LMFDB - Embedded newform 1040.6.a.q.1.5

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:	$1$
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			1.5
Root			$-5.93318$ 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)

$f(q)$	$=$	$q+7.05430 q^{3} -25.0000 q^{5} +185.746 q^{7} -193.237 q^{9} +353.912 q^{11} +169.000 q^{13} -176.358 q^{15} +634.652 q^{17} -1118.29 q^{19} +1310.31 q^{21} -3509.85 q^{23} +625.000 q^{25} -3077.35 q^{27} -3765.67 q^{29} -2906.63 q^{31} +2496.60 q^{33} -4643.65 q^{35} +283.305 q^{37} +1192.18 q^{39} -13563.6 q^{41} +5184.47 q^{43} +4830.92 q^{45} +6781.50 q^{47} +17694.6 q^{49} +4477.03 q^{51} +7664.43 q^{53} -8847.81 q^{55} -7888.78 q^{57} -2806.29 q^{59} -13764.7 q^{61} -35893.0 q^{63} -4225.00 q^{65} -67744.1 q^{67} -24759.5 q^{69} +66519.0 q^{71} +75902.7 q^{73} +4408.94 q^{75} +65737.9 q^{77} -101641. q^{79} +25248.0 q^{81} +50882.7 q^{83} -15866.3 q^{85} -26564.2 q^{87} -52439.2 q^{89} +31391.1 q^{91} -20504.2 q^{93} +27957.4 q^{95} -142557. q^{97} -68388.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	7.05430	0.452534	0.226267	−	0.974065i	$-0.427348\pi$
$3$	7.05430	0.452534	0.226267	+	0.974065i	$0.427348\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	185.746	1.43276	0.716382	−	0.697708i	$-0.245797\pi$
$7$	185.746	1.43276	0.716382	+	0.697708i	$0.245797\pi$
$8$	0	0
$9$	−193.237	−0.795213
$10$	0	0
$11$	353.912	0.881889	0.440945	−	0.897534i	$-0.354643\pi$
$11$	353.912	0.881889	0.440945	+	0.897534i	$0.354643\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−176.358	−0.202379
$16$	0	0
$17$	634.652	0.532615	0.266308	−	0.963888i	$-0.414196\pi$
$17$	634.652	0.532615	0.266308	+	0.963888i	$0.414196\pi$
$18$	0	0
$19$	−1118.29	−0.710677	−0.355338	−	0.934738i	$-0.615634\pi$
$19$	−1118.29	−0.710677	−0.355338	+	0.934738i	$0.615634\pi$
$20$	0	0
$21$	1310.31	0.648374
$22$	0	0
$23$	−3509.85	−1.38347	−0.691734	−	0.722153i	$-0.743153\pi$
$23$	−3509.85	−1.38347	−0.691734	+	0.722153i	$0.743153\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−3077.35	−0.812394
$28$	0	0
$29$	−3765.67	−0.831471	−0.415735	−	0.909486i	$-0.636476\pi$
$29$	−3765.67	−0.831471	−0.415735	+	0.909486i	$0.636476\pi$
$30$	0	0
$31$	−2906.63	−0.543232	−0.271616	−	0.962406i	$-0.587558\pi$
$31$	−2906.63	−0.543232	−0.271616	+	0.962406i	$0.587558\pi$
$32$	0	0
$33$	2496.60	0.399085
$34$	0	0
$35$	−4643.65	−0.640751
$36$	0	0
$37$	283.305	0.0340213	0.0170106	−	0.999855i	$-0.494585\pi$
$37$	283.305	0.0340213	0.0170106	+	0.999855i	$0.494585\pi$
$38$	0	0
$39$	1192.18	0.125510
$40$	0	0
$41$	−13563.6	−1.26013	−0.630064	−	0.776544i	$-0.716971\pi$
$41$	−13563.6	−1.26013	−0.630064	+	0.776544i	$0.716971\pi$
$42$	0	0
$43$	5184.47	0.427596	0.213798	−	0.976878i	$-0.431417\pi$
$43$	5184.47	0.427596	0.213798	+	0.976878i	$0.431417\pi$
$44$	0	0
$45$	4830.92	0.355630
$46$	0	0
$47$	6781.50	0.447797	0.223898	−	0.974612i	$-0.428122\pi$
$47$	6781.50	0.447797	0.223898	+	0.974612i	$0.428122\pi$
$48$	0	0
$49$	17694.6	1.05281
$50$	0	0
$51$	4477.03	0.241026
$52$	0	0
$53$	7664.43	0.374792	0.187396	−	0.982284i	$-0.439995\pi$
$53$	7664.43	0.374792	0.187396	+	0.982284i	$0.439995\pi$
$54$	0	0
$55$	−8847.81	−0.394393
$56$	0	0
$57$	−7888.78	−0.321605
$58$	0	0
$59$	−2806.29	−0.104955	−0.0524773	−	0.998622i	$-0.516712\pi$
$59$	−2806.29	−0.104955	−0.0524773	+	0.998622i	$0.516712\pi$
$60$	0	0
$61$	−13764.7	−0.473634	−0.236817	−	0.971554i	$-0.576104\pi$
$61$	−13764.7	−0.473634	−0.236817	+	0.971554i	$0.576104\pi$
$62$	0	0
$63$	−35893.0	−1.13935
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	−67744.1	−1.84368	−0.921838	−	0.387576i	$-0.873313\pi$
$67$	−67744.1	−1.84368	−0.921838	+	0.387576i	$0.873313\pi$
$68$	0	0
$69$	−24759.5	−0.626065
$70$	0	0
$71$	66519.0	1.56603	0.783014	−	0.622004i	$-0.213681\pi$
$71$	66519.0	1.56603	0.783014	+	0.622004i	$0.213681\pi$
$72$	0	0
$73$	75902.7	1.66706	0.833528	−	0.552478i	$-0.186318\pi$
$73$	75902.7	1.66706	0.833528	+	0.552478i	$0.186318\pi$
$74$	0	0
$75$	4408.94	0.0905067
$76$	0	0
$77$	65737.9	1.26354
$78$	0	0
$79$	−101641.	−1.83233	−0.916163	−	0.400806i	$-0.868730\pi$
$79$	−101641.	−1.83233	−0.916163	+	0.400806i	$0.868730\pi$
$80$	0	0
$81$	25248.0	0.427578
$82$	0	0
$83$	50882.7	0.810727	0.405363	−	0.914156i	$-0.367145\pi$
$83$	50882.7	0.810727	0.405363	+	0.914156i	$0.367145\pi$
$84$	0	0
$85$	−15866.3	−0.238193
$86$	0	0
$87$	−26564.2	−0.376269
$88$	0	0
$89$	−52439.2	−0.701748	−0.350874	−	0.936423i	$-0.614115\pi$
$89$	−52439.2	−0.701748	−0.350874	+	0.936423i	$0.614115\pi$
$90$	0	0
$91$	31391.1	0.397377
$92$	0	0
$93$	−20504.2	−0.245831
