LMFDB - Embedded newform 96.3.h.c.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,3,Mod(17,96)]

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, 1, 1]))

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

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

chi := DirichletCharacter("96.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	96.h (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:	$2.61581053786$
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} + 6x^{2} + 16 $ x^4 + 6*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$0.707107 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	96.17
Dual form			96.3.h.c.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 2.64575i) q^{3} -5.65685 q^{5} -4.00000 q^{7} +(-5.00000 - 7.48331i) q^{9} +O(q^{10})$	$q+(-1.41421 + 2.64575i) q^{3} -5.65685 q^{5} -4.00000 q^{7} +(-5.00000 - 7.48331i) q^{9} -8.48528 q^{11} +10.5830i q^{13} +(8.00000 - 14.9666i) q^{15} +14.9666i q^{17} -5.29150i q^{19} +(5.65685 - 10.5830i) q^{21} +29.9333i q^{23} +7.00000 q^{25} +(26.8701 - 2.64575i) q^{27} +16.9706 q^{29} +4.00000 q^{31} +(12.0000 - 22.4499i) q^{33} +22.6274 q^{35} -52.9150i q^{37} +(-28.0000 - 14.9666i) q^{39} +29.9333i q^{41} +5.29150i q^{43} +(28.2843 + 42.3320i) q^{45} -33.0000 q^{49} +(-39.5980 - 21.1660i) q^{51} -50.9117 q^{53} +48.0000 q^{55} +(14.0000 + 7.48331i) q^{57} -48.0833 q^{59} +95.2470i q^{61} +(20.0000 + 29.9333i) q^{63} -59.8665i q^{65} +47.6235i q^{67} +(-79.1960 - 42.3320i) q^{69} -89.7998i q^{71} -6.00000 q^{73} +(-9.89949 + 18.5203i) q^{75} +33.9411 q^{77} -124.000 q^{79} +(-31.0000 + 74.8331i) q^{81} +2.82843 q^{83} -84.6640i q^{85} +(-24.0000 + 44.8999i) q^{87} -104.766i q^{89} -42.3320i q^{91} +(-5.65685 + 10.5830i) q^{93} +29.9333i q^{95} +118.000 q^{97} +(42.4264 + 63.4980i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{7} - 20 q^{9}+O(q^{10}) $ 4 * q - 16 * q^7 - 20 * q^9	$ 4 q - 16 q^{7} - 20 q^{9} + 32 q^{15} + 28 q^{25} + 16 q^{31} + 48 q^{33} - 112 q^{39} - 132 q^{49} + 192 q^{55} + 56 q^{57} + 80 q^{63} - 24 q^{73} - 496 q^{79} - 124 q^{81} - 96 q^{87} + 472 q^{97}+O(q^{100}) $ 4 * q - 16 * q^7 - 20 * q^9 + 32 * q^15 + 28 * q^25 + 16 * q^31 + 48 * q^33 - 112 * q^39 - 132 * q^49 + 192 * q^55 + 56 * q^57 + 80 * q^63 - 24 * q^73 - 496 * q^79 - 124 * q^81 - 96 * q^87 + 472 * 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
$3$	−1.41421	+	2.64575i	−0.471405	+	0.881917i
$3$	−1.41421	+	2.64575i	−0.471405	+	0.881917i
$4$		0			0
$5$	−5.65685			−1.13137			−0.565685	−	0.824621i	$-0.691388\pi$
$5$	−5.65685			−1.13137			−0.565685	+	0.824621i	$0.691388\pi$
$6$		0			0
$7$	−4.00000			−0.571429			−0.285714	−	0.958315i	$-0.592231\pi$
$7$	−4.00000			−0.571429			−0.285714	+	0.958315i	$0.592231\pi$
$8$		0			0
$9$	−5.00000	−	7.48331i	−0.555556	−	0.831479i
$10$		0			0
$11$	−8.48528			−0.771389			−0.385695	−	0.922627i	$-0.626038\pi$
$11$	−8.48528			−0.771389			−0.385695	+	0.922627i	$0.626038\pi$
$12$		0			0
$13$			10.5830i			0.814077i	0.913411	+	0.407039i	$0.133439\pi$
$13$			10.5830i			0.814077i	−0.913411	+	0.407039i	$0.866561\pi$
$14$		0			0
$15$	8.00000	−	14.9666i	0.533333	−	0.997775i
$16$		0			0
$17$			14.9666i			0.880390i	0.897902	+	0.440195i	$0.145091\pi$
$17$			14.9666i			0.880390i	−0.897902	+	0.440195i	$0.854909\pi$
$18$		0			0
$19$		−	5.29150i		−	0.278500i	−0.990257	−	0.139250i	$-0.955531\pi$
$19$		−	5.29150i		−	0.278500i	0.990257	−	0.139250i	$-0.0444692\pi$
$20$		0			0
$21$	5.65685	−	10.5830i	0.269374	−	0.503953i
$22$		0			0
$23$			29.9333i			1.30145i	0.759315	+	0.650723i	$0.225534\pi$
$23$			29.9333i			1.30145i	−0.759315	+	0.650723i	$0.774466\pi$
$24$		0			0
$25$	7.00000			0.280000
$26$		0			0
$27$	26.8701	−	2.64575i	0.995187	−	0.0979908i
$28$		0			0
$29$	16.9706			0.585192			0.292596	−	0.956236i	$-0.405481\pi$
$29$	16.9706			0.585192			0.292596	+	0.956236i	$0.405481\pi$
$30$		0			0
$31$	4.00000			0.129032			0.0645161	−	0.997917i	$-0.479450\pi$
$31$	4.00000			0.129032			0.0645161	+	0.997917i	$0.479450\pi$
$32$		0			0
$33$	12.0000	−	22.4499i	0.363636	−	0.680301i
$34$		0			0
$35$	22.6274			0.646498
$36$		0			0
$37$		−	52.9150i		−	1.43014i	−0.699055	−	0.715068i	$-0.746396\pi$
$37$		−	52.9150i		−	1.43014i	0.699055	−	0.715068i	$-0.253604\pi$
$38$		0			0
$39$	−28.0000	−	14.9666i	−0.717949	−	0.383760i
$40$		0			0
$41$			29.9333i			0.730079i	0.930992	+	0.365040i	$0.118945\pi$
$41$			29.9333i			0.730079i	−0.930992	+	0.365040i	$0.881055\pi$
$42$		0			0
$43$			5.29150i			0.123058i	0.998105	+	0.0615291i	$0.0195977\pi$
$43$			5.29150i			0.123058i	−0.998105	+	0.0615291i	$0.980402\pi$
$44$		0			0
$45$	28.2843	+	42.3320i	0.628539	+	0.940712i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−33.0000			−0.673469
$50$		0			0
$51$	−39.5980	−	21.1660i	−0.776431	−	0.415020i
$52$		0			0
$53$	−50.9117			−0.960598			−0.480299	−	0.877105i	$-0.659472\pi$
$53$	−50.9117			−0.960598			−0.480299	+	0.877105i	$0.659472\pi$
$54$		0			0
$55$	48.0000			0.872727
$56$		0			0
$57$	14.0000	+	7.48331i	0.245614	+	0.131286i
$58$		0			0
$59$	−48.0833			−0.814971			−0.407485	−	0.913212i	$-0.633594\pi$
$59$	−48.0833			−0.814971			−0.407485	+	0.913212i	$0.633594\pi$
$60$		0			0
$61$			95.2470i			1.56143i	0.624889	+	0.780714i	$0.285144\pi$
$61$			95.2470i			1.56143i	−0.624889	+	0.780714i	$0.714856\pi$
$62$		0			0
$63$	20.0000	+	29.9333i	0.317460	+	0.475131i
$64$		0			0
$65$		−	59.8665i		−	0.921023i
$66$		0			0
$67$			47.6235i			0.710799i	0.934715	+	0.355399i	$0.115655\pi$
$67$			47.6235i			0.710799i	−0.934715	+	0.355399i	$0.884345\pi$
$68$		0			0
$69$	−79.1960	−	42.3320i	−1.14777	−	0.613508i
$70$		0			0
$71$		−	89.7998i		−	1.26479i	−0.774648	−	0.632393i	$-0.782073\pi$
