LMFDB - Embedded newform 1040.6.a.ba.1.3

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			1.3
Root			$15.6369$ 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)

$f(q)$	$=$	$q-11.6369 q^{3} +25.0000 q^{5} +99.8440 q^{7} -107.583 q^{9} -79.5242 q^{11} -169.000 q^{13} -290.922 q^{15} +297.235 q^{17} -2334.52 q^{19} -1161.87 q^{21} +1929.41 q^{23} +625.000 q^{25} +4079.69 q^{27} +1301.96 q^{29} -6959.16 q^{31} +925.412 q^{33} +2496.10 q^{35} -6365.45 q^{37} +1966.63 q^{39} -12767.4 q^{41} +14362.3 q^{43} -2689.59 q^{45} +29207.6 q^{47} -6838.17 q^{49} -3458.88 q^{51} +19699.8 q^{53} -1988.11 q^{55} +27166.4 q^{57} +20403.1 q^{59} -9434.91 q^{61} -10741.6 q^{63} -4225.00 q^{65} -4728.34 q^{67} -22452.3 q^{69} +36629.7 q^{71} -63544.1 q^{73} -7273.04 q^{75} -7940.02 q^{77} -70756.8 q^{79} -21332.0 q^{81} +80248.1 q^{83} +7430.88 q^{85} -15150.8 q^{87} -49045.6 q^{89} -16873.6 q^{91} +80982.8 q^{93} -58362.9 q^{95} -40294.0 q^{97} +8555.49 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−11.6369	−0.746505	−0.373252	−	0.927730i	$-0.621757\pi$
$3$	−11.6369	−0.746505	−0.373252	+	0.927730i	$0.621757\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	99.8440	0.770153	0.385076	−	0.922885i	$-0.374175\pi$
$7$	99.8440	0.770153	0.385076	+	0.922885i	$0.374175\pi$
$8$	0	0
$9$	−107.583	−0.442730
$10$	0	0
$11$	−79.5242	−0.198161	−0.0990804	−	0.995079i	$-0.531590\pi$
$11$	−79.5242	−0.198161	−0.0990804	+	0.995079i	$0.531590\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	−290.922	−0.333847
$16$	0	0
$17$	297.235	0.249447	0.124723	−	0.992192i	$-0.460196\pi$
$17$	297.235	0.249447	0.124723	+	0.992192i	$0.460196\pi$
$18$	0	0
$19$	−2334.52	−1.48359	−0.741793	−	0.670629i	$-0.766024\pi$
$19$	−2334.52	−1.48359	−0.741793	+	0.670629i	$0.766024\pi$
$20$	0	0
$21$	−1161.87	−0.574923
$22$	0	0
$23$	1929.41	0.760510	0.380255	−	0.924882i	$-0.375836\pi$
$23$	1929.41	0.760510	0.380255	+	0.924882i	$0.375836\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	4079.69	1.07701
$28$	0	0
$29$	1301.96	0.287478	0.143739	−	0.989616i	$-0.454087\pi$
$29$	1301.96	0.287478	0.143739	+	0.989616i	$0.454087\pi$
$30$	0	0
$31$	−6959.16	−1.30063	−0.650313	−	0.759666i	$-0.725362\pi$
$31$	−6959.16	−1.30063	−0.650313	+	0.759666i	$0.725362\pi$
$32$	0	0
$33$	925.412	0.147928
$34$	0	0
$35$	2496.10	0.344423
$36$	0	0
$37$	−6365.45	−0.764407	−0.382204	−	0.924078i	$-0.624835\pi$
$37$	−6365.45	−0.764407	−0.382204	+	0.924078i	$0.624835\pi$
$38$	0	0
$39$	1966.63	0.207043
$40$	0	0
$41$	−12767.4	−1.18615	−0.593077	−	0.805146i	$-0.702087\pi$
$41$	−12767.4	−1.18615	−0.593077	+	0.805146i	$0.702087\pi$
$42$	0	0
$43$	14362.3	1.18455	0.592273	−	0.805738i	$-0.298231\pi$
$43$	14362.3	1.18455	0.592273	+	0.805738i	$0.298231\pi$
$44$	0	0
$45$	−2689.59	−0.197995
$46$	0	0
$47$	29207.6	1.92864	0.964318	−	0.264745i	$-0.0852877\pi$
$47$	29207.6	1.92864	0.964318	+	0.264745i	$0.0852877\pi$
$48$	0	0
$49$	−6838.17	−0.406865
$50$	0	0
$51$	−3458.88	−0.186213
$52$	0	0
$53$	19699.8	0.963325	0.481663	−	0.876357i	$-0.340033\pi$
$53$	19699.8	0.963325	0.481663	+	0.876357i	$0.340033\pi$
$54$	0	0
$55$	−1988.11	−0.0886202
$56$	0	0
$57$	27166.4	1.10750
$58$	0	0
$59$	20403.1	0.763074	0.381537	−	0.924354i	$-0.375395\pi$
$59$	20403.1	0.763074	0.381537	+	0.924354i	$0.375395\pi$
$60$	0	0
$61$	−9434.91	−0.324648	−0.162324	−	0.986737i	$-0.551899\pi$
$61$	−9434.91	−0.324648	−0.162324	+	0.986737i	$0.551899\pi$
$62$	0	0
$63$	−10741.6	−0.340970
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	−4728.34	−0.128683	−0.0643416	−	0.997928i	$-0.520495\pi$
$67$	−4728.34	−0.128683	−0.0643416	+	0.997928i	$0.520495\pi$
$68$	0	0
$69$	−22452.3	−0.567725
$70$	0	0
$71$	36629.7	0.862358	0.431179	−	0.902266i	$-0.358098\pi$
$71$	36629.7	0.862358	0.431179	+	0.902266i	$0.358098\pi$
$72$	0	0
$73$	−63544.1	−1.39562	−0.697812	−	0.716281i	$-0.745843\pi$
$73$	−63544.1	−1.39562	−0.697812	+	0.716281i	$0.745843\pi$
$74$	0	0
$75$	−7273.04	−0.149301
$76$	0	0
$77$	−7940.02	−0.152614
$78$	0	0
$79$	−70756.8	−1.27556	−0.637779	−	0.770219i	$-0.720147\pi$
$79$	−70756.8	−1.27556	−0.637779	+	0.770219i	$0.720147\pi$
$80$	0	0
$81$	−21332.0	−0.361259
$82$	0	0
$83$	80248.1	1.27861	0.639307	−	0.768951i	$-0.279221\pi$
$83$	80248.1	1.27861	0.639307	+	0.768951i	$0.279221\pi$
$84$	0	0
$85$	7430.88	0.111556
$86$	0	0
$87$	−15150.8	−0.214604
$88$	0	0
$89$	−49045.6	−0.656334	−0.328167	−	0.944620i	$-0.606431\pi$
$89$	−49045.6	−0.656334	−0.328167	+	0.944620i	$0.606431\pi$
$90$	0	0
$91$	−16873.6	−0.213602
$92$	0	0
$93$	80982.8	0.970924
