LMFDB - Embedded newform 7360.2.a.cl.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.876604.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 8x^{2} + 18x - 16 $ x^5 - x^4 - 9x^3 + 8x^2 + 18*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3680)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.77328$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82032 q^{3} -1.00000 q^{5} -2.35036 q^{7} +4.95418 q^{9} +O(q^{10})$	$q+2.82032 q^{3} -1.00000 q^{5} -2.35036 q^{7} +4.95418 q^{9} -0.877575 q^{11} -5.73813 q^{13} -2.82032 q^{15} +7.91903 q^{17} -2.90715 q^{19} -6.62875 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.51141 q^{27} +4.15880 q^{29} -1.44907 q^{31} -2.47504 q^{33} +2.35036 q^{35} +4.79660 q^{37} -16.1833 q^{39} -9.18211 q^{41} -9.52793 q^{43} -4.95418 q^{45} +5.19278 q^{47} -1.47582 q^{49} +22.3342 q^{51} +4.98977 q^{53} +0.877575 q^{55} -8.19907 q^{57} -1.34914 q^{59} -1.86735 q^{61} -11.6441 q^{63} +5.73813 q^{65} -14.7524 q^{67} -2.82032 q^{69} +0.871992 q^{71} -4.75798 q^{73} +2.82032 q^{75} +2.06262 q^{77} -14.7263 q^{79} +0.681369 q^{81} -5.15415 q^{83} -7.91903 q^{85} +11.7291 q^{87} -1.70347 q^{89} +13.4867 q^{91} -4.08683 q^{93} +2.90715 q^{95} +0.421708 q^{97} -4.34767 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9	$ 5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9} - 3 q^{11} - 7 q^{13} + q^{15} + 9 q^{17} - q^{19} - 20 q^{21} - 5 q^{23} + 5 q^{25} - 4 q^{27} + 10 q^{29} + 21 q^{31} + 7 q^{33} - q^{35} - 8 q^{37} - 24 q^{39} - 13 q^{41} + 6 q^{43} - 6 q^{45} + 24 q^{49} + 17 q^{51} + 6 q^{53} + 3 q^{55} + 26 q^{57} - 18 q^{59} + 11 q^{61} + 4 q^{63} + 7 q^{65} - 38 q^{67} + q^{69} - 21 q^{71} - 12 q^{73} - q^{75} - 46 q^{77} - 18 q^{79} + 9 q^{81} - 20 q^{83} - 9 q^{85} - 6 q^{87} - 16 q^{89} + 3 q^{91} - 22 q^{93} + q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9 - 3 * q^11 - 7 * q^13 + q^15 + 9 * q^17 - q^19 - 20 * q^21 - 5 * q^23 + 5 * q^25 - 4 * q^27 + 10 * q^29 + 21 * q^31 + 7 * q^33 - q^35 - 8 * q^37 - 24 * q^39 - 13 * q^41 + 6 * q^43 - 6 * q^45 + 24 * q^49 + 17 * q^51 + 6 * q^53 + 3 * q^55 + 26 * q^57 - 18 * q^59 + 11 * q^61 + 4 * q^63 + 7 * q^65 - 38 * q^67 + q^69 - 21 * q^71 - 12 * q^73 - q^75 - 46 * q^77 - 18 * q^79 + 9 * q^81 - 20 * q^83 - 9 * q^85 - 6 * q^87 - 16 * q^89 + 3 * q^91 - 22 * q^93 + q^95 + 29 * q^97 - 29 * 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
$3$	2.82032	1.62831	0.814155	−	0.580648i	$-0.197201\pi$
$3$	2.82032	1.62831	0.814155	+	0.580648i	$0.197201\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.35036	−0.888352	−0.444176	−	0.895940i	$-0.646504\pi$
$7$	−2.35036	−0.888352	−0.444176	+	0.895940i	$0.646504\pi$
$8$	0	0
$9$	4.95418	1.65139
$10$	0	0
$11$	−0.877575	−0.264599	−0.132299	−	0.991210i	$-0.542236\pi$
$11$	−0.877575	−0.264599	−0.132299	+	0.991210i	$0.542236\pi$
$12$	0	0
$13$	−5.73813	−1.59147	−0.795735	−	0.605645i	$-0.792915\pi$
$13$	−5.73813	−1.59147	−0.795735	+	0.605645i	$0.792915\pi$
$14$	0	0
$15$	−2.82032	−0.728202
$16$	0	0
$17$	7.91903	1.92065	0.960323	−	0.278890i	$-0.0899666\pi$
$17$	7.91903	1.92065	0.960323	+	0.278890i	$0.0899666\pi$
$18$	0	0
$19$	−2.90715	−0.666945	−0.333473	−	0.942760i	$-0.608221\pi$
$19$	−2.90715	−0.666945	−0.333473	+	0.942760i	$0.608221\pi$
$20$	0	0
$21$	−6.62875	−1.44651
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.51141	1.06067
$28$	0	0
$29$	4.15880	0.772269	0.386134	−	0.922443i	$-0.373810\pi$
$29$	4.15880	0.772269	0.386134	+	0.922443i	$0.373810\pi$
$30$	0	0
$31$	−1.44907	−0.260260	−0.130130	−	0.991497i	$-0.541540\pi$
$31$	−1.44907	−0.260260	−0.130130	+	0.991497i	$0.541540\pi$
$32$	0	0
$33$	−2.47504	−0.430849
$34$	0	0
$35$	2.35036	0.397283
$36$	0	0
$37$	4.79660	0.788556	0.394278	−	0.918991i	$-0.370995\pi$
$37$	4.79660	0.788556	0.394278	+	0.918991i	$0.370995\pi$
$38$	0	0
$39$	−16.1833	−2.59141
$40$	0	0
$41$	−9.18211	−1.43401	−0.717003	−	0.697070i	$-0.754486\pi$
$41$	−9.18211	−1.43401	−0.717003	+	0.697070i	$0.754486\pi$
$42$	0	0
$43$	−9.52793	−1.45300	−0.726498	−	0.687169i	$-0.758853\pi$
$43$	−9.52793	−1.45300	−0.726498	+	0.687169i	$0.758853\pi$
$44$	0	0
$45$	−4.95418	−0.738526
$46$	0	0
$47$	5.19278	0.757445	0.378722	−	0.925510i	$-0.376364\pi$
$47$	5.19278	0.757445	0.378722	+	0.925510i	$0.376364\pi$
$48$	0	0
$49$	−1.47582	−0.210831
$50$	0	0
$51$	22.3342	3.12741
$52$	0	0
$53$	4.98977	0.685398	0.342699	−	0.939445i	$-0.388659\pi$
$53$	4.98977	0.685398	0.342699	+	0.939445i	$0.388659\pi$
$54$	0	0
$55$	0.877575	0.118332
$56$	0	0
$57$	−8.19907	−1.08599
$58$	0	0
$59$	−1.34914	−0.175643	−0.0878217	−	0.996136i	$-0.527991\pi$
$59$	−1.34914	−0.175643	−0.0878217	+	0.996136i	$0.527991\pi$
$60$	0	0
$61$	−1.86735	−0.239090	−0.119545	−	0.992829i	$-0.538144\pi$
