LMFDB - Embedded newform 1104.6.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, 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("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 3225x^{5} + 19410x^{4} + 2132445x^{3} - 10443621x^{2} - 341555347x - 181104660 $ x^7 - 3x^6 - 3225x^5 + 19410x^4 + 2132445x^3 - 10443621x^2 - 341555347x - 181104660
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$21.5006$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -57.0012 q^{5} -23.4975 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -57.0012 q^{5} -23.4975 q^{7} +81.0000 q^{9} +720.537 q^{11} +226.924 q^{13} +513.011 q^{15} +2126.62 q^{17} -2811.79 q^{19} +211.477 q^{21} +529.000 q^{23} +124.142 q^{25} -729.000 q^{27} +6226.03 q^{29} +8059.12 q^{31} -6484.83 q^{33} +1339.39 q^{35} -10685.2 q^{37} -2042.32 q^{39} +17237.1 q^{41} +14006.1 q^{43} -4617.10 q^{45} +6497.44 q^{47} -16254.9 q^{49} -19139.6 q^{51} -15653.8 q^{53} -41071.5 q^{55} +25306.1 q^{57} +848.107 q^{59} -44644.8 q^{61} -1903.30 q^{63} -12935.0 q^{65} -26295.0 q^{67} -4761.00 q^{69} -14740.8 q^{71} -54308.7 q^{73} -1117.28 q^{75} -16930.8 q^{77} +71642.0 q^{79} +6561.00 q^{81} -8890.20 q^{83} -121220. q^{85} -56034.3 q^{87} -53822.3 q^{89} -5332.15 q^{91} -72532.1 q^{93} +160275. q^{95} -154764. q^{97} +58363.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9}+O(q^{10}) $ 7 * q - 63 * q^3 - 104 * q^5 + 182 * q^7 + 567 * q^9	$ 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9} + 124 q^{11} + 294 q^{13} + 936 q^{15} + 428 q^{17} + 1826 q^{19} - 1638 q^{21} + 3703 q^{23} + 5501 q^{25} - 5103 q^{27} - 1682 q^{29} + 14420 q^{31} - 1116 q^{33} - 3312 q^{35} - 14218 q^{37} - 2646 q^{39} + 7294 q^{41} - 3630 q^{43} - 8424 q^{45} + 19808 q^{47} - 4325 q^{49} - 3852 q^{51} - 77412 q^{53} + 61136 q^{55} - 16434 q^{57} + 51076 q^{59} - 67186 q^{61} + 14742 q^{63} - 121800 q^{65} + 70870 q^{67} - 33327 q^{69} + 45784 q^{71} - 175522 q^{73} - 49509 q^{75} - 148632 q^{77} + 54538 q^{79} + 45927 q^{81} - 27612 q^{83} - 197552 q^{85} + 15138 q^{87} - 184360 q^{89} + 217412 q^{91} - 129780 q^{93} + 82608 q^{95} - 339766 q^{97} + 10044 q^{99}+O(q^{100}) $ 7 * q - 63 * q^3 - 104 * q^5 + 182 * q^7 + 567 * q^9 + 124 * q^11 + 294 * q^13 + 936 * q^15 + 428 * q^17 + 1826 * q^19 - 1638 * q^21 + 3703 * q^23 + 5501 * q^25 - 5103 * q^27 - 1682 * q^29 + 14420 * q^31 - 1116 * q^33 - 3312 * q^35 - 14218 * q^37 - 2646 * q^39 + 7294 * q^41 - 3630 * q^43 - 8424 * q^45 + 19808 * q^47 - 4325 * q^49 - 3852 * q^51 - 77412 * q^53 + 61136 * q^55 - 16434 * q^57 + 51076 * q^59 - 67186 * q^61 + 14742 * q^63 - 121800 * q^65 + 70870 * q^67 - 33327 * q^69 + 45784 * q^71 - 175522 * q^73 - 49509 * q^75 - 148632 * q^77 + 54538 * q^79 + 45927 * q^81 - 27612 * q^83 - 197552 * q^85 + 15138 * q^87 - 184360 * q^89 + 217412 * q^91 - 129780 * q^93 + 82608 * q^95 - 339766 * q^97 + 10044 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−57.0012	−1.01967	−0.509835	−	0.860272i	$-0.670293\pi$
$5$	−57.0012	−1.01967	−0.509835	+	0.860272i	$0.670293\pi$
$6$	0	0
$7$	−23.4975	−0.181249	−0.0906246	−	0.995885i	$-0.528886\pi$
$7$	−23.4975	−0.181249	−0.0906246	+	0.995885i	$0.528886\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	720.537	1.79546	0.897728	−	0.440551i	$-0.145217\pi$
$11$	720.537	1.79546	0.897728	+	0.440551i	$0.145217\pi$
$12$	0	0
$13$	226.924	0.372411	0.186206	−	0.982511i	$-0.440381\pi$
$13$	226.924	0.372411	0.186206	+	0.982511i	$0.440381\pi$
$14$	0	0
$15$	513.011	0.588706
$16$	0	0
$17$	2126.62	1.78471	0.892354	−	0.451337i	$-0.149053\pi$
$17$	2126.62	1.78471	0.892354	+	0.451337i	$0.149053\pi$
$18$	0	0
$19$	−2811.79	−1.78689	−0.893446	−	0.449170i	$-0.851720\pi$
$19$	−2811.79	−1.78689	−0.893446	+	0.449170i	$0.851720\pi$
$20$	0	0
$21$	211.477	0.104644
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	124.142	0.0397254
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	6226.03	1.37473	0.687363	−	0.726314i	$-0.258768\pi$
$29$	6226.03	1.37473	0.687363	+	0.726314i	$0.258768\pi$
$30$	0	0
$31$	8059.12	1.50620	0.753101	−	0.657905i	$-0.228557\pi$
$31$	8059.12	1.50620	0.753101	+	0.657905i	$0.228557\pi$
$32$	0	0
$33$	−6484.83	−1.03661
$34$	0	0
$35$	1339.39	0.184814
$36$	0	0
$37$	−10685.2	−1.28315	−0.641575	−	0.767060i	$-0.721719\pi$
$37$	−10685.2	−1.28315	−0.641575	+	0.767060i	$0.721719\pi$
$38$	0	0
$39$	−2042.32	−0.215012
$40$	0	0
$41$	17237.1	1.60142	0.800710	−	0.599052i	$-0.204456\pi$
$41$	17237.1	1.60142	0.800710	+	0.599052i	$0.204456\pi$
$42$	0	0
$43$	14006.1	1.15517	0.577587	−	0.816329i	$-0.303994\pi$
$43$	14006.1	1.15517	0.577587	+	0.816329i	$0.303994\pi$
$44$	0	0
$45$	−4617.10	−0.339890
$46$	0	0
$47$	6497.44	0.429040	0.214520	−	0.976720i	$-0.431181\pi$
$47$	6497.44	0.429040	0.214520	+	0.976720i	$0.431181\pi$
$48$	0	0
$49$	−16254.9	−0.967149
$50$	0	0
$51$	−19139.6	−1.03040
$52$	0	0
$53$	−15653.8	−0.765473	−0.382737	−	0.923858i	$-0.625018\pi$
