LMFDB - Embedded newform 2400.2.m.c.1199.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1199,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2400.1199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.m (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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} + 4x^{12} + 12x^{10} + 16x^{8} + 48x^{6} + 64x^{4} + 64x^{2} + 256 $ x^16 + x^14 + 4x^12 + 12x^10 + 16x^8 + 48x^6 + 64x^4 + 64x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1199.12
Root			$-1.15595 - 0.814732i$ of defining polynomial
Character	$\chi$	$=$	2400.1199
Dual form			2400.2.m.c.1199.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.887900 + 1.48716i) q^{3} +0.797253 q^{7} +(-1.42327 + 2.64089i) q^{9} +O(q^{10})$	$q+(0.887900 + 1.48716i) q^{3} +0.797253 q^{7} +(-1.42327 + 2.64089i) q^{9} +0.320548i q^{11} -4.30324 q^{13} -2.57305 q^{17} -6.10546 q^{19} +(0.707881 + 1.18564i) q^{21} -3.13115i q^{23} +(-5.19114 + 0.228229i) q^{27} -8.79516 q^{29} -9.90557i q^{31} +(-0.476705 + 0.284615i) q^{33} +8.49593 q^{37} +(-3.82085 - 6.39959i) q^{39} +5.28178i q^{41} -2.97431i q^{43} -6.56192i q^{47} -6.36439 q^{49} +(-2.28461 - 3.82653i) q^{51} +3.94862i q^{53} +(-5.42104 - 9.07977i) q^{57} +12.4786i q^{59} +8.83339i q^{61} +(-1.13470 + 2.10546i) q^{63} -4.66738i q^{67} +(4.65651 - 2.78015i) q^{69} +3.43077 q^{71} +1.43077i q^{73} +0.255558i q^{77} -2.89360i q^{79} +(-4.94862 - 7.51739i) q^{81} -3.37031 q^{83} +(-7.80922 - 13.0798i) q^{87} +13.7526i q^{89} -3.43077 q^{91} +(14.7311 - 8.79516i) q^{93} -4.26230i q^{97} +(-0.846533 - 0.456225i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{19} - 8 q^{21} + 32 q^{39} + 32 q^{49} - 40 q^{51} - 40 q^{69} + 48 q^{71} + 16 q^{81} - 48 q^{91} + 32 q^{99}+O(q^{100}) $ 16 * q - 8 * q^19 - 8 * q^21 + 32 * q^39 + 32 * q^49 - 40 * q^51 - 40 * q^69 + 48 * q^71 + 16 * q^81 - 48 * q^91 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$3$	0.887900	+	1.48716i	0.512629	+	0.858610i
$3$	0.887900	+	1.48716i	0.512629	+	0.858610i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.797253			0.301333			0.150667	−	0.988585i	$-0.451858\pi$
$7$	0.797253			0.301333			0.150667	+	0.988585i	$0.451858\pi$
$8$		0			0
$9$	−1.42327	+	2.64089i	−0.474422	+	0.880297i
$10$		0			0
$11$			0.320548i			0.0966489i	0.998832	+	0.0483245i	$0.0153881\pi$
$11$			0.320548i			0.0966489i	−0.998832	+	0.0483245i	$0.984612\pi$
$12$		0			0
$13$	−4.30324			−1.19350			−0.596752	−	0.802426i	$-0.703542\pi$
$13$	−4.30324			−1.19350			−0.596752	+	0.802426i	$0.703542\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.57305			−0.624057			−0.312029	−	0.950073i	$-0.601008\pi$
$17$	−2.57305			−0.624057			−0.312029	+	0.950073i	$0.601008\pi$
$18$		0			0
$19$	−6.10546			−1.40069			−0.700344	−	0.713805i	$-0.746970\pi$
$19$	−6.10546			−1.40069			−0.700344	+	0.713805i	$0.746970\pi$
$20$		0			0
$21$	0.707881	+	1.18564i	0.154472	+	0.258728i
$22$		0			0
$23$		−	3.13115i		−	0.652889i	−0.945216	−	0.326445i	$-0.894149\pi$
$23$		−	3.13115i		−	0.652889i	0.945216	−	0.326445i	$-0.105851\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−5.19114	+	0.228229i	−0.999035	+	0.0439227i
$28$		0			0
$29$	−8.79516			−1.63322			−0.816610	−	0.577190i	$-0.804149\pi$
$29$	−8.79516			−1.63322			−0.816610	+	0.577190i	$0.804149\pi$
$30$		0			0
$31$		−	9.90557i		−	1.77909i	−0.456845	−	0.889546i	$-0.651020\pi$
$31$		−	9.90557i		−	1.77909i	0.456845	−	0.889546i	$-0.348980\pi$
$32$		0			0
$33$	−0.476705	+	0.284615i	−0.0829837	+	0.0495451i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.49593			1.39672			0.698362	−	0.715745i	$-0.253913\pi$
$37$	8.49593			1.39672			0.698362	+	0.715745i	$0.253913\pi$
$38$		0			0
$39$	−3.82085	−	6.39959i	−0.611825	−	1.02475i
$40$		0			0
$41$			5.28178i			0.824876i	0.910986	+	0.412438i	$0.135323\pi$
$41$			5.28178i			0.824876i	−0.910986	+	0.412438i	$0.864677\pi$
$42$		0			0
$43$		−	2.97431i		−	0.453578i	−0.973944	−	0.226789i	$-0.927177\pi$
$43$		−	2.97431i		−	0.453578i	0.973944	−	0.226789i	$-0.0728228\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.56192i		−	0.957154i	−0.878046	−	0.478577i	$-0.841153\pi$
$47$		−	6.56192i		−	0.957154i	0.878046	−	0.478577i	$-0.158847\pi$
$48$		0			0
$49$	−6.36439			−0.909198
$50$		0			0
$51$	−2.28461	−	3.82653i	−0.319910	−	0.535822i
$52$		0			0
$53$			3.94862i			0.542385i	0.962525	+	0.271193i	$0.0874180\pi$
$53$			3.94862i			0.542385i	−0.962525	+	0.271193i	$0.912582\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.42104	−	9.07977i	−0.718034	−	1.20265i
$58$		0			0
$59$			12.4786i			1.62458i	0.583255	+	0.812289i	$0.301779\pi$
$59$			12.4786i			1.62458i	−0.583255	+	0.812289i	$0.698221\pi$
$60$		0			0
$61$			8.83339i			1.13100i	0.824749	+	0.565500i	$0.191317\pi$
$61$			8.83339i			1.13100i	−0.824749	+	0.565500i	$0.808683\pi$
$62$		0			0
$63$	−1.13470	+	2.10546i	−0.142959	+	0.265263i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	4.66738i		−	0.570211i	−0.958496	−	0.285106i	$-0.907971\pi$
$67$		−	4.66738i		−	0.570211i	0.958496	−	0.285106i	$-0.0920286\pi$
$68$		0			0
$69$	4.65651	−	2.78015i	0.560577	−	0.334690i
$70$		0			0
$71$	3.43077			0.407158			0.203579	−	0.979059i	$-0.434743\pi$
$71$	3.43077			0.407158			0.203579	+	0.979059i	$0.434743\pi$
$72$		0			0
$73$			1.43077i			0.167459i	0.996489	+	0.0837295i	$0.0266832\pi$
$73$			1.43077i			0.167459i	−0.996489	+	0.0837295i	$0.973317\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.255558i			0.0291235i
$78$		0			0
$79$		−	2.89360i		−	0.325556i	−0.986663	−	0.162778i	$-0.947955\pi$
$79$		−	2.89360i		−	0.325556i	0.986663	−	0.162778i	$-0.0520454\pi$
