LMFDB - Embedded newform 6030.2.a.bu.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.70292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 10x + 2 $ x^4 - x^3 - 8x^2 + 10x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2010)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.59676$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.45037 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.45037 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.50940 q^{11} +4.95977 q^{13} -1.45037 q^{14} +1.00000 q^{16} +3.19351 q^{17} +3.19351 q^{19} +1.00000 q^{20} +3.50940 q^{22} -4.15328 q^{23} +1.00000 q^{25} +4.95977 q^{26} -1.45037 q^{28} +3.25254 q^{29} +6.95977 q^{31} +1.00000 q^{32} +3.19351 q^{34} -1.45037 q^{35} -4.70291 q^{37} +3.19351 q^{38} +1.00000 q^{40} -8.21231 q^{41} -7.01880 q^{43} +3.50940 q^{44} -4.15328 q^{46} +8.66700 q^{47} -4.89642 q^{49} +1.00000 q^{50} +4.95977 q^{52} -0.900743 q^{53} +3.50940 q^{55} -1.45037 q^{56} +3.25254 q^{58} -8.21231 q^{59} +0.549629 q^{61} +6.95977 q^{62} +1.00000 q^{64} +4.95977 q^{65} -1.00000 q^{67} +3.19351 q^{68} -1.45037 q^{70} +6.64388 q^{71} +7.40582 q^{73} -4.70291 q^{74} +3.19351 q^{76} -5.08993 q^{77} -12.3656 q^{79} +1.00000 q^{80} -8.21231 q^{82} +17.0899 q^{83} +3.19351 q^{85} -7.01880 q^{86} +3.50940 q^{88} -14.1087 q^{89} -7.19351 q^{91} -4.15328 q^{92} +8.66700 q^{94} +3.19351 q^{95} -3.50940 q^{97} -4.89642 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 2 q^{13} + q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{19} + 4 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 2 q^{26} + q^{28} - 2 q^{29} + 10 q^{31} + 4 q^{32} + 2 q^{34} + q^{35} + 3 q^{37} + 2 q^{38} + 4 q^{40} - 6 q^{43} + 3 q^{44} + 12 q^{46} + 14 q^{47} + 13 q^{49} + 4 q^{50} + 2 q^{52} + 10 q^{53} + 3 q^{55} + q^{56} - 2 q^{58} + 9 q^{61} + 10 q^{62} + 4 q^{64} + 2 q^{65} - 4 q^{67} + 2 q^{68} + q^{70} + 9 q^{71} - 14 q^{73} + 3 q^{74} + 2 q^{76} + 23 q^{77} + 12 q^{79} + 4 q^{80} + 25 q^{83} + 2 q^{85} - 6 q^{86} + 3 q^{88} + 9 q^{89} - 18 q^{91} + 12 q^{92} + 14 q^{94} + 2 q^{95} - 3 q^{97} + 13 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 2 * q^13 + q^14 + 4 * q^16 + 2 * q^17 + 2 * q^19 + 4 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 2 * q^26 + q^28 - 2 * q^29 + 10 * q^31 + 4 * q^32 + 2 * q^34 + q^35 + 3 * q^37 + 2 * q^38 + 4 * q^40 - 6 * q^43 + 3 * q^44 + 12 * q^46 + 14 * q^47 + 13 * q^49 + 4 * q^50 + 2 * q^52 + 10 * q^53 + 3 * q^55 + q^56 - 2 * q^58 + 9 * q^61 + 10 * q^62 + 4 * q^64 + 2 * q^65 - 4 * q^67 + 2 * q^68 + q^70 + 9 * q^71 - 14 * q^73 + 3 * q^74 + 2 * q^76 + 23 * q^77 + 12 * q^79 + 4 * q^80 + 25 * q^83 + 2 * q^85 - 6 * q^86 + 3 * q^88 + 9 * q^89 - 18 * q^91 + 12 * q^92 + 14 * q^94 + 2 * q^95 - 3 * q^97 + 13 * 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
$7$	−1.45037	−0.548189	−0.274094	−	0.961703i	$-0.588378\pi$
$7$	−1.45037	−0.548189	−0.274094	+	0.961703i	$0.588378\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	3.50940	1.05812	0.529062	−	0.848583i	$-0.322544\pi$
$11$	3.50940	1.05812	0.529062	+	0.848583i	$0.322544\pi$
$12$	0	0
$13$	4.95977	1.37559	0.687797	−	0.725903i	$-0.258578\pi$
$13$	4.95977	1.37559	0.687797	+	0.725903i	$0.258578\pi$
$14$	−1.45037	−0.387628
$15$	0	0
$16$	1.00000	0.250000
$17$	3.19351	0.774540	0.387270	−	0.921966i	$-0.373418\pi$
$17$	3.19351	0.774540	0.387270	+	0.921966i	$0.373418\pi$
$18$	0	0
$19$	3.19351	0.732642	0.366321	−	0.930489i	$-0.380617\pi$
$19$	3.19351	0.732642	0.366321	+	0.930489i	$0.380617\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	3.50940	0.748207
$23$	−4.15328	−0.866019	−0.433010	−	0.901389i	$-0.642548\pi$
$23$	−4.15328	−0.866019	−0.433010	+	0.901389i	$0.642548\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.95977	0.972691
$27$	0	0
$28$	−1.45037	−0.274094
$29$	3.25254	0.603981	0.301991	−	0.953311i	$-0.402349\pi$
$29$	3.25254	0.603981	0.301991	+	0.953311i	$0.402349\pi$
$30$	0	0
$31$	6.95977	1.25001	0.625006	−	0.780620i	$-0.285097\pi$
$31$	6.95977	1.25001	0.625006	+	0.780620i	$0.285097\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.19351	0.547683
$35$	−1.45037	−0.245158
$36$	0	0
$37$	−4.70291	−0.773154	−0.386577	−	0.922257i	$-0.626343\pi$
$37$	−4.70291	−0.773154	−0.386577	+	0.922257i	$0.626343\pi$
$38$	3.19351	0.518056
$39$	0	0
$40$	1.00000	0.158114
$41$	−8.21231	−1.28255	−0.641274	−	0.767312i	$-0.721594\pi$
$41$	−8.21231	−1.28255	−0.641274	+	0.767312i	$0.721594\pi$
$42$	0	0
$43$	−7.01880	−1.07036	−0.535178	−	0.844739i	$-0.679756\pi$
$43$	−7.01880	−1.07036	−0.535178	+	0.844739i	$0.679756\pi$
$44$	3.50940	0.529062
$45$	0	0
$46$	−4.15328	−0.612368
$47$	8.66700	1.26421	0.632106	−	0.774882i	$-0.282191\pi$
$47$	8.66700	1.26421	0.632106	+	0.774882i	$0.282191\pi$
$48$	0	0
$49$	−4.89642	−0.699489
$50$	1.00000	0.141421
$51$	0	0
$52$	4.95977	0.687797
$53$	−0.900743	−0.123727	−0.0618633	−	0.998085i	$-0.519704\pi$
$53$	−0.900743	−0.123727	−0.0618633	+	0.998085i	$0.519704\pi$
$54$	0	0
$55$	3.50940	0.473207
$56$	−1.45037	−0.193814
$57$	0	0
$58$	3.25254	0.427079
