LMFDB - Embedded newform 2240.2.k.g.1791.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1791,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2240.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$16$
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} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} + \cdots + 5956 $ x^16 - 4x^15 - 6x^14 + 68x^13 - 126x^12 - 148x^11 + 1006x^10 - 1516x^9 - 179x^8 + 3992x^7 - 5596x^6 - 488x^5 + 16080x^4 - 33776x^3 + 37344x^2 - 22336*x + 5956
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.10
Root			$1.23885 - 0.371381i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.g.1791.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.742762 q^{3} +1.00000i q^{5} +(2.53467 - 0.758567i) q^{7} -2.44831 q^{9} +O(q^{10})$	$q+0.742762 q^{3} +1.00000i q^{5} +(2.53467 - 0.758567i) q^{7} -2.44831 q^{9} +2.35482i q^{11} -0.960570i q^{13} +0.742762i q^{15} -2.59165i q^{17} +1.82622 q^{19} +(1.88266 - 0.563434i) q^{21} -3.24313i q^{23} -1.00000 q^{25} -4.04679 q^{27} +9.30721 q^{29} +8.86838 q^{31} +1.74907i q^{33} +(0.758567 + 2.53467i) q^{35} +7.50984 q^{37} -0.713475i q^{39} +4.14017i q^{41} +9.65793i q^{43} -2.44831i q^{45} -5.52648 q^{47} +(5.84915 - 3.84544i) q^{49} -1.92497i q^{51} +5.21385 q^{53} -2.35482 q^{55} +1.35644 q^{57} -0.391319 q^{59} +13.8165i q^{61} +(-6.20566 + 1.85720i) q^{63} +0.960570 q^{65} +0.695969i q^{67} -2.40888i q^{69} -16.3335i q^{71} +11.7289i q^{73} -0.742762 q^{75} +(1.78629 + 5.96870i) q^{77} -1.05184i q^{79} +4.33911 q^{81} +1.49920 q^{83} +2.59165 q^{85} +6.91304 q^{87} -9.48370i q^{89} +(-0.728657 - 2.43473i) q^{91} +6.58709 q^{93} +1.82622i q^{95} -7.40940i q^{97} -5.76532i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 4 * q^7 + 16 * q^9	$ 16 q + 4 q^{7} + 16 q^{9} + 8 q^{19} - 4 q^{21} - 16 q^{25} - 48 q^{27} - 8 q^{29} - 16 q^{37} + 8 q^{47} - 4 q^{49} - 16 q^{53} - 8 q^{55} + 16 q^{57} + 8 q^{59} + 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} - 64 q^{83} + 16 q^{87} + 64 q^{91} + 48 q^{93}+O(q^{100}) $ 16 * q + 4 * q^7 + 16 * q^9 + 8 * q^19 - 4 * q^21 - 16 * q^25 - 48 * q^27 - 8 * q^29 - 16 * q^37 + 8 * q^47 - 4 * q^49 - 16 * q^53 - 8 * q^55 + 16 * q^57 + 8 * q^59 + 60 * q^63 + 8 * q^65 + 8 * q^77 + 40 * q^81 - 64 * q^83 + 16 * q^87 + 64 * q^91 + 48 * q^93

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.742762			0.428834			0.214417	−	0.976742i	$-0.431215\pi$
$3$	0.742762			0.428834			0.214417	+	0.976742i	$0.431215\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	2.53467	−	0.758567i	0.958017	−	0.286711i
$7$	2.53467	−	0.758567i	0.958017	−	0.286711i
$8$		0			0
$9$	−2.44831			−0.816102
$10$		0			0
$11$			2.35482i			0.710005i	0.934865	+	0.355003i	$0.115520\pi$
$11$			2.35482i			0.710005i	−0.934865	+	0.355003i	$0.884480\pi$
$12$		0			0
$13$		−	0.960570i		−	0.266414i	−0.991088	−	0.133207i	$-0.957472\pi$
$13$		−	0.960570i		−	0.266414i	0.991088	−	0.133207i	$-0.0425276\pi$
$14$		0			0
$15$			0.742762i			0.191780i
$16$		0			0
$17$		−	2.59165i		−	0.628566i	−0.949329	−	0.314283i	$-0.898236\pi$
$17$		−	2.59165i		−	0.628566i	0.949329	−	0.314283i	$-0.101764\pi$
$18$		0			0
$19$	1.82622			0.418963			0.209481	−	0.977813i	$-0.432822\pi$
$19$	1.82622			0.418963			0.209481	+	0.977813i	$0.432822\pi$
$20$		0			0
$21$	1.88266	−	0.563434i	0.410830	−	0.122951i
$22$		0			0
$23$		−	3.24313i		−	0.676240i	−0.941103	−	0.338120i	$-0.890209\pi$
$23$		−	3.24313i		−	0.676240i	0.941103	−	0.338120i	$-0.109791\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−4.04679			−0.778806
$28$		0			0
$29$	9.30721			1.72831			0.864153	−	0.503230i	$-0.167855\pi$
$29$	9.30721			1.72831			0.864153	+	0.503230i	$0.167855\pi$
$30$		0			0
$31$	8.86838			1.59281			0.796404	−	0.604765i	$-0.206733\pi$
$31$	8.86838			1.59281			0.796404	+	0.604765i	$0.206733\pi$
$32$		0			0
$33$			1.74907i			0.304474i
$34$		0			0
$35$	0.758567	+	2.53467i	0.128221	+	0.428438i
$36$		0			0
$37$	7.50984			1.23461			0.617305	−	0.786724i	$-0.288225\pi$
$37$	7.50984			1.23461			0.617305	+	0.786724i	$0.288225\pi$
$38$		0			0
$39$		−	0.713475i		−	0.114247i
$40$		0			0
$41$			4.14017i			0.646585i	0.946299	+	0.323293i	$0.104790\pi$
$41$			4.14017i			0.646585i	−0.946299	+	0.323293i	$0.895210\pi$
$42$		0			0
$43$			9.65793i			1.47282i	0.676535	+	0.736410i	$0.263481\pi$
$43$			9.65793i			1.47282i	−0.676535	+	0.736410i	$0.736519\pi$
$44$		0			0
$45$		−	2.44831i		−	0.364972i
$46$		0			0
$47$	−5.52648			−0.806121			−0.403060	−	0.915173i	$-0.632054\pi$
$47$	−5.52648			−0.806121			−0.403060	+	0.915173i	$0.632054\pi$
$48$		0			0
$49$	5.84915	−	3.84544i	0.835593	−	0.549349i
$50$		0			0
$51$		−	1.92497i		−	0.269550i
$52$		0			0
$53$	5.21385			0.716177			0.358088	−	0.933688i	$-0.383429\pi$
$53$	5.21385			0.716177			0.358088	+	0.933688i	$0.383429\pi$
$54$		0			0
$55$	−2.35482			−0.317524
$56$		0			0
$57$	1.35644			0.179665
$58$		0			0
$59$	−0.391319			−0.0509454			−0.0254727	−	0.999676i	$-0.508109\pi$
$59$	−0.391319			−0.0509454			−0.0254727	+	0.999676i	$0.508109\pi$
$60$		0			0
$61$			13.8165i			1.76902i	0.466517	+	0.884512i	$0.345509\pi$
$61$			13.8165i			1.76902i	−0.466517	+	0.884512i	$0.654491\pi$
$62$		0			0
$63$	−6.20566	+	1.85720i	−0.781839	+	0.233986i
$64$		0			0
$65$	0.960570			0.119144
$66$		0			0
$67$			0.695969i			0.0850262i	0.999096	+	0.0425131i	$0.0135364\pi$
$67$			0.695969i			0.0850262i	−0.999096	+	0.0425131i	$0.986464\pi$
$68$		0			0
$69$		−	2.40888i		−	0.289995i
$70$		0			0
