LMFDB - Embedded newform 1040.6.a.q.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1040,6,Mod(1,1040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1040.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-38,0,-150,0,-220] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$166.799172605$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 $ x^6 - 163x^4 - 8x^3 + 6120x^2 + 6624x - 19440
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-10.7882$ 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)

$f(q)$	$=$	$q-0.527294 q^{3} -25.0000 q^{5} -231.710 q^{7} -242.722 q^{9} +466.402 q^{11} +169.000 q^{13} +13.1824 q^{15} +882.905 q^{17} -167.308 q^{19} +122.180 q^{21} -3037.27 q^{23} +625.000 q^{25} +256.118 q^{27} +6278.22 q^{29} +4294.08 q^{31} -245.931 q^{33} +5792.76 q^{35} -7285.93 q^{37} -89.1127 q^{39} +18984.4 q^{41} -14295.7 q^{43} +6068.05 q^{45} -8375.45 q^{47} +36882.8 q^{49} -465.551 q^{51} +20546.0 q^{53} -11660.0 q^{55} +88.2207 q^{57} +118.006 q^{59} +24590.5 q^{61} +56241.2 q^{63} -4225.00 q^{65} +34789.8 q^{67} +1601.54 q^{69} -9033.72 q^{71} +30590.1 q^{73} -329.559 q^{75} -108070. q^{77} +14290.7 q^{79} +58846.4 q^{81} -65535.3 q^{83} -22072.6 q^{85} -3310.47 q^{87} -85188.3 q^{89} -39159.1 q^{91} -2264.24 q^{93} +4182.71 q^{95} +51071.6 q^{97} -113206. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 38 q^{3} - 150 q^{5} - 220 q^{7} + 518 q^{9} + 170 q^{11} + 1014 q^{13} + 950 q^{15} + 728 q^{17} - 1218 q^{19} - 396 q^{21} - 8954 q^{23} + 3750 q^{25} - 13112 q^{27} + 8364 q^{29} - 2862 q^{31}+ \cdots + 32270 q^{99}+O(q^{100}) $ 6 * q - 38 * q^3 - 150 * q^5 - 220 * q^7 + 518 * q^9 + 170 * q^11 + 1014 * q^13 + 950 * q^15 + 728 * q^17 - 1218 * q^19 - 396 * q^21 - 8954 * q^23 + 3750 * q^25 - 13112 * q^27 + 8364 * q^29 - 2862 * q^31 - 3772 * q^33 + 5500 * q^35 + 13840 * q^37 - 6422 * q^39 + 2248 * q^41 - 3822 * q^43 - 12950 * q^45 + 10860 * q^47 + 85902 * q^49 + 55028 * q^51 + 6584 * q^53 - 4250 * q^55 + 11692 * q^57 + 28874 * q^59 + 972 * q^61 + 125236 * q^63 - 25350 * q^65 - 60040 * q^67 + 153088 * q^69 + 74174 * q^71 + 48960 * q^73 - 23750 * q^75 + 231588 * q^77 - 214508 * q^79 + 116006 * q^81 - 123260 * q^83 - 18200 * q^85 - 228144 * q^87 + 94028 * q^89 - 37180 * q^91 + 192892 * q^93 + 30450 * q^95 + 246284 * q^97 + 32270 * 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
$3$	−0.527294	−0.0338259	−0.0169130	−	0.999857i	$-0.505384\pi$
$3$	−0.527294	−0.0338259	−0.0169130	+	0.999857i	$0.505384\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−231.710	−1.78731	−0.893656	−	0.448752i	$-0.851869\pi$
$7$	−231.710	−1.78731	−0.893656	+	0.448752i	$0.851869\pi$
$8$	0	0
$9$	−242.722	−0.998856
$10$	0	0
$11$	466.402	1.16219	0.581097	−	0.813835i	$-0.302624\pi$
$11$	466.402	1.16219	0.581097	+	0.813835i	$0.302624\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	13.1824	0.0151274
$16$	0	0
$17$	882.905	0.740955	0.370478	−	0.928841i	$-0.379194\pi$
$17$	882.905	0.740955	0.370478	+	0.928841i	$0.379194\pi$
$18$	0	0
$19$	−167.308	−0.106325	−0.0531623	−	0.998586i	$-0.516930\pi$
$19$	−167.308	−0.106325	−0.0531623	+	0.998586i	$0.516930\pi$
$20$	0	0
$21$	122.180	0.0604575
$22$	0	0
$23$	−3037.27	−1.19719	−0.598597	−	0.801051i	$-0.704275\pi$
$23$	−3037.27	−1.19719	−0.598597	+	0.801051i	$0.704275\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	256.118	0.0676132
$28$	0	0
$29$	6278.22	1.38625	0.693125	−	0.720818i	$-0.256233\pi$
$29$	6278.22	1.38625	0.693125	+	0.720818i	$0.256233\pi$
$30$	0	0
$31$	4294.08	0.802538	0.401269	−	0.915960i	$-0.368569\pi$
$31$	4294.08	0.802538	0.401269	+	0.915960i	$0.368569\pi$
$32$	0	0
$33$	−245.931	−0.0393123
$34$	0	0
$35$	5792.76	0.799311
$36$	0	0
$37$	−7285.93	−0.874945	−0.437472	−	0.899232i	$-0.644126\pi$
$37$	−7285.93	−0.874945	−0.437472	+	0.899232i	$0.644126\pi$
$38$	0	0
$39$	−89.1127	−0.00938162
$40$	0	0
$41$	18984.4	1.76375	0.881875	−	0.471482i	$-0.156281\pi$
$41$	18984.4	1.76375	0.881875	+	0.471482i	$0.156281\pi$
$42$	0	0
$43$	−14295.7	−1.17906	−0.589528	−	0.807748i	$-0.700686\pi$
$43$	−14295.7	−1.17906	−0.589528	+	0.807748i	$0.700686\pi$
$44$	0	0
$45$	6068.05	0.446702
$46$	0	0
$47$	−8375.45	−0.553049	−0.276524	−	0.961007i	$-0.589183\pi$
$47$	−8375.45	−0.553049	−0.276524	+	0.961007i	$0.589183\pi$
$48$	0	0
$49$	36882.8	2.19449
$50$	0	0
$51$	−465.551	−0.0250635
$52$	0	0
$53$	20546.0	1.00470	0.502351	−	0.864664i	$-0.332468\pi$
$53$	20546.0	1.00470	0.502351	+	0.864664i	$0.332468\pi$
$54$	0	0
$55$	−11660.0	−0.519749
$56$	0	0
$57$	88.2207	0.00359653
$58$	0	0
$59$	118.006	0.00441341	0.00220670	−	0.999998i	$-0.499298\pi$
$59$	118.006	0.00441341	0.00220670	+	0.999998i	$0.499298\pi$
$60$	0	0
$61$	24590.5	0.846143	0.423071	−	0.906096i	$-0.360952\pi$
$61$	24590.5	0.846143	0.423071	+	0.906096i	$0.360952\pi$
$62$	0	0
$63$	56241.2	1.78527
