LMFDB - Embedded newform 2760.2.k.d.2209.7

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 2 x^{12} + 112 x^{10} - 228 x^{9} + 232 x^{8} + 40 x^{7} + 1316 x^{6} - 2688 x^{5} + \cdots + 128 $ x^14 - 2x^13 + 2x^12 + 112x^10 - 228x^9 + 232x^8 + 40x^7 + 1316x^6 - 2688x^5 + 2752x^4 + 608x^3 + 256x^2 - 256x + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.7
Root			$2.20998 - 2.20998i$ of defining polynomial
Character	$\chi$	$=$	2760.2209
Dual form			2760.2.k.d.2209.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.20998 - 0.340571i) q^{5} +3.34134i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(2.20998 - 0.340571i) q^{5} +3.34134i q^{7} -1.00000 q^{9} -2.80370 q^{11} +4.69348i q^{13} +(-0.340571 - 2.20998i) q^{15} -1.44262i q^{17} -3.37462 q^{19} +3.34134 q^{21} -1.00000i q^{23} +(4.76802 - 1.50531i) q^{25} +1.00000i q^{27} +2.31379 q^{29} -5.27474 q^{31} +2.80370i q^{33} +(1.13796 + 7.38428i) q^{35} +7.36064i q^{37} +4.69348 q^{39} -4.50991 q^{41} +9.69135i q^{43} +(-2.20998 + 0.340571i) q^{45} -2.04968i q^{47} -4.16452 q^{49} -1.44262 q^{51} +0.361261i q^{53} +(-6.19612 + 0.954859i) q^{55} +3.37462i q^{57} -9.06304 q^{59} -8.84257 q^{61} -3.34134i q^{63} +(1.59846 + 10.3725i) q^{65} +5.39033i q^{67} -1.00000 q^{69} -1.63918 q^{71} +11.8851i q^{73} +(-1.50531 - 4.76802i) q^{75} -9.36810i q^{77} +4.60979 q^{79} +1.00000 q^{81} -5.25307i q^{83} +(-0.491316 - 3.18817i) q^{85} -2.31379i q^{87} -12.5643 q^{89} -15.6825 q^{91} +5.27474i q^{93} +(-7.45785 + 1.14930i) q^{95} -10.1248i q^{97} +2.80370 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} - 14 q^{9}+O(q^{10}) $ 14 * q + 2 * q^5 - 14 * q^9	$ 14 q + 2 q^{5} - 14 q^{9} + 8 q^{11} - 4 q^{15} + 8 q^{19} - 10 q^{21} - 10 q^{25} + 26 q^{29} - 18 q^{31} - 10 q^{35} + 12 q^{39} - 2 q^{41} - 2 q^{45} + 4 q^{49} - 6 q^{51} - 32 q^{55} - 10 q^{59} - 24 q^{61} - 36 q^{65} - 14 q^{69} - 38 q^{71} + 4 q^{79} + 14 q^{81} - 14 q^{85} - 16 q^{89} - 4 q^{91} - 8 q^{95} - 8 q^{99}+O(q^{100}) $ 14 * q + 2 * q^5 - 14 * q^9 + 8 * q^11 - 4 * q^15 + 8 * q^19 - 10 * q^21 - 10 * q^25 + 26 * q^29 - 18 * q^31 - 10 * q^35 + 12 * q^39 - 2 * q^41 - 2 * q^45 + 4 * q^49 - 6 * q^51 - 32 * q^55 - 10 * q^59 - 24 * q^61 - 36 * q^65 - 14 * q^69 - 38 * q^71 + 4 * q^79 + 14 * q^81 - 14 * q^85 - 16 * q^89 - 4 * q^91 - 8 * q^95 - 8 * q^99

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
$5$	2.20998	−	0.340571i	0.988333	−	0.152308i
$5$	2.20998	−	0.340571i	0.988333	−	0.152308i
$6$		0			0
$7$			3.34134i			1.26291i	0.775414	+	0.631453i	$0.217541\pi$
$7$			3.34134i			1.26291i	−0.775414	+	0.631453i	$0.782459\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.80370			−0.845347			−0.422673	−	0.906282i	$-0.638908\pi$
$11$	−2.80370			−0.845347			−0.422673	+	0.906282i	$0.638908\pi$
$12$		0			0
$13$			4.69348i			1.30174i	0.759191	+	0.650868i	$0.225595\pi$
$13$			4.69348i			1.30174i	−0.759191	+	0.650868i	$0.774405\pi$
$14$		0			0
$15$	−0.340571	−	2.20998i	−0.0879351	−	0.570614i
$16$		0			0
$17$		−	1.44262i		−	0.349887i	−0.984578	−	0.174944i	$-0.944026\pi$
$17$		−	1.44262i		−	0.349887i	0.984578	−	0.174944i	$-0.0559743\pi$
$18$		0			0
$19$	−3.37462			−0.774191			−0.387096	−	0.922040i	$-0.626522\pi$
$19$	−3.37462			−0.774191			−0.387096	+	0.922040i	$0.626522\pi$
$20$		0			0
$21$	3.34134			0.729139
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	4.76802	−	1.50531i	0.953604	−	0.301062i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	2.31379			0.429661			0.214830	−	0.976651i	$-0.431080\pi$
$29$	2.31379			0.429661			0.214830	+	0.976651i	$0.431080\pi$
$30$		0			0
$31$	−5.27474			−0.947372			−0.473686	−	0.880694i	$-0.657077\pi$
$31$	−5.27474			−0.947372			−0.473686	+	0.880694i	$0.657077\pi$
$32$		0			0
$33$			2.80370i			0.488061i
$34$		0			0
$35$	1.13796	+	7.38428i	0.192351	+	1.24817i
$36$		0			0
$37$			7.36064i			1.21008i	0.796195	+	0.605041i	$0.206843\pi$
$37$			7.36064i			1.21008i	−0.796195	+	0.605041i	$0.793157\pi$
$38$		0			0
$39$	4.69348			0.751558
$40$		0			0
$41$	−4.50991			−0.704330			−0.352165	−	0.935938i	$-0.614554\pi$
$41$	−4.50991			−0.704330			−0.352165	+	0.935938i	$0.614554\pi$
$42$		0			0
$43$			9.69135i			1.47792i	0.673751	+	0.738959i	$0.264682\pi$
$43$			9.69135i			1.47792i	−0.673751	+	0.738959i	$0.735318\pi$
$44$		0			0
$45$	−2.20998	+	0.340571i	−0.329444	+	0.0507694i
$46$		0			0
$47$		−	2.04968i		−	0.298976i	−0.988764	−	0.149488i	$-0.952237\pi$
$47$		−	2.04968i		−	0.298976i	0.988764	−	0.149488i	$-0.0477625\pi$
$48$		0			0
$49$	−4.16452			−0.594932
$50$		0			0
$51$	−1.44262			−0.202008
$52$		0			0
$53$			0.361261i			0.0496229i	0.999692	+	0.0248115i	$0.00789855\pi$
$53$			0.361261i			0.0496229i	−0.999692	+	0.0248115i	$0.992101\pi$
$54$		0			0
$55$	−6.19612	+	0.954859i	−0.835484	+	0.128753i
$56$		0			0
$57$			3.37462i			0.446980i
$58$		0			0
$59$	−9.06304			−1.17991			−0.589953	−	0.807437i	$-0.700854\pi$
$59$	−9.06304			−1.17991			−0.589953	+	0.807437i	$0.700854\pi$
$60$		0			0
$61$	−8.84257			−1.13217			−0.566087	−	0.824345i	$-0.691543\pi$
$61$	−8.84257			−1.13217			−0.566087	+	0.824345i	$0.691543\pi$
$62$		0			0
$63$		−	3.34134i		−	0.420969i
$64$		0			0
$65$	1.59846	+	10.3725i	0.198265	+	1.28655i
$66$		0			0
$67$			5.39033i			0.658534i	0.944237	+	0.329267i	$0.106802\pi$
$67$			5.39033i			0.658534i	−0.944237	+	0.329267i	$0.893198\pi$
$68$		0			0
$69$	−1.00000			−0.120386
$70$		0			0
$71$	−1.63918			−0.194534			−0.0972672	−	0.995258i	$-0.531010\pi$
$71$	−1.63918			−0.194534			−0.0972672	+	0.995258i	$0.531010\pi$
