LMFDB - Embedded newform 378.5.b.b.323.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,5,Mod(323,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("378.323");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	378.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:	$39.0738460457$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 $ x^8 + 92x^6 + 2949x^4 + 37548*x^2 + 142884
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.7
Root			$-5.12618i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.b.323.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +28.1791i q^{5} -18.5203 q^{7} -22.6274i q^{8} +O(q^{10})$	\(q+2.82843i q^{2} -8.00000 q^{4} +28.1791i q^{5} -18.5203 q^{7} -22.6274i q^{8} -79.7026 q^{10} -168.507i q^{11} +319.388 q^{13} -52.3832i q^{14} +64.0000 q^{16} -39.1240i q^{17} +457.590 q^{19} -225.433i q^{20} +476.611 q^{22} +435.106i q^{23} -169.064 q^{25} +903.365i q^{26} +148.162 q^{28} -135.719i q^{29} -267.371 q^{31} +181.019i q^{32} +110.659 q^{34} -521.885i q^{35} +812.186 q^{37} +1294.26i q^{38} +637.621 q^{40} -737.686i q^{41} -2092.00 q^{43} +1348.06i q^{44} -1230.66 q^{46} +2962.37i q^{47} +343.000 q^{49} -478.184i q^{50} -2555.10 q^{52} -3190.68i q^{53} +4748.39 q^{55} +419.066i q^{56} +383.871 q^{58} +5620.52i q^{59} +4862.90 q^{61} -756.240i q^{62} -512.000 q^{64} +9000.07i q^{65} -2008.97 q^{67} +312.992i q^{68} +1476.11 q^{70} +4306.53i q^{71} +6314.60 q^{73} +2297.21i q^{74} -3660.72 q^{76} +3120.80i q^{77} -3703.76 q^{79} +1803.46i q^{80} +2086.49 q^{82} +9033.39i q^{83} +1102.48 q^{85} -5917.06i q^{86} -3812.88 q^{88} -3711.45i q^{89} -5915.14 q^{91} -3480.84i q^{92} -8378.85 q^{94} +12894.5i q^{95} +13748.8 q^{97} +970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100}) $ 8 * q - 64 * q^4 - 32 * q^10 + 296 * q^13 + 512 * q^16 - 80 * q^19 + 800 * q^22 - 1464 * q^25 - 1024 * q^31 + 896 * q^34 + 904 * q^37 + 256 * q^40 - 3224 * q^43 - 4384 * q^46 + 2744 * q^49 - 2368 * q^52 + 8312 * q^55 + 2944 * q^58 + 128 * q^61 - 4096 * q^64 - 10152 * q^67 + 1568 * q^70 + 10632 * q^73 + 640 * q^76 - 4072 * q^79 - 3552 * q^82 + 13448 * q^85 - 6400 * q^88 - 6664 * q^91 - 384 * q^94 + 35616 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	28.1791i	1.12717i	0.826060	+	0.563583i	$0.190577\pi$
$5$	28.1791i	1.12717i	−0.826060	+	0.563583i	$0.809423\pi$
$6$	0	0
$7$	−18.5203	−0.377964
$7$	−18.5203	−0.377964
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−79.7026	−0.797026
$11$	− 168.507i	− 1.39262i	−0.717740	−	0.696311i	$-0.754823\pi$
$11$	− 168.507i	− 1.39262i	0.717740	−	0.696311i	$-0.245177\pi$
$12$	0	0
$13$	319.388	1.88987	0.944934	−	0.327261i	$-0.106126\pi$
$13$	319.388	1.88987	0.944934	+	0.327261i	$0.106126\pi$
$14$	− 52.3832i	− 0.267261i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 39.1240i	− 0.135377i	−0.997706	−	0.0676886i	$-0.978438\pi$
$17$	− 39.1240i	− 0.135377i	0.997706	−	0.0676886i	$-0.0215624\pi$
$18$	0	0
$19$	457.590	1.26756	0.633782	−	0.773512i	$-0.281502\pi$
$19$	457.590	1.26756	0.633782	+	0.773512i	$0.281502\pi$
$20$	− 225.433i	− 0.563583i
$21$	0	0
$22$	476.611	0.984733
$23$	435.106i	0.822506i	0.911521	+	0.411253i	$0.134909\pi$
$23$	435.106i	0.822506i	−0.911521	+	0.411253i	$0.865091\pi$
$24$	0	0
$25$	−169.064	−0.270502
$26$	903.365i	1.33634i
$27$	0	0
$28$	148.162	0.188982
$29$	− 135.719i	− 0.161378i	−0.996739	−	0.0806890i	$-0.974288\pi$
$29$	− 135.719i	− 0.161378i	0.996739	−	0.0806890i	$-0.0257121\pi$
$30$	0	0
$31$	−267.371	−0.278222	−0.139111	−	0.990277i	$-0.544424\pi$
$31$	−267.371	−0.278222	−0.139111	+	0.990277i	$0.544424\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	110.659	0.0957261
$35$	− 521.885i	− 0.426028i
$36$	0	0
$37$	812.186	0.593270	0.296635	−	0.954991i	$-0.404136\pi$
$37$	812.186	0.593270	0.296635	+	0.954991i	$0.404136\pi$
$38$	1294.26i	0.896303i
$39$	0	0
$40$	637.621	0.398513
$41$	− 737.686i	− 0.438837i	−0.975631	−	0.219419i	$-0.929584\pi$
$41$	− 737.686i	− 0.438837i	0.975631	−	0.219419i	$-0.0704161\pi$
$42$	0	0
$43$	−2092.00	−1.13142	−0.565710	−	0.824604i	$-0.691398\pi$
$43$	−2092.00	−1.13142	−0.565710	+	0.824604i	$0.691398\pi$
$44$	1348.06i	0.696311i
$45$	0	0
$46$	−1230.66	−0.581599
$47$	2962.37i	1.34105i	0.741889	+	0.670523i	$0.233930\pi$
$47$	2962.37i	1.34105i	−0.741889	+	0.670523i	$0.766070\pi$
$48$	0	0
$49$	343.000	0.142857
$50$	− 478.184i	− 0.191274i
$51$	0	0
$52$	−2555.10	−0.944934
$53$	− 3190.68i	− 1.13588i	−0.823071	−	0.567939i	$-0.807741\pi$
$53$	− 3190.68i	− 1.13588i	0.823071	−	0.567939i	$-0.192259\pi$
$54$	0	0
$55$	4748.39	1.56972
$56$	419.066i	0.133631i
$57$	0	0
$58$	383.871	0.114111
$59$	5620.52i	1.61463i	0.590122	+	0.807314i	$0.299080\pi$
$59$	5620.52i	1.61463i	−0.590122	+	0.807314i	$0.700920\pi$
$60$	0	0
$61$	4862.90	1.30688	0.653439	−	0.756979i	$-0.273325\pi$
$61$	4862.90	1.30688	0.653439	+	0.756979i	$0.273325\pi$
$62$	− 756.240i	− 0.196733i
$63$	0	0
$64$	−512.000	−0.125000
$65$	9000.07i	2.13019i
$66$	0	0
$67$	−2008.97	−0.447533	−0.223766	−	0.974643i	$-0.571835\pi$
$67$	−2008.97	−0.447533	−0.223766	+	0.974643i	$0.571835\pi$
$68$	312.992i	0.0676886i
$69$	0	0
