LMFDB - Embedded newform 192.3.e.e.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(65,192)]

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, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	192.e (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:	$5.23162107572$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	192.65
Dual form			192.3.e.e.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 + 2.44949i) q^{3} +2.82843i q^{5} -10.3923 q^{7} +(-3.00000 + 8.48528i) q^{9} +O(q^{10})$	\(q+(1.73205 + 2.44949i) q^{3} +2.82843i q^{5} -10.3923 q^{7} +(-3.00000 + 8.48528i) q^{9} +14.6969i q^{11} +6.00000 q^{13} +(-6.92820 + 4.89898i) q^{15} -22.6274i q^{17} -10.3923 q^{19} +(-18.0000 - 25.4558i) q^{21} +29.3939i q^{23} +17.0000 q^{25} +(-25.9808 + 7.34847i) q^{27} +31.1127i q^{29} +31.1769 q^{31} +(-36.0000 + 25.4558i) q^{33} -29.3939i q^{35} +38.0000 q^{37} +(10.3923 + 14.6969i) q^{39} +5.65685i q^{41} -10.3923 q^{43} +(-24.0000 - 8.48528i) q^{45} -58.7878i q^{47} +59.0000 q^{49} +(55.4256 - 39.1918i) q^{51} +14.1421i q^{53} -41.5692 q^{55} +(-18.0000 - 25.4558i) q^{57} +14.6969i q^{59} +22.0000 q^{61} +(31.1769 - 88.1816i) q^{63} +16.9706i q^{65} +114.315 q^{67} +(-72.0000 + 50.9117i) q^{69} -29.3939i q^{71} -30.0000 q^{73} +(29.4449 + 41.6413i) q^{75} -152.735i q^{77} +31.1769 q^{79} +(-63.0000 - 50.9117i) q^{81} -73.4847i q^{83} +64.0000 q^{85} +(-76.2102 + 53.8888i) q^{87} +5.65685i q^{89} -62.3538 q^{91} +(54.0000 + 76.3675i) q^{93} -29.3939i q^{95} +90.0000 q^{97} +(-124.708 - 44.0908i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 24 q^{13} - 72 q^{21} + 68 q^{25} - 144 q^{33} + 152 q^{37} - 96 q^{45} + 236 q^{49} - 72 q^{57} + 88 q^{61} - 288 q^{69} - 120 q^{73} - 252 q^{81} + 256 q^{85} + 216 q^{93} + 360 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 + 24 * q^13 - 72 * q^21 + 68 * q^25 - 144 * q^33 + 152 * q^37 - 96 * q^45 + 236 * q^49 - 72 * q^57 + 88 * q^61 - 288 * q^69 - 120 * q^73 - 252 * q^81 + 256 * q^85 + 216 * q^93 + 360 * q^97

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
$3$	1.73205	+	2.44949i	0.577350	+	0.816497i
$3$	1.73205	+	2.44949i	0.577350	+	0.816497i
$4$		0			0
$5$			2.82843i			0.565685i	0.959166	+	0.282843i	$0.0912774\pi$
$5$			2.82843i			0.565685i	−0.959166	+	0.282843i	$0.908723\pi$
$6$		0			0
$7$	−10.3923			−1.48461			−0.742307	−	0.670059i	$-0.766269\pi$
$7$	−10.3923			−1.48461			−0.742307	+	0.670059i	$0.766269\pi$
$8$		0			0
$9$	−3.00000	+	8.48528i	−0.333333	+	0.942809i
$10$		0			0
$11$			14.6969i			1.33609i	0.744123	+	0.668043i	$0.232868\pi$
$11$			14.6969i			1.33609i	−0.744123	+	0.668043i	$0.767132\pi$
$12$		0			0
$13$	6.00000			0.461538			0.230769	−	0.973009i	$-0.425876\pi$
$13$	6.00000			0.461538			0.230769	+	0.973009i	$0.425876\pi$
$14$		0			0
$15$	−6.92820	+	4.89898i	−0.461880	+	0.326599i
$16$		0			0
$17$		−	22.6274i		−	1.33102i	−0.746387	−	0.665512i	$-0.768213\pi$
$17$		−	22.6274i		−	1.33102i	0.746387	−	0.665512i	$-0.231787\pi$
$18$		0			0
$19$	−10.3923			−0.546963			−0.273482	−	0.961877i	$-0.588175\pi$
$19$	−10.3923			−0.546963			−0.273482	+	0.961877i	$0.588175\pi$
$20$		0			0
$21$	−18.0000	−	25.4558i	−0.857143	−	1.21218i
$22$		0			0
$23$			29.3939i			1.27799i	0.769209	+	0.638997i	$0.220651\pi$
$23$			29.3939i			1.27799i	−0.769209	+	0.638997i	$0.779349\pi$
$24$		0			0
$25$	17.0000			0.680000
$26$		0			0
$27$	−25.9808	+	7.34847i	−0.962250	+	0.272166i
$28$		0			0
$29$			31.1127i			1.07285i	0.843947	+	0.536426i	$0.180226\pi$
$29$			31.1127i			1.07285i	−0.843947	+	0.536426i	$0.819774\pi$
$30$		0			0
$31$	31.1769			1.00571			0.502853	−	0.864372i	$-0.332284\pi$
$31$	31.1769			1.00571			0.502853	+	0.864372i	$0.332284\pi$
$32$		0			0
$33$	−36.0000	+	25.4558i	−1.09091	+	0.771389i
$34$		0			0
$35$		−	29.3939i		−	0.839825i
$36$		0			0
$37$	38.0000			1.02703			0.513514	−	0.858082i	$-0.328344\pi$
$37$	38.0000			1.02703			0.513514	+	0.858082i	$0.328344\pi$
$38$		0			0
$39$	10.3923	+	14.6969i	0.266469	+	0.376845i
$40$		0			0
$41$			5.65685i			0.137972i	0.997618	+	0.0689860i	$0.0219764\pi$
$41$			5.65685i			0.137972i	−0.997618	+	0.0689860i	$0.978024\pi$
$42$		0			0
$43$	−10.3923			−0.241682			−0.120841	−	0.992672i	$-0.538559\pi$
$43$	−10.3923			−0.241682			−0.120841	+	0.992672i	$0.538559\pi$
$44$		0			0
$45$	−24.0000	−	8.48528i	−0.533333	−	0.188562i
$46$		0			0
$47$		−	58.7878i		−	1.25080i	−0.780303	−	0.625402i	$-0.784935\pi$
$47$		−	58.7878i		−	1.25080i	0.780303	−	0.625402i	$-0.215065\pi$
$48$		0			0
$49$	59.0000			1.20408
$50$		0			0
$51$	55.4256	−	39.1918i	1.08678	−	0.768467i
$52$		0			0
$53$			14.1421i			0.266833i	0.991060	+	0.133416i	$0.0425948\pi$
$53$			14.1421i			0.266833i	−0.991060	+	0.133416i	$0.957405\pi$
$54$		0			0
$55$	−41.5692			−0.755804
$56$		0			0
$57$	−18.0000	−	25.4558i	−0.315789	−	0.446594i
$58$		0			0
$59$			14.6969i			0.249101i	0.992213	+	0.124550i	$0.0397488\pi$
$59$			14.6969i			0.249101i	−0.992213	+	0.124550i	$0.960251\pi$
$60$		0			0
$61$	22.0000			0.360656			0.180328	−	0.983607i	$-0.442284\pi$
$61$	22.0000			0.360656			0.180328	+	0.983607i	$0.442284\pi$
$62$		0			0
$63$	31.1769	−	88.1816i	0.494872	−	1.39971i
$64$		0			0
$65$			16.9706i			0.261086i
$66$		0			0
$67$	114.315			1.70620			0.853100	−	0.521748i	$-0.174720\pi$
$67$	114.315			1.70620			0.853100	+	0.521748i	$0.174720\pi$
$68$		0			0
$69$	−72.0000	+	50.9117i	−1.04348	+	0.737851i
$70$		0			0
$71$		−	29.3939i		−	0.413998i	−0.978341	−	0.206999i	$-0.933630\pi$
$71$		−	29.3939i		−	0.413998i	0.978341	−	0.206999i	$-0.0663697\pi$
$72$		0			0
$73$	−30.0000			−0.410959			−0.205479	−	0.978661i	$-0.565875\pi$
$73$	−30.0000			−0.410959			−0.205479	+	0.978661i	$0.565875\pi$
$74$		0			0
$75$	29.4449	+	41.6413i	0.392598	+	0.555218i