$94$	0	0
$95$	27957.4	0.317824
$96$	0	0
$97$	−142557.	−1.53837	−0.769183	−	0.639028i	$-0.779337\pi$
$97$	−142557.	−1.53837	−0.769183	+	0.639028i	$0.779337\pi$
$98$	0	0
$99$	−68388.9	−0.701290
$100$	0	0
$101$	4751.74	0.0463499	0.0231750	−	0.999731i	$-0.492623\pi$
$101$	4751.74	0.0463499	0.0231750	+	0.999731i	$0.492623\pi$
$102$	0	0
$103$	59290.6	0.550672	0.275336	−	0.961348i	$-0.411211\pi$
$103$	59290.6	0.550672	0.275336	+	0.961348i	$0.411211\pi$
$104$	0	0
$105$	−32757.7	−0.289961
$106$	0	0
$107$	−157927.	−1.33351	−0.666756	−	0.745276i	$-0.732318\pi$
$107$	−157927.	−1.33351	−0.666756	+	0.745276i	$0.732318\pi$
$108$	0	0
$109$	58878.4	0.474668	0.237334	−	0.971428i	$-0.423726\pi$
$109$	58878.4	0.474668	0.237334	+	0.971428i	$0.423726\pi$
$110$	0	0
$111$	1998.52	0.0153958
$112$	0	0
$113$	179734.	1.32414	0.662069	−	0.749443i	$-0.269678\pi$
$113$	179734.	1.32414	0.662069	+	0.749443i	$0.269678\pi$
$114$	0	0
$115$	87746.2	0.618705
$116$	0	0
$117$	−32657.0	−0.220553
$118$	0	0
$119$	117884.	0.763112
$120$	0	0
$121$	−35797.0	−0.222271
$122$	0	0
$123$	−95681.5	−0.570250
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	123741.	0.680774	0.340387	−	0.940286i	$-0.389442\pi$
$127$	123741.	0.680774	0.340387	+	0.940286i	$0.389442\pi$
$128$	0	0
$129$	36572.8	0.193501
$130$	0	0
$131$	43205.0	0.219966	0.109983	−	0.993933i	$-0.464920\pi$
$131$	43205.0	0.219966	0.109983	+	0.993933i	$0.464920\pi$
$132$	0	0
$133$	−207719.	−1.01823
$134$	0	0
$135$	76933.6	0.363314
$136$	0	0
$137$	−188517.	−0.858120	−0.429060	−	0.903276i	$-0.641155\pi$
$137$	−188517.	−0.858120	−0.429060	+	0.903276i	$0.641155\pi$
$138$	0	0
$139$	−344148.	−1.51081	−0.755403	−	0.655260i	$-0.772559\pi$
$139$	−344148.	−1.51081	−0.755403	+	0.655260i	$0.772559\pi$
$140$	0	0
$141$	47838.7	0.202643
$142$	0	0
$143$	59811.2	0.244592
$144$	0	0
$145$	94141.7	0.371845
$146$	0	0
$147$	124823.	0.476433
$148$	0	0
$149$	−177809.	−0.656126	−0.328063	−	0.944656i	$-0.606396\pi$
$149$	−177809.	−0.656126	−0.328063	+	0.944656i	$0.606396\pi$
$150$	0	0
$151$	−554784.	−1.98008	−0.990038	−	0.140803i	$-0.955031\pi$
$151$	−554784.	−1.98008	−0.990038	+	0.140803i	$0.955031\pi$
$152$	0	0
$153$	−122638.	−0.423543
$154$	0	0
$155$	72665.7	0.242941
$156$	0	0
$157$	255896.	0.828542	0.414271	−	0.910154i	$-0.364037\pi$
$157$	255896.	0.828542	0.414271	+	0.910154i	$0.364037\pi$
$158$	0	0
$159$	54067.2	0.169606
$160$	0	0
$161$	−651941.	−1.98218
$162$	0	0
$163$	262686.	0.774404	0.387202	−	0.921995i	$-0.373442\pi$
$163$	262686.	0.774404	0.387202	+	0.921995i	$0.373442\pi$
$164$	0	0
$165$	−62415.1	−0.178476
$166$	0	0
$167$	−287069.	−0.796517	−0.398259	−	0.917273i	$-0.630385\pi$
$167$	−287069.	−0.796517	−0.398259	+	0.917273i	$0.630385\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	216096.	0.565140
$172$	0	0
$173$	719663.	1.82816	0.914080	−	0.405534i	$-0.132915\pi$
$173$	719663.	1.82816	0.914080	+	0.405534i	$0.132915\pi$
$174$	0	0
$175$	116091.	0.286553
$176$	0	0
$177$	−19796.4	−0.0474955
$178$	0	0
$179$	−779772.	−1.81901	−0.909505	−	0.415693i	$-0.863539\pi$
$179$	−779772.	−1.81901	−0.909505	+	0.415693i	$0.863539\pi$
$180$	0	0
$181$	336065.	0.762478	0.381239	−	0.924477i	$-0.375498\pi$
$181$	336065.	0.762478	0.381239	+	0.924477i	$0.375498\pi$
$182$	0	0
$183$	−97100.5	−0.214335
$184$	0	0
$185$	−7082.63	−0.0152148
$186$	0	0
$187$	224611.	0.469708
$188$	0	0
$189$	−571605.	−1.16397
$190$	0	0
$191$	−409479.	−0.812173	−0.406086	−	0.913835i	$-0.633107\pi$
$191$	−409479.	−0.812173	−0.406086	+	0.913835i	$0.633107\pi$
$192$	0	0
$193$	339565.	0.656189	0.328095	−	0.944645i	$-0.393593\pi$
$193$	339565.	0.656189	0.328095	+	0.944645i	$0.393593\pi$
$194$	0	0
$195$	−29804.4	−0.0561299
$196$	0	0
$197$	−871469.	−1.59988	−0.799938	−	0.600082i	$-0.795135\pi$
$197$	−871469.	−1.59988	−0.799938	+	0.600082i	$0.795135\pi$
$198$	0	0
$199$	−270952.	−0.485019	−0.242510	−	0.970149i	$-0.577971\pi$
$199$	−270952.	−0.485019	−0.242510	+	0.970149i	$0.577971\pi$
$200$	0	0
$201$	−477887.	−0.834325
$202$	0	0
$203$	−699458.	−1.19130
$204$	0	0
$205$	339089.	0.563546
$206$	0	0
$207$	678232.	1.10015
$208$	0	0
$209$	−395778.	−0.626738
$210$	0	0
$211$	−181455.	−0.280583	−0.140292	−	0.990110i	$-0.544804\pi$
$211$	−181455.	−0.280583	−0.140292	+	0.990110i	$0.544804\pi$
$212$	0	0
$213$	469245.	0.708681
$214$	0	0
$215$	−129612.	−0.191227
$216$	0	0
$217$	−539895.	−0.778323
$218$	0	0
$219$	535440.	0.754398
$220$	0	0
$221$	107256.	0.147721
$222$	0	0
$223$	−1.38761e6	−1.86855	−0.934276	−	0.356552i	$-0.883952\pi$
$223$	−1.38761e6	−1.86855	−0.934276	+	0.356552i	$0.883952\pi$
$224$	0	0
