LMFDB - Embedded newform 1040.6.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,6,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1040.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1040.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:	$166.799172605$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 1384x^{6} + 3672x^{5} + 603912x^{4} - 998448x^{3} - 83285728x^{2} + 113377312x + 442451152 $ x^8 - 4x^7 - 1384x^6 + 3672x^5 + 603912x^4 - 998448x^3 - 83285728x^2 + 113377312*x + 442451152
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$26.7009$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-22.7009 q^{3} +25.0000 q^{5} +147.624 q^{7} +272.331 q^{9} -137.651 q^{11} -169.000 q^{13} -567.523 q^{15} -1513.08 q^{17} +1627.36 q^{19} -3351.20 q^{21} -1653.47 q^{23} +625.000 q^{25} -665.847 q^{27} +2688.57 q^{29} +7589.65 q^{31} +3124.79 q^{33} +3690.60 q^{35} +5167.18 q^{37} +3836.45 q^{39} +3198.33 q^{41} +8182.03 q^{43} +6808.28 q^{45} +11704.4 q^{47} +4985.90 q^{49} +34348.3 q^{51} -40150.5 q^{53} -3441.27 q^{55} -36942.4 q^{57} -7504.29 q^{59} -25223.5 q^{61} +40202.7 q^{63} -4225.00 q^{65} +47087.0 q^{67} +37535.2 q^{69} -82033.3 q^{71} -27624.5 q^{73} -14188.1 q^{75} -20320.6 q^{77} +57175.9 q^{79} -51061.2 q^{81} -72698.2 q^{83} -37827.0 q^{85} -61032.9 q^{87} +68283.8 q^{89} -24948.5 q^{91} -172292. q^{93} +40683.9 q^{95} +29166.2 q^{97} -37486.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 28 q^{3} + 200 q^{5} - 80 q^{7} + 936 q^{9} + 444 q^{11} - 1352 q^{13} + 700 q^{15} - 3400 q^{17} + 1668 q^{19} - 3816 q^{21} + 2964 q^{23} + 5000 q^{25} + 14464 q^{27} - 8272 q^{29} + 8684 q^{31}+ \cdots + 535828 q^{99}+O(q^{100}) $ 8 * q + 28 * q^3 + 200 * q^5 - 80 * q^7 + 936 * q^9 + 444 * q^11 - 1352 * q^13 + 700 * q^15 - 3400 * q^17 + 1668 * q^19 - 3816 * q^21 + 2964 * q^23 + 5000 * q^25 + 14464 * q^27 - 8272 * q^29 + 8684 * q^31 + 5288 * q^33 - 2000 * q^35 + 11320 * q^37 - 4732 * q^39 + 280 * q^41 + 36268 * q^43 + 23400 * q^45 + 17936 * q^47 + 76456 * q^49 + 10744 * q^51 + 28936 * q^53 + 11100 * q^55 - 19640 * q^57 + 34668 * q^59 + 62976 * q^61 - 9040 * q^63 - 33800 * q^65 + 181688 * q^67 - 43600 * q^69 + 6036 * q^71 + 23000 * q^73 + 17500 * q^75 - 178792 * q^77 + 24392 * q^79 - 3992 * q^81 - 34416 * q^83 - 85000 * q^85 + 144144 * q^87 + 57328 * q^89 + 13520 * q^91 - 66648 * q^93 + 41700 * q^95 - 210176 * q^97 + 535828 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−22.7009	−1.45626	−0.728132	−	0.685437i	$-0.759611\pi$
$3$	−22.7009	−1.45626	−0.728132	+	0.685437i	$0.759611\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	147.624	1.13871	0.569354	−	0.822092i	$-0.307193\pi$
$7$	147.624	1.13871	0.569354	+	0.822092i	$0.307193\pi$
$8$	0	0
$9$	272.331	1.12070
$10$	0	0
$11$	−137.651	−0.343002	−0.171501	−	0.985184i	$-0.554862\pi$
$11$	−137.651	−0.343002	−0.171501	+	0.985184i	$0.554862\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	−567.523	−0.651261
$16$	0	0
$17$	−1513.08	−1.26981	−0.634906	−	0.772589i	$-0.718961\pi$
$17$	−1513.08	−1.26981	−0.634906	+	0.772589i	$0.718961\pi$
$18$	0	0
$19$	1627.36	1.03419	0.517093	−	0.855929i	$-0.327014\pi$
$19$	1627.36	1.03419	0.517093	+	0.855929i	$0.327014\pi$
$20$	0	0
$21$	−3351.20	−1.65826
$22$	0	0
$23$	−1653.47	−0.651742	−0.325871	−	0.945414i	$-0.605658\pi$
$23$	−1653.47	−0.651742	−0.325871	+	0.945414i	$0.605658\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−665.847	−0.175778
$28$	0	0
$29$	2688.57	0.593644	0.296822	−	0.954933i	$-0.404073\pi$
$29$	2688.57	0.593644	0.296822	+	0.954933i	$0.404073\pi$
$30$	0	0
$31$	7589.65	1.41846	0.709231	−	0.704977i	$-0.249043\pi$
$31$	7589.65	1.41846	0.709231	+	0.704977i	$0.249043\pi$
$32$	0	0
$33$	3124.79	0.499501
$34$	0	0
$35$	3690.60	0.509246
$36$	0	0
$37$	5167.18	0.620510	0.310255	−	0.950653i	$-0.399586\pi$
$37$	5167.18	0.620510	0.310255	+	0.950653i	$0.399586\pi$
$38$	0	0
$39$	3836.45	0.403895
$40$	0	0
$41$	3198.33	0.297142	0.148571	−	0.988902i	$-0.452533\pi$
$41$	3198.33	0.297142	0.148571	+	0.988902i	$0.452533\pi$
$42$	0	0
$43$	8182.03	0.674823	0.337411	−	0.941357i	$-0.390449\pi$
$43$	8182.03	0.674823	0.337411	+	0.941357i	$0.390449\pi$
$44$	0	0
$45$	6808.28	0.501194
$46$	0	0
$47$	11704.4	0.772865	0.386433	−	0.922318i	$-0.373707\pi$
$47$	11704.4	0.772865	0.386433	+	0.922318i	$0.373707\pi$
$48$	0	0
$49$	4985.90	0.296656
$50$	0	0
$51$	34348.3	1.84918
$52$	0	0
$53$	−40150.5	−1.96337	−0.981684	−	0.190516i	$-0.938984\pi$
$53$	−40150.5	−1.96337	−0.981684	+	0.190516i	$0.938984\pi$
$54$	0	0
$55$	−3441.27	−0.153395
$56$	0	0
$57$	−36942.4	−1.50605
$58$	0	0
$59$	−7504.29	−0.280659	−0.140330	−	0.990105i	$-0.544816\pi$
$59$	−7504.29	−0.280659	−0.140330	+	0.990105i	$0.544816\pi$
$60$	0	0
$61$	−25223.5	−0.867921	−0.433961	−	0.900932i	$-0.642884\pi$
