LMFDB - Embedded newform 4080.2.h.l.3841.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4080,2,Mod(3841,4080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4080.3841");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.h (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:	$32.5789640247$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2040)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3841.1
Root			$-2.70156i$ of defining polynomial
Character	$\chi$	$=$	4080.3841
Dual form			4080.2.h.l.3841.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -2.70156i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.00000i q^{5} -2.70156i q^{7} -1.00000 q^{9} +4.70156i q^{11} -2.00000 q^{13} +1.00000 q^{15} +(-1.00000 - 4.00000i) q^{17} +4.70156 q^{19} -2.70156 q^{21} +7.40312i q^{23} -1.00000 q^{25} +1.00000i q^{27} -6.70156i q^{29} -3.40312i q^{31} +4.70156 q^{33} +2.70156 q^{35} -10.7016i q^{37} +2.00000i q^{39} +10.1047i q^{41} -1.00000i q^{45} +3.29844 q^{47} -0.298438 q^{49} +(-4.00000 + 1.00000i) q^{51} +6.70156 q^{53} -4.70156 q^{55} -4.70156i q^{57} -4.00000 q^{59} -0.596876i q^{61} +2.70156i q^{63} -2.00000i q^{65} +13.4031 q^{67} +7.40312 q^{69} -15.4031i q^{71} -8.70156i q^{73} +1.00000i q^{75} +12.7016 q^{77} +2.00000i q^{79} +1.00000 q^{81} +(4.00000 - 1.00000i) q^{85} -6.70156 q^{87} -7.40312 q^{89} +5.40312i q^{91} -3.40312 q^{93} +4.70156i q^{95} -17.4031i q^{97} -4.70156i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 8 q^{13} + 4 q^{15} - 4 q^{17} + 6 q^{19} + 2 q^{21} - 4 q^{25} + 6 q^{33} - 2 q^{35} + 26 q^{47} - 14 q^{49} - 16 q^{51} + 14 q^{53} - 6 q^{55} - 16 q^{59} + 28 q^{67} + 4 q^{69} + 38 q^{77} + 4 q^{81} + 16 q^{85} - 14 q^{87} - 4 q^{89} + 12 q^{93}+O(q^{100}) $ 4 * q - 4 * q^9 - 8 * q^13 + 4 * q^15 - 4 * q^17 + 6 * q^19 + 2 * q^21 - 4 * q^25 + 6 * q^33 - 2 * q^35 + 26 * q^47 - 14 * q^49 - 16 * q^51 + 14 * q^53 - 6 * q^55 - 16 * q^59 + 28 * q^67 + 4 * q^69 + 38 * q^77 + 4 * q^81 + 16 * q^85 - 14 * q^87 - 4 * q^89 + 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1361$	$3061$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	2.70156i		−	1.02109i	−0.859850	−	0.510547i	$-0.829443\pi$
$7$		−	2.70156i		−	1.02109i	0.859850	−	0.510547i	$-0.170557\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			4.70156i			1.41757i	0.705422	+	0.708787i	$0.250757\pi$
$11$			4.70156i			1.41757i	−0.705422	+	0.708787i	$0.749243\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$18$		0			0
$19$	4.70156			1.07861			0.539306	−	0.842110i	$-0.318687\pi$
$19$	4.70156			1.07861			0.539306	+	0.842110i	$0.318687\pi$
$20$		0			0
$21$	−2.70156			−0.589529
$22$		0			0
$23$			7.40312i			1.54366i	0.635830	+	0.771829i	$0.280658\pi$
$23$			7.40312i			1.54366i	−0.635830	+	0.771829i	$0.719342\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	6.70156i		−	1.24445i	−0.782839	−	0.622224i	$-0.786229\pi$
$29$		−	6.70156i		−	1.24445i	0.782839	−	0.622224i	$-0.213771\pi$
$30$		0			0
$31$		−	3.40312i		−	0.611219i	−0.952157	−	0.305610i	$-0.901140\pi$
$31$		−	3.40312i		−	0.611219i	0.952157	−	0.305610i	$-0.0988602\pi$
$32$		0			0
$33$	4.70156			0.818437
$34$		0			0
$35$	2.70156			0.456647
$36$		0			0
$37$		−	10.7016i		−	1.75933i	−0.475598	−	0.879663i	$-0.657768\pi$
$37$		−	10.7016i		−	1.75933i	0.475598	−	0.879663i	$-0.342232\pi$
$38$		0			0
$39$			2.00000i			0.320256i
$40$		0			0
$41$			10.1047i			1.57809i	0.614337	+	0.789043i	$0.289423\pi$
$41$			10.1047i			1.57809i	−0.614337	+	0.789043i	$0.710577\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$	3.29844			0.481127			0.240563	−	0.970633i	$-0.422668\pi$
$47$	3.29844			0.481127			0.240563	+	0.970633i	$0.422668\pi$
$48$		0			0
$49$	−0.298438			−0.0426340
$50$		0			0
$51$	−4.00000	+	1.00000i	−0.560112	+	0.140028i
$52$		0			0
$53$	6.70156			0.920530			0.460265	−	0.887781i	$-0.347754\pi$
$53$	6.70156			0.920530			0.460265	+	0.887781i	$0.347754\pi$
$54$		0			0
$55$	−4.70156			−0.633959
$56$		0			0
$57$		−	4.70156i		−	0.622737i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$		−	0.596876i		−	0.0764221i	−0.999270	−	0.0382111i	$-0.987834\pi$
$61$		−	0.596876i		−	0.0764221i	0.999270	−	0.0382111i	$-0.0121659\pi$
$62$		0			0
$63$			2.70156i			0.340365i
$64$		0			0
$65$		−	2.00000i		−	0.248069i
$66$		0			0
$67$	13.4031			1.63745			0.818726	−	0.574184i	$-0.194681\pi$
$67$	13.4031			1.63745			0.818726	+	0.574184i	$0.194681\pi$
$68$		0			0
$69$	7.40312			0.891231
$70$		0			0
$71$		−	15.4031i		−	1.82801i	−0.405698	−	0.914007i	$-0.632971\pi$
$71$		−	15.4031i		−	1.82801i	0.405698	−	0.914007i	$-0.367029\pi$
$72$		0			0
$73$		−	8.70156i		−	1.01844i	−0.860636	−	0.509220i	$-0.829934\pi$
$73$		−	8.70156i		−	1.01844i	0.860636	−	0.509220i	$-0.170066\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	12.7016			1.44748
$78$		0			0
$79$			2.00000i			0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$			2.00000i			0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	4.00000	−	1.00000i	0.433861	−	0.108465i
$86$		0			0
$87$	−6.70156			−0.718483
