LMFDB - Embedded newform 4320.2.d.i.3889.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(3889,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4320.3889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.d (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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	20.0.71899280742356107798058983489536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3x^{18} + 8x^{16} - 24x^{14} + 56x^{12} - 92x^{10} + 224x^{8} - 384x^{6} + 512x^{4} - 768x^{2} + 1024 $ x^20 - 3x^18 + 8x^16 - 24x^14 + 56x^12 - 92x^10 + 224x^8 - 384x^6 + 512x^4 - 768*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3889.7
Root			$1.35662 - 0.399488i$ of defining polynomial
Character	$\chi$	$=$	4320.3889
Dual form			4320.2.d.i.3889.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.49578 - 1.66212i) q^{5} -1.43534i q^{7} +0.0984392i q^{11} +1.92218 q^{13} +2.03741i q^{17} -0.999558i q^{19} +4.30361i q^{23} +(-0.525254 + 4.97233i) q^{25} +3.56681i q^{29} -1.42267 q^{31} +(-2.38570 + 2.14696i) q^{35} +8.27330 q^{37} +5.11958 q^{41} +5.50577 q^{43} +9.00709i q^{47} +4.93980 q^{49} +4.93699 q^{53} +(0.163617 - 0.147244i) q^{55} -6.27246i q^{59} -12.8449i q^{61} +(-2.87517 - 3.19489i) q^{65} -1.89027 q^{67} -12.3259 q^{71} -5.62966i q^{73} +0.141294 q^{77} -2.48816 q^{79} -2.26930 q^{83} +(3.38640 - 3.04752i) q^{85} +13.3550 q^{89} -2.75898i q^{91} +(-1.66138 + 1.49512i) q^{95} +14.7457i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{13} + 20 q^{25} - 12 q^{31} - 32 q^{37} - 12 q^{43} - 52 q^{49} + 28 q^{55} - 36 q^{79} - 44 q^{85}+O(q^{100}) $ 20 * q - 4 * q^13 + 20 * q^25 - 12 * q^31 - 32 * q^37 - 12 * q^43 - 52 * q^49 + 28 * q^55 - 36 * q^79 - 44 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.49578	−	1.66212i	−0.668935	−	0.743321i
$5$	−1.49578	−	1.66212i	−0.668935	−	0.743321i
$6$		0			0
$7$		−	1.43534i		−	0.542507i	−0.962508	−	0.271254i	$-0.912562\pi$
$7$		−	1.43534i		−	0.542507i	0.962508	−	0.271254i	$-0.0874382\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.0984392i			0.0296805i	0.999890	+	0.0148403i	$0.00472398\pi$
$11$			0.0984392i			0.0296805i	−0.999890	+	0.0148403i	$0.995276\pi$
$12$		0			0
$13$	1.92218			0.533118			0.266559	−	0.963819i	$-0.414113\pi$
$13$	1.92218			0.533118			0.266559	+	0.963819i	$0.414113\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.03741i			0.494144i	0.968997	+	0.247072i	$0.0794684\pi$
$17$			2.03741i			0.494144i	−0.968997	+	0.247072i	$0.920532\pi$
$18$		0			0
$19$		−	0.999558i		−	0.229314i	−0.993405	−	0.114657i	$-0.963423\pi$
$19$		−	0.999558i		−	0.229314i	0.993405	−	0.114657i	$-0.0365770\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.30361i			0.897364i	0.893691	+	0.448682i	$0.148106\pi$
$23$			4.30361i			0.897364i	−0.893691	+	0.448682i	$0.851894\pi$
$24$		0			0
$25$	−0.525254	+	4.97233i	−0.105051	+	0.994467i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.56681i			0.662339i	0.943571	+	0.331170i	$0.107443\pi$
$29$			3.56681i			0.662339i	−0.943571	+	0.331170i	$0.892557\pi$
$30$		0			0
$31$	−1.42267			−0.255519			−0.127760	−	0.991805i	$-0.540779\pi$
$31$	−1.42267			−0.255519			−0.127760	+	0.991805i	$0.540779\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.38570	+	2.14696i	−0.403257	+	0.362902i
$36$		0			0
$37$	8.27330			1.36012			0.680061	−	0.733156i	$-0.261953\pi$
$37$	8.27330			1.36012			0.680061	+	0.733156i	$0.261953\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.11958			0.799544			0.399772	−	0.916615i	$-0.369089\pi$
$41$	5.11958			0.799544			0.399772	+	0.916615i	$0.369089\pi$
$42$		0			0
$43$	5.50577			0.839623			0.419811	−	0.907611i	$-0.362096\pi$
$43$	5.50577			0.839623			0.419811	+	0.907611i	$0.362096\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			9.00709i			1.31382i	0.753969	+	0.656910i	$0.228137\pi$
$47$			9.00709i			1.31382i	−0.753969	+	0.656910i	$0.771863\pi$
$48$		0			0
$49$	4.93980			0.705686
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.93699			0.678148			0.339074	−	0.940760i	$-0.389886\pi$
$53$	4.93699			0.678148			0.339074	+	0.940760i	$0.389886\pi$
$54$		0			0
$55$	0.163617	−	0.147244i	0.0220622	−	0.0198544i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	6.27246i		−	0.816605i	−0.912847	−	0.408303i	$-0.866121\pi$
$59$		−	6.27246i		−	0.816605i	0.912847	−	0.408303i	$-0.133879\pi$
$60$		0			0
$61$		−	12.8449i		−	1.64463i	−0.569035	−	0.822313i	$-0.692683\pi$
$61$		−	12.8449i		−	1.64463i	0.569035	−	0.822313i	$-0.307317\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.87517	−	3.19489i	−0.356621	−	0.396277i
$66$		0			0
$67$	−1.89027			−0.230933			−0.115466	−	0.993311i	$-0.536836\pi$
$67$	−1.89027			−0.230933			−0.115466	+	0.993311i	$0.536836\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.3259			−1.46281			−0.731406	−	0.681942i	$-0.761136\pi$
$71$	−12.3259			−1.46281			−0.731406	+	0.681942i	$0.761136\pi$
$72$		0			0
$73$		−	5.62966i		−	0.658902i	−0.944173	−	0.329451i	$-0.893136\pi$
$73$		−	5.62966i		−	0.658902i	0.944173	−	0.329451i	$-0.106864\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.141294			0.0161019
$78$		0			0
