LMFDB - Embedded newform 6030.2.d.k.2411.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(2411,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("6030.2411");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.14
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.k.2411.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.376360i q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.376360i q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.43766 q^{11} -3.04071i q^{13} -0.376360i q^{14} +1.00000 q^{16} +5.53492i q^{17} +4.42258 q^{19} -1.00000 q^{20} -3.43766 q^{22} +2.14184i q^{23} +1.00000 q^{25} +3.04071i q^{26} +0.376360i q^{28} -1.98502i q^{29} +2.86722i q^{31} -1.00000 q^{32} -5.53492i q^{34} -0.376360i q^{35} -0.233662 q^{37} -4.42258 q^{38} +1.00000 q^{40} +1.78768 q^{41} -9.77289i q^{43} +3.43766 q^{44} -2.14184i q^{46} -5.43908i q^{47} +6.85835 q^{49} -1.00000 q^{50} -3.04071i q^{52} +5.11246 q^{53} -3.43766 q^{55} -0.376360i q^{56} +1.98502i q^{58} +14.9707i q^{59} +11.6222i q^{61} -2.86722i q^{62} +1.00000 q^{64} +3.04071i q^{65} +(-7.52958 - 3.21021i) q^{67} +5.53492i q^{68} +0.376360i q^{70} +9.67551i q^{71} -9.52485 q^{73} +0.233662 q^{74} +4.42258 q^{76} +1.29380i q^{77} -5.92123i q^{79} -1.00000 q^{80} -1.78768 q^{82} +5.42789i q^{83} -5.53492i q^{85} +9.77289i q^{86} -3.43766 q^{88} -3.00820i q^{89} +1.14440 q^{91} +2.14184i q^{92} +5.43908i q^{94} -4.42258 q^{95} -11.2746i q^{97} -6.85835 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8}+O(q^{10}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^5 - 24 * q^8	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8} + 24 q^{10} - 12 q^{11} + 24 q^{16} + 4 q^{19} - 24 q^{20} + 12 q^{22} + 24 q^{25} - 24 q^{32} - 16 q^{37} - 4 q^{38} + 24 q^{40} - 8 q^{41} - 12 q^{44} - 20 q^{49} - 24 q^{50} - 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} + 16 q^{74} + 4 q^{76} - 24 q^{80} + 8 q^{82} + 12 q^{88} - 4 q^{95} + 20 q^{98}+O(q^{100}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^5 - 24 * q^8 + 24 * q^10 - 12 * q^11 + 24 * q^16 + 4 * q^19 - 24 * q^20 + 12 * q^22 + 24 * q^25 - 24 * q^32 - 16 * q^37 - 4 * q^38 + 24 * q^40 - 8 * q^41 - 12 * q^44 - 20 * q^49 - 24 * q^50 - 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 + 16 * q^74 + 4 * q^76 - 24 * q^80 + 8 * q^82 + 12 * q^88 - 4 * q^95 + 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times$.

$n$	$1207$	$3151$	$4691$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			0.376360i			0.142251i	0.997467	+	0.0711253i	$0.0226590\pi$
$7$			0.376360i			0.142251i	−0.997467	+	0.0711253i	$0.977341\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	3.43766			1.03649			0.518247	−	0.855231i	$-0.326585\pi$
$11$	3.43766			1.03649			0.518247	+	0.855231i	$0.326585\pi$
$12$		0			0
$13$		−	3.04071i		−	0.843343i	−0.906749	−	0.421671i	$-0.861444\pi$
$13$		−	3.04071i		−	0.843343i	0.906749	−	0.421671i	$-0.138556\pi$
$14$		−	0.376360i		−	0.100586i
$15$		0			0
$16$	1.00000			0.250000
$17$			5.53492i			1.34242i	0.741269	+	0.671208i	$0.234224\pi$
$17$			5.53492i			1.34242i	−0.741269	+	0.671208i	$0.765776\pi$
$18$		0			0
$19$	4.42258			1.01461			0.507305	−	0.861767i	$-0.330642\pi$
$19$	4.42258			1.01461			0.507305	+	0.861767i	$0.330642\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$	−3.43766			−0.732911
$23$			2.14184i			0.446605i	0.974749	+	0.223303i	$0.0716838\pi$
$23$			2.14184i			0.446605i	−0.974749	+	0.223303i	$0.928316\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$			3.04071i			0.596333i
$27$		0			0
$28$			0.376360i			0.0711253i
$29$		−	1.98502i		−	0.368609i	−0.982869	−	0.184305i	$-0.940997\pi$
$29$		−	1.98502i		−	0.368609i	0.982869	−	0.184305i	$-0.0590033\pi$
$30$		0			0
$31$			2.86722i			0.514969i	0.966282	+	0.257484i	$0.0828936\pi$
$31$			2.86722i			0.514969i	−0.966282	+	0.257484i	$0.917106\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$		−	5.53492i		−	0.949231i
$35$		−	0.376360i		−	0.0636164i
$36$		0			0
$37$	−0.233662			−0.0384138			−0.0192069	−	0.999816i	$-0.506114\pi$
$37$	−0.233662			−0.0384138			−0.0192069	+	0.999816i	$0.506114\pi$
$38$	−4.42258			−0.717438
$39$		0			0
$40$	1.00000			0.158114
$41$	1.78768			0.279188			0.139594	−	0.990209i	$-0.455420\pi$
$41$	1.78768			0.279188			0.139594	+	0.990209i	$0.455420\pi$
$42$		0			0
$43$		−	9.77289i		−	1.49035i	−0.666868	−	0.745176i	$-0.732365\pi$
$43$		−	9.77289i		−	1.49035i	0.666868	−	0.745176i	$-0.267635\pi$
$44$	3.43766			0.518247
$45$		0			0
$46$		−	2.14184i		−	0.315798i
$47$		−	5.43908i		−	0.793371i	−0.917955	−	0.396685i	$-0.870160\pi$
$47$		−	5.43908i		−	0.793371i	0.917955	−	0.396685i	$-0.129840\pi$
$48$		0			0
$49$	6.85835			0.979765
$50$	−1.00000			−0.141421
$51$		0			0
$52$		−	3.04071i		−	0.421671i
$53$	5.11246			0.702250			0.351125	−	0.936329i	$-0.385799\pi$
$53$	5.11246			0.702250			0.351125	+	0.936329i	$0.385799\pi$
$54$		0			0
$55$	−3.43766			−0.463534
$56$		−	0.376360i		−	0.0502932i
$57$		0			0
$58$			1.98502i			0.260646i
$59$			14.9707i			1.94902i	0.224352	+	0.974508i	$0.427973\pi$
$59$			14.9707i			1.94902i	−0.224352	+	0.974508i	$0.572027\pi$
$60$		0			0
$61$			11.6222i			1.48808i	0.668138	+	0.744038i	$0.267092\pi$
$61$			11.6222i			1.48808i	−0.668138	+	0.744038i	$0.732908\pi$
$62$		−	2.86722i		−	0.364138i
$63$		0			0
$64$	1.00000			0.125000
$65$			3.04071i			0.377154i
$66$		0			0
$67$	−7.52958	−	3.21021i	−0.919884	−	0.392190i
$67$	−7.52958	−	3.21021i	−0.919884	−	0.392190i
$68$			5.53492i			0.671208i
$69$		0			0
$70$			0.376360i			0.0449836i
$71$			9.67551i			1.14827i	0.818760	+	0.574136i	$0.194662\pi$
$71$			9.67551i			1.14827i	−0.818760	+	0.574136i	$0.805338\pi$
