LMFDB - Embedded newform 2400.2.m.d.1199.10

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			1199.10
Root			$-0.842022 + 1.13622i$ of defining polynomial
Character	$\chi$	$=$	2400.1199
Dual form			2400.2.m.d.1199.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.218455 + 1.71822i) q^{3} -3.64426 q^{7} +(-2.90455 + 0.750707i) q^{9} +O(q^{10})$	$q+(0.218455 + 1.71822i) q^{3} -3.64426 q^{7} +(-2.90455 + 0.750707i) q^{9} +5.07403i q^{11} +1.70594 q^{13} -4.08117 q^{17} -1.26422 q^{19} +(-0.796108 - 6.26164i) q^{21} -4.70066i q^{23} +(-1.92439 - 4.82667i) q^{27} -1.06377 q^{29} +4.86950i q^{31} +(-8.71829 + 1.10845i) q^{33} +7.56830 q^{37} +(0.372671 + 2.93118i) q^{39} +1.50141i q^{41} -3.43644i q^{43} -10.9176i q^{47} +6.28066 q^{49} +(-0.891553 - 7.01235i) q^{51} -8.87288i q^{53} +(-0.276176 - 2.17221i) q^{57} +0.788328i q^{59} -0.627594i q^{61} +(10.5850 - 2.73578i) q^{63} +4.18178i q^{67} +(8.07677 - 1.02688i) q^{69} -6.21689 q^{71} -4.21689i q^{73} -18.4911i q^{77} +0.992853i q^{79} +(7.87288 - 4.36094i) q^{81} -7.72544 q^{83} +(-0.232385 - 1.82779i) q^{87} -11.5742i q^{89} -6.21689 q^{91} +(-8.36687 + 1.06377i) q^{93} +7.40133i q^{97} +(-3.80911 - 14.7378i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.218455	+	1.71822i	0.126125	+	0.992014i
$3$	0.218455	+	1.71822i	0.126125	+	0.992014i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.64426			−1.37740			−0.688701	−	0.725045i	$-0.741819\pi$
$7$	−3.64426			−1.37740			−0.688701	+	0.725045i	$0.741819\pi$
$8$		0			0
$9$	−2.90455	+	0.750707i	−0.968185	+	0.250236i
$10$		0			0
$11$			5.07403i			1.52988i	0.644103	+	0.764938i	$0.277231\pi$
$11$			5.07403i			1.52988i	−0.644103	+	0.764938i	$0.722769\pi$
$12$		0			0
$13$	1.70594			0.473142			0.236571	−	0.971614i	$-0.423976\pi$
$13$	1.70594			0.473142			0.236571	+	0.971614i	$0.423976\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.08117			−0.989830			−0.494915	−	0.868941i	$-0.664801\pi$
$17$	−4.08117			−0.989830			−0.494915	+	0.868941i	$0.664801\pi$
$18$		0			0
$19$	−1.26422			−0.290033			−0.145016	−	0.989429i	$-0.546324\pi$
$19$	−1.26422			−0.290033			−0.145016	+	0.989429i	$0.546324\pi$
$20$		0			0
$21$	−0.796108	−	6.26164i	−0.173725	−	1.36640i
$22$		0			0
$23$		−	4.70066i		−	0.980156i	−0.871679	−	0.490078i	$-0.836968\pi$
$23$		−	4.70066i		−	0.980156i	0.871679	−	0.490078i	$-0.163032\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.92439	−	4.82667i	−0.370350	−	0.928892i
$28$		0			0
$29$	−1.06377			−0.197537			−0.0987683	−	0.995110i	$-0.531490\pi$
$29$	−1.06377			−0.197537			−0.0987683	+	0.995110i	$0.531490\pi$
$30$		0			0
$31$			4.86950i			0.874588i	0.899318	+	0.437294i	$0.144063\pi$
$31$			4.86950i			0.874588i	−0.899318	+	0.437294i	$0.855937\pi$
$32$		0			0
$33$	−8.71829	+	1.10845i	−1.51766	+	0.192956i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.56830			1.24422			0.622110	−	0.782930i	$-0.286276\pi$
$37$	7.56830			1.24422			0.622110	+	0.782930i	$0.286276\pi$
$38$		0			0
$39$	0.372671	+	2.93118i	0.0596751	+	0.469364i
$40$		0			0
$41$			1.50141i			0.234482i	0.993104	+	0.117241i	$0.0374049\pi$
$41$			1.50141i			0.234482i	−0.993104	+	0.117241i	$0.962595\pi$
$42$		0			0
$43$		−	3.43644i		−	0.524052i	−0.965061	−	0.262026i	$-0.915609\pi$
$43$		−	3.43644i		−	0.524052i	0.965061	−	0.262026i	$-0.0843906\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.9176i		−	1.59249i	−0.604975	−	0.796245i	$-0.706817\pi$
$47$		−	10.9176i		−	1.59249i	0.604975	−	0.796245i	$-0.293183\pi$
$48$		0			0
$49$	6.28066			0.897237
$50$		0			0
$51$	−0.891553	−	7.01235i	−0.124842	−	0.981926i
$52$		0			0
$53$		−	8.87288i		−	1.21878i	−0.792869	−	0.609392i	$-0.791414\pi$
$53$		−	8.87288i		−	1.21878i	0.792869	−	0.609392i	$-0.208586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.276176	−	2.17221i	−0.0365804	−	0.287717i
$58$		0			0
$59$			0.788328i			0.102632i	0.998682	+	0.0513158i	$0.0163415\pi$
$59$			0.788328i			0.102632i	−0.998682	+	0.0513158i	$0.983658\pi$
$60$		0			0
$61$		−	0.627594i		−	0.0803552i	−0.999193	−	0.0401776i	$-0.987208\pi$
$61$		−	0.627594i		−	0.0803552i	0.999193	−	0.0401776i	$-0.0127924\pi$
$62$		0			0
$63$	10.5850	−	2.73578i	1.33358	−	0.344675i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			4.18178i			0.510886i	0.966824	+	0.255443i	$0.0822213\pi$
$67$			4.18178i			0.510886i	−0.966824	+	0.255443i	$0.917779\pi$
$68$		0			0
$69$	8.07677	−	1.02688i	0.972329	−	0.123622i
$70$		0			0
$71$	−6.21689			−0.737810			−0.368905	−	0.929467i	$-0.620267\pi$
$71$	−6.21689			−0.737810			−0.368905	+	0.929467i	$0.620267\pi$
$72$		0			0
$73$		−	4.21689i		−	0.493550i	−0.969073	−	0.246775i	$-0.920629\pi$
$73$		−	4.21689i		−	0.493550i	0.969073	−	0.246775i	$-0.0793709\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	18.4911i		−	2.10726i
$78$		0			0
$79$			0.992853i			0.111705i	0.998439	+	0.0558524i	$0.0177876\pi$
$79$			0.992853i			0.111705i	−0.998439	+	0.0558524i	$0.982212\pi$
$80$		0			0
$81$	7.87288	−	4.36094i	0.874764	−	0.484549i
$82$		0			0
$83$	−7.72544			−0.847977			−0.423989	−	0.905668i	$-0.639370\pi$
$83$	−7.72544			−0.847977			−0.423989	+	0.905668i	$0.639370\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.232385	−	1.82779i	−0.0249143	−	0.195959i
$88$		0			0
$89$		−	11.5742i		−	1.22687i	−0.789747	−	0.613433i	$-0.789788\pi$
$89$		−	11.5742i		−	1.22687i	0.789747	−	0.613433i	$-0.210212\pi$
$90$		0			0
$91$	−6.21689			−0.651708
$92$		0			0