$88$		0			0
$89$	−7.40312			−0.784730			−0.392365	−	0.919810i	$-0.628343\pi$
$89$	−7.40312			−0.784730			−0.392365	+	0.919810i	$0.628343\pi$
$90$		0			0
$91$			5.40312i			0.566401i
$92$		0			0
$93$	−3.40312			−0.352888
$94$		0			0
$95$			4.70156i			0.482370i
$96$		0			0
$97$		−	17.4031i		−	1.76702i	−0.468413	−	0.883510i	$-0.655174\pi$
$97$		−	17.4031i		−	1.76702i	0.468413	−	0.883510i	$-0.344826\pi$
$98$		0			0
$99$		−	4.70156i		−	0.472525i
$100$		0			0
$101$	19.4031			1.93068			0.965342	−	0.260990i	$-0.0840490\pi$
$101$	19.4031			1.93068			0.965342	+	0.260990i	$0.0840490\pi$
$102$		0			0
$103$	−1.40312			−0.138254			−0.0691270	−	0.997608i	$-0.522021\pi$
$103$	−1.40312			−0.138254			−0.0691270	+	0.997608i	$0.522021\pi$
$104$		0			0
$105$		−	2.70156i		−	0.263645i
$106$		0			0
$107$		−	1.40312i		−	0.135645i	−0.997697	−	0.0678226i	$-0.978395\pi$
$107$		−	1.40312i		−	0.135645i	0.997697	−	0.0678226i	$-0.0216052\pi$
$108$		0			0
$109$			3.40312i			0.325960i	0.986629	+	0.162980i	$0.0521106\pi$
$109$			3.40312i			0.325960i	−0.986629	+	0.162980i	$0.947889\pi$
$110$		0			0
$111$	−10.7016			−1.01575
$112$		0			0
$113$		−	4.00000i		−	0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$		−	4.00000i		−	0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$		0			0
$115$	−7.40312			−0.690345
$116$		0			0
$117$	2.00000			0.184900
$118$		0			0
$119$	−10.8062	+	2.70156i	−0.990607	+	0.247652i
$120$		0			0
$121$	−11.1047			−1.00952
$122$		0			0
$123$	10.1047			0.911109
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.8062i			0.944146i	0.881559	+	0.472073i	$0.156494\pi$
$131$			10.8062i			0.944146i	−0.881559	+	0.472073i	$0.843506\pi$
$132$		0			0
$133$		−	12.7016i		−	1.10137i
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	−8.10469			−0.692430			−0.346215	−	0.938155i	$-0.612533\pi$
$137$	−8.10469			−0.692430			−0.346215	+	0.938155i	$0.612533\pi$
$138$		0			0
$139$		−	9.40312i		−	0.797563i	−0.917046	−	0.398781i	$-0.869433\pi$
$139$		−	9.40312i		−	0.797563i	0.917046	−	0.398781i	$-0.130567\pi$
$140$		0			0
$141$		−	3.29844i		−	0.277779i
$142$		0			0
$143$		−	9.40312i		−	0.786329i
$144$		0			0
$145$	6.70156			0.556534
$146$		0			0
$147$			0.298438i			0.0246147i
$148$		0			0
$149$	−22.2094			−1.81946			−0.909731	−	0.415197i	$-0.863713\pi$
$149$	−22.2094			−1.81946			−0.909731	+	0.415197i	$0.863713\pi$
$150$		0			0
$151$	4.70156			0.382608			0.191304	−	0.981531i	$-0.438728\pi$
$151$	4.70156			0.382608			0.191304	+	0.981531i	$0.438728\pi$
$152$		0			0
$153$	1.00000	+	4.00000i	0.0808452	+	0.323381i
$154$		0			0
$155$	3.40312			0.273346
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$		−	6.70156i		−	0.531468i
$160$		0			0
$161$	20.0000			1.57622
$162$		0			0
$163$		−	16.7016i		−	1.30817i	−0.756422	−	0.654084i	$-0.773054\pi$
$163$		−	16.7016i		−	1.30817i	0.756422	−	0.654084i	$-0.226946\pi$
$164$		0			0
$165$			4.70156i			0.366016i
$166$		0			0
$167$			8.59688i			0.665246i	0.943060	+	0.332623i	$0.107934\pi$
$167$			8.59688i			0.665246i	−0.943060	+	0.332623i	$0.892066\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−4.70156			−0.359537
$172$		0			0
$173$		−	18.0000i		−	1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$		−	18.0000i		−	1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$		0			0
$175$			2.70156i			0.204219i
$176$		0			0
$177$			4.00000i			0.300658i
$178$		0			0
$179$	10.8062			0.807697			0.403848	−	0.914826i	$-0.367672\pi$
$179$	10.8062			0.807697			0.403848	+	0.914826i	$0.367672\pi$
$180$		0			0
$181$		−	20.8062i		−	1.54652i	−0.634091	−	0.773258i	$-0.718626\pi$
$181$		−	20.8062i		−	1.54652i	0.634091	−	0.773258i	$-0.281374\pi$
$182$		0			0
$183$	−0.596876			−0.0441223
$184$		0			0
$185$	10.7016			0.786794
$186$		0			0
$187$	18.8062	−	4.70156i	1.37525	−	0.343812i
$188$		0			0
$189$	2.70156			0.196510
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	1.40312i		−	0.100999i	−0.998724	−	0.0504995i	$-0.983919\pi$
$193$		−	1.40312i		−	0.100999i	0.998724	−	0.0504995i	$-0.0160813\pi$
$194$		0			0
$195$	−2.00000			−0.143223
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$			22.2094i			1.57438i	0.616710	+	0.787190i	$0.288465\pi$
$199$			22.2094i			1.57438i	−0.616710	+	0.787190i	$0.711535\pi$
$200$		0			0
$201$		−	13.4031i		−	0.945383i
$202$		0			0
$203$	−18.1047			−1.27070
$204$		0			0
$205$	−10.1047			−0.705742
$206$		0			0
$207$		−	7.40312i		−	0.514553i
$208$		0			0
$209$			22.1047i			1.52901i
$210$		0			0
$211$			12.2094i			0.840528i	0.907402	+	0.420264i	$0.138063\pi$
$211$			12.2094i			0.840528i	−0.907402	+	0.420264i	$0.861937\pi$
$212$		0			0
$213$	−15.4031			−1.05540
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−9.19375			−0.624113
$218$		0			0
$219$	−8.70156			−0.587997
$220$		0			0
$221$	2.00000	+	8.00000i	0.134535	+	0.538138i
$222$		0			0
$223$	−1.40312			−0.0939601			−0.0469801	−	0.998896i	$-0.514960\pi$
$223$	−1.40312			−0.0939601			−0.0469801	+	0.998896i	$0.514960\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$		−	21.4031i		−	1.42058i	−0.703912	−	0.710288i	$-0.748565\pi$
