			0.237695
$130$		0			0
$131$	−5.39211			−0.471111			−0.235555	−	0.971861i	$-0.575691\pi$
$131$	−5.39211			−0.471111			−0.235555	+	0.971861i	$0.575691\pi$
$132$		0			0
$133$		−	20.4234i		−	1.77093i
$134$		0			0
$135$	−0.793869	+	2.09040i	−0.0683254	+	0.179913i
$136$		0			0
$137$		−	20.7356i		−	1.77156i	−0.464104	−	0.885781i	$-0.653624\pi$
$137$		−	20.7356i		−	1.77156i	0.464104	−	0.885781i	$-0.346376\pi$
$138$		0			0
$139$	−6.24481			−0.529678			−0.264839	−	0.964293i	$-0.585319\pi$
$139$	−6.24481			−0.529678			−0.264839	+	0.964293i	$0.585319\pi$
$140$		0			0
$141$	13.4124			1.12953
$142$		0			0
$143$		−	1.64942i		−	0.137931i
$144$		0			0
$145$	−16.0648	−	6.10093i	−1.33411	−	0.506655i
$146$		0			0
$147$		−	10.2923i		−	0.848892i
$148$		0			0
$149$	−18.6649			−1.52908			−0.764542	−	0.644574i	$-0.777035\pi$
$149$	−18.6649			−1.52908			−0.764542	+	0.644574i	$0.777035\pi$
$150$		0			0
$151$	11.0437			0.898727			0.449364	−	0.893349i	$-0.351651\pi$
$151$	11.0437			0.898727			0.449364	+	0.893349i	$0.351651\pi$
$152$		0			0
$153$		−	4.88896i		−	0.395249i
$154$		0			0
$155$	10.0591	+	3.82012i	0.807963	+	0.306839i
$156$		0			0
$157$		−	9.24592i		−	0.737905i	−0.929448	−	0.368952i	$-0.879717\pi$
$157$		−	9.24592i		−	0.737905i	0.929448	−	0.368952i	$-0.120283\pi$
$158$		0			0
$159$	−6.11730			−0.485133
$160$		0			0
$161$	4.15840			0.327728
$162$		0			0
$163$			18.0937i			1.41721i	0.705606	+	0.708604i	$0.250675\pi$
$163$			18.0937i			1.41721i	−0.705606	+	0.708604i	$0.749325\pi$
$164$		0			0
$165$	−1.79235	+	4.71957i	−0.139534	+	0.367418i
$166$		0			0
$167$			8.93329i			0.691279i	0.938367	+	0.345640i	$0.112338\pi$
$167$			8.93329i			0.691279i	−0.938367	+	0.345640i	$0.887662\pi$
$168$		0			0
$169$	12.4663			0.958944
$170$		0			0
$171$	4.91136			0.375581
$172$		0			0
$173$			6.00102i			0.456249i	0.973632	+	0.228124i	$0.0732593\pi$
$173$			6.00102i			0.456249i	−0.973632	+	0.228124i	$0.926741\pi$
$174$		0			0
$175$	−13.8018	+	15.5505i	−1.04331	+	1.17551i
$176$		0			0
$177$			13.5406i			1.01777i
$178$		0			0
$179$	12.7360			0.951934			0.475967	−	0.879463i	$-0.342098\pi$
$179$	12.7360			0.951934			0.475967	+	0.879463i	$0.342098\pi$
$180$		0			0
$181$	−22.1009			−1.64275			−0.821373	−	0.570392i	$-0.806791\pi$
$181$	−22.1009			−1.64275			−0.821373	+	0.570392i	$0.806791\pi$
$182$		0			0
$183$		−	2.27771i		−	0.168373i
$184$		0			0
$185$	1.13311	−	2.98368i	0.0833078	−	0.219365i
$186$		0			0
$187$		−	11.0380i		−	0.807177i
$188$		0			0
$189$	4.15840			0.302479
$190$		0			0
$191$	−5.65080			−0.408878			−0.204439	−	0.978879i	$-0.565537\pi$
$191$	−5.65080			−0.408878			−0.204439	+	0.978879i	$0.565537\pi$
$192$		0			0
$193$		−	15.2536i		−	1.09798i	−0.835829	−	0.548991i	$-0.815012\pi$
$193$		−	15.2536i		−	1.09798i	0.835829	−	0.548991i	$-0.184988\pi$
$194$		0			0
$195$	1.52717	+	0.579972i	0.109363	+	0.0415327i
$196$		0			0
$197$			7.46296i			0.531714i	0.964012	+	0.265857i	$0.0856549\pi$
$197$			7.46296i			0.531714i	−0.964012	+	0.265857i	$0.914345\pi$
$198$		0			0
$199$	−22.8848			−1.62226			−0.811130	−	0.584866i	$-0.801147\pi$
$199$	−22.8848			−1.62226			−0.811130	+	0.584866i	$0.801147\pi$
$200$		0			0
$201$	7.22429			0.509562
$202$		0			0
$203$			31.9575i			2.24298i
$204$		0			0
$205$	11.4730	+	4.35707i	0.801306	+	0.304311i
$206$		0			0
$207$			1.00000i			0.0695048i
$208$		0			0
$209$	11.0886			0.767011
$210$		0			0
$211$	0.678461			0.0467072			0.0233536	−	0.999727i	$-0.492566\pi$
$211$	0.678461			0.0467072			0.0233536	+	0.999727i	$0.492566\pi$
$212$		0			0
$213$		−	12.0946i		−	0.828711i
$214$		0			0
$215$	−2.14321	+	5.64344i	−0.146165	+	0.384880i
$216$		0			0
$217$		−	20.0103i		−	1.35839i
$218$		0			0
$219$	−10.5827			−0.715113
$220$		0			0
$221$	−3.57170			−0.240258
$222$		0			0
$223$		−	14.0322i		−	0.939665i	−0.882755	−	0.469833i	$-0.844314\pi$
$223$		−	14.0322i		−	0.939665i	0.882755	−	0.469833i	$-0.155686\pi$
$224$		0			0
$225$	−3.73954	−	3.31901i	−0.249303	−	0.221267i
$226$		0			0
$227$		−	1.83740i		−	0.121953i	−0.998139	−	0.0609764i	$-0.980579\pi$
$227$		−	1.83740i		−	0.121953i	0.998139	−	0.0609764i	$-0.0194214\pi$
$228$		0			0
$229$	9.14771			0.604498			0.302249	−	0.953229i	$-0.402263\pi$
$229$	9.14771			0.604498			0.302249	+	0.953229i	$0.402263\pi$
$230$		0			0
$231$	9.38856			0.617722
$232$		0			0
$233$		−	10.9526i		−	0.717528i	−0.933428	−	0.358764i	$-0.883198\pi$
$233$		−	10.9526i		−	0.717528i	0.933428	−	0.358764i	$-0.116802\pi$
$234$		0			0
$235$	−10.6477	+	28.0372i	−0.694578	+	1.82895i
