LMFDB - Embedded newform 8752.2.a.s.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8752,2,Mod(1,8752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8752 = 2^{4} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8752.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:	$69.8850718490$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.24960$ of defining polynomial
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.790850 q^{3} -3.96974 q^{5} +4.97706 q^{7} -2.37456 q^{9} +O(q^{10})$	$q+0.790850 q^{3} -3.96974 q^{5} +4.97706 q^{7} -2.37456 q^{9} -6.10795 q^{11} -0.944916 q^{13} -3.13947 q^{15} +0.884722 q^{17} +2.59993 q^{19} +3.93611 q^{21} +2.77074 q^{23} +10.7588 q^{25} -4.25047 q^{27} -1.93886 q^{29} +3.39568 q^{31} -4.83047 q^{33} -19.7576 q^{35} +2.71002 q^{37} -0.747287 q^{39} +1.37400 q^{41} +10.4149 q^{43} +9.42637 q^{45} +2.84781 q^{47} +17.7712 q^{49} +0.699683 q^{51} +6.58855 q^{53} +24.2470 q^{55} +2.05616 q^{57} -0.838365 q^{59} -9.71712 q^{61} -11.8183 q^{63} +3.75107 q^{65} -13.4835 q^{67} +2.19124 q^{69} +7.51364 q^{71} -12.4106 q^{73} +8.50862 q^{75} -30.3996 q^{77} +0.398622 q^{79} +3.76218 q^{81} -4.09663 q^{83} -3.51212 q^{85} -1.53335 q^{87} -15.7666 q^{89} -4.70291 q^{91} +2.68547 q^{93} -10.3211 q^{95} -3.97689 q^{97} +14.5037 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) $ 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9	$ 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) $ 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9 - 2 * q^11 - 25 * q^13 - 9 * q^15 - 30 * q^17 - 4 * q^19 - 16 * q^21 + 26 * q^23 + 31 * q^25 + 37 * q^27 - 18 * q^29 + 5 * q^31 - 10 * q^33 + 9 * q^35 - 18 * q^37 - 7 * q^39 - 17 * q^41 - 8 * q^43 - 44 * q^45 + 52 * q^47 + 29 * q^49 - 19 * q^51 - 60 * q^53 - 11 * q^55 + 4 * q^57 + 8 * q^59 - 26 * q^61 + q^63 - 6 * q^65 - 12 * q^67 - 38 * q^69 + q^71 - 2 * q^73 + 17 * q^75 - 73 * q^77 - 18 * q^79 + 18 * q^81 + 43 * q^83 + 51 * q^85 - 3 * q^87 - 28 * q^89 + q^91 - 60 * q^93 + 18 * q^95 - 34 * q^97 - 15 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	0.790850	0.456598	0.228299	−	0.973591i	$-0.426684\pi$
$3$	0.790850	0.456598	0.228299	+	0.973591i	$0.426684\pi$
$4$	0	0
$5$	−3.96974	−1.77532	−0.887660	−	0.460499i	$-0.847671\pi$
$5$	−3.96974	−1.77532	−0.887660	+	0.460499i	$0.847671\pi$
$6$	0	0
$7$	4.97706	1.88115	0.940576	−	0.339582i	$-0.110286\pi$
$7$	4.97706	1.88115	0.940576	+	0.339582i	$0.110286\pi$
$8$	0	0
$9$	−2.37456	−0.791519
$10$	0	0
$11$	−6.10795	−1.84162	−0.920808	−	0.390016i	$-0.872469\pi$
$11$	−6.10795	−1.84162	−0.920808	+	0.390016i	$0.872469\pi$
$12$	0	0
$13$	−0.944916	−0.262073	−0.131036	−	0.991378i	$-0.541830\pi$
$13$	−0.944916	−0.262073	−0.131036	+	0.991378i	$0.541830\pi$
$14$	0	0
$15$	−3.13947	−0.810607
$16$	0	0
$17$	0.884722	0.214577	0.107288	−	0.994228i	$-0.465783\pi$
$17$	0.884722	0.214577	0.107288	+	0.994228i	$0.465783\pi$
$18$	0	0
$19$	2.59993	0.596466	0.298233	−	0.954493i	$-0.403603\pi$
$19$	2.59993	0.596466	0.298233	+	0.954493i	$0.403603\pi$
$20$	0	0
$21$	3.93611	0.858930
$22$	0	0
$23$	2.77074	0.577740	0.288870	−	0.957368i	$-0.406720\pi$
$23$	2.77074	0.577740	0.288870	+	0.957368i	$0.406720\pi$
$24$	0	0
$25$	10.7588	2.15176
$26$	0	0
$27$	−4.25047	−0.818003
$28$	0	0
$29$	−1.93886	−0.360037	−0.180019	−	0.983663i	$-0.557616\pi$
$29$	−1.93886	−0.360037	−0.180019	+	0.983663i	$0.557616\pi$
$30$	0	0
$31$	3.39568	0.609881	0.304941	−	0.952371i	$-0.401363\pi$
$31$	3.39568	0.609881	0.304941	+	0.952371i	$0.401363\pi$
$32$	0	0
$33$	−4.83047	−0.840878
$34$	0	0
$35$	−19.7576	−3.33965
$36$	0	0
$37$	2.71002	0.445525	0.222762	−	0.974873i	$-0.428493\pi$
$37$	2.71002	0.445525	0.222762	+	0.974873i	$0.428493\pi$
$38$	0	0
$39$	−0.747287	−0.119662
$40$	0	0
$41$	1.37400	0.214583	0.107291	−	0.994228i	$-0.465782\pi$
$41$	1.37400	0.214583	0.107291	+	0.994228i	$0.465782\pi$
$42$	0	0
$43$	10.4149	1.58825	0.794126	−	0.607753i	$-0.207929\pi$
$43$	10.4149	1.58825	0.794126	+	0.607753i	$0.207929\pi$
$44$	0	0
$45$	9.42637	1.40520
$46$	0	0
$47$	2.84781	0.415395	0.207698	−	0.978193i	$-0.433403\pi$
$47$	2.84781	0.415395	0.207698	+	0.978193i	$0.433403\pi$
$48$	0	0
$49$	17.7712	2.53874
$50$	0	0
$51$	0.699683	0.0979752
$52$	0	0
$53$	6.58855	0.905007	0.452503	−	0.891763i	$-0.350531\pi$
$53$	6.58855	0.905007	0.452503	+	0.891763i	$0.350531\pi$
$54$	0	0
$55$	24.2470	3.26946
$56$	0	0
$57$	2.05616	0.272345
$58$	0	0
$59$	−0.838365	−0.109146	−0.0545729	−	0.998510i	$-0.517380\pi$
$59$	−0.838365	−0.109146	−0.0545729	+	0.998510i	$0.517380\pi$
$60$	0	0
$61$	−9.71712	−1.24415	−0.622075	−	0.782958i	$-0.713710\pi$
$61$	−9.71712	−1.24415	−0.622075	+	0.782958i	$0.713710\pi$