$59$	−8.21231	−1.06915	−0.534576	−	0.845120i	$-0.679529\pi$
$59$	−8.21231	−1.06915	−0.534576	+	0.845120i	$0.679529\pi$
$60$	0	0
$61$	0.549629	0.0703727	0.0351864	−	0.999381i	$-0.488798\pi$
$61$	0.549629	0.0703727	0.0351864	+	0.999381i	$0.488798\pi$
$62$	6.95977	0.883892
$63$	0	0
$64$	1.00000	0.125000
$65$	4.95977	0.615184
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	3.19351	0.387270
$69$	0	0
$70$	−1.45037	−0.173353
$71$	6.64388	0.788484	0.394242	−	0.919007i	$-0.371007\pi$
$71$	6.64388	0.788484	0.394242	+	0.919007i	$0.371007\pi$
$72$	0	0
$73$	7.40582	0.866786	0.433393	−	0.901205i	$-0.357316\pi$
$73$	7.40582	0.866786	0.433393	+	0.901205i	$0.357316\pi$
$74$	−4.70291	−0.546702
$75$	0	0
$76$	3.19351	0.366321
$77$	−5.08993	−0.580052
$78$	0	0
$79$	−12.3656	−1.39124	−0.695619	−	0.718411i	$-0.744870\pi$
$79$	−12.3656	−1.39124	−0.695619	+	0.718411i	$0.744870\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−8.21231	−0.906898
$83$	17.0899	1.87586	0.937932	−	0.346819i	$-0.112738\pi$
$83$	17.0899	1.87586	0.937932	+	0.346819i	$0.112738\pi$
$84$	0	0
$85$	3.19351	0.346385
$86$	−7.01880	−0.756857
$87$	0	0
$88$	3.50940	0.374103
$89$	−14.1087	−1.49552	−0.747761	−	0.663968i	$-0.768871\pi$
$89$	−14.1087	−1.49552	−0.747761	+	0.663968i	$0.768871\pi$
$90$	0	0
$91$	−7.19351	−0.754085
$92$	−4.15328	−0.433010
$93$	0	0
$94$	8.66700	0.893933
$95$	3.19351	0.327647
$96$	0	0
$97$	−3.50940	−0.356326	−0.178163	−	0.984001i	$-0.557015\pi$
$97$	−3.50940	−0.356326	−0.178163	+	0.984001i	$0.557015\pi$
$98$	−4.89642	−0.494613
$99$	0	0
$100$	1.00000	0.100000
$101$	1.68411	0.167575	0.0837877	−	0.996484i	$-0.473298\pi$
$101$	1.68411	0.167575	0.0837877	+	0.996484i	$0.473298\pi$
$102$	0	0
$103$	−11.0188	−1.08571	−0.542857	−	0.839825i	$-0.682658\pi$
$103$	−11.0188	−1.08571	−0.542857	+	0.839825i	$0.682658\pi$
$104$	4.95977	0.486346
$105$	0	0
$106$	−0.900743	−0.0874879
$107$	17.9195	1.73235	0.866174	−	0.499743i	$-0.166572\pi$
$107$	17.9195	1.73235	0.866174	+	0.499743i	$0.166572\pi$
$108$	0	0
$109$	5.83739	0.559121	0.279560	−	0.960128i	$-0.409811\pi$
$109$	5.83739	0.559121	0.279560	+	0.960128i	$0.409811\pi$
$110$	3.50940	0.334608
$111$	0	0
$112$	−1.45037	−0.137047
$113$	8.99568	0.846242	0.423121	−	0.906073i	$-0.360934\pi$
$113$	8.99568	0.846242	0.423121	+	0.906073i	$0.360934\pi$
$114$	0	0
$115$	−4.15328	−0.387296
$116$	3.25254	0.301991
$117$	0	0
$118$	−8.21231	−0.756005
$119$	−4.63178	−0.424594
$120$	0	0
$121$	1.31589	0.119626
$122$	0.549629	0.0497610
$123$	0	0
$124$	6.95977	0.625006
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.3254	1.35991	0.679953	−	0.733256i	$-0.262000\pi$
$127$	15.3254	1.35991	0.679953	+	0.733256i	$0.262000\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.95977	0.435001
$131$	0.886946	0.0774928	0.0387464	−	0.999249i	$-0.487664\pi$
$131$	0.886946	0.0774928	0.0387464	+	0.999249i	$0.487664\pi$
$132$	0	0
$133$	−4.63178	−0.401626
$134$	−1.00000	−0.0863868
$135$	0	0
$136$	3.19351	0.273841
$137$	8.70291	0.743540	0.371770	−	0.928325i	$-0.378751\pi$
$137$	8.70291	0.743540	0.371770	+	0.928325i	$0.378751\pi$
$138$	0	0
$139$	18.6456	1.58150	0.790749	−	0.612141i	$-0.209692\pi$
$139$	18.6456	1.58150	0.790749	+	0.612141i	$0.209692\pi$
$140$	−1.45037	−0.122579
$141$	0	0
$142$	6.64388	0.557542
$143$	17.4058	1.45555
$144$	0	0
$145$	3.25254	0.270109
$146$	7.40582	0.612910
$147$	0	0
$148$	−4.70291	−0.386577
$149$	3.25254	0.266458	0.133229	−	0.991085i	$-0.457465\pi$
$149$	3.25254	0.266458	0.133229	+	0.991085i	$0.457465\pi$
$150$	0	0
$151$	−14.2354	−1.15846	−0.579232	−	0.815163i	$-0.696647\pi$
$151$	−14.2354	−1.15846	−0.579232	+	0.815163i	$0.696647\pi$
$152$	3.19351	0.259028
$153$	0	0
$154$	−5.08993	−0.410159
$155$	6.95977	0.559022
$156$	0	0
$157$	−2.58554	−0.206348	−0.103174	−	0.994663i	$-0.532900\pi$
$157$	−2.58554	−0.206348	−0.103174	+	0.994663i	$0.532900\pi$
$158$	−12.3656	−0.983754
$159$	0	0
$160$	1.00000	0.0790569
$161$	6.02380	0.474742
$162$	0	0
$163$	−14.7029	−1.15162	−0.575810	−	0.817583i	$-0.695313\pi$
$163$	−14.7029	−1.15162	−0.575810	+	0.817583i	$0.695313\pi$
$164$	−8.21231	−0.641274
$165$	0	0
$166$	17.0899	1.32644
$167$	22.0728	1.70805	0.854023	−	0.520235i	$-0.174156\pi$
$167$	22.0728	1.70805	0.854023	+	0.520235i	$0.174156\pi$
$168$	0	0
$169$	11.5993	0.892256
$170$	3.19351	0.244931
$171$	0	0
$172$	−7.01880	−0.535178
$173$	6.77058	0.514758	0.257379	−	0.966311i	$-0.417141\pi$
$173$	6.77058	0.514758	0.257379	+	0.966311i	$0.417141\pi$
$174$	0	0
$175$	−1.45037	−0.109638
$176$	3.50940	0.264531
$177$	0	0
$178$	−14.1087	−1.05749
$179$	13.6268	1.01851	0.509256	−	0.860615i	$-0.329920\pi$
$179$	13.6268	1.01851	0.509256	+	0.860615i	$0.329920\pi$
$180$	0	0
$181$	−21.2073	−1.57633	−0.788163	−	0.615466i	$-0.788968\pi$
$181$	−21.2073	−1.57633	−0.788163	+	0.615466i	$0.788968\pi$
$182$	−7.19351	−0.533219
$183$	0	0
$184$	−4.15328	−0.306184
$185$	−4.70291	−0.345765
$186$	0	0
$187$	11.2073	0.819560
$188$	8.66700	0.632106
$189$	0	0
$190$	3.19351	0.231682
$191$	4.98120	0.360427	0.180213	−	0.983628i	$-0.442321\pi$
$191$	4.98120	0.360427	0.180213	+	0.983628i	$0.442321\pi$
$192$	0	0
