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Properties

Label 162.7.b.c.161.6
Level $162$
Weight $7$
Character 162.161
Analytic conductor $37.269$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,7,Mod(161,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.161"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.2687615464\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{42} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.6
Root \(-8.88570i\) of defining polynomial
Character \(\chi\) \(=\) 162.161
Dual form 162.7.b.c.161.7

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.65685i q^{2} -32.0000 q^{4} +233.541i q^{5} -191.150 q^{7} +181.019i q^{8} +1321.11 q^{10} +777.202i q^{11} -91.1604 q^{13} +1081.31i q^{14} +1024.00 q^{16} +7047.39i q^{17} +2731.10 q^{19} -7473.32i q^{20} +4396.52 q^{22} -19894.3i q^{23} -38916.6 q^{25} +515.681i q^{26} +6116.81 q^{28} -31297.4i q^{29} -12349.0 q^{31} -5792.62i q^{32} +39866.1 q^{34} -44641.5i q^{35} -27972.0 q^{37} -15449.4i q^{38} -42275.5 q^{40} -43218.3i q^{41} -38512.1 q^{43} -24870.5i q^{44} -112539. q^{46} +166012. i q^{47} -81110.5 q^{49} +220145. i q^{50} +2917.13 q^{52} -54741.5i q^{53} -181509. q^{55} -34601.9i q^{56} -177045. q^{58} -16283.7i q^{59} -58887.4 q^{61} +69856.5i q^{62} -32768.0 q^{64} -21289.7i q^{65} +295997. q^{67} -225517. i q^{68} -252530. q^{70} +157251. i q^{71} +80297.0 q^{73} +158234. i q^{74} -87395.0 q^{76} -148563. i q^{77} -376849. q^{79} +239146. i q^{80} -244480. q^{82} -847541. i q^{83} -1.64586e6 q^{85} +217857. i q^{86} -140689. q^{88} +1128.91i q^{89} +17425.3 q^{91} +636617. i q^{92} +939108. q^{94} +637824. i q^{95} -1.35030e6 q^{97} +458831. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 384 q^{4} - 480 q^{7} - 3360 q^{13} + 12288 q^{16} - 2820 q^{19} + 7200 q^{22} - 16188 q^{25} + 15360 q^{28} - 42960 q^{31} + 54720 q^{34} - 25536 q^{37} - 142860 q^{43} - 135072 q^{46} + 271908 q^{49}+ \cdots + 77748 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.65685i − 0.707107i
\(3\) 0 0
\(4\) −32.0000 −0.500000
\(5\) 233.541i 1.86833i 0.356841 + 0.934165i \(0.383854\pi\)
−0.356841 + 0.934165i \(0.616146\pi\)
\(6\) 0 0
\(7\) −191.150 −0.557290 −0.278645 − 0.960394i \(-0.589885\pi\)
−0.278645 + 0.960394i \(0.589885\pi\)
\(8\) 181.019i 0.353553i
\(9\) 0 0
\(10\) 1321.11 1.32111
\(11\) 777.202i 0.583924i 0.956430 + 0.291962i \(0.0943080\pi\)
−0.956430 + 0.291962i \(0.905692\pi\)
\(12\) 0 0
\(13\) −91.1604 −0.0414931 −0.0207466 − 0.999785i \(-0.506604\pi\)
−0.0207466 + 0.999785i \(0.506604\pi\)
\(14\) 1081.31i 0.394063i
\(15\) 0 0
\(16\) 1024.00 0.250000
\(17\) 7047.39i 1.43444i 0.696848 + 0.717219i \(0.254585\pi\)
−0.696848 + 0.717219i \(0.745415\pi\)
\(18\) 0 0
\(19\) 2731.10 0.398177 0.199088 − 0.979982i \(-0.436202\pi\)
0.199088 + 0.979982i \(0.436202\pi\)
\(20\) − 7473.32i − 0.934165i
\(21\) 0 0
\(22\) 4396.52 0.412896
\(23\) − 19894.3i − 1.63510i −0.575857 − 0.817550i \(-0.695332\pi\)
0.575857 − 0.817550i \(-0.304668\pi\)
\(24\) 0 0
\(25\) −38916.6 −2.49066
\(26\) 515.681i 0.0293401i
\(27\) 0 0
\(28\) 6116.81 0.278645
\(29\) − 31297.4i − 1.28326i −0.767015 − 0.641629i \(-0.778259\pi\)
0.767015 − 0.641629i \(-0.221741\pi\)
\(30\) 0 0
\(31\) −12349.0 −0.414521 −0.207261 − 0.978286i \(-0.566455\pi\)
