LMFDB - Embedded newform 5040.2.d.g.4591.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(4591,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("5040.4591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 328 x^{8} - 658 x^{7} + 1045 x^{6} - 1270 x^{5} + 1183 x^{4} + \cdots + 13 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 328x^8 - 658x^7 + 1045x^6 - 1270x^5 + 1183x^4 - 810x^3 + 376x^2 - 104*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1680)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4591.9
Root			$0.500000 + 1.18724i$ of defining polynomial
Character	$\chi$	$=$	5040.4591
Dual form			5040.2.d.g.4591.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +(-0.321212 - 2.62618i) q^{7} +O(q^{10})$	$q+1.00000i q^{5} +(-0.321212 - 2.62618i) q^{7} -1.68712i q^{11} +2.64242i q^{13} -5.53212i q^{17} +4.06801 q^{19} +8.17454i q^{23} -1.00000 q^{25} -7.02934 q^{29} +5.21924 q^{31} +(2.62618 - 0.321212i) q^{35} +4.93948 q^{37} +1.96688i q^{41} -0.922816i q^{43} +9.67177 q^{47} +(-6.79365 + 1.68712i) q^{49} +0.279755 q^{53} +1.68712 q^{55} +3.49659 q^{59} -14.1421i q^{61} -2.64242 q^{65} -15.3613i q^{67} +10.6112i q^{71} -0.380893i q^{73} +(-4.43068 + 0.541923i) q^{77} -0.0898605i q^{79} -14.4385 q^{83} +5.53212 q^{85} -5.54339i q^{89} +(6.93948 - 0.848778i) q^{91} +4.06801i q^{95} -10.8856i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{7}+O(q^{10}) $ 12 * q + 4 * q^7	$ 12 q + 4 q^{7} + 16 q^{19} - 12 q^{25} - 8 q^{29} - 32 q^{31} + 4 q^{35} - 16 q^{37} + 24 q^{47} - 4 q^{49} - 16 q^{53} + 24 q^{59} - 16 q^{65} - 24 q^{77} + 16 q^{83} - 8 q^{85} + 8 q^{91}+O(q^{100}) $ 12 * q + 4 * q^7 + 16 * q^19 - 12 * q^25 - 8 * q^29 - 32 * q^31 + 4 * q^35 - 16 * q^37 + 24 * q^47 - 4 * q^49 - 16 * q^53 + 24 * q^59 - 16 * q^65 - 24 * q^77 + 16 * q^83 - 8 * q^85 + 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−0.321212	−	2.62618i	−0.121407	−	0.992603i
$7$	−0.321212	−	2.62618i	−0.121407	−	0.992603i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.68712i		−	0.508686i	−0.967114	−	0.254343i	$-0.918141\pi$
$11$		−	1.68712i		−	0.508686i	0.967114	−	0.254343i	$-0.0818592\pi$
$12$		0			0
$13$			2.64242i			0.732876i	0.930442	+	0.366438i	$0.119423\pi$
$13$			2.64242i			0.732876i	−0.930442	+	0.366438i	$0.880577\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.53212i		−	1.34174i	−0.741577	−	0.670868i	$-0.765922\pi$
$17$		−	5.53212i		−	1.34174i	0.741577	−	0.670868i	$-0.234078\pi$
$18$		0			0
$19$	4.06801			0.933266			0.466633	−	0.884451i	$-0.345467\pi$
$19$	4.06801			0.933266			0.466633	+	0.884451i	$0.345467\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.17454i			1.70451i	0.523127	+	0.852255i	$0.324765\pi$
$23$			8.17454i			1.70451i	−0.523127	+	0.852255i	$0.675235\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.02934			−1.30532			−0.652658	−	0.757653i	$-0.726346\pi$
$29$	−7.02934			−1.30532			−0.652658	+	0.757653i	$0.726346\pi$
$30$		0			0
$31$	5.21924			0.937402			0.468701	−	0.883357i	$-0.344722\pi$
$31$	5.21924			0.937402			0.468701	+	0.883357i	$0.344722\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.62618	−	0.321212i	0.443905	−	0.0542947i
$36$		0			0
$37$	4.93948			0.812046			0.406023	−	0.913863i	$-0.366915\pi$
$37$	4.93948			0.812046			0.406023	+	0.913863i	$0.366915\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.96688i			0.307174i	0.988135	+	0.153587i	$0.0490826\pi$
$41$			1.96688i			0.307174i	−0.988135	+	0.153587i	$0.950917\pi$
$42$		0			0
$43$		−	0.922816i		−	0.140728i	−0.997521	−	0.0703641i	$-0.977584\pi$
$43$		−	0.922816i		−	0.140728i	0.997521	−	0.0703641i	$-0.0224161\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.67177			1.41077			0.705386	−	0.708823i	$-0.250774\pi$
$47$	9.67177			1.41077			0.705386	+	0.708823i	$0.250774\pi$
$48$		0			0
$49$	−6.79365	+	1.68712i	−0.970521	+	0.241017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.279755			0.0384273			0.0192136	−	0.999815i	$-0.493884\pi$
$53$	0.279755			0.0384273			0.0192136	+	0.999815i	$0.493884\pi$
$54$		0			0
$55$	1.68712			0.227491
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.49659			0.455217			0.227609	−	0.973753i	$-0.426909\pi$
$59$	3.49659			0.455217			0.227609	+	0.973753i	$0.426909\pi$
$60$		0			0
$61$		−	14.1421i		−	1.81070i	−0.424662	−	0.905352i	$-0.639607\pi$
$61$		−	14.1421i		−	1.81070i	0.424662	−	0.905352i	$-0.360393\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.64242			−0.327752
$66$		0			0
$67$		−	15.3613i		−	1.87668i	−0.345714	−	0.938340i	$-0.612363\pi$
$67$		−	15.3613i		−	1.87668i	0.345714	−	0.938340i	$-0.387637\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10.6112i			1.25932i	0.776870	+	0.629662i	$0.216806\pi$
$71$			10.6112i			1.25932i	−0.776870	+	0.629662i	$0.783194\pi$
$72$		0			0
$73$		−	0.380893i		−	0.0445802i	−0.999752	−	0.0222901i	$-0.992904\pi$
$73$		−	0.380893i		−	0.0445802i	0.999752	−	0.0222901i	$-0.00709574\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.43068	+	0.541923i	−0.504923	+	0.0617579i
$78$		0			0