$72$		0			0
$73$			11.8851i			1.39104i	0.718506	+	0.695521i	$0.244826\pi$
$73$			11.8851i			1.39104i	−0.718506	+	0.695521i	$0.755174\pi$
$74$		0			0
$75$	−1.50531	−	4.76802i	−0.173818	−	0.550564i
$76$		0			0
$77$		−	9.36810i		−	1.06759i
$78$		0			0
$79$	4.60979			0.518642			0.259321	−	0.965791i	$-0.416501\pi$
$79$	4.60979			0.518642			0.259321	+	0.965791i	$0.416501\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	5.25307i		−	0.576599i	−0.957540	−	0.288299i	$-0.906910\pi$
$83$		−	5.25307i		−	0.576599i	0.957540	−	0.288299i	$-0.0930898\pi$
$84$		0			0
$85$	−0.491316	−	3.18817i	−0.0532907	−	0.345805i
$86$		0			0
$87$		−	2.31379i		−	0.248065i
$88$		0			0
$89$	−12.5643			−1.33181			−0.665905	−	0.746037i	$-0.731954\pi$
$89$	−12.5643			−1.33181			−0.665905	+	0.746037i	$0.731954\pi$
$90$		0			0
$91$	−15.6825			−1.64397
$92$		0			0
$93$			5.27474i			0.546965i
$94$		0			0
$95$	−7.45785	+	1.14930i	−0.765159	+	0.117916i
$96$		0			0
$97$		−	10.1248i		−	1.02801i	−0.857786	−	0.514007i	$-0.828161\pi$
$97$		−	10.1248i		−	1.02801i	0.857786	−	0.514007i	$-0.171839\pi$
$98$		0			0
$99$	2.80370			0.281782
$100$		0			0
$101$	16.3073			1.62264			0.811320	−	0.584602i	$-0.198749\pi$
$101$	16.3073			1.62264			0.811320	+	0.584602i	$0.198749\pi$
$102$		0			0
$103$		−	0.0236876i		−	0.00233401i	−0.999999	−	0.00116700i	$-0.999629\pi$
$103$		−	0.0236876i		−	0.00233401i	0.999999	−	0.00116700i	$-0.000371469\pi$
$104$		0			0
$105$	7.38428	−	1.13796i	0.720632	−	0.111054i
$106$		0			0
$107$			5.84284i			0.564849i	0.959290	+	0.282424i	$0.0911386\pi$
$107$			5.84284i			0.564849i	−0.959290	+	0.282424i	$0.908861\pi$
$108$		0			0
$109$	15.7186			1.50557			0.752787	−	0.658265i	$-0.228709\pi$
$109$	15.7186			1.50557			0.752787	+	0.658265i	$0.228709\pi$
$110$		0			0
$111$	7.36064			0.698641
$112$		0			0
$113$			18.2481i			1.71664i	0.513118	+	0.858318i	$0.328490\pi$
$113$			18.2481i			1.71664i	−0.513118	+	0.858318i	$0.671510\pi$
$114$		0			0
$115$	−0.340571	−	2.20998i	−0.0317584	−	0.206082i
$116$		0			0
$117$		−	4.69348i		−	0.433912i
$118$		0			0
$119$	4.82029			0.441875
$120$		0			0
$121$	−3.13928			−0.285389
$122$		0			0
$123$			4.50991i			0.406645i
$124$		0			0
$125$	10.0246	−	4.95056i	0.896625	−	0.442792i
$126$		0			0
$127$			10.3868i			0.921677i	0.887484	+	0.460839i	$0.152451\pi$
$127$			10.3868i			0.921677i	−0.887484	+	0.460839i	$0.847549\pi$
$128$		0			0
$129$	9.69135			0.853276
$130$		0			0
$131$	−10.7584			−0.939969			−0.469985	−	0.882675i	$-0.655741\pi$
$131$	−10.7584			−0.939969			−0.469985	+	0.882675i	$0.655741\pi$
$132$		0			0
$133$		−	11.2757i		−	0.977731i
$134$		0			0
$135$	0.340571	+	2.20998i	0.0293117	+	0.190205i
$136$		0			0
$137$			1.31482i			0.112333i	0.998421	+	0.0561663i	$0.0178877\pi$
$137$			1.31482i			0.112333i	−0.998421	+	0.0561663i	$0.982112\pi$
$138$		0			0
$139$	10.5288			0.893039			0.446519	−	0.894774i	$-0.352663\pi$
$139$	10.5288			0.893039			0.446519	+	0.894774i	$0.352663\pi$
$140$		0			0
$141$	−2.04968			−0.172614
$142$		0			0
$143$		−	13.1591i		−	1.10042i
$144$		0			0
$145$	5.11344	−	0.788012i	0.424648	−	0.0654408i
$146$		0			0
$147$			4.16452i			0.343484i
$148$		0			0
$149$	8.26615			0.677190			0.338595	−	0.940932i	$-0.390048\pi$
$149$	8.26615			0.677190			0.338595	+	0.940932i	$0.390048\pi$
$150$		0			0
$151$	−19.6308			−1.59754			−0.798768	−	0.601639i	$-0.794514\pi$
$151$	−19.6308			−1.59754			−0.798768	+	0.601639i	$0.794514\pi$
$152$		0			0
$153$			1.44262i			0.116629i
$154$		0			0
$155$	−11.6571	+	1.79643i	−0.936319	+	0.144292i
$156$		0			0
$157$		−	19.8921i		−	1.58757i	−0.608201	−	0.793783i	$-0.708109\pi$
$157$		−	19.8921i		−	1.58757i	0.608201	−	0.793783i	$-0.291891\pi$
$158$		0			0
$159$	0.361261			0.0286498
$160$		0			0
$161$	3.34134			0.263334
$162$		0			0
$163$		−	10.1544i		−	0.795357i	−0.917525	−	0.397678i	$-0.869816\pi$
$163$		−	10.1544i		−	0.795357i	0.917525	−	0.397678i	$-0.130184\pi$
$164$		0			0
$165$	0.954859	+	6.19612i	0.0743357	+	0.482367i
$166$		0			0
$167$			17.7387i			1.37266i	0.727289	+	0.686331i	$0.240780\pi$
$167$			17.7387i			1.37266i	−0.727289	+	0.686331i	$0.759220\pi$
$168$		0			0
$169$	−9.02874			−0.694519
$170$		0			0
$171$	3.37462			0.258064
$172$		0			0
$173$			2.50785i			0.190669i	0.995445	+	0.0953343i	$0.0303920\pi$
$173$			2.50785i			0.190669i	−0.995445	+	0.0953343i	$0.969608\pi$
$174$		0			0
$175$	5.02975	+	15.9316i	0.380213	+	1.20431i
$176$		0			0
$177$			9.06304i			0.681220i
$178$		0			0
$179$	6.33269			0.473327			0.236664	−	0.971592i	$-0.423946\pi$
$179$	6.33269			0.473327			0.236664	+	0.971592i	$0.423946\pi$
$180$		0			0
$181$	−0.752057			−0.0559000			−0.0279500	−	0.999609i	$-0.508898\pi$
$181$	−0.752057			−0.0559000			−0.0279500	+	0.999609i	$0.508898\pi$
$182$		0			0
$183$			8.84257i			0.653661i
$184$		0			0
$185$	2.50682	+	16.2669i	0.184305	+	1.19596i
$186$		0			0
$187$			4.04468i			0.295776i
$188$		0			0
$189$	−3.34134			−0.243046
$190$		0			0
$191$	−2.23880			−0.161994			−0.0809968	−	0.996714i	$-0.525810\pi$
$191$	−2.23880			−0.161994			−0.0809968	+	0.996714i	$0.525810\pi$
$192$		0			0
$193$			12.5789i			0.905452i	0.891650	+	0.452726i	$0.149548\pi$
$193$			12.5789i			0.905452i	−0.891650	+	0.452726i	$0.850452\pi$
$194$		0			0
$195$	10.3725	−	1.59846i	0.742790	−	0.114468i
$196$		0			0
$197$		−	12.3492i		−	0.879847i	−0.898035	−	0.439924i	$-0.855006\pi$
$197$		−	12.3492i		−	0.879847i	0.898035	−	0.439924i	$-0.144994\pi$
$198$		0			0
$199$	−5.51668			−0.391067			−0.195533	−	0.980697i	$-0.562644\pi$
