LMFDB - Embedded newform 192.6.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,6,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("192.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	192.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:	$30.7936934041$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.56776$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -107.084 q^{5} +149.084 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -107.084 q^{5} +149.084 q^{7} +81.0000 q^{9} +434.505 q^{11} +392.505 q^{13} +963.758 q^{15} +803.495 q^{17} -854.842 q^{19} -1341.76 q^{21} -4592.53 q^{23} +8342.03 q^{25} -729.000 q^{27} -6798.60 q^{29} -4798.58 q^{31} -3910.55 q^{33} -15964.6 q^{35} +909.075 q^{37} -3532.55 q^{39} -2372.21 q^{41} +8664.95 q^{43} -8673.82 q^{45} +16878.7 q^{47} +5419.11 q^{49} -7231.45 q^{51} -10831.0 q^{53} -46528.7 q^{55} +7693.58 q^{57} -8305.81 q^{59} +35998.0 q^{61} +12075.8 q^{63} -42031.1 q^{65} -25077.1 q^{67} +41332.7 q^{69} -57850.9 q^{71} -68092.5 q^{73} -75078.3 q^{75} +64777.9 q^{77} -98723.8 q^{79} +6561.00 q^{81} -34859.3 q^{83} -86041.6 q^{85} +61187.4 q^{87} +34678.2 q^{89} +58516.4 q^{91} +43187.2 q^{93} +91540.1 q^{95} +39891.1 q^{97} +35194.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} - 36 q^{5} + 120 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 - 36 * q^5 + 120 * q^7 + 162 * q^9	$ 2 q - 18 q^{3} - 36 q^{5} + 120 q^{7} + 162 q^{9} - 200 q^{11} - 284 q^{13} + 324 q^{15} + 2676 q^{17} + 72 q^{19} - 1080 q^{21} - 3840 q^{23} + 10270 q^{25} - 1458 q^{27} - 10212 q^{29} - 10488 q^{31} + 1800 q^{33} - 18032 q^{35} - 13148 q^{37} + 2556 q^{39} + 4164 q^{41} - 5832 q^{43} - 2916 q^{45} - 1520 q^{47} - 10542 q^{49} - 24084 q^{51} - 9012 q^{53} - 91632 q^{55} - 648 q^{57} - 55096 q^{59} + 63444 q^{61} + 9720 q^{63} - 90120 q^{65} + 36792 q^{67} + 34560 q^{69} - 37664 q^{71} - 37836 q^{73} - 92430 q^{75} + 83232 q^{77} - 144888 q^{79} + 13122 q^{81} - 109272 q^{83} + 47064 q^{85} + 91908 q^{87} - 32556 q^{89} + 78192 q^{91} + 94392 q^{93} + 157424 q^{95} + 69092 q^{97} - 16200 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 - 36 * q^5 + 120 * q^7 + 162 * q^9 - 200 * q^11 - 284 * q^13 + 324 * q^15 + 2676 * q^17 + 72 * q^19 - 1080 * q^21 - 3840 * q^23 + 10270 * q^25 - 1458 * q^27 - 10212 * q^29 - 10488 * q^31 + 1800 * q^33 - 18032 * q^35 - 13148 * q^37 + 2556 * q^39 + 4164 * q^41 - 5832 * q^43 - 2916 * q^45 - 1520 * q^47 - 10542 * q^49 - 24084 * q^51 - 9012 * q^53 - 91632 * q^55 - 648 * q^57 - 55096 * q^59 + 63444 * q^61 + 9720 * q^63 - 90120 * q^65 + 36792 * q^67 + 34560 * q^69 - 37664 * q^71 - 37836 * q^73 - 92430 * q^75 + 83232 * q^77 - 144888 * q^79 + 13122 * q^81 - 109272 * q^83 + 47064 * q^85 + 91908 * q^87 - 32556 * q^89 + 78192 * q^91 + 94392 * q^93 + 157424 * q^95 + 69092 * q^97 - 16200 * 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
$5$	−107.084	−1.91558	−0.957790	−	0.287467i	$-0.907187\pi$
$5$	−107.084	−1.91558	−0.957790	+	0.287467i	$0.907187\pi$
$6$	0	0
$7$	149.084	1.14997	0.574985	−	0.818164i	$-0.305008\pi$
$7$	149.084	1.14997	0.574985	+	0.818164i	$0.305008\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	434.505	1.08271	0.541357	−	0.840793i	$-0.317911\pi$
$11$	434.505	1.08271	0.541357	+	0.840793i	$0.317911\pi$
$12$	0	0
$13$	392.505	0.644150	0.322075	−	0.946714i	$-0.395620\pi$
$13$	392.505	0.644150	0.322075	+	0.946714i	$0.395620\pi$
$14$	0	0
$15$	963.758	1.10596
$16$	0	0
$17$	803.495	0.674312	0.337156	−	0.941449i	$-0.390535\pi$
$17$	803.495	0.674312	0.337156	+	0.941449i	$0.390535\pi$
$18$	0	0
$19$	−854.842	−0.543253	−0.271626	−	0.962403i	$-0.587562\pi$
$19$	−854.842	−0.543253	−0.271626	+	0.962403i	$0.587562\pi$
$20$	0	0
$21$	−1341.76	−0.663936
$22$	0	0
$23$	−4592.53	−1.81022	−0.905112	−	0.425174i	$-0.860213\pi$
$23$	−4592.53	−1.81022	−0.905112	+	0.425174i	$0.860213\pi$
$24$	0	0
$25$	8342.03	2.66945
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−6798.60	−1.50115	−0.750576	−	0.660784i	$-0.770224\pi$
$29$	−6798.60	−1.50115	−0.750576	+	0.660784i	$0.770224\pi$
$30$	0	0
$31$	−4798.58	−0.896826	−0.448413	−	0.893826i	$-0.648011\pi$
$31$	−4798.58	−0.896826	−0.448413	+	0.893826i	$0.648011\pi$
$32$	0	0
$33$	−3910.55	−0.625105
$34$	0	0
$35$	−15964.6	−2.20286
$36$	0	0
$37$	909.075	0.109168	0.0545840	−	0.998509i	$-0.482617\pi$
$37$	909.075	0.109168	0.0545840	+	0.998509i	$0.482617\pi$
$38$	0	0
$39$	−3532.55	−0.371900
$40$	0	0
$41$	−2372.21	−0.220391	−0.110195	−	0.993910i	$-0.535148\pi$
$41$	−2372.21	−0.220391	−0.110195	+	0.993910i	$0.535148\pi$
$42$	0	0
$43$	8664.95	0.714652	0.357326	−	0.933980i	$-0.383688\pi$
$43$	8664.95	0.714652	0.357326	+	0.933980i	$0.383688\pi$
$44$	0	0
$45$	−8673.82	−0.638527
$46$	0	0
$47$	16878.7	1.11454	0.557268	−	0.830333i	$-0.311850\pi$
$47$	16878.7	1.11454	0.557268	+	0.830333i	$0.311850\pi$
$48$	0	0
$49$	5419.11	0.322432
$50$	0	0
$51$	−7231.45	−0.389314
$52$	0	0
$53$	−10831.0	−0.529637	−0.264818	−	0.964298i	$-0.585312\pi$
