LMFDB - Embedded newform 378.5.b.a.323.1

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:	8.0.5443747577856.29
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 $ x^8 + 24x^6 + 180x^4 + 488*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.1
Root			$0.982345i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.a.323.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -42.2558i q^{5} -18.5203 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} -42.2558i q^{5} -18.5203 q^{7} +22.6274i q^{8} -119.518 q^{10} -63.4392i q^{11} -99.9091 q^{13} +52.3832i q^{14} +64.0000 q^{16} -450.455i q^{17} -262.252 q^{19} +338.047i q^{20} -179.433 q^{22} +673.509i q^{23} -1160.56 q^{25} +282.586i q^{26} +148.162 q^{28} -498.837i q^{29} -201.115 q^{31} -181.019i q^{32} -1274.08 q^{34} +782.589i q^{35} +2517.87 q^{37} +741.761i q^{38} +956.140 q^{40} -1698.04i q^{41} -1359.85 q^{43} +507.513i q^{44} +1904.97 q^{46} +685.838i q^{47} +343.000 q^{49} +3282.55i q^{50} +799.273 q^{52} +3969.53i q^{53} -2680.68 q^{55} -419.066i q^{56} -1410.93 q^{58} +741.755i q^{59} -4675.57 q^{61} +568.838i q^{62} -512.000 q^{64} +4221.74i q^{65} +7531.63 q^{67} +3603.64i q^{68} +2213.50 q^{70} +8956.91i q^{71} -1465.30 q^{73} -7121.61i q^{74} +2098.02 q^{76} +1174.91i q^{77} -5557.36 q^{79} -2704.37i q^{80} -4802.77 q^{82} +2588.64i q^{83} -19034.3 q^{85} +3846.24i q^{86} +1435.46 q^{88} +8753.43i q^{89} +1850.34 q^{91} -5388.07i q^{92} +1939.84 q^{94} +11081.7i q^{95} +4336.45 q^{97} -970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) $ 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * 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
$5$	− 42.2558i	− 1.69023i	−0.534582	−	0.845117i	$-0.679531\pi$
$5$	− 42.2558i	− 1.69023i	0.534582	−	0.845117i	$-0.320469\pi$
$6$	0	0
$7$	−18.5203	−0.377964
$7$	−18.5203	−0.377964
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−119.518	−1.19518
$11$	− 63.4392i	− 0.524291i	−0.965028	−	0.262145i	$-0.915570\pi$
$11$	− 63.4392i	− 0.524291i	0.965028	−	0.262145i	$-0.0844300\pi$
$12$	0	0
$13$	−99.9091	−0.591178	−0.295589	−	0.955315i	$-0.595516\pi$
$13$	−99.9091	−0.591178	−0.295589	+	0.955315i	$0.595516\pi$
$14$	52.3832i	0.267261i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 450.455i	− 1.55867i	−0.626609	−	0.779334i	$-0.715558\pi$
$17$	− 450.455i	− 1.55867i	0.626609	−	0.779334i	$-0.284442\pi$
$18$	0	0
$19$	−262.252	−0.726460	−0.363230	−	0.931700i	$-0.618326\pi$
$19$	−262.252	−0.726460	−0.363230	+	0.931700i	$0.618326\pi$
$20$	338.047i	0.845117i
$21$	0	0
$22$	−179.433	−0.370730
$23$	673.509i	1.27317i	0.771205	+	0.636587i	$0.219654\pi$
$23$	673.509i	1.27317i	−0.771205	+	0.636587i	$0.780346\pi$
$24$	0	0
$25$	−1160.56	−1.85689
$26$	282.586i	0.418026i
$27$	0	0
$28$	148.162	0.188982
$29$	− 498.837i	− 0.593148i	−0.955010	−	0.296574i	$-0.904156\pi$
$29$	− 498.837i	− 0.593148i	0.955010	−	0.296574i	$-0.0958441\pi$
$30$	0	0
$31$	−201.115	−0.209276	−0.104638	−	0.994510i	$-0.533368\pi$
$31$	−201.115	−0.209276	−0.104638	+	0.994510i	$0.533368\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−1274.08	−1.10214
$35$	782.589i	0.638848i
$36$	0	0
$37$	2517.87	1.83920	0.919602	−	0.392852i	$-0.128512\pi$
$37$	2517.87	1.83920	0.919602	+	0.392852i	$0.128512\pi$
$38$	741.761i	0.513685i
$39$	0	0
$40$	956.140	0.597588
$41$	− 1698.04i	− 1.01013i	−0.863080	−	0.505067i	$-0.831468\pi$
$41$	− 1698.04i	− 1.01013i	0.863080	−	0.505067i	$-0.168532\pi$
$42$	0	0
$43$	−1359.85	−0.735452	−0.367726	−	0.929934i	$-0.619864\pi$
$43$	−1359.85	−0.735452	−0.367726	+	0.929934i	$0.619864\pi$
$44$	507.513i	0.262145i
$45$	0	0
$46$	1904.97	0.900270
$47$	685.838i	0.310474i	0.987877	+	0.155237i	$0.0496142\pi$
$47$	685.838i	0.310474i	−0.987877	+	0.155237i	$0.950386\pi$
$48$	0	0
$49$	343.000	0.142857
$50$	3282.55i	1.31302i
$51$	0	0
$52$	799.273	0.295589
$53$	3969.53i	1.41315i	0.707640	+	0.706573i	$0.249760\pi$
$53$	3969.53i	1.41315i	−0.707640	+	0.706573i	$0.750240\pi$
$54$	0	0
$55$	−2680.68	−0.886174
$56$	− 419.066i	− 0.133631i
$57$	0	0
$58$	−1410.93	−0.419419
$59$	741.755i	0.213087i	0.994308	+	0.106543i	$0.0339783\pi$
$59$	741.755i	0.213087i	−0.994308	+	0.106543i	$0.966022\pi$
$60$	0	0
$61$	−4675.57	−1.25654	−0.628268	−	0.777997i	$-0.716236\pi$
$61$	−4675.57	−1.25654	−0.628268	+	0.777997i	$0.716236\pi$
$62$	568.838i	0.147981i
$63$	0	0
$64$	−512.000	−0.125000
$65$	4221.74i	0.999229i
$66$	0	0
$67$	7531.63	1.67780	0.838899	−	0.544288i	$-0.183200\pi$
$67$	7531.63	1.67780	0.838899	+	0.544288i	$0.183200\pi$
$68$	3603.64i	0.779334i
$69$	0	0
$70$	2213.50	0.451734
$71$	8956.91i	1.77681i	0.459059	+	0.888406i	$0.348187\pi$
$71$	8956.91i	1.77681i	−0.459059	+	0.888406i	$0.651813\pi$
$72$	0	0
$73$	−1465.30	−0.274967	−0.137484	−	0.990504i	$-0.543901\pi$
$73$	−1465.30	−0.274967	−0.137484	+	0.990504i	$0.543901\pi$
$74$	− 7121.61i	− 1.30051i
$75$	0	0
$76$	2098.02	0.363230
$77$	1174.91i	0.198163i
$78$	0	0
$79$	−5557.36	−0.890460	−0.445230	−	0.895416i	$-0.646878\pi$