$93$	−8.36687	+	1.06377i	−0.867604	+	0.110308i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			7.40133i			0.751491i	0.926723	+	0.375745i	$0.122613\pi$
$97$			7.40133i			0.751491i	−0.926723	+	0.375745i	$0.877387\pi$
$98$		0			0
$99$	−3.80911	−	14.7378i	−0.382830	−	1.48120i
$100$		0			0
$101$	−10.1535			−1.01031			−0.505157	−	0.863027i	$-0.668566\pi$
$101$	−10.1535			−1.01031			−0.505157	+	0.863027i	$0.668566\pi$
$102$		0			0
$103$	−15.5400			−1.53120			−0.765599	−	0.643318i	$-0.777557\pi$
$103$	−15.5400			−1.53120			−0.765599	+	0.643318i	$0.777557\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.8707			−1.43760			−0.718801	−	0.695216i	$-0.755309\pi$
$107$	−14.8707			−1.43760			−0.718801	+	0.695216i	$0.755309\pi$
$108$		0			0
$109$		−	10.5376i		−	1.00932i	−0.863319	−	0.504659i	$-0.831618\pi$
$109$		−	10.5376i		−	1.00932i	0.863319	−	0.504659i	$-0.168382\pi$
$110$		0			0
$111$	1.65333	+	13.0040i	0.156927	+	1.23428i
$112$		0			0
$113$	9.94353			0.935409			0.467704	−	0.883885i	$-0.345081\pi$
$113$	9.94353			0.935409			0.467704	+	0.883885i	$0.345081\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.95499	+	1.28066i	−0.458089	+	0.118397i
$118$		0			0
$119$	14.8729			1.36339
$120$		0			0
$121$	−14.7458			−1.34052
$122$		0			0
$123$	−2.57976	+	0.327992i	−0.232609	+	0.0295740i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	5.62997			0.499579			0.249790	−	0.968300i	$-0.419639\pi$
$127$	5.62997			0.499579			0.249790	+	0.968300i	$0.419639\pi$
$128$		0			0
$129$	5.90455	−	0.750707i	0.519867	−	0.0660961i
$130$		0			0
$131$			1.66215i			0.145223i	0.997360	+	0.0726113i	$0.0231333\pi$
$131$			1.66215i			0.145223i	−0.997360	+	0.0726113i	$0.976867\pi$
$132$		0			0
$133$	4.60717			0.399492
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.3554			−1.14103			−0.570515	−	0.821287i	$-0.693256\pi$
$137$	−13.3554			−1.14103			−0.570515	+	0.821287i	$0.693256\pi$
$138$		0			0
$139$	14.3540			1.21749			0.608745	−	0.793366i	$-0.291673\pi$
$139$	14.3540			1.21749			0.608745	+	0.793366i	$0.291673\pi$
$140$		0			0
$141$	18.7588	−	2.38500i	1.57977	−	0.200853i
$142$		0			0
$143$			8.65598i			0.723850i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.37204	+	10.7916i	0.113164	+	0.890072i
$148$		0			0
$149$	18.2742			1.49708			0.748540	−	0.663089i	$-0.230755\pi$
$149$	18.2742			1.49708			0.748540	+	0.663089i	$0.230755\pi$
$150$		0			0
$151$		−	16.7652i		−	1.36433i	−0.731197	−	0.682166i	$-0.761038\pi$
$151$		−	16.7652i		−	1.36433i	0.731197	−	0.682166i	$-0.238962\pi$
$152$		0			0
$153$	11.8540	−	3.06377i	0.958339	−	0.247691i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.31311			0.503841			0.251920	−	0.967748i	$-0.418938\pi$
$157$	6.31311			0.503841			0.251920	+	0.967748i	$0.418938\pi$
$158$		0			0
$159$	15.2455	−	1.93832i	1.20905	−	0.153719i
$160$		0			0
$161$			17.1305i			1.35007i
$162$		0			0
$163$		−	2.69110i		−	0.210783i	−0.994431	−	0.105391i	$-0.966390\pi$
$163$		−	2.69110i		−	0.210783i	0.994431	−	0.105391i	$-0.0336096\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	16.6633i		−	1.28945i	−0.764416	−	0.644723i	$-0.776973\pi$
$167$		−	16.6633i		−	1.28945i	0.764416	−	0.644723i	$-0.223027\pi$
$168$		0			0
$169$	−10.0898			−0.776136
$170$		0			0
$171$	3.67201	−	0.949062i	0.280805	−	0.0725766i
$172$		0			0
$173$			11.7458i			0.893013i	0.894780	+	0.446507i	$0.147332\pi$
$173$			11.7458i			0.893013i	−0.894780	+	0.446507i	$0.852668\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.35452	+	0.172214i	−0.101812	+	0.0129444i
$178$		0			0
$179$		−	0.466861i		−	0.0348948i	−0.999848	−	0.0174474i	$-0.994446\pi$
$179$		−	0.466861i		−	0.0348948i	0.999848	−	0.0174474i	$-0.00555396\pi$
$180$		0			0
$181$			16.5628i			1.23110i	0.788098	+	0.615550i	$0.211066\pi$
$181$			16.5628i			1.23110i	−0.788098	+	0.615550i	$0.788934\pi$
$182$		0			0
$183$	1.07834	−	0.137101i	0.0797135	−	0.0101348i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	20.7080i		−	1.51432i
$188$		0			0
$189$	7.01300	+	17.5896i	0.510121	+	1.27946i
$190$		0			0
$191$	−6.49110			−0.469679			−0.234840	−	0.972034i	$-0.575456\pi$
$191$	−6.49110			−0.469679			−0.234840	+	0.972034i	$0.575456\pi$
$192$		0			0
$193$			11.4013i			0.820685i	0.911931	+	0.410343i	$0.134591\pi$
$193$			11.4013i			0.820685i	−0.911931	+	0.410343i	$0.865409\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.56091i		−	0.253704i	−0.991922	−	0.126852i	$-0.959513\pi$
$197$		−	3.56091i		−	0.253704i	0.991922	−	0.126852i	$-0.0404874\pi$
$198$		0			0
$199$			4.40473i			0.312243i	0.987738	+	0.156122i	$0.0498992\pi$
$199$			4.40473i			0.312243i	−0.987738	+	0.156122i	$0.950101\pi$
$200$		0			0
$201$	−7.18522	+	0.913531i	−0.506806	+	0.0644355i
$202$		0			0
$203$	3.87665			0.272087
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.52882	+	13.6533i	0.245270	+	0.948972i
$208$		0			0
$209$		−	6.41471i		−	0.443715i
$210$		0			0
$211$	16.1371			1.11092			0.555462	−	0.831542i	$-0.312541\pi$
$211$	16.1371			1.11092			0.555462	+	0.831542i	$0.312541\pi$
$212$		0			0
$213$	−1.35811	−	10.6820i	−0.0930563	−	0.731918i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	17.7458i		−	1.20466i
$218$		0			0
$219$	7.24555	−	0.921202i	0.489609	−	0.0622490i
$220$		0			0
$221$	−6.96224			−0.468331
$222$		0			0
$223$	6.09474			0.408134			0.204067	−	0.978957i	$-0.434584\pi$
$223$	6.09474			0.408134			0.204067	+	0.978957i	$0.434584\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.4673			−1.35846			−0.679230	−	0.733925i	$-0.737686\pi$
