LMFDB - Embedded newform 5796.2.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5796,2,Mod(1,5796)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5796, 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("5796.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5796.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:	$46.2812930115$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.6963152.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 $ x^5 - 2x^4 - 10x^3 + 10x^2 + 29x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 644)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.25688$ of defining polynomial
Character	$\chi$	$=$	5796.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04771 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+3.04771 q^{5} -1.00000 q^{7} -5.22523 q^{11} +3.38206 q^{13} +2.63895 q^{17} +3.39461 q^{19} +1.00000 q^{23} +4.28854 q^{25} +4.00190 q^{29} +6.97185 q^{31} -3.04771 q^{35} -5.11916 q^{37} +5.89583 q^{41} -7.90838 q^{43} -9.54893 q^{47} +1.00000 q^{49} +11.8200 q^{53} -15.9250 q^{55} -1.51979 q^{59} +1.13934 q^{61} +10.3076 q^{65} -8.80688 q^{67} +14.5333 q^{71} -13.2271 q^{73} +5.22523 q^{77} +14.4398 q^{79} +0.693795 q^{83} +8.04275 q^{85} -10.8511 q^{89} -3.38206 q^{91} +10.3458 q^{95} -3.23476 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} - 5 q^{7}+O(q^{10}) $ 5 * q - 2 * q^5 - 5 * q^7	$ 5 q - 2 q^{5} - 5 q^{7} - 2 q^{11} + 13 q^{13} - 4 q^{17} + 12 q^{19} + 5 q^{23} + 19 q^{25} - 13 q^{29} - 3 q^{31} + 2 q^{35} - 4 q^{37} - q^{41} - 8 q^{43} - 5 q^{47} + 5 q^{49} + 8 q^{53} - 2 q^{55} - 12 q^{59} + 20 q^{61} + 12 q^{65} - 12 q^{67} - 9 q^{71} - 9 q^{73} + 2 q^{77} - 8 q^{79} + 28 q^{83} + 16 q^{85} - 32 q^{89} - 13 q^{91} + 36 q^{95} + 4 q^{97}+O(q^{100}) $ 5 * q - 2 * q^5 - 5 * q^7 - 2 * q^11 + 13 * q^13 - 4 * q^17 + 12 * q^19 + 5 * q^23 + 19 * q^25 - 13 * q^29 - 3 * q^31 + 2 * q^35 - 4 * q^37 - q^41 - 8 * q^43 - 5 * q^47 + 5 * q^49 + 8 * q^53 - 2 * q^55 - 12 * q^59 + 20 * q^61 + 12 * q^65 - 12 * q^67 - 9 * q^71 - 9 * q^73 + 2 * q^77 - 8 * q^79 + 28 * q^83 + 16 * q^85 - 32 * q^89 - 13 * q^91 + 36 * q^95 + 4 * 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	0
$3$	0	0
$4$	0	0
$5$	3.04771	1.36298	0.681489	−	0.731828i	$-0.261333\pi$
$5$	3.04771	1.36298	0.681489	+	0.731828i	$0.261333\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.22523	−1.57546	−0.787732	−	0.616018i	$-0.788745\pi$
$11$	−5.22523	−1.57546	−0.787732	+	0.616018i	$0.788745\pi$
$12$	0	0
$13$	3.38206	0.938016	0.469008	−	0.883194i	$-0.344612\pi$
$13$	3.38206	0.938016	0.469008	+	0.883194i	$0.344612\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.63895	0.640039	0.320019	−	0.947411i	$-0.396311\pi$
$17$	2.63895	0.640039	0.320019	+	0.947411i	$0.396311\pi$
$18$	0	0
$19$	3.39461	0.778777	0.389388	−	0.921074i	$-0.372686\pi$
$19$	3.39461	0.778777	0.389388	+	0.921074i	$0.372686\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.28854	0.857708
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.00190	0.743134	0.371567	−	0.928406i	$-0.378821\pi$
$29$	4.00190	0.743134	0.371567	+	0.928406i	$0.378821\pi$
$30$	0	0
$31$	6.97185	1.25218	0.626091	−	0.779750i	$-0.284654\pi$
$31$	6.97185	1.25218	0.626091	+	0.779750i	$0.284654\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.04771	−0.515157
$36$	0	0
$37$	−5.11916	−0.841585	−0.420792	−	0.907157i	$-0.638248\pi$
$37$	−5.11916	−0.841585	−0.420792	+	0.907157i	$0.638248\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.89583	0.920774	0.460387	−	0.887718i	$-0.347711\pi$
$41$	5.89583	0.920774	0.460387	+	0.887718i	$0.347711\pi$
$42$	0	0
$43$	−7.90838	−1.20602	−0.603008	−	0.797735i	$-0.706031\pi$
$43$	−7.90838	−1.20602	−0.603008	+	0.797735i	$0.706031\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.54893	−1.39286	−0.696428	−	0.717627i	$-0.745228\pi$
$47$	−9.54893	−1.39286	−0.696428	+	0.717627i	$0.745228\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.8200	1.62360	0.811799	−	0.583937i	$-0.198488\pi$
$53$	11.8200	1.62360	0.811799	+	0.583937i	$0.198488\pi$
$54$	0	0
$55$	−15.9250	−2.14732
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.51979	−0.197860	−0.0989299	−	0.995094i	$-0.531542\pi$
$59$	−1.51979	−0.197860	−0.0989299	+	0.995094i	$0.531542\pi$
$60$	0	0
$61$	1.13934	0.145877	0.0729385	−	0.997336i	$-0.476762\pi$
$61$	1.13934	0.145877	0.0729385	+	0.997336i	$0.476762\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.3076	1.27849
$66$	0	0
$67$	−8.80688	−1.07593	−0.537966	−	0.842967i	$-0.680807\pi$