$61$	−1.86735	−0.239090	−0.119545	+	0.992829i	$0.538144\pi$
$62$	0	0
$63$	−11.6441	−1.46702
$64$	0	0
$65$	5.73813	0.711727
$66$	0	0
$67$	−14.7524	−1.80229	−0.901146	−	0.433516i	$-0.857273\pi$
$67$	−14.7524	−1.80229	−0.901146	+	0.433516i	$0.857273\pi$
$68$	0	0
$69$	−2.82032	−0.339526
$70$	0	0
$71$	0.871992	0.103486	0.0517432	−	0.998660i	$-0.483522\pi$
$71$	0.871992	0.103486	0.0517432	+	0.998660i	$0.483522\pi$
$72$	0	0
$73$	−4.75798	−0.556879	−0.278439	−	0.960454i	$-0.589817\pi$
$73$	−4.75798	−0.556879	−0.278439	+	0.960454i	$0.589817\pi$
$74$	0	0
$75$	2.82032	0.325662
$76$	0	0
$77$	2.06262	0.235057
$78$	0	0
$79$	−14.7263	−1.65684	−0.828419	−	0.560109i	$-0.810759\pi$
$79$	−14.7263	−1.65684	−0.828419	+	0.560109i	$0.810759\pi$
$80$	0	0
$81$	0.681369	0.0757077
$82$	0	0
$83$	−5.15415	−0.565742	−0.282871	−	0.959158i	$-0.591287\pi$
$83$	−5.15415	−0.565742	−0.282871	+	0.959158i	$0.591287\pi$
$84$	0	0
$85$	−7.91903	−0.858939
$86$	0	0
$87$	11.7291	1.25749
$88$	0	0
$89$	−1.70347	−0.180568	−0.0902839	−	0.995916i	$-0.528777\pi$
$89$	−1.70347	−0.180568	−0.0902839	+	0.995916i	$0.528777\pi$
$90$	0	0
$91$	13.4867	1.41379
$92$	0	0
$93$	−4.08683	−0.423785
$94$	0	0
$95$	2.90715	0.298267
$96$	0	0
$97$	0.421708	0.0428180	0.0214090	−	0.999771i	$-0.493185\pi$
$97$	0.421708	0.0428180	0.0214090	+	0.999771i	$0.493185\pi$
$98$	0	0
$99$	−4.34767	−0.436957
$100$	0	0
$101$	−8.21875	−0.817796	−0.408898	−	0.912580i	$-0.634087\pi$
$101$	−8.21875	−0.817796	−0.408898	+	0.912580i	$0.634087\pi$
$102$	0	0
$103$	11.5383	1.13691	0.568453	−	0.822716i	$-0.307542\pi$
$103$	11.5383	1.13691	0.568453	+	0.822716i	$0.307542\pi$
$104$	0	0
$105$	6.62875	0.646900
$106$	0	0
$107$	9.61957	0.929959	0.464979	−	0.885322i	$-0.346062\pi$
$107$	9.61957	0.929959	0.464979	+	0.885322i	$0.346062\pi$
$108$	0	0
$109$	−19.4174	−1.85985	−0.929924	−	0.367752i	$-0.880128\pi$
$109$	−19.4174	−1.85985	−0.929924	+	0.367752i	$0.880128\pi$
$110$	0	0
$111$	13.5279	1.28401
$112$	0	0
$113$	19.5701	1.84100	0.920498	−	0.390747i	$-0.127783\pi$
$113$	19.5701	1.84100	0.920498	+	0.390747i	$0.127783\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−28.4277	−2.62814
$118$	0	0
$119$	−18.6125	−1.70621
$120$	0	0
$121$	−10.2299	−0.929987
$122$	0	0
$123$	−25.8965	−2.33501
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.3562	−1.45138	−0.725690	−	0.688022i	$-0.758479\pi$
$127$	−16.3562	−1.45138	−0.725690	+	0.688022i	$0.758479\pi$
$128$	0	0
$129$	−26.8718	−2.36593
$130$	0	0
$131$	−20.3613	−1.77898	−0.889489	−	0.456956i	$-0.848940\pi$
$131$	−20.3613	−1.77898	−0.889489	+	0.456956i	$0.848940\pi$
$132$	0	0
$133$	6.83284	0.592482
$134$	0	0
$135$	−5.51141	−0.474347
$136$	0	0
$137$	−10.1367	−0.866040	−0.433020	−	0.901384i	$-0.642552\pi$
$137$	−10.1367	−0.866040	−0.433020	+	0.901384i	$0.642552\pi$
$138$	0	0
$139$	15.4113	1.30717	0.653583	−	0.756855i	$-0.273265\pi$
$139$	15.4113	1.30717	0.653583	+	0.756855i	$0.273265\pi$
$140$	0	0
$141$	14.6453	1.23335
$142$	0	0
$143$	5.03564	0.421101
$144$	0	0
$145$	−4.15880	−0.345369
$146$	0	0
$147$	−4.16227	−0.343298
$148$	0	0
$149$	−2.08444	−0.170764	−0.0853821	−	0.996348i	$-0.527211\pi$
$149$	−2.08444	−0.170764	−0.0853821	+	0.996348i	$0.527211\pi$
$150$	0	0
$151$	−1.22208	−0.0994511	−0.0497255	−	0.998763i	$-0.515835\pi$
$151$	−1.22208	−0.0994511	−0.0497255	+	0.998763i	$0.515835\pi$
$152$	0	0
$153$	39.2323	3.17174
$154$	0	0
$155$	1.44907	0.116392
$156$	0	0
$157$	−20.4610	−1.63296	−0.816482	−	0.577370i	$-0.804079\pi$
$157$	−20.4610	−1.63296	−0.816482	+	0.577370i	$0.804079\pi$
$158$	0	0
$159$	14.0727	1.11604
$160$	0	0
$161$	2.35036	0.185234
$162$	0	0
$163$	−6.95512	−0.544767	−0.272384	−	0.962189i	$-0.587812\pi$
$163$	−6.95512	−0.544767	−0.272384	+	0.962189i	$0.587812\pi$
$164$	0	0
$165$	2.47504	0.192681
$166$	0	0
$167$	0.646347	0.0500158	0.0250079	−	0.999687i	$-0.492039\pi$
$167$	0.646347	0.0500158	0.0250079	+	0.999687i	$0.492039\pi$
$168$	0	0
$169$	19.9261	1.53278
$170$	0	0
$171$	−14.4025	−1.10139
$172$	0	0
$173$	−5.76581	−0.438367	−0.219183	−	0.975684i	$-0.570339\pi$
$173$	−5.76581	−0.438367	−0.219183	+	0.975684i	$0.570339\pi$
$174$	0	0
$175$	−2.35036	−0.177670
$176$	0	0
$177$	−3.80501	−0.286002
$178$	0	0
$179$	12.8409	0.959772	0.479886	−	0.877331i	$-0.340678\pi$
$179$	12.8409	0.959772	0.479886	+	0.877331i	$0.340678\pi$
$180$	0	0
$181$	−4.45371	−0.331042	−0.165521	−	0.986206i	$-0.552931\pi$
$181$	−4.45371	−0.331042	−0.165521	+	0.986206i	$0.552931\pi$
$182$	0	0
$183$	−5.26652	−0.389312
$184$	0	0
$185$	−4.79660	−0.352653
$186$	0	0
$187$	−6.94954	−0.508201
$188$	0	0
$189$	−12.9538	−0.942249
$190$	0	0
$191$	11.2338	0.812852	0.406426	−	0.913684i	$-0.366775\pi$
$191$	11.2338	0.812852	0.406426	+	0.913684i	$0.366775\pi$
$192$	0	0
