LMFDB - Embedded newform 85.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [85,4,Mod(1,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("85.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$85 = 5 \cdot 17$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	85.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,3,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.01516235049$
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	$\chi$	$=$	85.1

$q$ -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} +10.0000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +30.0000 q^{6} -22.0000 q^{7} -21.0000 q^{8} +73.0000 q^{9} +15.0000 q^{10} -30.0000 q^{11} +10.0000 q^{12} -46.0000 q^{13} -66.0000 q^{14} +50.0000 q^{15} -71.0000 q^{16} +17.0000 q^{17} +219.000 q^{18} +104.000 q^{19} +5.00000 q^{20} -220.000 q^{21} -90.0000 q^{22} +42.0000 q^{23} -210.000 q^{24} +25.0000 q^{25} -138.000 q^{26} +460.000 q^{27} -22.0000 q^{28} -66.0000 q^{29} +150.000 q^{30} +194.000 q^{31} -45.0000 q^{32} -300.000 q^{33} +51.0000 q^{34} -110.000 q^{35} +73.0000 q^{36} +206.000 q^{37} +312.000 q^{38} -460.000 q^{39} -105.000 q^{40} -126.000 q^{41} -660.000 q^{42} -388.000 q^{43} -30.0000 q^{44} +365.000 q^{45} +126.000 q^{46} -540.000 q^{47} -710.000 q^{48} +141.000 q^{49} +75.0000 q^{50} +170.000 q^{51} -46.0000 q^{52} +78.0000 q^{53} +1380.00 q^{54} -150.000 q^{55} +462.000 q^{56} +1040.00 q^{57} -198.000 q^{58} +432.000 q^{59} +50.0000 q^{60} -610.000 q^{61} +582.000 q^{62} -1606.00 q^{63} +433.000 q^{64} -230.000 q^{65} -900.000 q^{66} +848.000 q^{67} +17.0000 q^{68} +420.000 q^{69} -330.000 q^{70} -174.000 q^{71} -1533.00 q^{72} +362.000 q^{73} +618.000 q^{74} +250.000 q^{75} +104.000 q^{76} +660.000 q^{77} -1380.00 q^{78} +398.000 q^{79} -355.000 q^{80} +2629.00 q^{81} -378.000 q^{82} +828.000 q^{83} -220.000 q^{84} +85.0000 q^{85} -1164.00 q^{86} -660.000 q^{87} +630.000 q^{88} +630.000 q^{89} +1095.00 q^{90} +1012.00 q^{91} +42.0000 q^{92} +1940.00 q^{93} -1620.00 q^{94} +520.000 q^{95} -450.000 q^{96} -1486.00 q^{97} +423.000 q^{98} -2190.00 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

)

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

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	10.0000	1.92450	0.962250	−	0.272166i	$-0.0877398\pi$
$3$	10.0000	1.92450	0.962250	+	0.272166i	$0.0877398\pi$
$4$	1.00000	0.125000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	30.0000	2.04124
$7$	−22.0000	−1.18789	−0.593944	−	0.804506i	$-0.702430\pi$
$7$	−22.0000	−1.18789	−0.593944	+	0.804506i	$0.702430\pi$
$8$	−21.0000	−0.928078
$9$	73.0000	2.70370
$10$	15.0000	0.474342
$11$	−30.0000	−0.822304	−0.411152	−	0.911567i	$-0.634873\pi$
$11$	−30.0000	−0.822304	−0.411152	+	0.911567i	$0.634873\pi$
$12$	10.0000	0.240563
$13$	−46.0000	−0.981393	−0.490696	−	0.871331i	$-0.663258\pi$
$13$	−46.0000	−0.981393	−0.490696	+	0.871331i	$0.663258\pi$
$14$	−66.0000	−1.25995
$15$	50.0000	0.860663
$16$	−71.0000	−1.10938
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	219.000	2.86771
$19$	104.000	1.25575	0.627875	−	0.778314i	$-0.283925\pi$
$19$	104.000	1.25575	0.627875	+	0.778314i	$0.283925\pi$
$20$	5.00000	0.0559017
$21$	−220.000	−2.28609
$22$	−90.0000	−0.872185
$23$	42.0000	0.380765	0.190383	−	0.981710i	$-0.439027\pi$
$23$	42.0000	0.380765	0.190383	+	0.981710i	$0.439027\pi$
$24$	−210.000	−1.78609
$25$	25.0000	0.200000
$26$	−138.000	−1.04092
$27$	460.000	3.27878
$28$	−22.0000	−0.148486
$29$	−66.0000	−0.422617	−0.211308	−	0.977419i	$-0.567772\pi$
$29$	−66.0000	−0.422617	−0.211308	+	0.977419i	$0.567772\pi$
$30$	150.000	0.912871
$31$	194.000	1.12398	0.561991	−	0.827143i	$-0.310036\pi$
$31$	194.000	1.12398	0.561991	+	0.827143i	$0.310036\pi$
$32$	−45.0000	−0.248592
$33$	−300.000	−1.58252
$34$	51.0000	0.257248
$35$	−110.000	−0.531240
$36$	73.0000	0.337963
$37$	206.000	0.915302	0.457651	−	0.889132i	$-0.348691\pi$
$37$	206.000	0.915302	0.457651	+	0.889132i	$0.348691\pi$
$38$	312.000	1.33192
$39$	−460.000	−1.88869
$40$	−105.000	−0.415049
$41$	−126.000	−0.479949	−0.239974	−	0.970779i	$-0.577139\pi$
$41$	−126.000	−0.479949	−0.239974	+	0.970779i	$0.577139\pi$
$42$	−660.000	−2.42477
$43$	−388.000	−1.37603	−0.688017	−	0.725695i	$-0.741518\pi$
$43$	−388.000	−1.37603	−0.688017	+	0.725695i	$0.741518\pi$
$44$	−30.0000	−0.102788
$45$	365.000	1.20913
$46$	126.000	0.403863
$47$	−540.000	−1.67590	−0.837948	−	0.545750i	$-0.816245\pi$
$47$	−540.000	−1.67590	−0.837948	+	0.545750i	$0.816245\pi$
$48$	−710.000	−2.13499
$49$	141.000	0.411079
$50$	75.0000	0.212132
$51$	170.000	0.466760
$52$	−46.0000	−0.122674
$53$	78.0000	0.202153	0.101077	−	0.994879i	$-0.467771\pi$
$53$	78.0000	0.202153	0.101077	+	0.994879i	$0.467771\pi$
$54$	1380.00	3.47767
$55$	−150.000	−0.367745
$56$	462.000	1.10245
$57$	1040.00	2.41669
$58$	−198.000	−0.448253
$59$	432.000	0.953248	0.476624	−	0.879107i	$-0.341860\pi$
$59$	432.000	0.953248	0.476624	+	0.879107i	$0.341860\pi$
$60$	50.0000	0.107583
$61$	−610.000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−610.000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	582.000	1.19216
$63$	−1606.00	−3.21170
$64$	433.000	0.845703
$65$	−230.000	−0.438892
$66$	−900.000	−1.67852
$67$	848.000	1.54626	0.773132	−	0.634245i	$-0.218689\pi$
$67$	848.000	1.54626	0.773132	+	0.634245i	$0.218689\pi$