$53$	−10831.0	−0.529637	−0.264818	+	0.964298i	$0.585312\pi$
$54$	0	0
$55$	−46528.7	−2.07402
$56$	0	0
$57$	7693.58	0.313647
$58$	0	0
$59$	−8305.81	−0.310636	−0.155318	−	0.987865i	$-0.549640\pi$
$59$	−8305.81	−0.310636	−0.155318	+	0.987865i	$0.549640\pi$
$60$	0	0
$61$	35998.0	1.23867	0.619333	−	0.785128i	$-0.287403\pi$
$61$	35998.0	1.23867	0.619333	+	0.785128i	$0.287403\pi$
$62$	0	0
$63$	12075.8	0.383323
$64$	0	0
$65$	−42031.1	−1.23392
$66$	0	0
$67$	−25077.1	−0.682481	−0.341240	−	0.939976i	$-0.610847\pi$
$67$	−25077.1	−0.682481	−0.341240	+	0.939976i	$0.610847\pi$
$68$	0	0
$69$	41332.7	1.04513
$70$	0	0
$71$	−57850.9	−1.36196	−0.680980	−	0.732302i	$-0.738446\pi$
$71$	−57850.9	−1.36196	−0.680980	+	0.732302i	$0.738446\pi$
$72$	0	0
$73$	−68092.5	−1.49552	−0.747760	−	0.663969i	$-0.768871\pi$
$73$	−68092.5	−1.49552	−0.747760	+	0.663969i	$0.768871\pi$
$74$	0	0
$75$	−75078.3	−1.54121
$76$	0	0
$77$	64777.9	1.24509
$78$	0	0
$79$	−98723.8	−1.77973	−0.889865	−	0.456223i	$-0.849202\pi$
$79$	−98723.8	−1.77973	−0.889865	+	0.456223i	$0.849202\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−34859.3	−0.555422	−0.277711	−	0.960665i	$-0.589576\pi$
$83$	−34859.3	−0.555422	−0.277711	+	0.960665i	$0.589576\pi$
$84$	0	0
$85$	−86041.6	−1.29170
$86$	0	0
$87$	61187.4	0.866690
$88$	0	0
$89$	34678.2	0.464068	0.232034	−	0.972708i	$-0.425462\pi$
$89$	34678.2	0.464068	0.232034	+	0.972708i	$0.425462\pi$
$90$	0	0
$91$	58516.4	0.740754
$92$	0	0
$93$	43187.2	0.517783
$94$	0	0
$95$	91540.1	1.04064
$96$	0	0
$97$	39891.1	0.430473	0.215237	−	0.976562i	$-0.430948\pi$
$97$	39891.1	0.430473	0.215237	+	0.976562i	$0.430948\pi$
$98$	0	0
$99$	35194.9	0.360904
$100$	0	0
$101$	−35341.5	−0.344732	−0.172366	−	0.985033i	$-0.555141\pi$
$101$	−35341.5	−0.344732	−0.172366	+	0.985033i	$0.555141\pi$
$102$	0	0
$103$	37097.4	0.344549	0.172274	−	0.985049i	$-0.444888\pi$
$103$	37097.4	0.344549	0.172274	+	0.985049i	$0.444888\pi$
$104$	0	0
$105$	143681.	1.27182
$106$	0	0
$107$	−153566.	−1.29669	−0.648345	−	0.761347i	$-0.724538\pi$
$107$	−153566.	−1.29669	−0.648345	+	0.761347i	$0.724538\pi$
$108$	0	0
$109$	106749.	0.860590	0.430295	−	0.902688i	$-0.358410\pi$
$109$	106749.	0.860590	0.430295	+	0.902688i	$0.358410\pi$
$110$	0	0
$111$	−8181.68	−0.0630282
$112$	0	0
$113$	−102232.	−0.753168	−0.376584	−	0.926383i	$-0.622901\pi$
$113$	−102232.	−0.753168	−0.376584	+	0.926383i	$0.622901\pi$
$114$	0	0
$115$	491787.	3.46763
$116$	0	0
$117$	31792.9	0.214717
$118$	0	0
$119$	119788.	0.775438
$120$	0	0
$121$	27743.9	0.172268
$122$	0	0
$123$	21349.9	0.127243
$124$	0	0
$125$	−558662.	−3.19797
$126$	0	0
$127$	−47290.4	−0.260174	−0.130087	−	0.991503i	$-0.541526\pi$
$127$	−47290.4	−0.260174	−0.130087	+	0.991503i	$0.541526\pi$
$128$	0	0
$129$	−77984.5	−0.412605
$130$	0	0
$131$	82047.2	0.417720	0.208860	−	0.977946i	$-0.433025\pi$
$131$	82047.2	0.417720	0.208860	+	0.977946i	$0.433025\pi$
$132$	0	0
$133$	−127444.	−0.624725
$134$	0	0
$135$	78064.4	0.368654
$136$	0	0
$137$	−68984.2	−0.314013	−0.157007	−	0.987598i	$-0.550184\pi$
$137$	−68984.2	−0.314013	−0.157007	+	0.987598i	$0.550184\pi$
$138$	0	0
$139$	−437887.	−1.92232	−0.961158	−	0.275998i	$-0.910992\pi$
$139$	−437887.	−1.92232	−0.961158	+	0.275998i	$0.910992\pi$
$140$	0	0
$141$	−151908.	−0.643477
$142$	0	0
$143$	170546.	0.697430
$144$	0	0
$145$	728023.	2.87558
$146$	0	0
$147$	−48772.0	−0.186156
$148$	0	0
$149$	27672.3	0.102113	0.0510564	−	0.998696i	$-0.483741\pi$
$149$	27672.3	0.102113	0.0510564	+	0.998696i	$0.483741\pi$
$150$	0	0
$151$	−84706.4	−0.302325	−0.151162	−	0.988509i	$-0.548302\pi$
$151$	−84706.4	−0.302325	−0.151162	+	0.988509i	$0.548302\pi$
$152$	0	0
$153$	65083.1	0.224771
$154$	0	0
$155$	513852.	1.71794
$156$	0	0
$157$	565956.	1.83246	0.916229	−	0.400656i	$-0.131218\pi$
$157$	565956.	1.83246	0.916229	+	0.400656i	$0.131218\pi$
$158$	0	0
$159$	97478.8	0.305786
$160$	0	0
$161$	−684673.	−2.08170
$162$	0	0
$163$	−26326.8	−0.0776121	−0.0388060	−	0.999247i	$-0.512355\pi$
$163$	−26326.8	−0.0776121	−0.0388060	+	0.999247i	$0.512355\pi$
$164$	0	0
$165$	418758.	1.19744
$166$	0	0
$167$	−268359.	−0.744603	−0.372301	−	0.928112i	$-0.621431\pi$
$167$	−268359.	−0.744603	−0.372301	+	0.928112i	$0.621431\pi$
$168$	0	0
$169$	−217233.	−0.585070
$170$	0	0
$171$	−69242.2	−0.181084
$172$	0	0
$173$	−103090.	−0.261879	−0.130939	−	0.991390i	$-0.541799\pi$
$173$	−103090.	−0.261879	−0.130939	+	0.991390i	$0.541799\pi$
$174$	0	0
$175$	1.24367e6	3.06979
$176$	0	0
$177$	74752.3	0.179346
$178$	0	0
$179$	671994.	1.56759	0.783796	−	0.621019i	$-0.213281\pi$
$179$	671994.	1.56759	0.783796	+	0.621019i	$0.213281\pi$
$180$	0	0
$181$	445972.	1.01184	0.505919	−	0.862581i	$-0.331154\pi$
$181$	445972.	1.01184	0.505919	+	0.862581i	$0.331154\pi$
$182$	0	0
$183$	−323982.	−0.715144
$184$	0	0
$185$	−97347.6	−0.209120
$186$	0	0
$187$	349123.	0.730086
$188$	0	0
$189$	−108682.	−0.221312
$190$	0	0
$191$	420397.	0.833827	0.416914	−	0.908946i	$-0.363112\pi$
$191$	420397.	0.833827	0.416914	+	0.908946i	$0.363112\pi$
$192$	0	0
$193$	−148098.	−0.286190	−0.143095	−	0.989709i	$-0.545705\pi$
