LMFDB - Embedded newform 1425.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,4,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1425.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1425.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:	$84.0777217582$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} +4.00000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} +4.00000 q^{7} +15.0000 q^{8} +9.00000 q^{9} -68.0000 q^{11} +21.0000 q^{12} -82.0000 q^{13} -4.00000 q^{14} +41.0000 q^{16} -86.0000 q^{17} -9.00000 q^{18} +19.0000 q^{19} -12.0000 q^{21} +68.0000 q^{22} +18.0000 q^{23} -45.0000 q^{24} +82.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +30.0000 q^{29} -298.000 q^{31} -161.000 q^{32} +204.000 q^{33} +86.0000 q^{34} -63.0000 q^{36} +34.0000 q^{37} -19.0000 q^{38} +246.000 q^{39} +52.0000 q^{41} +12.0000 q^{42} -482.000 q^{43} +476.000 q^{44} -18.0000 q^{46} +114.000 q^{47} -123.000 q^{48} -327.000 q^{49} +258.000 q^{51} +574.000 q^{52} -362.000 q^{53} +27.0000 q^{54} +60.0000 q^{56} -57.0000 q^{57} -30.0000 q^{58} -210.000 q^{59} -718.000 q^{61} +298.000 q^{62} +36.0000 q^{63} -167.000 q^{64} -204.000 q^{66} +904.000 q^{67} +602.000 q^{68} -54.0000 q^{69} -988.000 q^{71} +135.000 q^{72} +488.000 q^{73} -34.0000 q^{74} -133.000 q^{76} -272.000 q^{77} -246.000 q^{78} -530.000 q^{79} +81.0000 q^{81} -52.0000 q^{82} -1032.00 q^{83} +84.0000 q^{84} +482.000 q^{86} -90.0000 q^{87} -1020.00 q^{88} -880.000 q^{89} -328.000 q^{91} -126.000 q^{92} +894.000 q^{93} -114.000 q^{94} +483.000 q^{96} -246.000 q^{97} +327.000 q^{98} -612.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.00000	−0.875000
$5$	0	0
$5$	0	0
$6$	3.00000	0.204124
$7$	4.00000	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$7$	4.00000	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$8$	15.0000	0.662913
$9$	9.00000	0.333333
$10$	0	0
$11$	−68.0000	−1.86389	−0.931944	−	0.362602i	$-0.881889\pi$
$11$	−68.0000	−1.86389	−0.931944	+	0.362602i	$0.881889\pi$
$12$	21.0000	0.505181
$13$	−82.0000	−1.74944	−0.874720	−	0.484629i	$-0.838954\pi$
$13$	−82.0000	−1.74944	−0.874720	+	0.484629i	$0.838954\pi$
$14$	−4.00000	−0.0763604
$15$	0	0
$16$	41.0000	0.640625
$17$	−86.0000	−1.22694	−0.613472	−	0.789716i	$-0.710228\pi$
$17$	−86.0000	−1.22694	−0.613472	+	0.789716i	$0.710228\pi$
$18$	−9.00000	−0.117851
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−12.0000	−0.124696
$22$	68.0000	0.658984
$23$	18.0000	0.163185	0.0815926	−	0.996666i	$-0.473999\pi$
$23$	18.0000	0.163185	0.0815926	+	0.996666i	$0.473999\pi$
$24$	−45.0000	−0.382733
$25$	0	0
$26$	82.0000	0.618520
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	0	0
$31$	−298.000	−1.72653	−0.863264	−	0.504752i	$-0.831584\pi$
$31$	−298.000	−1.72653	−0.863264	+	0.504752i	$0.831584\pi$
$32$	−161.000	−0.889408
$33$	204.000	1.07612
$34$	86.0000	0.433791
$35$	0	0
$36$	−63.0000	−0.291667
$37$	34.0000	0.151069	0.0755347	−	0.997143i	$-0.475934\pi$
$37$	34.0000	0.151069	0.0755347	+	0.997143i	$0.475934\pi$
$38$	−19.0000	−0.0811107
$39$	246.000	1.01004
$40$	0	0
$41$	52.0000	0.198074	0.0990370	−	0.995084i	$-0.468424\pi$
$41$	52.0000	0.198074	0.0990370	+	0.995084i	$0.468424\pi$
$42$	12.0000	0.0440867
$43$	−482.000	−1.70940	−0.854701	−	0.519120i	$-0.826260\pi$
$43$	−482.000	−1.70940	−0.854701	+	0.519120i	$0.826260\pi$
$44$	476.000	1.63090
$45$	0	0
$46$	−18.0000	−0.0576947
$47$	114.000	0.353800	0.176900	−	0.984229i	$-0.443393\pi$
$47$	114.000	0.353800	0.176900	+	0.984229i	$0.443393\pi$
$48$	−123.000	−0.369865
$49$	−327.000	−0.953353
$50$	0	0
$51$	258.000	0.708377
$52$	574.000	1.53076
$53$	−362.000	−0.938199	−0.469099	−	0.883145i	$-0.655421\pi$
$53$	−362.000	−0.938199	−0.469099	+	0.883145i	$0.655421\pi$
$54$	27.0000	0.0680414
$55$	0	0
$56$	60.0000	0.143176
$57$	−57.0000	−0.132453
$58$	−30.0000	−0.0679171
$59$	−210.000	−0.463384	−0.231692	−	0.972789i	$-0.574426\pi$
$59$	−210.000	−0.463384	−0.231692	+	0.972789i	$0.574426\pi$
$60$	0	0
$61$	−718.000	−1.50706	−0.753529	−	0.657415i	$-0.771650\pi$
$61$	−718.000	−1.50706	−0.753529	+	0.657415i	$0.771650\pi$
$62$	298.000	0.610420
$63$	36.0000	0.0719932
$64$	−167.000	−0.326172
$65$	0	0
$66$	−204.000	−0.380465
$67$	904.000	1.64838	0.824188	−	0.566316i	$-0.191632\pi$
$67$	904.000	1.64838	0.824188	+	0.566316i	$0.191632\pi$
$68$	602.000	1.07358
$69$	−54.0000	−0.0942150
$70$	0	0
$71$	−988.000	−1.65147	−0.825733	−	0.564062i	$-0.809238\pi$
$71$	−988.000	−1.65147	−0.825733	+	0.564062i	$0.809238\pi$
$72$	135.000	0.220971
$73$	488.000	0.782412	0.391206	−	0.920303i	$-0.372058\pi$
$73$	488.000	0.782412	0.391206	+	0.920303i	$0.372058\pi$
$74$	−34.0000	−0.0534111
$75$	0	0
$76$	−133.000	−0.200739
$77$	−272.000	−0.402562
$78$	−246.000	−0.357103
$79$	−530.000	−0.754806	−0.377403	−	0.926049i	$-0.623183\pi$
$79$	−530.000	−0.754806	−0.377403	+	0.926049i	$0.623183\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−52.0000	−0.0700297
