LMFDB - Embedded newform 2400.4.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2400 = 2^{5} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2400.1

$q$ -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^{3} +12.0000 q^{7} +9.00000 q^{9} -60.0000 q^{11} +42.0000 q^{13} -10.0000 q^{17} -132.000 q^{19} +36.0000 q^{21} -48.0000 q^{23} +27.0000 q^{27} +226.000 q^{29} +252.000 q^{31} -180.000 q^{33} +362.000 q^{37} +126.000 q^{39} -94.0000 q^{41} -228.000 q^{43} -408.000 q^{47} -199.000 q^{49} -30.0000 q^{51} -346.000 q^{53} -396.000 q^{57} +300.000 q^{59} -466.000 q^{61} +108.000 q^{63} +204.000 q^{67} -144.000 q^{69} -1056.00 q^{71} -330.000 q^{73} -720.000 q^{77} -612.000 q^{79} +81.0000 q^{81} +564.000 q^{83} +678.000 q^{87} -1510.00 q^{89} +504.000 q^{91} +756.000 q^{93} -594.000 q^{97} -540.000 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	12.0000	0.647939	0.323970	−	0.946068i	$-0.394982\pi$
$7$	12.0000	0.647939	0.323970	+	0.946068i	$0.394982\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−60.0000	−1.64461	−0.822304	−	0.569049i	$-0.807311\pi$
$11$	−60.0000	−1.64461	−0.822304	+	0.569049i	$0.807311\pi$
$12$	0	0
$13$	42.0000	0.896054	0.448027	−	0.894020i	$-0.352127\pi$
$13$	42.0000	0.896054	0.448027	+	0.894020i	$0.352127\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−10.0000	−0.142668	−0.0713340	−	0.997452i	$-0.522726\pi$
$17$	−10.0000	−0.142668	−0.0713340	+	0.997452i	$0.522726\pi$
$18$	0	0
$19$	−132.000	−1.59384	−0.796918	−	0.604088i	$-0.793538\pi$
$19$	−132.000	−1.59384	−0.796918	+	0.604088i	$0.793538\pi$
$20$	0	0
$21$	36.0000	0.374088
$22$	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
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	226.000	1.44714	0.723571	−	0.690249i	$-0.242499\pi$
$29$	226.000	1.44714	0.723571	+	0.690249i	$0.242499\pi$
$30$	0	0
$31$	252.000	1.46002	0.730009	−	0.683438i	$-0.239516\pi$
$31$	252.000	1.46002	0.730009	+	0.683438i	$0.239516\pi$
$32$	0	0
$33$	−180.000	−0.949514
$34$	0	0
$35$	0	0
$36$	0	0
$37$	362.000	1.60844	0.804222	−	0.594329i	$-0.202582\pi$
$37$	362.000	1.60844	0.804222	+	0.594329i	$0.202582\pi$
$38$	0	0
$39$	126.000	0.517337
$40$	0	0
$41$	−94.0000	−0.358057	−0.179028	−	0.983844i	$-0.557295\pi$
$41$	−94.0000	−0.358057	−0.179028	+	0.983844i	$0.557295\pi$
$42$	0	0
$43$	−228.000	−0.808597	−0.404299	−	0.914627i	$-0.632484\pi$
$43$	−228.000	−0.808597	−0.404299	+	0.914627i	$0.632484\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−408.000	−1.26623	−0.633116	−	0.774057i	$-0.718224\pi$
$47$	−408.000	−1.26623	−0.633116	+	0.774057i	$0.718224\pi$
$48$	0	0
$49$	−199.000	−0.580175
$50$	0	0
$51$	−30.0000	−0.0823694
$52$	0	0
$53$	−346.000	−0.896731	−0.448366	−	0.893850i	$-0.647994\pi$
$53$	−346.000	−0.896731	−0.448366	+	0.893850i	$0.647994\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−396.000	−0.920201
$58$	0	0
$59$	300.000	0.661978	0.330989	−	0.943635i	$-0.392618\pi$
$59$	300.000	0.661978	0.330989	+	0.943635i	$0.392618\pi$
$60$	0	0
$61$	−466.000	−0.978118	−0.489059	−	0.872251i	$-0.662660\pi$
$61$	−466.000	−0.978118	−0.489059	+	0.872251i	$0.662660\pi$
$62$	0	0
$63$	108.000	0.215980
$64$	0	0
$65$	0	0
$66$	0	0
$67$	204.000	0.371979	0.185989	−	0.982552i	$-0.440451\pi$
$67$	204.000	0.371979	0.185989	+	0.982552i	$0.440451\pi$
$68$	0	0
$69$	−144.000	−0.251240
$70$	0	0
$71$	−1056.00	−1.76513	−0.882564	−	0.470192i	$-0.844185\pi$
$71$	−1056.00	−1.76513	−0.882564	+	0.470192i	$0.844185\pi$
$72$	0	0
$73$	−330.000	−0.529090	−0.264545	−	0.964373i	$-0.585222\pi$
$73$	−330.000	−0.529090	−0.264545	+	0.964373i	$0.585222\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−720.000	−1.06561
$78$	0	0
$79$	−612.000	−0.871587	−0.435794	−	0.900047i	$-0.643532\pi$
$79$	−612.000	−0.871587	−0.435794	+	0.900047i	$0.643532\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	564.000	0.745868	0.372934	−	0.927858i	$-0.378352\pi$
$83$	564.000	0.745868	0.372934	+	0.927858i	$0.378352\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	678.000	0.835508
$88$	0	0
$89$	−1510.00	−1.79842	−0.899212	−	0.437514i	$-0.855859\pi$
$89$	−1510.00	−1.79842	−0.899212	+	0.437514i	$0.855859\pi$
$90$	0	0
$91$	504.000	0.580589
$92$	0	0
$93$	756.000	0.842941
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−594.000	−0.621769	−0.310884	−	0.950448i	$-0.600625\pi$