$80$		0			0
$81$	−4.94862	−	7.51739i	−0.549847	−	0.835265i
$82$		0			0
$83$	−3.37031			−0.369939			−0.184970	−	0.982744i	$-0.559219\pi$
$83$	−3.37031			−0.369939			−0.184970	+	0.982744i	$0.559219\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−7.80922	−	13.0798i	−0.837236	−	1.40230i
$88$		0			0
$89$			13.7526i			1.45777i	0.684636	+	0.728885i	$0.259961\pi$
$89$			13.7526i			1.45777i	−0.684636	+	0.728885i	$0.740039\pi$
$90$		0			0
$91$	−3.43077			−0.359642
$92$		0			0
$93$	14.7311	−	8.79516i	1.52755	−	0.912015i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	4.26230i		−	0.432771i	−0.976308	−	0.216385i	$-0.930573\pi$
$97$		−	4.26230i		−	0.432771i	0.976308	−	0.216385i	$-0.0694267\pi$
$98$		0			0
$99$	−0.846533	−	0.456225i	−0.0850798	−	0.0458524i
$100$		0			0
$101$	−15.3130			−1.52370			−0.761851	−	0.647753i	$-0.775709\pi$
$101$	−15.3130			−1.52370			−0.761851	+	0.647753i	$0.775709\pi$
$102$		0			0
$103$	7.25936			0.715286			0.357643	−	0.933858i	$-0.383581\pi$
$103$	7.25936			0.715286			0.357643	+	0.933858i	$0.383581\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.2928			−1.28506			−0.642531	−	0.766260i	$-0.722116\pi$
$107$	−13.2928			−1.28506			−0.642531	+	0.766260i	$0.722116\pi$
$108$		0			0
$109$		−	3.41592i		−	0.327186i	−0.986528	−	0.163593i	$-0.947692\pi$
$109$		−	3.41592i		−	0.327186i	0.986528	−	0.163593i	$-0.0523084\pi$
$110$		0			0
$111$	7.54354	+	12.6348i	0.716001	+	1.19924i
$112$		0			0
$113$	−10.2261			−0.961992			−0.480996	−	0.876723i	$-0.659725\pi$
$113$	−10.2261			−0.961992			−0.480996	+	0.876723i	$0.659725\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.12465	−	11.3644i	0.566224	−	1.05064i
$118$		0			0
$119$	−2.05138			−0.188049
$120$		0			0
$121$	10.8972			0.990659
$122$		0			0
$123$	−7.85484	+	4.68970i	−0.708247	+	0.422856i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	4.98995			0.442786			0.221393	−	0.975185i	$-0.428940\pi$
$127$	4.98995			0.442786			0.221393	+	0.975185i	$0.428940\pi$
$128$		0			0
$129$	4.42327	−	2.64089i	0.389447	−	0.232518i
$130$		0			0
$131$			8.92702i			0.779958i	0.920824	+	0.389979i	$0.127518\pi$
$131$			8.92702i			0.779958i	−0.920824	+	0.389979i	$0.872482\pi$
$132$		0			0
$133$	−4.86760			−0.422074
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.61964			0.138375			0.0691877	−	0.997604i	$-0.477959\pi$
$137$	1.61964			0.138375			0.0691877	+	0.997604i	$0.477959\pi$
$138$		0			0
$139$	3.58761			0.304297			0.152148	−	0.988358i	$-0.451381\pi$
$139$	3.58761			0.304297			0.152148	+	0.988358i	$0.451381\pi$
$140$		0			0
$141$	9.75860	−	5.82633i	0.821822	−	0.490665i
$142$		0			0
$143$		−	1.37939i		−	0.115351i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.65094	−	9.46484i	−0.466082	−	0.780647i
$148$		0			0
$149$	−2.31367			−0.189543			−0.0947717	−	0.995499i	$-0.530212\pi$
$149$	−2.31367			−0.189543			−0.0947717	+	0.995499i	$0.530212\pi$
$150$		0			0
$151$			3.44347i			0.280225i	0.990136	+	0.140113i	$0.0447465\pi$
$151$			3.44347i			0.280225i	−0.990136	+	0.140113i	$0.955254\pi$
$152$		0			0
$153$	3.66214	−	6.79516i	0.296067	−	0.549356i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−9.17084			−0.731912			−0.365956	−	0.930632i	$-0.619258\pi$
$157$	−9.17084			−0.731912			−0.365956	+	0.930632i	$0.619258\pi$
$158$		0			0
$159$	−5.87222	+	3.50598i	−0.465697	+	0.278043i
$160$		0			0
$161$		−	2.49632i		−	0.196737i
$162$		0			0
$163$		−	10.6160i		−	0.831510i	−0.909477	−	0.415755i	$-0.863517\pi$
$163$		−	10.6160i		−	0.831510i	0.909477	−	0.415755i	$-0.136483\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			13.3353i			1.03192i	0.856613	+	0.515959i	$0.172564\pi$
$167$			13.3353i			1.03192i	−0.856613	+	0.515959i	$0.827436\pi$
$168$		0			0
$169$	5.51785			0.424450
$170$		0			0
$171$	8.68970	−	16.1239i	0.664518	−	1.23302i
$172$		0			0
$173$		−	13.8972i		−	1.05659i	−0.849061	−	0.528294i	$-0.822832\pi$
$173$		−	13.8972i		−	1.05659i	0.849061	−	0.528294i	$-0.177168\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−18.5577	+	11.0798i	−1.39488	+	0.832807i
$178$		0			0
$179$		−	5.18815i		−	0.387780i	−0.981023	−	0.193890i	$-0.937889\pi$
$179$		−	5.18815i		−	0.387780i	0.981023	−	0.193890i	$-0.0621105\pi$
$180$		0			0
$181$		−	2.59819i		−	0.193122i	−0.995327	−	0.0965610i	$-0.969216\pi$
$181$		−	2.59819i		−	0.193122i	0.995327	−	0.0965610i	$-0.0307843\pi$
$182$		0			0
$183$	−13.1366	+	7.84316i	−0.971087	+	0.579783i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	0.824788i		−	0.0603144i
$188$		0			0
$189$	−4.13865	+	0.181956i	−0.301043	+	0.0132354i
$190$		0			0
$191$	−12.2556			−0.886781			−0.443391	−	0.896329i	$-0.646225\pi$
$191$	−12.2556			−0.886781			−0.443391	+	0.896329i	$0.646225\pi$
$192$		0			0
$193$		−	8.26230i		−	0.594733i	−0.954763	−	0.297367i	$-0.903892\pi$
$193$		−	8.26230i		−	0.594733i	0.954763	−	0.297367i	$-0.0961083\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	10.8102i		−	0.770192i	−0.922876	−	0.385096i	$-0.874168\pi$
$197$		−	10.8102i		−	0.770192i	0.922876	−	0.385096i	$-0.125832\pi$
$198$		0			0
$199$			5.71287i			0.404975i	0.979285	+	0.202487i	$0.0649025\pi$
$199$			5.71287i			0.404975i	−0.979285	+	0.202487i	$0.935097\pi$
$200$		0			0
$201$	6.94112	−	4.14417i	0.489589	−	0.292307i
$202$		0			0
$203$	−7.01197			−0.492144
$204$		0			0
$205$		0			0
$206$		0			0
$207$	8.26902	+	4.45646i	0.574737	+	0.309745i
$208$		0			0
$209$		−	1.95709i		−	0.135375i
$210$		0			0
$211$	8.15684			0.561540			0.280770	−	0.959775i	$-0.409410\pi$
$211$	8.15684			0.561540			0.280770	+	0.959775i	$0.409410\pi$
$212$		0			0