$70$	1476.11	0.301248
$71$	4306.53i	0.854302i	0.904180	+	0.427151i	$0.140483\pi$
$71$	4306.53i	0.854302i	−0.904180	+	0.427151i	$0.859517\pi$
$72$	0	0
$73$	6314.60	1.18495	0.592475	−	0.805589i	$-0.298151\pi$
$73$	6314.60	1.18495	0.592475	+	0.805589i	$0.298151\pi$
$74$	2297.21i	0.419505i
$75$	0	0
$76$	−3660.72	−0.633782
$77$	3120.80i	0.526362i
$78$	0	0
$79$	−3703.76	−0.593457	−0.296728	−	0.954962i	$-0.595896\pi$
$79$	−3703.76	−0.593457	−0.296728	+	0.954962i	$0.595896\pi$
$80$	1803.46i	0.281791i
$81$	0	0
$82$	2086.49	0.310305
$83$	9033.39i	1.31128i	0.755075	+	0.655639i	$0.227601\pi$
$83$	9033.39i	1.31128i	−0.755075	+	0.655639i	$0.772399\pi$
$84$	0	0
$85$	1102.48	0.152592
$86$	− 5917.06i	− 0.800035i
$87$	0	0
$88$	−3812.88	−0.492366
$89$	− 3711.45i	− 0.468559i	−0.972169	−	0.234279i	$-0.924727\pi$
$89$	− 3711.45i	− 0.468559i	0.972169	−	0.234279i	$-0.0752730\pi$
$90$	0	0
$91$	−5915.14	−0.714303
$92$	− 3480.84i	− 0.411253i
$93$	0	0
$94$	−8378.85	−0.948263
$95$	12894.5i	1.42875i
$96$	0	0
$97$	13748.8	1.46124	0.730620	−	0.682784i	$-0.239231\pi$
$97$	13748.8	1.46124	0.730620	+	0.682784i	$0.239231\pi$
$98$	970.151i	0.101015i
$99$	0	0
$100$	1352.51	0.135251
$101$	− 17899.9i	− 1.75472i	−0.479835	−	0.877359i	$-0.659303\pi$
$101$	− 17899.9i	− 1.75472i	0.479835	−	0.877359i	$-0.340697\pi$
$102$	0	0
$103$	3979.89	0.375143	0.187571	−	0.982251i	$-0.439938\pi$
$103$	3979.89	0.375143	0.187571	+	0.982251i	$0.439938\pi$
$104$	− 7226.92i	− 0.668169i
$105$	0	0
$106$	9024.61	0.803187
$107$	8914.65i	0.778640i	0.921103	+	0.389320i	$0.127290\pi$
$107$	8914.65i	0.778640i	−0.921103	+	0.389320i	$0.872710\pi$
$108$	0	0
$109$	−13486.3	−1.13511	−0.567557	−	0.823334i	$-0.692111\pi$
$109$	−13486.3	−1.13511	−0.567557	+	0.823334i	$0.692111\pi$
$110$	13430.5i	1.10996i
$111$	0	0
$112$	−1185.30	−0.0944911
$113$	4042.36i	0.316576i	0.987393	+	0.158288i	$0.0505974\pi$
$113$	4042.36i	0.316576i	−0.987393	+	0.158288i	$0.949403\pi$
$114$	0	0
$115$	−12260.9	−0.927100
$116$	1085.75i	0.0806890i
$117$	0	0
$118$	−15897.2	−1.14171
$119$	724.586i	0.0511677i
$120$	0	0
$121$	−13753.7	−0.939397
$122$	13754.3i	0.924103i
$123$	0	0
$124$	2138.97	0.139111
$125$	12847.9i	0.822265i
$126$	0	0
$127$	18650.2	1.15631	0.578157	−	0.815926i	$-0.303772\pi$
$127$	18650.2	1.15631	0.578157	+	0.815926i	$0.303772\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	−25456.0	−1.50627
$131$	14164.3i	0.825375i	0.910873	+	0.412688i	$0.135410\pi$
$131$	14164.3i	0.825375i	−0.910873	+	0.412688i	$0.864590\pi$
$132$	0	0
$133$	−8474.69	−0.479094
$134$	− 5682.24i	− 0.316453i
$135$	0	0
$136$	−885.275	−0.0478630
$137$	27896.5i	1.48631i	0.669120	+	0.743154i	$0.266671\pi$
$137$	27896.5i	1.48631i	−0.669120	+	0.743154i	$0.733329\pi$
$138$	0	0
$139$	6374.52	0.329927	0.164963	−	0.986300i	$-0.447249\pi$
$139$	6374.52	0.329927	0.164963	+	0.986300i	$0.447249\pi$
$140$	4175.08i	0.213014i
$141$	0	0
$142$	−12180.7	−0.604082
$143$	− 53819.2i	− 2.63187i
$144$	0	0
$145$	3824.44	0.181900
$146$	17860.4i	0.837887i
$147$	0	0
$148$	−6497.49	−0.296635
$149$	− 32359.8i	− 1.45758i	−0.684736	−	0.728791i	$-0.740082\pi$
$149$	− 32359.8i	− 1.45758i	0.684736	−	0.728791i	$-0.259918\pi$
$150$	0	0
$151$	19569.8	0.858288	0.429144	−	0.903236i	$-0.358815\pi$
$151$	19569.8	0.858288	0.429144	+	0.903236i	$0.358815\pi$
$152$	− 10354.1i	− 0.448151i
$153$	0	0
$154$	−8826.95	−0.372194
$155$	− 7534.29i	− 0.313602i
$156$	0	0
$157$	−29337.2	−1.19020	−0.595100	−	0.803652i	$-0.702888\pi$
$157$	−29337.2	−1.19020	−0.595100	+	0.803652i	$0.702888\pi$
$158$	− 10475.8i	− 0.419637i
$159$	0	0
$160$	−5100.97	−0.199257
$161$	− 8058.27i	− 0.310878i
$162$	0	0
$163$	39562.6	1.48905	0.744526	−	0.667593i	$-0.232675\pi$
$163$	39562.6	1.48905	0.744526	+	0.667593i	$0.232675\pi$
$164$	5901.49i	0.219419i
$165$	0	0
$166$	−25550.3	−0.927214
$167$	26576.6i	0.952943i	0.879190	+	0.476471i	$0.158084\pi$
$167$	26576.6i	0.952943i	−0.879190	+	0.476471i	$0.841916\pi$
$168$	0	0
$169$	73447.5	2.57160
$170$	3118.28i	0.107899i
$171$	0	0
$172$	16736.0	0.565710
$173$	2650.12i	0.0885468i	0.999019	+	0.0442734i	$0.0140973\pi$
$173$	2650.12i	0.0885468i	−0.999019	+	0.0442734i	$0.985903\pi$
$174$	0	0
$175$	3131.10	0.102240
$176$	− 10784.5i	− 0.348156i
$177$	0	0
$178$	10497.6	0.331321
$179$	− 28123.7i	− 0.877741i	−0.898550	−	0.438871i	$-0.855379\pi$
$179$	− 28123.7i	− 0.877741i	0.898550	−	0.438871i	$-0.144621\pi$
$180$	0	0
$181$	6956.01	0.212326	0.106163	−	0.994349i	$-0.466143\pi$
$181$	6956.01	0.212326	0.106163	+	0.994349i	$0.466143\pi$
$182$	− 16730.6i	− 0.505089i
$183$	0	0
$184$	9845.31	0.290800
$185$	22886.7i	0.668713i
$186$	0	0
$187$	−6592.68	−0.188529
$188$	− 23699.0i	− 0.670523i
$189$	0	0
$190$	−36471.1	−1.01028
$191$	35193.7i	0.964713i	0.875975	+	0.482356i	$0.160219\pi$
$191$	35193.7i	0.964713i	−0.875975	+	0.482356i	$0.839781\pi$
$192$	0	0
$193$	37582.9	1.00897	0.504483	−	0.863422i	$-0.331683\pi$
$193$	37582.9	1.00897	0.504483	+	0.863422i	$0.331683\pi$
$194$	38887.5i	1.03325i
$195$	0	0
$196$	−2744.00	−0.0714286
$197$	− 40590.7i	− 1.04591i	−0.852361	−	0.522954i	$-0.824830\pi$
$197$	− 40590.7i	− 1.04591i	0.852361	−	0.522954i	$-0.175170\pi$
$198$	0	0