$97$	−594.000	−0.621769	−0.310884	+	0.950448i	$0.600625\pi$
$98$	0	0
$99$	−540.000	−0.548202
$100$	0	0
$101$	554.000	0.545793	0.272896	−	0.962043i	$-0.412018\pi$
$101$	554.000	0.545793	0.272896	+	0.962043i	$0.412018\pi$
$102$	0	0
$103$	1284.00	1.22831	0.614157	−	0.789184i	$-0.289496\pi$
$103$	1284.00	1.22831	0.614157	+	0.789184i	$0.289496\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1356.00	−1.22514	−0.612568	−	0.790418i	$-0.709863\pi$
$107$	−1356.00	−1.22514	−0.612568	+	0.790418i	$0.709863\pi$
$108$	0	0
$109$	390.000	0.342708	0.171354	−	0.985209i	$-0.445186\pi$
$109$	390.000	0.342708	0.171354	+	0.985209i	$0.445186\pi$
$110$	0	0
$111$	1086.00	0.928636
$112$	0	0
$113$	766.000	0.637692	0.318846	−	0.947807i	$-0.396705\pi$
$113$	766.000	0.637692	0.318846	+	0.947807i	$0.396705\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	378.000	0.298685
$118$	0	0
$119$	−120.000	−0.0924402
$120$	0	0
$121$	2269.00	1.70473
$122$	0	0
$123$	−282.000	−0.206724
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2388.00	−1.66851	−0.834255	−	0.551379i	$-0.814102\pi$
$127$	−2388.00	−1.66851	−0.834255	+	0.551379i	$0.814102\pi$
$128$	0	0
$129$	−684.000	−0.466844
$130$	0	0
$131$	−396.000	−0.264112	−0.132056	−	0.991242i	$-0.542158\pi$
$131$	−396.000	−0.264112	−0.132056	+	0.991242i	$0.542158\pi$
$132$	0	0
$133$	−1584.00	−1.03271
$134$	0	0
$135$	0	0
$136$	0	0
$137$	110.000	0.0685981	0.0342990	−	0.999412i	$-0.489080\pi$
$137$	110.000	0.0685981	0.0342990	+	0.999412i	$0.489080\pi$
$138$	0	0
$139$	732.000	0.446672	0.223336	−	0.974742i	$-0.428305\pi$
$139$	732.000	0.446672	0.223336	+	0.974742i	$0.428305\pi$
$140$	0	0
$141$	−1224.00	−0.731060
$142$	0	0
$143$	−2520.00	−1.47366
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−597.000	−0.334964
$148$	0	0
$149$	−1934.00	−1.06335	−0.531676	−	0.846948i	$-0.678438\pi$
$149$	−1934.00	−1.06335	−0.531676	+	0.846948i	$0.678438\pi$
$150$	0	0
$151$	1092.00	0.588515	0.294257	−	0.955726i	$-0.404928\pi$
$151$	1092.00	0.588515	0.294257	+	0.955726i	$0.404928\pi$
$152$	0	0
$153$	−90.0000	−0.0475560
$154$	0	0
$155$	0	0
$156$	0	0
$157$	578.000	0.293818	0.146909	−	0.989150i	$-0.453068\pi$
$157$	578.000	0.293818	0.146909	+	0.989150i	$0.453068\pi$
$158$	0	0
$159$	−1038.00	−0.517728
$160$	0	0
$161$	−576.000	−0.281958
$162$	0	0
$163$	2532.00	1.21670	0.608348	−	0.793670i	$-0.291832\pi$
$163$	2532.00	1.21670	0.608348	+	0.793670i	$0.291832\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−648.000	−0.300262	−0.150131	−	0.988666i	$-0.547970\pi$
$167$	−648.000	−0.300262	−0.150131	+	0.988666i	$0.547970\pi$
$168$	0	0
$169$	−433.000	−0.197087
$170$	0	0
$171$	−1188.00	−0.531279
$172$	0	0
$173$	−3338.00	−1.46696	−0.733478	−	0.679713i	$-0.762104\pi$
$173$	−3338.00	−1.46696	−0.733478	+	0.679713i	$0.762104\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	900.000	0.382193
$178$	0	0
$179$	−3804.00	−1.58840	−0.794202	−	0.607654i	$-0.792111\pi$
$179$	−3804.00	−1.58840	−0.794202	+	0.607654i	$0.792111\pi$
$180$	0	0
$181$	1854.00	0.761363	0.380682	−	0.924706i	$-0.375689\pi$
$181$	1854.00	0.761363	0.380682	+	0.924706i	$0.375689\pi$
$182$	0	0
$183$	−1398.00	−0.564717
$184$	0	0
$185$	0	0
$186$	0	0
$187$	600.000	0.234633
$188$	0	0
$189$	324.000	0.124696
$190$	0	0
$191$	−1344.00	−0.509154	−0.254577	−	0.967052i	$-0.581936\pi$
$191$	−1344.00	−0.509154	−0.254577	+	0.967052i	$0.581936\pi$
$192$	0	0
$193$	1262.00	0.470677	0.235339	−	0.971913i	$-0.424380\pi$
$193$	1262.00	0.470677	0.235339	+	0.971913i	$0.424380\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4294.00	1.55297	0.776484	−	0.630137i	$-0.217001\pi$
$197$	4294.00	1.55297	0.776484	+	0.630137i	$0.217001\pi$
$198$	0	0
$199$	−4308.00	−1.53460	−0.767302	−	0.641286i	$-0.778401\pi$
$199$	−4308.00	−1.53460	−0.767302	+	0.641286i	$0.778401\pi$
$200$	0	0
$201$	612.000	0.214762
$202$	0	0
$203$	2712.00	0.937661
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−432.000	−0.145054
$208$	0	0
$209$	7920.00	2.62123
$210$	0	0
$211$	−1212.00	−0.395438	−0.197719	−	0.980259i	$-0.563353\pi$
$211$	−1212.00	−0.395438	−0.197719	+	0.980259i	$0.563353\pi$
$212$	0	0
$213$	−3168.00	−1.01910
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3024.00	0.946002
$218$	0	0
$219$	−990.000	−0.305470
$220$	0	0
$221$	−420.000	−0.127838
$222$	0	0
$223$	2172.00	0.652233	0.326116	−	0.945330i	$-0.394260\pi$
