LMFDB - Embedded newform 275.6.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,6,Mod(1,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,4,15] 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:	$44.1055504486$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+4.00000 q^{2} +15.0000 q^{3} -16.0000 q^{4} +60.0000 q^{6} -10.0000 q^{7} -192.000 q^{8} -18.0000 q^{9} -121.000 q^{11} -240.000 q^{12} +1148.00 q^{13} -40.0000 q^{14} -256.000 q^{16} -686.000 q^{17} -72.0000 q^{18} -384.000 q^{19} -150.000 q^{21} -484.000 q^{22} -3709.00 q^{23} -2880.00 q^{24} +4592.00 q^{26} -3915.00 q^{27} +160.000 q^{28} -5424.00 q^{29} -6443.00 q^{31} +5120.00 q^{32} -1815.00 q^{33} -2744.00 q^{34} +288.000 q^{36} -12063.0 q^{37} -1536.00 q^{38} +17220.0 q^{39} -1528.00 q^{41} -600.000 q^{42} +4026.00 q^{43} +1936.00 q^{44} -14836.0 q^{46} -7168.00 q^{47} -3840.00 q^{48} -16707.0 q^{49} -10290.0 q^{51} -18368.0 q^{52} +29862.0 q^{53} -15660.0 q^{54} +1920.00 q^{56} -5760.00 q^{57} -21696.0 q^{58} -6461.00 q^{59} -16980.0 q^{61} -25772.0 q^{62} +180.000 q^{63} +28672.0 q^{64} -7260.00 q^{66} -29999.0 q^{67} +10976.0 q^{68} -55635.0 q^{69} +31023.0 q^{71} +3456.00 q^{72} -1924.00 q^{73} -48252.0 q^{74} +6144.00 q^{76} +1210.00 q^{77} +68880.0 q^{78} +65138.0 q^{79} -54351.0 q^{81} -6112.00 q^{82} +102714. q^{83} +2400.00 q^{84} +16104.0 q^{86} -81360.0 q^{87} +23232.0 q^{88} +17415.0 q^{89} -11480.0 q^{91} +59344.0 q^{92} -96645.0 q^{93} -28672.0 q^{94} +76800.0 q^{96} -66905.0 q^{97} -66828.0 q^{98} +2178.00 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	4.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	15.0000	0.962250	0.481125	−	0.876652i	$-0.340228\pi$
$3$	15.0000	0.962250	0.481125	+	0.876652i	$0.340228\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	60.0000	0.680414
$7$	−10.0000	−0.0771356	−0.0385678	−	0.999256i	$-0.512280\pi$
$7$	−10.0000	−0.0771356	−0.0385678	+	0.999256i	$0.512280\pi$
$8$	−192.000	−1.06066
$9$	−18.0000	−0.0740741
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−240.000	−0.481125
$13$	1148.00	1.88401	0.942006	−	0.335597i	$-0.108938\pi$
$13$	1148.00	1.88401	0.942006	+	0.335597i	$0.108938\pi$
$14$	−40.0000	−0.0545431
$15$	0	0
$16$	−256.000	−0.250000
$17$	−686.000	−0.575707	−0.287854	−	0.957674i	$-0.592942\pi$
$17$	−686.000	−0.575707	−0.287854	+	0.957674i	$0.592942\pi$
$18$	−72.0000	−0.0523783
$19$	−384.000	−0.244032	−0.122016	−	0.992528i	$-0.538936\pi$
$19$	−384.000	−0.244032	−0.122016	+	0.992528i	$0.538936\pi$
$20$	0	0
$21$	−150.000	−0.0742238
$22$	−484.000	−0.213201
$23$	−3709.00	−1.46197	−0.730983	−	0.682396i	$-0.760938\pi$
$23$	−3709.00	−1.46197	−0.730983	+	0.682396i	$0.760938\pi$
$24$	−2880.00	−1.02062
$25$	0	0
$26$	4592.00	1.33220
$27$	−3915.00	−1.03353
$28$	160.000	0.0385678
$29$	−5424.00	−1.19764	−0.598818	−	0.800885i	$-0.704363\pi$
$29$	−5424.00	−1.19764	−0.598818	+	0.800885i	$0.704363\pi$
$30$	0	0
$31$	−6443.00	−1.20416	−0.602080	−	0.798436i	$-0.705661\pi$
$31$	−6443.00	−1.20416	−0.602080	+	0.798436i	$0.705661\pi$
$32$	5120.00	0.883883
$33$	−1815.00	−0.290129
$34$	−2744.00	−0.407087
$35$	0	0
$36$	288.000	0.0370370
$37$	−12063.0	−1.44861	−0.724304	−	0.689481i	$-0.757839\pi$
$37$	−12063.0	−1.44861	−0.724304	+	0.689481i	$0.757839\pi$
$38$	−1536.00	−0.172557
$39$	17220.0	1.81289
$40$	0	0
$41$	−1528.00	−0.141959	−0.0709796	−	0.997478i	$-0.522613\pi$
$41$	−1528.00	−0.141959	−0.0709796	+	0.997478i	$0.522613\pi$
$42$	−600.000	−0.0524841
$43$	4026.00	0.332049	0.166025	−	0.986122i	$-0.446907\pi$
$43$	4026.00	0.332049	0.166025	+	0.986122i	$0.446907\pi$
$44$	1936.00	0.150756
$45$	0	0
$46$	−14836.0	−1.03377
$47$	−7168.00	−0.473318	−0.236659	−	0.971593i	$-0.576052\pi$
$47$	−7168.00	−0.473318	−0.236659	+	0.971593i	$0.576052\pi$
$48$	−3840.00	−0.240563
$49$	−16707.0	−0.994050
$50$	0	0
$51$	−10290.0	−0.553975
$52$	−18368.0	−0.942006
$53$	29862.0	1.46026	0.730128	−	0.683310i	$-0.239460\pi$
$53$	29862.0	1.46026	0.730128	+	0.683310i	$0.239460\pi$
$54$	−15660.0	−0.730815
$55$	0	0
$56$	1920.00	0.0818147
$57$	−5760.00	−0.234820
$58$	−21696.0	−0.846856
$59$	−6461.00	−0.241640	−0.120820	−	0.992674i	$-0.538552\pi$
$59$	−6461.00	−0.241640	−0.120820	+	0.992674i	$0.538552\pi$
$60$	0	0
$61$	−16980.0	−0.584269	−0.292135	−	0.956377i	$-0.594366\pi$
$61$	−16980.0	−0.584269	−0.292135	+	0.956377i	$0.594366\pi$
$62$	−25772.0	−0.851469
$63$	180.000	0.00571375
$64$	28672.0	0.875000
$65$	0	0
$66$	−7260.00	−0.205152
$67$	−29999.0	−0.816432	−0.408216	−	0.912885i	$-0.633849\pi$
$67$	−29999.0	−0.816432	−0.408216	+	0.912885i	$0.633849\pi$
$68$	10976.0	0.287854
$69$	−55635.0	−1.40678
$70$	0	0
$71$	31023.0	0.730362	0.365181	−	0.930937i	$-0.381007\pi$
$71$	31023.0	0.730362	0.365181	+	0.930937i	$0.381007\pi$
$72$	3456.00	0.0785674
$73$	−1924.00	−0.0422569	−0.0211285	−	0.999777i	$-0.506726\pi$
$73$	−1924.00	−0.0422569	−0.0211285	+	0.999777i	$0.506726\pi$
$74$	−48252.0	−1.02432
$75$	0	0
$76$	6144.00	0.122016
$77$	1210.00	0.0232573
$78$	68880.0	1.28191
$79$	65138.0	1.17427	0.587133	−	0.809490i	$-0.300256\pi$
$79$	65138.0	1.17427	0.587133	+	0.809490i	$0.300256\pi$
$80$	0	0
