LMFDB - Embedded newform 405.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+5.00000 q^{2} +17.0000 q^{4} -5.00000 q^{5} +9.00000 q^{7} +45.0000 q^{8} +O(q^{10})$

\(q+5.00000 q^{2} +17.0000 q^{4} -5.00000 q^{5} +9.00000 q^{7} +45.0000 q^{8} -25.0000 q^{10} +8.00000 q^{11} +43.0000 q^{13} +45.0000 q^{14} +89.0000 q^{16} +122.000 q^{17} -59.0000 q^{19} -85.0000 q^{20} +40.0000 q^{22} +213.000 q^{23} +25.0000 q^{25} +215.000 q^{26} +153.000 q^{28} -224.000 q^{29} -36.0000 q^{31} +85.0000 q^{32} +610.000 q^{34} -45.0000 q^{35} +206.000 q^{37} -295.000 q^{38} -225.000 q^{40} -413.000 q^{41} -392.000 q^{43} +136.000 q^{44} +1065.00 q^{46} +311.000 q^{47} -262.000 q^{49} +125.000 q^{50} +731.000 q^{52} +377.000 q^{53} -40.0000 q^{55} +405.000 q^{56} -1120.00 q^{58} -337.000 q^{59} +40.0000 q^{61} -180.000 q^{62} -287.000 q^{64} -215.000 q^{65} +348.000 q^{67} +2074.00 q^{68} -225.000 q^{70} -62.0000 q^{71} -1214.00 q^{73} +1030.00 q^{74} -1003.00 q^{76} +72.0000 q^{77} -294.000 q^{79} -445.000 q^{80} -2065.00 q^{82} -534.000 q^{83} -610.000 q^{85} -1960.00 q^{86} +360.000 q^{88} +810.000 q^{89} +387.000 q^{91} +3621.00 q^{92} +1555.00 q^{94} +295.000 q^{95} -928.000 q^{97} -1310.00 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	5.00000	1.76777	0.883883	−	0.467707i	$-0.154920\pi$
$2$	5.00000	1.76777	0.883883	+	0.467707i	$0.154920\pi$
$3$	0	0
$3$	0	0
$4$	17.0000	2.12500
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	9.00000	0.485954	0.242977	−	0.970032i	$-0.421876\pi$
$7$	9.00000	0.485954	0.242977	+	0.970032i	$0.421876\pi$
$8$	45.0000	1.98874
$9$	0	0
$10$	−25.0000	−0.790569
$11$	8.00000	0.219281	0.109640	−	0.993971i	$-0.465030\pi$
$11$	8.00000	0.219281	0.109640	+	0.993971i	$0.465030\pi$
$12$	0	0
$13$	43.0000	0.917389	0.458694	−	0.888594i	$-0.348317\pi$
$13$	43.0000	0.917389	0.458694	+	0.888594i	$0.348317\pi$
$14$	45.0000	0.859054
$15$	0	0
$16$	89.0000	1.39062
$17$	122.000	1.74055	0.870275	−	0.492566i	$-0.163941\pi$
$17$	122.000	1.74055	0.870275	+	0.492566i	$0.163941\pi$
$18$	0	0
$19$	−59.0000	−0.712396	−0.356198	−	0.934410i	$-0.615927\pi$
$19$	−59.0000	−0.712396	−0.356198	+	0.934410i	$0.615927\pi$
$20$	−85.0000	−0.950329
$21$	0	0
$22$	40.0000	0.387638
$23$	213.000	1.93102	0.965512	−	0.260357i	$-0.0838403\pi$
$23$	213.000	1.93102	0.965512	+	0.260357i	$0.0838403\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	215.000	1.62173
$27$	0	0
$28$	153.000	1.03265
$29$	−224.000	−1.43434	−0.717168	−	0.696900i	$-0.754562\pi$
$29$	−224.000	−1.43434	−0.717168	+	0.696900i	$0.754562\pi$
$30$	0	0
$31$	−36.0000	−0.208574	−0.104287	−	0.994547i	$-0.533256\pi$
$31$	−36.0000	−0.208574	−0.104287	+	0.994547i	$0.533256\pi$
$32$	85.0000	0.469563
$33$	0	0
$34$	610.000	3.07689
$35$	−45.0000	−0.217325
$36$	0	0
$37$	206.000	0.915302	0.457651	−	0.889132i	$-0.348691\pi$
$37$	206.000	0.915302	0.457651	+	0.889132i	$0.348691\pi$
$38$	−295.000	−1.25935
$39$	0	0
$40$	−225.000	−0.889391
$41$	−413.000	−1.57316	−0.786582	−	0.617485i	$-0.788152\pi$
$41$	−413.000	−1.57316	−0.786582	+	0.617485i	$0.788152\pi$
$42$	0	0
$43$	−392.000	−1.39022	−0.695110	−	0.718904i	$-0.744644\pi$
$43$	−392.000	−1.39022	−0.695110	+	0.718904i	$0.744644\pi$
$44$	136.000	0.465972
$45$	0	0
$46$	1065.00	3.41360
$47$	311.000	0.965192	0.482596	−	0.875843i	$-0.339694\pi$
$47$	311.000	0.965192	0.482596	+	0.875843i	$0.339694\pi$
$48$	0	0
$49$	−262.000	−0.763848
$50$	125.000	0.353553
$51$	0	0
$52$	731.000	1.94945
$53$	377.000	0.977074	0.488537	−	0.872543i	$-0.337531\pi$
$53$	377.000	0.977074	0.488537	+	0.872543i	$0.337531\pi$
$54$	0	0
$55$	−40.0000	−0.0980654
$56$	405.000	0.966436
$57$	0	0
$58$	−1120.00	−2.53557
$59$	−337.000	−0.743621	−0.371811	−	0.928309i	$-0.621263\pi$
$59$	−337.000	−0.743621	−0.371811	+	0.928309i	$0.621263\pi$
$60$	0	0
$61$	40.0000	0.0839586	0.0419793	−	0.999118i	$-0.486634\pi$
$61$	40.0000	0.0839586	0.0419793	+	0.999118i	$0.486634\pi$
$62$	−180.000	−0.368710
$63$	0	0
$64$	−287.000	−0.560547
$65$	−215.000	−0.410269
$66$	0	0
$67$	348.000	0.634552	0.317276	−	0.948333i	$-0.397232\pi$
$67$	348.000	0.634552	0.317276	+	0.948333i	$0.397232\pi$
$68$	2074.00	3.69867
$69$	0	0
$70$	−225.000	−0.384181
$71$	−62.0000	−0.103634	−0.0518172	−	0.998657i	$-0.516501\pi$
$71$	−62.0000	−0.103634	−0.0518172	+	0.998657i	$0.516501\pi$
$72$	0	0
$73$	−1214.00	−1.94641	−0.973205	−	0.229939i	$-0.926147\pi$
$73$	−1214.00	−1.94641	−0.973205	+	0.229939i	$0.926147\pi$
$74$	1030.00	1.61804
$75$	0	0
$76$	−1003.00	−1.51384
$77$	72.0000	0.106561
$78$	0	0
$79$	−294.000	−0.418704	−0.209352	−	0.977840i	$-0.567135\pi$
$79$	−294.000	−0.418704	−0.209352	+	0.977840i	$0.567135\pi$