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	34789.8	0.946813	0.473407	−	0.880844i	$-0.343024\pi$
$67$	34789.8	0.946813	0.473407	+	0.880844i	$0.343024\pi$
$68$	0	0
$69$	1601.54	0.0404962
$70$	0	0
$71$	−9033.72	−0.212677	−0.106339	−	0.994330i	$-0.533913\pi$
$71$	−9033.72	−0.212677	−0.106339	+	0.994330i	$0.533913\pi$
$72$	0	0
$73$	30590.1	0.671853	0.335926	−	0.941888i	$-0.390951\pi$
$73$	30590.1	0.671853	0.335926	+	0.941888i	$0.390951\pi$
$74$	0	0
$75$	−329.559	−0.00676519
$76$	0	0
$77$	−108070.	−2.07720
$78$	0	0
$79$	14290.7	0.257624	0.128812	−	0.991669i	$-0.458884\pi$
$79$	14290.7	0.257624	0.128812	+	0.991669i	$0.458884\pi$
$80$	0	0
$81$	58846.4	0.996569
$82$	0	0
$83$	−65535.3	−1.04419	−0.522096	−	0.852887i	$-0.674850\pi$
$83$	−65535.3	−1.04419	−0.522096	+	0.852887i	$0.674850\pi$
$84$	0	0
$85$	−22072.6	−0.331365
$86$	0	0
$87$	−3310.47	−0.0468912
$88$	0	0
$89$	−85188.3	−1.14000	−0.570000	−	0.821644i	$-0.693057\pi$
$89$	−85188.3	−1.14000	−0.570000	+	0.821644i	$0.693057\pi$
$90$	0	0
$91$	−39159.1	−0.495711
$92$	0	0
$93$	−2264.24	−0.0271466
$94$	0	0
$95$	4182.71	0.0475498
$96$	0	0
$97$	51071.6	0.551124	0.275562	−	0.961283i	$-0.411136\pi$
$97$	51071.6	0.551124	0.275562	+	0.961283i	$0.411136\pi$
$98$	0	0
$99$	−113206.	−1.16086
$100$	0	0
$101$	−149777.	−1.46098	−0.730488	−	0.682926i	$-0.760707\pi$
$101$	−149777.	−1.46098	−0.730488	+	0.682926i	$0.760707\pi$
$102$	0	0
$103$	−79535.6	−0.738700	−0.369350	−	0.929290i	$-0.620420\pi$
$103$	−79535.6	−0.738700	−0.369350	+	0.929290i	$0.620420\pi$
$104$	0	0
$105$	−3054.49	−0.0270374
$106$	0	0
$107$	−15175.1	−0.128136	−0.0640682	−	0.997946i	$-0.520408\pi$
$107$	−15175.1	−0.128136	−0.0640682	+	0.997946i	$0.520408\pi$
$108$	0	0
$109$	−112925.	−0.910384	−0.455192	−	0.890393i	$-0.650429\pi$
$109$	−112925.	−0.910384	−0.455192	+	0.890393i	$0.650429\pi$
$110$	0	0
$111$	3841.83	0.0295958
$112$	0	0
$113$	−30361.7	−0.223682	−0.111841	−	0.993726i	$-0.535675\pi$
$113$	−30361.7	−0.223682	−0.111841	+	0.993726i	$0.535675\pi$
$114$	0	0
$115$	75931.8	0.535401
$116$	0	0
$117$	−41020.0	−0.277033
$118$	0	0
$119$	−204578.	−1.32432
$120$	0	0
$121$	56479.5	0.350693
$122$	0	0
$123$	−10010.4	−0.0596605
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−4509.41	−0.0248091	−0.0124045	−	0.999923i	$-0.503949\pi$
$127$	−4509.41	−0.0248091	−0.0124045	+	0.999923i	$0.503949\pi$
$128$	0	0
$129$	7538.04	0.0398827
$130$	0	0
$131$	149727.	0.762292	0.381146	−	0.924515i	$-0.375529\pi$
$131$	149727.	0.762292	0.381146	+	0.924515i	$0.375529\pi$
$132$	0	0
$133$	38767.1	0.190035
$134$	0	0
$135$	−6402.96	−0.0302375
$136$	0	0
$137$	158516.	0.721558	0.360779	−	0.932651i	$-0.382511\pi$
$137$	158516.	0.721558	0.360779	+	0.932651i	$0.382511\pi$
$138$	0	0
$139$	−69849.6	−0.306639	−0.153319	−	0.988177i	$-0.548996\pi$
$139$	−69849.6	−0.306639	−0.153319	+	0.988177i	$0.548996\pi$
$140$	0	0
$141$	4416.33	0.0187074
$142$	0	0
$143$	78821.9	0.322334
$144$	0	0
$145$	−156955.	−0.619950
$146$	0	0
$147$	−19448.1	−0.0742306
$148$	0	0
$149$	−56811.4	−0.209638	−0.104819	−	0.994491i	$-0.533426\pi$
$149$	−56811.4	−0.209638	−0.104819	+	0.994491i	$0.533426\pi$
$150$	0	0
$151$	−25593.3	−0.0913447	−0.0456724	−	0.998956i	$-0.514543\pi$
$151$	−25593.3	−0.0913447	−0.0456724	+	0.998956i	$0.514543\pi$
$152$	0	0
$153$	−214301.	−0.740107
$154$	0	0
$155$	−107352.	−0.358906
$156$	0	0
$157$	−464261.	−1.50319	−0.751593	−	0.659627i	$-0.770714\pi$
$157$	−464261.	−1.50319	−0.751593	+	0.659627i	$0.770714\pi$
$158$	0	0
$159$	−10833.8	−0.0339850
$160$	0	0
$161$	703768.	2.13976
$162$	0	0
$163$	−233236.	−0.687584	−0.343792	−	0.939046i	$-0.611712\pi$
$163$	−233236.	−0.687584	−0.343792	+	0.939046i	$0.611712\pi$
$164$	0	0
$165$	6148.27	0.0175810
$166$	0	0
$167$	5473.80	0.0151879	0.00759395	−	0.999971i	$-0.497583\pi$
$167$	5473.80	0.0151879	0.00759395	+	0.999971i	$0.497583\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	40609.4	0.106203
$172$	0	0
$173$	−583869.	−1.48320	−0.741601	−	0.670841i	$-0.765933\pi$
$173$	−583869.	−1.48320	−0.741601	+	0.670841i	$0.765933\pi$
$174$	0	0
$175$	−144819.	−0.357463
$176$	0	0
$177$	−62.2238	−0.000149288 0
$178$	0	0
$179$	−495004.	−1.15472	−0.577359	−	0.816490i	$-0.695917\pi$
$179$	−495004.	−1.15472	−0.577359	+	0.816490i	$0.695917\pi$
$180$	0	0
$181$	648193.	1.47065	0.735323	−	0.677717i	$-0.237031\pi$
$181$	648193.	1.47065	0.735323	+	0.677717i	$0.237031\pi$
$182$	0	0
$183$	−12966.4	−0.0286216
$184$	0	0
$185$	182148.	0.391287
$186$	0	0
$187$	411789.	0.861133
$188$	0	0
$189$	−59345.3	−0.120846
$190$	0	0
$191$	882509.	1.75039	0.875197	−	0.483767i	$-0.160732\pi$
$191$	882509.	1.75039	0.875197	+	0.483767i	$0.160732\pi$
$192$	0	0
$193$	693680.	1.34050	0.670249	−	0.742137i	$-0.266187\pi$
$193$	693680.	1.34050	0.670249	+	0.742137i	$0.266187\pi$
$194$	0	0
$195$	2227.82	0.00419559
$196$	0	0
