LMFDB - Embedded newform 552.6.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,6,Mod(1,552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, 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("552.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	552.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:	$88.5318685368$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 374x^{3} + 1565x^{2} + 19136x - 84640 $ x^5 - 2x^4 - 374x^3 + 1565x^2 + 19136x - 84640
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.75056$ of defining polynomial
Character	$\chi$	$=$	552.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +5.87631 q^{5} +170.213 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +5.87631 q^{5} +170.213 q^{7} +81.0000 q^{9} -578.770 q^{11} -282.075 q^{13} +52.8868 q^{15} +151.453 q^{17} -237.501 q^{19} +1531.91 q^{21} +529.000 q^{23} -3090.47 q^{25} +729.000 q^{27} -6313.78 q^{29} -5132.58 q^{31} -5208.93 q^{33} +1000.22 q^{35} -4511.91 q^{37} -2538.68 q^{39} +4746.75 q^{41} -14485.2 q^{43} +475.981 q^{45} -11362.3 q^{47} +12165.3 q^{49} +1363.08 q^{51} +35953.7 q^{53} -3401.03 q^{55} -2137.51 q^{57} +29829.1 q^{59} -12324.1 q^{61} +13787.2 q^{63} -1657.56 q^{65} -55464.0 q^{67} +4761.00 q^{69} +23917.3 q^{71} +32752.5 q^{73} -27814.2 q^{75} -98513.9 q^{77} -39559.2 q^{79} +6561.00 q^{81} +21175.8 q^{83} +889.984 q^{85} -56824.0 q^{87} +65097.3 q^{89} -48012.8 q^{91} -46193.2 q^{93} -1395.63 q^{95} +19272.5 q^{97} -46880.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 45 q^{3} + 16 q^{5} - 134 q^{7} + 405 q^{9}+O(q^{10}) $ 5 * q + 45 * q^3 + 16 * q^5 - 134 * q^7 + 405 * q^9	$ 5 q + 45 q^{3} + 16 q^{5} - 134 q^{7} + 405 q^{9} - 632 q^{11} + 326 q^{13} + 144 q^{15} - 1044 q^{17} - 722 q^{19} - 1206 q^{21} + 2645 q^{23} - 4525 q^{25} + 3645 q^{27} - 7822 q^{29} - 2228 q^{31} - 5688 q^{33} - 3020 q^{35} - 18818 q^{37} + 2934 q^{39} - 4550 q^{41} - 2226 q^{43} + 1296 q^{45} - 16164 q^{47} - 24563 q^{49} - 9396 q^{51} - 8972 q^{53} - 37496 q^{55} - 6498 q^{57} - 56168 q^{59} - 61474 q^{61} - 10854 q^{63} - 32312 q^{65} - 58270 q^{67} + 23805 q^{69} - 75920 q^{71} + 7970 q^{73} - 40725 q^{75} - 86424 q^{77} - 64818 q^{79} + 32805 q^{81} - 92680 q^{83} - 18556 q^{85} - 70398 q^{87} - 52256 q^{89} - 80636 q^{91} - 20052 q^{93} - 132324 q^{95} + 42230 q^{97} - 51192 q^{99}+O(q^{100}) $ 5 * q + 45 * q^3 + 16 * q^5 - 134 * q^7 + 405 * q^9 - 632 * q^11 + 326 * q^13 + 144 * q^15 - 1044 * q^17 - 722 * q^19 - 1206 * q^21 + 2645 * q^23 - 4525 * q^25 + 3645 * q^27 - 7822 * q^29 - 2228 * q^31 - 5688 * q^33 - 3020 * q^35 - 18818 * q^37 + 2934 * q^39 - 4550 * q^41 - 2226 * q^43 + 1296 * q^45 - 16164 * q^47 - 24563 * q^49 - 9396 * q^51 - 8972 * q^53 - 37496 * q^55 - 6498 * q^57 - 56168 * q^59 - 61474 * q^61 - 10854 * q^63 - 32312 * q^65 - 58270 * q^67 + 23805 * q^69 - 75920 * q^71 + 7970 * q^73 - 40725 * q^75 - 86424 * q^77 - 64818 * q^79 + 32805 * q^81 - 92680 * q^83 - 18556 * q^85 - 70398 * q^87 - 52256 * q^89 - 80636 * q^91 - 20052 * q^93 - 132324 * q^95 + 42230 * q^97 - 51192 * 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
$5$	5.87631	0.105119	0.0525593	−	0.998618i	$-0.483262\pi$
$5$	5.87631	0.105119	0.0525593	+	0.998618i	$0.483262\pi$
$6$	0	0
$7$	170.213	1.31295	0.656473	−	0.754350i	$-0.272048\pi$
$7$	170.213	1.31295	0.656473	+	0.754350i	$0.272048\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−578.770	−1.44220	−0.721098	−	0.692833i	$-0.756362\pi$
$11$	−578.770	−1.44220	−0.721098	+	0.692833i	$0.756362\pi$
$12$	0	0
$13$	−282.075	−0.462921	−0.231460	−	0.972844i	$-0.574350\pi$
$13$	−282.075	−0.462921	−0.231460	+	0.972844i	$0.574350\pi$
$14$	0	0
$15$	52.8868	0.0606903
$16$	0	0
$17$	151.453	0.127103	0.0635514	−	0.997979i	$-0.479757\pi$
$17$	151.453	0.127103	0.0635514	+	0.997979i	$0.479757\pi$
$18$	0	0
$19$	−237.501	−0.150932	−0.0754660	−	0.997148i	$-0.524044\pi$
$19$	−237.501	−0.150932	−0.0754660	+	0.997148i	$0.524044\pi$
$20$	0	0
$21$	1531.91	0.758029
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−3090.47	−0.988950
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−6313.78	−1.39410	−0.697051	−	0.717022i	$-0.745505\pi$
$29$	−6313.78	−1.39410	−0.697051	+	0.717022i	$0.745505\pi$
$30$	0	0
$31$	−5132.58	−0.959249	−0.479624	−	0.877474i	$-0.659227\pi$
$31$	−5132.58	−0.959249	−0.479624	+	0.877474i	$0.659227\pi$
$32$	0	0
$33$	−5208.93	−0.832652
$34$	0	0
$35$	1000.22	0.138015
$36$	0	0
$37$	−4511.91	−0.541822	−0.270911	−	0.962604i	$-0.587325\pi$
$37$	−4511.91	−0.541822	−0.270911	+	0.962604i	$0.587325\pi$
$38$	0	0
$39$	−2538.68	−0.267267
$40$	0	0
$41$	4746.75	0.440998	0.220499	−	0.975387i	$-0.429231\pi$
$41$	4746.75	0.440998	0.220499	+	0.975387i	$0.429231\pi$
$42$	0	0
$43$	−14485.2	−1.19469	−0.597344	−	0.801985i	$-0.703777\pi$
$43$	−14485.2	−1.19469	−0.597344	+	0.801985i	$0.703777\pi$
$44$	0	0
$45$	475.981	0.0350396
$46$	0	0
$47$	−11362.3	−0.750276	−0.375138	−	0.926969i	$-0.622405\pi$
