LMFDB - Embedded newform 7448.2.a.bs.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 25x^{9} - 7x^{8} + 212x^{7} + 112x^{6} - 694x^{5} - 480x^{4} + 740x^{3} + 632x^{2} + 48x - 16 $ x^11 - 25x^9 - 7x^8 + 212x^7 + 112x^6 - 694x^5 - 480x^4 + 740x^3 + 632x^2 + 48*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.82026$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82026 q^{3} +3.85544 q^{5} +0.313338 q^{9} +O(q^{10})$	$q+1.82026 q^{3} +3.85544 q^{5} +0.313338 q^{9} +4.19083 q^{11} +3.06318 q^{13} +7.01790 q^{15} -3.97982 q^{17} +1.00000 q^{19} +4.41380 q^{23} +9.86445 q^{25} -4.89042 q^{27} -1.30749 q^{29} -2.76089 q^{31} +7.62839 q^{33} -6.65060 q^{37} +5.57577 q^{39} +7.13129 q^{41} +0.941270 q^{43} +1.20806 q^{45} -3.82965 q^{47} -7.24431 q^{51} -4.75686 q^{53} +16.1575 q^{55} +1.82026 q^{57} +14.2121 q^{59} +12.3094 q^{61} +11.8099 q^{65} +1.23465 q^{67} +8.03426 q^{69} +3.34049 q^{71} +10.1491 q^{73} +17.9558 q^{75} -4.37040 q^{79} -9.84183 q^{81} -3.65672 q^{83} -15.3440 q^{85} -2.37997 q^{87} +2.18611 q^{89} -5.02554 q^{93} +3.85544 q^{95} -15.2271 q^{97} +1.31315 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 3 q^{5} + 17 q^{9}+O(q^{10}) $ 11 * q - 3 * q^5 + 17 * q^9	$ 11 q - 3 q^{5} + 17 q^{9} - 6 q^{11} + 8 q^{13} + 4 q^{15} - 13 q^{17} + 11 q^{19} - 2 q^{23} + 38 q^{25} + 21 q^{27} + 16 q^{29} + 6 q^{31} - 8 q^{33} + 9 q^{37} + q^{39} + 4 q^{41} + 31 q^{43} - 33 q^{45} - 7 q^{47} - 21 q^{51} + 11 q^{55} - 34 q^{59} + 42 q^{65} - 14 q^{67} + 19 q^{69} - 16 q^{71} - 8 q^{73} - 2 q^{75} - 10 q^{79} + 51 q^{81} + 16 q^{83} + 34 q^{85} - 72 q^{87} + 20 q^{89} + 50 q^{93} - 3 q^{95} + 34 q^{99}+O(q^{100}) $ 11 * q - 3 * q^5 + 17 * q^9 - 6 * q^11 + 8 * q^13 + 4 * q^15 - 13 * q^17 + 11 * q^19 - 2 * q^23 + 38 * q^25 + 21 * q^27 + 16 * q^29 + 6 * q^31 - 8 * q^33 + 9 * q^37 + q^39 + 4 * q^41 + 31 * q^43 - 33 * q^45 - 7 * q^47 - 21 * q^51 + 11 * q^55 - 34 * q^59 + 42 * q^65 - 14 * q^67 + 19 * q^69 - 16 * q^71 - 8 * q^73 - 2 * q^75 - 10 * q^79 + 51 * q^81 + 16 * q^83 + 34 * q^85 - 72 * q^87 + 20 * q^89 + 50 * q^93 - 3 * q^95 + 34 * 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
$3$	1.82026	1.05093	0.525463	−	0.850816i	$-0.323892\pi$
$3$	1.82026	1.05093	0.525463	+	0.850816i	$0.323892\pi$
$4$	0	0
$5$	3.85544	1.72421	0.862103	−	0.506732i	$-0.169147\pi$
$5$	3.85544	1.72421	0.862103	+	0.506732i	$0.169147\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.313338	0.104446
$10$	0	0
$11$	4.19083	1.26358	0.631792	−	0.775138i	$-0.282320\pi$
$11$	4.19083	1.26358	0.631792	+	0.775138i	$0.282320\pi$
$12$	0	0
$13$	3.06318	0.849572	0.424786	−	0.905294i	$-0.360349\pi$
$13$	3.06318	0.849572	0.424786	+	0.905294i	$0.360349\pi$
$14$	0	0
$15$	7.01790	1.81201
$16$	0	0
$17$	−3.97982	−0.965249	−0.482625	−	0.875827i	$-0.660316\pi$
$17$	−3.97982	−0.965249	−0.482625	+	0.875827i	$0.660316\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.41380	0.920341	0.460171	−	0.887830i	$-0.347788\pi$
$23$	4.41380	0.920341	0.460171	+	0.887830i	$0.347788\pi$
$24$	0	0
$25$	9.86445	1.97289
$26$	0	0
$27$	−4.89042	−0.941161
$28$	0	0
$29$	−1.30749	−0.242795	−0.121398	−	0.992604i	$-0.538738\pi$
$29$	−1.30749	−0.242795	−0.121398	+	0.992604i	$0.538738\pi$
$30$	0	0
$31$	−2.76089	−0.495871	−0.247935	−	0.968777i	$-0.579752\pi$
$31$	−2.76089	−0.495871	−0.247935	+	0.968777i	$0.579752\pi$
$32$	0	0
$33$	7.62839	1.32793
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.65060	−1.09335	−0.546676	−	0.837344i	$-0.684107\pi$
$37$	−6.65060	−1.09335	−0.546676	+	0.837344i	$0.684107\pi$
$38$	0	0
$39$	5.57577	0.892838
$40$	0	0
$41$	7.13129	1.11372	0.556860	−	0.830606i	$-0.312006\pi$
$41$	7.13129	1.11372	0.556860	+	0.830606i	$0.312006\pi$
$42$	0	0
$43$	0.941270	0.143542	0.0717712	−	0.997421i	$-0.477135\pi$
$43$	0.941270	0.143542	0.0717712	+	0.997421i	$0.477135\pi$
$44$	0	0
$45$	1.20806	0.180086
$46$	0	0
$47$	−3.82965	−0.558612	−0.279306	−	0.960202i	$-0.590104\pi$
$47$	−3.82965	−0.558612	−0.279306	+	0.960202i	$0.590104\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−7.24431	−1.01441
$52$	0	0
$53$	−4.75686	−0.653405	−0.326702	−	0.945127i	$-0.605937\pi$
$53$	−4.75686	−0.653405	−0.326702	+	0.945127i	$0.605937\pi$
$54$	0	0
$55$	16.1575	2.17868
$56$	0	0
$57$	1.82026	0.241099
$58$	0	0
$59$	14.2121	1.85026	0.925129	−	0.379653i	$-0.123957\pi$
$59$	14.2121	1.85026	0.925129	+	0.379653i	$0.123957\pi$
$60$	0	0
$61$	12.3094	1.57605	0.788026	−	0.615642i	$-0.211103\pi$
$61$	12.3094	1.57605	0.788026	+	0.615642i	$0.211103\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	11.8099	1.46484
$66$	0	0
$67$	1.23465	0.150837	0.0754184	−	0.997152i	$-0.475971\pi$