$62$	0	0
$63$	−11.8183	−1.48897
$64$	0	0
$65$	3.75107	0.465263
$66$	0	0
$67$	−13.4835	−1.64727	−0.823633	−	0.567123i	$-0.808056\pi$
$67$	−13.4835	−1.64727	−0.823633	+	0.567123i	$0.808056\pi$
$68$	0	0
$69$	2.19124	0.263795
$70$	0	0
$71$	7.51364	0.891705	0.445853	−	0.895106i	$-0.352901\pi$
$71$	7.51364	0.891705	0.445853	+	0.895106i	$0.352901\pi$
$72$	0	0
$73$	−12.4106	−1.45255	−0.726274	−	0.687405i	$-0.758750\pi$
$73$	−12.4106	−1.45255	−0.726274	+	0.687405i	$0.758750\pi$
$74$	0	0
$75$	8.50862	0.982491
$76$	0	0
$77$	−30.3996	−3.46436
$78$	0	0
$79$	0.398622	0.0448485	0.0224243	−	0.999749i	$-0.492862\pi$
$79$	0.398622	0.0448485	0.0224243	+	0.999749i	$0.492862\pi$
$80$	0	0
$81$	3.76218	0.418020
$82$	0	0
$83$	−4.09663	−0.449664	−0.224832	−	0.974398i	$-0.572183\pi$
$83$	−4.09663	−0.449664	−0.224832	+	0.974398i	$0.572183\pi$
$84$	0	0
$85$	−3.51212	−0.380943
$86$	0	0
$87$	−1.53335	−0.164392
$88$	0	0
$89$	−15.7666	−1.67125	−0.835626	−	0.549298i	$-0.814895\pi$
$89$	−15.7666	−1.67125	−0.835626	+	0.549298i	$0.814895\pi$
$90$	0	0
$91$	−4.70291	−0.492999
$92$	0	0
$93$	2.68547	0.278470
$94$	0	0
$95$	−10.3211	−1.05892
$96$	0	0
$97$	−3.97689	−0.403792	−0.201896	−	0.979407i	$-0.564710\pi$
$97$	−3.97689	−0.403792	−0.201896	+	0.979407i	$0.564710\pi$
$98$	0	0
$99$	14.5037	1.45767
$100$	0	0
$101$	−1.75191	−0.174321	−0.0871605	−	0.996194i	$-0.527779\pi$
$101$	−1.75191	−0.174321	−0.0871605	+	0.996194i	$0.527779\pi$
$102$	0	0
$103$	9.27653	0.914044	0.457022	−	0.889455i	$-0.348916\pi$
$103$	9.27653	0.914044	0.457022	+	0.889455i	$0.348916\pi$
$104$	0	0
$105$	−15.6253	−1.52488
$106$	0	0
$107$	8.46764	0.818597	0.409299	−	0.912400i	$-0.365773\pi$
$107$	8.46764	0.818597	0.409299	+	0.912400i	$0.365773\pi$
$108$	0	0
$109$	−2.68117	−0.256810	−0.128405	−	0.991722i	$-0.540986\pi$
$109$	−2.68117	−0.256810	−0.128405	+	0.991722i	$0.540986\pi$
$110$	0	0
$111$	2.14322	0.203425
$112$	0	0
$113$	−17.8209	−1.67645	−0.838223	−	0.545327i	$-0.816405\pi$
$113$	−17.8209	−1.67645	−0.838223	+	0.545327i	$0.816405\pi$
$114$	0	0
$115$	−10.9991	−1.02567
$116$	0	0
$117$	2.24376	0.207435
$118$	0	0
$119$	4.40332	0.403652
$120$	0	0
$121$	26.3070	2.39155
$122$	0	0
$123$	1.08663	0.0979781
$124$	0	0
$125$	−22.8610	−2.04475
$126$	0	0
$127$	−14.9282	−1.32467	−0.662333	−	0.749210i	$-0.730433\pi$
$127$	−14.9282	−1.32467	−0.662333	+	0.749210i	$0.730433\pi$
$128$	0	0
$129$	8.23660	0.725192
$130$	0	0
$131$	−15.7075	−1.37237	−0.686185	−	0.727427i	$-0.740716\pi$
$131$	−15.7075	−1.37237	−0.686185	+	0.727427i	$0.740716\pi$
$132$	0	0
$133$	12.9400	1.12204
$134$	0	0
$135$	16.8732	1.45222
$136$	0	0
$137$	2.42982	0.207593	0.103797	−	0.994599i	$-0.466901\pi$
$137$	2.42982	0.207593	0.103797	+	0.994599i	$0.466901\pi$
$138$	0	0
$139$	−10.6270	−0.901368	−0.450684	−	0.892684i	$-0.648820\pi$
$139$	−10.6270	−0.901368	−0.450684	+	0.892684i	$0.648820\pi$
$140$	0	0
$141$	2.25219	0.189668
$142$	0	0
$143$	5.77150	0.482637
$144$	0	0
$145$	7.69677	0.639182
$146$	0	0
$147$	14.0543	1.15918
$148$	0	0
$149$	8.49250	0.695733	0.347866	−	0.937544i	$-0.386906\pi$
$149$	8.49250	0.695733	0.347866	+	0.937544i	$0.386906\pi$
$150$	0	0
$151$	−23.1852	−1.88679	−0.943393	−	0.331677i	$-0.892385\pi$
$151$	−23.1852	−1.88679	−0.943393	+	0.331677i	$0.892385\pi$
$152$	0	0
$153$	−2.10082	−0.169841
$154$	0	0
$155$	−13.4799	−1.08274
$156$	0	0
$157$	−10.9628	−0.874927	−0.437464	−	0.899236i	$-0.644123\pi$
$157$	−10.9628	−0.874927	−0.437464	+	0.899236i	$0.644123\pi$
$158$	0	0
$159$	5.21055	0.413224
$160$	0	0
$161$	13.7902	1.08682
$162$	0	0
$163$	10.0906	0.790355	0.395178	−	0.918605i	$-0.370683\pi$
$163$	10.0906	0.790355	0.395178	+	0.918605i	$0.370683\pi$
$164$	0	0
$165$	19.1757	1.49283
$166$	0	0
$167$	−6.93218	−0.536428	−0.268214	−	0.963359i	$-0.586433\pi$
$167$	−6.93218	−0.536428	−0.268214	+	0.963359i	$0.586433\pi$
$168$	0	0
$169$	−12.1071	−0.931318
$170$	0	0
$171$	−6.17369	−0.472114
$172$	0	0
$173$	−25.8915	−1.96850	−0.984249	−	0.176789i	$-0.943429\pi$
$173$	−25.8915	−1.96850	−0.984249	+	0.176789i	$0.943429\pi$
$174$	0	0
$175$	53.5473	4.04780
$176$	0	0
$177$	−0.663021	−0.0498357
$178$	0	0
$179$	−0.366164	−0.0273684	−0.0136842	−	0.999906i	$-0.504356\pi$
$179$	−0.366164	−0.0273684	−0.0136842	+	0.999906i	$0.504356\pi$
$180$	0	0
$181$	13.7369	1.02105	0.510527	−	0.859862i	$-0.329450\pi$
$181$	13.7369	1.02105	0.510527	+	0.859862i	$0.329450\pi$
$182$	0	0
$183$	−7.68479	−0.568076
$184$	0	0
$185$	−10.7581	−0.790949
$186$	0	0
$187$	−5.40384	−0.395168
$188$	0	0
$189$	−21.1549	−1.53879
$190$	0	0
$191$	−8.48775	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$191$	−8.48775	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$192$	0	0
