LMFDB - Embedded newform 162.5.b.a.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,5,Mod(161,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("162.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	162.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:	$16.7459340196$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.5.b.a.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +45.9483i q^{5} -80.5500 q^{7} -22.6274i q^{8} +O(q^{10})$	\(q+2.82843i q^{2} -8.00000 q^{4} +45.9483i q^{5} -80.5500 q^{7} -22.6274i q^{8} -129.962 q^{10} -113.110i q^{11} +70.2539 q^{13} -227.830i q^{14} +64.0000 q^{16} +256.030i q^{17} +192.435 q^{19} -367.587i q^{20} +319.923 q^{22} -543.928i q^{23} -1486.25 q^{25} +198.708i q^{26} +644.400 q^{28} -822.977i q^{29} +615.408 q^{31} +181.019i q^{32} -724.161 q^{34} -3701.14i q^{35} -2355.20 q^{37} +544.287i q^{38} +1039.69 q^{40} +502.171i q^{41} -164.719 q^{43} +904.879i q^{44} +1538.46 q^{46} -1041.23i q^{47} +4087.30 q^{49} -4203.75i q^{50} -562.031 q^{52} -182.004i q^{53} +5197.21 q^{55} +1822.64i q^{56} +2327.73 q^{58} +1811.76i q^{59} -2859.85 q^{61} +1740.64i q^{62} -512.000 q^{64} +3228.05i q^{65} -8895.85 q^{67} -2048.24i q^{68} +10468.4 q^{70} -3556.66i q^{71} -8008.40 q^{73} -6661.50i q^{74} -1539.48 q^{76} +9111.00i q^{77} -4026.40 q^{79} +2940.69i q^{80} -1420.35 q^{82} -1846.98i q^{83} -11764.1 q^{85} -465.896i q^{86} -2559.38 q^{88} +4204.70i q^{89} -5658.95 q^{91} +4351.42i q^{92} +2945.03 q^{94} +8842.05i q^{95} +13186.8 q^{97} +11560.6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} - 52 q^{7}+O(q^{10}) $ 4 * q - 32 * q^4 - 52 * q^7	$ 4 q - 32 q^{4} - 52 q^{7} - 312 q^{10} + 572 q^{13} + 256 q^{16} - 124 q^{19} + 864 q^{22} - 1892 q^{25} + 416 q^{28} + 3584 q^{31} - 1608 q^{34} - 9400 q^{37} + 2496 q^{40} + 3020 q^{43} - 2160 q^{46} + 9324 q^{49} - 4576 q^{52} + 11124 q^{55} - 2952 q^{58} - 4144 q^{61} - 2048 q^{64} - 12076 q^{67} + 18096 q^{70} - 1792 q^{73} + 992 q^{76} - 15004 q^{79} + 5376 q^{82} - 24048 q^{85} - 6912 q^{88} + 12220 q^{91} - 12912 q^{94} + 46304 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 - 52 * q^7 - 312 * q^10 + 572 * q^13 + 256 * q^16 - 124 * q^19 + 864 * q^22 - 1892 * q^25 + 416 * q^28 + 3584 * q^31 - 1608 * q^34 - 9400 * q^37 + 2496 * q^40 + 3020 * q^43 - 2160 * q^46 + 9324 * q^49 - 4576 * q^52 + 11124 * q^55 - 2952 * q^58 - 4144 * q^61 - 2048 * q^64 - 12076 * q^67 + 18096 * q^70 - 1792 * q^73 + 992 * q^76 - 15004 * q^79 + 5376 * q^82 - 24048 * q^85 - 6912 * q^88 + 12220 * q^91 - 12912 * q^94 + 46304 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	45.9483i	1.83793i	0.394335	+	0.918967i	$0.370975\pi$
$5$	45.9483i	1.83793i	−0.394335	+	0.918967i	$0.629025\pi$
$6$	0	0
$7$	−80.5500	−1.64388	−0.821939	−	0.569576i	$-0.807107\pi$
$7$	−80.5500	−1.64388	−0.821939	+	0.569576i	$0.807107\pi$
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−129.962	−1.29962
$11$	− 113.110i	− 0.934792i	−0.884048	−	0.467396i	$-0.845192\pi$
$11$	− 113.110i	− 0.934792i	0.884048	−	0.467396i	$-0.154808\pi$
$12$	0	0
$13$	70.2539	0.415703	0.207852	−	0.978160i	$-0.433353\pi$
$13$	70.2539	0.415703	0.207852	+	0.978160i	$0.433353\pi$
$14$	− 227.830i	− 1.16240i
$15$	0	0
$16$	64.0000	0.250000
$17$	256.030i	0.885916i	0.896542	+	0.442958i	$0.146071\pi$
$17$	256.030i	0.885916i	−0.896542	+	0.442958i	$0.853929\pi$
$18$	0	0
$19$	192.435	0.533060	0.266530	−	0.963827i	$-0.414123\pi$
$19$	192.435	0.533060	0.266530	+	0.963827i	$0.414123\pi$
$20$	− 367.587i	− 0.918967i
$21$	0	0
$22$	319.923	0.660998
$23$	− 543.928i	− 1.02822i	−0.857724	−	0.514110i	$-0.828122\pi$
$23$	− 543.928i	− 1.02822i	0.857724	−	0.514110i	$-0.171878\pi$
$24$	0	0
$25$	−1486.25	−2.37800
$26$	198.708i	0.293947i
$27$	0	0
$28$	644.400	0.821939
$29$	− 822.977i	− 0.978569i	−0.872124	−	0.489285i	$-0.837258\pi$
$29$	− 822.977i	− 0.978569i	0.872124	−	0.489285i	$-0.162742\pi$
$30$	0	0
$31$	615.408	0.640383	0.320191	−	0.947353i	$-0.396253\pi$
$31$	615.408	0.640383	0.320191	+	0.947353i	$0.396253\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	−724.161	−0.626437
$35$	− 3701.14i	− 3.02134i
$36$	0	0
$37$	−2355.20	−1.72038	−0.860189	−	0.509976i	$-0.829654\pi$
$37$	−2355.20	−1.72038	−0.860189	+	0.509976i	$0.829654\pi$
$38$	544.287i	0.376930i
$39$	0	0
$40$	1039.69	0.649808
$41$	502.171i	0.298733i	0.988782	+	0.149367i	$0.0477235\pi$
$41$	502.171i	0.298733i	−0.988782	+	0.149367i	$0.952277\pi$
$42$	0	0
$43$	−164.719	−0.0890854	−0.0445427	−	0.999007i	$-0.514183\pi$
$43$	−164.719	−0.0890854	−0.0445427	+	0.999007i	$0.514183\pi$
$44$	904.879i	0.467396i
$45$	0	0
$46$	1538.46	0.727061
$47$	− 1041.23i	− 0.471356i	−0.971831	−	0.235678i	$-0.924269\pi$
$47$	− 1041.23i	− 0.471356i	0.971831	−	0.235678i	$-0.0757310\pi$
$48$	0	0
$49$	4087.30	1.70233
$50$	− 4203.75i	− 1.68150i
$51$	0	0
$52$	−562.031	−0.207852
$53$	− 182.004i	− 0.0647930i	−0.999475	−	0.0323965i	$-0.989686\pi$
$53$	− 182.004i	− 0.0647930i	0.999475	−	0.0323965i	$-0.0103139\pi$
$54$	0	0
$55$	5197.21	1.71809
$56$	1822.64i	0.581198i
$57$	0	0
$58$	2327.73	0.691953
$59$	1811.76i	0.520473i	0.965545	+	0.260236i	$0.0838005\pi$
$59$	1811.76i	0.520473i	−0.965545	+	0.260236i	$0.916200\pi$
$60$	0	0
$61$	−2859.85	−0.768570	−0.384285	−	0.923214i	$-0.625552\pi$
