LMFDB - Embedded newform 1456.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, 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("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.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:	$11.6262185343$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} -3.44949 q^{5} +1.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+2.44949 q^{3} -3.44949 q^{5} +1.00000 q^{7} +3.00000 q^{9} -6.44949 q^{11} -1.00000 q^{13} -8.44949 q^{15} +2.44949 q^{17} -0.550510 q^{19} +2.44949 q^{21} -3.89898 q^{23} +6.89898 q^{25} -5.89898 q^{29} -3.44949 q^{31} -15.7980 q^{33} -3.44949 q^{35} -8.44949 q^{37} -2.44949 q^{39} +10.8990 q^{41} -1.00000 q^{43} -10.3485 q^{45} -3.44949 q^{47} +1.00000 q^{49} +6.00000 q^{51} +1.89898 q^{53} +22.2474 q^{55} -1.34847 q^{57} -14.0000 q^{59} -2.00000 q^{61} +3.00000 q^{63} +3.44949 q^{65} +6.89898 q^{67} -9.55051 q^{69} -12.4495 q^{71} +11.2474 q^{73} +16.8990 q^{75} -6.44949 q^{77} +2.10102 q^{79} -9.00000 q^{81} +4.34847 q^{83} -8.44949 q^{85} -14.4495 q^{87} -11.4495 q^{89} -1.00000 q^{91} -8.44949 q^{93} +1.89898 q^{95} -10.5505 q^{97} -19.3485 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7 + 6 * q^9	$ 2 q - 2 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 2 q^{13} - 12 q^{15} - 6 q^{19} + 2 q^{23} + 4 q^{25} - 2 q^{29} - 2 q^{31} - 12 q^{33} - 2 q^{35} - 12 q^{37} + 12 q^{41} - 2 q^{43} - 6 q^{45} - 2 q^{47} + 2 q^{49} + 12 q^{51} - 6 q^{53} + 20 q^{55} + 12 q^{57} - 28 q^{59} - 4 q^{61} + 6 q^{63} + 2 q^{65} + 4 q^{67} - 24 q^{69} - 20 q^{71} - 2 q^{73} + 24 q^{75} - 8 q^{77} + 14 q^{79} - 18 q^{81} - 6 q^{83} - 12 q^{85} - 24 q^{87} - 18 q^{89} - 2 q^{91} - 12 q^{93} - 6 q^{95} - 26 q^{97} - 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 2 * q^13 - 12 * q^15 - 6 * q^19 + 2 * q^23 + 4 * q^25 - 2 * q^29 - 2 * q^31 - 12 * q^33 - 2 * q^35 - 12 * q^37 + 12 * q^41 - 2 * q^43 - 6 * q^45 - 2 * q^47 + 2 * q^49 + 12 * q^51 - 6 * q^53 + 20 * q^55 + 12 * q^57 - 28 * q^59 - 4 * q^61 + 6 * q^63 + 2 * q^65 + 4 * q^67 - 24 * q^69 - 20 * q^71 - 2 * q^73 + 24 * q^75 - 8 * q^77 + 14 * q^79 - 18 * q^81 - 6 * q^83 - 12 * q^85 - 24 * q^87 - 18 * q^89 - 2 * q^91 - 12 * q^93 - 6 * q^95 - 26 * q^97 - 24 * 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
$3$	2.44949	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	0	0
$5$	−3.44949	−1.54266	−0.771329	−	0.636436i	$-0.780408\pi$
$5$	−3.44949	−1.54266	−0.771329	+	0.636436i	$0.780408\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−6.44949	−1.94459	−0.972297	−	0.233748i	$-0.924901\pi$
$11$	−6.44949	−1.94459	−0.972297	+	0.233748i	$0.924901\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−8.44949	−2.18165
$16$	0	0
$17$	2.44949	0.594089	0.297044	−	0.954864i	$-0.403999\pi$
$17$	2.44949	0.594089	0.297044	+	0.954864i	$0.403999\pi$
$18$	0	0
$19$	−0.550510	−0.126296	−0.0631479	−	0.998004i	$-0.520114\pi$
$19$	−0.550510	−0.126296	−0.0631479	+	0.998004i	$0.520114\pi$
$20$	0	0
$21$	2.44949	0.534522
$22$	0	0
$23$	−3.89898	−0.812993	−0.406497	−	0.913652i	$-0.633250\pi$
$23$	−3.89898	−0.812993	−0.406497	+	0.913652i	$0.633250\pi$
$24$	0	0
$25$	6.89898	1.37980
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.89898	−1.09541	−0.547706	−	0.836671i	$-0.684499\pi$
$29$	−5.89898	−1.09541	−0.547706	+	0.836671i	$0.684499\pi$
$30$	0	0
$31$	−3.44949	−0.619547	−0.309773	−	0.950810i	$-0.600253\pi$
$31$	−3.44949	−0.619547	−0.309773	+	0.950810i	$0.600253\pi$
$32$	0	0
$33$	−15.7980	−2.75007
$34$	0	0
$35$	−3.44949	−0.583070
$36$	0	0
$37$	−8.44949	−1.38909	−0.694544	−	0.719450i	$-0.744394\pi$
$37$	−8.44949	−1.38909	−0.694544	+	0.719450i	$0.744394\pi$
$38$	0	0
$39$	−2.44949	−0.392232
$40$	0	0
$41$	10.8990	1.70213	0.851067	−	0.525057i	$-0.175956\pi$
$41$	10.8990	1.70213	0.851067	+	0.525057i	$0.175956\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	−10.3485	−1.54266
$46$	0	0
$47$	−3.44949	−0.503160	−0.251580	−	0.967837i	$-0.580950\pi$
$47$	−3.44949	−0.503160	−0.251580	+	0.967837i	$0.580950\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	1.89898	0.260845	0.130422	−	0.991459i	$-0.458367\pi$
$53$	1.89898	0.260845	0.130422	+	0.991459i	$0.458367\pi$
$54$	0	0
$55$	22.2474	2.99985
$56$	0	0
$57$	−1.34847	−0.178609
$58$	0	0
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	3.44949	0.427857
$66$	0	0
$67$	6.89898	0.842844	0.421422	−	0.906865i	$-0.361531\pi$
$67$	6.89898	0.842844	0.421422	+	0.906865i	$0.361531\pi$
$68$	0	0
$69$	−9.55051	−1.14975
$70$	0	0
$71$	−12.4495	−1.47748	−0.738741	−	0.673989i	$-0.764579\pi$
$71$	−12.4495	−1.47748	−0.738741	+	0.673989i	$0.764579\pi$
$72$	0	0
