LMFDB - Embedded newform 560.6.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,6,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, 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("560.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.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:	$89.8149390953$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 232x^{2} + 60x + 5808 $ x^4 - 2x^3 - 232x^2 + 60*x + 5808
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.33412$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.9563 q^{3} +25.0000 q^{5} -49.0000 q^{7} -48.2216 q^{9} +O(q^{10})$	$q-13.9563 q^{3} +25.0000 q^{5} -49.0000 q^{7} -48.2216 q^{9} +473.121 q^{11} +22.8930 q^{13} -348.908 q^{15} -1781.22 q^{17} +2957.77 q^{19} +683.859 q^{21} -3336.13 q^{23} +625.000 q^{25} +4064.38 q^{27} -4717.44 q^{29} +1593.72 q^{31} -6603.02 q^{33} -1225.00 q^{35} +7789.36 q^{37} -319.502 q^{39} -5784.32 q^{41} +4478.15 q^{43} -1205.54 q^{45} -29535.1 q^{47} +2401.00 q^{49} +24859.3 q^{51} -24301.3 q^{53} +11828.0 q^{55} -41279.6 q^{57} +35855.1 q^{59} +12362.5 q^{61} +2362.86 q^{63} +572.326 q^{65} -30370.6 q^{67} +46560.0 q^{69} -9507.13 q^{71} +56604.5 q^{73} -8722.69 q^{75} -23182.9 q^{77} +75201.1 q^{79} -45005.8 q^{81} -36995.9 q^{83} -44530.6 q^{85} +65838.0 q^{87} -77946.0 q^{89} -1121.76 q^{91} -22242.4 q^{93} +73944.4 q^{95} -15751.9 q^{97} -22814.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9}+O(q^{10}) $ 4 * q + 13 * q^3 + 100 * q^5 - 196 * q^7 - 193 * q^9	$ 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9} + 595 q^{11} - 969 q^{13} + 325 q^{15} - 1315 q^{17} + 1090 q^{19} - 637 q^{21} + 1534 q^{23} + 2500 q^{25} - 173 q^{27} + 4099 q^{29} + 4820 q^{31} + 4149 q^{33} - 4900 q^{35} + 7692 q^{37} + 6371 q^{39} - 9722 q^{41} + 20610 q^{43} - 4825 q^{45} + 1661 q^{47} + 9604 q^{49} + 73361 q^{51} - 28898 q^{53} + 14875 q^{55} - 21246 q^{57} + 101872 q^{59} - 24742 q^{61} + 9457 q^{63} - 24225 q^{65} + 82060 q^{67} + 16914 q^{69} + 102784 q^{71} - 80652 q^{73} + 8125 q^{75} - 29155 q^{77} + 117801 q^{79} - 141052 q^{81} + 155440 q^{83} - 32875 q^{85} + 82519 q^{87} + 56426 q^{89} + 47481 q^{91} - 17332 q^{93} + 27250 q^{95} - 261031 q^{97} + 61686 q^{99}+O(q^{100}) $ 4 * q + 13 * q^3 + 100 * q^5 - 196 * q^7 - 193 * q^9 + 595 * q^11 - 969 * q^13 + 325 * q^15 - 1315 * q^17 + 1090 * q^19 - 637 * q^21 + 1534 * q^23 + 2500 * q^25 - 173 * q^27 + 4099 * q^29 + 4820 * q^31 + 4149 * q^33 - 4900 * q^35 + 7692 * q^37 + 6371 * q^39 - 9722 * q^41 + 20610 * q^43 - 4825 * q^45 + 1661 * q^47 + 9604 * q^49 + 73361 * q^51 - 28898 * q^53 + 14875 * q^55 - 21246 * q^57 + 101872 * q^59 - 24742 * q^61 + 9457 * q^63 - 24225 * q^65 + 82060 * q^67 + 16914 * q^69 + 102784 * q^71 - 80652 * q^73 + 8125 * q^75 - 29155 * q^77 + 117801 * q^79 - 141052 * q^81 + 155440 * q^83 - 32875 * q^85 + 82519 * q^87 + 56426 * q^89 + 47481 * q^91 - 17332 * q^93 + 27250 * q^95 - 261031 * q^97 + 61686 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−13.9563	−0.895297	−0.447649	−	0.894210i	$-0.647738\pi$
$3$	−13.9563	−0.895297	−0.447649	+	0.894210i	$0.647738\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−48.2216	−0.198443
$10$	0	0
$11$	473.121	1.17894	0.589468	−	0.807792i	$-0.299337\pi$
$11$	473.121	1.17894	0.589468	+	0.807792i	$0.299337\pi$
$12$	0	0
$13$	22.8930	0.0375703	0.0187852	−	0.999824i	$-0.494020\pi$
$13$	22.8930	0.0375703	0.0187852	+	0.999824i	$0.494020\pi$
$14$	0	0
$15$	−348.908	−0.400389
$16$	0	0
$17$	−1781.22	−1.49484	−0.747422	−	0.664349i	$-0.768709\pi$
$17$	−1781.22	−1.49484	−0.747422	+	0.664349i	$0.768709\pi$
$18$	0	0
$19$	2957.77	1.87967	0.939834	−	0.341632i	$-0.110980\pi$
$19$	2957.77	1.87967	0.939834	+	0.341632i	$0.110980\pi$
$20$	0	0
$21$	683.859	0.338391
$22$	0	0
$23$	−3336.13	−1.31499	−0.657496	−	0.753458i	$-0.728384\pi$
$23$	−3336.13	−1.31499	−0.657496	+	0.753458i	$0.728384\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	4064.38	1.07296
$28$	0	0
$29$	−4717.44	−1.04163	−0.520813	−	0.853671i	$-0.674371\pi$
$29$	−4717.44	−1.04163	−0.520813	+	0.853671i	$0.674371\pi$
$30$	0	0
$31$	1593.72	0.297856	0.148928	−	0.988848i	$-0.452418\pi$
$31$	1593.72	0.297856	0.148928	+	0.988848i	$0.452418\pi$
$32$	0	0
$33$	−6603.02	−1.05550
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	7789.36	0.935401	0.467700	−	0.883887i	$-0.345083\pi$
$37$	7789.36	0.935401	0.467700	+	0.883887i	$0.345083\pi$
$38$	0	0
$39$	−319.502	−0.0336366
$40$	0	0
$41$	−5784.32	−0.537394	−0.268697	−	0.963225i	$-0.586593\pi$
$41$	−5784.32	−0.537394	−0.268697	+	0.963225i	$0.586593\pi$
$42$	0	0
$43$	4478.15	0.369341	0.184671	−	0.982800i	$-0.440878\pi$
$43$	4478.15	0.369341	0.184671	+	0.982800i	$0.440878\pi$
$44$	0	0
$45$	−1205.54	−0.0887462
$46$	0	0
$47$	−29535.1	−1.95026	−0.975132	−	0.221627i	$-0.928863\pi$
$47$	−29535.1	−1.95026	−0.975132	+	0.221627i	$0.928863\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	24859.3	1.33833
