LMFDB - Embedded newform 1160.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,4,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1160.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:	$68.4422156067$
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} - 3 x^{10} - 200 x^{9} + 418 x^{8} + 13211 x^{7} - 14353 x^{6} - 314463 x^{5} + 64817 x^{4} + \cdots + 712233 $ x^11 - 3x^10 - 200x^9 + 418x^8 + 13211x^7 - 14353x^6 - 314463x^5 + 64817x^4 + 1676824x^3 - 292536x^2 - 2502333x + 712233
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{3}\cdot 43 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.75401$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.75401 q^{3} -5.00000 q^{5} +20.2215 q^{7} +49.6327 q^{9} +33.9939 q^{11} +39.9236 q^{13} +43.7701 q^{15} +113.921 q^{17} +88.7397 q^{19} -177.019 q^{21} -32.5916 q^{23} +25.0000 q^{25} -198.127 q^{27} +29.0000 q^{29} +231.747 q^{31} -297.583 q^{33} -101.108 q^{35} +150.130 q^{37} -349.492 q^{39} -467.305 q^{41} +487.375 q^{43} -248.164 q^{45} -494.646 q^{47} +65.9094 q^{49} -997.264 q^{51} -400.043 q^{53} -169.970 q^{55} -776.829 q^{57} +601.764 q^{59} +330.984 q^{61} +1003.65 q^{63} -199.618 q^{65} +822.435 q^{67} +285.307 q^{69} -717.511 q^{71} -243.999 q^{73} -218.850 q^{75} +687.408 q^{77} -1330.75 q^{79} +394.323 q^{81} +761.481 q^{83} -569.604 q^{85} -253.866 q^{87} +1483.31 q^{89} +807.315 q^{91} -2028.72 q^{93} -443.699 q^{95} -1341.77 q^{97} +1687.21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 8 q^{3} - 55 q^{5} + 16 q^{7} + 117 q^{9} + 26 q^{11} + 52 q^{13} - 40 q^{15} + 14 q^{17} + 146 q^{19} - 288 q^{21} - 80 q^{23} + 275 q^{25} + 224 q^{27} + 319 q^{29} - 38 q^{31} - 152 q^{33} - 80 q^{35}+ \cdots + 4690 q^{99}+O(q^{100}) $ 11 * q + 8 * q^3 - 55 * q^5 + 16 * q^7 + 117 * q^9 + 26 * q^11 + 52 * q^13 - 40 * q^15 + 14 * q^17 + 146 * q^19 - 288 * q^21 - 80 * q^23 + 275 * q^25 + 224 * q^27 + 319 * q^29 - 38 * q^31 - 152 * q^33 - 80 * q^35 + 376 * q^37 + 108 * q^39 - 146 * q^41 + 644 * q^43 - 585 * q^45 - 464 * q^47 + 393 * q^49 - 576 * q^51 - 188 * q^53 - 130 * q^55 + 1380 * q^57 + 672 * q^59 + 472 * q^61 + 628 * q^63 - 260 * q^65 + 1688 * q^67 + 138 * q^69 + 92 * q^71 + 1514 * q^73 + 200 * q^75 + 2884 * q^77 + 2130 * q^79 + 4067 * q^81 + 2584 * q^83 - 70 * q^85 + 232 * q^87 + 1606 * q^89 + 3208 * q^91 + 3540 * q^93 - 730 * q^95 + 3322 * q^97 + 4690 * q^99

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
$3$	−8.75401	−1.68471	−0.842355	−	0.538923i	$-0.818831\pi$
$3$	−8.75401	−1.68471	−0.842355	+	0.538923i	$0.818831\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	20.2215	1.09186	0.545929	−	0.837831i	$-0.316177\pi$
$7$	20.2215	1.09186	0.545929	+	0.837831i	$0.316177\pi$
$8$	0	0
$9$	49.6327	1.83825
$10$	0	0
$11$	33.9939	0.931778	0.465889	−	0.884843i	$-0.345735\pi$
$11$	33.9939	0.931778	0.465889	+	0.884843i	$0.345735\pi$
$12$	0	0
$13$	39.9236	0.851755	0.425878	−	0.904781i	$-0.359965\pi$
$13$	39.9236	0.851755	0.425878	+	0.904781i	$0.359965\pi$
$14$	0	0
$15$	43.7701	0.753425
$16$	0	0
$17$	113.921	1.62529	0.812643	−	0.582762i	$-0.198028\pi$
$17$	113.921	1.62529	0.812643	+	0.582762i	$0.198028\pi$
$18$	0	0
$19$	88.7397	1.07149	0.535744	−	0.844380i	$-0.320031\pi$
$19$	88.7397	1.07149	0.535744	+	0.844380i	$0.320031\pi$
$20$	0	0
$21$	−177.019	−1.83947
$22$	0	0
$23$	−32.5916	−0.295470	−0.147735	−	0.989027i	$-0.547198\pi$
$23$	−32.5916	−0.295470	−0.147735	+	0.989027i	$0.547198\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−198.127	−1.41221
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	231.747	1.34268	0.671339	−	0.741150i	$-0.265719\pi$
$31$	231.747	1.34268	0.671339	+	0.741150i	$0.265719\pi$
$32$	0	0
$33$	−297.583	−1.56978
$34$	0	0
$35$	−101.108	−0.488294
$36$	0	0
$37$	150.130	0.667062	0.333531	−	0.942739i	$-0.391760\pi$
$37$	150.130	0.667062	0.333531	+	0.942739i	$0.391760\pi$
$38$	0	0
$39$	−349.492	−1.43496
$40$	0	0
$41$	−467.305	−1.78002	−0.890010	−	0.455940i	$-0.849303\pi$
$41$	−467.305	−1.78002	−0.890010	+	0.455940i	$0.849303\pi$
$42$	0	0
$43$	487.375	1.72846	0.864232	−	0.503093i	$-0.167805\pi$
$43$	487.375	1.72846	0.864232	+	0.503093i	$0.167805\pi$
$44$	0	0
$45$	−248.164	−0.822090
$46$	0	0
$47$	−494.646	−1.53514	−0.767570	−	0.640965i	$-0.778534\pi$
$47$	−494.646	−1.53514	−0.767570	+	0.640965i	$0.778534\pi$
$48$	0	0
$49$	65.9094	0.192156
$50$	0	0
$51$	−997.264	−2.73813
$52$	0	0
$53$	−400.043	−1.03680	−0.518398	−	0.855140i	$-0.673471\pi$
$53$	−400.043	−1.03680	−0.518398	+	0.855140i	$0.673471\pi$
$54$	0	0
$55$	−169.970	−0.416704
$56$	0	0
$57$	−776.829	−1.80515
$58$	0	0
$59$	601.764	1.32785	0.663924	−	0.747800i	$-0.268890\pi$
