LMFDB - Embedded newform 4020.2.g.c.1609.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.g (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.32
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.17292 + 1.90375i) q^{5} -2.37828i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.17292 + 1.90375i) q^{5} -2.37828i q^{7} -1.00000 q^{9} -2.77093 q^{11} -0.720315i q^{13} +(-1.90375 + 1.17292i) q^{15} +3.36647i q^{17} -5.08557 q^{19} +2.37828 q^{21} -6.99724i q^{23} +(-2.24851 + 4.46590i) q^{25} -1.00000i q^{27} -10.0764 q^{29} +7.55706 q^{31} -2.77093i q^{33} +(4.52764 - 2.78954i) q^{35} -10.4627i q^{37} +0.720315 q^{39} +10.6294 q^{41} -7.57505i q^{43} +(-1.17292 - 1.90375i) q^{45} +0.513534i q^{47} +1.34379 q^{49} -3.36647 q^{51} +2.32405i q^{53} +(-3.25009 - 5.27515i) q^{55} -5.08557i q^{57} +9.22703 q^{59} +13.6979 q^{61} +2.37828i q^{63} +(1.37130 - 0.844874i) q^{65} +1.00000i q^{67} +6.99724 q^{69} -11.5610 q^{71} -4.90896i q^{73} +(-4.46590 - 2.24851i) q^{75} +6.59005i q^{77} -2.46242 q^{79} +1.00000 q^{81} -7.91192i q^{83} +(-6.40891 + 3.94861i) q^{85} -10.0764i q^{87} -3.49779 q^{89} -1.71311 q^{91} +7.55706i q^{93} +(-5.96498 - 9.68164i) q^{95} -12.5554i q^{97} +2.77093 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) $ 38 * q - 2 * q^5 - 38 * q^9	$ 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) $ 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.17292	+	1.90375i	0.524547	+	0.851382i
$5$	1.17292	+	1.90375i	0.524547	+	0.851382i
$6$		0			0
$7$		−	2.37828i		−	0.898905i	−0.893304	−	0.449453i	$-0.851619\pi$
$7$		−	2.37828i		−	0.898905i	0.893304	−	0.449453i	$-0.148381\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.77093			−0.835467			−0.417734	−	0.908570i	$-0.637175\pi$
$11$	−2.77093			−0.835467			−0.417734	+	0.908570i	$0.637175\pi$
$12$		0			0
$13$		−	0.720315i		−	0.199780i	−0.994999	−	0.0998898i	$-0.968151\pi$
$13$		−	0.720315i		−	0.199780i	0.994999	−	0.0998898i	$-0.0318490\pi$
$14$		0			0
$15$	−1.90375	+	1.17292i	−0.491545	+	0.302847i
$16$		0			0
$17$			3.36647i			0.816489i	0.912873	+	0.408245i	$0.133859\pi$
$17$			3.36647i			0.816489i	−0.912873	+	0.408245i	$0.866141\pi$
$18$		0			0
$19$	−5.08557			−1.16671			−0.583355	−	0.812217i	$-0.698260\pi$
$19$	−5.08557			−1.16671			−0.583355	+	0.812217i	$0.698260\pi$
$20$		0			0
$21$	2.37828			0.518983
$22$		0			0
$23$		−	6.99724i		−	1.45903i	−0.683967	−	0.729513i	$-0.739747\pi$
$23$		−	6.99724i		−	1.45903i	0.683967	−	0.729513i	$-0.260253\pi$
$24$		0			0
$25$	−2.24851	+	4.46590i	−0.449701	+	0.893179i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−10.0764			−1.87114			−0.935569	−	0.353144i	$-0.885113\pi$
$29$	−10.0764			−1.87114			−0.935569	+	0.353144i	$0.885113\pi$
$30$		0			0
$31$	7.55706			1.35729			0.678644	−	0.734468i	$-0.262568\pi$
$31$	7.55706			1.35729			0.678644	+	0.734468i	$0.262568\pi$
$32$		0			0
$33$		−	2.77093i		−	0.482357i
$34$		0			0
$35$	4.52764	−	2.78954i	0.765311	−	0.471518i
$36$		0			0
$37$		−	10.4627i		−	1.72006i	−0.510244	−	0.860030i	$-0.670445\pi$
$37$		−	10.4627i		−	1.72006i	0.510244	−	0.860030i	$-0.329555\pi$
$38$		0			0
$39$	0.720315			0.115343
$40$		0			0
$41$	10.6294			1.66003			0.830016	−	0.557740i	$-0.188331\pi$
$41$	10.6294			1.66003			0.830016	+	0.557740i	$0.188331\pi$
$42$		0			0
$43$		−	7.57505i		−	1.15518i	−0.816326	−	0.577592i	$-0.803993\pi$
$43$		−	7.57505i		−	1.15518i	0.816326	−	0.577592i	$-0.196007\pi$
$44$		0			0
$45$	−1.17292	−	1.90375i	−0.174849	−	0.283794i
$46$		0			0
$47$			0.513534i			0.0749066i	0.999298	+	0.0374533i	$0.0119245\pi$
$47$			0.513534i			0.0749066i	−0.999298	+	0.0374533i	$0.988075\pi$
$48$		0			0
$49$	1.34379			0.191970
$50$		0			0
$51$	−3.36647			−0.471400
$52$		0			0
$53$			2.32405i			0.319233i	0.987179	+	0.159617i	$0.0510258\pi$
$53$			2.32405i			0.319233i	−0.987179	+	0.159617i	$0.948974\pi$
$54$		0			0
$55$	−3.25009	−	5.27515i	−0.438242	−	0.711301i
$56$		0			0
$57$		−	5.08557i		−	0.673600i
$58$		0			0
$59$	9.22703			1.20126			0.600629	−	0.799528i	$-0.294917\pi$
$59$	9.22703			1.20126			0.600629	+	0.799528i	$0.294917\pi$
$60$		0			0
$61$	13.6979			1.75384			0.876919	−	0.480638i	$-0.159595\pi$
$61$	13.6979			1.75384			0.876919	+	0.480638i	$0.159595\pi$
$62$		0			0
$63$			2.37828i			0.299635i
$64$		0			0
$65$	1.37130	−	0.844874i	0.170089	−	0.104794i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	6.99724			0.842369
$70$		0			0
$71$	−11.5610			−1.37204			−0.686018	−	0.727585i	$-0.740643\pi$
$71$	−11.5610			−1.37204			−0.686018	+	0.727585i	$0.740643\pi$
$72$		0			0
$73$		−	4.90896i		−	0.574550i	−0.957848	−	0.287275i	$-0.907251\pi$
$73$		−	4.90896i		−	0.574550i	0.957848	−	0.287275i	$-0.0927494\pi$
$74$		0			0
$75$	−4.46590	−	2.24851i	−0.515677	−	0.259635i
$76$		0			0
$77$			6.59005i			0.751006i
$78$		0			0
$79$	−2.46242			−0.277044			−0.138522	−	0.990359i	$-0.544235\pi$
$79$	−2.46242			−0.277044			−0.138522	+	0.990359i	$0.544235\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.91192i		−	0.868446i	−0.900805	−	0.434223i	$-0.857023\pi$
$83$		−	7.91192i		−	0.868446i	0.900805	−	0.434223i	$-0.142977\pi$
$84$		0			0
$85$	−6.40891	+	3.94861i	−0.695144	+	0.428287i
$86$		0			0
$87$		−	10.0764i		−	1.08030i
$88$		0			0
$89$	−3.49779			−0.370765			−0.185382	−	0.982666i	$-0.559352\pi$
