LMFDB - Embedded newform 1078.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,6,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1078.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1078.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:	$172.893757758$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -1.00000 q^{3} +16.0000 q^{4} +51.0000 q^{5} +4.00000 q^{6} -64.0000 q^{8} -242.000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -1.00000 q^{3} +16.0000 q^{4} +51.0000 q^{5} +4.00000 q^{6} -64.0000 q^{8} -242.000 q^{9} -204.000 q^{10} -121.000 q^{11} -16.0000 q^{12} -692.000 q^{13} -51.0000 q^{15} +256.000 q^{16} +738.000 q^{17} +968.000 q^{18} -1424.00 q^{19} +816.000 q^{20} +484.000 q^{22} -1779.00 q^{23} +64.0000 q^{24} -524.000 q^{25} +2768.00 q^{26} +485.000 q^{27} -2064.00 q^{29} +204.000 q^{30} -6245.00 q^{31} -1024.00 q^{32} +121.000 q^{33} -2952.00 q^{34} -3872.00 q^{36} -14785.0 q^{37} +5696.00 q^{38} +692.000 q^{39} -3264.00 q^{40} -5304.00 q^{41} +17798.0 q^{43} -1936.00 q^{44} -12342.0 q^{45} +7116.00 q^{46} +17184.0 q^{47} -256.000 q^{48} +2096.00 q^{50} -738.000 q^{51} -11072.0 q^{52} -30726.0 q^{53} -1940.00 q^{54} -6171.00 q^{55} +1424.00 q^{57} +8256.00 q^{58} +34989.0 q^{59} -816.000 q^{60} +45940.0 q^{61} +24980.0 q^{62} +4096.00 q^{64} -35292.0 q^{65} -484.000 q^{66} +25343.0 q^{67} +11808.0 q^{68} +1779.00 q^{69} +13311.0 q^{71} +15488.0 q^{72} +53260.0 q^{73} +59140.0 q^{74} +524.000 q^{75} -22784.0 q^{76} -2768.00 q^{78} +77234.0 q^{79} +13056.0 q^{80} +58321.0 q^{81} +21216.0 q^{82} -55014.0 q^{83} +37638.0 q^{85} -71192.0 q^{86} +2064.00 q^{87} +7744.00 q^{88} -125415. q^{89} +49368.0 q^{90} -28464.0 q^{92} +6245.00 q^{93} -68736.0 q^{94} -72624.0 q^{95} +1024.00 q^{96} +88807.0 q^{97} +29282.0 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.0641500	−0.0320750	−	0.999485i	$-0.510212\pi$
$3$	−1.00000	−0.0641500	−0.0320750	+	0.999485i	$0.510212\pi$
$4$	16.0000	0.500000
$5$	51.0000	0.912316	0.456158	−	0.889899i	$-0.349225\pi$
$5$	51.0000	0.912316	0.456158	+	0.889899i	$0.349225\pi$
$6$	4.00000	0.0453609
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	−242.000	−0.995885
$10$	−204.000	−0.645105
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−16.0000	−0.0320750
$13$	−692.000	−1.13566	−0.567829	−	0.823146i	$-0.692217\pi$
$13$	−692.000	−1.13566	−0.567829	+	0.823146i	$0.692217\pi$
$14$	0	0
$15$	−51.0000	−0.0585251
$16$	256.000	0.250000
$17$	738.000	0.619347	0.309674	−	0.950843i	$-0.399780\pi$
$17$	738.000	0.619347	0.309674	+	0.950843i	$0.399780\pi$
$18$	968.000	0.704197
$19$	−1424.00	−0.904953	−0.452476	−	0.891776i	$-0.649459\pi$
$19$	−1424.00	−0.904953	−0.452476	+	0.891776i	$0.649459\pi$
$20$	816.000	0.456158
$21$	0	0
$22$	484.000	0.213201
$23$	−1779.00	−0.701223	−0.350612	−	0.936521i	$-0.614026\pi$
$23$	−1779.00	−0.701223	−0.350612	+	0.936521i	$0.614026\pi$
$24$	64.0000	0.0226805
$25$	−524.000	−0.167680
$26$	2768.00	0.803032
$27$	485.000	0.128036
$28$	0	0
$29$	−2064.00	−0.455737	−0.227869	−	0.973692i	$-0.573176\pi$
$29$	−2064.00	−0.455737	−0.227869	+	0.973692i	$0.573176\pi$
$30$	204.000	0.0413835
$31$	−6245.00	−1.16715	−0.583577	−	0.812058i	$-0.698347\pi$
$31$	−6245.00	−1.16715	−0.583577	+	0.812058i	$0.698347\pi$
$32$	−1024.00	−0.176777
$33$	121.000	0.0193420
$34$	−2952.00	−0.437944
$35$	0	0
$36$	−3872.00	−0.497942
$37$	−14785.0	−1.77549	−0.887743	−	0.460340i	$-0.847727\pi$
$37$	−14785.0	−1.77549	−0.887743	+	0.460340i	$0.847727\pi$
$38$	5696.00	0.639898
$39$	692.000	0.0728525
$40$	−3264.00	−0.322552
$41$	−5304.00	−0.492770	−0.246385	−	0.969172i	$-0.579243\pi$
$41$	−5304.00	−0.492770	−0.246385	+	0.969172i	$0.579243\pi$
$42$	0	0
$43$	17798.0	1.46791	0.733956	−	0.679197i	$-0.237672\pi$
$43$	17798.0	1.46791	0.733956	+	0.679197i	$0.237672\pi$
$44$	−1936.00	−0.150756
$45$	−12342.0	−0.908561
$46$	7116.00	0.495840
$47$	17184.0	1.13470	0.567348	−	0.823478i	$-0.307969\pi$
$47$	17184.0	1.13470	0.567348	+	0.823478i	$0.307969\pi$
$48$	−256.000	−0.0160375
$49$	0	0
$50$	2096.00	0.118568
$51$	−738.000	−0.0397311
$52$	−11072.0	−0.567829
$53$	−30726.0	−1.50251	−0.751253	−	0.660014i	$-0.770550\pi$
$53$	−30726.0	−1.50251	−0.751253	+	0.660014i	$0.770550\pi$
$54$	−1940.00	−0.0905352
$55$	−6171.00	−0.275074
$56$	0	0
$57$	1424.00	0.0580528
$58$	8256.00	0.322255
$59$	34989.0	1.30858	0.654292	−	0.756242i	$-0.272967\pi$
$59$	34989.0	1.30858	0.654292	+	0.756242i	$0.272967\pi$
$60$	−816.000	−0.0292625
$61$	45940.0	1.58076	0.790381	−	0.612616i	$-0.209883\pi$
$61$	45940.0	1.58076	0.790381	+	0.612616i	$0.209883\pi$
$62$	24980.0	0.825303
$63$	0	0
$64$	4096.00	0.125000
$65$	−35292.0	−1.03608
$66$	−484.000	−0.0136768
$67$	25343.0	0.689717	0.344859	−	0.938655i	$-0.387927\pi$
$67$	25343.0	0.689717	0.344859	+	0.938655i	$0.387927\pi$
$68$	11808.0	0.309674
$69$	1779.00	0.0449835
$70$	0	0
$71$	13311.0	0.313375	0.156688	−	0.987648i	$-0.449918\pi$
$71$	13311.0	0.313375	0.156688	+	0.987648i	$0.449918\pi$
$72$	15488.0	0.352098
$73$	53260.0	1.16975	0.584876	−	0.811123i	$-0.301143\pi$
$73$	53260.0	1.16975	0.584876	+	0.811123i	$0.301143\pi$
$74$	59140.0	1.25546
$75$	524.000	0.0107567
$76$	−22784.0	−0.452476
$77$	0	0
$78$	−2768.00	−0.0515145
$79$	77234.0	1.39233	0.696163	−	0.717884i	$-0.254889\pi$