$71$		−	16.3335i		−	1.93843i	−0.246214	−	0.969215i	$-0.579187\pi$
$71$		−	16.3335i		−	1.93843i	0.246214	−	0.969215i	$-0.420813\pi$
$72$		0			0
$73$			11.7289i			1.37276i	0.727243	+	0.686380i	$0.240801\pi$
$73$			11.7289i			1.37276i	−0.727243	+	0.686380i	$0.759199\pi$
$74$		0			0
$75$	−0.742762			−0.0857667
$76$		0			0
$77$	1.78629	+	5.96870i	0.203567	+	0.680197i
$78$		0			0
$79$		−	1.05184i		−	0.118342i	−0.998248	−	0.0591708i	$-0.981154\pi$
$79$		−	1.05184i		−	0.118342i	0.998248	−	0.0591708i	$-0.0188457\pi$
$80$		0			0
$81$	4.33911			0.482124
$82$		0			0
$83$	1.49920			0.164558			0.0822790	−	0.996609i	$-0.473780\pi$
$83$	1.49920			0.164558			0.0822790	+	0.996609i	$0.473780\pi$
$84$		0			0
$85$	2.59165			0.281103
$86$		0			0
$87$	6.91304			0.741156
$88$		0			0
$89$		−	9.48370i		−	1.00527i	−0.864499	−	0.502635i	$-0.832364\pi$
$89$		−	9.48370i		−	1.00527i	0.864499	−	0.502635i	$-0.167636\pi$
$90$		0			0
$91$	−0.728657	−	2.43473i	−0.0763840	−	0.255229i
$92$		0			0
$93$	6.58709			0.683050
$94$		0			0
$95$			1.82622i			0.187366i
$96$		0			0
$97$		−	7.40940i		−	0.752310i	−0.926557	−	0.376155i	$-0.877246\pi$
$97$		−	7.40940i		−	0.752310i	0.926557	−	0.376155i	$-0.122754\pi$
$98$		0			0
$99$		−	5.76532i		−	0.579436i
$100$		0			0
$101$			8.27256i			0.823150i	0.911376	+	0.411575i	$0.135021\pi$
$101$			8.27256i			0.823150i	−0.911376	+	0.411575i	$0.864979\pi$
$102$		0			0
$103$	−8.94812			−0.881684			−0.440842	−	0.897585i	$-0.645320\pi$
$103$	−8.94812			−0.881684			−0.440842	+	0.897585i	$0.645320\pi$
$104$		0			0
$105$	0.563434	+	1.88266i	0.0549856	+	0.183729i
$106$		0			0
$107$			2.83467i			0.274038i	0.990568	+	0.137019i	$0.0437521\pi$
$107$			2.83467i			0.274038i	−0.990568	+	0.137019i	$0.956248\pi$
$108$		0			0
$109$	6.91377			0.662219			0.331109	−	0.943592i	$-0.392577\pi$
$109$	6.91377			0.662219			0.331109	+	0.943592i	$0.392577\pi$
$110$		0			0
$111$	5.57802			0.529442
$112$		0			0
$113$	−0.934659			−0.0879253			−0.0439627	−	0.999033i	$-0.513998\pi$
$113$	−0.934659			−0.0879253			−0.0439627	+	0.999033i	$0.513998\pi$
$114$		0			0
$115$	3.24313			0.302424
$116$		0			0
$117$			2.35177i			0.217421i
$118$		0			0
$119$	−1.96594	−	6.56898i	−0.180217	−	0.602177i
$120$		0			0
$121$	5.45482			0.495893
$122$		0			0
$123$			3.07516i			0.277278i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			8.59120i			0.762346i	0.924504	+	0.381173i	$0.124480\pi$
$127$			8.59120i			0.762346i	−0.924504	+	0.381173i	$0.875520\pi$
$128$		0			0
$129$			7.17354i			0.631595i
$130$		0			0
$131$	−6.76374			−0.590951			−0.295475	−	0.955350i	$-0.595478\pi$
$131$	−6.76374			−0.590951			−0.295475	+	0.955350i	$0.595478\pi$
$132$		0			0
$133$	4.62886	−	1.38531i	0.401373	−	0.120121i
$134$		0			0
$135$		−	4.04679i		−	0.348292i
$136$		0			0
$137$	4.39724			0.375682			0.187841	−	0.982199i	$-0.439851\pi$
$137$	4.39724			0.375682			0.187841	+	0.982199i	$0.439851\pi$
$138$		0			0
$139$	−9.71702			−0.824187			−0.412093	−	0.911142i	$-0.635202\pi$
$139$	−9.71702			−0.824187			−0.412093	+	0.911142i	$0.635202\pi$
$140$		0			0
$141$	−4.10486			−0.345692
$142$		0			0
$143$	2.26197			0.189156
$144$		0			0
$145$			9.30721i			0.772922i
$146$		0			0
$147$	4.34453	−	2.85625i	0.358331	−	0.235579i
$148$		0			0
$149$	−6.09792			−0.499561			−0.249780	−	0.968303i	$-0.580358\pi$
$149$	−6.09792			−0.499561			−0.249780	+	0.968303i	$0.580358\pi$
$150$		0			0
$151$			15.1994i			1.23691i	0.785821	+	0.618454i	$0.212241\pi$
$151$			15.1994i			1.23691i	−0.785821	+	0.618454i	$0.787759\pi$
$152$		0			0
$153$			6.34514i			0.512974i
$154$		0			0
$155$			8.86838i			0.712325i
$156$		0			0
$157$		−	20.4181i		−	1.62954i	−0.579781	−	0.814772i	$-0.696862\pi$
$157$		−	20.4181i		−	1.62954i	0.579781	−	0.814772i	$-0.303138\pi$
$158$		0			0
$159$	3.87265			0.307121
$160$		0			0
$161$	−2.46013	−	8.22029i	−0.193886	−	0.647850i
$162$		0			0
$163$		−	10.6372i		−	0.833166i	−0.909098	−	0.416583i	$-0.863228\pi$
$163$		−	10.6372i		−	0.833166i	0.909098	−	0.416583i	$-0.136772\pi$
$164$		0			0
$165$	−1.74907			−0.136165
$166$		0			0
$167$	−0.633668			−0.0490347			−0.0245174	−	0.999699i	$-0.507805\pi$
$167$	−0.633668			−0.0490347			−0.0245174	+	0.999699i	$0.507805\pi$
$168$		0			0
$169$	12.0773			0.929023
$170$		0			0
$171$	−4.47113			−0.341916
$172$		0			0
$173$		−	12.6130i		−	0.958952i	−0.877555	−	0.479476i	$-0.840827\pi$
$173$		−	12.6130i		−	0.958952i	0.877555	−	0.479476i	$-0.159173\pi$
$174$		0			0
$175$	−2.53467	+	0.758567i	−0.191603	+	0.0573423i
$176$		0			0
$177$	−0.290657			−0.0218471
$178$		0			0
$179$		−	6.47707i		−	0.484119i	−0.970261	−	0.242059i	$-0.922177\pi$
$179$		−	6.47707i		−	0.484119i	0.970261	−	0.242059i	$-0.0778229\pi$
$180$		0			0
$181$			16.8594i			1.25315i	0.779361	+	0.626576i	$0.215544\pi$
$181$			16.8594i			1.25315i	−0.779361	+	0.626576i	$0.784456\pi$
$182$		0			0
$183$			10.2624i			0.758617i
$184$		0			0
$185$			7.50984i			0.552134i
$186$		0			0
$187$	6.10286			0.446285
$188$		0			0
$189$	−10.2573	+	3.06976i	−0.746109	+	0.223292i
$190$		0			0
$191$			11.4816i			0.830783i	0.909643	+	0.415391i	$0.136355\pi$
$191$			11.4816i			0.830783i	−0.909643	+	0.415391i	$0.863645\pi$
$192$		0			0
$193$	16.6299			1.19704			0.598521	−	0.801107i	$-0.295755\pi$
$193$	16.6299			1.19704			0.598521	+	0.801107i	$0.295755\pi$
$194$		0			0
$195$	0.713475			0.0510930
$196$		0			0
$197$	13.3094			0.948258			0.474129	−	0.880455i	$-0.342763\pi$
$197$	13.3094			0.948258			0.474129	+	0.880455i	$0.342763\pi$
$198$		0			0