$197$	−190019.	−0.348844	−0.174422	−	0.984671i	$-0.555806\pi$
$197$	−190019.	−0.348844	−0.174422	+	0.984671i	$0.555806\pi$
$198$	0	0
$199$	141002.	0.252403	0.126201	−	0.992005i	$-0.459721\pi$
$199$	141002.	0.252403	0.126201	+	0.992005i	$0.459721\pi$
$200$	0	0
$201$	−18344.4	−0.0320268
$202$	0	0
$203$	−1.45473e6	−2.47766
$204$	0	0
$205$	−474610.	−0.788773
$206$	0	0
$207$	737213.	1.19582
$208$	0	0
$209$	−78032.9	−0.123570
$210$	0	0
$211$	417539.	0.645640	0.322820	−	0.946460i	$-0.395369\pi$
$211$	417539.	0.645640	0.322820	+	0.946460i	$0.395369\pi$
$212$	0	0
$213$	4763.43	0.00719400
$214$	0	0
$215$	357393.	0.527290
$216$	0	0
$217$	−994982.	−1.43439
$218$	0	0
$219$	−16130.0	−0.0227260
$220$	0	0
$221$	149211.	0.205504
$222$	0	0
$223$	−289938.	−0.390429	−0.195215	−	0.980761i	$-0.562540\pi$
$223$	−289938.	−0.390429	−0.195215	+	0.980761i	$0.562540\pi$
$224$	0	0
$225$	−151701.	−0.199771
$226$	0	0
$227$	1.05748e6	1.36210	0.681049	−	0.732238i	$-0.261524\pi$
$227$	1.05748e6	1.36210	0.681049	+	0.732238i	$0.261524\pi$
$228$	0	0
$229$	−827977.	−1.04335	−0.521674	−	0.853145i	$-0.674692\pi$
$229$	−827977.	−1.04335	−0.521674	+	0.853145i	$0.674692\pi$
$230$	0	0
$231$	56984.8	0.0702633
$232$	0	0
$233$	339021.	0.409107	0.204554	−	0.978855i	$-0.434426\pi$
$233$	339021.	0.409107	0.204554	+	0.978855i	$0.434426\pi$
$234$	0	0
$235$	209386.	0.247331
$236$	0	0
$237$	−7535.42	−0.00871438
$238$	0	0
$239$	−1.66517e6	−1.88567	−0.942833	−	0.333267i	$-0.891849\pi$
$239$	−1.66517e6	−1.88567	−0.942833	+	0.333267i	$0.891849\pi$
$240$	0	0
$241$	−9484.31	−0.0105187	−0.00525937	−	0.999986i	$-0.501674\pi$
$241$	−9484.31	−0.0105187	−0.00525937	+	0.999986i	$0.501674\pi$
$242$	0	0
$243$	−93266.1	−0.101323
$244$	0	0
$245$	−922069.	−0.981405
$246$	0	0
$247$	−28275.1	−0.0294891
$248$	0	0
$249$	34556.4	0.0353207
$250$	0	0
$251$	−872880.	−0.874521	−0.437261	−	0.899335i	$-0.644051\pi$
$251$	−872880.	−0.874521	−0.437261	+	0.899335i	$0.644051\pi$
$252$	0	0
$253$	−1.41659e6	−1.39137
$254$	0	0
$255$	11638.8	0.0112087
$256$	0	0
$257$	−207242.	−0.195724	−0.0978622	−	0.995200i	$-0.531200\pi$
$257$	−207242.	−0.195724	−0.0978622	+	0.995200i	$0.531200\pi$
$258$	0	0
$259$	1.68823e6	1.56380
$260$	0	0
$261$	−1.52386e6	−1.38466
$262$	0	0
$263$	−2.22009e6	−1.97916	−0.989581	−	0.143976i	$-0.954011\pi$
$263$	−2.22009e6	−1.97916	−0.989581	+	0.143976i	$0.954011\pi$
$264$	0	0
$265$	−513650.	−0.449317
$266$	0	0
$267$	44919.3	0.0385616
$268$	0	0
$269$	939034.	0.791226	0.395613	−	0.918417i	$-0.370532\pi$
$269$	939034.	0.791226	0.395613	+	0.918417i	$0.370532\pi$
$270$	0	0
$271$	−404835.	−0.334853	−0.167427	−	0.985885i	$-0.553546\pi$
$271$	−404835.	−0.334853	−0.167427	+	0.985885i	$0.553546\pi$
$272$	0	0
$273$	20648.3	0.0167679
$274$	0	0
$275$	291501.	0.232439
$276$	0	0
$277$	484637.	0.379505	0.189752	−	0.981832i	$-0.439231\pi$
$277$	484637.	0.379505	0.189752	+	0.981832i	$0.439231\pi$
$278$	0	0
$279$	−1.04227e6	−0.801620
$280$	0	0
$281$	−2.25453e6	−1.70330	−0.851650	−	0.524111i	$-0.824398\pi$
$281$	−2.25453e6	−1.70330	−0.851650	+	0.524111i	$0.824398\pi$
$282$	0	0
$283$	271952.	0.201849	0.100924	−	0.994894i	$-0.467820\pi$
$283$	271952.	0.201849	0.100924	+	0.994894i	$0.467820\pi$
$284$	0	0
$285$	−2205.52	−0.00160842
$286$	0	0
$287$	−4.39889e6	−3.15237
$288$	0	0
$289$	−640335.	−0.450986
$290$	0	0
$291$	−26929.7	−0.0186423
$292$	0	0
$293$	−1.81834e6	−1.23739	−0.618694	−	0.785632i	$-0.712338\pi$
$293$	−1.81834e6	−1.23739	−0.618694	+	0.785632i	$0.712338\pi$
$294$	0	0
$295$	−2950.15	−0.00197374
$296$	0	0
$297$	119454.	0.0785796
$298$	0	0
$299$	−513299.	−0.332042
$300$	0	0
$301$	3.31247e6	2.10734
$302$	0	0
$303$	78976.7	0.0494188
$304$	0	0
$305$	−614764.	−0.378406
$306$	0	0
$307$	2.29926e6	1.39233	0.696166	−	0.717881i	$-0.254888\pi$
$307$	2.29926e6	1.39233	0.696166	+	0.717881i	$0.254888\pi$
$308$	0	0
$309$	41938.6	0.0249872
$310$	0	0
$311$	1.59531e6	0.935286	0.467643	−	0.883917i	$-0.345103\pi$
$311$	1.59531e6	0.935286	0.467643	+	0.883917i	$0.345103\pi$
$312$	0	0
$313$	−985636.	−0.568664	−0.284332	−	0.958726i	$-0.591772\pi$
$313$	−985636.	−0.568664	−0.284332	+	0.958726i	$0.591772\pi$
$314$	0	0
$315$	−1.40603e6	−0.798396
$316$	0	0
$317$	−1.82666e6	−1.02096	−0.510480	−	0.859890i	$-0.670532\pi$
$317$	−1.82666e6	−1.02096	−0.510480	+	0.859890i	$0.670532\pi$
$318$	0	0
$319$	2.92817e6	1.61109
$320$	0	0
$321$	8001.74	0.00433433
$322$	0	0
$323$	−147718.	−0.0787818
$324$	0	0
$325$	105625.	0.0554700
$326$	0	0
$327$	59544.8	0.0307946
$328$	0	0
$329$	1.94068e6	0.988472
$330$	0	0
$331$	−2.10290e6	−1.05499	−0.527495	−	0.849558i	$-0.676869\pi$
$331$	−2.10290e6	−1.05499	−0.527495	+	0.849558i	$0.676869\pi$
$332$	0	0
$333$	1.76846e6	0.873944
$334$	0	0
$335$	−869744.	−0.423428
$336$	0	0
$337$	−121239.	−0.0581526	−0.0290763	−	0.999577i	$-0.509257\pi$
$337$	−121239.	−0.0581526	−0.0290763	+	0.999577i	$0.509257\pi$
$338$	0	0