$236$		0			0
$237$		−	14.8635i		−	0.965486i
$238$		0			0
$239$	−21.0183			−1.35956			−0.679781	−	0.733415i	$-0.737925\pi$
$239$	−21.0183			−1.35956			−0.679781	+	0.733415i	$0.737925\pi$
$240$		0			0
$241$	−14.4088			−0.928153			−0.464076	−	0.885795i	$-0.653614\pi$
$241$	−14.4088			−0.928153			−0.464076	+	0.885795i	$0.653614\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	21.5150	+	8.17071i	1.37454	+	0.522008i
$246$		0			0
$247$		−	3.58806i		−	0.228303i
$248$		0			0
$249$	15.4631			0.979936
$250$		0			0
$251$	−1.53862			−0.0971171			−0.0485585	−	0.998820i	$-0.515463\pi$
$251$	−1.53862			−0.0971171			−0.0485585	+	0.998820i	$0.515463\pi$
$252$		0			0
$253$			2.25773i			0.141943i
$254$		0			0
$255$	10.2199	+	3.88119i	0.639994	+	0.243050i
$256$		0			0
$257$			29.0908i			1.81463i	0.420449	+	0.907316i	$0.361873\pi$
$257$			29.0908i			1.81463i	−0.420449	+	0.907316i	$0.638127\pi$
$258$		0			0
$259$	−5.93538			−0.368807
$260$		0			0
$261$	−7.68506			−0.475693
$262$		0			0
$263$		−	4.69550i		−	0.289537i	−0.989466	−	0.144768i	$-0.953756\pi$
$263$		−	4.69550i		−	0.289537i	0.989466	−	0.144768i	$-0.0462437\pi$
$264$		0			0
$265$	4.85633	−	12.7876i	0.298322	−	0.785536i
$266$		0			0
$267$			4.89024i			0.299278i
$268$		0			0
$269$	15.6075			0.951607			0.475804	−	0.879552i	$-0.342157\pi$
$269$	15.6075			0.951607			0.475804	+	0.879552i	$0.342157\pi$
$270$		0			0
$271$	13.0275			0.791362			0.395681	−	0.918388i	$-0.370509\pi$
$271$	13.0275			0.791362			0.395681	+	0.918388i	$0.370509\pi$
$272$		0			0
$273$		−	3.03797i		−	0.183867i
$274$		0			0
$275$	−8.44290	−	7.49344i	−0.509126	−	0.451871i
$276$		0			0
$277$		−	19.2081i		−	1.15410i	−0.816708	−	0.577050i	$-0.804204\pi$
$277$		−	19.2081i		−	1.15410i	0.816708	−	0.577050i	$-0.195796\pi$
$278$		0			0
$279$	4.81203			0.288088
$280$		0			0
$281$	3.14658			0.187709			0.0938547	−	0.995586i	$-0.470081\pi$
$281$	3.14658			0.187709			0.0938547	+	0.995586i	$0.470081\pi$
$282$		0			0
$283$			4.83477i			0.287397i	0.989621	+	0.143699i	$0.0458996\pi$
$283$			4.83477i			0.287397i	−0.989621	+	0.143699i	$0.954100\pi$
$284$		0			0
$285$	−3.89898	+	10.2667i	−0.230956	+	0.608148i
$286$		0			0
$287$		−	22.8229i		−	1.34720i
$288$		0			0
$289$	−6.90194			−0.405996
$290$		0			0
$291$	−2.40242			−0.140832
$292$		0			0
$293$			27.0591i			1.58081i	0.612586	+	0.790404i	$0.290129\pi$
$293$			27.0591i			1.58081i	−0.612586	+	0.790404i	$0.709871\pi$
$294$		0			0
$295$	−28.3052	−	10.7494i	−1.64799	−	0.625857i
$296$		0			0
$297$			2.25773i			0.131007i
$298$		0			0
$299$	0.730564			0.0422496
$300$		0			0
$301$	11.2264			0.647079
$302$		0			0
$303$			13.1031i			0.752752i
$304$		0			0
$305$	4.76133	+	1.80820i	0.272633	+	0.103537i
$306$		0			0
$307$		−	6.33286i		−	0.361435i	−0.983535	−	0.180718i	$-0.942158\pi$
$307$		−	6.33286i		−	0.361435i	0.983535	−	0.180718i	$-0.0578420\pi$
$308$		0			0
$309$	−9.15986			−0.521086
$310$		0			0
$311$	15.9547			0.904706			0.452353	−	0.891839i	$-0.350585\pi$
$311$	15.9547			0.904706			0.452353	+	0.891839i	$0.350585\pi$
$312$		0			0
$313$		−	8.19940i		−	0.463457i	−0.972780	−	0.231729i	$-0.925562\pi$
$313$		−	8.19940i		−	0.463457i	0.972780	−	0.231729i	$-0.0744382\pi$
$314$		0			0
$315$	−3.30122	+	8.69271i	−0.186003	+	0.489779i
$316$		0			0
$317$		−	19.6908i		−	1.10594i	−0.833200	−	0.552972i	$-0.813494\pi$
$317$		−	19.6908i		−	1.10594i	0.833200	−	0.552972i	$-0.186506\pi$
$318$		0			0
$319$	−17.3508			−0.971460
$320$		0			0
$321$	17.0677			0.952629
$322$		0			0
$323$		−	24.0115i		−	1.33603i
$324$		0			0
$325$	−2.42475	+	2.73198i	−0.134501	+	0.151543i
$326$		0			0
$327$			6.03908i			0.333962i
$328$		0			0
$329$	55.7740			3.07492
$330$		0			0
$331$	25.8233			1.41938			0.709689	−	0.704515i	$-0.248835\pi$
$331$	25.8233			1.41938			0.709689	+	0.704515i	$0.248835\pi$
$332$		0			0
$333$		−	1.42733i		−	0.0782169i
$334$		0			0
$335$	−5.73514	+	15.1017i	−0.313344	+	0.825091i
$336$		0			0
$337$			10.4548i			0.569510i	0.958600	+	0.284755i	$0.0919123\pi$
$337$			10.4548i			0.569510i	−0.958600	+	0.284755i	$0.908088\pi$
$338$		0			0
$339$	2.65951			0.144445
$340$		0			0
$341$	10.8643			0.588334
$342$		0			0
$343$		−	13.6905i		−	0.739220i
$344$		0			0
$345$	−2.09040	−	0.793869i	−0.112543	−	0.0427405i
$346$		0			0
$347$		−	7.82706i		−	0.420179i	−0.977682	−	0.210089i	$-0.932625\pi$
$347$		−	7.82706i		−	0.420179i	0.977682	−	0.210089i	$-0.0673755\pi$
$348$		0			0
$349$	−26.5144			−1.41928			−0.709641	−	0.704563i	$-0.751143\pi$
