LMFDB - Embedded newform 5040.2.t.u.1009.2

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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.2
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.u.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{5} +1.00000i q^{7} +O(q^{10})$	$q-2.23607 q^{5} +1.00000i q^{7} -2.00000 q^{11} -4.47214i q^{13} +6.47214i q^{17} +2.00000 q^{19} +4.00000i q^{23} +5.00000 q^{25} -8.47214 q^{29} +0.472136 q^{31} -2.23607i q^{35} -2.47214i q^{37} +3.52786 q^{41} +2.47214i q^{43} -6.47214i q^{47} -1.00000 q^{49} -2.00000i q^{53} +4.47214 q^{55} -3.52786 q^{61} +10.0000i q^{65} +1.52786i q^{67} +12.4721 q^{71} +7.52786i q^{73} -2.00000i q^{77} +8.94427 q^{79} -4.94427i q^{83} -14.4721i q^{85} -17.4164 q^{89} +4.47214 q^{91} -4.47214 q^{95} -3.52786i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{11} + 8 q^{19} + 20 q^{25} - 16 q^{29} - 16 q^{31} + 32 q^{41} - 4 q^{49} - 32 q^{61} + 32 q^{71} - 16 q^{89}+O(q^{100}) $ 4 * q - 8 * q^11 + 8 * q^19 + 20 * q^25 - 16 * q^29 - 16 * q^31 + 32 * q^41 - 4 * q^49 - 32 * q^61 + 32 * q^71 - 16 * q^89

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
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	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	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	− 4.47214i	− 1.24035i	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	− 4.47214i	− 1.24035i	0.784465	−	0.620174i	$-0.212938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.47214i	1.56972i	0.619671	+	0.784862i	$0.287266\pi$
$17$	6.47214i	1.56972i	−0.619671	+	0.784862i	$0.712734\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.47214	−1.57324	−0.786618	−	0.617440i	$-0.788170\pi$
$29$	−8.47214	−1.57324	−0.786618	+	0.617440i	$0.788170\pi$
$30$	0	0
$31$	0.472136	0.0847981	0.0423991	−	0.999101i	$-0.486500\pi$
$31$	0.472136	0.0847981	0.0423991	+	0.999101i	$0.486500\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.23607i	− 0.377964i
$36$	0	0
$37$	− 2.47214i	− 0.406417i	−0.979136	−	0.203208i	$-0.934863\pi$
$37$	− 2.47214i	− 0.406417i	0.979136	−	0.203208i	$-0.0651369\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.52786	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$41$	3.52786	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$42$	0	0
$43$	2.47214i	0.376997i	0.982073	+	0.188499i	$0.0603621\pi$
$43$	2.47214i	0.376997i	−0.982073	+	0.188499i	$0.939638\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.47214i	− 0.944058i	−0.881583	−	0.472029i	$-0.843522\pi$
$47$	− 6.47214i	− 0.944058i	0.881583	−	0.472029i	$-0.156478\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	4.47214	0.603023
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.0000i	1.24035i
$66$	0	0
$67$	1.52786i	0.186658i	0.995635	+	0.0933292i	$0.0297509\pi$
$67$	1.52786i	0.186658i	−0.995635	+	0.0933292i	$0.970249\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.4721	1.48017	0.740085	−	0.672513i	$-0.234785\pi$
$71$	12.4721	1.48017	0.740085	+	0.672513i	$0.234785\pi$
$72$	0	0
$73$	7.52786i	0.881070i	0.897735	+	0.440535i	$0.145211\pi$
$73$	7.52786i	0.881070i	−0.897735	+	0.440535i	$0.854789\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.00000i	− 0.227921i
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.94427i	− 0.542704i	−0.962480	−	0.271352i	$-0.912529\pi$
$83$	− 4.94427i	− 0.542704i	0.962480	−	0.271352i	$-0.0874708\pi$
$84$	0	0
$85$	− 14.4721i	− 1.56972i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.4164	−1.84614	−0.923068	−	0.384637i	$-0.874327\pi$
$89$	−17.4164	−1.84614	−0.923068	+	0.384637i	$0.874327\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.47214	−0.458831
$96$	0	0
$97$	− 3.52786i	− 0.358200i	−0.983831	−	0.179100i	$-0.942681\pi$
$97$	− 3.52786i	− 0.358200i	0.983831	−	0.179100i	$-0.0573186\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	0	0
$103$	− 12.9443i	− 1.27544i	−0.770270	−	0.637719i	$-0.779878\pi$