$71$		−	89.7998i		−	1.26479i	0.774648	−	0.632393i	$-0.217927\pi$
$72$		0			0
$73$	−6.00000			−0.0821918			−0.0410959	−	0.999155i	$-0.513085\pi$
$73$	−6.00000			−0.0821918			−0.0410959	+	0.999155i	$0.513085\pi$
$74$		0			0
$75$	−9.89949	+	18.5203i	−0.131993	+	0.246937i
$76$		0			0
$77$	33.9411			0.440794
$78$		0			0
$79$	−124.000			−1.56962			−0.784810	−	0.619736i	$-0.787240\pi$
$79$	−124.000			−1.56962			−0.784810	+	0.619736i	$0.787240\pi$
$80$		0			0
$81$	−31.0000	+	74.8331i	−0.382716	+	0.923866i
$82$		0			0
$83$	2.82843			0.0340774			0.0170387	−	0.999855i	$-0.494576\pi$
$83$	2.82843			0.0340774			0.0170387	+	0.999855i	$0.494576\pi$
$84$		0			0
$85$		−	84.6640i		−	0.996048i
$86$		0			0
$87$	−24.0000	+	44.8999i	−0.275862	+	0.516091i
$88$		0			0
$89$		−	104.766i		−	1.17715i	−0.808442	−	0.588575i	$-0.799689\pi$
$89$		−	104.766i		−	1.17715i	0.808442	−	0.588575i	$-0.200311\pi$
$90$		0			0
$91$		−	42.3320i		−	0.465187i
$92$		0			0
$93$	−5.65685	+	10.5830i	−0.0608264	+	0.113796i
$94$		0			0
$95$			29.9333i			0.315087i
$96$		0			0
$97$	118.000			1.21649			0.608247	−	0.793747i	$-0.291873\pi$
$97$	118.000			1.21649			0.608247	+	0.793747i	$0.291873\pi$
$98$		0			0
$99$	42.4264	+	63.4980i	0.428550	+	0.641394i
$100$		0			0
$101$	62.2254			0.616093			0.308047	−	0.951371i	$-0.400325\pi$
$101$	62.2254			0.616093			0.308047	+	0.951371i	$0.400325\pi$
$102$		0			0
$103$	108.000			1.04854			0.524272	−	0.851551i	$-0.324338\pi$
$103$	108.000			1.04854			0.524272	+	0.851551i	$0.324338\pi$
$104$		0			0
$105$	−32.0000	+	59.8665i	−0.304762	+	0.570157i
$106$		0			0
$107$	144.250			1.34813			0.674064	−	0.738673i	$-0.264547\pi$
$107$	144.250			1.34813			0.674064	+	0.738673i	$0.264547\pi$
$108$		0			0
$109$			52.9150i			0.485459i	0.970094	+	0.242729i	$0.0780427\pi$
$109$			52.9150i			0.485459i	−0.970094	+	0.242729i	$0.921957\pi$
$110$		0			0
$111$	140.000	+	74.8331i	1.26126	+	0.674173i
$112$		0			0
$113$			89.7998i			0.794688i	0.917670	+	0.397344i	$0.130068\pi$
$113$			89.7998i			0.794688i	−0.917670	+	0.397344i	$0.869932\pi$
$114$		0			0
$115$		−	169.328i		−	1.47242i
$116$		0			0
$117$	79.1960	−	52.9150i	0.676889	−	0.452265i
$118$		0			0
$119$		−	59.8665i		−	0.503080i
$120$		0			0
$121$	−49.0000			−0.404959
$122$		0			0
$123$	−79.1960	−	42.3320i	−0.643870	−	0.344163i
$124$		0			0
$125$	101.823			0.814587
$126$		0			0
$127$	−76.0000			−0.598425			−0.299213	−	0.954186i	$-0.596724\pi$
$127$	−76.0000			−0.598425			−0.299213	+	0.954186i	$0.596724\pi$
$128$		0			0
$129$	−14.0000	−	7.48331i	−0.108527	−	0.0580102i
$130$		0			0
$131$	−14.1421			−0.107955			−0.0539776	−	0.998542i	$-0.517190\pi$
$131$	−14.1421			−0.107955			−0.0539776	+	0.998542i	$0.517190\pi$
$132$		0			0
$133$			21.1660i			0.159143i
$134$		0			0
$135$	−152.000	+	14.9666i	−1.12593	+	0.110864i
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$			121.705i			0.875572i	0.899079	+	0.437786i	$0.144237\pi$
$139$			121.705i			0.875572i	−0.899079	+	0.437786i	$0.855763\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	89.7998i		−	0.627970i
$144$		0			0
$145$	−96.0000			−0.662069
$146$		0			0
$147$	46.6690	−	87.3098i	0.317477	−	0.593944i
$148$		0			0
$149$	−186.676			−1.25286			−0.626430	−	0.779478i	$-0.715485\pi$
$149$	−186.676			−1.25286			−0.626430	+	0.779478i	$0.715485\pi$
$150$		0			0
$151$	60.0000			0.397351			0.198675	−	0.980065i	$-0.436336\pi$
$151$	60.0000			0.397351			0.198675	+	0.980065i	$0.436336\pi$
$152$		0			0
$153$	112.000	−	74.8331i	0.732026	−	0.489106i
$154$		0			0
$155$	−22.6274			−0.145983
$156$		0			0
$157$		−	116.413i		−	0.741484i	−0.928736	−	0.370742i	$-0.879103\pi$
$157$		−	116.413i		−	0.741484i	0.928736	−	0.370742i	$-0.120897\pi$
$158$		0			0
$159$	72.0000	−	134.700i	0.452830	−	0.847168i
$160$		0			0
$161$		−	119.733i		−	0.743683i
$162$		0			0
$163$			291.033i			1.78548i	0.450576	+	0.892738i	$0.351219\pi$
$163$			291.033i			1.78548i	−0.450576	+	0.892738i	$0.648781\pi$
$164$		0			0
$165$	−67.8823	+	126.996i	−0.411408	+	0.769673i
$166$		0			0
$167$			329.266i			1.97165i	0.167772	+	0.985826i	$0.446343\pi$
$167$			329.266i			1.97165i	−0.167772	+	0.985826i	$0.553657\pi$
$168$		0			0
$169$	57.0000			0.337278
$170$		0			0
$171$	−39.5980	+	26.4575i	−0.231567	+	0.154722i
$172$		0			0
$173$	−186.676			−1.07905			−0.539527	−	0.841969i	$-0.681397\pi$
$173$	−186.676			−1.07905			−0.539527	+	0.841969i	$0.681397\pi$
$174$		0			0
$175$	−28.0000			−0.160000
$176$		0			0
$177$	68.0000	−	127.216i	0.384181	−	0.718736i
$178$		0			0
$179$	−319.612			−1.78554			−0.892772	−	0.450509i	$-0.851242\pi$
$179$	−319.612			−1.78554			−0.892772	+	0.450509i	$0.851242\pi$
$180$		0			0
$181$			116.413i			0.643166i	0.946881	+	0.321583i	$0.104215\pi$
$181$			116.413i			0.643166i	−0.946881	+	0.321583i	$0.895785\pi$
$182$		0			0
$183$	−252.000	−	134.700i	−1.37705	−	0.736064i
$184$		0			0
$185$			299.333i			1.61801i
$186$		0			0
$187$		−	126.996i		−	0.679123i
$188$		0			0
$189$	−107.480	+	10.5830i	−0.568678	+	0.0559947i
$190$		0			0
$191$		−	59.8665i		−	0.313437i	−0.987643	−	0.156719i	$-0.949908\pi$
$191$		−	59.8665i		−	0.313437i	0.987643	−	0.156719i	$-0.0500916\pi$
$192$		0			0
$193$	−102.000			−0.528497			−0.264249	−	0.964455i	$-0.585124\pi$
$193$	−102.000			−0.528497			−0.264249	+	0.964455i	$0.585124\pi$
$194$		0			0
$195$	158.392	+	84.6640i	0.812266	+	0.434175i
$196$		0			0
$197$	243.245			1.23474			0.617372	−	0.786671i	$-0.288197\pi$
$197$	243.245			1.23474			0.617372	+	0.786671i	$0.288197\pi$
$198$		0			0
$199$	188.000			0.944724			0.472362	−	0.881405i	$-0.343402\pi$
$199$	188.000			0.944724			0.472362	+	0.881405i	$0.343402\pi$
$200$		0			0
$201$	−126.000	−	67.3498i	−0.626866	−	0.335074i
$202$		0			0
$203$	−67.8823			−0.334395
$204$		0			0
$205$		−	169.328i		−	0.825991i