$199$	71304.5	1.80057	0.900286	−	0.435298i	$-0.143357\pi$
$199$	71304.5	1.80057	0.900286	+	0.435298i	$0.143357\pi$
$200$	3825.47i	0.0956368i
$201$	0	0
$202$	50628.5	1.24077
$203$	2513.55i	0.0609951i
$204$	0	0
$205$	20787.3	0.494642
$206$	11256.8i	0.265266i
$207$	0	0
$208$	20440.8	0.472467
$209$	− 77107.3i	− 1.76524i
$210$	0	0
$211$	−58607.1	−1.31639	−0.658196	−	0.752847i	$-0.728680\pi$
$211$	−58607.1	−1.31639	−0.658196	+	0.752847i	$0.728680\pi$
$212$	25525.5i	0.567939i
$213$	0	0
$214$	−25214.4	−0.550582
$215$	− 58950.6i	− 1.27530i
$216$	0	0
$217$	4951.79	0.105158
$218$	− 38145.0i	− 0.802646i
$219$	0	0
$220$	−37987.1	−0.784858
$221$	− 12495.7i	− 0.255845i
$222$	0	0
$223$	−31333.8	−0.630091	−0.315045	−	0.949077i	$-0.602020\pi$
$223$	−31333.8	−0.630091	−0.315045	+	0.949077i	$0.602020\pi$
$224$	− 3352.53i	− 0.0668153i
$225$	0	0
$226$	−11433.5	−0.223853
$227$	− 3437.19i	− 0.0667040i	−0.999444	−	0.0333520i	$-0.989382\pi$
$227$	− 3437.19i	− 0.0667040i	0.999444	−	0.0333520i	$-0.0106182\pi$
$228$	0	0
$229$	67814.1	1.29315	0.646576	−	0.762850i	$-0.276201\pi$
$229$	67814.1	1.29315	0.646576	+	0.762850i	$0.276201\pi$
$230$	− 34679.1i	− 0.655559i
$231$	0	0
$232$	−3070.97	−0.0570557
$233$	− 62389.0i	− 1.14920i	−0.818434	−	0.574601i	$-0.805157\pi$
$233$	− 62389.0i	− 1.14920i	0.818434	−	0.574601i	$-0.194843\pi$
$234$	0	0
$235$	−83477.0	−1.51158
$236$	− 44964.2i	− 0.807314i
$237$	0	0
$238$	−2049.44	−0.0361811
$239$	− 93904.5i	− 1.64396i	−0.569518	−	0.821979i	$-0.692870\pi$
$239$	− 93904.5i	− 1.64396i	0.569518	−	0.821979i	$-0.307130\pi$
$240$	0	0
$241$	−107318.	−1.84773	−0.923864	−	0.382722i	$-0.874987\pi$
$241$	−107318.	−1.84773	−0.923864	+	0.382722i	$0.874987\pi$
$242$	− 38901.4i	− 0.664254i
$243$	0	0
$244$	−38903.2	−0.653439
$245$	9665.44i	0.161024i
$246$	0	0
$247$	146149.	2.39553
$248$	6049.92i	0.0983663i
$249$	0	0
$250$	−36339.3	−0.581429
$251$	− 56349.2i	− 0.894417i	−0.894430	−	0.447209i	$-0.852418\pi$
$251$	− 56349.2i	− 0.894417i	0.894430	−	0.447209i	$-0.147582\pi$
$252$	0	0
$253$	73318.5	1.14544
$254$	52750.7i	0.817638i
$255$	0	0
$256$	4096.00	0.0625000
$257$	34716.3i	0.525614i	0.964848	+	0.262807i	$0.0846482\pi$
$257$	34716.3i	0.525614i	−0.964848	+	0.262807i	$0.915352\pi$
$258$	0	0
$259$	−15041.9	−0.224235
$260$	− 72000.6i	− 1.06510i
$261$	0	0
$262$	−40062.6	−0.583628
$263$	104358.i	1.50874i	0.656449	+	0.754371i	$0.272058\pi$
$263$	104358.i	1.50874i	−0.656449	+	0.754371i	$0.727942\pi$
$264$	0	0
$265$	89910.7	1.28032
$266$	− 23970.0i	− 0.338771i
$267$	0	0
$268$	16071.8	0.223766
$269$	− 81356.1i	− 1.12431i	−0.827032	−	0.562155i	$-0.809973\pi$
$269$	− 81356.1i	− 1.12431i	0.827032	−	0.562155i	$-0.190027\pi$
$270$	0	0
$271$	−125208.	−1.70488	−0.852439	−	0.522827i	$-0.824878\pi$
$271$	−125208.	−1.70488	−0.852439	+	0.522827i	$0.824878\pi$
$272$	− 2503.94i	− 0.0338443i
$273$	0	0
$274$	−78903.3	−1.05098
$275$	28488.4i	0.376707i
$276$	0	0
$277$	3328.93	0.0433855	0.0216928	−	0.999765i	$-0.493094\pi$
$277$	3328.93	0.0433855	0.0216928	+	0.999765i	$0.493094\pi$
$278$	18029.9i	0.233294i
$279$	0	0
$280$	−11808.9	−0.150624
$281$	151897.i	1.92370i	0.273584	+	0.961848i	$0.411791\pi$
$281$	151897.i	1.92370i	−0.273584	+	0.961848i	$0.588209\pi$
$282$	0	0
$283$	4034.61	0.0503765	0.0251883	−	0.999683i	$-0.491981\pi$
$283$	4034.61	0.0503765	0.0251883	+	0.999683i	$0.491981\pi$
$284$	− 34452.3i	− 0.427151i
$285$	0	0
$286$	152224.	1.86101
$287$	13662.1i	0.165865i
$288$	0	0
$289$	81990.3	0.981673
$290$	10817.1i	0.128622i
$291$	0	0
$292$	−50516.8	−0.592475
$293$	− 39529.1i	− 0.460449i	−0.973138	−	0.230225i	$-0.926054\pi$
$293$	− 39529.1i	− 0.460449i	0.973138	−	0.230225i	$-0.0739461\pi$
$294$	0	0
$295$	−158381.	−1.81995
$296$	− 18377.7i	− 0.209752i
$297$	0	0
$298$	91527.3	1.03067
$299$	138967.i	1.55443i
$300$	0	0
$301$	38744.3	0.427637
$302$	55351.8i	0.606902i
$303$	0	0
$304$	29285.8	0.316891
$305$	137032.i	1.47307i
$306$	0	0
$307$	−87173.0	−0.924922	−0.462461	−	0.886640i	$-0.653034\pi$
$307$	−87173.0	−0.924922	−0.462461	+	0.886640i	$0.653034\pi$
$308$	− 24966.4i	− 0.263181i
$309$	0	0
$310$	21310.2	0.221750
$311$	− 130723.i	− 1.35154i	−0.737110	−	0.675772i	$-0.763810\pi$
$311$	− 130723.i	− 1.35154i	0.737110	−	0.675772i	$-0.236190\pi$
$312$	0	0
$313$	−129850.	−1.32542	−0.662711	−	0.748876i	$-0.730594\pi$
$313$	−129850.	−1.32542	−0.662711	+	0.748876i	$0.730594\pi$
$314$	− 82978.2i	− 0.841598i
$315$	0	0
$316$	29630.1	0.296728
$317$	− 80859.0i	− 0.804656i	−0.915496	−	0.402328i	$-0.868201\pi$
$317$	− 80859.0i	− 0.804656i	0.915496	−	0.402328i	$-0.131799\pi$
$318$	0	0
$319$	−22869.6	−0.224739
$320$	− 14427.7i	− 0.140896i
$321$	0	0
$322$	22792.2	0.219824
$323$	− 17902.8i	− 0.171599i
$324$	0	0
$325$	−53996.8	−0.511213
$326$	111900.i	1.05292i
$327$	0	0
$328$	−16691.9	−0.155152
$329$	− 54863.9i	− 0.506868i
$330$	0	0
$331$	44166.4	0.403122	0.201561	−	0.979476i	$-0.435399\pi$
$331$	44166.4	0.403122	0.201561	+	0.979476i	$0.435399\pi$
$332$	− 72267.1i	− 0.655639i
$333$	0	0
$334$	−75170.0	−0.673832
$335$	− 56611.2i	− 0.504443i
$336$	0	0
$337$	−19903.7	−0.175256	−0.0876280	−	0.996153i	$-0.527929\pi$