$227$		−	21.4031i		−	1.42058i	0.703912	−	0.710288i	$-0.251435\pi$
$228$		0			0
$229$	9.50781			0.628294			0.314147	−	0.949374i	$-0.398282\pi$
$229$	9.50781			0.628294			0.314147	+	0.949374i	$0.398282\pi$
$230$		0			0
$231$		−	12.7016i		−	0.835701i
$232$		0			0
$233$		−	2.59688i		−	0.170127i	−0.996376	−	0.0850635i	$-0.972891\pi$
$233$		−	2.59688i		−	0.170127i	0.996376	−	0.0850635i	$-0.0271093\pi$
$234$		0			0
$235$			3.29844i			0.215166i
$236$		0			0
$237$	2.00000			0.129914
$238$		0			0
$239$	17.4031			1.12571			0.562857	−	0.826554i	$-0.309702\pi$
$239$	17.4031			1.12571			0.562857	+	0.826554i	$0.309702\pi$
$240$		0			0
$241$			5.40312i			0.348046i	0.984742	+	0.174023i	$0.0556767\pi$
$241$			5.40312i			0.348046i	−0.984742	+	0.174023i	$0.944323\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		−	0.298438i		−	0.0190665i
$246$		0			0
$247$	−9.40312			−0.598306
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−17.4031			−1.09848			−0.549238	−	0.835666i	$-0.685082\pi$
$251$	−17.4031			−1.09848			−0.549238	+	0.835666i	$0.685082\pi$
$252$		0			0
$253$	−34.8062			−2.18825
$254$		0			0
$255$	−1.00000	−	4.00000i	−0.0626224	−	0.250490i
$256$		0			0
$257$	16.8062			1.04834			0.524172	−	0.851612i	$-0.324375\pi$
$257$	16.8062			1.04834			0.524172	+	0.851612i	$0.324375\pi$
$258$		0			0
$259$	−28.9109			−1.79644
$260$		0			0
$261$			6.70156i			0.414816i
$262$		0			0
$263$	3.29844			0.203390			0.101695	−	0.994816i	$-0.467573\pi$
$263$	3.29844			0.203390			0.101695	+	0.994816i	$0.467573\pi$
$264$		0			0
$265$			6.70156i			0.411674i
$266$		0			0
$267$			7.40312i			0.453064i
$268$		0			0
$269$		−	24.1047i		−	1.46969i	−0.678236	−	0.734844i	$-0.737255\pi$
$269$		−	24.1047i		−	1.46969i	0.678236	−	0.734844i	$-0.262745\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$	5.40312			0.327012
$274$		0			0
$275$		−	4.70156i		−	0.283515i
$276$		0			0
$277$		−	7.19375i		−	0.432231i	−0.976368	−	0.216115i	$-0.930661\pi$
$277$		−	7.19375i		−	0.432231i	0.976368	−	0.216115i	$-0.0693387\pi$
$278$		0			0
$279$			3.40312i			0.203740i
$280$		0			0
$281$	16.8062			1.00258			0.501288	−	0.865280i	$-0.332860\pi$
$281$	16.8062			1.00258			0.501288	+	0.865280i	$0.332860\pi$
$282$		0			0
$283$		−	14.1047i		−	0.838437i	−0.907885	−	0.419218i	$-0.862304\pi$
$283$		−	14.1047i		−	0.838437i	0.907885	−	0.419218i	$-0.137696\pi$
$284$		0			0
$285$	4.70156			0.278497
$286$		0			0
$287$	27.2984			1.61138
$288$		0			0
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$	−17.4031			−1.02019
$292$		0			0
$293$	26.9109			1.57215			0.786077	−	0.618129i	$-0.212109\pi$
$293$	26.9109			1.57215			0.786077	+	0.618129i	$0.212109\pi$
$294$		0			0
$295$		−	4.00000i		−	0.232889i
$296$		0			0
$297$	−4.70156			−0.272812
$298$		0			0
$299$		−	14.8062i		−	0.856267i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	19.4031i		−	1.11468i
$304$		0			0
$305$	0.596876			0.0341770
$306$		0			0
$307$	24.0000			1.36975			0.684876	−	0.728659i	$-0.259856\pi$
$307$	24.0000			1.36975			0.684876	+	0.728659i	$0.259856\pi$
$308$		0			0
$309$			1.40312i			0.0798209i
$310$		0			0
$311$		−	13.2984i		−	0.754085i	−0.926196	−	0.377043i	$-0.876941\pi$
$311$		−	13.2984i		−	0.754085i	0.926196	−	0.377043i	$-0.123059\pi$
$312$		0			0
$313$			23.5078i			1.32874i	0.747404	+	0.664370i	$0.231300\pi$
$313$			23.5078i			1.32874i	−0.747404	+	0.664370i	$0.768700\pi$
$314$		0			0
$315$	−2.70156			−0.152216
$316$		0			0
$317$		−	23.6125i		−	1.32621i	−0.748526	−	0.663105i	$-0.769238\pi$
$317$		−	23.6125i		−	1.32621i	0.748526	−	0.663105i	$-0.230762\pi$
$318$		0			0
$319$	31.5078			1.76410
$320$		0			0
$321$	−1.40312			−0.0783148
$322$		0			0
$323$	−4.70156	−	18.8062i	−0.261602	−	1.04641i
$324$		0			0
$325$	2.00000			0.110940
$326$		0			0
$327$	3.40312			0.188193
$328$		0			0
$329$		−	8.91093i		−	0.491276i
$330$		0			0
$331$	3.50781			0.192807			0.0964034	−	0.995342i	$-0.469266\pi$
$331$	3.50781			0.192807			0.0964034	+	0.995342i	$0.469266\pi$
$332$		0			0
$333$			10.7016i			0.586442i
$334$		0			0
$335$			13.4031i			0.732291i
$336$		0			0
$337$			18.1047i			0.986225i	0.869966	+	0.493113i	$0.164141\pi$
$337$			18.1047i			0.986225i	−0.869966	+	0.493113i	$0.835859\pi$
$338$		0			0
$339$	−4.00000			−0.217250
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$		−	18.1047i		−	0.977561i
$344$		0			0
$345$			7.40312i			0.398571i
$346$		0			0
$347$			35.0156i			1.87974i	0.341536	+	0.939869i	$0.389053\pi$
$347$			35.0156i			1.87974i	−0.341536	+	0.939869i	$0.610947\pi$
$348$		0			0
$349$	−24.1047			−1.29029			−0.645147	−	0.764058i	$-0.723204\pi$
$349$	−24.1047			−1.29029			−0.645147	+	0.764058i	$0.723204\pi$
$350$		0			0
$351$		−	2.00000i		−	0.106752i
$352$		0			0
$353$	−24.1047			−1.28296			−0.641482	−	0.767139i	$-0.721680\pi$
$353$	−24.1047			−1.28296			−0.641482	+	0.767139i	$0.721680\pi$
$354$		0			0
$355$	15.4031			0.817513
$356$		0			0
$357$	2.70156	+	10.8062i	0.142982	+	0.571927i
$358$		0			0
$359$	4.20937			0.222162			0.111081	−	0.993811i	$-0.464569\pi$