$199$	19.1162			1.35511			0.677555	−	0.735472i	$-0.263039\pi$
$199$	19.1162			1.35511			0.677555	+	0.735472i	$0.263039\pi$
$200$		0			0
$201$			0.516939i			0.0364621i
$202$		0			0
$203$	23.5907	−	7.06014i	1.65575	−	0.495525i
$204$		0			0
$205$	−4.14017			−0.289162
$206$		0			0
$207$			7.94018i			0.551881i
$208$		0			0
$209$			4.30041i			0.297466i
$210$		0			0
$211$		−	9.64523i		−	0.664005i	−0.943279	−	0.332002i	$-0.892276\pi$
$211$		−	9.64523i		−	0.664005i	0.943279	−	0.332002i	$-0.107724\pi$
$212$		0			0
$213$		−	12.1319i		−	0.831264i
$214$		0			0
$215$	−9.65793			−0.658665
$216$		0			0
$217$	22.4785	−	6.72726i	1.52594	−	0.456676i
$218$		0			0
$219$			8.71175i			0.588685i
$220$		0			0
$221$	−2.48946			−0.167459
$222$		0			0
$223$	−24.8236			−1.66231			−0.831157	−	0.556038i	$-0.812321\pi$
$223$	−24.8236			−1.66231			−0.831157	+	0.556038i	$0.812321\pi$
$224$		0			0
$225$	2.44831			0.163220
$226$		0			0
$227$	−17.3561			−1.15196			−0.575981	−	0.817463i	$-0.695380\pi$
$227$	−17.3561			−1.15196			−0.575981	+	0.817463i	$0.695380\pi$
$228$		0			0
$229$			6.13404i			0.405349i	0.979246	+	0.202674i	$0.0649633\pi$
$229$			6.13404i			0.405349i	−0.979246	+	0.202674i	$0.935037\pi$
$230$		0			0
$231$	1.32679	+	4.43332i	0.0872962	+	0.291691i
$232$		0			0
$233$	−14.1543			−0.927281			−0.463641	−	0.886023i	$-0.653457\pi$
$233$	−14.1543			−0.927281			−0.463641	+	0.886023i	$0.653457\pi$
$234$		0			0
$235$		−	5.52648i		−	0.360508i
$236$		0			0
$237$		−	0.781269i		−	0.0507489i
$238$		0			0
$239$		−	0.370459i		−	0.0239630i	−0.999928	−	0.0119815i	$-0.996186\pi$
$239$		−	0.370459i		−	0.0239630i	0.999928	−	0.0119815i	$-0.00381392\pi$
$240$		0			0
$241$			19.8194i			1.27668i	0.769754	+	0.638340i	$0.220379\pi$
$241$			19.8194i			1.27668i	−0.769754	+	0.638340i	$0.779621\pi$
$242$		0			0
$243$	15.3633			0.985556
$244$		0			0
$245$	3.84544	+	5.84915i	0.245676	+	0.373689i
$246$		0			0
$247$		−	1.75421i		−	0.111618i
$248$		0			0
$249$	1.11354			0.0705680
$250$		0			0
$251$	−29.9284			−1.88907			−0.944533	−	0.328415i	$-0.893485\pi$
$251$	−29.9284			−1.88907			−0.944533	+	0.328415i	$0.893485\pi$
$252$		0			0
$253$	7.63700			0.480134
$254$		0			0
$255$	1.92497			0.120547
$256$		0			0
$257$		−	24.8900i		−	1.55259i	−0.630367	−	0.776297i	$-0.717096\pi$
$257$		−	24.8900i		−	1.55259i	0.630367	−	0.776297i	$-0.282904\pi$
$258$		0			0
$259$	19.0350	−	5.69671i	1.18278	−	0.353977i
$260$		0			0
$261$	−22.7869			−1.41047
$262$		0			0
$263$		−	27.8438i		−	1.71692i	−0.512876	−	0.858462i	$-0.671420\pi$
$263$		−	27.8438i		−	1.71692i	0.512876	−	0.858462i	$-0.328580\pi$
$264$		0			0
$265$			5.21385i			0.320284i
$266$		0			0
$267$		−	7.04413i		−	0.431094i
$268$		0			0
$269$		−	16.3592i		−	0.997438i	−0.866764	−	0.498719i	$-0.833804\pi$
$269$		−	16.3592i		−	0.997438i	0.866764	−	0.498719i	$-0.166196\pi$
$270$		0			0
$271$	−12.8401			−0.779982			−0.389991	−	0.920819i	$-0.627522\pi$
$271$	−12.8401			−0.779982			−0.389991	+	0.920819i	$0.627522\pi$
$272$		0			0
$273$	−0.541219	−	1.80843i	−0.0327560	−	0.109451i
$274$		0			0
$275$		−	2.35482i		−	0.142001i
$276$		0			0
$277$	−20.2109			−1.21436			−0.607178	−	0.794566i	$-0.707698\pi$
$277$	−20.2109			−1.21436			−0.607178	+	0.794566i	$0.707698\pi$
$278$		0			0
$279$	−21.7125			−1.29989
$280$		0			0
$281$	8.84677			0.527754			0.263877	−	0.964556i	$-0.414999\pi$
$281$	8.84677			0.527754			0.263877	+	0.964556i	$0.414999\pi$
$282$		0			0
$283$	7.13944			0.424396			0.212198	−	0.977227i	$-0.431938\pi$
$283$	7.13944			0.424396			0.212198	+	0.977227i	$0.431938\pi$
$284$		0			0
$285$			1.35644i			0.0803488i
$286$		0			0
$287$	3.14059	+	10.4940i	0.185383	+	0.619440i
$288$		0			0
$289$	10.2834			0.604904
$290$		0			0
$291$		−	5.50342i		−	0.322616i
$292$		0			0
$293$		−	29.5271i		−	1.72499i	−0.506062	−	0.862497i	$-0.668899\pi$
$293$		−	29.5271i		−	1.72499i	0.506062	−	0.862497i	$-0.331101\pi$
$294$		0			0
$295$		−	0.391319i		−	0.0227835i
$296$		0			0
$297$		−	9.52947i		−	0.552956i
$298$		0			0
$299$	−3.11526			−0.180160
$300$		0			0
$301$	7.32619	+	24.4797i	0.422274	+	1.41099i
$302$		0			0
$303$			6.14454i			0.352994i
$304$		0			0
$305$	−13.8165			−0.791132
$306$		0			0
$307$	−18.5381			−1.05802			−0.529012	−	0.848614i	$-0.677437\pi$
$307$	−18.5381			−1.05802			−0.529012	+	0.848614i	$0.677437\pi$
$308$		0			0
$309$	−6.64632			−0.378096
$310$		0			0
$311$	27.0703			1.53502			0.767508	−	0.641040i	$-0.221497\pi$
$311$	27.0703			1.53502			0.767508	+	0.641040i	$0.221497\pi$
$312$		0			0
$313$		−	9.94685i		−	0.562229i	−0.959674	−	0.281115i	$-0.909296\pi$
$313$		−	9.94685i		−	0.562229i	0.959674	−	0.281115i	$-0.0907041\pi$
$314$		0			0
$315$	−1.85720	−	6.20566i	−0.104642	−	0.349649i
$316$		0			0
$317$	−6.12820			−0.344194			−0.172097	−	0.985080i	$-0.555054\pi$
$317$	−6.12820			−0.344194			−0.172097	+	0.985080i	$0.555054\pi$
$318$		0			0
$319$			21.9168i			1.22711i
$320$		0			0
$321$			2.10548i			0.117517i
$322$		0			0
$323$		−	4.73290i		−	0.263346i
$324$		0			0
$325$			0.960570i			0.0532829i
$326$		0			0
$327$	5.13528			0.283982
$328$		0			0
$329$	−14.0078	+	4.19221i	−0.772277	+	0.231124i
$330$		0			0
$331$		−	2.39632i		−	0.131714i	−0.997829	−	0.0658569i	$-0.979022\pi$
$331$		−	2.39632i		−	0.131714i	0.997829	−	0.0658569i	$-0.0209781\pi$
$332$		0			0
$333$	−18.3864			−1.00757
$334$		0			0
$335$	−0.695969			−0.0380249
$336$		0			0
$337$	−1.80372			−0.0982550			−0.0491275	−	0.998793i	$-0.515644\pi$
$337$	−1.80372			−0.0982550			−0.0491275	+	0.998793i	$0.515644\pi$
$338$		0			0