$337$	−19903.7	−0.175256	−0.0876280	+	0.996153i	$0.527929\pi$
$338$	207741.i	1.81840i
$339$	0	0
$340$	−8819.84	−0.0762962
$341$	45054.0i	0.387458i
$342$	0	0
$343$	−6352.45	−0.0539949
$344$	47336.5i	0.400017i
$345$	0	0
$346$	−7495.66	−0.0626120
$347$	151074.i	1.25467i	0.778749	+	0.627335i	$0.215854\pi$
$347$	151074.i	1.25467i	−0.778749	+	0.627335i	$0.784146\pi$
$348$	0	0
$349$	−67141.3	−0.551237	−0.275619	−	0.961267i	$-0.588883\pi$
$349$	−67141.3	−0.551237	−0.275619	+	0.961267i	$0.588883\pi$
$350$	8856.09i	0.0722946i
$351$	0	0
$352$	30503.1	0.246183
$353$	− 48713.4i	− 0.390930i	−0.980711	−	0.195465i	$-0.937378\pi$
$353$	− 48713.4i	− 0.390930i	0.980711	−	0.195465i	$-0.0626216\pi$
$354$	0	0
$355$	−121354.	−0.962939
$356$	29691.6i	0.234279i
$357$	0	0
$358$	79545.9	0.620657
$359$	− 149704.i	− 1.16157i	−0.814057	−	0.580784i	$-0.802746\pi$
$359$	− 149704.i	− 1.16157i	0.814057	−	0.580784i	$-0.197254\pi$
$360$	0	0
$361$	79067.9	0.606716
$362$	19674.6i	0.150137i
$363$	0	0
$364$	47321.1	0.357152
$365$	177940.i	1.33564i
$366$	0	0
$367$	170863.	1.26857	0.634286	−	0.773098i	$-0.281294\pi$
$367$	170863.	1.26857	0.634286	+	0.773098i	$0.281294\pi$
$368$	27846.8i	0.205626i
$369$	0	0
$370$	−64733.4	−0.472851
$371$	59092.3i	0.429322i
$372$	0	0
$373$	75344.3	0.541542	0.270771	−	0.962644i	$-0.412721\pi$
$373$	75344.3	0.541542	0.270771	+	0.962644i	$0.412721\pi$
$374$	− 18646.9i	− 0.133310i
$375$	0	0
$376$	67030.8	0.474131
$377$	− 43346.9i	− 0.304983i
$378$	0	0
$379$	213599.	1.48703	0.743515	−	0.668719i	$-0.233157\pi$
$379$	213599.	1.48703	0.743515	+	0.668719i	$0.233157\pi$
$380$	− 103156.i	− 0.714377i
$381$	0	0
$382$	−99542.8	−0.682155
$383$	− 37077.1i	− 0.252760i	−0.991982	−	0.126380i	$-0.959664\pi$
$383$	− 37077.1i	− 0.252760i	0.991982	−	0.126380i	$-0.0403359\pi$
$384$	0	0
$385$	−87941.4	−0.593297
$386$	106301.i	0.713446i
$387$	0	0
$388$	−109991.	−0.730620
$389$	152168.i	1.00560i	0.864403	+	0.502799i	$0.167697\pi$
$389$	152168.i	1.00560i	−0.864403	+	0.502799i	$0.832303\pi$
$390$	0	0
$391$	17023.1	0.111348
$392$	− 7761.20i	− 0.0505076i
$393$	0	0
$394$	114808.	0.739569
$395$	− 104369.i	− 0.668924i
$396$	0	0
$397$	−273698.	−1.73656	−0.868281	−	0.496073i	$-0.834775\pi$
$397$	−273698.	−1.73656	−0.868281	+	0.496073i	$0.834775\pi$
$398$	201680.i	1.27320i
$399$	0	0
$400$	−10820.1	−0.0676254
$401$	206734.i	1.28565i	0.766014	+	0.642824i	$0.222237\pi$
$401$	206734.i	1.28565i	−0.766014	+	0.642824i	$0.777763\pi$
$402$	0	0
$403$	−85395.1	−0.525803
$404$	143199.i	0.877359i
$405$	0	0
$406$	−7109.39	−0.0431301
$407$	− 136859.i	− 0.826200i
$408$	0	0
$409$	−18110.7	−0.108265	−0.0541327	−	0.998534i	$-0.517239\pi$
$409$	−18110.7	−0.108265	−0.0541327	+	0.998534i	$0.517239\pi$
$410$	58795.5i	0.349765i
$411$	0	0
$412$	−31839.1	−0.187571
$413$	− 104094.i	− 0.610272i
$414$	0	0
$415$	−254553.	−1.47803
$416$	57815.4i	0.334085i
$417$	0	0
$418$	218092.	1.24821
$419$	31032.7i	0.176763i	0.996087	+	0.0883816i	$0.0281695\pi$
$419$	31032.7i	0.176763i	−0.996087	+	0.0883816i	$0.971830\pi$
$420$	0	0
$421$	104982.	0.592314	0.296157	−	0.955139i	$-0.404295\pi$
$421$	104982.	0.592314	0.296157	+	0.955139i	$0.404295\pi$
$422$	− 165766.i	− 0.930830i
$423$	0	0
$424$	−72196.9	−0.401594
$425$	6614.44i	0.0366197i
$426$	0	0
$427$	−90062.1	−0.493954
$428$	− 71317.2i	− 0.389320i
$429$	0	0
$430$	166738.	0.901771
$431$	279476.i	1.50449i	0.658882	+	0.752246i	$0.271029\pi$
$431$	279476.i	1.50449i	−0.658882	+	0.752246i	$0.728971\pi$
$432$	0	0
$433$	147044.	0.784280	0.392140	−	0.919905i	$-0.371735\pi$
$433$	147044.	0.784280	0.392140	+	0.919905i	$0.371735\pi$
$434$	14005.8i	0.0743580i
$435$	0	0
$436$	107890.	0.567557
$437$	199100.i	1.04258i
$438$	0	0
$439$	−191301.	−0.992631	−0.496316	−	0.868142i	$-0.665314\pi$
$439$	−191301.	−0.992631	−0.496316	+	0.868142i	$0.665314\pi$
$440$	− 107444.i	− 0.554978i
$441$	0	0
$442$	35343.2	0.180910
$443$	268170.i	1.36648i	0.730195	+	0.683239i	$0.239429\pi$
$443$	268170.i	1.36648i	−0.730195	+	0.683239i	$0.760571\pi$
$444$	0	0
$445$	104586.	0.528143
$446$	− 88625.3i	− 0.445541i
$447$	0	0
$448$	9482.37	0.0472456
$449$	71968.8i	0.356986i	0.983941	+	0.178493i	$0.0571222\pi$
$449$	71968.8i	0.356986i	−0.983941	+	0.178493i	$0.942878\pi$
$450$	0	0
$451$	−124305.	−0.611135
$452$	− 32338.9i	− 0.158288i
$453$	0	0
$454$	9721.85	0.0471669
$455$	− 166684.i	− 0.805138i
$456$	0	0
$457$	−261174.	−1.25054	−0.625270	−	0.780408i	$-0.715011\pi$
$457$	−261174.	−1.25054	−0.625270	+	0.780408i	$0.715011\pi$
$458$	191807.i	0.914396i
$459$	0	0
$460$	98087.2	0.463550
$461$	− 148535.i	− 0.698917i	−0.936952	−	0.349458i	$-0.886366\pi$
$461$	− 148535.i	− 0.698917i	0.936952	−	0.349458i	$-0.113634\pi$
$462$	0	0
$463$	−179727.	−0.838401	−0.419200	−	0.907894i	$-0.637689\pi$
$463$	−179727.	−0.838401	−0.419200	+	0.907894i	$0.637689\pi$
$464$	− 8686.01i	− 0.0403445i
$465$	0	0
$466$	176463.	0.812609
$467$	86089.6i	0.394745i	0.980329	+	0.197373i	$0.0632409\pi$
$467$	86089.6i	0.394745i	−0.980329	+	0.197373i	$0.936759\pi$
$468$	0	0
$469$	37206.7	0.169152
$470$	− 236109.i	− 1.06885i
$471$	0	0
$472$	127178.	0.570857
$473$	352517.i	1.57564i
$474$	0	0
$475$	−77361.8	−0.342878