$76$		0			0
$77$		−	152.735i		−	1.98357i
$78$		0			0
$79$	31.1769			0.394644			0.197322	−	0.980339i	$-0.436775\pi$
$79$	31.1769			0.394644			0.197322	+	0.980339i	$0.436775\pi$
$80$		0			0
$81$	−63.0000	−	50.9117i	−0.777778	−	0.628539i
$82$		0			0
$83$		−	73.4847i		−	0.885358i	−0.896680	−	0.442679i	$-0.854028\pi$
$83$		−	73.4847i		−	0.885358i	0.896680	−	0.442679i	$-0.145972\pi$
$84$		0			0
$85$	64.0000			0.752941
$86$		0			0
$87$	−76.2102	+	53.8888i	−0.875980	+	0.619411i
$88$		0			0
$89$			5.65685i			0.0635602i	0.999495	+	0.0317801i	$0.0101176\pi$
$89$			5.65685i			0.0635602i	−0.999495	+	0.0317801i	$0.989882\pi$
$90$		0			0
$91$	−62.3538			−0.685207
$92$		0			0
$93$	54.0000	+	76.3675i	0.580645	+	0.821156i
$94$		0			0
$95$		−	29.3939i		−	0.309409i
$96$		0			0
$97$	90.0000			0.927835			0.463918	−	0.885878i	$-0.346443\pi$
$97$	90.0000			0.927835			0.463918	+	0.885878i	$0.346443\pi$
$98$		0			0
$99$	−124.708	−	44.0908i	−1.25967	−	0.445362i
$100$		0			0
$101$			2.82843i			0.0280042i	0.999902	+	0.0140021i	$0.00445716\pi$
$101$			2.82843i			0.0280042i	−0.999902	+	0.0140021i	$0.995543\pi$
$102$		0			0
$103$	72.7461			0.706273			0.353137	−	0.935572i	$-0.385115\pi$
$103$	72.7461			0.706273			0.353137	+	0.935572i	$0.385115\pi$
$104$		0			0
$105$	72.0000	−	50.9117i	0.685714	−	0.484873i
$106$		0			0
$107$			73.4847i			0.686773i	0.939194	+	0.343386i	$0.111574\pi$
$107$			73.4847i			0.686773i	−0.939194	+	0.343386i	$0.888426\pi$
$108$		0			0
$109$	−138.000			−1.26606			−0.633028	−	0.774129i	$-0.718188\pi$
$109$	−138.000			−1.26606			−0.633028	+	0.774129i	$0.718188\pi$
$110$		0			0
$111$	65.8179	+	93.0806i	0.592954	+	0.838564i
$112$		0			0
$113$			192.333i			1.70206i	0.525115	+	0.851031i	$0.324022\pi$
$113$			192.333i			1.70206i	−0.525115	+	0.851031i	$0.675978\pi$
$114$		0			0
$115$	−83.1384			−0.722943
$116$		0			0
$117$	−18.0000	+	50.9117i	−0.153846	+	0.435143i
$118$		0			0
$119$			235.151i			1.97606i
$120$		0			0
$121$	−95.0000			−0.785124
$122$		0			0
$123$	−13.8564	+	9.79796i	−0.112654	+	0.0796582i
$124$		0			0
$125$			118.794i			0.950352i
$126$		0			0
$127$	−51.9615			−0.409146			−0.204573	−	0.978851i	$-0.565581\pi$
$127$	−51.9615			−0.409146			−0.204573	+	0.978851i	$0.565581\pi$
$128$		0			0
$129$	−18.0000	−	25.4558i	−0.139535	−	0.197332i
$130$		0			0
$131$			44.0908i			0.336571i	0.985738	+	0.168286i	$0.0538231\pi$
$131$			44.0908i			0.336571i	−0.985738	+	0.168286i	$0.946177\pi$
$132$		0			0
$133$	108.000			0.812030
$134$		0			0
$135$	−20.7846	−	73.4847i	−0.153960	−	0.544331i
$136$		0			0
$137$		−	39.5980i		−	0.289036i	−0.989502	−	0.144518i	$-0.953837\pi$
$137$		−	39.5980i		−	0.289036i	0.989502	−	0.144518i	$-0.0461632\pi$
$138$		0			0
$139$	−135.100			−0.971942			−0.485971	−	0.873975i	$-0.661534\pi$
$139$	−135.100			−0.971942			−0.485971	+	0.873975i	$0.661534\pi$
$140$		0			0
$141$	144.000	−	101.823i	1.02128	−	0.722152i
$142$		0			0
$143$			88.1816i			0.616655i
$144$		0			0
$145$	−88.0000			−0.606897
$146$		0			0
$147$	102.191	+	144.520i	0.695177	+	0.983129i
$148$		0			0
$149$		−	189.505i		−	1.27184i	−0.771754	−	0.635922i	$-0.780620\pi$
$149$		−	189.505i		−	1.27184i	0.771754	−	0.635922i	$-0.219380\pi$
$150$		0			0
$151$	−176.669			−1.16999			−0.584997	−	0.811035i	$-0.698904\pi$
$151$	−176.669			−1.16999			−0.584997	+	0.811035i	$0.698904\pi$
$152$		0			0
$153$	192.000	+	67.8823i	1.25490	+	0.443675i
$154$		0			0
$155$			88.1816i			0.568914i
$156$		0			0
$157$	−154.000			−0.980892			−0.490446	−	0.871472i	$-0.663166\pi$
$157$	−154.000			−0.980892			−0.490446	+	0.871472i	$0.663166\pi$
$158$		0			0
$159$	−34.6410	+	24.4949i	−0.217868	+	0.154056i
$160$		0			0
$161$		−	305.470i		−	1.89733i
$162$		0			0
$163$	72.7461			0.446295			0.223148	−	0.974785i	$-0.428367\pi$
$163$	72.7461			0.446295			0.223148	+	0.974785i	$0.428367\pi$
$164$		0			0
$165$	−72.0000	−	101.823i	−0.436364	−	0.617111i
$166$		0			0
$167$		−	264.545i		−	1.58410i	−0.610455	−	0.792051i	$-0.709014\pi$
$167$		−	264.545i		−	1.58410i	0.610455	−	0.792051i	$-0.290986\pi$
$168$		0			0
$169$	−133.000			−0.786982
$170$		0			0
$171$	31.1769	−	88.1816i	0.182321	−	0.515682i
$172$		0			0
$173$			223.446i			1.29159i	0.763509	+	0.645797i	$0.223475\pi$
$173$			223.446i			1.29159i	−0.763509	+	0.645797i	$0.776525\pi$
$174$		0			0
$175$	−176.669			−1.00954
$176$		0			0
$177$	−36.0000	+	25.4558i	−0.203390	+	0.143818i
$178$		0			0
$179$			102.879i			0.574741i	0.957820	+	0.287370i	$0.0927810\pi$
$179$			102.879i			0.574741i	−0.957820	+	0.287370i	$0.907219\pi$
$180$		0			0
$181$	−234.000			−1.29282			−0.646409	−	0.762991i	$-0.723730\pi$
$181$	−234.000			−1.29282			−0.646409	+	0.762991i	$0.723730\pi$
$182$		0			0
$183$	38.1051	+	53.8888i	0.208225	+	0.294474i
$184$		0			0
$185$			107.480i			0.580974i
$186$		0			0
$187$	332.554			1.77836
$188$		0			0
$189$	270.000	−	76.3675i	1.42857	−	0.404061i
$190$		0			0
$191$		−	117.576i		−	0.615579i	−0.951455	−	0.307789i	$-0.900411\pi$
$191$		−	117.576i		−	0.615579i	0.951455	−	0.307789i	$-0.0995892\pi$
$192$		0			0
$193$	82.0000			0.424870			0.212435	−	0.977175i	$-0.431861\pi$
$193$	82.0000			0.424870			0.212435	+	0.977175i	$0.431861\pi$
$194$		0			0
$195$	−41.5692	+	29.3939i	−0.213175	+	0.150738i
$196$		0			0
$197$			206.475i			1.04810i	0.851688	+	0.524049i	$0.175579\pi$
$197$			206.475i			1.04810i	−0.851688	+	0.524049i	$0.824421\pi$
$198$		0			0
$199$	322.161			1.61890			0.809451	−	0.587188i	$-0.199765\pi$
$199$	322.161			1.61890			0.809451	+	0.587188i	$0.199765\pi$
$200$		0			0
$201$	198.000	+	280.014i	0.985075	+	1.39311i
$202$		0			0
$203$		−	323.333i		−	1.59277i
$204$		0			0
$205$	−16.0000			−0.0780488
$206$		0			0
$207$	−249.415	−	88.1816i	−1.20490	−	0.425998i
$208$		0			0
$209$		−	152.735i		−	0.730790i