$59$	601.764	1.32785	0.663924	+	0.747800i	$0.268890\pi$
$60$	0	0
$61$	330.984	0.694724	0.347362	−	0.937731i	$-0.387078\pi$
$61$	330.984	0.694724	0.347362	+	0.937731i	$0.387078\pi$
$62$	0	0
$63$	1003.65	2.00711
$64$	0	0
$65$	−199.618	−0.380916
$66$	0	0
$67$	822.435	1.49965	0.749824	−	0.661638i	$-0.230138\pi$
$67$	822.435	1.49965	0.749824	+	0.661638i	$0.230138\pi$
$68$	0	0
$69$	285.307	0.497782
$70$	0	0
$71$	−717.511	−1.19934	−0.599668	−	0.800249i	$-0.704701\pi$
$71$	−717.511	−1.19934	−0.599668	+	0.800249i	$0.704701\pi$
$72$	0	0
$73$	−243.999	−0.391204	−0.195602	−	0.980683i	$-0.562666\pi$
$73$	−243.999	−0.391204	−0.195602	+	0.980683i	$0.562666\pi$
$74$	0	0
$75$	−218.850	−0.336942
$76$	0	0
$77$	687.408	1.01737
$78$	0	0
$79$	−1330.75	−1.89521	−0.947605	−	0.319444i	$-0.896504\pi$
$79$	−1330.75	−1.89521	−0.947605	+	0.319444i	$0.896504\pi$
$80$	0	0
$81$	394.323	0.540909
$82$	0	0
$83$	761.481	1.00703	0.503514	−	0.863987i	$-0.332040\pi$
$83$	761.481	1.00703	0.503514	+	0.863987i	$0.332040\pi$
$84$	0	0
$85$	−569.604	−0.726850
$86$	0	0
$87$	−253.866	−0.312843
$88$	0	0
$89$	1483.31	1.76664	0.883319	−	0.468773i	$-0.155304\pi$
$89$	1483.31	1.76664	0.883319	+	0.468773i	$0.155304\pi$
$90$	0	0
$91$	807.315	0.929996
$92$	0	0
$93$	−2028.72	−2.26203
$94$	0	0
$95$	−443.699	−0.479184
$96$	0	0
$97$	−1341.77	−1.40450	−0.702250	−	0.711930i	$-0.747821\pi$
$97$	−1341.77	−1.40450	−0.702250	+	0.711930i	$0.747821\pi$
$98$	0	0
$99$	1687.21	1.71284
$100$	0	0
$101$	33.9198	0.0334173	0.0167086	−	0.999860i	$-0.494681\pi$
$101$	33.9198	0.0334173	0.0167086	+	0.999860i	$0.494681\pi$
$102$	0	0
$103$	86.2177	0.0824784	0.0412392	−	0.999149i	$-0.486869\pi$
$103$	86.2177	0.0824784	0.0412392	+	0.999149i	$0.486869\pi$
$104$	0	0
$105$	885.097	0.822634
$106$	0	0
$107$	1645.15	1.48638	0.743192	−	0.669079i	$-0.233311\pi$
$107$	1645.15	1.48638	0.743192	+	0.669079i	$0.233311\pi$
$108$	0	0
$109$	−23.2824	−0.0204592	−0.0102296	−	0.999948i	$-0.503256\pi$
$109$	−23.2824	−0.0204592	−0.0102296	+	0.999948i	$0.503256\pi$
$110$	0	0
$111$	−1314.24	−1.12381
$112$	0	0
$113$	1062.77	0.884752	0.442376	−	0.896830i	$-0.354136\pi$
$113$	1062.77	0.884752	0.442376	+	0.896830i	$0.354136\pi$
$114$	0	0
$115$	162.958	0.132138
$116$	0	0
$117$	1981.52	1.56574
$118$	0	0
$119$	2303.65	1.77458
$120$	0	0
$121$	−175.413	−0.131790
$122$	0	0
$123$	4090.80	2.99882
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−369.524	−0.258188	−0.129094	−	0.991632i	$-0.541207\pi$
$127$	−369.524	−0.258188	−0.129094	+	0.991632i	$0.541207\pi$
$128$	0	0
$129$	−4266.48	−2.91196
$130$	0	0
$131$	−1773.75	−1.18300	−0.591502	−	0.806304i	$-0.701465\pi$
$131$	−1773.75	−1.18300	−0.591502	+	0.806304i	$0.701465\pi$
$132$	0	0
$133$	1794.45	1.16991
$134$	0	0
$135$	990.635	0.631558
$136$	0	0
$137$	58.3106	0.0363636	0.0181818	−	0.999835i	$-0.494212\pi$
$137$	58.3106	0.0363636	0.0181818	+	0.999835i	$0.494212\pi$
$138$	0	0
$139$	701.578	0.428108	0.214054	−	0.976822i	$-0.431333\pi$
$139$	701.578	0.428108	0.214054	+	0.976822i	$0.431333\pi$
$140$	0	0
$141$	4330.14	2.58627
$142$	0	0
$143$	1357.16	0.793646
$144$	0	0
$145$	−145.000	−0.0830455
$146$	0	0
$147$	−576.972	−0.323727
$148$	0	0
$149$	−1211.72	−0.666227	−0.333113	−	0.942887i	$-0.608099\pi$
$149$	−1211.72	−0.666227	−0.333113	+	0.942887i	$0.608099\pi$
$150$	0	0
$151$	2515.91	1.35590	0.677952	−	0.735106i	$-0.262868\pi$
$151$	2515.91	1.35590	0.677952	+	0.735106i	$0.262868\pi$
$152$	0	0
$153$	5654.20	2.98768
$154$	0	0
$155$	−1158.74	−0.600464
$156$	0	0
$157$	−869.155	−0.441823	−0.220911	−	0.975294i	$-0.570903\pi$
$157$	−869.155	−0.441823	−0.220911	+	0.975294i	$0.570903\pi$
$158$	0	0
$159$	3501.98	1.74670
$160$	0	0
$161$	−659.051	−0.322612
$162$	0	0
$163$	3865.09	1.85729	0.928643	−	0.370976i	$-0.120977\pi$
$163$	3865.09	1.85729	0.928643	+	0.370976i	$0.120977\pi$
$164$	0	0
$165$	1487.92	0.702025
$166$	0	0
$167$	3407.54	1.57894	0.789472	−	0.613787i	$-0.210355\pi$
$167$	3407.54	1.57894	0.789472	+	0.613787i	$0.210355\pi$
$168$	0	0
$169$	−603.106	−0.274513
$170$	0	0
$171$	4404.39	1.96966
$172$	0	0
$173$	−796.244	−0.349927	−0.174963	−	0.984575i	$-0.555981\pi$
$173$	−796.244	−0.349927	−0.174963	+	0.984575i	$0.555981\pi$
$174$	0	0
$175$	505.538	0.218372
$176$	0	0
$177$	−5267.85	−2.23704
$178$	0	0
$179$	−958.647	−0.400294	−0.200147	−	0.979766i	$-0.564142\pi$
$179$	−958.647	−0.400294	−0.200147	+	0.979766i	$0.564142\pi$
$180$	0	0
$181$	−4679.53	−1.92169	−0.960847	−	0.277080i	$-0.910633\pi$
$181$	−4679.53	−1.92169	−0.960847	+	0.277080i	$0.910633\pi$
$182$	0	0
$183$	−2897.44	−1.17041
$184$	0	0
$185$	−750.652	−0.298319
$186$	0	0
$187$	3872.61	1.51440
$188$	0	0
$189$	−4006.43	−1.54193
$190$	0	0
$191$	−1343.72	−0.509049	−0.254524	−	0.967066i	$-0.581919\pi$