$225$	−120773.	−0.159043
$226$	0	0
$227$	690397.	0.889271	0.444636	−	0.895711i	$-0.353333\pi$
$227$	690397.	0.889271	0.444636	+	0.895711i	$0.353333\pi$
$228$	0	0
$229$	−1.38257e6	−1.74221	−0.871104	−	0.491099i	$-0.836595\pi$
$229$	−1.38257e6	−1.74221	−0.871104	+	0.491099i	$0.836595\pi$
$230$	0	0
$231$	463735.	0.571794
$232$	0	0
$233$	−71911.6	−0.0867780	−0.0433890	−	0.999058i	$-0.513815\pi$
$233$	−71911.6	−0.0867780	−0.0433890	+	0.999058i	$0.513815\pi$
$234$	0	0
$235$	−169537.	−0.200261
$236$	0	0
$237$	−717009.	−0.829189
$238$	0	0
$239$	825442.	0.934743	0.467371	−	0.884061i	$-0.345201\pi$
$239$	825442.	0.934743	0.467371	+	0.884061i	$0.345201\pi$
$240$	0	0
$241$	−615086.	−0.682171	−0.341086	−	0.940032i	$-0.610795\pi$
$241$	−615086.	−0.682171	−0.341086	+	0.940032i	$0.610795\pi$
$242$	0	0
$243$	925902.	1.00589
$244$	0	0
$245$	−442365.	−0.470832
$246$	0	0
$247$	−188992.	−0.197106
$248$	0	0
$249$	358942.	0.366881
$250$	0	0
$251$	−622589.	−0.623759	−0.311879	−	0.950122i	$-0.600959\pi$
$251$	−622589.	−0.623759	−0.311879	+	0.950122i	$0.600959\pi$
$252$	0	0
$253$	−1.24218e6	−1.22007
$254$	0	0
$255$	−111926.	−0.107790
$256$	0	0
$257$	−1.04334e6	−0.985359	−0.492680	−	0.870211i	$-0.663983\pi$
$257$	−1.04334e6	−0.985359	−0.492680	+	0.870211i	$0.663983\pi$
$258$	0	0
$259$	52622.9	0.0487444
$260$	0	0
$261$	727666.	0.661197
$262$	0	0
$263$	1.31940e6	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$263$	1.31940e6	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$264$	0	0
$265$	−191611.	−0.167612
$266$	0	0
$267$	−369922.	−0.317564
$268$	0	0
$269$	369633.	0.311452	0.155726	−	0.987800i	$-0.450228\pi$
$269$	369633.	0.311452	0.155726	+	0.987800i	$0.450228\pi$
$270$	0	0
$271$	−749291.	−0.619765	−0.309883	−	0.950775i	$-0.600290\pi$
$271$	−749291.	−0.619765	−0.309883	+	0.950775i	$0.600290\pi$
$272$	0	0
$273$	221442.	0.179826
$274$	0	0
$275$	221195.	0.176378
$276$	0	0
$277$	1.64757e6	1.29016	0.645081	−	0.764114i	$-0.276824\pi$
$277$	1.64757e6	1.29016	0.645081	+	0.764114i	$0.276824\pi$
$278$	0	0
$279$	561668.	0.431986
$280$	0	0
$281$	1.21917e6	0.921080	0.460540	−	0.887639i	$-0.347656\pi$
$281$	1.21917e6	0.921080	0.460540	+	0.887639i	$0.347656\pi$
$282$	0	0
$283$	997517.	0.740379	0.370190	−	0.928956i	$-0.379293\pi$
$283$	997517.	0.740379	0.370190	+	0.928956i	$0.379293\pi$
$284$	0	0
$285$	197220.	0.143826
$286$	0	0
$287$	−2.51938e6	−1.80546
$288$	0	0
$289$	−1.01707e6	−0.716321
$290$	0	0
$291$	−1.00564e6	−0.696162
$292$	0	0
$293$	1.80793e6	1.23031	0.615153	−	0.788408i	$-0.289094\pi$
$293$	1.80793e6	1.23031	0.615153	+	0.788408i	$0.289094\pi$
$294$	0	0
$295$	70157.1	0.0469372
$296$	0	0
$297$	−1.08911e6	−0.716442
$298$	0	0
$299$	−593165.	−0.383705
$300$	0	0
$301$	962995.	0.612644
$302$	0	0
$303$	33520.2	0.0209749
$304$	0	0
$305$	344118.	0.211816
$306$	0	0
$307$	24494.5	0.0148328	0.00741638	−	0.999972i	$-0.497639\pi$
$307$	24494.5	0.0148328	0.00741638	+	0.999972i	$0.497639\pi$
$308$	0	0
$309$	418254.	0.249197
$310$	0	0
$311$	−1.48212e6	−0.868924	−0.434462	−	0.900690i	$-0.643061\pi$
$311$	−1.48212e6	−0.868924	−0.434462	+	0.900690i	$0.643061\pi$
$312$	0	0
$313$	−348766.	−0.201221	−0.100611	−	0.994926i	$-0.532080\pi$
$313$	−348766.	−0.201221	−0.100611	+	0.994926i	$0.532080\pi$
$314$	0	0
$315$	897325.	0.509534
$316$	0	0
$317$	−406486.	−0.227194	−0.113597	−	0.993527i	$-0.536237\pi$
$317$	−406486.	−0.227194	−0.113597	+	0.993527i	$0.536237\pi$
$318$	0	0
$319$	−1.33272e6	−0.733265
$320$	0	0
$321$	−1.11406e6	−0.603459
$322$	0	0
$323$	−709728.	−0.378517
$324$	0	0
$325$	105625.	0.0554700
$326$	0	0
$327$	415346.	0.214803
$328$	0	0
$329$	1.25964e6	0.641587
$330$	0	0
$331$	−1.58614e6	−0.795743	−0.397871	−	0.917441i	$-0.630251\pi$
$331$	−1.58614e6	−0.795743	−0.397871	+	0.917441i	$0.630251\pi$
$332$	0	0
$333$	−54745.0	−0.0270542
$334$	0	0
$335$	1.69360e6	0.824517
$336$	0	0
$337$	2.43587e6	1.16837	0.584185	−	0.811621i	$-0.301414\pi$
$337$	2.43587e6	1.16837	0.584185	+	0.811621i	$0.301414\pi$
$338$	0	0
$339$	1.26790e6	0.599217
$340$	0	0
$341$	−1.02869e6	−0.479071
$342$	0	0
$343$	164869.	0.0756664
$344$	0	0
$345$	618988.	0.279985
$346$	0	0
$347$	1.17786e6	0.525132	0.262566	−	0.964914i	$-0.415431\pi$
$347$	1.17786e6	0.525132	0.262566	+	0.964914i	$0.415431\pi$
$348$	0	0
$349$	−338854.	−0.148919	−0.0744594	−	0.997224i	$-0.523723\pi$
$349$	−338854.	−0.148919	−0.0744594	+	0.997224i	$0.523723\pi$
$350$	0	0
$351$	−520071.	−0.225318
$352$	0	0
$353$	3.25607e6	1.39077	0.695387	−	0.718635i	$-0.255233\pi$
$353$	3.25607e6	1.39077	0.695387	+	0.718635i	$0.255233\pi$
$354$	0	0
$355$	−1.66297e6	−0.700349
$356$	0	0