$68$	17.0000	0.0303170
$69$	420.000	0.732783
$70$	−330.000	−0.563465
$71$	−174.000	−0.290845	−0.145423	−	0.989370i	$-0.546454\pi$
$71$	−174.000	−0.290845	−0.145423	+	0.989370i	$0.546454\pi$
$72$	−1533.00	−2.50925
$73$	362.000	0.580396	0.290198	−	0.956967i	$-0.406279\pi$
$73$	362.000	0.580396	0.290198	+	0.956967i	$0.406279\pi$
$74$	618.000	0.970825
$75$	250.000	0.384900
$76$	104.000	0.156969
$77$	660.000	0.976805
$78$	−1380.00	−2.00326
$79$	398.000	0.566816	0.283408	−	0.958999i	$-0.408535\pi$
$79$	398.000	0.566816	0.283408	+	0.958999i	$0.408535\pi$
$80$	−355.000	−0.496128
$81$	2629.00	3.60631
$82$	−378.000	−0.509062
$83$	828.000	1.09500	0.547499	−	0.836806i	$-0.315580\pi$
$83$	828.000	1.09500	0.547499	+	0.836806i	$0.315580\pi$
$84$	−220.000	−0.285762
$85$	85.0000	0.108465
$86$	−1164.00	−1.45950
$87$	−660.000	−0.813327
$88$	630.000	0.763162
$89$	630.000	0.750336	0.375168	−	0.926957i	$-0.377585\pi$
$89$	630.000	0.750336	0.375168	+	0.926957i	$0.377585\pi$
$90$	1095.00	1.28248
$91$	1012.00	1.16578
$92$	42.0000	0.0475957
$93$	1940.00	2.16310
$94$	−1620.00	−1.77756
$95$	520.000	0.561588
$96$	−450.000	−0.478416
$97$	−1486.00	−1.55547	−0.777734	−	0.628593i	$-0.783631\pi$
$97$	−1486.00	−1.55547	−0.777734	+	0.628593i	$0.783631\pi$
$98$	423.000	0.436015
$99$	−2190.00	−2.22327
$100$	25.0000	0.0250000
$101$	−1230.00	−1.21178	−0.605889	−	0.795549i	$-0.707182\pi$
$101$	−1230.00	−1.21178	−0.605889	+	0.795549i	$0.707182\pi$
$102$	510.000	0.495074
$103$	−1276.00	−1.22066	−0.610330	−	0.792147i	$-0.708963\pi$
$103$	−1276.00	−1.22066	−0.610330	+	0.792147i	$0.708963\pi$
$104$	966.000	0.910809
$105$	−1100.00	−1.02237
$106$	234.000	0.214416
$107$	−414.000	−0.374046	−0.187023	−	0.982356i	$-0.559884\pi$
$107$	−414.000	−0.374046	−0.187023	+	0.982356i	$0.559884\pi$
$108$	460.000	0.409847
$109$	614.000	0.539546	0.269773	−	0.962924i	$-0.413051\pi$
$109$	614.000	0.539546	0.269773	+	0.962924i	$0.413051\pi$
$110$	−450.000	−0.390053
$111$	2060.00	1.76150
$112$	1562.00	1.31781
$113$	258.000	0.214784	0.107392	−	0.994217i	$-0.465750\pi$
$113$	258.000	0.214784	0.107392	+	0.994217i	$0.465750\pi$
$114$	3120.00	2.56329
$115$	210.000	0.170283
$116$	−66.0000	−0.0528271
$117$	−3358.00	−2.65339
$118$	1296.00	1.01107
$119$	−374.000	−0.288105
$120$	−1050.00	−0.798762
$121$	−431.000	−0.323817
$122$	−1830.00	−1.35804
$123$	−1260.00	−0.923662
$124$	194.000	0.140498
$125$	125.000	0.0894427
$126$	−4818.00	−3.40652
$127$	296.000	0.206817	0.103408	−	0.994639i	$-0.467025\pi$
$127$	296.000	0.206817	0.103408	+	0.994639i	$0.467025\pi$
$128$	1659.00	1.14560
$129$	−3880.00	−2.64818
$130$	−690.000	−0.465515
$131$	246.000	0.164070	0.0820348	−	0.996629i	$-0.473858\pi$
$131$	246.000	0.164070	0.0820348	+	0.996629i	$0.473858\pi$
$132$	−300.000	−0.197816
$133$	−2288.00	−1.49169
$134$	2544.00	1.64006
$135$	2300.00	1.46631
$136$	−357.000	−0.225092
$137$	1998.00	1.24599	0.622995	−	0.782226i	$-0.285916\pi$
$137$	1998.00	1.24599	0.622995	+	0.782226i	$0.285916\pi$
$138$	1260.00	0.777234
$139$	−562.000	−0.342937	−0.171468	−	0.985190i	$-0.554851\pi$
$139$	−562.000	−0.342937	−0.171468	+	0.985190i	$0.554851\pi$
$140$	−110.000	−0.0664050
$141$	−5400.00	−3.22526
$142$	−522.000	−0.308488
$143$	1380.00	0.807003
$144$	−5183.00	−2.99942
$145$	−330.000	−0.189000
$146$	1086.00	0.615603
$147$	1410.00	0.791121
$148$	206.000	0.114413
$149$	2670.00	1.46802	0.734010	−	0.679139i	$-0.237647\pi$
$149$	2670.00	1.46802	0.734010	+	0.679139i	$0.237647\pi$
$150$	750.000	0.408248
$151$	1244.00	0.670432	0.335216	−	0.942141i	$-0.391191\pi$
$151$	1244.00	0.670432	0.335216	+	0.942141i	$0.391191\pi$
$152$	−2184.00	−1.16543
$153$	1241.00	0.655744
$154$	1980.00	1.03606
$155$	970.000	0.502660
$156$	−460.000	−0.236086
$157$	−2794.00	−1.42029	−0.710145	−	0.704056i	$-0.751371\pi$
$157$	−2794.00	−1.42029	−0.710145	+	0.704056i	$0.751371\pi$
$158$	1194.00	0.601200
$159$	780.000	0.389044
$160$	−225.000	−0.111174
$161$	−924.000	−0.452307
$162$	7887.00	3.82507
$163$	−3418.00	−1.64244	−0.821222	−	0.570609i	$-0.806707\pi$
$163$	−3418.00	−1.64244	−0.821222	+	0.570609i	$0.806707\pi$
$164$	−126.000	−0.0599936
$165$	−1500.00	−0.707726
$166$	2484.00	1.16142
$167$	102.000	0.0472635	0.0236317	−	0.999721i	$-0.492477\pi$
$167$	102.000	0.0472635	0.0236317	+	0.999721i	$0.492477\pi$
$168$	4620.00	2.12167
$169$	−81.0000	−0.0368685
$170$	255.000	0.115045
$171$	7592.00	3.39517
$172$	−388.000	−0.172004
$173$	1806.00	0.793686	0.396843	−	0.917887i	$-0.370106\pi$
$173$	1806.00	0.793686	0.396843	+	0.917887i	$0.370106\pi$
$174$	−1980.00	−0.862663
$175$	−550.000	−0.237578
$176$	2130.00	0.912243
$177$	4320.00	1.83453
$178$	1890.00	0.795851
$179$	528.000	0.220472	0.110236	−	0.993905i	$-0.464839\pi$
$179$	528.000	0.220472	0.110236	+	0.993905i	$0.464839\pi$
$180$	365.000	0.151142
$181$	−4042.00	−1.65989	−0.829943	−	0.557848i	$-0.811627\pi$
$181$	−4042.00	−1.65989	−0.829943	+	0.557848i	$0.811627\pi$
$182$	3036.00	1.23650
$183$	−6100.00	−2.46407
$184$	−882.000	−0.353380
$185$	1030.00	0.409336
$186$	5820.00	2.29432
$187$	−510.000	−0.199438
$188$	−540.000	−0.209487
$189$	−10120.0	−3.89482
$190$	1560.00	0.595654
$191$	3144.00	1.19106	0.595528	−	0.803334i	$-0.296943\pi$
$191$	3144.00	1.19106	0.595528	+	0.803334i	$0.296943\pi$
$192$	4330.00	1.62756
$193$	−2014.00	−0.751145	−0.375572	−	0.926793i	$-0.622554\pi$
$193$	−2014.00	−0.751145	−0.375572	+	0.926793i	$0.622554\pi$
