LMFDB - Embedded newform 5376.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5376,2,Mod(1,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, 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("5376.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5376 = 2^{8} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5376.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:	$42.9275761266$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.19664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 2x + 2 $ x^4 - 2x^3 - 5x^2 + 2*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.512486$ of defining polynomial
Character	$\chi$	$=$	5376.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.10245 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.10245 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.67767 q^{11} -3.02497 q^{13} +4.10245 q^{15} -5.12742 q^{17} +2.78012 q^{19} -1.00000 q^{21} +7.12742 q^{23} +11.8301 q^{25} -1.00000 q^{27} +8.83006 q^{29} +1.42477 q^{31} +2.67767 q^{33} -4.10245 q^{35} +1.42477 q^{37} +3.02497 q^{39} -5.12742 q^{41} -2.39980 q^{43} -4.10245 q^{45} +9.56024 q^{47} +1.00000 q^{49} +5.12742 q^{51} -2.78012 q^{53} +10.9850 q^{55} -2.78012 q^{57} -4.00000 q^{59} +5.17992 q^{61} +1.00000 q^{63} +12.4098 q^{65} -0.244852 q^{67} -7.12742 q^{69} -4.27787 q^{71} +4.15495 q^{73} -11.8301 q^{75} -2.67767 q^{77} -6.25484 q^{79} +1.00000 q^{81} -9.35535 q^{83} +21.0350 q^{85} -8.83006 q^{87} -11.2824 q^{89} -3.02497 q^{91} -1.42477 q^{93} -11.4053 q^{95} +6.69460 q^{97} -2.67767 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9} - 6 q^{11} - 4 q^{13} + 2 q^{15} + 2 q^{17} - 8 q^{19} - 4 q^{21} + 6 q^{23} + 12 q^{25} - 4 q^{27} - 4 q^{31} + 6 q^{33} - 2 q^{35} - 4 q^{37} + 4 q^{39} + 2 q^{41} - 8 q^{43} - 2 q^{45} + 4 q^{49} - 2 q^{51} + 8 q^{53} - 4 q^{55} + 8 q^{57} - 16 q^{59} + 4 q^{63} - 8 q^{65} - 12 q^{67} - 6 q^{69} - 14 q^{71} + 4 q^{73} - 12 q^{75} - 6 q^{77} + 20 q^{79} + 4 q^{81} - 28 q^{83} + 20 q^{85} - 10 q^{89} - 4 q^{91} + 4 q^{93} - 20 q^{95} + 20 q^{97} - 6 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9 - 6 * q^11 - 4 * q^13 + 2 * q^15 + 2 * q^17 - 8 * q^19 - 4 * q^21 + 6 * q^23 + 12 * q^25 - 4 * q^27 - 4 * q^31 + 6 * q^33 - 2 * q^35 - 4 * q^37 + 4 * q^39 + 2 * q^41 - 8 * q^43 - 2 * q^45 + 4 * q^49 - 2 * q^51 + 8 * q^53 - 4 * q^55 + 8 * q^57 - 16 * q^59 + 4 * q^63 - 8 * q^65 - 12 * q^67 - 6 * q^69 - 14 * q^71 + 4 * q^73 - 12 * q^75 - 6 * q^77 + 20 * q^79 + 4 * q^81 - 28 * q^83 + 20 * q^85 - 10 * q^89 - 4 * q^91 + 4 * q^93 - 20 * q^95 + 20 * q^97 - 6 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.10245	−1.83467	−0.917335	−	0.398117i	$-0.869664\pi$
$5$	−4.10245	−1.83467	−0.917335	+	0.398117i	$0.869664\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.67767	−0.807349	−0.403674	−	0.914903i	$-0.632267\pi$
$11$	−2.67767	−0.807349	−0.403674	+	0.914903i	$0.632267\pi$
$12$	0	0
$13$	−3.02497	−0.838976	−0.419488	−	0.907761i	$-0.637790\pi$
$13$	−3.02497	−0.838976	−0.419488	+	0.907761i	$0.637790\pi$
$14$	0	0
$15$	4.10245	1.05925
$16$	0	0
$17$	−5.12742	−1.24358	−0.621791	−	0.783183i	$-0.713595\pi$
$17$	−5.12742	−1.24358	−0.621791	+	0.783183i	$0.713595\pi$
$18$	0	0
$19$	2.78012	0.637803	0.318902	−	0.947788i	$-0.396686\pi$
$19$	2.78012	0.637803	0.318902	+	0.947788i	$0.396686\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	7.12742	1.48617	0.743085	−	0.669197i	$-0.233362\pi$
$23$	7.12742	1.48617	0.743085	+	0.669197i	$0.233362\pi$
$24$	0	0
$25$	11.8301	2.36601
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.83006	1.63970	0.819851	−	0.572577i	$-0.194056\pi$
$29$	8.83006	1.63970	0.819851	+	0.572577i	$0.194056\pi$
$30$	0	0
$31$	1.42477	0.255897	0.127948	−	0.991781i	$-0.459161\pi$
$31$	1.42477	0.255897	0.127948	+	0.991781i	$0.459161\pi$
$32$	0	0
$33$	2.67767	0.466123
$34$	0	0
$35$	−4.10245	−0.693440
$36$	0	0
$37$	1.42477	0.234231	0.117116	−	0.993118i	$-0.462635\pi$
$37$	1.42477	0.234231	0.117116	+	0.993118i	$0.462635\pi$
$38$	0	0
$39$	3.02497	0.484383
$40$	0	0
$41$	−5.12742	−0.800768	−0.400384	−	0.916347i	$-0.631123\pi$
$41$	−5.12742	−0.800768	−0.400384	+	0.916347i	$0.631123\pi$
$42$	0	0
$43$	−2.39980	−0.365966	−0.182983	−	0.983116i	$-0.558575\pi$
$43$	−2.39980	−0.365966	−0.182983	+	0.983116i	$0.558575\pi$
$44$	0	0
$45$	−4.10245	−0.611557
$46$	0	0
$47$	9.56024	1.39450	0.697252	−	0.716826i	$-0.254406\pi$
$47$	9.56024	1.39450	0.697252	+	0.716826i	$0.254406\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.12742	0.717982
$52$	0	0
$53$	−2.78012	−0.381879	−0.190939	−	0.981602i	$-0.561153\pi$
$53$	−2.78012	−0.381879	−0.190939	+	0.981602i	$0.561153\pi$
$54$	0	0
$55$	10.9850	1.48122
$56$	0	0
$57$	−2.78012	−0.368236
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	5.17992	0.663221	0.331610	−	0.943416i	$-0.392408\pi$
