LMFDB - Embedded newform 1148.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 238 x^{13} + 602 x^{12} + 21013 x^{11} - 44923 x^{10} - 876344 x^{9} + \cdots - 45134496 $ x^15 - 3x^14 - 238x^13 + 602x^12 + 21013x^11 - 44923x^10 - 876344x^9 + 1512470x^8 + 18440602x^7 - 23548938x^6 - 192795335x^5 + 157072885x^4 + 930883767x^3 - 339053445x^2 - 1549251522x - 45134496
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.69288$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.69288 q^{3} +15.9636 q^{5} -7.00000 q^{7} +17.7947 q^{9} +O(q^{10})$	$q-6.69288 q^{3} +15.9636 q^{5} -7.00000 q^{7} +17.7947 q^{9} -11.6240 q^{11} -38.9493 q^{13} -106.842 q^{15} -115.789 q^{17} +81.7673 q^{19} +46.8502 q^{21} +159.440 q^{23} +129.835 q^{25} +61.6102 q^{27} -153.429 q^{29} -14.9860 q^{31} +77.7981 q^{33} -111.745 q^{35} +212.229 q^{37} +260.683 q^{39} +41.0000 q^{41} -94.7895 q^{43} +284.066 q^{45} -123.292 q^{47} +49.0000 q^{49} +774.961 q^{51} -91.8513 q^{53} -185.560 q^{55} -547.259 q^{57} -295.688 q^{59} -28.3290 q^{61} -124.563 q^{63} -621.769 q^{65} -237.166 q^{67} -1067.11 q^{69} -488.957 q^{71} +85.6885 q^{73} -868.971 q^{75} +81.3681 q^{77} +1007.05 q^{79} -892.806 q^{81} +663.957 q^{83} -1848.40 q^{85} +1026.88 q^{87} +686.613 q^{89} +272.645 q^{91} +100.299 q^{93} +1305.30 q^{95} -19.6544 q^{97} -206.845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * 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
$3$	−6.69288	−1.28805	−0.644023	−	0.765006i	$-0.722736\pi$
$3$	−6.69288	−1.28805	−0.644023	+	0.765006i	$0.722736\pi$
$4$	0	0
$5$	15.9636	1.42782	0.713912	−	0.700236i	$-0.246922\pi$
$5$	15.9636	1.42782	0.713912	+	0.700236i	$0.246922\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	17.7947	0.659062
$10$	0	0
$11$	−11.6240	−0.318615	−0.159308	−	0.987229i	$-0.550926\pi$
$11$	−11.6240	−0.318615	−0.159308	+	0.987229i	$0.550926\pi$
$12$	0	0
$13$	−38.9493	−0.830968	−0.415484	−	0.909601i	$-0.636388\pi$
$13$	−38.9493	−0.830968	−0.415484	+	0.909601i	$0.636388\pi$
$14$	0	0
$15$	−106.842	−1.83910
$16$	0	0
$17$	−115.789	−1.65194	−0.825969	−	0.563716i	$-0.809371\pi$
$17$	−115.789	−1.65194	−0.825969	+	0.563716i	$0.809371\pi$
$18$	0	0
$19$	81.7673	0.987300	0.493650	−	0.869661i	$-0.335662\pi$
$19$	81.7673	0.987300	0.493650	+	0.869661i	$0.335662\pi$
$20$	0	0
$21$	46.8502	0.486836
$22$	0	0
$23$	159.440	1.44546	0.722730	−	0.691130i	$-0.242887\pi$
$23$	159.440	1.44546	0.722730	+	0.691130i	$0.242887\pi$
$24$	0	0
$25$	129.835	1.03868
$26$	0	0
$27$	61.6102	0.439144
$28$	0	0
$29$	−153.429	−0.982449	−0.491225	−	0.871033i	$-0.663451\pi$
$29$	−153.429	−0.982449	−0.491225	+	0.871033i	$0.663451\pi$
$30$	0	0
$31$	−14.9860	−0.0868246	−0.0434123	−	0.999057i	$-0.513823\pi$
$31$	−14.9860	−0.0868246	−0.0434123	+	0.999057i	$0.513823\pi$
$32$	0	0
$33$	77.7981	0.410391
$34$	0	0
$35$	−111.745	−0.539667
$36$	0	0
$37$	212.229	0.942979	0.471490	−	0.881872i	$-0.343716\pi$
$37$	212.229	0.942979	0.471490	+	0.881872i	$0.343716\pi$
$38$	0	0
$39$	260.683	1.07032
$40$	0	0
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	−94.7895	−0.336169	−0.168084	−	0.985773i	$-0.553758\pi$
$43$	−94.7895	−0.336169	−0.168084	+	0.985773i	$0.553758\pi$
$44$	0	0
$45$	284.066	0.941024
$46$	0	0
$47$	−123.292	−0.382640	−0.191320	−	0.981528i	$-0.561277\pi$
$47$	−123.292	−0.382640	−0.191320	+	0.981528i	$0.561277\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	774.961	2.12777
$52$	0	0
$53$	−91.8513	−0.238052	−0.119026	−	0.992891i	$-0.537977\pi$
$53$	−91.8513	−0.238052	−0.119026	+	0.992891i	$0.537977\pi$
$54$	0	0
$55$	−185.560	−0.454927
$56$	0	0
$57$	−547.259	−1.27169
$58$	0	0
$59$	−295.688	−0.652463	−0.326232	−	0.945290i	$-0.605779\pi$
$59$	−295.688	−0.652463	−0.326232	+	0.945290i	$0.605779\pi$
$60$	0	0
$61$	−28.3290	−0.0594616	−0.0297308	−	0.999558i	$-0.509465\pi$
$61$	−28.3290	−0.0594616	−0.0297308	+	0.999558i	$0.509465\pi$
$62$	0	0
$63$	−124.563	−0.249102
$64$	0	0
$65$	−621.769	−1.18648
$66$	0	0
$67$	−237.166	−0.432454	−0.216227	−	0.976343i	$-0.569375\pi$
$67$	−237.166	−0.432454	−0.216227	+	0.976343i	$0.569375\pi$
$68$	0	0
$69$	−1067.11	−1.86182
$70$	0	0
$71$	−488.957	−0.817303	−0.408651	−	0.912691i	$-0.634001\pi$
$71$	−488.957	−0.817303	−0.408651	+	0.912691i	$0.634001\pi$
$72$	0	0
$73$	85.6885	0.137385	0.0686923	−	0.997638i	$-0.478117\pi$
$73$	85.6885	0.137385	0.0686923	+	0.997638i	$0.478117\pi$
$74$	0	0
$75$	−868.971	−1.33787
$76$	0	0
$77$	81.3681	0.120425
$78$	0	0