$89$	−3.49779			−0.370765			−0.185382	+	0.982666i	$0.559352\pi$
$90$		0			0
$91$	−1.71311			−0.179583
$92$		0			0
$93$			7.55706i			0.783630i
$94$		0			0
$95$	−5.96498	−	9.68164i	−0.611994	−	0.993315i
$96$		0			0
$97$		−	12.5554i		−	1.27481i	−0.770531	−	0.637403i	$-0.780009\pi$
$97$		−	12.5554i		−	1.27481i	0.770531	−	0.637403i	$-0.219991\pi$
$98$		0			0
$99$	2.77093			0.278489
$100$		0			0
$101$	−7.64280			−0.760487			−0.380243	−	0.924887i	$-0.624160\pi$
$101$	−7.64280			−0.760487			−0.380243	+	0.924887i	$0.624160\pi$
$102$		0			0
$103$		−	17.2781i		−	1.70246i	−0.524793	−	0.851230i	$-0.675857\pi$
$103$		−	17.2781i		−	1.70246i	0.524793	−	0.851230i	$-0.324143\pi$
$104$		0			0
$105$	2.78954	+	4.52764i	0.272231	+	0.441853i
$106$		0			0
$107$		−	5.37539i		−	0.519659i	−0.965655	−	0.259829i	$-0.916334\pi$
$107$		−	5.37539i		−	0.519659i	0.965655	−	0.259829i	$-0.0836663\pi$
$108$		0			0
$109$	−13.1319			−1.25781			−0.628906	−	0.777481i	$-0.716497\pi$
$109$	−13.1319			−1.25781			−0.628906	+	0.777481i	$0.716497\pi$
$110$		0			0
$111$	10.4627			0.993077
$112$		0			0
$113$			14.0631i			1.32295i	0.749969	+	0.661473i	$0.230068\pi$
$113$			14.0631i			1.32295i	−0.749969	+	0.661473i	$0.769932\pi$
$114$		0			0
$115$	13.3210	−	8.20722i	1.24219	−	0.765327i
$116$		0			0
$117$			0.720315i			0.0665932i
$118$		0			0
$119$	8.00641			0.733947
$120$		0			0
$121$	−3.32194			−0.301995
$122$		0			0
$123$			10.6294i			0.958420i
$124$		0			0
$125$	−11.1393	+	0.957565i	−0.996326	+	0.0856472i
$126$		0			0
$127$		−	9.35357i		−	0.829995i	−0.909822	−	0.414998i	$-0.863782\pi$
$127$		−	9.35357i		−	0.829995i	0.909822	−	0.414998i	$-0.136218\pi$
$128$		0			0
$129$	7.57505			0.666946
$130$		0			0
$131$	−6.05269			−0.528826			−0.264413	−	0.964409i	$-0.585178\pi$
$131$	−6.05269			−0.528826			−0.264413	+	0.964409i	$0.585178\pi$
$132$		0			0
$133$			12.0949i			1.04876i
$134$		0			0
$135$	1.90375	−	1.17292i	0.163848	−	0.100949i
$136$		0			0
$137$			7.84119i			0.669918i	0.942233	+	0.334959i	$0.108722\pi$
$137$			7.84119i			0.669918i	−0.942233	+	0.334959i	$0.891278\pi$
$138$		0			0
$139$	13.8995			1.17894			0.589469	−	0.807791i	$-0.299337\pi$
$139$	13.8995			1.17894			0.589469	+	0.807791i	$0.299337\pi$
$140$		0			0
$141$	−0.513534			−0.0432473
$142$		0			0
$143$			1.99594i			0.166909i
$144$		0			0
$145$	−11.8188	−	19.1829i	−0.981500	−	1.59305i
$146$		0			0
$147$			1.34379i			0.110834i
$148$		0			0
$149$	0.0593509			0.00486222			0.00243111	−	0.999997i	$-0.499226\pi$
$149$	0.0593509			0.00486222			0.00243111	+	0.999997i	$0.499226\pi$
$150$		0			0
$151$	−0.668221			−0.0543790			−0.0271895	−	0.999630i	$-0.508656\pi$
$151$	−0.668221			−0.0543790			−0.0271895	+	0.999630i	$0.508656\pi$
$152$		0			0
$153$		−	3.36647i		−	0.272163i
$154$		0			0
$155$	8.86384	+	14.3867i	0.711961	+	1.15557i
$156$		0			0
$157$		−	0.374589i		−	0.0298954i	−0.999888	−	0.0149477i	$-0.995242\pi$
$157$		−	0.374589i		−	0.0298954i	0.999888	−	0.0149477i	$-0.00475818\pi$
$158$		0			0
$159$	−2.32405			−0.184309
$160$		0			0
$161$	−16.6414			−1.31153
$162$		0			0
$163$		−	12.2079i		−	0.956196i	−0.878307	−	0.478098i	$-0.841326\pi$
$163$		−	12.2079i		−	0.956196i	0.878307	−	0.478098i	$-0.158674\pi$
$164$		0			0
$165$	5.27515	−	3.25009i	0.410670	−	0.253019i
$166$		0			0
$167$		−	14.8326i		−	1.14778i	−0.818933	−	0.573889i	$-0.805434\pi$
$167$		−	14.8326i		−	1.14778i	0.818933	−	0.573889i	$-0.194566\pi$
$168$		0			0
$169$	12.4811			0.960088
$170$		0			0
$171$	5.08557			0.388903
$172$		0			0
$173$		−	15.6634i		−	1.19086i	−0.803406	−	0.595431i	$-0.796981\pi$
$173$		−	15.6634i		−	1.19086i	0.803406	−	0.595431i	$-0.203019\pi$
$174$		0			0
$175$	10.6211	+	5.34757i	0.802883	+	0.404239i
$176$		0			0
$177$			9.22703i			0.693546i
$178$		0			0
$179$	15.1453			1.13201			0.566006	−	0.824401i	$-0.308488\pi$
$179$	15.1453			1.13201			0.566006	+	0.824401i	$0.308488\pi$
$180$		0			0
$181$	1.11499			0.0828767			0.0414384	−	0.999141i	$-0.486806\pi$
$181$	1.11499			0.0828767			0.0414384	+	0.999141i	$0.486806\pi$
$182$		0			0
$183$			13.6979i			1.01258i
$184$		0			0
$185$	19.9184	−	12.2720i	1.46443	−	0.902252i
$186$		0			0
$187$		−	9.32826i		−	0.682150i
$188$		0			0
$189$	−2.37828			−0.172994
$190$		0			0
$191$	0.189407			0.0137050			0.00685250	−	0.999977i	$-0.497819\pi$
$191$	0.189407			0.0137050			0.00685250	+	0.999977i	$0.497819\pi$
$192$		0			0
$193$			5.12381i			0.368820i	0.982849	+	0.184410i	$0.0590374\pi$
$193$			5.12381i			0.368820i	−0.982849	+	0.184410i	$0.940963\pi$
$194$		0			0
$195$	0.844874	+	1.37130i	0.0605027	+	0.0982007i
$196$		0			0
$197$			1.44327i			0.102829i	0.998677	+	0.0514144i	$0.0163729\pi$
$197$			1.44327i			0.102829i	−0.998677	+	0.0514144i	$0.983627\pi$
$198$		0			0
$199$	−17.0443			−1.20824			−0.604119	−	0.796894i	$-0.706475\pi$
$199$	−17.0443			−1.20824			−0.604119	+	0.796894i	$0.706475\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$			23.9645i			1.68198i
$204$		0			0
$205$	12.4675	+	20.2357i	0.870765	+	1.41332i
$206$		0			0
$207$			6.99724i			0.486342i
$208$		0			0
$209$	14.0918			0.974748
$210$		0			0
$211$	−22.2415			−1.53117			−0.765584	−	0.643336i	$-0.777550\pi$
$211$	−22.2415			−1.53117			−0.765584	+	0.643336i	$0.777550\pi$
$212$		0			0
$213$		−	11.5610i		−	0.792145i
$214$		0			0
$215$	14.4210	−	8.88494i	0.983502	−	0.605948i
$216$		0			0