$339$	16009.6	0.00756624
$340$	0	0
$341$	2.00276e6	0.932704
$342$	0	0
$343$	−4.65176e6	−2.13492
$344$	0	0
$345$	−40038.4	−0.0181104
$346$	0	0
$347$	−388296.	−0.173117	−0.0865584	−	0.996247i	$-0.527587\pi$
$347$	−388296.	−0.173117	−0.0865584	+	0.996247i	$0.527587\pi$
$348$	0	0
$349$	1.54959e6	0.681011	0.340506	−	0.940242i	$-0.389402\pi$
$349$	1.54959e6	0.681011	0.340506	+	0.940242i	$0.389402\pi$
$350$	0	0
$351$	43284.0	0.0187525
$352$	0	0
$353$	592554.	0.253100	0.126550	−	0.991960i	$-0.459610\pi$
$353$	592554.	0.253100	0.126550	+	0.991960i	$0.459610\pi$
$354$	0	0
$355$	225843.	0.0951121
$356$	0	0
$357$	107873.	0.0447963
$358$	0	0
$359$	−589168.	−0.241270	−0.120635	−	0.992697i	$-0.538493\pi$
$359$	−589168.	−0.241270	−0.120635	+	0.992697i	$0.538493\pi$
$360$	0	0
$361$	−2.44811e6	−0.988695
$362$	0	0
$363$	−29781.3	−0.0118625
$364$	0	0
$365$	−764753.	−0.300462
$366$	0	0
$367$	−3.28606e6	−1.27353	−0.636766	−	0.771057i	$-0.719728\pi$
$367$	−3.28606e6	−1.27353	−0.636766	+	0.771057i	$0.719728\pi$
$368$	0	0
$369$	−4.60793e6	−1.76173
$370$	0	0
$371$	−4.76072e6	−1.79572
$372$	0	0
$373$	4.94295e6	1.83956	0.919781	−	0.392432i	$-0.128366\pi$
$373$	4.94295e6	1.83956	0.919781	+	0.392432i	$0.128366\pi$
$374$	0	0
$375$	8238.97	0.00302548
$376$	0	0
$377$	1.06102e6	0.384476
$378$	0	0
$379$	588891.	0.210590	0.105295	−	0.994441i	$-0.466421\pi$
$379$	588891.	0.210590	0.105295	+	0.994441i	$0.466421\pi$
$380$	0	0
$381$	2377.79	0.000839190 0
$382$	0	0
$383$	2.27183e6	0.791370	0.395685	−	0.918386i	$-0.370507\pi$
$383$	2.27183e6	0.791370	0.395685	+	0.918386i	$0.370507\pi$
$384$	0	0
$385$	2.70175e6	0.928954
$386$	0	0
$387$	3.46988e6	1.17771
$388$	0	0
$389$	−5.34339e6	−1.79037	−0.895185	−	0.445694i	$-0.852957\pi$
$389$	−5.34339e6	−1.79037	−0.895185	+	0.445694i	$0.852957\pi$
$390$	0	0
$391$	−2.68163e6	−0.887067
$392$	0	0
$393$	−78950.1	−0.0257852
$394$	0	0
$395$	−357268.	−0.115213
$396$	0	0
$397$	5.07081e6	1.61473	0.807367	−	0.590050i	$-0.200892\pi$
$397$	5.07081e6	1.61473	0.807367	+	0.590050i	$0.200892\pi$
$398$	0	0
$399$	−20441.7	−0.00642812
$400$	0	0
$401$	4.70142e6	1.46005	0.730026	−	0.683420i	$-0.239508\pi$
$401$	4.70142e6	1.46005	0.730026	+	0.683420i	$0.239508\pi$
$402$	0	0
$403$	725699.	0.222584
$404$	0	0
$405$	−1.47116e6	−0.445679
$406$	0	0
$407$	−3.39817e6	−1.01686
$408$	0	0
$409$	−3.15579e6	−0.932824	−0.466412	−	0.884568i	$-0.654454\pi$
$409$	−3.15579e6	−0.932824	−0.466412	+	0.884568i	$0.654454\pi$
$410$	0	0
$411$	−83584.4	−0.0244074
$412$	0	0
$413$	−27343.2	−0.00788814
$414$	0	0
$415$	1.63838e6	0.466977
$416$	0	0
$417$	36831.3	0.0103723
$418$	0	0
$419$	682165.	0.189825	0.0949127	−	0.995486i	$-0.469743\pi$
$419$	682165.	0.189825	0.0949127	+	0.995486i	$0.469743\pi$
$420$	0	0
$421$	−706461.	−0.194260	−0.0971299	−	0.995272i	$-0.530966\pi$
$421$	−706461.	−0.194260	−0.0971299	+	0.995272i	$0.530966\pi$
$422$	0	0
$423$	2.03291e6	0.552416
$424$	0	0
$425$	551816.	0.148191
$426$	0	0
$427$	−5.69789e6	−1.51232
$428$	0	0
$429$	−41562.3	−0.0109033
$430$	0	0
$431$	−4.89113e6	−1.26828	−0.634142	−	0.773217i	$-0.718646\pi$
$431$	−4.89113e6	−1.26828	−0.634142	+	0.773217i	$0.718646\pi$
$432$	0	0
$433$	−4.98945e6	−1.27889	−0.639444	−	0.768837i	$-0.720836\pi$
$433$	−4.98945e6	−1.27889	−0.639444	+	0.768837i	$0.720836\pi$
$434$	0	0
$435$	82761.7	0.0209704
$436$	0	0
$437$	508162.	0.127291
$438$	0	0
$439$	5.44947e6	1.34956	0.674782	−	0.738017i	$-0.264238\pi$
$439$	5.44947e6	1.34956	0.674782	+	0.738017i	$0.264238\pi$
$440$	0	0
$441$	−8.95225e6	−2.19198
$442$	0	0
$443$	−1.00675e6	−0.243732	−0.121866	−	0.992547i	$-0.538888\pi$
$443$	−1.00675e6	−0.243732	−0.121866	+	0.992547i	$0.538888\pi$
$444$	0	0
$445$	2.12971e6	0.509824
$446$	0	0
$447$	29956.3	0.00709119
$448$	0	0
$449$	200413.	0.0469148	0.0234574	−	0.999725i	$-0.492533\pi$
$449$	200413.	0.0469148	0.0234574	+	0.999725i	$0.492533\pi$
$450$	0	0
$451$	8.85436e6	2.04982
$452$	0	0
$453$	13495.2	0.00308982
$454$	0	0
$455$	978977.	0.221689
$456$	0	0
$457$	647622.	0.145054	0.0725272	−	0.997366i	$-0.476894\pi$
$457$	647622.	0.145054	0.0725272	+	0.997366i	$0.476894\pi$
$458$	0	0
$459$	226128.	0.0500983
$460$	0	0
$461$	−2.89528e6	−0.634509	−0.317254	−	0.948340i	$-0.602761\pi$
$461$	−2.89528e6	−0.634509	−0.317254	+	0.948340i	$0.602761\pi$
$462$	0	0
$463$	−6.96850e6	−1.51073	−0.755364	−	0.655305i	$-0.772540\pi$
$463$	−6.96850e6	−1.51073	−0.755364	+	0.655305i	$0.772540\pi$
$464$	0	0
$465$	56606.0	0.0121403
$466$	0	0
$467$	−1.46098e6	−0.309994	−0.154997	−	0.987915i	$-0.549537\pi$
$467$	−1.46098e6	−0.309994	−0.154997	+	0.987915i	$0.549537\pi$
$468$	0	0
$469$	−8.06115e6	−1.69225
$470$	0	0
$471$	244802.	0.0508467
$472$	0	0
$473$	−6.66754e6	−1.37029
$474$	0	0
$475$	−104568.	−0.0212649
$476$	0	0
$477$	−4.98697e6	−1.00355
$478$	0	0
$479$	6.63657e6	1.32162	0.660808	−	0.750555i	$-0.270214\pi$
