LMFDB - Embedded newform 378.5.b.b.323.8

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			323.8
Root			$-4.82885i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.b.323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +30.1368i q^{5} +18.5203 q^{7} -22.6274i q^{8} +O(q^{10})$	\(q+2.82843i q^{2} -8.00000 q^{4} +30.1368i q^{5} +18.5203 q^{7} -22.6274i q^{8} -85.2398 q^{10} +11.8001i q^{11} -159.095 q^{13} +52.3832i q^{14} +64.0000 q^{16} -120.538i q^{17} -391.298 q^{19} -241.095i q^{20} -33.3758 q^{22} +371.311i q^{23} -283.229 q^{25} -449.990i q^{26} -148.162 q^{28} +216.816i q^{29} -930.283 q^{31} +181.019i q^{32} +340.933 q^{34} +558.142i q^{35} -672.478 q^{37} -1106.76i q^{38} +681.919 q^{40} -368.730i q^{41} +1278.43 q^{43} -94.4009i q^{44} -1050.23 q^{46} -1899.14i q^{47} +343.000 q^{49} -801.092i q^{50} +1272.76 q^{52} -1723.71i q^{53} -355.618 q^{55} -419.066i q^{56} -613.249 q^{58} -6318.71i q^{59} +2529.76 q^{61} -2631.24i q^{62} -512.000 q^{64} -4794.63i q^{65} -6171.39 q^{67} +964.305i q^{68} -1578.66 q^{70} -907.855i q^{71} -3649.04 q^{73} -1902.06i q^{74} +3130.38 q^{76} +218.541i q^{77} -6817.08 q^{79} +1928.76i q^{80} +1042.93 q^{82} -9100.53i q^{83} +3632.64 q^{85} +3615.96i q^{86} +267.006 q^{88} +8432.60i q^{89} -2946.49 q^{91} -2970.49i q^{92} +5371.59 q^{94} -11792.5i q^{95} -545.318 q^{97} +970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	30.1368i	1.20547i	0.797940	+	0.602737i	$0.205923\pi$
$5$	30.1368i	1.20547i	−0.797940	+	0.602737i	$0.794077\pi$
$6$	0	0
$7$	18.5203	0.377964
$7$	18.5203	0.377964
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−85.2398	−0.852398
$11$	11.8001i	0.0975216i	0.998810	+	0.0487608i	$0.0155272\pi$
$11$	11.8001i	0.0975216i	−0.998810	+	0.0487608i	$0.984473\pi$
$12$	0	0
$13$	−159.095	−0.941393	−0.470696	−	0.882295i	$-0.655997\pi$
$13$	−159.095	−0.941393	−0.470696	+	0.882295i	$0.655997\pi$
$14$	52.3832i	0.267261i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 120.538i	− 0.417087i	−0.978013	−	0.208543i	$-0.933128\pi$
$17$	− 120.538i	− 0.417087i	0.978013	−	0.208543i	$-0.0668723\pi$
$18$	0	0
$19$	−391.298	−1.08393	−0.541964	−	0.840402i	$-0.682319\pi$
$19$	−391.298	−1.08393	−0.541964	+	0.840402i	$0.682319\pi$
$20$	− 241.095i	− 0.602737i
$21$	0	0
$22$	−33.3758	−0.0689582
$23$	371.311i	0.701911i	0.936392	+	0.350955i	$0.114143\pi$
$23$	371.311i	0.701911i	−0.936392	+	0.350955i	$0.885857\pi$
$24$	0	0
$25$	−283.229	−0.453166
$26$	− 449.990i	− 0.665665i
$27$	0	0
$28$	−148.162	−0.188982
$29$	216.816i	0.257808i	0.991657	+	0.128904i	$0.0411459\pi$
$29$	216.816i	0.257808i	−0.991657	+	0.128904i	$0.958854\pi$
$30$	0	0
$31$	−930.283	−0.968037	−0.484018	−	0.875058i	$-0.660823\pi$
$31$	−930.283	−0.968037	−0.484018	+	0.875058i	$0.660823\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	340.933	0.294925
$35$	558.142i	0.455626i
$36$	0	0
$37$	−672.478	−0.491219	−0.245609	−	0.969369i	$-0.578988\pi$
$37$	−672.478	−0.491219	−0.245609	+	0.969369i	$0.578988\pi$
$38$	− 1106.76i	− 0.766453i
$39$	0	0
$40$	681.919	0.426199
$41$	− 368.730i	− 0.219351i	−0.993967	−	0.109676i	$-0.965019\pi$
$41$	− 368.730i	− 0.219351i	0.993967	−	0.109676i	$-0.0349812\pi$
$42$	0	0
$43$	1278.43	0.691419	0.345710	−	0.938342i	$-0.387638\pi$
$43$	1278.43	0.691419	0.345710	+	0.938342i	$0.387638\pi$
$44$	− 94.4009i	− 0.0487608i
$45$	0	0
$46$	−1050.23	−0.496326
$47$	− 1899.14i	− 0.859730i	−0.902893	−	0.429865i	$-0.858561\pi$
$47$	− 1899.14i	− 0.859730i	0.902893	−	0.429865i	$-0.141439\pi$
$48$	0	0
$49$	343.000	0.142857
$50$	− 801.092i	− 0.320437i
$51$	0	0
$52$	1272.76	0.470696
$53$	− 1723.71i	− 0.613638i	−0.951768	−	0.306819i	$-0.900735\pi$
$53$	− 1723.71i	− 0.613638i	0.951768	−	0.306819i	$-0.0992647\pi$
$54$	0	0
$55$	−355.618	−0.117560
$56$	− 419.066i	− 0.133631i
$57$	0	0
$58$	−613.249	−0.182298
$59$	− 6318.71i	− 1.81520i	−0.419837	−	0.907600i	$-0.637913\pi$
$59$	− 6318.71i	− 1.81520i	0.419837	−	0.907600i	$-0.362087\pi$
$60$	0	0
$61$	2529.76	0.679860	0.339930	−	0.940451i	$-0.389597\pi$
$61$	2529.76	0.679860	0.339930	+	0.940451i	$0.389597\pi$
$62$	− 2631.24i	− 0.684505i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 4794.63i	− 1.13482i
$66$	0	0
$67$	−6171.39	−1.37478	−0.687391	−	0.726288i	$-0.741244\pi$
$67$	−6171.39	−1.37478	−0.687391	+	0.726288i	$0.741244\pi$
$68$	964.305i	0.208543i
$69$	0	0
$70$	−1578.66	−0.322176
$71$	− 907.855i	− 0.180094i	−0.995938	−	0.0900471i	$-0.971298\pi$
$71$	− 907.855i	− 0.180094i	0.995938	−	0.0900471i	$-0.0287017\pi$
$72$	0	0
$73$	−3649.04	−0.684751	−0.342376	−	0.939563i	$-0.611232\pi$
$73$	−3649.04	−0.684751	−0.342376	+	0.939563i	$0.611232\pi$
$74$	− 1902.06i	− 0.347344i
$75$	0	0
$76$	3130.38	0.541964
$77$	218.541i	0.0368597i
$78$	0	0
$79$	−6817.08	−1.09231	−0.546153	−	0.837686i	$-0.683908\pi$
$79$	−6817.08	−1.09231	−0.546153	+	0.837686i	$0.683908\pi$
$80$	1928.76i	0.301368i
$81$	0	0
$82$	1042.93	0.155105
$83$	− 9100.53i	− 1.32102i	−0.750816	−	0.660512i	$-0.770339\pi$