$210$		0			0
$211$	−135.100			−0.640284			−0.320142	−	0.947370i	$-0.603731\pi$
$211$	−135.100			−0.640284			−0.320142	+	0.947370i	$0.603731\pi$
$212$		0			0
$213$	72.0000	−	50.9117i	0.338028	−	0.239022i
$214$		0			0
$215$		−	29.3939i		−	0.136716i
$216$		0			0
$217$	−324.000			−1.49309
$218$		0			0
$219$	−51.9615	−	73.4847i	−0.237267	−	0.335547i
$220$		0			0
$221$		−	135.765i		−	0.614319i
$222$		0			0
$223$	363.731			1.63108			0.815540	−	0.578701i	$-0.196440\pi$
$223$	363.731			1.63108			0.815540	+	0.578701i	$0.196440\pi$
$224$		0			0
$225$	−51.0000	+	144.250i	−0.226667	+	0.641110i
$226$		0			0
$227$		−	14.6969i		−	0.0647442i	−0.999476	−	0.0323721i	$-0.989694\pi$
$227$		−	14.6969i		−	0.0647442i	0.999476	−	0.0323721i	$-0.0103062\pi$
$228$		0			0
$229$	246.000			1.07424			0.537118	−	0.843507i	$-0.319513\pi$
$229$	246.000			1.07424			0.537118	+	0.843507i	$0.319513\pi$
$230$		0			0
$231$	374.123	−	264.545i	1.61958	−	1.14522i
$232$		0			0
$233$		−	209.304i		−	0.898299i	−0.893457	−	0.449149i	$-0.851727\pi$
$233$		−	209.304i		−	0.898299i	0.893457	−	0.449149i	$-0.148273\pi$
$234$		0			0
$235$	166.277			0.707561
$236$		0			0
$237$	54.0000	+	76.3675i	0.227848	+	0.322226i
$238$		0			0
$239$			411.514i			1.72182i	0.508760	+	0.860909i	$0.330104\pi$
$239$			411.514i			1.72182i	−0.508760	+	0.860909i	$0.669896\pi$
$240$		0			0
$241$	210.000			0.871369			0.435685	−	0.900099i	$-0.356506\pi$
$241$	210.000			0.871369			0.435685	+	0.900099i	$0.356506\pi$
$242$		0			0
$243$	15.5885	−	242.499i	0.0641500	−	0.997940i
$244$		0			0
$245$			166.877i			0.681131i
$246$		0			0
$247$	−62.3538			−0.252445
$248$		0			0
$249$	180.000	−	127.279i	0.722892	−	0.511162i
$250$		0			0
$251$		−	396.817i		−	1.58095i	−0.612497	−	0.790473i	$-0.709835\pi$
$251$		−	396.817i		−	1.58095i	0.612497	−	0.790473i	$-0.290165\pi$
$252$		0			0
$253$	−432.000			−1.70751
$254$		0			0
$255$	110.851	+	156.767i	0.434711	+	0.614774i
$256$		0			0
$257$			384.666i			1.49676i	0.663273	+	0.748378i	$0.269167\pi$
$257$			384.666i			1.49676i	−0.663273	+	0.748378i	$0.730833\pi$
$258$		0			0
$259$	−394.908			−1.52474
$260$		0			0
$261$	−264.000	−	93.3381i	−1.01149	−	0.357617i
$262$		0			0
$263$		−	499.696i		−	1.89998i	−0.312274	−	0.949992i	$-0.601091\pi$
$263$		−	499.696i		−	1.89998i	0.312274	−	0.949992i	$-0.398909\pi$
$264$		0			0
$265$	−40.0000			−0.150943
$266$		0			0
$267$	−13.8564	+	9.79796i	−0.0518967	+	0.0366965i
$268$		0			0
$269$			19.7990i			0.0736022i	0.999323	+	0.0368011i	$0.0117168\pi$
$269$			19.7990i			0.0736022i	−0.999323	+	0.0368011i	$0.988283\pi$
$270$		0			0
$271$	−135.100			−0.498524			−0.249262	−	0.968436i	$-0.580188\pi$
$271$	−135.100			−0.498524			−0.249262	+	0.968436i	$0.580188\pi$
$272$		0			0
$273$	−108.000	−	152.735i	−0.395604	−	0.559469i
$274$		0			0
$275$			249.848i			0.908538i
$276$		0			0
$277$	198.000			0.714801			0.357401	−	0.933951i	$-0.383663\pi$
$277$	198.000			0.714801			0.357401	+	0.933951i	$0.383663\pi$
$278$		0			0
$279$	−93.5307	+	264.545i	−0.335236	+	0.948190i
$280$		0			0
$281$		−	197.990i		−	0.704590i	−0.935889	−	0.352295i	$-0.885401\pi$
$281$		−	197.990i		−	0.704590i	0.935889	−	0.352295i	$-0.114599\pi$
$282$		0			0
$283$	−51.9615			−0.183610			−0.0918048	−	0.995777i	$-0.529264\pi$
$283$	−51.9615			−0.183610			−0.0918048	+	0.995777i	$0.529264\pi$
$284$		0			0
$285$	72.0000	−	50.9117i	0.252632	−	0.178638i
$286$		0			0
$287$		−	58.7878i		−	0.204835i
$288$		0			0
$289$	−223.000			−0.771626
$290$		0			0
$291$	155.885	+	220.454i	0.535686	+	0.757574i
$292$		0			0
$293$		−	359.210i		−	1.22597i	−0.790093	−	0.612987i	$-0.789968\pi$
$293$		−	359.210i		−	1.22597i	0.790093	−	0.612987i	$-0.210032\pi$
$294$		0			0
$295$	−41.5692			−0.140913
$296$		0			0
$297$	−108.000	−	381.838i	−0.363636	−	1.28565i
$298$		0			0
$299$			176.363i			0.589844i
$300$		0			0
$301$	108.000			0.358804
$302$		0			0
$303$	−6.92820	+	4.89898i	−0.0228654	+	0.0161682i
$304$		0			0
$305$			62.2254i			0.204018i
$306$		0			0
$307$	197.454			0.643172			0.321586	−	0.946880i	$-0.395784\pi$
$307$	197.454			0.643172			0.321586	+	0.946880i	$0.395784\pi$
$308$		0			0
$309$	126.000	+	178.191i	0.407767	+	0.576670i
$310$		0			0
$311$			264.545i			0.850627i	0.905046	+	0.425313i	$0.139836\pi$
$311$			264.545i			0.850627i	−0.905046	+	0.425313i	$0.860164\pi$
$312$		0			0
$313$	266.000			0.849840			0.424920	−	0.905231i	$-0.360302\pi$
$313$	266.000			0.849840			0.424920	+	0.905231i	$0.360302\pi$
$314$		0			0
$315$	249.415	+	88.1816i	0.791795	+	0.279942i
$316$		0			0
$317$		−	421.436i		−	1.32945i	−0.747088	−	0.664725i	$-0.768549\pi$
$317$		−	421.436i		−	1.32945i	0.747088	−	0.664725i	$-0.231451\pi$
$318$		0			0
$319$	−457.261			−1.43342
$320$		0			0
$321$	−180.000	+	127.279i	−0.560748	+	0.396508i
$322$		0			0
$323$			235.151i			0.728022i
$324$		0			0
$325$	102.000			0.313846
$326$		0			0
$327$	−239.023	−	338.030i	−0.730957	−	1.03373i
$328$		0			0
$329$			610.940i			1.85696i
$330$		0			0
$331$	197.454			0.596537			0.298269	−	0.954482i	$-0.403591\pi$
$331$	197.454			0.596537			0.298269	+	0.954482i	$0.403591\pi$
$332$		0			0
$333$	−114.000	+	322.441i	−0.342342	+	0.968290i
$334$		0			0
$335$			323.333i			0.965172i
$336$		0			0
$337$	522.000			1.54896			0.774481	−	0.632598i	$-0.218011\pi$
$337$	522.000			1.54896			0.774481	+	0.632598i	$0.218011\pi$
$338$		0			0
$339$	−471.118	+	333.131i	−1.38973	+	0.982686i
$340$		0			0
$341$			458.205i			1.34371i
$342$		0			0
$343$	−103.923			−0.302983
$344$		0			0
$345$	−144.000	−	203.647i	−0.417391	−	0.590280i
$346$		0			0
$347$			426.211i			1.22827i	0.789199	+	0.614137i	$0.210496\pi$
$347$			426.211i			1.22827i	−0.789199	+	0.614137i	$0.789504\pi$
$348$		0			0
$349$	166.000			0.475645			0.237822	−	0.971309i	$-0.423566\pi$