$359$	4.20937			0.222162			0.111081	+	0.993811i	$0.464569\pi$
$360$		0			0
$361$	3.10469			0.163405
$362$		0			0
$363$			11.1047i			0.582845i
$364$		0			0
$365$	8.70156			0.455461
$366$		0			0
$367$			10.2094i			0.532925i	0.963845	+	0.266462i	$0.0858548\pi$
$367$			10.2094i			0.532925i	−0.963845	+	0.266462i	$0.914145\pi$
$368$		0			0
$369$		−	10.1047i		−	0.526029i
$370$		0			0
$371$		−	18.1047i		−	0.939948i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$			13.4031i			0.690296i
$378$		0			0
$379$		−	24.2094i		−	1.24355i	−0.783195	−	0.621776i	$-0.786411\pi$
$379$		−	24.2094i		−	1.24355i	0.783195	−	0.621776i	$-0.213589\pi$
$380$		0			0
$381$		−	8.00000i		−	0.409852i
$382$		0			0
$383$	16.9109			0.864108			0.432054	−	0.901848i	$-0.357789\pi$
$383$	16.9109			0.864108			0.432054	+	0.901848i	$0.357789\pi$
$384$		0			0
$385$			12.7016i			0.647332i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	31.6125			1.60282			0.801409	−	0.598117i	$-0.204084\pi$
$389$	31.6125			1.60282			0.801409	+	0.598117i	$0.204084\pi$
$390$		0			0
$391$	29.6125	−	7.40312i	1.49757	−	0.374392i
$392$		0			0
$393$	10.8062			0.545103
$394$		0			0
$395$	−2.00000			−0.100631
$396$		0			0
$397$		−	8.10469i		−	0.406763i	−0.979100	−	0.203381i	$-0.934807\pi$
$397$		−	8.10469i		−	0.406763i	0.979100	−	0.203381i	$-0.0651931\pi$
$398$		0			0
$399$	−12.7016			−0.635873
$400$		0			0
$401$		−	31.7172i		−	1.58388i	−0.610599	−	0.791940i	$-0.709071\pi$
$401$		−	31.7172i		−	1.58388i	0.610599	−	0.791940i	$-0.290929\pi$
$402$		0			0
$403$			6.80625i			0.339043i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$	50.3141			2.49398
$408$		0			0
$409$	−13.2984			−0.657565			−0.328783	−	0.944406i	$-0.606638\pi$
$409$	−13.2984			−0.657565			−0.328783	+	0.944406i	$0.606638\pi$
$410$		0			0
$411$			8.10469i			0.399775i
$412$		0			0
$413$			10.8062i			0.531741i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−9.40312			−0.460473
$418$		0			0
$419$		−	8.70156i		−	0.425099i	−0.977150	−	0.212550i	$-0.931823\pi$
$419$		−	8.70156i		−	0.425099i	0.977150	−	0.212550i	$-0.0681767\pi$
$420$		0			0
$421$	20.3141			0.990047			0.495023	−	0.868880i	$-0.335160\pi$
$421$	20.3141			0.990047			0.495023	+	0.868880i	$0.335160\pi$
$422$		0			0
$423$	−3.29844			−0.160376
$424$		0			0
$425$	1.00000	+	4.00000i	0.0485071	+	0.194029i
$426$		0			0
$427$	−1.61250			−0.0780342
$428$		0			0
$429$	−9.40312			−0.453987
$430$		0			0
$431$		−	28.1047i		−	1.35376i	−0.736096	−	0.676878i	$-0.763333\pi$
$431$		−	28.1047i		−	1.35376i	0.736096	−	0.676878i	$-0.236667\pi$
$432$		0			0
$433$	−19.6125			−0.942516			−0.471258	−	0.881995i	$-0.656200\pi$
$433$	−19.6125			−0.942516			−0.471258	+	0.881995i	$0.656200\pi$
$434$		0			0
$435$		−	6.70156i		−	0.321315i
$436$		0			0
$437$			34.8062i			1.66501i
$438$		0			0
$439$		−	7.40312i		−	0.353332i	−0.984271	−	0.176666i	$-0.943469\pi$
$439$		−	7.40312i		−	0.353332i	0.984271	−	0.176666i	$-0.0565312\pi$
$440$		0			0
$441$	0.298438			0.0142113
$442$		0			0
$443$	12.0000			0.570137			0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000			0.570137			0.285069	+	0.958507i	$0.407984\pi$
$444$		0			0
$445$		−	7.40312i		−	0.350942i
$446$		0			0
$447$			22.2094i			1.05047i
$448$		0			0
$449$			32.2094i			1.52005i	0.649891	+	0.760027i	$0.274814\pi$
$449$			32.2094i			1.52005i	−0.649891	+	0.760027i	$0.725186\pi$
$450$		0			0
$451$	−47.5078			−2.23706
$452$		0			0
$453$		−	4.70156i		−	0.220899i
$454$		0			0
$455$	−5.40312			−0.253302
$456$		0			0
$457$	−34.2094			−1.60025			−0.800123	−	0.599835i	$-0.795233\pi$
$457$	−34.2094			−1.60025			−0.800123	+	0.599835i	$0.795233\pi$
$458$		0			0
$459$	4.00000	−	1.00000i	0.186704	−	0.0466760i
$460$		0			0
$461$	−33.0156			−1.53769			−0.768845	−	0.639435i	$-0.779168\pi$
$461$	−33.0156			−1.53769			−0.768845	+	0.639435i	$0.779168\pi$
$462$		0			0
$463$	18.8062			0.874000			0.437000	−	0.899461i	$-0.356041\pi$
$463$	18.8062			0.874000			0.437000	+	0.899461i	$0.356041\pi$
$464$		0			0
$465$		−	3.40312i		−	0.157816i
$466$		0			0
$467$	−38.3141			−1.77296			−0.886482	−	0.462764i	$-0.846858\pi$
$467$	−38.3141			−1.77296			−0.886482	+	0.462764i	$0.846858\pi$
$468$		0			0
$469$		−	36.2094i		−	1.67199i
$470$		0			0
$471$		−	2.00000i		−	0.0921551i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.70156			−0.215722
$476$		0			0
$477$	−6.70156			−0.306843
$478$		0			0
$479$		−	4.59688i		−	0.210037i	−0.994470	−	0.105018i	$-0.966510\pi$
$479$		−	4.59688i		−	0.210037i	0.994470	−	0.105018i	$-0.0334901\pi$
$480$		0			0
$481$			21.4031i			0.975898i
$482$		0			0
$483$		−	20.0000i		−	0.910032i
$484$		0			0
$485$	17.4031			0.790235
$486$		0			0
$487$			20.5969i			0.933334i	0.884433	+	0.466667i	$0.154545\pi$
$487$			20.5969i			0.933334i	−0.884433	+	0.466667i	$0.845455\pi$
$488$		0			0
$489$	−16.7016			−0.755271
$490$		0			0
$491$	−10.8062			−0.487679			−0.243840	−	0.969816i	$-0.578407\pi$
$491$	−10.8062			−0.487679			−0.243840	+	0.969816i	$0.578407\pi$
$492$		0			0
$493$	−26.8062	+	6.70156i	−1.20729	+	0.301823i
$494$		0			0
$495$	4.70156			0.211320