$223$	2172.00	0.652233	0.326116	+	0.945330i	$0.394260\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3948.00	−1.15435	−0.577176	−	0.816620i	$-0.695845\pi$
$227$	−3948.00	−1.15435	−0.577176	+	0.816620i	$0.695845\pi$
$228$	0	0
$229$	−3522.00	−1.01633	−0.508167	−	0.861259i	$-0.669677\pi$
$229$	−3522.00	−1.01633	−0.508167	+	0.861259i	$0.669677\pi$
$230$	0	0
$231$	−2160.00	−0.615228
$232$	0	0
$233$	2774.00	0.779960	0.389980	−	0.920823i	$-0.372482\pi$
$233$	2774.00	0.779960	0.389980	+	0.920823i	$0.372482\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1836.00	−0.503211
$238$	0	0
$239$	−2784.00	−0.753481	−0.376741	−	0.926319i	$-0.622955\pi$
$239$	−2784.00	−0.753481	−0.376741	+	0.926319i	$0.622955\pi$
$240$	0	0
$241$	−4686.00	−1.25250	−0.626249	−	0.779623i	$-0.715410\pi$
$241$	−4686.00	−1.25250	−0.626249	+	0.779623i	$0.715410\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5544.00	−1.42816
$248$	0	0
$249$	1692.00	0.430627
$250$	0	0
$251$	2484.00	0.624656	0.312328	−	0.949974i	$-0.398891\pi$
$251$	2484.00	0.624656	0.312328	+	0.949974i	$0.398891\pi$
$252$	0	0
$253$	2880.00	0.715668
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6658.00	−1.61601	−0.808005	−	0.589175i	$-0.799453\pi$
$257$	−6658.00	−1.61601	−0.808005	+	0.589175i	$0.799453\pi$
$258$	0	0
$259$	4344.00	1.04217
$260$	0	0
$261$	2034.00	0.482381
$262$	0	0
$263$	2904.00	0.680868	0.340434	−	0.940268i	$-0.389426\pi$
$263$	2904.00	0.680868	0.340434	+	0.940268i	$0.389426\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4530.00	−1.03832
$268$	0	0
$269$	−1006.00	−0.228018	−0.114009	−	0.993480i	$-0.536369\pi$
$269$	−1006.00	−0.228018	−0.114009	+	0.993480i	$0.536369\pi$
$270$	0	0
$271$	−876.000	−0.196359	−0.0981794	−	0.995169i	$-0.531302\pi$
$271$	−876.000	−0.196359	−0.0981794	+	0.995169i	$0.531302\pi$
$272$	0	0
$273$	1512.00	0.335203
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2718.00	−0.589562	−0.294781	−	0.955565i	$-0.595247\pi$
$277$	−2718.00	−0.589562	−0.294781	+	0.955565i	$0.595247\pi$
$278$	0	0
$279$	2268.00	0.486672
$280$	0	0
$281$	5354.00	1.13663	0.568315	−	0.822811i	$-0.307596\pi$
$281$	5354.00	1.13663	0.568315	+	0.822811i	$0.307596\pi$
$282$	0	0
$283$	−780.000	−0.163838	−0.0819191	−	0.996639i	$-0.526105\pi$
$283$	−780.000	−0.163838	−0.0819191	+	0.996639i	$0.526105\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1128.00	−0.231999
$288$	0	0
$289$	−4813.00	−0.979646
$290$	0	0
$291$	−1782.00	−0.358978
$292$	0	0
$293$	3350.00	0.667949	0.333975	−	0.942582i	$-0.391610\pi$
$293$	3350.00	0.667949	0.333975	+	0.942582i	$0.391610\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1620.00	−0.316505
$298$	0	0
$299$	−2016.00	−0.389927
$300$	0	0
$301$	−2736.00	−0.523922
$302$	0	0
$303$	1662.00	0.315114
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9636.00	−1.79139	−0.895693	−	0.444673i	$-0.853320\pi$
$307$	−9636.00	−1.79139	−0.895693	+	0.444673i	$0.853320\pi$
$308$	0	0
$309$	3852.00	0.709167
$310$	0	0
$311$	7560.00	1.37842	0.689209	−	0.724562i	$-0.257958\pi$
$311$	7560.00	1.37842	0.689209	+	0.724562i	$0.257958\pi$
$312$	0	0
$313$	3526.00	0.636745	0.318373	−	0.947966i	$-0.396864\pi$
$313$	3526.00	0.636745	0.318373	+	0.947966i	$0.396864\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7634.00	−1.35258	−0.676290	−	0.736635i	$-0.736414\pi$
$317$	−7634.00	−1.35258	−0.676290	+	0.736635i	$0.736414\pi$
$318$	0	0
$319$	−13560.0	−2.37998
$320$	0	0
$321$	−4068.00	−0.707332
$322$	0	0
$323$	1320.00	0.227389
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1170.00	0.197863
$328$	0	0
$329$	−4896.00	−0.820441
$330$	0	0
$331$	−7572.00	−1.25739	−0.628693	−	0.777654i	$-0.716410\pi$
$331$	−7572.00	−1.25739	−0.628693	+	0.777654i	$0.716410\pi$
$332$	0	0
$333$	3258.00	0.536148
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−162.000	−0.0261861	−0.0130930	−	0.999914i	$-0.504168\pi$
$337$	−162.000	−0.0261861	−0.0130930	+	0.999914i	$0.504168\pi$
$338$	0	0
$339$	2298.00	0.368172
$340$	0	0
$341$	−15120.0	−2.40116
$342$	0	0
$343$	−6504.00	−1.02386
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6636.00	1.02663	0.513313	−	0.858202i	$-0.328418\pi$
$347$	6636.00	1.02663	0.513313	+	0.858202i	$0.328418\pi$
$348$	0	0
$349$	4430.00	0.679463	0.339731	−	0.940523i	$-0.389664\pi$
$349$	4430.00	0.679463	0.339731	+	0.940523i	$0.389664\pi$
$350$	0	0
$351$	1134.00	0.172446
$352$	0	0