$213$	3.04618	+	5.10209i	0.208721	+	0.349590i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	7.89725i		−	0.536100i
$218$		0			0
$219$	−2.12778	+	1.27038i	−0.143782	+	0.0858444i
$220$		0			0
$221$	11.0725			0.744814
$222$		0			0
$223$	20.6084			1.38004			0.690020	−	0.723790i	$-0.257602\pi$
$223$	20.6084			1.38004			0.690020	+	0.723790i	$0.257602\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	27.0044			1.79235			0.896173	−	0.443706i	$-0.146336\pi$
$227$	27.0044			1.79235			0.896173	+	0.443706i	$0.146336\pi$
$228$		0			0
$229$			9.65112i			0.637764i	0.947794	+	0.318882i	$0.103307\pi$
$229$			9.65112i			0.637764i	−0.947794	+	0.318882i	$0.896693\pi$
$230$		0			0
$231$	−0.380055	+	0.226910i	−0.0250058	+	0.0149296i
$232$		0			0
$233$	−1.29086			−0.0845671			−0.0422836	−	0.999106i	$-0.513463\pi$
$233$	−1.29086			−0.0845671			−0.0422836	+	0.999106i	$0.513463\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.30324	−	2.56923i	0.279525	−	0.166889i
$238$		0			0
$239$	−4.21092			−0.272382			−0.136191	−	0.990683i	$-0.543486\pi$
$239$	−4.21092			−0.272382			−0.136191	+	0.990683i	$0.543486\pi$
$240$		0			0
$241$	−19.5686			−1.26052			−0.630261	−	0.776383i	$-0.717052\pi$
$241$	−19.5686			−1.26052			−0.630261	+	0.776383i	$0.717052\pi$
$242$		0			0
$243$	6.78564	−	14.0341i	0.435299	−	0.900286i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	26.2732			1.67173
$248$		0			0
$249$	−2.99250	−	5.01217i	−0.189642	−	0.317634i
$250$		0			0
$251$			17.5335i			1.10670i	0.832947	+	0.553352i	$0.186652\pi$
$251$			17.5335i			1.10670i	−0.832947	+	0.553352i	$0.813348\pi$
$252$		0			0
$253$	1.00368			0.0631011
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.16582			0.0727221			0.0363610	−	0.999339i	$-0.488423\pi$
$257$	1.16582			0.0727221			0.0363610	+	0.999339i	$0.488423\pi$
$258$		0			0
$259$	6.77341			0.420879
$260$		0			0
$261$	12.5179	−	23.2271i	0.774836	−	1.43772i
$262$		0			0
$263$		−	15.3867i		−	0.948785i	−0.880313	−	0.474392i	$-0.842668\pi$
$263$		−	15.3867i		−	0.948785i	0.880313	−	0.474392i	$-0.157332\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−20.4522	+	12.2109i	−1.25166	+	0.747296i
$268$		0			0
$269$	−2.82479			−0.172230			−0.0861152	−	0.996285i	$-0.527445\pi$
$269$	−2.82479			−0.172230			−0.0861152	+	0.996285i	$0.527445\pi$
$270$		0			0
$271$			3.89729i			0.236743i	0.992969	+	0.118372i	$0.0377674\pi$
$271$			3.89729i			0.236743i	−0.992969	+	0.118372i	$0.962233\pi$
$272$		0			0
$273$	−3.04618	−	5.10209i	−0.184363	−	0.308793i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−21.8450			−1.31254			−0.656269	−	0.754527i	$-0.727866\pi$
$277$	−21.8450			−1.31254			−0.656269	+	0.754527i	$0.727866\pi$
$278$		0			0
$279$	26.1595	+	14.0983i	1.56613	+	0.844041i
$280$		0			0
$281$			20.7201i			1.23606i	0.786155	+	0.618029i	$0.212069\pi$
$281$			20.7201i			1.23606i	−0.786155	+	0.618029i	$0.787931\pi$
$282$		0			0
$283$			1.23661i			0.0735087i	0.999324	+	0.0367544i	$0.0117019\pi$
$283$			1.23661i			0.0735087i	−0.999324	+	0.0367544i	$0.988298\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.21092i			0.248563i
$288$		0			0
$289$	−10.3794			−0.610553
$290$		0			0
$291$	6.33870	−	3.78449i	0.371581	−	0.221851i
$292$		0			0
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.0731584	−	1.66401i	−0.00424508	−	0.0965556i
$298$		0			0
$299$			13.4741i			0.779226i
$300$		0			0
$301$		−	2.37128i		−	0.136678i
$302$		0			0
$303$	−13.5964	−	22.7728i	−0.781094	−	1.30827i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	6.71875i		−	0.383460i	−0.981448	−	0.191730i	$-0.938590\pi$
$307$		−	6.71875i		−	0.383460i	0.981448	−	0.191730i	$-0.0614097\pi$
$308$		0			0
$309$	6.44559	+	10.7958i	0.366677	+	0.614152i
$310$		0			0
$311$	−22.0568			−1.25073			−0.625363	−	0.780334i	$-0.715049\pi$
$311$	−22.0568			−1.25073			−0.625363	+	0.780334i	$0.715049\pi$
$312$		0			0
$313$			11.0357i			0.623775i	0.950119	+	0.311888i	$0.100961\pi$
$313$			11.0357i			0.623775i	−0.950119	+	0.311888i	$0.899039\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			24.3705i			1.36878i	0.729115	+	0.684391i	$0.239932\pi$
$317$			24.3705i			1.36878i	−0.729115	+	0.684391i	$0.760068\pi$
$318$		0			0
$319$		−	2.81927i		−	0.157849i
$320$		0			0
$321$	−11.8027	−	19.7684i	−0.658760	−	1.10337i
$322$		0			0
$323$	15.7097			0.874110
$324$		0			0
$325$		0			0
$326$		0			0
$327$	5.08001	−	3.03300i	0.280925	−	0.167725i
$328$		0			0
$329$		−	5.23151i		−	0.288423i
$330$		0			0
$331$	−13.1925			−0.725128			−0.362564	−	0.931959i	$-0.618099\pi$
$331$	−13.1925			−0.725128			−0.362564	+	0.931959i	$0.618099\pi$
$332$		0			0
$333$	−12.0920	+	22.4368i	−0.662636	+	1.22953i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			27.4876i			1.49734i	0.662941	+	0.748672i	$0.269308\pi$
$337$			27.4876i			1.49734i	−0.662941	+	0.748672i	$0.730692\pi$
$338$		0			0
$339$	−9.07977	−	15.2078i	−0.493146	−	0.825976i
$340$		0			0
$341$	3.17521			0.171947
$342$		0			0
$343$	−10.6548			−0.575305
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.9731			0.589067			0.294533	−	0.955641i	$-0.404836\pi$
$347$	10.9731			0.589067			0.294533	+	0.955641i	$0.404836\pi$
$348$		0			0
$349$		−	31.6066i		−	1.69186i	−0.533291	−	0.845932i	$-0.679045\pi$
$349$		−	31.6066i		−	1.69186i	0.533291	−	0.845932i	$-0.320955\pi$
$350$		0			0
$351$	22.3387	−	0.982124i	1.19235	−	0.0524219i
$352$		0			0
$353$	21.4646			1.14244			0.571222	−	0.820795i	$-0.306469\pi$
$353$	21.4646			1.14244			0.571222	+	0.820795i	$0.306469\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.82142	−	3.05072i	−0.0963996	−	0.161461i
$358$		0			0