$53$	−15653.8	−0.765473	−0.382737	+	0.923858i	$0.625018\pi$
$54$	0	0
$55$	−41071.5	−1.83077
$56$	0	0
$57$	25306.1	1.03166
$58$	0	0
$59$	848.107	0.0317191	0.0158595	−	0.999874i	$-0.494952\pi$
$59$	848.107	0.0317191	0.0158595	+	0.999874i	$0.494952\pi$
$60$	0	0
$61$	−44644.8	−1.53619	−0.768097	−	0.640334i	$-0.778796\pi$
$61$	−44644.8	−1.53619	−0.768097	+	0.640334i	$0.778796\pi$
$62$	0	0
$63$	−1903.30	−0.0604164
$64$	0	0
$65$	−12935.0	−0.379736
$66$	0	0
$67$	−26295.0	−0.715626	−0.357813	−	0.933793i	$-0.616478\pi$
$67$	−26295.0	−0.715626	−0.357813	+	0.933793i	$0.616478\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	−14740.8	−0.347037	−0.173519	−	0.984831i	$-0.555514\pi$
$71$	−14740.8	−0.347037	−0.173519	+	0.984831i	$0.555514\pi$
$72$	0	0
$73$	−54308.7	−1.19279	−0.596393	−	0.802693i	$-0.703400\pi$
$73$	−54308.7	−1.19279	−0.596393	+	0.802693i	$0.703400\pi$
$74$	0	0
$75$	−1117.28	−0.0229355
$76$	0	0
$77$	−16930.8	−0.325425
$78$	0	0
$79$	71642.0	1.29152	0.645758	−	0.763542i	$-0.276541\pi$
$79$	71642.0	1.29152	0.645758	+	0.763542i	$0.276541\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−8890.20	−0.141650	−0.0708249	−	0.997489i	$-0.522563\pi$
$83$	−8890.20	−0.141650	−0.0708249	+	0.997489i	$0.522563\pi$
$84$	0	0
$85$	−121220.	−1.81981
$86$	0	0
$87$	−56034.3	−0.793698
$88$	0	0
$89$	−53822.3	−0.720257	−0.360128	−	0.932903i	$-0.617267\pi$
$89$	−53822.3	−0.720257	−0.360128	+	0.932903i	$0.617267\pi$
$90$	0	0
$91$	−5332.15	−0.0674993
$92$	0	0
$93$	−72532.1	−0.869606
$94$	0	0
$95$	160275.	1.82204
$96$	0	0
$97$	−154764.	−1.67009	−0.835045	−	0.550182i	$-0.814558\pi$
$97$	−154764.	−1.67009	−0.835045	+	0.550182i	$0.814558\pi$
$98$	0	0
$99$	58363.5	0.598485
$100$	0	0
$101$	100516.	0.980464	0.490232	−	0.871592i	$-0.336912\pi$
$101$	100516.	0.980464	0.490232	+	0.871592i	$0.336912\pi$
$102$	0	0
$103$	85368.4	0.792874	0.396437	−	0.918062i	$-0.370247\pi$
$103$	85368.4	0.792874	0.396437	+	0.918062i	$0.370247\pi$
$104$	0	0
$105$	−12054.5	−0.106703
$106$	0	0
$107$	−38065.9	−0.321423	−0.160711	−	0.987001i	$-0.551379\pi$
$107$	−38065.9	−0.321423	−0.160711	+	0.987001i	$0.551379\pi$
$108$	0	0
$109$	193254.	1.55798	0.778991	−	0.627036i	$-0.215732\pi$
$109$	193254.	1.55798	0.778991	+	0.627036i	$0.215732\pi$
$110$	0	0
$111$	96166.7	0.740828
$112$	0	0
$113$	−56460.9	−0.415960	−0.207980	−	0.978133i	$-0.566689\pi$
$113$	−56460.9	−0.415960	−0.207980	+	0.978133i	$0.566689\pi$
$114$	0	0
$115$	−30153.7	−0.212616
$116$	0	0
$117$	18380.9	0.124137
$118$	0	0
$119$	−49970.1	−0.323477
$120$	0	0
$121$	358123.	2.22366
$122$	0	0
$123$	−155134.	−0.924580
$124$	0	0
$125$	171053.	0.979162
$126$	0	0
$127$	171140.	0.941547	0.470773	−	0.882254i	$-0.343975\pi$
$127$	171140.	0.941547	0.470773	+	0.882254i	$0.343975\pi$
$128$	0	0
$129$	−126055.	−0.666940
$130$	0	0
$131$	−90404.1	−0.460267	−0.230134	−	0.973159i	$-0.573916\pi$
$131$	−90404.1	−0.460267	−0.230134	+	0.973159i	$0.573916\pi$
$132$	0	0
$133$	66069.9	0.323873
$134$	0	0
$135$	41553.9	0.196235
$136$	0	0
$137$	−328131.	−1.49364	−0.746820	−	0.665026i	$-0.768420\pi$
$137$	−328131.	−1.49364	−0.746820	+	0.665026i	$0.768420\pi$
$138$	0	0
$139$	256584.	1.12640	0.563199	−	0.826321i	$-0.309570\pi$
$139$	256584.	1.12640	0.563199	+	0.826321i	$0.309570\pi$
$140$	0	0
$141$	−58477.0	−0.247706
$142$	0	0
$143$	163507.	0.668648
$144$	0	0
$145$	−354891.	−1.40177
$146$	0	0
$147$	146294.	0.558384
$148$	0	0
$149$	−44916.7	−0.165746	−0.0828729	−	0.996560i	$-0.526410\pi$
$149$	−44916.7	−0.165746	−0.0828729	+	0.996560i	$0.526410\pi$
$150$	0	0
$151$	−303123.	−1.08187	−0.540937	−	0.841063i	$-0.681930\pi$
$151$	−303123.	−1.08187	−0.540937	+	0.841063i	$0.681930\pi$
$152$	0	0
$153$	172256.	0.594902
$154$	0	0
$155$	−459380.	−1.53583
$156$	0	0
$157$	353675.	1.14513	0.572566	−	0.819859i	$-0.305948\pi$
$157$	353675.	1.14513	0.572566	+	0.819859i	$0.305948\pi$
$158$	0	0
$159$	140884.	0.441946
$160$	0	0
$161$	−12430.2	−0.0377931
$162$	0	0
$163$	164347.	0.484500	0.242250	−	0.970214i	$-0.422115\pi$
$163$	164347.	0.484500	0.242250	+	0.970214i	$0.422115\pi$
$164$	0	0
$165$	369644.	1.05700
$166$	0	0
$167$	174090.	0.483041	0.241520	−	0.970396i	$-0.422354\pi$
$167$	174090.	0.483041	0.241520	+	0.970396i	$0.422354\pi$
$168$	0	0
$169$	−319798.	−0.861310
$170$	0	0
$171$	−227755.	−0.595631
$172$	0	0
$173$	−436028.	−1.10764	−0.553821	−	0.832636i	$-0.686831\pi$
$173$	−436028.	−1.10764	−0.553821	+	0.832636i	$0.686831\pi$
$174$	0	0
$175$	−2917.02	−0.00720020
$176$	0	0
$177$	−7632.96	−0.0183130
$178$	0	0
$179$	285491.	0.665978	0.332989	−	0.942931i	$-0.391943\pi$
$179$	285491.	0.665978	0.332989	+	0.942931i	$0.391943\pi$
$180$	0	0
$181$	192373.	0.436463	0.218231	−	0.975897i	$-0.429971\pi$
$181$	192373.	0.436463	0.218231	+	0.975897i	$0.429971\pi$
$182$	0	0
$183$	401803.	0.886922
$184$	0	0
$185$	609069.	1.30839
$186$	0	0
$187$	1.53231e6	3.20436
$188$	0	0
$189$	17129.7	0.0348814
$190$	0	0
$191$	472901.	0.937965	0.468983	−	0.883207i	$-0.344621\pi$