$61$	−25223.5	−0.867921	−0.433961	+	0.900932i	$0.642884\pi$
$62$	0	0
$63$	40202.7	1.27616
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	47087.0	1.28149	0.640743	−	0.767756i	$-0.278627\pi$
$67$	47087.0	1.28149	0.640743	+	0.767756i	$0.278627\pi$
$68$	0	0
$69$	37535.2	0.949109
$70$	0	0
$71$	−82033.3	−1.93128	−0.965638	−	0.259892i	$-0.916313\pi$
$71$	−82033.3	−1.93128	−0.965638	+	0.259892i	$0.916313\pi$
$72$	0	0
$73$	−27624.5	−0.606719	−0.303360	−	0.952876i	$-0.598108\pi$
$73$	−27624.5	−0.606719	−0.303360	+	0.952876i	$0.598108\pi$
$74$	0	0
$75$	−14188.1	−0.291253
$76$	0	0
$77$	−20320.6	−0.390579
$78$	0	0
$79$	57175.9	1.03073	0.515366	−	0.856970i	$-0.327656\pi$
$79$	57175.9	1.03073	0.515366	+	0.856970i	$0.327656\pi$
$80$	0	0
$81$	−51061.2	−0.864725
$82$	0	0
$83$	−72698.2	−1.15832	−0.579160	−	0.815214i	$-0.696619\pi$
$83$	−72698.2	−1.15832	−0.579160	+	0.815214i	$0.696619\pi$
$84$	0	0
$85$	−37827.0	−0.567877
$86$	0	0
$87$	−61032.9	−0.864502
$88$	0	0
$89$	68283.8	0.913782	0.456891	−	0.889523i	$-0.348963\pi$
$89$	68283.8	0.913782	0.456891	+	0.889523i	$0.348963\pi$
$90$	0	0
$91$	−24948.5	−0.315821
$92$	0	0
$93$	−172292.	−2.06565
$94$	0	0
$95$	40683.9	0.462502
$96$	0	0
$97$	29166.2	0.314739	0.157369	−	0.987540i	$-0.449699\pi$
$97$	29166.2	0.314739	0.157369	+	0.987540i	$0.449699\pi$
$98$	0	0
$99$	−37486.6	−0.384404
$100$	0	0
$101$	−54468.8	−0.531306	−0.265653	−	0.964069i	$-0.585587\pi$
$101$	−54468.8	−0.531306	−0.265653	+	0.964069i	$0.585587\pi$
$102$	0	0
$103$	−44395.2	−0.412328	−0.206164	−	0.978517i	$-0.566098\pi$
$103$	−44395.2	−0.412328	−0.206164	+	0.978517i	$0.566098\pi$
$104$	0	0
$105$	−83780.1	−0.741596
$106$	0	0
$107$	109705.	0.926337	0.463168	−	0.886270i	$-0.346713\pi$
$107$	109705.	0.926337	0.463168	+	0.886270i	$0.346713\pi$
$108$	0	0
$109$	164919.	1.32955	0.664773	−	0.747045i	$-0.268528\pi$
$109$	164919.	1.32955	0.664773	+	0.747045i	$0.268528\pi$
$110$	0	0
$111$	−117300.	−0.903627
$112$	0	0
$113$	26715.3	0.196818	0.0984089	−	0.995146i	$-0.468625\pi$
$113$	26715.3	0.196818	0.0984089	+	0.995146i	$0.468625\pi$
$114$	0	0
$115$	−41336.7	−0.291468
$116$	0	0
$117$	−46024.0	−0.310828
$118$	0	0
$119$	−223367.	−1.44595
$120$	0	0
$121$	−142103.	−0.882350
$122$	0	0
$123$	−72605.1	−0.432717
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−78433.4	−0.431511	−0.215755	−	0.976447i	$-0.569221\pi$
$127$	−78433.4	−0.431511	−0.215755	+	0.976447i	$0.569221\pi$
$128$	0	0
$129$	−185739.	−0.982720
$130$	0	0
$131$	110286.	0.561491	0.280746	−	0.959782i	$-0.409418\pi$
$131$	110286.	0.561491	0.280746	+	0.959782i	$0.409418\pi$
$132$	0	0
$133$	240237.	1.17764
$134$	0	0
$135$	−16646.2	−0.0786104
$136$	0	0
$137$	−314670.	−1.43236	−0.716182	−	0.697913i	$-0.754112\pi$
$137$	−314670.	−1.43236	−0.716182	+	0.697913i	$0.754112\pi$
$138$	0	0
$139$	−42243.6	−0.185449	−0.0927244	−	0.995692i	$-0.529558\pi$
$139$	−42243.6	−0.185449	−0.0927244	+	0.995692i	$0.529558\pi$
$140$	0	0
$141$	−265700.	−1.12550
$142$	0	0
$143$	23263.0	0.0951316
$144$	0	0
$145$	67214.2	0.265485
$146$	0	0
$147$	−113184.	−0.432009
$148$	0	0
$149$	336810.	1.24285	0.621425	−	0.783473i	$-0.286554\pi$
$149$	336810.	1.24285	0.621425	+	0.783473i	$0.286554\pi$
$150$	0	0
$151$	541317.	1.93201	0.966005	−	0.258524i	$-0.0832363\pi$
$151$	541317.	1.93201	0.966005	+	0.258524i	$0.0832363\pi$
$152$	0	0
$153$	−412059.	−1.42309
$154$	0	0
$155$	189741.	0.634355
$156$	0	0
$157$	139742.	0.452459	0.226230	−	0.974074i	$-0.427360\pi$
$157$	139742.	0.452459	0.226230	+	0.974074i	$0.427360\pi$
$158$	0	0
$159$	911454.	2.85918
$160$	0	0
$161$	−244092.	−0.742144
$162$	0	0
$163$	466661.	1.37573	0.687864	−	0.725840i	$-0.258549\pi$
$163$	466661.	1.37573	0.687864	+	0.725840i	$0.258549\pi$
$164$	0	0
$165$	78119.9	0.223384
$166$	0	0
$167$	−521176.	−1.44608	−0.723041	−	0.690805i	$-0.757256\pi$
$167$	−521176.	−1.44608	−0.723041	+	0.690805i	$0.757256\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	443180.	1.15902
$172$	0	0
$173$	301291.	0.765369	0.382684	−	0.923879i	$-0.375000\pi$
$173$	301291.	0.765369	0.382684	+	0.923879i	$0.375000\pi$
$174$	0	0
$175$	92265.1	0.227742
$176$	0	0
$177$	170354.	0.408714
$178$	0	0
$179$	−614256.	−1.43290	−0.716452	−	0.697636i	$-0.754235\pi$
$179$	−614256.	−1.43290	−0.716452	+	0.697636i	$0.754235\pi$
$180$	0	0
$181$	37018.5	0.0839889	0.0419945	−	0.999118i	$-0.486629\pi$
$181$	37018.5	0.0839889	0.0419945	+	0.999118i	$0.486629\pi$
$182$	0	0
$183$	572596.	1.26392
$184$	0	0
$185$	129179.	0.277501
$186$	0	0
$187$	208276.	0.435548
$188$	0	0
$189$	−98295.2	−0.200160
$190$	0	0
$191$	−598128.	−1.18634	−0.593172	−	0.805076i	$-0.702125\pi$
$191$	−598128.	−1.18634	−0.593172	+	0.805076i	$0.702125\pi$
$192$	0	0
$193$	−677994.	−1.31018	−0.655092	−	0.755549i	$-0.727370\pi$
$193$	−677994.	−1.31018	−0.655092	+	0.755549i	$0.727370\pi$