$339$	−0.694229			−0.0377054
$340$		0			0
$341$			20.8834i			1.13090i
$342$		0			0
$343$	11.9087	−	14.1839i	0.643008	−	0.765859i
$344$		0			0
$345$	2.40888			0.129689
$346$		0			0
$347$			21.3337i			1.14525i	0.819816	+	0.572627i	$0.194076\pi$
$347$			21.3337i			1.14525i	−0.819816	+	0.572627i	$0.805924\pi$
$348$		0			0
$349$		−	10.7074i		−	0.573154i	−0.958057	−	0.286577i	$-0.907483\pi$
$349$		−	10.7074i		−	0.573154i	0.958057	−	0.286577i	$-0.0925174\pi$
$350$		0			0
$351$			3.88723i			0.207485i
$352$		0			0
$353$			20.0938i			1.06948i	0.845016	+	0.534742i	$0.179591\pi$
$353$			20.0938i			1.06948i	−0.845016	+	0.534742i	$0.820409\pi$
$354$		0			0
$355$	16.3335			0.866893
$356$		0			0
$357$	−1.46022	−	4.87919i	−0.0772832	−	0.258234i
$358$		0			0
$359$		−	5.45276i		−	0.287786i	−0.989593	−	0.143893i	$-0.954038\pi$
$359$		−	5.45276i		−	0.287786i	0.989593	−	0.143893i	$-0.0459621\pi$
$360$		0			0
$361$	−15.6649			−0.824470
$362$		0			0
$363$	4.05163			0.212656
$364$		0			0
$365$	−11.7289			−0.613917
$366$		0			0
$367$	10.9534			0.571762			0.285881	−	0.958265i	$-0.407714\pi$
$367$	10.9534			0.571762			0.285881	+	0.958265i	$0.407714\pi$
$368$		0			0
$369$		−	10.1364i		−	0.527679i
$370$		0			0
$371$	13.2154	−	3.95505i	0.686110	−	0.205336i
$372$		0			0
$373$	−14.7390			−0.763159			−0.381579	−	0.924336i	$-0.624620\pi$
$373$	−14.7390			−0.763159			−0.381579	+	0.924336i	$0.624620\pi$
$374$		0			0
$375$		−	0.742762i		−	0.0383561i
$376$		0			0
$377$		−	8.94023i		−	0.460445i
$378$		0			0
$379$		−	12.0629i		−	0.619628i	−0.950797	−	0.309814i	$-0.899733\pi$
$379$		−	12.0629i		−	0.619628i	0.950797	−	0.309814i	$-0.100267\pi$
$380$		0			0
$381$			6.38122i			0.326920i
$382$		0			0
$383$	−18.0144			−0.920495			−0.460247	−	0.887791i	$-0.652239\pi$
$383$	−18.0144			−0.920495			−0.460247	+	0.887791i	$0.652239\pi$
$384$		0			0
$385$	−5.96870	+	1.78629i	−0.304193	+	0.0910377i
$386$		0			0
$387$		−	23.6456i		−	1.20197i
$388$		0			0
$389$	2.83811			0.143898			0.0719490	−	0.997408i	$-0.477078\pi$
$389$	2.83811			0.143898			0.0719490	+	0.997408i	$0.477078\pi$
$390$		0			0
$391$	−8.40505			−0.425062
$392$		0			0
$393$	−5.02385			−0.253420
$394$		0			0
$395$	1.05184			0.0529240
$396$		0			0
$397$		−	19.6549i		−	0.986451i	−0.869902	−	0.493225i	$-0.835818\pi$
$397$		−	19.6549i		−	0.986451i	0.869902	−	0.493225i	$-0.164182\pi$
$398$		0			0
$399$	3.43814	−	1.02895i	0.172122	−	0.0515121i
$400$		0			0
$401$	−17.9592			−0.896838			−0.448419	−	0.893823i	$-0.648013\pi$
$401$	−17.9592			−0.896838			−0.448419	+	0.893823i	$0.648013\pi$
$402$		0			0
$403$		−	8.51870i		−	0.424347i
$404$		0			0
$405$			4.33911i			0.215612i
$406$		0			0
$407$			17.6843i			0.876579i
$408$		0			0
$409$		−	28.9337i		−	1.43068i	−0.698777	−	0.715339i	$-0.746272\pi$
$409$		−	28.9337i		−	1.43068i	0.698777	−	0.715339i	$-0.253728\pi$
$410$		0			0
$411$	3.26610			0.161105
$412$		0			0
$413$	−0.991866	+	0.296842i	−0.0488066	+	0.0146066i
$414$		0			0
$415$			1.49920i			0.0735926i
$416$		0			0
$417$	−7.21743			−0.353439
$418$		0			0
$419$	−11.2184			−0.548052			−0.274026	−	0.961722i	$-0.588355\pi$
$419$	−11.2184			−0.548052			−0.274026	+	0.961722i	$0.588355\pi$
$420$		0			0
$421$	−19.9856			−0.974038			−0.487019	−	0.873391i	$-0.661916\pi$
$421$	−19.9856			−0.974038			−0.487019	+	0.873391i	$0.661916\pi$
$422$		0			0
$423$	13.5305			0.657876
$424$		0			0
$425$			2.59165i			0.125713i
$426$		0			0
$427$	10.4808	+	35.0204i	0.507199	+	1.69476i
$428$		0			0
$429$	1.68011			0.0811163
$430$		0			0
$431$			5.19790i			0.250374i	0.992133	+	0.125187i	$0.0399531\pi$
$431$			5.19790i			0.250374i	−0.992133	+	0.125187i	$0.960047\pi$
$432$		0			0
$433$			22.0625i			1.06026i	0.847918	+	0.530128i	$0.177856\pi$
$433$			22.0625i			1.06026i	−0.847918	+	0.530128i	$0.822144\pi$
$434$		0			0
$435$			6.91304i			0.331455i
$436$		0			0
$437$		−	5.92266i		−	0.283319i
$438$		0			0
$439$	−13.1076			−0.625591			−0.312796	−	0.949820i	$-0.601265\pi$
$439$	−13.1076			−0.625591			−0.312796	+	0.949820i	$0.601265\pi$
$440$		0			0
$441$	−14.3205	+	9.41481i	−0.681929	+	0.448324i
$442$		0			0
$443$			36.5977i			1.73881i	0.494099	+	0.869406i	$0.335498\pi$
$443$			36.5977i			1.73881i	−0.494099	+	0.869406i	$0.664502\pi$
$444$		0			0
$445$	9.48370			0.449571
$446$		0			0
$447$	−4.52930			−0.214228
$448$		0			0
$449$	−27.7301			−1.30867			−0.654333	−	0.756207i	$-0.727050\pi$
$449$	−27.7301			−1.30867			−0.654333	+	0.756207i	$0.727050\pi$
$450$		0			0
$451$	−9.74935			−0.459079
$452$		0			0
$453$			11.2895i			0.530428i
$454$		0			0
$455$	2.43473	−	0.728657i	0.114142	−	0.0341600i
$456$		0			0
$457$	9.45098			0.442098			0.221049	−	0.975263i	$-0.429052\pi$
$457$	9.45098			0.442098			0.221049	+	0.975263i	$0.429052\pi$
$458$		0			0
$459$			10.4879i			0.489531i
$460$		0			0
$461$			20.2537i			0.943311i	0.881783	+	0.471655i	$0.156343\pi$
$461$			20.2537i			0.943311i	−0.881783	+	0.471655i	$0.843657\pi$
$462$		0			0
$463$		−	9.65039i		−	0.448492i	−0.974533	−	0.224246i	$-0.928008\pi$
$463$		−	9.65039i		−	0.448492i	0.974533	−	0.224246i	$-0.0719919\pi$
$464$		0			0
$465$			6.58709i			0.305469i
$466$		0			0
$467$	10.8321			0.501250			0.250625	−	0.968084i	$-0.419364\pi$
$467$	10.8321			0.501250			0.250625	+	0.968084i	$0.419364\pi$
$468$		0			0
$469$	0.527939	+	1.76406i	0.0243780	+	0.0814565i
$470$		0			0
$471$		−	15.1658i		−	0.698804i
$472$		0			0
$473$	−22.7427			−1.04571
$474$		0			0
$475$	−1.82622			−0.0837925
$476$		0			0
$477$	−12.7651			−0.584473
$478$		0			0