$61$	−2859.85	−0.768570	−0.384285	+	0.923214i	$0.625552\pi$
$62$	1740.64i	0.452819i
$63$	0	0
$64$	−512.000	−0.125000
$65$	3228.05i	0.764035i
$66$	0	0
$67$	−8895.85	−1.98170	−0.990850	−	0.134970i	$-0.956906\pi$
$67$	−8895.85	−1.98170	−0.990850	+	0.134970i	$0.956906\pi$
$68$	− 2048.24i	− 0.442958i
$69$	0	0
$70$	10468.4	2.13641
$71$	− 3556.66i	− 0.705547i	−0.935709	−	0.352773i	$-0.885239\pi$
$71$	− 3556.66i	− 0.705547i	0.935709	−	0.352773i	$-0.114761\pi$
$72$	0	0
$73$	−8008.40	−1.50280	−0.751398	−	0.659849i	$-0.770620\pi$
$73$	−8008.40	−1.50280	−0.751398	+	0.659849i	$0.770620\pi$
$74$	− 6661.50i	− 1.21649i
$75$	0	0
$76$	−1539.48	−0.266530
$77$	9111.00i	1.53668i
$78$	0	0
$79$	−4026.40	−0.645152	−0.322576	−	0.946544i	$-0.604549\pi$
$79$	−4026.40	−0.645152	−0.322576	+	0.946544i	$0.604549\pi$
$80$	2940.69i	0.459483i
$81$	0	0
$82$	−1420.35	−0.211236
$83$	− 1846.98i	− 0.268106i	−0.990974	−	0.134053i	$-0.957201\pi$
$83$	− 1846.98i	− 0.268106i	0.990974	−	0.134053i	$-0.0427993\pi$
$84$	0	0
$85$	−11764.1	−1.62825
$86$	− 465.896i	− 0.0629929i
$87$	0	0
$88$	−2559.38	−0.330499
$89$	4204.70i	0.530830i	0.964134	+	0.265415i	$0.0855089\pi$
$89$	4204.70i	0.530830i	−0.964134	+	0.265415i	$0.914491\pi$
$90$	0	0
$91$	−5658.95	−0.683365
$92$	4351.42i	0.514110i
$93$	0	0
$94$	2945.03	0.333299
$95$	8842.05i	0.979728i
$96$	0	0
$97$	13186.8	1.40151	0.700755	−	0.713402i	$-0.252847\pi$
$97$	13186.8	1.40151	0.700755	+	0.713402i	$0.252847\pi$
$98$	11560.6i	1.20373i
$99$	0	0
$100$	11890.0	1.18900
$101$	9990.28i	0.979344i	0.871907	+	0.489672i	$0.162883\pi$
$101$	9990.28i	0.979344i	−0.871907	+	0.489672i	$0.837117\pi$
$102$	0	0
$103$	−3278.99	−0.309076	−0.154538	−	0.987987i	$-0.549389\pi$
$103$	−3278.99	−0.309076	−0.154538	+	0.987987i	$0.549389\pi$
$104$	− 1589.66i	− 0.146973i
$105$	0	0
$106$	514.784	0.0458156
$107$	20923.5i	1.82754i	0.406236	+	0.913768i	$0.366841\pi$
$107$	20923.5i	1.82754i	−0.406236	+	0.913768i	$0.633159\pi$
$108$	0	0
$109$	3340.77	0.281186	0.140593	−	0.990067i	$-0.455099\pi$
$109$	3340.77	0.281186	0.140593	+	0.990067i	$0.455099\pi$
$110$	14699.9i	1.21487i
$111$	0	0
$112$	−5155.20	−0.410969
$113$	− 8872.38i	− 0.694838i	−0.937710	−	0.347419i	$-0.887058\pi$
$113$	− 8872.38i	− 0.694838i	0.937710	−	0.347419i	$-0.112942\pi$
$114$	0	0
$115$	24992.6	1.88980
$116$	6583.81i	0.489285i
$117$	0	0
$118$	−5124.45	−0.368030
$119$	− 20623.2i	− 1.45634i
$120$	0	0
$121$	1847.16	0.126163
$122$	− 8088.88i	− 0.543461i
$123$	0	0
$124$	−4923.26	−0.320191
$125$	− 39573.0i	− 2.53267i
$126$	0	0
$127$	−15509.1	−0.961569	−0.480784	−	0.876839i	$-0.659648\pi$
$127$	−15509.1	−0.961569	−0.480784	+	0.876839i	$0.659648\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	−9130.30	−0.540254
$131$	13596.5i	0.792289i	0.918188	+	0.396145i	$0.129652\pi$
$131$	13596.5i	0.792289i	−0.918188	+	0.396145i	$0.870348\pi$
$132$	0	0
$133$	−15500.6	−0.876285
$134$	− 25161.3i	− 1.40127i
$135$	0	0
$136$	5793.29	0.313219
$137$	21664.7i	1.15428i	0.816644	+	0.577142i	$0.195832\pi$
$137$	21664.7i	1.15428i	−0.816644	+	0.577142i	$0.804168\pi$
$138$	0	0
$139$	15438.2	0.799040	0.399520	−	0.916725i	$-0.369177\pi$
$139$	15438.2	0.799040	0.399520	+	0.916725i	$0.369177\pi$
$140$	29609.1i	1.51067i
$141$	0	0
$142$	10059.8	0.498897
$143$	− 7946.41i	− 0.388596i
$144$	0	0
$145$	37814.4	1.79855
$146$	− 22651.2i	− 1.06264i
$147$	0	0
$148$	18841.6	0.860189
$149$	− 20472.0i	− 0.922120i	−0.887369	−	0.461060i	$-0.847469\pi$
$149$	− 20472.0i	− 0.922120i	0.887369	−	0.461060i	$-0.152531\pi$
$150$	0	0
$151$	−32004.7	−1.40365	−0.701827	−	0.712347i	$-0.747632\pi$
$151$	−32004.7	−1.40365	−0.701827	+	0.712347i	$0.747632\pi$
$152$	− 4354.30i	− 0.188465i
$153$	0	0
$154$	−25769.8	−1.08660
$155$	28277.0i	1.17698i
$156$	0	0
$157$	−24207.2	−0.982075	−0.491037	−	0.871139i	$-0.663382\pi$
$157$	−24207.2	−0.982075	−0.491037	+	0.871139i	$0.663382\pi$
$158$	− 11388.4i	− 0.456192i
$159$	0	0
$160$	−8317.54	−0.324904
$161$	43813.4i	1.69027i
$162$	0	0
$163$	21091.0	0.793821	0.396911	−	0.917857i	$-0.370082\pi$
$163$	21091.0	0.793821	0.396911	+	0.917857i	$0.370082\pi$
$164$	− 4017.37i	− 0.149367i
$165$	0	0
$166$	5224.06	0.189580
$167$	− 12967.4i	− 0.464964i	−0.972601	−	0.232482i	$-0.925315\pi$
$167$	− 12967.4i	− 0.464964i	0.972601	−	0.232482i	$-0.0746848\pi$
$168$	0	0
$169$	−23625.4	−0.827191
$170$	− 33274.0i	− 1.15135i
$171$	0	0
$172$	1317.75	0.0445427
$173$	− 41778.6i	− 1.39592i	−0.716135	−	0.697962i	$-0.754090\pi$
$173$	− 41778.6i	− 1.39592i	0.716135	−	0.697962i	$-0.245910\pi$
$174$	0	0
$175$	119717.	3.90914
$176$	− 7239.03i	− 0.233698i
$177$	0	0
$178$	−11892.7	−0.375353
$179$	13288.6i	0.414736i	0.978263	+	0.207368i	$0.0664897\pi$
$179$	13288.6i	0.414736i	−0.978263	+	0.207368i	$0.933510\pi$
$180$	0	0
$181$	−28910.7	−0.882475	−0.441237	−	0.897390i	$-0.645460\pi$
$181$	−28910.7	−0.882475	−0.441237	+	0.897390i	$0.645460\pi$
$182$	− 16005.9i	− 0.483212i
$183$	0	0
$184$	−12307.7	−0.363530
$185$	− 108217.i	− 3.16194i
$186$	0	0
$187$	28959.5	0.828148
$188$	8329.80i	0.235678i
$189$	0	0
$190$	−25009.1	−0.692773
$191$	− 50359.0i	− 1.38042i	−0.723610	−	0.690209i	$-0.757518\pi$
$191$	− 50359.0i	− 1.38042i	0.723610	−	0.690209i	$-0.242482\pi$
$192$	0	0
$193$	−3466.02	−0.0930500	−0.0465250	−	0.998917i	$-0.514815\pi$
$193$	−3466.02	−0.0930500	−0.0465250	+	0.998917i	$0.514815\pi$