$103$	− 12.9443i	− 1.27544i	0.770270	−	0.637719i	$-0.220122\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−14.9443	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$109$	−14.9443	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 14.9443i	− 1.40584i	−0.711269	−	0.702919i	$-0.751879\pi$
$113$	− 14.9443i	− 1.40584i	0.711269	−	0.702919i	$-0.248121\pi$
$114$	0	0
$115$	− 8.94427i	− 0.834058i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.47214	−0.593300
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.8885	1.21345	0.606724	−	0.794913i	$-0.292483\pi$
$131$	13.8885	1.21345	0.606724	+	0.794913i	$0.292483\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 19.8885i	− 1.69919i	−0.527433	−	0.849596i	$-0.676846\pi$
$137$	− 19.8885i	− 1.69919i	0.527433	−	0.849596i	$-0.323154\pi$
$138$	0	0
$139$	−2.94427	−0.249730	−0.124865	−	0.992174i	$-0.539850\pi$
$139$	−2.94427	−0.249730	−0.124865	+	0.992174i	$0.539850\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.94427i	0.747958i
$144$	0	0
$145$	18.9443	1.57324
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.4721	−1.02176	−0.510879	−	0.859653i	$-0.670680\pi$
$149$	−12.4721	−1.02176	−0.510879	+	0.859653i	$0.670680\pi$
$150$	0	0
$151$	17.8885	1.45575	0.727875	−	0.685710i	$-0.240508\pi$
$151$	17.8885	1.45575	0.727875	+	0.685710i	$0.240508\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.05573	−0.0847981
$156$	0	0
$157$	− 9.41641i	− 0.751511i	−0.926719	−	0.375756i	$-0.877383\pi$
$157$	− 9.41641i	− 0.751511i	0.926719	−	0.375756i	$-0.122617\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	− 23.4164i	− 1.83411i	−0.398755	−	0.917057i	$-0.630558\pi$
$163$	− 23.4164i	− 1.83411i	0.398755	−	0.917057i	$-0.369442\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 10.4721i	− 0.810358i	−0.914237	−	0.405179i	$-0.867209\pi$
$167$	− 10.4721i	− 0.810358i	0.914237	−	0.405179i	$-0.132791\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	0	0
$175$	5.00000i	0.377964i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.9443	1.71494	0.857468	−	0.514538i	$-0.172036\pi$
$179$	22.9443	1.71494	0.857468	+	0.514538i	$0.172036\pi$
$180$	0	0
$181$	16.4721	1.22436	0.612182	−	0.790717i	$-0.290292\pi$
$181$	16.4721	1.22436	0.612182	+	0.790717i	$0.290292\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.52786i	0.406417i
$186$	0	0
$187$	− 12.9443i	− 0.946579i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.4164	−1.26021	−0.630104	−	0.776511i	$-0.716988\pi$
$191$	−17.4164	−1.26021	−0.630104	+	0.776511i	$0.716988\pi$
$192$	0	0
$193$	12.9443i	0.931749i	0.884851	+	0.465875i	$0.154260\pi$
$193$	12.9443i	0.931749i	−0.884851	+	0.465875i	$0.845740\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19.8885i	− 1.41700i	−0.705711	−	0.708500i	$-0.749372\pi$
$197$	− 19.8885i	− 1.41700i	0.705711	−	0.708500i	$-0.250628\pi$
$198$	0	0
$199$	20.4721	1.45123	0.725616	−	0.688100i	$-0.241555\pi$
$199$	20.4721	1.45123	0.725616	+	0.688100i	$0.241555\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 8.47214i	− 0.594627i
$204$	0	0
$205$	−7.88854	−0.550960
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−25.8885	−1.78224	−0.891120	−	0.453767i	$-0.850080\pi$
$211$	−25.8885	−1.78224	−0.891120	+	0.453767i	$0.850080\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 5.52786i	− 0.376997i
$216$	0	0
$217$	0.472136i	0.0320507i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	28.9443	1.94700
$222$	0	0
$223$	− 20.9443i	− 1.40253i	−0.712900	−	0.701266i	$-0.752618\pi$
$223$	− 20.9443i	− 1.40253i	0.712900	−	0.701266i	$-0.247382\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.88854i	0.656326i	0.944621	+	0.328163i	$0.106429\pi$
$227$	9.88854i	0.656326i	−0.944621	+	0.328163i	$0.893571\pi$
$228$	0	0
$229$	26.3607	1.74196	0.870981	−	0.491316i	$-0.163484\pi$
$229$	26.3607	1.74196	0.870981	+	0.491316i	$0.163484\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.94427i	0.192886i	0.995339	+	0.0964428i	$0.0307465\pi$
$233$	2.94427i	0.192886i	−0.995339	+	0.0964428i	$0.969254\pi$
$234$	0	0
$235$	14.4721i	0.944058i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.5279	−1.00441	−0.502207	−	0.864747i	$-0.667478\pi$
$239$	−15.5279	−1.00441	−0.502207	+	0.864747i	$0.667478\pi$