$73$	11.2474	1.31641	0.658207	−	0.752837i	$-0.271315\pi$
$73$	11.2474	1.31641	0.658207	+	0.752837i	$0.271315\pi$
$74$	0	0
$75$	16.8990	1.95133
$76$	0	0
$77$	−6.44949	−0.734988
$78$	0	0
$79$	2.10102	0.236383	0.118192	−	0.992991i	$-0.462290\pi$
$79$	2.10102	0.236383	0.118192	+	0.992991i	$0.462290\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	4.34847	0.477307	0.238653	−	0.971105i	$-0.423294\pi$
$83$	4.34847	0.477307	0.238653	+	0.971105i	$0.423294\pi$
$84$	0	0
$85$	−8.44949	−0.916476
$86$	0	0
$87$	−14.4495	−1.54915
$88$	0	0
$89$	−11.4495	−1.21364	−0.606822	−	0.794838i	$-0.707556\pi$
$89$	−11.4495	−1.21364	−0.606822	+	0.794838i	$0.707556\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−8.44949	−0.876171
$94$	0	0
$95$	1.89898	0.194831
$96$	0	0
$97$	−10.5505	−1.07124	−0.535621	−	0.844458i	$-0.679922\pi$
$97$	−10.5505	−1.07124	−0.535621	+	0.844458i	$0.679922\pi$
$98$	0	0
$99$	−19.3485	−1.94459
$100$	0	0
$101$	5.34847	0.532193	0.266096	−	0.963946i	$-0.414266\pi$
$101$	5.34847	0.532193	0.266096	+	0.963946i	$0.414266\pi$
$102$	0	0
$103$	−1.79796	−0.177158	−0.0885791	−	0.996069i	$-0.528233\pi$
$103$	−1.79796	−0.177158	−0.0885791	+	0.996069i	$0.528233\pi$
$104$	0	0
$105$	−8.44949	−0.824586
$106$	0	0
$107$	12.6969	1.22746	0.613730	−	0.789516i	$-0.289668\pi$
$107$	12.6969	1.22746	0.613730	+	0.789516i	$0.289668\pi$
$108$	0	0
$109$	18.4495	1.76714	0.883570	−	0.468299i	$-0.155133\pi$
$109$	18.4495	1.76714	0.883570	+	0.468299i	$0.155133\pi$
$110$	0	0
$111$	−20.6969	−1.96447
$112$	0	0
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	0	0
$115$	13.4495	1.25417
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	2.44949	0.224544
$120$	0	0
$121$	30.5959	2.78145
$122$	0	0
$123$	26.6969	2.40718
$124$	0	0
$125$	−6.55051	−0.585895
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	0	0
$129$	−2.44949	−0.215666
$130$	0	0
$131$	−22.6969	−1.98304	−0.991520	−	0.129951i	$-0.958518\pi$
$131$	−22.6969	−1.98304	−0.991520	+	0.129951i	$0.958518\pi$
$132$	0	0
$133$	−0.550510	−0.0477353
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.34847	−0.115208	−0.0576038	−	0.998340i	$-0.518346\pi$
$137$	−1.34847	−0.115208	−0.0576038	+	0.998340i	$0.518346\pi$
$138$	0	0
$139$	19.1464	1.62398	0.811989	−	0.583672i	$-0.198385\pi$
$139$	19.1464	1.62398	0.811989	+	0.583672i	$0.198385\pi$
$140$	0	0
$141$	−8.44949	−0.711575
$142$	0	0
$143$	6.44949	0.539333
$144$	0	0
$145$	20.3485	1.68985
$146$	0	0
$147$	2.44949	0.202031
$148$	0	0
$149$	7.34847	0.602010	0.301005	−	0.953623i	$-0.402678\pi$
$149$	7.34847	0.602010	0.301005	+	0.953623i	$0.402678\pi$
$150$	0	0
$151$	13.3485	1.08628	0.543142	−	0.839641i	$-0.317235\pi$
$151$	13.3485	1.08628	0.543142	+	0.839641i	$0.317235\pi$
$152$	0	0
$153$	7.34847	0.594089
$154$	0	0
$155$	11.8990	0.955749
$156$	0	0
$157$	5.55051	0.442979	0.221489	−	0.975163i	$-0.428908\pi$
$157$	5.55051	0.442979	0.221489	+	0.975163i	$0.428908\pi$
$158$	0	0
$159$	4.65153	0.368890
$160$	0	0
$161$	−3.89898	−0.307283
$162$	0	0
$163$	−22.6969	−1.77776	−0.888881	−	0.458139i	$-0.848516\pi$
$163$	−22.6969	−1.77776	−0.888881	+	0.458139i	$0.848516\pi$
$164$	0	0
$165$	54.4949	4.24242
$166$	0	0
$167$	−22.3485	−1.72938	−0.864688	−	0.502309i	$-0.832484\pi$
$167$	−22.3485	−1.72938	−0.864688	+	0.502309i	$0.832484\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.65153	−0.126296
$172$	0	0
$173$	−11.3485	−0.862808	−0.431404	−	0.902159i	$-0.641982\pi$
$173$	−11.3485	−0.862808	−0.431404	+	0.902159i	$0.641982\pi$
$174$	0	0
$175$	6.89898	0.521514
$176$	0	0
$177$	−34.2929	−2.57761
$178$	0	0
$179$	22.5959	1.68890	0.844449	−	0.535636i	$-0.179928\pi$
$179$	22.5959	1.68890	0.844449	+	0.535636i	$0.179928\pi$
$180$	0	0
$181$	−8.44949	−0.628046	−0.314023	−	0.949415i	$-0.601677\pi$
$181$	−8.44949	−0.628046	−0.314023	+	0.949415i	$0.601677\pi$
$182$	0	0
$183$	−4.89898	−0.362143
$184$	0	0
$185$	29.1464	2.14289
$186$	0	0
$187$	−15.7980	−1.15526
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.69694	0.339859	0.169929	−	0.985456i	$-0.445646\pi$
$191$	4.69694	0.339859	0.169929	+	0.985456i	$0.445646\pi$
$192$	0	0
$193$	−12.6969	−0.913946	−0.456973	−	0.889481i	$-0.651066\pi$
$193$	−12.6969	−0.913946	−0.456973	+	0.889481i	$0.651066\pi$
$194$	0	0
$195$	8.44949	0.605081
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	9.14643	0.648373	0.324187	−	0.945993i	$-0.394909\pi$
$199$	9.14643	0.648373	0.324187	+	0.945993i	$0.394909\pi$
$200$	0	0
$201$	16.8990	1.19196
$202$	0	0
$203$	−5.89898	−0.414027
$204$	0	0
$205$	−37.5959	−2.62581
$206$	0	0
$207$	−11.6969	−0.812993
$208$	0	0
$209$	3.55051	0.245594
$210$	0	0
$211$	−8.79796	−0.605676	−0.302838	−	0.953042i	$-0.597934\pi$
