LMFDB - Embedded newform 1440.2.h.b.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(1151,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("1440.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1440.1151
Dual form			1440.2.h.b.1151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +3.41421i q^{7} +O(q^{10})$	$q+1.00000i q^{5} +3.41421i q^{7} +1.41421 q^{11} -6.24264 q^{13} +6.82843i q^{17} -5.65685i q^{19} -8.82843 q^{23} -1.00000 q^{25} +2.00000i q^{29} -8.82843i q^{31} -3.41421 q^{35} +7.41421 q^{37} -0.242641i q^{41} +1.17157i q^{43} -8.00000 q^{47} -4.65685 q^{49} +1.17157i q^{53} +1.41421i q^{55} -2.58579 q^{59} -8.82843 q^{61} -6.24264i q^{65} -11.3137i q^{67} -2.34315 q^{71} -4.82843 q^{73} +4.82843i q^{77} +10.4853i q^{79} +6.82843 q^{83} -6.82843 q^{85} +15.0711i q^{89} -21.3137i q^{91} +5.65685 q^{95} -6.48528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{13} - 24 q^{23} - 4 q^{25} - 8 q^{35} + 24 q^{37} - 32 q^{47} + 4 q^{49} - 16 q^{59} - 24 q^{61} - 32 q^{71} - 8 q^{73} + 16 q^{83} - 16 q^{85} + 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^13 - 24 * q^23 - 4 * q^25 - 8 * q^35 + 24 * q^37 - 32 * q^47 + 4 * q^49 - 16 * q^59 - 24 * q^61 - 32 * q^71 - 8 * q^73 + 16 * q^83 - 16 * q^85 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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	0
$3$	0	0
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	3.41421i	1.29045i	0.763992	+	0.645226i	$0.223237\pi$
$7$	3.41421i	1.29045i	−0.763992	+	0.645226i	$0.776763\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.82843i	1.65614i	0.560627	+	0.828068i	$0.310560\pi$
$17$	6.82843i	1.65614i	−0.560627	+	0.828068i	$0.689440\pi$
$18$	0	0
$19$	− 5.65685i	− 1.29777i	−0.760886	−	0.648886i	$-0.775235\pi$
$19$	− 5.65685i	− 1.29777i	0.760886	−	0.648886i	$-0.224765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	− 8.82843i	− 1.58563i	−0.609461	−	0.792816i	$-0.708614\pi$
$31$	− 8.82843i	− 1.58563i	0.609461	−	0.792816i	$-0.291386\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.41421	−0.577107
$36$	0	0
$37$	7.41421	1.21889	0.609445	−	0.792829i	$-0.291392\pi$
$37$	7.41421	1.21889	0.609445	+	0.792829i	$0.291392\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 0.242641i	− 0.0378941i	−0.999820	−	0.0189471i	$-0.993969\pi$
$41$	− 0.242641i	− 0.0378941i	0.999820	−	0.0189471i	$-0.00603140\pi$
$42$	0	0
$43$	1.17157i	0.178663i	0.996002	+	0.0893316i	$0.0284731\pi$
$43$	1.17157i	0.178663i	−0.996002	+	0.0893316i	$0.971527\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−4.65685	−0.665265
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.17157i	0.160928i	0.996758	+	0.0804640i	$0.0256402\pi$
$53$	1.17157i	0.160928i	−0.996758	+	0.0804640i	$0.974360\pi$
$54$	0	0
$55$	1.41421i	0.190693i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.58579	−0.336641	−0.168320	−	0.985732i	$-0.553834\pi$
$59$	−2.58579	−0.336641	−0.168320	+	0.985732i	$0.553834\pi$
$60$	0	0
$61$	−8.82843	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$61$	−8.82843	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.24264i	− 0.774304i
$66$	0	0
$67$	− 11.3137i	− 1.38219i	−0.722764	−	0.691095i	$-0.757129\pi$
$67$	− 11.3137i	− 1.38219i	0.722764	−	0.691095i	$-0.242871\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	−4.82843	−0.565125	−0.282562	−	0.959249i	$-0.591184\pi$
$73$	−4.82843	−0.565125	−0.282562	+	0.959249i	$0.591184\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.82843i	0.550250i
$78$	0	0
$79$	10.4853i	1.17969i	0.807518	+	0.589843i	$0.200810\pi$
$79$	10.4853i	1.17969i	−0.807518	+	0.589843i	$0.799190\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.82843	0.749517	0.374759	−	0.927122i	$-0.377726\pi$
$83$	6.82843	0.749517	0.374759	+	0.927122i	$0.377726\pi$
$84$	0	0
$85$	−6.82843	−0.740647
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.0711i	1.59753i	0.601643	+	0.798765i	$0.294513\pi$
$89$	15.0711i	1.59753i	−0.601643	+	0.798765i	$0.705487\pi$
$90$	0	0
$91$	− 21.3137i	− 2.23428i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	−6.48528	−0.658481	−0.329240	−	0.944246i	$-0.606793\pi$
$97$	−6.48528	−0.658481	−0.329240	+	0.944246i	$0.606793\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.8284i	1.67449i	0.546827	+	0.837246i	$0.315835\pi$
$101$	16.8284i	1.67449i	−0.546827	+	0.837246i	$0.684165\pi$
$102$	0	0
$103$	6.24264i	0.615106i	0.951531	+	0.307553i	$0.0995101\pi$