$79$	−5557.36	−0.890460	−0.445230	+	0.895416i	$0.646878\pi$
$80$	− 2704.37i	− 0.422558i
$81$	0	0
$82$	−4802.77	−0.714273
$83$	2588.64i	0.375765i	0.982192	+	0.187882i	$0.0601623\pi$
$83$	2588.64i	0.375765i	−0.982192	+	0.187882i	$0.939838\pi$
$84$	0	0
$85$	−19034.3	−2.63451
$86$	3846.24i	0.520043i
$87$	0	0
$88$	1435.46	0.185365
$89$	8753.43i	1.10509i	0.833483	+	0.552545i	$0.186343\pi$
$89$	8753.43i	1.10509i	−0.833483	+	0.552545i	$0.813657\pi$
$90$	0	0
$91$	1850.34	0.223444
$92$	− 5388.07i	− 0.636587i
$93$	0	0
$94$	1939.84	0.219538
$95$	11081.7i	1.22789i
$96$	0	0
$97$	4336.45	0.460883	0.230442	−	0.973086i	$-0.425983\pi$
$97$	4336.45	0.460883	0.230442	+	0.973086i	$0.425983\pi$
$98$	− 970.151i	− 0.101015i
$99$	0	0
$100$	9284.44	0.928444
$101$	− 1236.26i	− 0.121190i	−0.998162	−	0.0605950i	$-0.980700\pi$
$101$	− 1236.26i	− 0.121190i	0.998162	−	0.0605950i	$-0.0192998\pi$
$102$	0	0
$103$	−16956.4	−1.59831	−0.799153	−	0.601128i	$-0.794718\pi$
$103$	−16956.4	−1.59831	−0.799153	+	0.601128i	$0.794718\pi$
$104$	− 2260.68i	− 0.209013i
$105$	0	0
$106$	11227.5	0.999246
$107$	4735.53i	0.413619i	0.978381	+	0.206810i	$0.0663081\pi$
$107$	4735.53i	0.413619i	−0.978381	+	0.206810i	$0.933692\pi$
$108$	0	0
$109$	18182.9	1.53042	0.765209	−	0.643782i	$-0.222636\pi$
$109$	18182.9	1.53042	0.765209	+	0.643782i	$0.222636\pi$
$110$	7582.10i	0.626619i
$111$	0	0
$112$	−1185.30	−0.0944911
$113$	− 19988.6i	− 1.56540i	−0.622398	−	0.782701i	$-0.713841\pi$
$113$	− 19988.6i	− 1.56540i	0.622398	−	0.782701i	$-0.286159\pi$
$114$	0	0
$115$	28459.7	2.15196
$116$	3990.70i	0.296574i
$117$	0	0
$118$	2098.00	0.150675
$119$	8342.54i	0.589121i
$120$	0	0
$121$	10616.5	0.725119
$122$	13224.5i	0.888504i
$123$	0	0
$124$	1608.92	0.104638
$125$	22630.3i	1.44834i
$126$	0	0
$127$	9159.59	0.567896	0.283948	−	0.958840i	$-0.408356\pi$
$127$	9159.59	0.567896	0.283948	+	0.958840i	$0.408356\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	11940.9	0.706561
$131$	8015.64i	0.467084i	0.972347	+	0.233542i	$0.0750317\pi$
$131$	8015.64i	0.467084i	−0.972347	+	0.233542i	$0.924968\pi$
$132$	0	0
$133$	4856.98	0.274576
$134$	− 21302.7i	− 1.18638i
$135$	0	0
$136$	10192.6	0.551072
$137$	− 13447.2i	− 0.716459i	−0.933634	−	0.358229i	$-0.883381\pi$
$137$	− 13447.2i	− 0.716459i	0.933634	−	0.358229i	$-0.116619\pi$
$138$	0	0
$139$	−27654.5	−1.43132	−0.715658	−	0.698451i	$-0.753873\pi$
$139$	−27654.5	−1.43132	−0.715658	+	0.698451i	$0.753873\pi$
$140$	− 6260.71i	− 0.319424i
$141$	0	0
$142$	25334.0	1.25640
$143$	6338.15i	0.309949i
$144$	0	0
$145$	−21078.8	−1.00256
$146$	4144.49i	0.194431i
$147$	0	0
$148$	−20143.0	−0.919602
$149$	− 23453.9i	− 1.05643i	−0.849110	−	0.528217i	$-0.822861\pi$
$149$	− 23453.9i	− 1.05643i	0.849110	−	0.528217i	$-0.177139\pi$
$150$	0	0
$151$	2549.51	0.111816	0.0559078	−	0.998436i	$-0.482195\pi$
$151$	2549.51	0.111816	0.0559078	+	0.998436i	$0.482195\pi$
$152$	− 5934.09i	− 0.256842i
$153$	0	0
$154$	3323.15	0.140123
$155$	8498.27i	0.353726i
$156$	0	0
$157$	−17969.8	−0.729029	−0.364514	−	0.931198i	$-0.618765\pi$
$157$	−17969.8	−0.729029	−0.364514	+	0.931198i	$0.618765\pi$
$158$	15718.6i	0.629650i
$159$	0	0
$160$	−7649.12	−0.298794
$161$	− 12473.6i	− 0.481214i
$162$	0	0
$163$	−25505.1	−0.959956	−0.479978	−	0.877281i	$-0.659355\pi$
$163$	−25505.1	−0.959956	−0.479978	+	0.877281i	$0.659355\pi$
$164$	13584.3i	0.505067i
$165$	0	0
$166$	7321.79	0.265706
$167$	− 1848.54i	− 0.0662821i	−0.999451	−	0.0331410i	$-0.989449\pi$
$167$	− 1848.54i	− 0.0662821i	0.999451	−	0.0331410i	$-0.0105510\pi$
$168$	0	0
$169$	−18579.2	−0.650509
$170$	53837.3i	1.86288i
$171$	0	0
$172$	10878.8	0.367726
$173$	− 40682.6i	− 1.35930i	−0.733535	−	0.679652i	$-0.762131\pi$
$173$	− 40682.6i	− 1.35930i	0.733535	−	0.679652i	$-0.237869\pi$
$174$	0	0
$175$	21493.8	0.701838
$176$	− 4060.11i	− 0.131073i
$177$	0	0
$178$	24758.4	0.781417
$179$	38271.8i	1.19446i	0.802069	+	0.597232i	$0.203733\pi$
$179$	38271.8i	1.19446i	−0.802069	+	0.597232i	$0.796267\pi$
$180$	0	0
$181$	−46467.3	−1.41837	−0.709186	−	0.705022i	$-0.750937\pi$
$181$	−46467.3	−1.41837	−0.709186	+	0.705022i	$0.750937\pi$
$182$	− 5233.56i	− 0.157999i
$183$	0	0
$184$	−15239.8	−0.450135
$185$	− 106395.i	− 3.10868i
$186$	0	0
$187$	−28576.5	−0.817195
$188$	− 5486.70i	− 0.155237i
$189$	0	0
$190$	31343.7	0.868247
$191$	− 48153.9i	− 1.31997i	−0.751278	−	0.659986i	$-0.770562\pi$
$191$	− 48153.9i	− 1.31997i	0.751278	−	0.659986i	$-0.229438\pi$
$192$	0	0
$193$	−51645.0	−1.38648	−0.693240	−	0.720706i	$-0.743818\pi$
$193$	−51645.0	−1.38648	−0.693240	+	0.720706i	$0.743818\pi$
$194$	− 12265.3i	− 0.325894i
$195$	0	0
$196$	−2744.00	−0.0714286
$197$	− 4002.20i	− 0.103126i	−0.998670	−	0.0515628i	$-0.983580\pi$
$197$	− 4002.20i	− 0.103126i	0.998670	−	0.0515628i	$-0.0164202\pi$
$198$	0	0
$199$	22970.5	0.580048	0.290024	−	0.957019i	$-0.406337\pi$
$199$	22970.5	0.580048	0.290024	+	0.957019i	$0.406337\pi$
$200$	− 26260.4i	− 0.656509i
$201$	0	0
$202$	−3496.67	−0.0856942