$199$	−5.51668			−0.391067			−0.195533	+	0.980697i	$0.562644\pi$
$200$		0			0
$201$	5.39033			0.380205
$202$		0			0
$203$			7.73116i			0.542621i
$204$		0			0
$205$	−9.96681	+	1.53595i	−0.696112	+	0.107275i
$206$		0			0
$207$			1.00000i			0.0695048i
$208$		0			0
$209$	9.46142			0.654460
$210$		0			0
$211$	18.4941			1.27319			0.636593	−	0.771200i	$-0.280343\pi$
$211$	18.4941			1.27319			0.636593	+	0.771200i	$0.280343\pi$
$212$		0			0
$213$			1.63918i			0.112315i
$214$		0			0
$215$	3.30060	+	21.4177i	0.225099	+	1.46067i
$216$		0			0
$217$		−	17.6247i		−	1.19644i
$218$		0			0
$219$	11.8851			0.803118
$220$		0			0
$221$	6.77092			0.455461
$222$		0			0
$223$		−	1.00866i		−	0.0675447i	−0.999430	−	0.0337724i	$-0.989248\pi$
$223$		−	1.00866i		−	0.0675447i	0.999430	−	0.0337724i	$-0.0107521\pi$
$224$		0			0
$225$	−4.76802	+	1.50531i	−0.317868	+	0.100354i
$226$		0			0
$227$			7.25039i			0.481225i	0.970621	+	0.240613i	$0.0773483\pi$
$227$			7.25039i			0.481225i	−0.970621	+	0.240613i	$0.922652\pi$
$228$		0			0
$229$	11.7342			0.775420			0.387710	−	0.921781i	$-0.373266\pi$
$229$	11.7342			0.775420			0.387710	+	0.921781i	$0.373266\pi$
$230$		0			0
$231$	−9.36810			−0.616375
$232$		0			0
$233$		−	10.8525i		−	0.710971i	−0.934682	−	0.355485i	$-0.884316\pi$
$233$		−	10.8525i		−	0.710971i	0.934682	−	0.355485i	$-0.115684\pi$
$234$		0			0
$235$	−0.698061	−	4.52974i	−0.0455365	−	0.295488i
$236$		0			0
$237$		−	4.60979i		−	0.299438i
$238$		0			0
$239$	15.4333			0.998296			0.499148	−	0.866517i	$-0.333646\pi$
$239$	15.4333			0.998296			0.499148	+	0.866517i	$0.333646\pi$
$240$		0			0
$241$	−3.62017			−0.233196			−0.116598	−	0.993179i	$-0.537199\pi$
$241$	−3.62017			−0.233196			−0.116598	+	0.993179i	$0.537199\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−9.20351	+	1.41832i	−0.587991	+	0.0906129i
$246$		0			0
$247$		−	15.8387i		−	1.00779i
$248$		0			0
$249$	−5.25307			−0.332899
$250$		0			0
$251$	−7.12930			−0.449997			−0.224999	−	0.974359i	$-0.572238\pi$
$251$	−7.12930			−0.449997			−0.224999	+	0.974359i	$0.572238\pi$
$252$		0			0
$253$			2.80370i			0.176267i
$254$		0			0
$255$	−3.18817	+	0.491316i	−0.199651	+	0.0307674i
$256$		0			0
$257$		−	2.68739i		−	0.167635i	−0.996481	−	0.0838174i	$-0.973289\pi$
$257$		−	2.68739i		−	0.167635i	0.996481	−	0.0838174i	$-0.0267112\pi$
$258$		0			0
$259$	−24.5944			−1.52822
$260$		0			0
$261$	−2.31379			−0.143220
$262$		0			0
$263$			23.3856i			1.44202i	0.692925	+	0.721010i	$0.256322\pi$
$263$			23.3856i			1.44202i	−0.692925	+	0.721010i	$0.743678\pi$
$264$		0			0
$265$	0.123035	+	0.798378i	0.00755798	+	0.0490440i
$266$		0			0
$267$			12.5643i			0.768921i
$268$		0			0
$269$	21.9479			1.33819			0.669094	−	0.743178i	$-0.266682\pi$
$269$	21.9479			1.33819			0.669094	+	0.743178i	$0.266682\pi$
$270$		0			0
$271$	−13.0888			−0.795085			−0.397543	−	0.917584i	$-0.630137\pi$
$271$	−13.0888			−0.795085			−0.397543	+	0.917584i	$0.630137\pi$
$272$		0			0
$273$			15.6825i			0.949147i
$274$		0			0
$275$	−13.3681	+	4.22044i	−0.806127	+	0.254502i
$276$		0			0
$277$			30.6235i			1.83999i	0.391931	+	0.919995i	$0.371807\pi$
$277$			30.6235i			1.83999i	−0.391931	+	0.919995i	$0.628193\pi$
$278$		0			0
$279$	5.27474			0.315791
$280$		0			0
$281$	32.5432			1.94136			0.970681	−	0.240370i	$-0.0772688\pi$
$281$	32.5432			1.94136			0.970681	+	0.240370i	$0.0772688\pi$
$282$		0			0
$283$			8.50785i			0.505739i	0.967500	+	0.252870i	$0.0813744\pi$
$283$			8.50785i			0.505739i	−0.967500	+	0.252870i	$0.918626\pi$
$284$		0			0
$285$	1.14930	+	7.45785i	0.0680786	+	0.441765i
$286$		0			0
$287$		−	15.0691i		−	0.889502i
$288$		0			0
$289$	14.9188			0.877579
$290$		0			0
$291$	−10.1248			−0.593524
$292$		0			0
$293$		−	15.6357i		−	0.913448i	−0.889609	−	0.456724i	$-0.849023\pi$
$293$		−	15.6357i		−	0.913448i	0.889609	−	0.456724i	$-0.150977\pi$
$294$		0			0
$295$	−20.0291	+	3.08661i	−1.16614	+	0.179709i
$296$		0			0
$297$		−	2.80370i		−	0.162687i
$298$		0			0
$299$	4.69348			0.271431
$300$		0			0
$301$	−32.3821			−1.86647
$302$		0			0
$303$		−	16.3073i		−	0.936832i
$304$		0			0
$305$	−19.5419	+	3.01152i	−1.11897	+	0.172439i
$306$		0			0
$307$		−	19.0905i		−	1.08955i	−0.838582	−	0.544776i	$-0.816615\pi$
$307$		−	19.0905i		−	1.08955i	0.838582	−	0.544776i	$-0.183385\pi$
$308$		0			0
$309$	−0.0236876			−0.00134754
$310$		0			0
$311$	26.7760			1.51833			0.759164	−	0.650899i	$-0.225608\pi$
$311$	26.7760			1.51833			0.759164	+	0.650899i	$0.225608\pi$
$312$		0			0
$313$			14.5967i			0.825054i	0.910945	+	0.412527i	$0.135354\pi$
$313$			14.5967i			0.825054i	−0.910945	+	0.412527i	$0.864646\pi$
$314$		0			0
$315$	−1.13796	−	7.38428i	−0.0641169	−	0.416057i
$316$		0			0
$317$		−	15.3952i		−	0.864679i	−0.901711	−	0.432340i	$-0.857688\pi$
$317$		−	15.3952i		−	0.864679i	0.901711	−	0.432340i	$-0.142312\pi$
$318$		0			0
$319$	−6.48718			−0.363212
$320$		0			0
$321$	5.84284			0.326116
$322$		0			0
$323$			4.86830i			0.270880i
$324$		0			0
$325$	7.06515	+	22.3786i	0.391904	+	1.24134i
$326$		0			0
$327$		−	15.7186i		−	0.869243i
$328$		0			0
$329$	6.84865			0.377579
$330$		0			0
$331$	−6.17011			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$331$	−6.17011			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$332$		0			0
$333$		−	7.36064i		−	0.403361i
$334$		0			0
$335$	1.83579	+	11.9125i	0.100300	+	0.650851i
$336$		0			0
$337$			11.3421i			0.617845i	0.951087	+	0.308923i	$0.0999684\pi$
$337$			11.3421i			0.617845i	−0.951087	+	0.308923i	$0.900032\pi$
$338$		0			0
