LMFDB - Embedded newform 2720.2.c.f.1121.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2720,2,Mod(1121,2720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2720.1121"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2720 = 2^{5} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2720.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-12,0,0,0,-16,0,0,0,-4,0,0,0,24,0,0,0,-16,0, 0,0,0,0,0,0,8,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(35)] == traces)

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

Self dual:	no
Analytic conductor:	$21.7193093498$
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} + 30x^{14} + 337x^{12} + 1804x^{10} + 4944x^{8} + 6760x^{6} + 4048x^{4} + 880x^{2} + 16 $ x^16 + 30x^14 + 337x^12 + 1804x^10 + 4944x^8 + 6760x^6 + 4048x^4 + 880*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.7
Root			$-0.700592i$ of defining polynomial
Character	$\chi$	$=$	2720.1121
Dual form			2720.2.c.f.1121.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.700592i q^{3} -1.00000i q^{5} -2.22408i q^{7} +2.50917 q^{9} +3.74422i q^{11} -6.72768 q^{13} -0.700592 q^{15} +(1.62059 + 3.79127i) q^{17} +4.67985 q^{19} -1.55818 q^{21} +6.25760i q^{23} -1.00000 q^{25} -3.85968i q^{27} +9.29954i q^{29} +7.15049i q^{31} +2.62317 q^{33} -2.22408 q^{35} -5.00719i q^{37} +4.71336i q^{39} +1.90591i q^{41} +12.4708 q^{43} -2.50917i q^{45} +9.41849 q^{47} +2.05345 q^{49} +(2.65613 - 1.13537i) q^{51} -12.9941 q^{53} +3.74422 q^{55} -3.27866i q^{57} -6.01042 q^{59} +4.79708i q^{61} -5.58061i q^{63} +6.72768i q^{65} +0.213042 q^{67} +4.38402 q^{69} +4.16565i q^{71} -5.01604i q^{73} +0.700592i q^{75} +8.32746 q^{77} -12.3859i q^{79} +4.82345 q^{81} -2.24588 q^{83} +(3.79127 - 1.62059i) q^{85} +6.51518 q^{87} -6.35525 q^{89} +14.9629i q^{91} +5.00957 q^{93} -4.67985i q^{95} +0.913832i q^{97} +9.39489i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 12 q^{9} - 16 q^{13} - 4 q^{17} + 24 q^{21} - 16 q^{25} + 8 q^{33} + 12 q^{35} - 4 q^{43} + 24 q^{47} - 24 q^{49} + 20 q^{51} - 8 q^{53} - 24 q^{59} + 28 q^{67} + 16 q^{69} - 24 q^{77} + 56 q^{81}+ \cdots - 48 q^{93}+O(q^{100}) $ 16 * q - 12 * q^9 - 16 * q^13 - 4 * q^17 + 24 * q^21 - 16 * q^25 + 8 * q^33 + 12 * q^35 - 4 * q^43 + 24 * q^47 - 24 * q^49 + 20 * q^51 - 8 * q^53 - 24 * q^59 + 28 * q^67 + 16 * q^69 - 24 * q^77 + 56 * q^81 - 20 * q^83 + 8 * q^85 + 100 * q^87 - 12 * q^89 - 48 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2720\mathbb{Z}\right)^\times$.

$n$	$511$	$1601$	$1701$	$2177$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	0.700592i		−	0.404487i	−0.979335	−	0.202243i	$-0.935177\pi$
$3$		−	0.700592i		−	0.404487i	0.979335	−	0.202243i	$-0.0648232\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	2.22408i		−	0.840625i	−0.907379	−	0.420312i	$-0.861920\pi$
$7$		−	2.22408i		−	0.840625i	0.907379	−	0.420312i	$-0.138080\pi$
$8$		0			0
$9$	2.50917			0.836390
$10$		0			0
$11$			3.74422i			1.12893i	0.825459	+	0.564463i	$0.190917\pi$
$11$			3.74422i			1.12893i	−0.825459	+	0.564463i	$0.809083\pi$
$12$		0			0
$13$	−6.72768			−1.86592			−0.932961	−	0.359976i	$-0.882785\pi$
$13$	−6.72768			−1.86592			−0.932961	+	0.359976i	$0.882785\pi$
$14$		0			0
$15$	−0.700592			−0.180892
$16$		0			0
$17$	1.62059	+	3.79127i	0.393050	+	0.919517i
$17$	1.62059	+	3.79127i	0.393050	+	0.919517i
$18$		0			0
$19$	4.67985			1.07363			0.536815	−	0.843700i	$-0.319627\pi$
$19$	4.67985			1.07363			0.536815	+	0.843700i	$0.319627\pi$
$20$		0			0
$21$	−1.55818			−0.340022
$22$		0			0
$23$			6.25760i			1.30480i	0.757875	+	0.652399i	$0.226238\pi$
$23$			6.25760i			1.30480i	−0.757875	+	0.652399i	$0.773762\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	3.85968i		−	0.742796i
$28$		0			0
$29$			9.29954i			1.72688i	0.504450	+	0.863441i	$0.331695\pi$
$29$			9.29954i			1.72688i	−0.504450	+	0.863441i	$0.668305\pi$
$30$		0			0
$31$			7.15049i			1.28427i	0.766594	+	0.642133i	$0.221950\pi$
$31$			7.15049i			1.28427i	−0.766594	+	0.642133i	$0.778050\pi$
$32$		0			0
$33$	2.62317			0.456635
$34$		0			0
$35$	−2.22408			−0.375939
$36$		0			0
$37$		−	5.00719i		−	0.823177i	−0.911370	−	0.411589i	$-0.864974\pi$
$37$		−	5.00719i		−	0.823177i	0.911370	−	0.411589i	$-0.135026\pi$
$38$		0			0
$39$			4.71336i			0.754741i
$40$		0			0
$41$			1.90591i			0.297653i	0.988863	+	0.148827i	$0.0475496\pi$
$41$			1.90591i			0.297653i	−0.988863	+	0.148827i	$0.952450\pi$
$42$		0			0
$43$	12.4708			1.90177			0.950887	−	0.309537i	$-0.100174\pi$
$43$	12.4708			1.90177			0.950887	+	0.309537i	$0.100174\pi$
$44$		0			0
$45$		−	2.50917i		−	0.374045i
$46$		0			0
$47$	9.41849			1.37383			0.686914	−	0.726738i	$-0.258965\pi$
$47$	9.41849			1.37383			0.686914	+	0.726738i	$0.258965\pi$
$48$		0			0
$49$	2.05345			0.293350
$50$		0			0
$51$	2.65613	−	1.13537i	0.371933	−	0.158984i
$52$		0			0
$53$	−12.9941			−1.78488			−0.892441	−	0.451164i	$-0.851009\pi$
$53$	−12.9941			−1.78488			−0.892441	+	0.451164i	$0.851009\pi$
$54$		0			0
$55$	3.74422			0.504871
$56$		0			0
$57$		−	3.27866i		−	0.434270i
$58$		0			0
$59$	−6.01042			−0.782490			−0.391245	−	0.920287i	$-0.627956\pi$
$59$	−6.01042			−0.782490			−0.391245	+	0.920287i	$0.627956\pi$
$60$		0			0
$61$			4.79708i			0.614203i	0.951677	+	0.307101i	$0.0993591\pi$
$61$			4.79708i			0.614203i	−0.951677	+	0.307101i	$0.900641\pi$
$62$		0			0
