LMFDB - Embedded newform 96.3.e.b.65.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,3,Mod(65,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, 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("96.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	96.e (of order $2$, degree $1$, 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:	$2.61581053786$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.3
Root			$1.16372i$ of defining polynomial
Character	$\chi$	$=$	96.65
Dual form			96.3.e.b.65.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.64575 - 1.41421i) q^{3} -7.48331i q^{5} -5.29150 q^{7} +(5.00000 - 7.48331i) q^{9} +O(q^{10})$	$q+(2.64575 - 1.41421i) q^{3} -7.48331i q^{5} -5.29150 q^{7} +(5.00000 - 7.48331i) q^{9} +14.1421i q^{11} +10.0000 q^{13} +(-10.5830 - 19.7990i) q^{15} +26.4575 q^{19} +(-14.0000 + 7.48331i) q^{21} +16.9706i q^{23} -31.0000 q^{25} +(2.64575 - 26.8701i) q^{27} +37.4166i q^{29} -26.4575 q^{31} +(20.0000 + 37.4166i) q^{33} +39.5980i q^{35} +10.0000 q^{37} +(26.4575 - 14.1421i) q^{39} +14.9666i q^{41} -58.2065 q^{43} +(-56.0000 - 37.4166i) q^{45} +11.3137i q^{47} -21.0000 q^{49} -37.4166i q^{53} +105.830 q^{55} +(70.0000 - 37.4166i) q^{57} -98.9949i q^{59} +90.0000 q^{61} +(-26.4575 + 39.5980i) q^{63} -74.8331i q^{65} +5.29150 q^{67} +(24.0000 + 44.8999i) q^{69} +28.2843i q^{71} -30.0000 q^{73} +(-82.0183 + 43.8406i) q^{75} -74.8331i q^{77} -26.4575 q^{79} +(-31.0000 - 74.8331i) q^{81} -25.4558i q^{83} +(52.9150 + 98.9949i) q^{87} +74.8331i q^{89} -52.9150 q^{91} +(-70.0000 + 37.4166i) q^{93} -197.990i q^{95} +10.0000 q^{97} +(105.830 + 70.7107i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{9}+O(q^{10}) $ 4 * q + 20 * q^9	$ 4 q + 20 q^{9} + 40 q^{13} - 56 q^{21} - 124 q^{25} + 80 q^{33} + 40 q^{37} - 224 q^{45} - 84 q^{49} + 280 q^{57} + 360 q^{61} + 96 q^{69} - 120 q^{73} - 124 q^{81} - 280 q^{93} + 40 q^{97}+O(q^{100}) $ 4 * q + 20 * q^9 + 40 * q^13 - 56 * q^21 - 124 * q^25 + 80 * q^33 + 40 * q^37 - 224 * q^45 - 84 * q^49 + 280 * q^57 + 360 * q^61 + 96 * q^69 - 120 * q^73 - 124 * q^81 - 280 * q^93 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$65$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.64575	−	1.41421i	0.881917	−	0.471405i
$3$	2.64575	−	1.41421i	0.881917	−	0.471405i
$4$		0			0
$5$		−	7.48331i		−	1.49666i	−0.663325	−	0.748331i	$-0.730855\pi$
$5$		−	7.48331i		−	1.49666i	0.663325	−	0.748331i	$-0.269145\pi$
$6$		0			0
$7$	−5.29150			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−5.29150			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$		0			0
$9$	5.00000	−	7.48331i	0.555556	−	0.831479i
$10$		0			0
$11$			14.1421i			1.28565i	0.766014	+	0.642824i	$0.222237\pi$
$11$			14.1421i			1.28565i	−0.766014	+	0.642824i	$0.777763\pi$
$12$		0			0
$13$	10.0000			0.769231			0.384615	−	0.923077i	$-0.374334\pi$
$13$	10.0000			0.769231			0.384615	+	0.923077i	$0.374334\pi$
$14$		0			0
$15$	−10.5830	−	19.7990i	−0.705534	−	1.31993i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	26.4575			1.39250			0.696250	−	0.717799i	$-0.254850\pi$
$19$	26.4575			1.39250			0.696250	+	0.717799i	$0.254850\pi$
$20$		0			0
$21$	−14.0000	+	7.48331i	−0.666667	+	0.356348i
$22$		0			0
$23$			16.9706i			0.737851i	0.929459	+	0.368925i	$0.120274\pi$
$23$			16.9706i			0.737851i	−0.929459	+	0.368925i	$0.879726\pi$
$24$		0			0
$25$	−31.0000			−1.24000
$26$		0			0
$27$	2.64575	−	26.8701i	0.0979908	−	0.995187i
$28$		0			0
$29$			37.4166i			1.29023i	0.764087	+	0.645113i	$0.223190\pi$
$29$			37.4166i			1.29023i	−0.764087	+	0.645113i	$0.776810\pi$
$30$		0			0
$31$	−26.4575			−0.853468			−0.426734	−	0.904377i	$-0.640336\pi$
$31$	−26.4575			−0.853468			−0.426734	+	0.904377i	$0.640336\pi$
$32$		0			0
$33$	20.0000	+	37.4166i	0.606061	+	1.13384i
$34$		0			0
$35$			39.5980i			1.13137i
$36$		0			0
$37$	10.0000			0.270270			0.135135	−	0.990827i	$-0.456853\pi$
$37$	10.0000			0.270270			0.135135	+	0.990827i	$0.456853\pi$
$38$		0			0
$39$	26.4575	−	14.1421i	0.678398	−	0.362619i
$40$		0			0
$41$			14.9666i			0.365040i	0.983202	+	0.182520i	$0.0584254\pi$
$41$			14.9666i			0.365040i	−0.983202	+	0.182520i	$0.941575\pi$
$42$		0			0
$43$	−58.2065			−1.35364			−0.676820	−	0.736148i	$-0.736643\pi$
$43$	−58.2065			−1.35364			−0.676820	+	0.736148i	$0.736643\pi$
$44$		0			0
$45$	−56.0000	−	37.4166i	−1.24444	−	0.831479i
$46$		0			0
$47$			11.3137i			0.240717i	0.992730	+	0.120359i	$0.0384044\pi$
$47$			11.3137i			0.240717i	−0.992730	+	0.120359i	$0.961596\pi$
$48$		0			0
$49$	−21.0000			−0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	37.4166i		−	0.705973i	−0.935628	−	0.352987i	$-0.885166\pi$
$53$		−	37.4166i		−	0.705973i	0.935628	−	0.352987i	$-0.114834\pi$
$54$		0			0
$55$	105.830			1.92418
$56$		0			0
$57$	70.0000	−	37.4166i	1.22807	−	0.656431i
$58$		0			0
$59$		−	98.9949i		−	1.67788i	−0.544224	−	0.838940i	$-0.683176\pi$
$59$		−	98.9949i		−	1.67788i	0.544224	−	0.838940i	$-0.316824\pi$
$60$		0			0
$61$	90.0000			1.47541			0.737705	−	0.675123i	$-0.235910\pi$
$61$	90.0000			1.47541			0.737705	+	0.675123i	$0.235910\pi$
$62$		0			0
$63$	−26.4575	+	39.5980i	−0.419961	+	0.628539i
$64$		0			0
$65$		−	74.8331i		−	1.15128i
$66$		0			0
$67$	5.29150			0.0789777			0.0394888	−	0.999220i	$-0.487427\pi$
$67$	5.29150			0.0789777			0.0394888	+	0.999220i	$0.487427\pi$
$68$		0			0
$69$	24.0000	+	44.8999i	0.347826	+	0.650723i
$70$		0			0
$71$			28.2843i			0.398370i	0.979962	+	0.199185i	$0.0638295\pi$
$71$			28.2843i			0.398370i	−0.979962	+	0.199185i	$0.936171\pi$
$72$		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