$94$	0	0
$95$	−58362.9	−0.663480
$96$	0	0
$97$	−40294.0	−0.434821	−0.217411	−	0.976080i	$-0.569761\pi$
$97$	−40294.0	−0.434821	−0.217411	+	0.976080i	$0.569761\pi$
$98$	0	0
$99$	8555.49	0.0877318
$100$	0	0
$101$	51176.6	0.499193	0.249596	−	0.968350i	$-0.419702\pi$
$101$	51176.6	0.499193	0.249596	+	0.968350i	$0.419702\pi$
$102$	0	0
$103$	136900.	1.27149	0.635743	−	0.771901i	$-0.280694\pi$
$103$	136900.	1.27149	0.635743	+	0.771901i	$0.280694\pi$
$104$	0	0
$105$	−29046.8	−0.257113
$106$	0	0
$107$	−213659.	−1.80411	−0.902053	−	0.431625i	$-0.857940\pi$
$107$	−213659.	−1.80411	−0.902053	+	0.431625i	$0.857940\pi$
$108$	0	0
$109$	63847.0	0.514724	0.257362	−	0.966315i	$-0.417147\pi$
$109$	63847.0	0.514724	0.257362	+	0.966315i	$0.417147\pi$
$110$	0	0
$111$	74073.9	0.570634
$112$	0	0
$113$	−44289.7	−0.326292	−0.163146	−	0.986602i	$-0.552164\pi$
$113$	−44289.7	−0.326292	−0.163146	+	0.986602i	$0.552164\pi$
$114$	0	0
$115$	48235.3	0.340111
$116$	0	0
$117$	18181.6	0.122791
$118$	0	0
$119$	29677.1	0.192112
$120$	0	0
$121$	−154727.	−0.960732
$122$	0	0
$123$	148572.	0.885470
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	2513.86	0.0138303	0.00691516	−	0.999976i	$-0.497799\pi$
$127$	2513.86	0.0138303	0.00691516	+	0.999976i	$0.497799\pi$
$128$	0	0
$129$	−167132.	−0.884269
$130$	0	0
$131$	−4491.85	−0.0228690	−0.0114345	−	0.999935i	$-0.503640\pi$
$131$	−4491.85	−0.0228690	−0.0114345	+	0.999935i	$0.503640\pi$
$132$	0	0
$133$	−233087.	−1.14259
$134$	0	0
$135$	101992.	0.481651
$136$	0	0
$137$	44888.6	0.204331	0.102166	−	0.994767i	$-0.467423\pi$
$137$	44888.6	0.204331	0.102166	+	0.994767i	$0.467423\pi$
$138$	0	0
$139$	−168464.	−0.739555	−0.369778	−	0.929120i	$-0.620566\pi$
$139$	−168464.	−0.739555	−0.369778	+	0.929120i	$0.620566\pi$
$140$	0	0
$141$	−339884.	−1.43974
$142$	0	0
$143$	13439.6	0.0549599
$144$	0	0
$145$	32549.1	0.128564
$146$	0	0
$147$	79574.9	0.303726
$148$	0	0
$149$	37368.6	0.137893	0.0689463	−	0.997620i	$-0.478036\pi$
$149$	37368.6	0.137893	0.0689463	+	0.997620i	$0.478036\pi$
$150$	0	0
$151$	143539.	0.512302	0.256151	−	0.966637i	$-0.417546\pi$
$151$	143539.	0.512302	0.256151	+	0.966637i	$0.417546\pi$
$152$	0	0
$153$	−31977.6	−0.110438
$154$	0	0
$155$	−173979.	−0.581658
$156$	0	0
$157$	−403810.	−1.30746	−0.653729	−	0.756729i	$-0.726796\pi$
$157$	−403810.	−1.30746	−0.653729	+	0.756729i	$0.726796\pi$
$158$	0	0
$159$	−229244.	−0.719127
$160$	0	0
$161$	192640.	0.585709
$162$	0	0
$163$	−51521.0	−0.151885	−0.0759426	−	0.997112i	$-0.524197\pi$
$163$	−51521.0	−0.151885	−0.0759426	+	0.997112i	$0.524197\pi$
$164$	0	0
$165$	23135.3	0.0661554
$166$	0	0
$167$	557787.	1.54767	0.773833	−	0.633389i	$-0.218337\pi$
$167$	557787.	1.54767	0.773833	+	0.633389i	$0.218337\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	251155.	0.656829
$172$	0	0
$173$	398379.	1.01200	0.506001	−	0.862533i	$-0.331123\pi$
$173$	398379.	1.01200	0.506001	+	0.862533i	$0.331123\pi$
$174$	0	0
$175$	62402.5	0.154031
$176$	0	0
$177$	−237428.	−0.569638
$178$	0	0
$179$	408069.	0.951921	0.475961	−	0.879467i	$-0.342100\pi$
$179$	408069.	0.951921	0.475961	+	0.879467i	$0.342100\pi$
$180$	0	0
$181$	−343160.	−0.778575	−0.389288	−	0.921116i	$-0.627279\pi$
$181$	−343160.	−0.778575	−0.389288	+	0.921116i	$0.627279\pi$
$182$	0	0
$183$	109793.	0.242352
$184$	0	0
$185$	−159136.	−0.341853
$186$	0	0
$187$	−23637.4	−0.0494306
$188$	0	0
$189$	407333.	0.829459
$190$	0	0
$191$	96103.0	0.190613	0.0953067	−	0.995448i	$-0.469617\pi$
$191$	96103.0	0.190613	0.0953067	+	0.995448i	$0.469617\pi$
$192$	0	0
$193$	562867.	1.08771	0.543855	−	0.839179i	$-0.316964\pi$
$193$	562867.	1.08771	0.543855	+	0.839179i	$0.316964\pi$
$194$	0	0
$195$	49165.7	0.0925925
$196$	0	0
$197$	828181.	1.52041	0.760203	−	0.649685i	$-0.225099\pi$
$197$	828181.	1.52041	0.760203	+	0.649685i	$0.225099\pi$
$198$	0	0
$199$	712208.	1.27489	0.637447	−	0.770494i	$-0.279990\pi$
$199$	712208.	1.27489	0.637447	+	0.770494i	$0.279990\pi$
$200$	0	0
$201$	55023.0	0.0960626
$202$	0	0
$203$	129993.	0.221402
$204$	0	0
$205$	−319184.	−0.530464
$206$	0	0
$207$	−207573.	−0.336701
$208$	0	0
$209$	185651.	0.293989
$210$	0	0
$211$	838056.	1.29589	0.647943	−	0.761689i	$-0.275629\pi$
$211$	838056.	1.29589	0.647943	+	0.761689i	$0.275629\pi$
$212$	0	0
$213$	−426255.	−0.643755
$214$	0	0
$215$	359057.	0.529745
$216$	0	0
$217$	−694831.	−1.00168
$218$	0	0
$219$	739454.	1.04184
$220$	0	0
$221$	−50232.7	−0.0691841
$222$	0	0
$223$	850811.	1.14570	0.572850	−	0.819660i	$-0.305838\pi$
$223$	850811.	1.14570	0.572850	+	0.819660i	$0.305838\pi$
$224$	0	0