$63$		−	5.58061i		−	0.703090i
$64$		0			0
$65$			6.72768i			0.834466i
$66$		0			0
$67$	0.213042			0.0260272			0.0130136	−	0.999915i	$-0.495858\pi$
$67$	0.213042			0.0260272			0.0130136	+	0.999915i	$0.495858\pi$
$68$		0			0
$69$	4.38402			0.527774
$70$		0			0
$71$			4.16565i			0.494372i	0.968968	+	0.247186i	$0.0795059\pi$
$71$			4.16565i			0.494372i	−0.968968	+	0.247186i	$0.920494\pi$
$72$		0			0
$73$		−	5.01604i		−	0.587083i	−0.955946	−	0.293541i	$-0.905166\pi$
$73$		−	5.01604i		−	0.587083i	0.955946	−	0.293541i	$-0.0948338\pi$
$74$		0			0
$75$			0.700592i			0.0808974i
$76$		0			0
$77$	8.32746			0.949002
$78$		0			0
$79$		−	12.3859i		−	1.39352i	−0.717303	−	0.696762i	$-0.754623\pi$
$79$		−	12.3859i		−	1.39352i	0.717303	−	0.696762i	$-0.245377\pi$
$80$		0			0
$81$	4.82345			0.535939
$82$		0			0
$83$	−2.24588			−0.246517			−0.123259	−	0.992375i	$-0.539334\pi$
$83$	−2.24588			−0.246517			−0.123259	+	0.992375i	$0.539334\pi$
$84$		0			0
$85$	3.79127	−	1.62059i	0.411220	−	0.175777i
$86$		0			0
$87$	6.51518			0.698501
$88$		0			0
$89$	−6.35525			−0.673655			−0.336828	−	0.941566i	$-0.609354\pi$
$89$	−6.35525			−0.673655			−0.336828	+	0.941566i	$0.609354\pi$
$90$		0			0
$91$			14.9629i			1.56854i
$92$		0			0
$93$	5.00957			0.519468
$94$		0			0
$95$		−	4.67985i		−	0.480142i
$96$		0			0
$97$			0.913832i			0.0927856i	0.998923	+	0.0463928i	$0.0147726\pi$
$97$			0.913832i			0.0927856i	−0.998923	+	0.0463928i	$0.985227\pi$
$98$		0			0
$99$			9.39489i			0.944222i
$100$		0			0
$101$	8.37843			0.833685			0.416843	−	0.908979i	$-0.363137\pi$
$101$	8.37843			0.833685			0.416843	+	0.908979i	$0.363137\pi$
$102$		0			0
$103$	4.80111			0.473067			0.236534	−	0.971623i	$-0.423989\pi$
$103$	4.80111			0.473067			0.236534	+	0.971623i	$0.423989\pi$
$104$		0			0
$105$			1.55818i			0.152062i
$106$		0			0
$107$		−	4.52676i		−	0.437619i	−0.975768	−	0.218809i	$-0.929783\pi$
$107$		−	4.52676i		−	0.437619i	0.975768	−	0.218809i	$-0.0702173\pi$
$108$		0			0
$109$		−	11.8824i		−	1.13813i	−0.822292	−	0.569066i	$-0.807305\pi$
$109$		−	11.8824i		−	1.13813i	0.822292	−	0.569066i	$-0.192695\pi$
$110$		0			0
$111$	−3.50800			−0.332965
$112$		0			0
$113$		−	5.04468i		−	0.474564i	−0.971441	−	0.237282i	$-0.923744\pi$
$113$		−	5.04468i		−	0.474564i	0.971441	−	0.237282i	$-0.0762565\pi$
$114$		0			0
$115$	6.25760			0.583524
$116$		0			0
$117$	−16.8809			−1.56064
$118$		0			0
$119$	8.43209	−	3.60433i	0.772969	−	0.330408i
$120$		0			0
$121$	−3.01919			−0.274472
$122$		0			0
$123$	1.33527			0.120397
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	16.1088			1.42942			0.714711	−	0.699420i	$-0.246558\pi$
$127$	16.1088			1.42942			0.714711	+	0.699420i	$0.246558\pi$
$128$		0			0
$129$		−	8.73692i		−	0.769243i
$130$		0			0
$131$			0.946721i			0.0827154i	0.999144	+	0.0413577i	$0.0131683\pi$
$131$			0.946721i			0.0827154i	−0.999144	+	0.0413577i	$0.986832\pi$
$132$		0			0
$133$		−	10.4084i		−	0.902521i
$134$		0			0
$135$	−3.85968			−0.332188
$136$		0			0
$137$	−11.6397			−0.994449			−0.497224	−	0.867622i	$-0.665647\pi$
$137$	−11.6397			−0.994449			−0.497224	+	0.867622i	$0.665647\pi$
$138$		0			0
$139$			14.1812i			1.20284i	0.798934	+	0.601419i	$0.205398\pi$
$139$			14.1812i			1.20284i	−0.798934	+	0.601419i	$0.794602\pi$
$140$		0			0
$141$		−	6.59852i		−	0.555696i
$142$		0			0
$143$		−	25.1899i		−	2.10649i
$144$		0			0
$145$	9.29954			0.772285
$146$		0			0
$147$		−	1.43863i		−	0.118656i
$148$		0			0
$149$	−9.58924			−0.785581			−0.392791	−	0.919628i	$-0.628490\pi$
$149$	−9.58924			−0.785581			−0.392791	+	0.919628i	$0.628490\pi$
$150$		0			0
$151$	1.47726			0.120218			0.0601090	−	0.998192i	$-0.480855\pi$
$151$	1.47726			0.120218			0.0601090	+	0.998192i	$0.480855\pi$
$152$		0			0
$153$	4.06633	+	9.51293i	0.328744	+	0.769075i
$154$		0			0
$155$	7.15049			0.574341
$156$		0			0
$157$	−11.6698			−0.931351			−0.465675	−	0.884956i	$-0.654189\pi$
$157$	−11.6698			−0.931351			−0.465675	+	0.884956i	$0.654189\pi$
$158$		0			0
$159$			9.10359i			0.721961i
$160$		0			0
$161$	13.9174			1.09685
$162$		0			0
$163$			0.439965i			0.0344607i	0.999852	+	0.0172303i	$0.00548486\pi$
$163$			0.439965i			0.0344607i	−0.999852	+	0.0172303i	$0.994515\pi$
$164$		0			0
$165$		−	2.62317i		−	0.204214i
$166$		0			0
$167$			20.3369i			1.57371i	0.617135	+	0.786857i	$0.288293\pi$
$167$			20.3369i			1.57371i	−0.617135	+	0.786857i	$0.711707\pi$
$168$		0			0
$169$	32.2617			2.48167
$170$		0			0
$171$	11.7425			0.897974
$172$		0			0
$173$			18.9699i			1.44225i	0.692803	+	0.721127i	$0.256376\pi$
$173$			18.9699i			1.44225i	−0.692803	+	0.721127i	$0.743624\pi$
$174$		0			0
$175$			2.22408i			0.168125i
$176$		0			0
$177$			4.21085i			0.316507i
$178$		0			0
$179$	21.9259			1.63882			0.819408	−	0.573211i	$-0.194302\pi$
$179$	21.9259			1.63882			0.819408	+	0.573211i	$0.194302\pi$
$180$		0			0
$181$			1.09659i			0.0815088i	0.999169	+	0.0407544i	$0.0129761\pi$
$181$			1.09659i			0.0815088i	−0.999169	+	0.0407544i	$0.987024\pi$
$182$		0			0
$183$	3.36079			0.248437
$184$		0			0
$185$	−5.00719			−0.368136
$186$		0			0
$187$	−14.1953	+	6.06784i	−1.03807	+	0.443724i
$188$		0			0
$189$	−8.58426			−0.624413
$190$		0			0
$191$	6.85889			0.496292			0.248146	−	0.968723i	$-0.420179\pi$
$191$	6.85889			0.496292			0.248146	+	0.968723i	$0.420179\pi$
$192$		0			0
$193$		−	11.8313i		−	0.851639i	−0.904808	−	0.425819i	$-0.859986\pi$