$194$	37297.9i	0.991017i
$195$	0	0
$196$	−32698.4	−0.851166
$197$	− 7813.76i	− 0.201339i	−0.994920	−	0.100669i	$-0.967902\pi$
$197$	− 7813.76i	− 0.201339i	0.994920	−	0.100669i	$-0.0320984\pi$
$198$	0	0
$199$	29153.3	0.736176	0.368088	−	0.929791i	$-0.380013\pi$
$199$	29153.3	0.736176	0.368088	+	0.929791i	$0.380013\pi$
$200$	33630.0i	0.840750i
$201$	0	0
$202$	−28256.8	−0.692501
$203$	66290.8i	1.60865i
$204$	0	0
$205$	−23073.9	−0.549052
$206$	− 9274.37i	− 0.218550i
$207$	0	0
$208$	4496.25	0.103926
$209$	− 21766.2i	− 0.498300i
$210$	0	0
$211$	−47158.0	−1.05923	−0.529615	−	0.848238i	$-0.677664\pi$
$211$	−47158.0	−1.05923	−0.529615	+	0.848238i	$0.677664\pi$
$212$	1456.03i	0.0323965i
$213$	0	0
$214$	−59180.5	−1.29226
$215$	− 7568.56i	− 0.163733i
$216$	0	0
$217$	−49571.1	−1.05271
$218$	9449.13i	0.198829i
$219$	0	0
$220$	−41577.7	−0.859043
$221$	17987.1i	0.368278i
$222$	0	0
$223$	10053.3	0.202161	0.101081	−	0.994878i	$-0.467770\pi$
$223$	10053.3	0.202161	0.101081	+	0.994878i	$0.467770\pi$
$224$	− 14581.1i	− 0.290599i
$225$	0	0
$226$	25094.9	0.491324
$227$	− 29951.8i	− 0.581260i	−0.956835	−	0.290630i	$-0.906135\pi$
$227$	− 29951.8i	− 0.581260i	0.956835	−	0.290630i	$-0.0938649\pi$
$228$	0	0
$229$	−20133.4	−0.383926	−0.191963	−	0.981402i	$-0.561485\pi$
$229$	−20133.4	−0.383926	−0.191963	+	0.981402i	$0.561485\pi$
$230$	70689.7i	1.33629i
$231$	0	0
$232$	−18621.8	−0.345977
$233$	19079.1i	0.351436i	0.984441	+	0.175718i	$0.0562246\pi$
$233$	19079.1i	0.351436i	−0.984441	+	0.175718i	$0.943775\pi$
$234$	0	0
$235$	47842.6	0.866321
$236$	− 14494.1i	− 0.260236i
$237$	0	0
$238$	58331.2	1.02979
$239$	− 90259.6i	− 1.58015i	−0.613012	−	0.790073i	$-0.710042\pi$
$239$	− 90259.6i	− 1.58015i	0.613012	−	0.790073i	$-0.289958\pi$
$240$	0	0
$241$	30161.0	0.519291	0.259646	−	0.965704i	$-0.416394\pi$
$241$	30161.0	0.519291	0.259646	+	0.965704i	$0.416394\pi$
$242$	5224.54i	0.0892109i
$243$	0	0
$244$	22878.8	0.384285
$245$	187805.i	3.12877i
$246$	0	0
$247$	13519.3	0.221595
$248$	− 13925.1i	− 0.226409i
$249$	0	0
$250$	111929.	1.79087
$251$	16436.6i	0.260894i	0.991455	+	0.130447i	$0.0416412\pi$
$251$	16436.6i	0.260894i	−0.991455	+	0.130447i	$0.958359\pi$
$252$	0	0
$253$	−61523.6	−0.961172
$254$	− 43866.5i	− 0.679932i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 23244.0i	− 0.351920i	−0.984397	−	0.175960i	$-0.943697\pi$
$257$	− 23244.0i	− 0.351920i	0.984397	−	0.175960i	$-0.0563030\pi$
$258$	0	0
$259$	189711.	2.82809
$260$	− 25824.4i	− 0.382018i
$261$	0	0
$262$	−38456.6	−0.560233
$263$	− 30701.7i	− 0.443865i	−0.975062	−	0.221932i	$-0.928764\pi$
$263$	− 30701.7i	− 0.443865i	0.975062	−	0.221932i	$-0.0712364\pi$
$264$	0	0
$265$	8362.76	0.119085
$266$	− 43842.3i	− 0.619627i
$267$	0	0
$268$	71166.8	0.990850
$269$	98285.6i	1.35827i	0.734014	+	0.679134i	$0.237644\pi$
$269$	98285.6i	1.35827i	−0.734014	+	0.679134i	$0.762356\pi$
$270$	0	0
$271$	111736.	1.52144	0.760722	−	0.649078i	$-0.224845\pi$
$271$	111736.	1.52144	0.760722	+	0.649078i	$0.224845\pi$
$272$	16385.9i	0.221479i
$273$	0	0
$274$	−61277.2	−0.816202
$275$	168110.i	2.22294i
$276$	0	0
$277$	−64408.5	−0.839429	−0.419714	−	0.907656i	$-0.637870\pi$
$277$	−64408.5	−0.839429	−0.419714	+	0.907656i	$0.637870\pi$
$278$	43666.0i	0.565007i
$279$	0	0
$280$	−83747.2	−1.06820
$281$	124863.i	1.58133i	0.612251	+	0.790664i	$0.290264\pi$
$281$	124863.i	1.58133i	−0.612251	+	0.790664i	$0.709736\pi$
$282$	0	0
$283$	−125798.	−1.57072	−0.785362	−	0.619037i	$-0.787523\pi$
$283$	−125798.	−1.57072	−0.785362	+	0.619037i	$0.787523\pi$
$284$	28453.3i	0.352773i
$285$	0	0
$286$	22475.8	0.274779
$287$	− 40449.8i	− 0.491081i
$288$	0	0
$289$	17969.8	0.215153
$290$	106955.i	1.27176i
$291$	0	0
$292$	64067.2	0.751398
$293$	70372.5i	0.819725i	0.912147	+	0.409862i	$0.134423\pi$
$293$	70372.5i	0.819725i	−0.912147	+	0.409862i	$0.865577\pi$
$294$	0	0
$295$	−83247.6	−0.956594
$296$	53292.0i	0.608245i
$297$	0	0
$298$	57903.5	0.652037
$299$	− 38213.1i	− 0.427434i
$300$	0	0
$301$	13268.1	0.146446
$302$	− 90523.0i	− 0.992533i
$303$	0	0
$304$	12315.8	0.133265
$305$	− 131405.i	− 1.41258i
$306$	0	0
$307$	−26790.2	−0.284249	−0.142125	−	0.989849i	$-0.545393\pi$
$307$	−26790.2	−0.284249	−0.142125	+	0.989849i	$0.545393\pi$
$308$	− 72888.0i	− 0.768342i
$309$	0	0
$310$	−79979.3	−0.832251
$311$	127879.i	1.32215i	0.750321	+	0.661074i	$0.229899\pi$
$311$	127879.i	1.32215i	−0.750321	+	0.661074i	$0.770101\pi$
$312$	0	0
$313$	9564.22	0.0976249	0.0488125	−	0.998808i	$-0.484456\pi$
$313$	9564.22	0.0976249	0.0488125	+	0.998808i	$0.484456\pi$
$314$	− 68468.2i	− 0.694432i
$315$	0	0
$316$	32211.2	0.322576
$317$	38993.9i	0.388042i	0.980997	+	0.194021i	$0.0621529\pi$
$317$	38993.9i	0.388042i	−0.980997	+	0.194021i	$0.937847\pi$
$318$	0	0
$319$	−93086.8	−0.914759
$320$	− 23525.5i	− 0.229742i
$321$	0	0
$322$	−123923.	−1.19520
$323$	49269.0i	0.472246i
$324$	0	0
$325$	−104415.	−0.988542
$326$	59654.4i	0.561316i
$327$	0	0
$328$	11362.8	0.105618
$329$	83870.7i	0.774851i
$330$	0	0
$331$	50434.2	0.460330	0.230165	−	0.973152i	$-0.426073\pi$
$331$	50434.2	0.460330	0.230165	+	0.973152i	$0.426073\pi$
$332$	14775.9i	0.134053i
$333$	0	0
$334$	36677.3	0.328780
$335$	− 408749.i	− 3.64223i
$336$	0	0
$337$	−127374.	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$337$	−127374.	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$338$	− 66822.7i	− 0.584912i