$211$	−8.79796	−0.605676	−0.302838	+	0.953042i	$0.597934\pi$
$212$	0	0
$213$	−30.4949	−2.08948
$214$	0	0
$215$	3.44949	0.235253
$216$	0	0
$217$	−3.44949	−0.234167
$218$	0	0
$219$	27.5505	1.86169
$220$	0	0
$221$	−2.44949	−0.164771
$222$	0	0
$223$	−0.550510	−0.0368649	−0.0184324	−	0.999830i	$-0.505868\pi$
$223$	−0.550510	−0.0368649	−0.0184324	+	0.999830i	$0.505868\pi$
$224$	0	0
$225$	20.6969	1.37980
$226$	0	0
$227$	4.89898	0.325157	0.162578	−	0.986696i	$-0.448019\pi$
$227$	4.89898	0.325157	0.162578	+	0.986696i	$0.448019\pi$
$228$	0	0
$229$	14.6969	0.971201	0.485601	−	0.874181i	$-0.338601\pi$
$229$	14.6969	0.971201	0.485601	+	0.874181i	$0.338601\pi$
$230$	0	0
$231$	−15.7980	−1.03943
$232$	0	0
$233$	−2.79796	−0.183300	−0.0916502	−	0.995791i	$-0.529214\pi$
$233$	−2.79796	−0.183300	−0.0916502	+	0.995791i	$0.529214\pi$
$234$	0	0
$235$	11.8990	0.776204
$236$	0	0
$237$	5.14643	0.334296
$238$	0	0
$239$	−16.8990	−1.09310	−0.546552	−	0.837425i	$-0.684060\pi$
$239$	−16.8990	−1.09310	−0.546552	+	0.837425i	$0.684060\pi$
$240$	0	0
$241$	1.65153	0.106384	0.0531922	−	0.998584i	$-0.483060\pi$
$241$	1.65153	0.106384	0.0531922	+	0.998584i	$0.483060\pi$
$242$	0	0
$243$	−22.0454	−1.41421
$244$	0	0
$245$	−3.44949	−0.220380
$246$	0	0
$247$	0.550510	0.0350281
$248$	0	0
$249$	10.6515	0.675013
$250$	0	0
$251$	−8.44949	−0.533327	−0.266664	−	0.963790i	$-0.585921\pi$
$251$	−8.44949	−0.533327	−0.266664	+	0.963790i	$0.585921\pi$
$252$	0	0
$253$	25.1464	1.58094
$254$	0	0
$255$	−20.6969	−1.29609
$256$	0	0
$257$	−30.2474	−1.88678	−0.943392	−	0.331680i	$-0.892385\pi$
$257$	−30.2474	−1.88678	−0.943392	+	0.331680i	$0.892385\pi$
$258$	0	0
$259$	−8.44949	−0.525026
$260$	0	0
$261$	−17.6969	−1.09541
$262$	0	0
$263$	−0.797959	−0.0492043	−0.0246021	−	0.999697i	$-0.507832\pi$
$263$	−0.797959	−0.0492043	−0.0246021	+	0.999697i	$0.507832\pi$
$264$	0	0
$265$	−6.55051	−0.402395
$266$	0	0
$267$	−28.0454	−1.71635
$268$	0	0
$269$	−7.10102	−0.432957	−0.216478	−	0.976287i	$-0.569457\pi$
$269$	−7.10102	−0.432957	−0.216478	+	0.976287i	$0.569457\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−2.44949	−0.148250
$274$	0	0
$275$	−44.4949	−2.68314
$276$	0	0
$277$	1.89898	0.114099	0.0570493	−	0.998371i	$-0.481831\pi$
$277$	1.89898	0.114099	0.0570493	+	0.998371i	$0.481831\pi$
$278$	0	0
$279$	−10.3485	−0.619547
$280$	0	0
$281$	−18.0454	−1.07650	−0.538249	−	0.842786i	$-0.680914\pi$
$281$	−18.0454	−1.07650	−0.538249	+	0.842786i	$0.680914\pi$
$282$	0	0
$283$	3.10102	0.184337	0.0921683	−	0.995743i	$-0.470620\pi$
$283$	3.10102	0.184337	0.0921683	+	0.995743i	$0.470620\pi$
$284$	0	0
$285$	4.65153	0.275533
$286$	0	0
$287$	10.8990	0.643346
$288$	0	0
$289$	−11.0000	−0.647059
$290$	0	0
$291$	−25.8434	−1.51496
$292$	0	0
$293$	−8.55051	−0.499526	−0.249763	−	0.968307i	$-0.580353\pi$
$293$	−8.55051	−0.499526	−0.249763	+	0.968307i	$0.580353\pi$
$294$	0	0
$295$	48.2929	2.81172
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.89898	0.225484
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	13.1010	0.752634
$304$	0	0
$305$	6.89898	0.395034
$306$	0	0
$307$	−3.65153	−0.208404	−0.104202	−	0.994556i	$-0.533229\pi$
$307$	−3.65153	−0.208404	−0.104202	+	0.994556i	$0.533229\pi$
$308$	0	0
$309$	−4.40408	−0.250539
$310$	0	0
$311$	15.5505	0.881789	0.440894	−	0.897559i	$-0.354661\pi$
$311$	15.5505	0.881789	0.440894	+	0.897559i	$0.354661\pi$
$312$	0	0
$313$	−0.449490	−0.0254067	−0.0127033	−	0.999919i	$-0.504044\pi$
$313$	−0.449490	−0.0254067	−0.0127033	+	0.999919i	$0.504044\pi$
$314$	0	0
$315$	−10.3485	−0.583070
$316$	0	0
$317$	33.7980	1.89828	0.949141	−	0.314851i	$-0.101954\pi$
$317$	33.7980	1.89828	0.949141	+	0.314851i	$0.101954\pi$
$318$	0	0
$319$	38.0454	2.13013
$320$	0	0
$321$	31.1010	1.73589
$322$	0	0
$323$	−1.34847	−0.0750308
$324$	0	0
$325$	−6.89898	−0.382687
$326$	0	0
$327$	45.1918	2.49911
$328$	0	0
$329$	−3.44949	−0.190177
$330$	0	0
$331$	−21.5959	−1.18702	−0.593510	−	0.804827i	$-0.702258\pi$
$331$	−21.5959	−1.18702	−0.593510	+	0.804827i	$0.702258\pi$
$332$	0	0
$333$	−25.3485	−1.38909
$334$	0	0
$335$	−23.7980	−1.30022
$336$	0	0
$337$	−8.59592	−0.468249	−0.234125	−	0.972207i	$-0.575222\pi$
$337$	−8.59592	−0.468249	−0.234125	+	0.972207i	$0.575222\pi$
$338$	0	0
$339$	26.9444	1.46342
$340$	0	0
$341$	22.2474	1.20477
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	32.9444	1.77367
$346$	0	0
$347$	−9.79796	−0.525982	−0.262991	−	0.964798i	$-0.584709\pi$
$347$	−9.79796	−0.525982	−0.262991	+	0.964798i	$0.584709\pi$
$348$	0	0
$349$	−17.2474	−0.923235	−0.461617	−	0.887079i	$-0.652731\pi$
$349$	−17.2474	−0.923235	−0.461617	+	0.887079i	$0.652731\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.8990	−1.32524	−0.662619	−	0.748956i	$-0.730555\pi$