$349$	−26.5144			−1.41928			−0.709641	+	0.704563i	$0.751143\pi$
$350$		0			0
$351$	0.730564			0.0389946
$352$		0			0
$353$			13.5444i			0.720895i	0.932779	+	0.360448i	$0.117376\pi$
$353$			13.5444i			0.720895i	−0.932779	+	0.360448i	$0.882624\pi$
$354$		0			0
$355$	25.2826	+	9.60156i	1.34186	+	0.509598i
$356$		0			0
$357$		−	20.3302i		−	1.07599i
$358$		0			0
$359$	−30.9391			−1.63290			−0.816452	−	0.577413i	$-0.804062\pi$
$359$	−30.9391			−1.63290			−0.816452	+	0.577413i	$0.804062\pi$
$360$		0			0
$361$	5.12149			0.269552
$362$		0			0
$363$		−	5.90264i		−	0.309808i
$364$		0			0
$365$	8.40129	−	22.1221i	0.439744	−	1.15792i
$366$		0			0
$367$			27.4513i			1.43295i	0.697613	+	0.716474i	$0.254245\pi$
$367$			27.4513i			1.43295i	−0.697613	+	0.716474i	$0.745755\pi$
$368$		0			0
$369$	5.48840			0.285715
$370$		0			0
$371$	−25.4382			−1.32068
$372$		0			0
$373$			22.4093i			1.16031i	0.814506	+	0.580155i	$0.197008\pi$
$373$			22.4093i			1.16031i	−0.814506	+	0.580155i	$0.802992\pi$
$374$		0			0
$375$	9.90676	−	5.18228i	0.511583	−	0.267612i
$376$		0			0
$377$			5.61443i			0.289158i
$378$		0			0
$379$	−9.55380			−0.490746			−0.245373	−	0.969429i	$-0.578910\pi$
$379$	−9.55380			−0.490746			−0.245373	+	0.969429i	$0.578910\pi$
$380$		0			0
$381$	8.41075			0.430896
$382$		0			0
$383$			29.0404i			1.48389i	0.670459	+	0.741946i	$0.266097\pi$
$383$			29.0404i			1.48389i	−0.670459	+	0.741946i	$0.733903\pi$
$384$		0			0
$385$	−7.45328	+	19.6258i	−0.379855	+	1.00023i
$386$		0			0
$387$			2.69970i			0.137233i
$388$		0			0
$389$	−11.0396			−0.559731			−0.279866	−	0.960039i	$-0.590290\pi$
$389$	−11.0396			−0.559731			−0.279866	+	0.960039i	$0.590290\pi$
$390$		0			0
$391$	4.88896			0.247245
$392$		0			0
$393$		−	5.39211i		−	0.271996i
$394$		0			0
$395$	31.0706	+	11.7996i	1.56333	+	0.593705i
$396$		0			0
$397$			10.9183i			0.547973i	0.961734	+	0.273986i	$0.0883423\pi$
$397$			10.9183i			0.547973i	−0.961734	+	0.273986i	$0.911658\pi$
$398$		0			0
$399$	20.4234			1.02245
$400$		0			0
$401$	26.4326			1.31998			0.659991	−	0.751274i	$-0.270560\pi$
$401$	26.4326			1.31998			0.659991	+	0.751274i	$0.270560\pi$
$402$		0			0
$403$		−	3.51549i		−	0.175119i
$404$		0			0
$405$	−2.09040	−	0.793869i	−0.103873	−	0.0394477i
$406$		0			0
$407$		−	3.22252i		−	0.159734i
$408$		0			0
$409$	−5.59353			−0.276582			−0.138291	−	0.990392i	$-0.544161\pi$
$409$	−5.59353			−0.276582			−0.138291	+	0.990392i	$0.544161\pi$
$410$		0			0
$411$	20.7356			1.02281
$412$		0			0
$413$			56.3071i			2.77069i
$414$		0			0
$415$	−12.2757	+	32.3241i	−0.602590	+	1.58673i
$416$		0			0
$417$		−	6.24481i		−	0.305810i
$418$		0			0
$419$	15.5220			0.758301			0.379150	−	0.925335i	$-0.376216\pi$
$419$	15.5220			0.758301			0.379150	+	0.925335i	$0.376216\pi$
$420$		0			0
$421$	0.422250			0.0205792			0.0102896	−	0.999947i	$-0.496725\pi$
$421$	0.422250			0.0205792			0.0102896	+	0.999947i	$0.496725\pi$
$422$		0			0
$423$			13.4124i			0.652132i
$424$		0			0
$425$	−16.2265	+	18.2825i	−0.787101	+	0.886831i
$426$		0			0
$427$		−	9.47162i		−	0.458364i
$428$		0			0
$429$	1.64942			0.0796347
$430$		0			0
$431$	−29.0319			−1.39842			−0.699208	−	0.714918i	$-0.746464\pi$
$431$	−29.0319			−1.39842			−0.699208	+	0.714918i	$0.746464\pi$
$432$		0			0
$433$			3.79638i			0.182442i	0.995831	+	0.0912212i	$0.0290770\pi$
$433$			3.79638i			0.182442i	−0.995831	+	0.0912212i	$0.970923\pi$
$434$		0			0
$435$	6.10093	−	16.0648i	0.292517	−	0.770250i
$436$		0			0
$437$			4.91136i			0.234942i
$438$		0			0
$439$	25.8716			1.23478			0.617391	−	0.786657i	$-0.288190\pi$
$439$	25.8716			1.23478			0.617391	+	0.786657i	$0.288190\pi$
$440$		0			0
$441$	10.2923			0.490108
$442$		0			0
$443$			0.576756i			0.0274025i	0.999906	+	0.0137013i	$0.00436138\pi$
$443$			0.576756i			0.0274025i	−0.999906	+	0.0137013i	$0.995639\pi$
$444$		0			0
$445$	−10.2226	−	3.88221i	−0.484596	−	0.184034i
$446$		0			0
$447$		−	18.6649i		−	0.882817i
$448$		0			0
$449$	−15.0088			−0.708310			−0.354155	−	0.935187i	$-0.615231\pi$
$449$	−15.0088			−0.708310			−0.354155	+	0.935187i	$0.615231\pi$
$450$		0			0
$451$	12.3914			0.583486
$452$		0			0
$453$			11.0437i			0.518881i
$454$		0			0
$455$	6.35058	+	2.41175i	0.297720	+	0.113065i
$456$		0			0
$457$		−	12.1887i		−	0.570162i	−0.958503	−	0.285081i	$-0.907979\pi$
$457$		−	12.1887i		−	0.570162i	0.958503	−	0.285081i	$-0.0920205\pi$
$458$		0			0
$459$	4.88896			0.228197
$460$		0			0
$461$	−16.4007			−0.763855			−0.381927	−	0.924192i	$-0.624740\pi$
