LMFDB - Embedded newform 192.6.c.c.191.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,6,Mod(191,192)]

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([1, 0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	192.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$30.7936934041$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.2
Root			$0.500000 - 1.65831i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.6.c.c.191.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(12.0000 + 9.94987i) q^{3} +79.5990i q^{5} +179.098i q^{7} +(45.0000 + 238.797i) q^{9} +O(q^{10})$	\(q+(12.0000 + 9.94987i) q^{3} +79.5990i q^{5} +179.098i q^{7} +(45.0000 + 238.797i) q^{9} +648.000 q^{11} +242.000 q^{13} +(-792.000 + 955.188i) q^{15} +318.396i q^{17} -1253.68i q^{19} +(-1782.00 + 2149.17i) q^{21} -1296.00 q^{23} -3211.00 q^{25} +(-1836.00 + 3313.31i) q^{27} -1989.97i q^{29} -3402.86i q^{31} +(7776.00 + 6447.52i) q^{33} -14256.0 q^{35} +12058.0 q^{37} +(2904.00 + 2407.87i) q^{39} -14805.4i q^{41} +18088.9i q^{43} +(-19008.0 + 3581.95i) q^{45} -12960.0 q^{47} -15269.0 q^{49} +(-3168.00 + 3820.75i) q^{51} -26984.1i q^{53} +51580.1i q^{55} +(12474.0 - 15044.2i) q^{57} +8424.00 q^{59} +25762.0 q^{61} +(-42768.0 + 8059.40i) q^{63} +19263.0i q^{65} +10208.6i q^{67} +(-15552.0 - 12895.0i) q^{69} -55728.0 q^{71} +26026.0 q^{73} +(-38532.0 - 31949.0i) q^{75} +116055. i q^{77} -19163.5i q^{79} +(-54999.0 + 21491.7i) q^{81} -78408.0 q^{83} -25344.0 q^{85} +(19800.0 - 23879.7i) q^{87} +84215.7i q^{89} +43341.7i q^{91} +(33858.0 - 40834.3i) q^{93} +99792.0 q^{95} +103090. q^{97} +(29160.0 + 154740. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{3} + 90 q^{9}+O(q^{10}) $ 2 * q + 24 * q^3 + 90 * q^9	$ 2 q + 24 q^{3} + 90 q^{9} + 1296 q^{11} + 484 q^{13} - 1584 q^{15} - 3564 q^{21} - 2592 q^{23} - 6422 q^{25} - 3672 q^{27} + 15552 q^{33} - 28512 q^{35} + 24116 q^{37} + 5808 q^{39} - 38016 q^{45} - 25920 q^{47} - 30538 q^{49} - 6336 q^{51} + 24948 q^{57} + 16848 q^{59} + 51524 q^{61} - 85536 q^{63} - 31104 q^{69} - 111456 q^{71} + 52052 q^{73} - 77064 q^{75} - 109998 q^{81} - 156816 q^{83} - 50688 q^{85} + 39600 q^{87} + 67716 q^{93} + 199584 q^{95} + 206180 q^{97} + 58320 q^{99}+O(q^{100}) $ 2 * q + 24 * q^3 + 90 * q^9 + 1296 * q^11 + 484 * q^13 - 1584 * q^15 - 3564 * q^21 - 2592 * q^23 - 6422 * q^25 - 3672 * q^27 + 15552 * q^33 - 28512 * q^35 + 24116 * q^37 + 5808 * q^39 - 38016 * q^45 - 25920 * q^47 - 30538 * q^49 - 6336 * q^51 + 24948 * q^57 + 16848 * q^59 + 51524 * q^61 - 85536 * q^63 - 31104 * q^69 - 111456 * q^71 + 52052 * q^73 - 77064 * q^75 - 109998 * q^81 - 156816 * q^83 - 50688 * q^85 + 39600 * q^87 + 67716 * q^93 + 199584 * q^95 + 206180 * q^97 + 58320 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	12.0000	+	9.94987i	0.769800	+	0.638285i
$3$	12.0000	+	9.94987i	0.769800	+	0.638285i
$4$		0			0
$5$			79.5990i			1.42391i	0.702225	+	0.711955i	$0.252190\pi$
$5$			79.5990i			1.42391i	−0.702225	+	0.711955i	$0.747810\pi$
$6$		0			0
$7$			179.098i			1.38148i	0.723103	+	0.690741i	$0.242715\pi$
$7$			179.098i			1.38148i	−0.723103	+	0.690741i	$0.757285\pi$
$8$		0			0
$9$	45.0000	+	238.797i	0.185185	+	0.982704i
$10$		0			0
$11$	648.000			1.61471			0.807353	−	0.590069i	$-0.200900\pi$
$11$	648.000			1.61471			0.807353	+	0.590069i	$0.200900\pi$
$12$		0			0
$13$	242.000			0.397152			0.198576	−	0.980085i	$-0.436368\pi$
$13$	242.000			0.397152			0.198576	+	0.980085i	$0.436368\pi$
$14$		0			0
$15$	−792.000	+	955.188i	−0.908860	+	1.09613i
$16$		0			0
$17$			318.396i			0.267205i	0.991035	+	0.133603i	$0.0426546\pi$
$17$			318.396i			0.267205i	−0.991035	+	0.133603i	$0.957345\pi$
$18$		0			0
$19$		−	1253.68i		−	0.796717i	−0.917230	−	0.398359i	$-0.869580\pi$
$19$		−	1253.68i		−	0.796717i	0.917230	−	0.398359i	$-0.130420\pi$
$20$		0			0
$21$	−1782.00	+	2149.17i	−0.881778	+	1.06346i
$22$		0			0
$23$	−1296.00			−0.510841			−0.255420	−	0.966830i	$-0.582214\pi$
$23$	−1296.00			−0.510841			−0.255420	+	0.966830i	$0.582214\pi$
$24$		0			0
$25$	−3211.00			−1.02752
$26$		0			0
$27$	−1836.00	+	3313.31i	−0.484689	+	0.874686i
$28$		0			0
$29$		−	1989.97i		−	0.439392i	−0.975568	−	0.219696i	$-0.929493\pi$
$29$		−	1989.97i		−	0.439392i	0.975568	−	0.219696i	$-0.0705066\pi$
$30$		0			0
$31$		−	3402.86i		−	0.635974i	−0.948095	−	0.317987i	$-0.896993\pi$
$31$		−	3402.86i		−	0.635974i	0.948095	−	0.317987i	$-0.103007\pi$
$32$		0			0
$33$	7776.00	+	6447.52i	1.24300	+	1.03064i
$34$		0			0
$35$	−14256.0			−1.96711
$36$		0			0
$37$	12058.0			1.44801			0.724004	−	0.689796i	$-0.242300\pi$
$37$	12058.0			1.44801			0.724004	+	0.689796i	$0.242300\pi$
$38$		0			0
$39$	2904.00	+	2407.87i	0.305728	+	0.253496i
$40$		0			0
$41$		−	14805.4i		−	1.37550i	−0.725947	−	0.687750i	$-0.758598\pi$
