LMFDB - Embedded newform 390.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,6,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("390.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	390.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:	$62.5496897271$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} +36.0000 q^{6} +227.000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} +36.0000 q^{6} +227.000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -100.000 q^{10} -303.000 q^{11} -144.000 q^{12} +169.000 q^{13} -908.000 q^{14} -225.000 q^{15} +256.000 q^{16} -1203.00 q^{17} -324.000 q^{18} -34.0000 q^{19} +400.000 q^{20} -2043.00 q^{21} +1212.00 q^{22} -4395.00 q^{23} +576.000 q^{24} +625.000 q^{25} -676.000 q^{26} -729.000 q^{27} +3632.00 q^{28} -3246.00 q^{29} +900.000 q^{30} +1778.00 q^{31} -1024.00 q^{32} +2727.00 q^{33} +4812.00 q^{34} +5675.00 q^{35} +1296.00 q^{36} -7819.00 q^{37} +136.000 q^{38} -1521.00 q^{39} -1600.00 q^{40} +10659.0 q^{41} +8172.00 q^{42} -7876.00 q^{43} -4848.00 q^{44} +2025.00 q^{45} +17580.0 q^{46} +21102.0 q^{47} -2304.00 q^{48} +34722.0 q^{49} -2500.00 q^{50} +10827.0 q^{51} +2704.00 q^{52} -9723.00 q^{53} +2916.00 q^{54} -7575.00 q^{55} -14528.0 q^{56} +306.000 q^{57} +12984.0 q^{58} -50520.0 q^{59} -3600.00 q^{60} -23599.0 q^{61} -7112.00 q^{62} +18387.0 q^{63} +4096.00 q^{64} +4225.00 q^{65} -10908.0 q^{66} +55484.0 q^{67} -19248.0 q^{68} +39555.0 q^{69} -22700.0 q^{70} +20721.0 q^{71} -5184.00 q^{72} -40042.0 q^{73} +31276.0 q^{74} -5625.00 q^{75} -544.000 q^{76} -68781.0 q^{77} +6084.00 q^{78} -82753.0 q^{79} +6400.00 q^{80} +6561.00 q^{81} -42636.0 q^{82} -42048.0 q^{83} -32688.0 q^{84} -30075.0 q^{85} +31504.0 q^{86} +29214.0 q^{87} +19392.0 q^{88} +39441.0 q^{89} -8100.00 q^{90} +38363.0 q^{91} -70320.0 q^{92} -16002.0 q^{93} -84408.0 q^{94} -850.000 q^{95} +9216.00 q^{96} +151667. q^{97} -138888. q^{98} -24543.0 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	36.0000	0.408248
$7$	227.000	1.75098	0.875489	−	0.483238i	$-0.160539\pi$
$7$	227.000	1.75098	0.875489	+	0.483238i	$0.160539\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	−100.000	−0.316228
$11$	−303.000	−0.755024	−0.377512	−	0.926005i	$-0.623220\pi$
$11$	−303.000	−0.755024	−0.377512	+	0.926005i	$0.623220\pi$
$12$	−144.000	−0.288675
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−908.000	−1.23813
$15$	−225.000	−0.258199
$16$	256.000	0.250000
$17$	−1203.00	−1.00959	−0.504793	−	0.863240i	$-0.668431\pi$
$17$	−1203.00	−1.00959	−0.504793	+	0.863240i	$0.668431\pi$
$18$	−324.000	−0.235702
$19$	−34.0000	−0.0216070	−0.0108035	−	0.999942i	$-0.503439\pi$
$19$	−34.0000	−0.0216070	−0.0108035	+	0.999942i	$0.503439\pi$
$20$	400.000	0.223607
$21$	−2043.00	−1.01093
$22$	1212.00	0.533883
$23$	−4395.00	−1.73236	−0.866182	−	0.499728i	$-0.833433\pi$
$23$	−4395.00	−1.73236	−0.866182	+	0.499728i	$0.833433\pi$
$24$	576.000	0.204124
$25$	625.000	0.200000
$26$	−676.000	−0.196116
$27$	−729.000	−0.192450
$28$	3632.00	0.875489
$29$	−3246.00	−0.716727	−0.358363	−	0.933582i	$-0.616665\pi$
$29$	−3246.00	−0.716727	−0.358363	+	0.933582i	$0.616665\pi$
$30$	900.000	0.182574
$31$	1778.00	0.332298	0.166149	−	0.986101i	$-0.446867\pi$
$31$	1778.00	0.332298	0.166149	+	0.986101i	$0.446867\pi$
$32$	−1024.00	−0.176777
$33$	2727.00	0.435913
$34$	4812.00	0.713885
$35$	5675.00	0.783061
$36$	1296.00	0.166667
$37$	−7819.00	−0.938960	−0.469480	−	0.882943i	$-0.655559\pi$
$37$	−7819.00	−0.938960	−0.469480	+	0.882943i	$0.655559\pi$
$38$	136.000	0.0152785
$39$	−1521.00	−0.160128
$40$	−1600.00	−0.158114
$41$	10659.0	0.990277	0.495139	−	0.868814i	$-0.335117\pi$
$41$	10659.0	0.990277	0.495139	+	0.868814i	$0.335117\pi$
$42$	8172.00	0.714834
$43$	−7876.00	−0.649583	−0.324791	−	0.945786i	$-0.605294\pi$
$43$	−7876.00	−0.649583	−0.324791	+	0.945786i	$0.605294\pi$
$44$	−4848.00	−0.377512
$45$	2025.00	0.149071
$46$	17580.0	1.22497
$47$	21102.0	1.39341	0.696705	−	0.717358i	$-0.254649\pi$
$47$	21102.0	1.39341	0.696705	+	0.717358i	$0.254649\pi$
$48$	−2304.00	−0.144338
$49$	34722.0	2.06592
$50$	−2500.00	−0.141421
$51$	10827.0	0.582885
$52$	2704.00	0.138675
$53$	−9723.00	−0.475456	−0.237728	−	0.971332i	$-0.576403\pi$
$53$	−9723.00	−0.475456	−0.237728	+	0.971332i	$0.576403\pi$
$54$	2916.00	0.136083
$55$	−7575.00	−0.337657
$56$	−14528.0	−0.619064
$57$	306.000	0.0124748
$58$	12984.0	0.506802
$59$	−50520.0	−1.88944	−0.944720	−	0.327877i	$-0.893667\pi$
$59$	−50520.0	−1.88944	−0.944720	+	0.327877i	$0.893667\pi$
$60$	−3600.00	−0.129099
$61$	−23599.0	−0.812024	−0.406012	−	0.913868i	$-0.633081\pi$
$61$	−23599.0	−0.812024	−0.406012	+	0.913868i	$0.633081\pi$
$62$	−7112.00	−0.234970
$63$	18387.0	0.583659
$64$	4096.00	0.125000
$65$	4225.00	0.124035
$66$	−10908.0	−0.308237
$67$	55484.0	1.51001	0.755007	−	0.655717i	$-0.227634\pi$
$67$	55484.0	1.51001	0.755007	+	0.655717i	$0.227634\pi$
$68$	−19248.0	−0.504793
$69$	39555.0	1.00018
$70$	−22700.0	−0.553708
$71$	20721.0	0.487826	0.243913	−	0.969797i	$-0.421569\pi$
$71$	20721.0	0.487826	0.243913	+	0.969797i	$0.421569\pi$
$72$	−5184.00	−0.117851
$73$	−40042.0	−0.879445	−0.439722	−	0.898134i	$-0.644923\pi$
$73$	−40042.0	−0.879445	−0.439722	+	0.898134i	$0.644923\pi$
$74$	31276.0	0.663945
$75$	−5625.00	−0.115470
$76$	−544.000	−0.0108035
$77$	−68781.0	−1.32203
$78$	6084.00	0.113228
$79$	−82753.0	−1.49182	−0.745909	−	0.666048i	$-0.767985\pi$
$79$	−82753.0	−1.49182	−0.745909	+	0.666048i	$0.767985\pi$