$206$		0			0
$207$	224.000	−	149.666i	1.08213	−	0.723026i
$208$		0			0
$209$			44.8999i			0.214832i
$210$		0			0
$211$		−	248.701i		−	1.17868i	−0.807887	−	0.589338i	$-0.799389\pi$
$211$		−	248.701i		−	1.17868i	0.807887	−	0.589338i	$-0.200611\pi$
$212$		0			0
$213$	237.588	+	126.996i	1.11544	+	0.596226i
$214$		0			0
$215$		−	29.9333i		−	0.139224i
$216$		0			0
$217$	−16.0000			−0.0737327
$218$		0			0
$219$	8.48528	−	15.8745i	0.0387456	−	0.0724863i
$220$		0			0
$221$	−158.392			−0.716706
$222$		0			0
$223$	−188.000			−0.843049			−0.421525	−	0.906817i	$-0.638505\pi$
$223$	−188.000			−0.843049			−0.421525	+	0.906817i	$0.638505\pi$
$224$		0			0
$225$	−35.0000	−	52.3832i	−0.155556	−	0.232814i
$226$		0			0
$227$	387.495			1.70702			0.853512	−	0.521073i	$-0.174468\pi$
$227$	387.495			1.70702			0.853512	+	0.521073i	$0.174468\pi$
$228$		0			0
$229$			243.409i			1.06292i	0.847083	+	0.531461i	$0.178357\pi$
$229$			243.409i			1.06292i	−0.847083	+	0.531461i	$0.821643\pi$
$230$		0			0
$231$	−48.0000	+	89.7998i	−0.207792	+	0.388744i
$232$		0			0
$233$		−	104.766i		−	0.449641i	−0.974400	−	0.224821i	$-0.927820\pi$
$233$		−	104.766i		−	0.449641i	0.974400	−	0.224821i	$-0.0721796\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	175.362	−	328.073i	0.739926	−	1.38427i
$238$		0			0
$239$			359.199i			1.50293i	0.659776	+	0.751463i	$0.270651\pi$
$239$			359.199i			1.50293i	−0.659776	+	0.751463i	$0.729349\pi$
$240$		0			0
$241$	122.000			0.506224			0.253112	−	0.967437i	$-0.418546\pi$
$241$	122.000			0.506224			0.253112	+	0.967437i	$0.418546\pi$
$242$		0			0
$243$	−154.149	−	187.848i	−0.634359	−	0.773038i
$244$		0			0
$245$	186.676			0.761944
$246$		0			0
$247$	56.0000			0.226721
$248$		0			0
$249$	−4.00000	+	7.48331i	−0.0160643	+	0.0300535i
$250$		0			0
$251$	161.220			0.642312			0.321156	−	0.947026i	$-0.395929\pi$
$251$	161.220			0.642312			0.321156	+	0.947026i	$0.395929\pi$
$252$		0			0
$253$		−	253.992i		−	1.00392i
$254$		0			0
$255$	224.000	+	119.733i	0.878431	+	0.469541i
$256$		0			0
$257$			448.999i			1.74708i	0.486755	+	0.873539i	$0.338181\pi$
$257$			448.999i			1.74708i	−0.486755	+	0.873539i	$0.661819\pi$
$258$		0			0
$259$			211.660i			0.817220i
$260$		0			0
$261$	−84.8528	−	126.996i	−0.325107	−	0.486575i
$262$		0			0
$263$		−	209.533i		−	0.796703i	−0.917233	−	0.398351i	$-0.869582\pi$
$263$		−	209.533i		−	0.796703i	0.917233	−	0.398351i	$-0.130418\pi$
$264$		0			0
$265$	288.000			1.08679
$266$		0			0
$267$	277.186	+	148.162i	1.03815	+	0.554914i
$268$		0			0
$269$	−50.9117			−0.189263			−0.0946314	−	0.995512i	$-0.530167\pi$
$269$	−50.9117			−0.189263			−0.0946314	+	0.995512i	$0.530167\pi$
$270$		0			0
$271$	−348.000			−1.28413			−0.642066	−	0.766649i	$-0.721923\pi$
$271$	−348.000			−1.28413			−0.642066	+	0.766649i	$0.721923\pi$
$272$		0			0
$273$	112.000	+	59.8665i	0.410256	+	0.219291i
$274$		0			0
$275$	−59.3970			−0.215989
$276$		0			0
$277$			243.409i			0.878733i	0.898308	+	0.439367i	$0.144797\pi$
$277$			243.409i			0.878733i	−0.898308	+	0.439367i	$0.855203\pi$
$278$		0			0
$279$	−20.0000	−	29.9333i	−0.0716846	−	0.107288i
$280$		0			0
$281$		−	314.299i		−	1.11850i	−0.828998	−	0.559251i	$-0.811089\pi$
$281$		−	314.299i		−	1.11850i	0.828998	−	0.559251i	$-0.188911\pi$
$282$		0			0
$283$		−	89.9555i		−	0.317864i	−0.987290	−	0.158932i	$-0.949195\pi$
$283$		−	89.9555i		−	0.317864i	0.987290	−	0.158932i	$-0.0508051\pi$
$284$		0			0
$285$	−79.1960	−	42.3320i	−0.277881	−	0.148533i
$286$		0			0
$287$		−	119.733i		−	0.417188i
$288$		0			0
$289$	65.0000			0.224913
$290$		0			0
$291$	−166.877	+	312.199i	−0.573461	+	1.07285i
$292$		0			0
$293$	16.9706			0.0579200			0.0289600	−	0.999581i	$-0.490780\pi$
$293$	16.9706			0.0579200			0.0289600	+	0.999581i	$0.490780\pi$
$294$		0			0
$295$	272.000			0.922034
$296$		0			0
$297$	−228.000	+	22.4499i	−0.767677	+	0.0755890i
$298$		0			0
$299$	−316.784			−1.05948
$300$		0			0
$301$		−	21.1660i		−	0.0703190i
$302$		0			0
$303$	−88.0000	+	164.633i	−0.290429	+	0.543343i
$304$		0			0
$305$		−	538.799i		−	1.76655i
$306$		0			0
$307$		−	460.361i		−	1.49955i	−0.661695	−	0.749773i	$-0.730163\pi$
$307$		−	460.361i		−	1.49955i	0.661695	−	0.749773i	$-0.269837\pi$
$308$		0			0
$309$	−152.735	+	285.741i	−0.494288	+	0.924729i
$310$		0			0
$311$		−	149.666i		−	0.481242i	−0.970619	−	0.240621i	$-0.922649\pi$
$311$		−	149.666i		−	0.481242i	0.970619	−	0.240621i	$-0.0773511\pi$
$312$		0			0
$313$	−562.000			−1.79553			−0.897764	−	0.440478i	$-0.854809\pi$
$313$	−562.000			−1.79553			−0.897764	+	0.440478i	$0.854809\pi$
$314$		0			0
$315$	−113.137	−	169.328i	−0.359165	−	0.537549i
$316$		0			0
$317$	−5.65685			−0.0178450			−0.00892248	−	0.999960i	$-0.502840\pi$
$317$	−5.65685			−0.0178450			−0.00892248	+	0.999960i	$0.502840\pi$
$318$		0			0
$319$	−144.000			−0.451411
$320$		0			0
$321$	−204.000	+	381.649i	−0.635514	+	1.18894i
$322$		0			0
$323$	79.1960			0.245189
$324$		0			0
$325$			74.0810i			0.227942i
$326$		0			0
$327$	−140.000	−	74.8331i	−0.428135	−	0.228848i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			418.029i			1.26293i	0.775406	+	0.631463i	$0.217545\pi$
$331$			418.029i			1.26293i	−0.775406	+	0.631463i	$0.782455\pi$
$332$		0			0
$333$	−395.980	+	264.575i	−1.18913	+	0.794520i
$334$		0			0
$335$		−	269.399i		−	0.804177i
$336$		0			0
$337$	−50.0000			−0.148368			−0.0741840	−	0.997245i	$-0.523635\pi$
$337$	−50.0000			−0.148368			−0.0741840	+	0.997245i	$0.523635\pi$
$338$		0			0
$339$	−237.588	−	126.996i	−0.700849	−	0.374620i
$340$		0			0
$341$	−33.9411			−0.0995341
$342$		0			0
$343$	328.000			0.956268
$344$		0			0
$345$	448.000	+	239.466i	1.29855	+	0.694105i
$346$		0			0
$347$	−359.210			−1.03519			−0.517594	−	0.855626i	$-0.673172\pi$
