LMFDB - Embedded newform 304.4.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [304,4,Mod(1,304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("304.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,5,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.9365806417$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3221.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 2 $ x^3 - x^2 - 9*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.66246$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.573746 q^{3} +5.92862 q^{5} +31.1522 q^{7} -26.6708 q^{9} +19.0761 q^{11} +27.3853 q^{13} -3.40152 q^{15} -27.5575 q^{17} -19.0000 q^{19} -17.8735 q^{21} +162.542 q^{23} -89.8514 q^{25} +30.7934 q^{27} -66.9561 q^{29} +127.424 q^{31} -10.9448 q^{33} +184.690 q^{35} +273.481 q^{37} -15.7122 q^{39} +228.663 q^{41} -198.111 q^{43} -158.121 q^{45} +573.403 q^{47} +627.461 q^{49} +15.8110 q^{51} +610.969 q^{53} +113.095 q^{55} +10.9012 q^{57} -202.690 q^{59} -774.773 q^{61} -830.855 q^{63} +162.357 q^{65} +589.121 q^{67} -93.2580 q^{69} -392.429 q^{71} +7.44178 q^{73} +51.5519 q^{75} +594.263 q^{77} -1010.77 q^{79} +702.444 q^{81} -209.297 q^{83} -163.378 q^{85} +38.4158 q^{87} +705.222 q^{89} +853.113 q^{91} -73.1093 q^{93} -112.644 q^{95} -94.8567 q^{97} -508.775 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{3} + 2 q^{5} + 35 q^{7} + 48 q^{9} + 28 q^{11} - 109 q^{13} + 228 q^{15} - 123 q^{17} - 57 q^{19} + 25 q^{21} + 193 q^{23} + 187 q^{25} + 719 q^{27} - 297 q^{29} + 140 q^{31} + 30 q^{33} + 246 q^{35}+ \cdots - 86 q^{99}+O(q^{100}) $ 3 * q + 5 * q^3 + 2 * q^5 + 35 * q^7 + 48 * q^9 + 28 * q^11 - 109 * q^13 + 228 * q^15 - 123 * q^17 - 57 * q^19 + 25 * q^21 + 193 * q^23 + 187 * q^25 + 719 * q^27 - 297 * q^29 + 140 * q^31 + 30 * q^33 + 246 * q^35 + 38 * q^37 - 57 * q^39 + 736 * q^41 + 514 * q^43 + 1046 * q^45 - 134 * q^47 - 42 * q^49 - 929 * q^51 + 311 * q^53 + 130 * q^55 - 95 * q^57 - 199 * q^59 + 56 * q^61 - 508 * q^63 + 1156 * q^65 + 509 * q^67 + 621 * q^69 + 874 * q^71 + 203 * q^73 - 129 * q^75 + 616 * q^77 - 242 * q^79 + 3531 * q^81 + 62 * q^83 - 1430 * q^85 - 3237 * q^87 + 1764 * q^89 + 687 * q^91 - 2592 * q^93 - 38 * q^95 - 2178 * q^97 - 86 * 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
$3$	−0.573746	−0.110417	−0.0552087	−	0.998475i	$-0.517582\pi$
$3$	−0.573746	−0.110417	−0.0552087	+	0.998475i	$0.517582\pi$
$4$	0	0
$5$	5.92862	0.530272	0.265136	−	0.964211i	$-0.414583\pi$
$5$	5.92862	0.530272	0.265136	+	0.964211i	$0.414583\pi$
$6$	0	0
$7$	31.1522	1.68206	0.841031	−	0.540987i	$-0.181949\pi$
$7$	31.1522	1.68206	0.841031	+	0.540987i	$0.181949\pi$
$8$	0	0
$9$	−26.6708	−0.987808
$10$	0	0
$11$	19.0761	0.522879	0.261439	−	0.965220i	$-0.415803\pi$
$11$	19.0761	0.522879	0.261439	+	0.965220i	$0.415803\pi$
$12$	0	0
$13$	27.3853	0.584255	0.292128	−	0.956379i	$-0.405637\pi$
$13$	27.3853	0.584255	0.292128	+	0.956379i	$0.405637\pi$
$14$	0	0
$15$	−3.40152	−0.0585513
$16$	0	0
$17$	−27.5575	−0.393158	−0.196579	−	0.980488i	$-0.562983\pi$
$17$	−27.5575	−0.393158	−0.196579	+	0.980488i	$0.562983\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−17.8735	−0.185729
$22$	0	0
$23$	162.542	1.47358	0.736791	−	0.676120i	$-0.236340\pi$
$23$	162.542	1.47358	0.736791	+	0.676120i	$0.236340\pi$
$24$	0	0
$25$	−89.8514	−0.718812
$26$	0	0
$27$	30.7934	0.219489
$28$	0	0
$29$	−66.9561	−0.428739	−0.214369	−	0.976753i	$-0.568770\pi$
$29$	−66.9561	−0.428739	−0.214369	+	0.976753i	$0.568770\pi$
$30$	0	0
$31$	127.424	0.738262	0.369131	−	0.929377i	$-0.379655\pi$
$31$	127.424	0.738262	0.369131	+	0.929377i	$0.379655\pi$
$32$	0	0
$33$	−10.9448	−0.0577349
$34$	0	0
$35$	184.690	0.891950
$36$	0	0
$37$	273.481	1.21514	0.607568	−	0.794267i	$-0.292145\pi$
$37$	273.481	1.21514	0.607568	+	0.794267i	$0.292145\pi$
$38$	0	0
$39$	−15.7122	−0.0645120
$40$	0	0
$41$	228.663	0.871005	0.435503	−	0.900187i	$-0.356571\pi$
$41$	228.663	0.871005	0.435503	+	0.900187i	$0.356571\pi$
$42$	0	0
$43$	−198.111	−0.702597	−0.351298	−	0.936264i	$-0.614260\pi$
$43$	−198.111	−0.702597	−0.351298	+	0.936264i	$0.614260\pi$
$44$	0	0
$45$	−158.121	−0.523807
$46$	0	0
$47$	573.403	1.77956	0.889782	−	0.456387i	$-0.150857\pi$
$47$	573.403	1.77956	0.889782	+	0.456387i	$0.150857\pi$
$48$	0	0
$49$	627.461	1.82933
$50$	0	0
$51$	15.8110	0.0434115
$52$	0	0
$53$	610.969	1.58345	0.791727	−	0.610875i	$-0.209182\pi$
$53$	610.969	1.58345	0.791727	+	0.610875i	$0.209182\pi$
$54$	0	0
$55$	113.095	0.277268
$56$	0	0
$57$	10.9012	0.0253315
$58$	0	0
$59$	−202.690	−0.447253	−0.223627	−	0.974675i	$-0.571790\pi$
$59$	−202.690	−0.447253	−0.223627	+	0.974675i	$0.571790\pi$
$60$	0	0
$61$	−774.773	−1.62622	−0.813111	−	0.582108i	$-0.802228\pi$