$81$	−54351.0	−0.920439
$82$	−6112.00	−0.100380
$83$	102714.	1.63657	0.818285	−	0.574813i	$-0.194925\pi$
$83$	102714.	1.63657	0.818285	+	0.574813i	$0.194925\pi$
$84$	2400.00	0.0371119
$85$	0	0
$86$	16104.0	0.234794
$87$	−81360.0	−1.15243
$88$	23232.0	0.319801
$89$	17415.0	0.233050	0.116525	−	0.993188i	$-0.462825\pi$
$89$	17415.0	0.233050	0.116525	+	0.993188i	$0.462825\pi$
$90$	0	0
$91$	−11480.0	−0.145324
$92$	59344.0	0.730983
$93$	−96645.0	−1.15870
$94$	−28672.0	−0.334687
$95$	0	0
$96$	76800.0	0.850517
$97$	−66905.0	−0.721987	−0.360993	−	0.932568i	$-0.617562\pi$
$97$	−66905.0	−0.721987	−0.360993	+	0.932568i	$0.617562\pi$
$98$	−66828.0	−0.702900
$99$	2178.00	0.0223342
$100$	0	0
$101$	96730.0	0.943534	0.471767	−	0.881723i	$-0.343616\pi$
$101$	96730.0	0.943534	0.471767	+	0.881723i	$0.343616\pi$
$102$	−41160.0	−0.391719
$103$	95704.0	0.888868	0.444434	−	0.895812i	$-0.353405\pi$
$103$	95704.0	0.888868	0.444434	+	0.895812i	$0.353405\pi$
$104$	−220416.	−1.99830
$105$	0	0
$106$	119448.	1.03256
$107$	32658.0	0.275759	0.137880	−	0.990449i	$-0.455971\pi$
$107$	32658.0	0.275759	0.137880	+	0.990449i	$0.455971\pi$
$108$	62640.0	0.516764
$109$	−185438.	−1.49497	−0.747485	−	0.664279i	$-0.768739\pi$
$109$	−185438.	−1.49497	−0.747485	+	0.664279i	$0.768739\pi$
$110$	0	0
$111$	−180945.	−1.39392
$112$	2560.00	0.0192839
$113$	−72849.0	−0.536695	−0.268347	−	0.963322i	$-0.586478\pi$
$113$	−72849.0	−0.536695	−0.268347	+	0.963322i	$0.586478\pi$
$114$	−23040.0	−0.166043
$115$	0	0
$116$	86784.0	0.598818
$117$	−20664.0	−0.139556
$118$	−25844.0	−0.170866
$119$	6860.00	0.0444075
$120$	0	0
$121$	14641.0	0.0909091
$122$	−67920.0	−0.413141
$123$	−22920.0	−0.136600
$124$	103088.	0.602080
$125$	0	0
$126$	720.000	0.00404023
$127$	78184.0	0.430139	0.215069	−	0.976599i	$-0.431002\pi$
$127$	78184.0	0.430139	0.215069	+	0.976599i	$0.431002\pi$
$128$	−49152.0	−0.265165
$129$	60390.0	0.319515
$130$	0	0
$131$	−462.000	−0.00235214	−0.00117607	−	0.999999i	$-0.500374\pi$
$131$	−462.000	−0.00235214	−0.00117607	+	0.999999i	$0.500374\pi$
$132$	29040.0	0.145065
$133$	3840.00	0.0188236
$134$	−119996.	−0.577304
$135$	0	0
$136$	131712.	0.610630
$137$	−296233.	−1.34844	−0.674221	−	0.738530i	$-0.735520\pi$
$137$	−296233.	−1.34844	−0.674221	+	0.738530i	$0.735520\pi$
$138$	−222540.	−0.994742
$139$	−399818.	−1.75519	−0.877597	−	0.479398i	$-0.840855\pi$
$139$	−399818.	−1.75519	−0.877597	+	0.479398i	$0.840855\pi$
$140$	0	0
$141$	−107520.	−0.455451
$142$	124092.	0.516444
$143$	−138908.	−0.568051
$144$	4608.00	0.0185185
$145$	0	0
$146$	−7696.00	−0.0298802
$147$	−250605.	−0.956525
$148$	193008.	0.724304
$149$	72670.0	0.268157	0.134079	−	0.990971i	$-0.457193\pi$
$149$	72670.0	0.268157	0.134079	+	0.990971i	$0.457193\pi$
$150$	0	0
$151$	−303082.	−1.08173	−0.540864	−	0.841110i	$-0.681902\pi$
$151$	−303082.	−1.08173	−0.540864	+	0.841110i	$0.681902\pi$
$152$	73728.0	0.258835
$153$	12348.0	0.0426450
$154$	4840.00	0.0164454
$155$	0	0
$156$	−275520.	−0.906445
$157$	532987.	1.72571	0.862854	−	0.505453i	$-0.168674\pi$
$157$	532987.	1.72571	0.862854	+	0.505453i	$0.168674\pi$
$158$	260552.	0.830332
$159$	447930.	1.40513
$160$	0	0
$161$	37090.0	0.112770
$162$	−217404.	−0.650849
$163$	−282076.	−0.831567	−0.415783	−	0.909464i	$-0.636493\pi$
$163$	−282076.	−0.831567	−0.415783	+	0.909464i	$0.636493\pi$
$164$	24448.0	0.0709796
$165$	0	0
$166$	410856.	1.15723
$167$	573588.	1.59151	0.795754	−	0.605620i	$-0.207075\pi$
$167$	573588.	1.59151	0.795754	+	0.605620i	$0.207075\pi$
$168$	28800.0	0.0787262
$169$	946611.	2.54950
$170$	0	0
$171$	6912.00	0.0180765
$172$	−64416.0	−0.166025
$173$	386286.	0.981282	0.490641	−	0.871362i	$-0.336763\pi$
$173$	386286.	0.981282	0.490641	+	0.871362i	$0.336763\pi$
$174$	−325440.	−0.814888
$175$	0	0
$176$	30976.0	0.0753778
$177$	−96915.0	−0.232519
$178$	69660.0	0.164791
$179$	545079.	1.27153	0.635765	−	0.771882i	$-0.280685\pi$
$179$	545079.	1.27153	0.635765	+	0.771882i	$0.280685\pi$
$180$	0	0
$181$	−279485.	−0.634106	−0.317053	−	0.948408i	$-0.602693\pi$
$181$	−279485.	−0.634106	−0.317053	+	0.948408i	$0.602693\pi$
$182$	−45920.0	−0.102760
$183$	−254700.	−0.562213
$184$	712128.	1.55065
$185$	0	0
$186$	−386580.	−0.819327
$187$	83006.0	0.173582
$188$	114688.	0.236659
$189$	39150.0	0.0797218
$190$	0	0
$191$	−444437.	−0.881509	−0.440755	−	0.897628i	$-0.645289\pi$
$191$	−444437.	−0.881509	−0.440755	+	0.897628i	$0.645289\pi$
$192$	430080.	0.841969
$193$	18476.0	0.0357038	0.0178519	−	0.999841i	$-0.494317\pi$
$193$	18476.0	0.0357038	0.0178519	+	0.999841i	$0.494317\pi$
$194$	−267620.	−0.510522
$195$	0	0
$196$	267312.	0.497025
$197$	−270182.	−0.496010	−0.248005	−	0.968759i	$-0.579775\pi$
$197$	−270182.	−0.496010	−0.248005	+	0.968759i	$0.579775\pi$
$198$	8712.00	0.0157926
$199$	43320.0	0.0775453	0.0387727	−	0.999248i	$-0.487655\pi$
$199$	43320.0	0.0775453	0.0387727	+	0.999248i	$0.487655\pi$
$200$	0	0
$201$	−449985.	−0.785612
$202$	386920.	0.667180
$203$	54240.0	0.0923803
$204$	164640.	0.276987
$205$	0	0
$206$	382816.	0.628524
$207$	66762.0	0.108294
$208$	−293888.	−0.471003
$209$	46464.0	0.0735785
$210$	0	0
$211$	1.02968e6	1.59220	0.796100	−	0.605165i	$-0.206893\pi$