$193$		−	11.8313i		−	0.851639i	0.904808	−	0.425819i	$-0.140014\pi$
$194$		0			0
$195$	4.71336			0.337531
$196$		0			0
$197$			8.48740i			0.604702i	0.953197	+	0.302351i	$0.0977715\pi$
$197$			8.48740i			0.604702i	−0.953197	+	0.302351i	$0.902229\pi$
$198$		0			0
$199$			15.3125i			1.08548i	0.839901	+	0.542739i	$0.182613\pi$
$199$			15.3125i			1.08548i	−0.839901	+	0.542739i	$0.817387\pi$
$200$		0			0
$201$		−	0.149255i		−	0.0105277i
$202$		0			0
$203$	20.6830			1.45166
$204$		0			0
$205$	1.90591			0.133115
$206$		0			0
$207$			15.7014i			1.09132i
$208$		0			0
$209$			17.5224i			1.21205i
$210$		0			0
$211$		−	8.82891i		−	0.607807i	−0.952703	−	0.303904i	$-0.901710\pi$
$211$		−	8.82891i		−	0.607807i	0.952703	−	0.303904i	$-0.0982901\pi$
$212$		0			0
$213$	2.91842			0.199967
$214$		0			0
$215$		−	12.4708i		−	0.850499i
$216$		0			0
$217$	15.9033			1.07959
$218$		0			0
$219$	−3.51419			−0.237467
$220$		0			0
$221$	−10.9028	−	25.5064i	−0.733402	−	1.71575i
$222$		0			0
$223$	3.11863			0.208839			0.104419	−	0.994533i	$-0.466702\pi$
$223$	3.11863			0.208839			0.104419	+	0.994533i	$0.466702\pi$
$224$		0			0
$225$	−2.50917			−0.167278
$226$		0			0
$227$			24.9063i			1.65309i	0.562870	+	0.826545i	$0.309697\pi$
$227$			24.9063i			1.65309i	−0.562870	+	0.826545i	$0.690303\pi$
$228$		0			0
$229$	−10.3978			−0.687107			−0.343554	−	0.939133i	$-0.611631\pi$
$229$	−10.3978			−0.687107			−0.343554	+	0.939133i	$0.611631\pi$
$230$		0			0
$231$		−	5.83415i		−	0.383859i
$232$		0			0
$233$		−	3.28149i		−	0.214977i	−0.994206	−	0.107489i	$-0.965719\pi$
$233$		−	3.28149i		−	0.214977i	0.994206	−	0.107489i	$-0.0342809\pi$
$234$		0			0
$235$		−	9.41849i		−	0.614395i
$236$		0			0
$237$	−8.67747			−0.563662
$238$		0			0
$239$	26.1678			1.69266			0.846328	−	0.532663i	$-0.178809\pi$
$239$	26.1678			1.69266			0.846328	+	0.532663i	$0.178809\pi$
$240$		0			0
$241$			28.3317i			1.82501i	0.409071	+	0.912503i	$0.365853\pi$
$241$			28.3317i			1.82501i	−0.409071	+	0.912503i	$0.634147\pi$
$242$		0			0
$243$		−	14.9583i		−	0.959576i
$244$		0			0
$245$		−	2.05345i		−	0.131190i
$246$		0			0
$247$	−31.4845			−2.00331
$248$		0			0
$249$			1.57344i			0.0997130i
$250$		0			0
$251$	−10.6718			−0.673599			−0.336800	−	0.941576i	$-0.609344\pi$
$251$	−10.6718			−0.673599			−0.336800	+	0.941576i	$0.609344\pi$
$252$		0			0
$253$	−23.4298			−1.47302
$254$		0			0
$255$	−1.13537	−	2.65613i	−0.0710997	−	0.166333i
$256$		0			0
$257$	3.45918			0.215778			0.107889	−	0.994163i	$-0.465591\pi$
$257$	3.45918			0.215778			0.107889	+	0.994163i	$0.465591\pi$
$258$		0			0
$259$	−11.1364			−0.691983
$260$		0			0
$261$			23.3341i			1.44435i
$262$		0			0
$263$	−3.86722			−0.238463			−0.119232	−	0.992866i	$-0.538043\pi$
$263$	−3.86722			−0.238463			−0.119232	+	0.992866i	$0.538043\pi$
$264$		0			0
$265$			12.9941i			0.798223i
$266$		0			0
$267$			4.45244i			0.272485i
$268$		0			0
$269$		−	10.3456i		−	0.630781i	−0.948962	−	0.315390i	$-0.897865\pi$
$269$		−	10.3456i		−	0.630781i	0.948962	−	0.315390i	$-0.102135\pi$
$270$		0			0
$271$	9.67737			0.587858			0.293929	−	0.955827i	$-0.405037\pi$
$271$	9.67737			0.587858			0.293929	+	0.955827i	$0.405037\pi$
$272$		0			0
$273$	10.4829			0.634454
$274$		0			0
$275$		−	3.74422i		−	0.225785i
$276$		0			0
$277$		−	25.4004i		−	1.52616i	−0.646304	−	0.763080i	$-0.723686\pi$
$277$		−	25.4004i		−	1.52616i	0.646304	−	0.763080i	$-0.276314\pi$
$278$		0			0
$279$			17.9418i			1.07415i
$280$		0			0
$281$	−4.82400			−0.287776			−0.143888	−	0.989594i	$-0.545960\pi$
$281$	−4.82400			−0.287776			−0.143888	+	0.989594i	$0.545960\pi$
$282$		0			0
$283$		−	15.7626i		−	0.936991i	−0.883466	−	0.468496i	$-0.844796\pi$
$283$		−	15.7626i		−	0.936991i	0.883466	−	0.468496i	$-0.155204\pi$
$284$		0			0
$285$	−3.27866			−0.194211
$286$		0			0
$287$	4.23891			0.250215
$288$		0			0
$289$	−11.7474	+	12.2882i	−0.691023	+	0.722833i
$290$		0			0
$291$	0.640224			0.0375306
$292$		0			0
$293$	24.8782			1.45340			0.726700	−	0.686955i	$-0.241053\pi$
$293$	24.8782			1.45340			0.726700	+	0.686955i	$0.241053\pi$
$294$		0			0
$295$			6.01042i			0.349940i
$296$		0			0
$297$	14.4515			0.838561
$298$		0			0
$299$		−	42.0991i		−	2.43465i
$300$		0			0
$301$		−	27.7360i		−	1.59868i
$302$		0			0
$303$		−	5.86986i		−	0.337215i
$304$		0			0
$305$	4.79708			0.274680
$306$		0			0
$307$	−29.9398			−1.70876			−0.854378	−	0.519652i	$-0.826062\pi$
$307$	−29.9398			−1.70876			−0.854378	+	0.519652i	$0.826062\pi$
$308$		0			0
$309$		−	3.36362i		−	0.191349i
$310$		0			0
$311$		−	2.52136i		−	0.142973i	−0.997442	−	0.0714865i	$-0.977226\pi$
$311$		−	2.52136i		−	0.142973i	0.997442	−	0.0714865i	$-0.0227743\pi$
$312$		0			0
$313$			15.8801i			0.897594i	0.893634	+	0.448797i	$0.148147\pi$
$313$			15.8801i			0.897594i	−0.893634	+	0.448797i	$0.851853\pi$
$314$		0			0
$315$	−5.58061			−0.314432
$316$		0			0
$317$			4.28449i			0.240641i	0.992735	+	0.120321i	$0.0383922\pi$
$317$			4.28449i			0.240641i	−0.992735	+	0.120321i	$0.961608\pi$
$318$		0			0
$319$	−34.8195			−1.94952
$320$		0			0
$321$	−3.17141			−0.177011
$322$		0			0
$323$	7.58411	+	17.7425i	0.421991	+	0.987222i
$324$		0			0
$325$	6.72768			0.373185
$326$		0			0
$327$	−8.32474			−0.460359
$328$		0			0
$329$		−	20.9475i		−	1.15487i
$330$		0			0
$331$	−26.2786			−1.44440			−0.722200	−	0.691684i	$-0.756869\pi$