$80$	−445.000	−0.621906
$81$	0	0
$82$	−2065.00	−2.78099
$83$	−534.000	−0.706194	−0.353097	−	0.935587i	$-0.614871\pi$
$83$	−534.000	−0.706194	−0.353097	+	0.935587i	$0.614871\pi$
$84$	0	0
$85$	−610.000	−0.778398
$86$	−1960.00	−2.45758
$87$	0	0
$88$	360.000	0.436092
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	387.000	0.445809
$92$	3621.00	4.10343
$93$	0	0
$94$	1555.00	1.70623
$95$	295.000	0.318593
$96$	0	0
$97$	−928.000	−0.971383	−0.485691	−	0.874130i	$-0.661432\pi$
$97$	−928.000	−0.971383	−0.485691	+	0.874130i	$0.661432\pi$
$98$	−1310.00	−1.35031
$99$	0	0
$100$	425.000	0.425000
$101$	996.000	0.981245	0.490622	−	0.871372i	$-0.336769\pi$
$101$	996.000	0.981245	0.490622	+	0.871372i	$0.336769\pi$
$102$	0	0
$103$	−433.000	−0.414221	−0.207110	−	0.978318i	$-0.566406\pi$
$103$	−433.000	−0.414221	−0.207110	+	0.978318i	$0.566406\pi$
$104$	1935.00	1.82445
$105$	0	0
$106$	1885.00	1.72724
$107$	1686.00	1.52329	0.761644	−	0.647996i	$-0.224393\pi$
$107$	1686.00	1.52329	0.761644	+	0.647996i	$0.224393\pi$
$108$	0	0
$109$	656.000	0.576453	0.288227	−	0.957562i	$-0.406934\pi$
$109$	656.000	0.576453	0.288227	+	0.957562i	$0.406934\pi$
$110$	−200.000	−0.173357
$111$	0	0
$112$	801.000	0.675780
$113$	−1018.00	−0.847481	−0.423741	−	0.905784i	$-0.639283\pi$
$113$	−1018.00	−0.847481	−0.423741	+	0.905784i	$0.639283\pi$
$114$	0	0
$115$	−1065.00	−0.863581
$116$	−3808.00	−3.04796
$117$	0	0
$118$	−1685.00	−1.31455
$119$	1098.00	0.845828
$120$	0	0
$121$	−1267.00	−0.951916
$122$	200.000	0.148419
$123$	0	0
$124$	−612.000	−0.443220
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1361.00	−0.950939	−0.475469	−	0.879732i	$-0.657722\pi$
$127$	−1361.00	−0.950939	−0.475469	+	0.879732i	$0.657722\pi$
$128$	−2115.00	−1.46048
$129$	0	0
$130$	−1075.00	−0.725260
$131$	−1911.00	−1.27454	−0.637270	−	0.770640i	$-0.719937\pi$
$131$	−1911.00	−1.27454	−0.637270	+	0.770640i	$0.719937\pi$
$132$	0	0
$133$	−531.000	−0.346192
$134$	1740.00	1.12174
$135$	0	0
$136$	5490.00	3.46150
$137$	−654.000	−0.407847	−0.203923	−	0.978987i	$-0.565369\pi$
$137$	−654.000	−0.407847	−0.203923	+	0.978987i	$0.565369\pi$
$138$	0	0
$139$	−733.000	−0.447282	−0.223641	−	0.974672i	$-0.571794\pi$
$139$	−733.000	−0.447282	−0.223641	+	0.974672i	$0.571794\pi$
$140$	−765.000	−0.461816
$141$	0	0
$142$	−310.000	−0.183202
$143$	344.000	0.201166
$144$	0	0
$145$	1120.00	0.641455
$146$	−6070.00	−3.44080
$147$	0	0
$148$	3502.00	1.94502
$149$	1126.00	0.619097	0.309549	−	0.950884i	$-0.399822\pi$
$149$	1126.00	0.619097	0.309549	+	0.950884i	$0.399822\pi$
$150$	0	0
$151$	−2546.00	−1.37212	−0.686061	−	0.727544i	$-0.740662\pi$
$151$	−2546.00	−1.37212	−0.686061	+	0.727544i	$0.740662\pi$
$152$	−2655.00	−1.41677
$153$	0	0
$154$	360.000	0.188374
$155$	180.000	0.0932771
$156$	0	0
$157$	1223.00	0.621694	0.310847	−	0.950460i	$-0.399387\pi$
$157$	1223.00	0.621694	0.310847	+	0.950460i	$0.399387\pi$
$158$	−1470.00	−0.740170
$159$	0	0
$160$	−425.000	−0.209995
$161$	1917.00	0.938390
$162$	0	0
$163$	3176.00	1.52616	0.763078	−	0.646306i	$-0.223687\pi$
$163$	3176.00	1.52616	0.763078	+	0.646306i	$0.223687\pi$
$164$	−7021.00	−3.34298
$165$	0	0
$166$	−2670.00	−1.24839
$167$	−132.000	−0.0611645	−0.0305822	−	0.999532i	$-0.509736\pi$
$167$	−132.000	−0.0611645	−0.0305822	+	0.999532i	$0.509736\pi$
$168$	0	0
$169$	−348.000	−0.158398
$170$	−3050.00	−1.37603
$171$	0	0
$172$	−6664.00	−2.95422
$173$	993.000	0.436395	0.218198	−	0.975905i	$-0.429982\pi$
$173$	993.000	0.436395	0.218198	+	0.975905i	$0.429982\pi$
$174$	0	0
$175$	225.000	0.0971909
$176$	712.000	0.304938
$177$	0	0
$178$	4050.00	1.70540
$179$	3101.00	1.29486	0.647429	−	0.762126i	$-0.275844\pi$
$179$	3101.00	1.29486	0.647429	+	0.762126i	$0.275844\pi$
$180$	0	0
$181$	2846.00	1.16874	0.584369	−	0.811488i	$-0.301342\pi$
$181$	2846.00	1.16874	0.584369	+	0.811488i	$0.301342\pi$
$182$	1935.00	0.788086
$183$	0	0
$184$	9585.00	3.84030
$185$	−1030.00	−0.409336
$186$	0	0
$187$	976.000	0.381669
$188$	5287.00	2.05103
$189$	0	0
$190$	1475.00	0.563199
$191$	−3080.00	−1.16681	−0.583406	−	0.812181i	$-0.698280\pi$
$191$	−3080.00	−1.16681	−0.583406	+	0.812181i	$0.698280\pi$
$192$	0	0
$193$	2588.00	0.965224	0.482612	−	0.875834i	$-0.339688\pi$
$193$	2588.00	0.965224	0.482612	+	0.875834i	$0.339688\pi$
$194$	−4640.00	−1.71718
$195$	0	0
$196$	−4454.00	−1.62318
$197$	1335.00	0.482816	0.241408	−	0.970424i	$-0.422391\pi$
$197$	1335.00	0.482816	0.241408	+	0.970424i	$0.422391\pi$
$198$	0	0
$199$	−5204.00	−1.85378	−0.926889	−	0.375336i	$-0.877527\pi$
$199$	−5204.00	−1.85378	−0.926889	+	0.375336i	$0.877527\pi$
$200$	1125.00	0.397748
$201$	0	0
$202$	4980.00	1.73461
$203$	−2016.00	−0.697022
$204$	0	0