$193$	2.92455	0.210513	0.105257	−	0.994445i	$-0.466434\pi$
$193$	2.92455	0.210513	0.105257	+	0.994445i	$0.466434\pi$
$194$	−3.50940	−0.251960
$195$	0	0
$196$	−4.89642	−0.349744
$197$	−7.91954	−0.564244	−0.282122	−	0.959379i	$-0.591038\pi$
$197$	−7.91954	−0.564244	−0.282122	+	0.959379i	$0.591038\pi$
$198$	0	0
$199$	9.08993	0.644368	0.322184	−	0.946677i	$-0.395583\pi$
$199$	9.08993	0.644368	0.322184	+	0.946677i	$0.395583\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	1.68411	0.118494
$203$	−4.71739	−0.331096
$204$	0	0
$205$	−8.21231	−0.573573
$206$	−11.0188	−0.767716
$207$	0	0
$208$	4.95977	0.343898
$209$	11.2073	0.775226
$210$	0	0
$211$	−8.41083	−0.579025	−0.289513	−	0.957174i	$-0.593493\pi$
$211$	−8.41083	−0.579025	−0.289513	+	0.957174i	$0.593493\pi$
$212$	−0.900743	−0.0618633
$213$	0	0
$214$	17.9195	1.22495
$215$	−7.01880	−0.478678
$216$	0	0
$217$	−10.0943	−0.685243
$218$	5.83739	0.395358
$219$	0	0
$220$	3.50940	0.236604
$221$	15.8391	1.06545
$222$	0	0
$223$	−14.5513	−0.974429	−0.487214	−	0.873282i	$-0.661987\pi$
$223$	−14.5513	−0.974429	−0.487214	+	0.873282i	$0.661987\pi$
$224$	−1.45037	−0.0969070
$225$	0	0
$226$	8.99568	0.598384
$227$	14.2690	0.947064	0.473532	−	0.880776i	$-0.342979\pi$
$227$	14.2690	0.947064	0.473532	+	0.880776i	$0.342979\pi$
$228$	0	0
$229$	−7.56843	−0.500136	−0.250068	−	0.968228i	$-0.580453\pi$
$229$	−7.56843	−0.500136	−0.250068	+	0.968228i	$0.580453\pi$
$230$	−4.15328	−0.273859
$231$	0	0
$232$	3.25254	0.213540
$233$	−23.9102	−1.56641	−0.783205	−	0.621763i	$-0.786417\pi$
$233$	−23.9102	−1.56641	−0.783205	+	0.621763i	$0.786417\pi$
$234$	0	0
$235$	8.66700	0.565373
$236$	−8.21231	−0.534576
$237$	0	0
$238$	−4.63178	−0.300234
$239$	15.3254	0.991316	0.495658	−	0.868518i	$-0.334927\pi$
$239$	15.3254	0.991316	0.495658	+	0.868518i	$0.334927\pi$
$240$	0	0
$241$	17.9340	1.15523	0.577616	−	0.816309i	$-0.303983\pi$
$241$	17.9340	1.15523	0.577616	+	0.816309i	$0.303983\pi$
$242$	1.31589	0.0845885
$243$	0	0
$244$	0.549629	0.0351864
$245$	−4.89642	−0.312821
$246$	0	0
$247$	15.8391	1.00782
$248$	6.95977	0.441946
$249$	0	0
$250$	1.00000	0.0632456
$251$	14.7029	0.928040	0.464020	−	0.885825i	$-0.346407\pi$
$251$	14.7029	0.928040	0.464020	+	0.885825i	$0.346407\pi$
$252$	0	0
$253$	−14.5755	−0.916356
$254$	15.3254	0.961599
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.0856	1.00339	0.501696	−	0.865044i	$-0.332709\pi$
$257$	16.0856	1.00339	0.501696	+	0.865044i	$0.332709\pi$
$258$	0	0
$259$	6.82097	0.423834
$260$	4.95977	0.307592
$261$	0	0
$262$	0.886946	0.0547957
$263$	−8.27134	−0.510033	−0.255016	−	0.966937i	$-0.582081\pi$
$263$	−8.27134	−0.510033	−0.255016	+	0.966937i	$0.582081\pi$
$264$	0	0
$265$	−0.900743	−0.0553322
$266$	−4.63178	−0.283993
$267$	0	0
$268$	−1.00000	−0.0610847
$269$	−12.8655	−0.784424	−0.392212	−	0.919875i	$-0.628290\pi$
$269$	−12.8655	−0.784424	−0.392212	+	0.919875i	$0.628290\pi$
$270$	0	0
$271$	−20.2475	−1.22995	−0.614975	−	0.788546i	$-0.710834\pi$
$271$	−20.2475	−1.22995	−0.614975	+	0.788546i	$0.710834\pi$
$272$	3.19351	0.193635
$273$	0	0
$274$	8.70291	0.525762
$275$	3.50940	0.211625
$276$	0	0
$277$	−17.6413	−1.05996	−0.529980	−	0.848010i	$-0.677801\pi$
$277$	−17.6413	−1.05996	−0.529980	+	0.848010i	$0.677801\pi$
$278$	18.6456	1.11829
$279$	0	0
$280$	−1.45037	−0.0866763
$281$	17.8391	1.06419	0.532095	−	0.846684i	$-0.321405\pi$
$281$	17.8391	1.06419	0.532095	+	0.846684i	$0.321405\pi$
$282$	0	0
$283$	20.1550	1.19809	0.599044	−	0.800716i	$-0.295547\pi$
$283$	20.1550	1.19809	0.599044	+	0.800716i	$0.295547\pi$
$284$	6.64388	0.394242
$285$	0	0
$286$	17.4058	1.02923
$287$	11.9109	0.703078
$288$	0	0
$289$	−6.80149	−0.400087
$290$	3.25254	0.190996
$291$	0	0
$292$	7.40582	0.433393
$293$	10.6815	0.624019	0.312009	−	0.950079i	$-0.398998\pi$
$293$	10.6815	0.624019	0.312009	+	0.950079i	$0.398998\pi$
$294$	0	0
$295$	−8.21231	−0.478139
$296$	−4.70291	−0.273351
$297$	0	0
$298$	3.25254	0.188415
$299$	−20.5993	−1.19129
$300$	0	0
$301$	10.1799	0.586758
$302$	−14.2354	−0.819157
$303$	0	0
$304$	3.19351	0.183160
$305$	0.549629	0.0314716
$306$	0	0
$307$	−6.75957	−0.385789	−0.192894	−	0.981220i	$-0.561787\pi$
$307$	−6.75957	−0.385789	−0.192894	+	0.981220i	$0.561787\pi$
$308$	−5.08993	−0.290026
$309$	0	0
$310$	6.95977	0.395288
$311$	12.1181	0.687152	0.343576	−	0.939125i	$-0.388362\pi$
$311$	12.1181	0.687152	0.343576	+	0.939125i	$0.388362\pi$
$312$	0	0
$313$	13.9427	0.788086	0.394043	−	0.919092i	$-0.371076\pi$
$313$	13.9427	0.788086	0.394043	+	0.919092i	$0.371076\pi$
$314$	−2.58554	−0.145910
$315$	0	0
$316$	−12.3656	−0.695619
$317$	−22.1584	−1.24454	−0.622271	−	0.782802i	$-0.713790\pi$
$317$	−22.1584	−1.24454	−0.622271	+	0.782802i	$0.713790\pi$
$318$	0	0
$319$	11.4145	0.639087
$320$	1.00000	0.0559017
$321$	0	0
$322$	6.02380	0.335693
$323$	10.1985	0.567461
$324$	0	0
$325$	4.95977	0.275119
$326$	−14.7029	−0.814319
$327$	0	0
$328$	−8.21231	−0.453449
$329$	−12.5704	−0.693027
$330$	0	0
$331$	16.0607	0.882777	0.441389	−	0.897316i	$-0.354486\pi$
$331$	16.0607	0.882777	0.441389	+	0.897316i	$0.354486\pi$
$332$	17.0899	0.937932
$333$	0	0
$334$	22.0728	1.20777