$47$	−11362.3	−0.750276	−0.375138	+	0.926969i	$0.622405\pi$
$48$	0	0
$49$	12165.3	0.723825
$50$	0	0
$51$	1363.08	0.0733829
$52$	0	0
$53$	35953.7	1.75814	0.879072	−	0.476689i	$-0.158163\pi$
$53$	35953.7	1.75814	0.879072	+	0.476689i	$0.158163\pi$
$54$	0	0
$55$	−3401.03	−0.151602
$56$	0	0
$57$	−2137.51	−0.0871407
$58$	0	0
$59$	29829.1	1.11560	0.557801	−	0.829975i	$-0.311645\pi$
$59$	29829.1	1.11560	0.557801	+	0.829975i	$0.311645\pi$
$60$	0	0
$61$	−12324.1	−0.424064	−0.212032	−	0.977263i	$-0.568008\pi$
$61$	−12324.1	−0.424064	−0.212032	+	0.977263i	$0.568008\pi$
$62$	0	0
$63$	13787.2	0.437648
$64$	0	0
$65$	−1657.56	−0.0486616
$66$	0	0
$67$	−55464.0	−1.50947	−0.754735	−	0.656030i	$-0.772234\pi$
$67$	−55464.0	−1.50947	−0.754735	+	0.656030i	$0.772234\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	23917.3	0.563076	0.281538	−	0.959550i	$-0.409155\pi$
$71$	23917.3	0.563076	0.281538	+	0.959550i	$0.409155\pi$
$72$	0	0
$73$	32752.5	0.719345	0.359673	−	0.933079i	$-0.382888\pi$
$73$	32752.5	0.719345	0.359673	+	0.933079i	$0.382888\pi$
$74$	0	0
$75$	−27814.2	−0.570971
$76$	0	0
$77$	−98513.9	−1.89352
$78$	0	0
$79$	−39559.2	−0.713147	−0.356574	−	0.934267i	$-0.616055\pi$
$79$	−39559.2	−0.713147	−0.356574	+	0.934267i	$0.616055\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	21175.8	0.337400	0.168700	−	0.985667i	$-0.446043\pi$
$83$	21175.8	0.337400	0.168700	+	0.985667i	$0.446043\pi$
$84$	0	0
$85$	889.984	0.0133609
$86$	0	0
$87$	−56824.0	−0.804885
$88$	0	0
$89$	65097.3	0.871140	0.435570	−	0.900155i	$-0.356547\pi$
$89$	65097.3	0.871140	0.435570	+	0.900155i	$0.356547\pi$
$90$	0	0
$91$	−48012.8	−0.607790
$92$	0	0
$93$	−46193.2	−0.553822
$94$	0	0
$95$	−1395.63	−0.0158658
$96$	0	0
$97$	19272.5	0.207973	0.103987	−	0.994579i	$-0.466840\pi$
$97$	19272.5	0.207973	0.103987	+	0.994579i	$0.466840\pi$
$98$	0	0
$99$	−46880.4	−0.480732
$100$	0	0
$101$	−136495.	−1.33141	−0.665707	−	0.746213i	$-0.731870\pi$
$101$	−136495.	−1.33141	−0.665707	+	0.746213i	$0.731870\pi$
$102$	0	0
$103$	−146154.	−1.35743	−0.678716	−	0.734401i	$-0.737463\pi$
$103$	−146154.	−1.35743	−0.678716	+	0.734401i	$0.737463\pi$
$104$	0	0
$105$	9002.00	0.0796830
$106$	0	0
$107$	42598.4	0.359695	0.179847	−	0.983695i	$-0.442440\pi$
$107$	42598.4	0.359695	0.179847	+	0.983695i	$0.442440\pi$
$108$	0	0
$109$	−76565.0	−0.617254	−0.308627	−	0.951183i	$-0.599870\pi$
$109$	−76565.0	−0.617254	−0.308627	+	0.951183i	$0.599870\pi$
$110$	0	0
$111$	−40607.2	−0.312821
$112$	0	0
$113$	−188819.	−1.39107	−0.695535	−	0.718492i	$-0.744833\pi$
$113$	−188819.	−1.39107	−0.695535	+	0.718492i	$0.744833\pi$
$114$	0	0
$115$	3108.57	0.0219188
$116$	0	0
$117$	−22848.1	−0.154307
$118$	0	0
$119$	25779.2	0.166879
$120$	0	0
$121$	173923.	1.07993
$122$	0	0
$123$	42720.7	0.254610
$124$	0	0
$125$	−36524.0	−0.209076
$126$	0	0
$127$	304762.	1.67668	0.838341	−	0.545146i	$-0.183526\pi$
$127$	304762.	1.67668	0.838341	+	0.545146i	$0.183526\pi$
$128$	0	0
$129$	−130367.	−0.689753
$130$	0	0
$131$	−189494.	−0.964755	−0.482378	−	0.875963i	$-0.660227\pi$
$131$	−189494.	−0.964755	−0.482378	+	0.875963i	$0.660227\pi$
$132$	0	0
$133$	−40425.7	−0.198166
$134$	0	0
$135$	4283.83	0.0202301
$136$	0	0
$137$	256513.	1.16764	0.583818	−	0.811885i	$-0.301558\pi$
$137$	256513.	1.16764	0.583818	+	0.811885i	$0.301558\pi$
$138$	0	0
$139$	−50717.6	−0.222649	−0.111325	−	0.993784i	$-0.535509\pi$
$139$	−50717.6	−0.222649	−0.111325	+	0.993784i	$0.535509\pi$
$140$	0	0
$141$	−102261.	−0.433172
$142$	0	0
$143$	163257.	0.667622
$144$	0	0
$145$	−37101.7	−0.146546
$146$	0	0
$147$	109488.	0.417901
$148$	0	0
$149$	−307421.	−1.13441	−0.567203	−	0.823578i	$-0.691974\pi$
$149$	−307421.	−1.13441	−0.567203	+	0.823578i	$0.691974\pi$
$150$	0	0
$151$	25836.3	0.0922122	0.0461061	−	0.998937i	$-0.485319\pi$
$151$	25836.3	0.0922122	0.0461061	+	0.998937i	$0.485319\pi$
$152$	0	0
$153$	12267.7	0.0423676
$154$	0	0
$155$	−30160.6	−0.100835
$156$	0	0
$157$	−129009.	−0.417707	−0.208854	−	0.977947i	$-0.566973\pi$
$157$	−129009.	−0.417707	−0.208854	+	0.977947i	$0.566973\pi$
$158$	0	0
$159$	323584.	1.01506
$160$	0	0
$161$	90042.5	0.273768
$162$	0	0
$163$	−173649.	−0.511923	−0.255961	−	0.966687i	$-0.582392\pi$
$163$	−173649.	−0.511923	−0.255961	+	0.966687i	$0.582392\pi$
$164$	0	0
$165$	−30609.3	−0.0875273
$166$	0	0
$167$	293629.	0.814719	0.407360	−	0.913268i	$-0.366450\pi$
$167$	293629.	0.814719	0.407360	+	0.913268i	$0.366450\pi$
$168$	0	0
$169$	−291727.	−0.785704
$170$	0	0
$171$	−19237.6	−0.0503107
$172$	0	0
$173$	−726221.	−1.84482	−0.922409	−	0.386214i	$-0.873783\pi$
$173$	−726221.	−1.84482	−0.922409	+	0.386214i	$0.873783\pi$
$174$	0	0
$175$	−526037.	−1.29844
$176$	0	0
$177$	268462.	0.644093
$178$	0	0
$179$	−342180.	−0.798220	−0.399110	−	0.916903i	$-0.630681\pi$
$179$	−342180.	−0.798220	−0.399110	+	0.916903i	$0.630681\pi$
$180$	0	0
$181$	69746.0	0.158243	0.0791213	−	0.996865i	$-0.474789\pi$
$181$	69746.0	0.158243	0.0791213	+	0.996865i	$0.474789\pi$
$182$	0	0
$183$	−110917.	−0.244834
$184$	0	0
$185$	−26513.4	−0.0569556