$79$		−	0.0898605i		−	0.0101101i	−0.999987	−	0.00505505i	$-0.998391\pi$
$79$		−	0.0898605i		−	0.0101101i	0.999987	−	0.00505505i	$-0.00160908\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−14.4385			−1.58483			−0.792414	−	0.609984i	$-0.791176\pi$
$83$	−14.4385			−1.58483			−0.792414	+	0.609984i	$0.791176\pi$
$84$		0			0
$85$	5.53212			0.600042
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5.54339i		−	0.587599i	−0.955867	−	0.293799i	$-0.905080\pi$
$89$		−	5.54339i		−	0.587599i	0.955867	−	0.293799i	$-0.0949197\pi$
$90$		0			0
$91$	6.93948	−	0.848778i	0.727455	−	0.0889761i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			4.06801i			0.417369i
$96$		0			0
$97$		−	10.8856i		−	1.10527i	−0.833424	−	0.552633i	$-0.813623\pi$
$97$		−	10.8856i		−	1.10527i	0.833424	−	0.552633i	$-0.186377\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	6.21305i		−	0.618222i	−0.951026	−	0.309111i	$-0.899969\pi$
$101$		−	6.21305i		−	0.618222i	0.951026	−	0.309111i	$-0.100031\pi$
$102$		0			0
$103$	−1.06591			−0.105027			−0.0525134	−	0.998620i	$-0.516723\pi$
$103$	−1.06591			−0.105027			−0.0525134	+	0.998620i	$0.516723\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.03851i		−	0.390418i	−0.980762	−	0.195209i	$-0.937462\pi$
$107$		−	4.03851i		−	0.390418i	0.980762	−	0.195209i	$-0.0625385\pi$
$108$		0			0
$109$	20.0150			1.91709			0.958544	−	0.284944i	$-0.0919750\pi$
$109$	20.0150			1.91709			0.958544	+	0.284944i	$0.0919750\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.5822			−0.995490			−0.497745	−	0.867324i	$-0.665838\pi$
$113$	−10.5822			−0.995490			−0.497745	+	0.867324i	$0.665838\pi$
$114$		0			0
$115$	−8.17454			−0.762280
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.5283	+	1.77698i	−1.33181	+	0.162896i
$120$		0			0
$121$	8.15362			0.741239
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			6.40118i			0.568013i	0.958822	+	0.284006i	$0.0916637\pi$
$127$			6.40118i			0.568013i	−0.958822	+	0.284006i	$0.908336\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.78826			−0.156241			−0.0781205	−	0.996944i	$-0.524892\pi$
$131$	−1.78826			−0.156241			−0.0781205	+	0.996944i	$0.524892\pi$
$132$		0			0
$133$	−1.30669	−	10.6833i	−0.113305	−	0.926363i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.0227			0.856296			0.428148	−	0.903709i	$-0.359166\pi$
$137$	10.0227			0.856296			0.428148	+	0.903709i	$0.359166\pi$
$138$		0			0
$139$	8.84185			0.749956			0.374978	−	0.927034i	$-0.377650\pi$
$139$	8.84185			0.749956			0.374978	+	0.927034i	$0.377650\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.45809			0.372804
$144$		0			0
$145$		−	7.02934i		−	0.583755i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.74449			−0.142915			−0.0714573	−	0.997444i	$-0.522765\pi$
$149$	−1.74449			−0.142915			−0.0714573	+	0.997444i	$0.522765\pi$
$150$		0			0
$151$		−	10.1793i		−	0.828376i	−0.910191	−	0.414188i	$-0.864066\pi$
$151$		−	10.1793i		−	0.828376i	0.910191	−	0.414188i	$-0.135934\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			5.21924i			0.419219i
$156$		0			0
$157$		−	18.9265i		−	1.51050i	−0.655436	−	0.755251i	$-0.727515\pi$
$157$		−	18.9265i		−	1.51050i	0.655436	−	0.755251i	$-0.272485\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	21.4678	−	2.62576i	1.69190	−	0.206939i
$162$		0			0
$163$		−	2.07164i		−	0.162263i	−0.996703	−	0.0811315i	$-0.974147\pi$
$163$		−	2.07164i		−	0.162263i	0.996703	−	0.0811315i	$-0.0258534\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.8078			1.06848			0.534240	−	0.845333i	$-0.320598\pi$
$167$	13.8078			1.06848			0.534240	+	0.845333i	$0.320598\pi$
$168$		0			0
$169$	6.01760			0.462892
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	16.7178i		−	1.27103i	−0.772089	−	0.635514i	$-0.780788\pi$
$173$		−	16.7178i		−	1.27103i	0.772089	−	0.635514i	$-0.219212\pi$
$174$		0			0
$175$	0.321212	+	2.62618i	0.0242813	+	0.198521i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	0.538301i		−	0.0402345i	−0.999798	−	0.0201172i	$-0.993596\pi$
$179$		−	0.538301i		−	0.0402345i	0.999798	−	0.0201172i	$-0.00640395\pi$
$180$		0			0
$181$			0.0833689i			0.00619676i	0.999995	+	0.00309838i	$0.000986246\pi$
$181$			0.0833689i			0.00619676i	−0.999995	+	0.00309838i	$0.999014\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			4.93948i			0.363158i
$186$		0			0
$187$	−9.33335			−0.682522
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.6660i		−	0.916482i	−0.888828	−	0.458241i	$-0.848480\pi$
$191$		−	12.6660i		−	0.916482i	0.888828	−	0.458241i	$-0.151520\pi$
$192$		0			0
$193$	−26.3611			−1.89752			−0.948758	−	0.316004i	$-0.897659\pi$
$193$	−26.3611			−1.89752			−0.948758	+	0.316004i	$0.897659\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.4873			−1.10342			−0.551711	−	0.834036i	$-0.686025\pi$
$197$	−15.4873			−1.10342			−0.551711	+	0.834036i	$0.686025\pi$
$198$		0			0
$199$	22.7184			1.61047			0.805233	−	0.592959i	$-0.202040\pi$
$199$	22.7184			1.61047			0.805233	+	0.592959i	$0.202040\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.25791	+	18.4603i	0.158474	+	1.29566i
$204$		0			0