$461$	−16.4007			−0.763855			−0.381927	+	0.924192i	$0.624740\pi$
$462$		0			0
$463$		−	4.47331i		−	0.207892i	−0.994583	−	0.103946i	$-0.966853\pi$
$463$		−	4.47331i		−	0.207892i	0.994583	−	0.103946i	$-0.0331470\pi$
$464$		0			0
$465$	−3.82012	+	10.0591i	−0.177154	+	0.466478i
$466$		0			0
$467$		−	41.7098i		−	1.93010i	−0.262065	−	0.965050i	$-0.584403\pi$
$467$		−	41.7098i		−	1.93010i	0.262065	−	0.965050i	$-0.415597\pi$
$468$		0			0
$469$	30.0415			1.38719
$470$		0			0
$471$	9.24592			0.426029
$472$		0			0
$473$			6.09520i			0.280257i
$474$		0			0
$475$	−18.3663	−	16.3009i	−0.842702	−	0.747934i
$476$		0			0
$477$		−	6.11730i		−	0.280092i
$478$		0			0
$479$	−32.4549			−1.48290			−0.741451	−	0.671007i	$-0.765862\pi$
$479$	−32.4549			−1.48290			−0.741451	+	0.671007i	$0.765862\pi$
$480$		0			0
$481$	−1.04275			−0.0475454
$482$		0			0
$483$			4.15840i			0.189214i
$484$		0			0
$485$	1.90721	−	5.02202i	0.0866018	−	0.228038i
$486$		0			0
$487$			14.9402i			0.677007i	0.940965	+	0.338503i	$0.109921\pi$
$487$			14.9402i			0.677007i	−0.940965	+	0.338503i	$0.890079\pi$
$488$		0			0
$489$	−18.0937			−0.818225
$490$		0			0
$491$	23.9061			1.07887			0.539433	−	0.842028i	$-0.318639\pi$
$491$	23.9061			1.07887			0.539433	+	0.842028i	$0.318639\pi$
$492$		0			0
$493$			37.5720i			1.69216i
$494$		0			0
$495$	−4.71957	−	1.79235i	−0.212129	−	0.0805599i
$496$		0			0
$497$		−	50.2943i		−	2.25601i
$498$		0			0
$499$	−41.2637			−1.84722			−0.923608	−	0.383339i	$-0.874774\pi$
$499$	−41.2637			−1.84722			−0.923608	+	0.383339i	$0.874774\pi$
$500$		0			0
$501$	−8.93329			−0.399110
$502$		0			0
$503$			29.3541i			1.30883i	0.756134	+	0.654417i	$0.227086\pi$
$503$			29.3541i			1.30883i	−0.756134	+	0.654417i	$0.772914\pi$
$504$		0			0
$505$	−27.3907	−	10.4021i	−1.21887	−	0.462888i
$506$		0			0
$507$			12.4663i			0.553647i
$508$		0			0
$509$	−22.5214			−0.998242			−0.499121	−	0.866532i	$-0.666344\pi$
$509$	−22.5214			−0.998242			−0.499121	+	0.866532i	$0.666344\pi$
$510$		0			0
$511$	−44.0071			−1.94676
$512$		0			0
$513$			4.91136i			0.216842i
$514$		0			0
$515$	7.27173	−	19.1478i	0.320431	−	0.843752i
$516$		0			0
$517$			30.2816i			1.33178i
$518$		0			0
$519$	−6.00102			−0.263415
$520$		0			0
$521$	−10.0476			−0.440191			−0.220096	−	0.975478i	$-0.570637\pi$
$521$	−10.0476			−0.440191			−0.220096	+	0.975478i	$0.570637\pi$
$522$		0			0
$523$			38.1986i			1.67031i	0.550014	+	0.835155i	$0.314622\pi$
$523$			38.1986i			1.67031i	−0.550014	+	0.835155i	$0.685378\pi$
$524$		0			0
$525$	−15.5505	−	13.8018i	−0.678680	−	0.602358i
$526$		0			0
$527$		−	23.5258i		−	1.02480i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−13.5406			−0.587611
$532$		0			0
$533$		−	4.00963i		−	0.173676i
$534$		0			0
$535$	−13.5496	+	35.6784i	−0.585798	+	1.54251i
$536$		0			0
$537$			12.7360i			0.549599i
$538$		0			0
$539$	23.2372			1.00090
$540$		0			0
$541$	−24.4554			−1.05142			−0.525711	−	0.850663i	$-0.676201\pi$
$541$	−24.4554			−1.05142			−0.525711	+	0.850663i	$0.676201\pi$
$542$		0			0
$543$		−	22.1009i		−	0.948440i
$544$		0			0
$545$	−12.6241	−	4.79424i	−0.540757	−	0.205363i
$546$		0			0
$547$			27.6238i			1.18111i	0.806998	+	0.590554i	$0.201091\pi$
$547$			27.6238i			1.18111i	−0.806998	+	0.590554i	$0.798909\pi$
$548$		0			0
$549$	2.27771			0.0972103
$550$		0			0
$551$	−37.7441			−1.60795
$552$		0			0
$553$		−	61.8082i		−	2.62835i
$554$		0			0
$555$	2.98368	+	1.13311i	0.126650	+	0.0480978i
$556$		0			0
$557$			28.0134i			1.18696i	0.804847	+	0.593482i	$0.202247\pi$
$557$			28.0134i			1.18696i	−0.804847	+	0.593482i	$0.797753\pi$
$558$		0			0
$559$	1.97230			0.0834194
$560$		0			0
$561$	11.0380			0.466024
$562$		0			0
$563$		−	2.78613i		−	0.117421i	−0.998275	−	0.0587106i	$-0.981301\pi$
$563$		−	2.78613i		−	0.117421i	0.998275	−	0.0587106i	$-0.0186989\pi$
$564$		0			0
$565$	−2.11131	+	5.55945i	−0.0888233	+	0.233888i
$566$		0			0
$567$			4.15840i			0.174636i
$568$		0			0
$569$	19.1590			0.803187			0.401593	−	0.915818i	$-0.368457\pi$
$569$	19.1590			0.803187			0.401593	+	0.915818i	$0.368457\pi$
$570$		0			0
$571$	−5.03532			−0.210722			−0.105361	−	0.994434i	$-0.533600\pi$
$571$	−5.03532			−0.210722			−0.105361	+	0.994434i	$0.533600\pi$
$572$		0			0
$573$		−	5.65080i		−	0.236066i
$574$		0			0
$575$	3.31901	−	3.73954i	0.138412	−	0.155950i
$576$		0			0
$577$			5.26394i			0.219141i	0.993979	+	0.109570i	$0.0349475\pi$
$577$			5.26394i			0.219141i	−0.993979	+	0.109570i	$0.965053\pi$
$578$		0			0
$579$	15.2536			0.633920
$580$		0			0
$581$	64.3018			2.66769
