LMFDB - Embedded newform 5040.2.t.v.1009.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1009,5040)]

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([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("5040.1009");

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.t (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:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.6
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.v.1009.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.48119 + 1.67513i) q^{5} -1.00000i q^{7} +2.00000 q^{11} +1.35026i q^{13} -3.35026i q^{17} +5.35026 q^{19} +4.96239i q^{23} +(-0.612127 + 4.96239i) q^{25} +7.92478 q^{29} -4.57452 q^{31} +(1.67513 - 1.48119i) q^{35} +0.775746i q^{37} -3.73813 q^{41} +12.6253i q^{43} -9.92478i q^{47} -1.00000 q^{49} -8.57452i q^{53} +(2.96239 + 3.35026i) q^{55} +8.62530 q^{59} -8.70052 q^{61} +(-2.26187 + 2.00000i) q^{65} +9.92478i q^{67} +2.00000 q^{71} -9.35026i q^{73} -2.00000i q^{77} +10.7005 q^{79} +3.22425i q^{83} +(5.61213 - 4.96239i) q^{85} +1.03761 q^{89} +1.35026 q^{91} +(7.92478 + 8.96239i) q^{95} +18.4993i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} + 12 q^{11} + 12 q^{19} - 2 q^{25} + 4 q^{29} - 4 q^{31} - 4 q^{41} - 6 q^{49} - 4 q^{55} - 32 q^{59} - 12 q^{61} - 32 q^{65} + 12 q^{71} + 24 q^{79} + 32 q^{85} + 28 q^{89} - 12 q^{91} + 4 q^{95}+O(q^{100}) $ 6 * q - 2 * q^5 + 12 * q^11 + 12 * q^19 - 2 * q^25 + 4 * q^29 - 4 * q^31 - 4 * q^41 - 6 * q^49 - 4 * q^55 - 32 * q^59 - 12 * q^61 - 32 * q^65 + 12 * q^71 + 24 * q^79 + 32 * q^85 + 28 * q^89 - 12 * q^91 + 4 * q^95

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
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$			1.35026i			0.374495i	0.982313	+	0.187248i	$0.0599567\pi$
$13$			1.35026i			0.374495i	−0.982313	+	0.187248i	$0.940043\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.35026i		−	0.812558i	−0.913749	−	0.406279i	$-0.866826\pi$
$17$		−	3.35026i		−	0.812558i	0.913749	−	0.406279i	$-0.133174\pi$
$18$		0			0
$19$	5.35026			1.22743			0.613717	−	0.789526i	$-0.289674\pi$
$19$	5.35026			1.22743			0.613717	+	0.789526i	$0.289674\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.96239i			1.03473i	0.855765	+	0.517365i	$0.173087\pi$
$23$			4.96239i			1.03473i	−0.855765	+	0.517365i	$0.826913\pi$
$24$		0			0
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.92478			1.47159			0.735797	−	0.677202i	$-0.236808\pi$
$29$	7.92478			1.47159			0.735797	+	0.677202i	$0.236808\pi$
$30$		0			0
$31$	−4.57452			−0.821607			−0.410804	−	0.911724i	$-0.634752\pi$
$31$	−4.57452			−0.821607			−0.410804	+	0.911724i	$0.634752\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.67513	−	1.48119i	0.283149	−	0.250368i
$36$		0			0
$37$			0.775746i			0.127532i	0.997965	+	0.0637660i	$0.0203111\pi$
$37$			0.775746i			0.127532i	−0.997965	+	0.0637660i	$0.979689\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.73813			−0.583799			−0.291899	−	0.956449i	$-0.594287\pi$
$41$	−3.73813			−0.583799			−0.291899	+	0.956449i	$0.594287\pi$
$42$		0			0
$43$			12.6253i			1.92534i	0.270677	+	0.962670i	$0.412752\pi$
$43$			12.6253i			1.92534i	−0.270677	+	0.962670i	$0.587248\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	9.92478i		−	1.44768i	−0.689969	−	0.723839i	$-0.742376\pi$
$47$		−	9.92478i		−	1.44768i	0.689969	−	0.723839i	$-0.257624\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.57452i		−	1.17780i	−0.808206	−	0.588900i	$-0.799561\pi$
$53$		−	8.57452i		−	1.17780i	0.808206	−	0.588900i	$-0.200439\pi$
$54$		0			0
$55$	2.96239	+	3.35026i	0.399448	+	0.451749i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.62530			1.12292			0.561459	−	0.827504i	$-0.310240\pi$
$59$	8.62530			1.12292			0.561459	+	0.827504i	$0.310240\pi$
$60$		0			0
$61$	−8.70052			−1.11399			−0.556994	−	0.830517i	$-0.688045\pi$
$61$	−8.70052			−1.11399			−0.556994	+	0.830517i	$0.688045\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.26187	+	2.00000i	−0.280550	+	0.248069i
$66$		0			0
$67$			9.92478i			1.21250i	0.795272	+	0.606252i	$0.207328\pi$
$67$			9.92478i			1.21250i	−0.795272	+	0.606252i	$0.792672\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$		−	9.35026i		−	1.09437i	−0.837013	−	0.547183i	$-0.815700\pi$
$73$		−	9.35026i		−	1.09437i	0.837013	−	0.547183i	$-0.184300\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.00000i		−	0.227921i
$78$		0			0
$79$	10.7005			1.20390			0.601951	−	0.798533i	$-0.294390\pi$
$79$	10.7005			1.20390			0.601951	+	0.798533i	$0.294390\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.22425i			0.353908i	0.984219	+	0.176954i	$0.0566244\pi$
$83$			3.22425i			0.353908i	−0.984219	+	0.176954i	$0.943376\pi$
$84$		0			0
$85$	5.61213	−	4.96239i	0.608721	−	0.538247i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.03761			0.109987			0.0549933	−	0.998487i	$-0.482486\pi$
$89$	1.03761			0.109987			0.0549933	+	0.998487i	$0.482486\pi$
$90$		0			0
$91$	1.35026			0.141546
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.92478	+	8.96239i	0.813065	+	0.919522i
$96$		0			0
$97$			18.4993i			1.87832i	0.343482	+	0.939159i	$0.388394\pi$