$80$	6400.00	0.111803
$81$	6561.00	0.111111
$82$	−42636.0	−0.700232
$83$	−42048.0	−0.669962	−0.334981	−	0.942225i	$-0.608730\pi$
$83$	−42048.0	−0.669962	−0.334981	+	0.942225i	$0.608730\pi$
$84$	−32688.0	−0.505464
$85$	−30075.0	−0.451501
$86$	31504.0	0.459324
$87$	29214.0	0.413802
$88$	19392.0	0.266941
$89$	39441.0	0.527804	0.263902	−	0.964549i	$-0.414990\pi$
$89$	39441.0	0.527804	0.263902	+	0.964549i	$0.414990\pi$
$90$	−8100.00	−0.105409
$91$	38363.0	0.485634
$92$	−70320.0	−0.866182
$93$	−16002.0	−0.191852
$94$	−84408.0	−0.985290
$95$	−850.000	−0.00966295
$96$	9216.00	0.102062
$97$	151667.	1.63667	0.818336	−	0.574740i	$-0.194897\pi$
$97$	151667.	1.63667	0.818336	+	0.574740i	$0.194897\pi$
$98$	−138888.	−1.46083
$99$	−24543.0	−0.251675
$100$	10000.0	0.100000
$101$	−37236.0	−0.363211	−0.181606	−	0.983371i	$-0.558129\pi$
$101$	−37236.0	−0.363211	−0.181606	+	0.983371i	$0.558129\pi$
$102$	−43308.0	−0.412162
$103$	−73924.0	−0.686582	−0.343291	−	0.939229i	$-0.611542\pi$
$103$	−73924.0	−0.686582	−0.343291	+	0.939229i	$0.611542\pi$
$104$	−10816.0	−0.0980581
$105$	−51075.0	−0.452101
$106$	38892.0	0.336198
$107$	90621.0	0.765190	0.382595	−	0.923916i	$-0.375030\pi$
$107$	90621.0	0.765190	0.382595	+	0.923916i	$0.375030\pi$
$108$	−11664.0	−0.0962250
$109$	−189208.	−1.52536	−0.762682	−	0.646774i	$-0.776118\pi$
$109$	−189208.	−1.52536	−0.762682	+	0.646774i	$0.776118\pi$
$110$	30300.0	0.238760
$111$	70371.0	0.542109
$112$	58112.0	0.437745
$113$	−114414.	−0.842914	−0.421457	−	0.906848i	$-0.638481\pi$
$113$	−114414.	−0.842914	−0.421457	+	0.906848i	$0.638481\pi$
$114$	−1224.00	−0.00882103
$115$	−109875.	−0.774737
$116$	−51936.0	−0.358363
$117$	13689.0	0.0924500
$118$	202080.	1.33604
$119$	−273081.	−1.76776
$120$	14400.0	0.0912871
$121$	−69242.0	−0.429938
$122$	94396.0	0.574188
$123$	−95931.0	−0.571737
$124$	28448.0	0.166149
$125$	15625.0	0.0894427
$126$	−73548.0	−0.412710
$127$	76106.0	0.418706	0.209353	−	0.977840i	$-0.432864\pi$
$127$	76106.0	0.418706	0.209353	+	0.977840i	$0.432864\pi$
$128$	−16384.0	−0.0883883
$129$	70884.0	0.375037
$130$	−16900.0	−0.0877058
$131$	−225966.	−1.15044	−0.575221	−	0.817998i	$-0.695084\pi$
$131$	−225966.	−1.15044	−0.575221	+	0.817998i	$0.695084\pi$
$132$	43632.0	0.217957
$133$	−7718.00	−0.0378334
$134$	−221936.	−1.06774
$135$	−18225.0	−0.0860663
$136$	76992.0	0.356943
$137$	229566.	1.04498	0.522488	−	0.852647i	$-0.325004\pi$
$137$	229566.	1.04498	0.522488	+	0.852647i	$0.325004\pi$
$138$	−158220.	−0.707235
$139$	−266431.	−1.16963	−0.584814	−	0.811167i	$-0.698833\pi$
$139$	−266431.	−1.16963	−0.584814	+	0.811167i	$0.698833\pi$
$140$	90800.0	0.391531
$141$	−189918.	−0.804486
$142$	−82884.0	−0.344945
$143$	−51207.0	−0.209406
$144$	20736.0	0.0833333
$145$	−81150.0	−0.320530
$146$	160168.	0.621861
$147$	−312498.	−1.19276
$148$	−125104.	−0.469480
$149$	−80463.0	−0.296914	−0.148457	−	0.988919i	$-0.547431\pi$
$149$	−80463.0	−0.296914	−0.148457	+	0.988919i	$0.547431\pi$
$150$	22500.0	0.0816497
$151$	−257116.	−0.917670	−0.458835	−	0.888521i	$-0.651733\pi$
$151$	−257116.	−0.917670	−0.458835	+	0.888521i	$0.651733\pi$
$152$	2176.00	0.00763924
$153$	−97443.0	−0.336529
$154$	275124.	0.934817
$155$	44450.0	0.148608
$156$	−24336.0	−0.0800641
$157$	−400570.	−1.29697	−0.648484	−	0.761228i	$-0.724597\pi$
$157$	−400570.	−1.29697	−0.648484	+	0.761228i	$0.724597\pi$
$158$	331012.	1.05488
$159$	87507.0	0.274505
$160$	−25600.0	−0.0790569
$161$	−997665.	−3.03333
$162$	−26244.0	−0.0785674
$163$	608087.	1.79265	0.896327	−	0.443393i	$-0.146225\pi$
$163$	608087.	1.79265	0.896327	+	0.443393i	$0.146225\pi$
$164$	170544.	0.495139
$165$	68175.0	0.194946
$166$	168192.	0.473735
$167$	−699528.	−1.94095	−0.970474	−	0.241206i	$-0.922457\pi$
$167$	−699528.	−1.94095	−0.970474	+	0.241206i	$0.922457\pi$
$168$	130752.	0.357417
$169$	28561.0	0.0769231
$170$	120300.	0.319259
$171$	−2754.00	−0.00720234
$172$	−126016.	−0.324791
$173$	99774.0	0.253456	0.126728	−	0.991938i	$-0.459553\pi$
$173$	99774.0	0.253456	0.126728	+	0.991938i	$0.459553\pi$
$174$	−116856.	−0.292602
$175$	141875.	0.350196
$176$	−77568.0	−0.188756
$177$	454680.	1.09087
$178$	−157764.	−0.373214
$179$	449022.	1.04745	0.523727	−	0.851886i	$-0.324541\pi$
$179$	449022.	1.04745	0.523727	+	0.851886i	$0.324541\pi$
$180$	32400.0	0.0745356
$181$	−383731.	−0.870624	−0.435312	−	0.900280i	$-0.643362\pi$
$181$	−383731.	−0.870624	−0.435312	+	0.900280i	$0.643362\pi$
$182$	−153452.	−0.343395
$183$	212391.	0.468822
$184$	281280.	0.612483
$185$	−195475.	−0.419915
$186$	64008.0	0.135660
$187$	364509.	0.762262
$188$	337632.	0.696705
$189$	−165483.	−0.336976
$190$	3400.00	0.00683274
$191$	−726732.	−1.44142	−0.720711	−	0.693236i	$-0.756184\pi$
$191$	−726732.	−1.44142	−0.720711	+	0.693236i	$0.756184\pi$
$192$	−36864.0	−0.0721688
$193$	603905.	1.16701	0.583506	−	0.812109i	$-0.301680\pi$
$193$	603905.	1.16701	0.583506	+	0.812109i	$0.301680\pi$
$194$	−606668.	−1.15730
$195$	−38025.0	−0.0716115
$196$	555552.	1.03296
$197$	820140.	1.50564	0.752822	−	0.658224i	$-0.228692\pi$
$197$	820140.	1.50564	0.752822	+	0.658224i	$0.228692\pi$
$198$	98172.0	0.177961
$199$	−128284.	−0.229636	−0.114818	−	0.993387i	$-0.536628\pi$
$199$	−128284.	−0.229636	−0.114818	+	0.993387i	$0.536628\pi$
$200$	−40000.0	−0.0707107
$201$	−499356.	−0.871807
$202$	148944.	0.256829
$203$	−736842.	−1.25497
$204$	173232.	0.291442
$205$	266475.	0.442865
$206$	295696.	0.485487
$207$	−355995.	−0.577455