$79$	−2.48816			−0.279939			−0.139970	−	0.990156i	$-0.544701\pi$
$79$	−2.48816			−0.279939			−0.139970	+	0.990156i	$0.544701\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.26930			−0.249088			−0.124544	−	0.992214i	$-0.539747\pi$
$83$	−2.26930			−0.249088			−0.124544	+	0.992214i	$0.539747\pi$
$84$		0			0
$85$	3.38640	−	3.04752i	0.367307	−	0.330550i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.3550			1.41563			0.707815	−	0.706398i	$-0.249681\pi$
$89$	13.3550			1.41563			0.707815	+	0.706398i	$0.249681\pi$
$90$		0			0
$91$		−	2.75898i		−	0.289220i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.66138	+	1.49512i	−0.170454	+	0.153396i
$96$		0			0
$97$			14.7457i			1.49720i	0.663024	+	0.748598i	$0.269273\pi$
$97$			14.7457i			1.49720i	−0.663024	+	0.748598i	$0.730727\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			13.6218i			1.35542i	0.735328	+	0.677711i	$0.237028\pi$
$101$			13.6218i			1.35542i	−0.735328	+	0.677711i	$0.762972\pi$
$102$		0			0
$103$			10.8458i			1.06867i	0.845272	+	0.534336i	$0.179438\pi$
$103$			10.8458i			1.06867i	−0.845272	+	0.534336i	$0.820562\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.4915			−1.20760			−0.603798	−	0.797137i	$-0.706347\pi$
$107$	−12.4915			−1.20760			−0.603798	+	0.797137i	$0.706347\pi$
$108$		0			0
$109$		−	16.5739i		−	1.58749i	−0.608250	−	0.793745i	$-0.708128\pi$
$109$		−	16.5739i		−	1.58749i	0.608250	−	0.793745i	$-0.291872\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.15123i		−	0.860875i	−0.902620	−	0.430438i	$-0.858359\pi$
$113$		−	9.15123i		−	0.860875i	0.902620	−	0.430438i	$-0.141641\pi$
$114$		0			0
$115$	7.15309	−	6.43727i	0.667029	−	0.600279i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.92437			0.268077
$120$		0			0
$121$	10.9903			0.999119
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.05026	−	6.56451i	0.809480	−	0.587148i
$126$		0			0
$127$		−	10.9648i		−	0.972972i	−0.873688	−	0.486486i	$-0.838278\pi$
$127$		−	10.9648i		−	0.972972i	0.873688	−	0.486486i	$-0.161722\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	22.1895i		−	1.93870i	−0.245678	−	0.969352i	$-0.579011\pi$
$131$		−	22.1895i		−	1.93870i	0.245678	−	0.969352i	$-0.420989\pi$
$132$		0			0
$133$	−1.43471			−0.124405
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	10.7851i		−	0.921431i	−0.887548	−	0.460716i	$-0.847593\pi$
$137$		−	10.7851i		−	0.921431i	0.887548	−	0.460716i	$-0.152407\pi$
$138$		0			0
$139$		−	15.5743i		−	1.32100i	−0.750827	−	0.660499i	$-0.770345\pi$
$139$		−	15.5743i		−	1.32100i	0.750827	−	0.660499i	$-0.229655\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			0.189218i			0.0158232i
$144$		0			0
$145$	5.92844	−	5.33518i	0.492330	−	0.443062i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			4.17528i			0.342052i	0.985266	+	0.171026i	$0.0547083\pi$
$149$			4.17528i			0.342052i	−0.985266	+	0.171026i	$0.945292\pi$
$150$		0			0
$151$	3.18461			0.259160			0.129580	−	0.991569i	$-0.458637\pi$
$151$	3.18461			0.259160			0.129580	+	0.991569i	$0.458637\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.12801	+	2.36464i	0.170926	+	0.189933i
$156$		0			0
$157$	20.4342			1.63083			0.815414	−	0.578878i	$-0.196509\pi$
$157$	20.4342			1.63083			0.815414	+	0.578878i	$0.196509\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.17714			0.486827
$162$		0			0
$163$	19.5466			1.53101			0.765504	−	0.643432i	$-0.222490\pi$
$163$	19.5466			1.53101			0.765504	+	0.643432i	$0.222490\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.16218i			0.0899322i	0.998989	+	0.0449661i	$0.0143180\pi$
$167$			1.16218i			0.0899322i	−0.998989	+	0.0449661i	$0.985682\pi$
$168$		0			0
$169$	−9.30521			−0.715786
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.38888			0.561766			0.280883	−	0.959742i	$-0.409373\pi$
$173$	7.38888			0.561766			0.280883	+	0.959742i	$0.409373\pi$
$174$		0			0
$175$	7.13699	+	0.753918i	0.539506	+	0.0569909i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			18.4318i			1.37766i	0.724924	+	0.688828i	$0.241875\pi$
$179$			18.4318i			1.37766i	−0.724924	+	0.688828i	$0.758125\pi$
$180$		0			0
$181$			4.75079i			0.353123i	0.984290	+	0.176562i	$0.0564975\pi$
$181$			4.75079i			0.353123i	−0.984290	+	0.176562i	$0.943503\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.3751	−	13.7512i	−0.909833	−	1.01101i
$186$		0			0
$187$	−0.200561			−0.0146664
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.1635			−0.952479			−0.476240	−	0.879316i	$-0.658001\pi$
$191$	−13.1635			−0.952479			−0.476240	+	0.879316i	$0.658001\pi$
$192$		0			0
$193$		−	8.50034i		−	0.611868i	−0.952053	−	0.305934i	$-0.901031\pi$
$193$		−	8.50034i		−	0.611868i	0.952053	−	0.305934i	$-0.0989687\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.46451			0.318083			0.159042	−	0.987272i	$-0.449160\pi$
$197$	4.46451			0.318083			0.159042	+	0.987272i	$0.449160\pi$
$198$		0			0
$199$	19.2962			1.36787			0.683936	−	0.729542i	$-0.260267\pi$
$199$	19.2962			1.36787			0.683936	+	0.729542i	$0.260267\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.11958			0.359324
$204$		0			0
$205$	−7.65779	−	8.50933i	−0.534843	−	0.594317i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.0983957			0.00680617