$353$	−24.8990	−1.32524	−0.662619	+	0.748956i	$0.730555\pi$
$354$	0	0
$355$	42.9444	2.27925
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	25.8434	1.36396	0.681980	−	0.731370i	$-0.261119\pi$
$359$	25.8434	1.36396	0.681980	+	0.731370i	$0.261119\pi$
$360$	0	0
$361$	−18.6969	−0.984049
$362$	0	0
$363$	74.9444	3.93356
$364$	0	0
$365$	−38.7980	−2.03078
$366$	0	0
$367$	9.75255	0.509079	0.254540	−	0.967062i	$-0.418076\pi$
$367$	9.75255	0.509079	0.254540	+	0.967062i	$0.418076\pi$
$368$	0	0
$369$	32.6969	1.70213
$370$	0	0
$371$	1.89898	0.0985901
$372$	0	0
$373$	−28.4949	−1.47541	−0.737705	−	0.675123i	$-0.764090\pi$
$373$	−28.4949	−1.47541	−0.737705	+	0.675123i	$0.764090\pi$
$374$	0	0
$375$	−16.0454	−0.828581
$376$	0	0
$377$	5.89898	0.303813
$378$	0	0
$379$	−9.14643	−0.469820	−0.234910	−	0.972017i	$-0.575480\pi$
$379$	−9.14643	−0.469820	−0.234910	+	0.972017i	$0.575480\pi$
$380$	0	0
$381$	−24.4949	−1.25491
$382$	0	0
$383$	16.8990	0.863498	0.431749	−	0.901994i	$-0.357897\pi$
$383$	16.8990	0.863498	0.431749	+	0.901994i	$0.357897\pi$
$384$	0	0
$385$	22.2474	1.13383
$386$	0	0
$387$	−3.00000	−0.152499
$388$	0	0
$389$	−2.20204	−0.111648	−0.0558240	−	0.998441i	$-0.517779\pi$
$389$	−2.20204	−0.111648	−0.0558240	+	0.998441i	$0.517779\pi$
$390$	0	0
$391$	−9.55051	−0.482990
$392$	0	0
$393$	−55.5959	−2.80444
$394$	0	0
$395$	−7.24745	−0.364659
$396$	0	0
$397$	−15.9444	−0.800226	−0.400113	−	0.916466i	$-0.631029\pi$
$397$	−15.9444	−0.800226	−0.400113	+	0.916466i	$0.631029\pi$
$398$	0	0
$399$	−1.34847	−0.0675079
$400$	0	0
$401$	31.5959	1.57782	0.788912	−	0.614506i	$-0.210644\pi$
$401$	31.5959	1.57782	0.788912	+	0.614506i	$0.210644\pi$
$402$	0	0
$403$	3.44949	0.171831
$404$	0	0
$405$	31.0454	1.54266
$406$	0	0
$407$	54.4949	2.70121
$408$	0	0
$409$	21.9444	1.08508	0.542540	−	0.840030i	$-0.317463\pi$
$409$	21.9444	1.08508	0.542540	+	0.840030i	$0.317463\pi$
$410$	0	0
$411$	−3.30306	−0.162928
$412$	0	0
$413$	−14.0000	−0.688895
$414$	0	0
$415$	−15.0000	−0.736321
$416$	0	0
$417$	46.8990	2.29665
$418$	0	0
$419$	−5.75255	−0.281031	−0.140515	−	0.990079i	$-0.544876\pi$
$419$	−5.75255	−0.281031	−0.140515	+	0.990079i	$0.544876\pi$
$420$	0	0
$421$	10.2474	0.499430	0.249715	−	0.968319i	$-0.419663\pi$
$421$	10.2474	0.499430	0.249715	+	0.968319i	$0.419663\pi$
$422$	0	0
$423$	−10.3485	−0.503160
$424$	0	0
$425$	16.8990	0.819721
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	15.7980	0.762733
$430$	0	0
$431$	25.5959	1.23291	0.616456	−	0.787389i	$-0.288568\pi$
$431$	25.5959	1.23291	0.616456	+	0.787389i	$0.288568\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	49.8434	2.38981
$436$	0	0
$437$	2.14643	0.102678
$438$	0	0
$439$	−30.8990	−1.47473	−0.737364	−	0.675496i	$-0.763930\pi$
$439$	−30.8990	−1.47473	−0.737364	+	0.675496i	$0.763930\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	−10.5959	−0.503427	−0.251714	−	0.967802i	$-0.580994\pi$
$443$	−10.5959	−0.503427	−0.251714	+	0.967802i	$0.580994\pi$
$444$	0	0
$445$	39.4949	1.87224
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	20.8990	0.986284	0.493142	−	0.869949i	$-0.335848\pi$
$449$	20.8990	0.986284	0.493142	+	0.869949i	$0.335848\pi$
$450$	0	0
$451$	−70.2929	−3.30996
$452$	0	0
$453$	32.6969	1.53624
$454$	0	0
$455$	3.44949	0.161715
$456$	0	0
$457$	8.04541	0.376348	0.188174	−	0.982136i	$-0.439743\pi$
$457$	8.04541	0.376348	0.188174	+	0.982136i	$0.439743\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.8990	0.507616	0.253808	−	0.967255i	$-0.418317\pi$
$461$	10.8990	0.507616	0.253808	+	0.967255i	$0.418317\pi$
$462$	0	0
$463$	−11.3485	−0.527408	−0.263704	−	0.964604i	$-0.584944\pi$
$463$	−11.3485	−0.527408	−0.263704	+	0.964604i	$0.584944\pi$
$464$	0	0
$465$	29.1464	1.35163
$466$	0	0
$467$	−19.1464	−0.885991	−0.442996	−	0.896524i	$-0.646084\pi$
$467$	−19.1464	−0.885991	−0.442996	+	0.896524i	$0.646084\pi$
$468$	0	0
$469$	6.89898	0.318565
$470$	0	0
$471$	13.5959	0.626467
$472$	0	0
$473$	6.44949	0.296548
$474$	0	0
$475$	−3.79796	−0.174262
$476$	0	0
$477$	5.69694	0.260845
$478$	0	0
$479$	−16.3485	−0.746981	−0.373490	−	0.927634i	$-0.621839\pi$
$479$	−16.3485	−0.746981	−0.373490	+	0.927634i	$0.621839\pi$
$480$	0	0
$481$	8.44949	0.385264
$482$	0	0
$483$	−9.55051	−0.434563
$484$	0	0
$485$	36.3939	1.65256
$486$	0	0
$487$	−16.6969	−0.756611	−0.378305	−	0.925681i	$-0.623493\pi$
$487$	−16.6969	−0.756611	−0.378305	+	0.925681i	$0.623493\pi$
$488$	0	0
$489$	−55.5959	−2.51413
$490$	0	0
$491$	9.59592	0.433058	0.216529	−	0.976276i	$-0.430526\pi$
$491$	9.59592	0.433058	0.216529	+	0.976276i	$0.430526\pi$
$492$	0	0
$493$	−14.4495	−0.650772
$494$	0	0