$211$	1.02968e6	1.59220	0.796100	+	0.605165i	$0.206893\pi$
$212$	−477792.	−0.730128
$213$	465345.	0.702791
$214$	130632.	0.194991
$215$	0	0
$216$	751680.	1.09622
$217$	64430.0	0.0928835
$218$	−741752.	−1.05710
$219$	−28860.0	−0.0406617
$220$	0	0
$221$	−787528.	−1.08464
$222$	−723780.	−0.985653
$223$	−461281.	−0.621160	−0.310580	−	0.950547i	$-0.600523\pi$
$223$	−461281.	−0.621160	−0.310580	+	0.950547i	$0.600523\pi$
$224$	−51200.0	−0.0681789
$225$	0	0
$226$	−291396.	−0.379501
$227$	855570.	1.10202	0.551012	−	0.834497i	$-0.314242\pi$
$227$	855570.	1.10202	0.551012	+	0.834497i	$0.314242\pi$
$228$	92160.0	0.117410
$229$	−665805.	−0.838993	−0.419497	−	0.907757i	$-0.637793\pi$
$229$	−665805.	−0.838993	−0.419497	+	0.907757i	$0.637793\pi$
$230$	0	0
$231$	18150.0	0.0223793
$232$	1.04141e6	1.27028
$233$	−1.20934e6	−1.45934	−0.729671	−	0.683798i	$-0.760327\pi$
$233$	−1.20934e6	−1.45934	−0.729671	+	0.683798i	$0.760327\pi$
$234$	−82656.0	−0.0986813
$235$	0	0
$236$	103376.	0.120820
$237$	977070.	1.12994
$238$	27440.0	0.0314009
$239$	−571482.	−0.647154	−0.323577	−	0.946202i	$-0.604886\pi$
$239$	−571482.	−0.647154	−0.323577	+	0.946202i	$0.604886\pi$
$240$	0	0
$241$	−267080.	−0.296209	−0.148105	−	0.988972i	$-0.547317\pi$
$241$	−267080.	−0.296209	−0.148105	+	0.988972i	$0.547317\pi$
$242$	58564.0	0.0642824
$243$	136080.	0.147835
$244$	271680.	0.292135
$245$	0	0
$246$	−91680.0	−0.0965910
$247$	−440832.	−0.459760
$248$	1.23706e6	1.27720
$249$	1.54071e6	1.57479
$250$	0	0
$251$	1.38737e6	1.38998	0.694988	−	0.719022i	$-0.255410\pi$
$251$	1.38737e6	1.38998	0.694988	+	0.719022i	$0.255410\pi$
$252$	−2880.00	−0.00285687
$253$	448789.	0.440799
$254$	312736.	0.304154
$255$	0	0
$256$	−1.11411e6	−1.06250
$257$	885922.	0.836686	0.418343	−	0.908289i	$-0.362611\pi$
$257$	885922.	0.836686	0.418343	+	0.908289i	$0.362611\pi$
$258$	241560.	0.225931
$259$	120630.	0.111739
$260$	0	0
$261$	97632.0	0.0887137
$262$	−1848.00	−0.00166322
$263$	−1.44687e6	−1.28986	−0.644928	−	0.764243i	$-0.723113\pi$
$263$	−1.44687e6	−1.28986	−0.644928	+	0.764243i	$0.723113\pi$
$264$	348480.	0.307729
$265$	0	0
$266$	15360.0	0.0133103
$267$	261225.	0.224252
$268$	479984.	0.408216
$269$	−353878.	−0.298176	−0.149088	−	0.988824i	$-0.547634\pi$
$269$	−353878.	−0.298176	−0.149088	+	0.988824i	$0.547634\pi$
$270$	0	0
$271$	525260.	0.434461	0.217231	−	0.976120i	$-0.430298\pi$
$271$	525260.	0.434461	0.217231	+	0.976120i	$0.430298\pi$
$272$	175616.	0.143927
$273$	−172200.	−0.139838
$274$	−1.18493e6	−0.953492
$275$	0	0
$276$	890160.	0.703389
$277$	595610.	0.466404	0.233202	−	0.972428i	$-0.425080\pi$
$277$	595610.	0.466404	0.233202	+	0.972428i	$0.425080\pi$
$278$	−1.59927e6	−1.24111
$279$	115974.	0.0891970
$280$	0	0
$281$	732318.	0.553266	0.276633	−	0.960976i	$-0.410781\pi$
$281$	732318.	0.553266	0.276633	+	0.960976i	$0.410781\pi$
$282$	−430080.	−0.322052
$283$	−2.23380e6	−1.65798	−0.828989	−	0.559264i	$-0.811084\pi$
$283$	−2.23380e6	−1.65798	−0.828989	+	0.559264i	$0.811084\pi$
$284$	−496368.	−0.365181
$285$	0	0
$286$	−555632.	−0.401673
$287$	15280.0	0.0109501
$288$	−92160.0	−0.0654729
$289$	−949261.	−0.668561
$290$	0	0
$291$	−1.00358e6	−0.694732
$292$	30784.0	0.0211285
$293$	1.53108e6	1.04191	0.520953	−	0.853585i	$-0.325577\pi$
$293$	1.53108e6	1.04191	0.520953	+	0.853585i	$0.325577\pi$
$294$	−1.00242e6	−0.676365
$295$	0	0
$296$	2.31610e6	1.53648
$297$	473715.	0.311620
$298$	290680.	0.189616
$299$	−4.25793e6	−2.75436
$300$	0	0
$301$	−40260.0	−0.0256128
$302$	−1.21233e6	−0.764897
$303$	1.45095e6	0.907916
$304$	98304.0	0.0610081
$305$	0	0
$306$	49392.0	0.0301546
$307$	1.14268e6	0.691956	0.345978	−	0.938243i	$-0.387547\pi$
$307$	1.14268e6	0.691956	0.345978	+	0.938243i	$0.387547\pi$
$308$	−19360.0	−0.0116286
$309$	1.43556e6	0.855313
$310$	0	0
$311$	586956.	0.344116	0.172058	−	0.985087i	$-0.444958\pi$
$311$	586956.	0.344116	0.172058	+	0.985087i	$0.444958\pi$
$312$	−3.30624e6	−1.92286
$313$	233857.	0.134924	0.0674621	−	0.997722i	$-0.478510\pi$
$313$	233857.	0.134924	0.0674621	+	0.997722i	$0.478510\pi$
$314$	2.13195e6	1.22026
$315$	0	0
$316$	−1.04221e6	−0.587133
$317$	935503.	0.522874	0.261437	−	0.965221i	$-0.415804\pi$
$317$	935503.	0.522874	0.261437	+	0.965221i	$0.415804\pi$
$318$	1.79172e6	0.993579
$319$	656304.	0.361101
$320$	0	0
$321$	489870.	0.265349
$322$	148360.	0.0797402
$323$	263424.	0.140491
$324$	869616.	0.460219
$325$	0	0
$326$	−1.12830e6	−0.588007
$327$	−2.78157e6	−1.43854
$328$	293376.	0.150571
$329$	71680.0	0.0365097
$330$	0	0
$331$	−1.05823e6	−0.530897	−0.265449	−	0.964125i	$-0.585520\pi$
$331$	−1.05823e6	−0.530897	−0.265449	+	0.964125i	$0.585520\pi$
$332$	−1.64342e6	−0.818285
$333$	217134.	0.107304
$334$	2.29435e6	1.12537
$335$	0	0
$336$	38400.0	0.0185559
$337$	−506186.	−0.242793	−0.121396	−	0.992604i	$-0.538737\pi$
$337$	−506186.	−0.242793	−0.121396	+	0.992604i	$0.538737\pi$
$338$	3.78644e6	1.80277
$339$	−1.09274e6	−0.516435
$340$	0	0
$341$	779603.	0.363068
$342$	27648.0	0.0127820
$343$	335140.	0.153812
$344$	−772992.	−0.352192
$345$	0	0
$346$	1.54514e6	0.693871
$347$	−467636.	−0.208490	−0.104245	−	0.994552i	$-0.533243\pi$