$61$	−774.773	−1.62622	−0.813111	+	0.582108i	$0.802228\pi$
$62$	0	0
$63$	−830.855	−1.66155
$64$	0	0
$65$	162.357	0.309814
$66$	0	0
$67$	589.121	1.07422	0.537109	−	0.843513i	$-0.319516\pi$
$67$	589.121	1.07422	0.537109	+	0.843513i	$0.319516\pi$
$68$	0	0
$69$	−93.2580	−0.162709
$70$	0	0
$71$	−392.429	−0.655955	−0.327978	−	0.944686i	$-0.606367\pi$
$71$	−392.429	−0.655955	−0.327978	+	0.944686i	$0.606367\pi$
$72$	0	0
$73$	7.44178	0.0119314	0.00596571	−	0.999982i	$-0.498101\pi$
$73$	7.44178	0.0119314	0.00596571	+	0.999982i	$0.498101\pi$
$74$	0	0
$75$	51.5519	0.0793694
$76$	0	0
$77$	594.263	0.879514
$78$	0	0
$79$	−1010.77	−1.43950	−0.719751	−	0.694232i	$-0.755744\pi$
$79$	−1010.77	−1.43950	−0.719751	+	0.694232i	$0.755744\pi$
$80$	0	0
$81$	702.444	0.963573
$82$	0	0
$83$	−209.297	−0.276787	−0.138394	−	0.990377i	$-0.544194\pi$
$83$	−209.297	−0.276787	−0.138394	+	0.990377i	$0.544194\pi$
$84$	0	0
$85$	−163.378	−0.208481
$86$	0	0
$87$	38.4158	0.0473403
$88$	0	0
$89$	705.222	0.839926	0.419963	−	0.907541i	$-0.362043\pi$
$89$	705.222	0.839926	0.419963	+	0.907541i	$0.362043\pi$
$90$	0	0
$91$	853.113	0.982753
$92$	0	0
$93$	−73.1093	−0.0815170
$94$	0	0
$95$	−112.644	−0.121653
$96$	0	0
$97$	−94.8567	−0.0992912	−0.0496456	−	0.998767i	$-0.515809\pi$
$97$	−94.8567	−0.0992912	−0.0496456	+	0.998767i	$0.515809\pi$
$98$	0	0
$99$	−508.775	−0.516504
$100$	0	0
$101$	−531.493	−0.523619	−0.261810	−	0.965120i	$-0.584319\pi$
$101$	−531.493	−0.523619	−0.261810	+	0.965120i	$0.584319\pi$
$102$	0	0
$103$	1420.46	1.35885	0.679425	−	0.733745i	$-0.262229\pi$
$103$	1420.46	1.35885	0.679425	+	0.733745i	$0.262229\pi$
$104$	0	0
$105$	−105.965	−0.0984869
$106$	0	0
$107$	−514.719	−0.465044	−0.232522	−	0.972591i	$-0.574698\pi$
$107$	−514.719	−0.465044	−0.232522	+	0.972591i	$0.574698\pi$
$108$	0	0
$109$	−1882.00	−1.65379	−0.826893	−	0.562359i	$-0.809894\pi$
$109$	−1882.00	−1.65379	−0.826893	+	0.562359i	$0.809894\pi$
$110$	0	0
$111$	−156.909	−0.134172
$112$	0	0
$113$	−40.1054	−0.0333876	−0.0166938	−	0.999861i	$-0.505314\pi$
$113$	−40.1054	−0.0333876	−0.0166938	+	0.999861i	$0.505314\pi$
$114$	0	0
$115$	963.652	0.781400
$116$	0	0
$117$	−730.388	−0.577132
$118$	0	0
$119$	−858.478	−0.661316
$120$	0	0
$121$	−967.102	−0.726598
$122$	0	0
$123$	−131.195	−0.0961742
$124$	0	0
$125$	−1273.77	−0.911438
$126$	0	0
$127$	−7.35102	−0.00513620	−0.00256810	−	0.999997i	$-0.500817\pi$
$127$	−7.35102	−0.00513620	−0.00256810	+	0.999997i	$0.500817\pi$
$128$	0	0
$129$	113.665	0.0775790
$130$	0	0
$131$	−1476.80	−0.984951	−0.492476	−	0.870326i	$-0.663908\pi$
$131$	−1476.80	−0.984951	−0.492476	+	0.870326i	$0.663908\pi$
$132$	0	0
$133$	−591.892	−0.385892
$134$	0	0
$135$	182.562	0.116389
$136$	0	0
$137$	−847.892	−0.528761	−0.264381	−	0.964418i	$-0.585168\pi$
$137$	−847.892	−0.528761	−0.264381	+	0.964418i	$0.585168\pi$
$138$	0	0
$139$	−1543.72	−0.941988	−0.470994	−	0.882136i	$-0.656105\pi$
$139$	−1543.72	−0.941988	−0.470994	+	0.882136i	$0.656105\pi$
$140$	0	0
$141$	−328.988	−0.196495
$142$	0	0
$143$	522.405	0.305494
$144$	0	0
$145$	−396.957	−0.227348
$146$	0	0
$147$	−360.003	−0.201990
$148$	0	0
$149$	−1324.76	−0.728382	−0.364191	−	0.931324i	$-0.618654\pi$
$149$	−1324.76	−0.728382	−0.364191	+	0.931324i	$0.618654\pi$
$150$	0	0
$151$	1509.91	0.813740	0.406870	−	0.913486i	$-0.366620\pi$
$151$	1509.91	0.813740	0.406870	+	0.913486i	$0.366620\pi$
$152$	0	0
$153$	734.982	0.388364
$154$	0	0
$155$	755.452	0.391480
$156$	0	0
$157$	−1441.00	−0.732512	−0.366256	−	0.930514i	$-0.619360\pi$
$157$	−1441.00	−0.732512	−0.366256	+	0.930514i	$0.619360\pi$
$158$	0	0
$159$	−350.541	−0.174841
$160$	0	0
$161$	5063.55	2.47866
$162$	0	0
$163$	790.260	0.379742	0.189871	−	0.981809i	$-0.439193\pi$
$163$	790.260	0.379742	0.189871	+	0.981809i	$0.439193\pi$
$164$	0	0
$165$	−64.8878	−0.0306152
$166$	0	0
$167$	3422.87	1.58605	0.793023	−	0.609192i	$-0.208506\pi$
$167$	3422.87	1.58605	0.793023	+	0.609192i	$0.208506\pi$
$168$	0	0
$169$	−1447.05	−0.658646
$170$	0	0
$171$	506.745	0.226619
$172$	0	0
$173$	−3925.10	−1.72497	−0.862485	−	0.506083i	$-0.831093\pi$
$173$	−3925.10	−1.72497	−0.862485	+	0.506083i	$0.831093\pi$
$174$	0	0
$175$	−2799.07	−1.20909
$176$	0	0
$177$	116.292	0.0493846
$178$	0	0
$179$	−2976.77	−1.24298	−0.621492	−	0.783421i	$-0.713473\pi$
$179$	−2976.77	−1.24298	−0.621492	+	0.783421i	$0.713473\pi$
$180$	0	0
$181$	1316.60	0.540674	0.270337	−	0.962766i	$-0.412865\pi$
$181$	1316.60	0.540674	0.270337	+	0.962766i	$0.412865\pi$
$182$	0	0
$183$	444.523	0.179563
$184$	0	0
$185$	1621.37	0.644353
$186$	0	0
$187$	−525.690	−0.205574
$188$	0	0
$189$	959.283	0.369194
$190$	0	0
$191$	861.264	0.326277	0.163138	−	0.986603i	$-0.447838\pi$
$191$	861.264	0.326277	0.163138	+	0.986603i	$0.447838\pi$
