LMFDB - Embedded newform 80.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,4,Mod(1,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$80 = 2^{4} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-4,0,5,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.72015280046$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	80.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q-4.00000 q^{3} +5.00000 q^{5} +16.0000 q^{7} -11.0000 q^{9} +60.0000 q^{11} +86.0000 q^{13} -20.0000 q^{15} +18.0000 q^{17} -44.0000 q^{19} -64.0000 q^{21} -48.0000 q^{23} +25.0000 q^{25} +152.000 q^{27} -186.000 q^{29} -176.000 q^{31} -240.000 q^{33} +80.0000 q^{35} +254.000 q^{37} -344.000 q^{39} +186.000 q^{41} +100.000 q^{43} -55.0000 q^{45} -168.000 q^{47} -87.0000 q^{49} -72.0000 q^{51} -498.000 q^{53} +300.000 q^{55} +176.000 q^{57} +252.000 q^{59} -58.0000 q^{61} -176.000 q^{63} +430.000 q^{65} +1036.00 q^{67} +192.000 q^{69} -168.000 q^{71} +506.000 q^{73} -100.000 q^{75} +960.000 q^{77} -272.000 q^{79} -311.000 q^{81} -948.000 q^{83} +90.0000 q^{85} +744.000 q^{87} -1014.00 q^{89} +1376.00 q^{91} +704.000 q^{93} -220.000 q^{95} -766.000 q^{97} -660.000 q^{99} +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

)

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

$2$	0	0
$2$	0	0
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	16.0000	0.863919	0.431959	−	0.901893i	$-0.357822\pi$
$7$	16.0000	0.863919	0.431959	+	0.901893i	$0.357822\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	60.0000	1.64461	0.822304	−	0.569049i	$-0.192689\pi$
$11$	60.0000	1.64461	0.822304	+	0.569049i	$0.192689\pi$
$12$	0	0
$13$	86.0000	1.83478	0.917389	−	0.397992i	$-0.130293\pi$
$13$	86.0000	1.83478	0.917389	+	0.397992i	$0.130293\pi$
$14$	0	0
$15$	−20.0000	−0.344265
$16$	0	0
$17$	18.0000	0.256802	0.128401	−	0.991722i	$-0.459015\pi$
$17$	18.0000	0.256802	0.128401	+	0.991722i	$0.459015\pi$
$18$	0	0
$19$	−44.0000	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$19$	−44.0000	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$20$	0	0
$21$	−64.0000	−0.665045
$22$	0	0
$23$	−48.0000	−0.435161	−0.217580	−	0.976042i	$-0.569816\pi$
$23$	−48.0000	−0.435161	−0.217580	+	0.976042i	$0.569816\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	−186.000	−1.19101	−0.595506	−	0.803351i	$-0.703048\pi$
$29$	−186.000	−1.19101	−0.595506	+	0.803351i	$0.703048\pi$
$30$	0	0
$31$	−176.000	−1.01969	−0.509847	−	0.860265i	$-0.670298\pi$
$31$	−176.000	−1.01969	−0.509847	+	0.860265i	$0.670298\pi$
$32$	0	0
$33$	−240.000	−1.26602
$34$	0	0
$35$	80.0000	0.386356
$36$	0	0
$37$	254.000	1.12858	0.564288	−	0.825578i	$-0.309151\pi$
$37$	254.000	1.12858	0.564288	+	0.825578i	$0.309151\pi$
$38$	0	0
$39$	−344.000	−1.41241
$40$	0	0
$41$	186.000	0.708496	0.354248	−	0.935152i	$-0.384737\pi$
$41$	186.000	0.708496	0.354248	+	0.935152i	$0.384737\pi$
$42$	0	0
$43$	100.000	0.354648	0.177324	−	0.984153i	$-0.443256\pi$
$43$	100.000	0.354648	0.177324	+	0.984153i	$0.443256\pi$
$44$	0	0
$45$	−55.0000	−0.182198
$46$	0	0
$47$	−168.000	−0.521390	−0.260695	−	0.965421i	$-0.583952\pi$
$47$	−168.000	−0.521390	−0.260695	+	0.965421i	$0.583952\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	−72.0000	−0.197687
$52$	0	0
$53$	−498.000	−1.29067	−0.645335	−	0.763899i	$-0.723282\pi$
$53$	−498.000	−1.29067	−0.645335	+	0.763899i	$0.723282\pi$
$54$	0	0
$55$	300.000	0.735491
$56$	0	0
$57$	176.000	0.408978
$58$	0	0
$59$	252.000	0.556061	0.278031	−	0.960572i	$-0.410318\pi$
$59$	252.000	0.556061	0.278031	+	0.960572i	$0.410318\pi$
$60$	0	0
$61$	−58.0000	−0.121740	−0.0608700	−	0.998146i	$-0.519388\pi$
$61$	−58.0000	−0.121740	−0.0608700	+	0.998146i	$0.519388\pi$
$62$	0	0
$63$	−176.000	−0.351967
$64$	0	0
$65$	430.000	0.820537
$66$	0	0
$67$	1036.00	1.88907	0.944534	−	0.328414i	$-0.106514\pi$
$67$	1036.00	1.88907	0.944534	+	0.328414i	$0.106514\pi$
$68$	0	0
$69$	192.000	0.334987
$70$	0	0
$71$	−168.000	−0.280816	−0.140408	−	0.990094i	$-0.544841\pi$
$71$	−168.000	−0.280816	−0.140408	+	0.990094i	$0.544841\pi$
$72$	0	0
$73$	506.000	0.811272	0.405636	−	0.914035i	$-0.367050\pi$
$73$	506.000	0.811272	0.405636	+	0.914035i	$0.367050\pi$
$74$	0	0
$75$	−100.000	−0.153960
$76$	0	0
$77$	960.000	1.42081
$78$	0	0
$79$	−272.000	−0.387372	−0.193686	−	0.981064i	$-0.562044\pi$
$79$	−272.000	−0.387372	−0.193686	+	0.981064i	$0.562044\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	−948.000	−1.25369	−0.626846	−	0.779143i	$-0.715655\pi$
$83$	−948.000	−1.25369	−0.626846	+	0.779143i	$0.715655\pi$
$84$	0	0
$85$	90.0000	0.114846
$86$	0	0