$347$	−359.210			−1.03519			−0.517594	+	0.855626i	$0.673172\pi$
$348$		0			0
$349$		−	116.413i		−	0.333562i	−0.985994	−	0.166781i	$-0.946663\pi$
$349$		−	116.413i		−	0.333562i	0.985994	−	0.166781i	$-0.0533373\pi$
$350$		0			0
$351$	28.0000	+	284.366i	0.0797721	+	0.810159i
$352$		0			0
$353$		−	179.600i		−	0.508781i	−0.967102	−	0.254390i	$-0.918125\pi$
$353$		−	179.600i		−	0.508781i	0.967102	−	0.254390i	$-0.0818748\pi$
$354$		0			0
$355$			507.984i			1.43094i
$356$		0			0
$357$	158.392	+	84.6640i	0.443675	+	0.237154i
$358$		0			0
$359$		−	329.266i		−	0.917175i	−0.888649	−	0.458588i	$-0.848356\pi$
$359$		−	329.266i		−	0.917175i	0.888649	−	0.458588i	$-0.151644\pi$
$360$		0			0
$361$	333.000			0.922438
$362$		0			0
$363$	69.2965	−	129.642i	0.190899	−	0.357140i
$364$		0			0
$365$	33.9411			0.0929894
$366$		0			0
$367$	228.000			0.621253			0.310627	−	0.950532i	$-0.399461\pi$
$367$	228.000			0.621253			0.310627	+	0.950532i	$0.399461\pi$
$368$		0			0
$369$	224.000	−	149.666i	0.607046	−	0.405600i
$370$		0			0
$371$	203.647			0.548913
$372$		0			0
$373$		−	137.579i		−	0.368845i	−0.982847	−	0.184422i	$-0.940959\pi$
$373$		−	137.579i		−	0.368845i	0.982847	−	0.184422i	$-0.0590414\pi$
$374$		0			0
$375$	−144.000	+	269.399i	−0.384000	+	0.718398i
$376$		0			0
$377$			179.600i			0.476391i
$378$		0			0
$379$		−	164.037i		−	0.432814i	−0.976303	−	0.216407i	$-0.930566\pi$
$379$		−	164.037i		−	0.432814i	0.976303	−	0.216407i	$-0.0694338\pi$
$380$		0			0
$381$	107.480	−	201.077i	0.282100	−	0.527761i
$382$		0			0
$383$			179.600i			0.468928i	0.972125	+	0.234464i	$0.0753336\pi$
$383$			179.600i			0.468928i	−0.972125	+	0.234464i	$0.924666\pi$
$384$		0			0
$385$	−192.000			−0.498701
$386$		0			0
$387$	39.5980	−	26.4575i	0.102320	−	0.0683657i
$388$		0			0
$389$	−96.1665			−0.247215			−0.123607	−	0.992331i	$-0.539446\pi$
$389$	−96.1665			−0.247215			−0.123607	+	0.992331i	$0.539446\pi$
$390$		0			0
$391$	−448.000			−1.14578
$392$		0			0
$393$	20.0000	−	37.4166i	0.0508906	−	0.0952076i
$394$		0			0
$395$	701.450			1.77582
$396$		0			0
$397$			52.9150i			0.133287i	0.997777	+	0.0666436i	$0.0212291\pi$
$397$			52.9150i			0.133287i	−0.997777	+	0.0666436i	$0.978771\pi$
$398$		0			0
$399$	−56.0000	−	29.9333i	−0.140351	−	0.0750207i
$400$		0			0
$401$		−	134.700i		−	0.335909i	−0.985795	−	0.167955i	$-0.946284\pi$
$401$		−	134.700i		−	0.335909i	0.985795	−	0.167955i	$-0.0537162\pi$
$402$		0			0
$403$			42.3320i			0.105042i
$404$		0			0
$405$	175.362	−	423.320i	0.432994	−	1.04524i
$406$		0			0
$407$			448.999i			1.10319i
$408$		0			0
$409$	−158.000			−0.386308			−0.193154	−	0.981168i	$-0.561872\pi$
$409$	−158.000			−0.386308			−0.193154	+	0.981168i	$0.561872\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	192.333			0.465697
$414$		0			0
$415$	−16.0000			−0.0385542
$416$		0			0
$417$	−322.000	−	172.116i	−0.772182	−	0.412749i
$418$		0			0
$419$	330.926			0.789799			0.394900	−	0.918724i	$-0.370779\pi$
$419$	330.926			0.789799			0.394900	+	0.918724i	$0.370779\pi$
$420$		0			0
$421$		−	518.567i		−	1.23175i	−0.787843	−	0.615876i	$-0.788802\pi$
$421$		−	518.567i		−	1.23175i	0.787843	−	0.615876i	$-0.211198\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			104.766i			0.246509i
$426$		0			0
$427$		−	380.988i		−	0.892244i
$428$		0			0
$429$	237.588	+	126.996i	0.553818	+	0.296028i
$430$		0			0
$431$		−	359.199i		−	0.833409i	−0.909042	−	0.416704i	$-0.863185\pi$
$431$		−	359.199i		−	0.833409i	0.909042	−	0.416704i	$-0.136815\pi$
$432$		0			0
$433$	86.0000			0.198614			0.0993072	−	0.995057i	$-0.468337\pi$
$433$	86.0000			0.198614			0.0993072	+	0.995057i	$0.468337\pi$
$434$		0			0
$435$	135.765	−	253.992i	0.312102	−	0.583890i
$436$		0			0
$437$	158.392			0.362453
$438$		0			0
$439$	156.000			0.355353			0.177677	−	0.984089i	$-0.443142\pi$
$439$	156.000			0.355353			0.177677	+	0.984089i	$0.443142\pi$
$440$		0			0
$441$	165.000	+	246.949i	0.374150	+	0.559976i
$442$		0			0
$443$	−212.132			−0.478853			−0.239427	−	0.970914i	$-0.576959\pi$
$443$	−212.132			−0.478853			−0.239427	+	0.970914i	$0.576959\pi$
$444$		0			0
$445$			592.648i			1.33179i
$446$		0			0
$447$	264.000	−	493.899i	0.590604	−	1.10492i
$448$		0			0
$449$			493.899i			1.10000i	0.835166	+	0.549999i	$0.185372\pi$
$449$			493.899i			1.10000i	−0.835166	+	0.549999i	$0.814628\pi$
$450$		0			0
$451$		−	253.992i		−	0.563175i
$452$		0			0
$453$	−84.8528	+	158.745i	−0.187313	+	0.350431i
$454$		0			0
$455$			239.466i			0.526299i
$456$		0			0
$457$	194.000			0.424508			0.212254	−	0.977215i	$-0.431920\pi$
$457$	194.000			0.424508			0.212254	+	0.977215i	$0.431920\pi$
$458$		0			0
$459$	39.5980	+	402.154i	0.0862701	+	0.876153i
$460$		0			0
$461$	560.029			1.21481			0.607406	−	0.794391i	$-0.292210\pi$
$461$	560.029			1.21481			0.607406	+	0.794391i	$0.292210\pi$
$462$		0			0
$463$	404.000			0.872570			0.436285	−	0.899808i	$-0.356294\pi$
$463$	404.000			0.872570			0.436285	+	0.899808i	$0.356294\pi$
$464$		0			0
$465$	32.0000	−	59.8665i	0.0688172	−	0.128745i
$466$		0			0
$467$	−664.680			−1.42330			−0.711649	−	0.702535i	$-0.752051\pi$
$467$	−664.680			−1.42330			−0.711649	+	0.702535i	$0.752051\pi$
$468$		0			0
$469$		−	190.494i		−	0.406171i
$470$		0			0
$471$	308.000	+	164.633i	0.653928	+	0.349539i
$472$		0			0
$473$		−	44.8999i		−	0.0949258i
$474$		0			0
$475$		−	37.0405i		−	0.0779800i
$476$		0			0
$477$	254.558	+	380.988i	0.533665	+	0.798717i
$478$		0			0
$479$			239.466i			0.499929i	0.968255	+	0.249965i	$0.0804190\pi$
$479$			239.466i			0.499929i	−0.968255	+	0.249965i	$0.919581\pi$
$480$		0			0
$481$	560.000			1.16424
$482$		0			0
$483$	316.784	+	169.328i	0.655867	+	0.350576i
$484$		0			0
$485$	−667.509			−1.37631
$486$		0			0
$487$	−500.000			−1.02669			−0.513347	−	0.858181i	$-0.671595\pi$