$208$	43264.0	0.0693375
$209$	10302.0	0.0163138
$210$	204300.	0.319683
$211$	442916.	0.684881	0.342441	−	0.939539i	$-0.388746\pi$
$211$	442916.	0.684881	0.342441	+	0.939539i	$0.388746\pi$
$212$	−155568.	−0.237728
$213$	−186489.	−0.281646
$214$	−362484.	−0.541071
$215$	−196900.	−0.290502
$216$	46656.0	0.0680414
$217$	403606.	0.581846
$218$	756832.	1.07859
$219$	360378.	0.507748
$220$	−121200.	−0.168829
$221$	−203307.	−0.280009
$222$	−281484.	−0.383329
$223$	674816.	0.908706	0.454353	−	0.890822i	$-0.349871\pi$
$223$	674816.	0.908706	0.454353	+	0.890822i	$0.349871\pi$
$224$	−232448.	−0.309532
$225$	50625.0	0.0666667
$226$	457656.	0.596030
$227$	−705918.	−0.909263	−0.454632	−	0.890680i	$-0.650229\pi$
$227$	−705918.	−0.909263	−0.454632	+	0.890680i	$0.650229\pi$
$228$	4896.00	0.00623741
$229$	648098.	0.816680	0.408340	−	0.912830i	$-0.366108\pi$
$229$	648098.	0.816680	0.408340	+	0.912830i	$0.366108\pi$
$230$	439500.	0.547822
$231$	619029.	0.763275
$232$	207744.	0.253401
$233$	−913689.	−1.10258	−0.551288	−	0.834315i	$-0.685863\pi$
$233$	−913689.	−1.10258	−0.551288	+	0.834315i	$0.685863\pi$
$234$	−54756.0	−0.0653720
$235$	527550.	0.623152
$236$	−808320.	−0.944720
$237$	744777.	0.861302
$238$	1.09232e6	1.25000
$239$	347313.	0.393302	0.196651	−	0.980474i	$-0.436993\pi$
$239$	347313.	0.393302	0.196651	+	0.980474i	$0.436993\pi$
$240$	−57600.0	−0.0645497
$241$	−935878.	−1.03795	−0.518975	−	0.854789i	$-0.673686\pi$
$241$	−935878.	−1.03795	−0.518975	+	0.854789i	$0.673686\pi$
$242$	276968.	0.304012
$243$	−59049.0	−0.0641500
$244$	−377584.	−0.406012
$245$	868050.	0.923910
$246$	383724.	0.404279
$247$	−5746.00	−0.00599271
$248$	−113792.	−0.117485
$249$	378432.	0.386803
$250$	−62500.0	−0.0632456
$251$	−506172.	−0.507124	−0.253562	−	0.967319i	$-0.581602\pi$
$251$	−506172.	−0.507124	−0.253562	+	0.967319i	$0.581602\pi$
$252$	294192.	0.291830
$253$	1.33168e6	1.30798
$254$	−304424.	−0.296070
$255$	270675.	0.260674
$256$	65536.0	0.0625000
$257$	−1.51352e6	−1.42940	−0.714702	−	0.699429i	$-0.753438\pi$
$257$	−1.51352e6	−1.42940	−0.714702	+	0.699429i	$0.753438\pi$
$258$	−283536.	−0.265191
$259$	−1.77491e6	−1.64410
$260$	67600.0	0.0620174
$261$	−262926.	−0.238909
$262$	903864.	0.813486
$263$	290952.	0.259377	0.129689	−	0.991555i	$-0.458602\pi$
$263$	290952.	0.259377	0.129689	+	0.991555i	$0.458602\pi$
$264$	−174528.	−0.154119
$265$	−243075.	−0.212631
$266$	30872.0	0.0267523
$267$	−354969.	−0.304728
$268$	887744.	0.755007
$269$	−2.03832e6	−1.71748	−0.858740	−	0.512412i	$-0.828752\pi$
$269$	−2.03832e6	−1.71748	−0.858740	+	0.512412i	$0.828752\pi$
$270$	72900.0	0.0608581
$271$	947426.	0.783650	0.391825	−	0.920040i	$-0.371844\pi$
$271$	947426.	0.783650	0.391825	+	0.920040i	$0.371844\pi$
$272$	−307968.	−0.252397
$273$	−345267.	−0.280381
$274$	−918264.	−0.738909
$275$	−189375.	−0.151005
$276$	632880.	0.500091
$277$	575258.	0.450467	0.225234	−	0.974305i	$-0.427685\pi$
$277$	575258.	0.450467	0.225234	+	0.974305i	$0.427685\pi$
$278$	1.06572e6	0.827052
$279$	144018.	0.110766
$280$	−363200.	−0.276854
$281$	787302.	0.594806	0.297403	−	0.954752i	$-0.403879\pi$
$281$	787302.	0.594806	0.297403	+	0.954752i	$0.403879\pi$
$282$	759672.	0.568857
$283$	1.50426e6	1.11649	0.558246	−	0.829675i	$-0.311475\pi$
$283$	1.50426e6	1.11649	0.558246	+	0.829675i	$0.311475\pi$
$284$	331536.	0.243913
$285$	7650.00	0.00557891
$286$	204828.	0.148072
$287$	2.41959e6	1.73395
$288$	−82944.0	−0.0589256
$289$	27352.0	0.0192639
$290$	324600.	0.226649
$291$	−1.36500e6	−0.944933
$292$	−640672.	−0.439722
$293$	1.19266e6	0.811608	0.405804	−	0.913960i	$-0.366992\pi$
$293$	1.19266e6	0.811608	0.405804	+	0.913960i	$0.366992\pi$
$294$	1.24999e6	0.843410
$295$	−1.26300e6	−0.844984
$296$	500416.	0.331972
$297$	220887.	0.145304
$298$	321852.	0.209950
$299$	−742755.	−0.480471
$300$	−90000.0	−0.0577350
$301$	−1.78785e6	−1.13741
$302$	1.02846e6	0.648891
$303$	335124.	0.209700
$304$	−8704.00	−0.00540176
$305$	−589975.	−0.363148
$306$	389772.	0.237962
$307$	668447.	0.404782	0.202391	−	0.979305i	$-0.435129\pi$
$307$	668447.	0.404782	0.202391	+	0.979305i	$0.435129\pi$
$308$	−1.10050e6	−0.661016
$309$	665316.	0.396398
$310$	−177800.	−0.105082
$311$	1.08888e6	0.638380	0.319190	−	0.947691i	$-0.396589\pi$
$311$	1.08888e6	0.638380	0.319190	+	0.947691i	$0.396589\pi$
$312$	97344.0	0.0566139
$313$	−1.31413e6	−0.758189	−0.379095	−	0.925358i	$-0.623764\pi$
$313$	−1.31413e6	−0.758189	−0.379095	+	0.925358i	$0.623764\pi$
$314$	1.60228e6	0.917095
$315$	459675.	0.261020
$316$	−1.32405e6	−0.745909
$317$	−3.16211e6	−1.76737	−0.883687	−	0.468078i	$-0.844947\pi$
$317$	−3.16211e6	−1.76737	−0.883687	+	0.468078i	$0.844947\pi$
$318$	−350028.	−0.194104
$319$	983538.	0.541146
$320$	102400.	0.0559017
$321$	−815589.	−0.441783
$322$	3.99066e6	2.14489
$323$	40902.0	0.0218141
$324$	104976.	0.0555556
$325$	105625.	0.0554700
$326$	−2.43235e6	−1.26760
$327$	1.70287e6	0.880669
$328$	−682176.	−0.350116
$329$	4.79015e6	2.43983
$330$	−272700.	−0.137848
$331$	1.29757e6	0.650969	0.325484	−	0.945547i	$-0.394473\pi$
$331$	1.29757e6	0.650969	0.325484	+	0.945547i	$0.394473\pi$
$332$	−672768.	−0.334981
$333$	−633339.	−0.312987
$334$	2.79811e6	1.37246
$335$	1.38710e6	0.675298
$336$	−523008.	−0.252732
$337$	−4.01554e6	−1.92606	−0.963029	−	0.269399i	$-0.913175\pi$
$337$	−4.01554e6	−1.92606	−0.963029	+	0.269399i	$0.913175\pi$
$338$	−114244.	−0.0543928
$339$	1.02973e6	0.486656
$340$	−481200.	−0.225750
