LMFDB - Embedded newform 378.5.b.a.323.8

Newspace parameters

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

Self dual:	no
Analytic conductor:	$39.0738460457$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.5443747577856.29
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			323.8
Root			$-0.982345i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.a.323.1

$q$-expansion

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +42.2558i q^{5} -18.5203 q^{7} -22.6274i q^{8} +O(q^{10})$	\(q+2.82843i q^{2} -8.00000 q^{4} +42.2558i q^{5} -18.5203 q^{7} -22.6274i q^{8} -119.518 q^{10} +63.4392i q^{11} -99.9091 q^{13} -52.3832i q^{14} +64.0000 q^{16} +450.455i q^{17} -262.252 q^{19} -338.047i q^{20} -179.433 q^{22} -673.509i q^{23} -1160.56 q^{25} -282.586i q^{26} +148.162 q^{28} +498.837i q^{29} -201.115 q^{31} +181.019i q^{32} -1274.08 q^{34} -782.589i q^{35} +2517.87 q^{37} -741.761i q^{38} +956.140 q^{40} +1698.04i q^{41} -1359.85 q^{43} -507.513i q^{44} +1904.97 q^{46} -685.838i q^{47} +343.000 q^{49} -3282.55i q^{50} +799.273 q^{52} -3969.53i q^{53} -2680.68 q^{55} +419.066i q^{56} -1410.93 q^{58} -741.755i q^{59} -4675.57 q^{61} -568.838i q^{62} -512.000 q^{64} -4221.74i q^{65} +7531.63 q^{67} -3603.64i q^{68} +2213.50 q^{70} -8956.91i q^{71} -1465.30 q^{73} +7121.61i q^{74} +2098.02 q^{76} -1174.91i q^{77} -5557.36 q^{79} +2704.37i q^{80} -4802.77 q^{82} -2588.64i q^{83} -19034.3 q^{85} -3846.24i q^{86} +1435.46 q^{88} -8753.43i q^{89} +1850.34 q^{91} +5388.07i q^{92} +1939.84 q^{94} -11081.7i q^{95} +4336.45 q^{97} +970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.320469\pi$
$5$	42.2558i	1.69023i	−0.534582	+	0.845117i	$0.679531\pi$
$6$	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	$0.0844300\pi$
$11$	63.4392i	0.524291i	−0.965028	+	0.262145i	$0.915570\pi$
$12$	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	$0.284442\pi$
$17$	450.455i	1.55867i	−0.626609	+	0.779334i	$0.715558\pi$
$18$	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	$-0.780346\pi$
$23$	− 673.509i	− 1.27317i	0.771205	−	0.636587i	$-0.219654\pi$
$24$	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	$0.0958441\pi$
$29$	498.837i	0.593148i	−0.955010	+	0.296574i	$0.904156\pi$
$30$	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	$0.168532\pi$
$41$	1698.04i	1.01013i	−0.863080	+	0.505067i	$0.831468\pi$
$42$	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	$-0.950386\pi$
$47$	− 685.838i	− 0.310474i	0.987877	−	0.155237i	$-0.0496142\pi$
$48$	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	$-0.750240\pi$
$53$	− 3969.53i	− 1.41315i	0.707640	−	0.706573i	$-0.249760\pi$
$54$	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	$-0.966022\pi$
$59$	− 741.755i	− 0.213087i	0.994308	−	0.106543i	$-0.0339783\pi$
$60$	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	$-0.651813\pi$
$71$	− 8956.91i	− 1.77681i	0.459059	−	0.888406i	$-0.348187\pi$
$72$	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	$-0.939838\pi$
$83$	− 2588.64i	− 0.375765i	0.982192	−	0.187882i	$-0.0601623\pi$
$84$	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	$-0.813657\pi$
$89$	− 8753.43i	− 1.10509i	0.833483	−	0.552545i	$-0.186343\pi$
$90$	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	$0.0192998\pi$
$101$	1236.26i	0.121190i	−0.998162	+	0.0605950i	$0.980700\pi$
$102$	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	$-0.933692\pi$
$107$	− 4735.53i	− 0.413619i	0.978381	−	0.206810i	$-0.0663081\pi$
$108$	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	$0.286159\pi$
$113$	19988.6i	1.56540i	−0.622398	+	0.782701i	$0.713841\pi$
$114$	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	$-0.924968\pi$
$131$	− 8015.64i	− 0.467084i	0.972347	−	0.233542i	$-0.0750317\pi$
$132$	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	$0.116619\pi$
$137$	13447.2i	0.716459i	−0.933634	+	0.358229i	$0.883381\pi$
$138$	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	$0.177139\pi$
$149$	23453.9i	1.05643i	−0.849110	+	0.528217i	$0.822861\pi$
$150$	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	$0.0105510\pi$
$167$	1848.54i	0.0662821i	−0.999451	+	0.0331410i	$0.989449\pi$
$168$	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	$0.237869\pi$
$173$	40682.6i	1.35930i	−0.733535	+	0.679652i	$0.762131\pi$
$174$	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	$-0.796267\pi$
$179$	− 38271.8i	− 1.19446i	0.802069	−	0.597232i	$-0.203733\pi$
$180$	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	$0.229438\pi$
$191$	48153.9i	1.31997i	−0.751278	+	0.659986i	$0.770562\pi$