$349$	166.000			0.475645			0.237822	+	0.971309i	$0.423566\pi$
$350$		0			0
$351$	−155.885	+	44.0908i	−0.444116	+	0.125615i
$352$		0			0
$353$			22.6274i			0.0641003i	0.999486	+	0.0320502i	$0.0102036\pi$
$353$			22.6274i			0.0641003i	−0.999486	+	0.0320502i	$0.989796\pi$
$354$		0			0
$355$	83.1384			0.234193
$356$		0			0
$357$	−576.000	+	407.294i	−1.61345	+	1.14088i
$358$		0			0
$359$			88.1816i			0.245631i	0.992430	+	0.122816i	$0.0391924\pi$
$359$			88.1816i			0.245631i	−0.992430	+	0.122816i	$0.960808\pi$
$360$		0			0
$361$	−253.000			−0.700831
$362$		0			0
$363$	−164.545	−	232.702i	−0.453292	−	0.641051i
$364$		0			0
$365$		−	84.8528i		−	0.232473i
$366$		0			0
$367$	−135.100			−0.368120			−0.184060	−	0.982915i	$-0.558924\pi$
$367$	−135.100			−0.368120			−0.184060	+	0.982915i	$0.558924\pi$
$368$		0			0
$369$	−48.0000	−	16.9706i	−0.130081	−	0.0459907i
$370$		0			0
$371$		−	146.969i		−	0.396144i
$372$		0			0
$373$	118.000			0.316354			0.158177	−	0.987411i	$-0.449438\pi$
$373$	118.000			0.316354			0.158177	+	0.987411i	$0.449438\pi$
$374$		0			0
$375$	−290.985	+	205.757i	−0.775959	+	0.548686i
$376$		0			0
$377$			186.676i			0.495162i
$378$		0			0
$379$	−592.361			−1.56296			−0.781479	−	0.623931i	$-0.785535\pi$
$379$	−592.361			−1.56296			−0.781479	+	0.623931i	$0.785535\pi$
$380$		0			0
$381$	−90.0000	−	127.279i	−0.236220	−	0.334066i
$382$		0			0
$383$		−	352.727i		−	0.920957i	−0.887671	−	0.460478i	$-0.847678\pi$
$383$		−	352.727i		−	0.920957i	0.887671	−	0.460478i	$-0.152322\pi$
$384$		0			0
$385$	432.000			1.12208
$386$		0			0
$387$	31.1769	−	88.1816i	0.0805605	−	0.227860i
$388$		0			0
$389$		−	359.210i		−	0.923420i	−0.887031	−	0.461710i	$-0.847236\pi$
$389$		−	359.210i		−	0.923420i	0.887031	−	0.461710i	$-0.152764\pi$
$390$		0			0
$391$	665.108			1.70104
$392$		0			0
$393$	−108.000	+	76.3675i	−0.274809	+	0.194319i
$394$		0			0
$395$			88.1816i			0.223245i
$396$		0			0
$397$	214.000			0.539043			0.269521	−	0.962994i	$-0.413135\pi$
$397$	214.000			0.539043			0.269521	+	0.962994i	$0.413135\pi$
$398$		0			0
$399$	187.061	+	264.545i	0.468826	+	0.663020i
$400$		0			0
$401$		−	226.274i		−	0.564275i	−0.959374	−	0.282137i	$-0.908957\pi$
$401$		−	226.274i		−	0.564275i	0.959374	−	0.282137i	$-0.0910434\pi$
$402$		0			0
$403$	187.061			0.464172
$404$		0			0
$405$	144.000	−	178.191i	0.355556	−	0.439978i
$406$		0			0
$407$			558.484i			1.37220i
$408$		0			0
$409$	114.000			0.278729			0.139364	−	0.990241i	$-0.455494\pi$
$409$	114.000			0.278729			0.139364	+	0.990241i	$0.455494\pi$
$410$		0			0
$411$	96.9948	−	68.5857i	0.235997	−	0.166875i
$412$		0			0
$413$		−	152.735i		−	0.369819i
$414$		0			0
$415$	207.846			0.500834
$416$		0			0
$417$	−234.000	−	330.926i	−0.561151	−	0.793587i
$418$		0			0
$419$		−	367.423i		−	0.876906i	−0.898754	−	0.438453i	$-0.855527\pi$
$419$		−	367.423i		−	0.876906i	0.898754	−	0.438453i	$-0.144473\pi$
$420$		0			0
$421$	534.000			1.26841			0.634204	−	0.773166i	$-0.281328\pi$
$421$	534.000			1.26841			0.634204	+	0.773166i	$0.281328\pi$
$422$		0			0
$423$	498.831	+	176.363i	1.17927	+	0.416934i
$424$		0			0
$425$		−	384.666i		−	0.905097i
$426$		0			0
$427$	−228.631			−0.535435
$428$		0			0
$429$	−216.000	+	152.735i	−0.503497	+	0.356026i
$430$		0			0
$431$		−	293.939i		−	0.681993i	−0.940065	−	0.340996i	$-0.889236\pi$
$431$		−	293.939i		−	0.681993i	0.940065	−	0.340996i	$-0.110764\pi$
$432$		0			0
$433$	−614.000			−1.41801			−0.709007	−	0.705202i	$-0.750857\pi$
$433$	−614.000			−1.41801			−0.709007	+	0.705202i	$0.750857\pi$
$434$		0			0
$435$	−152.420	−	215.555i	−0.350392	−	0.495529i
$436$		0			0
$437$		−	305.470i		−	0.699016i
$438$		0			0
$439$	−342.946			−0.781198			−0.390599	−	0.920561i	$-0.627732\pi$
$439$	−342.946			−0.781198			−0.390599	+	0.920561i	$0.627732\pi$
$440$		0			0
$441$	−177.000	+	500.632i	−0.401361	+	1.13522i
$442$		0			0
$443$		−	514.393i		−	1.16116i	−0.814204	−	0.580579i	$-0.802826\pi$
$443$		−	514.393i		−	1.16116i	0.814204	−	0.580579i	$-0.197174\pi$
$444$		0			0
$445$	−16.0000			−0.0359551
$446$		0			0
$447$	464.190	−	328.232i	1.03846	−	0.734299i
$448$		0			0
$449$		−	214.960i		−	0.478754i	−0.970927	−	0.239377i	$-0.923057\pi$
$449$		−	214.960i		−	0.478754i	0.970927	−	0.239377i	$-0.0769432\pi$
$450$		0			0
$451$	−83.1384			−0.184342
$452$		0			0
$453$	−306.000	−	432.749i	−0.675497	−	0.955297i
$454$		0			0
$455$		−	176.363i		−	0.387612i
$456$		0			0
$457$	786.000			1.71991			0.859956	−	0.510368i	$-0.170491\pi$
$457$	786.000			1.71991			0.859956	+	0.510368i	$0.170491\pi$
$458$		0			0
$459$	166.277	+	587.878i	0.362259	+	1.28078i
$460$		0			0
$461$			811.759i			1.76086i	0.474172	+	0.880432i	$0.342748\pi$
$461$			811.759i			1.76086i	−0.474172	+	0.880432i	$0.657252\pi$
$462$		0			0
$463$	446.869			0.965160			0.482580	−	0.875852i	$-0.339700\pi$
$463$	446.869			0.965160			0.482580	+	0.875852i	$0.339700\pi$
$464$		0			0
$465$	−216.000	+	152.735i	−0.464516	+	0.328463i
$466$		0			0
$467$		−	426.211i		−	0.912658i	−0.889811	−	0.456329i	$-0.849164\pi$
$467$		−	426.211i		−	0.912658i	0.889811	−	0.456329i	$-0.150836\pi$
$468$		0			0
$469$	−1188.00			−2.53305
$470$		0			0
$471$	−266.736	−	377.221i	−0.566318	−	0.800895i
$472$		0			0
$473$		−	152.735i		−	0.322907i
$474$		0			0
$475$	−176.669			−0.371935
$476$		0			0
$477$	−120.000	−	42.4264i	−0.251572	−	0.0889442i
$478$		0			0
$479$		−	117.576i		−	0.245460i	−0.992440	−	0.122730i	$-0.960835\pi$
$479$		−	117.576i		−	0.245460i	0.992440	−	0.122730i	$-0.0391650\pi$
$480$		0			0
$481$	228.000			0.474012
$482$		0			0
$483$	748.246	−	529.090i	1.54916	−	1.09542i
$484$		0			0
$485$			254.558i			0.524863i
$486$		0			0
$487$	−426.084			−0.874917			−0.437458	−	0.899239i	$-0.644121\pi$
