LMFDB - Embedded newform 8752.2.a.s.1.8

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			1.8
Root			$1.98431$ 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)

$f(q)$	$=$	$q+0.150114 q^{3} +2.87852 q^{5} +2.68467 q^{7} -2.97747 q^{9} +O(q^{10})$	$q+0.150114 q^{3} +2.87852 q^{5} +2.68467 q^{7} -2.97747 q^{9} -0.368123 q^{11} -2.43844 q^{13} +0.432105 q^{15} -3.34729 q^{17} -0.377567 q^{19} +0.403005 q^{21} +0.915445 q^{23} +3.28588 q^{25} -0.897299 q^{27} -2.06894 q^{29} -5.88622 q^{31} -0.0552603 q^{33} +7.72786 q^{35} -7.95740 q^{37} -0.366043 q^{39} -1.24111 q^{41} -5.58190 q^{43} -8.57070 q^{45} +12.4681 q^{47} +0.207426 q^{49} -0.502474 q^{51} -11.7975 q^{53} -1.05965 q^{55} -0.0566780 q^{57} +13.7347 q^{59} +1.64414 q^{61} -7.99350 q^{63} -7.01909 q^{65} -5.77093 q^{67} +0.137421 q^{69} +7.47759 q^{71} -10.4002 q^{73} +0.493255 q^{75} -0.988288 q^{77} -0.641853 q^{79} +8.79770 q^{81} -15.7053 q^{83} -9.63525 q^{85} -0.310576 q^{87} +4.77332 q^{89} -6.54639 q^{91} -0.883602 q^{93} -1.08683 q^{95} -15.9649 q^{97} +1.09607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.150114	0.0866682	0.0433341	−	0.999061i	$-0.486202\pi$
$3$	0.150114	0.0866682	0.0433341	+	0.999061i	$0.486202\pi$
$4$	0	0
$5$	2.87852	1.28731	0.643657	−	0.765314i	$-0.277417\pi$
$5$	2.87852	1.28731	0.643657	+	0.765314i	$0.277417\pi$
$6$	0	0
$7$	2.68467	1.01471	0.507354	−	0.861738i	$-0.330624\pi$
$7$	2.68467	1.01471	0.507354	+	0.861738i	$0.330624\pi$
$8$	0	0
$9$	−2.97747	−0.992489
$10$	0	0
$11$	−0.368123	−0.110993	−0.0554967	−	0.998459i	$-0.517674\pi$
$11$	−0.368123	−0.110993	−0.0554967	+	0.998459i	$0.517674\pi$
$12$	0	0
$13$	−2.43844	−0.676301	−0.338150	−	0.941092i	$-0.609801\pi$
$13$	−2.43844	−0.676301	−0.338150	+	0.941092i	$0.609801\pi$
$14$	0	0
$15$	0.432105	0.111569
$16$	0	0
$17$	−3.34729	−0.811838	−0.405919	−	0.913909i	$-0.633048\pi$
$17$	−3.34729	−0.811838	−0.405919	+	0.913909i	$0.633048\pi$
$18$	0	0
$19$	−0.377567	−0.0866199	−0.0433099	−	0.999062i	$-0.513790\pi$
$19$	−0.377567	−0.0866199	−0.0433099	+	0.999062i	$0.513790\pi$
$20$	0	0
$21$	0.403005	0.0879429
$22$	0	0
$23$	0.915445	0.190884	0.0954418	−	0.995435i	$-0.469574\pi$
$23$	0.915445	0.190884	0.0954418	+	0.995435i	$0.469574\pi$
$24$	0	0
$25$	3.28588	0.657175
$26$	0	0
$27$	−0.897299	−0.172685
$28$	0	0
$29$	−2.06894	−0.384192	−0.192096	−	0.981376i	$-0.561529\pi$
$29$	−2.06894	−0.384192	−0.192096	+	0.981376i	$0.561529\pi$
$30$	0	0
$31$	−5.88622	−1.05720	−0.528598	−	0.848872i	$-0.677282\pi$
$31$	−5.88622	−1.05720	−0.528598	+	0.848872i	$0.677282\pi$
$32$	0	0
$33$	−0.0552603	−0.00961959
$34$	0	0
$35$	7.72786	1.30625
$36$	0	0
$37$	−7.95740	−1.30819	−0.654094	−	0.756413i	$-0.726950\pi$
$37$	−7.95740	−1.30819	−0.654094	+	0.756413i	$0.726950\pi$
$38$	0	0
$39$	−0.366043	−0.0586138
$40$	0	0
$41$	−1.24111	−0.193829	−0.0969145	−	0.995293i	$-0.530897\pi$
$41$	−1.24111	−0.193829	−0.0969145	+	0.995293i	$0.530897\pi$
$42$	0	0
$43$	−5.58190	−0.851232	−0.425616	−	0.904904i	$-0.639943\pi$
$43$	−5.58190	−0.851232	−0.425616	+	0.904904i	$0.639943\pi$
$44$	0	0
$45$	−8.57070	−1.27764
$46$	0	0
$47$	12.4681	1.81866	0.909329	−	0.416077i	$-0.136595\pi$
$47$	12.4681	1.81866	0.909329	+	0.416077i	$0.136595\pi$
$48$	0	0
$49$	0.207426	0.0296323
$50$	0	0
$51$	−0.502474	−0.0703605
$52$	0	0
$53$	−11.7975	−1.62052	−0.810258	−	0.586074i	$-0.800673\pi$
$53$	−11.7975	−1.62052	−0.810258	+	0.586074i	$0.800673\pi$
$54$	0	0
$55$	−1.05965	−0.142883
$56$	0	0
$57$	−0.0566780	−0.00750719
$58$	0	0
$59$	13.7347	1.78810	0.894051	−	0.447964i	$-0.147851\pi$
$59$	13.7347	1.78810	0.894051	+	0.447964i	$0.147851\pi$
$60$	0	0
$61$	1.64414	0.210511	0.105255	−	0.994445i	$-0.466434\pi$
$61$	1.64414	0.210511	0.105255	+	0.994445i	$0.466434\pi$
$62$	0	0
$63$	−7.99350	−1.00709
$64$	0	0
$65$	−7.01909	−0.870611
$66$	0	0
$67$	−5.77093	−0.705031	−0.352515	−	0.935806i	$-0.614674\pi$
$67$	−5.77093	−0.705031	−0.352515	+	0.935806i	$0.614674\pi$
$68$	0	0
$69$	0.137421	0.0165435
$70$	0	0
$71$	7.47759	0.887426	0.443713	−	0.896169i	$-0.353661\pi$
$71$	7.47759	0.887426	0.443713	+	0.896169i	$0.353661\pi$
$72$	0	0
$73$	−10.4002	−1.21725	−0.608625	−	0.793458i	$-0.708279\pi$
$73$	−10.4002	−1.21725	−0.608625	+	0.793458i	$0.708279\pi$
$74$	0	0
$75$	0.493255	0.0569562
$76$	0	0
$77$	−0.988288	−0.112626
$78$	0	0
$79$	−0.641853	−0.0722141	−0.0361070	−	0.999348i	$-0.511496\pi$
$79$	−0.641853	−0.0722141	−0.0361070	+	0.999348i	$0.511496\pi$
$80$	0	0
$81$	8.79770	0.977522
$82$	0	0