$83$	− 9100.53i	− 1.32102i	0.750816	−	0.660512i	$-0.229661\pi$
$84$	0	0
$85$	3632.64	0.502787
$86$	3615.96i	0.488907i
$87$	0	0
$88$	267.006	0.0344791
$89$	8432.60i	1.06459i	0.846560	+	0.532294i	$0.178670\pi$
$89$	8432.60i	1.06459i	−0.846560	+	0.532294i	$0.821330\pi$
$90$	0	0
$91$	−2946.49	−0.355813
$92$	− 2970.49i	− 0.350955i
$93$	0	0
$94$	5371.59	0.607921
$95$	− 11792.5i	− 1.30665i
$96$	0	0
$97$	−545.318	−0.0579571	−0.0289785	−	0.999580i	$-0.509225\pi$
$97$	−545.318	−0.0579571	−0.0289785	+	0.999580i	$0.509225\pi$
$98$	970.151i	0.101015i
$99$	0	0
$100$	2265.83	0.226583
$101$	− 7298.60i	− 0.715479i	−0.933822	−	0.357739i	$-0.883548\pi$
$101$	− 7298.60i	− 0.715479i	0.933822	−	0.357739i	$-0.116452\pi$
$102$	0	0
$103$	12988.8	1.22432	0.612159	−	0.790735i	$-0.290301\pi$
$103$	12988.8	1.22432	0.612159	+	0.790735i	$0.290301\pi$
$104$	3599.92i	0.332833i
$105$	0	0
$106$	4875.39	0.433908
$107$	− 1506.48i	− 0.131582i	−0.997833	−	0.0657910i	$-0.979043\pi$
$107$	− 1506.48i	− 0.131582i	0.997833	−	0.0657910i	$-0.0209571\pi$
$108$	0	0
$109$	−11205.6	−0.943149	−0.471575	−	0.881826i	$-0.656314\pi$
$109$	−11205.6	−0.943149	−0.471575	+	0.881826i	$0.656314\pi$
$110$	− 1005.84i	− 0.0831273i
$111$	0	0
$112$	1185.30	0.0944911
$113$	− 15734.3i	− 1.23223i	−0.787658	−	0.616113i	$-0.788706\pi$
$113$	− 15734.3i	− 1.23223i	0.787658	−	0.616113i	$-0.211294\pi$
$114$	0	0
$115$	−11190.1	−0.846135
$116$	− 1734.53i	− 0.128904i
$117$	0	0
$118$	17872.0	1.28354
$119$	− 2232.40i	− 0.157644i
$120$	0	0
$121$	14501.8	0.990490
$122$	7155.24i	0.480733i
$123$	0	0
$124$	7442.27	0.484018
$125$	10299.9i	0.659194i
$126$	0	0
$127$	14352.6	0.889864	0.444932	−	0.895564i	$-0.353228\pi$
$127$	14352.6	0.889864	0.444932	+	0.895564i	$0.353228\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	13561.3	0.802442
$131$	− 15941.9i	− 0.928963i	−0.885583	−	0.464481i	$-0.846241\pi$
$131$	− 15941.9i	− 0.928963i	0.885583	−	0.464481i	$-0.153759\pi$
$132$	0	0
$133$	−7246.94	−0.409686
$134$	− 17455.3i	− 0.972117i
$135$	0	0
$136$	−2727.47	−0.147462
$137$	27139.8i	1.44599i	0.690852	+	0.722996i	$0.257236\pi$
$137$	27139.8i	1.44599i	−0.690852	+	0.722996i	$0.742764\pi$
$138$	0	0
$139$	20223.3	1.04670	0.523351	−	0.852117i	$-0.324681\pi$
$139$	20223.3	1.04670	0.523351	+	0.852117i	$0.324681\pi$
$140$	− 4465.14i	− 0.227813i
$141$	0	0
$142$	2567.80	0.127346
$143$	− 1877.34i	− 0.0918061i
$144$	0	0
$145$	−6534.16	−0.310780
$146$	− 10321.0i	− 0.484192i
$147$	0	0
$148$	5379.83	0.245609
$149$	6130.36i	0.276130i	0.990423	+	0.138065i	$0.0440882\pi$
$149$	6130.36i	0.276130i	−0.990423	+	0.138065i	$0.955912\pi$
$150$	0	0
$151$	−24047.1	−1.05465	−0.527325	−	0.849664i	$-0.676805\pi$
$151$	−24047.1	−1.05465	−0.527325	+	0.849664i	$0.676805\pi$
$152$	8854.06i	0.383226i
$153$	0	0
$154$	−618.128	−0.0260637
$155$	− 28035.8i	− 1.16694i
$156$	0	0
$157$	−21025.6	−0.852999	−0.426499	−	0.904488i	$-0.640253\pi$
$157$	−21025.6	−0.852999	−0.426499	+	0.904488i	$0.640253\pi$
$158$	− 19281.6i	− 0.772377i
$159$	0	0
$160$	−5455.35	−0.213100
$161$	6876.77i	0.265297i
$162$	0	0
$163$	−16709.0	−0.628890	−0.314445	−	0.949276i	$-0.601818\pi$
$163$	−16709.0	−0.628890	−0.314445	+	0.949276i	$0.601818\pi$
$164$	2949.84i	0.109676i
$165$	0	0
$166$	25740.2	0.934105
$167$	28204.0i	1.01129i	0.862741	+	0.505647i	$0.168746\pi$
$167$	28204.0i	1.01129i	−0.862741	+	0.505647i	$0.831254\pi$
$168$	0	0
$169$	−3249.67	−0.113780
$170$	10274.6i	0.355524i
$171$	0	0
$172$	−10227.5	−0.345710
$173$	14376.2i	0.480343i	0.970731	+	0.240171i	$0.0772036\pi$
$173$	14376.2i	0.480343i	−0.970731	+	0.240171i	$0.922796\pi$
$174$	0	0
$175$	−5245.47	−0.171281
$176$	755.207i	0.0243804i
$177$	0	0
$178$	−23851.0	−0.752777
$179$	55016.8i	1.71708i	0.512750	+	0.858538i	$0.328627\pi$
$179$	55016.8i	1.71708i	−0.512750	+	0.858538i	$0.671373\pi$
$180$	0	0
$181$	−59154.3	−1.80563	−0.902816	−	0.430027i	$-0.858504\pi$
$181$	−59154.3	−1.80563	−0.902816	+	0.430027i	$0.858504\pi$
$182$	− 8333.92i	− 0.251598i
$183$	0	0
$184$	8401.80	0.248163
$185$	− 20266.4i	− 0.592151i
$186$	0	0
$187$	1422.36	0.0406750
$188$	15193.2i	0.429865i
$189$	0	0
$190$	33354.2	0.923938
$191$	− 1660.41i	− 0.0455143i	−0.999741	−	0.0227572i	$-0.992756\pi$
$191$	− 1660.41i	− 0.0455143i	0.999741	−	0.0227572i	$-0.00724446\pi$
$192$	0	0
$193$	−28950.8	−0.777224	−0.388612	−	0.921402i	$-0.627045\pi$
$193$	−28950.8	−0.777224	−0.388612	+	0.921402i	$0.627045\pi$
$194$	− 1542.39i	− 0.0409818i
$195$	0	0
$196$	−2744.00	−0.0714286
$197$	66964.1i	1.72548i	0.505650	+	0.862739i	$0.331253\pi$
$197$	66964.1i	1.72548i	−0.505650	+	0.862739i	$0.668747\pi$
$198$	0	0
$199$	−25164.4	−0.635449	−0.317725	−	0.948183i	$-0.602919\pi$
$199$	−25164.4	−0.635449	−0.317725	+	0.948183i	$0.602919\pi$
$200$	6408.74i	0.160218i
$201$	0	0
$202$	20643.5	0.505920
$203$	4015.49i	0.0974421i
$204$	0	0
$205$	11112.3	0.264422
$206$	36737.8i	0.865724i
$207$	0	0
$208$	−10182.1	−0.235348
$209$	− 4617.36i	− 0.105706i
$210$	0	0
$211$	−68772.5	−1.54472	−0.772361	−	0.635184i	$-0.780924\pi$