$79$	77234.0	1.39233	0.696163	+	0.717884i	$0.254889\pi$
$80$	13056.0	0.228079
$81$	58321.0	0.987671
$82$	21216.0	0.348441
$83$	−55014.0	−0.876553	−0.438276	−	0.898840i	$-0.644411\pi$
$83$	−55014.0	−0.876553	−0.438276	+	0.898840i	$0.644411\pi$
$84$	0	0
$85$	37638.0	0.565040
$86$	−71192.0	−1.03797
$87$	2064.00	0.0292356
$88$	7744.00	0.106600
$89$	−125415.	−1.67832	−0.839159	−	0.543886i	$-0.816953\pi$
$89$	−125415.	−1.67832	−0.839159	+	0.543886i	$0.816953\pi$
$90$	49368.0	0.642450
$91$	0	0
$92$	−28464.0	−0.350612
$93$	6245.00	0.0748730
$94$	−68736.0	−0.802351
$95$	−72624.0	−0.825603
$96$	1024.00	0.0113402
$97$	88807.0	0.958336	0.479168	−	0.877723i	$-0.340938\pi$
$97$	88807.0	0.958336	0.479168	+	0.877723i	$0.340938\pi$
$98$	0	0
$99$	29282.0	0.300271
$100$	−8384.00	−0.0838400
$101$	−1482.00	−0.0144559	−0.00722794	−	0.999974i	$-0.502301\pi$
$101$	−1482.00	−0.0144559	−0.00722794	+	0.999974i	$0.502301\pi$
$102$	2952.00	0.0280942
$103$	117496.	1.09126	0.545632	−	0.838025i	$-0.316290\pi$
$103$	117496.	1.09126	0.545632	+	0.838025i	$0.316290\pi$
$104$	44288.0	0.401516
$105$	0	0
$106$	122904.	1.06243
$107$	−79362.0	−0.670121	−0.335060	−	0.942197i	$-0.608757\pi$
$107$	−79362.0	−0.670121	−0.335060	+	0.942197i	$0.608757\pi$
$108$	7760.00	0.0640180
$109$	87842.0	0.708167	0.354084	−	0.935214i	$-0.384793\pi$
$109$	87842.0	0.708167	0.354084	+	0.935214i	$0.384793\pi$
$110$	24684.0	0.194506
$111$	14785.0	0.113897
$112$	0	0
$113$	−47247.0	−0.348079	−0.174040	−	0.984739i	$-0.555682\pi$
$113$	−47247.0	−0.348079	−0.174040	+	0.984739i	$0.555682\pi$
$114$	−5696.00	−0.0410495
$115$	−90729.0	−0.639737
$116$	−33024.0	−0.227869
$117$	167464.	1.13098
$118$	−139956.	−0.925308
$119$	0	0
$120$	3264.00	0.0206917
$121$	14641.0	0.0909091
$122$	−183760.	−1.11777
$123$	5304.00	0.0316112
$124$	−99920.0	−0.583577
$125$	−186099.	−1.06529
$126$	0	0
$127$	−239416.	−1.31718	−0.658588	−	0.752504i	$-0.728846\pi$
$127$	−239416.	−1.31718	−0.658588	+	0.752504i	$0.728846\pi$
$128$	−16384.0	−0.0883883
$129$	−17798.0	−0.0941666
$130$	141168.	0.732618
$131$	98142.0	0.499662	0.249831	−	0.968289i	$-0.419625\pi$
$131$	98142.0	0.499662	0.249831	+	0.968289i	$0.419625\pi$
$132$	1936.00	0.00967098
$133$	0	0
$134$	−101372.	−0.487704
$135$	24735.0	0.116809
$136$	−47232.0	−0.218972
$137$	400137.	1.82141	0.910704	−	0.413059i	$-0.135540\pi$
$137$	400137.	1.82141	0.910704	+	0.413059i	$0.135540\pi$
$138$	−7116.00	−0.0318081
$139$	−205766.	−0.903310	−0.451655	−	0.892193i	$-0.649166\pi$
$139$	−205766.	−0.903310	−0.451655	+	0.892193i	$0.649166\pi$
$140$	0	0
$141$	−17184.0	−0.0727908
$142$	−53244.0	−0.221590
$143$	83732.0	0.342414
$144$	−61952.0	−0.248971
$145$	−105264.	−0.415776
$146$	−213040.	−0.827140
$147$	0	0
$148$	−236560.	−0.887743
$149$	87726.0	0.323715	0.161857	−	0.986814i	$-0.448252\pi$
$149$	87726.0	0.323715	0.161857	+	0.986814i	$0.448252\pi$
$150$	−2096.00	−0.00760612
$151$	−432778.	−1.54462	−0.772312	−	0.635243i	$-0.780900\pi$
$151$	−432778.	−1.54462	−0.772312	+	0.635243i	$0.780900\pi$
$152$	91136.0	0.319949
$153$	−178596.	−0.616798
$154$	0	0
$155$	−318495.	−1.06481
$156$	11072.0	0.0364263
$157$	34075.0	0.110328	0.0551641	−	0.998477i	$-0.482432\pi$
$157$	34075.0	0.110328	0.0551641	+	0.998477i	$0.482432\pi$
$158$	−308936.	−0.984523
$159$	30726.0	0.0963858
$160$	−52224.0	−0.161276
$161$	0	0
$162$	−233284.	−0.698389
$163$	45020.0	0.132720	0.0663600	−	0.997796i	$-0.478861\pi$
$163$	45020.0	0.132720	0.0663600	+	0.997796i	$0.478861\pi$
$164$	−84864.0	−0.246385
$165$	6171.00	0.0176460
$166$	220056.	0.619816
$167$	−482556.	−1.33893	−0.669463	−	0.742845i	$-0.733476\pi$
$167$	−482556.	−1.33893	−0.669463	+	0.742845i	$0.733476\pi$
$168$	0	0
$169$	107571.	0.289720
$170$	−150552.	−0.399544
$171$	344608.	0.901229
$172$	284768.	0.733956
$173$	766254.	1.94651	0.973257	−	0.229719i	$-0.0737808\pi$
$173$	766254.	1.94651	0.973257	+	0.229719i	$0.0737808\pi$
$174$	−8256.00	−0.0206727
$175$	0	0
$176$	−30976.0	−0.0753778
$177$	−34989.0	−0.0839457
$178$	501660.	1.18675
$179$	303399.	0.707753	0.353876	−	0.935292i	$-0.384863\pi$
$179$	303399.	0.707753	0.353876	+	0.935292i	$0.384863\pi$
$180$	−197472.	−0.454281
$181$	285181.	0.647030	0.323515	−	0.946223i	$-0.395135\pi$
$181$	285181.	0.647030	0.323515	+	0.946223i	$0.395135\pi$
$182$	0	0
$183$	−45940.0	−0.101406
$184$	113856.	0.247920
$185$	−754035.	−1.61980
$186$	−24980.0	−0.0529432
$187$	−89298.0	−0.186740
$188$	274944.	0.567348
$189$	0	0
$190$	290496.	0.583789
$191$	767067.	1.52142	0.760711	−	0.649090i	$-0.224850\pi$
$191$	767067.	1.52142	0.760711	+	0.649090i	$0.224850\pi$
$192$	−4096.00	−0.00801875
$193$	411668.	0.795525	0.397763	−	0.917488i	$-0.369787\pi$
$193$	411668.	0.795525	0.397763	+	0.917488i	$0.369787\pi$
$194$	−355228.	−0.677646
$195$	35292.0	0.0664645
$196$	0	0
$197$	−759258.	−1.39387	−0.696937	−	0.717132i	$-0.745455\pi$
$197$	−759258.	−1.39387	−0.696937	+	0.717132i	$0.745455\pi$
$198$	−117128.	−0.212323
$199$	46600.0	0.0834167	0.0417084	−	0.999130i	$-0.486720\pi$
$199$	46600.0	0.0834167	0.0417084	+	0.999130i	$0.486720\pi$
$200$	33536.0	0.0592838
$201$	−25343.0	−0.0442454
$202$	5928.00	0.0102219
$203$	0	0
$204$	−11808.0	−0.0198656
$205$	−270504.	−0.449561
$206$	−469984.	−0.771641
$207$	430518.	0.698338