$52$	0	0
$53$	−24301.3	−1.18834	−0.594168	−	0.804341i	$-0.702519\pi$
$53$	−24301.3	−1.18834	−0.594168	+	0.804341i	$0.702519\pi$
$54$	0	0
$55$	11828.0	0.527236
$56$	0	0
$57$	−41279.6	−1.68286
$58$	0	0
$59$	35855.1	1.34098	0.670488	−	0.741921i	$-0.266085\pi$
$59$	35855.1	1.34098	0.670488	+	0.741921i	$0.266085\pi$
$60$	0	0
$61$	12362.5	0.425383	0.212692	−	0.977119i	$-0.431777\pi$
$61$	12362.5	0.425383	0.212692	+	0.977119i	$0.431777\pi$
$62$	0	0
$63$	2362.86	0.0750043
$64$	0	0
$65$	572.326	0.0168020
$66$	0	0
$67$	−30370.6	−0.826543	−0.413272	−	0.910608i	$-0.635614\pi$
$67$	−30370.6	−0.826543	−0.413272	+	0.910608i	$0.635614\pi$
$68$	0	0
$69$	46560.0	1.17731
$70$	0	0
$71$	−9507.13	−0.223823	−0.111911	−	0.993718i	$-0.535697\pi$
$71$	−9507.13	−0.223823	−0.111911	+	0.993718i	$0.535697\pi$
$72$	0	0
$73$	56604.5	1.24321	0.621604	−	0.783331i	$-0.286481\pi$
$73$	56604.5	1.24321	0.621604	+	0.783331i	$0.286481\pi$
$74$	0	0
$75$	−8722.69	−0.179059
$76$	0	0
$77$	−23182.9	−0.445596
$78$	0	0
$79$	75201.1	1.35568	0.677838	−	0.735211i	$-0.262917\pi$
$79$	75201.1	1.35568	0.677838	+	0.735211i	$0.262917\pi$
$80$	0	0
$81$	−45005.8	−0.762178
$82$	0	0
$83$	−36995.9	−0.589465	−0.294733	−	0.955580i	$-0.595231\pi$
$83$	−36995.9	−0.589465	−0.294733	+	0.955580i	$0.595231\pi$
$84$	0	0
$85$	−44530.6	−0.668515
$86$	0	0
$87$	65838.0	0.932564
$88$	0	0
$89$	−77946.0	−1.04308	−0.521541	−	0.853226i	$-0.674643\pi$
$89$	−77946.0	−1.04308	−0.521541	+	0.853226i	$0.674643\pi$
$90$	0	0
$91$	−1121.76	−0.0142002
$92$	0	0
$93$	−22242.4	−0.266670
$94$	0	0
$95$	73944.4	0.840613
$96$	0	0
$97$	−15751.9	−0.169982	−0.0849910	−	0.996382i	$-0.527086\pi$
$97$	−15751.9	−0.169982	−0.0849910	+	0.996382i	$0.527086\pi$
$98$	0	0
$99$	−22814.6	−0.233951
$100$	0	0
$101$	−91977.6	−0.897178	−0.448589	−	0.893738i	$-0.648073\pi$
$101$	−91977.6	−0.897178	−0.448589	+	0.893738i	$0.648073\pi$
$102$	0	0
$103$	−10835.5	−0.100637	−0.0503185	−	0.998733i	$-0.516024\pi$
$103$	−10835.5	−0.100637	−0.0503185	+	0.998733i	$0.516024\pi$
$104$	0	0
$105$	17096.5	0.151333
$106$	0	0
$107$	215276.	1.81776	0.908880	−	0.417058i	$-0.136939\pi$
$107$	215276.	1.81776	0.908880	+	0.417058i	$0.136939\pi$
$108$	0	0
$109$	20472.3	0.165044	0.0825222	−	0.996589i	$-0.473702\pi$
$109$	20472.3	0.165044	0.0825222	+	0.996589i	$0.473702\pi$
$110$	0	0
$111$	−108711.	−0.837462
$112$	0	0
$113$	72335.2	0.532910	0.266455	−	0.963847i	$-0.414148\pi$
$113$	72335.2	0.532910	0.266455	+	0.963847i	$0.414148\pi$
$114$	0	0
$115$	−83403.2	−0.588082
$116$	0	0
$117$	−1103.94	−0.00745555
$118$	0	0
$119$	87279.9	0.564998
$120$	0	0
$121$	62792.2	0.389890
$122$	0	0
$123$	80727.8	0.481127
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	324470.	1.78511	0.892556	−	0.450936i	$-0.148910\pi$
$127$	324470.	1.78511	0.892556	+	0.450936i	$0.148910\pi$
$128$	0	0
$129$	−62498.4	−0.330670
$130$	0	0
$131$	198917.	1.01273	0.506365	−	0.862319i	$-0.330989\pi$
$131$	198917.	1.01273	0.506365	+	0.862319i	$0.330989\pi$
$132$	0	0
$133$	−144931.	−0.710448
$134$	0	0
$135$	101609.	0.479843
$136$	0	0
$137$	424147.	1.93070	0.965351	−	0.260957i	$-0.0840379\pi$
$137$	424147.	1.93070	0.965351	+	0.260957i	$0.0840379\pi$
$138$	0	0
$139$	423775.	1.86036	0.930182	−	0.367099i	$-0.119649\pi$
$139$	423775.	1.86036	0.930182	+	0.367099i	$0.119649\pi$
$140$	0	0
$141$	412200.	1.74607
$142$	0	0
$143$	10831.2	0.0442930
$144$	0	0
$145$	−117936.	−0.465829
$146$	0	0
$147$	−33509.1	−0.127900
$148$	0	0
$149$	357312.	1.31850	0.659252	−	0.751922i	$-0.270873\pi$
$149$	357312.	1.31850	0.659252	+	0.751922i	$0.270873\pi$
$150$	0	0
$151$	118548.	0.423110	0.211555	−	0.977366i	$-0.432147\pi$
$151$	118548.	0.423110	0.211555	+	0.977366i	$0.432147\pi$
$152$	0	0
$153$	85893.4	0.296641
$154$	0	0
$155$	39842.9	0.133205
$156$	0	0
$157$	218538.	0.707584	0.353792	−	0.935324i	$-0.384892\pi$
$157$	218538.	0.707584	0.353792	+	0.935324i	$0.384892\pi$
$158$	0	0
$159$	339156.	1.06391
$160$	0	0
$161$	163470.	0.497020
$162$	0	0
$163$	−261352.	−0.770471	−0.385236	−	0.922818i	$-0.625880\pi$
$163$	−261352.	−0.770471	−0.385236	+	0.922818i	$0.625880\pi$
$164$	0	0
$165$	−165075.	−0.472033
$166$	0	0
$167$	−61318.5	−0.170138	−0.0850688	−	0.996375i	$-0.527111\pi$
$167$	−61318.5	−0.170138	−0.0850688	+	0.996375i	$0.527111\pi$
$168$	0	0
$169$	−370769.	−0.998588
$170$	0	0
$171$	−142628.	−0.373006
$172$	0	0
$173$	597307.	1.51734	0.758670	−	0.651476i	$-0.225850\pi$
$173$	597307.	1.51734	0.758670	+	0.651476i	$0.225850\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−500405.	−1.20057
$178$	0	0
$179$	−663337.	−1.54740	−0.773698	−	0.633555i	$-0.781595\pi$
$179$	−663337.	−1.54740	−0.773698	+	0.633555i	$0.781595\pi$
$180$	0	0
$181$	651691.	1.47858	0.739291	−	0.673386i	$-0.235161\pi$
$181$	651691.	1.47858	0.739291	+	0.673386i	$0.235161\pi$
$182$	0	0
$183$	−172534.	−0.380844
$184$	0	0
$185$	194734.	0.418324
$186$	0	0
$187$	−842734.	−1.76233
$188$	0	0
$189$	−199154.	−0.405542
$190$	0	0