$353$	−8402.00	−1.26684	−0.633418	−	0.773810i	$-0.718349\pi$
$353$	−8402.00	−1.26684	−0.633418	+	0.773810i	$0.718349\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−360.000	−0.0533704
$358$	0	0
$359$	−11520.0	−1.69360	−0.846800	−	0.531912i	$-0.821474\pi$
$359$	−11520.0	−1.69360	−0.846800	+	0.531912i	$0.821474\pi$
$360$	0	0
$361$	10565.0	1.54031
$362$	0	0
$363$	6807.00	0.984228
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7404.00	−1.05309	−0.526547	−	0.850146i	$-0.676514\pi$
$367$	−7404.00	−1.05309	−0.526547	+	0.850146i	$0.676514\pi$
$368$	0	0
$369$	−846.000	−0.119352
$370$	0	0
$371$	−4152.00	−0.581027
$372$	0	0
$373$	−1910.00	−0.265137	−0.132568	−	0.991174i	$-0.542322\pi$
$373$	−1910.00	−0.265137	−0.132568	+	0.991174i	$0.542322\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9492.00	1.29672
$378$	0	0
$379$	−10332.0	−1.40031	−0.700157	−	0.713989i	$-0.746887\pi$
$379$	−10332.0	−1.40031	−0.700157	+	0.713989i	$0.746887\pi$
$380$	0	0
$381$	−7164.00	−0.963315
$382$	0	0
$383$	6624.00	0.883735	0.441868	−	0.897080i	$-0.354316\pi$
$383$	6624.00	0.883735	0.441868	+	0.897080i	$0.354316\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2052.00	−0.269532
$388$	0	0
$389$	10210.0	1.33076	0.665382	−	0.746503i	$-0.268268\pi$
$389$	10210.0	1.33076	0.665382	+	0.746503i	$0.268268\pi$
$390$	0	0
$391$	480.000	0.0620835
$392$	0	0
$393$	−1188.00	−0.152485
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4066.00	0.514022	0.257011	−	0.966408i	$-0.417262\pi$
$397$	4066.00	0.514022	0.257011	+	0.966408i	$0.417262\pi$
$398$	0	0
$399$	−4752.00	−0.596234
$400$	0	0
$401$	−5510.00	−0.686175	−0.343088	−	0.939303i	$-0.611473\pi$
$401$	−5510.00	−0.686175	−0.343088	+	0.939303i	$0.611473\pi$
$402$	0	0
$403$	10584.0	1.30825
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21720.0	−2.64526
$408$	0	0
$409$	15450.0	1.86786	0.933928	−	0.357460i	$-0.116357\pi$
$409$	15450.0	1.86786	0.933928	+	0.357460i	$0.116357\pi$
$410$	0	0
$411$	330.000	0.0396051
$412$	0	0
$413$	3600.00	0.428921
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2196.00	0.257886
$418$	0	0
$419$	3084.00	0.359578	0.179789	−	0.983705i	$-0.442458\pi$
$419$	3084.00	0.359578	0.179789	+	0.983705i	$0.442458\pi$
$420$	0	0
$421$	10446.0	1.20928	0.604640	−	0.796499i	$-0.293317\pi$
$421$	10446.0	1.20928	0.604640	+	0.796499i	$0.293317\pi$
$422$	0	0
$423$	−3672.00	−0.422077
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5592.00	−0.633761
$428$	0	0
$429$	−7560.00	−0.850816
$430$	0	0
$431$	2184.00	0.244083	0.122041	−	0.992525i	$-0.461056\pi$
$431$	2184.00	0.244083	0.122041	+	0.992525i	$0.461056\pi$
$432$	0	0
$433$	110.000	0.0122085	0.00610423	−	0.999981i	$-0.498057\pi$
$433$	110.000	0.0122085	0.00610423	+	0.999981i	$0.498057\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6336.00	0.693574
$438$	0	0
$439$	2412.00	0.262229	0.131114	−	0.991367i	$-0.458144\pi$
$439$	2412.00	0.262229	0.131114	+	0.991367i	$0.458144\pi$
$440$	0	0
$441$	−1791.00	−0.193392
$442$	0	0
$443$	6540.00	0.701410	0.350705	−	0.936486i	$-0.385942\pi$
$443$	6540.00	0.701410	0.350705	+	0.936486i	$0.385942\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5802.00	−0.613927
$448$	0	0
$449$	−9670.00	−1.01638	−0.508191	−	0.861244i	$-0.669686\pi$
$449$	−9670.00	−1.01638	−0.508191	+	0.861244i	$0.669686\pi$
$450$	0	0
$451$	5640.00	0.588863
$452$	0	0
$453$	3276.00	0.339779
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6774.00	0.693379	0.346690	−	0.937980i	$-0.387306\pi$
$457$	6774.00	0.693379	0.346690	+	0.937980i	$0.387306\pi$
$458$	0	0
$459$	−270.000	−0.0274565
$460$	0	0
$461$	14602.0	1.47523	0.737617	−	0.675219i	$-0.235951\pi$
$461$	14602.0	1.47523	0.737617	+	0.675219i	$0.235951\pi$
$462$	0	0
$463$	−13620.0	−1.36712	−0.683558	−	0.729896i	$-0.739569\pi$
$463$	−13620.0	−1.36712	−0.683558	+	0.729896i	$0.739569\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8508.00	−0.843048	−0.421524	−	0.906817i	$-0.638505\pi$
$467$	−8508.00	−0.843048	−0.421524	+	0.906817i	$0.638505\pi$
$468$	0	0
$469$	2448.00	0.241019
$470$	0	0
$471$	1734.00	0.169636
$472$	0	0
$473$	13680.0	1.32982
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3114.00	−0.298910
$478$	0	0
$479$	−6312.00	−0.602093	−0.301047	−	0.953609i	$-0.597336\pi$
$479$	−6312.00	−0.602093	−0.301047	+	0.953609i	$0.597336\pi$
$480$	0	0
$481$	15204.0	1.44125
$482$	0	0