$347$	−467636.	−0.208490	−0.104245	+	0.994552i	$0.533243\pi$
$348$	1.30176e6	0.576213
$349$	304470.	0.133808	0.0669038	−	0.997759i	$-0.478688\pi$
$349$	304470.	0.133808	0.0669038	+	0.997759i	$0.478688\pi$
$350$	0	0
$351$	−4.49442e6	−1.94718
$352$	−619520.	−0.266501
$353$	−2.51868e6	−1.07581	−0.537906	−	0.843005i	$-0.680785\pi$
$353$	−2.51868e6	−1.07581	−0.537906	+	0.843005i	$0.680785\pi$
$354$	−387660.	−0.164416
$355$	0	0
$356$	−278640.	−0.116525
$357$	102900.	0.0427312
$358$	2.18032e6	0.899108
$359$	−3.01841e6	−1.23607	−0.618034	−	0.786151i	$-0.712071\pi$
$359$	−3.01841e6	−1.23607	−0.618034	+	0.786151i	$0.712071\pi$
$360$	0	0
$361$	−2.32864e6	−0.940448
$362$	−1.11794e6	−0.448381
$363$	219615.	0.0874773
$364$	183680.	0.0726622
$365$	0	0
$366$	−1.01880e6	−0.397545
$367$	−994429.	−0.385397	−0.192699	−	0.981258i	$-0.561724\pi$
$367$	−994429.	−0.385397	−0.192699	+	0.981258i	$0.561724\pi$
$368$	949504.	0.365491
$369$	27504.0	0.0105155
$370$	0	0
$371$	−298620.	−0.112638
$372$	1.54632e6	0.579351
$373$	−1.72896e6	−0.643446	−0.321723	−	0.946834i	$-0.604262\pi$
$373$	−1.72896e6	−0.643446	−0.321723	+	0.946834i	$0.604262\pi$
$374$	332024.	0.122741
$375$	0	0
$376$	1.37626e6	0.502030
$377$	−6.22675e6	−2.25636
$378$	156600.	0.0563718
$379$	454765.	0.162626	0.0813128	−	0.996689i	$-0.474089\pi$
$379$	454765.	0.162626	0.0813128	+	0.996689i	$0.474089\pi$
$380$	0	0
$381$	1.17276e6	0.413901
$382$	−1.77775e6	−0.623321
$383$	−2.27557e6	−0.792673	−0.396336	−	0.918105i	$-0.629719\pi$
$383$	−2.27557e6	−0.792673	−0.396336	+	0.918105i	$0.629719\pi$
$384$	−737280.	−0.255155
$385$	0	0
$386$	73904.0	0.0252464
$387$	−72468.0	−0.0245962
$388$	1.07048e6	0.360993
$389$	389781.	0.130601	0.0653005	−	0.997866i	$-0.479199\pi$
$389$	389781.	0.130601	0.0653005	+	0.997866i	$0.479199\pi$
$390$	0	0
$391$	2.54437e6	0.841665
$392$	3.20774e6	1.05435
$393$	−6930.00	−0.00226335
$394$	−1.08073e6	−0.350732
$395$	0	0
$396$	−34848.0	−0.0111671
$397$	1.61933e6	0.515655	0.257827	−	0.966191i	$-0.416993\pi$
$397$	1.61933e6	0.515655	0.257827	+	0.966191i	$0.416993\pi$
$398$	173280.	0.0548328
$399$	57600.0	0.0181130
$400$	0	0
$401$	−5.54368e6	−1.72162	−0.860810	−	0.508927i	$-0.830042\pi$
$401$	−5.54368e6	−1.72162	−0.860810	+	0.508927i	$0.830042\pi$
$402$	−1.79994e6	−0.555511
$403$	−7.39656e6	−2.26865
$404$	−1.54768e6	−0.471767
$405$	0	0
$406$	216960.	0.0653228
$407$	1.45962e6	0.436772
$408$	1.97568e6	0.587579
$409$	−2.70493e6	−0.799553	−0.399776	−	0.916613i	$-0.630912\pi$
$409$	−2.70493e6	−0.799553	−0.399776	+	0.916613i	$0.630912\pi$
$410$	0	0
$411$	−4.44350e6	−1.29754
$412$	−1.53126e6	−0.444434
$413$	64610.0	0.0186391
$414$	267048.	0.0765753
$415$	0	0
$416$	5.87776e6	1.66525
$417$	−5.99727e6	−1.68894
$418$	185856.	0.0520279
$419$	3.37337e6	0.938705	0.469353	−	0.883011i	$-0.344487\pi$
$419$	3.37337e6	0.938705	0.469353	+	0.883011i	$0.344487\pi$
$420$	0	0
$421$	−4.52551e6	−1.24441	−0.622204	−	0.782855i	$-0.713762\pi$
$421$	−4.52551e6	−1.24441	−0.622204	+	0.782855i	$0.713762\pi$
$422$	4.11874e6	1.12586
$423$	129024.	0.0350606
$424$	−5.73350e6	−1.54884
$425$	0	0
$426$	1.86138e6	0.496948
$427$	169800.	0.0450680
$428$	−522528.	−0.137880
$429$	−2.08362e6	−0.546607
$430$	0	0
$431$	−684534.	−0.177501	−0.0887507	−	0.996054i	$-0.528287\pi$
$431$	−684534.	−0.177501	−0.0887507	+	0.996054i	$0.528287\pi$
$432$	1.00224e6	0.258382
$433$	4.22591e6	1.08318	0.541589	−	0.840643i	$-0.317823\pi$
$433$	4.22591e6	1.08318	0.541589	+	0.840643i	$0.317823\pi$
$434$	257720.	0.0656786
$435$	0	0
$436$	2.96701e6	0.747485
$437$	1.42426e6	0.356767
$438$	−115440.	−0.0287522
$439$	−2.09185e6	−0.518047	−0.259023	−	0.965871i	$-0.583401\pi$
$439$	−2.09185e6	−0.518047	−0.259023	+	0.965871i	$0.583401\pi$
$440$	0	0
$441$	300726.	0.0736333
$442$	−3.15011e6	−0.766956
$443$	−1.56284e6	−0.378361	−0.189180	−	0.981942i	$-0.560583\pi$
$443$	−1.56284e6	−0.378361	−0.189180	+	0.981942i	$0.560583\pi$
$444$	2.89512e6	0.696962
$445$	0	0
$446$	−1.84512e6	−0.439226
$447$	1.09005e6	0.258034
$448$	−286720.	−0.0674937
$449$	−3.00449e6	−0.703324	−0.351662	−	0.936127i	$-0.614383\pi$
$449$	−3.00449e6	−0.703324	−0.351662	+	0.936127i	$0.614383\pi$
$450$	0	0
$451$	184888.	0.0428023
$452$	1.16558e6	0.268347
$453$	−4.54623e6	−1.04089
$454$	3.42228e6	0.779248
$455$	0	0
$456$	1.10592e6	0.249064
$457$	2.44552e6	0.547747	0.273874	−	0.961766i	$-0.411695\pi$
$457$	2.44552e6	0.547747	0.273874	+	0.961766i	$0.411695\pi$
$458$	−2.66322e6	−0.593258
$459$	2.68569e6	0.595010
$460$	0	0
$461$	7.79104e6	1.70743	0.853715	−	0.520741i	$-0.174344\pi$
$461$	7.79104e6	1.70743	0.853715	+	0.520741i	$0.174344\pi$
$462$	72600.0	0.0158246
$463$	1.05196e6	0.228059	0.114029	−	0.993477i	$-0.463624\pi$
$463$	1.05196e6	0.228059	0.114029	+	0.993477i	$0.463624\pi$
$464$	1.38854e6	0.299409
$465$	0	0
$466$	−4.83734e6	−1.03191
$467$	−3.97003e6	−0.842369	−0.421184	−	0.906975i	$-0.638385\pi$
$467$	−3.97003e6	−0.842369	−0.421184	+	0.906975i	$0.638385\pi$
$468$	330624.	0.0697782
$469$	299990.	0.0629759
$470$	0	0
$471$	7.99480e6	1.66056
$472$	1.24051e6	0.256298
$473$	−487146.	−0.100117
$474$	3.90828e6	0.798987
$475$	0	0