$192$	0	0
$193$	−1201.95	−0.448282	−0.224141	−	0.974557i	$-0.571958\pi$
$193$	−1201.95	−0.448282	−0.224141	+	0.974557i	$0.571958\pi$
$194$	0	0
$195$	−93.1517	−0.0342089
$196$	0	0
$197$	1143.30	0.413485	0.206742	−	0.978395i	$-0.433714\pi$
$197$	1143.30	0.413485	0.206742	+	0.978395i	$0.433714\pi$
$198$	0	0
$199$	−3776.93	−1.34542	−0.672711	−	0.739905i	$-0.734870\pi$
$199$	−3776.93	−1.34542	−0.672711	+	0.739905i	$0.734870\pi$
$200$	0	0
$201$	−338.006	−0.118612
$202$	0	0
$203$	−2085.83	−0.721166
$204$	0	0
$205$	1355.66	0.461870
$206$	0	0
$207$	−4335.13	−1.45562
$208$	0	0
$209$	−362.446	−0.119957
$210$	0	0
$211$	2328.89	0.759844	0.379922	−	0.925019i	$-0.375951\pi$
$211$	2328.89	0.759844	0.379922	+	0.925019i	$0.375951\pi$
$212$	0	0
$213$	225.155	0.0724289
$214$	0	0
$215$	−1174.53	−0.372568
$216$	0	0
$217$	3969.56	1.24180
$218$	0	0
$219$	−4.26969	−0.00131744
$220$	0	0
$221$	−754.671	−0.229704
$222$	0	0
$223$	3393.00	1.01889	0.509445	−	0.860504i	$-0.329851\pi$
$223$	3393.00	1.01889	0.509445	+	0.860504i	$0.329851\pi$
$224$	0	0
$225$	2396.41	0.710048
$226$	0	0
$227$	−55.0932	−0.0161087	−0.00805433	−	0.999968i	$-0.502564\pi$
$227$	−55.0932	−0.0161087	−0.00805433	+	0.999968i	$0.502564\pi$
$228$	0	0
$229$	−6838.40	−1.97334	−0.986669	−	0.162739i	$-0.947967\pi$
$229$	−6838.40	−1.97334	−0.986669	+	0.162739i	$0.947967\pi$
$230$	0	0
$231$	−340.956	−0.0971137
$232$	0	0
$233$	5813.50	1.63457	0.817285	−	0.576234i	$-0.195478\pi$
$233$	5813.50	1.63457	0.817285	+	0.576234i	$0.195478\pi$
$234$	0	0
$235$	3399.49	0.943653
$236$	0	0
$237$	579.926	0.158946
$238$	0	0
$239$	85.3888	0.0231102	0.0115551	−	0.999933i	$-0.496322\pi$
$239$	85.3888	0.0231102	0.0115551	+	0.999933i	$0.496322\pi$
$240$	0	0
$241$	−193.112	−0.0516159	−0.0258079	−	0.999667i	$-0.508216\pi$
$241$	−193.112	−0.0516159	−0.0258079	+	0.999667i	$0.508216\pi$
$242$	0	0
$243$	−1234.45	−0.325884
$244$	0	0
$245$	3719.98	0.970044
$246$	0	0
$247$	−520.321	−0.134037
$248$	0	0
$249$	120.083	0.0305621
$250$	0	0
$251$	2272.68	0.571516	0.285758	−	0.958302i	$-0.407755\pi$
$251$	2272.68	0.571516	0.285758	+	0.958302i	$0.407755\pi$
$252$	0	0
$253$	3100.67	0.770505
$254$	0	0
$255$	93.7375	0.0230199
$256$	0	0
$257$	4319.19	1.04834	0.524170	−	0.851614i	$-0.324376\pi$
$257$	4319.19	1.04834	0.524170	+	0.851614i	$0.324376\pi$
$258$	0	0
$259$	8519.55	2.04394
$260$	0	0
$261$	1785.77	0.423512
$262$	0	0
$263$	684.056	0.160383	0.0801915	−	0.996779i	$-0.474447\pi$
$263$	684.056	0.160383	0.0801915	+	0.996779i	$0.474447\pi$
$264$	0	0
$265$	3622.21	0.839662
$266$	0	0
$267$	−404.618	−0.0927425
$268$	0	0
$269$	8416.86	1.90775	0.953876	−	0.300201i	$-0.0970539\pi$
$269$	8416.86	1.90775	0.953876	+	0.300201i	$0.0970539\pi$
$270$	0	0
$271$	−1723.62	−0.386356	−0.193178	−	0.981164i	$-0.561880\pi$
$271$	−1723.62	−0.386356	−0.193178	+	0.981164i	$0.561880\pi$
$272$	0	0
$273$	−489.470	−0.108513
$274$	0	0
$275$	−1714.02	−0.375851
$276$	0	0
$277$	−4546.96	−0.986284	−0.493142	−	0.869949i	$-0.664152\pi$
$277$	−4546.96	−0.986284	−0.493142	+	0.869949i	$0.664152\pi$
$278$	0	0
$279$	−3398.52	−0.729261
$280$	0	0
$281$	−5850.05	−1.24194	−0.620969	−	0.783835i	$-0.713261\pi$
$281$	−5850.05	−1.24194	−0.620969	+	0.783835i	$0.713261\pi$
$282$	0	0
$283$	5975.96	1.25524	0.627622	−	0.778518i	$-0.284028\pi$
$283$	5975.96	1.25524	0.627622	+	0.778518i	$0.284028\pi$
$284$	0	0
$285$	64.6289	0.0134326
$286$	0	0
$287$	7123.37	1.46509
$288$	0	0
$289$	−4153.58	−0.845427
$290$	0	0
$291$	54.4237	0.0109635
$292$	0	0
$293$	−4618.01	−0.920775	−0.460387	−	0.887718i	$-0.652289\pi$
$293$	−4618.01	−0.920775	−0.460387	+	0.887718i	$0.652289\pi$
$294$	0	0
$295$	−1201.67	−0.237166
$296$	0	0
$297$	587.419	0.114766
$298$	0	0
$299$	4451.27	0.860948
$300$	0	0
$301$	−6171.60	−1.18181
$302$	0	0
$303$	304.942	0.0578167
$304$	0	0
$305$	−4593.34	−0.862340
$306$	0	0
$307$	−3550.30	−0.660020	−0.330010	−	0.943977i	$-0.607052\pi$
$307$	−3550.30	−0.660020	−0.330010	+	0.943977i	$0.607052\pi$
$308$	0	0
$309$	−814.980	−0.150041
$310$	0	0
$311$	7089.01	1.29254	0.646272	−	0.763107i	$-0.276327\pi$
$311$	7089.01	1.29254	0.646272	+	0.763107i	$0.276327\pi$
$312$	0	0
$313$	−4505.14	−0.813564	−0.406782	−	0.913525i	$-0.633349\pi$
$313$	−4505.14	−0.813564	−0.406782	+	0.913525i	$0.633349\pi$
$314$	0	0
$315$	−4925.83	−0.881076
$316$	0	0
$317$	4235.04	0.750358	0.375179	−	0.926952i	$-0.377581\pi$
$317$	4235.04	0.750358	0.375179	+	0.926952i	$0.377581\pi$
$318$	0	0
$319$	−1277.26	−0.224178
$320$	0	0
$321$	295.318	0.0513490
$322$	0	0
$323$	523.593	0.0901966
$324$	0	0
$325$	−2460.61	−0.419969
$326$	0	0
$327$	1079.79	0.182607
$328$	0	0
$329$	17862.8	2.99334
$330$	0	0
$331$	−6789.23	−1.12740	−0.563701	−	0.825979i	$-0.690623\pi$
$331$	−6789.23	−1.12740	−0.563701	+	0.825979i	$0.690623\pi$