$203$	9238.60i	0.224189i
$204$	0	0
$205$	−71751.9	−1.70736
$206$	47960.0i	1.13017i
$207$	0	0
$208$	−6394.18	−0.147794
$209$	16637.1i	0.380876i
$210$	0	0
$211$	−51005.4	−1.14565	−0.572824	−	0.819679i	$-0.694152\pi$
$211$	−51005.4	−1.14565	−0.572824	+	0.819679i	$0.694152\pi$
$212$	− 31756.2i	− 0.706573i
$213$	0	0
$214$	13394.1	0.292473
$215$	57461.6i	1.24309i
$216$	0	0
$217$	3724.70	0.0790991
$218$	− 51429.0i	− 1.08217i
$219$	0	0
$220$	21445.4	0.443087
$221$	45004.5i	0.921450i
$222$	0	0
$223$	−16479.0	−0.331375	−0.165688	−	0.986178i	$-0.552984\pi$
$223$	−16479.0	−0.331375	−0.165688	+	0.986178i	$0.552984\pi$
$224$	3352.53i	0.0668153i
$225$	0	0
$226$	−56536.3	−1.10691
$227$	− 101140.i	− 1.96278i	−0.192026	−	0.981390i	$-0.561506\pi$
$227$	− 101140.i	− 1.96278i	0.192026	−	0.981390i	$-0.438494\pi$
$228$	0	0
$229$	98918.5	1.88628	0.943140	−	0.332394i	$-0.107857\pi$
$229$	98918.5	1.88628	0.943140	+	0.332394i	$0.107857\pi$
$230$	− 80496.1i	− 1.52167i
$231$	0	0
$232$	11287.4	0.209709
$233$	− 8359.71i	− 0.153985i	−0.997032	−	0.0769926i	$-0.975468\pi$
$233$	− 8359.71i	− 0.153985i	0.997032	−	0.0769926i	$-0.0245318\pi$
$234$	0	0
$235$	28980.6	0.524774
$236$	− 5934.04i	− 0.106543i
$237$	0	0
$238$	23596.3	0.416571
$239$	− 29969.3i	− 0.524663i	−0.964978	−	0.262331i	$-0.915509\pi$
$239$	− 29969.3i	− 0.524663i	0.964978	−	0.262331i	$-0.0844913\pi$
$240$	0	0
$241$	−74966.2	−1.29072	−0.645359	−	0.763879i	$-0.723292\pi$
$241$	−74966.2	−1.29072	−0.645359	+	0.763879i	$0.723292\pi$
$242$	− 30027.9i	− 0.512737i
$243$	0	0
$244$	37404.5	0.628268
$245$	− 14493.8i	− 0.241462i
$246$	0	0
$247$	26201.4	0.429467
$248$	− 4550.71i	− 0.0739904i
$249$	0	0
$250$	64008.3	1.02413
$251$	− 92012.3i	− 1.46049i	−0.683186	−	0.730244i	$-0.739406\pi$
$251$	− 92012.3i	− 1.46049i	0.683186	−	0.730244i	$-0.260594\pi$
$252$	0	0
$253$	42726.8	0.667513
$254$	− 25907.2i	− 0.401563i
$255$	0	0
$256$	4096.00	0.0625000
$257$	30757.4i	0.465676i	0.972516	+	0.232838i	$0.0748011\pi$
$257$	30757.4i	0.465676i	−0.972516	+	0.232838i	$0.925199\pi$
$258$	0	0
$259$	−46631.6	−0.695153
$260$	− 33773.9i	− 0.499614i
$261$	0	0
$262$	22671.6	0.330279
$263$	− 37491.9i	− 0.542034i	−0.962575	−	0.271017i	$-0.912640\pi$
$263$	− 37491.9i	− 0.542034i	0.962575	−	0.271017i	$-0.0873599\pi$
$264$	0	0
$265$	167736.	2.38855
$266$	− 13737.6i	− 0.194155i
$267$	0	0
$268$	−60253.1	−0.838899
$269$	− 37506.9i	− 0.518330i	−0.965833	−	0.259165i	$-0.916553\pi$
$269$	− 37506.9i	− 0.518330i	0.965833	−	0.259165i	$-0.0834473\pi$
$270$	0	0
$271$	53171.4	0.724002	0.362001	−	0.932178i	$-0.382094\pi$
$271$	53171.4	0.724002	0.362001	+	0.932178i	$0.382094\pi$
$272$	− 28829.1i	− 0.389667i
$273$	0	0
$274$	−38034.5	−0.506613
$275$	73624.7i	0.973550i
$276$	0	0
$277$	−30111.4	−0.392438	−0.196219	−	0.980560i	$-0.562866\pi$
$277$	−30111.4	−0.392438	−0.196219	+	0.980560i	$0.562866\pi$
$278$	78218.6i	1.01209i
$279$	0	0
$280$	−17708.0	−0.225867
$281$	28226.0i	0.357467i	0.983898	+	0.178734i	$0.0572000\pi$
$281$	28226.0i	0.357467i	−0.983898	+	0.178734i	$0.942800\pi$
$282$	0	0
$283$	11670.3	0.145716	0.0728581	−	0.997342i	$-0.476788\pi$
$283$	11670.3	0.145716	0.0728581	+	0.997342i	$0.476788\pi$
$284$	− 71655.3i	− 0.888406i
$285$	0	0
$286$	17927.0	0.219167
$287$	31448.1i	0.381795i
$288$	0	0
$289$	−119389.	−1.42944
$290$	59619.8i	0.708916i
$291$	0	0
$292$	11722.4	0.137484
$293$	29595.2i	0.344735i	0.985033	+	0.172368i	$0.0551417\pi$
$293$	29595.2i	0.344735i	−0.985033	+	0.172368i	$0.944858\pi$
$294$	0	0
$295$	31343.5	0.360166
$296$	56972.9i	0.650257i
$297$	0	0
$298$	−66337.6	−0.747011
$299$	− 67289.6i	− 0.752672i
$300$	0	0
$301$	25184.8	0.277975
$302$	− 7211.09i	− 0.0790655i
$303$	0	0
$304$	−16784.1	−0.181615
$305$	197570.i	2.12384i
$306$	0	0
$307$	−96742.0	−1.02645	−0.513225	−	0.858254i	$-0.671550\pi$
$307$	−96742.0	−1.02645	−0.513225	+	0.858254i	$0.671550\pi$
$308$	− 9399.28i	− 0.0990816i
$309$	0	0
$310$	24036.7	0.250122
$311$	− 83000.7i	− 0.858146i	−0.903270	−	0.429073i	$-0.858840\pi$
$311$	− 83000.7i	− 0.858146i	0.903270	−	0.429073i	$-0.141160\pi$
$312$	0	0
$313$	148025.	1.51093	0.755467	−	0.655186i	$-0.227410\pi$
$313$	148025.	1.51093	0.755467	+	0.655186i	$0.227410\pi$
$314$	50826.3i	0.515501i
$315$	0	0
$316$	44458.9	0.445230
$317$	61615.0i	0.613152i	0.951846	+	0.306576i	$0.0991833\pi$
$317$	61615.0i	0.613152i	−0.951846	+	0.306576i	$0.900817\pi$
$318$	0	0
$319$	−31645.8	−0.310982
$320$	21635.0i	0.211279i
$321$	0	0
$322$	−35280.5	−0.340270
$323$	118133.i	1.13231i
$324$	0	0
$325$	115950.	1.09775
$326$	72139.2i	0.678791i
$327$	0	0
$328$	38422.2	0.357136
$329$	− 12701.9i	− 0.117348i
$330$	0	0
$331$	−100037.	−0.913067	−0.456534	−	0.889706i	$-0.650909\pi$
$331$	−100037.	−0.913067	−0.456534	+	0.889706i	$0.650909\pi$
$332$	− 20709.1i	− 0.187882i
$333$	0	0
$334$	−5228.46	−0.0468685
$335$	− 318255.i	− 2.83587i
$336$	0	0
$337$	−170263.	−1.49920	−0.749600	−	0.661891i	$-0.769754\pi$
$337$	−170263.	−1.49920	−0.749600	+	0.661891i	$0.769754\pi$