$341$	−538734.	−0.250893
$342$	11016.0	0.00509282
$343$	4.06670e6	1.86641
$344$	504064.	0.229662
$345$	988875.	0.447295
$346$	−399096.	−0.179220
$347$	1.85826e6	0.828480	0.414240	−	0.910168i	$-0.364047\pi$
$347$	1.85826e6	0.828480	0.414240	+	0.910168i	$0.364047\pi$
$348$	467424.	0.206901
$349$	−47716.0	−0.0209701	−0.0104850	−	0.999945i	$-0.503338\pi$
$349$	−47716.0	−0.0209701	−0.0104850	+	0.999945i	$0.503338\pi$
$350$	−567500.	−0.247626
$351$	−123201.	−0.0533761
$352$	310272.	0.133471
$353$	−1.30553e6	−0.557637	−0.278818	−	0.960344i	$-0.589943\pi$
$353$	−1.30553e6	−0.557637	−0.278818	+	0.960344i	$0.589943\pi$
$354$	−1.81872e6	−0.771361
$355$	518025.	0.218162
$356$	631056.	0.263902
$357$	2.45773e6	1.02062
$358$	−1.79609e6	−0.740662
$359$	982032.	0.402151	0.201076	−	0.979576i	$-0.435556\pi$
$359$	982032.	0.402151	0.201076	+	0.979576i	$0.435556\pi$
$360$	−129600.	−0.0527046
$361$	−2.47494e6	−0.999533
$362$	1.53492e6	0.615624
$363$	623178.	0.248225
$364$	613808.	0.242817
$365$	−1.00105e6	−0.393300
$366$	−849564.	−0.331508
$367$	3.81416e6	1.47820	0.739101	−	0.673595i	$-0.235251\pi$
$367$	3.81416e6	1.47820	0.739101	+	0.673595i	$0.235251\pi$
$368$	−1.12512e6	−0.433091
$369$	863379.	0.330092
$370$	781900.	0.296925
$371$	−2.20712e6	−0.832514
$372$	−256032.	−0.0959261
$373$	−2.28179e6	−0.849188	−0.424594	−	0.905384i	$-0.639583\pi$
$373$	−2.28179e6	−0.849188	−0.424594	+	0.905384i	$0.639583\pi$
$374$	−1.45804e6	−0.539001
$375$	−140625.	−0.0516398
$376$	−1.35053e6	−0.492645
$377$	−548574.	−0.198784
$378$	661932.	0.238278
$379$	2.07058e6	0.740446	0.370223	−	0.928943i	$-0.379281\pi$
$379$	2.07058e6	0.740446	0.370223	+	0.928943i	$0.379281\pi$
$380$	−13600.0	−0.00483148
$381$	−684954.	−0.241740
$382$	2.90693e6	1.01924
$383$	−623886.	−0.217324	−0.108662	−	0.994079i	$-0.534657\pi$
$383$	−623886.	−0.217324	−0.108662	+	0.994079i	$0.534657\pi$
$384$	147456.	0.0510310
$385$	−1.71953e6	−0.591230
$386$	−2.41562e6	−0.825202
$387$	−637956.	−0.216528
$388$	2.42667e6	0.818336
$389$	3.34450e6	1.12062	0.560308	−	0.828284i	$-0.310683\pi$
$389$	3.34450e6	1.12062	0.560308	+	0.828284i	$0.310683\pi$
$390$	152100.	0.0506370
$391$	5.28718e6	1.74897
$392$	−2.22221e6	−0.730415
$393$	2.03369e6	0.664208
$394$	−3.28056e6	−1.06465
$395$	−2.06882e6	−0.667162
$396$	−392688.	−0.125837
$397$	−738283.	−0.235097	−0.117548	−	0.993067i	$-0.537504\pi$
$397$	−738283.	−0.235097	−0.117548	+	0.993067i	$0.537504\pi$
$398$	513136.	0.162377
$399$	69462.0	0.0218431
$400$	160000.	0.0500000
$401$	1.01662e6	0.315717	0.157859	−	0.987462i	$-0.449541\pi$
$401$	1.01662e6	0.315717	0.157859	+	0.987462i	$0.449541\pi$
$402$	1.99742e6	0.616460
$403$	300482.	0.0921628
$404$	−595776.	−0.181606
$405$	164025.	0.0496904
$406$	2.94737e6	0.887400
$407$	2.36916e6	0.708937
$408$	−692928.	−0.206081
$409$	−5.73251e6	−1.69448	−0.847240	−	0.531211i	$-0.821737\pi$
$409$	−5.73251e6	−1.69448	−0.847240	+	0.531211i	$0.821737\pi$
$410$	−1.06590e6	−0.313153
$411$	−2.06609e6	−0.603317
$412$	−1.18278e6	−0.343291
$413$	−1.14680e7	−3.30837
$414$	1.42398e6	0.408322
$415$	−1.05120e6	−0.299616
$416$	−173056.	−0.0490290
$417$	2.39788e6	0.675285
$418$	−41208.0	−0.0115356
$419$	−2.33255e6	−0.649078	−0.324539	−	0.945872i	$-0.605209\pi$
$419$	−2.33255e6	−0.649078	−0.324539	+	0.945872i	$0.605209\pi$
$420$	−817200.	−0.226050
$421$	−4.67150e6	−1.28455	−0.642275	−	0.766475i	$-0.722009\pi$
$421$	−4.67150e6	−1.28455	−0.642275	+	0.766475i	$0.722009\pi$
$422$	−1.77166e6	−0.484284
$423$	1.70926e6	0.464470
$424$	622272.	0.168099
$425$	−751875.	−0.201917
$426$	745956.	0.199154
$427$	−5.35697e6	−1.42184
$428$	1.44994e6	0.382595
$429$	460863.	0.120901
$430$	787600.	0.205416
$431$	−6.31742e6	−1.63812	−0.819062	−	0.573705i	$-0.805506\pi$
$431$	−6.31742e6	−1.63812	−0.819062	+	0.573705i	$0.805506\pi$
$432$	−186624.	−0.0481125
$433$	−3.10269e6	−0.795277	−0.397638	−	0.917542i	$-0.630170\pi$
$433$	−3.10269e6	−0.795277	−0.397638	+	0.917542i	$0.630170\pi$
$434$	−1.61442e6	−0.411427
$435$	730350.	0.185058
$436$	−3.02733e6	−0.762682
$437$	149430.	0.0374312
$438$	−1.44151e6	−0.359032
$439$	284237.	0.0703914	0.0351957	−	0.999380i	$-0.488795\pi$
$439$	284237.	0.0703914	0.0351957	+	0.999380i	$0.488795\pi$
$440$	484800.	0.119380
$441$	2.81248e6	0.688642
$442$	813228.	0.197996
$443$	−7.69961e6	−1.86406	−0.932029	−	0.362383i	$-0.881963\pi$
$443$	−7.69961e6	−1.86406	−0.932029	+	0.362383i	$0.881963\pi$
$444$	1.12594e6	0.271054
$445$	986025.	0.236041
$446$	−2.69926e6	−0.642552
$447$	724167.	0.171423
$448$	929792.	0.218872
$449$	6.87080e6	1.60839	0.804195	−	0.594366i	$-0.202597\pi$
$449$	6.87080e6	1.60839	0.804195	+	0.594366i	$0.202597\pi$
$450$	−202500.	−0.0471405
$451$	−3.22968e6	−0.747683
$452$	−1.83062e6	−0.421457
$453$	2.31404e6	0.529817
$454$	2.82367e6	0.642946
$455$	959075.	0.217182
$456$	−19584.0	−0.00441051
$457$	−6.36679e6	−1.42604	−0.713018	−	0.701146i	$-0.752672\pi$
$457$	−6.36679e6	−1.42604	−0.713018	+	0.701146i	$0.752672\pi$
$458$	−2.59239e6	−0.577480
$459$	876987.	0.194295
$460$	−1.75800e6	−0.387368
$461$	−2.87009e6	−0.628988	−0.314494	−	0.949259i	$-0.601835\pi$
$461$	−2.87009e6	−0.628988	−0.314494	+	0.949259i	$0.601835\pi$
$462$	−2.47612e6	−0.539717
$463$	1.75549e6	0.380580	0.190290	−	0.981728i	$-0.439057\pi$
$463$	1.75549e6	0.380580	0.190290	+	0.981728i	$0.439057\pi$
$464$	−830976.	−0.179182
$465$	−400050.	−0.0857989
$466$	3.65476e6	0.779639
$467$	−2.79039e6	−0.592069	−0.296034	−	0.955177i	$-0.595664\pi$