$487$	−500.000			−1.02669			−0.513347	+	0.858181i	$0.671595\pi$
$488$		0			0
$489$	−770.000	−	411.582i	−1.57464	−	0.841682i
$490$		0			0
$491$	−115.966			−0.236182			−0.118091	−	0.993003i	$-0.537678\pi$
$491$	−115.966			−0.236182			−0.118091	+	0.993003i	$0.537678\pi$
$492$		0			0
$493$			253.992i			0.515197i
$494$		0			0
$495$	−240.000	−	359.199i	−0.484848	−	0.725655i
$496$		0			0
$497$			359.199i			0.722735i
$498$		0			0
$499$		−	333.365i		−	0.668065i	−0.942561	−	0.334033i	$-0.891590\pi$
$499$		−	333.365i		−	0.668065i	0.942561	−	0.334033i	$-0.108410\pi$
$500$		0			0
$501$	−871.156	−	465.652i	−1.73883	−	0.929446i
$502$		0			0
$503$			448.999i			0.892642i	0.894873	+	0.446321i	$0.147266\pi$
$503$			448.999i			0.892642i	−0.894873	+	0.446321i	$0.852734\pi$
$504$		0			0
$505$	−352.000			−0.697030
$506$		0			0
$507$	−80.6102	+	150.808i	−0.158994	+	0.297451i
$508$		0			0
$509$	−322.441			−0.633479			−0.316739	−	0.948513i	$-0.602588\pi$
$509$	−322.441			−0.633479			−0.316739	+	0.948513i	$0.602588\pi$
$510$		0			0
$511$	24.0000			0.0469667
$512$		0			0
$513$	−14.0000	−	142.183i	−0.0272904	−	0.277160i
$514$		0			0
$515$	−610.940			−1.18629
$516$		0			0
$517$		0			0
$518$		0			0
$519$	264.000	−	493.899i	0.508671	−	0.951635i
$520$		0			0
$521$		−	179.600i		−	0.344721i	−0.985034	−	0.172360i	$-0.944861\pi$
$521$		−	179.600i		−	0.344721i	0.985034	−	0.172360i	$-0.0551394\pi$
$522$		0			0
$523$			555.608i			1.06235i	0.847263	+	0.531174i	$0.178249\pi$
$523$			555.608i			1.06235i	−0.847263	+	0.531174i	$0.821751\pi$
$524$		0			0
$525$	39.5980	−	74.0810i	0.0754247	−	0.141107i
$526$		0			0
$527$			59.8665i			0.113599i
$528$		0			0
$529$	−367.000			−0.693762
$530$		0			0
$531$	240.416	+	359.822i	0.452761	+	0.677631i
$532$		0			0
$533$	−316.784			−0.594341
$534$		0			0
$535$	−816.000			−1.52523
$536$		0			0
$537$	452.000	−	845.615i	0.841713	−	1.57470i
$538$		0			0
$539$	280.014			0.519507
$540$		0			0
$541$			772.559i			1.42802i	0.700135	+	0.714011i	$0.253123\pi$
$541$			772.559i			1.42802i	−0.700135	+	0.714011i	$0.746877\pi$
$542$		0			0
$543$	−308.000	−	164.633i	−0.567219	−	0.303191i
$544$		0			0
$545$		−	299.333i		−	0.549234i
$546$		0			0
$547$			502.693i			0.919000i	0.888178	+	0.459500i	$0.151971\pi$
$547$			502.693i			0.919000i	−0.888178	+	0.459500i	$0.848029\pi$
$548$		0			0
$549$	712.764	−	476.235i	1.29829	−	0.867459i
$550$		0			0
$551$		−	89.7998i		−	0.162976i
$552$		0			0
$553$	496.000			0.896926
$554$		0			0
$555$	−791.960	−	423.320i	−1.42695	−	0.762739i
$556$		0			0
$557$	−526.087			−0.944502			−0.472251	−	0.881464i	$-0.656558\pi$
$557$	−526.087			−0.944502			−0.472251	+	0.881464i	$0.656558\pi$
$558$		0			0
$559$	−56.0000			−0.100179
$560$		0			0
$561$	336.000	+	179.600i	0.598930	+	0.320142i
$562$		0			0
$563$	−8.48528			−0.0150715			−0.00753577	−	0.999972i	$-0.502399\pi$
$563$	−8.48528			−0.0150715			−0.00753577	+	0.999972i	$0.502399\pi$
$564$		0			0
$565$		−	507.984i		−	0.899087i
$566$		0			0
$567$	124.000	−	299.333i	0.218695	−	0.527923i
$568$		0			0
$569$			359.199i			0.631281i	0.948879	+	0.315641i	$0.102219\pi$
$569$			359.199i			0.631281i	−0.948879	+	0.315641i	$0.897781\pi$
$570$		0			0
$571$		−	5.29150i		−	0.00926708i	−0.999989	−	0.00463354i	$-0.998525\pi$
$571$		−	5.29150i		−	0.00926708i	0.999989	−	0.00463354i	$-0.00147491\pi$
$572$		0			0
$573$	158.392	+	84.6640i	0.276426	+	0.147756i
$574$		0			0
$575$			209.533i			0.364405i
$576$		0			0
$577$	−18.0000			−0.0311958			−0.0155979	−	0.999878i	$-0.504965\pi$
$577$	−18.0000			−0.0311958			−0.0155979	+	0.999878i	$0.504965\pi$
$578$		0			0
$579$	144.250	−	269.867i	0.249136	−	0.466091i
$580$		0			0
$581$	−11.3137			−0.0194728
$582$		0			0
$583$	432.000			0.740995
$584$		0			0
$585$	−448.000	+	299.333i	−0.765812	+	0.511680i
$586$		0			0
$587$	121.622			0.207193			0.103597	−	0.994619i	$-0.466965\pi$
$587$	121.622			0.207193			0.103597	+	0.994619i	$0.466965\pi$
$588$		0			0
$589$		−	21.1660i		−	0.0359355i
$590$		0			0
$591$	−344.000	+	643.565i	−0.582064	+	1.08894i
$592$		0			0
$593$			718.398i			1.21146i	0.795669	+	0.605732i	$0.207120\pi$
$593$			718.398i			1.21146i	−0.795669	+	0.605732i	$0.792880\pi$
$594$		0			0
$595$			338.656i			0.569170i
$596$		0			0
$597$	−265.872	+	497.401i	−0.445347	+	0.833168i
$598$		0			0
$599$		−	688.465i		−	1.14936i	−0.818379	−	0.574679i	$-0.805127\pi$
$599$		−	688.465i		−	1.14936i	0.818379	−	0.574679i	$-0.194873\pi$
$600$		0			0
$601$	−358.000			−0.595674			−0.297837	−	0.954617i	$-0.596265\pi$
$601$	−358.000			−0.595674			−0.297837	+	0.954617i	$0.596265\pi$
$602$		0			0
$603$	356.382	−	238.118i	0.591015	−	0.394888i
$604$		0			0
$605$	277.186			0.458158
$606$		0			0
$607$	884.000			1.45634			0.728171	−	0.685395i	$-0.240371\pi$
$607$	884.000			1.45634			0.728171	+	0.685395i	$0.240371\pi$
$608$		0			0
$609$	96.0000	−	179.600i	0.157635	−	0.294909i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	391.571i		−	0.638778i	−0.947624	−	0.319389i	$-0.896522\pi$
$613$		−	391.571i		−	0.638778i	0.947624	−	0.319389i	$-0.103478\pi$
$614$		0			0
$615$	448.000	+	239.466i	0.728455	+	0.389376i
$616$		0			0
$617$			104.766i			0.169800i	0.996389	+	0.0848998i	$0.0270570\pi$
$617$			104.766i			0.169800i	−0.996389	+	0.0848998i	$0.972943\pi$
$618$		0			0
$619$		−	682.604i		−	1.10275i	−0.834257	−	0.551376i	$-0.814103\pi$
$619$		−	682.604i		−	1.10275i	0.834257	−	0.551376i	$-0.185897\pi$
$620$		0			0
$621$	79.1960	+	804.308i	0.127530	+	1.29518i
$622$		0			0
$623$			419.066i			0.672658i
$624$		0			0
$625$	−751.000			−1.20160
$626$		0			0
$627$	−118.794	−	63.4980i	−0.189464	−	0.101273i
$628$		0			0
$629$	791.960			1.25908
$630$		0			0
$631$	428.000			0.678288			0.339144	−	0.940734i	$-0.389863\pi$