$339$	0	0
$340$	94113.1	0.814127
$341$	− 69608.7i	− 0.598625i
$342$	0	0
$343$	−135831.	−1.15455
$344$	3727.17i	0.0314965i
$345$	0	0
$346$	118168.	0.987067
$347$	− 193817.i	− 1.60965i	−0.593510	−	0.804827i	$-0.702258\pi$
$347$	− 193817.i	− 1.60965i	0.593510	−	0.804827i	$-0.297742\pi$
$348$	0	0
$349$	104069.	0.854417	0.427209	−	0.904153i	$-0.359497\pi$
$349$	104069.	0.854417	0.427209	+	0.904153i	$0.359497\pi$
$350$	338612.i	2.76418i
$351$	0	0
$352$	20475.1	0.165250
$353$	160774.i	1.29022i	0.764088	+	0.645112i	$0.223189\pi$
$353$	160774.i	1.29022i	−0.764088	+	0.645112i	$0.776811\pi$
$354$	0	0
$355$	163423.	1.29675
$356$	− 33637.6i	− 0.265415i
$357$	0	0
$358$	−37585.7	−0.293263
$359$	21776.0i	0.168962i	0.996425	+	0.0844811i	$0.0269233\pi$
$359$	21776.0i	0.168962i	−0.996425	+	0.0844811i	$0.973077\pi$
$360$	0	0
$361$	−93289.9	−0.715847
$362$	− 81771.9i	− 0.624004i
$363$	0	0
$364$	45271.6	0.341683
$365$	− 367973.i	− 2.76204i
$366$	0	0
$367$	−251965.	−1.87072	−0.935360	−	0.353698i	$-0.884924\pi$
$367$	−251965.	−1.87072	−0.935360	+	0.353698i	$0.884924\pi$
$368$	− 34811.4i	− 0.257055i
$369$	0	0
$370$	306085.	2.23583
$371$	14660.4i	0.106512i
$372$	0	0
$373$	128026.	0.920196	0.460098	−	0.887868i	$-0.347814\pi$
$373$	128026.	0.920196	0.460098	+	0.887868i	$0.347814\pi$
$374$	81909.8i	0.585589i
$375$	0	0
$376$	−23560.2	−0.166649
$377$	− 57817.3i	− 0.406795i
$378$	0	0
$379$	92997.5	0.647430	0.323715	−	0.946155i	$-0.395068\pi$
$379$	92997.5	0.647430	0.323715	+	0.946155i	$0.395068\pi$
$380$	− 70736.4i	− 0.489864i
$381$	0	0
$382$	142437.	0.976103
$383$	278293.i	1.89716i	0.316537	+	0.948580i	$0.397480\pi$
$383$	278293.i	1.89716i	−0.316537	+	0.948580i	$0.602520\pi$
$384$	0	0
$385$	−418635.	−2.82432
$386$	− 9803.38i	− 0.0657963i
$387$	0	0
$388$	−105494.	−0.700755
$389$	− 138923.i	− 0.918071i	−0.888418	−	0.459035i	$-0.848195\pi$
$389$	− 138923.i	− 0.918071i	0.888418	−	0.459035i	$-0.151805\pi$
$390$	0	0
$391$	139262.	0.910916
$392$	− 92485.0i	− 0.601865i
$393$	0	0
$394$	22100.6	0.142368
$395$	− 185006.i	− 1.18575i
$396$	0	0
$397$	−1760.70	−0.0111713	−0.00558564	−	0.999984i	$-0.501778\pi$
$397$	−1760.70	−0.0111713	−0.00558564	+	0.999984i	$0.501778\pi$
$398$	82457.9i	0.520555i
$399$	0	0
$400$	−95120.0	−0.594500
$401$	41057.9i	0.255334i	0.991817	+	0.127667i	$0.0407489\pi$
$401$	41057.9i	0.255334i	−0.991817	+	0.127667i	$0.959251\pi$
$402$	0	0
$403$	43234.8	0.266209
$404$	− 79922.3i	− 0.489672i
$405$	0	0
$406$	−187499.	−1.13749
$407$	266396.i	1.60820i
$408$	0	0
$409$	−166714.	−0.996608	−0.498304	−	0.867002i	$-0.666044\pi$
$409$	−166714.	−0.996608	−0.498304	+	0.867002i	$0.666044\pi$
$410$	− 65262.9i	− 0.388238i
$411$	0	0
$412$	26231.9	0.154538
$413$	− 145938.i	− 0.855593i
$414$	0	0
$415$	84865.9	0.492762
$416$	12717.3i	0.0734867i
$417$	0	0
$418$	61564.2	0.352351
$419$	331708.i	1.88942i	0.327912	+	0.944708i	$0.393655\pi$
$419$	331708.i	1.88942i	−0.327912	+	0.944708i	$0.606345\pi$
$420$	0	0
$421$	−187827.	−1.05973	−0.529863	−	0.848083i	$-0.677757\pi$
$421$	−187827.	−1.05973	−0.529863	+	0.848083i	$0.677757\pi$
$422$	− 133383.i	− 0.748989i
$423$	0	0
$424$	−4118.27	−0.0229078
$425$	− 380524.i	− 2.10671i
$426$	0	0
$427$	230361.	1.26343
$428$	− 167388.i	− 0.913768i
$429$	0	0
$430$	21407.1	0.115777
$431$	6952.01i	0.0374245i	0.999825	+	0.0187122i	$0.00595664\pi$
$431$	6952.01i	0.0374245i	−0.999825	+	0.0187122i	$0.994043\pi$
$432$	0	0
$433$	−37941.0	−0.202364	−0.101182	−	0.994868i	$-0.532262\pi$
$433$	−37941.0	−0.202364	−0.101182	+	0.994868i	$0.532262\pi$
$434$	− 140208.i	− 0.744379i
$435$	0	0
$436$	−26726.2	−0.140593
$437$	− 104671.i	− 0.548102i
$438$	0	0
$439$	248405.	1.28893	0.644467	−	0.764632i	$-0.277079\pi$
$439$	248405.	1.28893	0.644467	+	0.764632i	$0.277079\pi$
$440$	− 117599.i	− 0.607435i
$441$	0	0
$442$	−50875.1	−0.260412
$443$	162849.i	0.829806i	0.909866	+	0.414903i	$0.136184\pi$
$443$	162849.i	0.829806i	−0.909866	+	0.414903i	$0.863816\pi$
$444$	0	0
$445$	−193199.	−0.975630
$446$	28435.0i	0.142950i
$447$	0	0
$448$	41241.6	0.205485
$449$	− 139250.i	− 0.690723i	−0.938470	−	0.345361i	$-0.887756\pi$
$449$	− 139250.i	− 0.690723i	0.938470	−	0.345361i	$-0.112244\pi$
$450$	0	0
$451$	56800.5	0.279254
$452$	70979.0i	0.347419i
$453$	0	0
$454$	84716.4	0.411013
$455$	− 260019.i	− 1.25598i
$456$	0	0
$457$	250539.	1.19962	0.599810	−	0.800143i	$-0.295243\pi$
$457$	250539.	1.19962	0.599810	+	0.800143i	$0.295243\pi$
$458$	− 56946.0i	− 0.271476i
$459$	0	0
$460$	−199941.	−0.944899
$461$	78186.5i	0.367900i	0.982936	+	0.183950i	$0.0588885\pi$
$461$	78186.5i	0.367900i	−0.982936	+	0.183950i	$0.941112\pi$
$462$	0	0
$463$	352591.	1.64479	0.822393	−	0.568920i	$-0.192639\pi$
$463$	352591.	1.64479	0.822393	+	0.568920i	$0.192639\pi$
$464$	− 52670.5i	− 0.244642i
$465$	0	0
$466$	−53963.8	−0.248502
$467$	− 38384.6i	− 0.176005i	−0.996120	−	0.0880023i	$-0.971952\pi$
$467$	− 38384.6i	− 0.176005i	0.996120	−	0.0880023i	$-0.0280483\pi$
$468$	0	0
$469$	716560.	3.25767
$470$	135319.i	0.612581i
$471$	0	0
$472$	40995.6	0.184015
$473$	18631.3i	0.0832764i
$474$	0	0
$475$	−286006.	−1.26762
$476$	164986.i	0.728169i
$477$	0	0
$478$	255293.	1.11733
$479$	52021.2i	0.226730i	0.993553	+	0.113365i	$0.0361630\pi$
$479$	52021.2i	0.226730i	−0.993553	+	0.113365i	$0.963837\pi$