$467$	−2.79039e6	−0.592069	−0.296034	+	0.955177i	$0.595664\pi$
$468$	219024.	0.0462250
$469$	1.25949e7	2.64400
$470$	−2.11020e6	−0.440635
$471$	3.60513e6	0.748805
$472$	3.23328e6	0.668018
$473$	2.38643e6	0.490451
$474$	−2.97911e6	−0.609032
$475$	−21250.0	−0.00432140
$476$	−4.36930e6	−0.883882
$477$	−787563.	−0.158485
$478$	−1.38925e6	−0.278107
$479$	3.28349e6	0.653878	0.326939	−	0.945045i	$-0.393983\pi$
$479$	3.28349e6	0.653878	0.326939	+	0.945045i	$0.393983\pi$
$480$	230400.	0.0456435
$481$	−1.32141e6	−0.260421
$482$	3.74351e6	0.733942
$483$	8.97898e6	1.75130
$484$	−1.10787e6	−0.214969
$485$	3.79168e6	0.731942
$486$	236196.	0.0453609
$487$	450209.	0.0860185	0.0430092	−	0.999075i	$-0.486306\pi$
$487$	450209.	0.0860185	0.0430092	+	0.999075i	$0.486306\pi$
$488$	1.51034e6	0.287094
$489$	−5.47278e6	−1.03499
$490$	−3.47220e6	−0.653303
$491$	−4.99459e6	−0.934967	−0.467484	−	0.884002i	$-0.654839\pi$
$491$	−4.99459e6	−0.934967	−0.467484	+	0.884002i	$0.654839\pi$
$492$	−1.53490e6	−0.285868
$493$	3.90494e6	0.723597
$494$	22984.0	0.00423749
$495$	−613575.	−0.112552
$496$	455168.	0.0830745
$497$	4.70367e6	0.854173
$498$	−1.51373e6	−0.273511
$499$	−4.46811e6	−0.803291	−0.401645	−	0.915795i	$-0.631562\pi$
$499$	−4.46811e6	−0.803291	−0.401645	+	0.915795i	$0.631562\pi$
$500$	250000.	0.0447214
$501$	6.29575e6	1.12061
$502$	2.02469e6	0.358591
$503$	717564.	0.126456	0.0632282	−	0.997999i	$-0.479860\pi$
$503$	717564.	0.126456	0.0632282	+	0.997999i	$0.479860\pi$
$504$	−1.17677e6	−0.206355
$505$	−930900.	−0.162433
$506$	−5.32674e6	−0.924880
$507$	−257049.	−0.0444116
$508$	1.21770e6	0.209353
$509$	−4.46348e6	−0.763622	−0.381811	−	0.924240i	$-0.624700\pi$
$509$	−4.46348e6	−0.763622	−0.381811	+	0.924240i	$0.624700\pi$
$510$	−1.08270e6	−0.184324
$511$	−9.08953e6	−1.53989
$512$	−262144.	−0.0441942
$513$	24786.0	0.00415827
$514$	6.05407e6	1.01074
$515$	−1.84810e6	−0.307049
$516$	1.13414e6	0.187518
$517$	−6.39391e6	−1.05206
$518$	7.09965e6	1.16255
$519$	−897966.	−0.146333
$520$	−270400.	−0.0438529
$521$	−758190.	−0.122372	−0.0611862	−	0.998126i	$-0.519488\pi$
$521$	−758190.	−0.122372	−0.0611862	+	0.998126i	$0.519488\pi$
$522$	1.05170e6	0.168934
$523$	6.07992e6	0.971949	0.485974	−	0.873973i	$-0.338465\pi$
$523$	6.07992e6	0.971949	0.485974	+	0.873973i	$0.338465\pi$
$524$	−3.61546e6	−0.575221
$525$	−1.27688e6	−0.202186
$526$	−1.16381e6	−0.183407
$527$	−2.13893e6	−0.335483
$528$	698112.	0.108978
$529$	1.28797e7	2.00109
$530$	972300.	0.150352
$531$	−4.09212e6	−0.629814
$532$	−123488.	−0.0189167
$533$	1.80137e6	0.274654
$534$	1.41988e6	0.215475
$535$	2.26552e6	0.342203
$536$	−3.55098e6	−0.533870
$537$	−4.04120e6	−0.604748
$538$	8.15328e6	1.21444
$539$	−1.05208e7	−1.55982
$540$	−291600.	−0.0430331
$541$	−4.56522e6	−0.670607	−0.335304	−	0.942110i	$-0.608839\pi$
$541$	−4.56522e6	−0.670607	−0.335304	+	0.942110i	$0.608839\pi$
$542$	−3.78970e6	−0.554124
$543$	3.45358e6	0.502655
$544$	1.23187e6	0.178471
$545$	−4.73020e6	−0.682163
$546$	1.38107e6	0.198259
$547$	−6.95788e6	−0.994280	−0.497140	−	0.867670i	$-0.665616\pi$
$547$	−6.95788e6	−0.994280	−0.497140	+	0.867670i	$0.665616\pi$
$548$	3.67306e6	0.522488
$549$	−1.91152e6	−0.270675
$550$	757500.	0.106777
$551$	110364.	0.0154863
$552$	−2.53152e6	−0.353617
$553$	−1.87849e7	−2.61214
$554$	−2.30103e6	−0.318528
$555$	1.75928e6	0.242438
$556$	−4.26290e6	−0.584814
$557$	9.83815e6	1.34362	0.671808	−	0.740725i	$-0.265518\pi$
$557$	9.83815e6	1.34362	0.671808	+	0.740725i	$0.265518\pi$
$558$	−576072.	−0.0783234
$559$	−1.33104e6	−0.180162
$560$	1.45280e6	0.195765
$561$	−3.28058e6	−0.440092
$562$	−3.14921e6	−0.420592
$563$	1.28406e7	1.70731	0.853656	−	0.520837i	$-0.174380\pi$
$563$	1.28406e7	1.70731	0.853656	+	0.520837i	$0.174380\pi$
$564$	−3.03869e6	−0.402243
$565$	−2.86035e6	−0.376962
$566$	−6.01702e6	−0.789479
$567$	1.48935e6	0.194553
$568$	−1.32614e6	−0.172472
$569$	989736.	0.128156	0.0640780	−	0.997945i	$-0.479589\pi$
$569$	989736.	0.128156	0.0640780	+	0.997945i	$0.479589\pi$
$570$	−30600.0	−0.00394488
$571$	2.80966e6	0.360631	0.180315	−	0.983609i	$-0.442288\pi$
$571$	2.80966e6	0.360631	0.180315	+	0.983609i	$0.442288\pi$
$572$	−819312.	−0.104703
$573$	6.54059e6	0.832205
$574$	−9.67837e6	−1.22609
$575$	−2.74688e6	−0.346473
$576$	331776.	0.0416667
$577$	5.72462e6	0.715826	0.357913	−	0.933755i	$-0.383488\pi$
$577$	5.72462e6	0.715826	0.357913	+	0.933755i	$0.383488\pi$
$578$	−109408.	−0.0136216
$579$	−5.43514e6	−0.673775
$580$	−1.29840e6	−0.160265
$581$	−9.54490e6	−1.17309
$582$	5.46001e6	0.668169
$583$	2.94607e6	0.358981
$584$	2.56269e6	0.310931
$585$	342225.	0.0413449
$586$	−4.77062e6	−0.573893
$587$	3.95725e6	0.474021	0.237011	−	0.971507i	$-0.423832\pi$
$587$	3.95725e6	0.474021	0.237011	+	0.971507i	$0.423832\pi$
$588$	−4.99997e6	−0.596381
$589$	−60452.0	−0.00717997
$590$	5.05200e6	0.597494
$591$	−7.38126e6	−0.869284
$592$	−2.00166e6	−0.234740
$593$	303684.	0.0354638	0.0177319	−	0.999843i	$-0.494355\pi$
$593$	303684.	0.0354638	0.0177319	+	0.999843i	$0.494355\pi$
$594$	−883548.	−0.102746
$595$	−6.82703e6	−0.790568
$596$	−1.28741e6	−0.148457
$597$	1.15456e6	0.132580
$598$	2.97102e6	0.339745
$599$	1.54884e7	1.76376	0.881881	−	0.471473i	$-0.156277\pi$
$599$	1.54884e7	1.76376	0.881881	+	0.471473i	$0.156277\pi$
$600$	360000.	0.0408248
$601$	4.84170e6	0.546779	0.273389	−	0.961903i	$-0.411855\pi$
$601$	4.84170e6	0.546779	0.273389	+	0.961903i	$0.411855\pi$
$602$	7.15141e6	0.804267