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−12.7636	−0.695279	−0.347640	−	0.937628i	$-0.613017\pi$
$337$	−12.7636	−0.695279	−0.347640	+	0.937628i	$0.613017\pi$
$338$	11.5993	0.630921
$339$	0	0
$340$	3.19351	0.173192
$341$	24.4246	1.32267
$342$	0	0
$343$	17.2542	0.931641
$344$	−7.01880	−0.378428
$345$	0	0
$346$	6.77058	0.363989
$347$	−35.2929	−1.89462	−0.947312	−	0.320313i	$-0.896212\pi$
$347$	−35.2929	−1.89462	−0.947312	+	0.320313i	$0.896212\pi$
$348$	0	0
$349$	−8.03760	−0.430243	−0.215121	−	0.976587i	$-0.569015\pi$
$349$	−8.03760	−0.430243	−0.215121	+	0.976587i	$0.569015\pi$
$350$	−1.45037	−0.0775256
$351$	0	0
$352$	3.50940	0.187052
$353$	−24.0943	−1.28241	−0.641204	−	0.767371i	$-0.721565\pi$
$353$	−24.0943	−1.28241	−0.641204	+	0.767371i	$0.721565\pi$
$354$	0	0
$355$	6.64388	0.352621
$356$	−14.1087	−0.747761
$357$	0	0
$358$	13.6268	0.720197
$359$	−28.8700	−1.52370	−0.761850	−	0.647754i	$-0.775709\pi$
$359$	−28.8700	−1.52370	−0.761850	+	0.647754i	$0.775709\pi$
$360$	0	0
$361$	−8.80149	−0.463236
$362$	−21.2073	−1.11463
$363$	0	0
$364$	−7.19351	−0.377042
$365$	7.40582	0.387638
$366$	0	0
$367$	25.4873	1.33043	0.665213	−	0.746654i	$-0.268341\pi$
$367$	25.4873	1.33043	0.665213	+	0.746654i	$0.268341\pi$
$368$	−4.15328	−0.216505
$369$	0	0
$370$	−4.70291	−0.244493
$371$	1.30641	0.0678255
$372$	0	0
$373$	−10.7613	−0.557197	−0.278598	−	0.960408i	$-0.589870\pi$
$373$	−10.7613	−0.557197	−0.278598	+	0.960408i	$0.589870\pi$
$374$	11.2073	0.579516
$375$	0	0
$376$	8.66700	0.446967
$377$	16.1319	0.830833
$378$	0	0
$379$	−24.4471	−1.25576	−0.627881	−	0.778310i	$-0.716077\pi$
$379$	−24.4471	−1.25576	−0.627881	+	0.778310i	$0.716077\pi$
$380$	3.19351	0.163824
$381$	0	0
$382$	4.98120	0.254860
$383$	4.50440	0.230164	0.115082	−	0.993356i	$-0.463287\pi$
$383$	4.50440	0.230164	0.115082	+	0.993356i	$0.463287\pi$
$384$	0	0
$385$	−5.08993	−0.259407
$386$	2.92455	0.148855
$387$	0	0
$388$	−3.50940	−0.178163
$389$	−19.8125	−1.00453	−0.502267	−	0.864713i	$-0.667500\pi$
$389$	−19.8125	−1.00453	−0.502267	+	0.864713i	$0.667500\pi$
$390$	0	0
$391$	−13.2636	−0.670767
$392$	−4.89642	−0.247307
$393$	0	0
$394$	−7.91954	−0.398981
$395$	−12.3656	−0.622181
$396$	0	0
$397$	−5.92023	−0.297128	−0.148564	−	0.988903i	$-0.547465\pi$
$397$	−5.92023	−0.297128	−0.148564	+	0.988903i	$0.547465\pi$
$398$	9.08993	0.455637
$399$	0	0
$400$	1.00000	0.0500000
$401$	−10.3784	−0.518272	−0.259136	−	0.965841i	$-0.583438\pi$
$401$	−10.3784	−0.518272	−0.259136	+	0.965841i	$0.583438\pi$
$402$	0	0
$403$	34.5189	1.71951
$404$	1.68411	0.0837877
$405$	0	0
$406$	−4.71739	−0.234120
$407$	−16.5044	−0.818093
$408$	0	0
$409$	0.0804569	0.00397834	0.00198917	−	0.999998i	$-0.499367\pi$
$409$	0.0804569	0.00397834	0.00198917	+	0.999998i	$0.499367\pi$
$410$	−8.21231	−0.405577
$411$	0	0
$412$	−11.0188	−0.542857
$413$	11.9109	0.586097
$414$	0	0
$415$	17.0899	0.838912
$416$	4.95977	0.243173
$417$	0	0
$418$	11.2073	0.548167
$419$	−28.9145	−1.41257	−0.706284	−	0.707929i	$-0.749630\pi$
$419$	−28.9145	−1.41257	−0.706284	+	0.707929i	$0.749630\pi$
$420$	0	0
$421$	7.21731	0.351750	0.175875	−	0.984412i	$-0.443724\pi$
$421$	7.21731	0.351750	0.175875	+	0.984412i	$0.443724\pi$
$422$	−8.41083	−0.409433
$423$	0	0
$424$	−0.900743	−0.0437439
$425$	3.19351	0.154908
$426$	0	0
$427$	−0.797166	−0.0385775
$428$	17.9195	0.866174
$429$	0	0
$430$	−7.01880	−0.338477
$431$	−14.8324	−0.714451	−0.357226	−	0.934018i	$-0.616277\pi$
$431$	−14.8324	−0.714451	−0.357226	+	0.934018i	$0.616277\pi$
$432$	0	0
$433$	1.83393	0.0881330	0.0440665	−	0.999029i	$-0.485969\pi$
$433$	1.83393	0.0881330	0.0440665	+	0.999029i	$0.485969\pi$
$434$	−10.0943	−0.484540
$435$	0	0
$436$	5.83739	0.279560
$437$	−13.2636	−0.634482
$438$	0	0
$439$	6.07113	0.289759	0.144880	−	0.989449i	$-0.453720\pi$
$439$	6.07113	0.289759	0.144880	+	0.989449i	$0.453720\pi$
$440$	3.50940	0.167304
$441$	0	0
$442$	15.8391	0.753389
$443$	10.3870	0.493502	0.246751	−	0.969079i	$-0.420637\pi$
$443$	10.3870	0.493502	0.246751	+	0.969079i	$0.420637\pi$
$444$	0	0
$445$	−14.1087	−0.668818
$446$	−14.5513	−0.689025
$447$	0	0
$448$	−1.45037	−0.0685236
$449$	−18.3877	−0.867769	−0.433885	−	0.900968i	$-0.642857\pi$
$449$	−18.3877	−0.867769	−0.433885	+	0.900968i	$0.642857\pi$
$450$	0	0
$451$	−28.8203	−1.35709
$452$	8.99568	0.423121
$453$	0	0
$454$	14.2690	0.669676
$455$	−7.19351	−0.337237
$456$	0	0
$457$	−9.31157	−0.435577	−0.217788	−	0.975996i	$-0.569884\pi$
$457$	−9.31157	−0.435577	−0.217788	+	0.975996i	$0.569884\pi$
$458$	−7.56843	−0.353649
$459$	0	0
$460$	−4.15328	−0.193648
$461$	24.2627	1.13003	0.565013	−	0.825082i	$-0.308871\pi$
$461$	24.2627	1.13003	0.565013	+	0.825082i	$0.308871\pi$
$462$	0	0
$463$	10.8193	0.502814	0.251407	−	0.967881i	$-0.419107\pi$
$463$	10.8193	0.502814	0.251407	+	0.967881i	$0.419107\pi$
$464$	3.25254	0.150995
$465$	0	0
$466$	−23.9102	−1.10762
$467$	19.5232	0.903426	0.451713	−	0.892163i	$-0.350813\pi$
$467$	19.5232	0.903426	0.451713	+	0.892163i	$0.350813\pi$
$468$	0	0
$469$	1.45037	0.0669719
$470$	8.66700	0.399779
$471$	0	0
$472$	−8.21231	−0.378002
$473$	−24.6318	−1.13257
$474$	0	0