$479$	11.0606			0.505371			0.252685	−	0.967548i	$-0.418686\pi$
$479$	11.0606			0.505371			0.252685	+	0.967548i	$0.418686\pi$
$480$		0			0
$481$		−	7.21373i		−	0.328918i
$482$		0			0
$483$	−1.82729	−	6.10572i	−0.0831447	−	0.277820i
$484$		0			0
$485$	7.40940			0.336443
$486$		0			0
$487$		−	38.9717i		−	1.76598i	−0.469395	−	0.882988i	$-0.655528\pi$
$487$		−	38.9717i		−	1.76598i	0.469395	−	0.882988i	$-0.344472\pi$
$488$		0			0
$489$		−	7.90087i		−	0.357290i
$490$		0			0
$491$			42.9855i			1.93991i	0.243287	+	0.969954i	$0.421774\pi$
$491$			42.9855i			1.93991i	−0.243287	+	0.969954i	$0.578226\pi$
$492$		0			0
$493$		−	24.1210i		−	1.08635i
$494$		0			0
$495$	5.76532			0.259132
$496$		0			0
$497$	−12.3901	−	41.4001i	−0.555770	−	1.85705i
$498$		0			0
$499$			6.19107i			0.277150i	0.990352	+	0.138575i	$0.0442523\pi$
$499$			6.19107i			0.277150i	−0.990352	+	0.138575i	$0.955748\pi$
$500$		0			0
$501$	−0.470664			−0.0210277
$502$		0			0
$503$	8.44782			0.376670			0.188335	−	0.982105i	$-0.439691\pi$
$503$	8.44782			0.376670			0.188335	+	0.982105i	$0.439691\pi$
$504$		0			0
$505$	−8.27256			−0.368124
$506$		0			0
$507$	8.97056			0.398397
$508$		0			0
$509$			26.8249i			1.18899i	0.804098	+	0.594496i	$0.202649\pi$
$509$			26.8249i			1.18899i	−0.804098	+	0.594496i	$0.797351\pi$
$510$		0			0
$511$	8.89713	+	29.7288i	0.393586	+	1.31513i
$512$		0			0
$513$	−7.39032			−0.326290
$514$		0			0
$515$		−	8.94812i		−	0.394301i
$516$		0			0
$517$		−	13.0139i		−	0.572350i
$518$		0			0
$519$		−	9.36849i		−	0.411231i
$520$		0			0
$521$			13.7907i			0.604184i	0.953279	+	0.302092i	$0.0976849\pi$
$521$			13.7907i			0.604184i	−0.953279	+	0.302092i	$0.902315\pi$
$522$		0			0
$523$	−1.39321			−0.0609210			−0.0304605	−	0.999536i	$-0.509697\pi$
$523$	−1.39321			−0.0609210			−0.0304605	+	0.999536i	$0.509697\pi$
$524$		0			0
$525$	−1.88266	+	0.563434i	−0.0821660	+	0.0245903i
$526$		0			0
$527$		−	22.9837i		−	1.00119i
$528$		0			0
$529$	12.4821			0.542699
$530$		0			0
$531$	0.958068			0.0415766
$532$		0			0
$533$	3.97692			0.172260
$534$		0			0
$535$	−2.83467			−0.122553
$536$		0			0
$537$		−	4.81092i		−	0.207606i
$538$		0			0
$539$	9.05532	+	13.7737i	0.390040	+	0.593275i
$540$		0			0
$541$	1.86528			0.0801946			0.0400973	−	0.999196i	$-0.487233\pi$
$541$	1.86528			0.0801946			0.0400973	+	0.999196i	$0.487233\pi$
$542$		0			0
$543$			12.5225i			0.537393i
$544$		0			0
$545$			6.91377i			0.296153i
$546$		0			0
$547$		−	17.8185i		−	0.761864i	−0.924603	−	0.380932i	$-0.875603\pi$
$547$		−	17.8185i		−	0.761864i	0.924603	−	0.380932i	$-0.124397\pi$
$548$		0			0
$549$		−	33.8271i		−	1.44370i
$550$		0			0
$551$	16.9970			0.724095
$552$		0			0
$553$	−0.797894	−	2.66608i	−0.0339299	−	0.113373i
$554$		0			0
$555$			5.57802i			0.236774i
$556$		0			0
$557$	5.49296			0.232744			0.116372	−	0.993206i	$-0.462873\pi$
$557$	5.49296			0.232744			0.116372	+	0.993206i	$0.462873\pi$
$558$		0			0
$559$	9.27712			0.392381
$560$		0			0
$561$	4.53297			0.191382
$562$		0			0
$563$	3.78670			0.159590			0.0797952	−	0.996811i	$-0.474573\pi$
$563$	3.78670			0.159590			0.0797952	+	0.996811i	$0.474573\pi$
$564$		0			0
$565$		−	0.934659i		−	0.0393214i
$566$		0			0
$567$	10.9982	−	3.29151i	0.461883	−	0.138230i
$568$		0			0
$569$	−31.6816			−1.32816			−0.664081	−	0.747661i	$-0.731177\pi$
$569$	−31.6816			−1.32816			−0.664081	+	0.747661i	$0.731177\pi$
$570$		0			0
$571$		−	12.2001i		−	0.510557i	−0.966868	−	0.255279i	$-0.917833\pi$
$571$		−	12.2001i		−	0.510557i	0.966868	−	0.255279i	$-0.0821672\pi$
$572$		0			0
$573$			8.52813i			0.356268i
$574$		0			0
$575$			3.24313i			0.135248i
$576$		0			0
$577$			29.0601i			1.20979i	0.796306	+	0.604894i	$0.206785\pi$
$577$			29.0601i			1.20979i	−0.796306	+	0.604894i	$0.793215\pi$
$578$		0			0
$579$	12.3520			0.513332
$580$		0			0
$581$	3.79997	−	1.13724i	0.157649	−	0.0471807i
$582$		0			0
$583$			12.2777i			0.508489i
$584$		0			0
$585$	−2.35177			−0.0972337
$586$		0			0
$587$	−43.5786			−1.79868			−0.899341	−	0.437248i	$-0.855953\pi$
$587$	−43.5786			−1.79868			−0.899341	+	0.437248i	$0.855953\pi$
$588$		0			0
$589$	16.1956			0.667327
$590$		0			0
$591$	9.88574			0.406645
$592$		0			0
$593$		−	21.1056i		−	0.866704i	−0.901225	−	0.433352i	$-0.857331\pi$
$593$		−	21.1056i		−	0.866704i	0.901225	−	0.433352i	$-0.142669\pi$
$594$		0			0
$595$	6.56898	−	1.96594i	0.269302	−	0.0805955i
$596$		0			0
$597$	14.1988			0.581117
$598$		0			0
$599$		−	36.0154i		−	1.47155i	−0.677228	−	0.735774i	$-0.736819\pi$
$599$		−	36.0154i		−	1.47155i	0.677228	−	0.735774i	$-0.263181\pi$
$600$		0			0
$601$			19.8990i			0.811697i	0.913940	+	0.405849i	$0.133024\pi$
$601$			19.8990i			0.811697i	−0.913940	+	0.405849i	$0.866976\pi$
$602$		0			0
$603$		−	1.70395i		−	0.0693900i
$604$		0			0
$605$			5.45482i			0.221770i
$606$		0			0
$607$	−39.8202			−1.61625			−0.808125	−	0.589011i	$-0.799517\pi$
$607$	−39.8202			−1.61625			−0.808125	+	0.589011i	$0.799517\pi$
$608$		0			0
$609$	17.5223	−	5.24400i	0.710040	−	0.212498i
$610$		0			0
$611$			5.30858i			0.214762i
$612$		0			0
$613$	28.5657			1.15376			0.576878	−	0.816830i	$-0.304271\pi$
$613$	28.5657			1.15376			0.576878	+	0.816830i	$0.304271\pi$
$614$		0			0
$615$	−3.07516			−0.124002
$616$		0			0
$617$	−37.3055			−1.50186			−0.750931	−	0.660381i	$-0.770395\pi$
$617$	−37.3055			−1.50186			−0.750931	+	0.660381i	$0.770395\pi$
$618$		0			0
$619$	−37.9012			−1.52338			−0.761690	−	0.647942i	$-0.775630\pi$
$619$	−37.9012			−1.52338			−0.761690	+	0.647942i	$0.775630\pi$