$41$		−	14805.4i		−	1.37550i	0.725947	−	0.687750i	$-0.241402\pi$
$42$		0			0
$43$			18088.9i			1.49190i	0.666001	+	0.745951i	$0.268005\pi$
$43$			18088.9i			1.49190i	−0.666001	+	0.745951i	$0.731995\pi$
$44$		0			0
$45$	−19008.0	+	3581.95i	−1.39928	+	0.263687i
$46$		0			0
$47$	−12960.0			−0.855777			−0.427888	−	0.903832i	$-0.640742\pi$
$47$	−12960.0			−0.855777			−0.427888	+	0.903832i	$0.640742\pi$
$48$		0			0
$49$	−15269.0			−0.908491
$50$		0			0
$51$	−3168.00	+	3820.75i	−0.170553	+	0.205695i
$52$		0			0
$53$		−	26984.1i		−	1.31952i	−0.751474	−	0.659762i	$-0.770657\pi$
$53$		−	26984.1i		−	1.31952i	0.751474	−	0.659762i	$-0.229343\pi$
$54$		0			0
$55$			51580.1i			2.29920i
$56$		0			0
$57$	12474.0	−	15044.2i	0.508532	−	0.613313i
$58$		0			0
$59$	8424.00			0.315056			0.157528	−	0.987514i	$-0.449647\pi$
$59$	8424.00			0.315056			0.157528	+	0.987514i	$0.449647\pi$
$60$		0			0
$61$	25762.0			0.886452			0.443226	−	0.896410i	$-0.353834\pi$
$61$	25762.0			0.886452			0.443226	+	0.896410i	$0.353834\pi$
$62$		0			0
$63$	−42768.0	+	8059.40i	−1.35759	+	0.255830i
$64$		0			0
$65$			19263.0i			0.565509i
$66$		0			0
$67$			10208.6i			0.277829i	0.990304	+	0.138915i	$0.0443614\pi$
$67$			10208.6i			0.277829i	−0.990304	+	0.138915i	$0.955639\pi$
$68$		0			0
$69$	−15552.0	−	12895.0i	−0.393245	−	0.326062i
$70$		0			0
$71$	−55728.0			−1.31198			−0.655991	−	0.754769i	$-0.727749\pi$
$71$	−55728.0			−1.31198			−0.655991	+	0.754769i	$0.727749\pi$
$72$		0			0
$73$	26026.0			0.571611			0.285805	−	0.958288i	$-0.407739\pi$
$73$	26026.0			0.571611			0.285805	+	0.958288i	$0.407739\pi$
$74$		0			0
$75$	−38532.0	−	31949.0i	−0.790985	−	0.655850i
$76$		0			0
$77$			116055.i			2.23069i
$78$		0			0
$79$		−	19163.5i		−	0.345467i	−0.984969	−	0.172733i	$-0.944740\pi$
$79$		−	19163.5i		−	0.345467i	0.984969	−	0.172733i	$-0.0552599\pi$
$80$		0			0
$81$	−54999.0	+	21491.7i	−0.931413	+	0.363964i
$82$		0			0
$83$	−78408.0			−1.24930			−0.624648	−	0.780907i	$-0.714757\pi$
$83$	−78408.0			−1.24930			−0.624648	+	0.780907i	$0.714757\pi$
$84$		0			0
$85$	−25344.0			−0.380477
$86$		0			0
$87$	19800.0	−	23879.7i	0.280458	−	0.338244i
$88$		0			0
$89$			84215.7i			1.12699i	0.826121	+	0.563493i	$0.190543\pi$
$89$			84215.7i			1.12699i	−0.826121	+	0.563493i	$0.809457\pi$
$90$		0			0
$91$			43341.7i			0.548658i
$92$		0			0
$93$	33858.0	−	40834.3i	0.405933	−	0.489573i
$94$		0			0
$95$	99792.0			1.13445
$96$		0			0
$97$	103090.			1.11247			0.556234	−	0.831026i	$-0.312246\pi$
$97$	103090.			1.11247			0.556234	+	0.831026i	$0.312246\pi$
$98$		0			0
$99$	29160.0	+	154740.i	0.299020	+	1.58678i
$100$		0			0
$101$		−	83340.1i		−	0.812926i	−0.913667	−	0.406463i	$-0.866762\pi$
$101$		−	83340.1i		−	0.812926i	0.913667	−	0.406463i	$-0.133238\pi$
$102$		0			0
$103$		−	45669.9i		−	0.424167i	−0.977252	−	0.212084i	$-0.931975\pi$
$103$		−	45669.9i		−	0.424167i	0.977252	−	0.212084i	$-0.0680249\pi$
$104$		0			0
$105$	−171072.	−	141845.i	−1.51428	−	1.25557i
$106$		0			0
$107$	21384.0			0.180563			0.0902817	−	0.995916i	$-0.471223\pi$
$107$	21384.0			0.180563			0.0902817	+	0.995916i	$0.471223\pi$
$108$		0			0
$109$	−58894.0			−0.474794			−0.237397	−	0.971413i	$-0.576294\pi$
$109$	−58894.0			−0.474794			−0.237397	+	0.971413i	$0.576294\pi$
$110$		0			0
$111$	144696.	+	119976.i	1.11468	+	0.924241i
$112$		0			0
$113$		−	79917.4i		−	0.588769i	−0.955687	−	0.294385i	$-0.904885\pi$
$113$		−	79917.4i		−	0.588769i	0.955687	−	0.294385i	$-0.0951147\pi$
$114$		0			0
$115$		−	103160.i		−	0.727391i
$116$		0			0
$117$	10890.0	+	57788.9i	0.0735467	+	0.390283i
$118$		0			0
$119$	−57024.0			−0.369139
$120$		0			0
$121$	258853.			1.60727
$122$		0			0
$123$	147312.	−	177665.i	0.877961	−	1.05886i
$124$		0			0
$125$		−	6845.51i		−	0.0391860i
$126$		0			0
$127$		−	140950.i		−	0.775453i	−0.921774	−	0.387727i	$-0.873260\pi$
$127$		−	140950.i		−	0.775453i	0.921774	−	0.387727i	$-0.126740\pi$
$128$		0			0
$129$	−179982.	+	217066.i	−0.952258	+	1.14847i
$130$		0			0
$131$	292248.			1.48790			0.743949	−	0.668236i	$-0.232950\pi$
$131$	292248.			1.48790			0.743949	+	0.668236i	$0.232950\pi$
$132$		0			0
$133$	224532.			1.10065
$134$		0			0
$135$	−263736.	−	146144.i	−1.24547	−	0.690154i
$136$		0			0
$137$		−	155536.i		−	0.707996i	−0.935246	−	0.353998i	$-0.884822\pi$
$137$		−	155536.i		−	0.707996i	0.935246	−	0.353998i	$-0.115178\pi$
$138$		0			0
$139$		−	398851.i		−	1.75095i	−0.483265	−	0.875474i	$-0.660549\pi$
$139$		−	398851.i		−	1.75095i	0.483265	−	0.875474i	$-0.339451\pi$
$140$		0			0
$141$	−155520.	−	128950.i	−0.658777	−	0.546229i
$142$		0			0
$143$	156816.			0.641284
$144$		0			0
$145$	158400.			0.625655
$146$		0			0
$147$	−183228.	−	151925.i	−0.699356	−	0.579876i
$148$		0			0
$149$		−	213564.i		−	0.788066i	−0.919096	−	0.394033i	$-0.871080\pi$
$149$		−	213564.i		−	0.788066i	0.919096	−	0.394033i	$-0.128920\pi$
