LMFDB - Embedded newform 3042.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 234)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.16228 q^{5} -3.16228 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.16228 q^{5} -3.16228 q^{7} -1.00000 q^{8} -2.16228 q^{10} -5.16228 q^{11} +3.16228 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.16228 q^{19} +2.16228 q^{20} +5.16228 q^{22} -0.837722 q^{23} -0.324555 q^{25} -3.16228 q^{28} +8.16228 q^{29} +6.32456 q^{31} -1.00000 q^{32} -3.00000 q^{34} -6.83772 q^{35} +10.1623 q^{37} -1.16228 q^{38} -2.16228 q^{40} -3.00000 q^{41} +2.83772 q^{43} -5.16228 q^{44} +0.837722 q^{46} -11.1623 q^{47} +3.00000 q^{49} +0.324555 q^{50} -6.48683 q^{53} -11.1623 q^{55} +3.16228 q^{56} -8.16228 q^{58} -10.3246 q^{59} -6.16228 q^{61} -6.32456 q^{62} +1.00000 q^{64} -9.16228 q^{67} +3.00000 q^{68} +6.83772 q^{70} +0.837722 q^{71} -1.00000 q^{73} -10.1623 q^{74} +1.16228 q^{76} +16.3246 q^{77} -4.00000 q^{79} +2.16228 q^{80} +3.00000 q^{82} -15.4868 q^{83} +6.48683 q^{85} -2.83772 q^{86} +5.16228 q^{88} -12.0000 q^{89} -0.837722 q^{92} +11.1623 q^{94} +2.51317 q^{95} -4.00000 q^{97} -3.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{16} + 6 q^{17} - 4 q^{19} - 2 q^{20} + 4 q^{22} - 8 q^{23} + 12 q^{25} + 10 q^{29} - 2 q^{32} - 6 q^{34} - 20 q^{35} + 14 q^{37} + 4 q^{38} + 2 q^{40} - 6 q^{41} + 12 q^{43} - 4 q^{44} + 8 q^{46} - 16 q^{47} + 6 q^{49} - 12 q^{50} + 6 q^{53} - 16 q^{55} - 10 q^{58} - 8 q^{59} - 6 q^{61} + 2 q^{64} - 12 q^{67} + 6 q^{68} + 20 q^{70} + 8 q^{71} - 2 q^{73} - 14 q^{74} - 4 q^{76} + 20 q^{77} - 8 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} - 6 q^{85} - 12 q^{86} + 4 q^{88} - 24 q^{89} - 8 q^{92} + 16 q^{94} + 24 q^{95} - 8 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^16 + 6 * q^17 - 4 * q^19 - 2 * q^20 + 4 * q^22 - 8 * q^23 + 12 * q^25 + 10 * q^29 - 2 * q^32 - 6 * q^34 - 20 * q^35 + 14 * q^37 + 4 * q^38 + 2 * q^40 - 6 * q^41 + 12 * q^43 - 4 * q^44 + 8 * q^46 - 16 * q^47 + 6 * q^49 - 12 * q^50 + 6 * q^53 - 16 * q^55 - 10 * q^58 - 8 * q^59 - 6 * q^61 + 2 * q^64 - 12 * q^67 + 6 * q^68 + 20 * q^70 + 8 * q^71 - 2 * q^73 - 14 * q^74 - 4 * q^76 + 20 * q^77 - 8 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 - 6 * q^85 - 12 * q^86 + 4 * q^88 - 24 * q^89 - 8 * q^92 + 16 * q^94 + 24 * q^95 - 8 * q^97 - 6 * 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
$5$	2.16228	0.967000	0.483500	−	0.875344i	$-0.339365\pi$
$5$	2.16228	0.967000	0.483500	+	0.875344i	$0.339365\pi$
$6$	0	0
$7$	−3.16228	−1.19523	−0.597614	−	0.801784i	$-0.703885\pi$
$7$	−3.16228	−1.19523	−0.597614	+	0.801784i	$0.703885\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.16228	−0.683772
$11$	−5.16228	−1.55649	−0.778243	−	0.627964i	$-0.783889\pi$
$11$	−5.16228	−1.55649	−0.778243	+	0.627964i	$0.783889\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	3.16228	0.845154
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	1.16228	0.266645	0.133322	−	0.991073i	$-0.457435\pi$
$19$	1.16228	0.266645	0.133322	+	0.991073i	$0.457435\pi$
$20$	2.16228	0.483500
$21$	0	0
$22$	5.16228	1.10060
$23$	−0.837722	−0.174677	−0.0873386	−	0.996179i	$-0.527836\pi$
$23$	−0.837722	−0.174677	−0.0873386	+	0.996179i	$0.527836\pi$
$24$	0	0
$25$	−0.324555	−0.0649111
$26$	0	0
$27$	0	0
$28$	−3.16228	−0.597614
$29$	8.16228	1.51570	0.757848	−	0.652431i	$-0.226251\pi$
$29$	8.16228	1.51570	0.757848	+	0.652431i	$0.226251\pi$
$30$	0	0
$31$	6.32456	1.13592	0.567962	−	0.823055i	$-0.307732\pi$
$31$	6.32456	1.13592	0.567962	+	0.823055i	$0.307732\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	−6.83772	−1.15579
$36$	0	0
$37$	10.1623	1.67067	0.835334	−	0.549743i	$-0.185274\pi$
$37$	10.1623	1.67067	0.835334	+	0.549743i	$0.185274\pi$
$38$	−1.16228	−0.188546
$39$	0	0
$40$	−2.16228	−0.341886
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	2.83772	0.432749	0.216374	−	0.976310i	$-0.430577\pi$
$43$	2.83772	0.432749	0.216374	+	0.976310i	$0.430577\pi$
$44$	−5.16228	−0.778243
$45$	0	0
$46$	0.837722	0.123515
$47$	−11.1623	−1.62819	−0.814093	−	0.580735i	$-0.802765\pi$
$47$	−11.1623	−1.62819	−0.814093	+	0.580735i	$0.802765\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0.324555	0.0458991
$51$	0	0
$52$	0	0
$53$	−6.48683	−0.891035	−0.445518	−	0.895273i	$-0.646980\pi$
$53$	−6.48683	−0.891035	−0.445518	+	0.895273i	$0.646980\pi$
$54$	0	0
$55$	−11.1623	−1.50512
$56$	3.16228	0.422577
$57$	0	0
$58$	−8.16228	−1.07176
$59$	−10.3246	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$59$	−10.3246	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$60$	0	0
$61$	−6.16228	−0.788999	−0.394499	−	0.918896i	$-0.629082\pi$
$61$	−6.16228	−0.788999	−0.394499	+	0.918896i	$0.629082\pi$
$62$	−6.32456	−0.803219