$83$	−15.7053	−1.72388	−0.861942	−	0.507006i	$-0.830752\pi$
$83$	−15.7053	−1.72388	−0.861942	+	0.507006i	$0.830752\pi$
$84$	0	0
$85$	−9.63525	−1.04509
$86$	0	0
$87$	−0.310576	−0.0332972
$88$	0	0
$89$	4.77332	0.505971	0.252986	−	0.967470i	$-0.418587\pi$
$89$	4.77332	0.505971	0.252986	+	0.967470i	$0.418587\pi$
$90$	0	0
$91$	−6.54639	−0.686248
$92$	0	0
$93$	−0.883602	−0.0916253
$94$	0	0
$95$	−1.08683	−0.111507
$96$	0	0
$97$	−15.9649	−1.62099	−0.810497	−	0.585742i	$-0.800803\pi$
$97$	−15.9649	−1.62099	−0.810497	+	0.585742i	$0.800803\pi$
$98$	0	0
$99$	1.09607	0.110160
$100$	0	0
$101$	13.4395	1.33728	0.668642	−	0.743585i	$-0.266876\pi$
$101$	13.4395	1.33728	0.668642	+	0.743585i	$0.266876\pi$
$102$	0	0
$103$	−8.31399	−0.819202	−0.409601	−	0.912265i	$-0.634332\pi$
$103$	−8.31399	−0.819202	−0.409601	+	0.912265i	$0.634332\pi$
$104$	0	0
$105$	1.16006	0.113210
$106$	0	0
$107$	−1.14891	−0.111069	−0.0555347	−	0.998457i	$-0.517686\pi$
$107$	−1.14891	−0.111069	−0.0555347	+	0.998457i	$0.517686\pi$
$108$	0	0
$109$	8.92809	0.855156	0.427578	−	0.903978i	$-0.359367\pi$
$109$	8.92809	0.855156	0.427578	+	0.903978i	$0.359367\pi$
$110$	0	0
$111$	−1.19451	−0.113378
$112$	0	0
$113$	−8.71949	−0.820260	−0.410130	−	0.912027i	$-0.634517\pi$
$113$	−8.71949	−0.820260	−0.410130	+	0.912027i	$0.634517\pi$
$114$	0	0
$115$	2.63513	0.245727
$116$	0	0
$117$	7.26037	0.671221
$118$	0	0
$119$	−8.98636	−0.823778
$120$	0	0
$121$	−10.8645	−0.987680
$122$	0	0
$123$	−0.186308	−0.0167988
$124$	0	0
$125$	−4.93414	−0.441323
$126$	0	0
$127$	2.33627	0.207310	0.103655	−	0.994613i	$-0.466946\pi$
$127$	2.33627	0.207310	0.103655	+	0.994613i	$0.466946\pi$
$128$	0	0
$129$	−0.837920	−0.0737747
$130$	0	0
$131$	1.82810	0.159722	0.0798610	−	0.996806i	$-0.474552\pi$
$131$	1.82810	0.159722	0.0798610	+	0.996806i	$0.474552\pi$
$132$	0	0
$133$	−1.01364	−0.0878939
$134$	0	0
$135$	−2.58289	−0.222300
$136$	0	0
$137$	−13.2541	−1.13238	−0.566188	−	0.824276i	$-0.691582\pi$
$137$	−13.2541	−1.13238	−0.566188	+	0.824276i	$0.691582\pi$
$138$	0	0
$139$	−11.3648	−0.963953	−0.481976	−	0.876184i	$-0.660081\pi$
$139$	−11.3648	−0.963953	−0.481976	+	0.876184i	$0.660081\pi$
$140$	0	0
$141$	1.87163	0.157620
$142$	0	0
$143$	0.897646	0.0750649
$144$	0	0
$145$	−5.95548	−0.494576
$146$	0	0
$147$	0.0311375	0.00256818
$148$	0	0
$149$	6.54097	0.535857	0.267929	−	0.963439i	$-0.413661\pi$
$149$	6.54097	0.535857	0.267929	+	0.963439i	$0.413661\pi$
$150$	0	0
$151$	−10.4153	−0.847586	−0.423793	−	0.905759i	$-0.639302\pi$
$151$	−10.4153	−0.847586	−0.423793	+	0.905759i	$0.639302\pi$
$152$	0	0
$153$	9.96645	0.805740
$154$	0	0
$155$	−16.9436	−1.36094
$156$	0	0
$157$	9.34946	0.746168	0.373084	−	0.927797i	$-0.378300\pi$
$157$	9.34946	0.746168	0.373084	+	0.927797i	$0.378300\pi$
$158$	0	0
$159$	−1.77097	−0.140447
$160$	0	0
$161$	2.45766	0.193691
$162$	0	0
$163$	−7.10285	−0.556338	−0.278169	−	0.960532i	$-0.589728\pi$
$163$	−7.10285	−0.556338	−0.278169	+	0.960532i	$0.589728\pi$
$164$	0	0
$165$	−0.159068	−0.0123834
$166$	0	0
$167$	10.3824	0.803411	0.401705	−	0.915769i	$-0.368418\pi$
$167$	10.3824	0.803411	0.401705	+	0.915769i	$0.368418\pi$
$168$	0	0
$169$	−7.05402	−0.542617
$170$	0	0
$171$	1.12419	0.0859692
$172$	0	0
$173$	−4.93534	−0.375227	−0.187614	−	0.982243i	$-0.560075\pi$
$173$	−4.93534	−0.375227	−0.187614	+	0.982243i	$0.560075\pi$
$174$	0	0
$175$	8.82148	0.666841
$176$	0	0
$177$	2.06176	0.154972
$178$	0	0
$179$	−17.2958	−1.29275	−0.646375	−	0.763020i	$-0.723716\pi$
$179$	−17.2958	−1.29275	−0.646375	+	0.763020i	$0.723716\pi$
$180$	0	0
$181$	17.2065	1.27895	0.639476	−	0.768811i	$-0.279152\pi$
$181$	17.2065	1.27895	0.639476	+	0.768811i	$0.279152\pi$
$182$	0	0
$183$	0.246808	0.0182446
$184$	0	0
$185$	−22.9055	−1.68405
$186$	0	0
$187$	1.23222	0.0901086
$188$	0	0
$189$	−2.40895	−0.175225
$190$	0	0
$191$	0.969679	0.0701635	0.0350817	−	0.999384i	$-0.488831\pi$
$191$	0.969679	0.0701635	0.0350817	+	0.999384i	$0.488831\pi$
$192$	0	0
$193$	7.21747	0.519525	0.259763	−	0.965673i	$-0.416356\pi$
$193$	7.21747	0.519525	0.259763	+	0.965673i	$0.416356\pi$
$194$	0	0
$195$	−1.05366	−0.0754543
$196$	0	0
$197$	−7.29036	−0.519417	−0.259708	−	0.965687i	$-0.583626\pi$
$197$	−7.29036	−0.519417	−0.259708	+	0.965687i	$0.583626\pi$
$198$	0	0
$199$	6.97295	0.494299	0.247150	−	0.968977i	$-0.420506\pi$
$199$	6.97295	0.494299	0.247150	+	0.968977i	$0.420506\pi$
$200$	0	0
$201$	−0.866295	−0.0611037
$202$	0	0
$203$	−5.55441	−0.389843
$204$	0	0
$205$	−3.57256	−0.249519
$206$	0	0
$207$	−2.72571	−0.189450