$476$	− 5796.69i	− 0.0255839i
$477$	0	0
$478$	265602.	1.16245
$479$	100240.i	0.436890i	0.975849	+	0.218445i	$0.0700984\pi$
$479$	100240.i	0.436890i	−0.975849	+	0.218445i	$0.929902\pi$
$480$	0	0
$481$	259402.	1.12120
$482$	− 303541.i	− 1.30654i
$483$	0	0
$484$	110030.	0.469698
$485$	387430.i	1.64706i
$486$	0	0
$487$	−79918.4	−0.336968	−0.168484	−	0.985704i	$-0.553887\pi$
$487$	−79918.4	−0.336968	−0.168484	+	0.985704i	$0.553887\pi$
$488$	− 110035.i	− 0.462051i
$489$	0	0
$490$	−27338.0	−0.113861
$491$	− 376135.i	− 1.56020i	−0.625654	−	0.780101i	$-0.715168\pi$
$491$	− 376135.i	− 1.56020i	0.625654	−	0.780101i	$-0.284832\pi$
$492$	0	0
$493$	−5309.86	−0.0218469
$494$	413371.i	1.69389i
$495$	0	0
$496$	−17111.8	−0.0695555
$497$	− 79758.1i	− 0.322896i
$498$	0	0
$499$	104632.	0.420205	0.210103	−	0.977679i	$-0.432620\pi$
$499$	104632.	0.420205	0.210103	+	0.977679i	$0.432620\pi$
$500$	− 102783.i	− 0.411133i
$501$	0	0
$502$	159380.	0.632449
$503$	− 8060.64i	− 0.0318591i	−0.999873	−	0.0159295i	$-0.994929\pi$
$503$	− 8060.64i	− 0.0318591i	0.999873	−	0.0159295i	$-0.00507075\pi$
$504$	0	0
$505$	504403.	1.97786
$506$	207376.i	0.809948i
$507$	0	0
$508$	−149202.	−0.578157
$509$	383177.i	1.47898i	0.673165	+	0.739492i	$0.264934\pi$
$509$	383177.i	1.47898i	−0.673165	+	0.739492i	$0.735066\pi$
$510$	0	0
$511$	−116948.	−0.447869
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	−98192.4	−0.371665
$515$	112150.i	0.422848i
$516$	0	0
$517$	499181.	1.86757
$518$	− 42544.9i	− 0.158558i
$519$	0	0
$520$	203648.	0.753137
$521$	− 165297.i	− 0.608962i	−0.952518	−	0.304481i	$-0.901517\pi$
$521$	− 165297.i	− 0.608962i	0.952518	−	0.304481i	$-0.0984831\pi$
$522$	0	0
$523$	346312.	1.26609	0.633044	−	0.774116i	$-0.281805\pi$
$523$	346312.	1.26609	0.633044	+	0.774116i	$0.281805\pi$
$524$	− 113314.i	− 0.412688i
$525$	0	0
$526$	−295169.	−1.06684
$527$	10460.6i	0.0376649i
$528$	0	0
$529$	90524.2	0.323484
$530$	254306.i	0.905325i
$531$	0	0
$532$	67797.5	0.239547
$533$	− 235608.i	− 0.829345i
$534$	0	0
$535$	−251207.	−0.877656
$536$	45457.9i	0.158227i
$537$	0	0
$538$	230110.	0.795007
$539$	− 57798.0i	− 0.198946i
$540$	0	0
$541$	138669.	0.473787	0.236894	−	0.971536i	$-0.423871\pi$
$541$	138669.	0.473787	0.236894	+	0.971536i	$0.423871\pi$
$542$	− 354142.i	− 1.20553i
$543$	0	0
$544$	7082.20	0.0239315
$545$	− 380032.i	− 1.27946i
$546$	0	0
$547$	−284471.	−0.950742	−0.475371	−	0.879785i	$-0.657686\pi$
$547$	−284471.	−0.950742	−0.475371	+	0.879785i	$0.657686\pi$
$548$	− 223172.i	− 0.743154i
$549$	0	0
$550$	−80577.5	−0.266372
$551$	− 62103.6i	− 0.204557i
$552$	0	0
$553$	68594.6	0.224305
$554$	9415.63i	0.0306782i
$555$	0	0
$556$	−50996.2	−0.164963
$557$	− 37910.6i	− 0.122194i	−0.998132	−	0.0610971i	$-0.980540\pi$
$557$	− 37910.6i	− 0.122194i	0.998132	−	0.0610971i	$-0.0194599\pi$
$558$	0	0
$559$	−668158.	−2.13823
$560$	− 33400.6i	− 0.106507i
$561$	0	0
$562$	−429629.	−1.36026
$563$	164337.i	0.518465i	0.965815	+	0.259233i	$0.0834696\pi$
$563$	164337.i	0.518465i	−0.965815	+	0.259233i	$0.916530\pi$
$564$	0	0
$565$	−113910.	−0.356833
$566$	11411.6i	0.0356216i
$567$	0	0
$568$	97445.7	0.302041
$569$	− 394027.i	− 1.21703i	−0.793542	−	0.608516i	$-0.791765\pi$
$569$	− 394027.i	− 1.21703i	0.793542	−	0.608516i	$-0.208235\pi$
$570$	0	0
$571$	−269585.	−0.826844	−0.413422	−	0.910540i	$-0.635667\pi$
$571$	−269585.	−0.826844	−0.413422	+	0.910540i	$0.635667\pi$
$572$	430553.i	1.31594i
$573$	0	0
$574$	−38642.3	−0.117284
$575$	− 73560.5i	− 0.222489i
$576$	0	0
$577$	11879.8	0.0356826	0.0178413	−	0.999841i	$-0.494321\pi$
$577$	11879.8	0.0356826	0.0178413	+	0.999841i	$0.494321\pi$
$578$	231904.i	0.694148i
$579$	0	0
$580$	−30595.5	−0.0909498
$581$	− 167301.i	− 0.495616i
$582$	0	0
$583$	−537653.	−1.58185
$584$	− 142883.i	− 0.418943i
$585$	0	0
$586$	111805.	0.325587
$587$	− 433502.i	− 1.25810i	−0.777365	−	0.629050i	$-0.783444\pi$
$587$	− 433502.i	− 1.25810i	0.777365	−	0.629050i	$-0.216556\pi$
$588$	0	0
$589$	−122347.	−0.352664
$590$	− 447970.i	− 1.28690i
$591$	0	0
$592$	51979.9	0.148317
$593$	− 65763.0i	− 0.187013i	−0.995619	−	0.0935066i	$-0.970192\pi$
$593$	− 65763.0i	− 0.187013i	0.995619	−	0.0935066i	$-0.0298076\pi$
$594$	0	0
$595$	−20418.2	−0.0576745
$596$	258878.i	0.728791i
$597$	0	0
$598$	−393059.	−1.09915
$599$	− 232771.i	− 0.648746i	−0.945929	−	0.324373i	$-0.894847\pi$
$599$	− 232771.i	− 0.648746i	0.945929	−	0.324373i	$-0.105153\pi$
$600$	0	0
$601$	−98549.6	−0.272839	−0.136419	−	0.990651i	$-0.543559\pi$
$601$	−98549.6	−0.272839	−0.136419	+	0.990651i	$0.543559\pi$
$602$	109585.i	0.302385i
$603$	0	0
$604$	−156559.	−0.429144
$605$	− 387568.i	− 1.05886i
$606$	0	0
$607$	58943.3	0.159977	0.0799884	−	0.996796i	$-0.474512\pi$
$607$	58943.3	0.159977	0.0799884	+	0.996796i	$0.474512\pi$
$608$	82832.7i	0.224076i
$609$	0	0
$610$	−387586.	−1.04162
$611$	946145.i	2.53440i
$612$	0	0
$613$	127556.	0.339453	0.169727	−	0.985491i	$-0.445712\pi$
$613$	127556.	0.339453	0.169727	+	0.985491i	$0.445712\pi$
$614$	− 246562.i	− 0.654019i
$615$	0	0
$616$	70615.6	0.186097
$617$	− 183225.i	− 0.481299i	−0.970612	−	0.240649i	$-0.922640\pi$
$617$	− 183225.i	− 0.481299i	0.970612	−	0.240649i	$-0.0773604\pi$