$193$	−20.3568	−1.46532	−0.732658	−	0.680597i	$-0.761721\pi$
$193$	−20.3568	−1.46532	−0.732658	+	0.680597i	$0.761721\pi$
$194$	0	0
$195$	16.1833	1.15891
$196$	0	0
$197$	8.68151	0.618532	0.309266	−	0.950976i	$-0.399917\pi$
$197$	8.68151	0.618532	0.309266	+	0.950976i	$0.399917\pi$
$198$	0	0
$199$	1.90089	0.134751	0.0673754	−	0.997728i	$-0.478537\pi$
$199$	1.90089	0.134751	0.0673754	+	0.997728i	$0.478537\pi$
$200$	0	0
$201$	−41.6064	−2.93469
$202$	0	0
$203$	−9.77466	−0.686047
$204$	0	0
$205$	9.18211	0.641307
$206$	0	0
$207$	−4.95418	−0.344339
$208$	0	0
$209$	2.55124	0.176473
$210$	0	0
$211$	5.06937	0.348990	0.174495	−	0.984658i	$-0.444171\pi$
$211$	5.06937	0.348990	0.174495	+	0.984658i	$0.444171\pi$
$212$	0	0
$213$	2.45929	0.168508
$214$	0	0
$215$	9.52793	0.649799
$216$	0	0
$217$	3.40583	0.231203
$218$	0	0
$219$	−13.4190	−0.906772
$220$	0	0
$221$	−45.4404	−3.05665
$222$	0	0
$223$	−24.7124	−1.65487	−0.827433	−	0.561564i	$-0.810200\pi$
$223$	−24.7124	−1.65487	−0.827433	+	0.561564i	$0.810200\pi$
$224$	0	0
$225$	4.95418	0.330279
$226$	0	0
$227$	−15.5676	−1.03326	−0.516630	−	0.856209i	$-0.672814\pi$
$227$	−15.5676	−1.03326	−0.516630	+	0.856209i	$0.672814\pi$
$228$	0	0
$229$	2.01198	0.132955	0.0664776	−	0.997788i	$-0.478824\pi$
$229$	2.01198	0.132955	0.0664776	+	0.997788i	$0.478824\pi$
$230$	0	0
$231$	5.81723	0.382745
$232$	0	0
$233$	8.32587	0.545446	0.272723	−	0.962093i	$-0.412076\pi$
$233$	8.32587	0.545446	0.272723	+	0.962093i	$0.412076\pi$
$234$	0	0
$235$	−5.19278	−0.338739
$236$	0	0
$237$	−41.5328	−2.69784
$238$	0	0
$239$	11.3478	0.734029	0.367014	−	0.930215i	$-0.380380\pi$
$239$	11.3478	0.734029	0.367014	+	0.930215i	$0.380380\pi$
$240$	0	0
$241$	30.9336	1.99261	0.996305	−	0.0858912i	$-0.0273737\pi$
$241$	30.9336	1.99261	0.996305	+	0.0858912i	$0.0273737\pi$
$242$	0	0
$243$	−14.6125	−0.937396
$244$	0	0
$245$	1.47582	0.0942864
$246$	0	0
$247$	16.6816	1.06142
$248$	0	0
$249$	−14.5363	−0.921203
$250$	0	0
$251$	2.68093	0.169219	0.0846094	−	0.996414i	$-0.473036\pi$
$251$	2.68093	0.169219	0.0846094	+	0.996414i	$0.473036\pi$
$252$	0	0
$253$	0.877575	0.0551727
$254$	0	0
$255$	−22.3342	−1.39862
$256$	0	0
$257$	4.14890	0.258801	0.129401	−	0.991592i	$-0.458695\pi$
$257$	4.14890	0.258801	0.129401	+	0.991592i	$0.458695\pi$
$258$	0	0
$259$	−11.2737	−0.700516
$260$	0	0
$261$	20.6034	1.27532
$262$	0	0
$263$	13.3018	0.820226	0.410113	−	0.912035i	$-0.365489\pi$
$263$	13.3018	0.820226	0.410113	+	0.912035i	$0.365489\pi$
$264$	0	0
$265$	−4.98977	−0.306519
$266$	0	0
$267$	−4.80433	−0.294020
$268$	0	0
$269$	31.2994	1.90836	0.954180	−	0.299233i	$-0.0967306\pi$
$269$	31.2994	1.90836	0.954180	+	0.299233i	$0.0967306\pi$
$270$	0	0
$271$	15.3002	0.929419	0.464710	−	0.885463i	$-0.346159\pi$
$271$	15.3002	0.929419	0.464710	+	0.885463i	$0.346159\pi$
$272$	0	0
$273$	38.0366	2.30208
$274$	0	0
$275$	−0.877575	−0.0529198
$276$	0	0
$277$	−7.65125	−0.459719	−0.229860	−	0.973224i	$-0.573827\pi$
$277$	−7.65125	−0.459719	−0.229860	+	0.973224i	$0.573827\pi$
$278$	0	0
$279$	−7.17895	−0.429792
$280$	0	0
$281$	−6.69568	−0.399431	−0.199715	−	0.979854i	$-0.564002\pi$
$281$	−6.69568	−0.399431	−0.199715	+	0.979854i	$0.564002\pi$
$282$	0	0
$283$	−1.26867	−0.0754147	−0.0377073	−	0.999289i	$-0.512005\pi$
$283$	−1.26867	−0.0754147	−0.0377073	+	0.999289i	$0.512005\pi$
$284$	0	0
$285$	8.19907	0.485671
$286$	0	0
$287$	21.5813	1.27390
$288$	0	0
$289$	45.7110	2.68888
$290$	0	0
$291$	1.18935	0.0697210
$292$	0	0
$293$	7.99071	0.466823	0.233411	−	0.972378i	$-0.425011\pi$
$293$	7.99071	0.466823	0.233411	+	0.972378i	$0.425011\pi$
$294$	0	0
$295$	1.34914	0.0785501
$296$	0	0
$297$	−4.83667	−0.280652
$298$	0	0
$299$	5.73813	0.331844
$300$	0	0
$301$	22.3941	1.29077
$302$	0	0
$303$	−23.1795	−1.33163
$304$	0	0
$305$	1.86735	0.106924
$306$	0	0
$307$	−20.2081	−1.15334	−0.576668	−	0.816978i	$-0.695647\pi$
$307$	−20.2081	−1.15334	−0.576668	+	0.816978i	$0.695647\pi$
$308$	0	0
$309$	32.5417	1.85123
$310$	0	0
$311$	4.80306	0.272357	0.136178	−	0.990684i	$-0.456518\pi$
$311$	4.80306	0.272357	0.136178	+	0.990684i	$0.456518\pi$
$312$	0	0
$313$	−11.0393	−0.623978	−0.311989	−	0.950086i	$-0.600995\pi$
$313$	−11.0393	−0.623978	−0.311989	+	0.950086i	$0.600995\pi$
$314$	0	0
$315$	11.6441	0.656071
$316$	0	0
$317$	−14.8001	−0.831256	−0.415628	−	0.909535i	$-0.636438\pi$
$317$	−14.8001	−0.831256	−0.415628	+	0.909535i	$0.636438\pi$
$318$	0	0
$319$	−3.64965	−0.204341
$320$	0	0
$321$	27.1302	1.51426
$322$	0	0
$323$	−23.0218	−1.28097
$324$	0	0
$325$	−5.73813	−0.318294
$326$	0	0
$327$	−54.7632	−3.02841
$328$	0	0
$329$	−12.2049	−0.672877
$330$	0	0
$331$	−34.3899	−1.89024	−0.945119	−	0.326726i	$-0.894055\pi$
$331$	−34.3899	−1.89024	−0.945119	+	0.326726i	$0.894055\pi$
$332$	0	0