$87$	744.000	0.916841
$88$	0	0
$89$	−1014.00	−1.20768	−0.603841	−	0.797104i	$-0.706364\pi$
$89$	−1014.00	−1.20768	−0.603841	+	0.797104i	$0.706364\pi$
$90$	0	0
$91$	1376.00	1.58510
$92$	0	0
$93$	704.000	0.784961
$94$	0	0
$95$	−220.000	−0.237595
$96$	0	0
$97$	−766.000	−0.801809	−0.400905	−	0.916120i	$-0.631304\pi$
$97$	−766.000	−0.801809	−0.400905	+	0.916120i	$0.631304\pi$
$98$	0	0
$99$	−660.000	−0.670025
$100$	0	0
$101$	−1314.00	−1.29453	−0.647267	−	0.762264i	$-0.724088\pi$
$101$	−1314.00	−1.29453	−0.647267	+	0.762264i	$0.724088\pi$
$102$	0	0
$103$	448.000	0.428570	0.214285	−	0.976771i	$-0.431258\pi$
$103$	448.000	0.428570	0.214285	+	0.976771i	$0.431258\pi$
$104$	0	0
$105$	−320.000	−0.297417
$106$	0	0
$107$	−1548.00	−1.39861	−0.699303	−	0.714826i	$-0.746506\pi$
$107$	−1548.00	−1.39861	−0.699303	+	0.714826i	$0.746506\pi$
$108$	0	0
$109$	278.000	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$109$	278.000	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$110$	0	0
$111$	−1016.00	−0.868779
$112$	0	0
$113$	−558.000	−0.464533	−0.232266	−	0.972652i	$-0.574614\pi$
$113$	−558.000	−0.464533	−0.232266	+	0.972652i	$0.574614\pi$
$114$	0	0
$115$	−240.000	−0.194610
$116$	0	0
$117$	−946.000	−0.747502
$118$	0	0
$119$	288.000	0.221856
$120$	0	0
$121$	2269.00	1.70473
$122$	0	0
$123$	−744.000	−0.545400
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−344.000	−0.240355	−0.120177	−	0.992752i	$-0.538346\pi$
$127$	−344.000	−0.240355	−0.120177	+	0.992752i	$0.538346\pi$
$128$	0	0
$129$	−400.000	−0.273008
$130$	0	0
$131$	−780.000	−0.520221	−0.260110	−	0.965579i	$-0.583759\pi$
$131$	−780.000	−0.520221	−0.260110	+	0.965579i	$0.583759\pi$
$132$	0	0
$133$	−704.000	−0.458982
$134$	0	0
$135$	760.000	0.484521
$136$	0	0
$137$	666.000	0.415330	0.207665	−	0.978200i	$-0.433414\pi$
$137$	666.000	0.415330	0.207665	+	0.978200i	$0.433414\pi$
$138$	0	0
$139$	−884.000	−0.539424	−0.269712	−	0.962941i	$-0.586928\pi$
$139$	−884.000	−0.539424	−0.269712	+	0.962941i	$0.586928\pi$
$140$	0	0
$141$	672.000	0.401366
$142$	0	0
$143$	5160.00	3.01749
$144$	0	0
$145$	−930.000	−0.532637
$146$	0	0
$147$	348.000	0.195255
$148$	0	0
$149$	−114.000	−0.0626795	−0.0313397	−	0.999509i	$-0.509977\pi$
$149$	−114.000	−0.0626795	−0.0313397	+	0.999509i	$0.509977\pi$
$150$	0	0
$151$	40.0000	0.0215573	0.0107787	−	0.999942i	$-0.496569\pi$
$151$	40.0000	0.0215573	0.0107787	+	0.999942i	$0.496569\pi$
$152$	0	0
$153$	−198.000	−0.104623
$154$	0	0
$155$	−880.000	−0.456021
$156$	0	0
$157$	−154.000	−0.0782837	−0.0391418	−	0.999234i	$-0.512462\pi$
$157$	−154.000	−0.0782837	−0.0391418	+	0.999234i	$0.512462\pi$
$158$	0	0
$159$	1992.00	0.993559
$160$	0	0
$161$	−768.000	−0.375943
$162$	0	0
$163$	−2180.00	−1.04755	−0.523775	−	0.851856i	$-0.675477\pi$
$163$	−2180.00	−1.04755	−0.523775	+	0.851856i	$0.675477\pi$
$164$	0	0
$165$	−1200.00	−0.566181
$166$	0	0
$167$	−3696.00	−1.71261	−0.856303	−	0.516474i	$-0.827244\pi$
$167$	−3696.00	−1.71261	−0.856303	+	0.516474i	$0.827244\pi$
$168$	0	0
$169$	5199.00	2.36641
$170$	0	0
$171$	484.000	0.216447
$172$	0	0
$173$	1302.00	0.572192	0.286096	−	0.958201i	$-0.407642\pi$
$173$	1302.00	0.572192	0.286096	+	0.958201i	$0.407642\pi$
$174$	0	0
$175$	400.000	0.172784
$176$	0	0
$177$	−1008.00	−0.428056
$178$	0	0
$179$	4308.00	1.79885	0.899427	−	0.437070i	$-0.143984\pi$
$179$	4308.00	1.79885	0.899427	+	0.437070i	$0.143984\pi$
$180$	0	0
$181$	1550.00	0.636523	0.318261	−	0.948003i	$-0.396901\pi$
$181$	1550.00	0.636523	0.318261	+	0.948003i	$0.396901\pi$
$182$	0	0
$183$	232.000	0.0937155
$184$	0	0
$185$	1270.00	0.504715
$186$	0	0
$187$	1080.00	0.422339
$188$	0	0
$189$	2432.00	0.935989
$190$	0	0
$191$	−48.0000	−0.0181841	−0.00909204	−	0.999959i	$-0.502894\pi$
$191$	−48.0000	−0.0181841	−0.00909204	+	0.999959i	$0.502894\pi$
$192$	0	0
$193$	1058.00	0.394593	0.197297	−	0.980344i	$-0.436784\pi$
$193$	1058.00	0.394593	0.197297	+	0.980344i	$0.436784\pi$
$194$	0	0
$195$	−1720.00	−0.631650
$196$	0	0
$197$	−3714.00	−1.34321	−0.671603	−	0.740911i	$-0.734394\pi$
$197$	−3714.00	−1.34321	−0.671603	+	0.740911i	$0.734394\pi$
$198$	0	0
$199$	1768.00	0.629800	0.314900	−	0.949125i	$-0.398029\pi$
$199$	1768.00	0.629800	0.314900	+	0.949125i	$0.398029\pi$
$200$	0	0
$201$	−4144.00	−1.45421
$202$	0	0
$203$	−2976.00	−1.02894
$204$	0	0
$205$	930.000	0.316849
$206$	0	0
$207$	528.000	0.177288
$208$	0	0
$209$	−2640.00	−0.873745
$210$	0	0
$211$	4036.00	1.31682	0.658412	−	0.752658i	$-0.271229\pi$
$211$	4036.00	1.31682	0.658412	+	0.752658i	$0.271229\pi$
$212$	0	0
$213$	672.000	0.216172
$214$	0	0
$215$	500.000	0.158603
