LMFDB - Embedded newform 5994.2.a.bb.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			$1.15778$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.33861 q^{5} +0.195962 q^{7} +1.00000 q^{8} -3.33861 q^{10} -5.08601 q^{11} -5.32840 q^{13} +0.195962 q^{14} +1.00000 q^{16} +0.739743 q^{17} +5.26526 q^{19} -3.33861 q^{20} -5.08601 q^{22} +2.50863 q^{23} +6.14633 q^{25} -5.32840 q^{26} +0.195962 q^{28} -6.13351 q^{29} -1.59981 q^{31} +1.00000 q^{32} +0.739743 q^{34} -0.654240 q^{35} -1.00000 q^{37} +5.26526 q^{38} -3.33861 q^{40} -5.25874 q^{41} +4.47515 q^{43} -5.08601 q^{44} +2.50863 q^{46} +13.4814 q^{47} -6.96160 q^{49} +6.14633 q^{50} -5.32840 q^{52} -9.76490 q^{53} +16.9802 q^{55} +0.195962 q^{56} -6.13351 q^{58} +0.821294 q^{59} +6.12609 q^{61} -1.59981 q^{62} +1.00000 q^{64} +17.7895 q^{65} +7.02877 q^{67} +0.739743 q^{68} -0.654240 q^{70} +10.4556 q^{71} +7.73574 q^{73} -1.00000 q^{74} +5.26526 q^{76} -0.996662 q^{77} -0.927623 q^{79} -3.33861 q^{80} -5.25874 q^{82} -8.85942 q^{83} -2.46971 q^{85} +4.47515 q^{86} -5.08601 q^{88} -10.6441 q^{89} -1.04416 q^{91} +2.50863 q^{92} +13.4814 q^{94} -17.5787 q^{95} +10.4087 q^{97} -6.96160 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q + 10 q^{2} + 10 q^{4} + q^{5} + q^{7} + 10 q^{8} + q^{10} + 3 q^{11} + 12 q^{13} + q^{14} + 10 q^{16} + 12 q^{17} + 24 q^{19} + q^{20} + 3 q^{22} + 3 q^{23} + 21 q^{25} + 12 q^{26} + q^{28} + 4 q^{29}+ \cdots + 35 q^{98}+O(q^{100})$ 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
$5$	−3.33861	−1.49307	−0.746536	−	0.665344i	$-0.768285\pi$
$5$	−3.33861	−1.49307	−0.746536	+	0.665344i	$0.768285\pi$
$6$	0	0
$7$	0.195962	0.0740665	0.0370333	−	0.999314i	$-0.488209\pi$
$7$	0.195962	0.0740665	0.0370333	+	0.999314i	$0.488209\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.33861	−1.05576
$11$	−5.08601	−1.53349	−0.766745	−	0.641952i	$-0.778125\pi$
$11$	−5.08601	−1.53349	−0.766745	+	0.641952i	$0.778125\pi$
$12$	0	0
$13$	−5.32840	−1.47783	−0.738917	−	0.673797i	$-0.764662\pi$
$13$	−5.32840	−1.47783	−0.738917	+	0.673797i	$0.764662\pi$
$14$	0.195962	0.0523729
$15$	0	0
$16$	1.00000	0.250000
$17$	0.739743	0.179414	0.0897070	−	0.995968i	$-0.471407\pi$
$17$	0.739743	0.179414	0.0897070	+	0.995968i	$0.471407\pi$
$18$	0	0
$19$	5.26526	1.20793	0.603966	−	0.797010i	$-0.293586\pi$
$19$	5.26526	1.20793	0.603966	+	0.797010i	$0.293586\pi$
$20$	−3.33861	−0.746536
$21$	0	0
$22$	−5.08601	−1.08434
$23$	2.50863	0.523085	0.261543	−	0.965192i	$-0.415769\pi$
$23$	2.50863	0.523085	0.261543	+	0.965192i	$0.415769\pi$
$24$	0	0
$25$	6.14633	1.22927
$26$	−5.32840	−1.04499
$27$	0	0
$28$	0.195962	0.0370333
$29$	−6.13351	−1.13896	−0.569482	−	0.822004i	$-0.692856\pi$
$29$	−6.13351	−1.13896	−0.569482	+	0.822004i	$0.692856\pi$
$30$	0	0
$31$	−1.59981	−0.287334	−0.143667	−	0.989626i	$-0.545889\pi$
$31$	−1.59981	−0.287334	−0.143667	+	0.989626i	$0.545889\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.739743	0.126865
$35$	−0.654240	−0.110587
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	5.26526	0.854137
$39$	0	0
$40$	−3.33861	−0.527881
$41$	−5.25874	−0.821277	−0.410638	−	0.911798i	$-0.634694\pi$
$41$	−5.25874	−0.821277	−0.410638	+	0.911798i	$0.634694\pi$
$42$	0	0
$43$	4.47515	0.682453	0.341227	−	0.939981i	$-0.389158\pi$
$43$	4.47515	0.682453	0.341227	+	0.939981i	$0.389158\pi$
$44$	−5.08601	−0.766745
$45$	0	0
$46$	2.50863	0.369877
$47$	13.4814	1.96647	0.983233	−	0.182352i	$-0.0583709\pi$
$47$	13.4814	1.96647	0.983233	+	0.182352i	$0.0583709\pi$
$48$	0	0
$49$	−6.96160	−0.994514
$50$	6.14633	0.869223
$51$	0	0
$52$	−5.32840	−0.738917
$53$	−9.76490	−1.34131	−0.670656	−	0.741768i	$-0.733987\pi$
$53$	−9.76490	−1.34131	−0.670656	+	0.741768i	$0.733987\pi$
$54$	0	0
$55$	16.9802	2.28961
$56$	0.195962	0.0261865
$57$	0	0
$58$	−6.13351	−0.805369
$59$	0.821294	0.106923	0.0534617	−	0.998570i	$-0.482974\pi$
$59$	0.821294	0.106923	0.0534617	+	0.998570i	$0.482974\pi$
$60$	0	0
$61$	6.12609	0.784366	0.392183	−	0.919887i	$-0.371720\pi$
$61$	6.12609	0.784366	0.392183	+	0.919887i	$0.371720\pi$
$62$	−1.59981	−0.203176
$63$	0	0
$64$	1.00000	0.125000
$65$	17.7895	2.20651
$66$	0	0
$67$	7.02877	0.858700	0.429350	−	0.903138i	$-0.358743\pi$
$67$	7.02877	0.858700	0.429350	+	0.903138i	$0.358743\pi$
$68$	0.739743	0.0897070
$69$	0	0
$70$	−0.654240	−0.0781966
$71$	10.4556	1.24085	0.620426	−	0.784265i	$-0.286960\pi$
$71$	10.4556	1.24085	0.620426	+	0.784265i	$0.286960\pi$
$72$	0	0
$73$	7.73574	0.905400	0.452700	−	0.891663i	$-0.350461\pi$
$73$	7.73574	0.905400	0.452700	+	0.891663i	$0.350461\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	5.26526	0.603966
$77$	−0.996662	−0.113580
$78$	0	0
$79$	−0.927623	−0.104366	−0.0521829	−	0.998638i	$-0.516618\pi$