$205$	−1.96688			−0.137373
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	6.86323i		−	0.474740i
$210$		0			0
$211$			10.3815i			0.714694i	0.933972	+	0.357347i	$0.116319\pi$
$211$			10.3815i			0.714694i	−0.933972	+	0.357347i	$0.883681\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.922816			0.0629356
$216$		0			0
$217$	−1.67648	−	13.7067i	−0.113807	−	0.930468i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	14.6182			0.983326
$222$		0			0
$223$	18.7011			1.25232			0.626159	−	0.779695i	$-0.284626\pi$
$223$	18.7011			1.25232			0.626159	+	0.779695i	$0.284626\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.1947			1.20763			0.603813	−	0.797126i	$-0.293647\pi$
$227$	18.1947			1.20763			0.603813	+	0.797126i	$0.293647\pi$
$228$		0			0
$229$		−	12.4761i		−	0.824447i	−0.911083	−	0.412223i	$-0.864752\pi$
$229$		−	12.4761i		−	0.824447i	0.911083	−	0.412223i	$-0.135248\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.0099			0.852307			0.426153	−	0.904651i	$-0.359868\pi$
$233$	13.0099			0.852307			0.426153	+	0.904651i	$0.359868\pi$
$234$		0			0
$235$			9.67177i			0.630916i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.5518i		−	1.26470i	−0.774682	−	0.632351i	$-0.782090\pi$
$239$		−	19.5518i		−	1.26470i	0.774682	−	0.632351i	$-0.217910\pi$
$240$		0			0
$241$			10.0894i			0.649915i	0.945729	+	0.324957i	$0.105350\pi$
$241$			10.0894i			0.649915i	−0.945729	+	0.324957i	$0.894650\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.68712	−	6.79365i	−0.107786	−	0.434030i
$246$		0			0
$247$			10.7494i			0.683969i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.78826			0.112874			0.0564370	−	0.998406i	$-0.482026\pi$
$251$	1.78826			0.112874			0.0564370	+	0.998406i	$0.482026\pi$
$252$		0			0
$253$	13.7914			0.867060
$254$		0			0
$255$		0			0
$256$		0			0
$257$			29.9808i			1.87015i	0.354451	+	0.935075i	$0.384668\pi$
$257$			29.9808i			1.87015i	−0.354451	+	0.935075i	$0.615332\pi$
$258$		0			0
$259$	−1.58662	−	12.9720i	−0.0985878	−	0.806039i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.18211i			0.134554i	0.997734	+	0.0672772i	$0.0214312\pi$
$263$			2.18211i			0.134554i	−0.997734	+	0.0672772i	$0.978569\pi$
$264$		0			0
$265$			0.279755i			0.0171852i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	10.5514i		−	0.643327i	−0.946854	−	0.321664i	$-0.895758\pi$
$269$		−	10.5514i		−	0.643327i	0.946854	−	0.321664i	$-0.104242\pi$
$270$		0			0
$271$	−24.8065			−1.50689			−0.753445	−	0.657511i	$-0.771609\pi$
$271$	−24.8065			−1.50689			−0.753445	+	0.657511i	$0.771609\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.68712i			0.101737i
$276$		0			0
$277$	−19.4216			−1.16693			−0.583467	−	0.812137i	$-0.698304\pi$
$277$	−19.4216			−1.16693			−0.583467	+	0.812137i	$0.698304\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.71255			0.102162			0.0510809	−	0.998695i	$-0.483733\pi$
$281$	1.71255			0.102162			0.0510809	+	0.998695i	$0.483733\pi$
$282$		0			0
$283$	26.2055			1.55775			0.778877	−	0.627176i	$-0.215790\pi$
$283$	26.2055			1.55775			0.778877	+	0.627176i	$0.215790\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.16537	−	0.631784i	0.304902	−	0.0372930i
$288$		0			0
$289$	−13.6043			−0.800253
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.57652i		−	0.559466i	−0.960078	−	0.279733i	$-0.909754\pi$
$293$		−	9.57652i		−	0.559466i	0.960078	−	0.279733i	$-0.0902460\pi$
$294$		0			0
$295$			3.49659i			0.203579i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−21.6006			−1.24919
$300$		0			0
$301$	−2.42348	+	0.296420i	−0.139687	+	0.0170853i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	14.1421			0.809772
$306$		0			0
$307$	−20.4864			−1.16922			−0.584610	−	0.811315i	$-0.698752\pi$
$307$	−20.4864			−1.16922			−0.584610	+	0.811315i	$0.698752\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	15.8688			0.899836			0.449918	−	0.893070i	$-0.351453\pi$
$311$	15.8688			0.899836			0.449918	+	0.893070i	$0.351453\pi$
$312$		0			0
$313$		−	12.6165i		−	0.713127i	−0.934271	−	0.356563i	$-0.883948\pi$
$313$		−	12.6165i		−	0.713127i	0.934271	−	0.356563i	$-0.116052\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−27.3466			−1.53594			−0.767969	−	0.640487i	$-0.778733\pi$
$317$	−27.3466			−1.53594			−0.767969	+	0.640487i	$0.778733\pi$
$318$		0			0
$319$			11.8593i			0.663996i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	22.5047i		−	1.25220i
$324$		0			0
$325$		−	2.64242i		−	0.146575i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.10669	−	25.3998i	−0.171277	−	1.40034i
$330$		0			0
$331$			19.2537i			1.05828i	0.848535	+	0.529139i	$0.177485\pi$
$331$			19.2537i			1.05828i	−0.848535	+	0.529139i	$0.822515\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15.3613			0.839277
$336$		0			0
$337$	7.85513			0.427896			0.213948	−	0.976845i	$-0.431368\pi$
$337$	7.85513			0.427896			0.213948	+	0.976845i	$0.431368\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	8.80548i		−	0.476843i
$342$		0			0
$343$	6.61288	+	17.2994i	0.357062	+	0.934081i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	13.0796i		−	0.702150i	−0.936347	−	0.351075i	$-0.885816\pi$
