LMFDB - Embedded newform 27.5.b.c.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,5,Mod(26,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("27.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	27.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.79098900326$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.5.b.c.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{2} +7.00000 q^{4} -33.0000i q^{5} -19.0000 q^{7} -69.0000i q^{8} +O(q^{10})$	\(q-3.00000i q^{2} +7.00000 q^{4} -33.0000i q^{5} -19.0000 q^{7} -69.0000i q^{8} -99.0000 q^{10} +123.000i q^{11} +302.000 q^{13} +57.0000i q^{14} -95.0000 q^{16} +414.000i q^{17} -304.000 q^{19} -231.000i q^{20} +369.000 q^{22} +300.000i q^{23} -464.000 q^{25} -906.000i q^{26} -133.000 q^{28} -678.000i q^{29} +239.000 q^{31} -819.000i q^{32} +1242.00 q^{34} +627.000i q^{35} +740.000 q^{37} +912.000i q^{38} -2277.00 q^{40} +228.000i q^{41} -982.000 q^{43} +861.000i q^{44} +900.000 q^{46} +2166.00i q^{47} -2040.00 q^{49} +1392.00i q^{50} +2114.00 q^{52} -1593.00i q^{53} +4059.00 q^{55} +1311.00i q^{56} -2034.00 q^{58} -2922.00i q^{59} -316.000 q^{61} -717.000i q^{62} -3977.00 q^{64} -9966.00i q^{65} +4622.00 q^{67} +2898.00i q^{68} +1881.00 q^{70} +1818.00i q^{71} -3031.00 q^{73} -2220.00i q^{74} -2128.00 q^{76} -2337.00i q^{77} -10450.0 q^{79} +3135.00i q^{80} +684.000 q^{82} +12633.0i q^{83} +13662.0 q^{85} +2946.00i q^{86} +8487.00 q^{88} -7002.00i q^{89} -5738.00 q^{91} +2100.00i q^{92} +6498.00 q^{94} +10032.0i q^{95} -6517.00 q^{97} +6120.00i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4} - 38 q^{7}+O(q^{10}) $ 2 * q + 14 * q^4 - 38 * q^7	$ 2 q + 14 q^{4} - 38 q^{7} - 198 q^{10} + 604 q^{13} - 190 q^{16} - 608 q^{19} + 738 q^{22} - 928 q^{25} - 266 q^{28} + 478 q^{31} + 2484 q^{34} + 1480 q^{37} - 4554 q^{40} - 1964 q^{43} + 1800 q^{46} - 4080 q^{49} + 4228 q^{52} + 8118 q^{55} - 4068 q^{58} - 632 q^{61} - 7954 q^{64} + 9244 q^{67} + 3762 q^{70} - 6062 q^{73} - 4256 q^{76} - 20900 q^{79} + 1368 q^{82} + 27324 q^{85} + 16974 q^{88} - 11476 q^{91} + 12996 q^{94} - 13034 q^{97}+O(q^{100}) $ 2 * q + 14 * q^4 - 38 * q^7 - 198 * q^10 + 604 * q^13 - 190 * q^16 - 608 * q^19 + 738 * q^22 - 928 * q^25 - 266 * q^28 + 478 * q^31 + 2484 * q^34 + 1480 * q^37 - 4554 * q^40 - 1964 * q^43 + 1800 * q^46 - 4080 * q^49 + 4228 * q^52 + 8118 * q^55 - 4068 * q^58 - 632 * q^61 - 7954 * q^64 + 9244 * q^67 + 3762 * q^70 - 6062 * q^73 - 4256 * q^76 - 20900 * q^79 + 1368 * q^82 + 27324 * q^85 + 16974 * q^88 - 11476 * q^91 + 12996 * q^94 - 13034 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 3.00000i	− 0.750000i	−0.927025	−	0.375000i	$-0.877643\pi$
$2$	− 3.00000i	− 0.750000i	0.927025	−	0.375000i	$-0.122357\pi$
$3$	0	0
$3$	0	0
$4$	7.00000	0.437500
$5$	− 33.0000i	− 1.32000i	−0.751266	−	0.660000i	$-0.770556\pi$
$5$	− 33.0000i	− 1.32000i	0.751266	−	0.660000i	$-0.229444\pi$
$6$	0	0
$7$	−19.0000	−0.387755	−0.193878	−	0.981026i	$-0.562106\pi$
$7$	−19.0000	−0.387755	−0.193878	+	0.981026i	$0.562106\pi$
$8$	− 69.0000i	− 1.07812i
$9$	0	0
$10$	−99.0000	−0.990000
$11$	123.000i	1.01653i	0.861201	+	0.508264i	$0.169713\pi$
$11$	123.000i	1.01653i	−0.861201	+	0.508264i	$0.830287\pi$
$12$	0	0
$13$	302.000	1.78698	0.893491	−	0.449081i	$-0.148248\pi$
$13$	302.000	1.78698	0.893491	+	0.449081i	$0.148248\pi$
$14$	57.0000i	0.290816i
$15$	0	0
$16$	−95.0000	−0.371094
$17$	414.000i	1.43253i	0.697830	+	0.716263i	$0.254149\pi$
$17$	414.000i	1.43253i	−0.697830	+	0.716263i	$0.745851\pi$
$18$	0	0
$19$	−304.000	−0.842105	−0.421053	−	0.907036i	$-0.638339\pi$
$19$	−304.000	−0.842105	−0.421053	+	0.907036i	$0.638339\pi$
$20$	− 231.000i	− 0.577500i
$21$	0	0
$22$	369.000	0.762397
$23$	300.000i	0.567108i	0.958956	+	0.283554i	$0.0915135\pi$
$23$	300.000i	0.567108i	−0.958956	+	0.283554i	$0.908487\pi$
$24$	0	0
$25$	−464.000	−0.742400
$26$	− 906.000i	− 1.34024i
$27$	0	0
$28$	−133.000	−0.169643
$29$	− 678.000i	− 0.806183i	−0.915160	−	0.403092i	$-0.867936\pi$
$29$	− 678.000i	− 0.806183i	0.915160	−	0.403092i	$-0.132064\pi$
$30$	0	0
$31$	239.000	0.248699	0.124350	−	0.992238i	$-0.460316\pi$
$31$	239.000	0.248699	0.124350	+	0.992238i	$0.460316\pi$
$32$	− 819.000i	− 0.799805i
$33$	0	0
$34$	1242.00	1.07439
$35$	627.000i	0.511837i
$36$	0	0
$37$	740.000	0.540541	0.270270	−	0.962784i	$-0.412887\pi$
$37$	740.000	0.540541	0.270270	+	0.962784i	$0.412887\pi$
$38$	912.000i	0.631579i
$39$	0	0
$40$	−2277.00	−1.42312
$41$	228.000i	0.135634i	0.997698	+	0.0678168i	$0.0216033\pi$
$41$	228.000i	0.135634i	−0.997698	+	0.0678168i	$0.978397\pi$
$42$	0	0
$43$	−982.000	−0.531098	−0.265549	−	0.964097i	$-0.585553\pi$
$43$	−982.000	−0.531098	−0.265549	+	0.964097i	$0.585553\pi$
$44$	861.000i	0.444731i
$45$	0	0
$46$	900.000	0.425331
$47$	2166.00i	0.980534i	0.871572	+	0.490267i	$0.163101\pi$
$47$	2166.00i	0.980534i	−0.871572	+	0.490267i	$0.836899\pi$
$48$	0	0
$49$	−2040.00	−0.849646
$50$	1392.00i	0.556800i
$51$	0	0
$52$	2114.00	0.781805
$53$	− 1593.00i	− 0.567106i	−0.958957	−	0.283553i	$-0.908487\pi$
$53$	− 1593.00i	− 0.567106i	0.958957	−	0.283553i	$-0.0915131\pi$
$54$	0	0
$55$	4059.00	1.34182
$56$	1311.00i	0.418048i
$57$	0	0
$58$	−2034.00	−0.604637
$59$	− 2922.00i	− 0.839414i	−0.907660	−	0.419707i	$-0.862133\pi$
$59$	− 2922.00i	− 0.839414i	0.907660	−	0.419707i	$-0.137867\pi$
$60$	0	0