$205$	2065.00	0.703541
$206$	−2165.00	−0.732246
$207$	0	0
$208$	3827.00	1.27574
$209$	−472.000	−0.156215
$210$	0	0
$211$	1637.00	0.534103	0.267051	−	0.963682i	$-0.413951\pi$
$211$	1637.00	0.534103	0.267051	+	0.963682i	$0.413951\pi$
$212$	6409.00	2.07628
$213$	0	0
$214$	8430.00	2.69282
$215$	1960.00	0.621725
$216$	0	0
$217$	−324.000	−0.101357
$218$	3280.00	1.01904
$219$	0	0
$220$	−680.000	−0.208389
$221$	5246.00	1.59676
$222$	0	0
$223$	−4480.00	−1.34530	−0.672652	−	0.739959i	$-0.734845\pi$
$223$	−4480.00	−1.34530	−0.672652	+	0.739959i	$0.734845\pi$
$224$	765.000	0.228186
$225$	0	0
$226$	−5090.00	−1.49815
$227$	3736.00	1.09237	0.546183	−	0.837666i	$-0.316080\pi$
$227$	3736.00	1.09237	0.546183	+	0.837666i	$0.316080\pi$
$228$	0	0
$229$	−1380.00	−0.398223	−0.199111	−	0.979977i	$-0.563806\pi$
$229$	−1380.00	−0.398223	−0.199111	+	0.979977i	$0.563806\pi$
$230$	−5325.00	−1.52661
$231$	0	0
$232$	−10080.0	−2.85252
$233$	2904.00	0.816512	0.408256	−	0.912867i	$-0.366137\pi$
$233$	2904.00	0.816512	0.408256	+	0.912867i	$0.366137\pi$
$234$	0	0
$235$	−1555.00	−0.431647
$236$	−5729.00	−1.58020
$237$	0	0
$238$	5490.00	1.49523
$239$	5966.00	1.61468	0.807340	−	0.590087i	$-0.200906\pi$
$239$	5966.00	1.61468	0.807340	+	0.590087i	$0.200906\pi$
$240$	0	0
$241$	−3218.00	−0.860123	−0.430061	−	0.902800i	$-0.641508\pi$
$241$	−3218.00	−0.860123	−0.430061	+	0.902800i	$0.641508\pi$
$242$	−6335.00	−1.68277
$243$	0	0
$244$	680.000	0.178412
$245$	1310.00	0.341603
$246$	0	0
$247$	−2537.00	−0.653544
$248$	−1620.00	−0.414799
$249$	0	0
$250$	−625.000	−0.158114
$251$	−6123.00	−1.53976	−0.769881	−	0.638187i	$-0.779685\pi$
$251$	−6123.00	−1.53976	−0.769881	+	0.638187i	$0.779685\pi$
$252$	0	0
$253$	1704.00	0.423437
$254$	−6805.00	−1.68104
$255$	0	0
$256$	−8279.00	−2.02124
$257$	1398.00	0.339318	0.169659	−	0.985503i	$-0.445733\pi$
$257$	1398.00	0.339318	0.169659	+	0.985503i	$0.445733\pi$
$258$	0	0
$259$	1854.00	0.444795
$260$	−3655.00	−0.871821
$261$	0	0
$262$	−9555.00	−2.25309
$263$	3211.00	0.752847	0.376423	−	0.926448i	$-0.377154\pi$
$263$	3211.00	0.752847	0.376423	+	0.926448i	$0.377154\pi$
$264$	0	0
$265$	−1885.00	−0.436961
$266$	−2655.00	−0.611987
$267$	0	0
$268$	5916.00	1.34842
$269$	−4018.00	−0.910713	−0.455356	−	0.890309i	$-0.650488\pi$
$269$	−4018.00	−0.910713	−0.455356	+	0.890309i	$0.650488\pi$
$270$	0	0
$271$	2314.00	0.518692	0.259346	−	0.965784i	$-0.416493\pi$
$271$	2314.00	0.518692	0.259346	+	0.965784i	$0.416493\pi$
$272$	10858.0	2.42045
$273$	0	0
$274$	−3270.00	−0.720978
$275$	200.000	0.0438562
$276$	0	0
$277$	−4347.00	−0.942909	−0.471455	−	0.881890i	$-0.656271\pi$
$277$	−4347.00	−0.942909	−0.471455	+	0.881890i	$0.656271\pi$
$278$	−3665.00	−0.790691
$279$	0	0
$280$	−2025.00	−0.432203
$281$	−1551.00	−0.329270	−0.164635	−	0.986355i	$-0.552645\pi$
$281$	−1551.00	−0.329270	−0.164635	+	0.986355i	$0.552645\pi$
$282$	0	0
$283$	4380.00	0.920014	0.460007	−	0.887915i	$-0.347847\pi$
$283$	4380.00	0.920014	0.460007	+	0.887915i	$0.347847\pi$
$284$	−1054.00	−0.220223
$285$	0	0
$286$	1720.00	0.355614
$287$	−3717.00	−0.764486
$288$	0	0
$289$	9971.00	2.02951
$290$	5600.00	1.13394
$291$	0	0
$292$	−20638.0	−4.13612
$293$	−5049.00	−1.00671	−0.503354	−	0.864080i	$-0.667901\pi$
$293$	−5049.00	−1.00671	−0.503354	+	0.864080i	$0.667901\pi$
$294$	0	0
$295$	1685.00	0.332558
$296$	9270.00	1.82030
$297$	0	0
$298$	5630.00	1.09442
$299$	9159.00	1.77150
$300$	0	0
$301$	−3528.00	−0.675583
$302$	−12730.0	−2.42559
$303$	0	0
$304$	−5251.00	−0.990676
$305$	−200.000	−0.0375474
$306$	0	0
$307$	5428.00	1.00910	0.504548	−	0.863384i	$-0.331659\pi$
$307$	5428.00	1.00910	0.504548	+	0.863384i	$0.331659\pi$
$308$	1224.00	0.226441
$309$	0	0
$310$	900.000	0.164892
$311$	18.0000	0.00328195	0.00164097	−	0.999999i	$-0.499478\pi$
$311$	18.0000	0.00328195	0.00164097	+	0.999999i	$0.499478\pi$
$312$	0	0
$313$	−2116.00	−0.382119	−0.191060	−	0.981578i	$-0.561192\pi$
$313$	−2116.00	−0.382119	−0.191060	+	0.981578i	$0.561192\pi$
$314$	6115.00	1.09901
$315$	0	0
$316$	−4998.00	−0.889745
$317$	−4415.00	−0.782243	−0.391122	−	0.920339i	$-0.627913\pi$
$317$	−4415.00	−0.782243	−0.391122	+	0.920339i	$0.627913\pi$
$318$	0	0
$319$	−1792.00	−0.314523
$320$	1435.00	0.250684
$321$	0	0
$322$	9585.00	1.65885
$323$	−7198.00	−1.23996
$324$	0	0
$325$	1075.00	0.183478
$326$	15880.0	2.69789
$327$	0	0
$328$	−18585.0	−3.12861
$329$	2799.00	0.469039
$330$	0	0
$331$	3480.00	0.577879	0.288940	−	0.957347i	$-0.406697\pi$
$331$	3480.00	0.577879	0.288940	+	0.957347i	$0.406697\pi$
$332$	−9078.00	−1.50066
$333$	0	0
$334$	−660.000	−0.108125
$335$	−1740.00	−0.283780
$336$	0	0
$337$	6322.00	1.02190	0.510951	−	0.859610i	$-0.329293\pi$