$97$			18.4993i			1.87832i	−0.343482	+	0.939159i	$0.611606\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	17.6629			1.75753			0.878763	−	0.477259i	$-0.158370\pi$
$101$	17.6629			1.75753			0.878763	+	0.477259i	$0.158370\pi$
$102$		0			0
$103$		−	6.70052i		−	0.660222i	−0.943942	−	0.330111i	$-0.892914\pi$
$103$		−	6.70052i		−	0.660222i	0.943942	−	0.330111i	$-0.107086\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.7381i		−	1.32812i	−0.747681	−	0.664058i	$-0.768833\pi$
$107$		−	13.7381i		−	1.32812i	0.747681	−	0.664058i	$-0.231167\pi$
$108$		0			0
$109$	2.77575			0.265868			0.132934	−	0.991125i	$-0.457560\pi$
$109$	2.77575			0.265868			0.132934	+	0.991125i	$0.457560\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.0508i		−	1.13364i	−0.823841	−	0.566821i	$-0.808173\pi$
$113$		−	12.0508i		−	1.13364i	0.823841	−	0.566821i	$-0.191827\pi$
$114$		0			0
$115$	−8.31265	+	7.35026i	−0.775159	+	0.685415i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.35026			−0.307118
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.21933	+	6.32487i	−0.824602	+	0.565713i
$126$		0			0
$127$			2.70052i			0.239633i	0.992796	+	0.119816i	$0.0382306\pi$
$127$			2.70052i			0.239633i	−0.992796	+	0.119816i	$0.961769\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	20.6253			1.80204			0.901020	−	0.433777i	$-0.142819\pi$
$131$	20.6253			1.80204			0.901020	+	0.433777i	$0.142819\pi$
$132$		0			0
$133$		−	5.35026i		−	0.463927i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			22.4993i			1.92224i	0.276124	+	0.961122i	$0.410950\pi$
$137$			22.4993i			1.92224i	−0.276124	+	0.961122i	$0.589050\pi$
$138$		0			0
$139$	−3.27504			−0.277785			−0.138893	−	0.990307i	$-0.544354\pi$
$139$	−3.27504			−0.277785			−0.138893	+	0.990307i	$0.544354\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.70052i			0.225829i
$144$		0			0
$145$	11.7381	+	13.2750i	0.974799	+	1.10243i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.44851			−0.364436			−0.182218	−	0.983258i	$-0.558328\pi$
$149$	−4.44851			−0.364436			−0.182218	+	0.983258i	$0.558328\pi$
$150$		0			0
$151$	−1.29948			−0.105750			−0.0528749	−	0.998601i	$-0.516838\pi$
$151$	−1.29948			−0.105750			−0.0528749	+	0.998601i	$0.516838\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.77575	−	7.66291i	−0.544241	−	0.615500i
$156$		0			0
$157$			2.64974i			0.211472i	0.994394	+	0.105736i	$0.0337199\pi$
$157$			2.64974i			0.211472i	−0.994394	+	0.105736i	$0.966280\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.96239			0.391091
$162$		0			0
$163$		−	5.29948i		−	0.415087i	−0.978226	−	0.207544i	$-0.933453\pi$
$163$		−	5.29948i		−	0.415087i	0.978226	−	0.207544i	$-0.0665469\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.5501i			1.12592i	0.826485	+	0.562959i	$0.190337\pi$
$167$			14.5501i			1.12592i	−0.826485	+	0.562959i	$0.809663\pi$
$168$		0			0
$169$	11.1768			0.859753
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4.49929i			0.342075i	0.985265	+	0.171037i	$0.0547119\pi$
$173$			4.49929i			0.342075i	−0.985265	+	0.171037i	$0.945288\pi$
$174$		0			0
$175$	4.96239	+	0.612127i	0.375121	+	0.0462724i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	10.6253			0.789772			0.394886	−	0.918730i	$-0.370784\pi$
$181$	10.6253			0.789772			0.394886	+	0.918730i	$0.370784\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.29948	+	1.14903i	−0.0955394	+	0.0844784i
$186$		0			0
$187$		−	6.70052i		−	0.489991i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.8496			−1.00212			−0.501059	−	0.865413i	$-0.667056\pi$
$191$	−13.8496			−1.00212			−0.501059	+	0.865413i	$0.667056\pi$
$192$		0			0
$193$			15.3258i			1.10318i	0.834116	+	0.551588i	$0.185978\pi$
$193$			15.3258i			1.10318i	−0.834116	+	0.551588i	$0.814022\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	0.574515i		−	0.0409325i	−0.999791	−	0.0204663i	$-0.993485\pi$
$197$		−	0.574515i		−	0.0409325i	0.999791	−	0.0204663i	$-0.00651507\pi$
$198$		0			0
$199$	−0.201231			−0.0142649			−0.00713244	−	0.999975i	$-0.502270\pi$
$199$	−0.201231			−0.0142649			−0.00713244	+	0.999975i	$0.502270\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	7.92478i		−	0.556210i
$204$		0			0
$205$	−5.53690	−	6.26187i	−0.386714	−	0.437348i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.7005			0.740171
$210$		0			0
$211$	−6.44851			−0.443934			−0.221967	−	0.975054i	$-0.571248\pi$
$211$	−6.44851			−0.443934			−0.221967	+	0.975054i	$0.571248\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−21.1490	+	18.7005i	−1.44235	+	1.27537i
$216$		0			0
$217$			4.57452i			0.310538i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.52373			0.304299
$222$		0			0
$223$		−	1.55149i		−	0.103896i	−0.998650	−	0.0519478i	$-0.983457\pi$
$223$		−	1.55149i		−	0.103896i	0.998650	−	0.0519478i	$-0.0165429\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			13.1490i			0.872732i	0.899769	+	0.436366i	$0.143735\pi$
$227$			13.1490i			0.872732i	−0.899769	+	0.436366i	$0.856265\pi$
$228$		0			0
$229$	−2.77575			−0.183426			−0.0917132	−	0.995785i	$-0.529234\pi$
