LMFDB - Embedded newform 8624.2.a.cc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} -3.41421 q^{5} -2.82843 q^{9} +O(q^{10})$	$q-0.414214 q^{3} -3.41421 q^{5} -2.82843 q^{9} -1.00000 q^{11} +1.82843 q^{13} +1.41421 q^{15} -7.65685 q^{17} +3.41421 q^{19} -2.24264 q^{23} +6.65685 q^{25} +2.41421 q^{27} -8.65685 q^{29} +4.00000 q^{31} +0.414214 q^{33} -6.58579 q^{37} -0.757359 q^{39} -2.58579 q^{41} -5.65685 q^{43} +9.65685 q^{45} -6.48528 q^{47} +3.17157 q^{51} -11.8995 q^{53} +3.41421 q^{55} -1.41421 q^{57} +8.41421 q^{59} +6.17157 q^{61} -6.24264 q^{65} -11.2426 q^{67} +0.928932 q^{69} -3.07107 q^{71} -6.58579 q^{73} -2.75736 q^{75} +4.75736 q^{79} +7.48528 q^{81} -16.1421 q^{83} +26.1421 q^{85} +3.58579 q^{87} -4.48528 q^{89} -1.65685 q^{93} -11.6569 q^{95} +1.82843 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5	$ 2 q + 2 q^{3} - 4 q^{5} - 2 q^{11} - 2 q^{13} - 4 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} + 8 q^{31} - 2 q^{33} - 16 q^{37} - 10 q^{39} - 8 q^{41} + 8 q^{45} + 4 q^{47} + 12 q^{51} - 4 q^{53} + 4 q^{55} + 14 q^{59} + 18 q^{61} - 4 q^{65} - 14 q^{67} + 16 q^{69} + 8 q^{71} - 16 q^{73} - 14 q^{75} + 18 q^{79} - 2 q^{81} - 4 q^{83} + 24 q^{85} + 10 q^{87} + 8 q^{89} + 8 q^{93} - 12 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^11 - 2 * q^13 - 4 * q^17 + 4 * q^19 + 4 * q^23 + 2 * q^25 + 2 * q^27 - 6 * q^29 + 8 * q^31 - 2 * q^33 - 16 * q^37 - 10 * q^39 - 8 * q^41 + 8 * q^45 + 4 * q^47 + 12 * q^51 - 4 * q^53 + 4 * q^55 + 14 * q^59 + 18 * q^61 - 4 * q^65 - 14 * q^67 + 16 * q^69 + 8 * q^71 - 16 * q^73 - 14 * q^75 + 18 * q^79 - 2 * q^81 - 4 * q^83 + 24 * q^85 + 10 * q^87 + 8 * q^89 + 8 * q^93 - 12 * q^95 - 2 * q^97

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
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	0	0
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.82843	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$13$	1.82843	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0	0
$19$	3.41421	0.783274	0.391637	−	0.920120i	$-0.371909\pi$
$19$	3.41421	0.783274	0.391637	+	0.920120i	$0.371909\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.24264	−0.467623	−0.233811	−	0.972282i	$-0.575120\pi$
$23$	−2.24264	−0.467623	−0.233811	+	0.972282i	$0.575120\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	−8.65685	−1.60754	−0.803769	−	0.594942i	$-0.797175\pi$
$29$	−8.65685	−1.60754	−0.803769	+	0.594942i	$0.797175\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0.414214	0.0721053
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.58579	−1.08270	−0.541348	−	0.840798i	$-0.682086\pi$
$37$	−6.58579	−1.08270	−0.541348	+	0.840798i	$0.682086\pi$
$38$	0	0
$39$	−0.757359	−0.121275
$40$	0	0
$41$	−2.58579	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$41$	−2.58579	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	0	0
$45$	9.65685	1.43956
$46$	0	0
$47$	−6.48528	−0.945976	−0.472988	−	0.881069i	$-0.656825\pi$
$47$	−6.48528	−0.945976	−0.472988	+	0.881069i	$0.656825\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.17157	0.444109
$52$	0	0
$53$	−11.8995	−1.63452	−0.817261	−	0.576268i	$-0.804508\pi$
$53$	−11.8995	−1.63452	−0.817261	+	0.576268i	$0.804508\pi$
$54$	0	0
$55$	3.41421	0.460372
$56$	0	0
$57$	−1.41421	−0.187317
$58$	0	0
$59$	8.41421	1.09544	0.547719	−	0.836663i	$-0.315496\pi$
$59$	8.41421	1.09544	0.547719	+	0.836663i	$0.315496\pi$
$60$	0	0
$61$	6.17157	0.790189	0.395094	−	0.918640i	$-0.370712\pi$
$61$	6.17157	0.790189	0.395094	+	0.918640i	$0.370712\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.24264	−0.774304
$66$	0	0
$67$	−11.2426	−1.37351	−0.686754	−	0.726890i	$-0.740965\pi$
$67$	−11.2426	−1.37351	−0.686754	+	0.726890i	$0.740965\pi$
$68$	0	0
$69$	0.928932	0.111830
$70$	0	0
$71$	−3.07107	−0.364469	−0.182234	−	0.983255i	$-0.558333\pi$
$71$	−3.07107	−0.364469	−0.182234	+	0.983255i	$0.558333\pi$
$72$	0	0
$73$	−6.58579	−0.770808	−0.385404	−	0.922748i	$-0.625938\pi$
$73$	−6.58579	−0.770808	−0.385404	+	0.922748i	$0.625938\pi$
$74$	0	0
$75$	−2.75736	−0.318392
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.75736	0.535245	0.267622	−	0.963524i	$-0.413762\pi$
$79$	4.75736	0.535245	0.267622	+	0.963524i	$0.413762\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−16.1421	−1.77183	−0.885915	−	0.463848i	$-0.846468\pi$
$83$	−16.1421	−1.77183	−0.885915	+	0.463848i	$0.846468\pi$
$84$	0	0
$85$	26.1421	2.83551
$86$	0	0
$87$	3.58579	0.384437
$88$	0	0