$211$	−68772.5	−1.54472	−0.772361	+	0.635184i	$0.780924\pi$
$212$	13789.7i	0.306819i
$213$	0	0
$214$	4260.97	0.0930425
$215$	38528.0i	0.833488i
$216$	0	0
$217$	−17229.1	−0.365884
$218$	− 31694.1i	− 0.666907i
$219$	0	0
$220$	2844.94	0.0587798
$221$	19177.0i	0.392642i
$222$	0	0
$223$	41818.0	0.840918	0.420459	−	0.907312i	$-0.361869\pi$
$223$	41818.0	0.840918	0.420459	+	0.907312i	$0.361869\pi$
$224$	3352.53i	0.0668153i
$225$	0	0
$226$	44503.3	0.871315
$227$	− 88775.6i	− 1.72283i	−0.507903	−	0.861414i	$-0.669579\pi$
$227$	− 88775.6i	− 1.72283i	0.507903	−	0.861414i	$-0.330421\pi$
$228$	0	0
$229$	−88748.9	−1.69236	−0.846179	−	0.532899i	$-0.821102\pi$
$229$	−88748.9	−1.69236	−0.846179	+	0.532899i	$0.821102\pi$
$230$	− 31650.5i	− 0.598308i
$231$	0	0
$232$	4905.99	0.0911488
$233$	27765.9i	0.511447i	0.966750	+	0.255723i	$0.0823136\pi$
$233$	27765.9i	0.511447i	−0.966750	+	0.255723i	$0.917686\pi$
$234$	0	0
$235$	57234.2	1.03638
$236$	50549.7i	0.907600i
$237$	0	0
$238$	6314.17	0.111471
$239$	30454.6i	0.533159i	0.963813	+	0.266579i	$0.0858934\pi$
$239$	30454.6i	0.533159i	−0.963813	+	0.266579i	$0.914107\pi$
$240$	0	0
$241$	41178.6	0.708985	0.354493	−	0.935059i	$-0.384654\pi$
$241$	41178.6	0.708985	0.354493	+	0.935059i	$0.384654\pi$
$242$	41017.2i	0.700382i
$243$	0	0
$244$	−20238.1	−0.339930
$245$	10336.9i	0.172210i
$246$	0	0
$247$	62253.7	1.02040
$248$	21049.9i	0.342253i
$249$	0	0
$250$	−29132.5	−0.466120
$251$	66882.6i	1.06161i	0.847493	+	0.530806i	$0.178111\pi$
$251$	66882.6i	1.06161i	−0.847493	+	0.530806i	$0.821889\pi$
$252$	0	0
$253$	−4381.51	−0.0684514
$254$	40595.4i	0.629229i
$255$	0	0
$256$	4096.00	0.0625000
$257$	64609.6i	0.978207i	0.872226	+	0.489104i	$0.162676\pi$
$257$	64609.6i	0.978207i	−0.872226	+	0.489104i	$0.837324\pi$
$258$	0	0
$259$	−12454.5	−0.185663
$260$	38357.0i	0.567412i
$261$	0	0
$262$	45090.6	0.656876
$263$	57042.8i	0.824688i	0.911028	+	0.412344i	$0.135290\pi$
$263$	57042.8i	0.824688i	−0.911028	+	0.412344i	$0.864710\pi$
$264$	0	0
$265$	51947.2	0.739725
$266$	− 20497.4i	− 0.289692i
$267$	0	0
$268$	49371.1	0.687391
$269$	− 41750.8i	− 0.576979i	−0.957483	−	0.288490i	$-0.906847\pi$
$269$	− 41750.8i	− 0.576979i	0.957483	−	0.288490i	$-0.0931531\pi$
$270$	0	0
$271$	−76655.5	−1.04377	−0.521885	−	0.853016i	$-0.674771\pi$
$271$	−76655.5	−1.04377	−0.521885	+	0.853016i	$0.674771\pi$
$272$	− 7714.44i	− 0.104272i
$273$	0	0
$274$	−76763.0	−1.02247
$275$	− 3342.13i	− 0.0441935i
$276$	0	0
$277$	19904.0	0.259406	0.129703	−	0.991553i	$-0.458598\pi$
$277$	19904.0	0.259406	0.129703	+	0.991553i	$0.458598\pi$
$278$	57200.3i	0.740131i
$279$	0	0
$280$	12629.3	0.161088
$281$	− 23149.0i	− 0.293170i	−0.989198	−	0.146585i	$-0.953172\pi$
$281$	− 23149.0i	− 0.293170i	0.989198	−	0.146585i	$-0.0468282\pi$
$282$	0	0
$283$	−125867.	−1.57158	−0.785792	−	0.618491i	$-0.787745\pi$
$283$	−125867.	−1.57158	−0.785792	+	0.618491i	$0.787745\pi$
$284$	7262.84i	0.0900471i
$285$	0	0
$286$	5309.93	0.0649167
$287$	− 6828.97i	− 0.0829070i
$288$	0	0
$289$	68991.6	0.826039
$290$	− 18481.4i	− 0.219755i
$291$	0	0
$292$	29192.3	0.342376
$293$	− 150189.i	− 1.74945i	−0.484616	−	0.874727i	$-0.661041\pi$
$293$	− 150189.i	− 1.74945i	0.484616	−	0.874727i	$-0.338959\pi$
$294$	0	0
$295$	190426.	2.18817
$296$	15216.4i	0.173672i
$297$	0	0
$298$	−17339.3	−0.195253
$299$	− 59073.8i	− 0.660773i
$300$	0	0
$301$	23676.9	0.261332
$302$	− 68015.4i	− 0.745750i
$303$	0	0
$304$	−25043.1	−0.270982
$305$	76238.9i	0.819553i
$306$	0	0
$307$	8930.45	0.0947538	0.0473769	−	0.998877i	$-0.484914\pi$
$307$	8930.45	0.0947538	0.0473769	+	0.998877i	$0.484914\pi$
$308$	− 1748.33i	− 0.0184298i
$309$	0	0
$310$	79297.2	0.825153
$311$	60394.3i	0.624418i	0.950013	+	0.312209i	$0.101069\pi$
$311$	60394.3i	0.624418i	−0.950013	+	0.312209i	$0.898931\pi$
$312$	0	0
$313$	−149441.	−1.52540	−0.762698	−	0.646755i	$-0.776126\pi$
$313$	−149441.	−1.52540	−0.762698	+	0.646755i	$0.776126\pi$
$314$	− 59469.3i	− 0.603161i
$315$	0	0
$316$	54536.6	0.546153
$317$	− 55635.4i	− 0.553647i	−0.960921	−	0.276823i	$-0.910718\pi$
$317$	− 55635.4i	− 0.553647i	0.960921	−	0.276823i	$-0.0892817\pi$
$318$	0	0
$319$	−2558.46	−0.0251418
$320$	− 15430.1i	− 0.150684i
$321$	0	0
$322$	−19450.4	−0.187594
$323$	47166.3i	0.452092i
$324$	0	0
$325$	45060.4	0.426607
$326$	− 47260.2i	− 0.444693i
$327$	0	0
$328$	−8343.40	−0.0775524
$329$	− 35172.6i	− 0.324948i
$330$	0	0
$331$	76613.6	0.699278	0.349639	−	0.936884i	$-0.386304\pi$
$331$	76613.6	0.699278	0.349639	+	0.936884i	$0.386304\pi$
$332$	72804.2i	0.660512i
$333$	0	0
$334$	−79772.8	−0.715092
$335$	− 185986.i	− 1.65726i
$336$	0	0
$337$	−177974.	−1.56710	−0.783549	−	0.621330i	$-0.786593\pi$
$337$	−177974.	−1.56710	−0.783549	+	0.621330i	$0.786593\pi$
$338$	− 9191.46i	− 0.0804547i
$339$	0	0
$340$	−29061.1	−0.251394
$341$	− 10977.4i	− 0.0944045i
$342$	0	0
$343$	6352.45	0.0539949
$344$	− 28927.7i	− 0.244454i
$345$	0	0
$346$	−40662.0	−0.339653
$347$	122160.i	1.01454i	0.861787	+	0.507271i	$0.169346\pi$