$67$	1.23465	0.150837	0.0754184	+	0.997152i	$0.475971\pi$
$68$	0	0
$69$	8.03426	0.967211
$70$	0	0
$71$	3.34049	0.396443	0.198222	−	0.980157i	$-0.436483\pi$
$71$	3.34049	0.396443	0.198222	+	0.980157i	$0.436483\pi$
$72$	0	0
$73$	10.1491	1.18786	0.593931	−	0.804516i	$-0.297575\pi$
$73$	10.1491	1.18786	0.593931	+	0.804516i	$0.297575\pi$
$74$	0	0
$75$	17.9558	2.07336
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.37040	−0.491708	−0.245854	−	0.969307i	$-0.579068\pi$
$79$	−4.37040	−0.491708	−0.245854	+	0.969307i	$0.579068\pi$
$80$	0	0
$81$	−9.84183	−1.09354
$82$	0	0
$83$	−3.65672	−0.401377	−0.200689	−	0.979655i	$-0.564318\pi$
$83$	−3.65672	−0.401377	−0.200689	+	0.979655i	$0.564318\pi$
$84$	0	0
$85$	−15.3440	−1.66429
$86$	0	0
$87$	−2.37997	−0.255160
$88$	0	0
$89$	2.18611	0.231728	0.115864	−	0.993265i	$-0.463036\pi$
$89$	2.18611	0.231728	0.115864	+	0.993265i	$0.463036\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.02554	−0.521124
$94$	0	0
$95$	3.85544	0.395560
$96$	0	0
$97$	−15.2271	−1.54608	−0.773038	−	0.634360i	$-0.781264\pi$
$97$	−15.2271	−1.54608	−0.773038	+	0.634360i	$0.781264\pi$
$98$	0	0
$99$	1.31315	0.131976
$100$	0	0
$101$	−17.7047	−1.76168	−0.880840	−	0.473414i	$-0.843021\pi$
$101$	−17.7047	−1.76168	−0.880840	+	0.473414i	$0.843021\pi$
$102$	0	0
$103$	7.23148	0.712539	0.356269	−	0.934383i	$-0.384049\pi$
$103$	7.23148	0.712539	0.356269	+	0.934383i	$0.384049\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.0821	−1.84474	−0.922370	−	0.386307i	$-0.873750\pi$
$107$	−19.0821	−1.84474	−0.922370	+	0.386307i	$0.873750\pi$
$108$	0	0
$109$	3.89101	0.372691	0.186346	−	0.982484i	$-0.440336\pi$
$109$	3.89101	0.372691	0.186346	+	0.982484i	$0.440336\pi$
$110$	0	0
$111$	−12.1058	−1.14903
$112$	0	0
$113$	13.2556	1.24699	0.623493	−	0.781829i	$-0.285713\pi$
$113$	13.2556	1.24699	0.623493	+	0.781829i	$0.285713\pi$
$114$	0	0
$115$	17.0172	1.58686
$116$	0	0
$117$	0.959809	0.0887344
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.56307	0.596643
$122$	0	0
$123$	12.9808	1.17044
$124$	0	0
$125$	18.7546	1.67746
$126$	0	0
$127$	1.30466	0.115770	0.0578850	−	0.998323i	$-0.481564\pi$
$127$	1.30466	0.115770	0.0578850	+	0.998323i	$0.481564\pi$
$128$	0	0
$129$	1.71335	0.150852
$130$	0	0
$131$	−13.3871	−1.16963	−0.584817	−	0.811165i	$-0.698834\pi$
$131$	−13.3871	−1.16963	−0.584817	+	0.811165i	$0.698834\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−18.8547	−1.62276
$136$	0	0
$137$	−4.60260	−0.393226	−0.196613	−	0.980481i	$-0.562994\pi$
$137$	−4.60260	−0.393226	−0.196613	+	0.980481i	$0.562994\pi$
$138$	0	0
$139$	−18.6169	−1.57906	−0.789532	−	0.613709i	$-0.789677\pi$
$139$	−18.6169	−1.57906	−0.789532	+	0.613709i	$0.789677\pi$
$140$	0	0
$141$	−6.97095	−0.587060
$142$	0	0
$143$	12.8373	1.07351
$144$	0	0
$145$	−5.04096	−0.418629
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.7162	1.04175	0.520876	−	0.853633i	$-0.325606\pi$
$149$	12.7162	1.04175	0.520876	+	0.853633i	$0.325606\pi$
$150$	0	0
$151$	9.86311	0.802648	0.401324	−	0.915936i	$-0.368550\pi$
$151$	9.86311	0.802648	0.401324	+	0.915936i	$0.368550\pi$
$152$	0	0
$153$	−1.24703	−0.100816
$154$	0	0
$155$	−10.6445	−0.854984
$156$	0	0
$157$	5.25745	0.419590	0.209795	−	0.977745i	$-0.432720\pi$
$157$	5.25745	0.419590	0.209795	+	0.977745i	$0.432720\pi$
$158$	0	0
$159$	−8.65871	−0.686680
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.86698	0.694515	0.347258	−	0.937770i	$-0.387113\pi$
$163$	8.86698	0.694515	0.347258	+	0.937770i	$0.387113\pi$
$164$	0	0
$165$	29.4108	2.28963
$166$	0	0
$167$	24.9530	1.93092	0.965460	−	0.260551i	$-0.0839042\pi$
$167$	24.9530	1.93092	0.965460	+	0.260551i	$0.0839042\pi$
$168$	0	0
$169$	−3.61696	−0.278227
$170$	0	0
$171$	0.313338	0.0239616
$172$	0	0
$173$	26.1510	1.98822	0.994111	−	0.108362i	$-0.0345606\pi$
$173$	26.1510	1.98822	0.994111	+	0.108362i	$0.0345606\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	25.8697	1.94448
$178$	0	0
$179$	−13.2486	−0.990250	−0.495125	−	0.868822i	$-0.664878\pi$
$179$	−13.2486	−0.990250	−0.495125	+	0.868822i	$0.664878\pi$
$180$	0	0
$181$	12.4709	0.926956	0.463478	−	0.886108i	$-0.346601\pi$
$181$	12.4709	0.926956	0.463478	+	0.886108i	$0.346601\pi$
$182$	0	0
$183$	22.4062	1.65631
$184$	0	0
$185$	−25.6410	−1.88517
$186$	0	0
$187$	−16.6788	−1.21967
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.61071	0.116547	0.0582734	−	0.998301i	$-0.481440\pi$
$191$	1.61071	0.116547	0.0582734	+	0.998301i	$0.481440\pi$
$192$	0	0
$193$	−7.35591	−0.529490	−0.264745	−	0.964318i	$-0.585288\pi$
$193$	−7.35591	−0.529490	−0.264745	+	0.964318i	$0.585288\pi$
$194$	0	0
$195$	21.4971	1.53944
$196$	0	0