$208$	−177152.	−0.283915
$209$	172304.	0.272854
$210$	0	0
$211$	−932428.	−1.44181	−0.720907	−	0.693032i	$-0.756274\pi$
$211$	−932428.	−1.44181	−0.720907	+	0.693032i	$0.756274\pi$
$212$	−491616.	−0.751253
$213$	−13311.0	−0.0201030
$214$	317448.	0.473847
$215$	907698.	1.33920
$216$	−31040.0	−0.0452676
$217$	0	0
$218$	−351368.	−0.500750
$219$	−53260.0	−0.0750397
$220$	−98736.0	−0.137537
$221$	−510696.	−0.703367
$222$	−59140.0	−0.0805376
$223$	−169745.	−0.228578	−0.114289	−	0.993448i	$-0.536459\pi$
$223$	−169745.	−0.228578	−0.114289	+	0.993448i	$0.536459\pi$
$224$	0	0
$225$	126808.	0.166990
$226$	188988.	0.246129
$227$	−198078.	−0.255136	−0.127568	−	0.991830i	$-0.540717\pi$
$227$	−198078.	−0.255136	−0.127568	+	0.991830i	$0.540717\pi$
$228$	22784.0	0.0290264
$229$	849997.	1.07110	0.535548	−	0.844505i	$-0.320105\pi$
$229$	849997.	1.07110	0.535548	+	0.844505i	$0.320105\pi$
$230$	362916.	0.452362
$231$	0	0
$232$	132096.	0.161128
$233$	−401832.	−0.484903	−0.242451	−	0.970164i	$-0.577952\pi$
$233$	−401832.	−0.484903	−0.242451	+	0.970164i	$0.577952\pi$
$234$	−669856.	−0.799727
$235$	876384.	1.03520
$236$	559824.	0.654292
$237$	−77234.0	−0.0893177
$238$	0	0
$239$	855174.	0.968411	0.484206	−	0.874954i	$-0.339109\pi$
$239$	855174.	0.968411	0.484206	+	0.874954i	$0.339109\pi$
$240$	−13056.0	−0.0146313
$241$	−1.12546e6	−1.24821	−0.624107	−	0.781339i	$-0.714537\pi$
$241$	−1.12546e6	−1.24821	−0.624107	+	0.781339i	$0.714537\pi$
$242$	−58564.0	−0.0642824
$243$	−176176.	−0.191395
$244$	735040.	0.790381
$245$	0	0
$246$	−21216.0	−0.0223525
$247$	985408.	1.02772
$248$	399680.	0.412651
$249$	55014.0	0.0562309
$250$	744396.	0.753276
$251$	1.19751e6	1.19976	0.599882	−	0.800088i	$-0.295214\pi$
$251$	1.19751e6	1.19976	0.599882	+	0.800088i	$0.295214\pi$
$252$	0	0
$253$	215259.	0.211427
$254$	957664.	0.931384
$255$	−37638.0	−0.0362473
$256$	65536.0	0.0625000
$257$	−37758.0	−0.0356596	−0.0178298	−	0.999841i	$-0.505676\pi$
$257$	−37758.0	−0.0356596	−0.0178298	+	0.999841i	$0.505676\pi$
$258$	71192.0	0.0665858
$259$	0	0
$260$	−564672.	−0.518040
$261$	499488.	0.453862
$262$	−392568.	−0.353315
$263$	−631254.	−0.562749	−0.281375	−	0.959598i	$-0.590790\pi$
$263$	−631254.	−0.562749	−0.281375	+	0.959598i	$0.590790\pi$
$264$	−7744.00	−0.00683842
$265$	−1.56703e6	−1.37076
$266$	0	0
$267$	125415.	0.107664
$268$	405488.	0.344859
$269$	1.08034e6	0.910292	0.455146	−	0.890417i	$-0.349587\pi$
$269$	1.08034e6	0.910292	0.455146	+	0.890417i	$0.349587\pi$
$270$	−98940.0	−0.0825967
$271$	816100.	0.675025	0.337513	−	0.941321i	$-0.390414\pi$
$271$	816100.	0.675025	0.337513	+	0.941321i	$0.390414\pi$
$272$	188928.	0.154837
$273$	0	0
$274$	−1.60055e6	−1.28793
$275$	63404.0	0.0505574
$276$	28464.0	0.0224917
$277$	1.68820e6	1.32198	0.660989	−	0.750396i	$-0.270137\pi$
$277$	1.68820e6	1.32198	0.660989	+	0.750396i	$0.270137\pi$
$278$	823064.	0.638736
$279$	1.51129e6	1.16235
$280$	0	0
$281$	−879042.	−0.664116	−0.332058	−	0.943259i	$-0.607743\pi$
$281$	−879042.	−0.664116	−0.332058	+	0.943259i	$0.607743\pi$
$282$	68736.0	0.0514709
$283$	−1.54027e6	−1.14322	−0.571611	−	0.820525i	$-0.693681\pi$
$283$	−1.54027e6	−1.14322	−0.571611	+	0.820525i	$0.693681\pi$
$284$	212976.	0.156688
$285$	72624.0	0.0529624
$286$	−334928.	−0.242123
$287$	0	0
$288$	247808.	0.176049
$289$	−875213.	−0.616409
$290$	421056.	0.293998
$291$	−88807.0	−0.0614773
$292$	852160.	0.584876
$293$	−720840.	−0.490535	−0.245267	−	0.969455i	$-0.578876\pi$
$293$	−720840.	−0.490535	−0.245267	+	0.969455i	$0.578876\pi$
$294$	0	0
$295$	1.78444e6	1.19384
$296$	946240.	0.627729
$297$	−58685.0	−0.0386043
$298$	−350904.	−0.228901
$299$	1.23107e6	0.796350
$300$	8384.00	0.00537834
$301$	0	0
$302$	1.73111e6	1.09221
$303$	1482.00	0.000927346 0
$304$	−364544.	−0.226238
$305$	2.34294e6	1.44215
$306$	714384.	0.436142
$307$	1.03905e6	0.629201	0.314601	−	0.949224i	$-0.398129\pi$
$307$	1.03905e6	0.629201	0.314601	+	0.949224i	$0.398129\pi$
$308$	0	0
$309$	−117496.	−0.0700046
$310$	1.27398e6	0.752937
$311$	1.25135e6	0.733630	0.366815	−	0.930294i	$-0.380448\pi$
$311$	1.25135e6	0.733630	0.366815	+	0.930294i	$0.380448\pi$
$312$	−44288.0	−0.0257573
$313$	1.44336e6	0.832749	0.416375	−	0.909193i	$-0.363301\pi$
$313$	1.44336e6	0.832749	0.416375	+	0.909193i	$0.363301\pi$
$314$	−136300.	−0.0780139
$315$	0	0
$316$	1.23574e6	0.696163
$317$	−2.01208e6	−1.12460	−0.562298	−	0.826934i	$-0.690083\pi$
$317$	−2.01208e6	−1.12460	−0.562298	+	0.826934i	$0.690083\pi$
$318$	−122904.	−0.0681551
$319$	249744.	0.137410
$320$	208896.	0.114039
$321$	79362.0	0.0429883
$322$	0	0
$323$	−1.05091e6	−0.560480
$324$	933136.	0.493836
$325$	362608.	0.190427
$326$	−180080.	−0.0938472
$327$	−87842.0	−0.0454290
$328$	339456.	0.174220
$329$	0	0
$330$	−24684.0	−0.0124776
$331$	2.01734e6	1.01207	0.506033	−	0.862514i	$-0.331112\pi$
$331$	2.01734e6	1.01207	0.506033	+	0.862514i	$0.331112\pi$
$332$	−880224.	−0.438276
$333$	3.57797e6	1.76818
$334$	1.93022e6	0.946764
$335$	1.29249e6	0.629240
$336$	0	0
$337$	264122.	0.126686	0.0633432	−	0.997992i	$-0.479824\pi$
$337$	264122.	0.126686	0.0633432	+	0.997992i	$0.479824\pi$
$338$	−430284.	−0.204863
$339$	47247.0	0.0223293
$340$	602208.	0.282520
$341$	755645.	0.351910
$342$	−1.37843e6	−0.637265
$343$	0	0
$344$	−1.13907e6	−0.518985