$631$	428.000			0.678288			0.339144	+	0.940734i	$0.389863\pi$
$632$		0			0
$633$	658.000	+	351.716i	1.03949	+	0.555633i
$634$		0			0
$635$	429.921			0.677041
$636$		0			0
$637$		−	349.239i		−	0.548256i
$638$		0			0
$639$	−672.000	+	448.999i	−1.05164	+	0.702659i
$640$		0			0
$641$		−	793.231i		−	1.23749i	−0.785592	−	0.618745i	$-0.787641\pi$
$641$		−	793.231i		−	1.23749i	0.785592	−	0.618745i	$-0.212359\pi$
$642$		0			0
$643$		−	851.932i		−	1.32493i	−0.749092	−	0.662467i	$-0.769510\pi$
$643$		−	851.932i		−	1.32493i	0.749092	−	0.662467i	$-0.230490\pi$
$644$		0			0
$645$	79.1960	+	42.3320i	0.122784	+	0.0656310i
$646$		0			0
$647$			448.999i			0.693970i	0.937871	+	0.346985i	$0.112795\pi$
$647$			448.999i			0.693970i	−0.937871	+	0.346985i	$0.887205\pi$
$648$		0			0
$649$	408.000			0.628659
$650$		0			0
$651$	22.6274	−	42.3320i	0.0347579	−	0.0650261i
$652$		0			0
$653$	1103.09			1.68926			0.844630	−	0.535351i	$-0.179821\pi$
$653$	1103.09			1.68926			0.844630	+	0.535351i	$0.179821\pi$
$654$		0			0
$655$	80.0000			0.122137
$656$		0			0
$657$	30.0000	+	44.8999i	0.0456621	+	0.0683408i
$658$		0			0
$659$	924.896			1.40348			0.701742	−	0.712431i	$-0.252406\pi$
$659$	924.896			1.40348			0.701742	+	0.712431i	$0.252406\pi$
$660$		0			0
$661$		−	433.903i		−	0.656435i	−0.944602	−	0.328217i	$-0.893552\pi$
$661$		−	433.903i		−	0.656435i	0.944602	−	0.328217i	$-0.106448\pi$
$662$		0			0
$663$	224.000	−	419.066i	0.337858	−	0.632075i
$664$		0			0
$665$		−	119.733i		−	0.180050i
$666$		0			0
$667$			507.984i			0.761596i
$668$		0			0
$669$	265.872	−	497.401i	0.397417	−	0.743500i
$670$		0			0
$671$		−	808.198i		−	1.20447i
$672$		0			0
$673$	566.000			0.841010			0.420505	−	0.907290i	$-0.361853\pi$
$673$	566.000			0.841010			0.420505	+	0.907290i	$0.361853\pi$
$674$		0			0
$675$	188.090	−	18.5203i	0.278652	−	0.0274374i
$676$		0			0
$677$	−797.616			−1.17816			−0.589082	−	0.808074i	$-0.700510\pi$
$677$	−797.616			−1.17816			−0.589082	+	0.808074i	$0.700510\pi$
$678$		0			0
$679$	−472.000			−0.695140
$680$		0			0
$681$	−548.000	+	1025.21i	−0.804699	+	1.50545i
$682$		0			0
$683$	−404.465			−0.592189			−0.296094	−	0.955159i	$-0.595684\pi$
$683$	−404.465			−0.592189			−0.296094	+	0.955159i	$0.595684\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−644.000	−	344.232i	−0.937409	−	0.501066i
$688$		0			0
$689$		−	538.799i		−	0.782001i
$690$		0			0
$691$			121.705i			0.176128i	0.996115	+	0.0880641i	$0.0280680\pi$
$691$			121.705i			0.176128i	−0.996115	+	0.0880641i	$0.971932\pi$
$692$		0			0
$693$	−169.706	−	253.992i	−0.244885	−	0.366511i
$694$		0			0
$695$		−	688.465i		−	0.990597i
$696$		0			0
$697$	−448.000			−0.642755
$698$		0			0
$699$	277.186	+	148.162i	0.396546	+	0.211963i
$700$		0			0
$701$	333.754			0.476112			0.238056	−	0.971251i	$-0.423490\pi$
$701$	333.754			0.476112			0.238056	+	0.971251i	$0.423490\pi$
$702$		0			0
$703$	−280.000			−0.398293
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−248.902			−0.352053
$708$		0			0
$709$			370.405i			0.522433i	0.965280	+	0.261217i	$0.0841237\pi$
$709$			370.405i			0.522433i	−0.965280	+	0.261217i	$0.915876\pi$
$710$		0			0
$711$	620.000	+	927.931i	0.872011	+	1.30511i
$712$		0			0
$713$			119.733i			0.167929i
$714$		0			0
$715$			507.984i			0.710467i
$716$		0			0
$717$	−950.352	−	507.984i	−1.32546	−	0.708486i
$718$		0			0
$719$			1017.73i			1.41548i	0.706473	+	0.707740i	$0.250285\pi$
$719$			1017.73i			1.41548i	−0.706473	+	0.707740i	$0.749715\pi$
$720$		0			0
$721$	−432.000			−0.599168
$722$		0			0
$723$	−172.534	+	322.782i	−0.238636	+	0.446448i
$724$		0			0
$725$	118.794			0.163854
$726$		0			0
$727$	−1140.00			−1.56809			−0.784044	−	0.620705i	$-0.786846\pi$
$727$	−1140.00			−1.56809			−0.784044	+	0.620705i	$0.786846\pi$
$728$		0			0
$729$	715.000	−	142.183i	0.980796	−	0.195038i
$730$		0			0
$731$	−79.1960			−0.108339
$732$		0			0
$733$		−	370.405i		−	0.505328i	−0.967554	−	0.252664i	$-0.918693\pi$
$733$		−	370.405i		−	0.505328i	0.967554	−	0.252664i	$-0.0813067\pi$
$734$		0			0
$735$	−264.000	+	493.899i	−0.359184	+	0.671971i
$736$		0			0
$737$		−	404.099i		−	0.548303i
$738$		0			0
$739$			5.29150i			0.00716036i	0.999994	+	0.00358018i	$0.00113961\pi$
$739$			5.29150i			0.00716036i	−0.999994	+	0.00358018i	$0.998860\pi$
$740$		0			0
$741$	−79.1960	+	148.162i	−0.106877	+	0.199949i
$742$		0			0
$743$		−	748.331i		−	1.00718i	−0.863944	−	0.503588i	$-0.832013\pi$
$743$		−	748.331i		−	1.00718i	0.863944	−	0.503588i	$-0.167987\pi$
$744$		0			0
$745$	1056.00			1.41745
$746$		0			0
$747$	−14.1421	−	21.1660i	−0.0189319	−	0.0283347i
$748$		0			0
$749$	−576.999			−0.770359
$750$		0			0
$751$	1108.00			1.47537			0.737683	−	0.675147i	$-0.235920\pi$
$751$	1108.00			1.47537			0.737683	+	0.675147i	$0.235920\pi$
$752$		0			0
$753$	−228.000	+	426.549i	−0.302789	+	0.566466i
$754$		0			0
$755$	−339.411			−0.449551
$756$		0			0
$757$			1047.72i			1.38404i	0.721879	+	0.692019i	$0.243279\pi$
$757$			1047.72i			1.38404i	−0.721879	+	0.692019i	$0.756721\pi$
$758$		0			0
$759$	672.000	+	359.199i	0.885375	+	0.473253i
$760$		0			0
$761$		−	1287.13i		−	1.69137i	−0.533685	−	0.845683i	$-0.679193\pi$
$761$		−	1287.13i		−	1.69137i	0.533685	−	0.845683i	$-0.320807\pi$
$762$		0			0
$763$		−	211.660i		−	0.277405i
$764$		0			0
$765$	−633.568	+	423.320i	−0.828193	+	0.553360i
$766$		0			0
$767$		−	508.865i		−	0.663449i
$768$		0			0
$769$	538.000			0.699610			0.349805	−	0.936823i	$-0.386248\pi$
$769$	538.000			0.699610			0.349805	+	0.936823i	$0.386248\pi$
$770$		0			0
$771$	−1187.94	−	634.980i	−1.54078	−	0.823580i
$772$		0			0
$773$	62.2254			0.0804986			0.0402493	−	0.999190i	$-0.487185\pi$
$773$	62.2254			0.0804986			0.0402493	+	0.999190i	$0.487185\pi$
$774$		0			0
$775$	28.0000			0.0361290
$776$		0			0
