LMFDB - Embedded newform 1458.3.b.c.1457.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,3,Mod(1457,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1458.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1458.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$39.7276225437$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.3
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -7.83103i q^{5} -0.901000 q^{7} +2.82843i q^{8} -11.0748 q^{10} +15.4771i q^{11} +0.909007 q^{13} +1.27421i q^{14} +4.00000 q^{16} -12.3174i q^{17} +10.8124 q^{19} +15.6621i q^{20} +21.8880 q^{22} +32.8204i q^{23} -36.3251 q^{25} -1.28553i q^{26} +1.80200 q^{28} +56.5740i q^{29} -49.5425 q^{31} -5.65685i q^{32} -17.4194 q^{34} +7.05576i q^{35} +34.8834 q^{37} -15.2910i q^{38} +22.1495 q^{40} +35.8136i q^{41} +41.5129 q^{43} -30.9542i q^{44} +46.4151 q^{46} -24.7597i q^{47} -48.1882 q^{49} +51.3714i q^{50} -1.81801 q^{52} +50.3340i q^{53} +121.202 q^{55} -2.54841i q^{56} +80.0077 q^{58} +66.7263i q^{59} -24.5276 q^{61} +70.0637i q^{62} -8.00000 q^{64} -7.11846i q^{65} +65.2216 q^{67} +24.6348i q^{68} +9.97835 q^{70} -28.2320i q^{71} +21.5521 q^{73} -49.3326i q^{74} -21.6247 q^{76} -13.9449i q^{77} +28.9712 q^{79} -31.3241i q^{80} +50.6481 q^{82} -0.148857i q^{83} -96.4578 q^{85} -58.7081i q^{86} -43.7759 q^{88} -91.5686i q^{89} -0.819015 q^{91} -65.6408i q^{92} -35.0155 q^{94} -84.6721i q^{95} -100.491 q^{97} +68.1484i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) $ 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times$.

$n$	$731$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 7.83103i	− 1.56621i	−0.621892	−	0.783103i	$-0.713636\pi$
$5$	− 7.83103i	− 1.56621i	0.621892	−	0.783103i	$-0.286364\pi$
$6$	0	0
$7$	−0.901000	−0.128714	−0.0643571	−	0.997927i	$-0.520500\pi$
$7$	−0.901000	−0.128714	−0.0643571	+	0.997927i	$0.520500\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−11.0748	−1.10748
$11$	15.4771i	1.40701i	0.710690	+	0.703505i	$0.248383\pi$
$11$	15.4771i	1.40701i	−0.710690	+	0.703505i	$0.751617\pi$
$12$	0	0
$13$	0.909007	0.0699236	0.0349618	−	0.999389i	$-0.488869\pi$
$13$	0.909007	0.0699236	0.0349618	+	0.999389i	$0.488869\pi$
$14$	1.27421i	0.0910148i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 12.3174i	− 0.724552i	−0.932071	−	0.362276i	$-0.882000\pi$
$17$	− 12.3174i	− 0.724552i	0.932071	−	0.362276i	$-0.118000\pi$
$18$	0	0
$19$	10.8124	0.569072	0.284536	−	0.958665i	$-0.408160\pi$
$19$	10.8124	0.569072	0.284536	+	0.958665i	$0.408160\pi$
$20$	15.6621i	0.783103i
$21$	0	0
$22$	21.8880	0.994907
$23$	32.8204i	1.42697i	0.700668	+	0.713487i	$0.252885\pi$
$23$	32.8204i	1.42697i	−0.700668	+	0.713487i	$0.747115\pi$
$24$	0	0
$25$	−36.3251	−1.45300
$26$	− 1.28553i	− 0.0494435i
$27$	0	0
$28$	1.80200	0.0643571
$29$	56.5740i	1.95083i	0.220381	+	0.975414i	$0.429270\pi$
$29$	56.5740i	1.95083i	−0.220381	+	0.975414i	$0.570730\pi$
$30$	0	0
$31$	−49.5425	−1.59815	−0.799073	−	0.601234i	$-0.794676\pi$
$31$	−49.5425	−1.59815	−0.799073	+	0.601234i	$0.794676\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−17.4194	−0.512335
$35$	7.05576i	0.201593i
$36$	0	0
$37$	34.8834	0.942795	0.471398	−	0.881921i	$-0.343750\pi$
$37$	34.8834	0.942795	0.471398	+	0.881921i	$0.343750\pi$
$38$	− 15.2910i	− 0.402395i
$39$	0	0
$40$	22.1495	0.553738
$41$	35.8136i	0.873503i	0.899582	+	0.436752i	$0.143871\pi$
$41$	35.8136i	0.873503i	−0.899582	+	0.436752i	$0.856129\pi$
$42$	0	0
$43$	41.5129	0.965417	0.482708	−	0.875781i	$-0.339653\pi$
$43$	41.5129	0.965417	0.482708	+	0.875781i	$0.339653\pi$
$44$	− 30.9542i	− 0.703505i
$45$	0	0
$46$	46.4151	1.00902
$47$	− 24.7597i	− 0.526802i	−0.964686	−	0.263401i	$-0.915156\pi$
$47$	− 24.7597i	− 0.526802i	0.964686	−	0.263401i	$-0.0848443\pi$
$48$	0	0
$49$	−48.1882	−0.983433
$50$	51.3714i	1.02743i
$51$	0	0
$52$	−1.81801	−0.0349618
$53$	50.3340i	0.949699i	0.880067	+	0.474850i	$0.157498\pi$
$53$	50.3340i	0.949699i	−0.880067	+	0.474850i	$0.842502\pi$
$54$	0	0
$55$	121.202	2.20367
$56$	− 2.54841i	− 0.0455074i
$57$	0	0
$58$	80.0077	1.37944
$59$	66.7263i	1.13095i	0.824764	+	0.565477i	$0.191308\pi$
$59$	66.7263i	1.13095i	−0.824764	+	0.565477i	$0.808692\pi$
$60$	0	0
$61$	−24.5276	−0.402093	−0.201046	−	0.979582i	$-0.564434\pi$
$61$	−24.5276	−0.402093	−0.201046	+	0.979582i	$0.564434\pi$
$62$	70.0637i	1.13006i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 7.11846i	− 0.109515i
$66$	0	0
$67$	65.2216	0.973456	0.486728	−	0.873554i	$-0.338190\pi$
$67$	65.2216	0.973456	0.486728	+	0.873554i	$0.338190\pi$
$68$	24.6348i	0.362276i
$69$	0	0
$70$	9.97835	0.142548
$71$	− 28.2320i	− 0.397634i	−0.980037	−	0.198817i	$-0.936290\pi$
$71$	− 28.2320i	− 0.397634i	0.980037	−	0.198817i	$-0.0637099\pi$
$72$	0	0
$73$	21.5521	0.295234	0.147617	−	0.989045i	$-0.452840\pi$
$73$	21.5521	0.295234	0.147617	+	0.989045i	$0.452840\pi$
$74$	− 49.3326i	− 0.666657i
$75$	0	0
$76$	−21.6247	−0.284536
$77$	− 13.9449i	− 0.181102i
$78$	0	0
$79$	28.9712	0.366724	0.183362	−	0.983045i	$-0.441302\pi$