$495$	66.7423	2.99985
$496$	0	0
$497$	−12.4495	−0.558436
$498$	0	0
$499$	−5.34847	−0.239430	−0.119715	−	0.992808i	$-0.538198\pi$
$499$	−5.34847	−0.239430	−0.119715	+	0.992808i	$0.538198\pi$
$500$	0	0
$501$	−54.7423	−2.44571
$502$	0	0
$503$	−2.40408	−0.107193	−0.0535964	−	0.998563i	$-0.517068\pi$
$503$	−2.40408	−0.107193	−0.0535964	+	0.998563i	$0.517068\pi$
$504$	0	0
$505$	−18.4495	−0.820992
$506$	0	0
$507$	2.44949	0.108786
$508$	0	0
$509$	−11.2474	−0.498534	−0.249267	−	0.968435i	$-0.580190\pi$
$509$	−11.2474	−0.498534	−0.249267	+	0.968435i	$0.580190\pi$
$510$	0	0
$511$	11.2474	0.497558
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.20204	0.273295
$516$	0	0
$517$	22.2474	0.978442
$518$	0	0
$519$	−27.7980	−1.22019
$520$	0	0
$521$	7.59592	0.332783	0.166392	−	0.986060i	$-0.446788\pi$
$521$	7.59592	0.332783	0.166392	+	0.986060i	$0.446788\pi$
$522$	0	0
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	0	0
$525$	16.8990	0.737532
$526$	0	0
$527$	−8.44949	−0.368066
$528$	0	0
$529$	−7.79796	−0.339042
$530$	0	0
$531$	−42.0000	−1.82264
$532$	0	0
$533$	−10.8990	−0.472087
$534$	0	0
$535$	−43.7980	−1.89355
$536$	0	0
$537$	55.3485	2.38846
$538$	0	0
$539$	−6.44949	−0.277799
$540$	0	0
$541$	18.6515	0.801892	0.400946	−	0.916102i	$-0.368682\pi$
$541$	18.6515	0.801892	0.400946	+	0.916102i	$0.368682\pi$
$542$	0	0
$543$	−20.6969	−0.888191
$544$	0	0
$545$	−63.6413	−2.72609
$546$	0	0
$547$	25.6969	1.09872	0.549361	−	0.835585i	$-0.314871\pi$
$547$	25.6969	1.09872	0.549361	+	0.835585i	$0.314871\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	3.24745	0.138346
$552$	0	0
$553$	2.10102	0.0893445
$554$	0	0
$555$	71.3939	3.03050
$556$	0	0
$557$	21.7980	0.923609	0.461805	−	0.886982i	$-0.347202\pi$
$557$	21.7980	0.923609	0.461805	+	0.886982i	$0.347202\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	−38.6969	−1.63379
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−37.9444	−1.59633
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	11.8990	0.498831	0.249416	−	0.968397i	$-0.419761\pi$
$569$	11.8990	0.498831	0.249416	+	0.968397i	$0.419761\pi$
$570$	0	0
$571$	−5.69694	−0.238409	−0.119205	−	0.992870i	$-0.538034\pi$
$571$	−5.69694	−0.238409	−0.119205	+	0.992870i	$0.538034\pi$
$572$	0	0
$573$	11.5051	0.480633
$574$	0	0
$575$	−26.8990	−1.12176
$576$	0	0
$577$	27.7980	1.15724	0.578622	−	0.815596i	$-0.303591\pi$
$577$	27.7980	1.15724	0.578622	+	0.815596i	$0.303591\pi$
$578$	0	0
$579$	−31.1010	−1.29251
$580$	0	0
$581$	4.34847	0.180405
$582$	0	0
$583$	−12.2474	−0.507237
$584$	0	0
$585$	10.3485	0.427857
$586$	0	0
$587$	−13.9444	−0.575546	−0.287773	−	0.957699i	$-0.592915\pi$
$587$	−13.9444	−0.575546	−0.287773	+	0.957699i	$0.592915\pi$
$588$	0	0
$589$	1.89898	0.0782461
$590$	0	0
$591$	44.0908	1.81365
$592$	0	0
$593$	−2.95459	−0.121331	−0.0606653	−	0.998158i	$-0.519322\pi$
$593$	−2.95459	−0.121331	−0.0606653	+	0.998158i	$0.519322\pi$
$594$	0	0
$595$	−8.44949	−0.346395
$596$	0	0
$597$	22.4041	0.916938
$598$	0	0
$599$	−15.6969	−0.641360	−0.320680	−	0.947188i	$-0.603911\pi$
$599$	−15.6969	−0.641360	−0.320680	+	0.947188i	$0.603911\pi$
$600$	0	0
$601$	−19.3939	−0.791093	−0.395546	−	0.918446i	$-0.629445\pi$
$601$	−19.3939	−0.791093	−0.395546	+	0.918446i	$0.629445\pi$
$602$	0	0
$603$	20.6969	0.842844
$604$	0	0
$605$	−105.540	−4.29082
$606$	0	0
$607$	−28.0454	−1.13833	−0.569164	−	0.822224i	$-0.692733\pi$
$607$	−28.0454	−1.13833	−0.569164	+	0.822224i	$0.692733\pi$
$608$	0	0
$609$	−14.4495	−0.585523
$610$	0	0
$611$	3.44949	0.139551
$612$	0	0
$613$	−25.7980	−1.04197	−0.520985	−	0.853566i	$-0.674435\pi$
$613$	−25.7980	−1.04197	−0.520985	+	0.853566i	$0.674435\pi$
$614$	0	0
$615$	−92.0908	−3.71346
$616$	0	0
$617$	5.59592	0.225283	0.112642	−	0.993636i	$-0.464069\pi$
$617$	5.59592	0.225283	0.112642	+	0.993636i	$0.464069\pi$
$618$	0	0
$619$	−45.3939	−1.82453	−0.912267	−	0.409596i	$-0.865670\pi$
$619$	−45.3939	−1.82453	−0.912267	+	0.409596i	$0.865670\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.4495	−0.458714
$624$	0	0
$625$	−11.8990	−0.475959
$626$	0	0
$627$	8.69694	0.347322
$628$	0	0
$629$	−20.6969	−0.825241
$630$	0	0
$631$	21.5959	0.859720	0.429860	−	0.902896i	$-0.358563\pi$
$631$	21.5959	0.859720	0.429860	+	0.902896i	$0.358563\pi$
$632$	0	0
$633$	−21.5505	−0.856556
$634$	0	0
$635$	34.4949	1.36889
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−37.3485	−1.47748
$640$	0	0
$641$	−40.3939	−1.59546	−0.797731	−	0.603013i	$-0.793967\pi$
$641$	−40.3939	−1.59546	−0.797731	+	0.603013i	$0.793967\pi$
$642$	0	0
$643$	−30.2929	−1.19463	−0.597317	−	0.802005i	$-0.703766\pi$
$643$	−30.2929	−1.19463	−0.597317	+	0.802005i	$0.703766\pi$
$644$	0	0