$582$		0			0
$583$		−	13.8112i		−	0.572003i
$584$		0			0
$585$	−0.579972	+	1.52717i	−0.0239789	+	0.0631407i
$586$		0			0
$587$		−	25.4879i		−	1.05200i	−0.850485	−	0.526000i	$-0.823691\pi$
$587$		−	25.4879i		−	1.05200i	0.850485	−	0.526000i	$-0.176309\pi$
$588$		0			0
$589$	23.6336			0.973806
$590$		0			0
$591$	−7.46296			−0.306985
$592$		0			0
$593$		−	24.4270i		−	1.00310i	−0.865129	−	0.501549i	$-0.832764\pi$
$593$		−	24.4270i		−	1.00310i	0.865129	−	0.501549i	$-0.167236\pi$
$594$		0			0
$595$	42.4983	+	16.1395i	1.74226	+	0.661657i
$596$		0			0
$597$		−	22.8848i		−	0.936612i
$598$		0			0
$599$	−9.74248			−0.398067			−0.199033	−	0.979993i	$-0.563780\pi$
$599$	−9.74248			−0.398067			−0.199033	+	0.979993i	$0.563780\pi$
$600$		0			0
$601$	−11.3597			−0.463372			−0.231686	−	0.972791i	$-0.574424\pi$
$601$	−11.3597			−0.463372			−0.231686	+	0.972791i	$0.574424\pi$
$602$		0			0
$603$			7.22429i			0.294196i
$604$		0			0
$605$	12.3389	+	4.68592i	0.501646	+	0.190510i
$606$		0			0
$607$			13.8162i			0.560781i	0.959886	+	0.280391i	$0.0904640\pi$
$607$			13.8162i			0.560781i	−0.959886	+	0.280391i	$0.909536\pi$
$608$		0			0
$609$	−31.9575			−1.29498
$610$		0			0
$611$	9.79860			0.396409
$612$		0			0
$613$			37.1039i			1.49861i	0.662224	+	0.749305i	$0.269613\pi$
$613$			37.1039i			1.49861i	−0.662224	+	0.749305i	$0.730387\pi$
$614$		0			0
$615$	−4.35707	+	11.4730i	−0.175694	+	0.462634i
$616$		0			0
$617$			3.64697i			0.146822i	0.997302	+	0.0734108i	$0.0233884\pi$
$617$			3.64697i			0.146822i	−0.997302	+	0.0734108i	$0.976612\pi$
$618$		0			0
$619$	−23.3451			−0.938318			−0.469159	−	0.883114i	$-0.655443\pi$
$619$	−23.3451			−0.938318			−0.469159	+	0.883114i	$0.655443\pi$
$620$		0			0
$621$	−1.00000			−0.0401286
$622$		0			0
$623$			20.3356i			0.814727i
$624$		0			0
$625$	2.96838	+	24.8231i	0.118735	+	0.992926i
$626$		0			0
$627$			11.0886i			0.442834i
$628$		0			0
$629$	−6.97814			−0.278237
$630$		0			0
$631$	40.9342			1.62957			0.814783	−	0.579766i	$-0.196856\pi$
$631$	40.9342			1.62957			0.814783	+	0.579766i	$0.196856\pi$
$632$		0			0
$633$			0.678461i			0.0269664i
$634$		0			0
$635$	−6.67704	+	17.5818i	−0.264970	+	0.697714i
$636$		0			0
$637$		−	7.51916i		−	0.297920i
$638$		0			0
$639$	12.0946			0.478457
$640$		0			0
$641$	−19.8841			−0.785373			−0.392686	−	0.919672i	$-0.628454\pi$
$641$	−19.8841			−0.785373			−0.392686	+	0.919672i	$0.628454\pi$
$642$		0			0
$643$		−	34.5876i		−	1.36400i	−0.731352	−	0.682000i	$-0.761110\pi$
$643$		−	34.5876i		−	1.36400i	0.731352	−	0.682000i	$-0.238890\pi$
$644$		0			0
$645$	−5.64344	−	2.14321i	−0.222210	−	0.0843886i
$646$		0			0
$647$			10.8866i			0.427997i	0.976834	+	0.213999i	$0.0686488\pi$
$647$			10.8866i			0.427997i	−0.976834	+	0.213999i	$0.931351\pi$
$648$		0			0
$649$	−30.5710			−1.20002
$650$		0			0
$651$	20.0103			0.784266
$652$		0			0
$653$		−	4.62404i		−	0.180953i	−0.995899	−	0.0904763i	$-0.971161\pi$
$653$		−	4.62404i		−	0.180953i	0.995899	−	0.0904763i	$-0.0288389\pi$
$654$		0			0
$655$	11.2717	+	4.28063i	0.440421	+	0.167258i
$656$		0			0
$657$		−	10.5827i		−	0.412871i
$658$		0			0
$659$	25.0478			0.975725			0.487862	−	0.872921i	$-0.337777\pi$
$659$	25.0478			0.975725			0.487862	+	0.872921i	$0.337777\pi$
$660$		0			0
$661$	19.9135			0.774547			0.387273	−	0.921965i	$-0.373417\pi$
$661$	19.9135			0.774547			0.387273	+	0.921965i	$0.373417\pi$
$662$		0			0
$663$		−	3.57170i		−	0.138713i
$664$		0			0
$665$	−16.2135	+	42.6931i	−0.628733	+	1.65557i
$666$		0			0
$667$		−	7.68506i		−	0.297567i
$668$		0			0
$669$	14.0322			0.542516
$670$		0			0
$671$	5.14247			0.198523
$672$		0			0
$673$			13.2309i			0.510016i	0.966939	+	0.255008i	$0.0820780\pi$
$673$			13.2309i			0.510016i	−0.966939	+	0.255008i	$0.917922\pi$
$674$		0			0
$675$	3.31901	−	3.73954i	0.127749	−	0.143935i
$676$		0			0
$677$			9.30620i			0.357666i	0.983879	+	0.178833i	$0.0572322\pi$
$677$			9.30620i			0.357666i	−0.983879	+	0.178833i	$0.942768\pi$
$678$		0			0
$679$	−9.99022			−0.383389
$680$		0			0
$681$	1.83740			0.0704095
$682$		0			0
$683$			0.444898i			0.0170236i	0.999964	+	0.00851178i	$0.00270942\pi$
$683$			0.444898i			0.0170236i	−0.999964	+	0.00851178i	$0.997291\pi$
$684$		0			0
$685$	−16.4613	+	43.3457i	−0.628956	+	1.65615i
$686$		0			0
$687$			9.14771i			0.349007i
$688$		0			0
$689$	−4.46908			−0.170258
$690$		0			0
$691$	−14.6646			−0.557866			−0.278933	−	0.960311i	$-0.589981\pi$
$691$	−14.6646			−0.557866			−0.278933	+	0.960311i	$0.589981\pi$
$692$		0			0