$72$		0			0
$73$	−9.52485			−1.11480			−0.557400	−	0.830244i	$-0.688201\pi$
$73$	−9.52485			−1.11480			−0.557400	+	0.830244i	$0.688201\pi$
$74$	0.233662			0.0271627
$75$		0			0
$76$	4.42258			0.507305
$77$			1.29380i			0.147442i
$78$		0			0
$79$		−	5.92123i		−	0.666191i	−0.942893	−	0.333095i	$-0.891907\pi$
$79$		−	5.92123i		−	0.666191i	0.942893	−	0.333095i	$-0.108093\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	−1.78768			−0.197416
$83$			5.42789i			0.595789i	0.954599	+	0.297894i	$0.0962843\pi$
$83$			5.42789i			0.595789i	−0.954599	+	0.297894i	$0.903716\pi$
$84$		0			0
$85$		−	5.53492i		−	0.600346i
$86$			9.77289i			1.05384i
$87$		0			0
$88$	−3.43766			−0.366456
$89$		−	3.00820i		−	0.318869i	−0.987209	−	0.159434i	$-0.949033\pi$
$89$		−	3.00820i		−	0.318869i	0.987209	−	0.159434i	$-0.0509670\pi$
$90$		0			0
$91$	1.14440			0.119966
$92$			2.14184i			0.223303i
$93$		0			0
$94$			5.43908i			0.560998i
$95$	−4.42258			−0.453747
$96$		0			0
$97$		−	11.2746i		−	1.14476i	−0.819987	−	0.572382i	$-0.806019\pi$
$97$		−	11.2746i		−	1.14476i	0.819987	−	0.572382i	$-0.193981\pi$
$98$	−6.85835			−0.692798
$99$		0			0
$100$	1.00000			0.100000
$101$	−13.3687			−1.33024			−0.665118	−	0.746738i	$-0.731619\pi$
$101$	−13.3687			−1.33024			−0.665118	+	0.746738i	$0.731619\pi$
$102$		0			0
$103$	2.94883			0.290557			0.145278	−	0.989391i	$-0.453592\pi$
$103$	2.94883			0.290557			0.145278	+	0.989391i	$0.453592\pi$
$104$			3.04071i			0.298167i
$105$		0			0
$106$	−5.11246			−0.496566
$107$		−	3.10842i		−	0.300503i	−0.988648	−	0.150251i	$-0.951992\pi$
$107$		−	3.10842i		−	0.300503i	0.988648	−	0.150251i	$-0.0480083\pi$
$108$		0			0
$109$		−	15.9580i		−	1.52850i	−0.644922	−	0.764248i	$-0.723110\pi$
$109$		−	15.9580i		−	1.52850i	0.644922	−	0.764248i	$-0.276890\pi$
$110$	3.43766			0.327768
$111$		0			0
$112$			0.376360i			0.0355627i
$113$	15.9113			1.49681			0.748407	−	0.663240i	$-0.230819\pi$
$113$	15.9113			1.49681			0.748407	+	0.663240i	$0.230819\pi$
$114$		0			0
$115$		−	2.14184i		−	0.199728i
$116$		−	1.98502i		−	0.184305i
$117$		0			0
$118$		−	14.9707i		−	1.37816i
$119$	−2.08312			−0.190960
$120$		0			0
$121$	0.817497			0.0743179
$122$		−	11.6222i		−	1.05223i
$123$		0			0
$124$			2.86722i			0.257484i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	16.0215			1.42168			0.710840	−	0.703353i	$-0.248315\pi$
$127$	16.0215			1.42168			0.710840	+	0.703353i	$0.248315\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$		−	3.04071i		−	0.266688i
$131$			6.67104i			0.582852i	0.956593	+	0.291426i	$0.0941297\pi$
$131$			6.67104i			0.582852i	−0.956593	+	0.291426i	$0.905870\pi$
$132$		0			0
$133$			1.66448i			0.144329i
$134$	7.52958	+	3.21021i	0.650456	+	0.277320i
$135$		0			0
$136$		−	5.53492i		−	0.474616i
$137$	−0.467078			−0.0399052			−0.0199526	−	0.999801i	$-0.506352\pi$
$137$	−0.467078			−0.0399052			−0.0199526	+	0.999801i	$0.506352\pi$
$138$		0			0
$139$			4.07129i			0.345322i	0.984981	+	0.172661i	$0.0552365\pi$
$139$			4.07129i			0.345322i	−0.984981	+	0.172661i	$0.944763\pi$
$140$		−	0.376360i		−	0.0318082i
$141$		0			0
$142$		−	9.67551i		−	0.811951i
$143$		−	10.4529i		−	0.874119i
$144$		0			0
$145$			1.98502i			0.164847i
$146$	9.52485			0.788282
$147$		0			0
$148$	−0.233662			−0.0192069
$149$		−	2.97696i		−	0.243882i	−0.992537	−	0.121941i	$-0.961088\pi$
$149$		−	2.97696i		−	0.243882i	0.992537	−	0.121941i	$-0.0389118\pi$
$150$		0			0
$151$	13.6341			1.10953			0.554765	−	0.832007i	$-0.312808\pi$
$151$	13.6341			1.10953			0.554765	+	0.832007i	$0.312808\pi$
$152$	−4.42258			−0.358719
$153$		0			0
$154$		−	1.29380i		−	0.104257i
$155$		−	2.86722i		−	0.230301i
$156$		0			0
$157$	−19.4022			−1.54847			−0.774233	−	0.632900i	$-0.781864\pi$
$157$	−19.4022			−1.54847			−0.774233	+	0.632900i	$0.781864\pi$
$158$			5.92123i			0.471068i
$159$		0			0
$160$	1.00000			0.0790569
$161$	−0.806104			−0.0635299
$162$		0			0
$163$	25.2344			1.97651			0.988255	−	0.152816i	$-0.0488342\pi$
$163$	25.2344			1.97651			0.988255	+	0.152816i	$0.0488342\pi$
$164$	1.78768			0.139594
$165$		0			0
$166$		−	5.42789i		−	0.421286i
$167$			12.3328i			0.954342i	0.878810	+	0.477171i	$0.158338\pi$
$167$			12.3328i			0.954342i	−0.878810	+	0.477171i	$0.841662\pi$
$168$		0			0
$169$	3.75405			0.288773
$170$			5.53492i			0.424509i
$171$		0			0
$172$		−	9.77289i		−	0.745176i
$173$			22.1257i			1.68219i	0.540891	+	0.841093i	$0.318087\pi$
$173$			22.1257i			1.68219i	−0.540891	+	0.841093i	$0.681913\pi$
$174$		0			0
$175$			0.376360i			0.0284501i
$176$	3.43766			0.259123
$177$		0			0
$178$			3.00820i			0.225474i
$179$	5.61772			0.419888			0.209944	−	0.977713i	$-0.432672\pi$
$179$	5.61772			0.419888			0.209944	+	0.977713i	$0.432672\pi$
$180$		0			0
$181$	6.31152			0.469132			0.234566	−	0.972100i	$-0.424633\pi$
$181$	6.31152			0.469132			0.234566	+	0.972100i	$0.424633\pi$
$182$	−1.14440			−0.0848288
$183$		0			0
$184$		−	2.14184i		−	0.157899i
$185$	0.233662			0.0171792
$186$		0			0
$187$			19.0272i			1.39140i
$188$		−	5.43908i		−	0.396685i
$189$		0			0
$190$	4.42258			0.320848
$191$	−21.1044			−1.52706			−0.763530	−	0.645773i	$-0.776535\pi$
$191$	−21.1044			−1.52706			−0.763530	+	0.645773i	$0.776535\pi$
$192$		0			0
$193$	−22.0302			−1.58577			−0.792884	−	0.609373i	$-0.791421\pi$
$193$	−22.0302			−1.58577			−0.792884	+	0.609373i	$0.791421\pi$
$194$			11.2746i			0.809471i
$195$		0			0
$196$	6.85835			0.489882
$197$	−11.5123			−0.820220			−0.410110	−	0.912036i	$-0.634510\pi$
$197$	−11.5123			−0.820220			−0.410110	+	0.912036i	$0.634510\pi$
$198$		0			0
$199$	22.2733			1.57891			0.789456	−	0.613807i	$-0.210363\pi$
$199$	22.2733			1.57891			0.789456	+	0.613807i	$0.210363\pi$
