LMFDB - Embedded newform 1521.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1521 = 3^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1521.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-3.00000 q^{2} +1.00000 q^{4} +9.00000 q^{5} +2.00000 q^{7} +21.0000 q^{8} -27.0000 q^{10} -30.0000 q^{11} -6.00000 q^{14} -71.0000 q^{16} +111.000 q^{17} -46.0000 q^{19} +9.00000 q^{20} +90.0000 q^{22} +6.00000 q^{23} -44.0000 q^{25} +2.00000 q^{28} +105.000 q^{29} -100.000 q^{31} +45.0000 q^{32} -333.000 q^{34} +18.0000 q^{35} +17.0000 q^{37} +138.000 q^{38} +189.000 q^{40} +231.000 q^{41} -514.000 q^{43} -30.0000 q^{44} -18.0000 q^{46} +162.000 q^{47} -339.000 q^{49} +132.000 q^{50} -639.000 q^{53} -270.000 q^{55} +42.0000 q^{56} -315.000 q^{58} -600.000 q^{59} +233.000 q^{61} +300.000 q^{62} +433.000 q^{64} +926.000 q^{67} +111.000 q^{68} -54.0000 q^{70} +930.000 q^{71} -253.000 q^{73} -51.0000 q^{74} -46.0000 q^{76} -60.0000 q^{77} -1324.00 q^{79} -639.000 q^{80} -693.000 q^{82} -810.000 q^{83} +999.000 q^{85} +1542.00 q^{86} -630.000 q^{88} -498.000 q^{89} +6.00000 q^{92} -486.000 q^{94} -414.000 q^{95} +1358.00 q^{97} +1017.00 q^{98} +O(q^{100})

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

)

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

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	9.00000	0.804984	0.402492	−	0.915423i	$-0.368144\pi$
$5$	9.00000	0.804984	0.402492	+	0.915423i	$0.368144\pi$
$6$	0	0
$7$	2.00000	0.107990	0.0539949	−	0.998541i	$-0.482805\pi$
$7$	2.00000	0.107990	0.0539949	+	0.998541i	$0.482805\pi$
$8$	21.0000	0.928078
$9$	0	0
$10$	−27.0000	−0.853815
$11$	−30.0000	−0.822304	−0.411152	−	0.911567i	$-0.634873\pi$
$11$	−30.0000	−0.822304	−0.411152	+	0.911567i	$0.634873\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−6.00000	−0.114541
$15$	0	0
$16$	−71.0000	−1.10938
$17$	111.000	1.58361	0.791807	−	0.610771i	$-0.209140\pi$
$17$	111.000	1.58361	0.791807	+	0.610771i	$0.209140\pi$
$18$	0	0
$19$	−46.0000	−0.555428	−0.277714	−	0.960664i	$-0.589577\pi$
$19$	−46.0000	−0.555428	−0.277714	+	0.960664i	$0.589577\pi$
$20$	9.00000	0.100623
$21$	0	0
$22$	90.0000	0.872185
$23$	6.00000	0.0543951	0.0271975	−	0.999630i	$-0.491342\pi$
$23$	6.00000	0.0543951	0.0271975	+	0.999630i	$0.491342\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	0	0
$27$	0	0
$28$	2.00000	0.0134987
$29$	105.000	0.672345	0.336173	−	0.941800i	$-0.390867\pi$
$29$	105.000	0.672345	0.336173	+	0.941800i	$0.390867\pi$
$30$	0	0
$31$	−100.000	−0.579372	−0.289686	−	0.957122i	$-0.593551\pi$
$31$	−100.000	−0.579372	−0.289686	+	0.957122i	$0.593551\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	−333.000	−1.67968
$35$	18.0000	0.0869302
$36$	0	0
$37$	17.0000	0.0755347	0.0377673	−	0.999287i	$-0.487975\pi$
$37$	17.0000	0.0755347	0.0377673	+	0.999287i	$0.487975\pi$
$38$	138.000	0.589120
$39$	0	0
$40$	189.000	0.747088
$41$	231.000	0.879906	0.439953	−	0.898021i	$-0.354995\pi$
$41$	231.000	0.879906	0.439953	+	0.898021i	$0.354995\pi$
$42$	0	0
$43$	−514.000	−1.82289	−0.911445	−	0.411422i	$-0.865032\pi$
$43$	−514.000	−1.82289	−0.911445	+	0.411422i	$0.865032\pi$
$44$	−30.0000	−0.102788
$45$	0	0
$46$	−18.0000	−0.0576947
$47$	162.000	0.502769	0.251384	−	0.967887i	$-0.419114\pi$
$47$	162.000	0.502769	0.251384	+	0.967887i	$0.419114\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	132.000	0.373352
$51$	0	0
$52$	0	0
$53$	−639.000	−1.65610	−0.828051	−	0.560653i	$-0.810550\pi$
$53$	−639.000	−1.65610	−0.828051	+	0.560653i	$0.810550\pi$
$54$	0	0
$55$	−270.000	−0.661942
$56$	42.0000	0.100223
$57$	0	0
$58$	−315.000	−0.713130
$59$	−600.000	−1.32396	−0.661978	−	0.749524i	$-0.730283\pi$
$59$	−600.000	−1.32396	−0.661978	+	0.749524i	$0.730283\pi$
$60$	0	0
$61$	233.000	0.489059	0.244529	−	0.969642i	$-0.421367\pi$
$61$	233.000	0.489059	0.244529	+	0.969642i	$0.421367\pi$
$62$	300.000	0.614517
$63$	0	0
$64$	433.000	0.845703
$65$	0	0
$66$	0	0
$67$	926.000	1.68849	0.844246	−	0.535957i	$-0.180049\pi$
$67$	926.000	1.68849	0.844246	+	0.535957i	$0.180049\pi$
$68$	111.000	0.197952
$69$	0	0
$70$	−54.0000	−0.0922033
$71$	930.000	1.55452	0.777258	−	0.629182i	$-0.216610\pi$
$71$	930.000	1.55452	0.777258	+	0.629182i	$0.216610\pi$
$72$	0	0
$73$	−253.000	−0.405636	−0.202818	−	0.979216i	$-0.565010\pi$
$73$	−253.000	−0.405636	−0.202818	+	0.979216i	$0.565010\pi$
$74$	−51.0000	−0.0801166
$75$	0	0
$76$	−46.0000	−0.0694284
$77$	−60.0000	−0.0888004
$78$	0	0
$79$	−1324.00	−1.88559	−0.942795	−	0.333373i	$-0.891813\pi$
$79$	−1324.00	−1.88559	−0.942795	+	0.333373i	$0.891813\pi$
$80$	−639.000	−0.893030
$81$	0	0
$82$	−693.000	−0.933281