$240$	0	0
$241$	−11.8885	−0.765808	−0.382904	−	0.923788i	$-0.625076\pi$
$241$	−11.8885	−0.765808	−0.382904	+	0.923788i	$0.625076\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.23607	0.142857
$246$	0	0
$247$	− 8.94427i	− 0.569110i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.4164i	1.46068i	0.683086	+	0.730338i	$0.260637\pi$
$257$	23.4164i	1.46068i	−0.683086	+	0.730338i	$0.739363\pi$
$258$	0	0
$259$	2.47214	0.153611
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	0	0
$265$	4.47214i	0.274721i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.3607	1.85112	0.925562	−	0.378597i	$-0.123593\pi$
$269$	30.3607	1.85112	0.925562	+	0.378597i	$0.123593\pi$
$270$	0	0
$271$	−24.4721	−1.48658	−0.743288	−	0.668971i	$-0.766735\pi$
$271$	−24.4721	−1.48658	−0.743288	+	0.668971i	$0.766735\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.0000	−0.603023
$276$	0	0
$277$	− 23.4164i	− 1.40696i	−0.710717	−	0.703478i	$-0.751629\pi$
$277$	− 23.4164i	− 1.40696i	0.710717	−	0.703478i	$-0.248371\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.94427	−0.175641	−0.0878203	−	0.996136i	$-0.527990\pi$
$281$	−2.94427	−0.175641	−0.0878203	+	0.996136i	$0.527990\pi$
$282$	0	0
$283$	− 16.9443i	− 1.00723i	−0.863927	−	0.503616i	$-0.832003\pi$
$283$	− 16.9443i	− 1.00723i	0.863927	−	0.503616i	$-0.167997\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.52786i	0.208243i
$288$	0	0
$289$	−24.8885	−1.46403
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 24.9443i	− 1.45726i	−0.684908	−	0.728630i	$-0.740157\pi$
$293$	− 24.9443i	− 1.45726i	0.684908	−	0.728630i	$-0.259843\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.8885	1.03452
$300$	0	0
$301$	−2.47214	−0.142492
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.88854	0.451697
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	− 25.4164i	− 1.43662i	−0.695723	−	0.718310i	$-0.744916\pi$
$313$	− 25.4164i	− 1.43662i	0.695723	−	0.718310i	$-0.255084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.94427i	0.390029i	0.980800	+	0.195015i	$0.0624754\pi$
$317$	6.94427i	0.390029i	−0.980800	+	0.195015i	$0.937525\pi$
$318$	0	0
$319$	16.9443	0.948697
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.9443i	0.720239i
$324$	0	0
$325$	− 22.3607i	− 1.24035i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.47214	0.356820
$330$	0	0
$331$	−33.8885	−1.86268	−0.931341	−	0.364147i	$-0.881361\pi$
$331$	−33.8885	−1.86268	−0.931341	+	0.364147i	$0.881361\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 3.41641i	− 0.186658i
$336$	0	0
$337$	11.0557i	0.602244i	0.953586	+	0.301122i	$0.0973611\pi$
$337$	11.0557i	0.602244i	−0.953586	+	0.301122i	$0.902639\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.944272	−0.0511352
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 30.8328i	− 1.65519i	−0.561324	−	0.827596i	$-0.689708\pi$
$347$	− 30.8328i	− 1.65519i	0.561324	−	0.827596i	$-0.310292\pi$
$348$	0	0
$349$	−25.4164	−1.36051	−0.680255	−	0.732976i	$-0.738131\pi$
$349$	−25.4164	−1.36051	−0.680255	+	0.732976i	$0.738131\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 17.5279i	− 0.932914i	−0.884544	−	0.466457i	$-0.845530\pi$
$353$	− 17.5279i	− 0.932914i	0.884544	−	0.466457i	$-0.154470\pi$
$354$	0	0
$355$	−27.8885	−1.48017
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.4164	−0.708091	−0.354045	−	0.935228i	$-0.615194\pi$
$359$	−13.4164	−0.708091	−0.354045	+	0.935228i	$0.615194\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 16.8328i	− 0.881070i
$366$	0	0
$367$	− 25.8885i	− 1.35137i	−0.737190	−	0.675685i	$-0.763848\pi$
$367$	− 25.8885i	− 1.35137i	0.737190	−	0.675685i	$-0.236152\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	− 16.3607i	− 0.847124i	−0.905867	−	0.423562i	$-0.860780\pi$
$373$	− 16.3607i	− 0.847124i	0.905867	−	0.423562i	$-0.139220\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.8885i	1.95136i
$378$	0	0
$379$	−5.88854	−0.302474	−0.151237	−	0.988498i	$-0.548326\pi$
$379$	−5.88854	−0.302474	−0.151237	+	0.988498i	$0.548326\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 20.3607i	− 1.04038i	−0.854050	−	0.520191i	$-0.825861\pi$