$193$	−148098.	−0.286190	−0.143095	+	0.989709i	$0.545705\pi$
$194$	0	0
$195$	378280.	0.712405
$196$	0	0
$197$	−399632.	−0.733660	−0.366830	−	0.930288i	$-0.619557\pi$
$197$	−399632.	−0.733660	−0.366830	+	0.930288i	$0.619557\pi$
$198$	0	0
$199$	−639043.	−1.14393	−0.571963	−	0.820280i	$-0.693818\pi$
$199$	−639043.	−1.14393	−0.571963	+	0.820280i	$0.693818\pi$
$200$	0	0
$201$	225694.	0.394030
$202$	0	0
$203$	−1.01356e6	−1.72628
$204$	0	0
$205$	254026.	0.422177
$206$	0	0
$207$	−371995.	−0.603408
$208$	0	0
$209$	−371434.	−0.588187
$210$	0	0
$211$	−840471.	−1.29962	−0.649810	−	0.760097i	$-0.725152\pi$
$211$	−840471.	−1.29962	−0.649810	+	0.760097i	$0.725152\pi$
$212$	0	0
$213$	520658.	0.786328
$214$	0	0
$215$	−927879.	−1.36897
$216$	0	0
$217$	−715392.	−1.03132
$218$	0	0
$219$	612832.	0.863439
$220$	0	0
$221$	315376.	0.434358
$222$	0	0
$223$	436150.	0.587319	0.293659	−	0.955910i	$-0.405127\pi$
$223$	436150.	0.587319	0.293659	+	0.955910i	$0.405127\pi$
$224$	0	0
$225$	675705.	0.889817
$226$	0	0
$227$	−656896.	−0.846119	−0.423060	−	0.906102i	$-0.639044\pi$
$227$	−656896.	−0.846119	−0.423060	+	0.906102i	$0.639044\pi$
$228$	0	0
$229$	−769622.	−0.969815	−0.484908	−	0.874565i	$-0.661147\pi$
$229$	−769622.	−0.969815	−0.484908	+	0.874565i	$0.661147\pi$
$230$	0	0
$231$	−583001.	−0.718852
$232$	0	0
$233$	−1.11414e6	−1.34446	−0.672232	−	0.740341i	$-0.734664\pi$
$233$	−1.11414e6	−1.34446	−0.672232	+	0.740341i	$0.734664\pi$
$234$	0	0
$235$	−1.80744e6	−2.13498
$236$	0	0
$237$	888515.	1.02753
$238$	0	0
$239$	−742802.	−0.841160	−0.420580	−	0.907256i	$-0.638173\pi$
$239$	−742802.	−0.841160	−0.420580	+	0.907256i	$0.638173\pi$
$240$	0	0
$241$	427097.	0.473679	0.236840	−	0.971549i	$-0.423888\pi$
$241$	427097.	0.473679	0.236840	+	0.971549i	$0.423888\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−580301.	−0.617644
$246$	0	0
$247$	−335530.	−0.349936
$248$	0	0
$249$	313734.	0.320673
$250$	0	0
$251$	−714389.	−0.715732	−0.357866	−	0.933773i	$-0.616495\pi$
$251$	−714389.	−0.715732	−0.357866	+	0.933773i	$0.616495\pi$
$252$	0	0
$253$	−1.99548e6	−1.95995
$254$	0	0
$255$	774374.	0.745763
$256$	0	0
$257$	2.01289e6	1.90102	0.950511	−	0.310691i	$-0.100561\pi$
$257$	2.01289e6	1.90102	0.950511	+	0.310691i	$0.100561\pi$
$258$	0	0
$259$	135529.	0.125540
$260$	0	0
$261$	−550687.	−0.500384
$262$	0	0
$263$	−1.13759e6	−1.01413	−0.507067	−	0.861907i	$-0.669270\pi$
$263$	−1.13759e6	−1.01413	−0.507067	+	0.861907i	$0.669270\pi$
$264$	0	0
$265$	1.15983e6	1.01456
$266$	0	0
$267$	−312104.	−0.267930
$268$	0	0
$269$	−1.15018e6	−0.969138	−0.484569	−	0.874753i	$-0.661023\pi$
$269$	−1.15018e6	−0.969138	−0.484569	+	0.874753i	$0.661023\pi$
$270$	0	0
$271$	271676.	0.224713	0.112356	−	0.993668i	$-0.464160\pi$
$271$	271676.	0.224713	0.112356	+	0.993668i	$0.464160\pi$
$272$	0	0
$273$	−526647.	−0.427674
$274$	0	0
$275$	3.62466e6	2.89025
$276$	0	0
$277$	1.20183e6	0.941115	0.470557	−	0.882369i	$-0.344053\pi$
$277$	1.20183e6	0.941115	0.470557	+	0.882369i	$0.344053\pi$
$278$	0	0
$279$	−388685.	−0.298942
$280$	0	0
$281$	1.14613e6	0.865900	0.432950	−	0.901418i	$-0.357473\pi$
$281$	1.14613e6	0.865900	0.432950	+	0.901418i	$0.357473\pi$
$282$	0	0
$283$	−552547.	−0.410113	−0.205056	−	0.978750i	$-0.565738\pi$
$283$	−552547.	−0.410113	−0.205056	+	0.978750i	$0.565738\pi$
$284$	0	0
$285$	−823861.	−0.600817
$286$	0	0
$287$	−353659.	−0.253443
$288$	0	0
$289$	−774253.	−0.545304
$290$	0	0
$291$	−359019.	−0.248534
$292$	0	0
$293$	−601434.	−0.409278	−0.204639	−	0.978837i	$-0.565602\pi$
$293$	−601434.	−0.409278	−0.204639	+	0.978837i	$0.565602\pi$
$294$	0	0
$295$	889421.	0.595048
$296$	0	0
$297$	−316754.	−0.208368
$298$	0	0
$299$	−1.80259e6	−1.16606
$300$	0	0
$301$	1.29181e6	0.821829
$302$	0	0
$303$	318074.	0.199031
$304$	0	0
$305$	−3.85482e6	−2.37277
$306$	0	0
$307$	561644.	0.340107	0.170053	−	0.985435i	$-0.445606\pi$
$307$	561644.	0.340107	0.170053	+	0.985435i	$0.445606\pi$
$308$	0	0
$309$	−333877.	−0.198925
$310$	0	0
$311$	1.83235e6	1.07425	0.537126	−	0.843502i	$-0.319510\pi$
$311$	1.83235e6	1.07425	0.537126	+	0.843502i	$0.319510\pi$
$312$	0	0
$313$	2.63782e6	1.52190	0.760948	−	0.648813i	$-0.224734\pi$
$313$	2.63782e6	1.52190	0.760948	+	0.648813i	$0.224734\pi$
$314$	0	0
$315$	−1.29313e6	−0.734287
$316$	0	0
$317$	2.11403e6	1.18158	0.590791	−	0.806825i	$-0.298816\pi$
$317$	2.11403e6	1.18158	0.590791	+	0.806825i	$0.298816\pi$
$318$	0	0
$319$	−2.95403e6	−1.62532
$320$	0	0
$321$	1.38209e6	0.748644
$322$	0	0
$323$	−686861.	−0.366322
$324$	0	0
$325$	3.27429e6	1.71953
$326$	0	0
$327$	−960738.	−0.496862
$328$	0	0
$329$	2.51634e6	1.28168
$330$	0	0
$331$	432931.	0.217194	0.108597	−	0.994086i	$-0.465364\pi$
$331$	432931.	0.217194	0.108597	+	0.994086i	$0.465364\pi$
$332$	0	0
$333$	73635.1	0.0363893
$334$	0	0
$335$	2.68536e6	1.30735
$336$	0	0
$337$	−188395.	−0.0903640	−0.0451820	−	0.998979i	$-0.514387\pi$
$337$	−188395.	−0.0903640	−0.0451820	+	0.998979i	$0.514387\pi$
$338$	0	0
$339$	920090.	0.434842
$340$	0	0
$341$	−2.08501e6	−0.971006
$342$	0	0