$332$	0	0
$333$	−7293.97	−1.20032
$334$	0	0
$335$	3492.68	0.569628
$336$	0	0
$337$	−10568.2	−1.70827	−0.854134	−	0.520053i	$-0.825912\pi$
$337$	−10568.2	−1.70827	−0.854134	+	0.520053i	$0.825912\pi$
$338$	0	0
$339$	23.0103	0.00368658
$340$	0	0
$341$	2430.76	0.386021
$342$	0	0
$343$	8861.60	1.39499
$344$	0	0
$345$	−552.891	−0.0862802
$346$	0	0
$347$	−7243.57	−1.12062	−0.560310	−	0.828283i	$-0.689318\pi$
$347$	−7243.57	−1.12062	−0.560310	+	0.828283i	$0.689318\pi$
$348$	0	0
$349$	5396.79	0.827746	0.413873	−	0.910335i	$-0.364176\pi$
$349$	5396.79	0.827746	0.413873	+	0.910335i	$0.364176\pi$
$350$	0	0
$351$	843.287	0.128237
$352$	0	0
$353$	5749.24	0.866859	0.433429	−	0.901188i	$-0.357303\pi$
$353$	5749.24	0.866859	0.433429	+	0.901188i	$0.357303\pi$
$354$	0	0
$355$	−2326.57	−0.347835
$356$	0	0
$357$	492.548	0.0730208
$358$	0	0
$359$	7397.70	1.08756	0.543782	−	0.839226i	$-0.316992\pi$
$359$	7397.70	1.08756	0.543782	+	0.839226i	$0.316992\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	554.871	0.0802291
$364$	0	0
$365$	44.1195	0.00632690
$366$	0	0
$367$	−5354.32	−0.761562	−0.380781	−	0.924665i	$-0.624345\pi$
$367$	−5354.32	−0.761562	−0.380781	+	0.924665i	$0.624345\pi$
$368$	0	0
$369$	−6098.64	−0.860386
$370$	0	0
$371$	19033.1	2.66347
$372$	0	0
$373$	−10159.4	−1.41028	−0.705140	−	0.709068i	$-0.749116\pi$
$373$	−10159.4	−1.41028	−0.705140	+	0.709068i	$0.749116\pi$
$374$	0	0
$375$	730.822	0.100639
$376$	0	0
$377$	−1833.61	−0.250493
$378$	0	0
$379$	−11073.5	−1.50081	−0.750403	−	0.660980i	$-0.770141\pi$
$379$	−11073.5	−1.50081	−0.750403	+	0.660980i	$0.770141\pi$
$380$	0	0
$381$	4.21762	0.000567126 0
$382$	0	0
$383$	−5530.71	−0.737875	−0.368937	−	0.929454i	$-0.620278\pi$
$383$	−5530.71	−0.737875	−0.368937	+	0.929454i	$0.620278\pi$
$384$	0	0
$385$	3523.16	0.466382
$386$	0	0
$387$	5283.79	0.694031
$388$	0	0
$389$	−3375.27	−0.439931	−0.219965	−	0.975508i	$-0.570594\pi$
$389$	−3375.27	−0.439931	−0.219965	+	0.975508i	$0.570594\pi$
$390$	0	0
$391$	−4479.26	−0.579350
$392$	0	0
$393$	847.308	0.108756
$394$	0	0
$395$	−5992.48	−0.763328
$396$	0	0
$397$	5697.65	0.720295	0.360147	−	0.932895i	$-0.382726\pi$
$397$	5697.65	0.720295	0.360147	+	0.932895i	$0.382726\pi$
$398$	0	0
$399$	339.596	0.0426092
$400$	0	0
$401$	−5261.73	−0.655258	−0.327629	−	0.944807i	$-0.606250\pi$
$401$	−5261.73	−0.655258	−0.327629	+	0.944807i	$0.606250\pi$
$402$	0	0
$403$	3489.56	0.431333
$404$	0	0
$405$	4164.53	0.510956
$406$	0	0
$407$	5216.96	0.635369
$408$	0	0
$409$	5749.47	0.695093	0.347546	−	0.937663i	$-0.387015\pi$
$409$	5749.47	0.695093	0.347546	+	0.937663i	$0.387015\pi$
$410$	0	0
$411$	486.475	0.0583845
$412$	0	0
$413$	−6314.23	−0.752308
$414$	0	0
$415$	−1240.84	−0.146773
$416$	0	0
$417$	885.702	0.104012
$418$	0	0
$419$	−14185.2	−1.65392	−0.826962	−	0.562257i	$-0.809933\pi$
$419$	−14185.2	−1.65392	−0.826962	+	0.562257i	$0.809933\pi$
$420$	0	0
$421$	14324.4	1.65826	0.829129	−	0.559057i	$-0.188837\pi$
$421$	14324.4	1.65826	0.829129	+	0.559057i	$0.188837\pi$
$422$	0	0
$423$	−15293.1	−1.75787
$424$	0	0
$425$	2476.08	0.282606
$426$	0	0
$427$	−24135.9	−2.73541
$428$	0	0
$429$	−299.728	−0.0337319
$430$	0	0
$431$	−16299.3	−1.82160	−0.910801	−	0.412845i	$-0.864535\pi$
$431$	−16299.3	−1.82160	−0.910801	+	0.412845i	$0.864535\pi$
$432$	0	0
$433$	−17412.0	−1.93248	−0.966242	−	0.257637i	$-0.917056\pi$
$433$	−17412.0	−1.93248	−0.966242	+	0.257637i	$0.917056\pi$
$434$	0	0
$435$	227.753	0.0251032
$436$	0	0
$437$	−3088.30	−0.338063
$438$	0	0
$439$	−3725.94	−0.405078	−0.202539	−	0.979274i	$-0.564919\pi$
$439$	−3725.94	−0.405078	−0.202539	+	0.979274i	$0.564919\pi$
$440$	0	0
$441$	−16734.9	−1.80703
$442$	0	0
$443$	1037.83	0.111307	0.0556534	−	0.998450i	$-0.482276\pi$
$443$	1037.83	0.111307	0.0556534	+	0.998450i	$0.482276\pi$
$444$	0	0
$445$	4181.00	0.445389
$446$	0	0
$447$	760.078	0.0804260
$448$	0	0
$449$	17335.5	1.82208	0.911038	−	0.412322i	$-0.135282\pi$
$449$	17335.5	1.82208	0.911038	+	0.412322i	$0.135282\pi$
$450$	0	0
$451$	4362.01	0.455430
$452$	0	0
$453$	−866.305	−0.0898511
$454$	0	0
$455$	5057.78	0.521127
$456$	0	0
$457$	−13910.6	−1.42387	−0.711936	−	0.702244i	$-0.752182\pi$
$457$	−13910.6	−1.42387	−0.711936	+	0.702244i	$0.752182\pi$
$458$	0	0
$459$	−848.590	−0.0862937
$460$	0	0
$461$	3352.50	0.338701	0.169351	−	0.985556i	$-0.445833\pi$
$461$	3352.50	0.338701	0.169351	+	0.985556i	$0.445833\pi$
$462$	0	0
$463$	−6267.27	−0.629081	−0.314541	−	0.949244i	$-0.601850\pi$
$463$	−6267.27	−0.629081	−0.314541	+	0.949244i	$0.601850\pi$
$464$	0	0
$465$	−433.437	−0.0432262
$466$	0	0
$467$	13761.7	1.36363	0.681815	−	0.731525i	$-0.261191\pi$
$467$	13761.7	1.36363	0.681815	+	0.731525i	$0.261191\pi$
$468$	0	0