$191$	770455.	1.52814	0.764071	−	0.645132i	$-0.223197\pi$
$191$	770455.	1.52814	0.764071	+	0.645132i	$0.223197\pi$
$192$	0	0
$193$	530263.	1.02470	0.512351	−	0.858776i	$-0.328775\pi$
$193$	530263.	1.02470	0.512351	+	0.858776i	$0.328775\pi$
$194$	0	0
$195$	−7987.55	−0.0150427
$196$	0	0
$197$	237902.	0.436749	0.218374	−	0.975865i	$-0.429925\pi$
$197$	237902.	0.436749	0.218374	+	0.975865i	$0.429925\pi$
$198$	0	0
$199$	94676.7	0.169477	0.0847385	−	0.996403i	$-0.472995\pi$
$199$	94676.7	0.169477	0.0847385	+	0.996403i	$0.472995\pi$
$200$	0	0
$201$	423861.	0.740002
$202$	0	0
$203$	231155.	0.393697
$204$	0	0
$205$	−144608.	−0.240330
$206$	0	0
$207$	160873.	0.260950
$208$	0	0
$209$	1.39938e6	2.21601
$210$	0	0
$211$	103131.	0.159472	0.0797358	−	0.996816i	$-0.474592\pi$
$211$	103131.	0.159472	0.0797358	+	0.996816i	$0.474592\pi$
$212$	0	0
$213$	132684.	0.200388
$214$	0	0
$215$	111954.	0.165174
$216$	0	0
$217$	−78092.0	−0.112579
$218$	0	0
$219$	−789990.	−1.11304
$220$	0	0
$221$	−40777.6	−0.0561618
$222$	0	0
$223$	−1.35072e6	−1.81887	−0.909436	−	0.415844i	$-0.863486\pi$
$223$	−1.35072e6	−1.81887	−0.909436	+	0.415844i	$0.863486\pi$
$224$	0	0
$225$	−30138.5	−0.0396885
$226$	0	0
$227$	1.16439e6	1.49980	0.749899	−	0.661553i	$-0.230102\pi$
$227$	1.16439e6	1.49980	0.749899	+	0.661553i	$0.230102\pi$
$228$	0	0
$229$	−367456.	−0.463038	−0.231519	−	0.972830i	$-0.574369\pi$
$229$	−367456.	−0.463038	−0.231519	+	0.972830i	$0.574369\pi$
$230$	0	0
$231$	323548.	0.398941
$232$	0	0
$233$	336123.	0.405610	0.202805	−	0.979219i	$-0.434994\pi$
$233$	336123.	0.405610	0.202805	+	0.979219i	$0.434994\pi$
$234$	0	0
$235$	−738376.	−0.872184
$236$	0	0
$237$	−1.04953e6	−1.21373
$238$	0	0
$239$	253077.	0.286588	0.143294	−	0.989680i	$-0.454231\pi$
$239$	253077.	0.286588	0.143294	+	0.989680i	$0.454231\pi$
$240$	0	0
$241$	881500.	0.977641	0.488821	−	0.872384i	$-0.337427\pi$
$241$	881500.	0.977641	0.488821	+	0.872384i	$0.337427\pi$
$242$	0	0
$243$	−359528.	−0.390587
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	67712.4	0.0706197
$248$	0	0
$249$	516326.	0.527747
$250$	0	0
$251$	−1.46515e6	−1.46790	−0.733952	−	0.679202i	$-0.762326\pi$
$251$	−1.46515e6	−1.46790	−0.733952	+	0.679202i	$0.762326\pi$
$252$	0	0
$253$	−1.57839e6	−1.55029
$254$	0	0
$255$	621482.	0.598520
$256$	0	0
$257$	−1.33894e6	−1.26453	−0.632265	−	0.774752i	$-0.717875\pi$
$257$	−1.33894e6	−1.26453	−0.632265	+	0.774752i	$0.717875\pi$
$258$	0	0
$259$	−381679.	−0.353548
$260$	0	0
$261$	227482.	0.206703
$262$	0	0
$263$	−273438.	−0.243764	−0.121882	−	0.992545i	$-0.538893\pi$
$263$	−273438.	−0.243764	−0.121882	+	0.992545i	$0.538893\pi$
$264$	0	0
$265$	−607532.	−0.531440
$266$	0	0
$267$	1.08784e6	0.933869
$268$	0	0
$269$	958012.	0.807217	0.403608	−	0.914932i	$-0.367756\pi$
$269$	958012.	0.807217	0.403608	+	0.914932i	$0.367756\pi$
$270$	0	0
$271$	4501.33	0.00372321	0.00186161	−	0.999998i	$-0.499407\pi$
$271$	4501.33	0.00372321	0.00186161	+	0.999998i	$0.499407\pi$
$272$	0	0
$273$	15655.6	0.0127134
$274$	0	0
$275$	295700.	0.235787
$276$	0	0
$277$	1.73335e6	1.35733	0.678667	−	0.734446i	$-0.262558\pi$
$277$	1.73335e6	1.35733	0.678667	+	0.734446i	$0.262558\pi$
$278$	0	0
$279$	−76851.4	−0.0591073
$280$	0	0
$281$	−1.71229e6	−1.29363	−0.646817	−	0.762645i	$-0.723900\pi$
$281$	−1.71229e6	−1.29363	−0.646817	+	0.762645i	$0.723900\pi$
$282$	0	0
$283$	769075.	0.570824	0.285412	−	0.958405i	$-0.407870\pi$
$283$	769075.	0.570824	0.285412	+	0.958405i	$0.407870\pi$
$284$	0	0
$285$	−1.03199e6	−0.752598
$286$	0	0
$287$	283432.	0.203116
$288$	0	0
$289$	1.75290e6	1.23456
$290$	0	0
$291$	219838.	0.152185
$292$	0	0
$293$	−1.73492e6	−1.18062	−0.590309	−	0.807177i	$-0.700994\pi$
$293$	−1.73492e6	−1.18062	−0.590309	+	0.807177i	$0.700994\pi$
$294$	0	0
$295$	896377.	0.599702
$296$	0	0
$297$	1.92294e6	1.26495
$298$	0	0
$299$	−76374.0	−0.0494046
$300$	0	0
$301$	−219429.	−0.139598
$302$	0	0
$303$	1.28367e6	0.803241
$304$	0	0
$305$	309061.	0.190237
$306$	0	0
$307$	2.29034e6	1.38693	0.693465	−	0.720490i	$-0.256083\pi$
$307$	2.29034e6	1.38693	0.693465	+	0.720490i	$0.256083\pi$
$308$	0	0
$309$	151224.	0.0901001
$310$	0	0
$311$	1.13571e6	0.665834	0.332917	−	0.942956i	$-0.391967\pi$
$311$	1.13571e6	0.665834	0.332917	+	0.942956i	$0.391967\pi$
$312$	0	0
$313$	231342.	0.133473	0.0667364	−	0.997771i	$-0.478741\pi$
$313$	231342.	0.133473	0.0667364	+	0.997771i	$0.478741\pi$
$314$	0	0
$315$	59071.4	0.0335429
$316$	0	0
$317$	−1.43914e6	−0.804366	−0.402183	−	0.915559i	$-0.631749\pi$
$317$	−1.43914e6	−0.804366	−0.402183	+	0.915559i	$0.631749\pi$
$318$	0	0
$319$	−2.23192e6	−1.22801
$320$	0	0
$321$	−3.00446e6	−1.62744
$322$	0	0
$323$	−5.26846e6	−2.80981
$324$	0	0
$325$	14308.1	0.00751406
$326$	0	0
$327$	−285718.	−0.147764
$328$	0	0
$329$	1.44722e6	0.737130
$330$	0	0
$331$	81561.8	0.0409182	0.0204591	−	0.999791i	$-0.493487\pi$
$331$	81561.8	0.0409182	0.0204591	+	0.999791i	$0.493487\pi$
$332$	0	0
$333$	−375615.	−0.185623
$334$	0	0