$645$	8.44949	0.332698
$646$	0	0
$647$	−12.4495	−0.489440	−0.244720	−	0.969594i	$-0.578696\pi$
$647$	−12.4495	−0.489440	−0.244720	+	0.969594i	$0.578696\pi$
$648$	0	0
$649$	90.2929	3.54430
$650$	0	0
$651$	−8.44949	−0.331162
$652$	0	0
$653$	−13.5959	−0.532049	−0.266025	−	0.963966i	$-0.585710\pi$
$653$	−13.5959	−0.532049	−0.266025	+	0.963966i	$0.585710\pi$
$654$	0	0
$655$	78.2929	3.05916
$656$	0	0
$657$	33.7423	1.31641
$658$	0	0
$659$	17.2020	0.670096	0.335048	−	0.942201i	$-0.391247\pi$
$659$	17.2020	0.670096	0.335048	+	0.942201i	$0.391247\pi$
$660$	0	0
$661$	−26.8434	−1.04409	−0.522043	−	0.852919i	$-0.674830\pi$
$661$	−26.8434	−1.04409	−0.522043	+	0.852919i	$0.674830\pi$
$662$	0	0
$663$	−6.00000	−0.233021
$664$	0	0
$665$	1.89898	0.0736393
$666$	0	0
$667$	23.0000	0.890564
$668$	0	0
$669$	−1.34847	−0.0521348
$670$	0	0
$671$	12.8990	0.497960
$672$	0	0
$673$	−9.40408	−0.362501	−0.181250	−	0.983437i	$-0.558014\pi$
$673$	−9.40408	−0.362501	−0.181250	+	0.983437i	$0.558014\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.8536	−0.494003	−0.247001	−	0.969015i	$-0.579445\pi$
$677$	−12.8536	−0.494003	−0.247001	+	0.969015i	$0.579445\pi$
$678$	0	0
$679$	−10.5505	−0.404891
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	−6.69694	−0.256251	−0.128126	−	0.991758i	$-0.540896\pi$
$683$	−6.69694	−0.256251	−0.128126	+	0.991758i	$0.540896\pi$
$684$	0	0
$685$	4.65153	0.177726
$686$	0	0
$687$	36.0000	1.37349
$688$	0	0
$689$	−1.89898	−0.0723454
$690$	0	0
$691$	−18.1464	−0.690323	−0.345161	−	0.938543i	$-0.612176\pi$
$691$	−18.1464	−0.690323	−0.345161	+	0.938543i	$0.612176\pi$
$692$	0	0
$693$	−19.3485	−0.734988
$694$	0	0
$695$	−66.0454	−2.50525
$696$	0	0
$697$	26.6969	1.01122
$698$	0	0
$699$	−6.85357	−0.259226
$700$	0	0
$701$	21.4949	0.811851	0.405926	−	0.913906i	$-0.366949\pi$
$701$	21.4949	0.811851	0.405926	+	0.913906i	$0.366949\pi$
$702$	0	0
$703$	4.65153	0.175436
$704$	0	0
$705$	29.1464	1.09772
$706$	0	0
$707$	5.34847	0.201150
$708$	0	0
$709$	15.8434	0.595010	0.297505	−	0.954720i	$-0.403845\pi$
$709$	15.8434	0.595010	0.297505	+	0.954720i	$0.403845\pi$
$710$	0	0
$711$	6.30306	0.236383
$712$	0	0
$713$	13.4495	0.503687
$714$	0	0
$715$	−22.2474	−0.832007
$716$	0	0
$717$	−41.3939	−1.54588
$718$	0	0
$719$	21.7526	0.811233	0.405617	−	0.914043i	$-0.367057\pi$
$719$	21.7526	0.811233	0.405617	+	0.914043i	$0.367057\pi$
$720$	0	0
$721$	−1.79796	−0.0669595
$722$	0	0
$723$	4.04541	0.150450
$724$	0	0
$725$	−40.6969	−1.51145
$726$	0	0
$727$	−46.9898	−1.74276	−0.871378	−	0.490613i	$-0.836773\pi$
$727$	−46.9898	−1.74276	−0.871378	+	0.490613i	$0.836773\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−2.44949	−0.0905977
$732$	0	0
$733$	−19.2474	−0.710921	−0.355460	−	0.934691i	$-0.615676\pi$
$733$	−19.2474	−0.710921	−0.355460	+	0.934691i	$0.615676\pi$
$734$	0	0
$735$	−8.44949	−0.311664
$736$	0	0
$737$	−44.4949	−1.63899
$738$	0	0
$739$	2.94439	0.108311	0.0541555	−	0.998533i	$-0.482753\pi$
$739$	2.94439	0.108311	0.0541555	+	0.998533i	$0.482753\pi$
$740$	0	0
$741$	1.34847	0.0495373
$742$	0	0
$743$	−21.3939	−0.784865	−0.392433	−	0.919781i	$-0.628366\pi$
$743$	−21.3939	−0.784865	−0.392433	+	0.919781i	$0.628366\pi$
$744$	0	0
$745$	−25.3485	−0.928696
$746$	0	0
$747$	13.0454	0.477307
$748$	0	0
$749$	12.6969	0.463936
$750$	0	0
$751$	49.2929	1.79872	0.899361	−	0.437207i	$-0.144032\pi$
$751$	49.2929	1.79872	0.899361	+	0.437207i	$0.144032\pi$
$752$	0	0
$753$	−20.6969	−0.754238
$754$	0	0
$755$	−46.0454	−1.67576
$756$	0	0
$757$	28.1010	1.02135	0.510674	−	0.859774i	$-0.329396\pi$
$757$	28.1010	1.02135	0.510674	+	0.859774i	$0.329396\pi$
$758$	0	0
$759$	61.5959	2.23579
$760$	0	0
$761$	−30.5505	−1.10746	−0.553728	−	0.832698i	$-0.686795\pi$
$761$	−30.5505	−1.10746	−0.553728	+	0.832698i	$0.686795\pi$
$762$	0	0
$763$	18.4495	0.667916
$764$	0	0
$765$	−25.3485	−0.916476
$766$	0	0
$767$	14.0000	0.505511
$768$	0	0
$769$	−21.2474	−0.766203	−0.383101	−	0.923706i	$-0.625144\pi$
$769$	−21.2474	−0.766203	−0.383101	+	0.923706i	$0.625144\pi$
$770$	0	0
$771$	−74.0908	−2.66832
$772$	0	0
$773$	−2.89898	−0.104269	−0.0521345	−	0.998640i	$-0.516602\pi$
$773$	−2.89898	−0.104269	−0.0521345	+	0.998640i	$0.516602\pi$
$774$	0	0
$775$	−23.7980	−0.854848
$776$	0	0
$777$	−20.6969	−0.742499
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	80.2929	2.87310
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.1464	−0.683365
$786$	0	0
$787$	14.3485	0.511468	0.255734	−	0.966747i	$-0.417683\pi$
$787$	14.3485	0.511468	0.255734	+	0.966747i	$0.417683\pi$
$788$	0	0
$789$	−1.95459	−0.0695853
$790$	0	0
$791$	11.0000	0.391115
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−16.0454	−0.569072