$476$	−109760.	−0.0222038
$477$	−537516.	−0.108167
$478$	−2.28593e6	−0.457607
$479$	−8.53908e6	−1.70048	−0.850241	−	0.526393i	$-0.823544\pi$
$479$	−8.53908e6	−1.70048	−0.850241	+	0.526393i	$0.823544\pi$
$480$	0	0
$481$	−1.38483e7	−2.72919
$482$	−1.06832e6	−0.209452
$483$	556350.	0.108513
$484$	−234256.	−0.0454545
$485$	0	0
$486$	544320.	0.104535
$487$	1.86487e6	0.356308	0.178154	−	0.984003i	$-0.442987\pi$
$487$	1.86487e6	0.356308	0.178154	+	0.984003i	$0.442987\pi$
$488$	3.26016e6	0.619711
$489$	−4.23114e6	−0.800175
$490$	0	0
$491$	5.15727e6	0.965420	0.482710	−	0.875780i	$-0.339653\pi$
$491$	5.15727e6	0.965420	0.482710	+	0.875780i	$0.339653\pi$
$492$	366720.	0.0683002
$493$	3.72086e6	0.689488
$494$	−1.76333e6	−0.325099
$495$	0	0
$496$	1.64941e6	0.301040
$497$	−310230.	−0.0563369
$498$	6.16284e6	1.11354
$499$	4.53340e6	0.815029	0.407514	−	0.913199i	$-0.366396\pi$
$499$	4.53340e6	0.815029	0.407514	+	0.913199i	$0.366396\pi$
$500$	0	0
$501$	8.60382e6	1.53143
$502$	5.54947e6	0.982861
$503$	−1.71163e6	−0.301641	−0.150821	−	0.988561i	$-0.548192\pi$
$503$	−1.71163e6	−0.301641	−0.150821	+	0.988561i	$0.548192\pi$
$504$	−34560.0	−0.00606035
$505$	0	0
$506$	1.79516e6	0.311692
$507$	1.41992e7	2.45326
$508$	−1.25094e6	−0.215069
$509$	9.73822e6	1.66604	0.833019	−	0.553244i	$-0.186610\pi$
$509$	9.73822e6	1.66604	0.833019	+	0.553244i	$0.186610\pi$
$510$	0	0
$511$	19240.0	0.00325951
$512$	−2.88358e6	−0.486136
$513$	1.50336e6	0.252214
$514$	3.54369e6	0.591627
$515$	0	0
$516$	−966240.	−0.159757
$517$	867328.	0.142711
$518$	482520.	0.0790116
$519$	5.79429e6	0.944239
$520$	0	0
$521$	4.30279e6	0.694474	0.347237	−	0.937777i	$-0.387120\pi$
$521$	4.30279e6	0.694474	0.347237	+	0.937777i	$0.387120\pi$
$522$	390528.	0.0627301
$523$	−2.62280e6	−0.419287	−0.209643	−	0.977778i	$-0.567230\pi$
$523$	−2.62280e6	−0.419287	−0.209643	+	0.977778i	$0.567230\pi$
$524$	7392.00	0.00117607
$525$	0	0
$526$	−5.78750e6	−0.912066
$527$	4.41990e6	0.693243
$528$	464640.	0.0725324
$529$	7.32034e6	1.13734
$530$	0	0
$531$	116298.	0.0178993
$532$	−61440.0	−0.00941179
$533$	−1.75414e6	−0.267453
$534$	1.04490e6	0.158570
$535$	0	0
$536$	5.75981e6	0.865956
$537$	8.17618e6	1.22353
$538$	−1.41551e6	−0.210842
$539$	2.02155e6	0.299717
$540$	0	0
$541$	−2.49634e6	−0.366700	−0.183350	−	0.983048i	$-0.558694\pi$
$541$	−2.49634e6	−0.366700	−0.183350	+	0.983048i	$0.558694\pi$
$542$	2.10104e6	0.307211
$543$	−4.19228e6	−0.610169
$544$	−3.51232e6	−0.508858
$545$	0	0
$546$	−688800.	−0.0988807
$547$	−1.14323e7	−1.63368	−0.816838	−	0.576868i	$-0.804275\pi$
$547$	−1.14323e7	−1.63368	−0.816838	+	0.576868i	$0.804275\pi$
$548$	4.73973e6	0.674221
$549$	305640.	0.0432792
$550$	0	0
$551$	2.08282e6	0.292262
$552$	1.06819e7	1.49211
$553$	−651380.	−0.0905778
$554$	2.38244e6	0.329798
$555$	0	0
$556$	6.39709e6	0.877597
$557$	9.81529e6	1.34049	0.670247	−	0.742138i	$-0.266188\pi$
$557$	9.81529e6	1.34049	0.670247	+	0.742138i	$0.266188\pi$
$558$	463896.	0.0630718
$559$	4.62185e6	0.625585
$560$	0	0
$561$	1.24509e6	0.167030
$562$	2.92927e6	0.391218
$563$	8.19192e6	1.08922	0.544609	−	0.838690i	$-0.316678\pi$
$563$	8.19192e6	1.08922	0.544609	+	0.838690i	$0.316678\pi$
$564$	1.72032e6	0.227725
$565$	0	0
$566$	−8.93522e6	−1.17237
$567$	543510.	0.0709986
$568$	−5.95642e6	−0.774665
$569$	−7.54286e6	−0.976687	−0.488344	−	0.872651i	$-0.662399\pi$
$569$	−7.54286e6	−0.976687	−0.488344	+	0.872651i	$0.662399\pi$
$570$	0	0
$571$	−8.69400e6	−1.11591	−0.557956	−	0.829871i	$-0.688414\pi$
$571$	−8.69400e6	−1.11591	−0.557956	+	0.829871i	$0.688414\pi$
$572$	2.22253e6	0.284025
$573$	−6.66656e6	−0.848233
$574$	61120.0	0.00774290
$575$	0	0
$576$	−516096.	−0.0648148
$577$	−2.03379e6	−0.254312	−0.127156	−	0.991883i	$-0.540585\pi$
$577$	−2.03379e6	−0.254312	−0.127156	+	0.991883i	$0.540585\pi$
$578$	−3.79704e6	−0.472744
$579$	277140.	0.0343560
$580$	0	0
$581$	−1.02714e6	−0.126238
$582$	−4.01430e6	−0.491250
$583$	−3.61330e6	−0.440284
$584$	369408.	0.0448202
$585$	0	0
$586$	6.12432e6	0.736739
$587$	−3.51780e6	−0.421381	−0.210691	−	0.977553i	$-0.567571\pi$
$587$	−3.51780e6	−0.421381	−0.210691	+	0.977553i	$0.567571\pi$
$588$	4.00968e6	0.478263
$589$	2.47411e6	0.293854
$590$	0	0
$591$	−4.05273e6	−0.477286
$592$	3.08813e6	0.362152
$593$	8.34535e6	0.974558	0.487279	−	0.873246i	$-0.337989\pi$
$593$	8.34535e6	0.974558	0.487279	+	0.873246i	$0.337989\pi$
$594$	1.89486e6	0.220349
$595$	0	0
$596$	−1.16272e6	−0.134079
$597$	649800.	0.0746180
$598$	−1.70317e7	−1.94763
$599$	6.15022e6	0.700364	0.350182	−	0.936682i	$-0.386120\pi$
$599$	6.15022e6	0.700364	0.350182	+	0.936682i	$0.386120\pi$
$600$	0	0
$601$	−6.86232e6	−0.774970	−0.387485	−	0.921876i	$-0.626656\pi$
$601$	−6.86232e6	−0.774970	−0.387485	+	0.921876i	$0.626656\pi$
$602$	−161040.	−0.0181110
$603$	539982.	0.0604764
$604$	4.84931e6	0.540864
$605$	0	0
$606$	5.80380e6	0.641994
$607$	9.45536e6	1.04161	0.520807	−	0.853675i	$-0.325631\pi$
$607$	9.45536e6	1.04161	0.520807	+	0.853675i	$0.325631\pi$
$608$	−1.96608e6	−0.215696
$609$	813600.	0.0888930
$610$	0	0
$611$	−8.22886e6	−0.891737
$612$	−197568.	−0.0213225
$613$	4.63658e6	0.498363	0.249182	−	0.968457i	$-0.419838\pi$