$339$	18.2481			0.991100
$340$		0			0
$341$	14.7888			0.800858
$342$		0			0
$343$			9.47428i			0.511563i
$344$		0			0
$345$	−2.20998	+	0.340571i	−0.118981	+	0.0183357i
$346$		0			0
$347$		−	22.2263i		−	1.19317i	−0.802549	−	0.596586i	$-0.796523\pi$
$347$		−	22.2263i		−	1.19317i	0.802549	−	0.596586i	$-0.203477\pi$
$348$		0			0
$349$	−32.6505			−1.74774			−0.873870	−	0.486160i	$-0.838397\pi$
$349$	−32.6505			−1.74774			−0.873870	+	0.486160i	$0.838397\pi$
$350$		0			0
$351$	−4.69348			−0.250519
$352$		0			0
$353$			19.6061i			1.04353i	0.853091	+	0.521763i	$0.174725\pi$
$353$			19.6061i			1.04353i	−0.853091	+	0.521763i	$0.825275\pi$
$354$		0			0
$355$	−3.62255	+	0.558257i	−0.192265	+	0.0296292i
$356$		0			0
$357$		−	4.82029i		−	0.255117i
$358$		0			0
$359$	−11.4304			−0.603274			−0.301637	−	0.953423i	$-0.597533\pi$
$359$	−11.4304			−0.603274			−0.301637	+	0.953423i	$0.597533\pi$
$360$		0			0
$361$	−7.61193			−0.400628
$362$		0			0
$363$			3.13928i			0.164769i
$364$		0			0
$365$	4.04771	+	26.2657i	0.211867	+	1.37481i
$366$		0			0
$367$		−	36.9457i		−	1.92855i	−0.264908	−	0.964274i	$-0.585342\pi$
$367$		−	36.9457i		−	1.92855i	0.264908	−	0.964274i	$-0.414658\pi$
$368$		0			0
$369$	4.50991			0.234777
$370$		0			0
$371$	−1.20709			−0.0626691
$372$		0			0
$373$			11.9365i			0.618048i	0.951054	+	0.309024i	$0.100002\pi$
$373$			11.9365i			0.618048i	−0.951054	+	0.309024i	$0.899998\pi$
$374$		0			0
$375$	−4.95056	−	10.0246i	−0.255646	−	0.517666i
$376$		0			0
$377$			10.8597i			0.559305i
$378$		0			0
$379$	−6.49401			−0.333575			−0.166787	−	0.985993i	$-0.553339\pi$
$379$	−6.49401			−0.333575			−0.166787	+	0.985993i	$0.553339\pi$
$380$		0			0
$381$	10.3868			0.532130
$382$		0			0
$383$			21.8904i			1.11855i	0.828983	+	0.559274i	$0.188920\pi$
$383$			21.8904i			1.11855i	−0.828983	+	0.559274i	$0.811080\pi$
$384$		0			0
$385$	−3.19050	−	20.7033i	−0.162603	−	1.05514i
$386$		0			0
$387$		−	9.69135i		−	0.492639i
$388$		0			0
$389$	−20.3414			−1.03135			−0.515675	−	0.856784i	$-0.672459\pi$
$389$	−20.3414			−1.03135			−0.515675	+	0.856784i	$0.672459\pi$
$390$		0			0
$391$	−1.44262			−0.0729566
$392$		0			0
$393$			10.7584i			0.542692i
$394$		0			0
$395$	10.1875	−	1.56996i	0.512591	−	0.0789934i
$396$		0			0
$397$		−	32.9504i		−	1.65373i	−0.562398	−	0.826866i	$-0.690121\pi$
$397$		−	32.9504i		−	1.65373i	0.562398	−	0.826866i	$-0.309879\pi$
$398$		0			0
$399$	−11.2757			−0.564493
$400$		0			0
$401$	7.66448			0.382746			0.191373	−	0.981517i	$-0.438706\pi$
$401$	7.66448			0.382746			0.191373	+	0.981517i	$0.438706\pi$
$402$		0			0
$403$		−	24.7569i		−	1.23323i
$404$		0			0
$405$	2.20998	−	0.340571i	0.109815	−	0.0169231i
$406$		0			0
$407$		−	20.6370i		−	1.02294i
$408$		0			0
$409$	−3.28798			−0.162580			−0.0812900	−	0.996690i	$-0.525904\pi$
$409$	−3.28798			−0.162580			−0.0812900	+	0.996690i	$0.525904\pi$
$410$		0			0
$411$	1.31482			0.0648552
$412$		0			0
$413$		−	30.2826i		−	1.49011i
$414$		0			0
$415$	−1.78904	−	11.6092i	−0.0878207	−	0.569872i
$416$		0			0
$417$		−	10.5288i		−	0.515596i
$418$		0			0
$419$	22.3730			1.09299			0.546496	−	0.837461i	$-0.315961\pi$
$419$	22.3730			1.09299			0.546496	+	0.837461i	$0.315961\pi$
$420$		0			0
$421$	16.3753			0.798081			0.399041	−	0.916933i	$-0.369343\pi$
$421$	16.3753			0.798081			0.399041	+	0.916933i	$0.369343\pi$
$422$		0			0
$423$			2.04968i			0.0996586i
$424$		0			0
$425$	−2.17160	−	6.87846i	−0.105338	−	0.333654i
$426$		0			0
$427$		−	29.5460i		−	1.42983i
$428$		0			0
$429$	−13.1591			−0.635327
$430$		0			0
$431$	0.824929			0.0397354			0.0198677	−	0.999803i	$-0.493675\pi$
$431$	0.824929			0.0397354			0.0198677	+	0.999803i	$0.493675\pi$
$432$		0			0
$433$			17.3870i			0.835564i	0.908547	+	0.417782i	$0.137192\pi$
$433$			17.3870i			0.835564i	−0.908547	+	0.417782i	$0.862808\pi$
$434$		0			0
$435$	−0.788012	−	5.11344i	−0.0377823	−	0.245171i
$436$		0			0
$437$			3.37462i			0.161430i
$438$		0			0
$439$	−11.0945			−0.529514			−0.264757	−	0.964315i	$-0.585292\pi$
$439$	−11.0945			−0.529514			−0.264757	+	0.964315i	$0.585292\pi$
$440$		0			0
$441$	4.16452			0.198311
$442$		0			0
$443$		−	25.9549i		−	1.23316i	−0.787294	−	0.616578i	$-0.788518\pi$
$443$		−	25.9549i		−	1.23316i	0.787294	−	0.616578i	$-0.211482\pi$
$444$		0			0
$445$	−27.7668	+	4.27903i	−1.31627	+	0.202845i
$446$		0			0
$447$		−	8.26615i		−	0.390976i
$448$		0			0
$449$	2.11511			0.0998180			0.0499090	−	0.998754i	$-0.484107\pi$
$449$	2.11511			0.0998180			0.0499090	+	0.998754i	$0.484107\pi$
$450$		0			0
$451$	12.6444			0.595403
$452$		0			0
$453$			19.6308i			0.922338i
$454$		0			0
$455$	−34.6580	+	5.34100i	−1.62479	+	0.250390i
$456$		0			0
$457$		−	18.0891i		−	0.846174i	−0.906089	−	0.423087i	$-0.860946\pi$
$457$		−	18.0891i		−	0.846174i	0.906089	−	0.423087i	$-0.139054\pi$
$458$		0			0
$459$	1.44262			0.0673359
$460$		0			0
$461$	36.5971			1.70450			0.852248	−	0.523137i	$-0.175239\pi$
$461$	36.5971			1.70450			0.852248	+	0.523137i	$0.175239\pi$
$462$		0			0
$463$		−	34.4795i		−	1.60240i	−0.598398	−	0.801199i	$-0.704196\pi$
$463$		−	34.4795i		−	1.60240i	0.598398	−	0.801199i	$-0.295804\pi$
$464$		0			0
$465$	1.79643	+	11.6571i	0.0833072	+	0.540584i
$466$		0			0
$467$		−	16.1724i		−	0.748368i	−0.927355	−	0.374184i	$-0.877923\pi$
$467$		−	16.1724i		−	0.748368i	0.927355	−	0.374184i	$-0.122077\pi$
$468$		0			0
$469$	−18.0109			−0.831666
$470$		0			0
$471$	−19.8921			−0.916582
$472$		0			0
$473$		−	27.1716i		−	1.24935i
$474$		0			0
$475$	−16.0903	+	5.07986i	−0.738272	+	0.233080i