$79$	1007.05	1.43420	0.717098	−	0.696972i	$-0.245470\pi$
$79$	1007.05	1.43420	0.717098	+	0.696972i	$0.245470\pi$
$80$	0	0
$81$	−892.806	−1.22470
$82$	0	0
$83$	663.957	0.878057	0.439029	−	0.898473i	$-0.355323\pi$
$83$	663.957	0.878057	0.439029	+	0.898473i	$0.355323\pi$
$84$	0	0
$85$	−1848.40	−2.35867
$86$	0	0
$87$	1026.88	1.26544
$88$	0	0
$89$	686.613	0.817762	0.408881	−	0.912588i	$-0.365919\pi$
$89$	686.613	0.817762	0.408881	+	0.912588i	$0.365919\pi$
$90$	0	0
$91$	272.645	0.314076
$92$	0	0
$93$	100.299	0.111834
$94$	0	0
$95$	1305.30	1.40969
$96$	0	0
$97$	−19.6544	−0.0205732	−0.0102866	−	0.999947i	$-0.503274\pi$
$97$	−19.6544	−0.0205732	−0.0102866	+	0.999947i	$0.503274\pi$
$98$	0	0
$99$	−206.845	−0.209987
$100$	0	0
$101$	−816.847	−0.804745	−0.402373	−	0.915476i	$-0.631814\pi$
$101$	−816.847	−0.804745	−0.402373	+	0.915476i	$0.631814\pi$
$102$	0	0
$103$	1599.27	1.52991	0.764956	−	0.644082i	$-0.222761\pi$
$103$	1599.27	1.52991	0.764956	+	0.644082i	$0.222761\pi$
$104$	0	0
$105$	747.895	0.695115
$106$	0	0
$107$	1251.67	1.13087	0.565436	−	0.824792i	$-0.308708\pi$
$107$	1251.67	1.13087	0.565436	+	0.824792i	$0.308708\pi$
$108$	0	0
$109$	1105.02	0.971025	0.485513	−	0.874230i	$-0.338633\pi$
$109$	1105.02	0.971025	0.485513	+	0.874230i	$0.338633\pi$
$110$	0	0
$111$	−1420.42	−1.21460
$112$	0	0
$113$	1112.62	0.926249	0.463125	−	0.886293i	$-0.346728\pi$
$113$	1112.62	0.926249	0.463125	+	0.886293i	$0.346728\pi$
$114$	0	0
$115$	2545.23	2.06386
$116$	0	0
$117$	−693.089	−0.547659
$118$	0	0
$119$	810.522	0.624373
$120$	0	0
$121$	−1195.88	−0.898484
$122$	0	0
$123$	−274.408	−0.201159
$124$	0	0
$125$	77.1845	0.0552287
$126$	0	0
$127$	1250.59	0.873798	0.436899	−	0.899511i	$-0.356077\pi$
$127$	1250.59	0.873798	0.436899	+	0.899511i	$0.356077\pi$
$128$	0	0
$129$	634.415	0.433001
$130$	0	0
$131$	1381.37	0.921302	0.460651	−	0.887581i	$-0.347616\pi$
$131$	1381.37	0.921302	0.460651	+	0.887581i	$0.347616\pi$
$132$	0	0
$133$	−572.371	−0.373164
$134$	0	0
$135$	983.517	0.627020
$136$	0	0
$137$	−511.464	−0.318959	−0.159479	−	0.987201i	$-0.550982\pi$
$137$	−511.464	−0.318959	−0.159479	+	0.987201i	$0.550982\pi$
$138$	0	0
$139$	−944.507	−0.576345	−0.288173	−	0.957578i	$-0.593048\pi$
$139$	−944.507	−0.576345	−0.288173	+	0.957578i	$0.593048\pi$
$140$	0	0
$141$	825.182	0.492857
$142$	0	0
$143$	452.746	0.264759
$144$	0	0
$145$	−2449.27	−1.40276
$146$	0	0
$147$	−327.951	−0.184007
$148$	0	0
$149$	437.209	0.240386	0.120193	−	0.992751i	$-0.461649\pi$
$149$	437.209	0.240386	0.120193	+	0.992751i	$0.461649\pi$
$150$	0	0
$151$	398.390	0.214705	0.107353	−	0.994221i	$-0.465763\pi$
$151$	398.390	0.214705	0.107353	+	0.994221i	$0.465763\pi$
$152$	0	0
$153$	−2060.43	−1.08873
$154$	0	0
$155$	−239.230	−0.123970
$156$	0	0
$157$	14.7596	0.00750285	0.00375142	−	0.999993i	$-0.498806\pi$
$157$	14.7596	0.00750285	0.00375142	+	0.999993i	$0.498806\pi$
$158$	0	0
$159$	614.750	0.306622
$160$	0	0
$161$	−1116.08	−0.546333
$162$	0	0
$163$	1065.34	0.511924	0.255962	−	0.966687i	$-0.417608\pi$
$163$	1065.34	0.511924	0.255962	+	0.966687i	$0.417608\pi$
$164$	0	0
$165$	1241.93	0.585966
$166$	0	0
$167$	2695.38	1.24895	0.624476	−	0.781044i	$-0.285312\pi$
$167$	2695.38	1.24895	0.624476	+	0.781044i	$0.285312\pi$
$168$	0	0
$169$	−679.956	−0.309493
$170$	0	0
$171$	1455.02	0.650692
$172$	0	0
$173$	3230.88	1.41988	0.709940	−	0.704262i	$-0.248722\pi$
$173$	3230.88	1.41988	0.709940	+	0.704262i	$0.248722\pi$
$174$	0	0
$175$	−908.845	−0.392584
$176$	0	0
$177$	1979.01	0.840403
$178$	0	0
$179$	−503.645	−0.210303	−0.105151	−	0.994456i	$-0.533533\pi$
$179$	−503.645	−0.210303	−0.105151	+	0.994456i	$0.533533\pi$
$180$	0	0
$181$	959.180	0.393897	0.196948	−	0.980414i	$-0.436897\pi$
$181$	959.180	0.393897	0.196948	+	0.980414i	$0.436897\pi$
$182$	0	0
$183$	189.603	0.0765893
$184$	0	0
$185$	3387.93	1.34641
$186$	0	0
$187$	1345.93	0.526333
$188$	0	0
$189$	−431.271	−0.165981
$190$	0	0
$191$	598.180	0.226611	0.113306	−	0.993560i	$-0.463856\pi$
$191$	598.180	0.226611	0.113306	+	0.993560i	$0.463856\pi$
$192$	0	0
$193$	1867.42	0.696476	0.348238	−	0.937406i	$-0.386780\pi$
$193$	1867.42	0.696476	0.348238	+	0.937406i	$0.386780\pi$
$194$	0	0
$195$	4161.42	1.52823
$196$	0	0
$197$	823.142	0.297698	0.148849	−	0.988860i	$-0.452443\pi$
$197$	823.142	0.297698	0.148849	+	0.988860i	$0.452443\pi$
$198$	0	0
$199$	3293.65	1.17327	0.586635	−	0.809851i	$-0.300452\pi$
$199$	3293.65	1.17327	0.586635	+	0.809851i	$0.300452\pi$
$200$	0	0