$210$		0			0
$211$		−	22.4861i		−	1.54801i	−0.633180	−	0.774005i	$-0.718251\pi$
$211$		−	22.4861i		−	1.54801i	0.633180	−	0.774005i	$-0.281749\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.23545	−	9.15123i	−0.561653	−	0.624109i
$216$		0			0
$217$			2.04201i			0.138621i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			3.91627i			0.263437i
$222$		0			0
$223$			27.6890i			1.85419i	0.374822	+	0.927097i	$0.377704\pi$
$223$			27.6890i			1.85419i	−0.374822	+	0.927097i	$0.622296\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.25244			0.547734			0.273867	−	0.961768i	$-0.411697\pi$
$227$	8.25244			0.547734			0.273867	+	0.961768i	$0.411697\pi$
$228$		0			0
$229$			6.48061i			0.428251i	0.976806	+	0.214125i	$0.0686902\pi$
$229$			6.48061i			0.428251i	−0.976806	+	0.214125i	$0.931310\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.27937i		−	0.542399i	−0.962523	−	0.271200i	$-0.912580\pi$
$233$		−	8.27937i		−	0.542399i	0.962523	−	0.271200i	$-0.0874204\pi$
$234$		0			0
$235$	14.9708	−	13.4727i	0.976589	−	0.878861i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	26.3117			1.70196			0.850980	−	0.525198i	$-0.176009\pi$
$239$	26.3117			1.70196			0.850980	+	0.525198i	$0.176009\pi$
$240$		0			0
$241$	−24.8023			−1.59765			−0.798827	−	0.601560i	$-0.794546\pi$
$241$	−24.8023			−1.59765			−0.798827	+	0.601560i	$0.794546\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.38888	−	8.21052i	−0.472058	−	0.524551i
$246$		0			0
$247$		−	1.92133i		−	0.122252i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	0.277559i		−	0.0175194i	−0.999962	−	0.00875968i	$-0.997212\pi$
$251$		−	0.277559i		−	0.0175194i	0.999962	−	0.00875968i	$-0.00278833\pi$
$252$		0			0
$253$	−0.423644			−0.0266343
$254$		0			0
$255$		0			0
$256$		0			0
$257$			14.2367i			0.888062i	0.896011	+	0.444031i	$0.146452\pi$
$257$			14.2367i			0.888062i	−0.896011	+	0.444031i	$0.853548\pi$
$258$		0			0
$259$		−	11.8750i		−	0.737876i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	17.1353i		−	1.05661i	−0.849056	−	0.528303i	$-0.822828\pi$
$263$		−	17.1353i		−	1.05661i	0.849056	−	0.528303i	$-0.177172\pi$
$264$		0			0
$265$	−7.38468	−	8.20585i	−0.453637	−	0.504082i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	14.2417i		−	0.868333i	−0.900833	−	0.434167i	$-0.857043\pi$
$269$		−	14.2417i		−	0.868333i	0.900833	−	0.434167i	$-0.142957\pi$
$270$		0			0
$271$	16.1177			0.979078			0.489539	−	0.871981i	$-0.337165\pi$
$271$	16.1177			0.979078			0.489539	+	0.871981i	$0.337165\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.489473	−	0.0517056i	−0.0295163	−	0.00311797i
$276$		0			0
$277$	15.6307			0.939156			0.469578	−	0.882891i	$-0.344406\pi$
$277$	15.6307			0.939156			0.469578	+	0.882891i	$0.344406\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.65195			0.337167			0.168583	−	0.985687i	$-0.446081\pi$
$281$	5.65195			0.337167			0.168583	+	0.985687i	$0.446081\pi$
$282$		0			0
$283$	16.5466			0.983593			0.491796	−	0.870710i	$-0.336340\pi$
$283$	16.5466			0.983593			0.491796	+	0.870710i	$0.336340\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	7.34833i		−	0.433758i
$288$		0			0
$289$	12.8490			0.755822
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−20.2544			−1.18328			−0.591638	−	0.806203i	$-0.701519\pi$
$293$	−20.2544			−1.18328			−0.591638	+	0.806203i	$0.701519\pi$
$294$		0			0
$295$	−10.4256	+	9.38226i	−0.606999	+	0.546256i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			8.27232i			0.478401i
$300$		0			0
$301$		−	7.90265i		−	0.455501i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−21.3498	+	19.2133i	−1.22248	+	1.10015i
$306$		0			0
$307$	−5.50772			−0.314342			−0.157171	−	0.987571i	$-0.550237\pi$
$307$	−5.50772			−0.314342			−0.157171	+	0.987571i	$0.550237\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.57632			−0.486318			−0.243159	−	0.969986i	$-0.578184\pi$
$311$	−8.57632			−0.486318			−0.243159	+	0.969986i	$0.578184\pi$
$312$		0			0
$313$			9.83297i			0.555793i	0.960611	+	0.277896i	$0.0896372\pi$
$313$			9.83297i			0.555793i	−0.960611	+	0.277896i	$0.910363\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.38888			−0.415001			−0.207500	−	0.978235i	$-0.566533\pi$
$317$	−7.38888			−0.415001			−0.207500	+	0.978235i	$0.566533\pi$
$318$		0			0
$319$	−0.351114			−0.0196586
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.03651			0.113314
$324$		0			0
$325$	−1.00964	+	9.55774i	−0.0560045	+	0.530168i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.9282			0.712757
$330$		0			0
$331$		−	5.64764i		−	0.310422i	−0.987881	−	0.155211i	$-0.950394\pi$
$331$		−	5.64764i		−	0.310422i	0.987881	−	0.155211i	$-0.0496058\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.82743	+	3.14184i	0.154479	+	0.171657i
$336$		0			0
$337$			7.91137i			0.430960i	0.976508	+	0.215480i	$0.0691315\pi$
$337$			7.91137i			0.430960i	−0.976508	+	0.215480i	$0.930868\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	0.140046i		−	0.00758394i
$342$		0			0
$343$		−	17.1377i		−	0.925347i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	25.8465			1.38751			0.693756	−	0.720210i	$-0.255954\pi$