$83$	−810.000	−1.07119	−0.535597	−	0.844474i	$-0.679913\pi$
$83$	−810.000	−1.07119	−0.535597	+	0.844474i	$0.679913\pi$
$84$	0	0
$85$	999.000	1.27479
$86$	1542.00	1.93347
$87$	0	0
$88$	−630.000	−0.763162
$89$	−498.000	−0.593122	−0.296561	−	0.955014i	$-0.595840\pi$
$89$	−498.000	−0.593122	−0.296561	+	0.955014i	$0.595840\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.00679938
$93$	0	0
$94$	−486.000	−0.533267
$95$	−414.000	−0.447111
$96$	0	0
$97$	1358.00	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	1358.00	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	1017.00	1.04829
$99$	0	0
$100$	−44.0000	−0.0440000
$101$	357.000	0.351711	0.175856	−	0.984416i	$-0.443731\pi$
$101$	357.000	0.351711	0.175856	+	0.984416i	$0.443731\pi$
$102$	0	0
$103$	1118.00	1.06951	0.534756	−	0.845006i	$-0.320403\pi$
$103$	1118.00	1.06951	0.534756	+	0.845006i	$0.320403\pi$
$104$	0	0
$105$	0	0
$106$	1917.00	1.75656
$107$	−714.000	−0.645093	−0.322547	−	0.946554i	$-0.604539\pi$
$107$	−714.000	−0.645093	−0.322547	+	0.946554i	$0.604539\pi$
$108$	0	0
$109$	2006.00	1.76275	0.881376	−	0.472416i	$-0.156618\pi$
$109$	2006.00	1.76275	0.881376	+	0.472416i	$0.156618\pi$
$110$	810.000	0.702095
$111$	0	0
$112$	−142.000	−0.119801
$113$	1119.00	0.931563	0.465782	−	0.884900i	$-0.345773\pi$
$113$	1119.00	0.931563	0.465782	+	0.884900i	$0.345773\pi$
$114$	0	0
$115$	54.0000	0.0437872
$116$	105.000	0.0840431
$117$	0	0
$118$	1800.00	1.40427
$119$	222.000	0.171014
$120$	0	0
$121$	−431.000	−0.323817
$122$	−699.000	−0.518725
$123$	0	0
$124$	−100.000	−0.0724215
$125$	−1521.00	−1.08834
$126$	0	0
$127$	−604.000	−0.422018	−0.211009	−	0.977484i	$-0.567675\pi$
$127$	−604.000	−0.422018	−0.211009	+	0.977484i	$0.567675\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	0	0
$131$	1584.00	1.05645	0.528224	−	0.849105i	$-0.322858\pi$
$131$	1584.00	1.05645	0.528224	+	0.849105i	$0.322858\pi$
$132$	0	0
$133$	−92.0000	−0.0599805
$134$	−2778.00	−1.79092
$135$	0	0
$136$	2331.00	1.46972
$137$	−717.000	−0.447135	−0.223567	−	0.974688i	$-0.571770\pi$
$137$	−717.000	−0.447135	−0.223567	+	0.974688i	$0.571770\pi$
$138$	0	0
$139$	−820.000	−0.500370	−0.250185	−	0.968198i	$-0.580492\pi$
$139$	−820.000	−0.500370	−0.250185	+	0.968198i	$0.580492\pi$
$140$	18.0000	0.0108663
$141$	0	0
$142$	−2790.00	−1.64881
$143$	0	0
$144$	0	0
$145$	945.000	0.541227
$146$	759.000	0.430242
$147$	0	0
$148$	17.0000	0.00944183
$149$	1749.00	0.961635	0.480818	−	0.876821i	$-0.340340\pi$
$149$	1749.00	0.961635	0.480818	+	0.876821i	$0.340340\pi$
$150$	0	0
$151$	−370.000	−0.199405	−0.0997026	−	0.995017i	$-0.531789\pi$
$151$	−370.000	−0.199405	−0.0997026	+	0.995017i	$0.531789\pi$
$152$	−966.000	−0.515480
$153$	0	0
$154$	180.000	0.0941871
$155$	−900.000	−0.466385
$156$	0	0
$157$	−2611.00	−1.32726	−0.663632	−	0.748059i	$-0.730986\pi$
$157$	−2611.00	−1.32726	−0.663632	+	0.748059i	$0.730986\pi$
$158$	3972.00	1.99997
$159$	0	0
$160$	405.000	0.200113
$161$	12.0000	0.00587411
$162$	0	0
$163$	−1636.00	−0.786144	−0.393072	−	0.919508i	$-0.628588\pi$
$163$	−1636.00	−0.786144	−0.393072	+	0.919508i	$0.628588\pi$
$164$	231.000	0.109988
$165$	0	0
$166$	2430.00	1.13617
$167$	−264.000	−0.122329	−0.0611645	−	0.998128i	$-0.519481\pi$
$167$	−264.000	−0.122329	−0.0611645	+	0.998128i	$0.519481\pi$
$168$	0	0
$169$	0	0
$170$	−2997.00	−1.35211
$171$	0	0
$172$	−514.000	−0.227861
$173$	−1410.00	−0.619655	−0.309827	−	0.950793i	$-0.600271\pi$
$173$	−1410.00	−0.619655	−0.309827	+	0.950793i	$0.600271\pi$
$174$	0	0
$175$	−88.0000	−0.0380124
$176$	2130.00	0.912243
$177$	0	0
$178$	1494.00	0.629101
$179$	474.000	0.197924	0.0989621	−	0.995091i	$-0.468448\pi$
$179$	474.000	0.197924	0.0989621	+	0.995091i	$0.468448\pi$
$180$	0	0
$181$	2249.00	0.923574	0.461787	−	0.886991i	$-0.347208\pi$
$181$	2249.00	0.923574	0.461787	+	0.886991i	$0.347208\pi$
$182$	0	0
$183$	0	0
$184$	126.000	0.0504828
$185$	153.000	0.0608042
$186$	0	0
$187$	−3330.00	−1.30221
$188$	162.000	0.0628461
$189$	0	0
$190$	1242.00	0.474232
$191$	−3444.00	−1.30471	−0.652354	−	0.757915i	$-0.726218\pi$
$191$	−3444.00	−1.30471	−0.652354	+	0.757915i	$0.726218\pi$
$192$	0	0
$193$	−4273.00	−1.59366	−0.796832	−	0.604201i	$-0.793493\pi$
$193$	−4273.00	−1.59366	−0.796832	+	0.604201i	$0.793493\pi$
$194$	−4074.00	−1.50771
$195$	0	0
$196$	−339.000	−0.123542
$197$	1986.00	0.718257	0.359129	−	0.933288i	$-0.383074\pi$
$197$	1986.00	0.718257	0.359129	+	0.933288i	$0.383074\pi$
$198$	0	0
$199$	−2386.00	−0.849945	−0.424973	−	0.905206i	$-0.639716\pi$
$199$	−2386.00	−0.849945	−0.424973	+	0.905206i	$0.639716\pi$
$200$	−924.000	−0.326683
$201$	0	0
$202$	−1071.00	−0.373046
$203$	210.000	0.0726065
$204$	0	0
$205$	2079.00	0.708311