$61$	5.17992	0.663221	0.331610	+	0.943416i	$0.392408\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	12.4098	1.53924
$66$	0	0
$67$	−0.244852	−0.0299135	−0.0149567	−	0.999888i	$-0.504761\pi$
$67$	−0.244852	−0.0299135	−0.0149567	+	0.999888i	$0.504761\pi$
$68$	0	0
$69$	−7.12742	−0.858040
$70$	0	0
$71$	−4.27787	−0.507690	−0.253845	−	0.967245i	$-0.581695\pi$
$71$	−4.27787	−0.507690	−0.253845	+	0.967245i	$0.581695\pi$
$72$	0	0
$73$	4.15495	0.486300	0.243150	−	0.969989i	$-0.421819\pi$
$73$	4.15495	0.486300	0.243150	+	0.969989i	$0.421819\pi$
$74$	0	0
$75$	−11.8301	−1.36602
$76$	0	0
$77$	−2.67767	−0.305149
$78$	0	0
$79$	−6.25484	−0.703724	−0.351862	−	0.936052i	$-0.614451\pi$
$79$	−6.25484	−0.703724	−0.351862	+	0.936052i	$0.614451\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.35535	−1.02688	−0.513441	−	0.858125i	$-0.671630\pi$
$83$	−9.35535	−1.02688	−0.513441	+	0.858125i	$0.671630\pi$
$84$	0	0
$85$	21.0350	2.28156
$86$	0	0
$87$	−8.83006	−0.946682
$88$	0	0
$89$	−11.2824	−1.19593	−0.597964	−	0.801523i	$-0.704024\pi$
$89$	−11.2824	−1.19593	−0.597964	+	0.801523i	$0.704024\pi$
$90$	0	0
$91$	−3.02497	−0.317103
$92$	0	0
$93$	−1.42477	−0.147742
$94$	0	0
$95$	−11.4053	−1.17016
$96$	0	0
$97$	6.69460	0.679733	0.339867	−	0.940474i	$-0.389618\pi$
$97$	6.69460	0.679733	0.339867	+	0.940474i	$0.389618\pi$
$98$	0	0
$99$	−2.67767	−0.269116
$100$	0	0
$101$	1.45779	0.145056	0.0725279	−	0.997366i	$-0.476893\pi$
$101$	1.45779	0.145056	0.0725279	+	0.997366i	$0.476893\pi$
$102$	0	0
$103$	8.13547	0.801611	0.400806	−	0.916163i	$-0.368730\pi$
$103$	8.13547	0.801611	0.400806	+	0.916163i	$0.368730\pi$
$104$	0	0
$105$	4.10245	0.400358
$106$	0	0
$107$	−5.32233	−0.514529	−0.257264	−	0.966341i	$-0.582821\pi$
$107$	−5.32233	−0.514529	−0.257264	+	0.966341i	$0.582821\pi$
$108$	0	0
$109$	13.5602	1.29884	0.649418	−	0.760432i	$-0.275013\pi$
$109$	13.5602	1.29884	0.649418	+	0.760432i	$0.275013\pi$
$110$	0	0
$111$	−1.42477	−0.135233
$112$	0	0
$113$	−4.15495	−0.390865	−0.195432	−	0.980717i	$-0.562611\pi$
$113$	−4.15495	−0.390865	−0.195432	+	0.980717i	$0.562611\pi$
$114$	0	0
$115$	−29.2398	−2.72663
$116$	0	0
$117$	−3.02497	−0.279659
$118$	0	0
$119$	−5.12742	−0.470030
$120$	0	0
$121$	−3.83006	−0.348188
$122$	0	0
$123$	5.12742	0.462324
$124$	0	0
$125$	−28.0200	−2.50618
$126$	0	0
$127$	0.694597	0.0616355	0.0308177	−	0.999525i	$-0.490189\pi$
$127$	0.694597	0.0616355	0.0308177	+	0.999525i	$0.490189\pi$
$128$	0	0
$129$	2.39980	0.211291
$130$	0	0
$131$	−12.2049	−1.06635	−0.533173	−	0.846006i	$-0.679001\pi$
$131$	−12.2049	−1.06635	−0.533173	+	0.846006i	$0.679001\pi$
$132$	0	0
$133$	2.78012	0.241067
$134$	0	0
$135$	4.10245	0.353082
$136$	0	0
$137$	15.1044	1.29045	0.645227	−	0.763991i	$-0.276763\pi$
$137$	15.1044	1.29045	0.645227	+	0.763991i	$0.276763\pi$
$138$	0	0
$139$	7.61018	0.645487	0.322744	−	0.946486i	$-0.395395\pi$
$139$	7.61018	0.645487	0.322744	+	0.946486i	$0.395395\pi$
$140$	0	0
$141$	−9.56024	−0.805117
$142$	0	0
$143$	8.09989	0.677347
$144$	0	0
$145$	−36.2249	−3.00831
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	5.21988	0.427629	0.213815	−	0.976874i	$-0.431411\pi$
$149$	5.21988	0.427629	0.213815	+	0.976874i	$0.431411\pi$
$150$	0	0
$151$	−5.56024	−0.452486	−0.226243	−	0.974071i	$-0.572644\pi$
$151$	−5.56024	−0.452486	−0.226243	+	0.974071i	$0.572644\pi$
$152$	0	0
$153$	−5.12742	−0.414527
$154$	0	0
$155$	−5.84505	−0.469486
$156$	0	0
$157$	0.519169	0.0414342	0.0207171	−	0.999785i	$-0.493405\pi$
$157$	0.519169	0.0414342	0.0207171	+	0.999785i	$0.493405\pi$
$158$	0	0
$159$	2.78012	0.220478
$160$	0	0
$161$	7.12742	0.561719
$162$	0	0
$163$	8.65464	0.677883	0.338942	−	0.940807i	$-0.389931\pi$
$163$	8.65464	0.677883	0.338942	+	0.940807i	$0.389931\pi$
$164$	0	0
$165$	−10.9850	−0.855182
$166$	0	0
$167$	−25.1044	−1.94264	−0.971318	−	0.237786i	$-0.923578\pi$
$167$	−25.1044	−1.94264	−0.971318	+	0.237786i	$0.923578\pi$
$168$	0	0
$169$	−3.84954	−0.296119
$170$	0	0
$171$	2.78012	0.212601
$172$	0	0
$173$	1.94750	0.148066	0.0740328	−	0.997256i	$-0.476413\pi$
$173$	1.94750	0.148066	0.0740328	+	0.997256i	$0.476413\pi$
$174$	0	0
$175$	11.8301	0.894269
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	22.6976	1.69650	0.848251	−	0.529595i	$-0.177656\pi$
$179$	22.6976	1.69650	0.848251	+	0.529595i	$0.177656\pi$
$180$	0	0
$181$	10.5353	0.783080	0.391540	−	0.920161i	$-0.371942\pi$
$181$	10.5353	0.783080	0.391540	+	0.920161i	$0.371942\pi$
$182$	0	0
$183$	−5.17992	−0.382911
$184$	0	0
$185$	−5.84505	−0.429737
$186$	0	0
$187$	13.7296	1.00400
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	4.27787	0.309536	0.154768	−	0.987951i	$-0.450537\pi$
$191$	4.27787	0.309536	0.154768	+	0.987951i	$0.450537\pi$
$192$	0	0
$193$	23.1400	1.66565	0.832825	−	0.553536i	$-0.186722\pi$
$193$	23.1400	1.66565	0.832825	+	0.553536i	$0.186722\pi$