$475$	3.19351	0.146528
$476$	−4.63178	−0.212297
$477$	0	0
$478$	15.3254	0.700966
$479$	23.8888	1.09151	0.545753	−	0.837946i	$-0.316244\pi$
$479$	23.8888	1.09151	0.545753	+	0.837946i	$0.316244\pi$
$480$	0	0
$481$	−23.3254	−1.06355
$482$	17.9340	0.816872
$483$	0	0
$484$	1.31589	0.0598131
$485$	−3.50940	−0.159354
$486$	0	0
$487$	−20.4263	−0.925605	−0.462802	−	0.886462i	$-0.653156\pi$
$487$	−20.4263	−0.925605	−0.462802	+	0.886462i	$0.653156\pi$
$488$	0.549629	0.0248805
$489$	0	0
$490$	−4.89642	−0.221198
$491$	8.04801	0.363202	0.181601	−	0.983372i	$-0.441872\pi$
$491$	8.04801	0.363202	0.181601	+	0.983372i	$0.441872\pi$
$492$	0	0
$493$	10.3870	0.467808
$494$	15.8391	0.712634
$495$	0	0
$496$	6.95977	0.312503
$497$	−9.63610	−0.432238
$498$	0	0
$499$	−35.5284	−1.59047	−0.795234	−	0.606303i	$-0.792652\pi$
$499$	−35.5284	−1.59047	−0.795234	+	0.606303i	$0.792652\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	14.7029	0.656223
$503$	−27.2456	−1.21482	−0.607410	−	0.794388i	$-0.707792\pi$
$503$	−27.2456	−1.21482	−0.607410	+	0.794388i	$0.707792\pi$
$504$	0	0
$505$	1.68411	0.0749420
$506$	−14.5755	−0.647961
$507$	0	0
$508$	15.3254	0.679953
$509$	−7.64820	−0.339001	−0.169500	−	0.985530i	$-0.554215\pi$
$509$	−7.64820	−0.339001	−0.169500	+	0.985530i	$0.554215\pi$
$510$	0	0
$511$	−10.7412	−0.475162
$512$	1.00000	0.0441942
$513$	0	0
$514$	16.0856	0.709506
$515$	−11.0188	−0.485546
$516$	0	0
$517$	30.4160	1.33769
$518$	6.82097	0.299696
$519$	0	0
$520$	4.95977	0.217500
$521$	−39.6557	−1.73735	−0.868675	−	0.495383i	$-0.835028\pi$
$521$	−39.6557	−1.73735	−0.868675	+	0.495383i	$0.835028\pi$
$522$	0	0
$523$	−34.5882	−1.51244	−0.756219	−	0.654319i	$-0.772956\pi$
$523$	−34.5882	−1.51244	−0.756219	+	0.654319i	$0.772956\pi$
$524$	0.886946	0.0387464
$525$	0	0
$526$	−8.27134	−0.360648
$527$	22.2261	0.968185
$528$	0	0
$529$	−5.75024	−0.250011
$530$	−0.900743	−0.0391258
$531$	0	0
$532$	−4.63178	−0.200813
$533$	−40.7312	−1.76426
$534$	0	0
$535$	17.9195	0.774729
$536$	−1.00000	−0.0431934
$537$	0	0
$538$	−12.8655	−0.554672
$539$	−17.1835	−0.740146
$540$	0	0
$541$	35.0025	1.50488	0.752438	−	0.658663i	$-0.228878\pi$
$541$	35.0025	1.50488	0.752438	+	0.658663i	$0.228878\pi$
$542$	−20.2475	−0.869706
$543$	0	0
$544$	3.19351	0.136921
$545$	5.83739	0.250046
$546$	0	0
$547$	31.3972	1.34245	0.671223	−	0.741255i	$-0.265769\pi$
$547$	31.3972	1.34245	0.671223	+	0.741255i	$0.265769\pi$
$548$	8.70291	0.371770
$549$	0	0
$550$	3.50940	0.149641
$551$	10.3870	0.442502
$552$	0	0
$553$	17.9347	0.762661
$554$	−17.6413	−0.749505
$555$	0	0
$556$	18.6456	0.790749
$557$	−36.6635	−1.55348	−0.776742	−	0.629819i	$-0.783129\pi$
$557$	−36.6635	−1.55348	−0.776742	+	0.629819i	$0.783129\pi$
$558$	0	0
$559$	−34.8116	−1.47238
$560$	−1.45037	−0.0612894
$561$	0	0
$562$	17.8391	0.752496
$563$	4.98120	0.209933	0.104966	−	0.994476i	$-0.466527\pi$
$563$	4.98120	0.209933	0.104966	+	0.994476i	$0.466527\pi$
$564$	0	0
$565$	8.99568	0.378451
$566$	20.1550	0.847177
$567$	0	0
$568$	6.64388	0.278771
$569$	13.8298	0.579774	0.289887	−	0.957061i	$-0.406382\pi$
$569$	13.8298	0.579774	0.289887	+	0.957061i	$0.406382\pi$
$570$	0	0
$571$	−6.38702	−0.267289	−0.133644	−	0.991029i	$-0.542668\pi$
$571$	−6.38702	−0.267289	−0.133644	+	0.991029i	$0.542668\pi$
$572$	17.4058	0.727774
$573$	0	0
$574$	11.9109	0.497151
$575$	−4.15328	−0.173204
$576$	0	0
$577$	26.3117	1.09537	0.547686	−	0.836684i	$-0.315509\pi$
$577$	26.3117	1.09537	0.547686	+	0.836684i	$0.315509\pi$
$578$	−6.80149	−0.282905
$579$	0	0
$580$	3.25254	0.135054
$581$	−24.7868	−1.02833
$582$	0	0
$583$	−3.16107	−0.130918
$584$	7.40582	0.306455
$585$	0	0
$586$	10.6815	0.441248
$587$	−15.7362	−0.649502	−0.324751	−	0.945800i	$-0.605281\pi$
$587$	−15.7362	−0.649502	−0.324751	+	0.945800i	$0.605281\pi$
$588$	0	0
$589$	22.2261	0.915811
$590$	−8.21231	−0.338096
$591$	0	0
$592$	−4.70291	−0.193288
$593$	33.1275	1.36038	0.680192	−	0.733034i	$-0.261896\pi$
$593$	33.1275	1.36038	0.680192	+	0.733034i	$0.261896\pi$
$594$	0	0
$595$	−4.63178	−0.189884
$596$	3.25254	0.133229
$597$	0	0
$598$	−20.5993	−0.842369
$599$	−32.3528	−1.32190	−0.660950	−	0.750430i	$-0.729846\pi$
$599$	−32.3528	−1.32190	−0.660950	+	0.750430i	$0.729846\pi$
$600$	0	0
$601$	6.53684	0.266643	0.133322	−	0.991073i	$-0.457436\pi$
$601$	6.53684	0.266643	0.133322	+	0.991073i	$0.457436\pi$
$602$	10.1799	0.414900
$603$	0	0
$604$	−14.2354	−0.579232
$605$	1.31589	0.0534985
$606$	0	0
$607$	38.7398	1.57240	0.786201	−	0.617971i	$-0.212045\pi$
$607$	38.7398	1.57240	0.786201	+	0.617971i	$0.212045\pi$
$608$	3.19351	0.129514
$609$	0	0
$610$	0.549629	0.0222538
$611$	42.9864	1.73904
$612$	0	0
$613$	17.6030	0.710977	0.355489	−	0.934681i	$-0.384314\pi$
$613$	17.6030	0.710977	0.355489	+	0.934681i	$0.384314\pi$
$614$	−6.75957	−0.272794
$615$	0	0
$616$	−5.08993	−0.205079
$617$	−7.26533	−0.292491	−0.146246	−	0.989248i	$-0.546719\pi$
$617$	−7.26533	−0.292491	−0.146246	+	0.989248i	$0.546719\pi$
$618$	0	0
$619$	−16.8065	−0.675510	−0.337755	−	0.941234i	$-0.609667\pi$
$619$	−16.8065	−0.675510	−0.337755	+	0.941234i	$0.609667\pi$
$620$	6.95977	0.279511
$621$	0	0