$89$	−4.48528	−0.475439	−0.237719	−	0.971334i	$-0.576400\pi$
$89$	−4.48528	−0.475439	−0.237719	+	0.971334i	$0.576400\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.65685	−0.171808
$94$	0	0
$95$	−11.6569	−1.19597
$96$	0	0
$97$	1.82843	0.185649	0.0928243	−	0.995683i	$-0.470411\pi$
$97$	1.82843	0.185649	0.0928243	+	0.995683i	$0.470411\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	11.8284	1.17697	0.588486	−	0.808507i	$-0.299724\pi$
$101$	11.8284	1.17697	0.588486	+	0.808507i	$0.299724\pi$
$102$	0	0
$103$	−10.5858	−1.04305	−0.521524	−	0.853236i	$-0.674636\pi$
$103$	−10.5858	−1.04305	−0.521524	+	0.853236i	$0.674636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.0711	−1.07028	−0.535140	−	0.844763i	$-0.679741\pi$
$107$	−11.0711	−1.07028	−0.535140	+	0.844763i	$0.679741\pi$
$108$	0	0
$109$	−0.485281	−0.0464815	−0.0232408	−	0.999730i	$-0.507398\pi$
$109$	−0.485281	−0.0464815	−0.0232408	+	0.999730i	$0.507398\pi$
$110$	0	0
$111$	2.72792	0.258923
$112$	0	0
$113$	−13.8284	−1.30087	−0.650434	−	0.759562i	$-0.725413\pi$
$113$	−13.8284	−1.30087	−0.650434	+	0.759562i	$0.725413\pi$
$114$	0	0
$115$	7.65685	0.714005
$116$	0	0
$117$	−5.17157	−0.478112
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.07107	0.0965749
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	9.72792	0.863213	0.431607	−	0.902062i	$-0.357947\pi$
$127$	9.72792	0.863213	0.431607	+	0.902062i	$0.357947\pi$
$128$	0	0
$129$	2.34315	0.206302
$130$	0	0
$131$	3.41421	0.298301	0.149151	−	0.988814i	$-0.452346\pi$
$131$	3.41421	0.298301	0.149151	+	0.988814i	$0.452346\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.24264	−0.709414
$136$	0	0
$137$	−5.34315	−0.456496	−0.228248	−	0.973603i	$-0.573300\pi$
$137$	−5.34315	−0.456496	−0.228248	+	0.973603i	$0.573300\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	2.68629	0.226227
$142$	0	0
$143$	−1.82843	−0.152901
$144$	0	0
$145$	29.5563	2.45452
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.34315	0.519651	0.259825	−	0.965656i	$-0.416335\pi$
$149$	6.34315	0.519651	0.259825	+	0.965656i	$0.416335\pi$
$150$	0	0
$151$	−9.72792	−0.791647	−0.395824	−	0.918327i	$-0.629541\pi$
$151$	−9.72792	−0.791647	−0.395824	+	0.918327i	$0.629541\pi$
$152$	0	0
$153$	21.6569	1.75085
$154$	0	0
$155$	−13.6569	−1.09694
$156$	0	0
$157$	6.34315	0.506238	0.253119	−	0.967435i	$-0.418544\pi$
$157$	6.34315	0.506238	0.253119	+	0.967435i	$0.418544\pi$
$158$	0	0
$159$	4.92893	0.390890
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.7279	1.23191	0.615953	−	0.787783i	$-0.288771\pi$
$163$	15.7279	1.23191	0.615953	+	0.787783i	$0.288771\pi$
$164$	0	0
$165$	−1.41421	−0.110096
$166$	0	0
$167$	11.7279	0.907534	0.453767	−	0.891120i	$-0.350080\pi$
$167$	11.7279	0.907534	0.453767	+	0.891120i	$0.350080\pi$
$168$	0	0
$169$	−9.65685	−0.742835
$170$	0	0
$171$	−9.65685	−0.738478
$172$	0	0
$173$	−4.17157	−0.317159	−0.158579	−	0.987346i	$-0.550691\pi$
$173$	−4.17157	−0.317159	−0.158579	+	0.987346i	$0.550691\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.48528	−0.261970
$178$	0	0
$179$	18.8995	1.41261	0.706307	−	0.707905i	$-0.250360\pi$
$179$	18.8995	1.41261	0.706307	+	0.707905i	$0.250360\pi$
$180$	0	0
$181$	3.65685	0.271812	0.135906	−	0.990722i	$-0.456606\pi$
$181$	3.65685	0.271812	0.135906	+	0.990722i	$0.456606\pi$
$182$	0	0
$183$	−2.55635	−0.188971
$184$	0	0
$185$	22.4853	1.65315
$186$	0	0
$187$	7.65685	0.559925
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.8284	0.928232	0.464116	−	0.885774i	$-0.346372\pi$
$191$	12.8284	0.928232	0.464116	+	0.885774i	$0.346372\pi$
$192$	0	0
$193$	2.10051	0.151198	0.0755988	−	0.997138i	$-0.475913\pi$
$193$	2.10051	0.151198	0.0755988	+	0.997138i	$0.475913\pi$
$194$	0	0
$195$	2.58579	0.185172
$196$	0	0
$197$	−17.4853	−1.24577	−0.622887	−	0.782312i	$-0.714041\pi$
$197$	−17.4853	−1.24577	−0.622887	+	0.782312i	$0.714041\pi$
$198$	0	0
$199$	−19.8995	−1.41064	−0.705319	−	0.708890i	$-0.749196\pi$
$199$	−19.8995	−1.41064	−0.705319	+	0.708890i	$0.749196\pi$
$200$	0	0
$201$	4.65685	0.328469
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.82843	0.616604
$206$	0	0
$207$	6.34315	0.440879
$208$	0	0
$209$	−3.41421	−0.236166
$210$	0	0
$211$	−4.58579	−0.315699	−0.157849	−	0.987463i	$-0.550456\pi$
$211$	−4.58579	−0.315699	−0.157849	+	0.987463i	$0.550456\pi$
$212$	0	0
$213$	1.27208	0.0871613
$214$	0	0
$215$	19.3137	1.31718
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.72792	0.184336
$220$	0	0
$221$	−14.0000	−0.941742
