LMFDB - Embedded newform 6030.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(1,6030)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("6030.1");

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.a (trivial)

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:	yes
Analytic conductor:	$48.1497924188$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15188.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 2 $ x^4 - x^3 - 7*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 670)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.09178$ of defining polynomial
Character	$\chi$	$=$	6030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.58057 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.58057 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.53669 q^{11} -3.51122 q^{13} -3.58057 q^{14} +1.00000 q^{16} +4.24987 q^{17} -4.93468 q^{19} +1.00000 q^{20} +2.53669 q^{22} -3.29375 q^{23} +1.00000 q^{25} -3.51122 q^{26} -3.58057 q^{28} +1.69478 q^{29} -4.67235 q^{31} +1.00000 q^{32} +4.24987 q^{34} -3.58057 q^{35} +9.34166 q^{37} -4.93468 q^{38} +1.00000 q^{40} -4.68480 q^{41} -7.22744 q^{43} +2.53669 q^{44} -3.29375 q^{46} -11.5622 q^{47} +5.82046 q^{49} +1.00000 q^{50} -3.51122 q^{52} -8.44589 q^{53} +2.53669 q^{55} -3.58057 q^{56} +1.69478 q^{58} +10.3457 q^{59} -0.268278 q^{61} -4.67235 q^{62} +1.00000 q^{64} -3.51122 q^{65} +1.00000 q^{67} +4.24987 q^{68} -3.58057 q^{70} -14.0928 q^{71} +1.66237 q^{73} +9.34166 q^{74} -4.93468 q^{76} -9.08279 q^{77} +0.934675 q^{79} +1.00000 q^{80} -4.68480 q^{82} +3.71028 q^{83} +4.24987 q^{85} -7.22744 q^{86} +2.53669 q^{88} +0.536689 q^{89} +12.5721 q^{91} -3.29375 q^{92} -11.5622 q^{94} -4.93468 q^{95} +9.47136 q^{97} +5.82046 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 5 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 5 q^{7} + 4 q^{8} + 4 q^{10} - 9 q^{11} - 12 q^{13} - 5 q^{14} + 4 q^{16} + 4 q^{19} + 4 q^{20} - 9 q^{22} - 6 q^{23} + 4 q^{25} - 12 q^{26} - 5 q^{28} - 18 q^{29} + 2 q^{31} + 4 q^{32} - 5 q^{35} + 9 q^{37} + 4 q^{38} + 4 q^{40} - 12 q^{41} - 16 q^{43} - 9 q^{44} - 6 q^{46} - 10 q^{47} + 15 q^{49} + 4 q^{50} - 12 q^{52} - 8 q^{53} - 9 q^{55} - 5 q^{56} - 18 q^{58} - 18 q^{59} - 11 q^{61} + 2 q^{62} + 4 q^{64} - 12 q^{65} + 4 q^{67} - 5 q^{70} - 27 q^{71} + 4 q^{73} + 9 q^{74} + 4 q^{76} - 25 q^{77} - 20 q^{79} + 4 q^{80} - 12 q^{82} - 9 q^{83} - 16 q^{86} - 9 q^{88} - 17 q^{89} - 4 q^{91} - 6 q^{92} - 10 q^{94} + 4 q^{95} - 5 q^{97} + 15 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 5 * q^7 + 4 * q^8 + 4 * q^10 - 9 * q^11 - 12 * q^13 - 5 * q^14 + 4 * q^16 + 4 * q^19 + 4 * q^20 - 9 * q^22 - 6 * q^23 + 4 * q^25 - 12 * q^26 - 5 * q^28 - 18 * q^29 + 2 * q^31 + 4 * q^32 - 5 * q^35 + 9 * q^37 + 4 * q^38 + 4 * q^40 - 12 * q^41 - 16 * q^43 - 9 * q^44 - 6 * q^46 - 10 * q^47 + 15 * q^49 + 4 * q^50 - 12 * q^52 - 8 * q^53 - 9 * q^55 - 5 * q^56 - 18 * q^58 - 18 * q^59 - 11 * q^61 + 2 * q^62 + 4 * q^64 - 12 * q^65 + 4 * q^67 - 5 * q^70 - 27 * q^71 + 4 * q^73 + 9 * q^74 + 4 * q^76 - 25 * q^77 - 20 * q^79 + 4 * q^80 - 12 * q^82 - 9 * q^83 - 16 * q^86 - 9 * q^88 - 17 * q^89 - 4 * q^91 - 6 * q^92 - 10 * q^94 + 4 * q^95 - 5 * q^97 + 15 * q^98

Display 100 coefficients 10 coefficients

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
$7$	−3.58057	−1.35333	−0.676664	−	0.736292i	$-0.736575\pi$
$7$	−3.58057	−1.35333	−0.676664	+	0.736292i	$0.736575\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	2.53669	0.764841	0.382420	−	0.923988i	$-0.375091\pi$
$11$	2.53669	0.764841	0.382420	+	0.923988i	$0.375091\pi$
$12$	0	0
$13$	−3.51122	−0.973836	−0.486918	−	0.873448i	$-0.661879\pi$
$13$	−3.51122	−0.973836	−0.486918	+	0.873448i	$0.661879\pi$
$14$	−3.58057	−0.956947
$15$	0	0
$16$	1.00000	0.250000
$17$	4.24987	1.03075	0.515373	−	0.856966i	$-0.327653\pi$
$17$	4.24987	1.03075	0.515373	+	0.856966i	$0.327653\pi$
$18$	0	0
$19$	−4.93468	−1.13209	−0.566046	−	0.824374i	$-0.691528\pi$
$19$	−4.93468	−1.13209	−0.566046	+	0.824374i	$0.691528\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	2.53669	0.540824
$23$	−3.29375	−0.686795	−0.343397	−	0.939190i	$-0.611578\pi$
$23$	−3.29375	−0.686795	−0.343397	+	0.939190i	$0.611578\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.51122	−0.688606
$27$	0	0
$28$	−3.58057	−0.676664
$29$	1.69478	0.314713	0.157356	−	0.987542i	$-0.449703\pi$
$29$	1.69478	0.314713	0.157356	+	0.987542i	$0.449703\pi$
$30$	0	0
$31$	−4.67235	−0.839179	−0.419589	−	0.907714i	$-0.637826\pi$
$31$	−4.67235	−0.839179	−0.419589	+	0.907714i	$0.637826\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.24987	0.728847
$35$	−3.58057	−0.605226
$36$	0	0
$37$	9.34166	1.53576	0.767879	−	0.640594i	$-0.221312\pi$
$37$	9.34166	1.53576	0.767879	+	0.640594i	$0.221312\pi$
$38$	−4.93468	−0.800510
$39$	0	0
$40$	1.00000	0.158114
$41$	−4.68480	−0.731643	−0.365822	−	0.930685i	$-0.619212\pi$
$41$	−4.68480	−0.731643	−0.365822	+	0.930685i	$0.619212\pi$
$42$	0	0
$43$	−7.22744	−1.10217	−0.551087	−	0.834448i	$-0.685787\pi$
$43$	−7.22744	−1.10217	−0.551087	+	0.834448i	$0.685787\pi$
$44$	2.53669	0.382420
$45$	0	0
$46$	−3.29375	−0.485637
$47$	−11.5622	−1.68651	−0.843257	−	0.537510i	$-0.819365\pi$