$347$	122160.i	1.01454i	−0.861787	+	0.507271i	$0.830654\pi$
$348$	0	0
$349$	60823.0	0.499363	0.249682	−	0.968328i	$-0.419674\pi$
$349$	60823.0	0.499363	0.249682	+	0.968328i	$0.419674\pi$
$350$	− 14836.4i	− 0.121114i
$351$	0	0
$352$	−2136.05	−0.0172395
$353$	− 96167.4i	− 0.771753i	−0.922550	−	0.385877i	$-0.873899\pi$
$353$	− 96167.4i	− 0.771753i	0.922550	−	0.385877i	$-0.126101\pi$
$354$	0	0
$355$	27359.9	0.217099
$356$	− 67460.8i	− 0.532294i
$357$	0	0
$358$	−155611.	−1.21416
$359$	200553.i	1.55611i	0.628196	+	0.778055i	$0.283794\pi$
$359$	200553.i	1.55611i	−0.628196	+	0.778055i	$0.716206\pi$
$360$	0	0
$361$	22793.1	0.174899
$362$	− 167314.i	− 1.27677i
$363$	0	0
$364$	23571.9	0.177906
$365$	− 109971.i	− 0.825450i
$366$	0	0
$367$	114564.	0.850581	0.425291	−	0.905057i	$-0.360172\pi$
$367$	114564.	0.850581	0.425291	+	0.905057i	$0.360172\pi$
$368$	23763.9i	0.175478i
$369$	0	0
$370$	57322.0	0.418714
$371$	− 31923.6i	− 0.231933i
$372$	0	0
$373$	87845.8	0.631398	0.315699	−	0.948859i	$-0.397761\pi$
$373$	87845.8	0.631398	0.315699	+	0.948859i	$0.397761\pi$
$374$	4023.05i	0.0287615i
$375$	0	0
$376$	−42972.7	−0.303961
$377$	− 34494.5i	− 0.242698i
$378$	0	0
$379$	277656.	1.93298	0.966492	−	0.256696i	$-0.0826339\pi$
$379$	277656.	1.93298	0.966492	+	0.256696i	$0.0826339\pi$
$380$	94339.8i	0.653323i
$381$	0	0
$382$	4696.34	0.0321835
$383$	92652.4i	0.631625i	0.948822	+	0.315812i	$0.102277\pi$
$383$	92652.4i	0.631625i	−0.948822	+	0.315812i	$0.897723\pi$
$384$	0	0
$385$	−6586.14	−0.0444334
$386$	− 81885.2i	− 0.549580i
$387$	0	0
$388$	4362.55	0.0289785
$389$	77310.8i	0.510906i	0.966821	+	0.255453i	$0.0822246\pi$
$389$	77310.8i	0.510906i	−0.966821	+	0.255453i	$0.917775\pi$
$390$	0	0
$391$	44757.1	0.292758
$392$	− 7761.20i	− 0.0505076i
$393$	0	0
$394$	−189403.	−1.22010
$395$	− 205445.i	− 1.31675i
$396$	0	0
$397$	−107227.	−0.680333	−0.340167	−	0.940365i	$-0.610483\pi$
$397$	−107227.	−0.680333	−0.340167	+	0.940365i	$0.610483\pi$
$398$	− 71175.8i	− 0.449331i
$399$	0	0
$400$	−18126.6	−0.113292
$401$	− 140836.i	− 0.875839i	−0.899014	−	0.437920i	$-0.855715\pi$
$401$	− 140836.i	− 0.875839i	0.899014	−	0.437920i	$-0.144285\pi$
$402$	0	0
$403$	148004.	0.911303
$404$	58388.8i	0.357739i
$405$	0	0
$406$	−11357.5	−0.0689020
$407$	− 7935.32i	− 0.0479044i
$408$	0	0
$409$	2186.60	0.0130714	0.00653572	−	0.999979i	$-0.497920\pi$
$409$	2186.60	0.0130714	0.00653572	+	0.999979i	$0.497920\pi$
$410$	31430.5i	0.186975i
$411$	0	0
$412$	−103910.	−0.612159
$413$	− 117024.i	− 0.686081i
$414$	0	0
$415$	274261.	1.59246
$416$	− 28799.3i	− 0.166416i
$417$	0	0
$418$	13059.9	0.0747457
$419$	− 147204.i	− 0.838480i	−0.907875	−	0.419240i	$-0.862297\pi$
$419$	− 147204.i	− 0.838480i	0.907875	−	0.419240i	$-0.137703\pi$
$420$	0	0
$421$	−3618.79	−0.0204173	−0.0102087	−	0.999948i	$-0.503250\pi$
$421$	−3618.79	−0.0204173	−0.0102087	+	0.999948i	$0.503250\pi$
$422$	− 194518.i	− 1.09228i
$423$	0	0
$424$	−39003.1	−0.216954
$425$	34139.9i	0.189010i
$426$	0	0
$427$	46851.8	0.256963
$428$	12051.9i	0.0657910i
$429$	0	0
$430$	−108974.	−0.589365
$431$	269119.i	1.44874i	0.689411	+	0.724370i	$0.257869\pi$
$431$	269119.i	1.44874i	−0.689411	+	0.724370i	$0.742131\pi$
$432$	0	0
$433$	−278335.	−1.48454	−0.742271	−	0.670100i	$-0.766251\pi$
$433$	−278335.	−1.48454	−0.742271	+	0.670100i	$0.766251\pi$
$434$	− 48731.2i	− 0.258719i
$435$	0	0
$436$	89644.5	0.471575
$437$	− 145293.i	− 0.760820i
$438$	0	0
$439$	12385.3	0.0642656	0.0321328	−	0.999484i	$-0.489770\pi$
$439$	12385.3	0.0642656	0.0321328	+	0.999484i	$0.489770\pi$
$440$	8046.72i	0.0415636i
$441$	0	0
$442$	−54240.9	−0.277640
$443$	− 190001.i	− 0.968164i	−0.875023	−	0.484082i	$-0.839154\pi$
$443$	− 190001.i	− 0.968164i	0.875023	−	0.484082i	$-0.160846\pi$
$444$	0	0
$445$	−254132.	−1.28333
$446$	118279.i	0.594619i
$447$	0	0
$448$	−9482.37	−0.0472456
$449$	− 75081.0i	− 0.372424i	−0.982510	−	0.186212i	$-0.940379\pi$
$449$	− 75081.0i	− 0.372424i	0.982510	−	0.186212i	$-0.0596211\pi$
$450$	0	0
$451$	4351.05	0.0213915
$452$	125874.i	0.616113i
$453$	0	0
$454$	251095.	1.21822
$455$	− 88797.8i	− 0.428923i
$456$	0	0
$457$	58972.2	0.282367	0.141184	−	0.989983i	$-0.454909\pi$
$457$	58972.2	0.282367	0.141184	+	0.989983i	$0.454909\pi$
$458$	− 251020.i	− 1.19668i
$459$	0	0
$460$	89521.0	0.423067
$461$	69046.3i	0.324892i	0.986717	+	0.162446i	$0.0519383\pi$
$461$	69046.3i	0.324892i	−0.986717	+	0.162446i	$0.948062\pi$
$462$	0	0
$463$	−112506.	−0.524824	−0.262412	−	0.964956i	$-0.584518\pi$
$463$	−112506.	−0.524824	−0.262412	+	0.964956i	$0.584518\pi$
$464$	13876.2i	0.0644519i
$465$	0	0
$466$	−78533.9	−0.361648
$467$	− 154153.i	− 0.706836i	−0.935465	−	0.353418i	$-0.885019\pi$
$467$	− 154153.i	− 0.706836i	0.935465	−	0.353418i	$-0.114981\pi$
$468$	0	0
$469$	−114296.	−0.519618
$470$	161883.i	0.732833i
$471$	0	0
$472$	−142976.	−0.641770
$473$	15085.7i	0.0674283i
$474$	0	0
$475$	110827.	0.491199
$476$	17859.2i	0.0788220i
$477$	0	0
$478$	−86138.5	−0.377000
$479$	165690.i	0.722144i	0.932538	+	0.361072i	$0.117589\pi$