$480$	0	0
$481$	−165462.	−0.715166
$482$	85308.1i	0.367195i
$483$	0	0
$484$	−14777.2	−0.0630816
$485$	605912.i	2.57588i
$486$	0	0
$487$	−253420.	−1.06852	−0.534261	−	0.845320i	$-0.679410\pi$
$487$	−253420.	−1.06852	−0.534261	+	0.845320i	$0.679410\pi$
$488$	64711.0i	0.271731i
$489$	0	0
$490$	−531192.	−2.21238
$491$	− 182553.i	− 0.757228i	−0.925555	−	0.378614i	$-0.876401\pi$
$491$	− 182553.i	− 0.757228i	0.925555	−	0.378614i	$-0.123599\pi$
$492$	0	0
$493$	210707.	0.866930
$494$	38238.3i	0.156691i
$495$	0	0
$496$	39386.1	0.160096
$497$	286489.i	1.15983i
$498$	0	0
$499$	34543.2	0.138727	0.0693636	−	0.997591i	$-0.477903\pi$
$499$	34543.2	0.138727	0.0693636	+	0.997591i	$0.477903\pi$
$500$	316584.i	1.26634i
$501$	0	0
$502$	−46489.7	−0.184480
$503$	466084.i	1.84216i	0.389368	+	0.921082i	$0.372693\pi$
$503$	466084.i	1.84216i	−0.389368	+	0.921082i	$0.627307\pi$
$504$	0	0
$505$	−459037.	−1.79997
$506$	− 174015.i	− 0.679651i
$507$	0	0
$508$	124073.	0.480784
$509$	− 157911.i	− 0.609506i	−0.952431	−	0.304753i	$-0.901426\pi$
$509$	− 157911.i	− 0.609506i	0.952431	−	0.304753i	$-0.0985739\pi$
$510$	0	0
$511$	645077.	2.47041
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	65743.9	0.248845
$515$	− 150664.i	− 0.568061i
$516$	0	0
$517$	−117773.	−0.440620
$518$	536584.i	1.99976i
$519$	0	0
$520$	73042.4	0.270127
$521$	− 39855.1i	− 0.146828i	−0.997302	−	0.0734139i	$-0.976611\pi$
$521$	− 39855.1i	− 0.146828i	0.997302	−	0.0734139i	$-0.0233894\pi$
$522$	0	0
$523$	−250487.	−0.915762	−0.457881	−	0.889014i	$-0.651391\pi$
$523$	−250487.	−0.915762	−0.457881	+	0.889014i	$0.651391\pi$
$524$	− 108772.i	− 0.396145i
$525$	0	0
$526$	86837.4	0.313860
$527$	157563.i	0.567325i
$528$	0	0
$529$	−16016.8	−0.0572352
$530$	23653.5i	0.0842060i
$531$	0	0
$532$	124005.	0.438142
$533$	35279.4i	0.124184i
$534$	0	0
$535$	−961398.	−3.35889
$536$	201290.i	0.700637i
$537$	0	0
$538$	−277994.	−0.960440
$539$	− 462314.i	− 1.59133i
$540$	0	0
$541$	−324441.	−1.10852	−0.554258	−	0.832345i	$-0.686998\pi$
$541$	−324441.	−1.10852	−0.554258	+	0.832345i	$0.686998\pi$
$542$	316038.i	1.07582i
$543$	0	0
$544$	−46346.3	−0.156609
$545$	153503.i	0.516801i
$546$	0	0
$547$	14074.4	0.0470387	0.0235193	−	0.999723i	$-0.492513\pi$
$547$	14074.4	0.0470387	0.0235193	+	0.999723i	$0.492513\pi$
$548$	− 173318.i	− 0.577142i
$549$	0	0
$550$	−475486.	−1.57185
$551$	− 158369.i	− 0.521636i
$552$	0	0
$553$	324326.	1.06055
$554$	− 182175.i	− 0.593566i
$555$	0	0
$556$	−123506.	−0.399520
$557$	94500.4i	0.304595i	0.988335	+	0.152298i	$0.0486672\pi$
$557$	94500.4i	0.304595i	−0.988335	+	0.152298i	$0.951333\pi$
$558$	0	0
$559$	−11572.1	−0.0370331
$560$	− 236873.i	− 0.755334i
$561$	0	0
$562$	−353166.	−1.11817
$563$	− 107077.i	− 0.337815i	−0.985632	−	0.168908i	$-0.945976\pi$
$563$	− 107077.i	− 0.337815i	0.985632	−	0.168908i	$-0.0540239\pi$
$564$	0	0
$565$	407671.	1.27707
$566$	− 355810.i	− 1.11067i
$567$	0	0
$568$	−80478.0	−0.249448
$569$	− 237865.i	− 0.734694i	−0.930084	−	0.367347i	$-0.880266\pi$
$569$	− 237865.i	− 0.734694i	0.930084	−	0.367347i	$-0.119734\pi$
$570$	0	0
$571$	244568.	0.750115	0.375057	−	0.927002i	$-0.377623\pi$
$571$	244568.	0.750115	0.375057	+	0.927002i	$0.377623\pi$
$572$	63571.3i	0.194298i
$573$	0	0
$574$	114409.	0.347247
$575$	808413.i	2.44511i
$576$	0	0
$577$	456171.	1.37018	0.685088	−	0.728461i	$-0.259764\pi$
$577$	456171.	1.37018	0.685088	+	0.728461i	$0.259764\pi$
$578$	50826.2i	0.152136i
$579$	0	0
$580$	−302515.	−0.899273
$581$	148775.i	0.440734i
$582$	0	0
$583$	−20586.4	−0.0605680
$584$	181209.i	0.531319i
$585$	0	0
$586$	−199044.	−0.579633
$587$	597028.i	1.73268i	0.499454	+	0.866341i	$0.333534\pi$
$587$	597028.i	1.73268i	−0.499454	+	0.866341i	$0.666466\pi$
$588$	0	0
$589$	118426.	0.341362
$590$	− 235460.i	− 0.676414i
$591$	0	0
$592$	−150733.	−0.430094
$593$	597802.i	1.70000i	0.526785	+	0.849999i	$0.323397\pi$
$593$	597802.i	1.70000i	−0.526785	+	0.849999i	$0.676603\pi$
$594$	0	0
$595$	947601.	2.67665
$596$	163776.i	0.461060i
$597$	0	0
$598$	108083.	0.302242
$599$	− 400872.i	− 1.11725i	−0.829419	−	0.558627i	$-0.811328\pi$
$599$	− 400872.i	− 1.11725i	0.829419	−	0.558627i	$-0.188672\pi$
$600$	0	0
$601$	77232.3	0.213821	0.106910	−	0.994269i	$-0.465904\pi$
$601$	77232.3	0.213821	0.106910	+	0.994269i	$0.465904\pi$
$602$	37527.9i	0.103553i
$603$	0	0
$604$	256038.	0.701827
$605$	84873.7i	0.231880i
$606$	0	0
$607$	−600706.	−1.63036	−0.815182	−	0.579204i	$-0.803363\pi$
$607$	−600706.	−1.63036	−0.815182	+	0.579204i	$0.803363\pi$
$608$	34834.4i	0.0942325i
$609$	0	0
$610$	371670.	0.998845
$611$	− 73150.1i	− 0.195944i
$612$	0	0
$613$	211376.	0.562516	0.281258	−	0.959632i	$-0.409248\pi$
$613$	211376.	0.562516	0.281258	+	0.959632i	$0.409248\pi$
$614$	− 75774.1i	− 0.200995i
$615$	0	0
$616$	206158.	0.543300
$617$	378327.i	0.993796i	0.867809	+	0.496898i	$0.165528\pi$
$617$	378327.i	0.993796i	−0.867809	+	0.496898i	$0.834472\pi$
$618$	0	0
$619$	−491923.	−1.28385	−0.641927	−	0.766765i	$-0.721865\pi$
$619$	−491923.	−1.28385	−0.641927	+	0.766765i	$0.721865\pi$
$620$	− 226216.i	− 0.588490i
$621$	0	0
$622$	−361698.	−0.934900
$623$	− 338689.i	− 0.872619i
$624$	0	0
$625$	889407.	2.27688
$626$	27051.7i	0.0690312i
$627$	0	0
$628$	193657.	0.491037
$629$	− 603000.i	− 1.52411i
$630$	0	0
$631$	−64515.4	−0.162033	−0.0810167	−	0.996713i	$-0.525817\pi$