$194$	0	0
$195$	−12.4098	−0.888683
$196$	0	0
$197$	−19.4747	−1.38752	−0.693758	−	0.720208i	$-0.744046\pi$
$197$	−19.4747	−1.38752	−0.693758	+	0.720208i	$0.744046\pi$
$198$	0	0
$199$	−17.5602	−1.24481	−0.622406	−	0.782694i	$-0.713845\pi$
$199$	−17.5602	−1.24481	−0.622406	+	0.782694i	$0.713845\pi$
$200$	0	0
$201$	0.244852	0.0172705
$202$	0	0
$203$	8.83006	0.619749
$204$	0	0
$205$	21.0350	1.46915
$206$	0	0
$207$	7.12742	0.495390
$208$	0	0
$209$	−7.44425	−0.514930
$210$	0	0
$211$	−23.9600	−1.64948	−0.824739	−	0.565514i	$-0.808678\pi$
$211$	−23.9600	−1.64948	−0.824739	+	0.565514i	$0.808678\pi$
$212$	0	0
$213$	4.27787	0.293115
$214$	0	0
$215$	9.84505	0.671427
$216$	0	0
$217$	1.42477	0.0967199
$218$	0	0
$219$	−4.15495	−0.280765
$220$	0	0
$221$	15.5103	1.04334
$222$	0	0
$223$	−2.74966	−0.184131	−0.0920653	−	0.995753i	$-0.529347\pi$
$223$	−2.74966	−0.184131	−0.0920653	+	0.995753i	$0.529347\pi$
$224$	0	0
$225$	11.8301	0.788671
$226$	0	0
$227$	8.30478	0.551208	0.275604	−	0.961271i	$-0.411122\pi$
$227$	8.30478	0.551208	0.275604	+	0.961271i	$0.411122\pi$
$228$	0	0
$229$	11.9245	0.787991	0.393995	−	0.919112i	$-0.371093\pi$
$229$	11.9245	0.787991	0.393995	+	0.919112i	$0.371093\pi$
$230$	0	0
$231$	2.67767	0.176178
$232$	0	0
$233$	3.56024	0.233239	0.116620	−	0.993177i	$-0.462794\pi$
$233$	3.56024	0.233239	0.116620	+	0.993177i	$0.462794\pi$
$234$	0	0
$235$	−39.2204	−2.55845
$236$	0	0
$237$	6.25484	0.406295
$238$	0	0
$239$	−26.9425	−1.74277	−0.871383	−	0.490604i	$-0.836776\pi$
$239$	−26.9425	−1.74277	−0.871383	+	0.490604i	$0.836776\pi$
$240$	0	0
$241$	25.8151	1.66290	0.831448	−	0.555603i	$-0.187513\pi$
$241$	25.8151	1.66290	0.831448	+	0.555603i	$0.187513\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.10245	−0.262096
$246$	0	0
$247$	−8.40978	−0.535102
$248$	0	0
$249$	9.35535	0.592871
$250$	0	0
$251$	−27.1543	−1.71397	−0.856983	−	0.515345i	$-0.827664\pi$
$251$	−27.1543	−1.71397	−0.856983	+	0.515345i	$0.827664\pi$
$252$	0	0
$253$	−19.0849	−1.19986
$254$	0	0
$255$	−21.0350	−1.31726
$256$	0	0
$257$	−31.3823	−1.95757	−0.978786	−	0.204887i	$-0.934317\pi$
$257$	−31.3823	−1.95757	−0.978786	+	0.204887i	$0.934317\pi$
$258$	0	0
$259$	1.42477	0.0885310
$260$	0	0
$261$	8.83006	0.546567
$262$	0	0
$263$	−11.0275	−0.679987	−0.339993	−	0.940428i	$-0.610425\pi$
$263$	−11.0275	−0.679987	−0.339993	+	0.940428i	$0.610425\pi$
$264$	0	0
$265$	11.4053	0.700621
$266$	0	0
$267$	11.2824	0.690470
$268$	0	0
$269$	−21.0019	−1.28051	−0.640255	−	0.768162i	$-0.721171\pi$
$269$	−21.0019	−1.28051	−0.640255	+	0.768162i	$0.721171\pi$
$270$	0	0
$271$	−28.6451	−1.74007	−0.870034	−	0.492991i	$-0.835903\pi$
$271$	−28.6451	−1.74007	−0.870034	+	0.492991i	$0.835903\pi$
$272$	0	0
$273$	3.02497	0.183080
$274$	0	0
$275$	−31.6771	−1.91020
$276$	0	0
$277$	2.96504	0.178152	0.0890761	−	0.996025i	$-0.471609\pi$
$277$	2.96504	0.178152	0.0890761	+	0.996025i	$0.471609\pi$
$278$	0	0
$279$	1.42477	0.0852989
$280$	0	0
$281$	−26.4098	−1.57548	−0.787738	−	0.616011i	$-0.788748\pi$
$281$	−26.4098	−1.57548	−0.787738	+	0.616011i	$0.788748\pi$
$282$	0	0
$283$	−11.2698	−0.669922	−0.334961	−	0.942232i	$-0.608723\pi$
$283$	−11.2698	−0.669922	−0.334961	+	0.942232i	$0.608723\pi$
$284$	0	0
$285$	11.4053	0.675591
$286$	0	0
$287$	−5.12742	−0.302662
$288$	0	0
$289$	9.29042	0.546495
$290$	0	0
$291$	−6.69460	−0.392444
$292$	0	0
$293$	10.5622	0.617049	0.308524	−	0.951216i	$-0.400165\pi$
$293$	10.5622	0.617049	0.308524	+	0.951216i	$0.400165\pi$
$294$	0	0
$295$	16.4098	0.955415
$296$	0	0
$297$	2.67767	0.155374
$298$	0	0
$299$	−21.5602	−1.24686
$300$	0	0
$301$	−2.39980	−0.138322
$302$	0	0
$303$	−1.45779	−0.0837480
$304$	0	0
$305$	−21.2503	−1.21679
$306$	0	0
$307$	−10.7801	−0.615254	−0.307627	−	0.951507i	$-0.599535\pi$
$307$	−10.7801	−0.615254	−0.307627	+	0.951507i	$0.599535\pi$
$308$	0	0
$309$	−8.13547	−0.462811
$310$	0	0
$311$	9.56024	0.542111	0.271056	−	0.962564i	$-0.412627\pi$
$311$	9.56024	0.542111	0.271056	+	0.962564i	$0.412627\pi$
$312$	0	0
$313$	−7.69909	−0.435178	−0.217589	−	0.976040i	$-0.569819\pi$
$313$	−7.69909	−0.435178	−0.217589	+	0.976040i	$0.569819\pi$
$314$	0	0
$315$	−4.10245	−0.231147
$316$	0	0
$317$	10.7801	0.605472	0.302736	−	0.953074i	$-0.402100\pi$
$317$	10.7801	0.605472	0.302736	+	0.953074i	$0.402100\pi$
$318$	0	0
$319$	−23.6440	−1.32381
$320$	0	0
$321$	5.32233	0.297063
$322$	0	0
$323$	−14.2548	−0.793160
$324$	0	0
$325$	−35.7856	−1.98503
$326$	0	0
$327$	−13.5602	−0.749883
$328$	0	0
$329$	9.56024	0.527073
$330$	0	0
$331$	−25.2104	−1.38569	−0.692844	−	0.721088i	$-0.743643\pi$
$331$	−25.2104	−1.38569	−0.692844	+	0.721088i	$0.743643\pi$
$332$	0	0
$333$	1.42477	0.0780770
$334$	0	0
$335$	1.00449	0.0548813
$336$	0	0
$337$	−23.9700	−1.30573	−0.652865	−	0.757474i	$-0.726433\pi$
$337$	−23.9700	−1.30573	−0.652865	+	0.757474i	$0.726433\pi$