$347$		−	13.0796i		−	0.702150i	0.936347	−	0.351075i	$-0.114184\pi$
$348$		0			0
$349$			16.4397i			0.879997i	0.897999	+	0.439998i	$0.145021\pi$
$349$			16.4397i			0.879997i	−0.897999	+	0.439998i	$0.854979\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			10.4705i			0.557289i	0.960394	+	0.278644i	$0.0898851\pi$
$353$			10.4705i			0.557289i	−0.960394	+	0.278644i	$0.910115\pi$
$354$		0			0
$355$	−10.6112			−0.563186
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	23.1058i		−	1.21948i	−0.792603	−	0.609738i	$-0.791275\pi$
$359$		−	23.1058i		−	1.21948i	0.792603	−	0.609738i	$-0.208725\pi$
$360$		0			0
$361$	−2.45126			−0.129014
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.380893			0.0199369
$366$		0			0
$367$	10.8088			0.564217			0.282108	−	0.959383i	$-0.408966\pi$
$367$	10.8088			0.564217			0.282108	+	0.959383i	$0.408966\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.0898605	−	0.734686i	−0.00466533	−	0.0381430i
$372$		0			0
$373$	21.4990			1.11318			0.556588	−	0.830789i	$-0.312110\pi$
$373$	21.4990			1.11318			0.556588	+	0.830789i	$0.312110\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	18.5745i		−	0.956635i
$378$		0			0
$379$			23.7362i			1.21924i	0.792692	+	0.609622i	$0.208679\pi$
$379$			23.7362i			1.21924i	−0.792692	+	0.609622i	$0.791321\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.7491			0.600351			0.300176	−	0.953884i	$-0.402955\pi$
$383$	11.7491			0.600351			0.300176	+	0.953884i	$0.402955\pi$
$384$		0			0
$385$	−0.541923	−	4.43068i	−0.0276190	−	0.225808i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.9191			1.06064			0.530320	−	0.847798i	$-0.322072\pi$
$389$	20.9191			1.06064			0.530320	+	0.847798i	$0.322072\pi$
$390$		0			0
$391$	45.2225			2.28700
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.0898605			0.00452137
$396$		0			0
$397$		−	3.71550i		−	0.186475i	−0.995644	−	0.0932377i	$-0.970278\pi$
$397$		−	3.71550i		−	0.186475i	0.995644	−	0.0932377i	$-0.0297217\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	33.0561			1.65074			0.825371	−	0.564591i	$-0.190966\pi$
$401$	33.0561			1.65074			0.825371	+	0.564591i	$0.190966\pi$
$402$		0			0
$403$			13.7914i			0.687000i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.33350i		−	0.413076i
$408$		0			0
$409$		−	14.6515i		−	0.724471i	−0.932087	−	0.362236i	$-0.882014\pi$
$409$		−	14.6515i		−	0.724471i	0.932087	−	0.362236i	$-0.117986\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.12315	−	9.18267i	−0.0552664	−	0.451850i
$414$		0			0
$415$		−	14.4385i		−	0.708757i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.0218345			0.00106669			0.000533343	−	1.00000i	$-0.499830\pi$
$419$	0.0218345			0.00106669			0.000533343		1.00000i	$0.499830\pi$
$420$		0			0
$421$	22.9206			1.11708			0.558542	−	0.829476i	$-0.311361\pi$
$421$	22.9206			1.11708			0.558542	+	0.829476i	$0.311361\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			5.53212i			0.268347i
$426$		0			0
$427$	−37.1396	+	4.54259i	−1.79731	+	0.219832i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			38.1985i			1.83996i	0.391967	+	0.919979i	$0.371795\pi$
$431$			38.1985i			1.83996i	−0.391967	+	0.919979i	$0.628205\pi$
$432$		0			0
$433$		−	35.8712i		−	1.72386i	−0.507026	−	0.861931i	$-0.669255\pi$
$433$		−	35.8712i		−	1.72386i	0.507026	−	0.861931i	$-0.330745\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			33.2541i			1.59076i
$438$		0			0
$439$	41.5602			1.98356			0.991779	−	0.127962i	$-0.0408435\pi$
$439$	41.5602			1.98356			0.991779	+	0.127962i	$0.0408435\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	17.5304i		−	0.832896i	−0.909159	−	0.416448i	$-0.863275\pi$
$443$		−	17.5304i		−	0.832896i	0.909159	−	0.416448i	$-0.136725\pi$
$444$		0			0
$445$	5.54339			0.262782
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.9056			−0.609055			−0.304528	−	0.952504i	$-0.598499\pi$
$449$	−12.9056			−0.609055			−0.304528	+	0.952504i	$0.598499\pi$
$450$		0			0
$451$	3.31836			0.156255
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.848778	+	6.93948i	0.0397913	+	0.325328i
$456$		0			0
$457$	−27.3239			−1.27816			−0.639079	−	0.769141i	$-0.720684\pi$
$457$	−27.3239			−1.27816			−0.639079	+	0.769141i	$0.720684\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			11.7144i			0.545594i	0.962072	+	0.272797i	$0.0879488\pi$
$461$			11.7144i			0.545594i	−0.962072	+	0.272797i	$0.912051\pi$
$462$		0			0
$463$			2.43763i			0.113286i	0.998394	+	0.0566431i	$0.0180397\pi$
$463$			2.43763i			0.113286i	−0.998394	+	0.0566431i	$0.981960\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.9974			0.601448			0.300724	−	0.953711i	$-0.402772\pi$
$467$	12.9974			0.601448			0.300724	+	0.953711i	$0.402772\pi$
$468$		0			0
$469$	−40.3415	+	4.93423i	−1.86280	+	0.227841i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.55690			−0.0715864
$474$		0			0
$475$	−4.06801			−0.186653
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.6250			−0.896690			−0.448345	−	0.893861i	$-0.647986\pi$