$197$	8.99752	0.641047	0.320523	−	0.947241i	$-0.396141\pi$
$197$	8.99752	0.641047	0.320523	+	0.947241i	$0.396141\pi$
$198$	0	0
$199$	−26.7271	−1.89463	−0.947316	−	0.320300i	$-0.896216\pi$
$199$	−26.7271	−1.89463	−0.947316	+	0.320300i	$0.896216\pi$
$200$	0	0
$201$	2.24739	0.158518
$202$	0	0
$203$	0	0
$204$	0	0
$205$	27.4943	1.92028
$206$	0	0
$207$	1.38301	0.0961260
$208$	0	0
$209$	4.19083	0.289886
$210$	0	0
$211$	−3.20799	−0.220847	−0.110424	−	0.993885i	$-0.535221\pi$
$211$	−3.20799	−0.220847	−0.110424	+	0.993885i	$0.535221\pi$
$212$	0	0
$213$	6.08056	0.416633
$214$	0	0
$215$	3.62902	0.247497
$216$	0	0
$217$	0	0
$218$	0	0
$219$	18.4740	1.24836
$220$	0	0
$221$	−12.1909	−0.820049
$222$	0	0
$223$	−2.50654	−0.167850	−0.0839252	−	0.996472i	$-0.526746\pi$
$223$	−2.50654	−0.167850	−0.0839252	+	0.996472i	$0.526746\pi$
$224$	0	0
$225$	3.09091	0.206060
$226$	0	0
$227$	−19.2125	−1.27518	−0.637589	−	0.770377i	$-0.720068\pi$
$227$	−19.2125	−1.27518	−0.637589	+	0.770377i	$0.720068\pi$
$228$	0	0
$229$	−21.2733	−1.40578	−0.702891	−	0.711297i	$-0.748108\pi$
$229$	−21.2733	−1.40578	−0.702891	+	0.711297i	$0.748108\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.1484	0.861380	0.430690	−	0.902500i	$-0.358270\pi$
$233$	13.1484	0.861380	0.430690	+	0.902500i	$0.358270\pi$
$234$	0	0
$235$	−14.7650	−0.963163
$236$	0	0
$237$	−7.95525	−0.516749
$238$	0	0
$239$	−23.9446	−1.54884	−0.774422	−	0.632669i	$-0.781959\pi$
$239$	−23.9446	−1.54884	−0.774422	+	0.632669i	$0.781959\pi$
$240$	0	0
$241$	−10.9362	−0.704460	−0.352230	−	0.935913i	$-0.614577\pi$
$241$	−10.9362	−0.704460	−0.352230	+	0.935913i	$0.614577\pi$
$242$	0	0
$243$	−3.24342	−0.208066
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.06318	0.194905
$248$	0	0
$249$	−6.65617	−0.421818
$250$	0	0
$251$	−3.82366	−0.241347	−0.120674	−	0.992692i	$-0.538505\pi$
$251$	−3.82366	−0.241347	−0.120674	+	0.992692i	$0.538505\pi$
$252$	0	0
$253$	18.4975	1.16293
$254$	0	0
$255$	−27.9300	−1.74905
$256$	0	0
$257$	−9.41167	−0.587084	−0.293542	−	0.955946i	$-0.594834\pi$
$257$	−9.41167	−0.587084	−0.293542	+	0.955946i	$0.594834\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.409687	−0.0253590
$262$	0	0
$263$	−31.6226	−1.94993	−0.974967	−	0.222349i	$-0.928627\pi$
$263$	−31.6226	−1.94993	−0.974967	+	0.222349i	$0.928627\pi$
$264$	0	0
$265$	−18.3398	−1.12660
$266$	0	0
$267$	3.97929	0.243529
$268$	0	0
$269$	16.0014	0.975622	0.487811	−	0.872949i	$-0.337796\pi$
$269$	16.0014	0.975622	0.487811	+	0.872949i	$0.337796\pi$
$270$	0	0
$271$	15.2324	0.925305	0.462652	−	0.886540i	$-0.346898\pi$
$271$	15.2324	0.925305	0.462652	+	0.886540i	$0.346898\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	41.3402	2.49291
$276$	0	0
$277$	−25.0290	−1.50385	−0.751924	−	0.659250i	$-0.770874\pi$
$277$	−25.0290	−1.50385	−0.751924	+	0.659250i	$0.770874\pi$
$278$	0	0
$279$	−0.865093	−0.0517917
$280$	0	0
$281$	13.0425	0.778048	0.389024	−	0.921228i	$-0.372812\pi$
$281$	13.0425	0.778048	0.389024	+	0.921228i	$0.372812\pi$
$282$	0	0
$283$	3.75533	0.223231	0.111616	−	0.993751i	$-0.464397\pi$
$283$	3.75533	0.223231	0.111616	+	0.993751i	$0.464397\pi$
$284$	0	0
$285$	7.01790	0.415705
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.16100	−0.0682941
$290$	0	0
$291$	−27.7172	−1.62481
$292$	0	0
$293$	18.1910	1.06273	0.531364	−	0.847144i	$-0.321680\pi$
$293$	18.1910	1.06273	0.531364	+	0.847144i	$0.321680\pi$
$294$	0	0
$295$	54.7940	3.19023
$296$	0	0
$297$	−20.4949	−1.18924
$298$	0	0
$299$	13.5203	0.781896
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−32.2270	−1.85140
$304$	0	0
$305$	47.4580	2.71744
$306$	0	0
$307$	4.85275	0.276961	0.138480	−	0.990365i	$-0.455778\pi$
$307$	4.85275	0.276961	0.138480	+	0.990365i	$0.455778\pi$
$308$	0	0
$309$	13.1632	0.748826
$310$	0	0
$311$	−31.2813	−1.77380	−0.886901	−	0.461960i	$-0.847147\pi$
$311$	−31.2813	−1.77380	−0.886901	+	0.461960i	$0.847147\pi$
$312$	0	0
$313$	30.5159	1.72486	0.862430	−	0.506176i	$-0.168941\pi$
$313$	30.5159	1.72486	0.862430	+	0.506176i	$0.168941\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.1774	1.41410	0.707052	−	0.707162i	$-0.250025\pi$
$317$	25.1774	1.41410	0.707052	+	0.707162i	$0.250025\pi$
$318$	0	0
$319$	−5.47948	−0.306792
$320$	0	0
$321$	−34.7344	−1.93869
$322$	0	0
$323$	−3.97982	−0.221443
$324$	0	0
$325$	30.2165	1.67611
$326$	0	0
$327$	7.08264	0.391671
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.74426	0.260768	0.130384	−	0.991464i	$-0.458379\pi$
$331$	4.74426	0.260768	0.130384	+	0.991464i	$0.458379\pi$
$332$	0	0
$333$	−2.08389	−0.114196
$334$	0	0
$335$	4.76013	0.260074
$336$	0	0
$337$	18.6177	1.01417	0.507085	−	0.861896i	$-0.330723\pi$