$222$	0	0
$223$	−11.4142	−0.764352	−0.382176	−	0.924089i	$-0.624825\pi$
$223$	−11.4142	−0.764352	−0.382176	+	0.924089i	$0.624825\pi$
$224$	0	0
$225$	−18.8284	−1.25523
$226$	0	0
$227$	23.1716	1.53795	0.768976	−	0.639278i	$-0.220767\pi$
$227$	23.1716	1.53795	0.768976	+	0.639278i	$0.220767\pi$
$228$	0	0
$229$	0.686292	0.0453514	0.0226757	−	0.999743i	$-0.492781\pi$
$229$	0.686292	0.0453514	0.0226757	+	0.999743i	$0.492781\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.41421	−0.0926482	−0.0463241	−	0.998926i	$-0.514751\pi$
$233$	−1.41421	−0.0926482	−0.0463241	+	0.998926i	$0.514751\pi$
$234$	0	0
$235$	22.1421	1.44439
$236$	0	0
$237$	−1.97056	−0.128002
$238$	0	0
$239$	22.2132	1.43685	0.718426	−	0.695603i	$-0.244863\pi$
$239$	22.2132	1.43685	0.718426	+	0.695603i	$0.244863\pi$
$240$	0	0
$241$	3.75736	0.242033	0.121016	−	0.992651i	$-0.461385\pi$
$241$	3.75736	0.242033	0.121016	+	0.992651i	$0.461385\pi$
$242$	0	0
$243$	−10.3431	−0.663513
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.24264	0.397210
$248$	0	0
$249$	6.68629	0.423727
$250$	0	0
$251$	−2.14214	−0.135210	−0.0676052	−	0.997712i	$-0.521536\pi$
$251$	−2.14214	−0.135210	−0.0676052	+	0.997712i	$0.521536\pi$
$252$	0	0
$253$	2.24264	0.140994
$254$	0	0
$255$	−10.8284	−0.678102
$256$	0	0
$257$	3.14214	0.196001	0.0980005	−	0.995186i	$-0.468755\pi$
$257$	3.14214	0.196001	0.0980005	+	0.995186i	$0.468755\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	24.4853	1.51560
$262$	0	0
$263$	−31.0416	−1.91411	−0.957054	−	0.289908i	$-0.906375\pi$
$263$	−31.0416	−1.91411	−0.957054	+	0.289908i	$0.906375\pi$
$264$	0	0
$265$	40.6274	2.49572
$266$	0	0
$267$	1.85786	0.113699
$268$	0	0
$269$	13.6569	0.832673	0.416337	−	0.909211i	$-0.363314\pi$
$269$	13.6569	0.832673	0.416337	+	0.909211i	$0.363314\pi$
$270$	0	0
$271$	26.5563	1.61318	0.806592	−	0.591109i	$-0.201310\pi$
$271$	26.5563	1.61318	0.806592	+	0.591109i	$0.201310\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.65685	−0.401423
$276$	0	0
$277$	−3.82843	−0.230028	−0.115014	−	0.993364i	$-0.536691\pi$
$277$	−3.82843	−0.230028	−0.115014	+	0.993364i	$0.536691\pi$
$278$	0	0
$279$	−11.3137	−0.677334
$280$	0	0
$281$	−16.7279	−0.997904	−0.498952	−	0.866630i	$-0.666282\pi$
$281$	−16.7279	−0.997904	−0.498952	+	0.866630i	$0.666282\pi$
$282$	0	0
$283$	20.5858	1.22370	0.611849	−	0.790975i	$-0.290426\pi$
$283$	20.5858	1.22370	0.611849	+	0.790975i	$0.290426\pi$
$284$	0	0
$285$	4.82843	0.286011
$286$	0	0
$287$	0	0
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	−0.757359	−0.0443972
$292$	0	0
$293$	−5.17157	−0.302127	−0.151063	−	0.988524i	$-0.548270\pi$
$293$	−5.17157	−0.302127	−0.151063	+	0.988524i	$0.548270\pi$
$294$	0	0
$295$	−28.7279	−1.67260
$296$	0	0
$297$	−2.41421	−0.140087
$298$	0	0
$299$	−4.10051	−0.237138
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.89949	−0.281469
$304$	0	0
$305$	−21.0711	−1.20653
$306$	0	0
$307$	9.89949	0.564994	0.282497	−	0.959268i	$-0.408837\pi$
$307$	9.89949	0.564994	0.282497	+	0.959268i	$0.408837\pi$
$308$	0	0
$309$	4.38478	0.249441
$310$	0	0
$311$	8.72792	0.494915	0.247458	−	0.968899i	$-0.420405\pi$
$311$	8.72792	0.494915	0.247458	+	0.968899i	$0.420405\pi$
$312$	0	0
$313$	−9.34315	−0.528106	−0.264053	−	0.964508i	$-0.585059\pi$
$313$	−9.34315	−0.528106	−0.264053	+	0.964508i	$0.585059\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.3137	1.75875	0.879377	−	0.476127i	$-0.157960\pi$
$317$	31.3137	1.75875	0.879377	+	0.476127i	$0.157960\pi$
$318$	0	0
$319$	8.65685	0.484691
$320$	0	0
$321$	4.58579	0.255954
$322$	0	0
$323$	−26.1421	−1.45459
$324$	0	0
$325$	12.1716	0.675157
$326$	0	0
$327$	0.201010	0.0111159
$328$	0	0
$329$	0	0
$330$	0	0
$331$	9.92893	0.545743	0.272872	−	0.962050i	$-0.412027\pi$
$331$	9.92893	0.545743	0.272872	+	0.962050i	$0.412027\pi$
$332$	0	0
$333$	18.6274	1.02078
$334$	0	0
$335$	38.3848	2.09718
$336$	0	0
$337$	−19.7574	−1.07625	−0.538126	−	0.842864i	$-0.680868\pi$
$337$	−19.7574	−1.07625	−0.538126	+	0.842864i	$0.680868\pi$
$338$	0	0
$339$	5.72792	0.311098
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.17157	−0.170752
$346$	0	0
$347$	−14.5858	−0.783006	−0.391503	−	0.920177i	$-0.628045\pi$
$347$	−14.5858	−0.783006	−0.391503	+	0.920177i	$0.628045\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	4.41421	0.235613
$352$	0	0
$353$	−25.3137	−1.34731	−0.673656	−	0.739045i	$-0.735277\pi$
$353$	−25.3137	−1.34731	−0.673656	+	0.739045i	$0.735277\pi$
$354$	0	0
$355$	10.4853	0.556501