$345$	90729.0	0.0410392
$346$	−3.06502e6	−1.37639
$347$	−1.71049e6	−0.762601	−0.381300	−	0.924451i	$-0.624524\pi$
$347$	−1.71049e6	−0.762601	−0.381300	+	0.924451i	$0.624524\pi$
$348$	33024.0	0.0146178
$349$	−218822.	−0.0961673	−0.0480836	−	0.998843i	$-0.515311\pi$
$349$	−218822.	−0.0961673	−0.0480836	+	0.998843i	$0.515311\pi$
$350$	0	0
$351$	−335620.	−0.145405
$352$	123904.	0.0533002
$353$	−3.68192e6	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$353$	−3.68192e6	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$354$	139956.	0.0593586
$355$	678861.	0.285897
$356$	−2.00664e6	−0.839159
$357$	0	0
$358$	−1.21360e6	−0.500457
$359$	1.88528e6	0.772042	0.386021	−	0.922490i	$-0.373849\pi$
$359$	1.88528e6	0.772042	0.386021	+	0.922490i	$0.373849\pi$
$360$	789888.	0.321225
$361$	−448323.	−0.181060
$362$	−1.14072e6	−0.457519
$363$	−14641.0	−0.00583182
$364$	0	0
$365$	2.71626e6	1.06718
$366$	183760.	0.0717048
$367$	3.11666e6	1.20788	0.603940	−	0.797029i	$-0.293596\pi$
$367$	3.11666e6	1.20788	0.603940	+	0.797029i	$0.293596\pi$
$368$	−455424.	−0.175306
$369$	1.28357e6	0.490742
$370$	3.01614e6	1.14537
$371$	0	0
$372$	99920.0	0.0374365
$373$	1.39441e6	0.518943	0.259471	−	0.965751i	$-0.416452\pi$
$373$	1.39441e6	0.518943	0.259471	+	0.965751i	$0.416452\pi$
$374$	357192.	0.132045
$375$	186099.	0.0683386
$376$	−1.09978e6	−0.401176
$377$	1.42829e6	0.517562
$378$	0	0
$379$	−4.26036e6	−1.52352	−0.761759	−	0.647860i	$-0.775664\pi$
$379$	−4.26036e6	−1.52352	−0.761759	+	0.647860i	$0.775664\pi$
$380$	−1.16198e6	−0.412801
$381$	239416.	0.0844969
$382$	−3.06827e6	−1.07581
$383$	−201765.	−0.0702828	−0.0351414	−	0.999382i	$-0.511188\pi$
$383$	−201765.	−0.0702828	−0.0351414	+	0.999382i	$0.511188\pi$
$384$	16384.0	0.00567012
$385$	0	0
$386$	−1.64667e6	−0.562521
$387$	−4.30712e6	−1.46187
$388$	1.42091e6	0.479168
$389$	1.94882e6	0.652977	0.326489	−	0.945201i	$-0.394135\pi$
$389$	1.94882e6	0.652977	0.326489	+	0.945201i	$0.394135\pi$
$390$	−141168.	−0.0469975
$391$	−1.31290e6	−0.434301
$392$	0	0
$393$	−98142.0	−0.0320534
$394$	3.03703e6	0.985618
$395$	3.93893e6	1.27024
$396$	468512.	0.150135
$397$	1.46826e6	0.467548	0.233774	−	0.972291i	$-0.424892\pi$
$397$	1.46826e6	0.467548	0.233774	+	0.972291i	$0.424892\pi$
$398$	−186400.	−0.0589845
$399$	0	0
$400$	−134144.	−0.0419200
$401$	2.24618e6	0.697563	0.348781	−	0.937204i	$-0.386596\pi$
$401$	2.24618e6	0.697563	0.348781	+	0.937204i	$0.386596\pi$
$402$	101372.	0.0312862
$403$	4.32154e6	1.32549
$404$	−23712.0	−0.00722794
$405$	2.97437e6	0.901068
$406$	0	0
$407$	1.78898e6	0.535329
$408$	47232.0	0.0140471
$409$	3.61488e6	1.06853	0.534263	−	0.845318i	$-0.320589\pi$
$409$	3.61488e6	1.06853	0.534263	+	0.845318i	$0.320589\pi$
$410$	1.08202e6	0.317888
$411$	−400137.	−0.116843
$412$	1.87994e6	0.545632
$413$	0	0
$414$	−1.72207e6	−0.493799
$415$	−2.80571e6	−0.799693
$416$	708608.	0.200758
$417$	205766.	0.0579473
$418$	−689216.	−0.192937
$419$	3.81239e6	1.06087	0.530435	−	0.847726i	$-0.322029\pi$
$419$	3.81239e6	1.06087	0.530435	+	0.847726i	$0.322029\pi$
$420$	0	0
$421$	1.97346e6	0.542655	0.271327	−	0.962487i	$-0.412537\pi$
$421$	1.97346e6	0.542655	0.271327	+	0.962487i	$0.412537\pi$
$422$	3.72971e6	1.01952
$423$	−4.15853e6	−1.13003
$424$	1.96646e6	0.531216
$425$	−386712.	−0.103852
$426$	53244.0	0.0142150
$427$	0	0
$428$	−1.26979e6	−0.335060
$429$	−83732.0	−0.0219659
$430$	−3.63079e6	−0.946957
$431$	−2.08359e6	−0.540280	−0.270140	−	0.962821i	$-0.587070\pi$
$431$	−2.08359e6	−0.540280	−0.270140	+	0.962821i	$0.587070\pi$
$432$	124160.	0.0320090
$433$	72691.0	0.0186321	0.00931603	−	0.999957i	$-0.497035\pi$
$433$	72691.0	0.0186321	0.00931603	+	0.999957i	$0.497035\pi$
$434$	0	0
$435$	105264.	0.0266721
$436$	1.40547e6	0.354084
$437$	2.53330e6	0.634574
$438$	213040.	0.0530611
$439$	−594392.	−0.147201	−0.0736007	−	0.997288i	$-0.523449\pi$
$439$	−594392.	−0.147201	−0.0736007	+	0.997288i	$0.523449\pi$
$440$	394944.	0.0972532
$441$	0	0
$442$	2.04278e6	0.497355
$443$	4.56651e6	1.10554	0.552770	−	0.833334i	$-0.313571\pi$
$443$	4.56651e6	1.10554	0.552770	+	0.833334i	$0.313571\pi$
$444$	236560.	0.0569487
$445$	−6.39616e6	−1.53116
$446$	678980.	0.161629
$447$	−87726.0	−0.0207663
$448$	0	0
$449$	−5.44382e6	−1.27435	−0.637174	−	0.770720i	$-0.719897\pi$
$449$	−5.44382e6	−1.27435	−0.637174	+	0.770720i	$0.719897\pi$
$450$	−507232.	−0.118080
$451$	641784.	0.148576
$452$	−755952.	−0.174040
$453$	432778.	0.0990877
$454$	792312.	0.180408
$455$	0	0
$456$	−91136.0	−0.0205247
$457$	6.70312e6	1.50137	0.750683	−	0.660662i	$-0.229724\pi$
$457$	6.70312e6	1.50137	0.750683	+	0.660662i	$0.229724\pi$
$458$	−3.39999e6	−0.757380
$459$	357930.	0.0792988
$460$	−1.45166e6	−0.319869
$461$	1.25994e6	0.276120	0.138060	−	0.990424i	$-0.455913\pi$
$461$	1.25994e6	0.276120	0.138060	+	0.990424i	$0.455913\pi$
$462$	0	0
$463$	−5.02308e6	−1.08897	−0.544487	−	0.838769i	$-0.683276\pi$
$463$	−5.02308e6	−1.08897	−0.544487	+	0.838769i	$0.683276\pi$
$464$	−528384.	−0.113934
$465$	318495.	0.0683078
$466$	1.60733e6	0.342878
$467$	2.35660e6	0.500028	0.250014	−	0.968242i	$-0.419565\pi$
$467$	2.35660e6	0.500028	0.250014	+	0.968242i	$0.419565\pi$
$468$	2.67942e6	0.565492
$469$	0	0
$470$	−3.50554e6	−0.731998
$471$	−34075.0	−0.00707756
$472$	−2.23930e6	−0.462654