$194$	−4458.00	−1.64982
$195$	−2300.00	−0.844648
$196$	141.000	0.0513848
$197$	−3666.00	−1.32585	−0.662923	−	0.748688i	$-0.730684\pi$
$197$	−3666.00	−1.32585	−0.662923	+	0.748688i	$0.730684\pi$
$198$	−6570.00	−2.35813
$199$	2090.00	0.744503	0.372252	−	0.928132i	$-0.378586\pi$
$199$	2090.00	0.744503	0.372252	+	0.928132i	$0.378586\pi$
$200$	−525.000	−0.185616
$201$	8480.00	2.97579
$202$	−3690.00	−1.28528
$203$	1452.00	0.502022
$204$	170.000	0.0583450
$205$	−630.000	−0.214640
$206$	−3828.00	−1.29471
$207$	3066.00	1.02948
$208$	3266.00	1.08873
$209$	−3120.00	−1.03261
$210$	−3300.00	−1.08439
$211$	−2542.00	−0.829377	−0.414688	−	0.909963i	$-0.636109\pi$
$211$	−2542.00	−0.829377	−0.414688	+	0.909963i	$0.636109\pi$
$212$	78.0000	0.0252692
$213$	−1740.00	−0.559732
$214$	−1242.00	−0.396735
$215$	−1940.00	−0.615381
$216$	−9660.00	−3.04296
$217$	−4268.00	−1.33516
$218$	1842.00	0.572275
$219$	3620.00	1.11697
$220$	−150.000	−0.0459682
$221$	−782.000	−0.238023
$222$	6180.00	1.86835
$223$	992.000	0.297889	0.148944	−	0.988846i	$-0.452412\pi$
$223$	992.000	0.297889	0.148944	+	0.988846i	$0.452412\pi$
$224$	990.000	0.295300
$225$	1825.00	0.540741
$226$	774.000	0.227813
$227$	−1098.00	−0.321043	−0.160522	−	0.987032i	$-0.551318\pi$
$227$	−1098.00	−0.321043	−0.160522	+	0.987032i	$0.551318\pi$
$228$	1040.00	0.302086
$229$	2954.00	0.852427	0.426214	−	0.904623i	$-0.359847\pi$
$229$	2954.00	0.852427	0.426214	+	0.904623i	$0.359847\pi$
$230$	630.000	0.180613
$231$	6600.00	1.87986
$232$	1386.00	0.392221
$233$	−678.000	−0.190632	−0.0953160	−	0.995447i	$-0.530386\pi$
$233$	−678.000	−0.190632	−0.0953160	+	0.995447i	$0.530386\pi$
$234$	−10074.0	−2.81435
$235$	−2700.00	−0.749483
$236$	432.000	0.119156
$237$	3980.00	1.09084
$238$	−1122.00	−0.305582
$239$	−2664.00	−0.721003	−0.360502	−	0.932759i	$-0.617395\pi$
$239$	−2664.00	−0.721003	−0.360502	+	0.932759i	$0.617395\pi$
$240$	−3550.00	−0.954798
$241$	3458.00	0.924271	0.462136	−	0.886809i	$-0.347083\pi$
$241$	3458.00	0.924271	0.462136	+	0.886809i	$0.347083\pi$
$242$	−1293.00	−0.343459
$243$	13870.0	3.66157
$244$	−610.000	−0.160046
$245$	705.000	0.183840
$246$	−3780.00	−0.979691
$247$	−4784.00	−1.23238
$248$	−4074.00	−1.04314
$249$	8280.00	2.10732
$250$	375.000	0.0948683
$251$	1092.00	0.274607	0.137304	−	0.990529i	$-0.456156\pi$
$251$	1092.00	0.274607	0.137304	+	0.990529i	$0.456156\pi$
$252$	−1606.00	−0.401462
$253$	−1260.00	−0.313105
$254$	888.000	0.219363
$255$	850.000	0.208741
$256$	1513.00	0.369385
$257$	−3714.00	−0.901451	−0.450726	−	0.892663i	$-0.648835\pi$
$257$	−3714.00	−0.901451	−0.450726	+	0.892663i	$0.648835\pi$
$258$	−11640.0	−2.80882
$259$	−4532.00	−1.08728
$260$	−230.000	−0.0548615
$261$	−4818.00	−1.14263
$262$	738.000	0.174022
$263$	3672.00	0.860932	0.430466	−	0.902607i	$-0.358349\pi$
$263$	3672.00	0.860932	0.430466	+	0.902607i	$0.358349\pi$
$264$	6300.00	1.46871
$265$	390.000	0.0904057
$266$	−6864.00	−1.58218
$267$	6300.00	1.44402
$268$	848.000	0.193283
$269$	−642.000	−0.145515	−0.0727573	−	0.997350i	$-0.523180\pi$
$269$	−642.000	−0.145515	−0.0727573	+	0.997350i	$0.523180\pi$
$270$	6900.00	1.55526
$271$	1424.00	0.319195	0.159597	−	0.987182i	$-0.448980\pi$
$271$	1424.00	0.319195	0.159597	+	0.987182i	$0.448980\pi$
$272$	−1207.00	−0.269063
$273$	10120.0	2.24355
$274$	5994.00	1.32157
$275$	−750.000	−0.164461
$276$	420.000	0.0915979
$277$	−4762.00	−1.03293	−0.516464	−	0.856309i	$-0.672752\pi$
$277$	−4762.00	−1.03293	−0.516464	+	0.856309i	$0.672752\pi$
$278$	−1686.00	−0.363739
$279$	14162.0	3.03891
$280$	2310.00	0.493032
$281$	−4806.00	−1.02029	−0.510146	−	0.860088i	$-0.670409\pi$
$281$	−4806.00	−1.02029	−0.510146	+	0.860088i	$0.670409\pi$
$282$	−16200.0	−3.42091
$283$	−298.000	−0.0625946	−0.0312973	−	0.999510i	$-0.509964\pi$
$283$	−298.000	−0.0625946	−0.0312973	+	0.999510i	$0.509964\pi$
$284$	−174.000	−0.0363556
$285$	5200.00	1.08078
$286$	4140.00	0.855956
$287$	2772.00	0.570125
$288$	−3285.00	−0.672120
$289$	289.000	0.0588235
$290$	−990.000	−0.200465
$291$	−14860.0	−2.99350
$292$	362.000	0.0725495
$293$	318.000	0.0634053	0.0317027	−	0.999497i	$-0.489907\pi$
$293$	318.000	0.0634053	0.0317027	+	0.999497i	$0.489907\pi$
$294$	4230.00	0.839111
$295$	2160.00	0.426305
$296$	−4326.00	−0.849472
$297$	−13800.0	−2.69615
$298$	8010.00	1.55707
$299$	−1932.00	−0.373680
$300$	250.000	0.0481125
$301$	8536.00	1.63457
$302$	3732.00	0.711101
$303$	−12300.0	−2.33207
$304$	−7384.00	−1.39310
$305$	−3050.00	−0.572598
$306$	3723.00	0.695522
$307$	1496.00	0.278115	0.139057	−	0.990284i	$-0.455593\pi$
$307$	1496.00	0.278115	0.139057	+	0.990284i	$0.455593\pi$
$308$	660.000	0.122101
$309$	−12760.0	−2.34916
$310$	2910.00	0.533151
$311$	6258.00	1.14102	0.570512	−	0.821289i	$-0.306745\pi$
$311$	6258.00	1.14102	0.570512	+	0.821289i	$0.306745\pi$
$312$	9660.00	1.75285
$313$	−2926.00	−0.528394	−0.264197	−	0.964469i	$-0.585107\pi$
$313$	−2926.00	−0.528394	−0.264197	+	0.964469i	$0.585107\pi$
$314$	−8382.00	−1.50644
$315$	−8030.00	−1.43632
$316$	398.000	0.0708521
$317$	−7290.00	−1.29163	−0.645816	−	0.763493i	$-0.723483\pi$
$317$	−7290.00	−1.29163	−0.645816	+	0.763493i	$0.723483\pi$
$318$	2340.00	0.412644
$319$	1980.00	0.347519
$320$	2165.00	0.378210
$321$	−4140.00	−0.719851
$322$	−2772.00	−0.479744
$323$	1768.00	0.304564
$324$	2629.00	0.450789
$325$	−1150.00	−0.196279