$83$	−1032.00	−1.36478	−0.682390	−	0.730988i	$-0.739059\pi$
$83$	−1032.00	−1.36478	−0.682390	+	0.730988i	$0.739059\pi$
$84$	84.0000	0.109109
$85$	0	0
$86$	482.000	0.604365
$87$	−90.0000	−0.110908
$88$	−1020.00	−1.23560
$89$	−880.000	−1.04809	−0.524044	−	0.851691i	$-0.675577\pi$
$89$	−880.000	−1.04809	−0.524044	+	0.851691i	$0.675577\pi$
$90$	0	0
$91$	−328.000	−0.377843
$92$	−126.000	−0.142787
$93$	894.000	0.996812
$94$	−114.000	−0.125087
$95$	0	0
$96$	483.000	0.513500
$97$	−246.000	−0.257500	−0.128750	−	0.991677i	$-0.541097\pi$
$97$	−246.000	−0.257500	−0.128750	+	0.991677i	$0.541097\pi$
$98$	327.000	0.337061
$99$	−612.000	−0.621296
$100$	0	0
$101$	−858.000	−0.845289	−0.422645	−	0.906296i	$-0.638898\pi$
$101$	−858.000	−0.845289	−0.422645	+	0.906296i	$0.638898\pi$
$102$	−258.000	−0.250449
$103$	1088.00	1.04081	0.520407	−	0.853918i	$-0.325780\pi$
$103$	1088.00	1.04081	0.520407	+	0.853918i	$0.325780\pi$
$104$	−1230.00	−1.15973
$105$	0	0
$106$	362.000	0.331703
$107$	−436.000	−0.393923	−0.196961	−	0.980411i	$-0.563107\pi$
$107$	−436.000	−0.393923	−0.196961	+	0.980411i	$0.563107\pi$
$108$	189.000	0.168394
$109$	−1680.00	−1.47628	−0.738141	−	0.674646i	$-0.764296\pi$
$109$	−1680.00	−1.47628	−0.738141	+	0.674646i	$0.764296\pi$
$110$	0	0
$111$	−102.000	−0.0872199
$112$	164.000	0.138362
$113$	−762.000	−0.634362	−0.317181	−	0.948365i	$-0.602736\pi$
$113$	−762.000	−0.634362	−0.317181	+	0.948365i	$0.602736\pi$
$114$	57.0000	0.0468293
$115$	0	0
$116$	−210.000	−0.168086
$117$	−738.000	−0.583146
$118$	210.000	0.163831
$119$	−344.000	−0.264995
$120$	0	0
$121$	3293.00	2.47408
$122$	718.000	0.532825
$123$	−156.000	−0.114358
$124$	2086.00	1.51071
$125$	0	0
$126$	−36.0000	−0.0254535
$127$	624.000	0.435992	0.217996	−	0.975950i	$-0.430048\pi$
$127$	624.000	0.435992	0.217996	+	0.975950i	$0.430048\pi$
$128$	1455.00	1.00473
$129$	1446.00	0.986924
$130$	0	0
$131$	1152.00	0.768326	0.384163	−	0.923265i	$-0.374490\pi$
$131$	1152.00	0.768326	0.384163	+	0.923265i	$0.374490\pi$
$132$	−1428.00	−0.941602
$133$	76.0000	0.0495491
$134$	−904.000	−0.582789
$135$	0	0
$136$	−1290.00	−0.813357
$137$	1254.00	0.782018	0.391009	−	0.920387i	$-0.372126\pi$
$137$	1254.00	0.782018	0.391009	+	0.920387i	$0.372126\pi$
$138$	54.0000	0.0333100
$139$	−1300.00	−0.793270	−0.396635	−	0.917976i	$-0.629822\pi$
$139$	−1300.00	−0.793270	−0.396635	+	0.917976i	$0.629822\pi$
$140$	0	0
$141$	−342.000	−0.204267
$142$	988.000	0.583881
$143$	5576.00	3.26076
$144$	369.000	0.213542
$145$	0	0
$146$	−488.000	−0.276624
$147$	981.000	0.550418
$148$	−238.000	−0.132186
$149$	1810.00	0.995174	0.497587	−	0.867414i	$-0.334219\pi$
$149$	1810.00	0.995174	0.497587	+	0.867414i	$0.334219\pi$
$150$	0	0
$151$	1662.00	0.895706	0.447853	−	0.894107i	$-0.352189\pi$
$151$	1662.00	0.895706	0.447853	+	0.894107i	$0.352189\pi$
$152$	285.000	0.152083
$153$	−774.000	−0.408982
$154$	272.000	0.142327
$155$	0	0
$156$	−1722.00	−0.883784
$157$	−706.000	−0.358885	−0.179442	−	0.983768i	$-0.557429\pi$
$157$	−706.000	−0.358885	−0.179442	+	0.983768i	$0.557429\pi$
$158$	530.000	0.266864
$159$	1086.00	0.541669
$160$	0	0
$161$	72.0000	0.0352447
$162$	−81.0000	−0.0392837
$163$	−1462.00	−0.702532	−0.351266	−	0.936276i	$-0.614249\pi$
$163$	−1462.00	−0.702532	−0.351266	+	0.936276i	$0.614249\pi$
$164$	−364.000	−0.173315
$165$	0	0
$166$	1032.00	0.482522
$167$	624.000	0.289141	0.144571	−	0.989494i	$-0.453820\pi$
$167$	624.000	0.289141	0.144571	+	0.989494i	$0.453820\pi$
$168$	−180.000	−0.0826625
$169$	4527.00	2.06054
$170$	0	0
$171$	171.000	0.0764719
$172$	3374.00	1.49573
$173$	3198.00	1.40543	0.702715	−	0.711471i	$-0.251971\pi$
$173$	3198.00	1.40543	0.702715	+	0.711471i	$0.251971\pi$
$174$	90.0000	0.0392120
$175$	0	0
$176$	−2788.00	−1.19405
$177$	630.000	0.267535
$178$	880.000	0.370555
$179$	−1710.00	−0.714030	−0.357015	−	0.934099i	$-0.616206\pi$
$179$	−1710.00	−0.714030	−0.357015	+	0.934099i	$0.616206\pi$
$180$	0	0
$181$	312.000	0.128126	0.0640629	−	0.997946i	$-0.479594\pi$
$181$	312.000	0.128126	0.0640629	+	0.997946i	$0.479594\pi$
$182$	328.000	0.133588
$183$	2154.00	0.870100
$184$	270.000	0.108178
$185$	0	0
$186$	−894.000	−0.352426
$187$	5848.00	2.28689
$188$	−798.000	−0.309575
$189$	−108.000	−0.0415653
$190$	0	0
$191$	2472.00	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$191$	2472.00	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$192$	501.000	0.188315
$193$	−2422.00	−0.903313	−0.451656	−	0.892192i	$-0.649167\pi$
$193$	−2422.00	−0.903313	−0.451656	+	0.892192i	$0.649167\pi$
$194$	246.000	0.0910401
$195$	0	0
$196$	2289.00	0.834184
$197$	3984.00	1.44085	0.720427	−	0.693531i	$-0.243946\pi$
$197$	3984.00	1.44085	0.720427	+	0.693531i	$0.243946\pi$
$198$	612.000	0.219661
$199$	−2000.00	−0.712443	−0.356222	−	0.934401i	$-0.615935\pi$
$199$	−2000.00	−0.712443	−0.356222	+	0.934401i	$0.615935\pi$
$200$	0	0
$201$	−2712.00	−0.951690
$202$	858.000	0.298855
$203$	120.000	0.0414894