$192$	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	$0.0164202\pi$
$197$	4002.20i	0.103126i	−0.998670	+	0.0515628i	$0.983580\pi$
$198$	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	$0.438494\pi$
$227$	101140.i	1.96278i	−0.192026	+	0.981390i	$0.561506\pi$
$228$	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	$0.0245318\pi$
$233$	8359.71i	0.153985i	−0.997032	+	0.0769926i	$0.975468\pi$
$234$	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	$0.0844913\pi$
$239$	29969.3i	0.524663i	−0.964978	+	0.262331i	$0.915509\pi$
$240$	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	$0.260594\pi$
$251$	92012.3i	1.46049i	−0.683186	+	0.730244i	$0.739406\pi$
$252$	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	$-0.925199\pi$
$257$	− 30757.4i	− 0.465676i	0.972516	−	0.232838i	$-0.0748011\pi$
$258$	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	$0.0873599\pi$
$263$	37491.9i	0.542034i	−0.962575	+	0.271017i	$0.912640\pi$
$264$	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	$0.0834473\pi$
$269$	37506.9i	0.518330i	−0.965833	+	0.259165i	$0.916553\pi$
$270$	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	$-0.942800\pi$
$281$	− 28226.0i	− 0.357467i	0.983898	−	0.178734i	$-0.0572000\pi$
$282$	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	$-0.944858\pi$
$293$	− 29595.2i	− 0.344735i	0.985033	−	0.172368i	$-0.0551417\pi$
$294$	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	$0.141160\pi$
$311$	83000.7i	0.858146i	−0.903270	+	0.429073i	$0.858840\pi$
$312$	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	$-0.900817\pi$
$317$	− 61615.0i	− 0.613152i	0.951846	−	0.306576i	$-0.0991833\pi$
$318$	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	$-0.718241\pi$
$347$	− 186399.i	− 1.54805i	0.633155	−	0.774025i	$-0.281759\pi$
$348$	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	$-0.992121\pi$
$353$	− 6167.83i	− 0.0494975i	0.999694	−	0.0247488i	$-0.00787858\pi$
$354$	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	$-0.989393\pi$
$359$	− 8588.08i	− 0.0666357i	0.999445	−	0.0333179i	$-0.0106074\pi$
$360$	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	$0.317518\pi$
$383$	246474.i	1.68025i	−0.542394	+	0.840125i	$0.682482\pi$
$384$	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	$-0.877457\pi$
$389$	− 113655.i	− 0.751083i	0.926805	−	0.375542i	$-0.122543\pi$
$390$	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	$0.0786625\pi$
$401$	78669.6i	0.489236i	−0.969620	+	0.244618i	$0.921337\pi$
$402$	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	$-0.898841\pi$
$419$	− 109718.i	− 0.624955i	0.949925	−	0.312477i	$-0.101159\pi$
$420$	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	$-0.802947\pi$
$431$	− 215583.i	− 1.16054i	0.814425	−	0.580269i	$-0.197053\pi$
$432$	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	$-0.744678\pi$
$443$	− 282139.i	− 1.43766i	0.695186	−	0.718830i	$-0.255322\pi$
$444$	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	$0.270463\pi$
$449$	302834.i	1.50214i	−0.660220	+	0.751072i	$0.729537\pi$
$450$	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	$-0.706688\pi$
$461$	− 338541.i	− 1.59298i	0.604653	−	0.796489i	$-0.293312\pi$
$462$	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	$-0.893253\pi$
$467$	− 143548.i	− 0.658208i	0.944294	−	0.329104i	$-0.106747\pi$
$468$	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	$0.255039\pi$
$479$	329574.i	1.43642i	−0.695825	+	0.718212i	$0.744961\pi$
$480$	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	$-0.805132\pi$
$491$	− 277082.i	− 1.14933i	0.818388	−	0.574666i	$-0.194868\pi$
$492$	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	$0.193257\pi$
$503$	288691.i	1.14103i	−0.821286	+	0.570516i	$0.806743\pi$
$504$	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	$-0.886374\pi$
$509$	− 181063.i	− 0.698866i	0.936961	−	0.349433i	$-0.113626\pi$
$510$	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	$-0.803845\pi$
$521$	− 313770.i	− 1.15594i	0.816057	−	0.577971i	$-0.196155\pi$
$522$	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	$0.0346247\pi$
$557$	67362.8i	0.217125i	−0.994090	+	0.108562i	$0.965375\pi$
$558$	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	$-0.912906\pi$
$563$	− 171298.i	− 0.540425i	0.962801	−	0.270212i	$-0.0870940\pi$
$564$	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	$0.0117111\pi$
$569$	23818.0i	0.0735666i	−0.999323	+	0.0367833i	$0.988289\pi$
$570$	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	$0.199891\pi$
$587$	404875.i	1.17502i	−0.809218	+	0.587509i	$0.800109\pi$