$79$	−0.927623	−0.104366	−0.0521829	+	0.998638i	$0.516618\pi$
$80$	−3.33861	−0.373268
$81$	0	0
$82$	−5.25874	−0.580730
$83$	−8.85942	−0.972448	−0.486224	−	0.873834i	$-0.661626\pi$
$83$	−8.85942	−0.972448	−0.486224	+	0.873834i	$0.661626\pi$
$84$	0	0
$85$	−2.46971	−0.267878
$86$	4.47515	0.482567
$87$	0	0
$88$	−5.08601	−0.542170
$89$	−10.6441	−1.12827	−0.564137	−	0.825682i	$-0.690791\pi$
$89$	−10.6441	−1.12827	−0.564137	+	0.825682i	$0.690791\pi$
$90$	0	0
$91$	−1.04416	−0.109458
$92$	2.50863	0.261543
$93$	0	0
$94$	13.4814	1.39050
$95$	−17.5787	−1.80353
$96$	0	0
$97$	10.4087	1.05684	0.528422	−	0.848982i	$-0.322784\pi$
$97$	10.4087	1.05684	0.528422	+	0.848982i	$0.322784\pi$
$98$	−6.96160	−0.703228
$99$	0	0
$100$	6.14633	0.614633
$101$	−2.91641	−0.290193	−0.145097	−	0.989417i	$-0.546349\pi$
$101$	−2.91641	−0.290193	−0.145097	+	0.989417i	$0.546349\pi$
$102$	0	0
$103$	12.6134	1.24284	0.621419	−	0.783478i	$-0.286557\pi$
$103$	12.6134	1.24284	0.621419	+	0.783478i	$0.286557\pi$
$104$	−5.32840	−0.522493
$105$	0	0
$106$	−9.76490	−0.948451
$107$	3.87889	0.374987	0.187493	−	0.982266i	$-0.439964\pi$
$107$	3.87889	0.374987	0.187493	+	0.982266i	$0.439964\pi$
$108$	0	0
$109$	0.536436	0.0513812	0.0256906	−	0.999670i	$-0.491822\pi$
$109$	0.536436	0.0513812	0.0256906	+	0.999670i	$0.491822\pi$
$110$	16.9802	1.61900
$111$	0	0
$112$	0.195962	0.0185166
$113$	19.0344	1.79061	0.895304	−	0.445456i	$-0.146959\pi$
$113$	19.0344	1.79061	0.895304	+	0.445456i	$0.146959\pi$
$114$	0	0
$115$	−8.37534	−0.781005
$116$	−6.13351	−0.569482
$117$	0	0
$118$	0.821294	0.0756063
$119$	0.144961	0.0132886
$120$	0	0
$121$	14.8675	1.35159
$122$	6.12609	0.554630
$123$	0	0
$124$	−1.59981	−0.143667
$125$	−3.82717	−0.342312
$126$	0	0
$127$	20.0651	1.78049	0.890247	−	0.455479i	$-0.150532\pi$
$127$	20.0651	1.78049	0.890247	+	0.455479i	$0.150532\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	17.7895	1.56024
$131$	8.42646	0.736223	0.368112	−	0.929782i	$-0.380004\pi$
$131$	8.42646	0.736223	0.368112	+	0.929782i	$0.380004\pi$
$132$	0	0
$133$	1.03179	0.0894674
$134$	7.02877	0.607193
$135$	0	0
$136$	0.739743	0.0634324
$137$	−20.8102	−1.77794	−0.888969	−	0.457968i	$-0.848577\pi$
$137$	−20.8102	−1.77794	−0.888969	+	0.457968i	$0.848577\pi$
$138$	0	0
$139$	−0.122437	−0.0103849	−0.00519246	−	0.999987i	$-0.501653\pi$
$139$	−0.122437	−0.0103849	−0.00519246	+	0.999987i	$0.501653\pi$
$140$	−0.654240	−0.0552934
$141$	0	0
$142$	10.4556	0.877414
$143$	27.1003	2.26624
$144$	0	0
$145$	20.4774	1.70056
$146$	7.73574	0.640214
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	6.84115	0.560449	0.280224	−	0.959935i	$-0.409591\pi$
$149$	6.84115	0.560449	0.280224	+	0.959935i	$0.409591\pi$
$150$	0	0
$151$	21.8371	1.77708	0.888538	−	0.458803i	$-0.151722\pi$
$151$	21.8371	1.77708	0.888538	+	0.458803i	$0.151722\pi$
$152$	5.26526	0.427069
$153$	0	0
$154$	−0.996662	−0.0803133
$155$	5.34115	0.429011
$156$	0	0
$157$	−19.8674	−1.58559	−0.792797	−	0.609486i	$-0.791376\pi$
$157$	−19.8674	−1.58559	−0.792797	+	0.609486i	$0.791376\pi$
$158$	−0.927623	−0.0737977
$159$	0	0
$160$	−3.33861	−0.263941
$161$	0.491595	0.0387431
$162$	0	0
$163$	−4.27443	−0.334799	−0.167399	−	0.985889i	$-0.553537\pi$
$163$	−4.27443	−0.334799	−0.167399	+	0.985889i	$0.553537\pi$
$164$	−5.25874	−0.410638
$165$	0	0
$166$	−8.85942	−0.687624
$167$	0.615260	0.0476102	0.0238051	−	0.999717i	$-0.492422\pi$
$167$	0.615260	0.0476102	0.0238051	+	0.999717i	$0.492422\pi$
$168$	0	0
$169$	15.3919	1.18399
$170$	−2.46971	−0.189418
$171$	0	0
$172$	4.47515	0.341227
$173$	20.9888	1.59575	0.797873	−	0.602826i	$-0.205959\pi$
$173$	20.9888	1.59575	0.797873	+	0.602826i	$0.205959\pi$
$174$	0	0
$175$	1.20445	0.0910475
$176$	−5.08601	−0.383372
$177$	0	0
$178$	−10.6441	−0.797810
$179$	−22.4414	−1.67735	−0.838673	−	0.544635i	$-0.816668\pi$
$179$	−22.4414	−1.67735	−0.838673	+	0.544635i	$0.816668\pi$
$180$	0	0
$181$	5.60462	0.416588	0.208294	−	0.978066i	$-0.433209\pi$
$181$	5.60462	0.416588	0.208294	+	0.978066i	$0.433209\pi$
$182$	−1.04416	−0.0773985
$183$	0	0
$184$	2.50863	0.184939
$185$	3.33861	0.245460
$186$	0	0
$187$	−3.76234	−0.275129
$188$	13.4814	0.983233
$189$	0	0
$190$	−17.5787	−1.27529
$191$	−4.12388	−0.298393	−0.149197	−	0.988808i	$-0.547669\pi$
$191$	−4.12388	−0.298393	−0.149197	+	0.988808i	$0.547669\pi$
$192$	0	0
$193$	−8.55099	−0.615513	−0.307757	−	0.951465i	$-0.599578\pi$
$193$	−8.55099	−0.615513	−0.307757	+	0.951465i	$0.599578\pi$
$194$	10.4087	0.747302
$195$	0	0
$196$	−6.96160	−0.497257
$197$	13.1755	0.938714	0.469357	−	0.883009i	$-0.344486\pi$
$197$	13.1755	0.938714	0.469357	+	0.883009i	$0.344486\pi$
$198$	0	0
$199$	18.5811	1.31718	0.658589	−	0.752503i	$-0.271154\pi$
$199$	18.5811	1.31718	0.658589	+	0.752503i	$0.271154\pi$