$201$	1587.32	0.557021
$202$	0	0
$203$	1074.00	0.371331
$204$	0	0
$205$	654.506	0.222989
$206$	0	0
$207$	2837.19	0.952648
$208$	0	0
$209$	−950.463	−0.314569
$210$	0	0
$211$	2253.14	0.735130	0.367565	−	0.929998i	$-0.380192\pi$
$211$	2253.14	0.735130	0.367565	+	0.929998i	$0.380192\pi$
$212$	0	0
$213$	3272.53	1.05272
$214$	0	0
$215$	−1513.18	−0.479990
$216$	0	0
$217$	104.902	0.0328166
$218$	0	0
$219$	−573.503	−0.176958
$220$	0	0
$221$	4509.89	1.37271
$222$	0	0
$223$	1885.18	0.566102	0.283051	−	0.959105i	$-0.408653\pi$
$223$	1885.18	0.566102	0.283051	+	0.959105i	$0.408653\pi$
$224$	0	0
$225$	2310.37	0.684555
$226$	0	0
$227$	−1692.20	−0.494780	−0.247390	−	0.968916i	$-0.579573\pi$
$227$	−1692.20	−0.494780	−0.247390	+	0.968916i	$0.579573\pi$
$228$	0	0
$229$	−5393.86	−1.55649	−0.778246	−	0.627960i	$-0.783890\pi$
$229$	−5393.86	−1.55649	−0.778246	+	0.627960i	$0.783890\pi$
$230$	0	0
$231$	−544.587	−0.155113
$232$	0	0
$233$	5372.83	1.51067	0.755334	−	0.655340i	$-0.227475\pi$
$233$	5372.83	1.51067	0.755334	+	0.655340i	$0.227475\pi$
$234$	0	0
$235$	−1968.19	−0.546342
$236$	0	0
$237$	−6740.04	−1.84731
$238$	0	0
$239$	4313.37	1.16740	0.583701	−	0.811969i	$-0.301604\pi$
$239$	4313.37	1.16740	0.583701	+	0.811969i	$0.301604\pi$
$240$	0	0
$241$	−2643.61	−0.706596	−0.353298	−	0.935511i	$-0.614940\pi$
$241$	−2643.61	−0.706596	−0.353298	+	0.935511i	$0.614940\pi$
$242$	0	0
$243$	4311.97	1.13833
$244$	0	0
$245$	782.214	0.203975
$246$	0	0
$247$	−3184.77	−0.820414
$248$	0	0
$249$	−4443.79	−1.13098
$250$	0	0
$251$	5687.71	1.43030	0.715149	−	0.698972i	$-0.246359\pi$
$251$	5687.71	1.43030	0.715149	+	0.698972i	$0.246359\pi$
$252$	0	0
$253$	−1853.33	−0.460546
$254$	0	0
$255$	12371.1	3.03808
$256$	0	0
$257$	−1030.90	−0.250216	−0.125108	−	0.992143i	$-0.539928\pi$
$257$	−1030.90	−0.250216	−0.125108	+	0.992143i	$0.539928\pi$
$258$	0	0
$259$	−1485.60	−0.356413
$260$	0	0
$261$	−2730.22	−0.647495
$262$	0	0
$263$	−365.574	−0.0857119	−0.0428560	−	0.999081i	$-0.513646\pi$
$263$	−365.574	−0.0857119	−0.0428560	+	0.999081i	$0.513646\pi$
$264$	0	0
$265$	−1466.27	−0.339896
$266$	0	0
$267$	−4595.42	−1.05331
$268$	0	0
$269$	2532.63	0.574042	0.287021	−	0.957924i	$-0.407335\pi$
$269$	2532.63	0.574042	0.287021	+	0.957924i	$0.407335\pi$
$270$	0	0
$271$	−3738.89	−0.838087	−0.419044	−	0.907966i	$-0.637635\pi$
$271$	−3738.89	−0.838087	−0.419044	+	0.907966i	$0.637635\pi$
$272$	0	0
$273$	−1824.78	−0.404545
$274$	0	0
$275$	−1509.20	−0.330940
$276$	0	0
$277$	915.631	0.198610	0.0993049	−	0.995057i	$-0.468338\pi$
$277$	915.631	0.198610	0.0993049	+	0.995057i	$0.468338\pi$
$278$	0	0
$279$	−266.671	−0.0572228
$280$	0	0
$281$	6036.48	1.28152	0.640759	−	0.767742i	$-0.278620\pi$
$281$	6036.48	1.28152	0.640759	+	0.767742i	$0.278620\pi$
$282$	0	0
$283$	3189.26	0.669901	0.334950	−	0.942236i	$-0.391280\pi$
$283$	3189.26	0.669901	0.334950	+	0.942236i	$0.391280\pi$
$284$	0	0
$285$	−8736.19	−1.81575
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	8494.07	1.72890
$290$	0	0
$291$	131.544	0.0264992
$292$	0	0
$293$	−7647.38	−1.52480	−0.762398	−	0.647109i	$-0.775978\pi$
$293$	−7647.38	−1.52480	−0.762398	+	0.647109i	$0.775978\pi$
$294$	0	0
$295$	−4720.24	−0.931603
$296$	0	0
$297$	−716.157	−0.139918
$298$	0	0
$299$	−6210.08	−1.20113
$300$	0	0
$301$	663.526	0.127060
$302$	0	0
$303$	5467.06	1.03655
$304$	0	0
$305$	−452.232	−0.0849007
$306$	0	0
$307$	−2757.19	−0.512577	−0.256288	−	0.966600i	$-0.582500\pi$
$307$	−2757.19	−0.512577	−0.256288	+	0.966600i	$0.582500\pi$
$308$	0	0
$309$	−10703.7	−1.97060
$310$	0	0
$311$	2651.67	0.483481	0.241740	−	0.970341i	$-0.422282\pi$
$311$	2651.67	0.483481	0.241740	+	0.970341i	$0.422282\pi$
$312$	0	0
$313$	−2103.42	−0.379847	−0.189924	−	0.981799i	$-0.560824\pi$
$313$	−2103.42	−0.379847	−0.189924	+	0.981799i	$0.560824\pi$
$314$	0	0
$315$	−1988.46	−0.355674
$316$	0	0
$317$	4464.77	0.791062	0.395531	−	0.918453i	$-0.370561\pi$
$317$	4464.77	0.791062	0.395531	+	0.918453i	$0.370561\pi$
$318$	0	0
$319$	1783.46	0.313024
$320$	0	0
$321$	−8377.26	−1.45661
$322$	0	0
$323$	−9467.74	−1.63096
$324$	0	0
$325$	−5056.98	−0.863110
$326$	0	0
$327$	−7395.77	−1.25072
$328$	0	0
$329$	863.047	0.144624
$330$	0	0
$331$	4410.42	0.732382	0.366191	−	0.930540i	$-0.380662\pi$
$331$	4410.42	0.732382	0.366191	+	0.930540i	$0.380662\pi$
$332$	0	0
$333$	3776.55	0.621482
$334$	0	0
$335$	−3786.01	−0.617469
$336$	0	0
$337$	2299.60	0.371713	0.185856	−	0.982577i	$-0.440494\pi$
$337$	2299.60	0.371713	0.185856	+	0.982577i	$0.440494\pi$