$479$	6.63657e6	1.32162	0.660808	+	0.750555i	$0.270214\pi$
$480$	0	0
$481$	−1.23132e6	−0.242666
$482$	0	0
$483$	−371093.	−0.0723793
$484$	0	0
$485$	−1.27679e6	−0.246470
$486$	0	0
$487$	6.08683e6	1.16297	0.581486	−	0.813557i	$-0.302472\pi$
$487$	6.08683e6	1.16297	0.581486	+	0.813557i	$0.302472\pi$
$488$	0	0
$489$	122984.	0.0232582
$490$	0	0
$491$	−1.91765e6	−0.358976	−0.179488	−	0.983760i	$-0.557444\pi$
$491$	−1.91765e6	−0.358976	−0.179488	+	0.983760i	$0.557444\pi$
$492$	0	0
$493$	5.54307e6	1.02715
$494$	0	0
$495$	2.83015e6	0.519154
$496$	0	0
$497$	2.09321e6	0.380121
$498$	0	0
$499$	−4.96658e6	−0.892907	−0.446453	−	0.894807i	$-0.647313\pi$
$499$	−4.96658e6	−0.892907	−0.446453	+	0.894807i	$0.647313\pi$
$500$	0	0
$501$	−2886.30	−0.000513745 0
$502$	0	0
$503$	2.49704e6	0.440054	0.220027	−	0.975494i	$-0.429385\pi$
$503$	2.49704e6	0.440054	0.220027	+	0.975494i	$0.429385\pi$
$504$	0	0
$505$	3.74444e6	0.653368
$506$	0	0
$507$	−15060.0	−0.00260199
$508$	0	0
$509$	8.69294e6	1.48721	0.743605	−	0.668619i	$-0.233114\pi$
$509$	8.69294e6	1.48721	0.743605	+	0.668619i	$0.233114\pi$
$510$	0	0
$511$	−7.08805e6	−1.20081
$512$	0	0
$513$	−42850.7	−0.00718894
$514$	0	0
$515$	1.98839e6	0.330357
$516$	0	0
$517$	−3.90632e6	−0.642750
$518$	0	0
$519$	307871.	0.0501707
$520$	0	0
$521$	−1.81970e6	−0.293702	−0.146851	−	0.989159i	$-0.546914\pi$
$521$	−1.81970e6	−0.293702	−0.146851	+	0.989159i	$0.546914\pi$
$522$	0	0
$523$	187933.	0.0300435	0.0150217	−	0.999887i	$-0.495218\pi$
$523$	187933.	0.0300435	0.0150217	+	0.999887i	$0.495218\pi$
$524$	0	0
$525$	76362.2	0.0120915
$526$	0	0
$527$	3.79126e6	0.594644
$528$	0	0
$529$	2.78869e6	0.433272
$530$	0	0
$531$	−28642.6	−0.00440836
$532$	0	0
$533$	3.20836e6	0.489177
$534$	0	0
$535$	379378.	0.0573043
$536$	0	0
$537$	261013.	0.0390594
$538$	0	0
$539$	1.72022e7	2.55042
$540$	0	0
$541$	−7.97826e6	−1.17197	−0.585983	−	0.810324i	$-0.699291\pi$
$541$	−7.97826e6	−1.17197	−0.585983	+	0.810324i	$0.699291\pi$
$542$	0	0
$543$	−341788.	−0.0497460
$544$	0	0
$545$	2.82313e6	0.407136
$546$	0	0
$547$	−4.31881e6	−0.617157	−0.308579	−	0.951199i	$-0.599853\pi$
$547$	−4.31881e6	−0.617157	−0.308579	+	0.951199i	$0.599853\pi$
$548$	0	0
$549$	−5.96866e6	−0.845174
$550$	0	0
$551$	−1.05040e6	−0.147392
$552$	0	0
$553$	−3.31131e6	−0.460455
$554$	0	0
$555$	−96045.7	−0.0132357
$556$	0	0
$557$	−9.23520e6	−1.26127	−0.630636	−	0.776079i	$-0.717206\pi$
$557$	−9.23520e6	−1.26127	−0.630636	+	0.776079i	$0.717206\pi$
$558$	0	0
$559$	−2.41597e6	−0.327011
$560$	0	0
$561$	−217134.	−0.0291286
$562$	0	0
$563$	8.89163e6	1.18225	0.591126	−	0.806579i	$-0.298683\pi$
$563$	8.89163e6	1.18225	0.591126	+	0.806579i	$0.298683\pi$
$564$	0	0
$565$	759044.	0.100034
$566$	0	0
$567$	−1.36353e7	−1.78118
$568$	0	0
$569$	−7.25901e6	−0.939932	−0.469966	−	0.882684i	$-0.655734\pi$
$569$	−7.25901e6	−0.939932	−0.469966	+	0.882684i	$0.655734\pi$
$570$	0	0
$571$	3.72513e6	0.478136	0.239068	−	0.971003i	$-0.423158\pi$
$571$	3.72513e6	0.478136	0.239068	+	0.971003i	$0.423158\pi$
$572$	0	0
$573$	−465342.	−0.0592087
$574$	0	0
$575$	−1.89830e6	−0.239439
$576$	0	0
$577$	1.17093e6	0.146417	0.0732086	−	0.997317i	$-0.476676\pi$
$577$	1.17093e6	0.146417	0.0732086	+	0.997317i	$0.476676\pi$
$578$	0	0
$579$	−365773.	−0.0453436
$580$	0	0
$581$	1.51852e7	1.86630
$582$	0	0
$583$	9.58269e6	1.16766
$584$	0	0
$585$	1.02550e6	0.123893
$586$	0	0
$587$	−8.45511e6	−1.01280	−0.506400	−	0.862298i	$-0.669024\pi$
$587$	−8.45511e6	−1.01280	−0.506400	+	0.862298i	$0.669024\pi$
$588$	0	0
$589$	−718435.	−0.0853295
$590$	0	0
$591$	100196.	0.0118000
$592$	0	0
$593$	1.34570e7	1.57149	0.785746	−	0.618550i	$-0.212279\pi$
$593$	1.34570e7	1.57149	0.785746	+	0.618550i	$0.212279\pi$
$594$	0	0
$595$	5.11446e6	0.592253
$596$	0	0
$597$	−74349.7	−0.00853775
$598$	0	0
$599$	−9.72026e6	−1.10691	−0.553453	−	0.832880i	$-0.686690\pi$
$599$	−9.72026e6	−1.10691	−0.553453	+	0.832880i	$0.686690\pi$
$600$	0	0
$601$	1.52752e7	1.72505	0.862525	−	0.506014i	$-0.168882\pi$
$601$	1.52752e7	1.72505	0.862525	+	0.506014i	$0.168882\pi$
$602$	0	0
$603$	−8.44424e6	−0.945730
$604$	0	0
$605$	−1.41199e6	−0.156835
$606$	0	0
$607$	3.44708e6	0.379735	0.189867	−	0.981810i	$-0.439194\pi$
$607$	3.44708e6	0.379735	0.189867	+	0.981810i	$0.439194\pi$
$608$	0	0
$609$	767070.	0.0838092
$610$	0	0
$611$	−1.41545e6	−0.153388
$612$	0	0
$613$	5.45977e6	0.586844	0.293422	−	0.955983i	$-0.405206\pi$
$613$	5.45977e6	0.586844	0.293422	+	0.955983i	$0.405206\pi$
$614$	0	0
$615$	250259.	0.0266810
$616$	0	0
$617$	8.48025e6	0.896801	0.448400	−	0.893833i	$-0.351994\pi$
$617$	8.48025e6	0.896801	0.448400	+	0.893833i	$0.351994\pi$
$618$	0	0
$619$	−8.43977e6	−0.885328	−0.442664	−	0.896688i	$-0.645966\pi$
$619$	−8.43977e6	−0.885328	−0.442664	+	0.896688i	$0.645966\pi$
$620$	0	0
$621$	−777901.	−0.0809460
$622$	0	0
$623$	1.97390e7	2.03754
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	41146.3	0.00417986