$186$	0	0
$187$	−87656.3	−0.183307
$188$	0	0
$189$	124085.	0.252676
$190$	0	0
$191$	−276911.	−0.549234	−0.274617	−	0.961554i	$-0.588551\pi$
$191$	−276911.	−0.549234	−0.274617	+	0.961554i	$0.588551\pi$
$192$	0	0
$193$	152163.	0.294047	0.147024	−	0.989133i	$-0.453031\pi$
$193$	152163.	0.294047	0.147024	+	0.989133i	$0.453031\pi$
$194$	0	0
$195$	−14918.1	−0.0280948
$196$	0	0
$197$	−143933.	−0.264238	−0.132119	−	0.991234i	$-0.542178\pi$
$197$	−143933.	−0.264238	−0.132119	+	0.991234i	$0.542178\pi$
$198$	0	0
$199$	−867915.	−1.55362	−0.776810	−	0.629735i	$-0.783163\pi$
$199$	−867915.	−1.55362	−0.776810	+	0.629735i	$0.783163\pi$
$200$	0	0
$201$	−499176.	−0.871492
$202$	0	0
$203$	−1.07468e6	−1.83038
$204$	0	0
$205$	27893.4	0.0463571
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	137458.	0.217674
$210$	0	0
$211$	488239.	0.754963	0.377482	−	0.926017i	$-0.376790\pi$
$211$	488239.	0.754963	0.377482	+	0.926017i	$0.376790\pi$
$212$	0	0
$213$	215256.	0.325092
$214$	0	0
$215$	−85119.8	−0.125584
$216$	0	0
$217$	−873629.	−1.25944
$218$	0	0
$219$	294772.	0.415314
$220$	0	0
$221$	−42721.1	−0.0588385
$222$	0	0
$223$	105814.	0.142489	0.0712443	−	0.997459i	$-0.477303\pi$
$223$	105814.	0.142489	0.0712443	+	0.997459i	$0.477303\pi$
$224$	0	0
$225$	−250328.	−0.329650
$226$	0	0
$227$	535711.	0.690027	0.345013	−	0.938598i	$-0.387874\pi$
$227$	535711.	0.690027	0.345013	+	0.938598i	$0.387874\pi$
$228$	0	0
$229$	−225043.	−0.283581	−0.141790	−	0.989897i	$-0.545286\pi$
$229$	−225043.	−0.283581	−0.141790	+	0.989897i	$0.545286\pi$
$230$	0	0
$231$	−886625.	−1.09323
$232$	0	0
$233$	1.08895e6	1.31407	0.657036	−	0.753859i	$-0.271810\pi$
$233$	1.08895e6	1.31407	0.657036	+	0.753859i	$0.271810\pi$
$234$	0	0
$235$	−66768.3	−0.0788680
$236$	0	0
$237$	−356032.	−0.411736
$238$	0	0
$239$	181916.	0.206004	0.103002	−	0.994681i	$-0.467155\pi$
$239$	181916.	0.206004	0.103002	+	0.994681i	$0.467155\pi$
$240$	0	0
$241$	−1.36730e6	−1.51642	−0.758211	−	0.652009i	$-0.773927\pi$
$241$	−1.36730e6	−1.51642	−0.758211	+	0.652009i	$0.773927\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	71487.3	0.0760876
$246$	0	0
$247$	66993.2	0.0698696
$248$	0	0
$249$	190582.	0.194798
$250$	0	0
$251$	710770.	0.712106	0.356053	−	0.934466i	$-0.384122\pi$
$251$	710770.	0.712106	0.356053	+	0.934466i	$0.384122\pi$
$252$	0	0
$253$	−306169.	−0.300719
$254$	0	0
$255$	8009.86	0.00771391
$256$	0	0
$257$	−1.22227e6	−1.15434	−0.577172	−	0.816623i	$-0.695844\pi$
$257$	−1.22227e6	−1.15434	−0.577172	+	0.816623i	$0.695844\pi$
$258$	0	0
$259$	−767985.	−0.711383
$260$	0	0
$261$	−511416.	−0.464700
$262$	0	0
$263$	362785.	0.323415	0.161708	−	0.986839i	$-0.448300\pi$
$263$	362785.	0.323415	0.161708	+	0.986839i	$0.448300\pi$
$264$	0	0
$265$	211275.	0.184814
$266$	0	0
$267$	585876.	0.502953
$268$	0	0
$269$	−493171.	−0.415544	−0.207772	−	0.978177i	$-0.566621\pi$
$269$	−493171.	−0.415544	−0.207772	+	0.978177i	$0.566621\pi$
$270$	0	0
$271$	1.47340e6	1.21870	0.609349	−	0.792902i	$-0.291431\pi$
$271$	1.47340e6	1.21870	0.609349	+	0.792902i	$0.291431\pi$
$272$	0	0
$273$	−432115.	−0.350908
$274$	0	0
$275$	1.78867e6	1.42626
$276$	0	0
$277$	−1.91598e6	−1.50035	−0.750173	−	0.661242i	$-0.770030\pi$
$277$	−1.91598e6	−1.50035	−0.750173	+	0.661242i	$0.770030\pi$
$278$	0	0
$279$	−415739.	−0.319750
$280$	0	0
$281$	−975478.	−0.736973	−0.368487	−	0.929633i	$-0.620124\pi$
$281$	−975478.	−0.736973	−0.368487	+	0.929633i	$0.620124\pi$
$282$	0	0
$283$	1.44233e6	1.07053	0.535266	−	0.844684i	$-0.320212\pi$
$283$	1.44233e6	1.07053	0.535266	+	0.844684i	$0.320212\pi$
$284$	0	0
$285$	−12560.7	−0.00916011
$286$	0	0
$287$	807956.	0.579006
$288$	0	0
$289$	−1.39692e6	−0.983845
$290$	0	0
$291$	173452.	0.120073
$292$	0	0
$293$	1.29196e6	0.879187	0.439593	−	0.898197i	$-0.355123\pi$
$293$	1.29196e6	0.879187	0.439593	+	0.898197i	$0.355123\pi$
$294$	0	0
$295$	175285.	0.117271
$296$	0	0
$297$	−421923.	−0.277551
$298$	0	0
$299$	−149218.	−0.0965257
$300$	0	0
$301$	−2.46557e6	−1.56856
$302$	0	0
$303$	−1.22845e6	−0.768692
$304$	0	0
$305$	−72420.4	−0.0445771
$306$	0	0
$307$	627499.	0.379986	0.189993	−	0.981785i	$-0.439153\pi$
$307$	627499.	0.379986	0.189993	+	0.981785i	$0.439153\pi$
$308$	0	0
$309$	−1.31539e6	−0.783714
$310$	0	0
$311$	2.51882e6	1.47671	0.738357	−	0.674410i	$-0.235602\pi$
$311$	2.51882e6	1.47671	0.738357	+	0.674410i	$0.235602\pi$
$312$	0	0
$313$	2.77303e6	1.59990	0.799950	−	0.600066i	$-0.204859\pi$
$313$	2.77303e6	1.59990	0.799950	+	0.600066i	$0.204859\pi$
$314$	0	0
$315$	81018.0	0.0460050
$316$	0	0
$317$	3.01447e6	1.68485	0.842427	−	0.538811i	$-0.181126\pi$
$317$	3.01447e6	1.68485	0.842427	+	0.538811i	$0.181126\pi$
$318$	0	0
$319$	3.65422e6	2.01057
$320$	0	0
$321$	383386.	0.207670
$322$	0	0
$323$	−35970.2	−0.0191839
$324$	0	0
$325$	871745.	0.457806
$326$	0	0
$327$	−689085.	−0.356372
$328$	0	0
$329$	−1.93400e6	−0.985071
$330$	0	0
$331$	2.45084e6	1.22955	0.614773	−	0.788704i	$-0.289247\pi$
$331$	2.45084e6	1.22955	0.614773	+	0.788704i	$0.289247\pi$