$337$	18.6177	1.01417	0.507085	+	0.861896i	$0.330723\pi$
$338$	0	0
$339$	24.1287	1.31049
$340$	0	0
$341$	−11.5704	−0.626574
$342$	0	0
$343$	0	0
$344$	0	0
$345$	30.9756	1.66767
$346$	0	0
$347$	15.8654	0.851699	0.425849	−	0.904794i	$-0.359975\pi$
$347$	15.8654	0.851699	0.425849	+	0.904794i	$0.359975\pi$
$348$	0	0
$349$	−19.2412	−1.02996	−0.514979	−	0.857203i	$-0.672200\pi$
$349$	−19.2412	−1.02996	−0.514979	+	0.857203i	$0.672200\pi$
$350$	0	0
$351$	−14.9802	−0.799584
$352$	0	0
$353$	−28.7574	−1.53060	−0.765300	−	0.643674i	$-0.777409\pi$
$353$	−28.7574	−1.53060	−0.765300	+	0.643674i	$0.777409\pi$
$354$	0	0
$355$	12.8791	0.683550
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.9140	−1.15658	−0.578289	−	0.815832i	$-0.696280\pi$
$359$	−21.9140	−1.15658	−0.578289	+	0.815832i	$0.696280\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.9465	0.627028
$364$	0	0
$365$	39.1293	2.04812
$366$	0	0
$367$	−16.3984	−0.855987	−0.427994	−	0.903782i	$-0.640779\pi$
$367$	−16.3984	−0.855987	−0.427994	+	0.903782i	$0.640779\pi$
$368$	0	0
$369$	2.23450	0.116324
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.52870	0.130931	0.0654655	−	0.997855i	$-0.479147\pi$
$373$	2.52870	0.130931	0.0654655	+	0.997855i	$0.479147\pi$
$374$	0	0
$375$	34.1382	1.76289
$376$	0	0
$377$	−4.00508	−0.206272
$378$	0	0
$379$	−10.4243	−0.535460	−0.267730	−	0.963494i	$-0.586274\pi$
$379$	−10.4243	−0.535460	−0.267730	+	0.963494i	$0.586274\pi$
$380$	0	0
$381$	2.37482	0.121666
$382$	0	0
$383$	−4.72139	−0.241252	−0.120626	−	0.992698i	$-0.538490\pi$
$383$	−4.72139	−0.241252	−0.120626	+	0.992698i	$0.538490\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.294936	0.0149924
$388$	0	0
$389$	4.31577	0.218818	0.109409	−	0.993997i	$-0.465104\pi$
$389$	4.31577	0.218818	0.109409	+	0.993997i	$0.465104\pi$
$390$	0	0
$391$	−17.5662	−0.888359
$392$	0	0
$393$	−24.3679	−1.22920
$394$	0	0
$395$	−16.8498	−0.847806
$396$	0	0
$397$	−30.1901	−1.51520	−0.757599	−	0.652721i	$-0.773627\pi$
$397$	−30.1901	−1.51520	−0.757599	+	0.652721i	$0.773627\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.33867	0.216663	0.108331	−	0.994115i	$-0.465449\pi$
$401$	4.33867	0.216663	0.108331	+	0.994115i	$0.465449\pi$
$402$	0	0
$403$	−8.45710	−0.421278
$404$	0	0
$405$	−37.9446	−1.88548
$406$	0	0
$407$	−27.8716	−1.38154
$408$	0	0
$409$	−25.6190	−1.26678	−0.633390	−	0.773833i	$-0.718337\pi$
$409$	−25.6190	−1.26678	−0.633390	+	0.773833i	$0.718337\pi$
$410$	0	0
$411$	−8.37791	−0.413252
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−14.0983	−0.692057
$416$	0	0
$417$	−33.8876	−1.65948
$418$	0	0
$419$	39.7850	1.94362	0.971812	−	0.235759i	$-0.0757576\pi$
$419$	39.7850	1.94362	0.971812	+	0.235759i	$0.0757576\pi$
$420$	0	0
$421$	−7.12318	−0.347163	−0.173581	−	0.984820i	$-0.555534\pi$
$421$	−7.12318	−0.347163	−0.173581	+	0.984820i	$0.555534\pi$
$422$	0	0
$423$	−1.19998	−0.0583448
$424$	0	0
$425$	−39.2588	−1.90433
$426$	0	0
$427$	0	0
$428$	0	0
$429$	23.3671	1.12817
$430$	0	0
$431$	−15.7513	−0.758714	−0.379357	−	0.925250i	$-0.623855\pi$
$431$	−15.7513	−0.758714	−0.379357	+	0.925250i	$0.623855\pi$
$432$	0	0
$433$	−31.7038	−1.52359	−0.761793	−	0.647821i	$-0.775681\pi$
$433$	−31.7038	−1.52359	−0.761793	+	0.647821i	$0.775681\pi$
$434$	0	0
$435$	−9.17585	−0.439948
$436$	0	0
$437$	4.41380	0.211141
$438$	0	0
$439$	7.77926	0.371284	0.185642	−	0.982617i	$-0.440564\pi$
$439$	7.77926	0.371284	0.185642	+	0.982617i	$0.440564\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.0201	1.33127	0.665636	−	0.746276i	$-0.268160\pi$
$443$	28.0201	1.33127	0.665636	+	0.746276i	$0.268160\pi$
$444$	0	0
$445$	8.42844	0.399546
$446$	0	0
$447$	23.1467	1.09480
$448$	0	0
$449$	36.2980	1.71301	0.856504	−	0.516141i	$-0.172632\pi$
$449$	36.2980	1.71301	0.856504	+	0.516141i	$0.172632\pi$
$450$	0	0
$451$	29.8860	1.40728
$452$	0	0
$453$	17.9534	0.843524
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.3401	1.46603	0.733014	−	0.680214i	$-0.238113\pi$
$457$	31.3401	1.46603	0.733014	+	0.680214i	$0.238113\pi$
$458$	0	0
$459$	19.4630	0.908455
$460$	0	0
$461$	−17.3923	−0.810040	−0.405020	−	0.914308i	$-0.632735\pi$
$461$	−17.3923	−0.810040	−0.405020	+	0.914308i	$0.632735\pi$
$462$	0	0
$463$	10.0838	0.468632	0.234316	−	0.972160i	$-0.424715\pi$
$463$	10.0838	0.468632	0.234316	+	0.972160i	$0.424715\pi$
$464$	0	0
$465$	−19.3757	−0.898525
$466$	0	0
$467$	−2.12411	−0.0982919	−0.0491459	−	0.998792i	$-0.515650\pi$
$467$	−2.12411	−0.0982919	−0.0491459	+	0.998792i	$0.515650\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.56992	0.440959
$472$	0	0
$473$	3.94471	0.181378
$474$	0	0
$475$	9.86445	0.452612
$476$	0	0