$191$	−1343.72	−0.509049	−0.254524	+	0.967066i	$0.581919\pi$
$192$	0	0
$193$	2895.86	1.08005	0.540023	−	0.841650i	$-0.318416\pi$
$193$	2895.86	1.08005	0.540023	+	0.841650i	$0.318416\pi$
$194$	0	0
$195$	1747.46	0.641734
$196$	0	0
$197$	−3151.36	−1.13972	−0.569860	−	0.821742i	$-0.693003\pi$
$197$	−3151.36	−1.13972	−0.569860	+	0.821742i	$0.693003\pi$
$198$	0	0
$199$	2922.77	1.04115	0.520577	−	0.853815i	$-0.325717\pi$
$199$	2922.77	1.04115	0.520577	+	0.853815i	$0.325717\pi$
$200$	0	0
$201$	−7199.60	−2.52647
$202$	0	0
$203$	586.424	0.202753
$204$	0	0
$205$	2336.53	0.796049
$206$	0	0
$207$	−1617.61	−0.543148
$208$	0	0
$209$	3016.61	0.998389
$210$	0	0
$211$	−3475.86	−1.13407	−0.567033	−	0.823695i	$-0.691909\pi$
$211$	−3475.86	−1.13407	−0.567033	+	0.823695i	$0.691909\pi$
$212$	0	0
$213$	6281.10	2.02053
$214$	0	0
$215$	−2436.87	−0.772993
$216$	0	0
$217$	4686.28	1.46602
$218$	0	0
$219$	2135.97	0.659066
$220$	0	0
$221$	4548.13	1.38434
$222$	0	0
$223$	2296.27	0.689549	0.344774	−	0.938686i	$-0.387955\pi$
$223$	2296.27	0.689549	0.344774	+	0.938686i	$0.387955\pi$
$224$	0	0
$225$	1240.82	0.367650
$226$	0	0
$227$	−976.842	−0.285618	−0.142809	−	0.989750i	$-0.545613\pi$
$227$	−976.842	−0.285618	−0.142809	+	0.989750i	$0.545613\pi$
$228$	0	0
$229$	−3657.24	−1.05536	−0.527680	−	0.849443i	$-0.676938\pi$
$229$	−3657.24	−1.05536	−0.527680	+	0.849443i	$0.676938\pi$
$230$	0	0
$231$	−6017.58	−1.71397
$232$	0	0
$233$	−3264.90	−0.917987	−0.458994	−	0.888440i	$-0.651790\pi$
$233$	−3264.90	−0.917987	−0.458994	+	0.888440i	$0.651790\pi$
$234$	0	0
$235$	2473.23	0.686535
$236$	0	0
$237$	11649.4	3.19288
$238$	0	0
$239$	3790.15	1.02579	0.512896	−	0.858451i	$-0.328573\pi$
$239$	3790.15	1.02579	0.512896	+	0.858451i	$0.328573\pi$
$240$	0	0
$241$	−577.658	−0.154399	−0.0771997	−	0.997016i	$-0.524598\pi$
$241$	−577.658	−0.154399	−0.0771997	+	0.997016i	$0.524598\pi$
$242$	0	0
$243$	1897.52	0.500931
$244$	0	0
$245$	−329.547	−0.0859346
$246$	0	0
$247$	3542.81	0.912646
$248$	0	0
$249$	−6666.01	−1.69655
$250$	0	0
$251$	−2331.95	−0.586419	−0.293209	−	0.956048i	$-0.594723\pi$
$251$	−2331.95	−0.586419	−0.293209	+	0.956048i	$0.594723\pi$
$252$	0	0
$253$	−1107.92	−0.275313
$254$	0	0
$255$	4986.32	1.22453
$256$	0	0
$257$	−4878.53	−1.18410	−0.592051	−	0.805900i	$-0.701682\pi$
$257$	−4878.53	−1.18410	−0.592051	+	0.805900i	$0.701682\pi$
$258$	0	0
$259$	3035.86	0.728338
$260$	0	0
$261$	1439.35	0.341354
$262$	0	0
$263$	4620.92	1.08341	0.541707	−	0.840567i	$-0.317778\pi$
$263$	4620.92	1.08341	0.541707	+	0.840567i	$0.317778\pi$
$264$	0	0
$265$	2000.22	0.463669
$266$	0	0
$267$	−12984.9	−2.97627
$268$	0	0
$269$	−1717.68	−0.389326	−0.194663	−	0.980870i	$-0.562361\pi$
$269$	−1717.68	−0.389326	−0.194663	+	0.980870i	$0.562361\pi$
$270$	0	0
$271$	1170.35	0.262339	0.131169	−	0.991360i	$-0.458127\pi$
$271$	1170.35	0.262339	0.131169	+	0.991360i	$0.458127\pi$
$272$	0	0
$273$	−7067.25	−1.56677
$274$	0	0
$275$	849.848	0.186356
$276$	0	0
$277$	−931.693	−0.202094	−0.101047	−	0.994882i	$-0.532219\pi$
$277$	−931.693	−0.202094	−0.101047	+	0.994882i	$0.532219\pi$
$278$	0	0
$279$	11502.2	2.46818
$280$	0	0
$281$	2345.63	0.497967	0.248983	−	0.968508i	$-0.419904\pi$
$281$	2345.63	0.497967	0.248983	+	0.968508i	$0.419904\pi$
$282$	0	0
$283$	−4409.55	−0.926221	−0.463111	−	0.886300i	$-0.653267\pi$
$283$	−4409.55	−0.926221	−0.463111	+	0.886300i	$0.653267\pi$
$284$	0	0
$285$	3884.14	0.807287
$286$	0	0
$287$	−9449.62	−1.94353
$288$	0	0
$289$	8064.94	1.64155
$290$	0	0
$291$	11745.9	2.36618
$292$	0	0
$293$	5722.09	1.14091	0.570457	−	0.821327i	$-0.306766\pi$
$293$	5722.09	1.14091	0.570457	+	0.821327i	$0.306766\pi$
$294$	0	0
$295$	−3008.82	−0.593831
$296$	0	0
$297$	−6735.11	−1.31586
$298$	0	0
$299$	−1301.17	−0.251668
$300$	0	0
$301$	9855.45	1.88724
$302$	0	0
$303$	−296.934	−0.0562984
$304$	0	0
$305$	−1654.92	−0.310690
$306$	0	0
$307$	−8543.77	−1.58833	−0.794167	−	0.607699i	$-0.792093\pi$
$307$	−8543.77	−1.58833	−0.794167	+	0.607699i	$0.792093\pi$
$308$	0	0
$309$	−754.750	−0.138952
$310$	0	0
$311$	−7379.96	−1.34559	−0.672796	−	0.739828i	$-0.734907\pi$
$311$	−7379.96	−1.34559	−0.672796	+	0.739828i	$0.734907\pi$
$312$	0	0
$313$	−5170.10	−0.933647	−0.466824	−	0.884350i	$-0.654602\pi$
$313$	−5170.10	−0.933647	−0.466824	+	0.884350i	$0.654602\pi$
$314$	0	0
$315$	−5018.24	−0.897606
$316$	0	0
$317$	−5298.42	−0.938766	−0.469383	−	0.882995i	$-0.655524\pi$
$317$	−5298.42	−0.938766	−0.469383	+	0.882995i	$0.655524\pi$
$318$	0	0
$319$	985.824	0.173027
$320$	0	0
$321$	−14401.7	−2.50413
$322$	0	0
$323$	10109.3	1.74147
$324$	0	0
$325$	998.090	0.170351
$326$	0	0
$327$	203.815	0.0344678
$328$	0	0
$329$	−10002.5	−1.67616
$330$	0	0