$208$	0	0
$209$	0.138991	0.00961423
$210$	0	0
$211$	−2.25378	−0.155157	−0.0775784	−	0.996986i	$-0.524719\pi$
$211$	−2.25378	−0.155157	−0.0775784	+	0.996986i	$0.524719\pi$
$212$	0	0
$213$	1.12249	0.0769116
$214$	0	0
$215$	−16.0676	−1.09580
$216$	0	0
$217$	−15.8025	−1.07275
$218$	0	0
$219$	−1.56121	−0.105497
$220$	0	0
$221$	8.16217	0.549047
$222$	0	0
$223$	5.31003	0.355586	0.177793	−	0.984068i	$-0.443104\pi$
$223$	5.31003	0.355586	0.177793	+	0.984068i	$0.443104\pi$
$224$	0	0
$225$	−9.78359	−0.652239
$226$	0	0
$227$	2.18872	0.145271	0.0726354	−	0.997359i	$-0.476859\pi$
$227$	2.18872	0.145271	0.0726354	+	0.997359i	$0.476859\pi$
$228$	0	0
$229$	4.87556	0.322186	0.161093	−	0.986939i	$-0.448498\pi$
$229$	4.87556	0.322186	0.161093	+	0.986939i	$0.448498\pi$
$230$	0	0
$231$	−0.148356	−0.00976108
$232$	0	0
$233$	−15.8464	−1.03813	−0.519066	−	0.854734i	$-0.673720\pi$
$233$	−15.8464	−1.03813	−0.519066	+	0.854734i	$0.673720\pi$
$234$	0	0
$235$	35.8897	2.34118
$236$	0	0
$237$	−0.0963509	−0.00625866
$238$	0	0
$239$	−24.8660	−1.60845	−0.804223	−	0.594328i	$-0.797418\pi$
$239$	−24.8660	−1.60845	−0.804223	+	0.594328i	$0.797418\pi$
$240$	0	0
$241$	17.9529	1.15645	0.578223	−	0.815879i	$-0.303746\pi$
$241$	17.9529	1.15645	0.578223	+	0.815879i	$0.303746\pi$
$242$	0	0
$243$	4.01255	0.257405
$244$	0	0
$245$	0.597081	0.0381461
$246$	0	0
$247$	0.920674	0.0585811
$248$	0	0
$249$	−2.35759	−0.149406
$250$	0	0
$251$	−1.86475	−0.117702	−0.0588509	−	0.998267i	$-0.518744\pi$
$251$	−1.86475	−0.117702	−0.0588509	+	0.998267i	$0.518744\pi$
$252$	0	0
$253$	−0.336997	−0.0211868
$254$	0	0
$255$	−1.44638	−0.0905760
$256$	0	0
$257$	2.57803	0.160813	0.0804066	−	0.996762i	$-0.474378\pi$
$257$	2.57803	0.160813	0.0804066	+	0.996762i	$0.474378\pi$
$258$	0	0
$259$	−21.3630	−1.32743
$260$	0	0
$261$	6.16019	0.381306
$262$	0	0
$263$	−5.06765	−0.312484	−0.156242	−	0.987719i	$-0.549938\pi$
$263$	−5.06765	−0.312484	−0.156242	+	0.987719i	$0.549938\pi$
$264$	0	0
$265$	−33.9594	−2.08611
$266$	0	0
$267$	0.716541	0.0438516
$268$	0	0
$269$	−10.2521	−0.625082	−0.312541	−	0.949904i	$-0.601180\pi$
$269$	−10.2521	−0.625082	−0.312541	+	0.949904i	$0.601180\pi$
$270$	0	0
$271$	26.7152	1.62283	0.811415	−	0.584470i	$-0.198698\pi$
$271$	26.7152	1.62283	0.811415	+	0.584470i	$0.198698\pi$
$272$	0	0
$273$	−0.982703	−0.0594759
$274$	0	0
$275$	−1.20961	−0.0729421
$276$	0	0
$277$	9.61574	0.577754	0.288877	−	0.957366i	$-0.406718\pi$
$277$	9.61574	0.577754	0.288877	+	0.957366i	$0.406718\pi$
$278$	0	0
$279$	17.5260	1.04926
$280$	0	0
$281$	22.9906	1.37150	0.685751	−	0.727837i	$-0.259474\pi$
$281$	22.9906	1.37150	0.685751	+	0.727837i	$0.259474\pi$
$282$	0	0
$283$	−8.48049	−0.504113	−0.252056	−	0.967713i	$-0.581107\pi$
$283$	−8.48049	−0.504113	−0.252056	+	0.967713i	$0.581107\pi$
$284$	0	0
$285$	−0.163149	−0.00966410
$286$	0	0
$287$	−3.33197	−0.196680
$288$	0	0
$289$	−5.79563	−0.340919
$290$	0	0
$291$	−2.39656	−0.140489
$292$	0	0
$293$	−11.3819	−0.664937	−0.332468	−	0.943114i	$-0.607881\pi$
$293$	−11.3819	−0.664937	−0.332468	+	0.943114i	$0.607881\pi$
$294$	0	0
$295$	39.5355	2.30185
$296$	0	0
$297$	0.330317	0.0191669
$298$	0	0
$299$	−2.23226	−0.129095
$300$	0	0
$301$	−14.9855	−0.863752
$302$	0	0
$303$	2.01746	0.115900
$304$	0	0
$305$	4.73269	0.270993
$306$	0	0
$307$	5.38784	0.307500	0.153750	−	0.988110i	$-0.450865\pi$
$307$	5.38784	0.307500	0.153750	+	0.988110i	$0.450865\pi$
$308$	0	0
$309$	−1.24804	−0.0709987
$310$	0	0
$311$	20.0317	1.13589	0.567947	−	0.823065i	$-0.307738\pi$
$311$	20.0317	1.13589	0.567947	+	0.823065i	$0.307738\pi$
$312$	0	0
$313$	27.9448	1.57953	0.789766	−	0.613408i	$-0.210202\pi$
$313$	27.9448	1.57953	0.789766	+	0.613408i	$0.210202\pi$
$314$	0	0
$315$	−23.0094	−1.29644
$316$	0	0
$317$	4.64368	0.260815	0.130408	−	0.991460i	$-0.458371\pi$
$317$	4.64368	0.260815	0.130408	+	0.991460i	$0.458371\pi$
$318$	0	0
$319$	0.761625	0.0426428
$320$	0	0
$321$	−0.172467	−0.00962618
$322$	0	0
$323$	1.26383	0.0703213
$324$	0	0
$325$	−8.01241	−0.444448
$326$	0	0
$327$	1.34023	0.0741148
$328$	0	0
$329$	33.4727	1.84541
$330$	0	0
$331$	−6.20064	−0.340818	−0.170409	−	0.985373i	$-0.554509\pi$
$331$	−6.20064	−0.340818	−0.170409	+	0.985373i	$0.554509\pi$
$332$	0	0
$333$	23.6929	1.29836
$334$	0	0
$335$	−16.6117	−0.907596
$336$	0	0
$337$	−5.35112	−0.291494	−0.145747	−	0.989322i	$-0.546559\pi$
$337$	−5.35112	−0.291494	−0.145747	+	0.989322i	$0.546559\pi$
$338$	0	0
$339$	−1.30891	−0.0710905
$340$	0	0
$341$	2.16686	0.117342