$75$	−82.0183	+	43.8406i	−1.09358	+	0.584542i
$76$		0			0
$77$		−	74.8331i		−	0.971859i
$78$		0			0
$79$	−26.4575			−0.334905			−0.167453	−	0.985880i	$-0.553554\pi$
$79$	−26.4575			−0.334905			−0.167453	+	0.985880i	$0.553554\pi$
$80$		0			0
$81$	−31.0000	−	74.8331i	−0.382716	−	0.923866i
$82$		0			0
$83$		−	25.4558i		−	0.306697i	−0.988172	−	0.153348i	$-0.950994\pi$
$83$		−	25.4558i		−	0.306697i	0.988172	−	0.153348i	$-0.0490057\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	52.9150	+	98.9949i	0.608219	+	1.13787i
$88$		0			0
$89$			74.8331i			0.840822i	0.907334	+	0.420411i	$0.138114\pi$
$89$			74.8331i			0.840822i	−0.907334	+	0.420411i	$0.861886\pi$
$90$		0			0
$91$	−52.9150			−0.581484
$92$		0			0
$93$	−70.0000	+	37.4166i	−0.752688	+	0.402329i
$94$		0			0
$95$		−	197.990i		−	2.08410i
$96$		0			0
$97$	10.0000			0.103093			0.0515464	−	0.998671i	$-0.483585\pi$
$97$	10.0000			0.103093			0.0515464	+	0.998671i	$0.483585\pi$
$98$		0			0
$99$	105.830	+	70.7107i	1.06899	+	0.714249i
$100$		0			0
$101$			112.250i			1.11138i	0.831388	+	0.555692i	$0.187546\pi$
$101$			112.250i			1.11138i	−0.831388	+	0.555692i	$0.812454\pi$
$102$		0			0
$103$	−47.6235			−0.462364			−0.231182	−	0.972910i	$-0.574259\pi$
$103$	−47.6235			−0.462364			−0.231182	+	0.972910i	$0.574259\pi$
$104$		0			0
$105$	56.0000	+	104.766i	0.533333	+	0.997775i
$106$		0			0
$107$			2.82843i			0.0264339i	0.999913	+	0.0132169i	$0.00420721\pi$
$107$			2.82843i			0.0264339i	−0.999913	+	0.0132169i	$0.995793\pi$
$108$		0			0
$109$	−70.0000			−0.642202			−0.321101	−	0.947045i	$-0.604053\pi$
$109$	−70.0000			−0.642202			−0.321101	+	0.947045i	$0.604053\pi$
$110$		0			0
$111$	26.4575	−	14.1421i	0.238356	−	0.127407i
$112$		0			0
$113$			149.666i			1.32448i	0.749292	+	0.662240i	$0.230394\pi$
$113$			149.666i			1.32448i	−0.749292	+	0.662240i	$0.769606\pi$
$114$		0			0
$115$	126.996			1.10431
$116$		0			0
$117$	50.0000	−	74.8331i	0.427350	−	0.639600i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−79.0000			−0.652893
$122$		0			0
$123$	21.1660	+	39.5980i	0.172081	+	0.321935i
$124$		0			0
$125$			44.8999i			0.359199i
$126$		0			0
$127$	−153.454			−1.20830			−0.604148	−	0.796872i	$-0.706486\pi$
$127$	−153.454			−1.20830			−0.604148	+	0.796872i	$0.706486\pi$
$128$		0			0
$129$	−154.000	+	82.3165i	−1.19380	+	0.638112i
$130$		0			0
$131$		−	70.7107i		−	0.539776i	−0.962892	−	0.269888i	$-0.913013\pi$
$131$		−	70.7107i		−	0.539776i	0.962892	−	0.269888i	$-0.0869867\pi$
$132$		0			0
$133$	−140.000			−1.05263
$134$		0			0
$135$	−201.077	−	19.7990i	−1.48946	−	0.146659i
$136$		0			0
$137$		−	224.499i		−	1.63868i	−0.573306	−	0.819341i	$-0.694340\pi$
$137$		−	224.499i		−	1.63868i	0.573306	−	0.819341i	$-0.305660\pi$
$138$		0			0
$139$	132.288			0.951709			0.475855	−	0.879524i	$-0.342139\pi$
$139$	132.288			0.951709			0.475855	+	0.879524i	$0.342139\pi$
$140$		0			0
$141$	16.0000	+	29.9333i	0.113475	+	0.212293i
$142$		0			0
$143$			141.421i			0.988961i
$144$		0			0
$145$	280.000			1.93103
$146$		0			0
$147$	−55.5608	+	29.6985i	−0.377964	+	0.202031i
$148$		0			0
$149$			22.4499i			0.150671i	0.997158	+	0.0753354i	$0.0240027\pi$
$149$			22.4499i			0.150671i	−0.997158	+	0.0753354i	$0.975997\pi$
$150$		0			0
$151$	79.3725			0.525646			0.262823	−	0.964844i	$-0.415346\pi$
$151$	79.3725			0.525646			0.262823	+	0.964844i	$0.415346\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			197.990i			1.27735i
$156$		0			0
$157$	170.000			1.08280			0.541401	−	0.840764i	$-0.317894\pi$
$157$	170.000			1.08280			0.541401	+	0.840764i	$0.317894\pi$
$158$		0			0
$159$	−52.9150	−	98.9949i	−0.332799	−	0.622610i
$160$		0			0
$161$		−	89.7998i		−	0.557763i
$162$		0			0
$163$	−100.539			−0.616801			−0.308400	−	0.951257i	$-0.599794\pi$
$163$	−100.539			−0.616801			−0.308400	+	0.951257i	$0.599794\pi$
$164$		0			0
$165$	280.000	−	149.666i	1.69697	−	0.907068i
$166$		0			0
$167$		−	243.245i		−	1.45656i	−0.685282	−	0.728278i	$-0.740321\pi$
$167$		−	243.245i		−	1.45656i	0.685282	−	0.728278i	$-0.259679\pi$
$168$		0			0
$169$	−69.0000			−0.408284
$170$		0			0
$171$	132.288	−	197.990i	0.773611	−	1.15784i
$172$		0			0
$173$		−	112.250i		−	0.648842i	−0.945913	−	0.324421i	$-0.894831\pi$
$173$		−	112.250i		−	0.648842i	0.945913	−	0.324421i	$-0.105169\pi$
$174$		0			0
$175$	164.037			0.937352
$176$		0			0
$177$	−140.000	−	261.916i	−0.790960	−	1.47975i
$178$		0			0
$179$			212.132i			1.18510i	0.805535	+	0.592548i	$0.201878\pi$
$179$			212.132i			1.18510i	−0.805535	+	0.592548i	$0.798122\pi$
$180$		0			0
$181$	−262.000			−1.44751			−0.723757	−	0.690055i	$-0.757586\pi$
$181$	−262.000			−1.44751			−0.723757	+	0.690055i	$0.757586\pi$
$182$		0			0
$183$	238.118	−	127.279i	1.30119	−	0.695515i
$184$		0			0
$185$		−	74.8331i		−	0.404504i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−14.0000	+	142.183i	−0.0740741	+	0.752291i
$190$		0			0
$191$			339.411i			1.77702i	0.458855	+	0.888511i	$0.348260\pi$
$191$			339.411i			1.77702i	−0.458855	+	0.888511i	$0.651740\pi$
$192$		0			0
$193$	210.000			1.08808			0.544041	−	0.839058i	$-0.316893\pi$
$193$	210.000			1.08808			0.544041	+	0.839058i	$0.316893\pi$
$194$		0			0
$195$	−105.830	−	197.990i	−0.542718	−	1.01533i
$196$		0			0
$197$		−	187.083i		−	0.949659i	−0.880078	−	0.474830i	$-0.842510\pi$
$197$		−	187.083i		−	0.949659i	0.880078	−	0.474830i	$-0.157490\pi$
$198$		0			0
$199$	−343.948			−1.72838			−0.864190	−	0.503165i	$-0.832169\pi$
$199$	−343.948			−1.72838			−0.864190	+	0.503165i	$0.832169\pi$
$200$		0			0
$201$	14.0000	−	7.48331i	0.0696517	−	0.0372304i
$202$		0			0
