LMFDB - Embedded newform 8624.2.a.cl.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.91223$ 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)

$f(q)$	$=$	$q-1.91223 q^{3} -3.56885 q^{5} +0.656620 q^{9} +O(q^{10})$	$q-1.91223 q^{3} -3.56885 q^{5} +0.656620 q^{9} +1.00000 q^{11} -5.91223 q^{13} +6.82446 q^{15} -1.65662 q^{17} -1.48108 q^{19} -3.34338 q^{23} +7.73669 q^{25} +4.48108 q^{27} +3.08007 q^{29} +7.08007 q^{31} -1.91223 q^{33} -4.51122 q^{37} +11.3055 q^{39} -1.28575 q^{41} -1.59899 q^{43} -2.34338 q^{45} +1.65662 q^{47} +3.16784 q^{51} +9.22547 q^{53} -3.56885 q^{55} +2.83216 q^{57} -8.85195 q^{59} -6.68676 q^{61} +21.0999 q^{65} +9.82446 q^{67} +6.39331 q^{69} +8.61878 q^{71} +4.56115 q^{73} -14.7943 q^{75} +6.39331 q^{79} -10.5387 q^{81} -0.167838 q^{83} +5.91223 q^{85} -5.88979 q^{87} -2.56885 q^{89} -13.5387 q^{93} +5.28575 q^{95} +9.73669 q^{97} +0.656620 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 2 q^{5}+O(q^{10}) $ 3 * q + q^3 - 2 * q^5	$ 3 q + q^{3} - 2 q^{5} + 3 q^{11} - 11 q^{13} + 7 q^{15} - 3 q^{17} + 11 q^{19} - 12 q^{23} + 3 q^{25} - 2 q^{27} - 9 q^{29} + 3 q^{31} + q^{33} - 4 q^{37} + 5 q^{39} - 5 q^{41} - 2 q^{43} - 9 q^{45} + 3 q^{47} - 2 q^{51} + 17 q^{53} - 2 q^{55} + 20 q^{57} - 8 q^{59} - 24 q^{61} + 15 q^{65} + 16 q^{67} - 3 q^{69} - 7 q^{71} - 20 q^{73} - 25 q^{75} - 3 q^{79} - 17 q^{81} + 11 q^{83} + 11 q^{85} - 30 q^{87} + q^{89} - 26 q^{93} + 17 q^{95} + 9 q^{97}+O(q^{100}) $ 3 * q + q^3 - 2 * q^5 + 3 * q^11 - 11 * q^13 + 7 * q^15 - 3 * q^17 + 11 * q^19 - 12 * q^23 + 3 * q^25 - 2 * q^27 - 9 * q^29 + 3 * q^31 + q^33 - 4 * q^37 + 5 * q^39 - 5 * q^41 - 2 * q^43 - 9 * q^45 + 3 * q^47 - 2 * q^51 + 17 * q^53 - 2 * q^55 + 20 * q^57 - 8 * q^59 - 24 * q^61 + 15 * q^65 + 16 * q^67 - 3 * q^69 - 7 * q^71 - 20 * q^73 - 25 * q^75 - 3 * q^79 - 17 * q^81 + 11 * q^83 + 11 * q^85 - 30 * q^87 + q^89 - 26 * q^93 + 17 * q^95 + 9 * 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
$3$	−1.91223	−1.10403	−0.552013	−	0.833835i	$-0.686140\pi$
$3$	−1.91223	−1.10403	−0.552013	+	0.833835i	$0.686140\pi$
$4$	0	0
$5$	−3.56885	−1.59604	−0.798019	−	0.602632i	$-0.794119\pi$
$5$	−3.56885	−1.59604	−0.798019	+	0.602632i	$0.794119\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.656620	0.218873
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.91223	−1.63976	−0.819879	−	0.572537i	$-0.805959\pi$
$13$	−5.91223	−1.63976	−0.819879	+	0.572537i	$0.805959\pi$
$14$	0	0
$15$	6.82446	1.76207
$16$	0	0
$17$	−1.65662	−0.401789	−0.200895	−	0.979613i	$-0.564385\pi$
$17$	−1.65662	−0.401789	−0.200895	+	0.979613i	$0.564385\pi$
$18$	0	0
$19$	−1.48108	−0.339783	−0.169891	−	0.985463i	$-0.554342\pi$
$19$	−1.48108	−0.339783	−0.169891	+	0.985463i	$0.554342\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.34338	−0.697143	−0.348571	−	0.937282i	$-0.613333\pi$
$23$	−3.34338	−0.697143	−0.348571	+	0.937282i	$0.613333\pi$
$24$	0	0
$25$	7.73669	1.54734
$26$	0	0
$27$	4.48108	0.862384
$28$	0	0
$29$	3.08007	0.571954	0.285977	−	0.958236i	$-0.407682\pi$
$29$	3.08007	0.571954	0.285977	+	0.958236i	$0.407682\pi$
$30$	0	0
$31$	7.08007	1.27162	0.635809	−	0.771847i	$-0.280667\pi$
$31$	7.08007	1.27162	0.635809	+	0.771847i	$0.280667\pi$
$32$	0	0
$33$	−1.91223	−0.332876
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.51122	−0.741640	−0.370820	−	0.928705i	$-0.620923\pi$
$37$	−4.51122	−0.741640	−0.370820	+	0.928705i	$0.620923\pi$
$38$	0	0
$39$	11.3055	1.81033
$40$	0	0
$41$	−1.28575	−0.200800	−0.100400	−	0.994947i	$-0.532012\pi$
$41$	−1.28575	−0.200800	−0.100400	+	0.994947i	$0.532012\pi$
$42$	0	0
$43$	−1.59899	−0.243843	−0.121922	−	0.992540i	$-0.538906\pi$
$43$	−1.59899	−0.243843	−0.121922	+	0.992540i	$0.538906\pi$
$44$	0	0
$45$	−2.34338	−0.349330
$46$	0	0
$47$	1.65662	0.241643	0.120821	−	0.992674i	$-0.461447\pi$
$47$	1.65662	0.241643	0.120821	+	0.992674i	$0.461447\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.16784	0.443586
$52$	0	0
$53$	9.22547	1.26722	0.633608	−	0.773654i	$-0.281573\pi$
$53$	9.22547	1.26722	0.633608	+	0.773654i	$0.281573\pi$
$54$	0	0
$55$	−3.56885	−0.481224
$56$	0	0
$57$	2.83216	0.375129
$58$	0	0
$59$	−8.85195	−1.15243	−0.576213	−	0.817300i	$-0.695470\pi$
$59$	−8.85195	−1.15243	−0.576213	+	0.817300i	$0.695470\pi$
$60$	0	0
$61$	−6.68676	−0.856152	−0.428076	−	0.903743i	$-0.640808\pi$
$61$	−6.68676	−0.856152	−0.428076	+	0.903743i	$0.640808\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21.0999	2.61712
$66$	0	0
$67$	9.82446	1.20025	0.600124	−	0.799907i	$-0.295118\pi$
$67$	9.82446	1.20025	0.600124	+	0.799907i	$0.295118\pi$
$68$	0	0
$69$	6.39331	0.769664
$70$	0	0
$71$	8.61878	1.02286	0.511430	−	0.859325i	$-0.329116\pi$
$71$	8.61878	1.02286	0.511430	+	0.859325i	$0.329116\pi$
$72$	0	0
$73$	4.56115	0.533842	0.266921	−	0.963718i	$-0.413994\pi$
$73$	4.56115	0.533842	0.266921	+	0.963718i	$0.413994\pi$
$74$	0	0