$331$	2759.40	0.458218	0.229109	−	0.973401i	$-0.426419\pi$
$331$	2759.40	0.458218	0.229109	+	0.973401i	$0.426419\pi$
$332$	0	0
$333$	7451.38	1.22623
$334$	0	0
$335$	−4112.17	−0.670663
$336$	0	0
$337$	9327.83	1.50777	0.753886	−	0.657005i	$-0.228177\pi$
$337$	9327.83	1.50777	0.753886	+	0.657005i	$0.228177\pi$
$338$	0	0
$339$	−9303.49	−1.49055
$340$	0	0
$341$	7878.00	1.25108
$342$	0	0
$343$	−5603.19	−0.882052
$344$	0	0
$345$	−1426.54	−0.222615
$346$	0	0
$347$	9199.14	1.42316	0.711579	−	0.702606i	$-0.247980\pi$
$347$	9199.14	1.42316	0.711579	+	0.702606i	$0.247980\pi$
$348$	0	0
$349$	1562.14	0.239598	0.119799	−	0.992798i	$-0.461775\pi$
$349$	1562.14	0.239598	0.119799	+	0.992798i	$0.461775\pi$
$350$	0	0
$351$	−7909.94	−1.20285
$352$	0	0
$353$	5984.23	0.902290	0.451145	−	0.892451i	$-0.351016\pi$
$353$	5984.23	0.902290	0.451145	+	0.892451i	$0.351016\pi$
$354$	0	0
$355$	3587.55	0.536360
$356$	0	0
$357$	−20166.2	−2.98966
$358$	0	0
$359$	−6116.97	−0.899280	−0.449640	−	0.893210i	$-0.648448\pi$
$359$	−6116.97	−0.899280	−0.449640	+	0.893210i	$0.648448\pi$
$360$	0	0
$361$	1015.74	0.148089
$362$	0	0
$363$	1535.57	0.222029
$364$	0	0
$365$	1220.00	0.174952
$366$	0	0
$367$	−13303.5	−1.89220	−0.946099	−	0.323877i	$-0.895014\pi$
$367$	−13303.5	−1.89220	−0.946099	+	0.323877i	$0.895014\pi$
$368$	0	0
$369$	−23193.6	−3.27212
$370$	0	0
$371$	−8089.47	−1.13203
$372$	0	0
$373$	11561.0	1.60484	0.802421	−	0.596758i	$-0.203545\pi$
$373$	11561.0	1.60484	0.802421	+	0.596758i	$0.203545\pi$
$374$	0	0
$375$	1094.25	0.150685
$376$	0	0
$377$	1157.78	0.158167
$378$	0	0
$379$	−7536.96	−1.02150	−0.510749	−	0.859730i	$-0.670632\pi$
$379$	−7536.96	−1.02150	−0.510749	+	0.859730i	$0.670632\pi$
$380$	0	0
$381$	3234.81	0.434973
$382$	0	0
$383$	5077.34	0.677388	0.338694	−	0.940897i	$-0.390015\pi$
$383$	5077.34	0.677388	0.338694	+	0.940897i	$0.390015\pi$
$384$	0	0
$385$	−3437.04	−0.454982
$386$	0	0
$387$	24189.7	3.17735
$388$	0	0
$389$	−8940.54	−1.16530	−0.582652	−	0.812722i	$-0.697985\pi$
$389$	−8940.54	−1.16530	−0.582652	+	0.812722i	$0.697985\pi$
$390$	0	0
$391$	−3712.86	−0.480223
$392$	0	0
$393$	15527.4	1.99302
$394$	0	0
$395$	6653.77	0.847564
$396$	0	0
$397$	6436.32	0.813676	0.406838	−	0.913500i	$-0.366631\pi$
$397$	6436.32	0.813676	0.406838	+	0.913500i	$0.366631\pi$
$398$	0	0
$399$	−15708.6	−1.97097
$400$	0	0
$401$	−6832.73	−0.850899	−0.425449	−	0.904982i	$-0.639884\pi$
$401$	−6832.73	−0.850899	−0.425449	+	0.904982i	$0.639884\pi$
$402$	0	0
$403$	9252.19	1.14363
$404$	0	0
$405$	−1971.61	−0.241902
$406$	0	0
$407$	5103.52	0.621553
$408$	0	0
$409$	−6981.52	−0.844044	−0.422022	−	0.906586i	$-0.638680\pi$
$409$	−6981.52	−0.844044	−0.422022	+	0.906586i	$0.638680\pi$
$410$	0	0
$411$	−510.452	−0.0612621
$412$	0	0
$413$	12168.6	1.44982
$414$	0	0
$415$	−3807.40	−0.450357
$416$	0	0
$417$	−6141.62	−0.721238
$418$	0	0
$419$	−9700.75	−1.13106	−0.565528	−	0.824729i	$-0.691328\pi$
$419$	−9700.75	−1.13106	−0.565528	+	0.824729i	$0.691328\pi$
$420$	0	0
$421$	7969.20	0.922554	0.461277	−	0.887256i	$-0.347392\pi$
$421$	7969.20	0.922554	0.461277	+	0.887256i	$0.347392\pi$
$422$	0	0
$423$	−24550.6	−2.82197
$424$	0	0
$425$	2848.02	0.325057
$426$	0	0
$427$	6693.00	0.758540
$428$	0	0
$429$	−11880.6	−1.33706
$430$	0	0
$431$	−1698.55	−0.189828	−0.0949142	−	0.995485i	$-0.530258\pi$
$431$	−1698.55	−0.189828	−0.0949142	+	0.995485i	$0.530258\pi$
$432$	0	0
$433$	5252.28	0.582929	0.291465	−	0.956582i	$-0.405857\pi$
$433$	5252.28	0.582929	0.291465	+	0.956582i	$0.405857\pi$
$434$	0	0
$435$	1269.33	0.139908
$436$	0	0
$437$	−2892.17	−0.316593
$438$	0	0
$439$	5682.28	0.617768	0.308884	−	0.951100i	$-0.400044\pi$
$439$	5682.28	0.617768	0.308884	+	0.951100i	$0.400044\pi$
$440$	0	0
$441$	3271.26	0.353230
$442$	0	0
$443$	−6717.45	−0.720442	−0.360221	−	0.932867i	$-0.617299\pi$
$443$	−6717.45	−0.720442	−0.360221	+	0.932867i	$0.617299\pi$
$444$	0	0
$445$	−7416.56	−0.790064
$446$	0	0
$447$	10607.4	1.12240
$448$	0	0
$449$	11142.6	1.17116	0.585581	−	0.810614i	$-0.300867\pi$
$449$	11142.6	1.17116	0.585581	+	0.810614i	$0.300867\pi$
$450$	0	0
$451$	−15885.5	−1.65858
$452$	0	0
$453$	−22024.3	−2.28431
$454$	0	0
$455$	−4036.58	−0.415907
$456$	0	0
$457$	14112.2	1.44451	0.722257	−	0.691625i	$-0.243105\pi$
$457$	14112.2	1.44451	0.722257	+	0.691625i	$0.243105\pi$
$458$	0	0
$459$	−22570.8	−2.29524
$460$	0	0
$461$	6100.16	0.616297	0.308149	−	0.951338i	$-0.400291\pi$
$461$	6100.16	0.616297	0.308149	+	0.951338i	$0.400291\pi$
$462$	0	0
$463$	−4635.93	−0.465335	−0.232667	−	0.972556i	$-0.574745\pi$
$463$	−4635.93	−0.465335	−0.232667	+	0.972556i	$0.574745\pi$
$464$	0	0
$465$	10143.6	1.01161
$466$	0	0
$467$	−4280.41	−0.424141	−0.212071	−	0.977254i	$-0.568021\pi$