$620$		0			0
$621$			13.1243i			0.526660i
$622$		0			0
$623$	−7.19402	−	24.0381i	−0.288222	−	0.963066i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			3.19418i			0.127563i
$628$		0			0
$629$		−	19.4628i		−	0.776034i
$630$		0			0
$631$		−	19.6267i		−	0.781328i	−0.920533	−	0.390664i	$-0.872246\pi$
$631$		−	19.6267i		−	0.781328i	0.920533	−	0.390664i	$-0.127754\pi$
$632$		0			0
$633$		−	7.16410i		−	0.284748i
$634$		0			0
$635$	−8.59120			−0.340931
$636$		0			0
$637$	−3.69382	−	5.61852i	−0.146354	−	0.222614i
$638$		0			0
$639$			39.9894i			1.58196i
$640$		0			0
$641$	−41.2938			−1.63101			−0.815504	−	0.578751i	$-0.803540\pi$
$641$	−41.2938			−1.63101			−0.815504	+	0.578751i	$0.803540\pi$
$642$		0			0
$643$	−41.2061			−1.62501			−0.812504	−	0.582955i	$-0.801896\pi$
$643$	−41.2061			−1.62501			−0.812504	+	0.582955i	$0.801896\pi$
$644$		0			0
$645$	−7.17354			−0.282458
$646$		0			0
$647$	21.3489			0.839312			0.419656	−	0.907683i	$-0.362151\pi$
$647$	21.3489			0.839312			0.419656	+	0.907683i	$0.362151\pi$
$648$		0			0
$649$		−	0.921486i		−	0.0361715i
$650$		0			0
$651$	16.6961	−	4.99675i	0.654373	−	0.195838i
$652$		0			0
$653$	−5.11170			−0.200036			−0.100018	−	0.994986i	$-0.531890\pi$
$653$	−5.11170			−0.200036			−0.100018	+	0.994986i	$0.531890\pi$
$654$		0			0
$655$		−	6.76374i		−	0.264281i
$656$		0			0
$657$		−	28.7158i		−	1.12031i
$658$		0			0
$659$		−	23.1212i		−	0.900674i	−0.892859	−	0.450337i	$-0.851304\pi$
$659$		−	23.1212i		−	0.900674i	0.892859	−	0.450337i	$-0.148696\pi$
$660$		0			0
$661$			19.4732i			0.757419i	0.925516	+	0.378709i	$0.123632\pi$
$661$			19.4732i			0.757419i	−0.925516	+	0.378709i	$0.876368\pi$
$662$		0			0
$663$	−1.84907			−0.0718121
$664$		0			0
$665$	1.38531	+	4.62886i	0.0537199	+	0.179500i
$666$		0			0
$667$		−	30.1845i		−	1.16875i
$668$		0			0
$669$	−18.4381			−0.712856
$670$		0			0
$671$	−32.5354			−1.25602
$672$		0			0
$673$	12.7564			0.491723			0.245862	−	0.969305i	$-0.420929\pi$
$673$	12.7564			0.491723			0.245862	+	0.969305i	$0.420929\pi$
$674$		0			0
$675$	4.04679			0.155761
$676$		0			0
$677$		−	30.2783i		−	1.16369i	−0.813300	−	0.581845i	$-0.802331\pi$
$677$		−	30.2783i		−	1.16369i	0.813300	−	0.581845i	$-0.197669\pi$
$678$		0			0
$679$	−5.62052	−	18.7804i	−0.215696	−	0.720726i
$680$		0			0
$681$	−12.8914			−0.494000
$682$		0			0
$683$			7.05360i			0.269899i	0.990852	+	0.134949i	$0.0430872\pi$
$683$			7.05360i			0.269899i	−0.990852	+	0.134949i	$0.956913\pi$
$684$		0			0
$685$			4.39724i			0.168010i
$686$		0			0
$687$			4.55613i			0.173827i
$688$		0			0
$689$		−	5.00827i		−	0.190800i
$690$		0			0
$691$	−28.5473			−1.08599			−0.542995	−	0.839736i	$-0.682710\pi$
$691$	−28.5473			−1.08599			−0.542995	+	0.839736i	$0.682710\pi$
$692$		0			0
$693$	−4.37338	−	14.6132i	−0.166131	−	0.555110i
$694$		0			0
$695$		−	9.71702i		−	0.368587i
$696$		0			0
$697$	10.7298			0.406422
$698$		0			0
$699$	−10.5133			−0.397650
$700$		0			0
$701$	−46.3708			−1.75140			−0.875700	−	0.482855i	$-0.839600\pi$
$701$	−46.3708			−1.75140			−0.875700	+	0.482855i	$0.839600\pi$
$702$		0			0
$703$	13.7146			0.517255
$704$		0			0
$705$		−	4.10486i		−	0.154598i
$706$		0			0
$707$	6.27529	+	20.9682i	0.236006	+	0.788592i
$708$		0			0
$709$	−9.88597			−0.371275			−0.185638	−	0.982618i	$-0.559435\pi$
$709$	−9.88597			−0.371275			−0.185638	+	0.982618i	$0.559435\pi$
$710$		0			0
$711$			2.57523i			0.0965788i
$712$		0			0
$713$		−	28.7613i		−	1.07712i
$714$		0			0
$715$			2.26197i			0.0845929i
$716$		0			0
$717$		−	0.275163i		−	0.0102761i
$718$		0			0
$719$	16.6211			0.619864			0.309932	−	0.950759i	$-0.399694\pi$
$719$	16.6211			0.619864			0.309932	+	0.950759i	$0.399694\pi$
$720$		0			0
$721$	−22.6806	+	6.78775i	−0.844669	+	0.252789i
$722$		0			0
$723$			14.7211i			0.547484i
$724$		0			0
$725$	−9.30721			−0.345661
$726$		0			0
$727$	−20.8209			−0.772205			−0.386103	−	0.922456i	$-0.626179\pi$
$727$	−20.8209			−0.772205			−0.386103	+	0.922456i	$0.626179\pi$
$728$		0			0
$729$	−1.60606			−0.0594838
$730$		0			0
$731$	25.0299			0.925765
$732$		0			0
$733$			23.9669i			0.885238i	0.896710	+	0.442619i	$0.145951\pi$
$733$			23.9669i			0.885238i	−0.896710	+	0.442619i	$0.854049\pi$
$734$		0			0
$735$	2.85625	+	4.34453i	0.105354	+	0.160250i
$736$		0			0
$737$	−1.63888			−0.0603690
$738$		0			0
$739$			43.0899i			1.58509i	0.609816	+	0.792543i	$0.291243\pi$
$739$			43.0899i			1.58509i	−0.609816	+	0.792543i	$0.708757\pi$
$740$		0			0
$741$		−	1.30296i		−	0.0478654i
$742$		0			0
$743$		−	1.13073i		−	0.0414824i	−0.999785	−	0.0207412i	$-0.993397\pi$
$743$		−	1.13073i		−	0.0414824i	0.999785	−	0.0207412i	$-0.00660260\pi$
$744$		0			0
$745$		−	6.09792i		−	0.223410i
$746$		0			0
$747$	−3.67049			−0.134296
$748$		0			0
$749$	2.15029	+	7.18496i	0.0785697	+	0.262533i
$750$		0			0
$751$		−	35.5478i		−	1.29716i	−0.761148	−	0.648578i	$-0.775364\pi$
$751$		−	35.5478i		−	1.29716i	0.761148	−	0.648578i	$-0.224636\pi$
$752$		0			0
$753$	−22.2297			−0.810096
$754$		0			0
$755$	−15.1994			−0.553162
$756$		0			0
$757$	52.7958			1.91890			0.959449	−	0.281883i	$-0.0909590\pi$
$757$	52.7958			1.91890			0.959449	+	0.281883i	$0.0909590\pi$
$758$		0			0
$759$	5.67247			0.205898
$760$		0			0
$761$			17.0143i			0.616769i	0.951262	+	0.308385i	$0.0997884\pi$
$761$			17.0143i			0.616769i	−0.951262	+	0.308385i	$0.900212\pi$
$762$		0			0
$763$	17.5242	−	5.24456i	0.634417	−	0.189866i
$764$		0			0
$765$	−6.34514			−0.229409
$766$		0			0
$767$			0.375889i			0.0135726i