$476$		0			0
$477$		−	0.361261i		−	0.0165410i
$478$		0			0
$479$	−6.15449			−0.281206			−0.140603	−	0.990066i	$-0.544904\pi$
$479$	−6.15449			−0.281206			−0.140603	+	0.990066i	$0.544904\pi$
$480$		0			0
$481$	−34.5470			−1.57521
$482$		0			0
$483$		−	3.34134i		−	0.152036i
$484$		0			0
$485$	−3.44820	−	22.3755i	−0.156575	−	1.01602i
$486$		0			0
$487$			5.17972i			0.234715i	0.993090	+	0.117358i	$0.0374424\pi$
$487$			5.17972i			0.234715i	−0.993090	+	0.117358i	$0.962558\pi$
$488$		0			0
$489$	−10.1544			−0.459199
$490$		0			0
$491$	29.3581			1.32491			0.662457	−	0.749100i	$-0.269514\pi$
$491$	29.3581			1.32491			0.662457	+	0.749100i	$0.269514\pi$
$492$		0			0
$493$		−	3.33793i		−	0.150333i
$494$		0			0
$495$	6.19612	−	0.954859i	0.278495	−	0.0429177i
$496$		0			0
$497$		−	5.47704i		−	0.245679i
$498$		0			0
$499$	−4.45774			−0.199556			−0.0997778	−	0.995010i	$-0.531813\pi$
$499$	−4.45774			−0.199556			−0.0997778	+	0.995010i	$0.531813\pi$
$500$		0			0
$501$	17.7387			0.792507
$502$		0			0
$503$			10.1488i			0.452514i	0.974068	+	0.226257i	$0.0726489\pi$
$503$			10.1488i			0.452514i	−0.974068	+	0.226257i	$0.927351\pi$
$504$		0			0
$505$	36.0389	−	5.55381i	1.60371	−	0.247141i
$506$		0			0
$507$			9.02874i			0.400980i
$508$		0			0
$509$	−39.5369			−1.75244			−0.876222	−	0.481908i	$-0.839944\pi$
$509$	−39.5369			−1.75244			−0.876222	+	0.481908i	$0.839944\pi$
$510$		0			0
$511$	−39.7120			−1.75675
$512$		0			0
$513$		−	3.37462i		−	0.148993i
$514$		0			0
$515$	−0.00806731	−	0.0523491i	−0.000355488	−	0.00230678i
$516$		0			0
$517$			5.74667i			0.252738i
$518$		0			0
$519$	2.50785			0.110083
$520$		0			0
$521$	23.5642			1.03237			0.516184	−	0.856478i	$-0.327352\pi$
$521$	23.5642			1.03237			0.516184	+	0.856478i	$0.327352\pi$
$522$		0			0
$523$		−	19.7786i		−	0.864856i	−0.901668	−	0.432428i	$-0.857657\pi$
$523$		−	19.7786i		−	0.864856i	0.901668	−	0.432428i	$-0.142343\pi$
$524$		0			0
$525$	15.9316	−	5.02975i	0.695310	−	0.219516i
$526$		0			0
$527$			7.60946i			0.331473i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	9.06304			0.393302
$532$		0			0
$533$		−	21.1672i		−	0.916852i
$534$		0			0
$535$	1.98990	+	12.9126i	0.0860311	+	0.558259i
$536$		0			0
$537$		−	6.33269i		−	0.273276i
$538$		0			0
$539$	11.6761			0.502924
$540$		0			0
$541$	6.55166			0.281678			0.140839	−	0.990033i	$-0.455020\pi$
$541$	6.55166			0.281678			0.140839	+	0.990033i	$0.455020\pi$
$542$		0			0
$543$			0.752057i			0.0322739i
$544$		0			0
$545$	34.7379	−	5.35332i	1.48801	−	0.229311i
$546$		0			0
$547$		−	28.0871i		−	1.20092i	−0.799656	−	0.600459i	$-0.794985\pi$
$547$		−	28.0871i		−	1.20092i	0.799656	−	0.600459i	$-0.205015\pi$
$548$		0			0
$549$	8.84257			0.377392
$550$		0			0
$551$	−7.80818			−0.332640
$552$		0			0
$553$			15.4029i			0.654996i
$554$		0			0
$555$	16.2669	−	2.50682i	0.690490	−	0.106409i
$556$		0			0
$557$		−	18.4110i		−	0.780101i	−0.920793	−	0.390051i	$-0.872458\pi$
$557$		−	18.4110i		−	0.780101i	0.920793	−	0.390051i	$-0.127542\pi$
$558$		0			0
$559$	−45.4861			−1.92386
$560$		0			0
$561$	4.04468			0.170766
$562$		0			0
$563$			1.71495i			0.0722764i	0.999347	+	0.0361382i	$0.0115057\pi$
$563$			1.71495i			0.0722764i	−0.999347	+	0.0361382i	$0.988494\pi$
$564$		0			0
$565$	6.21478	+	40.3279i	0.261458	+	1.69661i
$566$		0			0
$567$			3.34134i			0.140323i
$568$		0			0
$569$	−22.4437			−0.940888			−0.470444	−	0.882430i	$-0.655906\pi$
$569$	−22.4437			−0.940888			−0.470444	+	0.882430i	$0.655906\pi$
$570$		0			0
$571$	34.8428			1.45813			0.729063	−	0.684446i	$-0.239956\pi$
$571$	34.8428			1.45813			0.729063	+	0.684446i	$0.239956\pi$
$572$		0			0
$573$			2.23880i			0.0935270i
$574$		0			0
$575$	−1.50531	−	4.76802i	−0.0627758	−	0.198840i
$576$		0			0
$577$			8.00735i			0.333350i	0.986012	+	0.166675i	$0.0533031\pi$
$577$			8.00735i			0.333350i	−0.986012	+	0.166675i	$0.946697\pi$
$578$		0			0
$579$	12.5789			0.522763
$580$		0			0
$581$	17.5523			0.728190
$582$		0			0
$583$		−	1.01287i		−	0.0419486i
$584$		0			0
$585$	−1.59846	−	10.3725i	−0.0660884	−	0.428850i
$586$		0			0
$587$			39.4346i			1.62764i	0.581118	+	0.813819i	$0.302615\pi$
$587$			39.4346i			1.62764i	−0.581118	+	0.813819i	$0.697385\pi$
$588$		0			0
$589$	17.8003			0.733447
$590$		0			0
$591$	−12.3492			−0.507980
$592$		0			0
$593$		−	2.04096i		−	0.0838121i	−0.999122	−	0.0419060i	$-0.986657\pi$
$593$		−	2.04096i		−	0.0838121i	0.999122	−	0.0419060i	$-0.0133430\pi$
$594$		0			0
$595$	10.6527	−	1.64165i	0.436720	−	0.0673011i
$596$		0			0
$597$			5.51668i			0.225783i
$598$		0			0
$599$	32.9554			1.34652			0.673260	−	0.739406i	$-0.264894\pi$
$599$	32.9554			1.34652			0.673260	+	0.739406i	$0.264894\pi$
$600$		0			0
$601$	−21.5465			−0.878900			−0.439450	−	0.898267i	$-0.644827\pi$
$601$	−21.5465			−0.878900			−0.439450	+	0.898267i	$0.644827\pi$
$602$		0			0
$603$		−	5.39033i		−	0.219511i
$604$		0			0
$605$	−6.93774	+	1.06915i	−0.282059	+	0.0434670i
$606$		0			0
$607$			2.57475i			0.104506i	0.998634	+	0.0522530i	$0.0166402\pi$
$607$			2.57475i			0.104506i	−0.998634	+	0.0522530i	$0.983360\pi$
$608$		0			0
$609$	7.73116			0.313282
$610$		0			0
$611$	9.62011			0.389188
$612$		0			0
$613$			3.70180i			0.149514i	0.997202	+	0.0747571i	$0.0238181\pi$
$613$			3.70180i			0.149514i	−0.997202	+	0.0747571i	$0.976182\pi$
$614$		0			0
$615$	1.53595	+	9.96681i	0.0619353	+	0.401901i
$616$		0			0
$617$		−	3.20240i		−	0.128924i	−0.997920	−	0.0644620i	$-0.979467\pi$
$617$		−	3.20240i		−	0.128924i	0.997920	−	0.0644620i	$-0.0205331\pi$