$338$	52549.8i	0.459979i
$339$	0	0
$340$	152275.	1.31726
$341$	12758.5i	0.109722i
$342$	0	0
$343$	−6352.45	−0.0539949
$344$	− 30769.9i	− 0.260022i
$345$	0	0
$346$	−115068.	−0.961173
$347$	186399.i	1.54805i	0.633155	+	0.774025i	$0.281759\pi$
$347$	186399.i	1.54805i	−0.633155	+	0.774025i	$0.718241\pi$
$348$	0	0
$349$	−230470.	−1.89218	−0.946091	−	0.323901i	$-0.895005\pi$
$349$	−230470.	−1.89218	−0.946091	+	0.323901i	$0.895005\pi$
$350$	− 60793.6i	− 0.496274i
$351$	0	0
$352$	−11483.7	−0.0926824
$353$	6167.83i	0.0494975i	0.999694	+	0.0247488i	$0.00787858\pi$
$353$	6167.83i	0.0494975i	−0.999694	+	0.0247488i	$0.992121\pi$
$354$	0	0
$355$	378482.	3.00323
$356$	− 70027.4i	− 0.552545i
$357$	0	0
$358$	108249.	0.844613
$359$	8588.08i	0.0666357i	0.999445	+	0.0333179i	$0.0106074\pi$
$359$	8588.08i	0.0666357i	−0.999445	+	0.0333179i	$0.989393\pi$
$360$	0	0
$361$	−61544.9	−0.472256
$362$	131429.i	1.00294i
$363$	0	0
$364$	−14802.7	−0.111722
$365$	61917.5i	0.464759i
$366$	0	0
$367$	147820.	1.09749	0.548747	−	0.835988i	$-0.315105\pi$
$367$	147820.	1.09749	0.548747	+	0.835988i	$0.315105\pi$
$368$	43104.6i	0.318293i
$369$	0	0
$370$	−300930.	−2.19817
$371$	− 73516.7i	− 0.534119i
$372$	0	0
$373$	−142961.	−1.02754	−0.513771	−	0.857927i	$-0.671752\pi$
$373$	−142961.	−1.02754	−0.513771	+	0.857927i	$0.671752\pi$
$374$	80826.5i	0.577844i
$375$	0	0
$376$	−15518.7	−0.109769
$377$	49838.4i	0.350656i
$378$	0	0
$379$	252021.	1.75452	0.877260	−	0.480016i	$-0.159369\pi$
$379$	252021.	1.75452	0.877260	+	0.480016i	$0.159369\pi$
$380$	− 88653.4i	− 0.613943i
$381$	0	0
$382$	−136200.	−0.933362
$383$	− 246474.i	− 1.68025i	−0.542394	−	0.840125i	$-0.682482\pi$
$383$	− 246474.i	− 1.68025i	0.542394	−	0.840125i	$-0.317518\pi$
$384$	0	0
$385$	49646.8	0.334942
$386$	146074.i	0.980390i
$387$	0	0
$388$	−34691.6	−0.230442
$389$	113655.i	0.751083i	0.926805	+	0.375542i	$0.122543\pi$
$389$	113655.i	0.751083i	−0.926805	+	0.375542i	$0.877457\pi$
$390$	0	0
$391$	303385.	1.98445
$392$	7761.20i	0.0505076i
$393$	0	0
$394$	−11319.9	−0.0729208
$395$	234831.i	1.50509i
$396$	0	0
$397$	48018.2	0.304667	0.152333	−	0.988329i	$-0.451321\pi$
$397$	48018.2	0.304667	0.152333	+	0.988329i	$0.451321\pi$
$398$	− 64970.3i	− 0.410156i
$399$	0	0
$400$	−74275.5	−0.464222
$401$	− 78669.6i	− 0.489236i	−0.969620	−	0.244618i	$-0.921337\pi$
$401$	− 78669.6i	− 0.489236i	0.969620	−	0.244618i	$-0.0786625\pi$
$402$	0	0
$403$	20093.2	0.123720
$404$	9890.07i	0.0605950i
$405$	0	0
$406$	26130.7	0.158525
$407$	− 159732.i	− 0.964277i
$408$	0	0
$409$	−89356.3	−0.534169	−0.267085	−	0.963673i	$-0.586060\pi$
$409$	−89356.3	−0.534169	−0.267085	+	0.963673i	$0.586060\pi$
$410$	202945.i	1.20729i
$411$	0	0
$412$	135651.	0.799153
$413$	− 13737.5i	− 0.0805393i
$414$	0	0
$415$	109385.	0.635130
$416$	18085.5i	0.104506i
$417$	0	0
$418$	47056.7	0.269320
$419$	109718.i	0.624955i	0.949925	+	0.312477i	$0.101159\pi$
$419$	109718.i	0.624955i	−0.949925	+	0.312477i	$0.898841\pi$
$420$	0	0
$421$	146241.	0.825096	0.412548	−	0.910936i	$-0.364639\pi$
$421$	146241.	0.825096	0.412548	+	0.910936i	$0.364639\pi$
$422$	144265.i	0.810095i
$423$	0	0
$424$	−89820.2	−0.499623
$425$	522778.i	2.89427i
$426$	0	0
$427$	86592.7	0.474926
$428$	− 37884.2i	− 0.206810i
$429$	0	0
$430$	162526.	0.878994
$431$	215583.i	1.16054i	0.814425	+	0.580269i	$0.197053\pi$
$431$	215583.i	1.16054i	−0.814425	+	0.580269i	$0.802947\pi$
$432$	0	0
$433$	−171626.	−0.915390	−0.457695	−	0.889109i	$-0.651325\pi$
$433$	−171626.	−0.915390	−0.457695	+	0.889109i	$0.651325\pi$
$434$	− 10535.0i	− 0.0559315i
$435$	0	0
$436$	−145463.	−0.765209
$437$	− 176629.i	− 0.924909i
$438$	0	0
$439$	−148476.	−0.770420	−0.385210	−	0.922829i	$-0.625871\pi$
$439$	−148476.	−0.770420	−0.385210	+	0.922829i	$0.625871\pi$
$440$	− 60656.8i	− 0.313310i
$441$	0	0
$442$	127292.	0.651564
$443$	282139.i	1.43766i	0.695186	+	0.718830i	$0.255322\pi$
$443$	282139.i	1.43766i	−0.695186	+	0.718830i	$0.744678\pi$
$444$	0	0
$445$	369883.	1.86786
$446$	46609.5i	0.234318i
$447$	0	0
$448$	9482.37	0.0472456
$449$	− 302834.i	− 1.50214i	−0.660220	−	0.751072i	$-0.729537\pi$
$449$	− 302834.i	− 1.50214i	0.660220	−	0.751072i	$-0.270463\pi$
$450$	0	0
$451$	−107722.	−0.529604
$452$	159909.i	0.782701i
$453$	0	0
$454$	−286067.	−1.38789
$455$	− 78187.7i	− 0.377673i
$456$	0	0
$457$	−216256.	−1.03547	−0.517734	−	0.855542i	$-0.673224\pi$
$457$	−216256.	−1.03547	−0.517734	+	0.855542i	$0.673224\pi$
$458$	− 279784.i	− 1.33380i
$459$	0	0
$460$	−227677.	−1.07598
$461$	338541.i	1.59298i	0.604653	+	0.796489i	$0.293312\pi$
$461$	338541.i	1.59298i	−0.604653	+	0.796489i	$0.706688\pi$
$462$	0	0
$463$	−406862.	−1.89795	−0.948975	−	0.315351i	$-0.897878\pi$
$463$	−406862.	−1.89795	−0.948975	+	0.315351i	$0.897878\pi$
$464$	− 31925.6i	− 0.148287i
$465$	0	0
$466$	−23644.8	−0.108884
$467$	143548.i	0.658208i	0.944294	+	0.329104i	$0.106747\pi$
$467$	143548.i	0.658208i	−0.944294	+	0.329104i	$0.893253\pi$
$468$	0	0
$469$	−139488.	−0.634148
$470$	− 81969.6i	− 0.371071i