$227$	−20.4673			−1.35846			−0.679230	+	0.733925i	$0.737686\pi$
$228$		0			0
$229$			26.4728i			1.74937i	0.484693	+	0.874684i	$0.338931\pi$
$229$			26.4728i			1.74937i	−0.484693	+	0.874684i	$0.661069\pi$
$230$		0			0
$231$	31.7718	−	4.03947i	2.09043	−	0.265778i
$232$		0			0
$233$	−24.3773			−1.59701			−0.798504	−	0.601989i	$-0.794375\pi$
$233$	−24.3773			−1.59701			−0.798504	+	0.601989i	$0.794375\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.70594	+	0.216894i	−0.110813	+	0.0140888i
$238$		0			0
$239$	−5.47155			−0.353925			−0.176963	−	0.984218i	$-0.556627\pi$
$239$	−5.47155			−0.353925			−0.176963	+	0.984218i	$0.556627\pi$
$240$		0			0
$241$	24.6446			1.58750			0.793750	−	0.608244i	$-0.208126\pi$
$241$	24.6446			1.58750			0.793750	+	0.608244i	$0.208126\pi$
$242$		0			0
$243$	9.21292	+	12.5747i	0.591009	+	0.806665i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.15669			−0.137227
$248$		0			0
$249$	−1.68766	−	13.2740i	−0.106951	−	0.841205i
$250$		0			0
$251$		−	1.74973i		−	0.110442i	−0.998474	−	0.0552210i	$-0.982414\pi$
$251$		−	1.74973i		−	0.110442i	0.998474	−	0.0552210i	$-0.0175863\pi$
$252$		0			0
$253$	23.8513			1.49952
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.27646			−0.329137			−0.164568	−	0.986366i	$-0.552623\pi$
$257$	−5.27646			−0.329137			−0.164568	+	0.986366i	$0.552623\pi$
$258$		0			0
$259$	−27.5809			−1.71379
$260$		0			0
$261$	3.08977	−	0.798578i	0.191252	−	0.0494307i
$262$		0			0
$263$			1.79043i			0.110403i	0.998475	+	0.0552014i	$0.0175801\pi$
$263$			1.79043i			0.110403i	−0.998475	+	0.0552014i	$0.982420\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	19.8871	−	2.52845i	1.21707	−	0.154739i
$268$		0			0
$269$	−18.7080			−1.14065			−0.570323	−	0.821420i	$-0.693182\pi$
$269$	−18.7080			−1.14065			−0.570323	+	0.821420i	$0.693182\pi$
$270$		0			0
$271$		−	24.8441i		−	1.50917i	−0.656200	−	0.754587i	$-0.727837\pi$
$271$		−	24.8441i		−	1.50917i	0.656200	−	0.754587i	$-0.272163\pi$
$272$		0			0
$273$	−1.35811	−	10.6820i	−0.0821967	−	0.646503i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−29.2030			−1.75464			−0.877319	−	0.479908i	$-0.840670\pi$
$277$	−29.2030			−1.75464			−0.877319	+	0.479908i	$0.840670\pi$
$278$		0			0
$279$	−3.65557	−	14.1437i	−0.218853	−	0.846763i
$280$		0			0
$281$		−	22.9856i		−	1.37121i	−0.727975	−	0.685604i	$-0.759538\pi$
$281$		−	22.9856i		−	1.37121i	0.727975	−	0.685604i	$-0.240462\pi$
$282$		0			0
$283$			2.03511i			0.120975i	0.998169	+	0.0604875i	$0.0192655\pi$
$283$			2.03511i			0.120975i	−0.998169	+	0.0604875i	$0.980734\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	5.47155i		−	0.322975i
$288$		0			0
$289$	−0.344017			−0.0202363
$290$		0			0
$291$	−12.7171	+	1.61686i	−0.745490	+	0.0947818i
$292$		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
$297$	24.4906	−	9.76443i	1.42109	−	0.566590i
$298$		0			0
$299$		−	8.01905i		−	0.463753i
$300$		0			0
$301$			12.5233i			0.721830i
$302$		0			0
$303$	−2.21809	−	17.4460i	−0.127426	−	1.00225i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			19.0547i			1.08751i	0.839245	+	0.543753i	$0.182997\pi$
$307$			19.0547i			1.08751i	−0.839245	+	0.543753i	$0.817003\pi$
$308$		0			0
$309$	−3.39478	−	26.7011i	−0.193122	−	1.51897i
$310$		0			0
$311$	−26.0902			−1.47944			−0.739719	−	0.672916i	$-0.765042\pi$
$311$	−26.0902			−1.47944			−0.739719	+	0.672916i	$0.765042\pi$
$312$		0			0
$313$			20.1795i			1.14062i	0.821431	+	0.570308i	$0.193176\pi$
$313$			20.1795i			1.14062i	−0.821431	+	0.570308i	$0.806824\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7.81598i		−	0.438989i	−0.975614	−	0.219495i	$-0.929559\pi$
$317$		−	7.81598i		−	0.438989i	0.975614	−	0.219495i	$-0.0704408\pi$
$318$		0			0
$319$		−	5.39759i		−	0.302207i
$320$		0			0
$321$	−3.24857	−	25.5511i	−0.181318	−	1.42612i
$322$		0			0
$323$	5.15952			0.287083
$324$		0			0
$325$		0			0
$326$		0			0
$327$	18.1059	−	2.30199i	1.00126	−	0.127300i
$328$		0			0
$329$			39.7865i			2.19350i
$330$		0			0
$331$	10.0424			0.551982			0.275991	−	0.961160i	$-0.410994\pi$
$331$	10.0424			0.551982			0.275991	+	0.961160i	$0.410994\pi$
$332$		0			0
$333$	−21.9825	+	5.68157i	−1.20464	+	0.311348i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			17.8733i			0.973620i	0.873508	+	0.486810i	$0.161840\pi$
$337$			17.8733i			0.973620i	−0.873508	+	0.486810i	$0.838160\pi$
$338$		0			0
$339$	2.17221	+	17.0852i	0.117978	+	0.927939i
$340$		0			0
$341$	−24.7080			−1.33801
$342$		0			0
$343$	2.62146			0.141546
$344$		0			0
$345$		0			0
$346$		0			0
$347$	34.9886			1.87829			0.939143	−	0.343525i	$-0.111621\pi$
$347$	34.9886			1.87829			0.939143	+	0.343525i	$0.111621\pi$
$348$		0			0
$349$		−	8.24993i		−	0.441609i	−0.975318	−	0.220804i	$-0.929132\pi$
$349$		−	8.24993i		−	0.441609i	0.975318	−	0.220804i	$-0.0708682\pi$
$350$		0			0
$351$	−3.28290	−	8.23400i	−0.175228	−	0.439498i
$352$		0			0
$353$	0.935042			0.0497673			0.0248836	−	0.999690i	$-0.492078\pi$
$353$	0.935042			0.0497673			0.0248836	+	0.999690i	$0.492078\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	3.24906	+	25.5549i	0.171958	+	1.35251i
$358$		0			0
$359$	−32.5813			−1.71957			−0.859787	−	0.510653i	$-0.829404\pi$
$359$	−32.5813			−1.71957			−0.859787	+	0.510653i	$0.829404\pi$
$360$		0			0
$361$	−17.4017			−0.915881
$362$		0			0
$363$	−3.22128	−	25.3364i	−0.169074	−	1.32982i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.791919			−0.0413378			−0.0206689	−	0.999786i	$-0.506580\pi$