$204$	−1806.00	−0.619830
$205$	0	0
$206$	−1088.00	−0.367983
$207$	162.000	0.0543951
$208$	−3362.00	−1.12073
$209$	−1292.00	−0.427605
$210$	0	0
$211$	−4768.00	−1.55565	−0.777826	−	0.628479i	$-0.783678\pi$
$211$	−4768.00	−1.55565	−0.777826	+	0.628479i	$0.783678\pi$
$212$	2534.00	0.820924
$213$	2964.00	0.953474
$214$	436.000	0.139273
$215$	0	0
$216$	−405.000	−0.127578
$217$	−1192.00	−0.372895
$218$	1680.00	0.521945
$219$	−1464.00	−0.451726
$220$	0	0
$221$	7052.00	2.14647
$222$	102.000	0.0308369
$223$	3248.00	0.975346	0.487673	−	0.873026i	$-0.337846\pi$
$223$	3248.00	0.975346	0.487673	+	0.873026i	$0.337846\pi$
$224$	−644.000	−0.192094
$225$	0	0
$226$	762.000	0.224281
$227$	5444.00	1.59177	0.795883	−	0.605450i	$-0.207007\pi$
$227$	5444.00	1.59177	0.795883	+	0.605450i	$0.207007\pi$
$228$	399.000	0.115897
$229$	−3490.00	−1.00710	−0.503550	−	0.863966i	$-0.667973\pi$
$229$	−3490.00	−1.00710	−0.503550	+	0.863966i	$0.667973\pi$
$230$	0	0
$231$	816.000	0.232419
$232$	450.000	0.127345
$233$	−4202.00	−1.18147	−0.590734	−	0.806866i	$-0.701162\pi$
$233$	−4202.00	−1.18147	−0.590734	+	0.806866i	$0.701162\pi$
$234$	738.000	0.206173
$235$	0	0
$236$	1470.00	0.405461
$237$	1590.00	0.435787
$238$	344.000	0.0936899
$239$	6400.00	1.73214	0.866070	−	0.499922i	$-0.166638\pi$
$239$	6400.00	1.73214	0.866070	+	0.499922i	$0.166638\pi$
$240$	0	0
$241$	3142.00	0.839809	0.419905	−	0.907568i	$-0.362064\pi$
$241$	3142.00	0.839809	0.419905	+	0.907568i	$0.362064\pi$
$242$	−3293.00	−0.874719
$243$	−243.000	−0.0641500
$244$	5026.00	1.31867
$245$	0	0
$246$	156.000	0.0404317
$247$	−1558.00	−0.401349
$248$	−4470.00	−1.14454
$249$	3096.00	0.787956
$250$	0	0
$251$	−3988.00	−1.00287	−0.501435	−	0.865195i	$-0.667194\pi$
$251$	−3988.00	−1.00287	−0.501435	+	0.865195i	$0.667194\pi$
$252$	−252.000	−0.0629941
$253$	−1224.00	−0.304159
$254$	−624.000	−0.154147
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−606.000	−0.147087	−0.0735433	−	0.997292i	$-0.523431\pi$
$257$	−606.000	−0.147087	−0.0735433	+	0.997292i	$0.523431\pi$
$258$	−1446.00	−0.348930
$259$	136.000	0.0326279
$260$	0	0
$261$	270.000	0.0640329
$262$	−1152.00	−0.271644
$263$	−2262.00	−0.530346	−0.265173	−	0.964201i	$-0.585429\pi$
$263$	−2262.00	−0.530346	−0.265173	+	0.964201i	$0.585429\pi$
$264$	3060.00	0.713371
$265$	0	0
$266$	−76.0000	−0.0175183
$267$	2640.00	0.605114
$268$	−6328.00	−1.44233
$269$	1170.00	0.265190	0.132595	−	0.991170i	$-0.457669\pi$
$269$	1170.00	0.265190	0.132595	+	0.991170i	$0.457669\pi$
$270$	0	0
$271$	−7728.00	−1.73226	−0.866130	−	0.499818i	$-0.833400\pi$
$271$	−7728.00	−1.73226	−0.866130	+	0.499818i	$0.833400\pi$
$272$	−3526.00	−0.786012
$273$	984.000	0.218148
$274$	−1254.00	−0.276485
$275$	0	0
$276$	378.000	0.0824381
$277$	−3386.00	−0.734459	−0.367229	−	0.930130i	$-0.619694\pi$
$277$	−3386.00	−0.734459	−0.367229	+	0.930130i	$0.619694\pi$
$278$	1300.00	0.280463
$279$	−2682.00	−0.575509
$280$	0	0
$281$	−1708.00	−0.362600	−0.181300	−	0.983428i	$-0.558031\pi$
$281$	−1708.00	−0.362600	−0.181300	+	0.983428i	$0.558031\pi$
$282$	342.000	0.0722192
$283$	7378.00	1.54974	0.774870	−	0.632120i	$-0.217815\pi$
$283$	7378.00	1.54974	0.774870	+	0.632120i	$0.217815\pi$
$284$	6916.00	1.44503
$285$	0	0
$286$	−5576.00	−1.15285
$287$	208.000	0.0427800
$288$	−1449.00	−0.296469
$289$	2483.00	0.505394
$290$	0	0
$291$	738.000	0.148668
$292$	−3416.00	−0.684611
$293$	2038.00	0.406352	0.203176	−	0.979142i	$-0.434874\pi$
$293$	2038.00	0.406352	0.203176	+	0.979142i	$0.434874\pi$
$294$	−981.000	−0.194602
$295$	0	0
$296$	510.000	0.100146
$297$	1836.00	0.358705
$298$	−1810.00	−0.351847
$299$	−1476.00	−0.285483
$300$	0	0
$301$	−1928.00	−0.369196
$302$	−1662.00	−0.316680
$303$	2574.00	0.488028
$304$	779.000	0.146969
$305$	0	0
$306$	774.000	0.144597
$307$	−1636.00	−0.304142	−0.152071	−	0.988370i	$-0.548594\pi$
$307$	−1636.00	−0.304142	−0.152071	+	0.988370i	$0.548594\pi$
$308$	1904.00	0.352242
$309$	−3264.00	−0.600914
$310$	0	0
$311$	−1848.00	−0.336947	−0.168473	−	0.985706i	$-0.553884\pi$
$311$	−1848.00	−0.336947	−0.168473	+	0.985706i	$0.553884\pi$
$312$	3690.00	0.669568
$313$	4908.00	0.886315	0.443157	−	0.896444i	$-0.353858\pi$
$313$	4908.00	0.886315	0.443157	+	0.896444i	$0.353858\pi$
$314$	706.000	0.126885
$315$	0	0
$316$	3710.00	0.660455
$317$	−4526.00	−0.801910	−0.400955	−	0.916098i	$-0.631322\pi$
$317$	−4526.00	−0.801910	−0.400955	+	0.916098i	$0.631322\pi$
$318$	−1086.00	−0.191509
$319$	−2040.00	−0.358050
$320$	0	0
$321$	1308.00	0.227431
$322$	−72.0000	−0.0124609
$323$	−1634.00	−0.281480
$324$	−567.000	−0.0972222
$325$	0	0
$326$	1462.00	0.248382
$327$	5040.00	0.852332
$328$	780.000	0.131306
$329$	456.000	0.0764137
$330$	0	0
$331$	−6388.00	−1.06077	−0.530387	−	0.847756i	$-0.677953\pi$
$331$	−6388.00	−1.06077	−0.530387	+	0.847756i	$0.677953\pi$
$332$	7224.00	1.19418
$333$	306.000	0.0503564
$334$	−624.000	−0.102227
$335$	0	0
$336$	−492.000	−0.0798833