$47$	−11.5622	−1.68651	−0.843257	+	0.537510i	$0.819365\pi$
$48$	0	0
$49$	5.82046	0.831495
$50$	1.00000	0.141421
$51$	0	0
$52$	−3.51122	−0.486918
$53$	−8.44589	−1.16013	−0.580066	−	0.814569i	$-0.696973\pi$
$53$	−8.44589	−1.16013	−0.580066	+	0.814569i	$0.696973\pi$
$54$	0	0
$55$	2.53669	0.342047
$56$	−3.58057	−0.478473
$57$	0	0
$58$	1.69478	0.222536
$59$	10.3457	1.34689	0.673447	−	0.739236i	$-0.264813\pi$
$59$	10.3457	1.34689	0.673447	+	0.739236i	$0.264813\pi$
$60$	0	0
$61$	−0.268278	−0.0343495	−0.0171747	−	0.999853i	$-0.505467\pi$
$61$	−0.268278	−0.0343495	−0.0171747	+	0.999853i	$0.505467\pi$
$62$	−4.67235	−0.593389
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.51122	−0.435513
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	4.24987	0.515373
$69$	0	0
$70$	−3.58057	−0.427960
$71$	−14.0928	−1.67250	−0.836252	−	0.548346i	$-0.815258\pi$
$71$	−14.0928	−1.67250	−0.836252	+	0.548346i	$0.815258\pi$
$72$	0	0
$73$	1.66237	0.194566	0.0972829	−	0.995257i	$-0.468985\pi$
$73$	1.66237	0.194566	0.0972829	+	0.995257i	$0.468985\pi$
$74$	9.34166	1.08595
$75$	0	0
$76$	−4.93468	−0.566046
$77$	−9.08279	−1.03508
$78$	0	0
$79$	0.934675	0.105159	0.0525796	−	0.998617i	$-0.483256\pi$
$79$	0.934675	0.105159	0.0525796	+	0.998617i	$0.483256\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−4.68480	−0.517350
$83$	3.71028	0.407256	0.203628	−	0.979048i	$-0.434727\pi$
$83$	3.71028	0.407256	0.203628	+	0.979048i	$0.434727\pi$
$84$	0	0
$85$	4.24987	0.460964
$86$	−7.22744	−0.779355
$87$	0	0
$88$	2.53669	0.270412
$89$	0.536689	0.0568890	0.0284445	−	0.999595i	$-0.490945\pi$
$89$	0.536689	0.0568890	0.0284445	+	0.999595i	$0.490945\pi$
$90$	0	0
$91$	12.5721	1.31792
$92$	−3.29375	−0.343397
$93$	0	0
$94$	−11.5622	−1.19255
$95$	−4.93468	−0.506287
$96$	0	0
$97$	9.47136	0.961671	0.480836	−	0.876811i	$-0.340333\pi$
$97$	9.47136	0.961671	0.480836	+	0.876811i	$0.340333\pi$
$98$	5.82046	0.587955
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.6539	−1.06011	−0.530054	−	0.847964i	$-0.677828\pi$
$101$	−10.6539	−1.06011	−0.530054	+	0.847964i	$0.677828\pi$
$102$	0	0
$103$	−16.6723	−1.64278	−0.821388	−	0.570370i	$-0.806800\pi$
$103$	−16.6723	−1.64278	−0.821388	+	0.570370i	$0.806800\pi$
$104$	−3.51122	−0.344303
$105$	0	0
$106$	−8.44589	−0.820337
$107$	−3.67137	−0.354924	−0.177462	−	0.984128i	$-0.556789\pi$
$107$	−3.67137	−0.354924	−0.177462	+	0.984128i	$0.556789\pi$
$108$	0	0
$109$	−9.80899	−0.939531	−0.469766	−	0.882791i	$-0.655662\pi$
$109$	−9.80899	−0.939531	−0.469766	+	0.882791i	$0.655662\pi$
$110$	2.53669	0.241864
$111$	0	0
$112$	−3.58057	−0.338332
$113$	13.5038	1.27033	0.635164	−	0.772377i	$-0.280932\pi$
$113$	13.5038	1.27033	0.635164	+	0.772377i	$0.280932\pi$
$114$	0	0
$115$	−3.29375	−0.307144
$116$	1.69478	0.157356
$117$	0	0
$118$	10.3457	0.952397
$119$	−15.2170	−1.39494
$120$	0	0
$121$	−4.56521	−0.415019
$122$	−0.268278	−0.0242887
$123$	0	0
$124$	−4.67235	−0.419589
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.3332	−1.80428	−0.902141	−	0.431441i	$-0.858005\pi$
$127$	−20.3332	−1.80428	−0.902141	+	0.431441i	$0.858005\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.51122	−0.307954
$131$	3.86738	0.337895	0.168947	−	0.985625i	$-0.445963\pi$
$131$	3.86738	0.337895	0.168947	+	0.985625i	$0.445963\pi$
$132$	0	0
$133$	17.6689	1.53209
$134$	1.00000	0.0863868
$135$	0	0
$136$	4.24987	0.364424
$137$	−20.1652	−1.72283	−0.861413	−	0.507905i	$-0.830420\pi$
$137$	−20.1652	−1.72283	−0.861413	+	0.507905i	$0.830420\pi$
$138$	0	0
$139$	10.2160	0.866508	0.433254	−	0.901272i	$-0.357365\pi$
$139$	10.2160	0.866508	0.433254	+	0.901272i	$0.357365\pi$
$140$	−3.58057	−0.302613
$141$	0	0
$142$	−14.0928	−1.18264
$143$	−8.90686	−0.744829
$144$	0	0
$145$	1.69478	0.140744
$146$	1.66237	0.137579
$147$	0	0
$148$	9.34166	0.767879
$149$	14.4997	1.18787	0.593933	−	0.804515i	$-0.297574\pi$
$149$	14.4997	1.18787	0.593933	+	0.804515i	$0.297574\pi$
$150$	0	0
$151$	−7.50526	−0.610770	−0.305385	−	0.952229i	$-0.598785\pi$
$151$	−7.50526	−0.610770	−0.305385	+	0.952229i	$0.598785\pi$
$152$	−4.93468	−0.400255
$153$	0	0
$154$	−9.08279	−0.731912
$155$	−4.67235	−0.375292
$156$	0	0
$157$	−10.8774	−0.868108	−0.434054	−	0.900887i	$-0.642917\pi$
$157$	−10.8774	−0.868108	−0.434054	+	0.900887i	$0.642917\pi$
$158$	0.934675	0.0743588
$159$	0	0
$160$	1.00000	0.0790569
$161$	11.7935	0.929458
$162$	0	0
$163$	5.83044	0.456675	0.228338	−	0.973582i	$-0.426671\pi$
$163$	5.83044	0.456675	0.228338	+	0.973582i	$0.426671\pi$
$164$	−4.68480	−0.365822
$165$	0	0
$166$	3.71028	0.287973
$167$	−6.75720	−0.522888	−0.261444	−	0.965219i	$-0.584199\pi$
$167$	−6.75720	−0.522888	−0.261444	+	0.965219i	$0.584199\pi$
$168$	0	0
$169$	−0.671366	−0.0516435
$170$	4.24987	0.325950
$171$	0	0
$172$	−7.22744	−0.551087
$173$	17.7398	1.34873	0.674365	−	0.738398i	$-0.264417\pi$
$173$	17.7398	1.34873	0.674365	+	0.738398i	$0.264417\pi$
$174$	0	0
$175$	−3.58057	−0.270665
$176$	2.53669	0.191210
$177$	0	0
$178$	0.536689	0.0402266
$179$	−22.0405	−1.64738	−0.823691	−	0.567039i	$-0.808089\pi$
$179$	−22.0405	−1.64738	−0.823691	+	0.567039i	$0.808089\pi$
$180$	0	0
$181$	−18.4659	−1.37256	−0.686278	−	0.727339i	$-0.740757\pi$
$181$	−18.4659	−1.37256	−0.686278	+	0.727339i	$0.740757\pi$
$182$	12.5721	0.931909
$183$	0	0
$184$	−3.29375	−0.242819