$337$	6322.00	1.02190	0.510951	+	0.859610i	$0.329293\pi$
$338$	−1740.00	−0.280010
$339$	0	0
$340$	−10370.0	−1.65409
$341$	−288.000	−0.0457363
$342$	0	0
$343$	−5445.00	−0.857150
$344$	−17640.0	−2.76478
$345$	0	0
$346$	4965.00	0.771445
$347$	10034.0	1.55232	0.776158	−	0.630539i	$-0.217166\pi$
$347$	10034.0	1.55232	0.776158	+	0.630539i	$0.217166\pi$
$348$	0	0
$349$	−2510.00	−0.384978	−0.192489	−	0.981299i	$-0.561656\pi$
$349$	−2510.00	−0.384978	−0.192489	+	0.981299i	$0.561656\pi$
$350$	1125.00	0.171811
$351$	0	0
$352$	680.000	0.102966
$353$	−3726.00	−0.561799	−0.280899	−	0.959737i	$-0.590633\pi$
$353$	−3726.00	−0.561799	−0.280899	+	0.959737i	$0.590633\pi$
$354$	0	0
$355$	310.000	0.0463467
$356$	13770.0	2.05002
$357$	0	0
$358$	15505.0	2.28901
$359$	10710.0	1.57452	0.787259	−	0.616622i	$-0.211499\pi$
$359$	10710.0	1.57452	0.787259	+	0.616622i	$0.211499\pi$
$360$	0	0
$361$	−3378.00	−0.492492
$362$	14230.0	2.06606
$363$	0	0
$364$	6579.00	0.947344
$365$	6070.00	0.870461
$366$	0	0
$367$	−2160.00	−0.307224	−0.153612	−	0.988131i	$-0.549091\pi$
$367$	−2160.00	−0.307224	−0.153612	+	0.988131i	$0.549091\pi$
$368$	18957.0	2.68533
$369$	0	0
$370$	−5150.00	−0.723610
$371$	3393.00	0.474813
$372$	0	0
$373$	3394.00	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$373$	3394.00	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$374$	4880.00	0.674703
$375$	0	0
$376$	13995.0	1.91951
$377$	−9632.00	−1.31584
$378$	0	0
$379$	9031.00	1.22399	0.611994	−	0.790863i	$-0.290368\pi$
$379$	9031.00	1.22399	0.611994	+	0.790863i	$0.290368\pi$
$380$	5015.00	0.677011
$381$	0	0
$382$	−15400.0	−2.06265
$383$	10305.0	1.37483	0.687416	−	0.726264i	$-0.258745\pi$
$383$	10305.0	1.37483	0.687416	+	0.726264i	$0.258745\pi$
$384$	0	0
$385$	−360.000	−0.0476553
$386$	12940.0	1.70629
$387$	0	0
$388$	−15776.0	−2.06419
$389$	3480.00	0.453581	0.226790	−	0.973944i	$-0.427177\pi$
$389$	3480.00	0.453581	0.226790	+	0.973944i	$0.427177\pi$
$390$	0	0
$391$	25986.0	3.36104
$392$	−11790.0	−1.51909
$393$	0	0
$394$	6675.00	0.853507
$395$	1470.00	0.187250
$396$	0	0
$397$	3706.00	0.468511	0.234255	−	0.972175i	$-0.424735\pi$
$397$	3706.00	0.468511	0.234255	+	0.972175i	$0.424735\pi$
$398$	−26020.0	−3.27705
$399$	0	0
$400$	2225.00	0.278125
$401$	−2679.00	−0.333623	−0.166812	−	0.985989i	$-0.553347\pi$
$401$	−2679.00	−0.333623	−0.166812	+	0.985989i	$0.553347\pi$
$402$	0	0
$403$	−1548.00	−0.191343
$404$	16932.0	2.08514
$405$	0	0
$406$	−10080.0	−1.23217
$407$	1648.00	0.200708
$408$	0	0
$409$	−12499.0	−1.51109	−0.755545	−	0.655097i	$-0.772628\pi$
$409$	−12499.0	−1.51109	−0.755545	+	0.655097i	$0.772628\pi$
$410$	10325.0	1.24370
$411$	0	0
$412$	−7361.00	−0.880220
$413$	−3033.00	−0.361366
$414$	0	0
$415$	2670.00	0.315820
$416$	3655.00	0.430772
$417$	0	0
$418$	−2360.00	−0.276152
$419$	928.000	0.108200	0.0541000	−	0.998536i	$-0.482771\pi$
$419$	928.000	0.108200	0.0541000	+	0.998536i	$0.482771\pi$
$420$	0	0
$421$	7570.00	0.876340	0.438170	−	0.898892i	$-0.355627\pi$
$421$	7570.00	0.876340	0.438170	+	0.898892i	$0.355627\pi$
$422$	8185.00	0.944170
$423$	0	0
$424$	16965.0	1.94314
$425$	3050.00	0.348110
$426$	0	0
$427$	360.000	0.0408000
$428$	28662.0	3.23699
$429$	0	0
$430$	9800.00	1.09907
$431$	2460.00	0.274928	0.137464	−	0.990507i	$-0.456105\pi$
$431$	2460.00	0.274928	0.137464	+	0.990507i	$0.456105\pi$
$432$	0	0
$433$	1648.00	0.182905	0.0914525	−	0.995809i	$-0.470849\pi$
$433$	1648.00	0.182905	0.0914525	+	0.995809i	$0.470849\pi$
$434$	−1620.00	−0.179176
$435$	0	0
$436$	11152.0	1.22496
$437$	−12567.0	−1.37565
$438$	0	0
$439$	15826.0	1.72058	0.860289	−	0.509807i	$-0.170283\pi$
$439$	15826.0	1.72058	0.860289	+	0.509807i	$0.170283\pi$
$440$	−1800.00	−0.195026
$441$	0	0
$442$	26230.0	2.82270
$443$	−12774.0	−1.37000	−0.685001	−	0.728542i	$-0.740198\pi$
$443$	−12774.0	−1.37000	−0.685001	+	0.728542i	$0.740198\pi$
$444$	0	0
$445$	−4050.00	−0.431435
$446$	−22400.0	−2.37819
$447$	0	0
$448$	−2583.00	−0.272400
$449$	−8875.00	−0.932822	−0.466411	−	0.884568i	$-0.654453\pi$
$449$	−8875.00	−0.932822	−0.466411	+	0.884568i	$0.654453\pi$
$450$	0	0
$451$	−3304.00	−0.344965
$452$	−17306.0	−1.80090
$453$	0	0
$454$	18680.0	1.93105
$455$	−1935.00	−0.199372
$456$	0	0
$457$	11524.0	1.17958	0.589792	−	0.807555i	$-0.299210\pi$
$457$	11524.0	1.17958	0.589792	+	0.807555i	$0.299210\pi$
$458$	−6900.00	−0.703965
$459$	0	0
$460$	−18105.0	−1.83511
$461$	8544.00	0.863197	0.431598	−	0.902066i	$-0.357950\pi$
$461$	8544.00	0.863197	0.431598	+	0.902066i	$0.357950\pi$
$462$	0	0
$463$	2523.00	0.253248	0.126624	−	0.991951i	$-0.459586\pi$
$463$	2523.00	0.253248	0.126624	+	0.991951i	$0.459586\pi$
$464$	−19936.0	−1.99462
$465$	0	0