$347$	25.8465			1.38751			0.693756	+	0.720210i	$0.255954\pi$
$348$		0			0
$349$		−	13.2897i		−	0.711382i	−0.934604	−	0.355691i	$-0.884246\pi$
$349$		−	13.2897i		−	0.711382i	0.934604	−	0.355691i	$-0.115754\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			28.1761i			1.49966i	0.661629	+	0.749831i	$0.269865\pi$
$353$			28.1761i			1.49966i	−0.661629	+	0.749831i	$0.730135\pi$
$354$		0			0
$355$	18.4369	+	20.4870i	0.978527	+	1.08734i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	27.5477			1.45391			0.726956	−	0.686684i	$-0.240934\pi$
$359$	27.5477			1.45391			0.726956	+	0.686684i	$0.240934\pi$
$360$		0			0
$361$	18.0009			0.947415
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.35715	+	8.42077i	−0.489776	+	0.440763i
$366$		0			0
$367$			15.8778i			0.828815i	0.910091	+	0.414407i	$0.136011\pi$
$367$			15.8778i			0.828815i	−0.910091	+	0.414407i	$0.863989\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	7.08626i		−	0.367900i
$372$		0			0
$373$	4.85063			0.251156			0.125578	−	0.992084i	$-0.459921\pi$
$373$	4.85063			0.251156			0.125578	+	0.992084i	$0.459921\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.85605i			0.353105i
$378$		0			0
$379$			16.0191i			0.822846i	0.911445	+	0.411423i	$0.134968\pi$
$379$			16.0191i			0.822846i	−0.911445	+	0.411423i	$0.865032\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			6.83627i			0.349317i	0.984629	+	0.174658i	$0.0558821\pi$
$383$			6.83627i			0.349317i	−0.984629	+	0.174658i	$0.944118\pi$
$384$		0			0
$385$	−0.211345	−	0.234846i	−0.0107711	−	0.0119689i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	15.1004i		−	0.765623i	−0.923827	−	0.382811i	$-0.874956\pi$
$389$		−	15.1004i		−	0.765623i	0.923827	−	0.382811i	$-0.125044\pi$
$390$		0			0
$391$	−8.76820			−0.443427
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.72175	+	4.13560i	0.187261	+	0.208085i
$396$		0			0
$397$	25.1307			1.26127			0.630636	−	0.776079i	$-0.282794\pi$
$397$	25.1307			1.26127			0.630636	+	0.776079i	$0.282794\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−27.2424			−1.36042			−0.680211	−	0.733016i	$-0.738112\pi$
$401$	−27.2424			−1.36042			−0.680211	+	0.733016i	$0.738112\pi$
$402$		0			0
$403$	−2.73463			−0.136222
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.814417i			0.0403691i
$408$		0			0
$409$	−2.07322			−0.102514			−0.0512571	−	0.998685i	$-0.516323\pi$
$409$	−2.07322			−0.102514			−0.0512571	+	0.998685i	$0.516323\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.00311			−0.443014
$414$		0			0
$415$	3.39439	+	3.77184i	0.166624	+	0.185152i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			28.2449i			1.37985i	0.723880	+	0.689926i	$0.242357\pi$
$419$			28.2449i			1.37985i	−0.723880	+	0.689926i	$0.757643\pi$
$420$		0			0
$421$		−	4.94229i		−	0.240873i	−0.992721	−	0.120436i	$-0.961571\pi$
$421$		−	4.94229i		−	0.240873i	0.992721	−	0.120436i	$-0.0384294\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.1307	−	1.07016i	−0.491410	−	0.0519102i
$426$		0			0
$427$	−18.4369			−0.892222
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.01109			−0.241376			−0.120688	−	0.992691i	$-0.538510\pi$
$431$	−5.01109			−0.241376			−0.120688	+	0.992691i	$0.538510\pi$
$432$		0			0
$433$		−	7.79967i		−	0.374828i	−0.982281	−	0.187414i	$-0.939989\pi$
$433$		−	7.79967i		−	0.374828i	0.982281	−	0.187414i	$-0.0600106\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.30171			0.205779
$438$		0			0
$439$	−34.9376			−1.66748			−0.833739	−	0.552158i	$-0.813804\pi$
$439$	−34.9376			−1.66748			−0.833739	+	0.552158i	$0.813804\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.6112			−1.12180			−0.560900	−	0.827883i	$-0.689545\pi$
$443$	−23.6112			−1.12180			−0.560900	+	0.827883i	$0.689545\pi$
$444$		0			0
$445$	−19.9763	−	22.1976i	−0.946965	−	1.05227i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.39141			0.443208			0.221604	−	0.975137i	$-0.428871\pi$
$449$	9.39141			0.443208			0.221604	+	0.975137i	$0.428871\pi$
$450$		0			0
$451$			0.503967i			0.0237309i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.58575	+	4.12685i	−0.214983	+	0.193470i
$456$		0			0
$457$			31.0457i			1.45226i	0.687560	+	0.726128i	$0.258682\pi$
$457$			31.0457i			1.45226i	−0.687560	+	0.726128i	$0.741318\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	0.867775i		−	0.0404163i	−0.999796	−	0.0202081i	$-0.993567\pi$
$461$		−	0.867775i		−	0.0404163i	0.999796	−	0.0202081i	$-0.00643289\pi$
$462$		0			0
$463$		−	31.0297i		−	1.44207i	−0.692898	−	0.721035i	$-0.743667\pi$
$463$		−	31.0297i		−	1.44207i	0.692898	−	0.721035i	$-0.256333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−27.7060			−1.28208			−0.641040	−	0.767507i	$-0.721497\pi$
$467$	−27.7060			−1.28208			−0.641040	+	0.767507i	$0.721497\pi$
$468$		0			0
$469$			2.71317i			0.125283i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.541984i			0.0249204i
$474$		0			0
$475$	4.97014	+	0.525022i	0.228046	+	0.0240897i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.8989			1.41181			0.705903	−	0.708309i	$-0.250541\pi$
$479$	30.8989			1.41181			0.705903	+	0.708309i	$0.250541\pi$
$480$		0			0
$481$	15.9028			0.725105
$482$		0			0
$483$		0			0