$338$	0	0
$339$	−7446.61	−1.19305
$340$	0	0
$341$	174.197	0.0276637
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−17034.9	−2.65835
$346$	0	0
$347$	1204.38	0.186325	0.0931623	−	0.995651i	$-0.470302\pi$
$347$	1204.38	0.186325	0.0931623	+	0.995651i	$0.470302\pi$
$348$	0	0
$349$	3793.43	0.581827	0.290914	−	0.956749i	$-0.406041\pi$
$349$	3793.43	0.581827	0.290914	+	0.956749i	$0.406041\pi$
$350$	0	0
$351$	−2399.67	−0.364914
$352$	0	0
$353$	5027.67	0.758062	0.379031	−	0.925384i	$-0.376257\pi$
$353$	5027.67	0.758062	0.379031	+	0.925384i	$0.376257\pi$
$354$	0	0
$355$	−7805.49	−1.16696
$356$	0	0
$357$	−5424.73	−0.804222
$358$	0	0
$359$	−782.324	−0.115013	−0.0575063	−	0.998345i	$-0.518315\pi$
$359$	−782.324	−0.115013	−0.0575063	+	0.998345i	$0.518315\pi$
$360$	0	0
$361$	−173.115	−0.0252391
$362$	0	0
$363$	8003.90	1.15729
$364$	0	0
$365$	1367.89	0.196161
$366$	0	0
$367$	13921.1	1.98005	0.990023	−	0.140906i	$-0.0450014\pi$
$367$	13921.1	1.98005	0.990023	+	0.140906i	$0.0450014\pi$
$368$	0	0
$369$	729.582	0.102928
$370$	0	0
$371$	642.959	0.0899751
$372$	0	0
$373$	4349.31	0.603750	0.301875	−	0.953347i	$-0.402387\pi$
$373$	4349.31	0.603750	0.301875	+	0.953347i	$0.402387\pi$
$374$	0	0
$375$	−516.587	−0.0711372
$376$	0	0
$377$	5975.94	0.816384
$378$	0	0
$379$	−7850.59	−1.06400	−0.532002	−	0.846743i	$-0.678560\pi$
$379$	−7850.59	−1.06400	−0.532002	+	0.846743i	$0.678560\pi$
$380$	0	0
$381$	−8370.08	−1.12549
$382$	0	0
$383$	−4497.95	−0.600090	−0.300045	−	0.953925i	$-0.597002\pi$
$383$	−4497.95	−0.600090	−0.300045	+	0.953925i	$0.597002\pi$
$384$	0	0
$385$	1298.92	0.171946
$386$	0	0
$387$	−1686.75	−0.221556
$388$	0	0
$389$	−4368.46	−0.569382	−0.284691	−	0.958619i	$-0.591891\pi$
$389$	−4368.46	−0.569382	−0.284691	+	0.958619i	$0.591891\pi$
$390$	0	0
$391$	−18461.4	−2.38781
$392$	0	0
$393$	−9245.33	−1.18668
$394$	0	0
$395$	16076.0	2.04778
$396$	0	0
$397$	924.571	0.116884	0.0584419	−	0.998291i	$-0.481387\pi$
$397$	924.571	0.116884	0.0584419	+	0.998291i	$0.481387\pi$
$398$	0	0
$399$	3830.81	0.480653
$400$	0	0
$401$	−10276.2	−1.27973	−0.639863	−	0.768489i	$-0.721009\pi$
$401$	−10276.2	−1.27973	−0.639863	+	0.768489i	$0.721009\pi$
$402$	0	0
$403$	583.693	0.0721484
$404$	0	0
$405$	−14252.4	−1.74865
$406$	0	0
$407$	−2466.95	−0.300448
$408$	0	0
$409$	−15383.9	−1.85987	−0.929933	−	0.367729i	$-0.880135\pi$
$409$	−15383.9	−1.85987	−0.929933	+	0.367729i	$0.880135\pi$
$410$	0	0
$411$	3423.17	0.410833
$412$	0	0
$413$	2069.82	0.246608
$414$	0	0
$415$	10599.1	1.25371
$416$	0	0
$417$	6321.47	0.742359
$418$	0	0
$419$	180.048	0.0209926	0.0104963	−	0.999945i	$-0.496659\pi$
$419$	180.048	0.0209926	0.0104963	+	0.999945i	$0.496659\pi$
$420$	0	0
$421$	1301.89	0.150713	0.0753566	−	0.997157i	$-0.475991\pi$
$421$	1301.89	0.150713	0.0753566	+	0.997157i	$0.475991\pi$
$422$	0	0
$423$	−2193.95	−0.252183
$424$	0	0
$425$	−15033.5	−1.71583
$426$	0	0
$427$	198.303	0.0224744
$428$	0	0
$429$	−3030.18	−0.341022
$430$	0	0
$431$	−9587.96	−1.07154	−0.535772	−	0.844363i	$-0.679979\pi$
$431$	−9587.96	−1.07154	−0.535772	+	0.844363i	$0.679979\pi$
$432$	0	0
$433$	8871.27	0.984586	0.492293	−	0.870429i	$-0.336159\pi$
$433$	8871.27	0.984586	0.492293	+	0.870429i	$0.336159\pi$
$434$	0	0
$435$	16392.7	1.80683
$436$	0	0
$437$	13037.0	1.42710
$438$	0	0
$439$	−16688.8	−1.81438	−0.907189	−	0.420723i	$-0.861777\pi$
$439$	−16688.8	−1.81438	−0.907189	+	0.420723i	$0.861777\pi$
$440$	0	0
$441$	871.939	0.0941517
$442$	0	0
$443$	7854.94	0.842437	0.421218	−	0.906959i	$-0.361603\pi$
$443$	7854.94	0.842437	0.421218	+	0.906959i	$0.361603\pi$
$444$	0	0
$445$	10960.8	1.16762
$446$	0	0
$447$	−2926.19	−0.309628
$448$	0	0
$449$	16635.4	1.74849	0.874244	−	0.485487i	$-0.161358\pi$
$449$	16635.4	1.74849	0.874244	+	0.485487i	$0.161358\pi$
$450$	0	0
$451$	−476.584	−0.0497594
$452$	0	0
$453$	−2666.38	−0.276550
$454$	0	0
$455$	4352.38	0.448445
$456$	0	0
$457$	18128.1	1.85557	0.927784	−	0.373117i	$-0.121711\pi$
$457$	18128.1	1.85557	0.927784	+	0.373117i	$0.121711\pi$
$458$	0	0
$459$	−7133.77	−0.725438
$460$	0	0
$461$	−4859.18	−0.490921	−0.245460	−	0.969407i	$-0.578939\pi$
$461$	−4859.18	−0.490921	−0.245460	+	0.969407i	$0.578939\pi$
$462$	0	0
$463$	−13368.0	−1.34182	−0.670912	−	0.741537i	$-0.734097\pi$
$463$	−13368.0	−1.34182	−0.670912	+	0.741537i	$0.734097\pi$
$464$	0	0
$465$	1601.14	0.159679
$466$	0	0
$467$	−8131.04	−0.805695	−0.402848	−	0.915267i	$-0.631980\pi$
$467$	−8131.04	−0.805695	−0.402848	+	0.915267i	$0.631980\pi$