$150$		0			0
$151$			199336.i			0.711448i	0.934591	+	0.355724i	$0.115766\pi$
$151$			199336.i			0.711448i	−0.934591	+	0.355724i	$0.884234\pi$
$152$		0			0
$153$	−76032.0	+	14327.8i	−0.262584	+	0.0494825i
$154$		0			0
$155$	270864.			0.905570
$156$		0			0
$157$	195490.			0.632959			0.316479	−	0.948599i	$-0.397499\pi$
$157$	195490.			0.632959			0.316479	+	0.948599i	$0.397499\pi$
$158$		0			0
$159$	268488.	−	323809.i	0.842233	−	1.01577i
$160$		0			0
$161$		−	232111.i		−	0.705717i
$162$		0			0
$163$			450073.i			1.32682i	0.748254	+	0.663412i	$0.230892\pi$
$163$			450073.i			1.32682i	−0.748254	+	0.663412i	$0.769108\pi$
$164$		0			0
$165$	−513216.	+	618962.i	−1.46754	+	1.76992i
$166$		0			0
$167$	384912.			1.06800			0.533999	−	0.845485i	$-0.320689\pi$
$167$	384912.			1.06800			0.533999	+	0.845485i	$0.320689\pi$
$168$		0			0
$169$	−312729.			−0.842270
$170$		0			0
$171$	299376.	−	56415.8i	0.782937	−	0.147540i
$172$		0			0
$173$		−	48794.2i		−	0.123952i	−0.998078	−	0.0619759i	$-0.980260\pi$
$173$		−	48794.2i		−	0.123952i	0.998078	−	0.0619759i	$-0.0197402\pi$
$174$		0			0
$175$		−	575083.i		−	1.41950i
$176$		0			0
$177$	101088.	+	83817.7i	0.242531	+	0.201096i
$178$		0			0
$179$	−607176.			−1.41639			−0.708194	−	0.706018i	$-0.750490\pi$
$179$	−607176.			−1.41639			−0.708194	+	0.706018i	$0.750490\pi$
$180$		0			0
$181$	511850.			1.16130			0.580652	−	0.814152i	$-0.302798\pi$
$181$	511850.			1.16130			0.580652	+	0.814152i	$0.302798\pi$
$182$		0			0
$183$	309144.	+	256329.i	0.682391	+	0.565808i
$184$		0			0
$185$			959805.i			2.06183i
$186$		0			0
$187$			206321.i			0.431458i
$188$		0			0
$189$	−593406.	−	328823.i	−1.20836	−	0.669589i
$190$		0			0
$191$	186624.			0.370155			0.185078	−	0.982724i	$-0.440746\pi$
$191$	186624.			0.370155			0.185078	+	0.982724i	$0.440746\pi$
$192$		0			0
$193$	169730.			0.327994			0.163997	−	0.986461i	$-0.447561\pi$
$193$	169730.			0.327994			0.163997	+	0.986461i	$0.447561\pi$
$194$		0			0
$195$	−191664.	+	231155.i	−0.360956	+	0.435329i
$196$		0			0
$197$			715834.i			1.31416i	0.753823	+	0.657078i	$0.228208\pi$
$197$			715834.i			1.31416i	−0.753823	+	0.657078i	$0.771792\pi$
$198$		0			0
$199$			596216.i			1.06726i	0.845717	+	0.533631i	$0.179173\pi$
$199$			596216.i			1.06726i	−0.845717	+	0.533631i	$0.820827\pi$
$200$		0			0
$201$	−101574.	+	122503.i	−0.177334	+	0.213873i
$202$		0			0
$203$	356400.			0.607012
$204$		0			0
$205$	1.17850e6			1.95859
$206$		0			0
$207$	−58320.0	−	309481.i	−0.0946001	−	0.502005i
$208$		0			0
$209$		−	812387.i		−	1.28646i
$210$		0			0
$211$		−	317898.i		−	0.491567i	−0.969325	−	0.245783i	$-0.920955\pi$
$211$		−	317898.i		−	0.491567i	0.969325	−	0.245783i	$-0.0790452\pi$
$212$		0			0
$213$	−668736.	−	554487.i	−1.00996	−	0.837418i
$214$		0			0
$215$	−1.43986e6			−2.12433
$216$		0			0
$217$	609444.			0.878586
$218$		0			0
$219$	312312.	+	258955.i	0.440026	+	0.364850i
$220$		0			0
$221$			77051.8i			0.106121i
$222$		0			0
$223$			134144.i			0.180638i	0.995913	+	0.0903191i	$0.0287887\pi$
$223$			134144.i			0.180638i	−0.995913	+	0.0903191i	$0.971211\pi$
$224$		0			0
$225$	−144495.	−	766777.i	−0.190281	−	1.00975i
$226$		0			0
$227$	−734184.			−0.945671			−0.472836	−	0.881151i	$-0.656770\pi$
$227$	−734184.			−0.945671			−0.472836	+	0.881151i	$0.656770\pi$
$228$		0			0
$229$	−317062.			−0.399536			−0.199768	−	0.979843i	$-0.564019\pi$
$229$	−317062.			−0.399536			−0.199768	+	0.979843i	$0.564019\pi$
$230$		0			0
$231$	−1.15474e6	+	1.39266e6i	−1.42381	+	1.71718i
$232$		0			0
$233$			234180.i			0.282592i	0.989967	+	0.141296i	$0.0451270\pi$
$233$			234180.i			0.282592i	−0.989967	+	0.141296i	$0.954873\pi$
$234$		0			0
$235$		−	1.03160e6i		−	1.21855i
$236$		0			0
$237$	190674.	−	229961.i	0.220506	−	0.265940i
$238$		0			0
$239$	−85536.0			−0.0968622			−0.0484311	−	0.998827i	$-0.515422\pi$
$239$	−85536.0			−0.0968622			−0.0484311	+	0.998827i	$0.515422\pi$
$240$		0			0
$241$	−666446.			−0.739133			−0.369566	−	0.929204i	$-0.620494\pi$
$241$	−666446.			−0.739133			−0.369566	+	0.929204i	$0.620494\pi$
$242$		0			0
$243$	−873828.	−	289332.i	−0.949315	−	0.314327i
$244$		0			0
$245$		−	1.21540e6i		−	1.29361i
$246$		0			0
$247$		−	303392.i		−	0.316418i
$248$		0			0
$249$	−940896.	−	780150.i	−0.961708	−	0.797406i
$250$		0			0
$251$	832680.			0.834245			0.417123	−	0.908850i	$-0.363039\pi$
$251$	832680.			0.834245			0.417123	+	0.908850i	$0.363039\pi$
$252$		0			0
$253$	−839808.			−0.824857
$254$		0			0
$255$	−304128.	−	252170.i	−0.292891	−	0.242852i
$256$		0			0
$257$			953914.i			0.900900i	0.892802	+	0.450450i	$0.148736\pi$
$257$			953914.i			0.900900i	−0.892802	+	0.450450i	$0.851264\pi$
$258$		0			0
$259$			2.15956e6i			2.00040i
$260$		0			0
$261$	475200.	−	89548.9i	0.431793	−	0.0813690i
$262$		0			0