$383$	− 20.3607i	− 1.04038i	0.854050	−	0.520191i	$-0.174139\pi$
$384$	0	0
$385$	4.47214i	0.227921i
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.58359	−0.130993	−0.0654967	−	0.997853i	$-0.520863\pi$
$389$	−2.58359	−0.130993	−0.0654967	+	0.997853i	$0.520863\pi$
$390$	0	0
$391$	−25.8885	−1.30924
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−20.0000	−1.00631
$396$	0	0
$397$	7.52786i	0.377813i	0.981995	+	0.188906i	$0.0604943\pi$
$397$	7.52786i	0.377813i	−0.981995	+	0.188906i	$0.939506\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.8885	0.793436	0.396718	−	0.917941i	$-0.370149\pi$
$401$	15.8885	0.793436	0.396718	+	0.917941i	$0.370149\pi$
$402$	0	0
$403$	− 2.11146i	− 0.105179i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.94427i	0.245078i
$408$	0	0
$409$	7.88854	0.390063	0.195032	−	0.980797i	$-0.437519\pi$
$409$	7.88854	0.390063	0.195032	+	0.980797i	$0.437519\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.0557i	0.542704i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−10.9443	−0.533391	−0.266696	−	0.963781i	$-0.585932\pi$
$421$	−10.9443	−0.533391	−0.266696	+	0.963781i	$0.585932\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	32.3607i	1.56972i
$426$	0	0
$427$	− 3.52786i	− 0.170725i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.4164	−1.03159	−0.515796	−	0.856711i	$-0.672504\pi$
$431$	−21.4164	−1.03159	−0.515796	+	0.856711i	$0.672504\pi$
$432$	0	0
$433$	21.4164i	1.02921i	0.857428	+	0.514603i	$0.172061\pi$
$433$	21.4164i	1.02921i	−0.857428	+	0.514603i	$0.827939\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	31.5279	1.50474	0.752371	−	0.658739i	$-0.228910\pi$
$439$	31.5279	1.50474	0.752371	+	0.658739i	$0.228910\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 36.9443i	− 1.75528i	−0.479325	−	0.877638i	$-0.659118\pi$
$443$	− 36.9443i	− 1.75528i	0.479325	−	0.877638i	$-0.340882\pi$
$444$	0	0
$445$	38.9443	1.84614
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.05573	−0.427366	−0.213683	−	0.976903i	$-0.568546\pi$
$449$	−9.05573	−0.427366	−0.213683	+	0.976903i	$0.568546\pi$
$450$	0	0
$451$	−7.05573	−0.332241
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.0000	−0.468807
$456$	0	0
$457$	8.94427i	0.418395i	0.977873	+	0.209198i	$0.0670852\pi$
$457$	8.94427i	0.418395i	−0.977873	+	0.209198i	$0.932915\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.4721	0.580885	0.290443	−	0.956892i	$-0.406197\pi$
$461$	12.4721	0.580885	0.290443	+	0.956892i	$0.406197\pi$
$462$	0	0
$463$	34.8328i	1.61882i	0.587245	+	0.809409i	$0.300212\pi$
$463$	34.8328i	1.61882i	−0.587245	+	0.809409i	$0.699788\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.0557i	0.881794i	0.897558	+	0.440897i	$0.145340\pi$
$467$	19.0557i	0.881794i	−0.897558	+	0.440897i	$0.854660\pi$
$468$	0	0
$469$	−1.52786	−0.0705502
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 4.94427i	− 0.227338i
$474$	0	0
$475$	10.0000	0.458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−11.0557	−0.504098
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.88854i	0.358200i
$486$	0	0
$487$	13.8885i	0.629350i	0.949199	+	0.314675i	$0.101896\pi$
$487$	13.8885i	0.629350i	−0.949199	+	0.314675i	$0.898104\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.05573	0.0476443	0.0238222	−	0.999716i	$-0.492416\pi$
$491$	1.05573	0.0476443	0.0238222	+	0.999716i	$0.492416\pi$
$492$	0	0
$493$	− 54.8328i	− 2.46955i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.4721i	0.559452i
$498$	0	0
$499$	−5.88854	−0.263607	−0.131804	−	0.991276i	$-0.542077\pi$
$499$	−5.88854	−0.263607	−0.131804	+	0.991276i	$0.542077\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 39.4164i	− 1.75749i	−0.477291	−	0.878745i	$-0.658381\pi$
$503$	− 39.4164i	− 1.75749i	0.477291	−	0.878745i	$-0.341619\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.5836	−0.469109	−0.234555	−	0.972103i	$-0.575363\pi$
$509$	−10.5836	−0.469109	−0.234555	+	0.972103i	$0.575363\pi$
$510$	0	0
$511$	−7.52786	−0.333013
$512$	0	0
$513$	0	0