$103$	6.24264i	0.615106i	−0.951531	+	0.307553i	$0.900490\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.1421	−0.980477	−0.490239	−	0.871588i	$-0.663090\pi$
$107$	−10.1421	−0.980477	−0.490239	+	0.871588i	$0.663090\pi$
$108$	0	0
$109$	3.17157	0.303782	0.151891	−	0.988397i	$-0.451464\pi$
$109$	3.17157	0.303782	0.151891	+	0.988397i	$0.451464\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	0	0
$115$	− 8.82843i	− 0.823255i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−23.3137	−2.13716
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	15.4142i	1.36779i	0.729580	+	0.683895i	$0.239715\pi$
$127$	15.4142i	1.36779i	−0.729580	+	0.683895i	$0.760285\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.8995	1.21440	0.607202	−	0.794547i	$-0.292292\pi$
$131$	13.8995	1.21440	0.607202	+	0.794547i	$0.292292\pi$
$132$	0	0
$133$	19.3137	1.67471
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.65685i	0.483298i	0.970364	+	0.241649i	$0.0776882\pi$
$137$	5.65685i	0.483298i	−0.970364	+	0.241649i	$0.922312\pi$
$138$	0	0
$139$	8.34315i	0.707656i	0.935310	+	0.353828i	$0.115120\pi$
$139$	8.34315i	0.707656i	−0.935310	+	0.353828i	$0.884880\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.82843	−0.738270
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 4.82843i	− 0.395560i	−0.980246	−	0.197780i	$-0.936627\pi$
$149$	− 4.82843i	− 0.395560i	0.980246	−	0.197780i	$-0.0633732\pi$
$150$	0	0
$151$	10.9706i	0.892772i	0.894841	+	0.446386i	$0.147289\pi$
$151$	10.9706i	0.892772i	−0.894841	+	0.446386i	$0.852711\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.82843	0.709116
$156$	0	0
$157$	0.100505	0.00802118	0.00401059	−	0.999992i	$-0.498723\pi$
$157$	0.100505	0.00802118	0.00401059	+	0.999992i	$0.498723\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 30.1421i	− 2.37553i
$162$	0	0
$163$	− 8.48528i	− 0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$	− 8.48528i	− 0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.4853	1.12090	0.560452	−	0.828187i	$-0.310627\pi$
$167$	14.4853	1.12090	0.560452	+	0.828187i	$0.310627\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.485281i	0.0368953i	0.999830	+	0.0184476i	$0.00587240\pi$
$173$	0.485281i	0.0368953i	−0.999830	+	0.0184476i	$0.994128\pi$
$174$	0	0
$175$	− 3.41421i	− 0.258090i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.4142	1.00263	0.501313	−	0.865266i	$-0.332851\pi$
$179$	13.4142	1.00263	0.501313	+	0.865266i	$0.332851\pi$
$180$	0	0
$181$	−24.8284	−1.84548	−0.922741	−	0.385420i	$-0.874057\pi$
$181$	−24.8284	−1.84548	−0.922741	+	0.385420i	$0.874057\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.41421i	0.545104i
$186$	0	0
$187$	9.65685i	0.706179i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.17157	−0.0847720	−0.0423860	−	0.999101i	$-0.513496\pi$
$191$	−1.17157	−0.0847720	−0.0423860	+	0.999101i	$0.513496\pi$
$192$	0	0
$193$	4.34315	0.312626	0.156313	−	0.987708i	$-0.450039\pi$
$193$	4.34315	0.312626	0.156313	+	0.987708i	$0.450039\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	− 15.6569i	− 1.10988i	−0.831889	−	0.554942i	$-0.812740\pi$
$199$	− 15.6569i	− 1.10988i	0.831889	−	0.554942i	$-0.187260\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.82843	−0.479262
$204$	0	0
$205$	0.242641	0.0169468
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 8.00000i	− 0.553372i
$210$	0	0
$211$	− 6.00000i	− 0.413057i	−0.978441	−	0.206529i	$-0.933783\pi$
$211$	− 6.00000i	− 0.413057i	0.978441	−	0.206529i	$-0.0662166\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.17157	−0.0799006
$216$	0	0
$217$	30.1421	2.04618
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 42.6274i	− 2.86743i
$222$	0	0
$223$	− 4.10051i	− 0.274590i	−0.990530	−	0.137295i	$-0.956159\pi$
$223$	− 4.10051i	− 0.274590i	0.990530	−	0.137295i	$-0.0438409\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−28.2843	−1.87729	−0.938647	−	0.344881i	$-0.887919\pi$
$227$	−28.2843	−1.87729	−0.938647	+	0.344881i	$0.887919\pi$
$228$	0	0
$229$	7.65685	0.505979	0.252990	−	0.967469i	$-0.418586\pi$
$229$	7.65685	0.505979	0.252990	+	0.967469i	$0.418586\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.4853i	0.817938i	0.912548	+	0.408969i	$0.134112\pi$
$233$	12.4853i	0.817938i	−0.912548	+	0.408969i	$0.865888\pi$
$234$	0	0
$235$	− 8.00000i	− 0.521862i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.14214	−0.397302	−0.198651	−	0.980070i	$-0.563656\pi$