$326$	−10254.0	−1.74208
$327$	6140.00	1.03836
$328$	2646.00	0.445430
$329$	11880.0	1.99078
$330$	−4500.00	−0.750657
$331$	10184.0	1.69113	0.845564	−	0.533874i	$-0.179264\pi$
$331$	10184.0	1.69113	0.845564	+	0.533874i	$0.179264\pi$
$332$	828.000	0.136875
$333$	15038.0	2.47471
$334$	306.000	0.0501305
$335$	4240.00	0.691510
$336$	15620.0	2.53613
$337$	−7198.00	−1.16350	−0.581751	−	0.813367i	$-0.697632\pi$
$337$	−7198.00	−1.16350	−0.581751	+	0.813367i	$0.697632\pi$
$338$	−243.000	−0.0391049
$339$	2580.00	0.413352
$340$	85.0000	0.0135582
$341$	−5820.00	−0.924254
$342$	22776.0	3.60113
$343$	4444.00	0.699573
$344$	8148.00	1.27707
$345$	2100.00	0.327711
$346$	5418.00	0.841831
$347$	6174.00	0.955152	0.477576	−	0.878590i	$-0.341516\pi$
$347$	6174.00	0.955152	0.477576	+	0.878590i	$0.341516\pi$
$348$	−660.000	−0.101666
$349$	5798.00	0.889283	0.444642	−	0.895709i	$-0.353331\pi$
$349$	5798.00	0.889283	0.444642	+	0.895709i	$0.353331\pi$
$350$	−1650.00	−0.251989
$351$	−21160.0	−3.21777
$352$	1350.00	0.204418
$353$	10050.0	1.51532	0.757659	−	0.652650i	$-0.226343\pi$
$353$	10050.0	1.51532	0.757659	+	0.652650i	$0.226343\pi$
$354$	12960.0	1.94581
$355$	−870.000	−0.130070
$356$	630.000	0.0937919
$357$	−3740.00	−0.554459
$358$	1584.00	0.233846
$359$	6300.00	0.926187	0.463094	−	0.886309i	$-0.346739\pi$
$359$	6300.00	0.926187	0.463094	+	0.886309i	$0.346739\pi$
$360$	−7665.00	−1.12217
$361$	3957.00	0.576906
$362$	−12126.0	−1.76058
$363$	−4310.00	−0.623185
$364$	1012.00	0.145723
$365$	1810.00	0.259561
$366$	−18300.0	−2.61354
$367$	−8494.00	−1.20813	−0.604064	−	0.796936i	$-0.706453\pi$
$367$	−8494.00	−1.20813	−0.604064	+	0.796936i	$0.706453\pi$
$368$	−2982.00	−0.422412
$369$	−9198.00	−1.29764
$370$	3090.00	0.434166
$371$	−1716.00	−0.240136
$372$	1940.00	0.270388
$373$	−1942.00	−0.269579	−0.134789	−	0.990874i	$-0.543036\pi$
$373$	−1942.00	−0.269579	−0.134789	+	0.990874i	$0.543036\pi$
$374$	−1530.00	−0.211536
$375$	1250.00	0.172133
$376$	11340.0	1.55536
$377$	3036.00	0.414753
$378$	−30360.0	−4.13108
$379$	5318.00	0.720758	0.360379	−	0.932806i	$-0.382647\pi$
$379$	5318.00	0.720758	0.360379	+	0.932806i	$0.382647\pi$
$380$	520.000	0.0701985
$381$	2960.00	0.398019
$382$	9432.00	1.26331
$383$	−5472.00	−0.730042	−0.365021	−	0.930999i	$-0.618938\pi$
$383$	−5472.00	−0.730042	−0.365021	+	0.930999i	$0.618938\pi$
$384$	16590.0	2.20470
$385$	3300.00	0.436840
$386$	−6042.00	−0.796709
$387$	−28324.0	−3.72039
$388$	−1486.00	−0.194434
$389$	−9366.00	−1.22076	−0.610379	−	0.792109i	$-0.708983\pi$
$389$	−9366.00	−1.22076	−0.610379	+	0.792109i	$0.708983\pi$
$390$	−6900.00	−0.895885
$391$	714.000	0.0923492
$392$	−2961.00	−0.381513
$393$	2460.00	0.315752
$394$	−10998.0	−1.40627
$395$	1990.00	0.253488
$396$	−2190.00	−0.277908
$397$	13502.0	1.70692	0.853458	−	0.521161i	$-0.174501\pi$
$397$	13502.0	1.70692	0.853458	+	0.521161i	$0.174501\pi$
$398$	6270.00	0.789665
$399$	−22880.0	−2.87076
$400$	−1775.00	−0.221875
$401$	5466.00	0.680696	0.340348	−	0.940300i	$-0.389455\pi$
$401$	5466.00	0.680696	0.340348	+	0.940300i	$0.389455\pi$
$402$	25440.0	3.15630
$403$	−8924.00	−1.10307
$404$	−1230.00	−0.151472
$405$	13145.0	1.61279
$406$	4356.00	0.532475
$407$	−6180.00	−0.752657
$408$	−3570.00	−0.433190
$409$	6074.00	0.734328	0.367164	−	0.930156i	$-0.380329\pi$
$409$	6074.00	0.734328	0.367164	+	0.930156i	$0.380329\pi$
$410$	−1890.00	−0.227660
$411$	19980.0	2.39791
$412$	−1276.00	−0.152583
$413$	−9504.00	−1.13235
$414$	9198.00	1.09193
$415$	4140.00	0.489698
$416$	2070.00	0.243967
$417$	−5620.00	−0.659982
$418$	−9360.00	−1.09525
$419$	8922.00	1.04026	0.520129	−	0.854088i	$-0.325884\pi$
$419$	8922.00	1.04026	0.520129	+	0.854088i	$0.325884\pi$
$420$	−1100.00	−0.127796
$421$	8882.00	1.02822	0.514112	−	0.857723i	$-0.328122\pi$
$421$	8882.00	1.02822	0.514112	+	0.857723i	$0.328122\pi$
$422$	−7626.00	−0.879687
$423$	−39420.0	−4.53113
$424$	−1638.00	−0.187614
$425$	425.000	0.0485071
$426$	−5220.00	−0.593685
$427$	13420.0	1.52094
$428$	−414.000	−0.0467557
$429$	13800.0	1.55308
$430$	−5820.00	−0.652710
$431$	−3738.00	−0.417757	−0.208878	−	0.977942i	$-0.566981\pi$
$431$	−3738.00	−0.417757	−0.208878	+	0.977942i	$0.566981\pi$
$432$	−32660.0	−3.63740
$433$	−6754.00	−0.749599	−0.374800	−	0.927106i	$-0.622289\pi$
$433$	−6754.00	−0.749599	−0.374800	+	0.927106i	$0.622289\pi$
$434$	−12804.0	−1.41616
$435$	−3300.00	−0.363731
$436$	614.000	0.0674433
$437$	4368.00	0.478146
$438$	10860.0	1.18473
$439$	−7270.00	−0.790383	−0.395192	−	0.918599i	$-0.629322\pi$
$439$	−7270.00	−0.790383	−0.395192	+	0.918599i	$0.629322\pi$
$440$	3150.00	0.341296
$441$	10293.0	1.11144
$442$	−2346.00	−0.252461
$443$	−11136.0	−1.19433	−0.597164	−	0.802119i	$-0.703706\pi$
$443$	−11136.0	−1.19433	−0.597164	+	0.802119i	$0.703706\pi$
$444$	2060.00	0.220188
$445$	3150.00	0.335560
$446$	2976.00	0.315959
$447$	26700.0	2.82521
$448$	−9526.00	−1.00460
$449$	−1974.00	−0.207481	−0.103740	−	0.994604i	$-0.533081\pi$
$449$	−1974.00	−0.207481	−0.103740	+	0.994604i	$0.533081\pi$
$450$	5475.00	0.573542
$451$	3780.00	0.394664
$452$	258.000	0.0268480
$453$	12440.0	1.29025
$454$	−3294.00	−0.340518
$455$	5060.00	0.521355
$456$	−21840.0	−2.24288
$457$	10838.0	1.10937	0.554683	−	0.832062i	$-0.312840\pi$
$457$	10838.0	1.10937	0.554683	+	0.832062i	$0.312840\pi$
$458$	8862.00	0.904136
$459$	7820.00	0.795221
$460$	210.000	0.0212854