$359$	−34.3124			−1.81094			−0.905468	−	0.424414i	$-0.860480\pi$
$359$	−34.3124			−1.81094			−0.905468	+	0.424414i	$0.860480\pi$
$360$		0			0
$361$	18.2766			0.961929
$362$		0			0
$363$	9.67567	+	16.2059i	0.507841	+	0.850590i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−27.9901			−1.46107			−0.730536	−	0.682874i	$-0.760730\pi$
$367$	−27.9901			−1.46107			−0.730536	+	0.682874i	$0.760730\pi$
$368$		0			0
$369$	−13.9486	−	7.51739i	−0.726136	−	0.391340i
$370$		0			0
$371$			3.14805i			0.163439i
$372$		0			0
$373$	−13.9134			−0.720408			−0.360204	−	0.932873i	$-0.617293\pi$
$373$	−13.9134			−0.720408			−0.360204	+	0.932873i	$0.617293\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	37.8477			1.94925
$378$		0			0
$379$	7.79853			0.400583			0.200292	−	0.979736i	$-0.435811\pi$
$379$	7.79853			0.400583			0.200292	+	0.979736i	$0.435811\pi$
$380$		0			0
$381$	4.43058	+	7.42084i	0.226985	+	0.380181i
$382$		0			0
$383$			27.8386i			1.42248i	0.702947	+	0.711242i	$0.251867\pi$
$383$			27.8386i			1.42248i	−0.702947	+	0.711242i	$0.748133\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7.85484	+	4.23324i	0.399284	+	0.215188i
$388$		0			0
$389$	18.5246			0.939234			0.469617	−	0.882870i	$-0.344392\pi$
$389$	18.5246			0.939234			0.469617	+	0.882870i	$0.344392\pi$
$390$		0			0
$391$			8.05661i			0.407440i
$392$		0			0
$393$	−13.2759	+	7.92631i	−0.669679	+	0.399829i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.97814			0.249846			0.124923	−	0.992166i	$-0.460132\pi$
$397$	4.97814			0.249846			0.124923	+	0.992166i	$0.460132\pi$
$398$		0			0
$399$	−4.32194	−	7.23888i	−0.216368	−	0.362397i
$400$		0			0
$401$			16.8094i			0.839422i	0.907658	+	0.419711i	$0.137868\pi$
$401$			16.8094i			0.839422i	−0.907658	+	0.419711i	$0.862132\pi$
$402$		0			0
$403$			42.6260i			2.12335i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.72336i			0.134992i
$408$		0			0
$409$	17.6053			0.870527			0.435264	−	0.900303i	$-0.356655\pi$
$409$	17.6053			0.870527			0.435264	+	0.900303i	$0.356655\pi$
$410$		0			0
$411$	1.43808	+	2.40866i	0.0709353	+	0.118811i
$412$		0			0
$413$			9.94862i			0.489540i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.18544	+	5.33533i	0.155991	+	0.261272i
$418$		0			0
$419$		−	13.3408i		−	0.651741i	−0.945414	−	0.325870i	$-0.894343\pi$
$419$		−	13.3408i		−	0.651741i	0.945414	−	0.325870i	$-0.105657\pi$
$420$		0			0
$421$			16.7650i			0.817074i	0.912742	+	0.408537i	$0.133961\pi$
$421$			16.7650i			0.817074i	−0.912742	+	0.408537i	$0.866039\pi$
$422$		0			0
$423$	17.3293	+	9.33936i	0.842580	+	0.454095i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			7.04245i			0.340808i
$428$		0			0
$429$	2.05138	−	1.22476i	0.0990413	−	0.0591322i
$430$		0			0
$431$	−28.9911			−1.39645			−0.698225	−	0.715878i	$-0.746027\pi$
$431$	−28.9911			−1.39645			−0.698225	+	0.715878i	$0.746027\pi$
$432$		0			0
$433$			23.6484i			1.13647i	0.822866	+	0.568235i	$0.192374\pi$
$433$			23.6484i			1.13647i	−0.822866	+	0.568235i	$0.807626\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			19.1171i			0.914495i
$438$		0			0
$439$		−	7.85724i		−	0.375006i	−0.982264	−	0.187503i	$-0.939961\pi$
$439$		−	7.85724i		−	0.375006i	0.982264	−	0.187503i	$-0.0600394\pi$
$440$		0			0
$441$	9.05822	−	16.8077i	0.431344	−	0.800365i
$442$		0			0
$443$	−12.9805			−0.616721			−0.308360	−	0.951270i	$-0.599780\pi$
$443$	−12.9805			−0.616721			−0.308360	+	0.951270i	$0.599780\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.05431	−	3.44079i	−0.0971655	−	0.162744i
$448$		0			0
$449$			13.4847i			0.636383i	0.948026	+	0.318191i	$0.103075\pi$
$449$			13.4847i			0.636383i	−0.948026	+	0.318191i	$0.896925\pi$
$450$		0			0
$451$	−1.69307			−0.0797234
$452$		0			0
$453$	−5.12097	+	3.05745i	−0.240604	+	0.143652i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	28.1014i		−	1.31453i	−0.753660	−	0.657265i	$-0.771713\pi$
$457$		−	28.1014i		−	1.31453i	0.753660	−	0.657265i	$-0.228287\pi$
$458$		0			0
$459$	13.3571	−	0.587246i	0.623455	−	0.0274103i
$460$		0			0
$461$	−29.2170			−1.36077			−0.680386	−	0.732854i	$-0.738188\pi$
$461$	−29.2170			−1.36077			−0.680386	+	0.732854i	$0.738188\pi$
$462$		0			0
$463$	−14.1463			−0.657434			−0.328717	−	0.944429i	$-0.606616\pi$
$463$	−14.1463			−0.657434			−0.328717	+	0.944429i	$0.606616\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	28.9687			1.34051			0.670255	−	0.742131i	$-0.266185\pi$
$467$	28.9687			1.34051			0.670255	+	0.742131i	$0.266185\pi$
$468$		0			0
$469$		−	3.72108i		−	0.171824i
$470$		0			0
$471$	−8.14279	−	13.6385i	−0.375200	−	0.628427i
$472$		0			0
$473$	0.953410			0.0438379
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−10.4279	−	5.61995i	−0.477460	−	0.257320i
$478$		0			0
$479$	37.9040			1.73188			0.865939	−	0.500150i	$-0.166722\pi$
$479$	37.9040			1.73188			0.865939	+	0.500150i	$0.166722\pi$
$480$		0			0
$481$	−36.5600			−1.66699
$482$		0			0
$483$	3.71241	−	2.21648i	0.168921	−	0.100853i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−41.9180			−1.89949			−0.949743	−	0.313031i	$-0.898656\pi$
$487$	−41.9180			−1.89949			−0.949743	+	0.313031i	$0.898656\pi$
$488$		0			0
$489$	15.7877	−	9.42595i	0.713942	−	0.426256i
$490$		0			0
$491$		−	5.09691i		−	0.230020i	−0.993364	−	0.115010i	$-0.963310\pi$
$491$		−	5.09691i		−	0.230020i	0.993364	−	0.115010i	$-0.0366901\pi$
$492$		0			0
$493$	22.6304			1.01922
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.73519			0.122690
$498$		0			0
$499$	−27.3821			−1.22579			−0.612896	−	0.790164i	$-0.709995\pi$
$499$	−27.3821			−1.22579			−0.612896	+	0.790164i	$0.709995\pi$
$500$		0			0