$263$	−584496.			−0.521065			−0.260533	−	0.965465i	$-0.583898\pi$
$263$	−584496.			−0.521065			−0.260533	+	0.965465i	$0.583898\pi$
$264$		0			0
$265$	2.14790e6			1.87888
$266$		0			0
$267$	−837936.	+	1.01059e6i	−0.719337	+	0.867554i
$268$		0			0
$269$			452361.i			0.381158i	0.981672	+	0.190579i	$0.0610365\pi$
$269$			452361.i			0.381158i	−0.981672	+	0.190579i	$0.938964\pi$
$270$		0			0
$271$			1.17291e6i			0.970157i	0.874471	+	0.485078i	$0.161209\pi$
$271$			1.17291e6i			0.970157i	−0.874471	+	0.485078i	$0.838791\pi$
$272$		0			0
$273$	−431244.	+	520100.i	−0.350200	+	0.422357i
$274$		0			0
$275$	−2.08073e6			−1.65914
$276$		0			0
$277$	1.10244e6			0.863289			0.431645	−	0.902044i	$-0.357933\pi$
$277$	1.10244e6			0.863289			0.431645	+	0.902044i	$0.357933\pi$
$278$		0			0
$279$	812592.	−	153129.i	0.624974	−	0.117773i
$280$		0			0
$281$		−	1.77776e6i		−	1.34310i	−0.740959	−	0.671550i	$-0.765629\pi$
$281$		−	1.77776e6i		−	1.34310i	0.740959	−	0.671550i	$-0.234371\pi$
$282$		0			0
$283$			190023.i			0.141039i	0.997510	+	0.0705195i	$0.0224657\pi$
$283$			190023.i			0.141039i	−0.997510	+	0.0705195i	$0.977534\pi$
$284$		0			0
$285$	1.19750e6	+	992918.i	0.873303	+	0.724104i
$286$		0			0
$287$	2.65162e6			1.90023
$288$		0			0
$289$	1.31848e6			0.928601
$290$		0			0
$291$	1.23708e6	+	1.02573e6i	0.856378	+	0.710071i
$292$		0			0
$293$			2.55425e6i			1.73818i	0.494654	+	0.869090i	$0.335295\pi$
$293$			2.55425e6i			1.73818i	−0.494654	+	0.869090i	$0.664705\pi$
$294$		0			0
$295$			670542.i			0.448612i
$296$		0			0
$297$	−1.18973e6	+	2.14702e6i	−0.782630	+	1.41236i
$298$		0			0
$299$	−313632.			−0.202881
$300$		0			0
$301$	−3.23968e6			−2.06103
$302$		0			0
$303$	829224.	−	1.00008e6i	0.518878	−	0.625790i
$304$		0			0
$305$			2.05063e6i			1.26223i
$306$		0			0
$307$			2.86001e6i			1.73190i	0.500134	+	0.865948i	$0.333284\pi$
$307$			2.86001e6i			1.73190i	−0.500134	+	0.865948i	$0.666716\pi$
$308$		0			0
$309$	454410.	−	548039.i	0.270740	−	0.326524i
$310$		0			0
$311$	−2.90952e6			−1.70577			−0.852885	−	0.522099i	$-0.825149\pi$
$311$	−2.90952e6			−1.70577			−0.852885	+	0.522099i	$0.825149\pi$
$312$		0			0
$313$	−2.56639e6			−1.48068			−0.740341	−	0.672231i	$-0.765336\pi$
$313$	−2.56639e6			−1.48068			−0.740341	+	0.672231i	$0.765336\pi$
$314$		0			0
$315$	−641520.	−	3.40429e6i	−0.364279	−	1.93308i
$316$		0			0
$317$		−	2.89366e6i		−	1.61733i	−0.588267	−	0.808667i	$-0.700189\pi$
$317$		−	2.89366e6i		−	1.61733i	0.588267	−	0.808667i	$-0.299811\pi$
$318$		0			0
$319$		−	1.28950e6i		−	0.709489i
$320$		0			0
$321$	256608.	+	212768.i	0.138998	+	0.115251i
$322$		0			0
$323$	399168.			0.212887
$324$		0			0
$325$	−777062.			−0.408082
$326$		0			0
$327$	−706728.	−	585988.i	−0.365496	−	0.303053i
$328$		0			0
$329$		−	2.32111e6i		−	1.18224i
$330$		0			0
$331$		−	3.88338e6i		−	1.94823i	−0.226060	−	0.974113i	$-0.572585\pi$
$331$		−	3.88338e6i		−	1.94823i	0.226060	−	0.974113i	$-0.427415\pi$
$332$		0			0
$333$	542610.	+	2.87941e6i	0.268150	+	1.42296i
$334$		0			0
$335$	−812592.			−0.395604
$336$		0			0
$337$	515746.			0.247378			0.123689	−	0.992321i	$-0.460527\pi$
$337$	515746.			0.247378			0.123689	+	0.992321i	$0.460527\pi$
$338$		0			0
$339$	795168.	−	959009.i	0.375802	−	0.453235i
$340$		0			0
$341$		−	2.20505e6i		−	1.02691i
$342$		0			0
$343$			275452.i			0.126419i
$344$		0			0
$345$	1.02643e6	−	1.23792e6i	0.464283	−	0.559946i
$346$		0			0
$347$	1.41847e6			0.632408			0.316204	−	0.948691i	$-0.397592\pi$
$347$	1.41847e6			0.632408			0.316204	+	0.948691i	$0.397592\pi$
$348$		0			0
$349$	2.28138e6			1.00261			0.501307	−	0.865270i	$-0.332853\pi$
$349$	2.28138e6			1.00261			0.501307	+	0.865270i	$0.332853\pi$
$350$		0			0
$351$	−444312.	+	801821.i	−0.192495	+	0.347384i
$352$		0			0
$353$			21014.1i			0.00897583i	0.999990	+	0.00448792i	$0.00142855\pi$
$353$			21014.1i			0.00897583i	−0.999990	+	0.00448792i	$0.998571\pi$
$354$		0			0
$355$		−	4.43589e6i		−	1.86814i
$356$		0			0
$357$	−684288.	−	567382.i	−0.284164	−	0.235616i
$358$		0			0
$359$	1.98158e6			0.811477			0.405739	−	0.913989i	$-0.367014\pi$
$359$	1.98158e6			0.811477			0.405739	+	0.913989i	$0.367014\pi$
$360$		0			0
$361$	904375.			0.365242
$362$		0			0
$363$	3.10624e6	+	2.57555e6i	1.23728	+	1.02590i
$364$		0			0
$365$			2.07164e6i			0.813922i
$366$		0			0
$367$		−	1.22557e6i		−	0.474976i	−0.971390	−	0.237488i	$-0.923676\pi$
$367$		−	1.22557e6i		−	0.474976i	0.971390	−	0.237488i	$-0.0763240\pi$
$368$		0			0
$369$	3.53549e6	−	666244.i	1.35171	−	0.254722i
$370$		0			0
$371$	4.83278e6			1.82290
$372$		0			0
$373$	−324566.			−0.120790			−0.0603950	−	0.998175i	$-0.519236\pi$
$373$	−324566.			−0.120790			−0.0603950	+	0.998175i	$0.519236\pi$
$374$		0			0
$375$	68112.0	−	82146.2i	0.0250118	−	0.0301654i
$376$		0			0
$377$		−	481574.i		−	0.174506i
$378$		0			0