$79$	28.9712	0.366724	0.183362	+	0.983045i	$0.441302\pi$
$80$	− 31.3241i	− 0.391552i
$81$	0	0
$82$	50.6481	0.617660
$83$	− 0.148857i	− 0.00179346i	−1.00000		0.000896729i	$-0.999715\pi$
$83$	− 0.148857i	− 0.00179346i	1.00000		0.000896729i	$-0.000285438\pi$
$84$	0	0
$85$	−96.4578	−1.13480
$86$	− 58.7081i	− 0.682653i
$87$	0	0
$88$	−43.7759	−0.497453
$89$	− 91.5686i	− 1.02886i	−0.857532	−	0.514431i	$-0.828003\pi$
$89$	− 91.5686i	− 1.02886i	0.857532	−	0.514431i	$-0.171997\pi$
$90$	0	0
$91$	−0.819015	−0.00900017
$92$	− 65.6408i	− 0.713487i
$93$	0	0
$94$	−35.0155	−0.372505
$95$	− 84.6721i	− 0.891285i
$96$	0	0
$97$	−100.491	−1.03599	−0.517996	−	0.855383i	$-0.673322\pi$
$97$	−100.491	−1.03599	−0.517996	+	0.855383i	$0.673322\pi$
$98$	68.1484i	0.695392i
$99$	0	0
$100$	72.6502	0.726502
$101$	− 105.600i	− 1.04555i	−0.852472	−	0.522773i	$-0.824898\pi$
$101$	− 105.600i	− 1.04555i	0.852472	−	0.522773i	$-0.175102\pi$
$102$	0	0
$103$	86.9818	0.844484	0.422242	−	0.906483i	$-0.361243\pi$
$103$	86.9818	0.844484	0.422242	+	0.906483i	$0.361243\pi$
$104$	2.57106i	0.0247217i
$105$	0	0
$106$	71.1831	0.671539
$107$	24.5062i	0.229030i	0.993422	+	0.114515i	$0.0365313\pi$
$107$	24.5062i	0.229030i	−0.993422	+	0.114515i	$0.963469\pi$
$108$	0	0
$109$	−33.5716	−0.307996	−0.153998	−	0.988071i	$-0.549215\pi$
$109$	−33.5716	−0.307996	−0.153998	+	0.988071i	$0.549215\pi$
$110$	− 171.405i	− 1.55823i
$111$	0	0
$112$	−3.60400	−0.0321786
$113$	122.011i	1.07975i	0.841747	+	0.539873i	$0.181528\pi$
$113$	122.011i	1.07975i	−0.841747	+	0.539873i	$0.818472\pi$
$114$	0	0
$115$	257.018	2.23494
$116$	− 113.148i	− 0.975414i
$117$	0	0
$118$	94.3652	0.799705
$119$	11.0980i	0.0932602i
$120$	0	0
$121$	−118.541	−0.979679
$122$	34.6873i	0.284322i
$123$	0	0
$124$	99.0851	0.799073
$125$	88.6871i	0.709497i
$126$	0	0
$127$	104.463	0.822541	0.411271	−	0.911513i	$-0.365085\pi$
$127$	104.463	0.822541	0.411271	+	0.911513i	$0.365085\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−10.0670	−0.0774387
$131$	0.914037i	0.00697738i	0.999994	+	0.00348869i	$0.00111049\pi$
$131$	0.914037i	0.00697738i	−0.999994	+	0.00348869i	$0.998890\pi$
$132$	0	0
$133$	−9.74195	−0.0732477
$134$	− 92.2372i	− 0.688337i
$135$	0	0
$136$	34.8388	0.256168
$137$	14.0527i	0.102575i	0.998684	+	0.0512874i	$0.0163324\pi$
$137$	14.0527i	0.102575i	−0.998684	+	0.0512874i	$0.983668\pi$
$138$	0	0
$139$	186.126	1.33903	0.669516	−	0.742797i	$-0.266501\pi$
$139$	186.126	1.33903	0.669516	+	0.742797i	$0.266501\pi$
$140$	− 14.1115i	− 0.100797i
$141$	0	0
$142$	−39.9261	−0.281169
$143$	14.0688i	0.0983833i
$144$	0	0
$145$	443.033	3.05540
$146$	− 30.4792i	− 0.208762i
$147$	0	0
$148$	−69.7669	−0.471398
$149$	126.786i	0.850911i	0.904979	+	0.425456i	$0.139886\pi$
$149$	126.786i	0.850911i	−0.904979	+	0.425456i	$0.860114\pi$
$150$	0	0
$151$	−137.434	−0.910158	−0.455079	−	0.890451i	$-0.650389\pi$
$151$	−137.434	−0.910158	−0.455079	+	0.890451i	$0.650389\pi$
$152$	30.5820i	0.201197i
$153$	0	0
$154$	−19.7210	−0.128059
$155$	387.969i	2.50303i
$156$	0	0
$157$	23.2456	0.148061	0.0740306	−	0.997256i	$-0.476414\pi$
$157$	23.2456	0.148061	0.0740306	+	0.997256i	$0.476414\pi$
$158$	− 40.9714i	− 0.259313i
$159$	0	0
$160$	−44.2990	−0.276869
$161$	− 29.5712i	− 0.183672i
$162$	0	0
$163$	157.977	0.969187	0.484593	−	0.874740i	$-0.338968\pi$
$163$	157.977	0.969187	0.484593	+	0.874740i	$0.338968\pi$
$164$	− 71.6272i	− 0.436752i
$165$	0	0
$166$	−0.210516	−0.00126817
$167$	− 94.5113i	− 0.565936i	−0.959129	−	0.282968i	$-0.908681\pi$
$167$	− 94.5113i	− 0.565936i	0.959129	−	0.282968i	$-0.0913190\pi$
$168$	0	0
$169$	−168.174	−0.995111
$170$	136.412i	0.802423i
$171$	0	0
$172$	−83.0258	−0.482708
$173$	290.466i	1.67899i	0.543365	+	0.839496i	$0.317150\pi$
$173$	290.466i	1.67899i	−0.543365	+	0.839496i	$0.682850\pi$
$174$	0	0
$175$	32.7289	0.187022
$176$	61.9085i	0.351753i
$177$	0	0
$178$	−129.498	−0.727515
$179$	− 11.1409i	− 0.0622399i	−0.999516	−	0.0311200i	$-0.990093\pi$
$179$	− 11.1409i	− 0.0622399i	0.999516	−	0.0311200i	$-0.00990739\pi$
$180$	0	0
$181$	190.553	1.05278	0.526389	−	0.850244i	$-0.323546\pi$
$181$	190.553	1.05278	0.526389	+	0.850244i	$0.323546\pi$
$182$	1.15826i	0.00636408i
$183$	0	0
$184$	−92.8301	−0.504512
$185$	− 273.173i	− 1.47661i
$186$	0	0
$187$	190.638	1.01945
$188$	49.5194i	0.263401i
$189$	0	0
$190$	−119.744	−0.630233
$191$	17.1446i	0.0897623i	0.998992	+	0.0448812i	$0.0142909\pi$
$191$	17.1446i	0.0897623i	−0.998992	+	0.0448812i	$0.985709\pi$
$192$	0	0
$193$	154.166	0.798789	0.399395	−	0.916779i	$-0.369220\pi$
$193$	154.166	0.798789	0.399395	+	0.916779i	$0.369220\pi$
$194$	142.116i	0.732557i
$195$	0	0
$196$	96.3764	0.491716
$197$	123.799i	0.628423i	0.949353	+	0.314211i	$0.101740\pi$
$197$	123.799i	0.628423i	−0.949353	+	0.314211i	$0.898260\pi$
$198$	0	0
$199$	234.477	1.17828	0.589139	−	0.808032i	$-0.299467\pi$
$199$	234.477	1.17828	0.589139	+	0.808032i	$0.299467\pi$
$200$	− 102.743i	− 0.513714i
$201$	0	0
$202$	−149.341	−0.739313
$203$	− 50.9732i	− 0.251099i
$204$	0	0
$205$	280.458	1.36809
$206$	− 123.011i	− 0.597140i
$207$	0	0