$487$	−426.084			−0.874917			−0.437458	+	0.899239i	$0.644121\pi$
$488$		0			0
$489$	126.000	+	178.191i	0.257669	+	0.364399i
$490$		0			0
$491$			191.060i			0.389125i	0.980890	+	0.194562i	$0.0623286\pi$
$491$			191.060i			0.389125i	−0.980890	+	0.194562i	$0.937671\pi$
$492$		0			0
$493$	704.000			1.42799
$494$		0			0
$495$	124.708	−	352.727i	0.251935	−	0.712579i
$496$		0			0
$497$			305.470i			0.614628i
$498$		0			0
$499$	862.561			1.72858			0.864290	−	0.502994i	$-0.167768\pi$
$499$	862.561			1.72858			0.864290	+	0.502994i	$0.167768\pi$
$500$		0			0
$501$	648.000	−	458.205i	1.29341	−	0.914581i
$502$		0			0
$503$		−	205.757i		−	0.409060i	−0.978860	−	0.204530i	$-0.934433\pi$
$503$		−	205.757i		−	0.409060i	0.978860	−	0.204530i	$-0.0655666\pi$
$504$		0			0
$505$	−8.00000			−0.0158416
$506$		0			0
$507$	−230.363	−	325.782i	−0.454364	−	0.642568i
$508$		0			0
$509$			845.700i			1.66149i	0.556651	+	0.830746i	$0.312086\pi$
$509$			845.700i			1.66149i	−0.556651	+	0.830746i	$0.687914\pi$
$510$		0			0
$511$	311.769			0.610116
$512$		0			0
$513$	270.000	−	76.3675i	0.526316	−	0.148865i
$514$		0			0
$515$			205.757i			0.399528i
$516$		0			0
$517$	864.000			1.67118
$518$		0			0
$519$	−547.328	+	387.019i	−1.05458	+	0.745702i
$520$		0			0
$521$		−	39.5980i		−	0.0760038i	−0.999278	−	0.0380019i	$-0.987901\pi$
$521$		−	39.5980i		−	0.0760038i	0.999278	−	0.0380019i	$-0.0120993\pi$
$522$		0			0
$523$	654.715			1.25185			0.625923	−	0.779885i	$-0.284723\pi$
$523$	654.715			1.25185			0.625923	+	0.779885i	$0.284723\pi$
$524$		0			0
$525$	−306.000	−	432.749i	−0.582857	−	0.824284i
$526$		0			0
$527$		−	705.453i		−	1.33862i
$528$		0			0
$529$	−335.000			−0.633270
$530$		0			0
$531$	−124.708	−	44.0908i	−0.234854	−	0.0830336i
$532$		0			0
$533$			33.9411i			0.0636794i
$534$		0			0
$535$	−207.846			−0.388497
$536$		0			0
$537$	−252.000	+	178.191i	−0.469274	+	0.331827i
$538$		0			0
$539$			867.119i			1.60876i
$540$		0			0
$541$	−330.000			−0.609982			−0.304991	−	0.952355i	$-0.598653\pi$
$541$	−330.000			−0.609982			−0.304991	+	0.952355i	$0.598653\pi$
$542$		0			0
$543$	−405.300	−	573.181i	−0.746409	−	1.05558i
$544$		0			0
$545$		−	390.323i		−	0.716189i
$546$		0			0
$547$	−93.5307			−0.170989			−0.0854943	−	0.996339i	$-0.527247\pi$
$547$	−93.5307			−0.170989			−0.0854943	+	0.996339i	$0.527247\pi$
$548$		0			0
$549$	−66.0000	+	186.676i	−0.120219	+	0.340029i
$550$		0			0
$551$		−	323.333i		−	0.586811i
$552$		0			0
$553$	−324.000			−0.585895
$554$		0			0
$555$	−263.272	+	186.161i	−0.474363	+	0.335426i
$556$		0			0
$557$		−	613.769i		−	1.10192i	−0.834532	−	0.550959i	$-0.814262\pi$
$557$		−	613.769i		−	1.10192i	0.834532	−	0.550959i	$-0.185738\pi$
$558$		0			0
$559$	−62.3538			−0.111545
$560$		0			0
$561$	576.000	+	814.587i	1.02674	+	1.45203i
$562$		0			0
$563$		−	543.787i		−	0.965873i	−0.875655	−	0.482937i	$-0.839570\pi$
$563$		−	543.787i		−	0.965873i	0.875655	−	0.482937i	$-0.160430\pi$
$564$		0			0
$565$	−544.000			−0.962832
$566$		0			0
$567$	654.715	+	529.090i	1.15470	+	0.933139i
$568$		0			0
$569$		−	5.65685i		−	0.00994175i	−0.999988	−	0.00497087i	$-0.998418\pi$
$569$		−	5.65685i		−	0.00994175i	0.999988	−	0.00497087i	$-0.00158228\pi$
$570$		0			0
$571$	613.146			1.07381			0.536905	−	0.843642i	$-0.319593\pi$
$571$	613.146			1.07381			0.536905	+	0.843642i	$0.319593\pi$
$572$		0			0
$573$	288.000	−	203.647i	0.502618	−	0.355404i
$574$		0			0
$575$			499.696i			0.869036i
$576$		0			0
$577$	−502.000			−0.870017			−0.435009	−	0.900426i	$-0.643255\pi$
$577$	−502.000			−0.870017			−0.435009	+	0.900426i	$0.643255\pi$
$578$		0			0
$579$	142.028	+	200.858i	0.245299	+	0.346905i
$580$		0			0
$581$			763.675i			1.31442i
$582$		0			0
$583$	−207.846			−0.356511
$584$		0			0
$585$	−144.000	−	50.9117i	−0.246154	−	0.0870285i
$586$		0			0
$587$		−	44.0908i		−	0.0751121i	−0.999295	−	0.0375561i	$-0.988043\pi$
$587$		−	44.0908i		−	0.0751121i	0.999295	−	0.0375561i	$-0.0119573\pi$
$588$		0			0
$589$	−324.000			−0.550085
$590$		0			0
$591$	−505.759	+	357.626i	−0.855768	+	0.605119i
$592$		0			0
$593$		−	260.215i		−	0.438812i	−0.975634	−	0.219406i	$-0.929588\pi$
$593$		−	260.215i		−	0.438812i	0.975634	−	0.219406i	$-0.0704119\pi$
$594$		0			0
$595$	−665.108			−1.11783
$596$		0			0
$597$	558.000	+	789.131i	0.934673	+	1.32183i
$598$		0			0
$599$			734.847i			1.22679i	0.789776	+	0.613395i	$0.210197\pi$
$599$			734.847i			1.22679i	−0.789776	+	0.613395i	$0.789803\pi$
$600$		0			0
$601$	130.000			0.216306			0.108153	−	0.994134i	$-0.465506\pi$
$601$	130.000			0.216306			0.108153	+	0.994134i	$0.465506\pi$
$602$		0			0
$603$	−342.946	+	969.998i	−0.568733	+	1.60862i
$604$		0			0
$605$		−	268.701i		−	0.444133i
$606$		0			0
$607$	613.146			1.01013			0.505063	−	0.863083i	$-0.331469\pi$
$607$	613.146			1.01013			0.505063	+	0.863083i	$0.331469\pi$
$608$		0			0
$609$	792.000	−	560.029i	1.30049	−	0.919587i
$610$		0			0
$611$		−	352.727i		−	0.577294i
$612$		0			0
$613$	−794.000			−1.29527			−0.647635	−	0.761951i	$-0.724242\pi$
$613$	−794.000			−1.29527			−0.647635	+	0.761951i	$0.724242\pi$
$614$		0			0
$615$	−27.7128	−	39.1918i	−0.0450615	−	0.0637266i
$616$		0			0
$617$		−	639.225i		−	1.03602i	−0.855374	−	0.518010i	$-0.826673\pi$
$617$		−	639.225i		−	1.03602i	0.855374	−	0.518010i	$-0.173327\pi$
$618$		0			0
$619$	−966.484			−1.56136			−0.780682	−	0.624928i	$-0.785128\pi$
$619$	−966.484			−1.56136			−0.780682	+	0.624928i	$0.785128\pi$
$620$		0			0
$621$	−216.000	−	763.675i	−0.347826	−	1.22975i
$622$		0			0
$623$		−	58.7878i		−	0.0943624i
$624$		0			0
$625$	89.0000			0.142400
$626$		0			0
$627$	374.123	−	264.545i	0.596687	−	0.421922i
$628$		0			0
$629$		−	859.842i		−	1.36700i
$630$		0			0
$631$	−93.5307			−0.148226			−0.0741131	−	0.997250i	$-0.523613\pi$