$768$		0			0
$769$		−	39.7806i		−	1.43452i	−0.696804	−	0.717262i	$-0.745395\pi$
$769$		−	39.7806i		−	1.43452i	0.696804	−	0.717262i	$-0.254605\pi$
$770$		0			0
$771$		−	18.4873i		−	0.665805i
$772$		0			0
$773$			43.7797i			1.57465i	0.616541	+	0.787323i	$0.288533\pi$
$773$			43.7797i			1.57465i	−0.616541	+	0.787323i	$0.711467\pi$
$774$		0			0
$775$	−8.86838			−0.318562
$776$		0			0
$777$	14.1385	−	4.23130i	0.507215	−	0.151797i
$778$		0			0
$779$			7.56084i			0.270895i
$780$		0			0
$781$	38.4625			1.37630
$782$		0			0
$783$	−37.6643			−1.34601
$784$		0			0
$785$	20.4181			0.728755
$786$		0			0
$787$	48.2086			1.71845			0.859226	−	0.511596i	$-0.170945\pi$
$787$	48.2086			1.71845			0.859226	+	0.511596i	$0.170945\pi$
$788$		0			0
$789$		−	20.6813i		−	0.736275i
$790$		0			0
$791$	−2.36906	+	0.709002i	−0.0842340	+	0.0252092i
$792$		0			0
$793$	13.2717			0.471293
$794$		0			0
$795$			3.87265i			0.137349i
$796$		0			0
$797$		−	15.3809i		−	0.544819i	−0.962181	−	0.272409i	$-0.912180\pi$
$797$		−	15.3809i		−	0.544819i	0.962181	−	0.272409i	$-0.0878205\pi$
$798$		0			0
$799$			14.3227i			0.506700i
$800$		0			0
$801$			23.2190i			0.820403i
$802$		0			0
$803$	−27.6194			−0.974666
$804$		0			0
$805$	8.22029	−	2.46013i	0.289727	−	0.0867083i
$806$		0			0
$807$		−	12.1510i		−	0.427735i
$808$		0			0
$809$	32.8146			1.15370			0.576850	−	0.816850i	$-0.304282\pi$
$809$	32.8146			1.15370			0.576850	+	0.816850i	$0.304282\pi$
$810$		0			0
$811$	8.54594			0.300089			0.150044	−	0.988679i	$-0.452058\pi$
$811$	8.54594			0.300089			0.150044	+	0.988679i	$0.452058\pi$
$812$		0			0
$813$	−9.53715			−0.334482
$814$		0			0
$815$	10.6372			0.372603
$816$		0			0
$817$			17.6375i			0.617057i
$818$		0			0
$819$	1.78397	+	5.96097i	0.0623371	+	0.208293i
$820$		0			0
$821$	3.87902			0.135379			0.0676894	−	0.997706i	$-0.478437\pi$
$821$	3.87902			0.135379			0.0676894	+	0.997706i	$0.478437\pi$
$822$		0			0
$823$			33.5085i			1.16803i	0.811741	+	0.584017i	$0.198520\pi$
$823$			33.5085i			1.16803i	−0.811741	+	0.584017i	$0.801480\pi$
$824$		0			0
$825$		−	1.74907i		−	0.0608948i
$826$		0			0
$827$			14.6421i			0.509154i	0.967052	+	0.254577i	$0.0819363\pi$
$827$			14.6421i			0.509154i	−0.967052	+	0.254577i	$0.918064\pi$
$828$		0			0
$829$		−	22.7281i		−	0.789381i	−0.918814	−	0.394691i	$-0.870852\pi$
$829$		−	22.7281i		−	0.789381i	0.918814	−	0.394691i	$-0.129148\pi$
$830$		0			0
$831$	−15.0119			−0.520756
$832$		0			0
$833$	−9.96602	−	15.1589i	−0.345302	−	0.525226i
$834$		0			0
$835$		−	0.633668i		−	0.0219290i
$836$		0			0
$837$	−35.8885			−1.24049
$838$		0			0
$839$	−42.7485			−1.47584			−0.737921	−	0.674887i	$-0.764192\pi$
$839$	−42.7485			−1.47584			−0.737921	+	0.674887i	$0.764192\pi$
$840$		0			0
$841$	57.6241			1.98704
$842$		0			0
$843$	6.57105			0.226319
$844$		0			0
$845$			12.0773i			0.415472i
$846$		0			0
$847$	13.8262	−	4.13785i	0.475074	−	0.142178i
$848$		0			0
$849$	5.30290			0.181995
$850$		0			0
$851$		−	24.3554i		−	0.834892i
$852$		0			0
$853$			42.5977i			1.45852i	0.684237	+	0.729260i	$0.260135\pi$
$853$			42.5977i			1.45852i	−0.684237	+	0.729260i	$0.739865\pi$
$854$		0			0
$855$		−	4.47113i		−	0.152910i
$856$		0			0
$857$		−	22.7085i		−	0.775706i	−0.921721	−	0.387853i	$-0.873217\pi$
$857$		−	22.7085i		−	0.775706i	0.921721	−	0.387853i	$-0.126783\pi$
$858$		0			0
$859$	−41.9627			−1.43175			−0.715875	−	0.698228i	$-0.753972\pi$
$859$	−41.9627			−1.43175			−0.715875	+	0.698228i	$0.753972\pi$
$860$		0			0
$861$	2.33271	+	7.79452i	0.0794986	+	0.265637i
$862$		0			0
$863$			51.1857i			1.74238i	0.490944	+	0.871191i	$0.336652\pi$
$863$			51.1857i			1.74238i	−0.490944	+	0.871191i	$0.663348\pi$
$864$		0			0
$865$	12.6130			0.428857
$866$		0			0
$867$	7.63810			0.259403
$868$		0			0
$869$	2.47690			0.0840232
$870$		0			0
$871$	0.668528			0.0226522
$872$		0			0
$873$			18.1405i			0.613962i
$874$		0			0
$875$	−0.758567	−	2.53467i	−0.0256442	−	0.0856876i
$876$		0			0
$877$	14.5202			0.490313			0.245156	−	0.969484i	$-0.421161\pi$
$877$	14.5202			0.490313			0.245156	+	0.969484i	$0.421161\pi$
$878$		0			0
$879$		−	21.9316i		−	0.739736i
$880$		0			0
$881$			52.5955i			1.77199i	0.463697	+	0.885994i	$0.346523\pi$
$881$			52.5955i			1.77199i	−0.463697	+	0.885994i	$0.653477\pi$
$882$		0			0
$883$		−	4.02334i		−	0.135396i	−0.997706	−	0.0676981i	$-0.978435\pi$
$883$		−	4.02334i		−	0.135396i	0.997706	−	0.0676981i	$-0.0215655\pi$
$884$		0			0
$885$		−	0.290657i		−	0.00977032i
$886$		0			0
$887$	0.498760			0.0167467			0.00837336	−	0.999965i	$-0.497335\pi$
$887$	0.498760			0.0167467			0.00837336	+	0.999965i	$0.497335\pi$
$888$		0			0
$889$	6.51700	+	21.7759i	0.218573	+	0.730340i
$890$		0			0
$891$			10.2178i			0.342310i
$892$		0			0
$893$	−10.0926			−0.337734
$894$		0			0
$895$	6.47707			0.216505
$896$		0			0
$897$	−2.31389			−0.0772587
$898$		0			0
$899$	82.5398			2.75286
$900$		0			0
$901$		−	13.5124i		−	0.450165i
$902$		0			0
$903$	5.44161	+	18.1826i	0.181085	+	0.605079i
$904$		0			0
$905$	−16.8594			−0.560426
$906$		0			0
$907$		−	44.9774i		−	1.49345i	−0.665133	−	0.746725i	$-0.731625\pi$
$907$		−	44.9774i		−	1.49345i	0.665133	−	0.746725i	$-0.268375\pi$
$908$		0			0
$909$		−	20.2537i		−	0.671774i
$910$		0			0
$911$		−	53.8728i		−	1.78489i	−0.451160	−	0.892443i	$-0.648990\pi$
$911$		−	53.8728i		−	1.78489i	0.451160	−	0.892443i	$-0.351010\pi$
$912$		0			0
$913$			3.53034i			0.116837i
$914$		0			0
$915$	−10.2624			−0.339264
$916$		0			0
$917$	−17.1439	+	5.13075i	−0.566141	+	0.169432i
