LMFDB - Embedded newform 5994.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5994,2,Mod(1,5994)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5994, 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("5994.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5994 = 2 \cdot 3^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5994.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:	$47.8623309716$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 26x^{8} + 49x^{7} + 236x^{6} - 420x^{5} - 860x^{4} + 1461x^{3} + 993x^{2} - 1638x + 99 $ x^10 - 2x^9 - 26x^8 + 49x^7 + 236x^6 - 420x^5 - 860x^4 + 1461x^3 + 993x^2 - 1638*x + 99
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 666)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.14737$ of defining polynomial
Character	$\chi$	$=$	5994.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.49123 q^{5} +2.86107 q^{7} -1.00000 q^{8} +3.49123 q^{10} +5.43746 q^{11} +3.62848 q^{13} -2.86107 q^{14} +1.00000 q^{16} -4.84071 q^{17} +7.31418 q^{19} -3.49123 q^{20} -5.43746 q^{22} +6.69512 q^{23} +7.18868 q^{25} -3.62848 q^{26} +2.86107 q^{28} +5.81232 q^{29} +8.96928 q^{31} -1.00000 q^{32} +4.84071 q^{34} -9.98865 q^{35} -1.00000 q^{37} -7.31418 q^{38} +3.49123 q^{40} +0.660400 q^{41} -5.76568 q^{43} +5.43746 q^{44} -6.69512 q^{46} +7.40907 q^{47} +1.18572 q^{49} -7.18868 q^{50} +3.62848 q^{52} -3.88952 q^{53} -18.9834 q^{55} -2.86107 q^{56} -5.81232 q^{58} -1.56553 q^{59} +8.49269 q^{61} -8.96928 q^{62} +1.00000 q^{64} -12.6679 q^{65} +0.371075 q^{67} -4.84071 q^{68} +9.98865 q^{70} +0.584961 q^{71} -11.9922 q^{73} +1.00000 q^{74} +7.31418 q^{76} +15.5570 q^{77} -4.13993 q^{79} -3.49123 q^{80} -0.660400 q^{82} -2.85366 q^{83} +16.9000 q^{85} +5.76568 q^{86} -5.43746 q^{88} +3.31264 q^{89} +10.3813 q^{91} +6.69512 q^{92} -7.40907 q^{94} -25.5355 q^{95} +12.8349 q^{97} -1.18572 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{2} + 10 q^{4} - q^{5} + q^{7} - 10 q^{8} + q^{10} - 3 q^{11} + 12 q^{13} - q^{14} + 10 q^{16} - 12 q^{17} + 24 q^{19} - q^{20} + 3 q^{22} - 3 q^{23} + 21 q^{25} - 12 q^{26} + q^{28} - 4 q^{29}+ \cdots - 35 q^{98}+O(q^{100}) $ 10 * q - 10 * q^2 + 10 * q^4 - q^5 + q^7 - 10 * q^8 + q^10 - 3 * q^11 + 12 * q^13 - q^14 + 10 * q^16 - 12 * q^17 + 24 * q^19 - q^20 + 3 * q^22 - 3 * q^23 + 21 * q^25 - 12 * q^26 + q^28 - 4 * q^29 + 11 * q^31 - 10 * q^32 + 12 * q^34 + 7 * q^35 - 10 * q^37 - 24 * q^38 + q^40 + 3 * q^41 + 6 * q^43 - 3 * q^44 + 3 * q^46 - 8 * q^47 + 35 * q^49 - 21 * q^50 + 12 * q^52 + 10 * q^53 + 2 * q^55 - q^56 + 4 * q^58 - 10 * q^59 + 2 * q^61 - 11 * q^62 + 10 * q^64 - 26 * q^65 + 7 * q^67 - 12 * q^68 - 7 * q^70 - 15 * q^71 + 3 * q^73 + 10 * q^74 + 24 * q^76 - 9 * q^77 + 16 * q^79 - q^80 - 3 * q^82 + 23 * q^83 + 15 * q^85 - 6 * q^86 + 3 * q^88 - 21 * q^89 + 59 * q^91 - 3 * q^92 + 8 * q^94 + 6 * q^95 + 24 * q^97 - 35 * q^98

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
$5$	−3.49123	−1.56133	−0.780663	−	0.624953i	$-0.785118\pi$
$5$	−3.49123	−1.56133	−0.780663	+	0.624953i	$0.785118\pi$
$6$	0	0
$7$	2.86107	1.08138	0.540691	−	0.841221i	$-0.318163\pi$
$7$	2.86107	1.08138	0.540691	+	0.841221i	$0.318163\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.49123	1.10402
$11$	5.43746	1.63946	0.819729	−	0.572752i	$-0.194124\pi$
$11$	5.43746	1.63946	0.819729	+	0.572752i	$0.194124\pi$
$12$	0	0
$13$	3.62848	1.00636	0.503180	−	0.864182i	$-0.332163\pi$
$13$	3.62848	1.00636	0.503180	+	0.864182i	$0.332163\pi$
$14$	−2.86107	−0.764653
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.84071	−1.17404	−0.587022	−	0.809571i	$-0.699700\pi$
$17$	−4.84071	−1.17404	−0.587022	+	0.809571i	$0.699700\pi$
$18$	0	0
$19$	7.31418	1.67799	0.838994	−	0.544140i	$-0.183144\pi$
$19$	7.31418	1.67799	0.838994	+	0.544140i	$0.183144\pi$
$20$	−3.49123	−0.780663
$21$	0	0
$22$	−5.43746	−1.15927
$23$	6.69512	1.39603	0.698014	−	0.716084i	$-0.254067\pi$
$23$	6.69512	1.39603	0.698014	+	0.716084i	$0.254067\pi$
$24$	0	0
$25$	7.18868	1.43774
$26$	−3.62848	−0.711604
$27$	0	0
$28$	2.86107	0.540691
$29$	5.81232	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$29$	5.81232	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$30$	0	0
$31$	8.96928	1.61093	0.805465	−	0.592643i	$-0.201916\pi$
$31$	8.96928	1.61093	0.805465	+	0.592643i	$0.201916\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.84071	0.830174
$35$	−9.98865	−1.68839
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−7.31418	−1.18652
$39$	0	0
$40$	3.49123	0.552012
$41$	0.660400	0.103137	0.0515686	−	0.998669i	$-0.483578\pi$
$41$	0.660400	0.103137	0.0515686	+	0.998669i	$0.483578\pi$
$42$	0	0
$43$	−5.76568	−0.879258	−0.439629	−	0.898179i	$-0.644890\pi$
$43$	−5.76568	−0.879258	−0.439629	+	0.898179i	$0.644890\pi$
$44$	5.43746	0.819729
$45$	0	0
$46$	−6.69512	−0.987141
$47$	7.40907	1.08072	0.540362	−	0.841433i	$-0.318287\pi$
$47$	7.40907	1.08072	0.540362	+	0.841433i	$0.318287\pi$
$48$	0	0
$49$	1.18572	0.169388
$50$	−7.18868	−1.01663
$51$	0	0
$52$	3.62848	0.503180
$53$	−3.88952	−0.534266	−0.267133	−	0.963660i	$-0.586076\pi$
$53$	−3.88952	−0.534266	−0.267133	+	0.963660i	$0.586076\pi$