$603$	4.49420e6	0.503338
$604$	−4.11386e6	−0.458835
$605$	−1.73105e6	−0.192274
$606$	−1.34050e6	−0.148280
$607$	1.14343e7	1.25962	0.629809	−	0.776750i	$-0.283133\pi$
$607$	1.14343e7	1.25962	0.629809	+	0.776750i	$0.283133\pi$
$608$	34816.0	0.00381962
$609$	6.63158e6	0.724559
$610$	2.35990e6	0.256785
$611$	3.56624e6	0.386462
$612$	−1.55909e6	−0.168264
$613$	−7.15252e6	−0.768790	−0.384395	−	0.923169i	$-0.625590\pi$
$613$	−7.15252e6	−0.768790	−0.384395	+	0.923169i	$0.625590\pi$
$614$	−2.67379e6	−0.286224
$615$	−2.39828e6	−0.255688
$616$	4.40198e6	0.467409
$617$	1.62896e7	1.72266	0.861328	−	0.508049i	$-0.169633\pi$
$617$	1.62896e7	1.72266	0.861328	+	0.508049i	$0.169633\pi$
$618$	−2.66126e6	−0.280296
$619$	−1.16203e7	−1.21896	−0.609481	−	0.792800i	$-0.708622\pi$
$619$	−1.16203e7	−1.21896	−0.609481	+	0.792800i	$0.708622\pi$
$620$	711200.	0.0743041
$621$	3.20396e6	0.333394
$622$	−4.35552e6	−0.451403
$623$	8.95311e6	0.924174
$624$	−389376.	−0.0400320
$625$	390625.	0.0400000
$626$	5.25652e6	0.536121
$627$	−92718.0	−0.00941879
$628$	−6.40912e6	−0.648484
$629$	9.40626e6	0.947960
$630$	−1.83870e6	−0.184569
$631$	−1.55601e6	−0.155575	−0.0777873	−	0.996970i	$-0.524785\pi$
$631$	−1.55601e6	−0.155575	−0.0777873	+	0.996970i	$0.524785\pi$
$632$	5.29619e6	0.527438
$633$	−3.98624e6	−0.395416
$634$	1.26484e7	1.24972
$635$	1.90265e6	0.187251
$636$	1.40011e6	0.137252
$637$	5.86802e6	0.572984
$638$	−3.93415e6	−0.382648
$639$	1.67840e6	0.162609
$640$	−409600.	−0.0395285
$641$	1.39735e7	1.34326	0.671632	−	0.740885i	$-0.265594\pi$
$641$	1.39735e7	1.34326	0.671632	+	0.740885i	$0.265594\pi$
$642$	3.26236e6	0.312388
$643$	−9.95137e6	−0.949195	−0.474598	−	0.880203i	$-0.657406\pi$
$643$	−9.95137e6	−0.949195	−0.474598	+	0.880203i	$0.657406\pi$
$644$	−1.59626e7	−1.51667
$645$	1.77210e6	0.167722
$646$	−163608.	−0.0154249
$647$	9.91727e6	0.931390	0.465695	−	0.884945i	$-0.345804\pi$
$647$	9.91727e6	0.931390	0.465695	+	0.884945i	$0.345804\pi$
$648$	−419904.	−0.0392837
$649$	1.53076e7	1.42657
$650$	−422500.	−0.0392232
$651$	−3.63245e6	−0.335929
$652$	9.72939e6	0.896327
$653$	134466.	0.0123404	0.00617020	−	0.999981i	$-0.498036\pi$
$653$	134466.	0.0123404	0.00617020	+	0.999981i	$0.498036\pi$
$654$	−6.81149e6	−0.622727
$655$	−5.64915e6	−0.514493
$656$	2.72870e6	0.247569
$657$	−3.24340e6	−0.293148
$658$	−1.91606e7	−1.72522
$659$	−1.15900e7	−1.03961	−0.519805	−	0.854285i	$-0.673995\pi$
$659$	−1.15900e7	−1.03961	−0.519805	+	0.854285i	$0.673995\pi$
$660$	1.09080e6	0.0974732
$661$	−1.75970e7	−1.56651	−0.783256	−	0.621699i	$-0.786443\pi$
$661$	−1.75970e7	−1.56651	−0.783256	+	0.621699i	$0.786443\pi$
$662$	−5.19027e6	−0.460304
$663$	1.82976e6	0.161663
$664$	2.69107e6	0.236867
$665$	−192950.	−0.0169196
$666$	2.53336e6	0.221315
$667$	1.42662e7	1.24163
$668$	−1.11924e7	−0.970474
$669$	−6.07334e6	−0.524641
$670$	−5.54840e6	−0.477508
$671$	7.15050e6	0.613098
$672$	2.09203e6	0.178708
$673$	1.82155e7	1.55026	0.775129	−	0.631802i	$-0.217685\pi$
$673$	1.82155e7	1.55026	0.775129	+	0.631802i	$0.217685\pi$
$674$	1.60622e7	1.36193
$675$	−455625.	−0.0384900
$676$	456976.	0.0384615
$677$	−1.46036e6	−0.122458	−0.0612289	−	0.998124i	$-0.519502\pi$
$677$	−1.46036e6	−0.122458	−0.0612289	+	0.998124i	$0.519502\pi$
$678$	−4.11890e6	−0.344118
$679$	3.44284e7	2.86578
$680$	1.92480e6	0.159630
$681$	6.35326e6	0.524963
$682$	2.15494e6	0.177408
$683$	4.19221e6	0.343868	0.171934	−	0.985108i	$-0.444998\pi$
$683$	4.19221e6	0.343868	0.171934	+	0.985108i	$0.444998\pi$
$684$	−44064.0	−0.00360117
$685$	5.73915e6	0.467327
$686$	−1.62668e7	−1.31975
$687$	−5.83288e6	−0.471510
$688$	−2.01626e6	−0.162396
$689$	−1.64319e6	−0.131868
$690$	−3.95550e6	−0.316285
$691$	9.33096e6	0.743415	0.371707	−	0.928350i	$-0.378772\pi$
$691$	9.33096e6	0.743415	0.371707	+	0.928350i	$0.378772\pi$
$692$	1.59638e6	0.126728
$693$	−5.57126e6	−0.440677
$694$	−7.43303e6	−0.585824
$695$	−6.66078e6	−0.523074
$696$	−1.86970e6	−0.146301
$697$	−1.28228e7	−0.999770
$698$	190864.	0.0148281
$699$	8.22320e6	0.636573
$700$	2.27000e6	0.175098
$701$	−1.21583e6	−0.0934495	−0.0467248	−	0.998908i	$-0.514878\pi$
$701$	−1.21583e6	−0.0934495	−0.0467248	+	0.998908i	$0.514878\pi$
$702$	492804.	0.0377426
$703$	265846.	0.0202881
$704$	−1.24109e6	−0.0943780
$705$	−4.74795e6	−0.359777
$706$	5.22214e6	0.394309
$707$	−8.45257e6	−0.635975
$708$	7.27488e6	0.545435
$709$	−1.63186e7	−1.21918	−0.609591	−	0.792716i	$-0.708666\pi$
$709$	−1.63186e7	−1.21918	−0.609591	+	0.792716i	$0.708666\pi$
$710$	−2.07210e6	−0.154264
$711$	−6.70299e6	−0.497273
$712$	−2.52422e6	−0.186607
$713$	−7.81431e6	−0.575661
$714$	−9.83092e6	−0.721686
$715$	−1.28018e6	−0.0936492
$716$	7.18435e6	0.523727
$717$	−3.12582e6	−0.227073
$718$	−3.92813e6	−0.284364
$719$	2.45020e7	1.76758	0.883790	−	0.467883i	$-0.154983\pi$
$719$	2.45020e7	1.76758	0.883790	+	0.467883i	$0.154983\pi$
$720$	518400.	0.0372678
$721$	−1.67807e7	−1.20219
$722$	9.89977e6	0.706777
$723$	8.42290e6	0.599261
$724$	−6.13970e6	−0.435312
$725$	−2.02875e6	−0.143345
$726$	−2.49271e6	−0.175522
$727$	6.66067e6	0.467393	0.233696	−	0.972310i	$-0.424918\pi$
$727$	6.66067e6	0.467393	0.233696	+	0.972310i	$0.424918\pi$
$728$	−2.45523e6	−0.171698
$729$	531441.	0.0370370
$730$	4.00420e6	0.278105
$731$	9.47483e6	0.655810
$732$	3.39826e6	0.234411
$733$	−600391.	−0.0412738	−0.0206369	−	0.999787i	$-0.506569\pi$
$733$	−600391.	−0.0412738	−0.0206369	+	0.999787i	$0.506569\pi$
$734$	−1.52566e7	−1.04525