$75$	−14.7943	−1.70830
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.39331	0.719303	0.359652	−	0.933087i	$-0.382896\pi$
$79$	6.39331	0.719303	0.359652	+	0.933087i	$0.382896\pi$
$80$	0	0
$81$	−10.5387	−1.17097
$82$	0	0
$83$	−0.167838	−0.0184226	−0.00921130	−	0.999958i	$-0.502932\pi$
$83$	−0.167838	−0.0184226	−0.00921130	+	0.999958i	$0.502932\pi$
$84$	0	0
$85$	5.91223	0.641271
$86$	0	0
$87$	−5.88979	−0.631452
$88$	0	0
$89$	−2.56885	−0.272298	−0.136149	−	0.990688i	$-0.543473\pi$
$89$	−2.56885	−0.272298	−0.136149	+	0.990688i	$0.543473\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−13.5387	−1.40390
$94$	0	0
$95$	5.28575	0.542306
$96$	0	0
$97$	9.73669	0.988611	0.494305	−	0.869288i	$-0.335422\pi$
$97$	9.73669	0.988611	0.494305	+	0.869288i	$0.335422\pi$
$98$	0	0
$99$	0.656620	0.0659928
$100$	0	0
$101$	1.85460	0.184539	0.0922697	−	0.995734i	$-0.470588\pi$
$101$	1.85460	0.184539	0.0922697	+	0.995734i	$0.470588\pi$
$102$	0	0
$103$	3.16784	0.312136	0.156068	−	0.987746i	$-0.450118\pi$
$103$	3.16784	0.312136	0.156068	+	0.987746i	$0.450118\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.76683	0.460826	0.230413	−	0.973093i	$-0.425992\pi$
$107$	4.76683	0.460826	0.230413	+	0.973093i	$0.425992\pi$
$108$	0	0
$109$	−14.8821	−1.42545	−0.712723	−	0.701446i	$-0.752538\pi$
$109$	−14.8821	−1.42545	−0.712723	+	0.701446i	$0.752538\pi$
$110$	0	0
$111$	8.62648	0.818789
$112$	0	0
$113$	12.4432	1.17056	0.585281	−	0.810831i	$-0.300984\pi$
$113$	12.4432	1.17056	0.585281	+	0.810831i	$0.300984\pi$
$114$	0	0
$115$	11.9320	1.11267
$116$	0	0
$117$	−3.88209	−0.358899
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.45864	0.221688
$124$	0	0
$125$	−9.76683	−0.873571
$126$	0	0
$127$	6.62142	0.587556	0.293778	−	0.955874i	$-0.405087\pi$
$127$	6.62142	0.587556	0.293778	+	0.955874i	$0.405087\pi$
$128$	0	0
$129$	3.05763	0.269209
$130$	0	0
$131$	6.05763	0.529258	0.264629	−	0.964350i	$-0.414751\pi$
$131$	6.05763	0.529258	0.264629	+	0.964350i	$0.414751\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−15.9923	−1.37640
$136$	0	0
$137$	−7.42345	−0.634228	−0.317114	−	0.948387i	$-0.602714\pi$
$137$	−7.42345	−0.634228	−0.317114	+	0.948387i	$0.602714\pi$
$138$	0	0
$139$	10.8245	0.918119	0.459059	−	0.888406i	$-0.348187\pi$
$139$	10.8245	0.918119	0.459059	+	0.888406i	$0.348187\pi$
$140$	0	0
$141$	−3.16784	−0.266780
$142$	0	0
$143$	−5.91223	−0.494405
$144$	0	0
$145$	−10.9923	−0.912861
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	−16.4234	−1.33652	−0.668261	−	0.743927i	$-0.732961\pi$
$151$	−16.4234	−1.33652	−0.668261	+	0.743927i	$0.732961\pi$
$152$	0	0
$153$	−1.08777	−0.0879411
$154$	0	0
$155$	−25.2677	−2.02955
$156$	0	0
$157$	−11.4509	−0.913885	−0.456942	−	0.889496i	$-0.651055\pi$
$157$	−11.4509	−0.913885	−0.456942	+	0.889496i	$0.651055\pi$
$158$	0	0
$159$	−17.6412	−1.39904
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.93972	0.700213	0.350107	−	0.936710i	$-0.386145\pi$
$163$	8.93972	0.700213	0.350107	+	0.936710i	$0.386145\pi$
$164$	0	0
$165$	6.82446	0.531283
$166$	0	0
$167$	18.2178	1.40973	0.704867	−	0.709340i	$-0.251007\pi$
$167$	18.2178	1.40973	0.704867	+	0.709340i	$0.251007\pi$
$168$	0	0
$169$	21.9545	1.68880
$170$	0	0
$171$	−0.972507	−0.0743694
$172$	0	0
$173$	19.5611	1.48721	0.743603	−	0.668621i	$-0.233115\pi$
$173$	19.5611	1.48721	0.743603	+	0.668621i	$0.233115\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.9270	1.27231
$178$	0	0
$179$	−3.24791	−0.242760	−0.121380	−	0.992606i	$-0.538732\pi$
$179$	−3.24791	−0.242760	−0.121380	+	0.992606i	$0.538732\pi$
$180$	0	0
$181$	−10.3407	−0.768621	−0.384310	−	0.923204i	$-0.625561\pi$
$181$	−10.3407	−0.768621	−0.384310	+	0.923204i	$0.625561\pi$
$182$	0	0
$183$	12.7866	0.945214
$184$	0	0
$185$	16.0999	1.18369
$186$	0	0
$187$	−1.65662	−0.121144
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.0224	−0.725198	−0.362599	−	0.931945i	$-0.618111\pi$
$191$	−10.0224	−0.725198	−0.362599	+	0.931945i	$0.618111\pi$
$192$	0	0
$193$	25.3253	1.82296	0.911478	−	0.411348i	$-0.134942\pi$
$193$	25.3253	1.82296	0.911478	+	0.411348i	$0.134942\pi$
$194$	0	0
$195$	−40.3478	−2.88936
$196$	0	0
$197$	−24.5809	−1.75132	−0.875660	−	0.482929i	$-0.839573\pi$
$197$	−24.5809	−1.75132	−0.875660	+	0.482929i	$0.839573\pi$
$198$	0	0
$199$	−5.59128	−0.396356	−0.198178	−	0.980166i	$-0.563502\pi$
$199$	−5.59128	−0.396356	−0.198178	+	0.980166i	$0.563502\pi$
$200$	0	0
$201$	−18.7866	−1.32511
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.58864	0.320484
$206$	0	0
$207$	−2.19533	−0.152586
$208$	0	0
$209$	−1.48108	−0.102448
$210$	0	0
$211$	−12.0999	−0.832988	−0.416494	−	0.909138i	$-0.636741\pi$
$211$	−12.0999	−0.832988	−0.416494	+	0.909138i	$0.636741\pi$
$212$	0	0
$213$	−16.4811	−1.12926
$214$	0	0
$215$	5.70655	0.389183
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.72196	−0.589375
$220$	0	0
$221$	9.79432	0.658837
$222$	0	0