$200$	−1.00000			−0.0707107
$201$		0			0
$202$	13.3687			0.940619
$203$	0.747083			0.0524349
$204$		0			0
$205$	−1.78768			−0.124857
$206$	−2.94883			−0.205455
$207$		0			0
$208$		−	3.04071i		−	0.210836i
$209$	15.2033			1.05164
$210$		0			0
$211$	−4.70062			−0.323604			−0.161802	−	0.986823i	$-0.551731\pi$
$211$	−4.70062			−0.323604			−0.161802	+	0.986823i	$0.551731\pi$
$212$	5.11246			0.351125
$213$		0			0
$214$			3.10842i			0.212487i
$215$			9.77289i			0.666506i
$216$		0			0
$217$	−1.07911			−0.0732546
$218$			15.9580i			1.08081i
$219$		0			0
$220$	−3.43766			−0.231767
$221$	16.8301			1.13212
$222$		0			0
$223$	5.99269			0.401300			0.200650	−	0.979663i	$-0.435695\pi$
$223$	5.99269			0.401300			0.200650	+	0.979663i	$0.435695\pi$
$224$		−	0.376360i		−	0.0251466i
$225$		0			0
$226$	−15.9113			−1.05841
$227$			3.83479i			0.254524i	0.991869	+	0.127262i	$0.0406188\pi$
$227$			3.83479i			0.254524i	−0.991869	+	0.127262i	$0.959381\pi$
$228$		0			0
$229$			24.1780i			1.59773i	0.601513	+	0.798863i	$0.294565\pi$
$229$			24.1780i			1.59773i	−0.601513	+	0.798863i	$0.705435\pi$
$230$			2.14184i			0.141229i
$231$		0			0
$232$			1.98502i			0.130323i
$233$	−19.7372			−1.29303			−0.646514	−	0.762902i	$-0.723774\pi$
$233$	−19.7372			−1.29303			−0.646514	+	0.762902i	$0.723774\pi$
$234$		0			0
$235$			5.43908i			0.354806i
$236$			14.9707i			0.974508i
$237$		0			0
$238$	2.08312			0.135029
$239$	−6.33458			−0.409750			−0.204875	−	0.978788i	$-0.565679\pi$
$239$	−6.33458			−0.409750			−0.204875	+	0.978788i	$0.565679\pi$
$240$		0			0
$241$	26.0565			1.67845			0.839223	−	0.543788i	$-0.183010\pi$
$241$	26.0565			1.67845			0.839223	+	0.543788i	$0.183010\pi$
$242$	−0.817497			−0.0525507
$243$		0			0
$244$			11.6222i			0.744038i
$245$	−6.85835			−0.438164
$246$		0			0
$247$		−	13.4478i		−	0.855664i
$248$		−	2.86722i		−	0.182069i
$249$		0			0
$250$	1.00000			0.0632456
$251$	22.0399			1.39115			0.695573	−	0.718456i	$-0.255151\pi$
$251$	22.0399			1.39115			0.695573	+	0.718456i	$0.255151\pi$
$252$		0			0
$253$			7.36293i			0.462903i
$254$	−16.0215			−1.00528
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	8.21817i		−	0.512636i	−0.966593	−	0.256318i	$-0.917491\pi$
$257$		−	8.21817i		−	0.512636i	0.966593	−	0.256318i	$-0.0825094\pi$
$258$		0			0
$259$		−	0.0879411i		−	0.00546439i
$260$			3.04071i			0.188577i
$261$		0			0
$262$		−	6.67104i		−	0.412138i
$263$		−	12.4302i		−	0.766481i	−0.923649	−	0.383240i	$-0.874808\pi$
$263$		−	12.4302i		−	0.766481i	0.923649	−	0.383240i	$-0.125192\pi$
$264$		0			0
$265$	−5.11246			−0.314056
$266$		−	1.66448i		−	0.102056i
$267$		0			0
$268$	−7.52958	−	3.21021i	−0.459942	−	0.196095i
$269$		−	8.54258i		−	0.520850i	−0.965494	−	0.260425i	$-0.916137\pi$
$269$		−	8.54258i		−	0.520850i	0.965494	−	0.260425i	$-0.0838627\pi$
$270$		0			0
$271$		−	4.91221i		−	0.298395i	−0.988807	−	0.149198i	$-0.952331\pi$
$271$		−	4.91221i		−	0.298395i	0.988807	−	0.149198i	$-0.0476691\pi$
$272$			5.53492i			0.335604i
$273$		0			0
$274$	0.467078			0.0282172
$275$	3.43766			0.207299
$276$		0			0
$277$	2.89036			0.173665			0.0868326	−	0.996223i	$-0.472325\pi$
$277$	2.89036			0.173665			0.0868326	+	0.996223i	$0.472325\pi$
$278$		−	4.07129i		−	0.244180i
$279$		0			0
$280$			0.376360i			0.0224918i
$281$	26.9659			1.60865			0.804324	−	0.594191i	$-0.202528\pi$
$281$	26.9659			1.60865			0.804324	+	0.594191i	$0.202528\pi$
$282$		0			0
$283$	23.5140			1.39776			0.698882	−	0.715237i	$-0.253681\pi$
$283$	23.5140			1.39776			0.698882	+	0.715237i	$0.253681\pi$
$284$			9.67551i			0.574136i
$285$		0			0
$286$			10.4529i			0.618095i
$287$			0.672810i			0.0397147i
$288$		0			0
$289$	−13.6353			−0.802079
$290$		−	1.98502i		−	0.116564i
$291$		0			0
$292$	−9.52485			−0.557400
$293$			0.639293i			0.0373479i	0.999826	+	0.0186740i	$0.00594445\pi$
$293$			0.639293i			0.0373479i	−0.999826	+	0.0186740i	$0.994056\pi$
$294$		0			0
$295$		−	14.9707i		−	0.871627i
$296$	0.233662			0.0135813
$297$		0			0
$298$			2.97696i			0.172450i
$299$	6.51274			0.376641
$300$		0			0
$301$	3.67813			0.212004
$302$	−13.6341			−0.784556
$303$		0			0
$304$	4.42258			0.253652
$305$		−	11.6222i		−	0.665488i
$306$		0			0
$307$	−23.0507			−1.31558			−0.657788	−	0.753203i	$-0.728508\pi$
$307$	−23.0507			−1.31558			−0.657788	+	0.753203i	$0.728508\pi$
$308$			1.29380i			0.0737209i
$309$		0			0
$310$			2.86722i			0.162847i
$311$	22.1173			1.25416			0.627080	−	0.778955i	$-0.284250\pi$
$311$	22.1173			1.25416			0.627080	+	0.778955i	$0.284250\pi$
$312$		0			0
$313$			27.6920i			1.56525i	0.622496	+	0.782623i	$0.286119\pi$
$313$			27.6920i			1.56525i	−0.622496	+	0.782623i	$0.713881\pi$
$314$	19.4022			1.09493
$315$		0			0
$316$		−	5.92123i		−	0.333095i
$317$		−	11.6652i		−	0.655184i	−0.944819	−	0.327592i	$-0.893763\pi$
$317$		−	11.6652i		−	0.655184i	0.944819	−	0.327592i	$-0.106237\pi$
$318$		0			0
$319$		−	6.82383i		−	0.382061i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	0.806104			0.0449224
$323$			24.4786i			1.36203i
$324$		0			0
$325$		−	3.04071i		−	0.168669i
$326$	−25.2344			−1.39760
$327$		0			0
$328$	−1.78768			−0.0987079
$329$	2.04705			0.112858
$330$		0			0
$331$		−	12.2259i		−	0.671996i	−0.941863	−	0.335998i	$-0.890926\pi$
$331$		−	12.2259i		−	0.671996i	0.941863	−	0.335998i	$-0.109074\pi$
$332$			5.42789i			0.297894i
$333$		0			0
$334$		−	12.3328i		−	0.674822i
$335$	7.52958	+	3.21021i	0.411385	+	0.175393i
$336$		0			0
$337$			6.12563i			0.333684i	0.985984	+	0.166842i	$0.0533571\pi$
$337$			6.12563i			0.333684i	−0.985984	+	0.166842i	$0.946643\pi$
$338$	−3.75405			−0.204194
$339$		0			0
$340$		−	5.53492i		−	0.300173i
$341$			9.85654i			0.533761i
$342$		0			0
$343$			5.21573i			0.281623i