$483$	−1728.00	−0.162788
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10572.0	0.983702	0.491851	−	0.870679i	$-0.336320\pi$
$487$	10572.0	0.983702	0.491851	+	0.870679i	$0.336320\pi$
$488$	0	0
$489$	7596.00	0.702460
$490$	0	0
$491$	4332.00	0.398168	0.199084	−	0.979982i	$-0.436203\pi$
$491$	4332.00	0.398168	0.199084	+	0.979982i	$0.436203\pi$
$492$	0	0
$493$	−2260.00	−0.206461
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12672.0	−1.14370
$498$	0	0
$499$	3684.00	0.330498	0.165249	−	0.986252i	$-0.447157\pi$
$499$	3684.00	0.330498	0.165249	+	0.986252i	$0.447157\pi$
$500$	0	0
$501$	−1944.00	−0.173356
$502$	0	0
$503$	−11184.0	−0.991391	−0.495696	−	0.868496i	$-0.665087\pi$
$503$	−11184.0	−0.991391	−0.495696	+	0.868496i	$0.665087\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1299.00	−0.113788
$508$	0	0
$509$	12946.0	1.12735	0.563675	−	0.825997i	$-0.309387\pi$
$509$	12946.0	1.12735	0.563675	+	0.825997i	$0.309387\pi$
$510$	0	0
$511$	−3960.00	−0.342818
$512$	0	0
$513$	−3564.00	−0.306734
$514$	0	0
$515$	0	0
$516$	0	0
$517$	24480.0	2.08245
$518$	0	0
$519$	−10014.0	−0.846948
$520$	0	0
$521$	−17150.0	−1.44214	−0.721070	−	0.692862i	$-0.756349\pi$
$521$	−17150.0	−1.44214	−0.721070	+	0.692862i	$0.756349\pi$
$522$	0	0
$523$	7884.00	0.659165	0.329582	−	0.944127i	$-0.393092\pi$
$523$	7884.00	0.659165	0.329582	+	0.944127i	$0.393092\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2520.00	−0.208298
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	2700.00	0.220659
$532$	0	0
$533$	−3948.00	−0.320838
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11412.0	−0.917065
$538$	0	0
$539$	11940.0	0.954160
$540$	0	0
$541$	5910.00	0.469669	0.234834	−	0.972035i	$-0.424545\pi$
$541$	5910.00	0.469669	0.234834	+	0.972035i	$0.424545\pi$
$542$	0	0
$543$	5562.00	0.439573
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−972.000	−0.0759775	−0.0379888	−	0.999278i	$-0.512095\pi$
$547$	−972.000	−0.0759775	−0.0379888	+	0.999278i	$0.512095\pi$
$548$	0	0
$549$	−4194.00	−0.326039
$550$	0	0
$551$	−29832.0	−2.30651
$552$	0	0
$553$	−7344.00	−0.564735
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2458.00	−0.186982	−0.0934908	−	0.995620i	$-0.529803\pi$
$557$	−2458.00	−0.186982	−0.0934908	+	0.995620i	$0.529803\pi$
$558$	0	0
$559$	−9576.00	−0.724547
$560$	0	0
$561$	1800.00	0.135465
$562$	0	0
$563$	11316.0	0.847092	0.423546	−	0.905875i	$-0.360785\pi$
$563$	11316.0	0.847092	0.423546	+	0.905875i	$0.360785\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	972.000	0.0719932
$568$	0	0
$569$	1810.00	0.133355	0.0666776	−	0.997775i	$-0.478760\pi$
$569$	1810.00	0.133355	0.0666776	+	0.997775i	$0.478760\pi$
$570$	0	0
$571$	−10500.0	−0.769547	−0.384773	−	0.923011i	$-0.625720\pi$
$571$	−10500.0	−0.769547	−0.384773	+	0.923011i	$0.625720\pi$
$572$	0	0
$573$	−4032.00	−0.293960
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19438.0	1.40245	0.701226	−	0.712939i	$-0.252637\pi$
$577$	19438.0	1.40245	0.701226	+	0.712939i	$0.252637\pi$
$578$	0	0
$579$	3786.00	0.271746
$580$	0	0
$581$	6768.00	0.483277
$582$	0	0
$583$	20760.0	1.47477
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15084.0	−1.06062	−0.530309	−	0.847804i	$-0.677924\pi$
$587$	−15084.0	−1.06062	−0.530309	+	0.847804i	$0.677924\pi$
$588$	0	0
$589$	−33264.0	−2.32703
$590$	0	0
$591$	12882.0	0.896607
$592$	0	0
$593$	−5794.00	−0.401233	−0.200616	−	0.979670i	$-0.564294\pi$
$593$	−5794.00	−0.401233	−0.200616	+	0.979670i	$0.564294\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12924.0	−0.886004
$598$	0	0
$599$	−25152.0	−1.71566	−0.857832	−	0.513930i	$-0.828189\pi$
$599$	−25152.0	−1.71566	−0.857832	+	0.513930i	$0.828189\pi$
$600$	0	0
$601$	−11846.0	−0.804007	−0.402004	−	0.915638i	$-0.631686\pi$
$601$	−11846.0	−0.804007	−0.402004	+	0.915638i	$0.631686\pi$
$602$	0	0
$603$	1836.00	0.123993
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8940.00	0.597798	0.298899	−	0.954285i	$-0.403381\pi$
$607$	8940.00	0.597798	0.298899	+	0.954285i	$0.403381\pi$
$608$	0	0
$609$	8136.00	0.541359
$610$	0	0
$611$	−17136.0	−1.13461
$612$	0	0
$613$	4570.00	0.301110	0.150555	−	0.988602i	$-0.451894\pi$
$613$	4570.00	0.301110	0.150555	+	0.988602i	$0.451894\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17786.0	−1.16051	−0.580257	−	0.814433i	$-0.697048\pi$
$617$	−17786.0	−1.16051	−0.580257	+	0.814433i	$0.697048\pi$