$484$		0			0
$485$	24.5090	−	22.0563i	1.11290	−	1.00153i
$486$		0			0
$487$		−	21.5162i		−	0.974991i	−0.873125	−	0.487496i	$-0.837910\pi$
$487$		−	21.5162i		−	0.974991i	0.873125	−	0.487496i	$-0.162090\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	9.74298i		−	0.439695i	−0.975534	−	0.219847i	$-0.929444\pi$
$491$		−	9.74298i		−	0.439695i	0.975534	−	0.219847i	$-0.0705559\pi$
$492$		0			0
$493$	−7.26704			−0.327291
$494$		0			0
$495$		0			0
$496$		0			0
$497$			17.6918i			0.793586i
$498$		0			0
$499$			6.91181i			0.309415i	0.987960	+	0.154708i	$0.0494435\pi$
$499$			6.91181i			0.309415i	−0.987960	+	0.154708i	$0.950556\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			15.3213i			0.683142i	0.939856	+	0.341571i	$0.110959\pi$
$503$			15.3213i			0.683142i	−0.939856	+	0.341571i	$0.889041\pi$
$504$		0			0
$505$	22.6411	−	20.3753i	1.00751	−	0.906690i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			28.8669i			1.27950i	0.768582	+	0.639752i	$0.220963\pi$
$509$			28.8669i			1.27950i	−0.768582	+	0.639752i	$0.779037\pi$
$510$		0			0
$511$	−8.08048			−0.357459
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.0270	−	16.2230i	0.794365	−	0.714872i
$516$		0			0
$517$	−0.886651			−0.0389949
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.525190			−0.0230090			−0.0115045	−	0.999934i	$-0.503662\pi$
$521$	−0.525190			−0.0230090			−0.0115045	+	0.999934i	$0.503662\pi$
$522$		0			0
$523$	−11.1631			−0.488127			−0.244064	−	0.969759i	$-0.578481\pi$
$523$	−11.1631			−0.488127			−0.244064	+	0.969759i	$0.578481\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.89856i		−	0.126263i
$528$		0			0
$529$	4.47895			0.194737
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.84076			0.426251
$534$		0			0
$535$	18.6846	+	20.7623i	0.807804	+	0.897631i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			0.486270i			0.0209451i
$540$		0			0
$541$		−	4.30602i		−	0.185130i	−0.995707	−	0.0925651i	$-0.970493\pi$
$541$		−	4.30602i		−	0.185130i	0.995707	−	0.0925651i	$-0.0295066\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−27.5477	+	24.7910i	−1.18001	+	1.06193i
$546$		0			0
$547$	−40.0149			−1.71091			−0.855456	−	0.517875i	$-0.826723\pi$
$547$	−40.0149			−1.71091			−0.855456	+	0.517875i	$0.826723\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.56523			0.151884
$552$		0			0
$553$			3.57135i			0.151869i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.0641			−0.807771			−0.403886	−	0.914809i	$-0.632341\pi$
$557$	−19.0641			−0.807771			−0.403886	+	0.914809i	$0.632341\pi$
$558$		0			0
$559$	10.5831			0.447618
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.40993			0.270147			0.135073	−	0.990836i	$-0.456873\pi$
$563$	6.40993			0.270147			0.135073	+	0.990836i	$0.456873\pi$
$564$		0			0
$565$	−15.2104	+	13.6883i	−0.639906	+	0.575870i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	37.1532			1.55754			0.778771	−	0.627308i	$-0.215843\pi$
$569$	37.1532			1.55754			0.778771	+	0.627308i	$0.215843\pi$
$570$		0			0
$571$			28.7752i			1.20421i	0.798418	+	0.602103i	$0.205670\pi$
$571$			28.7752i			1.20421i	−0.798418	+	0.602103i	$0.794330\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−21.3990	−	2.26049i	−0.892399	−	0.0942689i
$576$		0			0
$577$		−	20.0173i		−	0.833332i	−0.909060	−	0.416666i	$-0.863198\pi$
$577$		−	20.0173i		−	0.833332i	0.909060	−	0.416666i	$-0.136802\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			3.25722i			0.135132i
$582$		0			0
$583$			0.485994i			0.0201278i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.91407			0.326649			0.163324	−	0.986572i	$-0.447778\pi$
$587$	7.91407			0.326649			0.163324	+	0.986572i	$0.447778\pi$
$588$		0			0
$589$			1.42204i			0.0585942i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			14.4728i			0.594325i	0.954827	+	0.297163i	$0.0960404\pi$
$593$			14.4728i			0.594325i	−0.954827	+	0.297163i	$0.903960\pi$
$594$		0			0
$595$	−4.37423	−	4.86064i	−0.179326	−	0.199267i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.6597			0.844133			0.422067	−	0.906565i	$-0.361305\pi$
$599$	20.6597			0.844133			0.422067	+	0.906565i	$0.361305\pi$
$600$		0			0
$601$	−32.3742			−1.32057			−0.660286	−	0.751014i	$-0.729565\pi$
$601$	−32.3742			−1.32057			−0.660286	+	0.751014i	$0.729565\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.4391	−	18.2672i	−0.668346	−	0.742666i
$606$		0			0
$607$		−	41.1110i		−	1.66865i	−0.551276	−	0.834323i	$-0.685859\pi$
$607$		−	41.1110i		−	1.66865i	0.551276	−	0.834323i	$-0.314141\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			17.3133i			0.700420i
$612$		0			0
$613$	9.54708			0.385603			0.192801	−	0.981238i	$-0.438243\pi$
$613$	9.54708			0.385603			0.192801	+	0.981238i	$0.438243\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	26.7796i		−	1.07811i	−0.842272	−	0.539053i	$-0.818782\pi$
$617$		−	26.7796i		−	1.07811i	0.842272	−	0.539053i	$-0.181218\pi$
$618$		0			0
$619$		−	19.1724i		−	0.770602i	−0.922791	−	0.385301i	$-0.874098\pi$
$619$		−	19.1724i		−	0.770602i	0.922791	−	0.385301i	$-0.125902\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	19.1690i		−	0.767990i
$624$		0			0