$471$	0	0
$472$	−16784.0	−0.0753376
$473$	86267.8i	0.385591i
$474$	0	0
$475$	304358.	1.34896
$476$	− 66740.3i	− 0.294560i
$477$	0	0
$478$	−84765.8	−0.370992
$479$	− 329574.i	− 1.43642i	−0.695825	−	0.718212i	$-0.744961\pi$
$479$	− 329574.i	− 1.43642i	0.695825	−	0.718212i	$-0.255039\pi$
$480$	0	0
$481$	−251558.	−1.08730
$482$	212037.i	0.912676i
$483$	0	0
$484$	−84931.8	−0.362560
$485$	− 183240.i	− 0.779000i
$486$	0	0
$487$	119437.	0.503594	0.251797	−	0.967780i	$-0.418978\pi$
$487$	119437.	0.503594	0.251797	+	0.967780i	$0.418978\pi$
$488$	− 105796.i	− 0.444252i
$489$	0	0
$490$	−40994.5	−0.170739
$491$	277082.i	1.14933i	0.818388	+	0.574666i	$0.194868\pi$
$491$	277082.i	1.14933i	−0.818388	+	0.574666i	$0.805132\pi$
$492$	0	0
$493$	−224704.	−0.924520
$494$	− 74108.6i	− 0.303679i
$495$	0	0
$496$	−12871.3	−0.0523191
$497$	− 165884.i	− 0.671572i
$498$	0	0
$499$	384780.	1.54529	0.772647	−	0.634836i	$-0.218932\pi$
$499$	384780.	1.54529	0.772647	+	0.634836i	$0.218932\pi$
$500$	− 181043.i	− 0.724171i
$501$	0	0
$502$	−260250.	−1.03272
$503$	− 288691.i	− 1.14103i	−0.821286	−	0.570516i	$-0.806743\pi$
$503$	− 288691.i	− 1.14103i	0.821286	−	0.570516i	$-0.193257\pi$
$504$	0	0
$505$	−52239.1	−0.204839
$506$	− 120850.i	− 0.472003i
$507$	0	0
$508$	−73276.8	−0.283948
$509$	181063.i	0.698866i	0.936961	+	0.349433i	$0.113626\pi$
$509$	181063.i	0.698866i	−0.936961	+	0.349433i	$0.886374\pi$
$510$	0	0
$511$	27137.7	0.103928
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	86995.1	0.329282
$515$	716508.i	2.70151i
$516$	0	0
$517$	43509.0	0.162779
$518$	131894.i	0.491548i
$519$	0	0
$520$	−95527.1	−0.353281
$521$	313770.i	1.15594i	0.816057	+	0.577971i	$0.196155\pi$
$521$	313770.i	1.15594i	−0.816057	+	0.577971i	$0.803845\pi$
$522$	0	0
$523$	10145.3	0.0370903	0.0185452	−	0.999828i	$-0.494097\pi$
$523$	10145.3	0.0370903	0.0185452	+	0.999828i	$0.494097\pi$
$524$	− 64125.1i	− 0.233542i
$525$	0	0
$526$	−106043.	−0.383276
$527$	90593.1i	0.326192i
$528$	0	0
$529$	−173773.	−0.620970
$530$	− 474428.i	− 1.68896i
$531$	0	0
$532$	−38855.8	−0.137288
$533$	169649.i	0.597169i
$534$	0	0
$535$	200104.	0.699113
$536$	170421.i	0.593191i
$537$	0	0
$538$	−106085.	−0.366515
$539$	− 21759.6i	− 0.0748987i
$540$	0	0
$541$	496514.	1.69643	0.848217	−	0.529649i	$-0.177676\pi$
$541$	496514.	1.69643	0.848217	+	0.529649i	$0.177676\pi$
$542$	− 150392.i	− 0.511947i
$543$	0	0
$544$	−81541.1	−0.275536
$545$	− 768333.i	− 2.58676i
$546$	0	0
$547$	−557220.	−1.86231	−0.931155	−	0.364623i	$-0.881198\pi$
$547$	−557220.	−1.86231	−0.931155	+	0.364623i	$0.881198\pi$
$548$	107578.i	0.358229i
$549$	0	0
$550$	208242.	0.688404
$551$	130821.i	0.430898i
$552$	0	0
$553$	102924.	0.336562
$554$	85167.9i	0.277496i
$555$	0	0
$556$	221236.	0.715658
$557$	− 67362.8i	− 0.217125i	−0.994090	−	0.108562i	$-0.965375\pi$
$557$	− 67362.8i	− 0.217125i	0.994090	−	0.108562i	$-0.0346247\pi$
$558$	0	0
$559$	135861.	0.434783
$560$	50085.7i	0.159712i
$561$	0	0
$562$	79835.1	0.252767
$563$	171298.i	0.540425i	0.962801	+	0.270212i	$0.0870940\pi$
$563$	171298.i	0.540425i	−0.962801	+	0.270212i	$0.912906\pi$
$564$	0	0
$565$	−844636.	−2.64589
$566$	− 33008.5i	− 0.103037i
$567$	0	0
$568$	−202672.	−0.628198
$569$	− 23818.0i	− 0.0735666i	−0.999323	−	0.0367833i	$-0.988289\pi$
$569$	− 23818.0i	− 0.0735666i	0.999323	−	0.0367833i	$-0.0117111\pi$
$570$	0	0
$571$	58622.1	0.179800	0.0898999	−	0.995951i	$-0.471345\pi$
$571$	58622.1	0.179800	0.0898999	+	0.995951i	$0.471345\pi$
$572$	− 50705.2i	− 0.154975i
$573$	0	0
$574$	88948.6	0.269970
$575$	− 781644.i	− 2.36414i
$576$	0	0
$577$	421423.	1.26580	0.632901	−	0.774232i	$-0.281864\pi$
$577$	421423.	1.26580	0.632901	+	0.774232i	$0.281864\pi$
$578$	337682.i	1.01077i
$579$	0	0
$580$	168630.	0.501279
$581$	− 47942.3i	− 0.142026i
$582$	0	0
$583$	251824.	0.740900
$584$	− 33156.0i	− 0.0972156i
$585$	0	0
$586$	83707.8	0.243765
$587$	− 404875.i	− 1.17502i	−0.809218	−	0.587509i	$-0.800109\pi$
$587$	− 404875.i	− 1.17502i	0.809218	−	0.587509i	$-0.199891\pi$
$588$	0	0
$589$	52742.7	0.152031
$590$	− 88652.8i	− 0.254676i
$591$	0	0
$592$	161144.	0.459801
$593$	541601.i	1.54018i	0.637938	+	0.770088i	$0.279788\pi$
$593$	541601.i	1.54018i	−0.637938	+	0.770088i	$0.720212\pi$
$594$	0	0
$595$	352521.	0.995752
$596$	187631.i	0.528217i
$597$	0	0
$598$	−190324.	−0.532219
$599$	236556.i	0.659295i	0.944104	+	0.329648i	$0.106930\pi$
$599$	236556.i	0.659295i	−0.944104	+	0.329648i	$0.893070\pi$
$600$	0	0
$601$	382066.	1.05776	0.528882	−	0.848695i	$-0.322611\pi$
$601$	382066.	1.05776	0.528882	+	0.848695i	$0.322611\pi$
$602$	− 71233.3i	− 0.196558i
$603$	0	0
$604$	−20396.1	−0.0559078
$605$	− 448608.i	− 1.22562i
$606$	0	0
$607$	376071.	1.02069	0.510344	−	0.859970i	$-0.329518\pi$
$607$	376071.	1.02069	0.510344	+	0.859970i	$0.329518\pi$
$608$	47472.7i	0.128421i
$609$	0	0
$610$	558812.	1.50178
$611$	− 68521.4i	− 0.183546i
$612$	0	0
$613$	−21902.9	−0.0582882	−0.0291441	−	0.999575i	$-0.509278\pi$