$461$	−15642.0	−1.58030	−0.790152	−	0.612910i	$-0.789999\pi$
$461$	−15642.0	−1.58030	−0.790152	+	0.612910i	$0.789999\pi$
$462$	19800.0	1.99389
$463$	2276.00	0.228455	0.114228	−	0.993455i	$-0.463561\pi$
$463$	2276.00	0.228455	0.114228	+	0.993455i	$0.463561\pi$
$464$	4686.00	0.468841
$465$	9700.00	0.967369
$466$	−2034.00	−0.202196
$467$	10164.0	1.00714	0.503569	−	0.863955i	$-0.332020\pi$
$467$	10164.0	1.00714	0.503569	+	0.863955i	$0.332020\pi$
$468$	−3358.00	−0.331674
$469$	−18656.0	−1.83679
$470$	−8100.00	−0.794947
$471$	−27940.0	−2.73335
$472$	−9072.00	−0.884688
$473$	11640.0	1.13152
$474$	11940.0	1.15701
$475$	2600.00	0.251150
$476$	−374.000	−0.0360132
$477$	5694.00	0.546563
$478$	−7992.00	−0.764740
$479$	14034.0	1.33868	0.669342	−	0.742954i	$-0.266576\pi$
$479$	14034.0	1.33868	0.669342	+	0.742954i	$0.266576\pi$
$480$	−2250.00	−0.213954
$481$	−9476.00	−0.898271
$482$	10374.0	0.980338
$483$	−9240.00	−0.870465
$484$	−431.000	−0.0404771
$485$	−7430.00	−0.695627
$486$	41610.0	3.88368
$487$	−11350.0	−1.05609	−0.528047	−	0.849215i	$-0.677075\pi$
$487$	−11350.0	−1.05609	−0.528047	+	0.849215i	$0.677075\pi$
$488$	12810.0	1.18828
$489$	−34180.0	−3.16089
$490$	2115.00	0.194992
$491$	7320.00	0.672804	0.336402	−	0.941718i	$-0.390790\pi$
$491$	7320.00	0.672804	0.336402	+	0.941718i	$0.390790\pi$
$492$	−1260.00	−0.115458
$493$	−1122.00	−0.102500
$494$	−14352.0	−1.30714
$495$	−10950.0	−0.994275
$496$	−13774.0	−1.24692
$497$	3828.00	0.345491
$498$	24840.0	2.23515
$499$	−7630.00	−0.684500	−0.342250	−	0.939609i	$-0.611189\pi$
$499$	−7630.00	−0.684500	−0.342250	+	0.939609i	$0.611189\pi$
$500$	125.000	0.0111803
$501$	1020.00	0.0909586
$502$	3276.00	0.291265
$503$	4314.00	0.382409	0.191205	−	0.981550i	$-0.438761\pi$
$503$	4314.00	0.382409	0.191205	+	0.981550i	$0.438761\pi$
$504$	33726.0	2.98071
$505$	−6150.00	−0.541924
$506$	−3780.00	−0.332098
$507$	−810.000	−0.0709534
$508$	296.000	0.0258521
$509$	−8058.00	−0.701699	−0.350849	−	0.936432i	$-0.614107\pi$
$509$	−8058.00	−0.701699	−0.350849	+	0.936432i	$0.614107\pi$
$510$	2550.00	0.221404
$511$	−7964.00	−0.689445
$512$	−8733.00	−0.753804
$513$	47840.0	4.11732
$514$	−11142.0	−0.956133
$515$	−6380.00	−0.545896
$516$	−3880.00	−0.331022
$517$	16200.0	1.37810
$518$	−13596.0	−1.15323
$519$	18060.0	1.52745
$520$	4830.00	0.407326
$521$	−3414.00	−0.287083	−0.143541	−	0.989644i	$-0.545849\pi$
$521$	−3414.00	−0.287083	−0.143541	+	0.989644i	$0.545849\pi$
$522$	−14454.0	−1.21194
$523$	15752.0	1.31699	0.658496	−	0.752584i	$-0.271193\pi$
$523$	15752.0	1.31699	0.658496	+	0.752584i	$0.271193\pi$
$524$	246.000	0.0205087
$525$	−5500.00	−0.457218
$526$	11016.0	0.913157
$527$	3298.00	0.272606
$528$	21300.0	1.75561
$529$	−10403.0	−0.855018
$530$	1170.00	0.0958897
$531$	31536.0	2.57730
$532$	−2288.00	−0.186461
$533$	5796.00	0.471018
$534$	18900.0	1.53162
$535$	−2070.00	−0.167278
$536$	−17808.0	−1.43505
$537$	5280.00	0.424299
$538$	−1926.00	−0.154342
$539$	−4230.00	−0.338032
$540$	2300.00	0.183289
$541$	−11554.0	−0.918198	−0.459099	−	0.888385i	$-0.651828\pi$
$541$	−11554.0	−0.918198	−0.459099	+	0.888385i	$0.651828\pi$
$542$	4272.00	0.338557
$543$	−40420.0	−3.19445
$544$	−765.000	−0.0602925
$545$	3070.00	0.241292
$546$	30360.0	2.37965
$547$	−8134.00	−0.635804	−0.317902	−	0.948124i	$-0.602978\pi$
$547$	−8134.00	−0.635804	−0.317902	+	0.948124i	$0.602978\pi$
$548$	1998.00	0.155749
$549$	−44530.0	−3.46174
$550$	−2250.00	−0.174437
$551$	−6864.00	−0.530701
$552$	−8820.00	−0.680080
$553$	−8756.00	−0.673315
$554$	−14286.0	−1.09558
$555$	10300.0	0.787767
$556$	−562.000	−0.0428671
$557$	10290.0	0.782767	0.391384	−	0.920228i	$-0.371997\pi$
$557$	10290.0	0.782767	0.391384	+	0.920228i	$0.371997\pi$
$558$	42486.0	3.22325
$559$	17848.0	1.35043
$560$	7810.00	0.589344
$561$	−5100.00	−0.383818
$562$	−14418.0	−1.08218
$563$	22428.0	1.67891	0.839456	−	0.543428i	$-0.182874\pi$
$563$	22428.0	1.67891	0.839456	+	0.543428i	$0.182874\pi$
$564$	−5400.00	−0.403158
$565$	1290.00	0.0960544
$566$	−894.000	−0.0663916
$567$	−57838.0	−4.28389
$568$	3654.00	0.269927
$569$	21978.0	1.61927	0.809636	−	0.586932i	$-0.199665\pi$
$569$	21978.0	1.61927	0.809636	+	0.586932i	$0.199665\pi$
$570$	15600.0	1.14634
$571$	8378.00	0.614025	0.307013	−	0.951705i	$-0.400671\pi$
$571$	8378.00	0.614025	0.307013	+	0.951705i	$0.400671\pi$
$572$	1380.00	0.100875
$573$	31440.0	2.29219
$574$	8316.00	0.604709
$575$	1050.00	0.0761531
$576$	31609.0	2.28653
$577$	−15658.0	−1.12972	−0.564862	−	0.825185i	$-0.691071\pi$
$577$	−15658.0	−1.12972	−0.564862	+	0.825185i	$0.691071\pi$
$578$	867.000	0.0623918
$579$	−20140.0	−1.44558
$580$	−330.000	−0.0236250
$581$	−18216.0	−1.30073
$582$	−44580.0	−3.17509
$583$	−2340.00	−0.166231
$584$	−7602.00	−0.538652
$585$	−16790.0	−1.18663
$586$	954.000	0.0672515
$587$	20028.0	1.40825	0.704126	−	0.710075i	$-0.251339\pi$
$587$	20028.0	1.40825	0.704126	+	0.710075i	$0.251339\pi$
$588$	1410.00	0.0988902
$589$	20176.0	1.41144
$590$	6480.00	0.452165
$591$	−36660.0	−2.55159
$592$	−14626.0	−1.01541
$593$	−21006.0	−1.45466	−0.727330	−	0.686288i	$-0.759239\pi$
$593$	−21006.0	−1.45466	−0.727330	+	0.686288i	$0.759239\pi$
$594$	−41400.0	−2.85970
$595$	−1870.00	−0.128845
$596$	2670.00	0.183502
$597$	20900.0	1.43280
$598$	−5796.00	−0.396348
$599$	9984.00	0.681027	0.340514	−	0.940240i	$-0.389399\pi$
$599$	9984.00	0.681027	0.340514	+	0.940240i	$0.389399\pi$