$693$			9.38856i			0.356642i
$694$		0			0
$695$	13.0542	+	4.95756i	0.495172	+	0.188051i
$696$		0			0
$697$		−	26.8326i		−	1.01636i
$698$		0			0
$699$	10.9526			0.414265
$700$		0			0
$701$	−26.1098			−0.986152			−0.493076	−	0.869986i	$-0.664128\pi$
$701$	−26.1098			−0.986152			−0.493076	+	0.869986i	$0.664128\pi$
$702$		0			0
$703$		−	7.01011i		−	0.264391i
$704$		0			0
$705$	−28.0372	−	10.6477i	−1.05594	−	0.401015i
$706$		0			0
$707$			54.4878i			2.04922i
$708$		0			0
$709$	15.0270			0.564350			0.282175	−	0.959363i	$-0.408944\pi$
$709$	15.0270			0.564350			0.282175	+	0.959363i	$0.408944\pi$
$710$		0			0
$711$	14.8635			0.557424
$712$		0			0
$713$			4.81203i			0.180212i
$714$		0			0
$715$	−1.30942	+	3.44795i	−0.0489696	+	0.128946i
$716$		0			0
$717$		−	21.0183i		−	0.784943i
$718$		0			0
$719$	−40.7253			−1.51880			−0.759398	−	0.650626i	$-0.774507\pi$
$719$	−40.7253			−1.51880			−0.759398	+	0.650626i	$0.774507\pi$
$720$		0			0
$721$	−38.0903			−1.41856
$722$		0			0
$723$		−	14.4088i		−	0.535869i
$724$		0			0
$725$	28.7386	+	25.5068i	1.06733	+	0.947298i
$726$		0			0
$727$			8.64116i			0.320483i	0.987078	+	0.160241i	$0.0512273\pi$
$727$			8.64116i			0.320483i	−0.987078	+	0.160241i	$0.948773\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	13.1987			0.488172
$732$		0			0
$733$			51.0113i			1.88415i	0.335409	+	0.942073i	$0.391125\pi$
$733$			51.0113i			1.88415i	−0.335409	+	0.942073i	$0.608875\pi$
$734$		0			0
$735$	−8.17071	+	21.5150i	−0.301381	+	0.793591i
$736$		0			0
$737$			16.3105i			0.600806i
$738$		0			0
$739$	31.8583			1.17193			0.585963	−	0.810338i	$-0.300716\pi$
$739$	31.8583			1.17193			0.585963	+	0.810338i	$0.300716\pi$
$740$		0			0
$741$	3.58806			0.131811
$742$		0			0
$743$		−	32.0739i		−	1.17668i	−0.808615	−	0.588338i	$-0.799782\pi$
$743$		−	32.0739i		−	1.17668i	0.808615	−	0.588338i	$-0.200218\pi$
$744$		0			0
$745$	39.0170	+	14.8174i	1.42947	+	0.542869i
$746$		0			0
$747$			15.4631i			0.565766i
$748$		0			0
$749$	70.9745			2.59335
$750$		0			0
$751$	39.7091			1.44900			0.724502	−	0.689273i	$-0.242070\pi$
$751$	39.7091			1.44900			0.724502	+	0.689273i	$0.242070\pi$
$752$		0			0
$753$		−	1.53862i		−	0.0560706i
$754$		0			0
$755$	−23.0859	−	8.76729i	−0.840180	−	0.319074i
$756$		0			0
$757$		−	19.7300i		−	0.717098i	−0.933511	−	0.358549i	$-0.883272\pi$
$757$		−	19.7300i		−	0.717098i	0.933511	−	0.358549i	$-0.116728\pi$
$758$		0			0
$759$	−2.25773			−0.0819506
$760$		0			0
$761$	−11.8950			−0.431193			−0.215596	−	0.976483i	$-0.569170\pi$
$761$	−11.8950			−0.431193			−0.215596	+	0.976483i	$0.569170\pi$
$762$		0			0
$763$			25.1129i			0.909148i
$764$		0			0
$765$	−3.88119	+	10.2199i	−0.140325	+	0.369501i
$766$		0			0
$767$			9.89225i			0.357189i
$768$		0			0
$769$	2.28009			0.0822220			0.0411110	−	0.999155i	$-0.486910\pi$
$769$	2.28009			0.0822220			0.0411110	+	0.999155i	$0.486910\pi$
$770$		0			0
$771$	−29.0908			−1.04768
$772$		0			0
$773$			10.5078i			0.377938i	0.981983	+	0.188969i	$0.0605146\pi$
$773$			10.5078i			0.377938i	−0.981983	+	0.188969i	$0.939485\pi$
$774$		0			0
$775$	−17.9948	−	15.9712i	−0.646392	−	0.573701i
$776$		0			0
$777$		−	5.93538i		−	0.212931i
$778$		0			0
$779$	26.9555			0.965782
$780$		0			0
$781$	27.3065			0.977103
$782$		0			0
$783$		−	7.68506i		−	0.274642i
$784$		0			0
$785$	−7.34005	+	19.3277i	−0.261978	+	0.689834i
$786$		0			0
$787$		−	16.8962i		−	0.602283i	−0.953579	−	0.301142i	$-0.902632\pi$
$787$		−	16.8962i		−	0.602283i	0.953579	−	0.301142i	$-0.0973677\pi$
$788$		0			0
$789$	4.69550			0.167164
$790$		0			0
$791$	11.0593			0.393224
$792$		0			0
$793$		−	1.66401i		−	0.0590908i
$794$		0			0
$795$	12.7876	+	4.85633i	0.453529	+	0.172236i
$796$		0			0
$797$		−	31.2230i		−	1.10598i	−0.833190	−	0.552988i	$-0.813488\pi$
$797$		−	31.2230i		−	1.10598i	0.833190	−	0.552988i	$-0.186512\pi$
$798$		0			0
$799$	65.5726			2.31979
$800$		0			0
$801$	−4.89024			−0.172788
$802$		0			0
$803$		−	23.8930i		−	0.843164i
$804$		0			0
$805$	−8.69271	−	3.30122i	−0.306378	−	0.116353i
$806$		0			0
$807$			15.6075i			0.549411i
$808$		0			0
$809$	−15.7601			−0.554097			−0.277048	−	0.960856i	$-0.589356\pi$
$809$	−15.7601			−0.554097			−0.277048	+	0.960856i	$0.589356\pi$
$810$		0			0
$811$	7.30047			0.256354			0.128177	−	0.991751i	$-0.459087\pi$
$811$	7.30047			0.256354			0.128177	+	0.991751i	$0.459087\pi$
$812$		0			0
$813$			13.0275i			0.456893i
$814$		0			0
$815$	14.3640	−	37.8231i	0.503150	−	1.32488i
$816$		0			0
$817$			13.2592i			0.463880i
$818$		0			0