$225$	−67239.7	−0.0885461
$226$	0	0
$227$	−731485.	−0.942196	−0.471098	−	0.882081i	$-0.656142\pi$
$227$	−731485.	−0.942196	−0.471098	+	0.882081i	$0.656142\pi$
$228$	0	0
$229$	1.16789e6	1.47168	0.735839	−	0.677157i	$-0.236788\pi$
$229$	1.16789e6	1.47168	0.735839	+	0.677157i	$0.236788\pi$
$230$	0	0
$231$	92396.9	0.113927
$232$	0	0
$233$	−122118.	−0.147364	−0.0736820	−	0.997282i	$-0.523475\pi$
$233$	−122118.	−0.147364	−0.0736820	+	0.997282i	$0.523475\pi$
$234$	0	0
$235$	730189.	0.862513
$236$	0	0
$237$	823387.	0.952211
$238$	0	0
$239$	363719.	0.411880	0.205940	−	0.978565i	$-0.433975\pi$
$239$	363719.	0.411880	0.205940	+	0.978565i	$0.433975\pi$
$240$	0	0
$241$	502433.	0.557232	0.278616	−	0.960403i	$-0.410124\pi$
$241$	502433.	0.557232	0.278616	+	0.960403i	$0.410124\pi$
$242$	0	0
$243$	−743127.	−0.807323
$244$	0	0
$245$	−170954.	−0.181955
$246$	0	0
$247$	394533.	0.411473
$248$	0	0
$249$	−933836.	−0.954492
$250$	0	0
$251$	−24195.1	−0.0242406	−0.0121203	−	0.999927i	$-0.503858\pi$
$251$	−24195.1	−0.0242406	−0.0121203	+	0.999927i	$0.503858\pi$
$252$	0	0
$253$	−153435.	−0.150703
$254$	0	0
$255$	−86472.1	−0.0832771
$256$	0	0
$257$	−712227.	−0.672644	−0.336322	−	0.941747i	$-0.609183\pi$
$257$	−712227.	−0.672644	−0.336322	+	0.941747i	$0.609183\pi$
$258$	0	0
$259$	−635552.	−0.588710
$260$	0	0
$261$	−140070.	−0.127275
$262$	0	0
$263$	2.02312e6	1.80357	0.901786	−	0.432184i	$-0.142257\pi$
$263$	2.02312e6	1.80357	0.901786	+	0.432184i	$0.142257\pi$
$264$	0	0
$265$	492496.	0.430812
$266$	0	0
$267$	570736.	0.489956
$268$	0	0
$269$	1.18862e6	1.00153	0.500765	−	0.865583i	$-0.333052\pi$
$269$	1.18862e6	1.00153	0.500765	+	0.865583i	$0.333052\pi$
$270$	0	0
$271$	−1.11841e6	−0.925075	−0.462538	−	0.886600i	$-0.653061\pi$
$271$	−1.11841e6	−0.925075	−0.462538	+	0.886600i	$0.653061\pi$
$272$	0	0
$273$	196356.	0.159455
$274$	0	0
$275$	−49702.6	−0.0396322
$276$	0	0
$277$	2.34259e6	1.83441	0.917207	−	0.398412i	$-0.130439\pi$
$277$	2.34259e6	1.83441	0.917207	+	0.398412i	$0.130439\pi$
$278$	0	0
$279$	748691.	0.575827
$280$	0	0
$281$	−2.20864e6	−1.66862	−0.834312	−	0.551293i	$-0.814135\pi$
$281$	−2.20864e6	−1.66862	−0.834312	+	0.551293i	$0.814135\pi$
$282$	0	0
$283$	2.59344e6	1.92491	0.962453	−	0.271449i	$-0.0875028\pi$
$283$	2.59344e6	1.92491	0.962453	+	0.271449i	$0.0875028\pi$
$284$	0	0
$285$	679161.	0.495291
$286$	0	0
$287$	−1.27474e6	−0.913520
$288$	0	0
$289$	−1.33151e6	−0.937776
$290$	0	0
$291$	468895.	0.324596
$292$	0	0
$293$	1.80233e6	1.22649	0.613247	−	0.789891i	$-0.289863\pi$
$293$	1.80233e6	1.22649	0.613247	+	0.789891i	$0.289863\pi$
$294$	0	0
$295$	510078.	0.341257
$296$	0	0
$297$	−324434.	−0.213420
$298$	0	0
$299$	−326070.	−0.210928
$300$	0	0
$301$	1.43399e6	0.912281
$302$	0	0
$303$	−595535.	−0.372650
$304$	0	0
$305$	−235873.	−0.145187
$306$	0	0
$307$	1.36590e6	0.827130	0.413565	−	0.910474i	$-0.364283\pi$
$307$	1.36590e6	0.827130	0.413565	+	0.910474i	$0.364283\pi$
$308$	0	0
$309$	−1.59309e6	−0.949171
$310$	0	0
$311$	1.45020e6	0.850212	0.425106	−	0.905144i	$-0.360237\pi$
$311$	1.45020e6	0.850212	0.425106	+	0.905144i	$0.360237\pi$
$312$	0	0
$313$	3.40623e6	1.96523	0.982613	−	0.185664i	$-0.0594437\pi$
$313$	3.40623e6	1.96523	0.982613	+	0.185664i	$0.0594437\pi$
$314$	0	0
$315$	−268539.	−0.152486
$316$	0	0
$317$	1.61062e6	0.900213	0.450107	−	0.892975i	$-0.351386\pi$
$317$	1.61062e6	0.900213	0.450107	+	0.892975i	$0.351386\pi$
$318$	0	0
$319$	−103538.	−0.0569668
$320$	0	0
$321$	2.48632e6	1.34677
$322$	0	0
$323$	−693900.	−0.370076
$324$	0	0
$325$	−105625.	−0.0554700
$326$	0	0
$327$	−742979.	−0.384244
$328$	0	0
$329$	2.91620e6	1.48535
$330$	0	0
$331$	−721558.	−0.361994	−0.180997	−	0.983484i	$-0.557932\pi$
$331$	−721558.	−0.361994	−0.180997	+	0.983484i	$0.557932\pi$
$332$	0	0
$333$	684817.	0.338426
$334$	0	0
$335$	−118209.	−0.0575489
$336$	0	0
$337$	297341.	0.142620	0.0713100	−	0.997454i	$-0.477282\pi$
$337$	297341.	0.142620	0.0713100	+	0.997454i	$0.477282\pi$
$338$	0	0
$339$	515393.	0.243579
$340$	0	0
$341$	553422.	0.257733
$342$	0	0
$343$	−2.36083e6	−1.08350
$344$	0	0
$345$	−561307.	−0.253894
$346$	0	0
$347$	−2.99246e6	−1.33415	−0.667075	−	0.744991i	$-0.732454\pi$
$347$	−2.99246e6	−1.33415	−0.667075	+	0.744991i	$0.732454\pi$
$348$	0	0
$349$	−1.00575e6	−0.442006	−0.221003	−	0.975273i	$-0.570933\pi$
$349$	−1.00575e6	−0.442006	−0.221003	+	0.975273i	$0.570933\pi$
$350$	0	0
$351$	−689468.	−0.298708
$352$	0	0
$353$	−233541.	−0.0997531	−0.0498765	−	0.998755i	$-0.515883\pi$
$353$	−233541.	−0.0997531	−0.0498765	+	0.998755i	$0.515883\pi$