$342$	0	0
$343$	−18.2358	−0.984640
$344$	0	0
$345$	0.395569	0.0212967
$346$	0	0
$347$	23.3962	1.25597	0.627987	−	0.778224i	$-0.283879\pi$
$347$	23.3962	1.25597	0.627987	+	0.778224i	$0.283879\pi$
$348$	0	0
$349$	15.4705	0.828119	0.414060	−	0.910250i	$-0.364111\pi$
$349$	15.4705	0.828119	0.414060	+	0.910250i	$0.364111\pi$
$350$	0	0
$351$	2.18801	0.116787
$352$	0	0
$353$	26.3066	1.40016	0.700079	−	0.714066i	$-0.253148\pi$
$353$	26.3066	1.40016	0.700079	+	0.714066i	$0.253148\pi$
$354$	0	0
$355$	21.5244	1.14240
$356$	0	0
$357$	−1.34898	−0.0713954
$358$	0	0
$359$	36.8814	1.94652	0.973262	−	0.229698i	$-0.0737740\pi$
$359$	36.8814	1.94652	0.973262	+	0.229698i	$0.0737740\pi$
$360$	0	0
$361$	−18.8574	−0.992497
$362$	0	0
$363$	−1.63091	−0.0856005
$364$	0	0
$365$	−29.9372	−1.56698
$366$	0	0
$367$	−20.8701	−1.08941	−0.544706	−	0.838627i	$-0.683359\pi$
$367$	−20.8701	−1.08941	−0.544706	+	0.838627i	$0.683359\pi$
$368$	0	0
$369$	3.69537	0.192373
$370$	0	0
$371$	−31.6724	−1.64435
$372$	0	0
$373$	−18.4400	−0.954786	−0.477393	−	0.878690i	$-0.658418\pi$
$373$	−18.4400	−0.954786	−0.477393	+	0.878690i	$0.658418\pi$
$374$	0	0
$375$	−0.740681	−0.0382486
$376$	0	0
$377$	5.04498	0.259830
$378$	0	0
$379$	−13.2372	−0.679947	−0.339974	−	0.940435i	$-0.610418\pi$
$379$	−13.2372	−0.679947	−0.339974	+	0.940435i	$0.610418\pi$
$380$	0	0
$381$	0.350706	0.0179672
$382$	0	0
$383$	3.10284	0.158548	0.0792738	−	0.996853i	$-0.474740\pi$
$383$	3.10284	0.158548	0.0792738	+	0.996853i	$0.474740\pi$
$384$	0	0
$385$	−2.84481	−0.144985
$386$	0	0
$387$	16.6199	0.844838
$388$	0	0
$389$	0.502131	0.0254591	0.0127295	−	0.999919i	$-0.495948\pi$
$389$	0.502131	0.0254591	0.0127295	+	0.999919i	$0.495948\pi$
$390$	0	0
$391$	−3.06426	−0.154966
$392$	0	0
$393$	0.274423	0.0138428
$394$	0	0
$395$	−1.84759	−0.0929622
$396$	0	0
$397$	−6.23481	−0.312916	−0.156458	−	0.987685i	$-0.550008\pi$
$397$	−6.23481	−0.312916	−0.156458	+	0.987685i	$0.550008\pi$
$398$	0	0
$399$	−0.152161	−0.00761760
$400$	0	0
$401$	−12.4398	−0.621216	−0.310608	−	0.950538i	$-0.600533\pi$
$401$	−12.4398	−0.621216	−0.310608	+	0.950538i	$0.600533\pi$
$402$	0	0
$403$	14.3532	0.714983
$404$	0	0
$405$	25.3244	1.25838
$406$	0	0
$407$	2.92931	0.145200
$408$	0	0
$409$	20.4619	1.01178	0.505888	−	0.862599i	$-0.331165\pi$
$409$	20.4619	1.01178	0.505888	+	0.862599i	$0.331165\pi$
$410$	0	0
$411$	−1.98962	−0.0981409
$412$	0	0
$413$	36.8730	1.81440
$414$	0	0
$415$	−45.2081	−2.21918
$416$	0	0
$417$	−1.70602	−0.0835440
$418$	0	0
$419$	−28.1633	−1.37587	−0.687933	−	0.725774i	$-0.741482\pi$
$419$	−28.1633	−1.37587	−0.687933	+	0.725774i	$0.741482\pi$
$420$	0	0
$421$	30.5281	1.48785	0.743924	−	0.668264i	$-0.232962\pi$
$421$	30.5281	1.48785	0.743924	+	0.668264i	$0.232962\pi$
$422$	0	0
$423$	−37.1233	−1.80500
$424$	0	0
$425$	−10.9988	−0.533520
$426$	0	0
$427$	4.41397	0.213607
$428$	0	0
$429$	0.134749	0.00650574
$430$	0	0
$431$	−25.7216	−1.23897	−0.619483	−	0.785010i	$-0.712658\pi$
$431$	−25.7216	−1.23897	−0.619483	+	0.785010i	$0.712658\pi$
$432$	0	0
$433$	23.1151	1.11084	0.555420	−	0.831570i	$-0.312557\pi$
$433$	23.1151	1.11084	0.555420	+	0.831570i	$0.312557\pi$
$434$	0	0
$435$	−0.893999	−0.0428640
$436$	0	0
$437$	−0.345642	−0.0165343
$438$	0	0
$439$	32.1881	1.53626	0.768128	−	0.640296i	$-0.221188\pi$
$439$	32.1881	1.53626	0.768128	+	0.640296i	$0.221188\pi$
$440$	0	0
$441$	−0.617605	−0.0294098
$442$	0	0
$443$	3.71749	0.176623	0.0883117	−	0.996093i	$-0.471853\pi$
$443$	3.71749	0.176623	0.0883117	+	0.996093i	$0.471853\pi$
$444$	0	0
$445$	13.7401	0.651343
$446$	0	0
$447$	0.981889	0.0464418
$448$	0	0
$449$	21.5153	1.01537	0.507686	−	0.861542i	$-0.330501\pi$
$449$	21.5153	1.01537	0.507686	+	0.861542i	$0.330501\pi$
$450$	0	0
$451$	0.456882	0.0215137
$452$	0	0
$453$	−1.56348	−0.0734588
$454$	0	0
$455$	−18.8439	−0.883416
$456$	0	0
$457$	−33.1622	−1.55126	−0.775630	−	0.631187i	$-0.782568\pi$
$457$	−33.1622	−1.55126	−0.775630	+	0.631187i	$0.782568\pi$
$458$	0	0
$459$	3.00352	0.140192
$460$	0	0
$461$	−24.3478	−1.13399	−0.566994	−	0.823722i	$-0.691894\pi$
$461$	−24.3478	−1.13399	−0.566994	+	0.823722i	$0.691894\pi$
$462$	0	0
$463$	−7.49196	−0.348181	−0.174090	−	0.984730i	$-0.555699\pi$
$463$	−7.49196	−0.348181	−0.174090	+	0.984730i	$0.555699\pi$
$464$	0	0
$465$	−2.54347	−0.117950
$466$	0	0
$467$	13.2796	0.614504	0.307252	−	0.951628i	$-0.400590\pi$
$467$	13.2796	0.614504	0.307252	+	0.951628i	$0.400590\pi$
$468$	0	0
$469$	−15.4930	−0.715401
$470$	0	0
$471$	1.40348	0.0646690
$472$	0	0