$819$	3.03797			0.106155
$820$		0			0
$821$	−39.1596			−1.36668			−0.683340	−	0.730101i	$-0.739473\pi$
$821$	−39.1596			−1.36668			−0.683340	+	0.730101i	$0.739473\pi$
$822$		0			0
$823$			48.2066i			1.68038i	0.542294	+	0.840189i	$0.317556\pi$
$823$			48.2066i			1.68038i	−0.542294	+	0.840189i	$0.682444\pi$
$824$		0			0
$825$	7.49344	−	8.44290i	0.260888	−	0.293944i
$826$		0			0
$827$			31.5470i			1.09700i	0.836152	+	0.548498i	$0.184800\pi$
$827$			31.5470i			1.09700i	−0.836152	+	0.548498i	$0.815200\pi$
$828$		0			0
$829$	−1.53447			−0.0532942			−0.0266471	−	0.999645i	$-0.508483\pi$
$829$	−1.53447			−0.0532942			−0.0266471	+	0.999645i	$0.508483\pi$
$830$		0			0
$831$	19.2081			0.666321
$832$		0			0
$833$		−	50.3185i		−	1.74343i
$834$		0			0
$835$	7.09187	−	18.6742i	0.245424	−	0.646246i
$836$		0			0
$837$			4.81203i			0.166328i
$838$		0			0
$839$	34.4098			1.18796			0.593979	−	0.804480i	$-0.297556\pi$
$839$	34.4098			1.18796			0.593979	+	0.804480i	$0.297556\pi$
$840$		0			0
$841$	30.0601			1.03656
$842$		0			0
$843$			3.14658i			0.108374i
$844$		0			0
$845$	−26.0595	−	9.89659i	−0.896474	−	0.340453i
$846$		0			0
$847$		−	24.5455i		−	0.843393i
$848$		0			0
$849$	−4.83477			−0.165929
$850$		0			0
$851$	1.42733			0.0489281
$852$		0			0
$853$		−	1.21406i		−	0.0415685i	−0.999784	−	0.0207842i	$-0.993384\pi$
$853$		−	1.21406i		−	0.0415685i	0.999784	−	0.0207842i	$-0.00661630\pi$
$854$		0			0
$855$	−10.2667	−	3.89898i	−0.351114	−	0.133342i
$856$		0			0
$857$		−	36.5652i		−	1.24904i	−0.781007	−	0.624522i	$-0.785294\pi$
$857$		−	36.5652i		−	1.24904i	0.781007	−	0.624522i	$-0.214706\pi$
$858$		0			0
$859$	−19.7187			−0.672794			−0.336397	−	0.941720i	$-0.609208\pi$
$859$	−19.7187			−0.672794			−0.336397	+	0.941720i	$0.609208\pi$
$860$		0			0
$861$	22.8229			0.777804
$862$		0			0
$863$			9.70535i			0.330374i	0.986262	+	0.165187i	$0.0528228\pi$
$863$			9.70535i			0.330374i	−0.986262	+	0.165187i	$0.947177\pi$
$864$		0			0
$865$	4.76402	−	12.5445i	0.161982	−	0.426527i
$866$		0			0
$867$		−	6.90194i		−	0.234402i
$868$		0			0
$869$	33.5578			1.13837
$870$		0			0
$871$	5.27780			0.178832
$872$		0			0
$873$		−	2.40242i		−	0.0813096i
$874$		0			0
$875$	41.1962	−	21.5500i	1.39269	−	0.728523i
$876$		0			0
$877$		−	37.5710i		−	1.26868i	−0.773054	−	0.634341i	$-0.781272\pi$
$877$		−	37.5710i		−	1.26868i	0.773054	−	0.634341i	$-0.218728\pi$
$878$		0			0
$879$	−27.0591			−0.912680
$880$		0			0
$881$	36.7833			1.23926			0.619631	−	0.784893i	$-0.287282\pi$
$881$	36.7833			1.23926			0.619631	+	0.784893i	$0.287282\pi$
$882$		0			0
$883$			9.94482i			0.334670i	0.985900	+	0.167335i	$0.0535161\pi$
$883$			9.94482i			0.334670i	−0.985900	+	0.167335i	$0.946484\pi$
$884$		0			0
$885$	10.7494	−	28.3052i	0.361339	−	0.951469i
$886$		0			0
$887$			31.1401i			1.04558i	0.852461	+	0.522791i	$0.175109\pi$
$887$			31.1401i			1.04558i	−0.852461	+	0.522791i	$0.824891\pi$
$888$		0			0
$889$	34.9753			1.17303
$890$		0			0
$891$	−2.25773			−0.0756369
$892$		0			0
$893$			65.8731i			2.20436i
$894$		0			0
$895$	−26.6234	−	10.1107i	−0.889921	−	0.337964i
$896$		0			0
$897$			0.730564i			0.0243928i
$898$		0			0
$899$	−36.9807			−1.23338
$900$		0			0
$901$	−29.9072			−0.996354
$902$		0			0
$903$			11.2264i			0.373591i
$904$		0			0
$905$	46.1997	+	17.5452i	1.53573	+	0.583222i
$906$		0			0
$907$		−	36.0496i		−	1.19701i	−0.801121	−	0.598503i	$-0.795762\pi$
$907$		−	36.0496i		−	1.19701i	0.801121	−	0.598503i	$-0.204238\pi$
$908$		0			0
$909$	−13.1031			−0.434601
$910$		0			0
$911$	7.09026			0.234911			0.117455	−	0.993078i	$-0.462526\pi$
$911$	7.09026			0.234911			0.117455	+	0.993078i	$0.462526\pi$
$912$		0			0
$913$			34.9116i			1.15541i
$914$		0			0
$915$	−1.80820	+	4.76133i	−0.0597774	+	0.157405i
$916$		0			0
$917$		−	22.4225i		−	0.740457i
$918$		0			0
$919$	−56.7530			−1.87211			−0.936054	−	0.351857i	$-0.885550\pi$
$919$	−56.7530			−1.87211			−0.936054	+	0.351857i	$0.885550\pi$
$920$		0			0
$921$	6.33286			0.208675
$922$		0			0
$923$		−	8.83591i		−	0.290838i
$924$		0			0
$925$	−4.73730	+	5.33754i	−0.155762	+	0.175497i
$926$		0			0
$927$		−	9.15986i		−	0.300849i
$928$		0			0
$929$	54.5724			1.79046			0.895231	−	0.445602i	$-0.147010\pi$
$929$	54.5724			1.79046			0.895231	+	0.445602i	$0.147010\pi$
$930$		0			0
$931$	50.5491			1.65668
$932$		0			0
$933$			15.9547i			0.522332i
$934$		0			0
$935$	−8.76271	+	23.0738i	−0.286571	+	0.754593i
$936$		0			0
$937$			12.5213i			0.409054i	0.978861	+	0.204527i	$0.0655657\pi$