$191$	472901.	0.937965	0.468983	+	0.883207i	$0.344621\pi$
$192$	0	0
$193$	190127.	0.367409	0.183705	−	0.982981i	$-0.441191\pi$
$193$	190127.	0.367409	0.183705	+	0.982981i	$0.441191\pi$
$194$	0	0
$195$	116415.	0.219241
$196$	0	0
$197$	225854.	0.414631	0.207316	−	0.978274i	$-0.433527\pi$
$197$	225854.	0.414631	0.207316	+	0.978274i	$0.433527\pi$
$198$	0	0
$199$	251597.	0.450373	0.225186	−	0.974316i	$-0.427701\pi$
$199$	251597.	0.450373	0.225186	+	0.974316i	$0.427701\pi$
$200$	0	0
$201$	236655.	0.413167
$202$	0	0
$203$	−146296.	−0.249168
$204$	0	0
$205$	−982538.	−1.63292
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	−2.02600e6	−3.20829
$210$	0	0
$211$	930590.	1.43897	0.719486	−	0.694507i	$-0.244377\pi$
$211$	930590.	1.43897	0.719486	+	0.694507i	$0.244377\pi$
$212$	0	0
$213$	132667.	0.200362
$214$	0	0
$215$	−798367.	−1.17790
$216$	0	0
$217$	−189369.	−0.272998
$218$	0	0
$219$	488779.	0.688655
$220$	0	0
$221$	482581.	0.664645
$222$	0	0
$223$	8445.42	0.0113726	0.00568629	−	0.999984i	$-0.498190\pi$
$223$	8445.42	0.0113726	0.00568629	+	0.999984i	$0.498190\pi$
$224$	0	0
$225$	10055.5	0.0132418
$226$	0	0
$227$	−473820.	−0.610308	−0.305154	−	0.952303i	$-0.598708\pi$
$227$	−473820.	−0.610308	−0.305154	+	0.952303i	$0.598708\pi$
$228$	0	0
$229$	−1.24614e6	−1.57029	−0.785144	−	0.619314i	$-0.787411\pi$
$229$	−1.24614e6	−1.57029	−0.785144	+	0.619314i	$0.787411\pi$
$230$	0	0
$231$	152377.	0.187884
$232$	0	0
$233$	209762.	0.253126	0.126563	−	0.991959i	$-0.459605\pi$
$233$	209762.	0.253126	0.126563	+	0.991959i	$0.459605\pi$
$234$	0	0
$235$	−370362.	−0.437479
$236$	0	0
$237$	−644778.	−0.745657
$238$	0	0
$239$	1.69750e6	1.92227	0.961135	−	0.276079i	$-0.0890352\pi$
$239$	1.69750e6	1.92227	0.961135	+	0.276079i	$0.0890352\pi$
$240$	0	0
$241$	732600.	0.812502	0.406251	−	0.913762i	$-0.366836\pi$
$241$	732600.	0.812502	0.406251	+	0.913762i	$0.366836\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	926548.	0.986172
$246$	0	0
$247$	−638063.	−0.665459
$248$	0	0
$249$	80011.8	0.0817816
$250$	0	0
$251$	−961120.	−0.962927	−0.481463	−	0.876466i	$-0.659895\pi$
$251$	−961120.	−0.962927	−0.481463	+	0.876466i	$0.659895\pi$
$252$	0	0
$253$	381164.	0.374378
$254$	0	0
$255$	1.09098e6	1.05067
$256$	0	0
$257$	1.98942e6	1.87886	0.939428	−	0.342747i	$-0.111357\pi$
$257$	1.98942e6	1.87886	0.939428	+	0.342747i	$0.111357\pi$
$258$	0	0
$259$	251075.	0.232570
$260$	0	0
$261$	504308.	0.458242
$262$	0	0
$263$	1.64450e6	1.46604	0.733019	−	0.680208i	$-0.238111\pi$
$263$	1.64450e6	1.46604	0.733019	+	0.680208i	$0.238111\pi$
$264$	0	0
$265$	892286.	0.780530
$266$	0	0
$267$	484401.	0.415840
$268$	0	0
$269$	−1.86743e6	−1.57349	−0.786744	−	0.617280i	$-0.788235\pi$
$269$	−1.86743e6	−1.57349	−0.786744	+	0.617280i	$0.788235\pi$
$270$	0	0
$271$	366104.	0.302818	0.151409	−	0.988471i	$-0.451619\pi$
$271$	366104.	0.302818	0.151409	+	0.988471i	$0.451619\pi$
$272$	0	0
$273$	47989.4	0.0389707
$274$	0	0
$275$	89448.8	0.0713252
$276$	0	0
$277$	1.87770e6	1.47037	0.735183	−	0.677868i	$-0.237096\pi$
$277$	1.87770e6	1.47037	0.735183	+	0.677868i	$0.237096\pi$
$278$	0	0
$279$	652789.	0.502067
$280$	0	0
$281$	−1.82396e6	−1.37800	−0.689002	−	0.724759i	$-0.741951\pi$
$281$	−1.82396e6	−1.37800	−0.689002	+	0.724759i	$0.741951\pi$
$282$	0	0
$283$	−2.27283e6	−1.68694	−0.843472	−	0.537173i	$-0.819492\pi$
$283$	−2.27283e6	−1.68694	−0.843472	+	0.537173i	$0.819492\pi$
$284$	0	0
$285$	−1.44248e6	−1.05196
$286$	0	0
$287$	−405029.	−0.290256
$288$	0	0
$289$	3.10264e6	2.18518
$290$	0	0
$291$	1.39287e6	0.964226
$292$	0	0
$293$	−45400.3	−0.0308951	−0.0154475	−	0.999881i	$-0.504917\pi$
$293$	−45400.3	−0.0308951	−0.0154475	+	0.999881i	$0.504917\pi$
$294$	0	0
$295$	−48343.1	−0.0323430
$296$	0	0
$297$	−525272.	−0.345536
$298$	0	0
$299$	120043.	0.0776531
$300$	0	0
$301$	−329109.	−0.209374
$302$	0	0
$303$	−904644.	−0.566071
$304$	0	0
$305$	2.54481e6	1.56641
$306$	0	0
$307$	1.63603e6	0.990705	0.495352	−	0.868692i	$-0.335039\pi$
$307$	1.63603e6	0.990705	0.495352	+	0.868692i	$0.335039\pi$
$308$	0	0
$309$	−768316.	−0.457766
$310$	0	0
$311$	1.86604e6	1.09400	0.547002	−	0.837131i	$-0.315769\pi$
$311$	1.86604e6	1.09400	0.547002	+	0.837131i	$0.315769\pi$
$312$	0	0
$313$	−296946.	−0.171324	−0.0856618	−	0.996324i	$-0.527300\pi$
$313$	−296946.	−0.171324	−0.0856618	+	0.996324i	$0.527300\pi$
$314$	0	0
$315$	108490.	0.0616048
$316$	0	0
$317$	−235701.	−0.131739	−0.0658693	−	0.997828i	$-0.520982\pi$
$317$	−235701.	−0.131739	−0.0658693	+	0.997828i	$0.520982\pi$
$318$	0	0
$319$	4.48608e6	2.46826
$320$	0	0
$321$	342593.	0.185574
$322$	0	0
$323$	−5.97959e6	−3.18908
$324$	0	0
$325$	28170.8	0.0147942
$326$	0	0
$327$	−1.73929e6	−0.899501
$328$	0	0
$329$	−152674.	−0.0777632
$330$	0	0
$331$	−905911.	−0.454481	−0.227240	−	0.973839i	$-0.572970\pi$
$331$	−905911.	−0.454481	−0.227240	+	0.973839i	$0.572970\pi$
$332$	0	0
$333$	−865500.	−0.427717
$334$	0	0
$335$	1.49885e6	0.729702
$336$	0	0