$344$			9.77289i			0.526919i
$345$		0			0
$346$		−	22.1257i		−	1.18948i
$347$	14.3857			0.772263			0.386132	−	0.922444i	$-0.373811\pi$
$347$	14.3857			0.772263			0.386132	+	0.922444i	$0.373811\pi$
$348$		0			0
$349$	19.9935			1.07023			0.535113	−	0.844781i	$-0.320269\pi$
$349$	19.9935			1.07023			0.535113	+	0.844781i	$0.320269\pi$
$350$		−	0.376360i		−	0.0201173i
$351$		0			0
$352$	−3.43766			−0.183228
$353$	32.4906			1.72930			0.864651	−	0.502373i	$-0.167540\pi$
$353$	32.4906			1.72930			0.864651	+	0.502373i	$0.167540\pi$
$354$		0			0
$355$		−	9.67551i		−	0.513523i
$356$		−	3.00820i		−	0.159434i
$357$		0			0
$358$	−5.61772			−0.296906
$359$		−	12.2208i		−	0.644992i	−0.946571	−	0.322496i	$-0.895478\pi$
$359$		−	12.2208i		−	0.644992i	0.946571	−	0.322496i	$-0.104522\pi$
$360$		0			0
$361$	0.559235			0.0294334
$362$	−6.31152			−0.331726
$363$		0			0
$364$	1.14440			0.0599830
$365$	9.52485			0.498554
$366$		0			0
$367$			29.2800i			1.52840i	0.644978	+	0.764201i	$0.276866\pi$
$367$			29.2800i			1.52840i	−0.644978	+	0.764201i	$0.723134\pi$
$368$			2.14184i			0.111651i
$369$		0			0
$370$	−0.233662			−0.0121475
$371$			1.92412i			0.0998956i
$372$		0			0
$373$			32.2843i			1.67162i	0.549020	+	0.835809i	$0.315001\pi$
$373$			32.2843i			1.67162i	−0.549020	+	0.835809i	$0.684999\pi$
$374$		−	19.0272i		−	0.983871i
$375$		0			0
$376$			5.43908i			0.280499i
$377$	−6.03588			−0.310864
$378$		0			0
$379$			25.7859i			1.32453i	0.749268	+	0.662267i	$0.230405\pi$
$379$			25.7859i			1.32453i	−0.749268	+	0.662267i	$0.769595\pi$
$380$	−4.42258			−0.226874
$381$		0			0
$382$	21.1044			1.07979
$383$	−9.87329			−0.504501			−0.252251	−	0.967662i	$-0.581171\pi$
$383$	−9.87329			−0.504501			−0.252251	+	0.967662i	$0.581171\pi$
$384$		0			0
$385$		−	1.29380i		−	0.0659380i
$386$	22.0302			1.12131
$387$		0			0
$388$		−	11.2746i		−	0.572382i
$389$			25.4056i			1.28811i	0.764977	+	0.644057i	$0.222750\pi$
$389$			25.4056i			1.28811i	−0.764977	+	0.644057i	$0.777250\pi$
$390$		0			0
$391$	−11.8549			−0.599530
$392$	−6.85835			−0.346399
$393$		0			0
$394$	11.5123			0.579983
$395$			5.92123i			0.297930i
$396$		0			0
$397$	14.2380			0.714582			0.357291	−	0.933993i	$-0.383700\pi$
$397$	14.2380			0.714582			0.357291	+	0.933993i	$0.383700\pi$
$398$	−22.2733			−1.11646
$399$		0			0
$400$	1.00000			0.0500000
$401$	0.656772			0.0327976			0.0163988	−	0.999866i	$-0.494780\pi$
$401$	0.656772			0.0327976			0.0163988	+	0.999866i	$0.494780\pi$
$402$		0			0
$403$	8.71841			0.434295
$404$	−13.3687			−0.665118
$405$		0			0
$406$	−0.747083			−0.0370771
$407$	−0.803251			−0.0398157
$408$		0			0
$409$			16.9201i			0.836645i	0.908299	+	0.418322i	$0.137382\pi$
$409$			16.9201i			0.836645i	−0.908299	+	0.418322i	$0.862618\pi$
$410$	1.78768			0.0882870
$411$		0			0
$412$	2.94883			0.145278
$413$	−5.63436			−0.277249
$414$		0			0
$415$		−	5.42789i		−	0.266445i
$416$			3.04071i			0.149083i
$417$		0			0
$418$	−15.2033			−0.743619
$419$		−	9.17201i		−	0.448082i	−0.974580	−	0.224041i	$-0.928075\pi$
$419$		−	9.17201i		−	0.448082i	0.974580	−	0.224041i	$-0.0719250\pi$
$420$		0			0
$421$	−12.3520			−0.601998			−0.300999	−	0.953624i	$-0.597320\pi$
$421$	−12.3520			−0.601998			−0.300999	+	0.953624i	$0.597320\pi$
$422$	4.70062			0.228823
$423$		0			0
$424$	−5.11246			−0.248283
$425$			5.53492i			0.268483i
$426$		0			0
$427$	−4.37415			−0.211680
$428$		−	3.10842i		−	0.150251i
$429$		0			0
$430$		−	9.77289i		−	0.471291i
$431$			31.2928i			1.50732i	0.657263	+	0.753661i	$0.271714\pi$
$431$			31.2928i			1.50732i	−0.657263	+	0.753661i	$0.728286\pi$
$432$		0			0
$433$		−	33.3900i		−	1.60462i	−0.596906	−	0.802311i	$-0.703603\pi$
$433$		−	33.3900i		−	1.60462i	0.596906	−	0.802311i	$-0.296397\pi$
$434$	1.07911			0.0517989
$435$		0			0
$436$		−	15.9580i		−	0.764248i
$437$			9.47248i			0.453130i
$438$		0			0
$439$	−34.0778			−1.62644			−0.813221	−	0.581955i	$-0.802288\pi$
$439$	−34.0778			−1.62644			−0.813221	+	0.581955i	$0.802288\pi$
$440$	3.43766			0.163884
$441$		0			0
$442$	−16.8301			−0.800527
$443$	7.75930			0.368656			0.184328	−	0.982865i	$-0.440989\pi$
$443$	7.75930			0.368656			0.184328	+	0.982865i	$0.440989\pi$
$444$		0			0
$445$			3.00820i			0.142602i
$446$	−5.99269			−0.283762
$447$		0			0
$448$			0.376360i			0.0177813i
$449$		−	19.9758i		−	0.942715i	−0.881942	−	0.471358i	$-0.843764\pi$
$449$		−	19.9758i		−	0.942715i	0.881942	−	0.471358i	$-0.156236\pi$
$450$		0			0
$451$	6.14542			0.289377
$452$	15.9113			0.748407
$453$		0			0
$454$		−	3.83479i		−	0.179975i
$455$	−1.14440			−0.0536505
$456$		0			0
$457$	37.9726			1.77629			0.888143	−	0.459568i	$-0.151996\pi$
$457$	37.9726			1.77629			0.888143	+	0.459568i	$0.151996\pi$
$458$		−	24.1780i		−	1.12976i
$459$		0			0
$460$		−	2.14184i		−	0.0998640i
$461$		−	13.7686i		−	0.641269i	−0.947203	−	0.320635i	$-0.896104\pi$
$461$		−	13.7686i		−	0.641269i	0.947203	−	0.320635i	$-0.103896\pi$
$462$		0			0
$463$		−	22.1473i		−	1.02927i	−0.857408	−	0.514637i	$-0.827927\pi$
$463$		−	22.1473i		−	1.02927i	0.857408	−	0.514637i	$-0.172073\pi$
$464$		−	1.98502i		−	0.0921523i
$465$		0			0
$466$	19.7372			0.914309
$467$			21.9398i			1.01525i	0.861578	+	0.507626i	$0.169477\pi$
$467$			21.9398i			1.01525i	−0.861578	+	0.507626i	$0.830523\pi$
$468$		0			0
$469$	1.20820	−	2.83383i	0.0557893	−	0.130854i
$470$		−	5.43908i		−	0.250886i
$471$		0			0
$472$		−	14.9707i		−	0.689081i
$473$		−	33.5959i		−	1.54474i
$474$		0			0
$475$	4.42258			0.202922
$476$	−2.08312			−0.0954798
$477$		0			0
$478$	6.33458			0.289737
$479$		−	11.9774i		−	0.547263i	−0.961835	−	0.273631i	$-0.911775\pi$
$479$		−	11.9774i		−	0.547263i	0.961835	−	0.273631i	$-0.0882248\pi$