$200$	6.14633	0.434611
$201$	0	0
$202$	−2.91641	−0.205198
$203$	−1.20193	−0.0843591
$204$	0	0
$205$	17.5569	1.22623
$206$	12.6134	0.878819
$207$	0	0
$208$	−5.32840	−0.369458
$209$	−26.7791	−1.85235
$210$	0	0
$211$	−4.80875	−0.331048	−0.165524	−	0.986206i	$-0.552932\pi$
$211$	−4.80875	−0.331048	−0.165524	+	0.986206i	$0.552932\pi$
$212$	−9.76490	−0.670656
$213$	0	0
$214$	3.87889	0.265156
$215$	−14.9408	−1.01895
$216$	0	0
$217$	−0.313501	−0.0212819
$218$	0.536436	0.0363320
$219$	0	0
$220$	16.9802	1.14481
$221$	−3.94165	−0.265144
$222$	0	0
$223$	5.28621	0.353991	0.176995	−	0.984212i	$-0.443362\pi$
$223$	5.28621	0.353991	0.176995	+	0.984212i	$0.443362\pi$
$224$	0.195962	0.0130932
$225$	0	0
$226$	19.0344	1.26615
$227$	6.52175	0.432864	0.216432	−	0.976298i	$-0.430558\pi$
$227$	6.52175	0.432864	0.216432	+	0.976298i	$0.430558\pi$
$228$	0	0
$229$	5.93818	0.392406	0.196203	−	0.980563i	$-0.437139\pi$
$229$	5.93818	0.392406	0.196203	+	0.980563i	$0.437139\pi$
$230$	−8.37534	−0.552254
$231$	0	0
$232$	−6.13351	−0.402685
$233$	18.4107	1.20613	0.603063	−	0.797693i	$-0.293947\pi$
$233$	18.4107	1.20613	0.603063	+	0.797693i	$0.293947\pi$
$234$	0	0
$235$	−45.0092	−2.93608
$236$	0.821294	0.0534617
$237$	0	0
$238$	0.144961	0.00939644
$239$	24.0032	1.55264	0.776320	−	0.630339i	$-0.217084\pi$
$239$	24.0032	1.55264	0.776320	+	0.630339i	$0.217084\pi$
$240$	0	0
$241$	−17.2454	−1.11087	−0.555437	−	0.831559i	$-0.687449\pi$
$241$	−17.2454	−1.11087	−0.555437	+	0.831559i	$0.687449\pi$
$242$	14.8675	0.955717
$243$	0	0
$244$	6.12609	0.392183
$245$	23.2421	1.48488
$246$	0	0
$247$	−28.0554	−1.78512
$248$	−1.59981	−0.101588
$249$	0	0
$250$	−3.82717	−0.242051
$251$	5.56820	0.351462	0.175731	−	0.984438i	$-0.443771\pi$
$251$	5.56820	0.351462	0.175731	+	0.984438i	$0.443771\pi$
$252$	0	0
$253$	−12.7589	−0.802146
$254$	20.0651	1.25900
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.8042	−0.798705	−0.399352	−	0.916798i	$-0.630765\pi$
$257$	−12.8042	−0.798705	−0.399352	+	0.916798i	$0.630765\pi$
$258$	0	0
$259$	−0.195962	−0.0121765
$260$	17.7895	1.10326
$261$	0	0
$262$	8.42646	0.520588
$263$	3.24586	0.200149	0.100074	−	0.994980i	$-0.468092\pi$
$263$	3.24586	0.200149	0.100074	+	0.994980i	$0.468092\pi$
$264$	0	0
$265$	32.6012	2.00268
$266$	1.03179	0.0632630
$267$	0	0
$268$	7.02877	0.429350
$269$	−20.4388	−1.24618	−0.623089	−	0.782151i	$-0.714122\pi$
$269$	−20.4388	−1.24618	−0.623089	+	0.782151i	$0.714122\pi$
$270$	0	0
$271$	−18.6519	−1.13303	−0.566513	−	0.824053i	$-0.691708\pi$
$271$	−18.6519	−1.13303	−0.566513	+	0.824053i	$0.691708\pi$
$272$	0.739743	0.0448535
$273$	0	0
$274$	−20.8102	−1.25719
$275$	−31.2603	−1.88507
$276$	0	0
$277$	−6.39476	−0.384224	−0.192112	−	0.981373i	$-0.561534\pi$
$277$	−6.39476	−0.384224	−0.192112	+	0.981373i	$0.561534\pi$
$278$	−0.122437	−0.00734325
$279$	0	0
$280$	−0.654240	−0.0390983
$281$	−20.2860	−1.21016	−0.605081	−	0.796164i	$-0.706859\pi$
$281$	−20.2860	−1.21016	−0.605081	+	0.796164i	$0.706859\pi$
$282$	0	0
$283$	21.7594	1.29346	0.646730	−	0.762719i	$-0.276136\pi$
$283$	21.7594	1.29346	0.646730	+	0.762719i	$0.276136\pi$
$284$	10.4556	0.620426
$285$	0	0
$286$	27.1003	1.60247
$287$	−1.03051	−0.0608291
$288$	0	0
$289$	−16.4528	−0.967811
$290$	20.4774	1.20248
$291$	0	0
$292$	7.73574	0.452700
$293$	29.9252	1.74825	0.874125	−	0.485700i	$-0.161435\pi$
$293$	29.9252	1.74825	0.874125	+	0.485700i	$0.161435\pi$
$294$	0	0
$295$	−2.74198	−0.159644
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	6.84115	0.396297
$299$	−13.3670	−0.773033
$300$	0	0
$301$	0.876957	0.0505469
$302$	21.8371	1.25658
$303$	0	0
$304$	5.26526	0.301983
$305$	−20.4527	−1.17112
$306$	0	0
$307$	−22.7730	−1.29972	−0.649861	−	0.760053i	$-0.725173\pi$
$307$	−22.7730	−1.29972	−0.649861	+	0.760053i	$0.725173\pi$
$308$	−0.996662	−0.0567901
$309$	0	0
$310$	5.34115	0.303357
$311$	−31.3806	−1.77943	−0.889716	−	0.456513i	$-0.849098\pi$
$311$	−31.3806	−1.77943	−0.889716	+	0.456513i	$0.849098\pi$
$312$	0	0
$313$	4.18723	0.236676	0.118338	−	0.992973i	$-0.462243\pi$
$313$	4.18723	0.236676	0.118338	+	0.992973i	$0.462243\pi$
$314$	−19.8674	−1.12118
$315$	0	0
$316$	−0.927623	−0.0521829
$317$	−9.71753	−0.545791	−0.272895	−	0.962044i	$-0.587981\pi$
$317$	−9.71753	−0.545791	−0.272895	+	0.962044i	$0.587981\pi$
$318$	0	0
$319$	31.1951	1.74659
$320$	−3.33861	−0.186634
$321$	0	0
$322$	0.491595	0.0273955
$323$	3.89493	0.216720
$324$	0	0
$325$	−32.7502	−1.81665
$326$	−4.27443	−0.236739
$327$	0	0
$328$	−5.25874	−0.290365
$329$	2.64184	0.145649
$330$	0	0
$331$	4.19467	0.230560	0.115280	−	0.993333i	$-0.463223\pi$
$331$	4.19467	0.230560	0.115280	+	0.993333i	$0.463223\pi$
$332$	−8.85942	−0.486224
$333$	0	0
$334$	0.615260	0.0336655
$335$	−23.4663	−1.28210