$331$	−26.2786			−1.44440			−0.722200	+	0.691684i	$0.756869\pi$
$332$		0			0
$333$		−	12.5639i		−	0.688498i
$334$		0			0
$335$		−	0.213042i		−	0.0116397i
$336$		0			0
$337$			7.46162i			0.406461i	0.979131	+	0.203230i	$0.0651440\pi$
$337$			7.46162i			0.406461i	−0.979131	+	0.203230i	$0.934856\pi$
$338$		0			0
$339$	−3.53426			−0.191955
$340$		0			0
$341$	−26.7730			−1.44984
$342$		0			0
$343$		−	20.1356i		−	1.08722i
$344$		0			0
$345$		−	4.38402i		−	0.236028i
$346$		0			0
$347$			10.7543i			0.577323i	0.957431	+	0.288662i	$0.0932102\pi$
$347$			10.7543i			0.577323i	−0.957431	+	0.288662i	$0.906790\pi$
$348$		0			0
$349$	−18.0592			−0.966687			−0.483343	−	0.875431i	$-0.660578\pi$
$349$	−18.0592			−0.966687			−0.483343	+	0.875431i	$0.660578\pi$
$350$		0			0
$351$			25.9667i			1.38600i
$352$		0			0
$353$	8.74508			0.465454			0.232727	−	0.972542i	$-0.425235\pi$
$353$	8.74508			0.465454			0.232727	+	0.972542i	$0.425235\pi$
$354$		0			0
$355$	4.16565			0.221090
$356$		0			0
$357$	−2.52516	−	5.90746i	−0.133646	−	0.312656i
$358$		0			0
$359$	−14.4023			−0.760123			−0.380061	−	0.924961i	$-0.624097\pi$
$359$	−14.4023			−0.760123			−0.380061	+	0.924961i	$0.624097\pi$
$360$		0			0
$361$	2.90098			0.152683
$362$		0			0
$363$			2.11522i			0.111020i
$364$		0			0
$365$	−5.01604			−0.262551
$366$		0			0
$367$		−	4.13473i		−	0.215831i	−0.994160	−	0.107915i	$-0.965582\pi$
$367$		−	4.13473i		−	0.215831i	0.994160	−	0.107915i	$-0.0344176\pi$
$368$		0			0
$369$			4.78226i			0.248954i
$370$		0			0
$371$			28.9001i			1.50042i
$372$		0			0
$373$	24.6724			1.27749			0.638746	−	0.769418i	$-0.279454\pi$
$373$	24.6724			1.27749			0.638746	+	0.769418i	$0.279454\pi$
$374$		0			0
$375$	0.700592			0.0361784
$376$		0			0
$377$		−	62.5643i		−	3.22223i
$378$		0			0
$379$			7.29145i			0.374537i	0.982309	+	0.187268i	$0.0599634\pi$
$379$			7.29145i			0.374537i	−0.982309	+	0.187268i	$0.940037\pi$
$380$		0			0
$381$		−	11.2857i		−	0.578182i
$382$		0			0
$383$	−26.0727			−1.33225			−0.666127	−	0.745839i	$-0.732049\pi$
$383$	−26.0727			−1.33225			−0.666127	+	0.745839i	$0.732049\pi$
$384$		0			0
$385$		−	8.32746i		−	0.424407i
$386$		0			0
$387$	31.2913			1.59063
$388$		0			0
$389$	27.2334			1.38079			0.690394	−	0.723433i	$-0.257437\pi$
$389$	27.2334			1.38079			0.690394	+	0.723433i	$0.257437\pi$
$390$		0			0
$391$	−23.7242	+	10.1410i	−1.19978	+	0.512852i
$392$		0			0
$393$	0.663265			0.0334573
$394$		0			0
$395$	−12.3859			−0.623203
$396$		0			0
$397$			29.6726i			1.48922i	0.667497	+	0.744612i	$0.267366\pi$
$397$			29.6726i			1.48922i	−0.667497	+	0.744612i	$0.732634\pi$
$398$		0			0
$399$	−7.29203			−0.365058
$400$		0			0
$401$		−	2.98582i		−	0.149105i	−0.997217	−	0.0745524i	$-0.976247\pi$
$401$		−	2.98582i		−	0.149105i	0.997217	−	0.0745524i	$-0.0237528\pi$
$402$		0			0
$403$		−	48.1062i		−	2.39634i
$404$		0			0
$405$		−	4.82345i		−	0.239679i
$406$		0			0
$407$	18.7480			0.929306
$408$		0			0
$409$	−18.6869			−0.924008			−0.462004	−	0.886878i	$-0.652869\pi$
$409$	−18.6869			−0.924008			−0.462004	+	0.886878i	$0.652869\pi$
$410$		0			0
$411$			8.15470i			0.402242i
$412$		0			0
$413$			13.3677i			0.657781i
$414$		0			0
$415$			2.24588i			0.110246i
$416$		0			0
$417$	9.93526			0.486532
$418$		0			0
$419$		−	12.3530i		−	0.603485i	−0.953389	−	0.301743i	$-0.902432\pi$
$419$		−	12.3530i		−	0.603485i	0.953389	−	0.301743i	$-0.0975683\pi$
$420$		0			0
$421$	23.8308			1.16144			0.580721	−	0.814102i	$-0.302771\pi$
$421$	23.8308			1.16144			0.580721	+	0.814102i	$0.302771\pi$
$422$		0			0
$423$	23.6326			1.14906
$424$		0			0
$425$	−1.62059	−	3.79127i	−0.0786101	−	0.183903i
$426$		0			0
$427$	10.6691			0.516314
$428$		0			0
$429$	−17.6479			−0.852047
$430$		0			0
$431$			16.1561i			0.778210i	0.921193	+	0.389105i	$0.127216\pi$
$431$			16.1561i			0.778210i	−0.921193	+	0.389105i	$0.872784\pi$
$432$		0			0
$433$	25.5062			1.22575			0.612875	−	0.790180i	$-0.290013\pi$
$433$	25.5062			1.22575			0.612875	+	0.790180i	$0.290013\pi$
$434$		0			0
$435$		−	6.51518i		−	0.312379i
$436$		0			0
$437$			29.2846i			1.40087i
$438$		0			0
$439$		−	35.8675i		−	1.71186i	−0.517091	−	0.855931i	$-0.672985\pi$
$439$		−	35.8675i		−	1.71186i	0.517091	−	0.855931i	$-0.327015\pi$
$440$		0			0
$441$	5.15245			0.245355
$442$		0			0
$443$	38.2707			1.81830			0.909148	−	0.416474i	$-0.136734\pi$
$443$	38.2707			1.81830			0.909148	+	0.416474i	$0.136734\pi$
$444$		0			0
$445$			6.35525i			0.301268i
$446$		0			0
$447$			6.71815i			0.317757i
$448$		0			0
$449$		−	1.97378i		−	0.0931485i	−0.998915	−	0.0465742i	$-0.985170\pi$
$449$		−	1.97378i		−	0.0931485i	0.998915	−	0.0465742i	$-0.0148304\pi$
$450$		0			0
$451$	−7.13615			−0.336028
$452$		0			0
$453$		−	1.03496i		−	0.0486266i
$454$		0			0
$455$	14.9629			0.701473
$456$		0			0
$457$	−7.85804			−0.367584			−0.183792	−	0.982965i	$-0.558837\pi$
$457$	−7.85804			−0.367584			−0.183792	+	0.982965i	$0.558837\pi$
$458$		0			0
$459$	14.6331	−	6.25495i	0.683013	−	0.291956i
$460$		0			0
$461$	5.14444			0.239600			0.119800	−	0.992798i	$-0.461775\pi$
$461$	5.14444			0.239600			0.119800	+	0.992798i	$0.461775\pi$
$462$		0			0
$463$	15.8456			0.736407			0.368203	−	0.929745i	$-0.379973\pi$
$463$	15.8456			0.736407			0.368203	+	0.929745i	$0.379973\pi$
$464$		0			0
$465$		−	5.00957i		−	0.232313i
$466$		0			0
$467$	−13.6018			−0.629417			−0.314708	−	0.949188i	$-0.601907\pi$