$193$	16.0739	1.15703	0.578513	−	0.815673i	$-0.303633\pi$
$193$	16.0739	1.15703	0.578513	+	0.815673i	$0.303633\pi$
$194$	0	0
$195$	2.96654	0.212438
$196$	0	0
$197$	−16.7983	−1.19683	−0.598415	−	0.801186i	$-0.704203\pi$
$197$	−16.7983	−1.19683	−0.598415	+	0.801186i	$0.704203\pi$
$198$	0	0
$199$	−8.79192	−0.623243	−0.311621	−	0.950206i	$-0.600872\pi$
$199$	−8.79192	−0.623243	−0.311621	+	0.950206i	$0.600872\pi$
$200$	0	0
$201$	−10.6634	−0.752138
$202$	0	0
$203$	−9.64983	−0.677285
$204$	0	0
$205$	−5.45443	−0.380954
$206$	0	0
$207$	−6.57928	−0.457292
$208$	0	0
$209$	−15.8803	−1.09846
$210$	0	0
$211$	14.8938	1.02533	0.512665	−	0.858589i	$-0.328658\pi$
$211$	14.8938	1.02533	0.512665	+	0.858589i	$0.328658\pi$
$212$	0	0
$213$	5.94217	0.407150
$214$	0	0
$215$	−41.3443	−2.81966
$216$	0	0
$217$	16.9005	1.14728
$218$	0	0
$219$	−9.81491	−0.663230
$220$	0	0
$221$	−0.835989	−0.0562347
$222$	0	0
$223$	−4.16716	−0.279054	−0.139527	−	0.990218i	$-0.544558\pi$
$223$	−4.16716	−0.279054	−0.139527	+	0.990218i	$0.544558\pi$
$224$	0	0
$225$	−25.5474	−1.70316
$226$	0	0
$227$	22.3637	1.48433	0.742166	−	0.670216i	$-0.233799\pi$
$227$	22.3637	1.48433	0.742166	+	0.670216i	$0.233799\pi$
$228$	0	0
$229$	9.45321	0.624686	0.312343	−	0.949969i	$-0.398886\pi$
$229$	9.45321	0.624686	0.312343	+	0.949969i	$0.398886\pi$
$230$	0	0
$231$	−24.0416	−1.58182
$232$	0	0
$233$	−10.2088	−0.668803	−0.334402	−	0.942431i	$-0.608534\pi$
$233$	−10.2088	−0.668803	−0.334402	+	0.942431i	$0.608534\pi$
$234$	0	0
$235$	−11.3050	−0.737460
$236$	0	0
$237$	0.315251	0.0204777
$238$	0	0
$239$	−10.1387	−0.655820	−0.327910	−	0.944709i	$-0.606344\pi$
$239$	−10.1387	−0.655820	−0.327910	+	0.944709i	$0.606344\pi$
$240$	0	0
$241$	1.70025	0.109523	0.0547613	−	0.998499i	$-0.482560\pi$
$241$	1.70025	0.109523	0.0547613	+	0.998499i	$0.482560\pi$
$242$	0	0
$243$	15.7267	1.00887
$244$	0	0
$245$	−70.5468	−4.50707
$246$	0	0
$247$	−2.45672	−0.156317
$248$	0	0
$249$	−3.23982	−0.205316
$250$	0	0
$251$	−2.42104	−0.152815	−0.0764074	−	0.997077i	$-0.524345\pi$
$251$	−2.42104	−0.152815	−0.0764074	+	0.997077i	$0.524345\pi$
$252$	0	0
$253$	−16.9236	−1.06397
$254$	0	0
$255$	−2.77756	−0.173937
$256$	0	0
$257$	5.01711	0.312958	0.156479	−	0.987681i	$-0.449986\pi$
$257$	5.01711	0.312958	0.156479	+	0.987681i	$0.449986\pi$
$258$	0	0
$259$	13.4879	0.838100
$260$	0	0
$261$	4.60393	0.284976
$262$	0	0
$263$	22.9041	1.41233	0.706164	−	0.708049i	$-0.250424\pi$
$263$	22.9041	1.41233	0.706164	+	0.708049i	$0.250424\pi$
$264$	0	0
$265$	−26.1548	−1.60668
$266$	0	0
$267$	−12.4690	−0.763090
$268$	0	0
$269$	−3.74942	−0.228606	−0.114303	−	0.993446i	$-0.536464\pi$
$269$	−3.74942	−0.228606	−0.114303	+	0.993446i	$0.536464\pi$
$270$	0	0
$271$	−10.2651	−0.623562	−0.311781	−	0.950154i	$-0.600925\pi$
$271$	−10.2651	−0.623562	−0.311781	+	0.950154i	$0.600925\pi$
$272$	0	0
$273$	−3.71930	−0.225102
$274$	0	0
$275$	−65.7143	−3.96272
$276$	0	0
$277$	−15.9783	−0.960044	−0.480022	−	0.877256i	$-0.659371\pi$
$277$	−15.9783	−0.960044	−0.480022	+	0.877256i	$0.659371\pi$
$278$	0	0
$279$	−8.06322	−0.482732
$280$	0	0
$281$	29.6055	1.76612	0.883059	−	0.469262i	$-0.155480\pi$
$281$	29.6055	1.76612	0.883059	+	0.469262i	$0.155480\pi$
$282$	0	0
$283$	−19.3104	−1.14788	−0.573941	−	0.818897i	$-0.694586\pi$
$283$	−19.3104	−1.14788	−0.573941	+	0.818897i	$0.694586\pi$
$284$	0	0
$285$	−8.16241	−0.483500
$286$	0	0
$287$	6.83849	0.403663
$288$	0	0
$289$	−16.2173	−0.953957
$290$	0	0
$291$	−3.14513	−0.184371
$292$	0	0
$293$	−8.44273	−0.493230	−0.246615	−	0.969114i	$-0.579318\pi$
$293$	−8.44273	−0.493230	−0.246615	+	0.969114i	$0.579318\pi$
$294$	0	0
$295$	3.32809	0.193769
$296$	0	0
$297$	25.9616	1.50645
$298$	0	0
$299$	−2.61812	−0.151410
$300$	0	0
$301$	51.8354	2.98775
$302$	0	0
$303$	−1.38549	−0.0795946
$304$	0	0
$305$	38.5744	2.20876
$306$	0	0
$307$	−18.4753	−1.05444	−0.527219	−	0.849729i	$-0.676765\pi$
$307$	−18.4753	−1.05444	−0.527219	+	0.849729i	$0.676765\pi$
$308$	0	0
$309$	7.33635	0.417350
$310$	0	0
$311$	−18.2445	−1.03455	−0.517276	−	0.855819i	$-0.673054\pi$
$311$	−18.2445	−1.03455	−0.517276	+	0.855819i	$0.673054\pi$
$312$	0	0
$313$	−10.9949	−0.621467	−0.310733	−	0.950497i	$-0.600575\pi$
$313$	−10.9949	−0.621467	−0.310733	+	0.950497i	$0.600575\pi$
$314$	0	0
$315$	46.9156	2.64340
$316$	0	0
$317$	−10.7948	−0.606294	−0.303147	−	0.952944i	$-0.598037\pi$
$317$	−10.7948	−0.606294	−0.303147	+	0.952944i	$0.598037\pi$
$318$	0	0
$319$	11.8425	0.663050
$320$	0	0
$321$	6.69663	0.373770
$322$	0	0
$323$	2.30022	0.127988
$324$	0	0
$325$	−10.1662	−0.563919
$326$	0	0
$327$	−2.12041	−0.117259
$328$	0	0
$329$	14.1737	0.781422
$330$	0	0