$333$	23.7632	1.30222
$334$	0	0
$335$	14.7524	0.806009
$336$	0	0
$337$	23.6951	1.29075	0.645377	−	0.763865i	$-0.276700\pi$
$337$	23.6951	1.29075	0.645377	+	0.763865i	$0.276700\pi$
$338$	0	0
$339$	55.1937	2.99771
$340$	0	0
$341$	1.27167	0.0688646
$342$	0	0
$343$	19.9212	1.07564
$344$	0	0
$345$	2.82032	0.151841
$346$	0	0
$347$	−13.1970	−0.708451	−0.354226	−	0.935160i	$-0.615255\pi$
$347$	−13.1970	−0.708451	−0.354226	+	0.935160i	$0.615255\pi$
$348$	0	0
$349$	−14.9130	−0.798275	−0.399137	−	0.916891i	$-0.630690\pi$
$349$	−14.9130	−0.798275	−0.399137	+	0.916891i	$0.630690\pi$
$350$	0	0
$351$	−31.6252	−1.68803
$352$	0	0
$353$	7.84633	0.417618	0.208809	−	0.977956i	$-0.433041\pi$
$353$	7.84633	0.417618	0.208809	+	0.977956i	$0.433041\pi$
$354$	0	0
$355$	−0.871992	−0.0462806
$356$	0	0
$357$	−52.4933	−2.77824
$358$	0	0
$359$	−17.8125	−0.940107	−0.470053	−	0.882638i	$-0.655765\pi$
$359$	−17.8125	−0.940107	−0.470053	+	0.882638i	$0.655765\pi$
$360$	0	0
$361$	−10.5485	−0.555184
$362$	0	0
$363$	−28.8514	−1.51431
$364$	0	0
$365$	4.75798	0.249044
$366$	0	0
$367$	−19.9627	−1.04205	−0.521023	−	0.853543i	$-0.674449\pi$
$367$	−19.9627	−1.04205	−0.521023	+	0.853543i	$0.674449\pi$
$368$	0	0
$369$	−45.4899	−2.36811
$370$	0	0
$371$	−11.7278	−0.608875
$372$	0	0
$373$	30.8089	1.59523	0.797613	−	0.603170i	$-0.206096\pi$
$373$	30.8089	1.59523	0.797613	+	0.603170i	$0.206096\pi$
$374$	0	0
$375$	−2.82032	−0.145640
$376$	0	0
$377$	−23.8637	−1.22904
$378$	0	0
$379$	4.77504	0.245277	0.122639	−	0.992451i	$-0.460864\pi$
$379$	4.77504	0.245277	0.122639	+	0.992451i	$0.460864\pi$
$380$	0	0
$381$	−46.1297	−2.36330
$382$	0	0
$383$	3.73288	0.190741	0.0953707	−	0.995442i	$-0.469596\pi$
$383$	3.73288	0.190741	0.0953707	+	0.995442i	$0.469596\pi$
$384$	0	0
$385$	−2.06262	−0.105121
$386$	0	0
$387$	−47.2031	−2.39947
$388$	0	0
$389$	34.5417	1.75133	0.875667	−	0.482916i	$-0.160422\pi$
$389$	34.5417	1.75133	0.875667	+	0.482916i	$0.160422\pi$
$390$	0	0
$391$	−7.91903	−0.400482
$392$	0	0
$393$	−57.4254	−2.89673
$394$	0	0
$395$	14.7263	0.740960
$396$	0	0
$397$	−7.10425	−0.356552	−0.178276	−	0.983981i	$-0.557052\pi$
$397$	−7.10425	−0.356552	−0.178276	+	0.983981i	$0.557052\pi$
$398$	0	0
$399$	19.2708	0.964745
$400$	0	0
$401$	13.5684	0.677576	0.338788	−	0.940863i	$-0.389983\pi$
$401$	13.5684	0.677576	0.338788	+	0.940863i	$0.389983\pi$
$402$	0	0
$403$	8.31494	0.414197
$404$	0	0
$405$	−0.681369	−0.0338575
$406$	0	0
$407$	−4.20938	−0.208651
$408$	0	0
$409$	−7.75354	−0.383388	−0.191694	−	0.981455i	$-0.561398\pi$
$409$	−7.75354	−0.383388	−0.191694	+	0.981455i	$0.561398\pi$
$410$	0	0
$411$	−28.5888	−1.41018
$412$	0	0
$413$	3.17097	0.156033
$414$	0	0
$415$	5.15415	0.253007
$416$	0	0
$417$	43.4646	2.12847
$418$	0	0
$419$	23.8708	1.16617	0.583083	−	0.812412i	$-0.301846\pi$
$419$	23.8708	1.16617	0.583083	+	0.812412i	$0.301846\pi$
$420$	0	0
$421$	−24.8597	−1.21159	−0.605794	−	0.795621i	$-0.707145\pi$
$421$	−24.8597	−1.21159	−0.605794	+	0.795621i	$0.707145\pi$
$422$	0	0
$423$	25.7260	1.25084
$424$	0	0
$425$	7.91903	0.384129
$426$	0	0
$427$	4.38894	0.212396
$428$	0	0
$429$	14.2021	0.685683
$430$	0	0
$431$	0.792091	0.0381537	0.0190768	−	0.999818i	$-0.493927\pi$
$431$	0.792091	0.0381537	0.0190768	+	0.999818i	$0.493927\pi$
$432$	0	0
$433$	16.8282	0.808710	0.404355	−	0.914602i	$-0.367496\pi$
$433$	16.8282	0.808710	0.404355	+	0.914602i	$0.367496\pi$
$434$	0	0
$435$	−11.7291	−0.562368
$436$	0	0
$437$	2.90715	0.139068
$438$	0	0
$439$	−11.1568	−0.532485	−0.266242	−	0.963906i	$-0.585782\pi$
$439$	−11.1568	−0.532485	−0.266242	+	0.963906i	$0.585782\pi$
$440$	0	0
$441$	−7.31146	−0.348165
$442$	0	0
$443$	−18.4575	−0.876941	−0.438471	−	0.898746i	$-0.644480\pi$
$443$	−18.4575	−0.876941	−0.438471	+	0.898746i	$0.644480\pi$
$444$	0	0
$445$	1.70347	0.0807524
$446$	0	0
$447$	−5.87879	−0.278057
$448$	0	0
$449$	8.63681	0.407596	0.203798	−	0.979013i	$-0.434671\pi$
$449$	8.63681	0.407596	0.203798	+	0.979013i	$0.434671\pi$
$450$	0	0
$451$	8.05799	0.379436
$452$	0	0
$453$	−3.44664	−0.161937
$454$	0	0
$455$	−13.4867	−0.632264
$456$	0	0
$457$	9.82303	0.459502	0.229751	−	0.973249i	$-0.426209\pi$
$457$	9.82303	0.459502	0.229751	+	0.973249i	$0.426209\pi$
$458$	0	0
$459$	43.6450	2.03717
$460$	0	0
$461$	−0.114588	−0.00533688	−0.00266844	−	0.999996i	$-0.500849\pi$
$461$	−0.114588	−0.00533688	−0.00266844	+	0.999996i	$0.500849\pi$
$462$	0	0
$463$	24.4551	1.13652	0.568262	−	0.822848i	$-0.307616\pi$
$463$	24.4551	1.13652	0.568262	+	0.822848i	$0.307616\pi$
$464$	0	0
$465$	4.08683	0.189522
$466$	0	0
$467$	−14.3370	−0.663439	−0.331720	−	0.943378i	$-0.607629\pi$
$467$	−14.3370	−0.663439	−0.331720	+	0.943378i	$0.607629\pi$
$468$	0	0
$469$	34.6734	1.60107
$470$	0	0
$471$	−57.7065	−2.65897