$217$		−	17.9728i		−	1.22007i
$218$		0			0
$219$	4.90896			0.331717
$220$		0			0
$221$	2.42492			0.163118
$222$		0			0
$223$			24.9462i			1.67052i	0.549855	+	0.835260i	$0.314683\pi$
$223$			24.9462i			1.67052i	−0.549855	+	0.835260i	$0.685317\pi$
$224$		0			0
$225$	2.24851	−	4.46590i	0.149900	−	0.297726i
$226$		0			0
$227$		−	9.95027i		−	0.660423i	−0.943907	−	0.330211i	$-0.892880\pi$
$227$		−	9.95027i		−	0.660423i	0.943907	−	0.330211i	$-0.107120\pi$
$228$		0			0
$229$	−28.8825			−1.90861			−0.954304	−	0.298838i	$-0.903401\pi$
$229$	−28.8825			−1.90861			−0.954304	+	0.298838i	$0.903401\pi$
$230$		0			0
$231$	−6.59005			−0.433593
$232$		0			0
$233$			0.721388i			0.0472597i	0.999721	+	0.0236298i	$0.00752231\pi$
$233$			0.721388i			0.0472597i	−0.999721	+	0.0236298i	$0.992478\pi$
$234$		0			0
$235$	−0.977638	+	0.602335i	−0.0637741	+	0.0392920i
$236$		0			0
$237$		−	2.46242i		−	0.159951i
$238$		0			0
$239$	0.356570			0.0230646			0.0115323	−	0.999934i	$-0.496329\pi$
$239$	0.356570			0.0230646			0.0115323	+	0.999934i	$0.496329\pi$
$240$		0			0
$241$	28.1224			1.81152			0.905762	−	0.423787i	$-0.139300\pi$
$241$	28.1224			1.81152			0.905762	+	0.423787i	$0.139300\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	1.57616	+	2.55823i	0.100697	+	0.163439i
$246$		0			0
$247$			3.66321i			0.233085i
$248$		0			0
$249$	7.91192			0.501398
$250$		0			0
$251$	26.2989			1.65997			0.829986	−	0.557784i	$-0.188348\pi$
$251$	26.2989			1.65997			0.829986	+	0.557784i	$0.188348\pi$
$252$		0			0
$253$			19.3889i			1.21897i
$254$		0			0
$255$	−3.94861	−	6.40891i	−0.247272	−	0.401342i
$256$		0			0
$257$			13.4382i			0.838252i	0.907928	+	0.419126i	$0.137663\pi$
$257$			13.4382i			0.838252i	−0.907928	+	0.419126i	$0.862337\pi$
$258$		0			0
$259$	−24.8833			−1.54617
$260$		0			0
$261$	10.0764			0.623713
$262$		0			0
$263$			25.4628i			1.57010i	0.619430	+	0.785052i	$0.287364\pi$
$263$			25.4628i			1.57010i	−0.619430	+	0.785052i	$0.712636\pi$
$264$		0			0
$265$	−4.42441	+	2.72593i	−0.271789	+	0.167453i
$266$		0			0
$267$		−	3.49779i		−	0.214061i
$268$		0			0
$269$	−13.0428			−0.795235			−0.397617	−	0.917551i	$-0.630163\pi$
$269$	−13.0428			−0.795235			−0.397617	+	0.917551i	$0.630163\pi$
$270$		0			0
$271$	1.15043			0.0698839			0.0349419	−	0.999389i	$-0.488875\pi$
$271$	1.15043			0.0698839			0.0349419	+	0.999389i	$0.488875\pi$
$272$		0			0
$273$		−	1.71311i		−	0.103682i
$274$		0			0
$275$	6.23045	−	12.3747i	0.375710	−	0.746222i
$276$		0			0
$277$		−	23.0377i		−	1.38420i	−0.721802	−	0.692100i	$-0.756686\pi$
$277$		−	23.0377i		−	1.38420i	0.721802	−	0.692100i	$-0.243314\pi$
$278$		0			0
$279$	−7.55706			−0.452429
$280$		0			0
$281$	23.1204			1.37925			0.689623	−	0.724169i	$-0.257776\pi$
$281$	23.1204			1.37925			0.689623	+	0.724169i	$0.257776\pi$
$282$		0			0
$283$			7.85938i			0.467192i	0.972334	+	0.233596i	$0.0750493\pi$
$283$			7.85938i			0.467192i	−0.972334	+	0.233596i	$0.924951\pi$
$284$		0			0
$285$	9.68164	−	5.96498i	0.573491	−	0.353335i
$286$		0			0
$287$		−	25.2797i		−	1.49221i
$288$		0			0
$289$	5.66686			0.333345
$290$		0			0
$291$	12.5554			0.736010
$292$		0			0
$293$		−	30.5975i		−	1.78753i	−0.448539	−	0.893763i	$-0.648055\pi$
$293$		−	30.5975i		−	1.78753i	0.448539	−	0.893763i	$-0.351945\pi$
$294$		0			0
$295$	10.8226	+	17.5659i	0.630116	+	1.02273i
$296$		0			0
$297$			2.77093i			0.160786i
$298$		0			0
$299$	−5.04022			−0.291483
$300$		0			0
$301$	−18.0156			−1.03840
$302$		0			0
$303$		−	7.64280i		−	0.439067i
$304$		0			0
$305$	16.0666	+	26.0774i	0.919970	+	1.49319i
$306$		0			0
$307$			2.70770i			0.154536i	0.997010	+	0.0772682i	$0.0246198\pi$
$307$			2.70770i			0.154536i	−0.997010	+	0.0772682i	$0.975380\pi$
$308$		0			0
$309$	17.2781			0.982916
$310$		0			0
$311$	26.2358			1.48770			0.743848	−	0.668349i	$-0.232999\pi$
$311$	26.2358			1.48770			0.743848	+	0.668349i	$0.232999\pi$
$312$		0			0
$313$		−	7.83735i		−	0.442993i	−0.975161	−	0.221496i	$-0.928906\pi$
$313$		−	7.83735i		−	0.442993i	0.975161	−	0.221496i	$-0.0710941\pi$
$314$		0			0
$315$	−4.52764	+	2.78954i	−0.255104	+	0.157173i
$316$		0			0
$317$		−	11.7895i		−	0.662166i	−0.943602	−	0.331083i	$-0.892586\pi$
$317$		−	11.7895i		−	0.662166i	0.943602	−	0.331083i	$-0.107414\pi$
$318$		0			0
$319$	27.9210			1.56327
$320$		0			0
$321$	5.37539			0.300025
$322$		0			0
$323$		−	17.1204i		−	0.952606i
$324$		0			0
$325$	3.21685	+	1.61963i	0.178439	+	0.0898411i
$326$		0			0
$327$		−	13.1319i		−	0.726198i
$328$		0			0
$329$	1.22133			0.0673339
$330$		0			0
$331$	30.3704			1.66931			0.834655	−	0.550774i	$-0.185667\pi$
$331$	30.3704			1.66931			0.834655	+	0.550774i	$0.185667\pi$
$332$		0			0
$333$			10.4627i			0.573353i
$334$		0			0
$335$	−1.90375	+	1.17292i	−0.104013	+	0.0640836i
$336$		0			0
$337$		−	13.5035i		−	0.735582i	−0.929909	−	0.367791i	$-0.880114\pi$
$337$		−	13.5035i		−	0.735582i	0.929909	−	0.367791i	$-0.119886\pi$
$338$		0			0
$339$	−14.0631			−0.763803
$340$		0			0
$341$	−20.9401			−1.13397
$342$		0			0
$343$		−	19.8439i		−	1.07147i
$344$		0			0
$345$	8.20722	+	13.3210i	0.441862	+	0.717177i
$346$		0			0
$347$			2.07645i			0.111470i	0.998446	+	0.0557348i	$0.0177501\pi$
$347$			2.07645i			0.111470i	−0.998446	+	0.0557348i	$0.982250\pi$
$348$		0			0
$349$	−29.6334			−1.58624			−0.793120	−	0.609065i	$-0.791545\pi$