$185$	9.34166	0.686812
$186$	0	0
$187$	10.7806	0.788356
$188$	−11.5622	−0.843257
$189$	0	0
$190$	−4.93468	−0.357999
$191$	−17.0395	−1.23293	−0.616467	−	0.787381i	$-0.711437\pi$
$191$	−17.0395	−1.23293	−0.616467	+	0.787381i	$0.711437\pi$
$192$	0	0
$193$	−1.61948	−0.116572	−0.0582862	−	0.998300i	$-0.518564\pi$
$193$	−1.61948	−0.116572	−0.0582862	+	0.998300i	$0.518564\pi$
$194$	9.47136	0.680004
$195$	0	0
$196$	5.82046	0.415747
$197$	2.79597	0.199205	0.0996024	−	0.995027i	$-0.468243\pi$
$197$	2.79597	0.199205	0.0996024	+	0.995027i	$0.468243\pi$
$198$	0	0
$199$	20.3826	1.44489	0.722443	−	0.691430i	$-0.243019\pi$
$199$	20.3826	1.44489	0.722443	+	0.691430i	$0.243019\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−10.6539	−0.749609
$203$	−6.06828	−0.425910
$204$	0	0
$205$	−4.68480	−0.327201
$206$	−16.6723	−1.16162
$207$	0	0
$208$	−3.51122	−0.243459
$209$	−12.5177	−0.865870
$210$	0	0
$211$	24.1253	1.66085	0.830427	−	0.557127i	$-0.188096\pi$
$211$	24.1253	1.66085	0.830427	+	0.557127i	$0.188096\pi$
$212$	−8.44589	−0.580066
$213$	0	0
$214$	−3.67137	−0.250969
$215$	−7.22744	−0.492908
$216$	0	0
$217$	16.7297	1.13568
$218$	−9.80899	−0.664349
$219$	0	0
$220$	2.53669	0.171024
$221$	−14.9222	−1.00378
$222$	0	0
$223$	−13.2489	−0.887211	−0.443606	−	0.896222i	$-0.646301\pi$
$223$	−13.2489	−0.887211	−0.443606	+	0.896222i	$0.646301\pi$
$224$	−3.58057	−0.239237
$225$	0	0
$226$	13.5038	0.898258
$227$	−12.2594	−0.813686	−0.406843	−	0.913498i	$-0.633370\pi$
$227$	−12.2594	−0.813686	−0.406843	+	0.913498i	$0.633370\pi$
$228$	0	0
$229$	9.88430	0.653173	0.326586	−	0.945167i	$-0.394102\pi$
$229$	9.88430	0.653173	0.326586	+	0.945167i	$0.394102\pi$
$230$	−3.29375	−0.217184
$231$	0	0
$232$	1.69478	0.111268
$233$	−7.99405	−0.523707	−0.261854	−	0.965108i	$-0.584334\pi$
$233$	−7.99405	−0.523707	−0.261854	+	0.965108i	$0.584334\pi$
$234$	0	0
$235$	−11.5622	−0.754232
$236$	10.3457	0.673447
$237$	0	0
$238$	−15.2170	−0.986369
$239$	26.2116	1.69549	0.847743	−	0.530407i	$-0.177961\pi$
$239$	26.2116	1.69549	0.847743	+	0.530407i	$0.177961\pi$
$240$	0	0
$241$	29.2385	1.88342	0.941710	−	0.336426i	$-0.109218\pi$
$241$	29.2385	1.88342	0.941710	+	0.336426i	$0.109218\pi$
$242$	−4.56521	−0.293463
$243$	0	0
$244$	−0.268278	−0.0171747
$245$	5.82046	0.371856
$246$	0	0
$247$	17.3267	1.10247
$248$	−4.67235	−0.296695
$249$	0	0
$250$	1.00000	0.0632456
$251$	15.0040	0.947046	0.473523	−	0.880782i	$-0.342982\pi$
$251$	15.0040	0.947046	0.473523	+	0.880782i	$0.342982\pi$
$252$	0	0
$253$	−8.35523	−0.525289
$254$	−20.3332	−1.27582
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.3537	−1.89341	−0.946707	−	0.322096i	$-0.895613\pi$
$257$	−30.3537	−1.89341	−0.946707	+	0.322096i	$0.895613\pi$
$258$	0	0
$259$	−33.4484	−2.07838
$260$	−3.51122	−0.217756
$261$	0	0
$262$	3.86738	0.238928
$263$	−14.8498	−0.915680	−0.457840	−	0.889035i	$-0.651377\pi$
$263$	−14.8498	−0.915680	−0.457840	+	0.889035i	$0.651377\pi$
$264$	0	0
$265$	−8.44589	−0.518827
$266$	17.6689	1.08335
$267$	0	0
$268$	1.00000	0.0610847
$269$	12.8898	0.785906	0.392953	−	0.919559i	$-0.371454\pi$
$269$	12.8898	0.785906	0.392953	+	0.919559i	$0.371454\pi$
$270$	0	0
$271$	20.2345	1.22916	0.614580	−	0.788855i	$-0.289326\pi$
$271$	20.2345	1.22916	0.614580	+	0.788855i	$0.289326\pi$
$272$	4.24987	0.257686
$273$	0	0
$274$	−20.1652	−1.21822
$275$	2.53669	0.152968
$276$	0	0
$277$	−22.0435	−1.32447	−0.662233	−	0.749298i	$-0.730391\pi$
$277$	−22.0435	−1.32447	−0.662233	+	0.749298i	$0.730391\pi$
$278$	10.2160	0.612714
$279$	0	0
$280$	−3.58057	−0.213980
$281$	−21.4425	−1.27915	−0.639576	−	0.768728i	$-0.720890\pi$
$281$	−21.4425	−1.27915	−0.639576	+	0.768728i	$0.720890\pi$
$282$	0	0
$283$	−2.61199	−0.155267	−0.0776335	−	0.996982i	$-0.524736\pi$
$283$	−2.61199	−0.155267	−0.0776335	+	0.996982i	$0.524736\pi$
$284$	−14.0928	−0.836252
$285$	0	0
$286$	−8.90686	−0.526674
$287$	16.7742	0.990152
$288$	0	0
$289$	1.06143	0.0624371
$290$	1.69478	0.0995210
$291$	0	0
$292$	1.66237	0.0972829
$293$	−1.14564	−0.0669290	−0.0334645	−	0.999440i	$-0.510654\pi$
$293$	−1.14564	−0.0669290	−0.0334645	+	0.999440i	$0.510654\pi$
$294$	0	0
$295$	10.3457	0.602349
$296$	9.34166	0.542973
$297$	0	0
$298$	14.4997	0.839948
$299$	11.5651	0.668825
$300$	0	0
$301$	25.8783	1.49160
$302$	−7.50526	−0.431879
$303$	0	0
$304$	−4.93468	−0.283023
$305$	−0.268278	−0.0153616
$306$	0	0
$307$	−20.6450	−1.17827	−0.589135	−	0.808035i	$-0.700531\pi$
$307$	−20.6450	−1.17827	−0.589135	+	0.808035i	$0.700531\pi$
$308$	−9.08279	−0.517540
$309$	0	0
$310$	−4.67235	−0.265372
$311$	26.1068	1.48038	0.740190	−	0.672398i	$-0.234736\pi$
$311$	26.1068	1.48038	0.740190	+	0.672398i	$0.234736\pi$
$312$	0	0
$313$	3.29828	0.186430	0.0932150	−	0.995646i	$-0.470286\pi$
$313$	3.29828	0.186430	0.0932150	+	0.995646i	$0.470286\pi$
$314$	−10.8774	−0.613845
$315$	0	0
$316$	0.934675	0.0525796
$317$	−32.7143	−1.83742	−0.918709	−	0.394935i	$-0.870767\pi$
$317$	−32.7143	−1.83742	−0.918709	+	0.394935i	$0.870767\pi$
$318$	0	0
$319$	4.29913	0.240705
$320$	1.00000	0.0559017
$321$	0	0
$322$	11.7935	0.657226
$323$	−20.9718	−1.16690
$324$	0	0
$325$	−3.51122	−0.194767
$326$	5.83044	0.322918
$327$	0	0
$328$	−4.68480	−0.258675
$329$	41.3991	2.28241
$330$	0	0
$331$	−10.8963	−0.598915	−0.299458	−	0.954110i	$-0.596806\pi$