$496$		0			0
$497$	−41.6125			−1.86658
$498$		0			0
$499$		−	1.40312i		−	0.0628125i	−0.999507	−	0.0314062i	$-0.990001\pi$
$499$		−	1.40312i		−	0.0628125i	0.999507	−	0.0314062i	$-0.00999856\pi$
$500$		0			0
$501$	8.59688			0.384080
$502$		0			0
$503$			22.0000i			0.980932i	0.871460	+	0.490466i	$0.163173\pi$
$503$			22.0000i			0.980932i	−0.871460	+	0.490466i	$0.836827\pi$
$504$		0			0
$505$			19.4031i			0.863428i
$506$		0			0
$507$			9.00000i			0.399704i
$508$		0			0
$509$	−15.6125			−0.692012			−0.346006	−	0.938232i	$-0.612462\pi$
$509$	−15.6125			−0.692012			−0.346006	+	0.938232i	$0.612462\pi$
$510$		0			0
$511$	−23.5078			−1.03992
$512$		0			0
$513$			4.70156i			0.207579i
$514$		0			0
$515$		−	1.40312i		−	0.0618290i
$516$		0			0
$517$			15.5078i			0.682033i
$518$		0			0
$519$	−18.0000			−0.790112
$520$		0			0
$521$			43.5078i			1.90611i	0.302794	+	0.953056i	$0.402080\pi$
$521$			43.5078i			1.90611i	−0.302794	+	0.953056i	$0.597920\pi$
$522$		0			0
$523$	−14.8062			−0.647432			−0.323716	−	0.946154i	$-0.604932\pi$
$523$	−14.8062			−0.647432			−0.323716	+	0.946154i	$0.604932\pi$
$524$		0			0
$525$	2.70156			0.117906
$526$		0			0
$527$	−13.6125	+	3.40312i	−0.592970	+	0.148242i
$528$		0			0
$529$	−31.8062			−1.38288
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		−	20.2094i		−	0.875365i
$534$		0			0
$535$	1.40312			0.0606624
$536$		0			0
$537$		−	10.8062i		−	0.466324i
$538$		0			0
$539$		−	1.40312i		−	0.0604368i
$540$		0			0
$541$			19.4031i			0.834205i	0.908859	+	0.417103i	$0.136954\pi$
$541$			19.4031i			0.834205i	−0.908859	+	0.417103i	$0.863046\pi$
$542$		0			0
$543$	−20.8062			−0.892882
$544$		0			0
$545$	−3.40312			−0.145774
$546$		0			0
$547$			8.91093i			0.381004i	0.981687	+	0.190502i	$0.0610116\pi$
$547$			8.91093i			0.381004i	−0.981687	+	0.190502i	$0.938988\pi$
$548$		0			0
$549$			0.596876i			0.0254740i
$550$		0			0
$551$		−	31.5078i		−	1.34228i
$552$		0			0
$553$	5.40312			0.229764
$554$		0			0
$555$		−	10.7016i		−	0.454256i
$556$		0			0
$557$	3.19375			0.135324			0.0676618	−	0.997708i	$-0.478446\pi$
$557$	3.19375			0.135324			0.0676618	+	0.997708i	$0.478446\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−4.70156	−	18.8062i	−0.198500	−	0.794000i
$562$		0			0
$563$	−11.2984			−0.476172			−0.238086	−	0.971244i	$-0.576520\pi$
$563$	−11.2984			−0.476172			−0.238086	+	0.971244i	$0.576520\pi$
$564$		0			0
$565$	4.00000			0.168281
$566$		0			0
$567$		−	2.70156i		−	0.113455i
$568$		0			0
$569$	43.8219			1.83711			0.918554	−	0.395295i	$-0.129358\pi$
$569$	43.8219			1.83711			0.918554	+	0.395295i	$0.129358\pi$
$570$		0			0
$571$		−	43.0156i		−	1.80015i	−0.435737	−	0.900074i	$-0.643512\pi$
$571$		−	43.0156i		−	1.80015i	0.435737	−	0.900074i	$-0.356488\pi$
$572$		0			0
$573$		−	8.00000i		−	0.334205i
$574$		0			0
$575$		−	7.40312i		−	0.308732i
$576$		0			0
$577$	32.8062			1.36574			0.682871	−	0.730539i	$-0.260731\pi$
$577$	32.8062			1.36574			0.682871	+	0.730539i	$0.260731\pi$
$578$		0			0
$579$	−1.40312			−0.0583119
$580$		0			0
$581$		0			0
$582$		0			0
$583$			31.5078i			1.30492i
$584$		0			0
$585$			2.00000i			0.0826898i
$586$		0			0
$587$	15.2984			0.631434			0.315717	−	0.948853i	$-0.397755\pi$
$587$	15.2984			0.631434			0.315717	+	0.948853i	$0.397755\pi$
$588$		0			0
$589$		−	16.0000i		−	0.659269i
$590$		0			0
$591$	−2.00000			−0.0822690
$592$		0			0
$593$	40.1047			1.64690			0.823451	−	0.567387i	$-0.192046\pi$
$593$	40.1047			1.64690			0.823451	+	0.567387i	$0.192046\pi$
$594$		0			0
$595$	−2.70156	−	10.8062i	−0.110753	−	0.443013i
$596$		0			0
$597$	22.2094			0.908969
$598$		0			0
$599$	−31.0156			−1.26726			−0.633632	−	0.773635i	$-0.718437\pi$
$599$	−31.0156			−1.26726			−0.633632	+	0.773635i	$0.718437\pi$
$600$		0			0
$601$			16.0000i			0.652654i	0.945257	+	0.326327i	$0.105811\pi$
$601$			16.0000i			0.652654i	−0.945257	+	0.326327i	$0.894189\pi$
$602$		0			0
$603$	−13.4031			−0.545817
$604$		0			0
$605$		−	11.1047i		−	0.451470i
$606$		0			0
$607$		−	46.9109i		−	1.90406i	−0.306011	−	0.952028i	$-0.598994\pi$
$607$		−	46.9109i		−	1.90406i	0.306011	−	0.952028i	$-0.401006\pi$
$608$		0			0
$609$			18.1047i			0.733639i
$610$		0			0
$611$	−6.59688			−0.266881
$612$		0			0
$613$	6.20937			0.250794			0.125397	−	0.992107i	$-0.459980\pi$
$613$	6.20937			0.250794			0.125397	+	0.992107i	$0.459980\pi$
$614$		0			0
$615$			10.1047i			0.407460i
$616$		0			0
$617$			44.0000i			1.77137i	0.464283	+	0.885687i	$0.346312\pi$
$617$			44.0000i			1.77137i	−0.464283	+	0.885687i	$0.653688\pi$
$618$		0			0
$619$			32.2094i			1.29460i	0.762234	+	0.647302i	$0.224103\pi$
$619$			32.2094i			1.29460i	−0.762234	+	0.647302i	$0.775897\pi$
$620$		0			0
$621$	−7.40312			−0.297077
$622$		0			0
$623$			20.0000i			0.801283i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	22.1047			0.882776
$628$		0			0
$629$	−42.8062	+	10.7016i	−1.70680	+	0.426699i
$630$		0			0
$631$	−19.2984			−0.768259			−0.384129	−	0.923279i	$-0.625498\pi$
$631$	−19.2984			−0.768259			−0.384129	+	0.923279i	$0.625498\pi$
$632$		0			0
$633$	12.2094			0.485279