$480$		0			0
$481$			0.710500i			0.0323960i
$482$	−26.0565			−1.18684
$483$		0			0
$484$	0.817497			0.0371589
$485$			11.2746i			0.511954i
$486$		0			0
$487$		−	35.4145i		−	1.60479i	−0.596796	−	0.802393i	$-0.703560\pi$
$487$		−	35.4145i		−	1.60479i	0.596796	−	0.802393i	$-0.296440\pi$
$488$		−	11.6222i		−	0.526114i
$489$		0			0
$490$	6.85835			0.309829
$491$			32.8192i			1.48111i	0.671995	+	0.740556i	$0.265438\pi$
$491$			32.8192i			1.48111i	−0.671995	+	0.740556i	$0.734562\pi$
$492$		0			0
$493$	10.9869			0.494827
$494$			13.4478i			0.605046i
$495$		0			0
$496$			2.86722i			0.128742i
$497$	−3.64148			−0.163342
$498$		0			0
$499$			10.5120i			0.470582i	0.971925	+	0.235291i	$0.0756043\pi$
$499$			10.5120i			0.470582i	−0.971925	+	0.235291i	$0.924396\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$	−22.0399			−0.983688
$503$	32.7203			1.45893			0.729463	−	0.684020i	$-0.239770\pi$
$503$	32.7203			1.45893			0.729463	+	0.684020i	$0.239770\pi$
$504$		0			0
$505$	13.3687			0.594900
$506$		−	7.36293i		−	0.327322i
$507$		0			0
$508$	16.0215			0.710840
$509$		−	3.78218i		−	0.167642i	−0.996481	−	0.0838211i	$-0.973288\pi$
$509$		−	3.78218i		−	0.167642i	0.996481	−	0.0838211i	$-0.0267124\pi$
$510$		0			0
$511$		−	3.58477i		−	0.158581i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			8.21817i			0.362488i
$515$	−2.94883			−0.129941
$516$		0			0
$517$		−	18.6977i		−	0.822323i
$518$			0.0879411i			0.00386391i
$519$		0			0
$520$		−	3.04071i		−	0.133344i
$521$	16.7651			0.734493			0.367247	−	0.930124i	$-0.380301\pi$
$521$	16.7651			0.734493			0.367247	+	0.930124i	$0.380301\pi$
$522$		0			0
$523$	−7.87419			−0.344314			−0.172157	−	0.985069i	$-0.555074\pi$
$523$	−7.87419			−0.344314			−0.172157	+	0.985069i	$0.555074\pi$
$524$			6.67104i			0.291426i
$525$		0			0
$526$			12.4302i			0.541984i
$527$	−15.8699			−0.691302
$528$		0			0
$529$	18.4125			0.800544
$530$	5.11246			0.222071
$531$		0			0
$532$			1.66448i			0.0721645i
$533$		−	5.43581i		−	0.235451i
$534$		0			0
$535$			3.10842i			0.134389i
$536$	7.52958	+	3.21021i	0.325228	+	0.138660i
$537$		0			0
$538$			8.54258i			0.368297i
$539$	23.5767			1.01552
$540$		0			0
$541$			21.3190i			0.916576i	0.888804	+	0.458288i	$0.151537\pi$
$541$			21.3190i			0.916576i	−0.888804	+	0.458288i	$0.848463\pi$
$542$			4.91221i			0.210997i
$543$		0			0
$544$		−	5.53492i		−	0.237308i
$545$			15.9580i			0.683564i
$546$		0			0
$547$		−	24.6979i		−	1.05600i	−0.849243	−	0.528002i	$-0.822941\pi$
$547$		−	24.6979i		−	1.05600i	0.849243	−	0.528002i	$-0.177059\pi$
$548$	−0.467078			−0.0199526
$549$		0			0
$550$	−3.43766			−0.146582
$551$		−	8.77892i		−	0.373995i
$552$		0			0
$553$	2.22852			0.0947661
$554$	−2.89036			−0.122800
$555$		0			0
$556$			4.07129i			0.172661i
$557$			9.07112i			0.384356i	0.981360	+	0.192178i	$0.0615551\pi$
$557$			9.07112i			0.384356i	−0.981360	+	0.192178i	$0.938445\pi$
$558$		0			0
$559$	−29.7166			−1.25688
$560$		−	0.376360i		−	0.0159041i
$561$		0			0
$562$	−26.9659			−1.13749
$563$	−6.60990			−0.278574			−0.139287	−	0.990252i	$-0.544481\pi$
$563$	−6.60990			−0.278574			−0.139287	+	0.990252i	$0.544481\pi$
$564$		0			0
$565$	−15.9113			−0.669395
$566$	−23.5140			−0.988369
$567$		0			0
$568$		−	9.67551i		−	0.405975i
$569$			18.0063i			0.754865i	0.926037	+	0.377432i	$0.123193\pi$
$569$			18.0063i			0.754865i	−0.926037	+	0.377432i	$0.876807\pi$
$570$		0			0
$571$	30.9277			1.29428			0.647142	−	0.762369i	$-0.275964\pi$
$571$	30.9277			1.29428			0.647142	+	0.762369i	$0.275964\pi$
$572$		−	10.4529i		−	0.437059i
$573$		0			0
$574$		−	0.672810i		−	0.0280825i
$575$			2.14184i			0.0893211i
$576$		0			0
$577$			28.8281i			1.20013i	0.799952	+	0.600064i	$0.204858\pi$
$577$			28.8281i			1.20013i	−0.799952	+	0.600064i	$0.795142\pi$
$578$	13.6353			0.567156
$579$		0			0
$580$			1.98502i			0.0824235i
$581$	−2.04284			−0.0847514
$582$		0			0
$583$	17.5749			0.727877
$584$	9.52485			0.394141
$585$		0			0
$586$		−	0.639293i		−	0.0264090i
$587$	−40.4095			−1.66788			−0.833939	−	0.551856i	$-0.813920\pi$
$587$	−40.4095			−1.66788			−0.833939	+	0.551856i	$0.813920\pi$
$588$		0			0
$589$			12.6805i			0.522492i
$590$			14.9707i			0.616333i
$591$		0			0
$592$	−0.233662			−0.00960346
$593$	33.6063			1.38004			0.690022	−	0.723788i	$-0.257601\pi$
$593$	33.6063			1.38004			0.690022	+	0.723788i	$0.257601\pi$
$594$		0			0
$595$	2.08312			0.0853997
$596$		−	2.97696i		−	0.121941i
$597$		0			0
$598$	−6.51274			−0.266326
$599$	9.00191			0.367808			0.183904	−	0.982944i	$-0.441126\pi$
$599$	9.00191			0.367808			0.183904	+	0.982944i	$0.441126\pi$
$600$		0			0
$601$	32.0170			1.30600			0.653001	−	0.757357i	$-0.273510\pi$
$601$	32.0170			1.30600			0.653001	+	0.757357i	$0.273510\pi$
$602$	−3.67813			−0.149909
$603$		0			0
$604$	13.6341			0.554765
$605$	−0.817497			−0.0332360
$606$		0			0
$607$	−14.0523			−0.570365			−0.285182	−	0.958473i	$-0.592054\pi$
$607$	−14.0523			−0.570365			−0.285182	+	0.958473i	$0.592054\pi$
$608$	−4.42258			−0.179359
$609$		0			0
$610$			11.6222i			0.470571i
$611$	−16.5387			−0.669083
$612$		0			0
$613$	26.7964			1.08230			0.541148	−	0.840928i	$-0.317990\pi$
$613$	26.7964			1.08230			0.541148	+	0.840928i	$0.317990\pi$
$614$	23.0507			0.930252
$615$		0			0
$616$		−	1.29380i		−	0.0521286i
$617$		−	7.96288i		−	0.320573i	−0.987071	−	0.160287i	$-0.948758\pi$
$617$		−	7.96288i		−	0.320573i	0.987071	−	0.160287i	$-0.0512419\pi$
$618$		0			0
$619$	−19.8434			−0.797575			−0.398788	−	0.917043i	$-0.630569\pi$
$619$	−19.8434			−0.797575			−0.398788	+	0.917043i	$0.630569\pi$
$620$		−	2.86722i		−	0.115150i
$621$		0			0
$622$	−22.1173			−0.886825
$623$	1.13217			0.0453593