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.7574	−0.567752	−0.283876	−	0.958861i	$-0.591620\pi$
$359$	−10.7574	−0.567752	−0.283876	+	0.958861i	$0.591620\pi$
$360$	0	0
$361$	−7.34315	−0.386481
$362$	0	0
$363$	−0.414214	−0.0217406
$364$	0	0
$365$	22.4853	1.17693
$366$	0	0
$367$	−14.7279	−0.768791	−0.384396	−	0.923168i	$-0.625590\pi$
$367$	−14.7279	−0.768791	−0.384396	+	0.923168i	$0.625590\pi$
$368$	0	0
$369$	7.31371	0.380736
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13.9706	0.723368	0.361684	−	0.932301i	$-0.382202\pi$
$373$	13.9706	0.723368	0.361684	+	0.932301i	$0.382202\pi$
$374$	0	0
$375$	2.34315	0.121000
$376$	0	0
$377$	−15.8284	−0.815205
$378$	0	0
$379$	25.8701	1.32886	0.664428	−	0.747352i	$-0.268675\pi$
$379$	25.8701	1.32886	0.664428	+	0.747352i	$0.268675\pi$
$380$	0	0
$381$	−4.02944	−0.206434
$382$	0	0
$383$	−30.3848	−1.55259	−0.776295	−	0.630370i	$-0.782903\pi$
$383$	−30.3848	−1.55259	−0.776295	+	0.630370i	$0.782903\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0000	0.813326
$388$	0	0
$389$	2.72792	0.138311	0.0691556	−	0.997606i	$-0.477970\pi$
$389$	2.72792	0.138311	0.0691556	+	0.997606i	$0.477970\pi$
$390$	0	0
$391$	17.1716	0.868404
$392$	0	0
$393$	−1.41421	−0.0713376
$394$	0	0
$395$	−16.2426	−0.817256
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.31371	−0.215416	−0.107708	−	0.994183i	$-0.534351\pi$
$401$	−4.31371	−0.215416	−0.107708	+	0.994183i	$0.534351\pi$
$402$	0	0
$403$	7.31371	0.364322
$404$	0	0
$405$	−25.5563	−1.26991
$406$	0	0
$407$	6.58579	0.326445
$408$	0	0
$409$	−22.7279	−1.12382	−0.561912	−	0.827197i	$-0.689934\pi$
$409$	−22.7279	−1.12382	−0.561912	+	0.827197i	$0.689934\pi$
$410$	0	0
$411$	2.21320	0.109169
$412$	0	0
$413$	0	0
$414$	0	0
$415$	55.1127	2.70538
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.14214	−0.104650	−0.0523251	−	0.998630i	$-0.516663\pi$
$419$	−2.14214	−0.104650	−0.0523251	+	0.998630i	$0.516663\pi$
$420$	0	0
$421$	23.3137	1.13624	0.568120	−	0.822946i	$-0.307671\pi$
$421$	23.3137	1.13624	0.568120	+	0.822946i	$0.307671\pi$
$422$	0	0
$423$	18.3431	0.891874
$424$	0	0
$425$	−50.9706	−2.47244
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.757359	0.0365657
$430$	0	0
$431$	20.4142	0.983318	0.491659	−	0.870788i	$-0.336391\pi$
$431$	20.4142	0.983318	0.491659	+	0.870788i	$0.336391\pi$
$432$	0	0
$433$	−2.14214	−0.102944	−0.0514722	−	0.998674i	$-0.516391\pi$
$433$	−2.14214	−0.102944	−0.0514722	+	0.998674i	$0.516391\pi$
$434$	0	0
$435$	−12.2426	−0.586990
$436$	0	0
$437$	−7.65685	−0.366277
$438$	0	0
$439$	−9.38478	−0.447911	−0.223955	−	0.974599i	$-0.571897\pi$
$439$	−9.38478	−0.447911	−0.223955	+	0.974599i	$0.571897\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.6274	1.55018	0.775088	−	0.631854i	$-0.217706\pi$
$443$	32.6274	1.55018	0.775088	+	0.631854i	$0.217706\pi$
$444$	0	0
$445$	15.3137	0.725939
$446$	0	0
$447$	−2.62742	−0.124273
$448$	0	0
$449$	33.6569	1.58837	0.794183	−	0.607679i	$-0.207899\pi$
$449$	33.6569	1.58837	0.794183	+	0.607679i	$0.207899\pi$
$450$	0	0
$451$	2.58579	0.121760
$452$	0	0
$453$	4.02944	0.189319
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.343146	−0.0160517	−0.00802584	−	0.999968i	$-0.502555\pi$
$457$	−0.343146	−0.0160517	−0.00802584	+	0.999968i	$0.502555\pi$
$458$	0	0
$459$	−18.4853	−0.862819
$460$	0	0
$461$	14.3137	0.666656	0.333328	−	0.942811i	$-0.391828\pi$
$461$	14.3137	0.666656	0.333328	+	0.942811i	$0.391828\pi$
$462$	0	0
$463$	−7.17157	−0.333291	−0.166646	−	0.986017i	$-0.553294\pi$
$463$	−7.17157	−0.333291	−0.166646	+	0.986017i	$0.553294\pi$
$464$	0	0
$465$	5.65685	0.262330
$466$	0	0
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.62742	−0.121065
$472$	0	0
$473$	5.65685	0.260102
$474$	0	0
$475$	22.7279	1.04283
$476$	0	0
$477$	33.6569	1.54104
$478$	0	0
$479$	26.0711	1.19122	0.595609	−	0.803275i	$-0.296911\pi$
$479$	26.0711	1.19122	0.595609	+	0.803275i	$0.296911\pi$
$480$	0	0
$481$	−12.0416	−0.549051
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.24264	−0.283464
$486$	0	0
$487$	−1.65685	−0.0750792	−0.0375396	−	0.999295i	$-0.511952\pi$
$487$	−1.65685	−0.0750792	−0.0375396	+	0.999295i	$0.511952\pi$
$488$	0	0
$489$	−6.51472	−0.294606
$490$	0	0
$491$	−24.8284	−1.12049	−0.560246	−	0.828327i	$-0.689293\pi$
$491$	−24.8284	−1.12049	−0.560246	+	0.828327i	$0.689293\pi$
$492$	0	0
$493$	66.2843	2.98529
$494$	0	0
$495$	−9.65685	−0.434043
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6.14214	0.274960	0.137480	−	0.990505i	$-0.456100\pi$