$600$	−5250.00	−0.357217
$601$	−12790.0	−0.868078	−0.434039	−	0.900894i	$-0.642912\pi$
$601$	−12790.0	−0.868078	−0.434039	+	0.900894i	$0.642912\pi$
$602$	25608.0	1.73373
$603$	61904.0	4.18064
$604$	1244.00	0.0838040
$605$	−2155.00	−0.144815
$606$	−36900.0	−2.47353
$607$	10310.0	0.689407	0.344703	−	0.938712i	$-0.387979\pi$
$607$	10310.0	0.689407	0.344703	+	0.938712i	$0.387979\pi$
$608$	−4680.00	−0.312170
$609$	14520.0	0.966141
$610$	−9150.00	−0.607332
$611$	24840.0	1.64471
$612$	1241.00	0.0819681
$613$	−18178.0	−1.19772	−0.598860	−	0.800854i	$-0.704379\pi$
$613$	−18178.0	−1.19772	−0.598860	+	0.800854i	$0.704379\pi$
$614$	4488.00	0.294985
$615$	−6300.00	−0.413074
$616$	−13860.0	−0.906551
$617$	22818.0	1.48885	0.744423	−	0.667708i	$-0.232725\pi$
$617$	22818.0	1.48885	0.744423	+	0.667708i	$0.232725\pi$
$618$	−38280.0	−2.49166
$619$	−25702.0	−1.66890	−0.834451	−	0.551082i	$-0.814215\pi$
$619$	−25702.0	−1.66890	−0.834451	+	0.551082i	$0.814215\pi$
$620$	970.000	0.0628325
$621$	19320.0	1.24845
$622$	18774.0	1.21024
$623$	−13860.0	−0.891315
$624$	32660.0	2.09527
$625$	625.000	0.0400000
$626$	−8778.00	−0.560446
$627$	−31200.0	−1.98725
$628$	−2794.00	−0.177536
$629$	3502.00	0.221993
$630$	−24090.0	−1.52344
$631$	23396.0	1.47604	0.738019	−	0.674780i	$-0.235761\pi$
$631$	23396.0	1.47604	0.738019	+	0.674780i	$0.235761\pi$
$632$	−8358.00	−0.526050
$633$	−25420.0	−1.59614
$634$	−21870.0	−1.36998
$635$	1480.00	0.0924914
$636$	780.000	0.0486305
$637$	−6486.00	−0.403430
$638$	5940.00	0.368600
$639$	−12702.0	−0.786359
$640$	8295.00	0.512326
$641$	90.0000	0.00554569	0.00277284	−	0.999996i	$-0.499117\pi$
$641$	90.0000	0.00554569	0.00277284	+	0.999996i	$0.499117\pi$
$642$	−12420.0	−0.763518
$643$	−18934.0	−1.16125	−0.580625	−	0.814171i	$-0.697192\pi$
$643$	−18934.0	−1.16125	−0.580625	+	0.814171i	$0.697192\pi$
$644$	−924.000	−0.0565384
$645$	−19400.0	−1.18430
$646$	5304.00	0.323039
$647$	−2124.00	−0.129062	−0.0645310	−	0.997916i	$-0.520555\pi$
$647$	−2124.00	−0.129062	−0.0645310	+	0.997916i	$0.520555\pi$
$648$	−55209.0	−3.34694
$649$	−12960.0	−0.783859
$650$	−3450.00	−0.208185
$651$	−42680.0	−2.56953
$652$	−3418.00	−0.205306
$653$	−7482.00	−0.448382	−0.224191	−	0.974545i	$-0.571974\pi$
$653$	−7482.00	−0.448382	−0.224191	+	0.974545i	$0.571974\pi$
$654$	18420.0	1.10134
$655$	1230.00	0.0733742
$656$	8946.00	0.532443
$657$	26426.0	1.56922
$658$	35640.0	2.11154
$659$	27996.0	1.65489	0.827443	−	0.561550i	$-0.189795\pi$
$659$	27996.0	1.65489	0.827443	+	0.561550i	$0.189795\pi$
$660$	−1500.00	−0.0884658
$661$	11198.0	0.658928	0.329464	−	0.944168i	$-0.393132\pi$
$661$	11198.0	0.658928	0.329464	+	0.944168i	$0.393132\pi$
$662$	30552.0	1.79371
$663$	−7820.00	−0.458075
$664$	−17388.0	−1.01624
$665$	−11440.0	−0.667104
$666$	45114.0	2.62482
$667$	−2772.00	−0.160918
$668$	102.000	0.00590793
$669$	9920.00	0.573288
$670$	12720.0	0.733457
$671$	18300.0	1.05285
$672$	9900.00	0.568305
$673$	−21334.0	−1.22194	−0.610970	−	0.791654i	$-0.709220\pi$
$673$	−21334.0	−1.22194	−0.610970	+	0.791654i	$0.709220\pi$
$674$	−21594.0	−1.23408
$675$	11500.0	0.655756
$676$	−81.0000	−0.00460856
$677$	11550.0	0.655691	0.327845	−	0.944731i	$-0.393678\pi$
$677$	11550.0	0.655691	0.327845	+	0.944731i	$0.393678\pi$
$678$	7740.00	0.438426
$679$	32692.0	1.84772
$680$	−1785.00	−0.100664
$681$	−10980.0	−0.617848
$682$	−17460.0	−0.980320
$683$	−25590.0	−1.43364	−0.716819	−	0.697260i	$-0.754402\pi$
$683$	−25590.0	−1.43364	−0.716819	+	0.697260i	$0.754402\pi$
$684$	7592.00	0.424397
$685$	9990.00	0.557224
$686$	13332.0	0.742009
$687$	29540.0	1.64050
$688$	27548.0	1.52654
$689$	−3588.00	−0.198392
$690$	6300.00	0.347590
$691$	−3358.00	−0.184869	−0.0924344	−	0.995719i	$-0.529465\pi$
$691$	−3358.00	−0.184869	−0.0924344	+	0.995719i	$0.529465\pi$
$692$	1806.00	0.0992107
$693$	48180.0	2.64099
$694$	18522.0	1.01309
$695$	−2810.00	−0.153366
$696$	13860.0	0.754830
$697$	−2142.00	−0.116405
$698$	17394.0	0.943227
$699$	−6780.00	−0.366871
$700$	−550.000	−0.0296972
$701$	−4758.00	−0.256358	−0.128179	−	0.991751i	$-0.540913\pi$
$701$	−4758.00	−0.256358	−0.128179	+	0.991751i	$0.540913\pi$
$702$	−63480.0	−3.41296
$703$	21424.0	1.14939
$704$	−12990.0	−0.695425
$705$	−27000.0	−1.44238
$706$	30150.0	1.60724
$707$	27060.0	1.43946
$708$	4320.00	0.229316
$709$	−12226.0	−0.647612	−0.323806	−	0.946123i	$-0.604963\pi$
$709$	−12226.0	−0.647612	−0.323806	+	0.946123i	$0.604963\pi$
$710$	−2610.00	−0.137960
$711$	29054.0	1.53250
$712$	−13230.0	−0.696370
$713$	8148.00	0.427973
$714$	−11220.0	−0.588092
$715$	6900.00	0.360903
$716$	528.000	0.0275591
$717$	−26640.0	−1.38757
$718$	18900.0	0.982370
$719$	−3774.00	−0.195753	−0.0978765	−	0.995199i	$-0.531205\pi$
$719$	−3774.00	−0.195753	−0.0978765	+	0.995199i	$0.531205\pi$
$720$	−25915.0	−1.34138
$721$	28072.0	1.45001
$722$	11871.0	0.611901
$723$	34580.0	1.77876
$724$	−4042.00	−0.207486
$725$	−1650.00	−0.0845234
$726$	−12930.0	−0.660988
$727$	3980.00	0.203040	0.101520	−	0.994834i	$-0.467629\pi$
$727$	3980.00	0.203040	0.101520	+	0.994834i	$0.467629\pi$
$728$	−21252.0	−1.08194
$729$	67717.0	3.44038
$730$	5430.00	0.275306
$731$	−6596.00	−0.333737
$732$	−6100.00	−0.308009
$733$	−18490.0	−0.931710	−0.465855	−	0.884861i	$-0.654253\pi$
$733$	−18490.0	−0.931710	−0.465855	+	0.884861i	$0.654253\pi$
$734$	−25482.0	−1.28141
$735$	7050.00	0.353800