$479$	−19.6250			−0.896690			−0.448345	+	0.893861i	$0.647986\pi$
$480$		0			0
$481$			13.0522i			0.595129i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.8856			0.494290
$486$		0			0
$487$		−	16.8370i		−	0.762959i	−0.924377	−	0.381479i	$-0.875415\pi$
$487$		−	16.8370i		−	0.762959i	0.924377	−	0.381479i	$-0.124585\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	21.4104i		−	0.966240i	−0.875554	−	0.483120i	$-0.839504\pi$
$491$		−	21.4104i		−	0.966240i	0.875554	−	0.483120i	$-0.160496\pi$
$492$		0			0
$493$			38.8871i			1.75139i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	27.8670	−	3.40846i	1.25001	−	0.152890i
$498$		0			0
$499$		−	10.6596i		−	0.477187i	−0.971120	−	0.238594i	$-0.923314\pi$
$499$		−	10.6596i		−	0.477187i	0.971120	−	0.238594i	$-0.0766864\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.35914			−0.0606009			−0.0303004	−	0.999541i	$-0.509646\pi$
$503$	−1.35914			−0.0606009			−0.0303004	+	0.999541i	$0.509646\pi$
$504$		0			0
$505$	6.21305			0.276477
$506$		0			0
$507$		0			0
$508$		0			0
$509$			11.4104i			0.505756i	0.967498	+	0.252878i	$0.0813771\pi$
$509$			11.4104i			0.505756i	−0.967498	+	0.252878i	$0.918623\pi$
$510$		0			0
$511$	−1.00029	+	0.122347i	−0.0442504	+	0.00541233i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	1.06591i		−	0.0469694i
$516$		0			0
$517$		−	16.3174i		−	0.717640i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	9.83474i		−	0.430868i	−0.976518	−	0.215434i	$-0.930883\pi$
$521$		−	9.83474i		−	0.430868i	0.976518	−	0.215434i	$-0.0691166\pi$
$522$		0			0
$523$	−13.0176			−0.569220			−0.284610	−	0.958643i	$-0.591864\pi$
$523$	−13.0176			−0.569220			−0.284610	+	0.958643i	$0.591864\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	28.8734i		−	1.25775i
$528$		0			0
$529$	−43.8231			−1.90535
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.19732			−0.225121
$534$		0			0
$535$	4.03851			0.174600
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.84638	+	11.4617i	0.122602	+	0.493690i
$540$		0			0
$541$	26.7644			1.15069			0.575346	−	0.817910i	$-0.304867\pi$
$541$	26.7644			1.15069			0.575346	+	0.817910i	$0.304867\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			20.0150i			0.857348i
$546$		0			0
$547$			16.6210i			0.710661i	0.934741	+	0.355330i	$0.115632\pi$
$547$			16.6210i			0.710661i	−0.934741	+	0.355330i	$0.884368\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−28.5955			−1.21821
$552$		0			0
$553$	−0.235990	+	0.0288643i	−0.0100353	+	0.00122743i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−35.6485			−1.51047			−0.755237	−	0.655452i	$-0.772478\pi$
$557$	−35.6485			−1.51047			−0.755237	+	0.655452i	$0.772478\pi$
$558$		0			0
$559$	2.43847			0.103136
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.1530			−0.807204			−0.403602	−	0.914935i	$-0.632242\pi$
$563$	−19.1530			−0.807204			−0.403602	+	0.914935i	$0.632242\pi$
$564$		0			0
$565$		−	10.5822i		−	0.445196i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.7579			−0.912140			−0.456070	−	0.889944i	$-0.650743\pi$
$569$	−21.7579			−0.912140			−0.456070	+	0.889944i	$0.650743\pi$
$570$		0			0
$571$		−	12.5160i		−	0.523777i	−0.965098	−	0.261888i	$-0.915655\pi$
$571$		−	12.5160i		−	0.523777i	0.965098	−	0.261888i	$-0.0843452\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	8.17454i		−	0.340902i
$576$		0			0
$577$			24.9404i			1.03828i	0.854688	+	0.519141i	$0.173748\pi$
$577$			24.9404i			1.03828i	−0.854688	+	0.519141i	$0.826252\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.63781	+	37.9180i	0.192409	+	1.57310i
$582$		0			0
$583$		−	0.471980i		−	0.0195474i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.76701			0.155481			0.0777406	−	0.996974i	$-0.475229\pi$
$587$	3.76701			0.155481			0.0777406	+	0.996974i	$0.475229\pi$
$588$		0			0
$589$	21.2319			0.874846
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	40.7552i		−	1.67362i	−0.547497	−	0.836808i	$-0.684419\pi$
$593$		−	40.7552i		−	1.67362i	0.547497	−	0.836808i	$-0.315581\pi$
$594$		0			0
$595$	−1.77698	−	14.5283i	−0.0728491	−	0.595604i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			26.3094i			1.07497i	0.843272	+	0.537486i	$0.180626\pi$
$599$			26.3094i			1.07497i	−0.843272	+	0.537486i	$0.819374\pi$
$600$		0			0
$601$		−	12.9465i		−	0.528101i	−0.964509	−	0.264050i	$-0.914942\pi$
$601$		−	12.9465i		−	0.528101i	0.964509	−	0.264050i	$-0.0850585\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			8.15362i			0.331492i
$606$		0			0
$607$	−8.09681			−0.328639			−0.164320	−	0.986407i	$-0.552543\pi$
$607$	−8.09681			−0.328639			−0.164320	+	0.986407i	$0.552543\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			25.5569i			1.03392i
$612$		0			0
$613$	−41.9476			−1.69425			−0.847125	−	0.531394i	$-0.821668\pi$
$613$	−41.9476			−1.69425			−0.847125	+	0.531394i	$0.821668\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−47.8424			−1.92606			−0.963032	−	0.269389i	$-0.913178\pi$
$617$	−47.8424			−1.92606			−0.963032	+	0.269389i	$0.913178\pi$
$618$		0			0