$379$			2.21168e6i			0.790904i	0.918487	+	0.395452i	$0.129412\pi$
$379$			2.21168e6i			0.790904i	−0.918487	+	0.395452i	$0.870588\pi$
$380$		0			0
$381$	1.40243e6	−	1.69140e6i	0.494960	−	0.596944i
$382$		0			0
$383$	2.66458e6			0.928178			0.464089	−	0.885789i	$-0.346382\pi$
$383$	2.66458e6			0.928178			0.464089	+	0.885789i	$0.346382\pi$
$384$		0			0
$385$	−9.23789e6			−3.17630
$386$		0			0
$387$	−4.31957e6	+	813999.i	−1.46610	+	0.276278i
$388$		0			0
$389$		−	417656.i		−	0.139941i	−0.997549	−	0.0699704i	$-0.977710\pi$
$389$		−	417656.i		−	0.139941i	0.997549	−	0.0699704i	$-0.0222905\pi$
$390$		0			0
$391$		−	412641.i		−	0.136499i
$392$		0			0
$393$	3.50698e6	+	2.90783e6i	1.14538	+	0.949703i
$394$		0			0
$395$	1.52539e6			0.491913
$396$		0			0
$397$	−2.01150e6			−0.640537			−0.320268	−	0.947327i	$-0.603773\pi$
$397$	−2.01150e6			−0.640537			−0.320268	+	0.947327i	$0.603773\pi$
$398$		0			0
$399$	2.69438e6	+	2.23407e6i	0.847281	+	0.702528i
$400$		0			0
$401$			872723.i			0.271029i	0.990775	+	0.135514i	$0.0432687\pi$
$401$			872723.i			0.271029i	−0.990775	+	0.135514i	$0.956731\pi$
$402$		0			0
$403$		−	823491.i		−	0.252579i
$404$		0			0
$405$	−1.71072e6	−	4.37787e6i	−0.518252	−	1.32625i
$406$		0			0
$407$	7.81358e6			2.33811
$408$		0			0
$409$	1.07411e6			0.317496			0.158748	−	0.987319i	$-0.449254\pi$
$409$	1.07411e6			0.317496			0.158748	+	0.987319i	$0.449254\pi$
$410$		0			0
$411$	1.54757e6	−	1.86644e6i	0.451903	−	0.545015i
$412$		0			0
$413$			1.50872e6i			0.435245i
$414$		0			0
$415$		−	6.24120e6i		−	1.77888i
$416$		0			0
$417$	3.96851e6	−	4.78621e6i	1.11760	−	1.34788i
$418$		0			0
$419$	−3.65796e6			−1.01790			−0.508949	−	0.860797i	$-0.669966\pi$
$419$	−3.65796e6			−1.01790			−0.508949	+	0.860797i	$0.669966\pi$
$420$		0			0
$421$	4.65190e6			1.27916			0.639580	−	0.768724i	$-0.279108\pi$
$421$	4.65190e6			1.27916			0.639580	+	0.768724i	$0.279108\pi$
$422$		0			0
$423$	−583200.	−	3.09481e6i	−0.158477	−	0.840975i
$424$		0			0
$425$		−	1.02237e6i		−	0.274559i
$426$		0			0
$427$			4.61392e6i			1.22462i
$428$		0			0
$429$	1.88179e6	+	1.56030e6i	0.493661	+	0.409322i
$430$		0			0
$431$	−4.98960e6			−1.29382			−0.646908	−	0.762568i	$-0.723938\pi$
$431$	−4.98960e6			−1.29382			−0.646908	+	0.762568i	$0.723938\pi$
$432$		0			0
$433$	−3.09888e6			−0.794300			−0.397150	−	0.917754i	$-0.630001\pi$
$433$	−3.09888e6			−0.794300			−0.397150	+	0.917754i	$0.630001\pi$
$434$		0			0
$435$	1.90080e6	+	1.57606e6i	0.481630	+	0.399346i
$436$		0			0
$437$			1.62477e6i			0.406995i
$438$		0			0
$439$		−	840864.i		−	0.208240i	−0.994565	−	0.104120i	$-0.966797\pi$
$439$		−	840864.i		−	0.208240i	0.994565	−	0.104120i	$-0.0332026\pi$
$440$		0			0
$441$	−687105.	−	3.64619e6i	−0.168239	−	0.892777i
$442$		0			0
$443$	−4.72457e6			−1.14381			−0.571904	−	0.820321i	$-0.693795\pi$
$443$	−4.72457e6			−1.14381			−0.571904	+	0.820321i	$0.693795\pi$
$444$		0			0
$445$	−6.70349e6			−1.60473
$446$		0			0
$447$	2.12494e6	−	2.56277e6i	0.503010	−	0.606653i
$448$		0			0
$449$			3.68512e6i			0.862651i	0.902196	+	0.431326i	$0.141954\pi$
$449$			3.68512e6i			0.862651i	−0.902196	+	0.431326i	$0.858046\pi$
$450$		0			0
$451$		−	9.59391e6i		−	2.22103i
$452$		0			0
$453$	−1.98337e6	+	2.39203e6i	−0.454106	+	0.547673i
$454$		0			0
$455$	−3.44995e6			−0.781240
$456$		0			0
$457$	−4.98364e6			−1.11624			−0.558118	−	0.829762i	$-0.688476\pi$
$457$	−4.98364e6			−1.11624			−0.558118	+	0.829762i	$0.688476\pi$
$458$		0			0
$459$	−1.05494e6	−	584575.i	−0.233721	−	0.129512i
$460$		0			0
$461$		−	1.24915e6i		−	0.273754i	−0.990588	−	0.136877i	$-0.956293\pi$
$461$		−	1.24915e6i		−	0.273754i	0.990588	−	0.136877i	$-0.0437066\pi$
$462$		0			0
$463$			6.43122e6i			1.39425i	0.716949	+	0.697125i	$0.245538\pi$
$463$			6.43122e6i			1.39425i	−0.716949	+	0.697125i	$0.754462\pi$
$464$		0			0
$465$	3.25037e6	+	2.69506e6i	0.697108	+	0.578012i
$466$		0			0
$467$	2.04703e6			0.434343			0.217171	−	0.976134i	$-0.430317\pi$
$467$	2.04703e6			0.434343			0.217171	+	0.976134i	$0.430317\pi$
$468$		0			0
$469$	−1.82833e6			−0.383816
$470$		0			0
$471$	2.34588e6	+	1.94510e6i	0.487252	+	0.404008i
$472$		0			0
$473$			1.17216e7i			2.40898i
$474$		0			0
$475$			4.02558e6i			0.818643i
$476$		0			0
$477$	6.44371e6	−	1.21428e6i	1.29670	−	0.244356i
$478$		0			0
$479$	−7.01395e6			−1.39677			−0.698384	−	0.715724i	$-0.746097\pi$
$479$	−7.01395e6			−1.39677			−0.698384	+	0.715724i	$0.746097\pi$
$480$		0			0
$481$	2.91804e6			0.575080
$482$		0			0
$483$	2.30947e6	−	2.78533e6i	0.450448	−	0.543261i
$484$		0			0
$485$			8.20586e6i			1.58405i
$486$		0			0
$487$			4.58508e6i			0.876041i	0.898965	+	0.438021i	$0.144320\pi$
$487$			4.58508e6i			0.876041i	−0.898965	+	0.438021i	$0.855680\pi$
$488$		0			0
$489$	−4.47817e6	+	5.40087e6i	−0.846892	+	1.02139i
$490$		0			0