$479$	165690.i	0.722144i	−0.932538	+	0.361072i	$0.882411\pi$
$480$	0	0
$481$	106988.	0.462430
$482$	116471.i	0.501328i
$483$	0	0
$484$	−116014.	−0.495245
$485$	− 16434.2i	− 0.0698657i
$486$	0	0
$487$	−154280.	−0.650508	−0.325254	−	0.945627i	$-0.605450\pi$
$487$	−154280.	−0.650508	−0.325254	+	0.945627i	$0.605450\pi$
$488$	− 57241.9i	− 0.240367i
$489$	0	0
$490$	−29237.3	−0.121771
$491$	− 433288.i	− 1.79727i	−0.438695	−	0.898636i	$-0.644559\pi$
$491$	− 433288.i	− 1.79727i	0.438695	−	0.898636i	$-0.355441\pi$
$492$	0	0
$493$	26134.6	0.107528
$494$	176080.i	0.721533i
$495$	0	0
$496$	−59538.1	−0.242009
$497$	− 16813.7i	− 0.0680692i
$498$	0	0
$499$	−298370.	−1.19827	−0.599134	−	0.800649i	$-0.704488\pi$
$499$	−298370.	−1.19827	−0.599134	+	0.800649i	$0.704488\pi$
$500$	− 82399.2i	− 0.329597i
$501$	0	0
$502$	−189173.	−0.750673
$503$	− 284631.i	− 1.12498i	−0.826803	−	0.562491i	$-0.809843\pi$
$503$	− 284631.i	− 1.12498i	0.826803	−	0.562491i	$-0.190157\pi$
$504$	0	0
$505$	219957.	0.862490
$506$	− 12392.8i	− 0.0484025i
$507$	0	0
$508$	−114821.	−0.444932
$509$	287223.i	1.10862i	0.832309	+	0.554312i	$0.187019\pi$
$509$	287223.i	1.10862i	−0.832309	+	0.554312i	$0.812981\pi$
$510$	0	0
$511$	−67581.2	−0.258812
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	−182744.	−0.691697
$515$	391441.i	1.47588i
$516$	0	0
$517$	22410.1	0.0838423
$518$	− 35226.6i	− 0.131284i
$519$	0	0
$520$	−108490.	−0.401221
$521$	− 458107.i	− 1.68769i	−0.536591	−	0.843843i	$-0.680288\pi$
$521$	− 458107.i	− 1.68769i	0.536591	−	0.843843i	$-0.319712\pi$
$522$	0	0
$523$	99411.5	0.363441	0.181720	−	0.983350i	$-0.441833\pi$
$523$	99411.5	0.363441	0.181720	+	0.983350i	$0.441833\pi$
$524$	127535.i	0.464481i
$525$	0	0
$526$	−161341.	−0.583142
$527$	112135.i	0.403755i
$528$	0	0
$529$	141969.	0.507321
$530$	146929.i	0.523064i
$531$	0	0
$532$	57975.5	0.204843
$533$	58663.2i	0.206496i
$534$	0	0
$535$	45400.6	0.158619
$536$	139643.i	0.486059i
$537$	0	0
$538$	118089.	0.407986
$539$	4047.44i	0.0139317i
$540$	0	0
$541$	−355697.	−1.21531	−0.607653	−	0.794202i	$-0.707889\pi$
$541$	−355697.	−1.21531	−0.607653	+	0.794202i	$0.707889\pi$
$542$	− 216815.i	− 0.738057i
$543$	0	0
$544$	21819.7	0.0737312
$545$	− 337700.i	− 1.13694i
$546$	0	0
$547$	356592.	1.19178	0.595892	−	0.803065i	$-0.296799\pi$
$547$	356592.	1.19178	0.595892	+	0.803065i	$0.296799\pi$
$548$	− 217119.i	− 0.722996i
$549$	0	0
$550$	9452.98	0.0312495
$551$	− 84839.7i	− 0.279445i
$552$	0	0
$553$	−126254.	−0.412853
$554$	56296.9i	0.183428i
$555$	0	0
$556$	−161787.	−0.523351
$557$	89649.4i	0.288960i	0.989508	+	0.144480i	$0.0461509\pi$
$557$	89649.4i	0.288960i	−0.989508	+	0.144480i	$0.953849\pi$
$558$	0	0
$559$	−203393.	−0.650897
$560$	35721.1i	0.113907i
$561$	0	0
$562$	65475.3	0.207303
$563$	146102.i	0.460934i	0.973080	+	0.230467i	$0.0740254\pi$
$563$	146102.i	0.460934i	−0.973080	+	0.230467i	$0.925975\pi$
$564$	0	0
$565$	474182.	1.48542
$566$	− 356005.i	− 1.11128i
$567$	0	0
$568$	−20542.4	−0.0636729
$569$	437298.i	1.35068i	0.737506	+	0.675341i	$0.236003\pi$
$569$	437298.i	1.35068i	−0.737506	+	0.675341i	$0.763997\pi$
$570$	0	0
$571$	429334.	1.31681	0.658405	−	0.752664i	$-0.271232\pi$
$571$	429334.	1.31681	0.658405	+	0.752664i	$0.271232\pi$
$572$	15018.7i	0.0459031i
$573$	0	0
$574$	19315.2	0.0586241
$575$	− 105166.i	− 0.318082i
$576$	0	0
$577$	−140587.	−0.422272	−0.211136	−	0.977457i	$-0.567716\pi$
$577$	−140587.	−0.422272	−0.211136	+	0.977457i	$0.567716\pi$
$578$	195138.i	0.584097i
$579$	0	0
$580$	52273.2	0.155390
$581$	− 168544.i	− 0.499300i
$582$	0	0
$583$	20340.0	0.0598430
$584$	82568.4i	0.242096i
$585$	0	0
$586$	424798.	1.23705
$587$	− 264968.i	− 0.768985i	−0.923128	−	0.384492i	$-0.874377\pi$
$587$	− 264968.i	− 0.768985i	0.923128	−	0.384492i	$-0.125623\pi$
$588$	0	0
$589$	364018.	1.04928
$590$	538606.i	1.54727i
$591$	0	0
$592$	−43038.6	−0.122805
$593$	510014.i	1.45035i	0.688565	+	0.725175i	$0.258241\pi$
$593$	510014.i	1.45035i	−0.688565	+	0.725175i	$0.741759\pi$
$594$	0	0
$595$	67277.4	0.190036
$596$	− 49042.8i	− 0.138065i
$597$	0	0
$598$	167086.	0.467237
$599$	590648.i	1.64617i	0.567917	+	0.823086i	$0.307749\pi$
$599$	590648.i	1.64617i	−0.567917	+	0.823086i	$0.692251\pi$
$600$	0	0
$601$	260749.	0.721896	0.360948	−	0.932586i	$-0.382453\pi$
$601$	260749.	0.721896	0.360948	+	0.932586i	$0.382453\pi$
$602$	66968.5i	0.184790i
$603$	0	0
$604$	192377.	0.527325
$605$	437037.i	1.19401i
$606$	0	0
$607$	432739.	1.17449	0.587245	−	0.809409i	$-0.300213\pi$
$607$	432739.	1.17449	0.587245	+	0.809409i	$0.300213\pi$
$608$	− 70832.5i	− 0.191613i
$609$	0	0
$610$	−215636.	−0.579511
$611$	302145.i	0.809344i
$612$	0	0
$613$	−509866.	−1.35686	−0.678430	−	0.734665i	$-0.737339\pi$
$613$	−509866.	−1.35686	−0.678430	+	0.734665i	$0.737339\pi$
$614$	25259.1i	0.0670010i
$615$	0	0
$616$	4945.02	0.0130319
$617$	400846.i	1.05295i	0.850191	+	0.526474i	$0.176486\pi$
$617$	400846.i	1.05295i	−0.850191	+	0.526474i	$0.823514\pi$
$618$	0	0