$619$	12.2585			0.492711			0.246356	−	0.969180i	$-0.420767\pi$
$619$	12.2585			0.492711			0.246356	+	0.969180i	$0.420767\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.5580	+	1.78060i	−0.583252	+	0.0713384i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	27.3258i		−	1.08955i
$630$		0			0
$631$			31.2870i			1.24552i	0.782415	+	0.622758i	$0.213988\pi$
$631$			31.2870i			1.24552i	−0.782415	+	0.622758i	$0.786012\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.40118			−0.254023
$636$		0			0
$637$	−4.45809	−	17.9517i	−0.176636	−	0.711272i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.4022			0.608351			0.304175	−	0.952616i	$-0.401619\pi$
$641$	15.4022			0.608351			0.304175	+	0.952616i	$0.401619\pi$
$642$		0			0
$643$	−34.3761			−1.35566			−0.677831	−	0.735218i	$-0.737080\pi$
$643$	−34.3761			−1.35566			−0.677831	+	0.735218i	$0.737080\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	47.0411			1.84937			0.924687	−	0.380727i	$-0.124326\pi$
$647$	47.0411			1.84937			0.924687	+	0.380727i	$0.124326\pi$
$648$		0			0
$649$		−	5.89917i		−	0.231563i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−26.7906			−1.04840			−0.524198	−	0.851597i	$-0.675635\pi$
$653$	−26.7906			−1.04840			−0.524198	+	0.851597i	$0.675635\pi$
$654$		0			0
$655$		−	1.78826i		−	0.0698731i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			0.888654i			0.0346171i	0.999850	+	0.0173085i	$0.00550975\pi$
$659$			0.888654i			0.0346171i	−0.999850	+	0.0173085i	$0.994490\pi$
$660$		0			0
$661$		−	28.2341i		−	1.09818i	−0.835764	−	0.549089i	$-0.814975\pi$
$661$		−	28.2341i		−	1.09818i	0.835764	−	0.549089i	$-0.185025\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.6833	−	1.30669i	0.414282	−	0.0506714i
$666$		0			0
$667$		−	57.4616i		−	2.22492i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−23.8593			−0.921080
$672$		0			0
$673$	19.0519			0.734395			0.367198	−	0.930143i	$-0.380317\pi$
$673$	19.0519			0.734395			0.367198	+	0.930143i	$0.380317\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	36.1925i		−	1.39099i	−0.718531	−	0.695495i	$-0.755185\pi$
$677$		−	36.1925i		−	1.39099i	0.718531	−	0.695495i	$-0.244815\pi$
$678$		0			0
$679$	−28.5876	+	3.49659i	−1.09709	+	0.134187i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	30.4770i		−	1.16617i	−0.812411	−	0.583085i	$-0.801846\pi$
$683$		−	30.4770i		−	1.16617i	0.812411	−	0.583085i	$-0.198154\pi$
$684$		0			0
$685$			10.0227i			0.382947i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			0.739231i			0.0281624i
$690$		0			0
$691$	−43.7242			−1.66335			−0.831673	−	0.555266i	$-0.812617\pi$
$691$	−43.7242			−1.66335			−0.831673	+	0.555266i	$0.812617\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			8.84185i			0.335390i
$696$		0			0
$697$	10.8810			0.412147
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.87319			−0.146288			−0.0731441	−	0.997321i	$-0.523303\pi$
$701$	−3.87319			−0.146288			−0.0731441	+	0.997321i	$0.523303\pi$
$702$		0			0
$703$	20.0939			0.757855
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−16.3166	+	1.99571i	−0.613649	+	0.0750562i
$708$		0			0
$709$	15.8066			0.593631			0.296815	−	0.954935i	$-0.404075\pi$
$709$	15.8066			0.593631			0.296815	+	0.954935i	$0.404075\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			42.6648i			1.59781i
$714$		0			0
$715$			4.45809i			0.166723i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11.5671			0.431380			0.215690	−	0.976462i	$-0.430800\pi$
$719$	11.5671			0.431380			0.215690	+	0.976462i	$0.430800\pi$
$720$		0			0
$721$	0.342382	+	2.79926i	0.0127510	+	0.104250i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.02934			0.261063
$726$		0			0
$727$	−11.6356			−0.431541			−0.215770	−	0.976444i	$-0.569226\pi$
$727$	−11.6356			−0.431541			−0.215770	+	0.976444i	$0.569226\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.10513			−0.188820
$732$		0			0
$733$			39.5670i			1.46144i	0.682678	+	0.730719i	$0.260815\pi$
$733$			39.5670i			1.46144i	−0.682678	+	0.730719i	$0.739185\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.9163			−0.954641
$738$		0			0
$739$		−	4.14668i		−	0.152538i	−0.997087	−	0.0762690i	$-0.975699\pi$
$739$		−	4.14668i		−	0.152538i	0.997087	−	0.0762690i	$-0.0243008\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	23.8410i		−	0.874643i	−0.899305	−	0.437321i	$-0.855927\pi$
$743$		−	23.8410i		−	0.874643i	0.899305	−	0.437321i	$-0.144073\pi$
$744$		0			0
$745$		−	1.74449i		−	0.0639133i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10.6059	+	1.29722i	−0.387530	+	0.0473993i
$750$		0			0
$751$			20.4817i			0.747388i	0.927552	+	0.373694i	$0.121909\pi$
$751$			20.4817i			0.747388i	−0.927552	+	0.373694i	$0.878091\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.1793			0.370461
$756$		0			0
$757$	12.3561			0.449092			0.224546	−	0.974464i	$-0.427910\pi$
$757$	12.3561			0.449092			0.224546	+	0.974464i	$0.427910\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	25.1367i		−	0.911203i	−0.890184	−	0.455602i	$-0.849424\pi$
$761$		−	25.1367i		−	0.911203i	0.890184	−	0.455602i	$-0.150576\pi$