$354$	0	0
$355$	915743.	0.385658
$356$	0	0
$357$	−345349.	−0.143413
$358$	0	0
$359$	−1.32413e6	−0.542244	−0.271122	−	0.962545i	$-0.587395\pi$
$359$	−1.32413e6	−0.542244	−0.271122	+	0.962545i	$0.587395\pi$
$360$	0	0
$361$	2.97387e6	1.20103
$362$	0	0
$363$	1.80054e6	0.717191
$364$	0	0
$365$	−1.58860e6	−0.624142
$366$	0	0
$367$	−1.80239e6	−0.698527	−0.349264	−	0.937024i	$-0.613568\pi$
$367$	−1.80239e6	−0.698527	−0.349264	+	0.937024i	$0.613568\pi$
$368$	0	0
$369$	1.37356e6	0.525147
$370$	0	0
$371$	1.96691e6	0.741908
$372$	0	0
$373$	−2.97050e6	−1.10550	−0.552749	−	0.833348i	$-0.686421\pi$
$373$	−2.97050e6	−1.10550	−0.552749	+	0.833348i	$0.686421\pi$
$374$	0	0
$375$	−181826.	−0.0667694
$376$	0	0
$377$	−220032.	−0.0797320
$378$	0	0
$379$	−2.82214e6	−1.00921	−0.504603	−	0.863351i	$-0.668361\pi$
$379$	−2.82214e6	−1.00921	−0.504603	+	0.863351i	$0.668361\pi$
$380$	0	0
$381$	−29253.5	−0.0103244
$382$	0	0
$383$	185821.	0.0647289	0.0323645	−	0.999476i	$-0.489696\pi$
$383$	185821.	0.0647289	0.0323645	+	0.999476i	$0.489696\pi$
$384$	0	0
$385$	−198500.	−0.0682511
$386$	0	0
$387$	−1.54514e6	−0.524434
$388$	0	0
$389$	4.38840e6	1.47039	0.735195	−	0.677856i	$-0.237091\pi$
$389$	4.38840e6	1.47039	0.735195	+	0.677856i	$0.237091\pi$
$390$	0	0
$391$	573489.	0.189707
$392$	0	0
$393$	52271.0	0.0170718
$394$	0	0
$395$	−1.76892e6	−0.570447
$396$	0	0
$397$	5.42974e6	1.72903	0.864515	−	0.502606i	$-0.167625\pi$
$397$	5.42974e6	1.72903	0.864515	+	0.502606i	$0.167625\pi$
$398$	0	0
$399$	2.71241e6	0.852948
$400$	0	0
$401$	−88016.8	−0.0273341	−0.0136670	−	0.999907i	$-0.504350\pi$
$401$	−88016.8	−0.0273341	−0.0136670	+	0.999907i	$0.504350\pi$
$402$	0	0
$403$	1.17610e6	0.360729
$404$	0	0
$405$	−533300.	−0.161560
$406$	0	0
$407$	506207.	0.151476
$408$	0	0
$409$	5.12199e6	1.51402	0.757008	−	0.653406i	$-0.226661\pi$
$409$	5.12199e6	1.51402	0.757008	+	0.653406i	$0.226661\pi$
$410$	0	0
$411$	−522362.	−0.152534
$412$	0	0
$413$	2.03713e6	0.587683
$414$	0	0
$415$	2.00620e6	0.571814
$416$	0	0
$417$	1.96039e6	0.552082
$418$	0	0
$419$	−883128.	−0.245747	−0.122874	−	0.992422i	$-0.539211\pi$
$419$	−883128.	−0.245747	−0.122874	+	0.992422i	$0.539211\pi$
$420$	0	0
$421$	5.44582e6	1.49747	0.748735	−	0.662869i	$-0.230661\pi$
$421$	5.44582e6	1.49747	0.748735	+	0.662869i	$0.230661\pi$
$422$	0	0
$423$	−3.14225e6	−0.853866
$424$	0	0
$425$	185772.	0.0498893
$426$	0	0
$427$	−942019.	−0.250029
$428$	0	0
$429$	−156395.	−0.0410278
$430$	0	0
$431$	−5.24811e6	−1.36085	−0.680424	−	0.732819i	$-0.738204\pi$
$431$	−5.24811e6	−1.36085	−0.680424	+	0.732819i	$0.738204\pi$
$432$	0	0
$433$	−3.34621e6	−0.857695	−0.428848	−	0.903377i	$-0.641080\pi$
$433$	−3.34621e6	−0.857695	−0.428848	+	0.903377i	$0.641080\pi$
$434$	0	0
$435$	−378769.	−0.0959736
$436$	0	0
$437$	−4.50424e6	−1.12828
$438$	0	0
$439$	−129149.	−0.0319838	−0.0159919	−	0.999872i	$-0.505091\pi$
$439$	−129149.	−0.0319838	−0.0159919	+	0.999872i	$0.505091\pi$
$440$	0	0
$441$	735674.	0.180131
$442$	0	0
$443$	2.97268e6	0.719680	0.359840	−	0.933014i	$-0.382831\pi$
$443$	2.97268e6	0.719680	0.359840	+	0.933014i	$0.382831\pi$
$444$	0	0
$445$	−1.22614e6	−0.293521
$446$	0	0
$447$	−434853.	−0.102937
$448$	0	0
$449$	−618002.	−0.144669	−0.0723343	−	0.997380i	$-0.523045\pi$
$449$	−618002.	−0.144669	−0.0723343	+	0.997380i	$0.523045\pi$
$450$	0	0
$451$	1.01531e6	0.235049
$452$	0	0
$453$	−1.67034e6	−0.382436
$454$	0	0
$455$	−421841.	−0.0955257
$456$	0	0
$457$	−4.45372e6	−0.997545	−0.498772	−	0.866733i	$-0.666216\pi$
$457$	−4.45372e6	−0.997545	−0.498772	+	0.866733i	$0.666216\pi$
$458$	0	0
$459$	1.21263e6	0.268655
$460$	0	0
$461$	−930948.	−0.204020	−0.102010	−	0.994783i	$-0.532527\pi$
$461$	−930948.	−0.204020	−0.102010	+	0.994783i	$0.532527\pi$
$462$	0	0
$463$	6.85746e6	1.48666	0.743329	−	0.668926i	$-0.233246\pi$
$463$	6.85746e6	1.48666	0.743329	+	0.668926i	$0.233246\pi$
$464$	0	0
$465$	2.02457e6	0.434211
$466$	0	0
$467$	749357.	0.159000	0.0794999	−	0.996835i	$-0.474668\pi$
$467$	749357.	0.159000	0.0794999	+	0.996835i	$0.474668\pi$
$468$	0	0
$469$	−472096.	−0.0991057
$470$	0	0
$471$	4.69908e6	0.976023
$472$	0	0
$473$	−1.14215e6	−0.234730
$474$	0	0
$475$	−1.45907e6	−0.296717
$476$	0	0
$477$	−2.11938e6	−0.426493
$478$	0	0
$479$	3.06170e6	0.609710	0.304855	−	0.952399i	$-0.401392\pi$
$479$	3.06170e6	0.609710	0.304855	+	0.952399i	$0.401392\pi$
$480$	0	0
$481$	1.07576e6	0.212008
$482$	0	0
$483$	−2.24173e6	−0.437235
$484$	0	0
$485$	−1.00735e6	−0.194458
$486$	0	0
$487$	−7.14247e6	−1.36466	−0.682332	−	0.731042i	$-0.739034\pi$