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	27.6920i		−	1.10680i
$627$		0			0
$628$	−19.4022			−0.774233
$629$		−	1.29330i		−	0.0515673i
$630$		0			0
$631$			31.7768i			1.26501i	0.774555	+	0.632507i	$0.217974\pi$
$631$			31.7768i			1.26501i	−0.774555	+	0.632507i	$0.782026\pi$
$632$			5.92123i			0.235534i
$633$		0			0
$634$			11.6652i			0.463285i
$635$	−16.0215			−0.635795
$636$		0			0
$637$		−	20.8543i		−	0.826277i
$638$			6.82383i			0.270158i
$639$		0			0
$640$	1.00000			0.0395285
$641$	30.4041			1.20089			0.600445	−	0.799666i	$-0.294990\pi$
$641$	30.4041			1.20089			0.600445	+	0.799666i	$0.294990\pi$
$642$		0			0
$643$	1.78762			0.0704969			0.0352484	−	0.999379i	$-0.488778\pi$
$643$	1.78762			0.0704969			0.0352484	+	0.999379i	$0.488778\pi$
$644$	−0.806104			−0.0317650
$645$		0			0
$646$		−	24.4786i		−	0.963099i
$647$	−36.3558			−1.42929			−0.714647	−	0.699485i	$-0.753413\pi$
$647$	−36.3558			−1.42929			−0.714647	+	0.699485i	$0.753413\pi$
$648$		0			0
$649$			51.4641i			2.02014i
$650$			3.04071i			0.119267i
$651$		0			0
$652$	25.2344			0.988255
$653$	−16.3825			−0.641098			−0.320549	−	0.947232i	$-0.603867\pi$
$653$	−16.3825			−0.641098			−0.320549	+	0.947232i	$0.603867\pi$
$654$		0			0
$655$		−	6.67104i		−	0.260659i
$656$	1.78768			0.0697970
$657$		0			0
$658$	−2.04705			−0.0798024
$659$		−	6.27010i		−	0.244249i	−0.992515	−	0.122124i	$-0.961029\pi$
$659$		−	6.27010i		−	0.244249i	0.992515	−	0.122124i	$-0.0389706\pi$
$660$		0			0
$661$			8.83448i			0.343622i	0.985130	+	0.171811i	$0.0549618\pi$
$661$			8.83448i			0.343622i	−0.985130	+	0.171811i	$0.945038\pi$
$662$			12.2259i			0.475173i
$663$		0			0
$664$		−	5.42789i		−	0.210643i
$665$		−	1.66448i		−	0.0645459i
$666$		0			0
$667$	4.25161			0.164623
$668$			12.3328i			0.477171i
$669$		0			0
$670$	−7.52958	−	3.21021i	−0.290893	−	0.124021i
$671$			39.9533i			1.54238i
$672$		0			0
$673$		−	0.599532i		−	0.0231102i	−0.999933	−	0.0115551i	$-0.996322\pi$
$673$		−	0.599532i		−	0.0231102i	0.999933	−	0.0115551i	$-0.00367819\pi$
$674$		−	6.12563i		−	0.235951i
$675$		0			0
$676$	3.75405			0.144387
$677$	38.3153			1.47258			0.736288	−	0.676669i	$-0.236577\pi$
$677$	38.3153			1.47258			0.736288	+	0.676669i	$0.236577\pi$
$678$		0			0
$679$	4.24332			0.162844
$680$			5.53492i			0.212255i
$681$		0			0
$682$		−	9.85654i		−	0.377426i
$683$	4.07044			0.155751			0.0778755	−	0.996963i	$-0.475186\pi$
$683$	4.07044			0.155751			0.0778755	+	0.996963i	$0.475186\pi$
$684$		0			0
$685$	0.467078			0.0178461
$686$		−	5.21573i		−	0.199137i
$687$		0			0
$688$		−	9.77289i		−	0.372588i
$689$		−	15.5455i		−	0.592237i
$690$		0			0
$691$	37.3072			1.41923			0.709617	−	0.704587i	$-0.248868\pi$
$691$	37.3072			1.41923			0.709617	+	0.704587i	$0.248868\pi$
$692$			22.1257i			0.841093i
$693$		0			0
$694$	−14.3857			−0.546073
$695$		−	4.07129i		−	0.154433i
$696$		0			0
$697$			9.89465i			0.374786i
$698$	−19.9935			−0.756764
$699$		0			0
$700$			0.376360i			0.0142251i
$701$	−1.96887			−0.0743632			−0.0371816	−	0.999309i	$-0.511838\pi$
$701$	−1.96887			−0.0743632			−0.0371816	+	0.999309i	$0.511838\pi$
$702$		0			0
$703$	−1.03339			−0.0389750
$704$	3.43766			0.129562
$705$		0			0
$706$	−32.4906			−1.22280
$707$		−	5.03145i		−	0.189227i
$708$		0			0
$709$	−15.2267			−0.571849			−0.285924	−	0.958252i	$-0.592301\pi$
$709$	−15.2267			−0.571849			−0.285924	+	0.958252i	$0.592301\pi$
$710$			9.67551i			0.363115i
$711$		0			0
$712$			3.00820i			0.112737i
$713$	−6.14114			−0.229988
$714$		0			0
$715$			10.4529i			0.390918i
$716$	5.61772			0.209944
$717$		0			0
$718$			12.2208i			0.456078i
$719$		−	51.2307i		−	1.91058i	−0.295666	−	0.955291i	$-0.595542\pi$
$719$		−	51.2307i		−	1.91058i	0.295666	−	0.955291i	$-0.404458\pi$
$720$		0			0
$721$			1.10982i			0.0413319i
$722$	−0.559235			−0.0208126
$723$		0			0
$724$	6.31152			0.234566
$725$		−	1.98502i		−	0.0737219i
$726$		0			0
$727$		−	24.9271i		−	0.924494i	−0.886751	−	0.462247i	$-0.847043\pi$
$727$		−	24.9271i		−	0.924494i	0.886751	−	0.462247i	$-0.152957\pi$
$728$	−1.14440			−0.0424144
$729$		0			0
$730$	−9.52485			−0.352531
$731$	54.0922			2.00067
$732$		0			0
$733$			29.1145i			1.07537i	0.843146	+	0.537685i	$0.180701\pi$
$733$			29.1145i			1.07537i	−0.843146	+	0.537685i	$0.819299\pi$
$734$		−	29.2800i		−	1.08074i
$735$		0			0
$736$		−	2.14184i		−	0.0789494i
$737$	−25.8841	−	11.0356i	−0.953454	−	0.406502i
$738$		0			0
$739$			45.2139i			1.66322i	0.555360	+	0.831610i	$0.312580\pi$
$739$			45.2139i			1.66322i	−0.555360	+	0.831610i	$0.687420\pi$
$740$	0.233662			0.00858959
$741$		0			0
$742$		−	1.92412i		−	0.0706368i
$743$		−	26.9715i		−	0.989488i	−0.869039	−	0.494744i	$-0.835262\pi$
$743$		−	26.9715i		−	0.989488i	0.869039	−	0.494744i	$-0.164738\pi$
$744$		0			0
$745$			2.97696i			0.109067i
$746$		−	32.2843i		−	1.18201i
$747$		0			0
$748$			19.0272i			0.695702i
$749$	1.16989			0.0427467
$750$		0			0
$751$	−15.8978			−0.580118			−0.290059	−	0.957009i	$-0.593675\pi$
$751$	−15.8978			−0.580118			−0.290059	+	0.957009i	$0.593675\pi$
$752$		−	5.43908i		−	0.198343i
$753$		0			0
$754$	6.03588			0.219814
$755$	−13.6341			−0.496197
$756$		0			0
$757$			36.6341i			1.33149i	0.746179	+	0.665745i	$0.231886\pi$
$757$			36.6341i			1.33149i	−0.746179	+	0.665745i	$0.768114\pi$
$758$		−	25.7859i		−	0.936587i
$759$		0			0
$760$	4.42258			0.160424
$761$			24.7576i			0.897461i	0.893667	+	0.448730i	$0.148124\pi$
$761$			24.7576i			0.897461i	−0.893667	+	0.448730i	$0.851876\pi$
$762$		0			0
$763$	6.00594			0.217430
$764$	−21.1044			−0.763530
$765$		0			0
$766$	9.87329			0.356736
$767$	45.5216			1.64369
$768$		0			0
$769$			27.3635i			0.986751i	0.869816	+	0.493376i	$0.164237\pi$