$735$	−7.81245e6	−0.533420
$736$	4.50048e6	0.306242
$737$	−1.68117e7	−1.14010
$738$	−3.45352e6	−0.233411
$739$	1.56807e7	1.05622	0.528111	−	0.849175i	$-0.322900\pi$
$739$	1.56807e7	1.05622	0.528111	+	0.849175i	$0.322900\pi$
$740$	−3.12760e6	−0.209958
$741$	51714.0	0.00345989
$742$	8.82848e6	0.588676
$743$	−9.67217e6	−0.642765	−0.321382	−	0.946950i	$-0.604147\pi$
$743$	−9.67217e6	−0.642765	−0.321382	+	0.946950i	$0.604147\pi$
$744$	1.02413e6	0.0678300
$745$	−2.01158e6	−0.132784
$746$	9.12717e6	0.600467
$747$	−3.40589e6	−0.223321
$748$	5.83214e6	0.381131
$749$	2.05710e7	1.33983
$750$	562500.	0.0365148
$751$	1.50220e7	0.971916	0.485958	−	0.873982i	$-0.338471\pi$
$751$	1.50220e7	0.971916	0.485958	+	0.873982i	$0.338471\pi$
$752$	5.40211e6	0.348353
$753$	4.55555e6	0.292788
$754$	2.19430e6	0.140562
$755$	−6.42790e6	−0.410395
$756$	−2.64773e6	−0.168488
$757$	2.51505e7	1.59517	0.797584	−	0.603207i	$-0.206111\pi$
$757$	2.51505e7	1.59517	0.797584	+	0.603207i	$0.206111\pi$
$758$	−8.28231e6	−0.523575
$759$	−1.19852e7	−0.755161
$760$	54400.0	0.00341637
$761$	2.61146e6	0.163464	0.0817319	−	0.996654i	$-0.473955\pi$
$761$	2.61146e6	0.163464	0.0817319	+	0.996654i	$0.473955\pi$
$762$	2.73982e6	0.170936
$763$	−4.29502e7	−2.67088
$764$	−1.16277e7	−0.720711
$765$	−2.43608e6	−0.150500
$766$	2.49554e6	0.153671
$767$	−8.53788e6	−0.524037
$768$	−589824.	−0.0360844
$769$	118052.	0.00719876	0.00359938	−	0.999994i	$-0.498854\pi$
$769$	118052.	0.00719876	0.00359938	+	0.999994i	$0.498854\pi$
$770$	6.87810e6	0.418063
$771$	1.36217e7	0.825266
$772$	9.66248e6	0.583506
$773$	−2.51165e6	−0.151185	−0.0755927	−	0.997139i	$-0.524085\pi$
$773$	−2.51165e6	−0.151185	−0.0755927	+	0.997139i	$0.524085\pi$
$774$	2.55182e6	0.153108
$775$	1.11125e6	0.0664596
$776$	−9.70669e6	−0.578651
$777$	1.59742e7	0.949220
$778$	−1.33780e7	−0.792395
$779$	−362406.	−0.0213969
$780$	−608400.	−0.0358057
$781$	−6.27846e6	−0.368320
$782$	−2.11487e7	−1.23671
$783$	2.36633e6	0.137934
$784$	8.88883e6	0.516481
$785$	−1.00142e7	−0.580022
$786$	−8.13478e6	−0.469666
$787$	1.85099e7	1.06529	0.532643	−	0.846340i	$-0.321199\pi$
$787$	1.85099e7	1.06529	0.532643	+	0.846340i	$0.321199\pi$
$788$	1.31222e7	0.752822
$789$	−2.61857e6	−0.149752
$790$	8.27530e6	0.471754
$791$	−2.59720e7	−1.47592
$792$	1.57075e6	0.0889805
$793$	−3.98823e6	−0.225215
$794$	2.95313e6	0.166238
$795$	2.18768e6	0.122762
$796$	−2.05254e6	−0.114818
$797$	8.51140e6	0.474630	0.237315	−	0.971433i	$-0.423733\pi$
$797$	8.51140e6	0.474630	0.237315	+	0.971433i	$0.423733\pi$
$798$	−277848.	−0.0154454
$799$	−2.53857e7	−1.40677
$800$	−640000.	−0.0353553
$801$	3.19472e6	0.175935
$802$	−4.06649e6	−0.223246
$803$	1.21327e7	0.664002
$804$	−7.98970e6	−0.435903
$805$	−2.49416e7	−1.35655
$806$	−1.20193e6	−0.0651690
$807$	1.83449e7	0.991587
$808$	2.38310e6	0.128415
$809$	−1.03927e7	−0.558287	−0.279143	−	0.960249i	$-0.590050\pi$
$809$	−1.03927e7	−0.558287	−0.279143	+	0.960249i	$0.590050\pi$
$810$	−656100.	−0.0351364
$811$	6.51888e6	0.348033	0.174017	−	0.984743i	$-0.444325\pi$
$811$	6.51888e6	0.348033	0.174017	+	0.984743i	$0.444325\pi$
$812$	−1.17895e7	−0.627486
$813$	−8.52683e6	−0.452440
$814$	−9.47663e6	−0.501294
$815$	1.52022e7	0.801700
$816$	2.77171e6	0.145721
$817$	267784.	0.0140356
$818$	2.29300e7	1.19818
$819$	3.10740e6	0.161878
$820$	4.26360e6	0.221433
$821$	−1.92093e7	−0.994613	−0.497306	−	0.867575i	$-0.665678\pi$
$821$	−1.92093e7	−0.994613	−0.497306	+	0.867575i	$0.665678\pi$
$822$	8.26438e6	0.426610
$823$	3.67906e6	0.189338	0.0946690	−	0.995509i	$-0.469821\pi$
$823$	3.67906e6	0.189338	0.0946690	+	0.995509i	$0.469821\pi$
$824$	4.73114e6	0.242743
$825$	1.70438e6	0.0871827
$826$	4.58722e7	2.33937
$827$	3.22079e7	1.63757	0.818783	−	0.574104i	$-0.194649\pi$
$827$	3.22079e7	1.63757	0.818783	+	0.574104i	$0.194649\pi$
$828$	−5.69592e6	−0.288727
$829$	3.21472e6	0.162464	0.0812319	−	0.996695i	$-0.474115\pi$
$829$	3.21472e6	0.162464	0.0812319	+	0.996695i	$0.474115\pi$
$830$	4.20480e6	0.211861
$831$	−5.17732e6	−0.260077
$832$	692224.	0.0346688
$833$	−4.17706e7	−2.08573
$834$	−9.59152e6	−0.477499
$835$	−1.74882e7	−0.868018
$836$	164832.	0.00815691
$837$	−1.29616e6	−0.0639507
$838$	9.33022e6	0.458967
$839$	2.46812e7	1.21049	0.605246	−	0.796039i	$-0.293075\pi$
$839$	2.46812e7	1.21049	0.605246	+	0.796039i	$0.293075\pi$
$840$	3.26880e6	0.159842
$841$	−9.97463e6	−0.486303
$842$	1.86860e7	0.908313
$843$	−7.08572e6	−0.343412
$844$	7.08666e6	0.342441
$845$	714025.	0.0344010
$846$	−6.83705e6	−0.328430
$847$	−1.57179e7	−0.752813
$848$	−2.48909e6	−0.118864
$849$	−1.35383e7	−0.644607
$850$	3.00750e6	0.142777
$851$	3.43645e7	1.62662
$852$	−2.98382e6	−0.140823
$853$	3.18924e7	1.50077	0.750386	−	0.661000i	$-0.229868\pi$
$853$	3.18924e7	1.50077	0.750386	+	0.661000i	$0.229868\pi$
$854$	2.14279e7	1.00539
$855$	−68850.0	−0.00322098
$856$	−5.79974e6	−0.270536
$857$	8.21152e6	0.381919	0.190960	−	0.981598i	$-0.438840\pi$
$857$	8.21152e6	0.381919	0.190960	+	0.981598i	$0.438840\pi$
$858$	−1.84345e6	−0.0854897
$859$	−1.86701e7	−0.863303	−0.431651	−	0.902041i	$-0.642069\pi$
$859$	−1.86701e7	−0.863303	−0.431651	+	0.902041i	$0.642069\pi$
$860$	−3.15040e6	−0.145251
$861$	−2.17763e7	−1.00110
$862$	2.52697e7	1.15833
$863$	2.24196e7	1.02471	0.512355	−	0.858774i	$-0.328773\pi$
$863$	2.24196e7	1.02471	0.512355	+	0.858774i	$0.328773\pi$
$864$	746496.	0.0340207
$865$	2.49435e6	0.113349
$866$	1.24108e7	0.562346
$867$	−246168.	−0.0111220
$868$	6.45770e6	0.290923