$208$	3.63603	0.0174809
$209$	167.344i	0.800691i
$210$	0	0
$211$	−53.2662	−0.252446	−0.126223	−	0.992002i	$-0.540286\pi$
$211$	−53.2662	−0.252446	−0.126223	+	0.992002i	$0.540286\pi$
$212$	− 100.668i	− 0.474850i
$213$	0	0
$214$	34.6569	0.161948
$215$	− 325.089i	− 1.51204i
$216$	0	0
$217$	44.6378	0.205704
$218$	47.4774i	0.217786i
$219$	0	0
$220$	−242.404	−1.10183
$221$	− 11.1966i	− 0.0506633i
$222$	0	0
$223$	−59.9197	−0.268698	−0.134349	−	0.990934i	$-0.542894\pi$
$223$	−59.9197	−0.268698	−0.134349	+	0.990934i	$0.542894\pi$
$224$	5.09683i	0.0227537i
$225$	0	0
$226$	172.550	0.763495
$227$	63.6721i	0.280494i	0.990117	+	0.140247i	$0.0447896\pi$
$227$	63.6721i	0.280494i	−0.990117	+	0.140247i	$0.955210\pi$
$228$	0	0
$229$	214.125	0.935043	0.467522	−	0.883982i	$-0.345147\pi$
$229$	214.125	0.935043	0.467522	+	0.883982i	$0.345147\pi$
$230$	− 363.478i	− 1.58034i
$231$	0	0
$232$	−160.015	−0.689722
$233$	215.572i	0.925202i	0.886567	+	0.462601i	$0.153084\pi$
$233$	215.572i	0.925202i	−0.886567	+	0.462601i	$0.846916\pi$
$234$	0	0
$235$	−193.894	−0.825081
$236$	− 133.453i	− 0.565477i
$237$	0	0
$238$	15.6949	0.0659449
$239$	− 41.8403i	− 0.175064i	−0.996162	−	0.0875319i	$-0.972102\pi$
$239$	− 41.8403i	− 0.175064i	0.996162	−	0.0875319i	$-0.0278980\pi$
$240$	0	0
$241$	−448.318	−1.86024	−0.930120	−	0.367255i	$-0.880297\pi$
$241$	−448.318	−1.86024	−0.930120	+	0.367255i	$0.880297\pi$
$242$	167.643i	0.692738i
$243$	0	0
$244$	49.0553	0.201046
$245$	377.363i	1.54026i
$246$	0	0
$247$	9.82852	0.0397916
$248$	− 140.127i	− 0.565030i
$249$	0	0
$250$	125.422	0.501690
$251$	− 256.659i	− 1.02254i	−0.859419	−	0.511272i	$-0.829174\pi$
$251$	− 256.659i	− 1.02254i	0.859419	−	0.511272i	$-0.170826\pi$
$252$	0	0
$253$	−507.965	−2.00777
$254$	− 147.733i	− 0.581624i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 392.701i	− 1.52802i	−0.645204	−	0.764010i	$-0.723228\pi$
$257$	− 392.701i	− 1.52802i	0.645204	−	0.764010i	$-0.276772\pi$
$258$	0	0
$259$	−31.4300	−0.121351
$260$	14.2369i	0.0547574i
$261$	0	0
$262$	1.29264	0.00493375
$263$	122.789i	0.466879i	0.972371	+	0.233439i	$0.0749980\pi$
$263$	122.789i	0.466879i	−0.972371	+	0.233439i	$0.925002\pi$
$264$	0	0
$265$	394.168	1.48742
$266$	13.7772i	0.0517940i
$267$	0	0
$268$	−130.443	−0.486728
$269$	392.295i	1.45835i	0.684329	+	0.729173i	$0.260095\pi$
$269$	392.295i	1.45835i	−0.684329	+	0.729173i	$0.739905\pi$
$270$	0	0
$271$	−43.9569	−0.162203	−0.0811013	−	0.996706i	$-0.525844\pi$
$271$	−43.9569	−0.162203	−0.0811013	+	0.996706i	$0.525844\pi$
$272$	− 49.2695i	− 0.181138i
$273$	0	0
$274$	19.8736	0.0725313
$275$	− 562.208i	− 2.04439i
$276$	0	0
$277$	−330.720	−1.19393	−0.596967	−	0.802266i	$-0.703628\pi$
$277$	−330.720	−1.19393	−0.596967	+	0.802266i	$0.703628\pi$
$278$	− 263.221i	− 0.946839i
$279$	0	0
$280$	−19.9567	−0.0712740
$281$	256.143i	0.911541i	0.890097	+	0.455770i	$0.150636\pi$
$281$	256.143i	0.911541i	−0.890097	+	0.455770i	$0.849364\pi$
$282$	0	0
$283$	−155.754	−0.550368	−0.275184	−	0.961392i	$-0.588739\pi$
$283$	−155.754	−0.550368	−0.275184	+	0.961392i	$0.588739\pi$
$284$	56.4640i	0.198817i
$285$	0	0
$286$	19.8963	0.0695675
$287$	− 32.2681i	− 0.112432i
$288$	0	0
$289$	137.282	0.475025
$290$	− 626.543i	− 2.16049i
$291$	0	0
$292$	−43.1042	−0.147617
$293$	204.468i	0.697844i	0.937152	+	0.348922i	$0.113452\pi$
$293$	204.468i	0.697844i	−0.937152	+	0.348922i	$0.886548\pi$
$294$	0	0
$295$	522.536	1.77131
$296$	98.6652i	0.333328i
$297$	0	0
$298$	179.302	0.601685
$299$	29.8340i	0.0997792i
$300$	0	0
$301$	−37.4031	−0.124263
$302$	194.361i	0.643579i
$303$	0	0
$304$	43.2495	0.142268
$305$	192.077i	0.629760i
$306$	0	0
$307$	−569.912	−1.85639	−0.928196	−	0.372092i	$-0.878641\pi$
$307$	−569.912	−1.85639	−0.928196	+	0.372092i	$0.878641\pi$
$308$	27.8898i	0.0905512i
$309$	0	0
$310$	548.671	1.76991
$311$	447.063i	1.43750i	0.695268	+	0.718751i	$0.255286\pi$
$311$	447.063i	1.43750i	−0.695268	+	0.718751i	$0.744714\pi$
$312$	0	0
$313$	−291.215	−0.930400	−0.465200	−	0.885206i	$-0.654018\pi$
$313$	−291.215	−0.930400	−0.465200	+	0.885206i	$0.654018\pi$
$314$	− 32.8743i	− 0.104695i
$315$	0	0
$316$	−57.9423	−0.183362
$317$	− 337.389i	− 1.06432i	−0.846644	−	0.532160i	$-0.821381\pi$
$317$	− 337.389i	− 1.06432i	0.846644	−	0.532160i	$-0.178619\pi$
$318$	0	0
$319$	−875.603	−2.74484
$320$	62.6483i	0.195776i
$321$	0	0
$322$	−41.8200	−0.129876
$323$	− 133.180i	− 0.412322i
$324$	0	0
$325$	−33.0197	−0.101599
$326$	− 223.414i	− 0.685318i
$327$	0	0
$328$	−101.296	−0.308830
$329$	22.3085i	0.0678070i
$330$	0	0
$331$	152.413	0.460463	0.230232	−	0.973136i	$-0.426052\pi$
$331$	152.413	0.460463	0.230232	+	0.973136i	$0.426052\pi$
$332$	0.297714i	0 0.000896729i
$333$	0	0
$334$	−133.659	−0.400177
$335$	− 510.752i	− 1.52463i
$336$	0	0
$337$	−233.503	−0.692888	−0.346444	−	0.938071i	$-0.612611\pi$
$337$	−233.503	−0.692888	−0.346444	+	0.938071i	$0.612611\pi$
$338$	237.834i	0.703650i
$339$	0	0
$340$	192.916	0.567399
$341$	− 766.776i	− 2.24861i
$342$	0	0
$343$	87.5666	0.255296
$344$	117.416i	0.341326i
$345$	0	0