$336$	0	0
$337$	17.3297	0.944010	0.472005	−	0.881596i	$-0.343530\pi$
$337$	17.3297	0.944010	0.472005	+	0.881596i	$0.343530\pi$
$338$	15.3919	0.837208
$339$	0	0
$340$	−2.46971	−0.133939
$341$	8.13665	0.440624
$342$	0	0
$343$	−2.73594	−0.147727
$344$	4.47515	0.241284
$345$	0	0
$346$	20.9888	1.12836
$347$	−0.604429	−0.0324474	−0.0162237	−	0.999868i	$-0.505164\pi$
$347$	−0.604429	−0.0324474	−0.0162237	+	0.999868i	$0.505164\pi$
$348$	0	0
$349$	15.8975	0.850971	0.425486	−	0.904965i	$-0.360103\pi$
$349$	15.8975	0.850971	0.425486	+	0.904965i	$0.360103\pi$
$350$	1.20445	0.0643803
$351$	0	0
$352$	−5.08601	−0.271085
$353$	−5.98441	−0.318518	−0.159259	−	0.987237i	$-0.550911\pi$
$353$	−5.98441	−0.318518	−0.159259	+	0.987237i	$0.550911\pi$
$354$	0	0
$355$	−34.9072	−1.85268
$356$	−10.6441	−0.564137
$357$	0	0
$358$	−22.4414	−1.18606
$359$	7.52687	0.397253	0.198627	−	0.980075i	$-0.436352\pi$
$359$	7.52687	0.397253	0.198627	+	0.980075i	$0.436352\pi$
$360$	0	0
$361$	8.72292	0.459101
$362$	5.60462	0.294572
$363$	0	0
$364$	−1.04416	−0.0547290
$365$	−25.8266	−1.35183
$366$	0	0
$367$	−18.5266	−0.967080	−0.483540	−	0.875322i	$-0.660649\pi$
$367$	−18.5266	−0.967080	−0.483540	+	0.875322i	$0.660649\pi$
$368$	2.50863	0.130771
$369$	0	0
$370$	3.33861	0.173566
$371$	−1.91355	−0.0993464
$372$	0	0
$373$	−29.4102	−1.52280	−0.761401	−	0.648282i	$-0.775488\pi$
$373$	−29.4102	−1.52280	−0.761401	+	0.648282i	$0.775488\pi$
$374$	−3.76234	−0.194546
$375$	0	0
$376$	13.4814	0.695251
$377$	32.6818	1.68320
$378$	0	0
$379$	10.0742	0.517475	0.258738	−	0.965948i	$-0.416694\pi$
$379$	10.0742	0.517475	0.258738	+	0.965948i	$0.416694\pi$
$380$	−17.5787	−0.901766
$381$	0	0
$382$	−4.12388	−0.210996
$383$	15.6220	0.798248	0.399124	−	0.916897i	$-0.369314\pi$
$383$	15.6220	0.798248	0.399124	+	0.916897i	$0.369314\pi$
$384$	0	0
$385$	3.32747	0.169584
$386$	−8.55099	−0.435234
$387$	0	0
$388$	10.4087	0.528422
$389$	16.6098	0.842151	0.421076	−	0.907026i	$-0.361653\pi$
$389$	16.6098	0.842151	0.421076	+	0.907026i	$0.361653\pi$
$390$	0	0
$391$	1.85574	0.0938488
$392$	−6.96160	−0.351614
$393$	0	0
$394$	13.1755	0.663771
$395$	3.09697	0.155826
$396$	0	0
$397$	23.9512	1.20207	0.601037	−	0.799221i	$-0.294754\pi$
$397$	23.9512	1.20207	0.601037	+	0.799221i	$0.294754\pi$
$398$	18.5811	0.931385
$399$	0	0
$400$	6.14633	0.307317
$401$	−28.2179	−1.40913	−0.704566	−	0.709638i	$-0.748858\pi$
$401$	−28.2179	−1.40913	−0.704566	+	0.709638i	$0.748858\pi$
$402$	0	0
$403$	8.52443	0.424632
$404$	−2.91641	−0.145097
$405$	0	0
$406$	−1.20193	−0.0596509
$407$	5.08601	0.252104
$408$	0	0
$409$	20.8364	1.03029	0.515147	−	0.857102i	$-0.327737\pi$
$409$	20.8364	1.03029	0.515147	+	0.857102i	$0.327737\pi$
$410$	17.5569	0.867073
$411$	0	0
$412$	12.6134	0.621419
$413$	0.160942	0.00791945
$414$	0	0
$415$	29.5782	1.45194
$416$	−5.32840	−0.261246
$417$	0	0
$418$	−26.7791	−1.30981
$419$	23.4768	1.14691	0.573457	−	0.819235i	$-0.305602\pi$
$419$	23.4768	1.14691	0.573457	+	0.819235i	$0.305602\pi$
$420$	0	0
$421$	−26.6544	−1.29905	−0.649527	−	0.760338i	$-0.725033\pi$
$421$	−26.6544	−1.29905	−0.649527	+	0.760338i	$0.725033\pi$
$422$	−4.80875	−0.234086
$423$	0	0
$424$	−9.76490	−0.474226
$425$	4.54671	0.220548
$426$	0	0
$427$	1.20048	0.0580952
$428$	3.87889	0.187493
$429$	0	0
$430$	−14.9408	−0.720508
$431$	27.9472	1.34617	0.673085	−	0.739565i	$-0.264969\pi$
$431$	27.9472	1.34617	0.673085	+	0.739565i	$0.264969\pi$
$432$	0	0
$433$	14.4980	0.696730	0.348365	−	0.937359i	$-0.386737\pi$
$433$	14.4980	0.696730	0.348365	+	0.937359i	$0.386737\pi$
$434$	−0.313501	−0.0150485
$435$	0	0
$436$	0.536436	0.0256906
$437$	13.2086	0.631852
$438$	0	0
$439$	10.2482	0.489122	0.244561	−	0.969634i	$-0.421356\pi$
$439$	10.2482	0.489122	0.244561	+	0.969634i	$0.421356\pi$
$440$	16.9802	0.809500
$441$	0	0
$442$	−3.94165	−0.187485
$443$	8.13881	0.386687	0.193343	−	0.981131i	$-0.438067\pi$
$443$	8.13881	0.386687	0.193343	+	0.981131i	$0.438067\pi$
$444$	0	0
$445$	35.5365	1.68459
$446$	5.28621	0.250309
$447$	0	0
$448$	0.195962	0.00925832
$449$	1.81755	0.0857754	0.0428877	−	0.999080i	$-0.486344\pi$
$449$	1.81755	0.0857754	0.0428877	+	0.999080i	$0.486344\pi$
$450$	0	0
$451$	26.7460	1.25942
$452$	19.0344	0.895304
$453$	0	0
$454$	6.52175	0.306081
$455$	3.48605	0.163429
$456$	0	0
$457$	−38.9876	−1.82376	−0.911880	−	0.410456i	$-0.865370\pi$
$457$	−38.9876	−1.82376	−0.911880	+	0.410456i	$0.865370\pi$
$458$	5.93818	0.277473
$459$	0	0
$460$	−8.37534	−0.390502
$461$	−14.6527	−0.682445	−0.341223	−	0.939983i	$-0.610841\pi$
$461$	−14.6527	−0.682445	−0.341223	+	0.939983i	$0.610841\pi$
$462$	0	0
$463$	25.3342	1.17738	0.588689	−	0.808359i	$-0.299644\pi$
$463$	25.3342	1.17738	0.588689	+	0.808359i	$0.299644\pi$
$464$	−6.13351	−0.284741