$203$		−	197.990i		−	0.975320i
$204$		0			0
$205$	112.000			0.546341
$206$		0			0
$207$	126.996	+	84.8528i	0.613508	+	0.409917i
$208$		0			0
$209$			374.166i			1.79027i
$210$		0			0
$211$	−291.033			−1.37930			−0.689651	−	0.724142i	$-0.742236\pi$
$211$	−291.033			−1.37930			−0.689651	+	0.724142i	$0.742236\pi$
$212$		0			0
$213$	40.0000	+	74.8331i	0.187793	+	0.351329i
$214$		0			0
$215$			435.578i			2.02594i
$216$		0			0
$217$	140.000			0.645161
$218$		0			0
$219$	−79.3725	+	42.4264i	−0.362432	+	0.193728i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	312.199			1.39999			0.699997	−	0.714146i	$-0.253185\pi$
$223$	312.199			1.39999			0.699997	+	0.714146i	$0.253185\pi$
$224$		0			0
$225$	−155.000	+	231.983i	−0.688889	+	1.03103i
$226$		0			0
$227$		−	59.3970i		−	0.261661i	−0.991405	−	0.130830i	$-0.958236\pi$
$227$		−	59.3970i		−	0.261661i	0.991405	−	0.130830i	$-0.0417643\pi$
$228$		0			0
$229$	−358.000			−1.56332			−0.781659	−	0.623706i	$-0.785626\pi$
$229$	−358.000			−1.56332			−0.781659	+	0.623706i	$0.785626\pi$
$230$		0			0
$231$	−105.830	−	197.990i	−0.458139	−	0.857099i
$232$		0			0
$233$			224.499i			0.963517i	0.876304	+	0.481758i	$0.160002\pi$
$233$			224.499i			0.963517i	−0.876304	+	0.481758i	$0.839998\pi$
$234$		0			0
$235$	84.6640			0.360273
$236$		0			0
$237$	−70.0000	+	37.4166i	−0.295359	+	0.157876i
$238$		0			0
$239$		−	169.706i		−	0.710065i	−0.934854	−	0.355033i	$-0.884470\pi$
$239$		−	169.706i		−	0.710065i	0.934854	−	0.355033i	$-0.115530\pi$
$240$		0			0
$241$	−110.000			−0.456432			−0.228216	−	0.973611i	$-0.573289\pi$
$241$	−110.000			−0.456432			−0.228216	+	0.973611i	$0.573289\pi$
$242$		0			0
$243$	−187.848	−	154.149i	−0.773038	−	0.634359i
$244$		0			0
$245$			157.150i			0.641427i
$246$		0			0
$247$	264.575			1.07115
$248$		0			0
$249$	−36.0000	−	67.3498i	−0.144578	−	0.270481i
$250$		0			0
$251$			296.985i			1.18321i	0.806229	+	0.591603i	$0.201505\pi$
$251$			296.985i			1.18321i	−0.806229	+	0.591603i	$0.798495\pi$
$252$		0			0
$253$	−240.000			−0.948617
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	299.333i		−	1.16472i	−0.812932	−	0.582359i	$-0.802130\pi$
$257$		−	299.333i		−	1.16472i	0.812932	−	0.582359i	$-0.197870\pi$
$258$		0			0
$259$	−52.9150			−0.204305
$260$		0			0
$261$	280.000	+	187.083i	1.07280	+	0.716793i
$262$		0			0
$263$		−	152.735i		−	0.580742i	−0.956914	−	0.290371i	$-0.906221\pi$
$263$		−	152.735i		−	0.580742i	0.956914	−	0.290371i	$-0.0937787\pi$
$264$		0			0
$265$	−280.000			−1.05660
$266$		0			0
$267$	105.830	+	197.990i	0.396367	+	0.741535i
$268$		0			0
$269$			426.549i			1.58568i	0.609427	+	0.792842i	$0.291399\pi$
$269$			426.549i			1.58568i	−0.609427	+	0.792842i	$0.708601\pi$
$270$		0			0
$271$	396.863			1.46444			0.732219	−	0.681069i	$-0.238485\pi$
$271$	396.863			1.46444			0.732219	+	0.681069i	$0.238485\pi$
$272$		0			0
$273$	−140.000	+	74.8331i	−0.512821	+	0.274114i
$274$		0			0
$275$		−	438.406i		−	1.59420i
$276$		0			0
$277$	490.000			1.76895			0.884477	−	0.466585i	$-0.154516\pi$
$277$	490.000			1.76895			0.884477	+	0.466585i	$0.154516\pi$
$278$		0			0
$279$	−132.288	+	197.990i	−0.474149	+	0.709641i
$280$		0			0
$281$			134.700i			0.479358i	0.970852	+	0.239679i	$0.0770422\pi$
$281$			134.700i			0.479358i	−0.970852	+	0.239679i	$0.922958\pi$
$282$		0			0
$283$	5.29150			0.0186979			0.00934894	−	0.999956i	$-0.497024\pi$
$283$	5.29150			0.0186979			0.00934894	+	0.999956i	$0.497024\pi$
$284$		0			0
$285$	−280.000	−	523.832i	−0.982456	−	1.83801i
$286$		0			0
$287$		−	79.1960i		−	0.275944i
$288$		0			0
$289$	289.000			1.00000
$290$		0			0
$291$	26.4575	−	14.1421i	0.0909193	−	0.0485984i
$292$		0			0
$293$			112.250i			0.383105i	0.981482	+	0.191552i	$0.0613522\pi$
$293$			112.250i			0.383105i	−0.981482	+	0.191552i	$0.938648\pi$
$294$		0			0
$295$	−740.810			−2.51122
$296$		0			0
$297$	380.000	+	37.4166i	1.27946	+	0.125982i
$298$		0			0
$299$			169.706i			0.567577i
$300$		0			0
$301$	308.000			1.02326
$302$		0			0
$303$	158.745	+	296.985i	0.523911	+	0.980148i
$304$		0			0
$305$		−	673.498i		−	2.20819i
$306$		0			0
$307$	47.6235			0.155125			0.0775627	−	0.996987i	$-0.475286\pi$
$307$	47.6235			0.155125			0.0775627	+	0.996987i	$0.475286\pi$
$308$		0			0
$309$	−126.000	+	67.3498i	−0.407767	+	0.217961i
$310$		0			0
$311$		−	254.558i		−	0.818516i	−0.912419	−	0.409258i	$-0.865788\pi$
$311$		−	254.558i		−	0.818516i	0.912419	−	0.409258i	$-0.134212\pi$
$312$		0			0
$313$	−230.000			−0.734824			−0.367412	−	0.930058i	$-0.619756\pi$
$313$	−230.000			−0.734824			−0.367412	+	0.930058i	$0.619756\pi$
$314$		0			0
$315$	296.324	+	197.990i	0.940712	+	0.628539i
$316$		0			0
$317$		−	561.249i		−	1.77050i	−0.465115	−	0.885250i	$-0.653987\pi$
$317$		−	561.249i		−	1.77050i	0.465115	−	0.885250i	$-0.346013\pi$
$318$		0			0
$319$	−529.150			−1.65878
$320$		0			0
$321$	4.00000	+	7.48331i	0.0124611	+	0.0233125i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−310.000			−0.953846
$326$		0			0
$327$	−185.203	+	98.9949i	−0.566369	+	0.302737i
$328$		0			0
$329$		−	59.8665i		−	0.181965i
$330$		0			0
$331$	132.288			0.399660			0.199830	−	0.979831i	$-0.435961\pi$
$331$	132.288			0.399660			0.199830	+	0.979831i	$0.435961\pi$
$332$		0			0
$333$	50.0000	−	74.8331i	0.150150	−	0.224724i
$334$		0			0
$335$		−	39.5980i		−	0.118203i
$336$		0			0
$337$	−70.0000			−0.207715			−0.103858	−	0.994592i	$-0.533119\pi$
$337$	−70.0000			−0.207715			−0.103858	+	0.994592i	$0.533119\pi$
$338$		0			0
$339$	211.660	+	395.980i	0.624366	+	1.16808i
$340$		0			0
$341$		−	374.166i		−	1.09726i
$342$		0			0
$343$	370.405			1.07990
$344$		0			0