$346$	410.781	1.18723
$347$	636.694i	1.83485i	0.397904	+	0.917427i	$0.369738\pi$
$347$	636.694i	1.83485i	−0.397904	+	0.917427i	$0.630262\pi$
$348$	0	0
$349$	−285.701	−0.818626	−0.409313	−	0.912394i	$-0.634232\pi$
$349$	−285.701	−0.818626	−0.409313	+	0.912394i	$0.634232\pi$
$350$	− 46.2857i	− 0.132245i
$351$	0	0
$352$	87.5518	0.248727
$353$	− 327.867i	− 0.928801i	−0.885625	−	0.464401i	$-0.846270\pi$
$353$	− 327.867i	− 0.928801i	0.885625	−	0.464401i	$-0.153730\pi$
$354$	0	0
$355$	−221.086	−0.622776
$356$	183.137i	0.514431i
$357$	0	0
$358$	−15.7557	−0.0440103
$359$	559.400i	1.55822i	0.626889	+	0.779109i	$0.284328\pi$
$359$	559.400i	1.55822i	−0.626889	+	0.779109i	$0.715672\pi$
$360$	0	0
$361$	−244.093	−0.676157
$362$	− 269.482i	− 0.744426i
$363$	0	0
$364$	1.63803	0.00450008
$365$	− 168.775i	− 0.462397i
$366$	0	0
$367$	479.700	1.30709	0.653543	−	0.756890i	$-0.273282\pi$
$367$	479.700	1.30709	0.653543	+	0.756890i	$0.273282\pi$
$368$	131.282i	0.356744i
$369$	0	0
$370$	−386.325	−1.04412
$371$	− 45.3510i	− 0.122240i
$372$	0	0
$373$	343.949	0.922116	0.461058	−	0.887370i	$-0.347470\pi$
$373$	343.949	0.922116	0.461058	+	0.887370i	$0.347470\pi$
$374$	− 269.602i	− 0.720862i
$375$	0	0
$376$	70.0310	0.186253
$377$	51.4262i	0.136409i
$378$	0	0
$379$	−104.156	−0.274817	−0.137408	−	0.990514i	$-0.543877\pi$
$379$	−104.156	−0.274817	−0.137408	+	0.990514i	$0.543877\pi$
$380$	169.344i	0.445642i
$381$	0	0
$382$	24.2461	0.0634716
$383$	− 64.8710i	− 0.169376i	−0.996408	−	0.0846880i	$-0.973011\pi$
$383$	− 64.8710i	− 0.169376i	0.996408	−	0.0846880i	$-0.0269894\pi$
$384$	0	0
$385$	−109.203	−0.283644
$386$	− 218.024i	− 0.564829i
$387$	0	0
$388$	200.982	0.517996
$389$	− 235.949i	− 0.606554i	−0.952903	−	0.303277i	$-0.901919\pi$
$389$	− 235.949i	− 0.606554i	0.952903	−	0.303277i	$-0.0980807\pi$
$390$	0	0
$391$	404.261	1.03392
$392$	− 136.297i	− 0.347696i
$393$	0	0
$394$	175.079	0.444362
$395$	− 226.874i	− 0.574365i
$396$	0	0
$397$	5.18698	0.0130654	0.00653272	−	0.999979i	$-0.497921\pi$
$397$	5.18698	0.0130654	0.00653272	+	0.999979i	$0.497921\pi$
$398$	− 331.601i	− 0.833168i
$399$	0	0
$400$	−145.300	−0.363251
$401$	− 160.833i	− 0.401079i	−0.979686	−	0.200539i	$-0.935731\pi$
$401$	− 160.833i	− 0.401079i	0.979686	−	0.200539i	$-0.0642694\pi$
$402$	0	0
$403$	−45.0345	−0.111748
$404$	211.200i	0.522773i
$405$	0	0
$406$	−72.0870	−0.177554
$407$	539.895i	1.32652i
$408$	0	0
$409$	−172.597	−0.421997	−0.210999	−	0.977486i	$-0.567672\pi$
$409$	−172.597	−0.421997	−0.210999	+	0.977486i	$0.567672\pi$
$410$	− 396.627i	− 0.967383i
$411$	0	0
$412$	−173.964	−0.422242
$413$	− 60.1204i	− 0.145570i
$414$	0	0
$415$	−1.16570	−0.00280893
$416$	− 5.14212i	− 0.0123609i
$417$	0	0
$418$	236.661	0.566174
$419$	− 286.337i	− 0.683382i	−0.939812	−	0.341691i	$-0.889000\pi$
$419$	− 286.337i	− 0.683382i	0.939812	−	0.341691i	$-0.111000\pi$
$420$	0	0
$421$	673.943	1.60081	0.800407	−	0.599457i	$-0.204617\pi$
$421$	673.943	1.60081	0.800407	+	0.599457i	$0.204617\pi$
$422$	75.3298i	0.178507i
$423$	0	0
$424$	−142.366	−0.335769
$425$	447.430i	1.05278i
$426$	0	0
$427$	22.0994	0.0517551
$428$	− 49.0123i	− 0.114515i
$429$	0	0
$430$	−459.745	−1.06918
$431$	− 539.138i	− 1.25090i	−0.780264	−	0.625450i	$-0.784915\pi$
$431$	− 539.138i	− 1.25090i	0.780264	−	0.625450i	$-0.215085\pi$
$432$	0	0
$433$	−802.991	−1.85448	−0.927241	−	0.374466i	$-0.877826\pi$
$433$	−802.991	−1.85448	−0.927241	+	0.374466i	$0.877826\pi$
$434$	− 63.1274i	− 0.145455i
$435$	0	0
$436$	67.1432	0.153998
$437$	354.867i	0.812051i
$438$	0	0
$439$	−78.2242	−0.178187	−0.0890936	−	0.996023i	$-0.528397\pi$
$439$	−78.2242	−0.178187	−0.0890936	+	0.996023i	$0.528397\pi$
$440$	342.811i	0.779115i
$441$	0	0
$442$	−15.8344	−0.0358243
$443$	546.293i	1.23317i	0.787289	+	0.616584i	$0.211484\pi$
$443$	546.293i	1.23317i	−0.787289	+	0.616584i	$0.788516\pi$
$444$	0	0
$445$	−717.077	−1.61141
$446$	84.7392i	0.189998i
$447$	0	0
$448$	7.20800	0.0160893
$449$	683.977i	1.52333i	0.647968	+	0.761667i	$0.275619\pi$
$449$	683.977i	1.52333i	−0.647968	+	0.761667i	$0.724381\pi$
$450$	0	0
$451$	−554.292	−1.22903
$452$	− 244.022i	− 0.539873i
$453$	0	0
$454$	90.0459	0.198339
$455$	6.41374i	0.0140961i
$456$	0	0
$457$	85.8233	0.187797	0.0938985	−	0.995582i	$-0.470067\pi$
$457$	85.8233	0.187797	0.0938985	+	0.995582i	$0.470067\pi$
$458$	− 302.818i	− 0.661175i
$459$	0	0
$460$	−514.035	−1.11747
$461$	5.69970i	0.0123638i	0.999981	+	0.00618189i	$0.00196777\pi$
$461$	5.69970i	0.0123638i	−0.999981	+	0.00618189i	$0.998032\pi$
$462$	0	0
$463$	−322.522	−0.696592	−0.348296	−	0.937385i	$-0.613240\pi$
$463$	−322.522	−0.696592	−0.348296	+	0.937385i	$0.613240\pi$
$464$	226.296i	0.487707i
$465$	0	0
$466$	304.865	0.654216
$467$	37.8579i	0.0810662i	0.999178	+	0.0405331i	$0.0129056\pi$
$467$	37.8579i	0.0810662i	−0.999178	+	0.0405331i	$0.987094\pi$
$468$	0	0
$469$	−58.7646	−0.125298
$470$	274.208i	0.583421i
$471$	0	0
$472$	−188.730	−0.399853
$473$	642.500i	1.35835i
$474$	0	0
$475$	−392.760	−0.826864
$476$	− 22.1959i	− 0.0466301i
$477$	0	0
$478$	−59.1711	−0.123789