$332$	0	0
$333$	−365465.	−0.180607
$334$	0	0
$335$	−325924.	−0.158673
$336$	0	0
$337$	−1.00042e6	−0.479851	−0.239926	−	0.970791i	$-0.577123\pi$
$337$	−1.00042e6	−0.479851	−0.239926	+	0.970791i	$0.577123\pi$
$338$	0	0
$339$	−1.69937e6	−0.803135
$340$	0	0
$341$	2.97058e6	1.38342
$342$	0	0
$343$	−790070.	−0.362602
$344$	0	0
$345$	27977.1	0.0126548
$346$	0	0
$347$	1.53410e6	0.683959	0.341979	−	0.939708i	$-0.388903\pi$
$347$	1.53410e6	0.683959	0.341979	+	0.939708i	$0.388903\pi$
$348$	0	0
$349$	−3.65507e6	−1.60632	−0.803160	−	0.595764i	$-0.796850\pi$
$349$	−3.65507e6	−1.60632	−0.803160	+	0.595764i	$0.796850\pi$
$350$	0	0
$351$	−205633.	−0.0890891
$352$	0	0
$353$	−2.93669e6	−1.25436	−0.627179	−	0.778875i	$-0.715790\pi$
$353$	−2.93669e6	−1.25436	−0.627179	+	0.778875i	$0.715790\pi$
$354$	0	0
$355$	140546.	0.0591898
$356$	0	0
$357$	232013.	0.0963477
$358$	0	0
$359$	3.67410e6	1.50458	0.752289	−	0.658834i	$-0.228950\pi$
$359$	3.67410e6	1.50458	0.752289	+	0.658834i	$0.228950\pi$
$360$	0	0
$361$	−2.41969e6	−0.977220
$362$	0	0
$363$	1.56531e6	0.623497
$364$	0	0
$365$	192464.	0.0756166
$366$	0	0
$367$	2.11999e6	0.821614	0.410807	−	0.911722i	$-0.365247\pi$
$367$	2.11999e6	0.821614	0.410807	+	0.911722i	$0.365247\pi$
$368$	0	0
$369$	384487.	0.146999
$370$	0	0
$371$	6.11978e6	2.30835
$372$	0	0
$373$	2.69632e6	1.00346	0.501729	−	0.865025i	$-0.332698\pi$
$373$	2.69632e6	1.00346	0.501729	+	0.865025i	$0.332698\pi$
$374$	0	0
$375$	−328716.	−0.120710
$376$	0	0
$377$	1.78096e6	0.645358
$378$	0	0
$379$	−4.09129e6	−1.46306	−0.731530	−	0.681809i	$-0.761193\pi$
$379$	−4.09129e6	−1.46306	−0.731530	+	0.681809i	$0.761193\pi$
$380$	0	0
$381$	2.74285e6	0.968033
$382$	0	0
$383$	−1.95010e6	−0.679298	−0.339649	−	0.940552i	$-0.610308\pi$
$383$	−1.95010e6	−0.679298	−0.339649	+	0.940552i	$0.610308\pi$
$384$	0	0
$385$	−578899.	−0.199045
$386$	0	0
$387$	−1.17330e6	−0.398229
$388$	0	0
$389$	2.79373e6	0.936074	0.468037	−	0.883709i	$-0.344961\pi$
$389$	2.79373e6	0.936074	0.468037	+	0.883709i	$0.344961\pi$
$390$	0	0
$391$	80118.6	0.0265028
$392$	0	0
$393$	−1.70545e6	−0.557002
$394$	0	0
$395$	−232462.	−0.0749651
$396$	0	0
$397$	3.17661e6	1.01155	0.505776	−	0.862665i	$-0.331206\pi$
$397$	3.17661e6	1.01155	0.505776	+	0.862665i	$0.331206\pi$
$398$	0	0
$399$	−363831.	−0.114411
$400$	0	0
$401$	−231970.	−0.0720396	−0.0360198	−	0.999351i	$-0.511468\pi$
$401$	−231970.	−0.0720396	−0.0360198	+	0.999351i	$0.511468\pi$
$402$	0	0
$403$	1.44777e6	0.444056
$404$	0	0
$405$	38554.5	0.0116799
$406$	0	0
$407$	2.61136e6	0.781413
$408$	0	0
$409$	−1.90264e6	−0.562404	−0.281202	−	0.959649i	$-0.590733\pi$
$409$	−1.90264e6	−0.562404	−0.281202	+	0.959649i	$0.590733\pi$
$410$	0	0
$411$	2.30861e6	0.674134
$412$	0	0
$413$	5.07728e6	1.46473
$414$	0	0
$415$	124436.	0.0354670
$416$	0	0
$417$	−456458.	−0.128547
$418$	0	0
$419$	−3.64335e6	−1.01383	−0.506916	−	0.861995i	$-0.669215\pi$
$419$	−3.64335e6	−1.01383	−0.506916	+	0.861995i	$0.669215\pi$
$420$	0	0
$421$	3.05108e6	0.838974	0.419487	−	0.907761i	$-0.362210\pi$
$421$	3.05108e6	0.838974	0.419487	+	0.907761i	$0.362210\pi$
$422$	0	0
$423$	−920345.	−0.250092
$424$	0	0
$425$	−468060.	−0.125698
$426$	0	0
$427$	−2.09772e6	−0.556773
$428$	0	0
$429$	1.46931e6	0.385452
$430$	0	0
$431$	6.81964e6	1.76835	0.884175	−	0.467156i	$-0.154721\pi$
$431$	6.81964e6	1.76835	0.884175	+	0.467156i	$0.154721\pi$
$432$	0	0
$433$	−1.52956e6	−0.392055	−0.196028	−	0.980598i	$-0.562804\pi$
$433$	−1.52956e6	−0.392055	−0.196028	+	0.980598i	$0.562804\pi$
$434$	0	0
$435$	−333916.	−0.0846084
$436$	0	0
$437$	−125638.	−0.0314715
$438$	0	0
$439$	−877236.	−0.217248	−0.108624	−	0.994083i	$-0.534644\pi$
$439$	−877236.	−0.217248	−0.108624	+	0.994083i	$0.534644\pi$
$440$	0	0
$441$	985392.	0.241275
$442$	0	0
$443$	−2.63189e6	−0.637175	−0.318587	−	0.947893i	$-0.603208\pi$
$443$	−2.63189e6	−0.637175	−0.318587	+	0.947893i	$0.603208\pi$
$444$	0	0
$445$	382532.	0.0915731
$446$	0	0
$447$	−2.76679e6	−0.654949
$448$	0	0
$449$	1.94984e6	0.456439	0.228220	−	0.973610i	$-0.426710\pi$
$449$	1.94984e6	0.456439	0.228220	+	0.973610i	$0.426710\pi$
$450$	0	0
$451$	−2.74727e6	−0.636005
$452$	0	0
$453$	232527.	0.0532387
$454$	0	0
$455$	−282138.	−0.0638900
$456$	0	0
$457$	526971.	0.118031	0.0590155	−	0.998257i	$-0.481204\pi$
$457$	526971.	0.118031	0.0590155	+	0.998257i	$0.481204\pi$
$458$	0	0
$459$	110409.	0.0244610
$460$	0	0
$461$	−5.14903e6	−1.12843	−0.564213	−	0.825629i	$-0.690820\pi$
$461$	−5.14903e6	−1.12843	−0.564213	+	0.825629i	$0.690820\pi$
$462$	0	0
$463$	5.81298e6	1.26022	0.630110	−	0.776506i	$-0.283010\pi$
$463$	5.81298e6	1.26022	0.630110	+	0.776506i	$0.283010\pi$
$464$	0	0
$465$	−271446.	−0.0582171
$466$	0	0
$467$	4.35947e6	0.925000	0.462500	−	0.886619i	$-0.346953\pi$
$467$	4.35947e6	0.925000	0.462500	+	0.886619i	$0.346953\pi$
$468$	0	0
$469$	−9.44067e6	−1.98185
$470$	0	0
$471$	−1.16108e6	−0.241163
$472$	0	0
$473$	8.38362e6	1.72297
$474$	0	0
$475$	733990.	0.149264
$476$	0	0
$477$	2.91225e6	0.586048
$478$	0	0