$54$	0	0
$55$	−18.9834	−2.55973
$56$	−2.86107	−0.382326
$57$	0	0
$58$	−5.81232	−0.763195
$59$	−1.56553	−0.203815	−0.101907	−	0.994794i	$-0.532495\pi$
$59$	−1.56553	−0.203815	−0.101907	+	0.994794i	$0.532495\pi$
$60$	0	0
$61$	8.49269	1.08738	0.543689	−	0.839287i	$-0.317027\pi$
$61$	8.49269	1.08738	0.543689	+	0.839287i	$0.317027\pi$
$62$	−8.96928	−1.13910
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.6679	−1.57125
$66$	0	0
$67$	0.371075	0.0453340	0.0226670	−	0.999743i	$-0.492784\pi$
$67$	0.371075	0.0453340	0.0226670	+	0.999743i	$0.492784\pi$
$68$	−4.84071	−0.587022
$69$	0	0
$70$	9.98865	1.19387
$71$	0.584961	0.0694221	0.0347110	−	0.999397i	$-0.488949\pi$
$71$	0.584961	0.0694221	0.0347110	+	0.999397i	$0.488949\pi$
$72$	0	0
$73$	−11.9922	−1.40358	−0.701792	−	0.712382i	$-0.747616\pi$
$73$	−11.9922	−1.40358	−0.701792	+	0.712382i	$0.747616\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	7.31418	0.838994
$77$	15.5570	1.77288
$78$	0	0
$79$	−4.13993	−0.465778	−0.232889	−	0.972503i	$-0.574818\pi$
$79$	−4.13993	−0.465778	−0.232889	+	0.972503i	$0.574818\pi$
$80$	−3.49123	−0.390331
$81$	0	0
$82$	−0.660400	−0.0729290
$83$	−2.85366	−0.313230	−0.156615	−	0.987660i	$-0.550058\pi$
$83$	−2.85366	−0.313230	−0.156615	+	0.987660i	$0.550058\pi$
$84$	0	0
$85$	16.9000	1.83306
$86$	5.76568	0.621729
$87$	0	0
$88$	−5.43746	−0.579636
$89$	3.31264	0.351139	0.175570	−	0.984467i	$-0.443823\pi$
$89$	3.31264	0.351139	0.175570	+	0.984467i	$0.443823\pi$
$90$	0	0
$91$	10.3813	1.08826
$92$	6.69512	0.698014
$93$	0	0
$94$	−7.40907	−0.764187
$95$	−25.5355	−2.61989
$96$	0	0
$97$	12.8349	1.30319	0.651593	−	0.758569i	$-0.274101\pi$
$97$	12.8349	1.30319	0.651593	+	0.758569i	$0.274101\pi$
$98$	−1.18572	−0.119776
$99$	0	0
$100$	7.18868	0.718868
$101$	−6.80546	−0.677169	−0.338584	−	0.940936i	$-0.609948\pi$
$101$	−6.80546	−0.677169	−0.338584	+	0.940936i	$0.609948\pi$
$102$	0	0
$103$	−12.3966	−1.22147	−0.610737	−	0.791833i	$-0.709127\pi$
$103$	−12.3966	−1.22147	−0.610737	+	0.791833i	$0.709127\pi$
$104$	−3.62848	−0.355802
$105$	0	0
$106$	3.88952	0.377783
$107$	0.144488	0.0139682	0.00698411	−	0.999976i	$-0.497777\pi$
$107$	0.144488	0.0139682	0.00698411	+	0.999976i	$0.497777\pi$
$108$	0	0
$109$	12.4288	1.19046	0.595232	−	0.803554i	$-0.297060\pi$
$109$	12.4288	1.19046	0.595232	+	0.803554i	$0.297060\pi$
$110$	18.9834	1.81000
$111$	0	0
$112$	2.86107	0.270346
$113$	4.15551	0.390917	0.195459	−	0.980712i	$-0.437380\pi$
$113$	4.15551	0.390917	0.195459	+	0.980712i	$0.437380\pi$
$114$	0	0
$115$	−23.3742	−2.17965
$116$	5.81232	0.539660
$117$	0	0
$118$	1.56553	0.144119
$119$	−13.8496	−1.26959
$120$	0	0
$121$	18.5660	1.68782
$122$	−8.49269	−0.768892
$123$	0	0
$124$	8.96928	0.805465
$125$	−7.64119	−0.683449
$126$	0	0
$127$	−19.5394	−1.73384	−0.866922	−	0.498445i	$-0.833905\pi$
$127$	−19.5394	−1.73384	−0.866922	+	0.498445i	$0.833905\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.6679	1.11104
$131$	−13.7848	−1.20439	−0.602193	−	0.798351i	$-0.705706\pi$
$131$	−13.7848	−1.20439	−0.602193	+	0.798351i	$0.705706\pi$
$132$	0	0
$133$	20.9264	1.81455
$134$	−0.371075	−0.0320560
$135$	0	0
$136$	4.84071	0.415087
$137$	17.1391	1.46429	0.732146	−	0.681147i	$-0.238519\pi$
$137$	17.1391	1.46429	0.732146	+	0.681147i	$0.238519\pi$
$138$	0	0
$139$	−1.76011	−0.149290	−0.0746452	−	0.997210i	$-0.523782\pi$
$139$	−1.76011	−0.149290	−0.0746452	+	0.997210i	$0.523782\pi$
$140$	−9.98865	−0.844195
$141$	0	0
$142$	−0.584961	−0.0490888
$143$	19.7297	1.64988
$144$	0	0
$145$	−20.2921	−1.68517
$146$	11.9922	0.992483
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−16.6486	−1.36391	−0.681954	−	0.731395i	$-0.738869\pi$
$149$	−16.6486	−1.36391	−0.681954	+	0.731395i	$0.738869\pi$
$150$	0	0
$151$	5.70123	0.463960	0.231980	−	0.972721i	$-0.425480\pi$
$151$	5.70123	0.463960	0.231980	+	0.972721i	$0.425480\pi$
$152$	−7.31418	−0.593259
$153$	0	0
$154$	−15.5570	−1.25362
$155$	−31.3138	−2.51519
$156$	0	0
$157$	−21.7894	−1.73898	−0.869491	−	0.493948i	$-0.835553\pi$
$157$	−21.7894	−1.73898	−0.869491	+	0.493948i	$0.835553\pi$
$158$	4.13993	0.329355
$159$	0	0
$160$	3.49123	0.276006
$161$	19.1552	1.50964
$162$	0	0
$163$	2.37983	0.186403	0.0932013	−	0.995647i	$-0.470290\pi$
$163$	2.37983	0.186403	0.0932013	+	0.995647i	$0.470290\pi$
$164$	0.660400	0.0515686
$165$	0	0
$166$	2.85366	0.221487
$167$	−4.22401	−0.326864	−0.163432	−	0.986555i	$-0.552256\pi$
$167$	−4.22401	−0.326864	−0.163432	+	0.986555i	$0.552256\pi$
$168$	0	0
$169$	0.165868	0.0127591
$170$	−16.9000	−1.29617
$171$	0	0
$172$	−5.76568	−0.439629
$173$	3.83472	0.291548	0.145774	−	0.989318i	$-0.453433\pi$
$173$	3.83472	0.291548	0.145774	+	0.989318i	$0.453433\pi$
$174$	0	0
$175$	20.5673	1.55474
$176$	5.43746	0.409864
$177$	0	0
$178$	−3.31264	−0.248293
$179$	4.36893	0.326549	0.163275	−	0.986581i	$-0.447794\pi$
$179$	4.36893	0.326549	0.163275	+	0.986581i	$0.447794\pi$
$180$	0	0
$181$	−3.71793	−0.276352	−0.138176	−	0.990408i	$-0.544124\pi$
$181$	−3.71793	−0.276352	−0.138176	+	0.990408i	$0.544124\pi$
$182$	−10.3813	−0.769516
$183$	0	0
$184$	−6.69512	−0.493571
$185$	3.49123	0.256680
$186$	0	0
$187$	−26.3212	−1.92479