$622$	12.1181	0.485890
$623$	20.4629	0.819829
$624$	0	0
$625$	1.00000	0.0400000
$626$	13.9427	0.557261
$627$	0	0
$628$	−2.58554	−0.103174
$629$	−15.0188	−0.598839
$630$	0	0
$631$	−6.65321	−0.264860	−0.132430	−	0.991192i	$-0.542278\pi$
$631$	−6.65321	−0.264860	−0.132430	+	0.991192i	$0.542278\pi$
$632$	−12.3656	−0.491877
$633$	0	0
$634$	−22.1584	−0.880024
$635$	15.3254	0.608169
$636$	0	0
$637$	−24.2851	−0.962212
$638$	11.4145	0.451903
$639$	0	0
$640$	1.00000	0.0395285
$641$	33.4421	1.32088	0.660441	−	0.750878i	$-0.270369\pi$
$641$	33.4421	1.32088	0.660441	+	0.750878i	$0.270369\pi$
$642$	0	0
$643$	−33.3029	−1.31334	−0.656670	−	0.754178i	$-0.728035\pi$
$643$	−33.3029	−1.31334	−0.656670	+	0.754178i	$0.728035\pi$
$644$	6.02380	0.237371
$645$	0	0
$646$	10.1985	0.401255
$647$	44.0645	1.73235	0.866177	−	0.499737i	$-0.166570\pi$
$647$	44.0645	1.73235	0.866177	+	0.499737i	$0.166570\pi$
$648$	0	0
$649$	−28.8203	−1.13130
$650$	4.95977	0.194538
$651$	0	0
$652$	−14.7029	−0.575810
$653$	−20.8835	−0.817233	−0.408616	−	0.912706i	$-0.633989\pi$
$653$	−20.8835	−0.817233	−0.408616	+	0.912706i	$0.633989\pi$
$654$	0	0
$655$	0.886946	0.0346558
$656$	−8.21231	−0.320637
$657$	0	0
$658$	−12.5704	−0.490044
$659$	−25.9676	−1.01155	−0.505776	−	0.862665i	$-0.668794\pi$
$659$	−25.9676	−1.01155	−0.505776	+	0.862665i	$0.668794\pi$
$660$	0	0
$661$	−14.0352	−0.545907	−0.272954	−	0.962027i	$-0.588001\pi$
$661$	−14.0352	−0.545907	−0.272954	+	0.962027i	$0.588001\pi$
$662$	16.0607	0.624218
$663$	0	0
$664$	17.0899	0.663218
$665$	−4.63178	−0.179613
$666$	0	0
$667$	−13.5087	−0.523060
$668$	22.0728	0.854023
$669$	0	0
$670$	−1.00000	−0.0386334
$671$	1.92887	0.0744631
$672$	0	0
$673$	19.8066	0.763490	0.381745	−	0.924268i	$-0.375323\pi$
$673$	19.8066	0.763490	0.381745	+	0.924268i	$0.375323\pi$
$674$	−12.7636	−0.491637
$675$	0	0
$676$	11.5993	0.446128
$677$	−21.8391	−0.839344	−0.419672	−	0.907676i	$-0.637855\pi$
$677$	−21.8391	−0.839344	−0.419672	+	0.907676i	$0.637855\pi$
$678$	0	0
$679$	5.08993	0.195334
$680$	3.19351	0.122466
$681$	0	0
$682$	24.4246	0.935267
$683$	13.5943	0.520173	0.260086	−	0.965585i	$-0.416249\pi$
$683$	13.5943	0.520173	0.260086	+	0.965585i	$0.416249\pi$
$684$	0	0
$685$	8.70291	0.332521
$686$	17.2542	0.658770
$687$	0	0
$688$	−7.01880	−0.267589
$689$	−4.46748	−0.170197
$690$	0	0
$691$	−44.4060	−1.68928	−0.844641	−	0.535332i	$-0.820186\pi$
$691$	−44.4060	−1.68928	−0.844641	+	0.535332i	$0.820186\pi$
$692$	6.77058	0.257379
$693$	0	0
$694$	−35.2929	−1.33970
$695$	18.6456	0.707267
$696$	0	0
$697$	−26.2261	−0.993385
$698$	−8.03760	−0.304228
$699$	0	0
$700$	−1.45037	−0.0548189
$701$	2.38634	0.0901308	0.0450654	−	0.998984i	$-0.485650\pi$
$701$	2.38634	0.0901308	0.0450654	+	0.998984i	$0.485650\pi$
$702$	0	0
$703$	−15.0188	−0.566445
$704$	3.50940	0.132265
$705$	0	0
$706$	−24.0943	−0.906799
$707$	−2.44259	−0.0918629
$708$	0	0
$709$	12.1543	0.456464	0.228232	−	0.973607i	$-0.426706\pi$
$709$	12.1543	0.456464	0.228232	+	0.973607i	$0.426706\pi$
$710$	6.64388	0.249340
$711$	0	0
$712$	−14.1087	−0.528747
$713$	−28.9059	−1.08253
$714$	0	0
$715$	17.4058	0.650941
$716$	13.6268	0.509256
$717$	0	0
$718$	−28.8700	−1.07742
$719$	1.62576	0.0606308	0.0303154	−	0.999540i	$-0.490349\pi$
$719$	1.62576	0.0606308	0.0303154	+	0.999540i	$0.490349\pi$
$720$	0	0
$721$	15.9814	0.595177
$722$	−8.80149	−0.327557
$723$	0	0
$724$	−21.2073	−0.788163
$725$	3.25254	0.120796
$726$	0	0
$727$	9.18072	0.340494	0.170247	−	0.985401i	$-0.445543\pi$
$727$	9.18072	0.340494	0.170247	+	0.985401i	$0.445543\pi$
$728$	−7.19351	−0.266609
$729$	0	0
$730$	7.40582	0.274102
$731$	−22.4146	−0.829035
$732$	0	0
$733$	−31.4220	−1.16060	−0.580299	−	0.814404i	$-0.697064\pi$
$733$	−31.4220	−1.16060	−0.580299	+	0.814404i	$0.697064\pi$
$734$	25.4873	0.940753
$735$	0	0
$736$	−4.15328	−0.153092
$737$	−3.50940	−0.129270
$738$	0	0
$739$	−29.0326	−1.06798	−0.533991	−	0.845490i	$-0.679308\pi$
$739$	−29.0326	−1.06798	−0.533991	+	0.845490i	$0.679308\pi$
$740$	−4.70291	−0.172882
$741$	0	0
$742$	1.30641	0.0479599
$743$	40.3452	1.48012	0.740060	−	0.672540i	$-0.234797\pi$
$743$	40.3452	1.48012	0.740060	+	0.672540i	$0.234797\pi$
$744$	0	0
$745$	3.25254	0.119164
$746$	−10.7613	−0.393998
$747$	0	0
$748$	11.2073	0.409780
$749$	−25.9900	−0.949654
$750$	0	0
$751$	10.0376	0.366277	0.183139	−	0.983087i	$-0.441374\pi$
$751$	10.0376	0.366277	0.183139	+	0.983087i	$0.441374\pi$
$752$	8.66700	0.316053
$753$	0	0
$754$	16.1319	0.587487
$755$	−14.2354	−0.518080
$756$	0	0
$757$	7.71825	0.280524	0.140262	−	0.990114i	$-0.455205\pi$
$757$	7.71825	0.280524	0.140262	+	0.990114i	$0.455205\pi$
$758$	−24.4471	−0.887957
$759$	0	0
$760$	3.19351	0.115841
$761$	37.4341	1.35699	0.678493	−	0.734607i	$-0.262634\pi$
$761$	37.4341	1.35699	0.678493	+	0.734607i	$0.262634\pi$
$762$	0	0
$763$	−8.46639	−0.306504
$764$	4.98120	0.180213
$765$	0	0
$766$	4.50440	0.162750
$767$	−40.7312	−1.47072
$768$	0	0
$769$	33.1645	1.19594	0.597970	−	0.801518i	$-0.295974\pi$
$769$	33.1645	1.19594	0.597970	+	0.801518i	$0.295974\pi$
$770$	−5.08993	−0.183428
$771$	0	0
$772$	2.92455	0.105257