$777$	−560.000	−	299.333i	−0.720721	−	0.385241i
$778$		0			0
$779$	158.392			0.203327
$780$		0			0
$781$			761.976i			0.975642i
$782$		0			0
$783$	456.000	−	44.8999i	0.582375	−	0.0573434i
$784$		0			0
$785$			658.532i			0.838894i
$786$		0			0
$787$			1476.33i			1.87589i	0.346778	+	0.937947i	$0.387276\pi$
$787$			1476.33i			1.87589i	−0.346778	+	0.937947i	$0.612724\pi$
$788$		0			0
$789$	554.372	+	296.324i	0.702626	+	0.375569i
$790$		0			0
$791$		−	359.199i		−	0.454108i
$792$		0			0
$793$	−1008.00			−1.27112
$794$		0			0
$795$	−407.294	+	761.976i	−0.512319	+	0.958461i
$796$		0			0
$797$	−1250.16			−1.56859			−0.784294	−	0.620389i	$-0.786975\pi$
$797$	−1250.16			−1.56859			−0.784294	+	0.620389i	$0.786975\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−784.000	+	523.832i	−0.978777	+	0.653973i
$802$		0			0
$803$	50.9117			0.0634019
$804$		0			0
$805$			677.312i			0.841382i
$806$		0			0
$807$	72.0000	−	134.700i	0.0892193	−	0.166914i
$808$		0			0
$809$			448.999i			0.555005i	0.960725	+	0.277502i	$0.0895066\pi$
$809$			448.999i			0.555005i	−0.960725	+	0.277502i	$0.910493\pi$
$810$		0			0
$811$		−	1391.67i		−	1.71599i	−0.513661	−	0.857993i	$-0.671711\pi$
$811$		−	1391.67i		−	1.71599i	0.513661	−	0.857993i	$-0.328289\pi$
$812$		0			0
$813$	492.146	−	920.721i	0.605346	−	1.13250i
$814$		0			0
$815$		−	1646.33i		−	2.02004i
$816$		0			0
$817$	28.0000			0.0342717
$818$		0			0
$819$	−316.784	+	211.660i	−0.386793	+	0.258437i
$820$		0			0
$821$	469.519			0.571887			0.285943	−	0.958247i	$-0.407693\pi$
$821$	469.519			0.571887			0.285943	+	0.958247i	$0.407693\pi$
$822$		0			0
$823$	−1268.00			−1.54070			−0.770352	−	0.637618i	$-0.779920\pi$
$823$	−1268.00			−1.54070			−0.770352	+	0.637618i	$0.779920\pi$
$824$		0			0
$825$	84.0000	−	157.150i	0.101818	−	0.190484i
$826$		0			0
$827$	−1145.51			−1.38514			−0.692571	−	0.721349i	$-0.743522\pi$
$827$	−1145.51			−1.38514			−0.692571	+	0.721349i	$0.743522\pi$
$828$		0			0
$829$		−	1217.05i		−	1.46809i	−0.679101	−	0.734044i	$-0.737630\pi$
$829$		−	1217.05i		−	1.46809i	0.679101	−	0.734044i	$-0.262370\pi$
$830$		0			0
$831$	−644.000	−	344.232i	−0.774970	−	0.414239i
$832$		0			0
$833$		−	493.899i		−	0.592916i
$834$		0			0
$835$		−	1862.61i		−	2.23067i
$836$		0			0
$837$	107.480	−	10.5830i	0.128411	−	0.0126440i
$838$		0			0
$839$			1227.26i			1.46277i	0.681965	+	0.731385i	$0.261126\pi$
$839$			1227.26i			1.46277i	−0.681965	+	0.731385i	$0.738874\pi$
$840$		0			0
$841$	−553.000			−0.657551
$842$		0			0
$843$	831.558	+	444.486i	0.986427	+	0.527267i
$844$		0			0
$845$	−322.441			−0.381587
$846$		0			0
$847$	196.000			0.231405
$848$		0			0
$849$	238.000	+	127.216i	0.280330	+	0.149843i
$850$		0			0
$851$	1583.92			1.86124
$852$		0			0
$853$			1047.72i			1.22827i	0.789200	+	0.614137i	$0.210496\pi$
$853$			1047.72i			1.22827i	−0.789200	+	0.614137i	$0.789504\pi$
$854$		0			0
$855$	224.000	−	149.666i	0.261988	−	0.175048i
$856$		0			0
$857$			359.199i			0.419135i	0.977794	+	0.209568i	$0.0672057\pi$
$857$			359.199i			0.419135i	−0.977794	+	0.209568i	$0.932794\pi$
$858$		0			0
$859$			513.276i			0.597527i	0.954327	+	0.298764i	$0.0965742\pi$
$859$			513.276i			0.597527i	−0.954327	+	0.298764i	$0.903426\pi$
$860$		0			0
$861$	316.784	+	169.328i	0.367925	+	0.196664i
$862$		0			0
$863$			538.799i			0.624332i	0.950028	+	0.312166i	$0.101055\pi$
$863$			538.799i			0.624332i	−0.950028	+	0.312166i	$0.898945\pi$
$864$		0			0
$865$	1056.00			1.22081
$866$		0			0
$867$	−91.9239	+	171.974i	−0.106025	+	0.198355i
$868$		0			0
$869$	1052.17			1.21079
$870$		0			0
$871$	−504.000			−0.578645
$872$		0			0
$873$	−590.000	−	883.031i	−0.675830	−	1.01149i
$874$		0			0
$875$	−407.294			−0.465478
$876$		0			0
$877$		−	1344.04i		−	1.53254i	−0.642516	−	0.766272i	$-0.722109\pi$
$877$		−	1344.04i		−	1.53254i	0.642516	−	0.766272i	$-0.277891\pi$
$878$		0			0
$879$	−24.0000	+	44.8999i	−0.0273038	+	0.0510806i
$880$		0			0
$881$			748.331i			0.849411i	0.905331	+	0.424706i	$0.139622\pi$
$881$			748.331i			0.849411i	−0.905331	+	0.424706i	$0.860378\pi$
$882$		0			0
$883$			418.029i			0.473419i	0.971581	+	0.236709i	$0.0760689\pi$
$883$			418.029i			0.473419i	−0.971581	+	0.236709i	$0.923931\pi$
$884$		0			0
$885$	−384.666	+	719.644i	−0.434651	+	0.813157i
$886$		0			0
$887$			1167.40i			1.31612i	0.752966	+	0.658059i	$0.228622\pi$
$887$			1167.40i			1.31612i	−0.752966	+	0.658059i	$0.771378\pi$
$888$		0			0
$889$	304.000			0.341957
$890$		0			0
$891$	263.044	−	634.980i	0.295223	−	0.712660i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	1808.00			2.02011
$896$		0			0
$897$	448.000	−	838.131i	0.499443	−	0.934372i
$898$		0			0
$899$	67.8823			0.0755086
$900$		0			0
$901$		−	761.976i		−	0.845701i
$902$		0			0
$903$	56.0000	+	29.9333i	0.0620155	+	0.0331487i
$904$		0			0
$905$		−	658.532i		−	0.727659i
$906$		0			0
$907$			597.940i			0.659250i	0.944112	+	0.329625i	$0.106922\pi$
$907$			597.940i			0.659250i	−0.944112	+	0.329625i	$0.893078\pi$
$908$		0			0
$909$	−311.127	−	465.652i	−0.342274	−	0.512269i
$910$		0			0
$911$			359.199i			0.394291i	0.980374	+	0.197146i	$0.0631671\pi$
$911$			359.199i			0.394291i	−0.980374	+	0.197146i	$0.936833\pi$
$912$		0			0
$913$	−24.0000			−0.0262870
$914$		0			0
$915$	1425.53	+	761.976i	1.55795	+	0.832761i
$916$		0			0
$917$	56.5685			0.0616887
$918$		0			0
$919$	780.000			0.848749			0.424374	−	0.905487i	$-0.360494\pi$
$919$	780.000			0.848749			0.424374	+	0.905487i	$0.360494\pi$
$920$		0			0
$921$	1218.00	+	651.048i	1.32248	+	0.706893i
$922$		0			0
$923$	950.352			1.02963
$924$		0			0
$925$		−	370.405i		−	0.400438i
$926$		0			0
$927$	−540.000	−	808.198i	−0.582524	−	0.871842i
$928$		0			0
$929$			1361.96i			1.46605i	0.680200	+	0.733027i	$0.261893\pi$