$762$		0			0
$763$	−6.42905	−	52.5630i	−0.232747	−	1.90291i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			9.23947i			0.333618i
$768$		0			0
$769$			46.7026i			1.68414i	0.539368	+	0.842070i	$0.318663\pi$
$769$			46.7026i			1.68414i	−0.539368	+	0.842070i	$0.681337\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			2.30873i			0.0830394i	0.999138	+	0.0415197i	$0.0132199\pi$
$773$			2.30873i			0.0830394i	−0.999138	+	0.0415197i	$0.986780\pi$
$774$		0			0
$775$	−5.21924			−0.187480
$776$		0			0
$777$		0			0
$778$		0			0
$779$			8.00128i			0.286675i
$780$		0			0
$781$	17.9025			0.640600
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.9265			0.675517
$786$		0			0
$787$	0.434256			0.0154795			0.00773977	−	0.999970i	$-0.497536\pi$
$787$	0.434256			0.0154795			0.00773977	+	0.999970i	$0.497536\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.39913	+	27.7908i	0.120859	+	0.988126i
$792$		0			0
$793$	37.3693			1.32702
$794$		0			0
$795$		0			0
$796$		0			0
$797$			31.9145i			1.13047i	0.824930	+	0.565235i	$0.191214\pi$
$797$			31.9145i			1.13047i	−0.824930	+	0.565235i	$0.808786\pi$
$798$		0			0
$799$		−	53.5053i		−	1.89288i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.642613			−0.0226773
$804$		0			0
$805$	2.62576	+	21.4678i	0.0925458	+	0.756641i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−7.49917			−0.263657			−0.131828	−	0.991273i	$-0.542085\pi$
$809$	−7.49917			−0.263657			−0.131828	+	0.991273i	$0.542085\pi$
$810$		0			0
$811$	−8.97307			−0.315087			−0.157544	−	0.987512i	$-0.550357\pi$
$811$	−8.97307			−0.315087			−0.157544	+	0.987512i	$0.550357\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.07164			0.0725663
$816$		0			0
$817$		−	3.75403i		−	0.131337i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	36.6347			1.27856			0.639281	−	0.768973i	$-0.279232\pi$
$821$	36.6347			1.27856			0.639281	+	0.768973i	$0.279232\pi$
$822$		0			0
$823$		−	14.4269i		−	0.502889i	−0.967872	−	0.251444i	$-0.919094\pi$
$823$		−	14.4269i		−	0.502889i	0.967872	−	0.251444i	$-0.0809055\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			31.4626i			1.09406i	0.837112	+	0.547031i	$0.184242\pi$
$827$			31.4626i			1.09406i	−0.837112	+	0.547031i	$0.815758\pi$
$828$		0			0
$829$			46.2807i			1.60740i	0.595038	+	0.803698i	$0.297137\pi$
$829$			46.2807i			1.60740i	−0.595038	+	0.803698i	$0.702863\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.33335	+	37.5832i	0.323381	+	1.30218i
$834$		0			0
$835$			13.8078i			0.477838i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	36.1922			1.24949			0.624746	−	0.780828i	$-0.285202\pi$
$839$	36.1922			1.24949			0.624746	+	0.780828i	$0.285202\pi$
$840$		0			0
$841$	20.4116			0.703850
$842$		0			0
$843$		0			0
$844$		0			0
$845$			6.01760i			0.207012i
$846$		0			0
$847$	−2.61904	−	21.4129i	−0.0899913	−	0.735756i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			40.3780i			1.38414i
$852$		0			0
$853$			46.1493i			1.58012i	0.613027	+	0.790062i	$0.289952\pi$
$853$			46.1493i			1.58012i	−0.613027	+	0.790062i	$0.710048\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	7.67833i		−	0.262287i	−0.991363	−	0.131143i	$-0.958135\pi$
$857$		−	7.67833i		−	0.262287i	0.991363	−	0.131143i	$-0.0418648\pi$
$858$		0			0
$859$	−15.3146			−0.522527			−0.261263	−	0.965268i	$-0.584139\pi$
$859$	−15.3146			−0.522527			−0.261263	+	0.965268i	$0.584139\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			19.0404i			0.648142i	0.946033	+	0.324071i	$0.105052\pi$
$863$			19.0404i			0.648142i	−0.946033	+	0.324071i	$0.894948\pi$
$864$		0			0
$865$	16.7178			0.568421
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.151606			−0.00514287
$870$		0			0
$871$	40.5910			1.37537
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.62618	+	0.321212i	−0.0887811	+	0.0108589i
$876$		0			0
$877$	−34.5743			−1.16749			−0.583746	−	0.811936i	$-0.698414\pi$
$877$	−34.5743			−1.16749			−0.583746	+	0.811936i	$0.698414\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	19.2873i		−	0.649804i	−0.945748	−	0.324902i	$-0.894669\pi$
$881$		−	19.2873i		−	0.649804i	0.945748	−	0.324902i	$-0.105331\pi$
$882$		0			0
$883$		−	1.02686i		−	0.0345565i	−0.999851	−	0.0172783i	$-0.994500\pi$
$883$		−	1.02686i		−	0.0345565i	0.999851	−	0.0172783i	$-0.00550011\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−26.1001			−0.876354			−0.438177	−	0.898889i	$-0.644376\pi$
$887$	−26.1001			−0.876354			−0.438177	+	0.898889i	$0.644376\pi$
$888$		0			0
$889$	16.8107	−	2.05614i	0.563811	−	0.0689606i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	39.3449			1.31663
$894$		0			0
$895$	0.538301			0.0179934
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−36.6878			−1.22361
$900$		0			0
$901$		−	1.54764i		−	0.0515592i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.0833689			−0.00277127
$906$		0			0
$907$			18.9339i			0.628690i	0.949309	+	0.314345i	$0.101785\pi$
$907$			18.9339i			0.628690i	−0.949309	+	0.314345i	$0.898215\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			55.6242i			1.84291i	0.388482	+	0.921456i	$0.373000\pi$