$499$	6.14214	0.274960	0.137480	+	0.990505i	$0.456100\pi$
$500$	0	0
$501$	−4.85786	−0.217033
$502$	0	0
$503$	38.2132	1.70384	0.851921	−	0.523670i	$-0.175437\pi$
$503$	38.2132	1.70384	0.851921	+	0.523670i	$0.175437\pi$
$504$	0	0
$505$	−40.3848	−1.79710
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	−13.3137	−0.590120	−0.295060	−	0.955479i	$-0.595340\pi$
$509$	−13.3137	−0.590120	−0.295060	+	0.955479i	$0.595340\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	8.24264	0.363921
$514$	0	0
$515$	36.1421	1.59261
$516$	0	0
$517$	6.48528	0.285222
$518$	0	0
$519$	1.72792	0.0758474
$520$	0	0
$521$	28.2843	1.23916	0.619578	−	0.784935i	$-0.287304\pi$
$521$	28.2843	1.23916	0.619578	+	0.784935i	$0.287304\pi$
$522$	0	0
$523$	−12.7279	−0.556553	−0.278277	−	0.960501i	$-0.589763\pi$
$523$	−12.7279	−0.556553	−0.278277	+	0.960501i	$0.589763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−30.6274	−1.33415
$528$	0	0
$529$	−17.9706	−0.781329
$530$	0	0
$531$	−23.7990	−1.03279
$532$	0	0
$533$	−4.72792	−0.204789
$534$	0	0
$535$	37.7990	1.63419
$536$	0	0
$537$	−7.82843	−0.337822
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−41.1421	−1.76884	−0.884419	−	0.466693i	$-0.845445\pi$
$541$	−41.1421	−1.76884	−0.884419	+	0.466693i	$0.845445\pi$
$542$	0	0
$543$	−1.51472	−0.0650028
$544$	0	0
$545$	1.65685	0.0709718
$546$	0	0
$547$	18.8701	0.806825	0.403413	−	0.915018i	$-0.367824\pi$
$547$	18.8701	0.806825	0.403413	+	0.915018i	$0.367824\pi$
$548$	0	0
$549$	−17.4558	−0.744997
$550$	0	0
$551$	−29.5563	−1.25914
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−9.31371	−0.395345
$556$	0	0
$557$	24.4853	1.03747	0.518737	−	0.854934i	$-0.326402\pi$
$557$	24.4853	1.03747	0.518737	+	0.854934i	$0.326402\pi$
$558$	0	0
$559$	−10.3431	−0.437468
$560$	0	0
$561$	−3.17157	−0.133904
$562$	0	0
$563$	5.07107	0.213720	0.106860	−	0.994274i	$-0.465920\pi$
$563$	5.07107	0.213720	0.106860	+	0.994274i	$0.465920\pi$
$564$	0	0
$565$	47.2132	1.98627
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	10.3848	0.434589	0.217295	−	0.976106i	$-0.430277\pi$
$571$	10.3848	0.434589	0.217295	+	0.976106i	$0.430277\pi$
$572$	0	0
$573$	−5.31371	−0.221983
$574$	0	0
$575$	−14.9289	−0.622580
$576$	0	0
$577$	−32.3137	−1.34524	−0.672619	−	0.739989i	$-0.734831\pi$
$577$	−32.3137	−1.34524	−0.672619	+	0.739989i	$0.734831\pi$
$578$	0	0
$579$	−0.870058	−0.0361584
$580$	0	0
$581$	0	0
$582$	0	0
$583$	11.8995	0.492827
$584$	0	0
$585$	17.6569	0.730021
$586$	0	0
$587$	44.8995	1.85320	0.926600	−	0.376048i	$-0.122717\pi$
$587$	44.8995	1.85320	0.926600	+	0.376048i	$0.122717\pi$
$588$	0	0
$589$	13.6569	0.562721
$590$	0	0
$591$	7.24264	0.297922
$592$	0	0
$593$	−35.6985	−1.46596	−0.732981	−	0.680250i	$-0.761871\pi$
$593$	−35.6985	−1.46596	−0.732981	+	0.680250i	$0.761871\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.24264	0.337349
$598$	0	0
$599$	2.62742	0.107353	0.0536767	−	0.998558i	$-0.482906\pi$
$599$	2.62742	0.107353	0.0536767	+	0.998558i	$0.482906\pi$
$600$	0	0
$601$	−35.9411	−1.46607	−0.733035	−	0.680191i	$-0.761897\pi$
$601$	−35.9411	−1.46607	−0.733035	+	0.680191i	$0.761897\pi$
$602$	0	0
$603$	31.7990	1.29495
$604$	0	0
$605$	−3.41421	−0.138808
$606$	0	0
$607$	9.02944	0.366494	0.183247	−	0.983067i	$-0.441339\pi$
$607$	9.02944	0.366494	0.183247	+	0.983067i	$0.441339\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.8579	−0.479718
$612$	0	0
$613$	−16.6274	−0.671575	−0.335788	−	0.941938i	$-0.609002\pi$
$613$	−16.6274	−0.671575	−0.335788	+	0.941938i	$0.609002\pi$
$614$	0	0
$615$	−3.65685	−0.147459
$616$	0	0
$617$	−8.02944	−0.323253	−0.161626	−	0.986852i	$-0.551674\pi$
$617$	−8.02944	−0.323253	−0.161626	+	0.986852i	$0.551674\pi$
$618$	0	0
$619$	−45.9411	−1.84653	−0.923265	−	0.384164i	$-0.874490\pi$
$619$	−45.9411	−1.84653	−0.923265	+	0.384164i	$0.874490\pi$
$620$	0	0
$621$	−5.41421	−0.217265
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	1.41421	0.0564782
$628$	0	0
$629$	50.4264	2.01063
$630$	0	0
$631$	−48.7279	−1.93983	−0.969914	−	0.243448i	$-0.921722\pi$
$631$	−48.7279	−1.93983	−0.969914	+	0.243448i	$0.921722\pi$
$632$	0	0
$633$	1.89949	0.0754981
$634$	0	0
$635$	−33.2132	−1.31803
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.68629	0.343624
$640$	0	0
$641$	−41.2843	−1.63063	−0.815315	−	0.579017i	$-0.803436\pi$
$641$	−41.2843	−1.63063	−0.815315	+	0.579017i	$0.803436\pi$
$642$	0	0
$643$	−4.41421	−0.174080	−0.0870398	−	0.996205i	$-0.527741\pi$