$345$	336.000	−	179.600i	0.973913	−	0.520578i
$346$		0			0
$347$		−	540.230i		−	1.55686i	−0.627733	−	0.778429i	$-0.716017\pi$
$347$		−	540.230i		−	1.55686i	0.627733	−	0.778429i	$-0.283983\pi$
$348$		0			0
$349$	298.000			0.853868			0.426934	−	0.904283i	$-0.359594\pi$
$349$	298.000			0.853868			0.426934	+	0.904283i	$0.359594\pi$
$350$		0			0
$351$	26.4575	−	268.701i	0.0753775	−	0.765529i
$352$		0			0
$353$			299.333i			0.847968i	0.905670	+	0.423984i	$0.139369\pi$
$353$			299.333i			0.847968i	−0.905670	+	0.423984i	$0.860631\pi$
$354$		0			0
$355$	211.660			0.596226
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	537.401i		−	1.49694i	−0.663169	−	0.748470i	$-0.730789\pi$
$359$		−	537.401i		−	1.49694i	0.663169	−	0.748470i	$-0.269211\pi$
$360$		0			0
$361$	339.000			0.939058
$362$		0			0
$363$	−209.014	+	111.723i	−0.575797	+	0.307777i
$364$		0			0
$365$			224.499i			0.615067i
$366$		0			0
$367$	−111.122			−0.302784			−0.151392	−	0.988474i	$-0.548375\pi$
$367$	−111.122			−0.302784			−0.151392	+	0.988474i	$0.548375\pi$
$368$		0			0
$369$	112.000	+	74.8331i	0.303523	+	0.202800i
$370$		0			0
$371$			197.990i			0.533665i
$372$		0			0
$373$	410.000			1.09920			0.549598	−	0.835429i	$-0.314781\pi$
$373$	410.000			1.09920			0.549598	+	0.835429i	$0.314781\pi$
$374$		0			0
$375$	63.4980	+	118.794i	0.169328	+	0.316784i
$376$		0			0
$377$			374.166i			0.992482i
$378$		0			0
$379$	661.438			1.74522			0.872609	−	0.488419i	$-0.162426\pi$
$379$	661.438			1.74522			0.872609	+	0.488419i	$0.162426\pi$
$380$		0			0
$381$	−406.000	+	217.016i	−1.06562	+	0.569596i
$382$		0			0
$383$		−	158.392i		−	0.413556i	−0.978388	−	0.206778i	$-0.933702\pi$
$383$		−	158.392i		−	0.413556i	0.978388	−	0.206778i	$-0.0662978\pi$
$384$		0			0
$385$	−560.000			−1.45455
$386$		0			0
$387$	−291.033	+	435.578i	−0.752022	+	1.12552i
$388$		0			0
$389$		−	246.949i		−	0.634831i	−0.948287	−	0.317416i	$-0.897185\pi$
$389$		−	246.949i		−	0.634831i	0.948287	−	0.317416i	$-0.102815\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−100.000	−	187.083i	−0.254453	−	0.476038i
$394$		0			0
$395$			197.990i			0.501240i
$396$		0			0
$397$	−230.000			−0.579345			−0.289673	−	0.957126i	$-0.593546\pi$
$397$	−230.000			−0.579345			−0.289673	+	0.957126i	$0.593546\pi$
$398$		0			0
$399$	−370.405	+	197.990i	−0.928334	+	0.496215i
$400$		0			0
$401$			299.333i			0.746465i	0.927738	+	0.373233i	$0.121751\pi$
$401$			299.333i			0.746465i	−0.927738	+	0.373233i	$0.878249\pi$
$402$		0			0
$403$	−264.575			−0.656514
$404$		0			0
$405$	−560.000	+	231.983i	−1.38272	+	0.572797i
$406$		0			0
$407$			141.421i			0.347473i
$408$		0			0
$409$	−590.000			−1.44254			−0.721271	−	0.692653i	$-0.756442\pi$
$409$	−590.000			−1.44254			−0.721271	+	0.692653i	$0.756442\pi$
$410$		0			0
$411$	−317.490	−	593.970i	−0.772482	−	1.44518i
$412$		0			0
$413$			523.832i			1.26836i
$414$		0			0
$415$	−190.494			−0.459022
$416$		0			0
$417$	350.000	−	187.083i	0.839329	−	0.448640i
$418$		0			0
$419$			98.9949i			0.236265i	0.992998	+	0.118132i	$0.0376907\pi$
$419$			98.9949i			0.236265i	−0.992998	+	0.118132i	$0.962309\pi$
$420$		0			0
$421$	−70.0000			−0.166271			−0.0831354	−	0.996538i	$-0.526493\pi$
$421$	−70.0000			−0.166271			−0.0831354	+	0.996538i	$0.526493\pi$
$422$		0			0
$423$	84.6640	+	56.5685i	0.200151	+	0.133732i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−476.235			−1.11531
$428$		0			0
$429$	200.000	+	374.166i	0.466200	+	0.872181i
$430$		0			0
$431$			509.117i			1.18125i	0.806948	+	0.590623i	$0.201118\pi$
$431$			509.117i			1.18125i	−0.806948	+	0.590623i	$0.798882\pi$
$432$		0			0
$433$	10.0000			0.0230947			0.0115473	−	0.999933i	$-0.496324\pi$
$433$	10.0000			0.0230947			0.0115473	+	0.999933i	$0.496324\pi$
$434$		0			0
$435$	740.810	−	395.980i	1.70301	−	0.910298i
$436$		0			0
$437$			448.999i			1.02746i
$438$		0			0
$439$	502.693			1.14509			0.572543	−	0.819875i	$-0.305957\pi$
$439$	502.693			1.14509			0.572543	+	0.819875i	$0.305957\pi$
$440$		0			0
$441$	−105.000	+	157.150i	−0.238095	+	0.356348i
$442$		0			0
$443$		−	313.955i		−	0.708703i	−0.935112	−	0.354351i	$-0.884702\pi$
$443$		−	313.955i		−	0.708703i	0.935112	−	0.354351i	$-0.115298\pi$
$444$		0			0
$445$	560.000			1.25843
$446$		0			0
$447$	31.7490	+	59.3970i	0.0710269	+	0.132879i
$448$		0			0
$449$			209.533i			0.466666i	0.972397	+	0.233333i	$0.0749631\pi$
$449$			209.533i			0.466666i	−0.972397	+	0.233333i	$0.925037\pi$
$450$		0			0
$451$	−211.660			−0.469313
$452$		0			0
$453$	210.000	−	112.250i	0.463576	−	0.247792i
$454$		0			0
$455$			395.980i			0.870285i
$456$		0			0
$457$	−430.000			−0.940919			−0.470460	−	0.882422i	$-0.655912\pi$
$457$	−430.000			−0.940919			−0.470460	+	0.882422i	$0.655912\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	112.250i		−	0.243492i	−0.992561	−	0.121746i	$-0.961151\pi$
$461$		−	112.250i		−	0.243492i	0.992561	−	0.121746i	$-0.0388493\pi$
$462$		0			0
$463$	100.539			0.217146			0.108573	−	0.994088i	$-0.465372\pi$
$463$	100.539			0.217146			0.108573	+	0.994088i	$0.465372\pi$
$464$		0			0
$465$	280.000	+	523.832i	0.602151	+	1.12652i
$466$		0			0
$467$		−	138.593i		−	0.296773i	−0.988929	−	0.148386i	$-0.952592\pi$
$467$		−	138.593i		−	0.296773i	0.988929	−	0.148386i	$-0.0474079\pi$
$468$		0			0
$469$	−28.0000			−0.0597015
$470$		0			0
$471$	449.778	−	240.416i	0.954942	−	0.510438i
$472$		0			0
$473$		−	823.165i		−	1.74031i
$474$		0			0
$475$	−820.183			−1.72670
$476$		0			0
$477$	−280.000	−	187.083i	−0.587002	−	0.392207i
$478$		0			0
$479$		−	339.411i		−	0.708583i	−0.935135	−	0.354291i	$-0.884722\pi$