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−9.16228	−1.11935	−0.559675	−	0.828712i	$-0.689074\pi$
$67$	−9.16228	−1.11935	−0.559675	+	0.828712i	$0.689074\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	6.83772	0.817264
$71$	0.837722	0.0994194	0.0497097	−	0.998764i	$-0.484170\pi$
$71$	0.837722	0.0994194	0.0497097	+	0.998764i	$0.484170\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−10.1623	−1.18134
$75$	0	0
$76$	1.16228	0.133322
$77$	16.3246	1.86036
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	2.16228	0.241750
$81$	0	0
$82$	3.00000	0.331295
$83$	−15.4868	−1.69990	−0.849950	−	0.526863i	$-0.823368\pi$
$83$	−15.4868	−1.69990	−0.849950	+	0.526863i	$0.823368\pi$
$84$	0	0
$85$	6.48683	0.703596
$86$	−2.83772	−0.305999
$87$	0	0
$88$	5.16228	0.550301
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	−0.837722	−0.0873386
$93$	0	0
$94$	11.1623	1.15130
$95$	2.51317	0.257845
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	−0.324555	−0.0324555
$101$	−14.1623	−1.40920	−0.704600	−	0.709605i	$-0.748873\pi$
$101$	−14.1623	−1.40920	−0.704600	+	0.709605i	$0.748873\pi$
$102$	0	0
$103$	15.8114	1.55794	0.778971	−	0.627060i	$-0.215742\pi$
$103$	15.8114	1.55794	0.778971	+	0.627060i	$0.215742\pi$
$104$	0	0
$105$	0	0
$106$	6.48683	0.630057
$107$	−13.8114	−1.33520	−0.667599	−	0.744521i	$-0.732678\pi$
$107$	−13.8114	−1.33520	−0.667599	+	0.744521i	$0.732678\pi$
$108$	0	0
$109$	−18.6491	−1.78626	−0.893130	−	0.449798i	$-0.851496\pi$
$109$	−18.6491	−1.78626	−0.893130	+	0.449798i	$0.851496\pi$
$110$	11.1623	1.06428
$111$	0	0
$112$	−3.16228	−0.298807
$113$	−19.3246	−1.81790	−0.908951	−	0.416904i	$-0.863115\pi$
$113$	−19.3246	−1.81790	−0.908951	+	0.416904i	$0.863115\pi$
$114$	0	0
$115$	−1.81139	−0.168913
$116$	8.16228	0.757848
$117$	0	0
$118$	10.3246	0.950452
$119$	−9.48683	−0.869657
$120$	0	0
$121$	15.6491	1.42265
$122$	6.16228	0.557906
$123$	0	0
$124$	6.32456	0.567962
$125$	−11.5132	−1.02977
$126$	0	0
$127$	−5.67544	−0.503614	−0.251807	−	0.967777i	$-0.581025\pi$
$127$	−5.67544	−0.503614	−0.251807	+	0.967777i	$0.581025\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	8.64911	0.755676	0.377838	−	0.925872i	$-0.376668\pi$
$131$	8.64911	0.755676	0.377838	+	0.925872i	$0.376668\pi$
$132$	0	0
$133$	−3.67544	−0.318701
$134$	9.16228	0.791500
$135$	0	0
$136$	−3.00000	−0.257248
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	6.32456	0.536442	0.268221	−	0.963357i	$-0.413564\pi$
$139$	6.32456	0.536442	0.268221	+	0.963357i	$0.413564\pi$
$140$	−6.83772	−0.577893
$141$	0	0
$142$	−0.837722	−0.0703001
$143$	0	0
$144$	0	0
$145$	17.6491	1.46568
$146$	1.00000	0.0827606
$147$	0	0
$148$	10.1623	0.835334
$149$	18.4868	1.51450	0.757250	−	0.653125i	$-0.226542\pi$
$149$	18.4868	1.51450	0.757250	+	0.653125i	$0.226542\pi$
$150$	0	0
$151$	7.16228	0.582858	0.291429	−	0.956592i	$-0.405869\pi$
$151$	7.16228	0.582858	0.291429	+	0.956592i	$0.405869\pi$
$152$	−1.16228	−0.0942732
$153$	0	0
$154$	−16.3246	−1.31547
$155$	13.6754	1.09844
$156$	0	0
$157$	8.48683	0.677323	0.338662	−	0.940908i	$-0.390026\pi$
$157$	8.48683	0.677323	0.338662	+	0.940908i	$0.390026\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−2.16228	−0.170943
$161$	2.64911	0.208779
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	15.4868	1.20201
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	0	0
$170$	−6.48683	−0.497517
$171$	0	0
$172$	2.83772	0.216374
$173$	2.64911	0.201408	0.100704	−	0.994916i	$-0.467890\pi$
$173$	2.64911	0.201408	0.100704	+	0.994916i	$0.467890\pi$
$174$	0	0
$175$	1.02633	0.0775836
$176$	−5.16228	−0.389121
$177$	0	0
$178$	12.0000	0.899438
$179$	3.48683	0.260618	0.130309	−	0.991473i	$-0.458403\pi$
$179$	3.48683	0.260618	0.130309	+	0.991473i	$0.458403\pi$
$180$	0	0
$181$	10.1623	0.755356	0.377678	−	0.925937i	$-0.376723\pi$
$181$	10.1623	0.755356	0.377678	+	0.925937i	$0.376723\pi$
$182$	0	0
$183$	0	0
$184$	0.837722	0.0617577
$185$	21.9737	1.61554
$186$	0	0
$187$	−15.4868	−1.13251
$188$	−11.1623	−0.814093
$189$	0	0
$190$	−2.51317	−0.182324
$191$	−1.67544	−0.121231	−0.0606155	−	0.998161i	$-0.519306\pi$
$191$	−1.67544	−0.121231	−0.0606155	+	0.998161i	$0.519306\pi$
$192$	0	0
$193$	17.9737	1.29377	0.646886	−	0.762586i	$-0.276071\pi$
$193$	17.9737	1.29377	0.646886	+	0.762586i	$0.276071\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	3.00000	0.214286
$197$	18.9737	1.35182	0.675909	−	0.736985i	$-0.263751\pi$
$197$	18.9737	1.35182	0.675909	+	0.736985i	$0.263751\pi$
$198$	0	0
$199$	−25.4868	−1.80671	−0.903357	−	0.428890i	$-0.858905\pi$
$199$	−25.4868	−1.80671	−0.903357	+	0.428890i	$0.858905\pi$
$200$	0.324555	0.0229495
$201$	0	0
$202$	14.1623	0.996454
$203$	−25.8114	−1.81160
$204$	0	0
$205$	−6.48683	−0.453060
$206$	−15.8114	−1.10163
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.48683	−0.445518
$213$	0	0