$194$	0	0
$195$	95911.3	0.180627
$196$	0	0
$197$	455876.	0.836915	0.418458	−	0.908236i	$-0.362571\pi$
$197$	455876.	0.836915	0.418458	+	0.908236i	$0.362571\pi$
$198$	0	0
$199$	194307.	0.347822	0.173911	−	0.984761i	$-0.444360\pi$
$199$	194307.	0.347822	0.173911	+	0.984761i	$0.444360\pi$
$200$	0	0
$201$	−1.06892e6	−1.86618
$202$	0	0
$203$	396897.	0.675987
$204$	0	0
$205$	79958.4	0.132886
$206$	0	0
$207$	−450291.	−0.730411
$208$	0	0
$209$	−224007.	−0.354728
$210$	0	0
$211$	829362.	1.28244	0.641221	−	0.767356i	$-0.278428\pi$
$211$	829362.	1.28244	0.641221	+	0.767356i	$0.278428\pi$
$212$	0	0
$213$	1.86223e6	2.81245
$214$	0	0
$215$	204551.	0.301790
$216$	0	0
$217$	1.12042e6	1.61521
$218$	0	0
$219$	627102.	0.883544
$220$	0	0
$221$	255711.	0.352183
$222$	0	0
$223$	31821.4	0.0428506	0.0214253	−	0.999770i	$-0.493180\pi$
$223$	31821.4	0.0428506	0.0214253	+	0.999770i	$0.493180\pi$
$224$	0	0
$225$	170207.	0.224141
$226$	0	0
$227$	223867.	0.288354	0.144177	−	0.989552i	$-0.453947\pi$
$227$	223867.	0.288354	0.144177	+	0.989552i	$0.453947\pi$
$228$	0	0
$229$	−192513.	−0.242590	−0.121295	−	0.992617i	$-0.538705\pi$
$229$	−192513.	−0.242590	−0.121295	+	0.992617i	$0.538705\pi$
$230$	0	0
$231$	461295.	0.568786
$232$	0	0
$233$	−600613.	−0.724778	−0.362389	−	0.932027i	$-0.618039\pi$
$233$	−600613.	−0.724778	−0.362389	+	0.932027i	$0.618039\pi$
$234$	0	0
$235$	292609.	0.345636
$236$	0	0
$237$	−1.29795e6	−1.50102
$238$	0	0
$239$	511423.	0.579143	0.289571	−	0.957156i	$-0.406487\pi$
$239$	511423.	0.579143	0.289571	+	0.957156i	$0.406487\pi$
$240$	0	0
$241$	−1.11145e6	−1.23268	−0.616338	−	0.787482i	$-0.711385\pi$
$241$	−1.11145e6	−1.23268	−0.616338	+	0.787482i	$0.711385\pi$
$242$	0	0
$243$	1.32094e6	1.43505
$244$	0	0
$245$	124647.	0.132669
$246$	0	0
$247$	−275023.	−0.286831
$248$	0	0
$249$	1.65031e6	1.68682
$250$	0	0
$251$	480575.	0.481478	0.240739	−	0.970590i	$-0.422610\pi$
$251$	480575.	0.481478	0.240739	+	0.970590i	$0.422610\pi$
$252$	0	0
$253$	227601.	0.223549
$254$	0	0
$255$	858707.	0.826979
$256$	0	0
$257$	−119512.	−0.112870	−0.0564349	−	0.998406i	$-0.517973\pi$
$257$	−119512.	−0.112870	−0.0564349	+	0.998406i	$0.517973\pi$
$258$	0	0
$259$	762800.	0.706580
$260$	0	0
$261$	732181.	0.665299
$262$	0	0
$263$	913483.	0.814350	0.407175	−	0.913350i	$-0.366514\pi$
$263$	913483.	0.814350	0.407175	+	0.913350i	$0.366514\pi$
$264$	0	0
$265$	−1.00376e6	−0.878045
$266$	0	0
$267$	−1.55010e6	−1.33071
$268$	0	0
$269$	−405994.	−0.342089	−0.171044	−	0.985263i	$-0.554714\pi$
$269$	−405994.	−0.342089	−0.171044	+	0.985263i	$0.554714\pi$
$270$	0	0
$271$	1.65395e6	1.36804	0.684020	−	0.729463i	$-0.260230\pi$
$271$	1.65395e6	1.36804	0.684020	+	0.729463i	$0.260230\pi$
$272$	0	0
$273$	566353.	0.459918
$274$	0	0
$275$	−86031.6	−0.0686004
$276$	0	0
$277$	198753.	0.155637	0.0778186	−	0.996968i	$-0.475205\pi$
$277$	198753.	0.155637	0.0778186	+	0.996968i	$0.475205\pi$
$278$	0	0
$279$	2.06690e6	1.58968
$280$	0	0
$281$	1.93766e6	1.46390	0.731949	−	0.681359i	$-0.238611\pi$
$281$	1.93766e6	1.46390	0.731949	+	0.681359i	$0.238611\pi$
$282$	0	0
$283$	511264.	0.379471	0.189736	−	0.981835i	$-0.439237\pi$
$283$	511264.	0.379471	0.189736	+	0.981835i	$0.439237\pi$
$284$	0	0
$285$	−923561.	−0.673525
$286$	0	0
$287$	472151.	0.338358
$288$	0	0
$289$	869554.	0.612424
$290$	0	0
$291$	−662099.	−0.458343
$292$	0	0
$293$	1.50838e6	1.02646	0.513228	−	0.858252i	$-0.328449\pi$
$293$	1.50838e6	1.02646	0.513228	+	0.858252i	$0.328449\pi$
$294$	0	0
$295$	−187607.	−0.125515
$296$	0	0
$297$	91654.3	0.0602923
$298$	0	0
$299$	279436.	0.180761
$300$	0	0
$301$	1.20786e6	0.768426
$302$	0	0
$303$	1.23649e6	0.773721
$304$	0	0
$305$	−630587.	−0.388146
$306$	0	0
$307$	−584805.	−0.354132	−0.177066	−	0.984199i	$-0.556661\pi$
$307$	−584805.	−0.354132	−0.177066	+	0.984199i	$0.556661\pi$
$308$	0	0
$309$	1.00781e6	0.600458
$310$	0	0
$311$	−921513.	−0.540257	−0.270129	−	0.962824i	$-0.587066\pi$
$311$	−921513.	−0.540257	−0.270129	+	0.962824i	$0.587066\pi$
$312$	0	0
$313$	1.53958e6	0.888263	0.444131	−	0.895962i	$-0.353512\pi$
$313$	1.53958e6	0.888263	0.444131	+	0.895962i	$0.353512\pi$
$314$	0	0
$315$	1.00507e6	0.570714
$316$	0	0
$317$	2.54339e6	1.42156	0.710779	−	0.703416i	$-0.248343\pi$
$317$	2.54339e6	1.42156	0.710779	+	0.703416i	$0.248343\pi$
$318$	0	0
$319$	−370083.	−0.203621
$320$	0	0
$321$	−2.49041e6	−1.34899
$322$	0	0
$323$	−2.46232e6	−1.31322
$324$	0	0
$325$	−105625.	−0.0554700
$326$	0	0
$327$	−3.74380e6	−1.93617
$328$	0	0
$329$	1.72785e6	0.880068
$330$	0	0
$331$	−247483.	−0.124158	−0.0620790	−	0.998071i	$-0.519773\pi$
$331$	−247483.	−0.124158	−0.0620790	+	0.998071i	$0.519773\pi$
$332$	0	0
$333$	1.40718e6	0.695409
$334$	0	0
$335$	1.17717e6	0.573098
$336$	0	0
$337$	1.06392e6	0.510312	0.255156	−	0.966900i	$-0.417873\pi$
$337$	1.06392e6	0.510312	0.255156	+	0.966900i	$0.417873\pi$