$491$	1.73210e6			0.324243			0.162121	−	0.986771i	$-0.448166\pi$
$491$	1.73210e6			0.324243			0.162121	+	0.986771i	$0.448166\pi$
$492$		0			0
$493$	633600.			0.117408
$494$		0			0
$495$	−1.23172e7	+	2.32111e6i	−2.25943	+	0.425777i
$496$		0			0
$497$		−	9.98076e6i		−	1.81248i
$498$		0			0
$499$			7.81171e6i			1.40441i	0.711974	+	0.702206i	$0.247801\pi$
$499$			7.81171e6i			1.40441i	−0.711974	+	0.702206i	$0.752199\pi$
$500$		0			0
$501$	4.61894e6	+	3.82983e6i	0.822145	+	0.681687i
$502$		0			0
$503$	1.49688e6			0.263795			0.131898	−	0.991263i	$-0.457893\pi$
$503$	1.49688e6			0.263795			0.131898	+	0.991263i	$0.457893\pi$
$504$		0			0
$505$	6.63379e6			1.15753
$506$		0			0
$507$	−3.75275e6	−	3.11161e6i	−0.648380	−	0.537608i
$508$		0			0
$509$		−	5.41663e6i		−	0.926691i	−0.886178	−	0.463345i	$-0.846649\pi$
$509$		−	5.41663e6i		−	0.926691i	0.886178	−	0.463345i	$-0.153351\pi$
$510$		0			0
$511$			4.66120e6i			0.789669i
$512$		0			0
$513$	4.15384e6	+	2.30176e6i	0.696878	+	0.386160i
$514$		0			0
$515$	3.63528e6			0.603976
$516$		0			0
$517$	−8.39808e6			−1.38183
$518$		0			0
$519$	485496.	−	585530.i	0.0791165	−	0.0954181i
$520$		0			0
$521$		−	1.00462e7i		−	1.62146i	−0.585418	−	0.810731i	$-0.699070\pi$
$521$		−	1.00462e7i		−	1.62146i	0.585418	−	0.810731i	$-0.300930\pi$
$522$		0			0
$523$			261662.i			0.0418298i	0.999781	+	0.0209149i	$0.00665791\pi$
$523$			261662.i			0.0418298i	−0.999781	+	0.0209149i	$0.993342\pi$
$524$		0			0
$525$	5.72200e6	−	6.90099e6i	0.906045	−	1.09273i
$526$		0			0
$527$	1.08346e6			0.169936
$528$		0			0
$529$	−4.75673e6			−0.739042
$530$		0			0
$531$	379080.	+	2.01163e6i	0.0583438	+	0.309607i
$532$		0			0
$533$		−	3.58291e6i		−	0.546283i
$534$		0			0
$535$			1.70214e6i			0.257106i
$536$		0			0
$537$	−7.28611e6	−	6.04132e6i	−1.09034	−	0.904058i
$538$		0			0
$539$	−9.89431e6			−1.46694
$540$		0			0
$541$	−9.21600e6			−1.35378			−0.676892	−	0.736083i	$-0.736673\pi$
$541$	−9.21600e6			−1.35378			−0.676892	+	0.736083i	$0.736673\pi$
$542$		0			0
$543$	6.14220e6	+	5.09284e6i	0.893973	+	0.741243i
$544$		0			0
$545$		−	4.68790e6i		−	0.676063i
$546$		0			0
$547$			4.34724e6i			0.621220i	0.950537	+	0.310610i	$0.100533\pi$
$547$			4.34724e6i			0.621220i	−0.950537	+	0.310610i	$0.899467\pi$
$548$		0			0
$549$	1.15929e6	+	6.15189e6i	0.164158	+	0.871119i
$550$		0			0
$551$	−2.49480e6			−0.350071
$552$		0			0
$553$	3.43213e6			0.477256
$554$		0			0
$555$	−9.54994e6	+	1.15177e7i	−1.31604	+	1.58720i
$556$		0			0
$557$		−	9.11544e6i		−	1.24491i	−0.782654	−	0.622457i	$-0.786134\pi$
$557$		−	9.11544e6i		−	1.24491i	0.782654	−	0.622457i	$-0.213866\pi$
$558$		0			0
$559$			4.37751e6i			0.592512i
$560$		0			0
$561$	−2.05286e6	+	2.47585e6i	−0.275393	+	0.332137i
$562$		0			0
$563$	891000.			0.118470			0.0592348	−	0.998244i	$-0.481134\pi$
$563$	891000.			0.118470			0.0592348	+	0.998244i	$0.481134\pi$
$564$		0			0
$565$	6.36134e6			0.838355
$566$		0			0
$567$	−3.84912e6	−	9.85020e6i	−0.502810	−	1.28673i
$568$		0			0
$569$			1.01544e7i			1.31485i	0.753521	+	0.657424i	$0.228354\pi$
$569$			1.01544e7i			1.31485i	−0.753521	+	0.657424i	$0.771646\pi$
$570$		0			0
$571$			4.46831e6i			0.573526i	0.958002	+	0.286763i	$0.0925792\pi$
$571$			4.46831e6i			0.573526i	−0.958002	+	0.286763i	$0.907421\pi$
$572$		0			0
$573$	2.23949e6	+	1.85689e6i	0.284946	+	0.236265i
$574$		0			0
$575$	4.16146e6			0.524899
$576$		0			0
$577$	4.65538e6			0.582124			0.291062	−	0.956704i	$-0.405991\pi$
$577$	4.65538e6			0.582124			0.291062	+	0.956704i	$0.405991\pi$
$578$		0			0
$579$	2.03676e6	+	1.68879e6i	0.252490	+	0.209353i
$580$		0			0
$581$		−	1.40427e7i		−	1.72588i
$582$		0			0
$583$		−	1.74857e7i		−	2.13064i
$584$		0			0
$585$	−4.59994e6	+	866833.i	−0.555728	+	0.104724i
$586$		0			0
$587$	−1.33067e7			−1.59395			−0.796975	−	0.604013i	$-0.793568\pi$
$587$	−1.33067e7			−1.59395			−0.796975	+	0.604013i	$0.793568\pi$
$588$		0			0
$589$	−4.26611e6			−0.506691
$590$		0			0
$591$	−7.12246e6	+	8.59001e6i	−0.838805	+	1.01164i
$592$		0			0
$593$		−	4.29102e6i		−	0.501100i	−0.968104	−	0.250550i	$-0.919389\pi$
$593$		−	4.29102e6i		−	0.501100i	0.968104	−	0.250550i	$-0.0806114\pi$
$594$		0			0
$595$		−	4.53905e6i		−	0.525621i
$596$		0			0
$597$	−5.93228e6	+	7.15460e6i	−0.681217	+	0.821579i
$598$		0			0
$599$	3.07800e6			0.350511			0.175255	−	0.984523i	$-0.443925\pi$
$599$	3.07800e6			0.350511			0.175255	+	0.984523i	$0.443925\pi$
$600$		0			0
$601$	7.36716e6			0.831982			0.415991	−	0.909369i	$-0.363435\pi$
$601$	7.36716e6			0.831982			0.415991	+	0.909369i	$0.363435\pi$
$602$		0			0
$603$	−2.43778e6	+	459386.i	−0.273024	+	0.0514499i
$604$		0			0
$605$			2.06044e7i			2.28861i
$606$		0			0
$607$			2.76187e6i			0.304250i	0.988361	+	0.152125i	$0.0486117\pi$
$607$			2.76187e6i			0.304250i	−0.988361	+	0.152125i	$0.951388\pi$