$337$	3.91605e6	1.87834	0.939170	−	0.343454i	$-0.111597\pi$
$337$	3.91605e6	1.87834	0.939170	+	0.343454i	$0.111597\pi$
$338$	0	0
$339$	508148.	0.240155
$340$	0	0
$341$	5.80689e6	2.70432
$342$	0	0
$343$	776871.	0.356544
$344$	0	0
$345$	271383.	0.122754
$346$	0	0
$347$	257742.	0.114911	0.0574554	−	0.998348i	$-0.481701\pi$
$347$	257742.	0.114911	0.0574554	+	0.998348i	$0.481701\pi$
$348$	0	0
$349$	−1.13821e6	−0.500219	−0.250109	−	0.968218i	$-0.580467\pi$
$349$	−1.13821e6	−0.500219	−0.250109	+	0.968218i	$0.580467\pi$
$350$	0	0
$351$	−165428.	−0.0716706
$352$	0	0
$353$	875628.	0.374010	0.187005	−	0.982359i	$-0.440122\pi$
$353$	875628.	0.374010	0.187005	+	0.982359i	$0.440122\pi$
$354$	0	0
$355$	840246.	0.353863
$356$	0	0
$357$	449731.	0.186759
$358$	0	0
$359$	−1.94041e6	−0.794617	−0.397308	−	0.917685i	$-0.630056\pi$
$359$	−1.94041e6	−0.794617	−0.397308	+	0.917685i	$0.630056\pi$
$360$	0	0
$361$	5.43005e6	2.19299
$362$	0	0
$363$	−3.22310e6	−1.28383
$364$	0	0
$365$	3.09567e6	1.21625
$366$	0	0
$367$	3.93047e6	1.52328	0.761640	−	0.648001i	$-0.224395\pi$
$367$	3.93047e6	1.52328	0.761640	+	0.648001i	$0.224395\pi$
$368$	0	0
$369$	1.39621e6	0.533807
$370$	0	0
$371$	367825.	0.138741
$372$	0	0
$373$	−5.28009e6	−1.96503	−0.982515	−	0.186182i	$-0.940389\pi$
$373$	−5.28009e6	−1.96503	−0.982515	+	0.186182i	$0.940389\pi$
$374$	0	0
$375$	−1.53947e6	−0.565320
$376$	0	0
$377$	1.41284e6	0.511963
$378$	0	0
$379$	−4.21160e6	−1.50608	−0.753041	−	0.657974i	$-0.771414\pi$
$379$	−4.21160e6	−1.50608	−0.753041	+	0.657974i	$0.771414\pi$
$380$	0	0
$381$	−1.54026e6	−0.543602
$382$	0	0
$383$	330225.	0.115030	0.0575152	−	0.998345i	$-0.481682\pi$
$383$	330225.	0.115030	0.0575152	+	0.998345i	$0.481682\pi$
$384$	0	0
$385$	965077.	0.331826
$386$	0	0
$387$	1.13450e6	0.385058
$388$	0	0
$389$	3.89699e6	1.30574	0.652868	−	0.757471i	$-0.273566\pi$
$389$	3.89699e6	1.30574	0.652868	+	0.757471i	$0.273566\pi$
$390$	0	0
$391$	1.12498e6	0.372137
$392$	0	0
$393$	813637.	0.265735
$394$	0	0
$395$	−4.08368e6	−1.31692
$396$	0	0
$397$	1.39569e6	0.444440	0.222220	−	0.974997i	$-0.428670\pi$
$397$	1.39569e6	0.444440	0.222220	+	0.974997i	$0.428670\pi$
$398$	0	0
$399$	−594629.	−0.186988
$400$	0	0
$401$	1.69474e6	0.526311	0.263155	−	0.964753i	$-0.415237\pi$
$401$	1.69474e6	0.526311	0.263155	+	0.964753i	$0.415237\pi$
$402$	0	0
$403$	1.82881e6	0.560927
$404$	0	0
$405$	−373985.	−0.113297
$406$	0	0
$407$	−7.69907e6	−2.30384
$408$	0	0
$409$	−3.26242e6	−0.964342	−0.482171	−	0.876077i	$-0.660152\pi$
$409$	−3.26242e6	−0.964342	−0.482171	+	0.876077i	$0.660152\pi$
$410$	0	0
$411$	2.95318e6	0.862354
$412$	0	0
$413$	−19928.4	−0.00574906
$414$	0	0
$415$	506752.	0.144436
$416$	0	0
$417$	−2.30925e6	−0.650327
$418$	0	0
$419$	2.59183e6	0.721225	0.360612	−	0.932716i	$-0.382568\pi$
$419$	2.59183e6	0.721225	0.360612	+	0.932716i	$0.382568\pi$
$420$	0	0
$421$	−7.06959e6	−1.94397	−0.971984	−	0.235049i	$-0.924475\pi$
$421$	−7.06959e6	−1.94397	−0.971984	+	0.235049i	$0.924475\pi$
$422$	0	0
$423$	526293.	0.143013
$424$	0	0
$425$	264002.	0.0708982
$426$	0	0
$427$	1.04904e6	0.278434
$428$	0	0
$429$	−1.47157e6	−0.386044
$430$	0	0
$431$	−543609.	−0.140959	−0.0704796	−	0.997513i	$-0.522453\pi$
$431$	−543609.	−0.140959	−0.0704796	+	0.997513i	$0.522453\pi$
$432$	0	0
$433$	−2.33713e6	−0.599050	−0.299525	−	0.954088i	$-0.596828\pi$
$433$	−2.33713e6	−0.599050	−0.299525	+	0.954088i	$0.596828\pi$
$434$	0	0
$435$	3.19402e6	0.809310
$436$	0	0
$437$	−1.48744e6	−0.372593
$438$	0	0
$439$	−3.30779e6	−0.819175	−0.409587	−	0.912271i	$-0.634327\pi$
$439$	−3.30779e6	−0.819175	−0.409587	+	0.912271i	$0.634327\pi$
$440$	0	0
$441$	−1.31664e6	−0.322383
$442$	0	0
$443$	4.59907e6	1.11342	0.556712	−	0.830706i	$-0.312063\pi$
$443$	4.59907e6	1.11342	0.556712	+	0.830706i	$0.312063\pi$
$444$	0	0
$445$	3.06794e6	0.734424
$446$	0	0
$447$	404251.	0.0956934
$448$	0	0
$449$	3.02862e6	0.708971	0.354486	−	0.935062i	$-0.384656\pi$
$449$	3.02862e6	0.708971	0.354486	+	0.935062i	$0.384656\pi$
$450$	0	0
$451$	1.24200e7	2.87528
$452$	0	0
$453$	2.72811e6	0.624620
$454$	0	0
$455$	303939.	0.0688269
$456$	0	0
$457$	490508.	0.109864	0.0549320	−	0.998490i	$-0.482506\pi$
$457$	490508.	0.109864	0.0549320	+	0.998490i	$0.482506\pi$
$458$	0	0
$459$	−1.55030e6	−0.343467
$460$	0	0
$461$	3.79243e6	0.831123	0.415562	−	0.909565i	$-0.363585\pi$
$461$	3.79243e6	0.831123	0.415562	+	0.909565i	$0.363585\pi$
$462$	0	0
$463$	−5.04126e6	−1.09292	−0.546458	−	0.837487i	$-0.684024\pi$
$463$	−5.04126e6	−1.09292	−0.546458	+	0.837487i	$0.684024\pi$
$464$	0	0
$465$	4.13442e6	0.886711
$466$	0	0
$467$	569444.	0.120825	0.0604127	−	0.998173i	$-0.480758\pi$
$467$	569444.	0.120825	0.0604127	+	0.998173i	$0.480758\pi$
$468$	0	0
$469$	617867.	0.129707
$470$	0	0
$471$	−3.18308e6	−0.661142
$472$	0	0
$473$	1.00919e7	2.07406
$474$	0	0
$475$	−349060.	−0.0709850
$476$	0	0
$477$	−1.26796e6	−0.255158
$478$	0	0
$479$	4.53576e6	0.903258	0.451629	−	0.892206i	$-0.350843\pi$