$343$	−1.69776e6	−0.779184
$344$	0	0
$345$	−4.42608e6	−2.00204
$346$	0	0
$347$	−2.75126e6	−1.22661	−0.613306	−	0.789845i	$-0.710161\pi$
$347$	−2.75126e6	−1.22661	−0.613306	+	0.789845i	$0.710161\pi$
$348$	0	0
$349$	−2.36608e6	−1.03984	−0.519919	−	0.854216i	$-0.674038\pi$
$349$	−2.36608e6	−1.03984	−0.519919	+	0.854216i	$0.674038\pi$
$350$	0	0
$351$	−286136.	−0.123967
$352$	0	0
$353$	−67235.9	−0.0287187	−0.0143593	−	0.999897i	$-0.504571\pi$
$353$	−67235.9	−0.0287187	−0.0143593	+	0.999897i	$0.504571\pi$
$354$	0	0
$355$	6.19492e6	2.60894
$356$	0	0
$357$	−1.07810e6	−0.447700
$358$	0	0
$359$	880277.	0.360482	0.180241	−	0.983622i	$-0.442312\pi$
$359$	880277.	0.360482	0.180241	+	0.983622i	$0.442312\pi$
$360$	0	0
$361$	−1.74534e6	−0.704876
$362$	0	0
$363$	−249695.	−0.0994589
$364$	0	0
$365$	7.29163e6	2.86479
$366$	0	0
$367$	2.71882e6	1.05370	0.526848	−	0.849959i	$-0.323374\pi$
$367$	2.71882e6	1.05370	0.526848	+	0.849959i	$0.323374\pi$
$368$	0	0
$369$	−192149.	−0.0734637
$370$	0	0
$371$	−1.61473e6	−0.609066
$372$	0	0
$373$	−111486.	−0.0414904	−0.0207452	−	0.999785i	$-0.506604\pi$
$373$	−111486.	−0.0414904	−0.0207452	+	0.999785i	$0.506604\pi$
$374$	0	0
$375$	5.02796e6	1.84635
$376$	0	0
$377$	−2.66849e6	−0.966967
$378$	0	0
$379$	2.62051e6	0.937104	0.468552	−	0.883436i	$-0.344776\pi$
$379$	2.62051e6	0.937104	0.468552	+	0.883436i	$0.344776\pi$
$380$	0	0
$381$	425614.	0.150212
$382$	0	0
$383$	1.82442e6	0.635519	0.317760	−	0.948171i	$-0.397070\pi$
$383$	1.82442e6	0.635519	0.317760	+	0.948171i	$0.397070\pi$
$384$	0	0
$385$	−6.93669e6	−2.38507
$386$	0	0
$387$	701861.	0.238217
$388$	0	0
$389$	162259.	0.0543668	0.0271834	−	0.999630i	$-0.491346\pi$
$389$	162259.	0.0543668	0.0271834	+	0.999630i	$0.491346\pi$
$390$	0	0
$391$	−3.69007e6	−1.22065
$392$	0	0
$393$	−738425.	−0.241171
$394$	0	0
$395$	1.05718e7	3.40922
$396$	0	0
$397$	−2.92783e6	−0.932330	−0.466165	−	0.884698i	$-0.654365\pi$
$397$	−2.92783e6	−0.932330	−0.466165	+	0.884698i	$0.654365\pi$
$398$	0	0
$399$	1.14699e6	0.360685
$400$	0	0
$401$	3.92259e6	1.21818	0.609090	−	0.793101i	$-0.291535\pi$
$401$	3.92259e6	1.21818	0.609090	+	0.793101i	$0.291535\pi$
$402$	0	0
$403$	−1.88347e6	−0.577691
$404$	0	0
$405$	−702580.	−0.212842
$406$	0	0
$407$	394998.	0.118198
$408$	0	0
$409$	−151551.	−0.0447973	−0.0223986	−	0.999749i	$-0.507130\pi$
$409$	−151551.	−0.0447973	−0.0223986	+	0.999749i	$0.507130\pi$
$410$	0	0
$411$	620858.	0.181296
$412$	0	0
$413$	−1.23826e6	−0.357222
$414$	0	0
$415$	3.73288e6	1.06396
$416$	0	0
$417$	3.94098e6	1.10985
$418$	0	0
$419$	−3.13904e6	−0.873497	−0.436748	−	0.899584i	$-0.643870\pi$
$419$	−3.13904e6	−0.873497	−0.436748	+	0.899584i	$0.643870\pi$
$420$	0	0
$421$	−2.32328e6	−0.638847	−0.319423	−	0.947612i	$-0.603489\pi$
$421$	−2.32328e6	−0.638847	−0.319423	+	0.947612i	$0.603489\pi$
$422$	0	0
$423$	1.36717e6	0.371512
$424$	0	0
$425$	6.70278e6	1.80004
$426$	0	0
$427$	5.36674e6	1.42443
$428$	0	0
$429$	−1.53491e6	−0.402661
$430$	0	0
$431$	−998677.	−0.258960	−0.129480	−	0.991582i	$-0.541331\pi$
$431$	−998677.	−0.258960	−0.129480	+	0.991582i	$0.541331\pi$
$432$	0	0
$433$	4.18444e6	1.07255	0.536275	−	0.844044i	$-0.319831\pi$
$433$	4.18444e6	1.07255	0.536275	+	0.844044i	$0.319831\pi$
$434$	0	0
$435$	−6.55221e6	−1.66022
$436$	0	0
$437$	3.92589e6	0.983409
$438$	0	0
$439$	3.30452e6	0.818364	0.409182	−	0.912453i	$-0.365814\pi$
$439$	3.30452e6	0.818364	0.409182	+	0.912453i	$0.365814\pi$
$440$	0	0
$441$	438948.	0.107477
$442$	0	0
$443$	−4.11097e6	−0.995257	−0.497629	−	0.867390i	$-0.665796\pi$
$443$	−4.11097e6	−0.995257	−0.497629	+	0.867390i	$0.665796\pi$
$444$	0	0
$445$	−3.71349e6	−0.888959
$446$	0	0
$447$	−249051.	−0.0589548
$448$	0	0
$449$	7.22741e6	1.69187	0.845934	−	0.533287i	$-0.179043\pi$
$449$	7.22741e6	1.69187	0.845934	+	0.533287i	$0.179043\pi$
$450$	0	0
$451$	−1.03074e6	−0.238620
$452$	0	0
$453$	762357.	0.174547
$454$	0	0
$455$	−6.26618e6	−1.41897
$456$	0	0
$457$	−6.11469e6	−1.36957	−0.684784	−	0.728746i	$-0.740103\pi$
$457$	−6.11469e6	−1.36957	−0.684784	+	0.728746i	$0.740103\pi$
$458$	0	0
$459$	−585748.	−0.129771
$460$	0	0
$461$	3.91568e6	0.858134	0.429067	−	0.903273i	$-0.358842\pi$
$461$	3.91568e6	0.858134	0.429067	+	0.903273i	$0.358842\pi$
$462$	0	0
$463$	−6.39830e6	−1.38711	−0.693557	−	0.720402i	$-0.743957\pi$
$463$	−6.39830e6	−1.38711	−0.693557	+	0.720402i	$0.743957\pi$
$464$	0	0
$465$	−4.62467e6	−0.991855
$466$	0	0
$467$	−3.54919e6	−0.753074	−0.376537	−	0.926402i	$-0.622885\pi$
$467$	−3.54919e6	−0.753074	−0.376537	+	0.926402i	$0.622885\pi$
$468$	0	0
$469$	−3.73860e6	−0.784833
$470$	0	0
$471$	−5.09361e6	−1.05797
$472$	0	0
$473$	3.76497e6	0.773764
$474$	0	0
$475$	−7.13112e6	−1.45019
$476$	0	0
$477$	−877309.	−0.176546
$478$	0	0
$479$	5.52909e6	1.10107	0.550535	−	0.834812i	$-0.314424\pi$
$479$	5.52909e6	1.10107	0.550535	+	0.834812i	$0.314424\pi$
$480$	0	0
$481$	356817.	0.0703206
$482$	0	0
$483$	6.16206e6	1.20187
$484$	0	0
$485$	−4.27170e6	−0.824606
$486$	0	0
$487$	−1.29274e6	−0.246996	−0.123498	−	0.992345i	$-0.539411\pi$