$467$	−13.6018			−0.629417			−0.314708	+	0.949188i	$0.601907\pi$
$468$		0			0
$469$		−	0.473822i		−	0.0218791i
$470$		0			0
$471$			8.17576i			0.376719i
$472$		0			0
$473$			46.6933i			2.14696i
$474$		0			0
$475$	−4.67985			−0.214726
$476$		0			0
$477$	−32.6045			−1.49286
$478$		0			0
$479$		−	22.6225i		−	1.03365i	−0.856092	−	0.516824i	$-0.827114\pi$
$479$		−	22.6225i		−	1.03365i	0.856092	−	0.516824i	$-0.172886\pi$
$480$		0			0
$481$			33.6868i			1.53599i
$482$		0			0
$483$		−	9.75043i		−	0.443660i
$484$		0			0
$485$	0.913832			0.0414950
$486$		0			0
$487$			4.67956i			0.212051i	0.994363	+	0.106025i	$0.0338125\pi$
$487$			4.67956i			0.212051i	−0.994363	+	0.106025i	$0.966187\pi$
$488$		0			0
$489$	0.308236			0.0139389
$490$		0			0
$491$	3.13010			0.141260			0.0706298	−	0.997503i	$-0.477499\pi$
$491$	3.13010			0.141260			0.0706298	+	0.997503i	$0.477499\pi$
$492$		0			0
$493$	−35.2570	+	15.0707i	−1.58790	+	0.678752i
$494$		0			0
$495$	9.39489			0.422269
$496$		0			0
$497$	9.26477			0.415582
$498$		0			0
$499$			8.52188i			0.381492i	0.981639	+	0.190746i	$0.0610906\pi$
$499$			8.52188i			0.381492i	−0.981639	+	0.190746i	$0.938909\pi$
$500$		0			0
$501$	14.2479			0.636547
$502$		0			0
$503$		−	31.5088i		−	1.40491i	−0.711729	−	0.702454i	$-0.752088\pi$
$503$		−	31.5088i		−	1.40491i	0.711729	−	0.702454i	$-0.247912\pi$
$504$		0			0
$505$		−	8.37843i		−	0.372835i
$506$		0			0
$507$		−	22.6023i		−	1.00380i
$508$		0			0
$509$	−12.3922			−0.549274			−0.274637	−	0.961548i	$-0.588558\pi$
$509$	−12.3922			−0.549274			−0.274637	+	0.961548i	$0.588558\pi$
$510$		0			0
$511$	−11.1561			−0.493516
$512$		0			0
$513$		−	18.0627i		−	0.797489i
$514$		0			0
$515$		−	4.80111i		−	0.211562i
$516$		0			0
$517$			35.2649i			1.55095i
$518$		0			0
$519$	13.2902			0.583373
$520$		0			0
$521$			27.3319i			1.19743i	0.800962	+	0.598716i	$0.204322\pi$
$521$			27.3319i			1.19743i	−0.800962	+	0.598716i	$0.795678\pi$
$522$		0			0
$523$	−43.3809			−1.89691			−0.948456	−	0.316907i	$-0.897356\pi$
$523$	−43.3809			−1.89691			−0.948456	+	0.316907i	$0.897356\pi$
$524$		0			0
$525$	1.55818			0.0680044
$526$		0			0
$527$	−27.1094	+	11.5880i	−1.18090	+	0.504781i
$528$		0			0
$529$	−16.1575			−0.702500
$530$		0			0
$531$	−15.0812			−0.654467
$532$		0			0
$533$		−	12.8224i		−	0.555398i
$534$		0			0
$535$	−4.52676			−0.195709
$536$		0			0
$537$		−	15.3611i		−	0.662880i
$538$		0			0
$539$			7.68857i			0.331170i
$540$		0			0
$541$		−	41.8291i		−	1.79837i	−0.437564	−	0.899187i	$-0.644159\pi$
$541$		−	41.8291i		−	1.79837i	0.437564	−	0.899187i	$-0.355841\pi$
$542$		0			0
$543$	0.768261			0.0329692
$544$		0			0
$545$	−11.8824			−0.508988
$546$		0			0
$547$		−	8.94213i		−	0.382338i	−0.981557	−	0.191169i	$-0.938772\pi$
$547$		−	8.94213i		−	0.382338i	0.981557	−	0.191169i	$-0.0612278\pi$
$548$		0			0
$549$			12.0367i			0.513713i
$550$		0			0
$551$			43.5204i			1.85403i
$552$		0			0
$553$	−27.5473			−1.17143
$554$		0			0
$555$			3.50800i			0.148906i
$556$		0			0
$557$	−9.48564			−0.401919			−0.200960	−	0.979600i	$-0.564406\pi$
$557$	−9.48564			−0.401919			−0.200960	+	0.979600i	$0.564406\pi$
$558$		0			0
$559$	−83.8994			−3.54856
$560$		0			0
$561$	4.25108	+	9.94514i	0.179481	+	0.419884i
$562$		0			0
$563$	−22.1943			−0.935379			−0.467690	−	0.883893i	$-0.654914\pi$
$563$	−22.1943			−0.935379			−0.467690	+	0.883893i	$0.654914\pi$
$564$		0			0
$565$	−5.04468			−0.212231
$566$		0			0
$567$		−	10.7278i		−	0.450524i
$568$		0			0
$569$	−28.3885			−1.19011			−0.595053	−	0.803686i	$-0.702869\pi$
$569$	−28.3885			−1.19011			−0.595053	+	0.803686i	$0.702869\pi$
$570$		0			0
$571$			14.0578i			0.588301i	0.955759	+	0.294150i	$0.0950366\pi$
$571$			14.0578i			0.588301i	−0.955759	+	0.294150i	$0.904963\pi$
$572$		0			0
$573$		−	4.80528i		−	0.200744i
$574$		0			0
$575$		−	6.25760i		−	0.260960i
$576$		0			0
$577$	17.7663			0.739622			0.369811	−	0.929107i	$-0.379422\pi$
$577$	17.7663			0.739622			0.369811	+	0.929107i	$0.379422\pi$
$578$		0			0
$579$	−8.28894			−0.344477
$580$		0			0
$581$			4.99502i			0.207229i
$582$		0			0
$583$		−	48.6529i		−	2.01500i
$584$		0			0
$585$			16.8809i			0.697939i
$586$		0			0
$587$	2.06718			0.0853217			0.0426608	−	0.999090i	$-0.486417\pi$
$587$	2.06718			0.0853217			0.0426608	+	0.999090i	$0.486417\pi$
$588$		0			0
$589$			33.4632i			1.37883i
$590$		0			0
$591$	5.94620			0.244594
$592$		0			0
$593$	4.28086			0.175794			0.0878969	−	0.996130i	$-0.471985\pi$
$593$	4.28086			0.175794			0.0878969	+	0.996130i	$0.471985\pi$
$594$		0			0
$595$	−3.60433	−	8.43209i	−0.147763	−	0.345682i
$596$		0			0
$597$	10.7278			0.439062
$598$		0			0
$599$	32.8147			1.34077			0.670386	−	0.742013i	$-0.266129\pi$
$599$	32.8147			1.34077			0.670386	+	0.742013i	$0.266129\pi$
$600$		0			0
$601$			14.2950i			0.583103i	0.956555	+	0.291552i	$0.0941715\pi$
$601$			14.2950i			0.583103i	−0.956555	+	0.291552i	$0.905828\pi$
$602$		0			0
$603$	0.534558			0.0217689
$604$		0			0
$605$			3.01919i			0.122748i
$606$		0			0
$607$		−	29.3858i		−	1.19273i	−0.802713	−	0.596366i	$-0.796611\pi$
$607$		−	29.3858i		−	1.19273i	0.802713	−	0.596366i	$-0.203389\pi$
$608$		0			0
$609$		−	14.4903i		−	0.587177i
$610$		0			0
$611$	−63.3646			−2.56346
$612$		0			0
$613$	−6.62320			−0.267508			−0.133754	−	0.991015i	$-0.542703\pi$
$613$	−6.62320			−0.267508			−0.133754	+	0.991015i	$0.542703\pi$