$473$	−2.15356e6	−0.442592
$474$	308936.	0.0631572
$475$	746176.	0.151743
$476$	0	0
$477$	7.43569e6	1.49632
$478$	−3.42070e6	−0.684770
$479$	6.72258e6	1.33874	0.669371	−	0.742928i	$-0.266563\pi$
$479$	6.72258e6	1.33874	0.669371	+	0.742928i	$0.266563\pi$
$480$	52224.0	0.0103459
$481$	1.02312e7	2.01634
$482$	4.50186e6	0.882620
$483$	0	0
$484$	234256.	0.0454545
$485$	4.52916e6	0.874305
$486$	704704.	0.135337
$487$	1.96001e6	0.374487	0.187243	−	0.982314i	$-0.440045\pi$
$487$	1.96001e6	0.374487	0.187243	+	0.982314i	$0.440045\pi$
$488$	−2.94016e6	−0.558884
$489$	−45020.0	−0.00851399
$490$	0	0
$491$	−579624.	−0.108503	−0.0542516	−	0.998527i	$-0.517277\pi$
$491$	−579624.	−0.108503	−0.0542516	+	0.998527i	$0.517277\pi$
$492$	84864.0	0.0158056
$493$	−1.52323e6	−0.282260
$494$	−3.94163e6	−0.726706
$495$	1.49338e6	0.273942
$496$	−1.59872e6	−0.291789
$497$	0	0
$498$	−220056.	−0.0397612
$499$	1.36905e6	0.246132	0.123066	−	0.992398i	$-0.460727\pi$
$499$	1.36905e6	0.246132	0.123066	+	0.992398i	$0.460727\pi$
$500$	−2.97758e6	−0.532646
$501$	482556.	0.0858921
$502$	−4.79005e6	−0.848361
$503$	−1.83343e6	−0.323105	−0.161552	−	0.986864i	$-0.551650\pi$
$503$	−1.83343e6	−0.323105	−0.161552	+	0.986864i	$0.551650\pi$
$504$	0	0
$505$	−75582.0	−0.0131883
$506$	−861036.	−0.149501
$507$	−107571.	−0.0185855
$508$	−3.83066e6	−0.658588
$509$	1.71266e6	0.293006	0.146503	−	0.989210i	$-0.453198\pi$
$509$	1.71266e6	0.293006	0.146503	+	0.989210i	$0.453198\pi$
$510$	150552.	0.0256307
$511$	0	0
$512$	−262144.	−0.0441942
$513$	−690640.	−0.115867
$514$	151032.	0.0252151
$515$	5.99230e6	0.995578
$516$	−284768.	−0.0470833
$517$	−2.07926e6	−0.342124
$518$	0	0
$519$	−766254.	−0.124869
$520$	2.25869e6	0.366309
$521$	789435.	0.127415	0.0637077	−	0.997969i	$-0.479707\pi$
$521$	789435.	0.127415	0.0637077	+	0.997969i	$0.479707\pi$
$522$	−1.99795e6	−0.320929
$523$	−627392.	−0.100296	−0.0501481	−	0.998742i	$-0.515969\pi$
$523$	−627392.	−0.100296	−0.0501481	+	0.998742i	$0.515969\pi$
$524$	1.57027e6	0.249831
$525$	0	0
$526$	2.52502e6	0.397924
$527$	−4.60881e6	−0.722873
$528$	30976.0	0.00483549
$529$	−3.27150e6	−0.508286
$530$	6.26810e6	0.969274
$531$	−8.46734e6	−1.30320
$532$	0	0
$533$	3.67037e6	0.559618
$534$	−501660.	−0.0761301
$535$	−4.04746e6	−0.611362
$536$	−1.62195e6	−0.243852
$537$	−303399.	−0.0454024
$538$	−4.32137e6	−0.643673
$539$	0	0
$540$	395760.	0.0584047
$541$	3.20895e6	0.471379	0.235689	−	0.971828i	$-0.424265\pi$
$541$	3.20895e6	0.471379	0.235689	+	0.971828i	$0.424265\pi$
$542$	−3.26440e6	−0.477315
$543$	−285181.	−0.0415070
$544$	−755712.	−0.109486
$545$	4.47994e6	0.646072
$546$	0	0
$547$	3.42658e6	0.489658	0.244829	−	0.969566i	$-0.421268\pi$
$547$	3.42658e6	0.489658	0.244829	+	0.969566i	$0.421268\pi$
$548$	6.40219e6	0.910704
$549$	−1.11175e7	−1.57426
$550$	−253616.	−0.0357495
$551$	2.93914e6	0.412421
$552$	−113856.	−0.0159041
$553$	0	0
$554$	−6.75279e6	−0.934779
$555$	754035.	0.103910
$556$	−3.29226e6	−0.451655
$557$	1.05198e7	1.43672	0.718358	−	0.695674i	$-0.244894\pi$
$557$	1.05198e7	1.43672	0.718358	+	0.695674i	$0.244894\pi$
$558$	−6.04516e6	−0.821906
$559$	−1.23162e7	−1.66705
$560$	0	0
$561$	89298.0	0.0119794
$562$	3.51617e6	0.469601
$563$	−5.47288e6	−0.727687	−0.363844	−	0.931460i	$-0.618536\pi$
$563$	−5.47288e6	−0.727687	−0.363844	+	0.931460i	$0.618536\pi$
$564$	−274944.	−0.0363954
$565$	−2.40960e6	−0.317558
$566$	6.16107e6	0.808379
$567$	0	0
$568$	−851904.	−0.110795
$569$	−1.17787e7	−1.52516	−0.762580	−	0.646893i	$-0.776068\pi$
$569$	−1.17787e7	−1.52516	−0.762580	+	0.646893i	$0.776068\pi$
$570$	−290496.	−0.0374501
$571$	−8.35628e6	−1.07256	−0.536281	−	0.844039i	$-0.680171\pi$
$571$	−8.35628e6	−1.07256	−0.536281	+	0.844039i	$0.680171\pi$
$572$	1.33971e6	0.171207
$573$	−767067.	−0.0975993
$574$	0	0
$575$	932196.	0.117581
$576$	−991232.	−0.124486
$577$	1.37758e7	1.72258	0.861288	−	0.508117i	$-0.169658\pi$
$577$	1.37758e7	1.72258	0.861288	+	0.508117i	$0.169658\pi$
$578$	3.50085e6	0.435867
$579$	−411668.	−0.0510330
$580$	−1.68422e6	−0.207888
$581$	0	0
$582$	355228.	0.0434710
$583$	3.71785e6	0.453023
$584$	−3.40864e6	−0.413570
$585$	8.54066e6	1.03182
$586$	2.88336e6	0.346860
$587$	1.27093e7	1.52239	0.761196	−	0.648522i	$-0.224612\pi$
$587$	1.27093e7	1.52239	0.761196	+	0.648522i	$0.224612\pi$
$588$	0	0
$589$	8.89288e6	1.05622
$590$	−7.13776e6	−0.844173
$591$	759258.	0.0894171
$592$	−3.78496e6	−0.443871
$593$	−1.00825e6	−0.117742	−0.0588711	−	0.998266i	$-0.518750\pi$
$593$	−1.00825e6	−0.117742	−0.0588711	+	0.998266i	$0.518750\pi$
$594$	234740.	0.0272974
$595$	0	0
$596$	1.40362e6	0.161857
$597$	−46600.0	−0.00535119
$598$	−4.92427e6	−0.563105
$599$	1.05100e7	1.19684	0.598421	−	0.801182i	$-0.295795\pi$
$599$	1.05100e7	1.19684	0.598421	+	0.801182i	$0.295795\pi$
$600$	−33536.0	−0.00380306
$601$	199390.	0.0225173	0.0112587	−	0.999937i	$-0.496416\pi$
$601$	199390.	0.0225173	0.0112587	+	0.999937i	$0.496416\pi$
$602$	0	0
$603$	−6.13301e6	−0.686879
$604$	−6.92445e6	−0.772312
$605$	746691.	0.0829378
$606$	−5928.00	−0.000655732 0
$607$	−16190.0	−0.00178351	−0.000891754	−	1.00000i	$-0.500284\pi$
$607$	−16190.0	−0.00178351	−0.000891754		1.00000i	$0.500284\pi$
$608$	1.45818e6	0.159975
$609$	0	0
$610$	−9.37176e6	−1.01976
$611$	−1.18913e7	−1.28863