$487$	−1.29274e6	−0.246996	−0.123498	+	0.992345i	$0.539411\pi$
$488$	0	0
$489$	236941.	0.0448094
$490$	0	0
$491$	5.06222e6	0.947626	0.473813	−	0.880625i	$-0.342877\pi$
$491$	5.06222e6	0.947626	0.473813	+	0.880625i	$0.342877\pi$
$492$	0	0
$493$	−5.46264e6	−1.01224
$494$	0	0
$495$	−3.76882e6	−0.691342
$496$	0	0
$497$	−8.62466e6	−1.56621
$498$	0	0
$499$	−2.89211e6	−0.519951	−0.259976	−	0.965615i	$-0.583715\pi$
$499$	−2.89211e6	−0.519951	−0.259976	+	0.965615i	$0.583715\pi$
$500$	0	0
$501$	2.41523e6	0.429897
$502$	0	0
$503$	4.20037e6	0.740231	0.370115	−	0.928986i	$-0.379318\pi$
$503$	4.20037e6	0.740231	0.370115	+	0.928986i	$0.379318\pi$
$504$	0	0
$505$	3.78452e6	0.660363
$506$	0	0
$507$	1.95509e6	0.337791
$508$	0	0
$509$	3.09356e6	0.529255	0.264627	−	0.964351i	$-0.414751\pi$
$509$	3.09356e6	0.529255	0.264627	+	0.964351i	$0.414751\pi$
$510$	0	0
$511$	−1.01515e7	−1.71980
$512$	0	0
$513$	623180.	0.104549
$514$	0	0
$515$	−3.97255e6	−0.660011
$516$	0	0
$517$	7.33388e6	1.20672
$518$	0	0
$519$	927807.	0.151196
$520$	0	0
$521$	6.99447e6	1.12891	0.564456	−	0.825463i	$-0.309086\pi$
$521$	6.99447e6	1.12891	0.564456	+	0.825463i	$0.309086\pi$
$522$	0	0
$523$	1.93493e6	0.309322	0.154661	−	0.987968i	$-0.450571\pi$
$523$	1.93493e6	0.309322	0.154661	+	0.987968i	$0.450571\pi$
$524$	0	0
$525$	−1.11930e7	−1.77234
$526$	0	0
$527$	−3.85563e6	−0.604741
$528$	0	0
$529$	1.46550e7	2.27691
$530$	0	0
$531$	−672770.	−0.103545
$532$	0	0
$533$	−931106.	−0.141965
$534$	0	0
$535$	1.64445e7	2.48391
$536$	0	0
$537$	−6.04795e6	−0.905049
$538$	0	0
$539$	2.35463e6	0.349101
$540$	0	0
$541$	8.23886e6	1.21025	0.605124	−	0.796131i	$-0.293124\pi$
$541$	8.23886e6	1.21025	0.605124	+	0.796131i	$0.293124\pi$
$542$	0	0
$543$	−4.01375e6	−0.584185
$544$	0	0
$545$	−1.14311e7	−1.64853
$546$	0	0
$547$	−9.16514e6	−1.30970	−0.654849	−	0.755760i	$-0.727268\pi$
$547$	−9.16514e6	−1.30970	−0.654849	+	0.755760i	$0.727268\pi$
$548$	0	0
$549$	2.91584e6	0.412889
$550$	0	0
$551$	5.81173e6	0.815505
$552$	0	0
$553$	−1.47182e7	−2.04664
$554$	0	0
$555$	876129.	0.120736
$556$	0	0
$557$	6.54035e6	0.893230	0.446615	−	0.894726i	$-0.352629\pi$
$557$	6.54035e6	0.893230	0.446615	+	0.894726i	$0.352629\pi$
$558$	0	0
$559$	3.40104e6	0.460344
$560$	0	0
$561$	−3.14210e6	−0.421515
$562$	0	0
$563$	4.50619e6	0.599155	0.299577	−	0.954072i	$-0.403154\pi$
$563$	4.50619e6	0.599155	0.299577	+	0.954072i	$0.403154\pi$
$564$	0	0
$565$	1.09475e7	1.44275
$566$	0	0
$567$	978142.	0.127774
$568$	0	0
$569$	3.55292e6	0.460050	0.230025	−	0.973185i	$-0.426119\pi$
$569$	3.55292e6	0.460050	0.230025	+	0.973185i	$0.426119\pi$
$570$	0	0
$571$	−1.19688e7	−1.53624	−0.768122	−	0.640304i	$-0.778808\pi$
$571$	−1.19688e7	−1.53624	−0.768122	+	0.640304i	$0.778808\pi$
$572$	0	0
$573$	−3.78357e6	−0.481410
$574$	0	0
$575$	−3.83110e7	−4.83230
$576$	0	0
$577$	−1.21327e7	−1.51711	−0.758554	−	0.651611i	$-0.774094\pi$
$577$	−1.21327e7	−1.51711	−0.758554	+	0.651611i	$0.774094\pi$
$578$	0	0
$579$	1.33288e6	0.165232
$580$	0	0
$581$	−5.19697e6	−0.638719
$582$	0	0
$583$	−4.70612e6	−0.573445
$584$	0	0
$585$	−3.40452e6	−0.411307
$586$	0	0
$587$	−1.03620e7	−1.24122	−0.620612	−	0.784118i	$-0.713116\pi$
$587$	−1.03620e7	−1.24122	−0.620612	+	0.784118i	$0.713116\pi$
$588$	0	0
$589$	4.10203e6	0.487204
$590$	0	0
$591$	3.59669e6	0.423579
$592$	0	0
$593$	−1.01261e7	−1.18251	−0.591254	−	0.806486i	$-0.701367\pi$
$593$	−1.01261e7	−1.18251	−0.591254	+	0.806486i	$0.701367\pi$
$594$	0	0
$595$	−1.28274e7	−1.48541
$596$	0	0
$597$	5.75139e6	0.660446
$598$	0	0
$599$	313236.	0.0356701	0.0178350	−	0.999841i	$-0.494323\pi$
$599$	313236.	0.0356701	0.0178350	+	0.999841i	$0.494323\pi$
$600$	0	0
$601$	−4.89443e6	−0.552734	−0.276367	−	0.961052i	$-0.589131\pi$
$601$	−4.89443e6	−0.552734	−0.276367	+	0.961052i	$0.589131\pi$
$602$	0	0
$603$	−2.03125e6	−0.227494
$604$	0	0
$605$	−2.97094e6	−0.329993
$606$	0	0
$607$	7.48909e6	0.825006	0.412503	−	0.910956i	$-0.364655\pi$
$607$	7.48909e6	0.825006	0.412503	+	0.910956i	$0.364655\pi$
$608$	0	0
$609$	9.12208e6	0.996668
$610$	0	0
$611$	6.62497e6	0.717928
$612$	0	0
$613$	−1.20289e7	−1.29293	−0.646466	−	0.762942i	$-0.723754\pi$
$613$	−1.20289e7	−1.29293	−0.646466	+	0.762942i	$0.723754\pi$
$614$	0	0
$615$	−2.28624e6	−0.243744
$616$	0	0
$617$	−4.27115e6	−0.451681	−0.225841	−	0.974164i	$-0.572513\pi$
$617$	−4.27115e6	−0.451681	−0.225841	+	0.974164i	$0.572513\pi$
$618$	0	0
$619$	1.05328e7	1.10488	0.552442	−	0.833552i	$-0.313696\pi$
$619$	1.05328e7	1.10488	0.552442	+	0.833552i	$0.313696\pi$
$620$	0	0
$621$	3.34795e6	0.348378
$622$	0	0
$623$	5.16997e6	0.533664
$624$	0	0
$625$	3.37550e7	3.45651
$626$	0	0
$627$	3.34290e6	0.339590
$628$	0	0
$629$	730437.	0.0736133
$630$	0	0
$631$	1.47931e7	1.47906	0.739532	−	0.673122i	$-0.235047\pi$
$631$	1.47931e7	1.47906	0.739532	+	0.673122i	$0.235047\pi$
$632$	0	0
$633$	7.56424e6	0.750336
$634$	0	0
$635$	5.06406e6	0.498384
$636$	0	0
$637$	2.12703e6	0.207694
$638$	0	0
$639$	−4.68592e6	−0.453986
$640$	0	0
$641$	−1.86457e6	−0.179240	−0.0896198	−	0.995976i	$-0.528565\pi$