$634$		0			0
$635$			8.00000i			0.317470i
$636$		0			0
$637$	0.596876			0.0236491
$638$		0			0
$639$			15.4031i			0.609338i
$640$		0			0
$641$			18.1047i			0.715092i	0.933896	+	0.357546i	$0.116386\pi$
$641$			18.1047i			0.715092i	−0.933896	+	0.357546i	$0.883614\pi$
$642$		0			0
$643$			12.9109i			0.509158i	0.967052	+	0.254579i	$0.0819368\pi$
$643$			12.9109i			0.509158i	−0.967052	+	0.254579i	$0.918063\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$		−	18.8062i		−	0.738210i
$650$		0			0
$651$			9.19375i			0.360332i
$652$		0			0
$653$		−	31.4031i		−	1.22890i	−0.788956	−	0.614450i	$-0.789378\pi$
$653$		−	31.4031i		−	1.22890i	0.788956	−	0.614450i	$-0.210622\pi$
$654$		0			0
$655$	−10.8062			−0.422235
$656$		0			0
$657$			8.70156i			0.339480i
$658$		0			0
$659$	29.4031			1.14538			0.572692	−	0.819771i	$-0.305899\pi$
$659$	29.4031			1.14538			0.572692	+	0.819771i	$0.305899\pi$
$660$		0			0
$661$	7.89531			0.307092			0.153546	−	0.988141i	$-0.450931\pi$
$661$	7.89531			0.307092			0.153546	+	0.988141i	$0.450931\pi$
$662$		0			0
$663$	8.00000	−	2.00000i	0.310694	−	0.0776736i
$664$		0			0
$665$	12.7016			0.492545
$666$		0			0
$667$	49.6125			1.92100
$668$		0			0
$669$			1.40312i			0.0542479i
$670$		0			0
$671$	2.80625			0.108334
$672$		0			0
$673$			3.79063i			0.146118i	0.997328	+	0.0730590i	$0.0232761\pi$
$673$			3.79063i			0.146118i	−0.997328	+	0.0730590i	$0.976724\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	12.5969i		−	0.484137i	−0.970259	−	0.242069i	$-0.922174\pi$
$677$		−	12.5969i		−	0.484137i	0.970259	−	0.242069i	$-0.0778259\pi$
$678$		0			0
$679$	−47.0156			−1.80429
$680$		0			0
$681$	−21.4031			−0.820170
$682$		0			0
$683$			44.4187i			1.69964i	0.527076	+	0.849818i	$0.323288\pi$
$683$			44.4187i			1.69964i	−0.527076	+	0.849818i	$0.676712\pi$
$684$		0			0
$685$		−	8.10469i		−	0.309664i
$686$		0			0
$687$		−	9.50781i		−	0.362746i
$688$		0			0
$689$	−13.4031			−0.510618
$690$		0			0
$691$			49.4031i			1.87938i	0.342023	+	0.939692i	$0.388888\pi$
$691$			49.4031i			1.87938i	−0.342023	+	0.939692i	$0.611112\pi$
$692$		0			0
$693$	−12.7016			−0.482492
$694$		0			0
$695$	9.40312			0.356681
$696$		0			0
$697$	40.4187	−	10.1047i	1.53097	−	0.382742i
$698$		0			0
$699$	−2.59688			−0.0982229
$700$		0			0
$701$	35.4031			1.33716			0.668579	−	0.743641i	$-0.266903\pi$
$701$	35.4031			1.33716			0.668579	+	0.743641i	$0.266903\pi$
$702$		0			0
$703$		−	50.3141i		−	1.89763i
$704$		0			0
$705$	3.29844			0.124226
$706$		0			0
$707$		−	52.4187i		−	1.97141i
$708$		0			0
$709$			30.4187i			1.14240i	0.820811	+	0.571200i	$0.193522\pi$
$709$			30.4187i			1.14240i	−0.820811	+	0.571200i	$0.806478\pi$
$710$		0			0
$711$		−	2.00000i		−	0.0750059i
$712$		0			0
$713$	25.1938			0.943513
$714$		0			0
$715$	9.40312			0.351657
$716$		0			0
$717$		−	17.4031i		−	0.649932i
$718$		0			0
$719$			21.7172i			0.809914i	0.914336	+	0.404957i	$0.132714\pi$
$719$			21.7172i			0.809914i	−0.914336	+	0.404957i	$0.867286\pi$
$720$		0			0
$721$			3.79063i			0.141170i
$722$		0			0
$723$	5.40312			0.200944
$724$		0			0
$725$			6.70156i			0.248890i
$726$		0			0
$727$	28.2094			1.04623			0.523114	−	0.852263i	$-0.324770\pi$
$727$	28.2094			1.04623			0.523114	+	0.852263i	$0.324770\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−23.4031			−0.864414			−0.432207	−	0.901774i	$-0.642265\pi$
$733$	−23.4031			−0.864414			−0.432207	+	0.901774i	$0.642265\pi$
$734$		0			0
$735$	−0.298438			−0.0110080
$736$		0			0
$737$			63.0156i			2.32121i
$738$		0			0
$739$	32.4187			1.19254			0.596271	−	0.802783i	$-0.296648\pi$
$739$	32.4187			1.19254			0.596271	+	0.802783i	$0.296648\pi$
$740$		0			0
$741$			9.40312i			0.345432i
$742$		0			0
$743$			7.19375i			0.263913i	0.991255	+	0.131957i	$0.0421259\pi$
$743$			7.19375i			0.263913i	−0.991255	+	0.131957i	$0.957874\pi$
$744$		0			0
$745$		−	22.2094i		−	0.813688i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.79063			−0.138507
$750$		0			0
$751$			34.0000i			1.24068i	0.784334	+	0.620339i	$0.213005\pi$
$751$			34.0000i			1.24068i	−0.784334	+	0.620339i	$0.786995\pi$
$752$		0			0
$753$			17.4031i			0.634205i
$754$		0			0
$755$			4.70156i			0.171107i
$756$		0			0
$757$	0.387503			0.0140840			0.00704202	−	0.999975i	$-0.497758\pi$
$757$	0.387503			0.0140840			0.00704202	+	0.999975i	$0.497758\pi$
$758$		0			0
$759$			34.8062i			1.26339i
$760$		0			0
$761$	−10.0000			−0.362500			−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000			−0.362500			−0.181250	+	0.983437i	$0.558014\pi$
$762$		0			0
$763$	9.19375			0.332836
$764$		0			0
$765$	−4.00000	+	1.00000i	−0.144620	+	0.0361551i
$766$		0			0
$767$	8.00000			0.288863
$768$		0			0
$769$	−4.31406			−0.155569			−0.0777845	−	0.996970i	$-0.524785\pi$
$769$	−4.31406			−0.155569			−0.0777845	+	0.996970i	$0.524785\pi$
$770$		0			0
$771$		−	16.8062i		−	0.605262i
$772$		0			0
$773$	4.10469			0.147635			0.0738177	−	0.997272i	$-0.476482\pi$
$773$	4.10469			0.147635			0.0738177	+	0.997272i	$0.476482\pi$
$774$		0			0
$775$			3.40312i			0.122244i
$776$		0			0