$357$	831590.	0.345334
$358$	0	0
$359$	−2.81818e6	−1.15407	−0.577036	−	0.816719i	$-0.695791\pi$
$359$	−2.81818e6	−1.15407	−0.577036	+	0.816719i	$0.695791\pi$
$360$	0	0
$361$	−1.22552e6	−0.494939
$362$	0	0
$363$	−252522.	−0.100585
$364$	0	0
$365$	−1.89757e6	−0.745530
$366$	0	0
$367$	−3.09661e6	−1.20011	−0.600056	−	0.799958i	$-0.704855\pi$
$367$	−3.09661e6	−1.20011	−0.600056	+	0.799958i	$0.704855\pi$
$368$	0	0
$369$	2.62098e6	1.00207
$370$	0	0
$371$	1.42364e6	0.536988
$372$	0	0
$373$	−4.21455e6	−1.56848	−0.784240	−	0.620457i	$-0.786947\pi$
$373$	−4.21455e6	−1.56848	−0.784240	+	0.620457i	$0.786947\pi$
$374$	0	0
$375$	−110223.	−0.0404758
$376$	0	0
$377$	−636398.	−0.230609
$378$	0	0
$379$	1.26649e6	0.452903	0.226452	−	0.974022i	$-0.427288\pi$
$379$	1.26649e6	0.452903	0.226452	+	0.974022i	$0.427288\pi$
$380$	0	0
$381$	872903.	0.308073
$382$	0	0
$383$	−5.66939e6	−1.97487	−0.987436	−	0.158017i	$-0.949490\pi$
$383$	−5.66939e6	−1.97487	−0.987436	+	0.158017i	$0.949490\pi$
$384$	0	0
$385$	−1.64345e6	−0.565072
$386$	0	0
$387$	−1.00183e6	−0.340030
$388$	0	0
$389$	−5.49114e6	−1.83988	−0.919938	−	0.392063i	$-0.871762\pi$
$389$	−5.49114e6	−1.83988	−0.919938	+	0.392063i	$0.871762\pi$
$390$	0	0
$391$	−2.22753e6	−0.736856
$392$	0	0
$393$	304781.	0.0995420
$394$	0	0
$395$	2.54103e6	0.819441
$396$	0	0
$397$	−1.53688e6	−0.489398	−0.244699	−	0.969599i	$-0.578689\pi$
$397$	−1.53688e6	−0.489398	−0.244699	+	0.969599i	$0.578689\pi$
$398$	0	0
$399$	−1.46531e6	−0.460784
$400$	0	0
$401$	30028.4	0.00932547	0.00466273	−	0.999989i	$-0.498516\pi$
$401$	30028.4	0.00932547	0.00466273	+	0.999989i	$0.498516\pi$
$402$	0	0
$403$	−491220.	−0.150666
$404$	0	0
$405$	−631201.	−0.191219
$406$	0	0
$407$	100265.	0.0300030
$408$	0	0
$409$	−5.54413e6	−1.63880	−0.819398	−	0.573225i	$-0.805692\pi$
$409$	−5.54413e6	−1.63880	−0.819398	+	0.573225i	$0.805692\pi$
$410$	0	0
$411$	−1.32985e6	−0.388328
$412$	0	0
$413$	−521257.	−0.150375
$414$	0	0
$415$	−1.27207e6	−0.362568
$416$	0	0
$417$	−2.42773e6	−0.683691
$418$	0	0
$419$	1.29360e6	0.359968	0.179984	−	0.983670i	$-0.442395\pi$
$419$	1.29360e6	0.359968	0.179984	+	0.983670i	$0.442395\pi$
$420$	0	0
$421$	−3.68620e6	−1.01362	−0.506809	−	0.862058i	$-0.669175\pi$
$421$	−3.68620e6	−1.01362	−0.506809	+	0.862058i	$0.669175\pi$
$422$	0	0
$423$	−1.31044e6	−0.356094
$424$	0	0
$425$	396658.	0.106523
$426$	0	0
$427$	−2.55674e6	−0.678606
$428$	0	0
$429$	421926.	0.110686
$430$	0	0
$431$	−1.22645e6	−0.318021	−0.159010	−	0.987277i	$-0.550830\pi$
$431$	−1.22645e6	−0.318021	−0.159010	+	0.987277i	$0.550830\pi$
$432$	0	0
$433$	1.02459e6	0.262621	0.131311	−	0.991341i	$-0.458081\pi$
$433$	1.02459e6	0.262621	0.131311	+	0.991341i	$0.458081\pi$
$434$	0	0
$435$	664104.	0.168272
$436$	0	0
$437$	3.92504e6	0.983198
$438$	0	0
$439$	5.04951e6	1.25051	0.625256	−	0.780420i	$-0.284995\pi$
$439$	5.04951e6	1.25051	0.625256	+	0.780420i	$0.284995\pi$
$440$	0	0
$441$	−3.41925e6	−0.837210
$442$	0	0
$443$	−6.30848e6	−1.52727	−0.763634	−	0.645649i	$-0.776587\pi$
$443$	−6.30848e6	−1.52727	−0.763634	+	0.645649i	$0.776587\pi$
$444$	0	0
$445$	1.31098e6	0.313831
$446$	0	0
$447$	−1.25432e6	−0.296919
$448$	0	0
$449$	1.16391e6	0.272461	0.136231	−	0.990677i	$-0.456501\pi$
$449$	1.16391e6	0.272461	0.136231	+	0.990677i	$0.456501\pi$
$450$	0	0
$451$	−4.80032e6	−1.11129
$452$	0	0
$453$	−3.91361e6	−0.896050
$454$	0	0
$455$	−784777.	−0.177712
$456$	0	0
$457$	1.48156e6	0.331840	0.165920	−	0.986139i	$-0.446941\pi$
$457$	1.48156e6	0.331840	0.165920	+	0.986139i	$0.446941\pi$
$458$	0	0
$459$	−1.95304e6	−0.432693
$460$	0	0
$461$	−5.65392e6	−1.23908	−0.619538	−	0.784967i	$-0.712680\pi$
$461$	−5.65392e6	−1.23908	−0.619538	+	0.784967i	$0.712680\pi$
$462$	0	0
$463$	−2.09215e6	−0.453566	−0.226783	−	0.973945i	$-0.572821\pi$
$463$	−2.09215e6	−0.453566	−0.226783	+	0.973945i	$0.572821\pi$
$464$	0	0
$465$	512606.	0.109939
$466$	0	0
$467$	7.48481e6	1.58814	0.794069	−	0.607827i	$-0.207959\pi$
$467$	7.48481e6	1.58814	0.794069	+	0.607827i	$0.207959\pi$
$468$	0	0
$469$	−1.25832e7	−2.64155
$470$	0	0
$471$	1.80517e6	0.374943
$472$	0	0
$473$	1.83485e6	0.377092
$474$	0	0
$475$	−698934.	−0.142135
$476$	0	0
$477$	−1.48105e6	−0.298040
$478$	0	0
$479$	−2.54779e6	−0.507371	−0.253685	−	0.967287i	$-0.581643\pi$
$479$	−2.54779e6	−0.507371	−0.253685	+	0.967287i	$0.581643\pi$
$480$	0	0
$481$	47878.6	0.00943580
$482$	0	0
$483$	−4.59899e6	−0.897004
$484$	0	0
$485$	3.56393e6	0.687978
$486$	0	0
$487$	−3.67779e6	−0.702692	−0.351346	−	0.936246i	$-0.614276\pi$
$487$	−3.67779e6	−0.702692	−0.351346	+	0.936246i	$0.614276\pi$
$488$	0	0