$469$	18352.4	1.80690
$470$	0	0
$471$	826.768	0.0808821
$472$	0	0
$473$	−3779.19	−0.367373
$474$	0	0
$475$	1707.18	0.164907
$476$	0	0
$477$	−16295.1	−1.56415
$478$	0	0
$479$	−7667.87	−0.731427	−0.365714	−	0.930727i	$-0.619175\pi$
$479$	−7667.87	−0.731427	−0.365714	+	0.930727i	$0.619175\pi$
$480$	0	0
$481$	7489.37	0.709950
$482$	0	0
$483$	−2905.19	−0.273687
$484$	0	0
$485$	−562.370	−0.0526513
$486$	0	0
$487$	15252.9	1.41925	0.709626	−	0.704579i	$-0.248864\pi$
$487$	15252.9	1.41925	0.709626	+	0.704579i	$0.248864\pi$
$488$	0	0
$489$	−453.409	−0.0419302
$490$	0	0
$491$	5428.00	0.498905	0.249452	−	0.968387i	$-0.419749\pi$
$491$	5428.00	0.498905	0.249452	+	0.968387i	$0.419749\pi$
$492$	0	0
$493$	1845.14	0.168562
$494$	0	0
$495$	−3016.34	−0.273887
$496$	0	0
$497$	−12225.1	−1.10336
$498$	0	0
$499$	−9828.19	−0.881704	−0.440852	−	0.897580i	$-0.645324\pi$
$499$	−9828.19	−0.881704	−0.440852	+	0.897580i	$0.645324\pi$
$500$	0	0
$501$	−1963.86	−0.175127
$502$	0	0
$503$	−9759.46	−0.865115	−0.432558	−	0.901606i	$-0.642389\pi$
$503$	−9759.46	−0.865115	−0.432558	+	0.901606i	$0.642389\pi$
$504$	0	0
$505$	−3151.02	−0.277661
$506$	0	0
$507$	830.236	0.0727260
$508$	0	0
$509$	8668.77	0.754885	0.377442	−	0.926033i	$-0.376804\pi$
$509$	8668.77	0.754885	0.377442	+	0.926033i	$0.376804\pi$
$510$	0	0
$511$	231.828	0.0200694
$512$	0	0
$513$	−585.075	−0.0503542
$514$	0	0
$515$	8421.34	0.720560
$516$	0	0
$517$	10938.3	0.930496
$518$	0	0
$519$	2252.01	0.190467
$520$	0	0
$521$	−10423.1	−0.876473	−0.438237	−	0.898860i	$-0.644397\pi$
$521$	−10423.1	−0.876473	−0.438237	+	0.898860i	$0.644397\pi$
$522$	0	0
$523$	−8623.79	−0.721017	−0.360509	−	0.932756i	$-0.617397\pi$
$523$	−8623.79	−0.721017	−0.360509	+	0.932756i	$0.617397\pi$
$524$	0	0
$525$	1605.96	0.133504
$526$	0	0
$527$	−3511.50	−0.290253
$528$	0	0
$529$	14253.0	1.17145
$530$	0	0
$531$	5405.90	0.441800
$532$	0	0
$533$	6262.02	0.508889
$534$	0	0
$535$	−3051.57	−0.246600
$536$	0	0
$537$	1707.91	0.137247
$538$	0	0
$539$	11969.5	0.956519
$540$	0	0
$541$	10370.4	0.824140	0.412070	−	0.911152i	$-0.364806\pi$
$541$	10370.4	0.824140	0.412070	+	0.911152i	$0.364806\pi$
$542$	0	0
$543$	−755.393	−0.0596999
$544$	0	0
$545$	−11157.7	−0.876957
$546$	0	0
$547$	15332.3	1.19847	0.599235	−	0.800573i	$-0.295472\pi$
$547$	15332.3	1.19847	0.599235	+	0.800573i	$0.295472\pi$
$548$	0	0
$549$	20663.8	1.60640
$550$	0	0
$551$	1272.17	0.0983595
$552$	0	0
$553$	−31487.8	−2.42133
$554$	0	0
$555$	−930.253	−0.0711478
$556$	0	0
$557$	5182.38	0.394227	0.197114	−	0.980381i	$-0.436843\pi$
$557$	5182.38	0.394227	0.197114	+	0.980381i	$0.436843\pi$
$558$	0	0
$559$	−5425.33	−0.410496
$560$	0	0
$561$	301.613	0.0226989
$562$	0	0
$563$	18805.5	1.40774	0.703871	−	0.710328i	$-0.251453\pi$
$563$	18805.5	1.40774	0.703871	+	0.710328i	$0.251453\pi$
$564$	0	0
$565$	−237.770	−0.0177045
$566$	0	0
$567$	21882.7	1.62079
$568$	0	0
$569$	108.440	0.00798951	0.00399476	−	0.999992i	$-0.498728\pi$
$569$	108.440	0.00798951	0.00399476	+	0.999992i	$0.498728\pi$
$570$	0	0
$571$	18908.5	1.38581	0.692903	−	0.721031i	$-0.256331\pi$
$571$	18908.5	1.38581	0.692903	+	0.721031i	$0.256331\pi$
$572$	0	0
$573$	−494.147	−0.0360266
$574$	0	0
$575$	−14604.7	−1.05923
$576$	0	0
$577$	−1480.28	−0.106803	−0.0534013	−	0.998573i	$-0.517006\pi$
$577$	−1480.28	−0.106803	−0.0534013	+	0.998573i	$0.517006\pi$
$578$	0	0
$579$	689.615	0.0494981
$580$	0	0
$581$	−6520.07	−0.465573
$582$	0	0
$583$	11654.9	0.827955
$584$	0	0
$585$	−4330.20	−0.306037
$586$	0	0
$587$	−21653.8	−1.52257	−0.761284	−	0.648418i	$-0.775431\pi$
$587$	−21653.8	−1.52257	−0.761284	+	0.648418i	$0.775431\pi$
$588$	0	0
$589$	−2421.07	−0.169369
$590$	0	0
$591$	−655.962	−0.0456560
$592$	0	0
$593$	−25573.6	−1.77096	−0.885481	−	0.464676i	$-0.846171\pi$
$593$	−25573.6	−1.77096	−0.885481	+	0.464676i	$0.846171\pi$
$594$	0	0
$595$	−5089.59	−0.350677
$596$	0	0
$597$	2167.00	0.148558
$598$	0	0
$599$	19638.0	1.33955	0.669773	−	0.742566i	$-0.266392\pi$
$599$	19638.0	1.33955	0.669773	+	0.742566i	$0.266392\pi$
$600$	0	0
$601$	4867.13	0.330340	0.165170	−	0.986265i	$-0.447183\pi$
$601$	4867.13	0.330340	0.165170	+	0.986265i	$0.447183\pi$
$602$	0	0
$603$	−15712.3	−1.06112
$604$	0	0
$605$	−5733.58	−0.385295
$606$	0	0
$607$	12459.9	0.833166	0.416583	−	0.909098i	$-0.363228\pi$
$607$	12459.9	0.833166	0.416583	+	0.909098i	$0.363228\pi$
$608$	0	0
$609$	1196.74	0.0796293
$610$	0	0
$611$	15702.8	1.03972
$612$	0	0
$613$	−22281.6	−1.46810	−0.734051	−	0.679094i	$-0.762373\pi$
$613$	−22281.6	−1.46810	−0.734051	+	0.679094i	$0.762373\pi$
$614$	0	0
$615$	−777.804	−0.0509985
$616$	0	0
$617$	22452.8	1.46502	0.732508	−	0.680759i	$-0.238350\pi$
$617$	22452.8	1.46502	0.732508	+	0.680759i	$0.238350\pi$
$618$	0	0