$223$	−15.6265	−1.04643	−0.523213	−	0.852202i	$-0.675267\pi$
$223$	−15.6265	−1.04643	−0.523213	+	0.852202i	$0.675267\pi$
$224$	0	0
$225$	5.08007	0.338671
$226$	0	0
$227$	15.6687	1.03997	0.519984	−	0.854176i	$-0.325938\pi$
$227$	15.6687	1.03997	0.519984	+	0.854176i	$0.325938\pi$
$228$	0	0
$229$	−5.57149	−0.368175	−0.184087	−	0.982910i	$-0.558933\pi$
$229$	−5.57149	−0.368175	−0.184087	+	0.982910i	$0.558933\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.2754	−1.26277	−0.631387	−	0.775468i	$-0.717514\pi$
$233$	−19.2754	−1.26277	−0.631387	+	0.775468i	$0.717514\pi$
$234$	0	0
$235$	−5.91223	−0.385671
$236$	0	0
$237$	−12.2255	−0.794130
$238$	0	0
$239$	22.1575	1.43325	0.716624	−	0.697459i	$-0.245686\pi$
$239$	22.1575	1.43325	0.716624	+	0.697459i	$0.245686\pi$
$240$	0	0
$241$	−19.8744	−1.28022	−0.640111	−	0.768283i	$-0.721112\pi$
$241$	−19.8744	−1.28022	−0.640111	+	0.768283i	$0.721112\pi$
$242$	0	0
$243$	6.70919	0.430395
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.75648	0.557161
$248$	0	0
$249$	0.320945	0.0203390
$250$	0	0
$251$	−22.1076	−1.39542	−0.697708	−	0.716382i	$-0.745797\pi$
$251$	−22.1076	−1.39542	−0.697708	+	0.716382i	$0.745797\pi$
$252$	0	0
$253$	−3.34338	−0.210196
$254$	0	0
$255$	−11.3055	−0.707980
$256$	0	0
$257$	29.1196	1.81643	0.908217	−	0.418500i	$-0.137444\pi$
$257$	29.1196	1.81643	0.908217	+	0.418500i	$0.137444\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.02243	0.125186
$262$	0	0
$263$	15.5035	0.955988	0.477994	−	0.878363i	$-0.341364\pi$
$263$	15.5035	0.955988	0.477994	+	0.878363i	$0.341364\pi$
$264$	0	0
$265$	−32.9243	−2.02252
$266$	0	0
$267$	4.91223	0.300624
$268$	0	0
$269$	1.70655	0.104050	0.0520251	−	0.998646i	$-0.483432\pi$
$269$	1.70655	0.104050	0.0520251	+	0.998646i	$0.483432\pi$
$270$	0	0
$271$	20.5284	1.24701	0.623505	−	0.781820i	$-0.285708\pi$
$271$	20.5284	1.24701	0.623505	+	0.781820i	$0.285708\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.73669	0.466540
$276$	0	0
$277$	−26.6610	−1.60190	−0.800952	−	0.598728i	$-0.795673\pi$
$277$	−26.6610	−1.60190	−0.800952	+	0.598728i	$0.795673\pi$
$278$	0	0
$279$	4.64892	0.278323
$280$	0	0
$281$	15.7444	0.939232	0.469616	−	0.882871i	$-0.344392\pi$
$281$	15.7444	0.939232	0.469616	+	0.882871i	$0.344392\pi$
$282$	0	0
$283$	16.0697	0.955246	0.477623	−	0.878565i	$-0.341499\pi$
$283$	16.0697	0.955246	0.477623	+	0.878565i	$0.341499\pi$
$284$	0	0
$285$	−10.1076	−0.598720
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.2556	−0.838565
$290$	0	0
$291$	−18.6188	−1.09145
$292$	0	0
$293$	15.3357	0.895920	0.447960	−	0.894054i	$-0.352151\pi$
$293$	15.3357	0.895920	0.447960	+	0.894054i	$0.352151\pi$
$294$	0	0
$295$	31.5913	1.83932
$296$	0	0
$297$	4.48108	0.260019
$298$	0	0
$299$	19.7668	1.14315
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.54641	−0.203736
$304$	0	0
$305$	23.8640	1.36645
$306$	0	0
$307$	16.4707	0.940034	0.470017	−	0.882657i	$-0.344248\pi$
$307$	16.4707	0.940034	0.470017	+	0.882657i	$0.344248\pi$
$308$	0	0
$309$	−6.05763	−0.344607
$310$	0	0
$311$	−21.6291	−1.22648	−0.613238	−	0.789898i	$-0.710133\pi$
$311$	−21.6291	−1.22648	−0.613238	+	0.789898i	$0.710133\pi$
$312$	0	0
$313$	16.3882	0.926319	0.463159	−	0.886275i	$-0.346716\pi$
$313$	16.3882	0.926319	0.463159	+	0.886275i	$0.346716\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.46129	0.250571	0.125285	−	0.992121i	$-0.460015\pi$
$317$	4.46129	0.250571	0.125285	+	0.992121i	$0.460015\pi$
$318$	0	0
$319$	3.08007	0.172451
$320$	0	0
$321$	−9.11526	−0.508764
$322$	0	0
$323$	2.45359	0.136521
$324$	0	0
$325$	−45.7411	−2.53726
$326$	0	0
$327$	28.4580	1.57373
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.0396	−1.04651	−0.523255	−	0.852176i	$-0.675282\pi$
$331$	−19.0396	−1.04651	−0.523255	+	0.852176i	$0.675282\pi$
$332$	0	0
$333$	−2.96216	−0.162325
$334$	0	0
$335$	−35.0620	−1.91564
$336$	0	0
$337$	27.0147	1.47159	0.735793	−	0.677206i	$-0.236810\pi$
$337$	27.0147	1.47159	0.735793	+	0.677206i	$0.236810\pi$
$338$	0	0
$339$	−23.7943	−1.29233
$340$	0	0
$341$	7.08007	0.383407
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−22.8168	−1.22841
$346$	0	0
$347$	20.2178	1.08535	0.542673	−	0.839944i	$-0.317412\pi$
$347$	20.2178	1.08535	0.542673	+	0.839944i	$0.317412\pi$
$348$	0	0
$349$	−12.0224	−0.643546	−0.321773	−	0.946817i	$-0.604279\pi$
$349$	−12.0224	−0.643546	−0.321773	+	0.946817i	$0.604279\pi$
$350$	0	0
$351$	−26.4932	−1.41410
$352$	0	0
$353$	−10.7591	−0.572650	−0.286325	−	0.958133i	$-0.592434\pi$
$353$	−10.7591	−0.572650	−0.286325	+	0.958133i	$0.592434\pi$
$354$	0	0
$355$	−30.7591	−1.63252
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.2901	−1.28198	−0.640992	−	0.767548i	$-0.721477\pi$
$359$	−24.2901	−1.28198	−0.640992	+	0.767548i	$0.721477\pi$
$360$	0	0
$361$	−16.8064	−0.884548
$362$	0	0
$363$	−1.91223	−0.100366
$364$	0	0
$365$	−16.2780	−0.852032
$366$	0	0