$67$	−8.80688	−1.07593	−0.537966	+	0.842967i	$0.680807\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.5333	1.72479	0.862394	−	0.506237i	$-0.168964\pi$
$71$	14.5333	1.72479	0.862394	+	0.506237i	$0.168964\pi$
$72$	0	0
$73$	−13.2271	−1.54812	−0.774059	−	0.633114i	$-0.781777\pi$
$73$	−13.2271	−1.54812	−0.774059	+	0.633114i	$0.781777\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.22523	0.595470
$78$	0	0
$79$	14.4398	1.62461	0.812303	−	0.583236i	$-0.198214\pi$
$79$	14.4398	1.62461	0.812303	+	0.583236i	$0.198214\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.693795	0.0761538	0.0380769	−	0.999275i	$-0.487877\pi$
$83$	0.693795	0.0761538	0.0380769	+	0.999275i	$0.487877\pi$
$84$	0	0
$85$	8.04275	0.872359
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.8511	−1.15021	−0.575106	−	0.818079i	$-0.695039\pi$
$89$	−10.8511	−1.15021	−0.575106	+	0.818079i	$0.695039\pi$
$90$	0	0
$91$	−3.38206	−0.354537
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.3458	1.06146
$96$	0	0
$97$	−3.23476	−0.328440	−0.164220	−	0.986424i	$-0.552511\pi$
$97$	−3.23476	−0.328440	−0.164220	+	0.986424i	$0.552511\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.32672	0.828540	0.414270	−	0.910154i	$-0.364037\pi$
$101$	8.32672	0.828540	0.414270	+	0.910154i	$0.364037\pi$
$102$	0	0
$103$	0.418345	0.0412208	0.0206104	−	0.999788i	$-0.493439\pi$
$103$	0.418345	0.0412208	0.0206104	+	0.999788i	$0.493439\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.9084	1.15123	0.575613	−	0.817722i	$-0.304763\pi$
$107$	11.9084	1.15123	0.575613	+	0.817722i	$0.304763\pi$
$108$	0	0
$109$	17.9154	1.71598	0.857992	−	0.513663i	$-0.171712\pi$
$109$	17.9154	1.71598	0.857992	+	0.513663i	$0.171712\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.6375	1.18884	0.594418	−	0.804156i	$-0.297383\pi$
$113$	12.6375	1.18884	0.594418	+	0.804156i	$0.297383\pi$
$114$	0	0
$115$	3.04771	0.284201
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.63895	−0.241912
$120$	0	0
$121$	16.3030	1.48209
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.16832	−0.193940
$126$	0	0
$127$	−3.22333	−0.286024	−0.143012	−	0.989721i	$-0.545679\pi$
$127$	−3.22333	−0.286024	−0.143012	+	0.989721i	$0.545679\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.0910	1.23114	0.615569	−	0.788083i	$-0.288926\pi$
$131$	14.0910	1.23114	0.615569	+	0.788083i	$0.288926\pi$
$132$	0	0
$133$	−3.39461	−0.294350
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.67250	−0.228327	−0.114164	−	0.993462i	$-0.536419\pi$
$137$	−2.67250	−0.228327	−0.114164	+	0.993462i	$0.536419\pi$
$138$	0	0
$139$	6.23315	0.528689	0.264344	−	0.964428i	$-0.414844\pi$
$139$	6.23315	0.528689	0.264344	+	0.964428i	$0.414844\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.6721	−1.47781
$144$	0	0
$145$	12.1966	1.01287
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.58165	0.457267	0.228633	−	0.973513i	$-0.426574\pi$
$149$	5.58165	0.457267	0.228633	+	0.973513i	$0.426574\pi$
$150$	0	0
$151$	−11.8325	−0.962916	−0.481458	−	0.876469i	$-0.659893\pi$
$151$	−11.8325	−0.962916	−0.481458	+	0.876469i	$0.659893\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	21.2482	1.70670
$156$	0	0
$157$	17.3015	1.38081	0.690406	−	0.723422i	$-0.257432\pi$
$157$	17.3015	1.38081	0.690406	+	0.723422i	$0.257432\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	14.3212	1.12172	0.560861	−	0.827910i	$-0.310470\pi$
$163$	14.3212	1.12172	0.560861	+	0.827910i	$0.310470\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.28358	−0.486238	−0.243119	−	0.969996i	$-0.578171\pi$
$167$	−6.28358	−0.486238	−0.243119	+	0.969996i	$0.578171\pi$
$168$	0	0
$169$	−1.56164	−0.120126
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.60539	0.198084	0.0990421	−	0.995083i	$-0.468422\pi$
$173$	2.60539	0.198084	0.0990421	+	0.995083i	$0.468422\pi$
$174$	0	0
$175$	−4.28854	−0.324183
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−25.2341	−1.88609	−0.943044	−	0.332667i	$-0.892051\pi$
$179$	−25.2341	−1.88609	−0.943044	+	0.332667i	$0.892051\pi$
$180$	0	0
$181$	−1.36485	−0.101448	−0.0507242	−	0.998713i	$-0.516153\pi$
$181$	−1.36485	−0.101448	−0.0507242	+	0.998713i	$0.516153\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−15.6017	−1.14706
$186$	0	0
$187$	−13.7891	−1.00836
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.1467	1.60248	0.801239	−	0.598344i	$-0.204174\pi$
$191$	22.1467	1.60248	0.801239	+	0.598344i	$0.204174\pi$
$192$	0	0
$193$	18.3653	1.32197	0.660983	−	0.750401i	$-0.270139\pi$
$193$	18.3653	1.32197	0.660983	+	0.750401i	$0.270139\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.07831	0.0768261	0.0384131	−	0.999262i	$-0.487770\pi$