$501$	−19.8317	+	11.8404i	−0.886016	+	0.528992i
$502$		0			0
$503$			21.7572i			0.970104i	0.874485	+	0.485052i	$0.161199\pi$
$503$			21.7572i			0.970104i	−0.874485	+	0.485052i	$0.838801\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.89930	+	8.20591i	0.217586	+	0.364437i
$508$		0			0
$509$	25.0061			1.10837			0.554187	−	0.832392i	$-0.313029\pi$
$509$	25.0061			1.10837			0.554187	+	0.832392i	$0.313029\pi$
$510$		0			0
$511$			1.14069i			0.0504610i
$512$		0			0
$513$	31.6943	−	1.39344i	1.39934	−	0.0615220i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.10341			0.0925079
$518$		0			0
$519$	20.6674	−	12.3394i	0.907197	−	0.541638i
$520$		0			0
$521$			26.3235i			1.15325i	0.817007	+	0.576627i	$0.195631\pi$
$521$			26.3235i			1.15325i	−0.817007	+	0.576627i	$0.804369\pi$
$522$		0			0
$523$		−	23.5435i		−	1.02949i	−0.857344	−	0.514744i	$-0.827887\pi$
$523$		−	23.5435i		−	1.02949i	0.857344	−	0.514744i	$-0.172113\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			25.4876i			1.11026i
$528$		0			0
$529$	13.1959			0.573735
$530$		0			0
$531$	−32.9547	−	17.7604i	−1.43011	−	0.770736i
$532$		0			0
$533$		−	22.7288i		−	0.984492i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	7.71558	−	4.60656i	0.332952	−	0.198788i
$538$		0			0
$539$		−	2.04009i		−	0.0878730i
$540$		0			0
$541$		−	18.2576i		−	0.784955i	−0.919762	−	0.392478i	$-0.871618\pi$
$541$		−	18.2576i		−	0.784955i	0.919762	−	0.392478i	$-0.128382\pi$
$542$		0			0
$543$	3.86391	−	2.30693i	0.165816	−	0.0990000i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			6.40508i			0.273862i	0.990581	+	0.136931i	$0.0437238\pi$
$547$			6.40508i			0.273862i	−0.990581	+	0.136931i	$0.956276\pi$
$548$		0			0
$549$	−23.3280	−	12.5723i	−0.995616	−	0.536571i
$550$		0			0
$551$	53.6985			2.28763
$552$		0			0
$553$		−	2.30693i		−	0.0981008i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			7.79582i			0.330319i	0.986267	+	0.165160i	$0.0528140\pi$
$557$			7.79582i			0.330319i	−0.986267	+	0.165160i	$0.947186\pi$
$558$		0			0
$559$			12.7992i			0.541347i
$560$		0			0
$561$	1.22659	−	0.732329i	0.0517866	−	0.0309190i
$562$		0			0
$563$	−4.37399			−0.184342			−0.0921709	−	0.995743i	$-0.529381\pi$
$563$	−4.37399			−0.184342			−0.0921709	+	0.995743i	$0.529381\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−3.94531	−	5.99326i	−0.165687	−	0.251693i
$568$		0			0
$569$		−	21.8198i		−	0.914735i	−0.889278	−	0.457368i	$-0.848792\pi$
$569$		−	21.8198i		−	0.914735i	0.889278	−	0.457368i	$-0.151208\pi$
$570$		0			0
$571$	−1.42806			−0.0597625			−0.0298813	−	0.999553i	$-0.509513\pi$
$571$	−1.42806			−0.0597625			−0.0298813	+	0.999553i	$0.509513\pi$
$572$		0			0
$573$	−10.8817	−	18.2259i	−0.454590	−	0.761399i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			43.3548i			1.80488i	0.430812	+	0.902442i	$0.358227\pi$
$577$			43.3548i			1.80488i	−0.430812	+	0.902442i	$0.641773\pi$
$578$		0			0
$579$	12.2873	−	7.33609i	0.510644	−	0.304878i
$580$		0			0
$581$	−2.68699			−0.111475
$582$		0			0
$583$	−1.26572			−0.0524209
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.88810			0.284302			0.142151	−	0.989845i	$-0.454598\pi$
$587$	6.88810			0.284302			0.142151	+	0.989845i	$0.454598\pi$
$588$		0			0
$589$			60.4781i			2.49196i
$590$		0			0
$591$	16.0764	−	9.59835i	0.661295	−	0.394823i
$592$		0			0
$593$	0.894469			0.0367314			0.0183657	−	0.999831i	$-0.494154\pi$
$593$	0.894469			0.0367314			0.0183657	+	0.999831i	$0.494154\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−8.49593	+	5.07246i	−0.347715	+	0.207602i
$598$		0			0
$599$	11.5836			0.473292			0.236646	−	0.971596i	$-0.423952\pi$
$599$	11.5836			0.473292			0.236646	+	0.971596i	$0.423952\pi$
$600$		0			0
$601$	24.9480			1.01765			0.508824	−	0.860870i	$-0.330080\pi$
$601$	24.9480			1.01765			0.508824	+	0.860870i	$0.330080\pi$
$602$		0			0
$603$	12.3260	+	6.64292i	0.501955	+	0.270521i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	21.8741			0.887843			0.443922	−	0.896066i	$-0.353587\pi$
$607$	21.8741			0.887843			0.443922	+	0.896066i	$0.353587\pi$
$608$		0			0
$609$	−6.22593	−	10.4279i	−0.252287	−	0.422560i
$610$		0			0
$611$			28.2375i			1.14237i
$612$		0			0
$613$	25.8339			1.04342			0.521711	−	0.853122i	$-0.325294\pi$
$613$	25.8339			1.04342			0.521711	+	0.853122i	$0.325294\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	27.5641			1.10969			0.554845	−	0.831954i	$-0.312778\pi$
$617$	27.5641			1.10969			0.554845	+	0.831954i	$0.312778\pi$
$618$		0			0
$619$	22.2136			0.892841			0.446421	−	0.894823i	$-0.352699\pi$
$619$	22.2136			0.892841			0.446421	+	0.894823i	$0.352699\pi$
$620$		0			0
$621$	0.714619	+	16.2542i	0.0286767	+	0.652259i
$622$		0			0
$623$			10.9643i			0.439275i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	2.91050	−	1.73770i	0.116234	−	0.0693972i
$628$		0			0
$629$	−21.8605			−0.871635
$630$		0			0
$631$		−	26.2225i		−	1.04390i	−0.852975	−	0.521951i	$-0.825204\pi$
$631$		−	26.2225i		−	1.04390i	0.852975	−	0.521951i	$-0.174796\pi$
$632$		0			0
$633$	7.24246	+	12.1305i	0.287862	+	0.482144i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	27.3875			1.08513
$638$		0			0
$639$	−4.88290	+	9.06030i	−0.193165	+	0.358420i
$640$		0			0
$641$		−	47.0436i		−	1.85811i	−0.369940	−	0.929056i	$-0.620622\pi$
$641$		−	47.0436i		−	1.85811i	0.369940	−	0.929056i	$-0.379378\pi$
$642$		0			0
$643$		−	18.1696i		−	0.716538i	−0.933618	−	0.358269i	$-0.883367\pi$
$643$		−	18.1696i		−	0.716538i	0.933618	−	0.358269i	$-0.116633\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			36.9324i			1.45196i	0.687715	+	0.725981i	$0.258614\pi$