$367$	−10.8442	−0.566065	−0.283033	−	0.959110i	$-0.591340\pi$
$367$	−10.8442	−0.566065	−0.283033	+	0.959110i	$0.591340\pi$
$368$	0	0
$369$	−0.844248	−0.0439498
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−33.0242	−1.70993	−0.854963	−	0.518688i	$-0.826421\pi$
$373$	−33.0242	−1.70993	−0.854963	+	0.518688i	$0.826421\pi$
$374$	0	0
$375$	18.6764	0.964446
$376$	0	0
$377$	−18.2101	−0.937866
$378$	0	0
$379$	21.9320	1.12657	0.563286	−	0.826262i	$-0.309537\pi$
$379$	21.9320	1.12657	0.563286	+	0.826262i	$0.309537\pi$
$380$	0	0
$381$	−12.6617	−0.648677
$382$	0	0
$383$	−36.4630	−1.86317	−0.931587	−	0.363519i	$-0.881575\pi$
$383$	−36.4630	−1.86317	−0.931587	+	0.363519i	$0.881575\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.04993	−0.0533709
$388$	0	0
$389$	19.4760	0.987473	0.493737	−	0.869611i	$-0.335631\pi$
$389$	19.4760	0.987473	0.493737	+	0.869611i	$0.335631\pi$
$390$	0	0
$391$	5.53871	0.280105
$392$	0	0
$393$	−11.5836	−0.584314
$394$	0	0
$395$	−22.8168	−1.14804
$396$	0	0
$397$	−7.83987	−0.393472	−0.196736	−	0.980457i	$-0.563034\pi$
$397$	−7.83987	−0.393472	−0.196736	+	0.980457i	$0.563034\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.4459	−1.22077	−0.610385	−	0.792105i	$-0.708985\pi$
$401$	−24.4459	−1.22077	−0.610385	+	0.792105i	$0.708985\pi$
$402$	0	0
$403$	−41.8590	−2.08514
$404$	0	0
$405$	37.6111	1.86891
$406$	0	0
$407$	−4.51122	−0.223613
$408$	0	0
$409$	−2.35373	−0.116384	−0.0581922	−	0.998305i	$-0.518534\pi$
$409$	−2.35373	−0.116384	−0.0581922	+	0.998305i	$0.518534\pi$
$410$	0	0
$411$	14.1953	0.700204
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.598988	0.0294032
$416$	0	0
$417$	−20.6988	−1.01363
$418$	0	0
$419$	−9.29081	−0.453886	−0.226943	−	0.973908i	$-0.572873\pi$
$419$	−9.29081	−0.453886	−0.226943	+	0.973908i	$0.572873\pi$
$420$	0	0
$421$	−39.0319	−1.90230	−0.951149	−	0.308733i	$-0.900095\pi$
$421$	−39.0319	−1.90230	−0.951149	+	0.308733i	$0.900095\pi$
$422$	0	0
$423$	1.08777	0.0528892
$424$	0	0
$425$	−12.8168	−0.621704
$426$	0	0
$427$	0	0
$428$	0	0
$429$	11.3055	0.545836
$430$	0	0
$431$	3.80202	0.183137	0.0915685	−	0.995799i	$-0.470812\pi$
$431$	3.80202	0.183137	0.0915685	+	0.995799i	$0.470812\pi$
$432$	0	0
$433$	8.22041	0.395048	0.197524	−	0.980298i	$-0.436710\pi$
$433$	8.22041	0.395048	0.197524	+	0.980298i	$0.436710\pi$
$434$	0	0
$435$	21.0198	1.00782
$436$	0	0
$437$	4.95181	0.236877
$438$	0	0
$439$	−4.54136	−0.216747	−0.108374	−	0.994110i	$-0.534564\pi$
$439$	−4.54136	−0.216747	−0.108374	+	0.994110i	$0.534564\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.7444	−0.653016	−0.326508	−	0.945194i	$-0.605872\pi$
$443$	−13.7444	−0.653016	−0.326508	+	0.945194i	$0.605872\pi$
$444$	0	0
$445$	9.16784	0.434597
$446$	0	0
$447$	1.91223	0.0904453
$448$	0	0
$449$	21.5662	1.01777	0.508886	−	0.860834i	$-0.330057\pi$
$449$	21.5662	1.01777	0.508886	+	0.860834i	$0.330057\pi$
$450$	0	0
$451$	−1.28575	−0.0605435
$452$	0	0
$453$	31.4054	1.47555
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.30554	−0.154627	−0.0773133	−	0.997007i	$-0.524634\pi$
$457$	−3.30554	−0.154627	−0.0773133	+	0.997007i	$0.524634\pi$
$458$	0	0
$459$	−7.42345	−0.346497
$460$	0	0
$461$	−32.1524	−1.49749	−0.748744	−	0.662859i	$-0.769343\pi$
$461$	−32.1524	−1.49749	−0.748744	+	0.662859i	$0.769343\pi$
$462$	0	0
$463$	−5.82181	−0.270563	−0.135281	−	0.990807i	$-0.543194\pi$
$463$	−5.82181	−0.270563	−0.135281	+	0.990807i	$0.543194\pi$
$464$	0	0
$465$	48.3176	2.24068
$466$	0	0
$467$	6.01473	0.278329	0.139164	−	0.990269i	$-0.455558\pi$
$467$	6.01473	0.278329	0.139164	+	0.990269i	$0.455558\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	21.8968	1.00895
$472$	0	0
$473$	−1.59899	−0.0735216
$474$	0	0
$475$	−11.4586	−0.525759
$476$	0	0
$477$	6.05763	0.277360
$478$	0	0
$479$	−17.1351	−0.782921	−0.391460	−	0.920195i	$-0.628030\pi$
$479$	−17.1351	−0.782921	−0.391460	+	0.920195i	$0.628030\pi$
$480$	0	0
$481$	26.6714	1.21611
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−34.7488	−1.57786
$486$	0	0
$487$	−5.71425	−0.258937	−0.129469	−	0.991584i	$-0.541327\pi$
$487$	−5.71425	−0.258937	−0.129469	+	0.991584i	$0.541327\pi$
$488$	0	0
$489$	−17.0948	−0.773054
$490$	0	0
$491$	−24.0673	−1.08614	−0.543071	−	0.839687i	$-0.682739\pi$
$491$	−24.0673	−1.08614	−0.543071	+	0.839687i	$0.682739\pi$
$492$	0	0
$493$	−5.10250	−0.229805
$494$	0	0
$495$	−2.34338	−0.105327
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−10.7893	−0.482994	−0.241497	−	0.970402i	$-0.577638\pi$
$499$	−10.7893	−0.482994	−0.241497	+	0.970402i	$0.577638\pi$
$500$	0	0
$501$	−34.8365	−1.55638
$502$	0	0
$503$	−28.0121	−1.24900	−0.624499	−	0.781026i	$-0.714697\pi$
$503$	−28.0121	−1.24900	−0.624499	+	0.781026i	$0.714697\pi$
$504$	0	0
$505$	−6.61878	−0.294532
$506$	0	0
$507$	−41.9819	−1.86448
$508$	0	0
$509$	−1.91487	−0.0848753	−0.0424377	−	0.999099i	$-0.513512\pi$