$239$	−6.14214	−0.397302	−0.198651	+	0.980070i	$0.563656\pi$
$240$	0	0
$241$	−14.3431	−0.923923	−0.461962	−	0.886900i	$-0.652854\pi$
$241$	−14.3431	−0.923923	−0.461962	+	0.886900i	$0.652854\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.65685i	− 0.297516i
$246$	0	0
$247$	35.3137i	2.24696i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.7279	1.30833	0.654167	−	0.756350i	$-0.273019\pi$
$251$	20.7279	1.30833	0.654167	+	0.756350i	$0.273019\pi$
$252$	0	0
$253$	−12.4853	−0.784943
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.97056i	0.559568i	0.960063	+	0.279784i	$0.0902629\pi$
$257$	8.97056i	0.559568i	−0.960063	+	0.279784i	$0.909737\pi$
$258$	0	0
$259$	25.3137i	1.57292i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.8284	1.53099	0.765493	−	0.643444i	$-0.222495\pi$
$263$	24.8284	1.53099	0.765493	+	0.643444i	$0.222495\pi$
$264$	0	0
$265$	−1.17157	−0.0719691
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.9706i	0.668887i	0.942416	+	0.334444i	$0.108548\pi$
$269$	10.9706i	0.668887i	−0.942416	+	0.334444i	$0.891452\pi$
$270$	0	0
$271$	1.31371i	0.0798021i	0.999204	+	0.0399011i	$0.0127043\pi$
$271$	1.31371i	0.0798021i	−0.999204	+	0.0399011i	$0.987296\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	32.8701	1.97497	0.987485	−	0.157712i	$-0.0504117\pi$
$277$	32.8701	1.97497	0.987485	+	0.157712i	$0.0504117\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1.89949i	− 0.113314i	−0.998394	−	0.0566572i	$-0.981956\pi$
$281$	− 1.89949i	− 0.113314i	0.998394	−	0.0566572i	$-0.0180442\pi$
$282$	0	0
$283$	− 15.3137i	− 0.910305i	−0.890413	−	0.455153i	$-0.849585\pi$
$283$	− 15.3137i	− 0.910305i	0.890413	−	0.455153i	$-0.150415\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.828427	0.0489005
$288$	0	0
$289$	−29.6274	−1.74279
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	− 2.58579i	− 0.150550i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	55.1127	3.18725
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.82843i	− 0.505514i
$306$	0	0
$307$	19.3137i	1.10229i	0.834409	+	0.551146i	$0.185809\pi$
$307$	19.3137i	1.10229i	−0.834409	+	0.551146i	$0.814191\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.4853	1.61525	0.807626	−	0.589695i	$-0.200752\pi$
$311$	28.4853	1.61525	0.807626	+	0.589695i	$0.200752\pi$
$312$	0	0
$313$	22.2843	1.25958	0.629791	−	0.776765i	$-0.283141\pi$
$313$	22.2843	1.25958	0.629791	+	0.776765i	$0.283141\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.31371i	0.298448i	0.988803	+	0.149224i	$0.0476775\pi$
$317$	5.31371i	0.298448i	−0.988803	+	0.149224i	$0.952323\pi$
$318$	0	0
$319$	2.82843i	0.158362i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	38.6274	2.14929
$324$	0	0
$325$	6.24264	0.346279
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 27.3137i	− 1.50585i
$330$	0	0
$331$	− 8.68629i	− 0.477442i	−0.971088	−	0.238721i	$-0.923272\pi$
$331$	− 8.68629i	− 0.477442i	0.971088	−	0.238721i	$-0.0767281\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3137	0.618134
$336$	0	0
$337$	11.1716	0.608554	0.304277	−	0.952584i	$-0.401585\pi$
$337$	11.1716	0.608554	0.304277	+	0.952584i	$0.401585\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 12.4853i	− 0.676116i
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.9706	−0.696296	−0.348148	−	0.937440i	$-0.613189\pi$
$347$	−12.9706	−0.696296	−0.348148	+	0.937440i	$0.613189\pi$
$348$	0	0
$349$	3.17157	0.169770	0.0848852	−	0.996391i	$-0.472948\pi$
$349$	3.17157	0.169770	0.0848852	+	0.996391i	$0.472948\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 32.4853i	− 1.72902i	−0.502618	−	0.864509i	$-0.667630\pi$
$353$	− 32.4853i	− 1.72902i	0.502618	−	0.864509i	$-0.332370\pi$
$354$	0	0
$355$	− 2.34315i	− 0.124361i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.1127	−1.64207	−0.821033	−	0.570881i	$-0.806602\pi$
$359$	−31.1127	−1.64207	−0.821033	+	0.570881i	$0.806602\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.82843i	− 0.252731i
$366$	0	0
$367$	33.0711i	1.72630i	0.504951	+	0.863148i	$0.331510\pi$
$367$	33.0711i	1.72630i	−0.504951	+	0.863148i	$0.668490\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	17.0711	0.883906	0.441953	−	0.897038i	$-0.354286\pi$
$373$	17.0711	0.883906	0.441953	+	0.897038i	$0.354286\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.4853i	− 0.643025i
$378$	0	0
$379$	29.3137i	1.50574i	0.658167	+	0.752872i	$0.271332\pi$