$61$	−316.000	−0.0849234	−0.0424617	−	0.999098i	$-0.513520\pi$
$61$	−316.000	−0.0849234	−0.0424617	+	0.999098i	$0.513520\pi$
$62$	− 717.000i	− 0.186524i
$63$	0	0
$64$	−3977.00	−0.970947
$65$	− 9966.00i	− 2.35882i
$66$	0	0
$67$	4622.00	1.02963	0.514814	−	0.857302i	$-0.327861\pi$
$67$	4622.00	1.02963	0.514814	+	0.857302i	$0.327861\pi$
$68$	2898.00i	0.626730i
$69$	0	0
$70$	1881.00	0.383878
$71$	1818.00i	0.360643i	0.983608	+	0.180321i	$0.0577138\pi$
$71$	1818.00i	0.360643i	−0.983608	+	0.180321i	$0.942286\pi$
$72$	0	0
$73$	−3031.00	−0.568775	−0.284387	−	0.958709i	$-0.591790\pi$
$73$	−3031.00	−0.568775	−0.284387	+	0.958709i	$0.591790\pi$
$74$	− 2220.00i	− 0.405405i
$75$	0	0
$76$	−2128.00	−0.368421
$77$	− 2337.00i	− 0.394164i
$78$	0	0
$79$	−10450.0	−1.67441	−0.837206	−	0.546888i	$-0.815812\pi$
$79$	−10450.0	−1.67441	−0.837206	+	0.546888i	$0.815812\pi$
$80$	3135.00i	0.489844i
$81$	0	0
$82$	684.000	0.101725
$83$	12633.0i	1.83379i	0.399125	+	0.916897i	$0.369314\pi$
$83$	12633.0i	1.83379i	−0.399125	+	0.916897i	$0.630686\pi$
$84$	0	0
$85$	13662.0	1.89093
$86$	2946.00i	0.398323i
$87$	0	0
$88$	8487.00	1.09595
$89$	− 7002.00i	− 0.883979i	−0.897020	−	0.441990i	$-0.854273\pi$
$89$	− 7002.00i	− 0.883979i	0.897020	−	0.441990i	$-0.145727\pi$
$90$	0	0
$91$	−5738.00	−0.692911
$92$	2100.00i	0.248110i
$93$	0	0
$94$	6498.00	0.735401
$95$	10032.0i	1.11158i
$96$	0	0
$97$	−6517.00	−0.692635	−0.346317	−	0.938117i	$-0.612568\pi$
$97$	−6517.00	−0.692635	−0.346317	+	0.938117i	$0.612568\pi$
$98$	6120.00i	0.637234i
$99$	0	0
$100$	−3248.00	−0.324800
$101$	5919.00i	0.580237i	0.956991	+	0.290119i	$0.0936948\pi$
$101$	5919.00i	0.580237i	−0.956991	+	0.290119i	$0.906305\pi$
$102$	0	0
$103$	−7654.00	−0.721463	−0.360731	−	0.932670i	$-0.617473\pi$
$103$	−7654.00	−0.721463	−0.360731	+	0.932670i	$0.617473\pi$
$104$	− 20838.0i	− 1.92659i
$105$	0	0
$106$	−4779.00	−0.425329
$107$	− 513.000i	− 0.0448074i	−0.999749	−	0.0224037i	$-0.992868\pi$
$107$	− 513.000i	− 0.0448074i	0.999749	−	0.0224037i	$-0.00713192\pi$
$108$	0	0
$109$	2324.00	0.195606	0.0978032	−	0.995206i	$-0.468818\pi$
$109$	2324.00	0.195606	0.0978032	+	0.995206i	$0.468818\pi$
$110$	− 12177.0i	− 1.00636i
$111$	0	0
$112$	1805.00	0.143893
$113$	− 4920.00i	− 0.385308i	−0.981267	−	0.192654i	$-0.938290\pi$
$113$	− 4920.00i	− 0.385308i	0.981267	−	0.192654i	$-0.0617095\pi$
$114$	0	0
$115$	9900.00	0.748582
$116$	− 4746.00i	− 0.352705i
$117$	0	0
$118$	−8766.00	−0.629560
$119$	− 7866.00i	− 0.555469i
$120$	0	0
$121$	−488.000	−0.0333311
$122$	948.000i	0.0636926i
$123$	0	0
$124$	1673.00	0.108806
$125$	− 5313.00i	− 0.340032i
$126$	0	0
$127$	24995.0	1.54969	0.774847	−	0.632149i	$-0.217827\pi$
$127$	24995.0	1.54969	0.774847	+	0.632149i	$0.217827\pi$
$128$	− 1173.00i	− 0.0715942i
$129$	0	0
$130$	−29898.0	−1.76911
$131$	28461.0i	1.65847i	0.558900	+	0.829235i	$0.311223\pi$
$131$	28461.0i	1.65847i	−0.558900	+	0.829235i	$0.688777\pi$
$132$	0	0
$133$	5776.00	0.326531
$134$	− 13866.0i	− 0.772221i
$135$	0	0
$136$	28566.0	1.54444
$137$	2454.00i	0.130748i	0.997861	+	0.0653738i	$0.0208240\pi$
$137$	2454.00i	0.130748i	−0.997861	+	0.0653738i	$0.979176\pi$
$138$	0	0
$139$	−11884.0	−0.615082	−0.307541	−	0.951535i	$-0.599506\pi$
$139$	−11884.0	−0.615082	−0.307541	+	0.951535i	$0.599506\pi$
$140$	4389.00i	0.223929i
$141$	0	0
$142$	5454.00	0.270482
$143$	37146.0i	1.81652i
$144$	0	0
$145$	−22374.0	−1.06416
$146$	9093.00i	0.426581i
$147$	0	0
$148$	5180.00	0.236486
$149$	− 21993.0i	− 0.990631i	−0.868713	−	0.495316i	$-0.835052\pi$
$149$	− 21993.0i	− 0.990631i	0.868713	−	0.495316i	$-0.164948\pi$
$150$	0	0
$151$	−2683.00	−0.117670	−0.0588351	−	0.998268i	$-0.518739\pi$
$151$	−2683.00	−0.117670	−0.0588351	+	0.998268i	$0.518739\pi$
$152$	20976.0i	0.907895i
$153$	0	0
$154$	−7011.00	−0.295623
$155$	− 7887.00i	− 0.328283i
$156$	0	0
$157$	−32116.0	−1.30293	−0.651467	−	0.758677i	$-0.725846\pi$
$157$	−32116.0	−1.30293	−0.651467	+	0.758677i	$0.725846\pi$
$158$	31350.0i	1.25581i
$159$	0	0
$160$	−27027.0	−1.05574
$161$	− 5700.00i	− 0.219899i
$162$	0	0
$163$	22790.0	0.857767	0.428883	−	0.903360i	$-0.358907\pi$
$163$	22790.0	0.857767	0.428883	+	0.903360i	$0.358907\pi$
$164$	1596.00i	0.0593397i
$165$	0	0
$166$	37899.0	1.37534
$167$	− 36078.0i	− 1.29363i	−0.762648	−	0.646814i	$-0.776101\pi$
$167$	− 36078.0i	− 1.29363i	0.762648	−	0.646814i	$-0.223899\pi$
$168$	0	0
$169$	62643.0	2.19331
$170$	− 40986.0i	− 1.41820i
$171$	0	0
$172$	−6874.00	−0.232355
$173$	19725.0i	0.659060i	0.944145	+	0.329530i	$0.106890\pi$
$173$	19725.0i	0.659060i	−0.944145	+	0.329530i	$0.893110\pi$
$174$	0	0
$175$	8816.00	0.287869
$176$	− 11685.0i	− 0.377228i
$177$	0	0
$178$	−21006.0	−0.662984
$179$	− 48915.0i	− 1.52664i	−0.646022	−	0.763319i	$-0.723569\pi$
$179$	− 48915.0i	− 1.52664i	0.646022	−	0.763319i	$-0.276431\pi$
$180$	0	0
$181$	−49552.0	−1.51253	−0.756265	−	0.654265i	$-0.772978\pi$
$181$	−49552.0	−1.51253	−0.756265	+	0.654265i	$0.772978\pi$
$182$	17214.0i	0.519684i
$183$	0	0
$184$	20700.0	0.611413
$185$	− 24420.0i	− 0.713514i
$186$	0	0
$187$	−50922.0	−1.45620
$188$	15162.0i	0.428984i
$189$	0	0
$190$	30096.0	0.833684
$191$	− 45390.0i	− 1.24421i	−0.782934	−	0.622105i	$-0.786278\pi$
$191$	− 45390.0i	− 1.24421i	0.782934	−	0.622105i	$-0.213722\pi$
$192$	0	0
$193$	35447.0	0.951623	0.475811	−	0.879547i	$-0.342154\pi$
$193$	35447.0	0.951623	0.475811	+	0.879547i	$0.342154\pi$