$613$	−21902.9	−0.0582882	−0.0291441	+	0.999575i	$0.509278\pi$
$614$	273628.i	0.725810i
$615$	0	0
$616$	−26585.2	−0.0700613
$617$	− 358506.i	− 0.941729i	−0.882206	−	0.470864i	$-0.843942\pi$
$617$	− 358506.i	− 0.941729i	0.882206	−	0.470864i	$-0.156058\pi$
$618$	0	0
$619$	448702.	1.17105	0.585526	−	0.810653i	$-0.300888\pi$
$619$	448702.	1.17105	0.585526	+	0.810653i	$0.300888\pi$
$620$	− 67986.1i	− 0.176863i
$621$	0	0
$622$	−234761.	−0.606801
$623$	− 162116.i	− 0.417685i
$624$	0	0
$625$	230917.	0.591147
$626$	− 418677.i	− 1.06839i
$627$	0	0
$628$	143759.	0.364514
$629$	− 1.13419e6i	− 2.86671i
$630$	0	0
$631$	302074.	0.758672	0.379336	−	0.925259i	$-0.376152\pi$
$631$	302074.	0.758672	0.379336	+	0.925259i	$0.376152\pi$
$632$	− 125749.i	− 0.314825i
$633$	0	0
$634$	174274.	0.433564
$635$	− 387046.i	− 0.959877i
$636$	0	0
$637$	−34268.8	−0.0844540
$638$	89507.9i	0.219897i
$639$	0	0
$640$	61193.0	0.149397
$641$	− 757749.i	− 1.84421i	−0.386944	−	0.922103i	$-0.626469\pi$
$641$	− 757749.i	− 1.84421i	0.386944	−	0.922103i	$-0.373531\pi$
$642$	0	0
$643$	−211423.	−0.511364	−0.255682	−	0.966761i	$-0.582300\pi$
$643$	−211423.	−0.511364	−0.255682	+	0.966761i	$0.582300\pi$
$644$	99788.4i	0.240607i
$645$	0	0
$646$	334130.	0.800664
$647$	− 361157.i	− 0.862756i	−0.902171	−	0.431378i	$-0.858028\pi$
$647$	− 361157.i	− 0.862756i	0.902171	−	0.431378i	$-0.141972\pi$
$648$	0	0
$649$	47056.3	0.111719
$650$	− 327956.i	− 0.776228i
$651$	0	0
$652$	204041.	0.479978
$653$	472486.i	1.10806i	0.832497	+	0.554030i	$0.186911\pi$
$653$	472486.i	1.10806i	−0.832497	+	0.554030i	$0.813089\pi$
$654$	0	0
$655$	338707.	0.789482
$656$	− 108674.i	− 0.252534i
$657$	0	0
$658$	−35926.4	−0.0829777
$659$	− 479228.i	− 1.10350i	−0.834010	−	0.551749i	$-0.813961\pi$
$659$	− 479228.i	− 1.10350i	0.834010	−	0.551749i	$-0.186039\pi$
$660$	0	0
$661$	−653224.	−1.49506	−0.747531	−	0.664227i	$-0.768761\pi$
$661$	−653224.	−1.49506	−0.747531	+	0.664227i	$0.768761\pi$
$662$	282946.i	0.645636i
$663$	0	0
$664$	−58574.3	−0.132853
$665$	− 205236.i	− 0.464098i
$666$	0	0
$667$	335971.	0.755180
$668$	14788.3i	0.0331410i
$669$	0	0
$670$	−900162.	−2.00526
$671$	296614.i	0.658790i
$672$	0	0
$673$	−852808.	−1.88287	−0.941436	−	0.337190i	$-0.890523\pi$
$673$	−852808.	−1.88287	−0.941436	+	0.337190i	$0.890523\pi$
$674$	481575.i	1.06009i
$675$	0	0
$676$	148633.	0.325254
$677$	344306.i	0.751220i	0.926778	+	0.375610i	$0.122567\pi$
$677$	344306.i	0.751220i	−0.926778	+	0.375610i	$0.877433\pi$
$678$	0	0
$679$	−80312.2	−0.174198
$680$	− 430698.i	− 0.931441i
$681$	0	0
$682$	36086.6	0.0775850
$683$	189722.i	0.406701i	0.979106	+	0.203350i	$0.0651831\pi$
$683$	189722.i	0.406701i	−0.979106	+	0.203350i	$0.934817\pi$
$684$	0	0
$685$	−568223.	−1.21098
$686$	17967.4i	0.0381802i
$687$	0	0
$688$	−87030.4	−0.183863
$689$	− 396592.i	− 0.835421i
$690$	0	0
$691$	323591.	0.677704	0.338852	−	0.940840i	$-0.389961\pi$
$691$	323591.	0.677704	0.338852	+	0.940840i	$0.389961\pi$
$692$	325461.i	0.679652i
$693$	0	0
$694$	527216.	1.09464
$695$	1.16856e6i	2.41926i
$696$	0	0
$697$	−764889.	−1.57446
$698$	651867.i	1.33797i
$699$	0	0
$700$	−171950.	−0.350919
$701$	− 705697.i	− 1.43609i	−0.695996	−	0.718046i	$-0.745037\pi$
$701$	− 705697.i	− 1.43609i	0.695996	−	0.718046i	$-0.254963\pi$
$702$	0	0
$703$	−660316.	−1.33611
$704$	32480.9i	0.0655363i
$705$	0	0
$706$	17445.3	0.0350000
$707$	22895.8i	0.0458055i
$708$	0	0
$709$	278238.	0.553509	0.276755	−	0.960941i	$-0.410741\pi$
$709$	278238.	0.553509	0.276755	+	0.960941i	$0.410741\pi$
$710$	− 1.07051e6i	− 2.12360i
$711$	0	0
$712$	−198067.	−0.390709
$713$	− 135452.i	− 0.266445i
$714$	0	0
$715$	267824.	0.523886
$716$	− 306174.i	− 0.597232i
$717$	0	0
$718$	24290.8	0.0471186
$719$	− 482776.i	− 0.933873i	−0.884291	−	0.466937i	$-0.845358\pi$
$719$	− 482776.i	− 0.933873i	0.884291	−	0.466937i	$-0.154642\pi$
$720$	0	0
$721$	314037.	0.604103
$722$	174075.i	0.333935i
$723$	0	0
$724$	371738.	0.709186
$725$	578928.i	1.10141i
$726$	0	0
$727$	130521.	0.246952	0.123476	−	0.992348i	$-0.460596\pi$
$727$	130521.	0.246952	0.123476	+	0.992348i	$0.460596\pi$
$728$	41868.5i	0.0789995i
$729$	0	0
$730$	175129.	0.328634
$731$	612551.i	1.14633i
$732$	0	0
$733$	−342877.	−0.638161	−0.319081	−	0.947728i	$-0.603374\pi$
$733$	−342877.	−0.638161	−0.319081	+	0.947728i	$0.603374\pi$
$734$	− 418099.i	− 0.776046i
$735$	0	0
$736$	121918.	0.225067
$737$	− 477801.i	− 0.879654i
$738$	0	0
$739$	−693065.	−1.26907	−0.634534	−	0.772895i	$-0.718808\pi$
$739$	−693065.	−1.26907	−0.634534	+	0.772895i	$0.718808\pi$
$740$	851157.i	1.55434i
$741$	0	0
$742$	−207937.	−0.377679
$743$	− 788374.i	− 1.42809i	−0.700102	−	0.714043i	$-0.746862\pi$
$743$	− 788374.i	− 1.42809i	0.700102	−	0.714043i	$-0.253138\pi$
$744$	0	0
$745$	−991063.	−1.78562
$746$	404354.i	0.726582i
$747$	0	0
$748$	228612.	0.408598
$749$	− 87703.2i	− 0.156333i
$750$	0	0
$751$	−346405.	−0.614192	−0.307096	−	0.951678i	$-0.599357\pi$
$751$	−346405.	−0.614192	−0.307096	+	0.951678i	$0.599357\pi$
