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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 161.8
Root \(-7.20150i\) of defining polynomial
Character \(\chi\) \(=\) 162.161
Dual form 162.7.b.c.161.5

$q$-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)
 
\(f(q)\) \(=\) \(q+5.65685i q^{2} -32.0000 q^{4} -45.6802i q^{5} +490.195 q^{7} -181.019i q^{8} +258.406 q^{10} +1008.44i q^{11} -933.602 q^{13} +2772.96i q^{14} +1024.00 q^{16} -8090.59i q^{17} -7727.36 q^{19} +1461.77i q^{20} -5704.61 q^{22} -13680.9i q^{23} +13538.3 q^{25} -5281.25i q^{26} -15686.2 q^{28} +2268.65i q^{29} +34125.1 q^{31} +5792.62i q^{32} +45767.3 q^{34} -22392.2i q^{35} +92058.0 q^{37} -43712.6i q^{38} -8269.00 q^{40} -35820.6i q^{41} -69141.8 q^{43} -32270.1i q^{44} +77390.9 q^{46} -15254.9i q^{47} +122642. q^{49} +76584.3i q^{50} +29875.3 q^{52} +236591. i q^{53} +46065.9 q^{55} -88734.7i q^{56} -12833.4 q^{58} -256217. i q^{59} +39839.2 q^{61} +193041. i q^{62} -32768.0 q^{64} +42647.1i q^{65} +320409. q^{67} +258899. i q^{68} +126669. q^{70} -404593. i q^{71} +393719. q^{73} +520759. i q^{74} +247276. q^{76} +494333. i q^{77} +898368. q^{79} -46776.5i q^{80} +202632. q^{82} -178882. i q^{83} -369580. q^{85} -391125. i q^{86} +182548. q^{88} -826458. i q^{89} -457647. q^{91} +437789. i q^{92} +86294.8 q^{94} +352988. i q^{95} +635962. q^{97} +693768. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 384 q^{4} - 480 q^{7} - 3360 q^{13} + 12288 q^{16} - 2820 q^{19} + 7200 q^{22} - 16188 q^{25} + 15360 q^{28} - 42960 q^{31} + 54720 q^{34} - 25536 q^{37} - 142860 q^{43} - 135072 q^{46} + 271908 q^{49}+ \cdots + 77748 q^{97}+O(q^{100}) \) 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
\(5\) − 45.6802i − 0.365442i −0.983165 − 0.182721i \(-0.941510\pi\)
0.983165 − 0.182721i \(-0.0584905\pi\)
\(6\) 0 0
\(7\) 490.195 1.42914 0.714570 − 0.699564i \(-0.246623\pi\)
0.714570 + 0.699564i \(0.246623\pi\)
\(8\) − 181.019i − 0.353553i
\(9\) 0 0
\(10\) 258.406 0.258406
\(11\) 1008.44i 0.757657i 0.925467 + 0.378829i \(0.123673\pi\)
−0.925467 + 0.378829i \(0.876327\pi\)
\(12\) 0 0
\(13\) −933.602 −0.424944 −0.212472 − 0.977167i \(-0.568151\pi\)
−0.212472 + 0.977167i \(0.568151\pi\)
\(14\) 2772.96i 1.01055i
\(15\) 0 0
\(16\) 1024.00 0.250000
\(17\) − 8090.59i − 1.64677i −0.567482 − 0.823386i \(-0.692082\pi\)
0.567482 − 0.823386i \(-0.307918\pi\)
\(18\) 0 0
\(19\) −7727.36 −1.12660 −0.563301 − 0.826252i \(-0.690469\pi\)
−0.563301 + 0.826252i \(0.690469\pi\)
\(20\) 1461.77i 0.182721i
\(21\) 0 0
\(22\) −5704.61 −0.535745
\(23\) − 13680.9i − 1.12443i −0.826992 − 0.562213i \(-0.809950\pi\)
0.826992 − 0.562213i \(-0.190050\pi\)
\(24\) 0 0
\(25\) 13538.3 0.866452
\(26\) − 5281.25i − 0.300481i
\(27\) 0 0
\(28\) −15686.2 −0.714570
\(29\) 2268.65i 0.0930192i 0.998918 + 0.0465096i \(0.0148098\pi\)
−0.998918 + 0.0465096i \(0.985190\pi\)
\(30\) 0 0
\(31\) 34125.1 1.14548 0.572742 − 0.819735i \(-0.305880\pi\)
0.572742 + 0.819735i \(0.305880\pi\)
\(32\) 5792.62i 0.176777i
\(33\) 0 0
\(34\) 45767.3 1.16444
\(35\) − 22392.2i − 0.522267i
\(36\) 0 0