$379$	29.3137i	1.50574i	−0.658167	+	0.752872i	$0.728668\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.6274	−0.747426	−0.373713	−	0.927544i	$-0.621916\pi$
$383$	−14.6274	−0.747426	−0.373713	+	0.927544i	$0.621916\pi$
$384$	0	0
$385$	−4.82843	−0.246079
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.65685i	0.388218i	0.980980	+	0.194109i	$0.0621815\pi$
$389$	7.65685i	0.388218i	−0.980980	+	0.194109i	$0.937818\pi$
$390$	0	0
$391$	− 60.2843i	− 3.04871i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.4853	−0.527572
$396$	0	0
$397$	−16.3848	−0.822328	−0.411164	−	0.911561i	$-0.634878\pi$
$397$	−16.3848	−0.822328	−0.411164	+	0.911561i	$0.634878\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.7279i	0.835353i	0.908596	+	0.417676i	$0.137155\pi$
$401$	16.7279i	0.835353i	−0.908596	+	0.417676i	$0.862845\pi$
$402$	0	0
$403$	55.1127i	2.74536i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.4853	0.519736
$408$	0	0
$409$	11.6569	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$409$	11.6569	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8.82843i	− 0.434418i
$414$	0	0
$415$	6.82843i	0.335194i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.2426	−1.77057	−0.885284	−	0.465050i	$-0.846036\pi$
$419$	−36.2426	−1.77057	−0.885284	+	0.465050i	$0.846036\pi$
$420$	0	0
$421$	27.9411	1.36177	0.680884	−	0.732392i	$-0.261596\pi$
$421$	27.9411	1.36177	0.680884	+	0.732392i	$0.261596\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 6.82843i	− 0.331227i
$426$	0	0
$427$	− 30.1421i	− 1.45868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.4853	0.601395	0.300697	−	0.953720i	$-0.402781\pi$
$431$	12.4853	0.601395	0.300697	+	0.953720i	$0.402781\pi$
$432$	0	0
$433$	7.17157	0.344644	0.172322	−	0.985041i	$-0.444873\pi$
$433$	7.17157	0.344644	0.172322	+	0.985041i	$0.444873\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	49.9411i	2.38901i
$438$	0	0
$439$	− 4.34315i	− 0.207287i	−0.994615	−	0.103644i	$-0.966950\pi$
$439$	− 4.34315i	− 0.207287i	0.994615	−	0.103644i	$-0.0330501\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.3137	−0.727576	−0.363788	−	0.931482i	$-0.618517\pi$
$443$	−15.3137	−0.727576	−0.363788	+	0.931482i	$0.618517\pi$
$444$	0	0
$445$	−15.0711	−0.714437
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.2132i	0.812341i	0.913797	+	0.406171i	$0.133136\pi$
$449$	17.2132i	0.812341i	−0.913797	+	0.406171i	$0.866864\pi$
$450$	0	0
$451$	− 0.343146i	− 0.0161581i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	21.3137	0.999202
$456$	0	0
$457$	32.1421	1.50355	0.751773	−	0.659422i	$-0.229199\pi$
$457$	32.1421	1.50355	0.751773	+	0.659422i	$0.229199\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 33.7990i	− 1.57418i	−0.616841	−	0.787088i	$-0.711588\pi$
$461$	− 33.7990i	− 1.57418i	0.616841	−	0.787088i	$-0.288412\pi$
$462$	0	0
$463$	− 20.3848i	− 0.947361i	−0.880697	−	0.473680i	$-0.842925\pi$
$463$	− 20.3848i	− 0.947361i	0.880697	−	0.473680i	$-0.157075\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.7990	−0.545992	−0.272996	−	0.962015i	$-0.588015\pi$
$467$	−11.7990	−0.545992	−0.272996	+	0.962015i	$0.588015\pi$
$468$	0	0
$469$	38.6274	1.78365
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.65685i	0.0761822i
$474$	0	0
$475$	5.65685i	0.259554i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−46.2843	−2.11038
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 6.48528i	− 0.294481i
$486$	0	0
$487$	− 6.04163i	− 0.273772i	−0.990587	−	0.136886i	$-0.956291\pi$
$487$	− 6.04163i	− 0.273772i	0.990587	−	0.136886i	$-0.0437095\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.9289	−0.583475	−0.291737	−	0.956498i	$-0.594233\pi$
$491$	−12.9289	−0.583475	−0.291737	+	0.956498i	$0.594233\pi$
$492$	0	0
$493$	−13.6569	−0.615074
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	− 12.9706i	− 0.580642i	−0.956929	−	0.290321i	$-0.906238\pi$
$499$	− 12.9706i	− 0.580642i	0.956929	−	0.290321i	$-0.0937621\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−16.8284	−0.748855
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 20.1421i	− 0.892784i	−0.894837	−	0.446392i	$-0.852709\pi$
$509$	− 20.1421i	− 0.892784i	0.894837	−	0.446392i	$-0.147291\pi$
$510$	0	0
$511$	− 16.4853i	− 0.729266i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.24264	−0.275084
$516$	0	0
$517$	−11.3137	−0.497576
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.27208i	0.318596i	0.987231	+	0.159298i	$0.0509230\pi$