$473$	2.05483	0.0944811
$474$	0	0
$475$	−1.24064	−0.0569244
$476$	0	0
$477$	35.1268	1.60834
$478$	0	0
$479$	−12.1494	−0.555122	−0.277561	−	0.960708i	$-0.589526\pi$
$479$	−12.1494	−0.555122	−0.277561	+	0.960708i	$0.589526\pi$
$480$	0	0
$481$	19.4036	0.884729
$482$	0	0
$483$	0.368929	0.0167869
$484$	0	0
$485$	−45.9554	−2.08673
$486$	0	0
$487$	10.7245	0.485974	0.242987	−	0.970029i	$-0.421873\pi$
$487$	10.7245	0.485974	0.242987	+	0.970029i	$0.421873\pi$
$488$	0	0
$489$	−1.06623	−0.0482168
$490$	0	0
$491$	28.6725	1.29397	0.646985	−	0.762503i	$-0.276030\pi$
$491$	28.6725	1.29397	0.646985	+	0.762503i	$0.276030\pi$
$492$	0	0
$493$	6.92534	0.311902
$494$	0	0
$495$	3.15507	0.141810
$496$	0	0
$497$	20.0748	0.900478
$498$	0	0
$499$	10.6822	0.478203	0.239102	−	0.970995i	$-0.423147\pi$
$499$	10.6822	0.478203	0.239102	+	0.970995i	$0.423147\pi$
$500$	0	0
$501$	1.55853	0.0696301
$502$	0	0
$503$	−33.7672	−1.50561	−0.752803	−	0.658245i	$-0.771299\pi$
$503$	−33.7672	−1.50561	−0.752803	+	0.658245i	$0.771299\pi$
$504$	0	0
$505$	38.6860	1.72150
$506$	0	0
$507$	−1.05891	−0.0470276
$508$	0	0
$509$	−16.5306	−0.732704	−0.366352	−	0.930476i	$-0.619393\pi$
$509$	−16.5306	−0.732704	−0.366352	+	0.930476i	$0.619393\pi$
$510$	0	0
$511$	−27.9210	−1.23515
$512$	0	0
$513$	0.338791	0.0149580
$514$	0	0
$515$	−23.9320	−1.05457
$516$	0	0
$517$	−4.58980	−0.201859
$518$	0	0
$519$	−0.740863	−0.0325203
$520$	0	0
$521$	−30.3577	−1.33000	−0.664999	−	0.746845i	$-0.731568\pi$
$521$	−30.3577	−1.33000	−0.664999	+	0.746845i	$0.731568\pi$
$522$	0	0
$523$	16.6911	0.729851	0.364925	−	0.931037i	$-0.381095\pi$
$523$	16.6911	0.729851	0.364925	+	0.931037i	$0.381095\pi$
$524$	0	0
$525$	1.32422	0.0577939
$526$	0	0
$527$	19.7029	0.858272
$528$	0	0
$529$	−22.1620	−0.963563
$530$	0	0
$531$	−40.8945	−1.77467
$532$	0	0
$533$	3.02637	0.131087
$534$	0	0
$535$	−3.30716	−0.142981
$536$	0	0
$537$	−2.59634	−0.112040
$538$	0	0
$539$	−0.0763585	−0.00328899
$540$	0	0
$541$	−36.1017	−1.55213	−0.776067	−	0.630650i	$-0.782788\pi$
$541$	−36.1017	−1.55213	−0.776067	+	0.630650i	$0.782788\pi$
$542$	0	0
$543$	2.58294	0.110844
$544$	0	0
$545$	25.6997	1.10085
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	−4.89537	−0.208929
$550$	0	0
$551$	0.781163	0.0332787
$552$	0	0
$553$	−1.72316	−0.0732762
$554$	0	0
$555$	−3.43844	−0.145953
$556$	0	0
$557$	2.18856	0.0927321	0.0463661	−	0.998925i	$-0.485236\pi$
$557$	2.18856	0.0927321	0.0463661	+	0.998925i	$0.485236\pi$
$558$	0	0
$559$	13.6111	0.575689
$560$	0	0
$561$	0.184973	0.00780955
$562$	0	0
$563$	23.1881	0.977260	0.488630	−	0.872491i	$-0.337497\pi$
$563$	23.1881	0.977260	0.488630	+	0.872491i	$0.337497\pi$
$564$	0	0
$565$	−25.0992	−1.05593
$566$	0	0
$567$	23.6189	0.991900
$568$	0	0
$569$	−7.55092	−0.316551	−0.158276	−	0.987395i	$-0.550593\pi$
$569$	−7.55092	−0.316551	−0.158276	+	0.987395i	$0.550593\pi$
$570$	0	0
$571$	2.22139	0.0929621	0.0464810	−	0.998919i	$-0.485199\pi$
$571$	2.22139	0.0929621	0.0464810	+	0.998919i	$0.485199\pi$
$572$	0	0
$573$	0.145562	0.00608094
$574$	0	0
$575$	3.00804	0.125444
$576$	0	0
$577$	23.8362	0.992314	0.496157	−	0.868233i	$-0.334744\pi$
$577$	23.8362	0.992314	0.496157	+	0.868233i	$0.334744\pi$
$578$	0	0
$579$	1.08344	0.0450263
$580$	0	0
$581$	−42.1636	−1.74924
$582$	0	0
$583$	4.34295	0.179866
$584$	0	0
$585$	20.8991	0.864072
$586$	0	0
$587$	−29.6166	−1.22241	−0.611204	−	0.791473i	$-0.709315\pi$
$587$	−29.6166	−1.22241	−0.611204	+	0.791473i	$0.709315\pi$
$588$	0	0
$589$	2.22244	0.0915742
$590$	0	0
$591$	−1.09438	−0.0450169
$592$	0	0
$593$	−40.2054	−1.65104	−0.825520	−	0.564373i	$-0.809118\pi$
$593$	−40.2054	−1.65104	−0.825520	+	0.564373i	$0.809118\pi$
$594$	0	0
$595$	−25.8674	−1.06046
$596$	0	0
$597$	1.04673	0.0428400
$598$	0	0
$599$	24.9569	1.01971	0.509856	−	0.860259i	$-0.329699\pi$
$599$	24.9569	1.01971	0.509856	+	0.860259i	$0.329699\pi$
$600$	0	0
$601$	−30.2445	−1.23370	−0.616850	−	0.787081i	$-0.711591\pi$
$601$	−30.2445	−1.23370	−0.616850	+	0.787081i	$0.711591\pi$
$602$	0	0
$603$	17.1827	0.699735
$604$	0	0
$605$	−31.2736	−1.27145
$606$	0	0
$607$	−18.6504	−0.756998	−0.378499	−	0.925602i	$-0.623560\pi$
$607$	−18.6504	−0.756998	−0.378499	+	0.925602i	$0.623560\pi$
$608$	0	0
$609$	−0.833793	−0.0337870
$610$	0	0
$611$	−30.4027	−1.22996
$612$	0	0
$613$	12.7148	0.513545	0.256773	−	0.966472i	$-0.417341\pi$
$613$	12.7148	0.513545	0.256773	+	0.966472i	$0.417341\pi$
$614$	0	0
$615$	−0.536291	−0.0216253
$616$	0	0
$617$	−9.36487	−0.377016	−0.188508	−	0.982072i	$-0.560365\pi$