$338$	0	0
$339$	−606462.	−0.286619
$340$	0	0
$341$	−1.04472e6	−0.486535
$342$	0	0
$343$	−1.74508e6	−0.800904
$344$	0	0
$345$	938380.	0.424454
$346$	0	0
$347$	−947067.	−0.422238	−0.211119	−	0.977460i	$-0.567711\pi$
$347$	−947067.	−0.422238	−0.211119	+	0.977460i	$0.567711\pi$
$348$	0	0
$349$	3.33542e6	1.46584	0.732921	−	0.680314i	$-0.238156\pi$
$349$	3.33542e6	1.46584	0.732921	+	0.680314i	$0.238156\pi$
$350$	0	0
$351$	112528.	0.0487521
$352$	0	0
$353$	2.27052e6	0.969812	0.484906	−	0.874566i	$-0.338854\pi$
$353$	2.27052e6	0.969812	0.484906	+	0.874566i	$0.338854\pi$
$354$	0	0
$355$	−2.05083e6	−0.863693
$356$	0	0
$357$	5.07064e6	2.10568
$358$	0	0
$359$	4.41146e6	1.80654	0.903268	−	0.429077i	$-0.141161\pi$
$359$	4.41146e6	1.80654	0.903268	+	0.429077i	$0.141161\pi$
$360$	0	0
$361$	172187.	0.0695396
$362$	0	0
$363$	3.22587e6	1.28493
$364$	0	0
$365$	−690614.	−0.271333
$366$	0	0
$367$	1.14080e6	0.442123	0.221062	−	0.975260i	$-0.429048\pi$
$367$	1.14080e6	0.442123	0.221062	+	0.975260i	$0.429048\pi$
$368$	0	0
$369$	871007.	0.333009
$370$	0	0
$371$	−5.92719e6	−2.23570
$372$	0	0
$373$	3.71773e6	1.38358	0.691792	−	0.722097i	$-0.256822\pi$
$373$	3.71773e6	1.38358	0.691792	+	0.722097i	$0.256822\pi$
$374$	0	0
$375$	−354702.	−0.130252
$376$	0	0
$377$	−454368.	−0.164647
$378$	0	0
$379$	−4.85248e6	−1.73526	−0.867631	−	0.497208i	$-0.834359\pi$
$379$	−4.85248e6	−1.73526	−0.867631	+	0.497208i	$0.834359\pi$
$380$	0	0
$381$	1.78051e6	0.628394
$382$	0	0
$383$	−1.33616e6	−0.465436	−0.232718	−	0.972544i	$-0.574762\pi$
$383$	−1.33616e6	−0.465436	−0.232718	+	0.972544i	$0.574762\pi$
$384$	0	0
$385$	−508014.	−0.174672
$386$	0	0
$387$	2.22822e6	0.756277
$388$	0	0
$389$	2.27174e6	0.761176	0.380588	−	0.924745i	$-0.375722\pi$
$389$	2.27174e6	0.761176	0.380588	+	0.924745i	$0.375722\pi$
$390$	0	0
$391$	2.50183e6	0.827590
$392$	0	0
$393$	−2.50360e6	−0.817679
$394$	0	0
$395$	1.42940e6	0.460957
$396$	0	0
$397$	−3.35406e6	−1.06806	−0.534029	−	0.845466i	$-0.679323\pi$
$397$	−3.35406e6	−1.06806	−0.534029	+	0.845466i	$0.679323\pi$
$398$	0	0
$399$	−5.45360e6	−1.71495
$400$	0	0
$401$	403581.	0.125334	0.0626671	−	0.998034i	$-0.480039\pi$
$401$	403581.	0.125334	0.0626671	+	0.998034i	$0.480039\pi$
$402$	0	0
$403$	−1.28265e6	−0.393410
$404$	0	0
$405$	−1.27653e6	−0.386717
$406$	0	0
$407$	−711265.	−0.212836
$408$	0	0
$409$	1.13451e6	0.335351	0.167676	−	0.985842i	$-0.446374\pi$
$409$	1.13451e6	0.335351	0.167676	+	0.985842i	$0.446374\pi$
$410$	0	0
$411$	7.14329e6	2.08590
$412$	0	0
$413$	−1.10781e6	−0.319589
$414$	0	0
$415$	−1.81745e6	−0.518016
$416$	0	0
$417$	958968.	0.270062
$418$	0	0
$419$	−4.83882e6	−1.34649	−0.673247	−	0.739418i	$-0.735101\pi$
$419$	−4.83882e6	−1.34649	−0.673247	+	0.739418i	$0.735101\pi$
$420$	0	0
$421$	6.42204e6	1.76591	0.882953	−	0.469462i	$-0.155552\pi$
$421$	6.42204e6	1.76591	0.882953	+	0.469462i	$0.155552\pi$
$422$	0	0
$423$	3.18747e6	0.866154
$424$	0	0
$425$	−945675.	−0.253963
$426$	0	0
$427$	−3.72359e6	−0.988309
$428$	0	0
$429$	−528090.	−0.138537
$430$	0	0
$431$	4.27098e6	1.10747	0.553737	−	0.832691i	$-0.313201\pi$
$431$	4.27098e6	1.10747	0.553737	+	0.832691i	$0.313201\pi$
$432$	0	0
$433$	−2.39291e6	−0.613347	−0.306673	−	0.951815i	$-0.599216\pi$
$433$	−2.39291e6	−0.613347	−0.306673	+	0.951815i	$0.599216\pi$
$434$	0	0
$435$	−1.52582e6	−0.386617
$436$	0	0
$437$	−2.69078e6	−0.674022
$438$	0	0
$439$	4.46618e6	1.10605	0.553025	−	0.833165i	$-0.313473\pi$
$439$	4.46618e6	1.10605	0.553025	+	0.833165i	$0.313473\pi$
$440$	0	0
$441$	1.35782e6	0.332464
$442$	0	0
$443$	3.43498e6	0.831601	0.415801	−	0.909456i	$-0.363501\pi$
$443$	3.43498e6	0.831601	0.415801	+	0.909456i	$0.363501\pi$
$444$	0	0
$445$	1.70709e6	0.408656
$446$	0	0
$447$	−7.64589e6	−1.80992
$448$	0	0
$449$	527773.	0.123547	0.0617733	−	0.998090i	$-0.480324\pi$
$449$	527773.	0.123547	0.0617733	+	0.998090i	$0.480324\pi$
$450$	0	0
$451$	−440253.	−0.101920
$452$	0	0
$453$	−1.22884e7	−2.81352
$454$	0	0
$455$	−623712.	−0.141239
$456$	0	0
$457$	−1.69933e6	−0.380616	−0.190308	−	0.981724i	$-0.560949\pi$
$457$	−1.69933e6	−0.380616	−0.190308	+	0.981724i	$0.560949\pi$
$458$	0	0
$459$	1.00748e6	0.223205
$460$	0	0
$461$	4.81607e6	1.05546	0.527729	−	0.849413i	$-0.323044\pi$
$461$	4.81607e6	1.05546	0.527729	+	0.849413i	$0.323044\pi$
$462$	0	0
$463$	−2.59596e6	−0.562790	−0.281395	−	0.959592i	$-0.590797\pi$
$463$	−2.59596e6	−0.562790	−0.281395	+	0.959592i	$0.590797\pi$
$464$	0	0
$465$	−4.30730e6	−0.923789
$466$	0	0
$467$	2.17723e6	0.461968	0.230984	−	0.972958i	$-0.425806\pi$
$467$	2.17723e6	0.461968	0.230984	+	0.972958i	$0.425806\pi$
$468$	0	0
$469$	6.95117e6	1.45924
$470$	0	0
$471$	−3.17228e6	−0.658900
$472$	0	0
$473$	−1.12626e6	−0.231465
$474$	0	0
$475$	1.01710e6	0.206837
$476$	0	0
$477$	−1.09342e7	−2.20036
$478$	0	0