$619$	−7314.16	−0.474929	−0.237464	−	0.971396i	$-0.576316\pi$
$619$	−7314.16	−0.474929	−0.237464	+	0.971396i	$0.576316\pi$
$620$	0	0
$621$	5005.23	0.323435
$622$	0	0
$623$	21969.2	1.41281
$624$	0	0
$625$	3679.71	0.235502
$626$	0	0
$627$	207.952	0.0132453
$628$	0	0
$629$	−7536.47	−0.477740
$630$	0	0
$631$	−21331.4	−1.34578	−0.672892	−	0.739741i	$-0.734948\pi$
$631$	−21331.4	−1.34578	−0.672892	+	0.739741i	$0.734948\pi$
$632$	0	0
$633$	−1336.19	−0.0839000
$634$	0	0
$635$	−43.5814	−0.00272358
$636$	0	0
$637$	17183.2	1.06880
$638$	0	0
$639$	10466.4	0.647958
$640$	0	0
$641$	−6385.58	−0.393472	−0.196736	−	0.980457i	$-0.563034\pi$
$641$	−6385.58	−0.393472	−0.196736	+	0.980457i	$0.563034\pi$
$642$	0	0
$643$	−6668.66	−0.408999	−0.204499	−	0.978867i	$-0.565557\pi$
$643$	−6668.66	−0.408999	−0.204499	+	0.978867i	$0.565557\pi$
$644$	0	0
$645$	673.880	0.0411380
$646$	0	0
$647$	32039.4	1.94683	0.973415	−	0.229050i	$-0.0735620\pi$
$647$	32039.4	1.94683	0.973415	+	0.229050i	$0.0735620\pi$
$648$	0	0
$649$	−3866.53	−0.233859
$650$	0	0
$651$	−2277.52	−0.137117
$652$	0	0
$653$	7067.12	0.423519	0.211759	−	0.977322i	$-0.432081\pi$
$653$	7067.12	0.423519	0.211759	+	0.977322i	$0.432081\pi$
$654$	0	0
$655$	−8755.39	−0.522292
$656$	0	0
$657$	−198.478	−0.0117860
$658$	0	0
$659$	−12722.0	−0.752013	−0.376007	−	0.926617i	$-0.622703\pi$
$659$	−12722.0	−0.752013	−0.376007	+	0.926617i	$0.622703\pi$
$660$	0	0
$661$	27275.7	1.60499	0.802497	−	0.596656i	$-0.203504\pi$
$661$	27275.7	1.60499	0.802497	+	0.596656i	$0.203504\pi$
$662$	0	0
$663$	432.989	0.0253634
$664$	0	0
$665$	−3509.11	−0.204627
$666$	0	0
$667$	−10883.2	−0.631782
$668$	0	0
$669$	−1946.72	−0.112503
$670$	0	0
$671$	−14779.7	−0.850317
$672$	0	0
$673$	32696.1	1.87273	0.936363	−	0.351035i	$-0.114170\pi$
$673$	32696.1	1.87273	0.936363	+	0.351035i	$0.114170\pi$
$674$	0	0
$675$	−2766.83	−0.157771
$676$	0	0
$677$	6789.23	0.385423	0.192711	−	0.981255i	$-0.438272\pi$
$677$	6789.23	0.385423	0.192711	+	0.981255i	$0.438272\pi$
$678$	0	0
$679$	−2955.00	−0.167014
$680$	0	0
$681$	31.6095	0.00177868
$682$	0	0
$683$	26262.7	1.47133	0.735663	−	0.677348i	$-0.236871\pi$
$683$	26262.7	1.47133	0.735663	+	0.677348i	$0.236871\pi$
$684$	0	0
$685$	−5026.83	−0.280387
$686$	0	0
$687$	3923.51	0.217891
$688$	0	0
$689$	16731.6	0.925141
$690$	0	0
$691$	26306.0	1.44823	0.724114	−	0.689680i	$-0.242249\pi$
$691$	26306.0	1.44823	0.724114	+	0.689680i	$0.242249\pi$
$692$	0	0
$693$	−15849.5	−0.868791
$694$	0	0
$695$	−9152.12	−0.499510
$696$	0	0
$697$	−6301.40	−0.342442
$698$	0	0
$699$	−3335.47	−0.180485
$700$	0	0
$701$	1698.09	0.0914922	0.0457461	−	0.998953i	$-0.485433\pi$
$701$	1698.09	0.0914922	0.0457461	+	0.998953i	$0.485433\pi$
$702$	0	0
$703$	−5196.14	−0.278771
$704$	0	0
$705$	−1950.44	−0.104196
$706$	0	0
$707$	−16557.2	−0.880760
$708$	0	0
$709$	22627.0	1.19856	0.599278	−	0.800541i	$-0.295454\pi$
$709$	22627.0	1.19856	0.599278	+	0.800541i	$0.295454\pi$
$710$	0	0
$711$	26958.1	1.42195
$712$	0	0
$713$	20711.9	1.08789
$714$	0	0
$715$	3097.14	0.161995
$716$	0	0
$717$	−48.9915	−0.00255177
$718$	0	0
$719$	−9824.54	−0.509587	−0.254794	−	0.966995i	$-0.582008\pi$
$719$	−9824.54	−0.509587	−0.254794	+	0.966995i	$0.582008\pi$
$720$	0	0
$721$	44250.3	2.28567
$722$	0	0
$723$	110.797	0.00569929
$724$	0	0
$725$	6016.10	0.308183
$726$	0	0
$727$	−4969.04	−0.253496	−0.126748	−	0.991935i	$-0.540454\pi$
$727$	−4969.04	−0.253496	−0.126748	+	0.991935i	$0.540454\pi$
$728$	0	0
$729$	−18257.7	−0.927589
$730$	0	0
$731$	5459.45	0.276231
$732$	0	0
$733$	−27512.0	−1.38633	−0.693164	−	0.720780i	$-0.743784\pi$
$733$	−27512.0	−1.38633	−0.693164	+	0.720780i	$0.743784\pi$
$734$	0	0
$735$	−2134.32	−0.107110
$736$	0	0
$737$	11238.1	0.561686
$738$	0	0
$739$	8175.74	0.406968	0.203484	−	0.979078i	$-0.434774\pi$
$739$	8175.74	0.406968	0.203484	+	0.979078i	$0.434774\pi$
$740$	0	0
$741$	298.532	0.0148001
$742$	0	0
$743$	−3518.94	−0.173751	−0.0868757	−	0.996219i	$-0.527688\pi$
$743$	−3518.94	−0.173751	−0.0868757	+	0.996219i	$0.527688\pi$
$744$	0	0
$745$	−7854.02	−0.386240
$746$	0	0
$747$	5582.12	0.273413
$748$	0	0
$749$	−16034.6	−0.782233
$750$	0	0
$751$	−7506.56	−0.364738	−0.182369	−	0.983230i	$-0.558377\pi$
$751$	−7506.56	−0.364738	−0.182369	+	0.983230i	$0.558377\pi$
$752$	0	0
$753$	−1303.94	−0.0631053
$754$	0	0
$755$	8951.69	0.431504
$756$	0	0
$757$	17229.2	0.827222	0.413611	−	0.910454i	$-0.364267\pi$
$757$	17229.2	0.827222	0.413611	+	0.910454i	$0.364267\pi$
$758$	0	0
$759$	−1779.00	−0.0850772
$760$	0	0
$761$	6430.58	0.306318	0.153159	−	0.988202i	$-0.451055\pi$
$761$	6430.58	0.306318	0.153159	+	0.988202i	$0.451055\pi$
$762$	0	0
$763$	−58628.4	−2.78177
$764$	0	0
$765$	4357.43	0.205939
$766$	0	0
$767$	−5550.72	−0.261310
$768$	0	0