$472$	0	0
$473$	8.36147	0.384461
$474$	0	0
$475$	−2.90715	−0.133389
$476$	0	0
$477$	24.7202	1.13186
$478$	0	0
$479$	−12.0033	−0.548445	−0.274223	−	0.961666i	$-0.588420\pi$
$479$	−12.0033	−0.548445	−0.274223	+	0.961666i	$0.588420\pi$
$480$	0	0
$481$	−27.5235	−1.25496
$482$	0	0
$483$	6.62875	0.301619
$484$	0	0
$485$	−0.421708	−0.0191488
$486$	0	0
$487$	−18.4391	−0.835556	−0.417778	−	0.908549i	$-0.637191\pi$
$487$	−18.4391	−0.835556	−0.417778	+	0.908549i	$0.637191\pi$
$488$	0	0
$489$	−19.6156	−0.887050
$490$	0	0
$491$	−26.3680	−1.18997	−0.594985	−	0.803736i	$-0.702842\pi$
$491$	−26.3680	−1.18997	−0.594985	+	0.803736i	$0.702842\pi$
$492$	0	0
$493$	32.9336	1.48326
$494$	0	0
$495$	4.34767	0.195413
$496$	0	0
$497$	−2.04949	−0.0919324
$498$	0	0
$499$	−31.8584	−1.42618	−0.713089	−	0.701074i	$-0.752704\pi$
$499$	−31.8584	−1.42618	−0.713089	+	0.701074i	$0.752704\pi$
$500$	0	0
$501$	1.82290	0.0814413
$502$	0	0
$503$	16.6509	0.742426	0.371213	−	0.928548i	$-0.378942\pi$
$503$	16.6509	0.742426	0.371213	+	0.928548i	$0.378942\pi$
$504$	0	0
$505$	8.21875	0.365729
$506$	0	0
$507$	56.1979	2.49584
$508$	0	0
$509$	21.5639	0.955801	0.477901	−	0.878414i	$-0.341398\pi$
$509$	21.5639	0.955801	0.477901	+	0.878414i	$0.341398\pi$
$510$	0	0
$511$	11.1829	0.494705
$512$	0	0
$513$	−16.0225	−0.707410
$514$	0	0
$515$	−11.5383	−0.508440
$516$	0	0
$517$	−4.55705	−0.200419
$518$	0	0
$519$	−16.2614	−0.713797
$520$	0	0
$521$	29.4732	1.29124	0.645622	−	0.763658i	$-0.276598\pi$
$521$	29.4732	1.29124	0.645622	+	0.763658i	$0.276598\pi$
$522$	0	0
$523$	−19.2026	−0.839671	−0.419836	−	0.907600i	$-0.637912\pi$
$523$	−19.2026	−0.839671	−0.419836	+	0.907600i	$0.637912\pi$
$524$	0	0
$525$	−6.62875	−0.289302
$526$	0	0
$527$	−11.4752	−0.499868
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.68390	−0.290057
$532$	0	0
$533$	52.6881	2.28218
$534$	0	0
$535$	−9.61957	−0.415890
$536$	0	0
$537$	36.2153	1.56281
$538$	0	0
$539$	1.29514	0.0557856
$540$	0	0
$541$	−17.8485	−0.767367	−0.383683	−	0.923465i	$-0.625345\pi$
$541$	−17.8485	−0.767367	−0.383683	+	0.923465i	$0.625345\pi$
$542$	0	0
$543$	−12.5609	−0.539039
$544$	0	0
$545$	19.4174	0.831749
$546$	0	0
$547$	18.4080	0.787068	0.393534	−	0.919310i	$-0.371252\pi$
$547$	18.4080	0.787068	0.393534	+	0.919310i	$0.371252\pi$
$548$	0	0
$549$	−9.25119	−0.394831
$550$	0	0
$551$	−12.0902	−0.515061
$552$	0	0
$553$	34.6121	1.47185
$554$	0	0
$555$	−13.5279	−0.574229
$556$	0	0
$557$	42.0835	1.78314	0.891568	−	0.452887i	$-0.149606\pi$
$557$	42.0835	1.78314	0.891568	+	0.452887i	$0.149606\pi$
$558$	0	0
$559$	54.6725	2.31240
$560$	0	0
$561$	−19.5999	−0.827508
$562$	0	0
$563$	27.5152	1.15963	0.579813	−	0.814750i	$-0.303126\pi$
$563$	27.5152	1.15963	0.579813	+	0.814750i	$0.303126\pi$
$564$	0	0
$565$	−19.5701	−0.823319
$566$	0	0
$567$	−1.60146	−0.0672551
$568$	0	0
$569$	30.5878	1.28231	0.641154	−	0.767412i	$-0.278456\pi$
$569$	30.5878	1.28231	0.641154	+	0.767412i	$0.278456\pi$
$570$	0	0
$571$	21.8061	0.912555	0.456278	−	0.889837i	$-0.349182\pi$
$571$	21.8061	0.912555	0.456278	+	0.889837i	$0.349182\pi$
$572$	0	0
$573$	31.6830	1.32357
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	33.1609	1.38051	0.690253	−	0.723568i	$-0.257499\pi$
$577$	33.1609	1.38051	0.690253	+	0.723568i	$0.257499\pi$
$578$	0	0
$579$	−57.4127	−2.38599
$580$	0	0
$581$	12.1141	0.502578
$582$	0	0
$583$	−4.37890	−0.181356
$584$	0	0
$585$	28.4277	1.17534
$586$	0	0
$587$	8.66288	0.357555	0.178778	−	0.983890i	$-0.442786\pi$
$587$	8.66288	0.357555	0.178778	+	0.983890i	$0.442786\pi$
$588$	0	0
$589$	4.21266	0.173580
$590$	0	0
$591$	24.4846	1.00716
$592$	0	0
$593$	−9.80864	−0.402793	−0.201396	−	0.979510i	$-0.564548\pi$
$593$	−9.80864	−0.402793	−0.201396	+	0.979510i	$0.564548\pi$
$594$	0	0
$595$	18.6125	0.763040
$596$	0	0
$597$	5.36112	0.219416
$598$	0	0
$599$	−38.6203	−1.57798	−0.788991	−	0.614404i	$-0.789396\pi$
$599$	−38.6203	−1.57798	−0.788991	+	0.614404i	$0.789396\pi$
$600$	0	0
$601$	38.0251	1.55108	0.775539	−	0.631300i	$-0.217478\pi$
$601$	38.0251	1.55108	0.775539	+	0.631300i	$0.217478\pi$
$602$	0	0
$603$	−73.0860	−2.97629
$604$	0	0
$605$	10.2299	0.415903
$606$	0	0
$607$	−43.1194	−1.75016	−0.875081	−	0.483976i	$-0.839192\pi$
$607$	−43.1194	−1.75016	−0.875081	+	0.483976i	$0.839192\pi$
$608$	0	0
$609$	−27.5676	−1.11710
$610$	0	0
$611$	−29.7968	−1.20545
$612$	0	0
$613$	23.1078	0.933315	0.466658	−	0.884438i	$-0.345458\pi$
$613$	23.1078	0.933315	0.466658	+	0.884438i	$0.345458\pi$
$614$	0	0
$615$	25.8965	1.04425
$616$	0	0
$617$	27.0402	1.08860	0.544298	−	0.838892i	$-0.316796\pi$
$617$	27.0402	1.08860	0.544298	+	0.838892i	$0.316796\pi$
$618$	0	0
$619$	35.4148	1.42344	0.711720	−	0.702463i	$-0.247916\pi$
$619$	35.4148	1.42344	0.711720	+	0.702463i	$0.247916\pi$