$929$			1361.96i			1.46605i	−0.680200	+	0.733027i	$0.738107\pi$
$930$		0			0
$931$			174.620i			0.187561i
$932$		0			0
$933$	395.980	+	211.660i	0.424416	+	0.226860i
$934$		0			0
$935$			718.398i			0.768340i
$936$		0			0
$937$	566.000			0.604055			0.302028	−	0.953299i	$-0.402336\pi$
$937$	566.000			0.604055			0.302028	+	0.953299i	$0.402336\pi$
$938$		0			0
$939$	794.788	−	1486.91i	0.846420	−	1.58351i
$940$		0			0
$941$	424.264			0.450865			0.225433	−	0.974259i	$-0.427620\pi$
$941$	424.264			0.450865			0.225433	+	0.974259i	$0.427620\pi$
$942$		0			0
$943$	−896.000			−0.950159
$944$		0			0
$945$	608.000	−	59.8665i	0.643386	−	0.0633508i
$946$		0			0
$947$	−466.690			−0.492809			−0.246405	−	0.969167i	$-0.579249\pi$
$947$	−466.690			−0.492809			−0.246405	+	0.969167i	$0.579249\pi$
$948$		0			0
$949$		−	63.4980i		−	0.0669105i
$950$		0			0
$951$	8.00000	−	14.9666i	0.00841220	−	0.0157378i
$952$		0			0
$953$			897.998i			0.942285i	0.882057	+	0.471143i	$0.156158\pi$
$953$			897.998i			0.942285i	−0.882057	+	0.471143i	$0.843842\pi$
$954$		0			0
$955$			338.656i			0.354614i
$956$		0			0
$957$	203.647	−	380.988i	0.212797	−	0.398107i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−945.000			−0.983351
$962$		0			0
$963$	−721.249	−	1079.47i	−0.748960	−	1.12094i
$964$		0			0
$965$	576.999			0.597927
$966$		0			0
$967$	−692.000			−0.715615			−0.357808	−	0.933795i	$-0.616476\pi$
$967$	−692.000			−0.715615			−0.357808	+	0.933795i	$0.616476\pi$
$968$		0			0
$969$	−112.000	+	209.533i	−0.115583	+	0.216236i
$970$		0			0
$971$	−449.720			−0.463151			−0.231576	−	0.972817i	$-0.574388\pi$
$971$	−449.720			−0.463151			−0.231576	+	0.972817i	$0.574388\pi$
$972$		0			0
$973$		−	486.818i		−	0.500327i
$974$		0			0
$975$	−196.000	−	104.766i	−0.201026	−	0.107453i
$976$		0			0
$977$		−	883.031i		−	0.903819i	−0.892064	−	0.451909i	$-0.850743\pi$
$977$		−	883.031i		−	0.903819i	0.892064	−	0.451909i	$-0.149257\pi$
$978$		0			0
$979$			888.972i			0.908041i
$980$		0			0
$981$	395.980	−	264.575i	0.403649	−	0.269699i
$982$		0			0
$983$			329.266i			0.334960i	0.985875	+	0.167480i	$0.0535630\pi$
$983$			329.266i			0.334960i	−0.985875	+	0.167480i	$0.946437\pi$
$984$		0			0
$985$	−1376.00			−1.39695
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−158.392			−0.160154
$990$		0			0
$991$	340.000			0.343088			0.171544	−	0.985176i	$-0.445124\pi$
$991$	340.000			0.343088			0.171544	+	0.985176i	$0.445124\pi$
$992$		0			0
$993$	−1106.00	−	591.182i	−1.11380	−	0.595349i
$994$		0			0
$995$	−1063.49			−1.06883
$996$		0			0
$997$			1555.70i			1.56038i	0.625541	+	0.780191i	$0.284878\pi$
$997$			1555.70i			1.56038i	−0.625541	+	0.780191i	$0.715122\pi$
$998$		0			0
$999$	−140.000	−	1421.83i	−0.140140	−	1.42325i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.h.c.17.2		4
3.2	odd	2	inner	96.3.h.c.17.4		4
4.3	odd	2		24.3.h.c.5.4	yes	4
8.3	odd	2		24.3.h.c.5.2	yes	4
8.5	even	2	inner	96.3.h.c.17.3		4
12.11	even	2		24.3.h.c.5.1	✓	4
16.3	odd	4		768.3.e.i.257.4		4
16.5	even	4		768.3.e.l.257.4		4
16.11	odd	4		768.3.e.i.257.1		4
16.13	even	4		768.3.e.l.257.1		4
24.5	odd	2	inner	96.3.h.c.17.1		4
24.11	even	2		24.3.h.c.5.3	yes	4
48.5	odd	4		768.3.e.l.257.3		4
48.11	even	4		768.3.e.i.257.2		4
48.29	odd	4		768.3.e.l.257.2		4
48.35	even	4		768.3.e.i.257.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.h.c.5.1	✓	4	12.11	even	2
24.3.h.c.5.2	yes	4	8.3	odd	2
24.3.h.c.5.3	yes	4	24.11	even	2
24.3.h.c.5.4	yes	4	4.3	odd	2
96.3.h.c.17.1		4	24.5	odd	2	inner
96.3.h.c.17.2		4	1.1	even	1	trivial
96.3.h.c.17.3		4	8.5	even	2	inner
96.3.h.c.17.4		4	3.2	odd	2	inner
768.3.e.i.257.1		4	16.11	odd	4
768.3.e.i.257.2		4	48.11	even	4
768.3.e.i.257.3		4	48.35	even	4
768.3.e.i.257.4		4	16.3	odd	4
768.3.e.l.257.1		4	16.13	even	4
768.3.e.l.257.2		4	48.29	odd	4
768.3.e.l.257.3		4	48.5	odd	4
768.3.e.l.257.4		4	16.5	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:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	96.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 2.64575i) q^{3} -5.65685 q^{5} -4.00000 q^{7} +(-5.00000 - 7.48331i) q^{9} +O(q^{10})\)	\(q+(-1.41421 + 2.64575i) q^{3} -5.65685 q^{5} -4.00000 q^{7} +(-5.00000 - 7.48331i) q^{9} -8.48528 q^{11} +10.5830i q^{13} +(8.00000 - 14.9666i) q^{15} +14.9666i q^{17} -5.29150i q^{19} +(5.65685 - 10.5830i) q^{21} +29.9333i q^{23} +7.00000 q^{25} +(26.8701 - 2.64575i) q^{27} +16.9706 q^{29} +4.00000 q^{31} +(12.0000 - 22.4499i) q^{33} +22.6274 q^{35} -52.9150i q^{37} +(-28.0000 - 14.9666i) q^{39} +29.9333i q^{41} +5.29150i q^{43} +(28.2843 + 42.3320i) q^{45} -33.0000 q^{49} +(-39.5980 - 21.1660i) q^{51} -50.9117 q^{53} +48.0000 q^{55} +(14.0000 + 7.48331i) q^{57} -48.0833 q^{59} +95.2470i q^{61} +(20.0000 + 29.9333i) q^{63} -59.8665i q^{65} +47.6235i q^{67} +(-79.1960 - 42.3320i) q^{69} -89.7998i q^{71} -6.00000 q^{73} +(-9.89949 + 18.5203i) q^{75} +33.9411 q^{77} -124.000 q^{79} +(-31.0000 + 74.8331i) q^{81} +2.82843 q^{83} -84.6640i q^{85} +(-24.0000 + 44.8999i) q^{87} -104.766i q^{89} -42.3320i q^{91} +(-5.65685 + 10.5830i) q^{93} +29.9333i q^{95} +118.000 q^{97} +(42.4264 + 63.4980i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{7} - 20 q^{9}+O(q^{10}) \) 4 * q - 16 * q^7 - 20 * q^9	\( 4 q - 16 q^{7} - 20 q^{9} + 32 q^{15} + 28 q^{25} + 16 q^{31} + 48 q^{33} - 112 q^{39} - 132 q^{49} + 192 q^{55} + 56 q^{57} + 80 q^{63} - 24 q^{73} - 496 q^{79} - 124 q^{81} - 96 q^{87} + 472 q^{97}+O(q^{100}) \) 4 * q - 16 * q^7 - 20 * q^9 + 32 * q^15 + 28 * q^25 + 16 * q^31 + 48 * q^33 - 112 * q^39 - 132 * q^49 + 192 * q^55 + 56 * q^57 + 80 * q^63 - 24 * q^73 - 496 * q^79 - 124 * q^81 - 96 * q^87 + 472 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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