$614$		0			0
$615$		−	1.33527i		−	0.0538431i
$616$		0			0
$617$		−	2.78191i		−	0.111996i	−0.998431	−	0.0559978i	$-0.982166\pi$
$617$		−	2.78191i		−	0.111996i	0.998431	−	0.0559978i	$-0.0178340\pi$
$618$		0			0
$619$			39.4925i			1.58734i	0.608351	+	0.793668i	$0.291831\pi$
$619$			39.4925i			1.58734i	−0.608351	+	0.793668i	$0.708169\pi$
$620$		0			0
$621$	24.1523			0.969199
$622$		0			0
$623$			14.1346i			0.566291i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	12.2760			0.490258
$628$		0			0
$629$	18.9836	−	8.11460i	0.756926	−	0.323550i
$630$		0			0
$631$	−30.5437			−1.21592			−0.607962	−	0.793966i	$-0.708013\pi$
$631$	−30.5437			−1.21592			−0.607962	+	0.793966i	$0.708013\pi$
$632$		0			0
$633$	−6.18546			−0.245850
$634$		0			0
$635$		−	16.1088i		−	0.639257i
$636$		0			0
$637$	−13.8149			−0.547368
$638$		0			0
$639$			10.4523i			0.413488i
$640$		0			0
$641$			8.76788i			0.346311i	0.984895	+	0.173155i	$0.0553963\pi$
$641$			8.76788i			0.346311i	−0.984895	+	0.173155i	$0.944604\pi$
$642$		0			0
$643$		−	35.2679i		−	1.39083i	−0.718607	−	0.695416i	$-0.755220\pi$
$643$		−	35.2679i		−	1.39083i	0.718607	−	0.695416i	$-0.244780\pi$
$644$		0			0
$645$	−8.73692			−0.344016
$646$		0			0
$647$	−49.6312			−1.95121			−0.975603	−	0.219544i	$-0.929543\pi$
$647$	−49.6312			−1.95121			−0.975603	+	0.219544i	$0.929543\pi$
$648$		0			0
$649$		−	22.5043i		−	0.883373i
$650$		0			0
$651$		−	11.1417i		−	0.436678i
$652$		0			0
$653$		−	47.8632i		−	1.87303i	−0.350627	−	0.936515i	$-0.614032\pi$
$653$		−	47.8632i		−	1.87303i	0.350627	−	0.936515i	$-0.385968\pi$
$654$		0			0
$655$	0.946721			0.0369915
$656$		0			0
$657$		−	12.5861i		−	0.491030i
$658$		0			0
$659$	5.12850			0.199778			0.0998889	−	0.994999i	$-0.468151\pi$
$659$	5.12850			0.199778			0.0998889	+	0.994999i	$0.468151\pi$
$660$		0			0
$661$	14.6546			0.569998			0.284999	−	0.958528i	$-0.408007\pi$
$661$	14.6546			0.569998			0.284999	+	0.958528i	$0.408007\pi$
$662$		0			0
$663$	−17.8696	+	7.63842i	−0.693998	+	0.296651i
$664$		0			0
$665$	−10.4084			−0.403620
$666$		0			0
$667$	−58.1928			−2.25323
$668$		0			0
$669$		−	2.18489i		−	0.0844726i
$670$		0			0
$671$	−17.9613			−0.693389
$672$		0			0
$673$		−	22.4658i		−	0.865995i	−0.901395	−	0.432997i	$-0.857456\pi$
$673$		−	22.4658i		−	0.865995i	0.901395	−	0.432997i	$-0.142544\pi$
$674$		0			0
$675$			3.85968i			0.148559i
$676$		0			0
$677$			40.0309i			1.53851i	0.638939	+	0.769257i	$0.279374\pi$
$677$			40.0309i			1.53851i	−0.638939	+	0.769257i	$0.720626\pi$
$678$		0			0
$679$	2.03244			0.0779979
$680$		0			0
$681$	17.4492			0.668654
$682$		0			0
$683$		−	29.3124i		−	1.12161i	−0.827949	−	0.560804i	$-0.810492\pi$
$683$		−	29.3124i		−	1.12161i	0.827949	−	0.560804i	$-0.189508\pi$
$684$		0			0
$685$			11.6397i			0.444731i
$686$		0			0
$687$			7.28463i			0.277926i
$688$		0			0
$689$	87.4204			3.33045
$690$		0			0
$691$			15.1987i			0.578187i	0.957301	+	0.289094i	$0.0933539\pi$
$691$			15.1987i			0.578187i	−0.957301	+	0.289094i	$0.906646\pi$
$692$		0			0
$693$	20.8950			0.793736
$694$		0			0
$695$	14.1812			0.537925
$696$		0			0
$697$	−7.22581	+	3.08870i	−0.273697	+	0.116993i
$698$		0			0
$699$	−2.29898			−0.0869555
$700$		0			0
$701$	15.3696			0.580500			0.290250	−	0.956951i	$-0.406262\pi$
$701$	15.3696			0.580500			0.290250	+	0.956951i	$0.406262\pi$
$702$		0			0
$703$		−	23.4329i		−	0.883789i
$704$		0			0
$705$	−6.59852			−0.248515
$706$		0			0
$707$		−	18.6343i		−	0.700817i
$708$		0			0
$709$			0.0958787i			0.00360080i	0.999998	+	0.00180040i	$0.000573085\pi$
$709$			0.0958787i			0.00360080i	−0.999998	+	0.00180040i	$0.999427\pi$
$710$		0			0
$711$		−	31.0784i		−	1.16553i
$712$		0			0
$713$	−44.7448			−1.67571
$714$		0			0
$715$	−25.1899			−0.942050
$716$		0			0
$717$		−	18.3330i		−	0.684657i
$718$		0			0
$719$		−	35.8162i		−	1.33572i	−0.744288	−	0.667859i	$-0.767211\pi$
$719$		−	35.8162i		−	1.33572i	0.744288	−	0.667859i	$-0.232789\pi$
$720$		0			0
$721$		−	10.6781i		−	0.397672i
$722$		0			0
$723$	19.8490			0.738191
$724$		0			0
$725$		−	9.29954i		−	0.345376i
$726$		0			0
$727$	2.96901			0.110115			0.0550573	−	0.998483i	$-0.482466\pi$
$727$	2.96901			0.110115			0.0550573	+	0.998483i	$0.482466\pi$
$728$		0			0
$729$	3.99068			0.147803
$730$		0			0
$731$	20.2100	+	47.2800i	0.747493	+	1.74871i
$732$		0			0
$733$	−3.23027			−0.119313			−0.0596564	−	0.998219i	$-0.519001\pi$
$733$	−3.23027			−0.119313			−0.0596564	+	0.998219i	$0.519001\pi$
$734$		0			0
$735$	−1.43863			−0.0530647
$736$		0			0
$737$			0.797675i			0.0293827i
$738$		0			0
$739$	7.14036			0.262663			0.131331	−	0.991339i	$-0.458075\pi$
$739$	7.14036			0.262663			0.131331	+	0.991339i	$0.458075\pi$
$740$		0			0
$741$			22.0578i			0.810314i
$742$		0			0
$743$		−	13.8353i		−	0.507570i	−0.967261	−	0.253785i	$-0.918325\pi$
$743$		−	13.8353i		−	0.507570i	0.967261	−	0.253785i	$-0.0816755\pi$
$744$		0			0
$745$			9.58924i			0.351323i
$746$		0			0
$747$	−5.63529			−0.206185
$748$		0			0
$749$	−10.0679			−0.367873
$750$		0			0
$751$			34.8612i			1.27210i	0.771647	+	0.636051i	$0.219433\pi$
$751$			34.8612i			1.27210i	−0.771647	+	0.636051i	$0.780567\pi$
$752$		0			0
$753$			7.47659i			0.272462i
$754$		0			0
$755$		−	1.47726i		−	0.0537631i
$756$		0			0
$757$	6.23021			0.226441			0.113220	−	0.993570i	$-0.463883\pi$
$757$	6.23021			0.226441			0.113220	+	0.993570i	$0.463883\pi$