$467$	−4280.41	−0.424141	−0.212071	+	0.977254i	$0.568021\pi$
$468$	0	0
$469$	16630.9	1.63740
$470$	0	0
$471$	7608.60	0.744343
$472$	0	0
$473$	16567.8	1.61054
$474$	0	0
$475$	2218.49	0.214298
$476$	0	0
$477$	−19855.2	−1.90589
$478$	0	0
$479$	−1717.52	−0.163832	−0.0819160	−	0.996639i	$-0.526104\pi$
$479$	−1717.52	−0.163832	−0.0819160	+	0.996639i	$0.526104\pi$
$480$	0	0
$481$	5993.75	0.568173
$482$	0	0
$483$	5769.34	0.543507
$484$	0	0
$485$	6708.87	0.628112
$486$	0	0
$487$	19402.3	1.80534	0.902672	−	0.430330i	$-0.141603\pi$
$487$	19402.3	1.80534	0.902672	+	0.430330i	$0.141603\pi$
$488$	0	0
$489$	−33835.1	−3.12899
$490$	0	0
$491$	−3585.22	−0.329529	−0.164764	−	0.986333i	$-0.552686\pi$
$491$	−3585.22	−0.329529	−0.164764	+	0.986333i	$0.552686\pi$
$492$	0	0
$493$	3303.70	0.301808
$494$	0	0
$495$	−8436.05	−0.766005
$496$	0	0
$497$	−14509.2	−1.30951
$498$	0	0
$499$	−2952.68	−0.264890	−0.132445	−	0.991190i	$-0.542283\pi$
$499$	−2952.68	−0.264890	−0.132445	+	0.991190i	$0.542283\pi$
$500$	0	0
$501$	−29829.7	−2.66006
$502$	0	0
$503$	−3703.50	−0.328292	−0.164146	−	0.986436i	$-0.552487\pi$
$503$	−3703.50	−0.328292	−0.164146	+	0.986436i	$0.552487\pi$
$504$	0	0
$505$	−169.599	−0.0149447
$506$	0	0
$507$	5279.60	0.462475
$508$	0	0
$509$	9806.65	0.853973	0.426986	−	0.904258i	$-0.359575\pi$
$509$	9806.65	0.853973	0.426986	+	0.904258i	$0.359575\pi$
$510$	0	0
$511$	−4934.03	−0.427140
$512$	0	0
$513$	−17581.7	−1.51316
$514$	0	0
$515$	−431.088	−0.0368855
$516$	0	0
$517$	−16815.0	−1.43041
$518$	0	0
$519$	6970.33	0.589525
$520$	0	0
$521$	1436.91	0.120830	0.0604149	−	0.998173i	$-0.480758\pi$
$521$	1436.91	0.120830	0.0604149	+	0.998173i	$0.480758\pi$
$522$	0	0
$523$	13099.2	1.09520	0.547598	−	0.836741i	$-0.315542\pi$
$523$	13099.2	1.09520	0.547598	+	0.836741i	$0.315542\pi$
$524$	0	0
$525$	−4425.48	−0.367893
$526$	0	0
$527$	26400.8	2.18224
$528$	0	0
$529$	−11104.8	−0.912697
$530$	0	0
$531$	29867.2	2.44091
$532$	0	0
$533$	−18656.5	−1.51614
$534$	0	0
$535$	−8225.77	−0.664731
$536$	0	0
$537$	8392.01	0.674380
$538$	0	0
$539$	2240.52	0.179046
$540$	0	0
$541$	6178.32	0.490992	0.245496	−	0.969398i	$-0.421049\pi$
$541$	6178.32	0.490992	0.245496	+	0.969398i	$0.421049\pi$
$542$	0	0
$543$	40964.6	3.23750
$544$	0	0
$545$	116.412	0.00914963
$546$	0	0
$547$	22268.3	1.74063	0.870313	−	0.492499i	$-0.163916\pi$
$547$	22268.3	1.74063	0.870313	+	0.492499i	$0.163916\pi$
$548$	0	0
$549$	16427.6	1.27708
$550$	0	0
$551$	2573.45	0.198970
$552$	0	0
$553$	−26909.9	−2.06930
$554$	0	0
$555$	6571.22	0.502581
$556$	0	0
$557$	−5242.87	−0.398829	−0.199414	−	0.979915i	$-0.563904\pi$
$557$	−5242.87	−0.398829	−0.199414	+	0.979915i	$0.563904\pi$
$558$	0	0
$559$	19457.8	1.47223
$560$	0	0
$561$	−33900.9	−2.55133
$562$	0	0
$563$	−21298.2	−1.59434	−0.797169	−	0.603757i	$-0.793670\pi$
$563$	−21298.2	−1.59434	−0.797169	+	0.603757i	$0.793670\pi$
$564$	0	0
$565$	−5313.85	−0.395673
$566$	0	0
$567$	7973.80	0.590596
$568$	0	0
$569$	−5685.99	−0.418927	−0.209463	−	0.977817i	$-0.567172\pi$
$569$	−5685.99	−0.418927	−0.209463	+	0.977817i	$0.567172\pi$
$570$	0	0
$571$	−12511.3	−0.916955	−0.458478	−	0.888706i	$-0.651605\pi$
$571$	−12511.3	−0.916955	−0.458478	+	0.888706i	$0.651605\pi$
$572$	0	0
$573$	11763.0	0.857600
$574$	0	0
$575$	−814.790	−0.0590941
$576$	0	0
$577$	15403.9	1.11139	0.555696	−	0.831386i	$-0.312452\pi$
$577$	15403.9	1.11139	0.555696	+	0.831386i	$0.312452\pi$
$578$	0	0
$579$	−25350.4	−1.81956
$580$	0	0
$581$	15398.3	1.09953
$582$	0	0
$583$	−13599.0	−0.966063
$584$	0	0
$585$	−9907.58	−0.700219
$586$	0	0
$587$	10911.3	0.767219	0.383609	−	0.923495i	$-0.374681\pi$
$587$	10911.3	0.767219	0.383609	+	0.923495i	$0.374681\pi$
$588$	0	0
$589$	20565.2	1.43867
$590$	0	0
$591$	27587.0	1.92010
$592$	0	0
$593$	−4112.34	−0.284779	−0.142389	−	0.989811i	$-0.545478\pi$
$593$	−4112.34	−0.284779	−0.142389	+	0.989811i	$0.545478\pi$
$594$	0	0
$595$	−11518.2	−0.793617
$596$	0	0
$597$	−25585.9	−1.75404
$598$	0	0
$599$	19002.0	1.29616	0.648080	−	0.761572i	$-0.275572\pi$
$599$	19002.0	1.29616	0.648080	+	0.761572i	$0.275572\pi$
$600$	0	0
$601$	3813.27	0.258813	0.129407	−	0.991592i	$-0.458693\pi$
$601$	3813.27	0.258813	0.129407	+	0.991592i	$0.458693\pi$
$602$	0	0
$603$	40819.7	2.75672
$604$	0	0
$605$	877.065	0.0589384
$606$	0	0
$607$	11543.4	0.771883	0.385941	−	0.922523i	$-0.373877\pi$
$607$	11543.4	0.771883	0.385941	+	0.922523i	$0.373877\pi$
$608$	0	0
$609$	−5133.56	−0.341580
$610$	0	0
$611$	−19748.1	−1.30756
$612$	0	0
$613$	1035.60	0.0682338	0.0341169	−	0.999418i	$-0.489138\pi$
$613$	1035.60	0.0682338	0.0341169	+	0.999418i	$0.489138\pi$
$614$	0	0
$615$	−20454.0	−1.34111
$616$	0	0
$617$	−19833.8	−1.29413	−0.647065	−	0.762434i	$-0.724004\pi$