$477$	−1.49050	−0.0682455
$478$	0	0
$479$	−3.13216	−0.143112	−0.0715559	−	0.997437i	$-0.522796\pi$
$479$	−3.13216	−0.143112	−0.0715559	+	0.997437i	$0.522796\pi$
$480$	0	0
$481$	−20.3720	−0.928882
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−58.7072	−2.66575
$486$	0	0
$487$	3.53918	0.160376	0.0801878	−	0.996780i	$-0.474448\pi$
$487$	3.53918	0.160376	0.0801878	+	0.996780i	$0.474448\pi$
$488$	0	0
$489$	16.1402	0.729885
$490$	0	0
$491$	−19.4301	−0.876869	−0.438434	−	0.898763i	$-0.644467\pi$
$491$	−19.4301	−0.876869	−0.438434	+	0.898763i	$0.644467\pi$
$492$	0	0
$493$	5.20359	0.234358
$494$	0	0
$495$	5.06276	0.227554
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.07116	0.406081	0.203041	−	0.979170i	$-0.434918\pi$
$499$	9.07116	0.406081	0.203041	+	0.979170i	$0.434918\pi$
$500$	0	0
$501$	45.4209	2.02925
$502$	0	0
$503$	−9.20620	−0.410484	−0.205242	−	0.978711i	$-0.565798\pi$
$503$	−9.20620	−0.410484	−0.205242	+	0.978711i	$0.565798\pi$
$504$	0	0
$505$	−68.2593	−3.03750
$506$	0	0
$507$	−6.58379	−0.292396
$508$	0	0
$509$	20.1553	0.893369	0.446684	−	0.894692i	$-0.352605\pi$
$509$	20.1553	0.893369	0.446684	+	0.894692i	$0.352605\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.89042	−0.215917
$514$	0	0
$515$	27.8806	1.22856
$516$	0	0
$517$	−16.0494	−0.705853
$518$	0	0
$519$	47.6015	2.08948
$520$	0	0
$521$	−22.5787	−0.989192	−0.494596	−	0.869123i	$-0.664684\pi$
$521$	−22.5787	−0.989192	−0.494596	+	0.869123i	$0.664684\pi$
$522$	0	0
$523$	−17.9527	−0.785017	−0.392509	−	0.919748i	$-0.628393\pi$
$523$	−17.9527	−0.785017	−0.392509	+	0.919748i	$0.628393\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.9879	0.478639
$528$	0	0
$529$	−3.51835	−0.152972
$530$	0	0
$531$	4.45319	0.193252
$532$	0	0
$533$	21.8444	0.946185
$534$	0	0
$535$	−73.5701	−3.18071
$536$	0	0
$537$	−24.1159	−1.04068
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−20.9664	−0.901418	−0.450709	−	0.892671i	$-0.648829\pi$
$541$	−20.9664	−0.901418	−0.450709	+	0.892671i	$0.648829\pi$
$542$	0	0
$543$	22.7003	0.974162
$544$	0	0
$545$	15.0016	0.642597
$546$	0	0
$547$	43.7726	1.87158	0.935791	−	0.352555i	$-0.114687\pi$
$547$	43.7726	1.87158	0.935791	+	0.352555i	$0.114687\pi$
$548$	0	0
$549$	3.85699	0.164612
$550$	0	0
$551$	−1.30749	−0.0557010
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−46.6733	−1.98117
$556$	0	0
$557$	−13.4598	−0.570310	−0.285155	−	0.958481i	$-0.592045\pi$
$557$	−13.4598	−0.570310	−0.285155	+	0.958481i	$0.592045\pi$
$558$	0	0
$559$	2.88328	0.121950
$560$	0	0
$561$	−30.3597	−1.28179
$562$	0	0
$563$	−31.0386	−1.30812	−0.654060	−	0.756443i	$-0.726935\pi$
$563$	−31.0386	−1.30812	−0.654060	+	0.756443i	$0.726935\pi$
$564$	0	0
$565$	51.1063	2.15006
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.6546	−0.446666	−0.223333	−	0.974742i	$-0.571694\pi$
$569$	−10.6546	−0.446666	−0.223333	+	0.974742i	$0.571694\pi$
$570$	0	0
$571$	7.23552	0.302797	0.151399	−	0.988473i	$-0.451622\pi$
$571$	7.23552	0.302797	0.151399	+	0.988473i	$0.451622\pi$
$572$	0	0
$573$	2.93191	0.122482
$574$	0	0
$575$	43.5397	1.81573
$576$	0	0
$577$	−19.3719	−0.806464	−0.403232	−	0.915098i	$-0.632113\pi$
$577$	−19.3719	−0.806464	−0.403232	+	0.915098i	$0.632113\pi$
$578$	0	0
$579$	−13.3897	−0.556455
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−19.9352	−0.825631
$584$	0	0
$585$	3.70049	0.152996
$586$	0	0
$587$	19.9453	0.823231	0.411616	−	0.911357i	$-0.364965\pi$
$587$	19.9453	0.823231	0.411616	+	0.911357i	$0.364965\pi$
$588$	0	0
$589$	−2.76089	−0.113761
$590$	0	0
$591$	16.3778	0.673693
$592$	0	0
$593$	24.7738	1.01734	0.508670	−	0.860962i	$-0.330138\pi$
$593$	24.7738	1.01734	0.508670	+	0.860962i	$0.330138\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−48.6502	−1.99112
$598$	0	0
$599$	−40.6363	−1.66035	−0.830177	−	0.557499i	$-0.811761\pi$
$599$	−40.6363	−1.66035	−0.830177	+	0.557499i	$0.811761\pi$
$600$	0	0
$601$	−3.27442	−0.133567	−0.0667833	−	0.997768i	$-0.521274\pi$
$601$	−3.27442	−0.133567	−0.0667833	+	0.997768i	$0.521274\pi$
$602$	0	0
$603$	0.386864	0.0157543
$604$	0	0
$605$	25.3036	1.02874
$606$	0	0
$607$	−26.3608	−1.06995	−0.534976	−	0.844867i	$-0.679679\pi$
$607$	−26.3608	−1.06995	−0.534976	+	0.844867i	$0.679679\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.7309	−0.474581
$612$	0	0
$613$	24.4643	0.988105	0.494052	−	0.869432i	$-0.335515\pi$
$613$	24.4643	0.988105	0.494052	+	0.869432i	$0.335515\pi$
$614$	0	0
$615$	50.0467	2.01808
$616$	0	0
$617$	36.1045	1.45351	0.726757	−	0.686895i	$-0.241027\pi$
$617$	36.1045	1.45351	0.726757	+	0.686895i	$0.241027\pi$
$618$	0	0
$619$	30.0182	1.20653	0.603267	−	0.797539i	$-0.293865\pi$