$617$	−9.36487	−0.377016	−0.188508	+	0.982072i	$0.560365\pi$
$618$	0	0
$619$	−4.28108	−0.172071	−0.0860355	−	0.996292i	$-0.527420\pi$
$619$	−4.28108	−0.172071	−0.0860355	+	0.996292i	$0.527420\pi$
$620$	0	0
$621$	−0.821429	−0.0329628
$622$	0	0
$623$	12.8148	0.513413
$624$	0	0
$625$	−30.6324	−1.22530
$626$	0	0
$627$	0.0208645	0.000833248 0
$628$	0	0
$629$	26.6358	1.06204
$630$	0	0
$631$	39.5061	1.57271	0.786357	−	0.617772i	$-0.211965\pi$
$631$	39.5061	1.57271	0.786357	+	0.617772i	$0.211965\pi$
$632$	0	0
$633$	−0.338324	−0.0134472
$634$	0	0
$635$	6.72500	0.266873
$636$	0	0
$637$	−0.505796	−0.0200404
$638$	0	0
$639$	−22.2643	−0.880760
$640$	0	0
$641$	18.2383	0.720369	0.360184	−	0.932881i	$-0.382714\pi$
$641$	18.2383	0.720369	0.360184	+	0.932881i	$0.382714\pi$
$642$	0	0
$643$	36.2895	1.43112	0.715559	−	0.698552i	$-0.246172\pi$
$643$	36.2895	1.43112	0.715559	+	0.698552i	$0.246172\pi$
$644$	0	0
$645$	−2.41197	−0.0949712
$646$	0	0
$647$	33.5029	1.31714	0.658568	−	0.752522i	$-0.271163\pi$
$647$	33.5029	1.31714	0.658568	+	0.752522i	$0.271163\pi$
$648$	0	0
$649$	−5.05606	−0.198468
$650$	0	0
$651$	−2.37218	−0.0929729
$652$	0	0
$653$	−15.9755	−0.625170	−0.312585	−	0.949890i	$-0.601195\pi$
$653$	−15.9755	−0.625170	−0.312585	+	0.949890i	$0.601195\pi$
$654$	0	0
$655$	5.26223	0.205612
$656$	0	0
$657$	30.9662	1.20811
$658$	0	0
$659$	−40.3580	−1.57213	−0.786063	−	0.618147i	$-0.787884\pi$
$659$	−40.3580	−1.57213	−0.786063	+	0.618147i	$0.787884\pi$
$660$	0	0
$661$	−15.5296	−0.604032	−0.302016	−	0.953303i	$-0.597660\pi$
$661$	−15.5296	−0.604032	−0.302016	+	0.953303i	$0.597660\pi$
$662$	0	0
$663$	1.22525	0.0475849
$664$	0	0
$665$	−2.91779	−0.113147
$666$	0	0
$667$	−1.89400	−0.0733360
$668$	0	0
$669$	0.797108	0.0308180
$670$	0	0
$671$	−0.605246	−0.0233653
$672$	0	0
$673$	−43.0979	−1.66130	−0.830652	−	0.556792i	$-0.812032\pi$
$673$	−43.0979	−1.66130	−0.830652	+	0.556792i	$0.812032\pi$
$674$	0	0
$675$	−2.94842	−0.113485
$676$	0	0
$677$	−23.0443	−0.885663	−0.442831	−	0.896605i	$-0.646026\pi$
$677$	−23.0443	−0.885663	−0.442831	+	0.896605i	$0.646026\pi$
$678$	0	0
$679$	−42.8605	−1.64484
$680$	0	0
$681$	0.328557	0.0125903
$682$	0	0
$683$	45.7310	1.74985	0.874923	−	0.484261i	$-0.160912\pi$
$683$	45.7310	1.74985	0.874923	+	0.484261i	$0.160912\pi$
$684$	0	0
$685$	−38.1522	−1.45772
$686$	0	0
$687$	0.731888	0.0279233
$688$	0	0
$689$	28.7676	1.09596
$690$	0	0
$691$	−2.00105	−0.0761236	−0.0380618	−	0.999275i	$-0.512118\pi$
$691$	−2.00105	−0.0761236	−0.0380618	+	0.999275i	$0.512118\pi$
$692$	0	0
$693$	2.94259	0.111780
$694$	0	0
$695$	−32.7139	−1.24091
$696$	0	0
$697$	4.15436	0.157358
$698$	0	0
$699$	−2.37876	−0.0899729
$700$	0	0
$701$	−36.9644	−1.39613	−0.698063	−	0.716036i	$-0.745954\pi$
$701$	−36.9644	−1.39613	−0.698063	+	0.716036i	$0.745954\pi$
$702$	0	0
$703$	3.00445	0.113315
$704$	0	0
$705$	5.38753	0.202906
$706$	0	0
$707$	36.0806	1.35695
$708$	0	0
$709$	−16.4717	−0.618609	−0.309305	−	0.950963i	$-0.600096\pi$
$709$	−16.4717	−0.618609	−0.309305	+	0.950963i	$0.600096\pi$
$710$	0	0
$711$	1.91110	0.0716717
$712$	0	0
$713$	−5.38851	−0.201801
$714$	0	0
$715$	2.58389	0.0966321
$716$	0	0
$717$	−3.73272	−0.139401
$718$	0	0
$719$	−1.45386	−0.0542200	−0.0271100	−	0.999632i	$-0.508630\pi$
$719$	−1.45386	−0.0542200	−0.0271100	+	0.999632i	$0.508630\pi$
$720$	0	0
$721$	−22.3203	−0.831250
$722$	0	0
$723$	2.69497	0.100227
$724$	0	0
$725$	−6.79828	−0.252482
$726$	0	0
$727$	−33.1204	−1.22837	−0.614184	−	0.789163i	$-0.710515\pi$
$727$	−33.1204	−1.22837	−0.614184	+	0.789163i	$0.710515\pi$
$728$	0	0
$729$	−25.7908	−0.955213
$730$	0	0
$731$	18.6843	0.691062
$732$	0	0
$733$	5.43027	0.200572	0.100286	−	0.994959i	$-0.468024\pi$
$733$	5.43027	0.200572	0.100286	+	0.994959i	$0.468024\pi$
$734$	0	0
$735$	0.0896300	0.00330605
$736$	0	0
$737$	2.12441	0.0782537
$738$	0	0
$739$	14.1862	0.521847	0.260924	−	0.965359i	$-0.415973\pi$
$739$	14.1862	0.521847	0.260924	+	0.965359i	$0.415973\pi$
$740$	0	0
$741$	0.138206	0.00507712
$742$	0	0
$743$	18.3288	0.672417	0.336209	−	0.941788i	$-0.390855\pi$
$743$	18.3288	0.672417	0.336209	+	0.941788i	$0.390855\pi$
$744$	0	0
$745$	18.8283	0.689816
$746$	0	0
$747$	46.7621	1.71094
$748$	0	0
$749$	−3.08444	−0.112703
$750$	0	0
$751$	4.60177	0.167921	0.0839605	−	0.996469i	$-0.473243\pi$
$751$	4.60177	0.167921	0.0839605	+	0.996469i	$0.473243\pi$
$752$	0	0
$753$	−0.279924	−0.0102010
$754$	0	0
$755$	−29.9807	−1.09111
$756$	0	0
$757$	31.7533	1.15409	0.577047	−	0.816711i	$-0.304205\pi$