$465$	0	0
$466$	18.4107	0.852861
$467$	−6.52767	−0.302065	−0.151032	−	0.988529i	$-0.548260\pi$
$467$	−6.52767	−0.302065	−0.151032	+	0.988529i	$0.548260\pi$
$468$	0	0
$469$	1.37737	0.0636010
$470$	−45.0092	−2.07612
$471$	0	0
$472$	0.821294	0.0378031
$473$	−22.7606	−1.04653
$474$	0	0
$475$	32.3620	1.48487
$476$	0.144961	0.00664428
$477$	0	0
$478$	24.0032	1.09788
$479$	19.4006	0.886438	0.443219	−	0.896413i	$-0.353836\pi$
$479$	19.4006	0.886438	0.443219	+	0.896413i	$0.353836\pi$
$480$	0	0
$481$	5.32840	0.242954
$482$	−17.2454	−0.785506
$483$	0	0
$484$	14.8675	0.675794
$485$	−34.7507	−1.57795
$486$	0	0
$487$	−11.5204	−0.522040	−0.261020	−	0.965333i	$-0.584059\pi$
$487$	−11.5204	−0.522040	−0.261020	+	0.965333i	$0.584059\pi$
$488$	6.12609	0.277315
$489$	0	0
$490$	23.2421	1.04997
$491$	−30.0958	−1.35821	−0.679103	−	0.734043i	$-0.737631\pi$
$491$	−30.0958	−1.35821	−0.679103	+	0.734043i	$0.737631\pi$
$492$	0	0
$493$	−4.53722	−0.204346
$494$	−28.0554	−1.26227
$495$	0	0
$496$	−1.59981	−0.0718336
$497$	2.04890	0.0919055
$498$	0	0
$499$	19.6116	0.877934	0.438967	−	0.898503i	$-0.355345\pi$
$499$	19.6116	0.877934	0.438967	+	0.898503i	$0.355345\pi$
$500$	−3.82717	−0.171156
$501$	0	0
$502$	5.56820	0.248521
$503$	−19.1959	−0.855903	−0.427951	−	0.903802i	$-0.640765\pi$
$503$	−19.1959	−0.855903	−0.427951	+	0.903802i	$0.640765\pi$
$504$	0	0
$505$	9.73675	0.433280
$506$	−12.7589	−0.567203
$507$	0	0
$508$	20.0651	0.890247
$509$	29.4055	1.30337	0.651687	−	0.758488i	$-0.274062\pi$
$509$	29.4055	1.30337	0.651687	+	0.758488i	$0.274062\pi$
$510$	0	0
$511$	1.51591	0.0670598
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.8042	−0.564769
$515$	−42.1113	−1.85565
$516$	0	0
$517$	−68.5666	−3.01556
$518$	−0.195962	−0.00861006
$519$	0	0
$520$	17.7895	0.780120
$521$	−1.28738	−0.0564013	−0.0282006	−	0.999602i	$-0.508978\pi$
$521$	−1.28738	−0.0564013	−0.0282006	+	0.999602i	$0.508978\pi$
$522$	0	0
$523$	−4.27752	−0.187043	−0.0935215	−	0.995617i	$-0.529812\pi$
$523$	−4.27752	−0.187043	−0.0935215	+	0.995617i	$0.529812\pi$
$524$	8.42646	0.368112
$525$	0	0
$526$	3.24586	0.141526
$527$	−1.18345	−0.0515518
$528$	0	0
$529$	−16.7068	−0.726382
$530$	32.6012	1.41611
$531$	0	0
$532$	1.03179	0.0447337
$533$	28.0207	1.21371
$534$	0	0
$535$	−12.9501	−0.559882
$536$	7.02877	0.303596
$537$	0	0
$538$	−20.4388	−0.881180
$539$	35.4067	1.52508
$540$	0	0
$541$	5.96148	0.256304	0.128152	−	0.991755i	$-0.459095\pi$
$541$	5.96148	0.256304	0.128152	+	0.991755i	$0.459095\pi$
$542$	−18.6519	−0.801170
$543$	0	0
$544$	0.739743	0.0317162
$545$	−1.79095	−0.0767159
$546$	0	0
$547$	28.8625	1.23407	0.617036	−	0.786935i	$-0.288333\pi$
$547$	28.8625	1.23407	0.617036	+	0.786935i	$0.288333\pi$
$548$	−20.8102	−0.888969
$549$	0	0
$550$	−31.2603	−1.33294
$551$	−32.2945	−1.37579
$552$	0	0
$553$	−0.181778	−0.00773001
$554$	−6.39476	−0.271687
$555$	0	0
$556$	−0.122437	−0.00519246
$557$	27.3289	1.15796	0.578981	−	0.815341i	$-0.303451\pi$
$557$	27.3289	1.15796	0.578981	+	0.815341i	$0.303451\pi$
$558$	0	0
$559$	−23.8454	−1.00855
$560$	−0.654240	−0.0276467
$561$	0	0
$562$	−20.2860	−0.855714
$563$	31.1305	1.31199	0.655996	−	0.754764i	$-0.272249\pi$
$563$	31.1305	1.31199	0.655996	+	0.754764i	$0.272249\pi$
$564$	0	0
$565$	−63.5485	−2.67351
$566$	21.7594	0.914615
$567$	0	0
$568$	10.4556	0.438707
$569$	−20.9407	−0.877879	−0.438939	−	0.898517i	$-0.644646\pi$
$569$	−20.9407	−0.877879	−0.438939	+	0.898517i	$0.644646\pi$
$570$	0	0
$571$	37.5254	1.57039	0.785194	−	0.619250i	$-0.212563\pi$
$571$	37.5254	1.57039	0.785194	+	0.619250i	$0.212563\pi$
$572$	27.1003	1.13312
$573$	0	0
$574$	−1.03051	−0.0430127
$575$	15.4189	0.643012
$576$	0	0
$577$	−17.9434	−0.746993	−0.373496	−	0.927632i	$-0.621841\pi$
$577$	−17.9434	−0.746993	−0.373496	+	0.927632i	$0.621841\pi$
$578$	−16.4528	−0.684345
$579$	0	0
$580$	20.4774	0.850278
$581$	−1.73611	−0.0720258
$582$	0	0
$583$	49.6644	2.05689
$584$	7.73574	0.320107
$585$	0	0
$586$	29.9252	1.23620
$587$	−37.8461	−1.56207	−0.781037	−	0.624484i	$-0.785309\pi$
$587$	−37.8461	−1.56207	−0.781037	+	0.624484i	$0.785309\pi$
$588$	0	0
$589$	−8.42341	−0.347081
$590$	−2.74198	−0.112886
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−14.5823	−0.598823	−0.299412	−	0.954124i	$-0.596790\pi$
$593$	−14.5823	−0.598823	−0.299412	+	0.954124i	$0.596790\pi$
$594$	0	0
$595$	−0.483969	−0.0198408
$596$	6.84115	0.280224
$597$	0	0
$598$	−13.3670	−0.546617
$599$	0.156940	0.00641241	0.00320620	−	0.999995i	$-0.498979\pi$
$599$	0.156940	0.00641241	0.00320620	+	0.999995i	$0.498979\pi$
$600$	0	0
$601$	36.9757	1.50827	0.754135	−	0.656720i	$-0.228056\pi$
$601$	36.9757	1.50827	0.754135	+	0.656720i	$0.228056\pi$
$602$	0.876957	0.0357421
$603$	0	0
$604$	21.8371	0.888538