$619$	30.0182	1.20653	0.603267	+	0.797539i	$0.293865\pi$
$620$	0	0
$621$	−21.5853	−0.866190
$622$	0	0
$623$	0	0
$624$	0	0
$625$	22.9851	0.919403
$626$	0	0
$627$	7.62839	0.304649
$628$	0	0
$629$	26.4682	1.05536
$630$	0	0
$631$	−18.6960	−0.744276	−0.372138	−	0.928177i	$-0.621375\pi$
$631$	−18.6960	−0.744276	−0.372138	+	0.928177i	$0.621375\pi$
$632$	0	0
$633$	−5.83937	−0.232094
$634$	0	0
$635$	5.03005	0.199611
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.04670	0.0414069
$640$	0	0
$641$	13.2707	0.524163	0.262081	−	0.965046i	$-0.415591\pi$
$641$	13.2707	0.524163	0.262081	+	0.965046i	$0.415591\pi$
$642$	0	0
$643$	−35.1813	−1.38741	−0.693707	−	0.720257i	$-0.744024\pi$
$643$	−35.1813	−1.38741	−0.693707	+	0.720257i	$0.744024\pi$
$644$	0	0
$645$	6.60574	0.260101
$646$	0	0
$647$	12.9481	0.509043	0.254522	−	0.967067i	$-0.418082\pi$
$647$	12.9481	0.509043	0.254522	+	0.967067i	$0.418082\pi$
$648$	0	0
$649$	59.5605	2.33796
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.2446	−0.987898	−0.493949	−	0.869491i	$-0.664447\pi$
$653$	−25.2446	−0.987898	−0.493949	+	0.869491i	$0.664447\pi$
$654$	0	0
$655$	−51.6131	−2.01669
$656$	0	0
$657$	3.18010	0.124067
$658$	0	0
$659$	34.1816	1.33152	0.665762	−	0.746164i	$-0.268107\pi$
$659$	34.1816	1.33152	0.665762	+	0.746164i	$0.268107\pi$
$660$	0	0
$661$	27.3212	1.06267	0.531336	−	0.847161i	$-0.321690\pi$
$661$	27.3212	1.06267	0.531336	+	0.847161i	$0.321690\pi$
$662$	0	0
$663$	−22.1906	−0.861811
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.77101	−0.223455
$668$	0	0
$669$	−4.56255	−0.176398
$670$	0	0
$671$	51.5864	1.99147
$672$	0	0
$673$	47.4313	1.82834	0.914171	−	0.405329i	$-0.132843\pi$
$673$	47.4313	1.82834	0.914171	+	0.405329i	$0.132843\pi$
$674$	0	0
$675$	−48.2413	−1.85681
$676$	0	0
$677$	7.38178	0.283705	0.141852	−	0.989888i	$-0.454694\pi$
$677$	7.38178	0.283705	0.141852	+	0.989888i	$0.454694\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−34.9717	−1.34012
$682$	0	0
$683$	−33.4430	−1.27966	−0.639831	−	0.768516i	$-0.720995\pi$
$683$	−33.4430	−1.27966	−0.639831	+	0.768516i	$0.720995\pi$
$684$	0	0
$685$	−17.7450	−0.678003
$686$	0	0
$687$	−38.7230	−1.47737
$688$	0	0
$689$	−14.5711	−0.555114
$690$	0	0
$691$	32.5382	1.23781	0.618906	−	0.785465i	$-0.287576\pi$
$691$	32.5382	1.23781	0.618906	+	0.785465i	$0.287576\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−71.7764	−2.72263
$696$	0	0
$697$	−28.3813	−1.07502
$698$	0	0
$699$	23.9335	0.905247
$700$	0	0
$701$	−22.2323	−0.839703	−0.419851	−	0.907593i	$-0.637918\pi$
$701$	−22.2323	−0.839703	−0.419851	+	0.907593i	$0.637918\pi$
$702$	0	0
$703$	−6.65060	−0.250832
$704$	0	0
$705$	−26.8761	−1.01221
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.54166	0.0578982	0.0289491	−	0.999581i	$-0.490784\pi$
$709$	1.54166	0.0578982	0.0289491	+	0.999581i	$0.490784\pi$
$710$	0	0
$711$	−1.36941	−0.0513569
$712$	0	0
$713$	−12.1860	−0.456371
$714$	0	0
$715$	49.4933	1.85094
$716$	0	0
$717$	−43.5853	−1.62772
$718$	0	0
$719$	7.61063	0.283828	0.141914	−	0.989879i	$-0.454674\pi$
$719$	7.61063	0.283828	0.141914	+	0.989879i	$0.454674\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−19.9066	−0.740336
$724$	0	0
$725$	−12.8977	−0.479008
$726$	0	0
$727$	−43.3840	−1.60902	−0.804511	−	0.593938i	$-0.797573\pi$
$727$	−43.3840	−1.60902	−0.804511	+	0.593938i	$0.797573\pi$
$728$	0	0
$729$	23.6216	0.874875
$730$	0	0
$731$	−3.74609	−0.138554
$732$	0	0
$733$	−29.5732	−1.09231	−0.546156	−	0.837684i	$-0.683909\pi$
$733$	−29.5732	−1.09231	−0.546156	+	0.837684i	$0.683909\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.17422	0.190595
$738$	0	0
$739$	−11.0938	−0.408092	−0.204046	−	0.978961i	$-0.565409\pi$
$739$	−11.0938	−0.408092	−0.204046	+	0.978961i	$0.565409\pi$
$740$	0	0
$741$	5.57577	0.204831
$742$	0	0
$743$	7.95211	0.291734	0.145867	−	0.989304i	$-0.453403\pi$
$743$	7.95211	0.291734	0.145867	+	0.989304i	$0.453403\pi$
$744$	0	0
$745$	49.0266	1.79619
$746$	0	0
$747$	−1.14579	−0.0419222
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−18.8022	−0.686101	−0.343050	−	0.939317i	$-0.611460\pi$
$751$	−18.8022	−0.686101	−0.343050	+	0.939317i	$0.611460\pi$
$752$	0	0
$753$	−6.96004	−0.253638
$754$	0	0
$755$	38.0267	1.38393
$756$	0	0
$757$	19.2555	0.699852	0.349926	−	0.936777i	$-0.386207\pi$
$757$	19.2555	0.699852	0.349926	+	0.936777i	$0.386207\pi$
$758$	0	0
$759$	33.6702	1.22215
$760$	0	0
$761$	34.4096	1.24735	0.623673	−	0.781686i	$-0.285640\pi$
$761$	34.4096	1.24735	0.623673	+	0.781686i	$0.285640\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.80785	−0.173828
$766$	0	0
$767$	43.5342	1.57193
$768$	0	0
$769$	−5.90346	−0.212884	−0.106442	−	0.994319i	$-0.533946\pi$