$479$	4.53576e6	0.903258	0.451629	+	0.892206i	$0.350843\pi$
$480$	0	0
$481$	−2.42473e6	−0.477860
$482$	0	0
$483$	111872.	0.0218198
$484$	0	0
$485$	8.82172e6	1.70294
$486$	0	0
$487$	2.28421e6	0.436430	0.218215	−	0.975901i	$-0.429977\pi$
$487$	2.28421e6	0.436430	0.218215	+	0.975901i	$0.429977\pi$
$488$	0	0
$489$	−1.47913e6	−0.279726
$490$	0	0
$491$	−8.39683e6	−1.57185	−0.785926	−	0.618321i	$-0.787813\pi$
$491$	−8.39683e6	−1.57185	−0.785926	+	0.618321i	$0.787813\pi$
$492$	0	0
$493$	1.32404e7	2.45348
$494$	0	0
$495$	−3.32679e6	−0.610257
$496$	0	0
$497$	346372.	0.0629002
$498$	0	0
$499$	−6.71399e6	−1.20706	−0.603531	−	0.797340i	$-0.706240\pi$
$499$	−6.71399e6	−1.20706	−0.603531	+	0.797340i	$0.706240\pi$
$500$	0	0
$501$	−1.56681e6	−0.278884
$502$	0	0
$503$	3.75326e6	0.661437	0.330719	−	0.943729i	$-0.392709\pi$
$503$	3.75326e6	0.661437	0.330719	+	0.943729i	$0.392709\pi$
$504$	0	0
$505$	−5.72954e6	−0.999749
$506$	0	0
$507$	2.87818e6	0.497277
$508$	0	0
$509$	−1.91065e6	−0.326879	−0.163440	−	0.986553i	$-0.552259\pi$
$509$	−1.91065e6	−0.326879	−0.163440	+	0.986553i	$0.552259\pi$
$510$	0	0
$511$	1.27612e6	0.216192
$512$	0	0
$513$	2.04979e6	0.343888
$514$	0	0
$515$	−4.86611e6	−0.808469
$516$	0	0
$517$	4.68165e6	0.770322
$518$	0	0
$519$	3.92425e6	0.639497
$520$	0	0
$521$	3.72565e6	0.601322	0.300661	−	0.953731i	$-0.402793\pi$
$521$	3.72565e6	0.601322	0.300661	+	0.953731i	$0.402793\pi$
$522$	0	0
$523$	5.69582e6	0.910546	0.455273	−	0.890352i	$-0.349542\pi$
$523$	5.69582e6	0.910546	0.455273	+	0.890352i	$0.349542\pi$
$524$	0	0
$525$	26253.2	0.00415704
$526$	0	0
$527$	1.71387e7	2.68813
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	68696.6	0.0105730
$532$	0	0
$533$	3.91152e6	0.596387
$534$	0	0
$535$	2.16980e6	0.327745
$536$	0	0
$537$	−2.56942e6	−0.384503
$538$	0	0
$539$	−1.17122e7	−1.73647
$540$	0	0
$541$	5.32753e6	0.782587	0.391294	−	0.920266i	$-0.372028\pi$
$541$	5.32753e6	0.782587	0.391294	+	0.920266i	$0.372028\pi$
$542$	0	0
$543$	−1.73136e6	−0.251992
$544$	0	0
$545$	−1.10157e7	−1.58863
$546$	0	0
$547$	−1.22755e7	−1.75417	−0.877086	−	0.480334i	$-0.840515\pi$
$547$	−1.22755e7	−1.75417	−0.877086	+	0.480334i	$0.840515\pi$
$548$	0	0
$549$	−3.61623e6	−0.512064
$550$	0	0
$551$	−1.75063e7	−2.45649
$552$	0	0
$553$	−1.68341e6	−0.234086
$554$	0	0
$555$	−5.48162e6	−0.755399
$556$	0	0
$557$	2.70889e6	0.369959	0.184980	−	0.982742i	$-0.440778\pi$
$557$	2.70889e6	0.369959	0.184980	+	0.982742i	$0.440778\pi$
$558$	0	0
$559$	3.17834e6	0.430200
$560$	0	0
$561$	−1.37908e7	−1.85004
$562$	0	0
$563$	3.93505e6	0.523214	0.261607	−	0.965175i	$-0.415748\pi$
$563$	3.93505e6	0.523214	0.261607	+	0.965175i	$0.415748\pi$
$564$	0	0
$565$	3.21834e6	0.424142
$566$	0	0
$567$	−154167.	−0.0201388
$568$	0	0
$569$	−3.14609e6	−0.407372	−0.203686	−	0.979036i	$-0.565292\pi$
$569$	−3.14609e6	−0.407372	−0.203686	+	0.979036i	$0.565292\pi$
$570$	0	0
$571$	6.28305e6	0.806455	0.403227	−	0.915100i	$-0.367888\pi$
$571$	6.28305e6	0.806455	0.403227	+	0.915100i	$0.367888\pi$
$572$	0	0
$573$	−4.25611e6	−0.541534
$574$	0	0
$575$	65671.0	0.00828332
$576$	0	0
$577$	−8.37736e6	−1.04753	−0.523766	−	0.851862i	$-0.675473\pi$
$577$	−8.37736e6	−1.04753	−0.523766	+	0.851862i	$0.675473\pi$
$578$	0	0
$579$	−1.71114e6	−0.212124
$580$	0	0
$581$	208897.	0.0256739
$582$	0	0
$583$	−1.12791e7	−1.37437
$584$	0	0
$585$	−1.04773e6	−0.126579
$586$	0	0
$587$	3.65884e6	0.438276	0.219138	−	0.975694i	$-0.429675\pi$
$587$	3.65884e6	0.438276	0.219138	+	0.975694i	$0.429675\pi$
$588$	0	0
$589$	−2.26605e7	−2.69142
$590$	0	0
$591$	−2.03268e6	−0.239387
$592$	0	0
$593$	−4.97828e6	−0.581357	−0.290679	−	0.956821i	$-0.593881\pi$
$593$	−4.97828e6	−0.581357	−0.290679	+	0.956821i	$0.593881\pi$
$594$	0	0
$595$	2.84836e6	0.329839
$596$	0	0
$597$	−2.26437e6	−0.260023
$598$	0	0
$599$	−5.13126e6	−0.584329	−0.292164	−	0.956368i	$-0.594375\pi$
$599$	−5.13126e6	−0.584329	−0.292164	+	0.956368i	$0.594375\pi$
$600$	0	0
$601$	1.07269e7	1.21140	0.605699	−	0.795694i	$-0.292893\pi$
$601$	1.07269e7	1.21140	0.605699	+	0.795694i	$0.292893\pi$
$602$	0	0
$603$	−2.12990e6	−0.238542
$604$	0	0
$605$	−2.04134e7	−2.26740
$606$	0	0
$607$	−3.54359e6	−0.390366	−0.195183	−	0.980767i	$-0.562530\pi$
$607$	−3.54359e6	−0.390366	−0.195183	+	0.980767i	$0.562530\pi$
$608$	0	0
$609$	1.31666e6	0.143857
$610$	0	0
$611$	1.47443e6	0.159779
$612$	0	0
$613$	2.01438e6	0.216516	0.108258	−	0.994123i	$-0.465473\pi$
$613$	2.01438e6	0.216516	0.108258	+	0.994123i	$0.465473\pi$
$614$	0	0
$615$	8.84284e6	0.942766
$616$	0	0
$617$	1.45094e7	1.53439	0.767196	−	0.641413i	$-0.221652\pi$
$617$	1.45094e7	1.53439	0.767196	+	0.641413i	$0.221652\pi$
$618$	0	0
$619$	1.10230e7	1.15630	0.578152	−	0.815929i	$-0.303774\pi$
$619$	1.10230e7	1.15630	0.578152	+	0.815929i	$0.303774\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	1.26469e6	0.130546
$624$	0	0
$625$	−1.01382e7	−1.03815
$626$	0	0
$627$	1.82340e7	1.85230
$628$	0	0
$629$	−2.27233e7	−2.29005
$630$	0	0