$216$	0	0
$217$	−2816.00	−0.880933
$218$	0	0
$219$	−2024.00	−0.624517
$220$	0	0
$221$	1548.00	0.471175
$222$	0	0
$223$	−680.000	−0.204198	−0.102099	−	0.994774i	$-0.532556\pi$
$223$	−680.000	−0.204198	−0.102099	+	0.994774i	$0.532556\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	−2388.00	−0.698225	−0.349113	−	0.937081i	$-0.613517\pi$
$227$	−2388.00	−0.698225	−0.349113	+	0.937081i	$0.613517\pi$
$228$	0	0
$229$	−3874.00	−1.11791	−0.558954	−	0.829198i	$-0.688797\pi$
$229$	−3874.00	−1.11791	−0.558954	+	0.829198i	$0.688797\pi$
$230$	0	0
$231$	−3840.00	−1.09374
$232$	0	0
$233$	3162.00	0.889054	0.444527	−	0.895766i	$-0.353372\pi$
$233$	3162.00	0.889054	0.444527	+	0.895766i	$0.353372\pi$
$234$	0	0
$235$	−840.000	−0.233173
$236$	0	0
$237$	1088.00	0.298199
$238$	0	0
$239$	−5424.00	−1.46799	−0.733995	−	0.679155i	$-0.762346\pi$
$239$	−5424.00	−1.46799	−0.733995	+	0.679155i	$0.762346\pi$
$240$	0	0
$241$	−3886.00	−1.03867	−0.519335	−	0.854571i	$-0.673820\pi$
$241$	−3886.00	−1.03867	−0.519335	+	0.854571i	$0.673820\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	−435.000	−0.113433
$246$	0	0
$247$	−3784.00	−0.974778
$248$	0	0
$249$	3792.00	0.965093
$250$	0	0
$251$	5100.00	1.28251	0.641253	−	0.767329i	$-0.278415\pi$
$251$	5100.00	1.28251	0.641253	+	0.767329i	$0.278415\pi$
$252$	0	0
$253$	−2880.00	−0.715668
$254$	0	0
$255$	−360.000	−0.0884081
$256$	0	0
$257$	2178.00	0.528638	0.264319	−	0.964435i	$-0.414853\pi$
$257$	2178.00	0.528638	0.264319	+	0.964435i	$0.414853\pi$
$258$	0	0
$259$	4064.00	0.974999
$260$	0	0
$261$	2046.00	0.485227
$262$	0	0
$263$	6144.00	1.44051	0.720257	−	0.693707i	$-0.244024\pi$
$263$	6144.00	1.44051	0.720257	+	0.693707i	$0.244024\pi$
$264$	0	0
$265$	−2490.00	−0.577206
$266$	0	0
$267$	4056.00	0.929675
$268$	0	0
$269$	822.000	0.186313	0.0931566	−	0.995651i	$-0.470304\pi$
$269$	822.000	0.186313	0.0931566	+	0.995651i	$0.470304\pi$
$270$	0	0
$271$	−8480.00	−1.90082	−0.950412	−	0.310994i	$-0.899338\pi$
$271$	−8480.00	−1.90082	−0.950412	+	0.310994i	$0.899338\pi$
$272$	0	0
$273$	−5504.00	−1.22021
$274$	0	0
$275$	1500.00	0.328921
$276$	0	0
$277$	−1138.00	−0.246844	−0.123422	−	0.992354i	$-0.539387\pi$
$277$	−1138.00	−0.246844	−0.123422	+	0.992354i	$0.539387\pi$
$278$	0	0
$279$	1936.00	0.415431
$280$	0	0
$281$	5706.00	1.21136	0.605679	−	0.795709i	$-0.292902\pi$
$281$	5706.00	1.21136	0.605679	+	0.795709i	$0.292902\pi$
$282$	0	0
$283$	3028.00	0.636028	0.318014	−	0.948086i	$-0.396984\pi$
$283$	3028.00	0.636028	0.318014	+	0.948086i	$0.396984\pi$
$284$	0	0
$285$	880.000	0.182901
$286$	0	0
$287$	2976.00	0.612083
$288$	0	0
$289$	−4589.00	−0.934053
$290$	0	0
$291$	3064.00	0.617233
$292$	0	0
$293$	3390.00	0.675925	0.337962	−	0.941160i	$-0.390262\pi$
$293$	3390.00	0.675925	0.337962	+	0.941160i	$0.390262\pi$
$294$	0	0
$295$	1260.00	0.248678
$296$	0	0
$297$	9120.00	1.78180
$298$	0	0
$299$	−4128.00	−0.798423
$300$	0	0
$301$	1600.00	0.306387
$302$	0	0
$303$	5256.00	0.996532
$304$	0	0
$305$	−290.000	−0.0544438
$306$	0	0
$307$	4156.00	0.772624	0.386312	−	0.922368i	$-0.373749\pi$
$307$	4156.00	0.772624	0.386312	+	0.922368i	$0.373749\pi$
$308$	0	0
$309$	−1792.00	−0.329914
$310$	0	0
$311$	−6552.00	−1.19463	−0.597315	−	0.802007i	$-0.703766\pi$
$311$	−6552.00	−1.19463	−0.597315	+	0.802007i	$0.703766\pi$
$312$	0	0
$313$	−1366.00	−0.246680	−0.123340	−	0.992364i	$-0.539361\pi$
$313$	−1366.00	−0.246680	−0.123340	+	0.992364i	$0.539361\pi$
$314$	0	0
$315$	−880.000	−0.157404
$316$	0	0
$317$	2598.00	0.460310	0.230155	−	0.973154i	$-0.426077\pi$
$317$	2598.00	0.460310	0.230155	+	0.973154i	$0.426077\pi$
$318$	0	0
$319$	−11160.0	−1.95875
$320$	0	0
$321$	6192.00	1.07665
$322$	0	0
$323$	−792.000	−0.136434
$324$	0	0
$325$	2150.00	0.366956
$326$	0	0
$327$	−1112.00	−0.188054
$328$	0	0
$329$	−2688.00	−0.450438
$330$	0	0
$331$	3292.00	0.546661	0.273330	−	0.961920i	$-0.411875\pi$
$331$	3292.00	0.546661	0.273330	+	0.961920i	$0.411875\pi$
$332$	0	0
$333$	−2794.00	−0.459791
$334$	0	0
$335$	5180.00	0.844817
$336$	0	0
$337$	6194.00	1.00121	0.500606	−	0.865675i	$-0.333110\pi$
$337$	6194.00	1.00121	0.500606	+	0.865675i	$0.333110\pi$
$338$	0	0
$339$	2232.00	0.357598
$340$	0	0
$341$	−10560.0	−1.67700
$342$	0	0
$343$	−6880.00	−1.08305
$344$	0	0
$345$	960.000	0.149811
$346$	0	0
$347$	10020.0	1.55015	0.775075	−	0.631870i	$-0.217712\pi$
$347$	10020.0	1.55015	0.775075	+	0.631870i	$0.217712\pi$
$348$	0	0
$349$	−3130.00	−0.480072	−0.240036	−	0.970764i	$-0.577159\pi$
$349$	−3130.00	−0.480072	−0.240036	+	0.970764i	$0.577159\pi$
$350$	0	0
$351$	13072.0	1.98784
$352$	0	0