$769$	−18106.8	−0.849087	−0.424544	−	0.905407i	$-0.639565\pi$
$769$	−18106.8	−0.849087	−0.424544	+	0.905407i	$0.639565\pi$
$770$	0	0
$771$	−2478.12	−0.115755
$772$	0	0
$773$	1139.92	0.0530401	0.0265201	−	0.999648i	$-0.491557\pi$
$773$	1139.92	0.0530401	0.0265201	+	0.999648i	$0.491557\pi$
$774$	0	0
$775$	−11449.3	−0.530671
$776$	0	0
$777$	−4888.06	−0.225686
$778$	0	0
$779$	−4344.60	−0.199822
$780$	0	0
$781$	−7486.03	−0.342985
$782$	0	0
$783$	−2061.81	−0.0941034
$784$	0	0
$785$	−8543.14	−0.388430
$786$	0	0
$787$	−26478.1	−1.19929	−0.599645	−	0.800266i	$-0.704691\pi$
$787$	−26478.1	−1.19929	−0.599645	+	0.800266i	$0.704691\pi$
$788$	0	0
$789$	−392.475	−0.0177091
$790$	0	0
$791$	−1249.37	−0.0561601
$792$	0	0
$793$	−21217.4	−0.950129
$794$	0	0
$795$	−2078.23	−0.0927133
$796$	0	0
$797$	37952.0	1.68674	0.843368	−	0.537337i	$-0.180570\pi$
$797$	37952.0	1.68674	0.843368	+	0.537337i	$0.180570\pi$
$798$	0	0
$799$	−15801.6	−0.699649
$800$	0	0
$801$	−18808.9	−0.829686
$802$	0	0
$803$	141.960	0.00623869
$804$	0	0
$805$	30019.9	1.31436
$806$	0	0
$807$	−4829.14	−0.210649
$808$	0	0
$809$	4455.38	0.193625	0.0968126	−	0.995303i	$-0.469135\pi$
$809$	4455.38	0.193625	0.0968126	+	0.995303i	$0.469135\pi$
$810$	0	0
$811$	−11239.3	−0.486639	−0.243319	−	0.969946i	$-0.578236\pi$
$811$	−11239.3	−0.486639	−0.243319	+	0.969946i	$0.578236\pi$
$812$	0	0
$813$	988.921	0.0426605
$814$	0	0
$815$	4685.16	0.201367
$816$	0	0
$817$	3764.11	0.161187
$818$	0	0
$819$	−22753.2	−0.970772
$820$	0	0
$821$	−36910.1	−1.56903	−0.784515	−	0.620110i	$-0.787088\pi$
$821$	−36910.1	−1.56903	−0.784515	+	0.620110i	$0.787088\pi$
$822$	0	0
$823$	19625.1	0.831213	0.415607	−	0.909544i	$-0.363569\pi$
$823$	19625.1	0.831213	0.415607	+	0.909544i	$0.363569\pi$
$824$	0	0
$825$	983.410	0.0415005
$826$	0	0
$827$	43857.2	1.84409	0.922047	−	0.387078i	$-0.126516\pi$
$827$	43857.2	1.84409	0.922047	+	0.387078i	$0.126516\pi$
$828$	0	0
$829$	−1531.44	−0.0641607	−0.0320803	−	0.999485i	$-0.510213\pi$
$829$	−1531.44	−0.0641607	−0.0320803	+	0.999485i	$0.510213\pi$
$830$	0	0
$831$	2608.80	0.108903
$832$	0	0
$833$	−17291.3	−0.719216
$834$	0	0
$835$	20292.9	0.841035
$836$	0	0
$837$	3923.84	0.162040
$838$	0	0
$839$	−9312.73	−0.383207	−0.191604	−	0.981472i	$-0.561369\pi$
$839$	−9312.73	−0.383207	−0.191604	+	0.981472i	$0.561369\pi$
$840$	0	0
$841$	−19905.9	−0.816183
$842$	0	0
$843$	3356.44	0.137132
$844$	0	0
$845$	−8578.98	−0.349262
$846$	0	0
$847$	−30127.4	−1.22218
$848$	0	0
$849$	−3428.68	−0.138601
$850$	0	0
$851$	44452.3	1.79060
$852$	0	0
$853$	−38161.3	−1.53179	−0.765895	−	0.642965i	$-0.777704\pi$
$853$	−38161.3	−1.53179	−0.765895	+	0.642965i	$0.777704\pi$
$854$	0	0
$855$	3004.30	0.120170
$856$	0	0
$857$	27799.4	1.10806	0.554031	−	0.832496i	$-0.313089\pi$
$857$	27799.4	1.10806	0.554031	+	0.832496i	$0.313089\pi$
$858$	0	0
$859$	−23218.5	−0.922239	−0.461120	−	0.887338i	$-0.652552\pi$
$859$	−23218.5	−0.922239	−0.461120	+	0.887338i	$0.652552\pi$
$860$	0	0
$861$	−4087.01	−0.161771
$862$	0	0
$863$	−20440.3	−0.806252	−0.403126	−	0.915144i	$-0.632076\pi$
$863$	−20440.3	−0.806252	−0.403126	+	0.915144i	$0.632076\pi$
$864$	0	0
$865$	−23270.4	−0.914703
$866$	0	0
$867$	2383.10	0.0933499
$868$	0	0
$869$	−19281.6	−0.752685
$870$	0	0
$871$	16133.3	0.627617
$872$	0	0
$873$	2529.91	0.0980806
$874$	0	0
$875$	−39680.9	−1.53309
$876$	0	0
$877$	−948.929	−0.0365371	−0.0182686	−	0.999833i	$-0.505815\pi$
$877$	−948.929	−0.0365371	−0.0182686	+	0.999833i	$0.505815\pi$
$878$	0	0
$879$	2649.56	0.101670
$880$	0	0
$881$	21701.4	0.829897	0.414948	−	0.909845i	$-0.363800\pi$
$881$	21701.4	0.829897	0.414948	+	0.909845i	$0.363800\pi$
$882$	0	0
$883$	32250.2	1.22911	0.614556	−	0.788873i	$-0.289335\pi$
$883$	32250.2	1.22911	0.614556	+	0.788873i	$0.289335\pi$
$884$	0	0
$885$	689.453	0.0261872
$886$	0	0
$887$	−1895.59	−0.0717561	−0.0358781	−	0.999356i	$-0.511423\pi$
$887$	−1895.59	−0.0717561	−0.0358781	+	0.999356i	$0.511423\pi$
$888$	0	0
$889$	−229.000	−0.00863940
$890$	0	0
$891$	13399.9	0.503831
$892$	0	0
$893$	−10894.7	−0.408260
$894$	0	0
$895$	−17648.1	−0.659119
$896$	0	0
$897$	−2553.90	−0.0950637
$898$	0	0
$899$	−8531.84	−0.316522
$900$	0	0
$901$	−16836.8	−0.622547
$902$	0	0
$903$	3540.93	0.130493
$904$	0	0
$905$	7805.61	0.286704
$906$	0	0
$907$	24616.4	0.901186	0.450593	−	0.892730i	$-0.351213\pi$
$907$	24616.4	0.901186	0.450593	+	0.892730i	$0.351213\pi$
$908$	0	0
$909$	14175.4	0.517235
$910$	0	0
$911$	2259.34	0.0821682	0.0410841	−	0.999156i	$-0.486919\pi$
$911$	2259.34	0.0821682	0.0410841	+	0.999156i	$0.486919\pi$
$912$	0	0
$913$	−3992.57	−0.144726
$914$	0	0
$915$	2635.41	0.0952174
$916$	0	0
$917$	−46005.6	−1.65675
$918$	0	0
$919$	13807.0	0.495595	0.247798	−	0.968812i	$-0.420293\pi$