$487$	−7.14247e6	−1.36466	−0.682332	+	0.731042i	$0.739034\pi$
$488$	0	0
$489$	599543.	0.113383
$490$	0	0
$491$	2.50239e6	0.468437	0.234218	−	0.972184i	$-0.424747\pi$
$491$	2.50239e6	0.468437	0.234218	+	0.972184i	$0.424747\pi$
$492$	0	0
$493$	386990.	0.0717104
$494$	0	0
$495$	213887.	0.0392348
$496$	0	0
$497$	3.65726e6	0.664148
$498$	0	0
$499$	7.90328e6	1.42088	0.710438	−	0.703760i	$-0.248497\pi$
$499$	7.90328e6	1.42088	0.710438	+	0.703760i	$0.248497\pi$
$500$	0	0
$501$	−6.49089e6	−1.15534
$502$	0	0
$503$	350720.	0.0618075	0.0309037	−	0.999522i	$-0.490161\pi$
$503$	350720.	0.0618075	0.0309037	+	0.999522i	$0.490161\pi$
$504$	0	0
$505$	1.27942e6	0.223246
$506$	0	0
$507$	−332360.	−0.0574235
$508$	0	0
$509$	7.12004e6	1.21811	0.609057	−	0.793126i	$-0.291548\pi$
$509$	7.12004e6	1.21811	0.609057	+	0.793126i	$0.291548\pi$
$510$	0	0
$511$	−6.34450e6	−1.07484
$512$	0	0
$513$	−9.52410e6	−1.59783
$514$	0	0
$515$	3.42251e6	0.568626
$516$	0	0
$517$	−2.32271e6	−0.382180
$518$	0	0
$519$	−4.63588e6	−0.755464
$520$	0	0
$521$	2.66156e6	0.429578	0.214789	−	0.976660i	$-0.431094\pi$
$521$	2.66156e6	0.429578	0.214789	+	0.976660i	$0.431094\pi$
$522$	0	0
$523$	4.16549e6	0.665904	0.332952	−	0.942944i	$-0.391955\pi$
$523$	4.16549e6	0.665904	0.332952	+	0.942944i	$0.391955\pi$
$524$	0	0
$525$	−726169.	−0.114985
$526$	0	0
$527$	−2.06851e6	−0.324437
$528$	0	0
$529$	−2.71372e6	−0.421624
$530$	0	0
$531$	−2.19504e6	−0.337836
$532$	0	0
$533$	2.15768e6	0.328980
$534$	0	0
$535$	−5.34148e6	−0.806821
$536$	0	0
$537$	−4.74864e6	−0.710614
$538$	0	0
$539$	543800.	0.0806246
$540$	0	0
$541$	1.20870e7	1.77552	0.887758	−	0.460311i	$-0.152262\pi$
$541$	1.20870e7	1.77552	0.887758	+	0.460311i	$0.152262\pi$
$542$	0	0
$543$	3.99331e6	0.581210
$544$	0	0
$545$	1.59618e6	0.230192
$546$	0	0
$547$	−1.03523e7	−1.47934	−0.739671	−	0.672968i	$-0.765019\pi$
$547$	−1.03523e7	−1.47934	−0.739671	+	0.672968i	$0.765019\pi$
$548$	0	0
$549$	1.01504e6	0.143732
$550$	0	0
$551$	−3.03946e6	−0.426498
$552$	0	0
$553$	−7.06464e6	−0.982375
$554$	0	0
$555$	1.85185e6	0.255195
$556$	0	0
$557$	−8.99816e6	−1.22890	−0.614449	−	0.788957i	$-0.710622\pi$
$557$	−8.99816e6	−1.22890	−0.614449	+	0.788957i	$0.710622\pi$
$558$	0	0
$559$	−2.42722e6	−0.328534
$560$	0	0
$561$	275065.	0.0369002
$562$	0	0
$563$	4.23529e6	0.563134	0.281567	−	0.959542i	$-0.409146\pi$
$563$	4.23529e6	0.563134	0.281567	+	0.959542i	$0.409146\pi$
$564$	0	0
$565$	−1.10724e6	−0.145922
$566$	0	0
$567$	−2.12987e6	−0.278225
$568$	0	0
$569$	−9.51598e6	−1.23218	−0.616088	−	0.787677i	$-0.711283\pi$
$569$	−9.51598e6	−1.23218	−0.616088	+	0.787677i	$0.711283\pi$
$570$	0	0
$571$	−2.55583e6	−0.328051	−0.164026	−	0.986456i	$-0.552448\pi$
$571$	−2.55583e6	−0.328051	−0.164026	+	0.986456i	$0.552448\pi$
$572$	0	0
$573$	−1.11834e6	−0.142294
$574$	0	0
$575$	1.20588e6	0.152102
$576$	0	0
$577$	1.14047e6	0.142609	0.0713043	−	0.997455i	$-0.477284\pi$
$577$	1.14047e6	0.142609	0.0713043	+	0.997455i	$0.477284\pi$
$578$	0	0
$579$	−6.55001e6	−0.811980
$580$	0	0
$581$	8.01229e6	0.984729
$582$	0	0
$583$	−1.56661e6	−0.190893
$584$	0	0
$585$	454540.	0.0549139
$586$	0	0
$587$	8.35974e6	1.00138	0.500688	−	0.865628i	$-0.333080\pi$
$587$	8.35974e6	1.00138	0.500688	+	0.865628i	$0.333080\pi$
$588$	0	0
$589$	1.62463e7	1.92959
$590$	0	0
$591$	−9.63743e6	−1.13499
$592$	0	0
$593$	−8.51631e6	−0.994523	−0.497261	−	0.867601i	$-0.665661\pi$
$593$	−8.51631e6	−0.994523	−0.497261	+	0.867601i	$0.665661\pi$
$594$	0	0
$595$	741929.	0.0859152
$596$	0	0
$597$	−8.28786e6	−0.951715
$598$	0	0
$599$	6.94083e6	0.790395	0.395197	−	0.918596i	$-0.370676\pi$
$599$	6.94083e6	0.790395	0.395197	+	0.918596i	$0.370676\pi$
$600$	0	0
$601$	−8.31102e6	−0.938574	−0.469287	−	0.883046i	$-0.655489\pi$
$601$	−8.31102e6	−0.938574	−0.469287	+	0.883046i	$0.655489\pi$
$602$	0	0
$603$	508691.	0.0569719
$604$	0	0
$605$	−3.86817e6	−0.429653
$606$	0	0
$607$	1.13341e7	1.24858	0.624288	−	0.781194i	$-0.285389\pi$
$607$	1.13341e7	1.24858	0.624288	+	0.781194i	$0.285389\pi$
$608$	0	0
$609$	−1.51271e6	−0.165278
$610$	0	0
$611$	−4.93608e6	−0.534908
$612$	0	0
$613$	−1.77962e6	−0.191283	−0.0956414	−	0.995416i	$-0.530490\pi$
$613$	−1.77962e6	−0.191283	−0.0956414	+	0.995416i	$0.530490\pi$
$614$	0	0
$615$	3.71430e6	0.395994
$616$	0	0
$617$	5.77125e6	0.610319	0.305160	−	0.952301i	$-0.401290\pi$
$617$	5.77125e6	0.610319	0.305160	+	0.952301i	$0.401290\pi$
$618$	0	0
$619$	1.32777e7	1.39283	0.696413	−	0.717641i	$-0.254778\pi$
$619$	1.32777e7	1.39283	0.696413	+	0.717641i	$0.254778\pi$
$620$	0	0
$621$	7.87140e6	0.819074
$622$	0	0
$623$	−4.89691e6	−0.505477