$331$	−10.8963	−0.598915	−0.299458	+	0.954110i	$0.596806\pi$
$332$	3.71028	0.203628
$333$	0	0
$334$	−6.75720	−0.369737
$335$	1.00000	0.0546358
$336$	0	0
$337$	−19.0629	−1.03842	−0.519211	−	0.854646i	$-0.673774\pi$
$337$	−19.0629	−1.03842	−0.519211	+	0.854646i	$0.673774\pi$
$338$	−0.671366	−0.0365175
$339$	0	0
$340$	4.24987	0.230482
$341$	−11.8523	−0.641838
$342$	0	0
$343$	4.22342	0.228043
$344$	−7.22744	−0.389678
$345$	0	0
$346$	17.7398	0.953696
$347$	14.3472	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$347$	14.3472	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$348$	0	0
$349$	−28.5078	−1.52599	−0.762994	−	0.646406i	$-0.776271\pi$
$349$	−28.5078	−1.52599	−0.762994	+	0.646406i	$0.776271\pi$
$350$	−3.58057	−0.191389
$351$	0	0
$352$	2.53669	0.135206
$353$	24.8483	1.32254	0.661272	−	0.750146i	$-0.270017\pi$
$353$	24.8483	1.32254	0.661272	+	0.750146i	$0.270017\pi$
$354$	0	0
$355$	−14.0928	−0.747966
$356$	0.536689	0.0284445
$357$	0	0
$358$	−22.0405	−1.16487
$359$	−11.7766	−0.621544	−0.310772	−	0.950484i	$-0.600588\pi$
$359$	−11.7766	−0.621544	−0.310772	+	0.950484i	$0.600588\pi$
$360$	0	0
$361$	5.35102	0.281633
$362$	−18.4659	−0.970544
$363$	0	0
$364$	12.5721	0.658959
$365$	1.66237	0.0870124
$366$	0	0
$367$	24.3566	1.27141	0.635703	−	0.771933i	$-0.280710\pi$
$367$	24.3566	1.27141	0.635703	+	0.771933i	$0.280710\pi$
$368$	−3.29375	−0.171699
$369$	0	0
$370$	9.34166	0.485650
$371$	30.2411	1.57004
$372$	0	0
$373$	−13.6980	−0.709253	−0.354627	−	0.935008i	$-0.615392\pi$
$373$	−13.6980	−0.709253	−0.354627	+	0.935008i	$0.615392\pi$
$374$	10.7806	0.557452
$375$	0	0
$376$	−11.5622	−0.596273
$377$	−5.95074	−0.306479
$378$	0	0
$379$	34.2160	1.75756	0.878778	−	0.477230i	$-0.158359\pi$
$379$	34.2160	1.75756	0.878778	+	0.477230i	$0.158359\pi$
$380$	−4.93468	−0.253144
$381$	0	0
$382$	−17.0395	−0.871816
$383$	−24.7173	−1.26300	−0.631499	−	0.775377i	$-0.717560\pi$
$383$	−24.7173	−1.26300	−0.631499	+	0.775377i	$0.717560\pi$
$384$	0	0
$385$	−9.08279	−0.462902
$386$	−1.61948	−0.0824292
$387$	0	0
$388$	9.47136	0.480836
$389$	38.5098	1.95252	0.976261	−	0.216595i	$-0.0694952\pi$
$389$	38.5098	1.95252	0.976261	+	0.216595i	$0.0694952\pi$
$390$	0	0
$391$	−13.9980	−0.707911
$392$	5.82046	0.293978
$393$	0	0
$394$	2.79597	0.140859
$395$	0.934675	0.0470286
$396$	0	0
$397$	31.4679	1.57933	0.789665	−	0.613539i	$-0.210255\pi$
$397$	31.4679	1.57933	0.789665	+	0.613539i	$0.210255\pi$
$398$	20.3826	1.02169
$399$	0	0
$400$	1.00000	0.0500000
$401$	−2.10920	−0.105329	−0.0526643	−	0.998612i	$-0.516771\pi$
$401$	−2.10920	−0.105329	−0.0526643	+	0.998612i	$0.516771\pi$
$402$	0	0
$403$	16.4056	0.817223
$404$	−10.6539	−0.530054
$405$	0	0
$406$	−6.06828	−0.301164
$407$	23.6969	1.17461
$408$	0	0
$409$	28.9722	1.43258	0.716292	−	0.697800i	$-0.245838\pi$
$409$	28.9722	1.43258	0.716292	+	0.697800i	$0.245838\pi$
$410$	−4.68480	−0.231366
$411$	0	0
$412$	−16.6723	−0.821388
$413$	−37.0434	−1.82279
$414$	0	0
$415$	3.71028	0.182130
$416$	−3.51122	−0.172152
$417$	0	0
$418$	−12.5177	−0.612263
$419$	−0.795972	−0.0388858	−0.0194429	−	0.999811i	$-0.506189\pi$
$419$	−0.795972	−0.0388858	−0.0194429	+	0.999811i	$0.506189\pi$
$420$	0	0
$421$	1.83348	0.0893586	0.0446793	−	0.999001i	$-0.485773\pi$
$421$	1.83348	0.0893586	0.0446793	+	0.999001i	$0.485773\pi$
$422$	24.1253	1.17440
$423$	0	0
$424$	−8.44589	−0.410169
$425$	4.24987	0.206149
$426$	0	0
$427$	0.960587	0.0464861
$428$	−3.67137	−0.177462
$429$	0	0
$430$	−7.22744	−0.348538
$431$	7.16006	0.344888	0.172444	−	0.985019i	$-0.444834\pi$
$431$	7.16006	0.344888	0.172444	+	0.985019i	$0.444834\pi$
$432$	0	0
$433$	16.0834	0.772917	0.386458	−	0.922307i	$-0.373698\pi$
$433$	16.0834	0.772917	0.386458	+	0.922307i	$0.373698\pi$
$434$	16.7297	0.803050
$435$	0	0
$436$	−9.80899	−0.469766
$437$	16.2536	0.777515
$438$	0	0
$439$	3.00694	0.143513	0.0717566	−	0.997422i	$-0.477140\pi$
$439$	3.00694	0.143513	0.0717566	+	0.997422i	$0.477140\pi$
$440$	2.53669	0.120932
$441$	0	0
$442$	−14.9222	−0.709778
$443$	3.52218	0.167344	0.0836719	−	0.996493i	$-0.473335\pi$
$443$	3.52218	0.167344	0.0836719	+	0.996493i	$0.473335\pi$
$444$	0	0
$445$	0.536689	0.0254415
$446$	−13.2489	−0.627353
$447$	0	0
$448$	−3.58057	−0.169166
$449$	−22.7037	−1.07145	−0.535727	−	0.844392i	$-0.679962\pi$
$449$	−22.7037	−1.07145	−0.535727	+	0.844392i	$0.679962\pi$
$450$	0	0
$451$	−11.8839	−0.559590
$452$	13.5038	0.635164
$453$	0	0
$454$	−12.2594	−0.575363
$455$	12.5721	0.589391
$456$	0	0
$457$	−1.69769	−0.0794146	−0.0397073	−	0.999211i	$-0.512643\pi$
$457$	−1.69769	−0.0794146	−0.0397073	+	0.999211i	$0.512643\pi$
$458$	9.88430	0.461863
$459$	0	0
$460$	−3.29375	−0.153572
$461$	7.14118	0.332598	0.166299	−	0.986075i	$-0.446818\pi$
$461$	7.14118	0.332598	0.166299	+	0.986075i	$0.446818\pi$
$462$	0	0
$463$	−22.6465	−1.05247	−0.526236	−	0.850338i	$-0.676397\pi$
$463$	−22.6465	−1.05247	−0.526236	+	0.850338i	$0.676397\pi$
$464$	1.69478	0.0786782
$465$	0	0
$466$	−7.99405	−0.370317
$467$	−21.1876	−0.980445	−0.490222	−	0.871597i	$-0.663084\pi$
$467$	−21.1876	−0.980445	−0.490222	+	0.871597i	$0.663084\pi$
$468$	0	0
$469$	−3.58057	−0.165335
$470$	−11.5622	−0.533323
$471$	0	0
$472$	10.3457	0.476199
$473$	−18.3338	−0.842988
$474$	0	0