$349$	−29.6334			−1.58624			−0.793120	+	0.609065i	$0.791545\pi$
$350$		0			0
$351$	−0.720315			−0.0384476
$352$		0			0
$353$		−	22.1611i		−	1.17952i	−0.807579	−	0.589759i	$-0.799223\pi$
$353$		−	22.1611i		−	1.17952i	0.807579	−	0.589759i	$-0.200777\pi$
$354$		0			0
$355$	−13.5601	−	22.0092i	−0.719697	−	1.16813i
$356$		0			0
$357$			8.00641i			0.423744i
$358$		0			0
$359$	−13.1318			−0.693071			−0.346535	−	0.938037i	$-0.612642\pi$
$359$	−13.1318			−0.693071			−0.346535	+	0.938037i	$0.612642\pi$
$360$		0			0
$361$	6.86303			0.361212
$362$		0			0
$363$		−	3.32194i		−	0.174357i
$364$		0			0
$365$	9.34541	−	5.75783i	0.489161	−	0.301378i
$366$		0			0
$367$			3.55491i			0.185565i	0.995686	+	0.0927824i	$0.0295761\pi$
$367$			3.55491i			0.185565i	−0.995686	+	0.0927824i	$0.970424\pi$
$368$		0			0
$369$	−10.6294			−0.553344
$370$		0			0
$371$	5.52725			0.286960
$372$		0			0
$373$			13.3162i			0.689485i	0.938697	+	0.344742i	$0.112034\pi$
$373$			13.3162i			0.689485i	−0.938697	+	0.344742i	$0.887966\pi$
$374$		0			0
$375$	−0.957565	−	11.1393i	−0.0494484	−	0.575229i
$376$		0			0
$377$			7.25818i			0.373815i
$378$		0			0
$379$	−19.1125			−0.981742			−0.490871	−	0.871232i	$-0.663321\pi$
$379$	−19.1125			−0.981742			−0.490871	+	0.871232i	$0.663321\pi$
$380$		0			0
$381$	9.35357			0.479198
$382$		0			0
$383$			7.39217i			0.377722i	0.982004	+	0.188861i	$0.0604796\pi$
$383$			7.39217i			0.377722i	−0.982004	+	0.188861i	$0.939520\pi$
$384$		0			0
$385$	−12.5458	+	7.72962i	−0.639392	+	0.393938i
$386$		0			0
$387$			7.57505i			0.385061i
$388$		0			0
$389$	−14.5282			−0.736608			−0.368304	−	0.929705i	$-0.620061\pi$
$389$	−14.5282			−0.736608			−0.368304	+	0.929705i	$0.620061\pi$
$390$		0			0
$391$	23.5560			1.19128
$392$		0			0
$393$		−	6.05269i		−	0.305318i
$394$		0			0
$395$	−2.88822	−	4.68782i	−0.145322	−	0.235870i
$396$		0			0
$397$			9.89164i			0.496447i	0.968703	+	0.248223i	$0.0798467\pi$
$397$			9.89164i			0.496447i	−0.968703	+	0.248223i	$0.920153\pi$
$398$		0			0
$399$	−12.0949			−0.605503
$400$		0			0
$401$	−7.50802			−0.374933			−0.187466	−	0.982271i	$-0.560028\pi$
$401$	−7.50802			−0.374933			−0.187466	+	0.982271i	$0.560028\pi$
$402$		0			0
$403$		−	5.44346i		−	0.271158i
$404$		0			0
$405$	1.17292	+	1.90375i	0.0582830	+	0.0945979i
$406$		0			0
$407$			28.9915i			1.43705i
$408$		0			0
$409$	−18.7809			−0.928654			−0.464327	−	0.885664i	$-0.653704\pi$
$409$	−18.7809			−0.928654			−0.464327	+	0.885664i	$0.653704\pi$
$410$		0			0
$411$	−7.84119			−0.386777
$412$		0			0
$413$		−	21.9445i		−	1.07982i
$414$		0			0
$415$	15.0623	−	9.28007i	0.739379	−	0.455541i
$416$		0			0
$417$			13.8995i			0.680661i
$418$		0			0
$419$	−10.1865			−0.497643			−0.248821	−	0.968549i	$-0.580043\pi$
$419$	−10.1865			−0.497643			−0.248821	+	0.968549i	$0.580043\pi$
$420$		0			0
$421$	−17.3680			−0.846464			−0.423232	−	0.906021i	$-0.639104\pi$
$421$	−17.3680			−0.846464			−0.423232	+	0.906021i	$0.639104\pi$
$422$		0			0
$423$		−	0.513534i		−	0.0249689i
$424$		0			0
$425$	−15.0343	−	7.56953i	−0.729271	−	0.367176i
$426$		0			0
$427$		−	32.5775i		−	1.57653i
$428$		0			0
$429$	−1.99594			−0.0963651
$430$		0			0
$431$	−14.7039			−0.708260			−0.354130	−	0.935196i	$-0.615223\pi$
$431$	−14.7039			−0.708260			−0.354130	+	0.935196i	$0.615223\pi$
$432$		0			0
$433$			26.1141i			1.25496i	0.778632	+	0.627481i	$0.215914\pi$
$433$			26.1141i			1.25496i	−0.778632	+	0.627481i	$0.784086\pi$
$434$		0			0
$435$	19.1829	−	11.8188i	0.919749	−	0.566669i
$436$		0			0
$437$			35.5850i			1.70226i
$438$		0			0
$439$	2.69256			0.128509			0.0642543	−	0.997934i	$-0.479533\pi$
$439$	2.69256			0.128509			0.0642543	+	0.997934i	$0.479533\pi$
$440$		0			0
$441$	−1.34379			−0.0639899
$442$		0			0
$443$		−	23.0794i		−	1.09654i	−0.836302	−	0.548268i	$-0.815287\pi$
$443$		−	23.0794i		−	1.09654i	0.836302	−	0.548268i	$-0.184713\pi$
$444$		0			0
$445$	−4.10263	−	6.65890i	−0.194483	−	0.315662i
$446$		0			0
$447$			0.0593509i			0.00280720i
$448$		0			0
$449$	−20.3397			−0.959888			−0.479944	−	0.877299i	$-0.659343\pi$
$449$	−20.3397			−0.959888			−0.479944	+	0.877299i	$0.659343\pi$
$450$		0			0
$451$	−29.4533			−1.38690
$452$		0			0
$453$		−	0.668221i		−	0.0313957i
$454$		0			0
$455$	−2.00935	−	3.26133i	−0.0941996	−	0.152894i
$456$		0			0
$457$		−	7.04045i		−	0.329338i	−0.986349	−	0.164669i	$-0.947344\pi$
$457$		−	7.04045i		−	0.329338i	0.986349	−	0.164669i	$-0.0526557\pi$
$458$		0			0
$459$	3.36647			0.157133
$460$		0			0
$461$	15.3191			0.713482			0.356741	−	0.934203i	$-0.383888\pi$
$461$	15.3191			0.713482			0.356741	+	0.934203i	$0.383888\pi$
$462$		0			0
$463$			33.8371i			1.57254i	0.617882	+	0.786271i	$0.287991\pi$
$463$			33.8371i			1.57254i	−0.617882	+	0.786271i	$0.712009\pi$
$464$		0			0
$465$	−14.3867	+	8.86384i	−0.667168	+	0.411051i
$466$		0			0
$467$		−	23.8753i		−	1.10482i	−0.833573	−	0.552410i	$-0.813708\pi$
$467$		−	23.8753i		−	1.10482i	0.833573	−	0.552410i	$-0.186292\pi$
$468$		0			0
$469$	2.37828			0.109819
$470$		0			0
$471$	0.374589			0.0172601
$472$		0			0
$473$			20.9899i			0.965118i
$474$		0			0
$475$	11.4349	−	22.7116i	0.524671	−	1.04208i
$476$		0			0
$477$		−	2.32405i		−	0.106411i
$478$		0			0
$479$	29.1876			1.33362			0.666808	−	0.745230i	$-0.267660\pi$
$479$	29.1876			1.33362			0.666808	+	0.745230i	$0.267660\pi$