$521$	7.27208i	0.318596i	−0.987231	+	0.159298i	$0.949077\pi$
$522$	0	0
$523$	27.7990i	1.21556i	0.794104	+	0.607782i	$0.207941\pi$
$523$	27.7990i	1.21556i	−0.794104	+	0.607782i	$0.792059\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.2843	2.62602
$528$	0	0
$529$	54.9411	2.38874
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.51472i	0.0656097i
$534$	0	0
$535$	− 10.1421i	− 0.438483i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.58579	−0.283670
$540$	0	0
$541$	−4.14214	−0.178084	−0.0890422	−	0.996028i	$-0.528381\pi$
$541$	−4.14214	−0.178084	−0.0890422	+	0.996028i	$0.528381\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.17157i	0.135855i
$546$	0	0
$547$	− 4.20101i	− 0.179622i	−0.995959	−	0.0898111i	$-0.971374\pi$
$547$	− 4.20101i	− 0.179622i	0.995959	−	0.0898111i	$-0.0286263\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.3137	0.481980
$552$	0	0
$553$	−35.7990	−1.52233
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 20.4853i	− 0.867989i	−0.900915	−	0.433995i	$-0.857104\pi$
$557$	− 20.4853i	− 0.867989i	0.900915	−	0.433995i	$-0.142896\pi$
$558$	0	0
$559$	− 7.31371i	− 0.309337i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.3431	−0.941651	−0.470826	−	0.882226i	$-0.656044\pi$
$563$	−22.3431	−0.941651	−0.470826	+	0.882226i	$0.656044\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.5858i	1.44991i	0.688795	+	0.724956i	$0.258140\pi$
$569$	34.5858i	1.44991i	−0.688795	+	0.724956i	$0.741860\pi$
$570$	0	0
$571$	34.6274i	1.44911i	0.689216	+	0.724556i	$0.257955\pi$
$571$	34.6274i	1.44911i	−0.689216	+	0.724556i	$0.742045\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.82843	0.368171
$576$	0	0
$577$	−27.4558	−1.14300	−0.571501	−	0.820601i	$-0.693639\pi$
$577$	−27.4558	−1.14300	−0.571501	+	0.820601i	$0.693639\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	23.3137i	0.967216i
$582$	0	0
$583$	1.65685i	0.0686199i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.4558	−1.54597	−0.772984	−	0.634425i	$-0.781237\pi$
$587$	−37.4558	−1.54597	−0.772984	+	0.634425i	$0.781237\pi$
$588$	0	0
$589$	−49.9411	−2.05779
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 14.6274i	− 0.600676i	−0.953833	−	0.300338i	$-0.902901\pi$
$593$	− 14.6274i	− 0.600676i	0.953833	−	0.300338i	$-0.0970995\pi$
$594$	0	0
$595$	− 23.3137i	− 0.955769i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.8579	0.566217	0.283108	−	0.959088i	$-0.408634\pi$
$599$	13.8579	0.566217	0.283108	+	0.959088i	$0.408634\pi$
$600$	0	0
$601$	45.9411	1.87398	0.936989	−	0.349359i	$-0.113601\pi$
$601$	45.9411	1.87398	0.936989	+	0.349359i	$0.113601\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 9.00000i	− 0.365902i
$606$	0	0
$607$	− 10.0416i	− 0.407577i	−0.979015	−	0.203789i	$-0.934674\pi$
$607$	− 10.0416i	− 0.407577i	0.979015	−	0.203789i	$-0.0653255\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	49.9411	2.02040
$612$	0	0
$613$	−21.0711	−0.851052	−0.425526	−	0.904946i	$-0.639911\pi$
$613$	−21.0711	−0.851052	−0.425526	+	0.904946i	$0.639911\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.1716i	0.530268i	0.964212	+	0.265134i	$0.0854161\pi$
$617$	13.1716i	0.530268i	−0.964212	+	0.265134i	$0.914584\pi$
$618$	0	0
$619$	26.9706i	1.08404i	0.840366	+	0.542019i	$0.182340\pi$
$619$	26.9706i	1.08404i	−0.840366	+	0.542019i	$0.817660\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−51.4558	−2.06153
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	50.6274i	2.01865i
$630$	0	0
$631$	− 16.1421i	− 0.642608i	−0.946976	−	0.321304i	$-0.895879\pi$
$631$	− 16.1421i	− 0.642608i	0.946976	−	0.321304i	$-0.104121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−15.4142	−0.611694
$636$	0	0
$637$	29.0711	1.15184
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14.8701i	0.587332i	0.955908	+	0.293666i	$0.0948753\pi$
$641$	14.8701i	0.587332i	−0.955908	+	0.293666i	$0.905125\pi$
$642$	0	0
$643$	1.17157i	0.0462023i	0.999733	+	0.0231012i	$0.00735398\pi$
$643$	1.17157i	0.0462023i	−0.999733	+	0.0231012i	$0.992646\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.0294	−0.590868	−0.295434	−	0.955363i	$-0.595464\pi$
$647$	−15.0294	−0.590868	−0.295434	+	0.955363i	$0.595464\pi$
$648$	0	0
$649$	−3.65685	−0.143544
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.2843i	1.18512i	0.805528	+	0.592558i	$0.201882\pi$
$653$	30.2843i	1.18512i	−0.805528	+	0.592558i	$0.798118\pi$