\(37\) 92058.0 1.81743 0.908713 − 0.417422i \(-0.137066\pi\)
0.908713 + 0.417422i \(0.137066\pi\)
\(38\) − 43712.6i − 0.796628i
\(39\) 0 0
\(40\) −8269.00 −0.129203
\(41\) − 35820.6i − 0.519734i −0.965644 − 0.259867i \(-0.916321\pi\)
0.965644 − 0.259867i \(-0.0836788\pi\)
\(42\) 0 0
\(43\) −69141.8 −0.869631 −0.434816 − 0.900520i \(-0.643186\pi\)
−0.434816 + 0.900520i \(0.643186\pi\)
\(44\) − 32270.1i − 0.378829i
\(45\) 0 0
\(46\) 77390.9 0.795090
\(47\) − 15254.9i − 0.146932i −0.997298 − 0.0734659i \(-0.976594\pi\)
0.997298 − 0.0734659i \(-0.0234060\pi\)
\(48\) 0 0
\(49\) 122642. 1.04244
\(50\) 76584.3i 0.612674i
\(51\) 0 0
\(52\) 29875.3 0.212472
\(53\) 236591.i 1.58917i 0.607153 + 0.794585i \(0.292312\pi\)
−0.607153 + 0.794585i \(0.707688\pi\)
\(54\) 0 0
\(55\) 46065.9 0.276880
\(56\) − 88734.7i − 0.505277i
\(57\) 0 0
\(58\) −12833.4 −0.0657745
\(59\) − 256217.i − 1.24753i −0.781611 − 0.623766i \(-0.785602\pi\)
0.781611 − 0.623766i \(-0.214398\pi\)
\(60\) 0 0
\(61\) 39839.2 0.175518 0.0877589 − 0.996142i \(-0.472029\pi\)
0.0877589 + 0.996142i \(0.472029\pi\)
\(62\) 193041.i 0.809980i
\(63\) 0 0
\(64\) −32768.0 −0.125000
\(65\) 42647.1i 0.155292i
\(66\) 0 0
\(67\) 320409. 1.06532 0.532660 − 0.846330i \(-0.321193\pi\)
0.532660 + 0.846330i \(0.321193\pi\)
\(68\) 258899.i 0.823386i
\(69\) 0 0
\(70\) 126669. 0.369299
\(71\) − 404593.i − 1.13043i −0.824944 − 0.565215i \(-0.808793\pi\)
0.824944 − 0.565215i \(-0.191207\pi\)
\(72\) 0 0
\(73\) 393719. 1.01209 0.506044 − 0.862508i \(-0.331107\pi\)
0.506044 + 0.862508i \(0.331107\pi\)
\(74\) 520759.i 1.28511i
\(75\) 0 0
\(76\) 247276. 0.563301
\(77\) 494333.i 1.08280i
\(78\) 0 0
\(79\) 898368. 1.82210 0.911052 − 0.412292i \(-0.135272\pi\)
0.911052 + 0.412292i \(0.135272\pi\)
\(80\) − 46776.5i − 0.0913604i
\(81\) 0 0
\(82\) 202632. 0.367508
\(83\) − 178882.i − 0.312847i −0.987690 − 0.156424i \(-0.950003\pi\)
0.987690 − 0.156424i \(-0.0499965\pi\)
\(84\) 0 0
\(85\) −369580. −0.601799
\(86\) − 391125.i − 0.614922i
\(87\) 0 0
\(88\) 182548. 0.267872
\(89\) − 826458.i − 1.17233i −0.810191 − 0.586166i \(-0.800636\pi\)
0.810191 − 0.586166i \(-0.199364\pi\)
\(90\) 0 0
\(91\) −457647. −0.607304
\(92\) 437789.i 0.562213i
\(93\) 0 0
\(94\) 86294.8 0.103897
\(95\) 352988.i 0.411707i
\(96\) 0 0
\(97\) 635962. 0.696813 0.348406 − 0.937344i \(-0.386723\pi\)
0.348406 + 0.937344i \(0.386723\pi\)
\(98\) 693768.i 0.737116i
\(99\) 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.161.8 12
3.2 odd 2 inner 162.7.b.c.161.5 12
9.2 odd 6 18.7.d.a.5.2 ✓ 12
9.4 even 3 18.7.d.a.11.2 yes 12
9.5 odd 6 54.7.d.a.35.6 12
9.7 even 3 54.7.d.a.17.6 12
36.7 odd 6 432.7.q.b.17.5 12
36.11 even 6 144.7.q.c.113.3 12
36.23 even 6 432.7.q.b.305.5 12
36.31 odd 6 144.7.q.c.65.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.2 ✓ 12 9.2 odd 6
18.7.d.a.11.2 yes 12 9.4 even 3
54.7.d.a.17.6 12 9.7 even 3
54.7.d.a.35.6 12 9.5 odd 6
144.7.q.c.65.3 12 36.31 odd 6
144.7.q.c.113.3 12 36.11 even 6
162.7.b.c.161.5 12 3.2 odd 2 inner
162.7.b.c.161.8 12 1.1 even 1 trivial
432.7.q.b.17.5 12 36.7 odd 6
432.7.q.b.305.5 12 36.23 even 6