$631$	−93.5307			−0.148226			−0.0741131	+	0.997250i	$0.523613\pi$
$632$		0			0
$633$	−234.000	−	330.926i	−0.369668	−	0.522790i
$634$		0			0
$635$		−	146.969i		−	0.231448i
$636$		0			0
$637$	354.000			0.555730
$638$		0			0
$639$	249.415	+	88.1816i	0.390321	+	0.137999i
$640$		0			0
$641$			622.254i			0.970755i	0.874305	+	0.485378i	$0.161318\pi$
$641$			622.254i			0.970755i	−0.874305	+	0.485378i	$0.838682\pi$
$642$		0			0
$643$	−592.361			−0.921246			−0.460623	−	0.887596i	$-0.652374\pi$
$643$	−592.361			−0.921246			−0.460623	+	0.887596i	$0.652374\pi$
$644$		0			0
$645$	72.0000	−	50.9117i	0.111628	−	0.0789328i
$646$		0			0
$647$			1263.94i			1.95353i	0.214303	+	0.976767i	$0.431252\pi$
$647$			1263.94i			1.95353i	−0.214303	+	0.976767i	$0.568748\pi$
$648$		0			0
$649$	−216.000			−0.332820
$650$		0			0
$651$	−561.184	−	793.635i	−0.862035	−	1.21910i
$652$		0			0
$653$		−	591.141i		−	0.905270i	−0.891696	−	0.452635i	$-0.850484\pi$
$653$		−	591.141i		−	0.905270i	0.891696	−	0.452635i	$-0.149516\pi$
$654$		0			0
$655$	−124.708			−0.190393
$656$		0			0
$657$	90.0000	−	254.558i	0.136986	−	0.387456i
$658$		0			0
$659$			1161.06i			1.76185i	0.473257	+	0.880924i	$0.343078\pi$
$659$			1161.06i			1.76185i	−0.473257	+	0.880924i	$0.656922\pi$
$660$		0			0
$661$	−442.000			−0.668684			−0.334342	−	0.942452i	$-0.608514\pi$
$661$	−442.000			−0.668684			−0.334342	+	0.942452i	$0.608514\pi$
$662$		0			0
$663$	332.554	−	235.151i	0.501589	−	0.354677i
$664$		0			0
$665$			305.470i			0.459354i
$666$		0			0
$667$	−914.523			−1.37110
$668$		0			0
$669$	630.000	+	890.955i	0.941704	+	1.33177i
$670$		0			0
$671$			323.333i			0.481867i
$672$		0			0
$673$	−742.000			−1.10253			−0.551263	−	0.834332i	$-0.685854\pi$
$673$	−742.000			−1.10253			−0.551263	+	0.834332i	$0.685854\pi$
$674$		0			0
$675$	−441.673	+	124.924i	−0.654330	+	0.185073i
$676$		0			0
$677$			2.82843i			0.00417788i	0.999998	+	0.00208894i	$0.000664931\pi$
$677$			2.82843i			0.00417788i	−0.999998	+	0.00208894i	$0.999335\pi$
$678$		0			0
$679$	−935.307			−1.37748
$680$		0			0
$681$	36.0000	−	25.4558i	0.0528634	−	0.0373801i
$682$		0			0
$683$			249.848i			0.365810i	0.983131	+	0.182905i	$0.0585500\pi$
$683$			249.848i			0.365810i	−0.983131	+	0.182905i	$0.941450\pi$
$684$		0			0
$685$	112.000			0.163504
$686$		0			0
$687$	426.084	+	602.574i	0.620210	+	0.877110i
$688$		0			0
$689$			84.8528i			0.123154i
$690$		0			0
$691$	322.161			0.466225			0.233112	−	0.972450i	$-0.425109\pi$
$691$	322.161			0.466225			0.233112	+	0.972450i	$0.425109\pi$
$692$		0			0
$693$	1296.00	+	458.205i	1.87013	+	0.661191i
$694$		0			0
$695$		−	382.120i		−	0.549814i
$696$		0			0
$697$	128.000			0.183644
$698$		0			0
$699$	512.687	−	362.524i	0.733458	−	0.518633i
$700$		0			0
$701$			234.759i			0.334892i	0.985881	+	0.167446i	$0.0535520\pi$
$701$			234.759i			0.334892i	−0.985881	+	0.167446i	$0.946448\pi$
$702$		0			0
$703$	−394.908			−0.561746
$704$		0			0
$705$	288.000	+	407.294i	0.408511	+	0.577721i
$706$		0			0
$707$		−	29.3939i		−	0.0415755i
$708$		0			0
$709$	102.000			0.143865			0.0719323	−	0.997410i	$-0.477083\pi$
$709$	102.000			0.143865			0.0719323	+	0.997410i	$0.477083\pi$
$710$		0			0
$711$	−93.5307	+	264.545i	−0.131548	+	0.372074i
$712$		0			0
$713$			916.410i			1.28529i
$714$		0			0
$715$	−249.415			−0.348833
$716$		0			0
$717$	−1008.00	+	712.764i	−1.40586	+	0.994092i
$718$		0			0
$719$		−	529.090i		−	0.735869i	−0.929852	−	0.367934i	$-0.880065\pi$
$719$		−	529.090i		−	0.735869i	0.929852	−	0.367934i	$-0.119935\pi$
$720$		0			0
$721$	−756.000			−1.04854
$722$		0			0
$723$	363.731	+	514.393i	0.503085	+	0.711470i
$724$		0			0
$725$			528.916i			0.729539i
$726$		0			0
$727$	1070.41			1.47236			0.736181	−	0.676785i	$-0.236627\pi$
$727$	1070.41			1.47236			0.736181	+	0.676785i	$0.236627\pi$
$728$		0			0
$729$	621.000	−	381.838i	0.851852	−	0.523783i
$730$		0			0
$731$			235.151i			0.321684i
$732$		0			0
$733$	−378.000			−0.515689			−0.257844	−	0.966186i	$-0.583012\pi$
$733$	−378.000			−0.515689			−0.257844	+	0.966186i	$0.583012\pi$
$734$		0			0
$735$	−408.764	+	289.040i	−0.556141	+	0.393251i
$736$		0			0
$737$			1680.09i			2.27963i
$738$		0			0
$739$	1111.98			1.50470			0.752352	−	0.658761i	$-0.228919\pi$
$739$	1111.98			1.50470			0.752352	+	0.658761i	$0.228919\pi$
$740$		0			0
$741$	−108.000	−	152.735i	−0.145749	−	0.206120i
$742$		0			0
$743$			205.757i			0.276928i	0.990368	+	0.138464i	$0.0442164\pi$
$743$			205.757i			0.276928i	−0.990368	+	0.138464i	$0.955784\pi$
$744$		0			0
$745$	536.000			0.719463
$746$		0			0
$747$	623.538	+	220.454i	0.834723	+	0.295119i
$748$		0			0
$749$		−	763.675i		−	1.01959i
$750$		0			0
$751$	−218.238			−0.290597			−0.145299	−	0.989388i	$-0.546414\pi$
$751$	−218.238			−0.290597			−0.145299	+	0.989388i	$0.546414\pi$
$752$		0			0
$753$	972.000	−	687.308i	1.29084	−	0.912759i
$754$		0			0
$755$		−	499.696i		−	0.661849i
$756$		0			0
$757$	−1098.00			−1.45046			−0.725231	−	0.688505i	$-0.758267\pi$
$757$	−1098.00			−1.45046			−0.725231	+	0.688505i	$0.758267\pi$
$758$		0			0
$759$	−748.246	−	1058.18i	−0.985831	−	1.39418i
$760$		0			0
$761$		−	1250.16i		−	1.64279i	−0.570358	−	0.821396i	$-0.693196\pi$
$761$		−	1250.16i		−	1.64279i	0.570358	−	0.821396i	$-0.306804\pi$
$762$		0			0
$763$	1434.14			1.87960
$764$		0			0
$765$	−192.000	+	543.058i	−0.250980	+	0.709880i
$766$		0			0
$767$			88.1816i			0.114970i
$768$		0			0
$769$	−334.000			−0.434330			−0.217165	−	0.976135i	$-0.569681\pi$
$769$	−334.000			−0.434330			−0.217165	+	0.976135i	$0.569681\pi$
$770$		0			0
$771$	−942.236	+	666.261i	−1.22210	+	0.864152i
$772$		0			0
$773$		−	562.857i		−	0.728146i	−0.931370	−	0.364073i	$-0.881386\pi$
$773$		−	562.857i		−	0.728146i	0.931370	−	0.364073i	$-0.118614\pi$
$774$		0			0