$337$	−286.000	−0.0462297	−0.0231149	−	0.999733i	$-0.507358\pi$
$337$	−286.000	−0.0462297	−0.0231149	+	0.999733i	$0.507358\pi$
$338$	−4527.00	−0.728510
$339$	2286.00	0.366249
$340$	0	0
$341$	20264.0	3.21806
$342$	−171.000	−0.0270369
$343$	−2680.00	−0.421885
$344$	−7230.00	−1.13318
$345$	0	0
$346$	−3198.00	−0.496895
$347$	8904.00	1.37750	0.688749	−	0.725000i	$-0.258160\pi$
$347$	8904.00	1.37750	0.688749	+	0.725000i	$0.258160\pi$
$348$	630.000	0.0970447
$349$	7670.00	1.17641	0.588203	−	0.808713i	$-0.299836\pi$
$349$	7670.00	1.17641	0.588203	+	0.808713i	$0.299836\pi$
$350$	0	0
$351$	2214.00	0.336680
$352$	10948.0	1.65776
$353$	−2.00000	−0.000301556 0	−0.000150778	−	1.00000i	$-0.500048\pi$
$353$	−2.00000	−0.000301556 0	−0.000150778		1.00000i	$0.500048\pi$
$354$	−630.000	−0.0945879
$355$	0	0
$356$	6160.00	0.917077
$357$	1032.00	0.152995
$358$	1710.00	0.252448
$359$	−1120.00	−0.164656	−0.0823278	−	0.996605i	$-0.526235\pi$
$359$	−1120.00	−0.164656	−0.0823278	+	0.996605i	$0.526235\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−312.000	−0.0452993
$363$	−9879.00	−1.42841
$364$	2296.00	0.330613
$365$	0	0
$366$	−2154.00	−0.307627
$367$	1984.00	0.282191	0.141095	−	0.989996i	$-0.454938\pi$
$367$	1984.00	0.282191	0.141095	+	0.989996i	$0.454938\pi$
$368$	738.000	0.104541
$369$	468.000	0.0660247
$370$	0	0
$371$	−1448.00	−0.202632
$372$	−6258.00	−0.872210
$373$	2058.00	0.285682	0.142841	−	0.989746i	$-0.454376\pi$
$373$	2058.00	0.285682	0.142841	+	0.989746i	$0.454376\pi$
$374$	−5848.00	−0.808537
$375$	0	0
$376$	1710.00	0.234539
$377$	−2460.00	−0.336065
$378$	108.000	0.0146956
$379$	−10640.0	−1.44206	−0.721029	−	0.692905i	$-0.756331\pi$
$379$	−10640.0	−1.44206	−0.721029	+	0.692905i	$0.756331\pi$
$380$	0	0
$381$	−1872.00	−0.251720
$382$	−2472.00	−0.331096
$383$	−5892.00	−0.786076	−0.393038	−	0.919522i	$-0.628576\pi$
$383$	−5892.00	−0.786076	−0.393038	+	0.919522i	$0.628576\pi$
$384$	−4365.00	−0.580079
$385$	0	0
$386$	2422.00	0.319369
$387$	−4338.00	−0.569801
$388$	1722.00	0.225313
$389$	990.000	0.129036	0.0645180	−	0.997917i	$-0.479449\pi$
$389$	990.000	0.129036	0.0645180	+	0.997917i	$0.479449\pi$
$390$	0	0
$391$	−1548.00	−0.200219
$392$	−4905.00	−0.631990
$393$	−3456.00	−0.443593
$394$	−3984.00	−0.509419
$395$	0	0
$396$	4284.00	0.543634
$397$	−526.000	−0.0664967	−0.0332483	−	0.999447i	$-0.510585\pi$
$397$	−526.000	−0.0664967	−0.0332483	+	0.999447i	$0.510585\pi$
$398$	2000.00	0.251887
$399$	−228.000	−0.0286072
$400$	0	0
$401$	9612.00	1.19701	0.598504	−	0.801120i	$-0.295762\pi$
$401$	9612.00	1.19701	0.598504	+	0.801120i	$0.295762\pi$
$402$	2712.00	0.336473
$403$	24436.0	3.02046
$404$	6006.00	0.739628
$405$	0	0
$406$	−120.000	−0.0146687
$407$	−2312.00	−0.281576
$408$	3870.00	0.469592
$409$	−9110.00	−1.10137	−0.550685	−	0.834713i	$-0.685634\pi$
$409$	−9110.00	−1.10137	−0.550685	+	0.834713i	$0.685634\pi$
$410$	0	0
$411$	−3762.00	−0.451498
$412$	−7616.00	−0.910712
$413$	−840.000	−0.100082
$414$	−162.000	−0.0192316
$415$	0	0
$416$	13202.0	1.55596
$417$	3900.00	0.457995
$418$	1292.00	0.151181
$419$	13920.0	1.62300	0.811499	−	0.584353i	$-0.198652\pi$
$419$	13920.0	1.62300	0.811499	+	0.584353i	$0.198652\pi$
$420$	0	0
$421$	−9588.00	−1.10995	−0.554977	−	0.831866i	$-0.687273\pi$
$421$	−9588.00	−1.10995	−0.554977	+	0.831866i	$0.687273\pi$
$422$	4768.00	0.550006
$423$	1026.00	0.117933
$424$	−5430.00	−0.621944
$425$	0	0
$426$	−2964.00	−0.337104
$427$	−2872.00	−0.325494
$428$	3052.00	0.344682
$429$	−16728.0	−1.88260
$430$	0	0
$431$	−4968.00	−0.555221	−0.277610	−	0.960694i	$-0.589542\pi$
$431$	−4968.00	−0.555221	−0.277610	+	0.960694i	$0.589542\pi$
$432$	−1107.00	−0.123288
$433$	−11342.0	−1.25880	−0.629402	−	0.777080i	$-0.716700\pi$
$433$	−11342.0	−1.25880	−0.629402	+	0.777080i	$0.716700\pi$
$434$	1192.00	0.131838
$435$	0	0
$436$	11760.0	1.29175
$437$	342.000	0.0374373
$438$	1464.00	0.159709
$439$	3710.00	0.403345	0.201673	−	0.979453i	$-0.435362\pi$
$439$	3710.00	0.403345	0.201673	+	0.979453i	$0.435362\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	−7052.00	−0.758890
$443$	−10772.0	−1.15529	−0.577645	−	0.816288i	$-0.696028\pi$
$443$	−10772.0	−1.15529	−0.577645	+	0.816288i	$0.696028\pi$
$444$	714.000	0.0763174
$445$	0	0
$446$	−3248.00	−0.344837
$447$	−5430.00	−0.574564
$448$	−668.000	−0.0704465
$449$	−1720.00	−0.180784	−0.0903918	−	0.995906i	$-0.528812\pi$
$449$	−1720.00	−0.180784	−0.0903918	+	0.995906i	$0.528812\pi$
$450$	0	0
$451$	−3536.00	−0.369188
$452$	5334.00	0.555067
$453$	−4986.00	−0.517136
$454$	−5444.00	−0.562774
$455$	0	0
$456$	−855.000	−0.0878049
$457$	−1456.00	−0.149035	−0.0745173	−	0.997220i	$-0.523742\pi$
$457$	−1456.00	−0.149035	−0.0745173	+	0.997220i	$0.523742\pi$
$458$	3490.00	0.356063
$459$	2322.00	0.236126
$460$	0	0
$461$	−6218.00	−0.628202	−0.314101	−	0.949390i	$-0.601703\pi$
$461$	−6218.00	−0.628202	−0.314101	+	0.949390i	$0.601703\pi$
$462$	−816.000	−0.0821726
$463$	−13892.0	−1.39442	−0.697209	−	0.716867i	$-0.745575\pi$