$647$			36.9324i			1.45196i	−0.687715	+	0.725981i	$0.741386\pi$
$648$		0			0
$649$	−4.00000			−0.157014
$650$		0			0
$651$	11.7444	−	7.01197i	0.460301	−	0.274821i
$652$		0			0
$653$			1.11710i			0.0437155i	0.999761	+	0.0218577i	$0.00695809\pi$
$653$			1.11710i			0.0437155i	−0.999761	+	0.0218577i	$0.993042\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−3.77851	−	2.03637i	−0.147414	−	0.0794463i
$658$		0			0
$659$		−	30.7865i		−	1.19927i	−0.800273	−	0.599636i	$-0.795312\pi$
$659$		−	30.7865i		−	1.19927i	0.800273	−	0.599636i	$-0.204688\pi$
$660$		0			0
$661$		−	11.5686i		−	0.449966i	−0.974363	−	0.224983i	$-0.927767\pi$
$661$		−	11.5686i		−	0.449966i	0.974363	−	0.224983i	$-0.0722326\pi$
$662$		0			0
$663$	9.83124	+	16.4665i	0.381814	+	0.639505i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			27.5389i			1.06631i
$668$		0			0
$669$	18.2982	+	30.6479i	0.707449	+	1.18492i
$670$		0			0
$671$	−2.83153			−0.109310
$672$		0			0
$673$		−	30.4072i		−	1.17211i	−0.810271	−	0.586056i	$-0.800680\pi$
$673$		−	30.4072i		−	1.17211i	0.810271	−	0.586056i	$-0.199320\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			29.9608i			1.15149i	0.817631	+	0.575743i	$0.195287\pi$
$677$			29.9608i			1.15149i	−0.817631	+	0.575743i	$0.804713\pi$
$678$		0			0
$679$		−	3.39813i		−	0.130408i
$680$		0			0
$681$	23.9772	+	40.1598i	0.918809	+	1.53893i
$682$		0			0
$683$	30.8345			1.17985			0.589925	−	0.807458i	$-0.299157\pi$
$683$	30.8345			1.17985			0.589925	+	0.807458i	$0.299157\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−14.3527	+	8.56923i	−0.547590	+	0.326936i
$688$		0			0
$689$		−	16.9919i		−	0.647339i
$690$		0			0
$691$	19.2293			0.731517			0.365758	−	0.930710i	$-0.380810\pi$
$691$	19.2293			0.731517			0.365758	+	0.930710i	$0.380810\pi$
$692$		0			0
$693$	−0.674901	−	0.363727i	−0.0256374	−	0.0138169i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	13.5903i		−	0.514770i
$698$		0			0
$699$	−1.14616	−	1.91971i	−0.0433516	−	0.0726102i
$700$		0			0
$701$	−25.8972			−0.978126			−0.489063	−	0.872249i	$-0.662661\pi$
$701$	−25.8972			−0.978126			−0.489063	+	0.872249i	$0.662661\pi$
$702$		0			0
$703$	−51.8716			−1.95637
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.2083			−0.459142
$708$		0			0
$709$			6.00828i			0.225646i	0.993615	+	0.112823i	$0.0359893\pi$
$709$			6.00828i			0.225646i	−0.993615	+	0.112823i	$0.964011\pi$
$710$		0			0
$711$	7.64169	+	4.11837i	0.286586	+	0.154451i
$712$		0			0
$713$	−31.0158			−1.16155
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−3.73888	−	6.26230i	−0.139631	−	0.233870i
$718$		0			0
$719$	−3.34264			−0.124659			−0.0623297	−	0.998056i	$-0.519853\pi$
$719$	−3.34264			−0.124659			−0.0623297	+	0.998056i	$0.519853\pi$
$720$		0			0
$721$	5.78755			0.215540
$722$		0			0
$723$	−17.3749	−	29.1015i	−0.646181	−	1.08230i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.4120			−0.571600			−0.285800	−	0.958289i	$-0.592259\pi$
$727$	−15.4120			−0.571600			−0.285800	+	0.958289i	$0.592259\pi$
$728$		0			0
$729$	26.8958	−	2.36954i	0.996142	−	0.0877607i
$730$		0			0
$731$			7.65306i			0.283059i
$732$		0			0
$733$	−42.7008			−1.57719			−0.788594	−	0.614914i	$-0.789191\pi$
$733$	−42.7008			−1.57719			−0.788594	+	0.614914i	$0.789191\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.49612			0.0551103
$738$		0			0
$739$	30.4546			1.12029			0.560145	−	0.828395i	$-0.310746\pi$
$739$	30.4546			1.12029			0.560145	+	0.828395i	$0.310746\pi$
$740$		0			0
$741$	23.3280	+	39.0724i	0.856976	+	1.43536i
$742$		0			0
$743$		−	49.8954i		−	1.83048i	−0.402906	−	0.915242i	$-0.632000\pi$
$743$		−	49.8954i		−	1.83048i	0.402906	−	0.915242i	$-0.368000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.79685	−	8.90062i	0.175507	−	0.325657i
$748$		0			0
$749$	−10.5977			−0.387232
$750$		0			0
$751$		−	3.81321i		−	0.139146i	−0.997577	−	0.0695730i	$-0.977836\pi$
$751$		−	3.81321i		−	0.139146i	0.997577	−	0.0695730i	$-0.0221637\pi$
$752$		0			0
$753$	−26.0750	+	15.5680i	−0.950228	+	0.567329i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.1793			1.38765			0.693825	−	0.720144i	$-0.255924\pi$
$757$	38.1793			1.38765			0.693825	+	0.720144i	$0.255924\pi$
$758$		0			0
$759$	0.891171	+	1.49263i	0.0323475	+	0.0541792i
$760$		0			0
$761$			10.7460i			0.389543i	0.980849	+	0.194772i	$0.0623966\pi$
$761$			10.7460i			0.389543i	−0.980849	+	0.194772i	$0.937603\pi$
$762$		0			0
$763$		−	2.72336i		−	0.0985921i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	53.6985i		−	1.93894i
$768$		0			0
$769$	−1.13172			−0.0408109			−0.0204055	−	0.999792i	$-0.506496\pi$
$769$	−1.13172			−0.0408109			−0.0204055	+	0.999792i	$0.506496\pi$
$770$		0			0
$771$	1.03513	+	1.73376i	0.0372795	+	0.0624399i
$772$		0			0
$773$			13.7144i			0.493274i	0.969108	+	0.246637i	$0.0793255\pi$
$773$			13.7144i			0.493274i	−0.969108	+	0.246637i	$0.920675\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	6.01411	+	10.0731i	0.215755	+	0.361371i
$778$		0			0
$779$		−	32.2477i		−	1.15539i
$780$		0			0
$781$			1.09973i			0.0393513i
$782$		0			0
$783$	45.6569	−	2.00731i	1.63164	−	0.0717354i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			32.6374i			1.16340i	0.813405	+	0.581698i	$0.197612\pi$
$787$			32.6374i			1.16340i	−0.813405	+	0.581698i	$0.802388\pi$
$788$		0			0
$789$	22.8824	−	13.6619i	0.814636	−	0.486375i
$790$		0			0
$791$	−8.15281			−0.289880
$792$		0			0
$793$		−	38.0122i		−	1.34985i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	39.1293i		−	1.38603i	−0.720923	−	0.693015i	$-0.756282\pi$
$797$		−	39.1293i		−	1.38603i	0.720923	−	0.693015i	$-0.243718\pi$
$798$		0			0