$618$	0	0
$619$	15804.0	1.02620	0.513099	−	0.858330i	$-0.328497\pi$
$619$	15804.0	1.02620	0.513099	+	0.858330i	$0.328497\pi$
$620$	0	0
$621$	−1296.00	−0.0837467
$622$	0	0
$623$	−18120.0	−1.16527
$624$	0	0
$625$	0	0
$626$	0	0
$627$	23760.0	1.51337
$628$	0	0
$629$	−3620.00	−0.229474
$630$	0	0
$631$	18468.0	1.16513	0.582567	−	0.812783i	$-0.302048\pi$
$631$	18468.0	1.16513	0.582567	+	0.812783i	$0.302048\pi$
$632$	0	0
$633$	−3636.00	−0.228307
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8358.00	−0.519868
$638$	0	0
$639$	−9504.00	−0.588376
$640$	0	0
$641$	−7814.00	−0.481489	−0.240744	−	0.970589i	$-0.577392\pi$
$641$	−7814.00	−0.481489	−0.240744	+	0.970589i	$0.577392\pi$
$642$	0	0
$643$	5364.00	0.328982	0.164491	−	0.986379i	$-0.447402\pi$
$643$	5364.00	0.328982	0.164491	+	0.986379i	$0.447402\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3936.00	−0.239166	−0.119583	−	0.992824i	$-0.538156\pi$
$647$	−3936.00	−0.239166	−0.119583	+	0.992824i	$0.538156\pi$
$648$	0	0
$649$	−18000.0	−1.08869
$650$	0	0
$651$	9072.00	0.546175
$652$	0	0
$653$	−7610.00	−0.456053	−0.228026	−	0.973655i	$-0.573227\pi$
$653$	−7610.00	−0.456053	−0.228026	+	0.973655i	$0.573227\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2970.00	−0.176363
$658$	0	0
$659$	13620.0	0.805098	0.402549	−	0.915398i	$-0.368124\pi$
$659$	13620.0	0.805098	0.402549	+	0.915398i	$0.368124\pi$
$660$	0	0
$661$	8710.00	0.512526	0.256263	−	0.966607i	$-0.417509\pi$
$661$	8710.00	0.512526	0.256263	+	0.966607i	$0.417509\pi$
$662$	0	0
$663$	−1260.00	−0.0738075
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10848.0	−0.629739
$668$	0	0
$669$	6516.00	0.376567
$670$	0	0
$671$	27960.0	1.60862
$672$	0	0
$673$	12094.0	0.692703	0.346352	−	0.938105i	$-0.387420\pi$
$673$	12094.0	0.692703	0.346352	+	0.938105i	$0.387420\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16466.0	−0.934771	−0.467385	−	0.884054i	$-0.654804\pi$
$677$	−16466.0	−0.934771	−0.467385	+	0.884054i	$0.654804\pi$
$678$	0	0
$679$	−7128.00	−0.402868
$680$	0	0
$681$	−11844.0	−0.666466
$682$	0	0
$683$	16428.0	0.920351	0.460176	−	0.887828i	$-0.347786\pi$
$683$	16428.0	0.920351	0.460176	+	0.887828i	$0.347786\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10566.0	−0.586780
$688$	0	0
$689$	−14532.0	−0.803520
$690$	0	0
$691$	−13332.0	−0.733970	−0.366985	−	0.930227i	$-0.619610\pi$
$691$	−13332.0	−0.733970	−0.366985	+	0.930227i	$0.619610\pi$
$692$	0	0
$693$	−6480.00	−0.355202
$694$	0	0
$695$	0	0
$696$	0	0
$697$	940.000	0.0510833
$698$	0	0
$699$	8322.00	0.450310
$700$	0	0
$701$	−19118.0	−1.03007	−0.515033	−	0.857170i	$-0.672221\pi$
$701$	−19118.0	−1.03007	−0.515033	+	0.857170i	$0.672221\pi$
$702$	0	0
$703$	−47784.0	−2.56360
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6648.00	0.353640
$708$	0	0
$709$	798.000	0.0422701	0.0211351	−	0.999777i	$-0.493272\pi$
$709$	798.000	0.0422701	0.0211351	+	0.999777i	$0.493272\pi$
$710$	0	0
$711$	−5508.00	−0.290529
$712$	0	0
$713$	−12096.0	−0.635342
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8352.00	−0.435023
$718$	0	0
$719$	8856.00	0.459351	0.229675	−	0.973267i	$-0.426234\pi$
$719$	8856.00	0.459351	0.229675	+	0.973267i	$0.426234\pi$
$720$	0	0
$721$	15408.0	0.795872
$722$	0	0
$723$	−14058.0	−0.723130
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−13764.0	−0.702171	−0.351086	−	0.936343i	$-0.614187\pi$
$727$	−13764.0	−0.702171	−0.351086	+	0.936343i	$0.614187\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	2280.00	0.115361
$732$	0	0
$733$	20538.0	1.03491	0.517455	−	0.855711i	$-0.326880\pi$
$733$	20538.0	1.03491	0.517455	+	0.855711i	$0.326880\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12240.0	−0.611759
$738$	0	0
$739$	−15900.0	−0.791463	−0.395731	−	0.918366i	$-0.629509\pi$
$739$	−15900.0	−0.791463	−0.395731	+	0.918366i	$0.629509\pi$
$740$	0	0
$741$	−16632.0	−0.824550
$742$	0	0
$743$	−20856.0	−1.02979	−0.514894	−	0.857254i	$-0.672169\pi$
$743$	−20856.0	−1.02979	−0.514894	+	0.857254i	$0.672169\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5076.00	0.248623
$748$	0	0
$749$	−16272.0	−0.793813
$750$	0	0
$751$	10332.0	0.502024	0.251012	−	0.967984i	$-0.419237\pi$
$751$	10332.0	0.502024	0.251012	+	0.967984i	$0.419237\pi$
$752$	0	0
$753$	7452.00	0.360645
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13806.0	−0.662863	−0.331432	−	0.943479i	$-0.607532\pi$