$335$	−759264.	−0.369641
$336$	0	0
$337$	−880659.	−0.422409	−0.211204	−	0.977442i	$-0.567739\pi$
$337$	−880659.	−0.422409	−0.211204	+	0.977442i	$0.567739\pi$
$338$	0	0
$339$	−1.00953e6	−0.477113
$340$	0	0
$341$	754020.	0.351153
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.16400e6	0.526508
$346$	0	0
$347$	4.19657e6	1.87099	0.935493	−	0.353346i	$-0.114956\pi$
$347$	4.19657e6	1.87099	0.935493	+	0.353346i	$0.114956\pi$
$348$	0	0
$349$	−456804.	−0.200755	−0.100377	−	0.994949i	$-0.532005\pi$
$349$	−456804.	−0.200755	−0.100377	+	0.994949i	$0.532005\pi$
$350$	0	0
$351$	93045.9	0.0403115
$352$	0	0
$353$	−4.40223e6	−1.88034	−0.940169	−	0.340709i	$-0.889333\pi$
$353$	−4.40223e6	−1.88034	−0.940169	+	0.340709i	$0.889333\pi$
$354$	0	0
$355$	−237678.	−0.100096
$356$	0	0
$357$	−1.21811e6	−0.505841
$358$	0	0
$359$	−276721.	−0.113320	−0.0566599	−	0.998394i	$-0.518045\pi$
$359$	−276721.	−0.113320	−0.0566599	+	0.998394i	$0.518045\pi$
$360$	0	0
$361$	6.27233e6	2.53315
$362$	0	0
$363$	−876347.	−0.349068
$364$	0	0
$365$	1.41511e6	0.555980
$366$	0	0
$367$	44862.8	0.0173869	0.00869344	−	0.999962i	$-0.497233\pi$
$367$	44862.8	0.0173869	0.00869344	+	0.999962i	$0.497233\pi$
$368$	0	0
$369$	278929.	0.106642
$370$	0	0
$371$	1.19076e6	0.449149
$372$	0	0
$373$	−1.03702e6	−0.385935	−0.192967	−	0.981205i	$-0.561811\pi$
$373$	−1.03702e6	−0.385935	−0.192967	+	0.981205i	$0.561811\pi$
$374$	0	0
$375$	−218067.	−0.0800778
$376$	0	0
$377$	−107997.	−0.0391342
$378$	0	0
$379$	205535.	0.0735001	0.0367500	−	0.999324i	$-0.488299\pi$
$379$	205535.	0.0735001	0.0367500	+	0.999324i	$0.488299\pi$
$380$	0	0
$381$	−4.52841e6	−1.59821
$382$	0	0
$383$	−1.43359e6	−0.499375	−0.249687	−	0.968326i	$-0.580328\pi$
$383$	−1.43359e6	−0.499375	−0.249687	+	0.968326i	$0.580328\pi$
$384$	0	0
$385$	−579573.	−0.199277
$386$	0	0
$387$	−215943.	−0.0732930
$388$	0	0
$389$	2.66566e6	0.893163	0.446582	−	0.894743i	$-0.352641\pi$
$389$	2.66566e6	0.893163	0.446582	+	0.894743i	$0.352641\pi$
$390$	0	0
$391$	5.94239e6	1.96571
$392$	0	0
$393$	−2.77615e6	−0.906695
$394$	0	0
$395$	1.88003e6	0.606277
$396$	0	0
$397$	−4.94698e6	−1.57530	−0.787651	−	0.616122i	$-0.788703\pi$
$397$	−4.94698e6	−1.57530	−0.787651	+	0.616122i	$0.788703\pi$
$398$	0	0
$399$	2.02270e6	0.636062
$400$	0	0
$401$	67297.3	0.0208995	0.0104498	−	0.999945i	$-0.496674\pi$
$401$	67297.3	0.0208995	0.0104498	+	0.999945i	$0.496674\pi$
$402$	0	0
$403$	36485.0	0.0111905
$404$	0	0
$405$	−1.12515e6	−0.340856
$406$	0	0
$407$	3.68531e6	1.10278
$408$	0	0
$409$	4.30687e6	1.27307	0.636537	−	0.771246i	$-0.280366\pi$
$409$	4.30687e6	1.27307	0.636537	+	0.771246i	$0.280366\pi$
$410$	0	0
$411$	−5.91953e6	−1.72855
$412$	0	0
$413$	−1.75690e6	−0.506841
$414$	0	0
$415$	−924897.	−0.263617
$416$	0	0
$417$	−5.91433e6	−1.66558
$418$	0	0
$419$	4.40325e6	1.22529	0.612645	−	0.790358i	$-0.290106\pi$
$419$	4.40325e6	1.22529	0.612645	+	0.790358i	$0.290106\pi$
$420$	0	0
$421$	1.49685e6	0.411599	0.205800	−	0.978594i	$-0.434021\pi$
$421$	1.49685e6	0.411599	0.205800	+	0.978594i	$0.434021\pi$
$422$	0	0
$423$	1.42423e6	0.387015
$424$	0	0
$425$	−1.11326e6	−0.298969
$426$	0	0
$427$	−605761.	−0.160780
$428$	0	0
$429$	−151163.	−0.0396554
$430$	0	0
$431$	−3.10009e6	−0.803862	−0.401931	−	0.915670i	$-0.631661\pi$
$431$	−3.10009e6	−0.803862	−0.401931	+	0.915670i	$0.631661\pi$
$432$	0	0
$433$	−435162.	−0.111540	−0.0557701	−	0.998444i	$-0.517761\pi$
$433$	−435162.	−0.111540	−0.0557701	+	0.998444i	$0.517761\pi$
$434$	0	0
$435$	1.64595e6	0.417055
$436$	0	0
$437$	−9.86751e6	−2.47175
$438$	0	0
$439$	−6.80310e6	−1.68479	−0.842395	−	0.538860i	$-0.818855\pi$
$439$	−6.80310e6	−1.68479	−0.842395	+	0.538860i	$0.818855\pi$
$440$	0	0
$441$	−115780.	−0.0283489
$442$	0	0
$443$	−2.49011e6	−0.602850	−0.301425	−	0.953490i	$-0.597462\pi$
$443$	−2.49011e6	−0.602850	−0.301425	+	0.953490i	$0.597462\pi$
$444$	0	0
$445$	−1.94865e6	−0.466481
$446$	0	0
$447$	−4.98675e6	−1.18045
$448$	0	0
$449$	−186239.	−0.0435968	−0.0217984	−	0.999762i	$-0.506939\pi$
$449$	−186239.	−0.0435968	−0.0217984	+	0.999762i	$0.506939\pi$
$450$	0	0
$451$	−2.73668e6	−0.633553
$452$	0	0
$453$	−1.65450e6	−0.378809
$454$	0	0
$455$	−28044.0	−0.00635054
$456$	0	0
$457$	2.11539e6	0.473805	0.236902	−	0.971533i	$-0.423868\pi$
$457$	2.11539e6	0.473805	0.236902	+	0.971533i	$0.423868\pi$
$458$	0	0
$459$	−7.23956e6	−1.60391
$460$	0	0
$461$	−5.86679e6	−1.28573	−0.642863	−	0.765981i	$-0.722253\pi$
$461$	−5.86679e6	−1.28573	−0.642863	+	0.765981i	$0.722253\pi$
$462$	0	0
$463$	546789.	0.118541	0.0592704	−	0.998242i	$-0.481123\pi$
$463$	546789.	0.118541	0.0592704	+	0.998242i	$0.481123\pi$
$464$	0	0
$465$	−556059.	−0.119258
$466$	0	0
$467$	1.66105e6	0.352444	0.176222	−	0.984350i	$-0.443612\pi$
$467$	1.66105e6	0.352444	0.176222	+	0.984350i	$0.443612\pi$
$468$	0	0
$469$	1.48816e6	0.312404
$470$	0	0
$471$	−3.04998e6	−0.633498
$472$	0	0
$473$	2.11871e6	0.435430
$474$	0	0
$475$	1.84861e6	0.375933
$476$	0	0