$799$			16.8842i			0.597319i
$800$		0			0
$801$	−36.3191	−	19.5736i	−1.28327	−	0.691599i
$802$		0			0
$803$	−0.458631			−0.0161847
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−2.50813	−	4.20090i	−0.0882903	−	0.147879i
$808$		0			0
$809$			25.5481i			0.898222i	0.893476	+	0.449111i	$0.148259\pi$
$809$			25.5481i			0.898222i	−0.893476	+	0.449111i	$0.851741\pi$
$810$		0			0
$811$	−5.63898			−0.198011			−0.0990057	−	0.995087i	$-0.531566\pi$
$811$	−5.63898			−0.198011			−0.0990057	+	0.995087i	$0.531566\pi$
$812$		0			0
$813$	−5.79587	+	3.46040i	−0.203270	+	0.121362i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			18.1595i			0.635322i
$818$		0			0
$819$	4.88290	−	9.06030i	0.170622	−	0.316592i
$820$		0			0
$821$	26.8248			0.936192			0.468096	−	0.883678i	$-0.344940\pi$
$821$	26.8248			0.936192			0.468096	+	0.883678i	$0.344940\pi$
$822$		0			0
$823$	8.35909			0.291379			0.145690	−	0.989330i	$-0.453460\pi$
$823$	8.35909			0.291379			0.145690	+	0.989330i	$0.453460\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.27366			0.218156			0.109078	−	0.994033i	$-0.465210\pi$
$827$	6.27366			0.218156			0.109078	+	0.994033i	$0.465210\pi$
$828$		0			0
$829$			8.61230i			0.299118i	0.988753	+	0.149559i	$0.0477853\pi$
$829$			8.61230i			0.299118i	−0.988753	+	0.149559i	$0.952215\pi$
$830$		0			0
$831$	−19.3962	−	32.4869i	−0.672845	−	1.12696i
$832$		0			0
$833$	16.3759			0.567392
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.26074	+	51.4212i	0.0781426	+	1.77738i
$838$		0			0
$839$	2.93969			0.101490			0.0507448	−	0.998712i	$-0.483840\pi$
$839$	2.93969			0.101490			0.0507448	+	0.998712i	$0.483840\pi$
$840$		0			0
$841$	48.3548			1.66741
$842$		0			0
$843$	−30.8140	+	18.3974i	−1.06129	+	0.633640i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	8.68787			0.298519
$848$		0			0
$849$	−1.83903	+	1.09798i	−0.0631153	+	0.0376827i
$850$		0			0
$851$		−	26.6020i		−	0.911906i
$852$		0			0
$853$	34.2193			1.17165			0.585824	−	0.810439i	$-0.300771\pi$
$853$	34.2193			1.17165			0.585824	+	0.810439i	$0.300771\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.519916			0.0177600			0.00888000	−	0.999961i	$-0.497173\pi$
$857$	0.519916			0.0177600			0.00888000	+	0.999961i	$0.497173\pi$
$858$		0			0
$859$	−32.9724			−1.12500			−0.562502	−	0.826796i	$-0.690161\pi$
$859$	−32.9724			−1.12500			−0.562502	+	0.826796i	$0.690161\pi$
$860$		0			0
$861$	−6.26230	+	3.73888i	−0.213418	+	0.127421i
$862$		0			0
$863$		−	21.8453i		−	0.743623i	−0.928308	−	0.371811i	$-0.878737\pi$
$863$		−	21.8453i		−	0.743623i	0.928308	−	0.371811i	$-0.121263\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−9.21587	−	15.4358i	−0.312987	−	0.524227i
$868$		0			0
$869$	0.927539			0.0314646
$870$		0			0
$871$			20.0848i			0.680549i
$872$		0			0
$873$	11.2563	+	6.06638i	0.380967	+	0.205316i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−37.2187			−1.25679			−0.628393	−	0.777896i	$-0.716287\pi$
$877$	−37.2187			−1.25679			−0.628393	+	0.777896i	$0.716287\pi$
$878$		0			0
$879$	−8.92294	+	5.32740i	−0.300963	+	0.179689i
$880$		0			0
$881$			17.2984i			0.582796i	0.956602	+	0.291398i	$0.0941205\pi$
$881$			17.2984i			0.582796i	−0.956602	+	0.291398i	$0.905880\pi$
$882$		0			0
$883$			51.3234i			1.72717i	0.504203	+	0.863585i	$0.331786\pi$
$883$			51.3234i			1.72717i	−0.504203	+	0.863585i	$0.668214\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	15.3867i		−	0.516635i	−0.966060	−	0.258318i	$-0.916832\pi$
$887$		−	15.3867i		−	0.516635i	0.966060	−	0.258318i	$-0.0831681\pi$
$888$		0			0
$889$	3.97825			0.133426
$890$		0			0
$891$	2.40968	−	1.58627i	0.0807275	−	0.0531421i
$892$		0			0
$893$			40.0635i			1.34067i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−20.0381	+	11.9636i	−0.669051	+	0.399454i
$898$		0			0
$899$			87.1211i			2.90565i
$900$		0			0
$901$		−	10.1600i		−	0.338479i
$902$		0			0
$903$	3.52646	−	2.10546i	0.117353	−	0.0700653i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	34.3304i		−	1.13992i	−0.821671	−	0.569962i	$-0.806958\pi$
$907$		−	34.3304i		−	1.13992i	0.821671	−	0.569962i	$-0.193042\pi$
$908$		0			0
$909$	21.7945	−	40.4400i	0.722878	−	1.34131i
$910$		0			0
$911$	20.7856			0.688657			0.344328	−	0.938849i	$-0.388107\pi$
$911$	20.7856			0.688657			0.344328	+	0.938849i	$0.388107\pi$
$912$		0			0
$913$		−	1.08035i		−	0.0357542i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			7.11710i			0.235027i
$918$		0			0
$919$			35.6290i			1.17529i	0.809119	+	0.587645i	$0.199945\pi$
$919$			35.6290i			1.17529i	−0.809119	+	0.587645i	$0.800055\pi$
$920$		0			0
$921$	9.99184	−	5.96558i	0.329242	−	0.196573i
$922$		0			0
$923$	−14.7634			−0.485944
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−10.3320	+	19.1712i	−0.339347	+	0.629664i
$928$		0			0
$929$		−	44.9041i		−	1.47325i	−0.676299	−	0.736627i	$-0.736417\pi$
$929$		−	44.9041i		−	1.47325i	0.676299	−	0.736627i	$-0.263583\pi$
$930$		0			0
$931$	38.8575			1.27350
$932$		0			0
$933$	−19.5842	−	32.8019i	−0.641159	−	1.07389i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	56.5086i		−	1.84606i	−0.384731	−	0.923029i	$-0.625706\pi$
$937$		−	56.5086i		−	1.84606i	0.384731	−	0.923029i	$-0.374294\pi$
$938$		0			0
$939$	−16.4118	+	9.79861i	−0.535580	+	0.319765i
$940$		0			0
$941$	29.6334			0.966022			0.483011	−	0.875614i	$-0.339543\pi$
$941$	29.6334			0.966022			0.483011	+	0.875614i	$0.339543\pi$
$942$		0			0
$943$	16.5380			0.538553
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−29.2810			−0.951504			−0.475752	−	0.879580i	$-0.657824\pi$
$947$	−29.2810			−0.951504			−0.475752	+	0.879580i	$0.657824\pi$