$480$		0			0
$481$	−7.53645			−0.343633
$482$		0			0
$483$		−	16.6414i		−	0.757210i
$484$		0			0
$485$	23.9023	−	14.7265i	1.08535	−	0.668696i
$486$		0			0
$487$		−	28.4217i		−	1.28791i	−0.765064	−	0.643954i	$-0.777293\pi$
$487$		−	28.4217i		−	1.28791i	0.765064	−	0.643954i	$-0.222707\pi$
$488$		0			0
$489$	12.2079			0.552060
$490$		0			0
$491$	−7.33763			−0.331142			−0.165571	−	0.986198i	$-0.552947\pi$
$491$	−7.33763			−0.331142			−0.165571	+	0.986198i	$0.552947\pi$
$492$		0			0
$493$		−	33.9219i		−	1.52776i
$494$		0			0
$495$	3.25009	+	5.27515i	0.146081	+	0.237100i
$496$		0			0
$497$			27.4952i			1.23333i
$498$		0			0
$499$	−17.4402			−0.780728			−0.390364	−	0.920661i	$-0.627651\pi$
$499$	−17.4402			−0.780728			−0.390364	+	0.920661i	$0.627651\pi$
$500$		0			0
$501$	14.8326			0.662670
$502$		0			0
$503$			20.0264i			0.892934i	0.894800	+	0.446467i	$0.147318\pi$
$503$			20.0264i			0.892934i	−0.894800	+	0.446467i	$0.852682\pi$
$504$		0			0
$505$	−8.96441	−	14.5499i	−0.398911	−	0.647464i
$506$		0			0
$507$			12.4811i			0.554307i
$508$		0			0
$509$	−15.3450			−0.680157			−0.340078	−	0.940397i	$-0.610454\pi$
$509$	−15.3450			−0.680157			−0.340078	+	0.940397i	$0.610454\pi$
$510$		0			0
$511$	−11.6749			−0.516466
$512$		0			0
$513$			5.08557i			0.224533i
$514$		0			0
$515$	32.8931	−	20.2658i	1.44944	−	0.893020i
$516$		0			0
$517$		−	1.42297i		−	0.0625820i
$518$		0			0
$519$	15.6634			0.687545
$520$		0			0
$521$	39.5178			1.73131			0.865654	−	0.500643i	$-0.166903\pi$
$521$	39.5178			1.73131			0.865654	+	0.500643i	$0.166903\pi$
$522$		0			0
$523$		−	20.3082i		−	0.888017i	−0.896023	−	0.444008i	$-0.853556\pi$
$523$		−	20.3082i		−	0.888017i	0.896023	−	0.444008i	$-0.146444\pi$
$524$		0			0
$525$	−5.34757	+	10.6211i	−0.233387	+	0.463545i
$526$		0			0
$527$			25.4406i			1.10821i
$528$		0			0
$529$	−25.9614			−1.12876
$530$		0			0
$531$	−9.22703			−0.400419
$532$		0			0
$533$		−	7.65651i		−	0.331640i
$534$		0			0
$535$	10.2334	−	6.30492i	0.442428	−	0.272585i
$536$		0			0
$537$			15.1453i			0.653567i
$538$		0			0
$539$	−3.72354			−0.160384
$540$		0			0
$541$	−21.4449			−0.921988			−0.460994	−	0.887403i	$-0.652507\pi$
$541$	−21.4449			−0.921988			−0.460994	+	0.887403i	$0.652507\pi$
$542$		0			0
$543$			1.11499i			0.0478489i
$544$		0			0
$545$	−15.4028	−	24.9999i	−0.659782	−	1.07088i
$546$		0			0
$547$		−	0.981250i		−	0.0419552i	−0.999780	−	0.0209776i	$-0.993322\pi$
$547$		−	0.981250i		−	0.0419552i	0.999780	−	0.0209776i	$-0.00667787\pi$
$548$		0			0
$549$	−13.6979			−0.584613
$550$		0			0
$551$	51.2442			2.18308
$552$		0			0
$553$			5.85632i			0.249036i
$554$		0			0
$555$	12.2720	+	19.9184i	0.520915	+	0.845487i
$556$		0			0
$557$			44.7526i			1.89623i	0.317927	+	0.948115i	$0.397013\pi$
$557$			44.7526i			1.89623i	−0.317927	+	0.948115i	$0.602987\pi$
$558$		0			0
$559$	−5.45642			−0.230782
$560$		0			0
$561$	9.32826			0.393840
$562$		0			0
$563$			28.6701i			1.20830i	0.796870	+	0.604151i	$0.206487\pi$
$563$			28.6701i			1.20830i	−0.796870	+	0.604151i	$0.793513\pi$
$564$		0			0
$565$	−26.7726	+	16.4949i	−1.12633	+	0.693947i
$566$		0			0
$567$		−	2.37828i		−	0.0998783i
$568$		0			0
$569$	−20.0606			−0.840985			−0.420492	−	0.907296i	$-0.638143\pi$
$569$	−20.0606			−0.840985			−0.420492	+	0.907296i	$0.638143\pi$
$570$		0			0
$571$	−5.70606			−0.238791			−0.119396	−	0.992847i	$-0.538096\pi$
$571$	−5.70606			−0.238791			−0.119396	+	0.992847i	$0.538096\pi$
$572$		0			0
$573$			0.189407i			0.00791259i
$574$		0			0
$575$	31.2489	+	15.7333i	1.30317	+	0.656125i
$576$		0			0
$577$			41.7552i			1.73829i	0.494555	+	0.869146i	$0.335331\pi$
$577$			41.7552i			1.73829i	−0.494555	+	0.869146i	$0.664669\pi$
$578$		0			0
$579$	−5.12381			−0.212938
$580$		0			0
$581$	−18.8168			−0.780651
$582$		0			0
$583$		−	6.43979i		−	0.266709i
$584$		0			0
$585$	−1.37130	+	0.844874i	−0.0566962	+	0.0349312i
$586$		0			0
$587$		−	32.1527i		−	1.32709i	−0.748139	−	0.663543i	$-0.769052\pi$
$587$		−	32.1527i		−	1.32709i	0.748139	−	0.663543i	$-0.230948\pi$
$588$		0			0
$589$	−38.4319			−1.58356
$590$		0			0
$591$	−1.44327			−0.0593682
$592$		0			0
$593$			39.5633i			1.62467i	0.583192	+	0.812335i	$0.301804\pi$
$593$			39.5633i			1.62467i	−0.583192	+	0.812335i	$0.698196\pi$
$594$		0			0
$595$	9.39090	+	15.2422i	0.384989	+	0.624869i
$596$		0			0
$597$		−	17.0443i		−	0.697576i
$598$		0			0
$599$	−4.07245			−0.166396			−0.0831978	−	0.996533i	$-0.526513\pi$
$599$	−4.07245			−0.166396			−0.0831978	+	0.996533i	$0.526513\pi$
$600$		0			0
$601$	−26.9987			−1.10130			−0.550651	−	0.834736i	$-0.685620\pi$
$601$	−26.9987			−1.10130			−0.550651	+	0.834736i	$0.685620\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	−3.89638	−	6.32414i	−0.158410	−	0.257113i
$606$		0			0
$607$		−	21.2983i		−	0.864471i	−0.901761	−	0.432236i	$-0.857725\pi$
$607$		−	21.2983i		−	0.864471i	0.901761	−	0.432236i	$-0.142275\pi$
$608$		0			0
$609$	−23.9645			−0.971089
$610$		0			0
$611$	0.369906			0.0149648
$612$		0			0
$613$		−	1.31314i		−	0.0530374i	−0.999648	−	0.0265187i	$-0.991558\pi$
$613$		−	1.31314i		−	0.0530374i	0.999648	−	0.0265187i	$-0.00844216\pi$
$614$		0			0
$615$	−20.2357	+	12.4675i	−0.815981	+	0.502736i
$616$		0			0
$617$			12.9225i			0.520240i	0.965576	+	0.260120i	$0.0837622\pi$
$617$			12.9225i			0.520240i	−0.965576	+	0.260120i	$0.916238\pi$