$758$		0			0
$759$			16.4147i			0.595817i
$760$		0			0
$761$	−39.9351			−1.44765			−0.723823	−	0.689986i	$-0.757617\pi$
$761$	−39.9351			−1.44765			−0.723823	+	0.689986i	$0.757617\pi$
$762$		0			0
$763$	−26.4275			−0.956742
$764$		0			0
$765$	9.51293	−	4.06633i	0.343941	−	0.147019i
$766$		0			0
$767$	40.4362			1.46007
$768$		0			0
$769$	12.8468			0.463267			0.231634	−	0.972803i	$-0.425593\pi$
$769$	12.8468			0.463267			0.231634	+	0.972803i	$0.425593\pi$
$770$		0			0
$771$		−	2.42347i		−	0.0872792i
$772$		0			0
$773$	−29.7697			−1.07074			−0.535371	−	0.844617i	$-0.679828\pi$
$773$	−29.7697			−1.07074			−0.535371	+	0.844617i	$0.679828\pi$
$774$		0			0
$775$		−	7.15049i		−	0.256853i
$776$		0			0
$777$			7.80209i			0.279898i
$778$		0			0
$779$			8.91937i			0.319570i
$780$		0			0
$781$	−15.5971			−0.558109
$782$		0			0
$783$	35.8933			1.28272
$784$		0			0
$785$			11.6698i			0.416513i
$786$		0			0
$787$		−	37.6997i		−	1.34385i	−0.740620	−	0.671924i	$-0.765468\pi$
$787$		−	37.6997i		−	1.34385i	0.740620	−	0.671924i	$-0.234532\pi$
$788$		0			0
$789$			2.70935i			0.0964553i
$790$		0			0
$791$	−11.2198			−0.398930
$792$		0			0
$793$		−	32.2732i		−	1.14605i
$794$		0			0
$795$	9.10359			0.322871
$796$		0			0
$797$	42.9591			1.52169			0.760845	−	0.648934i	$-0.224785\pi$
$797$	42.9591			1.52169			0.760845	+	0.648934i	$0.224785\pi$
$798$		0			0
$799$	15.2635	+	35.7080i	0.539984	+	1.26326i
$800$		0			0
$801$	−15.9464			−0.563439
$802$		0			0
$803$	18.7811			0.662772
$804$		0			0
$805$		−	13.9174i		−	0.490525i
$806$		0			0
$807$	−7.24803			−0.255143
$808$		0			0
$809$		−	35.0353i		−	1.23178i	−0.787833	−	0.615888i	$-0.788797\pi$
$809$		−	35.0353i		−	1.23178i	0.787833	−	0.615888i	$-0.211203\pi$
$810$		0			0
$811$		−	7.51429i		−	0.263863i	−0.991259	−	0.131931i	$-0.957882\pi$
$811$		−	7.51429i		−	0.263863i	0.991259	−	0.131931i	$-0.0421178\pi$
$812$		0			0
$813$		−	6.77988i		−	0.237781i
$814$		0			0
$815$	0.439965			0.0154113
$816$		0			0
$817$	58.3613			2.04180
$818$		0			0
$819$			37.5445i			1.31191i
$820$		0			0
$821$		−	31.1545i		−	1.08730i	−0.839312	−	0.543650i	$-0.817042\pi$
$821$		−	31.1545i		−	1.08730i	0.839312	−	0.543650i	$-0.182958\pi$
$822$		0			0
$823$			36.0412i			1.25632i	0.778086	+	0.628158i	$0.216191\pi$
$823$			36.0412i			1.25632i	−0.778086	+	0.628158i	$0.783809\pi$
$824$		0			0
$825$	−2.62317			−0.0913271
$826$		0			0
$827$			54.1571i			1.88323i	0.336694	+	0.941614i	$0.390691\pi$
$827$			54.1571i			1.88323i	−0.336694	+	0.941614i	$0.609309\pi$
$828$		0			0
$829$	−25.4545			−0.884072			−0.442036	−	0.896997i	$-0.645744\pi$
$829$	−25.4545			−0.884072			−0.442036	+	0.896997i	$0.645744\pi$
$830$		0			0
$831$	−17.7953			−0.617312
$832$		0			0
$833$	3.32780	+	7.78517i	0.115301	+	0.269740i
$834$		0			0
$835$	20.3369			0.703787
$836$		0			0
$837$	27.5986			0.953947
$838$		0			0
$839$			13.2362i			0.456965i	0.973548	+	0.228482i	$0.0733763\pi$
$839$			13.2362i			0.456965i	−0.973548	+	0.228482i	$0.926624\pi$
$840$		0			0
$841$	−57.4815			−1.98212
$842$		0			0
$843$			3.37966i			0.116402i
$844$		0			0
$845$		−	32.2617i		−	1.10984i
$846$		0			0
$847$			6.71493i			0.230728i
$848$		0			0
$849$	−11.0432			−0.379001
$850$		0			0
$851$	31.3330			1.07408
$852$		0			0
$853$			31.8398i			1.09017i	0.838380	+	0.545087i	$0.183503\pi$
$853$			31.8398i			1.09017i	−0.838380	+	0.545087i	$0.816497\pi$
$854$		0			0
$855$		−	11.7425i		−	0.401586i
$856$		0			0
$857$		−	22.8762i		−	0.781437i	−0.920510	−	0.390718i	$-0.872227\pi$
$857$		−	22.8762i		−	0.781437i	0.920510	−	0.390718i	$-0.127773\pi$
$858$		0			0
$859$	22.5897			0.770751			0.385376	−	0.922760i	$-0.374072\pi$
$859$	22.5897			0.770751			0.385376	+	0.922760i	$0.374072\pi$
$860$		0			0
$861$		−	2.96974i		−	0.101209i
$862$		0			0
$863$	13.7660			0.468600			0.234300	−	0.972164i	$-0.424720\pi$
$863$	13.7660			0.468600			0.234300	+	0.972164i	$0.424720\pi$
$864$		0			0
$865$	18.9699			0.644996
$866$		0			0
$867$	8.60899	+	8.23012i	0.292377	+	0.279510i
$868$		0			0
$869$	46.3756			1.57318
$870$		0			0
$871$	−1.43328			−0.0485647
$872$		0			0
$873$			2.29296i			0.0776050i
$874$		0			0
$875$	2.22408			0.0751878
$876$		0			0
$877$			16.5833i			0.559977i	0.960003	+	0.279988i	$0.0903306\pi$
$877$			16.5833i			0.559977i	−0.960003	+	0.279988i	$0.909669\pi$
$878$		0			0
$879$		−	17.4295i		−	0.587881i
$880$		0			0
$881$		−	40.9275i		−	1.37888i	−0.724341	−	0.689442i	$-0.757856\pi$
$881$		−	40.9275i		−	1.37888i	0.724341	−	0.689442i	$-0.242144\pi$
$882$		0			0
$883$	11.7019			0.393799			0.196899	−	0.980424i	$-0.436913\pi$
$883$	11.7019			0.393799			0.196899	+	0.980424i	$0.436913\pi$
$884$		0			0
$885$	4.21085			0.141546
$886$		0			0
$887$			0.644527i			0.0216411i	0.999941	+	0.0108205i	$0.00344435\pi$
$887$			0.644527i			0.0216411i	−0.999941	+	0.0108205i	$0.996556\pi$
$888$		0			0
$889$		−	35.8272i		−	1.20161i
$890$		0			0
$891$			18.0601i			0.605035i
$892$		0			0
$893$	44.0771			1.47498
$894$		0			0
$895$		−	21.9259i		−	0.732901i
$896$		0			0
$897$	−29.4943			−0.984786
$898$		0			0
$899$	−66.4962			−2.21777
$900$		0			0
$901$	−21.0581	−	49.2642i	−0.701549	−	1.64123i
$902$		0			0
$903$	−19.4317			−0.646645
$904$		0			0
$905$	1.09659			0.0364518
$906$		0			0
$907$		−	19.2262i		−	0.638396i	−0.947688	−	0.319198i	$-0.896587\pi$
$907$		−	19.2262i		−	0.638396i	0.947688	−	0.319198i	$-0.103413\pi$