$736$	−1890.00	−0.0946553
$737$	−25440.0	−1.27150
$738$	−27594.0	−1.37635
$739$	−24448.0	−1.21696	−0.608481	−	0.793569i	$-0.708221\pi$
$739$	−24448.0	−1.21696	−0.608481	+	0.793569i	$0.708221\pi$
$740$	1030.00	0.0511670
$741$	−47840.0	−2.37172
$742$	−5148.00	−0.254702
$743$	−32274.0	−1.59356	−0.796782	−	0.604267i	$-0.793466\pi$
$743$	−32274.0	−1.59356	−0.796782	+	0.604267i	$0.793466\pi$
$744$	−40740.0	−2.00753
$745$	13350.0	0.656518
$746$	−5826.00	−0.285932
$747$	60444.0	2.96055
$748$	−510.000	−0.0249297
$749$	9108.00	0.444325
$750$	3750.00	0.182574
$751$	9890.00	0.480548	0.240274	−	0.970705i	$-0.422763\pi$
$751$	9890.00	0.480548	0.240274	+	0.970705i	$0.422763\pi$
$752$	38340.0	1.85920
$753$	10920.0	0.528482
$754$	9108.00	0.439912
$755$	6220.00	0.299826
$756$	−10120.0	−0.486853
$757$	13394.0	0.643082	0.321541	−	0.946896i	$-0.395799\pi$
$757$	13394.0	0.643082	0.321541	+	0.946896i	$0.395799\pi$
$758$	15954.0	0.764479
$759$	−12600.0	−0.602571
$760$	−10920.0	−0.521197
$761$	30198.0	1.43847	0.719236	−	0.694766i	$-0.244492\pi$
$761$	30198.0	1.43847	0.719236	+	0.694766i	$0.244492\pi$
$762$	8880.00	0.422163
$763$	−13508.0	−0.640921
$764$	3144.00	0.148882
$765$	6205.00	0.293258
$766$	−16416.0	−0.774327
$767$	−19872.0	−0.935510
$768$	15130.0	0.710881
$769$	8174.00	0.383306	0.191653	−	0.981463i	$-0.438615\pi$
$769$	8174.00	0.383306	0.191653	+	0.981463i	$0.438615\pi$
$770$	9900.00	0.463339
$771$	−37140.0	−1.73484
$772$	−2014.00	−0.0938931
$773$	7842.00	0.364886	0.182443	−	0.983216i	$-0.441599\pi$
$773$	7842.00	0.364886	0.182443	+	0.983216i	$0.441599\pi$
$774$	−84972.0	−3.94607
$775$	4850.00	0.224796
$776$	31206.0	1.44360
$777$	−45320.0	−2.09247
$778$	−28098.0	−1.29481
$779$	−13104.0	−0.602695
$780$	−2300.00	−0.105581
$781$	5220.00	0.239163
$782$	2142.00	0.0979511
$783$	−30360.0	−1.38567
$784$	−10011.0	−0.456040
$785$	−13970.0	−0.635173
$786$	7380.00	0.334906
$787$	−28294.0	−1.28154	−0.640770	−	0.767733i	$-0.721385\pi$
$787$	−28294.0	−1.28154	−0.640770	+	0.767733i	$0.721385\pi$
$788$	−3666.00	−0.165731
$789$	36720.0	1.65687
$790$	5970.00	0.268865
$791$	−5676.00	−0.255139
$792$	45990.0	2.06336
$793$	28060.0	1.25654
$794$	40506.0	1.81046
$795$	3900.00	0.173986
$796$	2090.00	0.0930629
$797$	−2970.00	−0.131998	−0.0659992	−	0.997820i	$-0.521023\pi$
$797$	−2970.00	−0.131998	−0.0659992	+	0.997820i	$0.521023\pi$
$798$	−68640.0	−3.04490
$799$	−9180.00	−0.406464
$800$	−1125.00	−0.0497184
$801$	45990.0	2.02869
$802$	16398.0	0.721987
$803$	−10860.0	−0.477262
$804$	8480.00	0.371973
$805$	−4620.00	−0.202278
$806$	−26772.0	−1.16998
$807$	−6420.00	−0.280043
$808$	25830.0	1.12462
$809$	8034.00	0.349148	0.174574	−	0.984644i	$-0.444145\pi$
$809$	8034.00	0.349148	0.174574	+	0.984644i	$0.444145\pi$
$810$	39435.0	1.71062
$811$	21110.0	0.914023	0.457011	−	0.889461i	$-0.348920\pi$
$811$	21110.0	0.914023	0.457011	+	0.889461i	$0.348920\pi$
$812$	1452.00	0.0627527
$813$	14240.0	0.614291
$814$	−18540.0	−0.798313
$815$	−17090.0	−0.734523
$816$	−12070.0	−0.517812
$817$	−40352.0	−1.72795
$818$	18222.0	0.778872
$819$	73876.0	3.15194
$820$	−630.000	−0.0268299
$821$	6702.00	0.284898	0.142449	−	0.989802i	$-0.454502\pi$
$821$	6702.00	0.284898	0.142449	+	0.989802i	$0.454502\pi$
$822$	59940.0	2.54337
$823$	6290.00	0.266410	0.133205	−	0.991088i	$-0.457473\pi$
$823$	6290.00	0.266410	0.133205	+	0.991088i	$0.457473\pi$
$824$	26796.0	1.13287
$825$	−7500.00	−0.316505
$826$	−28512.0	−1.20104
$827$	−38346.0	−1.61236	−0.806180	−	0.591671i	$-0.798469\pi$
$827$	−38346.0	−1.61236	−0.806180	+	0.591671i	$0.798469\pi$
$828$	3066.00	0.128685
$829$	−7066.00	−0.296034	−0.148017	−	0.988985i	$-0.547289\pi$
$829$	−7066.00	−0.296034	−0.148017	+	0.988985i	$0.547289\pi$
$830$	12420.0	0.519403
$831$	−47620.0	−1.98787
$832$	−19918.0	−0.829967
$833$	2397.00	0.0997012
$834$	−16860.0	−0.700017
$835$	510.000	0.0211369
$836$	−3120.00	−0.129076
$837$	89240.0	3.68529
$838$	26766.0	1.10336
$839$	−2370.00	−0.0975226	−0.0487613	−	0.998810i	$-0.515527\pi$
$839$	−2370.00	−0.0975226	−0.0487613	+	0.998810i	$0.515527\pi$
$840$	23100.0	0.948840
$841$	−20033.0	−0.821395
$842$	26646.0	1.09060
$843$	−48060.0	−1.96355
$844$	−2542.00	−0.103672
$845$	−405.000	−0.0164881
$846$	−118260.	−4.80598
$847$	9482.00	0.384658
$848$	−5538.00	−0.224264
$849$	−2980.00	−0.120463
$850$	1275.00	0.0514496
$851$	8652.00	0.348516
$852$	−1740.00	−0.0699665
$853$	16742.0	0.672022	0.336011	−	0.941858i	$-0.390922\pi$
$853$	16742.0	0.672022	0.336011	+	0.941858i	$0.390922\pi$
$854$	40260.0	1.61320
$855$	37960.0	1.51837
$856$	8694.00	0.347143
$857$	210.000	0.00837044	0.00418522	−	0.999991i	$-0.498668\pi$
$857$	210.000	0.00837044	0.00418522	+	0.999991i	$0.498668\pi$
$858$	41400.0	1.64729
$859$	26528.0	1.05369	0.526847	−	0.849960i	$-0.323374\pi$
$859$	26528.0	1.05369	0.526847	+	0.849960i	$0.323374\pi$
$860$	−1940.00	−0.0769226
$861$	27720.0	1.09721
$862$	−11214.0	−0.443098
$863$	21108.0	0.832589	0.416295	−	0.909230i	$-0.363328\pi$
$863$	21108.0	0.832589	0.416295	+	0.909230i	$0.363328\pi$
$864$	−20700.0	−0.815079
$865$	9030.00	0.354947
$866$	−20262.0	−0.795070
$867$	2890.00	0.113206
$868$	−4268.00	−0.166896
$869$	−11940.0	−0.466095
$870$	−9900.00	−0.385795
$871$	−39008.0	−1.51749
$872$	−12894.0	−0.500741
$873$	−108478.	−4.20553
$874$	13104.0	0.507150
$875$	−2750.00	−0.106248
$876$	3620.00	0.139622