$752$	43893.6i	0.0776186i
$753$	0	0
$754$	140964.	0.247951
$755$	− 107732.i	− 0.188994i
$756$	0	0
$757$	−774595.	−1.35171	−0.675854	−	0.737036i	$-0.736225\pi$
$757$	−774595.	−1.35171	−0.675854	+	0.737036i	$0.736225\pi$
$758$	− 712823.i	− 1.24063i
$759$	0	0
$760$	−250750.	−0.434124
$761$	956099.i	1.65095i	0.564439	+	0.825475i	$0.309092\pi$
$761$	956099.i	1.65095i	−0.564439	+	0.825475i	$0.690908\pi$
$762$	0	0
$763$	−336752.	−0.578443
$764$	385231.i	0.659986i
$765$	0	0
$766$	−697134.	−1.18812
$767$	− 74108.1i	− 0.125972i
$768$	0	0
$769$	62358.6	0.105449	0.0527247	−	0.998609i	$-0.483209\pi$
$769$	62358.6	0.105449	0.0527247	+	0.998609i	$0.483209\pi$
$770$	− 140422.i	− 0.236840i
$771$	0	0
$772$	413160.	0.693240
$773$	− 664455.i	− 1.11201i	−0.831181	−	0.556003i	$-0.812334\pi$
$773$	− 664455.i	− 1.11201i	0.831181	−	0.556003i	$-0.187666\pi$
$774$	0	0
$775$	233405.	0.388603
$776$	98122.7i	0.162947i
$777$	0	0
$778$	321464.	0.531096
$779$	445313.i	0.733822i
$780$	0	0
$781$	568219.	0.931566
$782$	− 858103.i	− 1.40322i
$783$	0	0
$784$	21952.0	0.0357143
$785$	759330.i	1.23223i
$786$	0	0
$787$	206026.	0.332639	0.166320	−	0.986072i	$-0.446812\pi$
$787$	206026.	0.332639	0.166320	+	0.986072i	$0.446812\pi$
$788$	32017.6i	0.0515628i
$789$	0	0
$790$	664202.	1.06426
$791$	370194.i	0.591666i
$792$	0	0
$793$	467132.	0.742836
$794$	− 135816.i	− 0.215432i
$795$	0	0
$796$	−183764.	−0.290024
$797$	44688.5i	0.0703524i	0.999381	+	0.0351762i	$0.0111992\pi$
$797$	44688.5i	0.0703524i	−0.999381	+	0.0351762i	$0.988801\pi$
$798$	0	0
$799$	308939.	0.483926
$800$	210083.i	0.328255i
$801$	0	0
$802$	−222511.	−0.345942
$803$	92957.5i	0.144163i
$804$	0	0
$805$	−527080.	−0.813364
$806$	− 56832.1i	− 0.0874830i
$807$	0	0
$808$	27973.3	0.0428471
$809$	377340.i	0.576549i	0.957548	+	0.288274i	$0.0930815\pi$
$809$	377340.i	0.576549i	−0.957548	+	0.288274i	$0.906918\pi$
$810$	0	0
$811$	289441.	0.440067	0.220034	−	0.975492i	$-0.429383\pi$
$811$	289441.	0.440067	0.220034	+	0.975492i	$0.429383\pi$
$812$	− 73908.8i	− 0.112094i
$813$	0	0
$814$	−451789.	−0.681847
$815$	1.07774e6i	1.62255i
$816$	0	0
$817$	356624.	0.534276
$818$	252738.i	0.377715i
$819$	0	0
$820$	574015.	0.853681
$821$	− 549071.i	− 0.814596i	−0.913295	−	0.407298i	$-0.866471\pi$
$821$	− 549071.i	− 0.814596i	0.913295	−	0.407298i	$-0.133529\pi$
$822$	0	0
$823$	262600.	0.387700	0.193850	−	0.981031i	$-0.437903\pi$
$823$	262600.	0.387700	0.193850	+	0.981031i	$0.437903\pi$
$824$	− 383680.i	− 0.565086i
$825$	0	0
$826$	−38855.5	−0.0569499
$827$	− 49767.0i	− 0.0727664i	−0.999338	−	0.0363832i	$-0.988416\pi$
$827$	− 49767.0i	− 0.0727664i	0.999338	−	0.0363832i	$-0.0115837\pi$
$828$	0	0
$829$	−738419.	−1.07447	−0.537234	−	0.843433i	$-0.680531\pi$
$829$	−738419.	−1.07447	−0.537234	+	0.843433i	$0.680531\pi$
$830$	− 309388.i	− 0.449105i
$831$	0	0
$832$	51153.4	0.0738972
$833$	− 154506.i	− 0.222667i
$834$	0	0
$835$	−78111.6	−0.112032
$836$	− 133096.i	− 0.190438i
$837$	0	0
$838$	310329.	0.441910
$839$	1.18604e6i	1.68490i	0.538776	+	0.842449i	$0.318887\pi$
$839$	1.18604e6i	1.68490i	−0.538776	+	0.842449i	$0.681113\pi$
$840$	0	0
$841$	458442.	0.648176
$842$	− 413632.i	− 0.583431i
$843$	0	0
$844$	408043.	0.572824
$845$	785079.i	1.09951i
$846$	0	0
$847$	−196620.	−0.274069
$848$	254050.i	0.353287i
$849$	0	0
$850$	1.47864e6	2.04656
$851$	1.69581e6i	2.34162i
$852$	0	0
$853$	470813.	0.647069	0.323534	−	0.946216i	$-0.395129\pi$
$853$	470813.	0.647069	0.323534	+	0.946216i	$0.395129\pi$
$854$	− 244921.i	− 0.335823i
$855$	0	0
$856$	−107153.	−0.146236
$857$	− 27792.1i	− 0.0378408i	−0.999821	−	0.0189204i	$-0.993977\pi$
$857$	− 27792.1i	− 0.0378408i	0.999821	−	0.0189204i	$-0.00602291\pi$
$858$	0	0
$859$	−897338.	−1.21610	−0.608051	−	0.793898i	$-0.708048\pi$
$859$	−897338.	−1.21610	−0.608051	+	0.793898i	$0.708048\pi$
$860$	− 459693.i	− 0.621543i
$861$	0	0
$862$	609760.	0.820624
$863$	− 327757.i	− 0.440078i	−0.975491	−	0.220039i	$-0.929382\pi$
$863$	− 327757.i	− 0.440078i	0.975491	−	0.220039i	$-0.0706185\pi$
$864$	0	0
$865$	−1.71908e6	−2.29754
$866$	485430.i	0.647278i
$867$	0	0
$868$	−29797.6	−0.0395495
$869$	352555.i	0.466860i
$870$	0	0
$871$	−752478.	−0.991877
$872$	411432.i	0.541084i
$873$	0	0
$874$	−499582.	−0.654010
$875$	− 419120.i	− 0.547422i
$876$	0	0
$877$	−675535.	−0.878312	−0.439156	−	0.898411i	$-0.644722\pi$
$877$	−675535.	−0.878312	−0.439156	+	0.898411i	$0.644722\pi$
$878$	419954.i	0.544770i
$879$	0	0
$880$	−171563.	−0.221543
$881$	− 105642.i	− 0.136108i	−0.997682	−	0.0680540i	$-0.978321\pi$
$881$	− 105642.i	− 0.136108i	0.997682	−	0.0680540i	$-0.0216790\pi$
$882$	0	0
$883$	−214939.	−0.275672	−0.137836	−	0.990455i	$-0.544015\pi$
$883$	−214939.	−0.275672	−0.137836	+	0.990455i	$0.544015\pi$
$884$	− 360036.i	− 0.460725i
$885$	0	0
$886$	798011.	1.01658
$887$	− 1.38541e6i	− 1.76089i	−0.474152	−	0.880443i	$-0.657245\pi$
$887$	− 1.38541e6i	− 1.76089i	0.474152	−	0.880443i	$-0.342755\pi$
$888$	0	0
$889$	−169638.	−0.214645
$890$	− 1.04619e6i	− 1.32078i