$479$	−589971.	−0.117488	−0.0587438	−	0.998273i	$-0.518709\pi$
$479$	−589971.	−0.117488	−0.0587438	+	0.998273i	$0.518709\pi$
$480$	0	0
$481$	−873253.	−0.172099
$482$	0	0
$483$	5.54110e6	1.08076
$484$	0	0
$485$	729154.	0.140755
$486$	0	0
$487$	−5.54614e6	−1.05966	−0.529832	−	0.848103i	$-0.677745\pi$
$487$	−5.54614e6	−1.05966	−0.529832	+	0.848103i	$0.677745\pi$
$488$	0	0
$489$	−1.05936e7	−2.00342
$490$	0	0
$491$	1.72697e6	0.323282	0.161641	−	0.986850i	$-0.448321\pi$
$491$	1.72697e6	0.323282	0.161641	+	0.986850i	$0.448321\pi$
$492$	0	0
$493$	−4.06802e6	−0.753816
$494$	0	0
$495$	−937164.	−0.171911
$496$	0	0
$497$	−1.21101e7	−2.19916
$498$	0	0
$499$	4.21295e6	0.757416	0.378708	−	0.925516i	$-0.376368\pi$
$499$	4.21295e6	0.757416	0.378708	+	0.925516i	$0.376368\pi$
$500$	0	0
$501$	1.18312e7	2.10588
$502$	0	0
$503$	7.74311e6	1.36457	0.682285	−	0.731087i	$-0.260986\pi$
$503$	7.74311e6	1.36457	0.682285	+	0.731087i	$0.260986\pi$
$504$	0	0
$505$	−1.36172e6	−0.237607
$506$	0	0
$507$	−648361.	−0.112020
$508$	0	0
$509$	6.55567e6	1.12156	0.560780	−	0.827965i	$-0.310501\pi$
$509$	6.55567e6	1.12156	0.560780	+	0.827965i	$0.310501\pi$
$510$	0	0
$511$	−4.07805e6	−0.690876
$512$	0	0
$513$	−1.08357e6	−0.181787
$514$	0	0
$515$	−1.10988e6	−0.184399
$516$	0	0
$517$	−1.61111e6	−0.265094
$518$	0	0
$519$	−6.83958e6	−1.11458
$520$	0	0
$521$	6.32460e6	1.02079	0.510397	−	0.859939i	$-0.329498\pi$
$521$	6.32460e6	1.02079	0.510397	+	0.859939i	$0.329498\pi$
$522$	0	0
$523$	3.89046e6	0.621938	0.310969	−	0.950420i	$-0.399347\pi$
$523$	3.89046e6	0.621938	0.310969	+	0.950420i	$0.399347\pi$
$524$	0	0
$525$	−2.09450e6	−0.331652
$526$	0	0
$527$	−1.14837e7	−1.80118
$528$	0	0
$529$	−3.70239e6	−0.575232
$530$	0	0
$531$	−2.04365e6	−0.314536
$532$	0	0
$533$	−540518.	−0.0824124
$534$	0	0
$535$	2.74264e6	0.414270
$536$	0	0
$537$	1.39442e7	2.08669
$538$	0	0
$539$	−686312.	−0.101754
$540$	0	0
$541$	986681.	0.144938	0.0724692	−	0.997371i	$-0.476912\pi$
$541$	986681.	0.144938	0.0724692	+	0.997371i	$0.476912\pi$
$542$	0	0
$543$	−840353.	−0.122310
$544$	0	0
$545$	4.12296e6	0.594591
$546$	0	0
$547$	8.88626e6	1.26985	0.634923	−	0.772576i	$-0.281032\pi$
$547$	8.88626e6	1.26985	0.634923	+	0.772576i	$0.281032\pi$
$548$	0	0
$549$	−6.86914e6	−0.972684
$550$	0	0
$551$	4.37525e6	0.613938
$552$	0	0
$553$	8.44055e6	1.17370
$554$	0	0
$555$	−2.93249e6	−0.404114
$556$	0	0
$557$	290968.	0.0397381	0.0198691	−	0.999803i	$-0.493675\pi$
$557$	290968.	0.0397381	0.0198691	+	0.999803i	$0.493675\pi$
$558$	0	0
$559$	−1.38276e6	−0.187162
$560$	0	0
$561$	−4.72806e6	−0.634273
$562$	0	0
$563$	−6.66482e6	−0.886171	−0.443086	−	0.896479i	$-0.646116\pi$
$563$	−6.66482e6	−0.886171	−0.443086	+	0.896479i	$0.646116\pi$
$564$	0	0
$565$	667883.	0.0880196
$566$	0	0
$567$	−7.53786e6	−0.984670
$568$	0	0
$569$	3.61160e6	0.467648	0.233824	−	0.972279i	$-0.424876\pi$
$569$	3.61160e6	0.467648	0.233824	+	0.972279i	$0.424876\pi$
$570$	0	0
$571$	4.60236e6	0.590731	0.295366	−	0.955384i	$-0.404559\pi$
$571$	4.60236e6	0.590731	0.295366	+	0.955384i	$0.404559\pi$
$572$	0	0
$573$	1.35781e7	1.72763
$574$	0	0
$575$	−1.03342e6	−0.130348
$576$	0	0
$577$	−5.43584e6	−0.679715	−0.339858	−	0.940477i	$-0.610379\pi$
$577$	−5.43584e6	−0.679715	−0.339858	+	0.940477i	$0.610379\pi$
$578$	0	0
$579$	1.53911e7	1.90797
$580$	0	0
$581$	−1.07320e7	−1.31899
$582$	0	0
$583$	5.52675e6	0.673439
$584$	0	0
$585$	−1.15060e6	−0.139006
$586$	0	0
$587$	1.04917e7	1.25676	0.628379	−	0.777907i	$-0.283719\pi$
$587$	1.04917e7	1.25676	0.628379	+	0.777907i	$0.283719\pi$
$588$	0	0
$589$	1.23511e7	1.46695
$590$	0	0
$591$	−1.03488e7	−1.21877
$592$	0	0
$593$	−1.31698e7	−1.53795	−0.768974	−	0.639280i	$-0.779233\pi$
$593$	−1.31698e7	−1.53795	−0.768974	+	0.639280i	$0.779233\pi$
$594$	0	0
$595$	−5.58418e6	−0.646647
$596$	0	0
$597$	−4.41095e6	−0.506520
$598$	0	0
$599$	1.40030e7	1.59461	0.797306	−	0.603576i	$-0.206258\pi$
$599$	1.40030e7	1.59461	0.797306	+	0.603576i	$0.206258\pi$
$600$	0	0
$601$	1.68332e7	1.90099	0.950494	−	0.310743i	$-0.100578\pi$
$601$	1.68332e7	1.90099	0.950494	+	0.310743i	$0.100578\pi$
$602$	0	0
$603$	1.28233e7	1.43617
$604$	0	0
$605$	−3.55258e6	−0.394599
$606$	0	0
$607$	−6.67520e6	−0.735347	−0.367674	−	0.929955i	$-0.619846\pi$
$607$	−6.67520e6	−0.735347	−0.367674	+	0.929955i	$0.619846\pi$
$608$	0	0
$609$	−9.00993e6	−0.984415
$610$	0	0
$611$	−1.97804e6	−0.214354
$612$	0	0
$613$	−1.63152e6	−0.175364	−0.0876819	−	0.996149i	$-0.527946\pi$
$613$	−1.63152e6	−0.175364	−0.0876819	+	0.996149i	$0.527946\pi$
$614$	0	0
$615$	−1.81513e6	−0.193517
$616$	0	0
$617$	−2.44624e6	−0.258694	−0.129347	−	0.991599i	$-0.541288\pi$
$617$	−2.44624e6	−0.258694	−0.129347	+	0.991599i	$0.541288\pi$
$618$	0	0
$619$	1.50386e7	1.57754	0.788772	−	0.614686i	$-0.210717\pi$
$619$	1.50386e7	1.57754	0.788772	+	0.614686i	$0.210717\pi$
$620$	0	0