$479$		−	339.411i		−	0.708583i	0.935135	−	0.354291i	$-0.115278\pi$
$480$		0			0
$481$	100.000			0.207900
$482$		0			0
$483$	−126.996	−	237.588i	−0.262932	−	0.491900i
$484$		0			0
$485$		−	74.8331i		−	0.154295i
$486$		0			0
$487$	375.697			0.771451			0.385726	−	0.922614i	$-0.373951\pi$
$487$	375.697			0.771451			0.385726	+	0.922614i	$0.373951\pi$
$488$		0			0
$489$	−266.000	+	142.183i	−0.543967	+	0.290763i
$490$		0			0
$491$			523.259i			1.06570i	0.846210	+	0.532850i	$0.178879\pi$
$491$			523.259i			1.06570i	−0.846210	+	0.532850i	$0.821121\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	529.150	−	791.960i	1.06899	−	1.59992i
$496$		0			0
$497$		−	149.666i		−	0.301139i
$498$		0			0
$499$	555.608			1.11344			0.556721	−	0.830699i	$-0.312059\pi$
$499$	555.608			1.11344			0.556721	+	0.830699i	$0.312059\pi$
$500$		0			0
$501$	−344.000	−	643.565i	−0.686627	−	1.28456i
$502$		0			0
$503$			831.558i			1.65320i	0.562793	+	0.826598i	$0.309727\pi$
$503$			831.558i			1.65320i	−0.562793	+	0.826598i	$0.690273\pi$
$504$		0			0
$505$	840.000			1.66337
$506$		0			0
$507$	−182.557	+	97.5807i	−0.360073	+	0.192467i
$508$		0			0
$509$		−	561.249i		−	1.10265i	−0.834291	−	0.551325i	$-0.814123\pi$
$509$		−	561.249i		−	1.10265i	0.834291	−	0.551325i	$-0.185877\pi$
$510$		0			0
$511$	158.745			0.310656
$512$		0			0
$513$	70.0000	−	710.915i	0.136452	−	1.38580i
$514$		0			0
$515$			356.382i			0.692004i
$516$		0			0
$517$	−160.000			−0.309478
$518$		0			0
$519$	−158.745	−	296.985i	−0.305867	−	0.572225i
$520$		0			0
$521$		−	224.499i		−	0.430901i	−0.976515	−	0.215451i	$-0.930878\pi$
$521$		−	224.499i		−	0.430901i	0.976515	−	0.215451i	$-0.0691220\pi$
$522$		0			0
$523$	−904.847			−1.73011			−0.865054	−	0.501678i	$-0.832716\pi$
$523$	−904.847			−1.73011			−0.865054	+	0.501678i	$0.832716\pi$
$524$		0			0
$525$	434.000	−	231.983i	0.826667	−	0.441872i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	241.000			0.455577
$530$		0			0
$531$	−740.810	−	494.975i	−1.39512	−	0.932156i
$532$		0			0
$533$			149.666i			0.280800i
$534$		0			0
$535$	21.1660			0.0395626
$536$		0			0
$537$	300.000	+	561.249i	0.558659	+	1.04516i
$538$		0			0
$539$		−	296.985i		−	0.550992i
$540$		0			0
$541$	−518.000			−0.957486			−0.478743	−	0.877955i	$-0.658907\pi$
$541$	−518.000			−0.957486			−0.478743	+	0.877955i	$0.658907\pi$
$542$		0			0
$543$	−693.187	+	370.524i	−1.27659	+	0.682365i
$544$		0			0
$545$			523.832i			0.961160i
$546$		0			0
$547$	1000.09			1.82833			0.914163	−	0.405347i	$-0.132849\pi$
$547$	1000.09			1.82833			0.914163	+	0.405347i	$0.132849\pi$
$548$		0			0
$549$	450.000	−	673.498i	0.819672	−	1.22677i
$550$		0			0
$551$			989.949i			1.79664i
$552$		0			0
$553$	140.000			0.253165
$554$		0			0
$555$	−105.830	−	197.990i	−0.190685	−	0.356739i
$556$		0			0
$557$		−	411.582i		−	0.738927i	−0.929245	−	0.369463i	$-0.879541\pi$
$557$		−	411.582i		−	0.738927i	0.929245	−	0.369463i	$-0.120459\pi$
$558$		0			0
$559$	−582.065			−1.04126
$560$		0			0
$561$		0			0
$562$		0			0
$563$			653.367i			1.16051i	0.814435	+	0.580255i	$0.197047\pi$
$563$			653.367i			1.16051i	−0.814435	+	0.580255i	$0.802953\pi$
$564$		0			0
$565$	1120.00			1.98230
$566$		0			0
$567$	164.037	+	395.980i	0.289306	+	0.698377i
$568$		0			0
$569$		−	613.632i		−	1.07844i	−0.842165	−	0.539220i	$-0.818719\pi$
$569$		−	613.632i		−	1.07844i	0.842165	−	0.539220i	$-0.181281\pi$
$570$		0			0
$571$	343.948			0.602360			0.301180	−	0.953567i	$-0.402619\pi$
$571$	343.948			0.602360			0.301180	+	0.953567i	$0.402619\pi$
$572$		0			0
$573$	480.000	+	897.998i	0.837696	+	1.56719i
$574$		0			0
$575$		−	526.087i		−	0.914935i
$576$		0			0
$577$	−390.000			−0.675910			−0.337955	−	0.941162i	$-0.609735\pi$
$577$	−390.000			−0.675910			−0.337955	+	0.941162i	$0.609735\pi$
$578$		0			0
$579$	555.608	−	296.985i	0.959599	−	0.512927i
$580$		0			0
$581$			134.700i			0.231841i
$582$		0			0
$583$	529.150			0.907633
$584$		0			0
$585$	−560.000	−	374.166i	−0.957265	−	0.639600i
$586$		0			0
$587$		−	653.367i		−	1.11306i	−0.830827	−	0.556530i	$-0.812132\pi$
$587$		−	653.367i		−	1.11306i	0.830827	−	0.556530i	$-0.187868\pi$
$588$		0			0
$589$	−700.000			−1.18846
$590$		0			0
$591$	−264.575	−	494.975i	−0.447674	−	0.837521i
$592$		0			0
$593$		−	448.999i		−	0.757165i	−0.925568	−	0.378583i	$-0.876412\pi$
$593$		−	448.999i		−	0.757165i	0.925568	−	0.378583i	$-0.123588\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−910.000	+	486.415i	−1.52429	+	0.814766i
$598$		0			0
$599$		−	480.833i		−	0.802726i	−0.915919	−	0.401363i	$-0.868537\pi$
$599$		−	480.833i		−	0.802726i	0.915919	−	0.401363i	$-0.131463\pi$
$600$		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
$603$	26.4575	−	39.5980i	0.0438765	−	0.0656683i
$604$		0			0
$605$			591.182i			0.977160i
$606$		0			0
$607$	−153.454			−0.252807			−0.126403	−	0.991979i	$-0.540343\pi$
$607$	−153.454			−0.252807			−0.126403	+	0.991979i	$0.540343\pi$
$608$		0			0
$609$	−280.000	−	523.832i	−0.459770	−	0.860151i
$610$		0			0
$611$			113.137i			0.185167i
$612$		0			0
$613$	−630.000			−1.02773			−0.513866	−	0.857870i	$-0.671787\pi$
$613$	−630.000			−1.02773			−0.513866	+	0.857870i	$0.671787\pi$
$614$		0			0
$615$	296.324	−	158.392i	0.481828	−	0.257548i
$616$		0			0
$617$			1122.50i			1.81928i	0.415395	+	0.909641i	$0.363643\pi$
$617$			1122.50i			1.81928i	−0.415395	+	0.909641i	$0.636357\pi$
$618$		0			0
$619$	555.608			0.897589			0.448795	−	0.893635i	$-0.351853\pi$
$619$	555.608			0.897589			0.448795	+	0.893635i	$0.351853\pi$
$620$		0			0
$621$	456.000	+	44.8999i	0.734300	+	0.0723026i
$622$		0			0
$623$		−	395.980i		−	0.635602i