$631$	−64515.4	−0.162033	−0.0810167	+	0.996713i	$0.525817\pi$
$632$	91106.9i	0.228096i
$633$	0	0
$634$	−110291.	−0.274387
$635$	− 712619.i	− 1.76730i
$636$	0	0
$637$	287149.	0.707665
$638$	− 263289.i	− 0.646832i
$639$	0	0
$640$	66540.3	0.162452
$641$	429607.i	1.04558i	0.852463	+	0.522788i	$0.175108\pi$
$641$	429607.i	1.04558i	−0.852463	+	0.522788i	$0.824892\pi$
$642$	0	0
$643$	500950.	1.21164	0.605818	−	0.795603i	$-0.292846\pi$
$643$	500950.	1.21164	0.605818	+	0.795603i	$0.292846\pi$
$644$	− 350507.i	− 0.845133i
$645$	0	0
$646$	−139354.	−0.333928
$647$	− 175906.i	− 0.420216i	−0.977678	−	0.210108i	$-0.932618\pi$
$647$	− 175906.i	− 0.420216i	0.977678	−	0.210108i	$-0.0673815\pi$
$648$	0	0
$649$	204929.	0.486534
$650$	− 295330.i	− 0.699005i
$651$	0	0
$652$	−168728.	−0.396911
$653$	52557.8i	0.123257i	0.998099	+	0.0616284i	$0.0196294\pi$
$653$	52557.8i	0.123257i	−0.998099	+	0.0616284i	$0.980371\pi$
$654$	0	0
$655$	−624735.	−1.45617
$656$	32138.9i	0.0746833i
$657$	0	0
$658$	−237222.	−0.547902
$659$	− 381919.i	− 0.879428i	−0.898138	−	0.439714i	$-0.855080\pi$
$659$	− 381919.i	− 0.879428i	0.898138	−	0.439714i	$-0.144920\pi$
$660$	0	0
$661$	711240.	1.62785	0.813923	−	0.580972i	$-0.197328\pi$
$661$	711240.	1.62785	0.813923	+	0.580972i	$0.197328\pi$
$662$	142650.i	0.325503i
$663$	0	0
$664$	−41792.5	−0.0947899
$665$	− 712227.i	− 1.61055i
$666$	0	0
$667$	−447640.	−1.00618
$668$	103739.i	0.232482i
$669$	0	0
$670$	1.15612e6	2.57545
$671$	323477.i	0.718454i
$672$	0	0
$673$	−170774.	−0.377044	−0.188522	−	0.982069i	$-0.560370\pi$
$673$	−170774.	−0.377044	−0.188522	+	0.982069i	$0.560370\pi$
$674$	− 360268.i	− 0.793060i
$675$	0	0
$676$	189003.	0.413595
$677$	− 258544.i	− 0.564102i	−0.959399	−	0.282051i	$-0.908985\pi$
$677$	− 258544.i	− 0.564102i	0.959399	−	0.282051i	$-0.0910148\pi$
$678$	0	0
$679$	−1.06220e6	−2.30391
$680$	266192.i	0.575675i
$681$	0	0
$682$	196883.	0.423292
$683$	− 402556.i	− 0.862948i	−0.902125	−	0.431474i	$-0.857994\pi$
$683$	− 402556.i	− 0.862948i	0.902125	−	0.431474i	$-0.142006\pi$
$684$	0	0
$685$	−995459.	−2.12150
$686$	− 384189.i	− 0.816389i
$687$	0	0
$688$	−10542.0	−0.0222714
$689$	− 12786.5i	− 0.0269347i
$690$	0	0
$691$	79523.0	0.166547	0.0832735	−	0.996527i	$-0.473462\pi$
$691$	79523.0	0.166547	0.0832735	+	0.996527i	$0.473462\pi$
$692$	334229.i	0.697962i
$693$	0	0
$694$	548197.	1.13820
$695$	709362.i	1.46858i
$696$	0	0
$697$	−128571.	−0.264653
$698$	294351.i	0.604164i
$699$	0	0
$700$	−957739.	−1.95457
$701$	− 371798.i	− 0.756609i	−0.925681	−	0.378305i	$-0.876507\pi$
$701$	− 371798.i	− 0.756609i	0.925681	−	0.378305i	$-0.123493\pi$
$702$	0	0
$703$	−453221.	−0.917064
$704$	57912.3i	0.116849i
$705$	0	0
$706$	−454736.	−0.912326
$707$	− 804717.i	− 1.60992i
$708$	0	0
$709$	−933206.	−1.85646	−0.928228	−	0.372011i	$-0.878668\pi$
$709$	−933206.	−1.85646	−0.928228	+	0.372011i	$0.878668\pi$
$710$	462229.i	0.916939i
$711$	0	0
$712$	95141.6	0.187677
$713$	− 334738.i	− 0.658454i
$714$	0	0
$715$	365124.	0.714214
$716$	− 106308.i	− 0.207368i
$717$	0	0
$718$	−61591.9	−0.119474
$719$	− 180424.i	− 0.349010i	−0.984656	−	0.174505i	$-0.944168\pi$
$719$	− 180424.i	− 0.349010i	0.984656	−	0.174505i	$-0.0558325\pi$
$720$	0	0
$721$	264122.	0.508083
$722$	− 263864.i	− 0.506181i
$723$	0	0
$724$	231286.	0.441237
$725$	1.22315e6i	2.32704i
$726$	0	0
$727$	−455249.	−0.861350	−0.430675	−	0.902507i	$-0.641725\pi$
$727$	−455249.	−0.861350	−0.430675	+	0.902507i	$0.641725\pi$
$728$	128047.i	0.241606i
$729$	0	0
$730$	1.04078e6	1.95306
$731$	− 42173.0i	− 0.0789222i
$732$	0	0
$733$	550038.	1.02373	0.511864	−	0.859066i	$-0.328955\pi$
$733$	550038.	1.02373	0.511864	+	0.859066i	$0.328955\pi$
$734$	− 712666.i	− 1.32280i
$735$	0	0
$736$	98461.5	0.181765
$737$	1.00621e6i	1.85248i
$738$	0	0
$739$	663313.	1.21459	0.607295	−	0.794477i	$-0.292255\pi$
$739$	663313.	1.21459	0.607295	+	0.794477i	$0.292255\pi$
$740$	865739.i	1.58097i
$741$	0	0
$742$	−41465.8	−0.0753152
$743$	− 322338.i	− 0.583893i	−0.956435	−	0.291947i	$-0.905697\pi$
$743$	− 322338.i	− 0.583893i	0.956435	−	0.291947i	$-0.0943030\pi$
$744$	0	0
$745$	940654.	1.69480
$746$	362112.i	0.650677i
$747$	0	0
$748$	−231676.	−0.414074
$749$	− 1.68538e6i	− 3.00425i
$750$	0	0
$751$	328672.	0.582751	0.291375	−	0.956609i	$-0.405887\pi$
$751$	328672.	0.582751	0.291375	+	0.956609i	$0.405887\pi$
$752$	− 66638.4i	− 0.117839i
$753$	0	0
$754$	163532.	0.287647
$755$	− 1.47056e6i	− 2.57982i
$756$	0	0
$757$	530625.	0.925968	0.462984	−	0.886367i	$-0.346779\pi$
$757$	530625.	0.925968	0.462984	+	0.886367i	$0.346779\pi$
$758$	263037.i	0.457802i
$759$	0	0
$760$	200073.	0.346386
$761$	− 1.08627e6i	− 1.87573i	−0.347002	−	0.937864i	$-0.612800\pi$
$761$	− 1.08627e6i	− 1.87573i	0.347002	−	0.937864i	$-0.387200\pi$
$762$	0	0
$763$	−269099.	−0.462235
$764$	402872.i	0.690209i
$765$	0	0
$766$	−787130.	−1.34150
$767$	127283.i	0.216362i
$768$	0	0
$769$	−640074.	−1.08237	−0.541187	−	0.840902i	$-0.682025\pi$
$769$	−640074.	−1.08237	−0.541187	+	0.840902i	$0.682025\pi$
$770$	− 1.18408e6i	− 1.99710i
$771$	0	0
$772$	27728.2	0.0465250
$773$	924647.i	1.54745i	0.633521	+	0.773726i	$0.281609\pi$
$773$	924647.i	1.54745i	−0.633521	+	0.773726i	$0.718391\pi$
$774$	0	0
$775$	−914650.	−1.52283
$776$	− 298383.i	− 0.495509i
$777$	0	0
$778$	392935.	0.649174