$479$	5.97865e6	1.19060	0.595298	−	0.803505i	$-0.297034\pi$
$479$	5.97865e6	1.19060	0.595298	+	0.803505i	$0.297034\pi$
$480$	0	0
$481$	1.27270e6	0.250821
$482$	0	0
$483$	810382.	0.158060
$484$	0	0
$485$	113251.	0.0218619
$486$	0	0
$487$	3.65636e6	0.698596	0.349298	−	0.937012i	$-0.386420\pi$
$487$	3.65636e6	0.698596	0.349298	+	0.937012i	$0.386420\pi$
$488$	0	0
$489$	−1.56285e6	−0.295559
$490$	0	0
$491$	996556.	0.186551	0.0932756	−	0.995640i	$-0.470266\pi$
$491$	996556.	0.186551	0.0932756	+	0.995640i	$0.470266\pi$
$492$	0	0
$493$	−956240.	−0.177194
$494$	0	0
$495$	−275484.	−0.0505339
$496$	0	0
$497$	4.07103e6	0.739288
$498$	0	0
$499$	3.80025e6	0.683221	0.341611	−	0.939842i	$-0.389028\pi$
$499$	3.80025e6	0.683221	0.341611	+	0.939842i	$0.389028\pi$
$500$	0	0
$501$	2.64266e6	0.470378
$502$	0	0
$503$	−4.59084e6	−0.809043	−0.404522	−	0.914528i	$-0.632562\pi$
$503$	−4.59084e6	−0.809043	−0.404522	+	0.914528i	$0.632562\pi$
$504$	0	0
$505$	−802087.	−0.139956
$506$	0	0
$507$	−2.62554e6	−0.453627
$508$	0	0
$509$	5.88850e6	1.00742	0.503709	−	0.863873i	$-0.331968\pi$
$509$	5.88850e6	1.00742	0.503709	+	0.863873i	$0.331968\pi$
$510$	0	0
$511$	5.57489e6	0.944461
$512$	0	0
$513$	−173138.	−0.0290469
$514$	0	0
$515$	−858848.	−0.142692
$516$	0	0
$517$	6.57614e6	1.08204
$518$	0	0
$519$	−6.53599e6	−1.06511
$520$	0	0
$521$	−6.82288e6	−1.10122	−0.550609	−	0.834763i	$-0.685604\pi$
$521$	−6.82288e6	−1.10122	−0.550609	+	0.834763i	$0.685604\pi$
$522$	0	0
$523$	3.84731e6	0.615039	0.307519	−	0.951542i	$-0.400501\pi$
$523$	3.84731e6	0.615039	0.307519	+	0.951542i	$0.400501\pi$
$524$	0	0
$525$	−4.73433e6	−0.749653
$526$	0	0
$527$	−777343.	−0.121923
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	2.41615e6	0.371868
$532$	0	0
$533$	−1.33894e6	−0.204147
$534$	0	0
$535$	250322.	0.0378106
$536$	0	0
$537$	−3.07962e6	−0.460852
$538$	0	0
$539$	−7.04093e6	−1.04390
$540$	0	0
$541$	−5.96401e6	−0.876083	−0.438041	−	0.898955i	$-0.644328\pi$
$541$	−5.96401e6	−0.876083	−0.438041	+	0.898955i	$0.644328\pi$
$542$	0	0
$543$	627714.	0.0913614
$544$	0	0
$545$	−449920.	−0.0648850
$546$	0	0
$547$	2.17290e6	0.310508	0.155254	−	0.987875i	$-0.450380\pi$
$547$	2.17290e6	0.310508	0.155254	+	0.987875i	$0.450380\pi$
$548$	0	0
$549$	−998254.	−0.141355
$550$	0	0
$551$	1.49953e6	0.210415
$552$	0	0
$553$	−6.73347e6	−0.936324
$554$	0	0
$555$	−238621.	−0.0328833
$556$	0	0
$557$	6.57494e6	0.897953	0.448976	−	0.893544i	$-0.351789\pi$
$557$	6.57494e6	0.897953	0.448976	+	0.893544i	$0.351789\pi$
$558$	0	0
$559$	4.08593e6	0.553046
$560$	0	0
$561$	−788907.	−0.105832
$562$	0	0
$563$	−945706.	−0.125743	−0.0628717	−	0.998022i	$-0.520026\pi$
$563$	−945706.	−0.125743	−0.0628717	+	0.998022i	$0.520026\pi$
$564$	0	0
$565$	−1.10956e6	−0.146228
$566$	0	0
$567$	1.11676e6	0.145883
$568$	0	0
$569$	−5.74312e6	−0.743648	−0.371824	−	0.928303i	$-0.621268\pi$
$569$	−5.74312e6	−0.743648	−0.371824	+	0.928303i	$0.621268\pi$
$570$	0	0
$571$	−7.12543e6	−0.914578	−0.457289	−	0.889318i	$-0.651179\pi$
$571$	−7.12543e6	−0.914578	−0.457289	+	0.889318i	$0.651179\pi$
$572$	0	0
$573$	−2.49220e6	−0.317100
$574$	0	0
$575$	−1.63486e6	−0.206210
$576$	0	0
$577$	−1.02533e7	−1.28211	−0.641054	−	0.767496i	$-0.721503\pi$
$577$	−1.02533e7	−1.28211	−0.641054	+	0.767496i	$0.721503\pi$
$578$	0	0
$579$	1.36947e6	0.169768
$580$	0	0
$581$	3.60439e6	0.442988
$582$	0	0
$583$	−2.08089e7	−2.53559
$584$	0	0
$585$	−134263.	−0.0162205
$586$	0	0
$587$	3.46183e6	0.414678	0.207339	−	0.978269i	$-0.433520\pi$
$587$	3.46183e6	0.414678	0.207339	+	0.978269i	$0.433520\pi$
$588$	0	0
$589$	1.21899e6	0.144781
$590$	0	0
$591$	−1.29540e6	−0.152558
$592$	0	0
$593$	1.69068e7	1.97435	0.987175	−	0.159641i	$-0.0510337\pi$
$593$	1.69068e7	1.97435	0.987175	+	0.159641i	$0.0510337\pi$
$594$	0	0
$595$	151487.	0.0175421
$596$	0	0
$597$	−7.81124e6	−0.896983
$598$	0	0
$599$	−9.19548e6	−1.04715	−0.523573	−	0.851981i	$-0.675401\pi$
$599$	−9.19548e6	−1.04715	−0.523573	+	0.851981i	$0.675401\pi$
$600$	0	0
$601$	−3.52100e6	−0.397630	−0.198815	−	0.980037i	$-0.563709\pi$
$601$	−3.52100e6	−0.397630	−0.198815	+	0.980037i	$0.563709\pi$
$602$	0	0
$603$	−4.49258e6	−0.503156
$604$	0	0
$605$	1.02203e6	0.113521
$606$	0	0
$607$	1.28793e7	1.41880	0.709399	−	0.704807i	$-0.248966\pi$
$607$	1.28793e7	1.41880	0.709399	+	0.704807i	$0.248966\pi$
$608$	0	0
$609$	−9.67216e6	−1.05677
$610$	0	0
$611$	3.20502e6	0.347318
$612$	0	0
$613$	−2.62103e6	−0.281722	−0.140861	−	0.990029i	$-0.544987\pi$
$613$	−2.62103e6	−0.281722	−0.140861	+	0.990029i	$0.544987\pi$
$614$	0	0
$615$	251040.	0.0267643
$616$	0	0
$617$	2.28450e6	0.241589	0.120795	−	0.992678i	$-0.461456\pi$
$617$	2.28450e6	0.241589	0.120795	+	0.992678i	$0.461456\pi$
$618$	0	0
$619$	−1.63051e7	−1.71040	−0.855198	−	0.518301i	$-0.826565\pi$
$619$	−1.63051e7	−1.71040	−0.855198	+	0.518301i	$0.826565\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	1.10804e7	1.14376
$624$	0	0
$625$	9.44309e6	0.966972
$626$	0	0
$627$	1.23713e6	0.125674
$628$	0	0
$629$	−683342.	−0.0688671