$777$			28.9109i			1.03717i
$778$		0			0
$779$			47.5078i			1.70214i
$780$		0			0
$781$	72.4187			2.59135
$782$		0			0
$783$	6.70156			0.239494
$784$		0			0
$785$			2.00000i			0.0713831i
$786$		0			0
$787$			14.1047i			0.502778i	0.967886	+	0.251389i	$0.0808873\pi$
$787$			14.1047i			0.502778i	−0.967886	+	0.251389i	$0.919113\pi$
$788$		0			0
$789$		−	3.29844i		−	0.117427i
$790$		0			0
$791$	−10.8062			−0.384226
$792$		0			0
$793$			1.19375i			0.0423914i
$794$		0			0
$795$	6.70156			0.237680
$796$		0			0
$797$	−2.70156			−0.0956942			−0.0478471	−	0.998855i	$-0.515236\pi$
$797$	−2.70156			−0.0956942			−0.0478471	+	0.998855i	$0.515236\pi$
$798$		0			0
$799$	−3.29844	−	13.1938i	−0.116690	−	0.466761i
$800$		0			0
$801$	7.40312			0.261577
$802$		0			0
$803$	40.9109			1.44372
$804$		0			0
$805$			20.0000i			0.704907i
$806$		0			0
$807$	−24.1047			−0.848525
$808$		0			0
$809$			27.0156i			0.949819i	0.880035	+	0.474909i	$0.157519\pi$
$809$			27.0156i			0.949819i	−0.880035	+	0.474909i	$0.842481\pi$
$810$		0			0
$811$			25.1938i			0.884672i	0.896849	+	0.442336i	$0.145850\pi$
$811$			25.1938i			0.884672i	−0.896849	+	0.442336i	$0.854150\pi$
$812$		0			0
$813$			16.0000i			0.561144i
$814$		0			0
$815$	16.7016			0.585030
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	5.40312i		−	0.188800i
$820$		0			0
$821$		−	19.6125i		−	0.684481i	−0.939612	−	0.342240i	$-0.888814\pi$
$821$		−	19.6125i		−	0.684481i	0.939612	−	0.342240i	$-0.111186\pi$
$822$		0			0
$823$			2.70156i			0.0941705i	0.998891	+	0.0470853i	$0.0149932\pi$
$823$			2.70156i			0.0941705i	−0.998891	+	0.0470853i	$0.985007\pi$
$824$		0			0
$825$	−4.70156			−0.163687
$826$		0			0
$827$			2.80625i			0.0975828i	0.998809	+	0.0487914i	$0.0155370\pi$
$827$			2.80625i			0.0975828i	−0.998809	+	0.0487914i	$0.984463\pi$
$828$		0			0
$829$	−10.4922			−0.364409			−0.182204	−	0.983261i	$-0.558323\pi$
$829$	−10.4922			−0.364409			−0.182204	+	0.983261i	$0.558323\pi$
$830$		0			0
$831$	−7.19375			−0.249548
$832$		0			0
$833$	0.298438	+	1.19375i	0.0103403	+	0.0413610i
$834$		0			0
$835$	−8.59688			−0.297507
$836$		0			0
$837$	3.40312			0.117629
$838$		0			0
$839$			31.8953i			1.10115i	0.834786	+	0.550574i	$0.185591\pi$
$839$			31.8953i			1.10115i	−0.834786	+	0.550574i	$0.814409\pi$
$840$		0			0
$841$	−15.9109			−0.548653
$842$		0			0
$843$		−	16.8062i		−	0.578838i
$844$		0			0
$845$		−	9.00000i		−	0.309609i
$846$		0			0
$847$			30.0000i			1.03081i
$848$		0			0
$849$	−14.1047			−0.484072
$850$		0			0
$851$	79.2250			2.71580
$852$		0			0
$853$		−	34.0000i		−	1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$		−	34.0000i		−	1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$		0			0
$855$		−	4.70156i		−	0.160790i
$856$		0			0
$857$		−	18.8062i		−	0.642409i	−0.947010	−	0.321205i	$-0.895912\pi$
$857$		−	18.8062i		−	0.642409i	0.947010	−	0.321205i	$-0.104088\pi$
$858$		0			0
$859$	−29.8953			−1.02001			−0.510007	−	0.860170i	$-0.670357\pi$
$859$	−29.8953			−1.02001			−0.510007	+	0.860170i	$0.670357\pi$
$860$		0			0
$861$		−	27.2984i		−	0.930328i
$862$		0			0
$863$	−14.1047			−0.480129			−0.240065	−	0.970757i	$-0.577169\pi$
$863$	−14.1047			−0.480129			−0.240065	+	0.970757i	$0.577169\pi$
$864$		0			0
$865$	18.0000			0.612018
$866$		0			0
$867$	8.00000	+	15.0000i	0.271694	+	0.509427i
$868$		0			0
$869$	−9.40312			−0.318979
$870$		0			0
$871$	−26.8062			−0.908295
$872$		0			0
$873$			17.4031i			0.589007i
$874$		0			0
$875$	−2.70156			−0.0913295
$876$		0			0
$877$		−	20.1047i		−	0.678887i	−0.940626	−	0.339444i	$-0.889761\pi$
$877$		−	20.1047i		−	0.678887i	0.940626	−	0.339444i	$-0.110239\pi$
$878$		0			0
$879$		−	26.9109i		−	0.907683i
$880$		0			0
$881$			12.7016i			0.427927i	0.976842	+	0.213963i	$0.0686373\pi$
$881$			12.7016i			0.427927i	−0.976842	+	0.213963i	$0.931363\pi$
$882$		0			0
$883$	−21.4031			−0.720272			−0.360136	−	0.932900i	$-0.617270\pi$
$883$	−21.4031			−0.720272			−0.360136	+	0.932900i	$0.617270\pi$
$884$		0			0
$885$	−4.00000			−0.134459
$886$		0			0
$887$			14.0000i			0.470074i	0.971986	+	0.235037i	$0.0755211\pi$
$887$			14.0000i			0.470074i	−0.971986	+	0.235037i	$0.924479\pi$
$888$		0			0
$889$		−	21.6125i		−	0.724860i
$890$		0			0
$891$			4.70156i			0.157508i
$892$		0			0
$893$	15.5078			0.518949
$894$		0			0
$895$			10.8062i			0.361213i
$896$		0			0
$897$	−14.8062			−0.494366
$898$		0			0
$899$	−22.8062			−0.760631
$900$		0			0
$901$	−6.70156	−	26.8062i	−0.223261	−	0.893046i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	20.8062			0.691623
$906$		0			0
$907$		−	6.10469i		−	0.202703i	−0.994851	−	0.101351i	$-0.967683\pi$
$907$		−	6.10469i		−	0.202703i	0.994851	−	0.101351i	$-0.0323166\pi$
$908$		0			0
$909$	−19.4031			−0.643561
$910$		0			0
$911$			44.5234i			1.47513i	0.675278	+	0.737564i	$0.264024\pi$
$911$			44.5234i			1.47513i	−0.675278	+	0.737564i	$0.735976\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	0.596876i		−	0.0197321i
$916$		0			0
$917$	29.1938			0.964063
$918$		0			0
$919$	8.91093			0.293945			0.146972	−	0.989141i	$-0.453047\pi$