$479$	717.727i	1.49839i	0.662351	+	0.749193i	$0.269559\pi$
$479$	717.727i	1.49839i	−0.662351	+	0.749193i	$0.730441\pi$
$480$	0	0
$481$	31.7093	0.0659236
$482$	634.017i	1.31539i
$483$	0	0
$484$	237.082	0.489840
$485$	786.950i	1.62258i
$486$	0	0
$487$	52.1558	0.107096	0.0535480	−	0.998565i	$-0.482947\pi$
$487$	52.1558	0.107096	0.0535480	+	0.998565i	$0.482947\pi$
$488$	− 69.3747i	− 0.142161i
$489$	0	0
$490$	533.672	1.08913
$491$	79.0118i	0.160920i	0.996758	+	0.0804600i	$0.0256389\pi$
$491$	79.0118i	0.160920i	−0.996758	+	0.0804600i	$0.974361\pi$
$492$	0	0
$493$	696.844	1.41348
$494$	− 13.8996i	− 0.0281369i
$495$	0	0
$496$	−198.170	−0.399537
$497$	25.4370i	0.0511811i
$498$	0	0
$499$	−29.7879	−0.0596952	−0.0298476	−	0.999554i	$-0.509502\pi$
$499$	−29.7879	−0.0596952	−0.0298476	+	0.999554i	$0.509502\pi$
$500$	− 177.374i	− 0.354748i
$501$	0	0
$502$	−362.970	−0.723048
$503$	168.679i	0.335345i	0.985843	+	0.167673i	$0.0536252\pi$
$503$	168.679i	0.335345i	−0.985843	+	0.167673i	$0.946375\pi$
$504$	0	0
$505$	−826.958	−1.63754
$506$	718.372i	1.41971i
$507$	0	0
$508$	−208.925	−0.411271
$509$	− 348.472i	− 0.684621i	−0.939587	−	0.342310i	$-0.888791\pi$
$509$	− 348.472i	− 0.684621i	0.939587	−	0.342310i	$-0.111209\pi$
$510$	0	0
$511$	−19.4184	−0.0380008
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−555.364	−1.08047
$515$	− 681.158i	− 1.32264i
$516$	0	0
$517$	383.209	0.741217
$518$	44.4487i	0.0858083i
$519$	0	0
$520$	20.1341	0.0387193
$521$	− 110.818i	− 0.212702i	−0.994329	−	0.106351i	$-0.966083\pi$
$521$	− 110.818i	− 0.212702i	0.994329	−	0.106351i	$-0.0339167\pi$
$522$	0	0
$523$	−560.394	−1.07150	−0.535750	−	0.844377i	$-0.679971\pi$
$523$	−560.394	−1.07150	−0.535750	+	0.844377i	$0.679971\pi$
$524$	− 1.82807i	− 0.00348869i
$525$	0	0
$526$	173.650	0.330133
$527$	610.234i	1.15794i
$528$	0	0
$529$	−548.179	−1.03626
$530$	− 557.437i	− 1.05177i
$531$	0	0
$532$	19.4839	0.0366239
$533$	32.5548i	0.0610785i
$534$	0	0
$535$	191.909	0.358708
$536$	184.474i	0.344169i
$537$	0	0
$538$	554.789	1.03121
$539$	− 745.814i	− 1.38370i
$540$	0	0
$541$	−313.055	−0.578660	−0.289330	−	0.957229i	$-0.593433\pi$
$541$	−313.055	−0.578660	−0.289330	+	0.957229i	$0.593433\pi$
$542$	62.1645i	0.114695i
$543$	0	0
$544$	−69.6776	−0.128084
$545$	262.900i	0.482386i
$546$	0	0
$547$	−736.583	−1.34659	−0.673293	−	0.739375i	$-0.735121\pi$
$547$	−736.583	−1.34659	−0.673293	+	0.739375i	$0.735121\pi$
$548$	− 28.1055i	− 0.0512874i
$549$	0	0
$550$	−795.082	−1.44560
$551$	611.699i	1.11016i
$552$	0	0
$553$	−26.1030	−0.0472026
$554$	467.708i	0.844238i
$555$	0	0
$556$	−372.251	−0.669516
$557$	638.093i	1.14559i	0.819699	+	0.572794i	$0.194141\pi$
$557$	638.093i	1.14559i	−0.819699	+	0.572794i	$0.805859\pi$
$558$	0	0
$559$	37.7355	0.0675054
$560$	28.2230i	0.0503983i
$561$	0	0
$562$	362.241	0.644557
$563$	− 348.761i	− 0.619469i	−0.950823	−	0.309734i	$-0.899760\pi$
$563$	− 348.761i	− 0.619469i	0.950823	−	0.309734i	$-0.100240\pi$
$564$	0	0
$565$	955.474	1.69110
$566$	220.270i	0.389169i
$567$	0	0
$568$	79.8521	0.140585
$569$	547.210i	0.961705i	0.876802	+	0.480852i	$0.159673\pi$
$569$	547.210i	0.961705i	−0.876802	+	0.480852i	$0.840327\pi$
$570$	0	0
$571$	697.589	1.22170	0.610849	−	0.791747i	$-0.290828\pi$
$571$	697.589	1.22170	0.610849	+	0.791747i	$0.290828\pi$
$572$	− 28.1376i	− 0.0491916i
$573$	0	0
$574$	−45.6340	−0.0795017
$575$	− 1192.20i	− 2.07340i
$576$	0	0
$577$	387.767	0.672040	0.336020	−	0.941855i	$-0.390919\pi$
$577$	387.767	0.672040	0.336020	+	0.941855i	$0.390919\pi$
$578$	− 194.146i	− 0.335893i
$579$	0	0
$580$	−886.066	−1.52770
$581$	0.134120i	0 0.000230844i
$582$	0	0
$583$	−779.026	−1.33624
$584$	60.9585i	0.104381i
$585$	0	0
$586$	289.162	0.493450
$587$	− 357.223i	− 0.608556i	−0.952583	−	0.304278i	$-0.901585\pi$
$587$	− 357.223i	− 0.608556i	0.952583	−	0.304278i	$-0.0984152\pi$
$588$	0	0
$589$	−535.672	−0.909461
$590$	− 738.977i	− 1.25250i
$591$	0	0
$592$	139.534	0.235699
$593$	828.411i	1.39698i	0.715618	+	0.698492i	$0.246145\pi$
$593$	828.411i	1.39698i	−0.715618	+	0.698492i	$0.753855\pi$
$594$	0	0
$595$	86.9085	0.146065
$596$	− 253.572i	− 0.425456i
$597$	0	0
$598$	42.1916	0.0705545
$599$	876.695i	1.46360i	0.681521	+	0.731799i	$0.261319\pi$
$599$	876.695i	1.46360i	−0.681521	+	0.731799i	$0.738681\pi$
$600$	0	0
$601$	819.381	1.36336	0.681682	−	0.731649i	$-0.261249\pi$
$601$	819.381	1.36336	0.681682	+	0.731649i	$0.261249\pi$
$602$	52.8960i	0.0878672i
$603$	0	0
$604$	274.868	0.455079
$605$	928.300i	1.53438i
$606$	0	0
$607$	579.308	0.954379	0.477190	−	0.878800i	$-0.341655\pi$
$607$	579.308	0.954379	0.477190	+	0.878800i	$0.341655\pi$
$608$	− 61.1640i	− 0.100599i
$609$	0	0
$610$	271.638	0.445308
$611$	− 22.5067i	− 0.0368359i
$612$	0	0
$613$	327.562	0.534359	0.267179	−	0.963647i	$-0.413908\pi$
$613$	327.562	0.534359	0.267179	+	0.963647i	$0.413908\pi$
$614$	805.978i	1.31267i
$615$	0	0
$616$	39.4421	0.0640294
$617$	1066.42i	1.72840i	0.503148	+	0.864201i	$0.332175\pi$
$617$	1066.42i	1.72840i	−0.503148	+	0.864201i	$0.667825\pi$
$618$	0	0