$206$	−3354.00	−1.13439
$207$	0	0
$208$	0	0
$209$	1380.00	0.456730
$210$	0	0
$211$	−1600.00	−0.522031	−0.261016	−	0.965335i	$-0.584057\pi$
$211$	−1600.00	−0.522031	−0.261016	+	0.965335i	$0.584057\pi$
$212$	−639.000	−0.207013
$213$	0	0
$214$	2142.00	0.684225
$215$	−4626.00	−1.46740
$216$	0	0
$217$	−200.000	−0.0625663
$218$	−6018.00	−1.86968
$219$	0	0
$220$	−270.000	−0.0827427
$221$	0	0
$222$	0	0
$223$	−3832.00	−1.15072	−0.575358	−	0.817902i	$-0.695137\pi$
$223$	−3832.00	−1.15072	−0.575358	+	0.817902i	$0.695137\pi$
$224$	90.0000	0.0268454
$225$	0	0
$226$	−3357.00	−0.988072
$227$	1398.00	0.408760	0.204380	−	0.978892i	$-0.434482\pi$
$227$	1398.00	0.408760	0.204380	+	0.978892i	$0.434482\pi$
$228$	0	0
$229$	4466.00	1.28874	0.644370	−	0.764714i	$-0.277120\pi$
$229$	4466.00	1.28874	0.644370	+	0.764714i	$0.277120\pi$
$230$	−162.000	−0.0464433
$231$	0	0
$232$	2205.00	0.623989
$233$	1638.00	0.460553	0.230277	−	0.973125i	$-0.426037\pi$
$233$	1638.00	0.460553	0.230277	+	0.973125i	$0.426037\pi$
$234$	0	0
$235$	1458.00	0.404721
$236$	−600.000	−0.165494
$237$	0	0
$238$	−666.000	−0.181388
$239$	594.000	0.160764	0.0803821	−	0.996764i	$-0.474386\pi$
$239$	594.000	0.160764	0.0803821	+	0.996764i	$0.474386\pi$
$240$	0	0
$241$	2303.00	0.615557	0.307779	−	0.951458i	$-0.400414\pi$
$241$	2303.00	0.615557	0.307779	+	0.951458i	$0.400414\pi$
$242$	1293.00	0.343459
$243$	0	0
$244$	233.000	0.0611324
$245$	−3051.00	−0.795597
$246$	0	0
$247$	0	0
$248$	−2100.00	−0.537702
$249$	0	0
$250$	4563.00	1.15436
$251$	−6324.00	−1.59031	−0.795154	−	0.606407i	$-0.792610\pi$
$251$	−6324.00	−1.59031	−0.795154	+	0.606407i	$0.792610\pi$
$252$	0	0
$253$	−180.000	−0.0447293
$254$	1812.00	0.447618
$255$	0	0
$256$	1513.00	0.369385
$257$	−7833.00	−1.90120	−0.950601	−	0.310414i	$-0.899532\pi$
$257$	−7833.00	−1.90120	−0.950601	+	0.310414i	$0.899532\pi$
$258$	0	0
$259$	34.0000	0.00815698
$260$	0	0
$261$	0	0
$262$	−4752.00	−1.12053
$263$	3030.00	0.710410	0.355205	−	0.934788i	$-0.384411\pi$
$263$	3030.00	0.710410	0.355205	+	0.934788i	$0.384411\pi$
$264$	0	0
$265$	−5751.00	−1.33314
$266$	276.000	0.0636190
$267$	0	0
$268$	926.000	0.211061
$269$	534.000	0.121036	0.0605178	−	0.998167i	$-0.480725\pi$
$269$	534.000	0.121036	0.0605178	+	0.998167i	$0.480725\pi$
$270$	0	0
$271$	−3688.00	−0.826679	−0.413340	−	0.910577i	$-0.635638\pi$
$271$	−3688.00	−0.826679	−0.413340	+	0.910577i	$0.635638\pi$
$272$	−7881.00	−1.75682
$273$	0	0
$274$	2151.00	0.474258
$275$	1320.00	0.289451
$276$	0	0
$277$	1865.00	0.404538	0.202269	−	0.979330i	$-0.435168\pi$
$277$	1865.00	0.404538	0.202269	+	0.979330i	$0.435168\pi$
$278$	2460.00	0.530723
$279$	0	0
$280$	378.000	0.0806779
$281$	−2997.00	−0.636249	−0.318125	−	0.948049i	$-0.603053\pi$
$281$	−2997.00	−0.636249	−0.318125	+	0.948049i	$0.603053\pi$
$282$	0	0
$283$	−4114.00	−0.864141	−0.432071	−	0.901840i	$-0.642217\pi$
$283$	−4114.00	−0.864141	−0.432071	+	0.901840i	$0.642217\pi$
$284$	930.000	0.194315
$285$	0	0
$286$	0	0
$287$	462.000	0.0950209
$288$	0	0
$289$	7408.00	1.50784
$290$	−2835.00	−0.574058
$291$	0	0
$292$	−253.000	−0.0507045
$293$	4665.00	0.930144	0.465072	−	0.885273i	$-0.346028\pi$
$293$	4665.00	0.930144	0.465072	+	0.885273i	$0.346028\pi$
$294$	0	0
$295$	−5400.00	−1.06576
$296$	357.000	0.0701020
$297$	0	0
$298$	−5247.00	−1.01997
$299$	0	0
$300$	0	0
$301$	−1028.00	−0.196854
$302$	1110.00	0.211501
$303$	0	0
$304$	3266.00	0.616177
$305$	2097.00	0.393685
$306$	0	0
$307$	1502.00	0.279230	0.139615	−	0.990206i	$-0.455413\pi$
$307$	1502.00	0.279230	0.139615	+	0.990206i	$0.455413\pi$
$308$	−60.0000	−0.0111001
$309$	0	0
$310$	2700.00	0.494676
$311$	−2106.00	−0.383988	−0.191994	−	0.981396i	$-0.561495\pi$
$311$	−2106.00	−0.383988	−0.191994	+	0.981396i	$0.561495\pi$
$312$	0	0
$313$	−3898.00	−0.703923	−0.351962	−	0.936014i	$-0.614485\pi$
$313$	−3898.00	−0.703923	−0.351962	+	0.936014i	$0.614485\pi$
$314$	7833.00	1.40778
$315$	0	0
$316$	−1324.00	−0.235699
$317$	−9351.00	−1.65680	−0.828398	−	0.560140i	$-0.810747\pi$
$317$	−9351.00	−1.65680	−0.828398	+	0.560140i	$0.810747\pi$
$318$	0	0
$319$	−3150.00	−0.552872
$320$	3897.00	0.680778
$321$	0	0
$322$	−36.0000	−0.00623044
$323$	−5106.00	−0.879583
$324$	0	0
$325$	0	0
$326$	4908.00	0.833831
$327$	0	0
$328$	4851.00	0.816621
$329$	324.000	0.0542939
$330$	0	0
$331$	−9172.00	−1.52308	−0.761539	−	0.648119i	$-0.775556\pi$
$331$	−9172.00	−1.52308	−0.761539	+	0.648119i	$0.775556\pi$
$332$	−810.000	−0.133899
$333$	0	0
$334$	792.000	0.129749
$335$	8334.00	1.35921
$336$	0	0
$337$	−11089.0	−1.79245	−0.896226	−	0.443598i	$-0.853702\pi$
$337$	−11089.0	−1.79245	−0.896226	+	0.443598i	$0.853702\pi$
$338$	0	0
$339$	0	0