$619$	435350.	1.13621	0.568103	−	0.822957i	$-0.307678\pi$
$619$	435350.	1.13621	0.568103	+	0.822957i	$0.307678\pi$
$620$	224286.i	0.583471i
$621$	0	0
$622$	−170821.	−0.441530
$623$	156174.i	0.402376i
$624$	0	0
$625$	−487424.	−1.24781
$626$	− 422684.i	− 1.07862i
$627$	0	0
$628$	168205.	0.426499
$629$	81059.3i	0.204881i
$630$	0	0
$631$	396490.	0.995803	0.497901	−	0.867234i	$-0.334104\pi$
$631$	396490.	0.995803	0.497901	+	0.867234i	$0.334104\pi$
$632$	154253.i	0.386188i
$633$	0	0
$634$	157361.	0.391487
$635$	432543.i	1.07271i
$636$	0	0
$637$	−54569.7	−0.134485
$638$	− 7236.41i	− 0.0177779i
$639$	0	0
$640$	43642.8	0.106550
$641$	568080.i	1.38259i	0.722572	+	0.691295i	$0.242960\pi$
$641$	568080.i	1.38259i	−0.722572	+	0.691295i	$0.757040\pi$
$642$	0	0
$643$	296340.	0.716751	0.358376	−	0.933577i	$-0.383331\pi$
$643$	296340.	0.716751	0.358376	+	0.933577i	$0.383331\pi$
$644$	− 55014.2i	− 0.132649i
$645$	0	0
$646$	−133406.	−0.319677
$647$	− 36510.4i	− 0.0872184i	−0.999049	−	0.0436092i	$-0.986114\pi$
$647$	− 36510.4i	− 0.0872184i	0.999049	−	0.0436092i	$-0.0138856\pi$
$648$	0	0
$649$	74561.5	0.177021
$650$	127450.i	0.301657i
$651$	0	0
$652$	133672.	0.314445
$653$	596493.i	1.39888i	0.714693	+	0.699438i	$0.246566\pi$
$653$	596493.i	1.39888i	−0.714693	+	0.699438i	$0.753434\pi$
$654$	0	0
$655$	480439.	1.11984
$656$	− 23598.7i	− 0.0548378i
$657$	0	0
$658$	99483.3	0.229773
$659$	− 746970.i	− 1.72002i	−0.510280	−	0.860008i	$-0.670458\pi$
$659$	− 746970.i	− 1.72002i	0.510280	−	0.860008i	$-0.329542\pi$
$660$	0	0
$661$	−147076.	−0.336619	−0.168309	−	0.985734i	$-0.553831\pi$
$661$	−147076.	−0.336619	−0.168309	+	0.985734i	$0.553831\pi$
$662$	216696.i	0.494464i
$663$	0	0
$664$	−205921.	−0.467052
$665$	− 218400.i	− 0.493866i
$666$	0	0
$667$	−80506.2	−0.180958
$668$	− 225632.i	− 0.505647i
$669$	0	0
$670$	526049.	1.17186
$671$	29851.4i	0.0663010i
$672$	0	0
$673$	−452502.	−0.999058	−0.499529	−	0.866297i	$-0.666494\pi$
$673$	−452502.	−0.999058	−0.499529	+	0.866297i	$0.666494\pi$
$674$	− 503386.i	− 1.10811i
$675$	0	0
$676$	25997.4	0.0568900
$677$	772073.i	1.68454i	0.539056	+	0.842270i	$0.318781\pi$
$677$	772073.i	1.68454i	−0.539056	+	0.842270i	$0.681219\pi$
$678$	0	0
$679$	−10099.4	−0.0219057
$680$	− 82197.2i	− 0.177762i
$681$	0	0
$682$	31048.9	0.0667541
$683$	− 404022.i	− 0.866092i	−0.901372	−	0.433046i	$-0.857439\pi$
$683$	− 404022.i	− 0.866092i	0.901372	−	0.433046i	$-0.142561\pi$
$684$	0	0
$685$	−817909.	−1.74311
$686$	17967.4i	0.0381802i
$687$	0	0
$688$	81819.8	0.172855
$689$	274234.i	0.577674i
$690$	0	0
$691$	−615946.	−1.28999	−0.644995	−	0.764187i	$-0.723141\pi$
$691$	−615946.	−1.28999	−0.644995	+	0.764187i	$0.723141\pi$
$692$	− 115009.i	− 0.240171i
$693$	0	0
$694$	−345520.	−0.717389
$695$	609468.i	1.26177i
$696$	0	0
$697$	−44446.0	−0.0914886
$698$	172033.i	0.353103i
$699$	0	0
$700$	41963.8	0.0856403
$701$	− 510387.i	− 1.03864i	−0.854581	−	0.519319i	$-0.826186\pi$
$701$	− 510387.i	− 1.03864i	0.854581	−	0.519319i	$-0.173814\pi$
$702$	0	0
$703$	263139.	0.532446
$704$	− 6041.66i	− 0.0121902i
$705$	0	0
$706$	272003.	0.545712
$707$	− 135172.i	− 0.270425i
$708$	0	0
$709$	−682535.	−1.35779	−0.678895	−	0.734235i	$-0.737541\pi$
$709$	−682535.	−1.35779	−0.678895	+	0.734235i	$0.737541\pi$
$710$	77385.4i	0.153512i
$711$	0	0
$712$	190808.	0.376389
$713$	− 345424.i	− 0.679475i
$714$	0	0
$715$	56577.2	0.110670
$716$	− 440135.i	− 0.858538i
$717$	0	0
$718$	−567250.	−1.10034
$719$	− 108955.i	− 0.210760i	−0.994432	−	0.105380i	$-0.966394\pi$
$719$	− 108955.i	− 0.210760i	0.994432	−	0.105380i	$-0.0336059\pi$
$720$	0	0
$721$	240556.	0.462749
$722$	64468.5i	0.123673i
$723$	0	0
$724$	473235.	0.902816
$725$	− 61408.6i	− 0.116830i
$726$	0	0
$727$	−146049.	−0.276331	−0.138165	−	0.990409i	$-0.544121\pi$
$727$	−146049.	−0.276331	−0.138165	+	0.990409i	$0.544121\pi$
$728$	66671.4i	0.125799i
$729$	0	0
$730$	311044.	0.583681
$731$	− 154100.i	− 0.288382i
$732$	0	0
$733$	−925797.	−1.72309	−0.861544	−	0.507682i	$-0.830502\pi$
$733$	−925797.	−1.72309	−0.861544	+	0.507682i	$0.830502\pi$
$734$	324036.i	0.601452i
$735$	0	0
$736$	−67214.4	−0.124081
$737$	− 72823.1i	− 0.134071i
$738$	0	0
$739$	40824.3	0.0747532	0.0373766	−	0.999301i	$-0.488100\pi$
$739$	40824.3	0.0747532	0.0373766	+	0.999301i	$0.488100\pi$
$740$	162131.i	0.296076i
$741$	0	0
$742$	90293.4	0.164002
$743$	106961.i	0.193753i	0.995296	+	0.0968766i	$0.0308852\pi$
$743$	106961.i	0.193753i	−0.995296	+	0.0968766i	$0.969115\pi$
$744$	0	0
$745$	−184750.	−0.332867
$746$	248465.i	0.446466i
$747$	0	0
$748$	−11378.9	−0.0203375
$749$	− 27900.4i	− 0.0497333i
$750$	0	0
$751$	−111510.	−0.197713	−0.0988564	−	0.995102i	$-0.531518\pi$
$751$	−111510.	−0.197713	−0.0988564	+	0.995102i	$0.531518\pi$
$752$	− 121545.i	− 0.214933i
$753$	0	0
$754$	97565.1	0.171614
$755$	− 724703.i	− 1.27135i
$756$	0	0
$757$	−78268.3	−0.136582	−0.0682911	−	0.997665i	$-0.521755\pi$
$757$	−78268.3	−0.136582	−0.0682911	+	0.997665i	$0.521755\pi$
$758$	785329.i	1.36683i
$759$	0	0