$331$	−2.70530	−0.148697	−0.0743483	−	0.997232i	$-0.523688\pi$
$331$	−2.70530	−0.148697	−0.0743483	+	0.997232i	$0.523688\pi$
$332$	0	0
$333$	−6.43510	−0.352641
$334$	0	0
$335$	53.5258	2.92443
$336$	0	0
$337$	−6.92073	−0.376996	−0.188498	−	0.982074i	$-0.560362\pi$
$337$	−6.92073	−0.376996	−0.188498	+	0.982074i	$0.560362\pi$
$338$	0	0
$339$	−14.0936	−0.765461
$340$	0	0
$341$	−20.7406	−1.12317
$342$	0	0
$343$	53.6087	2.89460
$344$	0	0
$345$	−8.69866	−0.468320
$346$	0	0
$347$	17.6570	0.947877	0.473939	−	0.880558i	$-0.342832\pi$
$347$	17.6570	0.947877	0.473939	+	0.880558i	$0.342832\pi$
$348$	0	0
$349$	2.79807	0.149777	0.0748886	−	0.997192i	$-0.476140\pi$
$349$	2.79807	0.149777	0.0748886	+	0.997192i	$0.476140\pi$
$350$	0	0
$351$	4.01634	0.214376
$352$	0	0
$353$	3.67373	0.195533	0.0977665	−	0.995209i	$-0.468830\pi$
$353$	3.67373	0.195533	0.0977665	+	0.995209i	$0.468830\pi$
$354$	0	0
$355$	−29.8272	−1.58306
$356$	0	0
$357$	3.48237	0.184306
$358$	0	0
$359$	30.6246	1.61631	0.808153	−	0.588972i	$-0.200467\pi$
$359$	30.6246	1.61631	0.808153	+	0.588972i	$0.200467\pi$
$360$	0	0
$361$	−12.2403	−0.644228
$362$	0	0
$363$	20.8049	1.09198
$364$	0	0
$365$	49.2667	2.57874
$366$	0	0
$367$	16.2490	0.848189	0.424095	−	0.905618i	$-0.360592\pi$
$367$	16.2490	0.848189	0.424095	+	0.905618i	$0.360592\pi$
$368$	0	0
$369$	−3.26264	−0.169846
$370$	0	0
$371$	32.7916	1.70246
$372$	0	0
$373$	3.35997	0.173973	0.0869863	−	0.996210i	$-0.472276\pi$
$373$	3.35997	0.173973	0.0869863	+	0.996210i	$0.472276\pi$
$374$	0	0
$375$	−18.0796	−0.933629
$376$	0	0
$377$	1.83206	0.0943559
$378$	0	0
$379$	13.0292	0.669265	0.334632	−	0.942349i	$-0.391388\pi$
$379$	13.0292	0.669265	0.334632	+	0.942349i	$0.391388\pi$
$380$	0	0
$381$	−11.8060	−0.604839
$382$	0	0
$383$	−16.4540	−0.840762	−0.420381	−	0.907348i	$-0.638104\pi$
$383$	−16.4540	−0.840762	−0.420381	+	0.907348i	$0.638104\pi$
$384$	0	0
$385$	120.679	6.15035
$386$	0	0
$387$	−24.7307	−1.25713
$388$	0	0
$389$	−31.8937	−1.61708	−0.808538	−	0.588444i	$-0.799741\pi$
$389$	−31.8937	−1.61708	−0.808538	+	0.588444i	$0.799741\pi$
$390$	0	0
$391$	2.45134	0.123969
$392$	0	0
$393$	−12.4223	−0.626621
$394$	0	0
$395$	−1.58243	−0.0796205
$396$	0	0
$397$	31.4614	1.57900	0.789502	−	0.613748i	$-0.210339\pi$
$397$	31.4614	1.57900	0.789502	+	0.613748i	$0.210339\pi$
$398$	0	0
$399$	10.2336	0.512322
$400$	0	0
$401$	−3.09593	−0.154603	−0.0773016	−	0.997008i	$-0.524630\pi$
$401$	−3.09593	−0.154603	−0.0773016	+	0.997008i	$0.524630\pi$
$402$	0	0
$403$	−3.20863	−0.159833
$404$	0	0
$405$	−14.9349	−0.742120
$406$	0	0
$407$	−16.5527	−0.820485
$408$	0	0
$409$	12.6949	0.627721	0.313860	−	0.949469i	$-0.398378\pi$
$409$	12.6949	0.627721	0.313860	+	0.949469i	$0.398378\pi$
$410$	0	0
$411$	1.92162	0.0947867
$412$	0	0
$413$	−4.17259	−0.205320
$414$	0	0
$415$	16.2626	0.798298
$416$	0	0
$417$	−8.40435	−0.411563
$418$	0	0
$419$	19.0216	0.929266	0.464633	−	0.885503i	$-0.346186\pi$
$419$	19.0216	0.929266	0.464633	+	0.885503i	$0.346186\pi$
$420$	0	0
$421$	−27.0312	−1.31742	−0.658710	−	0.752397i	$-0.728897\pi$
$421$	−27.0312	−1.31742	−0.658710	+	0.752397i	$0.728897\pi$
$422$	0	0
$423$	−6.76227	−0.328793
$424$	0	0
$425$	9.51857	0.461719
$426$	0	0
$427$	−48.3627	−2.34044
$428$	0	0
$429$	4.56439	0.220371
$430$	0	0
$431$	−1.88408	−0.0907530	−0.0453765	−	0.998970i	$-0.514449\pi$
$431$	−1.88408	−0.0907530	−0.0453765	+	0.998970i	$0.514449\pi$
$432$	0	0
$433$	15.6410	0.751660	0.375830	−	0.926689i	$-0.377358\pi$
$433$	15.6410	0.751660	0.375830	+	0.926689i	$0.377358\pi$
$434$	0	0
$435$	6.08699	0.291849
$436$	0	0
$437$	7.20375	0.344602
$438$	0	0
$439$	−28.8504	−1.37696	−0.688478	−	0.725257i	$-0.741721\pi$
$439$	−28.8504	−1.37696	−0.688478	+	0.725257i	$0.741721\pi$
$440$	0	0
$441$	−42.1986	−2.00946
$442$	0	0
$443$	28.5293	1.35547	0.677733	−	0.735308i	$-0.262962\pi$
$443$	28.5293	1.35547	0.677733	+	0.735308i	$0.262962\pi$
$444$	0	0
$445$	62.5892	2.96701
$446$	0	0
$447$	6.71630	0.317670
$448$	0	0
$449$	0.158303	0.00747076	0.00373538	−	0.999993i	$-0.498811\pi$
$449$	0.158303	0.00747076	0.00373538	+	0.999993i	$0.498811\pi$
$450$	0	0
$451$	−8.39233	−0.395179
$452$	0	0
$453$	−18.3360	−0.861502
$454$	0	0
$455$	18.6693	0.875231
$456$	0	0
$457$	20.8707	0.976288	0.488144	−	0.872763i	$-0.337674\pi$
$457$	20.8707	0.976288	0.488144	+	0.872763i	$0.337674\pi$
$458$	0	0
$459$	−3.76049	−0.175524
$460$	0	0
$461$	25.4051	1.18323	0.591616	−	0.806220i	$-0.298490\pi$
$461$	25.4051	1.18323	0.591616	+	0.806220i	$0.298490\pi$
$462$	0	0
$463$	−9.62944	−0.447518	−0.223759	−	0.974645i	$-0.571833\pi$
$463$	−9.62944	−0.447518	−0.223759	+	0.974645i	$0.571833\pi$
$464$	0	0
$465$	−10.6606	−0.494374
$466$	0	0