$617$	−19833.8	−1.29413	−0.647065	+	0.762434i	$0.724004\pi$
$618$	0	0
$619$	25107.7	1.63031	0.815157	−	0.579241i	$-0.196651\pi$
$619$	25107.7	1.63031	0.815157	+	0.579241i	$0.196651\pi$
$620$	0	0
$621$	6457.27	0.417265
$622$	0	0
$623$	29994.8	1.92892
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−26407.5	−1.68200
$628$	0	0
$629$	17103.0	1.08417
$630$	0	0
$631$	−2334.04	−0.147253	−0.0736265	−	0.997286i	$-0.523457\pi$
$631$	−2334.04	−0.147253	−0.0736265	+	0.997286i	$0.523457\pi$
$632$	0	0
$633$	30427.7	1.91057
$634$	0	0
$635$	1847.62	0.115465
$636$	0	0
$637$	2631.34	0.163670
$638$	0	0
$639$	−35612.0	−2.20468
$640$	0	0
$641$	−4443.82	−0.273823	−0.136911	−	0.990583i	$-0.543718\pi$
$641$	−4443.82	−0.273823	−0.136911	+	0.990583i	$0.543718\pi$
$642$	0	0
$643$	−1157.81	−0.0710102	−0.0355051	−	0.999369i	$-0.511304\pi$
$643$	−1157.81	−0.0710102	−0.0355051	+	0.999369i	$0.511304\pi$
$644$	0	0
$645$	21332.4	1.30227
$646$	0	0
$647$	−23478.7	−1.42665	−0.713325	−	0.700834i	$-0.752812\pi$
$647$	−23478.7	−1.42665	−0.713325	+	0.700834i	$0.752812\pi$
$648$	0	0
$649$	20456.3	1.23726
$650$	0	0
$651$	−41023.8	−2.46981
$652$	0	0
$653$	46.1166	0.00276368	0.00138184	−	0.999999i	$-0.499560\pi$
$653$	46.1166	0.00276368	0.00138184	+	0.999999i	$0.499560\pi$
$654$	0	0
$655$	8868.76	0.529055
$656$	0	0
$657$	−12110.3	−0.719131
$658$	0	0
$659$	−10953.4	−0.647474	−0.323737	−	0.946147i	$-0.604939\pi$
$659$	−10953.4	−0.647474	−0.323737	+	0.946147i	$0.604939\pi$
$660$	0	0
$661$	8628.34	0.507721	0.253861	−	0.967241i	$-0.418300\pi$
$661$	8628.34	0.507721	0.253861	+	0.967241i	$0.418300\pi$
$662$	0	0
$663$	−39814.4	−2.33222
$664$	0	0
$665$	−8972.26	−0.523202
$666$	0	0
$667$	−945.156	−0.0548674
$668$	0	0
$669$	−20101.5	−1.16169
$670$	0	0
$671$	11251.4	0.647328
$672$	0	0
$673$	10831.3	0.620380	0.310190	−	0.950675i	$-0.399607\pi$
$673$	10831.3	0.620380	0.310190	+	0.950675i	$0.399607\pi$
$674$	0	0
$675$	−4953.17	−0.282441
$676$	0	0
$677$	12794.5	0.726342	0.363171	−	0.931723i	$-0.381694\pi$
$677$	12794.5	0.726342	0.363171	+	0.931723i	$0.381694\pi$
$678$	0	0
$679$	−27132.7	−1.53352
$680$	0	0
$681$	8551.29	0.481184
$682$	0	0
$683$	738.760	0.0413878	0.0206939	−	0.999786i	$-0.493412\pi$
$683$	738.760	0.0413878	0.0206939	+	0.999786i	$0.493412\pi$
$684$	0	0
$685$	−291.553	−0.0162623
$686$	0	0
$687$	32015.5	1.77798
$688$	0	0
$689$	−15971.2	−0.883096
$690$	0	0
$691$	−19190.3	−1.05649	−0.528245	−	0.849092i	$-0.677150\pi$
$691$	−19190.3	−1.05649	−0.528245	+	0.849092i	$0.677150\pi$
$692$	0	0
$693$	34117.9	1.87018
$694$	0	0
$695$	−3507.89	−0.191456
$696$	0	0
$697$	−53235.8	−2.89304
$698$	0	0
$699$	28581.0	1.54654
$700$	0	0
$701$	12515.4	0.674322	0.337161	−	0.941447i	$-0.390533\pi$
$701$	12515.4	0.674322	0.337161	+	0.941447i	$0.390533\pi$
$702$	0	0
$703$	13322.5	0.714750
$704$	0	0
$705$	−21650.7	−1.15661
$706$	0	0
$707$	685.909	0.0364870
$708$	0	0
$709$	13814.4	0.731752	0.365876	−	0.930664i	$-0.380769\pi$
$709$	13814.4	0.731752	0.365876	+	0.930664i	$0.380769\pi$
$710$	0	0
$711$	−66049.0	−3.48387
$712$	0	0
$713$	−7553.01	−0.396722
$714$	0	0
$715$	−6785.80	−0.354929
$716$	0	0
$717$	−33179.0	−1.72816
$718$	0	0
$719$	29419.8	1.52597	0.762985	−	0.646416i	$-0.223733\pi$
$719$	29419.8	1.52597	0.762985	+	0.646416i	$0.223733\pi$
$720$	0	0
$721$	1743.45	0.0900548
$722$	0	0
$723$	5056.83	0.260118
$724$	0	0
$725$	725.000	0.0371391
$726$	0	0
$727$	−26045.8	−1.32873	−0.664364	−	0.747409i	$-0.731297\pi$
$727$	−26045.8	−1.32873	−0.664364	+	0.747409i	$0.731297\pi$
$728$	0	0
$729$	−27257.7	−1.38483
$730$	0	0
$731$	55522.1	2.80925
$732$	0	0
$733$	−24216.1	−1.22025	−0.610125	−	0.792305i	$-0.708881\pi$
$733$	−24216.1	−1.22025	−0.610125	+	0.792305i	$0.708881\pi$
$734$	0	0
$735$	2884.86	0.144775
$736$	0	0
$737$	27957.8	1.39734
$738$	0	0
$739$	8915.31	0.443782	0.221891	−	0.975071i	$-0.428777\pi$
$739$	8915.31	0.443782	0.221891	+	0.975071i	$0.428777\pi$
$740$	0	0
$741$	−31013.8	−1.53754
$742$	0	0
$743$	−22073.3	−1.08989	−0.544947	−	0.838470i	$-0.683450\pi$
$743$	−22073.3	−1.08989	−0.544947	+	0.838470i	$0.683450\pi$
$744$	0	0
$745$	6058.59	0.297946
$746$	0	0
$747$	37794.3	1.85117
$748$	0	0
$749$	33267.5	1.62292
$750$	0	0
$751$	27646.1	1.34330	0.671651	−	0.740867i	$-0.265585\pi$
$751$	27646.1	1.34330	0.671651	+	0.740867i	$0.265585\pi$
$752$	0	0
$753$	20413.9	0.987946
$754$	0	0
$755$	−12579.5	−0.606379
$756$	0	0
$757$	32415.6	1.55636	0.778181	−	0.628040i	$-0.216143\pi$
$757$	32415.6	1.55636	0.778181	+	0.628040i	$0.216143\pi$
$758$	0	0
$759$	9698.71	0.463822
$760$	0	0
$761$	20254.3	0.964806	0.482403	−	0.875949i	$-0.339764\pi$
$761$	20254.3	0.964806	0.482403	+	0.875949i	$0.339764\pi$
$762$	0	0
$763$	−470.806	−0.0223385