$911$			55.6242i			1.84291i	−0.388482	+	0.921456i	$0.627000\pi$
$912$		0			0
$913$			24.3594i			0.806180i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.574410	+	4.69629i	0.0189687	+	0.155085i
$918$		0			0
$919$			18.3140i			0.604123i	0.953288	+	0.302062i	$0.0976749\pi$
$919$			18.3140i			0.604123i	−0.953288	+	0.302062i	$0.902325\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−28.0394			−0.922928
$924$		0			0
$925$	−4.93948			−0.162409
$926$		0			0
$927$		0			0
$928$		0			0
$929$			50.5518i			1.65855i	0.558840	+	0.829276i	$0.311247\pi$
$929$			50.5518i			1.65855i	−0.558840	+	0.829276i	$0.688753\pi$
$930$		0			0
$931$	−27.6366	+	6.86323i	−0.905754	+	0.224933i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	9.33335i		−	0.305233i
$936$		0			0
$937$			11.8586i			0.387404i	0.981060	+	0.193702i	$0.0620495\pi$
$937$			11.8586i			0.387404i	−0.981060	+	0.193702i	$0.937951\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			22.8012i			0.743298i	0.928373	+	0.371649i	$0.121207\pi$
$941$			22.8012i			0.743298i	−0.928373	+	0.371649i	$0.878793\pi$
$942$		0			0
$943$	−16.0783			−0.523581
$944$		0			0
$945$		0			0
$946$		0			0
$947$			14.0946i			0.458012i	0.973425	+	0.229006i	$0.0735476\pi$
$947$			14.0946i			0.458012i	−0.973425	+	0.229006i	$0.926452\pi$
$948$		0			0
$949$	1.00648			0.0326718
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21.3897			−0.692881			−0.346440	−	0.938072i	$-0.612610\pi$
$953$	−21.3897			−0.692881			−0.346440	+	0.938072i	$0.612610\pi$
$954$		0			0
$955$	12.6660			0.409863
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.21941	−	26.3214i	−0.103960	−	0.849962i
$960$		0			0
$961$	−3.75958			−0.121277
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	26.3611i		−	0.848595i
$966$		0			0
$967$		−	6.12889i		−	0.197092i	−0.995132	−	0.0985460i	$-0.968581\pi$
$967$		−	6.12889i		−	0.197092i	0.995132	−	0.0985460i	$-0.0314192\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.1735			−0.647397			−0.323699	−	0.946160i	$-0.604926\pi$
$971$	−20.1735			−0.647397			−0.323699	+	0.946160i	$0.604926\pi$
$972$		0			0
$973$	−2.84011	−	23.2203i	−0.0910497	−	0.744408i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.7224			−0.662970			−0.331485	−	0.943461i	$-0.607550\pi$
$977$	−20.7224			−0.662970			−0.331485	+	0.943461i	$0.607550\pi$
$978$		0			0
$979$	−9.35237			−0.298903
$980$		0			0
$981$		0			0
$982$		0			0
$983$	34.7908			1.10965			0.554827	−	0.831966i	$-0.312785\pi$
$983$	34.7908			1.10965			0.554827	+	0.831966i	$0.312785\pi$
$984$		0			0
$985$		−	15.4873i		−	0.493465i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.54360			0.239872
$990$		0			0
$991$		−	20.5467i		−	0.652687i	−0.945251	−	0.326343i	$-0.894183\pi$
$991$		−	20.5467i		−	0.652687i	0.945251	−	0.326343i	$-0.105817\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			22.7184i			0.720222i
$996$		0			0
$997$			37.4749i			1.18684i	0.804892	+	0.593422i	$0.202223\pi$
$997$			37.4749i			1.18684i	−0.804892	+	0.593422i	$0.797777\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.d.g.4591.9		12
3.2	odd	2		1680.2.d.c.1231.3	✓	12
4.3	odd	2		5040.2.d.f.4591.10		12
7.6	odd	2		5040.2.d.f.4591.4		12
12.11	even	2		1680.2.d.d.1231.4	yes	12
21.20	even	2		1680.2.d.d.1231.10	yes	12
28.27	even	2	inner	5040.2.d.g.4591.3		12
84.83	odd	2		1680.2.d.c.1231.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.d.c.1231.3	✓	12	3.2	odd	2
1680.2.d.c.1231.9	yes	12	84.83	odd	2
1680.2.d.d.1231.4	yes	12	12.11	even	2
1680.2.d.d.1231.10	yes	12	21.20	even	2
5040.2.d.f.4591.4		12	7.6	odd	2
5040.2.d.f.4591.10		12	4.3	odd	2
5040.2.d.g.4591.3		12	28.27	even	2	inner
5040.2.d.g.4591.9		12	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:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +(-0.321212 - 2.62618i) q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} +(-0.321212 - 2.62618i) q^{7} -1.68712i q^{11} +2.64242i q^{13} -5.53212i q^{17} +4.06801 q^{19} +8.17454i q^{23} -1.00000 q^{25} -7.02934 q^{29} +5.21924 q^{31} +(2.62618 - 0.321212i) q^{35} +4.93948 q^{37} +1.96688i q^{41} -0.922816i q^{43} +9.67177 q^{47} +(-6.79365 + 1.68712i) q^{49} +0.279755 q^{53} +1.68712 q^{55} +3.49659 q^{59} -14.1421i q^{61} -2.64242 q^{65} -15.3613i q^{67} +10.6112i q^{71} -0.380893i q^{73} +(-4.43068 + 0.541923i) q^{77} -0.0898605i q^{79} -14.4385 q^{83} +5.53212 q^{85} -5.54339i q^{89} +(6.93948 - 0.848778i) q^{91} +4.06801i q^{95} -10.8856i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{7}+O(q^{10}) \) 12 * q + 4 * q^7	\( 12 q + 4 q^{7} + 16 q^{19} - 12 q^{25} - 8 q^{29} - 32 q^{31} + 4 q^{35} - 16 q^{37} + 24 q^{47} - 4 q^{49} - 16 q^{53} + 24 q^{59} - 16 q^{65} - 24 q^{77} + 16 q^{83} - 8 q^{85} + 8 q^{91}+O(q^{100}) \) 12 * q + 4 * q^7 + 16 * q^19 - 12 * q^25 - 8 * q^29 - 32 * q^31 + 4 * q^35 - 16 * q^37 + 24 * q^47 - 4 * q^49 - 16 * q^53 + 24 * q^59 - 16 * q^65 - 24 * q^77 + 16 * q^83 - 8 * q^85 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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