$643$	−4.41421	−0.174080	−0.0870398	+	0.996205i	$0.527741\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	46.1838	1.81567	0.907836	−	0.419326i	$-0.137734\pi$
$647$	46.1838	1.81567	0.907836	+	0.419326i	$0.137734\pi$
$648$	0	0
$649$	−8.41421	−0.330287
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.3848	0.719452	0.359726	−	0.933058i	$-0.382870\pi$
$653$	18.3848	0.719452	0.359726	+	0.933058i	$0.382870\pi$
$654$	0	0
$655$	−11.6569	−0.455471
$656$	0	0
$657$	18.6274	0.726725
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−14.9706	−0.582287	−0.291144	−	0.956679i	$-0.594036\pi$
$661$	−14.9706	−0.582287	−0.291144	+	0.956679i	$0.594036\pi$
$662$	0	0
$663$	5.79899	0.225214
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19.4142	0.751721
$668$	0	0
$669$	4.72792	0.182792
$670$	0	0
$671$	−6.17157	−0.238251
$672$	0	0
$673$	−25.5563	−0.985125	−0.492562	−	0.870277i	$-0.663940\pi$
$673$	−25.5563	−0.985125	−0.492562	+	0.870277i	$0.663940\pi$
$674$	0	0
$675$	16.0711	0.618576
$676$	0	0
$677$	−12.6863	−0.487574	−0.243787	−	0.969829i	$-0.578390\pi$
$677$	−12.6863	−0.487574	−0.243787	+	0.969829i	$0.578390\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−9.59798	−0.367795
$682$	0	0
$683$	−16.4142	−0.628072	−0.314036	−	0.949411i	$-0.601681\pi$
$683$	−16.4142	−0.628072	−0.314036	+	0.949411i	$0.601681\pi$
$684$	0	0
$685$	18.2426	0.697015
$686$	0	0
$687$	−0.284271	−0.0108456
$688$	0	0
$689$	−21.7574	−0.828889
$690$	0	0
$691$	−13.9289	−0.529882	−0.264941	−	0.964265i	$-0.585352\pi$
$691$	−13.9289	−0.529882	−0.264941	+	0.964265i	$0.585352\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.7990	0.749940
$698$	0	0
$699$	0.585786	0.0221565
$700$	0	0
$701$	36.1127	1.36396	0.681979	−	0.731372i	$-0.261120\pi$
$701$	36.1127	1.36396	0.681979	+	0.731372i	$0.261120\pi$
$702$	0	0
$703$	−22.4853	−0.848048
$704$	0	0
$705$	−9.17157	−0.345421
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36.0416	1.35357	0.676786	−	0.736180i	$-0.263372\pi$
$709$	36.0416	1.35357	0.676786	+	0.736180i	$0.263372\pi$
$710$	0	0
$711$	−13.4558	−0.504634
$712$	0	0
$713$	−8.97056	−0.335950
$714$	0	0
$715$	6.24264	0.233462
$716$	0	0
$717$	−9.20101	−0.343618
$718$	0	0
$719$	18.4853	0.689385	0.344692	−	0.938716i	$-0.387983\pi$
$719$	18.4853	0.689385	0.344692	+	0.938716i	$0.387983\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.55635	−0.0578812
$724$	0	0
$725$	−57.6274	−2.14023
$726$	0	0
$727$	48.4264	1.79604	0.898018	−	0.439959i	$-0.145007\pi$
$727$	48.4264	1.79604	0.898018	+	0.439959i	$0.145007\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	43.3137	1.60202
$732$	0	0
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.2426	0.414128
$738$	0	0
$739$	−32.4264	−1.19282	−0.596412	−	0.802678i	$-0.703408\pi$
$739$	−32.4264	−1.19282	−0.596412	+	0.802678i	$0.703408\pi$
$740$	0	0
$741$	−2.58579	−0.0949912
$742$	0	0
$743$	−9.31371	−0.341687	−0.170843	−	0.985298i	$-0.554649\pi$
$743$	−9.31371	−0.341687	−0.170843	+	0.985298i	$0.554649\pi$
$744$	0	0
$745$	−21.6569	−0.793446
$746$	0	0
$747$	45.6569	1.67050
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−34.3431	−1.25320	−0.626600	−	0.779341i	$-0.715554\pi$
$751$	−34.3431	−1.25320	−0.626600	+	0.779341i	$0.715554\pi$
$752$	0	0
$753$	0.887302	0.0323351
$754$	0	0
$755$	33.2132	1.20875
$756$	0	0
$757$	19.6569	0.714441	0.357220	−	0.934020i	$-0.383725\pi$
$757$	19.6569	0.714441	0.357220	+	0.934020i	$0.383725\pi$
$758$	0	0
$759$	−0.928932	−0.0337181
$760$	0	0
$761$	−2.97056	−0.107683	−0.0538414	−	0.998549i	$-0.517147\pi$
$761$	−2.97056	−0.107683	−0.0538414	+	0.998549i	$0.517147\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−73.9411	−2.67335
$766$	0	0
$767$	15.3848	0.555512
$768$	0	0
$769$	−10.9706	−0.395609	−0.197804	−	0.980242i	$-0.563381\pi$
$769$	−10.9706	−0.395609	−0.197804	+	0.980242i	$0.563381\pi$
$770$	0	0
$771$	−1.30152	−0.0468729
$772$	0	0
$773$	45.9411	1.65239	0.826194	−	0.563386i	$-0.190502\pi$
$773$	45.9411	1.65239	0.826194	+	0.563386i	$0.190502\pi$
$774$	0	0
$775$	26.6274	0.956485
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.82843	−0.316311
$780$	0	0
$781$	3.07107	0.109891
$782$	0	0
$783$	−20.8995	−0.746887
$784$	0	0
$785$	−21.6569	−0.772966
$786$	0	0
$787$	17.5563	0.625816	0.312908	−	0.949783i	$-0.398697\pi$
$787$	17.5563	0.625816	0.312908	+	0.949783i	$0.398697\pi$
$788$	0	0
$789$	12.8579	0.457752
$790$	0	0
$791$	0	0
$792$	0	0
$793$	11.2843	0.400716
$794$	0	0
$795$	−16.8284	−0.596843
$796$	0	0