$188$	7.40907	0.540362
$189$	0	0
$190$	25.5355	1.85254
$191$	5.53343	0.400385	0.200193	−	0.979757i	$-0.435843\pi$
$191$	5.53343	0.400385	0.200193	+	0.979757i	$0.435843\pi$
$192$	0	0
$193$	−23.0916	−1.66217	−0.831084	−	0.556147i	$-0.812279\pi$
$193$	−23.0916	−1.66217	−0.831084	+	0.556147i	$0.812279\pi$
$194$	−12.8349	−0.921492
$195$	0	0
$196$	1.18572	0.0846941
$197$	−14.5861	−1.03922	−0.519609	−	0.854404i	$-0.673922\pi$
$197$	−14.5861	−1.03922	−0.519609	+	0.854404i	$0.673922\pi$
$198$	0	0
$199$	19.8939	1.41024	0.705121	−	0.709087i	$-0.250893\pi$
$199$	19.8939	1.41024	0.705121	+	0.709087i	$0.250893\pi$
$200$	−7.18868	−0.508317
$201$	0	0
$202$	6.80546	0.478831
$203$	16.6295	1.16716
$204$	0	0
$205$	−2.30561	−0.161031
$206$	12.3966	0.863713
$207$	0	0
$208$	3.62848	0.251590
$209$	39.7706	2.75099
$210$	0	0
$211$	−19.4285	−1.33752	−0.668758	−	0.743481i	$-0.733174\pi$
$211$	−19.4285	−1.33752	−0.668758	+	0.743481i	$0.733174\pi$
$212$	−3.88952	−0.267133
$213$	0	0
$214$	−0.144488	−0.00987702
$215$	20.1293	1.37281
$216$	0	0
$217$	25.6617	1.74203
$218$	−12.4288	−0.841786
$219$	0	0
$220$	−18.9834	−1.27986
$221$	−17.5644	−1.18151
$222$	0	0
$223$	13.2705	0.888657	0.444328	−	0.895864i	$-0.353442\pi$
$223$	13.2705	0.888657	0.444328	+	0.895864i	$0.353442\pi$
$224$	−2.86107	−0.191163
$225$	0	0
$226$	−4.15551	−0.276420
$227$	8.11658	0.538717	0.269358	−	0.963040i	$-0.413188\pi$
$227$	8.11658	0.538717	0.269358	+	0.963040i	$0.413188\pi$
$228$	0	0
$229$	17.7272	1.17145	0.585723	−	0.810511i	$-0.300811\pi$
$229$	17.7272	1.17145	0.585723	+	0.810511i	$0.300811\pi$
$230$	23.3742	1.54125
$231$	0	0
$232$	−5.81232	−0.381598
$233$	−9.99766	−0.654968	−0.327484	−	0.944857i	$-0.606201\pi$
$233$	−9.99766	−0.654968	−0.327484	+	0.944857i	$0.606201\pi$
$234$	0	0
$235$	−25.8668	−1.68736
$236$	−1.56553	−0.101907
$237$	0	0
$238$	13.8496	0.897736
$239$	16.5064	1.06771	0.533856	−	0.845575i	$-0.320742\pi$
$239$	16.5064	1.06771	0.533856	+	0.845575i	$0.320742\pi$
$240$	0	0
$241$	16.9919	1.09454	0.547272	−	0.836955i	$-0.315666\pi$
$241$	16.9919	1.09454	0.547272	+	0.836955i	$0.315666\pi$
$242$	−18.5660	−1.19347
$243$	0	0
$244$	8.49269	0.543689
$245$	−4.13961	−0.264470
$246$	0	0
$247$	26.5394	1.68866
$248$	−8.96928	−0.569550
$249$	0	0
$250$	7.64119	0.483271
$251$	22.2583	1.40493	0.702467	−	0.711716i	$-0.252082\pi$
$251$	22.2583	1.40493	0.702467	+	0.711716i	$0.252082\pi$
$252$	0	0
$253$	36.4045	2.28873
$254$	19.5394	1.22601
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.59811	−0.473957	−0.236979	−	0.971515i	$-0.576157\pi$
$257$	−7.59811	−0.473957	−0.236979	+	0.971515i	$0.576157\pi$
$258$	0	0
$259$	−2.86107	−0.177778
$260$	−12.6679	−0.785627
$261$	0	0
$262$	13.7848	0.851629
$263$	−27.5780	−1.70053	−0.850265	−	0.526355i	$-0.823558\pi$
$263$	−27.5780	−1.70053	−0.850265	+	0.526355i	$0.823558\pi$
$264$	0	0
$265$	13.5792	0.834163
$266$	−20.9264	−1.28308
$267$	0	0
$268$	0.371075	0.0226670
$269$	5.98578	0.364959	0.182480	−	0.983210i	$-0.441588\pi$
$269$	5.98578	0.364959	0.182480	+	0.983210i	$0.441588\pi$
$270$	0	0
$271$	−17.3044	−1.05117	−0.525583	−	0.850742i	$-0.676153\pi$
$271$	−17.3044	−1.05117	−0.525583	+	0.850742i	$0.676153\pi$
$272$	−4.84071	−0.293511
$273$	0	0
$274$	−17.1391	−1.03541
$275$	39.0882	2.35711
$276$	0	0
$277$	−4.11281	−0.247115	−0.123558	−	0.992337i	$-0.539430\pi$
$277$	−4.11281	−0.247115	−0.123558	+	0.992337i	$0.539430\pi$
$278$	1.76011	0.105564
$279$	0	0
$280$	9.98865	0.596936
$281$	−5.24793	−0.313065	−0.156533	−	0.987673i	$-0.550032\pi$
$281$	−5.24793	−0.313065	−0.156533	+	0.987673i	$0.550032\pi$
$282$	0	0
$283$	1.87337	0.111360	0.0556801	−	0.998449i	$-0.482267\pi$
$283$	1.87337	0.111360	0.0556801	+	0.998449i	$0.482267\pi$
$284$	0.584961	0.0347110
$285$	0	0
$286$	−19.7297	−1.16664
$287$	1.88945	0.111531
$288$	0	0
$289$	6.43244	0.378379
$290$	20.2921	1.19160
$291$	0	0
$292$	−11.9922	−0.701792
$293$	8.40851	0.491231	0.245615	−	0.969367i	$-0.421010\pi$
$293$	8.40851	0.491231	0.245615	+	0.969367i	$0.421010\pi$
$294$	0	0
$295$	5.46563	0.318221
$296$	1.00000	0.0581238
$297$	0	0
$298$	16.6486	0.964428
$299$	24.2931	1.40491
$300$	0	0
$301$	−16.4960	−0.950814
$302$	−5.70123	−0.328069
$303$	0	0
$304$	7.31418	0.419497
$305$	−29.6499	−1.69775
$306$	0	0
$307$	0.609988	0.0348138	0.0174069	−	0.999848i	$-0.494459\pi$
$307$	0.609988	0.0348138	0.0174069	+	0.999848i	$0.494459\pi$
$308$	15.5570	0.886440
$309$	0	0
$310$	31.3138	1.77850
$311$	2.93104	0.166204	0.0831021	−	0.996541i	$-0.473517\pi$
$311$	2.93104	0.166204	0.0831021	+	0.996541i	$0.473517\pi$
$312$	0	0
$313$	13.6998	0.774359	0.387179	−	0.922004i	$-0.373449\pi$
$313$	13.6998	0.774359	0.387179	+	0.922004i	$0.373449\pi$
$314$	21.7894	1.22965
$315$	0	0
$316$	−4.13993	−0.232889
$317$	4.46543	0.250803	0.125402	−	0.992106i	$-0.459978\pi$
$317$	4.46543	0.250803	0.125402	+	0.992106i	$0.459978\pi$
$318$	0	0
$319$	31.6043	1.76950
$320$	−3.49123	−0.195166
$321$	0	0
$322$	−19.1552	−1.06748
$323$	−35.4058	−1.97003
$324$	0	0
$325$	26.0840	1.44688
$326$	−2.37983	−0.131806
$327$	0	0
$328$	−0.660400	−0.0364645
$329$	21.1979	1.16868
$330$	0	0
$331$	−30.0470	−1.65153	−0.825765	−	0.564014i	$-0.809256\pi$