$338$	0	0
$339$	4.15495	0.225666
$340$	0	0
$341$	−3.81508	−0.206598
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	29.2398	1.57422
$346$	0	0
$347$	−6.98757	−0.375112	−0.187556	−	0.982254i	$-0.560057\pi$
$347$	−6.98757	−0.375112	−0.187556	+	0.982254i	$0.560057\pi$
$348$	0	0
$349$	20.8900	1.11822	0.559108	−	0.829095i	$-0.311144\pi$
$349$	20.8900	1.11822	0.559108	+	0.829095i	$0.311144\pi$
$350$	0	0
$351$	3.02497	0.161461
$352$	0	0
$353$	3.38225	0.180019	0.0900096	−	0.995941i	$-0.471310\pi$
$353$	3.38225	0.180019	0.0900096	+	0.995941i	$0.471310\pi$
$354$	0	0
$355$	17.5497	0.931444
$356$	0	0
$357$	5.12742	0.271372
$358$	0	0
$359$	20.6877	1.09185	0.545926	−	0.837833i	$-0.316178\pi$
$359$	20.6877	1.09185	0.545926	+	0.837833i	$0.316178\pi$
$360$	0	0
$361$	−11.2709	−0.593207
$362$	0	0
$363$	3.83006	0.201026
$364$	0	0
$365$	−17.0455	−0.892200
$366$	0	0
$367$	9.56024	0.499040	0.249520	−	0.968370i	$-0.419727\pi$
$367$	9.56024	0.499040	0.249520	+	0.968370i	$0.419727\pi$
$368$	0	0
$369$	−5.12742	−0.266923
$370$	0	0
$371$	−2.78012	−0.144337
$372$	0	0
$373$	1.95006	0.100970	0.0504850	−	0.998725i	$-0.483923\pi$
$373$	1.95006	0.100970	0.0504850	+	0.998725i	$0.483923\pi$
$374$	0	0
$375$	28.0200	1.44694
$376$	0	0
$377$	−26.7107	−1.37567
$378$	0	0
$379$	−32.3698	−1.66273	−0.831363	−	0.555730i	$-0.812439\pi$
$379$	−32.3698	−1.66273	−0.831363	+	0.555730i	$0.812439\pi$
$380$	0	0
$381$	−0.694597	−0.0355853
$382$	0	0
$383$	17.8451	0.911840	0.455920	−	0.890021i	$-0.349310\pi$
$383$	17.8451	0.911840	0.455920	+	0.890021i	$0.349310\pi$
$384$	0	0
$385$	10.9850	0.559848
$386$	0	0
$387$	−2.39980	−0.121989
$388$	0	0
$389$	6.11937	0.310264	0.155132	−	0.987894i	$-0.450420\pi$
$389$	6.11937	0.310264	0.155132	+	0.987894i	$0.450420\pi$
$390$	0	0
$391$	−36.5453	−1.84817
$392$	0	0
$393$	12.2049	0.615655
$394$	0	0
$395$	25.6601	1.29110
$396$	0	0
$397$	−3.71957	−0.186680	−0.0933399	−	0.995634i	$-0.529754\pi$
$397$	−3.71957	−0.186680	−0.0933399	+	0.995634i	$0.529754\pi$
$398$	0	0
$399$	−2.78012	−0.139180
$400$	0	0
$401$	−32.5258	−1.62426	−0.812130	−	0.583477i	$-0.801692\pi$
$401$	−32.5258	−1.62426	−0.812130	+	0.583477i	$0.801692\pi$
$402$	0	0
$403$	−4.30990	−0.214691
$404$	0	0
$405$	−4.10245	−0.203852
$406$	0	0
$407$	−3.81508	−0.189106
$408$	0	0
$409$	1.13436	0.0560903	0.0280452	−	0.999607i	$-0.491072\pi$
$409$	1.13436	0.0560903	0.0280452	+	0.999607i	$0.491072\pi$
$410$	0	0
$411$	−15.1044	−0.745044
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	38.3798	1.88399
$416$	0	0
$417$	−7.61018	−0.372672
$418$	0	0
$419$	1.56024	0.0762227	0.0381113	−	0.999273i	$-0.487866\pi$
$419$	1.56024	0.0762227	0.0381113	+	0.999273i	$0.487866\pi$
$420$	0	0
$421$	−31.8845	−1.55396	−0.776978	−	0.629528i	$-0.783248\pi$
$421$	−31.8845	−1.55396	−0.776978	+	0.629528i	$0.783248\pi$
$422$	0	0
$423$	9.56024	0.464835
$424$	0	0
$425$	−60.6577	−2.94233
$426$	0	0
$427$	5.17992	0.250674
$428$	0	0
$429$	−8.09989	−0.391066
$430$	0	0
$431$	−8.87258	−0.427377	−0.213689	−	0.976902i	$-0.568548\pi$
$431$	−8.87258	−0.427377	−0.213689	+	0.976902i	$0.568548\pi$
$432$	0	0
$433$	11.5602	0.555550	0.277775	−	0.960646i	$-0.410403\pi$
$433$	11.5602	0.555550	0.277775	+	0.960646i	$0.410403\pi$
$434$	0	0
$435$	36.2249	1.73685
$436$	0	0
$437$	19.8151	0.947884
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−5.87807	−0.279276	−0.139638	−	0.990203i	$-0.544594\pi$
$443$	−5.87807	−0.279276	−0.139638	+	0.990203i	$0.544594\pi$
$444$	0	0
$445$	46.2853	2.19413
$446$	0	0
$447$	−5.21988	−0.246892
$448$	0	0
$449$	−11.8451	−0.559003	−0.279501	−	0.960145i	$-0.590169\pi$
$449$	−11.8451	−0.559003	−0.279501	+	0.960145i	$0.590169\pi$
$450$	0	0
$451$	13.7296	0.646499
$452$	0	0
$453$	5.56024	0.261243
$454$	0	0
$455$	12.4098	0.581780
$456$	0	0
$457$	28.2088	1.31955	0.659775	−	0.751463i	$-0.270652\pi$
$457$	28.2088	1.31955	0.659775	+	0.751463i	$0.270652\pi$
$458$	0	0
$459$	5.12742	0.239327
$460$	0	0
$461$	11.0519	0.514737	0.257369	−	0.966313i	$-0.417145\pi$
$461$	11.0519	0.514737	0.257369	+	0.966313i	$0.417145\pi$
$462$	0	0
$463$	22.2548	1.03427	0.517135	−	0.855904i	$-0.326999\pi$
$463$	22.2548	1.03427	0.517135	+	0.855904i	$0.326999\pi$
$464$	0	0
$465$	5.84505	0.271058
$466$	0	0
$467$	21.4552	0.992830	0.496415	−	0.868085i	$-0.334649\pi$
$467$	21.4552	0.992830	0.496415	+	0.868085i	$0.334649\pi$
$468$	0	0
$469$	−0.244852	−0.0113062
$470$	0	0
$471$	−0.519169	−0.0239221
$472$	0	0
$473$	6.42588	0.295462
$474$	0	0
$475$	32.8890	1.50905
$476$	0	0
$477$	−2.78012	−0.127293
$478$	0	0
$479$	−8.09989	−0.370093	−0.185047	−	0.982730i	$-0.559244\pi$
$479$	−8.09989	−0.370093	−0.185047	+	0.982730i	$0.559244\pi$
$480$	0	0
$481$	−4.30990	−0.196514
$482$	0	0
$483$	−7.12742	−0.324309
$484$	0	0
$485$	−27.4642	−1.24709
$486$	0	0
$487$	−16.4098	−0.743598	−0.371799	−	0.928313i	$-0.621259\pi$