$197$	1.07831	0.0768261	0.0384131	+	0.999262i	$0.487770\pi$
$198$	0	0
$199$	−5.55335	−0.393666	−0.196833	−	0.980437i	$-0.563066\pi$
$199$	−5.55335	−0.393666	−0.196833	+	0.980437i	$0.563066\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.00190	−0.280878
$204$	0	0
$205$	17.9688	1.25499
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−17.7376	−1.22693
$210$	0	0
$211$	−19.3543	−1.33240	−0.666201	−	0.745772i	$-0.732081\pi$
$211$	−19.3543	−1.33240	−0.666201	+	0.745772i	$0.732081\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−24.1024	−1.64377
$216$	0	0
$217$	−6.97185	−0.473280
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.92509	0.600367
$222$	0	0
$223$	−2.69234	−0.180293	−0.0901464	−	0.995929i	$-0.528733\pi$
$223$	−2.69234	−0.180293	−0.0901464	+	0.995929i	$0.528733\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.26663	0.548675	0.274338	−	0.961633i	$-0.411541\pi$
$227$	8.26663	0.548675	0.274338	+	0.961633i	$0.411541\pi$
$228$	0	0
$229$	26.2369	1.73378	0.866891	−	0.498499i	$-0.166115\pi$
$229$	26.2369	1.73378	0.866891	+	0.498499i	$0.166115\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.17735	−0.0771310	−0.0385655	−	0.999256i	$-0.512279\pi$
$233$	−1.17735	−0.0771310	−0.0385655	+	0.999256i	$0.512279\pi$
$234$	0	0
$235$	−29.1024	−1.89843
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.56911	−0.360236	−0.180118	−	0.983645i	$-0.557648\pi$
$239$	−5.56911	−0.360236	−0.180118	+	0.983645i	$0.557648\pi$
$240$	0	0
$241$	8.12518	0.523389	0.261694	−	0.965151i	$-0.415719\pi$
$241$	8.12518	0.523389	0.261694	+	0.965151i	$0.415719\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.04771	0.194711
$246$	0	0
$247$	11.4808	0.730505
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.78542	−0.175814	−0.0879071	−	0.996129i	$-0.528018\pi$
$251$	−2.78542	−0.175814	−0.0879071	+	0.996129i	$0.528018\pi$
$252$	0	0
$253$	−5.22523	−0.328507
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.1150	1.50425	0.752126	−	0.659020i	$-0.229029\pi$
$257$	24.1150	1.50425	0.752126	+	0.659020i	$0.229029\pi$
$258$	0	0
$259$	5.11916	0.318089
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.40905	−0.210211	−0.105106	−	0.994461i	$-0.533518\pi$
$263$	−3.40905	−0.210211	−0.105106	+	0.994461i	$0.533518\pi$
$264$	0	0
$265$	36.0239	2.21293
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.29044	−0.322564	−0.161282	−	0.986908i	$-0.551563\pi$
$269$	−5.29044	−0.322564	−0.161282	+	0.986908i	$0.551563\pi$
$270$	0	0
$271$	16.2206	0.985331	0.492666	−	0.870219i	$-0.336023\pi$
$271$	16.2206	0.985331	0.492666	+	0.870219i	$0.336023\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−22.4086	−1.35129
$276$	0	0
$277$	−22.2211	−1.33513	−0.667567	−	0.744550i	$-0.732664\pi$
$277$	−22.2211	−1.33513	−0.667567	+	0.744550i	$0.732664\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.74964	0.342995	0.171497	−	0.985185i	$-0.445140\pi$
$281$	5.74964	0.342995	0.171497	+	0.985185i	$0.445140\pi$
$282$	0	0
$283$	−14.0108	−0.832857	−0.416428	−	0.909169i	$-0.636718\pi$
$283$	−14.0108	−0.832857	−0.416428	+	0.909169i	$0.636718\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.89583	−0.348020
$288$	0	0
$289$	−10.0360	−0.590350
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−15.7616	−0.920801	−0.460400	−	0.887711i	$-0.652294\pi$
$293$	−15.7616	−0.920801	−0.460400	+	0.887711i	$0.652294\pi$
$294$	0	0
$295$	−4.63188	−0.269678
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.38206	0.195590
$300$	0	0
$301$	7.90838	0.455831
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.47237	0.198827
$306$	0	0
$307$	11.4944	0.656018	0.328009	−	0.944675i	$-0.393622\pi$
$307$	11.4944	0.656018	0.328009	+	0.944675i	$0.393622\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.41105	−0.250128	−0.125064	−	0.992149i	$-0.539914\pi$
$311$	−4.41105	−0.250128	−0.125064	+	0.992149i	$0.539914\pi$
$312$	0	0
$313$	−6.20154	−0.350532	−0.175266	−	0.984521i	$-0.556079\pi$
$313$	−6.20154	−0.350532	−0.175266	+	0.984521i	$0.556079\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.50769	0.534005	0.267003	−	0.963696i	$-0.413967\pi$
$317$	9.50769	0.534005	0.267003	+	0.963696i	$0.413967\pi$
$318$	0	0
$319$	−20.9108	−1.17078
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.95819	0.498447
$324$	0	0
$325$	14.5041	0.804544
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.54893	0.526450
$330$	0	0
$331$	10.0087	0.550130	0.275065	−	0.961426i	$-0.411301\pi$
$331$	10.0087	0.550130	0.275065	+	0.961426i	$0.411301\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−26.8408	−1.46647