$624$		0			0
$625$	−439.000			−0.702400
$626$		0			0
$627$	529.150	+	989.949i	0.843940	+	1.57887i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−978.928			−1.55139			−0.775696	−	0.631107i	$-0.782601\pi$
$631$	−978.928			−1.55139			−0.775696	+	0.631107i	$0.782601\pi$
$632$		0			0
$633$	−770.000	+	411.582i	−1.21643	+	0.650209i
$634$		0			0
$635$			1148.34i			1.80841i
$636$		0			0
$637$	−210.000			−0.329670
$638$		0			0
$639$	211.660	+	141.421i	0.331236	+	0.221317i
$640$		0			0
$641$			389.132i			0.607071i	0.952820	+	0.303535i	$0.0981671\pi$
$641$			389.132i			0.607071i	−0.952820	+	0.303535i	$0.901833\pi$
$642$		0			0
$643$	−269.867			−0.419699			−0.209850	−	0.977734i	$-0.567297\pi$
$643$	−269.867			−0.419699			−0.209850	+	0.977734i	$0.567297\pi$
$644$		0			0
$645$	616.000	+	1152.43i	0.955039	+	1.78671i
$646$		0			0
$647$		−	1170.97i		−	1.80984i	−0.425578	−	0.904922i	$-0.639929\pi$
$647$		−	1170.97i		−	1.80984i	0.425578	−	0.904922i	$-0.360071\pi$
$648$		0			0
$649$	1400.00			2.15716
$650$		0			0
$651$	370.405	−	197.990i	0.568979	−	0.304132i
$652$		0			0
$653$		−	710.915i		−	1.08869i	−0.838861	−	0.544345i	$-0.816778\pi$
$653$		−	710.915i		−	1.08869i	0.838861	−	0.544345i	$-0.183222\pi$
$654$		0			0
$655$	−529.150			−0.807863
$656$		0			0
$657$	−150.000	+	224.499i	−0.228311	+	0.341704i
$658$		0			0
$659$		−	127.279i		−	0.193140i	−0.995326	−	0.0965700i	$-0.969213\pi$
$659$		−	127.279i		−	0.193140i	0.995326	−	0.0965700i	$-0.0307872\pi$
$660$		0			0
$661$	170.000			0.257186			0.128593	−	0.991697i	$-0.458954\pi$
$661$	170.000			0.257186			0.128593	+	0.991697i	$0.458954\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			1047.66i			1.57543i
$666$		0			0
$667$	−634.980			−0.951994
$668$		0			0
$669$	826.000	−	441.516i	1.23468	−	0.659963i
$670$		0			0
$671$			1272.79i			1.89686i
$672$		0			0
$673$	−310.000			−0.460624			−0.230312	−	0.973117i	$-0.573975\pi$
$673$	−310.000			−0.460624			−0.230312	+	0.973117i	$0.573975\pi$
$674$		0			0
$675$	−82.0183	+	832.972i	−0.121509	+	1.23403i
$676$		0			0
$677$			1309.58i			1.93439i	0.254041	+	0.967194i	$0.418240\pi$
$677$			1309.58i			1.93439i	−0.254041	+	0.967194i	$0.581760\pi$
$678$		0			0
$679$	−52.9150			−0.0779308
$680$		0			0
$681$	−84.0000	−	157.150i	−0.123348	−	0.230763i
$682$		0			0
$683$		−	1162.48i		−	1.70203i	−0.525145	−	0.851013i	$-0.675989\pi$
$683$		−	1162.48i		−	1.70203i	0.525145	−	0.851013i	$-0.324011\pi$
$684$		0			0
$685$	−1680.00			−2.45255
$686$		0			0
$687$	−947.179	+	506.288i	−1.37872	+	0.736956i
$688$		0			0
$689$		−	374.166i		−	0.543056i
$690$		0			0
$691$	26.4575			0.0382887			0.0191444	−	0.999817i	$-0.493906\pi$
$691$	26.4575			0.0382887			0.0191444	+	0.999817i	$0.493906\pi$
$692$		0			0
$693$	−560.000	−	374.166i	−0.808081	−	0.539922i
$694$		0			0
$695$		−	989.949i		−	1.42439i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	317.490	+	593.970i	0.454206	+	0.849742i
$700$		0			0
$701$		−	22.4499i		−	0.0320256i	−0.999872	−	0.0160128i	$-0.994903\pi$
$701$		−	22.4499i		−	0.0320256i	0.999872	−	0.0160128i	$-0.00509725\pi$
$702$		0			0
$703$	264.575			0.376352
$704$		0			0
$705$	224.000	−	119.733i	0.317730	−	0.169834i
$706$		0			0
$707$		−	593.970i		−	0.840127i
$708$		0			0
$709$	−182.000			−0.256700			−0.128350	−	0.991729i	$-0.540968\pi$
$709$	−182.000			−0.256700			−0.128350	+	0.991729i	$0.540968\pi$
$710$		0			0
$711$	−132.288	+	197.990i	−0.186058	+	0.278467i
$712$		0			0
$713$		−	448.999i		−	0.629732i
$714$		0			0
$715$	1058.30			1.48014
$716$		0			0
$717$	−240.000	−	448.999i	−0.334728	−	0.626219i
$718$		0			0
$719$		−	395.980i		−	0.550737i	−0.961339	−	0.275368i	$-0.911200\pi$
$719$		−	395.980i		−	0.550737i	0.961339	−	0.275368i	$-0.0887998\pi$
$720$		0			0
$721$	252.000			0.349515
$722$		0			0
$723$	−291.033	+	155.563i	−0.402535	+	0.215164i
$724$		0			0
$725$		−	1159.91i		−	1.59988i
$726$		0			0
$727$	−47.6235			−0.0655069			−0.0327535	−	0.999463i	$-0.510428\pi$
$727$	−47.6235			−0.0655069			−0.0327535	+	0.999463i	$0.510428\pi$
$728$		0			0
$729$	−715.000	−	142.183i	−0.980796	−	0.195038i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	650.000			0.886767			0.443383	−	0.896332i	$-0.353778\pi$
$733$	650.000			0.886767			0.443383	+	0.896332i	$0.353778\pi$
$734$		0			0
$735$	222.243	+	415.779i	0.302372	+	0.565685i
$736$		0			0
$737$			74.8331i			0.101538i
$738$		0			0
$739$	343.948			0.465423			0.232712	−	0.972546i	$-0.425240\pi$
$739$	343.948			0.465423			0.232712	+	0.972546i	$0.425240\pi$
$740$		0			0
$741$	700.000	−	374.166i	0.944669	−	0.504947i
$742$		0			0
$743$			661.852i			0.890783i	0.895336	+	0.445392i	$0.146936\pi$
$743$			661.852i			0.890783i	−0.895336	+	0.445392i	$0.853064\pi$
$744$		0			0
$745$	168.000			0.225503
$746$		0			0
$747$	−190.494	−	127.279i	−0.255012	−	0.170387i
$748$		0			0
$749$		−	14.9666i		−	0.0199821i
$750$		0			0
$751$	608.523			0.810283			0.405142	−	0.914254i	$-0.367222\pi$
$751$	608.523			0.810283			0.405142	+	0.914254i	$0.367222\pi$
$752$		0			0
$753$	420.000	+	785.748i	0.557769	+	1.04349i
$754$		0			0
$755$		−	593.970i		−	0.786715i
$756$		0			0
$757$	1050.00			1.38705			0.693527	−	0.720431i	$-0.256056\pi$
$757$	1050.00			1.38705			0.693527	+	0.720431i	$0.256056\pi$
$758$		0			0
$759$	−634.980	+	339.411i	−0.836601	+	0.447182i
$760$		0			0
$761$		−	673.498i		−	0.885018i	−0.896764	−	0.442509i	$-0.854089\pi$
$761$		−	673.498i		−	0.885018i	0.896764	−	0.442509i	$-0.145911\pi$
$762$		0			0
$763$	370.405			0.485459
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	989.949i		−	1.29068i
$768$		0			0
$769$	1138.00			1.47984			0.739922	−	0.672693i	$-0.234862\pi$