$630$	0	0
$631$	−8.74116e6	−0.873969	−0.436984	−	0.899469i	$-0.643953\pi$
$631$	−8.74116e6	−0.873969	−0.436984	+	0.899469i	$0.643953\pi$
$632$	0	0
$633$	4.39415e6	0.435878
$634$	0	0
$635$	1.79087e6	0.176251
$636$	0	0
$637$	−3.43154e6	−0.335074
$638$	0	0
$639$	1.93730e6	0.187692
$640$	0	0
$641$	2.29734e6	0.220842	0.110421	−	0.993885i	$-0.464780\pi$
$641$	2.29734e6	0.220842	0.110421	+	0.993885i	$0.464780\pi$
$642$	0	0
$643$	1.39916e6	0.133457	0.0667284	−	0.997771i	$-0.478744\pi$
$643$	1.39916e6	0.133457	0.0667284	+	0.997771i	$0.478744\pi$
$644$	0	0
$645$	−766078.	−0.0725059
$646$	0	0
$647$	1.65900e6	0.155807	0.0779033	−	0.996961i	$-0.475177\pi$
$647$	1.65900e6	0.155807	0.0779033	+	0.996961i	$0.475177\pi$
$648$	0	0
$649$	−1.72642e7	−1.60892
$650$	0	0
$651$	−7.86266e6	−0.727139
$652$	0	0
$653$	−8.16894e6	−0.749692	−0.374846	−	0.927087i	$-0.622304\pi$
$653$	−8.16894e6	−0.749692	−0.374846	+	0.927087i	$0.622304\pi$
$654$	0	0
$655$	−1.11353e6	−0.101414
$656$	0	0
$657$	2.65295e6	0.239782
$658$	0	0
$659$	−757154.	−0.0679157	−0.0339579	−	0.999423i	$-0.510811\pi$
$659$	−757154.	−0.0679157	−0.0339579	+	0.999423i	$0.510811\pi$
$660$	0	0
$661$	1.12044e7	0.997439	0.498719	−	0.866764i	$-0.333804\pi$
$661$	1.12044e7	0.997439	0.498719	+	0.866764i	$0.333804\pi$
$662$	0	0
$663$	−384490.	−0.0339704
$664$	0	0
$665$	−237554.	−0.0208309
$666$	0	0
$667$	−3.33999e6	−0.290690
$668$	0	0
$669$	952324.	0.0822658
$670$	0	0
$671$	7.13283e6	0.611583
$672$	0	0
$673$	1.86212e7	1.58478	0.792390	−	0.610014i	$-0.208836\pi$
$673$	1.86212e7	1.58478	0.792390	+	0.610014i	$0.208836\pi$
$674$	0	0
$675$	−2.25295e6	−0.190324
$676$	0	0
$677$	1.58636e7	1.33024	0.665120	−	0.746736i	$-0.268380\pi$
$677$	1.58636e7	1.33024	0.665120	+	0.746736i	$0.268380\pi$
$678$	0	0
$679$	3.28042e6	0.273058
$680$	0	0
$681$	4.82140e6	0.398387
$682$	0	0
$683$	2.14752e7	1.76152	0.880758	−	0.473567i	$-0.157034\pi$
$683$	2.14752e7	1.76152	0.880758	+	0.473567i	$0.157034\pi$
$684$	0	0
$685$	1.50735e6	0.122740
$686$	0	0
$687$	−2.02539e6	−0.163725
$688$	0	0
$689$	−1.01417e7	−0.813881
$690$	0	0
$691$	−1.19081e7	−0.948739	−0.474369	−	0.880326i	$-0.657324\pi$
$691$	−1.19081e7	−0.948739	−0.474369	+	0.880326i	$0.657324\pi$
$692$	0	0
$693$	−7.97963e6	−0.631175
$694$	0	0
$695$	−298032.	−0.0234046
$696$	0	0
$697$	718909.	0.0560521
$698$	0	0
$699$	9.80058e6	0.758680
$700$	0	0
$701$	−9.03876e6	−0.694726	−0.347363	−	0.937731i	$-0.612923\pi$
$701$	−9.03876e6	−0.694726	−0.347363	+	0.937731i	$0.612923\pi$
$702$	0	0
$703$	1.07158e6	0.0817783
$704$	0	0
$705$	−600915.	−0.0455344
$706$	0	0
$707$	−2.32332e7	−1.74807
$708$	0	0
$709$	4.05652e6	0.303067	0.151533	−	0.988452i	$-0.451579\pi$
$709$	4.05652e6	0.303067	0.151533	+	0.988452i	$0.451579\pi$
$710$	0	0
$711$	−3.20429e6	−0.237716
$712$	0	0
$713$	−2.71513e6	−0.200017
$714$	0	0
$715$	959347.	0.0701796
$716$	0	0
$717$	1.63725e6	0.118937
$718$	0	0
$719$	3.92098e6	0.282861	0.141430	−	0.989948i	$-0.454830\pi$
$719$	3.92098e6	0.282861	0.141430	+	0.989948i	$0.454830\pi$
$720$	0	0
$721$	−2.48773e7	−1.78223
$722$	0	0
$723$	−1.23057e7	−0.875507
$724$	0	0
$725$	1.95125e7	1.37870
$726$	0	0
$727$	1.50660e7	1.05721	0.528606	−	0.848867i	$-0.322715\pi$
$727$	1.50660e7	1.05721	0.528606	+	0.848867i	$0.322715\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.19383e6	−0.151848
$732$	0	0
$733$	−7.96942e6	−0.547856	−0.273928	−	0.961750i	$-0.588323\pi$
$733$	−7.96942e6	−0.547856	−0.273928	+	0.961750i	$0.588323\pi$
$734$	0	0
$735$	643386.	0.0439292
$736$	0	0
$737$	3.21009e7	2.17695
$738$	0	0
$739$	−1.83156e7	−1.23370	−0.616852	−	0.787079i	$-0.711592\pi$
$739$	−1.83156e7	−1.23370	−0.616852	+	0.787079i	$0.711592\pi$
$740$	0	0
$741$	602939.	0.0403392
$742$	0	0
$743$	3.40919e6	0.226558	0.113279	−	0.993563i	$-0.463865\pi$
$743$	3.40919e6	0.226558	0.113279	+	0.993563i	$0.463865\pi$
$744$	0	0
$745$	−1.80650e6	−0.119247
$746$	0	0
$747$	1.71524e6	0.112467
$748$	0	0
$749$	7.25079e6	0.472259
$750$	0	0
$751$	1.20178e7	0.777543	0.388772	−	0.921334i	$-0.372899\pi$
$751$	1.20178e7	0.777543	0.388772	+	0.921334i	$0.372899\pi$
$752$	0	0
$753$	6.39693e6	0.411134
$754$	0	0
$755$	151822.	0.00969322
$756$	0	0
$757$	−1.72306e7	−1.09285	−0.546425	−	0.837508i	$-0.684012\pi$
$757$	−1.72306e7	−1.09285	−0.546425	+	0.837508i	$0.684012\pi$
$758$	0	0
$759$	−2.75552e6	−0.173620
$760$	0	0
$761$	1.12918e6	0.0706810	0.0353405	−	0.999375i	$-0.488748\pi$
$761$	1.12918e6	0.0706810	0.0353405	+	0.999375i	$0.488748\pi$
$762$	0	0
$763$	−1.30323e7	−0.810421
$764$	0	0
$765$	72088.7	0.00445363
$766$	0	0
$767$	−8.41404e6	−0.516436
$768$	0	0
$769$	−3.26472e7	−1.99081	−0.995406	−	0.0957473i	$-0.969476\pi$
$769$	−3.26472e7	−1.99081	−0.995406	+	0.0957473i	$0.969476\pi$
$770$	0	0
$771$	−1.10004e7	−0.666461
$772$	0	0
$773$	3.15109e6	0.189676	0.0948381	−	0.995493i	$-0.469767\pi$
$773$	3.15109e6	0.189676	0.0948381	+	0.995493i	$0.469767\pi$
$774$	0	0
$775$	1.58621e7	0.948649
$776$	0	0
$777$	−6.91186e6	−0.410717
$778$	0	0
$779$	−1.12736e6	−0.0665607
$780$	0	0