$340$	999.000	0.159348
$341$	3000.00	0.476420
$342$	0	0
$343$	−1364.00	−0.214720
$344$	−10794.0	−1.69178
$345$	0	0
$346$	4230.00	0.657243
$347$	−9762.00	−1.51024	−0.755118	−	0.655589i	$-0.772420\pi$
$347$	−9762.00	−1.51024	−0.755118	+	0.655589i	$0.772420\pi$
$348$	0	0
$349$	−8290.00	−1.27150	−0.635750	−	0.771895i	$-0.719309\pi$
$349$	−8290.00	−1.27150	−0.635750	+	0.771895i	$0.719309\pi$
$350$	264.000	0.0403183
$351$	0	0
$352$	−1350.00	−0.204418
$353$	−12405.0	−1.87040	−0.935200	−	0.354119i	$-0.884781\pi$
$353$	−12405.0	−1.87040	−0.935200	+	0.354119i	$0.884781\pi$
$354$	0	0
$355$	8370.00	1.25136
$356$	−498.000	−0.0741403
$357$	0	0
$358$	−1422.00	−0.209930
$359$	1098.00	0.161421	0.0807106	−	0.996738i	$-0.474281\pi$
$359$	1098.00	0.161421	0.0807106	+	0.996738i	$0.474281\pi$
$360$	0	0
$361$	−4743.00	−0.691500
$362$	−6747.00	−0.979598
$363$	0	0
$364$	0	0
$365$	−2277.00	−0.326530
$366$	0	0
$367$	−5734.00	−0.815565	−0.407783	−	0.913079i	$-0.633698\pi$
$367$	−5734.00	−0.815565	−0.407783	+	0.913079i	$0.633698\pi$
$368$	−426.000	−0.0603445
$369$	0	0
$370$	−459.000	−0.0644926
$371$	−1278.00	−0.178842
$372$	0	0
$373$	−8971.00	−1.24531	−0.622655	−	0.782496i	$-0.713946\pi$
$373$	−8971.00	−1.24531	−0.622655	+	0.782496i	$0.713946\pi$
$374$	9990.00	1.38120
$375$	0	0
$376$	3402.00	0.466608
$377$	0	0
$378$	0	0
$379$	7244.00	0.981792	0.490896	−	0.871218i	$-0.336669\pi$
$379$	7244.00	0.981792	0.490896	+	0.871218i	$0.336669\pi$
$380$	−414.000	−0.0558888
$381$	0	0
$382$	10332.0	1.38385
$383$	6312.00	0.842110	0.421055	−	0.907035i	$-0.361660\pi$
$383$	6312.00	0.842110	0.421055	+	0.907035i	$0.361660\pi$
$384$	0	0
$385$	−540.000	−0.0714830
$386$	12819.0	1.69034
$387$	0	0
$388$	1358.00	0.177686
$389$	−3627.00	−0.472741	−0.236370	−	0.971663i	$-0.575958\pi$
$389$	−3627.00	−0.472741	−0.236370	+	0.971663i	$0.575958\pi$
$390$	0	0
$391$	666.000	0.0861408
$392$	−7119.00	−0.917255
$393$	0	0
$394$	−5958.00	−0.761827
$395$	−11916.0	−1.51787
$396$	0	0
$397$	−3898.00	−0.492783	−0.246392	−	0.969170i	$-0.579245\pi$
$397$	−3898.00	−0.492783	−0.246392	+	0.969170i	$0.579245\pi$
$398$	7158.00	0.901503
$399$	0	0
$400$	3124.00	0.390500
$401$	5703.00	0.710210	0.355105	−	0.934826i	$-0.384445\pi$
$401$	5703.00	0.710210	0.355105	+	0.934826i	$0.384445\pi$
$402$	0	0
$403$	0	0
$404$	357.000	0.0439639
$405$	0	0
$406$	−630.000	−0.0770108
$407$	−510.000	−0.0621124
$408$	0	0
$409$	6311.00	0.762980	0.381490	−	0.924373i	$-0.375411\pi$
$409$	6311.00	0.762980	0.381490	+	0.924373i	$0.375411\pi$
$410$	−6237.00	−0.751277
$411$	0	0
$412$	1118.00	0.133689
$413$	−1200.00	−0.142974
$414$	0	0
$415$	−7290.00	−0.862294
$416$	0	0
$417$	0	0
$418$	−4140.00	−0.484435
$419$	2328.00	0.271433	0.135716	−	0.990748i	$-0.456666\pi$
$419$	2328.00	0.271433	0.135716	+	0.990748i	$0.456666\pi$
$420$	0	0
$421$	2045.00	0.236739	0.118370	−	0.992970i	$-0.462233\pi$
$421$	2045.00	0.236739	0.118370	+	0.992970i	$0.462233\pi$
$422$	4800.00	0.553697
$423$	0	0
$424$	−13419.0	−1.53699
$425$	−4884.00	−0.557432
$426$	0	0
$427$	466.000	0.0528134
$428$	−714.000	−0.0806367
$429$	0	0
$430$	13878.0	1.55641
$431$	−5034.00	−0.562597	−0.281298	−	0.959620i	$-0.590765\pi$
$431$	−5034.00	−0.562597	−0.281298	+	0.959620i	$0.590765\pi$
$432$	0	0
$433$	4283.00	0.475353	0.237676	−	0.971344i	$-0.423614\pi$
$433$	4283.00	0.475353	0.237676	+	0.971344i	$0.423614\pi$
$434$	600.000	0.0663616
$435$	0	0
$436$	2006.00	0.220344
$437$	−276.000	−0.0302125
$438$	0	0
$439$	−1306.00	−0.141986	−0.0709931	−	0.997477i	$-0.522617\pi$
$439$	−1306.00	−0.141986	−0.0709931	+	0.997477i	$0.522617\pi$
$440$	−5670.00	−0.614333
$441$	0	0
$442$	0	0
$443$	5796.00	0.621617	0.310808	−	0.950473i	$-0.399400\pi$
$443$	5796.00	0.621617	0.310808	+	0.950473i	$0.399400\pi$
$444$	0	0
$445$	−4482.00	−0.477454
$446$	11496.0	1.22052
$447$	0	0
$448$	866.000	0.0913274
$449$	−2706.00	−0.284419	−0.142209	−	0.989837i	$-0.545421\pi$
$449$	−2706.00	−0.284419	−0.142209	+	0.989837i	$0.545421\pi$
$450$	0	0
$451$	−6930.00	−0.723550
$452$	1119.00	0.116445
$453$	0	0
$454$	−4194.00	−0.433555
$455$	0	0
$456$	0	0
$457$	−829.000	−0.0848555	−0.0424278	−	0.999100i	$-0.513509\pi$
$457$	−829.000	−0.0848555	−0.0424278	+	0.999100i	$0.513509\pi$
$458$	−13398.0	−1.36692
$459$	0	0
$460$	54.0000	0.00547340
$461$	5493.00	0.554956	0.277478	−	0.960732i	$-0.410502\pi$
$461$	5493.00	0.554956	0.277478	+	0.960732i	$0.410502\pi$
$462$	0	0
$463$	−15346.0	−1.54037	−0.770183	−	0.637823i	$-0.779835\pi$
$463$	−15346.0	−1.54037	−0.770183	+	0.637823i	$0.779835\pi$
$464$	−7455.00	−0.745883
$465$	0	0
$466$	−4914.00	−0.488491
$467$	9594.00	0.950658	0.475329	−	0.879808i	$-0.342329\pi$