$487$	−16.4098	−0.743598	−0.371799	+	0.928313i	$0.621259\pi$
$488$	0	0
$489$	−8.65464	−0.391376
$490$	0	0
$491$	39.5971	1.78699	0.893497	−	0.449070i	$-0.148245\pi$
$491$	39.5971	1.78699	0.893497	+	0.449070i	$0.148245\pi$
$492$	0	0
$493$	−45.2754	−2.03910
$494$	0	0
$495$	10.9850	0.493740
$496$	0	0
$497$	−4.27787	−0.191889
$498$	0	0
$499$	−1.14946	−0.0514568	−0.0257284	−	0.999669i	$-0.508191\pi$
$499$	−1.14946	−0.0514568	−0.0257284	+	0.999669i	$0.508191\pi$
$500$	0	0
$501$	25.1044	1.12158
$502$	0	0
$503$	9.00449	0.401490	0.200745	−	0.979643i	$-0.435664\pi$
$503$	9.00449	0.401490	0.200745	+	0.979643i	$0.435664\pi$
$504$	0	0
$505$	−5.98052	−0.266130
$506$	0	0
$507$	3.84954	0.170964
$508$	0	0
$509$	36.5122	1.61838	0.809188	−	0.587550i	$-0.199907\pi$
$509$	36.5122	1.61838	0.809188	+	0.587550i	$0.199907\pi$
$510$	0	0
$511$	4.15495	0.183804
$512$	0	0
$513$	−2.78012	−0.122745
$514$	0	0
$515$	−33.3753	−1.47069
$516$	0	0
$517$	−25.5992	−1.12585
$518$	0	0
$519$	−1.94750	−0.0854857
$520$	0	0
$521$	18.0929	0.792666	0.396333	−	0.918107i	$-0.370282\pi$
$521$	18.0929	0.792666	0.396333	+	0.918107i	$0.370282\pi$
$522$	0	0
$523$	14.7107	0.643254	0.321627	−	0.946866i	$-0.395770\pi$
$523$	14.7107	0.643254	0.321627	+	0.946866i	$0.395770\pi$
$524$	0	0
$525$	−11.8301	−0.516306
$526$	0	0
$527$	−7.30540	−0.318228
$528$	0	0
$529$	27.8001	1.20870
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	15.5103	0.671825
$534$	0	0
$535$	21.8346	0.943990
$536$	0	0
$537$	−22.6976	−0.979476
$538$	0	0
$539$	−2.67767	−0.115336
$540$	0	0
$541$	−37.7157	−1.62152	−0.810762	−	0.585376i	$-0.800947\pi$
$541$	−37.7157	−1.62152	−0.810762	+	0.585376i	$0.800947\pi$
$542$	0	0
$543$	−10.5353	−0.452112
$544$	0	0
$545$	−55.6302	−2.38293
$546$	0	0
$547$	−23.1144	−0.988299	−0.494149	−	0.869377i	$-0.664520\pi$
$547$	−23.1144	−0.988299	−0.494149	+	0.869377i	$0.664520\pi$
$548$	0	0
$549$	5.17992	0.221074
$550$	0	0
$551$	24.5486	1.04581
$552$	0	0
$553$	−6.25484	−0.265983
$554$	0	0
$555$	5.84505	0.248109
$556$	0	0
$557$	−11.1899	−0.474131	−0.237066	−	0.971494i	$-0.576186\pi$
$557$	−11.1899	−0.474131	−0.237066	+	0.971494i	$0.576186\pi$
$558$	0	0
$559$	7.25933	0.307037
$560$	0	0
$561$	−13.7296	−0.579662
$562$	0	0
$563$	−2.81058	−0.118452	−0.0592260	−	0.998245i	$-0.518863\pi$
$563$	−2.81058	−0.118452	−0.0592260	+	0.998245i	$0.518863\pi$
$564$	0	0
$565$	17.0455	0.717108
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	20.0699	0.841374	0.420687	−	0.907206i	$-0.361789\pi$
$569$	20.0699	0.841374	0.420687	+	0.907206i	$0.361789\pi$
$570$	0	0
$571$	2.46584	0.103192	0.0515961	−	0.998668i	$-0.483569\pi$
$571$	2.46584	0.103192	0.0515961	+	0.998668i	$0.483569\pi$
$572$	0	0
$573$	−4.27787	−0.178711
$574$	0	0
$575$	84.3178	3.51630
$576$	0	0
$577$	4.84954	0.201889	0.100945	−	0.994892i	$-0.467814\pi$
$577$	4.84954	0.201889	0.100945	+	0.994892i	$0.467814\pi$
$578$	0	0
$579$	−23.1400	−0.961664
$580$	0	0
$581$	−9.35535	−0.388125
$582$	0	0
$583$	7.44425	0.308309
$584$	0	0
$585$	12.4098	0.513081
$586$	0	0
$587$	−19.0155	−0.784853	−0.392426	−	0.919783i	$-0.628364\pi$
$587$	−19.0155	−0.784853	−0.392426	+	0.919783i	$0.628364\pi$
$588$	0	0
$589$	3.96104	0.163212
$590$	0	0
$591$	19.4747	0.801083
$592$	0	0
$593$	24.0379	0.987118	0.493559	−	0.869712i	$-0.335696\pi$
$593$	24.0379	0.987118	0.493559	+	0.869712i	$0.335696\pi$
$594$	0	0
$595$	21.0350	0.862349
$596$	0	0
$597$	17.5602	0.718693
$598$	0	0
$599$	33.9631	1.38769	0.693847	−	0.720122i	$-0.255914\pi$
$599$	33.9631	1.38769	0.693847	+	0.720122i	$0.255914\pi$
$600$	0	0
$601$	−10.0999	−0.411983	−0.205992	−	0.978554i	$-0.566042\pi$
$601$	−10.0999	−0.411983	−0.205992	+	0.978554i	$0.566042\pi$
$602$	0	0
$603$	−0.244852	−0.00997115
$604$	0	0
$605$	15.7126	0.638809
$606$	0	0
$607$	−8.17105	−0.331653	−0.165826	−	0.986155i	$-0.553029\pi$
$607$	−8.17105	−0.331653	−0.165826	+	0.986155i	$0.553029\pi$
$608$	0	0
$609$	−8.83006	−0.357812
$610$	0	0
$611$	−28.9195	−1.16996
$612$	0	0
$613$	23.0206	0.929793	0.464896	−	0.885365i	$-0.346092\pi$
$613$	23.0206	0.929793	0.464896	+	0.885365i	$0.346092\pi$
$614$	0	0
$615$	−21.0350	−0.848211
$616$	0	0
$617$	−31.1044	−1.25222	−0.626108	−	0.779737i	$-0.715353\pi$
$617$	−31.1044	−1.25222	−0.626108	+	0.779737i	$0.715353\pi$
$618$	0	0
$619$	−40.9195	−1.64469	−0.822346	−	0.568988i	$-0.807335\pi$
$619$	−40.9195	−1.64469	−0.822346	+	0.568988i	$0.807335\pi$
$620$	0	0
$621$	−7.12742	−0.286013
$622$	0	0
$623$	−11.2824	−0.452018
$624$	0	0
$625$	55.8001	2.23200
$626$	0	0
$627$	7.44425	0.297295
$628$	0	0
$629$	−7.30540	−0.291286
$630$	0	0
$631$	−2.29380	−0.0913147	−0.0456573	−	0.998957i	$-0.514538\pi$
$631$	−2.29380	−0.0913147	−0.0456573	+	0.998957i	$0.514538\pi$
$632$	0	0
$633$	23.9600	0.952326
$634$	0	0
$635$	−2.84954	−0.113081
$636$	0	0
$637$	−3.02497	−0.119854
$638$	0	0
$639$	−4.27787	−0.169230