$514$	0	0
$515$	28.9443i	1.27544i
$516$	0	0
$517$	12.9443i	0.569288i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.41641	0.412540	0.206270	−	0.978495i	$-0.433867\pi$
$521$	9.41641	0.412540	0.206270	+	0.978495i	$0.433867\pi$
$522$	0	0
$523$	7.05573i	0.308525i	0.988030	+	0.154263i	$0.0493002\pi$
$523$	7.05573i	0.308525i	−0.988030	+	0.154263i	$0.950700\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.05573i	0.133110i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 15.7771i	− 0.683382i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−22.9443	−0.986451	−0.493226	−	0.869901i	$-0.664182\pi$
$541$	−22.9443	−0.986451	−0.493226	+	0.869901i	$0.664182\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	33.4164	1.43140
$546$	0	0
$547$	37.3050i	1.59504i	0.603289	+	0.797522i	$0.293856\pi$
$547$	37.3050i	1.59504i	−0.603289	+	0.797522i	$0.706144\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.9443	−0.721850
$552$	0	0
$553$	8.94427i	0.380349i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.05573i	0.214218i	0.994247	+	0.107109i	$0.0341594\pi$
$557$	5.05573i	0.214218i	−0.994247	+	0.107109i	$0.965841\pi$
$558$	0	0
$559$	11.0557	0.467607
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.8328i	1.46803i	0.679134	+	0.734014i	$0.262355\pi$
$563$	34.8328i	1.46803i	−0.679134	+	0.734014i	$0.737645\pi$
$564$	0	0
$565$	33.4164i	1.40584i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	10.1115	0.423151	0.211576	−	0.977362i	$-0.432141\pi$
$571$	10.1115	0.423151	0.211576	+	0.977362i	$0.432141\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	20.0000i	0.834058i
$576$	0	0
$577$	− 24.4721i	− 1.01879i	−0.860533	−	0.509394i	$-0.829870\pi$
$577$	− 24.4721i	− 1.01879i	0.860533	−	0.509394i	$-0.170130\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.94427	0.205123
$582$	0	0
$583$	4.00000i	0.165663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.0557i	0.621416i	0.950505	+	0.310708i	$0.100566\pi$
$587$	15.0557i	0.621416i	−0.950505	+	0.310708i	$0.899434\pi$
$588$	0	0
$589$	0.944272	0.0389080
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.3050i	0.874890i	0.899245	+	0.437445i	$0.144116\pi$
$593$	21.3050i	0.874890i	−0.899245	+	0.437445i	$0.855884\pi$
$594$	0	0
$595$	14.4721	0.593300
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.3607	−0.750197	−0.375099	−	0.926985i	$-0.622391\pi$
$599$	−18.3607	−0.750197	−0.375099	+	0.926985i	$0.622391\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.6525	0.636364
$606$	0	0
$607$	− 4.94427i	− 0.200682i	−0.994953	−	0.100341i	$-0.968007\pi$
$607$	− 4.94427i	− 0.200682i	0.994953	−	0.100341i	$-0.0319933\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−28.9443	−1.17096
$612$	0	0
$613$	− 7.41641i	− 0.299546i	−0.988720	−	0.149773i	$-0.952146\pi$
$613$	− 7.41641i	− 0.299546i	0.988720	−	0.149773i	$-0.0478543\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 47.8885i	− 1.92792i	−0.266047	−	0.963960i	$-0.585718\pi$
$617$	− 47.8885i	− 1.92792i	0.266047	−	0.963960i	$-0.414282\pi$
$618$	0	0
$619$	11.8885	0.477841	0.238920	−	0.971039i	$-0.423206\pi$
$619$	11.8885	0.477841	0.238920	+	0.971039i	$0.423206\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 17.4164i	− 0.697774i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.0000	0.637962
$630$	0	0
$631$	−23.0557	−0.917834	−0.458917	−	0.888479i	$-0.651762\pi$
$631$	−23.0557	−0.917834	−0.458917	+	0.888479i	$0.651762\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 8.94427i	− 0.354943i
$636$	0	0
$637$	4.47214i	0.177192i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	7.05573i	0.278251i	0.990275	+	0.139125i	$0.0444291\pi$
$643$	7.05573i	0.278251i	−0.990275	+	0.139125i	$0.955571\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.2492i	1.50373i	0.659316	+	0.751866i	$0.270846\pi$
$647$	38.2492i	1.50373i	−0.659316	+	0.751866i	$0.729154\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.05573i	− 0.0413138i	−0.999787	−	0.0206569i	$-0.993424\pi$
$653$	− 1.05573i	− 0.0413138i	0.999787	−	0.0206569i	$-0.00657577\pi$
$654$	0	0
$655$	−31.0557	−1.21345