$229$	−2.77575			−0.183426			−0.0917132	+	0.995785i	$0.529234\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.0507852i			0.00332705i	0.999999	+	0.00166353i	$0.000529517\pi$
$233$			0.0507852i			0.00332705i	−0.999999	+	0.00166353i	$0.999470\pi$
$234$		0			0
$235$	16.6253	−	14.7005i	1.08452	−	0.958956i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.84955			−0.378376			−0.189188	−	0.981941i	$-0.560586\pi$
$239$	−5.84955			−0.378376			−0.189188	+	0.981941i	$0.560586\pi$
$240$		0			0
$241$	−0.0752228			−0.00484553			−0.00242276	−	0.999997i	$-0.500771\pi$
$241$	−0.0752228			−0.00484553			−0.00242276	+	0.999997i	$0.500771\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.48119	−	1.67513i	−0.0946300	−	0.107020i
$246$		0			0
$247$			7.22425i			0.459668i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.2243			1.21342			0.606712	−	0.794922i	$-0.292488\pi$
$251$	19.2243			1.21342			0.606712	+	0.794922i	$0.292488\pi$
$252$		0			0
$253$			9.92478i			0.623965i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			7.35026i			0.458497i	0.973368	+	0.229248i	$0.0736268\pi$
$257$			7.35026i			0.458497i	−0.973368	+	0.229248i	$0.926373\pi$
$258$		0			0
$259$	0.775746			0.0482025
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	12.9624i		−	0.799295i	−0.916669	−	0.399648i	$-0.869133\pi$
$263$		−	12.9624i		−	0.799295i	0.916669	−	0.399648i	$-0.130867\pi$
$264$		0			0
$265$	14.3634	−	12.7005i	0.882339	−	0.780187i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.11142			0.250678			0.125339	−	0.992114i	$-0.459998\pi$
$269$	4.11142			0.250678			0.125339	+	0.992114i	$0.459998\pi$
$270$		0			0
$271$	16.4241			0.997691			0.498846	−	0.866691i	$-0.333757\pi$
$271$	16.4241			0.997691			0.498846	+	0.866691i	$0.333757\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.22425	+	9.92478i	−0.0738253	+	0.598487i
$276$		0			0
$277$			11.0738i			0.665361i	0.943040	+	0.332680i	$0.107953\pi$
$277$			11.0738i			0.665361i	−0.943040	+	0.332680i	$0.892047\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.3733			−0.857438			−0.428719	−	0.903438i	$-0.641035\pi$
$281$	−14.3733			−0.857438			−0.428719	+	0.903438i	$0.641035\pi$
$282$		0			0
$283$		−	1.14903i		−	0.0683028i	−0.999417	−	0.0341514i	$-0.989127\pi$
$283$		−	1.14903i		−	0.0683028i	0.999417	−	0.0341514i	$-0.0108728\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			3.73813i			0.220655i
$288$		0			0
$289$	5.77575			0.339750
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	0.649738i		−	0.0379581i	−0.999820	−	0.0189791i	$-0.993958\pi$
$293$		−	0.649738i		−	0.0379581i	0.999820	−	0.0189791i	$-0.00604158\pi$
$294$		0			0
$295$	12.7757	+	14.4485i	0.743833	+	0.841225i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.70052			−0.387501
$300$		0			0
$301$	12.6253			0.727710
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.8872	−	14.5745i	−0.737917	−	0.834534i
$306$		0			0
$307$			24.1016i			1.37555i	0.725924	+	0.687775i	$0.241412\pi$
$307$			24.1016i			1.37555i	−0.725924	+	0.687775i	$0.758588\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.25202			0.467929			0.233964	−	0.972245i	$-0.424830\pi$
$311$	8.25202			0.467929			0.233964	+	0.972245i	$0.424830\pi$
$312$		0			0
$313$		−	14.9018i		−	0.842297i	−0.906992	−	0.421148i	$-0.861627\pi$
$313$		−	14.9018i		−	0.842297i	0.906992	−	0.421148i	$-0.138373\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			10.1260i			0.568733i	0.958716	+	0.284367i	$0.0917833\pi$
$317$			10.1260i			0.568733i	−0.958716	+	0.284367i	$0.908217\pi$
$318$		0			0
$319$	15.8496			0.887405
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	17.9248i		−	0.997361i
$324$		0			0
$325$	−6.70052	−	0.826531i	−0.371678	−	0.0458477i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.92478			−0.547171
$330$		0			0
$331$	−27.8496			−1.53075			−0.765375	−	0.643585i	$-0.777446\pi$
$331$	−27.8496			−1.53075			−0.765375	+	0.643585i	$0.777446\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−16.6253	+	14.7005i	−0.908337	+	0.803175i
$336$		0			0
$337$			3.84955i			0.209699i	0.994488	+	0.104849i	$0.0334360\pi$
$337$			3.84955i			0.209699i	−0.994488	+	0.104849i	$0.966564\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−9.14903			−0.495448
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	9.58769i		−	0.514694i	−0.966319	−	0.257347i	$-0.917152\pi$
$347$		−	9.58769i		−	0.514694i	0.966319	−	0.257347i	$-0.0828484\pi$
$348$		0			0
$349$	15.1490			0.810909			0.405455	−	0.914115i	$-0.367113\pi$
$349$	15.1490			0.810909			0.405455	+	0.914115i	$0.367113\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.3488i			1.08306i	0.840681	+	0.541530i	$0.182155\pi$
$353$			20.3488i			1.08306i	−0.840681	+	0.541530i	$0.817845\pi$
$354$		0			0
$355$	2.96239	+	3.35026i	0.157227	+	0.177813i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31.4010			1.65728			0.828642	−	0.559779i	$-0.189114\pi$
$359$	31.4010			1.65728			0.828642	+	0.559779i	$0.189114\pi$
$360$		0			0
$361$	9.62530			0.506595
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.6629	−	13.8496i	0.819834	−	0.724919i
$366$		0			0
$367$		−	29.4010i		−	1.53472i	−0.641215	−	0.767361i	$-0.721569\pi$