$621$	1.10096e6	0.114562
$622$	0	0
$623$	1.00803e7	1.04053
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	5.08515e6	0.516577
$628$	0	0
$629$	−7.81835e6	−0.787932
$630$	0	0
$631$	−1.03549e7	−1.03532	−0.517659	−	0.855587i	$-0.673196\pi$
$631$	−1.03549e7	−1.03532	−0.517659	+	0.855587i	$0.673196\pi$
$632$	0	0
$633$	−1.88273e7	−1.86758
$634$	0	0
$635$	−1.96084e6	−0.192978
$636$	0	0
$637$	−842616.	−0.0822775
$638$	0	0
$639$	−2.23402e7	−2.16439
$640$	0	0
$641$	7.97904e6	0.767018	0.383509	−	0.923537i	$-0.374716\pi$
$641$	7.97904e6	0.767018	0.383509	+	0.923537i	$0.374716\pi$
$642$	0	0
$643$	3.41189e6	0.325437	0.162719	−	0.986672i	$-0.447974\pi$
$643$	3.41189e6	0.325437	0.162719	+	0.986672i	$0.447974\pi$
$644$	0	0
$645$	−4.64349e6	−0.439486
$646$	0	0
$647$	−2.15098e6	−0.202011	−0.101006	−	0.994886i	$-0.532206\pi$
$647$	−2.15098e6	−0.202011	−0.101006	+	0.994886i	$0.532206\pi$
$648$	0	0
$649$	1.03297e6	0.0962666
$650$	0	0
$651$	−2.54345e7	−2.35218
$652$	0	0
$653$	−4.06049e6	−0.372645	−0.186322	−	0.982489i	$-0.559657\pi$
$653$	−4.06049e6	−0.372645	−0.186322	+	0.982489i	$0.559657\pi$
$654$	0	0
$655$	2.75715e6	0.251106
$656$	0	0
$657$	−7.52303e6	−0.679954
$658$	0	0
$659$	1.12408e7	1.00828	0.504142	−	0.863621i	$-0.331809\pi$
$659$	1.12408e7	1.00828	0.504142	+	0.863621i	$0.331809\pi$
$660$	0	0
$661$	1.59322e7	1.41831	0.709155	−	0.705053i	$-0.249077\pi$
$661$	1.59322e7	1.41831	0.709155	+	0.705053i	$0.249077\pi$
$662$	0	0
$663$	−5.80486e6	−0.512871
$664$	0	0
$665$	6.00592e6	0.526654
$666$	0	0
$667$	−4.44545e6	−0.386903
$668$	0	0
$669$	−722375.	−0.0624018
$670$	0	0
$671$	3.47203e6	0.297699
$672$	0	0
$673$	−663601.	−0.0564767	−0.0282383	−	0.999601i	$-0.508990\pi$
$673$	−663601.	−0.0564767	−0.0282383	+	0.999601i	$0.508990\pi$
$674$	0	0
$675$	−416155.	−0.0351557
$676$	0	0
$677$	6.74152e6	0.565309	0.282654	−	0.959222i	$-0.408785\pi$
$677$	6.74152e6	0.565309	0.282654	+	0.959222i	$0.408785\pi$
$678$	0	0
$679$	4.30563e6	0.358395
$680$	0	0
$681$	−5.08198e6	−0.419919
$682$	0	0
$683$	−1.81453e7	−1.48838	−0.744188	−	0.667971i	$-0.767163\pi$
$683$	−1.81453e7	−1.48838	−0.744188	+	0.667971i	$0.767163\pi$
$684$	0	0
$685$	−7.86674e6	−0.640573
$686$	0	0
$687$	4.37023e6	0.353275
$688$	0	0
$689$	6.78544e6	0.544540
$690$	0	0
$691$	−1.81716e7	−1.44776	−0.723881	−	0.689925i	$-0.757644\pi$
$691$	−1.81716e7	−1.44776	−0.723881	+	0.689925i	$0.757644\pi$
$692$	0	0
$693$	−5.53392e6	−0.437724
$694$	0	0
$695$	−1.05609e6	−0.0829352
$696$	0	0
$697$	−4.83934e6	−0.377315
$698$	0	0
$699$	1.36345e7	1.05547
$700$	0	0
$701$	2.27659e7	1.74981	0.874904	−	0.484296i	$-0.160924\pi$
$701$	2.27659e7	1.74981	0.874904	+	0.484296i	$0.160924\pi$
$702$	0	0
$703$	8.40883e6	0.641723
$704$	0	0
$705$	−6.64250e6	−0.503337
$706$	0	0
$707$	−8.04091e6	−0.605002
$708$	0	0
$709$	1.78660e7	1.33479	0.667394	−	0.744705i	$-0.267410\pi$
$709$	1.78660e7	1.33479	0.667394	+	0.744705i	$0.267410\pi$
$710$	0	0
$711$	1.55708e7	1.15515
$712$	0	0
$713$	−1.25492e7	−0.924471
$714$	0	0
$715$	581574.	0.0425441
$716$	0	0
$717$	−1.16098e7	−0.843385
$718$	0	0
$719$	2.02589e7	1.46148	0.730741	−	0.682655i	$-0.239175\pi$
$719$	2.02589e7	1.46148	0.730741	+	0.682655i	$0.239175\pi$
$720$	0	0
$721$	−6.55380e6	−0.469521
$722$	0	0
$723$	2.52310e7	1.79510
$724$	0	0
$725$	1.68035e6	0.118729
$726$	0	0
$727$	−3.97292e6	−0.278788	−0.139394	−	0.990237i	$-0.544515\pi$
$727$	−3.97292e6	−0.278788	−0.139394	+	0.990237i	$0.544515\pi$
$728$	0	0
$729$	−1.75786e7	−1.22508
$730$	0	0
$731$	−1.23801e7	−0.856898
$732$	0	0
$733$	7.73407e6	0.531677	0.265839	−	0.964018i	$-0.414351\pi$
$733$	7.73407e6	0.531677	0.265839	+	0.964018i	$0.414351\pi$
$734$	0	0
$735$	−2.82961e6	−0.193200
$736$	0	0
$737$	−6.48155e6	−0.439552
$738$	0	0
$739$	1.05854e7	0.713009	0.356504	−	0.934294i	$-0.383968\pi$
$739$	1.05854e7	0.713009	0.356504	+	0.934294i	$0.383968\pi$
$740$	0	0
$741$	6.24327e6	0.417702
$742$	0	0
$743$	6.43652e6	0.427739	0.213870	−	0.976862i	$-0.431393\pi$
$743$	6.43652e6	0.427739	0.213870	+	0.976862i	$0.431393\pi$
$744$	0	0
$745$	8.42024e6	0.555820
$746$	0	0
$747$	−1.97980e7	−1.29813
$748$	0	0
$749$	1.61952e7	1.05483
$750$	0	0
$751$	−1.15224e7	−0.745493	−0.372747	−	0.927933i	$-0.621584\pi$
$751$	−1.15224e7	−0.745493	−0.372747	+	0.927933i	$0.621584\pi$
$752$	0	0
$753$	−1.09095e7	−0.701159
$754$	0	0
$755$	1.35329e7	0.864021
$756$	0	0
$757$	−2.37710e7	−1.50768	−0.753838	−	0.657061i	$-0.771799\pi$
$757$	−2.37710e7	−1.50768	−0.753838	+	0.657061i	$0.771799\pi$
$758$	0	0
$759$	−5.16674e6	−0.325546
$760$	0	0
$761$	1.04140e7	0.651861	0.325930	−	0.945394i	$-0.394323\pi$
$761$	1.04140e7	0.651861	0.325930	+	0.945394i	$0.394323\pi$
$762$	0	0
$763$	2.43460e7	1.51396
$764$	0	0
$765$	−1.03015e7	−0.636423
$766$	0	0
$767$	1.26822e6	0.0778409
$768$	0	0
$769$	1.62159e7	0.988840	0.494420	−	0.869223i	$-0.335380\pi$