$628$	0	0
$629$	−6.43279e6	−0.648295
$630$	0	0
$631$	1.46417e7	1.46392	0.731962	−	0.681346i	$-0.238605\pi$
$631$	1.46417e7	1.46392	0.731962	+	0.681346i	$0.238605\pi$
$632$	0	0
$633$	−220166.	−0.0218394
$634$	0	0
$635$	112735.	0.0110950
$636$	0	0
$637$	6.23319e6	0.608641
$638$	0	0
$639$	2.19268e6	0.212434
$640$	0	0
$641$	−8.49886e6	−0.816988	−0.408494	−	0.912761i	$-0.633946\pi$
$641$	−8.49886e6	−0.816988	−0.408494	+	0.912761i	$0.633946\pi$
$642$	0	0
$643$	−1.07051e7	−1.02108	−0.510542	−	0.859853i	$-0.670555\pi$
$643$	−1.07051e7	−1.02108	−0.510542	+	0.859853i	$0.670555\pi$
$644$	0	0
$645$	−188451.	−0.0178361
$646$	0	0
$647$	−1.94772e7	−1.82922	−0.914609	−	0.404340i	$-0.867501\pi$
$647$	−1.94772e7	−1.82922	−0.914609	+	0.404340i	$0.867501\pi$
$648$	0	0
$649$	55038.2	0.00512923
$650$	0	0
$651$	524648.	0.0485194
$652$	0	0
$653$	1.36819e7	1.25563	0.627817	−	0.778361i	$-0.283949\pi$
$653$	1.36819e7	1.25563	0.627817	+	0.778361i	$0.283949\pi$
$654$	0	0
$655$	−3.74317e6	−0.340908
$656$	0	0
$657$	−7.42490e6	−0.671084
$658$	0	0
$659$	7.40016e6	0.663785	0.331893	−	0.943317i	$-0.392313\pi$
$659$	7.40016e6	0.663785	0.331893	+	0.943317i	$0.392313\pi$
$660$	0	0
$661$	1.88654e7	1.67943	0.839715	−	0.543028i	$-0.182722\pi$
$661$	1.88654e7	1.67943	0.839715	+	0.543028i	$0.182722\pi$
$662$	0	0
$663$	−78678.1	−0.00695136
$664$	0	0
$665$	−969178.	−0.0849864
$666$	0	0
$667$	−1.90687e7	−1.65961
$668$	0	0
$669$	152882.	0.0132066
$670$	0	0
$671$	1.14691e7	0.983381
$672$	0	0
$673$	−1.68803e7	−1.43662	−0.718311	−	0.695722i	$-0.755085\pi$
$673$	−1.68803e7	−1.43662	−0.718311	+	0.695722i	$0.755085\pi$
$674$	0	0
$675$	160074.	0.0135226
$676$	0	0
$677$	−1.07388e7	−0.900504	−0.450252	−	0.892902i	$-0.648666\pi$
$677$	−1.07388e7	−0.900504	−0.450252	+	0.892902i	$0.648666\pi$
$678$	0	0
$679$	−1.18338e7	−0.985032
$680$	0	0
$681$	−557604.	−0.0460742
$682$	0	0
$683$	3.08525e6	0.253069	0.126535	−	0.991962i	$-0.459615\pi$
$683$	3.08525e6	0.253069	0.126535	+	0.991962i	$0.459615\pi$
$684$	0	0
$685$	−3.96290e6	−0.322690
$686$	0	0
$687$	436587.	0.0352922
$688$	0	0
$689$	3.47227e6	0.278654
$690$	0	0
$691$	1.14232e7	0.910110	0.455055	−	0.890463i	$-0.349620\pi$
$691$	1.14232e7	0.910110	0.455055	+	0.890463i	$0.349620\pi$
$692$	0	0
$693$	2.62310e7	2.07483
$694$	0	0
$695$	1.74624e6	0.137133
$696$	0	0
$697$	1.67614e7	1.30686
$698$	0	0
$699$	−178764.	−0.0138384
$700$	0	0
$701$	−1.28448e7	−0.987264	−0.493632	−	0.869671i	$-0.664331\pi$
$701$	−1.28448e7	−0.987264	−0.493632	+	0.869671i	$0.664331\pi$
$702$	0	0
$703$	1.21900e6	0.0930282
$704$	0	0
$705$	−110408.	−0.00836620
$706$	0	0
$707$	3.47050e7	2.61122
$708$	0	0
$709$	1.28235e7	0.958055	0.479028	−	0.877800i	$-0.340989\pi$
$709$	1.28235e7	0.958055	0.479028	+	0.877800i	$0.340989\pi$
$710$	0	0
$711$	−3.46867e6	−0.257329
$712$	0	0
$713$	−1.30423e7	−0.960793
$714$	0	0
$715$	−1.97055e6	−0.144152
$716$	0	0
$717$	878036.	0.0637844
$718$	0	0
$719$	−7.59185e6	−0.547678	−0.273839	−	0.961776i	$-0.588294\pi$
$719$	−7.59185e6	−0.547678	−0.273839	+	0.961776i	$0.588294\pi$
$720$	0	0
$721$	1.84292e7	1.32029
$722$	0	0
$723$	5001.02	0.000355806 0
$724$	0	0
$725$	3.92389e6	0.277250
$726$	0	0
$727$	−2.64094e6	−0.185320	−0.0926600	−	0.995698i	$-0.529537\pi$
$727$	−2.64094e6	−0.185320	−0.0926600	+	0.995698i	$0.529537\pi$
$728$	0	0
$729$	−1.42505e7	−0.993141
$730$	0	0
$731$	−1.26218e7	−0.873628
$732$	0	0
$733$	−2.36492e6	−0.162576	−0.0812881	−	0.996691i	$-0.525903\pi$
$733$	−2.36492e6	−0.162576	−0.0812881	+	0.996691i	$0.525903\pi$
$734$	0	0
$735$	486201.	0.0331969
$736$	0	0
$737$	1.62260e7	1.10038
$738$	0	0
$739$	6.44734e6	0.434280	0.217140	−	0.976140i	$-0.430327\pi$
$739$	6.44734e6	0.434280	0.217140	+	0.976140i	$0.430327\pi$
$740$	0	0
$741$	14909.3	0.000997498 0
$742$	0	0
$743$	−7.54303e6	−0.501273	−0.250636	−	0.968081i	$-0.580640\pi$
$743$	−7.54303e6	−0.501273	−0.250636	+	0.968081i	$0.580640\pi$
$744$	0	0
$745$	1.42028e6	0.0937529
$746$	0	0
$747$	1.59069e7	1.04300
$748$	0	0
$749$	3.51623e6	0.229020
$750$	0	0
$751$	2.33272e7	1.50925	0.754627	−	0.656154i	$-0.227818\pi$
$751$	2.33272e7	1.50925	0.754627	+	0.656154i	$0.227818\pi$
$752$	0	0
$753$	460264.	0.0295815
$754$	0	0
$755$	639832.	0.0408506
$756$	0	0
$757$	−2.12561e7	−1.34817	−0.674085	−	0.738654i	$-0.735462\pi$
$757$	−2.12561e7	−1.34817	−0.674085	+	0.738654i	$0.735462\pi$
$758$	0	0
$759$	746959.	0.0470644
$760$	0	0
$761$	−5.46782e6	−0.342257	−0.171129	−	0.985249i	$-0.554741\pi$
$761$	−5.46782e6	−0.342257	−0.171129	+	0.985249i	$0.554741\pi$
$762$	0	0
$763$	2.61660e7	1.62714
$764$	0	0
$765$	5.35751e6	0.330986
$766$	0	0
$767$	19943.0	0.00122406
$768$	0	0
$769$	1.12535e7	0.686234	0.343117	−	0.939293i	$-0.388517\pi$
$769$	1.12535e7	0.686234	0.343117	+	0.939293i	$0.388517\pi$
$770$	0	0
$771$	109277.	0.00662056
$772$	0	0