$919$	13807.0	0.495595	0.247798	+	0.968812i	$0.420293\pi$
$920$	0	0
$921$	2036.97	0.0728778
$922$	0	0
$923$	−10746.8	−0.383245
$924$	0	0
$925$	−24572.7	−0.873454
$926$	0	0
$927$	−37884.7	−1.34228
$928$	0	0
$929$	−29318.1	−1.03541	−0.517705	−	0.855559i	$-0.673214\pi$
$929$	−29318.1	−1.03541	−0.517705	+	0.855559i	$0.673214\pi$
$930$	0	0
$931$	−11921.8	−0.419678
$932$	0	0
$933$	−4067.29	−0.142719
$934$	0	0
$935$	−3116.62	−0.109010
$936$	0	0
$937$	3320.96	0.115785	0.0578927	−	0.998323i	$-0.481562\pi$
$937$	3320.96	0.115785	0.0578927	+	0.998323i	$0.481562\pi$
$938$	0	0
$939$	2584.80	0.0898316
$940$	0	0
$941$	23508.3	0.814399	0.407200	−	0.913339i	$-0.366505\pi$
$941$	23508.3	0.814399	0.407200	+	0.913339i	$0.366505\pi$
$942$	0	0
$943$	37167.5	1.28350
$944$	0	0
$945$	5687.23	0.195773
$946$	0	0
$947$	7784.67	0.267126	0.133563	−	0.991040i	$-0.457358\pi$
$947$	7784.67	0.267126	0.133563	+	0.991040i	$0.457358\pi$
$948$	0	0
$949$	203.795	0.00697100
$950$	0	0
$951$	−2429.84	−0.0828526
$952$	0	0
$953$	30032.6	1.02083	0.510415	−	0.859928i	$-0.329492\pi$
$953$	30032.6	1.02083	0.510415	+	0.859928i	$0.329492\pi$
$954$	0	0
$955$	5106.11	0.173015
$956$	0	0
$957$	732.824	0.0247532
$958$	0	0
$959$	−26413.7	−0.889409
$960$	0	0
$961$	−13554.0	−0.454970
$962$	0	0
$963$	13728.0	0.459374
$964$	0	0
$965$	−7125.91	−0.237711
$966$	0	0
$967$	41844.9	1.39156	0.695781	−	0.718254i	$-0.255059\pi$
$967$	41844.9	1.39156	0.695781	+	0.718254i	$0.255059\pi$
$968$	0	0
$969$	−300.409	−0.00995928
$970$	0	0
$971$	−9767.47	−0.322815	−0.161407	−	0.986888i	$-0.551603\pi$
$971$	−9767.47	−0.322815	−0.161407	+	0.986888i	$0.551603\pi$
$972$	0	0
$973$	−48090.2	−1.58448
$974$	0	0
$975$	1411.76	0.0463719
$976$	0	0
$977$	−34528.2	−1.13066	−0.565331	−	0.824864i	$-0.691251\pi$
$977$	−34528.2	−1.13066	−0.565331	+	0.824864i	$0.691251\pi$
$978$	0	0
$979$	13452.9	0.439179
$980$	0	0
$981$	50194.4	1.63362
$982$	0	0
$983$	−37096.6	−1.20366	−0.601830	−	0.798625i	$-0.705561\pi$
$983$	−37096.6	−1.20366	−0.601830	+	0.798625i	$0.705561\pi$
$984$	0	0
$985$	6778.18	0.219260
$986$	0	0
$987$	−10248.7	−0.330517
$988$	0	0
$989$	−32201.4	−1.03533
$990$	0	0
$991$	44556.2	1.42823	0.714114	−	0.700029i	$-0.246830\pi$
$991$	44556.2	1.42823	0.714114	+	0.700029i	$0.246830\pi$
$992$	0	0
$993$	3895.29	0.124485
$994$	0	0
$995$	−22392.0	−0.713440
$996$	0	0
$997$	−53251.1	−1.69156	−0.845778	−	0.533536i	$-0.820863\pi$
$997$	−53251.1	−1.69156	−0.845778	+	0.533536i	$0.820863\pi$
$998$	0	0
$999$	8421.42	0.266709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.4.a.j.1.2		3
4.3	odd	2		152.4.a.b.1.2	✓	3
8.3	odd	2		1216.4.a.x.1.2		3
8.5	even	2		1216.4.a.q.1.2		3
12.11	even	2		1368.4.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.b.1.2	✓	3	4.3	odd	2
304.4.a.j.1.2		3	1.1	even	1	trivial
1216.4.a.q.1.2		3	8.5	even	2
1216.4.a.x.1.2		3	8.3	odd	2
1368.4.a.e.1.2		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-0.573746 q^{3} +5.92862 q^{5} +31.1522 q^{7} -26.6708 q^{9} +19.0761 q^{11} +27.3853 q^{13} -3.40152 q^{15} -27.5575 q^{17} -19.0000 q^{19} -17.8735 q^{21} +162.542 q^{23} -89.8514 q^{25} +30.7934 q^{27} -66.9561 q^{29} +127.424 q^{31} -10.9448 q^{33} +184.690 q^{35} +273.481 q^{37} -15.7122 q^{39} +228.663 q^{41} -198.111 q^{43} -158.121 q^{45} +573.403 q^{47} +627.461 q^{49} +15.8110 q^{51} +610.969 q^{53} +113.095 q^{55} +10.9012 q^{57} -202.690 q^{59} -774.773 q^{61} -830.855 q^{63} +162.357 q^{65} +589.121 q^{67} -93.2580 q^{69} -392.429 q^{71} +7.44178 q^{73} +51.5519 q^{75} +594.263 q^{77} -1010.77 q^{79} +702.444 q^{81} -209.297 q^{83} -163.378 q^{85} +38.4158 q^{87} +705.222 q^{89} +853.113 q^{91} -73.1093 q^{93} -112.644 q^{95} -94.8567 q^{97} -508.775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{3} + 2 q^{5} + 35 q^{7} + 48 q^{9} + 28 q^{11} - 109 q^{13} + 228 q^{15} - 123 q^{17} - 57 q^{19} + 25 q^{21} + 193 q^{23} + 187 q^{25} + 719 q^{27} - 297 q^{29} + 140 q^{31} + 30 q^{33} + 246 q^{35}+ \cdots - 86 q^{99}+O(q^{100}) \) 3 * q + 5 * q^3 + 2 * q^5 + 35 * q^7 + 48 * q^9 + 28 * q^11 - 109 * q^13 + 228 * q^15 - 123 * q^17 - 57 * q^19 + 25 * q^21 + 193 * q^23 + 187 * q^25 + 719 * q^27 - 297 * q^29 + 140 * q^31 + 30 * q^33 + 246 * q^35 + 38 * q^37 - 57 * q^39 + 736 * q^41 + 514 * q^43 + 1046 * q^45 - 134 * q^47 - 42 * q^49 - 929 * q^51 + 311 * q^53 + 130 * q^55 - 95 * q^57 - 199 * q^59 + 56 * q^61 - 508 * q^63 + 1156 * q^65 + 509 * q^67 + 621 * q^69 + 874 * q^71 + 203 * q^73 - 129 * q^75 + 616 * q^77 - 242 * q^79 + 3531 * q^81 + 62 * q^83 - 1430 * q^85 - 3237 * q^87 + 1764 * q^89 + 687 * q^91 - 2592 * q^93 - 38 * q^95 - 2178 * q^97 - 86 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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