$760$	−266833.	−0.461969
$761$	799900.i	1.38123i	0.723222	+	0.690616i	$0.242660\pi$
$761$	799900.i	1.38123i	−0.723222	+	0.690616i	$0.757340\pi$
$762$	0	0
$763$	−207530.	−0.356477
$764$	13283.3i	0.0227572i
$765$	0	0
$766$	−262061.	−0.446626
$767$	1.00528e6i	1.70882i
$768$	0	0
$769$	−696974.	−1.17859	−0.589296	−	0.807917i	$-0.700595\pi$
$769$	−696974.	−1.17859	−0.589296	+	0.807917i	$0.700595\pi$
$770$	− 18628.4i	− 0.0314191i
$771$	0	0
$772$	231606.	0.388612
$773$	− 946901.i	− 1.58469i	−0.610071	−	0.792347i	$-0.708859\pi$
$773$	− 946901.i	− 1.58469i	0.610071	−	0.792347i	$-0.291141\pi$
$774$	0	0
$775$	263483.	0.438681
$776$	12339.1i	0.0204909i
$777$	0	0
$778$	−218668.	−0.361265
$779$	144283.i	0.237761i
$780$	0	0
$781$	10712.8	0.0175631
$782$	126592.i	0.207011i
$783$	0	0
$784$	21952.0	0.0357143
$785$	− 633644.i	− 1.02827i
$786$	0	0
$787$	693165.	1.11915	0.559574	−	0.828780i	$-0.310965\pi$
$787$	693165.	1.11915	0.559574	+	0.828780i	$0.310965\pi$
$788$	− 535712.i	− 0.862739i
$789$	0	0
$790$	581087.	0.931079
$791$	− 291403.i	− 0.465738i
$792$	0	0
$793$	−402473.	−0.640015
$794$	− 303283.i	− 0.481068i
$795$	0	0
$796$	201315.	0.317725
$797$	366791.i	0.577434i	0.957414	+	0.288717i	$0.0932287\pi$
$797$	366791.i	0.577434i	−0.957414	+	0.288717i	$0.906771\pi$
$798$	0	0
$799$	−228919.	−0.358582
$800$	− 51269.9i	− 0.0801092i
$801$	0	0
$802$	398344.	0.619312
$803$	− 43059.1i	− 0.0667781i
$804$	0	0
$805$	−207244.	−0.319809
$806$	418618.i	0.644388i
$807$	0	0
$808$	−165148.	−0.252960
$809$	17053.8i	0.0260569i	0.999915	+	0.0130285i	$0.00414721\pi$
$809$	17053.8i	0.0260569i	−0.999915	+	0.0130285i	$0.995853\pi$
$810$	0	0
$811$	127087.	0.193223	0.0966116	−	0.995322i	$-0.469200\pi$
$811$	127087.	0.193223	0.0966116	+	0.995322i	$0.469200\pi$
$812$	− 32123.9i	− 0.0487211i
$813$	0	0
$814$	22444.5	0.0338735
$815$	− 503556.i	− 0.758111i
$816$	0	0
$817$	−500249.	−0.749449
$818$	6184.64i	0.00924290i
$819$	0	0
$820$	−88898.8	−0.132211
$821$	− 870988.i	− 1.29219i	−0.763258	−	0.646094i	$-0.776401\pi$
$821$	− 870988.i	− 1.29219i	0.763258	−	0.646094i	$-0.223599\pi$
$822$	0	0
$823$	590520.	0.871836	0.435918	−	0.899986i	$-0.356424\pi$
$823$	590520.	0.871836	0.435918	+	0.899986i	$0.356424\pi$
$824$	− 293903.i	− 0.432862i
$825$	0	0
$826$	330994.	0.485132
$827$	− 254863.i	− 0.372645i	−0.982489	−	0.186323i	$-0.940343\pi$
$827$	− 254863.i	− 0.372645i	0.982489	−	0.186323i	$-0.0596570\pi$
$828$	0	0
$829$	102666.	0.149388	0.0746942	−	0.997206i	$-0.476202\pi$
$829$	102666.	0.149388	0.0746942	+	0.997206i	$0.476202\pi$
$830$	775728.i	1.12604i
$831$	0	0
$832$	81456.8	0.117674
$833$	− 41344.6i	− 0.0595838i
$834$	0	0
$835$	−849978.	−1.21909
$836$	36938.9i	0.0528532i
$837$	0	0
$838$	416357.	0.592895
$839$	− 428504.i	− 0.608739i	−0.952554	−	0.304369i	$-0.901554\pi$
$839$	− 428504.i	− 0.608739i	0.952554	−	0.304369i	$-0.0984458\pi$
$840$	0	0
$841$	660272.	0.933535
$842$	− 10235.5i	− 0.0144372i
$843$	0	0
$844$	550180.	0.772361
$845$	− 97934.8i	− 0.137159i
$846$	0	0
$847$	268576.	0.374370
$848$	− 110317.i	− 0.153410i
$849$	0	0
$850$	−96562.1	−0.133650
$851$	− 249698.i	− 0.344792i
$852$	0	0
$853$	248624.	0.341700	0.170850	−	0.985297i	$-0.445349\pi$
$853$	248624.	0.341700	0.170850	+	0.985297i	$0.445349\pi$
$854$	132517.i	0.181700i
$855$	0	0
$856$	−34087.8	−0.0465212
$857$	− 222103.i	− 0.302407i	−0.988503	−	0.151204i	$-0.951685\pi$
$857$	− 222103.i	− 0.302407i	0.988503	−	0.151204i	$-0.0483149\pi$
$858$	0	0
$859$	−893111.	−1.21037	−0.605187	−	0.796084i	$-0.706902\pi$
$859$	−893111.	−1.21037	−0.605187	+	0.796084i	$0.706902\pi$
$860$	− 308224.i	− 0.416744i
$861$	0	0
$862$	−761185.	−1.02441
$863$	− 153799.i	− 0.206505i	−0.994655	−	0.103253i	$-0.967075\pi$
$863$	− 153799.i	− 0.206505i	0.994655	−	0.103253i	$-0.0329250\pi$
$864$	0	0
$865$	−433252.	−0.579040
$866$	− 787251.i	− 1.04973i
$867$	0	0
$868$	137833.	0.182942
$869$	− 80442.3i	− 0.106523i
$870$	0	0
$871$	981840.	1.29421
$872$	253553.i	0.333454i
$873$	0	0
$874$	410951.	0.537981
$875$	190757.i	0.249152i
$876$	0	0
$877$	−845361.	−1.09912	−0.549558	−	0.835456i	$-0.685204\pi$
$877$	−845361.	−1.09912	−0.549558	+	0.835456i	$0.685204\pi$
$878$	35031.0i	0.0454426i
$879$	0	0
$880$	−22759.6	−0.0293899
$881$	603302.i	0.777290i	0.921388	+	0.388645i	$0.127057\pi$
$881$	603302.i	0.777290i	−0.921388	+	0.388645i	$0.872943\pi$
$882$	0	0
$883$	−642582.	−0.824152	−0.412076	−	0.911149i	$-0.635196\pi$
$883$	−642582.	−0.824152	−0.412076	+	0.911149i	$0.635196\pi$
$884$	− 153416.i	− 0.196321i
$885$	0	0
$886$	537404.	0.684595
$887$	− 516432.i	− 0.656396i	−0.944609	−	0.328198i	$-0.893559\pi$
$887$	− 516432.i	− 0.656396i	0.944609	−	0.328198i	$-0.106441\pi$
$888$	0	0
$889$	265814.	0.336337
$890$	− 718793.i	− 0.907453i
$891$	0	0
$892$	−334544.	−0.420459
$893$	743131.i	0.931886i
$894$	0	0
$895$	−1.65803e6	−2.06989
$896$	− 26820.2i	− 0.0334077i
$897$	0	0
$898$	212361.	0.263343
$899$	− 201701.i	− 0.249567i
$900$	0	0
$901$	−207773.	−0.255940
$902$	12306.6i	0.0151261i
$903$	0	0