$608$		0			0
$609$	4.27680e6	+	3.54614e6i	0.467278	+	0.387447i
$610$		0			0
$611$	−3.13632e6			−0.339874
$612$		0			0
$613$	1.65456e7			1.77840			0.889202	−	0.457515i	$-0.151260\pi$
$613$	1.65456e7			1.77840			0.889202	+	0.457515i	$0.151260\pi$
$614$		0			0
$615$	1.41420e7	+	1.17259e7i	1.50772	+	1.25014i
$616$		0			0
$617$		−	7.52577e6i		−	0.795862i	−0.917415	−	0.397931i	$-0.869728\pi$
$617$		−	7.52577e6i		−	0.795862i	0.917415	−	0.397931i	$-0.130272\pi$
$618$		0			0
$619$		−	1.14169e7i		−	1.19763i	−0.800887	−	0.598816i	$-0.795638\pi$
$619$		−	1.14169e7i		−	1.19763i	0.800887	−	0.598816i	$-0.204362\pi$
$620$		0			0
$621$	2.37946e6	−	4.29405e6i	0.247599	−	0.446825i
$622$		0			0
$623$	−1.50828e7			−1.55691
$624$		0			0
$625$	−9.48948e6			−0.971723
$626$		0			0
$627$	8.08315e6	−	9.74865e6i	0.821130	−	0.990320i
$628$		0			0
$629$			3.83922e6i			0.386916i
$630$		0			0
$631$		−	1.69023e7i		−	1.68995i	−0.534806	−	0.844975i	$-0.679615\pi$
$631$		−	1.69023e7i		−	1.68995i	0.534806	−	0.844975i	$-0.320385\pi$
$632$		0			0
$633$	3.16305e6	−	3.81478e6i	0.313759	−	0.378408i
$634$		0			0
$635$	1.12195e7			1.10418
$636$		0			0
$637$	−3.69510e6			−0.360809
$638$		0			0
$639$	−2.50776e6	−	1.33077e7i	−0.242959	−	1.28929i
$640$		0			0
$641$			1.09089e7i			1.04866i	0.851515	+	0.524331i	$0.175684\pi$
$641$			1.09089e7i			1.04866i	−0.851515	+	0.524331i	$0.824316\pi$
$642$		0			0
$643$		−	4.25805e6i		−	0.406147i	−0.979164	−	0.203073i	$-0.934907\pi$
$643$		−	4.25805e6i		−	0.406147i	0.979164	−	0.203073i	$-0.0650930\pi$
$644$		0			0
$645$	−1.72783e7	−	1.43264e7i	−1.63531	−	1.35593i
$646$		0			0
$647$	−1.18338e7			−1.11138			−0.555690	−	0.831390i	$-0.687546\pi$
$647$	−1.18338e7			−1.11138			−0.555690	+	0.831390i	$0.687546\pi$
$648$		0			0
$649$	5.45875e6			0.508723
$650$		0			0
$651$	7.31333e6	+	6.06389e6i	0.676336	+	0.560788i
$652$		0			0
$653$			1.55813e7i			1.42995i	0.699152	+	0.714973i	$0.253561\pi$
$653$			1.55813e7i			1.42995i	−0.699152	+	0.714973i	$0.746439\pi$
$654$		0			0
$655$			2.32626e7i			2.11863i
$656$		0			0
$657$	1.17117e6	+	6.21493e6i	0.105854	+	0.561724i
$658$		0			0
$659$	1.04710e7			0.939239			0.469619	−	0.882869i	$-0.344391\pi$
$659$	1.04710e7			0.939239			0.469619	+	0.882869i	$0.344391\pi$
$660$		0			0
$661$	8.68177e6			0.772867			0.386433	−	0.922317i	$-0.373707\pi$
$661$	8.68177e6			0.772867			0.386433	+	0.922317i	$0.373707\pi$
$662$		0			0
$663$	−766656.	+	924622.i	−0.0677356	+	0.0816922i
$664$		0			0
$665$			1.78725e7i			1.56723i
$666$		0			0
$667$			2.57901e6i			0.224460i
$668$		0			0
$669$	−1.33472e6	+	1.60973e6i	−0.115299	+	0.139055i
$670$		0			0
$671$	1.66938e7			1.43136
$672$		0			0
$673$	−1.94048e7			−1.65147			−0.825737	−	0.564055i	$-0.809241\pi$
$673$	−1.94048e7			−1.65147			−0.825737	+	0.564055i	$0.809241\pi$
$674$		0			0
$675$	5.89540e6	−	1.06390e7i	0.498028	−	0.898758i
$676$		0			0
$677$		−	8.69914e6i		−	0.729465i	−0.931112	−	0.364732i	$-0.881160\pi$
$677$		−	8.69914e6i		−	0.729465i	0.931112	−	0.364732i	$-0.118840\pi$
$678$		0			0
$679$			1.84632e7i			1.53685i
$680$		0			0
$681$	−8.81021e6	−	7.30504e6i	−0.727978	−	0.603608i
$682$		0			0
$683$	1.25511e7			1.02951			0.514755	−	0.857337i	$-0.327883\pi$
$683$	1.25511e7			1.02951			0.514755	+	0.857337i	$0.327883\pi$
$684$		0			0
$685$	1.23805e7			1.00812
$686$		0			0
$687$	−3.80474e6	−	3.15473e6i	−0.307563	−	0.255017i
$688$		0			0
$689$		−	6.53014e6i		−	0.524052i
$690$		0			0
$691$		−	6.02180e6i		−	0.479768i	−0.970802	−	0.239884i	$-0.922891\pi$
$691$		−	6.02180e6i		−	0.479768i	0.970802	−	0.239884i	$-0.0771094\pi$
$692$		0			0
$693$	−2.77137e7	+	5.22249e6i	−2.19210	+	0.413090i
$694$		0			0
$695$	3.17481e7			2.49319
$696$		0			0
$697$	4.71398e6			0.367541
$698$		0			0
$699$	−2.33006e6	+	2.81016e6i	−0.180374	+	0.217540i
$700$		0			0
$701$		−	4.60918e6i		−	0.354265i	−0.984187	−	0.177133i	$-0.943318\pi$
$701$		−	4.60918e6i		−	0.354265i	0.984187	−	0.177133i	$-0.0566822\pi$
$702$		0			0
$703$		−	1.51169e7i		−	1.15365i
$704$		0			0
$705$	1.02643e7	−	1.23792e7i	0.777781	−	0.938039i
$706$		0			0
$707$	1.49260e7			1.12304
$708$		0			0
$709$	5.92601e6			0.442738			0.221369	−	0.975190i	$-0.428947\pi$
$709$	5.92601e6			0.442738			0.221369	+	0.975190i	$0.428947\pi$
$710$		0			0
$711$	4.57618e6	−	862356.i	0.339491	−	0.0639753i
$712$		0			0
$713$			4.41010e6i			0.324881i
$714$		0			0
$715$			1.24824e7i			0.913131i
$716$		0			0
$717$	−1.02643e6	−	851072.i	−0.0745645	−	0.0618256i
$718$		0			0
$719$	−1.25012e7			−0.901841			−0.450921	−	0.892564i	$-0.648904\pi$
$719$	−1.25012e7			−0.901841			−0.450921	+	0.892564i	$0.648904\pi$
$720$		0			0
$721$	8.17938e6			0.585979
$722$		0			0
$723$	−7.99735e6	−	6.63105e6i	−0.568985	−	0.471777i
$724$		0			0
$725$			6.38981e6i			0.451485i
$726$		0			0
$727$		−	1.56959e7i		−	1.10142i	−0.834698	−	0.550708i	$-0.814358\pi$