$612$	−2.85754e6	−0.308399
$613$	−1.15253e7	−1.23880	−0.619402	−	0.785074i	$-0.712625\pi$
$613$	−1.15253e7	−1.23880	−0.619402	+	0.785074i	$0.712625\pi$
$614$	−4.15619e6	−0.444913
$615$	270504.	0.0288394
$616$	0	0
$617$	1.69974e7	1.79750	0.898751	−	0.438459i	$-0.144476\pi$
$617$	1.69974e7	1.79750	0.898751	+	0.438459i	$0.144476\pi$
$618$	469984.	0.0495008
$619$	1.84875e7	1.93933	0.969663	−	0.244445i	$-0.0786058\pi$
$619$	1.84875e7	1.93933	0.969663	+	0.244445i	$0.0786058\pi$
$620$	−5.09592e6	−0.532407
$621$	−862815.	−0.0897819
$622$	−5.00539e6	−0.518755
$623$	0	0
$624$	177152.	0.0182131
$625$	−7.85355e6	−0.804203
$626$	−5.77344e6	−0.588842
$627$	−172304.	−0.0175036
$628$	545200.	0.0551641
$629$	−1.09113e7	−1.09964
$630$	0	0
$631$	−4.54281e6	−0.454204	−0.227102	−	0.973871i	$-0.572925\pi$
$631$	−4.54281e6	−0.454204	−0.227102	+	0.973871i	$0.572925\pi$
$632$	−4.94298e6	−0.492261
$633$	932428.	0.0924924
$634$	8.04832e6	0.795210
$635$	−1.22102e7	−1.20168
$636$	491616.	0.0481929
$637$	0	0
$638$	−998976.	−0.0971635
$639$	−3.22126e6	−0.312086
$640$	−835584.	−0.0806381
$641$	1.84286e7	1.77153	0.885764	−	0.464136i	$-0.153635\pi$
$641$	1.84286e7	1.77153	0.885764	+	0.464136i	$0.153635\pi$
$642$	−317448.	−0.0303973
$643$	−9.66604e6	−0.921979	−0.460989	−	0.887406i	$-0.652505\pi$
$643$	−9.66604e6	−0.921979	−0.460989	+	0.887406i	$0.652505\pi$
$644$	0	0
$645$	−907698.	−0.0859097
$646$	4.20365e6	0.396319
$647$	4.51430e6	0.423965	0.211982	−	0.977273i	$-0.432008\pi$
$647$	4.51430e6	0.423965	0.211982	+	0.977273i	$0.432008\pi$
$648$	−3.73254e6	−0.349195
$649$	−4.23367e6	−0.394553
$650$	−1.45043e6	−0.134652
$651$	0	0
$652$	720320.	0.0663600
$653$	−5.37235e6	−0.493039	−0.246519	−	0.969138i	$-0.579287\pi$
$653$	−5.37235e6	−0.493039	−0.246519	+	0.969138i	$0.579287\pi$
$654$	351368.	0.0321231
$655$	5.00524e6	0.455850
$656$	−1.35782e6	−0.123192
$657$	−1.28889e7	−1.16494
$658$	0	0
$659$	9.87956e6	0.886184	0.443092	−	0.896476i	$-0.353881\pi$
$659$	9.87956e6	0.886184	0.443092	+	0.896476i	$0.353881\pi$
$660$	98736.0	0.00882299
$661$	−1.08052e7	−0.961898	−0.480949	−	0.876748i	$-0.659708\pi$
$661$	−1.08052e7	−0.961898	−0.480949	+	0.876748i	$0.659708\pi$
$662$	−8.06935e6	−0.715638
$663$	510696.	0.0451210
$664$	3.52090e6	0.309908
$665$	0	0
$666$	−1.43119e7	−1.25029
$667$	3.67186e6	0.319574
$668$	−7.72090e6	−0.669463
$669$	169745.	0.0146633
$670$	−5.16997e6	−0.444940
$671$	−5.55874e6	−0.476618
$672$	0	0
$673$	1.13275e7	0.964042	0.482021	−	0.876160i	$-0.339903\pi$
$673$	1.13275e7	0.964042	0.482021	+	0.876160i	$0.339903\pi$
$674$	−1.05649e6	−0.0895808
$675$	−254140.	−0.0214691
$676$	1.72114e6	0.144860
$677$	1.20595e7	1.01125	0.505624	−	0.862754i	$-0.331262\pi$
$677$	1.20595e7	1.01125	0.505624	+	0.862754i	$0.331262\pi$
$678$	−188988.	−0.0157892
$679$	0	0
$680$	−2.40883e6	−0.199772
$681$	198078.	0.0163670
$682$	−3.02258e6	−0.248838
$683$	−5.14166e6	−0.421747	−0.210873	−	0.977513i	$-0.567631\pi$
$683$	−5.14166e6	−0.421747	−0.210873	+	0.977513i	$0.567631\pi$
$684$	5.51373e6	0.450614
$685$	2.04070e7	1.66170
$686$	0	0
$687$	−849997.	−0.0687109
$688$	4.55629e6	0.366978
$689$	2.12624e7	1.70633
$690$	−362916.	−0.0290191
$691$	−1.31243e7	−1.04563	−0.522817	−	0.852445i	$-0.675119\pi$
$691$	−1.31243e7	−1.04563	−0.522817	+	0.852445i	$0.675119\pi$
$692$	1.22601e7	0.973257
$693$	0	0
$694$	6.84197e6	0.539240
$695$	−1.04941e7	−0.824104
$696$	−132096.	−0.0103363
$697$	−3.91435e6	−0.305195
$698$	875288.	0.0680005
$699$	401832.	0.0311065
$700$	0	0
$701$	3.65956e6	0.281277	0.140638	−	0.990061i	$-0.455084\pi$
$701$	3.65956e6	0.281277	0.140638	+	0.990061i	$0.455084\pi$
$702$	1.34248e6	0.102817
$703$	2.10538e7	1.60673
$704$	−495616.	−0.0376889
$705$	−876384.	−0.0664082
$706$	1.47277e7	1.11204
$707$	0	0
$708$	−559824.	−0.0419728
$709$	1.02252e7	0.763935	0.381968	−	0.924176i	$-0.375247\pi$
$709$	1.02252e7	0.763935	0.381968	+	0.924176i	$0.375247\pi$
$710$	−2.71544e6	−0.202160
$711$	−1.86906e7	−1.38660
$712$	8.02656e6	0.593375
$713$	1.11099e7	0.818436
$714$	0	0
$715$	4.27033e6	0.312390
$716$	4.85438e6	0.353876
$717$	−855174.	−0.0621236
$718$	−7.54114e6	−0.545916
$719$	−2.41683e7	−1.74351	−0.871753	−	0.489945i	$-0.837017\pi$
$719$	−2.41683e7	−1.74351	−0.871753	+	0.489945i	$0.837017\pi$
$720$	−3.15955e6	−0.227140
$721$	0	0
$722$	1.79329e6	0.128029
$723$	1.12546e6	0.0800730
$724$	4.56290e6	0.323515
$725$	1.08154e6	0.0764181
$726$	58564.0	0.00412372
$727$	−1.68246e7	−1.18062	−0.590310	−	0.807177i	$-0.700994\pi$
$727$	−1.68246e7	−1.18062	−0.590310	+	0.807177i	$0.700994\pi$
$728$	0	0
$729$	−1.39958e7	−0.975393
$730$	−1.08650e7	−0.754613
$731$	1.31349e7	0.909147
$732$	−735040.	−0.0507030
$733$	5.04168e6	0.346590	0.173295	−	0.984870i	$-0.444559\pi$
$733$	5.04168e6	0.346590	0.173295	+	0.984870i	$0.444559\pi$
$734$	−1.24666e7	−0.854101
$735$	0	0
$736$	1.82170e6	0.123960
$737$	−3.06650e6	−0.207958
$738$	−5.13427e6	−0.347007
$739$	−6.26375e6	−0.421913	−0.210957	−	0.977495i	$-0.567658\pi$
$739$	−6.26375e6	−0.421913	−0.210957	+	0.977495i	$0.567658\pi$
$740$	−1.20646e7	−0.809901
$741$	−985408.	−0.0659281
$742$	0	0
$743$	3.63976e6	0.241880	0.120940	−	0.992660i	$-0.461409\pi$
$743$	3.63976e6	0.241880	0.120940	+	0.992660i	$0.461409\pi$
$744$	−399680.	−0.0264716
$745$	4.47403e6	0.295330