$214$	13.8114	0.944127
$215$	6.13594	0.418468
$216$	0	0
$217$	−20.0000	−1.35769
$218$	18.6491	1.26308
$219$	0	0
$220$	−11.1623	−0.752561
$221$	0	0
$222$	0	0
$223$	−14.3246	−0.959243	−0.479622	−	0.877475i	$-0.659226\pi$
$223$	−14.3246	−0.959243	−0.479622	+	0.877475i	$0.659226\pi$
$224$	3.16228	0.211289
$225$	0	0
$226$	19.3246	1.28545
$227$	−3.48683	−0.231429	−0.115715	−	0.993283i	$-0.536916\pi$
$227$	−3.48683	−0.231429	−0.115715	+	0.993283i	$0.536916\pi$
$228$	0	0
$229$	9.67544	0.639371	0.319686	−	0.947524i	$-0.396423\pi$
$229$	9.67544	0.639371	0.319686	+	0.947524i	$0.396423\pi$
$230$	1.81139	0.119439
$231$	0	0
$232$	−8.16228	−0.535880
$233$	−8.64911	−0.566622	−0.283311	−	0.959028i	$-0.591433\pi$
$233$	−8.64911	−0.566622	−0.283311	+	0.959028i	$0.591433\pi$
$234$	0	0
$235$	−24.1359	−1.57446
$236$	−10.3246	−0.672071
$237$	0	0
$238$	9.48683	0.614940
$239$	2.51317	0.162563	0.0812816	−	0.996691i	$-0.474099\pi$
$239$	2.51317	0.162563	0.0812816	+	0.996691i	$0.474099\pi$
$240$	0	0
$241$	0.675445	0.0435092	0.0217546	−	0.999763i	$-0.493075\pi$
$241$	0.675445	0.0435092	0.0217546	+	0.999763i	$0.493075\pi$
$242$	−15.6491	−1.00596
$243$	0	0
$244$	−6.16228	−0.394499
$245$	6.48683	0.414429
$246$	0	0
$247$	0	0
$248$	−6.32456	−0.401610
$249$	0	0
$250$	11.5132	0.728157
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	4.32456	0.271882
$254$	5.67544	0.356109
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.9737	−0.996410	−0.498205	−	0.867059i	$-0.666007\pi$
$257$	−15.9737	−0.996410	−0.498205	+	0.867059i	$0.666007\pi$
$258$	0	0
$259$	−32.1359	−1.99683
$260$	0	0
$261$	0	0
$262$	−8.64911	−0.534344
$263$	21.4868	1.32493	0.662467	−	0.749091i	$-0.269509\pi$
$263$	21.4868	1.32493	0.662467	+	0.749091i	$0.269509\pi$
$264$	0	0
$265$	−14.0263	−0.861631
$266$	3.67544	0.225356
$267$	0	0
$268$	−9.16228	−0.559675
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	6.32456	0.384189	0.192095	−	0.981376i	$-0.438472\pi$
$271$	6.32456	0.384189	0.192095	+	0.981376i	$0.438472\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−3.00000	−0.181237
$275$	1.67544	0.101033
$276$	0	0
$277$	−20.8114	−1.25044	−0.625218	−	0.780451i	$-0.714990\pi$
$277$	−20.8114	−1.25044	−0.625218	+	0.780451i	$0.714990\pi$
$278$	−6.32456	−0.379322
$279$	0	0
$280$	6.83772	0.408632
$281$	13.3246	0.794876	0.397438	−	0.917629i	$-0.369899\pi$
$281$	13.3246	0.794876	0.397438	+	0.917629i	$0.369899\pi$
$282$	0	0
$283$	−9.16228	−0.544641	−0.272320	−	0.962207i	$-0.587791\pi$
$283$	−9.16228	−0.544641	−0.272320	+	0.962207i	$0.587791\pi$
$284$	0.837722	0.0497097
$285$	0	0
$286$	0	0
$287$	9.48683	0.559990
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−17.6491	−1.03639
$291$	0	0
$292$	−1.00000	−0.0585206
$293$	−8.16228	−0.476845	−0.238423	−	0.971161i	$-0.576630\pi$
$293$	−8.16228	−0.476845	−0.238423	+	0.971161i	$0.576630\pi$
$294$	0	0
$295$	−22.3246	−1.29979
$296$	−10.1623	−0.590670
$297$	0	0
$298$	−18.4868	−1.07091
$299$	0	0
$300$	0	0
$301$	−8.97367	−0.517234
$302$	−7.16228	−0.412143
$303$	0	0
$304$	1.16228	0.0666612
$305$	−13.3246	−0.762962
$306$	0	0
$307$	−7.48683	−0.427296	−0.213648	−	0.976911i	$-0.568535\pi$
$307$	−7.48683	−0.427296	−0.213648	+	0.976911i	$0.568535\pi$
$308$	16.3246	0.930178
$309$	0	0
$310$	−13.6754	−0.776713
$311$	−2.51317	−0.142509	−0.0712543	−	0.997458i	$-0.522700\pi$
$311$	−2.51317	−0.142509	−0.0712543	+	0.997458i	$0.522700\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	−8.48683	−0.478940
$315$	0	0
$316$	−4.00000	−0.225018
$317$	6.48683	0.364337	0.182168	−	0.983267i	$-0.441688\pi$
$317$	6.48683	0.364337	0.182168	+	0.983267i	$0.441688\pi$
$318$	0	0
$319$	−42.1359	−2.35916
$320$	2.16228	0.120875
$321$	0	0
$322$	−2.64911	−0.147629
$323$	3.48683	0.194013
$324$	0	0
$325$	0	0
$326$	16.0000	0.886158
$327$	0	0
$328$	3.00000	0.165647
$329$	35.2982	1.94605
$330$	0	0
$331$	26.9737	1.48261	0.741303	−	0.671170i	$-0.234208\pi$
$331$	26.9737	1.48261	0.741303	+	0.671170i	$0.234208\pi$
$332$	−15.4868	−0.849950
$333$	0	0
$334$	12.0000	0.656611
$335$	−19.8114	−1.08241
$336$	0	0
$337$	11.0000	0.599208	0.299604	−	0.954064i	$-0.403145\pi$
$337$	11.0000	0.599208	0.299604	+	0.954064i	$0.403145\pi$
$338$	0	0
$339$	0	0
$340$	6.48683	0.351798
$341$	−32.6491	−1.76805
$342$	0	0
$343$	12.6491	0.682988
$344$	−2.83772	−0.153000
$345$	0	0
$346$	−2.64911	−0.142417
$347$	−3.48683	−0.187183	−0.0935915	−	0.995611i	$-0.529835\pi$
$347$	−3.48683	−0.187183	−0.0935915	+	0.995611i	$0.529835\pi$
$348$	0	0
$349$	−30.6491	−1.64061	−0.820305	−	0.571927i	$-0.806196\pi$
$349$	−30.6491	−1.64061	−0.820305	+	0.571927i	$0.806196\pi$
$350$	−1.02633	−0.0548599
$351$	0	0
$352$	5.16228	0.275150
$353$	−16.6754	−0.887544	−0.443772	−	0.896140i	$-0.646360\pi$
$353$	−16.6754	−0.887544	−0.443772	+	0.896140i	$0.646360\pi$
$354$	0	0
$355$	1.81139	0.0961385