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.1115	0.471795	0.235898	−	0.971778i	$-0.424197\pi$
$659$	12.1115	0.471795	0.235898	+	0.971778i	$0.424197\pi$
$660$	0	0
$661$	48.2492	1.87668	0.938339	−	0.345717i	$-0.112364\pi$
$661$	48.2492	1.87668	0.938339	+	0.345717i	$0.112364\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 4.47214i	− 0.173422i
$666$	0	0
$667$	− 33.8885i	− 1.31217i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.05573	0.272383
$672$	0	0
$673$	21.8885i	0.843741i	0.906656	+	0.421871i	$0.138626\pi$
$673$	21.8885i	0.843741i	−0.906656	+	0.421871i	$0.861374\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.11146i	0.0811499i	0.999176	+	0.0405749i	$0.0129189\pi$
$677$	2.11146i	0.0811499i	−0.999176	+	0.0405749i	$0.987081\pi$
$678$	0	0
$679$	3.52786	0.135387
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 14.1115i	− 0.539960i	−0.962866	−	0.269980i	$-0.912983\pi$
$683$	− 14.1115i	− 0.539960i	0.962866	−	0.269980i	$-0.0870171\pi$
$684$	0	0
$685$	44.4721i	1.69919i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.94427	−0.340750
$690$	0	0
$691$	48.8328	1.85769	0.928844	−	0.370471i	$-0.120804\pi$
$691$	48.8328	1.85769	0.928844	+	0.370471i	$0.120804\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.58359	0.249730
$696$	0	0
$697$	22.8328i	0.864855i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0.472136	0.0178323	0.00891616	−	0.999960i	$-0.497162\pi$
$701$	0.472136	0.0178323	0.00891616	+	0.999960i	$0.497162\pi$
$702$	0	0
$703$	− 4.94427i	− 0.186477i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.47214i	− 0.168192i
$708$	0	0
$709$	−35.8885	−1.34782	−0.673911	−	0.738812i	$-0.735387\pi$
$709$	−35.8885	−1.34782	−0.673911	+	0.738812i	$0.735387\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.88854i	0.0707265i
$714$	0	0
$715$	− 20.0000i	− 0.747958i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.94427	−0.333565	−0.166783	−	0.985994i	$-0.553338\pi$
$719$	−8.94427	−0.333565	−0.166783	+	0.985994i	$0.553338\pi$
$720$	0	0
$721$	12.9443	0.482070
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−42.3607	−1.57324
$726$	0	0
$727$	− 19.0557i	− 0.706738i	−0.935484	−	0.353369i	$-0.885036\pi$
$727$	− 19.0557i	− 0.706738i	0.935484	−	0.353369i	$-0.114964\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	− 5.63932i	− 0.208293i	−0.994562	−	0.104147i	$-0.966789\pi$
$733$	− 5.63932i	− 0.208293i	0.994562	−	0.104147i	$-0.0332111\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.05573i	− 0.112559i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.83282i	0.103926i	0.998649	+	0.0519630i	$0.0165478\pi$
$743$	2.83282i	0.103926i	−0.998649	+	0.0519630i	$0.983452\pi$
$744$	0	0
$745$	27.8885	1.02176
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.8328	−0.979143	−0.489572	−	0.871963i	$-0.662847\pi$
$751$	−26.8328	−0.979143	−0.489572	+	0.871963i	$0.662847\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−40.0000	−1.45575
$756$	0	0
$757$	30.4721i	1.10753i	0.832673	+	0.553764i	$0.186809\pi$
$757$	30.4721i	1.10753i	−0.832673	+	0.553764i	$0.813191\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24.4721	0.887114	0.443557	−	0.896246i	$-0.353716\pi$
$761$	24.4721	0.887114	0.443557	+	0.896246i	$0.353716\pi$
$762$	0	0
$763$	− 14.9443i	− 0.541019i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1.05573	−0.0380705	−0.0190353	−	0.999819i	$-0.506059\pi$
$769$	−1.05573	−0.0380705	−0.0190353	+	0.999819i	$0.506059\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16.9443i	0.609443i	0.952441	+	0.304722i	$0.0985634\pi$
$773$	16.9443i	0.609443i	−0.952441	+	0.304722i	$0.901437\pi$
$774$	0	0
$775$	2.36068	0.0847981
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.05573	0.252798
$780$	0	0
$781$	−24.9443	−0.892576
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.0557i	0.751511i
$786$	0	0
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.9443	0.531357
$792$	0	0
$793$	15.7771i	0.560261i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.0000i	0.991811i	0.868377	+	0.495905i	$0.165164\pi$