$769$			27.3635i			0.986751i	−0.869816	+	0.493376i	$0.835763\pi$
$770$			1.29380i			0.0466252i
$771$		0			0
$772$	−22.0302			−0.792884
$773$			47.7179i			1.71629i	0.513405	+	0.858146i	$0.328384\pi$
$773$			47.7179i			1.71629i	−0.513405	+	0.858146i	$0.671616\pi$
$774$		0			0
$775$			2.86722i			0.102994i
$776$			11.2746i			0.404735i
$777$		0			0
$778$		−	25.4056i		−	0.910835i
$779$	7.90615			0.283267
$780$		0			0
$781$			33.2611i			1.19018i
$782$	11.8549			0.423932
$783$		0			0
$784$	6.85835			0.244941
$785$	19.4022			0.692495
$786$		0			0
$787$			16.9236i			0.603262i	0.953425	+	0.301631i	$0.0975310\pi$
$787$			16.9236i			0.603262i	−0.953425	+	0.301631i	$0.902469\pi$
$788$	−11.5123			−0.410110
$789$		0			0
$790$		−	5.92123i		−	0.210668i
$791$			5.98839i			0.212923i
$792$		0			0
$793$	35.3399			1.25496
$794$	−14.2380			−0.505286
$795$		0			0
$796$	22.2733			0.789456
$797$		−	12.0993i		−	0.428581i	−0.976770	−	0.214290i	$-0.931256\pi$
$797$		−	12.0993i		−	0.428581i	0.976770	−	0.214290i	$-0.0687439\pi$
$798$		0			0
$799$	30.1049			1.06503
$800$	−1.00000			−0.0353553
$801$		0			0
$802$	−0.656772			−0.0231914
$803$	−32.7432			−1.15548
$804$		0			0
$805$	0.806104			0.0284114
$806$	−8.71841			−0.307093
$807$		0			0
$808$	13.3687			0.470309
$809$	−6.06751			−0.213322			−0.106661	−	0.994295i	$-0.534016\pi$
$809$	−6.06751			−0.213322			−0.106661	+	0.994295i	$0.534016\pi$
$810$		0			0
$811$		−	24.1310i		−	0.847355i	−0.905813	−	0.423678i	$-0.860739\pi$
$811$		−	24.1310i		−	0.847355i	0.905813	−	0.423678i	$-0.139261\pi$
$812$	0.747083			0.0262175
$813$		0			0
$814$	0.803251			0.0281539
$815$	−25.2344			−0.883922
$816$		0			0
$817$		−	43.2214i		−	1.51213i
$818$		−	16.9201i		−	0.591597i
$819$		0			0
$820$	−1.78768			−0.0624284
$821$		−	25.1969i		−	0.879378i	−0.898150	−	0.439689i	$-0.855089\pi$
$821$		−	25.1969i		−	0.879378i	0.898150	−	0.439689i	$-0.144911\pi$
$822$		0			0
$823$	−24.6599			−0.859590			−0.429795	−	0.902927i	$-0.641414\pi$
$823$	−24.6599			−0.859590			−0.429795	+	0.902927i	$0.641414\pi$
$824$	−2.94883			−0.102727
$825$		0			0
$826$	5.63436			0.196045
$827$		−	3.59850i		−	0.125132i	−0.998041	−	0.0625661i	$-0.980072\pi$
$827$		−	3.59850i		−	0.125132i	0.998041	−	0.0625661i	$-0.0199284\pi$
$828$		0			0
$829$	−31.4690			−1.09296			−0.546482	−	0.837471i	$-0.684033\pi$
$829$	−31.4690			−1.09296			−0.546482	+	0.837471i	$0.684033\pi$
$830$			5.42789i			0.188405i
$831$		0			0
$832$		−	3.04071i		−	0.105418i
$833$			37.9604i			1.31525i
$834$		0			0
$835$		−	12.3328i		−	0.426795i
$836$	15.2033			0.525818
$837$		0			0
$838$			9.17201i			0.316842i
$839$		−	33.3273i		−	1.15059i	−0.817948	−	0.575293i	$-0.804888\pi$
$839$		−	33.3273i		−	1.15059i	0.817948	−	0.575293i	$-0.195112\pi$
$840$		0			0
$841$	25.0597			0.864127
$842$	12.3520			0.425677
$843$		0			0
$844$	−4.70062			−0.161802
$845$	−3.75405			−0.129143
$846$		0			0
$847$			0.307673i			0.0105718i
$848$	5.11246			0.175563
$849$		0			0
$850$		−	5.53492i		−	0.189846i
$851$		−	0.500468i		−	0.0171558i
$852$		0			0
$853$	−23.2990			−0.797744			−0.398872	−	0.917007i	$-0.630598\pi$
$853$	−23.2990			−0.797744			−0.398872	+	0.917007i	$0.630598\pi$
$854$	4.37415			0.149680
$855$		0			0
$856$			3.10842i			0.106244i
$857$	0.0337033			0.00115128			0.000575642	−	1.00000i	$-0.499817\pi$
$857$	0.0337033			0.00115128			0.000575642		1.00000i	$0.499817\pi$
$858$		0			0
$859$	−30.6695			−1.04643			−0.523216	−	0.852200i	$-0.675268\pi$
$859$	−30.6695			−1.04643			−0.523216	+	0.852200i	$0.675268\pi$
$860$			9.77289i			0.333253i
$861$		0			0
$862$		−	31.2928i		−	1.06584i
$863$			51.6214i			1.75721i	0.477547	+	0.878606i	$0.341526\pi$
$863$			51.6214i			1.75721i	−0.477547	+	0.878606i	$0.658474\pi$
$864$		0			0
$865$		−	22.1257i		−	0.752296i
$866$			33.3900i			1.13464i
$867$		0			0
$868$	−1.07911			−0.0366273
$869$		−	20.3552i		−	0.690502i
$870$		0			0
$871$	−9.76134	+	22.8953i	−0.330750	+	0.775778i
$872$			15.9580i			0.540405i
$873$		0			0
$874$		−	9.47248i		−	0.320411i
$875$		−	0.376360i		−	0.0127233i
$876$		0			0
$877$	−53.4684			−1.80550			−0.902751	−	0.430164i	$-0.858456\pi$
$877$	−53.4684			−1.80550			−0.902751	+	0.430164i	$0.858456\pi$
$878$	34.0778			1.15007
$879$		0			0
$880$	−3.43766			−0.115883
$881$		−	38.3896i		−	1.29338i	−0.762753	−	0.646690i	$-0.776153\pi$
$881$		−	38.3896i		−	1.29338i	0.762753	−	0.646690i	$-0.223847\pi$
$882$		0			0
$883$			13.1299i			0.441857i	0.975290	+	0.220928i	$0.0709087\pi$
$883$			13.1299i			0.441857i	−0.975290	+	0.220928i	$0.929091\pi$
$884$	16.8301			0.566058
$885$		0			0
$886$	−7.75930			−0.260679
$887$		−	5.47248i		−	0.183748i	−0.995771	−	0.0918739i	$-0.970714\pi$
$887$		−	5.47248i		−	0.183748i	0.995771	−	0.0918739i	$-0.0292857\pi$
$888$		0			0
$889$			6.02986i			0.202235i
$890$		−	3.00820i		−	0.100835i
$891$		0			0
$892$	5.99269			0.200650
$893$		−	24.0548i		−	0.804962i
$894$		0			0
$895$	−5.61772			−0.187780
$896$		−	0.376360i		−	0.0125733i
$897$		0			0
$898$			19.9758i			0.666600i
$899$	5.69150			0.189822
$900$		0			0
$901$			28.2971i			0.942712i
$902$	−6.14542			−0.204620
$903$		0			0
$904$	−15.9113			−0.529203
$905$	−6.31152			−0.209802
$906$		0			0
$907$	6.49727			0.215738			0.107869	−	0.994165i	$-0.465597\pi$
$907$	6.49727			0.215738			0.107869	+	0.994165i	$0.465597\pi$
$908$			3.83479i			0.127262i
$909$		0			0
$910$	1.14440			0.0379366
$911$		−	11.1986i		−	0.371025i	−0.982642	−	0.185513i	$-0.940605\pi$
$911$		−	11.1986i		−	0.371025i	0.982642	−	0.185513i	$-0.0593946\pi$
$912$		0			0
$913$			18.6592i			0.617531i
$914$	−37.9726			−1.25602
$915$		0			0
$916$			24.1780i			0.798863i
$917$	−2.51071			−0.0829111
$918$		0			0
$919$			41.5804i			1.37161i	0.727785	+	0.685805i	$0.240550\pi$