$463$	−13892.0	−1.39442	−0.697209	+	0.716867i	$0.745575\pi$
$464$	1230.00	0.123063
$465$	0	0
$466$	4202.00	0.417712
$467$	6004.00	0.594929	0.297465	−	0.954733i	$-0.403859\pi$
$467$	6004.00	0.594929	0.297465	+	0.954733i	$0.403859\pi$
$468$	5166.00	0.510253
$469$	3616.00	0.356016
$470$	0	0
$471$	2118.00	0.207202
$472$	−3150.00	−0.307183
$473$	32776.0	3.18614
$474$	−1590.00	−0.154074
$475$	0	0
$476$	2408.00	0.231871
$477$	−3258.00	−0.312733
$478$	−6400.00	−0.612404
$479$	4800.00	0.457866	0.228933	−	0.973442i	$-0.426476\pi$
$479$	4800.00	0.457866	0.228933	+	0.973442i	$0.426476\pi$
$480$	0	0
$481$	−2788.00	−0.264287
$482$	−3142.00	−0.296917
$483$	−216.000	−0.0203485
$484$	−23051.0	−2.16482
$485$	0	0
$486$	243.000	0.0226805
$487$	−12616.0	−1.17389	−0.586946	−	0.809626i	$-0.699670\pi$
$487$	−12616.0	−1.17389	−0.586946	+	0.809626i	$0.699670\pi$
$488$	−10770.0	−0.999047
$489$	4386.00	0.405607
$490$	0	0
$491$	4232.00	0.388977	0.194488	−	0.980905i	$-0.437695\pi$
$491$	4232.00	0.388977	0.194488	+	0.980905i	$0.437695\pi$
$492$	1092.00	0.100063
$493$	−2580.00	−0.235694
$494$	1558.00	0.141898
$495$	0	0
$496$	−12218.0	−1.10606
$497$	−3952.00	−0.356683
$498$	−3096.00	−0.278584
$499$	6340.00	0.568772	0.284386	−	0.958710i	$-0.408210\pi$
$499$	6340.00	0.568772	0.284386	+	0.958710i	$0.408210\pi$
$500$	0	0
$501$	−1872.00	−0.166936
$502$	3988.00	0.354568
$503$	14878.0	1.31884	0.659421	−	0.751774i	$-0.270802\pi$
$503$	14878.0	1.31884	0.659421	+	0.751774i	$0.270802\pi$
$504$	540.000	0.0477252
$505$	0	0
$506$	1224.00	0.107536
$507$	−13581.0	−1.18965
$508$	−4368.00	−0.381493
$509$	−6090.00	−0.530323	−0.265162	−	0.964204i	$-0.585425\pi$
$509$	−6090.00	−0.530323	−0.265162	+	0.964204i	$0.585425\pi$
$510$	0	0
$511$	1952.00	0.168985
$512$	−11521.0	−0.994455
$513$	−513.000	−0.0441511
$514$	606.000	0.0520029
$515$	0	0
$516$	−10122.0	−0.863559
$517$	−7752.00	−0.659444
$518$	−136.000	−0.0115357
$519$	−9594.00	−0.811426
$520$	0	0
$521$	12612.0	1.06054	0.530270	−	0.847829i	$-0.322090\pi$
$521$	12612.0	1.06054	0.530270	+	0.847829i	$0.322090\pi$
$522$	−270.000	−0.0226390
$523$	8828.00	0.738091	0.369045	−	0.929411i	$-0.379685\pi$
$523$	8828.00	0.738091	0.369045	+	0.929411i	$0.379685\pi$
$524$	−8064.00	−0.672285
$525$	0	0
$526$	2262.00	0.187505
$527$	25628.0	2.11836
$528$	8364.00	0.689387
$529$	−11843.0	−0.973371
$530$	0	0
$531$	−1890.00	−0.154461
$532$	−532.000	−0.0433555
$533$	−4264.00	−0.346518
$534$	−2640.00	−0.213940
$535$	0	0
$536$	13560.0	1.09273
$537$	5130.00	0.412246
$538$	−1170.00	−0.0937589
$539$	22236.0	1.77694
$540$	0	0
$541$	−20938.0	−1.66395	−0.831973	−	0.554816i	$-0.812789\pi$
$541$	−20938.0	−1.66395	−0.831973	+	0.554816i	$0.812789\pi$
$542$	7728.00	0.612447
$543$	−936.000	−0.0739735
$544$	13846.0	1.09125
$545$	0	0
$546$	−984.000	−0.0771269
$547$	−6316.00	−0.493698	−0.246849	−	0.969054i	$-0.579395\pi$
$547$	−6316.00	−0.493698	−0.246849	+	0.969054i	$0.579395\pi$
$548$	−8778.00	−0.684266
$549$	−6462.00	−0.502352
$550$	0	0
$551$	570.000	0.0440704
$552$	−810.000	−0.0624563
$553$	−2120.00	−0.163023
$554$	3386.00	0.259670
$555$	0	0
$556$	9100.00	0.694111
$557$	−14776.0	−1.12402	−0.562010	−	0.827130i	$-0.689972\pi$
$557$	−14776.0	−1.12402	−0.562010	+	0.827130i	$0.689972\pi$
$558$	2682.00	0.203473
$559$	39524.0	2.99050
$560$	0	0
$561$	−17544.0	−1.32034
$562$	1708.00	0.128199
$563$	−7932.00	−0.593773	−0.296886	−	0.954913i	$-0.595948\pi$
$563$	−7932.00	−0.593773	−0.296886	+	0.954913i	$0.595948\pi$
$564$	2394.00	0.178733
$565$	0	0
$566$	−7378.00	−0.547916
$567$	324.000	0.0239977
$568$	−14820.0	−1.09478
$569$	−8280.00	−0.610045	−0.305023	−	0.952345i	$-0.598664\pi$
$569$	−8280.00	−0.610045	−0.305023	+	0.952345i	$0.598664\pi$
$570$	0	0
$571$	−1868.00	−0.136906	−0.0684530	−	0.997654i	$-0.521806\pi$
$571$	−1868.00	−0.136906	−0.0684530	+	0.997654i	$0.521806\pi$
$572$	−39032.0	−2.85316
$573$	−7416.00	−0.540677
$574$	−208.000	−0.0151250
$575$	0	0
$576$	−1503.00	−0.108724
$577$	−1656.00	−0.119480	−0.0597402	−	0.998214i	$-0.519027\pi$
$577$	−1656.00	−0.119480	−0.0597402	+	0.998214i	$0.519027\pi$
$578$	−2483.00	−0.178684
$579$	7266.00	0.521528
$580$	0	0
$581$	−4128.00	−0.294765
$582$	−738.000	−0.0525620
$583$	24616.0	1.74870
$584$	7320.00	0.518671
$585$	0	0
$586$	−2038.00	−0.143667
$587$	20664.0	1.45297	0.726486	−	0.687181i	$-0.241152\pi$
$587$	20664.0	1.45297	0.726486	+	0.687181i	$0.241152\pi$
$588$	−6867.00	−0.481616
$589$	−5662.00	−0.396093
$590$	0	0
$591$	−11952.0	−0.831877
$592$	1394.00	0.0967788
$593$	−11702.0	−0.810360	−0.405180	−	0.914237i	$-0.632791\pi$
$593$	−11702.0	−0.810360	−0.405180	+	0.914237i	$0.632791\pi$
$594$	−1836.00	−0.126822
$595$	0	0
$596$	−12670.0	−0.870778
$597$	6000.00	0.411329
$598$	1476.00	0.100933
$599$	22680.0	1.54704	0.773522	−	0.633769i	$-0.218493\pi$
$599$	22680.0	1.54704	0.773522	+	0.633769i	$0.218493\pi$
$600$	0	0
$601$	13742.0	0.932692	0.466346	−	0.884602i	$-0.345570\pi$