$631$	1.69788e7	1.69759	0.848795	−	0.528722i	$-0.177328\pi$
$631$	1.69788e7	1.69759	0.848795	+	0.528722i	$0.177328\pi$
$632$	0	0
$633$	−8.37531e6	−0.830791
$634$	0	0
$635$	−9.75519e6	−0.960066
$636$	0	0
$637$	−3.68863e6	−0.360177
$638$	0	0
$639$	−1.19401e6	−0.115679
$640$	0	0
$641$	1.77216e7	1.70356	0.851779	−	0.523901i	$-0.175524\pi$
$641$	1.77216e7	1.70356	0.851779	+	0.523901i	$0.175524\pi$
$642$	0	0
$643$	−1.27710e7	−1.21814	−0.609069	−	0.793118i	$-0.708457\pi$
$643$	−1.27710e7	−1.21814	−0.609069	+	0.793118i	$0.708457\pi$
$644$	0	0
$645$	7.18531e6	0.680058
$646$	0	0
$647$	−5.11739e6	−0.480605	−0.240302	−	0.970698i	$-0.577247\pi$
$647$	−5.11739e6	−0.480605	−0.240302	+	0.970698i	$0.577247\pi$
$648$	0	0
$649$	611092.	0.0569502
$650$	0	0
$651$	1.70432e6	0.157615
$652$	0	0
$653$	1.84748e7	1.69549	0.847747	−	0.530401i	$-0.177959\pi$
$653$	1.84748e7	1.69549	0.847747	+	0.530401i	$0.177959\pi$
$654$	0	0
$655$	5.15315e6	0.469320
$656$	0	0
$657$	−4.39901e6	−0.397595
$658$	0	0
$659$	−273786.	−0.0245583	−0.0122791	−	0.999925i	$-0.503909\pi$
$659$	−273786.	−0.0245583	−0.0122791	+	0.999925i	$0.503909\pi$
$660$	0	0
$661$	−3.03392e6	−0.270085	−0.135043	−	0.990840i	$-0.543117\pi$
$661$	−3.03392e6	−0.270085	−0.135043	+	0.990840i	$0.543117\pi$
$662$	0	0
$663$	−4.34323e6	−0.383733
$664$	0	0
$665$	−3.76607e6	−0.330243
$666$	0	0
$667$	3.29357e6	0.286650
$668$	0	0
$669$	−76008.8	−0.00656597
$670$	0	0
$671$	−3.21682e7	−2.75817
$672$	0	0
$673$	−1.60711e6	−0.136775	−0.0683877	−	0.997659i	$-0.521785\pi$
$673$	−1.60711e6	−0.136775	−0.0683877	+	0.997659i	$0.521785\pi$
$674$	0	0
$675$	−90499.4	−0.00764515
$676$	0	0
$677$	−8.74733e6	−0.733507	−0.366753	−	0.930318i	$-0.619531\pi$
$677$	−8.74733e6	−0.733507	−0.366753	+	0.930318i	$0.619531\pi$
$678$	0	0
$679$	3.63656e6	0.302702
$680$	0	0
$681$	4.26438e6	0.352361
$682$	0	0
$683$	−3.24520e6	−0.266189	−0.133095	−	0.991103i	$-0.542491\pi$
$683$	−3.24520e6	−0.266189	−0.133095	+	0.991103i	$0.542491\pi$
$684$	0	0
$685$	1.87039e7	1.52302
$686$	0	0
$687$	1.12153e7	0.906606
$688$	0	0
$689$	−3.55223e6	−0.285071
$690$	0	0
$691$	1.89550e7	1.51018	0.755089	−	0.655622i	$-0.227593\pi$
$691$	1.89550e7	1.51018	0.755089	+	0.655622i	$0.227593\pi$
$692$	0	0
$693$	−1.37140e6	−0.108475
$694$	0	0
$695$	−1.46256e7	−1.14855
$696$	0	0
$697$	3.66568e7	2.85807
$698$	0	0
$699$	−1.88786e6	−0.146142
$700$	0	0
$701$	−2.20729e7	−1.69654	−0.848272	−	0.529561i	$-0.822357\pi$
$701$	−2.20729e7	−1.69654	−0.848272	+	0.529561i	$0.822357\pi$
$702$	0	0
$703$	3.00445e7	2.29285
$704$	0	0
$705$	3.33326e6	0.252579
$706$	0	0
$707$	−2.36187e6	−0.177708
$708$	0	0
$709$	−9.26963e6	−0.692543	−0.346272	−	0.938134i	$-0.612552\pi$
$709$	−9.26963e6	−0.692543	−0.346272	+	0.938134i	$0.612552\pi$
$710$	0	0
$711$	5.80300e6	0.430505
$712$	0	0
$713$	4.26327e6	0.314065
$714$	0	0
$715$	−9.32013e6	−0.681800
$716$	0	0
$717$	−1.52775e7	−1.10982
$718$	0	0
$719$	4.51073e6	0.325405	0.162703	−	0.986675i	$-0.447979\pi$
$719$	4.51073e6	0.325405	0.162703	+	0.986675i	$0.447979\pi$
$720$	0	0
$721$	−2.00594e6	−0.143708
$722$	0	0
$723$	−6.59340e6	−0.469098
$724$	0	0
$725$	772911.	0.0546115
$726$	0	0
$727$	9.68176e6	0.679389	0.339694	−	0.940536i	$-0.389676\pi$
$727$	9.68176e6	0.679389	0.339694	+	0.940536i	$0.389676\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.97857e7	2.06165
$732$	0	0
$733$	−1.46161e7	−1.00478	−0.502390	−	0.864641i	$-0.667546\pi$
$733$	−1.46161e7	−1.00478	−0.502390	+	0.864641i	$0.667546\pi$
$734$	0	0
$735$	−8.33893e6	−0.569367
$736$	0	0
$737$	−1.89465e7	−1.28488
$738$	0	0
$739$	−2.33852e7	−1.57518	−0.787589	−	0.616201i	$-0.788671\pi$
$739$	−2.33852e7	−1.57518	−0.787589	+	0.616201i	$0.788671\pi$
$740$	0	0
$741$	5.74257e6	0.384203
$742$	0	0
$743$	2.46171e7	1.63593	0.817967	−	0.575266i	$-0.195101\pi$
$743$	2.46171e7	1.63593	0.817967	+	0.575266i	$0.195101\pi$
$744$	0	0
$745$	2.56031e6	0.169006
$746$	0	0
$747$	−720106.	−0.0472166
$748$	0	0
$749$	894453.	0.0582577
$750$	0	0
$751$	3.99719e6	0.258616	0.129308	−	0.991605i	$-0.458724\pi$
$751$	3.99719e6	0.258616	0.129308	+	0.991605i	$0.458724\pi$
$752$	0	0
$753$	8.65008e6	0.555946
$754$	0	0
$755$	1.72784e7	1.10315
$756$	0	0
$757$	1.89680e7	1.20305	0.601524	−	0.798855i	$-0.294561\pi$
$757$	1.89680e7	1.20305	0.601524	+	0.798855i	$0.294561\pi$
$758$	0	0
$759$	−3.43048e6	−0.216147
$760$	0	0
$761$	−2.92639e7	−1.83177	−0.915885	−	0.401442i	$-0.868509\pi$
$761$	−2.92639e7	−1.83177	−0.915885	+	0.401442i	$0.868509\pi$
$762$	0	0
$763$	−4.54098e6	−0.282383
$764$	0	0
$765$	−9.81880e6	−0.606604
$766$	0	0
$767$	192456.	0.0118125
$768$	0	0
$769$	8.32935e6	0.507920	0.253960	−	0.967215i	$-0.418267\pi$
$769$	8.32935e6	0.507920	0.253960	+	0.967215i	$0.418267\pi$
$770$	0	0
$771$	−1.79048e7	−1.08476
$772$	0	0
$773$	−1.35469e7	−0.815437	−0.407719	−	0.913108i	$-0.633676\pi$
$773$	−1.35469e7	−0.815437	−0.407719	+	0.913108i	$0.633676\pi$
$774$	0	0
$775$	1.00047e6	0.0598345
$776$	0	0
$777$	−2.25967e6	−0.134274