$467$	9594.00	0.950658	0.475329	+	0.879808i	$0.342329\pi$
$468$	0	0
$469$	1852.00	0.182340
$470$	−4374.00	−0.429271
$471$	0	0
$472$	−12600.0	−1.22873
$473$	15420.0	1.49897
$474$	0	0
$475$	2024.00	0.195511
$476$	222.000	0.0213768
$477$	0	0
$478$	−1782.00	−0.170516
$479$	12840.0	1.22479	0.612395	−	0.790552i	$-0.290206\pi$
$479$	12840.0	1.22479	0.612395	+	0.790552i	$0.290206\pi$
$480$	0	0
$481$	0	0
$482$	−6909.00	−0.652897
$483$	0	0
$484$	−431.000	−0.0404771
$485$	12222.0	1.14427
$486$	0	0
$487$	−14086.0	−1.31067	−0.655336	−	0.755337i	$-0.727473\pi$
$487$	−14086.0	−1.31067	−0.655336	+	0.755337i	$0.727473\pi$
$488$	4893.00	0.453885
$489$	0	0
$490$	9153.00	0.843858
$491$	−11694.0	−1.07483	−0.537416	−	0.843317i	$-0.680600\pi$
$491$	−11694.0	−1.07483	−0.537416	+	0.843317i	$0.680600\pi$
$492$	0	0
$493$	11655.0	1.06474
$494$	0	0
$495$	0	0
$496$	7100.00	0.642741
$497$	1860.00	0.167872
$498$	0	0
$499$	−3688.00	−0.330857	−0.165428	−	0.986222i	$-0.552901\pi$
$499$	−3688.00	−0.330857	−0.165428	+	0.986222i	$0.552901\pi$
$500$	−1521.00	−0.136042
$501$	0	0
$502$	18972.0	1.68678
$503$	4746.00	0.420703	0.210352	−	0.977626i	$-0.432539\pi$
$503$	4746.00	0.420703	0.210352	+	0.977626i	$0.432539\pi$
$504$	0	0
$505$	3213.00	0.283122
$506$	540.000	0.0474425
$507$	0	0
$508$	−604.000	−0.0527523
$509$	14505.0	1.26311	0.631555	−	0.775331i	$-0.282417\pi$
$509$	14505.0	1.26311	0.631555	+	0.775331i	$0.282417\pi$
$510$	0	0
$511$	−506.000	−0.0438045
$512$	8733.00	0.753804
$513$	0	0
$514$	23499.0	2.01653
$515$	10062.0	0.860941
$516$	0	0
$517$	−4860.00	−0.413429
$518$	−102.000	−0.00865178
$519$	0	0
$520$	0	0
$521$	−5085.00	−0.427597	−0.213798	−	0.976878i	$-0.568584\pi$
$521$	−5085.00	−0.427597	−0.213798	+	0.976878i	$0.568584\pi$
$522$	0	0
$523$	−10882.0	−0.909821	−0.454911	−	0.890537i	$-0.650329\pi$
$523$	−10882.0	−0.909821	−0.454911	+	0.890537i	$0.650329\pi$
$524$	1584.00	0.132056
$525$	0	0
$526$	−9090.00	−0.753503
$527$	−11100.0	−0.917502
$528$	0	0
$529$	−12131.0	−0.997041
$530$	17253.0	1.41400
$531$	0	0
$532$	−92.0000	−0.00749757
$533$	0	0
$534$	0	0
$535$	−6426.00	−0.519290
$536$	19446.0	1.56705
$537$	0	0
$538$	−1602.00	−0.128378
$539$	10170.0	0.812714
$540$	0	0
$541$	−4699.00	−0.373430	−0.186715	−	0.982414i	$-0.559784\pi$
$541$	−4699.00	−0.373430	−0.186715	+	0.982414i	$0.559784\pi$
$542$	11064.0	0.876826
$543$	0	0
$544$	4995.00	0.393674
$545$	18054.0	1.41899
$546$	0	0
$547$	8270.00	0.646434	0.323217	−	0.946325i	$-0.395236\pi$
$547$	8270.00	0.646434	0.323217	+	0.946325i	$0.395236\pi$
$548$	−717.000	−0.0558918
$549$	0	0
$550$	−3960.00	−0.307009
$551$	−4830.00	−0.373439
$552$	0	0
$553$	−2648.00	−0.203625
$554$	−5595.00	−0.429077
$555$	0	0
$556$	−820.000	−0.0625463
$557$	22785.0	1.73327	0.866635	−	0.498943i	$-0.166278\pi$
$557$	22785.0	1.73327	0.866635	+	0.498943i	$0.166278\pi$
$558$	0	0
$559$	0	0
$560$	−1278.00	−0.0964381
$561$	0	0
$562$	8991.00	0.674844
$563$	11928.0	0.892905	0.446452	−	0.894807i	$-0.352687\pi$
$563$	11928.0	0.892905	0.446452	+	0.894807i	$0.352687\pi$
$564$	0	0
$565$	10071.0	0.749894
$566$	12342.0	0.916560
$567$	0	0
$568$	19530.0	1.44271
$569$	7962.00	0.586616	0.293308	−	0.956018i	$-0.405244\pi$
$569$	7962.00	0.586616	0.293308	+	0.956018i	$0.405244\pi$
$570$	0	0
$571$	20618.0	1.51110	0.755549	−	0.655093i	$-0.227370\pi$
$571$	20618.0	1.51110	0.755549	+	0.655093i	$0.227370\pi$
$572$	0	0
$573$	0	0
$574$	−1386.00	−0.100785
$575$	−264.000	−0.0191471
$576$	0	0
$577$	−3493.00	−0.252020	−0.126010	−	0.992029i	$-0.540217\pi$
$577$	−3493.00	−0.252020	−0.126010	+	0.992029i	$0.540217\pi$
$578$	−22224.0	−1.59930
$579$	0	0
$580$	945.000	0.0676534
$581$	−1620.00	−0.115678
$582$	0	0
$583$	19170.0	1.36182
$584$	−5313.00	−0.376461
$585$	0	0
$586$	−13995.0	−0.986567
$587$	−10416.0	−0.732392	−0.366196	−	0.930538i	$-0.619340\pi$
$587$	−10416.0	−0.732392	−0.366196	+	0.930538i	$0.619340\pi$
$588$	0	0
$589$	4600.00	0.321799
$590$	16200.0	1.13041
$591$	0	0
$592$	−1207.00	−0.0837963
$593$	−2061.00	−0.142724	−0.0713618	−	0.997450i	$-0.522734\pi$
$593$	−2061.00	−0.142724	−0.0713618	+	0.997450i	$0.522734\pi$
$594$	0	0
$595$	1998.00	0.137664
$596$	1749.00	0.120204
$597$	0	0
$598$	0	0
$599$	−12456.0	−0.849647	−0.424823	−	0.905276i	$-0.639664\pi$
$599$	−12456.0	−0.849647	−0.424823	+	0.905276i	$0.639664\pi$
$600$	0	0
$601$	−781.000	−0.0530077	−0.0265039	−	0.999649i	$-0.508437\pi$
$601$	−781.000	−0.0530077	−0.0265039	+	0.999649i	$0.508437\pi$
$602$	3084.00	0.208795
$603$	0	0
$604$	−370.000	−0.0249256
$605$	−3879.00	−0.260667
$606$	0	0
$607$	19304.0	1.29082	0.645408	−	0.763838i	$-0.276687\pi$
$607$	19304.0	1.29082	0.645408	+	0.763838i	$0.276687\pi$
$608$	−2070.00	−0.138075
$609$	0	0