$477$	1.17184e6	0.235816
$478$	0	0
$479$	181346.	0.0361136	0.0180568	−	0.999837i	$-0.494252\pi$
$479$	181346.	0.0361136	0.0180568	+	0.999837i	$0.494252\pi$
$480$	0	0
$481$	178322.	0.0351433
$482$	0	0
$483$	−2.28144e6	−0.444981
$484$	0	0
$485$	−393797.	−0.0760183
$486$	0	0
$487$	254424.	0.0486111	0.0243056	−	0.999705i	$-0.492263\pi$
$487$	254424.	0.0486111	0.0243056	+	0.999705i	$0.492263\pi$
$488$	0	0
$489$	3.64750e6	0.689801
$490$	0	0
$491$	7.01599e6	1.31336	0.656682	−	0.754168i	$-0.271959\pi$
$491$	7.01599e6	1.31336	0.656682	+	0.754168i	$0.271959\pi$
$492$	0	0
$493$	8.40282e6	1.55707
$494$	0	0
$495$	−570365.	−0.104626
$496$	0	0
$497$	465850.	0.0845970
$498$	0	0
$499$	2.72217e6	0.489400	0.244700	−	0.969599i	$-0.421311\pi$
$499$	2.72217e6	0.489400	0.244700	+	0.969599i	$0.421311\pi$
$500$	0	0
$501$	855780.	0.152324
$502$	0	0
$503$	1.91147e6	0.336859	0.168430	−	0.985714i	$-0.446130\pi$
$503$	1.91147e6	0.336859	0.168430	+	0.985714i	$0.446130\pi$
$504$	0	0
$505$	−2.29944e6	−0.401230
$506$	0	0
$507$	5.17456e6	0.894034
$508$	0	0
$509$	2.07515e6	0.355022	0.177511	−	0.984119i	$-0.443195\pi$
$509$	2.07515e6	0.355022	0.177511	+	0.984119i	$0.443195\pi$
$510$	0	0
$511$	−2.77362e6	−0.469889
$512$	0	0
$513$	1.20215e7	2.01681
$514$	0	0
$515$	−270889.	−0.0450063
$516$	0	0
$517$	−1.39736e7	−2.29924
$518$	0	0
$519$	−8.33620e6	−1.35847
$520$	0	0
$521$	−7.09369e6	−1.14493	−0.572464	−	0.819930i	$-0.694012\pi$
$521$	−7.09369e6	−1.14493	−0.572464	+	0.819930i	$0.694012\pi$
$522$	0	0
$523$	−9.40296e6	−1.50318	−0.751589	−	0.659631i	$-0.770712\pi$
$523$	−9.40296e6	−1.50318	−0.751589	+	0.659631i	$0.770712\pi$
$524$	0	0
$525$	427412.	0.0676781
$526$	0	0
$527$	−2.83876e6	−0.445249
$528$	0	0
$529$	4.69340e6	0.729202
$530$	0	0
$531$	−1.72899e6	−0.266107
$532$	0	0
$533$	−132421.	−0.0201901
$534$	0	0
$535$	5.38190e6	0.812927
$536$	0	0
$537$	9.25773e6	1.38538
$538$	0	0
$539$	1.13596e6	0.168419
$540$	0	0
$541$	−3.70767e6	−0.544638	−0.272319	−	0.962207i	$-0.587791\pi$
$541$	−3.70767e6	−0.544638	−0.272319	+	0.962207i	$0.587791\pi$
$542$	0	0
$543$	−9.09520e6	−1.32377
$544$	0	0
$545$	511808.	0.0738101
$546$	0	0
$547$	2.60109e6	0.371696	0.185848	−	0.982579i	$-0.440497\pi$
$547$	2.60109e6	0.371696	0.185848	+	0.982579i	$0.440497\pi$
$548$	0	0
$549$	−596137.	−0.0844142
$550$	0	0
$551$	−1.39531e7	−1.95791
$552$	0	0
$553$	−3.68485e6	−0.512398
$554$	0	0
$555$	−2.71777e6	−0.374524
$556$	0	0
$557$	1.23876e7	1.69180	0.845899	−	0.533343i	$-0.179065\pi$
$557$	1.23876e7	1.69180	0.845899	+	0.533343i	$0.179065\pi$
$558$	0	0
$559$	102518.	0.0138763
$560$	0	0
$561$	1.17614e7	1.57781
$562$	0	0
$563$	1.18095e6	0.157023	0.0785113	−	0.996913i	$-0.474983\pi$
$563$	1.18095e6	0.157023	0.0785113	+	0.996913i	$0.474983\pi$
$564$	0	0
$565$	1.80838e6	0.238324
$566$	0	0
$567$	2.20529e6	0.288076
$568$	0	0
$569$	4.06064e6	0.525792	0.262896	−	0.964824i	$-0.415322\pi$
$569$	4.06064e6	0.525792	0.262896	+	0.964824i	$0.415322\pi$
$570$	0	0
$571$	−868360.	−0.111458	−0.0557288	−	0.998446i	$-0.517748\pi$
$571$	−868360.	−0.111458	−0.0557288	+	0.998446i	$0.517748\pi$
$572$	0	0
$573$	−1.07527e7	−1.36814
$574$	0	0
$575$	−2.08508e6	−0.262998
$576$	0	0
$577$	−3.42545e6	−0.428330	−0.214165	−	0.976798i	$-0.568703\pi$
$577$	−3.42545e6	−0.428330	−0.214165	+	0.976798i	$0.568703\pi$
$578$	0	0
$579$	−7.40051e6	−0.917414
$580$	0	0
$581$	1.81280e6	0.222797
$582$	0	0
$583$	−1.14974e7	−1.40097
$584$	0	0
$585$	−27598.4	−0.00333422
$586$	0	0
$587$	7.37523e6	0.883446	0.441723	−	0.897151i	$-0.354367\pi$
$587$	7.37523e6	0.883446	0.441723	+	0.897151i	$0.354367\pi$
$588$	0	0
$589$	4.71385e6	0.559870
$590$	0	0
$591$	−3.32023e6	−0.391020
$592$	0	0
$593$	−681768.	−0.0796159	−0.0398080	−	0.999207i	$-0.512675\pi$
$593$	−681768.	−0.0796159	−0.0398080	+	0.999207i	$0.512675\pi$
$594$	0	0
$595$	2.18200e6	0.252675
$596$	0	0
$597$	−1.32134e6	−0.151732
$598$	0	0
$599$	1.65581e7	1.88558	0.942788	−	0.333392i	$-0.108193\pi$
$599$	1.65581e7	1.88558	0.942788	+	0.333392i	$0.108193\pi$
$600$	0	0
$601$	6.78009e6	0.765684	0.382842	−	0.923814i	$-0.374945\pi$
$601$	6.78009e6	0.765684	0.382842	+	0.923814i	$0.374945\pi$
$602$	0	0
$603$	1.46452e6	0.164021
$604$	0	0
$605$	1.56981e6	0.174364
$606$	0	0
$607$	−9.45843e6	−1.04195	−0.520975	−	0.853572i	$-0.674432\pi$
$607$	−9.45843e6	−1.04195	−0.520975	+	0.853572i	$0.674432\pi$
$608$	0	0
$609$	−3.22606e6	−0.352476
$610$	0	0
$611$	−676147.	−0.0732720
$612$	0	0
$613$	2.87909e6	0.309460	0.154730	−	0.987957i	$-0.450549\pi$
$613$	2.87909e6	0.309460	0.154730	+	0.987957i	$0.450549\pi$
$614$	0	0
$615$	2.01819e6	0.215167
$616$	0	0
$617$	−3.88639e6	−0.410992	−0.205496	−	0.978658i	$-0.565881\pi$
$617$	−3.88639e6	−0.410992	−0.205496	+	0.978658i	$0.565881\pi$
$618$	0	0
$619$	1.42098e7	1.49060	0.745302	−	0.666727i	$-0.232305\pi$
$619$	1.42098e7	1.49060	0.745302	+	0.666727i	$0.232305\pi$
$620$	0	0
$621$	−1.35593e7	−1.41094
$622$	0	0
$623$	3.81935e6	0.394248
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.95302e7	−1.98399