$764$	0	0
$765$	−28271.0	−1.33613
$766$	0	0
$767$	24024.6	1.13100
$768$	0	0
$769$	−14158.3	−0.663928	−0.331964	−	0.943292i	$-0.607711\pi$
$769$	−14158.3	−0.663928	−0.331964	+	0.943292i	$0.607711\pi$
$770$	0	0
$771$	42706.7	1.99487
$772$	0	0
$773$	2423.96	0.112786	0.0563931	−	0.998409i	$-0.482040\pi$
$773$	2423.96	0.112786	0.0563931	+	0.998409i	$0.482040\pi$
$774$	0	0
$775$	5793.68	0.268536
$776$	0	0
$777$	−26576.0	−1.22704
$778$	0	0
$779$	−41468.6	−1.90727
$780$	0	0
$781$	−24391.0	−1.11751
$782$	0	0
$783$	−5745.68	−0.262240
$784$	0	0
$785$	4345.78	0.197589
$786$	0	0
$787$	9820.85	0.444823	0.222411	−	0.974953i	$-0.428607\pi$
$787$	9820.85	0.444823	0.222411	+	0.974953i	$0.428607\pi$
$788$	0	0
$789$	−40451.6	−1.82524
$790$	0	0
$791$	21490.8	0.966024
$792$	0	0
$793$	13214.1	0.591735
$794$	0	0
$795$	−17509.9	−0.781148
$796$	0	0
$797$	4153.45	0.184596	0.0922978	−	0.995731i	$-0.470579\pi$
$797$	4153.45	0.184596	0.0922978	+	0.995731i	$0.470579\pi$
$798$	0	0
$799$	−56350.5	−2.49504
$800$	0	0
$801$	73620.8	3.24752
$802$	0	0
$803$	−8294.49	−0.364516
$804$	0	0
$805$	3295.26	0.144276
$806$	0	0
$807$	15036.6	0.655902
$808$	0	0
$809$	6883.12	0.299132	0.149566	−	0.988752i	$-0.452212\pi$
$809$	6883.12	0.299132	0.149566	+	0.988752i	$0.452212\pi$
$810$	0	0
$811$	22778.1	0.986249	0.493125	−	0.869959i	$-0.335855\pi$
$811$	22778.1	0.986249	0.493125	+	0.869959i	$0.335855\pi$
$812$	0	0
$813$	−10245.3	−0.441965
$814$	0	0
$815$	−19325.5	−0.830603
$816$	0	0
$817$	43249.5	1.85203
$818$	0	0
$819$	40069.3	1.70956
$820$	0	0
$821$	9157.92	0.389298	0.194649	−	0.980873i	$-0.437643\pi$
$821$	9157.92	0.389298	0.194649	+	0.980873i	$0.437643\pi$
$822$	0	0
$823$	−33128.1	−1.40313	−0.701564	−	0.712607i	$-0.747514\pi$
$823$	−33128.1	−1.40313	−0.701564	+	0.712607i	$0.747514\pi$
$824$	0	0
$825$	−7439.58	−0.313955
$826$	0	0
$827$	−33683.5	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$827$	−33683.5	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$828$	0	0
$829$	−22625.9	−0.947926	−0.473963	−	0.880545i	$-0.657177\pi$
$829$	−22625.9	−0.947926	−0.473963	+	0.880545i	$0.657177\pi$
$830$	0	0
$831$	8156.05	0.340470
$832$	0	0
$833$	7508.45	0.312308
$834$	0	0
$835$	−17037.7	−0.706125
$836$	0	0
$837$	−45915.4	−1.89614
$838$	0	0
$839$	−32784.1	−1.34903	−0.674513	−	0.738263i	$-0.735646\pi$
$839$	−32784.1	−1.34903	−0.674513	+	0.738263i	$0.735646\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−20533.7	−0.838930
$844$	0	0
$845$	3015.53	0.122766
$846$	0	0
$847$	−3547.11	−0.143896
$848$	0	0
$849$	38601.3	1.56041
$850$	0	0
$851$	−4892.99	−0.197097
$852$	0	0
$853$	−30236.5	−1.21369	−0.606846	−	0.794820i	$-0.707565\pi$
$853$	−30236.5	−1.21369	−0.606846	+	0.794820i	$0.707565\pi$
$854$	0	0
$855$	−22022.0	−0.880860
$856$	0	0
$857$	−35600.8	−1.41902	−0.709510	−	0.704695i	$-0.751084\pi$
$857$	−35600.8	−1.41902	−0.709510	+	0.704695i	$0.751084\pi$
$858$	0	0
$859$	−41550.4	−1.65038	−0.825192	−	0.564852i	$-0.808933\pi$
$859$	−41550.4	−1.65038	−0.825192	+	0.564852i	$0.808933\pi$
$860$	0	0
$861$	82722.1	3.27429
$862$	0	0
$863$	−26752.2	−1.05522	−0.527609	−	0.849487i	$-0.676912\pi$
$863$	−26752.2	−1.05522	−0.527609	+	0.849487i	$0.676912\pi$
$864$	0	0
$865$	3981.22	0.156492
$866$	0	0
$867$	−70600.6	−2.76554
$868$	0	0
$869$	−45237.6	−1.76591
$870$	0	0
$871$	32834.6	1.27733
$872$	0	0
$873$	−66595.9	−2.58182
$874$	0	0
$875$	−2527.69	−0.0976588
$876$	0	0
$877$	40055.1	1.54226	0.771131	−	0.636676i	$-0.219691\pi$
$877$	40055.1	1.54226	0.771131	+	0.636676i	$0.219691\pi$
$878$	0	0
$879$	−50091.2	−1.92211
$880$	0	0
$881$	−6192.02	−0.236793	−0.118396	−	0.992966i	$-0.537775\pi$
$881$	−6192.02	−0.236793	−0.118396	+	0.992966i	$0.537775\pi$
$882$	0	0
$883$	49830.2	1.89912	0.949559	−	0.313589i	$-0.101531\pi$
$883$	49830.2	1.89912	0.949559	+	0.313589i	$0.101531\pi$
$884$	0	0
$885$	26339.2	1.00043
$886$	0	0
$887$	4553.13	0.172355	0.0861776	−	0.996280i	$-0.472535\pi$
$887$	4553.13	0.172355	0.0861776	+	0.996280i	$0.472535\pi$
$888$	0	0
$889$	−7472.33	−0.281905
$890$	0	0
$891$	13404.6	0.504007
$892$	0	0
$893$	−43894.8	−1.64489
$894$	0	0
$895$	4793.24	0.179017
$896$	0	0
$897$	11390.5	0.423988
$898$	0	0
$899$	6720.67	0.249329
$900$	0	0
$901$	−45573.2	−1.68509
$902$	0	0
$903$	−86274.8	−3.17945
$904$	0	0
$905$	23397.6	0.859408
$906$	0	0
$907$	24108.9	0.882606	0.441303	−	0.897358i	$-0.354516\pi$
$907$	24108.9	0.882606	0.441303	+	0.897358i	$0.354516\pi$
$908$	0	0
$909$	1683.53	0.0614293
$910$	0	0
$911$	18934.6	0.688619	0.344309	−	0.938856i	$-0.388113\pi$
$911$	18934.6	0.688619	0.344309	+	0.938856i	$0.388113\pi$
$912$	0	0
$913$	25885.7	0.938327
$914$	0	0
$915$	14487.2	0.523423
$916$	0	0
$917$	−35867.9	−1.29167
$918$	0	0