$769$	1138.00			1.47984			0.739922	+	0.672693i	$0.234862\pi$
$770$		0			0
$771$	−423.320	−	791.960i	−0.549053	−	1.02718i
$772$		0			0
$773$		−	785.748i		−	1.01649i	−0.861212	−	0.508246i	$-0.830294\pi$
$773$		−	785.748i		−	1.01649i	0.861212	−	0.508246i	$-0.169706\pi$
$774$		0			0
$775$	820.183			1.05830
$776$		0			0
$777$	−140.000	+	74.8331i	−0.180180	+	0.0963104i
$778$		0			0
$779$			395.980i			0.508318i
$780$		0			0
$781$	−400.000			−0.512164
$782$		0			0
$783$	1005.39	+	98.9949i	1.28402	+	0.126430i
$784$		0			0
$785$		−	1272.16i		−	1.62059i
$786$		0			0
$787$	−904.847			−1.14974			−0.574871	−	0.818244i	$-0.694948\pi$
$787$	−904.847			−1.14974			−0.574871	+	0.818244i	$0.694948\pi$
$788$		0			0
$789$	−216.000	−	404.099i	−0.273764	−	0.512166i
$790$		0			0
$791$		−	791.960i		−	1.00121i
$792$		0			0
$793$	900.000			1.13493
$794$		0			0
$795$	−740.810	+	395.980i	−0.931837	+	0.498088i
$796$		0			0
$797$			1234.75i			1.54924i	0.632425	+	0.774622i	$0.282060\pi$
$797$			1234.75i			1.54924i	−0.632425	+	0.774622i	$0.717940\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	560.000	+	374.166i	0.699126	+	0.467123i
$802$		0			0
$803$		−	424.264i		−	0.528349i
$804$		0			0
$805$	−672.000			−0.834783
$806$		0			0
$807$	603.231	+	1128.54i	0.747499	+	1.39844i
$808$		0			0
$809$		−	823.165i		−	1.01751i	−0.860912	−	0.508754i	$-0.830106\pi$
$809$		−	823.165i		−	1.01751i	0.860912	−	0.508754i	$-0.169894\pi$
$810$		0			0
$811$	26.4575			0.0326233			0.0163117	−	0.999867i	$-0.494808\pi$
$811$	26.4575			0.0326233			0.0163117	+	0.999867i	$0.494808\pi$
$812$		0			0
$813$	1050.00	−	561.249i	1.29151	−	0.690343i
$814$		0			0
$815$			752.362i			0.923143i
$816$		0			0
$817$	−1540.00			−1.88494
$818$		0			0
$819$	−264.575	+	395.980i	−0.323047	+	0.483492i
$820$		0			0
$821$		−	97.2831i		−	0.118493i	−0.998243	−	0.0592467i	$-0.981130\pi$
$821$		−	97.2831i		−	0.118493i	0.998243	−	0.0592467i	$-0.0188699\pi$
$822$		0			0
$823$	−894.264			−1.08659			−0.543295	−	0.839542i	$-0.682824\pi$
$823$	−894.264			−1.08659			−0.543295	+	0.839542i	$0.682824\pi$
$824$		0			0
$825$	−620.000	−	1159.91i	−0.751515	−	1.40596i
$826$		0			0
$827$			285.671i			0.345431i	0.984972	+	0.172715i	$0.0552541\pi$
$827$			285.671i			0.345431i	−0.984972	+	0.172715i	$0.944746\pi$
$828$		0			0
$829$	170.000			0.205066			0.102533	−	0.994730i	$-0.467305\pi$
$829$	170.000			0.205066			0.102533	+	0.994730i	$0.467305\pi$
$830$		0			0
$831$	1296.42	−	692.965i	1.56007	−	0.833892i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1820.28			−2.17997
$836$		0			0
$837$	−70.0000	+	710.915i	−0.0836320	+	0.849361i
$838$		0			0
$839$			707.107i			0.842797i	0.906876	+	0.421399i	$0.138461\pi$
$839$			707.107i			0.842797i	−0.906876	+	0.421399i	$0.861539\pi$
$840$		0			0
$841$	−559.000			−0.664685
$842$		0			0
$843$	190.494	+	356.382i	0.225972	+	0.422754i
$844$		0			0
$845$			516.349i			0.611064i
$846$		0			0
$847$	418.029			0.493540
$848$		0			0
$849$	14.0000	−	7.48331i	0.0164900	−	0.00881427i
$850$		0			0
$851$			169.706i			0.199419i
$852$		0			0
$853$	570.000			0.668230			0.334115	−	0.942532i	$-0.391563\pi$
$853$	570.000			0.668230			0.334115	+	0.942532i	$0.391563\pi$
$854$		0			0
$855$	−1481.62	−	989.949i	−1.73289	−	1.15784i
$856$		0			0
$857$			523.832i			0.611239i	0.952154	+	0.305620i	$0.0988636\pi$
$857$			523.832i			0.611239i	−0.952154	+	0.305620i	$0.901136\pi$
$858$		0			0
$859$	−1455.16			−1.69402			−0.847010	−	0.531577i	$-0.821600\pi$
$859$	−1455.16			−1.69402			−0.847010	+	0.531577i	$0.821600\pi$
$860$		0			0
$861$	−112.000	−	209.533i	−0.130081	−	0.243360i
$862$		0			0
$863$			181.019i			0.209756i	0.994485	+	0.104878i	$0.0334452\pi$
$863$			181.019i			0.209756i	−0.994485	+	0.104878i	$0.966555\pi$
$864$		0			0
$865$	−840.000			−0.971098
$866$		0			0
$867$	764.622	−	408.708i	0.881917	−	0.471405i
$868$		0			0
$869$		−	374.166i		−	0.430570i
$870$		0			0
$871$	52.9150			0.0607520
$872$		0			0
$873$	50.0000	−	74.8331i	0.0572738	−	0.0857195i
$874$		0			0
$875$		−	237.588i		−	0.271529i
$876$		0			0
$877$	490.000			0.558723			0.279361	−	0.960186i	$-0.409877\pi$
$877$	490.000			0.558723			0.279361	+	0.960186i	$0.409877\pi$
$878$		0			0
$879$	158.745	+	296.985i	0.180597	+	0.337867i
$880$		0			0
$881$			987.798i			1.12122i	0.828079	+	0.560612i	$0.189434\pi$
$881$			987.798i			1.12122i	−0.828079	+	0.560612i	$0.810566\pi$
$882$		0			0
$883$	957.762			1.08467			0.542334	−	0.840163i	$-0.317541\pi$
$883$	957.762			1.08467			0.542334	+	0.840163i	$0.317541\pi$
$884$		0			0
$885$	−1960.00	+	1047.66i	−2.21469	+	1.18380i
$886$		0			0
$887$			1374.62i			1.54974i	0.632123	+	0.774868i	$0.282184\pi$
$887$			1374.62i			1.54974i	−0.632123	+	0.774868i	$0.717816\pi$
$888$		0			0
$889$	812.000			0.913386
$890$		0			0
$891$	1058.30	−	438.406i	1.18777	−	0.492038i
$892$		0			0
$893$			299.333i			0.335199i
$894$		0			0
$895$	1587.45			1.77369
$896$		0			0
$897$	240.000	+	448.999i	0.267559	+	0.500556i
$898$		0			0
$899$		−	989.949i		−	1.10117i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	814.891	−	435.578i	0.902427	−	0.482367i
$904$		0			0
$905$			1960.63i			2.16644i
$906$		0			0
$907$	534.442			0.589241			0.294621	−	0.955614i	$-0.404807\pi$
$907$	534.442			0.589241			0.294621	+	0.955614i	$0.404807\pi$
$908$		0			0
$909$	840.000	+	561.249i	0.924092	+	0.617435i
$910$		0			0
$911$			169.706i			0.186285i	0.995653	+	0.0931425i	$0.0296912\pi$
$911$			169.706i			0.186285i	−0.995653	+	0.0931425i	$0.970309\pi$
$912$		0			0
$913$	360.000			0.394304
$914$		0			0
$915$	−952.470	−	1781.91i	−1.04095	−	1.94744i