$331$	−30.0470	−1.65153	−0.825765	+	0.564014i	$0.809256\pi$
$332$	−2.85366	−0.156615
$333$	0	0
$334$	4.22401	0.231128
$335$	−1.29551	−0.0707812
$336$	0	0
$337$	3.62683	0.197566	0.0987829	−	0.995109i	$-0.468505\pi$
$337$	3.62683	0.197566	0.0987829	+	0.995109i	$0.468505\pi$
$338$	−0.165868	−0.00902204
$339$	0	0
$340$	16.9000	0.916532
$341$	48.7701	2.64105
$342$	0	0
$343$	−16.6351	−0.898209
$344$	5.76568	0.310865
$345$	0	0
$346$	−3.83472	−0.206156
$347$	25.2388	1.35489	0.677446	−	0.735572i	$-0.263087\pi$
$347$	25.2388	1.35489	0.677446	+	0.735572i	$0.263087\pi$
$348$	0	0
$349$	28.8966	1.54680	0.773401	−	0.633917i	$-0.218554\pi$
$349$	28.8966	1.54680	0.773401	+	0.633917i	$0.218554\pi$
$350$	−20.5673	−1.09937
$351$	0	0
$352$	−5.43746	−0.289818
$353$	−4.43635	−0.236123	−0.118062	−	0.993006i	$-0.537668\pi$
$353$	−4.43635	−0.236123	−0.118062	+	0.993006i	$0.537668\pi$
$354$	0	0
$355$	−2.04223	−0.108390
$356$	3.31264	0.175570
$357$	0	0
$358$	−4.36893	−0.230905
$359$	22.3860	1.18149	0.590743	−	0.806860i	$-0.298835\pi$
$359$	22.3860	1.18149	0.590743	+	0.806860i	$0.298835\pi$
$360$	0	0
$361$	34.4973	1.81565
$362$	3.71793	0.195410
$363$	0	0
$364$	10.3813	0.544130
$365$	41.8676	2.19145
$366$	0	0
$367$	−10.4628	−0.546152	−0.273076	−	0.961992i	$-0.588041\pi$
$367$	−10.4628	−0.546152	−0.273076	+	0.961992i	$0.588041\pi$
$368$	6.69512	0.349007
$369$	0	0
$370$	−3.49123	−0.181500
$371$	−11.1282	−0.577746
$372$	0	0
$373$	−13.1068	−0.678646	−0.339323	−	0.940670i	$-0.610198\pi$
$373$	−13.1068	−0.678646	−0.339323	+	0.940670i	$0.610198\pi$
$374$	26.3212	1.36104
$375$	0	0
$376$	−7.40907	−0.382094
$377$	21.0899	1.08618
$378$	0	0
$379$	−10.8284	−0.556219	−0.278110	−	0.960549i	$-0.589708\pi$
$379$	−10.8284	−0.556219	−0.278110	+	0.960549i	$0.589708\pi$
$380$	−25.5355	−1.30994
$381$	0	0
$382$	−5.53343	−0.283115
$383$	23.3409	1.19266	0.596331	−	0.802739i	$-0.296625\pi$
$383$	23.3409	1.19266	0.596331	+	0.802739i	$0.296625\pi$
$384$	0	0
$385$	−54.3129	−2.76804
$386$	23.0916	1.17533
$387$	0	0
$388$	12.8349	0.651593
$389$	10.9206	0.553696	0.276848	−	0.960914i	$-0.410710\pi$
$389$	10.9206	0.553696	0.276848	+	0.960914i	$0.410710\pi$
$390$	0	0
$391$	−32.4091	−1.63900
$392$	−1.18572	−0.0598878
$393$	0	0
$394$	14.5861	0.734838
$395$	14.4534	0.727231
$396$	0	0
$397$	−1.28277	−0.0643805	−0.0321902	−	0.999482i	$-0.510248\pi$
$397$	−1.28277	−0.0643805	−0.0321902	+	0.999482i	$0.510248\pi$
$398$	−19.8939	−0.997192
$399$	0	0
$400$	7.18868	0.359434
$401$	−5.44679	−0.272000	−0.136000	−	0.990709i	$-0.543425\pi$
$401$	−5.44679	−0.272000	−0.136000	+	0.990709i	$0.543425\pi$
$402$	0	0
$403$	32.5448	1.62117
$404$	−6.80546	−0.338584
$405$	0	0
$406$	−16.6295	−0.825306
$407$	−5.43746	−0.269525
$408$	0	0
$409$	4.04403	0.199965	0.0999823	−	0.994989i	$-0.468121\pi$
$409$	4.04403	0.199965	0.0999823	+	0.994989i	$0.468121\pi$
$410$	2.30561	0.113866
$411$	0	0
$412$	−12.3966	−0.610737
$413$	−4.47909	−0.220402
$414$	0	0
$415$	9.96277	0.489053
$416$	−3.62848	−0.177901
$417$	0	0
$418$	−39.7706	−1.94524
$419$	11.3722	0.555570	0.277785	−	0.960643i	$-0.410400\pi$
$419$	11.3722	0.555570	0.277785	+	0.960643i	$0.410400\pi$
$420$	0	0
$421$	36.4875	1.77829	0.889145	−	0.457626i	$-0.151300\pi$
$421$	36.4875	1.77829	0.889145	+	0.457626i	$0.151300\pi$
$422$	19.4285	0.945766
$423$	0	0
$424$	3.88952	0.188892
$425$	−34.7983	−1.68797
$426$	0	0
$427$	24.2982	1.17587
$428$	0.144488	0.00698411
$429$	0	0
$430$	−20.1293	−0.970722
$431$	−21.7781	−1.04901	−0.524506	−	0.851407i	$-0.675750\pi$
$431$	−21.7781	−1.04901	−0.524506	+	0.851407i	$0.675750\pi$
$432$	0	0
$433$	15.7500	0.756897	0.378448	−	0.925622i	$-0.376458\pi$
$433$	15.7500	0.756897	0.378448	+	0.925622i	$0.376458\pi$
$434$	−25.6617	−1.23180
$435$	0	0
$436$	12.4288	0.595232
$437$	48.9693	2.34252
$438$	0	0
$439$	−29.8767	−1.42594	−0.712968	−	0.701196i	$-0.752650\pi$
$439$	−29.8767	−1.42594	−0.712968	+	0.701196i	$0.752650\pi$
$440$	18.9834	0.905000
$441$	0	0
$442$	17.5644	0.835454
$443$	−33.1876	−1.57679	−0.788395	−	0.615169i	$-0.789088\pi$
$443$	−33.1876	−1.57679	−0.788395	+	0.615169i	$0.789088\pi$
$444$	0	0
$445$	−11.5652	−0.548243
$446$	−13.2705	−0.628375
$447$	0	0
$448$	2.86107	0.135173
$449$	−33.4784	−1.57994	−0.789972	−	0.613144i	$-0.789905\pi$
$449$	−33.4784	−1.57994	−0.789972	+	0.613144i	$0.789905\pi$
$450$	0	0
$451$	3.59090	0.169089
$452$	4.15551	0.195459
$453$	0	0
$454$	−8.11658	−0.380930
$455$	−36.2436	−1.69913
$456$	0	0
$457$	15.5789	0.728750	0.364375	−	0.931252i	$-0.381283\pi$
$457$	15.5789	0.728750	0.364375	+	0.931252i	$0.381283\pi$
$458$	−17.7272	−0.828338
$459$	0	0
$460$	−23.3742	−1.08983
$461$	−19.6290	−0.914212	−0.457106	−	0.889412i	$-0.651114\pi$
$461$	−19.6290	−0.914212	−0.457106	+	0.889412i	$0.651114\pi$
$462$	0	0
$463$	26.0267	1.20956	0.604782	−	0.796391i	$-0.293260\pi$
$463$	26.0267	1.20956	0.604782	+	0.796391i	$0.293260\pi$
$464$	5.81232	0.269830
$465$	0	0
$466$	9.99766	0.463132
$467$	27.4418	1.26986	0.634929	−	0.772571i	$-0.281029\pi$
$467$	27.4418	1.26986	0.634929	+	0.772571i	$0.281029\pi$
$468$	0	0
$469$	1.06167	0.0490234
$470$	25.8668	1.19314
$471$	0	0
$472$	1.56553	0.0720594