$367$	−0.791919			−0.0413378			−0.0206689	+	0.999786i	$0.506580\pi$
$368$		0			0
$369$	−1.12712	−	4.36094i	−0.0586757	−	0.227022i
$370$		0			0
$371$			32.3351i			1.67876i
$372$		0			0
$373$	−18.7335			−0.969982			−0.484991	−	0.874519i	$-0.661177\pi$
$373$	−18.7335			−0.969982			−0.484991	+	0.874519i	$0.661177\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.81472			−0.0934630
$378$		0			0
$379$	8.88244			0.456260			0.228130	−	0.973631i	$-0.426739\pi$
$379$	8.88244			0.456260			0.228130	+	0.973631i	$0.426739\pi$
$380$		0			0
$381$	1.22990	+	9.67352i	0.0630095	+	0.495590i
$382$		0			0
$383$		−	3.48418i		−	0.178033i	−0.996030	−	0.0890167i	$-0.971628\pi$
$383$		−	3.48418i		−	0.178033i	0.996030	−	0.0890167i	$-0.0283724\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.57976	+	9.98132i	0.131137	+	0.507379i
$388$		0			0
$389$	−24.8027			−1.25754			−0.628772	−	0.777590i	$-0.716442\pi$
$389$	−24.8027			−1.25754			−0.628772	+	0.777590i	$0.716442\pi$
$390$		0			0
$391$			19.1842i			0.970188i
$392$		0			0
$393$	−2.85594	+	0.363105i	−0.144063	+	0.0183162i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−15.5873			−0.782306			−0.391153	−	0.920326i	$-0.627924\pi$
$397$	−15.5873			−0.782306			−0.391153	+	0.920326i	$0.627924\pi$
$398$		0			0
$399$	1.00646	+	7.91612i	0.0503860	+	0.396302i
$400$		0			0
$401$			26.3852i			1.31761i	0.752313	+	0.658806i	$0.228938\pi$
$401$			26.3852i			1.31761i	−0.752313	+	0.658806i	$0.771062\pi$
$402$		0			0
$403$			8.30708i			0.413805i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			38.4017i			1.90350i
$408$		0			0
$409$	0.497143			0.0245821			0.0122911	−	0.999924i	$-0.496088\pi$
$409$	0.497143			0.0245821			0.0122911	+	0.999924i	$0.496088\pi$
$410$		0			0
$411$	−2.91756	−	22.9475i	−0.143912	−	1.13192i
$412$		0			0
$413$		−	2.87288i		−	0.141365i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.13570	+	24.6633i	0.153556	+	1.20777i
$418$		0			0
$419$			11.0240i			0.538556i	0.963063	+	0.269278i	$0.0867850\pi$
$419$			11.0240i			0.538556i	−0.963063	+	0.269278i	$0.913215\pi$
$420$		0			0
$421$		−	11.0971i		−	0.540840i	−0.962742	−	0.270420i	$-0.912837\pi$
$421$		−	11.0971i		−	0.540840i	0.962742	−	0.270420i	$-0.0871626\pi$
$422$		0			0
$423$	8.19589	+	31.7106i	0.398498	+	1.54182i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			2.28712i			0.110681i
$428$		0			0
$429$	−14.8729	+	1.89094i	−0.718069	+	0.0912956i
$430$		0			0
$431$	6.84000			0.329471			0.164736	−	0.986338i	$-0.447323\pi$
$431$	6.84000			0.329471			0.164736	+	0.986338i	$0.447323\pi$
$432$		0			0
$433$		−	38.6378i		−	1.85681i	−0.371567	−	0.928406i	$-0.621179\pi$
$433$		−	38.6378i		−	1.85681i	0.371567	−	0.928406i	$-0.378821\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5.94269i			0.284277i
$438$		0			0
$439$			4.07908i			0.194684i	0.995251	+	0.0973420i	$0.0310341\pi$
$439$			4.07908i			0.194684i	−0.995251	+	0.0973420i	$0.968966\pi$
$440$		0			0
$441$	−18.2425	+	4.71494i	−0.868692	+	0.224521i
$442$		0			0
$443$	12.7140			0.604059			0.302030	−	0.953299i	$-0.402336\pi$
$443$	12.7140			0.604059			0.302030	+	0.953299i	$0.402336\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	3.99209	+	31.3991i	0.188819	+	1.48513i
$448$		0			0
$449$			31.2985i			1.47707i	0.674217	+	0.738533i	$0.264481\pi$
$449$			31.2985i			1.47707i	−0.674217	+	0.738533i	$0.735519\pi$
$450$		0			0
$451$	−7.61822			−0.358728
$452$		0			0
$453$	28.8063	−	3.66244i	1.35344	−	0.172077i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	29.1097i		−	1.36170i	−0.732425	−	0.680848i	$-0.761612\pi$
$457$		−	29.1097i		−	1.36170i	0.732425	−	0.680848i	$-0.238388\pi$
$458$		0			0
$459$	7.85379	+	19.6985i	0.366583	+	0.919446i
$460$		0			0
$461$	−0.00687071			−0.000320001			−0.000160001	−	1.00000i	$-0.500051\pi$
$461$	−0.00687071			−0.000320001			−0.000160001		1.00000i	$0.500051\pi$
$462$		0			0
$463$	−17.9904			−0.836086			−0.418043	−	0.908427i	$-0.637284\pi$
$463$	−17.9904			−0.836086			−0.418043	+	0.908427i	$0.637284\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.99927			−0.185064			−0.0925321	−	0.995710i	$-0.529496\pi$
$467$	−3.99927			−0.185064			−0.0925321	+	0.995710i	$0.529496\pi$
$468$		0			0
$469$		−	15.2395i		−	0.703695i
$470$		0			0
$471$	1.37913	+	10.8473i	0.0635470	+	0.499817i
$472$		0			0
$473$	17.4366			0.801735
$474$		0			0
$475$		0			0
$476$		0			0
$477$	6.66093	+	25.7718i	0.304983	+	1.18001i
$478$		0			0
$479$	−34.1467			−1.56020			−0.780101	−	0.625654i	$-0.784832\pi$
$479$	−34.1467			−1.56020			−0.780101	+	0.625654i	$0.784832\pi$
$480$		0			0
$481$	12.9111			0.588693
$482$		0			0
$483$	−29.4339	+	3.74223i	−1.33929	+	0.170278i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−0.851820			−0.0385997			−0.0192998	−	0.999814i	$-0.506144\pi$
$487$	−0.851820			−0.0385997			−0.0192998	+	0.999814i	$0.506144\pi$
$488$		0			0
$489$	4.62389	−	0.587884i	0.209100	−	0.0265850i
$490$		0			0
$491$		−	6.09115i		−	0.274890i	−0.990509	−	0.137445i	$-0.956111\pi$
$491$		−	6.09115i		−	0.274890i	0.990509	−	0.137445i	$-0.0438890\pi$
$492$		0			0
$493$	4.34142			0.195528
$494$		0			0
$495$		0			0
$496$		0			0
$497$	22.6560			1.01626
$498$		0			0
$499$	13.1375			0.588116			0.294058	−	0.955788i	$-0.404994\pi$
$499$	13.1375			0.588116			0.294058	+	0.955788i	$0.404994\pi$
$500$		0			0
$501$	28.6312	−	3.64018i	1.27915	−	0.162631i
$502$		0			0
$503$		−	27.6064i		−	1.23091i	−0.788172	−	0.615455i	$-0.788972\pi$