$353$	4194.00	0.632363	0.316181	−	0.948699i	$-0.397599\pi$
$353$	4194.00	0.632363	0.316181	+	0.948699i	$0.397599\pi$
$354$	0	0
$355$	−840.000	−0.125585
$356$	0	0
$357$	−1152.00	−0.170785
$358$	0	0
$359$	4104.00	0.603345	0.301672	−	0.953412i	$-0.402455\pi$
$359$	4104.00	0.603345	0.301672	+	0.953412i	$0.402455\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	−9076.00	−1.31230
$364$	0	0
$365$	2530.00	0.362812
$366$	0	0
$367$	−7496.00	−1.06618	−0.533090	−	0.846059i	$-0.678969\pi$
$367$	−7496.00	−1.06618	−0.533090	+	0.846059i	$0.678969\pi$
$368$	0	0
$369$	−2046.00	−0.288646
$370$	0	0
$371$	−7968.00	−1.11503
$372$	0	0
$373$	−5842.00	−0.810958	−0.405479	−	0.914104i	$-0.632895\pi$
$373$	−5842.00	−0.810958	−0.405479	+	0.914104i	$0.632895\pi$
$374$	0	0
$375$	−500.000	−0.0688530
$376$	0	0
$377$	−15996.0	−2.18524
$378$	0	0
$379$	412.000	0.0558391	0.0279195	−	0.999610i	$-0.491112\pi$
$379$	412.000	0.0558391	0.0279195	+	0.999610i	$0.491112\pi$
$380$	0	0
$381$	1376.00	0.185025
$382$	0	0
$383$	−2568.00	−0.342607	−0.171304	−	0.985218i	$-0.554798\pi$
$383$	−2568.00	−0.342607	−0.171304	+	0.985218i	$0.554798\pi$
$384$	0	0
$385$	4800.00	0.635404
$386$	0	0
$387$	−1100.00	−0.144486
$388$	0	0
$389$	13086.0	1.70562	0.852810	−	0.522221i	$-0.174896\pi$
$389$	13086.0	1.70562	0.852810	+	0.522221i	$0.174896\pi$
$390$	0	0
$391$	−864.000	−0.111750
$392$	0	0
$393$	3120.00	0.400466
$394$	0	0
$395$	−1360.00	−0.173238
$396$	0	0
$397$	10454.0	1.32159	0.660795	−	0.750566i	$-0.270219\pi$
$397$	10454.0	1.32159	0.660795	+	0.750566i	$0.270219\pi$
$398$	0	0
$399$	2816.00	0.353324
$400$	0	0
$401$	−10830.0	−1.34869	−0.674345	−	0.738417i	$-0.735574\pi$
$401$	−10830.0	−1.34869	−0.674345	+	0.738417i	$0.735574\pi$
$402$	0	0
$403$	−15136.0	−1.87091
$404$	0	0
$405$	−1555.00	−0.190787
$406$	0	0
$407$	15240.0	1.85607
$408$	0	0
$409$	−8566.00	−1.03560	−0.517801	−	0.855501i	$-0.673249\pi$
$409$	−8566.00	−1.03560	−0.517801	+	0.855501i	$0.673249\pi$
$410$	0	0
$411$	−2664.00	−0.319721
$412$	0	0
$413$	4032.00	0.480392
$414$	0	0
$415$	−4740.00	−0.560669
$416$	0	0
$417$	3536.00	0.415249
$418$	0	0
$419$	−13884.0	−1.61880	−0.809401	−	0.587257i	$-0.800208\pi$
$419$	−13884.0	−1.61880	−0.809401	+	0.587257i	$0.800208\pi$
$420$	0	0
$421$	4286.00	0.496168	0.248084	−	0.968738i	$-0.420199\pi$
$421$	4286.00	0.496168	0.248084	+	0.968738i	$0.420199\pi$
$422$	0	0
$423$	1848.00	0.212418
$424$	0	0
$425$	450.000	0.0513605
$426$	0	0
$427$	−928.000	−0.105173
$428$	0	0
$429$	−20640.0	−2.32286
$430$	0	0
$431$	−6336.00	−0.708108	−0.354054	−	0.935225i	$-0.615197\pi$
$431$	−6336.00	−0.708108	−0.354054	+	0.935225i	$0.615197\pi$
$432$	0	0
$433$	−8974.00	−0.995988	−0.497994	−	0.867180i	$-0.665930\pi$
$433$	−8974.00	−0.995988	−0.497994	+	0.867180i	$0.665930\pi$
$434$	0	0
$435$	3720.00	0.410024
$436$	0	0
$437$	2112.00	0.231191
$438$	0	0
$439$	2968.00	0.322676	0.161338	−	0.986899i	$-0.448419\pi$
$439$	2968.00	0.322676	0.161338	+	0.986899i	$0.448419\pi$
$440$	0	0
$441$	957.000	0.103337
$442$	0	0
$443$	12372.0	1.32689	0.663444	−	0.748226i	$-0.269094\pi$
$443$	12372.0	1.32689	0.663444	+	0.748226i	$0.269094\pi$
$444$	0	0
$445$	−5070.00	−0.540092
$446$	0	0
$447$	456.000	0.0482507
$448$	0	0
$449$	11394.0	1.19759	0.598793	−	0.800904i	$-0.295647\pi$
$449$	11394.0	1.19759	0.598793	+	0.800904i	$0.295647\pi$
$450$	0	0
$451$	11160.0	1.16520
$452$	0	0
$453$	−160.000	−0.0165948
$454$	0	0
$455$	6880.00	0.708878
$456$	0	0
$457$	−358.000	−0.0366445	−0.0183222	−	0.999832i	$-0.505832\pi$
$457$	−358.000	−0.0366445	−0.0183222	+	0.999832i	$0.505832\pi$
$458$	0	0
$459$	2736.00	0.278226
$460$	0	0
$461$	−7530.00	−0.760753	−0.380376	−	0.924832i	$-0.624206\pi$
$461$	−7530.00	−0.760753	−0.380376	+	0.924832i	$0.624206\pi$
$462$	0	0
$463$	13768.0	1.38197	0.690986	−	0.722868i	$-0.257177\pi$
$463$	13768.0	1.38197	0.690986	+	0.722868i	$0.257177\pi$
$464$	0	0
$465$	3520.00	0.351045
$466$	0	0
$467$	−13380.0	−1.32581	−0.662904	−	0.748704i	$-0.730676\pi$
$467$	−13380.0	−1.32581	−0.662904	+	0.748704i	$0.730676\pi$
$468$	0	0
$469$	16576.0	1.63200
$470$	0	0
$471$	616.000	0.0602628
$472$	0	0
$473$	6000.00	0.583256
$474$	0	0
$475$	−1100.00	−0.106256
$476$	0	0
$477$	5478.00	0.525829
$478$	0	0
$479$	6336.00	0.604383	0.302191	−	0.953247i	$-0.402282\pi$
$479$	6336.00	0.604383	0.302191	+	0.953247i	$0.402282\pi$
$480$	0	0
$481$	21844.0	2.07069
$482$	0	0
$483$	3072.00	0.289401
$484$	0	0
$485$	−3830.00	−0.358580
$486$	0	0
$487$	5008.00	0.465984	0.232992	−	0.972479i	$-0.425148\pi$
$487$	5008.00	0.465984	0.232992	+	0.972479i	$0.425148\pi$
$488$	0	0
$489$	8720.00	0.806405