$473$	−31.3507	−1.44151
$474$	0	0
$475$	52.5793	2.41251
$476$	−13.8496	−0.634795
$477$	0	0
$478$	−16.5064	−0.754987
$479$	−20.6312	−0.942662	−0.471331	−	0.881956i	$-0.656226\pi$
$479$	−20.6312	−0.942662	−0.471331	+	0.881956i	$0.656226\pi$
$480$	0	0
$481$	−3.62848	−0.165444
$482$	−16.9919	−0.773959
$483$	0	0
$484$	18.5660	0.843910
$485$	−44.8096	−2.03470
$486$	0	0
$487$	−1.04823	−0.0474997	−0.0237499	−	0.999718i	$-0.507561\pi$
$487$	−1.04823	−0.0474997	−0.0237499	+	0.999718i	$0.507561\pi$
$488$	−8.49269	−0.384446
$489$	0	0
$490$	4.13961	0.187009
$491$	−33.0819	−1.49296	−0.746482	−	0.665406i	$-0.768259\pi$
$491$	−33.0819	−1.49296	−0.746482	+	0.665406i	$0.768259\pi$
$492$	0	0
$493$	−28.1357	−1.26717
$494$	−26.5394	−1.19406
$495$	0	0
$496$	8.96928	0.402733
$497$	1.67361	0.0750718
$498$	0	0
$499$	−6.65247	−0.297805	−0.148903	−	0.988852i	$-0.547574\pi$
$499$	−6.65247	−0.297805	−0.148903	+	0.988852i	$0.547574\pi$
$500$	−7.64119	−0.341724
$501$	0	0
$502$	−22.2583	−0.993438
$503$	−17.2357	−0.768500	−0.384250	−	0.923229i	$-0.625540\pi$
$503$	−17.2357	−0.768500	−0.384250	+	0.923229i	$0.625540\pi$
$504$	0	0
$505$	23.7594	1.05728
$506$	−36.4045	−1.61838
$507$	0	0
$508$	−19.5394	−0.866922
$509$	21.2969	0.943970	0.471985	−	0.881606i	$-0.343538\pi$
$509$	21.2969	0.943970	0.471985	+	0.881606i	$0.343538\pi$
$510$	0	0
$511$	−34.3106	−1.51781
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	7.59811	0.335138
$515$	43.2794	1.90712
$516$	0	0
$517$	40.2866	1.77180
$518$	2.86107	0.125708
$519$	0	0
$520$	12.6679	0.555522
$521$	−20.1473	−0.882668	−0.441334	−	0.897343i	$-0.645495\pi$
$521$	−20.1473	−0.882668	−0.441334	+	0.897343i	$0.645495\pi$
$522$	0	0
$523$	−4.80666	−0.210181	−0.105090	−	0.994463i	$-0.533513\pi$
$523$	−4.80666	−0.210181	−0.105090	+	0.994463i	$0.533513\pi$
$524$	−13.7848	−0.602193
$525$	0	0
$526$	27.5780	1.20246
$527$	−43.4176	−1.89130
$528$	0	0
$529$	21.8246	0.948896
$530$	−13.5792	−0.589842
$531$	0	0
$532$	20.9264	0.907274
$533$	2.39625	0.103793
$534$	0	0
$535$	−0.504442	−0.0218089
$536$	−0.371075	−0.0160280
$537$	0	0
$538$	−5.98578	−0.258065
$539$	6.44729	0.277705
$540$	0	0
$541$	−19.3496	−0.831904	−0.415952	−	0.909386i	$-0.636552\pi$
$541$	−19.3496	−0.831904	−0.415952	+	0.909386i	$0.636552\pi$
$542$	17.3044	0.743287
$543$	0	0
$544$	4.84071	0.207544
$545$	−43.3918	−1.85870
$546$	0	0
$547$	23.5743	1.00796	0.503982	−	0.863714i	$-0.331868\pi$
$547$	23.5743	1.00796	0.503982	+	0.863714i	$0.331868\pi$
$548$	17.1391	0.732146
$549$	0	0
$550$	−39.0882	−1.66673
$551$	42.5124	1.81109
$552$	0	0
$553$	−11.8446	−0.503685
$554$	4.11281	0.174737
$555$	0	0
$556$	−1.76011	−0.0746452
$557$	13.9116	0.589453	0.294727	−	0.955582i	$-0.404771\pi$
$557$	13.9116	0.589453	0.294727	+	0.955582i	$0.404771\pi$
$558$	0	0
$559$	−20.9207	−0.884850
$560$	−9.98865	−0.422097
$561$	0	0
$562$	5.24793	0.221370
$563$	−39.4046	−1.66071	−0.830354	−	0.557236i	$-0.811862\pi$
$563$	−39.4046	−1.66071	−0.830354	+	0.557236i	$0.811862\pi$
$564$	0	0
$565$	−14.5078	−0.610349
$566$	−1.87337	−0.0787436
$567$	0	0
$568$	−0.584961	−0.0245444
$569$	−9.12190	−0.382410	−0.191205	−	0.981550i	$-0.561240\pi$
$569$	−9.12190	−0.382410	−0.191205	+	0.981550i	$0.561240\pi$
$570$	0	0
$571$	−20.3135	−0.850095	−0.425047	−	0.905171i	$-0.639743\pi$
$571$	−20.3135	−0.850095	−0.425047	+	0.905171i	$0.639743\pi$
$572$	19.7297	0.824941
$573$	0	0
$574$	−1.88945	−0.0788641
$575$	48.1291	2.00712
$576$	0	0
$577$	−11.2761	−0.469428	−0.234714	−	0.972064i	$-0.575415\pi$
$577$	−11.2761	−0.469428	−0.234714	+	0.972064i	$0.575415\pi$
$578$	−6.43244	−0.267554
$579$	0	0
$580$	−20.2921	−0.842585
$581$	−8.16451	−0.338721
$582$	0	0
$583$	−21.1491	−0.875906
$584$	11.9922	0.496242
$585$	0	0
$586$	−8.40851	−0.347352
$587$	48.2944	1.99333	0.996663	−	0.0816316i	$-0.0260131\pi$
$587$	48.2944	1.99333	0.996663	+	0.0816316i	$0.0260131\pi$
$588$	0	0
$589$	65.6030	2.70312
$590$	−5.46563	−0.225016
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	9.83676	0.403947	0.201974	−	0.979391i	$-0.435264\pi$
$593$	9.83676	0.403947	0.201974	+	0.979391i	$0.435264\pi$
$594$	0	0
$595$	48.3521	1.98224
$596$	−16.6486	−0.681954
$597$	0	0
$598$	−24.2931	−0.993419
$599$	4.67625	0.191066	0.0955332	−	0.995426i	$-0.469544\pi$
$599$	4.67625	0.191066	0.0955332	+	0.995426i	$0.469544\pi$
$600$	0	0
$601$	−8.37334	−0.341556	−0.170778	−	0.985310i	$-0.554628\pi$
$601$	−8.37334	−0.341556	−0.170778	+	0.985310i	$0.554628\pi$
$602$	16.4960	0.672327
$603$	0	0
$604$	5.70123	0.231980
$605$	−64.8182	−2.63524
$606$	0	0
$607$	−38.4154	−1.55923	−0.779616	−	0.626258i	$-0.784586\pi$
$607$	−38.4154	−1.55923	−0.779616	+	0.626258i	$0.784586\pi$
$608$	−7.31418	−0.296629
$609$	0	0
$610$	29.6499	1.20049
$611$	26.8837	1.08760
$612$	0	0
$613$	18.6423	0.752955	0.376477	−	0.926426i	$-0.377135\pi$
$613$	18.6423	0.752955	0.376477	+	0.926426i	$0.377135\pi$
$614$	−0.609988	−0.0246171
$615$	0	0
$616$	−15.5570	−0.626808
$617$	15.6409	0.629678	0.314839	−	0.949145i	$-0.398050\pi$
$617$	15.6409	0.629678	0.314839	+	0.949145i	$0.398050\pi$
$618$	0	0
$619$	16.3358	0.656591	0.328296	−	0.944575i	$-0.393526\pi$
$619$	16.3358	0.656591	0.328296	+	0.944575i	$0.393526\pi$