$618$	0	0
$619$	−171376.	−0.447268	−0.223634	−	0.974673i	$-0.571792\pi$
$619$	−171376.	−0.447268	−0.223634	+	0.974673i	$0.571792\pi$
$620$	60274.3i	0.156801i
$621$	0	0
$622$	369740.	0.955686
$623$	68737.1i	0.177099i
$624$	0	0
$625$	−467707.	−1.19733
$626$	− 367272.i	− 0.937214i
$627$	0	0
$628$	234698.	0.595100
$629$	− 31776.0i	− 0.0803151i
$630$	0	0
$631$	−195310.	−0.490530	−0.245265	−	0.969456i	$-0.578875\pi$
$631$	−195310.	−0.490530	−0.245265	+	0.969456i	$0.578875\pi$
$632$	83806.6i	0.209819i
$633$	0	0
$634$	228704.	0.568977
$635$	525546.i	1.30336i
$636$	0	0
$637$	109550.	0.269981
$638$	− 64685.0i	− 0.158914i
$639$	0	0
$640$	40807.7	0.0996283
$641$	− 10697.3i	− 0.0260351i	−0.999915	−	0.0130175i	$-0.995856\pi$
$641$	− 10697.3i	− 0.0260351i	0.999915	−	0.0130175i	$-0.00414373\pi$
$642$	0	0
$643$	−171673.	−0.415221	−0.207611	−	0.978212i	$-0.566569\pi$
$643$	−171673.	−0.415221	−0.207611	+	0.978212i	$0.566569\pi$
$644$	64466.1i	0.155439i
$645$	0	0
$646$	50636.6	0.121339
$647$	− 814856.i	− 1.94658i	−0.229580	−	0.973290i	$-0.573735\pi$
$647$	− 814856.i	− 1.94658i	0.229580	−	0.973290i	$-0.426265\pi$
$648$	0	0
$649$	947099.	2.24857
$650$	− 152726.i	− 0.361482i
$651$	0	0
$652$	−316501.	−0.744526
$653$	− 741201.i	− 1.73824i	−0.494601	−	0.869120i	$-0.664686\pi$
$653$	− 741201.i	− 1.73824i	0.494601	−	0.869120i	$-0.335314\pi$
$654$	0	0
$655$	−399137.	−0.930334
$656$	− 47211.9i	− 0.109709i
$657$	0	0
$658$	155178.	0.358410
$659$	− 548473.i	− 1.26294i	−0.775398	−	0.631472i	$-0.782451\pi$
$659$	− 548473.i	− 1.26294i	0.775398	−	0.631472i	$-0.217549\pi$
$660$	0	0
$661$	426168.	0.975389	0.487695	−	0.873014i	$-0.337838\pi$
$661$	426168.	0.975389	0.487695	+	0.873014i	$0.337838\pi$
$662$	124922.i	0.285050i
$663$	0	0
$664$	204402.	0.463607
$665$	− 238809.i	− 0.540018i
$666$	0	0
$667$	59052.0	0.132734
$668$	− 212613.i	− 0.476471i
$669$	0	0
$670$	160121.	0.356695
$671$	− 819433.i	− 1.81999i
$672$	0	0
$673$	332282.	0.733630	0.366815	−	0.930294i	$-0.380448\pi$
$673$	332282.	0.733630	0.366815	+	0.930294i	$0.380448\pi$
$674$	− 56296.0i	− 0.123925i
$675$	0	0
$676$	−587580.	−1.28580
$677$	− 371163.i	− 0.809818i	−0.914357	−	0.404909i	$-0.867303\pi$
$677$	− 371163.i	− 0.809818i	0.914357	−	0.404909i	$-0.132697\pi$
$678$	0	0
$679$	−254632.	−0.552297
$680$	− 24946.3i	− 0.0539496i
$681$	0	0
$682$	−127432.	−0.273974
$683$	688730.i	1.47641i	0.674575	+	0.738206i	$0.264327\pi$
$683$	688730.i	1.47641i	−0.674575	+	0.738206i	$0.735673\pi$
$684$	0	0
$685$	−786100.	−1.67532
$686$	− 17967.4i	− 0.0381802i
$687$	0	0
$688$	−133888.	−0.282855
$689$	− 1.01906e6i	− 2.14666i
$690$	0	0
$691$	28665.8	0.0600354	0.0300177	−	0.999549i	$-0.490444\pi$
$691$	28665.8	0.0600354	0.0300177	+	0.999549i	$0.490444\pi$
$692$	− 21200.9i	− 0.0442734i
$693$	0	0
$694$	−427301.	−0.887186
$695$	179628.i	0.371882i
$696$	0	0
$697$	−28861.2	−0.0594085
$698$	− 189904.i	− 0.389784i
$699$	0	0
$700$	−25048.8	−0.0511200
$701$	347036.i	0.706218i	0.935582	+	0.353109i	$0.114876\pi$
$701$	347036.i	0.706218i	−0.935582	+	0.353109i	$0.885124\pi$
$702$	0	0
$703$	371648.	0.752007
$704$	86275.7i	0.174078i
$705$	0	0
$706$	137782.	0.276429
$707$	331510.i	0.663221i
$708$	0	0
$709$	117651.	0.234047	0.117024	−	0.993129i	$-0.462665\pi$
$709$	117651.	0.234047	0.117024	+	0.993129i	$0.462665\pi$
$710$	− 343242.i	− 0.680901i
$711$	0	0
$712$	−83980.6	−0.165661
$713$	− 116335.i	− 0.228839i
$714$	0	0
$715$	1.51658e6	2.96656
$716$	224990.i	0.438871i
$717$	0	0
$718$	423427.	0.821353
$719$	376532.i	0.728356i	0.931329	+	0.364178i	$0.118650\pi$
$719$	376532.i	0.728356i	−0.931329	+	0.364178i	$0.881350\pi$
$720$	0	0
$721$	−73708.6	−0.141791
$722$	223638.i	0.429013i
$723$	0	0
$724$	−55648.0	−0.106163
$725$	22945.1i	0.0436530i
$726$	0	0
$727$	618700.	1.17061	0.585304	−	0.810814i	$-0.300975\pi$
$727$	618700.	1.17061	0.585304	+	0.810814i	$0.300975\pi$
$728$	133844.i	0.252544i
$729$	0	0
$730$	−503290.	−0.944437
$731$	81847.2i	0.153168i
$732$	0	0
$733$	−454163.	−0.845287	−0.422643	−	0.906296i	$-0.638898\pi$
$733$	−454163.	−0.845287	−0.422643	+	0.906296i	$0.638898\pi$
$734$	483273.i	0.897016i
$735$	0	0
$736$	−78762.5	−0.145400
$737$	338527.i	0.623244i
$738$	0	0
$739$	200302.	0.366771	0.183386	−	0.983041i	$-0.441294\pi$
$739$	200302.	0.366771	0.183386	+	0.983041i	$0.441294\pi$
$740$	− 183094.i	− 0.334356i
$741$	0	0
$742$	−167138.	−0.303576
$743$	− 854018.i	− 1.54700i	−0.633799	−	0.773498i	$-0.718505\pi$
$743$	− 854018.i	− 1.54700i	0.633799	−	0.773498i	$-0.281495\pi$
$744$	0	0
$745$	911871.	1.64294
$746$	213106.i	0.382928i
$747$	0	0
$748$	52741.4	0.0942646
$749$	− 165102.i	− 0.294298i
$750$	0	0
$751$	−692616.	−1.22804	−0.614020	−	0.789290i	$-0.710449\pi$
$751$	−692616.	−1.22804	−0.614020	+	0.789290i	$0.710449\pi$
$752$	189592.i	0.335261i
$753$	0	0
$754$	122604.	0.215656
$755$	551461.i	0.967433i
$756$	0	0
$757$	154839.	0.270202	0.135101	−	0.990832i	$-0.456864\pi$
$757$	154839.	0.270202	0.135101	+	0.990832i	$0.456864\pi$
$758$	604148.i	1.05149i
$759$	0	0
$760$	291769.	0.505141
$761$	913167.i	1.57682i	0.615153	+	0.788408i	$0.289094\pi$
$761$	913167.i	1.57682i	−0.615153	+	0.788408i	$0.710906\pi$