$797$	3.65685	0.129532	0.0647662	−	0.997900i	$-0.479370\pi$
$797$	3.65685	0.129532	0.0647662	+	0.997900i	$0.479370\pi$
$798$	0	0
$799$	49.6569	1.75673
$800$	0	0
$801$	12.6863	0.448248
$802$	0	0
$803$	6.58579	0.232407
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.65685	−0.199131
$808$	0	0
$809$	−1.37258	−0.0482574	−0.0241287	−	0.999709i	$-0.507681\pi$
$809$	−1.37258	−0.0482574	−0.0241287	+	0.999709i	$0.507681\pi$
$810$	0	0
$811$	−32.2843	−1.13365	−0.566827	−	0.823837i	$-0.691829\pi$
$811$	−32.2843	−1.13365	−0.566827	+	0.823837i	$0.691829\pi$
$812$	0	0
$813$	−11.0000	−0.385787
$814$	0	0
$815$	−53.6985	−1.88098
$816$	0	0
$817$	−19.3137	−0.675701
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.5147	−0.436767	−0.218383	−	0.975863i	$-0.570078\pi$
$821$	−12.5147	−0.436767	−0.218383	+	0.975863i	$0.570078\pi$
$822$	0	0
$823$	−7.45584	−0.259894	−0.129947	−	0.991521i	$-0.541481\pi$
$823$	−7.45584	−0.259894	−0.129947	+	0.991521i	$0.541481\pi$
$824$	0	0
$825$	2.75736	0.0959989
$826$	0	0
$827$	−49.3553	−1.71625	−0.858127	−	0.513438i	$-0.828372\pi$
$827$	−49.3553	−1.71625	−0.858127	+	0.513438i	$0.828372\pi$
$828$	0	0
$829$	30.2426	1.05037	0.525185	−	0.850988i	$-0.323996\pi$
$829$	30.2426	1.05037	0.525185	+	0.850988i	$0.323996\pi$
$830$	0	0
$831$	1.58579	0.0550103
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−40.0416	−1.38570
$836$	0	0
$837$	9.65685	0.333790
$838$	0	0
$839$	1.51472	0.0522939	0.0261469	−	0.999658i	$-0.491676\pi$
$839$	1.51472	0.0522939	0.0261469	+	0.999658i	$0.491676\pi$
$840$	0	0
$841$	45.9411	1.58418
$842$	0	0
$843$	6.92893	0.238645
$844$	0	0
$845$	32.9706	1.13422
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.52691	−0.292643
$850$	0	0
$851$	14.7696	0.506294
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	32.9706	1.12757
$856$	0	0
$857$	10.6274	0.363026	0.181513	−	0.983389i	$-0.441901\pi$
$857$	10.6274	0.363026	0.181513	+	0.983389i	$0.441901\pi$
$858$	0	0
$859$	−19.7279	−0.673108	−0.336554	−	0.941664i	$-0.609261\pi$
$859$	−19.7279	−0.673108	−0.336554	+	0.941664i	$0.609261\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.21320	0.245540	0.122770	−	0.992435i	$-0.460822\pi$
$863$	7.21320	0.245540	0.122770	+	0.992435i	$0.460822\pi$
$864$	0	0
$865$	14.2426	0.484264
$866$	0	0
$867$	−17.2426	−0.585591
$868$	0	0
$869$	−4.75736	−0.161382
$870$	0	0
$871$	−20.5563	−0.696525
$872$	0	0
$873$	−5.17157	−0.175031
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25.6274	−0.865376	−0.432688	−	0.901544i	$-0.642435\pi$
$877$	−25.6274	−0.865376	−0.432688	+	0.901544i	$0.642435\pi$
$878$	0	0
$879$	2.14214	0.0722524
$880$	0	0
$881$	−44.4558	−1.49776	−0.748878	−	0.662708i	$-0.769407\pi$
$881$	−44.4558	−1.49776	−0.748878	+	0.662708i	$0.769407\pi$
$882$	0	0
$883$	9.72792	0.327371	0.163685	−	0.986513i	$-0.447662\pi$
$883$	9.72792	0.327371	0.163685	+	0.986513i	$0.447662\pi$
$884$	0	0
$885$	11.8995	0.399997
$886$	0	0
$887$	22.8995	0.768890	0.384445	−	0.923148i	$-0.374393\pi$
$887$	22.8995	0.768890	0.384445	+	0.923148i	$0.374393\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−7.48528	−0.250766
$892$	0	0
$893$	−22.1421	−0.740958
$894$	0	0
$895$	−64.5269	−2.15690
$896$	0	0
$897$	1.69848	0.0567108
$898$	0	0
$899$	−34.6274	−1.15489
$900$	0	0
$901$	91.1127	3.03540
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.4853	−0.415025
$906$	0	0
$907$	−33.3137	−1.10616	−0.553082	−	0.833127i	$-0.686548\pi$
$907$	−33.3137	−1.10616	−0.553082	+	0.833127i	$0.686548\pi$
$908$	0	0
$909$	−33.4558	−1.10966
$910$	0	0
$911$	13.5147	0.447763	0.223881	−	0.974616i	$-0.428127\pi$
$911$	13.5147	0.447763	0.223881	+	0.974616i	$0.428127\pi$
$912$	0	0
$913$	16.1421	0.534227
$914$	0	0
$915$	8.72792	0.288536
$916$	0	0
$917$	0	0
$918$	0	0
$919$	6.14214	0.202610	0.101305	−	0.994855i	$-0.467698\pi$
$919$	6.14214	0.202610	0.101305	+	0.994855i	$0.467698\pi$
$920$	0	0
$921$	−4.10051	−0.135116
$922$	0	0
$923$	−5.61522	−0.184827
$924$	0	0
$925$	−43.8406	−1.44147
$926$	0	0
$927$	29.9411	0.983396
$928$	0	0
$929$	10.4558	0.343045	0.171523	−	0.985180i	$-0.445131\pi$
$929$	10.4558	0.343045	0.171523	+	0.985180i	$0.445131\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.61522	−0.118357
$934$	0	0
$935$	−26.1421	−0.854939
$936$	0	0
$937$	−16.5858	−0.541834	−0.270917	−	0.962603i	$-0.587327\pi$
$937$	−16.5858	−0.541834	−0.270917	+	0.962603i	$0.587327\pi$
$938$	0	0
$939$	3.87006	0.126295
$940$	0	0
$941$	13.3431	0.434974	0.217487	−	0.976063i	$-0.430214\pi$