$475$	−4.93468	−0.226418
$476$	−15.2170	−0.697468
$477$	0	0
$478$	26.2116	1.19889
$479$	−11.6764	−0.533507	−0.266754	−	0.963765i	$-0.585951\pi$
$479$	−11.6764	−0.533507	−0.266754	+	0.963765i	$0.585951\pi$
$480$	0	0
$481$	−32.8006	−1.49558
$482$	29.2385	1.33178
$483$	0	0
$484$	−4.56521	−0.207509
$485$	9.47136	0.430073
$486$	0	0
$487$	7.94770	0.360145	0.180072	−	0.983653i	$-0.442367\pi$
$487$	7.94770	0.360145	0.180072	+	0.983653i	$0.442367\pi$
$488$	−0.268278	−0.0121444
$489$	0	0
$490$	5.82046	0.262942
$491$	−31.7433	−1.43256	−0.716278	−	0.697816i	$-0.754156\pi$
$491$	−31.7433	−1.43256	−0.716278	+	0.697816i	$0.754156\pi$
$492$	0	0
$493$	7.20261	0.324389
$494$	17.3267	0.779565
$495$	0	0
$496$	−4.67235	−0.209795
$497$	50.4601	2.26344
$498$	0	0
$499$	−30.4938	−1.36509	−0.682545	−	0.730844i	$-0.739127\pi$
$499$	−30.4938	−1.36509	−0.682545	+	0.730844i	$0.739127\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	15.0040	0.669663
$503$	41.5662	1.85335	0.926673	−	0.375868i	$-0.122655\pi$
$503$	41.5662	1.85335	0.926673	+	0.375868i	$0.122655\pi$
$504$	0	0
$505$	−10.6539	−0.474094
$506$	−8.35523	−0.371435
$507$	0	0
$508$	−20.3332	−0.902141
$509$	25.5254	1.13139	0.565696	−	0.824614i	$-0.308608\pi$
$509$	25.5254	1.13139	0.565696	+	0.824614i	$0.308608\pi$
$510$	0	0
$511$	−5.95223	−0.263311
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.3537	−1.33885
$515$	−16.6723	−0.734672
$516$	0	0
$517$	−29.3296	−1.28992
$518$	−33.4484	−1.46964
$519$	0	0
$520$	−3.51122	−0.153977
$521$	0.962208	0.0421551	0.0210776	−	0.999778i	$-0.493290\pi$
$521$	0.962208	0.0421551	0.0210776	+	0.999778i	$0.493290\pi$
$522$	0	0
$523$	34.6206	1.51385	0.756926	−	0.653500i	$-0.226700\pi$
$523$	34.6206	1.51385	0.756926	+	0.653500i	$0.226700\pi$
$524$	3.86738	0.168947
$525$	0	0
$526$	−14.8498	−0.647483
$527$	−19.8569	−0.864980
$528$	0	0
$529$	−12.1512	−0.528313
$530$	−8.44589	−0.366866
$531$	0	0
$532$	17.6689	0.766046
$533$	16.4493	0.712500
$534$	0	0
$535$	−3.67137	−0.158727
$536$	1.00000	0.0431934
$537$	0	0
$538$	12.8898	0.555719
$539$	14.7647	0.635961
$540$	0	0
$541$	28.3060	1.21697	0.608484	−	0.793566i	$-0.291778\pi$
$541$	28.3060	1.21697	0.608484	+	0.793566i	$0.291778\pi$
$542$	20.2345	0.869147
$543$	0	0
$544$	4.24987	0.182212
$545$	−9.80899	−0.420171
$546$	0	0
$547$	−35.0444	−1.49839	−0.749194	−	0.662350i	$-0.769559\pi$
$547$	−35.0444	−1.49839	−0.749194	+	0.662350i	$0.769559\pi$
$548$	−20.1652	−0.861413
$549$	0	0
$550$	2.53669	0.108165
$551$	−8.36319	−0.356284
$552$	0	0
$553$	−3.34667	−0.142315
$554$	−22.0435	−0.936539
$555$	0	0
$556$	10.2160	0.433254
$557$	13.4574	0.570207	0.285103	−	0.958497i	$-0.407972\pi$
$557$	13.4574	0.570207	0.285103	+	0.958497i	$0.407972\pi$
$558$	0	0
$559$	25.3771	1.07334
$560$	−3.58057	−0.151307
$561$	0	0
$562$	−21.4425	−0.904496
$563$	−28.9761	−1.22120	−0.610598	−	0.791941i	$-0.709071\pi$
$563$	−28.9761	−1.22120	−0.610598	+	0.791941i	$0.709071\pi$
$564$	0	0
$565$	13.5038	0.568108
$566$	−2.61199	−0.109790
$567$	0	0
$568$	−14.0928	−0.591319
$569$	18.7895	0.787696	0.393848	−	0.919176i	$-0.371144\pi$
$569$	18.7895	0.787696	0.393848	+	0.919176i	$0.371144\pi$
$570$	0	0
$571$	33.3842	1.39709	0.698543	−	0.715568i	$-0.253832\pi$
$571$	33.3842	1.39709	0.698543	+	0.715568i	$0.253832\pi$
$572$	−8.90686	−0.372415
$573$	0	0
$574$	16.7742	0.700143
$575$	−3.29375	−0.137359
$576$	0	0
$577$	−3.82252	−0.159134	−0.0795668	−	0.996830i	$-0.525354\pi$
$577$	−3.82252	−0.159134	−0.0795668	+	0.996830i	$0.525354\pi$
$578$	1.06143	0.0441497
$579$	0	0
$580$	1.69478	0.0703719
$581$	−13.2849	−0.551150
$582$	0	0
$583$	−21.4246	−0.887316
$584$	1.66237	0.0687894
$585$	0	0
$586$	−1.14564	−0.0473260
$587$	20.1448	0.831464	0.415732	−	0.909487i	$-0.363525\pi$
$587$	20.1448	0.831464	0.415732	+	0.909487i	$0.363525\pi$
$588$	0	0
$589$	23.0565	0.950028
$590$	10.3457	0.425925
$591$	0	0
$592$	9.34166	0.383940
$593$	29.8446	1.22557	0.612785	−	0.790250i	$-0.290049\pi$
$593$	29.8446	1.22557	0.612785	+	0.790250i	$0.290049\pi$
$594$	0	0
$595$	−15.2170	−0.623835
$596$	14.4997	0.593933
$597$	0	0
$598$	11.5651	0.472931
$599$	11.2260	0.458683	0.229341	−	0.973346i	$-0.426343\pi$
$599$	11.2260	0.458683	0.229341	+	0.973346i	$0.426343\pi$
$600$	0	0
$601$	13.4448	0.548424	0.274212	−	0.961669i	$-0.411583\pi$
$601$	13.4448	0.548424	0.274212	+	0.961669i	$0.411583\pi$
$602$	25.8783	1.05472
$603$	0	0
$604$	−7.50526	−0.305385
$605$	−4.56521	−0.185602
$606$	0	0
$607$	−2.62431	−0.106518	−0.0532588	−	0.998581i	$-0.516961\pi$
$607$	−2.62431	−0.106518	−0.0532588	+	0.998581i	$0.516961\pi$
$608$	−4.93468	−0.200128
$609$	0	0
$610$	−0.268278	−0.0108623
$611$	40.5972	1.64239
$612$	0	0
$613$	45.8569	1.85214	0.926071	−	0.377348i	$-0.123164\pi$
$613$	45.8569	1.85214	0.926071	+	0.377348i	$0.123164\pi$
$614$	−20.6450	−0.833162
$615$	0	0
$616$	−9.08279	−0.365956
$617$	3.85784	0.155311	0.0776554	−	0.996980i	$-0.475257\pi$
$617$	3.85784	0.155311	0.0776554	+	0.996980i	$0.475257\pi$
$618$	0	0
$619$	15.6141	0.627582	0.313791	−	0.949492i	$-0.398401\pi$
$619$	15.6141	0.627582	0.313791	+	0.949492i	$0.398401\pi$
$620$	−4.67235	−0.187646
$621$	0	0
$622$	26.1068	1.04679
$623$	−1.92165	−0.0769894
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.29828	0.131826
$627$	0	0
$628$	−10.8774	−0.434054