$194$	19551.0i	0.519476i
$195$	0	0
$196$	−14280.0	−0.371720
$197$	35739.0i	0.920895i	0.887687	+	0.460447i	$0.152311\pi$
$197$	35739.0i	0.920895i	−0.887687	+	0.460447i	$0.847689\pi$
$198$	0	0
$199$	−31255.0	−0.789248	−0.394624	−	0.918843i	$-0.629125\pi$
$199$	−31255.0	−0.789248	−0.394624	+	0.918843i	$0.629125\pi$
$200$	32016.0i	0.800400i
$201$	0	0
$202$	17757.0	0.435178
$203$	12882.0i	0.312602i
$204$	0	0
$205$	7524.00	0.179036
$206$	22962.0i	0.541097i
$207$	0	0
$208$	−28690.0	−0.663138
$209$	− 37392.0i	− 0.856024i
$210$	0	0
$211$	−15052.0	−0.338088	−0.169044	−	0.985609i	$-0.554068\pi$
$211$	−15052.0	−0.338088	−0.169044	+	0.985609i	$0.554068\pi$
$212$	− 11151.0i	− 0.248109i
$213$	0	0
$214$	−1539.00	−0.0336056
$215$	32406.0i	0.701049i
$216$	0	0
$217$	−4541.00	−0.0964344
$218$	− 6972.00i	− 0.146705i
$219$	0	0
$220$	28413.0	0.587045
$221$	125028.i	2.55990i
$222$	0	0
$223$	50174.0	1.00895	0.504474	−	0.863427i	$-0.331686\pi$
$223$	50174.0	1.00895	0.504474	+	0.863427i	$0.331686\pi$
$224$	15561.0i	0.310128i
$225$	0	0
$226$	−14760.0	−0.288981
$227$	19266.0i	0.373887i	0.982371	+	0.186943i	$0.0598581\pi$
$227$	19266.0i	0.373887i	−0.982371	+	0.186943i	$0.940142\pi$
$228$	0	0
$229$	34214.0	0.652428	0.326214	−	0.945296i	$-0.394227\pi$
$229$	34214.0	0.652428	0.326214	+	0.945296i	$0.394227\pi$
$230$	− 29700.0i	− 0.561437i
$231$	0	0
$232$	−46782.0	−0.869166
$233$	− 37386.0i	− 0.688648i	−0.938851	−	0.344324i	$-0.888108\pi$
$233$	− 37386.0i	− 0.688648i	0.938851	−	0.344324i	$-0.111892\pi$
$234$	0	0
$235$	71478.0	1.29431
$236$	− 20454.0i	− 0.367244i
$237$	0	0
$238$	−23598.0	−0.416602
$239$	− 61800.0i	− 1.08191i	−0.841050	−	0.540957i	$-0.818062\pi$
$239$	− 61800.0i	− 1.08191i	0.841050	−	0.540957i	$-0.181938\pi$
$240$	0	0
$241$	41390.0	0.712625	0.356313	−	0.934367i	$-0.384034\pi$
$241$	41390.0	0.712625	0.356313	+	0.934367i	$0.384034\pi$
$242$	1464.00i	0.0249983i
$243$	0	0
$244$	−2212.00	−0.0371540
$245$	67320.0i	1.12153i
$246$	0	0
$247$	−91808.0	−1.50483
$248$	− 16491.0i	− 0.268129i
$249$	0	0
$250$	−15939.0	−0.255024
$251$	− 82818.0i	− 1.31455i	−0.753651	−	0.657275i	$-0.771709\pi$
$251$	− 82818.0i	− 1.31455i	0.753651	−	0.657275i	$-0.228291\pi$
$252$	0	0
$253$	−36900.0	−0.576481
$254$	− 74985.0i	− 1.16227i
$255$	0	0
$256$	−67151.0	−1.02464
$257$	− 19590.0i	− 0.296598i	−0.988943	−	0.148299i	$-0.952620\pi$
$257$	− 19590.0i	− 0.296598i	0.988943	−	0.148299i	$-0.0473798\pi$
$258$	0	0
$259$	−14060.0	−0.209597
$260$	− 69762.0i	− 1.03198i
$261$	0	0
$262$	85383.0	1.24385
$263$	16692.0i	0.241322i	0.992694	+	0.120661i	$0.0385014\pi$
$263$	16692.0i	0.241322i	−0.992694	+	0.120661i	$0.961499\pi$
$264$	0	0
$265$	−52569.0	−0.748580
$266$	− 17328.0i	− 0.244898i
$267$	0	0
$268$	32354.0	0.450462
$269$	120906.i	1.67087i	0.549587	+	0.835436i	$0.314785\pi$
$269$	120906.i	1.67087i	−0.549587	+	0.835436i	$0.685215\pi$
$270$	0	0
$271$	73739.0	1.00406	0.502029	−	0.864851i	$-0.332587\pi$
$271$	73739.0	1.00406	0.502029	+	0.864851i	$0.332587\pi$
$272$	− 39330.0i	− 0.531601i
$273$	0	0
$274$	7362.00	0.0980606
$275$	− 57072.0i	− 0.754671i
$276$	0	0
$277$	11996.0	0.156342	0.0781712	−	0.996940i	$-0.475092\pi$
$277$	11996.0	0.156342	0.0781712	+	0.996940i	$0.475092\pi$
$278$	35652.0i	0.461312i
$279$	0	0
$280$	43263.0	0.551824
$281$	51126.0i	0.647484i	0.946145	+	0.323742i	$0.104941\pi$
$281$	51126.0i	0.647484i	−0.946145	+	0.323742i	$0.895059\pi$
$282$	0	0
$283$	−1048.00	−0.0130854	−0.00654272	−	0.999979i	$-0.502083\pi$
$283$	−1048.00	−0.0130854	−0.00654272	+	0.999979i	$0.502083\pi$
$284$	12726.0i	0.157781i
$285$	0	0
$286$	111438.	1.36239
$287$	− 4332.00i	− 0.0525926i
$288$	0	0
$289$	−87875.0	−1.05213
$290$	67122.0i	0.798121i
$291$	0	0
$292$	−21217.0	−0.248839
$293$	64182.0i	0.747615i	0.927506	+	0.373807i	$0.121948\pi$
$293$	64182.0i	0.747615i	−0.927506	+	0.373807i	$0.878052\pi$
$294$	0	0
$295$	−96426.0	−1.10803
$296$	− 51060.0i	− 0.582770i
$297$	0	0
$298$	−65979.0	−0.742973
$299$	90600.0i	1.01341i
$300$	0	0
$301$	18658.0	0.205936
$302$	8049.00i	0.0882527i
$303$	0	0
$304$	28880.0	0.312500
$305$	10428.0i	0.112099i
$306$	0	0
$307$	154154.	1.63560	0.817802	−	0.575500i	$-0.195193\pi$
$307$	154154.	1.63560	0.817802	+	0.575500i	$0.195193\pi$
$308$	− 16359.0i	− 0.172447i
$309$	0	0
$310$	−23661.0	−0.246212
$311$	94080.0i	0.972695i	0.873766	+	0.486347i	$0.161671\pi$
$311$	94080.0i	0.972695i	−0.873766	+	0.486347i	$0.838329\pi$
$312$	0	0
$313$	−25903.0	−0.264400	−0.132200	−	0.991223i	$-0.542204\pi$
$313$	−25903.0	−0.264400	−0.132200	+	0.991223i	$0.542204\pi$
$314$	96348.0i	0.977200i
$315$	0	0
$316$	−73150.0	−0.732555
$317$	− 96843.0i	− 0.963717i	−0.876249	−	0.481859i	$-0.839962\pi$
$317$	− 96843.0i	− 0.963717i	0.876249	−	0.481859i	$-0.160038\pi$
$318$	0	0
$319$	83394.0	0.819508
$320$	131241.i	1.28165i
$321$	0	0
$322$	−17100.0	−0.164924
$323$	− 125856.i	− 1.20634i
$324$	0	0
$325$	−140128.	−1.32666
$326$	− 68370.0i	− 0.643325i
$327$	0	0
$328$	15732.0	0.146230
$329$	− 41154.0i	− 0.380207i
$330$	0	0
$331$	−164854.	−1.50468	−0.752339	−	0.658776i	$-0.771074\pi$
$331$	−164854.	−1.50468	−0.752339	+	0.658776i	$0.771074\pi$
$332$	88431.0i	0.802284i
$333$	0	0
$334$	−108234.	−0.970221
$335$	− 152526.i	− 1.35911i
$336$	0	0
$337$	148694.	1.30928	0.654642	−	0.755939i	$-0.272820\pi$
$337$	148694.	1.30928	0.654642	+	0.755939i	$0.272820\pi$
$338$	− 187929.i	− 1.64498i
$339$	0	0