$588$	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	$-0.720212\pi$
$593$	− 541601.i	− 1.54018i	0.637938	−	0.770088i	$-0.279788\pi$
$594$	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	$-0.893070\pi$
$599$	− 236556.i	− 0.659295i	0.944104	−	0.329648i	$-0.106930\pi$
$600$	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	$0.156058\pi$
$617$	358506.i	0.941729i	−0.882206	+	0.470864i	$0.843942\pi$
$618$	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	$0.373531\pi$
$641$	757749.i	1.84421i	−0.386944	+	0.922103i	$0.626469\pi$
$642$	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	$0.141972\pi$
$647$	361157.i	0.862756i	−0.902171	+	0.431378i	$0.858028\pi$
$648$	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	$-0.813089\pi$
$653$	− 472486.i	− 1.10806i	0.832497	−	0.554030i	$-0.186911\pi$
$654$	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	$0.186039\pi$
$659$	479228.i	1.10350i	−0.834010	+	0.551749i	$0.813961\pi$
$660$	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	$-0.877433\pi$
$677$	− 344306.i	− 0.751220i	0.926778	−	0.375610i	$-0.122567\pi$
$678$	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	$-0.934817\pi$
$683$	− 189722.i	− 0.406701i	0.979106	−	0.203350i	$-0.0651831\pi$
$684$	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	$0.254963\pi$
$701$	705697.i	1.43609i	−0.695996	+	0.718046i	$0.745037\pi$
$702$	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	$0.154642\pi$
$719$	482776.i	0.933873i	−0.884291	+	0.466937i	$0.845358\pi$
$720$	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	$0.253138\pi$
$743$	788374.i	1.42809i	−0.700102	+	0.714043i	$0.746862\pi$
$744$	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	$-0.690908\pi$
$761$	− 956099.i	− 1.65095i	0.564439	−	0.825475i	$-0.309092\pi$
$762$	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	$0.187666\pi$
$773$	664455.i	1.11201i	−0.831181	+	0.556003i	$0.812334\pi$
$774$	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	$-0.988801\pi$
$797$	− 44688.5i	− 0.0703524i	0.999381	−	0.0351762i	$-0.0111992\pi$
$798$	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	$-0.906918\pi$
$809$	− 377340.i	− 0.576549i	0.957548	−	0.288274i	$-0.0930815\pi$
$810$	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	$0.133529\pi$
$821$	549071.i	0.814596i	−0.913295	+	0.407298i	$0.866471\pi$
$822$	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	$0.0115837\pi$
$827$	49767.0i	0.0727664i	−0.999338	+	0.0363832i	$0.988416\pi$
$828$	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	$-0.681113\pi$
$839$	− 1.18604e6i	− 1.68490i	0.538776	−	0.842449i	$-0.318887\pi$
$840$	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	$0.00602291\pi$
$857$	27792.1i	0.0378408i	−0.999821	+	0.0189204i	$0.993977\pi$
$858$	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	$0.0706185\pi$
$863$	327757.i	0.440078i	−0.975491	+	0.220039i	$0.929382\pi$
$864$	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	$0.0216790\pi$
$881$	105642.i	0.136108i	−0.997682	+	0.0680540i	$0.978321\pi$
$882$	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	$0.342755\pi$
$887$	1.38541e6i	1.76089i	−0.474152	+	0.880443i	$0.657245\pi$
$888$	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	$0.0558350\pi$
$911$	289663.i	0.349025i	−0.984655	+	0.174513i	$0.944165\pi$
$912$	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	$-0.979309\pi$
$929$	− 112122.i	− 0.129915i	0.997888	−	0.0649576i	$-0.0206912\pi$
$930$	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	$0.370662\pi$
$941$	1.62677e6i	1.83716i	−0.395239	+	0.918578i	$0.629338\pi$
$942$	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	$0.116489\pi$
$947$	641839.i	0.715692i	−0.933781	+	0.357846i	$0.883511\pi$
$948$	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	$-0.772759\pi$
$953$	− 1.18936e6i	− 1.30957i	0.755816	−	0.654785i	$-0.227241\pi$
$954$	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	$0.0681932\pi$
$971$	400896.i	0.425200i	−0.977139	+	0.212600i	$0.931807\pi$
$972$	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	$0.0926753\pi$
$977$	547999.i	0.574104i	−0.957915	+	0.287052i	$0.907325\pi$
$978$	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	$0.250721\pi$
$983$	1.36963e6i	1.41742i	−0.705502	+	0.708708i	$0.749279\pi$
$984$	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.323.8	yes	8
3.2	odd	2	inner	378.5.b.a.323.1	✓	8

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

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

Fields

Representations

Groups

Database

Properties

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