$778$	0	0
$779$	−4.84671e7	−2.86157
$780$	0	0
$781$	−1.06213e7	−0.623090
$782$	0	0
$783$	−4.53878e6	−0.264566
$784$	0	0
$785$	−2.01599e7	−1.16766
$786$	0	0
$787$	4.45513e6	0.256404	0.128202	−	0.991748i	$-0.459079\pi$
$787$	4.45513e6	0.256404	0.128202	+	0.991748i	$0.459079\pi$
$788$	0	0
$789$	−1.48005e7	−0.846417
$790$	0	0
$791$	1.32669e6	0.0753925
$792$	0	0
$793$	−1.01310e7	−0.572096
$794$	0	0
$795$	−8.03057e6	−0.450639
$796$	0	0
$797$	2.43397e7	1.35728	0.678641	−	0.734470i	$-0.262569\pi$
$797$	2.43397e7	1.35728	0.678641	+	0.734470i	$0.262569\pi$
$798$	0	0
$799$	1.38176e7	0.765711
$800$	0	0
$801$	−4.35961e6	−0.240086
$802$	0	0
$803$	−3.91315e7	−2.14159
$804$	0	0
$805$	708535.	0.0385364
$806$	0	0
$807$	1.68069e7	0.908454
$808$	0	0
$809$	3.23897e7	1.73994	0.869972	−	0.493101i	$-0.164137\pi$
$809$	3.23897e7	1.73994	0.869972	+	0.493101i	$0.164137\pi$
$810$	0	0
$811$	2.74080e7	1.46327	0.731635	−	0.681696i	$-0.238758\pi$
$811$	2.74080e7	1.46327	0.731635	+	0.681696i	$0.238758\pi$
$812$	0	0
$813$	−3.29494e6	−0.174832
$814$	0	0
$815$	−9.36801e6	−0.494030
$816$	0	0
$817$	−3.93823e7	−2.06417
$818$	0	0
$819$	−431904.	−0.0224998
$820$	0	0
$821$	2.13453e6	0.110521	0.0552605	−	0.998472i	$-0.482401\pi$
$821$	2.13453e6	0.110521	0.0552605	+	0.998472i	$0.482401\pi$
$822$	0	0
$823$	1.41978e7	0.730671	0.365335	−	0.930876i	$-0.380954\pi$
$823$	1.41978e7	0.730671	0.365335	+	0.930876i	$0.380954\pi$
$824$	0	0
$825$	−805039.	−0.0411796
$826$	0	0
$827$	2.21015e7	1.12372	0.561859	−	0.827233i	$-0.310086\pi$
$827$	2.21015e7	1.12372	0.561859	+	0.827233i	$0.310086\pi$
$828$	0	0
$829$	1.11617e6	0.0564085	0.0282043	−	0.999602i	$-0.491021\pi$
$829$	1.11617e6	0.0564085	0.0282043	+	0.999602i	$0.491021\pi$
$830$	0	0
$831$	−1.68993e7	−0.848917
$832$	0	0
$833$	−3.45679e7	−1.72608
$834$	0	0
$835$	−9.92337e6	−0.492542
$836$	0	0
$837$	−5.87510e6	−0.289869
$838$	0	0
$839$	1.84394e7	0.904360	0.452180	−	0.891927i	$-0.350646\pi$
$839$	1.84394e7	0.904360	0.452180	+	0.891927i	$0.350646\pi$
$840$	0	0
$841$	1.82523e7	0.889872
$842$	0	0
$843$	1.64157e7	0.795591
$844$	0	0
$845$	1.82289e7	0.878251
$846$	0	0
$847$	−8.41498e6	−0.403037
$848$	0	0
$849$	2.04555e7	0.973957
$850$	0	0
$851$	−5.65246e6	−0.267555
$852$	0	0
$853$	−3.58844e7	−1.68863	−0.844313	−	0.535850i	$-0.819991\pi$
$853$	−3.58844e7	−1.68863	−0.844313	+	0.535850i	$0.819991\pi$
$854$	0	0
$855$	1.29823e7	0.607347
$856$	0	0
$857$	3.02059e6	0.140488	0.0702442	−	0.997530i	$-0.477622\pi$
$857$	3.02059e6	0.140488	0.0702442	+	0.997530i	$0.477622\pi$
$858$	0	0
$859$	−204775.	−0.00946879	−0.00473440	−	0.999989i	$-0.501507\pi$
$859$	−204775.	−0.00946879	−0.00473440	+	0.999989i	$0.501507\pi$
$860$	0	0
$861$	3.64526e6	0.167579
$862$	0	0
$863$	7.83608e6	0.358156	0.179078	−	0.983835i	$-0.442689\pi$
$863$	7.83608e6	0.358156	0.179078	+	0.983835i	$0.442689\pi$
$864$	0	0
$865$	2.48541e7	1.12943
$866$	0	0
$867$	−2.79238e7	−1.26161
$868$	0	0
$869$	5.16207e7	2.31886
$870$	0	0
$871$	−5.96698e6	−0.266507
$872$	0	0
$873$	−1.25359e7	−0.556696
$874$	0	0
$875$	−4.01931e6	−0.177472
$876$	0	0
$877$	1.15148e7	0.505544	0.252772	−	0.967526i	$-0.418658\pi$
$877$	1.15148e7	0.505544	0.252772	+	0.967526i	$0.418658\pi$
$878$	0	0
$879$	408602.	0.0178373
$880$	0	0
$881$	−2.29988e7	−0.998309	−0.499154	−	0.866513i	$-0.666356\pi$
$881$	−2.29988e7	−0.998309	−0.499154	+	0.866513i	$0.666356\pi$
$882$	0	0
$883$	−1.62308e7	−0.700546	−0.350273	−	0.936648i	$-0.613911\pi$
$883$	−1.62308e7	−0.700546	−0.350273	+	0.936648i	$0.613911\pi$
$884$	0	0
$885$	435088.	0.0186732
$886$	0	0
$887$	−1.52413e7	−0.650449	−0.325224	−	0.945637i	$-0.605440\pi$
$887$	−1.52413e7	−0.650449	−0.325224	+	0.945637i	$0.605440\pi$
$888$	0	0
$889$	−4.02136e6	−0.170655
$890$	0	0
$891$	4.72744e6	0.199495
$892$	0	0
$893$	−1.82694e7	−0.766649
$894$	0	0
$895$	−1.62733e7	−0.679077
$896$	0	0
$897$	−1.08039e6	−0.0448330
$898$	0	0
$899$	5.01763e7	2.07062
$900$	0	0
$901$	−3.32896e7	−1.36615
$902$	0	0
$903$	2.96198e6	0.120882
$904$	0	0
$905$	−1.09655e7	−0.445048
$906$	0	0
$907$	−1.38217e7	−0.557884	−0.278942	−	0.960308i	$-0.589984\pi$
$907$	−1.38217e7	−0.557884	−0.278942	+	0.960308i	$0.589984\pi$
$908$	0	0
$909$	8.14179e6	0.326821
$910$	0	0
$911$	1.89745e7	0.757487	0.378744	−	0.925502i	$-0.376356\pi$
$911$	1.89745e7	0.757487	0.378744	+	0.925502i	$0.376356\pi$
$912$	0	0
$913$	−6.40572e6	−0.254326
$914$	0	0
$915$	−2.29033e7	−0.904367
$916$	0	0
$917$	2.12427e6	0.0834231
$918$	0	0
$919$	6.48820e6	0.253417	0.126708	−	0.991940i	$-0.459559\pi$
$919$	6.48820e6	0.253417	0.126708	+	0.991940i	$0.459559\pi$
$920$	0	0
$921$	−1.47242e7	−0.571984
$922$	0	0
$923$	−3.34505e6	−0.129241
$924$	0	0
$925$	−1.32648e6	−0.0509737
$926$	0	0
$927$	6.91484e6	0.264291
$928$	0	0
$929$	1.56610e6	0.0595361	0.0297680	−	0.999557i	$-0.490523\pi$
$929$	1.56610e6	0.0595361	0.0297680	+	0.999557i	$0.490523\pi$
$930$	0	0
$931$	4.57052e7	1.72819
$932$	0	0
$933$	−1.67943e7	−0.631624
$934$	0	0