$654$	0	0
$655$	13.8995i	0.543098i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.04163	0.157440	0.0787198	−	0.996897i	$-0.474917\pi$
$659$	4.04163	0.157440	0.0787198	+	0.996897i	$0.474917\pi$
$660$	0	0
$661$	−28.1421	−1.09460	−0.547301	−	0.836936i	$-0.684345\pi$
$661$	−28.1421	−1.09460	−0.547301	+	0.836936i	$0.684345\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	19.3137i	0.748953i
$666$	0	0
$667$	− 17.6569i	− 0.683676i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.4853	−0.481989
$672$	0	0
$673$	−42.7696	−1.64865	−0.824323	−	0.566120i	$-0.808444\pi$
$673$	−42.7696	−1.64865	−0.824323	+	0.566120i	$0.808444\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.68629i	0.256975i	0.991711	+	0.128488i	$0.0410122\pi$
$677$	6.68629i	0.256975i	−0.991711	+	0.128488i	$0.958988\pi$
$678$	0	0
$679$	− 22.1421i	− 0.849737i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19.7990	0.757587	0.378794	−	0.925481i	$-0.376339\pi$
$683$	19.7990	0.757587	0.378794	+	0.925481i	$0.376339\pi$
$684$	0	0
$685$	−5.65685	−0.216137
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 7.31371i	− 0.278630i
$690$	0	0
$691$	− 29.6569i	− 1.12820i	−0.825707	−	0.564100i	$-0.809223\pi$
$691$	− 29.6569i	− 1.12820i	0.825707	−	0.564100i	$-0.190777\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.34315	−0.316474
$696$	0	0
$697$	1.65685	0.0627578
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.5147i	0.812600i	0.913740	+	0.406300i	$0.133181\pi$
$701$	21.5147i	0.812600i	−0.913740	+	0.406300i	$0.866819\pi$
$702$	0	0
$703$	− 41.9411i	− 1.58184i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−57.4558	−2.16085
$708$	0	0
$709$	−31.6569	−1.18890	−0.594449	−	0.804133i	$-0.702630\pi$
$709$	−31.6569	−1.18890	−0.594449	+	0.804133i	$0.702630\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	77.9411i	2.91892i
$714$	0	0
$715$	− 8.82843i	− 0.330164i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.7990	−1.03673	−0.518364	−	0.855160i	$-0.673459\pi$
$719$	−27.7990	−1.03673	−0.518364	+	0.855160i	$0.673459\pi$
$720$	0	0
$721$	−21.3137	−0.793764
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	0	0
$727$	9.75736i	0.361880i	0.983494	+	0.180940i	$0.0579141\pi$
$727$	9.75736i	0.361880i	−0.983494	+	0.180940i	$0.942086\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−2.24264	−0.0828338	−0.0414169	−	0.999142i	$-0.513187\pi$
$733$	−2.24264	−0.0828338	−0.0414169	+	0.999142i	$0.513187\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	12.0000i	0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$	12.0000i	0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.02944	0.257885	0.128943	−	0.991652i	$-0.458842\pi$
$743$	7.02944	0.257885	0.128943	+	0.991652i	$0.458842\pi$
$744$	0	0
$745$	4.82843	0.176900
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 34.6274i	− 1.26526i
$750$	0	0
$751$	− 21.7990i	− 0.795456i	−0.917503	−	0.397728i	$-0.869799\pi$
$751$	− 21.7990i	− 0.795456i	0.917503	−	0.397728i	$-0.130201\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.9706	−0.399260
$756$	0	0
$757$	−3.41421	−0.124092	−0.0620459	−	0.998073i	$-0.519763\pi$
$757$	−3.41421	−0.124092	−0.0620459	+	0.998073i	$0.519763\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 38.8701i	− 1.40904i	−0.709685	−	0.704519i	$-0.751163\pi$
$761$	− 38.8701i	− 1.40904i	0.709685	−	0.704519i	$-0.248837\pi$
$762$	0	0
$763$	10.8284i	0.392015i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.1421	0.582859
$768$	0	0
$769$	−31.5980	−1.13945	−0.569726	−	0.821835i	$-0.692951\pi$
$769$	−31.5980	−1.13945	−0.569726	+	0.821835i	$0.692951\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 43.2548i	− 1.55577i	−0.628408	−	0.777884i	$-0.716293\pi$
$773$	− 43.2548i	− 1.55577i	0.628408	−	0.777884i	$-0.283707\pi$
$774$	0	0
$775$	8.82843i	0.317126i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.37258	−0.0491779
$780$	0	0
$781$	−3.31371	−0.118574
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.100505i	0.00358718i
$786$	0	0
$787$	26.1421i	0.931866i	0.884820	+	0.465933i	$0.154281\pi$
$787$	26.1421i	0.931866i	−0.884820	+	0.465933i	$0.845719\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.6569	−0.485582
$792$	0	0
$793$	55.1127	1.95711
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 0.485281i	− 0.0171895i	−0.999963	−	0.00859477i	$-0.997264\pi$
$797$	− 0.485281i	− 0.0171895i	0.999963	−	0.00859477i	$-0.00273584\pi$