$610$	−6291.00	−0.417566
$611$	0	0
$612$	0	0
$613$	12041.0	0.793363	0.396681	−	0.917956i	$-0.370162\pi$
$613$	12041.0	0.793363	0.396681	+	0.917956i	$0.370162\pi$
$614$	−4506.00	−0.296168
$615$	0	0
$616$	−1260.00	−0.0824137
$617$	−9717.00	−0.634022	−0.317011	−	0.948422i	$-0.602679\pi$
$617$	−9717.00	−0.634022	−0.317011	+	0.948422i	$0.602679\pi$
$618$	0	0
$619$	−21040.0	−1.36619	−0.683093	−	0.730332i	$-0.739366\pi$
$619$	−21040.0	−1.36619	−0.683093	+	0.730332i	$0.739366\pi$
$620$	−900.000	−0.0582982
$621$	0	0
$622$	6318.00	0.407281
$623$	−996.000	−0.0640512
$624$	0	0
$625$	−8189.00	−0.524096
$626$	11694.0	0.746623
$627$	0	0
$628$	−2611.00	−0.165908
$629$	1887.00	0.119618
$630$	0	0
$631$	−5068.00	−0.319737	−0.159868	−	0.987138i	$-0.551107\pi$
$631$	−5068.00	−0.319737	−0.159868	+	0.987138i	$0.551107\pi$
$632$	−27804.0	−1.74997
$633$	0	0
$634$	28053.0	1.75730
$635$	−5436.00	−0.339718
$636$	0	0
$637$	0	0
$638$	9450.00	0.586409
$639$	0	0
$640$	−14931.0	−0.922187
$641$	−10185.0	−0.627587	−0.313794	−	0.949491i	$-0.601600\pi$
$641$	−10185.0	−0.627587	−0.313794	+	0.949491i	$0.601600\pi$
$642$	0	0
$643$	25928.0	1.59020	0.795101	−	0.606476i	$-0.207418\pi$
$643$	25928.0	1.59020	0.795101	+	0.606476i	$0.207418\pi$
$644$	12.0000	0.000734264 0
$645$	0	0
$646$	15318.0	0.932939
$647$	−23160.0	−1.40729	−0.703643	−	0.710554i	$-0.748444\pi$
$647$	−23160.0	−1.40729	−0.703643	+	0.710554i	$0.748444\pi$
$648$	0	0
$649$	18000.0	1.08869
$650$	0	0
$651$	0	0
$652$	−1636.00	−0.0982680
$653$	−16626.0	−0.996364	−0.498182	−	0.867073i	$-0.665999\pi$
$653$	−16626.0	−0.996364	−0.498182	+	0.867073i	$0.665999\pi$
$654$	0	0
$655$	14256.0	0.850424
$656$	−16401.0	−0.976146
$657$	0	0
$658$	−972.000	−0.0575874
$659$	14808.0	0.875323	0.437661	−	0.899140i	$-0.355807\pi$
$659$	14808.0	0.875323	0.437661	+	0.899140i	$0.355807\pi$
$660$	0	0
$661$	4853.00	0.285567	0.142784	−	0.989754i	$-0.454395\pi$
$661$	4853.00	0.285567	0.142784	+	0.989754i	$0.454395\pi$
$662$	27516.0	1.61547
$663$	0	0
$664$	−17010.0	−0.994151
$665$	−828.000	−0.0482834
$666$	0	0
$667$	630.000	0.0365723
$668$	−264.000	−0.0152911
$669$	0	0
$670$	−25002.0	−1.44166
$671$	−6990.00	−0.402155
$672$	0	0
$673$	−16165.0	−0.925877	−0.462938	−	0.886391i	$-0.653205\pi$
$673$	−16165.0	−0.925877	−0.462938	+	0.886391i	$0.653205\pi$
$674$	33267.0	1.90118
$675$	0	0
$676$	0	0
$677$	25686.0	1.45819	0.729094	−	0.684414i	$-0.239942\pi$
$677$	25686.0	1.45819	0.729094	+	0.684414i	$0.239942\pi$
$678$	0	0
$679$	2716.00	0.153506
$680$	20979.0	1.18310
$681$	0	0
$682$	−9000.00	−0.505319
$683$	19056.0	1.06758	0.533790	−	0.845617i	$-0.320767\pi$
$683$	19056.0	1.06758	0.533790	+	0.845617i	$0.320767\pi$
$684$	0	0
$685$	−6453.00	−0.359936
$686$	4092.00	0.227745
$687$	0	0
$688$	36494.0	2.02227
$689$	0	0
$690$	0	0
$691$	−16390.0	−0.902323	−0.451161	−	0.892442i	$-0.648990\pi$
$691$	−16390.0	−0.902323	−0.451161	+	0.892442i	$0.648990\pi$
$692$	−1410.00	−0.0774569
$693$	0	0
$694$	29286.0	1.60185
$695$	−7380.00	−0.402790
$696$	0	0
$697$	25641.0	1.39343
$698$	24870.0	1.34863
$699$	0	0
$700$	−88.0000	−0.00475155
$701$	27846.0	1.50033	0.750163	−	0.661253i	$-0.229975\pi$
$701$	27846.0	1.50033	0.750163	+	0.661253i	$0.229975\pi$
$702$	0	0
$703$	−782.000	−0.0419540
$704$	−12990.0	−0.695425
$705$	0	0
$706$	37215.0	1.98386
$707$	714.000	0.0379812
$708$	0	0
$709$	−12283.0	−0.650632	−0.325316	−	0.945605i	$-0.605471\pi$
$709$	−12283.0	−0.650632	−0.325316	+	0.945605i	$0.605471\pi$
$710$	−25110.0	−1.32727
$711$	0	0
$712$	−10458.0	−0.550464
$713$	−600.000	−0.0315150
$714$	0	0
$715$	0	0
$716$	474.000	0.0247405
$717$	0	0
$718$	−3294.00	−0.171213
$719$	−25512.0	−1.32328	−0.661639	−	0.749822i	$-0.730139\pi$
$719$	−25512.0	−1.32328	−0.661639	+	0.749822i	$0.730139\pi$
$720$	0	0
$721$	2236.00	0.115497
$722$	14229.0	0.733447
$723$	0	0
$724$	2249.00	0.115447
$725$	−4620.00	−0.236666
$726$	0	0
$727$	6110.00	0.311702	0.155851	−	0.987781i	$-0.450188\pi$
$727$	6110.00	0.311702	0.155851	+	0.987781i	$0.450188\pi$
$728$	0	0
$729$	0	0
$730$	6831.00	0.346338
$731$	−57054.0	−2.88676
$732$	0	0
$733$	−27127.0	−1.36693	−0.683464	−	0.729984i	$-0.739527\pi$
$733$	−27127.0	−1.36693	−0.683464	+	0.729984i	$0.739527\pi$
$734$	17202.0	0.865037
$735$	0	0
$736$	270.000	0.0135222
$737$	−27780.0	−1.38845
$738$	0	0
$739$	−880.000	−0.0438042	−0.0219021	−	0.999760i	$-0.506972\pi$
$739$	−880.000	−0.0438042	−0.0219021	+	0.999760i	$0.506972\pi$
$740$	153.000	0.00760053
$741$	0	0
$742$	3834.00	0.189691
$743$	21876.0	1.08015	0.540076	−	0.841616i	$-0.318396\pi$
$743$	21876.0	1.08015	0.540076	+	0.841616i	$0.318396\pi$
$744$	0	0
$745$	15741.0	0.774102
$746$	26913.0	1.32085
$747$	0	0