$640$	0	0
$641$	27.0654	1.06902	0.534510	−	0.845162i	$-0.320496\pi$
$641$	27.0654	1.06902	0.534510	+	0.845162i	$0.320496\pi$
$642$	0	0
$643$	31.9895	1.26154	0.630771	−	0.775969i	$-0.282739\pi$
$643$	31.9895	1.26154	0.630771	+	0.775969i	$0.282739\pi$
$644$	0	0
$645$	−9.84505	−0.387649
$646$	0	0
$647$	−12.5947	−0.495149	−0.247575	−	0.968869i	$-0.579634\pi$
$647$	−12.5947	−0.495149	−0.247575	+	0.968869i	$0.579634\pi$
$648$	0	0
$649$	10.7107	0.420432
$650$	0	0
$651$	−1.42477	−0.0558412
$652$	0	0
$653$	21.8346	0.854452	0.427226	−	0.904145i	$-0.359491\pi$
$653$	21.8346	0.854452	0.427226	+	0.904145i	$0.359491\pi$
$654$	0	0
$655$	50.0699	1.95639
$656$	0	0
$657$	4.15495	0.162100
$658$	0	0
$659$	24.7865	0.965547	0.482773	−	0.875745i	$-0.339629\pi$
$659$	24.7865	0.965547	0.482773	+	0.875745i	$0.339629\pi$
$660$	0	0
$661$	−3.09101	−0.120227	−0.0601133	−	0.998192i	$-0.519146\pi$
$661$	−3.09101	−0.120227	−0.0601133	+	0.998192i	$0.519146\pi$
$662$	0	0
$663$	−15.5103	−0.602370
$664$	0	0
$665$	−11.4053	−0.442278
$666$	0	0
$667$	62.9356	2.43687
$668$	0	0
$669$	2.74966	0.106308
$670$	0	0
$671$	−13.8701	−0.535451
$672$	0	0
$673$	−27.6496	−1.06581	−0.532907	−	0.846174i	$-0.678901\pi$
$673$	−27.6496	−1.06581	−0.532907	+	0.846174i	$0.678901\pi$
$674$	0	0
$675$	−11.8301	−0.455339
$676$	0	0
$677$	−5.84249	−0.224545	−0.112273	−	0.993677i	$-0.535813\pi$
$677$	−5.84249	−0.224545	−0.112273	+	0.993677i	$0.535813\pi$
$678$	0	0
$679$	6.69460	0.256915
$680$	0	0
$681$	−8.30478	−0.318240
$682$	0	0
$683$	15.5433	0.594748	0.297374	−	0.954761i	$-0.403889\pi$
$683$	15.5433	0.594748	0.297374	+	0.954761i	$0.403889\pi$
$684$	0	0
$685$	−61.9649	−2.36756
$686$	0	0
$687$	−11.9245	−0.454947
$688$	0	0
$689$	8.40978	0.320387
$690$	0	0
$691$	−42.9304	−1.63315	−0.816575	−	0.577239i	$-0.804130\pi$
$691$	−42.9304	−1.63315	−0.816575	+	0.577239i	$0.804130\pi$
$692$	0	0
$693$	−2.67767	−0.101716
$694$	0	0
$695$	−31.2204	−1.18426
$696$	0	0
$697$	26.2904	0.995820
$698$	0	0
$699$	−3.56024	−0.134661
$700$	0	0
$701$	5.21476	0.196959	0.0984795	−	0.995139i	$-0.468602\pi$
$701$	5.21476	0.196959	0.0984795	+	0.995139i	$0.468602\pi$
$702$	0	0
$703$	3.96104	0.149393
$704$	0	0
$705$	39.2204	1.47712
$706$	0	0
$707$	1.45779	0.0548260
$708$	0	0
$709$	−41.6601	−1.56458	−0.782289	−	0.622915i	$-0.785948\pi$
$709$	−41.6601	−1.56458	−0.782289	+	0.622915i	$0.785948\pi$
$710$	0	0
$711$	−6.25484	−0.234575
$712$	0	0
$713$	10.1549	0.380306
$714$	0	0
$715$	−33.2294	−1.24271
$716$	0	0
$717$	26.9425	1.00619
$718$	0	0
$719$	−14.3547	−0.535341	−0.267670	−	0.963511i	$-0.586254\pi$
$719$	−14.3547	−0.535341	−0.267670	+	0.963511i	$0.586254\pi$
$720$	0	0
$721$	8.13547	0.302981
$722$	0	0
$723$	−25.8151	−0.960073
$724$	0	0
$725$	104.460	3.87955
$726$	0	0
$727$	53.4647	1.98290	0.991448	−	0.130501i	$-0.0416587\pi$
$727$	53.4647	1.98290	0.991448	+	0.130501i	$0.0416587\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.3048	0.455109
$732$	0	0
$733$	−49.1149	−1.81410	−0.907049	−	0.421025i	$-0.861671\pi$
$733$	−49.1149	−1.81410	−0.907049	+	0.421025i	$0.861671\pi$
$734$	0	0
$735$	4.10245	0.151321
$736$	0	0
$737$	0.655634	0.0241506
$738$	0	0
$739$	−7.75515	−0.285278	−0.142639	−	0.989775i	$-0.545559\pi$
$739$	−7.75515	−0.285278	−0.142639	+	0.989775i	$0.545559\pi$
$740$	0	0
$741$	8.40978	0.308941
$742$	0	0
$743$	30.9886	1.13686	0.568430	−	0.822732i	$-0.307551\pi$
$743$	30.9886	1.13686	0.568430	+	0.822732i	$0.307551\pi$
$744$	0	0
$745$	−21.4143	−0.784558
$746$	0	0
$747$	−9.35535	−0.342294
$748$	0	0
$749$	−5.32233	−0.194474
$750$	0	0
$751$	34.9105	1.27390	0.636951	−	0.770905i	$-0.280196\pi$
$751$	34.9105	1.27390	0.636951	+	0.770905i	$0.280196\pi$
$752$	0	0
$753$	27.1543	0.989559
$754$	0	0
$755$	22.8106	0.830162
$756$	0	0
$757$	9.25034	0.336209	0.168105	−	0.985769i	$-0.446235\pi$
$757$	9.25034	0.336209	0.168105	+	0.985769i	$0.446235\pi$
$758$	0	0
$759$	19.0849	0.692738
$760$	0	0
$761$	11.8381	0.429131	0.214566	−	0.976710i	$-0.431166\pi$
$761$	11.8381	0.429131	0.214566	+	0.976710i	$0.431166\pi$
$762$	0	0
$763$	13.5602	0.490914
$764$	0	0
$765$	21.0350	0.760520
$766$	0	0
$767$	12.0999	0.436902
$768$	0	0
$769$	−20.0699	−0.723740	−0.361870	−	0.932229i	$-0.617861\pi$
$769$	−20.0699	−0.723740	−0.361870	+	0.932229i	$0.617861\pi$
$770$	0	0
$771$	31.3823	1.13020
$772$	0	0
$773$	−18.8521	−0.678063	−0.339032	−	0.940775i	$-0.610099\pi$
$773$	−18.8521	−0.678063	−0.339032	+	0.940775i	$0.610099\pi$
$774$	0	0
$775$	16.8551	0.605455
$776$	0	0
$777$	−1.42477	−0.0511134
$778$	0	0
$779$	−14.2548	−0.510733
$780$	0	0
$781$	11.4547	0.409883
$782$	0	0
$783$	−8.83006	−0.315561
$784$	0	0
$785$	−2.12986	−0.0760181
$786$	0	0
$787$	25.9111	0.923631	0.461815	−	0.886976i	$-0.347198\pi$
$787$	25.9111	0.923631	0.461815	+	0.886976i	$0.347198\pi$
$788$	0	0
$789$	11.0275	0.392590
$790$	0	0
$791$	−4.15495	−0.147733