$798$	0	0
$799$	− 54.6274i	− 1.93258i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.82843	−0.240970
$804$	0	0
$805$	30.1421	1.06237
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 10.1005i	− 0.355115i	−0.984110	−	0.177557i	$-0.943180\pi$
$809$	− 10.1005i	− 0.355115i	0.984110	−	0.177557i	$-0.0568195\pi$
$810$	0	0
$811$	− 12.6274i	− 0.443409i	−0.975114	−	0.221704i	$-0.928838\pi$
$811$	− 12.6274i	− 0.443409i	0.975114	−	0.221704i	$-0.0711620\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.48528	0.297226
$816$	0	0
$817$	6.62742	0.231864
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.31371i	0.185450i	0.995692	+	0.0927249i	$0.0295577\pi$
$821$	5.31371i	0.185450i	−0.995692	+	0.0927249i	$0.970442\pi$
$822$	0	0
$823$	− 36.8701i	− 1.28521i	−0.766198	−	0.642605i	$-0.777854\pi$
$823$	− 36.8701i	− 1.28521i	0.766198	−	0.642605i	$-0.222146\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.6274	−1.76049	−0.880244	−	0.474522i	$-0.842621\pi$
$827$	−50.6274	−1.76049	−0.880244	+	0.474522i	$0.842621\pi$
$828$	0	0
$829$	13.5147	0.469386	0.234693	−	0.972070i	$-0.424592\pi$
$829$	13.5147	0.469386	0.234693	+	0.972070i	$0.424592\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 31.7990i	− 1.10177i
$834$	0	0
$835$	14.4853i	0.501284i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.4853	−0.845326	−0.422663	−	0.906287i	$-0.638905\pi$
$839$	−24.4853	−0.845326	−0.422663	+	0.906287i	$0.638905\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25.9706i	0.893415i
$846$	0	0
$847$	− 30.7279i	− 1.05582i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−65.4558	−2.24380
$852$	0	0
$853$	47.2132	1.61655	0.808275	−	0.588806i	$-0.200402\pi$
$853$	47.2132	1.61655	0.808275	+	0.588806i	$0.200402\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.4853i	1.65623i	0.560561	+	0.828113i	$0.310585\pi$
$857$	48.4853i	1.65623i	−0.560561	+	0.828113i	$0.689415\pi$
$858$	0	0
$859$	10.0000i	0.341196i	0.985341	+	0.170598i	$0.0545699\pi$
$859$	10.0000i	0.341196i	−0.985341	+	0.170598i	$0.945430\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50.7696	1.72822	0.864108	−	0.503307i	$-0.167883\pi$
$863$	50.7696	1.72822	0.864108	+	0.503307i	$0.167883\pi$
$864$	0	0
$865$	−0.485281	−0.0165001
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.8284i	0.503020i
$870$	0	0
$871$	70.6274i	2.39312i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.41421	0.115421
$876$	0	0
$877$	4.58579	0.154851	0.0774255	−	0.996998i	$-0.475330\pi$
$877$	4.58579	0.154851	0.0774255	+	0.996998i	$0.475330\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	25.4142i	0.856227i	0.903725	+	0.428113i	$0.140822\pi$
$881$	25.4142i	0.856227i	−0.903725	+	0.428113i	$0.859178\pi$
$882$	0	0
$883$	− 33.4558i	− 1.12588i	−0.826498	−	0.562939i	$-0.809670\pi$
$883$	− 33.4558i	− 1.12588i	0.826498	−	0.562939i	$-0.190330\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.1421	0.944920	0.472460	−	0.881352i	$-0.343366\pi$
$887$	28.1421	0.944920	0.472460	+	0.881352i	$0.343366\pi$
$888$	0	0
$889$	−52.6274	−1.76507
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45.2548i	1.51440i
$894$	0	0
$895$	13.4142i	0.448388i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.6569	0.588889
$900$	0	0
$901$	−8.00000	−0.266519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 24.8284i	− 0.825325i
$906$	0	0
$907$	31.1127i	1.03308i	0.856263	+	0.516540i	$0.172780\pi$
$907$	31.1127i	1.03308i	−0.856263	+	0.516540i	$0.827220\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.2843	−1.20215	−0.601076	−	0.799192i	$-0.705261\pi$
$911$	−36.2843	−1.20215	−0.601076	+	0.799192i	$0.705261\pi$
$912$	0	0
$913$	9.65685	0.319595
$914$	0	0
$915$	0	0
$916$	0	0
$917$	47.4558i	1.56713i
$918$	0	0
$919$	− 26.4853i	− 0.873669i	−0.899542	−	0.436834i	$-0.856100\pi$
$919$	− 26.4853i	− 0.873669i	0.899542	−	0.436834i	$-0.143900\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.6274	0.481467
$924$	0	0
$925$	−7.41421	−0.243778
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 32.7279i	− 1.07377i	−0.843656	−	0.536884i	$-0.819601\pi$
$929$	− 32.7279i	− 1.07377i	0.843656	−	0.536884i	$-0.180399\pi$
$930$	0	0
$931$	26.3431i	0.863362i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.65685	−0.315813
$936$	0	0
$937$	−42.2843	−1.38137	−0.690683	−	0.723157i	$-0.742690\pi$
$937$	−42.2843	−1.38137	−0.690683	+	0.723157i	$0.742690\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.9706i	1.53120i	0.643319	+	0.765598i	$0.277557\pi$