$773$	54.1967	1.94932	0.974660	−	0.223690i	$-0.0718104\pi$
$773$	54.1967	1.94932	0.974660	+	0.223690i	$0.0718104\pi$
$774$	0	0
$775$	6.95977	0.250002
$776$	−3.50940	−0.125980
$777$	0	0
$778$	−19.8125	−0.710313
$779$	−26.2261	−0.939648
$780$	0	0
$781$	23.3160	0.834314
$782$	−13.2636	−0.474304
$783$	0	0
$784$	−4.89642	−0.174872
$785$	−2.58554	−0.0922818
$786$	0	0
$787$	−53.2021	−1.89645	−0.948224	−	0.317602i	$-0.897122\pi$
$787$	−53.2021	−1.89645	−0.948224	+	0.317602i	$0.897122\pi$
$788$	−7.91954	−0.282122
$789$	0	0
$790$	−12.3656	−0.439948
$791$	−13.0471	−0.463901
$792$	0	0
$793$	2.72603	0.0968042
$794$	−5.92023	−0.210101
$795$	0	0
$796$	9.08993	0.322184
$797$	17.8183	0.631158	0.315579	−	0.948899i	$-0.397801\pi$
$797$	17.8183	0.631158	0.315579	+	0.948899i	$0.397801\pi$
$798$	0	0
$799$	27.6782	0.979183
$800$	1.00000	0.0353553
$801$	0	0
$802$	−10.3784	−0.366473
$803$	25.9900	0.917167
$804$	0	0
$805$	6.02380	0.212311
$806$	34.5189	1.21588
$807$	0	0
$808$	1.68411	0.0592468
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	54.7819	1.92365	0.961826	−	0.273660i	$-0.0882344\pi$
$811$	54.7819	1.92365	0.961826	+	0.273660i	$0.0882344\pi$
$812$	−4.71739	−0.165548
$813$	0	0
$814$	−16.5044	−0.578479
$815$	−14.7029	−0.515021
$816$	0	0
$817$	−22.4146	−0.784188
$818$	0.0804569	0.00281311
$819$	0	0
$820$	−8.21231	−0.286786
$821$	33.6672	1.17499	0.587496	−	0.809227i	$-0.300114\pi$
$821$	33.6672	1.17499	0.587496	+	0.809227i	$0.300114\pi$
$822$	0	0
$823$	39.4434	1.37491	0.687456	−	0.726226i	$-0.258728\pi$
$823$	39.4434	1.37491	0.687456	+	0.726226i	$0.258728\pi$
$824$	−11.0188	−0.383858
$825$	0	0
$826$	11.9109	0.414433
$827$	40.2268	1.39882	0.699411	−	0.714719i	$-0.253446\pi$
$827$	40.2268	1.39882	0.699411	+	0.714719i	$0.253446\pi$
$828$	0	0
$829$	−19.2878	−0.669892	−0.334946	−	0.942237i	$-0.608718\pi$
$829$	−19.2878	−0.669892	−0.334946	+	0.942237i	$0.608718\pi$
$830$	17.0899	0.593200
$831$	0	0
$832$	4.95977	0.171949
$833$	−15.6368	−0.541782
$834$	0	0
$835$	22.0728	0.763862
$836$	11.2073	0.387613
$837$	0	0
$838$	−28.9145	−0.998836
$839$	−20.5079	−0.708010	−0.354005	−	0.935244i	$-0.615180\pi$
$839$	−20.5079	−0.708010	−0.354005	+	0.935244i	$0.615180\pi$
$840$	0	0
$841$	−18.4210	−0.635206
$842$	7.21731	0.248725
$843$	0	0
$844$	−8.41083	−0.289513
$845$	11.5993	0.399029
$846$	0	0
$847$	−1.90853	−0.0655778
$848$	−0.900743	−0.0309316
$849$	0	0
$850$	3.19351	0.109537
$851$	19.5325	0.669566
$852$	0	0
$853$	−54.3342	−1.86037	−0.930183	−	0.367096i	$-0.880352\pi$
$853$	−54.3342	−1.86037	−0.930183	+	0.367096i	$0.880352\pi$
$854$	−0.797166	−0.0272784
$855$	0	0
$856$	17.9195	0.612477
$857$	6.35526	0.217092	0.108546	−	0.994091i	$-0.465381\pi$
$857$	6.35526	0.217092	0.108546	+	0.994091i	$0.465381\pi$
$858$	0	0
$859$	−20.7602	−0.708331	−0.354165	−	0.935183i	$-0.615235\pi$
$859$	−20.7602	−0.708331	−0.354165	+	0.935183i	$0.615235\pi$
$860$	−7.01880	−0.239339
$861$	0	0
$862$	−14.8324	−0.505193
$863$	41.0916	1.39878	0.699388	−	0.714743i	$-0.253456\pi$
$863$	41.0916	1.39878	0.699388	+	0.714743i	$0.253456\pi$
$864$	0	0
$865$	6.77058	0.230207
$866$	1.83393	0.0623195
$867$	0	0
$868$	−10.0943	−0.342621
$869$	−43.3958	−1.47210
$870$	0	0
$871$	−4.95977	−0.168055
$872$	5.83739	0.197679
$873$	0	0
$874$	−13.2636	−0.448646
$875$	−1.45037	−0.0490315
$876$	0	0
$877$	−36.5803	−1.23523	−0.617614	−	0.786481i	$-0.711901\pi$
$877$	−36.5803	−1.23523	−0.617614	+	0.786481i	$0.711901\pi$
$878$	6.07113	0.204891
$879$	0	0
$880$	3.50940	0.118302
$881$	−18.8828	−0.636177	−0.318088	−	0.948061i	$-0.603041\pi$
$881$	−18.8828	−0.636177	−0.318088	+	0.948061i	$0.603041\pi$
$882$	0	0
$883$	−24.1181	−0.811637	−0.405819	−	0.913954i	$-0.633014\pi$
$883$	−24.1181	−0.811637	−0.405819	+	0.913954i	$0.633014\pi$
$884$	15.8391	0.532726
$885$	0	0
$886$	10.3870	0.348959
$887$	−40.1104	−1.34678	−0.673388	−	0.739289i	$-0.735162\pi$
$887$	−40.1104	−1.34678	−0.673388	+	0.739289i	$0.735162\pi$
$888$	0	0
$889$	−22.2275	−0.745486
$890$	−14.1087	−0.472926
$891$	0	0
$892$	−14.5513	−0.487214
$893$	27.6782	0.926215
$894$	0	0
$895$	13.6268	0.455493
$896$	−1.45037	−0.0484535
$897$	0	0
$898$	−18.3877	−0.613606
$899$	22.6369	0.754984
$900$	0	0
$901$	−2.87653	−0.0958312
$902$	−28.8203	−0.959611
$903$	0	0
$904$	8.99568	0.299192
$905$	−21.2073	−0.704955
$906$	0	0
$907$	−13.5423	−0.449663	−0.224832	−	0.974398i	$-0.572183\pi$
$907$	−13.5423	−0.449663	−0.224832	+	0.974398i	$0.572183\pi$
$908$	14.2690	0.473532
$909$	0	0
$910$	−7.19351	−0.238463
$911$	47.4931	1.57352	0.786759	−	0.617261i	$-0.211758\pi$
$911$	47.4931	1.57352	0.786759	+	0.617261i	$0.211758\pi$
$912$	0	0
$913$	59.9754	1.98490
$914$	−9.31157	−0.307999
$915$	0	0
$916$	−7.56843	−0.250068
$917$	−1.28640	−0.0424807
$918$	0	0
$919$	4.48365	0.147902	0.0739510	−	0.997262i	$-0.476439\pi$
$919$	4.48365	0.147902	0.0739510	+	0.997262i	$0.476439\pi$
$920$	−4.15328	−0.136930
$921$	0	0
$922$	24.2627	0.799049
$923$	32.9521	1.08463
$924$	0	0
$925$	−4.70291	−0.154631
$926$	10.8193	0.355543
$927$	0	0
$928$	3.25254	0.106770
$929$	−6.01516	−0.197351	−0.0986755	−	0.995120i	$-0.531461\pi$
$929$	−6.01516	−0.197351	−0.0986755	+	0.995120i	$0.531461\pi$