$619$	−889.551	−1.43708	−0.718539	−	0.695487i	$-0.755189\pi$
$619$	−889.551	−1.43708	−0.718539	+	0.695487i	$0.755189\pi$
$620$	− 775.938i	− 1.25151i
$621$	0	0
$622$	632.242	1.01647
$623$	82.5033i	0.132429i
$624$	0	0
$625$	−213.616	−0.341785
$626$	411.840i	0.657892i
$627$	0	0
$628$	−46.4912	−0.0740306
$629$	− 429.672i	− 0.683104i
$630$	0	0
$631$	954.681	1.51297	0.756483	−	0.654014i	$-0.226916\pi$
$631$	954.681	1.51297	0.756483	+	0.654014i	$0.226916\pi$
$632$	81.9428i	0.129656i
$633$	0	0
$634$	−477.140	−0.752587
$635$	− 818.051i	− 1.28827i
$636$	0	0
$637$	−43.8034	−0.0687652
$638$	1238.29i	1.94089i
$639$	0	0
$640$	88.5980	0.138434
$641$	− 709.946i	− 1.10756i	−0.832663	−	0.553780i	$-0.813185\pi$
$641$	− 709.946i	− 1.10756i	0.832663	−	0.553780i	$-0.186815\pi$
$642$	0	0
$643$	49.6295	0.0771842	0.0385921	−	0.999255i	$-0.487713\pi$
$643$	49.6295	0.0771842	0.0385921	+	0.999255i	$0.487713\pi$
$644$	59.1424i	0.0918360i
$645$	0	0
$646$	−188.345	−0.291556
$647$	− 918.622i	− 1.41982i	−0.704294	−	0.709909i	$-0.748736\pi$
$647$	− 918.622i	− 1.41982i	0.704294	−	0.709909i	$-0.251264\pi$
$648$	0	0
$649$	−1032.73	−1.59126
$650$	46.6970i	0.0718415i
$651$	0	0
$652$	−315.955	−0.484593
$653$	347.677i	0.532430i	0.963914	+	0.266215i	$0.0857731\pi$
$653$	347.677i	0.532430i	−0.963914	+	0.266215i	$0.914227\pi$
$654$	0	0
$655$	7.15785	0.0109280
$656$	143.254i	0.218376i
$657$	0	0
$658$	31.5490	0.0479468
$659$	− 1282.98i	− 1.94685i	−0.228999	−	0.973427i	$-0.573545\pi$
$659$	− 1282.98i	− 1.94685i	0.228999	−	0.973427i	$-0.426455\pi$
$660$	0	0
$661$	134.467	0.203430	0.101715	−	0.994814i	$-0.467567\pi$
$661$	134.467	0.203430	0.101715	+	0.994814i	$0.467567\pi$
$662$	− 215.545i	− 0.325597i
$663$	0	0
$664$	0.421031	0.000634083 0
$665$	76.2895i	0.114721i
$666$	0	0
$667$	−1856.78	−2.78378
$668$	189.023i	0.282968i
$669$	0	0
$670$	−722.313	−1.07808
$671$	− 379.617i	− 0.565749i
$672$	0	0
$673$	−556.202	−0.826451	−0.413226	−	0.910629i	$-0.635598\pi$
$673$	−556.202	−0.826451	−0.413226	+	0.910629i	$0.635598\pi$
$674$	330.223i	0.489946i
$675$	0	0
$676$	336.347	0.497555
$677$	− 341.527i	− 0.504472i	−0.967666	−	0.252236i	$-0.918834\pi$
$677$	− 341.527i	− 0.504472i	0.967666	−	0.252236i	$-0.0811659\pi$
$678$	0	0
$679$	90.5426	0.133347
$680$	− 272.824i	− 0.401212i
$681$	0	0
$682$	−1084.38	−1.59001
$683$	209.738i	0.307084i	0.988142	+	0.153542i	$0.0490680\pi$
$683$	209.738i	0.307084i	−0.988142	+	0.153542i	$0.950932\pi$
$684$	0	0
$685$	110.047	0.160653
$686$	− 123.838i	− 0.180522i
$687$	0	0
$688$	166.052	0.241354
$689$	45.7540i	0.0664064i
$690$	0	0
$691$	−291.333	−0.421611	−0.210805	−	0.977528i	$-0.567609\pi$
$691$	−291.333	−0.421611	−0.210805	+	0.977528i	$0.567609\pi$
$692$	− 580.931i	− 0.839496i
$693$	0	0
$694$	900.422	1.29744
$695$	− 1457.56i	− 2.09720i
$696$	0	0
$697$	441.130	0.632898
$698$	404.042i	0.578856i
$699$	0	0
$700$	−65.4578	−0.0935111
$701$	445.528i	0.635560i	0.948164	+	0.317780i	$0.102937\pi$
$701$	445.528i	0.635560i	−0.948164	+	0.317780i	$0.897063\pi$
$702$	0	0
$703$	377.173	0.536519
$704$	− 123.817i	− 0.175876i
$705$	0	0
$706$	−463.674	−0.656762
$707$	95.1457i	0.134577i
$708$	0	0
$709$	−146.607	−0.206780	−0.103390	−	0.994641i	$-0.532969\pi$
$709$	−146.607	−0.206780	−0.103390	+	0.994641i	$0.532969\pi$
$710$	312.662i	0.440369i
$711$	0	0
$712$	258.995	0.363757
$713$	− 1626.01i	− 2.28051i
$714$	0	0
$715$	110.173	0.154089
$716$	22.2819i	0.0311200i
$717$	0	0
$718$	791.111	1.10183
$719$	− 511.293i	− 0.711117i	−0.934654	−	0.355559i	$-0.884291\pi$
$719$	− 511.293i	− 0.711117i	0.934654	−	0.355559i	$-0.115709\pi$
$720$	0	0
$721$	−78.3706	−0.108697
$722$	345.199i	0.478115i
$723$	0	0
$724$	−381.105	−0.526389
$725$	− 2055.06i	− 2.83456i
$726$	0	0
$727$	248.843	0.342287	0.171143	−	0.985246i	$-0.445254\pi$
$727$	248.843	0.342287	0.171143	+	0.985246i	$0.445254\pi$
$728$	− 2.31653i	− 0.00318204i
$729$	0	0
$730$	−238.684	−0.326964
$731$	− 511.330i	− 0.699494i
$732$	0	0
$733$	843.120	1.15023	0.575116	−	0.818072i	$-0.304957\pi$
$733$	843.120	1.15023	0.575116	+	0.818072i	$0.304957\pi$
$734$	− 678.399i	− 0.924249i
$735$	0	0
$736$	185.660	0.252256
$737$	1009.44i	1.36966i
$738$	0	0
$739$	−679.277	−0.919184	−0.459592	−	0.888130i	$-0.652004\pi$
$739$	−679.277	−0.919184	−0.459592	+	0.888130i	$0.652004\pi$
$740$	546.347i	0.738306i
$741$	0	0
$742$	−64.1360	−0.0864366
$743$	− 629.416i	− 0.847128i	−0.905866	−	0.423564i	$-0.860779\pi$
$743$	− 629.416i	− 0.847128i	0.905866	−	0.423564i	$-0.139221\pi$
$744$	0	0
$745$	992.864	1.33270
$746$	− 486.418i	− 0.652035i
$747$	0	0
$748$	−381.275	−0.509726
$749$	− 22.0801i	− 0.0294794i
$750$	0	0
$751$	713.489	0.950052	0.475026	−	0.879972i	$-0.342439\pi$
$751$	713.489	0.950052	0.475026	+	0.879972i	$0.342439\pi$
$752$	− 99.0388i	− 0.131701i
$753$	0	0
$754$	72.7276	0.0964557
$755$	1076.25i	1.42550i
$756$	0	0
$757$	1327.86	1.75411	0.877055	−	0.480390i	$-0.159505\pi$
$757$	1327.86	1.75411	0.877055	+	0.480390i	$0.159505\pi$
$758$	147.298i	0.194325i
$759$	0	0
$760$	239.489	0.315117
$761$	− 942.331i	− 1.23828i	−0.785281	−	0.619140i	$-0.787481\pi$