−0.207261 + 0.978286i \(0.566455\pi\)
\(32\) − 5792.62i − 0.176777i
\(33\) 0 0
\(34\) 39866.1 1.01430
\(35\) − 44641.5i − 1.04120i
\(36\) 0 0
\(37\) −27972.0 −0.552228 −0.276114 − 0.961125i \(-0.589047\pi\)
−0.276114 + 0.961125i \(0.589047\pi\)
\(38\) − 15449.4i − 0.281554i
\(39\) 0 0
\(40\) −42275.5 −0.660555
\(41\) − 43218.3i − 0.627070i −0.949577 − 0.313535i \(-0.898487\pi\)
0.949577 − 0.313535i \(-0.101513\pi\)
\(42\) 0 0
\(43\) −38512.1 −0.484386 −0.242193 − 0.970228i \(-0.577867\pi\)
−0.242193 + 0.970228i \(0.577867\pi\)
\(44\) − 24870.5i − 0.291962i
\(45\) 0 0
\(46\) −112539. −1.15619
\(47\) 166012.i 1.59899i 0.600670 + 0.799497i \(0.294901\pi\)
−0.600670 + 0.799497i \(0.705099\pi\)
\(48\) 0 0
\(49\) −81110.5 −0.689428
\(50\) 220145.i 1.76116i
\(51\) 0 0
\(52\) 2917.13 0.0207466
\(53\) − 54741.5i − 0.367696i −0.982955 − 0.183848i \(-0.941145\pi\)
0.982955 − 0.183848i \(-0.0588554\pi\)
\(54\) 0 0
\(55\) −181509. −1.09096
\(56\) − 34601.9i − 0.197032i
\(57\) 0 0
\(58\) −177045. −0.907400
\(59\) − 16283.7i − 0.0792860i −0.999214 − 0.0396430i \(-0.987378\pi\)
0.999214 − 0.0396430i \(-0.0126221\pi\)
\(60\) 0 0
\(61\) −58887.4 −0.259438 −0.129719 − 0.991551i \(-0.541407\pi\)
−0.129719 + 0.991551i \(0.541407\pi\)
\(62\) 69856.5i 0.293111i
\(63\) 0 0
\(64\) −32768.0 −0.125000
\(65\) − 21289.7i − 0.0775229i
\(66\) 0 0
\(67\) 295997. 0.984152 0.492076 − 0.870552i \(-0.336238\pi\)
0.492076 + 0.870552i \(0.336238\pi\)
\(68\) − 225517.i − 0.717219i
\(69\) 0 0
\(70\) −252530. −0.736240
\(71\) 157251.i 0.439358i 0.975572 + 0.219679i \(0.0705010\pi\)
−0.975572 + 0.219679i \(0.929499\pi\)
\(72\) 0 0
\(73\) 80297.0 0.206410 0.103205 − 0.994660i \(-0.467090\pi\)
0.103205 + 0.994660i \(0.467090\pi\)
\(74\) 158234.i 0.390484i
\(75\) 0 0
\(76\) −87395.0 −0.199088
\(77\) − 148563.i − 0.325415i
\(78\) 0 0
\(79\) −376849. −0.764338 −0.382169 − 0.924092i \(-0.624823\pi\)
−0.382169 + 0.924092i \(0.624823\pi\)
\(80\) 239146.i 0.467083i
\(81\) 0 0
\(82\) −244480. −0.443406
\(83\) − 847541.i − 1.48227i −0.671358 − 0.741134i \(-0.734289\pi\)
0.671358 − 0.741134i \(-0.265711\pi\)
\(84\) 0 0
\(85\) −1.64586e6 −2.68000
\(86\) 217857.i 0.342513i
\(87\) 0 0
\(88\) −140689. −0.206448
\(89\) 1128.91i 0.00160136i 1.00000 0.000800679i \(0.000254864\pi\)
−1.00000 0.000800679i \(0.999745\pi\)
\(90\) 0 0
\(91\) 17425.3 0.0231237
\(92\) 636617.i 0.817550i
\(93\) 0 0
\(94\) 939108. 1.13066
\(95\) 637824.i 0.743926i
\(96\) 0 0
\(97\) −1.35030e6 −1.47951 −0.739753 − 0.672879i \(-0.765057\pi\)
−0.739753 + 0.672879i \(0.765057\pi\)
\(98\) 458831.i 0.487499i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 162.7.b.c.161.6 12
3.2 odd 2 inner 162.7.b.c.161.7 12
9.2 odd 6 18.7.d.a.5.5 ✓ 12
9.4 even 3 18.7.d.a.11.5 yes 12
9.5 odd 6 54.7.d.a.35.1 12
9.7 even 3 54.7.d.a.17.1 12
36.7 odd 6 432.7.q.b.17.1 12
36.11 even 6 144.7.q.c.113.4 12
36.23 even 6 432.7.q.b.305.1 12
36.31 odd 6 144.7.q.c.65.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.5 ✓ 12 9.2 odd 6
18.7.d.a.11.5 yes 12 9.4 even 3
54.7.d.a.17.1 12 9.7 even 3
54.7.d.a.35.1 12 9.5 odd 6
144.7.q.c.65.4 12 36.31 odd 6
144.7.q.c.113.4 12 36.11 even 6
162.7.b.c.161.6 12 1.1 even 1 trivial
162.7.b.c.161.7 12 3.2 odd 2 inner
432.7.q.b.17.1 12 36.7 odd 6
432.7.q.b.305.1 12 36.23 even 6