$769$	1.62159e7	0.988840	0.494420	+	0.869223i	$0.335380\pi$
$770$	0	0
$771$	2.71302e6	0.164368
$772$	0	0
$773$	−1.81491e7	−1.09246	−0.546232	−	0.837634i	$-0.683938\pi$
$773$	−1.81491e7	−1.09246	−0.546232	+	0.837634i	$0.683938\pi$
$774$	0	0
$775$	4.74353e6	0.283692
$776$	0	0
$777$	−1.73163e7	−1.02897
$778$	0	0
$779$	5.20483e6	0.307300
$780$	0	0
$781$	1.12919e7	0.662431
$782$	0	0
$783$	−1.79017e6	−0.104350
$784$	0	0
$785$	3.49356e6	0.202346
$786$	0	0
$787$	5.88857e6	0.338901	0.169451	−	0.985539i	$-0.445801\pi$
$787$	5.88857e6	0.338901	0.169451	+	0.985539i	$0.445801\pi$
$788$	0	0
$789$	−2.07369e7	−1.18591
$790$	0	0
$791$	3.94383e6	0.224118
$792$	0	0
$793$	4.26277e6	0.240718
$794$	0	0
$795$	2.27863e7	1.27867
$796$	0	0
$797$	−288198.	−0.0160711	−0.00803553	−	0.999968i	$-0.502558\pi$
$797$	−288198.	−0.0160711	−0.00803553	+	0.999968i	$0.502558\pi$
$798$	0	0
$799$	−1.77097e7	−0.981394
$800$	0	0
$801$	1.85958e7	1.02408
$802$	0	0
$803$	3.80254e6	0.208106
$804$	0	0
$805$	−6.10229e6	−0.331897
$806$	0	0
$807$	9.21643e6	0.498171
$808$	0	0
$809$	3.20143e6	0.171978	0.0859890	−	0.996296i	$-0.472595\pi$
$809$	3.20143e6	0.171978	0.0859890	+	0.996296i	$0.472595\pi$
$810$	0	0
$811$	4.36938e6	0.233275	0.116637	−	0.993175i	$-0.462788\pi$
$811$	4.36938e6	0.233275	0.116637	+	0.993175i	$0.462788\pi$
$812$	0	0
$813$	−3.75462e7	−1.99223
$814$	0	0
$815$	1.16665e7	0.615244
$816$	0	0
$817$	1.33151e7	0.697892
$818$	0	0
$819$	−6.79425e6	−0.353942
$820$	0	0
$821$	−2.60548e7	−1.34906	−0.674528	−	0.738250i	$-0.735653\pi$
$821$	−2.60548e7	−1.34906	−0.674528	+	0.738250i	$0.735653\pi$
$822$	0	0
$823$	−6.18846e6	−0.318480	−0.159240	−	0.987240i	$-0.550904\pi$
$823$	−6.18846e6	−0.318480	−0.159240	+	0.987240i	$0.550904\pi$
$824$	0	0
$825$	1.95300e6	0.0999003
$826$	0	0
$827$	2.87318e7	1.46083	0.730415	−	0.683004i	$-0.239327\pi$
$827$	2.87318e7	1.46083	0.730415	+	0.683004i	$0.239327\pi$
$828$	0	0
$829$	−1.25204e7	−0.632747	−0.316374	−	0.948635i	$-0.602465\pi$
$829$	−1.25204e7	−0.632747	−0.316374	+	0.948635i	$0.602465\pi$
$830$	0	0
$831$	−4.51186e6	−0.226649
$832$	0	0
$833$	−7.54406e6	−0.376697
$834$	0	0
$835$	−1.30294e7	−0.646708
$836$	0	0
$837$	−5.05355e6	−0.249335
$838$	0	0
$839$	−3.18538e7	−1.56227	−0.781135	−	0.624362i	$-0.785359\pi$
$839$	−3.18538e7	−1.56227	−0.781135	+	0.624362i	$0.785359\pi$
$840$	0	0
$841$	−1.32828e7	−0.647587
$842$	0	0
$843$	−4.39865e7	−2.13182
$844$	0	0
$845$	714025.	0.0344010
$846$	0	0
$847$	−2.09779e7	−1.00474
$848$	0	0
$849$	−1.16062e7	−0.552610
$850$	0	0
$851$	−8.54375e6	−0.404413
$852$	0	0
$853$	−3.85797e7	−1.81546	−0.907728	−	0.419559i	$-0.862185\pi$
$853$	−3.85797e7	−1.81546	−0.907728	+	0.419559i	$0.862185\pi$
$854$	0	0
$855$	1.10795e7	0.518328
$856$	0	0
$857$	−1.88205e7	−0.875343	−0.437671	−	0.899135i	$-0.644197\pi$
$857$	−1.88205e7	−0.875343	−0.437671	+	0.899135i	$0.644197\pi$
$858$	0	0
$859$	1.70447e7	0.788144	0.394072	−	0.919080i	$-0.371066\pi$
$859$	1.70447e7	0.788144	0.394072	+	0.919080i	$0.371066\pi$
$860$	0	0
$861$	−1.07183e7	−0.492739
$862$	0	0
$863$	−2.59489e7	−1.18602	−0.593011	−	0.805194i	$-0.702061\pi$
$863$	−2.59489e7	−1.18602	−0.593011	+	0.805194i	$0.702061\pi$
$864$	0	0
$865$	7.53227e6	0.342283
$866$	0	0
$867$	−1.97397e7	−0.891851
$868$	0	0
$869$	−7.87030e6	−0.353543
$870$	0	0
$871$	−7.95770e6	−0.355420
$872$	0	0
$873$	7.94286e6	0.352729
$874$	0	0
$875$	2.30663e6	0.101849
$876$	0	0
$877$	−1.89248e7	−0.830868	−0.415434	−	0.909623i	$-0.636370\pi$
$877$	−1.89248e7	−0.830868	−0.415434	+	0.909623i	$0.636370\pi$
$878$	0	0
$879$	−3.42415e7	−1.49479
$880$	0	0
$881$	−1.33947e7	−0.581425	−0.290713	−	0.956810i	$-0.593892\pi$
$881$	−1.33947e7	−0.581425	−0.290713	+	0.956810i	$0.593892\pi$
$882$	0	0
$883$	2.31251e6	0.0998117	0.0499059	−	0.998754i	$-0.484108\pi$
$883$	2.31251e6	0.0998117	0.0499059	+	0.998754i	$0.484108\pi$
$884$	0	0
$885$	4.25885e6	0.182782
$886$	0	0
$887$	2.16186e7	0.922611	0.461306	−	0.887241i	$-0.347381\pi$
$887$	2.16186e7	0.922611	0.461306	+	0.887241i	$0.347381\pi$
$888$	0	0
$889$	−1.15787e7	−0.491365
$890$	0	0
$891$	7.02860e6	0.296602
$892$	0	0
$893$	1.90472e7	0.799286
$894$	0	0
$895$	−1.53564e7	−0.640814
$896$	0	0
$897$	−6.34345e6	−0.263235
$898$	0	0
$899$	2.04053e7	0.842060
$900$	0	0
$901$	6.07510e7	2.49311
$902$	0	0
$903$	−2.74196e7	−1.11903
$904$	0	0
$905$	925461.	0.0375610
$906$	0	0
$907$	−2.52097e7	−1.01753	−0.508767	−	0.860904i	$-0.669898\pi$
$907$	−2.52097e7	−1.01753	−0.508767	+	0.860904i	$0.669898\pi$
$908$	0	0
$909$	−1.48336e7	−0.595437
$910$	0	0
$911$	1.76590e7	0.704968	0.352484	−	0.935818i	$-0.385337\pi$
$911$	1.76590e7	0.704968	0.352484	+	0.935818i	$0.385337\pi$
$912$	0	0
$913$	1.00069e7	0.397306
$914$	0	0
$915$	1.43149e7	0.565243
$916$	0	0
$917$	1.62809e7	0.639374
$918$	0	0
$919$	1.60007e7	0.624958	0.312479	−	0.949925i	$-0.398841\pi$