$916$		0			0
$917$			374.166i			0.408032i
$918$		0			0
$919$	−555.608			−0.604579			−0.302289	−	0.953216i	$-0.597751\pi$
$919$	−555.608			−0.604579			−0.302289	+	0.953216i	$0.597751\pi$
$920$		0			0
$921$	126.000	−	67.3498i	0.136808	−	0.0731269i
$922$		0			0
$923$			282.843i			0.306438i
$924$		0			0
$925$	−310.000			−0.335135
$926$		0			0
$927$	−238.118	+	356.382i	−0.256869	+	0.384446i
$928$		0			0
$929$			89.7998i			0.0966628i	0.998831	+	0.0483314i	$0.0153904\pi$
$929$			89.7998i			0.0966628i	−0.998831	+	0.0483314i	$0.984610\pi$
$930$		0			0
$931$	−555.608			−0.596786
$932$		0			0
$933$	−360.000	−	673.498i	−0.385852	−	0.721863i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1270.00			−1.35539			−0.677695	−	0.735343i	$-0.737021\pi$
$937$	−1270.00			−1.35539			−0.677695	+	0.735343i	$0.737021\pi$
$938$		0			0
$939$	−608.523	+	325.269i	−0.648054	+	0.346399i
$940$		0			0
$941$			187.083i			0.198813i	0.995047	+	0.0994064i	$0.0316944\pi$
$941$			187.083i			0.198813i	−0.995047	+	0.0994064i	$0.968306\pi$
$942$		0			0
$943$	−253.992			−0.269345
$944$		0			0
$945$	1064.00	+	104.766i	1.12593	+	0.110864i
$946$		0			0
$947$			280.014i			0.295686i	0.989011	+	0.147843i	$0.0472330\pi$
$947$			280.014i			0.295686i	−0.989011	+	0.147843i	$0.952767\pi$
$948$		0			0
$949$	−300.000			−0.316122
$950$		0			0
$951$	−793.725	−	1484.92i	−0.834622	−	1.56143i
$952$		0			0
$953$			224.499i			0.235571i	0.993039	+	0.117786i	$0.0375796\pi$
$953$			224.499i			0.235571i	−0.993039	+	0.117786i	$0.962420\pi$
$954$		0			0
$955$	2539.92			2.65960
$956$		0			0
$957$	−1400.00	+	748.331i	−1.46290	+	0.781956i
$958$		0			0
$959$			1187.94i			1.23873i
$960$		0			0
$961$	−261.000			−0.271592
$962$		0			0
$963$	21.1660	+	14.1421i	0.0219792	+	0.0146855i
$964$		0			0
$965$		−	1571.50i		−	1.62849i
$966$		0			0
$967$	206.369			0.213411			0.106706	−	0.994291i	$-0.465970\pi$
$967$	206.369			0.213411			0.106706	+	0.994291i	$0.465970\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1711.20i			1.76231i	0.472832	+	0.881153i	$0.343232\pi$
$971$			1711.20i			1.76231i	−0.472832	+	0.881153i	$0.656768\pi$
$972$		0			0
$973$	−700.000			−0.719424
$974$		0			0
$975$	−820.183	+	438.406i	−0.841213	+	0.449647i
$976$		0			0
$977$			598.665i			0.612759i	0.951910	+	0.306379i	$0.0991176\pi$
$977$			598.665i			0.612759i	−0.951910	+	0.306379i	$0.900882\pi$
$978$		0			0
$979$	−1058.30			−1.08100
$980$		0			0
$981$	−350.000	+	523.832i	−0.356779	+	0.533978i
$982$		0			0
$983$		−	1544.32i		−	1.57103i	−0.618843	−	0.785514i	$-0.712398\pi$
$983$		−	1544.32i		−	1.57103i	0.618843	−	0.785514i	$-0.287602\pi$
$984$		0			0
$985$	−1400.00			−1.42132
$986$		0			0
$987$	−84.6640	−	158.392i	−0.0857792	−	0.160478i
$988$		0			0
$989$		−	987.798i		−	0.998784i
$990$		0			0
$991$	608.523			0.614049			0.307025	−	0.951702i	$-0.400667\pi$
$991$	608.523			0.614049			0.307025	+	0.951702i	$0.400667\pi$
$992$		0			0
$993$	350.000	−	187.083i	0.352467	−	0.188402i
$994$		0			0
$995$			2573.87i			2.58680i
$996$		0			0
$997$	810.000			0.812437			0.406219	−	0.913776i	$-0.366847\pi$
$997$	810.000			0.812437			0.406219	+	0.913776i	$0.366847\pi$
$998$		0			0
$999$	26.4575	−	268.701i	0.0264840	−	0.268970i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.e.b.65.1	✓	4	12.11	even	2	inner
96.3.e.b.65.2	yes	4	4.3	odd	2	inner
96.3.e.b.65.3	yes	4	1.1	even	1	trivial
96.3.e.b.65.4	yes	4	3.2	odd	2	inner
192.3.e.f.65.1		4	24.5	odd	2
192.3.e.f.65.2		4	8.5	even	2
192.3.e.f.65.3		4	8.3	odd	2
192.3.e.f.65.4		4	24.11	even	2
768.3.h.e.641.1		8	16.3	odd	4
768.3.h.e.641.2		8	16.5	even	4
768.3.h.e.641.3		8	48.11	even	4
768.3.h.e.641.4		8	48.29	odd	4
768.3.h.e.641.5		8	48.5	odd	4
768.3.h.e.641.6		8	48.35	even	4
768.3.h.e.641.7		8	16.13	even	4
768.3.h.e.641.8		8	16.11	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:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	96.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.64575 - 1.41421i) q^{3} -7.48331i q^{5} -5.29150 q^{7} +(5.00000 - 7.48331i) q^{9} +O(q^{10})\)	\(q+(2.64575 - 1.41421i) q^{3} -7.48331i q^{5} -5.29150 q^{7} +(5.00000 - 7.48331i) q^{9} +14.1421i q^{11} +10.0000 q^{13} +(-10.5830 - 19.7990i) q^{15} +26.4575 q^{19} +(-14.0000 + 7.48331i) q^{21} +16.9706i q^{23} -31.0000 q^{25} +(2.64575 - 26.8701i) q^{27} +37.4166i q^{29} -26.4575 q^{31} +(20.0000 + 37.4166i) q^{33} +39.5980i q^{35} +10.0000 q^{37} +(26.4575 - 14.1421i) q^{39} +14.9666i q^{41} -58.2065 q^{43} +(-56.0000 - 37.4166i) q^{45} +11.3137i q^{47} -21.0000 q^{49} -37.4166i q^{53} +105.830 q^{55} +(70.0000 - 37.4166i) q^{57} -98.9949i q^{59} +90.0000 q^{61} +(-26.4575 + 39.5980i) q^{63} -74.8331i q^{65} +5.29150 q^{67} +(24.0000 + 44.8999i) q^{69} +28.2843i q^{71} -30.0000 q^{73} +(-82.0183 + 43.8406i) q^{75} -74.8331i q^{77} -26.4575 q^{79} +(-31.0000 - 74.8331i) q^{81} -25.4558i q^{83} +(52.9150 + 98.9949i) q^{87} +74.8331i q^{89} -52.9150 q^{91} +(-70.0000 + 37.4166i) q^{93} -197.990i q^{95} +10.0000 q^{97} +(105.830 + 70.7107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{9}+O(q^{10}) \) 4 * q + 20 * q^9	\( 4 q + 20 q^{9} + 40 q^{13} - 56 q^{21} - 124 q^{25} + 80 q^{33} + 40 q^{37} - 224 q^{45} - 84 q^{49} + 280 q^{57} + 360 q^{61} + 96 q^{69} - 120 q^{73} - 124 q^{81} - 280 q^{93} + 40 q^{97}+O(q^{100}) \) 4 * q + 20 * q^9 + 40 * q^13 - 56 * q^21 - 124 * q^25 + 80 * q^33 + 40 * q^37 - 224 * q^45 - 84 * q^49 + 280 * q^57 + 360 * q^61 + 96 * q^69 - 120 * q^73 - 124 * q^81 - 280 * q^93 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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