$908$		0			0
$909$	21.0229			0.697286
$910$		0			0
$911$		−	33.0601i		−	1.09533i	−0.836697	−	0.547665i	$-0.815517\pi$
$911$		−	33.0601i		−	1.09533i	0.836697	−	0.547665i	$-0.184483\pi$
$912$		0			0
$913$		−	8.40907i		−	0.278300i
$914$		0			0
$915$		−	3.36079i		−	0.111104i
$916$		0			0
$917$	2.10559			0.0695326
$918$		0			0
$919$	−8.40732			−0.277332			−0.138666	−	0.990339i	$-0.544281\pi$
$919$	−8.40732			−0.277332			−0.138666	+	0.990339i	$0.544281\pi$
$920$		0			0
$921$			20.9756i			0.691170i
$922$		0			0
$923$		−	28.0252i		−	0.922460i
$924$		0			0
$925$			5.00719i			0.164635i
$926$		0			0
$927$	12.0468			0.395669
$928$		0			0
$929$			22.4176i			0.735498i	0.929925	+	0.367749i	$0.119871\pi$
$929$			22.4176i			0.735498i	−0.929925	+	0.367749i	$0.880129\pi$
$930$		0			0
$931$	9.60983			0.314949
$932$		0			0
$933$	−1.76644			−0.0578307
$934$		0			0
$935$	6.06784	+	14.1953i	0.198440	+	0.464237i
$936$		0			0
$937$	−32.3858			−1.05800			−0.528999	−	0.848622i	$-0.677433\pi$
$937$	−32.3858			−1.05800			−0.528999	+	0.848622i	$0.677433\pi$
$938$		0			0
$939$	11.1254			0.363065
$940$		0			0
$941$		−	29.5171i		−	0.962231i	−0.876657	−	0.481115i	$-0.840232\pi$
$941$		−	29.5171i		−	0.962231i	0.876657	−	0.481115i	$-0.159768\pi$
$942$		0			0
$943$	−11.9264			−0.388378
$944$		0			0
$945$			8.58426i			0.279246i
$946$		0			0
$947$			45.7242i			1.48584i	0.669381	+	0.742919i	$0.266559\pi$
$947$			45.7242i			1.48584i	−0.669381	+	0.742919i	$0.733441\pi$
$948$		0			0
$949$			33.7463i			1.09545i
$950$		0			0
$951$	3.00168			0.0973362
$952$		0			0
$953$	−24.2760			−0.786376			−0.393188	−	0.919458i	$-0.628628\pi$
$953$	−24.2760			−0.786376			−0.393188	+	0.919458i	$0.628628\pi$
$954$		0			0
$955$		−	6.85889i		−	0.221948i
$956$		0			0
$957$			24.3943i			0.788555i
$958$		0			0
$959$			25.8877i			0.835958i
$960$		0			0
$961$	−20.1294			−0.649337
$962$		0			0
$963$		−	11.3584i		−	0.366020i
$964$		0			0
$965$	−11.8313			−0.380864
$966$		0			0
$967$	45.5601			1.46511			0.732557	−	0.680705i	$-0.238327\pi$
$967$	45.5601			1.46511			0.732557	+	0.680705i	$0.238327\pi$
$968$		0			0
$969$	12.4303	−	5.31336i	0.399318	−	0.170690i
$970$		0			0
$971$	16.8044			0.539280			0.269640	−	0.962961i	$-0.413095\pi$
$971$	16.8044			0.539280			0.269640	+	0.962961i	$0.413095\pi$
$972$		0			0
$973$	31.5403			1.01113
$974$		0			0
$975$		−	4.71336i		−	0.150948i
$976$		0			0
$977$	−17.2559			−0.552064			−0.276032	−	0.961148i	$-0.589020\pi$
$977$	−17.2559			−0.552064			−0.276032	+	0.961148i	$0.589020\pi$
$978$		0			0
$979$		−	23.7955i		−	0.760506i
$980$		0			0
$981$		−	29.8151i		−	0.951922i
$982$		0			0
$983$		−	7.24700i		−	0.231144i	−0.993299	−	0.115572i	$-0.963130\pi$
$983$		−	7.24700i		−	0.231144i	0.993299	−	0.115572i	$-0.0368700\pi$
$984$		0			0
$985$	8.48740			0.270431
$986$		0			0
$987$	−14.6757			−0.467132
$988$		0			0
$989$			78.0370i			2.48143i
$990$		0			0
$991$		−	26.9695i		−	0.856714i	−0.903610	−	0.428357i	$-0.859093\pi$
$991$		−	26.9695i		−	0.856714i	0.903610	−	0.428357i	$-0.140907\pi$
$992$		0			0
$993$			18.4105i			0.584241i
$994$		0			0
$995$	15.3125			0.485440
$996$		0			0
$997$		−	41.4864i		−	1.31389i	−0.753939	−	0.656944i	$-0.771849\pi$
$997$		−	41.4864i		−	1.31389i	0.753939	−	0.656944i	$-0.228151\pi$
$998$		0			0
$999$	−19.3262			−0.611453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2720.2.c.f.1121.7	yes	16
4.3	odd	2		2720.2.c.e.1121.10	yes	16
17.16	even	2	inner	2720.2.c.f.1121.10	yes	16
68.67	odd	2		2720.2.c.e.1121.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2720.2.c.e.1121.7	✓	16	68.67	odd	2
2720.2.c.e.1121.10	yes	16	4.3	odd	2
2720.2.c.f.1121.7	yes	16	1.1	even	1	trivial
2720.2.c.f.1121.10	yes	16	17.16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2720 = 2^{5} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2720.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.700592i q^{3} -1.00000i q^{5} -2.22408i q^{7} +2.50917 q^{9} +3.74422i q^{11} -6.72768 q^{13} -0.700592 q^{15} +(1.62059 + 3.79127i) q^{17} +4.67985 q^{19} -1.55818 q^{21} +6.25760i q^{23} -1.00000 q^{25} -3.85968i q^{27} +9.29954i q^{29} +7.15049i q^{31} +2.62317 q^{33} -2.22408 q^{35} -5.00719i q^{37} +4.71336i q^{39} +1.90591i q^{41} +12.4708 q^{43} -2.50917i q^{45} +9.41849 q^{47} +2.05345 q^{49} +(2.65613 - 1.13537i) q^{51} -12.9941 q^{53} +3.74422 q^{55} -3.27866i q^{57} -6.01042 q^{59} +4.79708i q^{61} -5.58061i q^{63} +6.72768i q^{65} +0.213042 q^{67} +4.38402 q^{69} +4.16565i q^{71} -5.01604i q^{73} +0.700592i q^{75} +8.32746 q^{77} -12.3859i q^{79} +4.82345 q^{81} -2.24588 q^{83} +(3.79127 - 1.62059i) q^{85} +6.51518 q^{87} -6.35525 q^{89} +14.9629i q^{91} +5.00957 q^{93} -4.67985i q^{95} +0.913832i q^{97} +9.39489i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 12 q^{9} - 16 q^{13} - 4 q^{17} + 24 q^{21} - 16 q^{25} + 8 q^{33} + 12 q^{35} - 4 q^{43} + 24 q^{47} - 24 q^{49} + 20 q^{51} - 8 q^{53} - 24 q^{59} + 28 q^{67} + 16 q^{69} - 24 q^{77} + 56 q^{81}+ \cdots - 48 q^{93}+O(q^{100}) \) 16 * q - 12 * q^9 - 16 * q^13 - 4 * q^17 + 24 * q^21 - 16 * q^25 + 8 * q^33 + 12 * q^35 - 4 * q^43 + 24 * q^47 - 24 * q^49 + 20 * q^51 - 8 * q^53 - 24 * q^59 + 28 * q^67 + 16 * q^69 - 24 * q^77 + 56 * q^81 - 20 * q^83 + 8 * q^85 + 100 * q^87 - 12 * q^89 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more