$629$	39.7009	1.58298
$630$	0	0
$631$	0.268412	0.0106853	0.00534265	−	0.999986i	$-0.498299\pi$
$631$	0.268412	0.0106853	0.00534265	+	0.999986i	$0.498299\pi$
$632$	0.934675	0.0371794
$633$	0	0
$634$	−32.7143	−1.29925
$635$	−20.3332	−0.806900
$636$	0	0
$637$	−20.4369	−0.809739
$638$	4.29913	0.170204
$639$	0	0
$640$	1.00000	0.0395285
$641$	1.58849	0.0627415	0.0313708	−	0.999508i	$-0.490013\pi$
$641$	1.58849	0.0627415	0.0313708	+	0.999508i	$0.490013\pi$
$642$	0	0
$643$	10.9283	0.430970	0.215485	−	0.976507i	$-0.430867\pi$
$643$	10.9283	0.430970	0.215485	+	0.976507i	$0.430867\pi$
$644$	11.7935	0.464729
$645$	0	0
$646$	−20.9718	−0.825122
$647$	18.2783	0.718594	0.359297	−	0.933223i	$-0.383016\pi$
$647$	18.2783	0.718594	0.359297	+	0.933223i	$0.383016\pi$
$648$	0	0
$649$	26.2438	1.03016
$650$	−3.51122	−0.137721
$651$	0	0
$652$	5.83044	0.228338
$653$	18.9318	0.740857	0.370429	−	0.928861i	$-0.379211\pi$
$653$	18.9318	0.740857	0.370429	+	0.928861i	$0.379211\pi$
$654$	0	0
$655$	3.86738	0.151111
$656$	−4.68480	−0.182911
$657$	0	0
$658$	41.3991	1.61391
$659$	−7.04585	−0.274467	−0.137234	−	0.990539i	$-0.543821\pi$
$659$	−7.04585	−0.274467	−0.137234	+	0.990539i	$0.543821\pi$
$660$	0	0
$661$	0.454886	0.0176930	0.00884652	−	0.999961i	$-0.497184\pi$
$661$	0.454886	0.0176930	0.00884652	+	0.999961i	$0.497184\pi$
$662$	−10.8963	−0.423497
$663$	0	0
$664$	3.71028	0.143987
$665$	17.6689	0.685172
$666$	0	0
$667$	−5.58219	−0.216143
$668$	−6.75720	−0.261444
$669$	0	0
$670$	1.00000	0.0386334
$671$	−0.680538	−0.0262719
$672$	0	0
$673$	3.37102	0.129943	0.0649717	−	0.997887i	$-0.479304\pi$
$673$	3.37102	0.129943	0.0649717	+	0.997887i	$0.479304\pi$
$674$	−19.0629	−0.734275
$675$	0	0
$676$	−0.671366	−0.0258218
$677$	−7.47123	−0.287143	−0.143571	−	0.989640i	$-0.545859\pi$
$677$	−7.47123	−0.287143	−0.143571	+	0.989640i	$0.545859\pi$
$678$	0	0
$679$	−33.9129	−1.30146
$680$	4.24987	0.162975
$681$	0	0
$682$	−11.8523	−0.453848
$683$	−16.4349	−0.628865	−0.314433	−	0.949280i	$-0.601814\pi$
$683$	−16.4349	−0.628865	−0.314433	+	0.949280i	$0.601814\pi$
$684$	0	0
$685$	−20.1652	−0.770471
$686$	4.22342	0.161251
$687$	0	0
$688$	−7.22744	−0.275544
$689$	29.6553	1.12978
$690$	0	0
$691$	10.0993	0.384194	0.192097	−	0.981376i	$-0.438471\pi$
$691$	10.0993	0.384194	0.192097	+	0.981376i	$0.438471\pi$
$692$	17.7398	0.674365
$693$	0	0
$694$	14.3472	0.544611
$695$	10.2160	0.387514
$696$	0	0
$697$	−19.9098	−0.754138
$698$	−28.5078	−1.07904
$699$	0	0
$700$	−3.58057	−0.135333
$701$	−13.0350	−0.492323	−0.246162	−	0.969229i	$-0.579169\pi$
$701$	−13.0350	−0.492323	−0.246162	+	0.969229i	$0.579169\pi$
$702$	0	0
$703$	−46.0980	−1.73862
$704$	2.53669	0.0956051
$705$	0	0
$706$	24.8483	0.935180
$707$	38.1472	1.43467
$708$	0	0
$709$	−1.72577	−0.0648126	−0.0324063	−	0.999475i	$-0.510317\pi$
$709$	−1.72577	−0.0648126	−0.0324063	+	0.999475i	$0.510317\pi$
$710$	−14.0928	−0.528892
$711$	0	0
$712$	0.536689	0.0201133
$713$	15.3896	0.576344
$714$	0	0
$715$	−8.90686	−0.333098
$716$	−22.0405	−0.823691
$717$	0	0
$718$	−11.7766	−0.439498
$719$	−27.7099	−1.03340	−0.516702	−	0.856165i	$-0.672840\pi$
$719$	−27.7099	−1.03340	−0.516702	+	0.856165i	$0.672840\pi$
$720$	0	0
$721$	59.6965	2.22321
$722$	5.35102	0.199144
$723$	0	0
$724$	−18.4659	−0.686278
$725$	1.69478	0.0629426
$726$	0	0
$727$	−29.6585	−1.09997	−0.549986	−	0.835174i	$-0.685367\pi$
$727$	−29.6585	−1.09997	−0.549986	+	0.835174i	$0.685367\pi$
$728$	12.5721	0.465955
$729$	0	0
$730$	1.66237	0.0615271
$731$	−30.7157	−1.13606
$732$	0	0
$733$	17.9232	0.662009	0.331004	−	0.943629i	$-0.392612\pi$
$733$	17.9232	0.662009	0.331004	+	0.943629i	$0.392612\pi$
$734$	24.3566	0.899020
$735$	0	0
$736$	−3.29375	−0.121409
$737$	2.53669	0.0934402
$738$	0	0
$739$	42.1512	1.55056	0.775278	−	0.631621i	$-0.217610\pi$
$739$	42.1512	1.55056	0.775278	+	0.631621i	$0.217610\pi$
$740$	9.34166	0.343406
$741$	0	0
$742$	30.2411	1.11018
$743$	−48.9778	−1.79682	−0.898410	−	0.439158i	$-0.855277\pi$
$743$	−48.9778	−1.79682	−0.898410	+	0.439158i	$0.855277\pi$
$744$	0	0
$745$	14.4997	0.531230
$746$	−13.6980	−0.501518
$747$	0	0
$748$	10.7806	0.394178
$749$	13.1456	0.480329
$750$	0	0
$751$	6.84692	0.249848	0.124924	−	0.992166i	$-0.460131\pi$
$751$	6.84692	0.249848	0.124924	+	0.992166i	$0.460131\pi$
$752$	−11.5622	−0.421629
$753$	0	0
$754$	−5.95074	−0.216713
$755$	−7.50526	−0.273144
$756$	0	0
$757$	14.9008	0.541578	0.270789	−	0.962639i	$-0.412715\pi$
$757$	14.9008	0.541578	0.270789	+	0.962639i	$0.412715\pi$
$758$	34.2160	1.24278
$759$	0	0
$760$	−4.93468	−0.178999
$761$	−42.4121	−1.53744	−0.768719	−	0.639586i	$-0.779106\pi$
$761$	−42.4121	−1.53744	−0.768719	+	0.639586i	$0.779106\pi$
$762$	0	0
$763$	35.1218	1.27149
$764$	−17.0395	−0.616467
$765$	0	0
$766$	−24.7173	−0.893074
$767$	−36.3259	−1.31165
$768$	0	0
$769$	1.51758	0.0547254	0.0273627	−	0.999626i	$-0.491289\pi$
$769$	1.51758	0.0547254	0.0273627	+	0.999626i	$0.491289\pi$
$770$	−9.08279	−0.327321
$771$	0	0
$772$	−1.61948	−0.0582862
$773$	−5.74560	−0.206655	−0.103327	−	0.994647i	$-0.532949\pi$
$773$	−5.74560	−0.206655	−0.103327	+	0.994647i	$0.532949\pi$
$774$	0	0
$775$	−4.67235	−0.167836
$776$	9.47136	0.340002
$777$	0	0
$778$	38.5098	1.38064
$779$	23.1180	0.828287
$780$	0	0
$781$	−35.7490	−1.27920