$356$	−12.0000	−0.635999
$357$	0	0
$358$	−3.48683	−0.184285
$359$	28.4605	1.50209	0.751044	−	0.660252i	$-0.229551\pi$
$359$	28.4605	1.50209	0.751044	+	0.660252i	$0.229551\pi$
$360$	0	0
$361$	−17.6491	−0.928901
$362$	−10.1623	−0.534117
$363$	0	0
$364$	0	0
$365$	−2.16228	−0.113179
$366$	0	0
$367$	−6.51317	−0.339985	−0.169992	−	0.985445i	$-0.554374\pi$
$367$	−6.51317	−0.339985	−0.169992	+	0.985445i	$0.554374\pi$
$368$	−0.837722	−0.0436693
$369$	0	0
$370$	−21.9737	−1.14236
$371$	20.5132	1.06499
$372$	0	0
$373$	2.48683	0.128763	0.0643817	−	0.997925i	$-0.479492\pi$
$373$	2.48683	0.128763	0.0643817	+	0.997925i	$0.479492\pi$
$374$	15.4868	0.800805
$375$	0	0
$376$	11.1623	0.575651
$377$	0	0
$378$	0	0
$379$	30.3246	1.55767	0.778834	−	0.627230i	$-0.215811\pi$
$379$	30.3246	1.55767	0.778834	+	0.627230i	$0.215811\pi$
$380$	2.51317	0.128923
$381$	0	0
$382$	1.67544	0.0857232
$383$	6.97367	0.356338	0.178169	−	0.984000i	$-0.442983\pi$
$383$	6.97367	0.356338	0.178169	+	0.984000i	$0.442983\pi$
$384$	0	0
$385$	35.2982	1.79896
$386$	−17.9737	−0.914836
$387$	0	0
$388$	−4.00000	−0.203069
$389$	6.48683	0.328895	0.164448	−	0.986386i	$-0.447416\pi$
$389$	6.48683	0.328895	0.164448	+	0.986386i	$0.447416\pi$
$390$	0	0
$391$	−2.51317	−0.127096
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−18.9737	−0.955879
$395$	−8.64911	−0.435184
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	25.4868	1.27754
$399$	0	0
$400$	−0.324555	−0.0162278
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	0	0
$404$	−14.1623	−0.704600
$405$	0	0
$406$	25.8114	1.28100
$407$	−52.4605	−2.60037
$408$	0	0
$409$	27.3246	1.35111	0.675556	−	0.737309i	$-0.263904\pi$
$409$	27.3246	1.35111	0.675556	+	0.737309i	$0.263904\pi$
$410$	6.48683	0.320362
$411$	0	0
$412$	15.8114	0.778971
$413$	32.6491	1.60656
$414$	0	0
$415$	−33.4868	−1.64380
$416$	0	0
$417$	0	0
$418$	6.00000	0.293470
$419$	−6.97367	−0.340686	−0.170343	−	0.985385i	$-0.554488\pi$
$419$	−6.97367	−0.340686	−0.170343	+	0.985385i	$0.554488\pi$
$420$	0	0
$421$	−30.1623	−1.47002	−0.735010	−	0.678057i	$-0.762822\pi$
$421$	−30.1623	−1.47002	−0.735010	+	0.678057i	$0.762822\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	6.48683	0.315028
$425$	−0.973666	−0.0472297
$426$	0	0
$427$	19.4868	0.943034
$428$	−13.8114	−0.667599
$429$	0	0
$430$	−6.13594	−0.295901
$431$	−9.48683	−0.456965	−0.228482	−	0.973548i	$-0.573376\pi$
$431$	−9.48683	−0.456965	−0.228482	+	0.973548i	$0.573376\pi$
$432$	0	0
$433$	3.32456	0.159768	0.0798840	−	0.996804i	$-0.474545\pi$
$433$	3.32456	0.159768	0.0798840	+	0.996804i	$0.474545\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	−18.6491	−0.893130
$437$	−0.973666	−0.0465768
$438$	0	0
$439$	−6.51317	−0.310857	−0.155428	−	0.987847i	$-0.549676\pi$
$439$	−6.51317	−0.310857	−0.155428	+	0.987847i	$0.549676\pi$
$440$	11.1623	0.532141
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−25.9473	−1.23002
$446$	14.3246	0.678287
$447$	0	0
$448$	−3.16228	−0.149404
$449$	−32.6491	−1.54081	−0.770403	−	0.637557i	$-0.779945\pi$
$449$	−32.6491	−1.54081	−0.770403	+	0.637557i	$0.779945\pi$
$450$	0	0
$451$	15.4868	0.729246
$452$	−19.3246	−0.908951
$453$	0	0
$454$	3.48683	0.163645
$455$	0	0
$456$	0	0
$457$	9.32456	0.436184	0.218092	−	0.975928i	$-0.430017\pi$
$457$	9.32456	0.436184	0.218092	+	0.975928i	$0.430017\pi$
$458$	−9.67544	−0.452104
$459$	0	0
$460$	−1.81139	−0.0844564
$461$	0.486833	0.0226741	0.0113370	−	0.999936i	$-0.496391\pi$
$461$	0.486833	0.0226741	0.0113370	+	0.999936i	$0.496391\pi$
$462$	0	0
$463$	8.83772	0.410724	0.205362	−	0.978686i	$-0.434163\pi$
$463$	8.83772	0.410724	0.205362	+	0.978686i	$0.434163\pi$
$464$	8.16228	0.378924
$465$	0	0
$466$	8.64911	0.400662
$467$	−37.8114	−1.74970	−0.874851	−	0.484392i	$-0.839041\pi$
$467$	−37.8114	−1.74970	−0.874851	+	0.484392i	$0.839041\pi$
$468$	0	0
$469$	28.9737	1.33788
$470$	24.1359	1.11331
$471$	0	0
$472$	10.3246	0.475226
$473$	−14.6491	−0.673567
$474$	0	0
$475$	−0.377223	−0.0173082
$476$	−9.48683	−0.434828
$477$	0	0
$478$	−2.51317	−0.114950
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	0	0
$482$	−0.675445	−0.0307657
$483$	0	0
$484$	15.6491	0.711323
$485$	−8.64911	−0.392736
$486$	0	0
$487$	5.48683	0.248632	0.124316	−	0.992243i	$-0.460326\pi$
$487$	5.48683	0.248632	0.124316	+	0.992243i	$0.460326\pi$
$488$	6.16228	0.278953
$489$	0	0
$490$	−6.48683	−0.293045
$491$	25.8114	1.16485	0.582426	−	0.812884i	$-0.302104\pi$
$491$	25.8114	1.16485	0.582426	+	0.812884i	$0.302104\pi$
$492$	0	0
$493$	24.4868	1.10283
$494$	0	0
$495$	0	0
$496$	6.32456	0.283981
$497$	−2.64911	−0.118829
$498$	0	0
$499$	−12.6491	−0.566252	−0.283126	−	0.959083i	$-0.591371\pi$
$499$	−12.6491	−0.566252	−0.283126	+	0.959083i	$0.591371\pi$
$500$	−11.5132	−0.514884
$501$	0	0
$502$	−12.0000	−0.535586