$336$	0	0
$337$	16.5389	0.900929	0.450464	−	0.892794i	$-0.351258\pi$
$337$	16.5389	0.900929	0.450464	+	0.892794i	$0.351258\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−36.4295	−1.97277
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−34.8917	−1.87308	−0.936541	−	0.350558i	$-0.885992\pi$
$347$	−34.8917	−1.87308	−0.936541	+	0.350558i	$0.885992\pi$
$348$	0	0
$349$	6.92337	0.370599	0.185300	−	0.982682i	$-0.440674\pi$
$349$	6.92337	0.370599	0.185300	+	0.982682i	$0.440674\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.6050	−1.14992	−0.574959	−	0.818182i	$-0.694982\pi$
$353$	−21.6050	−1.14992	−0.574959	+	0.818182i	$0.694982\pi$
$354$	0	0
$355$	44.2934	2.35085
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.2085	1.38323	0.691616	−	0.722265i	$-0.256899\pi$
$359$	26.2085	1.38323	0.691616	+	0.722265i	$0.256899\pi$
$360$	0	0
$361$	−7.47664	−0.393507
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−40.3124	−2.11005
$366$	0	0
$367$	16.5650	0.864688	0.432344	−	0.901709i	$-0.357687\pi$
$367$	16.5650	0.864688	0.432344	+	0.901709i	$0.357687\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.8200	−0.613662
$372$	0	0
$373$	15.6116	0.808340	0.404170	−	0.914684i	$-0.367560\pi$
$373$	15.6116	0.808340	0.404170	+	0.914684i	$0.367560\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.5347	0.697071
$378$	0	0
$379$	−9.80933	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$379$	−9.80933	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.4925	0.944921	0.472461	−	0.881352i	$-0.343366\pi$
$383$	18.4925	0.944921	0.472461	+	0.881352i	$0.343366\pi$
$384$	0	0
$385$	15.9250	0.811612
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.3030	−0.573084	−0.286542	−	0.958068i	$-0.592506\pi$
$389$	−11.3030	−0.573084	−0.286542	+	0.958068i	$0.592506\pi$
$390$	0	0
$391$	2.63895	0.133457
$392$	0	0
$393$	0	0
$394$	0	0
$395$	44.0084	2.21430
$396$	0	0
$397$	−1.20204	−0.0603285	−0.0301643	−	0.999545i	$-0.509603\pi$
$397$	−1.20204	−0.0603285	−0.0301643	+	0.999545i	$0.509603\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.01127	0.150376	0.0751878	−	0.997169i	$-0.476044\pi$
$401$	3.01127	0.150376	0.0751878	+	0.997169i	$0.476044\pi$
$402$	0	0
$403$	23.5793	1.17457
$404$	0	0
$405$	0	0
$406$	0	0
$407$	26.7488	1.32589
$408$	0	0
$409$	−33.4725	−1.65511	−0.827553	−	0.561387i	$-0.810268\pi$
$409$	−33.4725	−1.65511	−0.827553	+	0.561387i	$0.810268\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.51979	0.0747839
$414$	0	0
$415$	2.11449	0.103796
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.40028	0.312674	0.156337	−	0.987704i	$-0.450031\pi$
$419$	6.40028	0.312674	0.156337	+	0.987704i	$0.450031\pi$
$420$	0	0
$421$	−8.01082	−0.390423	−0.195212	−	0.980761i	$-0.562539\pi$
$421$	−8.01082	−0.390423	−0.195212	+	0.980761i	$0.562539\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11.3172	0.548967
$426$	0	0
$427$	−1.13934	−0.0551363
$428$	0	0
$429$	0	0
$430$	0	0
$431$	27.6084	1.32985	0.664925	−	0.746910i	$-0.268463\pi$
$431$	27.6084	1.32985	0.664925	+	0.746910i	$0.268463\pi$
$432$	0	0
$433$	−14.8797	−0.715074	−0.357537	−	0.933899i	$-0.616383\pi$
$433$	−14.8797	−0.715074	−0.357537	+	0.933899i	$0.616383\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.39461	0.162386
$438$	0	0
$439$	−36.1931	−1.72740	−0.863702	−	0.504004i	$-0.831860\pi$
$439$	−36.1931	−1.72740	−0.863702	+	0.504004i	$0.831860\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.5546	0.501465	0.250733	−	0.968056i	$-0.419329\pi$
$443$	10.5546	0.501465	0.250733	+	0.968056i	$0.419329\pi$
$444$	0	0
$445$	−33.0710	−1.56771
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−39.9560	−1.88564	−0.942821	−	0.333301i	$-0.891838\pi$
$449$	−39.9560	−1.88564	−0.942821	+	0.333301i	$0.891838\pi$
$450$	0	0
$451$	−30.8071	−1.45065
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.3076	−0.483226
$456$	0	0
$457$	10.1870	0.476530	0.238265	−	0.971200i	$-0.423421\pi$
$457$	10.1870	0.476530	0.238265	+	0.971200i	$0.423421\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.70956	0.312495	0.156248	−	0.987718i	$-0.450060\pi$
$461$	6.70956	0.312495	0.156248	+	0.987718i	$0.450060\pi$
$462$	0	0
$463$	16.6512	0.773848	0.386924	−	0.922112i	$-0.373538\pi$
$463$	16.6512	0.773848	0.386924	+	0.922112i	$0.373538\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−39.1863	−1.81332	−0.906662	−	0.421857i	$-0.861378\pi$
$467$	−39.1863	−1.81332	−0.906662	+	0.421857i	$0.861378\pi$
$468$	0	0
$469$	8.80688	0.406664
$470$	0	0
$471$	0	0
$472$	0	0
$473$	41.3230	1.90004