$625$	−24.4482	−	5.22348i	−0.977929	−	0.208939i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			16.8561i			0.672095i
$630$		0			0
$631$	9.42121			0.375052			0.187526	−	0.982260i	$-0.439953\pi$
$631$	9.42121			0.375052			0.187526	+	0.982260i	$0.439953\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18.2248	+	16.4010i	−0.723230	+	0.650856i
$636$		0			0
$637$	9.49520			0.376214
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−36.5224			−1.44255			−0.721275	−	0.692649i	$-0.756443\pi$
$641$	−36.5224			−1.44255			−0.721275	+	0.692649i	$0.756443\pi$
$642$		0			0
$643$	−11.6563			−0.459681			−0.229840	−	0.973228i	$-0.573820\pi$
$643$	−11.6563			−0.459681			−0.229840	+	0.973228i	$0.573820\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	40.2063i		−	1.58067i	−0.612673	−	0.790337i	$-0.709906\pi$
$647$		−	40.2063i		−	1.58067i	0.612673	−	0.790337i	$-0.290094\pi$
$648$		0			0
$649$	0.617456			0.0242373
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.69902			−0.340419			−0.170209	−	0.985408i	$-0.554444\pi$
$653$	−8.69902			−0.340419			−0.170209	+	0.985408i	$0.554444\pi$
$654$		0			0
$655$	−36.8815	+	33.1907i	−1.44108	+	1.29687i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	22.2085i		−	0.865122i	−0.901605	−	0.432561i	$-0.857610\pi$
$659$		−	22.2085i		−	0.865122i	0.901605	−	0.432561i	$-0.142390\pi$
$660$		0			0
$661$			32.1865i			1.25191i	0.779859	+	0.625956i	$0.215291\pi$
$661$			32.1865i			1.25191i	−0.779859	+	0.625956i	$0.784709\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.14601	+	2.38465i	0.0832187	+	0.0924726i
$666$		0			0
$667$	−15.3501			−0.594360
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.26445			0.0488134
$672$		0			0
$673$			39.8887i			1.53760i	0.639492	+	0.768798i	$0.279145\pi$
$673$			39.8887i			1.53760i	−0.639492	+	0.768798i	$0.720855\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.46451			−0.171585			−0.0857925	−	0.996313i	$-0.527342\pi$
$677$	−4.46451			−0.171585			−0.0857925	+	0.996313i	$0.527342\pi$
$678$		0			0
$679$	21.1650			0.812240
$680$		0			0
$681$		0			0
$682$		0			0
$683$	42.7166			1.63450			0.817252	−	0.576281i	$-0.195497\pi$
$683$	42.7166			1.63450			0.817252	+	0.576281i	$0.195497\pi$
$684$		0			0
$685$	−17.9260	+	16.1322i	−0.684919	+	0.616378i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.48981			0.361533
$690$		0			0
$691$			14.4881i			0.551153i	0.961279	+	0.275577i	$0.0888687\pi$
$691$			14.4881i			0.551153i	−0.961279	+	0.275577i	$0.911131\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−25.8863	+	23.2959i	−0.981925	+	0.883662i
$696$		0			0
$697$			10.4307i			0.395090i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	27.2562i		−	1.02945i	−0.857355	−	0.514726i	$-0.827894\pi$
$701$		−	27.2562i		−	1.02945i	0.857355	−	0.514726i	$-0.172106\pi$
$702$		0			0
$703$		−	8.26964i		−	0.311895i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.5520			0.735327
$708$		0			0
$709$			18.4360i			0.692379i	0.938165	+	0.346190i	$0.112525\pi$
$709$			18.4360i			0.692379i	−0.938165	+	0.346190i	$0.887475\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	6.12261i		−	0.229294i
$714$		0			0
$715$	0.314502	−	0.283030i	0.0117617	−	0.0105847i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.2329			0.754561			0.377280	−	0.926099i	$-0.376859\pi$
$719$	20.2329			0.754561			0.377280	+	0.926099i	$0.376859\pi$
$720$		0			0
$721$	15.5674			0.579762
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.7354	−	1.87348i	−0.658675	−	0.0695793i
$726$		0			0
$727$		−	3.89252i		−	0.144366i	−0.997391	−	0.0721828i	$-0.977003\pi$
$727$		−	3.89252i		−	0.144366i	0.997391	−	0.0721828i	$-0.0229965\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			11.2175i			0.414894i
$732$		0			0
$733$	−35.3194			−1.30455			−0.652276	−	0.757982i	$-0.726186\pi$
$733$	−35.3194			−1.30455			−0.652276	+	0.757982i	$0.726186\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	0.186076i		−	0.00685421i
$738$		0			0
$739$			38.5052i			1.41644i	0.705993	+	0.708219i	$0.250501\pi$
$739$			38.5052i			1.41644i	−0.705993	+	0.708219i	$0.749499\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	19.4158i		−	0.712295i	−0.934430	−	0.356148i	$-0.884090\pi$
$743$		−	19.4158i		−	0.712295i	0.934430	−	0.356148i	$-0.115910\pi$
$744$		0			0
$745$	6.93980	−	6.24532i	0.254255	−	0.228811i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			17.9295i			0.655130i
$750$		0			0
$751$	−36.4832			−1.33129			−0.665646	−	0.746267i	$-0.731844\pi$
$751$	−36.4832			−1.33129			−0.665646	+	0.746267i	$0.731844\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.76349	−	5.29319i	−0.173361	−	0.192639i
$756$		0			0
$757$	−10.9870			−0.399329			−0.199664	−	0.979864i	$-0.563985\pi$
$757$	−10.9870			−0.399329			−0.199664	+	0.979864i	$0.563985\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.1991			0.442216			0.221108	−	0.975249i	$-0.429033\pi$
$761$	12.1991			0.442216			0.221108	+	0.975249i	$0.429033\pi$
$762$		0			0
$763$	−23.7892			−0.861225
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	12.0568i		−	0.435347i
$768$		0			0
$769$	−13.2424			−0.477531			−0.238766	−	0.971077i	$-0.576743\pi$