$467$	36.4715	1.68770	0.843849	−	0.536581i	$-0.180284\pi$
$467$	36.4715	1.68770	0.843849	+	0.536581i	$0.180284\pi$
$468$	0	0
$469$	−67.1080	−3.09876
$470$	0	0
$471$	−8.66994	−0.399490
$472$	0	0
$473$	−63.6135	−2.92495
$474$	0	0
$475$	27.9722	1.28345
$476$	0	0
$477$	−15.6449	−0.716330
$478$	0	0
$479$	−19.9600	−0.911996	−0.455998	−	0.889981i	$-0.650718\pi$
$479$	−19.9600	−0.911996	−0.455998	+	0.889981i	$0.650718\pi$
$480$	0	0
$481$	−2.56074	−0.116760
$482$	0	0
$483$	10.9060	0.496238
$484$	0	0
$485$	15.7872	0.716861
$486$	0	0
$487$	12.9846	0.588390	0.294195	−	0.955745i	$-0.404949\pi$
$487$	12.9846	0.588390	0.294195	+	0.955745i	$0.404949\pi$
$488$	0	0
$489$	7.98014	0.360874
$490$	0	0
$491$	−39.2860	−1.77295	−0.886476	−	0.462774i	$-0.846854\pi$
$491$	−39.2860	−1.77295	−0.886476	+	0.462774i	$0.846854\pi$
$492$	0	0
$493$	−1.71535	−0.0772556
$494$	0	0
$495$	−57.5758	−2.58784
$496$	0	0
$497$	37.3959	1.67743
$498$	0	0
$499$	−18.9542	−0.848506	−0.424253	−	0.905544i	$-0.639463\pi$
$499$	−18.9542	−0.848506	−0.424253	+	0.905544i	$0.639463\pi$
$500$	0	0
$501$	−5.48231	−0.244932
$502$	0	0
$503$	−19.9072	−0.887617	−0.443808	−	0.896122i	$-0.646373\pi$
$503$	−19.9072	−0.887617	−0.443808	+	0.896122i	$0.646373\pi$
$504$	0	0
$505$	6.95460	0.309476
$506$	0	0
$507$	−9.57493	−0.425238
$508$	0	0
$509$	−13.6549	−0.605244	−0.302622	−	0.953111i	$-0.597862\pi$
$509$	−13.6549	−0.605244	−0.302622	+	0.953111i	$0.597862\pi$
$510$	0	0
$511$	−61.7682	−2.73246
$512$	0	0
$513$	−11.0509	−0.487911
$514$	0	0
$515$	−36.8254	−1.62272
$516$	0	0
$517$	−17.3943	−0.764998
$518$	0	0
$519$	−20.4763	−0.898811
$520$	0	0
$521$	−7.74427	−0.339283	−0.169641	−	0.985506i	$-0.554261\pi$
$521$	−7.74427	−0.339283	−0.169641	+	0.985506i	$0.554261\pi$
$522$	0	0
$523$	−39.9635	−1.74748	−0.873740	−	0.486393i	$-0.838312\pi$
$523$	−39.9635	−1.74748	−0.873740	+	0.486393i	$0.838312\pi$
$524$	0	0
$525$	42.3479	1.84821
$526$	0	0
$527$	3.00423	0.130866
$528$	0	0
$529$	−15.3230	−0.666217
$530$	0	0
$531$	1.99074	0.0863909
$532$	0	0
$533$	−1.29832	−0.0562363
$534$	0	0
$535$	−33.6143	−1.45327
$536$	0	0
$537$	−0.289581	−0.0124963
$538$	0	0
$539$	−108.545	−4.67538
$540$	0	0
$541$	−16.7943	−0.722042	−0.361021	−	0.932558i	$-0.617572\pi$
$541$	−16.7943	−0.722042	−0.361021	+	0.932558i	$0.617572\pi$
$542$	0	0
$543$	10.8638	0.466211
$544$	0	0
$545$	10.6436	0.455920
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	23.0738	0.984768
$550$	0	0
$551$	−5.04091	−0.214750
$552$	0	0
$553$	1.98397	0.0843669
$554$	0	0
$555$	−8.50802	−0.361146
$556$	0	0
$557$	−20.5063	−0.868882	−0.434441	−	0.900700i	$-0.643054\pi$
$557$	−20.5063	−0.868882	−0.434441	+	0.900700i	$0.643054\pi$
$558$	0	0
$559$	−9.84118	−0.416237
$560$	0	0
$561$	−4.27363	−0.180433
$562$	0	0
$563$	32.4055	1.36573	0.682865	−	0.730545i	$-0.260734\pi$
$563$	32.4055	1.36573	0.682865	+	0.730545i	$0.260734\pi$
$564$	0	0
$565$	70.7442	2.97623
$566$	0	0
$567$	18.7246	0.786360
$568$	0	0
$569$	−8.66151	−0.363110	−0.181555	−	0.983381i	$-0.558113\pi$
$569$	−8.66151	−0.363110	−0.181555	+	0.983381i	$0.558113\pi$
$570$	0	0
$571$	−24.3017	−1.01699	−0.508496	−	0.861064i	$-0.669798\pi$
$571$	−24.3017	−1.01699	−0.508496	+	0.861064i	$0.669798\pi$
$572$	0	0
$573$	−6.71254	−0.280420
$574$	0	0
$575$	29.8099	1.24316
$576$	0	0
$577$	21.4414	0.892617	0.446309	−	0.894879i	$-0.352738\pi$
$577$	21.4414	0.892617	0.446309	+	0.894879i	$0.352738\pi$
$578$	0	0
$579$	12.7121	0.528295
$580$	0	0
$581$	−20.3892	−0.845887
$582$	0	0
$583$	−40.2425	−1.66667
$584$	0	0
$585$	−8.90713	−0.368264
$586$	0	0
$587$	−7.24339	−0.298967	−0.149483	−	0.988764i	$-0.547761\pi$
$587$	−7.24339	−0.298967	−0.149483	+	0.988764i	$0.547761\pi$
$588$	0	0
$589$	8.82853	0.363773
$590$	0	0
$591$	−13.2849	−0.546470
$592$	0	0
$593$	−30.4235	−1.24934	−0.624672	−	0.780887i	$-0.714767\pi$
$593$	−30.4235	−1.24934	−0.624672	+	0.780887i	$0.714767\pi$
$594$	0	0
$595$	−17.4800	−0.716611
$596$	0	0
$597$	−6.95309	−0.284571
$598$	0	0
$599$	−33.3767	−1.36373	−0.681866	−	0.731477i	$-0.738832\pi$
$599$	−33.3767	−1.36373	−0.681866	+	0.731477i	$0.738832\pi$
$600$	0	0
$601$	−15.6192	−0.637119	−0.318559	−	0.947903i	$-0.603199\pi$
$601$	−15.6192	−0.637119	−0.318559	+	0.947903i	$0.603199\pi$
$602$	0	0
$603$	32.0172	1.30384
$604$	0	0
$605$	−104.432	−4.24577
$606$	0	0
$607$	34.4069	1.39653	0.698267	−	0.715837i	$-0.253955\pi$
$607$	34.4069	1.39653	0.698267	+	0.715837i	$0.253955\pi$
$608$	0	0
$609$	−7.63157	−0.309247
$610$	0	0
$611$	−2.69094	−0.108864
$612$	0	0
$613$	7.54454	0.304721	0.152361	−	0.988325i	$-0.451312\pi$
$613$	7.54454	0.304721	0.152361	+	0.988325i	$0.451312\pi$
$614$	0	0
$615$	−4.31363	−0.173943
$616$	0	0
$617$	24.6440	0.992131	0.496065	−	0.868285i	$-0.334778\pi$