$757$	−13806.0	−0.662863	−0.331432	+	0.943479i	$0.607532\pi$
$758$	0	0
$759$	8640.00	0.413191
$760$	0	0
$761$	15554.0	0.740909	0.370455	−	0.928851i	$-0.379202\pi$
$761$	15554.0	0.740909	0.370455	+	0.928851i	$0.379202\pi$
$762$	0	0
$763$	4680.00	0.222054
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12600.0	0.593168
$768$	0	0
$769$	13106.0	0.614583	0.307292	−	0.951615i	$-0.400577\pi$
$769$	13106.0	0.614583	0.307292	+	0.951615i	$0.400577\pi$
$770$	0	0
$771$	−19974.0	−0.933004
$772$	0	0
$773$	−18874.0	−0.878203	−0.439101	−	0.898438i	$-0.644703\pi$
$773$	−18874.0	−0.878203	−0.439101	+	0.898438i	$0.644703\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	13032.0	0.601699
$778$	0	0
$779$	12408.0	0.570684
$780$	0	0
$781$	63360.0	2.90294
$782$	0	0
$783$	6102.00	0.278503
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15444.0	0.699516	0.349758	−	0.936840i	$-0.386264\pi$
$787$	15444.0	0.699516	0.349758	+	0.936840i	$0.386264\pi$
$788$	0	0
$789$	8712.00	0.393099
$790$	0	0
$791$	9192.00	0.413186
$792$	0	0
$793$	−19572.0	−0.876447
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39286.0	1.74602	0.873012	−	0.487698i	$-0.162163\pi$
$797$	39286.0	1.74602	0.873012	+	0.487698i	$0.162163\pi$
$798$	0	0
$799$	4080.00	0.180651
$800$	0	0
$801$	−13590.0	−0.599474
$802$	0	0
$803$	19800.0	0.870145
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3018.00	−0.131646
$808$	0	0
$809$	20018.0	0.869957	0.434979	−	0.900441i	$-0.356756\pi$
$809$	20018.0	0.869957	0.434979	+	0.900441i	$0.356756\pi$
$810$	0	0
$811$	8388.00	0.363184	0.181592	−	0.983374i	$-0.441875\pi$
$811$	8388.00	0.363184	0.181592	+	0.983374i	$0.441875\pi$
$812$	0	0
$813$	−2628.00	−0.113368
$814$	0	0
$815$	0	0
$816$	0	0
$817$	30096.0	1.28877
$818$	0	0
$819$	4536.00	0.193530
$820$	0	0
$821$	−37942.0	−1.61289	−0.806446	−	0.591307i	$-0.798612\pi$
$821$	−37942.0	−1.61289	−0.806446	+	0.591307i	$0.798612\pi$
$822$	0	0
$823$	11628.0	0.492499	0.246249	−	0.969206i	$-0.420802\pi$
$823$	11628.0	0.492499	0.246249	+	0.969206i	$0.420802\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32388.0	1.36184	0.680920	−	0.732358i	$-0.261580\pi$
$827$	32388.0	1.36184	0.680920	+	0.732358i	$0.261580\pi$
$828$	0	0
$829$	9846.00	0.412504	0.206252	−	0.978499i	$-0.433873\pi$
$829$	9846.00	0.412504	0.206252	+	0.978499i	$0.433873\pi$
$830$	0	0
$831$	−8154.00	−0.340384
$832$	0	0
$833$	1990.00	0.0827724
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6804.00	0.280980
$838$	0	0
$839$	−16848.0	−0.693275	−0.346637	−	0.937999i	$-0.612677\pi$
$839$	−16848.0	−0.693275	−0.346637	+	0.937999i	$0.612677\pi$
$840$	0	0
$841$	26687.0	1.09422
$842$	0	0
$843$	16062.0	0.656233
$844$	0	0
$845$	0	0
$846$	0	0
$847$	27228.0	1.10456
$848$	0	0
$849$	−2340.00	−0.0945920
$850$	0	0
$851$	−17376.0	−0.699931
$852$	0	0
$853$	−18214.0	−0.731108	−0.365554	−	0.930790i	$-0.619121\pi$
$853$	−18214.0	−0.731108	−0.365554	+	0.930790i	$0.619121\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2446.00	0.0974956	0.0487478	−	0.998811i	$-0.484477\pi$
$857$	2446.00	0.0974956	0.0487478	+	0.998811i	$0.484477\pi$
$858$	0	0
$859$	26244.0	1.04241	0.521207	−	0.853430i	$-0.325482\pi$
$859$	26244.0	1.04241	0.521207	+	0.853430i	$0.325482\pi$
$860$	0	0
$861$	−3384.00	−0.133945
$862$	0	0
$863$	25248.0	0.995889	0.497944	−	0.867209i	$-0.334088\pi$
$863$	25248.0	0.995889	0.497944	+	0.867209i	$0.334088\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−14439.0	−0.565599
$868$	0	0
$869$	36720.0	1.43342
$870$	0	0
$871$	8568.00	0.333313
$872$	0	0
$873$	−5346.00	−0.207256
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.0000	0.00130912	0.000654560	−	1.00000i	$-0.499792\pi$
$877$	34.0000	0.00130912	0.000654560		1.00000i	$0.499792\pi$
$878$	0	0
$879$	10050.0	0.385641
$880$	0	0
$881$	−19022.0	−0.727432	−0.363716	−	0.931510i	$-0.618492\pi$
$881$	−19022.0	−0.727432	−0.363716	+	0.931510i	$0.618492\pi$
$882$	0	0
$883$	12852.0	0.489812	0.244906	−	0.969547i	$-0.421243\pi$
$883$	12852.0	0.489812	0.244906	+	0.969547i	$0.421243\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40104.0	1.51811	0.759053	−	0.651028i	$-0.225662\pi$
$887$	40104.0	1.51811	0.759053	+	0.651028i	$0.225662\pi$
$888$	0	0
$889$	−28656.0	−1.08109
$890$	0	0
$891$	−4860.00	−0.182734
$892$	0	0
$893$	53856.0	2.01817
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6048.00	−0.225125