$935$	−8.73434e7	−3.26739
$936$	0	0
$937$	2.18431e7	0.812767	0.406383	−	0.913703i	$-0.366790\pi$
$937$	2.18431e7	0.812767	0.406383	+	0.913703i	$0.366790\pi$
$938$	0	0
$939$	2.67252e6	0.0989137
$940$	0	0
$941$	1.85836e7	0.684157	0.342079	−	0.939671i	$-0.388869\pi$
$941$	1.85836e7	0.684157	0.342079	+	0.939671i	$0.388869\pi$
$942$	0	0
$943$	9.11844e6	0.333919
$944$	0	0
$945$	−976412.	−0.0355675
$946$	0	0
$947$	2.54892e7	0.923593	0.461796	−	0.886986i	$-0.347205\pi$
$947$	2.54892e7	0.923593	0.461796	+	0.886986i	$0.347205\pi$
$948$	0	0
$949$	−1.23240e7	−0.444207
$950$	0	0
$951$	2.12131e6	0.0760593
$952$	0	0
$953$	−5.38540e7	−1.92082	−0.960409	−	0.278595i	$-0.910131\pi$
$953$	−5.38540e7	−1.92082	−0.960409	+	0.278595i	$0.910131\pi$
$954$	0	0
$955$	−2.69559e7	−0.956414
$956$	0	0
$957$	−4.03748e7	−1.42505
$958$	0	0
$959$	7.71026e6	0.270721
$960$	0	0
$961$	3.63202e7	1.26864
$962$	0	0
$963$	−3.08334e6	−0.107141
$964$	0	0
$965$	−1.08375e7	−0.374636
$966$	0	0
$967$	−4.91627e6	−0.169071	−0.0845355	−	0.996420i	$-0.526941\pi$
$967$	−4.91627e6	−0.169071	−0.0845355	+	0.996420i	$0.526941\pi$
$968$	0	0
$969$	5.38164e7	1.84122
$970$	0	0
$971$	−3.71800e7	−1.26550	−0.632748	−	0.774358i	$-0.718073\pi$
$971$	−3.71800e7	−1.26550	−0.632748	+	0.774358i	$0.718073\pi$
$972$	0	0
$973$	−6.02907e6	−0.204159
$974$	0	0
$975$	−253537.	−0.00854142
$976$	0	0
$977$	−4.33194e7	−1.45193	−0.725966	−	0.687731i	$-0.758607\pi$
$977$	−4.33194e7	−1.45193	−0.725966	+	0.687731i	$0.758607\pi$
$978$	0	0
$979$	−3.87810e7	−1.29319
$980$	0	0
$981$	1.56536e7	0.519327
$982$	0	0
$983$	−1.68420e7	−0.555917	−0.277959	−	0.960593i	$-0.589658\pi$
$983$	−1.68420e7	−0.555917	−0.277959	+	0.960593i	$0.589658\pi$
$984$	0	0
$985$	−1.28740e7	−0.422787
$986$	0	0
$987$	1.37406e6	0.0448966
$988$	0	0
$989$	7.40925e6	0.240870
$990$	0	0
$991$	2.41728e7	0.781884	0.390942	−	0.920415i	$-0.372149\pi$
$991$	2.41728e7	0.781884	0.390942	+	0.920415i	$0.372149\pi$
$992$	0	0
$993$	8.15320e6	0.262395
$994$	0	0
$995$	−1.43413e7	−0.459231
$996$	0	0
$997$	3.15819e7	1.00624	0.503118	−	0.864218i	$-0.332186\pi$
$997$	3.15819e7	1.00624	0.503118	+	0.864218i	$0.332186\pi$
$998$	0	0
$999$	7.78950e6	0.246943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.x.1.3		7
4.3	odd	2		552.6.a.g.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.g.1.3	✓	7	4.3	odd	2
1104.6.a.x.1.3		7	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -57.0012 q^{5} -23.4975 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -57.0012 q^{5} -23.4975 q^{7} +81.0000 q^{9} +720.537 q^{11} +226.924 q^{13} +513.011 q^{15} +2126.62 q^{17} -2811.79 q^{19} +211.477 q^{21} +529.000 q^{23} +124.142 q^{25} -729.000 q^{27} +6226.03 q^{29} +8059.12 q^{31} -6484.83 q^{33} +1339.39 q^{35} -10685.2 q^{37} -2042.32 q^{39} +17237.1 q^{41} +14006.1 q^{43} -4617.10 q^{45} +6497.44 q^{47} -16254.9 q^{49} -19139.6 q^{51} -15653.8 q^{53} -41071.5 q^{55} +25306.1 q^{57} +848.107 q^{59} -44644.8 q^{61} -1903.30 q^{63} -12935.0 q^{65} -26295.0 q^{67} -4761.00 q^{69} -14740.8 q^{71} -54308.7 q^{73} -1117.28 q^{75} -16930.8 q^{77} +71642.0 q^{79} +6561.00 q^{81} -8890.20 q^{83} -121220. q^{85} -56034.3 q^{87} -53822.3 q^{89} -5332.15 q^{91} -72532.1 q^{93} +160275. q^{95} -154764. q^{97} +58363.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9}+O(q^{10}) \) 7 * q - 63 * q^3 - 104 * q^5 + 182 * q^7 + 567 * q^9	\( 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9} + 124 q^{11} + 294 q^{13} + 936 q^{15} + 428 q^{17} + 1826 q^{19} - 1638 q^{21} + 3703 q^{23} + 5501 q^{25} - 5103 q^{27} - 1682 q^{29} + 14420 q^{31} - 1116 q^{33} - 3312 q^{35} - 14218 q^{37} - 2646 q^{39} + 7294 q^{41} - 3630 q^{43} - 8424 q^{45} + 19808 q^{47} - 4325 q^{49} - 3852 q^{51} - 77412 q^{53} + 61136 q^{55} - 16434 q^{57} + 51076 q^{59} - 67186 q^{61} + 14742 q^{63} - 121800 q^{65} + 70870 q^{67} - 33327 q^{69} + 45784 q^{71} - 175522 q^{73} - 49509 q^{75} - 148632 q^{77} + 54538 q^{79} + 45927 q^{81} - 27612 q^{83} - 197552 q^{85} + 15138 q^{87} - 184360 q^{89} + 217412 q^{91} - 129780 q^{93} + 82608 q^{95} - 339766 q^{97} + 10044 q^{99}+O(q^{100}) \) 7 * q - 63 * q^3 - 104 * q^5 + 182 * q^7 + 567 * q^9 + 124 * q^11 + 294 * q^13 + 936 * q^15 + 428 * q^17 + 1826 * q^19 - 1638 * q^21 + 3703 * q^23 + 5501 * q^25 - 5103 * q^27 - 1682 * q^29 + 14420 * q^31 - 1116 * q^33 - 3312 * q^35 - 14218 * q^37 - 2646 * q^39 + 7294 * q^41 - 3630 * q^43 - 8424 * q^45 + 19808 * q^47 - 4325 * q^49 - 3852 * q^51 - 77412 * q^53 + 61136 * q^55 - 16434 * q^57 + 51076 * q^59 - 67186 * q^61 + 14742 * q^63 - 121800 * q^65 + 70870 * q^67 - 33327 * q^69 + 45784 * q^71 - 175522 * q^73 - 49509 * q^75 - 148632 * q^77 + 54538 * q^79 + 45927 * q^81 - 27612 * q^83 - 197552 * q^85 + 15138 * q^87 - 184360 * q^89 + 217412 * q^91 - 129780 * q^93 + 82608 * q^95 - 339766 * q^97 + 10044 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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