$775$	530.008			0.683881
$776$		0			0
$777$	−684.000	−	967.322i	−0.880309	−	1.24494i
$778$		0			0
$779$		−	58.7878i		−	0.0754657i
$780$		0			0
$781$	432.000			0.553137
$782$		0			0
$783$	−228.631	−	808.332i	−0.291993	−	1.03235i
$784$		0			0
$785$		−	435.578i		−	0.554876i
$786$		0			0
$787$	−841.777			−1.06960			−0.534801	−	0.844978i	$-0.679613\pi$
$787$	−841.777			−1.06960			−0.534801	+	0.844978i	$0.679613\pi$
$788$		0			0
$789$	1224.00	−	865.499i	1.55133	−	1.09696i
$790$		0			0
$791$		−	1998.78i		−	2.52691i
$792$		0			0
$793$	132.000			0.166456
$794$		0			0
$795$	−69.2820	−	97.9796i	−0.0871472	−	0.123245i
$796$		0			0
$797$			393.151i			0.493289i	0.969106	+	0.246645i	$0.0793280\pi$
$797$			393.151i			0.493289i	−0.969106	+	0.246645i	$0.920672\pi$
$798$		0			0
$799$	−1330.22			−1.66485
$800$		0			0
$801$	−48.0000	−	16.9706i	−0.0599251	−	0.0211867i
$802$		0			0
$803$		−	440.908i		−	0.549076i
$804$		0			0
$805$	864.000			1.07329
$806$		0			0
$807$	−48.4974	+	34.2929i	−0.0600959	+	0.0424942i
$808$		0			0
$809$			1227.54i			1.51735i	0.651468	+	0.758676i	$0.274153\pi$
$809$			1227.54i			1.51735i	−0.651468	+	0.758676i	$0.725847\pi$
$810$		0			0
$811$	1153.55			1.42237			0.711187	−	0.703003i	$-0.248158\pi$
$811$	1153.55			1.42237			0.711187	+	0.703003i	$0.248158\pi$
$812$		0			0
$813$	−234.000	−	330.926i	−0.287823	−	0.407043i
$814$		0			0
$815$			205.757i			0.252463i
$816$		0			0
$817$	108.000			0.132191
$818$		0			0
$819$	187.061	−	529.090i	0.228402	−	0.646019i
$820$		0			0
$821$		−	370.524i		−	0.451308i	−0.974207	−	0.225654i	$-0.927548\pi$
$821$		−	370.524i		−	0.451308i	0.974207	−	0.225654i	$-0.0724519\pi$
$822$		0			0
$823$	405.300			0.492466			0.246233	−	0.969211i	$-0.420807\pi$
$823$	405.300			0.492466			0.246233	+	0.969211i	$0.420807\pi$
$824$		0			0
$825$	−612.000	+	432.749i	−0.741818	+	0.524545i
$826$		0			0
$827$		−	1043.48i		−	1.26177i	−0.775877	−	0.630884i	$-0.782692\pi$
$827$		−	1043.48i		−	1.26177i	0.775877	−	0.630884i	$-0.217308\pi$
$828$		0			0
$829$	486.000			0.586248			0.293124	−	0.956074i	$-0.405305\pi$
$829$	486.000			0.586248			0.293124	+	0.956074i	$0.405305\pi$
$830$		0			0
$831$	342.946	+	484.999i	0.412691	+	0.583633i
$832$		0			0
$833$		−	1335.02i		−	1.60266i
$834$		0			0
$835$	748.246			0.896103
$836$		0			0
$837$	−810.000	+	229.103i	−0.967742	+	0.273719i
$838$		0			0
$839$		−	264.545i		−	0.315310i	−0.987494	−	0.157655i	$-0.949607\pi$
$839$		−	264.545i		−	0.315310i	0.987494	−	0.157655i	$-0.0503934\pi$
$840$		0			0
$841$	−127.000			−0.151011
$842$		0			0
$843$	484.974	−	342.929i	0.575296	−	0.406795i
$844$		0			0
$845$		−	376.181i		−	0.445184i
$846$		0			0
$847$	987.269			1.16561
$848$		0			0
$849$	−90.0000	−	127.279i	−0.106007	−	0.149917i
$850$		0			0
$851$			1116.97i			1.31254i
$852$		0			0
$853$	−1642.00			−1.92497			−0.962485	−	0.271334i	$-0.912535\pi$
$853$	−1642.00			−1.92497			−0.962485	+	0.271334i	$0.912535\pi$
$854$		0			0
$855$	249.415	+	88.1816i	0.291714	+	0.103136i
$856$		0			0
$857$			1057.83i			1.23434i	0.786829	+	0.617171i	$0.211721\pi$
$857$			1057.83i			1.23434i	−0.786829	+	0.617171i	$0.788279\pi$
$858$		0			0
$859$	−93.5307			−0.108883			−0.0544416	−	0.998517i	$-0.517338\pi$
$859$	−93.5307			−0.108883			−0.0544416	+	0.998517i	$0.517338\pi$
$860$		0			0
$861$	144.000	−	101.823i	0.167247	−	0.118262i
$862$		0			0
$863$			235.151i			0.272481i	0.990676	+	0.136240i	$0.0435020\pi$
$863$			235.151i			0.272481i	−0.990676	+	0.136240i	$0.956498\pi$
$864$		0			0
$865$	−632.000			−0.730636
$866$		0			0
$867$	−386.247	−	546.236i	−0.445499	−	0.630030i
$868$		0			0
$869$			458.205i			0.527279i
$870$		0			0
$871$	685.892			0.787477
$872$		0			0
$873$	−270.000	+	763.675i	−0.309278	+	0.874771i
$874$		0			0
$875$		−	1234.54i		−	1.41091i
$876$		0			0
$877$	550.000			0.627138			0.313569	−	0.949565i	$-0.398475\pi$
$877$	550.000			0.627138			0.313569	+	0.949565i	$0.398475\pi$
$878$		0			0
$879$	879.882	−	622.170i	1.00100	−	0.707816i
$880$		0			0
$881$		−	1210.57i		−	1.37408i	−0.726618	−	0.687041i	$-0.758909\pi$
$881$		−	1210.57i		−	1.37408i	0.726618	−	0.687041i	$-0.241091\pi$
$882$		0			0
$883$	−841.777			−0.953314			−0.476657	−	0.879089i	$-0.658152\pi$
$883$	−841.777			−0.953314			−0.476657	+	0.879089i	$0.658152\pi$
$884$		0			0
$885$	−72.0000	−	101.823i	−0.0813559	−	0.115055i
$886$		0			0
$887$		−	1146.36i		−	1.29240i	−0.763167	−	0.646201i	$-0.776357\pi$
$887$		−	1146.36i		−	1.29240i	0.763167	−	0.646201i	$-0.223643\pi$
$888$		0			0
$889$	540.000			0.607424
$890$		0			0
$891$	748.246	−	925.907i	0.839782	−	1.03918i
$892$		0			0
$893$			610.940i			0.684144i
$894$		0			0
$895$	−290.985			−0.325122
$896$		0			0
$897$	−432.000	+	305.470i	−0.481605	+	0.340546i
$898$		0			0
$899$			969.998i			1.07897i
$900$		0			0
$901$	320.000			0.355161
$902$		0			0
$903$	187.061	+	264.545i	0.207156	+	0.292962i
$904$		0			0
$905$		−	661.852i		−	0.731328i
$906$		0			0
$907$	−1506.88			−1.66139			−0.830697	−	0.556725i	$-0.812058\pi$
$907$	−1506.88			−1.66139			−0.830697	+	0.556725i	$0.812058\pi$
$908$		0			0
$909$	−24.0000	−	8.48528i	−0.0264026	−	0.00933474i
$910$		0			0
$911$			1469.69i			1.61328i	0.591046	+	0.806638i	$0.298715\pi$
$911$			1469.69i			1.61328i	−0.591046	+	0.806638i	$0.701285\pi$
$912$		0			0
$913$	1080.00			1.18291
$914$		0			0
$915$	−152.420	+	107.778i	−0.166580	+	0.117790i
$916$		0			0
$917$		−	458.205i		−	0.499679i
$918$		0			0
$919$	1070.41			1.16475			0.582376	−	0.812919i	$-0.302123\pi$
$919$	1070.41			1.16475			0.582376	+	0.812919i	$0.302123\pi$
$920$		0			0
$921$	342.000	+	483.661i	0.371336	+	0.525148i
$922$		0			0
$923$		−	176.363i		−	0.191076i
$924$		0			0
$925$	646.000			0.698378
$926$		0			0
$927$	−218.238	+	617.271i	−0.235424	+	0.665881i