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−2.16039e6	−0.219464
$628$	0	0
$629$	−1.89204e6	−0.190679
$630$	0	0
$631$	1.17159e6	0.117139	0.0585695	−	0.998283i	$-0.481346\pi$
$631$	1.17159e6	0.117139	0.0585695	+	0.998283i	$0.481346\pi$
$632$	0	0
$633$	−9.75234e6	−0.967385
$634$	0	0
$635$	62846.5	0.00618510
$636$	0	0
$637$	1.15565e6	0.112844
$638$	0	0
$639$	−3.94075e6	−0.381792
$640$	0	0
$641$	1.17828e7	1.13267	0.566337	−	0.824174i	$-0.308360\pi$
$641$	1.17828e7	1.13267	0.566337	+	0.824174i	$0.308360\pi$
$642$	0	0
$643$	−4.39280e6	−0.419000	−0.209500	−	0.977809i	$-0.567184\pi$
$643$	−4.39280e6	−0.419000	−0.209500	+	0.977809i	$0.567184\pi$
$644$	0	0
$645$	−4.17829e6	−0.395457
$646$	0	0
$647$	9.82885e6	0.923086	0.461543	−	0.887118i	$-0.347296\pi$
$647$	9.82885e6	0.923086	0.461543	+	0.887118i	$0.347296\pi$
$648$	0	0
$649$	−1.62254e6	−0.151211
$650$	0	0
$651$	8.08565e6	0.747760
$652$	0	0
$653$	2.24847e6	0.206350	0.103175	−	0.994663i	$-0.467100\pi$
$653$	2.24847e6	0.206350	0.103175	+	0.994663i	$0.467100\pi$
$654$	0	0
$655$	−112296.	−0.0102273
$656$	0	0
$657$	6.83630e6	0.617885
$658$	0	0
$659$	7.71969e6	0.692447	0.346224	−	0.938152i	$-0.387464\pi$
$659$	7.71969e6	0.692447	0.346224	+	0.938152i	$0.387464\pi$
$660$	0	0
$661$	1.75441e7	1.56181	0.780903	−	0.624652i	$-0.214759\pi$
$661$	1.75441e7	1.56181	0.780903	+	0.624652i	$0.214759\pi$
$662$	0	0
$663$	584551.	0.0516463
$664$	0	0
$665$	−5.82719e6	−0.510981
$666$	0	0
$667$	2.51202e6	0.218630
$668$	0	0
$669$	−9.90077e6	−0.855270
$670$	0	0
$671$	750304.	0.0643326
$672$	0	0
$673$	−3.06858e6	−0.261156	−0.130578	−	0.991438i	$-0.541683\pi$
$673$	−3.06858e6	−0.261156	−0.130578	+	0.991438i	$0.541683\pi$
$674$	0	0
$675$	2.54981e6	0.215401
$676$	0	0
$677$	−1.37167e7	−1.15021	−0.575106	−	0.818079i	$-0.695039\pi$
$677$	−1.37167e7	−1.15021	−0.575106	+	0.818079i	$0.695039\pi$
$678$	0	0
$679$	−4.02311e6	−0.334879
$680$	0	0
$681$	8.51219e6	0.703354
$682$	0	0
$683$	1.10195e7	0.903877	0.451939	−	0.892049i	$-0.350733\pi$
$683$	1.10195e7	0.903877	0.451939	+	0.892049i	$0.350733\pi$
$684$	0	0
$685$	1.12221e6	0.0913797
$686$	0	0
$687$	−1.35906e7	−1.09861
$688$	0	0
$689$	−3.32927e6	−0.267178
$690$	0	0
$691$	3.78853e6	0.301840	0.150920	−	0.988546i	$-0.451776\pi$
$691$	3.78853e6	0.301840	0.150920	+	0.988546i	$0.451776\pi$
$692$	0	0
$693$	854215.	0.0675669
$694$	0	0
$695$	−4.21161e6	−0.330739
$696$	0	0
$697$	−3.79491e6	−0.295882
$698$	0	0
$699$	1.42107e6	0.110008
$700$	0	0
$701$	−1.42400e7	−1.09450	−0.547248	−	0.836970i	$-0.684325\pi$
$701$	−1.42400e7	−1.09450	−0.547248	+	0.836970i	$0.684325\pi$
$702$	0	0
$703$	1.48602e7	1.13406
$704$	0	0
$705$	−8.49710e6	−0.643870
$706$	0	0
$707$	5.10968e6	0.384455
$708$	0	0
$709$	1.61344e6	0.120542	0.0602710	−	0.998182i	$-0.480804\pi$
$709$	1.61344e6	0.120542	0.0602710	+	0.998182i	$0.480804\pi$
$710$	0	0
$711$	7.61226e6	0.564728
$712$	0	0
$713$	−1.34271e7	−0.989140
$714$	0	0
$715$	335990.	0.0245788
$716$	0	0
$717$	−4.23254e6	−0.307471
$718$	0	0
$719$	−1.59531e7	−1.15086	−0.575431	−	0.817850i	$-0.695166\pi$
$719$	−1.59531e7	−1.15086	−0.575431	+	0.817850i	$0.695166\pi$
$720$	0	0
$721$	1.36687e7	0.979239
$722$	0	0
$723$	−5.84675e6	−0.415976
$724$	0	0
$725$	813728.	0.0574955
$726$	0	0
$727$	−1.22874e7	−0.862230	−0.431115	−	0.902297i	$-0.641880\pi$
$727$	−1.22874e7	−0.862230	−0.431115	+	0.902297i	$0.641880\pi$
$728$	0	0
$729$	1.38313e7	0.963930
$730$	0	0
$731$	4.26897e6	0.295481
$732$	0	0
$733$	−1.38897e7	−0.954843	−0.477422	−	0.878674i	$-0.658429\pi$
$733$	−1.38897e7	−0.954843	−0.477422	+	0.878674i	$0.658429\pi$
$734$	0	0
$735$	1.98937e6	0.135831
$736$	0	0
$737$	376018.	0.0255000
$738$	0	0
$739$	−2.20837e7	−1.48751	−0.743757	−	0.668450i	$-0.766958\pi$
$739$	−2.20837e7	−1.48751	−0.743757	+	0.668450i	$0.766958\pi$
$740$	0	0
$741$	−4.59113e6	−0.307167
$742$	0	0
$743$	2.23745e7	1.48690	0.743449	−	0.668793i	$-0.233189\pi$
$743$	2.23745e7	1.48690	0.743449	+	0.668793i	$0.233189\pi$
$744$	0	0
$745$	934214.	0.0616674
$746$	0	0
$747$	−8.63337e6	−0.566081
$748$	0	0
$749$	−2.13326e7	−1.38944
$750$	0	0
$751$	−2.41424e7	−1.56200	−0.781001	−	0.624530i	$-0.785291\pi$
$751$	−2.41424e7	−1.56200	−0.781001	+	0.624530i	$0.785291\pi$
$752$	0	0
$753$	281555.	0.0180957
$754$	0	0
$755$	3.58846e6	0.229109
$756$	0	0
$757$	−1.48180e7	−0.939833	−0.469917	−	0.882711i	$-0.655716\pi$
$757$	−1.48180e7	−0.939833	−0.469917	+	0.882711i	$0.655716\pi$
$758$	0	0
$759$	1.78550e6	0.112501
$760$	0	0
$761$	1.14020e7	0.713706	0.356853	−	0.934161i	$-0.383850\pi$