$617$	24.6440	0.992131	0.496065	+	0.868285i	$0.334778\pi$
$618$	0	0
$619$	37.1366	1.49264	0.746322	−	0.665585i	$-0.231818\pi$
$619$	37.1366	1.49264	0.746322	+	0.665585i	$0.231818\pi$
$620$	0	0
$621$	−11.7770	−0.472593
$622$	0	0
$623$	−78.4712	−3.14388
$624$	0	0
$625$	36.9581	1.47833
$626$	0	0
$627$	−12.5589	−0.501555
$628$	0	0
$629$	2.39762	0.0955992
$630$	0	0
$631$	17.4883	0.696197	0.348099	−	0.937458i	$-0.386827\pi$
$631$	17.4883	0.696197	0.348099	+	0.937458i	$0.386827\pi$
$632$	0	0
$633$	11.7788	0.468163
$634$	0	0
$635$	59.2611	2.35171
$636$	0	0
$637$	−16.7923	−0.665333
$638$	0	0
$639$	−17.8416	−0.705801
$640$	0	0
$641$	−48.0566	−1.89812	−0.949061	−	0.315094i	$-0.897964\pi$
$641$	−48.0566	−1.89812	−0.949061	+	0.315094i	$0.897964\pi$
$642$	0	0
$643$	−11.4373	−0.451044	−0.225522	−	0.974238i	$-0.572409\pi$
$643$	−11.4373	−0.451044	−0.225522	+	0.974238i	$0.572409\pi$
$644$	0	0
$645$	−32.6971	−1.28745
$646$	0	0
$647$	4.77121	0.187576	0.0937878	−	0.995592i	$-0.470102\pi$
$647$	4.77121	0.187576	0.0937878	+	0.995592i	$0.470102\pi$
$648$	0	0
$649$	5.12069	0.201005
$650$	0	0
$651$	13.3658	0.523845
$652$	0	0
$653$	−23.7882	−0.930904	−0.465452	−	0.885073i	$-0.654108\pi$
$653$	−23.7882	−0.930904	−0.465452	+	0.885073i	$0.654108\pi$
$654$	0	0
$655$	62.3546	2.43640
$656$	0	0
$657$	29.4696	1.14972
$658$	0	0
$659$	−18.8698	−0.735062	−0.367531	−	0.930011i	$-0.619797\pi$
$659$	−18.8698	−0.735062	−0.367531	+	0.930011i	$0.619797\pi$
$660$	0	0
$661$	−31.2126	−1.21403	−0.607015	−	0.794691i	$-0.707633\pi$
$661$	−31.2126	−1.21403	−0.607015	+	0.794691i	$0.707633\pi$
$662$	0	0
$663$	−0.661142	−0.0256766
$664$	0	0
$665$	−51.3686	−1.99199
$666$	0	0
$667$	−5.37208	−0.208008
$668$	0	0
$669$	−3.29560	−0.127415
$670$	0	0
$671$	59.3517	2.29125
$672$	0	0
$673$	47.1177	1.81625	0.908127	−	0.418695i	$-0.137512\pi$
$673$	47.1177	1.81625	0.908127	+	0.418695i	$0.137512\pi$
$674$	0	0
$675$	−45.7300	−1.76015
$676$	0	0
$677$	−2.36717	−0.0909779	−0.0454889	−	0.998965i	$-0.514485\pi$
$677$	−2.36717	−0.0909779	−0.0454889	+	0.998965i	$0.514485\pi$
$678$	0	0
$679$	−19.7932	−0.759595
$680$	0	0
$681$	17.6864	0.677743
$682$	0	0
$683$	5.92218	0.226606	0.113303	−	0.993560i	$-0.463857\pi$
$683$	5.92218	0.226606	0.113303	+	0.993560i	$0.463857\pi$
$684$	0	0
$685$	−9.64575	−0.368545
$686$	0	0
$687$	7.47607	0.285230
$688$	0	0
$689$	−6.22563	−0.237177
$690$	0	0
$691$	14.5963	0.555270	0.277635	−	0.960687i	$-0.410449\pi$
$691$	14.5963	0.555270	0.277635	+	0.960687i	$0.410449\pi$
$692$	0	0
$693$	72.1857	2.74211
$694$	0	0
$695$	42.1863	1.60022
$696$	0	0
$697$	1.21561	0.0460445
$698$	0	0
$699$	−8.07366	−0.305374
$700$	0	0
$701$	−0.810679	−0.0306189	−0.0153095	−	0.999883i	$-0.504873\pi$
$701$	−0.810679	−0.0306189	−0.0153095	+	0.999883i	$0.504873\pi$
$702$	0	0
$703$	7.04588	0.265740
$704$	0	0
$705$	−8.94059	−0.336722
$706$	0	0
$707$	−8.71934	−0.327925
$708$	0	0
$709$	−14.7383	−0.553508	−0.276754	−	0.960941i	$-0.589259\pi$
$709$	−14.7383	−0.553508	−0.276754	+	0.960941i	$0.589259\pi$
$710$	0	0
$711$	−0.946551	−0.0354984
$712$	0	0
$713$	9.40854	0.352353
$714$	0	0
$715$	−22.9114	−0.856836
$716$	0	0
$717$	−8.01821	−0.299446
$718$	0	0
$719$	−5.18877	−0.193508	−0.0967542	−	0.995308i	$-0.530846\pi$
$719$	−5.18877	−0.193508	−0.0967542	+	0.995308i	$0.530846\pi$
$720$	0	0
$721$	46.1699	1.71946
$722$	0	0
$723$	1.34464	0.0500078
$724$	0	0
$725$	−20.8598	−0.774715
$726$	0	0
$727$	2.93624	0.108899	0.0544496	−	0.998517i	$-0.482660\pi$
$727$	2.93624	0.108899	0.0544496	+	0.998517i	$0.482660\pi$
$728$	0	0
$729$	1.15094	0.0426274
$730$	0	0
$731$	9.21427	0.340802
$732$	0	0
$733$	−4.53301	−0.167430	−0.0837152	−	0.996490i	$-0.526679\pi$
$733$	−4.53301	−0.167430	−0.0837152	+	0.996490i	$0.526679\pi$
$734$	0	0
$735$	−55.7920	−2.05792
$736$	0	0
$737$	82.3563	3.03363
$738$	0	0
$739$	33.7934	1.24311	0.621555	−	0.783371i	$-0.286501\pi$
$739$	33.7934	1.24311	0.621555	+	0.783371i	$0.286501\pi$
$740$	0	0
$741$	−1.94290	−0.0713741
$742$	0	0
$743$	−6.05916	−0.222289	−0.111144	−	0.993804i	$-0.535452\pi$
$743$	−6.05916	−0.222289	−0.111144	+	0.993804i	$0.535452\pi$
$744$	0	0
$745$	−33.7130	−1.23515
$746$	0	0
$747$	9.72769	0.355917
$748$	0	0
$749$	42.1440	1.53991
$750$	0	0
$751$	−38.1075	−1.39056	−0.695280	−	0.718739i	$-0.744720\pi$
$751$	−38.1075	−1.39056	−0.695280	+	0.718739i	$0.744720\pi$
$752$	0	0
$753$	−1.91468	−0.0697748
$754$	0	0
$755$	92.0392	3.34965
$756$	0	0
$757$	3.62120	0.131615	0.0658074	−	0.997832i	$-0.479038\pi$
$757$	3.62120	0.131615	0.0658074	+	0.997832i	$0.479038\pi$
$758$	0	0
$759$	−13.3840	−0.485808
$760$	0	0
$761$	33.9368	1.23021	0.615105	−	0.788446i	$-0.289114\pi$