$796$	0	0
$797$	−20.2020	−0.715593	−0.357797	−	0.933800i	$-0.616472\pi$
$797$	−20.2020	−0.715593	−0.357797	+	0.933800i	$0.616472\pi$
$798$	0	0
$799$	−8.44949	−0.298921
$800$	0	0
$801$	−34.3485	−1.21364
$802$	0	0
$803$	−72.5403	−2.55989
$804$	0	0
$805$	13.4495	0.474032
$806$	0	0
$807$	−17.3939	−0.612293
$808$	0	0
$809$	−15.4949	−0.544772	−0.272386	−	0.962188i	$-0.587813\pi$
$809$	−15.4949	−0.544772	−0.272386	+	0.962188i	$0.587813\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	48.9898	1.71815
$814$	0	0
$815$	78.2929	2.74248
$816$	0	0
$817$	0.550510	0.0192599
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	−46.4949	−1.62268	−0.811342	−	0.584572i	$-0.801262\pi$
$821$	−46.4949	−1.62268	−0.811342	+	0.584572i	$0.801262\pi$
$822$	0	0
$823$	−46.8990	−1.63480	−0.817398	−	0.576074i	$-0.804584\pi$
$823$	−46.8990	−1.63480	−0.817398	+	0.576074i	$0.804584\pi$
$824$	0	0
$825$	−108.990	−3.79454
$826$	0	0
$827$	4.89898	0.170354	0.0851771	−	0.996366i	$-0.472854\pi$
$827$	4.89898	0.170354	0.0851771	+	0.996366i	$0.472854\pi$
$828$	0	0
$829$	−19.1464	−0.664983	−0.332491	−	0.943106i	$-0.607889\pi$
$829$	−19.1464	−0.664983	−0.332491	+	0.943106i	$0.607889\pi$
$830$	0	0
$831$	4.65153	0.161360
$832$	0	0
$833$	2.44949	0.0848698
$834$	0	0
$835$	77.0908	2.66784
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.49490	0.155181	0.0775905	−	0.996985i	$-0.475277\pi$
$839$	4.49490	0.155181	0.0775905	+	0.996985i	$0.475277\pi$
$840$	0	0
$841$	5.79796	0.199930
$842$	0	0
$843$	−44.2020	−1.52240
$844$	0	0
$845$	−3.44949	−0.118666
$846$	0	0
$847$	30.5959	1.05129
$848$	0	0
$849$	7.59592	0.260691
$850$	0	0
$851$	32.9444	1.12932
$852$	0	0
$853$	−38.5505	−1.31994	−0.659972	−	0.751290i	$-0.729432\pi$
$853$	−38.5505	−1.31994	−0.659972	+	0.751290i	$0.729432\pi$
$854$	0	0
$855$	5.69694	0.194831
$856$	0	0
$857$	22.4949	0.768411	0.384206	−	0.923248i	$-0.374475\pi$
$857$	22.4949	0.768411	0.384206	+	0.923248i	$0.374475\pi$
$858$	0	0
$859$	−11.7980	−0.402541	−0.201271	−	0.979536i	$-0.564507\pi$
$859$	−11.7980	−0.402541	−0.201271	+	0.979536i	$0.564507\pi$
$860$	0	0
$861$	26.6969	0.909829
$862$	0	0
$863$	3.59592	0.122406	0.0612032	−	0.998125i	$-0.480506\pi$
$863$	3.59592	0.122406	0.0612032	+	0.998125i	$0.480506\pi$
$864$	0	0
$865$	39.1464	1.33102
$866$	0	0
$867$	−26.9444	−0.915079
$868$	0	0
$869$	−13.5505	−0.459670
$870$	0	0
$871$	−6.89898	−0.233763
$872$	0	0
$873$	−31.6515	−1.07124
$874$	0	0
$875$	−6.55051	−0.221448
$876$	0	0
$877$	−20.9444	−0.707242	−0.353621	−	0.935389i	$-0.615050\pi$
$877$	−20.9444	−0.707242	−0.353621	+	0.935389i	$0.615050\pi$
$878$	0	0
$879$	−20.9444	−0.706437
$880$	0	0
$881$	−18.4949	−0.623109	−0.311554	−	0.950228i	$-0.600850\pi$
$881$	−18.4949	−0.623109	−0.311554	+	0.950228i	$0.600850\pi$
$882$	0	0
$883$	29.1918	0.982383	0.491192	−	0.871051i	$-0.336561\pi$
$883$	29.1918	0.982383	0.491192	+	0.871051i	$0.336561\pi$
$884$	0	0
$885$	118.293	3.97637
$886$	0	0
$887$	−32.8990	−1.10464	−0.552320	−	0.833632i	$-0.686257\pi$
$887$	−32.8990	−1.10464	−0.552320	+	0.833632i	$0.686257\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	58.0454	1.94459
$892$	0	0
$893$	1.89898	0.0635469
$894$	0	0
$895$	−77.9444	−2.60539
$896$	0	0
$897$	9.55051	0.318882
$898$	0	0
$899$	20.3485	0.678659
$900$	0	0
$901$	4.65153	0.154965
$902$	0	0
$903$	−2.44949	−0.0815139
$904$	0	0
$905$	29.1464	0.968860
$906$	0	0
$907$	−5.20204	−0.172731	−0.0863655	−	0.996264i	$-0.527525\pi$
$907$	−5.20204	−0.172731	−0.0863655	+	0.996264i	$0.527525\pi$
$908$	0	0
$909$	16.0454	0.532193
$910$	0	0
$911$	−32.3939	−1.07326	−0.536629	−	0.843819i	$-0.680302\pi$
$911$	−32.3939	−1.07326	−0.536629	+	0.843819i	$0.680302\pi$
$912$	0	0
$913$	−28.0454	−0.928168
$914$	0	0
$915$	16.8990	0.558663
$916$	0	0
$917$	−22.6969	−0.749519
$918$	0	0
$919$	19.5959	0.646410	0.323205	−	0.946329i	$-0.395240\pi$
$919$	19.5959	0.646410	0.323205	+	0.946329i	$0.395240\pi$
$920$	0	0
$921$	−8.94439	−0.294728
$922$	0	0
$923$	12.4495	0.409780
$924$	0	0
$925$	−58.2929	−1.91666
$926$	0	0
$927$	−5.39388	−0.177158
$928$	0	0
$929$	−9.24745	−0.303399	−0.151699	−	0.988427i	$-0.548475\pi$
$929$	−9.24745	−0.303399	−0.151699	+	0.988427i	$0.548475\pi$
$930$	0	0
$931$	−0.550510	−0.0180422
$932$	0	0
$933$	38.0908	1.24704
$934$	0	0
$935$	54.4949	1.78217
$936$	0	0
$937$	50.9444	1.66428	0.832140	−	0.554565i	$-0.187115\pi$
$937$	50.9444	1.66428	0.832140	+	0.554565i	$0.187115\pi$
$938$	0	0
$939$	−1.10102	−0.0359304
$940$	0	0
$941$	25.2474	0.823043	0.411522	−	0.911400i	$-0.364997\pi$
$941$	25.2474	0.823043	0.411522	+	0.911400i	$0.364997\pi$
$942$	0	0
$943$	−42.4949	−1.38382
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.8888	1.55618	0.778088	−	0.628155i	$-0.216190\pi$