$641$	−1.86457e6	−0.179240	−0.0896198	+	0.995976i	$0.528565\pi$
$642$	0	0
$643$	−6.14368e6	−0.586004	−0.293002	−	0.956112i	$-0.594654\pi$
$643$	−6.14368e6	−0.586004	−0.293002	+	0.956112i	$0.594654\pi$
$644$	0	0
$645$	8.35092e6	0.790378
$646$	0	0
$647$	−5.58318e6	−0.524350	−0.262175	−	0.965020i	$-0.584440\pi$
$647$	−5.58318e6	−0.524350	−0.262175	+	0.965020i	$0.584440\pi$
$648$	0	0
$649$	−3.60892e6	−0.336330
$650$	0	0
$651$	6.43853e6	0.595435
$652$	0	0
$653$	−9.55163e6	−0.876586	−0.438293	−	0.898832i	$-0.644417\pi$
$653$	−9.55163e6	−0.876586	−0.438293	+	0.898832i	$0.644417\pi$
$654$	0	0
$655$	−8.78596e6	−0.800177
$656$	0	0
$657$	−5.51549e6	−0.498507
$658$	0	0
$659$	−2.48421e6	−0.222831	−0.111415	−	0.993774i	$-0.535538\pi$
$659$	−2.48421e6	−0.222831	−0.111415	+	0.993774i	$0.535538\pi$
$660$	0	0
$661$	−1.32567e7	−1.18013	−0.590066	−	0.807355i	$-0.700898\pi$
$661$	−1.32567e7	−1.18013	−0.590066	+	0.807355i	$0.700898\pi$
$662$	0	0
$663$	−2.83838e6	−0.250777
$664$	0	0
$665$	1.36472e7	1.19671
$666$	0	0
$667$	3.12228e7	2.71742
$668$	0	0
$669$	−3.92535e6	−0.339089
$670$	0	0
$671$	1.56413e7	1.34112
$672$	0	0
$673$	−1.13191e7	−0.963327	−0.481663	−	0.876356i	$-0.659967\pi$
$673$	−1.13191e7	−0.963327	−0.481663	+	0.876356i	$0.659967\pi$
$674$	0	0
$675$	−6.08134e6	−0.513736
$676$	0	0
$677$	5.27180e6	0.442066	0.221033	−	0.975266i	$-0.429057\pi$
$677$	5.27180e6	0.442066	0.221033	+	0.975266i	$0.429057\pi$
$678$	0	0
$679$	5.94713e6	0.495031
$680$	0	0
$681$	5.91206e6	0.488507
$682$	0	0
$683$	9.64351e6	0.791013	0.395506	−	0.918463i	$-0.370569\pi$
$683$	9.64351e6	0.791013	0.395506	+	0.918463i	$0.370569\pi$
$684$	0	0
$685$	7.38712e6	0.601518
$686$	0	0
$687$	6.92660e6	0.559923
$688$	0	0
$689$	−4.25122e6	−0.341166
$690$	0	0
$691$	1.01345e7	0.807437	0.403719	−	0.914883i	$-0.367717\pi$
$691$	1.01345e7	0.807437	0.403719	+	0.914883i	$0.367717\pi$
$692$	0	0
$693$	5.24701e6	0.415029
$694$	0	0
$695$	4.68908e7	3.68235
$696$	0	0
$697$	−1.90606e6	−0.148612
$698$	0	0
$699$	1.00272e7	0.776227
$700$	0	0
$701$	2.04002e7	1.56797	0.783987	−	0.620777i	$-0.213183\pi$
$701$	2.04002e7	1.56797	0.783987	+	0.620777i	$0.213183\pi$
$702$	0	0
$703$	−777116.	−0.0593059
$704$	0	0
$705$	1.62670e7	1.23263
$706$	0	0
$707$	−5.26887e6	−0.396432
$708$	0	0
$709$	−1.99117e7	−1.48762	−0.743810	−	0.668391i	$-0.766983\pi$
$709$	−1.99117e7	−1.48762	−0.743810	+	0.668391i	$0.766983\pi$
$710$	0	0
$711$	−7.99663e6	−0.593244
$712$	0	0
$713$	2.20376e7	1.62346
$714$	0	0
$715$	−1.82628e7	−1.33598
$716$	0	0
$717$	6.68522e6	0.485644
$718$	0	0
$719$	1.41435e7	1.02032	0.510159	−	0.860080i	$-0.329587\pi$
$719$	1.41435e7	1.02032	0.510159	+	0.860080i	$0.329587\pi$
$720$	0	0
$721$	5.53064e6	0.396221
$722$	0	0
$723$	−3.84388e6	−0.273479
$724$	0	0
$725$	−5.67141e7	−4.00725
$726$	0	0
$727$	−1.28420e7	−0.901146	−0.450573	−	0.892740i	$-0.648780\pi$
$727$	−1.28420e7	−0.901146	−0.450573	+	0.892740i	$0.648780\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	6.96224e6	0.481899
$732$	0	0
$733$	1.88331e6	0.129468	0.0647339	−	0.997903i	$-0.479380\pi$
$733$	1.88331e6	0.129468	0.0647339	+	0.997903i	$0.479380\pi$
$734$	0	0
$735$	5.22271e6	0.356597
$736$	0	0
$737$	−1.08961e7	−0.738931
$738$	0	0
$739$	1.31690e6	0.0887036	0.0443518	−	0.999016i	$-0.485878\pi$
$739$	1.31690e6	0.0887036	0.0443518	+	0.999016i	$0.485878\pi$
$740$	0	0
$741$	3.01977e6	0.202036
$742$	0	0
$743$	−2.87356e7	−1.90962	−0.954812	−	0.297210i	$-0.903944\pi$
$743$	−2.87356e7	−1.90962	−0.954812	+	0.297210i	$0.903944\pi$
$744$	0	0
$745$	−2.96327e6	−0.195605
$746$	0	0
$747$	−2.82360e6	−0.185141
$748$	0	0
$749$	−2.28943e7	−1.49115
$750$	0	0
$751$	7.33862e6	0.474804	0.237402	−	0.971411i	$-0.423704\pi$
$751$	7.33862e6	0.474804	0.237402	+	0.971411i	$0.423704\pi$
$752$	0	0
$753$	6.42950e6	0.413228
$754$	0	0
$755$	9.07072e6	0.579128
$756$	0	0
$757$	2.05944e7	1.30620	0.653100	−	0.757272i	$-0.273468\pi$
$757$	2.05944e7	1.30620	0.653100	+	0.757272i	$0.273468\pi$
$758$	0	0
$759$	1.79593e7	1.13158
$760$	0	0
$761$	−2.08295e7	−1.30382	−0.651911	−	0.758296i	$-0.726032\pi$
$761$	−2.08295e7	−1.30382	−0.651911	+	0.758296i	$0.726032\pi$
$762$	0	0
$763$	1.59145e7	0.989653
$764$	0	0
$765$	−6.96937e6	−0.430566
$766$	0	0
$767$	−3.26007e6	−0.200096
$768$	0	0
$769$	7.24343e6	0.441701	0.220851	−	0.975308i	$-0.429117\pi$
$769$	7.24343e6	0.441701	0.220851	+	0.975308i	$0.429117\pi$
$770$	0	0
$771$	−1.81160e7	−1.09756
$772$	0	0
$773$	7.56360e6	0.455281	0.227641	−	0.973745i	$-0.426899\pi$
$773$	7.56360e6	0.455281	0.227641	+	0.973745i	$0.426899\pi$
$774$	0	0
$775$	−4.00299e7	−2.39403
$776$	0	0
$777$	−1.21976e6	−0.0724806
$778$	0	0
$779$	2.02787e6	0.119728
$780$	0	0
$781$	−2.51365e7	−1.47461
$782$	0	0
$783$	4.95618e6	0.288897
$784$	0	0
$785$	−6.06050e7	−3.51022
$786$	0	0
$787$	1.74475e7	1.00415	0.502074	−	0.864825i	$-0.332571\pi$
$787$	1.74475e7	1.00415	0.502074	+	0.864825i	$0.332571\pi$
$788$	0	0
$789$	1.02383e7	0.585510
$790$	0	0
$791$	−1.52412e7	−0.866121
$792$	0	0
$793$	1.41294e7	0.797887
$794$	0	0
$795$	−1.04384e7	−0.585758
$796$	0	0