$761$	− 942.331i	− 1.23828i	0.785281	−	0.619140i	$-0.212519\pi$
$762$	0	0
$763$	30.2480	0.0396435
$764$	− 34.2892i	− 0.0448812i
$765$	0	0
$766$	−91.7415	−0.119767
$767$	60.6547i	0.0790804i
$768$	0	0
$769$	1063.98	1.38359	0.691793	−	0.722096i	$-0.256821\pi$
$769$	1063.98	1.38359	0.691793	+	0.722096i	$0.256821\pi$
$770$	154.436i	0.200566i
$771$	0	0
$772$	−308.333	−0.399395
$773$	− 289.008i	− 0.373879i	−0.982371	−	0.186939i	$-0.940143\pi$
$773$	− 289.008i	− 0.373879i	0.982371	−	0.186939i	$-0.0598568\pi$
$774$	0	0
$775$	1799.64	2.32211
$776$	− 284.232i	− 0.366279i
$777$	0	0
$778$	−333.683	−0.428898
$779$	387.230i	0.497086i
$780$	0	0
$781$	436.950	0.559475
$782$	− 571.712i	− 0.731090i
$783$	0	0
$784$	−192.753	−0.245858
$785$	− 182.037i	− 0.231895i
$786$	0	0
$787$	−669.765	−0.851036	−0.425518	−	0.904950i	$-0.639908\pi$
$787$	−669.765	−0.851036	−0.425518	+	0.904950i	$0.639908\pi$
$788$	− 247.599i	− 0.314211i
$789$	0	0
$790$	−320.849	−0.406137
$791$	− 109.932i	− 0.138979i
$792$	0	0
$793$	−22.2958	−0.0281158
$794$	− 7.33550i	− 0.00923867i
$795$	0	0
$796$	−468.955	−0.589139
$797$	− 281.788i	− 0.353561i	−0.984250	−	0.176781i	$-0.943432\pi$
$797$	− 281.788i	− 0.353561i	0.984250	−	0.176781i	$-0.0565683\pi$
$798$	0	0
$799$	−304.975	−0.381696
$800$	205.486i	0.256857i
$801$	0	0
$802$	−227.452	−0.283605
$803$	333.564i	0.415397i
$804$	0	0
$805$	−231.573	−0.287668
$806$	63.6884i	0.0790179i
$807$	0	0
$808$	298.682	0.369656
$809$	− 747.542i	− 0.924032i	−0.886872	−	0.462016i	$-0.847126\pi$
$809$	− 747.542i	− 0.924032i	0.886872	−	0.462016i	$-0.152874\pi$
$810$	0	0
$811$	−1416.36	−1.74644	−0.873218	−	0.487330i	$-0.837971\pi$
$811$	−1416.36	−1.74644	−0.873218	+	0.487330i	$0.837971\pi$
$812$	101.946i	0.125550i
$813$	0	0
$814$	763.527	0.937994
$815$	− 1237.13i	− 1.51795i
$816$	0	0
$817$	448.853	0.549392
$818$	244.089i	0.298397i
$819$	0	0
$820$	−560.915	−0.684043
$821$	− 92.9879i	− 0.113262i	−0.998395	−	0.0566309i	$-0.981964\pi$
$821$	− 92.9879i	− 0.113262i	0.998395	−	0.0566309i	$-0.0180358\pi$
$822$	0	0
$823$	−37.5271	−0.0455980	−0.0227990	−	0.999740i	$-0.507258\pi$
$823$	−37.5271	−0.0455980	−0.0227990	+	0.999740i	$0.507258\pi$
$824$	246.022i	0.298570i
$825$	0	0
$826$	−85.0231	−0.102933
$827$	− 1314.39i	− 1.58935i	−0.607038	−	0.794673i	$-0.707642\pi$
$827$	− 1314.39i	− 1.58935i	0.607038	−	0.794673i	$-0.292358\pi$
$828$	0	0
$829$	55.4345	0.0668691	0.0334346	−	0.999441i	$-0.489355\pi$
$829$	55.4345	0.0668691	0.0334346	+	0.999441i	$0.489355\pi$
$830$	1.64855i	0.00198621i
$831$	0	0
$832$	−7.27206	−0.00874045
$833$	593.552i	0.712548i
$834$	0	0
$835$	−740.121	−0.886373
$836$	− 334.689i	− 0.400345i
$837$	0	0
$838$	−404.942	−0.483224
$839$	− 563.657i	− 0.671820i	−0.941894	−	0.335910i	$-0.890956\pi$
$839$	− 563.657i	− 0.671820i	0.941894	−	0.335910i	$-0.109044\pi$
$840$	0	0
$841$	−2359.62	−2.80573
$842$	− 953.099i	− 1.13195i
$843$	0	0
$844$	106.532	0.126223
$845$	1316.97i	1.55855i
$846$	0	0
$847$	106.806	0.126099
$848$	201.336i	0.237425i
$849$	0	0
$850$	632.761	0.744425
$851$	1144.89i	1.34534i
$852$	0	0
$853$	474.801	0.556625	0.278313	−	0.960491i	$-0.410225\pi$
$853$	474.801	0.556625	0.278313	+	0.960491i	$0.410225\pi$
$854$	− 31.2533i	− 0.0365964i
$855$	0	0
$856$	−69.3139	−0.0809742
$857$	− 744.299i	− 0.868494i	−0.900794	−	0.434247i	$-0.857015\pi$
$857$	− 744.299i	− 0.868494i	0.900794	−	0.434247i	$-0.142985\pi$
$858$	0	0
$859$	417.865	0.486455	0.243228	−	0.969969i	$-0.421794\pi$
$859$	417.865	0.486455	0.243228	+	0.969969i	$0.421794\pi$
$860$	650.178i	0.756021i
$861$	0	0
$862$	−762.457	−0.884521
$863$	816.761i	0.946420i	0.880950	+	0.473210i	$0.156905\pi$
$863$	816.761i	0.946420i	−0.880950	+	0.473210i	$0.843095\pi$
$864$	0	0
$865$	2274.65	2.62965
$866$	1135.60i	1.31132i
$867$	0	0
$868$	−89.2756	−0.102852
$869$	448.390i	0.515984i
$870$	0	0
$871$	59.2868	0.0680676
$872$	− 94.9548i	− 0.108893i
$873$	0	0
$874$	501.857	0.574207
$875$	− 79.9071i	− 0.0913224i
$876$	0	0
$877$	−551.919	−0.629327	−0.314663	−	0.949203i	$-0.601892\pi$
$877$	−551.919	−0.629327	−0.314663	+	0.949203i	$0.601892\pi$
$878$	110.626i	0.125997i
$879$	0	0
$880$	484.807	0.550917
$881$	1409.72i	1.60013i	0.599910	+	0.800067i	$0.295203\pi$
$881$	1409.72i	1.60013i	−0.599910	+	0.800067i	$0.704797\pi$
$882$	0	0
$883$	1609.41	1.82266	0.911328	−	0.411680i	$-0.135058\pi$
$883$	1609.41	1.82266	0.911328	+	0.411680i	$0.135058\pi$
$884$	22.3932i	0.0253316i
$885$	0	0
$886$	772.575	0.871981
$887$	1184.61i	1.33552i	0.744376	+	0.667761i	$0.232747\pi$
$887$	1184.61i	1.33552i	−0.744376	+	0.667761i	$0.767253\pi$
$888$	0	0
$889$	−94.1209	−0.105873
$890$	1014.10i	1.13944i
$891$	0	0
$892$	119.839	0.134349
$893$	− 267.711i	− 0.299789i
$894$	0	0
$895$	−87.2451	−0.0974806
$896$	− 10.1937i	− 0.0113768i
$897$	0	0
$898$	967.290	1.07716
$899$	− 2802.82i	− 3.11771i
$900$	0	0
$901$	619.984	0.688106
$902$	783.887i	0.869054i
$903$	0	0
$904$	−345.100	−0.381748
$905$	− 1492.22i	− 1.64887i
$906$	0	0
$907$	397.992	0.438801	0.219400	−	0.975635i	$-0.429590\pi$