$769$	−5.90346	−0.212884	−0.106442	+	0.994319i	$0.533946\pi$
$770$	0	0
$771$	−17.1317	−0.616982
$772$	0	0
$773$	28.4983	1.02501	0.512506	−	0.858684i	$-0.328717\pi$
$773$	28.4983	1.02501	0.512506	+	0.858684i	$0.328717\pi$
$774$	0	0
$775$	−27.2347	−0.978298
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.13129	0.255505
$780$	0	0
$781$	13.9994	0.500939
$782$	0	0
$783$	6.39418	0.228509
$784$	0	0
$785$	20.2698	0.723461
$786$	0	0
$787$	−29.8483	−1.06398	−0.531988	−	0.846752i	$-0.678555\pi$
$787$	−29.8483	−1.06398	−0.531988	+	0.846752i	$0.678555\pi$
$788$	0	0
$789$	−57.5613	−2.04924
$790$	0	0
$791$	0	0
$792$	0	0
$793$	37.7057	1.33897
$794$	0	0
$795$	−33.3832	−1.18398
$796$	0	0
$797$	−52.1913	−1.84871	−0.924355	−	0.381533i	$-0.875396\pi$
$797$	−52.1913	−1.84871	−0.924355	+	0.381533i	$0.875396\pi$
$798$	0	0
$799$	15.2413	0.539200
$800$	0	0
$801$	0.684992	0.0242030
$802$	0	0
$803$	42.5332	1.50096
$804$	0	0
$805$	0	0
$806$	0	0
$807$	29.1266	1.02531
$808$	0	0
$809$	−10.2475	−0.360283	−0.180142	−	0.983641i	$-0.557656\pi$
$809$	−10.2475	−0.360283	−0.180142	+	0.983641i	$0.557656\pi$
$810$	0	0
$811$	7.95529	0.279348	0.139674	−	0.990198i	$-0.455395\pi$
$811$	7.95529	0.279348	0.139674	+	0.990198i	$0.455395\pi$
$812$	0	0
$813$	27.7270	0.972427
$814$	0	0
$815$	34.1861	1.19749
$816$	0	0
$817$	0.941270	0.0329309
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.1258	1.50510	0.752550	−	0.658535i	$-0.228824\pi$
$821$	43.1258	1.50510	0.752550	+	0.658535i	$0.228824\pi$
$822$	0	0
$823$	9.24429	0.322236	0.161118	−	0.986935i	$-0.448490\pi$
$823$	9.24429	0.322236	0.161118	+	0.986935i	$0.448490\pi$
$824$	0	0
$825$	75.2499	2.61986
$826$	0	0
$827$	−42.1419	−1.46542	−0.732708	−	0.680543i	$-0.761744\pi$
$827$	−42.1419	−1.46542	−0.732708	+	0.680543i	$0.761744\pi$
$828$	0	0
$829$	−43.1709	−1.49939	−0.749693	−	0.661786i	$-0.769799\pi$
$829$	−43.1709	−1.49939	−0.749693	+	0.661786i	$0.769799\pi$
$830$	0	0
$831$	−45.5593	−1.58043
$832$	0	0
$833$	0	0
$834$	0	0
$835$	96.2048	3.32931
$836$	0	0
$837$	13.5019	0.466695
$838$	0	0
$839$	−25.1710	−0.869001	−0.434500	−	0.900672i	$-0.643075\pi$
$839$	−25.1710	−0.869001	−0.434500	+	0.900672i	$0.643075\pi$
$840$	0	0
$841$	−27.2905	−0.941050
$842$	0	0
$843$	23.7406	0.817671
$844$	0	0
$845$	−13.9450	−0.479722
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6.83567	0.234600
$850$	0	0
$851$	−29.3544	−1.00626
$852$	0	0
$853$	18.4525	0.631802	0.315901	−	0.948792i	$-0.397693\pi$
$853$	18.4525	0.631802	0.315901	+	0.948792i	$0.397693\pi$
$854$	0	0
$855$	1.20806	0.0413147
$856$	0	0
$857$	56.3954	1.92643	0.963216	−	0.268729i	$-0.0866035\pi$
$857$	56.3954	1.92643	0.963216	+	0.268729i	$0.0866035\pi$
$858$	0	0
$859$	−24.3404	−0.830482	−0.415241	−	0.909711i	$-0.636303\pi$
$859$	−24.3404	−0.830482	−0.415241	+	0.909711i	$0.636303\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.3952	0.932545	0.466273	−	0.884641i	$-0.345597\pi$
$863$	27.3952	0.932545	0.466273	+	0.884641i	$0.345597\pi$
$864$	0	0
$865$	100.824	3.42811
$866$	0	0
$867$	−2.11332	−0.0717721
$868$	0	0
$869$	−18.3156	−0.621314
$870$	0	0
$871$	3.78196	0.128147
$872$	0	0
$873$	−4.77122	−0.161481
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.5904	1.16803	0.584017	−	0.811741i	$-0.301480\pi$
$877$	34.5904	1.16803	0.584017	+	0.811741i	$0.301480\pi$
$878$	0	0
$879$	33.1122	1.11685
$880$	0	0
$881$	13.1505	0.443053	0.221526	−	0.975154i	$-0.428896\pi$
$881$	13.1505	0.443053	0.221526	+	0.975154i	$0.428896\pi$
$882$	0	0
$883$	−7.72413	−0.259938	−0.129969	−	0.991518i	$-0.541488\pi$
$883$	−7.72413	−0.259938	−0.129969	+	0.991518i	$0.541488\pi$
$884$	0	0
$885$	99.7391	3.35269
$886$	0	0
$887$	−20.9567	−0.703659	−0.351829	−	0.936064i	$-0.614440\pi$
$887$	−20.9567	−0.703659	−0.351829	+	0.936064i	$0.614440\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−41.2455	−1.38178
$892$	0	0
$893$	−3.82965	−0.128154
$894$	0	0
$895$	−51.0794	−1.70740
$896$	0	0
$897$	24.6103	0.821715
$898$	0	0
$899$	3.60985	0.120395
$900$	0	0
$901$	18.9315	0.630698
$902$	0	0
$903$	0	0
$904$	0	0
$905$	48.0809	1.59826
$906$	0	0
$907$	−1.72830	−0.0573874	−0.0286937	−	0.999588i	$-0.509135\pi$
$907$	−1.72830	−0.0573874	−0.0286937	+	0.999588i	$0.509135\pi$
$908$	0	0
$909$	−5.54754	−0.184000
$910$	0	0
$911$	−30.5274	−1.01142	−0.505709	−	0.862704i	$-0.668769\pi$
$911$	−30.5274	−1.01142	−0.505709	+	0.862704i	$0.668769\pi$
$912$	0	0
$913$	−15.3247	−0.507174
$914$	0	0
$915$	86.3858	2.85583
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−39.0623	−1.28855	−0.644273	−	0.764796i	$-0.722840\pi$
$919$	−39.0623	−1.28855	−0.644273	+	0.764796i	$0.722840\pi$
$920$	0	0