$761$	1.14020e7	0.713706	0.356853	+	0.934161i	$0.383850\pi$
$762$	0	0
$763$	6.37474e6	0.396416
$764$	0	0
$765$	−799440.	−0.0493892
$766$	0	0
$767$	−3.44813e6	−0.211639
$768$	0	0
$769$	−2.30707e7	−1.40684	−0.703422	−	0.710773i	$-0.748345\pi$
$769$	−2.30707e7	−1.40684	−0.703422	+	0.710773i	$0.748345\pi$
$770$	0	0
$771$	8.28809e6	0.502132
$772$	0	0
$773$	−1.66600e7	−1.00283	−0.501414	−	0.865208i	$-0.667186\pi$
$773$	−1.66600e7	−1.00283	−0.501414	+	0.865208i	$0.667186\pi$
$774$	0	0
$775$	−4.34948e6	−0.260125
$776$	0	0
$777$	7.39583e6	0.439475
$778$	0	0
$779$	2.98056e7	1.75976
$780$	0	0
$781$	−2.91295e6	−0.170886
$782$	0	0
$783$	5.31161e6	0.309615
$784$	0	0
$785$	−1.00952e7	−0.584713
$786$	0	0
$787$	−1.11136e7	−0.639617	−0.319808	−	0.947482i	$-0.603619\pi$
$787$	−1.11136e7	−0.639617	−0.319808	+	0.947482i	$0.603619\pi$
$788$	0	0
$789$	−2.35428e7	−1.34637
$790$	0	0
$791$	−4.42206e6	−0.251295
$792$	0	0
$793$	1.59450e6	0.0900412
$794$	0	0
$795$	−5.73111e6	−0.321603
$796$	0	0
$797$	2.62504e7	1.46383	0.731913	−	0.681398i	$-0.238628\pi$
$797$	2.62504e7	1.46383	0.731913	+	0.681398i	$0.238628\pi$
$798$	0	0
$799$	8.68151e6	0.481092
$800$	0	0
$801$	5.27649e6	0.290579
$802$	0	0
$803$	5.05330e6	0.276558
$804$	0	0
$805$	4.81600e6	0.261937
$806$	0	0
$807$	−1.38319e7	−0.747647
$808$	0	0
$809$	−1.18334e7	−0.635680	−0.317840	−	0.948144i	$-0.602958\pi$
$809$	−1.18334e7	−0.635680	−0.317840	+	0.948144i	$0.602958\pi$
$810$	0	0
$811$	−1.94078e7	−1.03615	−0.518076	−	0.855334i	$-0.673352\pi$
$811$	−1.94078e7	−1.03615	−0.518076	+	0.855334i	$0.673352\pi$
$812$	0	0
$813$	1.30148e7	0.690573
$814$	0	0
$815$	−1.28803e6	−0.0679251
$816$	0	0
$817$	−3.35289e7	−1.75738
$818$	0	0
$819$	1.81532e6	0.0945681
$820$	0	0
$821$	2.99379e7	1.55011	0.775056	−	0.631892i	$-0.217722\pi$
$821$	2.99379e7	1.55011	0.775056	+	0.631892i	$0.217722\pi$
$822$	0	0
$823$	1.05061e7	0.540684	0.270342	−	0.962764i	$-0.412863\pi$
$823$	1.05061e7	0.540684	0.270342	+	0.962764i	$0.412863\pi$
$824$	0	0
$825$	578383.	0.0295856
$826$	0	0
$827$	−1.30147e6	−0.0661715	−0.0330858	−	0.999453i	$-0.510533\pi$
$827$	−1.30147e6	−0.0661715	−0.0330858	+	0.999453i	$0.510533\pi$
$828$	0	0
$829$	5.25708e6	0.265679	0.132840	−	0.991138i	$-0.457590\pi$
$829$	5.25708e6	0.265679	0.132840	+	0.991138i	$0.457590\pi$
$830$	0	0
$831$	−2.72604e7	−1.36940
$832$	0	0
$833$	−2.03255e6	−0.101491
$834$	0	0
$835$	1.39447e7	0.692138
$836$	0	0
$837$	−2.83912e7	−1.40078
$838$	0	0
$839$	−3.50995e6	−0.172146	−0.0860728	−	0.996289i	$-0.527432\pi$
$839$	−3.50995e6	−0.172146	−0.0860728	+	0.996289i	$0.527432\pi$
$840$	0	0
$841$	−1.88160e7	−0.917357
$842$	0	0
$843$	2.57016e7	1.24564
$844$	0	0
$845$	714025.	0.0344010
$846$	0	0
$847$	−1.54486e7	−0.739911
$848$	0	0
$849$	−3.01795e7	−1.43695
$850$	0	0
$851$	−1.22816e7	−0.581340
$852$	0	0
$853$	2.12343e7	0.999231	0.499616	−	0.866247i	$-0.333475\pi$
$853$	2.12343e7	0.999231	0.499616	+	0.866247i	$0.333475\pi$
$854$	0	0
$855$	6.27888e6	0.293743
$856$	0	0
$857$	3.69643e7	1.71921	0.859607	−	0.510955i	$-0.170708\pi$
$857$	3.69643e7	1.71921	0.859607	+	0.510955i	$0.170708\pi$
$858$	0	0
$859$	1.12909e7	0.522089	0.261045	−	0.965327i	$-0.415933\pi$
$859$	1.12909e7	0.522089	0.261045	+	0.965327i	$0.415933\pi$
$860$	0	0
$861$	1.48340e7	0.681947
$862$	0	0
$863$	−1.91140e7	−0.873625	−0.436813	−	0.899553i	$-0.643893\pi$
$863$	−1.91140e7	−0.873625	−0.436813	+	0.899553i	$0.643893\pi$
$864$	0	0
$865$	9.95947e6	0.452581
$866$	0	0
$867$	1.54946e7	0.700055
$868$	0	0
$869$	5.62688e6	0.252766
$870$	0	0
$871$	799089.	0.0356903
$872$	0	0
$873$	4.33497e6	0.192509
$874$	0	0
$875$	1.56006e6	0.0688846
$876$	0	0
$877$	−2.04090e7	−0.896029	−0.448014	−	0.894026i	$-0.647869\pi$
$877$	−2.04090e7	−0.896029	−0.448014	+	0.894026i	$0.647869\pi$
$878$	0	0
$879$	−2.09735e7	−0.915583
$880$	0	0
$881$	−3.33290e7	−1.44671	−0.723356	−	0.690476i	$-0.757401\pi$
$881$	−3.33290e7	−1.44671	−0.723356	+	0.690476i	$0.757401\pi$
$882$	0	0
$883$	1.72858e7	0.746082	0.373041	−	0.927815i	$-0.378315\pi$
$883$	1.72858e7	0.746082	0.373041	+	0.927815i	$0.378315\pi$
$884$	0	0
$885$	−5.93570e6	−0.254750
$886$	0	0
$887$	−1.87501e7	−0.800192	−0.400096	−	0.916473i	$-0.631023\pi$
$887$	−1.87501e7	−0.800192	−0.400096	+	0.916473i	$0.631023\pi$
$888$	0	0
$889$	250994.	0.0106515
$890$	0	0
$891$	1.69641e6	0.0715875
$892$	0	0
$893$	−6.81855e7	−2.86130
$894$	0	0
$895$	1.02017e7	0.425712
$896$	0	0
$897$	3.79444e6	0.157459
$898$	0	0
$899$	−9.06058e6	−0.373901
$900$	0	0
$901$	5.85548e6	0.240298
$902$	0	0
$903$	−1.66871e7	−0.681022
$904$	0	0
$905$	−8.57901e6	−0.348189
$906$	0	0