$605$	−49.6367	−2.01802
$606$	0	0
$607$	40.9855	1.66355	0.831776	−	0.555112i	$-0.187324\pi$
$607$	40.9855	1.66355	0.831776	+	0.555112i	$0.187324\pi$
$608$	5.26526	0.213534
$609$	0	0
$610$	−20.4527	−0.828104
$611$	−71.8344	−2.90611
$612$	0	0
$613$	3.72829	0.150584	0.0752921	−	0.997162i	$-0.476011\pi$
$613$	3.72829	0.150584	0.0752921	+	0.997162i	$0.476011\pi$
$614$	−22.7730	−0.919042
$615$	0	0
$616$	−0.996662	−0.0401567
$617$	−16.5328	−0.665584	−0.332792	−	0.943000i	$-0.607991\pi$
$617$	−16.5328	−0.665584	−0.332792	+	0.943000i	$0.607991\pi$
$618$	0	0
$619$	24.5417	0.986415	0.493208	−	0.869912i	$-0.335824\pi$
$619$	24.5417	0.986415	0.493208	+	0.869912i	$0.335824\pi$
$620$	5.34115	0.214506
$621$	0	0
$622$	−31.3806	−1.25825
$623$	−2.08584	−0.0835673
$624$	0	0
$625$	−17.9542	−0.718170
$626$	4.18723	0.167355
$627$	0	0
$628$	−19.8674	−0.792797
$629$	−0.739743	−0.0294955
$630$	0	0
$631$	6.74272	0.268423	0.134212	−	0.990953i	$-0.457150\pi$
$631$	6.74272	0.268423	0.134212	+	0.990953i	$0.457150\pi$
$632$	−0.927623	−0.0368989
$633$	0	0
$634$	−9.71753	−0.385932
$635$	−66.9897	−2.65841
$636$	0	0
$637$	37.0942	1.46973
$638$	31.1951	1.23502
$639$	0	0
$640$	−3.33861	−0.131970
$641$	33.4562	1.32144	0.660719	−	0.750633i	$-0.270251\pi$
$641$	33.4562	1.32144	0.660719	+	0.750633i	$0.270251\pi$
$642$	0	0
$643$	44.9939	1.77439	0.887193	−	0.461398i	$-0.152652\pi$
$643$	44.9939	1.77439	0.887193	+	0.461398i	$0.152652\pi$
$644$	0.491595	0.0193716
$645$	0	0
$646$	3.89493	0.153244
$647$	4.93692	0.194090	0.0970451	−	0.995280i	$-0.469061\pi$
$647$	4.93692	0.194090	0.0970451	+	0.995280i	$0.469061\pi$
$648$	0	0
$649$	−4.17711	−0.163966
$650$	−32.7502	−1.28457
$651$	0	0
$652$	−4.27443	−0.167399
$653$	36.7104	1.43659	0.718294	−	0.695740i	$-0.244923\pi$
$653$	36.7104	1.43659	0.718294	+	0.695740i	$0.244923\pi$
$654$	0	0
$655$	−28.1327	−1.09923
$656$	−5.25874	−0.205319
$657$	0	0
$658$	2.64184	0.102990
$659$	−43.6251	−1.69939	−0.849696	−	0.527273i	$-0.823215\pi$
$659$	−43.6251	−1.69939	−0.849696	+	0.527273i	$0.823215\pi$
$660$	0	0
$661$	−13.6589	−0.531268	−0.265634	−	0.964074i	$-0.585581\pi$
$661$	−13.6589	−0.531268	−0.265634	+	0.964074i	$0.585581\pi$
$662$	4.19467	0.163031
$663$	0	0
$664$	−8.85942	−0.343812
$665$	−3.44474	−0.133581
$666$	0	0
$667$	−15.3867	−0.595776
$668$	0.615260	0.0238051
$669$	0	0
$670$	−23.4663	−0.906583
$671$	−31.1574	−1.20282
$672$	0	0
$673$	−33.0580	−1.27429	−0.637146	−	0.770743i	$-0.719885\pi$
$673$	−33.0580	−1.27429	−0.637146	+	0.770743i	$0.719885\pi$
$674$	17.3297	0.667516
$675$	0	0
$676$	15.3919	0.591996
$677$	−17.1146	−0.657767	−0.328884	−	0.944370i	$-0.606672\pi$
$677$	−17.1146	−0.657767	−0.328884	+	0.944370i	$0.606672\pi$
$678$	0	0
$679$	2.03971	0.0782768
$680$	−2.46971	−0.0947092
$681$	0	0
$682$	8.13665	0.311568
$683$	−0.957620	−0.0366423	−0.0183212	−	0.999832i	$-0.505832\pi$
$683$	−0.957620	−0.0366423	−0.0183212	+	0.999832i	$0.505832\pi$
$684$	0	0
$685$	69.4773	2.65459
$686$	−2.73594	−0.104459
$687$	0	0
$688$	4.47515	0.170613
$689$	52.0314	1.98224
$690$	0	0
$691$	−9.99795	−0.380340	−0.190170	−	0.981751i	$-0.560904\pi$
$691$	−9.99795	−0.380340	−0.190170	+	0.981751i	$0.560904\pi$
$692$	20.9888	0.797873
$693$	0	0
$694$	−0.604429	−0.0229438
$695$	0.408768	0.0155055
$696$	0	0
$697$	−3.89011	−0.147349
$698$	15.8975	0.601728
$699$	0	0
$700$	1.20445	0.0455238
$701$	3.05328	0.115321	0.0576604	−	0.998336i	$-0.481636\pi$
$701$	3.05328	0.115321	0.0576604	+	0.998336i	$0.481636\pi$
$702$	0	0
$703$	−5.26526	−0.198583
$704$	−5.08601	−0.191686
$705$	0	0
$706$	−5.98441	−0.225226
$707$	−0.571504	−0.0214936
$708$	0	0
$709$	−47.9650	−1.80136	−0.900682	−	0.434479i	$-0.856933\pi$
$709$	−47.9650	−1.80136	−0.900682	+	0.434479i	$0.856933\pi$
$710$	−34.9072	−1.31004
$711$	0	0
$712$	−10.6441	−0.398905
$713$	−4.01333	−0.150300
$714$	0	0
$715$	−90.4774	−3.38366
$716$	−22.4414	−0.838673
$717$	0	0
$718$	7.52687	0.280900
$719$	32.0932	1.19687	0.598437	−	0.801170i	$-0.295789\pi$
$719$	32.0932	1.19687	0.598437	+	0.801170i	$0.295789\pi$
$720$	0	0
$721$	2.47175	0.0920527
$722$	8.72292	0.324633
$723$	0	0
$724$	5.60462	0.208294
$725$	−37.6986	−1.40009
$726$	0	0
$727$	23.7998	0.882686	0.441343	−	0.897339i	$-0.354502\pi$
$727$	23.7998	0.882686	0.441343	+	0.897339i	$0.354502\pi$
$728$	−1.04416	−0.0386992
$729$	0	0
$730$	−25.8266	−0.955887
$731$	3.31046	0.122442
$732$	0	0
$733$	6.63216	0.244964	0.122482	−	0.992471i	$-0.460915\pi$
$733$	6.63216	0.244964	0.122482	+	0.992471i	$0.460915\pi$
$734$	−18.5266	−0.683829
$735$	0	0
$736$	2.50863	0.0924693
$737$	−35.7484	−1.31681
$738$	0	0
$739$	−42.4986	−1.56334	−0.781669	−	0.623694i	$-0.785631\pi$
$739$	−42.4986	−1.56334	−0.781669	+	0.623694i	$0.785631\pi$
$740$	3.33861	0.122730
$741$	0	0