$466$	14520.0	1.44340
$467$	2902.00	0.287556	0.143778	−	0.989610i	$-0.454075\pi$
$467$	2902.00	0.287556	0.143778	+	0.989610i	$0.454075\pi$
$468$	0	0
$469$	3132.00	0.308363
$470$	−7775.00	−0.763051
$471$	0	0
$472$	−15165.0	−1.47887
$473$	−3136.00	−0.304849
$474$	0	0
$475$	−1475.00	−0.142479
$476$	18666.0	1.79738
$477$	0	0
$478$	29830.0	2.85438
$479$	−4362.00	−0.416085	−0.208043	−	0.978120i	$-0.566709\pi$
$479$	−4362.00	−0.416085	−0.208043	+	0.978120i	$0.566709\pi$
$480$	0	0
$481$	8858.00	0.839688
$482$	−16090.0	−1.52050
$483$	0	0
$484$	−21539.0	−2.02282
$485$	4640.00	0.434416
$486$	0	0
$487$	5723.00	0.532513	0.266257	−	0.963902i	$-0.414213\pi$
$487$	5723.00	0.532513	0.266257	+	0.963902i	$0.414213\pi$
$488$	1800.00	0.166972
$489$	0	0
$490$	6550.00	0.603875
$491$	13339.0	1.22603	0.613015	−	0.790071i	$-0.289957\pi$
$491$	13339.0	1.22603	0.613015	+	0.790071i	$0.289957\pi$
$492$	0	0
$493$	−27328.0	−2.49653
$494$	−12685.0	−1.15531
$495$	0	0
$496$	−3204.00	−0.290048
$497$	−558.000	−0.0503616
$498$	0	0
$499$	−19637.0	−1.76167	−0.880835	−	0.473424i	$-0.843018\pi$
$499$	−19637.0	−1.76167	−0.880835	+	0.473424i	$0.843018\pi$
$500$	−2125.00	−0.190066
$501$	0	0
$502$	−30615.0	−2.72194
$503$	5416.00	0.480094	0.240047	−	0.970761i	$-0.422837\pi$
$503$	5416.00	0.480094	0.240047	+	0.970761i	$0.422837\pi$
$504$	0	0
$505$	−4980.00	−0.438826
$506$	8520.00	0.748538
$507$	0	0
$508$	−23137.0	−2.02074
$509$	6110.00	0.532065	0.266032	−	0.963964i	$-0.414287\pi$
$509$	6110.00	0.532065	0.266032	+	0.963964i	$0.414287\pi$
$510$	0	0
$511$	−10926.0	−0.945867
$512$	−24475.0	−2.11260
$513$	0	0
$514$	6990.00	0.599836
$515$	2165.00	0.185245
$516$	0	0
$517$	2488.00	0.211648
$518$	9270.00	0.786294
$519$	0	0
$520$	−9675.00	−0.815917
$521$	−20375.0	−1.71333	−0.856665	−	0.515873i	$-0.827468\pi$
$521$	−20375.0	−1.71333	−0.856665	+	0.515873i	$0.827468\pi$
$522$	0	0
$523$	−19010.0	−1.58939	−0.794693	−	0.607011i	$-0.792368\pi$
$523$	−19010.0	−1.58939	−0.794693	+	0.607011i	$0.792368\pi$
$524$	−32487.0	−2.70840
$525$	0	0
$526$	16055.0	1.33086
$527$	−4392.00	−0.363033
$528$	0	0
$529$	33202.0	2.72886
$530$	−9425.00	−0.772445
$531$	0	0
$532$	−9027.00	−0.735658
$533$	−17759.0	−1.44320
$534$	0	0
$535$	−8430.00	−0.681235
$536$	15660.0	1.26196
$537$	0	0
$538$	−20090.0	−1.60993
$539$	−2096.00	−0.167497
$540$	0	0
$541$	3288.00	0.261298	0.130649	−	0.991429i	$-0.458294\pi$
$541$	3288.00	0.261298	0.130649	+	0.991429i	$0.458294\pi$
$542$	11570.0	0.916926
$543$	0	0
$544$	10370.0	0.817298
$545$	−3280.00	−0.257798
$546$	0	0
$547$	3256.00	0.254509	0.127255	−	0.991870i	$-0.459383\pi$
$547$	3256.00	0.254509	0.127255	+	0.991870i	$0.459383\pi$
$548$	−11118.0	−0.866674
$549$	0	0
$550$	1000.00	0.0775275
$551$	13216.0	1.02182
$552$	0	0
$553$	−2646.00	−0.203471
$554$	−21735.0	−1.66684
$555$	0	0
$556$	−12461.0	−0.950475
$557$	−213.000	−0.0162031	−0.00810153	−	0.999967i	$-0.502579\pi$
$557$	−213.000	−0.0162031	−0.00810153	+	0.999967i	$0.502579\pi$
$558$	0	0
$559$	−16856.0	−1.27537
$560$	−4005.00	−0.302218
$561$	0	0
$562$	−7755.00	−0.582073
$563$	−17388.0	−1.30163	−0.650814	−	0.759237i	$-0.725572\pi$
$563$	−17388.0	−1.30163	−0.650814	+	0.759237i	$0.725572\pi$
$564$	0	0
$565$	5090.00	0.379005
$566$	21900.0	1.62637
$567$	0	0
$568$	−2790.00	−0.206102
$569$	4353.00	0.320716	0.160358	−	0.987059i	$-0.448735\pi$
$569$	4353.00	0.320716	0.160358	+	0.987059i	$0.448735\pi$
$570$	0	0
$571$	−1924.00	−0.141010	−0.0705052	−	0.997511i	$-0.522461\pi$
$571$	−1924.00	−0.141010	−0.0705052	+	0.997511i	$0.522461\pi$
$572$	5848.00	0.427478
$573$	0	0
$574$	−18585.0	−1.35143
$575$	5325.00	0.386205
$576$	0	0
$577$	16832.0	1.21443	0.607214	−	0.794538i	$-0.292287\pi$
$577$	16832.0	1.21443	0.607214	+	0.794538i	$0.292287\pi$
$578$	49855.0	3.58771
$579$	0	0
$580$	19040.0	1.36309
$581$	−4806.00	−0.343178
$582$	0	0
$583$	3016.00	0.214254
$584$	−54630.0	−3.87090
$585$	0	0
$586$	−25245.0	−1.77963
$587$	2106.00	0.148082	0.0740408	−	0.997255i	$-0.476410\pi$
$587$	2106.00	0.148082	0.0740408	+	0.997255i	$0.476410\pi$
$588$	0	0
$589$	2124.00	0.148587
$590$	8425.00	0.587884
$591$	0	0
$592$	18334.0	1.27284
$593$	−4694.00	−0.325058	−0.162529	−	0.986704i	$-0.551965\pi$
$593$	−4694.00	−0.325058	−0.162529	+	0.986704i	$0.551965\pi$
$594$	0	0
$595$	−5490.00	−0.378266
$596$	19142.0	1.31558
$597$	0	0
$598$	45795.0	3.13160
$599$	−13226.0	−0.902170	−0.451085	−	0.892481i	$-0.648963\pi$
$599$	−13226.0	−0.902170	−0.451085	+	0.892481i	$0.648963\pi$
$600$	0	0
$601$	−13291.0	−0.902082	−0.451041	−	0.892503i	$-0.648947\pi$
$601$	−13291.0	−0.902082	−0.451041	+	0.892503i	$0.648947\pi$
$602$	−17640.0	−1.19427
$603$	0	0
$604$	−43282.0	−2.91576