$620$	0	0
$621$	−5.51141	−0.221165
$622$	0	0
$623$	4.00377	0.160408
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.19530	0.287353
$628$	0	0
$629$	37.9844	1.51454
$630$	0	0
$631$	36.3867	1.44853	0.724266	−	0.689520i	$-0.242179\pi$
$631$	36.3867	1.44853	0.724266	+	0.689520i	$0.242179\pi$
$632$	0	0
$633$	14.2972	0.568264
$634$	0	0
$635$	16.3562	0.649077
$636$	0	0
$637$	8.46842	0.335531
$638$	0	0
$639$	4.32001	0.170897
$640$	0	0
$641$	−30.2166	−1.19348	−0.596742	−	0.802433i	$-0.703538\pi$
$641$	−30.2166	−1.19348	−0.596742	+	0.802433i	$0.703538\pi$
$642$	0	0
$643$	37.8506	1.49268	0.746342	−	0.665563i	$-0.231808\pi$
$643$	37.8506	1.49268	0.746342	+	0.665563i	$0.231808\pi$
$644$	0	0
$645$	26.8718	1.05808
$646$	0	0
$647$	−6.62132	−0.260311	−0.130155	−	0.991494i	$-0.541548\pi$
$647$	−6.62132	−0.260311	−0.130155	+	0.991494i	$0.541548\pi$
$648$	0	0
$649$	1.18397	0.0464750
$650$	0	0
$651$	9.60552	0.376470
$652$	0	0
$653$	27.1215	1.06135	0.530674	−	0.847576i	$-0.321939\pi$
$653$	27.1215	1.06135	0.530674	+	0.847576i	$0.321939\pi$
$654$	0	0
$655$	20.3613	0.795583
$656$	0	0
$657$	−23.5719	−0.919626
$658$	0	0
$659$	−2.90499	−0.113162	−0.0565812	−	0.998398i	$-0.518020\pi$
$659$	−2.90499	−0.113162	−0.0565812	+	0.998398i	$0.518020\pi$
$660$	0	0
$661$	2.89070	0.112435	0.0562175	−	0.998419i	$-0.482096\pi$
$661$	2.89070	0.112435	0.0562175	+	0.998419i	$0.482096\pi$
$662$	0	0
$663$	−128.156	−4.97717
$664$	0	0
$665$	−6.83284	−0.264966
$666$	0	0
$667$	−4.15880	−0.161029
$668$	0	0
$669$	−69.6969	−2.69464
$670$	0	0
$671$	1.63874	0.0632628
$672$	0	0
$673$	−7.38695	−0.284746	−0.142373	−	0.989813i	$-0.545473\pi$
$673$	−7.38695	−0.284746	−0.142373	+	0.989813i	$0.545473\pi$
$674$	0	0
$675$	5.51141	0.212134
$676$	0	0
$677$	−20.5240	−0.788803	−0.394401	−	0.918938i	$-0.629048\pi$
$677$	−20.5240	−0.788803	−0.394401	+	0.918938i	$0.629048\pi$
$678$	0	0
$679$	−0.991166	−0.0380374
$680$	0	0
$681$	−43.9056	−1.68247
$682$	0	0
$683$	23.7070	0.907123	0.453561	−	0.891225i	$-0.350153\pi$
$683$	23.7070	0.907123	0.453561	+	0.891225i	$0.350153\pi$
$684$	0	0
$685$	10.1367	0.387305
$686$	0	0
$687$	5.67442	0.216492
$688$	0	0
$689$	−28.6320	−1.09079
$690$	0	0
$691$	7.88226	0.299855	0.149928	−	0.988697i	$-0.452096\pi$
$691$	7.88226	0.299855	0.149928	+	0.988697i	$0.452096\pi$
$692$	0	0
$693$	10.2186	0.388171
$694$	0	0
$695$	−15.4113	−0.584583
$696$	0	0
$697$	−72.7134	−2.75422
$698$	0	0
$699$	23.4816	0.888155
$700$	0	0
$701$	−37.9992	−1.43521	−0.717604	−	0.696451i	$-0.754761\pi$
$701$	−37.9992	−1.43521	−0.717604	+	0.696451i	$0.754761\pi$
$702$	0	0
$703$	−13.9444	−0.525924
$704$	0	0
$705$	−14.6453	−0.551573
$706$	0	0
$707$	19.3170	0.726491
$708$	0	0
$709$	−35.5890	−1.33657	−0.668287	−	0.743904i	$-0.732972\pi$
$709$	−35.5890	−1.33657	−0.668287	+	0.743904i	$0.732972\pi$
$710$	0	0
$711$	−72.9567	−2.73609
$712$	0	0
$713$	1.44907	0.0542681
$714$	0	0
$715$	−5.03564	−0.188322
$716$	0	0
$717$	32.0044	1.19523
$718$	0	0
$719$	26.3080	0.981122	0.490561	−	0.871407i	$-0.336792\pi$
$719$	26.3080	0.981122	0.490561	+	0.871407i	$0.336792\pi$
$720$	0	0
$721$	−27.1192	−1.00997
$722$	0	0
$723$	87.2426	3.24459
$724$	0	0
$725$	4.15880	0.154454
$726$	0	0
$727$	−9.75222	−0.361690	−0.180845	−	0.983512i	$-0.557883\pi$
$727$	−9.75222	−0.361690	−0.180845	+	0.983512i	$0.557883\pi$
$728$	0	0
$729$	−43.2561	−1.60208
$730$	0	0
$731$	−75.4519	−2.79069
$732$	0	0
$733$	12.8277	0.473803	0.236902	−	0.971534i	$-0.423868\pi$
$733$	12.8277	0.473803	0.236902	+	0.971534i	$0.423868\pi$
$734$	0	0
$735$	4.16227	0.153527
$736$	0	0
$737$	12.9463	0.476884
$738$	0	0
$739$	54.2742	1.99651	0.998255	−	0.0590573i	$-0.0188095\pi$
$739$	54.2742	1.99651	0.998255	+	0.0590573i	$0.0188095\pi$
$740$	0	0
$741$	47.0473	1.72833
$742$	0	0
$743$	−49.0964	−1.80117	−0.900587	−	0.434676i	$-0.856863\pi$
$743$	−49.0964	−1.80117	−0.900587	+	0.434676i	$0.856863\pi$
$744$	0	0
$745$	2.08444	0.0763681
$746$	0	0
$747$	−25.5346	−0.934262
$748$	0	0
$749$	−22.6094	−0.826131
$750$	0	0
$751$	18.8308	0.687145	0.343573	−	0.939126i	$-0.388363\pi$
$751$	18.8308	0.687145	0.343573	+	0.939126i	$0.388363\pi$
$752$	0	0
$753$	7.56107	0.275541
$754$	0	0
$755$	1.22208	0.0444759
$756$	0	0
$757$	4.56756	0.166011	0.0830055	−	0.996549i	$-0.473548\pi$
$757$	4.56756	0.166011	0.0830055	+	0.996549i	$0.473548\pi$
$758$	0	0
$759$	2.47504	0.0898382
$760$	0	0
$761$	42.8027	1.55160	0.775798	−	0.630981i	$-0.217348\pi$
$761$	42.8027	1.55160	0.775798	+	0.630981i	$0.217348\pi$
$762$	0	0
$763$	45.6378	1.65220
$764$	0	0
$765$	−39.2323	−1.41845
$766$	0	0
$767$	7.74155	0.279531
$768$	0	0
$769$	14.6239	0.527351	0.263675	−	0.964612i	$-0.415065\pi$
$769$	14.6239	0.527351	0.263675	+	0.964612i	$0.415065\pi$
$770$	0	0
$771$	11.7012	0.421408