$503$		−	27.6064i		−	1.23091i	0.788172	−	0.615455i	$-0.211028\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.20416	−	17.3364i	−0.0978902	−	0.769938i
$508$		0			0
$509$	−5.46468			−0.242218			−0.121109	−	0.992639i	$-0.538645\pi$
$509$	−5.46468			−0.242218			−0.121109	+	0.992639i	$0.538645\pi$
$510$		0			0
$511$			15.3675i			0.679817i
$512$		0			0
$513$	2.43287	+	6.10199i	0.107414	+	0.269409i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	55.3960			2.43631
$518$		0			0
$519$	−20.1818	+	2.56592i	−0.885882	+	0.112631i
$520$		0			0
$521$			39.1312i			1.71437i	0.515010	+	0.857184i	$0.327788\pi$
$521$			39.1312i			1.71437i	−0.515010	+	0.857184i	$0.672212\pi$
$522$		0			0
$523$			14.3467i			0.627336i	0.949533	+	0.313668i	$0.101558\pi$
$523$			14.3467i			0.627336i	−0.949533	+	0.313668i	$0.898442\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	19.8733i		−	0.865694i
$528$		0			0
$529$	0.903770			0.0392944
$530$		0			0
$531$	−0.591804	−	2.28974i	−0.0256821	−	0.0993664i
$532$		0			0
$533$			2.56132i			0.110943i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.802169	−	0.101988i	0.0346161	−	0.00440111i
$538$		0			0
$539$			31.8682i			1.37266i
$540$		0			0
$541$		−	29.8846i		−	1.28484i	−0.766352	−	0.642420i	$-0.777930\pi$
$541$		−	29.8846i		−	1.28484i	0.766352	−	0.642420i	$-0.222070\pi$
$542$		0			0
$543$	−28.4585	+	3.61822i	−1.22127	+	0.155273i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	2.78046i		−	0.118884i	−0.998232	−	0.0594418i	$-0.981068\pi$
$547$		−	2.78046i		−	0.118884i	0.998232	−	0.0594418i	$-0.0189321\pi$
$548$		0			0
$549$	0.471140	+	1.82288i	0.0201078	+	0.0777987i
$550$		0			0
$551$	1.34484			0.0572921
$552$		0			0
$553$		−	3.61822i		−	0.153862i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			39.3640i			1.66791i	0.551836	+	0.833953i	$0.313927\pi$
$557$			39.3640i			1.66791i	−0.551836	+	0.833953i	$0.686073\pi$
$558$		0			0
$559$		−	5.86236i		−	0.247951i
$560$		0			0
$561$	35.5809	−	4.52376i	1.50223	−	0.190994i
$562$		0			0
$563$	16.1259			0.679624			0.339812	−	0.940493i	$-0.389637\pi$
$563$	16.1259			0.679624			0.339812	+	0.940493i	$0.389637\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−28.6908	+	15.8924i	−1.20490	+	0.667419i
$568$		0			0
$569$		−	8.55906i		−	0.358814i	−0.983775	−	0.179407i	$-0.942582\pi$
$569$		−	8.55906i		−	0.358814i	0.983775	−	0.179407i	$-0.0574180\pi$
$570$		0			0
$571$	−34.6984			−1.45208			−0.726042	−	0.687650i	$-0.758642\pi$
$571$	−34.6984			−1.45208			−0.726042	+	0.687650i	$0.758642\pi$
$572$		0			0
$573$	−1.41801	−	11.1531i	−0.0592383	−	0.465929i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			32.8684i			1.36833i	0.729328	+	0.684165i	$0.239833\pi$
$577$			32.8684i			1.36833i	−0.729328	+	0.684165i	$0.760167\pi$
$578$		0			0
$579$	−19.5900	+	2.49068i	−0.814132	+	0.103509i
$580$		0			0
$581$	28.1535			1.16801
$582$		0			0
$583$	45.0212			1.86459
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.4302			−0.636872			−0.318436	−	0.947944i	$-0.603158\pi$
$587$	−15.4302			−0.636872			−0.318436	+	0.947944i	$0.603158\pi$
$588$		0			0
$589$		−	6.15614i		−	0.253659i
$590$		0			0
$591$	6.11843	−	0.777899i	0.251678	−	0.0319985i
$592$		0			0
$593$	14.0511			0.577008			0.288504	−	0.957479i	$-0.406842\pi$
$593$	14.0511			0.577008			0.288504	+	0.957479i	$0.406842\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.56830	+	0.962236i	−0.309750	+	0.0393817i
$598$		0			0
$599$	30.0200			1.22658			0.613291	−	0.789857i	$-0.289845\pi$
$599$	30.0200			1.22658			0.613291	+	0.789857i	$0.289845\pi$
$600$		0			0
$601$	−29.3006			−1.19520			−0.597599	−	0.801795i	$-0.703878\pi$
$601$	−29.3006			−1.19520			−0.597599	+	0.801795i	$0.703878\pi$
$602$		0			0
$603$	−3.13929	−	12.1462i	−0.127842	−	0.494632i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−38.9265			−1.57998			−0.789989	−	0.613122i	$-0.789914\pi$
$607$	−38.9265			−1.57998			−0.789989	+	0.613122i	$0.789914\pi$
$608$		0			0
$609$	0.846874	+	6.66093i	0.0343171	+	0.269915i
$610$		0			0
$611$		−	18.6247i		−	0.753474i
$612$		0			0
$613$	−28.9092			−1.16763			−0.583816	−	0.811886i	$-0.698441\pi$
$613$	−28.9092			−1.16763			−0.583816	+	0.811886i	$0.698441\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	26.5340			1.06822			0.534109	−	0.845416i	$-0.320647\pi$
$617$	26.5340			1.06822			0.534109	+	0.845416i	$0.320647\pi$
$618$		0			0
$619$	−17.9531			−0.721595			−0.360798	−	0.932644i	$-0.617495\pi$
$619$	−17.9531			−0.721595			−0.360798	+	0.932644i	$0.617495\pi$
$620$		0			0
$621$	−22.6885	+	9.04593i	−0.910459	+	0.363001i
$622$		0			0
$623$			42.1795i			1.68989i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	11.0219	−	1.40133i	0.440171	−	0.0559635i
$628$		0			0
$629$	−30.8875			−1.23157
$630$		0			0
$631$			33.8875i			1.34904i	0.738257	+	0.674520i	$0.235649\pi$
$631$			33.8875i			1.34904i	−0.738257	+	0.674520i	$0.764351\pi$
$632$		0			0
$633$	3.52523	+	27.7271i	0.140115	+	1.10205i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	10.7144			0.424521
$638$		0			0
$639$	18.0573	−	4.66707i	0.714336	−	0.184626i
$640$		0			0
$641$		−	16.1456i		−	0.637711i	−0.947803	−	0.318856i	$-0.896702\pi$
$641$		−	16.1456i		−	0.637711i	0.947803	−	0.318856i	$-0.103298\pi$
$642$		0			0
$643$		−	41.9604i		−	1.65476i	−0.561645	−	0.827378i	$-0.689831\pi$
$643$		−	41.9604i		−	1.65476i	0.561645	−	0.827378i	$-0.310169\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9.10158i			0.357820i	0.983865	+	0.178910i	$0.0572571\pi$
$647$			9.10158i			0.357820i	−0.983865	+	0.178910i	$0.942743\pi$