$490$	0	0
$491$	−12900.0	−1.18568	−0.592840	−	0.805320i	$-0.701993\pi$
$491$	−12900.0	−1.18568	−0.592840	+	0.805320i	$0.701993\pi$
$492$	0	0
$493$	−3348.00	−0.305855
$494$	0	0
$495$	−3300.00	−0.299644
$496$	0	0
$497$	−2688.00	−0.242602
$498$	0	0
$499$	8116.00	0.728100	0.364050	−	0.931379i	$-0.381394\pi$
$499$	8116.00	0.728100	0.364050	+	0.931379i	$0.381394\pi$
$500$	0	0
$501$	14784.0	1.31836
$502$	0	0
$503$	4944.00	0.438255	0.219127	−	0.975696i	$-0.429679\pi$
$503$	4944.00	0.438255	0.219127	+	0.975696i	$0.429679\pi$
$504$	0	0
$505$	−6570.00	−0.578933
$506$	0	0
$507$	−20796.0	−1.82166
$508$	0	0
$509$	−5466.00	−0.475985	−0.237992	−	0.971267i	$-0.576489\pi$
$509$	−5466.00	−0.475985	−0.237992	+	0.971267i	$0.576489\pi$
$510$	0	0
$511$	8096.00	0.700873
$512$	0	0
$513$	−6688.00	−0.575599
$514$	0	0
$515$	2240.00	0.191663
$516$	0	0
$517$	−10080.0	−0.857481
$518$	0	0
$519$	−5208.00	−0.440474
$520$	0	0
$521$	10074.0	0.847121	0.423560	−	0.905868i	$-0.360780\pi$
$521$	10074.0	0.847121	0.423560	+	0.905868i	$0.360780\pi$
$522$	0	0
$523$	13828.0	1.15613	0.578065	−	0.815991i	$-0.303808\pi$
$523$	13828.0	1.15613	0.578065	+	0.815991i	$0.303808\pi$
$524$	0	0
$525$	−1600.00	−0.133009
$526$	0	0
$527$	−3168.00	−0.261860
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	−2772.00	−0.226543
$532$	0	0
$533$	15996.0	1.29993
$534$	0	0
$535$	−7740.00	−0.625475
$536$	0	0
$537$	−17232.0	−1.38476
$538$	0	0
$539$	−5220.00	−0.417145
$540$	0	0
$541$	−15226.0	−1.21001	−0.605006	−	0.796221i	$-0.706829\pi$
$541$	−15226.0	−1.21001	−0.605006	+	0.796221i	$0.706829\pi$
$542$	0	0
$543$	−6200.00	−0.489995
$544$	0	0
$545$	1390.00	0.109250
$546$	0	0
$547$	13228.0	1.03398	0.516991	−	0.855991i	$-0.327052\pi$
$547$	13228.0	1.03398	0.516991	+	0.855991i	$0.327052\pi$
$548$	0	0
$549$	638.000	0.0495978
$550$	0	0
$551$	8184.00	0.632759
$552$	0	0
$553$	−4352.00	−0.334658
$554$	0	0
$555$	−5080.00	−0.388530
$556$	0	0
$557$	−8490.00	−0.645840	−0.322920	−	0.946426i	$-0.604664\pi$
$557$	−8490.00	−0.645840	−0.322920	+	0.946426i	$0.604664\pi$
$558$	0	0
$559$	8600.00	0.650700
$560$	0	0
$561$	−4320.00	−0.325117
$562$	0	0
$563$	10284.0	0.769838	0.384919	−	0.922950i	$-0.374229\pi$
$563$	10284.0	0.769838	0.384919	+	0.922950i	$0.374229\pi$
$564$	0	0
$565$	−2790.00	−0.207745
$566$	0	0
$567$	−4976.00	−0.368558
$568$	0	0
$569$	1770.00	0.130408	0.0652041	−	0.997872i	$-0.479230\pi$
$569$	1770.00	0.130408	0.0652041	+	0.997872i	$0.479230\pi$
$570$	0	0
$571$	−6068.00	−0.444725	−0.222362	−	0.974964i	$-0.571377\pi$
$571$	−6068.00	−0.444725	−0.222362	+	0.974964i	$0.571377\pi$
$572$	0	0
$573$	192.000	0.0139981
$574$	0	0
$575$	−1200.00	−0.0870321
$576$	0	0
$577$	21506.0	1.55166	0.775829	−	0.630943i	$-0.217332\pi$
$577$	21506.0	1.55166	0.775829	+	0.630943i	$0.217332\pi$
$578$	0	0
$579$	−4232.00	−0.303758
$580$	0	0
$581$	−15168.0	−1.08309
$582$	0	0
$583$	−29880.0	−2.12265
$584$	0	0
$585$	−4730.00	−0.334293
$586$	0	0
$587$	−12108.0	−0.851364	−0.425682	−	0.904873i	$-0.639966\pi$
$587$	−12108.0	−0.851364	−0.425682	+	0.904873i	$0.639966\pi$
$588$	0	0
$589$	7744.00	0.541742
$590$	0	0
$591$	14856.0	1.03400
$592$	0	0
$593$	15474.0	1.07157	0.535785	−	0.844354i	$-0.320016\pi$
$593$	15474.0	1.07157	0.535785	+	0.844354i	$0.320016\pi$
$594$	0	0
$595$	1440.00	0.0992172
$596$	0	0
$597$	−7072.00	−0.484820
$598$	0	0
$599$	2520.00	0.171894	0.0859469	−	0.996300i	$-0.472608\pi$
$599$	2520.00	0.171894	0.0859469	+	0.996300i	$0.472608\pi$
$600$	0	0
$601$	−12790.0	−0.868078	−0.434039	−	0.900894i	$-0.642912\pi$
$601$	−12790.0	−0.868078	−0.434039	+	0.900894i	$0.642912\pi$
$602$	0	0
$603$	−11396.0	−0.769620
$604$	0	0
$605$	11345.0	0.762380
$606$	0	0
$607$	−11576.0	−0.774062	−0.387031	−	0.922067i	$-0.626499\pi$
$607$	−11576.0	−0.774062	−0.387031	+	0.922067i	$0.626499\pi$
$608$	0	0
$609$	11904.0	0.792076
$610$	0	0
$611$	−14448.0	−0.956634
$612$	0	0
$613$	20126.0	1.32607	0.663035	−	0.748588i	$-0.269268\pi$
$613$	20126.0	1.32607	0.663035	+	0.748588i	$0.269268\pi$
$614$	0	0
$615$	−3720.00	−0.243910
$616$	0	0
$617$	−27942.0	−1.82318	−0.911590	−	0.411100i	$-0.865145\pi$
$617$	−27942.0	−1.82318	−0.911590	+	0.411100i	$0.865145\pi$
$618$	0	0
$619$	22540.0	1.46358	0.731792	−	0.681528i	$-0.238684\pi$
$619$	22540.0	1.46358	0.731792	+	0.681528i	$0.238684\pi$
$620$	0	0
$621$	−7296.00	−0.471463
$622$	0	0
$623$	−16224.0	−1.04334
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	10560.0	0.672609
$628$	0	0
$629$	4572.00	0.289821
$630$	0	0
$631$	5128.00	0.323522	0.161761	−	0.986830i	$-0.448283\pi$
$631$	5128.00	0.323522	0.161761	+	0.986830i	$0.448283\pi$