$618$		0			0
$619$	3.47314			0.139597			0.0697986	−	0.997561i	$-0.477764\pi$
$619$	3.47314			0.139597			0.0697986	+	0.997561i	$0.477764\pi$
$620$		0			0
$621$	−6.99724			−0.280790
$622$		0			0
$623$			8.31871i			0.333282i
$624$		0			0
$625$	−14.8884	−	20.0832i	−0.595538	−	0.803327i
$626$		0			0
$627$			14.0918i			0.562771i
$628$		0			0
$629$	35.2224			1.40441
$630$		0			0
$631$	−24.3398			−0.968954			−0.484477	−	0.874804i	$-0.660990\pi$
$631$	−24.3398			−0.968954			−0.484477	+	0.874804i	$0.660990\pi$
$632$		0			0
$633$		−	22.2415i		−	0.884020i
$634$		0			0
$635$	17.8068	−	10.9710i	0.706643	−	0.435372i
$636$		0			0
$637$		−	0.967950i		−	0.0383516i
$638$		0			0
$639$	11.5610			0.457345
$640$		0			0
$641$	10.0349			0.396356			0.198178	−	0.980166i	$-0.436498\pi$
$641$	10.0349			0.396356			0.198178	+	0.980166i	$0.436498\pi$
$642$		0			0
$643$		−	42.2175i		−	1.66490i	−0.554103	−	0.832448i	$-0.686939\pi$
$643$		−	42.2175i		−	1.66490i	0.554103	−	0.832448i	$-0.313061\pi$
$644$		0			0
$645$	8.88494	+	14.4210i	0.349844	+	0.567825i
$646$		0			0
$647$		−	1.43176i		−	0.0562884i	−0.999604	−	0.0281442i	$-0.991040\pi$
$647$		−	1.43176i		−	0.0562884i	0.999604	−	0.0281442i	$-0.00895976\pi$
$648$		0			0
$649$	−25.5675			−1.00361
$650$		0			0
$651$	17.9728			0.704409
$652$		0			0
$653$		−	11.6570i		−	0.456173i	−0.973641	−	0.228087i	$-0.926753\pi$
$653$		−	11.6570i		−	0.456173i	0.973641	−	0.228087i	$-0.0732469\pi$
$654$		0			0
$655$	−7.09934	−	11.5228i	−0.277394	−	0.450233i
$656$		0			0
$657$			4.90896i			0.191517i
$658$		0			0
$659$	2.62294			0.102175			0.0510877	−	0.998694i	$-0.483731\pi$
$659$	2.62294			0.102175			0.0510877	+	0.998694i	$0.483731\pi$
$660$		0			0
$661$	−6.25395			−0.243251			−0.121625	−	0.992576i	$-0.538811\pi$
$661$	−6.25395			−0.243251			−0.121625	+	0.992576i	$0.538811\pi$
$662$		0			0
$663$			2.42492i			0.0941762i
$664$		0			0
$665$	−23.0256	+	14.1864i	−0.892896	+	0.550125i
$666$		0			0
$667$			70.5069i			2.73004i
$668$		0			0
$669$	−24.9462			−0.964475
$670$		0			0
$671$	−37.9560			−1.46527
$672$		0			0
$673$		−	10.6011i		−	0.408644i	−0.978904	−	0.204322i	$-0.934501\pi$
$673$		−	10.6011i		−	0.408644i	0.978904	−	0.204322i	$-0.0654990\pi$
$674$		0			0
$675$	4.46590	+	2.24851i	0.171892	+	0.0865450i
$676$		0			0
$677$			1.25826i			0.0483589i	0.999708	+	0.0241794i	$0.00769731\pi$
$677$			1.25826i			0.0483589i	−0.999708	+	0.0241794i	$0.992303\pi$
$678$		0			0
$679$	−29.8602			−1.14593
$680$		0			0
$681$	9.95027			0.381295
$682$		0			0
$683$			4.31165i			0.164981i	0.996592	+	0.0824904i	$0.0262874\pi$
$683$			4.31165i			0.164981i	−0.996592	+	0.0824904i	$0.973713\pi$
$684$		0			0
$685$	−14.9276	+	9.19710i	−0.570356	+	0.351403i
$686$		0			0
$687$		−	28.8825i		−	1.10194i
$688$		0			0
$689$	1.67405			0.0637762
$690$		0			0
$691$	−45.4992			−1.73087			−0.865436	−	0.501019i	$-0.832959\pi$
$691$	−45.4992			−1.73087			−0.865436	+	0.501019i	$0.832959\pi$
$692$		0			0
$693$		−	6.59005i		−	0.250335i
$694$		0			0
$695$	16.3030	+	26.4611i	0.618409	+	1.00373i
$696$		0			0
$697$			35.7835i			1.35540i
$698$		0			0
$699$	−0.721388			−0.0272854
$700$		0			0
$701$	13.2773			0.501475			0.250738	−	0.968055i	$-0.419327\pi$
$701$	13.2773			0.501475			0.250738	+	0.968055i	$0.419327\pi$
$702$		0			0
$703$			53.2089i			2.00681i
$704$		0			0
$705$	−0.602335	−	0.977638i	−0.0226853	−	0.0368200i
$706$		0			0
$707$			18.1767i			0.683605i
$708$		0			0
$709$	−13.0531			−0.490220			−0.245110	−	0.969495i	$-0.578824\pi$
$709$	−13.0531			−0.490220			−0.245110	+	0.969495i	$0.578824\pi$
$710$		0			0
$711$	2.46242			0.0923479
$712$		0			0
$713$		−	52.8785i		−	1.98032i
$714$		0			0
$715$	−3.79977	+	2.34109i	−0.142103	+	0.0875517i
$716$		0			0
$717$			0.356570i			0.0133164i
$718$		0			0
$719$	35.1616			1.31131			0.655653	−	0.755062i	$-0.272393\pi$
$719$	35.1616			1.31131			0.655653	+	0.755062i	$0.272393\pi$
$720$		0			0
$721$	−41.0921			−1.53035
$722$		0			0
$723$			28.1224i			1.04588i
$724$		0			0
$725$	22.6568	−	45.0001i	0.841453	−	1.67126i
$726$		0			0
$727$		−	6.79429i		−	0.251986i	−0.992031	−	0.125993i	$-0.959788\pi$
$727$		−	6.79429i		−	0.251986i	0.992031	−	0.125993i	$-0.0402117\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	25.5012			0.943195
$732$		0			0
$733$			7.30768i			0.269915i	0.990851	+	0.134958i	$0.0430898\pi$
$733$			7.30768i			0.269915i	−0.990851	+	0.134958i	$0.956910\pi$
$734$		0			0
$735$	−2.55823	+	1.57616i	−0.0943617	+	0.0581375i
$736$		0			0
$737$		−	2.77093i		−	0.102069i
$738$		0			0
$739$	15.7162			0.578130			0.289065	−	0.957310i	$-0.406656\pi$
$739$	15.7162			0.578130			0.289065	+	0.957310i	$0.406656\pi$
$740$		0			0
$741$	−3.66321			−0.134572
$742$		0			0
$743$			5.72768i			0.210128i	0.994465	+	0.105064i	$0.0335048\pi$
$743$			5.72768i			0.210128i	−0.994465	+	0.105064i	$0.966495\pi$
$744$		0			0
$745$	0.0696140	+	0.112989i	0.00255046	+	0.00413960i
$746$		0			0
$747$			7.91192i			0.289482i
$748$		0			0
$749$	−12.7842			−0.467124
$750$		0			0
$751$	39.2342			1.43167			0.715837	−	0.698267i	$-0.246045\pi$
$751$	39.2342			1.43167			0.715837	+	0.698267i	$0.246045\pi$
$752$		0			0
$753$			26.2989i			0.958385i
$754$		0			0
$755$	−0.783771	−	1.27212i	−0.0285244	−	0.0462973i
$756$		0			0
$757$		−	33.6988i		−	1.22481i	−0.790546	−	0.612403i	$-0.790203\pi$