$773$	1.16249e7	0.699747	0.349873	−	0.936797i	$-0.386225\pi$
$773$	1.16249e7	0.699747	0.349873	+	0.936797i	$0.386225\pi$
$774$	0	0
$775$	2.68380e6	0.160508
$776$	0	0
$777$	−890192.	−0.0528970
$778$	0	0
$779$	−3.17625e6	−0.187530
$780$	0	0
$781$	−4.21334e6	−0.247172
$782$	0	0
$783$	1.60797e6	0.0937287
$784$	0	0
$785$	1.16065e7	0.672245
$786$	0	0
$787$	2.48768e7	1.43172	0.715859	−	0.698245i	$-0.246035\pi$
$787$	2.48768e7	1.43172	0.715859	+	0.698245i	$0.246035\pi$
$788$	0	0
$789$	1.17064e6	0.0669470
$790$	0	0
$791$	7.03514e6	0.399789
$792$	0	0
$793$	4.15580e6	0.234678
$794$	0	0
$795$	270845.	0.0151986
$796$	0	0
$797$	−1.88084e7	−1.04883	−0.524416	−	0.851462i	$-0.675716\pi$
$797$	−1.88084e7	−1.04883	−0.524416	+	0.851462i	$0.675716\pi$
$798$	0	0
$799$	−7.39473e6	−0.409784
$800$	0	0
$801$	2.06771e7	1.13870
$802$	0	0
$803$	1.42673e7	0.780823
$804$	0	0
$805$	−1.75942e7	−0.956930
$806$	0	0
$807$	−495147.	−0.0267640
$808$	0	0
$809$	1.38160e7	0.742184	0.371092	−	0.928596i	$-0.378983\pi$
$809$	1.38160e7	0.742184	0.371092	+	0.928596i	$0.378983\pi$
$810$	0	0
$811$	−3.04514e7	−1.62575	−0.812877	−	0.582435i	$-0.802100\pi$
$811$	−3.04514e7	−1.62575	−0.812877	+	0.582435i	$0.802100\pi$
$812$	0	0
$813$	213467.	0.0113267
$814$	0	0
$815$	5.83089e6	0.307497
$816$	0	0
$817$	2.39179e6	0.125363
$818$	0	0
$819$	9.50477e6	0.495144
$820$	0	0
$821$	−5.01920e6	−0.259882	−0.129941	−	0.991522i	$-0.541479\pi$
$821$	−5.01920e6	−0.259882	−0.129941	+	0.991522i	$0.541479\pi$
$822$	0	0
$823$	−3.70421e7	−1.90632	−0.953162	−	0.302462i	$-0.902192\pi$
$823$	−3.70421e7	−1.90632	−0.953162	+	0.302462i	$0.902192\pi$
$824$	0	0
$825$	−153707.	−0.00786245
$826$	0	0
$827$	1.52758e7	0.776674	0.388337	−	0.921517i	$-0.373050\pi$
$827$	1.52758e7	0.776674	0.388337	+	0.921517i	$0.373050\pi$
$828$	0	0
$829$	1.13644e7	0.574327	0.287163	−	0.957882i	$-0.407288\pi$
$829$	1.13644e7	0.574327	0.287163	+	0.957882i	$0.407288\pi$
$830$	0	0
$831$	−255546.	−0.0128371
$832$	0	0
$833$	3.25640e7	1.62602
$834$	0	0
$835$	−136845.	−0.00679223
$836$	0	0
$837$	1.09979e6	0.0542621
$838$	0	0
$839$	−2.16145e7	−1.06008	−0.530042	−	0.847971i	$-0.677824\pi$
$839$	−2.16145e7	−1.06008	−0.530042	+	0.847971i	$0.677824\pi$
$840$	0	0
$841$	1.89049e7	0.921688
$842$	0	0
$843$	1.18880e6	0.0576157
$844$	0	0
$845$	−714025.	−0.0344010
$846$	0	0
$847$	−1.30869e7	−0.626799
$848$	0	0
$849$	−143399.	−0.00682772
$850$	0	0
$851$	2.21294e7	1.04748
$852$	0	0
$853$	−3.22268e7	−1.51651	−0.758253	−	0.651960i	$-0.773947\pi$
$853$	−3.22268e7	−1.51651	−0.758253	+	0.651960i	$0.773947\pi$
$854$	0	0
$855$	−1.01524e6	−0.0474954
$856$	0	0
$857$	−2.07998e7	−0.967404	−0.483702	−	0.875233i	$-0.660708\pi$
$857$	−2.07998e7	−0.967404	−0.483702	+	0.875233i	$0.660708\pi$
$858$	0	0
$859$	−2.66909e7	−1.23418	−0.617092	−	0.786891i	$-0.711689\pi$
$859$	−2.66909e7	−1.23418	−0.617092	+	0.786891i	$0.711689\pi$
$860$	0	0
$861$	2.31951e6	0.106632
$862$	0	0
$863$	6.44627e6	0.294633	0.147316	−	0.989089i	$-0.452936\pi$
$863$	6.44627e6	0.294633	0.147316	+	0.989089i	$0.452936\pi$
$864$	0	0
$865$	1.45967e7	0.663308
$866$	0	0
$867$	337645.	0.0152550
$868$	0	0
$869$	6.66522e6	0.299409
$870$	0	0
$871$	5.87947e6	0.262599
$872$	0	0
$873$	−1.23962e7	−0.550494
$874$	0	0
$875$	3.62048e6	0.159862
$876$	0	0
$877$	−2.93830e7	−1.29002	−0.645012	−	0.764173i	$-0.723148\pi$
$877$	−2.93830e7	−1.29002	−0.645012	+	0.764173i	$0.723148\pi$
$878$	0	0
$879$	958799.	0.0418558
$880$	0	0
$881$	6.33454e6	0.274964	0.137482	−	0.990504i	$-0.456099\pi$
$881$	6.33454e6	0.274964	0.137482	+	0.990504i	$0.456099\pi$
$882$	0	0
$883$	1.53999e7	0.664686	0.332343	−	0.943159i	$-0.392161\pi$
$883$	1.53999e7	0.664686	0.332343	+	0.943159i	$0.392161\pi$
$884$	0	0
$885$	1555.60	6.67634e−5 0
$886$	0	0
$887$	823263.	0.0351342	0.0175671	−	0.999846i	$-0.494408\pi$
$887$	823263.	0.0351342	0.0175671	+	0.999846i	$0.494408\pi$
$888$	0	0
$889$	1.04488e6	0.0443416
$890$	0	0
$891$	2.74461e7	1.15821
$892$	0	0
$893$	1.40128e6	0.0588027
$894$	0	0
$895$	1.23751e7	0.516406
$896$	0	0
$897$	270660.	0.0112316
$898$	0	0
$899$	2.69591e7	1.11252
$900$	0	0
$901$	1.81402e7	0.744440
$902$	0	0
$903$	−1.74664e6	−0.0712828
$904$	0	0
$905$	−1.62048e7	−0.657693
$906$	0	0
$907$	3.29987e6	0.133192	0.0665961	−	0.997780i	$-0.478786\pi$
$907$	3.29987e6	0.133192	0.0665961	+	0.997780i	$0.478786\pi$
$908$	0	0
$909$	3.63543e7	1.45930
$910$	0	0
$911$	3.58804e7	1.43239	0.716196	−	0.697900i	$-0.245882\pi$
$911$	3.58804e7	1.43239	0.716196	+	0.697900i	$0.245882\pi$
$912$	0	0
$913$	−3.05658e7	−1.21355
$914$	0	0
$915$	324161.	0.0127999
$916$	0	0
$917$	−3.46933e7	−1.36246
$918$	0	0
$919$	−4.29501e7	−1.67755	−0.838776	−	0.544477i	$-0.816728\pi$
$919$	−4.29501e7	−1.67755	−0.838776	+	0.544477i	$0.816728\pi$
$920$	0	0
$921$	−1.21239e6	−0.0470969
$922$	0	0
$923$	−1.52670e6	−0.0589860
$924$	0	0
$925$	−4.55371e6	−0.174989
$926$	0	0
$927$	1.93050e7	0.737855