$601$	13742.0	0.932692	0.466346	+	0.884602i	$0.345570\pi$
$602$	1928.00	0.130531
$603$	8136.00	0.549459
$604$	−11634.0	−0.783743
$605$	0	0
$606$	−2574.00	−0.172544
$607$	4184.00	0.279775	0.139887	−	0.990167i	$-0.455326\pi$
$607$	4184.00	0.279775	0.139887	+	0.990167i	$0.455326\pi$
$608$	−3059.00	−0.204044
$609$	−360.000	−0.0239539
$610$	0	0
$611$	−9348.00	−0.618952
$612$	5418.00	0.357859
$613$	1098.00	0.0723455	0.0361728	−	0.999346i	$-0.488483\pi$
$613$	1098.00	0.0723455	0.0361728	+	0.999346i	$0.488483\pi$
$614$	1636.00	0.107530
$615$	0	0
$616$	−4080.00	−0.266863
$617$	−21426.0	−1.39802	−0.699010	−	0.715112i	$-0.746376\pi$
$617$	−21426.0	−1.39802	−0.699010	+	0.715112i	$0.746376\pi$
$618$	3264.00	0.212455
$619$	8020.00	0.520761	0.260380	−	0.965506i	$-0.416152\pi$
$619$	8020.00	0.520761	0.260380	+	0.965506i	$0.416152\pi$
$620$	0	0
$621$	−486.000	−0.0314050
$622$	1848.00	0.119129
$623$	−3520.00	−0.226366
$624$	10086.0	0.647056
$625$	0	0
$626$	−4908.00	−0.313360
$627$	3876.00	0.246878
$628$	4942.00	0.314024
$629$	−2924.00	−0.185354
$630$	0	0
$631$	6292.00	0.396958	0.198479	−	0.980105i	$-0.436400\pi$
$631$	6292.00	0.396958	0.198479	+	0.980105i	$0.436400\pi$
$632$	−7950.00	−0.500370
$633$	14304.0	0.898156
$634$	4526.00	0.283518
$635$	0	0
$636$	−7602.00	−0.473961
$637$	26814.0	1.66783
$638$	2040.00	0.126590
$639$	−8892.00	−0.550488
$640$	0	0
$641$	19332.0	1.19121	0.595607	−	0.803276i	$-0.296912\pi$
$641$	19332.0	1.19121	0.595607	+	0.803276i	$0.296912\pi$
$642$	−1308.00	−0.0804091
$643$	−20702.0	−1.26968	−0.634842	−	0.772642i	$-0.718935\pi$
$643$	−20702.0	−1.26968	−0.634842	+	0.772642i	$0.718935\pi$
$644$	−504.000	−0.0308391
$645$	0	0
$646$	1634.00	0.0995184
$647$	11754.0	0.714215	0.357108	−	0.934063i	$-0.383763\pi$
$647$	11754.0	0.714215	0.357108	+	0.934063i	$0.383763\pi$
$648$	1215.00	0.0736570
$649$	14280.0	0.863697
$650$	0	0
$651$	3576.00	0.215291
$652$	10234.0	0.614715
$653$	448.000	0.0268478	0.0134239	−	0.999910i	$-0.495727\pi$
$653$	448.000	0.0268478	0.0134239	+	0.999910i	$0.495727\pi$
$654$	−5040.00	−0.301345
$655$	0	0
$656$	2132.00	0.126891
$657$	4392.00	0.260804
$658$	−456.000	−0.0270163
$659$	−28910.0	−1.70891	−0.854457	−	0.519523i	$-0.826110\pi$
$659$	−28910.0	−1.70891	−0.854457	+	0.519523i	$0.826110\pi$
$660$	0	0
$661$	−8748.00	−0.514762	−0.257381	−	0.966310i	$-0.582860\pi$
$661$	−8748.00	−0.514762	−0.257381	+	0.966310i	$0.582860\pi$
$662$	6388.00	0.375040
$663$	−21156.0	−1.23926
$664$	−15480.0	−0.904730
$665$	0	0
$666$	−306.000	−0.0178037
$667$	540.000	0.0313477
$668$	−4368.00	−0.252998
$669$	−9744.00	−0.563116
$670$	0	0
$671$	48824.0	2.80899
$672$	1932.00	0.110906
$673$	−5002.00	−0.286498	−0.143249	−	0.989687i	$-0.545755\pi$
$673$	−5002.00	−0.286498	−0.143249	+	0.989687i	$0.545755\pi$
$674$	286.000	0.0163447
$675$	0	0
$676$	−31689.0	−1.80297
$677$	−26706.0	−1.51609	−0.758046	−	0.652201i	$-0.773846\pi$
$677$	−26706.0	−1.51609	−0.758046	+	0.652201i	$0.773846\pi$
$678$	−2286.00	−0.129489
$679$	−984.000	−0.0556148
$680$	0	0
$681$	−16332.0	−0.919007
$682$	−20264.0	−1.13775
$683$	22668.0	1.26994	0.634968	−	0.772538i	$-0.281013\pi$
$683$	22668.0	1.26994	0.634968	+	0.772538i	$0.281013\pi$
$684$	−1197.00	−0.0669129
$685$	0	0
$686$	2680.00	0.149159
$687$	10470.0	0.581449
$688$	−19762.0	−1.09509
$689$	29684.0	1.64132
$690$	0	0
$691$	−21228.0	−1.16867	−0.584335	−	0.811512i	$-0.698645\pi$
$691$	−21228.0	−1.16867	−0.584335	+	0.811512i	$0.698645\pi$
$692$	−22386.0	−1.22975
$693$	−2448.00	−0.134187
$694$	−8904.00	−0.487019
$695$	0	0
$696$	−1350.00	−0.0735224
$697$	−4472.00	−0.243026
$698$	−7670.00	−0.415922
$699$	12606.0	0.682121
$700$	0	0
$701$	−22158.0	−1.19386	−0.596930	−	0.802293i	$-0.703613\pi$
$701$	−22158.0	−1.19386	−0.596930	+	0.802293i	$0.703613\pi$
$702$	−2214.00	−0.119034
$703$	646.000	0.0346577
$704$	11356.0	0.607948
$705$	0	0
$706$	2.00000	0.000106616 0
$707$	−3432.00	−0.182565
$708$	−4410.00	−0.234093
$709$	−10850.0	−0.574725	−0.287363	−	0.957822i	$-0.592779\pi$
$709$	−10850.0	−0.574725	−0.287363	+	0.957822i	$0.592779\pi$
$710$	0	0
$711$	−4770.00	−0.251602
$712$	−13200.0	−0.694791
$713$	−5364.00	−0.281744
$714$	−1032.00	−0.0540919
$715$	0	0
$716$	11970.0	0.624776
$717$	−19200.0	−1.00005
$718$	1120.00	0.0582145
$719$	23160.0	1.20128	0.600641	−	0.799519i	$-0.294912\pi$
$719$	23160.0	1.20128	0.600641	+	0.799519i	$0.294912\pi$
$720$	0	0
$721$	4352.00	0.224795
$722$	−361.000	−0.0186081
$723$	−9426.00	−0.484864
$724$	−2184.00	−0.112110
$725$	0	0
$726$	9879.00	0.505019
$727$	7144.00	0.364452	0.182226	−	0.983257i	$-0.441670\pi$
$727$	7144.00	0.364452	0.182226	+	0.983257i	$0.441670\pi$
$728$	−4920.00	−0.250477
$729$	729.000	0.0370370
$730$	0	0
$731$	41452.0	2.09734
$732$	−15078.0	−0.761337
$733$	−7242.00	−0.364924	−0.182462	−	0.983213i	$-0.558407\pi$
$733$	−7242.00	−0.364924	−0.182462	+	0.983213i	$0.558407\pi$
$734$	−1984.00	−0.0997694
$735$	0	0
$736$	−2898.00	−0.145138