$937$			12.5213i			0.409054i	−0.978861	+	0.204527i	$0.934434\pi$
$938$		0			0
$939$	8.19940			0.267577
$940$		0			0
$941$	−34.5743			−1.12709			−0.563545	−	0.826085i	$-0.690563\pi$
$941$	−34.5743			−1.12709			−0.563545	+	0.826085i	$0.690563\pi$
$942$		0			0
$943$			5.48840i			0.178727i
$944$		0			0
$945$	−8.69271	−	3.30122i	−0.282774	−	0.107389i
$946$		0			0
$947$		−	46.5130i		−	1.51147i	−0.654878	−	0.755734i	$-0.727280\pi$
$947$		−	46.5130i		−	1.51147i	0.654878	−	0.755734i	$-0.272720\pi$
$948$		0			0
$949$	−7.73135			−0.250970
$950$		0			0
$951$	19.6908			0.638516
$952$		0			0
$953$			4.47068i			0.144820i	0.997375	+	0.0724098i	$0.0230689\pi$
$953$			4.47068i			0.144820i	−0.997375	+	0.0724098i	$0.976931\pi$
$954$		0			0
$955$	11.8124	+	4.48600i	0.382242	+	0.145163i
$956$		0			0
$957$		−	17.3508i		−	0.560872i
$958$		0			0
$959$	86.2268			2.78441
$960$		0			0
$961$	−7.84440			−0.253045
$962$		0			0
$963$			17.0677i			0.550000i
$964$		0			0
$965$	−12.1094	+	31.8862i	−0.389815	+	1.02645i
$966$		0			0
$967$		−	15.5416i		−	0.499785i	−0.968274	−	0.249893i	$-0.919605\pi$
$967$		−	15.5416i		−	0.499785i	0.968274	−	0.249893i	$-0.0803953\pi$
$968$		0			0
$969$	24.0115			0.771359
$970$		0			0
$971$	−9.70784			−0.311539			−0.155770	−	0.987793i	$-0.549786\pi$
$971$	−9.70784			−0.311539			−0.155770	+	0.987793i	$0.549786\pi$
$972$		0			0
$973$		−	25.9684i		−	0.832509i
$974$		0			0
$975$	−2.73198	−	2.42475i	−0.0874932	−	0.0776540i
$976$		0			0
$977$		−	0.232574i		−	0.00744069i	−0.999993	−	0.00372034i	$-0.998816\pi$
$977$		−	0.232574i		−	0.00744069i	0.999993	−	0.00372034i	$-0.00118422\pi$
$978$		0			0
$979$	−11.0409			−0.352868
$980$		0			0
$981$	−6.03908			−0.192813
$982$		0			0
$983$		−	18.1348i		−	0.578410i	−0.957267	−	0.289205i	$-0.906609\pi$
$983$		−	18.1348i		−	0.578410i	0.957267	−	0.289205i	$-0.0933910\pi$
$984$		0			0
$985$	5.92462	−	15.6006i	0.188774	−	0.497076i
$986$		0			0
$987$			55.7740i			1.77531i
$988$		0			0
$989$	−2.69970			−0.0858453
$990$		0			0
$991$	−40.7854			−1.29559			−0.647796	−	0.761814i	$-0.724309\pi$
$991$	−40.7854			−1.29559			−0.647796	+	0.761814i	$0.724309\pi$
$992$		0			0
$993$			25.8233i			0.819479i
$994$		0			0
$995$	47.8384	+	18.1675i	1.51658	+	0.575949i
$996$		0			0
$997$			20.8820i			0.661339i	0.943747	+	0.330669i	$0.107275\pi$
$997$			20.8820i			0.661339i	−0.943747	+	0.330669i	$0.892725\pi$
$998$		0			0
$999$	1.42733			0.0451586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.k.e.2209.9	yes	16
5.4	even	2	inner	2760.2.k.e.2209.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.k.e.2209.1	✓	16	5.4	even	2	inner
2760.2.k.e.2209.9	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.09040 - 0.793869i) q^{5} +4.15840i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-2.09040 - 0.793869i) q^{5} +4.15840i q^{7} -1.00000 q^{9} -2.25773 q^{11} +0.730564i q^{13} +(0.793869 - 2.09040i) q^{15} +4.88896i q^{17} -4.91136 q^{19} -4.15840 q^{21} -1.00000i q^{23} +(3.73954 + 3.31901i) q^{25} -1.00000i q^{27} +7.68506 q^{29} -4.81203 q^{31} -2.25773i q^{33} +(3.30122 - 8.69271i) q^{35} +1.42733i q^{37} -0.730564 q^{39} -5.48840 q^{41} -2.69970i q^{43} +(2.09040 + 0.793869i) q^{45} -13.4124i q^{47} -10.2923 q^{49} -4.88896 q^{51} +6.11730i q^{53} +(4.71957 + 1.79235i) q^{55} -4.91136i q^{57} +13.5406 q^{59} -2.27771 q^{61} -4.15840i q^{63} +(0.579972 - 1.52717i) q^{65} -7.22429i q^{67} +1.00000 q^{69} -12.0946 q^{71} +10.5827i q^{73} +(-3.31901 + 3.73954i) q^{75} -9.38856i q^{77} -14.8635 q^{79} +1.00000 q^{81} -15.4631i q^{83} +(3.88119 - 10.2199i) q^{85} +7.68506i q^{87} +4.89024 q^{89} -3.03797 q^{91} -4.81203i q^{93} +(10.2667 + 3.89898i) q^{95} +2.40242i q^{97} +2.25773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^5 - 16 * q^9	\( 16 q - 2 q^{5} - 16 q^{9} - 2 q^{15} - 8 q^{19} + 6 q^{21} + 22 q^{29} + 30 q^{31} + 2 q^{35} - 4 q^{39} - 22 q^{41} + 2 q^{45} - 38 q^{49} + 2 q^{51} + 12 q^{55} - 6 q^{59} - 4 q^{61} + 20 q^{65} + 16 q^{69} - 10 q^{71} - 4 q^{75} - 84 q^{79} + 16 q^{81} + 22 q^{85} + 16 q^{89} + 36 q^{91} + 76 q^{95}+O(q^{100}) \) 16 * q - 2 * q^5 - 16 * q^9 - 2 * q^15 - 8 * q^19 + 6 * q^21 + 22 * q^29 + 30 * q^31 + 2 * q^35 - 4 * q^39 - 22 * q^41 + 2 * q^45 - 38 * q^49 + 2 * q^51 + 12 * q^55 - 6 * q^59 - 4 * q^61 + 20 * q^65 + 16 * q^69 - 10 * q^71 - 4 * q^75 - 84 * q^79 + 16 * q^81 + 22 * q^85 + 16 * q^89 + 36 * q^91 + 76 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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