$761$	33.9368	1.23021	0.615105	+	0.788446i	$0.289114\pi$
$762$	0	0
$763$	−13.3444	−0.483099
$764$	0	0
$765$	8.33972	0.301523
$766$	0	0
$767$	0.792185	0.0286041
$768$	0	0
$769$	−22.6616	−0.817199	−0.408599	−	0.912714i	$-0.633983\pi$
$769$	−22.6616	−0.817199	−0.408599	+	0.912714i	$0.633983\pi$
$770$	0	0
$771$	3.96778	0.142896
$772$	0	0
$773$	−1.85144	−0.0665915	−0.0332958	−	0.999446i	$-0.510600\pi$
$773$	−1.85144	−0.0665915	−0.0332958	+	0.999446i	$0.510600\pi$
$774$	0	0
$775$	36.5335	1.31232
$776$	0	0
$777$	10.6669	0.382674
$778$	0	0
$779$	3.57231	0.127991
$780$	0	0
$781$	−45.8929	−1.64218
$782$	0	0
$783$	8.24106	0.294512
$784$	0	0
$785$	43.5195	1.55328
$786$	0	0
$787$	−1.88604	−0.0672302	−0.0336151	−	0.999435i	$-0.510702\pi$
$787$	−1.88604	−0.0672302	−0.0336151	+	0.999435i	$0.510702\pi$
$788$	0	0
$789$	18.1137	0.644865
$790$	0	0
$791$	−88.6956	−3.15365
$792$	0	0
$793$	9.18187	0.326058
$794$	0	0
$795$	−20.6845	−0.733605
$796$	0	0
$797$	−12.7479	−0.451555	−0.225777	−	0.974179i	$-0.572492\pi$
$797$	−12.7479	−0.451555	−0.225777	+	0.974179i	$0.572492\pi$
$798$	0	0
$799$	2.51952	0.0891341
$800$	0	0
$801$	37.4386	1.32283
$802$	0	0
$803$	75.8032	2.67504
$804$	0	0
$805$	−54.7433	−1.92945
$806$	0	0
$807$	−2.96523	−0.104381
$808$	0	0
$809$	−5.11274	−0.179754	−0.0898772	−	0.995953i	$-0.528647\pi$
$809$	−5.11274	−0.179754	−0.0898772	+	0.995953i	$0.528647\pi$
$810$	0	0
$811$	16.5806	0.582225	0.291113	−	0.956689i	$-0.405975\pi$
$811$	16.5806	0.582225	0.291113	+	0.956689i	$0.405975\pi$
$812$	0	0
$813$	−8.11817	−0.284717
$814$	0	0
$815$	−40.0570	−1.40313
$816$	0	0
$817$	27.0780	0.947338
$818$	0	0
$819$	11.1673	0.390218
$820$	0	0
$821$	36.7875	1.28389	0.641946	−	0.766750i	$-0.278127\pi$
$821$	36.7875	1.28389	0.641946	+	0.766750i	$0.278127\pi$
$822$	0	0
$823$	−35.7536	−1.24629	−0.623145	−	0.782106i	$-0.714146\pi$
$823$	−35.7536	−1.24629	−0.623145	+	0.782106i	$0.714146\pi$
$824$	0	0
$825$	−51.9702	−1.80937
$826$	0	0
$827$	43.6084	1.51641	0.758206	−	0.652015i	$-0.226076\pi$
$827$	43.6084	1.51641	0.758206	+	0.652015i	$0.226076\pi$
$828$	0	0
$829$	−41.5254	−1.44224	−0.721118	−	0.692812i	$-0.756372\pi$
$829$	−41.5254	−1.44224	−0.721118	+	0.692812i	$0.756372\pi$
$830$	0	0
$831$	−12.6365	−0.438354
$832$	0	0
$833$	15.7225	0.544754
$834$	0	0
$835$	27.5189	0.952332
$836$	0	0
$837$	−14.4332	−0.498885
$838$	0	0
$839$	−1.67546	−0.0578433	−0.0289216	−	0.999582i	$-0.509207\pi$
$839$	−1.67546	−0.0578433	−0.0289216	+	0.999582i	$0.509207\pi$
$840$	0	0
$841$	−25.2408	−0.870373
$842$	0	0
$843$	23.4135	0.806405
$844$	0	0
$845$	48.0622	1.65339
$846$	0	0
$847$	130.932	4.49887
$848$	0	0
$849$	−15.2716	−0.524120
$850$	0	0
$851$	7.50877	0.257397
$852$	0	0
$853$	−16.7442	−0.573311	−0.286655	−	0.958034i	$-0.592543\pi$
$853$	−16.7442	−0.573311	−0.286655	+	0.958034i	$0.592543\pi$
$854$	0	0
$855$	24.5079	0.838154
$856$	0	0
$857$	−30.4030	−1.03855	−0.519273	−	0.854608i	$-0.673797\pi$
$857$	−30.4030	−1.03855	−0.519273	+	0.854608i	$0.673797\pi$
$858$	0	0
$859$	−38.6690	−1.31937	−0.659685	−	0.751542i	$-0.729310\pi$
$859$	−38.6690	−1.31937	−0.659685	+	0.751542i	$0.729310\pi$
$860$	0	0
$861$	5.40822	0.184312
$862$	0	0
$863$	−21.7245	−0.739511	−0.369756	−	0.929129i	$-0.620559\pi$
$863$	−21.7245	−0.739511	−0.369756	+	0.929129i	$0.620559\pi$
$864$	0	0
$865$	102.783	3.49471
$866$	0	0
$867$	−12.8254	−0.435574
$868$	0	0
$869$	−2.43477	−0.0825938
$870$	0	0
$871$	12.7407	0.431703
$872$	0	0
$873$	9.44335	0.319609
$874$	0	0
$875$	−113.781	−3.84649
$876$	0	0
$877$	−25.2290	−0.851923	−0.425961	−	0.904741i	$-0.640064\pi$
$877$	−25.2290	−0.851923	−0.425961	+	0.904741i	$0.640064\pi$
$878$	0	0
$879$	−6.67694	−0.225208
$880$	0	0
$881$	−8.90432	−0.299994	−0.149997	−	0.988686i	$-0.547926\pi$
$881$	−8.90432	−0.299994	−0.149997	+	0.988686i	$0.547926\pi$
$882$	0	0
$883$	51.5248	1.73395	0.866974	−	0.498353i	$-0.166062\pi$
$883$	51.5248	1.73395	0.866974	+	0.498353i	$0.166062\pi$
$884$	0	0
$885$	2.63202	0.0884744
$886$	0	0
$887$	−46.1037	−1.54801	−0.774005	−	0.633179i	$-0.781750\pi$
$887$	−46.1037	−1.54801	−0.774005	+	0.633179i	$0.781750\pi$
$888$	0	0
$889$	−74.2987	−2.49190
$890$	0	0
$891$	−22.9792	−0.769833
$892$	0	0
$893$	7.40411	0.247769
$894$	0	0
$895$	1.45357	0.0485876
$896$	0	0
$897$	−2.07054	−0.0691333
$898$	0	0
$899$	−6.58374	−0.219580
$900$	0	0
$901$	5.82904	0.194193
$902$	0	0
$903$	40.9941	1.36420
$904$	0	0
$905$	−54.5318	−1.81270
$906$	0	0
$907$	−14.9725	−0.497152	−0.248576	−	0.968612i	$-0.579963\pi$
$907$	−14.9725	−0.497152	−0.248576	+	0.968612i	$0.579963\pi$
$908$	0	0
$909$	4.16000	0.137978
$910$	0	0
$911$	−19.9974	−0.662543	−0.331271	−	0.943536i	$-0.607478\pi$