$921$	8.83325	0.291065
$922$	0	0
$923$	10.2325	0.336807
$924$	0	0
$925$	−65.6045	−2.15706
$926$	0	0
$927$	2.26590	0.0744218
$928$	0	0
$929$	9.90256	0.324892	0.162446	−	0.986717i	$-0.448062\pi$
$929$	9.90256	0.324892	0.162446	+	0.986717i	$0.448062\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−56.9401	−1.86414
$934$	0	0
$935$	−64.3041	−2.10297
$936$	0	0
$937$	−40.0515	−1.30843	−0.654213	−	0.756311i	$-0.727000\pi$
$937$	−40.0515	−1.30843	−0.654213	+	0.756311i	$0.727000\pi$
$938$	0	0
$939$	55.5468	1.81270
$940$	0	0
$941$	40.4678	1.31921	0.659606	−	0.751611i	$-0.270723\pi$
$941$	40.4678	1.31921	0.659606	+	0.751611i	$0.270723\pi$
$942$	0	0
$943$	31.4761	1.02500
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.63179	0.248000	0.124000	−	0.992282i	$-0.460428\pi$
$947$	7.63179	0.248000	0.124000	+	0.992282i	$0.460428\pi$
$948$	0	0
$949$	31.0885	1.00917
$950$	0	0
$951$	45.8294	1.48612
$952$	0	0
$953$	−3.41287	−0.110554	−0.0552768	−	0.998471i	$-0.517604\pi$
$953$	−3.41287	−0.110554	−0.0552768	+	0.998471i	$0.517604\pi$
$954$	0	0
$955$	6.21000	0.200951
$956$	0	0
$957$	−9.97407	−0.322416
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.3775	−0.754112
$962$	0	0
$963$	−5.97916	−0.192676
$964$	0	0
$965$	−28.3603	−0.912950
$966$	0	0
$967$	16.5267	0.531463	0.265732	−	0.964047i	$-0.414386\pi$
$967$	16.5267	0.531463	0.265732	+	0.964047i	$0.414386\pi$
$968$	0	0
$969$	−7.24431	−0.232721
$970$	0	0
$971$	−5.34763	−0.171614	−0.0858068	−	0.996312i	$-0.527347\pi$
$971$	−5.34763	−0.171614	−0.0858068	+	0.996312i	$0.527347\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	55.0019	1.76147
$976$	0	0
$977$	52.0300	1.66459	0.832293	−	0.554336i	$-0.187028\pi$
$977$	52.0300	1.66459	0.832293	+	0.554336i	$0.187028\pi$
$978$	0	0
$979$	9.16163	0.292807
$980$	0	0
$981$	1.21920	0.0389261
$982$	0	0
$983$	−8.82293	−0.281408	−0.140704	−	0.990052i	$-0.544937\pi$
$983$	−8.82293	−0.281408	−0.140704	+	0.990052i	$0.544937\pi$
$984$	0	0
$985$	34.6894	1.10530
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.15458	0.132108
$990$	0	0
$991$	−31.8970	−1.01324	−0.506621	−	0.862169i	$-0.669106\pi$
$991$	−31.8970	−1.01324	−0.506621	+	0.862169i	$0.669106\pi$
$992$	0	0
$993$	8.63577	0.274048
$994$	0	0
$995$	−103.045	−3.26674
$996$	0	0
$997$	53.1722	1.68398	0.841990	−	0.539494i	$-0.181384\pi$
$997$	53.1722	1.68398	0.841990	+	0.539494i	$0.181384\pi$
$998$	0	0
$999$	32.5242	1.02902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bs.1.9		11
7.3	odd	6		1064.2.q.o.457.9	yes	22
7.5	odd	6		1064.2.q.o.305.9	✓	22
7.6	odd	2		7448.2.a.bt.1.3		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.o.305.9	✓	22	7.5	odd	6
1064.2.q.o.457.9	yes	22	7.3	odd	6
7448.2.a.bs.1.9		11	1.1	even	1	trivial
7448.2.a.bt.1.3		11	7.6	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82026 q^{3} +3.85544 q^{5} +0.313338 q^{9} +O(q^{10})\)	\(q+1.82026 q^{3} +3.85544 q^{5} +0.313338 q^{9} +4.19083 q^{11} +3.06318 q^{13} +7.01790 q^{15} -3.97982 q^{17} +1.00000 q^{19} +4.41380 q^{23} +9.86445 q^{25} -4.89042 q^{27} -1.30749 q^{29} -2.76089 q^{31} +7.62839 q^{33} -6.65060 q^{37} +5.57577 q^{39} +7.13129 q^{41} +0.941270 q^{43} +1.20806 q^{45} -3.82965 q^{47} -7.24431 q^{51} -4.75686 q^{53} +16.1575 q^{55} +1.82026 q^{57} +14.2121 q^{59} +12.3094 q^{61} +11.8099 q^{65} +1.23465 q^{67} +8.03426 q^{69} +3.34049 q^{71} +10.1491 q^{73} +17.9558 q^{75} -4.37040 q^{79} -9.84183 q^{81} -3.65672 q^{83} -15.3440 q^{85} -2.37997 q^{87} +2.18611 q^{89} -5.02554 q^{93} +3.85544 q^{95} -15.2271 q^{97} +1.31315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 3 q^{5} + 17 q^{9}+O(q^{10}) \) 11 * q - 3 * q^5 + 17 * q^9	\( 11 q - 3 q^{5} + 17 q^{9} - 6 q^{11} + 8 q^{13} + 4 q^{15} - 13 q^{17} + 11 q^{19} - 2 q^{23} + 38 q^{25} + 21 q^{27} + 16 q^{29} + 6 q^{31} - 8 q^{33} + 9 q^{37} + q^{39} + 4 q^{41} + 31 q^{43} - 33 q^{45} - 7 q^{47} - 21 q^{51} + 11 q^{55} - 34 q^{59} + 42 q^{65} - 14 q^{67} + 19 q^{69} - 16 q^{71} - 8 q^{73} - 2 q^{75} - 10 q^{79} + 51 q^{81} + 16 q^{83} + 34 q^{85} - 72 q^{87} + 20 q^{89} + 50 q^{93} - 3 q^{95} + 34 q^{99}+O(q^{100}) \) 11 * q - 3 * q^5 + 17 * q^9 - 6 * q^11 + 8 * q^13 + 4 * q^15 - 13 * q^17 + 11 * q^19 - 2 * q^23 + 38 * q^25 + 21 * q^27 + 16 * q^29 + 6 * q^31 - 8 * q^33 + 9 * q^37 + q^39 + 4 * q^41 + 31 * q^43 - 33 * q^45 - 7 * q^47 - 21 * q^51 + 11 * q^55 - 34 * q^59 + 42 * q^65 - 14 * q^67 + 19 * q^69 - 16 * q^71 - 8 * q^73 - 2 * q^75 - 10 * q^79 + 51 * q^81 + 16 * q^83 + 34 * q^85 - 72 * q^87 + 20 * q^89 + 50 * q^93 - 3 * q^95 + 34 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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