$367$		−	29.4010i		−	1.53472i	0.641215	−	0.767361i	$-0.278431\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.57452			−0.445167
$372$		0			0
$373$			16.0000i			0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$			16.0000i			0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			10.7005i			0.551105i
$378$		0			0
$379$	10.7005			0.549649			0.274824	−	0.961494i	$-0.411380\pi$
$379$	10.7005			0.549649			0.274824	+	0.961494i	$0.411380\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	16.7757i		−	0.857201i	−0.903494	−	0.428600i	$-0.859007\pi$
$383$		−	16.7757i		−	0.857201i	0.903494	−	0.428600i	$-0.140993\pi$
$384$		0			0
$385$	3.35026	−	2.96239i	0.170745	−	0.150977i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−29.3258			−1.48688			−0.743439	−	0.668804i	$-0.766807\pi$
$389$	−29.3258			−1.48688			−0.743439	+	0.668804i	$0.766807\pi$
$390$		0			0
$391$	16.6253			0.840778
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.8496	+	17.9248i	0.797478	+	0.901893i
$396$		0			0
$397$		−	18.3488i		−	0.920902i	−0.887685	−	0.460451i	$-0.847688\pi$
$397$		−	18.3488i		−	0.920902i	0.887685	−	0.460451i	$-0.152312\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	37.3258			1.86396			0.931981	−	0.362506i	$-0.118079\pi$
$401$	37.3258			1.86396			0.931981	+	0.362506i	$0.118079\pi$
$402$		0			0
$403$		−	6.17679i		−	0.307688i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.55149i			0.0769046i
$408$		0			0
$409$	22.3733			1.10629			0.553144	−	0.833086i	$-0.313428\pi$
$409$	22.3733			1.10629			0.553144	+	0.833086i	$0.313428\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	8.62530i		−	0.424423i
$414$		0			0
$415$	−5.40105	+	4.77575i	−0.265127	+	0.234432i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−23.4763			−1.14689			−0.573445	−	0.819244i	$-0.694394\pi$
$419$	−23.4763			−1.14689			−0.573445	+	0.819244i	$0.694394\pi$
$420$		0			0
$421$	−25.2243			−1.22935			−0.614677	−	0.788779i	$-0.710714\pi$
$421$	−25.2243			−1.22935			−0.614677	+	0.788779i	$0.710714\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	16.6253	+	2.05079i	0.806446	+	0.0994777i
$426$		0			0
$427$			8.70052i			0.421048i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−19.4010			−0.934516			−0.467258	−	0.884121i	$-0.654758\pi$
$431$	−19.4010			−0.934516			−0.467258	+	0.884121i	$0.654758\pi$
$432$		0			0
$433$			6.49929i			0.312336i	0.987731	+	0.156168i	$0.0499141\pi$
$433$			6.49929i			0.312336i	−0.987731	+	0.156168i	$0.950086\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			26.5501i			1.27006i
$438$		0			0
$439$	−14.6497			−0.699194			−0.349597	−	0.936900i	$-0.613681\pi$
$439$	−14.6497			−0.699194			−0.349597	+	0.936900i	$0.613681\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			19.1392i			0.909330i	0.890663	+	0.454665i	$0.150241\pi$
$443$			19.1392i			0.909330i	−0.890663	+	0.454665i	$0.849759\pi$
$444$		0			0
$445$	1.53690	+	1.73813i	0.0728562	+	0.0823955i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−32.8021			−1.54803			−0.774013	−	0.633169i	$-0.781754\pi$
$449$	−32.8021			−1.54803			−0.774013	+	0.633169i	$0.781754\pi$
$450$		0			0
$451$	−7.47627			−0.352044
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	+	2.26187i	0.0937614	+	0.106038i
$456$		0			0
$457$			18.7005i			0.874774i	0.899273	+	0.437387i	$0.144096\pi$
$457$			18.7005i			0.874774i	−0.899273	+	0.437387i	$0.855904\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.96239			0.324271			0.162135	−	0.986769i	$-0.448162\pi$
$461$	6.96239			0.324271			0.162135	+	0.986769i	$0.448162\pi$
$462$		0			0
$463$		−	5.29948i		−	0.246288i	−0.992389	−	0.123144i	$-0.960702\pi$
$463$		−	5.29948i		−	0.246288i	0.992389	−	0.123144i	$-0.0392976\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			13.1490i			0.608465i	0.952598	+	0.304232i	$0.0983999\pi$
$467$			13.1490i			0.608465i	−0.952598	+	0.304232i	$0.901600\pi$
$468$		0			0
$469$	9.92478			0.458284
$470$		0			0
$471$		0			0
$472$		0			0
$473$			25.2506i			1.16102i
$474$		0			0
$475$	−3.27504	+	26.5501i	−0.150269	+	1.21820i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.14903			−0.235265			−0.117633	−	0.993057i	$-0.537531\pi$
$479$	−5.14903			−0.235265			−0.117633	+	0.993057i	$0.537531\pi$
$480$		0			0
$481$	−1.04746			−0.0477601
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−30.9887	+	27.4010i	−1.40713	+	1.24422i
$486$		0			0
$487$		−	22.1768i		−	1.00493i	−0.864599	−	0.502463i	$-0.832427\pi$
$487$		−	22.1768i		−	1.00493i	0.864599	−	0.502463i	$-0.167573\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.00000			0.0902587			0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000			0.0902587			0.0451294	+	0.998981i	$0.485630\pi$
$492$		0			0
$493$		−	26.5501i		−	1.19576i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	2.00000i		−	0.0897123i
$498$		0			0
$499$	−6.55008			−0.293222			−0.146611	−	0.989194i	$-0.546837\pi$
$499$	−6.55008			−0.293222			−0.146611	+	0.989194i	$0.546837\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			8.77575i			0.391291i	0.980675	+	0.195646i	$0.0626802\pi$
$503$			8.77575i			0.391291i	−0.980675	+	0.195646i	$0.937320\pi$