$762$	0	0
$763$	249769.	0.429032
$764$	− 281549.i	− 0.482356i
$765$	0	0
$766$	104870.	0.178728
$767$	1.79513e6i	3.05144i
$768$	0	0
$769$	−1.00329e6	−1.69657	−0.848286	−	0.529538i	$-0.822365\pi$
$769$	−1.00329e6	−1.69657	−0.848286	+	0.529538i	$0.822365\pi$
$770$	− 248736.i	− 0.419524i
$771$	0	0
$772$	−300664.	−0.504483
$773$	474439.i	0.794001i	0.917818	+	0.397001i	$0.129949\pi$
$773$	474439.i	0.794001i	−0.917818	+	0.397001i	$0.870051\pi$
$774$	0	0
$775$	45202.8	0.0752595
$776$	− 311100.i	− 0.516627i
$777$	0	0
$778$	−430397.	−0.711066
$779$	− 337558.i	− 0.556254i
$780$	0	0
$781$	725682.	1.18972
$782$	48148.5i	0.0787353i
$783$	0	0
$784$	21952.0	0.0357143
$785$	− 826698.i	− 1.34155i
$786$	0	0
$787$	590744.	0.953783	0.476892	−	0.878962i	$-0.341763\pi$
$787$	590744.	0.953783	0.476892	+	0.878962i	$0.341763\pi$
$788$	324725.i	0.522954i
$789$	0	0
$790$	295200.	0.473000
$791$	− 74865.5i	− 0.119654i
$792$	0	0
$793$	1.55315e6	2.46983
$794$	− 774134.i	− 1.22793i
$795$	0	0
$796$	−570436.	−0.900286
$797$	− 600262.i	− 0.944984i	−0.881335	−	0.472492i	$-0.843355\pi$
$797$	− 600262.i	− 0.944984i	0.881335	−	0.472492i	$-0.156645\pi$
$798$	0	0
$799$	115900.	0.181547
$800$	− 30603.8i	− 0.0478184i
$801$	0	0
$802$	−584731.	−0.909091
$803$	− 1.06406e6i	− 1.65019i
$804$	0	0
$805$	227075.	0.350411
$806$	− 241534.i	− 0.371799i
$807$	0	0
$808$	−405028.	−0.620386
$809$	− 649360.i	− 0.992176i	−0.868272	−	0.496088i	$-0.834769\pi$
$809$	− 649360.i	− 0.992176i	0.868272	−	0.496088i	$-0.165231\pi$
$810$	0	0
$811$	−1.13293e6	−1.72250	−0.861251	−	0.508179i	$-0.830319\pi$
$811$	−1.13293e6	−1.72250	−0.861251	+	0.508179i	$0.830319\pi$
$812$	− 20108.4i	− 0.0304976i
$813$	0	0
$814$	387096.	0.584212
$815$	1.11484e6i	1.67841i
$816$	0	0
$817$	−957277.	−1.43415
$818$	− 51224.9i	− 0.0765552i
$819$	0	0
$820$	−166299.	−0.247321
$821$	463203.i	0.687204i	0.939115	+	0.343602i	$0.111647\pi$
$821$	463203.i	0.687204i	−0.939115	+	0.343602i	$0.888353\pi$
$822$	0	0
$823$	1.00476e6	1.48341	0.741705	−	0.670726i	$-0.234017\pi$
$823$	1.00476e6	1.48341	0.741705	+	0.670726i	$0.234017\pi$
$824$	− 90054.6i	− 0.132633i
$825$	0	0
$826$	294421.	0.431528
$827$	57967.0i	0.0847559i	0.999102	+	0.0423779i	$0.0134934\pi$
$827$	57967.0i	0.0847559i	−0.999102	+	0.0423779i	$0.986507\pi$
$828$	0	0
$829$	1.08695e6	1.58162	0.790809	−	0.612063i	$-0.209660\pi$
$829$	1.08695e6	1.58162	0.790809	+	0.612063i	$0.209660\pi$
$830$	− 719985.i	− 1.04512i
$831$	0	0
$832$	−163527.	−0.236234
$833$	− 13419.5i	− 0.0193396i
$834$	0	0
$835$	−748906.	−1.07412
$836$	616858.i	0.882618i
$837$	0	0
$838$	−87773.8	−0.124990
$839$	− 1.04936e6i	− 1.49073i	−0.666655	−	0.745367i	$-0.732274\pi$
$839$	− 1.04936e6i	− 1.49073i	0.666655	−	0.745367i	$-0.267726\pi$
$840$	0	0
$841$	688861.	0.973957
$842$	296935.i	0.418829i
$843$	0	0
$844$	468857.	0.658196
$845$	2.06969e6i	2.89862i
$846$	0	0
$847$	254722.	0.355059
$848$	− 204204.i	− 0.283970i
$849$	0	0
$850$	−18708.5	−0.0258941
$851$	353387.i	0.487968i
$852$	0	0
$853$	731878.	1.00587	0.502933	−	0.864325i	$-0.332254\pi$
$853$	731878.	1.00587	0.502933	+	0.864325i	$0.332254\pi$
$854$	− 254734.i	− 0.349278i
$855$	0	0
$856$	201715.	0.275291
$857$	704349.i	0.959017i	0.877537	+	0.479509i	$0.159185\pi$
$857$	704349.i	0.959017i	−0.877537	+	0.479509i	$0.840815\pi$
$858$	0	0
$859$	−1.16702e6	−1.58159	−0.790793	−	0.612084i	$-0.790331\pi$
$859$	−1.16702e6	−1.58159	−0.790793	+	0.612084i	$0.790331\pi$
$860$	471605.i	0.637649i
$861$	0	0
$862$	−790477.	−1.06384
$863$	126671.i	0.170081i	0.996377	+	0.0850405i	$0.0271020\pi$
$863$	126671.i	0.170081i	−0.996377	+	0.0850405i	$0.972898\pi$
$864$	0	0
$865$	−74678.0	−0.0998069
$866$	415903.i	0.554570i
$867$	0	0
$868$	−39614.3	−0.0525790
$869$	624111.i	0.826461i
$870$	0	0
$871$	−641642.	−0.845778
$872$	305160.i	0.401323i
$873$	0	0
$874$	−563140.	−0.737214
$875$	− 237946.i	− 0.310787i
$876$	0	0
$877$	−891593.	−1.15922	−0.579612	−	0.814892i	$-0.696796\pi$
$877$	−891593.	−1.15922	−0.579612	+	0.814892i	$0.696796\pi$
$878$	− 541081.i	− 0.701896i
$879$	0	0
$880$	303897.	0.392429
$881$	494760.i	0.637445i	0.947848	+	0.318723i	$0.103254\pi$
$881$	494760.i	0.637445i	−0.947848	+	0.318723i	$0.896746\pi$
$882$	0	0
$883$	−104454.	−0.133969	−0.0669847	−	0.997754i	$-0.521338\pi$
$883$	−104454.	−0.133969	−0.0669847	+	0.997754i	$0.521338\pi$
$884$	99965.8i	0.127922i
$885$	0	0
$886$	−758499.	−0.966246
$887$	− 176125.i	− 0.223859i	−0.993716	−	0.111929i	$-0.964297\pi$
$887$	− 176125.i	− 0.223859i	0.993716	−	0.111929i	$-0.0357031\pi$
$888$	0	0
$889$	−345406.	−0.437046
$890$	295813.i	0.373454i
$891$	0	0
$892$	250670.	0.315045
$893$	1.35555e6i	1.69986i
$894$	0	0
$895$	792502.	0.989360
$896$	26820.2i	0.0334077i
$897$	0	0
$898$	−203559.	−0.252427
$899$	36287.3i	0.0448989i
$900$	0	0
$901$	−124832.	−0.153772
$902$	− 351589.i	− 0.432138i
$903$	0	0
$904$	91468.1	0.111926
$905$	196014.i	0.239326i
$906$	0	0
$907$	734712.	0.893105	0.446553	−	0.894757i	$-0.352652\pi$
$907$	734712.	0.893105	0.446553	+	0.894757i	$0.352652\pi$
$908$	27497.5i	0.0333520i
$909$	0	0
$910$	471452.	0.569318
$911$	− 628633.i	− 0.757462i	−0.925507	−	0.378731i	$-0.876361\pi$