$509$	−1.91487	−0.0848753	−0.0424377	+	0.999099i	$0.513512\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−6.63683	−0.293023
$514$	0	0
$515$	−11.3055	−0.498181
$516$	0	0
$517$	1.65662	0.0728581
$518$	0	0
$519$	−37.4054	−1.64191
$520$	0	0
$521$	1.57920	0.0691859	0.0345930	−	0.999401i	$-0.488987\pi$
$521$	1.57920	0.0691859	0.0345930	+	0.999401i	$0.488987\pi$
$522$	0	0
$523$	8.96986	0.392225	0.196112	−	0.980581i	$-0.437168\pi$
$523$	8.96986	0.392225	0.196112	+	0.980581i	$0.437168\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.7290	−0.510923
$528$	0	0
$529$	−11.8218	−0.513992
$530$	0	0
$531$	−5.81237	−0.252235
$532$	0	0
$533$	7.60163	0.329263
$534$	0	0
$535$	−17.0121	−0.735497
$536$	0	0
$537$	6.21074	0.268013
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.1025	0.778287	0.389144	−	0.921177i	$-0.372771\pi$
$541$	18.1025	0.778287	0.389144	+	0.921177i	$0.372771\pi$
$542$	0	0
$543$	19.7739	0.848577
$544$	0	0
$545$	53.1119	2.27507
$546$	0	0
$547$	−22.6885	−0.970090	−0.485045	−	0.874489i	$-0.661197\pi$
$547$	−22.6885	−0.970090	−0.485045	+	0.874489i	$0.661197\pi$
$548$	0	0
$549$	−4.39066	−0.187389
$550$	0	0
$551$	−4.56182	−0.194340
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−30.7866	−1.30682
$556$	0	0
$557$	38.3555	1.62517	0.812587	−	0.582840i	$-0.198059\pi$
$557$	38.3555	1.62517	0.812587	+	0.582840i	$0.198059\pi$
$558$	0	0
$559$	9.45359	0.399844
$560$	0	0
$561$	3.16784	0.133746
$562$	0	0
$563$	41.7739	1.76056	0.880279	−	0.474456i	$-0.157355\pi$
$563$	41.7739	1.76056	0.880279	+	0.474456i	$0.157355\pi$
$564$	0	0
$565$	−44.4080	−1.86826
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.8700	−0.497616	−0.248808	−	0.968553i	$-0.580039\pi$
$569$	−11.8700	−0.497616	−0.248808	+	0.968553i	$0.580039\pi$
$570$	0	0
$571$	−19.8013	−0.828661	−0.414330	−	0.910127i	$-0.635984\pi$
$571$	−19.8013	−0.828661	−0.414330	+	0.910127i	$0.635984\pi$
$572$	0	0
$573$	19.1652	0.800637
$574$	0	0
$575$	−25.8667	−1.07872
$576$	0	0
$577$	28.6791	1.19392	0.596962	−	0.802269i	$-0.296374\pi$
$577$	28.6791	1.19392	0.596962	+	0.802269i	$0.296374\pi$
$578$	0	0
$579$	−48.4278	−2.01259
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.22547	0.382080
$584$	0	0
$585$	13.8546	0.572817
$586$	0	0
$587$	−1.01209	−0.0417733	−0.0208866	−	0.999782i	$-0.506649\pi$
$587$	−1.01209	−0.0417733	−0.0208866	+	0.999782i	$0.506649\pi$
$588$	0	0
$589$	−10.4861	−0.432074
$590$	0	0
$591$	47.0044	1.93350
$592$	0	0
$593$	14.2332	0.584486	0.292243	−	0.956344i	$-0.405598\pi$
$593$	14.2332	0.584486	0.292243	+	0.956344i	$0.405598\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.6918	0.437587
$598$	0	0
$599$	26.5457	1.08463	0.542315	−	0.840175i	$-0.317548\pi$
$599$	26.5457	1.08463	0.542315	+	0.840175i	$0.317548\pi$
$600$	0	0
$601$	−12.1558	−0.495843	−0.247922	−	0.968780i	$-0.579748\pi$
$601$	−12.1558	−0.495843	−0.247922	+	0.968780i	$0.579748\pi$
$602$	0	0
$603$	6.45094	0.262703
$604$	0	0
$605$	−3.56885	−0.145094
$606$	0	0
$607$	−13.9672	−0.566912	−0.283456	−	0.958985i	$-0.591481\pi$
$607$	−13.9672	−0.566912	−0.283456	+	0.958985i	$0.591481\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.79432	−0.396236
$612$	0	0
$613$	4.64189	0.187484	0.0937421	−	0.995597i	$-0.470117\pi$
$613$	4.64189	0.187484	0.0937421	+	0.995597i	$0.470117\pi$
$614$	0	0
$615$	−8.77453	−0.353823
$616$	0	0
$617$	−26.3960	−1.06266	−0.531331	−	0.847165i	$-0.678308\pi$
$617$	−26.3960	−1.06266	−0.531331	+	0.847165i	$0.678308\pi$
$618$	0	0
$619$	−14.6815	−0.590098	−0.295049	−	0.955482i	$-0.595336\pi$
$619$	−14.6815	−0.590098	−0.295049	+	0.955482i	$0.595336\pi$
$620$	0	0
$621$	−14.9819	−0.601205
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.82710	−0.153084
$626$	0	0
$627$	2.83216	0.113106
$628$	0	0
$629$	7.47338	0.297983
$630$	0	0
$631$	30.1498	1.20024	0.600122	−	0.799908i	$-0.295119\pi$
$631$	30.1498	1.20024	0.600122	+	0.799908i	$0.295119\pi$
$632$	0	0
$633$	23.1377	0.919641
$634$	0	0
$635$	−23.6309	−0.937762
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.65927	0.223877
$640$	0	0
$641$	−16.1782	−0.639000	−0.319500	−	0.947586i	$-0.603515\pi$
$641$	−16.1782	−0.639000	−0.319500	+	0.947586i	$0.603515\pi$
$642$	0	0
$643$	2.33568	0.0921101	0.0460550	−	0.998939i	$-0.485335\pi$
$643$	2.33568	0.0921101	0.0460550	+	0.998939i	$0.485335\pi$
$644$	0	0
$645$	−10.9122	−0.429669
$646$	0	0
$647$	−14.9622	−0.588223	−0.294112	−	0.955771i	$-0.595024\pi$
$647$	−14.9622	−0.588223	−0.294112	+	0.955771i	$0.595024\pi$
$648$	0	0
$649$	−8.85195	−0.347470
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.8898	−0.895747	−0.447873	−	0.894097i	$-0.647818\pi$
$653$	−22.8898	−0.895747	−0.447873	+	0.894097i	$0.647818\pi$
$654$	0	0
$655$	−21.6188	−0.844716
$656$	0	0
$657$	2.99494	0.116844
$658$	0	0
$659$	2.20568	0.0859211	0.0429606	−	0.999077i	$-0.486321\pi$
$659$	2.20568	0.0859211	0.0429606	+	0.999077i	$0.486321\pi$
$660$	0	0