$489$	1.85306e6	0.350444
$490$	0	0
$491$	7.23294e6	1.35398	0.676988	−	0.735994i	$-0.263285\pi$
$491$	7.23294e6	1.35398	0.676988	+	0.735994i	$0.263285\pi$
$492$	0	0
$493$	−2.38989e6	−0.442854
$494$	0	0
$495$	1.70972e6	0.313627
$496$	0	0
$497$	1.23556e7	2.24375
$498$	0	0
$499$	−875124.	−0.157332	−0.0786662	−	0.996901i	$-0.525066\pi$
$499$	−875124.	−0.157332	−0.0786662	+	0.996901i	$0.525066\pi$
$500$	0	0
$501$	−2.02507e6	−0.360451
$502$	0	0
$503$	2.95982e6	0.521609	0.260805	−	0.965392i	$-0.416012\pi$
$503$	2.95982e6	0.521609	0.260805	+	0.965392i	$0.416012\pi$
$504$	0	0
$505$	−118793.	−0.0207283
$506$	0	0
$507$	201478.	0.0348103
$508$	0	0
$509$	−1.12208e7	−1.91968	−0.959842	−	0.280540i	$-0.909487\pi$
$509$	−1.12208e7	−1.91968	−0.959842	+	0.280540i	$0.909487\pi$
$510$	0	0
$511$	1.40986e7	2.38850
$512$	0	0
$513$	3.44138e6	0.577350
$514$	0	0
$515$	−1.48226e6	−0.246268
$516$	0	0
$517$	2.40006e6	0.394907
$518$	0	0
$519$	5.07672e6	0.827304
$520$	0	0
$521$	8.92586e6	1.44064	0.720321	−	0.693641i	$-0.243995\pi$
$521$	8.92586e6	1.44064	0.720321	+	0.693641i	$0.243995\pi$
$522$	0	0
$523$	−6.44897e6	−1.03095	−0.515473	−	0.856906i	$-0.672384\pi$
$523$	−6.44897e6	−1.03095	−0.515473	+	0.856906i	$0.672384\pi$
$524$	0	0
$525$	818943.	0.129675
$526$	0	0
$527$	−1.84470e6	−0.289334
$528$	0	0
$529$	5.88270e6	0.913982
$530$	0	0
$531$	542278.	0.0834614
$532$	0	0
$533$	−2.29224e6	−0.349496
$534$	0	0
$535$	3.94818e6	0.596365
$536$	0	0
$537$	−5.50075e6	−0.823163
$538$	0	0
$539$	6.26234e6	0.928463
$540$	0	0
$541$	6.01652e6	0.883796	0.441898	−	0.897065i	$-0.354305\pi$
$541$	6.01652e6	0.883796	0.441898	+	0.897065i	$0.354305\pi$
$542$	0	0
$543$	2.37071e6	0.345047
$544$	0	0
$545$	−1.47196e6	−0.212278
$546$	0	0
$547$	−928354.	−0.132662	−0.0663308	−	0.997798i	$-0.521129\pi$
$547$	−928354.	−0.132662	−0.0663308	+	0.997798i	$0.521129\pi$
$548$	0	0
$549$	2.65985e6	0.376640
$550$	0	0
$551$	4.21113e6	0.590907
$552$	0	0
$553$	−1.88795e7	−2.62529
$554$	0	0
$555$	−49963.0	−0.00688520
$556$	0	0
$557$	652347.	0.0890924	0.0445462	−	0.999007i	$-0.485816\pi$
$557$	652347.	0.0890924	0.0445462	+	0.999007i	$0.485816\pi$
$558$	0	0
$559$	876176.	0.118594
$560$	0	0
$561$	1.58448e6	0.212558
$562$	0	0
$563$	992674.	0.131988	0.0659942	−	0.997820i	$-0.478978\pi$
$563$	992674.	0.131988	0.0659942	+	0.997820i	$0.478978\pi$
$564$	0	0
$565$	−4.49334e6	−0.592173
$566$	0	0
$567$	4.68972e6	0.612618
$568$	0	0
$569$	8.79703e6	1.13908	0.569541	−	0.821963i	$-0.307121\pi$
$569$	8.79703e6	1.13908	0.569541	+	0.821963i	$0.307121\pi$
$570$	0	0
$571$	6.18261e6	0.793563	0.396782	−	0.917913i	$-0.370127\pi$
$571$	6.18261e6	0.793563	0.396782	+	0.917913i	$0.370127\pi$
$572$	0	0
$573$	−2.88859e6	−0.367535
$574$	0	0
$575$	−2.19366e6	−0.276693
$576$	0	0
$577$	1.42671e7	1.78401	0.892003	−	0.452030i	$-0.149300\pi$
$577$	1.42671e7	1.78401	0.892003	+	0.452030i	$0.149300\pi$
$578$	0	0
$579$	2.39539e6	0.296948
$580$	0	0
$581$	9.45125e6	1.16158
$582$	0	0
$583$	2.71254e6	0.330525
$584$	0	0
$585$	816426.	0.0986341
$586$	0	0
$587$	7.84422e6	0.939625	0.469812	−	0.882766i	$-0.344322\pi$
$587$	7.84422e6	0.939625	0.469812	+	0.882766i	$0.344322\pi$
$588$	0	0
$589$	3.25047e6	0.386062
$590$	0	0
$591$	−6.14761e6	−0.723998
$592$	0	0
$593$	−1.85555e6	−0.216689	−0.108344	−	0.994113i	$-0.534555\pi$
$593$	−1.85555e6	−0.216689	−0.108344	+	0.994113i	$0.534555\pi$
$594$	0	0
$595$	−2.94710e6	−0.341274
$596$	0	0
$597$	−1.91137e6	−0.219487
$598$	0	0
$599$	1.54479e7	1.75915	0.879573	−	0.475764i	$-0.157828\pi$
$599$	1.54479e7	1.75915	0.879573	+	0.475764i	$0.157828\pi$
$600$	0	0
$601$	431785.	0.0487619	0.0243810	−	0.999703i	$-0.492239\pi$
$601$	431785.	0.0487619	0.0243810	+	0.999703i	$0.492239\pi$
$602$	0	0
$603$	1.30907e7	1.46612
$604$	0	0
$605$	894924.	0.0994026
$606$	0	0
$607$	−1.21071e7	−1.33373	−0.666863	−	0.745180i	$-0.732363\pi$
$607$	−1.21071e7	−1.33373	−0.666863	+	0.745180i	$0.732363\pi$
$608$	0	0
$609$	−4.93419e6	−0.539104
$610$	0	0
$611$	1.14607e6	0.124197
$612$	0	0
$613$	9.44151e6	1.01482	0.507411	−	0.861704i	$-0.330602\pi$
$613$	9.44151e6	1.01482	0.507411	+	0.861704i	$0.330602\pi$
$614$	0	0
$615$	2.39204e6	0.255023
$616$	0	0
$617$	−9.98389e6	−1.05581	−0.527906	−	0.849303i	$-0.677023\pi$
$617$	−9.98389e6	−1.05581	−0.527906	+	0.849303i	$0.677023\pi$
$618$	0	0
$619$	6.44689e6	0.676275	0.338138	−	0.941097i	$-0.390203\pi$
$619$	6.44689e6	0.676275	0.338138	+	0.941097i	$0.390203\pi$
$620$	0	0
$621$	1.08010e7	1.12392
$622$	0	0
$623$	−9.74037e6	−1.00544
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−2.79194e6	−0.283620
$628$	0	0
$629$	179800.	0.0181202
$630$	0	0