$632$	0	0
$633$	−16144.0	−1.01369
$634$	0	0
$635$	−1720.00	−0.107490
$636$	0	0
$637$	−7482.00	−0.465381
$638$	0	0
$639$	1848.00	0.114406
$640$	0	0
$641$	−12798.0	−0.788597	−0.394298	−	0.918982i	$-0.629012\pi$
$641$	−12798.0	−0.788597	−0.394298	+	0.918982i	$0.629012\pi$
$642$	0	0
$643$	21148.0	1.29704	0.648519	−	0.761198i	$-0.275389\pi$
$643$	21148.0	1.29704	0.648519	+	0.761198i	$0.275389\pi$
$644$	0	0
$645$	−2000.00	−0.122093
$646$	0	0
$647$	−16464.0	−1.00041	−0.500206	−	0.865906i	$-0.666742\pi$
$647$	−16464.0	−1.00041	−0.500206	+	0.865906i	$0.666742\pi$
$648$	0	0
$649$	15120.0	0.914502
$650$	0	0
$651$	11264.0	0.678143
$652$	0	0
$653$	−24234.0	−1.45230	−0.726148	−	0.687538i	$-0.758691\pi$
$653$	−24234.0	−1.45230	−0.726148	+	0.687538i	$0.758691\pi$
$654$	0	0
$655$	−3900.00	−0.232650
$656$	0	0
$657$	−5566.00	−0.330518
$658$	0	0
$659$	22836.0	1.34987	0.674935	−	0.737877i	$-0.264172\pi$
$659$	22836.0	1.34987	0.674935	+	0.737877i	$0.264172\pi$
$660$	0	0
$661$	26318.0	1.54864	0.774320	−	0.632794i	$-0.218092\pi$
$661$	26318.0	1.54864	0.774320	+	0.632794i	$0.218092\pi$
$662$	0	0
$663$	−6192.00	−0.362711
$664$	0	0
$665$	−3520.00	−0.205263
$666$	0	0
$667$	8928.00	0.518281
$668$	0	0
$669$	2720.00	0.157192
$670$	0	0
$671$	−3480.00	−0.200214
$672$	0	0
$673$	28802.0	1.64968	0.824841	−	0.565365i	$-0.191265\pi$
$673$	28802.0	1.64968	0.824841	+	0.565365i	$0.191265\pi$
$674$	0	0
$675$	3800.00	0.216685
$676$	0	0
$677$	2526.00	0.143400	0.0717002	−	0.997426i	$-0.477158\pi$
$677$	2526.00	0.143400	0.0717002	+	0.997426i	$0.477158\pi$
$678$	0	0
$679$	−12256.0	−0.692698
$680$	0	0
$681$	9552.00	0.537494
$682$	0	0
$683$	23076.0	1.29279	0.646397	−	0.763001i	$-0.276275\pi$
$683$	23076.0	1.29279	0.646397	+	0.763001i	$0.276275\pi$
$684$	0	0
$685$	3330.00	0.185741
$686$	0	0
$687$	15496.0	0.860567
$688$	0	0
$689$	−42828.0	−2.36809
$690$	0	0
$691$	−7868.00	−0.433159	−0.216579	−	0.976265i	$-0.569490\pi$
$691$	−7868.00	−0.433159	−0.216579	+	0.976265i	$0.569490\pi$
$692$	0	0
$693$	−10560.0	−0.578847
$694$	0	0
$695$	−4420.00	−0.241238
$696$	0	0
$697$	3348.00	0.181943
$698$	0	0
$699$	−12648.0	−0.684394
$700$	0	0
$701$	21510.0	1.15895	0.579473	−	0.814991i	$-0.303258\pi$
$701$	21510.0	1.15895	0.579473	+	0.814991i	$0.303258\pi$
$702$	0	0
$703$	−11176.0	−0.599589
$704$	0	0
$705$	3360.00	0.179496
$706$	0	0
$707$	−21024.0	−1.11837
$708$	0	0
$709$	30014.0	1.58984	0.794922	−	0.606712i	$-0.207512\pi$
$709$	30014.0	1.58984	0.794922	+	0.606712i	$0.207512\pi$
$710$	0	0
$711$	2992.00	0.157818
$712$	0	0
$713$	8448.00	0.443731
$714$	0	0
$715$	25800.0	1.34946
$716$	0	0
$717$	21696.0	1.13006
$718$	0	0
$719$	816.000	0.0423250	0.0211625	−	0.999776i	$-0.493263\pi$
$719$	816.000	0.0423250	0.0211625	+	0.999776i	$0.493263\pi$
$720$	0	0
$721$	7168.00	0.370250
$722$	0	0
$723$	15544.0	0.799568
$724$	0	0
$725$	−4650.00	−0.238202
$726$	0	0
$727$	9952.00	0.507702	0.253851	−	0.967243i	$-0.418303\pi$
$727$	9952.00	0.507702	0.253851	+	0.967243i	$0.418303\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	1800.00	0.0910744
$732$	0	0
$733$	−33946.0	−1.71054	−0.855269	−	0.518185i	$-0.826608\pi$
$733$	−33946.0	−1.71054	−0.855269	+	0.518185i	$0.826608\pi$
$734$	0	0
$735$	1740.00	0.0873209
$736$	0	0
$737$	62160.0	3.10677
$738$	0	0
$739$	−23420.0	−1.16579	−0.582895	−	0.812548i	$-0.698080\pi$
$739$	−23420.0	−1.16579	−0.582895	+	0.812548i	$0.698080\pi$
$740$	0	0
$741$	15136.0	0.750384
$742$	0	0
$743$	14592.0	0.720496	0.360248	−	0.932857i	$-0.382692\pi$
$743$	14592.0	0.720496	0.360248	+	0.932857i	$0.382692\pi$
$744$	0	0
$745$	−570.000	−0.0280311
$746$	0	0
$747$	10428.0	0.510764
$748$	0	0
$749$	−24768.0	−1.20828
$750$	0	0
$751$	−9056.00	−0.440024	−0.220012	−	0.975497i	$-0.570610\pi$
$751$	−9056.00	−0.440024	−0.220012	+	0.975497i	$0.570610\pi$
$752$	0	0
$753$	−20400.0	−0.987274
$754$	0	0
$755$	200.000	0.00964072
$756$	0	0
$757$	−17554.0	−0.842815	−0.421408	−	0.906871i	$-0.638464\pi$
$757$	−17554.0	−0.842815	−0.421408	+	0.906871i	$0.638464\pi$
$758$	0	0
$759$	11520.0	0.550922
$760$	0	0
$761$	−36438.0	−1.73571	−0.867856	−	0.496816i	$-0.834502\pi$
$761$	−36438.0	−1.73571	−0.867856	+	0.496816i	$0.834502\pi$
$762$	0	0
$763$	4448.00	0.211046
$764$	0	0
$765$	−990.000	−0.0467889
$766$	0	0
$767$	21672.0	1.02025
$768$	0	0
$769$	−9022.00	−0.423071	−0.211536	−	0.977370i	$-0.567846\pi$
$769$	−9022.00	−0.423071	−0.211536	+	0.977370i	$0.567846\pi$
$770$	0	0
$771$	−8712.00	−0.406946
$772$	0	0
$773$	1470.00	0.0683987	0.0341994	−	0.999415i	$-0.489112\pi$
$773$	1470.00	0.0683987	0.0341994	+	0.999415i	$0.489112\pi$
$774$	0	0