$757$		−	33.6988i		−	1.22481i	0.790546	−	0.612403i	$-0.209797\pi$
$758$		0			0
$759$	−19.3889			−0.703771
$760$		0			0
$761$	−47.0023			−1.70383			−0.851916	−	0.523679i	$-0.824559\pi$
$761$	−47.0023			−1.70383			−0.851916	+	0.523679i	$0.824559\pi$
$762$		0			0
$763$			31.2314i			1.13065i
$764$		0			0
$765$	6.40891	−	3.94861i	0.231715	−	0.142762i
$766$		0			0
$767$		−	6.64637i		−	0.239987i
$768$		0			0
$769$	46.0445			1.66041			0.830203	−	0.557462i	$-0.188225\pi$
$769$	46.0445			1.66041			0.830203	+	0.557462i	$0.188225\pi$
$770$		0			0
$771$	−13.4382			−0.483965
$772$		0			0
$773$			52.2738i			1.88016i	0.340956	+	0.940079i	$0.389249\pi$
$773$			52.2738i			1.88016i	−0.340956	+	0.940079i	$0.610751\pi$
$774$		0			0
$775$	−16.9921	+	33.7490i	−0.610373	+	1.21230i
$776$		0			0
$777$		−	24.8833i		−	0.892682i
$778$		0			0
$779$	−54.0565			−1.93678
$780$		0			0
$781$	32.0347			1.14629
$782$		0			0
$783$			10.0764i			0.360101i
$784$		0			0
$785$	0.713122	−	0.439364i	0.0254524	−	0.0156816i
$786$		0			0
$787$			36.8463i			1.31343i	0.754139	+	0.656715i	$0.228055\pi$
$787$			36.8463i			1.31343i	−0.754139	+	0.656715i	$0.771945\pi$
$788$		0			0
$789$	−25.4628			−0.906500
$790$		0			0
$791$	33.4460			1.18920
$792$		0			0
$793$		−	9.86682i		−	0.350381i
$794$		0			0
$795$	−2.72593	−	4.42441i	−0.0966789	−	0.156918i
$796$		0			0
$797$			54.7014i			1.93762i	0.247802	+	0.968811i	$0.420292\pi$
$797$			54.7014i			1.93762i	−0.247802	+	0.968811i	$0.579708\pi$
$798$		0			0
$799$	−1.72880			−0.0611604
$800$		0			0
$801$	3.49779			0.123588
$802$		0			0
$803$			13.6024i			0.480018i
$804$		0			0
$805$	−19.5191	−	31.6810i	−0.687957	−	1.11661i
$806$		0			0
$807$		−	13.0428i		−	0.459129i
$808$		0			0
$809$	−23.4389			−0.824067			−0.412034	−	0.911169i	$-0.635181\pi$
$809$	−23.4389			−0.824067			−0.412034	+	0.911169i	$0.635181\pi$
$810$		0			0
$811$	−31.1032			−1.09218			−0.546091	−	0.837726i	$-0.683885\pi$
$811$	−31.1032			−1.09218			−0.546091	+	0.837726i	$0.683885\pi$
$812$		0			0
$813$			1.15043i			0.0403475i
$814$		0			0
$815$	23.2407	−	14.3189i	0.814088	−	0.501570i
$816$		0			0
$817$			38.5234i			1.34776i
$818$		0			0
$819$	1.71311			0.0598610
$820$		0			0
$821$	−20.1313			−0.702588			−0.351294	−	0.936265i	$-0.614258\pi$
$821$	−20.1313			−0.702588			−0.351294	+	0.936265i	$0.614258\pi$
$822$		0			0
$823$		−	35.2268i		−	1.22793i	−0.789334	−	0.613964i	$-0.789574\pi$
$823$		−	35.2268i		−	1.22793i	0.789334	−	0.613964i	$-0.210426\pi$
$824$		0			0
$825$	12.3747	+	6.23045i	0.430831	+	0.216917i
$826$		0			0
$827$			42.3723i			1.47343i	0.676204	+	0.736714i	$0.263624\pi$
$827$			42.3723i			1.47343i	−0.676204	+	0.736714i	$0.736376\pi$
$828$		0			0
$829$	−4.39630			−0.152690			−0.0763449	−	0.997081i	$-0.524325\pi$
$829$	−4.39630			−0.152690			−0.0763449	+	0.997081i	$0.524325\pi$
$830$		0			0
$831$	23.0377			0.799168
$832$		0			0
$833$			4.52382i			0.156741i
$834$		0			0
$835$	28.2374	−	17.3974i	0.977196	−	0.602063i
$836$		0			0
$837$		−	7.55706i		−	0.261210i
$838$		0			0
$839$	−6.86188			−0.236898			−0.118449	−	0.992960i	$-0.537792\pi$
$839$	−6.86188			−0.236898			−0.118449	+	0.992960i	$0.537792\pi$
$840$		0			0
$841$	72.5335			2.50116
$842$		0			0
$843$			23.1204i			0.796308i
$844$		0			0
$845$	14.6394	+	23.7609i	0.503611	+	0.817401i
$846$		0			0
$847$			7.90051i			0.271465i
$848$		0			0
$849$	−7.85938			−0.269733
$850$		0			0
$851$	−73.2101			−2.50961
$852$		0			0
$853$		−	32.9115i		−	1.12687i	−0.826161	−	0.563433i	$-0.809480\pi$
$853$		−	32.9115i		−	1.12687i	0.826161	−	0.563433i	$-0.190520\pi$
$854$		0			0
$855$	5.96498	+	9.68164i	0.203998	+	0.331105i
$856$		0			0
$857$		−	14.0105i		−	0.478591i	−0.970947	−	0.239295i	$-0.923084\pi$
$857$		−	14.0105i		−	0.478591i	0.970947	−	0.239295i	$-0.0769164\pi$
$858$		0			0
$859$	48.0337			1.63889			0.819444	−	0.573160i	$-0.194282\pi$
$859$	48.0337			1.63889			0.819444	+	0.573160i	$0.194282\pi$
$860$		0			0
$861$	25.2797			0.861528
$862$		0			0
$863$		−	38.4258i		−	1.30803i	−0.756482	−	0.654014i	$-0.773084\pi$
$863$		−	38.4258i		−	1.30803i	0.756482	−	0.654014i	$-0.226916\pi$
$864$		0			0
$865$	29.8191	−	18.3719i	1.01388	−	0.624663i
$866$		0			0
$867$			5.66686i			0.192457i
$868$		0			0
$869$	6.82319			0.231461
$870$		0			0
$871$	0.720315			0.0244070
$872$		0			0
$873$			12.5554i			0.424935i
$874$		0			0
$875$	2.27736	+	26.4923i	0.0769887	+	0.895602i
$876$		0			0
$877$			5.67108i			0.191499i	0.995405	+	0.0957495i	$0.0305248\pi$
$877$			5.67108i			0.191499i	−0.995405	+	0.0957495i	$0.969475\pi$
$878$		0			0
$879$	30.5975			1.03203
$880$		0			0
$881$	44.1998			1.48913			0.744564	−	0.667551i	$-0.232657\pi$
$881$	44.1998			1.48913			0.744564	+	0.667551i	$0.232657\pi$
$882$		0			0
$883$			46.9735i			1.58078i	0.612602	+	0.790391i	$0.290123\pi$
$883$			46.9735i			1.58078i	−0.612602	+	0.790391i	$0.709877\pi$
$884$		0			0
$885$	−17.5659	+	10.8226i	−0.590472	+	0.363797i
$886$		0			0
$887$		−	57.1559i		−	1.91911i	−0.281525	−	0.959554i	$-0.590840\pi$
$887$		−	57.1559i		−	1.91911i	0.281525	−	0.959554i	$-0.409160\pi$
$888$		0			0
$889$	−22.2454			−0.746087
$890$		0			0
$891$	−2.77093			−0.0928297
$892$		0			0
$893$		−	2.61161i		−	0.0873942i
$894$		0			0
$895$	17.7642	+	28.8328i	0.593793	+	0.963774i
$896$		0			0
$897$		−	5.04022i		−	0.168288i