$468$	0	0
$469$	1660.16	0.163452
$470$	0	0
$471$	−98.7845	−0.00966401
$472$	0	0
$473$	1101.83	0.107109
$474$	0	0
$475$	10616.3	1.02549
$476$	0	0
$477$	−1634.46	−0.156891
$478$	0	0
$479$	1313.11	0.125255	0.0626277	−	0.998037i	$-0.480052\pi$
$479$	1313.11	0.125255	0.0626277	+	0.998037i	$0.480052\pi$
$480$	0	0
$481$	−8266.16	−0.783585
$482$	0	0
$483$	7469.80	0.703702
$484$	0	0
$485$	−313.753	−0.0293749
$486$	0	0
$487$	−15953.5	−1.48444	−0.742221	−	0.670155i	$-0.766228\pi$
$487$	−15953.5	−1.48444	−0.742221	+	0.670155i	$0.766228\pi$
$488$	0	0
$489$	−7130.18	−0.659382
$490$	0	0
$491$	16314.2	1.49949	0.749745	−	0.661727i	$-0.230176\pi$
$491$	16314.2	1.49949	0.749745	+	0.661727i	$0.230176\pi$
$492$	0	0
$493$	17765.4	1.62294
$494$	0	0
$495$	−3301.99	−0.299825
$496$	0	0
$497$	3422.70	0.308911
$498$	0	0
$499$	7381.43	0.662201	0.331101	−	0.943595i	$-0.392580\pi$
$499$	7381.43	0.662201	0.331101	+	0.943595i	$0.392580\pi$
$500$	0	0
$501$	−18039.9	−1.60871
$502$	0	0
$503$	312.521	0.0277030	0.0138515	−	0.999904i	$-0.495591\pi$
$503$	312.521	0.0277030	0.0138515	+	0.999904i	$0.495591\pi$
$504$	0	0
$505$	−13039.8	−1.14903
$506$	0	0
$507$	4550.86	0.398641
$508$	0	0
$509$	−4167.02	−0.362868	−0.181434	−	0.983403i	$-0.558074\pi$
$509$	−4167.02	−0.362868	−0.181434	+	0.983403i	$0.558074\pi$
$510$	0	0
$511$	−599.820	−0.0519265
$512$	0	0
$513$	5037.69	0.433566
$514$	0	0
$515$	25530.1	2.18444
$516$	0	0
$517$	1433.15	0.121915
$518$	0	0
$519$	−21623.9	−1.82887
$520$	0	0
$521$	−5637.16	−0.474028	−0.237014	−	0.971506i	$-0.576169\pi$
$521$	−5637.16	−0.474028	−0.237014	+	0.971506i	$0.576169\pi$
$522$	0	0
$523$	12319.8	1.03003	0.515017	−	0.857180i	$-0.327785\pi$
$523$	12319.8	1.03003	0.515017	+	0.857180i	$0.327785\pi$
$524$	0	0
$525$	6082.79	0.505667
$526$	0	0
$527$	1735.21	0.143429
$528$	0	0
$529$	13254.2	1.08936
$530$	0	0
$531$	−5261.68	−0.430014
$532$	0	0
$533$	−1596.92	−0.129775
$534$	0	0
$535$	19981.1	1.61468
$536$	0	0
$537$	3370.84	0.270880
$538$	0	0
$539$	−569.576	−0.0455165
$540$	0	0
$541$	10290.9	0.817818	0.408909	−	0.912575i	$-0.365909\pi$
$541$	10290.9	0.817818	0.408909	+	0.912575i	$0.365909\pi$
$542$	0	0
$543$	−6419.68	−0.507357
$544$	0	0
$545$	17640.0	1.38645
$546$	0	0
$547$	17277.2	1.35049	0.675245	−	0.737593i	$-0.264038\pi$
$547$	17277.2	1.35049	0.675245	+	0.737593i	$0.264038\pi$
$548$	0	0
$549$	−504.106	−0.0391889
$550$	0	0
$551$	−12545.5	−0.969972
$552$	0	0
$553$	−7049.32	−0.542075
$554$	0	0
$555$	−22675.0	−1.73424
$556$	0	0
$557$	19024.3	1.44719	0.723594	−	0.690226i	$-0.242489\pi$
$557$	19024.3	1.44719	0.723594	+	0.690226i	$0.242489\pi$
$558$	0	0
$559$	3691.98	0.279345
$560$	0	0
$561$	−9008.16	−0.677941
$562$	0	0
$563$	−23453.3	−1.75567	−0.877833	−	0.478967i	$-0.841011\pi$
$563$	−23453.3	−1.75567	−0.877833	+	0.478967i	$0.841011\pi$
$564$	0	0
$565$	17761.3	1.32252
$566$	0	0
$567$	6249.64	0.462893
$568$	0	0
$569$	−14449.0	−1.06456	−0.532279	−	0.846569i	$-0.678664\pi$
$569$	−14449.0	−1.06456	−0.532279	+	0.846569i	$0.678664\pi$
$570$	0	0
$571$	20923.5	1.53348	0.766742	−	0.641955i	$-0.221877\pi$
$571$	20923.5	1.53348	0.766742	+	0.641955i	$0.221877\pi$
$572$	0	0
$573$	−4003.55	−0.291886
$574$	0	0
$575$	20700.9	1.50137
$576$	0	0
$577$	51.3171	0.00370253	0.00185127	−	0.999998i	$-0.499411\pi$
$577$	51.3171	0.00370253	0.00185127	+	0.999998i	$0.499411\pi$
$578$	0	0
$579$	−12498.4	−0.897093
$580$	0	0
$581$	−4647.70	−0.331875
$582$	0	0
$583$	1067.68	0.0758470
$584$	0	0
$585$	−11064.2	−0.781961
$586$	0	0
$587$	−12134.3	−0.853210	−0.426605	−	0.904438i	$-0.640291\pi$
$587$	−12134.3	−0.853210	−0.426605	+	0.904438i	$0.640291\pi$
$588$	0	0
$589$	−1225.36	−0.0857219
$590$	0	0
$591$	−5509.19	−0.383448
$592$	0	0
$593$	−22017.6	−1.52471	−0.762357	−	0.647157i	$-0.775958\pi$
$593$	−22017.6	−1.52471	−0.762357	+	0.647157i	$0.775958\pi$
$594$	0	0
$595$	12938.8	0.891495
$596$	0	0
$597$	−22044.0	−1.51123
$598$	0	0
$599$	−21423.9	−1.46136	−0.730681	−	0.682720i	$-0.760797\pi$
$599$	−21423.9	−1.46136	−0.730681	+	0.682720i	$0.760797\pi$
$600$	0	0
$601$	19272.7	1.30807	0.654035	−	0.756464i	$-0.273075\pi$
$601$	19272.7	1.30807	0.654035	+	0.756464i	$0.273075\pi$
$602$	0	0
$603$	−4220.29	−0.285014
$604$	0	0
$605$	−19090.5	−1.28288
$606$	0	0
$607$	12569.3	0.840479	0.420240	−	0.907413i	$-0.361946\pi$
$607$	12569.3	0.840479	0.420240	+	0.907413i	$0.361946\pi$
$608$	0	0
$609$	−7188.17	−0.478291
$610$	0	0
$611$	4802.15	0.317961
$612$	0	0