$918$		0			0
$919$			53.7071i			1.77163i	0.464035	+	0.885817i	$0.346401\pi$
$919$			53.7071i			1.77163i	−0.464035	+	0.885817i	$0.653599\pi$
$920$		0			0
$921$	−13.7694			−0.453717
$922$		0			0
$923$	−15.6895			−0.516426
$924$		0			0
$925$	−7.50984			−0.246922
$926$		0			0
$927$	21.9077			0.719544
$928$		0			0
$929$			23.3690i			0.766711i	0.923601	+	0.383356i	$0.125232\pi$
$929$			23.3690i			0.766711i	−0.923601	+	0.383356i	$0.874768\pi$
$930$		0			0
$931$	10.6818	−	7.02261i	0.350082	−	0.230157i
$932$		0			0
$933$	20.1068			0.658266
$934$		0			0
$935$			6.10286i			0.199585i
$936$		0			0
$937$		−	9.35564i		−	0.305635i	−0.988254	−	0.152818i	$-0.951165\pi$
$937$		−	9.35564i		−	0.305635i	0.988254	−	0.152818i	$-0.0488347\pi$
$938$		0			0
$939$		−	7.38814i		−	0.241103i
$940$		0			0
$941$		−	9.54945i		−	0.311303i	−0.987812	−	0.155652i	$-0.950252\pi$
$941$		−	9.54945i		−	0.311303i	0.987812	−	0.155652i	$-0.0497477\pi$
$942$		0			0
$943$	13.4271			0.437247
$944$		0			0
$945$	−3.06976	−	10.2573i	−0.0998594	−	0.333670i
$946$		0			0
$947$		−	38.8445i		−	1.26228i	−0.775670	−	0.631139i	$-0.782588\pi$
$947$		−	38.8445i		−	1.26228i	0.775670	−	0.631139i	$-0.217412\pi$
$948$		0			0
$949$	11.2664			0.365723
$950$		0			0
$951$	−4.55179			−0.147602
$952$		0			0
$953$	−6.03716			−0.195563			−0.0977814	−	0.995208i	$-0.531175\pi$
$953$	−6.03716			−0.195563			−0.0977814	+	0.995208i	$0.531175\pi$
$954$		0			0
$955$	−11.4816			−0.371537
$956$		0			0
$957$			16.2790i			0.526224i
$958$		0			0
$959$	11.1456	−	3.33560i	0.359909	−	0.107712i
$960$		0			0
$961$	47.6481			1.53704
$962$		0			0
$963$		−	6.94013i		−	0.223643i
$964$		0			0
$965$			16.6299i			0.535334i
$966$		0			0
$967$			38.8147i			1.24820i	0.781346	+	0.624099i	$0.214534\pi$
$967$			38.8147i			1.24820i	−0.781346	+	0.624099i	$0.785466\pi$
$968$		0			0
$969$		−	3.51542i		−	0.112932i
$970$		0			0
$971$	53.3500			1.71208			0.856041	−	0.516907i	$-0.172917\pi$
$971$	53.3500			1.71208			0.856041	+	0.516907i	$0.172917\pi$
$972$		0			0
$973$	−24.6295	+	7.37101i	−0.789585	+	0.236304i
$974$		0			0
$975$			0.713475i			0.0228495i
$976$		0			0
$977$	58.2222			1.86269			0.931347	−	0.364132i	$-0.118634\pi$
$977$	58.2222			1.86269			0.931347	+	0.364132i	$0.118634\pi$
$978$		0			0
$979$	22.3324			0.713747
$980$		0			0
$981$	−16.9270			−0.540438
$982$		0			0
$983$	31.4443			1.00292			0.501459	−	0.865182i	$-0.332797\pi$
$983$	31.4443			1.00292			0.501459	+	0.865182i	$0.332797\pi$
$984$		0			0
$985$			13.3094i			0.424074i
$986$		0			0
$987$	−10.4045	+	3.11381i	−0.331179	+	0.0991137i
$988$		0			0
$989$	31.3220			0.995980
$990$		0			0
$991$			33.1199i			1.05209i	0.850457	+	0.526044i	$0.176325\pi$
$991$			33.1199i			1.05209i	−0.850457	+	0.526044i	$0.823675\pi$
$992$		0			0
$993$		−	1.77990i		−	0.0564833i
$994$		0			0
$995$			19.1162i			0.606024i
$996$		0			0
$997$		−	1.65823i		−	0.0525168i	−0.999655	−	0.0262584i	$-0.991641\pi$
$997$		−	1.65823i		−	0.0525168i	0.999655	−	0.0262584i	$-0.00835927\pi$
$998$		0			0
$999$	−30.3907			−0.961521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.g.1791.10		16
4.3	odd	2		2240.2.k.f.1791.8		16
7.6	odd	2		2240.2.k.f.1791.7		16
8.3	odd	2		1120.2.k.a.671.9	✓	16
8.5	even	2		1120.2.k.b.671.7	yes	16
28.27	even	2	inner	2240.2.k.g.1791.9		16
56.13	odd	2		1120.2.k.a.671.10	yes	16
56.27	even	2		1120.2.k.b.671.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.k.a.671.9	✓	16	8.3	odd	2
1120.2.k.a.671.10	yes	16	56.13	odd	2
1120.2.k.b.671.7	yes	16	8.5	even	2
1120.2.k.b.671.8	yes	16	56.27	even	2
2240.2.k.f.1791.7		16	7.6	odd	2
2240.2.k.f.1791.8		16	4.3	odd	2
2240.2.k.g.1791.9		16	28.27	even	2	inner
2240.2.k.g.1791.10		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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.742762 q^{3} +1.00000i q^{5} +(2.53467 - 0.758567i) q^{7} -2.44831 q^{9} +O(q^{10})\)	\(q+0.742762 q^{3} +1.00000i q^{5} +(2.53467 - 0.758567i) q^{7} -2.44831 q^{9} +2.35482i q^{11} -0.960570i q^{13} +0.742762i q^{15} -2.59165i q^{17} +1.82622 q^{19} +(1.88266 - 0.563434i) q^{21} -3.24313i q^{23} -1.00000 q^{25} -4.04679 q^{27} +9.30721 q^{29} +8.86838 q^{31} +1.74907i q^{33} +(0.758567 + 2.53467i) q^{35} +7.50984 q^{37} -0.713475i q^{39} +4.14017i q^{41} +9.65793i q^{43} -2.44831i q^{45} -5.52648 q^{47} +(5.84915 - 3.84544i) q^{49} -1.92497i q^{51} +5.21385 q^{53} -2.35482 q^{55} +1.35644 q^{57} -0.391319 q^{59} +13.8165i q^{61} +(-6.20566 + 1.85720i) q^{63} +0.960570 q^{65} +0.695969i q^{67} -2.40888i q^{69} -16.3335i q^{71} +11.7289i q^{73} -0.742762 q^{75} +(1.78629 + 5.96870i) q^{77} -1.05184i q^{79} +4.33911 q^{81} +1.49920 q^{83} +2.59165 q^{85} +6.91304 q^{87} -9.48370i q^{89} +(-0.728657 - 2.43473i) q^{91} +6.58709 q^{93} +1.82622i q^{95} -7.40940i q^{97} -5.76532i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 4 * q^7 + 16 * q^9	\( 16 q + 4 q^{7} + 16 q^{9} + 8 q^{19} - 4 q^{21} - 16 q^{25} - 48 q^{27} - 8 q^{29} - 16 q^{37} + 8 q^{47} - 4 q^{49} - 16 q^{53} - 8 q^{55} + 16 q^{57} + 8 q^{59} + 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} - 64 q^{83} + 16 q^{87} + 64 q^{91} + 48 q^{93}+O(q^{100}) \) 16 * q + 4 * q^7 + 16 * q^9 + 8 * q^19 - 4 * q^21 - 16 * q^25 - 48 * q^27 - 8 * q^29 - 16 * q^37 + 8 * q^47 - 4 * q^49 - 16 * q^53 - 8 * q^55 + 16 * q^57 + 8 * q^59 + 60 * q^63 + 8 * q^65 + 8 * q^77 + 40 * q^81 - 64 * q^83 + 16 * q^87 + 64 * q^91 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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