$775$	−4400.00	−0.203939
$776$	0	0
$777$	−16256.0	−0.750554
$778$	0	0
$779$	−8184.00	−0.376409
$780$	0	0
$781$	−10080.0	−0.461832
$782$	0	0
$783$	−28272.0	−1.29037
$784$	0	0
$785$	−770.000	−0.0350095
$786$	0	0
$787$	−5252.00	−0.237883	−0.118941	−	0.992901i	$-0.537950\pi$
$787$	−5252.00	−0.237883	−0.118941	+	0.992901i	$0.537950\pi$
$788$	0	0
$789$	−24576.0	−1.10891
$790$	0	0
$791$	−8928.00	−0.401319
$792$	0	0
$793$	−4988.00	−0.223366
$794$	0	0
$795$	9960.00	0.444333
$796$	0	0
$797$	12294.0	0.546394	0.273197	−	0.961958i	$-0.411919\pi$
$797$	12294.0	0.546394	0.273197	+	0.961958i	$0.411919\pi$
$798$	0	0
$799$	−3024.00	−0.133894
$800$	0	0
$801$	11154.0	0.492019
$802$	0	0
$803$	30360.0	1.33422
$804$	0	0
$805$	−3840.00	−0.168127
$806$	0	0
$807$	−3288.00	−0.143424
$808$	0	0
$809$	15546.0	0.675610	0.337805	−	0.941216i	$-0.390316\pi$
$809$	15546.0	0.675610	0.337805	+	0.941216i	$0.390316\pi$
$810$	0	0
$811$	−19364.0	−0.838424	−0.419212	−	0.907888i	$-0.637694\pi$
$811$	−19364.0	−0.838424	−0.419212	+	0.907888i	$0.637694\pi$
$812$	0	0
$813$	33920.0	1.46326
$814$	0	0
$815$	−10900.0	−0.468479
$816$	0	0
$817$	−4400.00	−0.188417
$818$	0	0
$819$	−15136.0	−0.645781
$820$	0	0
$821$	−7314.00	−0.310914	−0.155457	−	0.987843i	$-0.549685\pi$
$821$	−7314.00	−0.310914	−0.155457	+	0.987843i	$0.549685\pi$
$822$	0	0
$823$	−11984.0	−0.507577	−0.253789	−	0.967260i	$-0.581677\pi$
$823$	−11984.0	−0.507577	−0.253789	+	0.967260i	$0.581677\pi$
$824$	0	0
$825$	−6000.00	−0.253204
$826$	0	0
$827$	−13500.0	−0.567643	−0.283822	−	0.958877i	$-0.591602\pi$
$827$	−13500.0	−0.567643	−0.283822	+	0.958877i	$0.591602\pi$
$828$	0	0
$829$	−44602.0	−1.86863	−0.934313	−	0.356453i	$-0.883986\pi$
$829$	−44602.0	−1.86863	−0.934313	+	0.356453i	$0.883986\pi$
$830$	0	0
$831$	4552.00	0.190021
$832$	0	0
$833$	−1566.00	−0.0651365
$834$	0	0
$835$	−18480.0	−0.765900
$836$	0	0
$837$	−26752.0	−1.10476
$838$	0	0
$839$	−35448.0	−1.45864	−0.729321	−	0.684172i	$-0.760164\pi$
$839$	−35448.0	−1.45864	−0.729321	+	0.684172i	$0.760164\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	0	0
$843$	−22824.0	−0.932503
$844$	0	0
$845$	25995.0	1.05829
$846$	0	0
$847$	36304.0	1.47275
$848$	0	0
$849$	−12112.0	−0.489615
$850$	0	0
$851$	−12192.0	−0.491112
$852$	0	0
$853$	12590.0	0.505362	0.252681	−	0.967550i	$-0.418688\pi$
$853$	12590.0	0.505362	0.252681	+	0.967550i	$0.418688\pi$
$854$	0	0
$855$	2420.00	0.0967980
$856$	0	0
$857$	24906.0	0.992734	0.496367	−	0.868113i	$-0.334667\pi$
$857$	24906.0	0.992734	0.496367	+	0.868113i	$0.334667\pi$
$858$	0	0
$859$	−23204.0	−0.921665	−0.460833	−	0.887487i	$-0.652449\pi$
$859$	−23204.0	−0.921665	−0.460833	+	0.887487i	$0.652449\pi$
$860$	0	0
$861$	−11904.0	−0.471181
$862$	0	0
$863$	−19848.0	−0.782890	−0.391445	−	0.920202i	$-0.628025\pi$
$863$	−19848.0	−0.782890	−0.391445	+	0.920202i	$0.628025\pi$
$864$	0	0
$865$	6510.00	0.255892
$866$	0	0
$867$	18356.0	0.719034
$868$	0	0
$869$	−16320.0	−0.637075
$870$	0	0
$871$	89096.0	3.46602
$872$	0	0
$873$	8426.00	0.326663
$874$	0	0
$875$	2000.00	0.0772712
$876$	0	0
$877$	27542.0	1.06046	0.530232	−	0.847852i	$-0.322105\pi$
$877$	27542.0	1.06046	0.530232	+	0.847852i	$0.322105\pi$
$878$	0	0
$879$	−13560.0	−0.520327
$880$	0	0
$881$	−20718.0	−0.792290	−0.396145	−	0.918188i	$-0.629652\pi$
$881$	−20718.0	−0.792290	−0.396145	+	0.918188i	$0.629652\pi$
$882$	0	0
$883$	−25172.0	−0.959349	−0.479675	−	0.877446i	$-0.659245\pi$
$883$	−25172.0	−0.959349	−0.479675	+	0.877446i	$0.659245\pi$
$884$	0	0
$885$	−5040.00	−0.191432
$886$	0	0
$887$	12864.0	0.486957	0.243478	−	0.969906i	$-0.421711\pi$
$887$	12864.0	0.486957	0.243478	+	0.969906i	$0.421711\pi$
$888$	0	0
$889$	−5504.00	−0.207647
$890$	0	0
$891$	−18660.0	−0.701609
$892$	0	0
$893$	7392.00	0.277003
$894$	0	0
$895$	21540.0	0.804472
$896$	0	0
$897$	16512.0	0.614626
$898$	0	0
$899$	32736.0	1.21447
$900$	0	0
$901$	−8964.00	−0.331447
$902$	0	0
$903$	−6400.00	−0.235857
$904$	0	0
$905$	7750.00	0.284662
$906$	0	0
$907$	23092.0	0.845377	0.422689	−	0.906275i	$-0.361086\pi$
$907$	23092.0	0.845377	0.422689	+	0.906275i	$0.361086\pi$
$908$	0	0
$909$	14454.0	0.527403
$910$	0	0
$911$	14208.0	0.516720	0.258360	−	0.966049i	$-0.416818\pi$
$911$	14208.0	0.516720	0.258360	+	0.966049i	$0.416818\pi$
$912$	0	0
$913$	−56880.0	−2.06183
$914$	0	0
$915$	1160.00	0.0419108
$916$	0	0
$917$	−12480.0	−0.449428
$918$	0	0
$919$	26584.0	0.954217	0.477108	−	0.878844i	$-0.341685\pi$
$919$	26584.0	0.954217	0.477108	+	0.878844i	$0.341685\pi$
$920$	0	0
$921$	−16624.0	−0.594766
$922$	0	0
$923$	−14448.0	−0.515235
$924$	0	0
$925$	6350.00	0.225715