$613$	6947.01	0.457728	0.228864	−	0.973458i	$-0.426499\pi$
$613$	6947.01	0.457728	0.228864	+	0.973458i	$0.426499\pi$
$614$	0	0
$615$	−4380.53	−0.287220
$616$	0	0
$617$	5635.15	0.367687	0.183843	−	0.982956i	$-0.441146\pi$
$617$	5635.15	0.367687	0.183843	+	0.982956i	$0.441146\pi$
$618$	0	0
$619$	−21467.2	−1.39393	−0.696963	−	0.717107i	$-0.745466\pi$
$619$	−21467.2	−1.39393	−0.696963	+	0.717107i	$0.745466\pi$
$620$	0	0
$621$	9823.14	0.634765
$622$	0	0
$623$	−4806.29	−0.309085
$624$	0	0
$625$	−14997.2	−0.959823
$626$	0	0
$627$	6361.34	0.405179
$628$	0	0
$629$	−24573.8	−1.55774
$630$	0	0
$631$	−2878.85	−0.181625	−0.0908123	−	0.995868i	$-0.528946\pi$
$631$	−2878.85	−0.181625	−0.0908123	+	0.995868i	$0.528946\pi$
$632$	0	0
$633$	−15080.0	−0.946881
$634$	0	0
$635$	19963.9	1.24763
$636$	0	0
$637$	−1908.51	−0.118710
$638$	0	0
$639$	−8700.83	−0.538653
$640$	0	0
$641$	8478.86	0.522457	0.261228	−	0.965277i	$-0.415872\pi$
$641$	8478.86	0.522457	0.261228	+	0.965277i	$0.415872\pi$
$642$	0	0
$643$	−24804.2	−1.52128	−0.760638	−	0.649176i	$-0.775114\pi$
$643$	−24804.2	−1.52128	−0.760638	+	0.649176i	$0.775114\pi$
$644$	0	0
$645$	10127.5	0.618249
$646$	0	0
$647$	−26734.3	−1.62447	−0.812236	−	0.583329i	$-0.801750\pi$
$647$	−26734.3	−1.62447	−0.812236	+	0.583329i	$0.801750\pi$
$648$	0	0
$649$	3437.08	0.207885
$650$	0	0
$651$	−702.096	−0.0422693
$652$	0	0
$653$	−24311.7	−1.45695	−0.728476	−	0.685071i	$-0.759771\pi$
$653$	−24311.7	−1.45695	−0.728476	+	0.685071i	$0.759771\pi$
$654$	0	0
$655$	22051.5	1.31546
$656$	0	0
$657$	1524.80	0.0905450
$658$	0	0
$659$	24176.5	1.42911	0.714554	−	0.699580i	$-0.246630\pi$
$659$	24176.5	1.42911	0.714554	+	0.699580i	$0.246630\pi$
$660$	0	0
$661$	−22263.2	−1.31005	−0.655023	−	0.755609i	$-0.727341\pi$
$661$	−22263.2	−1.31005	−0.655023	+	0.755609i	$0.727341\pi$
$662$	0	0
$663$	−30184.2	−1.76811
$664$	0	0
$665$	−9137.07	−0.532813
$666$	0	0
$667$	−24462.7	−1.42009
$668$	0	0
$669$	−12617.3	−0.729165
$670$	0	0
$671$	329.297	0.0189454
$672$	0	0
$673$	10555.4	0.604575	0.302288	−	0.953217i	$-0.402250\pi$
$673$	10555.4	0.604575	0.302288	+	0.953217i	$0.402250\pi$
$674$	0	0
$675$	7999.16	0.456130
$676$	0	0
$677$	−16568.7	−0.940602	−0.470301	−	0.882506i	$-0.655855\pi$
$677$	−16568.7	−0.940602	−0.470301	+	0.882506i	$0.655855\pi$
$678$	0	0
$679$	137.580	0.00777593
$680$	0	0
$681$	11325.7	0.637299
$682$	0	0
$683$	27254.2	1.52687	0.763435	−	0.645885i	$-0.223511\pi$
$683$	27254.2	1.52687	0.763435	+	0.645885i	$0.223511\pi$
$684$	0	0
$685$	−8164.79	−0.455417
$686$	0	0
$687$	36100.5	2.00483
$688$	0	0
$689$	3577.54	0.197813
$690$	0	0
$691$	−27444.7	−1.51092	−0.755459	−	0.655196i	$-0.772586\pi$
$691$	−27444.7	−1.51092	−0.755459	+	0.655196i	$0.772586\pi$
$692$	0	0
$693$	1447.92	0.0793678
$694$	0	0
$695$	−15077.7	−0.822920
$696$	0	0
$697$	−4747.34	−0.257989
$698$	0	0
$699$	−35959.7	−1.94581
$700$	0	0
$701$	26330.8	1.41869	0.709343	−	0.704863i	$-0.248992\pi$
$701$	26330.8	1.41869	0.709343	+	0.704863i	$0.248992\pi$
$702$	0	0
$703$	17353.4	0.931003
$704$	0	0
$705$	13172.8	0.703713
$706$	0	0
$707$	5717.93	0.304165
$708$	0	0
$709$	25952.8	1.37472	0.687361	−	0.726316i	$-0.258769\pi$
$709$	25952.8	1.37472	0.687361	+	0.726316i	$0.258769\pi$
$710$	0	0
$711$	17920.1	0.945224
$712$	0	0
$713$	−2389.37	−0.125502
$714$	0	0
$715$	7227.44	0.378029
$716$	0	0
$717$	−28868.9	−1.50367
$718$	0	0
$719$	−21733.5	−1.12729	−0.563647	−	0.826016i	$-0.690602\pi$
$719$	−21733.5	−1.12729	−0.563647	+	0.826016i	$0.690602\pi$
$720$	0	0
$721$	−11194.9	−0.578252
$722$	0	0
$723$	17693.4	0.910128
$724$	0	0
$725$	−19920.4	−1.02045
$726$	0	0
$727$	−1620.00	−0.0826443	−0.0413222	−	0.999146i	$-0.513157\pi$
$727$	−1620.00	−0.0826443	−0.0413222	+	0.999146i	$0.513157\pi$
$728$	0	0
$729$	−4753.75	−0.241516
$730$	0	0
$731$	10975.6	0.555330
$732$	0	0
$733$	−17729.7	−0.893400	−0.446700	−	0.894684i	$-0.647401\pi$
$733$	−17729.7	−0.893400	−0.446700	+	0.894684i	$0.647401\pi$
$734$	0	0
$735$	−5235.27	−0.262729
$736$	0	0
$737$	2756.82	0.137787
$738$	0	0
$739$	9110.07	0.453477	0.226738	−	0.973956i	$-0.427194\pi$
$739$	9110.07	0.453477	0.226738	+	0.973956i	$0.427194\pi$
$740$	0	0
$741$	21315.3	1.05673
$742$	0	0
$743$	14464.9	0.714222	0.357111	−	0.934062i	$-0.383762\pi$
$743$	14464.9	0.714222	0.357111	+	0.934062i	$0.383762\pi$
$744$	0	0
$745$	6979.41	0.343229
$746$	0	0
$747$	11814.9	0.578694
$748$	0	0
$749$	−8761.67	−0.427429
$750$	0	0
$751$	−18828.1	−0.914841	−0.457421	−	0.889250i	$-0.651227\pi$