$340$	95634.0	0.827284
$341$	29397.0i	0.252810i
$342$	0	0
$343$	84379.0	0.717210
$344$	67758.0i	0.572590i
$345$	0	0
$346$	59175.0	0.494295
$347$	− 107673.i	− 0.894227i	−0.894477	−	0.447114i	$-0.852452\pi$
$347$	− 107673.i	− 0.894227i	0.894477	−	0.447114i	$-0.147548\pi$
$348$	0	0
$349$	127520.	1.04695	0.523477	−	0.852040i	$-0.324635\pi$
$349$	127520.	1.04695	0.523477	+	0.852040i	$0.324635\pi$
$350$	− 26448.0i	− 0.215902i
$351$	0	0
$352$	100737.	0.813025
$353$	− 142104.i	− 1.14040i	−0.821506	−	0.570200i	$-0.806866\pi$
$353$	− 142104.i	− 1.14040i	0.821506	−	0.570200i	$-0.193134\pi$
$354$	0	0
$355$	59994.0	0.476048
$356$	− 49014.0i	− 0.386741i
$357$	0	0
$358$	−146745.	−1.14498
$359$	− 19422.0i	− 0.150697i	−0.997157	−	0.0753486i	$-0.975993\pi$
$359$	− 19422.0i	− 0.150697i	0.997157	−	0.0753486i	$-0.0240069\pi$
$360$	0	0
$361$	−37905.0	−0.290859
$362$	148656.i	1.13440i
$363$	0	0
$364$	−40166.0	−0.303149
$365$	100023.i	0.750783i
$366$	0	0
$367$	−151345.	−1.12366	−0.561831	−	0.827252i	$-0.689903\pi$
$367$	−151345.	−1.12366	−0.561831	+	0.827252i	$0.689903\pi$
$368$	− 28500.0i	− 0.210450i
$369$	0	0
$370$	−73260.0	−0.535135
$371$	30267.0i	0.219898i
$372$	0	0
$373$	237506.	1.70709	0.853546	−	0.521018i	$-0.174447\pi$
$373$	237506.	1.70709	0.853546	+	0.521018i	$0.174447\pi$
$374$	152766.i	1.09215i
$375$	0	0
$376$	149454.	1.05714
$377$	− 204756.i	− 1.44063i
$378$	0	0
$379$	−261952.	−1.82366	−0.911829	−	0.410571i	$-0.865330\pi$
$379$	−261952.	−1.82366	−0.911829	+	0.410571i	$0.865330\pi$
$380$	70224.0i	0.486316i
$381$	0	0
$382$	−136170.	−0.933157
$383$	− 87162.0i	− 0.594196i	−0.954847	−	0.297098i	$-0.903981\pi$
$383$	− 87162.0i	− 0.594196i	0.954847	−	0.297098i	$-0.0960188\pi$
$384$	0	0
$385$	−77121.0	−0.520297
$386$	− 106341.i	− 0.713717i
$387$	0	0
$388$	−45619.0	−0.303028
$389$	239343.i	1.58169i	0.612016	+	0.790845i	$0.290359\pi$
$389$	239343.i	1.58169i	−0.612016	+	0.790845i	$0.709641\pi$
$390$	0	0
$391$	−124200.	−0.812397
$392$	140760.i	0.916025i
$393$	0	0
$394$	107217.	0.690671
$395$	344850.i	2.21022i
$396$	0	0
$397$	217154.	1.37780	0.688901	−	0.724855i	$-0.258093\pi$
$397$	217154.	1.37780	0.688901	+	0.724855i	$0.258093\pi$
$398$	93765.0i	0.591936i
$399$	0	0
$400$	44080.0	0.275500
$401$	− 256200.i	− 1.59327i	−0.604458	−	0.796637i	$-0.706610\pi$
$401$	− 256200.i	− 1.59327i	0.604458	−	0.796637i	$-0.293390\pi$
$402$	0	0
$403$	72178.0	0.444421
$404$	41433.0i	0.253854i
$405$	0	0
$406$	38646.0	0.234451
$407$	91020.0i	0.549475i
$408$	0	0
$409$	−199291.	−1.19135	−0.595677	−	0.803224i	$-0.703116\pi$
$409$	−199291.	−1.19135	−0.595677	+	0.803224i	$0.703116\pi$
$410$	− 22572.0i	− 0.134277i
$411$	0	0
$412$	−53578.0	−0.315640
$413$	55518.0i	0.325487i
$414$	0	0
$415$	416889.	2.42061
$416$	− 247338.i	− 1.42924i
$417$	0	0
$418$	−112176.	−0.642018
$419$	251274.i	1.43126i	0.698478	+	0.715632i	$0.253861\pi$
$419$	251274.i	1.43126i	−0.698478	+	0.715632i	$0.746139\pi$
$420$	0	0
$421$	−30412.0	−0.171586	−0.0857928	−	0.996313i	$-0.527342\pi$
$421$	−30412.0	−0.171586	−0.0857928	+	0.996313i	$0.527342\pi$
$422$	45156.0i	0.253566i
$423$	0	0
$424$	−109917.	−0.611411
$425$	− 192096.i	− 1.06351i
$426$	0	0
$427$	6004.00	0.0329295
$428$	− 3591.00i	− 0.0196032i
$429$	0	0
$430$	97218.0	0.525787
$431$	161730.i	0.870635i	0.900277	+	0.435317i	$0.143364\pi$
$431$	161730.i	0.870635i	−0.900277	+	0.435317i	$0.856636\pi$
$432$	0	0
$433$	−213541.	−1.13895	−0.569476	−	0.822008i	$-0.692854\pi$
$433$	−213541.	−1.13895	−0.569476	+	0.822008i	$0.692854\pi$
$434$	13623.0i	0.0723258i
$435$	0	0
$436$	16268.0	0.0855778
$437$	− 91200.0i	− 0.477564i
$438$	0	0
$439$	66725.0	0.346226	0.173113	−	0.984902i	$-0.444617\pi$
$439$	66725.0	0.346226	0.173113	+	0.984902i	$0.444617\pi$
$440$	− 280071.i	− 1.44665i
$441$	0	0
$442$	375084.	1.91992
$443$	− 274170.i	− 1.39705i	−0.715585	−	0.698526i	$-0.753840\pi$
$443$	− 274170.i	− 1.39705i	0.715585	−	0.698526i	$-0.246160\pi$
$444$	0	0
$445$	−231066.	−1.16685
$446$	− 150522.i	− 0.756711i
$447$	0	0
$448$	75563.0	0.376490
$449$	− 233784.i	− 1.15964i	−0.814746	−	0.579819i	$-0.803123\pi$
$449$	− 233784.i	− 1.15964i	0.814746	−	0.579819i	$-0.196877\pi$
$450$	0	0
$451$	−28044.0	−0.137875
$452$	− 34440.0i	− 0.168572i
$453$	0	0
$454$	57798.0	0.280415
$455$	189354.i	0.914643i
$456$	0	0
$457$	−90667.0	−0.434127	−0.217064	−	0.976157i	$-0.569648\pi$
$457$	−90667.0	−0.434127	−0.217064	+	0.976157i	$0.569648\pi$
$458$	− 102642.i	− 0.489321i
$459$	0	0
$460$	69300.0	0.327505
$461$	201957.i	0.950292i	0.879907	+	0.475146i	$0.157605\pi$
$461$	201957.i	0.950292i	−0.879907	+	0.475146i	$0.842395\pi$
$462$	0	0
$463$	−323977.	−1.51131	−0.755653	−	0.654973i	$-0.772680\pi$
$463$	−323977.	−1.51131	−0.755653	+	0.654973i	$0.772680\pi$
$464$	64410.0i	0.299170i
$465$	0	0
$466$	−112158.	−0.516486
$467$	76941.0i	0.352796i	0.984319	+	0.176398i	$0.0564446\pi$
$467$	76941.0i	0.352796i	−0.984319	+	0.176398i	$0.943555\pi$
$468$	0	0
$469$	−87818.0	−0.399244
$470$	− 214434.i	− 0.970729i
$471$	0	0
$472$	−201618.	−0.904993
$473$	− 120786.i	− 0.539876i
$474$	0	0
$475$	141056.	0.625179
$476$	− 55062.0i	− 0.243018i
$477$	0	0
$478$	−185400.	−0.811435
$479$	193218.i	0.842125i	0.907032	+	0.421062i	$0.138343\pi$
$479$	193218.i	0.842125i	−0.907032	+	0.421062i	$0.861657\pi$
$480$	0	0
$481$	223480.	0.965936
$482$	− 124170.i	− 0.534469i
$483$	0	0
$484$	−3416.00	−0.0145823