$605$	6335.00	0.425710
$606$	0	0
$607$	6040.00	0.403881	0.201941	−	0.979398i	$-0.435275\pi$
$607$	6040.00	0.403881	0.201941	+	0.979398i	$0.435275\pi$
$608$	−5015.00	−0.334515
$609$	0	0
$610$	−1000.00	−0.0663751
$611$	13373.0	0.885456
$612$	0	0
$613$	−23.0000	−0.00151543	−0.000757717	−	1.00000i	$-0.500241\pi$
$613$	−23.0000	−0.00151543	−0.000757717		1.00000i	$0.500241\pi$
$614$	27140.0	1.78385
$615$	0	0
$616$	3240.00	0.211921
$617$	3018.00	0.196921	0.0984604	−	0.995141i	$-0.468608\pi$
$617$	3018.00	0.196921	0.0984604	+	0.995141i	$0.468608\pi$
$618$	0	0
$619$	−9439.00	−0.612901	−0.306450	−	0.951887i	$-0.599141\pi$
$619$	−9439.00	−0.612901	−0.306450	+	0.951887i	$0.599141\pi$
$620$	3060.00	0.198214
$621$	0	0
$622$	90.0000	0.00580172
$623$	7290.00	0.468808
$624$	0	0
$625$	625.000	0.0400000
$626$	−10580.0	−0.675498
$627$	0	0
$628$	20791.0	1.32110
$629$	25132.0	1.59313
$630$	0	0
$631$	−9800.00	−0.618275	−0.309138	−	0.951017i	$-0.600040\pi$
$631$	−9800.00	−0.618275	−0.309138	+	0.951017i	$0.600040\pi$
$632$	−13230.0	−0.832692
$633$	0	0
$634$	−22075.0	−1.38282
$635$	6805.00	0.425273
$636$	0	0
$637$	−11266.0	−0.700746
$638$	−8960.00	−0.556003
$639$	0	0
$640$	10575.0	0.653146
$641$	−23442.0	−1.44447	−0.722233	−	0.691649i	$-0.756884\pi$
$641$	−23442.0	−1.44447	−0.722233	+	0.691649i	$0.756884\pi$
$642$	0	0
$643$	31308.0	1.92017	0.960083	−	0.279715i	$-0.0902398\pi$
$643$	31308.0	1.92017	0.960083	+	0.279715i	$0.0902398\pi$
$644$	32589.0	1.99408
$645$	0	0
$646$	−35990.0	−2.19196
$647$	−712.000	−0.0432637	−0.0216318	−	0.999766i	$-0.506886\pi$
$647$	−712.000	−0.0432637	−0.0216318	+	0.999766i	$0.506886\pi$
$648$	0	0
$649$	−2696.00	−0.163062
$650$	5375.00	0.324346
$651$	0	0
$652$	53992.0	3.24308
$653$	−31478.0	−1.88642	−0.943208	−	0.332203i	$-0.892208\pi$
$653$	−31478.0	−1.88642	−0.943208	+	0.332203i	$0.892208\pi$
$654$	0	0
$655$	9555.00	0.569992
$656$	−36757.0	−2.18768
$657$	0	0
$658$	13995.0	0.829152
$659$	16121.0	0.952936	0.476468	−	0.879192i	$-0.341917\pi$
$659$	16121.0	0.952936	0.476468	+	0.879192i	$0.341917\pi$
$660$	0	0
$661$	−19160.0	−1.12744	−0.563720	−	0.825966i	$-0.690630\pi$
$661$	−19160.0	−1.12744	−0.563720	+	0.825966i	$0.690630\pi$
$662$	17400.0	1.02156
$663$	0	0
$664$	−24030.0	−1.40444
$665$	2655.00	0.154822
$666$	0	0
$667$	−47712.0	−2.76974
$668$	−2244.00	−0.129975
$669$	0	0
$670$	−8700.00	−0.501657
$671$	320.000	0.0184105
$672$	0	0
$673$	−13422.0	−0.768767	−0.384383	−	0.923174i	$-0.625586\pi$
$673$	−13422.0	−0.768767	−0.384383	+	0.923174i	$0.625586\pi$
$674$	31610.0	1.80649
$675$	0	0
$676$	−5916.00	−0.336595
$677$	−25905.0	−1.47062	−0.735310	−	0.677731i	$-0.762964\pi$
$677$	−25905.0	−1.47062	−0.735310	+	0.677731i	$0.762964\pi$
$678$	0	0
$679$	−8352.00	−0.472048
$680$	−27450.0	−1.54803
$681$	0	0
$682$	−1440.00	−0.0808511
$683$	9246.00	0.517992	0.258996	−	0.965878i	$-0.416608\pi$
$683$	9246.00	0.517992	0.258996	+	0.965878i	$0.416608\pi$
$684$	0	0
$685$	3270.00	0.182395
$686$	−27225.0	−1.51524
$687$	0	0
$688$	−34888.0	−1.93327
$689$	16211.0	0.896357
$690$	0	0
$691$	25039.0	1.37848	0.689239	−	0.724534i	$-0.257945\pi$
$691$	25039.0	1.37848	0.689239	+	0.724534i	$0.257945\pi$
$692$	16881.0	0.927340
$693$	0	0
$694$	50170.0	2.74413
$695$	3665.00	0.200031
$696$	0	0
$697$	−50386.0	−2.73817
$698$	−12550.0	−0.680551
$699$	0	0
$700$	3825.00	0.206531
$701$	−32930.0	−1.77425	−0.887125	−	0.461530i	$-0.847301\pi$
$701$	−32930.0	−1.77425	−0.887125	+	0.461530i	$0.847301\pi$
$702$	0	0
$703$	−12154.0	−0.652058
$704$	−2296.00	−0.122917
$705$	0	0
$706$	−18630.0	−0.993129
$707$	8964.00	0.476840
$708$	0	0
$709$	1882.00	0.0996897	0.0498448	−	0.998757i	$-0.484127\pi$
$709$	1882.00	0.0996897	0.0498448	+	0.998757i	$0.484127\pi$
$710$	1550.00	0.0819302
$711$	0	0
$712$	36450.0	1.91857
$713$	−7668.00	−0.402761
$714$	0	0
$715$	−1720.00	−0.0899641
$716$	52717.0	2.75157
$717$	0	0
$718$	53550.0	2.78338
$719$	1962.00	0.101767	0.0508833	−	0.998705i	$-0.483796\pi$
$719$	1962.00	0.101767	0.0508833	+	0.998705i	$0.483796\pi$
$720$	0	0
$721$	−3897.00	−0.201292
$722$	−16890.0	−0.870610
$723$	0	0
$724$	48382.0	2.48357
$725$	−5600.00	−0.286867
$726$	0	0
$727$	−13741.0	−0.700998	−0.350499	−	0.936563i	$-0.613988\pi$
$727$	−13741.0	−0.700998	−0.350499	+	0.936563i	$0.613988\pi$
$728$	17415.0	0.886597
$729$	0	0
$730$	30350.0	1.53877
$731$	−47824.0	−2.41975
$732$	0	0
$733$	−32458.0	−1.63556	−0.817779	−	0.575533i	$-0.804795\pi$
$733$	−32458.0	−1.63556	−0.817779	+	0.575533i	$0.804795\pi$
$734$	−10800.0	−0.543100
$735$	0	0
$736$	18105.0	0.906738
$737$	2784.00	0.139145
$738$	0	0
$739$	19612.0	0.976237	0.488118	−	0.872777i	$-0.337683\pi$
$739$	19612.0	0.976237	0.488118	+	0.872777i	$0.337683\pi$