$613$	4.63658e6	0.498363	0.249182	+	0.968457i	$0.419838\pi$
$614$	4.57072e6	0.489287
$615$	0	0
$616$	−232320.	−0.0246680
$617$	−6.05704e6	−0.640542	−0.320271	−	0.947326i	$-0.603774\pi$
$617$	−6.05704e6	−0.640542	−0.320271	+	0.947326i	$0.603774\pi$
$618$	5.74224e6	0.604798
$619$	−5.63994e6	−0.591626	−0.295813	−	0.955246i	$-0.595591\pi$
$619$	−5.63994e6	−0.591626	−0.295813	+	0.955246i	$0.595591\pi$
$620$	0	0
$621$	1.45207e7	1.51098
$622$	2.34782e6	0.243327
$623$	−174150.	−0.0179764
$624$	−4.40832e6	−0.453223
$625$	0	0
$626$	935428.	0.0954057
$627$	696960.	0.0708009
$628$	−8.52779e6	−0.862854
$629$	8.27522e6	0.833975
$630$	0	0
$631$	1.12616e6	0.112597	0.0562987	−	0.998414i	$-0.482070\pi$
$631$	1.12616e6	0.112597	0.0562987	+	0.998414i	$0.482070\pi$
$632$	−1.25065e7	−1.24550
$633$	1.54453e7	1.53210
$634$	3.74201e6	0.369728
$635$	0	0
$636$	−7.16688e6	−0.702566
$637$	−1.91796e7	−1.87280
$638$	2.62522e6	0.255337
$639$	−558414.	−0.0541009
$640$	0	0
$641$	−1.42020e7	−1.36522	−0.682611	−	0.730782i	$-0.739156\pi$
$641$	−1.42020e7	−1.36522	−0.682611	+	0.730782i	$0.739156\pi$
$642$	1.95948e6	0.187630
$643$	−1.60794e6	−0.153371	−0.0766853	−	0.997055i	$-0.524434\pi$
$643$	−1.60794e6	−0.153371	−0.0766853	+	0.997055i	$0.524434\pi$
$644$	−593440.	−0.0563848
$645$	0	0
$646$	1.05370e6	0.0993423
$647$	−3.10236e6	−0.291361	−0.145680	−	0.989332i	$-0.546537\pi$
$647$	−3.10236e6	−0.291361	−0.145680	+	0.989332i	$0.546537\pi$
$648$	1.04354e7	0.976273
$649$	781781.	0.0728573
$650$	0	0
$651$	966450.	0.0893772
$652$	4.51322e6	0.415783
$653$	−6.88852e6	−0.632183	−0.316091	−	0.948729i	$-0.602371\pi$
$653$	−6.88852e6	−0.632183	−0.316091	+	0.948729i	$0.602371\pi$
$654$	−1.11263e7	−1.01720
$655$	0	0
$656$	391168.	0.0354898
$657$	34632.0	0.00313014
$658$	286720.	0.0258163
$659$	−1.24134e7	−1.11347	−0.556735	−	0.830690i	$-0.687946\pi$
$659$	−1.24134e7	−1.11347	−0.556735	+	0.830690i	$0.687946\pi$
$660$	0	0
$661$	−8.10994e6	−0.721961	−0.360980	−	0.932573i	$-0.617558\pi$
$661$	−8.10994e6	−0.721961	−0.360980	+	0.932573i	$0.617558\pi$
$662$	−4.23292e6	−0.375401
$663$	−1.18129e7	−1.04369
$664$	−1.97211e7	−1.73584
$665$	0	0
$666$	868536.	0.0758756
$667$	2.01176e7	1.75090
$668$	−9.17741e6	−0.795754
$669$	−6.91922e6	−0.597711
$670$	0	0
$671$	2.05458e6	0.176164
$672$	−768000.	−0.0656052
$673$	−1.78063e7	−1.51543	−0.757717	−	0.652584i	$-0.773685\pi$
$673$	−1.78063e7	−1.51543	−0.757717	+	0.652584i	$0.773685\pi$
$674$	−2.02474e6	−0.171680
$675$	0	0
$676$	−1.51458e7	−1.27475
$677$	−1.55179e7	−1.30125	−0.650626	−	0.759398i	$-0.725493\pi$
$677$	−1.55179e7	−1.30125	−0.650626	+	0.759398i	$0.725493\pi$
$678$	−4.37094e6	−0.365175
$679$	669050.	0.0556909
$680$	0	0
$681$	1.28336e7	1.06042
$682$	3.11841e6	0.256728
$683$	2.18106e6	0.178902	0.0894510	−	0.995991i	$-0.471489\pi$
$683$	2.18106e6	0.178902	0.0894510	+	0.995991i	$0.471489\pi$
$684$	−110592.	−0.00903823
$685$	0	0
$686$	1.34056e6	0.108762
$687$	−9.98708e6	−0.807321
$688$	−1.03066e6	−0.0830123
$689$	3.42816e7	2.75114
$690$	0	0
$691$	2.29892e7	1.83159	0.915795	−	0.401647i	$-0.131562\pi$
$691$	2.29892e7	1.83159	0.915795	+	0.401647i	$0.131562\pi$
$692$	−6.18058e6	−0.490641
$693$	−21780.0	−0.00172276
$694$	−1.87054e6	−0.147424
$695$	0	0
$696$	1.56211e7	1.22233
$697$	1.04821e6	0.0817270
$698$	1.21788e6	0.0946163
$699$	−1.81400e7	−1.40425
$700$	0	0
$701$	2.34092e6	0.179925	0.0899626	−	0.995945i	$-0.471325\pi$
$701$	2.34092e6	0.179925	0.0899626	+	0.995945i	$0.471325\pi$
$702$	−1.79777e7	−1.37686
$703$	4.63219e6	0.353507
$704$	−3.46931e6	−0.263822
$705$	0	0
$706$	−1.00747e7	−0.760715
$707$	−967300.	−0.0727801
$708$	1.55064e6	0.116259
$709$	−1.92694e7	−1.43964	−0.719820	−	0.694161i	$-0.755775\pi$
$709$	−1.92694e7	−1.43964	−0.719820	+	0.694161i	$0.755775\pi$
$710$	0	0
$711$	−1.17248e6	−0.0869827
$712$	−3.34368e6	−0.247186
$713$	2.38971e7	1.76044
$714$	411600.	0.0302155
$715$	0	0
$716$	−8.72126e6	−0.635765
$717$	−8.57223e6	−0.622724
$718$	−1.20736e7	−0.874032
$719$	−2.14665e7	−1.54860	−0.774300	−	0.632819i	$-0.781898\pi$
$719$	−2.14665e7	−1.54860	−0.774300	+	0.632819i	$0.781898\pi$
$720$	0	0
$721$	−957040.	−0.0685633
$722$	−9.31457e6	−0.664997
$723$	−4.00620e6	−0.285028
$724$	4.47176e6	0.317053
$725$	0	0
$726$	878460.	0.0618558
$727$	1.67705e7	1.17682	0.588411	−	0.808562i	$-0.299754\pi$
$727$	1.67705e7	1.17682	0.588411	+	0.808562i	$0.299754\pi$
$728$	2.20416e6	0.154140
$729$	1.52485e7	1.06269
$730$	0	0
$731$	−2.76184e6	−0.191163
$732$	4.07520e6	0.281107
$733$	1.75373e7	1.20560	0.602798	−	0.797894i	$-0.294052\pi$
$733$	1.75373e7	1.20560	0.602798	+	0.797894i	$0.294052\pi$
$734$	−3.97772e6	−0.272517
$735$	0	0
$736$	−1.89901e7	−1.29221
$737$	3.62988e6	0.246163
$738$	110016.	0.00743558
$739$	1.47387e7	0.992766	0.496383	−	0.868104i	$-0.334661\pi$
$739$	1.47387e7	0.992766	0.496383	+	0.868104i	$0.334661\pi$
$740$	0	0
$741$	−6.61248e6	−0.442404
$742$	−1.19448e6	−0.0796469
$743$	4.80946e6	0.319613	0.159806	−	0.987148i	$-0.448913\pi$
$743$	4.80946e6	0.319613	0.159806	+	0.987148i	$0.448913\pi$
$744$	1.85558e7	1.22899
$745$	0	0
$746$	−6.91583e6	−0.454985
$747$	−1.84885e6	−0.121227