$485$	215061.i	0.914278i
$486$	0	0
$487$	−34882.0	−0.147077	−0.0735383	−	0.997292i	$-0.523429\pi$
$487$	−34882.0	−0.147077	−0.0735383	+	0.997292i	$0.523429\pi$
$488$	21804.0i	0.0915580i
$489$	0	0
$490$	201960.	0.841150
$491$	217047.i	0.900307i	0.892951	+	0.450154i	$0.148631\pi$
$491$	217047.i	0.900307i	−0.892951	+	0.450154i	$0.851369\pi$
$492$	0	0
$493$	280692.	1.15488
$494$	275424.i	1.12862i
$495$	0	0
$496$	−22705.0	−0.0922907
$497$	− 34542.0i	− 0.139841i
$498$	0	0
$499$	464810.	1.86670	0.933350	−	0.358969i	$-0.116871\pi$
$499$	464810.	1.86670	0.933350	+	0.358969i	$0.116871\pi$
$500$	− 37191.0i	− 0.148764i
$501$	0	0
$502$	−248454.	−0.985913
$503$	− 167580.i	− 0.662348i	−0.943570	−	0.331174i	$-0.892555\pi$
$503$	− 167580.i	− 0.662348i	0.943570	−	0.331174i	$-0.107445\pi$
$504$	0	0
$505$	195327.	0.765913
$506$	110700.i	0.432361i
$507$	0	0
$508$	174965.	0.677991
$509$	35697.0i	0.137783i	0.997624	+	0.0688916i	$0.0219463\pi$
$509$	35697.0i	0.137783i	−0.997624	+	0.0688916i	$0.978054\pi$
$510$	0	0
$511$	57589.0	0.220545
$512$	182685.i	0.696888i
$513$	0	0
$514$	−58770.0	−0.222448
$515$	252582.i	0.952331i
$516$	0	0
$517$	−266418.	−0.996741
$518$	42180.0i	0.157198i
$519$	0	0
$520$	−687654.	−2.54310
$521$	− 42750.0i	− 0.157493i	−0.996895	−	0.0787464i	$-0.974908\pi$
$521$	− 42750.0i	− 0.157493i	0.996895	−	0.0787464i	$-0.0250917\pi$
$522$	0	0
$523$	−176434.	−0.645028	−0.322514	−	0.946565i	$-0.604528\pi$
$523$	−176434.	−0.645028	−0.322514	+	0.946565i	$0.604528\pi$
$524$	199227.i	0.725581i
$525$	0	0
$526$	50076.0	0.180991
$527$	98946.0i	0.356268i
$528$	0	0
$529$	189841.	0.678389
$530$	157707.i	0.561435i
$531$	0	0
$532$	40432.0	0.142857
$533$	68856.0i	0.242375i
$534$	0	0
$535$	−16929.0	−0.0591458
$536$	− 318918.i	− 1.11007i
$537$	0	0
$538$	362718.	1.25315
$539$	− 250920.i	− 0.863690i
$540$	0	0
$541$	−323836.	−1.10645	−0.553223	−	0.833033i	$-0.686602\pi$
$541$	−323836.	−1.10645	−0.553223	+	0.833033i	$0.686602\pi$
$542$	− 221217.i	− 0.753043i
$543$	0	0
$544$	339066.	1.14574
$545$	− 76692.0i	− 0.258200i
$546$	0	0
$547$	−223390.	−0.746602	−0.373301	−	0.927710i	$-0.621774\pi$
$547$	−223390.	−0.746602	−0.373301	+	0.927710i	$0.621774\pi$
$548$	17178.0i	0.0572020i
$549$	0	0
$550$	−171216.	−0.566003
$551$	206112.i	0.678891i
$552$	0	0
$553$	198550.	0.649261
$554$	− 35988.0i	− 0.117257i
$555$	0	0
$556$	−83188.0	−0.269098
$557$	− 585027.i	− 1.88567i	−0.333261	−	0.942835i	$-0.608149\pi$
$557$	− 585027.i	− 1.88567i	0.333261	−	0.942835i	$-0.391851\pi$
$558$	0	0
$559$	−296564.	−0.949063
$560$	− 59565.0i	− 0.189939i
$561$	0	0
$562$	153378.	0.485613
$563$	− 84075.0i	− 0.265247i	−0.991167	−	0.132623i	$-0.957660\pi$
$563$	− 84075.0i	− 0.265247i	0.991167	−	0.132623i	$-0.0423401\pi$
$564$	0	0
$565$	−162360.	−0.508607
$566$	3144.00i	0.00981408i
$567$	0	0
$568$	125442.	0.388818
$569$	− 637392.i	− 1.96871i	−0.176192	−	0.984356i	$-0.556378\pi$
$569$	− 637392.i	− 1.96871i	0.176192	−	0.984356i	$-0.443622\pi$
$570$	0	0
$571$	80726.0	0.247595	0.123797	−	0.992308i	$-0.460493\pi$
$571$	80726.0	0.247595	0.123797	+	0.992308i	$0.460493\pi$
$572$	260022.i	0.794727i
$573$	0	0
$574$	−12996.0	−0.0394445
$575$	− 139200.i	− 0.421021i
$576$	0	0
$577$	261182.	0.784498	0.392249	−	0.919859i	$-0.371697\pi$
$577$	261182.	0.784498	0.392249	+	0.919859i	$0.371697\pi$
$578$	263625.i	0.789098i
$579$	0	0
$580$	−156618.	−0.465571
$581$	− 240027.i	− 0.711063i
$582$	0	0
$583$	195939.	0.576479
$584$	209139.i	0.613210i
$585$	0	0
$586$	192546.	0.560711
$587$	− 391305.i	− 1.13564i	−0.823154	−	0.567818i	$-0.807788\pi$
$587$	− 391305.i	− 1.13564i	0.823154	−	0.567818i	$-0.192212\pi$
$588$	0	0
$589$	−72656.0	−0.209431
$590$	289278.i	0.831020i
$591$	0	0
$592$	−70300.0	−0.200591
$593$	302670.i	0.860716i	0.902658	+	0.430358i	$0.141613\pi$
$593$	302670.i	0.860716i	−0.902658	+	0.430358i	$0.858387\pi$
$594$	0	0
$595$	−259578.	−0.733219
$596$	− 153951.i	− 0.433401i
$597$	0	0
$598$	271800.	0.760059
$599$	− 291498.i	− 0.812422i	−0.913779	−	0.406211i	$-0.866850\pi$
$599$	− 291498.i	− 0.812422i	0.913779	−	0.406211i	$-0.133150\pi$
$600$	0	0
$601$	402173.	1.11343	0.556716	−	0.830703i	$-0.312061\pi$
$601$	402173.	1.11343	0.556716	+	0.830703i	$0.312061\pi$
$602$	− 55974.0i	− 0.154452i
$603$	0	0
$604$	−18781.0	−0.0514807
$605$	16104.0i	0.0439970i
$606$	0	0
$607$	−378670.	−1.02774	−0.513870	−	0.857868i	$-0.671789\pi$
$607$	−378670.	−1.02774	−0.513870	+	0.857868i	$0.671789\pi$
$608$	248976.i	0.673520i
$609$	0	0
$610$	31284.0	0.0840742
$611$	654132.i	1.75220i
$612$	0	0
$613$	287570.	0.765284	0.382642	−	0.923897i	$-0.375014\pi$
$613$	287570.	0.765284	0.382642	+	0.923897i	$0.375014\pi$
$614$	− 462462.i	− 1.22670i
$615$	0	0
$616$	−161253.	−0.424958
$617$	576264.i	1.51374i	0.653566	+	0.756870i	$0.273272\pi$
$617$	576264.i	1.51374i	−0.653566	+	0.756870i	$0.726728\pi$
$618$	0	0
$619$	223262.	0.582685	0.291342	−	0.956619i	$-0.405898\pi$
$619$	223262.	0.582685	0.291342	+	0.956619i	$0.405898\pi$
$620$	− 55209.0i	− 0.143624i
$621$	0	0
$622$	282240.	0.729521
$623$	133038.i	0.342767i
$624$	0	0
$625$	−465329.	−1.19124
$626$	77709.0i	0.198300i
$627$	0	0
$628$	−224812.	−0.570033
$629$	306360.i	0.774338i
$630$	0	0
$631$	43373.0	0.108933	0.0544667	−	0.998516i	$-0.482654\pi$
$631$	43373.0	0.108933	0.0544667	+	0.998516i	$0.482654\pi$
$632$	721050.i	1.80522i
$633$	0	0
$634$	−290529.	−0.722788