$474$	0	0
$475$	14.5579	0.667963
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.2704	1.65724	0.828619	−	0.559813i	$-0.189127\pi$
$479$	36.2704	1.65724	0.828619	+	0.559813i	$0.189127\pi$
$480$	0	0
$481$	−17.3133	−0.789420
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.85861	−0.447656
$486$	0	0
$487$	23.7517	1.07629	0.538146	−	0.842851i	$-0.319125\pi$
$487$	23.7517	1.07629	0.538146	+	0.842851i	$0.319125\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.3752	1.00978	0.504889	−	0.863185i	$-0.331534\pi$
$491$	22.3752	1.00978	0.504889	+	0.863185i	$0.331534\pi$
$492$	0	0
$493$	10.5608	0.475635
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.5333	−0.651909
$498$	0	0
$499$	16.2296	0.726535	0.363268	−	0.931685i	$-0.381661\pi$
$499$	16.2296	0.726535	0.363268	+	0.931685i	$0.381661\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.57708	0.114906	0.0574532	−	0.998348i	$-0.481702\pi$
$503$	2.57708	0.114906	0.0574532	+	0.998348i	$0.481702\pi$
$504$	0	0
$505$	25.3774	1.12928
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−33.4091	−1.48083	−0.740417	−	0.672148i	$-0.765372\pi$
$509$	−33.4091	−1.48083	−0.740417	+	0.672148i	$0.765372\pi$
$510$	0	0
$511$	13.2271	0.585134
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.27500	0.0561830
$516$	0	0
$517$	49.8953	2.19439
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.4303	−0.544580	−0.272290	−	0.962215i	$-0.587781\pi$
$521$	−12.4303	−0.544580	−0.272290	+	0.962215i	$0.587781\pi$
$522$	0	0
$523$	1.05262	0.0460279	0.0230140	−	0.999735i	$-0.492674\pi$
$523$	1.05262	0.0460279	0.0230140	+	0.999735i	$0.492674\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.3984	0.801445
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	19.9401	0.863701
$534$	0	0
$535$	36.2933	1.56910
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.22523	−0.225066
$540$	0	0
$541$	29.4265	1.26514	0.632572	−	0.774502i	$-0.281999\pi$
$541$	29.4265	1.26514	0.632572	+	0.774502i	$0.281999\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	54.6009	2.33885
$546$	0	0
$547$	−36.3851	−1.55571	−0.777857	−	0.628441i	$-0.783693\pi$
$547$	−36.3851	−1.55571	−0.777857	+	0.628441i	$0.783693\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.5849	0.578735
$552$	0	0
$553$	−14.4398	−0.614043
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.2666	0.943467	0.471734	−	0.881741i	$-0.343628\pi$
$557$	22.2666	0.943467	0.471734	+	0.881741i	$0.343628\pi$
$558$	0	0
$559$	−26.7466	−1.13126
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.0300	−0.928453	−0.464227	−	0.885717i	$-0.653668\pi$
$563$	−22.0300	−0.928453	−0.464227	+	0.885717i	$0.653668\pi$
$564$	0	0
$565$	38.5154	1.62036
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−27.9918	−1.17348	−0.586738	−	0.809777i	$-0.699588\pi$
$569$	−27.9918	−1.17348	−0.586738	+	0.809777i	$0.699588\pi$
$570$	0	0
$571$	19.7000	0.824421	0.412210	−	0.911089i	$-0.364757\pi$
$571$	19.7000	0.824421	0.412210	+	0.911089i	$0.364757\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.28854	0.178845
$576$	0	0
$577$	11.9624	0.498000	0.249000	−	0.968503i	$-0.419898\pi$
$577$	11.9624	0.498000	0.249000	+	0.968503i	$0.419898\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.693795	−0.0287834
$582$	0	0
$583$	−61.7620	−2.55792
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.3278	0.797743	0.398872	−	0.917007i	$-0.369402\pi$
$587$	19.3278	0.797743	0.398872	+	0.917007i	$0.369402\pi$
$588$	0	0
$589$	23.6667	0.975170
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.45045	−0.347018	−0.173509	−	0.984832i	$-0.555511\pi$
$593$	−8.45045	−0.347018	−0.173509	+	0.984832i	$0.555511\pi$
$594$	0	0
$595$	−8.04275	−0.329721
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.56581	−0.227413	−0.113706	−	0.993514i	$-0.536272\pi$
$599$	−5.56581	−0.227413	−0.113706	+	0.993514i	$0.536272\pi$
$600$	0	0
$601$	−18.2258	−0.743445	−0.371722	−	0.928344i	$-0.621233\pi$
$601$	−18.2258	−0.743445	−0.371722	+	0.928344i	$0.621233\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	49.6868	2.02005
$606$	0	0
$607$	28.3740	1.15166	0.575832	−	0.817568i	$-0.304678\pi$
$607$	28.3740	1.15166	0.575832	+	0.817568i	$0.304678\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.2951	−1.30652
$612$	0	0
$613$	15.8620	0.640660	0.320330	−	0.947306i	$-0.396206\pi$
$613$	15.8620	0.640660	0.320330	+	0.947306i	$0.396206\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.9140	0.600417	0.300208	−	0.953874i	$-0.402944\pi$
$617$	14.9140	0.600417	0.300208	+	0.953874i	$0.402944\pi$
$618$	0	0