$877$	16022.0	0.616904	0.308452	−	0.951240i	$-0.400189\pi$
$877$	16022.0	0.616904	0.308452	+	0.951240i	$0.400189\pi$
$878$	−21810.0	−0.838328
$879$	3180.00	0.122024
$880$	10650.0	0.407968
$881$	11034.0	0.421958	0.210979	−	0.977491i	$-0.432335\pi$
$881$	11034.0	0.421958	0.210979	+	0.977491i	$0.432335\pi$
$882$	30879.0	1.17885
$883$	23384.0	0.891205	0.445603	−	0.895231i	$-0.352989\pi$
$883$	23384.0	0.891205	0.445603	+	0.895231i	$0.352989\pi$
$884$	−782.000	−0.0297528
$885$	21600.0	0.820425
$886$	−33408.0	−1.26678
$887$	534.000	0.0202142	0.0101071	−	0.999949i	$-0.496783\pi$
$887$	534.000	0.0202142	0.0101071	+	0.999949i	$0.496783\pi$
$888$	−43260.0	−1.63481
$889$	−6512.00	−0.245675
$890$	9450.00	0.355915
$891$	−78870.0	−2.96548
$892$	992.000	0.0372361
$893$	−56160.0	−2.10450
$894$	80100.0	2.99658
$895$	2640.00	0.0985983
$896$	−36498.0	−1.36084
$897$	−19320.0	−0.719148
$898$	−5922.00	−0.220066
$899$	−12804.0	−0.475014
$900$	1825.00	0.0675926
$901$	1326.00	0.0490294
$902$	11340.0	0.418604
$903$	85360.0	3.14574
$904$	−5418.00	−0.199336
$905$	−20210.0	−0.742324
$906$	37320.0	1.36851
$907$	30014.0	1.09879	0.549393	−	0.835564i	$-0.314859\pi$
$907$	30014.0	1.09879	0.549393	+	0.835564i	$0.314859\pi$
$908$	−1098.00	−0.0401304
$909$	−89790.0	−3.27629
$910$	15180.0	0.552980
$911$	45246.0	1.64552	0.822759	−	0.568390i	$-0.192434\pi$
$911$	45246.0	1.64552	0.822759	+	0.568390i	$0.192434\pi$
$912$	−73840.0	−2.68102
$913$	−24840.0	−0.900421
$914$	32514.0	1.17666
$915$	−30500.0	−1.10197
$916$	2954.00	0.106553
$917$	−5412.00	−0.194896
$918$	23460.0	0.843459
$919$	24176.0	0.867783	0.433892	−	0.900965i	$-0.357140\pi$
$919$	24176.0	0.867783	0.433892	+	0.900965i	$0.357140\pi$
$920$	−4410.00	−0.158036
$921$	14960.0	0.535232
$922$	−46926.0	−1.67617
$923$	8004.00	0.285433
$924$	6600.00	0.234983
$925$	5150.00	0.183060
$926$	6828.00	0.242313
$927$	−93148.0	−3.30030
$928$	2970.00	0.105059
$929$	−5070.00	−0.179054	−0.0895271	−	0.995984i	$-0.528536\pi$
$929$	−5070.00	−0.179054	−0.0895271	+	0.995984i	$0.528536\pi$
$930$	29100.0	1.02605
$931$	14664.0	0.516212
$932$	−678.000	−0.0238290
$933$	62580.0	2.19590
$934$	30492.0	1.06823
$935$	−2550.00	−0.0891914
$936$	70518.0	2.46256
$937$	27530.0	0.959836	0.479918	−	0.877313i	$-0.340666\pi$
$937$	27530.0	0.959836	0.479918	+	0.877313i	$0.340666\pi$
$938$	−55968.0	−1.94821
$939$	−29260.0	−1.01689
$940$	−2700.00	−0.0936854
$941$	27726.0	0.960512	0.480256	−	0.877128i	$-0.340544\pi$
$941$	27726.0	0.960512	0.480256	+	0.877128i	$0.340544\pi$
$942$	−83820.0	−2.89915
$943$	−5292.00	−0.182748
$944$	−30672.0	−1.05751
$945$	−50600.0	−1.74182
$946$	34920.0	1.20016
$947$	−50406.0	−1.72965	−0.864823	−	0.502077i	$-0.832569\pi$
$947$	−50406.0	−1.72965	−0.864823	+	0.502077i	$0.832569\pi$
$948$	3980.00	0.136355
$949$	−16652.0	−0.569596
$950$	7800.00	0.266385
$951$	−72900.0	−2.48575
$952$	7854.00	0.267384
$953$	−19746.0	−0.671181	−0.335591	−	0.942008i	$-0.608936\pi$
$953$	−19746.0	−0.671181	−0.335591	+	0.942008i	$0.608936\pi$
$954$	17082.0	0.579717
$955$	15720.0	0.532657
$956$	−2664.00	−0.0901254
$957$	19800.0	0.668802
$958$	42102.0	1.41989
$959$	−43956.0	−1.48010
$960$	21650.0	0.727865
$961$	7845.00	0.263335
$962$	−28428.0	−0.952760
$963$	−30222.0	−1.01131
$964$	3458.00	0.115534
$965$	−10070.0	−0.335922
$966$	−27720.0	−0.923267
$967$	−16720.0	−0.556028	−0.278014	−	0.960577i	$-0.589676\pi$
$967$	−16720.0	−0.556028	−0.278014	+	0.960577i	$0.589676\pi$
$968$	9051.00	0.300527
$969$	17680.0	0.586134
$970$	−22290.0	−0.737823
$971$	28128.0	0.929630	0.464815	−	0.885408i	$-0.346121\pi$
$971$	28128.0	0.929630	0.464815	+	0.885408i	$0.346121\pi$
$972$	13870.0	0.457696
$973$	12364.0	0.407371
$974$	−34050.0	−1.12016
$975$	−11500.0	−0.377738
$976$	43310.0	1.42041
$977$	−60174.0	−1.97046	−0.985229	−	0.171244i	$-0.945221\pi$
$977$	−60174.0	−1.97046	−0.985229	+	0.171244i	$0.945221\pi$
$978$	−102540.	−3.35263
$979$	−18900.0	−0.617004
$980$	705.000	0.0229800
$981$	44822.0	1.45877
$982$	21960.0	0.713617
$983$	40566.0	1.31623	0.658115	−	0.752917i	$-0.271354\pi$
$983$	40566.0	1.31623	0.658115	+	0.752917i	$0.271354\pi$
$984$	26460.0	0.857230
$985$	−18330.0	−0.592936
$986$	−3366.00	−0.108717
$987$	118800.	3.83125
$988$	−4784.00	−0.154048
$989$	−16296.0	−0.523946
$990$	−32850.0	−1.05459
$991$	−26422.0	−0.846945	−0.423472	−	0.905909i	$-0.639189\pi$
$991$	−26422.0	−0.846945	−0.423472	+	0.905909i	$0.639189\pi$
$992$	−8730.00	−0.279413
$993$	101840.	3.25458
$994$	11484.0	0.366449
$995$	10450.0	0.332952
$996$	8280.00	0.263416
$997$	55982.0	1.77830	0.889151	−	0.457613i	$-0.151296\pi$
$997$	55982.0	1.77830	0.889151	+	0.457613i	$0.151296\pi$
$998$	−22890.0	−0.726022
$999$	94760.0	3.00107

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	85.4.a.c.1.1	✓	1
3.2	odd	2		765.4.a.a.1.1		1
4.3	odd	2		1360.4.a.a.1.1		1
5.2	odd	4		425.4.b.d.324.2		2
5.3	odd	4		425.4.b.d.324.1		2
5.4	even	2		425.4.a.a.1.1		1
17.16	even	2		1445.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.a.c.1.1	✓	1	1.1	even	1	trivial
425.4.a.a.1.1		1	5.4	even	2
425.4.b.d.324.1		2	5.3	odd	4
425.4.b.d.324.2		2	5.2	odd	4
765.4.a.a.1.1		1	3.2	odd	2
1360.4.a.a.1.1		1	4.3	odd	2
1445.4.a.f.1.1		1	17.16	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	85.4.a.c.1.1

Level	$85$
Weight	$4$
Character	85.1
Self dual	yes
Analytic conductor	$5.015$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$