$740$	−17510.0	−0.869838
$741$	0	0
$742$	16965.0	0.839359
$743$	−36736.0	−1.81388	−0.906940	−	0.421259i	$-0.861588\pi$
$743$	−36736.0	−1.81388	−0.906940	+	0.421259i	$0.861588\pi$
$744$	0	0
$745$	−5630.00	−0.276869
$746$	16970.0	0.832863
$747$	0	0
$748$	16592.0	0.811048
$749$	15174.0	0.740248
$750$	0	0
$751$	−3746.00	−0.182015	−0.0910076	−	0.995850i	$-0.529009\pi$
$751$	−3746.00	−0.182015	−0.0910076	+	0.995850i	$0.529009\pi$
$752$	27679.0	1.34222
$753$	0	0
$754$	−48160.0	−2.32611
$755$	12730.0	0.613632
$756$	0	0
$757$	5725.00	0.274873	0.137436	−	0.990511i	$-0.456114\pi$
$757$	5725.00	0.274873	0.137436	+	0.990511i	$0.456114\pi$
$758$	45155.0	2.16372
$759$	0	0
$760$	13275.0	0.633599
$761$	−37323.0	−1.77787	−0.888934	−	0.458035i	$-0.848553\pi$
$761$	−37323.0	−1.77787	−0.888934	+	0.458035i	$0.848553\pi$
$762$	0	0
$763$	5904.00	0.280130
$764$	−52360.0	−2.47947
$765$	0	0
$766$	51525.0	2.43038
$767$	−14491.0	−0.682190
$768$	0	0
$769$	−24586.0	−1.15292	−0.576459	−	0.817126i	$-0.695566\pi$
$769$	−24586.0	−1.15292	−0.576459	+	0.817126i	$0.695566\pi$
$770$	−1800.00	−0.0842435
$771$	0	0
$772$	43996.0	2.05110
$773$	3078.00	0.143219	0.0716093	−	0.997433i	$-0.477187\pi$
$773$	3078.00	0.143219	0.0716093	+	0.997433i	$0.477187\pi$
$774$	0	0
$775$	−900.000	−0.0417148
$776$	−41760.0	−1.93183
$777$	0	0
$778$	17400.0	0.801825
$779$	24367.0	1.12072
$780$	0	0
$781$	−496.000	−0.0227251
$782$	129930.	5.94154
$783$	0	0
$784$	−23318.0	−1.06223
$785$	−6115.00	−0.278030
$786$	0	0
$787$	41038.0	1.85876	0.929382	−	0.369120i	$-0.120341\pi$
$787$	41038.0	1.85876	0.929382	+	0.369120i	$0.120341\pi$
$788$	22695.0	1.02598
$789$	0	0
$790$	7350.00	0.331014
$791$	−9162.00	−0.411837
$792$	0	0
$793$	1720.00	0.0770227
$794$	18530.0	0.828218
$795$	0	0
$796$	−88468.0	−3.93928
$797$	−9362.00	−0.416084	−0.208042	−	0.978120i	$-0.566709\pi$
$797$	−9362.00	−0.416084	−0.208042	+	0.978120i	$0.566709\pi$
$798$	0	0
$799$	37942.0	1.67996
$800$	2125.00	0.0939126
$801$	0	0
$802$	−13395.0	−0.589768
$803$	−9712.00	−0.426811
$804$	0	0
$805$	−9585.00	−0.419661
$806$	−7740.00	−0.338250
$807$	0	0
$808$	44820.0	1.95144
$809$	45115.0	1.96064	0.980321	−	0.197411i	$-0.0632535\pi$
$809$	45115.0	1.96064	0.980321	+	0.197411i	$0.0632535\pi$
$810$	0	0
$811$	−13512.0	−0.585044	−0.292522	−	0.956259i	$-0.594494\pi$
$811$	−13512.0	−0.585044	−0.292522	+	0.956259i	$0.594494\pi$
$812$	−34272.0	−1.48117
$813$	0	0
$814$	8240.00	0.354806
$815$	−15880.0	−0.682518
$816$	0	0
$817$	23128.0	0.990387
$818$	−62495.0	−2.67125
$819$	0	0
$820$	35105.0	1.49502
$821$	4530.00	0.192568	0.0962839	−	0.995354i	$-0.469304\pi$
$821$	4530.00	0.192568	0.0962839	+	0.995354i	$0.469304\pi$
$822$	0	0
$823$	30884.0	1.30808	0.654039	−	0.756461i	$-0.273073\pi$
$823$	30884.0	1.30808	0.654039	+	0.756461i	$0.273073\pi$
$824$	−19485.0	−0.823777
$825$	0	0
$826$	−15165.0	−0.638811
$827$	−12088.0	−0.508272	−0.254136	−	0.967168i	$-0.581791\pi$
$827$	−12088.0	−0.508272	−0.254136	+	0.967168i	$0.581791\pi$
$828$	0	0
$829$	−14112.0	−0.591230	−0.295615	−	0.955307i	$-0.595525\pi$
$829$	−14112.0	−0.591230	−0.295615	+	0.955307i	$0.595525\pi$
$830$	13350.0	0.558295
$831$	0	0
$832$	−12341.0	−0.514239
$833$	−31964.0	−1.32952
$834$	0	0
$835$	660.000	0.0273536
$836$	−8024.00	−0.331957
$837$	0	0
$838$	4640.00	0.191272
$839$	−3860.00	−0.158834	−0.0794172	−	0.996841i	$-0.525306\pi$
$839$	−3860.00	−0.158834	−0.0794172	+	0.996841i	$0.525306\pi$
$840$	0	0
$841$	25787.0	1.05732
$842$	37850.0	1.54917
$843$	0	0
$844$	27829.0	1.13497
$845$	1740.00	0.0708377
$846$	0	0
$847$	−11403.0	−0.462588
$848$	33553.0	1.35874
$849$	0	0
$850$	15250.0	0.615377
$851$	43878.0	1.76747
$852$	0	0
$853$	2862.00	0.114880	0.0574402	−	0.998349i	$-0.481706\pi$
$853$	2862.00	0.114880	0.0574402	+	0.998349i	$0.481706\pi$
$854$	1800.00	0.0721250
$855$	0	0
$856$	75870.0	3.02942
$857$	32534.0	1.29678	0.648390	−	0.761308i	$-0.275443\pi$
$857$	32534.0	1.29678	0.648390	+	0.761308i	$0.275443\pi$
$858$	0	0
$859$	−500.000	−0.0198600	−0.00993002	−	0.999951i	$-0.503161\pi$
$859$	−500.000	−0.0198600	−0.00993002	+	0.999951i	$0.503161\pi$
$860$	33320.0	1.32117
$861$	0	0
$862$	12300.0	0.486009
$863$	36595.0	1.44346	0.721731	−	0.692173i	$-0.243347\pi$
$863$	36595.0	1.44346	0.721731	+	0.692173i	$0.243347\pi$
$864$	0	0
$865$	−4965.00	−0.195162
$866$	8240.00	0.323333
$867$	0	0
$868$	−5508.00	−0.215384
$869$	−2352.00	−0.0918137
$870$	0	0
$871$	14964.0	0.582131
$872$	29520.0	1.14641
$873$	0	0
$874$	−62835.0	−2.43184
$875$	−1125.00	−0.0434651
$876$	0	0
$877$	40143.0	1.54565	0.772824	−	0.634621i	$-0.218844\pi$
$877$	40143.0	1.54565	0.772824	+	0.634621i	$0.218844\pi$
$878$	79130.0	3.04158