$772$	0	0
$773$	−10.7175	−0.385481	−0.192740	−	0.981250i	$-0.561737\pi$
$773$	−10.7175	−0.385481	−0.192740	+	0.981250i	$0.561737\pi$
$774$	0	0
$775$	−1.44907	−0.0520521
$776$	0	0
$777$	−31.7955	−1.14066
$778$	0	0
$779$	26.6938	0.956403
$780$	0	0
$781$	−0.765239	−0.0273824
$782$	0	0
$783$	22.9208	0.819123
$784$	0	0
$785$	20.4610	0.730284
$786$	0	0
$787$	−26.0515	−0.928636	−0.464318	−	0.885669i	$-0.653700\pi$
$787$	−26.0515	−0.928636	−0.464318	+	0.885669i	$0.653700\pi$
$788$	0	0
$789$	37.5154	1.33558
$790$	0	0
$791$	−45.9967	−1.63545
$792$	0	0
$793$	10.7151	0.380504
$794$	0	0
$795$	−14.0727	−0.499109
$796$	0	0
$797$	−42.9478	−1.52129	−0.760645	−	0.649168i	$-0.775117\pi$
$797$	−42.9478	−1.52129	−0.760645	+	0.649168i	$0.775117\pi$
$798$	0	0
$799$	41.1217	1.45478
$800$	0	0
$801$	−8.43932	−0.298189
$802$	0	0
$803$	4.17548	0.147349
$804$	0	0
$805$	−2.35036	−0.0828392
$806$	0	0
$807$	88.2743	3.10740
$808$	0	0
$809$	−24.3715	−0.856855	−0.428427	−	0.903576i	$-0.640932\pi$
$809$	−24.3715	−0.856855	−0.428427	+	0.903576i	$0.640932\pi$
$810$	0	0
$811$	18.6174	0.653744	0.326872	−	0.945069i	$-0.394005\pi$
$811$	18.6174	0.653744	0.326872	+	0.945069i	$0.394005\pi$
$812$	0	0
$813$	43.1513	1.51338
$814$	0	0
$815$	6.95512	0.243627
$816$	0	0
$817$	27.6991	0.969069
$818$	0	0
$819$	66.8153	2.33472
$820$	0	0
$821$	30.8026	1.07502	0.537509	−	0.843258i	$-0.319365\pi$
$821$	30.8026	1.07502	0.537509	+	0.843258i	$0.319365\pi$
$822$	0	0
$823$	14.9016	0.519439	0.259719	−	0.965684i	$-0.416370\pi$
$823$	14.9016	0.519439	0.259719	+	0.965684i	$0.416370\pi$
$824$	0	0
$825$	−2.47504	−0.0861698
$826$	0	0
$827$	−18.5826	−0.646181	−0.323091	−	0.946368i	$-0.604722\pi$
$827$	−18.5826	−0.646181	−0.323091	+	0.946368i	$0.604722\pi$
$828$	0	0
$829$	42.3038	1.46927	0.734635	−	0.678462i	$-0.237353\pi$
$829$	42.3038	1.46927	0.734635	+	0.678462i	$0.237353\pi$
$830$	0	0
$831$	−21.5789	−0.748565
$832$	0	0
$833$	−11.6870	−0.404931
$834$	0	0
$835$	−0.646347	−0.0223678
$836$	0	0
$837$	−7.98641	−0.276051
$838$	0	0
$839$	−32.9008	−1.13586	−0.567931	−	0.823076i	$-0.692256\pi$
$839$	−32.9008	−1.13586	−0.567931	+	0.823076i	$0.692256\pi$
$840$	0	0
$841$	−11.7044	−0.403601
$842$	0	0
$843$	−18.8839	−0.650397
$844$	0	0
$845$	−19.9261	−0.685479
$846$	0	0
$847$	24.0438	0.826156
$848$	0	0
$849$	−3.57805	−0.122798
$850$	0	0
$851$	−4.79660	−0.164425
$852$	0	0
$853$	−0.828492	−0.0283670	−0.0141835	−	0.999899i	$-0.504515\pi$
$853$	−0.828492	−0.0283670	−0.0141835	+	0.999899i	$0.504515\pi$
$854$	0	0
$855$	14.4025	0.492556
$856$	0	0
$857$	−23.7967	−0.812881	−0.406440	−	0.913677i	$-0.633230\pi$
$857$	−23.7967	−0.812881	−0.406440	+	0.913677i	$0.633230\pi$
$858$	0	0
$859$	−33.2006	−1.13279	−0.566395	−	0.824134i	$-0.691662\pi$
$859$	−33.2006	−1.13279	−0.566395	+	0.824134i	$0.691662\pi$
$860$	0	0
$861$	60.8660	2.07431
$862$	0	0
$863$	46.4866	1.58242	0.791212	−	0.611542i	$-0.209451\pi$
$863$	46.4866	1.58242	0.791212	+	0.611542i	$0.209451\pi$
$864$	0	0
$865$	5.76581	0.196044
$866$	0	0
$867$	128.919	4.37833
$868$	0	0
$869$	12.9234	0.438397
$870$	0	0
$871$	84.6511	2.86829
$872$	0	0
$873$	2.08922	0.0707094
$874$	0	0
$875$	2.35036	0.0794566
$876$	0	0
$877$	−43.4577	−1.46746	−0.733732	−	0.679439i	$-0.762223\pi$
$877$	−43.4577	−1.46746	−0.733732	+	0.679439i	$0.762223\pi$
$878$	0	0
$879$	22.5363	0.760132
$880$	0	0
$881$	−37.4140	−1.26051	−0.630255	−	0.776388i	$-0.717050\pi$
$881$	−37.4140	−1.26051	−0.630255	+	0.776388i	$0.717050\pi$
$882$	0	0
$883$	−54.7078	−1.84106	−0.920531	−	0.390669i	$-0.872244\pi$
$883$	−54.7078	−1.84106	−0.920531	+	0.390669i	$0.872244\pi$
$884$	0	0
$885$	3.80501	0.127904
$886$	0	0
$887$	−36.4308	−1.22323	−0.611614	−	0.791156i	$-0.709479\pi$
$887$	−36.4308	−1.22323	−0.611614	+	0.791156i	$0.709479\pi$
$888$	0	0
$889$	38.4430	1.28934
$890$	0	0
$891$	−0.597953	−0.0200322
$892$	0	0
$893$	−15.0962	−0.505174
$894$	0	0
$895$	−12.8409	−0.429223
$896$	0	0
$897$	16.1833	0.540346
$898$	0	0
$899$	−6.02638	−0.200991
$900$	0	0
$901$	39.5142	1.31641
$902$	0	0
$903$	63.1583	2.10178
$904$	0	0
$905$	4.45371	0.148046
$906$	0	0
$907$	3.39066	0.112585	0.0562925	−	0.998414i	$-0.482072\pi$
$907$	3.39066	0.112585	0.0562925	+	0.998414i	$0.482072\pi$
$908$	0	0
$909$	−40.7172	−1.35050
$910$	0	0
$911$	−37.1336	−1.23029	−0.615146	−	0.788413i	$-0.710903\pi$
$911$	−37.1336	−1.23029	−0.615146	+	0.788413i	$0.710903\pi$
$912$	0	0
$913$	4.52316	0.149695
$914$	0	0
$915$	5.26652	0.174106
$916$	0	0
$917$	47.8564	1.58036
$918$	0	0
$919$	45.0283	1.48535	0.742673	−	0.669654i	$-0.233558\pi$
$919$	45.0283	1.48535	0.742673	+	0.669654i	$0.233558\pi$
$920$	0	0
$921$	−56.9932	−1.87799
$922$	0	0
$923$	−5.00360	−0.164696
$924$	0	0
$925$	4.79660	0.157711
$926$	0	0
$927$	57.1630	1.87748