$737$	−61472.0	−3.07239
$738$	−468.000	−0.0233432
$739$	−3380.00	−0.168248	−0.0841240	−	0.996455i	$-0.526809\pi$
$739$	−3380.00	−0.168248	−0.0841240	+	0.996455i	$0.526809\pi$
$740$	0	0
$741$	4674.00	0.231719
$742$	1448.00	0.0716412
$743$	3848.00	0.189999	0.0949996	−	0.995477i	$-0.469715\pi$
$743$	3848.00	0.189999	0.0949996	+	0.995477i	$0.469715\pi$
$744$	13410.0	0.660799
$745$	0	0
$746$	−2058.00	−0.101004
$747$	−9288.00	−0.454927
$748$	−40936.0	−2.00103
$749$	−1744.00	−0.0850793
$750$	0	0
$751$	−22558.0	−1.09608	−0.548038	−	0.836453i	$-0.684625\pi$
$751$	−22558.0	−1.09608	−0.548038	+	0.836453i	$0.684625\pi$
$752$	4674.00	0.226653
$753$	11964.0	0.579007
$754$	2460.00	0.118817
$755$	0	0
$756$	756.000	0.0363696
$757$	−15866.0	−0.761770	−0.380885	−	0.924623i	$-0.624381\pi$
$757$	−15866.0	−0.761770	−0.380885	+	0.924623i	$0.624381\pi$
$758$	10640.0	0.509845
$759$	3672.00	0.175606
$760$	0	0
$761$	462.000	0.0220072	0.0110036	−	0.999939i	$-0.496497\pi$
$761$	462.000	0.0220072	0.0110036	+	0.999939i	$0.496497\pi$
$762$	1872.00	0.0889966
$763$	−6720.00	−0.318847
$764$	−17304.0	−0.819420
$765$	0	0
$766$	5892.00	0.277920
$767$	17220.0	0.810663
$768$	357.000	0.0167736
$769$	−22210.0	−1.04150	−0.520750	−	0.853709i	$-0.674348\pi$
$769$	−22210.0	−1.04150	−0.520750	+	0.853709i	$0.674348\pi$
$770$	0	0
$771$	1818.00	0.0849205
$772$	16954.0	0.790399
$773$	658.000	0.0306166	0.0153083	−	0.999883i	$-0.495127\pi$
$773$	658.000	0.0306166	0.0153083	+	0.999883i	$0.495127\pi$
$774$	4338.00	0.201455
$775$	0	0
$776$	−3690.00	−0.170700
$777$	−408.000	−0.0188377
$778$	−990.000	−0.0456211
$779$	988.000	0.0454413
$780$	0	0
$781$	67184.0	3.07815
$782$	1548.00	0.0707882
$783$	−810.000	−0.0369694
$784$	−13407.0	−0.610742
$785$	0	0
$786$	3456.00	0.156834
$787$	−36016.0	−1.63130	−0.815649	−	0.578547i	$-0.803620\pi$
$787$	−36016.0	−1.63130	−0.815649	+	0.578547i	$0.803620\pi$
$788$	−27888.0	−1.26075
$789$	6786.00	0.306195
$790$	0	0
$791$	−3048.00	−0.137009
$792$	−9180.00	−0.411865
$793$	58876.0	2.63650
$794$	526.000	0.0235101
$795$	0	0
$796$	14000.0	0.623388
$797$	−5346.00	−0.237597	−0.118799	−	0.992918i	$-0.537904\pi$
$797$	−5346.00	−0.237597	−0.118799	+	0.992918i	$0.537904\pi$
$798$	228.000	0.0101142
$799$	−9804.00	−0.434093
$800$	0	0
$801$	−7920.00	−0.349363
$802$	−9612.00	−0.423206
$803$	−33184.0	−1.45833
$804$	18984.0	0.832729
$805$	0	0
$806$	−24436.0	−1.06789
$807$	−3510.00	−0.153108
$808$	−12870.0	−0.560353
$809$	−24170.0	−1.05040	−0.525199	−	0.850979i	$-0.676009\pi$
$809$	−24170.0	−1.05040	−0.525199	+	0.850979i	$0.676009\pi$
$810$	0	0
$811$	13932.0	0.603229	0.301614	−	0.953430i	$-0.402474\pi$
$811$	13932.0	0.603229	0.301614	+	0.953430i	$0.402474\pi$
$812$	−840.000	−0.0363032
$813$	23184.0	1.00012
$814$	2312.00	0.0995523
$815$	0	0
$816$	10578.0	0.453804
$817$	−9158.00	−0.392164
$818$	9110.00	0.389393
$819$	−2952.00	−0.125948
$820$	0	0
$821$	14202.0	0.603719	0.301859	−	0.953352i	$-0.402393\pi$
$821$	14202.0	0.603719	0.301859	+	0.953352i	$0.402393\pi$
$822$	3762.00	0.159629
$823$	2688.00	0.113849	0.0569245	−	0.998378i	$-0.481871\pi$
$823$	2688.00	0.113849	0.0569245	+	0.998378i	$0.481871\pi$
$824$	16320.0	0.689969
$825$	0	0
$826$	840.000	0.0353842
$827$	−4236.00	−0.178114	−0.0890569	−	0.996027i	$-0.528385\pi$
$827$	−4236.00	−0.178114	−0.0890569	+	0.996027i	$0.528385\pi$
$828$	−1134.00	−0.0475957
$829$	−20800.0	−0.871428	−0.435714	−	0.900085i	$-0.643504\pi$
$829$	−20800.0	−0.871428	−0.435714	+	0.900085i	$0.643504\pi$
$830$	0	0
$831$	10158.0	0.424040
$832$	13694.0	0.570618
$833$	28122.0	1.16971
$834$	−3900.00	−0.161926
$835$	0	0
$836$	9044.00	0.374155
$837$	8046.00	0.332271
$838$	−13920.0	−0.573817
$839$	1020.00	0.0419718	0.0209859	−	0.999780i	$-0.493319\pi$
$839$	1020.00	0.0419718	0.0209859	+	0.999780i	$0.493319\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	9588.00	0.392428
$843$	5124.00	0.209347
$844$	33376.0	1.36120
$845$	0	0
$846$	−1026.00	−0.0416958
$847$	13172.0	0.534351
$848$	−14842.0	−0.601033
$849$	−22134.0	−0.894743
$850$	0	0
$851$	612.000	0.0246523
$852$	−20748.0	−0.834290
$853$	−32102.0	−1.28857	−0.644286	−	0.764785i	$-0.722845\pi$
$853$	−32102.0	−1.28857	−0.644286	+	0.764785i	$0.722845\pi$
$854$	2872.00	0.115079
$855$	0	0
$856$	−6540.00	−0.261136
$857$	31434.0	1.25293	0.626467	−	0.779448i	$-0.284500\pi$
$857$	31434.0	1.25293	0.626467	+	0.779448i	$0.284500\pi$
$858$	16728.0	0.665600
$859$	25660.0	1.01922	0.509609	−	0.860406i	$-0.329790\pi$
$859$	25660.0	1.01922	0.509609	+	0.860406i	$0.329790\pi$
$860$	0	0
$861$	−624.000	−0.0246990
$862$	4968.00	0.196300
$863$	−14212.0	−0.560582	−0.280291	−	0.959915i	$-0.590431\pi$
$863$	−14212.0	−0.560582	−0.280291	+	0.959915i	$0.590431\pi$
$864$	4347.00	0.171167
$865$	0	0
$866$	11342.0	0.445054
$867$	−7449.00	−0.291789
$868$	8344.00	0.326283
$869$	36040.0	1.40687
$870$	0	0
$871$	−74128.0	−2.88373
$872$	−25200.0	−0.978646
$873$	−2214.00	−0.0858334