$879$	0	0
$880$	−3560.00	−0.136372
$881$	−16414.0	−0.627698	−0.313849	−	0.949473i	$-0.601619\pi$
$881$	−16414.0	−0.627698	−0.313849	+	0.949473i	$0.601619\pi$
$882$	0	0
$883$	−33478.0	−1.27591	−0.637953	−	0.770076i	$-0.720218\pi$
$883$	−33478.0	−1.27591	−0.637953	+	0.770076i	$0.720218\pi$
$884$	89182.0	3.39312
$885$	0	0
$886$	−63870.0	−2.42184
$887$	3633.00	0.137524	0.0687622	−	0.997633i	$-0.478095\pi$
$887$	3633.00	0.137524	0.0687622	+	0.997633i	$0.478095\pi$
$888$	0	0
$889$	−12249.0	−0.462113
$890$	−20250.0	−0.762676
$891$	0	0
$892$	−76160.0	−2.85877
$893$	−18349.0	−0.687599
$894$	0	0
$895$	−15505.0	−0.579078
$896$	−19035.0	−0.709726
$897$	0	0
$898$	−44375.0	−1.64901
$899$	8064.00	0.299165
$900$	0	0
$901$	45994.0	1.70065
$902$	−16520.0	−0.609818
$903$	0	0
$904$	−45810.0	−1.68542
$905$	−14230.0	−0.522675
$906$	0	0
$907$	20466.0	0.749242	0.374621	−	0.927178i	$-0.377773\pi$
$907$	20466.0	0.749242	0.374621	+	0.927178i	$0.377773\pi$
$908$	63512.0	2.32128
$909$	0	0
$910$	−9675.00	−0.352443
$911$	15074.0	0.548215	0.274108	−	0.961699i	$-0.411618\pi$
$911$	15074.0	0.548215	0.274108	+	0.961699i	$0.411618\pi$
$912$	0	0
$913$	−4272.00	−0.154855
$914$	57620.0	2.08523
$915$	0	0
$916$	−23460.0	−0.846223
$917$	−17199.0	−0.619369
$918$	0	0
$919$	−36848.0	−1.32264	−0.661318	−	0.750105i	$-0.730003\pi$
$919$	−36848.0	−1.32264	−0.661318	+	0.750105i	$0.730003\pi$
$920$	−47925.0	−1.71744
$921$	0	0
$922$	42720.0	1.52593
$923$	−2666.00	−0.0950731
$924$	0	0
$925$	5150.00	0.183060
$926$	12615.0	0.447683
$927$	0	0
$928$	−19040.0	−0.673511
$929$	−35174.0	−1.24222	−0.621110	−	0.783724i	$-0.713318\pi$
$929$	−35174.0	−1.24222	−0.621110	+	0.783724i	$0.713318\pi$
$930$	0	0
$931$	15458.0	0.544163
$932$	49368.0	1.73509
$933$	0	0
$934$	14510.0	0.508332
$935$	−4880.00	−0.170688
$936$	0	0
$937$	−10092.0	−0.351858	−0.175929	−	0.984403i	$-0.556293\pi$
$937$	−10092.0	−0.351858	−0.175929	+	0.984403i	$0.556293\pi$
$938$	15660.0	0.545114
$939$	0	0
$940$	−26435.0	−0.917250
$941$	12910.0	0.447241	0.223621	−	0.974676i	$-0.428212\pi$
$941$	12910.0	0.447241	0.223621	+	0.974676i	$0.428212\pi$
$942$	0	0
$943$	−87969.0	−3.03782
$944$	−29993.0	−1.03410
$945$	0	0
$946$	−15680.0	−0.538901
$947$	48060.0	1.64914	0.824572	−	0.565756i	$-0.191416\pi$
$947$	48060.0	1.64914	0.824572	+	0.565756i	$0.191416\pi$
$948$	0	0
$949$	−52202.0	−1.78561
$950$	−7375.00	−0.251870
$951$	0	0
$952$	49410.0	1.68213
$953$	−6316.00	−0.214686	−0.107343	−	0.994222i	$-0.534234\pi$
$953$	−6316.00	−0.214686	−0.107343	+	0.994222i	$0.534234\pi$
$954$	0	0
$955$	15400.0	0.521814
$956$	101422.	3.43119
$957$	0	0
$958$	−21810.0	−0.735542
$959$	−5886.00	−0.198195
$960$	0	0
$961$	−28495.0	−0.956497
$962$	44290.0	1.48437
$963$	0	0
$964$	−54706.0	−1.82776
$965$	−12940.0	−0.431661
$966$	0	0
$967$	7760.00	0.258061	0.129030	−	0.991641i	$-0.458814\pi$
$967$	7760.00	0.258061	0.129030	+	0.991641i	$0.458814\pi$
$968$	−57015.0	−1.89311
$969$	0	0
$970$	23200.0	0.767945
$971$	−22437.0	−0.741542	−0.370771	−	0.928724i	$-0.620907\pi$
$971$	−22437.0	−0.741542	−0.370771	+	0.928724i	$0.620907\pi$
$972$	0	0
$973$	−6597.00	−0.217359
$974$	28615.0	0.941359
$975$	0	0
$976$	3560.00	0.116755
$977$	3506.00	0.114807	0.0574037	−	0.998351i	$-0.481718\pi$
$977$	3506.00	0.114807	0.0574037	+	0.998351i	$0.481718\pi$
$978$	0	0
$979$	6480.00	0.211544
$980$	22270.0	0.725907
$981$	0	0
$982$	66695.0	2.16734
$983$	30648.0	0.994425	0.497212	−	0.867629i	$-0.334357\pi$
$983$	30648.0	0.994425	0.497212	+	0.867629i	$0.334357\pi$
$984$	0	0
$985$	−6675.00	−0.215922
$986$	−136640.	−4.41329
$987$	0	0
$988$	−43129.0	−1.38878
$989$	−83496.0	−2.68455
$990$	0	0
$991$	19232.0	0.616473	0.308236	−	0.951310i	$-0.400261\pi$
$991$	19232.0	0.616473	0.308236	+	0.951310i	$0.400261\pi$
$992$	−3060.00	−0.0979386
$993$	0	0
$994$	−2790.00	−0.0890276
$995$	26020.0	0.829035
$996$	0	0
$997$	38415.0	1.22028	0.610138	−	0.792295i	$-0.291114\pi$
$997$	38415.0	1.22028	0.610138	+	0.792295i	$0.291114\pi$
$998$	−98185.0	−3.11422
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.b.1.1	yes	1
3.2	odd	2		405.4.a.a.1.1	✓	1
5.4	even	2		2025.4.a.a.1.1		1
9.2	odd	6		405.4.e.m.271.1		2
9.4	even	3		405.4.e.a.136.1		2
9.5	odd	6		405.4.e.m.136.1		2
9.7	even	3		405.4.e.a.271.1		2
15.14	odd	2		2025.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.a.1.1	✓	1	3.2	odd	2
405.4.a.b.1.1	yes	1	1.1	even	1	trivial
405.4.e.a.136.1		2	9.4	even	3
405.4.e.a.271.1		2	9.7	even	3
405.4.e.m.136.1		2	9.5	odd	6
405.4.e.m.271.1		2	9.2	odd	6
2025.4.a.a.1.1		1	5.4	even	2
2025.4.a.f.1.1		1	15.14	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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