$757$	31.7533	1.15409	0.577047	+	0.816711i	$0.304205\pi$
$758$	0	0
$759$	−0.0505878	−0.00183622
$760$	0	0
$761$	10.3955	0.376837	0.188418	−	0.982089i	$-0.439664\pi$
$761$	10.3955	0.376837	0.188418	+	0.982089i	$0.439664\pi$
$762$	0	0
$763$	23.9689	0.867734
$764$	0	0
$765$	28.6886	1.03724
$766$	0	0
$767$	−33.4912	−1.20930
$768$	0	0
$769$	31.1538	1.12344	0.561718	−	0.827329i	$-0.310141\pi$
$769$	31.1538	1.12344	0.561718	+	0.827329i	$0.310141\pi$
$770$	0	0
$771$	0.386998	0.0139374
$772$	0	0
$773$	3.74734	0.134782	0.0673912	−	0.997727i	$-0.478532\pi$
$773$	3.74734	0.134782	0.0673912	+	0.997727i	$0.478532\pi$
$774$	0	0
$775$	−19.3414	−0.694764
$776$	0	0
$777$	−3.20687	−0.115046
$778$	0	0
$779$	0.468603	0.0167894
$780$	0	0
$781$	−2.75267	−0.0984984
$782$	0	0
$783$	1.85646	0.0663444
$784$	0	0
$785$	26.9126	0.960552
$786$	0	0
$787$	−50.4938	−1.79991	−0.899955	−	0.435983i	$-0.856401\pi$
$787$	−50.4938	−1.79991	−0.899955	+	0.435983i	$0.856401\pi$
$788$	0	0
$789$	−0.760723	−0.0270825
$790$	0	0
$791$	−23.4089	−0.832325
$792$	0	0
$793$	−4.00913	−0.142368
$794$	0	0
$795$	−5.09778	−0.180799
$796$	0	0
$797$	−48.9797	−1.73495	−0.867475	−	0.497481i	$-0.834258\pi$
$797$	−48.9797	−1.73495	−0.867475	+	0.497481i	$0.834258\pi$
$798$	0	0
$799$	−41.7344	−1.47646
$800$	0	0
$801$	−14.2124	−0.502171
$802$	0	0
$803$	3.82855	0.135107
$804$	0	0
$805$	7.07444	0.249341
$806$	0	0
$807$	−1.53898	−0.0541747
$808$	0	0
$809$	−24.6489	−0.866611	−0.433305	−	0.901247i	$-0.642653\pi$
$809$	−24.6489	−0.866611	−0.433305	+	0.901247i	$0.642653\pi$
$810$	0	0
$811$	−31.9192	−1.12083	−0.560417	−	0.828210i	$-0.689359\pi$
$811$	−31.9192	−1.12083	−0.560417	+	0.828210i	$0.689359\pi$
$812$	0	0
$813$	4.01031	0.140648
$814$	0	0
$815$	−20.4457	−0.716182
$816$	0	0
$817$	2.10754	0.0737336
$818$	0	0
$819$	19.4916	0.681093
$820$	0	0
$821$	10.1931	0.355740	0.177870	−	0.984054i	$-0.443079\pi$
$821$	10.1931	0.355740	0.177870	+	0.984054i	$0.443079\pi$
$822$	0	0
$823$	−34.6165	−1.20666	−0.603328	−	0.797493i	$-0.706159\pi$
$823$	−34.6165	−1.20666	−0.603328	+	0.797493i	$0.706159\pi$
$824$	0	0
$825$	−0.181579	−0.00632176
$826$	0	0
$827$	−37.5580	−1.30602	−0.653010	−	0.757349i	$-0.726494\pi$
$827$	−37.5580	−1.30602	−0.653010	+	0.757349i	$0.726494\pi$
$828$	0	0
$829$	24.0820	0.836401	0.418200	−	0.908355i	$-0.362661\pi$
$829$	24.0820	0.836401	0.418200	+	0.908355i	$0.362661\pi$
$830$	0	0
$831$	1.44345	0.0500729
$832$	0	0
$833$	−0.694317	−0.0240567
$834$	0	0
$835$	29.8858	1.03424
$836$	0	0
$837$	5.28170	0.182562
$838$	0	0
$839$	34.4662	1.18991	0.594953	−	0.803761i	$-0.297171\pi$
$839$	34.4662	1.18991	0.594953	+	0.803761i	$0.297171\pi$
$840$	0	0
$841$	−24.7195	−0.852396
$842$	0	0
$843$	3.45120	0.118866
$844$	0	0
$845$	−20.3051	−0.698518
$846$	0	0
$847$	−29.1675	−1.00221
$848$	0	0
$849$	−1.27304	−0.0436905
$850$	0	0
$851$	−7.28457	−0.249712
$852$	0	0
$853$	8.14117	0.278748	0.139374	−	0.990240i	$-0.455491\pi$
$853$	8.14117	0.278748	0.139374	+	0.990240i	$0.455491\pi$
$854$	0	0
$855$	3.23601	0.110669
$856$	0	0
$857$	57.4729	1.96324	0.981618	−	0.190854i	$-0.0611258\pi$
$857$	57.4729	1.96324	0.981618	+	0.190854i	$0.0611258\pi$
$858$	0	0
$859$	−38.5381	−1.31490	−0.657451	−	0.753497i	$-0.728365\pi$
$859$	−38.5381	−1.31490	−0.657451	+	0.753497i	$0.728365\pi$
$860$	0	0
$861$	−0.500174	−0.0170459
$862$	0	0
$863$	−41.0444	−1.39717	−0.698584	−	0.715529i	$-0.746186\pi$
$863$	−41.0444	−1.39717	−0.698584	+	0.715529i	$0.746186\pi$
$864$	0	0
$865$	−14.2065	−0.483035
$866$	0	0
$867$	−0.870003	−0.0295469
$868$	0	0
$869$	0.236281	0.00801528
$870$	0	0
$871$	14.0720	0.476813
$872$	0	0
$873$	47.5351	1.60882
$874$	0	0
$875$	−13.2465	−0.447814
$876$	0	0
$877$	9.31493	0.314543	0.157271	−	0.987555i	$-0.449730\pi$
$877$	9.31493	0.314543	0.157271	+	0.987555i	$0.449730\pi$
$878$	0	0
$879$	−1.70858	−0.0576288
$880$	0	0
$881$	−29.9630	−1.00948	−0.504739	−	0.863272i	$-0.668411\pi$
$881$	−29.9630	−1.00948	−0.504739	+	0.863272i	$0.668411\pi$
$882$	0	0
$883$	45.4522	1.52959	0.764794	−	0.644275i	$-0.222841\pi$
$883$	45.4522	1.52959	0.764794	+	0.644275i	$0.222841\pi$
$884$	0	0
$885$	5.93483	0.199497
$886$	0	0
$887$	31.0703	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$887$	31.0703	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$888$	0	0
$889$	6.27210	0.210359
$890$	0	0
$891$	−3.23864	−0.108498
$892$	0	0
$893$	−4.70754	−0.157532
$894$	0	0
$895$	−49.7863	−1.66417
$896$	0	0
$897$	−0.335092	−0.0111884
$898$	0	0
$899$	12.1782	0.406167