$503$	31.8114	1.41840	0.709200	−	0.705007i	$-0.249056\pi$
$503$	31.8114	1.41840	0.709200	+	0.705007i	$0.249056\pi$
$504$	0	0
$505$	−30.6228	−1.36270
$506$	−4.32456	−0.192250
$507$	0	0
$508$	−5.67544	−0.251807
$509$	21.8377	0.967940	0.483970	−	0.875085i	$-0.339194\pi$
$509$	21.8377	0.967940	0.483970	+	0.875085i	$0.339194\pi$
$510$	0	0
$511$	3.16228	0.139891
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	15.9737	0.704568
$515$	34.1886	1.50653
$516$	0	0
$517$	57.6228	2.53425
$518$	32.1359	1.41197
$519$	0	0
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	1.16228	0.0508229	0.0254114	−	0.999677i	$-0.491910\pi$
$523$	1.16228	0.0508229	0.0254114	+	0.999677i	$0.491910\pi$
$524$	8.64911	0.377838
$525$	0	0
$526$	−21.4868	−0.936870
$527$	18.9737	0.826506
$528$	0	0
$529$	−22.2982	−0.969488
$530$	14.0263	0.609265
$531$	0	0
$532$	−3.67544	−0.159351
$533$	0	0
$534$	0	0
$535$	−29.8641	−1.29114
$536$	9.16228	0.395750
$537$	0	0
$538$	6.00000	0.258678
$539$	−15.4868	−0.667065
$540$	0	0
$541$	−34.4868	−1.48270	−0.741352	−	0.671116i	$-0.765815\pi$
$541$	−34.4868	−1.48270	−0.741352	+	0.671116i	$0.765815\pi$
$542$	−6.32456	−0.271663
$543$	0	0
$544$	−3.00000	−0.128624
$545$	−40.3246	−1.72731
$546$	0	0
$547$	−2.18861	−0.0935783	−0.0467891	−	0.998905i	$-0.514899\pi$
$547$	−2.18861	−0.0935783	−0.0467891	+	0.998905i	$0.514899\pi$
$548$	3.00000	0.128154
$549$	0	0
$550$	−1.67544	−0.0714412
$551$	9.48683	0.404153
$552$	0	0
$553$	12.6491	0.537895
$554$	20.8114	0.884191
$555$	0	0
$556$	6.32456	0.268221
$557$	−0.486833	−0.0206278	−0.0103139	−	0.999947i	$-0.503283\pi$
$557$	−0.486833	−0.0206278	−0.0103139	+	0.999947i	$0.503283\pi$
$558$	0	0
$559$	0	0
$560$	−6.83772	−0.288947
$561$	0	0
$562$	−13.3246	−0.562062
$563$	−15.3509	−0.646963	−0.323481	−	0.946235i	$-0.604853\pi$
$563$	−15.3509	−0.646963	−0.323481	+	0.946235i	$0.604853\pi$
$564$	0	0
$565$	−41.7851	−1.75791
$566$	9.16228	0.385119
$567$	0	0
$568$	−0.837722	−0.0351500
$569$	23.2982	0.976712	0.488356	−	0.872644i	$-0.337597\pi$
$569$	23.2982	0.976712	0.488356	+	0.872644i	$0.337597\pi$
$570$	0	0
$571$	8.13594	0.340479	0.170239	−	0.985403i	$-0.445546\pi$
$571$	8.13594	0.340479	0.170239	+	0.985403i	$0.445546\pi$
$572$	0	0
$573$	0	0
$574$	−9.48683	−0.395973
$575$	0.271887	0.0113385
$576$	0	0
$577$	25.6491	1.06779	0.533893	−	0.845552i	$-0.320728\pi$
$577$	25.6491	1.06779	0.533893	+	0.845552i	$0.320728\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	17.6491	0.732839
$581$	48.9737	2.03177
$582$	0	0
$583$	33.4868	1.38688
$584$	1.00000	0.0413803
$585$	0	0
$586$	8.16228	0.337181
$587$	−3.35089	−0.138306	−0.0691530	−	0.997606i	$-0.522030\pi$
$587$	−3.35089	−0.138306	−0.0691530	+	0.997606i	$0.522030\pi$
$588$	0	0
$589$	7.35089	0.302888
$590$	22.3246	0.919087
$591$	0	0
$592$	10.1623	0.417667
$593$	−13.3246	−0.547174	−0.273587	−	0.961847i	$-0.588210\pi$
$593$	−13.3246	−0.547174	−0.273587	+	0.961847i	$0.588210\pi$
$594$	0	0
$595$	−20.5132	−0.840958
$596$	18.4868	0.757250
$597$	0	0
$598$	0	0
$599$	−39.6228	−1.61894	−0.809471	−	0.587159i	$-0.800246\pi$
$599$	−39.6228	−1.61894	−0.809471	+	0.587159i	$0.800246\pi$
$600$	0	0
$601$	16.2982	0.664818	0.332409	−	0.943135i	$-0.392139\pi$
$601$	16.2982	0.664818	0.332409	+	0.943135i	$0.392139\pi$
$602$	8.97367	0.365739
$603$	0	0
$604$	7.16228	0.291429
$605$	33.8377	1.37570
$606$	0	0
$607$	−33.2982	−1.35153	−0.675767	−	0.737116i	$-0.736187\pi$
$607$	−33.2982	−1.35153	−0.675767	+	0.737116i	$0.736187\pi$
$608$	−1.16228	−0.0471366
$609$	0	0
$610$	13.3246	0.539495
$611$	0	0
$612$	0	0
$613$	1.51317	0.0611162	0.0305581	−	0.999533i	$-0.490272\pi$
$613$	1.51317	0.0611162	0.0305581	+	0.999533i	$0.490272\pi$
$614$	7.48683	0.302144
$615$	0	0
$616$	−16.3246	−0.657735
$617$	−36.6228	−1.47438	−0.737189	−	0.675687i	$-0.763847\pi$
$617$	−36.6228	−1.47438	−0.737189	+	0.675687i	$0.763847\pi$
$618$	0	0
$619$	−0.649111	−0.0260900	−0.0130450	−	0.999915i	$-0.504152\pi$
$619$	−0.649111	−0.0260900	−0.0130450	+	0.999915i	$0.504152\pi$
$620$	13.6754	0.549219
$621$	0	0
$622$	2.51317	0.100769
$623$	37.9473	1.52033
$624$	0	0
$625$	−23.2719	−0.930875
$626$	4.00000	0.159872
$627$	0	0
$628$	8.48683	0.338662
$629$	30.4868	1.21559
$630$	0	0
$631$	25.2982	1.00711	0.503553	−	0.863964i	$-0.332026\pi$
$631$	25.2982	1.00711	0.503553	+	0.863964i	$0.332026\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−6.48683	−0.257625
$635$	−12.2719	−0.486995
$636$	0	0
$637$	0	0
$638$	42.1359	1.66818
$639$	0	0
$640$	−2.16228	−0.0854715
$641$	−41.6491	−1.64504	−0.822520	−	0.568735i	$-0.807433\pi$
$641$	−41.6491	−1.64504	−0.822520	+	0.568735i	$0.807433\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	2.64911	0.104390
$645$	0	0
$646$	−3.48683	−0.137188
$647$	15.3509	0.603506	0.301753	−	0.953386i	$-0.402428\pi$
$647$	15.3509	0.603506	0.301753	+	0.953386i	$0.402428\pi$