$919$			41.5804i			1.37161i	−0.727785	+	0.685805i	$0.759450\pi$
$920$			2.14184i			0.0706145i
$921$		0			0
$922$			13.7686i			0.453446i
$923$	29.4205			0.968387
$924$		0			0
$925$	−0.233662			−0.00768276
$926$			22.1473i			0.727806i
$927$		0			0
$928$			1.98502i			0.0651615i
$929$	26.4267			0.867033			0.433516	−	0.901146i	$-0.357273\pi$
$929$	26.4267			0.867033			0.433516	+	0.901146i	$0.357273\pi$
$930$		0			0
$931$	30.3316			0.994079
$932$	−19.7372			−0.646514
$933$		0			0
$934$		−	21.9398i		−	0.717891i
$935$		−	19.0272i		−	0.622255i
$936$		0			0
$937$		−	57.6694i		−	1.88398i	−0.335643	−	0.941989i	$-0.608954\pi$
$937$		−	57.6694i		−	1.88398i	0.335643	−	0.941989i	$-0.391046\pi$
$938$	−1.20820	+	2.83383i	−0.0394490	+	0.0925279i
$939$		0			0
$940$			5.43908i			0.177403i
$941$	−8.60422			−0.280490			−0.140245	−	0.990117i	$-0.544789\pi$
$941$	−8.60422			−0.280490			−0.140245	+	0.990117i	$0.544789\pi$
$942$		0			0
$943$			3.82892i			0.124687i
$944$			14.9707i			0.487254i
$945$		0			0
$946$			33.5959i			1.09230i
$947$		−	17.9829i		−	0.584366i	−0.956362	−	0.292183i	$-0.905618\pi$
$947$		−	17.9829i		−	0.584366i	0.956362	−	0.292183i	$-0.0943816\pi$
$948$		0			0
$949$			28.9624i			0.940158i
$950$	−4.42258			−0.143488
$951$		0			0
$952$	2.08312			0.0675144
$953$		−	31.3779i		−	1.01643i	−0.861231	−	0.508214i	$-0.830306\pi$
$953$		−	31.3779i		−	1.01643i	0.861231	−	0.508214i	$-0.169694\pi$
$954$		0			0
$955$	21.1044			0.682922
$956$	−6.33458			−0.204875
$957$		0			0
$958$			11.9774i			0.386973i
$959$		−	0.175790i		−	0.00567654i
$960$		0			0
$961$	22.7790			0.734807
$962$		−	0.710500i		−	0.0229074i
$963$		0			0
$964$	26.0565			0.839223
$965$	22.0302			0.709177
$966$		0			0
$967$	16.4244			0.528174			0.264087	−	0.964499i	$-0.414929\pi$
$967$	16.4244			0.528174			0.264087	+	0.964499i	$0.414929\pi$
$968$	−0.817497			−0.0262753
$969$		0			0
$970$		−	11.2746i		−	0.362006i
$971$			7.51780i			0.241258i	0.992698	+	0.120629i	$0.0384911\pi$
$971$			7.51780i			0.241258i	−0.992698	+	0.120629i	$0.961509\pi$
$972$		0			0
$973$	−1.53227			−0.0491223
$974$			35.4145i			1.13475i
$975$		0			0
$976$			11.6222i			0.372019i
$977$		−	6.18265i		−	0.197800i	−0.995097	−	0.0989002i	$-0.968468\pi$
$977$		−	6.18265i		−	0.197800i	0.995097	−	0.0989002i	$-0.0315325\pi$
$978$		0			0
$979$		−	10.3412i		−	0.330505i
$980$	−6.85835			−0.219082
$981$		0			0
$982$		−	32.8192i		−	1.04730i
$983$	51.7651			1.65105			0.825525	−	0.564366i	$-0.190879\pi$
$983$	51.7651			1.65105			0.825525	+	0.564366i	$0.190879\pi$
$984$		0			0
$985$	11.5123			0.366814
$986$	−10.9869			−0.349895
$987$		0			0
$988$		−	13.4478i		−	0.427832i
$989$	20.9320			0.665599
$990$		0			0
$991$		−	30.4370i		−	0.966864i	−0.875382	−	0.483432i	$-0.839390\pi$
$991$		−	30.4370i		−	0.966864i	0.875382	−	0.483432i	$-0.160610\pi$
$992$		−	2.86722i		−	0.0910344i
$993$		0			0
$994$	3.64148			0.115501
$995$	−22.2733			−0.706111
$996$		0			0
$997$	−4.34218			−0.137518			−0.0687591	−	0.997633i	$-0.521904\pi$
$997$	−4.34218			−0.137518			−0.0687591	+	0.997633i	$0.521904\pi$
$998$		−	10.5120i		−	0.332752i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.k.2411.14	yes	24
3.2	odd	2		6030.2.d.l.2411.14	yes	24
67.66	odd	2		6030.2.d.l.2411.11	yes	24
201.200	even	2	inner	6030.2.d.k.2411.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.11	✓	24	201.200	even	2	inner
6030.2.d.k.2411.14	yes	24	1.1	even	1	trivial
6030.2.d.l.2411.11	yes	24	67.66	odd	2
6030.2.d.l.2411.14	yes	24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.376360i q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.376360i q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.43766 q^{11} -3.04071i q^{13} -0.376360i q^{14} +1.00000 q^{16} +5.53492i q^{17} +4.42258 q^{19} -1.00000 q^{20} -3.43766 q^{22} +2.14184i q^{23} +1.00000 q^{25} +3.04071i q^{26} +0.376360i q^{28} -1.98502i q^{29} +2.86722i q^{31} -1.00000 q^{32} -5.53492i q^{34} -0.376360i q^{35} -0.233662 q^{37} -4.42258 q^{38} +1.00000 q^{40} +1.78768 q^{41} -9.77289i q^{43} +3.43766 q^{44} -2.14184i q^{46} -5.43908i q^{47} +6.85835 q^{49} -1.00000 q^{50} -3.04071i q^{52} +5.11246 q^{53} -3.43766 q^{55} -0.376360i q^{56} +1.98502i q^{58} +14.9707i q^{59} +11.6222i q^{61} -2.86722i q^{62} +1.00000 q^{64} +3.04071i q^{65} +(-7.52958 - 3.21021i) q^{67} +5.53492i q^{68} +0.376360i q^{70} +9.67551i q^{71} -9.52485 q^{73} +0.233662 q^{74} +4.42258 q^{76} +1.29380i q^{77} -5.92123i q^{79} -1.00000 q^{80} -1.78768 q^{82} +5.42789i q^{83} -5.53492i q^{85} +9.77289i q^{86} -3.43766 q^{88} -3.00820i q^{89} +1.14440 q^{91} +2.14184i q^{92} +5.43908i q^{94} -4.42258 q^{95} -11.2746i q^{97} -6.85835 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8}+O(q^{10}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^5 - 24 * q^8	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8} + 24 q^{10} - 12 q^{11} + 24 q^{16} + 4 q^{19} - 24 q^{20} + 12 q^{22} + 24 q^{25} - 24 q^{32} - 16 q^{37} - 4 q^{38} + 24 q^{40} - 8 q^{41} - 12 q^{44} - 20 q^{49} - 24 q^{50} - 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} + 16 q^{74} + 4 q^{76} - 24 q^{80} + 8 q^{82} + 12 q^{88} - 4 q^{95} + 20 q^{98}+O(q^{100}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^5 - 24 * q^8 + 24 * q^10 - 12 * q^11 + 24 * q^16 + 4 * q^19 - 24 * q^20 + 12 * q^22 + 24 * q^25 - 24 * q^32 - 16 * q^37 - 4 * q^38 + 24 * q^40 - 8 * q^41 - 12 * q^44 - 20 * q^49 - 24 * q^50 - 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 + 16 * q^74 + 4 * q^76 - 24 * q^80 + 8 * q^82 + 12 * q^88 - 4 * q^95 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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