$797$	2.86812e7	1.59938	0.799689	−	0.600414i	$-0.204998\pi$
$797$	2.86812e7	1.59938	0.799689	+	0.600414i	$0.204998\pi$
$798$	0	0
$799$	1.35619e7	0.751544
$800$	0	0
$801$	2.80893e6	0.154689
$802$	0	0
$803$	−2.95866e7	−1.61922
$804$	0	0
$805$	7.33177e7	3.98767
$806$	0	0
$807$	1.03516e7	0.559532
$808$	0	0
$809$	1.18731e7	0.637812	0.318906	−	0.947786i	$-0.396685\pi$
$809$	1.18731e7	0.637812	0.318906	+	0.947786i	$0.396685\pi$
$810$	0	0
$811$	−7.63302e6	−0.407516	−0.203758	−	0.979021i	$-0.565316\pi$
$811$	−7.63302e6	−0.407516	−0.203758	+	0.979021i	$0.565316\pi$
$812$	0	0
$813$	−2.44508e6	−0.129738
$814$	0	0
$815$	2.81919e6	0.148672
$816$	0	0
$817$	−7.40717e6	−0.388237
$818$	0	0
$819$	4.73983e6	0.246918
$820$	0	0
$821$	−1.42730e7	−0.739021	−0.369511	−	0.929227i	$-0.620475\pi$
$821$	−1.42730e7	−0.739021	−0.369511	+	0.929227i	$0.620475\pi$
$822$	0	0
$823$	1.71695e7	0.883604	0.441802	−	0.897113i	$-0.354339\pi$
$823$	1.71695e7	0.883604	0.441802	+	0.897113i	$0.354339\pi$
$824$	0	0
$825$	−3.26219e7	−1.66869
$826$	0	0
$827$	2.21323e7	1.12529	0.562644	−	0.826699i	$-0.309784\pi$
$827$	2.21323e7	1.12529	0.562644	+	0.826699i	$0.309784\pi$
$828$	0	0
$829$	−1.55903e7	−0.787896	−0.393948	−	0.919133i	$-0.628891\pi$
$829$	−1.55903e7	−0.787896	−0.393948	+	0.919133i	$0.628891\pi$
$830$	0	0
$831$	−1.08164e7	−0.543353
$832$	0	0
$833$	4.35422e6	0.217419
$834$	0	0
$835$	2.87370e7	1.42635
$836$	0	0
$837$	3.49816e6	0.172594
$838$	0	0
$839$	6.48897e6	0.318252	0.159126	−	0.987258i	$-0.449132\pi$
$839$	6.48897e6	0.318252	0.159126	+	0.987258i	$0.449132\pi$
$840$	0	0
$841$	2.57098e7	1.25346
$842$	0	0
$843$	−1.03152e7	−0.499928
$844$	0	0
$845$	2.32622e7	1.12075
$846$	0	0
$847$	4.13618e6	0.198103
$848$	0	0
$849$	4.97292e6	0.236779
$850$	0	0
$851$	−4.17495e6	−0.197619
$852$	0	0
$853$	2.78176e7	1.30902	0.654512	−	0.756052i	$-0.272874\pi$
$853$	2.78176e7	1.30902	0.654512	+	0.756052i	$0.272874\pi$
$854$	0	0
$855$	7.41475e6	0.346882
$856$	0	0
$857$	−1.23818e7	−0.575880	−0.287940	−	0.957648i	$-0.592970\pi$
$857$	−1.23818e7	−0.575880	−0.287940	+	0.957648i	$0.592970\pi$
$858$	0	0
$859$	1.64820e7	0.762126	0.381063	−	0.924549i	$-0.375558\pi$
$859$	1.64820e7	0.762126	0.381063	+	0.924549i	$0.375558\pi$
$860$	0	0
$861$	3.18293e6	0.146325
$862$	0	0
$863$	−6.92492e6	−0.316510	−0.158255	−	0.987398i	$-0.550587\pi$
$863$	−6.92492e6	−0.316510	−0.158255	+	0.987398i	$0.550587\pi$
$864$	0	0
$865$	1.10393e7	0.501650
$866$	0	0
$867$	6.96828e6	0.314831
$868$	0	0
$869$	−4.28960e7	−1.92694
$870$	0	0
$871$	−9.84290e6	−0.439620
$872$	0	0
$873$	3.23118e6	0.143491
$874$	0	0
$875$	−8.32877e7	−3.67757
$876$	0	0
$877$	1.07951e7	0.473943	0.236972	−	0.971517i	$-0.423845\pi$
$877$	1.07951e7	0.473943	0.236972	+	0.971517i	$0.423845\pi$
$878$	0	0
$879$	5.41290e6	0.236297
$880$	0	0
$881$	1.27616e7	0.553942	0.276971	−	0.960878i	$-0.410669\pi$
$881$	1.27616e7	0.553942	0.276971	+	0.960878i	$0.410669\pi$
$882$	0	0
$883$	−6.93416e6	−0.299290	−0.149645	−	0.988740i	$-0.547813\pi$
$883$	−6.93416e6	−0.299290	−0.149645	+	0.988740i	$0.547813\pi$
$884$	0	0
$885$	−8.00479e6	−0.343551
$886$	0	0
$887$	2.18558e7	0.932734	0.466367	−	0.884591i	$-0.345563\pi$
$887$	2.18558e7	0.932734	0.466367	+	0.884591i	$0.345563\pi$
$888$	0	0
$889$	−7.05026e6	−0.299192
$890$	0	0
$891$	2.85079e6	0.120301
$892$	0	0
$893$	−1.44286e7	−0.605474
$894$	0	0
$895$	−7.19600e7	−3.00285
$896$	0	0
$897$	1.62233e7	0.673223
$898$	0	0
$899$	3.26236e7	1.34627
$900$	0	0
$901$	−8.70263e6	−0.357140
$902$	0	0
$903$	−1.16263e7	−0.474483
$904$	0	0
$905$	−4.77565e7	−1.93826
$906$	0	0
$907$	−4.22111e7	−1.70376	−0.851880	−	0.523738i	$-0.824537\pi$
$907$	−4.22111e7	−1.70376	−0.851880	+	0.523738i	$0.824537\pi$
$908$	0	0
$909$	−2.86266e6	−0.114911
$910$	0	0
$911$	1.45953e7	0.582661	0.291331	−	0.956622i	$-0.405902\pi$
$911$	1.45953e7	0.582661	0.291331	+	0.956622i	$0.405902\pi$
$912$	0	0
$913$	−1.51466e7	−0.601363
$914$	0	0
$915$	3.46934e7	1.36992
$916$	0	0
$917$	1.22319e7	0.480366
$918$	0	0
$919$	−4.21908e6	−0.164789	−0.0823946	−	0.996600i	$-0.526257\pi$
$919$	−4.21908e6	−0.164789	−0.0823946	+	0.996600i	$0.526257\pi$
$920$	0	0
$921$	−5.05480e6	−0.196361
$922$	0	0
$923$	−2.27068e7	−0.877307
$924$	0	0
$925$	7.58354e6	0.291419
$926$	0	0
$927$	3.00489e6	0.114850
$928$	0	0
$929$	2.76343e7	1.05053	0.525265	−	0.850939i	$-0.323966\pi$
$929$	2.76343e7	1.05053	0.525265	+	0.850939i	$0.323966\pi$
$930$	0	0
$931$	−4.63248e6	−0.175162
$932$	0	0
$933$	−1.64911e7	−0.620220
$934$	0	0
$935$	−3.73855e7	−1.39854
$936$	0	0
$937$	−2.61923e7	−0.974597	−0.487298	−	0.873236i	$-0.662018\pi$
$937$	−2.61923e7	−0.974597	−0.487298	+	0.873236i	$0.662018\pi$
$938$	0	0
$939$	−2.37404e7	−0.878667
$940$	0	0
$941$	3.59191e7	1.32236	0.661182	−	0.750226i	$-0.270055\pi$
$941$	3.59191e7	1.32236	0.661182	+	0.750226i	$0.270055\pi$
$942$	0	0
$943$	1.08944e7	0.398957
$944$	0	0
$945$	1.16382e7	0.423941
$946$	0	0
$947$	−3.21442e7	−1.16474	−0.582369	−	0.812924i	$-0.697874\pi$
$947$	−3.21442e7	−1.16474	−0.582369	+	0.812924i	$0.697874\pi$
$948$	0	0