$628$	0	0
$629$	−1.38746e7	−1.39828
$630$	0	0
$631$	−7.34544e6	−0.734419	−0.367210	−	0.930138i	$-0.619687\pi$
$631$	−7.34544e6	−0.734419	−0.367210	+	0.930138i	$0.619687\pi$
$632$	0	0
$633$	−1.43933e6	−0.142775
$634$	0	0
$635$	8.11176e6	0.798327
$636$	0	0
$637$	54966.2	0.00536719
$638$	0	0
$639$	458449.	0.0444159
$640$	0	0
$641$	1.11128e7	1.06826	0.534131	−	0.845402i	$-0.320639\pi$
$641$	1.11128e7	1.06826	0.534131	+	0.845402i	$0.320639\pi$
$642$	0	0
$643$	−4.31469e6	−0.411549	−0.205775	−	0.978599i	$-0.565971\pi$
$643$	−4.31469e6	−0.411549	−0.205775	+	0.978599i	$0.565971\pi$
$644$	0	0
$645$	−1.56246e6	−0.147880
$646$	0	0
$647$	−2.97655e6	−0.279546	−0.139773	−	0.990184i	$-0.544637\pi$
$647$	−2.97655e6	−0.279546	−0.139773	+	0.990184i	$0.544637\pi$
$648$	0	0
$649$	1.69638e7	1.58092
$650$	0	0
$651$	1.08988e6	0.100792
$652$	0	0
$653$	1.30910e7	1.20141	0.600705	−	0.799471i	$-0.294887\pi$
$653$	1.30910e7	1.20141	0.600705	+	0.799471i	$0.294887\pi$
$654$	0	0
$655$	4.97293e6	0.452907
$656$	0	0
$657$	−2.72956e6	−0.246706
$658$	0	0
$659$	2.06932e7	1.85615	0.928076	−	0.372390i	$-0.121462\pi$
$659$	2.06932e7	1.85615	0.928076	+	0.372390i	$0.121462\pi$
$660$	0	0
$661$	1.52116e7	1.35416	0.677080	−	0.735910i	$-0.263245\pi$
$661$	1.52116e7	1.35416	0.677080	+	0.735910i	$0.263245\pi$
$662$	0	0
$663$	569105.	0.0502815
$664$	0	0
$665$	−3.62327e6	−0.317722
$666$	0	0
$667$	1.57380e7	1.36973
$668$	0	0
$669$	1.88510e7	1.62843
$670$	0	0
$671$	5.84894e6	0.501500
$672$	0	0
$673$	7.38700e6	0.628681	0.314340	−	0.949310i	$-0.398217\pi$
$673$	7.38700e6	0.628681	0.314340	+	0.949310i	$0.398217\pi$
$674$	0	0
$675$	2.54024e6	0.214593
$676$	0	0
$677$	−4.81532e6	−0.403788	−0.201894	−	0.979407i	$-0.564710\pi$
$677$	−4.81532e6	−0.403788	−0.201894	+	0.979407i	$0.564710\pi$
$678$	0	0
$679$	771842.	0.0642472
$680$	0	0
$681$	−1.62505e7	−1.34276
$682$	0	0
$683$	−1.65574e6	−0.135813	−0.0679063	−	0.997692i	$-0.521632\pi$
$683$	−1.65574e6	−0.135813	−0.0679063	+	0.997692i	$0.521632\pi$
$684$	0	0
$685$	1.06037e7	0.863436
$686$	0	0
$687$	5.12832e6	0.414556
$688$	0	0
$689$	−556330.	−0.0446462
$690$	0	0
$691$	8.91133e6	0.709982	0.354991	−	0.934870i	$-0.384484\pi$
$691$	8.91133e6	0.709982	0.354991	+	0.934870i	$0.384484\pi$
$692$	0	0
$693$	1.11792e6	0.0884252
$694$	0	0
$695$	1.05944e7	0.831980
$696$	0	0
$697$	1.03032e7	0.803321
$698$	0	0
$699$	−4.69103e6	−0.363141
$700$	0	0
$701$	−2.15507e7	−1.65640	−0.828201	−	0.560432i	$-0.810635\pi$
$701$	−2.15507e7	−1.65640	−0.828201	+	0.560432i	$0.810635\pi$
$702$	0	0
$703$	2.30392e7	1.75824
$704$	0	0
$705$	1.03050e7	0.780864
$706$	0	0
$707$	4.50690e6	0.339102
$708$	0	0
$709$	−3.59129e6	−0.268309	−0.134154	−	0.990960i	$-0.542832\pi$
$709$	−3.59129e6	−0.268309	−0.134154	+	0.990960i	$0.542832\pi$
$710$	0	0
$711$	−3.62631e6	−0.269024
$712$	0	0
$713$	−5.31684e6	−0.391678
$714$	0	0
$715$	270779.	0.0198084
$716$	0	0
$717$	−3.53202e6	−0.256581
$718$	0	0
$719$	1.19801e7	0.864251	0.432125	−	0.901814i	$-0.357764\pi$
$719$	1.19801e7	0.864251	0.432125	+	0.901814i	$0.357764\pi$
$720$	0	0
$721$	530942.	0.0380372
$722$	0	0
$723$	−1.23025e7	−0.875280
$724$	0	0
$725$	−2.94840e6	−0.208325
$726$	0	0
$727$	2.23234e7	1.56648	0.783238	−	0.621722i	$-0.213567\pi$
$727$	2.23234e7	1.56648	0.783238	+	0.621722i	$0.213567\pi$
$728$	0	0
$729$	1.59541e7	1.11187
$730$	0	0
$731$	−7.97659e6	−0.552108
$732$	0	0
$733$	−2.12772e7	−1.46270	−0.731349	−	0.682003i	$-0.761109\pi$
$733$	−2.12772e7	−1.46270	−0.731349	+	0.682003i	$0.761109\pi$
$734$	0	0
$735$	−837727.	−0.0571985
$736$	0	0
$737$	−1.43689e7	−0.974442
$738$	0	0
$739$	−683399.	−0.0460324	−0.0230162	−	0.999735i	$-0.507327\pi$
$739$	−683399.	−0.0460324	−0.0230162	+	0.999735i	$0.507327\pi$
$740$	0	0
$741$	−945015.	−0.0632256
$742$	0	0
$743$	8.89112e6	0.590860	0.295430	−	0.955364i	$-0.404537\pi$
$743$	8.89112e6	0.590860	0.295430	+	0.955364i	$0.404537\pi$
$744$	0	0
$745$	8.93280e6	0.589653
$746$	0	0
$747$	1.78400e6	0.116975
$748$	0	0
$749$	−1.05485e7	−0.687049
$750$	0	0
$751$	−2.33218e6	−0.150891	−0.0754454	−	0.997150i	$-0.524038\pi$
$751$	−2.33218e6	−0.150891	−0.0754454	+	0.997150i	$0.524038\pi$
$752$	0	0
$753$	2.04481e7	1.31421
$754$	0	0
$755$	2.96371e6	0.189220
$756$	0	0
$757$	2.09657e6	0.132975	0.0664875	−	0.997787i	$-0.478821\pi$
$757$	2.09657e6	0.132975	0.0664875	+	0.997787i	$0.478821\pi$
$758$	0	0
$759$	2.20285e7	1.38797
$760$	0	0
$761$	−5.23202e6	−0.327497	−0.163749	−	0.986502i	$-0.552359\pi$
$761$	−5.23202e6	−0.327497	−0.163749	+	0.986502i	$0.552359\pi$
$762$	0	0
$763$	−1.00314e6	−0.0623809
$764$	0	0
$765$	2.14733e6	0.132662
$766$	0	0
$767$	820832.	0.0503809
$768$	0	0
$769$	1.41419e7	0.862366	0.431183	−	0.902265i	$-0.358096\pi$
$769$	1.41419e7	0.862366	0.431183	+	0.902265i	$0.358096\pi$
$770$	0	0
$771$	1.86867e7	1.13213
$772$	0	0
$773$	−1.14012e7	−0.686282	−0.343141	−	0.939284i	$-0.611491\pi$
$773$	−1.14012e7	−0.686282	−0.343141	+	0.939284i	$0.611491\pi$
$774$	0	0