$661$	−0.682377	−0.0265414	−0.0132707	−	0.999912i	$-0.504224\pi$
$661$	−0.682377	−0.0265414	−0.0132707	+	0.999912i	$0.504224\pi$
$662$	0	0
$663$	−18.7290	−0.727373
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.2978	−0.398734
$668$	0	0
$669$	29.8814	1.15528
$670$	0	0
$671$	−6.68676	−0.258139
$672$	0	0
$673$	−10.8865	−0.419643	−0.209821	−	0.977740i	$-0.567288\pi$
$673$	−10.8865	−0.419643	−0.209821	+	0.977740i	$0.567288\pi$
$674$	0	0
$675$	34.6687	1.33440
$676$	0	0
$677$	45.6654	1.75506	0.877532	−	0.479519i	$-0.159189\pi$
$677$	45.6654	1.75506	0.877532	+	0.479519i	$0.159189\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−29.9622	−1.14815
$682$	0	0
$683$	−6.48372	−0.248093	−0.124046	−	0.992276i	$-0.539587\pi$
$683$	−6.48372	−0.248093	−0.124046	+	0.992276i	$0.539587\pi$
$684$	0	0
$685$	26.4932	1.01225
$686$	0	0
$687$	10.6540	0.406475
$688$	0	0
$689$	−54.5431	−2.07793
$690$	0	0
$691$	−10.9468	−0.416434	−0.208217	−	0.978083i	$-0.566766\pi$
$691$	−10.9468	−0.416434	−0.208217	+	0.978083i	$0.566766\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−38.6309	−1.46535
$696$	0	0
$697$	2.13000	0.0806793
$698$	0	0
$699$	36.8590	1.39413
$700$	0	0
$701$	−0.914874	−0.0345543	−0.0172772	−	0.999851i	$-0.505500\pi$
$701$	−0.914874	−0.0345543	−0.0172772	+	0.999851i	$0.505500\pi$
$702$	0	0
$703$	6.68147	0.251996
$704$	0	0
$705$	11.3055	0.425791
$706$	0	0
$707$	0	0
$708$	0	0
$709$	44.3123	1.66418	0.832092	−	0.554637i	$-0.187143\pi$
$709$	44.3123	1.66418	0.832092	+	0.554637i	$0.187143\pi$
$710$	0	0
$711$	4.19798	0.157436
$712$	0	0
$713$	−23.6714	−0.886499
$714$	0	0
$715$	21.0999	0.789090
$716$	0	0
$717$	−42.3702	−1.58234
$718$	0	0
$719$	5.20236	0.194015	0.0970076	−	0.995284i	$-0.469073\pi$
$719$	5.20236	0.194015	0.0970076	+	0.995284i	$0.469073\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	38.0044	1.41340
$724$	0	0
$725$	23.8295	0.885006
$726$	0	0
$727$	−50.0871	−1.85763	−0.928814	−	0.370547i	$-0.879170\pi$
$727$	−50.0871	−1.85763	−0.928814	+	0.370547i	$0.879170\pi$
$728$	0	0
$729$	18.7866	0.695801
$730$	0	0
$731$	2.64892	0.0979737
$732$	0	0
$733$	−47.0697	−1.73856	−0.869280	−	0.494320i	$-0.835417\pi$
$733$	−47.0697	−1.73856	−0.869280	+	0.494320i	$0.835417\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.82446	0.361889
$738$	0	0
$739$	−46.9914	−1.72861	−0.864303	−	0.502971i	$-0.832240\pi$
$739$	−46.9914	−1.72861	−0.864303	+	0.502971i	$0.832240\pi$
$740$	0	0
$741$	−16.7444	−0.615121
$742$	0	0
$743$	5.19533	0.190598	0.0952991	−	0.995449i	$-0.469619\pi$
$743$	5.19533	0.190598	0.0952991	+	0.995449i	$0.469619\pi$
$744$	0	0
$745$	3.56885	0.130753
$746$	0	0
$747$	−0.110206	−0.00403222
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−33.4536	−1.22074	−0.610369	−	0.792117i	$-0.708979\pi$
$751$	−33.4536	−1.22074	−0.610369	+	0.792117i	$0.708979\pi$
$752$	0	0
$753$	42.2747	1.54058
$754$	0	0
$755$	58.6128	2.13314
$756$	0	0
$757$	40.0440	1.45542	0.727711	−	0.685884i	$-0.240584\pi$
$757$	40.0440	1.45542	0.727711	+	0.685884i	$0.240584\pi$
$758$	0	0
$759$	6.39331	0.232062
$760$	0	0
$761$	−7.55850	−0.273995	−0.136998	−	0.990571i	$-0.543745\pi$
$761$	−7.55850	−0.273995	−0.136998	+	0.990571i	$0.543745\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.88209	0.140357
$766$	0	0
$767$	52.3348	1.88970
$768$	0	0
$769$	51.5407	1.85860	0.929302	−	0.369320i	$-0.120409\pi$
$769$	51.5407	1.85860	0.929302	+	0.369320i	$0.120409\pi$
$770$	0	0
$771$	−55.6834	−2.00539
$772$	0	0
$773$	−15.4657	−0.556262	−0.278131	−	0.960543i	$-0.589715\pi$
$773$	−15.4657	−0.556262	−0.278131	+	0.960543i	$0.589715\pi$
$774$	0	0
$775$	54.7763	1.96762
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.90429	0.0682284
$780$	0	0
$781$	8.61878	0.308404
$782$	0	0
$783$	13.8020	0.493244
$784$	0	0
$785$	40.8667	1.45859
$786$	0	0
$787$	−21.9518	−0.782497	−0.391249	−	0.920285i	$-0.627957\pi$
$787$	−21.9518	−0.782497	−0.391249	+	0.920285i	$0.627957\pi$
$788$	0	0
$789$	−29.6463	−1.05544
$790$	0	0
$791$	0	0
$792$	0	0
$793$	39.5337	1.40388
$794$	0	0
$795$	62.9588	2.23292
$796$	0	0
$797$	42.6258	1.50988	0.754942	−	0.655792i	$-0.227665\pi$
$797$	42.6258	1.50988	0.754942	+	0.655792i	$0.227665\pi$
$798$	0	0
$799$	−2.74439	−0.0970896
$800$	0	0
$801$	−1.68676	−0.0595987
$802$	0	0
$803$	4.56115	0.160959
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.26331	−0.114874
$808$	0	0
$809$	3.96745	0.139488	0.0697440	−	0.997565i	$-0.477782\pi$
$809$	3.96745	0.139488	0.0697440	+	0.997565i	$0.477782\pi$
$810$	0	0
$811$	46.9217	1.64764	0.823821	−	0.566850i	$-0.191838\pi$
$811$	46.9217	1.64764	0.823821	+	0.566850i	$0.191838\pi$
$812$	0	0
$813$	−39.2549	−1.37673
$814$	0	0
$815$	−31.9045	−1.11757
$816$	0	0
$817$	2.36823	0.0828538
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.6000	−1.76595	−0.882977	−	0.469416i	$-0.844464\pi$
$821$	−50.6000	−1.76595	−0.882977	+	0.469416i	$0.844464\pi$
$822$	0	0