$797$	28.0000i	0.991811i	−0.868377	+	0.495905i	$0.834836\pi$
$798$	0	0
$799$	41.8885	1.48191
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 15.0557i	− 0.531305i
$804$	0	0
$805$	8.94427	0.315244
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.8328	−0.873075	−0.436538	−	0.899686i	$-0.643795\pi$
$809$	−24.8328	−0.873075	−0.436538	+	0.899686i	$0.643795\pi$
$810$	0	0
$811$	−13.0557	−0.458449	−0.229224	−	0.973374i	$-0.573619\pi$
$811$	−13.0557	−0.458449	−0.229224	+	0.973374i	$0.573619\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	52.3607i	1.83411i
$816$	0	0
$817$	4.94427i	0.172978i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.3050	−0.673747	−0.336874	−	0.941550i	$-0.609370\pi$
$821$	−19.3050	−0.673747	−0.336874	+	0.941550i	$0.609370\pi$
$822$	0	0
$823$	− 15.0557i	− 0.524810i	−0.964958	−	0.262405i	$-0.915484\pi$
$823$	− 15.0557i	− 0.524810i	0.964958	−	0.262405i	$-0.0845156\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.7214i	1.13783i	0.822395	+	0.568917i	$0.192637\pi$
$827$	32.7214i	1.13783i	−0.822395	+	0.568917i	$0.807363\pi$
$828$	0	0
$829$	−3.52786	−0.122528	−0.0612639	−	0.998122i	$-0.519513\pi$
$829$	−3.52786	−0.122528	−0.0612639	+	0.998122i	$0.519513\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 6.47214i	− 0.224246i
$834$	0	0
$835$	23.4164i	0.810358i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.9443	−1.13736	−0.568681	−	0.822558i	$-0.692546\pi$
$839$	−32.9443	−1.13736	−0.568681	+	0.822558i	$0.692546\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.6525	0.538462
$846$	0	0
$847$	− 7.00000i	− 0.240523i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.88854	0.338975
$852$	0	0
$853$	− 31.5279i	− 1.07949i	−0.841827	−	0.539747i	$-0.818520\pi$
$853$	− 31.5279i	− 1.07949i	0.841827	−	0.539747i	$-0.181480\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 16.5836i	− 0.566485i	−0.959048	−	0.283242i	$-0.908590\pi$
$857$	− 16.5836i	− 0.566485i	0.959048	−	0.283242i	$-0.0914101\pi$
$858$	0	0
$859$	−21.0557	−0.718412	−0.359206	−	0.933258i	$-0.616952\pi$
$859$	−21.0557	−0.718412	−0.359206	+	0.933258i	$0.616952\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.8328i	0.641077i	0.947236	+	0.320538i	$0.103864\pi$
$863$	18.8328i	0.641077i	−0.947236	+	0.320538i	$0.896136\pi$
$864$	0	0
$865$	26.8328i	0.912343i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−17.8885	−0.606827
$870$	0	0
$871$	6.83282	0.231521
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 11.1803i	− 0.377964i
$876$	0	0
$877$	− 49.3050i	− 1.66491i	−0.554093	−	0.832455i	$-0.686935\pi$
$877$	− 49.3050i	− 1.66491i	0.554093	−	0.832455i	$-0.313065\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−27.5279	−0.927437	−0.463719	−	0.885983i	$-0.653485\pi$
$881$	−27.5279	−0.927437	−0.463719	+	0.885983i	$0.653485\pi$
$882$	0	0
$883$	16.3607i	0.550581i	0.961361	+	0.275290i	$0.0887740\pi$
$883$	16.3607i	0.550581i	−0.961361	+	0.275290i	$0.911226\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.58359i	0.153902i	0.997035	+	0.0769510i	$0.0245185\pi$
$887$	4.58359i	0.153902i	−0.997035	+	0.0769510i	$0.975482\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 12.9443i	− 0.433164i
$894$	0	0
$895$	−51.3050	−1.71494
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	12.9443	0.431236
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−36.8328	−1.22436
$906$	0	0
$907$	− 16.3607i	− 0.543247i	−0.962404	−	0.271624i	$-0.912439\pi$
$907$	− 16.3607i	− 0.543247i	0.962404	−	0.271624i	$-0.0875606\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.4164	−0.842083	−0.421042	−	0.907041i	$-0.638335\pi$
$911$	−25.4164	−0.842083	−0.421042	+	0.907041i	$0.638335\pi$
$912$	0	0
$913$	9.88854i	0.327263i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.8885i	0.458640i
$918$	0	0
$919$	−36.7214	−1.21133	−0.605663	−	0.795721i	$-0.707092\pi$
$919$	−36.7214	−1.21133	−0.605663	+	0.795721i	$0.707092\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 55.7771i	− 1.83593i
$924$	0	0
$925$	− 12.3607i	− 0.406417i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.3050	0.633375	0.316687	−	0.948530i	$-0.397429\pi$