$648$	0	0
$649$	53.2982	2.09214
$650$	0	0
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	18.7018	0.730739
$656$	−3.00000	−0.117130
$657$	0	0
$658$	−35.2982	−1.37607
$659$	42.9737	1.67402	0.837008	−	0.547190i	$-0.184303\pi$
$659$	42.9737	1.67402	0.837008	+	0.547190i	$0.184303\pi$
$660$	0	0
$661$	7.51317	0.292228	0.146114	−	0.989268i	$-0.453323\pi$
$661$	7.51317	0.292228	0.146114	+	0.989268i	$0.453323\pi$
$662$	−26.9737	−1.04836
$663$	0	0
$664$	15.4868	0.601006
$665$	−7.94733	−0.308184
$666$	0	0
$667$	−6.83772	−0.264758
$668$	−12.0000	−0.464294
$669$	0	0
$670$	19.8114	0.765381
$671$	31.8114	1.22807
$672$	0	0
$673$	27.3246	1.05328	0.526642	−	0.850087i	$-0.323451\pi$
$673$	27.3246	1.05328	0.526642	+	0.850087i	$0.323451\pi$
$674$	−11.0000	−0.423704
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	12.6491	0.485428
$680$	−6.48683	−0.248759
$681$	0	0
$682$	32.6491	1.25020
$683$	−27.6228	−1.05696	−0.528478	−	0.848947i	$-0.677237\pi$
$683$	−27.6228	−1.05696	−0.528478	+	0.848947i	$0.677237\pi$
$684$	0	0
$685$	6.48683	0.247849
$686$	−12.6491	−0.482945
$687$	0	0
$688$	2.83772	0.108187
$689$	0	0
$690$	0	0
$691$	2.83772	0.107952	0.0539760	−	0.998542i	$-0.482811\pi$
$691$	2.83772	0.107952	0.0539760	+	0.998542i	$0.482811\pi$
$692$	2.64911	0.100704
$693$	0	0
$694$	3.48683	0.132358
$695$	13.6754	0.518739
$696$	0	0
$697$	−9.00000	−0.340899
$698$	30.6491	1.16009
$699$	0	0
$700$	1.02633	0.0387918
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	11.8114	0.445475
$704$	−5.16228	−0.194561
$705$	0	0
$706$	16.6754	0.627589
$707$	44.7851	1.68432
$708$	0	0
$709$	−11.4605	−0.430408	−0.215204	−	0.976569i	$-0.569042\pi$
$709$	−11.4605	−0.430408	−0.215204	+	0.976569i	$0.569042\pi$
$710$	−1.81139	−0.0679802
$711$	0	0
$712$	12.0000	0.449719
$713$	−5.29822	−0.198420
$714$	0	0
$715$	0	0
$716$	3.48683	0.130309
$717$	0	0
$718$	−28.4605	−1.06214
$719$	25.6754	0.957533	0.478766	−	0.877942i	$-0.341084\pi$
$719$	25.6754	0.957533	0.478766	+	0.877942i	$0.341084\pi$
$720$	0	0
$721$	−50.0000	−1.86210
$722$	17.6491	0.656832
$723$	0	0
$724$	10.1623	0.377678
$725$	−2.64911	−0.0983855
$726$	0	0
$727$	−15.1623	−0.562338	−0.281169	−	0.959658i	$-0.590722\pi$
$727$	−15.1623	−0.562338	−0.281169	+	0.959658i	$0.590722\pi$
$728$	0	0
$729$	0	0
$730$	2.16228	0.0800295
$731$	8.51317	0.314871
$732$	0	0
$733$	−35.4605	−1.30976	−0.654882	−	0.755731i	$-0.727282\pi$
$733$	−35.4605	−1.30976	−0.654882	+	0.755731i	$0.727282\pi$
$734$	6.51317	0.240405
$735$	0	0
$736$	0.837722	0.0308789
$737$	47.2982	1.74225
$738$	0	0
$739$	23.6228	0.868978	0.434489	−	0.900677i	$-0.356929\pi$
$739$	23.6228	0.868978	0.434489	+	0.900677i	$0.356929\pi$
$740$	21.9737	0.807768
$741$	0	0
$742$	−20.5132	−0.753062
$743$	10.3246	0.378771	0.189386	−	0.981903i	$-0.439350\pi$
$743$	10.3246	0.378771	0.189386	+	0.981903i	$0.439350\pi$
$744$	0	0
$745$	39.9737	1.46452
$746$	−2.48683	−0.0910494
$747$	0	0
$748$	−15.4868	−0.566255
$749$	43.6754	1.59587
$750$	0	0
$751$	−30.7851	−1.12336	−0.561681	−	0.827354i	$-0.689845\pi$
$751$	−30.7851	−1.12336	−0.561681	+	0.827354i	$0.689845\pi$
$752$	−11.1623	−0.407046
$753$	0	0
$754$	0	0
$755$	15.4868	0.563624
$756$	0	0
$757$	−1.35089	−0.0490989	−0.0245495	−	0.999699i	$-0.507815\pi$
$757$	−1.35089	−0.0490989	−0.0245495	+	0.999699i	$0.507815\pi$
$758$	−30.3246	−1.10144
$759$	0	0
$760$	−2.51317	−0.0911621
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	58.9737	2.13499
$764$	−1.67544	−0.0606155
$765$	0	0
$766$	−6.97367	−0.251969
$767$	0	0
$768$	0	0
$769$	28.6491	1.03311	0.516557	−	0.856253i	$-0.327214\pi$
$769$	28.6491	1.03311	0.516557	+	0.856253i	$0.327214\pi$
$770$	−35.2982	−1.27206
$771$	0	0
$772$	17.9737	0.646886
$773$	2.64911	0.0952819	0.0476409	−	0.998865i	$-0.484830\pi$
$773$	2.64911	0.0952819	0.0476409	+	0.998865i	$0.484830\pi$
$774$	0	0
$775$	−2.05267	−0.0737340
$776$	4.00000	0.143592
$777$	0	0
$778$	−6.48683	−0.232564
$779$	−3.48683	−0.124929
$780$	0	0
$781$	−4.32456	−0.154745
$782$	2.51317	0.0898707
$783$	0	0
$784$	3.00000	0.107143
$785$	18.3509	0.654971
$786$	0	0
$787$	−46.9737	−1.67443	−0.837215	−	0.546874i	$-0.815818\pi$
$787$	−46.9737	−1.67443	−0.837215	+	0.546874i	$0.815818\pi$
$788$	18.9737	0.675909
$789$	0	0
$790$	8.64911	0.307722
$791$	61.1096	2.17281
$792$	0	0
$793$	0	0
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−25.4868	−0.903357
$797$	−18.9737	−0.672082	−0.336041	−	0.941847i	$-0.609088\pi$
$797$	−18.9737	−0.672082	−0.336041	+	0.941847i	$0.609088\pi$
$798$	0	0
$799$	−33.4868	−1.18468
$800$	0.324555	0.0114748
$801$	0	0
$802$	−15.0000	−0.529668
$803$	5.16228	0.182173
$804$	0	0
$805$	5.72811	0.201889
$806$	0	0
$807$	0	0
$808$	14.1623	0.498227
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	0	0
$811$	38.9737	1.36855	0.684275	−	0.729224i	$-0.260119\pi$