$779$	96635.0i	0.159243i
$780$	0	0
$781$	−402293.	−0.659540
$782$	393892.i	0.644115i
$783$	0	0
$784$	261587.	0.425583
$785$	− 1.11228e6i	− 1.80499i
$786$	0	0
$787$	453454.	0.732123	0.366062	−	0.930591i	$-0.380706\pi$
$787$	453454.	0.732123	0.366062	+	0.930591i	$0.380706\pi$
$788$	62510.1i	0.100669i
$789$	0	0
$790$	523277.	0.838450
$791$	714670.i	1.14223i
$792$	0	0
$793$	−200915.	−0.319497
$794$	− 4980.00i	− 0.00789929i
$795$	0	0
$796$	−233226.	−0.368088
$797$	957183.i	1.50688i	0.657518	+	0.753439i	$0.271606\pi$
$797$	957183.i	1.50688i	−0.657518	+	0.753439i	$0.728394\pi$
$798$	0	0
$799$	266585.	0.417582
$800$	− 269040.i	− 0.420375i
$801$	0	0
$802$	−116129.	−0.180548
$803$	905829.i	1.40480i
$804$	0	0
$805$	−2.01315e6	−3.10660
$806$	122286.i	0.188238i
$807$	0	0
$808$	226054.	0.346250
$809$	− 464039.i	− 0.709018i	−0.935053	−	0.354509i	$-0.884648\pi$
$809$	− 464039.i	− 0.709018i	0.935053	−	0.354509i	$-0.115352\pi$
$810$	0	0
$811$	−732128.	−1.11313	−0.556564	−	0.830804i	$-0.687881\pi$
$811$	−732128.	−1.11313	−0.556564	+	0.830804i	$0.687881\pi$
$812$	− 530326.i	− 0.804324i
$813$	0	0
$814$	−753482.	−1.13717
$815$	969098.i	1.45899i
$816$	0	0
$817$	−31697.6	−0.0474879
$818$	− 471537.i	− 0.704708i
$819$	0	0
$820$	184591.	0.274526
$821$	1.26636e6i	1.87876i	0.342876	+	0.939381i	$0.388599\pi$
$821$	1.26636e6i	1.87876i	−0.342876	+	0.939381i	$0.611401\pi$
$822$	0	0
$823$	−172258.	−0.254320	−0.127160	−	0.991882i	$-0.540586\pi$
$823$	−172258.	−0.254320	−0.127160	+	0.991882i	$0.540586\pi$
$824$	74195.0i	0.109275i
$825$	0	0
$826$	412774.	0.604996
$827$	− 135383.i	− 0.197949i	−0.995090	−	0.0989746i	$-0.968444\pi$
$827$	− 135383.i	− 0.197949i	0.995090	−	0.0989746i	$-0.0315563\pi$
$828$	0	0
$829$	290151.	0.422197	0.211099	−	0.977465i	$-0.432296\pi$
$829$	290151.	0.422197	0.211099	+	0.977465i	$0.432296\pi$
$830$	240037.i	0.348435i
$831$	0	0
$832$	−35970.0	−0.0519629
$833$	1.04647e6i	1.50812i
$834$	0	0
$835$	595830.	0.854574
$836$	174130.i	0.249150i
$837$	0	0
$838$	−938212.	−1.33602
$839$	534767.i	0.759697i	0.925049	+	0.379849i	$0.124024\pi$
$839$	534767.i	0.759697i	−0.925049	+	0.379849i	$0.875976\pi$
$840$	0	0
$841$	29990.2	0.0424021
$842$	− 531255.i	− 0.749339i
$843$	0	0
$844$	377264.	0.529615
$845$	− 1.08555e6i	− 1.52032i
$846$	0	0
$847$	−148788.	−0.207397
$848$	− 11648.2i	− 0.0161983i
$849$	0	0
$850$	1.07628e6	1.48967
$851$	1.28106e6i	1.76893i
$852$	0	0
$853$	−939343.	−1.29100	−0.645500	−	0.763760i	$-0.723351\pi$
$853$	−939343.	−1.29100	−0.645500	+	0.763760i	$0.723351\pi$
$854$	651559.i	0.893383i
$855$	0	0
$856$	473444.	0.646132
$857$	− 671520.i	− 0.914318i	−0.889385	−	0.457159i	$-0.848867\pi$
$857$	− 671520.i	− 0.914318i	0.889385	−	0.457159i	$-0.151133\pi$
$858$	0	0
$859$	−1.14276e6	−1.54871	−0.774356	−	0.632751i	$-0.781926\pi$
$859$	−1.14276e6	−1.54871	−0.774356	+	0.632751i	$0.781926\pi$
$860$	60548.5i	0.0818666i
$861$	0	0
$862$	−19663.2	−0.0264631
$863$	1.09963e6i	1.47647i	0.674545	+	0.738234i	$0.264340\pi$
$863$	1.09963e6i	1.47647i	−0.674545	+	0.738234i	$0.735660\pi$
$864$	0	0
$865$	1.91966e6	2.56562
$866$	− 107313.i	− 0.143093i
$867$	0	0
$868$	396569.	0.526355
$869$	455425.i	0.603084i
$870$	0	0
$871$	−624968.	−0.823799
$872$	− 75593.0i	− 0.0994143i
$873$	0	0
$874$	296053.	0.387567
$875$	3.18760e6i	4.16340i
$876$	0	0
$877$	779731.	1.01379	0.506893	−	0.862009i	$-0.330794\pi$
$877$	779731.	1.01379	0.506893	+	0.862009i	$0.330794\pi$
$878$	702594.i	0.911414i
$879$	0	0
$880$	332621.	0.429522
$881$	622180.i	0.801611i	0.916163	+	0.400806i	$0.131270\pi$
$881$	622180.i	0.801611i	−0.916163	+	0.400806i	$0.868730\pi$
$882$	0	0
$883$	−1.39076e6	−1.78373	−0.891865	−	0.452301i	$-0.850603\pi$
$883$	−1.39076e6	−1.78373	−0.891865	+	0.452301i	$0.850603\pi$
$884$	− 143897.i	− 0.184139i
$885$	0	0
$886$	−460605.	−0.586761
$887$	− 957517.i	− 1.21702i	−0.793545	−	0.608512i	$-0.791767\pi$
$887$	− 957517.i	− 1.21702i	0.793545	−	0.608512i	$-0.208233\pi$
$888$	0	0
$889$	1.24926e6	1.58070
$890$	− 546450.i	− 0.689874i
$891$	0	0
$892$	−80426.2	−0.101081
$893$	− 200368.i	− 0.251261i
$894$	0	0
$895$	−610587.	−0.762257
$896$	116649.i	0.145300i
$897$	0	0
$898$	393860.	0.488415
$899$	− 506466.i	− 0.626659i
$900$	0	0
$901$	46598.3	0.0574012
$902$	160656.i	0.197462i
$903$	0	0
$904$	−200759.	−0.245662
$905$	− 1.32840e6i	− 1.62193i
$906$	0	0
$907$	219571.	0.266907	0.133454	−	0.991055i	$-0.457393\pi$
$907$	219571.	0.266907	0.133454	+	0.991055i	$0.457393\pi$
$908$	239614.i	0.290630i
$909$	0	0
$910$	735445.	0.888112
$911$	− 1.52237e6i	− 1.83436i	−0.398475	−	0.917179i	$-0.630460\pi$
$911$	− 1.52237e6i	− 1.83436i	0.398475	−	0.917179i	$-0.369540\pi$
$912$	0	0
$913$	−208912.	−0.250624
$914$	708632.i	0.848259i
$915$	0	0
$916$	161068.	0.191963
$917$	− 1.09520e6i	− 1.30243i
$918$	0	0
$919$	−239088.	−0.283091	−0.141546	−	0.989932i	$-0.545207\pi$
$919$	−239088.	−0.283091	−0.141546	+	0.989932i	$0.545207\pi$
$920$	− 565518.i	− 0.668145i
$921$	0	0
$922$	−221145.	−0.260145
$923$	− 249869.i	− 0.293298i
$924$	0	0
$925$	3.50041e6	4.09106
$926$	997278.i	1.16304i
$927$	0	0
$928$	148975.	0.172988
$929$	− 36734.8i	− 0.0425643i	−0.999774	−	0.0212822i	$-0.993225\pi$
$929$	− 36734.8i	− 0.0425643i	0.999774	−	0.0212822i	$-0.00677483\pi$
$930$	0	0
$931$	786538.	0.907445
$932$	− 152633.i	− 0.175718i
$933$	0	0