$869$	2.50742e7	1.12636
$870$	−2.92140e6	−0.130856
$871$	9.37680e6	0.418802
$872$	1.21093e7	0.539297
$873$	1.22850e7	0.545557
$874$	−597720.	−0.0264679
$875$	3.54688e6	0.156612
$876$	5.76605e6	0.253874
$877$	3.77896e7	1.65910	0.829552	−	0.558430i	$-0.188596\pi$
$877$	3.77896e7	1.65910	0.829552	+	0.558430i	$0.188596\pi$
$878$	−1.13695e6	−0.0497742
$879$	−1.07339e7	−0.468582
$880$	−1.93920e6	−0.0844143
$881$	−2.69877e7	−1.17146	−0.585728	−	0.810508i	$-0.699191\pi$
$881$	−2.69877e7	−1.17146	−0.585728	+	0.810508i	$0.699191\pi$
$882$	−1.12499e7	−0.486943
$883$	1.30211e7	0.562013	0.281007	−	0.959706i	$-0.409332\pi$
$883$	1.30211e7	0.562013	0.281007	+	0.959706i	$0.409332\pi$
$884$	−3.25291e6	−0.140004
$885$	1.13670e7	0.487852
$886$	3.07984e7	1.31809
$887$	3.62737e7	1.54804	0.774021	−	0.633160i	$-0.218243\pi$
$887$	3.62737e7	1.54804	0.774021	+	0.633160i	$0.218243\pi$
$888$	−4.50374e6	−0.191664
$889$	1.72761e7	0.733146
$890$	−3.94410e6	−0.166906
$891$	−1.98798e6	−0.0838916
$892$	1.07971e7	0.454353
$893$	−717468.	−0.0301074
$894$	−2.89667e6	−0.121215
$895$	1.12256e7	0.468436
$896$	−3.71917e6	−0.154766
$897$	6.68480e6	0.277400
$898$	−2.74832e7	−1.13730
$899$	−5.77139e6	−0.238167
$900$	810000.	0.0333333
$901$	1.16968e7	0.480014
$902$	1.29187e7	0.528692
$903$	1.60907e7	0.656681
$904$	7.32250e6	0.298015
$905$	−9.59328e6	−0.389355
$906$	−9.25618e6	−0.374637
$907$	1.45468e7	0.587151	0.293576	−	0.955936i	$-0.405155\pi$
$907$	1.45468e7	0.587151	0.293576	+	0.955936i	$0.405155\pi$
$908$	−1.12947e7	−0.454632
$909$	−3.01612e6	−0.121070
$910$	−3.83630e6	−0.153571
$911$	2.49587e7	0.996381	0.498190	−	0.867068i	$-0.333998\pi$
$911$	2.49587e7	0.996381	0.498190	+	0.867068i	$0.333998\pi$
$912$	78336.0	0.00311870
$913$	1.27405e7	0.505838
$914$	2.54672e7	1.00836
$915$	5.30978e6	0.209664
$916$	1.03696e7	0.408340
$917$	−5.12943e7	−2.01440
$918$	−3.50795e6	−0.137387
$919$	−1.92844e7	−0.753214	−0.376607	−	0.926373i	$-0.622909\pi$
$919$	−1.92844e7	−0.753214	−0.376607	+	0.926373i	$0.622909\pi$
$920$	7.03200e6	0.273911
$921$	−6.01602e6	−0.233701
$922$	1.14803e7	0.444762
$923$	3.50185e6	0.135299
$924$	9.90446e6	0.381638
$925$	−4.88688e6	−0.187792
$926$	−7.02196e6	−0.269111
$927$	−5.98784e6	−0.228861
$928$	3.32390e6	0.126701
$929$	2.41569e7	0.918338	0.459169	−	0.888349i	$-0.348147\pi$
$929$	2.41569e7	0.918338	0.459169	+	0.888349i	$0.348147\pi$
$930$	1.60020e6	0.0606690
$931$	−1.18055e6	−0.0446385
$932$	−1.46190e7	−0.551288
$933$	−9.79992e6	−0.368569
$934$	1.11615e7	0.418656
$935$	9.11272e6	0.340894
$936$	−876096.	−0.0326860
$937$	−3.66903e7	−1.36522	−0.682610	−	0.730783i	$-0.739155\pi$
$937$	−3.66903e7	−1.36522	−0.682610	+	0.730783i	$0.739155\pi$
$938$	−5.03795e7	−1.86959
$939$	1.18272e7	0.437741
$940$	8.44080e6	0.311576
$941$	−1.42226e7	−0.523605	−0.261802	−	0.965121i	$-0.584317\pi$
$941$	−1.42226e7	−0.523605	−0.261802	+	0.965121i	$0.584317\pi$
$942$	−1.44205e7	−0.529485
$943$	−4.68463e7	−1.71552
$944$	−1.29331e7	−0.472360
$945$	−4.13708e6	−0.150700
$946$	−9.54571e6	−0.346801
$947$	3.58939e7	1.30061	0.650303	−	0.759675i	$-0.274642\pi$
$947$	3.58939e7	1.30061	0.650303	+	0.759675i	$0.274642\pi$
$948$	1.19164e7	0.430651
$949$	−6.76710e6	−0.243914
$950$	85000.0	0.00305569
$951$	2.84590e7	1.02039
$952$	1.74772e7	0.624999
$953$	3.86510e6	0.137857	0.0689285	−	0.997622i	$-0.478042\pi$
$953$	3.86510e6	0.137857	0.0689285	+	0.997622i	$0.478042\pi$
$954$	3.15025e6	0.112066
$955$	−1.81683e7	−0.644623
$956$	5.55701e6	0.196651
$957$	−8.85184e6	−0.312431
$958$	−1.31340e7	−0.462362
$959$	5.21115e7	1.82973
$960$	−921600.	−0.0322749
$961$	−2.54679e7	−0.889578
$962$	5.28564e6	0.184145
$963$	7.34030e6	0.255063
$964$	−1.49740e7	−0.518975
$965$	1.50976e7	0.521904
$966$	−3.59159e7	−1.23835
$967$	−5.17522e7	−1.77976	−0.889882	−	0.456192i	$-0.849213\pi$
$967$	−5.17522e7	−1.77976	−0.889882	+	0.456192i	$0.849213\pi$
$968$	4.43149e6	0.152006
$969$	−368118.	−0.0125944
$970$	−1.51667e7	−0.517561
$971$	−2.46544e7	−0.839162	−0.419581	−	0.907718i	$-0.637823\pi$
$971$	−2.46544e7	−0.839162	−0.419581	+	0.907718i	$0.637823\pi$
$972$	−944784.	−0.0320750
$973$	−6.04798e7	−2.04799
$974$	−1.80084e6	−0.0608243
$975$	−950625.	−0.0320256
$976$	−6.04134e6	−0.203006
$977$	−3.16778e7	−1.06174	−0.530871	−	0.847453i	$-0.678135\pi$
$977$	−3.16778e7	−1.06174	−0.530871	+	0.847453i	$0.678135\pi$
$978$	2.18911e7	0.731848
$979$	−1.19506e7	−0.398505
$980$	1.38888e7	0.461955
$981$	−1.53258e7	−0.508454
$982$	1.99784e7	0.661122
$983$	5.69022e7	1.87822	0.939108	−	0.343622i	$-0.111654\pi$
$983$	5.69022e7	1.87822	0.939108	+	0.343622i	$0.111654\pi$
$984$	6.13958e6	0.202140
$985$	2.05035e7	0.673345
$986$	−1.56198e7	−0.511660
$987$	−4.31114e7	−1.40864
$988$	−91936.0	−0.00299635
$989$	3.46150e7	1.12531
$990$	2.45430e6	0.0795865
$991$	−1.82693e7	−0.590931	−0.295465	−	0.955353i	$-0.595475\pi$
$991$	−1.82693e7	−0.590931	−0.295465	+	0.955353i	$0.595475\pi$
$992$	−1.82067e6	−0.0587425
$993$	−1.16781e7	−0.375837
$994$	−1.88147e7	−0.603991
$995$	−3.20710e6	−0.102696
$996$	6.05491e6	0.193401
$997$	−3.49313e7	−1.11295	−0.556477	−	0.830863i	$-0.687847\pi$
$997$	−3.49313e7	−1.11295	−0.556477	+	0.830863i	$0.687847\pi$
$998$	1.78725e7	0.568012
$999$	5.70005e6	0.180703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.6.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.6.a.b.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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