$919$	14881.3	0.534155	0.267078	−	0.963675i	$-0.413942\pi$
$919$	14881.3	0.534155	0.267078	+	0.963675i	$0.413942\pi$
$920$	0	0
$921$	74792.3	2.67588
$922$	0	0
$923$	−28645.6	−1.02154
$924$	0	0
$925$	3753.26	0.133412
$926$	0	0
$927$	4279.22	0.151616
$928$	0	0
$929$	21932.9	0.774592	0.387296	−	0.921955i	$-0.373409\pi$
$929$	21932.9	0.774592	0.387296	+	0.921955i	$0.373409\pi$
$930$	0	0
$931$	5848.78	0.205893
$932$	0	0
$933$	64604.3	2.26693
$934$	0	0
$935$	−19363.1	−0.677262
$936$	0	0
$937$	14132.3	0.492722	0.246361	−	0.969178i	$-0.420765\pi$
$937$	14132.3	0.492722	0.246361	+	0.969178i	$0.420765\pi$
$938$	0	0
$939$	45259.2	1.57292
$940$	0	0
$941$	19998.9	0.692821	0.346411	−	0.938083i	$-0.387400\pi$
$941$	19998.9	0.692821	0.346411	+	0.938083i	$0.387400\pi$
$942$	0	0
$943$	15230.2	0.525943
$944$	0	0
$945$	20032.1	0.689572
$946$	0	0
$947$	26226.6	0.899948	0.449974	−	0.893042i	$-0.351433\pi$
$947$	26226.6	0.899948	0.449974	+	0.893042i	$0.351433\pi$
$948$	0	0
$949$	−9741.32	−0.333210
$950$	0	0
$951$	46382.4	1.58155
$952$	0	0
$953$	12989.1	0.441510	0.220755	−	0.975329i	$-0.429148\pi$
$953$	12989.1	0.441510	0.220755	+	0.975329i	$0.429148\pi$
$954$	0	0
$955$	6718.61	0.227653
$956$	0	0
$957$	−8629.91	−0.291500
$958$	0	0
$959$	1179.13	0.0397039
$960$	0	0
$961$	23915.8	0.802787
$962$	0	0
$963$	81653.4	2.73234
$964$	0	0
$965$	−14479.3	−0.483011
$966$	0	0
$967$	−3837.04	−0.127602	−0.0638008	−	0.997963i	$-0.520322\pi$
$967$	−3837.04	−0.127602	−0.0638008	+	0.997963i	$0.520322\pi$
$968$	0	0
$969$	−88496.9	−2.93388
$970$	0	0
$971$	−9063.90	−0.299562	−0.149781	−	0.988719i	$-0.547857\pi$
$971$	−9063.90	−0.299562	−0.149781	+	0.988719i	$0.547857\pi$
$972$	0	0
$973$	14187.0	0.467434
$974$	0	0
$975$	−8737.29	−0.286992
$976$	0	0
$977$	5605.42	0.183555	0.0917775	−	0.995780i	$-0.470745\pi$
$977$	5605.42	0.183555	0.0917775	+	0.995780i	$0.470745\pi$
$978$	0	0
$979$	50423.6	1.64611
$980$	0	0
$981$	−1155.57	−0.0376091
$982$	0	0
$983$	−18475.9	−0.599480	−0.299740	−	0.954021i	$-0.596900\pi$
$983$	−18475.9	−0.599480	−0.299740	+	0.954021i	$0.596900\pi$
$984$	0	0
$985$	15756.8	0.509698
$986$	0	0
$987$	87561.9	2.82384
$988$	0	0
$989$	−15884.3	−0.510710
$990$	0	0
$991$	−41350.4	−1.32547	−0.662733	−	0.748855i	$-0.730604\pi$
$991$	−41350.4	−1.32547	−0.662733	+	0.748855i	$0.730604\pi$
$992$	0	0
$993$	−24155.8	−0.771964
$994$	0	0
$995$	−14613.8	−0.465618
$996$	0	0
$997$	−25636.3	−0.814354	−0.407177	−	0.913349i	$-0.633487\pi$
$997$	−25636.3	−0.814354	−0.407177	+	0.913349i	$0.633487\pi$
$998$	0	0
$999$	−29744.9	−0.942029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.h.1.1	✓	11
4.3	odd	2		2320.4.a.z.1.11		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.h.1.1	✓	11	1.1	even	1	trivial
2320.4.a.z.1.11		11	4.3	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:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q-8.75401 q^{3} -5.00000 q^{5} +20.2215 q^{7} +49.6327 q^{9} +33.9939 q^{11} +39.9236 q^{13} +43.7701 q^{15} +113.921 q^{17} +88.7397 q^{19} -177.019 q^{21} -32.5916 q^{23} +25.0000 q^{25} -198.127 q^{27} +29.0000 q^{29} +231.747 q^{31} -297.583 q^{33} -101.108 q^{35} +150.130 q^{37} -349.492 q^{39} -467.305 q^{41} +487.375 q^{43} -248.164 q^{45} -494.646 q^{47} +65.9094 q^{49} -997.264 q^{51} -400.043 q^{53} -169.970 q^{55} -776.829 q^{57} +601.764 q^{59} +330.984 q^{61} +1003.65 q^{63} -199.618 q^{65} +822.435 q^{67} +285.307 q^{69} -717.511 q^{71} -243.999 q^{73} -218.850 q^{75} +687.408 q^{77} -1330.75 q^{79} +394.323 q^{81} +761.481 q^{83} -569.604 q^{85} -253.866 q^{87} +1483.31 q^{89} +807.315 q^{91} -2028.72 q^{93} -443.699 q^{95} -1341.77 q^{97} +1687.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 8 q^{3} - 55 q^{5} + 16 q^{7} + 117 q^{9} + 26 q^{11} + 52 q^{13} - 40 q^{15} + 14 q^{17} + 146 q^{19} - 288 q^{21} - 80 q^{23} + 275 q^{25} + 224 q^{27} + 319 q^{29} - 38 q^{31} - 152 q^{33} - 80 q^{35}+ \cdots + 4690 q^{99}+O(q^{100}) \) 11 * q + 8 * q^3 - 55 * q^5 + 16 * q^7 + 117 * q^9 + 26 * q^11 + 52 * q^13 - 40 * q^15 + 14 * q^17 + 146 * q^19 - 288 * q^21 - 80 * q^23 + 275 * q^25 + 224 * q^27 + 319 * q^29 - 38 * q^31 - 152 * q^33 - 80 * q^35 + 376 * q^37 + 108 * q^39 - 146 * q^41 + 644 * q^43 - 585 * q^45 - 464 * q^47 + 393 * q^49 - 576 * q^51 - 188 * q^53 - 130 * q^55 + 1380 * q^57 + 672 * q^59 + 472 * q^61 + 628 * q^63 - 260 * q^65 + 1688 * q^67 + 138 * q^69 + 92 * q^71 + 1514 * q^73 + 200 * q^75 + 2884 * q^77 + 2130 * q^79 + 4067 * q^81 + 2584 * q^83 - 70 * q^85 + 232 * q^87 + 1606 * q^89 + 3208 * q^91 + 3540 * q^93 - 730 * q^95 + 3322 * q^97 + 4690 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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