$823$	−0.398599	−0.0138943	−0.00694714	−	0.999976i	$-0.502211\pi$
$823$	−0.398599	−0.0138943	−0.00694714	+	0.999976i	$0.502211\pi$
$824$	0	0
$825$	−14.7943	−0.515072
$826$	0	0
$827$	20.4234	0.710193	0.355096	−	0.934830i	$-0.384448\pi$
$827$	20.4234	0.710193	0.355096	+	0.934830i	$0.384448\pi$
$828$	0	0
$829$	−23.2633	−0.807968	−0.403984	−	0.914766i	$-0.632375\pi$
$829$	−23.2633	−0.807968	−0.403984	+	0.914766i	$0.632375\pi$
$830$	0	0
$831$	50.9819	1.76854
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−65.0165	−2.24999
$836$	0	0
$837$	31.7263	1.09662
$838$	0	0
$839$	16.4861	0.569165	0.284582	−	0.958652i	$-0.408145\pi$
$839$	16.4861	0.569165	0.284582	+	0.958652i	$0.408145\pi$
$840$	0	0
$841$	−19.5132	−0.672869
$842$	0	0
$843$	−30.1069	−1.03694
$844$	0	0
$845$	−78.3521	−2.69540
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−30.7290	−1.05462
$850$	0	0
$851$	15.0827	0.517029
$852$	0	0
$853$	−14.8315	−0.507820	−0.253910	−	0.967228i	$-0.581717\pi$
$853$	−14.8315	−0.507820	−0.253910	+	0.967228i	$0.581717\pi$
$854$	0	0
$855$	3.47073	0.118696
$856$	0	0
$857$	−5.03188	−0.171886	−0.0859428	−	0.996300i	$-0.527390\pi$
$857$	−5.03188	−0.171886	−0.0859428	+	0.996300i	$0.527390\pi$
$858$	0	0
$859$	56.0363	1.91193	0.955966	−	0.293477i	$-0.0948123\pi$
$859$	56.0363	1.91193	0.955966	+	0.293477i	$0.0948123\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.4760	1.24166	0.620829	−	0.783946i	$-0.286796\pi$
$863$	36.4760	1.24166	0.620829	+	0.783946i	$0.286796\pi$
$864$	0	0
$865$	−69.8108	−2.37364
$866$	0	0
$867$	27.2600	0.925798
$868$	0	0
$869$	6.39331	0.216878
$870$	0	0
$871$	−58.0844	−1.96812
$872$	0	0
$873$	6.39331	0.216381
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.61613	−0.223411	−0.111705	−	0.993741i	$-0.535631\pi$
$877$	−6.61613	−0.223411	−0.111705	+	0.993741i	$0.535631\pi$
$878$	0	0
$879$	−29.3253	−0.989119
$880$	0	0
$881$	−22.5286	−0.759008	−0.379504	−	0.925190i	$-0.623905\pi$
$881$	−22.5286	−0.759008	−0.379504	+	0.925190i	$0.623905\pi$
$882$	0	0
$883$	51.1652	1.72185	0.860923	−	0.508735i	$-0.169887\pi$
$883$	51.1652	1.72185	0.860923	+	0.508735i	$0.169887\pi$
$884$	0	0
$885$	−60.4098	−2.03065
$886$	0	0
$887$	6.33809	0.212812	0.106406	−	0.994323i	$-0.466066\pi$
$887$	6.33809	0.212812	0.106406	+	0.994323i	$0.466066\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−10.5387	−0.353060
$892$	0	0
$893$	−2.45359	−0.0821061
$894$	0	0
$895$	11.5913	0.387454
$896$	0	0
$897$	−37.7987	−1.26206
$898$	0	0
$899$	21.8071	0.727307
$900$	0	0
$901$	−15.2831	−0.509154
$902$	0	0
$903$	0	0
$904$	0	0
$905$	36.9045	1.22675
$906$	0	0
$907$	6.88474	0.228604	0.114302	−	0.993446i	$-0.463537\pi$
$907$	6.88474	0.228604	0.114302	+	0.993446i	$0.463537\pi$
$908$	0	0
$909$	1.21777	0.0403908
$910$	0	0
$911$	41.0818	1.36110	0.680550	−	0.732701i	$-0.261741\pi$
$911$	41.0818	1.36110	0.680550	+	0.732701i	$0.261741\pi$
$912$	0	0
$913$	−0.167838	−0.00555462
$914$	0	0
$915$	−45.6335	−1.50860
$916$	0	0
$917$	0	0
$918$	0	0
$919$	11.6265	0.383522	0.191761	−	0.981442i	$-0.438580\pi$
$919$	11.6265	0.383522	0.191761	+	0.981442i	$0.438580\pi$
$920$	0	0
$921$	−31.4958	−1.03782
$922$	0	0
$923$	−50.9562	−1.67724
$924$	0	0
$925$	−34.9019	−1.14757
$926$	0	0
$927$	2.08007	0.0683184
$928$	0	0
$929$	24.7668	0.812573	0.406287	−	0.913746i	$-0.366823\pi$
$929$	24.7668	0.812573	0.406287	+	0.913746i	$0.366823\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	41.3598	1.35406
$934$	0	0
$935$	5.91223	0.193351
$936$	0	0
$937$	1.15046	0.0375839	0.0187920	−	0.999823i	$-0.494018\pi$
$937$	1.15046	0.0375839	0.0187920	+	0.999823i	$0.494018\pi$
$938$	0	0
$939$	−31.3381	−1.02268
$940$	0	0
$941$	10.1102	0.329583	0.164792	−	0.986328i	$-0.447305\pi$
$941$	10.1102	0.329583	0.164792	+	0.986328i	$0.447305\pi$
$942$	0	0
$943$	4.29874	0.139986
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.4553	0.794691	0.397346	−	0.917669i	$-0.369931\pi$
$947$	24.4553	0.794691	0.397346	+	0.917669i	$0.369931\pi$
$948$	0	0
$949$	−26.9665	−0.875371
$950$	0	0
$951$	−8.53101	−0.276637
$952$	0	0
$953$	−16.1696	−0.523784	−0.261892	−	0.965097i	$-0.584346\pi$
$953$	−16.1696	−0.523784	−0.261892	+	0.965097i	$0.584346\pi$
$954$	0	0
$955$	35.7686	1.15744
$956$	0	0
$957$	−5.88979	−0.190390
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.1274	0.617011
$962$	0	0
$963$	3.13000	0.100863
$964$	0	0
$965$	−90.3823	−2.90951
$966$	0	0
$967$	1.55941	0.0501472	0.0250736	−	0.999686i	$-0.492018\pi$
$967$	1.55941	0.0501472	0.0250736	+	0.999686i	$0.492018\pi$
$968$	0	0
$969$	−4.69182	−0.150723
$970$	0	0
$971$	9.04728	0.290341	0.145171	−	0.989407i	$-0.453627\pi$
$971$	9.04728	0.290341	0.145171	+	0.989407i	$0.453627\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	87.4674	2.80120
$976$	0	0
$977$	−6.75648	−0.216159	−0.108079	−	0.994142i	$-0.534470\pi$
$977$	−6.75648	−0.216159	−0.108079	+	0.994142i	$0.534470\pi$
$978$	0	0
$979$	−2.56885	−0.0821008