$751$	−18828.1	−0.914841	−0.457421	+	0.889250i	$0.651227\pi$
$752$	0	0
$753$	−38067.1	−1.84229
$754$	0	0
$755$	6359.72	0.306561
$756$	0	0
$757$	24136.6	1.15886	0.579431	−	0.815021i	$-0.303275\pi$
$757$	24136.6	1.15886	0.579431	+	0.815021i	$0.303275\pi$
$758$	0	0
$759$	12404.2	0.593204
$760$	0	0
$761$	−1091.82	−0.0520085	−0.0260043	−	0.999662i	$-0.508278\pi$
$761$	−1091.82	−0.0520085	−0.0260043	+	0.999662i	$0.508278\pi$
$762$	0	0
$763$	−7735.14	−0.367013
$764$	0	0
$765$	−32891.7	−1.55451
$766$	0	0
$767$	11516.8	0.542176
$768$	0	0
$769$	32638.4	1.53052	0.765261	−	0.643720i	$-0.222610\pi$
$769$	32638.4	1.53052	0.765261	+	0.643720i	$0.222610\pi$
$770$	0	0
$771$	6899.67	0.322290
$772$	0	0
$773$	−30138.4	−1.40233	−0.701167	−	0.712998i	$-0.747337\pi$
$773$	−30138.4	−1.40233	−0.701167	+	0.712998i	$0.747337\pi$
$774$	0	0
$775$	−1945.71	−0.0901830
$776$	0	0
$777$	9942.97	0.459076
$778$	0	0
$779$	3352.46	0.154190
$780$	0	0
$781$	5683.64	0.260405
$782$	0	0
$783$	−9452.78	−0.431436
$784$	0	0
$785$	235.616	0.0107127
$786$	0	0
$787$	36157.7	1.63772	0.818858	−	0.573997i	$-0.194608\pi$
$787$	36157.7	1.63772	0.818858	+	0.573997i	$0.194608\pi$
$788$	0	0
$789$	2446.74	0.110401
$790$	0	0
$791$	−7788.31	−0.350089
$792$	0	0
$793$	1103.39	0.0494107
$794$	0	0
$795$	9813.59	0.437802
$796$	0	0
$797$	15298.5	0.679925	0.339963	−	0.940439i	$-0.389586\pi$
$797$	15298.5	0.679925	0.339963	+	0.940439i	$0.389586\pi$
$798$	0	0
$799$	14275.9	0.632096
$800$	0	0
$801$	12218.0	0.538956
$802$	0	0
$803$	−996.044	−0.0437729
$804$	0	0
$805$	−17816.6	−0.780067
$806$	0	0
$807$	−16950.6	−0.739392
$808$	0	0
$809$	12345.1	0.536501	0.268251	−	0.963349i	$-0.413554\pi$
$809$	12345.1	0.536501	0.268251	+	0.963349i	$0.413554\pi$
$810$	0	0
$811$	−17871.6	−0.773806	−0.386903	−	0.922120i	$-0.626455\pi$
$811$	−17871.6	−0.773806	−0.386903	+	0.922120i	$0.626455\pi$
$812$	0	0
$813$	25024.0	1.07949
$814$	0	0
$815$	17006.6	0.730938
$816$	0	0
$817$	−7750.68	−0.331900
$818$	0	0
$819$	4851.63	0.206996
$820$	0	0
$821$	−37660.9	−1.60094	−0.800471	−	0.599371i	$-0.795417\pi$
$821$	−37660.9	−1.60094	−0.800471	+	0.599371i	$0.795417\pi$
$822$	0	0
$823$	−23125.4	−0.979467	−0.489733	−	0.871872i	$-0.662906\pi$
$823$	−23125.4	−0.979467	−0.489733	+	0.871872i	$0.662906\pi$
$824$	0	0
$825$	10100.9	0.426265
$826$	0	0
$827$	−2098.97	−0.0882568	−0.0441284	−	0.999026i	$-0.514051\pi$
$827$	−2098.97	−0.0882568	−0.0441284	+	0.999026i	$0.514051\pi$
$828$	0	0
$829$	10151.1	0.425286	0.212643	−	0.977130i	$-0.431793\pi$
$829$	10151.1	0.425286	0.212643	+	0.977130i	$0.431793\pi$
$830$	0	0
$831$	−6128.21	−0.255819
$832$	0	0
$833$	−5673.66	−0.235991
$834$	0	0
$835$	43027.9	1.78328
$836$	0	0
$837$	−923.289	−0.0381285
$838$	0	0
$839$	−40698.7	−1.67470	−0.837352	−	0.546665i	$-0.815897\pi$
$839$	−40698.7	−1.67470	−0.837352	+	0.546665i	$0.815897\pi$
$840$	0	0
$841$	−848.569	−0.0347931
$842$	0	0
$843$	−40401.5	−1.65065
$844$	0	0
$845$	−10854.5	−0.441901
$846$	0	0
$847$	8371.18	0.339595
$848$	0	0
$849$	−21345.4	−0.862863
$850$	0	0
$851$	33837.8	1.36304
$852$	0	0
$853$	−5214.46	−0.209308	−0.104654	−	0.994509i	$-0.533374\pi$
$853$	−5214.46	−0.209308	−0.104654	+	0.994509i	$0.533374\pi$
$854$	0	0
$855$	23227.3	0.929073
$856$	0	0
$857$	−31027.3	−1.23672	−0.618361	−	0.785894i	$-0.712203\pi$
$857$	−31027.3	−1.23672	−0.618361	+	0.785894i	$0.712203\pi$
$858$	0	0
$859$	−6583.19	−0.261485	−0.130742	−	0.991416i	$-0.541736\pi$
$859$	−6583.19	−0.261485	−0.130742	+	0.991416i	$0.541736\pi$
$860$	0	0
$861$	1920.86	0.0760309
$862$	0	0
$863$	313.309	0.0123583	0.00617913	−	0.999981i	$-0.498033\pi$
$863$	313.309	0.0123583	0.00617913	+	0.999981i	$0.498033\pi$
$864$	0	0
$865$	51576.3	2.02734
$866$	0	0
$867$	−56849.8	−2.22690
$868$	0	0
$869$	−11705.9	−0.456957
$870$	0	0
$871$	9237.44	0.359356
$872$	0	0
$873$	−349.743	−0.0135590
$874$	0	0
$875$	−540.292	−0.0208745
$876$	0	0
$877$	−15744.0	−0.606201	−0.303101	−	0.952959i	$-0.598022\pi$
$877$	−15744.0	−0.606201	−0.303101	+	0.952959i	$0.598022\pi$
$878$	0	0
$879$	51183.0	1.96401
$880$	0	0
$881$	48090.3	1.83905	0.919525	−	0.393030i	$-0.128573\pi$
$881$	48090.3	1.83905	0.919525	+	0.393030i	$0.128573\pi$
$882$	0	0
$883$	16474.7	0.627881	0.313941	−	0.949443i	$-0.398351\pi$
$883$	16474.7	0.627881	0.313941	+	0.949443i	$0.398351\pi$
$884$	0	0
$885$	31592.0	1.19995
$886$	0	0
$887$	28266.1	1.06999	0.534995	−	0.844855i	$-0.320313\pi$
$887$	28266.1	1.06999	0.534995	+	0.844855i	$0.320313\pi$
$888$	0	0
$889$	−8754.16	−0.330264