$907$	−3.48113e7	−1.40508	−0.702542	−	0.711643i	$-0.747952\pi$
$907$	−3.48113e7	−1.40508	−0.702542	+	0.711643i	$0.747952\pi$
$908$	0	0
$909$	−5.50576e6	−0.221008
$910$	0	0
$911$	3.48528e6	0.139137	0.0695684	−	0.997577i	$-0.477838\pi$
$911$	3.48528e6	0.139137	0.0695684	+	0.997577i	$0.477838\pi$
$912$	0	0
$913$	−6.38167e6	−0.253371
$914$	0	0
$915$	2.74482e6	0.108383
$916$	0	0
$917$	−448484.	−0.0176126
$918$	0	0
$919$	−3.52355e7	−1.37623	−0.688116	−	0.725601i	$-0.741562\pi$
$919$	−3.52355e7	−1.37623	−0.688116	+	0.725601i	$0.741562\pi$
$920$	0	0
$921$	−1.58948e7	−0.617457
$922$	0	0
$923$	−6.19042e6	−0.239175
$924$	0	0
$925$	−3.97841e6	−0.152881
$926$	0	0
$927$	−1.47282e7	−0.562925
$928$	0	0
$929$	3.53806e7	1.34501	0.672506	−	0.740092i	$-0.265218\pi$
$929$	3.53806e7	1.34501	0.672506	+	0.740092i	$0.265218\pi$
$930$	0	0
$931$	1.59638e7	0.603619
$932$	0	0
$933$	−1.68758e7	−0.634688
$934$	0	0
$935$	−590935.	−0.0221060
$936$	0	0
$937$	−1.91406e7	−0.712209	−0.356104	−	0.934446i	$-0.615895\pi$
$937$	−1.91406e7	−0.712209	−0.356104	+	0.934446i	$0.615895\pi$
$938$	0	0
$939$	−3.96378e7	−1.46705
$940$	0	0
$941$	1.85564e7	0.683157	0.341579	−	0.939853i	$-0.389038\pi$
$941$	1.85564e7	0.683157	0.341579	+	0.939853i	$0.389038\pi$
$942$	0	0
$943$	−2.46335e7	−0.902083
$944$	0	0
$945$	1.01833e7	0.370945
$946$	0	0
$947$	3.21634e6	0.116543	0.0582716	−	0.998301i	$-0.481441\pi$
$947$	3.21634e6	0.116543	0.0582716	+	0.998301i	$0.481441\pi$
$948$	0	0
$949$	1.07390e7	0.387076
$950$	0	0
$951$	−1.87426e7	−0.672014
$952$	0	0
$953$	2.39773e7	0.855202	0.427601	−	0.903968i	$-0.359359\pi$
$953$	2.39773e7	0.855202	0.427601	+	0.903968i	$0.359359\pi$
$954$	0	0
$955$	2.40257e6	0.0852449
$956$	0	0
$957$	1.20485e6	0.0425260
$958$	0	0
$959$	4.48186e6	0.157366
$960$	0	0
$961$	1.98008e7	0.691630
$962$	0	0
$963$	2.29862e7	0.798732
$964$	0	0
$965$	1.40717e7	0.486438
$966$	0	0
$967$	−1.59207e6	−0.0547514	−0.0273757	−	0.999625i	$-0.508715\pi$
$967$	−1.59207e6	−0.0547514	−0.0273757	+	0.999625i	$0.508715\pi$
$968$	0	0
$969$	8.07482e6	0.276263
$970$	0	0
$971$	−1.59047e6	−0.0541350	−0.0270675	−	0.999634i	$-0.508617\pi$
$971$	−1.59047e6	−0.0541350	−0.0270675	+	0.999634i	$0.508617\pi$
$972$	0	0
$973$	−1.68201e7	−0.569571
$974$	0	0
$975$	1.22914e6	0.0414086
$976$	0	0
$977$	−8.91327e6	−0.298745	−0.149372	−	0.988781i	$-0.547725\pi$
$977$	−8.91327e6	−0.298745	−0.149372	+	0.988781i	$0.547725\pi$
$978$	0	0
$979$	3.90031e6	0.130060
$980$	0	0
$981$	−6.86889e6	−0.227884
$982$	0	0
$983$	1.35469e7	0.447154	0.223577	−	0.974686i	$-0.428227\pi$
$983$	1.35469e7	0.447154	0.223577	+	0.974686i	$0.428227\pi$
$984$	0	0
$985$	2.07045e7	0.679946
$986$	0	0
$987$	−3.39354e7	−1.10882
$988$	0	0
$989$	2.77107e7	0.900859
$990$	0	0
$991$	−4.24496e7	−1.37306	−0.686529	−	0.727102i	$-0.740867\pi$
$991$	−4.24496e7	−1.37306	−0.686529	+	0.727102i	$0.740867\pi$
$992$	0	0
$993$	8.39667e6	0.270230
$994$	0	0
$995$	1.78052e7	0.570150
$996$	0	0
$997$	2.15543e6	0.0686747	0.0343374	−	0.999410i	$-0.489068\pi$
$997$	2.15543e6	0.0686747	0.0343374	+	0.999410i	$0.489068\pi$
$998$	0	0
$999$	−2.59691e7	−0.823271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.ba.1.3		8
4.3	odd	2		520.6.a.d.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.6.a.d.1.6	✓	8	4.3	odd	2
1040.6.a.ba.1.3		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)

\(f(q)\)	\(=\)	\(q-11.6369 q^{3} +25.0000 q^{5} +99.8440 q^{7} -107.583 q^{9} -79.5242 q^{11} -169.000 q^{13} -290.922 q^{15} +297.235 q^{17} -2334.52 q^{19} -1161.87 q^{21} +1929.41 q^{23} +625.000 q^{25} +4079.69 q^{27} +1301.96 q^{29} -6959.16 q^{31} +925.412 q^{33} +2496.10 q^{35} -6365.45 q^{37} +1966.63 q^{39} -12767.4 q^{41} +14362.3 q^{43} -2689.59 q^{45} +29207.6 q^{47} -6838.17 q^{49} -3458.88 q^{51} +19699.8 q^{53} -1988.11 q^{55} +27166.4 q^{57} +20403.1 q^{59} -9434.91 q^{61} -10741.6 q^{63} -4225.00 q^{65} -4728.34 q^{67} -22452.3 q^{69} +36629.7 q^{71} -63544.1 q^{73} -7273.04 q^{75} -7940.02 q^{77} -70756.8 q^{79} -21332.0 q^{81} +80248.1 q^{83} +7430.88 q^{85} -15150.8 q^{87} -49045.6 q^{89} -16873.6 q^{91} +80982.8 q^{93} -58362.9 q^{95} -40294.0 q^{97} +8555.49 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 28 q^{3} + 200 q^{5} - 80 q^{7} + 936 q^{9} + 444 q^{11} - 1352 q^{13} + 700 q^{15} - 3400 q^{17} + 1668 q^{19} - 3816 q^{21} + 2964 q^{23} + 5000 q^{25} + 14464 q^{27} - 8272 q^{29} + 8684 q^{31}+ \cdots + 535828 q^{99}+O(q^{100}) \) 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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