$900$	0	0
$901$	39.4898	1.31560
$902$	0	0
$903$	−2.24953	−0.0748598
$904$	0	0
$905$	49.5293	1.64641
$906$	0	0
$907$	−27.4635	−0.911910	−0.455955	−	0.890003i	$-0.650702\pi$
$907$	−27.4635	−0.911910	−0.455955	+	0.890003i	$0.650702\pi$
$908$	0	0
$909$	−40.0158	−1.32724
$910$	0	0
$911$	8.25357	0.273453	0.136727	−	0.990609i	$-0.456342\pi$
$911$	8.25357	0.273453	0.136727	+	0.990609i	$0.456342\pi$
$912$	0	0
$913$	5.78150	0.191340
$914$	0	0
$915$	0.710442	0.0234865
$916$	0	0
$917$	4.90784	0.162071
$918$	0	0
$919$	40.7751	1.34505	0.672524	−	0.740075i	$-0.265210\pi$
$919$	40.7751	1.34505	0.672524	+	0.740075i	$0.265210\pi$
$920$	0	0
$921$	0.808789	0.0266505
$922$	0	0
$923$	−18.2336	−0.600167
$924$	0	0
$925$	−26.1470	−0.859710
$926$	0	0
$927$	24.7546	0.813048
$928$	0	0
$929$	−49.6880	−1.63021	−0.815104	−	0.579314i	$-0.803321\pi$
$929$	−49.6880	−1.63021	−0.815104	+	0.579314i	$0.803321\pi$
$930$	0	0
$931$	−0.0783174	−0.00256675
$932$	0	0
$933$	3.00704	0.0984459
$934$	0	0
$935$	3.54696	0.115998
$936$	0	0
$937$	−11.8945	−0.388575	−0.194288	−	0.980945i	$-0.562239\pi$
$937$	−11.8945	−0.388575	−0.194288	+	0.980945i	$0.562239\pi$
$938$	0	0
$939$	4.19489	0.136895
$940$	0	0
$941$	−23.3867	−0.762386	−0.381193	−	0.924496i	$-0.624487\pi$
$941$	−23.3867	−0.762386	−0.381193	+	0.924496i	$0.624487\pi$
$942$	0	0
$943$	−1.13617	−0.0369988
$944$	0	0
$945$	−6.93421	−0.225570
$946$	0	0
$947$	−3.24137	−0.105330	−0.0526651	−	0.998612i	$-0.516772\pi$
$947$	−3.24137	−0.105330	−0.0526651	+	0.998612i	$0.516772\pi$
$948$	0	0
$949$	25.3602	0.823228
$950$	0	0
$951$	0.697080	0.0226044
$952$	0	0
$953$	55.4588	1.79649	0.898244	−	0.439498i	$-0.144844\pi$
$953$	55.4588	1.79649	0.898244	+	0.439498i	$0.144844\pi$
$954$	0	0
$955$	2.79124	0.0903224
$956$	0	0
$957$	0.114330	0.00369577
$958$	0	0
$959$	−35.5829	−1.14903
$960$	0	0
$961$	3.64761	0.117665
$962$	0	0
$963$	3.42084	0.110235
$964$	0	0
$965$	20.7756	0.668792
$966$	0	0
$967$	19.5302	0.628048	0.314024	−	0.949415i	$-0.398323\pi$
$967$	19.5302	0.628048	0.314024	+	0.949415i	$0.398323\pi$
$968$	0	0
$969$	0.189718	0.00609462
$970$	0	0
$971$	14.9189	0.478770	0.239385	−	0.970925i	$-0.423054\pi$
$971$	14.9189	0.478770	0.239385	+	0.970925i	$0.423054\pi$
$972$	0	0
$973$	−30.5108	−0.978130
$974$	0	0
$975$	−1.20277	−0.0385195
$976$	0	0
$977$	−38.6671	−1.23707	−0.618536	−	0.785757i	$-0.712274\pi$
$977$	−38.6671	−1.23707	−0.618536	+	0.785757i	$0.712274\pi$
$978$	0	0
$979$	−1.75717	−0.0561594
$980$	0	0
$981$	−26.5831	−0.848733
$982$	0	0
$983$	43.8743	1.39937	0.699687	−	0.714449i	$-0.253323\pi$
$983$	43.8743	1.39937	0.699687	+	0.714449i	$0.253323\pi$
$984$	0	0
$985$	−20.9854	−0.668652
$986$	0	0
$987$	5.02470	0.159938
$988$	0	0
$989$	−5.10992	−0.162486
$990$	0	0
$991$	−7.55457	−0.239979	−0.119989	−	0.992775i	$-0.538286\pi$
$991$	−7.55457	−0.239979	−0.119989	+	0.992775i	$0.538286\pi$
$992$	0	0
$993$	−0.930801	−0.0295381
$994$	0	0
$995$	20.0718	0.636318
$996$	0	0
$997$	−0.872251	−0.0276245	−0.0138122	−	0.999905i	$-0.504397\pi$
$997$	−0.872251	−0.0276245	−0.0138122	+	0.999905i	$0.504397\pi$
$998$	0	0
$999$	7.14017	0.225905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.8		18
4.3	odd	2		547.2.a.b.1.4	✓	18
12.11	even	2		4923.2.a.l.1.15		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.4	✓	18	4.3	odd	2
4923.2.a.l.1.15		18	12.11	even	2
8752.2.a.s.1.8		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)

\(f(q)\)	\(=\)	\(q+0.150114 q^{3} +2.87852 q^{5} +2.68467 q^{7} -2.97747 q^{9} +O(q^{10})\)	\(q+0.150114 q^{3} +2.87852 q^{5} +2.68467 q^{7} -2.97747 q^{9} -0.368123 q^{11} -2.43844 q^{13} +0.432105 q^{15} -3.34729 q^{17} -0.377567 q^{19} +0.403005 q^{21} +0.915445 q^{23} +3.28588 q^{25} -0.897299 q^{27} -2.06894 q^{29} -5.88622 q^{31} -0.0552603 q^{33} +7.72786 q^{35} -7.95740 q^{37} -0.366043 q^{39} -1.24111 q^{41} -5.58190 q^{43} -8.57070 q^{45} +12.4681 q^{47} +0.207426 q^{49} -0.502474 q^{51} -11.7975 q^{53} -1.05965 q^{55} -0.0566780 q^{57} +13.7347 q^{59} +1.64414 q^{61} -7.99350 q^{63} -7.01909 q^{65} -5.77093 q^{67} +0.137421 q^{69} +7.47759 q^{71} -10.4002 q^{73} +0.493255 q^{75} -0.988288 q^{77} -0.641853 q^{79} +8.79770 q^{81} -15.7053 q^{83} -9.63525 q^{85} -0.310576 q^{87} +4.77332 q^{89} -6.54639 q^{91} -0.883602 q^{93} -1.08683 q^{95} -15.9649 q^{97} +1.09607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 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

Introduction

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