$648$		0			0
$649$	−4.00000			−0.157014
$650$		0			0
$651$	30.4911	−	3.87665i	1.19504	−	0.151938i
$652$		0			0
$653$		−	12.0573i		−	0.471839i	−0.971773	−	0.235919i	$-0.924190\pi$
$653$		−	12.0573i		−	0.471839i	0.971773	−	0.235919i	$-0.0758102\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	3.16565	+	12.2482i	0.123504	+	0.477848i
$658$		0			0
$659$		−	12.1916i		−	0.474916i	−0.971398	−	0.237458i	$-0.923686\pi$
$659$		−	12.1916i		−	0.474916i	0.971398	−	0.237458i	$-0.0763142\pi$
$660$		0			0
$661$		−	22.0284i		−	0.856806i	−0.903588	−	0.428403i	$-0.859076\pi$
$661$		−	22.0284i		−	0.856806i	0.903588	−	0.428403i	$-0.140924\pi$
$662$		0			0
$663$	−1.52094	−	11.9626i	−0.0590682	−	0.464591i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			5.00041i			0.193617i
$668$		0			0
$669$	1.33143	+	10.4721i	0.0514759	+	0.404875i
$670$		0			0
$671$	3.18443			0.122934
$672$		0			0
$673$			25.3258i			0.976238i	0.872777	+	0.488119i	$0.162317\pi$
$673$			25.3258i			0.976238i	−0.872777	+	0.488119i	$0.837683\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	21.9435i		−	0.843358i	−0.906745	−	0.421679i	$-0.861441\pi$
$677$		−	21.9435i		−	0.843358i	0.906745	−	0.421679i	$-0.138559\pi$
$678$		0			0
$679$		−	26.9724i		−	1.03511i
$680$		0			0
$681$	−4.47118	−	35.1673i	−0.171336	−	1.34761i
$682$		0			0
$683$	−16.0383			−0.613687			−0.306844	−	0.951760i	$-0.599273\pi$
$683$	−16.0383			−0.613687			−0.306844	+	0.951760i	$0.599273\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−45.4860	+	5.78311i	−1.73540	+	0.220639i
$688$		0			0
$689$		−	15.1366i		−	0.576658i
$690$		0			0
$691$	23.0993			0.878740			0.439370	−	0.898306i	$-0.355202\pi$
$691$	23.0993			0.878740			0.439370	+	0.898306i	$0.355202\pi$
$692$		0			0
$693$	13.8814	+	53.7084i	0.527311	+	2.04021i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	6.12753i		−	0.232097i
$698$		0			0
$699$	−5.32534	−	41.8855i	−0.201423	−	1.58426i
$700$		0			0
$701$	0.254246			0.00960274			0.00480137	−	0.999988i	$-0.498472\pi$
$701$	0.254246			0.00960274			0.00480137	+	0.999988i	$0.498472\pi$
$702$		0			0
$703$	−9.56802			−0.360865
$704$		0			0
$705$		0			0
$706$		0			0
$707$	37.0022			1.39161
$708$		0			0
$709$			19.9746i			0.750163i	0.926992	+	0.375082i	$0.122385\pi$
$709$			19.9746i			0.750163i	−0.926992	+	0.375082i	$0.877615\pi$
$710$		0			0
$711$	−0.745342	−	2.88380i	−0.0279525	−	0.108151i
$712$		0			0
$713$	22.8899			0.857233
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1.19529	−	9.40133i	−0.0446389	−	0.351099i
$718$		0			0
$719$	−33.7978			−1.26044			−0.630222	−	0.776415i	$-0.717036\pi$
$719$	−33.7978			−1.26044			−0.630222	+	0.776415i	$0.717036\pi$
$720$		0			0
$721$	56.6317			2.10908
$722$		0			0
$723$	5.38374	+	42.3449i	0.200224	+	1.57482i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	27.0308			1.00252			0.501258	−	0.865298i	$-0.332871\pi$
$727$	27.0308			1.00252			0.501258	+	0.865298i	$0.332871\pi$
$728$		0			0
$729$	−19.5934	+	18.5768i	−0.725682	+	0.688030i
$730$		0			0
$731$			14.0247i			0.518722i
$732$		0			0
$733$	−15.8811			−0.586583			−0.293291	−	0.956023i	$-0.594751\pi$
$733$	−15.8811			−0.586583			−0.293291	+	0.956023i	$0.594751\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.2185			−0.781592
$738$		0			0
$739$	−14.1753			−0.521446			−0.260723	−	0.965414i	$-0.583961\pi$
$739$	−14.1753			−0.521446			−0.260723	+	0.965414i	$0.583961\pi$
$740$		0			0
$741$	−0.471140	−	3.70567i	−0.0173078	−	0.136131i
$742$		0			0
$743$			29.5744i			1.08498i	0.840063	+	0.542489i	$0.182518\pi$
$743$			29.5744i			1.08498i	−0.840063	+	0.542489i	$0.817482\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	22.4390	−	5.79954i	0.820999	−	0.212194i
$748$		0			0
$749$	54.1926			1.98016
$750$		0			0
$751$			7.58573i			0.276807i	0.990376	+	0.138404i	$0.0441971\pi$
$751$			7.58573i			0.276807i	−0.990376	+	0.138404i	$0.955803\pi$
$752$		0			0
$753$	3.00642	−	0.382237i	0.109560	−	0.0139295i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−31.1258			−1.13129			−0.565643	−	0.824650i	$-0.691372\pi$
$757$	−31.1258			−1.13129			−0.565643	+	0.824650i	$0.691372\pi$
$758$		0			0
$759$	5.21043	+	40.9817i	0.189127	+	1.48754i
$760$		0			0
$761$		−	8.24575i		−	0.298908i	−0.988769	−	0.149454i	$-0.952248\pi$
$761$		−	8.24575i		−	0.298908i	0.988769	−	0.149454i	$-0.0477516\pi$
$762$		0			0
$763$			38.4017i			1.39024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1.34484i			0.0485594i
$768$		0			0
$769$	26.3262			0.949347			0.474674	−	0.880162i	$-0.342566\pi$
$769$	26.3262			0.949347			0.474674	+	0.880162i	$0.342566\pi$
$770$		0			0
$771$	−1.15267	−	9.06612i	−0.0415124	−	0.326508i
$772$		0			0
$773$			27.2417i			0.979817i	0.871774	+	0.489909i	$0.162970\pi$
$773$			27.2417i			0.979817i	−0.871774	+	0.489909i	$0.837030\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−6.02518	−	47.3900i	−0.216152	−	1.70011i
$778$		0			0
$779$		−	1.89812i		−	0.0680074i
$780$		0			0
$781$		−	31.5447i		−	1.12876i
$782$		0			0
$783$	2.04711	+	5.13445i	0.0731577	+	0.183490i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	26.9324i		−	0.960037i	−0.877258	−	0.480019i	$-0.840630\pi$
$787$		−	26.9324i		−	0.960037i	0.877258	−	0.480019i	$-0.159370\pi$
$788$		0			0
$789$	−3.07636	+	0.391129i	−0.109521	+	0.0139246i
$790$		0			0
$791$	−36.2369			−1.28843
$792$		0			0
$793$		−	1.07064i		−	0.0380195i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			13.1279i			0.465016i	0.972595	+	0.232508i	$0.0746931\pi$