$504$		0			0
$505$	26.1622	+	29.5877i	1.16420	+	1.31663i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.1392			0.582384			0.291192	−	0.956665i	$-0.405948\pi$
$509$	13.1392			0.582384			0.291192	+	0.956665i	$0.405948\pi$
$510$		0			0
$511$	−9.35026			−0.413631
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.2243	−	9.92478i	0.494600	−	0.437338i
$516$		0			0
$517$		−	19.8496i		−	0.872982i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	37.6629			1.65004			0.825021	−	0.565102i	$-0.191163\pi$
$521$	37.6629			1.65004			0.825021	+	0.565102i	$0.191163\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			15.3258i			0.667603i
$528$		0			0
$529$	−1.62530			−0.0706652
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5.04746i		−	0.218630i
$534$		0			0
$535$	23.0132	−	20.3488i	0.994946	−	0.879757i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	−22.4749			−0.966269			−0.483135	−	0.875546i	$-0.660502\pi$
$541$	−22.4749			−0.966269			−0.483135	+	0.875546i	$0.660502\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.11142	+	4.64974i	0.176114	+	0.199173i
$546$		0			0
$547$		−	25.9248i		−	1.10846i	−0.832362	−	0.554232i	$-0.813012\pi$
$547$		−	25.9248i		−	1.10846i	0.832362	−	0.554232i	$-0.186988\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	42.3996			1.80629
$552$		0			0
$553$		−	10.7005i		−	0.455033i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	28.5256i		−	1.20867i	−0.796730	−	0.604335i	$-0.793439\pi$
$557$		−	28.5256i		−	1.20867i	0.796730	−	0.604335i	$-0.206561\pi$
$558$		0			0
$559$	−17.0475			−0.721031
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	11.6267i		−	0.490008i	−0.969522	−	0.245004i	$-0.921211\pi$
$563$		−	11.6267i		−	0.490008i	0.969522	−	0.245004i	$-0.0787892\pi$
$564$		0			0
$565$	20.1866	−	17.8496i	0.849258	−	0.750936i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.32582			0.390959			0.195479	−	0.980708i	$-0.437374\pi$
$569$	9.32582			0.390959			0.195479	+	0.980708i	$0.437374\pi$
$570$		0			0
$571$	19.6991			0.824382			0.412191	−	0.911097i	$-0.364764\pi$
$571$	19.6991			0.824382			0.412191	+	0.911097i	$0.364764\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.6253	−	3.03761i	−1.02695	−	0.126677i
$576$		0			0
$577$		−	32.7974i		−	1.36537i	−0.730712	−	0.682686i	$-0.760812\pi$
$577$		−	32.7974i		−	1.36537i	0.730712	−	0.682686i	$-0.239188\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.22425			0.133765
$582$		0			0
$583$		−	17.1490i		−	0.710240i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.8218i		−	0.776859i	−0.921479	−	0.388429i	$-0.873018\pi$
$587$		−	18.8218i		−	0.776859i	0.921479	−	0.388429i	$-0.126982\pi$
$588$		0			0
$589$	−24.4749			−1.00847
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	33.7499i		−	1.38594i	−0.720965	−	0.692971i	$-0.756301\pi$
$593$		−	33.7499i		−	1.38594i	0.720965	−	0.692971i	$-0.243699\pi$
$594$		0			0
$595$	−4.96239	−	5.61213i	−0.203438	−	0.230075i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.2981			0.829356			0.414678	−	0.909968i	$-0.363894\pi$
$599$	20.2981			0.829356			0.414678	+	0.909968i	$0.363894\pi$
$600$		0			0
$601$	−13.8496			−0.564935			−0.282468	−	0.959277i	$-0.591153\pi$
$601$	−13.8496			−0.564935			−0.282468	+	0.959277i	$0.591153\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.3684	−	11.7259i	−0.421534	−	0.476726i
$606$		0			0
$607$		−	25.2506i		−	1.02489i	−0.858720	−	0.512445i	$-0.828740\pi$
$607$		−	25.2506i		−	1.02489i	0.858720	−	0.512445i	$-0.171260\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.4010			0.542148
$612$		0			0
$613$		−	9.14903i		−	0.369526i	−0.982783	−	0.184763i	$-0.940848\pi$
$613$		−	9.14903i		−	0.369526i	0.982783	−	0.184763i	$-0.0591517\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15.9492i			0.642091i	0.947064	+	0.321046i	$0.104034\pi$
$617$			15.9492i			0.642091i	−0.947064	+	0.321046i	$0.895966\pi$
$618$		0			0
$619$	11.1735			0.449100			0.224550	−	0.974463i	$-0.427909\pi$
$619$	11.1735			0.449100			0.224550	+	0.974463i	$0.427909\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1.03761i		−	0.0415710i
$624$		0			0
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.59895			0.103627
$630$		0			0
$631$	14.5501			0.579229			0.289615	−	0.957143i	$-0.406473\pi$
$631$	14.5501			0.579229			0.289615	+	0.957143i	$0.406473\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.52373	+	4.00000i	−0.179519	+	0.158735i
$636$		0			0
$637$		−	1.35026i		−	0.0534993i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	38.7269			1.52962			0.764810	−	0.644256i	$-0.222833\pi$
$641$	38.7269			1.52962			0.764810	+	0.644256i	$0.222833\pi$
$642$		0			0
$643$			11.9511i			0.471306i	0.971837	+	0.235653i	$0.0757229\pi$
$643$			11.9511i			0.471306i	−0.971837	+	0.235653i	$0.924277\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			14.5501i			0.572023i	0.958226	+	0.286011i	$0.0923295\pi$
$647$			14.5501i			0.572023i	−0.958226	+	0.286011i	$0.907671\pi$
$648$		0			0
$649$	17.2506			0.677145
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	49.9756i		−	1.95569i	−0.209319	−	0.977847i	$-0.567125\pi$