$748$	−1.32810e6	−0.0867912
$749$	−326580.	−0.0212709
$750$	0	0
$751$	8.29317e6	0.536563	0.268282	−	0.963341i	$-0.413544\pi$
$751$	8.29317e6	0.536563	0.268282	+	0.963341i	$0.413544\pi$
$752$	1.83501e6	0.118330
$753$	2.08105e7	1.33750
$754$	−2.49070e7	−1.59549
$755$	0	0
$756$	−626400.	−0.0398609
$757$	352294.	0.0223442	0.0111721	−	0.999938i	$-0.496444\pi$
$757$	352294.	0.0223442	0.0111721	+	0.999938i	$0.496444\pi$
$758$	1.81906e6	0.114994
$759$	6.73184e6	0.424159
$760$	0	0
$761$	1.68985e7	1.05776	0.528878	−	0.848698i	$-0.322613\pi$
$761$	1.68985e7	1.05776	0.528878	+	0.848698i	$0.322613\pi$
$762$	4.69104e6	0.292672
$763$	1.85438e6	0.115315
$764$	7.11099e6	0.440755
$765$	0	0
$766$	−9.10229e6	−0.560504
$767$	−7.41723e6	−0.455253
$768$	−1.67117e7	−1.02239
$769$	−36652.0	−0.00223502	−0.00111751	−	0.999999i	$-0.500356\pi$
$769$	−36652.0	−0.00223502	−0.00111751	+	0.999999i	$0.500356\pi$
$770$	0	0
$771$	1.32888e7	0.805102
$772$	−295616.	−0.0178519
$773$	3.17462e7	1.91093	0.955463	−	0.295113i	$-0.0953571\pi$
$773$	3.17462e7	1.91093	0.955463	+	0.295113i	$0.0953571\pi$
$774$	−289872.	−0.0173922
$775$	0	0
$776$	1.28458e7	0.765783
$777$	1.80945e6	0.107521
$778$	1.55912e6	0.0923489
$779$	586752.	0.0346426
$780$	0	0
$781$	−3.75378e6	−0.220212
$782$	1.01775e7	0.595147
$783$	2.12350e7	1.23779
$784$	4.27699e6	0.248513
$785$	0	0
$786$	−27720.0	−0.00160043
$787$	−2.01985e7	−1.16247	−0.581236	−	0.813735i	$-0.697431\pi$
$787$	−2.01985e7	−1.16247	−0.581236	+	0.813735i	$0.697431\pi$
$788$	4.32291e6	0.248005
$789$	−2.17031e7	−1.24116
$790$	0	0
$791$	728490.	0.0413983
$792$	−418176.	−0.0236890
$793$	−1.94930e7	−1.10077
$794$	6.47732e6	0.364623
$795$	0	0
$796$	−693120.	−0.0387727
$797$	−1.55660e7	−0.868023	−0.434011	−	0.900907i	$-0.642902\pi$
$797$	−1.55660e7	−0.868023	−0.434011	+	0.900907i	$0.642902\pi$
$798$	230400.	0.0128078
$799$	4.91725e6	0.272493
$800$	0	0
$801$	−313470.	−0.0172629
$802$	−2.21747e7	−1.21737
$803$	232804.	0.0127409
$804$	7.19976e6	0.392806
$805$	0	0
$806$	−2.95863e7	−1.60418
$807$	−5.30817e6	−0.286920
$808$	−1.85722e7	−1.00077
$809$	−2.91667e7	−1.56681	−0.783404	−	0.621513i	$-0.786518\pi$
$809$	−2.91667e7	−1.56681	−0.783404	+	0.621513i	$0.786518\pi$
$810$	0	0
$811$	−1.65215e7	−0.882057	−0.441029	−	0.897493i	$-0.645386\pi$
$811$	−1.65215e7	−0.882057	−0.441029	+	0.897493i	$0.645386\pi$
$812$	−867840.	−0.0461902
$813$	7.87890e6	0.418061
$814$	5.83849e6	0.308844
$815$	0	0
$816$	2.63424e6	0.138494
$817$	−1.54598e6	−0.0810307
$818$	−1.08197e7	−0.565369
$819$	206640.	0.0107648
$820$	0	0
$821$	5.56614e6	0.288202	0.144101	−	0.989563i	$-0.453971\pi$
$821$	5.56614e6	0.288202	0.144101	+	0.989563i	$0.453971\pi$
$822$	−1.77740e7	−0.917498
$823$	−1.18801e7	−0.611391	−0.305696	−	0.952129i	$-0.598889\pi$
$823$	−1.18801e7	−0.611391	−0.305696	+	0.952129i	$0.598889\pi$
$824$	−1.83752e7	−0.942786
$825$	0	0
$826$	258440.	0.0131798
$827$	1.32856e7	0.675489	0.337745	−	0.941238i	$-0.390336\pi$
$827$	1.32856e7	0.675489	0.337745	+	0.941238i	$0.390336\pi$
$828$	−1.06819e6	−0.0541469
$829$	−653987.	−0.0330509	−0.0165254	−	0.999863i	$-0.505260\pi$
$829$	−653987.	−0.0330509	−0.0165254	+	0.999863i	$0.505260\pi$
$830$	0	0
$831$	8.93415e6	0.448798
$832$	3.29155e7	1.64851
$833$	1.14610e7	0.572282
$834$	−2.39891e7	−1.19426
$835$	0	0
$836$	−743424.	−0.0367892
$837$	2.52243e7	1.24453
$838$	1.34935e7	0.663765
$839$	2.47747e7	1.21508	0.607538	−	0.794290i	$-0.292157\pi$
$839$	2.47747e7	1.21508	0.607538	+	0.794290i	$0.292157\pi$
$840$	0	0
$841$	8.90863e6	0.434331
$842$	−1.81021e7	−0.879929
$843$	1.09848e7	0.532380
$844$	−1.64749e7	−0.796100
$845$	0	0
$846$	516096.	0.0247916
$847$	−146410.	−0.00701233
$848$	−7.64467e6	−0.365064
$849$	−3.35071e7	−1.59539
$850$	0	0
$851$	4.47417e7	2.11782
$852$	−7.44552e6	−0.351395
$853$	−2.71291e7	−1.27662	−0.638311	−	0.769779i	$-0.720367\pi$
$853$	−2.71291e7	−1.27662	−0.638311	+	0.769779i	$0.720367\pi$
$854$	679200.	0.0318679
$855$	0	0
$856$	−6.27034e6	−0.292487
$857$	2.84232e7	1.32197	0.660984	−	0.750400i	$-0.270139\pi$
$857$	2.84232e7	1.32197	0.660984	+	0.750400i	$0.270139\pi$
$858$	−8.33448e6	−0.386510
$859$	2.65922e7	1.22962	0.614810	−	0.788675i	$-0.289233\pi$
$859$	2.65922e7	1.22962	0.614810	+	0.788675i	$0.289233\pi$
$860$	0	0
$861$	229200.	0.0105368
$862$	−2.73814e6	−0.125512
$863$	2.22500e7	1.01696	0.508479	−	0.861074i	$-0.330208\pi$
$863$	2.22500e7	1.01696	0.508479	+	0.861074i	$0.330208\pi$
$864$	−2.00448e7	−0.913519
$865$	0	0
$866$	1.69036e7	0.765923
$867$	−1.42389e7	−0.643323
$868$	−1.03088e6	−0.0464418
$869$	−7.88170e6	−0.354055
$870$	0	0
$871$	−3.44389e7	−1.53817
$872$	3.56041e7	1.58566
$873$	1.20429e6	0.0534805
$874$	5.69702e6	0.252272
$875$	0	0
$876$	461760.	0.0203309
$877$	−2.83428e7	−1.24435	−0.622176	−	0.782877i	$-0.713751\pi$
$877$	−2.83428e7	−1.24435	−0.622176	+	0.782877i	$0.713751\pi$
$878$	−8.36739e6	−0.366314
$879$	2.29662e7	1.00258
$880$	0	0
$881$	3.66445e7	1.59063	0.795315	−	0.606196i	$-0.207305\pi$
$881$	3.66445e7	1.59063	0.795315	+	0.606196i	$0.207305\pi$
$882$	1.20290e6	0.0520666
$883$	−1.68772e7	−0.728447	−0.364223	−	0.931312i	$-0.618666\pi$