$631$	−4.35897e6	−0.435823	−0.217912	−	0.975969i	$-0.569924\pi$
$631$	−4.35897e6	−0.435823	−0.217912	+	0.975969i	$0.569924\pi$
$632$	0	0
$633$	−1.28003e6	−0.126973
$634$	0	0
$635$	−3.09351e6	−0.304451
$636$	0	0
$637$	2.99039e6	0.291997
$638$	0	0
$639$	−1.28539e7	−1.24533
$640$	0	0
$641$	1.63272e7	1.56952	0.784758	−	0.619803i	$-0.212787\pi$
$641$	1.63272e7	1.56952	0.784758	+	0.619803i	$0.212787\pi$
$642$	0	0
$643$	−1.31929e7	−1.25838	−0.629192	−	0.777250i	$-0.716614\pi$
$643$	−1.31929e7	−1.25838	−0.629192	+	0.777250i	$0.716614\pi$
$644$	0	0
$645$	−914321.	−0.0865365
$646$	0	0
$647$	9.42830e6	0.885468	0.442734	−	0.896653i	$-0.354009\pi$
$647$	9.42830e6	0.885468	0.442734	+	0.896653i	$0.354009\pi$
$648$	0	0
$649$	−993180.	−0.0925584
$650$	0	0
$651$	−3.80858e6	−0.352217
$652$	0	0
$653$	1.60701e7	1.47481	0.737406	−	0.675450i	$-0.236050\pi$
$653$	1.60701e7	1.47481	0.737406	+	0.675450i	$0.236050\pi$
$654$	0	0
$655$	−1.08012e6	−0.0983718
$656$	0	0
$657$	−1.46672e7	−1.32566
$658$	0	0
$659$	−6.63639e6	−0.595276	−0.297638	−	0.954679i	$-0.596199\pi$
$659$	−6.63639e6	−0.595276	−0.297638	+	0.954679i	$0.596199\pi$
$660$	0	0
$661$	2.01198e7	1.79110	0.895552	−	0.444956i	$-0.146781\pi$
$661$	2.01198e7	1.79110	0.895552	+	0.444956i	$0.146781\pi$
$662$	0	0
$663$	756618.	0.0668486
$664$	0	0
$665$	5.19297e6	0.455367
$666$	0	0
$667$	1.32169e7	1.15031
$668$	0	0
$669$	−9.78861e6	−0.845582
$670$	0	0
$671$	−4.87151e6	−0.417693
$672$	0	0
$673$	−9.52533e6	−0.810667	−0.405334	−	0.914169i	$-0.632845\pi$
$673$	−9.52533e6	−0.810667	−0.405334	+	0.914169i	$0.632845\pi$
$674$	0	0
$675$	−1.92334e6	−0.162479
$676$	0	0
$677$	−3.37825e6	−0.283283	−0.141641	−	0.989918i	$-0.545238\pi$
$677$	−3.37825e6	−0.283283	−0.141641	+	0.989918i	$0.545238\pi$
$678$	0	0
$679$	−2.64794e7	−2.20412
$680$	0	0
$681$	4.87027e6	0.402425
$682$	0	0
$683$	8.86253e6	0.726953	0.363476	−	0.931603i	$-0.381590\pi$
$683$	8.86253e6	0.726953	0.363476	+	0.931603i	$0.381590\pi$
$684$	0	0
$685$	4.71291e6	0.383763
$686$	0	0
$687$	−9.75310e6	−0.788407
$688$	0	0
$689$	1.29529e6	0.103949
$690$	0	0
$691$	−25025.8	−0.00199385	−0.000996925	−	1.00000i	$-0.500317\pi$
$691$	−25025.8	−0.00199385	−0.000996925		1.00000i	$0.500317\pi$
$692$	0	0
$693$	−1.27030e7	−1.00478
$694$	0	0
$695$	8.60371e6	0.675653
$696$	0	0
$697$	−8.60815e6	−0.671163
$698$	0	0
$699$	−507286.	−0.0392699
$700$	0	0
$701$	−2.15506e7	−1.65640	−0.828198	−	0.560436i	$-0.810634\pi$
$701$	−2.15506e7	−1.65640	−0.828198	+	0.560436i	$0.810634\pi$
$702$	0	0
$703$	−316819.	−0.0241781
$704$	0	0
$705$	−1.19597e6	−0.0906248
$706$	0	0
$707$	882617.	0.0664085
$708$	0	0
$709$	2.07938e7	1.55352	0.776761	−	0.629796i	$-0.216861\pi$
$709$	2.07938e7	1.55352	0.776761	+	0.629796i	$0.216861\pi$
$710$	0	0
$711$	1.96409e7	1.45709
$712$	0	0
$713$	1.02018e7	0.751544
$714$	0	0
$715$	−1.49528e6	−0.109385
$716$	0	0
$717$	5.82292e6	0.423002
$718$	0	0
$719$	3.65717e6	0.263829	0.131915	−	0.991261i	$-0.457887\pi$
$719$	3.65717e6	0.263829	0.131915	+	0.991261i	$0.457887\pi$
$720$	0	0
$721$	1.10130e7	0.788982
$722$	0	0
$723$	−4.33900e6	−0.308705
$724$	0	0
$725$	−2.35354e6	−0.166294
$726$	0	0
$727$	8.36880e6	0.587256	0.293628	−	0.955920i	$-0.405137\pi$
$727$	8.36880e6	0.587256	0.293628	+	0.955920i	$0.405137\pi$
$728$	0	0
$729$	396319.	0.0276202
$730$	0	0
$731$	3.29034e6	0.227744
$732$	0	0
$733$	1.81111e7	1.24504	0.622522	−	0.782602i	$-0.286108\pi$
$733$	1.81111e7	1.24504	0.622522	+	0.782602i	$0.286108\pi$
$734$	0	0
$735$	−3.12058e6	−0.213067
$736$	0	0
$737$	−2.39755e7	−1.62592
$738$	0	0
$739$	−7.31705e6	−0.492861	−0.246431	−	0.969160i	$-0.579258\pi$
$739$	−7.31705e6	−0.492861	−0.246431	+	0.969160i	$0.579258\pi$
$740$	0	0
$741$	−1.33320e6	−0.0891972
$742$	0	0
$743$	−1.04179e7	−0.692322	−0.346161	−	0.938175i	$-0.612515\pi$
$743$	−1.04179e7	−0.692322	−0.346161	+	0.938175i	$0.612515\pi$
$744$	0	0
$745$	4.44522e6	0.293428
$746$	0	0
$747$	−9.83240e6	−0.644701
$748$	0	0
$749$	−2.93343e7	−1.91061
$750$	0	0
$751$	1.16729e6	0.0755229	0.0377615	−	0.999287i	$-0.487977\pi$
$751$	1.16729e6	0.0755229	0.0377615	+	0.999287i	$0.487977\pi$
$752$	0	0
$753$	−4.39193e6	−0.282272
$754$	0	0
$755$	1.38696e7	0.885517
$756$	0	0
$757$	4.75104e6	0.301334	0.150667	−	0.988585i	$-0.451858\pi$
$757$	4.75104e6	0.301334	0.150667	+	0.988585i	$0.451858\pi$
$758$	0	0
$759$	−8.76271e6	−0.552120
$760$	0	0
$761$	−5.92209e6	−0.370692	−0.185346	−	0.982673i	$-0.559341\pi$
$761$	−5.92209e6	−0.370692	−0.185346	+	0.982673i	$0.559341\pi$
$762$	0	0
$763$	1.09364e7	0.680087
$764$	0	0
$765$	3.06595e6	0.189414
$766$	0	0
$767$	−474262.	−0.0291092