$898$		0			0
$899$	−76.1478			−2.53967
$900$		0			0
$901$	−7.82386			−0.260651
$902$		0			0
$903$		−	18.0156i		−	0.599521i
$904$		0			0
$905$	1.30780	+	2.12266i	0.0434727	+	0.0705597i
$906$		0			0
$907$		−	38.1649i		−	1.26725i	−0.773642	−	0.633623i	$-0.781567\pi$
$907$		−	38.1649i		−	1.26725i	0.773642	−	0.633623i	$-0.218433\pi$
$908$		0			0
$909$	7.64280			0.253496
$910$		0			0
$911$	−21.9785			−0.728181			−0.364091	−	0.931364i	$-0.618620\pi$
$911$	−21.9785			−0.728181			−0.364091	+	0.931364i	$0.618620\pi$
$912$		0			0
$913$			21.9234i			0.725558i
$914$		0			0
$915$	−26.0774	+	16.0666i	−0.862091	+	0.531145i
$916$		0			0
$917$			14.3950i			0.475365i
$918$		0			0
$919$	−0.487617			−0.0160850			−0.00804250	−	0.999968i	$-0.502560\pi$
$919$	−0.487617			−0.0160850			−0.00804250	+	0.999968i	$0.502560\pi$
$920$		0			0
$921$	−2.70770			−0.0892217
$922$		0			0
$923$			8.32755i			0.274105i
$924$		0			0
$925$	46.7254	+	23.5255i	1.53632	+	0.773513i
$926$		0			0
$927$			17.2781i			0.567487i
$928$		0			0
$929$	57.4850			1.88602			0.943011	−	0.332760i	$-0.107980\pi$
$929$	57.4850			1.88602			0.943011	+	0.332760i	$0.107980\pi$
$930$		0			0
$931$	−6.83392			−0.223973
$932$		0			0
$933$			26.2358i			0.858921i
$934$		0			0
$935$	17.7587	−	10.9413i	0.580770	−	0.357820i
$936$		0			0
$937$		−	5.12013i		−	0.167267i	−0.996497	−	0.0836337i	$-0.973347\pi$
$937$		−	5.12013i		−	0.167267i	0.996497	−	0.0836337i	$-0.0266526\pi$
$938$		0			0
$939$	7.83735			0.255762
$940$		0			0
$941$	42.6449			1.39018			0.695092	−	0.718921i	$-0.255364\pi$
$941$	42.6449			1.39018			0.695092	+	0.718921i	$0.255364\pi$
$942$		0			0
$943$		−	74.3764i		−	2.42203i
$944$		0			0
$945$	−2.78954	−	4.52764i	−0.0907437	−	0.147284i
$946$		0			0
$947$			5.18927i			0.168629i	0.996439	+	0.0843143i	$0.0268700\pi$
$947$			5.18927i			0.168629i	−0.996439	+	0.0843143i	$0.973130\pi$
$948$		0			0
$949$	−3.53600			−0.114783
$950$		0			0
$951$	11.7895			0.382302
$952$		0			0
$953$			3.05769i			0.0990484i	0.998773	+	0.0495242i	$0.0157705\pi$
$953$			3.05769i			0.0990484i	−0.998773	+	0.0495242i	$0.984230\pi$
$954$		0			0
$955$	0.222160	+	0.360583i	0.00718892	+	0.0116682i
$956$		0			0
$957$			27.9210i			0.902557i
$958$		0			0
$959$	18.6485			0.602192
$960$		0			0
$961$	26.1091			0.842228
$962$		0			0
$963$			5.37539i			0.173220i
$964$		0			0
$965$	−9.75444	+	6.00984i	−0.314007	+	0.193463i
$966$		0			0
$967$			18.2072i			0.585505i	0.956188	+	0.292753i	$0.0945712\pi$
$967$			18.2072i			0.585505i	−0.956188	+	0.292753i	$0.905429\pi$
$968$		0			0
$969$	17.1204			0.549988
$970$		0			0
$971$	−16.1652			−0.518767			−0.259384	−	0.965774i	$-0.583519\pi$
$971$	−16.1652			−0.518767			−0.259384	+	0.965774i	$0.583519\pi$
$972$		0			0
$973$		−	33.0569i		−	1.05975i
$974$		0			0
$975$	−1.61963	+	3.21685i	−0.0518698	+	0.103022i
$976$		0			0
$977$		−	41.1608i		−	1.31685i	−0.752647	−	0.658425i	$-0.771223\pi$
$977$		−	41.1608i		−	1.31685i	0.752647	−	0.658425i	$-0.228777\pi$
$978$		0			0
$979$	9.69212			0.309762
$980$		0			0
$981$	13.1319			0.419271
$982$		0			0
$983$			14.0301i			0.447491i	0.974648	+	0.223746i	$0.0718285\pi$
$983$			14.0301i			0.447491i	−0.974648	+	0.223746i	$0.928172\pi$
$984$		0			0
$985$	−2.74762	+	1.69284i	−0.0875465	+	0.0539385i
$986$		0			0
$987$			1.22133i			0.0388752i
$988$		0			0
$989$	−53.0044			−1.68544
$990$		0			0
$991$	−0.351627			−0.0111698			−0.00558489	−	0.999984i	$-0.501778\pi$
$991$	−0.351627			−0.0111698			−0.00558489	+	0.999984i	$0.501778\pi$
$992$		0			0
$993$			30.3704i			0.963776i
$994$		0			0
$995$	−19.9916	−	32.4480i	−0.633777	−	1.02867i
$996$		0			0
$997$		−	24.3690i		−	0.771775i	−0.922546	−	0.385887i	$-0.873895\pi$
$997$		−	24.3690i		−	0.771775i	0.922546	−	0.385887i	$-0.126105\pi$
$998$		0			0
$999$	−10.4627			−0.331026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.32	yes	38
5.4	even	2	inner	4020.2.g.c.1609.13	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.13	✓	38	5.4	even	2	inner
4020.2.g.c.1609.32	yes	38	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.17292 + 1.90375i) q^{5} -2.37828i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.17292 + 1.90375i) q^{5} -2.37828i q^{7} -1.00000 q^{9} -2.77093 q^{11} -0.720315i q^{13} +(-1.90375 + 1.17292i) q^{15} +3.36647i q^{17} -5.08557 q^{19} +2.37828 q^{21} -6.99724i q^{23} +(-2.24851 + 4.46590i) q^{25} -1.00000i q^{27} -10.0764 q^{29} +7.55706 q^{31} -2.77093i q^{33} +(4.52764 - 2.78954i) q^{35} -10.4627i q^{37} +0.720315 q^{39} +10.6294 q^{41} -7.57505i q^{43} +(-1.17292 - 1.90375i) q^{45} +0.513534i q^{47} +1.34379 q^{49} -3.36647 q^{51} +2.32405i q^{53} +(-3.25009 - 5.27515i) q^{55} -5.08557i q^{57} +9.22703 q^{59} +13.6979 q^{61} +2.37828i q^{63} +(1.37130 - 0.844874i) q^{65} +1.00000i q^{67} +6.99724 q^{69} -11.5610 q^{71} -4.90896i q^{73} +(-4.46590 - 2.24851i) q^{75} +6.59005i q^{77} -2.46242 q^{79} +1.00000 q^{81} -7.91192i q^{83} +(-6.40891 + 3.94861i) q^{85} -10.0764i q^{87} -3.49779 q^{89} -1.71311 q^{91} +7.55706i q^{93} +(-5.96498 - 9.68164i) q^{95} -12.5554i q^{97} +2.77093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 38 * q - 2 * q^5 - 38 * q^9	\( 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) \) 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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