$618$		0			0
$619$	−33.6612			−1.35296			−0.676479	−	0.736461i	$-0.736495\pi$
$619$	−33.6612			−1.35296			−0.676479	+	0.736461i	$0.736495\pi$
$620$		0			0
$621$	1.00000			0.0401286
$622$		0			0
$623$		−	41.9814i		−	1.68195i
$624$		0			0
$625$	20.4681	−	14.3547i	0.818723	−	0.574189i
$626$		0			0
$627$		−	9.46142i		−	0.377853i
$628$		0			0
$629$	10.6186			0.423392
$630$		0			0
$631$	24.9687			0.993989			0.496994	−	0.867754i	$-0.334437\pi$
$631$	24.9687			0.993989			0.496994	+	0.867754i	$0.334437\pi$
$632$		0			0
$633$		−	18.4941i		−	0.735075i
$634$		0			0
$635$	3.53744	+	22.9546i	0.140379	+	0.910924i
$636$		0			0
$637$		−	19.5461i		−	0.774444i
$638$		0			0
$639$	1.63918			0.0648448
$640$		0			0
$641$	−32.8142			−1.29608			−0.648042	−	0.761605i	$-0.724412\pi$
$641$	−32.8142			−1.29608			−0.648042	+	0.761605i	$0.724412\pi$
$642$		0			0
$643$			25.2162i			0.994430i	0.867627	+	0.497215i	$0.165644\pi$
$643$			25.2162i			0.994430i	−0.867627	+	0.497215i	$0.834356\pi$
$644$		0			0
$645$	21.4177	−	3.30060i	0.843321	−	0.129961i
$646$		0			0
$647$		−	26.3783i		−	1.03704i	−0.855066	−	0.518520i	$-0.826483\pi$
$647$		−	26.3783i		−	1.03704i	0.855066	−	0.518520i	$-0.173517\pi$
$648$		0			0
$649$	25.4100			0.997431
$650$		0			0
$651$	−17.6247			−0.690766
$652$		0			0
$653$		−	10.3702i		−	0.405815i	−0.979198	−	0.202908i	$-0.934961\pi$
$653$		−	10.3702i		−	0.405815i	0.979198	−	0.202908i	$-0.0650392\pi$
$654$		0			0
$655$	−23.7759	+	3.66402i	−0.929003	+	0.143165i
$656$		0			0
$657$		−	11.8851i		−	0.463680i
$658$		0			0
$659$	1.95218			0.0760462			0.0380231	−	0.999277i	$-0.487894\pi$
$659$	1.95218			0.0760462			0.0380231	+	0.999277i	$0.487894\pi$
$660$		0			0
$661$	22.9086			0.891042			0.445521	−	0.895272i	$-0.353019\pi$
$661$	22.9086			0.891042			0.445521	+	0.895272i	$0.353019\pi$
$662$		0			0
$663$		−	6.77092i		−	0.262961i
$664$		0			0
$665$	−3.84019	−	24.9192i	−0.148916	−	0.966324i
$666$		0			0
$667$		−	2.31379i		−	0.0895905i
$668$		0			0
$669$	−1.00866			−0.0389970
$670$		0			0
$671$	24.7919			0.957080
$672$		0			0
$673$		−	43.6003i		−	1.68067i	−0.542070	−	0.840333i	$-0.682359\pi$
$673$		−	43.6003i		−	1.68067i	0.542070	−	0.840333i	$-0.317641\pi$
$674$		0			0
$675$	1.50531	+	4.76802i	0.0579395	+	0.183521i
$676$		0			0
$677$		−	34.4622i		−	1.32449i	−0.749287	−	0.662245i	$-0.769604\pi$
$677$		−	34.4622i		−	1.32449i	0.749287	−	0.662245i	$-0.230396\pi$
$678$		0			0
$679$	33.8302			1.29829
$680$		0			0
$681$	7.25039			0.277835
$682$		0			0
$683$			25.2917i			0.967760i	0.875134	+	0.483880i	$0.160773\pi$
$683$			25.2917i			0.967760i	−0.875134	+	0.483880i	$0.839227\pi$
$684$		0			0
$685$	0.447790	+	2.90572i	0.0171092	+	0.111022i
$686$		0			0
$687$		−	11.7342i		−	0.447689i
$688$		0			0
$689$	−1.69557			−0.0645960
$690$		0			0
$691$	44.7272			1.70150			0.850752	−	0.525567i	$-0.176147\pi$
$691$	44.7272			1.70150			0.850752	+	0.525567i	$0.176147\pi$
$692$		0			0
$693$			9.36810i			0.355865i
$694$		0			0
$695$	23.2684	−	3.58580i	0.882620	−	0.136017i
$696$		0			0
$697$			6.50610i			0.246436i
$698$		0			0
$699$	−10.8525			−0.410479
$700$		0			0
$701$	−47.8600			−1.80765			−0.903823	−	0.427906i	$-0.859251\pi$
$701$	−47.8600			−1.80765			−0.903823	+	0.427906i	$0.859251\pi$
$702$		0			0
$703$		−	24.8394i		−	0.936834i
$704$		0			0
$705$	−4.52974	+	0.698061i	−0.170600	+	0.0262905i
$706$		0			0
$707$			54.4883i			2.04924i
$708$		0			0
$709$	28.9605			1.08763			0.543816	−	0.839204i	$-0.316979\pi$
$709$	28.9605			1.08763			0.543816	+	0.839204i	$0.316979\pi$
$710$		0			0
$711$	−4.60979			−0.172881
$712$		0			0
$713$			5.27474i			0.197541i
$714$		0			0
$715$	−4.48161	−	29.0813i	−0.167603	−	1.08758i
$716$		0			0
$717$		−	15.4333i		−	0.576366i
$718$		0			0
$719$	−25.6049			−0.954902			−0.477451	−	0.878658i	$-0.658439\pi$
$719$	−25.6049			−0.954902			−0.477451	+	0.878658i	$0.658439\pi$
$720$		0			0
$721$	0.0791482			0.00294763
$722$		0			0
$723$			3.62017i			0.134636i
$724$		0			0
$725$	11.0322	−	3.48298i	0.409726	−	0.129355i
$726$		0			0
$727$		−	11.1509i		−	0.413563i	−0.978387	−	0.206781i	$-0.933701\pi$
$727$		−	11.1509i		−	0.413563i	0.978387	−	0.206781i	$-0.0662988\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	13.9810			0.517105
$732$		0			0
$733$			12.2966i			0.454186i	0.973873	+	0.227093i	$0.0729221\pi$
$733$			12.2966i			0.454186i	−0.973873	+	0.227093i	$0.927078\pi$
$734$		0			0
$735$	1.41832	+	9.20351i	0.0523154	+	0.339477i
$736$		0			0
$737$		−	15.1129i		−	0.556689i
$738$		0			0
$739$	15.8510			0.583087			0.291544	−	0.956558i	$-0.405831\pi$
$739$	15.8510			0.583087			0.291544	+	0.956558i	$0.405831\pi$
$740$		0			0
$741$	−15.8387			−0.581850
$742$		0			0
$743$			27.8382i			1.02128i	0.859793	+	0.510642i	$0.170592\pi$
$743$			27.8382i			1.02128i	−0.859793	+	0.510642i	$0.829408\pi$
$744$		0			0
$745$	18.2680	−	2.81521i	0.669289	−	0.103141i
$746$		0			0
$747$			5.25307i			0.192200i
$748$		0			0
$749$	−19.5229			−0.713351
$750$		0			0
$751$	−37.6401			−1.37351			−0.686754	−	0.726890i	$-0.740965\pi$
$751$	−37.6401			−1.37351			−0.686754	+	0.726890i	$0.740965\pi$
$752$		0			0
$753$			7.12930i			0.259806i
$754$		0			0
$755$	−43.3838	+	6.68570i	−1.57890	+	0.243318i
$756$		0			0
$757$			46.9956i			1.70808i	0.520204	+	0.854042i	$0.325856\pi$
$757$			46.9956i			1.70808i	−0.520204	+	0.854042i	$0.674144\pi$
$758$		0			0
$759$	2.80370			0.101768
$760$		0			0
$761$	−6.28971			−0.228002			−0.114001	−	0.993481i	$-0.536367\pi$
$761$	−6.28971			−0.228002			−0.114001	+	0.993481i	$0.536367\pi$