$941$	46.9706i	1.53120i	−0.643319	+	0.765598i	$0.722443\pi$
$942$	0	0
$943$	2.14214i	0.0697575i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.9706	1.46135	0.730673	−	0.682727i	$-0.239206\pi$
$947$	44.9706	1.46135	0.730673	+	0.682727i	$0.239206\pi$
$948$	0	0
$949$	30.1421	0.978455
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.34315i	− 0.0759019i	−0.999280	−	0.0379510i	$-0.987917\pi$
$953$	− 2.34315i	− 0.0759019i	0.999280	−	0.0379510i	$-0.0120831\pi$
$954$	0	0
$955$	− 1.17157i	− 0.0379112i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.3137	−0.623672
$960$	0	0
$961$	−46.9411	−1.51423
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.34315i	0.139811i
$966$	0	0
$967$	19.8995i	0.639925i	0.947430	+	0.319962i	$0.103670\pi$
$967$	19.8995i	0.639925i	−0.947430	+	0.319962i	$0.896330\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.6985	−0.953070	−0.476535	−	0.879156i	$-0.658107\pi$
$971$	−29.6985	−0.953070	−0.476535	+	0.879156i	$0.658107\pi$
$972$	0	0
$973$	−28.4853	−0.913196
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.4853i	0.783354i	0.920103	+	0.391677i	$0.128105\pi$
$977$	24.4853i	0.783354i	−0.920103	+	0.391677i	$0.871895\pi$
$978$	0	0
$979$	21.3137i	0.681189i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.3137	−0.616012	−0.308006	−	0.951384i	$-0.599662\pi$
$983$	−19.3137	−0.616012	−0.308006	+	0.951384i	$0.599662\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 10.3431i	− 0.328893i
$990$	0	0
$991$	− 0.343146i	− 0.0109004i	−0.999985	−	0.00545019i	$-0.998265\pi$
$991$	− 0.343146i	− 0.0109004i	0.999985	−	0.00545019i	$-0.00173486\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.6569	0.496356
$996$	0	0
$997$	−34.2426	−1.08448	−0.542238	−	0.840225i	$-0.682423\pi$
$997$	−34.2426	−1.08448	−0.542238	+	0.840225i	$0.682423\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.h.b.1151.4	yes	4
3.2	odd	2		1440.2.h.c.1151.2	yes	4
4.3	odd	2		1440.2.h.c.1151.3	yes	4
5.2	odd	4		7200.2.o.c.7199.2		4
5.3	odd	4		7200.2.o.l.7199.3		4
5.4	even	2		7200.2.h.f.1151.1		4
8.3	odd	2		2880.2.h.c.1151.1		4
8.5	even	2		2880.2.h.b.1151.2		4
12.11	even	2	inner	1440.2.h.b.1151.1	✓	4
15.2	even	4		7200.2.o.d.7199.2		4
15.8	even	4		7200.2.o.k.7199.3		4
15.14	odd	2		7200.2.h.e.1151.1		4
20.3	even	4		7200.2.o.d.7199.1		4
20.7	even	4		7200.2.o.k.7199.4		4
20.19	odd	2		7200.2.h.e.1151.4		4
24.5	odd	2		2880.2.h.c.1151.4		4
24.11	even	2		2880.2.h.b.1151.3		4
60.23	odd	4		7200.2.o.c.7199.1		4
60.47	odd	4		7200.2.o.l.7199.4		4
60.59	even	2		7200.2.h.f.1151.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.h.b.1151.1	✓	4	12.11	even	2	inner
1440.2.h.b.1151.4	yes	4	1.1	even	1	trivial
1440.2.h.c.1151.2	yes	4	3.2	odd	2
1440.2.h.c.1151.3	yes	4	4.3	odd	2
2880.2.h.b.1151.2		4	8.5	even	2
2880.2.h.b.1151.3		4	24.11	even	2
2880.2.h.c.1151.1		4	8.3	odd	2
2880.2.h.c.1151.4		4	24.5	odd	2
7200.2.h.e.1151.1		4	15.14	odd	2
7200.2.h.e.1151.4		4	20.19	odd	2
7200.2.h.f.1151.1		4	5.4	even	2
7200.2.h.f.1151.4		4	60.59	even	2
7200.2.o.c.7199.1		4	60.23	odd	4
7200.2.o.c.7199.2		4	5.2	odd	4
7200.2.o.d.7199.1		4	20.3	even	4
7200.2.o.d.7199.2		4	15.2	even	4
7200.2.o.k.7199.3		4	15.8	even	4
7200.2.o.k.7199.4		4	20.7	even	4
7200.2.o.l.7199.3		4	5.3	odd	4
7200.2.o.l.7199.4		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +3.41421i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} +3.41421i q^{7} +1.41421 q^{11} -6.24264 q^{13} +6.82843i q^{17} -5.65685i q^{19} -8.82843 q^{23} -1.00000 q^{25} +2.00000i q^{29} -8.82843i q^{31} -3.41421 q^{35} +7.41421 q^{37} -0.242641i q^{41} +1.17157i q^{43} -8.00000 q^{47} -4.65685 q^{49} +1.17157i q^{53} +1.41421i q^{55} -2.58579 q^{59} -8.82843 q^{61} -6.24264i q^{65} -11.3137i q^{67} -2.34315 q^{71} -4.82843 q^{73} +4.82843i q^{77} +10.4853i q^{79} +6.82843 q^{83} -6.82843 q^{85} +15.0711i q^{89} -21.3137i q^{91} +5.65685 q^{95} -6.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{13} - 24 q^{23} - 4 q^{25} - 8 q^{35} + 24 q^{37} - 32 q^{47} + 4 q^{49} - 16 q^{59} - 24 q^{61} - 32 q^{71} - 8 q^{73} + 16 q^{83} - 16 q^{85} + 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^13 - 24 * q^23 - 4 * q^25 - 8 * q^35 + 24 * q^37 - 32 * q^47 + 4 * q^49 - 16 * q^59 - 24 * q^61 - 32 * q^71 - 8 * q^73 + 16 * q^83 - 16 * q^85 + 8 * q^97

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