$635$	− 824835.i	− 2.04559i
$636$	0	0
$637$	−616080.	−1.51830
$638$	− 250182.i	− 0.614631i
$639$	0	0
$640$	−38709.0	−0.0945044
$641$	423420.i	1.03052i	0.857035	+	0.515259i	$0.172304\pi$
$641$	423420.i	1.03052i	−0.857035	+	0.515259i	$0.827696\pi$
$642$	0	0
$643$	−546088.	−1.32081	−0.660406	−	0.750909i	$-0.729616\pi$
$643$	−546088.	−1.32081	−0.660406	+	0.750909i	$0.729616\pi$
$644$	− 39900.0i	− 0.0962058i
$645$	0	0
$646$	−377568.	−0.904753
$647$	− 418932.i	− 1.00077i	−0.865803	−	0.500386i	$-0.833192\pi$
$647$	− 418932.i	− 1.00077i	0.865803	−	0.500386i	$-0.166808\pi$
$648$	0	0
$649$	359406.	0.853289
$650$	420384.i	0.994992i
$651$	0	0
$652$	159530.	0.375273
$653$	703209.i	1.64914i	0.565758	+	0.824571i	$0.308583\pi$
$653$	703209.i	1.64914i	−0.565758	+	0.824571i	$0.691417\pi$
$654$	0	0
$655$	939213.	2.18918
$656$	− 21660.0i	− 0.0503328i
$657$	0	0
$658$	−123462.	−0.285155
$659$	102021.i	0.234919i	0.993078	+	0.117460i	$0.0374751\pi$
$659$	102021.i	0.234919i	−0.993078	+	0.117460i	$0.962525\pi$
$660$	0	0
$661$	230720.	0.528059	0.264029	−	0.964515i	$-0.414948\pi$
$661$	230720.	0.528059	0.264029	+	0.964515i	$0.414948\pi$
$662$	494562.i	1.12851i
$663$	0	0
$664$	871677.	1.97706
$665$	− 190608.i	− 0.431020i
$666$	0	0
$667$	203400.	0.457193
$668$	− 252546.i	− 0.565962i
$669$	0	0
$670$	−457578.	−1.01933
$671$	− 38868.0i	− 0.0863271i
$672$	0	0
$673$	−469369.	−1.03630	−0.518149	−	0.855291i	$-0.673379\pi$
$673$	−469369.	−1.03630	−0.518149	+	0.855291i	$0.673379\pi$
$674$	− 446082.i	− 0.981963i
$675$	0	0
$676$	438501.	0.959571
$677$	− 343146.i	− 0.748689i	−0.927290	−	0.374345i	$-0.877868\pi$
$677$	− 343146.i	− 0.748689i	0.927290	−	0.374345i	$-0.122132\pi$
$678$	0	0
$679$	123823.	0.268573
$680$	− 942678.i	− 2.03866i
$681$	0	0
$682$	88191.0	0.189608
$683$	24642.0i	0.0528244i	0.999651	+	0.0264122i	$0.00840824\pi$
$683$	24642.0i	0.0528244i	−0.999651	+	0.0264122i	$0.991592\pi$
$684$	0	0
$685$	80982.0	0.172587
$686$	− 253137.i	− 0.537907i
$687$	0	0
$688$	93290.0	0.197087
$689$	− 481086.i	− 1.01341i
$690$	0	0
$691$	−266500.	−0.558137	−0.279069	−	0.960271i	$-0.590026\pi$
$691$	−266500.	−0.558137	−0.279069	+	0.960271i	$0.590026\pi$
$692$	138075.i	0.288339i
$693$	0	0
$694$	−323019.	−0.670670
$695$	392172.i	0.811908i
$696$	0	0
$697$	−94392.0	−0.194299
$698$	− 382560.i	− 0.785215i
$699$	0	0
$700$	61712.0	0.125943
$701$	690309.i	1.40478i	0.711794	+	0.702389i	$0.247883\pi$
$701$	690309.i	1.40478i	−0.711794	+	0.702389i	$0.752117\pi$
$702$	0	0
$703$	−224960.	−0.455192
$704$	− 489171.i	− 0.986996i
$705$	0	0
$706$	−426312.	−0.855299
$707$	− 112461.i	− 0.224990i
$708$	0	0
$709$	−105184.	−0.209246	−0.104623	−	0.994512i	$-0.533364\pi$
$709$	−105184.	−0.209246	−0.104623	+	0.994512i	$0.533364\pi$
$710$	− 179982.i	− 0.357036i
$711$	0	0
$712$	−483138.	−0.953040
$713$	71700.0i	0.141039i
$714$	0	0
$715$	1.22582e6	2.39781
$716$	− 342405.i	− 0.667904i
$717$	0	0
$718$	−58266.0	−0.113023
$719$	− 704988.i	− 1.36372i	−0.731485	−	0.681858i	$-0.761172\pi$
$719$	− 704988.i	− 1.36372i	0.731485	−	0.681858i	$-0.238828\pi$
$720$	0	0
$721$	145426.	0.279751
$722$	113715.i	0.218144i
$723$	0	0
$724$	−346864.	−0.661732
$725$	314592.i	0.598510i
$726$	0	0
$727$	126089.	0.238566	0.119283	−	0.992860i	$-0.461940\pi$
$727$	126089.	0.238566	0.119283	+	0.992860i	$0.461940\pi$
$728$	395922.i	0.747045i
$729$	0	0
$730$	300069.	0.563087
$731$	− 406548.i	− 0.760812i
$732$	0	0
$733$	97736.0	0.181906	0.0909529	−	0.995855i	$-0.471009\pi$
$733$	97736.0	0.181906	0.0909529	+	0.995855i	$0.471009\pi$
$734$	454035.i	0.842747i
$735$	0	0
$736$	245700.	0.453575
$737$	568506.i	1.04665i
$738$	0	0
$739$	−857158.	−1.56954	−0.784769	−	0.619788i	$-0.787219\pi$
$739$	−857158.	−1.56954	−0.784769	+	0.619788i	$0.787219\pi$
$740$	− 170940.i	− 0.312162i
$741$	0	0
$742$	90801.0	0.164924
$743$	909966.i	1.64834i	0.566340	+	0.824171i	$0.308359\pi$
$743$	909966.i	1.64834i	−0.566340	+	0.824171i	$0.691641\pi$
$744$	0	0
$745$	−725769.	−1.30763
$746$	− 712518.i	− 1.28032i
$747$	0	0
$748$	−356454.	−0.637089
$749$	9747.00i	0.0173743i
$750$	0	0
$751$	61223.0	0.108551	0.0542756	−	0.998526i	$-0.482715\pi$
$751$	61223.0	0.108551	0.0542756	+	0.998526i	$0.482715\pi$
$752$	− 205770.i	− 0.363870i
$753$	0	0
$754$	−614268.	−1.08048
$755$	88539.0i	0.155325i
$756$	0	0
$757$	782570.	1.36562	0.682812	−	0.730594i	$-0.260757\pi$
$757$	782570.	1.36562	0.682812	+	0.730594i	$0.260757\pi$
$758$	785856.i	1.36774i
$759$	0	0
$760$	692208.	1.19842
$761$	701400.i	1.21115i	0.795790	+	0.605573i	$0.207056\pi$
$761$	701400.i	1.21115i	−0.795790	+	0.605573i	$0.792944\pi$
$762$	0	0
$763$	−44156.0	−0.0758474
$764$	− 317730.i	− 0.544342i
$765$	0	0
$766$	−261486.	−0.445647
$767$	− 882444.i	− 1.50002i
$768$	0	0
$769$	−85045.0	−0.143812	−0.0719062	−	0.997411i	$-0.522908\pi$
$769$	−85045.0	−0.143812	−0.0719062	+	0.997411i	$0.522908\pi$
$770$	231363.i	0.390223i
$771$	0	0
$772$	248129.	0.416335
$773$	643122.i	1.07630i	0.842848	+	0.538151i	$0.180877\pi$
$773$	643122.i	1.07630i	−0.842848	+	0.538151i	$0.819123\pi$
$774$	0	0
$775$	−110896.	−0.184634
$776$	449673.i	0.746747i
$777$	0	0
$778$	718029.	1.18627
$779$	− 69312.0i	− 0.114218i
$780$	0	0
$781$	−223614.	−0.366604
$782$	372600.i	0.609297i
$783$	0	0
$784$	193800.	0.315298
$785$	1.05983e6i	1.71987i
$786$	0	0
$787$	1.06855e6	1.72522	0.862610	−	0.505869i	$-0.168828\pi$