$762$		0			0
$763$			52.5213i			1.90140i
$764$		0			0
$765$	0.491316	+	3.18817i	0.0177636	+	0.115268i
$766$		0			0
$767$		−	42.5372i		−	1.53593i
$768$		0			0
$769$	−4.69437			−0.169283			−0.0846417	−	0.996411i	$-0.526975\pi$
$769$	−4.69437			−0.169283			−0.0846417	+	0.996411i	$0.526975\pi$
$770$		0			0
$771$	−2.68739			−0.0967840
$772$		0			0
$773$			7.28638i			0.262073i	0.991378	+	0.131037i	$0.0418305\pi$
$773$			7.28638i			0.262073i	−0.991378	+	0.131037i	$0.958169\pi$
$774$		0			0
$775$	−25.1501	+	7.94013i	−0.903418	+	0.285218i
$776$		0			0
$777$			24.5944i			0.882318i
$778$		0			0
$779$	15.2192			0.545286
$780$		0			0
$781$	4.59576			0.164449
$782$		0			0
$783$			2.31379i			0.0826883i
$784$		0			0
$785$	−6.77470	−	43.9612i	−0.241799	−	1.56904i
$786$		0			0
$787$		−	8.15046i		−	0.290533i	−0.989393	−	0.145266i	$-0.953596\pi$
$787$		−	8.15046i		−	0.290533i	0.989393	−	0.145266i	$-0.0464039\pi$
$788$		0			0
$789$	23.3856			0.832550
$790$		0			0
$791$	−60.9730			−2.16795
$792$		0			0
$793$		−	41.5024i		−	1.47379i
$794$		0			0
$795$	0.798378	−	0.123035i	0.0283156	−	0.00436360i
$796$		0			0
$797$			1.60202i			0.0567466i	0.999597	+	0.0283733i	$0.00903271\pi$
$797$			1.60202i			0.0567466i	−0.999597	+	0.0283733i	$0.990967\pi$
$798$		0			0
$799$	−2.95691			−0.104608
$800$		0			0
$801$	12.5643			0.443937
$802$		0			0
$803$		−	33.3221i		−	1.17591i
$804$		0			0
$805$	7.38428	−	1.13796i	0.260262	−	0.0401079i
$806$		0			0
$807$		−	21.9479i		−	0.772603i
$808$		0			0
$809$	−27.7065			−0.974108			−0.487054	−	0.873372i	$-0.661929\pi$
$809$	−27.7065			−0.974108			−0.487054	+	0.873372i	$0.661929\pi$
$810$		0			0
$811$	4.45154			0.156315			0.0781574	−	0.996941i	$-0.475096\pi$
$811$	4.45154			0.156315			0.0781574	+	0.996941i	$0.475096\pi$
$812$		0			0
$813$			13.0888i			0.459043i
$814$		0			0
$815$	−3.45831	−	22.4411i	−0.121139	−	0.786077i
$816$		0			0
$817$		−	32.7046i		−	1.14419i
$818$		0			0
$819$	15.6825			0.547990
$820$		0			0
$821$	18.7300			0.653681			0.326840	−	0.945080i	$-0.394016\pi$
$821$	18.7300			0.653681			0.326840	+	0.945080i	$0.394016\pi$
$822$		0			0
$823$			33.4227i			1.16504i	0.812815	+	0.582522i	$0.197934\pi$
$823$			33.4227i			1.16504i	−0.812815	+	0.582522i	$0.802066\pi$
$824$		0			0
$825$	4.22044	+	13.3681i	0.146937	+	0.465417i
$826$		0			0
$827$			26.0089i			0.904418i	0.891912	+	0.452209i	$0.149364\pi$
$827$			26.0089i			0.904418i	−0.891912	+	0.452209i	$0.850636\pi$
$828$		0			0
$829$	−50.2919			−1.74671			−0.873355	−	0.487084i	$-0.838061\pi$
$829$	−50.2919			−1.74671			−0.873355	+	0.487084i	$0.838061\pi$
$830$		0			0
$831$	30.6235			1.06232
$832$		0			0
$833$			6.00783i			0.208159i
$834$		0			0
$835$	6.04130	+	39.2022i	0.209068	+	1.35665i
$836$		0			0
$837$		−	5.27474i		−	0.182322i
$838$		0			0
$839$	−7.57018			−0.261352			−0.130676	−	0.991425i	$-0.541715\pi$
$839$	−7.57018			−0.261352			−0.130676	+	0.991425i	$0.541715\pi$
$840$		0			0
$841$	−23.6464			−0.815392
$842$		0			0
$843$		−	32.5432i		−	1.12085i
$844$		0			0
$845$	−19.9533	+	3.07493i	−0.686416	+	0.105781i
$846$		0			0
$847$		−	10.4894i		−	0.360419i
$848$		0			0
$849$	8.50785			0.291989
$850$		0			0
$851$	7.36064			0.252319
$852$		0			0
$853$		−	5.25619i		−	0.179968i	−0.995943	−	0.0899842i	$-0.971318\pi$
$853$		−	5.25619i		−	0.179968i	0.995943	−	0.0899842i	$-0.0286817\pi$
$854$		0			0
$855$	7.45785	−	1.14930i	0.255053	−	0.0393052i
$856$		0			0
$857$		−	36.1016i		−	1.23321i	−0.787274	−	0.616603i	$-0.788508\pi$
$857$		−	36.1016i		−	1.23321i	0.787274	−	0.616603i	$-0.211492\pi$
$858$		0			0
$859$	55.2401			1.88477			0.942384	−	0.334532i	$-0.108578\pi$
$859$	55.2401			1.88477			0.942384	+	0.334532i	$0.108578\pi$
$860$		0			0
$861$	−15.0691			−0.513554
$862$		0			0
$863$			21.4303i			0.729497i	0.931106	+	0.364748i	$0.118845\pi$
$863$			21.4303i			0.729497i	−0.931106	+	0.364748i	$0.881155\pi$
$864$		0			0
$865$	0.854103	+	5.54231i	0.0290404	+	0.188444i
$866$		0			0
$867$		−	14.9188i		−	0.506670i
$868$		0			0
$869$	−12.9245			−0.438432
$870$		0			0
$871$	−25.2994			−0.857237
$872$		0			0
$873$			10.1248i			0.342671i
$874$		0			0
$875$	16.5415	+	33.4954i	0.559204	+	1.13235i
$876$		0			0
$877$		−	2.00338i		−	0.0676494i	−0.999428	−	0.0338247i	$-0.989231\pi$
$877$		−	2.00338i		−	0.0676494i	0.999428	−	0.0338247i	$-0.0107688\pi$
$878$		0			0
$879$	−15.6357			−0.527379
$880$		0			0
$881$	−11.6145			−0.391303			−0.195652	−	0.980673i	$-0.562682\pi$
$881$	−11.6145			−0.391303			−0.195652	+	0.980673i	$0.562682\pi$
$882$		0			0
$883$		−	2.89026i		−	0.0972650i	−0.998817	−	0.0486325i	$-0.984514\pi$
$883$		−	2.89026i		−	0.0972650i	0.998817	−	0.0486325i	$-0.0154863\pi$
$884$		0			0
$885$	3.08661	+	20.0291i	0.103755	+	0.673272i
$886$		0			0
$887$			21.3383i			0.716470i	0.933631	+	0.358235i	$0.116621\pi$
$887$			21.3383i			0.716470i	−0.933631	+	0.358235i	$0.883379\pi$
$888$		0			0
$889$	−34.7057			−1.16399
$890$		0			0
$891$	−2.80370			−0.0939274
$892$		0			0
$893$			6.91688i			0.231465i
$894$		0			0
$895$	13.9951	−	2.15673i	0.467805	−	0.0720916i
$896$		0			0
$897$		−	4.69348i		−	0.156711i
$898$		0			0
$899$	−12.2047			−0.407048
$900$		0			0
$901$	0.521163			0.0173624
$902$		0			0
$903$			32.3821i			1.07761i
$904$		0			0
$905$	−1.66203	+	0.256129i	−0.0552478	+	0.00851402i
$906$		0			0
$907$		−	20.2386i		−	0.672012i	−0.941860	−	0.336006i	$-0.890924\pi$
$907$		−	20.2386i		−	0.672012i	0.941860	−	0.336006i	$-0.109076\pi$
$908$		0			0
$909$	−16.3073			−0.540880
$910$		0			0