$746$	−5.57766e6	−0.366948
$747$	1.33134e7	0.872945
$748$	−1.42877e6	−0.0933701
$749$	0	0
$750$	−744396.	−0.0483227
$751$	−1.87370e7	−1.21227	−0.606135	−	0.795362i	$-0.707281\pi$
$751$	−1.87370e7	−1.21227	−0.606135	+	0.795362i	$0.707281\pi$
$752$	4.39910e6	0.283674
$753$	−1.19751e6	−0.0769649
$754$	−5.71315e6	−0.365972
$755$	−2.20717e7	−1.40918
$756$	0	0
$757$	489242.	0.0310302	0.0155151	−	0.999880i	$-0.495061\pi$
$757$	489242.	0.0310302	0.0155151	+	0.999880i	$0.495061\pi$
$758$	1.70414e7	1.07729
$759$	−215259.	−0.0135630
$760$	4.64794e6	0.291895
$761$	−1.46969e7	−0.919952	−0.459976	−	0.887931i	$-0.652142\pi$
$761$	−1.46969e7	−0.919952	−0.459976	+	0.887931i	$0.652142\pi$
$762$	−957664.	−0.0597483
$763$	0	0
$764$	1.22731e7	0.760711
$765$	−9.10840e6	−0.562715
$766$	807060.	0.0496974
$767$	−2.42124e7	−1.48610
$768$	−65536.0	−0.00400938
$769$	−2.42072e7	−1.47615	−0.738073	−	0.674721i	$-0.764264\pi$
$769$	−2.42072e7	−1.47615	−0.738073	+	0.674721i	$0.764264\pi$
$770$	0	0
$771$	37758.0	0.00228756
$772$	6.58669e6	0.397763
$773$	−1.35260e7	−0.814181	−0.407091	−	0.913388i	$-0.633457\pi$
$773$	−1.35260e7	−0.814181	−0.407091	+	0.913388i	$0.633457\pi$
$774$	1.72285e7	1.03370
$775$	3.27238e6	0.195708
$776$	−5.68365e6	−0.338823
$777$	0	0
$778$	−7.79528e6	−0.461725
$779$	7.55290e6	0.445933
$780$	564672.	0.0332323
$781$	−1.61063e6	−0.0944862
$782$	5.25161e6	0.307097
$783$	−1.00104e6	−0.0583508
$784$	0	0
$785$	1.73782e6	0.100654
$786$	392568.	0.0226651
$787$	−1.42094e7	−0.817786	−0.408893	−	0.912582i	$-0.634085\pi$
$787$	−1.42094e7	−0.817786	−0.408893	+	0.912582i	$0.634085\pi$
$788$	−1.21481e7	−0.696937
$789$	631254.	0.0361004
$790$	−1.57557e7	−0.898196
$791$	0	0
$792$	−1.87405e6	−0.106162
$793$	−3.17905e7	−1.79521
$794$	−5.87303e6	−0.330606
$795$	1.56703e6	0.0879343
$796$	745600.	0.0417084
$797$	7.93333e6	0.442395	0.221197	−	0.975229i	$-0.429003\pi$
$797$	7.93333e6	0.442395	0.221197	+	0.975229i	$0.429003\pi$
$798$	0	0
$799$	1.26818e7	0.702771
$800$	536576.	0.0296419
$801$	3.03504e7	1.67141
$802$	−8.98471e6	−0.493251
$803$	−6.44446e6	−0.352694
$804$	−405488.	−0.0221227
$805$	0	0
$806$	−1.72862e7	−0.937262
$807$	−1.08034e6	−0.0583952
$808$	94848.0	0.00511093
$809$	−1.04685e7	−0.562359	−0.281180	−	0.959655i	$-0.590726\pi$
$809$	−1.04685e7	−0.562359	−0.281180	+	0.959655i	$0.590726\pi$
$810$	−1.18975e7	−0.637151
$811$	−1.19147e7	−0.636110	−0.318055	−	0.948072i	$-0.603030\pi$
$811$	−1.19147e7	−0.636110	−0.318055	+	0.948072i	$0.603030\pi$
$812$	0	0
$813$	−816100.	−0.0433029
$814$	−7.15594e6	−0.378535
$815$	2.29602e6	0.121083
$816$	−188928.	−0.00993278
$817$	−2.53444e7	−1.32839
$818$	−1.44595e7	−0.755562
$819$	0	0
$820$	−4.32806e6	−0.224781
$821$	−1.86112e6	−0.0963645	−0.0481822	−	0.998839i	$-0.515343\pi$
$821$	−1.86112e6	−0.0963645	−0.0481822	+	0.998839i	$0.515343\pi$
$822$	1.60055e6	0.0826208
$823$	2.30153e7	1.18445	0.592225	−	0.805773i	$-0.298250\pi$
$823$	2.30153e7	1.18445	0.592225	+	0.805773i	$0.298250\pi$
$824$	−7.51974e6	−0.385820
$825$	−63404.0	−0.00324326
$826$	0	0
$827$	−1.68351e7	−0.855959	−0.427980	−	0.903788i	$-0.640775\pi$
$827$	−1.68351e7	−0.855959	−0.427980	+	0.903788i	$0.640775\pi$
$828$	6.88829e6	0.349169
$829$	2.35299e7	1.18914	0.594570	−	0.804044i	$-0.297322\pi$
$829$	2.35299e7	1.18914	0.594570	+	0.804044i	$0.297322\pi$
$830$	1.12229e7	0.565468
$831$	−1.68820e6	−0.0848049
$832$	−2.83443e6	−0.141957
$833$	0	0
$834$	−823064.	−0.0409750
$835$	−2.46104e7	−1.22152
$836$	2.75686e6	0.136427
$837$	−3.02882e6	−0.149438
$838$	−1.52496e7	−0.750148
$839$	−2.91549e7	−1.42990	−0.714952	−	0.699173i	$-0.753552\pi$
$839$	−2.91549e7	−1.42990	−0.714952	+	0.699173i	$0.753552\pi$
$840$	0	0
$841$	−1.62511e7	−0.792303
$842$	−7.89385e6	−0.383715
$843$	879042.	0.0426030
$844$	−1.49188e7	−0.720907
$845$	5.48612e6	0.264316
$846$	1.66341e7	0.799050
$847$	0	0
$848$	−7.86586e6	−0.375627
$849$	1.54027e6	0.0733377
$850$	1.54685e6	0.0734345
$851$	2.63025e7	1.24501
$852$	−212976.	−0.0100515
$853$	9.49052e6	0.446599	0.223299	−	0.974750i	$-0.428317\pi$
$853$	9.49052e6	0.446599	0.223299	+	0.974750i	$0.428317\pi$
$854$	0	0
$855$	1.75750e7	0.822205
$856$	5.07917e6	0.236924
$857$	1.81553e6	0.0844405	0.0422203	−	0.999108i	$-0.486557\pi$
$857$	1.81553e6	0.0844405	0.0422203	+	0.999108i	$0.486557\pi$
$858$	334928.	0.0155322
$859$	1.07812e7	0.498522	0.249261	−	0.968436i	$-0.419812\pi$
$859$	1.07812e7	0.498522	0.249261	+	0.968436i	$0.419812\pi$
$860$	1.45232e7	0.669600
$861$	0	0
$862$	8.33436e6	0.382036
$863$	−2.83355e7	−1.29510	−0.647550	−	0.762023i	$-0.724206\pi$
$863$	−2.83355e7	−1.29510	−0.647550	+	0.762023i	$0.724206\pi$
$864$	−496640.	−0.0226338
$865$	3.90790e7	1.77584
$866$	−290764.	−0.0131749
$867$	875213.	0.0395427
$868$	0	0
$869$	−9.34531e6	−0.419802
$870$	−421056.	−0.0188600
$871$	−1.75374e7	−0.783283
$872$	−5.62189e6	−0.250375
$873$	−2.14913e7	−0.954392
$874$	−1.01332e7	−0.448712
$875$	0	0
$876$	−852160.	−0.0375198
$877$	−2.68919e7	−1.18065	−0.590326	−	0.807165i	$-0.701001\pi$
$877$	−2.68919e7	−1.18065	−0.590326	+	0.807165i	$0.701001\pi$
$878$	2.37757e6	0.104087
$879$	720840.	0.0314678
$880$	−1.57978e6	−0.0687684
$881$	1.92132e7	0.833989	0.416995	−	0.908909i	$-0.363083\pi$
$881$	1.92132e7	0.833989	0.416995	+	0.908909i	$0.363083\pi$