$948$		0			0
$949$		−	6.15695i		−	0.199863i
$950$		0			0
$951$	−36.2427	+	21.6385i	−1.17525	+	0.701678i
$952$		0			0
$953$	40.2311			1.30321			0.651606	−	0.758557i	$-0.274095\pi$
$953$	40.2311			1.30321			0.651606	+	0.758557i	$0.274095\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	4.19270	−	2.50323i	0.135531	−	0.0809180i
$958$		0			0
$959$	1.29127			0.0416972
$960$		0			0
$961$	−67.1203			−2.16517
$962$		0			0
$963$	18.9192	−	35.1048i	0.609662	−	1.13124i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	34.4522			1.10791			0.553954	−	0.832547i	$-0.313118\pi$
$967$	34.4522			1.10791			0.553954	+	0.832547i	$0.313118\pi$
$968$		0			0
$969$	13.9486	+	23.3627i	0.448094	+	0.750519i
$970$		0			0
$971$		−	26.9133i		−	0.863688i	−0.901948	−	0.431844i	$-0.857863\pi$
$971$		−	26.9133i		−	0.863688i	0.901948	−	0.431844i	$-0.142137\pi$
$972$		0			0
$973$	2.86023			0.0916948
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−41.9916			−1.34343			−0.671715	−	0.740810i	$-0.734442\pi$
$977$	−41.9916			−1.34343			−0.671715	+	0.740810i	$0.734442\pi$
$978$		0			0
$979$	−4.40837			−0.140892
$980$		0			0
$981$	9.02109	+	4.86177i	0.288021	+	0.155224i
$982$		0			0
$983$			3.83856i			0.122431i	0.998125	+	0.0612156i	$0.0194977\pi$
$983$			3.83856i			0.122431i	−0.998125	+	0.0612156i	$0.980502\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.78007	−	4.64506i	0.247642	−	0.147854i
$988$		0			0
$989$	−9.31301			−0.296137
$990$		0			0
$991$			48.3440i			1.53570i	0.640630	+	0.767850i	$0.278673\pi$
$991$			48.3440i			1.53570i	−0.640630	+	0.767850i	$0.721327\pi$
$992$		0			0
$993$	−11.7137	−	19.6194i	−0.371722	−	0.622602i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−23.5236			−0.744998			−0.372499	−	0.928032i	$-0.621499\pi$
$997$	−23.5236			−0.744998			−0.372499	+	0.928032i	$0.621499\pi$
$998$		0			0
$999$	−44.1036	+	1.93902i	−1.39538	+	0.0613479i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.m.c.1199.12		16
3.2	odd	2		2400.2.m.d.1199.6		16
4.3	odd	2		600.2.m.d.299.4		16
5.2	odd	4		2400.2.b.e.2351.7		8
5.3	odd	4		480.2.b.a.431.2		8
5.4	even	2	inner	2400.2.m.c.1199.5		16
8.3	odd	2		2400.2.m.d.1199.12		16
8.5	even	2		600.2.m.c.299.3		16
12.11	even	2		600.2.m.c.299.13		16
15.2	even	4		2400.2.b.f.2351.8		8
15.8	even	4		480.2.b.b.431.1		8
15.14	odd	2		2400.2.m.d.1199.11		16
20.3	even	4		120.2.b.b.11.6	yes	8
20.7	even	4		600.2.b.e.251.3		8
20.19	odd	2		600.2.m.d.299.13		16
24.5	odd	2		600.2.m.d.299.14		16
24.11	even	2	inner	2400.2.m.c.1199.6		16
40.3	even	4		480.2.b.b.431.2		8
40.13	odd	4		120.2.b.a.11.4	yes	8
40.19	odd	2		2400.2.m.d.1199.5		16
40.27	even	4		2400.2.b.f.2351.7		8
40.29	even	2		600.2.m.c.299.14		16
40.37	odd	4		600.2.b.f.251.5		8
60.23	odd	4		120.2.b.a.11.3	✓	8
60.47	odd	4		600.2.b.f.251.6		8
60.59	even	2		600.2.m.c.299.4		16
120.29	odd	2		600.2.m.d.299.3		16
120.53	even	4		120.2.b.b.11.5	yes	8
120.59	even	2	inner	2400.2.m.c.1199.11		16
120.77	even	4		600.2.b.e.251.4		8
120.83	odd	4		480.2.b.a.431.1		8
120.107	odd	4		2400.2.b.e.2351.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.b.a.11.3	✓	8	60.23	odd	4
120.2.b.a.11.4	yes	8	40.13	odd	4
120.2.b.b.11.5	yes	8	120.53	even	4
120.2.b.b.11.6	yes	8	20.3	even	4
480.2.b.a.431.1		8	120.83	odd	4
480.2.b.a.431.2		8	5.3	odd	4
480.2.b.b.431.1		8	15.8	even	4
480.2.b.b.431.2		8	40.3	even	4
600.2.b.e.251.3		8	20.7	even	4
600.2.b.e.251.4		8	120.77	even	4
600.2.b.f.251.5		8	40.37	odd	4
600.2.b.f.251.6		8	60.47	odd	4
600.2.m.c.299.3		16	8.5	even	2
600.2.m.c.299.4		16	60.59	even	2
600.2.m.c.299.13		16	12.11	even	2
600.2.m.c.299.14		16	40.29	even	2
600.2.m.d.299.3		16	120.29	odd	2
600.2.m.d.299.4		16	4.3	odd	2
600.2.m.d.299.13		16	20.19	odd	2
600.2.m.d.299.14		16	24.5	odd	2
2400.2.b.e.2351.7		8	5.2	odd	4
2400.2.b.e.2351.8		8	120.107	odd	4
2400.2.b.f.2351.7		8	40.27	even	4
2400.2.b.f.2351.8		8	15.2	even	4
2400.2.m.c.1199.5		16	5.4	even	2	inner
2400.2.m.c.1199.6		16	24.11	even	2	inner
2400.2.m.c.1199.11		16	120.59	even	2	inner
2400.2.m.c.1199.12		16	1.1	even	1	trivial
2400.2.m.d.1199.5		16	40.19	odd	2
2400.2.m.d.1199.6		16	3.2	odd	2
2400.2.m.d.1199.11		16	15.14	odd	2
2400.2.m.d.1199.12		16	8.3	odd	2

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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.887900 + 1.48716i) q^{3} +0.797253 q^{7} +(-1.42327 + 2.64089i) q^{9} +O(q^{10})\)	\(q+(0.887900 + 1.48716i) q^{3} +0.797253 q^{7} +(-1.42327 + 2.64089i) q^{9} +0.320548i q^{11} -4.30324 q^{13} -2.57305 q^{17} -6.10546 q^{19} +(0.707881 + 1.18564i) q^{21} -3.13115i q^{23} +(-5.19114 + 0.228229i) q^{27} -8.79516 q^{29} -9.90557i q^{31} +(-0.476705 + 0.284615i) q^{33} +8.49593 q^{37} +(-3.82085 - 6.39959i) q^{39} +5.28178i q^{41} -2.97431i q^{43} -6.56192i q^{47} -6.36439 q^{49} +(-2.28461 - 3.82653i) q^{51} +3.94862i q^{53} +(-5.42104 - 9.07977i) q^{57} +12.4786i q^{59} +8.83339i q^{61} +(-1.13470 + 2.10546i) q^{63} -4.66738i q^{67} +(4.65651 - 2.78015i) q^{69} +3.43077 q^{71} +1.43077i q^{73} +0.255558i q^{77} -2.89360i q^{79} +(-4.94862 - 7.51739i) q^{81} -3.37031 q^{83} +(-7.80922 - 13.0798i) q^{87} +13.7526i q^{89} -3.43077 q^{91} +(14.7311 - 8.79516i) q^{93} -4.26230i q^{97} +(-0.846533 - 0.456225i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{19} - 8 q^{21} + 32 q^{39} + 32 q^{49} - 40 q^{51} - 40 q^{69} + 48 q^{71} + 16 q^{81} - 48 q^{91} + 32 q^{99}+O(q^{100}) \) 16 * q - 8 * q^19 - 8 * q^21 + 32 * q^39 + 32 * q^49 - 40 * q^51 - 40 * q^69 + 48 * q^71 + 16 * q^81 - 48 * q^91 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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