$934$	108568.	0.124454
$935$	1.33064e6i	1.52208i
$936$	0	0
$937$	−849585.	−0.967671	−0.483835	−	0.875159i	$-0.660757\pi$
$937$	−849585.	−0.967671	−0.483835	+	0.875159i	$0.660757\pi$
$938$	2.02674e6i	2.30352i
$939$	0	0
$940$	−382740.	−0.433160
$941$	− 115282.i	− 0.130191i	−0.997879	−	0.0650956i	$-0.979265\pi$
$941$	− 115282.i	− 0.130191i	0.997879	−	0.0650956i	$-0.0207352\pi$
$942$	0	0
$943$	273145.	0.307163
$944$	115953.i	0.130118i
$945$	0	0
$946$	−52697.4	−0.0588853
$947$	51123.6i	0.0570061i	0.999594	+	0.0285031i	$0.00907404\pi$
$947$	51123.6i	0.0570061i	−0.999594	+	0.0285031i	$0.990926\pi$
$948$	0	0
$949$	−562621.	−0.624717
$950$	− 808947.i	− 0.896340i
$951$	0	0
$952$	−466650.	−0.514893
$953$	407821.i	0.449038i	0.974470	+	0.224519i	$0.0720812\pi$
$953$	407821.i	0.449038i	−0.974470	+	0.224519i	$0.927919\pi$
$954$	0	0
$955$	2.31391e6	2.53712
$956$	722076.i	0.790073i
$957$	0	0
$958$	−147138.	−0.160322
$959$	− 1.74509e6i	− 1.89750i
$960$	0	0
$961$	−544794.	−0.589910
$962$	− 467996.i	− 0.505699i
$963$	0	0
$964$	−241288.	−0.259646
$965$	− 159258.i	− 0.171020i
$966$	0	0
$967$	1.27909e6	1.36788	0.683941	−	0.729538i	$-0.260265\pi$
$967$	1.27909e6	1.36788	0.683941	+	0.729538i	$0.260265\pi$
$968$	− 41796.4i	− 0.0446054i
$969$	0	0
$970$	−1.71378e6	−1.82142
$971$	− 806048.i	− 0.854914i	−0.904036	−	0.427457i	$-0.859410\pi$
$971$	− 806048.i	− 0.854914i	0.904036	−	0.427457i	$-0.140590\pi$
$972$	0	0
$973$	−1.24355e6	−1.31352
$974$	− 716781.i	− 0.755559i
$975$	0	0
$976$	−183030.	−0.192143
$977$	1.36715e6i	1.43228i	0.697956	+	0.716141i	$0.254093\pi$
$977$	1.36715e6i	1.43228i	−0.697956	+	0.716141i	$0.745907\pi$
$978$	0	0
$979$	475593.	0.496216
$980$	− 1.50244e6i	− 1.56439i
$981$	0	0
$982$	516339.	0.535441
$983$	− 1.44380e6i	− 1.49417i	−0.664731	−	0.747083i	$-0.731454\pi$
$983$	− 1.44380e6i	− 1.49417i	0.664731	−	0.747083i	$-0.268546\pi$
$984$	0	0
$985$	359029.	0.370047
$986$	595968.i	0.613012i
$987$	0	0
$988$	−108154.	−0.110797
$989$	89595.3i	0.0915994i
$990$	0	0
$991$	−476821.	−0.485521	−0.242761	−	0.970086i	$-0.578053\pi$
$991$	−476821.	−0.485521	−0.242761	+	0.970086i	$0.578053\pi$
$992$	111401.i	0.113205i
$993$	0	0
$994$	−810313.	−0.820125
$995$	1.33955e6i	1.35304i
$996$	0	0
$997$	−1.02549e6	−1.03167	−0.515834	−	0.856689i	$-0.672518\pi$
$997$	−1.02549e6	−1.03167	−0.515834	+	0.856689i	$0.672518\pi$
$998$	97702.9i	0.0980949i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.5.b.a.161.4	yes	4
3.2	odd	2	inner	162.5.b.a.161.1	✓	4
4.3	odd	2		1296.5.e.b.161.4		4
9.2	odd	6		162.5.d.d.53.2		8
9.4	even	3		162.5.d.d.107.2		8
9.5	odd	6		162.5.d.d.107.3		8
9.7	even	3		162.5.d.d.53.3		8
12.11	even	2		1296.5.e.b.161.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.5.b.a.161.1	✓	4	3.2	odd	2	inner
162.5.b.a.161.4	yes	4	1.1	even	1	trivial
162.5.d.d.53.2		8	9.2	odd	6
162.5.d.d.53.3		8	9.7	even	3
162.5.d.d.107.2		8	9.4	even	3
162.5.d.d.107.3		8	9.5	odd	6
1296.5.e.b.161.1		4	12.11	even	2
1296.5.e.b.161.4		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +45.9483i q^{5} -80.5500 q^{7} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +45.9483i q^{5} -80.5500 q^{7} -22.6274i q^{8} -129.962 q^{10} -113.110i q^{11} +70.2539 q^{13} -227.830i q^{14} +64.0000 q^{16} +256.030i q^{17} +192.435 q^{19} -367.587i q^{20} +319.923 q^{22} -543.928i q^{23} -1486.25 q^{25} +198.708i q^{26} +644.400 q^{28} -822.977i q^{29} +615.408 q^{31} +181.019i q^{32} -724.161 q^{34} -3701.14i q^{35} -2355.20 q^{37} +544.287i q^{38} +1039.69 q^{40} +502.171i q^{41} -164.719 q^{43} +904.879i q^{44} +1538.46 q^{46} -1041.23i q^{47} +4087.30 q^{49} -4203.75i q^{50} -562.031 q^{52} -182.004i q^{53} +5197.21 q^{55} +1822.64i q^{56} +2327.73 q^{58} +1811.76i q^{59} -2859.85 q^{61} +1740.64i q^{62} -512.000 q^{64} +3228.05i q^{65} -8895.85 q^{67} -2048.24i q^{68} +10468.4 q^{70} -3556.66i q^{71} -8008.40 q^{73} -6661.50i q^{74} -1539.48 q^{76} +9111.00i q^{77} -4026.40 q^{79} +2940.69i q^{80} -1420.35 q^{82} -1846.98i q^{83} -11764.1 q^{85} -465.896i q^{86} -2559.38 q^{88} +4204.70i q^{89} -5658.95 q^{91} +4351.42i q^{92} +2945.03 q^{94} +8842.05i q^{95} +13186.8 q^{97} +11560.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} - 52 q^{7}+O(q^{10}) \) 4 * q - 32 * q^4 - 52 * q^7	\( 4 q - 32 q^{4} - 52 q^{7} - 312 q^{10} + 572 q^{13} + 256 q^{16} - 124 q^{19} + 864 q^{22} - 1892 q^{25} + 416 q^{28} + 3584 q^{31} - 1608 q^{34} - 9400 q^{37} + 2496 q^{40} + 3020 q^{43} - 2160 q^{46} + 9324 q^{49} - 4576 q^{52} + 11124 q^{55} - 2952 q^{58} - 4144 q^{61} - 2048 q^{64} - 12076 q^{67} + 18096 q^{70} - 1792 q^{73} + 992 q^{76} - 15004 q^{79} + 5376 q^{82} - 24048 q^{85} - 6912 q^{88} + 12220 q^{91} - 12912 q^{94} + 46304 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 - 52 * q^7 - 312 * q^10 + 572 * q^13 + 256 * q^16 - 124 * q^19 + 864 * q^22 - 1892 * q^25 + 416 * q^28 + 3584 * q^31 - 1608 * q^34 - 9400 * q^37 + 2496 * q^40 + 3020 * q^43 - 2160 * q^46 + 9324 * q^49 - 4576 * q^52 + 11124 * q^55 - 2952 * q^58 - 4144 * q^61 - 2048 * q^64 - 12076 * q^67 + 18096 * q^70 - 1792 * q^73 + 992 * q^76 - 15004 * q^79 + 5376 * q^82 - 24048 * q^85 - 6912 * q^88 + 12220 * q^91 - 12912 * q^94 + 46304 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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