$797$			13.1279i			0.465016i	−0.972595	+	0.232508i	$0.925307\pi$
$798$		0			0
$799$			44.5565i			1.57629i
$800$		0			0
$801$	8.68886	+	33.6180i	0.307006	+	1.18783i
$802$		0			0
$803$	21.3966			0.755071
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−4.08685	−	32.1444i	−0.143864	−	1.13154i
$808$		0			0
$809$		−	29.5632i		−	1.03939i	−0.854353	−	0.519693i	$-0.826046\pi$
$809$		−	29.5632i		−	1.03939i	0.854353	−	0.519693i	$-0.173954\pi$
$810$		0			0
$811$	−29.2269			−1.02629			−0.513147	−	0.858301i	$-0.671520\pi$
$811$	−29.2269			−1.02629			−0.513147	+	0.858301i	$0.671520\pi$
$812$		0			0
$813$	42.6877	−	5.42733i	1.49712	−	0.190345i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			4.34443i			0.151992i
$818$		0			0
$819$	18.0573	−	4.66707i	0.630973	−	0.163081i
$820$		0			0
$821$	−5.29201			−0.184692			−0.0923462	−	0.995727i	$-0.529437\pi$
$821$	−5.29201			−0.184692			−0.0923462	+	0.995727i	$0.529437\pi$
$822$		0			0
$823$	16.0047			0.557890			0.278945	−	0.960307i	$-0.410015\pi$
$823$	16.0047			0.557890			0.278945	+	0.960307i	$0.410015\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.13539			−0.143802			−0.0719009	−	0.997412i	$-0.522907\pi$
$827$	−4.13539			−0.143802			−0.0719009	+	0.997412i	$0.522907\pi$
$828$		0			0
$829$		−	22.5879i		−	0.784512i	−0.919856	−	0.392256i	$-0.871695\pi$
$829$		−	22.5879i		−	0.784512i	0.919856	−	0.392256i	$-0.128305\pi$
$830$		0			0
$831$	−6.37954	−	50.1771i	−0.221304	−	1.74063i
$832$		0			0
$833$	−25.6325			−0.888113
$834$		0			0
$835$		0			0
$836$		0			0
$837$	23.5035	−	9.37084i	0.812399	−	0.323904i
$838$		0			0
$839$	32.0329			1.10590			0.552949	−	0.833215i	$-0.313502\pi$
$839$	32.0329			1.10590			0.552949	+	0.833215i	$0.313502\pi$
$840$		0			0
$841$	−27.8684			−0.960979
$842$		0			0
$843$	39.4944	−	5.02133i	1.36026	−	0.172944i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	53.7374			1.84644
$848$		0			0
$849$	−3.49677	+	0.444581i	−0.120009	+	0.0152580i
$850$		0			0
$851$		−	35.5760i		−	1.21953i
$852$		0			0
$853$	−10.3607			−0.354745			−0.177372	−	0.984144i	$-0.556760\pi$
$853$	−10.3607			−0.354745			−0.177372	+	0.984144i	$0.556760\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.1893			0.621334			0.310667	−	0.950519i	$-0.399448\pi$
$857$	18.1893			0.621334			0.310667	+	0.950519i	$0.399448\pi$
$858$		0			0
$859$	27.2650			0.930271			0.465136	−	0.885239i	$-0.346006\pi$
$859$	27.2650			0.930271			0.465136	+	0.885239i	$0.346006\pi$
$860$		0			0
$861$	9.40133	−	1.19529i	0.320396	−	0.0407353i
$862$		0			0
$863$		−	12.4082i		−	0.422381i	−0.977445	−	0.211191i	$-0.932266\pi$
$863$		−	12.4082i		−	0.422381i	0.977445	−	0.211191i	$-0.0677341\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.0751522	−	0.591096i	−0.00255230	−	0.0200747i
$868$		0			0
$869$	−5.03776			−0.170894
$870$		0			0
$871$			7.13386i			0.241722i
$872$		0			0
$873$	−5.55623	−	21.4976i	−0.188050	−	0.727582i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.19349			−0.310442			−0.155221	−	0.987880i	$-0.549609\pi$
$877$	−9.19349			−0.310442			−0.155221	+	0.987880i	$0.549609\pi$
$878$		0			0
$879$	−10.3093	+	1.31073i	−0.347725	+	0.0442098i
$880$		0			0
$881$		−	38.4479i		−	1.29534i	−0.761920	−	0.647671i	$-0.775743\pi$
$881$		−	38.4479i		−	1.29534i	0.761920	−	0.647671i	$-0.224257\pi$
$882$		0			0
$883$		−	5.12406i		−	0.172438i	−0.996276	−	0.0862192i	$-0.972521\pi$
$883$		−	5.12406i		−	0.172438i	0.996276	−	0.0862192i	$-0.0274785\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1.79043i			0.0601169i	0.999548	+	0.0300584i	$0.00956934\pi$
$887$			1.79043i			0.0601169i	−0.999548	+	0.0300584i	$0.990431\pi$
$888$		0			0
$889$	−20.5171			−0.688121
$890$		0			0
$891$	22.1275	+	39.9472i	0.741300	+	1.33828i
$892$		0			0
$893$			13.8022i			0.461874i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	13.7785	−	1.75180i	0.460050	−	0.0584909i
$898$		0			0
$899$		−	5.18002i		−	0.172763i
$900$		0			0
$901$			36.2118i			1.20639i
$902$		0			0
$903$	−21.5178	+	2.73578i	−0.716066	+	0.0910409i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			34.5506i			1.14724i	0.819123	+	0.573618i	$0.194460\pi$
$907$			34.5506i			1.14724i	−0.819123	+	0.573618i	$0.805540\pi$
$908$		0			0
$909$	29.4915	−	7.62234i	0.978172	−	0.252817i
$910$		0			0
$911$	52.6515			1.74442			0.872211	−	0.489130i	$-0.162686\pi$
$911$	52.6515			1.74442			0.872211	+	0.489130i	$0.162686\pi$
$912$		0			0
$913$		−	39.1991i		−	1.29730i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	6.05731i		−	0.200030i
$918$		0			0
$919$			13.0595i			0.430794i	0.976527	+	0.215397i	$0.0691046\pi$
$919$			13.0595i			0.430794i	−0.976527	+	0.215397i	$0.930895\pi$
$920$		0			0
$921$	−32.7401	+	4.16259i	−1.07882	+	0.137162i
$922$		0			0
$923$	−10.6056			−0.349089
$924$		0			0
$925$		0			0
$926$		0			0
$927$	45.1367	−	11.6660i	1.48248	−	0.383160i
$928$		0			0
$929$		−	20.9794i		−	0.688313i	−0.938912	−	0.344156i	$-0.888165\pi$
$929$		−	20.9794i		−	0.688313i	0.938912	−	0.344156i	$-0.111835\pi$
$930$		0			0
$931$	−7.94016			−0.260228
$932$		0			0
$933$	−5.69953	−	44.8287i	−0.186594	−	1.46762i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	5.78393i		−	0.188953i	−0.995527	−	0.0944764i	$-0.969882\pi$
$937$		−	5.78393i		−	0.188953i	0.995527	−	0.0944764i	$-0.0301177\pi$
$938$		0			0
$939$	−34.6729	+	4.40832i	−1.13151	+	0.143860i
$940$		0			0
$941$	−42.0131			−1.36959			−0.684794	−	0.728737i	$-0.740108\pi$
$941$	−42.0131			−1.36959			−0.684794	+	0.728737i	$0.740108\pi$