$748$	−3330.00	−0.162777
$749$	−1428.00	−0.0696635
$750$	0	0
$751$	11798.0	0.573256	0.286628	−	0.958042i	$-0.407466\pi$
$751$	11798.0	0.573256	0.286628	+	0.958042i	$0.407466\pi$
$752$	−11502.0	−0.557759
$753$	0	0
$754$	0	0
$755$	−3330.00	−0.160518
$756$	0	0
$757$	−8074.00	−0.387655	−0.193827	−	0.981036i	$-0.562090\pi$
$757$	−8074.00	−0.387655	−0.193827	+	0.981036i	$0.562090\pi$
$758$	−21732.0	−1.04135
$759$	0	0
$760$	−8694.00	−0.414953
$761$	−19554.0	−0.931448	−0.465724	−	0.884930i	$-0.654206\pi$
$761$	−19554.0	−0.931448	−0.465724	+	0.884930i	$0.654206\pi$
$762$	0	0
$763$	4012.00	0.190359
$764$	−3444.00	−0.163088
$765$	0	0
$766$	−18936.0	−0.893193
$767$	0	0
$768$	0	0
$769$	14030.0	0.657913	0.328956	−	0.944345i	$-0.393303\pi$
$769$	14030.0	0.657913	0.328956	+	0.944345i	$0.393303\pi$
$770$	1620.00	0.0758192
$771$	0	0
$772$	−4273.00	−0.199208
$773$	−36042.0	−1.67703	−0.838513	−	0.544882i	$-0.816574\pi$
$773$	−36042.0	−1.67703	−0.838513	+	0.544882i	$0.816574\pi$
$774$	0	0
$775$	4400.00	0.203939
$776$	28518.0	1.31925
$777$	0	0
$778$	10881.0	0.501417
$779$	−10626.0	−0.488724
$780$	0	0
$781$	−27900.0	−1.27828
$782$	−1998.00	−0.0913662
$783$	0	0
$784$	24069.0	1.09644
$785$	−23499.0	−1.06843
$786$	0	0
$787$	28628.0	1.29667	0.648334	−	0.761356i	$-0.275466\pi$
$787$	28628.0	1.29667	0.648334	+	0.761356i	$0.275466\pi$
$788$	1986.00	0.0897821
$789$	0	0
$790$	35748.0	1.60995
$791$	2238.00	0.100599
$792$	0	0
$793$	0	0
$794$	11694.0	0.522676
$795$	0	0
$796$	−2386.00	−0.106243
$797$	37434.0	1.66371	0.831857	−	0.554990i	$-0.187278\pi$
$797$	37434.0	1.66371	0.831857	+	0.554990i	$0.187278\pi$
$798$	0	0
$799$	17982.0	0.796192
$800$	−1980.00	−0.0875045
$801$	0	0
$802$	−17109.0	−0.753292
$803$	7590.00	0.333556
$804$	0	0
$805$	108.000	0.00472857
$806$	0	0
$807$	0	0
$808$	7497.00	0.326415
$809$	−37569.0	−1.63270	−0.816351	−	0.577556i	$-0.804006\pi$
$809$	−37569.0	−1.63270	−0.816351	+	0.577556i	$0.804006\pi$
$810$	0	0
$811$	5516.00	0.238832	0.119416	−	0.992844i	$-0.461898\pi$
$811$	5516.00	0.238832	0.119416	+	0.992844i	$0.461898\pi$
$812$	210.000	0.00907581
$813$	0	0
$814$	1530.00	0.0658802
$815$	−14724.0	−0.632833
$816$	0	0
$817$	23644.0	1.01248
$818$	−18933.0	−0.809263
$819$	0	0
$820$	2079.00	0.0885388
$821$	−8778.00	−0.373148	−0.186574	−	0.982441i	$-0.559738\pi$
$821$	−8778.00	−0.373148	−0.186574	+	0.982441i	$0.559738\pi$
$822$	0	0
$823$	−3088.00	−0.130791	−0.0653955	−	0.997859i	$-0.520831\pi$
$823$	−3088.00	−0.130791	−0.0653955	+	0.997859i	$0.520831\pi$
$824$	23478.0	0.992591
$825$	0	0
$826$	3600.00	0.151647
$827$	−13176.0	−0.554020	−0.277010	−	0.960867i	$-0.589343\pi$
$827$	−13176.0	−0.554020	−0.277010	+	0.960867i	$0.589343\pi$
$828$	0	0
$829$	−2359.00	−0.0988317	−0.0494158	−	0.998778i	$-0.515736\pi$
$829$	−2359.00	−0.0988317	−0.0494158	+	0.998778i	$0.515736\pi$
$830$	21870.0	0.914601
$831$	0	0
$832$	0	0
$833$	−37629.0	−1.56515
$834$	0	0
$835$	−2376.00	−0.0984729
$836$	1380.00	0.0570913
$837$	0	0
$838$	−6984.00	−0.287898
$839$	2676.00	0.110114	0.0550571	−	0.998483i	$-0.482466\pi$
$839$	2676.00	0.110114	0.0550571	+	0.998483i	$0.482466\pi$
$840$	0	0
$841$	−13364.0	−0.547952
$842$	−6135.00	−0.251100
$843$	0	0
$844$	−1600.00	−0.0652539
$845$	0	0
$846$	0	0
$847$	−862.000	−0.0349689
$848$	45369.0	1.83724
$849$	0	0
$850$	14652.0	0.591246
$851$	102.000	0.00410871
$852$	0	0
$853$	2477.00	0.0994266	0.0497133	−	0.998764i	$-0.484169\pi$
$853$	2477.00	0.0994266	0.0497133	+	0.998764i	$0.484169\pi$
$854$	−1398.00	−0.0560171
$855$	0	0
$856$	−14994.0	−0.598697
$857$	17199.0	0.685539	0.342769	−	0.939420i	$-0.388635\pi$
$857$	17199.0	0.685539	0.342769	+	0.939420i	$0.388635\pi$
$858$	0	0
$859$	24338.0	0.966708	0.483354	−	0.875425i	$-0.339418\pi$
$859$	24338.0	0.966708	0.483354	+	0.875425i	$0.339418\pi$
$860$	−4626.00	−0.183425
$861$	0	0
$862$	15102.0	0.596724
$863$	−25146.0	−0.991865	−0.495933	−	0.868361i	$-0.665174\pi$
$863$	−25146.0	−0.991865	−0.495933	+	0.868361i	$0.665174\pi$
$864$	0	0
$865$	−12690.0	−0.498813
$866$	−12849.0	−0.504188
$867$	0	0
$868$	−200.000	−0.00782079
$869$	39720.0	1.55053
$870$	0	0
$871$	0	0
$872$	42126.0	1.63597
$873$	0	0
$874$	828.000	0.0320452
$875$	−3042.00	−0.117530
$876$	0	0
$877$	18089.0	0.696490	0.348245	−	0.937403i	$-0.386778\pi$
$877$	18089.0	0.696490	0.348245	+	0.937403i	$0.386778\pi$
$878$	3918.00	0.150599
$879$	0	0
$880$	19170.0	0.734342
$881$	15099.0	0.577410	0.288705	−	0.957418i	$-0.406775\pi$
$881$	15099.0	0.577410	0.288705	+	0.957418i	$0.406775\pi$
$882$	0	0
$883$	33488.0	1.27629	0.638143	−	0.769918i	$-0.279703\pi$
$883$	33488.0	1.27629	0.638143	+	0.769918i	$0.279703\pi$
$884$	0	0
$885$	0	0