$882$	0	0
$883$	1.15931e7	0.500378	0.250189	−	0.968197i	$-0.419507\pi$
$883$	1.15931e7	0.500378	0.250189	+	0.968197i	$0.419507\pi$
$884$	−8.17114e6	−0.351683
$885$	−1.78444e6	−0.0765850
$886$	−1.82660e7	−0.781735
$887$	−1.31857e7	−0.562721	−0.281361	−	0.959602i	$-0.590786\pi$
$887$	−1.31857e7	−0.562721	−0.281361	+	0.959602i	$0.590786\pi$
$888$	−946240.	−0.0402688
$889$	0	0
$890$	2.55847e7	1.08269
$891$	−7.05684e6	−0.297794
$892$	−2.71592e6	−0.114289
$893$	−2.44700e7	−1.02685
$894$	350904.	0.0146840
$895$	1.54733e7	0.645694
$896$	0	0
$897$	−1.23107e6	−0.0510859
$898$	2.17753e7	0.901100
$899$	1.28897e7	0.531916
$900$	2.02893e6	0.0834950
$901$	−2.26758e7	−0.930573
$902$	−2.56714e6	−0.105059
$903$	0	0
$904$	3.02381e6	0.123065
$905$	1.45442e7	0.590295
$906$	−1.73111e6	−0.0700656
$907$	2.98195e6	0.120360	0.0601800	−	0.998188i	$-0.480833\pi$
$907$	2.98195e6	0.120360	0.0601800	+	0.998188i	$0.480833\pi$
$908$	−3.16925e6	−0.127568
$909$	358644.	0.0143964
$910$	0	0
$911$	2.96579e7	1.18398	0.591989	−	0.805946i	$-0.298343\pi$
$911$	2.96579e7	1.18398	0.591989	+	0.805946i	$0.298343\pi$
$912$	364544.	0.0145132
$913$	6.65669e6	0.264291
$914$	−2.68125e7	−1.06163
$915$	−2.34294e6	−0.0925142
$916$	1.36000e7	0.535548
$917$	0	0
$918$	−1.43172e6	−0.0560727
$919$	3.18057e7	1.24227	0.621135	−	0.783704i	$-0.286672\pi$
$919$	3.18057e7	1.24227	0.621135	+	0.783704i	$0.286672\pi$
$920$	5.80666e6	0.226181
$921$	−1.03905e6	−0.0403633
$922$	−5.03976e6	−0.195246
$923$	−9.21121e6	−0.355887
$924$	0	0
$925$	7.74734e6	0.297713
$926$	2.00923e7	0.770021
$927$	−2.84340e7	−1.08677
$928$	2.11354e6	0.0805638
$929$	2.33444e7	0.887451	0.443725	−	0.896163i	$-0.353657\pi$
$929$	2.33444e7	0.887451	0.443725	+	0.896163i	$0.353657\pi$
$930$	−1.27398e6	−0.0483009
$931$	0	0
$932$	−6.42931e6	−0.242451
$933$	−1.25135e6	−0.0470624
$934$	−9.42642e6	−0.353573
$935$	−4.55420e6	−0.170366
$936$	−1.07177e7	−0.399864
$937$	−2.07372e7	−0.771616	−0.385808	−	0.922579i	$-0.626077\pi$
$937$	−2.07372e7	−0.771616	−0.385808	+	0.922579i	$0.626077\pi$
$938$	0	0
$939$	−1.44336e6	−0.0534209
$940$	1.40221e7	0.517601
$941$	−2.69193e7	−0.991036	−0.495518	−	0.868598i	$-0.665022\pi$
$941$	−2.69193e7	−0.991036	−0.495518	+	0.868598i	$0.665022\pi$
$942$	136300.	0.00500459
$943$	9.43582e6	0.345542
$944$	8.95718e6	0.327146
$945$	0	0
$946$	8.61423e6	0.312960
$947$	1.01896e7	0.369216	0.184608	−	0.982812i	$-0.440898\pi$
$947$	1.01896e7	0.369216	0.184608	+	0.982812i	$0.440898\pi$
$948$	−1.23574e6	−0.0446589
$949$	−3.68559e7	−1.32844
$950$	−2.98470e6	−0.107298
$951$	2.01208e6	0.0721429
$952$	0	0
$953$	1.03924e7	0.370665	0.185333	−	0.982676i	$-0.440664\pi$
$953$	1.03924e7	0.370665	0.185333	+	0.982676i	$0.440664\pi$
$954$	−2.97428e7	−1.05806
$955$	3.91204e7	1.38802
$956$	1.36828e7	0.484206
$957$	−249744.	−0.00881486
$958$	−2.68903e7	−0.946634
$959$	0	0
$960$	−208896.	−0.00731564
$961$	1.03709e7	0.362249
$962$	−4.09249e7	−1.42577
$963$	1.92056e7	0.667363
$964$	−1.80074e7	−0.624107
$965$	2.09951e7	0.725770
$966$	0	0
$967$	−8.18877e6	−0.281613	−0.140806	−	0.990037i	$-0.544970\pi$
$967$	−8.18877e6	−0.281613	−0.140806	+	0.990037i	$0.544970\pi$
$968$	−937024.	−0.0321412
$969$	1.05091e6	0.0359548
$970$	−1.81166e7	−0.618227
$971$	1.73274e7	0.589775	0.294887	−	0.955532i	$-0.404718\pi$
$971$	1.73274e7	0.589775	0.294887	+	0.955532i	$0.404718\pi$
$972$	−2.81882e6	−0.0956976
$973$	0	0
$974$	−7.84005e6	−0.264802
$975$	−362608.	−0.0122159
$976$	1.17606e7	0.395190
$977$	−438963.	−0.0147127	−0.00735634	−	0.999973i	$-0.502342\pi$
$977$	−438963.	−0.0147127	−0.00735634	+	0.999973i	$0.502342\pi$
$978$	180080.	0.00602030
$979$	1.51752e7	0.506032
$980$	0	0
$981$	−2.12578e7	−0.705253
$982$	2.31850e6	0.0767234
$983$	2.79124e7	0.921326	0.460663	−	0.887575i	$-0.347612\pi$
$983$	2.79124e7	0.921326	0.460663	+	0.887575i	$0.347612\pi$
$984$	−339456.	−0.0111762
$985$	−3.87222e7	−1.27165
$986$	6.09293e6	0.199588
$987$	0	0
$988$	1.57665e7	0.513859
$989$	−3.16626e7	−1.02933
$990$	−5.97353e6	−0.193706
$991$	−4.26846e7	−1.38066	−0.690331	−	0.723494i	$-0.742535\pi$
$991$	−4.26846e7	−1.38066	−0.690331	+	0.723494i	$0.742535\pi$
$992$	6.39488e6	0.206326
$993$	−2.01734e6	−0.0649240
$994$	0	0
$995$	2.37660e6	0.0761024
$996$	880224.	0.0281154
$997$	2.21044e7	0.704273	0.352137	−	0.935949i	$-0.385455\pi$
$997$	2.21044e7	0.704273	0.352137	+	0.935949i	$0.385455\pi$
$998$	−5.47621e6	−0.174042
$999$	−7.17072e6	−0.227326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.6.a.a.1.1		1
7.6	odd	2		22.6.a.b.1.1	✓	1
21.20	even	2		198.6.a.i.1.1		1
28.27	even	2		176.6.a.b.1.1		1
35.13	even	4		550.6.b.f.199.2		2
35.27	even	4		550.6.b.f.199.1		2
35.34	odd	2		550.6.a.f.1.1		1
56.13	odd	2		704.6.a.e.1.1		1
56.27	even	2		704.6.a.f.1.1		1
77.76	even	2		242.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.6.a.b.1.1	✓	1	7.6	odd	2
176.6.a.b.1.1		1	28.27	even	2
198.6.a.i.1.1		1	21.20	even	2
242.6.a.d.1.1		1	77.76	even	2
550.6.a.f.1.1		1	35.34	odd	2
550.6.b.f.199.1		2	35.27	even	4
550.6.b.f.199.2		2	35.13	even	4
704.6.a.e.1.1		1	56.13	odd	2
704.6.a.f.1.1		1	56.27	even	2
1078.6.a.a.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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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