$781$	−1.38426e7	−0.812066
$782$	0	0
$783$	−4.60274e6	−0.268295
$784$	0	0
$785$	−758099.	−0.0439088
$786$	0	0
$787$	−2.77484e7	−1.59699	−0.798493	−	0.602005i	$-0.794369\pi$
$787$	−2.77484e7	−1.59699	−0.798493	+	0.602005i	$0.794369\pi$
$788$	0	0
$789$	3.26507e6	0.186724
$790$	0	0
$791$	−3.21394e7	−1.82640
$792$	0	0
$793$	3.47633e6	0.196308
$794$	0	0
$795$	1.90148e6	0.106702
$796$	0	0
$797$	−6.72665e6	−0.375105	−0.187553	−	0.982255i	$-0.560056\pi$
$797$	−6.72665e6	−0.375105	−0.187553	+	0.982255i	$0.560056\pi$
$798$	0	0
$799$	−1.72085e6	−0.0953621
$800$	0	0
$801$	5.27288e6	0.290380
$802$	0	0
$803$	−1.89562e7	−1.03744
$804$	0	0
$805$	529118.	0.0287781
$806$	0	0
$807$	−4.43854e6	−0.239914
$808$	0	0
$809$	−1.45612e7	−0.782216	−0.391108	−	0.920345i	$-0.627908\pi$
$809$	−1.45612e7	−0.782216	−0.391108	+	0.920345i	$0.627908\pi$
$810$	0	0
$811$	−1.60819e7	−0.858587	−0.429294	−	0.903165i	$-0.641237\pi$
$811$	−1.60819e7	−0.858587	−0.429294	+	0.903165i	$0.641237\pi$
$812$	0	0
$813$	1.32606e7	0.703616
$814$	0	0
$815$	−1.02042e6	−0.0538126
$816$	0	0
$817$	3.44026e6	0.180317
$818$	0	0
$819$	−3.88903e6	−0.202597
$820$	0	0
$821$	−2.56712e7	−1.32919	−0.664596	−	0.747203i	$-0.731396\pi$
$821$	−2.56712e7	−1.32919	−0.664596	+	0.747203i	$0.731396\pi$
$822$	0	0
$823$	−2.31576e7	−1.19178	−0.595888	−	0.803067i	$-0.703200\pi$
$823$	−2.31576e7	−1.19178	−0.595888	+	0.803067i	$0.703200\pi$
$824$	0	0
$825$	1.60980e7	0.823451
$826$	0	0
$827$	−2.18123e7	−1.10902	−0.554508	−	0.832179i	$-0.687093\pi$
$827$	−2.18123e7	−1.10902	−0.554508	+	0.832179i	$0.687093\pi$
$828$	0	0
$829$	1.17878e7	0.595728	0.297864	−	0.954608i	$-0.403726\pi$
$829$	1.17878e7	0.595728	0.297864	+	0.954608i	$0.403726\pi$
$830$	0	0
$831$	−1.72438e7	−0.866225
$832$	0	0
$833$	1.84247e6	0.0920003
$834$	0	0
$835$	1.72546e6	0.0856422
$836$	0	0
$837$	−3.74165e6	−0.184607
$838$	0	0
$839$	1.24870e7	0.612425	0.306213	−	0.951963i	$-0.400938\pi$
$839$	1.24870e7	0.612425	0.306213	+	0.951963i	$0.400938\pi$
$840$	0	0
$841$	1.93526e7	0.943518
$842$	0	0
$843$	−8.77930e6	−0.425492
$844$	0	0
$845$	−1.71428e6	−0.0825922
$846$	0	0
$847$	2.96040e7	1.41789
$848$	0	0
$849$	1.29810e7	0.618072
$850$	0	0
$851$	−2.38680e6	−0.112978
$852$	0	0
$853$	−2.06486e7	−0.971667	−0.485834	−	0.874051i	$-0.661484\pi$
$853$	−2.06486e7	−0.971667	−0.485834	+	0.874051i	$0.661484\pi$
$854$	0	0
$855$	−113046.	−0.00528859
$856$	0	0
$857$	6.92785e6	0.322216	0.161108	−	0.986937i	$-0.448493\pi$
$857$	6.92785e6	0.322216	0.161108	+	0.986937i	$0.448493\pi$
$858$	0	0
$859$	1.40746e7	0.650807	0.325404	−	0.945575i	$-0.394500\pi$
$859$	1.40746e7	0.650807	0.325404	+	0.945575i	$0.394500\pi$
$860$	0	0
$861$	7.27161e6	0.334289
$862$	0	0
$863$	−2.91778e7	−1.33360	−0.666800	−	0.745237i	$-0.732336\pi$
$863$	−2.91778e7	−1.33360	−0.666800	+	0.745237i	$0.732336\pi$
$864$	0	0
$865$	−4.26750e6	−0.193925
$866$	0	0
$867$	−1.25723e7	−0.568023
$868$	0	0
$869$	2.28956e7	1.02850
$870$	0	0
$871$	1.56450e7	0.698765
$872$	0	0
$873$	1.56107e6	0.0693245
$874$	0	0
$875$	−6.21685e6	−0.274505
$876$	0	0
$877$	−3.45203e6	−0.151557	−0.0757784	−	0.997125i	$-0.524144\pi$
$877$	−3.45203e6	−0.151557	−0.0757784	+	0.997125i	$0.524144\pi$
$878$	0	0
$879$	1.16277e7	0.507599
$880$	0	0
$881$	−1.30918e7	−0.568275	−0.284138	−	0.958784i	$-0.591707\pi$
$881$	−1.30918e7	−0.568275	−0.284138	+	0.958784i	$0.591707\pi$
$882$	0	0
$883$	1.26698e7	0.546851	0.273426	−	0.961893i	$-0.411843\pi$
$883$	1.26698e7	0.546851	0.273426	+	0.961893i	$0.411843\pi$
$884$	0	0
$885$	1.57756e6	0.0677062
$886$	0	0
$887$	−3.68692e7	−1.57346	−0.786728	−	0.617300i	$-0.788227\pi$
$887$	−3.68692e7	−1.57346	−0.786728	+	0.617300i	$0.788227\pi$
$888$	0	0
$889$	5.18743e7	2.20139
$890$	0	0
$891$	−3.79731e6	−0.160244
$892$	0	0
$893$	2.69855e6	0.113241
$894$	0	0
$895$	−2.01076e6	−0.0839078
$896$	0	0
$897$	−1.34296e6	−0.0557291
$898$	0	0
$899$	3.24059e7	1.33729
$900$	0	0
$901$	5.44530e6	0.223465
$902$	0	0
$903$	−2.21901e7	−0.905608
$904$	0	0
$905$	409850.	0.0166342
$906$	0	0
$907$	3.09310e7	1.24846	0.624231	−	0.781240i	$-0.285412\pi$
$907$	3.09310e7	1.24846	0.624231	+	0.781240i	$0.285412\pi$
$908$	0	0
$909$	−1.10561e7	−0.443804
$910$	0	0
$911$	−1.76155e7	−0.703232	−0.351616	−	0.936144i	$-0.614368\pi$
$911$	−1.76155e7	−0.703232	−0.351616	+	0.936144i	$0.614368\pi$
$912$	0	0
$913$	−1.22559e7	−0.486597
$914$	0	0
$915$	−651784.	−0.0257366
$916$	0	0
$917$	−3.22543e7	−1.26667
$918$	0	0
$919$	3.80576e7	1.48646	0.743229	−	0.669038i	$-0.233294\pi$
$919$	3.80576e7	1.48646	0.743229	+	0.669038i	$0.233294\pi$
$920$	0	0
$921$	5.64750e6	0.219385
$922$	0	0
$923$	−6.74649e6	−0.260660
$924$	0	0
$925$	1.39439e7	0.535835
$926$	0	0
$927$	−1.18385e7	−0.452478
$928$	0	0
$929$	5.71846e6	0.217390	0.108695	−	0.994075i	$-0.465333\pi$
$929$	5.71846e6	0.217390	0.108695	+	0.994075i	$0.465333\pi$
$930$	0	0
$931$	−2.88928e6	−0.109248
$932$	0	0
$933$	2.26694e7	0.852581
$934$	0	0
$935$	−515096.	−0.0192690
$936$	0	0
$937$	2.02614e7	0.753912	0.376956	−	0.926231i	$-0.376971\pi$