$619$	32.3397	1.29984	0.649920	−	0.760002i	$-0.274802\pi$
$619$	32.3397	1.29984	0.649920	+	0.760002i	$0.274802\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.8511	0.434739
$624$	0	0
$625$	−28.0511	−1.12204
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.5092	−0.538647
$630$	0	0
$631$	−15.3840	−0.612426	−0.306213	−	0.951963i	$-0.599062\pi$
$631$	−15.3840	−0.612426	−0.306213	+	0.951963i	$0.599062\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.82377	−0.389844
$636$	0	0
$637$	3.38206	0.134002
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−37.0193	−1.46217	−0.731087	−	0.682284i	$-0.760987\pi$
$641$	−37.0193	−1.46217	−0.731087	+	0.682284i	$0.760987\pi$
$642$	0	0
$643$	24.7183	0.974796	0.487398	−	0.873180i	$-0.337946\pi$
$643$	24.7183	0.974796	0.487398	+	0.873180i	$0.337946\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.3207	0.798887	0.399444	−	0.916758i	$-0.369203\pi$
$647$	20.3207	0.798887	0.399444	+	0.916758i	$0.369203\pi$
$648$	0	0
$649$	7.94124	0.311721
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.7950	0.539839	0.269919	−	0.962883i	$-0.413003\pi$
$653$	13.7950	0.539839	0.269919	+	0.962883i	$0.413003\pi$
$654$	0	0
$655$	42.9453	1.67801
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−39.4316	−1.53604	−0.768019	−	0.640427i	$-0.778757\pi$
$659$	−39.4316	−1.53604	−0.768019	+	0.640427i	$0.778757\pi$
$660$	0	0
$661$	38.9211	1.51385	0.756927	−	0.653499i	$-0.226700\pi$
$661$	38.9211	1.51385	0.756927	+	0.653499i	$0.226700\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10.3458	−0.401192
$666$	0	0
$667$	4.00190	0.154954
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.95329	−0.229824
$672$	0	0
$673$	−43.7150	−1.68509	−0.842544	−	0.538627i	$-0.818943\pi$
$673$	−43.7150	−1.68509	−0.842544	+	0.538627i	$0.818943\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.6777	−0.640977	−0.320489	−	0.947252i	$-0.603847\pi$
$677$	−16.6777	−0.640977	−0.320489	+	0.947252i	$0.603847\pi$
$678$	0	0
$679$	3.23476	0.124139
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−48.7312	−1.86465	−0.932324	−	0.361625i	$-0.882222\pi$
$683$	−48.7312	−1.86465	−0.932324	+	0.361625i	$0.882222\pi$
$684$	0	0
$685$	−8.14502	−0.311205
$686$	0	0
$687$	0	0
$688$	0	0
$689$	39.9759	1.52296
$690$	0	0
$691$	−41.7882	−1.58970	−0.794849	−	0.606807i	$-0.792450\pi$
$691$	−41.7882	−1.58970	−0.794849	+	0.606807i	$0.792450\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.9968	0.720591
$696$	0	0
$697$	15.5588	0.589331
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.1499	1.06321	0.531604	−	0.846993i	$-0.321590\pi$
$701$	28.1499	1.06321	0.531604	+	0.846993i	$0.321590\pi$
$702$	0	0
$703$	−17.3775	−0.655406
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.32672	−0.313159
$708$	0	0
$709$	22.1680	0.832536	0.416268	−	0.909242i	$-0.363338\pi$
$709$	22.1680	0.832536	0.416268	+	0.909242i	$0.363338\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.97185	0.261098
$714$	0	0
$715$	−53.8593	−2.01422
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.2623	0.830243	0.415122	−	0.909766i	$-0.363739\pi$
$719$	22.2623	0.830243	0.415122	+	0.909766i	$0.363739\pi$
$720$	0	0
$721$	−0.418345	−0.0155800
$722$	0	0
$723$	0	0
$724$	0	0
$725$	17.1623	0.637392
$726$	0	0
$727$	−21.6618	−0.803392	−0.401696	−	0.915773i	$-0.631579\pi$
$727$	−21.6618	−0.803392	−0.401696	+	0.915773i	$0.631579\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−20.8698	−0.771897
$732$	0	0
$733$	−4.50596	−0.166431	−0.0832156	−	0.996532i	$-0.526519\pi$
$733$	−4.50596	−0.166431	−0.0832156	+	0.996532i	$0.526519\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.0179	1.69509
$738$	0	0
$739$	−15.7692	−0.580079	−0.290040	−	0.957015i	$-0.593669\pi$
$739$	−15.7692	−0.580079	−0.290040	+	0.957015i	$0.593669\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30.8619	−1.13222	−0.566108	−	0.824331i	$-0.691551\pi$
$743$	−30.8619	−1.13222	−0.566108	+	0.824331i	$0.691551\pi$
$744$	0	0
$745$	17.0113	0.623245
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11.9084	−0.435123
$750$	0	0
$751$	38.1808	1.39324	0.696618	−	0.717442i	$-0.254687\pi$
$751$	38.1808	1.39324	0.696618	+	0.717442i	$0.254687\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−36.0621	−1.31243
$756$	0	0
$757$	−15.6733	−0.569655	−0.284828	−	0.958579i	$-0.591936\pi$
$757$	−15.6733	−0.569655	−0.284828	+	0.958579i	$0.591936\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−41.8363	−1.51657	−0.758283	−	0.651926i	$-0.773961\pi$
$761$	−41.8363	−1.51657	−0.758283	+	0.651926i	$0.773961\pi$
$762$	0	0
$763$	−17.9154	−0.648581