$775$	996072.	0.0595712
$776$	0	0
$777$	5.32683e6	0.316531
$778$	0	0
$779$	−1.71087e7	−1.01012
$780$	0	0
$781$	−4.49802e6	−0.263872
$782$	0	0
$783$	−1.91735e7	−1.11762
$784$	0	0
$785$	5.46345e6	0.316441
$786$	0	0
$787$	−1.97329e7	−1.13568	−0.567839	−	0.823140i	$-0.692220\pi$
$787$	−1.97329e7	−1.13568	−0.567839	+	0.823140i	$0.692220\pi$
$788$	0	0
$789$	3.81618e6	0.218241
$790$	0	0
$791$	−3.54443e6	−0.201421
$792$	0	0
$793$	283014.	0.0159818
$794$	0	0
$795$	8.47890e6	0.475797
$796$	0	0
$797$	−2.27816e7	−1.27040	−0.635198	−	0.772349i	$-0.719082\pi$
$797$	−2.27816e7	−1.27040	−0.635198	+	0.772349i	$0.719082\pi$
$798$	0	0
$799$	5.26085e7	2.91534
$800$	0	0
$801$	3.75868e6	0.206992
$802$	0	0
$803$	2.67808e7	1.46566
$804$	0	0
$805$	4.08675e6	0.222274
$806$	0	0
$807$	−1.33703e7	−0.722699
$808$	0	0
$809$	2.84077e7	1.52603	0.763017	−	0.646378i	$-0.223717\pi$
$809$	2.84077e7	1.52603	0.763017	+	0.646378i	$0.223717\pi$
$810$	0	0
$811$	1.44008e7	0.768838	0.384419	−	0.923159i	$-0.374402\pi$
$811$	1.44008e7	0.768838	0.384419	+	0.923159i	$0.374402\pi$
$812$	0	0
$813$	−62822.0	−0.00333338
$814$	0	0
$815$	−6.53379e6	−0.344565
$816$	0	0
$817$	1.32454e7	0.694238
$818$	0	0
$819$	54092.9	0.00281793
$820$	0	0
$821$	−1.29199e7	−0.668961	−0.334481	−	0.942403i	$-0.608561\pi$
$821$	−1.29199e7	−0.668961	−0.334481	+	0.942403i	$0.608561\pi$
$822$	0	0
$823$	−2.04860e7	−1.05428	−0.527142	−	0.849777i	$-0.676737\pi$
$823$	−2.04860e7	−1.05428	−0.527142	+	0.849777i	$0.676737\pi$
$824$	0	0
$825$	−4.12689e6	−0.211100
$826$	0	0
$827$	1.95049e7	0.991701	0.495851	−	0.868408i	$-0.334856\pi$
$827$	1.95049e7	0.991701	0.495851	+	0.868408i	$0.334856\pi$
$828$	0	0
$829$	−2.30466e7	−1.16472	−0.582360	−	0.812931i	$-0.697870\pi$
$829$	−2.30466e7	−1.16472	−0.582360	+	0.812931i	$0.697870\pi$
$830$	0	0
$831$	−2.41912e7	−1.21522
$832$	0	0
$833$	−4.27672e6	−0.213549
$834$	0	0
$835$	−1.53296e6	−0.0760879
$836$	0	0
$837$	6.47746e6	0.319588
$838$	0	0
$839$	723933.	0.0355053	0.0177527	−	0.999842i	$-0.494349\pi$
$839$	723933.	0.0355053	0.0177527	+	0.999842i	$0.494349\pi$
$840$	0	0
$841$	1.74310e6	0.0849830
$842$	0	0
$843$	2.38972e7	1.15819
$844$	0	0
$845$	−9.26922e6	−0.446582
$846$	0	0
$847$	−3.07682e6	−0.147365
$848$	0	0
$849$	−1.07334e7	−0.511057
$850$	0	0
$851$	−2.59863e7	−1.23004
$852$	0	0
$853$	6.46977e6	0.304450	0.152225	−	0.988346i	$-0.451356\pi$
$853$	6.46977e6	0.304450	0.152225	+	0.988346i	$0.451356\pi$
$854$	0	0
$855$	−3.56571e6	−0.166813
$856$	0	0
$857$	1.07017e7	0.497740	0.248870	−	0.968537i	$-0.419941\pi$
$857$	1.07017e7	0.497740	0.248870	+	0.968537i	$0.419941\pi$
$858$	0	0
$859$	−1.06996e7	−0.494748	−0.247374	−	0.968920i	$-0.579568\pi$
$859$	−1.06996e7	−0.494748	−0.247374	+	0.968920i	$0.579568\pi$
$860$	0	0
$861$	−3.95566e6	−0.181849
$862$	0	0
$863$	−1.56531e7	−0.715439	−0.357720	−	0.933829i	$-0.616446\pi$
$863$	−1.56531e7	−0.715439	−0.357720	+	0.933829i	$0.616446\pi$
$864$	0	0
$865$	1.49327e7	0.678575
$866$	0	0
$867$	−2.44640e7	−1.10530
$868$	0	0
$869$	3.55792e7	1.59826
$870$	0	0
$871$	−695274.	−0.0310535
$872$	0	0
$873$	759580.	0.0337317
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	7.20997e6	0.316544	0.158272	−	0.987396i	$-0.449408\pi$
$877$	7.20997e6	0.316544	0.158272	+	0.987396i	$0.449408\pi$
$878$	0	0
$879$	2.42130e7	1.05700
$880$	0	0
$881$	−1.42075e7	−0.616705	−0.308352	−	0.951272i	$-0.599778\pi$
$881$	−1.42075e7	−0.616705	−0.308352	+	0.951272i	$0.599778\pi$
$882$	0	0
$883$	1.79016e7	0.772661	0.386331	−	0.922360i	$-0.373742\pi$
$883$	1.79016e7	0.772661	0.386331	+	0.922360i	$0.373742\pi$
$884$	0	0
$885$	−1.25101e7	−0.536912
$886$	0	0
$887$	−1.01802e7	−0.434456	−0.217228	−	0.976121i	$-0.569701\pi$
$887$	−1.01802e7	−0.434456	−0.217228	+	0.976121i	$0.569701\pi$
$888$	0	0
$889$	−1.58990e7	−0.674709
$890$	0	0
$891$	−2.12932e7	−0.898559
$892$	0	0
$893$	−8.73580e7	−3.66585
$894$	0	0
$895$	−1.65834e7	−0.692017
$896$	0	0
$897$	1.06590e6	0.0442319
$898$	0	0
$899$	−7.51826e6	−0.310254
$900$	0	0
$901$	4.32860e7	1.77638
$902$	0	0
$903$	3.06242e6	0.124982
$904$	0	0
$905$	1.62923e7	0.661242
$906$	0	0
$907$	−1.28796e7	−0.519857	−0.259928	−	0.965628i	$-0.583699\pi$
$907$	−1.28796e7	−0.519857	−0.259928	+	0.965628i	$0.583699\pi$
$908$	0	0
$909$	4.43530e6	0.178038
$910$	0	0
$911$	3.33658e7	1.33201	0.666003	−	0.745949i	$-0.268004\pi$
$911$	3.33658e7	1.33201	0.666003	+	0.745949i	$0.268004\pi$
$912$	0	0
$913$	−1.75035e7	−0.694942
$914$	0	0
$915$	−4.31336e6	−0.170319
$916$	0	0
$917$	−9.74693e6	−0.382776
$918$	0	0
$919$	3.33616e7	1.30304	0.651521	−	0.758631i	$-0.274131\pi$
$919$	3.33616e7	1.30304	0.651521	+	0.758631i	$0.274131\pi$
$920$	0	0
$921$	−3.19647e7	−1.24172
$922$	0	0
$923$	−217647.	−0.00840908
$924$	0	0
$925$	4.86835e6	0.187080
$926$	0	0
$927$	522507.	0.0199707
$928$	0	0
$929$	−3.31001e7	−1.25832	−0.629159	−	0.777276i	$-0.716601\pi$
$929$	−3.31001e7	−1.25832	−0.629159	+	0.777276i	$0.716601\pi$
$930$	0	0
$931$	7.10162e6	0.268524
$932$	0	0