$890$	0	0
$891$	10378.0	0.390208
$892$	0	0
$893$	−10081.3	−0.377780
$894$	0	0
$895$	−8039.97	−0.300275
$896$	0	0
$897$	41563.3	1.54711
$898$	0	0
$899$	2299.28	0.0853008
$900$	0	0
$901$	10635.4	0.393247
$902$	0	0
$903$	−4440.90	−0.163659
$904$	0	0
$905$	15311.9	0.562415
$906$	0	0
$907$	22348.0	0.818140	0.409070	−	0.912503i	$-0.365853\pi$
$907$	22348.0	0.818140	0.409070	+	0.912503i	$0.365853\pi$
$908$	0	0
$909$	−14535.5	−0.530377
$910$	0	0
$911$	−45803.6	−1.66580	−0.832898	−	0.553427i	$-0.813320\pi$
$911$	−45803.6	−1.66580	−0.832898	+	0.553427i	$0.813320\pi$
$912$	0	0
$913$	−7717.84	−0.279763
$914$	0	0
$915$	3026.74	0.109356
$916$	0	0
$917$	−9669.57	−0.348220
$918$	0	0
$919$	2408.67	0.0864578	0.0432289	−	0.999065i	$-0.486236\pi$
$919$	2408.67	0.0864578	0.0432289	+	0.999065i	$0.486236\pi$
$920$	0	0
$921$	18453.5	0.660222
$922$	0	0
$923$	19044.5	0.679152
$924$	0	0
$925$	27554.8	0.979454
$926$	0	0
$927$	28458.5	1.00831
$928$	0	0
$929$	10730.1	0.378947	0.189474	−	0.981886i	$-0.439322\pi$
$929$	10730.1	0.378947	0.189474	+	0.981886i	$0.439322\pi$
$930$	0	0
$931$	4006.60	0.141043
$932$	0	0
$933$	−17747.3	−0.622745
$934$	0	0
$935$	21485.8	0.751510
$936$	0	0
$937$	3650.20	0.127264	0.0636322	−	0.997973i	$-0.479732\pi$
$937$	3650.20	0.127264	0.0636322	+	0.997973i	$0.479732\pi$
$938$	0	0
$939$	14077.9	0.489261
$940$	0	0
$941$	13957.4	0.483526	0.241763	−	0.970335i	$-0.422274\pi$
$941$	13957.4	0.483526	0.241763	+	0.970335i	$0.422274\pi$
$942$	0	0
$943$	6537.05	0.225743
$944$	0	0
$945$	−6884.62	−0.236991
$946$	0	0
$947$	9223.12	0.316485	0.158242	−	0.987400i	$-0.449417\pi$
$947$	9223.12	0.316485	0.158242	+	0.987400i	$0.449417\pi$
$948$	0	0
$949$	−3337.50	−0.114162
$950$	0	0
$951$	−29882.2	−1.01892
$952$	0	0
$953$	−48171.9	−1.63740	−0.818700	−	0.574222i	$-0.805305\pi$
$953$	−48171.9	−1.63740	−0.818700	+	0.574222i	$0.805305\pi$
$954$	0	0
$955$	9549.08	0.323561
$956$	0	0
$957$	−11936.5	−0.403189
$958$	0	0
$959$	3580.25	0.120555
$960$	0	0
$961$	−29566.4	−0.992461
$962$	0	0
$963$	22273.0	0.745314
$964$	0	0
$965$	29810.7	0.994445
$966$	0	0
$967$	−13721.7	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$967$	−13721.7	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$968$	0	0
$969$	63366.5	2.10075
$970$	0	0
$971$	−27512.1	−0.909274	−0.454637	−	0.890677i	$-0.650231\pi$
$971$	−27512.1	−0.909274	−0.454637	+	0.890677i	$0.650231\pi$
$972$	0	0
$973$	6611.55	0.217838
$974$	0	0
$975$	33845.8	1.11172
$976$	0	0
$977$	−54923.3	−1.79852	−0.899258	−	0.437418i	$-0.855893\pi$
$977$	−54923.3	−1.79852	−0.899258	+	0.437418i	$0.855893\pi$
$978$	0	0
$979$	−7981.19	−0.260552
$980$	0	0
$981$	19663.5	0.639966
$982$	0	0
$983$	4805.74	0.155930	0.0779651	−	0.996956i	$-0.475158\pi$
$983$	4805.74	0.155930	0.0779651	+	0.996956i	$0.475158\pi$
$984$	0	0
$985$	13140.3	0.425060
$986$	0	0
$987$	−5776.27	−0.186283
$988$	0	0
$989$	−15113.3	−0.485919
$990$	0	0
$991$	−24473.4	−0.784482	−0.392241	−	0.919862i	$-0.628300\pi$
$991$	−24473.4	−0.784482	−0.392241	+	0.919862i	$0.628300\pi$
$992$	0	0
$993$	−29518.4	−0.943342
$994$	0	0
$995$	52578.4	1.67522
$996$	0	0
$997$	−739.003	−0.0234749	−0.0117374	−	0.999931i	$-0.503736\pi$
$997$	−739.003	−0.0234749	−0.0117374	+	0.999931i	$0.503736\pi$
$998$	0	0
$999$	13075.5	0.414103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.d.1.2	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.d.1.2	✓	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q-6.69288 q^{3} +15.9636 q^{5} -7.00000 q^{7} +17.7947 q^{9} +O(q^{10})\)	\(q-6.69288 q^{3} +15.9636 q^{5} -7.00000 q^{7} +17.7947 q^{9} -11.6240 q^{11} -38.9493 q^{13} -106.842 q^{15} -115.789 q^{17} +81.7673 q^{19} +46.8502 q^{21} +159.440 q^{23} +129.835 q^{25} +61.6102 q^{27} -153.429 q^{29} -14.9860 q^{31} +77.7981 q^{33} -111.745 q^{35} +212.229 q^{37} +260.683 q^{39} +41.0000 q^{41} -94.7895 q^{43} +284.066 q^{45} -123.292 q^{47} +49.0000 q^{49} +774.961 q^{51} -91.8513 q^{53} -185.560 q^{55} -547.259 q^{57} -295.688 q^{59} -28.3290 q^{61} -124.563 q^{63} -621.769 q^{65} -237.166 q^{67} -1067.11 q^{69} -488.957 q^{71} +85.6885 q^{73} -868.971 q^{75} +81.3681 q^{77} +1007.05 q^{79} -892.806 q^{81} +663.957 q^{83} -1848.40 q^{85} +1026.88 q^{87} +686.613 q^{89} +272.645 q^{91} +100.299 q^{93} +1305.30 q^{95} -19.6544 q^{97} -206.845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * q^99

Introduction

L-functions

Modular forms

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