$787$	1.06855e6	1.72522	0.862610	+	0.505869i	$0.168828\pi$
$788$	250173.i	0.402891i
$789$	0	0
$790$	1.03455e6	1.65767
$791$	93480.0i	0.149405i
$792$	0	0
$793$	−95432.0	−0.151757
$794$	− 651462.i	− 1.03335i
$795$	0	0
$796$	−218785.	−0.345296
$797$	− 538935.i	− 0.848437i	−0.905560	−	0.424219i	$-0.860549\pi$
$797$	− 538935.i	− 0.848437i	0.905560	−	0.424219i	$-0.139451\pi$
$798$	0	0
$799$	−896724.	−1.40464
$800$	380016.i	0.593775i
$801$	0	0
$802$	−768600.	−1.19496
$803$	− 372813.i	− 0.578176i
$804$	0	0
$805$	−188100.	−0.290267
$806$	− 216534.i	− 0.333316i
$807$	0	0
$808$	408411.	0.625568
$809$	− 459594.i	− 0.702227i	−0.936333	−	0.351113i	$-0.885803\pi$
$809$	− 459594.i	− 0.702227i	0.936333	−	0.351113i	$-0.114197\pi$
$810$	0	0
$811$	−961360.	−1.46165	−0.730827	−	0.682563i	$-0.760865\pi$
$811$	−961360.	−1.46165	−0.730827	+	0.682563i	$0.760865\pi$
$812$	90174.0i	0.136763i
$813$	0	0
$814$	273060.	0.412106
$815$	− 752070.i	− 1.13225i
$816$	0	0
$817$	298528.	0.447240
$818$	597873.i	0.893516i
$819$	0	0
$820$	52668.0	0.0783284
$821$	105666.i	0.156765i	0.996923	+	0.0783825i	$0.0249755\pi$
$821$	105666.i	0.156765i	−0.996923	+	0.0783825i	$0.975024\pi$
$822$	0	0
$823$	−493555.	−0.728678	−0.364339	−	0.931266i	$-0.618705\pi$
$823$	−493555.	−0.728678	−0.364339	+	0.931266i	$0.618705\pi$
$824$	528126.i	0.777827i
$825$	0	0
$826$	166554.	0.244115
$827$	192870.i	0.282003i	0.990009	+	0.141001i	$0.0450322\pi$
$827$	192870.i	0.282003i	−0.990009	+	0.141001i	$0.954968\pi$
$828$	0	0
$829$	577226.	0.839918	0.419959	−	0.907543i	$-0.362044\pi$
$829$	577226.	0.839918	0.419959	+	0.907543i	$0.362044\pi$
$830$	− 1.25067e6i	− 1.81546i
$831$	0	0
$832$	−1.20105e6	−1.73507
$833$	− 844560.i	− 1.21714i
$834$	0	0
$835$	−1.19057e6	−1.70759
$836$	− 261744.i	− 0.374511i
$837$	0	0
$838$	753822.	1.07345
$839$	− 70986.0i	− 0.100844i	−0.998728	−	0.0504219i	$-0.983943\pi$
$839$	− 70986.0i	− 0.100844i	0.998728	−	0.0504219i	$-0.0160566\pi$
$840$	0	0
$841$	247597.	0.350069
$842$	91236.0i	0.128689i
$843$	0	0
$844$	−105364.	−0.147913
$845$	− 2.06722e6i	− 2.89516i
$846$	0	0
$847$	9272.00	0.0129243
$848$	151335.i	0.210449i
$849$	0	0
$850$	−576288.	−0.797630
$851$	222000.i	0.306545i
$852$	0	0
$853$	81206.0	0.111607	0.0558033	−	0.998442i	$-0.482228\pi$
$853$	81206.0	0.111607	0.0558033	+	0.998442i	$0.482228\pi$
$854$	− 18012.0i	− 0.0246971i
$855$	0	0
$856$	−35397.0	−0.0483080
$857$	1.08044e6i	1.47109i	0.677478	+	0.735543i	$0.263073\pi$
$857$	1.08044e6i	1.47109i	−0.677478	+	0.735543i	$0.736927\pi$
$858$	0	0
$859$	503348.	0.682153	0.341077	−	0.940035i	$-0.389208\pi$
$859$	503348.	0.682153	0.341077	+	0.940035i	$0.389208\pi$
$860$	226842.i	0.306709i
$861$	0	0
$862$	485190.	0.652976
$863$	− 548100.i	− 0.735933i	−0.929839	−	0.367966i	$-0.880054\pi$
$863$	− 548100.i	− 0.735933i	0.929839	−	0.367966i	$-0.119946\pi$
$864$	0	0
$865$	650925.	0.869959
$866$	640623.i	0.854214i
$867$	0	0
$868$	−31787.0	−0.0421901
$869$	− 1.28535e6i	− 1.70209i
$870$	0	0
$871$	1.39584e6	1.83993
$872$	− 160356.i	− 0.210888i
$873$	0	0
$874$	−273600.	−0.358173
$875$	100947.i	0.131849i
$876$	0	0
$877$	700034.	0.910165	0.455082	−	0.890449i	$-0.349610\pi$
$877$	700034.	0.910165	0.455082	+	0.890449i	$0.349610\pi$
$878$	− 200175.i	− 0.259669i
$879$	0	0
$880$	−385605.	−0.497940
$881$	806634.i	1.03926i	0.854391	+	0.519631i	$0.173930\pi$
$881$	806634.i	1.03926i	−0.854391	+	0.519631i	$0.826070\pi$
$882$	0	0
$883$	342704.	0.439539	0.219770	−	0.975552i	$-0.429469\pi$
$883$	342704.	0.439539	0.219770	+	0.975552i	$0.429469\pi$
$884$	875196.i	1.11996i
$885$	0	0
$886$	−822510.	−1.04779
$887$	16122.0i	0.0204914i	0.999948	+	0.0102457i	$0.00326137\pi$
$887$	16122.0i	0.0204914i	−0.999948	+	0.0102457i	$0.996739\pi$
$888$	0	0
$889$	−474905.	−0.600901
$890$	693198.i	0.875140i
$891$	0	0
$892$	351218.	0.441415
$893$	− 658464.i	− 0.825713i
$894$	0	0
$895$	−1.61420e6	−2.01516
$896$	22287.0i	0.0277610i
$897$	0	0
$898$	−701352.	−0.869728
$899$	− 162042.i	− 0.200497i
$900$	0	0
$901$	659502.	0.812394
$902$	84132.0i	0.103407i
$903$	0	0
$904$	−339480.	−0.415410
$905$	1.63522e6i	1.99654i
$906$	0	0
$907$	1.56950e6	1.90786	0.953931	−	0.300028i	$-0.0969959\pi$
$907$	1.56950e6	1.90786	0.953931	+	0.300028i	$0.0969959\pi$
$908$	134862.i	0.163575i
$909$	0	0
$910$	568062.	0.685982
$911$	− 500898.i	− 0.603549i	−0.953379	−	0.301775i	$-0.902421\pi$
$911$	− 500898.i	− 0.603549i	0.953379	−	0.301775i	$-0.0975790\pi$
$912$	0	0
$913$	−1.55386e6	−1.86410
$914$	272001.i	0.325595i
$915$	0	0
$916$	239498.	0.285437
$917$	− 540759.i	− 0.643080i
$918$	0	0
$919$	1.03715e6	1.22804	0.614019	−	0.789291i	$-0.289552\pi$
$919$	1.03715e6	1.22804	0.614019	+	0.789291i	$0.289552\pi$
$920$	− 683100.i	− 0.807065i
$921$	0	0
$922$	605871.	0.712719
$923$	549036.i	0.644462i
$924$	0	0
$925$	−343360.	−0.401297
$926$	971931.i	1.13348i
$927$	0	0
$928$	−555282.	−0.644789
$929$	128076.i	0.148401i	0.997243	+	0.0742004i	$0.0236405\pi$
$929$	128076.i	0.148401i	−0.997243	+	0.0742004i	$0.976360\pi$
$930$	0	0
$931$	620160.	0.715491
$932$	− 261702.i	− 0.301283i
$933$	0	0
$934$	230823.	0.264597
$935$	1.68043e6i	1.92219i
$936$	0	0
$937$	−879451.	−1.00169	−0.500844	−	0.865538i	$-0.666977\pi$
$937$	−879451.	−1.00169	−0.500844	+	0.865538i	$0.666977\pi$
$938$	263454.i	0.299433i
$939$	0	0
$940$	500346.	0.566258
$941$	718257.i	0.811149i	0.914062	+	0.405574i	$0.132929\pi$