$886$	−17388.0	−0.659324
$887$	39768.0	1.50539	0.752694	−	0.658371i	$-0.228754\pi$
$887$	39768.0	1.50539	0.752694	+	0.658371i	$0.228754\pi$
$888$	0	0
$889$	−1208.00	−0.0455737
$890$	13446.0	0.506417
$891$	0	0
$892$	−3832.00	−0.143840
$893$	−7452.00	−0.279252
$894$	0	0
$895$	4266.00	0.159326
$896$	−3318.00	−0.123713
$897$	0	0
$898$	8118.00	0.301672
$899$	−10500.0	−0.389538
$900$	0	0
$901$	−70929.0	−2.62263
$902$	20790.0	0.767440
$903$	0	0
$904$	23499.0	0.864563
$905$	20241.0	0.743463
$906$	0	0
$907$	32156.0	1.17720	0.588601	−	0.808424i	$-0.299679\pi$
$907$	32156.0	1.17720	0.588601	+	0.808424i	$0.299679\pi$
$908$	1398.00	0.0510950
$909$	0	0
$910$	0	0
$911$	−11520.0	−0.418962	−0.209481	−	0.977813i	$-0.567177\pi$
$911$	−11520.0	−0.418962	−0.209481	+	0.977813i	$0.567177\pi$
$912$	0	0
$913$	24300.0	0.880846
$914$	2487.00	0.0900029
$915$	0	0
$916$	4466.00	0.161093
$917$	3168.00	0.114086
$918$	0	0
$919$	4952.00	0.177749	0.0888745	−	0.996043i	$-0.471673\pi$
$919$	4952.00	0.177749	0.0888745	+	0.996043i	$0.471673\pi$
$920$	1134.00	0.0406379
$921$	0	0
$922$	−16479.0	−0.588619
$923$	0	0
$924$	0	0
$925$	−748.000	−0.0265882
$926$	46038.0	1.63380
$927$	0	0
$928$	4725.00	0.167140
$929$	−8781.00	−0.310113	−0.155057	−	0.987906i	$-0.549556\pi$
$929$	−8781.00	−0.310113	−0.155057	+	0.987906i	$0.549556\pi$
$930$	0	0
$931$	15594.0	0.548950
$932$	1638.00	0.0575692
$933$	0	0
$934$	−28782.0	−1.00833
$935$	−29970.0	−1.04826
$936$	0	0
$937$	50039.0	1.74461	0.872307	−	0.488959i	$-0.162623\pi$
$937$	50039.0	1.74461	0.872307	+	0.488959i	$0.162623\pi$
$938$	−5556.00	−0.193401
$939$	0	0
$940$	1458.00	0.0505901
$941$	50670.0	1.75536	0.877681	−	0.479246i	$-0.159090\pi$
$941$	50670.0	1.75536	0.877681	+	0.479246i	$0.159090\pi$
$942$	0	0
$943$	1386.00	0.0478625
$944$	42600.0	1.46876
$945$	0	0
$946$	−46260.0	−1.58990
$947$	−42384.0	−1.45438	−0.727188	−	0.686438i	$-0.759173\pi$
$947$	−42384.0	−1.45438	−0.727188	+	0.686438i	$0.759173\pi$
$948$	0	0
$949$	0	0
$950$	−6072.00	−0.207370
$951$	0	0
$952$	4662.00	0.158715
$953$	50538.0	1.71782	0.858912	−	0.512123i	$-0.171141\pi$
$953$	50538.0	1.71782	0.858912	+	0.512123i	$0.171141\pi$
$954$	0	0
$955$	−30996.0	−1.05027
$956$	594.000	0.0200955
$957$	0	0
$958$	−38520.0	−1.29909
$959$	−1434.00	−0.0482860
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	0	0
$964$	2303.00	0.0769446
$965$	−38457.0	−1.28288
$966$	0	0
$967$	−6886.00	−0.228996	−0.114498	−	0.993423i	$-0.536526\pi$
$967$	−6886.00	−0.228996	−0.114498	+	0.993423i	$0.536526\pi$
$968$	−9051.00	−0.300527
$969$	0	0
$970$	−36666.0	−1.21368
$971$	−9060.00	−0.299433	−0.149716	−	0.988729i	$-0.547836\pi$
$971$	−9060.00	−0.299433	−0.149716	+	0.988729i	$0.547836\pi$
$972$	0	0
$973$	−1640.00	−0.0540349
$974$	42258.0	1.39018
$975$	0	0
$976$	−16543.0	−0.542550
$977$	28311.0	0.927072	0.463536	−	0.886078i	$-0.346581\pi$
$977$	28311.0	0.927072	0.463536	+	0.886078i	$0.346581\pi$
$978$	0	0
$979$	14940.0	0.487727
$980$	−3051.00	−0.0994496
$981$	0	0
$982$	35082.0	1.14003
$983$	−4284.00	−0.139001	−0.0695007	−	0.997582i	$-0.522141\pi$
$983$	−4284.00	−0.139001	−0.0695007	+	0.997582i	$0.522141\pi$
$984$	0	0
$985$	17874.0	0.578186
$986$	−34965.0	−1.12932
$987$	0	0
$988$	0	0
$989$	−3084.00	−0.0991562
$990$	0	0
$991$	−2458.00	−0.0787901	−0.0393950	−	0.999224i	$-0.512543\pi$
$991$	−2458.00	−0.0787901	−0.0393950	+	0.999224i	$0.512543\pi$
$992$	−4500.00	−0.144027
$993$	0	0
$994$	−5580.00	−0.178055
$995$	−21474.0	−0.684193
$996$	0	0
$997$	24101.0	0.765583	0.382792	−	0.923835i	$-0.374963\pi$
$997$	24101.0	0.765583	0.382792	+	0.923835i	$0.374963\pi$
$998$	11064.0	0.350927
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.c.1.1		1
3.2	odd	2		507.4.a.e.1.1		1
13.3	even	3		117.4.g.b.100.1		2
13.9	even	3		117.4.g.b.55.1		2
13.12	even	2		1521.4.a.j.1.1		1
39.5	even	4		507.4.b.c.337.1		2
39.8	even	4		507.4.b.c.337.2		2
39.29	odd	6		39.4.e.a.22.1	yes	2
39.35	odd	6		39.4.e.a.16.1	✓	2
39.38	odd	2		507.4.a.a.1.1		1
156.35	even	6		624.4.q.b.289.1		2
156.107	even	6		624.4.q.b.529.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.a.16.1	✓	2	39.35	odd	6
39.4.e.a.22.1	yes	2	39.29	odd	6
117.4.g.b.55.1		2	13.9	even	3
117.4.g.b.100.1		2	13.3	even	3
507.4.a.a.1.1		1	39.38	odd	2
507.4.a.e.1.1		1	3.2	odd	2
507.4.b.c.337.1		2	39.5	even	4
507.4.b.c.337.2		2	39.8	even	4
624.4.q.b.289.1		2	156.35	even	6
624.4.q.b.529.1		2	156.107	even	6
1521.4.a.c.1.1		1	1.1	even	1	trivial
1521.4.a.j.1.1		1	13.12	even	2

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1521.4.a.c.1.1

Level	$1521$
Weight	$4$
Character	1521.1
Self dual	yes
Analytic conductor	$89.742$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$