$768$	0	0
$769$	−5.07027e6	−0.309183	−0.154591	−	0.987979i	$-0.549406\pi$
$769$	−5.07027e6	−0.309183	−0.154591	+	0.987979i	$0.549406\pi$
$770$	0	0
$771$	−7.36006e6	−0.445908
$772$	0	0
$773$	2.31839e7	1.39552	0.697761	−	0.716330i	$-0.254180\pi$
$773$	2.31839e7	1.39552	0.697761	+	0.716330i	$0.254180\pi$
$774$	0	0
$775$	−1.81664e6	−0.108646
$776$	0	0
$777$	371217.	0.0220585
$778$	0	0
$779$	1.51681e7	0.895543
$780$	0	0
$781$	2.35419e7	1.38106
$782$	0	0
$783$	1.15883e7	0.675482
$784$	0	0
$785$	−6.39740e6	−0.370535
$786$	0	0
$787$	6.07650e6	0.349717	0.174858	−	0.984594i	$-0.444053\pi$
$787$	6.07650e6	0.349717	0.174858	+	0.984594i	$0.444053\pi$
$788$	0	0
$789$	9.30742e6	0.532276
$790$	0	0
$791$	3.33848e7	1.89718
$792$	0	0
$793$	−2.32624e6	−0.131362
$794$	0	0
$795$	−1.35168e6	−0.0758501
$796$	0	0
$797$	2.78805e7	1.55473	0.777363	−	0.629052i	$-0.216557\pi$
$797$	2.78805e7	1.55473	0.777363	+	0.629052i	$0.216557\pi$
$798$	0	0
$799$	4.30389e6	0.238503
$800$	0	0
$801$	1.01332e7	0.558039
$802$	0	0
$803$	2.68629e7	1.47016
$804$	0	0
$805$	1.62985e7	0.886459
$806$	0	0
$807$	2.60750e6	0.140942
$808$	0	0
$809$	6.10438e6	0.327922	0.163961	−	0.986467i	$-0.447573\pi$
$809$	6.10438e6	0.327922	0.163961	+	0.986467i	$0.447573\pi$
$810$	0	0
$811$	2.23956e7	1.19567	0.597835	−	0.801619i	$-0.296028\pi$
$811$	2.23956e7	1.19567	0.597835	+	0.801619i	$0.296028\pi$
$812$	0	0
$813$	−5.28572e6	−0.280465
$814$	0	0
$815$	−6.56714e6	−0.346324
$816$	0	0
$817$	−5.79777e6	−0.303882
$818$	0	0
$819$	−6.06591e6	−0.316000
$820$	0	0
$821$	−2.42967e7	−1.25803	−0.629014	−	0.777394i	$-0.716541\pi$
$821$	−2.42967e7	−1.25803	−0.629014	+	0.777394i	$0.716541\pi$
$822$	0	0
$823$	−3.64578e7	−1.87625	−0.938127	−	0.346293i	$-0.887440\pi$
$823$	−3.64578e7	−1.87625	−0.938127	+	0.346293i	$0.887440\pi$
$824$	0	0
$825$	1.56038e6	0.0798169
$826$	0	0
$827$	2.81247e7	1.42996	0.714981	−	0.699143i	$-0.246435\pi$
$827$	2.81247e7	1.42996	0.714981	+	0.699143i	$0.246435\pi$
$828$	0	0
$829$	2.68734e7	1.35812	0.679058	−	0.734085i	$-0.262389\pi$
$829$	2.68734e7	1.35812	0.679058	+	0.734085i	$0.262389\pi$
$830$	0	0
$831$	1.16224e7	0.583841
$832$	0	0
$833$	1.12299e7	0.560743
$834$	0	0
$835$	7.17673e6	0.356213
$836$	0	0
$837$	8.94470e6	0.441319
$838$	0	0
$839$	−3.46774e7	−1.70076	−0.850378	−	0.526172i	$-0.823627\pi$
$839$	−3.46774e7	−1.70076	−0.850378	+	0.526172i	$0.823627\pi$
$840$	0	0
$841$	−6.33089e6	−0.308656
$842$	0	0
$843$	8.60037e6	0.416819
$844$	0	0
$845$	−714025.	−0.0344010
$846$	0	0
$847$	−6.64914e6	−0.318462
$848$	0	0
$849$	7.03678e6	0.335046
$850$	0	0
$851$	−994359.	−0.0470673
$852$	0	0
$853$	1.54571e6	0.0727368	0.0363684	−	0.999338i	$-0.488421\pi$
$853$	1.54571e6	0.0727368	0.0363684	+	0.999338i	$0.488421\pi$
$854$	0	0
$855$	−5.40239e6	−0.252738
$856$	0	0
$857$	−1.27926e7	−0.594987	−0.297493	−	0.954724i	$-0.596151\pi$
$857$	−1.27926e7	−0.594987	−0.297493	+	0.954724i	$0.596151\pi$
$858$	0	0
$859$	−2.66940e6	−0.123433	−0.0617165	−	0.998094i	$-0.519657\pi$
$859$	−2.66940e6	−0.123433	−0.0617165	+	0.998094i	$0.519657\pi$
$860$	0	0
$861$	−1.77725e7	−0.817033
$862$	0	0
$863$	3.37798e7	1.54394	0.771970	−	0.635659i	$-0.219272\pi$
$863$	3.37798e7	1.54394	0.771970	+	0.635659i	$0.219272\pi$
$864$	0	0
$865$	−1.79916e7	−0.817578
$866$	0	0
$867$	−7.17474e6	−0.324159
$868$	0	0
$869$	−3.59721e7	−1.61591
$870$	0	0
$871$	−1.14488e7	−0.511344
$872$	0	0
$873$	2.75473e7	1.22333
$874$	0	0
$875$	−2.90228e6	−0.128150
$876$	0	0
$877$	3.97308e7	1.74433	0.872164	−	0.489214i	$-0.162716\pi$
$877$	3.97308e7	1.74433	0.872164	+	0.489214i	$0.162716\pi$
$878$	0	0
$879$	1.27537e7	0.556755
$880$	0	0
$881$	−1.67058e7	−0.725149	−0.362574	−	0.931955i	$-0.618102\pi$
$881$	−1.67058e7	−0.725149	−0.362574	+	0.931955i	$0.618102\pi$
$882$	0	0
$883$	1.67930e7	0.724816	0.362408	−	0.932020i	$-0.381955\pi$
$883$	1.67930e7	0.724816	0.362408	+	0.932020i	$0.381955\pi$
$884$	0	0
$885$	494910.	0.0212406
$886$	0	0
$887$	−8.61531e6	−0.367673	−0.183837	−	0.982957i	$-0.558852\pi$
$887$	−8.61531e6	−0.367673	−0.183837	+	0.982957i	$0.558852\pi$
$888$	0	0
$889$	2.29843e7	0.975388
$890$	0	0
$891$	8.93559e6	0.377076
$892$	0	0
$893$	−7.58371e6	−0.318239
$894$	0	0
$895$	1.94943e7	0.813486
$896$	0	0
$897$	−4.18436e6	−0.173639
$898$	0	0
$899$	1.09454e7	0.451682
$900$	0	0
$901$	4.86425e6	0.199620
$902$	0	0
$903$	6.79326e6	0.277242
$904$	0	0
$905$	−8.40163e6	−0.340990
$906$	0	0
$907$	−7.34436e6	−0.296439	−0.148220	−	0.988954i	$-0.547354\pi$
$907$	−7.34436e6	−0.296439	−0.148220	+	0.988954i	$0.547354\pi$
$908$	0	0
$909$	−918211.	−0.0368581
$910$	0	0
$911$	−3.63225e7	−1.45004	−0.725019	−	0.688729i	$-0.758169\pi$