$926$	0	0
$927$	−4928.00	−0.174603
$928$	0	0
$929$	162.000	0.00572126	0.00286063	−	0.999996i	$-0.499089\pi$
$929$	162.000	0.00572126	0.00286063	+	0.999996i	$0.499089\pi$
$930$	0	0
$931$	3828.00	0.134756
$932$	0	0
$933$	26208.0	0.919626
$934$	0	0
$935$	5400.00	0.188876
$936$	0	0
$937$	−29734.0	−1.03668	−0.518339	−	0.855175i	$-0.673449\pi$
$937$	−29734.0	−1.03668	−0.518339	+	0.855175i	$0.673449\pi$
$938$	0	0
$939$	5464.00	0.189894
$940$	0	0
$941$	17142.0	0.593850	0.296925	−	0.954901i	$-0.404039\pi$
$941$	17142.0	0.593850	0.296925	+	0.954901i	$0.404039\pi$
$942$	0	0
$943$	−8928.00	−0.308309
$944$	0	0
$945$	12160.0	0.418587
$946$	0	0
$947$	−26436.0	−0.907133	−0.453566	−	0.891223i	$-0.649848\pi$
$947$	−26436.0	−0.907133	−0.453566	+	0.891223i	$0.649848\pi$
$948$	0	0
$949$	43516.0	1.48850
$950$	0	0
$951$	−10392.0	−0.354347
$952$	0	0
$953$	27882.0	0.947730	0.473865	−	0.880598i	$-0.342858\pi$
$953$	27882.0	0.947730	0.473865	+	0.880598i	$0.342858\pi$
$954$	0	0
$955$	−240.000	−0.00813217
$956$	0	0
$957$	44640.0	1.50784
$958$	0	0
$959$	10656.0	0.358811
$960$	0	0
$961$	1185.00	0.0397771
$962$	0	0
$963$	17028.0	0.569802
$964$	0	0
$965$	5290.00	0.176467
$966$	0	0
$967$	−12656.0	−0.420879	−0.210439	−	0.977607i	$-0.567489\pi$
$967$	−12656.0	−0.420879	−0.210439	+	0.977607i	$0.567489\pi$
$968$	0	0
$969$	3168.00	0.105027
$970$	0	0
$971$	−2916.00	−0.0963737	−0.0481869	−	0.998838i	$-0.515344\pi$
$971$	−2916.00	−0.0963737	−0.0481869	+	0.998838i	$0.515344\pi$
$972$	0	0
$973$	−14144.0	−0.466018
$974$	0	0
$975$	−8600.00	−0.282482
$976$	0	0
$977$	−6894.00	−0.225751	−0.112875	−	0.993609i	$-0.536006\pi$
$977$	−6894.00	−0.225751	−0.112875	+	0.993609i	$0.536006\pi$
$978$	0	0
$979$	−60840.0	−1.98616
$980$	0	0
$981$	−3058.00	−0.0995254
$982$	0	0
$983$	45264.0	1.46866	0.734332	−	0.678790i	$-0.237495\pi$
$983$	45264.0	1.46866	0.734332	+	0.678790i	$0.237495\pi$
$984$	0	0
$985$	−18570.0	−0.600700
$986$	0	0
$987$	10752.0	0.346748
$988$	0	0
$989$	−4800.00	−0.154329
$990$	0	0
$991$	−52016.0	−1.66735	−0.833674	−	0.552256i	$-0.813767\pi$
$991$	−52016.0	−1.66735	−0.833674	+	0.552256i	$0.813767\pi$
$992$	0	0
$993$	−13168.0	−0.420820
$994$	0	0
$995$	8840.00	0.281655
$996$	0	0
$997$	−13858.0	−0.440208	−0.220104	−	0.975476i	$-0.570640\pi$
$997$	−13858.0	−0.440208	−0.220104	+	0.975476i	$0.570640\pi$
$998$	0	0
$999$	38608.0	1.22273

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	80.4.a.c.1.1		1
3.2	odd	2		720.4.a.k.1.1		1
4.3	odd	2		20.4.a.a.1.1	✓	1
5.2	odd	4		400.4.c.j.49.2		2
5.3	odd	4		400.4.c.j.49.1		2
5.4	even	2		400.4.a.o.1.1		1
8.3	odd	2		320.4.a.d.1.1		1
8.5	even	2		320.4.a.k.1.1		1
12.11	even	2		180.4.a.a.1.1		1
16.3	odd	4		1280.4.d.n.641.2		2
16.5	even	4		1280.4.d.c.641.2		2
16.11	odd	4		1280.4.d.n.641.1		2
16.13	even	4		1280.4.d.c.641.1		2
20.3	even	4		100.4.c.a.49.2		2
20.7	even	4		100.4.c.a.49.1		2
20.19	odd	2		100.4.a.a.1.1		1
28.3	even	6		980.4.i.n.961.1		2
28.11	odd	6		980.4.i.e.961.1		2
28.19	even	6		980.4.i.n.361.1		2
28.23	odd	6		980.4.i.e.361.1		2
28.27	even	2		980.4.a.c.1.1		1
36.7	odd	6		1620.4.i.d.1081.1		2
36.11	even	6		1620.4.i.j.1081.1		2
36.23	even	6		1620.4.i.j.541.1		2
36.31	odd	6		1620.4.i.d.541.1		2
40.19	odd	2		1600.4.a.bl.1.1		1
40.29	even	2		1600.4.a.p.1.1		1
44.43	even	2		2420.4.a.d.1.1		1
60.23	odd	4		900.4.d.k.649.2		2
60.47	odd	4		900.4.d.k.649.1		2
60.59	even	2		900.4.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.a.a.1.1	✓	1	4.3	odd	2
80.4.a.c.1.1		1	1.1	even	1	trivial
100.4.a.a.1.1		1	20.19	odd	2
100.4.c.a.49.1		2	20.7	even	4
100.4.c.a.49.2		2	20.3	even	4
180.4.a.a.1.1		1	12.11	even	2
320.4.a.d.1.1		1	8.3	odd	2
320.4.a.k.1.1		1	8.5	even	2
400.4.a.o.1.1		1	5.4	even	2
400.4.c.j.49.1		2	5.3	odd	4
400.4.c.j.49.2		2	5.2	odd	4
720.4.a.k.1.1		1	3.2	odd	2
900.4.a.m.1.1		1	60.59	even	2
900.4.d.k.649.1		2	60.47	odd	4
900.4.d.k.649.2		2	60.23	odd	4
980.4.a.c.1.1		1	28.27	even	2
980.4.i.e.361.1		2	28.23	odd	6
980.4.i.e.961.1		2	28.11	odd	6
980.4.i.n.361.1		2	28.19	even	6
980.4.i.n.961.1		2	28.3	even	6
1280.4.d.c.641.1		2	16.13	even	4
1280.4.d.c.641.2		2	16.5	even	4
1280.4.d.n.641.1		2	16.11	odd	4
1280.4.d.n.641.2		2	16.3	odd	4
1600.4.a.p.1.1		1	40.29	even	2
1600.4.a.bl.1.1		1	40.19	odd	2
1620.4.i.d.541.1		2	36.31	odd	6
1620.4.i.d.1081.1		2	36.7	odd	6
1620.4.i.j.541.1		2	36.23	even	6
1620.4.i.j.1081.1		2	36.11	even	6
2420.4.a.d.1.1		1	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	80.4.a.c.1.1

Level	$80$
Weight	$4$
Character	80.1
Self dual	yes
Analytic conductor	$4.720$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$