$653$		−	49.9756i		−	1.95569i	0.209319	−	0.977847i	$-0.432875\pi$
$654$		0			0
$655$	30.5501	+	34.5501i	1.19369	+	1.34998i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16.9525			0.660377			0.330189	−	0.943915i	$-0.392888\pi$
$659$	16.9525			0.660377			0.330189	+	0.943915i	$0.392888\pi$
$660$		0			0
$661$	−15.6531			−0.608834			−0.304417	−	0.952539i	$-0.598462\pi$
$661$	−15.6531			−0.608834			−0.304417	+	0.952539i	$0.598462\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.96239	−	7.92478i	0.347547	−	0.307310i
$666$		0			0
$667$			39.3258i			1.52270i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.4010			−0.671760
$672$		0			0
$673$			26.0263i			1.00324i	0.865088	+	0.501621i	$0.167263\pi$
$673$			26.0263i			1.00324i	−0.865088	+	0.501621i	$0.832737\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	35.4518i		−	1.36252i	−0.732039	−	0.681262i	$-0.761431\pi$
$677$		−	35.4518i		−	1.36252i	0.732039	−	0.681262i	$-0.238569\pi$
$678$		0			0
$679$	18.4993			0.709938
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	23.6629i		−	0.905436i	−0.891654	−	0.452718i	$-0.850454\pi$
$683$		−	23.6629i		−	0.905436i	0.891654	−	0.452718i	$-0.149546\pi$
$684$		0			0
$685$	−37.6893	+	33.3258i	−1.44003	+	1.27331i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11.5778			0.441081
$690$		0			0
$691$	0.574515			0.0218556			0.0109278	−	0.999940i	$-0.496522\pi$
$691$	0.574515			0.0218556			0.0109278	+	0.999940i	$0.496522\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.85097	−	5.48612i	−0.184008	−	0.208100i
$696$		0			0
$697$			12.5237i			0.474370i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−42.7269			−1.61377			−0.806886	−	0.590707i	$-0.798849\pi$
$701$	−42.7269			−1.61377			−0.806886	+	0.590707i	$0.798849\pi$
$702$		0			0
$703$			4.15045i			0.156537i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	17.6629i		−	0.664282i
$708$		0			0
$709$	−27.2506			−1.02342			−0.511709	−	0.859159i	$-0.670987\pi$
$709$	−27.2506			−1.02342			−0.511709	+	0.859159i	$0.670987\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	22.7005i		−	0.850141i
$714$		0			0
$715$	−4.52373	+	4.00000i	−0.169178	+	0.149592i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.7005			0.399062			0.199531	−	0.979891i	$-0.436058\pi$
$719$	10.7005			0.399062			0.199531	+	0.979891i	$0.436058\pi$
$720$		0			0
$721$	−6.70052			−0.249541
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.85097	+	39.3258i	−0.180160	+	1.46052i
$726$		0			0
$727$			39.9511i			1.48171i	0.671668	+	0.740853i	$0.265578\pi$
$727$			39.9511i			1.48171i	−0.671668	+	0.740853i	$0.734422\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	42.2981			1.56445
$732$		0			0
$733$		−	30.3488i		−	1.12096i	−0.828168	−	0.560480i	$-0.810617\pi$
$733$		−	30.3488i		−	1.12096i	0.828168	−	0.560480i	$-0.189383\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			19.8496i			0.731168i
$738$		0			0
$739$	37.2506			1.37029			0.685143	−	0.728409i	$-0.259740\pi$
$739$	37.2506			1.37029			0.685143	+	0.728409i	$0.259740\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			26.3634i			0.967181i	0.875294	+	0.483590i	$0.160668\pi$
$743$			26.3634i			0.967181i	−0.875294	+	0.483590i	$0.839332\pi$
$744$		0			0
$745$	−6.58910	−	7.45183i	−0.241406	−	0.273014i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13.7381			−0.501981
$750$		0			0
$751$	−50.6516			−1.84830			−0.924152	−	0.382024i	$-0.875227\pi$
$751$	−50.6516			−1.84830			−0.924152	+	0.382024i	$0.875227\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.92478	−	2.17679i	−0.0700498	−	0.0792216i
$756$		0			0
$757$		−	38.9525i		−	1.41575i	−0.706336	−	0.707877i	$-0.749653\pi$
$757$		−	38.9525i		−	1.41575i	0.706336	−	0.707877i	$-0.250347\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−48.2130			−1.74772			−0.873860	−	0.486178i	$-0.838391\pi$
$761$	−48.2130			−1.74772			−0.873860	+	0.486178i	$0.838391\pi$
$762$		0			0
$763$		−	2.77575i		−	0.100489i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			11.6464i			0.420528i
$768$		0			0
$769$	−4.44851			−0.160417			−0.0802086	−	0.996778i	$-0.525559\pi$
$769$	−4.44851			−0.160417			−0.0802086	+	0.996778i	$0.525559\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	39.3014i		−	1.41357i	−0.707427	−	0.706786i	$-0.750144\pi$
$773$		−	39.3014i		−	1.41357i	0.707427	−	0.706786i	$-0.249856\pi$
$774$		0			0
$775$	2.80018	−	22.7005i	0.100586	−	0.815427i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20.0000			−0.716574
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.43866	+	3.92478i	−0.158423	+	0.140081i
$786$		0			0
$787$			0.897015i			0.0319751i	0.999872	+	0.0159876i	$0.00508922\pi$
$787$			0.897015i			0.0319751i	−0.999872	+	0.0159876i	$0.994911\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0508			−0.428477
$792$		0			0
$793$		−	11.7480i		−	0.417183i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.19982i			0.113343i	0.998393	+	0.0566717i	$0.0180488\pi$
$797$			3.19982i			0.113343i	−0.998393	+	0.0566717i	$0.981951\pi$
$798$		0			0
$799$	−33.2506			−1.17632
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	18.7005i		−	0.659927i