$947$	47.8888	1.55618	0.778088	+	0.628155i	$0.216190\pi$
$948$	0	0
$949$	−11.2474	−0.365108
$950$	0	0
$951$	82.7878	2.68458
$952$	0	0
$953$	9.89898	0.320659	0.160330	−	0.987064i	$-0.448744\pi$
$953$	9.89898	0.320659	0.160330	+	0.987064i	$0.448744\pi$
$954$	0	0
$955$	−16.2020	−0.524286
$956$	0	0
$957$	93.1918	3.01246
$958$	0	0
$959$	−1.34847	−0.0435443
$960$	0	0
$961$	−19.1010	−0.616162
$962$	0	0
$963$	38.0908	1.22746
$964$	0	0
$965$	43.7980	1.40991
$966$	0	0
$967$	38.0454	1.22346	0.611729	−	0.791067i	$-0.290474\pi$
$967$	38.0454	1.22346	0.611729	+	0.791067i	$0.290474\pi$
$968$	0	0
$969$	−3.30306	−0.106110
$970$	0	0
$971$	3.30306	0.106000	0.0530001	−	0.998595i	$-0.483122\pi$
$971$	3.30306	0.106000	0.0530001	+	0.998595i	$0.483122\pi$
$972$	0	0
$973$	19.1464	0.613806
$974$	0	0
$975$	−16.8990	−0.541200
$976$	0	0
$977$	10.0454	0.321381	0.160691	−	0.987005i	$-0.448628\pi$
$977$	10.0454	0.321381	0.160691	+	0.987005i	$0.448628\pi$
$978$	0	0
$979$	73.8434	2.36004
$980$	0	0
$981$	55.3485	1.76714
$982$	0	0
$983$	−9.04541	−0.288504	−0.144252	−	0.989541i	$-0.546078\pi$
$983$	−9.04541	−0.288504	−0.144252	+	0.989541i	$0.546078\pi$
$984$	0	0
$985$	−62.0908	−1.97838
$986$	0	0
$987$	−8.44949	−0.268950
$988$	0	0
$989$	3.89898	0.123980
$990$	0	0
$991$	−23.3939	−0.743131	−0.371565	−	0.928407i	$-0.621179\pi$
$991$	−23.3939	−0.743131	−0.371565	+	0.928407i	$0.621179\pi$
$992$	0	0
$993$	−52.8990	−1.67870
$994$	0	0
$995$	−31.5505	−1.00022
$996$	0	0
$997$	24.8990	0.788559	0.394279	−	0.918991i	$-0.370994\pi$
$997$	24.8990	0.788559	0.394279	+	0.918991i	$0.370994\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.p.1.2		2
4.3	odd	2		364.2.a.c.1.1	✓	2
8.3	odd	2		5824.2.a.bm.1.2		2
8.5	even	2		5824.2.a.bn.1.1		2
12.11	even	2		3276.2.a.q.1.2		2
20.19	odd	2		9100.2.a.v.1.2		2
28.3	even	6		2548.2.j.l.1353.1		4
28.11	odd	6		2548.2.j.m.1353.2		4
28.19	even	6		2548.2.j.l.1145.1		4
28.23	odd	6		2548.2.j.m.1145.2		4
28.27	even	2		2548.2.a.m.1.2		2
52.31	even	4		4732.2.g.f.337.2		4
52.47	even	4		4732.2.g.f.337.1		4
52.51	odd	2		4732.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.a.c.1.1	✓	2	4.3	odd	2
1456.2.a.p.1.2		2	1.1	even	1	trivial
2548.2.a.m.1.2		2	28.27	even	2
2548.2.j.l.1145.1		4	28.19	even	6
2548.2.j.l.1353.1		4	28.3	even	6
2548.2.j.m.1145.2		4	28.23	odd	6
2548.2.j.m.1353.2		4	28.11	odd	6
3276.2.a.q.1.2		2	12.11	even	2
4732.2.a.i.1.1		2	52.51	odd	2
4732.2.g.f.337.1		4	52.47	even	4
4732.2.g.f.337.2		4	52.31	even	4
5824.2.a.bm.1.2		2	8.3	odd	2
5824.2.a.bn.1.1		2	8.5	even	2
9100.2.a.v.1.2		2	20.19	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} -3.44949 q^{5} +1.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{3} -3.44949 q^{5} +1.00000 q^{7} +3.00000 q^{9} -6.44949 q^{11} -1.00000 q^{13} -8.44949 q^{15} +2.44949 q^{17} -0.550510 q^{19} +2.44949 q^{21} -3.89898 q^{23} +6.89898 q^{25} -5.89898 q^{29} -3.44949 q^{31} -15.7980 q^{33} -3.44949 q^{35} -8.44949 q^{37} -2.44949 q^{39} +10.8990 q^{41} -1.00000 q^{43} -10.3485 q^{45} -3.44949 q^{47} +1.00000 q^{49} +6.00000 q^{51} +1.89898 q^{53} +22.2474 q^{55} -1.34847 q^{57} -14.0000 q^{59} -2.00000 q^{61} +3.00000 q^{63} +3.44949 q^{65} +6.89898 q^{67} -9.55051 q^{69} -12.4495 q^{71} +11.2474 q^{73} +16.8990 q^{75} -6.44949 q^{77} +2.10102 q^{79} -9.00000 q^{81} +4.34847 q^{83} -8.44949 q^{85} -14.4495 q^{87} -11.4495 q^{89} -1.00000 q^{91} -8.44949 q^{93} +1.89898 q^{95} -10.5505 q^{97} -19.3485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7 + 6 * q^9	\( 2 q - 2 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 2 q^{13} - 12 q^{15} - 6 q^{19} + 2 q^{23} + 4 q^{25} - 2 q^{29} - 2 q^{31} - 12 q^{33} - 2 q^{35} - 12 q^{37} + 12 q^{41} - 2 q^{43} - 6 q^{45} - 2 q^{47} + 2 q^{49} + 12 q^{51} - 6 q^{53} + 20 q^{55} + 12 q^{57} - 28 q^{59} - 4 q^{61} + 6 q^{63} + 2 q^{65} + 4 q^{67} - 24 q^{69} - 20 q^{71} - 2 q^{73} + 24 q^{75} - 8 q^{77} + 14 q^{79} - 18 q^{81} - 6 q^{83} - 12 q^{85} - 24 q^{87} - 18 q^{89} - 2 q^{91} - 12 q^{93} - 6 q^{95} - 26 q^{97} - 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 2 * q^13 - 12 * q^15 - 6 * q^19 + 2 * q^23 + 4 * q^25 - 2 * q^29 - 2 * q^31 - 12 * q^33 - 2 * q^35 - 12 * q^37 + 12 * q^41 - 2 * q^43 - 6 * q^45 - 2 * q^47 + 2 * q^49 + 12 * q^51 - 6 * q^53 + 20 * q^55 + 12 * q^57 - 28 * q^59 - 4 * q^61 + 6 * q^63 + 2 * q^65 + 4 * q^67 - 24 * q^69 - 20 * q^71 - 2 * q^73 + 24 * q^75 - 8 * q^77 + 14 * q^79 - 18 * q^81 - 6 * q^83 - 12 * q^85 - 24 * q^87 - 18 * q^89 - 2 * q^91 - 12 * q^93 - 6 * q^95 - 26 * q^97 - 24 * q^99

Introduction

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