$928$	0	0
$929$	18.9370	0.621302	0.310651	−	0.950524i	$-0.399453\pi$
$929$	18.9370	0.621302	0.310651	+	0.950524i	$0.399453\pi$
$930$	0	0
$931$	4.29041	0.140613
$932$	0	0
$933$	13.5461	0.443481
$934$	0	0
$935$	6.94954	0.227274
$936$	0	0
$937$	−17.6815	−0.577629	−0.288814	−	0.957385i	$-0.593261\pi$
$937$	−17.6815	−0.577629	−0.288814	+	0.957385i	$0.593261\pi$
$938$	0	0
$939$	−31.1343	−1.01603
$940$	0	0
$941$	42.6664	1.39088	0.695442	−	0.718582i	$-0.255209\pi$
$941$	42.6664	1.39088	0.695442	+	0.718582i	$0.255209\pi$
$942$	0	0
$943$	9.18211	0.299011
$944$	0	0
$945$	12.9538	0.421387
$946$	0	0
$947$	−20.1540	−0.654918	−0.327459	−	0.944865i	$-0.606192\pi$
$947$	−20.1540	−0.654918	−0.327459	+	0.944865i	$0.606192\pi$
$948$	0	0
$949$	27.3019	0.886256
$950$	0	0
$951$	−41.7410	−1.35354
$952$	0	0
$953$	−19.4794	−0.630999	−0.315500	−	0.948926i	$-0.602172\pi$
$953$	−19.4794	−0.630999	−0.315500	+	0.948926i	$0.602172\pi$
$954$	0	0
$955$	−11.2338	−0.363518
$956$	0	0
$957$	−10.2932	−0.332731
$958$	0	0
$959$	23.8250	0.769348
$960$	0	0
$961$	−28.9002	−0.932265
$962$	0	0
$963$	47.6571	1.53573
$964$	0	0
$965$	20.3568	0.655310
$966$	0	0
$967$	34.8042	1.11923	0.559614	−	0.828753i	$-0.310949\pi$
$967$	34.8042	1.11923	0.559614	+	0.828753i	$0.310949\pi$
$968$	0	0
$969$	−64.9287	−2.08581
$970$	0	0
$971$	20.5044	0.658019	0.329009	−	0.944327i	$-0.393285\pi$
$971$	20.5044	0.658019	0.329009	+	0.944327i	$0.393285\pi$
$972$	0	0
$973$	−36.2220	−1.16122
$974$	0	0
$975$	−16.1833	−0.518281
$976$	0	0
$977$	6.92613	0.221587	0.110793	−	0.993843i	$-0.464661\pi$
$977$	6.92613	0.221587	0.110793	+	0.993843i	$0.464661\pi$
$978$	0	0
$979$	1.49493	0.0477780
$980$	0	0
$981$	−96.1972	−3.07134
$982$	0	0
$983$	−27.5372	−0.878300	−0.439150	−	0.898414i	$-0.644720\pi$
$983$	−27.5372	−0.878300	−0.439150	+	0.898414i	$0.644720\pi$
$984$	0	0
$985$	−8.68151	−0.276616
$986$	0	0
$987$	−34.4216	−1.09565
$988$	0	0
$989$	9.52793	0.302971
$990$	0	0
$991$	9.27005	0.294473	0.147236	−	0.989101i	$-0.452962\pi$
$991$	9.27005	0.294473	0.147236	+	0.989101i	$0.452962\pi$
$992$	0	0
$993$	−96.9903	−3.07789
$994$	0	0
$995$	−1.90089	−0.0602624
$996$	0	0
$997$	−42.5729	−1.34830	−0.674149	−	0.738596i	$-0.735489\pi$
$997$	−42.5729	−1.34830	−0.674149	+	0.738596i	$0.735489\pi$
$998$	0	0
$999$	26.4360	0.836399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.cl.1.5		5
4.3	odd	2		7360.2.a.cq.1.1		5
8.3	odd	2		3680.2.a.x.1.5	✓	5
8.5	even	2		3680.2.a.ba.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.x.1.5	✓	5	8.3	odd	2
3680.2.a.ba.1.1	yes	5	8.5	even	2
7360.2.a.cl.1.5		5	1.1	even	1	trivial
7360.2.a.cq.1.1		5	4.3	odd	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82032 q^{3} -1.00000 q^{5} -2.35036 q^{7} +4.95418 q^{9} +O(q^{10})\)	\(q+2.82032 q^{3} -1.00000 q^{5} -2.35036 q^{7} +4.95418 q^{9} -0.877575 q^{11} -5.73813 q^{13} -2.82032 q^{15} +7.91903 q^{17} -2.90715 q^{19} -6.62875 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.51141 q^{27} +4.15880 q^{29} -1.44907 q^{31} -2.47504 q^{33} +2.35036 q^{35} +4.79660 q^{37} -16.1833 q^{39} -9.18211 q^{41} -9.52793 q^{43} -4.95418 q^{45} +5.19278 q^{47} -1.47582 q^{49} +22.3342 q^{51} +4.98977 q^{53} +0.877575 q^{55} -8.19907 q^{57} -1.34914 q^{59} -1.86735 q^{61} -11.6441 q^{63} +5.73813 q^{65} -14.7524 q^{67} -2.82032 q^{69} +0.871992 q^{71} -4.75798 q^{73} +2.82032 q^{75} +2.06262 q^{77} -14.7263 q^{79} +0.681369 q^{81} -5.15415 q^{83} -7.91903 q^{85} +11.7291 q^{87} -1.70347 q^{89} +13.4867 q^{91} -4.08683 q^{93} +2.90715 q^{95} +0.421708 q^{97} -4.34767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9	\( 5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9} - 3 q^{11} - 7 q^{13} + q^{15} + 9 q^{17} - q^{19} - 20 q^{21} - 5 q^{23} + 5 q^{25} - 4 q^{27} + 10 q^{29} + 21 q^{31} + 7 q^{33} - q^{35} - 8 q^{37} - 24 q^{39} - 13 q^{41} + 6 q^{43} - 6 q^{45} + 24 q^{49} + 17 q^{51} + 6 q^{53} + 3 q^{55} + 26 q^{57} - 18 q^{59} + 11 q^{61} + 4 q^{63} + 7 q^{65} - 38 q^{67} + q^{69} - 21 q^{71} - 12 q^{73} - q^{75} - 46 q^{77} - 18 q^{79} + 9 q^{81} - 20 q^{83} - 9 q^{85} - 6 q^{87} - 16 q^{89} + 3 q^{91} - 22 q^{93} + q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9 - 3 * q^11 - 7 * q^13 + q^15 + 9 * q^17 - q^19 - 20 * q^21 - 5 * q^23 + 5 * q^25 - 4 * q^27 + 10 * q^29 + 21 * q^31 + 7 * q^33 - q^35 - 8 * q^37 - 24 * q^39 - 13 * q^41 + 6 * q^43 - 6 * q^45 + 24 * q^49 + 17 * q^51 + 6 * q^53 + 3 * q^55 + 26 * q^57 - 18 * q^59 + 11 * q^61 + 4 * q^63 + 7 * q^65 - 38 * q^67 + q^69 - 21 * q^71 - 12 * q^73 - q^75 - 46 * q^77 - 18 * q^79 + 9 * q^81 - 20 * q^83 - 9 * q^85 - 6 * q^87 - 16 * q^89 + 3 * q^91 - 22 * q^93 + q^95 + 29 * q^97 - 29 * q^99

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