$874$	−342.000	−0.0132361
$875$	0	0
$876$	10248.0	0.395260
$877$	46294.0	1.78248	0.891241	−	0.453529i	$-0.149835\pi$
$877$	46294.0	1.78248	0.891241	+	0.453529i	$0.149835\pi$
$878$	−3710.00	−0.142604
$879$	−6114.00	−0.234608
$880$	0	0
$881$	−7558.00	−0.289030	−0.144515	−	0.989503i	$-0.546162\pi$
$881$	−7558.00	−0.289030	−0.144515	+	0.989503i	$0.546162\pi$
$882$	2943.00	0.112354
$883$	−27202.0	−1.03672	−0.518358	−	0.855164i	$-0.673457\pi$
$883$	−27202.0	−1.03672	−0.518358	+	0.855164i	$0.673457\pi$
$884$	−49364.0	−1.87816
$885$	0	0
$886$	10772.0	0.408456
$887$	−36196.0	−1.37017	−0.685086	−	0.728462i	$-0.740235\pi$
$887$	−36196.0	−1.37017	−0.685086	+	0.728462i	$0.740235\pi$
$888$	−1530.00	−0.0578192
$889$	2496.00	0.0941655
$890$	0	0
$891$	−5508.00	−0.207099
$892$	−22736.0	−0.853428
$893$	2166.00	0.0811673
$894$	5430.00	0.203139
$895$	0	0
$896$	5820.00	0.217001
$897$	4428.00	0.164823
$898$	1720.00	0.0639166
$899$	−8940.00	−0.331664
$900$	0	0
$901$	31132.0	1.15112
$902$	3536.00	0.130528
$903$	5784.00	0.213156
$904$	−11430.0	−0.420527
$905$	0	0
$906$	4986.00	0.182835
$907$	2424.00	0.0887405	0.0443702	−	0.999015i	$-0.485872\pi$
$907$	2424.00	0.0887405	0.0443702	+	0.999015i	$0.485872\pi$
$908$	−38108.0	−1.39280
$909$	−7722.00	−0.281763
$910$	0	0
$911$	37252.0	1.35479	0.677395	−	0.735619i	$-0.263109\pi$
$911$	37252.0	1.35479	0.677395	+	0.735619i	$0.263109\pi$
$912$	−2337.00	−0.0848529
$913$	70176.0	2.54380
$914$	1456.00	0.0526917
$915$	0	0
$916$	24430.0	0.881212
$917$	4608.00	0.165943
$918$	−2322.00	−0.0834830
$919$	−45980.0	−1.65042	−0.825212	−	0.564823i	$-0.808945\pi$
$919$	−45980.0	−1.65042	−0.825212	+	0.564823i	$0.808945\pi$
$920$	0	0
$921$	4908.00	0.175596
$922$	6218.00	0.222103
$923$	81016.0	2.88914
$924$	−5712.00	−0.203367
$925$	0	0
$926$	13892.0	0.493002
$927$	9792.00	0.346938
$928$	−4830.00	−0.170854
$929$	−34470.0	−1.21736	−0.608678	−	0.793417i	$-0.708300\pi$
$929$	−34470.0	−1.21736	−0.608678	+	0.793417i	$0.708300\pi$
$930$	0	0
$931$	−6213.00	−0.218714
$932$	29414.0	1.03378
$933$	5544.00	0.194536
$934$	−6004.00	−0.210339
$935$	0	0
$936$	−11070.0	−0.386575
$937$	12764.0	0.445018	0.222509	−	0.974931i	$-0.428575\pi$
$937$	12764.0	0.445018	0.222509	+	0.974931i	$0.428575\pi$
$938$	−3616.00	−0.125871
$939$	−14724.0	−0.511714
$940$	0	0
$941$	−55538.0	−1.92400	−0.962002	−	0.273044i	$-0.911970\pi$
$941$	−55538.0	−1.92400	−0.962002	+	0.273044i	$0.911970\pi$
$942$	−2118.00	−0.0732571
$943$	936.000	0.0323228
$944$	−8610.00	−0.296856
$945$	0	0
$946$	−32776.0	−1.12647
$947$	8604.00	0.295240	0.147620	−	0.989044i	$-0.452839\pi$
$947$	8604.00	0.295240	0.147620	+	0.989044i	$0.452839\pi$
$948$	−11130.0	−0.381314
$949$	−40016.0	−1.36878
$950$	0	0
$951$	13578.0	0.462983
$952$	−5160.00	−0.175669
$953$	18018.0	0.612445	0.306223	−	0.951960i	$-0.400935\pi$
$953$	18018.0	0.612445	0.306223	+	0.951960i	$0.400935\pi$
$954$	3258.00	0.110568
$955$	0	0
$956$	−44800.0	−1.51562
$957$	6120.00	0.206720
$958$	−4800.00	−0.161880
$959$	5016.00	0.168900
$960$	0	0
$961$	59013.0	1.98090
$962$	2788.00	0.0934394
$963$	−3924.00	−0.131308
$964$	−21994.0	−0.734833
$965$	0	0
$966$	216.000	0.00719429
$967$	−45496.0	−1.51298	−0.756491	−	0.654005i	$-0.773088\pi$
$967$	−45496.0	−1.51298	−0.756491	+	0.654005i	$0.773088\pi$
$968$	49395.0	1.64010
$969$	4902.00	0.162513
$970$	0	0
$971$	37722.0	1.24671	0.623356	−	0.781938i	$-0.285769\pi$
$971$	37722.0	1.24671	0.623356	+	0.781938i	$0.285769\pi$
$972$	1701.00	0.0561313
$973$	−5200.00	−0.171330
$974$	12616.0	0.415034
$975$	0	0
$976$	−29438.0	−0.965458
$977$	43474.0	1.42360	0.711800	−	0.702383i	$-0.247880\pi$
$977$	43474.0	1.42360	0.711800	+	0.702383i	$0.247880\pi$
$978$	−4386.00	−0.143404
$979$	59840.0	1.95352
$980$	0	0
$981$	−15120.0	−0.492094
$982$	−4232.00	−0.137524
$983$	−13032.0	−0.422845	−0.211422	−	0.977395i	$-0.567810\pi$
$983$	−13032.0	−0.422845	−0.211422	+	0.977395i	$0.567810\pi$
$984$	−2340.00	−0.0758094
$985$	0	0
$986$	2580.00	0.0833306
$987$	−1368.00	−0.0441174
$988$	10906.0	0.351180
$989$	−8676.00	−0.278949
$990$	0	0
$991$	−52978.0	−1.69819	−0.849093	−	0.528244i	$-0.822851\pi$
$991$	−52978.0	−1.69819	−0.849093	+	0.528244i	$0.822851\pi$
$992$	47978.0	1.53559
$993$	19164.0	0.612438
$994$	3952.00	0.126106
$995$	0	0
$996$	−21672.0	−0.689461
$997$	1774.00	0.0563522	0.0281761	−	0.999603i	$-0.491030\pi$
$997$	1774.00	0.0563522	0.0281761	+	0.999603i	$0.491030\pi$
$998$	−6340.00	−0.201091
$999$	−918.000	−0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.4.a.b.1.1		1
5.2	odd	4		285.4.c.a.229.1	✓	2
5.3	odd	4		285.4.c.a.229.2	yes	2
5.4	even	2		1425.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.4.c.a.229.1	✓	2	5.2	odd	4
285.4.c.a.229.2	yes	2	5.3	odd	4
1425.4.a.b.1.1		1	1.1	even	1	trivial
1425.4.a.d.1.1		1	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1425.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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