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.14003	−0.185596
$768$	0	0
$769$	5.39928	0.194703	0.0973515	−	0.995250i	$-0.468963\pi$
$769$	5.39928	0.194703	0.0973515	+	0.995250i	$0.468963\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−37.1956	−1.33783	−0.668917	−	0.743337i	$-0.733242\pi$
$773$	−37.1956	−1.33783	−0.668917	+	0.743337i	$0.733242\pi$
$774$	0	0
$775$	29.8991	1.07401
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.0140	0.717077
$780$	0	0
$781$	−75.9399	−2.71734
$782$	0	0
$783$	0	0
$784$	0	0
$785$	52.7301	1.88202
$786$	0	0
$787$	−38.9942	−1.38999	−0.694997	−	0.719013i	$-0.744594\pi$
$787$	−38.9942	−1.38999	−0.694997	+	0.719013i	$0.744594\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.6375	−0.449338
$792$	0	0
$793$	3.85331	0.136835
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.8311	−0.914986	−0.457493	−	0.889213i	$-0.651253\pi$
$797$	−25.8311	−0.914986	−0.457493	+	0.889213i	$0.651253\pi$
$798$	0	0
$799$	−25.1991	−0.891482
$800$	0	0
$801$	0	0
$802$	0	0
$803$	69.1147	2.43901
$804$	0	0
$805$	−3.04771	−0.107418
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.7612	−0.940875	−0.470437	−	0.882433i	$-0.655904\pi$
$809$	−26.7612	−0.940875	−0.470437	+	0.882433i	$0.655904\pi$
$810$	0	0
$811$	30.3856	1.06698	0.533492	−	0.845805i	$-0.320880\pi$
$811$	30.3856	1.06698	0.533492	+	0.845805i	$0.320880\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	43.6469	1.52888
$816$	0	0
$817$	−26.8458	−0.939217
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.5069	0.576095	0.288048	−	0.957616i	$-0.406994\pi$
$821$	16.5069	0.576095	0.288048	+	0.957616i	$0.406994\pi$
$822$	0	0
$823$	−19.9562	−0.695631	−0.347816	−	0.937563i	$-0.613076\pi$
$823$	−19.9562	−0.695631	−0.347816	+	0.937563i	$0.613076\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.8443	1.28120	0.640601	−	0.767874i	$-0.278685\pi$
$827$	36.8443	1.28120	0.640601	+	0.767874i	$0.278685\pi$
$828$	0	0
$829$	5.80916	0.201760	0.100880	−	0.994899i	$-0.467834\pi$
$829$	5.80916	0.201760	0.100880	+	0.994899i	$0.467834\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.63895	0.0914341
$834$	0	0
$835$	−19.1505	−0.662732
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.2733	0.423722	0.211861	−	0.977300i	$-0.432048\pi$
$839$	12.2733	0.423722	0.211861	+	0.977300i	$0.432048\pi$
$840$	0	0
$841$	−12.9848	−0.447752
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.75942	−0.163729
$846$	0	0
$847$	−16.3030	−0.560177
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.11916	−0.175483
$852$	0	0
$853$	−13.2560	−0.453878	−0.226939	−	0.973909i	$-0.572872\pi$
$853$	−13.2560	−0.453878	−0.226939	+	0.973909i	$0.572872\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.1066	0.652670	0.326335	−	0.945254i	$-0.394186\pi$
$857$	19.1066	0.652670	0.326335	+	0.945254i	$0.394186\pi$
$858$	0	0
$859$	−46.1878	−1.57591	−0.787953	−	0.615735i	$-0.788859\pi$
$859$	−46.1878	−1.57591	−0.787953	+	0.615735i	$0.788859\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.5341	−0.392625	−0.196313	−	0.980541i	$-0.562897\pi$
$863$	−11.5341	−0.392625	−0.196313	+	0.980541i	$0.562897\pi$
$864$	0	0
$865$	7.94048	0.269984
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−75.4512	−2.55951
$870$	0	0
$871$	−29.7854	−1.00924
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.16832	0.0733026
$876$	0	0
$877$	−16.9695	−0.573020	−0.286510	−	0.958077i	$-0.592495\pi$
$877$	−16.9695	−0.573020	−0.286510	+	0.958077i	$0.592495\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−11.4505	−0.385775	−0.192888	−	0.981221i	$-0.561785\pi$
$881$	−11.4505	−0.385775	−0.192888	+	0.981221i	$0.561785\pi$
$882$	0	0
$883$	17.3634	0.584325	0.292162	−	0.956369i	$-0.405625\pi$
$883$	17.3634	0.584325	0.292162	+	0.956369i	$0.405625\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.7378	−0.763460	−0.381730	−	0.924274i	$-0.624672\pi$
$887$	−22.7378	−0.763460	−0.381730	+	0.924274i	$0.624672\pi$
$888$	0	0
$889$	3.22333	0.108107
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−32.4149	−1.08472
$894$	0	0
$895$	−76.9064	−2.57070
$896$	0	0
$897$	0	0
$898$	0	0
$899$	27.9006	0.930539
$900$	0	0
$901$	31.1923	1.03917
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.15966	−0.138272
$906$	0	0
$907$	−41.5359	−1.37918	−0.689588	−	0.724202i	$-0.742208\pi$
$907$	−41.5359	−1.37918	−0.689588	+	0.724202i	$0.742208\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.92749	−0.0969921	−0.0484961	−	0.998823i	$-0.515443\pi$
$911$	−2.92749	−0.0969921	−0.0484961	+	0.998823i	$0.515443\pi$
$912$	0	0
$913$	−3.62523	−0.119978