$883$	−1.68772e7	−0.728447	−0.364223	+	0.931312i	$0.618666\pi$
$884$	1.26004e7	0.542320
$885$	0	0
$886$	−6.25137e6	−0.267541
$887$	−2.73941e6	−0.116909	−0.0584544	−	0.998290i	$-0.518617\pi$
$887$	−2.73941e6	−0.116909	−0.0584544	+	0.998290i	$0.518617\pi$
$888$	3.47414e7	1.47848
$889$	−781840.	−0.0331790
$890$	0	0
$891$	6.57647e6	0.277523
$892$	7.38050e6	0.310580
$893$	2.75251e6	0.115505
$894$	4.36020e6	0.182458
$895$	0	0
$896$	491520.	0.0204537
$897$	−6.38690e7	−2.65038
$898$	−1.20180e7	−0.497325
$899$	3.49468e7	1.44214
$900$	0	0
$901$	−2.04853e7	−0.840681
$902$	739552.	0.0302658
$903$	−603900.	−0.0246460
$904$	1.39870e7	0.569251
$905$	0	0
$906$	−1.81849e7	−0.736022
$907$	3.13286e7	1.26451	0.632255	−	0.774760i	$-0.282129\pi$
$907$	3.13286e7	1.26451	0.632255	+	0.774760i	$0.282129\pi$
$908$	−1.36891e7	−0.551012
$909$	−1.74114e6	−0.0698914
$910$	0	0
$911$	−2.49762e7	−0.997081	−0.498541	−	0.866866i	$-0.666131\pi$
$911$	−2.49762e7	−0.997081	−0.498541	+	0.866866i	$0.666131\pi$
$912$	1.47456e6	0.0587050
$913$	−1.24284e7	−0.493444
$914$	9.78206e6	0.387316
$915$	0	0
$916$	1.06529e7	0.419497
$917$	4620.00	0.000181434 0
$918$	1.07428e7	0.420736
$919$	−1.10613e7	−0.432032	−0.216016	−	0.976390i	$-0.569306\pi$
$919$	−1.10613e7	−0.432032	−0.216016	+	0.976390i	$0.569306\pi$
$920$	0	0
$921$	1.71402e7	0.665835
$922$	3.11641e7	1.20734
$923$	3.56144e7	1.37601
$924$	−290400.	−0.0111897
$925$	0	0
$926$	4.20784e6	0.161262
$927$	−1.72267e6	−0.0658420
$928$	−2.77709e7	−1.05857
$929$	−2.01739e7	−0.766919	−0.383460	−	0.923558i	$-0.625267\pi$
$929$	−2.01739e7	−0.766919	−0.383460	+	0.923558i	$0.625267\pi$
$930$	0	0
$931$	6.41549e6	0.242580
$932$	1.93494e7	0.729671
$933$	8.80434e6	0.331126
$934$	−1.58801e7	−0.595645
$935$	0	0
$936$	3.96749e6	0.148022
$937$	−9.10734e6	−0.338877	−0.169439	−	0.985541i	$-0.554195\pi$
$937$	−9.10734e6	−0.338877	−0.169439	+	0.985541i	$0.554195\pi$
$938$	1.19996e6	0.0445307
$939$	3.50785e6	0.129831
$940$	0	0
$941$	−3.67709e7	−1.35372	−0.676861	−	0.736110i	$-0.736660\pi$
$941$	−3.67709e7	−1.35372	−0.676861	+	0.736110i	$0.736660\pi$
$942$	3.19792e7	1.17420
$943$	5.66735e6	0.207540
$944$	1.65402e6	0.0604101
$945$	0	0
$946$	−1.94858e6	−0.0707932
$947$	4.95743e7	1.79631	0.898156	−	0.439677i	$-0.144907\pi$
$947$	4.95743e7	1.79631	0.898156	+	0.439677i	$0.144907\pi$
$948$	−1.56331e7	−0.564969
$949$	−2.20875e6	−0.0796125
$950$	0	0
$951$	1.40325e7	0.503136
$952$	−1.31712e6	−0.0471013
$953$	−3.53787e7	−1.26186	−0.630928	−	0.775841i	$-0.717326\pi$
$953$	−3.53787e7	−1.26186	−0.630928	+	0.775841i	$0.717326\pi$
$954$	−2.15006e6	−0.0764857
$955$	0	0
$956$	9.14371e6	0.323577
$957$	9.84456e6	0.347469
$958$	−3.41563e7	−1.20242
$959$	2.96233e6	0.104013
$960$	0	0
$961$	1.28831e7	0.449999
$962$	−5.53933e7	−1.92983
$963$	−587844.	−0.0204266
$964$	4.27328e6	0.148105
$965$	0	0
$966$	2.22540e6	0.0767300
$967$	−2.78059e7	−0.956248	−0.478124	−	0.878292i	$-0.658683\pi$
$967$	−2.78059e7	−0.956248	−0.478124	+	0.878292i	$0.658683\pi$
$968$	−2.81107e6	−0.0964237
$969$	3.95136e6	0.135188
$970$	0	0
$971$	1.56835e7	0.533821	0.266910	−	0.963721i	$-0.413997\pi$
$971$	1.56835e7	0.533821	0.266910	+	0.963721i	$0.413997\pi$
$972$	−2.17728e6	−0.0739177
$973$	3.99818e6	0.135388
$974$	7.45947e6	0.251948
$975$	0	0
$976$	4.34688e6	0.146067
$977$	−2.01140e7	−0.674157	−0.337079	−	0.941476i	$-0.609439\pi$
$977$	−2.01140e7	−0.674157	−0.337079	+	0.941476i	$0.609439\pi$
$978$	−1.69246e7	−0.565810
$979$	−2.10722e6	−0.0702671
$980$	0	0
$981$	3.33788e6	0.110739
$982$	2.06291e7	0.682655
$983$	2.09269e7	0.690750	0.345375	−	0.938465i	$-0.387752\pi$
$983$	2.09269e7	0.690750	0.345375	+	0.938465i	$0.387752\pi$
$984$	4.40064e6	0.144887
$985$	0	0
$986$	1.48835e7	0.487541
$987$	1.07520e6	0.0351315
$988$	7.05331e6	0.229880
$989$	−1.49324e7	−0.485445
$990$	0	0
$991$	−3.18663e7	−1.03074	−0.515368	−	0.856969i	$-0.672345\pi$
$991$	−3.18663e7	−1.03074	−0.515368	+	0.856969i	$0.672345\pi$
$992$	−3.29882e7	−1.06434
$993$	−1.58735e7	−0.510856
$994$	−1.24092e6	−0.0398362
$995$	0	0
$996$	−2.46514e7	−0.787395
$997$	−1.38913e6	−0.0442595	−0.0221297	−	0.999755i	$-0.507045\pi$
$997$	−1.38913e6	−0.0442595	−0.0221297	+	0.999755i	$0.507045\pi$
$998$	1.81336e7	0.576313
$999$	4.72266e7	1.49718

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.6.a.a.1.1		1
5.2	odd	4		275.6.b.a.199.2		2
5.3	odd	4		275.6.b.a.199.1		2
5.4	even	2		11.6.a.a.1.1	✓	1
15.14	odd	2		99.6.a.c.1.1		1
20.19	odd	2		176.6.a.c.1.1		1
35.34	odd	2		539.6.a.c.1.1		1
40.19	odd	2		704.6.a.c.1.1		1
40.29	even	2		704.6.a.h.1.1		1
55.54	odd	2		121.6.a.b.1.1		1
165.164	even	2		1089.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.a.a.1.1	✓	1	5.4	even	2
99.6.a.c.1.1		1	15.14	odd	2
121.6.a.b.1.1		1	55.54	odd	2
176.6.a.c.1.1		1	20.19	odd	2
275.6.a.a.1.1		1	1.1	even	1	trivial
275.6.b.a.199.1		2	5.3	odd	4
275.6.b.a.199.2		2	5.2	odd	4
539.6.a.c.1.1		1	35.34	odd	2
704.6.a.c.1.1		1	40.19	odd	2
704.6.a.h.1.1		1	40.29	even	2
1089.6.a.c.1.1		1	165.164	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

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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