$811$	38.9737	1.36855	0.684275	+	0.729224i	$0.260119\pi$
$812$	−25.8114	−0.905802
$813$	0	0
$814$	52.4605	1.83874
$815$	−34.5964	−1.21186
$816$	0	0
$817$	3.29822	0.115390
$818$	−27.3246	−0.955381
$819$	0	0
$820$	−6.48683	−0.226530
$821$	14.6491	0.511257	0.255629	−	0.966775i	$-0.417718\pi$
$821$	14.6491	0.511257	0.255629	+	0.966775i	$0.417718\pi$
$822$	0	0
$823$	2.70178	0.0941781	0.0470890	−	0.998891i	$-0.485006\pi$
$823$	2.70178	0.0941781	0.0470890	+	0.998891i	$0.485006\pi$
$824$	−15.8114	−0.550816
$825$	0	0
$826$	−32.6491	−1.13601
$827$	−30.9737	−1.07706	−0.538530	−	0.842606i	$-0.681020\pi$
$827$	−30.9737	−1.07706	−0.538530	+	0.842606i	$0.681020\pi$
$828$	0	0
$829$	5.83772	0.202752	0.101376	−	0.994848i	$-0.467675\pi$
$829$	5.83772	0.202752	0.101376	+	0.994848i	$0.467675\pi$
$830$	33.4868	1.16234
$831$	0	0
$832$	0	0
$833$	9.00000	0.311832
$834$	0	0
$835$	−25.9473	−0.897944
$836$	−6.00000	−0.207514
$837$	0	0
$838$	6.97367	0.240901
$839$	20.6491	0.712886	0.356443	−	0.934317i	$-0.383989\pi$
$839$	20.6491	0.712886	0.356443	+	0.934317i	$0.383989\pi$
$840$	0	0
$841$	37.6228	1.29734
$842$	30.1623	1.03946
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	−49.4868	−1.70039
$848$	−6.48683	−0.222759
$849$	0	0
$850$	0.973666	0.0333965
$851$	−8.51317	−0.291828
$852$	0	0
$853$	10.8641	0.371978	0.185989	−	0.982552i	$-0.440451\pi$
$853$	10.8641	0.371978	0.185989	+	0.982552i	$0.440451\pi$
$854$	−19.4868	−0.666826
$855$	0	0
$856$	13.8114	0.472064
$857$	0.350889	0.0119862	0.00599308	−	0.999982i	$-0.498092\pi$
$857$	0.350889	0.0119862	0.00599308	+	0.999982i	$0.498092\pi$
$858$	0	0
$859$	47.4868	1.62023	0.810115	−	0.586271i	$-0.199405\pi$
$859$	47.4868	1.62023	0.810115	+	0.586271i	$0.199405\pi$
$860$	6.13594	0.209234
$861$	0	0
$862$	9.48683	0.323123
$863$	−16.1886	−0.551067	−0.275533	−	0.961292i	$-0.588854\pi$
$863$	−16.1886	−0.551067	−0.275533	+	0.961292i	$0.588854\pi$
$864$	0	0
$865$	5.72811	0.194762
$866$	−3.32456	−0.112973
$867$	0	0
$868$	−20.0000	−0.678844
$869$	20.6491	0.700473
$870$	0	0
$871$	0	0
$872$	18.6491	0.631539
$873$	0	0
$874$	0.973666	0.0329347
$875$	36.4078	1.23081
$876$	0	0
$877$	−0.864056	−0.0291771	−0.0145886	−	0.999894i	$-0.504644\pi$
$877$	−0.864056	−0.0291771	−0.0145886	+	0.999894i	$0.504644\pi$
$878$	6.51317	0.219809
$879$	0	0
$880$	−11.1623	−0.376280
$881$	27.9737	0.942457	0.471228	−	0.882011i	$-0.343811\pi$
$881$	27.9737	0.942457	0.471228	+	0.882011i	$0.343811\pi$
$882$	0	0
$883$	−36.6491	−1.23334	−0.616670	−	0.787221i	$-0.711519\pi$
$883$	−36.6491	−1.23334	−0.616670	+	0.787221i	$0.711519\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.6754	−0.862097	−0.431049	−	0.902329i	$-0.641856\pi$
$887$	−25.6754	−0.862097	−0.431049	+	0.902329i	$0.641856\pi$
$888$	0	0
$889$	17.9473	0.601934
$890$	25.9473	0.869757
$891$	0	0
$892$	−14.3246	−0.479622
$893$	−12.9737	−0.434147
$894$	0	0
$895$	7.53950	0.252018
$896$	3.16228	0.105644
$897$	0	0
$898$	32.6491	1.08951
$899$	51.6228	1.72172
$900$	0	0
$901$	−19.4605	−0.648323
$902$	−15.4868	−0.515655
$903$	0	0
$904$	19.3246	0.642725
$905$	21.9737	0.730429
$906$	0	0
$907$	−52.2719	−1.73566	−0.867830	−	0.496862i	$-0.834486\pi$
$907$	−52.2719	−1.73566	−0.867830	+	0.496862i	$0.834486\pi$
$908$	−3.48683	−0.115715
$909$	0	0
$910$	0	0
$911$	−25.9473	−0.859673	−0.429837	−	0.902907i	$-0.641429\pi$
$911$	−25.9473	−0.859673	−0.429837	+	0.902907i	$0.641429\pi$
$912$	0	0
$913$	79.9473	2.64587
$914$	−9.32456	−0.308429
$915$	0	0
$916$	9.67544	0.319686
$917$	−27.3509	−0.903206
$918$	0	0
$919$	28.6491	0.945047	0.472523	−	0.881318i	$-0.343343\pi$
$919$	28.6491	0.945047	0.472523	+	0.881318i	$0.343343\pi$
$920$	1.81139	0.0597197
$921$	0	0
$922$	−0.486833	−0.0160330
$923$	0	0
$924$	0	0
$925$	−3.29822	−0.108445
$926$	−8.83772	−0.290426
$927$	0	0
$928$	−8.16228	−0.267940
$929$	46.9473	1.54029	0.770146	−	0.637868i	$-0.220183\pi$
$929$	46.9473	1.54029	0.770146	+	0.637868i	$0.220183\pi$
$930$	0	0
$931$	3.48683	0.114276
$932$	−8.64911	−0.283311
$933$	0	0
$934$	37.8114	1.23723
$935$	−33.4868	−1.09514
$936$	0	0
$937$	−36.2982	−1.18581	−0.592906	−	0.805272i	$-0.702019\pi$
$937$	−36.2982	−1.18581	−0.592906	+	0.805272i	$0.702019\pi$
$938$	−28.9737	−0.946024
$939$	0	0
$940$	−24.1359	−0.787228
$941$	−52.5964	−1.71460	−0.857298	−	0.514821i	$-0.827858\pi$
$941$	−52.5964	−1.71460	−0.857298	+	0.514821i	$0.827858\pi$
$942$	0	0
$943$	2.51317	0.0818400
$944$	−10.3246	−0.336036
$945$	0	0
$946$	14.6491	0.476284
$947$	18.9737	0.616561	0.308281	−	0.951295i	$-0.400246\pi$
$947$	18.9737	0.616561	0.308281	+	0.951295i	$0.400246\pi$
$948$	0	0
$949$	0	0
$950$	0.377223	0.0122387
$951$	0	0
$952$	9.48683	0.307470
$953$	−23.2982	−0.754703	−0.377352	−	0.926070i	$-0.623165\pi$
$953$	−23.2982	−0.754703	−0.377352	+	0.926070i	$0.623165\pi$
$954$	0	0
$955$	−3.62278	−0.117230