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.0910	−0.465326
$918$	0	0
$919$	13.9648	0.460658	0.230329	−	0.973113i	$-0.426020\pi$
$919$	13.9648	0.460658	0.230329	+	0.973113i	$0.426020\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	49.1527	1.61788
$924$	0	0
$925$	−21.9537	−0.721834
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23.3541	0.766222	0.383111	−	0.923702i	$-0.374853\pi$
$929$	23.3541	0.766222	0.383111	+	0.923702i	$0.374853\pi$
$930$	0	0
$931$	3.39461	0.111254
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−42.0252	−1.37437
$936$	0	0
$937$	11.1156	0.363130	0.181565	−	0.983379i	$-0.441884\pi$
$937$	11.1156	0.363130	0.181565	+	0.983379i	$0.441884\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.13868	−0.167516	−0.0837580	−	0.996486i	$-0.526692\pi$
$941$	−5.13868	−0.167516	−0.0837580	+	0.996486i	$0.526692\pi$
$942$	0	0
$943$	5.89583	0.191995
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.4147	−0.955851	−0.477925	−	0.878400i	$-0.658611\pi$
$947$	−29.4147	−0.955851	−0.477925	+	0.878400i	$0.658611\pi$
$948$	0	0
$949$	−44.7350	−1.45216
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.67396	0.183797	0.0918987	−	0.995768i	$-0.470706\pi$
$953$	5.67396	0.183797	0.0918987	+	0.995768i	$0.470706\pi$
$954$	0	0
$955$	67.4967	2.18414
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.67250	0.0862997
$960$	0	0
$961$	17.6067	0.567959
$962$	0	0
$963$	0	0
$964$	0	0
$965$	55.9723	1.80181
$966$	0	0
$967$	−0.898731	−0.0289013	−0.0144506	−	0.999896i	$-0.504600\pi$
$967$	−0.898731	−0.0289013	−0.0144506	+	0.999896i	$0.504600\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−34.2084	−1.09780	−0.548899	−	0.835889i	$-0.684953\pi$
$971$	−34.2084	−1.09780	−0.548899	+	0.835889i	$0.684953\pi$
$972$	0	0
$973$	−6.23315	−0.199825
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.19697	−0.262244	−0.131122	−	0.991366i	$-0.541858\pi$
$977$	−8.19697	−0.262244	−0.131122	+	0.991366i	$0.541858\pi$
$978$	0	0
$979$	56.6994	1.81212
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.12997	0.0360406	0.0180203	−	0.999838i	$-0.494264\pi$
$983$	1.12997	0.0360406	0.0180203	+	0.999838i	$0.494264\pi$
$984$	0	0
$985$	3.28637	0.104712
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.90838	−0.251472
$990$	0	0
$991$	14.6766	0.466218	0.233109	−	0.972451i	$-0.425110\pi$
$991$	14.6766	0.466218	0.233109	+	0.972451i	$0.425110\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.9250	−0.536558
$996$	0	0
$997$	−1.64207	−0.0520049	−0.0260024	−	0.999662i	$-0.508278\pi$
$997$	−1.64207	−0.0520049	−0.0260024	+	0.999662i	$0.508278\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5796.2.a.t.1.4		5
3.2	odd	2		644.2.a.d.1.5	✓	5
12.11	even	2		2576.2.a.bb.1.1		5
21.20	even	2		4508.2.a.f.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.d.1.5	✓	5	3.2	odd	2
2576.2.a.bb.1.1		5	12.11	even	2
4508.2.a.f.1.1		5	21.20	even	2
5796.2.a.t.1.4		5	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:	\( N \)	\(=\)	\( 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5796.a (trivial)

\(f(q)\)	\(=\)	\(q+3.04771 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+3.04771 q^{5} -1.00000 q^{7} -5.22523 q^{11} +3.38206 q^{13} +2.63895 q^{17} +3.39461 q^{19} +1.00000 q^{23} +4.28854 q^{25} +4.00190 q^{29} +6.97185 q^{31} -3.04771 q^{35} -5.11916 q^{37} +5.89583 q^{41} -7.90838 q^{43} -9.54893 q^{47} +1.00000 q^{49} +11.8200 q^{53} -15.9250 q^{55} -1.51979 q^{59} +1.13934 q^{61} +10.3076 q^{65} -8.80688 q^{67} +14.5333 q^{71} -13.2271 q^{73} +5.22523 q^{77} +14.4398 q^{79} +0.693795 q^{83} +8.04275 q^{85} -10.8511 q^{89} -3.38206 q^{91} +10.3458 q^{95} -3.23476 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} - 5 q^{7}+O(q^{10}) \) 5 * q - 2 * q^5 - 5 * q^7	\( 5 q - 2 q^{5} - 5 q^{7} - 2 q^{11} + 13 q^{13} - 4 q^{17} + 12 q^{19} + 5 q^{23} + 19 q^{25} - 13 q^{29} - 3 q^{31} + 2 q^{35} - 4 q^{37} - q^{41} - 8 q^{43} - 5 q^{47} + 5 q^{49} + 8 q^{53} - 2 q^{55} - 12 q^{59} + 20 q^{61} + 12 q^{65} - 12 q^{67} - 9 q^{71} - 9 q^{73} + 2 q^{77} - 8 q^{79} + 28 q^{83} + 16 q^{85} - 32 q^{89} - 13 q^{91} + 36 q^{95} + 4 q^{97}+O(q^{100}) \) 5 * q - 2 * q^5 - 5 * q^7 - 2 * q^11 + 13 * q^13 - 4 * q^17 + 12 * q^19 + 5 * q^23 + 19 * q^25 - 13 * q^29 - 3 * q^31 + 2 * q^35 - 4 * q^37 - q^41 - 8 * q^43 - 5 * q^47 + 5 * q^49 + 8 * q^53 - 2 * q^55 - 12 * q^59 + 20 * q^61 + 12 * q^65 - 12 * q^67 - 9 * q^71 - 9 * q^73 + 2 * q^77 - 8 * q^79 + 28 * q^83 + 16 * q^85 - 32 * q^89 - 13 * q^91 + 36 * q^95 + 4 * q^97

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