$742$	−1.91355	−0.0702485
$743$	−43.9188	−1.61122	−0.805612	−	0.592443i	$-0.798164\pi$
$743$	−43.9188	−1.61122	−0.805612	+	0.592443i	$0.798164\pi$
$744$	0	0
$745$	−22.8399	−0.836791
$746$	−29.4102	−1.07678
$747$	0	0
$748$	−3.76234	−0.137565
$749$	0.760114	0.0277740
$750$	0	0
$751$	10.6148	0.387340	0.193670	−	0.981067i	$-0.437961\pi$
$751$	10.6148	0.387340	0.193670	+	0.981067i	$0.437961\pi$
$752$	13.4814	0.491617
$753$	0	0
$754$	32.6818	1.19020
$755$	−72.9055	−2.65330
$756$	0	0
$757$	40.1639	1.45978	0.729892	−	0.683563i	$-0.239570\pi$
$757$	40.1639	1.45978	0.729892	+	0.683563i	$0.239570\pi$
$758$	10.0742	0.365910
$759$	0	0
$760$	−17.5787	−0.637645
$761$	17.1467	0.621568	0.310784	−	0.950481i	$-0.399408\pi$
$761$	17.1467	0.621568	0.310784	+	0.950481i	$0.399408\pi$
$762$	0	0
$763$	0.105121	0.00380563
$764$	−4.12388	−0.149197
$765$	0	0
$766$	15.6220	0.564446
$767$	−4.37619	−0.158015
$768$	0	0
$769$	11.9471	0.430824	0.215412	−	0.976523i	$-0.430891\pi$
$769$	11.9471	0.430824	0.215412	+	0.976523i	$0.430891\pi$
$770$	3.32747	0.119914
$771$	0	0
$772$	−8.55099	−0.307757
$773$	−50.8932	−1.83050	−0.915250	−	0.402886i	$-0.868007\pi$
$773$	−50.8932	−1.83050	−0.915250	+	0.402886i	$0.868007\pi$
$774$	0	0
$775$	−9.83297	−0.353211
$776$	10.4087	0.373651
$777$	0	0
$778$	16.6098	0.595491
$779$	−27.6886	−0.992047
$780$	0	0
$781$	−53.1773	−1.90283
$782$	1.85574	0.0663611
$783$	0	0
$784$	−6.96160	−0.248629
$785$	66.3297	2.36741
$786$	0	0
$787$	−12.0089	−0.428071	−0.214036	−	0.976826i	$-0.568661\pi$
$787$	−12.0089	−0.428071	−0.214036	+	0.976826i	$0.568661\pi$
$788$	13.1755	0.469357
$789$	0	0
$790$	3.09697	0.110185
$791$	3.73001	0.132624
$792$	0	0
$793$	−32.6423	−1.15916
$794$	23.9512	0.849995
$795$	0	0
$796$	18.5811	0.658589
$797$	−1.21781	−0.0431372	−0.0215686	−	0.999767i	$-0.506866\pi$
$797$	−1.21781	−0.0431372	−0.0215686	+	0.999767i	$0.506866\pi$
$798$	0	0
$799$	9.97278	0.352812
$800$	6.14633	0.217306
$801$	0	0
$802$	−28.2179	−0.996407
$803$	−39.3440	−1.38842
$804$	0	0
$805$	−1.64125	−0.0578463
$806$	8.52443	0.300260
$807$	0	0
$808$	−2.91641	−0.102599
$809$	−31.5509	−1.10927	−0.554635	−	0.832094i	$-0.687142\pi$
$809$	−31.5509	−1.10927	−0.554635	+	0.832094i	$0.687142\pi$
$810$	0	0
$811$	−11.0562	−0.388235	−0.194118	−	0.980978i	$-0.562184\pi$
$811$	−11.0562	−0.388235	−0.194118	+	0.980978i	$0.562184\pi$
$812$	−1.20193	−0.0421796
$813$	0	0
$814$	5.08601	0.178264
$815$	14.2707	0.499879
$816$	0	0
$817$	23.5628	0.824358
$818$	20.8364	0.728527
$819$	0	0
$820$	17.5569	0.613113
$821$	19.6171	0.684642	0.342321	−	0.939583i	$-0.388787\pi$
$821$	19.6171	0.684642	0.342321	+	0.939583i	$0.388787\pi$
$822$	0	0
$823$	−40.3719	−1.40728	−0.703638	−	0.710559i	$-0.748442\pi$
$823$	−40.3719	−1.40728	−0.703638	+	0.710559i	$0.748442\pi$
$824$	12.6134	0.439410
$825$	0	0
$826$	0.160942	0.00559989
$827$	−44.3183	−1.54110	−0.770549	−	0.637381i	$-0.780018\pi$
$827$	−44.3183	−1.54110	−0.770549	+	0.637381i	$0.780018\pi$
$828$	0	0
$829$	−14.1163	−0.490279	−0.245139	−	0.969488i	$-0.578834\pi$
$829$	−14.1163	−0.490279	−0.245139	+	0.969488i	$0.578834\pi$
$830$	29.5782	1.02667
$831$	0	0
$832$	−5.32840	−0.184729
$833$	−5.14979	−0.178430
$834$	0	0
$835$	−2.05411	−0.0710855
$836$	−26.7791	−0.926176
$837$	0	0
$838$	23.4768	0.810991
$839$	−4.20041	−0.145014	−0.0725071	−	0.997368i	$-0.523100\pi$
$839$	−4.20041	−0.145014	−0.0725071	+	0.997368i	$0.523100\pi$
$840$	0	0
$841$	8.61994	0.297239
$842$	−26.6544	−0.918570
$843$	0	0
$844$	−4.80875	−0.165524
$845$	−51.3876	−1.76779
$846$	0	0
$847$	2.91345	0.100107
$848$	−9.76490	−0.335328
$849$	0	0
$850$	4.54671	0.155951
$851$	−2.50863	−0.0859947
$852$	0	0
$853$	−15.9537	−0.546243	−0.273121	−	0.961980i	$-0.588056\pi$
$853$	−15.9537	−0.546243	−0.273121	+	0.961980i	$0.588056\pi$
$854$	1.20048	0.0410795
$855$	0	0
$856$	3.87889	0.132578
$857$	−0.986693	−0.0337048	−0.0168524	−	0.999858i	$-0.505365\pi$
$857$	−0.986693	−0.0337048	−0.0168524	+	0.999858i	$0.505365\pi$
$858$	0	0
$859$	6.41493	0.218874	0.109437	−	0.993994i	$-0.465095\pi$
$859$	6.41493	0.218874	0.109437	+	0.993994i	$0.465095\pi$
$860$	−14.9408	−0.509476
$861$	0	0
$862$	27.9472	0.951886
$863$	−19.6447	−0.668714	−0.334357	−	0.942446i	$-0.608519\pi$
$863$	−19.6447	−0.668714	−0.334357	+	0.942446i	$0.608519\pi$
$864$	0	0
$865$	−70.0733	−2.38257
$866$	14.4980	0.492663
$867$	0	0
$868$	−0.313501	−0.0106409
$869$	4.71790	0.160044
$870$	0	0
$871$	−37.4521	−1.26902
$872$	0.536436	0.0181660
$873$	0	0
$874$	13.2086	0.446787
$875$	−0.749978	−0.0253539
$876$	0	0
$877$	−25.0212	−0.844905	−0.422452	−	0.906385i	$-0.638831\pi$
$877$	−25.0212	−0.844905	−0.422452	+	0.906385i	$0.638831\pi$
$878$	10.2482	0.345861
$879$	0	0
$880$	16.9802	0.572403
$881$	20.0574	0.675751	0.337875	−	0.941191i	$-0.390292\pi$