$904$	−356026.	−0.435658
$905$	− 1.78272e6i	− 2.17664i
$906$	0	0
$907$	332852.	0.404610	0.202305	−	0.979323i	$-0.435157\pi$
$907$	332852.	0.404610	0.202305	+	0.979323i	$0.435157\pi$
$908$	710205.i	0.861414i
$909$	0	0
$910$	251158.	0.303294
$911$	− 128205.i	− 0.154478i	−0.997013	−	0.0772390i	$-0.975390\pi$
$911$	− 128205.i	− 0.154478i	0.997013	−	0.0772390i	$-0.0246104\pi$
$912$	0	0
$913$	107387.	0.128828
$914$	166798.i	0.199664i
$915$	0	0
$916$	709991.	0.846179
$917$	− 295249.i	− 0.351115i
$918$	0	0
$919$	1.30265e6	1.54240	0.771201	−	0.636592i	$-0.219656\pi$
$919$	1.30265e6	1.54240	0.771201	+	0.636592i	$0.219656\pi$
$920$	253204.i	0.299154i
$921$	0	0
$922$	−195292.	−0.229733
$923$	144435.i	0.169539i
$924$	0	0
$925$	190465.	0.222604
$926$	− 318215.i	− 0.371106i
$927$	0	0
$928$	−39247.9	−0.0455744
$929$	− 1.34170e6i	− 1.55462i	−0.629117	−	0.777311i	$-0.716583\pi$
$929$	− 1.34170e6i	− 1.55462i	0.629117	−	0.777311i	$-0.283417\pi$
$930$	0	0
$931$	−134215.	−0.154847
$932$	− 222127.i	− 0.255723i
$933$	0	0
$934$	436011.	0.499809
$935$	42865.5i	0.0490326i
$936$	0	0
$937$	−1.49619e6	−1.70415	−0.852074	−	0.523421i	$-0.824656\pi$
$937$	−1.49619e6	−1.70415	−0.852074	+	0.523421i	$0.824656\pi$
$938$	− 323277.i	− 0.367426i
$939$	0	0
$940$	−457874.	−0.518191
$941$	− 778807.i	− 0.879530i	−0.898113	−	0.439765i	$-0.855062\pi$
$941$	− 778807.i	− 0.879530i	0.898113	−	0.439765i	$-0.144938\pi$
$942$	0	0
$943$	136913.	0.153965
$944$	− 404397.i	− 0.453800i
$945$	0	0
$946$	−42668.7	−0.0476790
$947$	1.36722e6i	1.52454i	0.647258	+	0.762271i	$0.275916\pi$
$947$	1.36722e6i	1.52454i	−0.647258	+	0.762271i	$0.724084\pi$
$948$	0	0
$949$	580545.	0.644620
$950$	313466.i	0.347330i
$951$	0	0
$952$	−50513.4	−0.0557356
$953$	656858.i	0.723245i	0.932325	+	0.361623i	$0.117777\pi$
$953$	656858.i	0.723245i	−0.932325	+	0.361623i	$0.882223\pi$
$954$	0	0
$955$	50039.5	0.0548663
$956$	− 243636.i	− 0.266579i
$957$	0	0
$958$	−468641.	−0.510633
$959$	502637.i	0.546534i
$960$	0	0
$961$	−58093.7	−0.0629046
$962$	302608.i	0.326987i
$963$	0	0
$964$	−329429.	−0.354493
$965$	− 872486.i	− 0.936923i
$966$	0	0
$967$	817206.	0.873934	0.436967	−	0.899478i	$-0.356053\pi$
$967$	817206.	0.873934	0.436967	+	0.899478i	$0.356053\pi$
$968$	− 328137.i	− 0.350191i
$969$	0	0
$970$	46482.8	0.0494025
$971$	− 1.02544e6i	− 1.08761i	−0.839211	−	0.543805i	$-0.816983\pi$
$971$	− 1.02544e6i	− 1.08761i	0.839211	−	0.543805i	$-0.183017\pi$
$972$	0	0
$973$	374542.	0.395617
$974$	− 436370.i	− 0.459978i
$975$	0	0
$976$	161905.	0.169965
$977$	− 233580.i	− 0.244707i	−0.992487	−	0.122354i	$-0.960956\pi$
$977$	− 233580.i	− 0.244707i	0.992487	−	0.122354i	$-0.0390442\pi$
$978$	0	0
$979$	−99505.6	−0.103820
$980$	− 82695.5i	− 0.0861052i
$981$	0	0
$982$	1.22552e6	1.27086
$983$	512053.i	0.529917i	0.964260	+	0.264958i	$0.0853582\pi$
$983$	512053.i	0.529917i	−0.964260	+	0.264958i	$0.914642\pi$
$984$	0	0
$985$	−2.01808e6	−2.08002
$986$	73919.9i	0.0760339i
$987$	0	0
$988$	−498029.	−0.510201
$989$	474696.i	0.485315i
$990$	0	0
$991$	153330.	0.156128	0.0780639	−	0.996948i	$-0.475126\pi$
$991$	153330.	0.156128	0.0780639	+	0.996948i	$0.475126\pi$
$992$	− 168399.i	− 0.171126i
$993$	0	0
$994$	47556.3	0.0481322
$995$	− 758376.i	− 0.766017i
$996$	0	0
$997$	731704.	0.736114	0.368057	−	0.929803i	$-0.380023\pi$
$997$	731704.	0.736114	0.368057	+	0.929803i	$0.380023\pi$
$998$	− 843917.i	− 0.847303i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.5.b.b.323.8	yes	8
3.2	odd	2	inner	378.5.b.b.323.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.5.b.b.323.1	✓	8	3.2	odd	2	inner
378.5.b.b.323.8	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)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +30.1368i q^{5} +18.5203 q^{7} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +30.1368i q^{5} +18.5203 q^{7} -22.6274i q^{8} -85.2398 q^{10} +11.8001i q^{11} -159.095 q^{13} +52.3832i q^{14} +64.0000 q^{16} -120.538i q^{17} -391.298 q^{19} -241.095i q^{20} -33.3758 q^{22} +371.311i q^{23} -283.229 q^{25} -449.990i q^{26} -148.162 q^{28} +216.816i q^{29} -930.283 q^{31} +181.019i q^{32} +340.933 q^{34} +558.142i q^{35} -672.478 q^{37} -1106.76i q^{38} +681.919 q^{40} -368.730i q^{41} +1278.43 q^{43} -94.4009i q^{44} -1050.23 q^{46} -1899.14i q^{47} +343.000 q^{49} -801.092i q^{50} +1272.76 q^{52} -1723.71i q^{53} -355.618 q^{55} -419.066i q^{56} -613.249 q^{58} -6318.71i q^{59} +2529.76 q^{61} -2631.24i q^{62} -512.000 q^{64} -4794.63i q^{65} -6171.39 q^{67} +964.305i q^{68} -1578.66 q^{70} -907.855i q^{71} -3649.04 q^{73} -1902.06i q^{74} +3130.38 q^{76} +218.541i q^{77} -6817.08 q^{79} +1928.76i q^{80} +1042.93 q^{82} -9100.53i q^{83} +3632.64 q^{85} +3615.96i q^{86} +267.006 q^{88} +8432.60i q^{89} -2946.49 q^{91} -2970.49i q^{92} +5371.59 q^{94} -11792.5i q^{95} -545.318 q^{97} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 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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