$928$		0			0
$929$		−	667.509i		−	0.718524i	−0.933237	−	0.359262i	$-0.883028\pi$
$929$		−	667.509i		−	0.718524i	0.933237	−	0.359262i	$-0.116972\pi$
$930$		0			0
$931$	−613.146			−0.658589
$932$		0			0
$933$	−648.000	+	458.205i	−0.694534	+	0.491110i
$934$		0			0
$935$			940.604i			1.00599i
$936$		0			0
$937$	1402.00			1.49626			0.748132	−	0.663550i	$-0.230951\pi$
$937$	1402.00			1.49626			0.748132	+	0.663550i	$0.230951\pi$
$938$		0			0
$939$	460.726	+	651.564i	0.490656	+	0.693892i
$940$		0			0
$941$			1694.23i			1.80045i	0.435420	+	0.900227i	$0.356600\pi$
$941$			1694.23i			1.80045i	−0.435420	+	0.900227i	$0.643400\pi$
$942$		0			0
$943$	−166.277			−0.176328
$944$		0			0
$945$	216.000	+	763.675i	0.228571	+	0.808122i
$946$		0			0
$947$		−	1190.45i		−	1.25708i	−0.777778	−	0.628539i	$-0.783653\pi$
$947$		−	1190.45i		−	1.25708i	0.777778	−	0.628539i	$-0.216347\pi$
$948$		0			0
$949$	−180.000			−0.189673
$950$		0			0
$951$	1032.30	−	729.948i	1.08549	−	0.767558i
$952$		0			0
$953$			401.637i			0.421445i	0.977546	+	0.210722i	$0.0675816\pi$
$953$			401.637i			0.421445i	−0.977546	+	0.210722i	$0.932418\pi$
$954$		0			0
$955$	332.554			0.348224
$956$		0			0
$957$	−792.000	−	1120.06i	−0.827586	−	1.17038i
$958$		0			0
$959$			411.514i			0.429108i
$960$		0			0
$961$	11.0000			0.0114464
$962$		0			0
$963$	−623.538	−	220.454i	−0.647496	−	0.228924i
$964$		0			0
$965$			231.931i			0.240343i
$966$		0			0
$967$	−1423.75			−1.47233			−0.736166	−	0.676801i	$-0.763366\pi$
$967$	−1423.75			−1.47233			−0.736166	+	0.676801i	$0.763366\pi$
$968$		0			0
$969$	−576.000	+	407.294i	−0.594427	+	0.420324i
$970$		0			0
$971$		−	808.332i		−	0.832473i	−0.909256	−	0.416237i	$-0.863349\pi$
$971$		−	808.332i		−	0.832473i	0.909256	−	0.416237i	$-0.136651\pi$
$972$		0			0
$973$	1404.00			1.44296
$974$		0			0
$975$	176.669	+	249.848i	0.181199	+	0.256254i
$976$		0			0
$977$			565.685i			0.579002i	0.957178	+	0.289501i	$0.0934894\pi$
$977$			565.685i			0.579002i	−0.957178	+	0.289501i	$0.906511\pi$
$978$		0			0
$979$	−83.1384			−0.0849218
$980$		0			0
$981$	414.000	−	1170.97i	0.422018	−	1.19365i
$982$		0			0
$983$		−	558.484i		−	0.568142i	−0.958803	−	0.284071i	$-0.908315\pi$
$983$		−	558.484i		−	0.568142i	0.958803	−	0.284071i	$-0.0916852\pi$
$984$		0			0
$985$	−584.000			−0.592893
$986$		0			0
$987$	−1496.49	+	1058.18i	−1.51620	+	1.07212i
$988$		0			0
$989$		−	305.470i		−	0.308868i
$990$		0			0
$991$	−550.792			−0.555794			−0.277897	−	0.960611i	$-0.589637\pi$
$991$	−550.792			−0.555794			−0.277897	+	0.960611i	$0.589637\pi$
$992$		0			0
$993$	342.000	+	483.661i	0.344411	+	0.487071i
$994$		0			0
$995$			911.210i			0.915789i
$996$		0			0
$997$	1606.00			1.61083			0.805416	−	0.592710i	$-0.201942\pi$
$997$	1606.00			1.61083			0.805416	+	0.592710i	$0.201942\pi$
$998$		0			0
$999$	−987.269	+	279.242i	−0.988257	+	0.279521i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.e.e.65.4		4
3.2	odd	2	inner	192.3.e.e.65.3		4
4.3	odd	2	inner	192.3.e.e.65.1		4
8.3	odd	2		96.3.e.a.65.4	yes	4
8.5	even	2		96.3.e.a.65.1	✓	4
12.11	even	2	inner	192.3.e.e.65.2		4
16.3	odd	4		768.3.h.f.641.6		8
16.5	even	4		768.3.h.f.641.5		8
16.11	odd	4		768.3.h.f.641.3		8
16.13	even	4		768.3.h.f.641.4		8
24.5	odd	2		96.3.e.a.65.2	yes	4
24.11	even	2		96.3.e.a.65.3	yes	4
48.5	odd	4		768.3.h.f.641.2		8
48.11	even	4		768.3.h.f.641.8		8
48.29	odd	4		768.3.h.f.641.7		8
48.35	even	4		768.3.h.f.641.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.e.a.65.1	✓	4	8.5	even	2
96.3.e.a.65.2	yes	4	24.5	odd	2
96.3.e.a.65.3	yes	4	24.11	even	2
96.3.e.a.65.4	yes	4	8.3	odd	2
192.3.e.e.65.1		4	4.3	odd	2	inner
192.3.e.e.65.2		4	12.11	even	2	inner
192.3.e.e.65.3		4	3.2	odd	2	inner
192.3.e.e.65.4		4	1.1	even	1	trivial
768.3.h.f.641.1		8	48.35	even	4
768.3.h.f.641.2		8	48.5	odd	4
768.3.h.f.641.3		8	16.11	odd	4
768.3.h.f.641.4		8	16.13	even	4
768.3.h.f.641.5		8	16.5	even	4
768.3.h.f.641.6		8	16.3	odd	4
768.3.h.f.641.7		8	48.29	odd	4
768.3.h.f.641.8		8	48.11	even	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 + 2.44949i) q^{3} +2.82843i q^{5} -10.3923 q^{7} +(-3.00000 + 8.48528i) q^{9} +O(q^{10})\)	\(q+(1.73205 + 2.44949i) q^{3} +2.82843i q^{5} -10.3923 q^{7} +(-3.00000 + 8.48528i) q^{9} +14.6969i q^{11} +6.00000 q^{13} +(-6.92820 + 4.89898i) q^{15} -22.6274i q^{17} -10.3923 q^{19} +(-18.0000 - 25.4558i) q^{21} +29.3939i q^{23} +17.0000 q^{25} +(-25.9808 + 7.34847i) q^{27} +31.1127i q^{29} +31.1769 q^{31} +(-36.0000 + 25.4558i) q^{33} -29.3939i q^{35} +38.0000 q^{37} +(10.3923 + 14.6969i) q^{39} +5.65685i q^{41} -10.3923 q^{43} +(-24.0000 - 8.48528i) q^{45} -58.7878i q^{47} +59.0000 q^{49} +(55.4256 - 39.1918i) q^{51} +14.1421i q^{53} -41.5692 q^{55} +(-18.0000 - 25.4558i) q^{57} +14.6969i q^{59} +22.0000 q^{61} +(31.1769 - 88.1816i) q^{63} +16.9706i q^{65} +114.315 q^{67} +(-72.0000 + 50.9117i) q^{69} -29.3939i q^{71} -30.0000 q^{73} +(29.4449 + 41.6413i) q^{75} -152.735i q^{77} +31.1769 q^{79} +(-63.0000 - 50.9117i) q^{81} -73.4847i q^{83} +64.0000 q^{85} +(-76.2102 + 53.8888i) q^{87} +5.65685i q^{89} -62.3538 q^{91} +(54.0000 + 76.3675i) q^{93} -29.3939i q^{95} +90.0000 q^{97} +(-124.708 - 44.0908i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 24 q^{13} - 72 q^{21} + 68 q^{25} - 144 q^{33} + 152 q^{37} - 96 q^{45} + 236 q^{49} - 72 q^{57} + 88 q^{61} - 288 q^{69} - 120 q^{73} - 252 q^{81} + 256 q^{85} + 216 q^{93} + 360 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 + 24 * q^13 - 72 * q^21 + 68 * q^25 - 144 * q^33 + 152 * q^37 - 96 * q^45 + 236 * q^49 - 72 * q^57 + 88 * q^61 - 288 * q^69 - 120 * q^73 - 252 * q^81 + 256 * q^85 + 216 * q^93 + 360 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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