$727$		−	1.56959e7i		−	1.10142i	0.834698	−	0.550708i	$-0.185642\pi$
$728$		0			0
$729$	−7.60712e6	−	1.21665e7i	−0.530153	−	0.847902i
$730$		0			0
$731$	−5.75942e6			−0.398644
$732$		0			0
$733$	1.47330e7			1.01282			0.506410	−	0.862293i	$-0.330972\pi$
$733$	1.47330e7			1.01282			0.506410	+	0.862293i	$0.330972\pi$
$734$		0			0
$735$	1.20930e7	−	1.45848e7i	0.825691	−	0.995821i
$736$		0			0
$737$			6.61515e6i			0.448612i
$738$		0			0
$739$			6.73569e6i			0.453702i	0.973929	+	0.226851i	$0.0728431\pi$
$739$			6.73569e6i			0.453702i	−0.973929	+	0.226851i	$0.927157\pi$
$740$		0			0
$741$	3.01871e6	−	3.64070e6i	0.201965	−	0.243579i
$742$		0			0
$743$	1.46409e7			0.972962			0.486481	−	0.873691i	$-0.338280\pi$
$743$	1.46409e7			0.972962			0.486481	+	0.873691i	$0.338280\pi$
$744$		0			0
$745$	1.69995e7			1.12214
$746$		0			0
$747$	−3.52836e6	−	1.87236e7i	−0.231351	−	1.22769i
$748$		0			0
$749$			3.82983e6i			0.249445i
$750$		0			0
$751$		−	2.66574e7i		−	1.72472i	−0.506295	−	0.862360i	$-0.668985\pi$
$751$		−	2.66574e7i		−	1.72472i	0.506295	−	0.862360i	$-0.331015\pi$
$752$		0			0
$753$	9.99216e6	+	8.28506e6i	0.642202	+	0.532486i
$754$		0			0
$755$	−1.58669e7			−1.01304
$756$		0			0
$757$	2.47914e7			1.57239			0.786197	−	0.617976i	$-0.212047\pi$
$757$	2.47914e7			1.57239			0.786197	+	0.617976i	$0.212047\pi$
$758$		0			0
$759$	−1.00777e7	−	8.35598e6i	−0.634975	−	0.526494i
$760$		0			0
$761$		−	3.05085e7i		−	1.90968i	−0.297126	−	0.954838i	$-0.596028\pi$
$761$		−	3.05085e7i		−	1.90968i	0.297126	−	0.954838i	$-0.403972\pi$
$762$		0			0
$763$		−	1.05478e7i		−	0.655918i
$764$		0			0
$765$	−1.14048e6	−	6.05207e6i	−0.0704586	−	0.373896i
$766$		0			0
$767$	2.03861e6			0.125125
$768$		0			0
$769$	−2.46605e6			−0.150378			−0.0751892	−	0.997169i	$-0.523956\pi$
$769$	−2.46605e6			−0.150378			−0.0751892	+	0.997169i	$0.523956\pi$
$770$		0			0
$771$	−9.49133e6	+	1.14470e7i	−0.575031	+	0.693513i
$772$		0			0
$773$		−	1.72684e7i		−	1.03945i	−0.854333	−	0.519726i	$-0.826034\pi$
$773$		−	1.72684e7i		−	1.03945i	0.854333	−	0.519726i	$-0.173966\pi$
$774$		0			0
$775$			1.09266e7i			0.653476i
$776$		0			0
$777$	−2.14874e7	+	2.59147e7i	−1.27682	+	1.53991i
$778$		0			0
$779$	−1.85613e7			−1.09589
$780$		0			0
$781$	−3.61117e7			−2.11846
$782$		0			0
$783$	6.59340e6	+	3.65359e6i	0.384331	+	0.212969i
$784$		0			0
$785$			1.55608e7i			0.901276i
$786$		0			0
$787$		−	2.74648e7i		−	1.58067i

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(12.0000 + 9.94987i) q^{3} +79.5990i q^{5} +179.098i q^{7} +(45.0000 + 238.797i) q^{9} +O(q^{10})\)	\(q+(12.0000 + 9.94987i) q^{3} +79.5990i q^{5} +179.098i q^{7} +(45.0000 + 238.797i) q^{9} +648.000 q^{11} +242.000 q^{13} +(-792.000 + 955.188i) q^{15} +318.396i q^{17} -1253.68i q^{19} +(-1782.00 + 2149.17i) q^{21} -1296.00 q^{23} -3211.00 q^{25} +(-1836.00 + 3313.31i) q^{27} -1989.97i q^{29} -3402.86i q^{31} +(7776.00 + 6447.52i) q^{33} -14256.0 q^{35} +12058.0 q^{37} +(2904.00 + 2407.87i) q^{39} -14805.4i q^{41} +18088.9i q^{43} +(-19008.0 + 3581.95i) q^{45} -12960.0 q^{47} -15269.0 q^{49} +(-3168.00 + 3820.75i) q^{51} -26984.1i q^{53} +51580.1i q^{55} +(12474.0 - 15044.2i) q^{57} +8424.00 q^{59} +25762.0 q^{61} +(-42768.0 + 8059.40i) q^{63} +19263.0i q^{65} +10208.6i q^{67} +(-15552.0 - 12895.0i) q^{69} -55728.0 q^{71} +26026.0 q^{73} +(-38532.0 - 31949.0i) q^{75} +116055. i q^{77} -19163.5i q^{79} +(-54999.0 + 21491.7i) q^{81} -78408.0 q^{83} -25344.0 q^{85} +(19800.0 - 23879.7i) q^{87} +84215.7i q^{89} +43341.7i q^{91} +(33858.0 - 40834.3i) q^{93} +99792.0 q^{95} +103090. q^{97} +(29160.0 + 154740. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{3} + 90 q^{9}+O(q^{10}) \) 2 * q + 24 * q^3 + 90 * q^9	\( 2 q + 24 q^{3} + 90 q^{9} + 1296 q^{11} + 484 q^{13} - 1584 q^{15} - 3564 q^{21} - 2592 q^{23} - 6422 q^{25} - 3672 q^{27} + 15552 q^{33} - 28512 q^{35} + 24116 q^{37} + 5808 q^{39} - 38016 q^{45} - 25920 q^{47} - 30538 q^{49} - 6336 q^{51} + 24948 q^{57} + 16848 q^{59} + 51524 q^{61} - 85536 q^{63} - 31104 q^{69} - 111456 q^{71} + 52052 q^{73} - 77064 q^{75} - 109998 q^{81} - 156816 q^{83} - 50688 q^{85} + 39600 q^{87} + 67716 q^{93} + 199584 q^{95} + 206180 q^{97} + 58320 q^{99}+O(q^{100}) \) 2 * q + 24 * q^3 + 90 * q^9 + 1296 * q^11 + 484 * q^13 - 1584 * q^15 - 3564 * q^21 - 2592 * q^23 - 6422 * q^25 - 3672 * q^27 + 15552 * q^33 - 28512 * q^35 + 24116 * q^37 + 5808 * q^39 - 38016 * q^45 - 25920 * q^47 - 30538 * q^49 - 6336 * q^51 + 24948 * q^57 + 16848 * q^59 + 51524 * q^61 - 85536 * q^63 - 31104 * q^69 - 111456 * q^71 + 52052 * q^73 - 77064 * q^75 - 109998 * q^81 - 156816 * q^83 - 50688 * q^85 + 39600 * q^87 + 67716 * q^93 + 199584 * q^95 + 206180 * q^97 + 58320 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more