$769$	−13.2424			−0.477531			−0.238766	+	0.971077i	$0.576743\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.66412			0.239692			0.119846	−	0.992793i	$-0.461760\pi$
$773$	6.66412			0.239692			0.119846	+	0.992793i	$0.461760\pi$
$774$		0			0
$775$	0.747264	−	7.07399i	0.0268425	−	0.254105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	5.11732i		−	0.183347i
$780$		0			0
$781$		−	1.21335i		−	0.0434170i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−30.5652	−	33.9640i	−1.09092	−	1.21223i
$786$		0			0
$787$	0.163075			0.00581301			0.00290650	−	0.999996i	$-0.499075\pi$
$787$	0.163075			0.00581301			0.00290650	+	0.999996i	$0.499075\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−13.1351			−0.467031
$792$		0			0
$793$		−	24.6903i		−	0.876780i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	46.0750			1.63206			0.816031	−	0.578009i	$-0.196170\pi$
$797$	46.0750			1.63206			0.816031	+	0.578009i	$0.196170\pi$
$798$		0			0
$799$	−18.3511			−0.649216
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.554180			0.0195566
$804$		0			0
$805$	−9.23967	−	10.2671i	−0.325656	−	0.361868i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	48.0231			1.68840			0.844201	−	0.536026i	$-0.180075\pi$
$809$	48.0231			1.68840			0.844201	+	0.536026i	$0.180075\pi$
$810$		0			0
$811$			19.0427i			0.668680i	0.942453	+	0.334340i	$0.108513\pi$
$811$			19.0427i			0.668680i	−0.942453	+	0.334340i	$0.891487\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−29.2375	−	32.4887i	−1.02414	−	1.13803i
$816$		0			0
$817$		−	5.50334i		−	0.192538i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	26.1910i		−	0.914073i	−0.889448	−	0.457037i	$-0.848911\pi$
$821$		−	26.1910i		−	0.914073i	0.889448	−	0.457037i	$-0.151089\pi$
$822$		0			0
$823$			11.9142i			0.415302i	0.978203	+	0.207651i	$0.0665819\pi$
$823$			11.9142i			0.415302i	−0.978203	+	0.207651i	$0.933418\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.2443			−0.703965			−0.351982	−	0.936007i	$-0.614492\pi$
$827$	−20.2443			−0.703965			−0.351982	+	0.936007i	$0.614492\pi$
$828$		0			0
$829$			32.5375i			1.13007i	0.825066	+	0.565037i	$0.191138\pi$
$829$			32.5375i			1.13007i	−0.825066	+	0.565037i	$0.808862\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			10.0644i			0.348710i
$834$		0			0
$835$	1.93168	−	1.73837i	0.0668484	−	0.0601588i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	40.6959			1.40498			0.702489	−	0.711695i	$-0.252072\pi$
$839$	40.6959			1.40498			0.702489	+	0.711695i	$0.252072\pi$
$840$		0			0
$841$	16.2779			0.561307
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.9186	+	15.4663i	0.478814	+	0.532058i
$846$		0			0
$847$		−	15.7748i		−	0.542029i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			35.6050i			1.22052i
$852$		0			0
$853$	−18.5346			−0.634611			−0.317306	−	0.948323i	$-0.602778\pi$
$853$	−18.5346			−0.634611			−0.317306	+	0.948323i	$0.602778\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	46.9004i		−	1.60209i	−0.598607	−	0.801043i	$-0.704279\pi$
$857$		−	46.9004i		−	1.60209i	0.598607	−	0.801043i	$-0.295721\pi$
$858$		0			0
$859$			32.2074i			1.09890i	0.835526	+	0.549452i	$0.185163\pi$
$859$			32.2074i			1.09890i	−0.835526	+	0.549452i	$0.814837\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			23.5848i			0.802836i	0.915895	+	0.401418i	$0.131483\pi$
$863$			23.5848i			0.802836i	−0.915895	+	0.401418i	$0.868517\pi$
$864$		0			0
$865$	−11.0522	−	12.2812i	−0.375785	−	0.417572i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	0.244932i		−	0.00830875i
$870$		0			0
$871$	−3.63344			−0.123114
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.42230	−	12.9902i	−0.318532	−	0.439149i
$876$		0			0
$877$	6.16767			0.208267			0.104134	−	0.994563i	$-0.466793\pi$
$877$	6.16767			0.208267			0.104134	+	0.994563i	$0.466793\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−48.3515			−1.62900			−0.814502	−	0.580161i	$-0.802990\pi$
$881$	−48.3515			−1.62900			−0.814502	+	0.580161i	$0.802990\pi$
$882$		0			0
$883$	−7.17454			−0.241443			−0.120721	−	0.992686i	$-0.538521\pi$
$883$	−7.17454			−0.241443			−0.120721	+	0.992686i	$0.538521\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			33.8797i			1.13757i	0.822486	+	0.568785i	$0.192586\pi$
$887$			33.8797i			1.13757i	−0.822486	+	0.568785i	$0.807414\pi$
$888$		0			0
$889$	−15.7383			−0.527845
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9.00311			0.301278
$894$		0			0
$895$	30.6358	−	27.5700i	1.02404	−	0.921564i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	5.07439i		−	0.169240i
$900$		0			0
$901$			10.0587i			0.335103i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.89635	−	7.10615i	0.262484	−	0.236217i
$906$		0			0
$907$	−25.4054			−0.843573			−0.421787	−	0.906695i	$-0.638597\pi$
$907$	−25.4054			−0.843573			−0.421787	+	0.906695i	$0.638597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.7243			−0.421574			−0.210787	−	0.977532i	$-0.567603\pi$
$911$	−12.7243			−0.421574			−0.210787	+	0.977532i	$0.567603\pi$
$912$		0			0
$913$		−	0.223388i		−	0.00739307i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−31.8494			−1.05176
$918$		0			0
$919$	−10.1070			−0.333399			−0.166699	−	0.986008i	$-0.553311\pi$