$804$		0			0
$805$	7.35026	+	8.31265i	0.259063	+	0.292982i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4.44851			0.156401			0.0782006	−	0.996938i	$-0.475083\pi$
$809$	4.44851			0.156401			0.0782006	+	0.996938i	$0.475083\pi$
$810$		0			0
$811$	−37.6747			−1.32294			−0.661468	−	0.749973i	$-0.730066\pi$
$811$	−37.6747			−1.32294			−0.661468	+	0.749973i	$0.730066\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.87732	−	7.84955i	0.310959	−	0.274958i
$816$		0			0
$817$			67.5487i			2.36323i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.749399			0.0261542			0.0130771	−	0.999914i	$-0.495837\pi$
$821$	0.749399			0.0261542			0.0130771	+	0.999914i	$0.495837\pi$
$822$		0			0
$823$			26.3996i			0.920233i	0.887858	+	0.460117i	$0.152192\pi$
$823$			26.3996i			0.920233i	−0.887858	+	0.460117i	$0.847808\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	5.43724i		−	0.189071i	−0.995521	−	0.0945357i	$-0.969863\pi$
$827$		−	5.43724i		−	0.189071i	0.995521	−	0.0945357i	$-0.0301367\pi$
$828$		0			0
$829$	−22.7757			−0.791034			−0.395517	−	0.918459i	$-0.629435\pi$
$829$	−22.7757			−0.791034			−0.395517	+	0.918459i	$0.629435\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			3.35026i			0.116080i
$834$		0			0
$835$	−24.3733	+	21.5515i	−0.843472	+	0.745820i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.8496			−0.547187			−0.273594	−	0.961845i	$-0.588212\pi$
$839$	−15.8496			−0.547187			−0.273594	+	0.961845i	$0.588212\pi$
$840$		0			0
$841$	33.8021			1.16559
$842$		0			0
$843$		0			0
$844$		0			0
$845$	16.5550	+	18.7226i	0.569509	+	0.644077i
$846$		0			0
$847$			7.00000i			0.240523i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.84955			−0.131961
$852$		0			0
$853$		−	21.0494i		−	0.720717i	−0.932814	−	0.360358i	$-0.882654\pi$
$853$		−	21.0494i		−	0.720717i	0.932814	−	0.360358i	$-0.117346\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			50.1524i			1.71317i	0.516004	+	0.856586i	$0.327419\pi$
$857$			50.1524i			1.71317i	−0.516004	+	0.856586i	$0.672581\pi$
$858$		0			0
$859$	−5.35026			−0.182549			−0.0912743	−	0.995826i	$-0.529094\pi$
$859$	−5.35026			−0.182549			−0.0912743	+	0.995826i	$0.529094\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	33.6893i		−	1.14680i	−0.819277	−	0.573398i	$-0.805625\pi$
$863$		−	33.6893i		−	1.14680i	0.819277	−	0.573398i	$-0.194375\pi$
$864$		0			0
$865$	−7.53690	+	6.66433i	−0.256262	+	0.226594i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21.4010			0.725981
$870$		0			0
$871$	−13.4010			−0.454077
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.32487	+	9.21933i	0.213820	+	0.311670i
$876$		0			0
$877$			21.5026i			0.726092i	0.931771	+	0.363046i	$0.118263\pi$
$877$			21.5026i			0.726092i	−0.931771	+	0.363046i	$0.881737\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−32.3634			−1.09035			−0.545176	−	0.838322i	$-0.683537\pi$
$881$	−32.3634			−1.09035			−0.545176	+	0.838322i	$0.683537\pi$
$882$		0			0
$883$			2.59895i			0.0874617i	0.999043	+	0.0437309i	$0.0139244\pi$
$883$			2.59895i			0.0874617i	−0.999043	+	0.0437309i	$0.986076\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	38.2784i		−	1.28526i	−0.766176	−	0.642631i	$-0.777843\pi$
$887$		−	38.2784i		−	1.28526i	0.766176	−	0.642631i	$-0.222157\pi$
$888$		0			0
$889$	2.70052			0.0905727
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	53.1002i		−	1.77693i
$894$		0			0
$895$	−14.8119	−	16.7513i	−0.495109	−	0.559934i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−36.2520			−1.20907
$900$		0			0
$901$	−28.7269			−0.957031
$902$		0			0
$903$		0			0
$904$		0			0
$905$	15.7381	+	17.7988i	0.523153	+	0.591651i
$906$		0			0
$907$		−	49.9972i		−	1.66013i	−0.557668	−	0.830064i	$-0.688304\pi$
$907$		−	49.9972i		−	1.66013i	0.557668	−	0.830064i	$-0.311696\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.9525			−0.826715			−0.413357	−	0.910569i	$-0.635644\pi$
$911$	−24.9525			−0.826715			−0.413357	+	0.910569i	$0.635644\pi$
$912$		0			0
$913$			6.44851i			0.213414i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	20.6253i		−	0.681107i
$918$		0			0
$919$	−11.6991			−0.385918			−0.192959	−	0.981207i	$-0.561808\pi$
$919$	−11.6991			−0.385918			−0.192959	+	0.981207i	$0.561808\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.70052i			0.0888888i
$924$		0			0
$925$	−3.84955	−	0.474855i	−0.126573	−	0.0156131i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.7090			0.777866			0.388933	−	0.921266i	$-0.372844\pi$
$929$	23.7090			0.777866			0.388933	+	0.921266i	$0.372844\pi$
$930$		0			0
$931$	−5.35026			−0.175348
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.2243	−	9.92478i	0.367072	−	0.324575i
$936$		0			0
$937$			19.9003i			0.650116i	0.945694	+	0.325058i	$0.105384\pi$
$937$			19.9003i			0.650116i	−0.945694	+	0.325058i	$0.894616\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.28821			0.204990			0.102495	−	0.994734i	$-0.467317\pi$
$941$	6.28821			0.204990			0.102495	+	0.994734i	$0.467317\pi$
$942$		0			0
$943$		−	18.5501i		−	0.604074i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			40.0362i			1.30100i	0.759506	+	0.650501i	$0.225441\pi$