$928$	0	0
$929$	−4.75514e6	−0.180769	−0.0903845	−	0.995907i	$-0.528810\pi$
$929$	−4.75514e6	−0.180769	−0.0903845	+	0.995907i	$0.528810\pi$
$930$	0	0
$931$	−6.17080e6	−0.233328
$932$	0	0
$933$	−841198.	−0.0316369
$934$	0	0
$935$	−1.02947e7	−0.385110
$936$	0	0
$937$	4.86133e7	1.80886	0.904432	−	0.426617i	$-0.140295\pi$
$937$	4.86133e7	1.80886	0.904432	+	0.426617i	$0.140295\pi$
$938$	0	0
$939$	519720.	0.0192356
$940$	0	0
$941$	825014.	0.0303730	0.0151865	−	0.999885i	$-0.495166\pi$
$941$	825014.	0.0303730	0.0151865	+	0.999885i	$0.495166\pi$
$942$	0	0
$943$	−5.76608e7	−2.11155
$944$	0	0
$945$	1.48363e6	0.0540439
$946$	0	0
$947$	2.90200e7	1.05153	0.525766	−	0.850630i	$-0.323779\pi$
$947$	2.90200e7	1.05153	0.525766	+	0.850630i	$0.323779\pi$
$948$	0	0
$949$	5.16973e6	0.186338
$950$	0	0
$951$	963185.	0.0345349
$952$	0	0
$953$	−1.85279e7	−0.660838	−0.330419	−	0.943834i	$-0.607190\pi$
$953$	−1.85279e7	−0.660838	−0.330419	+	0.943834i	$0.607190\pi$
$954$	0	0
$955$	−2.20627e7	−0.782800
$956$	0	0
$957$	−1.54401e6	−0.0544966
$958$	0	0
$959$	−3.67298e7	−1.28965
$960$	0	0
$961$	−1.01901e7	−0.355933
$962$	0	0
$963$	3.68333e6	0.127990
$964$	0	0
$965$	−1.73420e7	−0.599489
$966$	0	0
$967$	−1.43655e7	−0.494032	−0.247016	−	0.969011i	$-0.579450\pi$
$967$	−1.43655e7	−0.494032	−0.247016	+	0.969011i	$0.579450\pi$
$968$	0	0
$969$	77890.6	0.00266487
$970$	0	0
$971$	2.66582e7	0.907368	0.453684	−	0.891163i	$-0.350110\pi$
$971$	2.66582e7	0.907368	0.453684	+	0.891163i	$0.350110\pi$
$972$	0	0
$973$	1.61849e7	0.548059
$974$	0	0
$975$	−55695.4	−0.00187632
$976$	0	0
$977$	−1.84613e6	−0.0618767	−0.0309383	−	0.999521i	$-0.509850\pi$
$977$	−1.84613e6	−0.0618767	−0.0309383	+	0.999521i	$0.509850\pi$
$978$	0	0
$979$	−3.97320e7	−1.32490
$980$	0	0
$981$	2.74094e7	0.909342
$982$	0	0
$983$	3.85226e7	1.27154	0.635772	−	0.771877i	$-0.280682\pi$
$983$	3.85226e7	1.27154	0.635772	+	0.771877i	$0.280682\pi$
$984$	0	0
$985$	4.75048e6	0.156008
$986$	0	0
$987$	−1.02331e6	−0.0334360
$988$	0	0
$989$	4.34200e7	1.41156
$990$	0	0
$991$	−4.70175e7	−1.52081	−0.760406	−	0.649448i	$-0.775000\pi$
$991$	−4.70175e7	−1.52081	−0.760406	+	0.649448i	$0.775000\pi$
$992$	0	0
$993$	1.10885e6	0.0356860
$994$	0	0
$995$	−3.52506e6	−0.112878
$996$	0	0
$997$	1.02621e6	0.0326961	0.0163481	−	0.999866i	$-0.494796\pi$
$997$	1.02621e6	0.0326961	0.0163481	+	0.999866i	$0.494796\pi$
$998$	0	0
$999$	−1.86606e6	−0.0591578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.q.1.4		6
4.3	odd	2		65.6.a.d.1.1	✓	6
12.11	even	2		585.6.a.m.1.6		6
20.3	even	4		325.6.b.g.274.12		12
20.7	even	4		325.6.b.g.274.1		12
20.19	odd	2		325.6.a.g.1.6		6
52.51	odd	2		845.6.a.h.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.1	✓	6	4.3	odd	2
325.6.a.g.1.6		6	20.19	odd	2
325.6.b.g.274.1		12	20.7	even	4
325.6.b.g.274.12		12	20.3	even	4
585.6.a.m.1.6		6	12.11	even	2
845.6.a.h.1.6		6	52.51	odd	2
1040.6.a.q.1.4		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q-0.527294 q^{3} -25.0000 q^{5} -231.710 q^{7} -242.722 q^{9} +466.402 q^{11} +169.000 q^{13} +13.1824 q^{15} +882.905 q^{17} -167.308 q^{19} +122.180 q^{21} -3037.27 q^{23} +625.000 q^{25} +256.118 q^{27} +6278.22 q^{29} +4294.08 q^{31} -245.931 q^{33} +5792.76 q^{35} -7285.93 q^{37} -89.1127 q^{39} +18984.4 q^{41} -14295.7 q^{43} +6068.05 q^{45} -8375.45 q^{47} +36882.8 q^{49} -465.551 q^{51} +20546.0 q^{53} -11660.0 q^{55} +88.2207 q^{57} +118.006 q^{59} +24590.5 q^{61} +56241.2 q^{63} -4225.00 q^{65} +34789.8 q^{67} +1601.54 q^{69} -9033.72 q^{71} +30590.1 q^{73} -329.559 q^{75} -108070. q^{77} +14290.7 q^{79} +58846.4 q^{81} -65535.3 q^{83} -22072.6 q^{85} -3310.47 q^{87} -85188.3 q^{89} -39159.1 q^{91} -2264.24 q^{93} +4182.71 q^{95} +51071.6 q^{97} -113206. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 38 q^{3} - 150 q^{5} - 220 q^{7} + 518 q^{9} + 170 q^{11} + 1014 q^{13} + 950 q^{15} + 728 q^{17} - 1218 q^{19} - 396 q^{21} - 8954 q^{23} + 3750 q^{25} - 13112 q^{27} + 8364 q^{29} - 2862 q^{31}+ \cdots + 32270 q^{99}+O(q^{100}) \) 6 * q - 38 * q^3 - 150 * q^5 - 220 * q^7 + 518 * q^9 + 170 * q^11 + 1014 * q^13 + 950 * q^15 + 728 * q^17 - 1218 * q^19 - 396 * q^21 - 8954 * q^23 + 3750 * q^25 - 13112 * q^27 + 8364 * q^29 - 2862 * q^31 - 3772 * q^33 + 5500 * q^35 + 13840 * q^37 - 6422 * q^39 + 2248 * q^41 - 3822 * q^43 - 12950 * q^45 + 10860 * q^47 + 85902 * q^49 + 55028 * q^51 + 6584 * q^53 - 4250 * q^55 + 11692 * q^57 + 28874 * q^59 + 972 * q^61 + 125236 * q^63 - 25350 * q^65 - 60040 * q^67 + 153088 * q^69 + 74174 * q^71 + 48960 * q^73 - 23750 * q^75 + 231588 * q^77 - 214508 * q^79 + 116006 * q^81 - 123260 * q^83 - 18200 * q^85 - 228144 * q^87 + 94028 * q^89 - 37180 * q^91 + 192892 * q^93 + 30450 * q^95 + 246284 * q^97 + 32270 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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