$949$	−2.67267e7	−0.963339
$950$	0	0
$951$	−1.90263e7	−0.682187
$952$	0	0
$953$	5.06599e7	1.80689	0.903446	−	0.428701i	$-0.141029\pi$
$953$	5.06599e7	1.80689	0.903446	+	0.428701i	$0.141029\pi$
$954$	0	0
$955$	−4.50179e7	−1.59726
$956$	0	0
$957$	2.65863e7	0.938377
$958$	0	0
$959$	−1.02845e7	−0.361106
$960$	0	0
$961$	−5.60279e6	−0.195702
$962$	0	0
$963$	−1.24389e7	−0.432230
$964$	0	0
$965$	1.58589e7	0.548221
$966$	0	0
$967$	−4.48263e7	−1.54158	−0.770790	−	0.637089i	$-0.780138\pi$
$967$	−4.48263e7	−1.54158	−0.770790	+	0.637089i	$0.780138\pi$
$968$	0	0
$969$	6.18175e6	0.211496
$970$	0	0
$971$	2.77374e7	0.944100	0.472050	−	0.881572i	$-0.343514\pi$
$971$	2.77374e7	0.944100	0.472050	+	0.881572i	$0.343514\pi$
$972$	0	0
$973$	−6.52820e7	−2.21061
$974$	0	0
$975$	−2.94686e7	−0.992770
$976$	0	0
$977$	3.60277e7	1.20754	0.603768	−	0.797160i	$-0.293666\pi$
$977$	3.60277e7	1.20754	0.603768	+	0.797160i	$0.293666\pi$
$978$	0	0
$979$	1.50679e7	0.502452
$980$	0	0
$981$	8.64664e6	0.286863
$982$	0	0
$983$	−2.49497e7	−0.823534	−0.411767	−	0.911289i	$-0.635088\pi$
$983$	−2.49497e7	−0.823534	−0.411767	+	0.911289i	$0.635088\pi$
$984$	0	0
$985$	4.27943e7	1.40539
$986$	0	0
$987$	−2.26471e7	−0.739980
$988$	0	0
$989$	−3.97940e7	−1.29368
$990$	0	0
$991$	1.88166e7	0.608636	0.304318	−	0.952571i	$-0.401572\pi$
$991$	1.88166e7	0.608636	0.304318	+	0.952571i	$0.401572\pi$
$992$	0	0
$993$	−3.89638e6	−0.125397
$994$	0	0
$995$	6.84315e7	2.19128
$996$	0	0
$997$	−4.25880e7	−1.35691	−0.678453	−	0.734644i	$-0.737349\pi$
$997$	−4.25880e7	−1.35691	−0.678453	+	0.734644i	$0.737349\pi$
$998$	0	0
$999$	−662716.	−0.0210094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.6.a.q.1.1		2
3.2	odd	2		576.6.a.bn.1.2		2
4.3	odd	2		192.6.a.r.1.1		2
8.3	odd	2		96.6.a.g.1.2	✓	2
8.5	even	2		96.6.a.h.1.2	yes	2
12.11	even	2		576.6.a.bm.1.2		2
16.3	odd	4		768.6.d.z.385.4		4
16.5	even	4		768.6.d.s.385.3		4
16.11	odd	4		768.6.d.z.385.1		4
16.13	even	4		768.6.d.s.385.2		4
24.5	odd	2		288.6.a.o.1.1		2
24.11	even	2		288.6.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.6.a.g.1.2	✓	2	8.3	odd	2
96.6.a.h.1.2	yes	2	8.5	even	2
192.6.a.q.1.1		2	1.1	even	1	trivial
192.6.a.r.1.1		2	4.3	odd	2
288.6.a.n.1.1		2	24.11	even	2
288.6.a.o.1.1		2	24.5	odd	2
576.6.a.bm.1.2		2	12.11	even	2
576.6.a.bn.1.2		2	3.2	odd	2
768.6.d.s.385.2		4	16.13	even	4
768.6.d.s.385.3		4	16.5	even	4
768.6.d.z.385.1		4	16.11	odd	4
768.6.d.z.385.4		4	16.3	odd	4

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -107.084 q^{5} +149.084 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -107.084 q^{5} +149.084 q^{7} +81.0000 q^{9} +434.505 q^{11} +392.505 q^{13} +963.758 q^{15} +803.495 q^{17} -854.842 q^{19} -1341.76 q^{21} -4592.53 q^{23} +8342.03 q^{25} -729.000 q^{27} -6798.60 q^{29} -4798.58 q^{31} -3910.55 q^{33} -15964.6 q^{35} +909.075 q^{37} -3532.55 q^{39} -2372.21 q^{41} +8664.95 q^{43} -8673.82 q^{45} +16878.7 q^{47} +5419.11 q^{49} -7231.45 q^{51} -10831.0 q^{53} -46528.7 q^{55} +7693.58 q^{57} -8305.81 q^{59} +35998.0 q^{61} +12075.8 q^{63} -42031.1 q^{65} -25077.1 q^{67} +41332.7 q^{69} -57850.9 q^{71} -68092.5 q^{73} -75078.3 q^{75} +64777.9 q^{77} -98723.8 q^{79} +6561.00 q^{81} -34859.3 q^{83} -86041.6 q^{85} +61187.4 q^{87} +34678.2 q^{89} +58516.4 q^{91} +43187.2 q^{93} +91540.1 q^{95} +39891.1 q^{97} +35194.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} - 36 q^{5} + 120 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 - 36 * q^5 + 120 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} - 36 q^{5} + 120 q^{7} + 162 q^{9} - 200 q^{11} - 284 q^{13} + 324 q^{15} + 2676 q^{17} + 72 q^{19} - 1080 q^{21} - 3840 q^{23} + 10270 q^{25} - 1458 q^{27} - 10212 q^{29} - 10488 q^{31} + 1800 q^{33} - 18032 q^{35} - 13148 q^{37} + 2556 q^{39} + 4164 q^{41} - 5832 q^{43} - 2916 q^{45} - 1520 q^{47} - 10542 q^{49} - 24084 q^{51} - 9012 q^{53} - 91632 q^{55} - 648 q^{57} - 55096 q^{59} + 63444 q^{61} + 9720 q^{63} - 90120 q^{65} + 36792 q^{67} + 34560 q^{69} - 37664 q^{71} - 37836 q^{73} - 92430 q^{75} + 83232 q^{77} - 144888 q^{79} + 13122 q^{81} - 109272 q^{83} + 47064 q^{85} + 91908 q^{87} - 32556 q^{89} + 78192 q^{91} + 94392 q^{93} + 157424 q^{95} + 69092 q^{97} - 16200 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 - 36 * q^5 + 120 * q^7 + 162 * q^9 - 200 * q^11 - 284 * q^13 + 324 * q^15 + 2676 * q^17 + 72 * q^19 - 1080 * q^21 - 3840 * q^23 + 10270 * q^25 - 1458 * q^27 - 10212 * q^29 - 10488 * q^31 + 1800 * q^33 - 18032 * q^35 - 13148 * q^37 + 2556 * q^39 + 4164 * q^41 - 5832 * q^43 - 2916 * q^45 - 1520 * q^47 - 10542 * q^49 - 24084 * q^51 - 9012 * q^53 - 91632 * q^55 - 648 * q^57 - 55096 * q^59 + 63444 * q^61 + 9720 * q^63 - 90120 * q^65 + 36792 * q^67 + 34560 * q^69 - 37664 * q^71 - 37836 * q^73 - 92430 * q^75 + 83232 * q^77 - 144888 * q^79 + 13122 * q^81 - 109272 * q^83 + 47064 * q^85 + 91908 * q^87 - 32556 * q^89 + 78192 * q^91 + 94392 * q^93 + 157424 * q^95 + 69092 * q^97 - 16200 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more