$941$	13.3431	0.434974	0.217487	+	0.976063i	$0.430214\pi$
$942$	0	0
$943$	5.79899	0.188841
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.1716	−0.947949	−0.473974	−	0.880539i	$-0.657181\pi$
$947$	−29.1716	−0.947949	−0.473974	+	0.880539i	$0.657181\pi$
$948$	0	0
$949$	−12.0416	−0.390888
$950$	0	0
$951$	−12.9706	−0.420599
$952$	0	0
$953$	−25.5563	−0.827851	−0.413926	−	0.910311i	$-0.635843\pi$
$953$	−25.5563	−0.827851	−0.413926	+	0.910311i	$0.635843\pi$
$954$	0	0
$955$	−43.7990	−1.41730
$956$	0	0
$957$	−3.58579	−0.115912
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	31.3137	1.00907
$964$	0	0
$965$	−7.17157	−0.230861
$966$	0	0
$967$	−20.2843	−0.652298	−0.326149	−	0.945318i	$-0.605751\pi$
$967$	−20.2843	−0.652298	−0.326149	+	0.945318i	$0.605751\pi$
$968$	0	0
$969$	10.8284	0.347859
$970$	0	0
$971$	−47.7279	−1.53166	−0.765831	−	0.643042i	$-0.777672\pi$
$971$	−47.7279	−1.53166	−0.765831	+	0.643042i	$0.777672\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5.04163	−0.161461
$976$	0	0
$977$	40.0000	1.27971	0.639857	−	0.768494i	$-0.278994\pi$
$977$	40.0000	1.27971	0.639857	+	0.768494i	$0.278994\pi$
$978$	0	0
$979$	4.48528	0.143350
$980$	0	0
$981$	1.37258	0.0438232
$982$	0	0
$983$	52.4264	1.67214	0.836071	−	0.548621i	$-0.184847\pi$
$983$	52.4264	1.67214	0.836071	+	0.548621i	$0.184847\pi$
$984$	0	0
$985$	59.6985	1.90215
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.6863	0.403401
$990$	0	0
$991$	38.3848	1.21933	0.609666	−	0.792658i	$-0.291303\pi$
$991$	38.3848	1.21933	0.609666	+	0.792658i	$0.291303\pi$
$992$	0	0
$993$	−4.11270	−0.130513
$994$	0	0
$995$	67.9411	2.15388
$996$	0	0
$997$	18.8284	0.596302	0.298151	−	0.954519i	$-0.403630\pi$
$997$	18.8284	0.596302	0.298151	+	0.954519i	$0.403630\pi$
$998$	0	0
$999$	−15.8995	−0.503038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cc.1.1		2
4.3	odd	2		1078.2.a.t.1.2		2
7.2	even	3		1232.2.q.f.529.2		4
7.4	even	3		1232.2.q.f.177.2		4
7.6	odd	2		8624.2.a.bh.1.2		2
12.11	even	2		9702.2.a.cx.1.2		2
28.3	even	6		1078.2.e.m.177.2		4
28.11	odd	6		154.2.e.e.23.1	✓	4
28.19	even	6		1078.2.e.m.67.2		4
28.23	odd	6		154.2.e.e.67.1	yes	4
28.27	even	2		1078.2.a.x.1.1		2
84.11	even	6		1386.2.k.t.793.1		4
84.23	even	6		1386.2.k.t.991.1		4
84.83	odd	2		9702.2.a.ch.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.e.23.1	✓	4	28.11	odd	6
154.2.e.e.67.1	yes	4	28.23	odd	6
1078.2.a.t.1.2		2	4.3	odd	2
1078.2.a.x.1.1		2	28.27	even	2
1078.2.e.m.67.2		4	28.19	even	6
1078.2.e.m.177.2		4	28.3	even	6
1232.2.q.f.177.2		4	7.4	even	3
1232.2.q.f.529.2		4	7.2	even	3
1386.2.k.t.793.1		4	84.11	even	6
1386.2.k.t.991.1		4	84.23	even	6
8624.2.a.bh.1.2		2	7.6	odd	2
8624.2.a.cc.1.1		2	1.1	even	1	trivial
9702.2.a.ch.1.1		2	84.83	odd	2
9702.2.a.cx.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} -3.41421 q^{5} -2.82843 q^{9} +O(q^{10})\)	\(q-0.414214 q^{3} -3.41421 q^{5} -2.82843 q^{9} -1.00000 q^{11} +1.82843 q^{13} +1.41421 q^{15} -7.65685 q^{17} +3.41421 q^{19} -2.24264 q^{23} +6.65685 q^{25} +2.41421 q^{27} -8.65685 q^{29} +4.00000 q^{31} +0.414214 q^{33} -6.58579 q^{37} -0.757359 q^{39} -2.58579 q^{41} -5.65685 q^{43} +9.65685 q^{45} -6.48528 q^{47} +3.17157 q^{51} -11.8995 q^{53} +3.41421 q^{55} -1.41421 q^{57} +8.41421 q^{59} +6.17157 q^{61} -6.24264 q^{65} -11.2426 q^{67} +0.928932 q^{69} -3.07107 q^{71} -6.58579 q^{73} -2.75736 q^{75} +4.75736 q^{79} +7.48528 q^{81} -16.1421 q^{83} +26.1421 q^{85} +3.58579 q^{87} -4.48528 q^{89} -1.65685 q^{93} -11.6569 q^{95} +1.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5	\( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{11} - 2 q^{13} - 4 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} + 8 q^{31} - 2 q^{33} - 16 q^{37} - 10 q^{39} - 8 q^{41} + 8 q^{45} + 4 q^{47} + 12 q^{51} - 4 q^{53} + 4 q^{55} + 14 q^{59} + 18 q^{61} - 4 q^{65} - 14 q^{67} + 16 q^{69} + 8 q^{71} - 16 q^{73} - 14 q^{75} + 18 q^{79} - 2 q^{81} - 4 q^{83} + 24 q^{85} + 10 q^{87} + 8 q^{89} + 8 q^{93} - 12 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^11 - 2 * q^13 - 4 * q^17 + 4 * q^19 + 4 * q^23 + 2 * q^25 + 2 * q^27 - 6 * q^29 + 8 * q^31 - 2 * q^33 - 16 * q^37 - 10 * q^39 - 8 * q^41 + 8 * q^45 + 4 * q^47 + 12 * q^51 - 4 * q^53 + 4 * q^55 + 14 * q^59 + 18 * q^61 - 4 * q^65 - 14 * q^67 + 16 * q^69 + 8 * q^71 - 16 * q^73 - 14 * q^75 + 18 * q^79 - 2 * q^81 - 4 * q^83 + 24 * q^85 + 10 * q^87 + 8 * q^89 + 8 * q^93 - 12 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more