$898$	0	0
$899$	56952.0	2.11285
$900$	0	0
$901$	3460.00	0.127935
$902$	0	0
$903$	−8208.00	−0.302486
$904$	0	0
$905$	0	0
$906$	0	0
$907$	42540.0	1.55735	0.778676	−	0.627427i	$-0.215892\pi$
$907$	42540.0	1.55735	0.778676	+	0.627427i	$0.215892\pi$
$908$	0	0
$909$	4986.00	0.181931
$910$	0	0
$911$	18528.0	0.673831	0.336915	−	0.941535i	$-0.390616\pi$
$911$	18528.0	0.673831	0.336915	+	0.941535i	$0.390616\pi$
$912$	0	0
$913$	−33840.0	−1.22666
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4752.00	−0.171129
$918$	0	0
$919$	−15756.0	−0.565552	−0.282776	−	0.959186i	$-0.591255\pi$
$919$	−15756.0	−0.565552	−0.282776	+	0.959186i	$0.591255\pi$
$920$	0	0
$921$	−28908.0	−1.03426
$922$	0	0
$923$	−44352.0	−1.58165
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11556.0	0.409438
$928$	0	0
$929$	−15542.0	−0.548887	−0.274444	−	0.961603i	$-0.588494\pi$
$929$	−15542.0	−0.548887	−0.274444	+	0.961603i	$0.588494\pi$
$930$	0	0
$931$	26268.0	0.924703
$932$	0	0
$933$	22680.0	0.795831
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29702.0	1.03556	0.517781	−	0.855513i	$-0.326758\pi$
$937$	29702.0	1.03556	0.517781	+	0.855513i	$0.326758\pi$
$938$	0	0
$939$	10578.0	0.367625
$940$	0	0
$941$	2890.00	0.100118	0.0500591	−	0.998746i	$-0.484059\pi$
$941$	2890.00	0.100118	0.0500591	+	0.998746i	$0.484059\pi$
$942$	0	0
$943$	4512.00	0.155812
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9180.00	0.315005	0.157503	−	0.987519i	$-0.449656\pi$
$947$	9180.00	0.315005	0.157503	+	0.987519i	$0.449656\pi$
$948$	0	0
$949$	−13860.0	−0.474093
$950$	0	0
$951$	−22902.0	−0.780913
$952$	0	0
$953$	−37906.0	−1.28845	−0.644227	−	0.764835i	$-0.722821\pi$
$953$	−37906.0	−1.28845	−0.644227	+	0.764835i	$0.722821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−40680.0	−1.37408
$958$	0	0
$959$	1320.00	0.0444474
$960$	0	0
$961$	33713.0	1.13165
$962$	0	0
$963$	−12204.0	−0.408378
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−41916.0	−1.39393	−0.696964	−	0.717106i	$-0.745466\pi$
$967$	−41916.0	−1.39393	−0.696964	+	0.717106i	$0.745466\pi$
$968$	0	0
$969$	3960.00	0.131283
$970$	0	0
$971$	7764.00	0.256600	0.128300	−	0.991735i	$-0.459048\pi$
$971$	7764.00	0.256600	0.128300	+	0.991735i	$0.459048\pi$
$972$	0	0
$973$	8784.00	0.289416
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32666.0	−1.06968	−0.534840	−	0.844953i	$-0.679628\pi$
$977$	−32666.0	−1.06968	−0.534840	+	0.844953i	$0.679628\pi$
$978$	0	0
$979$	90600.0	2.95770
$980$	0	0
$981$	3510.00	0.114236
$982$	0	0
$983$	−53016.0	−1.72019	−0.860096	−	0.510133i	$-0.829596\pi$
$983$	−53016.0	−1.72019	−0.860096	+	0.510133i	$0.829596\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−14688.0	−0.473682
$988$	0	0
$989$	10944.0	0.351870
$990$	0	0
$991$	−17844.0	−0.571981	−0.285991	−	0.958232i	$-0.592323\pi$
$991$	−17844.0	−0.571981	−0.285991	+	0.958232i	$0.592323\pi$
$992$	0	0
$993$	−22716.0	−0.725952
$994$	0	0
$995$	0	0
$996$	0	0
$997$	55834.0	1.77360	0.886801	−	0.462152i	$-0.152923\pi$
$997$	55834.0	1.77360	0.886801	+	0.462152i	$0.152923\pi$
$998$	0	0
$999$	9774.00	0.309545

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	2400.4.a.t.1.1		1
4.3	odd	2		2400.4.a.c.1.1		1
5.4	even	2		96.4.a.b.1.1	✓	1
15.14	odd	2		288.4.a.e.1.1		1
20.19	odd	2		96.4.a.e.1.1	yes	1
40.19	odd	2		192.4.a.d.1.1		1
40.29	even	2		192.4.a.j.1.1		1
60.59	even	2		288.4.a.f.1.1		1
80.19	odd	4		768.4.d.d.385.2		2
80.29	even	4		768.4.d.m.385.1		2
80.59	odd	4		768.4.d.d.385.1		2
80.69	even	4		768.4.d.m.385.2		2
120.29	odd	2		576.4.a.n.1.1		1
120.59	even	2		576.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.4.a.b.1.1	✓	1	5.4	even	2
96.4.a.e.1.1	yes	1	20.19	odd	2
192.4.a.d.1.1		1	40.19	odd	2
192.4.a.j.1.1		1	40.29	even	2
288.4.a.e.1.1		1	15.14	odd	2
288.4.a.f.1.1		1	60.59	even	2
576.4.a.n.1.1		1	120.29	odd	2
576.4.a.o.1.1		1	120.59	even	2
768.4.d.d.385.1		2	80.59	odd	4
768.4.d.d.385.2		2	80.19	odd	4
768.4.d.m.385.1		2	80.29	even	4
768.4.d.m.385.2		2	80.69	even	4
2400.4.a.c.1.1		1	4.3	odd	2
2400.4.a.t.1.1		1	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

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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	2400.4.a.t.1.1

Level	$2400$
Weight	$4$
Character	2400.1
Self dual	yes
Analytic conductor	$141.605$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$