$620$	−31.3138	−1.25759
$621$	0	0
$622$	−2.93104	−0.117524
$623$	9.47770	0.379716
$624$	0	0
$625$	−9.26626	−0.370650
$626$	−13.6998	−0.547554
$627$	0	0
$628$	−21.7894	−0.869491
$629$	4.84071	0.193012
$630$	0	0
$631$	26.0065	1.03530	0.517650	−	0.855592i	$-0.326807\pi$
$631$	26.0065	1.03530	0.517650	+	0.855592i	$0.326807\pi$
$632$	4.13993	0.164677
$633$	0	0
$634$	−4.46543	−0.177345
$635$	68.2166	2.70709
$636$	0	0
$637$	4.30235	0.170465
$638$	−31.6043	−1.25123
$639$	0	0
$640$	3.49123	0.138003
$641$	35.3060	1.39450	0.697251	−	0.716827i	$-0.254406\pi$
$641$	35.3060	1.39450	0.697251	+	0.716827i	$0.254406\pi$
$642$	0	0
$643$	45.2085	1.78285	0.891425	−	0.453167i	$-0.149706\pi$
$643$	45.2085	1.78285	0.891425	+	0.453167i	$0.149706\pi$
$644$	19.1552	0.754820
$645$	0	0
$646$	35.4058	1.39302
$647$	−7.32633	−0.288028	−0.144014	−	0.989576i	$-0.546001\pi$
$647$	−7.32633	−0.288028	−0.144014	+	0.989576i	$0.546001\pi$
$648$	0	0
$649$	−8.51252	−0.334146
$650$	−26.0840	−1.02310
$651$	0	0
$652$	2.37983	0.0932013
$653$	−23.2699	−0.910620	−0.455310	−	0.890333i	$-0.650472\pi$
$653$	−23.2699	−0.910620	−0.455310	+	0.890333i	$0.650472\pi$
$654$	0	0
$655$	48.1260	1.88044
$656$	0.660400	0.0257843
$657$	0	0
$658$	−21.1979	−0.826379
$659$	−24.7534	−0.964257	−0.482128	−	0.876101i	$-0.660136\pi$
$659$	−24.7534	−0.964257	−0.482128	+	0.876101i	$0.660136\pi$
$660$	0	0
$661$	−15.8312	−0.615763	−0.307881	−	0.951425i	$-0.599620\pi$
$661$	−15.8312	−0.615763	−0.307881	+	0.951425i	$0.599620\pi$
$662$	30.0470	1.16781
$663$	0	0
$664$	2.85366	0.110743
$665$	−73.0588	−2.83310
$666$	0	0
$667$	38.9142	1.50676
$668$	−4.22401	−0.163432
$669$	0	0
$670$	1.29551	0.0500499
$671$	46.1787	1.78271
$672$	0	0
$673$	10.2447	0.394903	0.197452	−	0.980313i	$-0.436733\pi$
$673$	10.2447	0.394903	0.197452	+	0.980313i	$0.436733\pi$
$674$	−3.62683	−0.139700
$675$	0	0
$676$	0.165868	0.00637955
$677$	25.9343	0.996735	0.498367	−	0.866966i	$-0.333933\pi$
$677$	25.9343	0.996735	0.498367	+	0.866966i	$0.333933\pi$
$678$	0	0
$679$	36.7215	1.40924
$680$	−16.9000	−0.648086
$681$	0	0
$682$	−48.7701	−1.86750
$683$	24.9994	0.956574	0.478287	−	0.878204i	$-0.341258\pi$
$683$	24.9994	0.956574	0.478287	+	0.878204i	$0.341258\pi$
$684$	0	0
$685$	−59.8366	−2.28624
$686$	16.6351	0.635130
$687$	0	0
$688$	−5.76568	−0.219815
$689$	−14.1130	−0.537664
$690$	0	0
$691$	−0.0335560	−0.00127653	−0.000638266	−	1.00000i	$-0.500203\pi$
$691$	−0.0335560	−0.00127653	−0.000638266		1.00000i	$0.500203\pi$
$692$	3.83472	0.145774
$693$	0	0
$694$	−25.2388	−0.958053
$695$	6.14494	0.233091
$696$	0	0
$697$	−3.19680	−0.121088
$698$	−28.8966	−1.09375
$699$	0	0
$700$	20.5673	0.777372
$701$	−9.20230	−0.347566	−0.173783	−	0.984784i	$-0.555599\pi$
$701$	−9.20230	−0.347566	−0.173783	+	0.984784i	$0.555599\pi$
$702$	0	0
$703$	−7.31418	−0.275860
$704$	5.43746	0.204932
$705$	0	0
$706$	4.43635	0.166964
$707$	−19.4709	−0.732279
$708$	0	0
$709$	−36.9470	−1.38758	−0.693788	−	0.720180i	$-0.744059\pi$
$709$	−36.9470	−1.38758	−0.693788	+	0.720180i	$0.744059\pi$
$710$	2.04223	0.0766436
$711$	0	0
$712$	−3.31264	−0.124147
$713$	60.0504	2.24890
$714$	0	0
$715$	−68.8810	−2.57600
$716$	4.36893	0.163275
$717$	0	0
$718$	−22.3860	−0.835437
$719$	18.1986	0.678695	0.339347	−	0.940661i	$-0.389794\pi$
$719$	18.1986	0.678695	0.339347	+	0.940661i	$0.389794\pi$
$720$	0	0
$721$	−35.4676	−1.32088
$722$	−34.4973	−1.28386
$723$	0	0
$724$	−3.71793	−0.138176
$725$	41.7829	1.55178
$726$	0	0
$727$	−40.9131	−1.51738	−0.758692	−	0.651450i	$-0.774161\pi$
$727$	−40.9131	−1.51738	−0.758692	+	0.651450i	$0.774161\pi$
$728$	−10.3813	−0.384758
$729$	0	0
$730$	−41.8676	−1.54959
$731$	27.9100	1.03229
$732$	0	0
$733$	31.2487	1.15420	0.577099	−	0.816674i	$-0.304185\pi$
$733$	31.2487	1.15420	0.577099	+	0.816674i	$0.304185\pi$
$734$	10.4628	0.386188
$735$	0	0
$736$	−6.69512	−0.246785
$737$	2.01771	0.0743232
$738$	0	0
$739$	53.2276	1.95801	0.979004	−	0.203841i	$-0.0653427\pi$
$739$	53.2276	1.95801	0.979004	+	0.203841i	$0.0653427\pi$
$740$	3.49123	0.128340
$741$	0	0
$742$	11.1282	0.408528
$743$	−14.3198	−0.525343	−0.262672	−	0.964885i	$-0.584604\pi$
$743$	−14.3198	−0.525343	−0.262672	+	0.964885i	$0.584604\pi$
$744$	0	0
$745$	58.1241	2.12950
$746$	13.1068	0.479875
$747$	0	0
$748$	−26.3212	−0.962397
$749$	0.413391	0.0151050
$750$	0	0
$751$	13.6154	0.496835	0.248417	−	0.968653i	$-0.420090\pi$
$751$	13.6154	0.496835	0.248417	+	0.968653i	$0.420090\pi$
$752$	7.40907	0.270181
$753$	0	0
$754$	−21.0899	−0.768049
$755$	−19.9043	−0.724392
$756$	0	0
$757$	−1.08985	−0.0396112	−0.0198056	−	0.999804i	$-0.506305\pi$
$757$	−1.08985	−0.0396112	−0.0198056	+	0.999804i	$0.506305\pi$
$758$	10.8284	0.393307
$759$	0	0
$760$	25.5355	0.926270
$761$	−0.380018	−0.0137757	−0.00688783	−	0.999976i	$-0.502192\pi$
$761$	−0.380018	−0.0137757	−0.00688783	+	0.999976i	$0.502192\pi$
$762$	0	0
$763$	35.5597	1.28735
$764$	5.53343	0.200193
$765$	0	0
$766$	−23.3409	−0.843339
$767$	−5.68050	−0.205111
$768$	0	0
$769$	−36.7259	−1.32437	−0.662185	−	0.749340i	$-0.730371\pi$
$769$	−36.7259	−1.32437	−0.662185	+	0.749340i	$0.730371\pi$
$770$	54.3129	1.95730
$771$	0	0
$772$	−23.0916	−0.831084