$782$	−13.9980	−0.500569
$783$	0	0
$784$	5.82046	0.207874
$785$	−10.8774	−0.388230
$786$	0	0
$787$	27.5268	0.981224	0.490612	−	0.871378i	$-0.336773\pi$
$787$	27.5268	0.981224	0.490612	+	0.871378i	$0.336773\pi$
$788$	2.79597	0.0996024
$789$	0	0
$790$	0.934675	0.0332543
$791$	−48.3512	−1.71917
$792$	0	0
$793$	0.941982	0.0334508
$794$	31.4679	1.11675
$795$	0	0
$796$	20.3826	0.722443
$797$	−31.9292	−1.13099	−0.565494	−	0.824752i	$-0.691315\pi$
$797$	−31.9292	−1.13099	−0.565494	+	0.824752i	$0.691315\pi$
$798$	0	0
$799$	−49.1377	−1.73837
$800$	1.00000	0.0353553
$801$	0	0
$802$	−2.10920	−0.0744785
$803$	4.21692	0.148812
$804$	0	0
$805$	11.7935	0.415666
$806$	16.4056	0.577864
$807$	0	0
$808$	−10.6539	−0.374805
$809$	8.97805	0.315651	0.157826	−	0.987467i	$-0.449552\pi$
$809$	8.97805	0.315651	0.157826	+	0.987467i	$0.449552\pi$
$810$	0	0
$811$	−22.3288	−0.784069	−0.392035	−	0.919950i	$-0.628229\pi$
$811$	−22.3288	−0.784069	−0.392035	+	0.919950i	$0.628229\pi$
$812$	−6.06828	−0.212955
$813$	0	0
$814$	23.6969	0.830575
$815$	5.83044	0.204231
$816$	0	0
$817$	35.6651	1.24776
$818$	28.9722	1.01299
$819$	0	0
$820$	−4.68480	−0.163600
$821$	−40.1168	−1.40009	−0.700043	−	0.714101i	$-0.746836\pi$
$821$	−40.1168	−1.40009	−0.700043	+	0.714101i	$0.746836\pi$
$822$	0	0
$823$	−11.3705	−0.396352	−0.198176	−	0.980166i	$-0.563502\pi$
$823$	−11.3705	−0.396352	−0.198176	+	0.980166i	$0.563502\pi$
$824$	−16.6723	−0.580809
$825$	0	0
$826$	−37.0434	−1.28891
$827$	9.65493	0.335735	0.167867	−	0.985810i	$-0.446312\pi$
$827$	9.65493	0.335735	0.167867	+	0.985810i	$0.446312\pi$
$828$	0	0
$829$	−53.0785	−1.84349	−0.921746	−	0.387795i	$-0.873237\pi$
$829$	−53.0785	−1.84349	−0.921746	+	0.387795i	$0.873237\pi$
$830$	3.71028	0.128786
$831$	0	0
$832$	−3.51122	−0.121729
$833$	24.7362	0.857060
$834$	0	0
$835$	−6.75720	−0.233842
$836$	−12.5177	−0.432935
$837$	0	0
$838$	−0.795972	−0.0274964
$839$	−17.3389	−0.598605	−0.299302	−	0.954158i	$-0.596754\pi$
$839$	−17.3389	−0.598605	−0.299302	+	0.954158i	$0.596754\pi$
$840$	0	0
$841$	−26.1277	−0.900956
$842$	1.83348	0.0631860
$843$	0	0
$844$	24.1253	0.830427
$845$	−0.671366	−0.0230957
$846$	0	0
$847$	16.3460	0.561656
$848$	−8.44589	−0.290033
$849$	0	0
$850$	4.24987	0.145769
$851$	−30.7691	−1.05475
$852$	0	0
$853$	−0.0199594	−0.000683397 0	−0.000341698	−	1.00000i	$-0.500109\pi$
$853$	−0.0199594	−0.000683397 0	−0.000341698		1.00000i	$0.500109\pi$
$854$	0.960587	0.0328706
$855$	0	0
$856$	−3.67137	−0.125485
$857$	−22.9858	−0.785179	−0.392589	−	0.919714i	$-0.628421\pi$
$857$	−22.9858	−0.785179	−0.392589	+	0.919714i	$0.628421\pi$
$858$	0	0
$859$	34.6729	1.18302	0.591512	−	0.806296i	$-0.298531\pi$
$859$	34.6729	1.18302	0.591512	+	0.806296i	$0.298531\pi$
$860$	−7.22744	−0.246454
$861$	0	0
$862$	7.16006	0.243873
$863$	2.25530	0.0767713	0.0383856	−	0.999263i	$-0.487778\pi$
$863$	2.25530	0.0767713	0.0383856	+	0.999263i	$0.487778\pi$
$864$	0	0
$865$	17.7398	0.603170
$866$	16.0834	0.546535
$867$	0	0
$868$	16.7297	0.567842
$869$	2.37098	0.0804300
$870$	0	0
$871$	−3.51122	−0.118973
$872$	−9.80899	−0.332174
$873$	0	0
$874$	16.2536	0.549786
$875$	−3.58057	−0.121045
$876$	0	0
$877$	18.1058	0.611391	0.305695	−	0.952129i	$-0.401111\pi$
$877$	18.1058	0.611391	0.305695	+	0.952129i	$0.401111\pi$
$878$	3.00694	0.101479
$879$	0	0
$880$	2.53669	0.0855118
$881$	−15.9836	−0.538500	−0.269250	−	0.963070i	$-0.586776\pi$
$881$	−15.9836	−0.538500	−0.269250	+	0.963070i	$0.586776\pi$
$882$	0	0
$883$	18.3813	0.618579	0.309290	−	0.950968i	$-0.399909\pi$
$883$	18.3813	0.618579	0.309290	+	0.950968i	$0.399909\pi$
$884$	−14.9222	−0.501889
$885$	0	0
$886$	3.52218	0.118330
$887$	−13.8913	−0.466426	−0.233213	−	0.972426i	$-0.574924\pi$
$887$	−13.8913	−0.466426	−0.233213	+	0.972426i	$0.574924\pi$
$888$	0	0
$889$	72.8045	2.44178
$890$	0.536689	0.0179899
$891$	0	0
$892$	−13.2489	−0.443606
$893$	57.0555	1.90929
$894$	0	0
$895$	−22.0405	−0.736731
$896$	−3.58057	−0.119618
$897$	0	0
$898$	−22.7037	−0.757632
$899$	−7.91861	−0.264100
$900$	0	0
$901$	−35.8940	−1.19580
$902$	−11.8839	−0.395690
$903$	0	0
$904$	13.5038	0.449129
$905$	−18.4659	−0.613826
$906$	0	0
$907$	−41.1079	−1.36497	−0.682483	−	0.730902i	$-0.739100\pi$
$907$	−41.1079	−1.36497	−0.682483	+	0.730902i	$0.739100\pi$
$908$	−12.2594	−0.406843
$909$	0	0
$910$	12.5721	0.416762
$911$	−15.9031	−0.526894	−0.263447	−	0.964674i	$-0.584859\pi$
$911$	−15.9031	−0.526894	−0.263447	+	0.964674i	$0.584859\pi$
$912$	0	0
$913$	9.41182	0.311486
$914$	−1.69769	−0.0561546
$915$	0	0
$916$	9.88430	0.326586
$917$	−13.8474	−0.457282
$918$	0	0
$919$	39.1402	1.29112	0.645558	−	0.763712i	$-0.276625\pi$
$919$	39.1402	1.29112	0.645558	+	0.763712i	$0.276625\pi$
$920$	−3.29375	−0.108592
$921$	0	0
$922$	7.14118	0.235182
$923$	49.4827	1.62874
$924$	0	0
$925$	9.34166	0.307152
$926$	−22.6465	−0.744210
$927$	0	0
$928$	1.69478	0.0556339
$929$	14.2594	0.467836	0.233918	−	0.972256i	$-0.424845\pi$
$929$	14.2594	0.467836	0.233918	+	0.972256i	$0.424845\pi$
$930$	0	0
$931$	−28.7221	−0.941328
$932$	−7.99405	−0.261854
$933$	0	0
$934$	−21.1876	−0.693279
$935$	10.7806	0.352564
$936$	0	0
$937$	−4.04088	−0.132010	−0.0660049	−	0.997819i	$-0.521025\pi$
$937$	−4.04088	−0.132010	−0.0660049	+	0.997819i	$0.521025\pi$
$938$	−3.58057	−0.116910
$939$	0	0