$941$	718257.i	0.811149i	−0.914062	+	0.405574i	$0.867071\pi$
$942$	0	0
$943$	−68400.0	−0.0769188
$944$	277590.i	0.311501i
$945$	0	0
$946$	−362358.	−0.404907
$947$	73005.0i	0.0814053i	0.999171	+	0.0407026i	$0.0129596\pi$
$947$	73005.0i	0.0814053i	−0.999171	+	0.0407026i	$0.987040\pi$
$948$	0	0
$949$	−915362.	−1.01639
$950$	− 423168.i	− 0.468884i
$951$	0	0
$952$	−542754.	−0.598865
$953$	309168.i	0.340415i	0.985408	+	0.170208i	$0.0544438\pi$
$953$	309168.i	0.340415i	−0.985408	+	0.170208i	$0.945556\pi$
$954$	0	0
$955$	−1.49787e6	−1.64236
$956$	− 432600.i	− 0.473337i
$957$	0	0
$958$	579654.	0.631594
$959$	− 46626.0i	− 0.0506980i
$960$	0	0
$961$	−866400.	−0.938149
$962$	− 670440.i	− 0.724452i
$963$	0	0
$964$	289730.	0.311774
$965$	− 1.16975e6i	− 1.25614i
$966$	0	0
$967$	−366187.	−0.391607	−0.195803	−	0.980643i	$-0.562731\pi$
$967$	−366187.	−0.391607	−0.195803	+	0.980643i	$0.562731\pi$
$968$	33672.0i	0.0359350i
$969$	0	0
$970$	645183.	0.685708
$971$	− 1.43410e6i	− 1.52105i	−0.649311	−	0.760523i	$-0.724943\pi$
$971$	− 1.43410e6i	− 1.52105i	0.649311	−	0.760523i	$-0.275057\pi$
$972$	0	0
$973$	225796.	0.238501
$974$	104646.i	0.110307i
$975$	0	0
$976$	30020.0	0.0315145
$977$	− 311802.i	− 0.326655i	−0.986572	−	0.163328i	$-0.947777\pi$
$977$	− 311802.i	− 0.326655i	0.986572	−	0.163328i	$-0.0522228\pi$
$978$	0	0
$979$	861246.	0.898591
$980$	471240.i	0.490671i
$981$	0	0
$982$	651141.	0.675231
$983$	690162.i	0.714240i	0.934059	+	0.357120i	$0.116241\pi$
$983$	690162.i	0.714240i	−0.934059	+	0.357120i	$0.883759\pi$
$984$	0	0
$985$	1.17939e6	1.21558
$986$	− 842076.i	− 0.866159i
$987$	0	0
$988$	−642656.	−0.658362
$989$	− 294600.i	− 0.301190i
$990$	0	0
$991$	981875.	0.999790	0.499895	−	0.866086i	$-0.333372\pi$
$991$	981875.	0.999790	0.499895	+	0.866086i	$0.333372\pi$
$992$	− 195741.i	− 0.198911i
$993$	0	0
$994$	−103626.	−0.104881
$995$	1.03142e6i	1.04181i
$996$	0	0
$997$	241946.	0.243404	0.121702	−	0.992567i	$-0.461165\pi$
$997$	241946.	0.243404	0.121702	+	0.992567i	$0.461165\pi$
$998$	− 1.39443e6i	− 1.40002i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.b.c.26.1	✓	2
3.2	odd	2	inner	27.5.b.c.26.2	yes	2
4.3	odd	2		432.5.e.e.161.1		2
5.2	odd	4		675.5.d.d.674.1		2
5.3	odd	4		675.5.d.a.674.2		2
5.4	even	2		675.5.c.h.26.2		2
9.2	odd	6		81.5.d.b.53.2		4
9.4	even	3		81.5.d.b.26.2		4
9.5	odd	6		81.5.d.b.26.1		4
9.7	even	3		81.5.d.b.53.1		4
12.11	even	2		432.5.e.e.161.2		2
15.2	even	4		675.5.d.a.674.1		2
15.8	even	4		675.5.d.d.674.2		2
15.14	odd	2		675.5.c.h.26.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.b.c.26.1	✓	2	1.1	even	1	trivial
27.5.b.c.26.2	yes	2	3.2	odd	2	inner
81.5.d.b.26.1		4	9.5	odd	6
81.5.d.b.26.2		4	9.4	even	3
81.5.d.b.53.1		4	9.7	even	3
81.5.d.b.53.2		4	9.2	odd	6
432.5.e.e.161.1		2	4.3	odd	2
432.5.e.e.161.2		2	12.11	even	2
675.5.c.h.26.1		2	15.14	odd	2
675.5.c.h.26.2		2	5.4	even	2
675.5.d.a.674.1		2	15.2	even	4
675.5.d.a.674.2		2	5.3	odd	4
675.5.d.d.674.1		2	5.2	odd	4
675.5.d.d.674.2		2	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	27.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{2} +7.00000 q^{4} -33.0000i q^{5} -19.0000 q^{7} -69.0000i q^{8} +O(q^{10})\)	\(q-3.00000i q^{2} +7.00000 q^{4} -33.0000i q^{5} -19.0000 q^{7} -69.0000i q^{8} -99.0000 q^{10} +123.000i q^{11} +302.000 q^{13} +57.0000i q^{14} -95.0000 q^{16} +414.000i q^{17} -304.000 q^{19} -231.000i q^{20} +369.000 q^{22} +300.000i q^{23} -464.000 q^{25} -906.000i q^{26} -133.000 q^{28} -678.000i q^{29} +239.000 q^{31} -819.000i q^{32} +1242.00 q^{34} +627.000i q^{35} +740.000 q^{37} +912.000i q^{38} -2277.00 q^{40} +228.000i q^{41} -982.000 q^{43} +861.000i q^{44} +900.000 q^{46} +2166.00i q^{47} -2040.00 q^{49} +1392.00i q^{50} +2114.00 q^{52} -1593.00i q^{53} +4059.00 q^{55} +1311.00i q^{56} -2034.00 q^{58} -2922.00i q^{59} -316.000 q^{61} -717.000i q^{62} -3977.00 q^{64} -9966.00i q^{65} +4622.00 q^{67} +2898.00i q^{68} +1881.00 q^{70} +1818.00i q^{71} -3031.00 q^{73} -2220.00i q^{74} -2128.00 q^{76} -2337.00i q^{77} -10450.0 q^{79} +3135.00i q^{80} +684.000 q^{82} +12633.0i q^{83} +13662.0 q^{85} +2946.00i q^{86} +8487.00 q^{88} -7002.00i q^{89} -5738.00 q^{91} +2100.00i q^{92} +6498.00 q^{94} +10032.0i q^{95} -6517.00 q^{97} +6120.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4} - 38 q^{7}+O(q^{10}) \) 2 * q + 14 * q^4 - 38 * q^7	\( 2 q + 14 q^{4} - 38 q^{7} - 198 q^{10} + 604 q^{13} - 190 q^{16} - 608 q^{19} + 738 q^{22} - 928 q^{25} - 266 q^{28} + 478 q^{31} + 2484 q^{34} + 1480 q^{37} - 4554 q^{40} - 1964 q^{43} + 1800 q^{46} - 4080 q^{49} + 4228 q^{52} + 8118 q^{55} - 4068 q^{58} - 632 q^{61} - 7954 q^{64} + 9244 q^{67} + 3762 q^{70} - 6062 q^{73} - 4256 q^{76} - 20900 q^{79} + 1368 q^{82} + 27324 q^{85} + 16974 q^{88} - 11476 q^{91} + 12996 q^{94} - 13034 q^{97}+O(q^{100}) \) 2 * q + 14 * q^4 - 38 * q^7 - 198 * q^10 + 604 * q^13 - 190 * q^16 - 608 * q^19 + 738 * q^22 - 928 * q^25 - 266 * q^28 + 478 * q^31 + 2484 * q^34 + 1480 * q^37 - 4554 * q^40 - 1964 * q^43 + 1800 * q^46 - 4080 * q^49 + 4228 * q^52 + 8118 * q^55 - 4068 * q^58 - 632 * q^61 - 7954 * q^64 + 9244 * q^67 + 3762 * q^70 - 6062 * q^73 - 4256 * q^76 - 20900 * q^79 + 1368 * q^82 + 27324 * q^85 + 16974 * q^88 - 11476 * q^91 + 12996 * q^94 - 13034 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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