$929$	19.3050	0.633375	0.316687	+	0.948530i	$0.397429\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	28.9443i	0.946579i
$936$	0	0
$937$	15.3050i	0.499991i	0.968247	+	0.249995i	$0.0804291\pi$
$937$	15.3050i	0.499991i	−0.968247	+	0.249995i	$0.919571\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−35.5279	−1.15818	−0.579088	−	0.815265i	$-0.696591\pi$
$941$	−35.5279	−1.15818	−0.579088	+	0.815265i	$0.696591\pi$
$942$	0	0
$943$	14.1115i	0.459532i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.0000i	0.519930i	0.965618	+	0.259965i	$0.0837111\pi$
$947$	16.0000i	0.519930i	−0.965618	+	0.259965i	$0.916289\pi$
$948$	0	0
$949$	33.6656	1.09283
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 18.0000i	− 0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$	− 18.0000i	− 0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$	0	0
$955$	38.9443	1.26021
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.8885	0.642235
$960$	0	0
$961$	−30.7771	−0.992809
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 28.9443i	− 0.931749i
$966$	0	0
$967$	25.8885i	0.832519i	0.909246	+	0.416260i	$0.136659\pi$
$967$	25.8885i	0.832519i	−0.909246	+	0.416260i	$0.863341\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.8885	−1.08754	−0.543768	−	0.839236i	$-0.683003\pi$
$971$	−33.8885	−1.08754	−0.543768	+	0.839236i	$0.683003\pi$
$972$	0	0
$973$	− 2.94427i	− 0.0943890i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 52.8328i	− 1.69027i	−0.534552	−	0.845136i	$-0.679520\pi$
$977$	− 52.8328i	− 1.69027i	0.534552	−	0.845136i	$-0.320480\pi$
$978$	0	0
$979$	34.8328	1.11326
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 25.5279i	− 0.814212i	−0.913381	−	0.407106i	$-0.866538\pi$
$983$	− 25.5279i	− 0.814212i	0.913381	−	0.407106i	$-0.133462\pi$
$984$	0	0
$985$	44.4721i	1.41700i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.88854	−0.314437
$990$	0	0
$991$	56.9443	1.80889	0.904447	−	0.426586i	$-0.140284\pi$
$991$	56.9443	1.80889	0.904447	+	0.426586i	$0.140284\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−45.7771	−1.45123
$996$	0	0
$997$	5.63932i	0.178599i	0.996005	+	0.0892995i	$0.0284628\pi$
$997$	5.63932i	0.178599i	−0.996005	+	0.0892995i	$0.971537\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.u.1009.2		4
3.2	odd	2		1680.2.t.h.1009.4		4
4.3	odd	2		2520.2.t.f.1009.1		4
5.4	even	2	inner	5040.2.t.u.1009.1		4
12.11	even	2		840.2.t.c.169.2	✓	4
15.2	even	4		8400.2.a.db.1.1		2
15.8	even	4		8400.2.a.cz.1.2		2
15.14	odd	2		1680.2.t.h.1009.2		4
20.19	odd	2		2520.2.t.f.1009.2		4
60.23	odd	4		4200.2.a.bk.1.2		2
60.47	odd	4		4200.2.a.bj.1.1		2
60.59	even	2		840.2.t.c.169.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.c.169.2	✓	4	12.11	even	2
840.2.t.c.169.4	yes	4	60.59	even	2
1680.2.t.h.1009.2		4	15.14	odd	2
1680.2.t.h.1009.4		4	3.2	odd	2
2520.2.t.f.1009.1		4	4.3	odd	2
2520.2.t.f.1009.2		4	20.19	odd	2
4200.2.a.bj.1.1		2	60.47	odd	4
4200.2.a.bk.1.2		2	60.23	odd	4
5040.2.t.u.1009.1		4	5.4	even	2	inner
5040.2.t.u.1009.2		4	1.1	even	1	trivial
8400.2.a.cz.1.2		2	15.8	even	4
8400.2.a.db.1.1		2	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)

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-2.23607 q^{5} +1.00000i q^{7} -2.00000 q^{11} -4.47214i q^{13} +6.47214i q^{17} +2.00000 q^{19} +4.00000i q^{23} +5.00000 q^{25} -8.47214 q^{29} +0.472136 q^{31} -2.23607i q^{35} -2.47214i q^{37} +3.52786 q^{41} +2.47214i q^{43} -6.47214i q^{47} -1.00000 q^{49} -2.00000i q^{53} +4.47214 q^{55} -3.52786 q^{61} +10.0000i q^{65} +1.52786i q^{67} +12.4721 q^{71} +7.52786i q^{73} -2.00000i q^{77} +8.94427 q^{79} -4.94427i q^{83} -14.4721i q^{85} -17.4164 q^{89} +4.47214 q^{91} -4.47214 q^{95} -3.52786i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{11} + 8 q^{19} + 20 q^{25} - 16 q^{29} - 16 q^{31} + 32 q^{41} - 4 q^{49} - 32 q^{61} + 32 q^{71} - 16 q^{89}+O(q^{100}) \) 4 * q - 8 * q^11 + 8 * q^19 + 20 * q^25 - 16 * q^29 - 16 * q^31 + 32 * q^41 - 4 * q^49 - 32 * q^61 + 32 * q^71 - 16 * q^89

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