$919$	−10.1070			−0.333399			−0.166699	+	0.986008i	$0.553311\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−23.6926			−0.779851
$924$		0			0
$925$	−4.34559	+	41.1376i	−0.142882	+	1.35260i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.2928			0.928257			0.464129	−	0.885768i	$-0.346368\pi$
$929$	28.2928			0.928257			0.464129	+	0.885768i	$0.346368\pi$
$930$		0			0
$931$		−	4.93762i		−	0.161824i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.299996	+	0.333355i	0.00981091	+	0.0109019i
$936$		0			0
$937$		−	21.5800i		−	0.704987i	−0.935814	−	0.352493i	$-0.885334\pi$
$937$		−	21.5800i		−	0.704987i	0.935814	−	0.352493i	$-0.114666\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	25.5990i		−	0.834503i	−0.908791	−	0.417252i	$-0.862993\pi$
$941$		−	25.5990i		−	0.834503i	0.908791	−	0.417252i	$-0.137007\pi$
$942$		0			0
$943$			22.0327i			0.717482i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26.0177			0.845461			0.422730	−	0.906256i	$-0.361072\pi$
$947$	26.0177			0.845461			0.422730	+	0.906256i	$0.361072\pi$
$948$		0			0
$949$		−	10.8212i		−	0.351272i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	58.0689i		−	1.88104i	−0.339745	−	0.940518i	$-0.610341\pi$
$953$		−	58.0689i		−	1.88104i	0.339745	−	0.940518i	$-0.389659\pi$
$954$		0			0
$955$	19.6898	+	21.8793i	0.637147	+	0.707997i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15.4802			−0.499883
$960$		0			0
$961$	−28.9760			−0.934710
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.1285	+	12.7147i	−0.454814	+	0.409300i
$966$		0			0
$967$			29.7133i			0.955516i	0.878491	+	0.477758i	$0.158550\pi$
$967$			29.7133i			0.955516i	−0.878491	+	0.477758i	$0.841450\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	29.8380i		−	0.957547i	−0.877938	−	0.478774i	$-0.841081\pi$
$971$		−	29.8380i		−	0.957547i	0.877938	−	0.478774i	$-0.158919\pi$
$972$		0			0
$973$	−22.3545			−0.716651
$974$		0			0
$975$		0			0
$976$		0			0
$977$			15.4533i			0.494394i	0.968965	+	0.247197i	$0.0795094\pi$
$977$			15.4533i			0.494394i	−0.968965	+	0.247197i	$0.920491\pi$
$978$		0			0
$979$			1.31466i			0.0420167i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			5.17515i			0.165062i	0.996589	+	0.0825309i	$0.0263003\pi$
$983$			5.17515i			0.165062i	−0.996589	+	0.0825309i	$0.973700\pi$
$984$		0			0
$985$	−6.67795	−	7.42053i	−0.212777	−	0.236438i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			23.6947i			0.753448i
$990$		0			0
$991$	18.2392			0.579388			0.289694	−	0.957119i	$-0.406446\pi$
$991$	18.2392			0.579388			0.289694	+	0.957119i	$0.406446\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−28.8630	−	32.0725i	−0.915018	−	1.01677i
$996$		0			0
$997$	−40.5033			−1.28275			−0.641377	−	0.767226i	$-0.721636\pi$
$997$	−40.5033			−1.28275			−0.641377	+	0.767226i	$0.721636\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.d.i.3889.7		20
3.2	odd	2	inner	4320.2.d.i.3889.14		20
4.3	odd	2		1080.2.d.i.109.4	yes	20
5.4	even	2		4320.2.d.j.3889.13		20
8.3	odd	2		1080.2.d.j.109.18	yes	20
8.5	even	2		4320.2.d.j.3889.14		20
12.11	even	2		1080.2.d.i.109.17	yes	20
15.14	odd	2		4320.2.d.j.3889.8		20
20.19	odd	2		1080.2.d.j.109.17	yes	20
24.5	odd	2		4320.2.d.j.3889.7		20
24.11	even	2		1080.2.d.j.109.3	yes	20
40.19	odd	2		1080.2.d.i.109.3	✓	20
40.29	even	2	inner	4320.2.d.i.3889.8		20
60.59	even	2		1080.2.d.j.109.4	yes	20
120.29	odd	2	inner	4320.2.d.i.3889.13		20
120.59	even	2		1080.2.d.i.109.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.d.i.109.3	✓	20	40.19	odd	2
1080.2.d.i.109.4	yes	20	4.3	odd	2
1080.2.d.i.109.17	yes	20	12.11	even	2
1080.2.d.i.109.18	yes	20	120.59	even	2
1080.2.d.j.109.3	yes	20	24.11	even	2
1080.2.d.j.109.4	yes	20	60.59	even	2
1080.2.d.j.109.17	yes	20	20.19	odd	2
1080.2.d.j.109.18	yes	20	8.3	odd	2
4320.2.d.i.3889.7		20	1.1	even	1	trivial
4320.2.d.i.3889.8		20	40.29	even	2	inner
4320.2.d.i.3889.13		20	120.29	odd	2	inner
4320.2.d.i.3889.14		20	3.2	odd	2	inner
4320.2.d.j.3889.7		20	24.5	odd	2
4320.2.d.j.3889.8		20	15.14	odd	2
4320.2.d.j.3889.13		20	5.4	even	2
4320.2.d.j.3889.14		20	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.49578 - 1.66212i) q^{5} -1.43534i q^{7} +0.0984392i q^{11} +1.92218 q^{13} +2.03741i q^{17} -0.999558i q^{19} +4.30361i q^{23} +(-0.525254 + 4.97233i) q^{25} +3.56681i q^{29} -1.42267 q^{31} +(-2.38570 + 2.14696i) q^{35} +8.27330 q^{37} +5.11958 q^{41} +5.50577 q^{43} +9.00709i q^{47} +4.93980 q^{49} +4.93699 q^{53} +(0.163617 - 0.147244i) q^{55} -6.27246i q^{59} -12.8449i q^{61} +(-2.87517 - 3.19489i) q^{65} -1.89027 q^{67} -12.3259 q^{71} -5.62966i q^{73} +0.141294 q^{77} -2.48816 q^{79} -2.26930 q^{83} +(3.38640 - 3.04752i) q^{85} +13.3550 q^{89} -2.75898i q^{91} +(-1.66138 + 1.49512i) q^{95} +14.7457i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{13} + 20 q^{25} - 12 q^{31} - 32 q^{37} - 12 q^{43} - 52 q^{49} + 28 q^{55} - 36 q^{79} - 44 q^{85}+O(q^{100}) \) 20 * q - 4 * q^13 + 20 * q^25 - 12 * q^31 - 32 * q^37 - 12 * q^43 - 52 * q^49 + 28 * q^55 - 36 * q^79 - 44 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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