$907$	397.992	0.438801	0.219400	+	0.975635i	$0.429590\pi$
$908$	− 127.344i	− 0.140247i
$909$	0	0
$910$	9.07039	0.00996746
$911$	811.418i	0.890690i	0.895359	+	0.445345i	$0.146919\pi$
$911$	811.418i	0.890690i	−0.895359	+	0.445345i	$0.853081\pi$
$912$	0	0
$913$	2.30388	0.00252342
$914$	− 121.372i	− 0.132793i
$915$	0	0
$916$	−428.250	−0.467522
$917$	− 0.823547i	0 0.000898089i
$918$	0	0
$919$	1571.44	1.70994	0.854972	−	0.518674i	$-0.173574\pi$
$919$	1571.44	1.70994	0.854972	+	0.518674i	$0.173574\pi$
$920$	726.956i	0.790169i
$921$	0	0
$922$	8.06059	0.00874251
$923$	− 25.6631i	− 0.0278040i
$924$	0	0
$925$	−1267.14	−1.36988
$926$	456.115i	0.492565i
$927$	0	0
$928$	320.031	0.344861
$929$	517.657i	0.557219i	0.960404	+	0.278610i	$0.0898736\pi$
$929$	517.657i	0.557219i	−0.960404	+	0.278610i	$0.910126\pi$
$930$	0	0
$931$	−521.029	−0.559644
$932$	− 431.144i	− 0.462601i
$933$	0	0
$934$	53.5392	0.0573225
$935$	− 1492.89i	− 1.59667i
$936$	0	0
$937$	343.474	0.366568	0.183284	−	0.983060i	$-0.441327\pi$
$937$	343.474	0.366568	0.183284	+	0.983060i	$0.441327\pi$
$938$	83.1057i	0.0885989i
$939$	0	0
$940$	387.788	0.412541
$941$	837.765i	0.890293i	0.895458	+	0.445146i	$0.146848\pi$
$941$	837.765i	0.890293i	−0.895458	+	0.445146i	$0.853152\pi$
$942$	0	0
$943$	−1175.42	−1.24647
$944$	266.905i	0.282738i
$945$	0	0
$946$	908.633	0.960500
$947$	1444.18i	1.52501i	0.646984	+	0.762504i	$0.276030\pi$
$947$	1444.18i	1.52501i	−0.646984	+	0.762504i	$0.723970\pi$
$948$	0	0
$949$	19.5910	0.0206438
$950$	555.447i	0.584681i
$951$	0	0
$952$	−31.3898	−0.0329725
$953$	1307.20i	1.37167i	0.727757	+	0.685835i	$0.240563\pi$
$953$	1307.20i	1.37167i	−0.727757	+	0.685835i	$0.759437\pi$
$954$	0	0
$955$	134.260	0.140586
$956$	83.6805i	0.0875319i
$957$	0	0
$958$	1015.02	1.05952
$959$	− 12.6615i	− 0.0132028i
$960$	0	0
$961$	1493.46	1.55407
$962$	− 44.8437i	− 0.0466151i
$963$	0	0
$964$	896.636	0.930120
$965$	− 1207.28i	− 1.25107i
$966$	0	0
$967$	677.964	0.701101	0.350550	−	0.936544i	$-0.385995\pi$
$967$	677.964	0.701101	0.350550	+	0.936544i	$0.385995\pi$
$968$	− 335.285i	− 0.346369i
$969$	0	0
$970$	1112.92	1.14734
$971$	− 172.492i	− 0.177644i	−0.996048	−	0.0888219i	$-0.971690\pi$
$971$	− 172.492i	− 0.177644i	0.996048	−	0.0888219i	$-0.0283102\pi$
$972$	0	0
$973$	−167.699	−0.172353
$974$	− 73.7594i	− 0.0757284i
$975$	0	0
$976$	−98.1106	−0.100523
$977$	− 1647.98i	− 1.68677i	−0.537308	−	0.843386i	$-0.680559\pi$
$977$	− 1647.98i	− 1.68677i	0.537308	−	0.843386i	$-0.319441\pi$
$978$	0	0
$979$	1417.22	1.44762
$980$	− 754.727i	− 0.770129i
$981$	0	0
$982$	111.740	0.113788
$983$	− 775.987i	− 0.789407i	−0.918809	−	0.394703i	$-0.870847\pi$
$983$	− 775.987i	− 0.789407i	0.918809	−	0.394703i	$-0.129153\pi$
$984$	0	0
$985$	969.476	0.984240
$986$	− 985.486i	− 0.999478i
$987$	0	0
$988$	−19.6570	−0.0198958
$989$	1362.47i	1.37762i
$990$	0	0
$991$	383.377	0.386858	0.193429	−	0.981114i	$-0.438039\pi$
$991$	383.377	0.386858	0.193429	+	0.981114i	$0.438039\pi$
$992$	280.255i	0.282515i
$993$	0	0
$994$	35.9734	0.0361905
$995$	− 1836.20i	− 1.84543i
$996$	0	0
$997$	1548.91	1.55357	0.776787	−	0.629764i	$-0.216848\pi$
$997$	1548.91	1.55357	0.776787	+	0.629764i	$0.216848\pi$
$998$	42.1265i	0.0422109i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.3		36
3.2	odd	2	inner	1458.3.b.c.1457.34		36
27.2	odd	18		162.3.f.a.17.6		36
27.13	even	9		162.3.f.a.143.6		36
27.14	odd	18		54.3.f.a.47.1	yes	36
27.25	even	9		54.3.f.a.23.1	✓	36
108.79	odd	18		432.3.bc.c.401.6		36
108.95	even	18		432.3.bc.c.209.6		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.23.1	✓	36	27.25	even	9
54.3.f.a.47.1	yes	36	27.14	odd	18
162.3.f.a.17.6		36	27.2	odd	18
162.3.f.a.143.6		36	27.13	even	9
432.3.bc.c.209.6		36	108.95	even	18
432.3.bc.c.401.6		36	108.79	odd	18
1458.3.b.c.1457.3		36	1.1	even	1	trivial
1458.3.b.c.1457.34		36	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1458.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -7.83103i q^{5} -0.901000 q^{7} +2.82843i q^{8} -11.0748 q^{10} +15.4771i q^{11} +0.909007 q^{13} +1.27421i q^{14} +4.00000 q^{16} -12.3174i q^{17} +10.8124 q^{19} +15.6621i q^{20} +21.8880 q^{22} +32.8204i q^{23} -36.3251 q^{25} -1.28553i q^{26} +1.80200 q^{28} +56.5740i q^{29} -49.5425 q^{31} -5.65685i q^{32} -17.4194 q^{34} +7.05576i q^{35} +34.8834 q^{37} -15.2910i q^{38} +22.1495 q^{40} +35.8136i q^{41} +41.5129 q^{43} -30.9542i q^{44} +46.4151 q^{46} -24.7597i q^{47} -48.1882 q^{49} +51.3714i q^{50} -1.81801 q^{52} +50.3340i q^{53} +121.202 q^{55} -2.54841i q^{56} +80.0077 q^{58} +66.7263i q^{59} -24.5276 q^{61} +70.0637i q^{62} -8.00000 q^{64} -7.11846i q^{65} +65.2216 q^{67} +24.6348i q^{68} +9.97835 q^{70} -28.2320i q^{71} +21.5521 q^{73} -49.3326i q^{74} -21.6247 q^{76} -13.9449i q^{77} +28.9712 q^{79} -31.3241i q^{80} +50.6481 q^{82} -0.148857i q^{83} -96.4578 q^{85} -58.7081i q^{86} -43.7759 q^{88} -91.5686i q^{89} -0.819015 q^{91} -65.6408i q^{92} -35.0155 q^{94} -84.6721i q^{95} -100.491 q^{97} +68.1484i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more