$792$	0	0
$793$	−15.6691	−0.556427
$794$	0	0
$795$	−11.4053	−0.404504
$796$	0	0
$797$	−11.1318	−0.394309	−0.197154	−	0.980372i	$-0.563170\pi$
$797$	−11.1318	−0.394309	−0.197154	+	0.980372i	$0.563170\pi$
$798$	0	0
$799$	−49.0193	−1.73418
$800$	0	0
$801$	−11.2824	−0.398643
$802$	0	0
$803$	−11.1256	−0.392614
$804$	0	0
$805$	−29.2398	−1.03057
$806$	0	0
$807$	21.0019	0.739303
$808$	0	0
$809$	−33.8151	−1.18887	−0.594437	−	0.804142i	$-0.702625\pi$
$809$	−33.8151	−1.18887	−0.594437	+	0.804142i	$0.702625\pi$
$810$	0	0
$811$	32.5097	1.14157	0.570784	−	0.821100i	$-0.306639\pi$
$811$	32.5097	1.14157	0.570784	+	0.821100i	$0.306639\pi$
$812$	0	0
$813$	28.6451	1.00463
$814$	0	0
$815$	−35.5052	−1.24369
$816$	0	0
$817$	−6.67173	−0.233414
$818$	0	0
$819$	−3.02497	−0.105701
$820$	0	0
$821$	40.7501	1.42219	0.711095	−	0.703096i	$-0.248200\pi$
$821$	40.7501	1.42219	0.711095	+	0.703096i	$0.248200\pi$
$822$	0	0
$823$	−34.2159	−1.19269	−0.596345	−	0.802728i	$-0.703381\pi$
$823$	−34.2159	−1.19269	−0.596345	+	0.802728i	$0.703381\pi$
$824$	0	0
$825$	31.6771	1.10285
$826$	0	0
$827$	37.2424	1.29505	0.647523	−	0.762046i	$-0.275805\pi$
$827$	37.2424	1.29505	0.647523	+	0.762046i	$0.275805\pi$
$828$	0	0
$829$	11.2959	0.392323	0.196162	−	0.980572i	$-0.437152\pi$
$829$	11.2959	0.392323	0.196162	+	0.980572i	$0.437152\pi$
$830$	0	0
$831$	−2.96504	−0.102856
$832$	0	0
$833$	−5.12742	−0.177655
$834$	0	0
$835$	102.989	3.56409
$836$	0	0
$837$	−1.42477	−0.0492473
$838$	0	0
$839$	−38.8746	−1.34210	−0.671051	−	0.741412i	$-0.734157\pi$
$839$	−38.8746	−1.34210	−0.671051	+	0.741412i	$0.734157\pi$
$840$	0	0
$841$	48.9700	1.68862
$842$	0	0
$843$	26.4098	0.909601
$844$	0	0
$845$	15.7925	0.543280
$846$	0	0
$847$	−3.83006	−0.131603
$848$	0	0
$849$	11.2698	0.386779
$850$	0	0
$851$	10.1549	0.348107
$852$	0	0
$853$	−25.5507	−0.874841	−0.437420	−	0.899257i	$-0.644108\pi$
$853$	−25.5507	−0.874841	−0.437420	+	0.899257i	$0.644108\pi$
$854$	0	0
$855$	−11.4053	−0.390053
$856$	0	0
$857$	−20.7038	−0.707227	−0.353613	−	0.935392i	$-0.615047\pi$
$857$	−20.7038	−0.707227	−0.353613	+	0.935392i	$0.615047\pi$
$858$	0	0
$859$	9.66863	0.329889	0.164945	−	0.986303i	$-0.447255\pi$
$859$	9.66863	0.329889	0.164945	+	0.986303i	$0.447255\pi$
$860$	0	0
$861$	5.12742	0.174742
$862$	0	0
$863$	−31.5981	−1.07561	−0.537806	−	0.843068i	$-0.680747\pi$
$863$	−31.5981	−1.07561	−0.537806	+	0.843068i	$0.680747\pi$
$864$	0	0
$865$	−7.98950	−0.271651
$866$	0	0
$867$	−9.29042	−0.315519
$868$	0	0
$869$	16.7484	0.568151
$870$	0	0
$871$	0.740671	0.0250967
$872$	0	0
$873$	6.69460	0.226578
$874$	0	0
$875$	−28.0200	−0.947248
$876$	0	0
$877$	4.50967	0.152281	0.0761404	−	0.997097i	$-0.475740\pi$
$877$	4.50967	0.152281	0.0761404	+	0.997097i	$0.475740\pi$
$878$	0	0
$879$	−10.5622	−0.356253
$880$	0	0
$881$	25.8471	0.870811	0.435405	−	0.900234i	$-0.356605\pi$
$881$	25.8471	0.870811	0.435405	+	0.900234i	$0.356605\pi$
$882$	0	0
$883$	−58.5786	−1.97133	−0.985663	−	0.168725i	$-0.946035\pi$
$883$	−58.5786	−1.97133	−0.985663	+	0.168725i	$0.946035\pi$
$884$	0	0
$885$	−16.4098	−0.551609
$886$	0	0
$887$	−20.2249	−0.679084	−0.339542	−	0.940591i	$-0.610272\pi$
$887$	−20.2249	−0.679084	−0.339542	+	0.940591i	$0.610272\pi$
$888$	0	0
$889$	0.694597	0.0232960
$890$	0	0
$891$	−2.67767	−0.0897054
$892$	0	0
$893$	26.5786	0.889419
$894$	0	0
$895$	−93.1158	−3.11252
$896$	0	0
$897$	21.5602	0.719875
$898$	0	0
$899$	12.5808	0.419594
$900$	0	0
$901$	14.2548	0.474897
$902$	0	0
$903$	2.39980	0.0798604
$904$	0	0
$905$	−43.2204	−1.43669
$906$	0	0
$907$	45.3103	1.50450	0.752251	−	0.658876i	$-0.228968\pi$
$907$	45.3103	1.50450	0.752251	+	0.658876i	$0.228968\pi$
$908$	0	0
$909$	1.45779	0.0483520
$910$	0	0
$911$	12.5488	0.415761	0.207880	−	0.978154i	$-0.433344\pi$
$911$	12.5488	0.415761	0.207880	+	0.978154i	$0.433344\pi$
$912$	0	0
$913$	25.0506	0.829053
$914$	0	0
$915$	21.2503	0.702515
$916$	0	0
$917$	−12.2049	−0.403041
$918$	0	0
$919$	−43.9400	−1.44945	−0.724724	−	0.689039i	$-0.758033\pi$
$919$	−43.9400	−1.44945	−0.724724	+	0.689039i	$0.758033\pi$
$920$	0	0
$921$	10.7801	0.355217
$922$	0	0
$923$	12.9404	0.425940
$924$	0	0
$925$	16.8551	0.554194
$926$	0	0
$927$	8.13547	0.267204
$928$	0	0
$929$	−53.7370	−1.76305	−0.881527	−	0.472134i	$-0.843484\pi$
$929$	−53.7370	−1.76305	−0.881527	+	0.472134i	$0.843484\pi$
$930$	0	0
$931$	2.78012	0.0911147
$932$	0	0
$933$	−9.56024	−0.312988
$934$	0	0
$935$	−56.3247	−1.84202
$936$	0	0
$937$	12.7107	0.415240	0.207620	−	0.978210i	$-0.433428\pi$
$937$	12.7107	0.415240	0.207620	+	0.978210i	$0.433428\pi$
$938$	0	0
$939$	7.69909	0.251250
$940$	0	0
$941$	4.02253	0.131131	0.0655653	−	0.997848i	$-0.479115\pi$
$941$	4.02253	0.131131	0.0655653	+	0.997848i	$0.479115\pi$
$942$	0	0
$943$	−36.5453	−1.19008
$944$	0	0
$945$	4.10245	0.133453
$946$	0	0
$947$	43.1413	1.40190	0.700951	−	0.713209i	$-0.252759\pi$