$911$	− 628633.i	− 0.757462i	0.925507	−	0.378731i	$-0.123639\pi$
$912$	0	0
$913$	1.52219e6	1.82611
$914$	− 738712.i	− 0.884266i
$915$	0	0
$916$	−542513.	−0.646576
$917$	− 262326.i	− 0.311963i
$918$	0	0
$919$	−1.34955e6	−1.59793	−0.798965	−	0.601377i	$-0.794619\pi$
$919$	−1.34955e6	−1.59793	−0.798965	+	0.601377i	$0.794619\pi$
$920$	277432.i	0.327779i
$921$	0	0
$922$	420119.	0.494209
$923$	1.37545e6i	1.61452i
$924$	0	0
$925$	−137311.	−0.160480
$926$	− 508345.i	− 0.592839i
$927$	0	0
$928$	24567.7	0.0285279
$929$	1.52629e6i	1.76851i	0.467007	+	0.884254i	$0.345332\pi$
$929$	1.52629e6i	1.76851i	−0.467007	+	0.884254i	$0.654668\pi$
$930$	0	0
$931$	156953.	0.181080
$932$	499112.i	0.574601i
$933$	0	0
$934$	−243498.	−0.279127
$935$	− 185776.i	− 0.212504i
$936$	0	0
$937$	542424.	0.617817	0.308908	−	0.951092i	$-0.400036\pi$
$937$	542424.	0.617817	0.308908	+	0.951092i	$0.400036\pi$
$938$	105237.i	0.119608i
$939$	0	0
$940$	667816.	0.755790
$941$	− 620071.i	− 0.700265i	−0.936700	−	0.350132i	$-0.886137\pi$
$941$	− 620071.i	− 0.700265i	0.936700	−	0.350132i	$-0.113863\pi$
$942$	0	0
$943$	320971.	0.360946
$944$	359713.i	0.403657i
$945$	0	0
$946$	−997067.	−1.11415
$947$	− 620123.i	− 0.691477i	−0.938331	−	0.345738i	$-0.887628\pi$
$947$	− 620123.i	− 0.691477i	0.938331	−	0.345738i	$-0.112372\pi$
$948$	0	0
$949$	2.01681e6	2.23940
$950$	− 218812.i	− 0.242451i
$951$	0	0
$952$	16395.5	0.0180905
$953$	− 568767.i	− 0.626251i	−0.949712	−	0.313126i	$-0.898624\pi$
$953$	− 568767.i	− 0.626251i	0.949712	−	0.313126i	$-0.101376\pi$
$954$	0	0
$955$	−991727.	−1.08739
$956$	751236.i	0.821979i
$957$	0	0
$958$	−283523.	−0.308928
$959$	− 516651.i	− 0.561772i
$960$	0	0
$961$	−852034.	−0.922593
$962$	733700.i	0.792809i
$963$	0	0
$964$	858543.	0.923864
$965$	1.05905e6i	1.13727i
$966$	0	0
$967$	747986.	0.799909	0.399954	−	0.916535i	$-0.369026\pi$
$967$	747986.	0.799909	0.399954	+	0.916535i	$0.369026\pi$
$968$	311211.i	0.332127i
$969$	0	0
$970$	−1.09582e6	−1.16465
$971$	− 1.50806e6i	− 1.59949i	−0.600343	−	0.799743i	$-0.704969\pi$
$971$	− 1.50806e6i	− 1.59949i	0.600343	−	0.799743i	$-0.295031\pi$
$972$	0	0
$973$	−118058.	−0.124701
$974$	− 226043.i	− 0.238272i
$975$	0	0
$976$	311225.	0.326720
$977$	− 1.24528e6i	− 1.30460i	−0.757960	−	0.652301i	$-0.773804\pi$
$977$	− 1.24528e6i	− 1.30460i	0.757960	−	0.652301i	$-0.226196\pi$
$978$	0	0
$979$	−625407.	−0.652525
$980$	− 77323.5i	− 0.0805118i
$981$	0	0
$982$	1.06387e6	1.10323
$983$	753418.i	0.779703i	0.920878	+	0.389851i	$0.127474\pi$
$983$	753418.i	0.779703i	−0.920878	+	0.389851i	$0.872526\pi$
$984$	0	0
$985$	1.14381e6	1.17891
$986$	− 15018.6i	− 0.0154481i
$987$	0	0
$988$	−1.16919e6	−1.19776
$989$	− 910239.i	− 0.930599i
$990$	0	0
$991$	−77288.4	−0.0786986	−0.0393493	−	0.999226i	$-0.512528\pi$
$991$	−77288.4	−0.0786986	−0.0393493	+	0.999226i	$0.512528\pi$
$992$	− 48399.4i	− 0.0491832i
$993$	0	0
$994$	225590.	0.228322
$995$	2.00930e6i	2.02954i
$996$	0	0
$997$	1.32860e6	1.33661	0.668305	−	0.743888i	$-0.267020\pi$
$997$	1.32860e6	1.33661	0.668305	+	0.743888i	$0.267020\pi$
$998$	295943.i	0.297130i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.5.b.b.323.7	yes	8
3.2	odd	2	inner	378.5.b.b.323.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.5.b.b.323.2	✓	8	3.2	odd	2	inner
378.5.b.b.323.7	yes	8	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	378.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +28.1791i q^{5} -18.5203 q^{7} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +28.1791i q^{5} -18.5203 q^{7} -22.6274i q^{8} -79.7026 q^{10} -168.507i q^{11} +319.388 q^{13} -52.3832i q^{14} +64.0000 q^{16} -39.1240i q^{17} +457.590 q^{19} -225.433i q^{20} +476.611 q^{22} +435.106i q^{23} -169.064 q^{25} +903.365i q^{26} +148.162 q^{28} -135.719i q^{29} -267.371 q^{31} +181.019i q^{32} +110.659 q^{34} -521.885i q^{35} +812.186 q^{37} +1294.26i q^{38} +637.621 q^{40} -737.686i q^{41} -2092.00 q^{43} +1348.06i q^{44} -1230.66 q^{46} +2962.37i q^{47} +343.000 q^{49} -478.184i q^{50} -2555.10 q^{52} -3190.68i q^{53} +4748.39 q^{55} +419.066i q^{56} +383.871 q^{58} +5620.52i q^{59} +4862.90 q^{61} -756.240i q^{62} -512.000 q^{64} +9000.07i q^{65} -2008.97 q^{67} +312.992i q^{68} +1476.11 q^{70} +4306.53i q^{71} +6314.60 q^{73} +2297.21i q^{74} -3660.72 q^{76} +3120.80i q^{77} -3703.76 q^{79} +1803.46i q^{80} +2086.49 q^{82} +9033.39i q^{83} +1102.48 q^{85} -5917.06i q^{86} -3812.88 q^{88} -3711.45i q^{89} -5915.14 q^{91} -3480.84i q^{92} -8378.85 q^{94} +12894.5i q^{95} +13748.8 q^{97} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100}) \) 8 * q - 64 * q^4 - 32 * q^10 + 296 * q^13 + 512 * q^16 - 80 * q^19 + 800 * q^22 - 1464 * q^25 - 1024 * q^31 + 896 * q^34 + 904 * q^37 + 256 * q^40 - 3224 * q^43 - 4384 * q^46 + 2744 * q^49 - 2368 * q^52 + 8312 * q^55 + 2944 * q^58 + 128 * q^61 - 4096 * q^64 - 10152 * q^67 + 1568 * q^70 + 10632 * q^73 + 640 * q^76 - 4072 * q^79 - 3552 * q^82 + 13448 * q^85 - 6400 * q^88 - 6664 * q^91 - 384 * q^94 + 35616 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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