$937$	2.02614e7	0.753912	0.376956	+	0.926231i	$0.376971\pi$
$938$	0	0
$939$	2.49572e7	0.923703
$940$	0	0
$941$	−5.25897e6	−0.193609	−0.0968047	−	0.995303i	$-0.530862\pi$
$941$	−5.25897e6	−0.193609	−0.0968047	+	0.995303i	$0.530862\pi$
$942$	0	0
$943$	2.51103e6	0.0919544
$944$	0	0
$945$	729162.	0.0265610
$946$	0	0
$947$	−1.60418e7	−0.581270	−0.290635	−	0.956834i	$-0.593867\pi$
$947$	−1.60418e7	−0.581270	−0.290635	+	0.956834i	$0.593867\pi$
$948$	0	0
$949$	−9.23867e6	−0.333000
$950$	0	0
$951$	2.71302e7	0.972751
$952$	0	0
$953$	−2.12140e7	−0.756641	−0.378321	−	0.925675i	$-0.623498\pi$
$953$	−2.12140e7	−0.756641	−0.378321	+	0.925675i	$0.623498\pi$
$954$	0	0
$955$	−1.62722e6	−0.0577347
$956$	0	0
$957$	3.28880e7	1.16080
$958$	0	0
$959$	4.36617e7	1.53304
$960$	0	0
$961$	−2.28581e6	−0.0798422
$962$	0	0
$963$	3.45047e6	0.119898
$964$	0	0
$965$	894160.	0.0309099
$966$	0	0
$967$	3.65309e7	1.25630	0.628152	−	0.778091i	$-0.283812\pi$
$967$	3.65309e7	1.25630	0.628152	+	0.778091i	$0.283812\pi$
$968$	0	0
$969$	−323732.	−0.0110758
$970$	0	0
$971$	−7.22481e6	−0.245911	−0.122956	−	0.992412i	$-0.539237\pi$
$971$	−7.22481e6	−0.245911	−0.122956	+	0.992412i	$0.539237\pi$
$972$	0	0
$973$	−8.63277e6	−0.292326
$974$	0	0
$975$	7.84570e6	0.264314
$976$	0	0
$977$	3.99076e7	1.33758	0.668790	−	0.743451i	$-0.266813\pi$
$977$	3.99076e7	1.33758	0.668790	+	0.743451i	$0.266813\pi$
$978$	0	0
$979$	−3.76764e7	−1.25635
$980$	0	0
$981$	−6.20177e6	−0.205751
$982$	0	0
$983$	−1.53144e7	−0.505493	−0.252746	−	0.967533i	$-0.581334\pi$
$983$	−1.53144e7	−0.505493	−0.252746	+	0.967533i	$0.581334\pi$
$984$	0	0
$985$	−845798.	−0.0277764
$986$	0	0
$987$	−1.74060e7	−0.568731
$988$	0	0
$989$	−7.66269e6	−0.249110
$990$	0	0
$991$	3.41013e7	1.10303	0.551515	−	0.834165i	$-0.314050\pi$
$991$	3.41013e7	1.10303	0.551515	+	0.834165i	$0.314050\pi$
$992$	0	0
$993$	2.20576e7	0.709879
$994$	0	0
$995$	−5.10014e6	−0.163314
$996$	0	0
$997$	9.17915e6	0.292459	0.146229	−	0.989251i	$-0.453286\pi$
$997$	9.17915e6	0.292459	0.146229	+	0.989251i	$0.453286\pi$
$998$	0	0
$999$	−3.28919e6	−0.104274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.6.a.b.1.3	✓	5
4.3	odd	2		1104.6.a.q.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.b.1.3	✓	5	1.1	even	1	trivial
1104.6.a.q.1.3		5	4.3	odd	2

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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	552.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +5.87631 q^{5} +170.213 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +5.87631 q^{5} +170.213 q^{7} +81.0000 q^{9} -578.770 q^{11} -282.075 q^{13} +52.8868 q^{15} +151.453 q^{17} -237.501 q^{19} +1531.91 q^{21} +529.000 q^{23} -3090.47 q^{25} +729.000 q^{27} -6313.78 q^{29} -5132.58 q^{31} -5208.93 q^{33} +1000.22 q^{35} -4511.91 q^{37} -2538.68 q^{39} +4746.75 q^{41} -14485.2 q^{43} +475.981 q^{45} -11362.3 q^{47} +12165.3 q^{49} +1363.08 q^{51} +35953.7 q^{53} -3401.03 q^{55} -2137.51 q^{57} +29829.1 q^{59} -12324.1 q^{61} +13787.2 q^{63} -1657.56 q^{65} -55464.0 q^{67} +4761.00 q^{69} +23917.3 q^{71} +32752.5 q^{73} -27814.2 q^{75} -98513.9 q^{77} -39559.2 q^{79} +6561.00 q^{81} +21175.8 q^{83} +889.984 q^{85} -56824.0 q^{87} +65097.3 q^{89} -48012.8 q^{91} -46193.2 q^{93} -1395.63 q^{95} +19272.5 q^{97} -46880.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 45 q^{3} + 16 q^{5} - 134 q^{7} + 405 q^{9}+O(q^{10}) \) 5 * q + 45 * q^3 + 16 * q^5 - 134 * q^7 + 405 * q^9	\( 5 q + 45 q^{3} + 16 q^{5} - 134 q^{7} + 405 q^{9} - 632 q^{11} + 326 q^{13} + 144 q^{15} - 1044 q^{17} - 722 q^{19} - 1206 q^{21} + 2645 q^{23} - 4525 q^{25} + 3645 q^{27} - 7822 q^{29} - 2228 q^{31} - 5688 q^{33} - 3020 q^{35} - 18818 q^{37} + 2934 q^{39} - 4550 q^{41} - 2226 q^{43} + 1296 q^{45} - 16164 q^{47} - 24563 q^{49} - 9396 q^{51} - 8972 q^{53} - 37496 q^{55} - 6498 q^{57} - 56168 q^{59} - 61474 q^{61} - 10854 q^{63} - 32312 q^{65} - 58270 q^{67} + 23805 q^{69} - 75920 q^{71} + 7970 q^{73} - 40725 q^{75} - 86424 q^{77} - 64818 q^{79} + 32805 q^{81} - 92680 q^{83} - 18556 q^{85} - 70398 q^{87} - 52256 q^{89} - 80636 q^{91} - 20052 q^{93} - 132324 q^{95} + 42230 q^{97} - 51192 q^{99}+O(q^{100}) \) 5 * q + 45 * q^3 + 16 * q^5 - 134 * q^7 + 405 * q^9 - 632 * q^11 + 326 * q^13 + 144 * q^15 - 1044 * q^17 - 722 * q^19 - 1206 * q^21 + 2645 * q^23 - 4525 * q^25 + 3645 * q^27 - 7822 * q^29 - 2228 * q^31 - 5688 * q^33 - 3020 * q^35 - 18818 * q^37 + 2934 * q^39 - 4550 * q^41 - 2226 * q^43 + 1296 * q^45 - 16164 * q^47 - 24563 * q^49 - 9396 * q^51 - 8972 * q^53 - 37496 * q^55 - 6498 * q^57 - 56168 * q^59 - 61474 * q^61 - 10854 * q^63 - 32312 * q^65 - 58270 * q^67 + 23805 * q^69 - 75920 * q^71 + 7970 * q^73 - 40725 * q^75 - 86424 * q^77 - 64818 * q^79 + 32805 * q^81 - 92680 * q^83 - 18556 * q^85 - 70398 * q^87 - 52256 * q^89 - 80636 * q^91 - 20052 * q^93 - 132324 * q^95 + 42230 * q^97 - 51192 * q^99

Introduction

L-functions

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