$980$	0	0
$981$	−9.77188	−0.311992
$982$	0	0
$983$	−26.4305	−0.843001	−0.421501	−	0.906828i	$-0.638496\pi$
$983$	−26.4305	−0.843001	−0.421501	+	0.906828i	$0.638496\pi$
$984$	0	0
$985$	87.7257	2.79517
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.34602	0.169994
$990$	0	0
$991$	0.634185	0.0201456	0.0100728	−	0.999949i	$-0.496794\pi$
$991$	0.634185	0.0201456	0.0100728	+	0.999949i	$0.496794\pi$
$992$	0	0
$993$	36.4080	1.15537
$994$	0	0
$995$	19.9545	0.632599
$996$	0	0
$997$	10.1274	0.320736	0.160368	−	0.987057i	$-0.448732\pi$
$997$	10.1274	0.320736	0.160368	+	0.987057i	$0.448732\pi$
$998$	0	0
$999$	−20.2151	−0.639578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cl.1.1		3
4.3	odd	2		539.2.a.h.1.2		3
7.2	even	3		1232.2.q.k.529.3		6
7.4	even	3		1232.2.q.k.177.3		6
7.6	odd	2		8624.2.a.ck.1.3		3
12.11	even	2		4851.2.a.bo.1.2		3
28.3	even	6		539.2.e.l.177.2		6
28.11	odd	6		77.2.e.b.23.2	✓	6
28.19	even	6		539.2.e.l.67.2		6
28.23	odd	6		77.2.e.b.67.2	yes	6
28.27	even	2		539.2.a.i.1.2		3
44.43	even	2		5929.2.a.v.1.2		3
84.11	even	6		693.2.i.g.100.2		6
84.23	even	6		693.2.i.g.298.2		6
84.83	odd	2		4851.2.a.bn.1.2		3
308.39	even	30		847.2.n.d.366.2		24
308.51	even	30		847.2.n.d.753.2		24
308.79	even	30		847.2.n.d.81.2		24
308.95	even	30		847.2.n.d.632.2		24
308.107	even	30		847.2.n.d.130.2		24
308.123	even	30		847.2.n.d.807.2		24
308.135	odd	30		847.2.n.e.130.2		24
308.151	even	30		847.2.n.d.9.2		24
308.163	odd	30		847.2.n.e.81.2		24
308.179	odd	30		847.2.n.e.9.2		24
308.191	odd	30		847.2.n.e.753.2		24
308.207	odd	30		847.2.n.e.807.2		24
308.219	even	6		847.2.e.d.606.2		6
308.235	odd	30		847.2.n.e.632.2		24
308.247	odd	30		847.2.n.e.487.2		24
308.263	even	6		847.2.e.d.485.2		6
308.291	odd	30		847.2.n.e.366.2		24
308.303	even	30		847.2.n.d.487.2		24
308.307	odd	2		5929.2.a.w.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.b.23.2	✓	6	28.11	odd	6
77.2.e.b.67.2	yes	6	28.23	odd	6
539.2.a.h.1.2		3	4.3	odd	2
539.2.a.i.1.2		3	28.27	even	2
539.2.e.l.67.2		6	28.19	even	6
539.2.e.l.177.2		6	28.3	even	6
693.2.i.g.100.2		6	84.11	even	6
693.2.i.g.298.2		6	84.23	even	6
847.2.e.d.485.2		6	308.263	even	6
847.2.e.d.606.2		6	308.219	even	6
847.2.n.d.9.2		24	308.151	even	30
847.2.n.d.81.2		24	308.79	even	30
847.2.n.d.130.2		24	308.107	even	30
847.2.n.d.366.2		24	308.39	even	30
847.2.n.d.487.2		24	308.303	even	30
847.2.n.d.632.2		24	308.95	even	30
847.2.n.d.753.2		24	308.51	even	30
847.2.n.d.807.2		24	308.123	even	30
847.2.n.e.9.2		24	308.179	odd	30
847.2.n.e.81.2		24	308.163	odd	30
847.2.n.e.130.2		24	308.135	odd	30
847.2.n.e.366.2		24	308.291	odd	30
847.2.n.e.487.2		24	308.247	odd	30
847.2.n.e.632.2		24	308.235	odd	30
847.2.n.e.753.2		24	308.191	odd	30
847.2.n.e.807.2		24	308.207	odd	30
1232.2.q.k.177.3		6	7.4	even	3
1232.2.q.k.529.3		6	7.2	even	3
4851.2.a.bn.1.2		3	84.83	odd	2
4851.2.a.bo.1.2		3	12.11	even	2
5929.2.a.v.1.2		3	44.43	even	2
5929.2.a.w.1.2		3	308.307	odd	2
8624.2.a.ck.1.3		3	7.6	odd	2
8624.2.a.cl.1.1		3	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91223 q^{3} -3.56885 q^{5} +0.656620 q^{9} +O(q^{10})\)	\(q-1.91223 q^{3} -3.56885 q^{5} +0.656620 q^{9} +1.00000 q^{11} -5.91223 q^{13} +6.82446 q^{15} -1.65662 q^{17} -1.48108 q^{19} -3.34338 q^{23} +7.73669 q^{25} +4.48108 q^{27} +3.08007 q^{29} +7.08007 q^{31} -1.91223 q^{33} -4.51122 q^{37} +11.3055 q^{39} -1.28575 q^{41} -1.59899 q^{43} -2.34338 q^{45} +1.65662 q^{47} +3.16784 q^{51} +9.22547 q^{53} -3.56885 q^{55} +2.83216 q^{57} -8.85195 q^{59} -6.68676 q^{61} +21.0999 q^{65} +9.82446 q^{67} +6.39331 q^{69} +8.61878 q^{71} +4.56115 q^{73} -14.7943 q^{75} +6.39331 q^{79} -10.5387 q^{81} -0.167838 q^{83} +5.91223 q^{85} -5.88979 q^{87} -2.56885 q^{89} -13.5387 q^{93} +5.28575 q^{95} +9.73669 q^{97} +0.656620 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 2 q^{5}+O(q^{10}) \) 3 * q + q^3 - 2 * q^5	\( 3 q + q^{3} - 2 q^{5} + 3 q^{11} - 11 q^{13} + 7 q^{15} - 3 q^{17} + 11 q^{19} - 12 q^{23} + 3 q^{25} - 2 q^{27} - 9 q^{29} + 3 q^{31} + q^{33} - 4 q^{37} + 5 q^{39} - 5 q^{41} - 2 q^{43} - 9 q^{45} + 3 q^{47} - 2 q^{51} + 17 q^{53} - 2 q^{55} + 20 q^{57} - 8 q^{59} - 24 q^{61} + 15 q^{65} + 16 q^{67} - 3 q^{69} - 7 q^{71} - 20 q^{73} - 25 q^{75} - 3 q^{79} - 17 q^{81} + 11 q^{83} + 11 q^{85} - 30 q^{87} + q^{89} - 26 q^{93} + 17 q^{95} + 9 q^{97}+O(q^{100}) \) 3 * q + q^3 - 2 * q^5 + 3 * q^11 - 11 * q^13 + 7 * q^15 - 3 * q^17 + 11 * q^19 - 12 * q^23 + 3 * q^25 - 2 * q^27 - 9 * q^29 + 3 * q^31 + q^33 - 4 * q^37 + 5 * q^39 - 5 * q^41 - 2 * q^43 - 9 * q^45 + 3 * q^47 - 2 * q^51 + 17 * q^53 - 2 * q^55 + 20 * q^57 - 8 * q^59 - 24 * q^61 + 15 * q^65 + 16 * q^67 - 3 * q^69 - 7 * q^71 - 20 * q^73 - 25 * q^75 - 3 * q^79 - 17 * q^81 + 11 * q^83 + 11 * q^85 - 30 * q^87 + q^89 - 26 * q^93 + 17 * q^95 + 9 * q^97

Introduction

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