$942$		0			0
$943$	7.05764			0.229829
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23.5854			−0.766421			−0.383211	−	0.923661i	$-0.625182\pi$
$947$	−23.5854			−0.766421			−0.383211	+	0.923661i	$0.625182\pi$
$948$		0			0
$949$		−	7.19377i		−	0.233520i
$950$		0			0
$951$	13.4296	−	1.70744i	0.435484	−	0.0553675i
$952$		0			0
$953$	−31.8648			−1.03220			−0.516102	−	0.856527i	$-0.672617\pi$
$953$	−31.8648			−1.03220			−0.516102	+	0.856527i	$0.672617\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	9.27423	−	1.17913i	0.299793	−	0.0381159i
$958$		0			0
$959$	48.6706			1.57166
$960$		0			0
$961$	7.28794			0.235095
$962$		0			0
$963$	43.1927	−	11.1635i	1.39186	−	0.359739i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−11.1038			−0.357073			−0.178537	−	0.983933i	$-0.557136\pi$
$967$	−11.1038			−0.357073			−0.178537	+	0.983933i	$0.557136\pi$
$968$		0			0
$969$	1.12712	+	8.86519i	0.0362084	+	0.284791i
$970$		0			0
$971$			47.5500i			1.52595i	0.646427	+	0.762976i	$0.276263\pi$
$971$			47.5500i			1.52595i	−0.646427	+	0.762976i	$0.723737\pi$
$972$		0			0
$973$	−52.3097			−1.67697
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−51.9896			−1.66329			−0.831647	−	0.555304i	$-0.812602\pi$
$977$	−51.9896			−1.66329			−0.831647	+	0.555304i	$0.812602\pi$
$978$		0			0
$979$	58.7279			1.87695
$980$		0			0
$981$	7.91064	+	30.6070i	0.252567	+	0.977206i
$982$		0			0
$983$		−	27.4842i		−	0.876609i	−0.898826	−	0.438305i	$-0.855579\pi$
$983$		−	27.4842i		−	0.876609i	0.898826	−	0.438305i	$-0.144421\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−68.3619	+	8.69155i	−2.17598	+	0.276655i
$988$		0			0
$989$	−16.1535			−0.513653
$990$		0			0
$991$			24.4556i			0.776858i	0.921479	+	0.388429i	$0.126982\pi$
$991$			24.4556i			0.776858i	−0.921479	+	0.388429i	$0.873018\pi$
$992$		0			0
$993$	2.19382	+	17.2551i	0.0696188	+	0.547575i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−39.1729			−1.24062			−0.620309	−	0.784358i	$-0.712993\pi$
$997$	−39.1729			−1.24062			−0.620309	+	0.784358i	$0.712993\pi$
$998$		0			0
$999$	−14.5644	−	36.5296i	−0.460797	−	1.15575i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.m.d.1199.10		16
3.2	odd	2		2400.2.m.c.1199.8		16
4.3	odd	2		600.2.m.c.299.11		16
5.2	odd	4		480.2.b.b.431.7		8
5.3	odd	4		2400.2.b.f.2351.2		8
5.4	even	2	inner	2400.2.m.d.1199.7		16
8.3	odd	2		2400.2.m.c.1199.10		16
8.5	even	2		600.2.m.d.299.12		16
12.11	even	2		600.2.m.d.299.6		16
15.2	even	4		480.2.b.a.431.8		8
15.8	even	4		2400.2.b.e.2351.1		8
15.14	odd	2		2400.2.m.c.1199.9		16
20.3	even	4		600.2.b.f.251.1		8
20.7	even	4		120.2.b.a.11.8	yes	8
20.19	odd	2		600.2.m.c.299.6		16
24.5	odd	2		600.2.m.c.299.5		16
24.11	even	2	inner	2400.2.m.d.1199.8		16
40.3	even	4		2400.2.b.e.2351.2		8
40.13	odd	4		600.2.b.e.251.7		8
40.19	odd	2		2400.2.m.c.1199.7		16
40.27	even	4		480.2.b.a.431.7		8
40.29	even	2		600.2.m.d.299.5		16
40.37	odd	4		120.2.b.b.11.2	yes	8
60.23	odd	4		600.2.b.e.251.8		8
60.47	odd	4		120.2.b.b.11.1	yes	8
60.59	even	2		600.2.m.d.299.11		16
120.29	odd	2		600.2.m.c.299.12		16
120.53	even	4		600.2.b.f.251.2		8
120.59	even	2	inner	2400.2.m.d.1199.9		16
120.77	even	4		120.2.b.a.11.7	✓	8
120.83	odd	4		2400.2.b.f.2351.1		8
120.107	odd	4		480.2.b.b.431.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.b.a.11.7	✓	8	120.77	even	4
120.2.b.a.11.8	yes	8	20.7	even	4
120.2.b.b.11.1	yes	8	60.47	odd	4
120.2.b.b.11.2	yes	8	40.37	odd	4
480.2.b.a.431.7		8	40.27	even	4
480.2.b.a.431.8		8	15.2	even	4
480.2.b.b.431.7		8	5.2	odd	4
480.2.b.b.431.8		8	120.107	odd	4
600.2.b.e.251.7		8	40.13	odd	4
600.2.b.e.251.8		8	60.23	odd	4
600.2.b.f.251.1		8	20.3	even	4
600.2.b.f.251.2		8	120.53	even	4
600.2.m.c.299.5		16	24.5	odd	2
600.2.m.c.299.6		16	20.19	odd	2
600.2.m.c.299.11		16	4.3	odd	2
600.2.m.c.299.12		16	120.29	odd	2
600.2.m.d.299.5		16	40.29	even	2
600.2.m.d.299.6		16	12.11	even	2
600.2.m.d.299.11		16	60.59	even	2
600.2.m.d.299.12		16	8.5	even	2
2400.2.b.e.2351.1		8	15.8	even	4
2400.2.b.e.2351.2		8	40.3	even	4
2400.2.b.f.2351.1		8	120.83	odd	4
2400.2.b.f.2351.2		8	5.3	odd	4
2400.2.m.c.1199.7		16	40.19	odd	2
2400.2.m.c.1199.8		16	3.2	odd	2
2400.2.m.c.1199.9		16	15.14	odd	2
2400.2.m.c.1199.10		16	8.3	odd	2
2400.2.m.d.1199.7		16	5.4	even	2	inner
2400.2.m.d.1199.8		16	24.11	even	2	inner
2400.2.m.d.1199.9		16	120.59	even	2	inner
2400.2.m.d.1199.10		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(0.218455 + 1.71822i) q^{3} -3.64426 q^{7} +(-2.90455 + 0.750707i) q^{9} +O(q^{10})\)	\(q+(0.218455 + 1.71822i) q^{3} -3.64426 q^{7} +(-2.90455 + 0.750707i) q^{9} +5.07403i q^{11} +1.70594 q^{13} -4.08117 q^{17} -1.26422 q^{19} +(-0.796108 - 6.26164i) q^{21} -4.70066i q^{23} +(-1.92439 - 4.82667i) q^{27} -1.06377 q^{29} +4.86950i q^{31} +(-8.71829 + 1.10845i) q^{33} +7.56830 q^{37} +(0.372671 + 2.93118i) q^{39} +1.50141i q^{41} -3.43644i q^{43} -10.9176i q^{47} +6.28066 q^{49} +(-0.891553 - 7.01235i) q^{51} -8.87288i q^{53} +(-0.276176 - 2.17221i) q^{57} +0.788328i q^{59} -0.627594i q^{61} +(10.5850 - 2.73578i) q^{63} +4.18178i q^{67} +(8.07677 - 1.02688i) q^{69} -6.21689 q^{71} -4.21689i q^{73} -18.4911i q^{77} +0.992853i q^{79} +(7.87288 - 4.36094i) q^{81} -7.72544 q^{83} +(-0.232385 - 1.82779i) q^{87} -11.5742i q^{89} -6.21689 q^{91} +(-8.36687 + 1.06377i) q^{93} +7.40133i q^{97} +(-3.80911 - 14.7378i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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