$911$	−19.9974	−0.662543	−0.331271	+	0.943536i	$0.607478\pi$
$912$	0	0
$913$	25.0220	0.828109
$914$	0	0
$915$	30.5066	1.00852
$916$	0	0
$917$	−78.1772	−2.58164
$918$	0	0
$919$	42.9780	1.41771	0.708857	−	0.705353i	$-0.249211\pi$
$919$	42.9780	1.41771	0.708857	+	0.705353i	$0.249211\pi$
$920$	0	0
$921$	−14.6112	−0.481454
$922$	0	0
$923$	−7.09976	−0.233692
$924$	0	0
$925$	29.1566	0.958664
$926$	0	0
$927$	−22.0276	−0.723483
$928$	0	0
$929$	23.1037	0.758007	0.379003	−	0.925395i	$-0.376267\pi$
$929$	23.1037	0.758007	0.379003	+	0.925395i	$0.376267\pi$
$930$	0	0
$931$	46.2038	1.51427
$932$	0	0
$933$	−14.4287	−0.472374
$934$	0	0
$935$	21.4518	0.701550
$936$	0	0
$937$	0.0952572	0.00311192	0.00155596	−	0.999999i	$-0.499505\pi$
$937$	0.0952572	0.00311192	0.00155596	+	0.999999i	$0.499505\pi$
$938$	0	0
$939$	−8.69529	−0.283760
$940$	0	0
$941$	−31.3460	−1.02185	−0.510925	−	0.859625i	$-0.670697\pi$
$941$	−31.3460	−1.02185	−0.510925	+	0.859625i	$0.670697\pi$
$942$	0	0
$943$	3.80700	0.123973
$944$	0	0
$945$	83.9792	2.73184
$946$	0	0
$947$	−46.5491	−1.51264	−0.756321	−	0.654200i	$-0.773005\pi$
$947$	−46.5491	−1.51264	−0.756321	+	0.654200i	$0.773005\pi$
$948$	0	0
$949$	11.7270	0.380673
$950$	0	0
$951$	−8.53703	−0.276832
$952$	0	0
$953$	−23.5470	−0.762761	−0.381380	−	0.924418i	$-0.624551\pi$
$953$	−23.5470	−0.762761	−0.381380	+	0.924418i	$0.624551\pi$
$954$	0	0
$955$	33.6942	1.09032
$956$	0	0
$957$	9.36561	0.302747
$958$	0	0
$959$	12.0934	0.390515
$960$	0	0
$961$	−19.4694	−0.628045
$962$	0	0
$963$	−20.1069	−0.647935
$964$	0	0
$965$	−63.8093	−2.05409
$966$	0	0
$967$	23.9508	0.770204	0.385102	−	0.922874i	$-0.374166\pi$
$967$	23.9508	0.770204	0.385102	+	0.922874i	$0.374166\pi$
$968$	0	0
$969$	1.81913	0.0584389
$970$	0	0
$971$	43.5907	1.39889	0.699447	−	0.714685i	$-0.253430\pi$
$971$	43.5907	1.39889	0.699447	+	0.714685i	$0.253430\pi$
$972$	0	0
$973$	−52.8911	−1.69561
$974$	0	0
$975$	−8.03993	−0.257484
$976$	0	0
$977$	−17.2616	−0.552246	−0.276123	−	0.961122i	$-0.589050\pi$
$977$	−17.2616	−0.552246	−0.276123	+	0.961122i	$0.589050\pi$
$978$	0	0
$979$	96.3014	3.07781
$980$	0	0
$981$	6.36660	0.203270
$982$	0	0
$983$	−45.3830	−1.44749	−0.723747	−	0.690066i	$-0.757582\pi$
$983$	−45.3830	−1.44749	−0.723747	+	0.690066i	$0.757582\pi$
$984$	0	0
$985$	66.6849	2.12476
$986$	0	0
$987$	11.2093	0.356795
$988$	0	0
$989$	28.8569	0.917596
$990$	0	0
$991$	−38.7136	−1.22978	−0.614889	−	0.788613i	$-0.710799\pi$
$991$	−38.7136	−1.22978	−0.614889	+	0.788613i	$0.710799\pi$
$992$	0	0
$993$	−2.13948	−0.0678945
$994$	0	0
$995$	34.9016	1.10646
$996$	0	0
$997$	58.6255	1.85669	0.928343	−	0.371724i	$-0.121233\pi$
$997$	58.6255	1.85669	0.928343	+	0.371724i	$0.121233\pi$
$998$	0	0
$999$	−11.5189	−0.364441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.10		18
4.3	odd	2		547.2.a.b.1.17	✓	18
12.11	even	2		4923.2.a.l.1.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.17	✓	18	4.3	odd	2
4923.2.a.l.1.2		18	12.11	even	2
8752.2.a.s.1.10		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8752 = 2^{4} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8752.a (trivial)

\(f(q)\)	\(=\)	\(q+0.790850 q^{3} -3.96974 q^{5} +4.97706 q^{7} -2.37456 q^{9} +O(q^{10})\)	\(q+0.790850 q^{3} -3.96974 q^{5} +4.97706 q^{7} -2.37456 q^{9} -6.10795 q^{11} -0.944916 q^{13} -3.13947 q^{15} +0.884722 q^{17} +2.59993 q^{19} +3.93611 q^{21} +2.77074 q^{23} +10.7588 q^{25} -4.25047 q^{27} -1.93886 q^{29} +3.39568 q^{31} -4.83047 q^{33} -19.7576 q^{35} +2.71002 q^{37} -0.747287 q^{39} +1.37400 q^{41} +10.4149 q^{43} +9.42637 q^{45} +2.84781 q^{47} +17.7712 q^{49} +0.699683 q^{51} +6.58855 q^{53} +24.2470 q^{55} +2.05616 q^{57} -0.838365 q^{59} -9.71712 q^{61} -11.8183 q^{63} +3.75107 q^{65} -13.4835 q^{67} +2.19124 q^{69} +7.51364 q^{71} -12.4106 q^{73} +8.50862 q^{75} -30.3996 q^{77} +0.398622 q^{79} +3.76218 q^{81} -4.09663 q^{83} -3.51212 q^{85} -1.53335 q^{87} -15.7666 q^{89} -4.70291 q^{91} +2.68547 q^{93} -10.3211 q^{95} -3.97689 q^{97} +14.5037 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) \) 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9 - 2 * q^11 - 25 * q^13 - 9 * q^15 - 30 * q^17 - 4 * q^19 - 16 * q^21 + 26 * q^23 + 31 * q^25 + 37 * q^27 - 18 * q^29 + 5 * q^31 - 10 * q^33 + 9 * q^35 - 18 * q^37 - 7 * q^39 - 17 * q^41 - 8 * q^43 - 44 * q^45 + 52 * q^47 + 29 * q^49 - 19 * q^51 - 60 * q^53 - 11 * q^55 + 4 * q^57 + 8 * q^59 - 26 * q^61 + q^63 - 6 * q^65 - 12 * q^67 - 38 * q^69 + q^71 - 2 * q^73 + 17 * q^75 - 73 * q^77 - 18 * q^79 + 18 * q^81 + 43 * q^83 + 51 * q^85 - 3 * q^87 - 28 * q^89 + q^91 - 60 * q^93 + 18 * q^95 - 34 * q^97 - 15 * q^99

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