$940$	−11.5622	−0.377116
$941$	10.6324	0.346605	0.173303	−	0.984869i	$-0.444556\pi$
$941$	10.6324	0.346605	0.173303	+	0.984869i	$0.444556\pi$
$942$	0	0
$943$	15.4306	0.502489
$944$	10.3457	0.336723
$945$	0	0
$946$	−18.3338	−0.596083
$947$	−58.4346	−1.89887	−0.949435	−	0.313965i	$-0.898343\pi$
$947$	−58.4346	−1.89887	−0.949435	+	0.313965i	$0.898343\pi$
$948$	0	0
$949$	−5.83694	−0.189475
$950$	−4.93468	−0.160102
$951$	0	0
$952$	−15.2170	−0.493185
$953$	−48.2523	−1.56305	−0.781523	−	0.623876i	$-0.785557\pi$
$953$	−48.2523	−1.56305	−0.781523	+	0.623876i	$0.785557\pi$
$954$	0	0
$955$	−17.0395	−0.551385
$956$	26.2116	0.847743
$957$	0	0
$958$	−11.6764	−0.377247
$959$	72.2027	2.33155
$960$	0	0
$961$	−9.16915	−0.295779
$962$	−32.8006	−1.05753
$963$	0	0
$964$	29.2385	0.941710
$965$	−1.61948	−0.0521328
$966$	0	0
$967$	−18.3133	−0.588915	−0.294458	−	0.955665i	$-0.595139\pi$
$967$	−18.3133	−0.588915	−0.294458	+	0.955665i	$0.595139\pi$
$968$	−4.56521	−0.146731
$969$	0	0
$970$	9.47136	0.304107
$971$	−7.99391	−0.256537	−0.128268	−	0.991739i	$-0.540942\pi$
$971$	−7.99391	−0.256537	−0.128268	+	0.991739i	$0.540942\pi$
$972$	0	0
$973$	−36.5790	−1.17267
$974$	7.94770	0.254661
$975$	0	0
$976$	−0.268278	−0.00858737
$977$	8.69697	0.278241	0.139120	−	0.990275i	$-0.455572\pi$
$977$	8.69697	0.278241	0.139120	+	0.990275i	$0.455572\pi$
$978$	0	0
$979$	1.36141	0.0435110
$980$	5.82046	0.185928
$981$	0	0
$982$	−31.7433	−1.01297
$983$	−52.0629	−1.66055	−0.830275	−	0.557354i	$-0.811817\pi$
$983$	−52.0629	−1.66055	−0.830275	+	0.557354i	$0.811817\pi$
$984$	0	0
$985$	2.79597	0.0890871
$986$	7.20261	0.229378
$987$	0	0
$988$	17.3267	0.551236
$989$	23.8054	0.756968
$990$	0	0
$991$	−17.4593	−0.554614	−0.277307	−	0.960781i	$-0.589442\pi$
$991$	−17.4593	−0.554614	−0.277307	+	0.960781i	$0.589442\pi$
$992$	−4.67235	−0.148347
$993$	0	0
$994$	50.4601	1.60050
$995$	20.3826	0.646173
$996$	0	0
$997$	15.9378	0.504754	0.252377	−	0.967629i	$-0.418788\pi$
$997$	15.9378	0.504754	0.252377	+	0.967629i	$0.418788\pi$
$998$	−30.4938	−0.965264
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.a.bt.1.1		4
3.2	odd	2		670.2.a.j.1.1	✓	4
12.11	even	2		5360.2.a.be.1.4		4
15.2	even	4		3350.2.c.m.2949.4		8
15.8	even	4		3350.2.c.m.2949.5		8
15.14	odd	2		3350.2.a.n.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
670.2.a.j.1.1	✓	4	3.2	odd	2
3350.2.a.n.1.4		4	15.14	odd	2
3350.2.c.m.2949.4		8	15.2	even	4
3350.2.c.m.2949.5		8	15.8	even	4
5360.2.a.be.1.4		4	12.11	even	2
6030.2.a.bt.1.1		4	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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.58057 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.58057 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.53669 q^{11} -3.51122 q^{13} -3.58057 q^{14} +1.00000 q^{16} +4.24987 q^{17} -4.93468 q^{19} +1.00000 q^{20} +2.53669 q^{22} -3.29375 q^{23} +1.00000 q^{25} -3.51122 q^{26} -3.58057 q^{28} +1.69478 q^{29} -4.67235 q^{31} +1.00000 q^{32} +4.24987 q^{34} -3.58057 q^{35} +9.34166 q^{37} -4.93468 q^{38} +1.00000 q^{40} -4.68480 q^{41} -7.22744 q^{43} +2.53669 q^{44} -3.29375 q^{46} -11.5622 q^{47} +5.82046 q^{49} +1.00000 q^{50} -3.51122 q^{52} -8.44589 q^{53} +2.53669 q^{55} -3.58057 q^{56} +1.69478 q^{58} +10.3457 q^{59} -0.268278 q^{61} -4.67235 q^{62} +1.00000 q^{64} -3.51122 q^{65} +1.00000 q^{67} +4.24987 q^{68} -3.58057 q^{70} -14.0928 q^{71} +1.66237 q^{73} +9.34166 q^{74} -4.93468 q^{76} -9.08279 q^{77} +0.934675 q^{79} +1.00000 q^{80} -4.68480 q^{82} +3.71028 q^{83} +4.24987 q^{85} -7.22744 q^{86} +2.53669 q^{88} +0.536689 q^{89} +12.5721 q^{91} -3.29375 q^{92} -11.5622 q^{94} -4.93468 q^{95} +9.47136 q^{97} +5.82046 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 5 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 5 q^{7} + 4 q^{8} + 4 q^{10} - 9 q^{11} - 12 q^{13} - 5 q^{14} + 4 q^{16} + 4 q^{19} + 4 q^{20} - 9 q^{22} - 6 q^{23} + 4 q^{25} - 12 q^{26} - 5 q^{28} - 18 q^{29} + 2 q^{31} + 4 q^{32} - 5 q^{35} + 9 q^{37} + 4 q^{38} + 4 q^{40} - 12 q^{41} - 16 q^{43} - 9 q^{44} - 6 q^{46} - 10 q^{47} + 15 q^{49} + 4 q^{50} - 12 q^{52} - 8 q^{53} - 9 q^{55} - 5 q^{56} - 18 q^{58} - 18 q^{59} - 11 q^{61} + 2 q^{62} + 4 q^{64} - 12 q^{65} + 4 q^{67} - 5 q^{70} - 27 q^{71} + 4 q^{73} + 9 q^{74} + 4 q^{76} - 25 q^{77} - 20 q^{79} + 4 q^{80} - 12 q^{82} - 9 q^{83} - 16 q^{86} - 9 q^{88} - 17 q^{89} - 4 q^{91} - 6 q^{92} - 10 q^{94} + 4 q^{95} - 5 q^{97} + 15 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 5 * q^7 + 4 * q^8 + 4 * q^10 - 9 * q^11 - 12 * q^13 - 5 * q^14 + 4 * q^16 + 4 * q^19 + 4 * q^20 - 9 * q^22 - 6 * q^23 + 4 * q^25 - 12 * q^26 - 5 * q^28 - 18 * q^29 + 2 * q^31 + 4 * q^32 - 5 * q^35 + 9 * q^37 + 4 * q^38 + 4 * q^40 - 12 * q^41 - 16 * q^43 - 9 * q^44 - 6 * q^46 - 10 * q^47 + 15 * q^49 + 4 * q^50 - 12 * q^52 - 8 * q^53 - 9 * q^55 - 5 * q^56 - 18 * q^58 - 18 * q^59 - 11 * q^61 + 2 * q^62 + 4 * q^64 - 12 * q^65 + 4 * q^67 - 5 * q^70 - 27 * q^71 + 4 * q^73 + 9 * q^74 + 4 * q^76 - 25 * q^77 - 20 * q^79 + 4 * q^80 - 12 * q^82 - 9 * q^83 - 16 * q^86 - 9 * q^88 - 17 * q^89 - 4 * q^91 - 6 * q^92 - 10 * q^94 + 4 * q^95 - 5 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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