$911$	−34.1368			−1.13100			−0.565502	−	0.824747i	$-0.691318\pi$
$911$	−34.1368			−1.13100			−0.565502	+	0.824747i	$0.691318\pi$
$912$		0			0
$913$			14.7280i			0.487426i
$914$		0			0
$915$	3.01152	+	19.5419i	0.0995579	+	0.646035i
$916$		0			0
$917$		−	35.9476i		−	1.18709i
$918$		0			0
$919$	29.4334			0.970919			0.485460	−	0.874259i	$-0.338652\pi$
$919$	29.4334			0.970919			0.485460	+	0.874259i	$0.338652\pi$
$920$		0			0
$921$	−19.0905			−0.629053
$922$		0			0
$923$		−	7.69344i		−	0.253233i
$924$		0			0
$925$	11.0801	+	35.0957i	0.364310	+	1.15394i
$926$		0			0
$927$			0.0236876i			0.000778003i
$928$		0			0
$929$	−40.9217			−1.34260			−0.671299	−	0.741186i	$-0.734264\pi$
$929$	−40.9217			−1.34260			−0.671299	+	0.741186i	$0.734264\pi$
$930$		0			0
$931$	14.0537			0.460591
$932$		0			0
$933$		−	26.7760i		−	0.876607i
$934$		0			0
$935$	1.37750	+	8.93866i	0.0450491	+	0.292325i
$936$		0			0
$937$			35.6319i			1.16404i	0.813173	+	0.582022i	$0.197738\pi$
$937$			35.6319i			1.16404i	−0.813173	+	0.582022i	$0.802262\pi$
$938$		0			0
$939$	14.5967			0.476345
$940$		0			0
$941$	57.3870			1.87076			0.935381	−	0.353640i	$-0.115056\pi$
$941$	57.3870			1.87076			0.935381	+	0.353640i	$0.115056\pi$
$942$		0			0
$943$			4.50991i			0.146863i
$944$		0			0
$945$	−7.38428	+	1.13796i	−0.240211	+	0.0370179i
$946$		0			0
$947$		−	47.5124i		−	1.54395i	−0.635655	−	0.771973i	$-0.719270\pi$
$947$		−	47.5124i		−	1.54395i	0.635655	−	0.771973i	$-0.280730\pi$
$948$		0			0
$949$	−55.7823			−1.81077
$950$		0			0
$951$	−15.3952			−0.499223
$952$		0			0
$953$			20.1754i			0.653546i	0.945103	+	0.326773i	$0.105961\pi$
$953$			20.1754i			0.653546i	−0.945103	+	0.326773i	$0.894039\pi$
$954$		0			0
$955$	−4.94769	+	0.762470i	−0.160104	+	0.0246729i
$956$		0			0
$957$			6.48718i			0.209701i
$958$		0			0
$959$	−4.39325			−0.141865
$960$		0			0
$961$	−3.17710			−0.102487
$962$		0			0
$963$		−	5.84284i		−	0.188283i
$964$		0			0
$965$	4.28402	+	27.7992i	0.137908	+	0.894888i
$966$		0			0
$967$		−	59.0165i		−	1.89784i	−0.315514	−	0.948921i	$-0.602177\pi$
$967$		−	59.0165i		−	1.89784i	0.315514	−	0.948921i	$-0.397823\pi$
$968$		0			0
$969$	4.86830			0.156392
$970$		0			0
$971$	42.7879			1.37313			0.686564	−	0.727069i	$-0.259118\pi$
$971$	42.7879			1.37313			0.686564	+	0.727069i	$0.259118\pi$
$972$		0			0
$973$			35.1802i			1.12782i
$974$		0			0
$975$	22.3786	−	7.06515i	0.716689	−	0.226266i
$976$		0			0
$977$			25.7758i			0.824639i	0.911039	+	0.412320i	$0.135281\pi$
$977$			25.7758i			0.824639i	−0.911039	+	0.412320i	$0.864719\pi$
$978$		0			0
$979$	35.2264			1.12584
$980$		0			0
$981$	−15.7186			−0.501858
$982$		0			0
$983$			30.7054i			0.979349i	0.871905	+	0.489674i	$0.162884\pi$
$983$			30.7054i			0.979349i	−0.871905	+	0.489674i	$0.837116\pi$
$984$		0			0
$985$	−4.20580	−	27.2916i	−0.134008	−	0.869582i
$986$		0			0
$987$		−	6.84865i		−	0.217995i
$988$		0			0
$989$	9.69135			0.308167
$990$		0			0
$991$	−12.2407			−0.388839			−0.194420	−	0.980918i	$-0.562282\pi$
$991$	−12.2407			−0.388839			−0.194420	+	0.980918i	$0.562282\pi$
$992$		0			0
$993$			6.17011i			0.195803i
$994$		0			0
$995$	−12.1917	+	1.87882i	−0.386504	+	0.0595627i
$996$		0			0
$997$			20.9959i			0.664946i	0.943113	+	0.332473i	$0.107883\pi$
$997$			20.9959i			0.664946i	−0.943113	+	0.332473i	$0.892117\pi$
$998$		0			0
$999$	−7.36064			−0.232880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.k.d.2209.7	✓	14
5.4	even	2	inner	2760.2.k.d.2209.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.k.d.2209.7	✓	14	1.1	even	1	trivial
2760.2.k.d.2209.14	yes	14	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.20998 - 0.340571i) q^{5} +3.34134i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(2.20998 - 0.340571i) q^{5} +3.34134i q^{7} -1.00000 q^{9} -2.80370 q^{11} +4.69348i q^{13} +(-0.340571 - 2.20998i) q^{15} -1.44262i q^{17} -3.37462 q^{19} +3.34134 q^{21} -1.00000i q^{23} +(4.76802 - 1.50531i) q^{25} +1.00000i q^{27} +2.31379 q^{29} -5.27474 q^{31} +2.80370i q^{33} +(1.13796 + 7.38428i) q^{35} +7.36064i q^{37} +4.69348 q^{39} -4.50991 q^{41} +9.69135i q^{43} +(-2.20998 + 0.340571i) q^{45} -2.04968i q^{47} -4.16452 q^{49} -1.44262 q^{51} +0.361261i q^{53} +(-6.19612 + 0.954859i) q^{55} +3.37462i q^{57} -9.06304 q^{59} -8.84257 q^{61} -3.34134i q^{63} +(1.59846 + 10.3725i) q^{65} +5.39033i q^{67} -1.00000 q^{69} -1.63918 q^{71} +11.8851i q^{73} +(-1.50531 - 4.76802i) q^{75} -9.36810i q^{77} +4.60979 q^{79} +1.00000 q^{81} -5.25307i q^{83} +(-0.491316 - 3.18817i) q^{85} -2.31379i q^{87} -12.5643 q^{89} -15.6825 q^{91} +5.27474i q^{93} +(-7.45785 + 1.14930i) q^{95} -10.1248i q^{97} +2.80370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} - 14 q^{9}+O(q^{10}) \) 14 * q + 2 * q^5 - 14 * q^9	\( 14 q + 2 q^{5} - 14 q^{9} + 8 q^{11} - 4 q^{15} + 8 q^{19} - 10 q^{21} - 10 q^{25} + 26 q^{29} - 18 q^{31} - 10 q^{35} + 12 q^{39} - 2 q^{41} - 2 q^{45} + 4 q^{49} - 6 q^{51} - 32 q^{55} - 10 q^{59} - 24 q^{61} - 36 q^{65} - 14 q^{69} - 38 q^{71} + 4 q^{79} + 14 q^{81} - 14 q^{85} - 16 q^{89} - 4 q^{91} - 8 q^{95} - 8 q^{99}+O(q^{100}) \) 14 * q + 2 * q^5 - 14 * q^9 + 8 * q^11 - 4 * q^15 + 8 * q^19 - 10 * q^21 - 10 * q^25 + 26 * q^29 - 18 * q^31 - 10 * q^35 + 12 * q^39 - 2 * q^41 - 2 * q^45 + 4 * q^49 - 6 * q^51 - 32 * q^55 - 10 * q^59 - 24 * q^61 - 36 * q^65 - 14 * q^69 - 38 * q^71 + 4 * q^79 + 14 * q^81 - 14 * q^85 - 16 * q^89 - 4 * q^91 - 8 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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