$947$			40.0362i			1.30100i	−0.759506	+	0.650501i	$0.774559\pi$
$948$		0			0
$949$	12.6253			0.409835
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	40.9478i		−	1.32643i	−0.748429	−	0.663215i	$-0.769192\pi$
$953$		−	40.9478i		−	1.32643i	0.748429	−	0.663215i	$-0.230808\pi$
$954$		0			0
$955$	−20.5139	−	23.1998i	−0.663814	−	0.750728i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	22.4993			0.726540
$960$		0			0
$961$	−10.0738			−0.324962
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−25.6728	+	22.7005i	−0.826435	+	0.730756i
$966$		0			0
$967$			38.2784i			1.23095i	0.788157	+	0.615475i	$0.211036\pi$
$967$			38.2784i			1.23095i	−0.788157	+	0.615475i	$0.788964\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.7269			−0.921889			−0.460945	−	0.887429i	$-0.652489\pi$
$971$	−28.7269			−0.921889			−0.460945	+	0.887429i	$0.652489\pi$
$972$		0			0
$973$			3.27504i			0.104993i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	41.3014i		−	1.32135i	−0.750673	−	0.660674i	$-0.770270\pi$
$977$		−	41.3014i		−	1.32135i	0.750673	−	0.660674i	$-0.229730\pi$
$978$		0			0
$979$	2.07522			0.0663244
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	24.0000i		−	0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$		−	24.0000i		−	0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$		0			0
$985$	0.962389	−	0.850969i	0.0306643	−	0.0271141i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−62.6516			−1.99221
$990$		0			0
$991$	−25.1002			−0.797333			−0.398666	−	0.917096i	$-0.630527\pi$
$991$	−25.1002			−0.797333			−0.398666	+	0.917096i	$0.630527\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.298062	−	0.337088i	−0.00944920	−	0.0106864i
$996$		0			0
$997$		−	32.7974i		−	1.03870i	−0.854561	−	0.519351i	$-0.826174\pi$
$997$		−	32.7974i		−	1.03870i	0.854561	−	0.519351i	$-0.173826\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.v.1009.6		6
3.2	odd	2		1680.2.t.k.1009.4		6
4.3	odd	2		315.2.d.e.64.4		6
5.4	even	2	inner	5040.2.t.v.1009.5		6
12.11	even	2		105.2.d.b.64.3	✓	6
15.2	even	4		8400.2.a.dj.1.3		3
15.8	even	4		8400.2.a.dg.1.1		3
15.14	odd	2		1680.2.t.k.1009.1		6
20.3	even	4		1575.2.a.x.1.2		3
20.7	even	4		1575.2.a.w.1.2		3
20.19	odd	2		315.2.d.e.64.3		6
28.27	even	2		2205.2.d.l.1324.4		6
60.23	odd	4		525.2.a.j.1.2		3
60.47	odd	4		525.2.a.k.1.2		3
60.59	even	2		105.2.d.b.64.4	yes	6
84.11	even	6		735.2.q.e.79.4		12
84.23	even	6		735.2.q.e.214.3		12
84.47	odd	6		735.2.q.f.214.3		12
84.59	odd	6		735.2.q.f.79.4		12
84.83	odd	2		735.2.d.b.589.3		6
140.139	even	2		2205.2.d.l.1324.3		6
420.59	odd	6		735.2.q.f.79.3		12
420.83	even	4		3675.2.a.bi.1.2		3
420.167	even	4		3675.2.a.bj.1.2		3
420.179	even	6		735.2.q.e.79.3		12
420.299	odd	6		735.2.q.f.214.4		12
420.359	even	6		735.2.q.e.214.4		12
420.419	odd	2		735.2.d.b.589.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.3	✓	6	12.11	even	2
105.2.d.b.64.4	yes	6	60.59	even	2
315.2.d.e.64.3		6	20.19	odd	2
315.2.d.e.64.4		6	4.3	odd	2
525.2.a.j.1.2		3	60.23	odd	4
525.2.a.k.1.2		3	60.47	odd	4
735.2.d.b.589.3		6	84.83	odd	2
735.2.d.b.589.4		6	420.419	odd	2
735.2.q.e.79.3		12	420.179	even	6
735.2.q.e.79.4		12	84.11	even	6
735.2.q.e.214.3		12	84.23	even	6
735.2.q.e.214.4		12	420.359	even	6
735.2.q.f.79.3		12	420.59	odd	6
735.2.q.f.79.4		12	84.59	odd	6
735.2.q.f.214.3		12	84.47	odd	6
735.2.q.f.214.4		12	420.299	odd	6
1575.2.a.w.1.2		3	20.7	even	4
1575.2.a.x.1.2		3	20.3	even	4
1680.2.t.k.1009.1		6	15.14	odd	2
1680.2.t.k.1009.4		6	3.2	odd	2
2205.2.d.l.1324.3		6	140.139	even	2
2205.2.d.l.1324.4		6	28.27	even	2
3675.2.a.bi.1.2		3	420.83	even	4
3675.2.a.bj.1.2		3	420.167	even	4
5040.2.t.v.1009.5		6	5.4	even	2	inner
5040.2.t.v.1009.6		6	1.1	even	1	trivial
8400.2.a.dg.1.1		3	15.8	even	4
8400.2.a.dj.1.3		3	15.2	even	4

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.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.48119 + 1.67513i) q^{5} -1.00000i q^{7} +2.00000 q^{11} +1.35026i q^{13} -3.35026i q^{17} +5.35026 q^{19} +4.96239i q^{23} +(-0.612127 + 4.96239i) q^{25} +7.92478 q^{29} -4.57452 q^{31} +(1.67513 - 1.48119i) q^{35} +0.775746i q^{37} -3.73813 q^{41} +12.6253i q^{43} -9.92478i q^{47} -1.00000 q^{49} -8.57452i q^{53} +(2.96239 + 3.35026i) q^{55} +8.62530 q^{59} -8.70052 q^{61} +(-2.26187 + 2.00000i) q^{65} +9.92478i q^{67} +2.00000 q^{71} -9.35026i q^{73} -2.00000i q^{77} +10.7005 q^{79} +3.22425i q^{83} +(5.61213 - 4.96239i) q^{85} +1.03761 q^{89} +1.35026 q^{91} +(7.92478 + 8.96239i) q^{95} +18.4993i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} + 12 q^{11} + 12 q^{19} - 2 q^{25} + 4 q^{29} - 4 q^{31} - 4 q^{41} - 6 q^{49} - 4 q^{55} - 32 q^{59} - 12 q^{61} - 32 q^{65} + 12 q^{71} + 24 q^{79} + 32 q^{85} + 28 q^{89} - 12 q^{91} + 4 q^{95}+O(q^{100}) \) 6 * q - 2 * q^5 + 12 * q^11 + 12 * q^19 - 2 * q^25 + 4 * q^29 - 4 * q^31 - 4 * q^41 - 6 * q^49 - 4 * q^55 - 32 * q^59 - 12 * q^61 - 32 * q^65 + 12 * q^71 + 24 * q^79 + 32 * q^85 + 28 * q^89 - 12 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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