$891$	0	0
$892$	131832.	0.165688
$893$	− 179862.i	− 0.225547i
$894$	0	0
$895$	1.61721e6	2.01892
$896$	− 26820.2i	− 0.0334077i
$897$	0	0
$898$	−856544.	−1.06218
$899$	100324.i	0.124132i
$900$	0	0
$901$	1.78809e6	2.20263
$902$	304684.i	0.374487i
$903$	0	0
$904$	452291.	0.553453
$905$	1.96351e6i	2.39738i
$906$	0	0
$907$	−1.24618e6	−1.51483	−0.757417	−	0.652932i	$-0.773539\pi$
$907$	−1.24618e6	−1.51483	−0.757417	+	0.652932i	$0.773539\pi$
$908$	809121.i	0.981390i
$909$	0	0
$910$	−221148.	−0.267055
$911$	− 289663.i	− 0.349025i	−0.984655	−	0.174513i	$-0.944165\pi$
$911$	− 289663.i	− 0.349025i	0.984655	−	0.174513i	$-0.0558350\pi$
$912$	0	0
$913$	164221.	0.197010
$914$	611665.i	0.732186i
$915$	0	0
$916$	−791348.	−0.943140
$917$	− 148452.i	− 0.176541i
$918$	0	0
$919$	781183.	0.924958	0.462479	−	0.886630i	$-0.346960\pi$
$919$	781183.	0.924958	0.462479	+	0.886630i	$0.346960\pi$
$920$	643969.i	0.760833i
$921$	0	0
$922$	957539.	1.12641
$923$	− 894876.i	− 1.05041i
$924$	0	0
$925$	−2.92213e6	−3.41520
$926$	1.15078e6i	1.34205i
$927$	0	0
$928$	−90299.2	−0.104855
$929$	112122.i	0.129915i	0.997888	+	0.0649576i	$0.0206912\pi$
$929$	112122.i	0.129915i	−0.997888	+	0.0649576i	$0.979309\pi$
$930$	0	0
$931$	−89952.4	−0.103780
$932$	66877.6i	0.0769926i
$933$	0	0
$934$	406015.	0.465424
$935$	1.20752e6i	1.38125i
$936$	0	0
$937$	405756.	0.462152	0.231076	−	0.972936i	$-0.425775\pi$
$937$	405756.	0.462152	0.231076	+	0.972936i	$0.425775\pi$
$938$	394531.i	0.448410i
$939$	0	0
$940$	−231845.	−0.262387
$941$	− 1.62677e6i	− 1.83716i	−0.395239	−	0.918578i	$-0.629338\pi$
$941$	− 1.62677e6i	− 1.83716i	0.395239	−	0.918578i	$-0.370662\pi$
$942$	0	0
$943$	1.14364e6	1.28608
$944$	47472.3i	0.0532717i
$945$	0	0
$946$	244002.	0.272654
$947$	− 641839.i	− 0.715692i	−0.933781	−	0.357846i	$-0.883511\pi$
$947$	− 641839.i	− 0.715692i	0.933781	−	0.357846i	$-0.116489\pi$
$948$	0	0
$949$	146397.	0.162555
$950$	− 860854.i	− 0.953855i
$951$	0	0
$952$	−188770.	−0.208286
$953$	1.18936e6i	1.30957i	0.755816	+	0.654785i	$0.227241\pi$
$953$	1.18936e6i	1.30957i	−0.755816	+	0.654785i	$0.772759\pi$
$954$	0	0
$955$	−2.03478e6	−2.23106
$956$	239754.i	0.262331i
$957$	0	0
$958$	−932177.	−1.01570
$959$	249046.i	0.270796i
$960$	0	0
$961$	−883074.	−0.956203
$962$	711513.i	0.768835i
$963$	0	0
$964$	599730.	0.645359
$965$	2.18230e6i	2.34348i
$966$	0	0
$967$	1.68998e6	1.80730	0.903648	−	0.428275i	$-0.140879\pi$
$967$	1.68998e6	1.80730	0.903648	+	0.428275i	$0.140879\pi$
$968$	240223.i	0.256368i
$969$	0	0
$970$	−518282.	−0.550836
$971$	− 400896.i	− 0.425200i	−0.977139	−	0.212600i	$-0.931807\pi$
$971$	− 400896.i	− 0.425200i	0.977139	−	0.212600i	$-0.0681932\pi$
$972$	0	0
$973$	512168.	0.540987
$974$	− 337819.i	− 0.356095i
$975$	0	0
$976$	−299236.	−0.314134
$977$	− 547999.i	− 0.574104i	−0.957915	−	0.287052i	$-0.907325\pi$
$977$	− 547999.i	− 0.574104i	0.957915	−	0.287052i	$-0.0926753\pi$
$978$	0	0
$979$	555310.	0.579389
$980$	115950.i	0.120731i
$981$	0	0
$982$	783707.	0.812701
$983$	− 1.36963e6i	− 1.41742i	−0.705502	−	0.708708i	$-0.749279\pi$
$983$	− 1.36963e6i	− 1.41742i	0.705502	−	0.708708i	$-0.250721\pi$
$984$	0	0
$985$	−169116.	−0.174306
$986$	635558.i	0.653735i
$987$	0	0
$988$	−209611.	−0.214734
$989$	− 915871.i	− 0.936358i
$990$	0	0
$991$	−406038.	−0.413446	−0.206723	−	0.978399i	$-0.566280\pi$
$991$	−406038.	−0.413446	−0.206723	+	0.978399i	$0.566280\pi$
$992$	36405.6i	0.0369952i
$993$	0	0
$994$	−469192.	−0.474873
$995$	− 970637.i	− 0.980417i
$996$	0	0
$997$	301551.	0.303368	0.151684	−	0.988429i	$-0.451530\pi$
$997$	301551.	0.303368	0.151684	+	0.988429i	$0.451530\pi$
$998$	− 1.08832e6i	− 1.09269i
$999$	0	0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -42.2558i q^{5} -18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -42.2558i q^{5} -18.5203 q^{7} +22.6274i q^{8} -119.518 q^{10} -63.4392i q^{11} -99.9091 q^{13} +52.3832i q^{14} +64.0000 q^{16} -450.455i q^{17} -262.252 q^{19} +338.047i q^{20} -179.433 q^{22} +673.509i q^{23} -1160.56 q^{25} +282.586i q^{26} +148.162 q^{28} -498.837i q^{29} -201.115 q^{31} -181.019i q^{32} -1274.08 q^{34} +782.589i q^{35} +2517.87 q^{37} +741.761i q^{38} +956.140 q^{40} -1698.04i q^{41} -1359.85 q^{43} +507.513i q^{44} +1904.97 q^{46} +685.838i q^{47} +343.000 q^{49} +3282.55i q^{50} +799.273 q^{52} +3969.53i q^{53} -2680.68 q^{55} -419.066i q^{56} -1410.93 q^{58} +741.755i q^{59} -4675.57 q^{61} +568.838i q^{62} -512.000 q^{64} +4221.74i q^{65} +7531.63 q^{67} +3603.64i q^{68} +2213.50 q^{70} +8956.91i q^{71} -1465.30 q^{73} -7121.61i q^{74} +2098.02 q^{76} +1174.91i q^{77} -5557.36 q^{79} -2704.37i q^{80} -4802.77 q^{82} +2588.64i q^{83} -19034.3 q^{85} +3846.24i q^{86} +1435.46 q^{88} +8753.43i q^{89} +1850.34 q^{91} -5388.07i q^{92} +1939.84 q^{94} +11081.7i q^{95} +4336.45 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) \) 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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