$773$	32.3346	1.16299	0.581497	−	0.813548i	$-0.302467\pi$
$773$	32.3346	1.16299	0.581497	+	0.813548i	$0.302467\pi$
$774$	0	0
$775$	64.4773	2.31609
$776$	−12.8349	−0.460746
$777$	0	0
$778$	−10.9206	−0.391522
$779$	4.83029	0.173063
$780$	0	0
$781$	3.18070	0.113815
$782$	32.4091	1.15895
$783$	0	0
$784$	1.18572	0.0423470
$785$	76.0717	2.71512
$786$	0	0
$787$	24.8290	0.885059	0.442529	−	0.896754i	$-0.354081\pi$
$787$	24.8290	0.885059	0.442529	+	0.896754i	$0.354081\pi$
$788$	−14.5861	−0.519609
$789$	0	0
$790$	−14.4534	−0.514230
$791$	11.8892	0.422731
$792$	0	0
$793$	30.8156	1.09429
$794$	1.28277	0.0455239
$795$	0	0
$796$	19.8939	0.705121
$797$	44.2527	1.56751	0.783756	−	0.621069i	$-0.213301\pi$
$797$	44.2527	1.56751	0.783756	+	0.621069i	$0.213301\pi$
$798$	0	0
$799$	−35.8651	−1.26882
$800$	−7.18868	−0.254158
$801$	0	0
$802$	5.44679	0.192333
$803$	−65.2073	−2.30111
$804$	0	0
$805$	−66.8752	−2.35704
$806$	−32.5448	−1.14634
$807$	0	0
$808$	6.80546	0.239415
$809$	46.5615	1.63701	0.818507	−	0.574497i	$-0.194802\pi$
$809$	46.5615	1.63701	0.818507	+	0.574497i	$0.194802\pi$
$810$	0	0
$811$	−43.6323	−1.53214	−0.766068	−	0.642759i	$-0.777789\pi$
$811$	−43.6323	−1.53214	−0.766068	+	0.642759i	$0.777789\pi$
$812$	16.6295	0.583579
$813$	0	0
$814$	5.43746	0.190583
$815$	−8.30852	−0.291035
$816$	0	0
$817$	−42.1713	−1.47539
$818$	−4.04403	−0.141396
$819$	0	0
$820$	−2.30561	−0.0805153
$821$	−25.5524	−0.891783	−0.445892	−	0.895087i	$-0.647113\pi$
$821$	−25.5524	−0.891783	−0.445892	+	0.895087i	$0.647113\pi$
$822$	0	0
$823$	−28.7281	−1.00140	−0.500699	−	0.865622i	$-0.666924\pi$
$823$	−28.7281	−1.00140	−0.500699	+	0.865622i	$0.666924\pi$
$824$	12.3966	0.431857
$825$	0	0
$826$	4.47909	0.155848
$827$	−10.3886	−0.361248	−0.180624	−	0.983552i	$-0.557812\pi$
$827$	−10.3886	−0.361248	−0.180624	+	0.983552i	$0.557812\pi$
$828$	0	0
$829$	−43.4017	−1.50740	−0.753702	−	0.657216i	$-0.771734\pi$
$829$	−43.4017	−1.50740	−0.753702	+	0.657216i	$0.771734\pi$
$830$	−9.96277	−0.345813
$831$	0	0
$832$	3.62848	0.125795
$833$	−5.73971	−0.198869
$834$	0	0
$835$	14.7470	0.510341
$836$	39.7706	1.37550
$837$	0	0
$838$	−11.3722	−0.392847
$839$	−43.9907	−1.51873	−0.759363	−	0.650667i	$-0.774489\pi$
$839$	−43.9907	−1.51873	−0.759363	+	0.650667i	$0.774489\pi$
$840$	0	0
$841$	4.78307	0.164934
$842$	−36.4875	−1.25744
$843$	0	0
$844$	−19.4285	−0.668758
$845$	−0.579084	−0.0199211
$846$	0	0
$847$	53.1187	1.82518
$848$	−3.88952	−0.133567
$849$	0	0
$850$	34.7983	1.19357
$851$	−6.69512	−0.229506
$852$	0	0
$853$	11.3844	0.389794	0.194897	−	0.980824i	$-0.437563\pi$
$853$	11.3844	0.389794	0.194897	+	0.980824i	$0.437563\pi$
$854$	−24.2982	−0.831467
$855$	0	0
$856$	−0.144488	−0.00493851
$857$	−20.7779	−0.709760	−0.354880	−	0.934912i	$-0.615478\pi$
$857$	−20.7779	−0.709760	−0.354880	+	0.934912i	$0.615478\pi$
$858$	0	0
$859$	−0.522296	−0.0178205	−0.00891025	−	0.999960i	$-0.502836\pi$
$859$	−0.522296	−0.0178205	−0.00891025	+	0.999960i	$0.502836\pi$
$860$	20.1293	0.686404
$861$	0	0
$862$	21.7781	0.741764
$863$	−21.6000	−0.735273	−0.367637	−	0.929969i	$-0.619833\pi$
$863$	−21.6000	−0.735273	−0.367637	+	0.929969i	$0.619833\pi$
$864$	0	0
$865$	−13.3879	−0.455202
$866$	−15.7500	−0.535207
$867$	0	0
$868$	25.6617	0.871016
$869$	−22.5107	−0.763624
$870$	0	0
$871$	1.34644	0.0456223
$872$	−12.4288	−0.420893
$873$	0	0
$874$	−48.9693	−1.65641
$875$	−21.8620	−0.739070
$876$	0	0
$877$	−30.1710	−1.01880	−0.509401	−	0.860529i	$-0.670133\pi$
$877$	−30.1710	−1.01880	−0.509401	+	0.860529i	$0.670133\pi$
$878$	29.8767	1.00829
$879$	0	0
$880$	−18.9834	−0.639931
$881$	7.64922	0.257709	0.128854	−	0.991664i	$-0.458870\pi$
$881$	7.64922	0.257709	0.128854	+	0.991664i	$0.458870\pi$
$882$	0	0
$883$	−1.50660	−0.0507012	−0.0253506	−	0.999679i	$-0.508070\pi$
$883$	−1.50660	−0.0507012	−0.0253506	+	0.999679i	$0.508070\pi$
$884$	−17.5644	−0.590755
$885$	0	0
$886$	33.1876	1.11496
$887$	−39.8124	−1.33677	−0.668385	−	0.743816i	$-0.733014\pi$
$887$	−39.8124	−1.33677	−0.668385	+	0.743816i	$0.733014\pi$
$888$	0	0
$889$	−55.9036	−1.87495
$890$	11.5652	0.387666
$891$	0	0
$892$	13.2705	0.444328
$893$	54.1913	1.81344
$894$	0	0
$895$	−15.2529	−0.509850
$896$	−2.86107	−0.0955816
$897$	0	0
$898$	33.4784	1.11719
$899$	52.1323	1.73871
$900$	0	0
$901$	18.8280	0.627252
$902$	−3.59090	−0.119564
$903$	0	0
$904$	−4.15551	−0.138210
$905$	12.9802	0.431475
$906$	0	0
$907$	−9.53619	−0.316644	−0.158322	−	0.987388i	$-0.550608\pi$
$907$	−9.53619	−0.316644	−0.158322	+	0.987388i	$0.550608\pi$
$908$	8.11658	0.269358
$909$	0	0
$910$	36.2436	1.20146
$911$	5.26468	0.174427	0.0872133	−	0.996190i	$-0.472204\pi$
$911$	5.26468	0.174427	0.0872133	+	0.996190i	$0.472204\pi$
$912$	0	0
$913$	−15.5167	−0.513526
$914$	−15.5789	−0.515304
$915$	0	0
$916$	17.7272	0.585723
$917$	−39.4393	−1.30240
$918$	0	0
$919$	−37.6756	−1.24280	−0.621401	−	0.783492i	$-0.713436\pi$
$919$	−37.6756	−1.24280	−0.621401	+	0.783492i	$0.713436\pi$
$920$	23.3742	0.770624
$921$	0	0
$922$	19.6290	0.646445
$923$	2.12252	0.0698636
$924$	0	0
$925$	−7.18868	−0.236362
$926$	−26.0267	−0.855291
$927$	0	0
$928$	−5.81232	−0.190799