$947$	43.1413	1.40190	0.700951	+	0.713209i	$0.252759\pi$
$948$	0	0
$949$	−12.5686	−0.407994
$950$	0	0
$951$	−10.7801	−0.349569
$952$	0	0
$953$	10.2387	0.331665	0.165833	−	0.986154i	$-0.446969\pi$
$953$	10.2387	0.331665	0.165833	+	0.986154i	$0.446969\pi$
$954$	0	0
$955$	−17.5497	−0.567896
$956$	0	0
$957$	23.6440	0.764303
$958$	0	0
$959$	15.1044	0.487746
$960$	0	0
$961$	−28.9700	−0.934517
$962$	0	0
$963$	−5.32233	−0.171510
$964$	0	0
$965$	−94.9304	−3.05592
$966$	0	0
$967$	46.9356	1.50935	0.754673	−	0.656101i	$-0.227796\pi$
$967$	46.9356	1.50935	0.754673	+	0.656101i	$0.227796\pi$
$968$	0	0
$969$	14.2548	0.457931
$970$	0	0
$971$	17.5602	0.563535	0.281767	−	0.959483i	$-0.409079\pi$
$971$	17.5602	0.563535	0.281767	+	0.959483i	$0.409079\pi$
$972$	0	0
$973$	7.61018	0.243971
$974$	0	0
$975$	35.7856	1.14606
$976$	0	0
$977$	−41.5303	−1.32867	−0.664335	−	0.747435i	$-0.731285\pi$
$977$	−41.5303	−1.32867	−0.664335	+	0.747435i	$0.731285\pi$
$978$	0	0
$979$	30.2105	0.965532
$980$	0	0
$981$	13.5602	0.432945
$982$	0	0
$983$	−50.4797	−1.61005	−0.805026	−	0.593239i	$-0.797849\pi$
$983$	−50.4797	−1.61005	−0.805026	+	0.593239i	$0.797849\pi$
$984$	0	0
$985$	79.8940	2.54563
$986$	0	0
$987$	−9.56024	−0.304306
$988$	0	0
$989$	−17.1044	−0.543888
$990$	0	0
$991$	−1.46035	−0.0463896	−0.0231948	−	0.999731i	$-0.507384\pi$
$991$	−1.46035	−0.0463896	−0.0231948	+	0.999731i	$0.507384\pi$
$992$	0	0
$993$	25.2104	0.800027
$994$	0	0
$995$	72.0399	2.28382
$996$	0	0
$997$	−21.8796	−0.692935	−0.346467	−	0.938062i	$-0.612619\pi$
$997$	−21.8796	−0.692935	−0.346467	+	0.938062i	$0.612619\pi$
$998$	0	0
$999$	−1.42477	−0.0450778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.a.bl.1.1		4
4.3	odd	2		5376.2.a.bp.1.1		4
8.3	odd	2		5376.2.a.bm.1.4		4
8.5	even	2		5376.2.a.bq.1.4		4
16.3	odd	4		168.2.c.b.85.7	✓	8
16.5	even	4		672.2.c.b.337.5		8
16.11	odd	4		168.2.c.b.85.8	yes	8
16.13	even	4		672.2.c.b.337.4		8
48.5	odd	4		2016.2.c.e.1009.8		8
48.11	even	4		504.2.c.f.253.1		8
48.29	odd	4		2016.2.c.e.1009.1		8
48.35	even	4		504.2.c.f.253.2		8
112.13	odd	4		4704.2.c.c.2353.5		8
112.27	even	4		1176.2.c.c.589.8		8
112.69	odd	4		4704.2.c.c.2353.4		8
112.83	even	4		1176.2.c.c.589.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.7	✓	8	16.3	odd	4
168.2.c.b.85.8	yes	8	16.11	odd	4
504.2.c.f.253.1		8	48.11	even	4
504.2.c.f.253.2		8	48.35	even	4
672.2.c.b.337.4		8	16.13	even	4
672.2.c.b.337.5		8	16.5	even	4
1176.2.c.c.589.7		8	112.83	even	4
1176.2.c.c.589.8		8	112.27	even	4
2016.2.c.e.1009.1		8	48.29	odd	4
2016.2.c.e.1009.8		8	48.5	odd	4
4704.2.c.c.2353.4		8	112.69	odd	4
4704.2.c.c.2353.5		8	112.13	odd	4
5376.2.a.bl.1.1		4	1.1	even	1	trivial
5376.2.a.bm.1.4		4	8.3	odd	2
5376.2.a.bp.1.1		4	4.3	odd	2
5376.2.a.bq.1.4		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5376.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.10245 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.10245 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.67767 q^{11} -3.02497 q^{13} +4.10245 q^{15} -5.12742 q^{17} +2.78012 q^{19} -1.00000 q^{21} +7.12742 q^{23} +11.8301 q^{25} -1.00000 q^{27} +8.83006 q^{29} +1.42477 q^{31} +2.67767 q^{33} -4.10245 q^{35} +1.42477 q^{37} +3.02497 q^{39} -5.12742 q^{41} -2.39980 q^{43} -4.10245 q^{45} +9.56024 q^{47} +1.00000 q^{49} +5.12742 q^{51} -2.78012 q^{53} +10.9850 q^{55} -2.78012 q^{57} -4.00000 q^{59} +5.17992 q^{61} +1.00000 q^{63} +12.4098 q^{65} -0.244852 q^{67} -7.12742 q^{69} -4.27787 q^{71} +4.15495 q^{73} -11.8301 q^{75} -2.67767 q^{77} -6.25484 q^{79} +1.00000 q^{81} -9.35535 q^{83} +21.0350 q^{85} -8.83006 q^{87} -11.2824 q^{89} -3.02497 q^{91} -1.42477 q^{93} -11.4053 q^{95} +6.69460 q^{97} -2.67767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9} - 6 q^{11} - 4 q^{13} + 2 q^{15} + 2 q^{17} - 8 q^{19} - 4 q^{21} + 6 q^{23} + 12 q^{25} - 4 q^{27} - 4 q^{31} + 6 q^{33} - 2 q^{35} - 4 q^{37} + 4 q^{39} + 2 q^{41} - 8 q^{43} - 2 q^{45} + 4 q^{49} - 2 q^{51} + 8 q^{53} - 4 q^{55} + 8 q^{57} - 16 q^{59} + 4 q^{63} - 8 q^{65} - 12 q^{67} - 6 q^{69} - 14 q^{71} + 4 q^{73} - 12 q^{75} - 6 q^{77} + 20 q^{79} + 4 q^{81} - 28 q^{83} + 20 q^{85} - 10 q^{89} - 4 q^{91} + 4 q^{93} - 20 q^{95} + 20 q^{97} - 6 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9 - 6 * q^11 - 4 * q^13 + 2 * q^15 + 2 * q^17 - 8 * q^19 - 4 * q^21 + 6 * q^23 + 12 * q^25 - 4 * q^27 - 4 * q^31 + 6 * q^33 - 2 * q^35 - 4 * q^37 + 4 * q^39 + 2 * q^41 - 8 * q^43 - 2 * q^45 + 4 * q^49 - 2 * q^51 + 8 * q^53 - 4 * q^55 + 8 * q^57 - 16 * q^59 + 4 * q^63 - 8 * q^65 - 12 * q^67 - 6 * q^69 - 14 * q^71 + 4 * q^73 - 12 * q^75 - 6 * q^77 + 20 * q^79 + 4 * q^81 - 28 * q^83 + 20 * q^85 - 10 * q^89 - 4 * q^91 + 4 * q^93 - 20 * q^95 + 20 * q^97 - 6 * q^99

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