$919$	8.91093			0.293945			0.146972	+	0.989141i	$0.453047\pi$
$920$		0			0
$921$		−	24.0000i		−	0.790827i
$922$		0			0
$923$			30.8062i			1.01400i
$924$		0			0
$925$			10.7016i			0.351865i
$926$		0			0
$927$	1.40312			0.0460846
$928$		0			0
$929$		−	5.89531i		−	0.193419i	−0.995313	−	0.0967095i	$-0.969168\pi$
$929$		−	5.89531i		−	0.193419i	0.995313	−	0.0967095i	$-0.0308318\pi$
$930$		0			0
$931$	−1.40312			−0.0459855
$932$		0			0
$933$	−13.2984			−0.435371
$934$		0			0
$935$	4.70156	+	18.8062i	0.153758	+	0.615030i
$936$		0			0
$937$	27.6125			0.902061			0.451030	−	0.892509i	$-0.351057\pi$
$937$	27.6125			0.902061			0.451030	+	0.892509i	$0.351057\pi$
$938$		0			0
$939$	23.5078			0.767148
$940$		0			0
$941$			46.4187i			1.51321i	0.653873	+	0.756604i	$0.273143\pi$
$941$			46.4187i			1.51321i	−0.653873	+	0.756604i	$0.726857\pi$
$942$		0			0
$943$	−74.8062			−2.43603
$944$		0			0
$945$			2.70156i			0.0878818i
$946$		0			0
$947$			17.4031i			0.565526i	0.959190	+	0.282763i	$0.0912509\pi$
$947$			17.4031i			0.565526i	−0.959190	+	0.282763i	$0.908749\pi$
$948$		0			0
$949$			17.4031i			0.564929i
$950$		0			0
$951$	−23.6125			−0.765688
$952$		0			0
$953$	3.89531			0.126182			0.0630908	−	0.998008i	$-0.479904\pi$
$953$	3.89531			0.126182			0.0630908	+	0.998008i	$0.479904\pi$
$954$		0			0
$955$			8.00000i			0.258874i
$956$		0			0
$957$		−	31.5078i		−	1.01850i
$958$		0			0
$959$			21.8953i			0.707037i
$960$		0			0
$961$	19.4187			0.626411
$962$		0			0
$963$			1.40312i			0.0452150i
$964$		0			0
$965$	1.40312			0.0451682
$966$		0			0
$967$	14.5969			0.469404			0.234702	−	0.972067i	$-0.424589\pi$
$967$	14.5969			0.469404			0.234702	+	0.972067i	$0.424589\pi$
$968$		0			0
$969$	−18.8062	+	4.70156i	−0.604144	+	0.151036i
$970$		0			0
$971$	−34.5969			−1.11027			−0.555133	−	0.831761i	$-0.687333\pi$
$971$	−34.5969			−1.11027			−0.555133	+	0.831761i	$0.687333\pi$
$972$		0			0
$973$	−25.4031			−0.814387
$974$		0			0
$975$		−	2.00000i		−	0.0640513i
$976$		0			0
$977$	47.6125			1.52326			0.761629	−	0.648013i	$-0.224400\pi$
$977$	47.6125			1.52326			0.761629	+	0.648013i	$0.224400\pi$
$978$		0			0
$979$		−	34.8062i		−	1.11241i
$980$		0			0
$981$		−	3.40312i		−	0.108653i
$982$		0			0
$983$			20.5969i			0.656938i	0.944515	+	0.328469i	$0.106533\pi$
$983$			20.5969i			0.656938i	−0.944515	+	0.328469i	$0.893467\pi$
$984$		0			0
$985$	2.00000			0.0637253
$986$		0			0
$987$	−8.91093			−0.283638
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	8.80625i		−	0.279740i	−0.990170	−	0.139870i	$-0.955332\pi$
$991$		−	8.80625i		−	0.279740i	0.990170	−	0.139870i	$-0.0446684\pi$
$992$		0			0
$993$		−	3.50781i		−	0.111317i
$994$		0			0
$995$	−22.2094			−0.704084
$996$		0			0
$997$			57.7172i			1.82792i	0.405803	+	0.913961i	$0.366992\pi$
$997$			57.7172i			1.82792i	−0.405803	+	0.913961i	$0.633008\pi$
$998$		0			0
$999$	10.7016			0.338582

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.h.l.3841.1		4
4.3	odd	2		2040.2.h.g.1801.4	yes	4
12.11	even	2		6120.2.h.j.1801.2		4
17.16	even	2	inner	4080.2.h.l.3841.4		4
68.67	odd	2		2040.2.h.g.1801.1	✓	4
204.203	even	2		6120.2.h.j.1801.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.h.g.1801.1	✓	4	68.67	odd	2
2040.2.h.g.1801.4	yes	4	4.3	odd	2
4080.2.h.l.3841.1		4	1.1	even	1	trivial
4080.2.h.l.3841.4		4	17.16	even	2	inner
6120.2.h.j.1801.2		4	12.11	even	2
6120.2.h.j.1801.3		4	204.203	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:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -2.70156i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.00000i q^{5} -2.70156i q^{7} -1.00000 q^{9} +4.70156i q^{11} -2.00000 q^{13} +1.00000 q^{15} +(-1.00000 - 4.00000i) q^{17} +4.70156 q^{19} -2.70156 q^{21} +7.40312i q^{23} -1.00000 q^{25} +1.00000i q^{27} -6.70156i q^{29} -3.40312i q^{31} +4.70156 q^{33} +2.70156 q^{35} -10.7016i q^{37} +2.00000i q^{39} +10.1047i q^{41} -1.00000i q^{45} +3.29844 q^{47} -0.298438 q^{49} +(-4.00000 + 1.00000i) q^{51} +6.70156 q^{53} -4.70156 q^{55} -4.70156i q^{57} -4.00000 q^{59} -0.596876i q^{61} +2.70156i q^{63} -2.00000i q^{65} +13.4031 q^{67} +7.40312 q^{69} -15.4031i q^{71} -8.70156i q^{73} +1.00000i q^{75} +12.7016 q^{77} +2.00000i q^{79} +1.00000 q^{81} +(4.00000 - 1.00000i) q^{85} -6.70156 q^{87} -7.40312 q^{89} +5.40312i q^{91} -3.40312 q^{93} +4.70156i q^{95} -17.4031i q^{97} -4.70156i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 8 q^{13} + 4 q^{15} - 4 q^{17} + 6 q^{19} + 2 q^{21} - 4 q^{25} + 6 q^{33} - 2 q^{35} + 26 q^{47} - 14 q^{49} - 16 q^{51} + 14 q^{53} - 6 q^{55} - 16 q^{59} + 28 q^{67} + 4 q^{69} + 38 q^{77} + 4 q^{81} + 16 q^{85} - 14 q^{87} - 4 q^{89} + 12 q^{93}+O(q^{100}) \) 4 * q - 4 * q^9 - 8 * q^13 + 4 * q^15 - 4 * q^17 + 6 * q^19 + 2 * q^21 - 4 * q^25 + 6 * q^33 - 2 * q^35 + 26 * q^47 - 14 * q^49 - 16 * q^51 + 14 * q^53 - 6 * q^55 - 16 * q^59 + 28 * q^67 + 4 * q^69 + 38 * q^77 + 4 * q^81 + 16 * q^85 - 14 * q^87 - 4 * q^89 + 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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