$881$	20.0574	0.675751	0.337875	+	0.941191i	$0.390292\pi$
$882$	0	0
$883$	−10.4184	−0.350608	−0.175304	−	0.984514i	$-0.556091\pi$
$883$	−10.4184	−0.350608	−0.175304	+	0.984514i	$0.556091\pi$
$884$	−3.94165	−0.132572
$885$	0	0
$886$	8.13881	0.273429
$887$	−28.0688	−0.942459	−0.471229	−	0.882011i	$-0.656190\pi$
$887$	−28.0688	−0.942459	−0.471229	+	0.882011i	$0.656190\pi$
$888$	0	0
$889$	3.93200	0.131875
$890$	35.5365	1.19119
$891$	0	0
$892$	5.28621	0.176995
$893$	70.9831	2.37536
$894$	0	0
$895$	74.9230	2.50440
$896$	0.195962	0.00654662
$897$	0	0
$898$	1.81755	0.0606524
$899$	9.81245	0.327264
$900$	0	0
$901$	−7.22352	−0.240650
$902$	26.7460	0.890544
$903$	0	0
$904$	19.0344	0.633075
$905$	−18.7116	−0.621996
$906$	0	0
$907$	−6.81634	−0.226333	−0.113166	−	0.993576i	$-0.536099\pi$
$907$	−6.81634	−0.226333	−0.113166	+	0.993576i	$0.536099\pi$
$908$	6.52175	0.216432
$909$	0	0
$910$	3.48605	0.115562
$911$	46.1659	1.52954	0.764772	−	0.644301i	$-0.222851\pi$
$911$	46.1659	1.52954	0.764772	+	0.644301i	$0.222851\pi$
$912$	0	0
$913$	45.0591	1.49124
$914$	−38.9876	−1.28959
$915$	0	0
$916$	5.93818	0.196203
$917$	1.65126	0.0545295
$918$	0	0
$919$	26.8038	0.884175	0.442087	−	0.896972i	$-0.354238\pi$
$919$	26.8038	0.884175	0.442087	+	0.896972i	$0.354238\pi$
$920$	−8.37534	−0.276127
$921$	0	0
$922$	−14.6527	−0.482562
$923$	−55.7117	−1.83377
$924$	0	0
$925$	−6.14633	−0.202090
$926$	25.3342	0.832533
$927$	0	0
$928$	−6.13351	−0.201342
$929$	25.9178	0.850334	0.425167	−	0.905115i	$-0.360215\pi$
$929$	25.9178	0.850334	0.425167	+	0.905115i	$0.360215\pi$
$930$	0	0
$931$	−36.6546	−1.20131
$932$	18.4107	0.603063
$933$	0	0
$934$	−6.52767	−0.213592
$935$	12.5610	0.410788
$936$	0	0
$937$	8.04348	0.262769	0.131384	−	0.991331i	$-0.458058\pi$
$937$	8.04348	0.262769	0.131384	+	0.991331i	$0.458058\pi$
$938$	1.37737	0.0449727
$939$	0	0
$940$	−45.0092	−1.46804
$941$	57.0300	1.85913	0.929563	−	0.368664i	$-0.120185\pi$
$941$	57.0300	1.85913	0.929563	+	0.368664i	$0.120185\pi$
$942$	0	0
$943$	−13.1922	−0.429598
$944$	0.821294	0.0267309
$945$	0	0
$946$	−22.7606	−0.740012
$947$	−11.9823	−0.389372	−0.194686	−	0.980866i	$-0.562369\pi$
$947$	−11.9823	−0.389372	−0.194686	+	0.980866i	$0.562369\pi$
$948$	0	0
$949$	−41.2191	−1.33803
$950$	32.3620	1.04996
$951$	0	0
$952$	0.144961	0.00469822
$953$	53.7478	1.74106	0.870532	−	0.492112i	$-0.163775\pi$
$953$	53.7478	1.74106	0.870532	+	0.492112i	$0.163775\pi$
$954$	0	0
$955$	13.7680	0.445523
$956$	24.0032	0.776320
$957$	0	0
$958$	19.4006	0.626806
$959$	−4.07800	−0.131686
$960$	0	0
$961$	−28.4406	−0.917439
$962$	5.32840	0.171795
$963$	0	0
$964$	−17.2454	−0.555437
$965$	28.5484	0.919006
$966$	0	0
$967$	−54.0756	−1.73895	−0.869476	−	0.493975i	$-0.835544\pi$
$967$	−54.0756	−1.73895	−0.869476	+	0.493975i	$0.835544\pi$
$968$	14.8675	0.477859
$969$	0	0
$970$	−34.7507	−1.11578
$971$	−16.9965	−0.545442	−0.272721	−	0.962093i	$-0.587924\pi$
$971$	−16.9965	−0.545442	−0.272721	+	0.962093i	$0.587924\pi$
$972$	0	0
$973$	−0.0239929	−0.000769176 0
$974$	−11.5204	−0.369138
$975$	0	0
$976$	6.12609	0.196091
$977$	−14.3965	−0.460585	−0.230293	−	0.973121i	$-0.573968\pi$
$977$	−14.3965	−0.460585	−0.230293	+	0.973121i	$0.573968\pi$
$978$	0	0
$979$	54.1360	1.73019
$980$	23.2421	0.742441
$981$	0	0
$982$	−30.0958	−0.960397
$983$	−46.2710	−1.47582	−0.737908	−	0.674901i	$-0.764186\pi$
$983$	−46.2710	−1.47582	−0.737908	+	0.674901i	$0.764186\pi$
$984$	0	0
$985$	−43.9878	−1.40157
$986$	−4.53722	−0.144494
$987$	0	0
$988$	−28.0554	−0.892561
$989$	11.2265	0.356981
$990$	0	0
$991$	50.4576	1.60284	0.801419	−	0.598103i	$-0.204079\pi$
$991$	50.4576	1.60284	0.801419	+	0.598103i	$0.204079\pi$
$992$	−1.59981	−0.0507940
$993$	0	0
$994$	2.04890	0.0649870
$995$	−62.0350	−1.96664
$996$	0	0
$997$	51.1368	1.61952	0.809759	−	0.586762i	$-0.199598\pi$
$997$	51.1368	1.61952	0.809759	+	0.586762i	$0.199598\pi$
$998$	19.6116	0.620793
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5994.2.a.bb.1.2		10
3.2	odd	2		5994.2.a.ba.1.9		10
9.2	odd	6		1998.2.e.e.1333.2		20
9.4	even	3		666.2.e.e.223.1	✓	20
9.5	odd	6		1998.2.e.e.667.2		20
9.7	even	3		666.2.e.e.445.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
666.2.e.e.223.1	✓	20	9.4	even	3
666.2.e.e.445.1	yes	20	9.7	even	3
1998.2.e.e.667.2		20	9.5	odd	6
1998.2.e.e.1333.2		20	9.2	odd	6
5994.2.a.ba.1.9		10	3.2	odd	2
5994.2.a.bb.1.2		10	1.1	even	1	trivial

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

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Modular forms

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5994.2.a.bb.1.2

Level	$5994$
Weight	$2$
Character	5994.1
Self dual	yes
Analytic conductor	$47.862$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$1$