$929$	2.88639	0.0946995	0.0473497	−	0.998878i	$-0.484922\pi$
$929$	2.88639	0.0946995	0.0473497	+	0.998878i	$0.484922\pi$
$930$	0	0
$931$	8.67255	0.284231
$932$	−9.99766	−0.327484
$933$	0	0
$934$	−27.4418	−0.897925
$935$	91.8932	3.00523
$936$	0	0
$937$	23.2591	0.759842	0.379921	−	0.925019i	$-0.375951\pi$
$937$	23.2591	0.759842	0.379921	+	0.925019i	$0.375951\pi$
$938$	−1.06167	−0.0346648
$939$	0	0
$940$	−25.8668	−0.843681
$941$	26.7089	0.870686	0.435343	−	0.900265i	$-0.356627\pi$
$941$	26.7089	0.870686	0.435343	+	0.900265i	$0.356627\pi$
$942$	0	0
$943$	4.42146	0.143982
$944$	−1.56553	−0.0509537
$945$	0	0
$946$	31.3507	1.01930
$947$	54.1066	1.75823	0.879115	−	0.476610i	$-0.158135\pi$
$947$	54.1066	1.75823	0.879115	+	0.476610i	$0.158135\pi$
$948$	0	0
$949$	−43.5135	−1.41251
$950$	−52.5793	−1.70590
$951$	0	0
$952$	13.8496	0.448868
$953$	35.2174	1.14080	0.570402	−	0.821366i	$-0.306787\pi$
$953$	35.2174	1.14080	0.570402	+	0.821366i	$0.306787\pi$
$954$	0	0
$955$	−19.3185	−0.625131
$956$	16.5064	0.533856
$957$	0	0
$958$	20.6312	0.666563
$959$	49.0362	1.58346
$960$	0	0
$961$	49.4480	1.59510
$962$	3.62848	0.116987
$963$	0	0
$964$	16.9919	0.547272
$965$	80.6180	2.59518
$966$	0	0
$967$	−14.2979	−0.459790	−0.229895	−	0.973215i	$-0.573838\pi$
$967$	−14.2979	−0.459790	−0.229895	+	0.973215i	$0.573838\pi$
$968$	−18.5660	−0.596734
$969$	0	0
$970$	44.8096	1.43875
$971$	10.0236	0.321672	0.160836	−	0.986981i	$-0.448581\pi$
$971$	10.0236	0.321672	0.160836	+	0.986981i	$0.448581\pi$
$972$	0	0
$973$	−5.03579	−0.161440
$974$	1.04823	0.0335874
$975$	0	0
$976$	8.49269	0.271845
$977$	−0.00109169	−3.49262e−5 0	−1.74631e−5	−	1.00000i	$-0.500006\pi$
$977$	−0.00109169	−3.49262e−5 0	−1.74631e−5		1.00000i	$0.500006\pi$
$978$	0	0
$979$	18.0124	0.575678
$980$	−4.13961	−0.132235
$981$	0	0
$982$	33.0819	1.05568
$983$	28.0299	0.894016	0.447008	−	0.894530i	$-0.352490\pi$
$983$	28.0299	0.894016	0.447008	+	0.894530i	$0.352490\pi$
$984$	0	0
$985$	50.9235	1.62256
$986$	28.1357	0.896025
$987$	0	0
$988$	26.5394	0.844330
$989$	−38.6019	−1.22747
$990$	0	0
$991$	−25.5039	−0.810157	−0.405078	−	0.914282i	$-0.632756\pi$
$991$	−25.5039	−0.810157	−0.405078	+	0.914282i	$0.632756\pi$
$992$	−8.96928	−0.284775
$993$	0	0
$994$	−1.67361	−0.0530838
$995$	−69.4542	−2.20185
$996$	0	0
$997$	−43.2073	−1.36839	−0.684195	−	0.729300i	$-0.739846\pi$
$997$	−43.2073	−1.36839	−0.684195	+	0.729300i	$0.739846\pi$
$998$	6.65247	0.210580
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5994.2.a.ba.1.2		10
3.2	odd	2		5994.2.a.bb.1.9		10
9.2	odd	6		666.2.e.e.445.8	yes	20
9.4	even	3		1998.2.e.e.667.9		20
9.5	odd	6		666.2.e.e.223.8	✓	20
9.7	even	3		1998.2.e.e.1333.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
666.2.e.e.223.8	✓	20	9.5	odd	6
666.2.e.e.445.8	yes	20	9.2	odd	6
1998.2.e.e.667.9		20	9.4	even	3
1998.2.e.e.1333.9		20	9.7	even	3
5994.2.a.ba.1.2		10	1.1	even	1	trivial
5994.2.a.bb.1.9		10	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5994 = 2 \cdot 3^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5994.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.49123 q^{5} +2.86107 q^{7} -1.00000 q^{8} +3.49123 q^{10} +5.43746 q^{11} +3.62848 q^{13} -2.86107 q^{14} +1.00000 q^{16} -4.84071 q^{17} +7.31418 q^{19} -3.49123 q^{20} -5.43746 q^{22} +6.69512 q^{23} +7.18868 q^{25} -3.62848 q^{26} +2.86107 q^{28} +5.81232 q^{29} +8.96928 q^{31} -1.00000 q^{32} +4.84071 q^{34} -9.98865 q^{35} -1.00000 q^{37} -7.31418 q^{38} +3.49123 q^{40} +0.660400 q^{41} -5.76568 q^{43} +5.43746 q^{44} -6.69512 q^{46} +7.40907 q^{47} +1.18572 q^{49} -7.18868 q^{50} +3.62848 q^{52} -3.88952 q^{53} -18.9834 q^{55} -2.86107 q^{56} -5.81232 q^{58} -1.56553 q^{59} +8.49269 q^{61} -8.96928 q^{62} +1.00000 q^{64} -12.6679 q^{65} +0.371075 q^{67} -4.84071 q^{68} +9.98865 q^{70} +0.584961 q^{71} -11.9922 q^{73} +1.00000 q^{74} +7.31418 q^{76} +15.5570 q^{77} -4.13993 q^{79} -3.49123 q^{80} -0.660400 q^{82} -2.85366 q^{83} +16.9000 q^{85} +5.76568 q^{86} -5.43746 q^{88} +3.31264 q^{89} +10.3813 q^{91} +6.69512 q^{92} -7.40907 q^{94} -25.5355 q^{95} +12.8349 q^{97} -1.18572 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 10 q^{4} - q^{5} + q^{7} - 10 q^{8} + q^{10} - 3 q^{11} + 12 q^{13} - q^{14} + 10 q^{16} - 12 q^{17} + 24 q^{19} - q^{20} + 3 q^{22} - 3 q^{23} + 21 q^{25} - 12 q^{26} + q^{28} - 4 q^{29}+ \cdots - 35 q^{98}+O(q^{100}) \) 10 * q - 10 * q^2 + 10 * q^4 - q^5 + q^7 - 10 * q^8 + q^10 - 3 * q^11 + 12 * q^13 - q^14 + 10 * q^16 - 12 * q^17 + 24 * q^19 - q^20 + 3 * q^22 - 3 * q^23 + 21 * q^25 - 12 * q^26 + q^28 - 4 * q^29 + 11 * q^31 - 10 * q^32 + 12 * q^34 + 7 * q^35 - 10 * q^37 - 24 * q^38 + q^40 + 3 * q^41 + 6 * q^43 - 3 * q^44 + 3 * q^46 - 8 * q^47 + 35 * q^49 - 21 * q^50 + 12 * q^52 + 10 * q^53 + 2 * q^55 - q^56 + 4 * q^58 - 10 * q^59 + 2 * q^61 - 11 * q^62 + 10 * q^64 - 26 * q^65 + 7 * q^67 - 12 * q^68 - 7 * q^70 - 15 * q^71 + 3 * q^73 + 10 * q^74 + 24 * q^76 - 9 * q^77 + 16 * q^79 - q^80 - 3 * q^82 + 23 * q^83 + 15 * q^85 - 6 * q^86 + 3 * q^88 - 21 * q^89 + 59 * q^91 - 3 * q^92 + 8 * q^94 + 6 * q^95 + 24 * q^97 - 35 * q^98

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