$919$	1.60007e7	0.624958	0.312479	+	0.949925i	$0.398841\pi$
$920$	0	0
$921$	1.32756e7	0.515710
$922$	0	0
$923$	1.38636e7	0.535639
$924$	0	0
$925$	3.22949e6	0.124102
$926$	0	0
$927$	−1.20902e7	−0.462098
$928$	0	0
$929$	−3.95210e7	−1.50241	−0.751204	−	0.660070i	$-0.770527\pi$
$929$	−3.95210e7	−1.50241	−0.751204	+	0.660070i	$0.770527\pi$
$930$	0	0
$931$	8.11382e6	0.306797
$932$	0	0
$933$	2.09192e7	0.786757
$934$	0	0
$935$	5.20691e6	0.194783
$936$	0	0
$937$	−2.98037e7	−1.10897	−0.554487	−	0.832192i	$-0.687086\pi$
$937$	−2.98037e7	−1.10897	−0.554487	+	0.832192i	$0.687086\pi$
$938$	0	0
$939$	−3.49499e7	−1.29354
$940$	0	0
$941$	−2.02939e7	−0.747120	−0.373560	−	0.927606i	$-0.621863\pi$
$941$	−2.02939e7	−0.747120	−0.373560	+	0.927606i	$0.621863\pi$
$942$	0	0
$943$	−5.28834e6	−0.193660
$944$	0	0
$945$	−2.45738e6	−0.0895143
$946$	0	0
$947$	−2.42851e7	−0.879963	−0.439982	−	0.898007i	$-0.645015\pi$
$947$	−2.42851e7	−0.879963	−0.439982	+	0.898007i	$0.645015\pi$
$948$	0	0
$949$	4.66855e6	0.168274
$950$	0	0
$951$	−5.77372e7	−2.07016
$952$	0	0
$953$	−2.78038e7	−0.991679	−0.495840	−	0.868414i	$-0.665140\pi$
$953$	−2.78038e7	−0.991679	−0.495840	+	0.868414i	$0.665140\pi$
$954$	0	0
$955$	−1.49532e7	−0.530549
$956$	0	0
$957$	8.40122e6	0.296526
$958$	0	0
$959$	−4.64529e7	−1.63105
$960$	0	0
$961$	2.89736e7	1.01203
$962$	0	0
$963$	2.98762e7	1.03815
$964$	0	0
$965$	−1.69498e7	−0.585932
$966$	0	0
$967$	−4.14481e7	−1.42541	−0.712704	−	0.701465i	$-0.752530\pi$
$967$	−4.14481e7	−1.42541	−0.712704	+	0.701465i	$0.752530\pi$
$968$	0	0
$969$	5.58969e7	1.91240
$970$	0	0
$971$	5.41023e7	1.84148	0.920742	−	0.390172i	$-0.127585\pi$
$971$	5.41023e7	1.84148	0.920742	+	0.390172i	$0.127585\pi$
$972$	0	0
$973$	−6.23617e6	−0.211172
$974$	0	0
$975$	2.39778e6	0.0807790
$976$	0	0
$977$	5.24528e7	1.75805	0.879027	−	0.476771i	$-0.158193\pi$
$977$	5.24528e7	1.75805	0.879027	+	0.476771i	$0.158193\pi$
$978$	0	0
$979$	−9.39930e6	−0.313429
$980$	0	0
$981$	4.49125e7	1.49003
$982$	0	0
$983$	1.07316e7	0.354227	0.177114	−	0.984190i	$-0.443324\pi$
$983$	1.07316e7	0.354227	0.177114	+	0.984190i	$0.443324\pi$
$984$	0	0
$985$	1.13969e7	0.374280
$986$	0	0
$987$	−3.92237e7	−1.28161
$988$	0	0
$989$	−1.35287e7	−0.439810
$990$	0	0
$991$	−1.29690e7	−0.419490	−0.209745	−	0.977756i	$-0.567263\pi$
$991$	−1.29690e7	−0.419490	−0.209745	+	0.977756i	$0.567263\pi$
$992$	0	0
$993$	5.61808e6	0.180807
$994$	0	0
$995$	4.85768e6	0.155551
$996$	0	0
$997$	−1.02028e7	−0.325074	−0.162537	−	0.986702i	$-0.551968\pi$
$997$	−1.02028e7	−0.325074	−0.162537	+	0.986702i	$0.551968\pi$
$998$	0	0
$999$	−3.44055e6	−0.109072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.ba.1.1		8
4.3	odd	2		520.6.a.d.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.6.a.d.1.8	✓	8	4.3	odd	2
1040.6.a.ba.1.1		8	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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q-22.7009 q^{3} +25.0000 q^{5} +147.624 q^{7} +272.331 q^{9} -137.651 q^{11} -169.000 q^{13} -567.523 q^{15} -1513.08 q^{17} +1627.36 q^{19} -3351.20 q^{21} -1653.47 q^{23} +625.000 q^{25} -665.847 q^{27} +2688.57 q^{29} +7589.65 q^{31} +3124.79 q^{33} +3690.60 q^{35} +5167.18 q^{37} +3836.45 q^{39} +3198.33 q^{41} +8182.03 q^{43} +6808.28 q^{45} +11704.4 q^{47} +4985.90 q^{49} +34348.3 q^{51} -40150.5 q^{53} -3441.27 q^{55} -36942.4 q^{57} -7504.29 q^{59} -25223.5 q^{61} +40202.7 q^{63} -4225.00 q^{65} +47087.0 q^{67} +37535.2 q^{69} -82033.3 q^{71} -27624.5 q^{73} -14188.1 q^{75} -20320.6 q^{77} +57175.9 q^{79} -51061.2 q^{81} -72698.2 q^{83} -37827.0 q^{85} -61032.9 q^{87} +68283.8 q^{89} -24948.5 q^{91} -172292. q^{93} +40683.9 q^{95} +29166.2 q^{97} -37486.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 28 q^{3} + 200 q^{5} - 80 q^{7} + 936 q^{9} + 444 q^{11} - 1352 q^{13} + 700 q^{15} - 3400 q^{17} + 1668 q^{19} - 3816 q^{21} + 2964 q^{23} + 5000 q^{25} + 14464 q^{27} - 8272 q^{29} + 8684 q^{31}+ \cdots + 535828 q^{99}+O(q^{100}) \) 8 * q + 28 * q^3 + 200 * q^5 - 80 * q^7 + 936 * q^9 + 444 * q^11 - 1352 * q^13 + 700 * q^15 - 3400 * q^17 + 1668 * q^19 - 3816 * q^21 + 2964 * q^23 + 5000 * q^25 + 14464 * q^27 - 8272 * q^29 + 8684 * q^31 + 5288 * q^33 - 2000 * q^35 + 11320 * q^37 - 4732 * q^39 + 280 * q^41 + 36268 * q^43 + 23400 * q^45 + 17936 * q^47 + 76456 * q^49 + 10744 * q^51 + 28936 * q^53 + 11100 * q^55 - 19640 * q^57 + 34668 * q^59 + 62976 * q^61 - 9040 * q^63 - 33800 * q^65 + 181688 * q^67 - 43600 * q^69 + 6036 * q^71 + 23000 * q^73 + 17500 * q^75 - 178792 * q^77 + 24392 * q^79 - 3992 * q^81 - 34416 * q^83 - 85000 * q^85 + 144144 * q^87 + 57328 * q^89 + 13520 * q^91 - 66648 * q^93 + 41700 * q^95 - 210176 * q^97 + 535828 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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