$930$	0	0
$931$	−15.6368	−0.512475
$932$	−23.9102	−0.783205
$933$	0	0
$934$	19.5232	0.638819
$935$	11.2073	0.366518
$936$	0	0
$937$	39.4017	1.28720	0.643598	−	0.765364i	$-0.277441\pi$
$937$	39.4017	1.28720	0.643598	+	0.765364i	$0.277441\pi$
$938$	1.45037	0.0473563
$939$	0	0
$940$	8.66700	0.282686
$941$	31.9900	1.04284	0.521422	−	0.853299i	$-0.325402\pi$
$941$	31.9900	1.04284	0.521422	+	0.853299i	$0.325402\pi$
$942$	0	0
$943$	34.1081	1.11071
$944$	−8.21231	−0.267288
$945$	0	0
$946$	−24.6318	−0.800848
$947$	17.7680	0.577381	0.288690	−	0.957422i	$-0.406780\pi$
$947$	17.7680	0.577381	0.288690	+	0.957422i	$0.406780\pi$
$948$	0	0
$949$	36.7312	1.19234
$950$	3.19351	0.103611
$951$	0	0
$952$	−4.63178	−0.150117
$953$	−41.0226	−1.32885	−0.664426	−	0.747354i	$-0.731324\pi$
$953$	−41.0226	−1.32885	−0.664426	+	0.747354i	$0.731324\pi$
$954$	0	0
$955$	4.98120	0.161188
$956$	15.3254	0.495658
$957$	0	0
$958$	23.8888	0.771812
$959$	−12.6225	−0.407600
$960$	0	0
$961$	17.4384	0.562530
$962$	−23.3254	−0.752040
$963$	0	0
$964$	17.9340	0.577616
$965$	2.92455	0.0941445
$966$	0	0
$967$	47.3081	1.52133	0.760663	−	0.649147i	$-0.224874\pi$
$967$	47.3081	1.52133	0.760663	+	0.649147i	$0.224874\pi$
$968$	1.31589	0.0422943
$969$	0	0
$970$	−3.50940	−0.112680
$971$	−0.494667	−0.0158746	−0.00793731	−	0.999968i	$-0.502527\pi$
$971$	−0.494667	−0.0158746	−0.00793731	+	0.999968i	$0.502527\pi$
$972$	0	0
$973$	−27.0430	−0.866959
$974$	−20.4263	−0.654501
$975$	0	0
$976$	0.549629	0.0175932
$977$	29.2760	0.936622	0.468311	−	0.883564i	$-0.344863\pi$
$977$	29.2760	0.936622	0.468311	+	0.883564i	$0.344863\pi$
$978$	0	0
$979$	−49.5132	−1.58245
$980$	−4.89642	−0.156410
$981$	0	0
$982$	8.04801	0.256822
$983$	−16.9626	−0.541021	−0.270511	−	0.962717i	$-0.587193\pi$
$983$	−16.9626	−0.541021	−0.270511	+	0.962717i	$0.587193\pi$
$984$	0	0
$985$	−7.91954	−0.252338
$986$	10.3870	0.330790
$987$	0	0
$988$	15.8391	0.503908
$989$	29.1511	0.926950
$990$	0	0
$991$	37.6996	1.19757	0.598784	−	0.800911i	$-0.295651\pi$
$991$	37.6996	1.19757	0.598784	+	0.800911i	$0.295651\pi$
$992$	6.95977	0.220973
$993$	0	0
$994$	−9.63610	−0.305638
$995$	9.08993	0.288170
$996$	0	0
$997$	27.4427	0.869120	0.434560	−	0.900643i	$-0.356904\pi$
$997$	27.4427	0.869120	0.434560	+	0.900643i	$0.356904\pi$
$998$	−35.5284	−1.12463
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.a.bu.1.2		4
3.2	odd	2		2010.2.a.r.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2010.2.a.r.1.2	✓	4	3.2	odd	2
6030.2.a.bu.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.45037 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.45037 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.50940 q^{11} +4.95977 q^{13} -1.45037 q^{14} +1.00000 q^{16} +3.19351 q^{17} +3.19351 q^{19} +1.00000 q^{20} +3.50940 q^{22} -4.15328 q^{23} +1.00000 q^{25} +4.95977 q^{26} -1.45037 q^{28} +3.25254 q^{29} +6.95977 q^{31} +1.00000 q^{32} +3.19351 q^{34} -1.45037 q^{35} -4.70291 q^{37} +3.19351 q^{38} +1.00000 q^{40} -8.21231 q^{41} -7.01880 q^{43} +3.50940 q^{44} -4.15328 q^{46} +8.66700 q^{47} -4.89642 q^{49} +1.00000 q^{50} +4.95977 q^{52} -0.900743 q^{53} +3.50940 q^{55} -1.45037 q^{56} +3.25254 q^{58} -8.21231 q^{59} +0.549629 q^{61} +6.95977 q^{62} +1.00000 q^{64} +4.95977 q^{65} -1.00000 q^{67} +3.19351 q^{68} -1.45037 q^{70} +6.64388 q^{71} +7.40582 q^{73} -4.70291 q^{74} +3.19351 q^{76} -5.08993 q^{77} -12.3656 q^{79} +1.00000 q^{80} -8.21231 q^{82} +17.0899 q^{83} +3.19351 q^{85} -7.01880 q^{86} +3.50940 q^{88} -14.1087 q^{89} -7.19351 q^{91} -4.15328 q^{92} +8.66700 q^{94} +3.19351 q^{95} -3.50940 q^{97} -4.89642 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 2 q^{13} + q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{19} + 4 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 2 q^{26} + q^{28} - 2 q^{29} + 10 q^{31} + 4 q^{32} + 2 q^{34} + q^{35} + 3 q^{37} + 2 q^{38} + 4 q^{40} - 6 q^{43} + 3 q^{44} + 12 q^{46} + 14 q^{47} + 13 q^{49} + 4 q^{50} + 2 q^{52} + 10 q^{53} + 3 q^{55} + q^{56} - 2 q^{58} + 9 q^{61} + 10 q^{62} + 4 q^{64} + 2 q^{65} - 4 q^{67} + 2 q^{68} + q^{70} + 9 q^{71} - 14 q^{73} + 3 q^{74} + 2 q^{76} + 23 q^{77} + 12 q^{79} + 4 q^{80} + 25 q^{83} + 2 q^{85} - 6 q^{86} + 3 q^{88} + 9 q^{89} - 18 q^{91} + 12 q^{92} + 14 q^{94} + 2 q^{95} - 3 q^{97} + 13 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 2 * q^13 + q^14 + 4 * q^16 + 2 * q^17 + 2 * q^19 + 4 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 2 * q^26 + q^28 - 2 * q^29 + 10 * q^31 + 4 * q^32 + 2 * q^34 + q^35 + 3 * q^37 + 2 * q^38 + 4 * q^40 - 6 * q^43 + 3 * q^44 + 12 * q^46 + 14 * q^47 + 13 * q^49 + 4 * q^50 + 2 * q^52 + 10 * q^53 + 3 * q^55 + q^56 - 2 * q^58 + 9 * q^61 + 10 * q^62 + 4 * q^64 + 2 * q^65 - 4 * q^67 + 2 * q^68 + q^70 + 9 * q^71 - 14 * q^73 + 3 * q^74 + 2 * q^76 + 23 * q^77 + 12 * q^79 + 4 * q^80 + 25 * q^83 + 2 * q^85 - 6 * q^86 + 3 * q^88 + 9 * q^89 - 18 * q^91 + 12 * q^92 + 14 * q^94 + 2 * q^95 - 3 * q^97 + 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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