$956$	2.51317	0.0812816
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−9.48683	−0.306346
$960$	0	0
$961$	9.00000	0.290323
$962$	0	0
$963$	0	0
$964$	0.675445	0.0217546
$965$	38.8641	1.25108
$966$	0	0
$967$	53.4868	1.72002	0.860010	−	0.510277i	$-0.170457\pi$
$967$	53.4868	1.72002	0.860010	+	0.510277i	$0.170457\pi$
$968$	−15.6491	−0.502981
$969$	0	0
$970$	8.64911	0.277706
$971$	−18.9737	−0.608894	−0.304447	−	0.952529i	$-0.598472\pi$
$971$	−18.9737	−0.608894	−0.304447	+	0.952529i	$0.598472\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	−5.48683	−0.175809
$975$	0	0
$976$	−6.16228	−0.197250
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	0	0
$979$	61.9473	1.97985
$980$	6.48683	0.207214
$981$	0	0
$982$	−25.8114	−0.823674
$983$	39.6228	1.26377	0.631885	−	0.775062i	$-0.282281\pi$
$983$	39.6228	1.26377	0.631885	+	0.775062i	$0.282281\pi$
$984$	0	0
$985$	41.0263	1.30721
$986$	−24.4868	−0.779820
$987$	0	0
$988$	0	0
$989$	−2.37722	−0.0755913
$990$	0	0
$991$	−42.7851	−1.35911	−0.679556	−	0.733624i	$-0.737828\pi$
$991$	−42.7851	−1.35911	−0.679556	+	0.733624i	$0.737828\pi$
$992$	−6.32456	−0.200805
$993$	0	0
$994$	2.64911	0.0840247
$995$	−55.1096	−1.74709
$996$	0	0
$997$	−2.53950	−0.0804268	−0.0402134	−	0.999191i	$-0.512804\pi$
$997$	−2.53950	−0.0804268	−0.0402134	+	0.999191i	$0.512804\pi$
$998$	12.6491	0.400401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.r.1.2		2
3.2	odd	2		3042.2.a.w.1.1		2
13.3	even	3		234.2.h.e.217.2	yes	4
13.5	odd	4		3042.2.b.j.1351.3		4
13.8	odd	4		3042.2.b.j.1351.2		4
13.9	even	3		234.2.h.e.55.2	yes	4
13.12	even	2		3042.2.a.x.1.1		2
39.5	even	4		3042.2.b.k.1351.2		4
39.8	even	4		3042.2.b.k.1351.3		4
39.29	odd	6		234.2.h.d.217.1	yes	4
39.35	odd	6		234.2.h.d.55.1	✓	4
39.38	odd	2		3042.2.a.q.1.2		2
52.3	odd	6		1872.2.t.n.1153.2		4
52.35	odd	6		1872.2.t.n.289.2		4
156.35	even	6		1872.2.t.p.289.1		4
156.107	even	6		1872.2.t.p.1153.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.h.d.55.1	✓	4	39.35	odd	6
234.2.h.d.217.1	yes	4	39.29	odd	6
234.2.h.e.55.2	yes	4	13.9	even	3
234.2.h.e.217.2	yes	4	13.3	even	3
1872.2.t.n.289.2		4	52.35	odd	6
1872.2.t.n.1153.2		4	52.3	odd	6
1872.2.t.p.289.1		4	156.35	even	6
1872.2.t.p.1153.1		4	156.107	even	6
3042.2.a.q.1.2		2	39.38	odd	2
3042.2.a.r.1.2		2	1.1	even	1	trivial
3042.2.a.w.1.1		2	3.2	odd	2
3042.2.a.x.1.1		2	13.12	even	2
3042.2.b.j.1351.2		4	13.8	odd	4
3042.2.b.j.1351.3		4	13.5	odd	4
3042.2.b.k.1351.2		4	39.5	even	4
3042.2.b.k.1351.3		4	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.16228 q^{5} -3.16228 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.16228 q^{5} -3.16228 q^{7} -1.00000 q^{8} -2.16228 q^{10} -5.16228 q^{11} +3.16228 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.16228 q^{19} +2.16228 q^{20} +5.16228 q^{22} -0.837722 q^{23} -0.324555 q^{25} -3.16228 q^{28} +8.16228 q^{29} +6.32456 q^{31} -1.00000 q^{32} -3.00000 q^{34} -6.83772 q^{35} +10.1623 q^{37} -1.16228 q^{38} -2.16228 q^{40} -3.00000 q^{41} +2.83772 q^{43} -5.16228 q^{44} +0.837722 q^{46} -11.1623 q^{47} +3.00000 q^{49} +0.324555 q^{50} -6.48683 q^{53} -11.1623 q^{55} +3.16228 q^{56} -8.16228 q^{58} -10.3246 q^{59} -6.16228 q^{61} -6.32456 q^{62} +1.00000 q^{64} -9.16228 q^{67} +3.00000 q^{68} +6.83772 q^{70} +0.837722 q^{71} -1.00000 q^{73} -10.1623 q^{74} +1.16228 q^{76} +16.3246 q^{77} -4.00000 q^{79} +2.16228 q^{80} +3.00000 q^{82} -15.4868 q^{83} +6.48683 q^{85} -2.83772 q^{86} +5.16228 q^{88} -12.0000 q^{89} -0.837722 q^{92} +11.1623 q^{94} +2.51317 q^{95} -4.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{16} + 6 q^{17} - 4 q^{19} - 2 q^{20} + 4 q^{22} - 8 q^{23} + 12 q^{25} + 10 q^{29} - 2 q^{32} - 6 q^{34} - 20 q^{35} + 14 q^{37} + 4 q^{38} + 2 q^{40} - 6 q^{41} + 12 q^{43} - 4 q^{44} + 8 q^{46} - 16 q^{47} + 6 q^{49} - 12 q^{50} + 6 q^{53} - 16 q^{55} - 10 q^{58} - 8 q^{59} - 6 q^{61} + 2 q^{64} - 12 q^{67} + 6 q^{68} + 20 q^{70} + 8 q^{71} - 2 q^{73} - 14 q^{74} - 4 q^{76} + 20 q^{77} - 8 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} - 6 q^{85} - 12 q^{86} + 4 q^{88} - 24 q^{89} - 8 q^{92} + 16 q^{94} + 24 q^{95} - 8 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^16 + 6 * q^17 - 4 * q^19 - 2 * q^20 + 4 * q^22 - 8 * q^23 + 12 * q^25 + 10 * q^29 - 2 * q^32 - 6 * q^34 - 20 * q^35 + 14 * q^37 + 4 * q^38 + 2 * q^40 - 6 * q^41 + 12 * q^43 - 4 * q^44 + 8 * q^46 - 16 * q^47 + 6 * q^49 - 12 * q^50 + 6 * q^53 - 16 * q^55 - 10 * q^58 - 8 * q^59 - 6 * q^61 + 2 * q^64 - 12 * q^67 + 6 * q^68 + 20 * q^70 + 8 * q^71 - 2 * q^73 - 14 * q^74 - 4 * q^76 + 20 * q^77 - 8 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 - 6 * q^85 - 12 * q^86 + 4 * q^88 - 24 * q^89 - 8 * q^92 + 16 * q^94 + 24 * q^95 - 8 * q^97 - 6 * q^98

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