$911$	−3.63225e7	−1.45004	−0.725019	+	0.688729i	$0.758169\pi$
$912$	0	0
$913$	1.80080e7	0.714971
$914$	0	0
$915$	2.42751e6	0.0958537
$916$	0	0
$917$	8.02515e6	0.315159
$918$	0	0
$919$	−2.25278e7	−0.879892	−0.439946	−	0.898024i	$-0.645002\pi$
$919$	−2.25278e7	−0.879892	−0.439946	+	0.898024i	$0.645002\pi$
$920$	0	0
$921$	172791.	0.00671232
$922$	0	0
$923$	1.12417e7	0.434338
$924$	0	0
$925$	177066.	0.00680425
$926$	0	0
$927$	−1.14571e7	−0.437901
$928$	0	0
$929$	3.93312e7	1.49520	0.747598	−	0.664152i	$-0.231207\pi$
$929$	3.93312e7	1.49520	0.747598	+	0.664152i	$0.231207\pi$
$930$	0	0
$931$	−1.97878e7	−0.748209
$932$	0	0
$933$	−1.04553e7	−0.393217
$934$	0	0
$935$	−5.61528e6	−0.210060
$936$	0	0
$937$	−1.36354e7	−0.507362	−0.253681	−	0.967288i	$-0.581641\pi$
$937$	−1.36354e7	−0.507362	−0.253681	+	0.967288i	$0.581641\pi$
$938$	0	0
$939$	−2.46030e6	−0.0910593
$940$	0	0
$941$	2.28438e7	0.840995	0.420498	−	0.907294i	$-0.361856\pi$
$941$	2.28438e7	0.840995	0.420498	+	0.907294i	$0.361856\pi$
$942$	0	0
$943$	4.76061e7	1.74334
$944$	0	0
$945$	1.42901e7	0.520543
$946$	0	0
$947$	6.46944e6	0.234418	0.117209	−	0.993107i	$-0.462605\pi$
$947$	6.46944e6	0.234418	0.117209	+	0.993107i	$0.462605\pi$
$948$	0	0
$949$	1.28276e7	0.462358
$950$	0	0
$951$	−2.86748e6	−0.102813
$952$	0	0
$953$	2.58068e6	0.0920452	0.0460226	−	0.998940i	$-0.485345\pi$
$953$	2.58068e6	0.0920452	0.0460226	+	0.998940i	$0.485345\pi$
$954$	0	0
$955$	1.02370e7	0.363215
$956$	0	0
$957$	−9.40139e6	−0.331827
$958$	0	0
$959$	−3.50162e7	−1.22948
$960$	0	0
$961$	−2.01807e7	−0.704899
$962$	0	0
$963$	3.05173e7	1.06043
$964$	0	0
$965$	−8.48912e6	−0.293457
$966$	0	0
$967$	1.88591e6	0.0648568	0.0324284	−	0.999474i	$-0.489676\pi$
$967$	1.88591e6	0.0648568	0.0324284	+	0.999474i	$0.489676\pi$
$968$	0	0
$969$	−5.00663e6	−0.171292
$970$	0	0
$971$	2.49003e7	0.847534	0.423767	−	0.905771i	$-0.360708\pi$
$971$	2.49003e7	0.847534	0.423767	+	0.905771i	$0.360708\pi$
$972$	0	0
$973$	−6.39242e7	−2.16463
$974$	0	0
$975$	745110.	0.0251020
$976$	0	0
$977$	1.01689e7	0.340829	0.170415	−	0.985372i	$-0.445489\pi$
$977$	1.01689e7	0.340829	0.170415	+	0.985372i	$0.445489\pi$
$978$	0	0
$979$	−1.85589e7	−0.618864
$980$	0	0
$981$	−1.13775e7	−0.377462
$982$	0	0
$983$	−4.29289e6	−0.141699	−0.0708494	−	0.997487i	$-0.522571\pi$
$983$	−4.29289e6	−0.141699	−0.0708494	+	0.997487i	$0.522571\pi$
$984$	0	0
$985$	2.17867e7	0.715487
$986$	0	0
$987$	8.88586e6	0.290340
$988$	0	0
$989$	−1.81967e7	−0.591565
$990$	0	0
$991$	2.64765e7	0.856398	0.428199	−	0.903684i	$-0.359148\pi$
$991$	2.64765e7	0.856398	0.428199	+	0.903684i	$0.359148\pi$
$992$	0	0
$993$	−1.11891e7	−0.360100
$994$	0	0
$995$	6.77379e6	0.216907
$996$	0	0
$997$	4.21089e6	0.134164	0.0670820	−	0.997747i	$-0.478631\pi$
$997$	4.21089e6	0.134164	0.0670820	+	0.997747i	$0.478631\pi$
$998$	0	0
$999$	−871829.	−0.0276387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.q.1.5		6
4.3	odd	2		65.6.a.d.1.2	✓	6
12.11	even	2		585.6.a.m.1.5		6
20.3	even	4		325.6.b.g.274.9		12
20.7	even	4		325.6.b.g.274.4		12
20.19	odd	2		325.6.a.g.1.5		6
52.51	odd	2		845.6.a.h.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.2	✓	6	4.3	odd	2
325.6.a.g.1.5		6	20.19	odd	2
325.6.b.g.274.4		12	20.7	even	4
325.6.b.g.274.9		12	20.3	even	4
585.6.a.m.1.5		6	12.11	even	2
845.6.a.h.1.5		6	52.51	odd	2
1040.6.a.q.1.5		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+7.05430 q^{3} -25.0000 q^{5} +185.746 q^{7} -193.237 q^{9} +353.912 q^{11} +169.000 q^{13} -176.358 q^{15} +634.652 q^{17} -1118.29 q^{19} +1310.31 q^{21} -3509.85 q^{23} +625.000 q^{25} -3077.35 q^{27} -3765.67 q^{29} -2906.63 q^{31} +2496.60 q^{33} -4643.65 q^{35} +283.305 q^{37} +1192.18 q^{39} -13563.6 q^{41} +5184.47 q^{43} +4830.92 q^{45} +6781.50 q^{47} +17694.6 q^{49} +4477.03 q^{51} +7664.43 q^{53} -8847.81 q^{55} -7888.78 q^{57} -2806.29 q^{59} -13764.7 q^{61} -35893.0 q^{63} -4225.00 q^{65} -67744.1 q^{67} -24759.5 q^{69} +66519.0 q^{71} +75902.7 q^{73} +4408.94 q^{75} +65737.9 q^{77} -101641. q^{79} +25248.0 q^{81} +50882.7 q^{83} -15866.3 q^{85} -26564.2 q^{87} -52439.2 q^{89} +31391.1 q^{91} -20504.2 q^{93} +27957.4 q^{95} -142557. q^{97} -68388.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 38 q^{3} - 150 q^{5} - 220 q^{7} + 518 q^{9} + 170 q^{11} + 1014 q^{13} + 950 q^{15} + 728 q^{17} - 1218 q^{19} - 396 q^{21} - 8954 q^{23} + 3750 q^{25} - 13112 q^{27} + 8364 q^{29} - 2862 q^{31}+ \cdots + 32270 q^{99}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

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