$933$	−1.58503e7	−0.596119
$934$	0	0
$935$	−2.10683e7	−0.788136
$936$	0	0
$937$	3.64876e7	1.35768	0.678838	−	0.734288i	$-0.262484\pi$
$937$	3.64876e7	1.35768	0.678838	+	0.734288i	$0.262484\pi$
$938$	0	0
$939$	−3.22867e6	−0.119498
$940$	0	0
$941$	−1.90067e7	−0.699732	−0.349866	−	0.936800i	$-0.613773\pi$
$941$	−1.90067e7	−0.699732	−0.349866	+	0.936800i	$0.613773\pi$
$942$	0	0
$943$	1.92972e7	0.706669
$944$	0	0
$945$	−4.97886e6	−0.181364
$946$	0	0
$947$	4.04213e7	1.46466	0.732328	−	0.680952i	$-0.238434\pi$
$947$	4.04213e7	1.46466	0.732328	+	0.680952i	$0.238434\pi$
$948$	0	0
$949$	1.29585e6	0.0467077
$950$	0	0
$951$	2.00850e7	0.720147
$952$	0	0
$953$	1.05318e7	0.375639	0.187820	−	0.982204i	$-0.439858\pi$
$953$	1.05318e7	0.375639	0.187820	+	0.982204i	$0.439858\pi$
$954$	0	0
$955$	1.92614e7	0.683406
$956$	0	0
$957$	3.11493e7	1.09943
$958$	0	0
$959$	−2.07832e7	−0.729736
$960$	0	0
$961$	−2.60892e7	−0.911282
$962$	0	0
$963$	−1.03810e7	−0.360721
$964$	0	0
$965$	1.32566e7	0.458261
$966$	0	0
$967$	−3.11101e7	−1.06988	−0.534940	−	0.844890i	$-0.679666\pi$
$967$	−3.11101e7	−1.06988	−0.534940	+	0.844890i	$0.679666\pi$
$968$	0	0
$969$	7.35282e7	2.51562
$970$	0	0
$971$	−1.47706e7	−0.502747	−0.251373	−	0.967890i	$-0.580882\pi$
$971$	−1.47706e7	−0.502747	−0.251373	+	0.967890i	$0.580882\pi$
$972$	0	0
$973$	−2.07650e7	−0.703152
$974$	0	0
$975$	−199689.	−0.00672732
$976$	0	0
$977$	4.13172e7	1.38482	0.692412	−	0.721502i	$-0.256548\pi$
$977$	4.13172e7	1.38482	0.692412	+	0.721502i	$0.256548\pi$
$978$	0	0
$979$	−3.68779e7	−1.22973
$980$	0	0
$981$	−987207.	−0.0327518
$982$	0	0
$983$	8.93062e6	0.294780	0.147390	−	0.989078i	$-0.452913\pi$
$983$	8.93062e6	0.294780	0.147390	+	0.989078i	$0.452913\pi$
$984$	0	0
$985$	5.94754e6	0.195320
$986$	0	0
$987$	−2.01978e7	−0.659951
$988$	0	0
$989$	−1.49397e7	−0.485680
$990$	0	0
$991$	−4.58196e7	−1.48206	−0.741032	−	0.671470i	$-0.765663\pi$
$991$	−4.58196e7	−1.48206	−0.741032	+	0.671470i	$0.765663\pi$
$992$	0	0
$993$	−1.13830e6	−0.0366340
$994$	0	0
$995$	2.36692e6	0.0757924
$996$	0	0
$997$	2.46900e7	0.786654	0.393327	−	0.919399i	$-0.371324\pi$
$997$	2.46900e7	0.786654	0.393327	+	0.919399i	$0.371324\pi$
$998$	0	0
$999$	3.16589e7	1.00365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.x.1.1		4
4.3	odd	2		280.6.a.g.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.g.1.4	✓	4	4.3	odd	2
560.6.a.x.1.1		4	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-13.9563 q^{3} +25.0000 q^{5} -49.0000 q^{7} -48.2216 q^{9} +O(q^{10})\)	\(q-13.9563 q^{3} +25.0000 q^{5} -49.0000 q^{7} -48.2216 q^{9} +473.121 q^{11} +22.8930 q^{13} -348.908 q^{15} -1781.22 q^{17} +2957.77 q^{19} +683.859 q^{21} -3336.13 q^{23} +625.000 q^{25} +4064.38 q^{27} -4717.44 q^{29} +1593.72 q^{31} -6603.02 q^{33} -1225.00 q^{35} +7789.36 q^{37} -319.502 q^{39} -5784.32 q^{41} +4478.15 q^{43} -1205.54 q^{45} -29535.1 q^{47} +2401.00 q^{49} +24859.3 q^{51} -24301.3 q^{53} +11828.0 q^{55} -41279.6 q^{57} +35855.1 q^{59} +12362.5 q^{61} +2362.86 q^{63} +572.326 q^{65} -30370.6 q^{67} +46560.0 q^{69} -9507.13 q^{71} +56604.5 q^{73} -8722.69 q^{75} -23182.9 q^{77} +75201.1 q^{79} -45005.8 q^{81} -36995.9 q^{83} -44530.6 q^{85} +65838.0 q^{87} -77946.0 q^{89} -1121.76 q^{91} -22242.4 q^{93} +73944.4 q^{95} -15751.9 q^{97} -22814.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9}+O(q^{10}) \) 4 * q + 13 * q^3 + 100 * q^5 - 196 * q^7 - 193 * q^9	\( 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9} + 595 q^{11} - 969 q^{13} + 325 q^{15} - 1315 q^{17} + 1090 q^{19} - 637 q^{21} + 1534 q^{23} + 2500 q^{25} - 173 q^{27} + 4099 q^{29} + 4820 q^{31} + 4149 q^{33} - 4900 q^{35} + 7692 q^{37} + 6371 q^{39} - 9722 q^{41} + 20610 q^{43} - 4825 q^{45} + 1661 q^{47} + 9604 q^{49} + 73361 q^{51} - 28898 q^{53} + 14875 q^{55} - 21246 q^{57} + 101872 q^{59} - 24742 q^{61} + 9457 q^{63} - 24225 q^{65} + 82060 q^{67} + 16914 q^{69} + 102784 q^{71} - 80652 q^{73} + 8125 q^{75} - 29155 q^{77} + 117801 q^{79} - 141052 q^{81} + 155440 q^{83} - 32875 q^{85} + 82519 q^{87} + 56426 q^{89} + 47481 q^{91} - 17332 q^{93} + 27250 q^{95} - 261031 q^{97} + 61686 q^{99}+O(q^{100}) \) 4 * q + 13 * q^3 + 100 * q^5 - 196 * q^7 - 193 * q^9 + 595 * q^11 - 969 * q^13 + 325 * q^15 - 1315 * q^17 + 1090 * q^19 - 637 * q^21 + 1534 * q^23 + 2500 * q^25 - 173 * q^27 + 4099 * q^29 + 4820 * q^31 + 4149 * q^33 - 4900 * q^35 + 7692 * q^37 + 6371 * q^39 - 9722 * q^41 + 20610 * q^43 - 4825 * q^45 + 1661 * q^47 + 9604 * q^49 + 73361 * q^51 - 28898 * q^53 + 14875 * q^55 - 21246 * q^57 + 101872 * q^59 - 24742 * q^61 + 9457 * q^63 - 24225 * q^65 + 82060 * q^67 + 16914 * q^69 + 102784 * q^71 - 80652 * q^73 + 8125 * q^75 - 29155 * q^77 + 117801 * q^79 - 141052 * q^81 + 155440 * q^83 - 32875 * q^85 + 82519 * q^87 + 56426 * q^89 + 47481 * q^91 - 17332 * q^93 + 27250 * q^95 - 261031 * q^97 + 61686 * q^99

Introduction

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