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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(109,864)] 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("864.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 109.11
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.791979 + 1.17165i) q^{2} +(-0.745540 - 1.85585i) q^{4} +(2.17386 - 0.900442i) q^{5} +(-1.13840 + 1.13840i) q^{7} +(2.76486 + 0.596278i) q^{8} +(-0.666644 + 3.26014i) q^{10} +(-1.16219 - 2.80577i) q^{11} +(5.29700 + 2.19409i) q^{13} +(-0.432223 - 2.23541i) q^{14} +(-2.88834 + 2.76722i) q^{16} -1.37204i q^{17} +(-0.435446 - 0.180368i) q^{19} +(-3.29178 - 3.36303i) q^{20} +(4.20782 + 0.860429i) q^{22} +(-0.900399 - 0.900399i) q^{23} +(0.379330 - 0.379330i) q^{25} +(-6.76583 + 4.46858i) q^{26} +(2.96143 + 1.26398i) q^{28} +(2.18893 - 5.28455i) q^{29} +3.64689 q^{31} +(-0.954714 - 5.57571i) q^{32} +(1.60755 + 1.08662i) q^{34} +(-1.44966 + 3.49980i) q^{35} +(10.4431 - 4.32566i) q^{37} +(0.556192 - 0.367344i) q^{38} +(6.54733 - 1.19337i) q^{40} +(3.37936 + 3.37936i) q^{41} +(-0.290826 - 0.702117i) q^{43} +(-4.34062 + 4.24866i) q^{44} +(1.76805 - 0.341858i) q^{46} -9.15553i q^{47} +4.40807i q^{49} +(0.144022 + 0.744865i) q^{50} +(0.122770 - 11.4662i) q^{52} +(0.279738 + 0.675348i) q^{53} +(-5.05286 - 5.05286i) q^{55} +(-3.82633 + 2.46872i) q^{56} +(4.45807 + 6.74992i) q^{58} +(9.51130 - 3.93971i) q^{59} +(-3.14636 + 7.59599i) q^{61} +(-2.88826 + 4.27289i) q^{62} +(7.28891 + 3.29725i) q^{64} +13.4906 q^{65} +(2.00791 - 4.84753i) q^{67} +(-2.54629 + 1.02291i) q^{68} +(-2.95244 - 4.47026i) q^{70} +(3.53207 - 3.53207i) q^{71} +(-4.79474 - 4.79474i) q^{73} +(-3.20251 + 15.6615i) q^{74} +(-0.0100924 + 0.942593i) q^{76} +(4.51714 + 1.87106i) q^{77} +14.3012i q^{79} +(-3.78712 + 8.61632i) q^{80} +(-6.63581 + 1.28305i) q^{82} +(7.87490 + 3.26189i) q^{83} +(-1.23544 - 2.98262i) q^{85} +(1.05296 + 0.215314i) q^{86} +(-1.54027 - 8.45055i) q^{88} +(-1.44497 + 1.44497i) q^{89} +(-8.52789 + 3.53237i) q^{91} +(-0.999720 + 2.34229i) q^{92} +(10.7271 + 7.25099i) q^{94} -1.10901 q^{95} +8.99231 q^{97} +(-5.16473 - 3.49110i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.791979 + 1.17165i −0.560013 + 0.828484i
\(3\) 0 0
\(4\) −0.745540 − 1.85585i −0.372770 − 0.927924i
\(5\) 2.17386 − 0.900442i 0.972179 − 0.402690i 0.160656 − 0.987010i \(-0.448639\pi\)
0.811523 + 0.584321i \(0.198639\pi\)
\(6\) 0 0
\(7\) −1.13840 + 1.13840i −0.430276 + 0.430276i −0.888722 − 0.458446i \(-0.848406\pi\)
0.458446 + 0.888722i \(0.348406\pi\)
\(8\) 2.76486 + 0.596278i 0.977526 + 0.210816i
\(9\) 0 0
\(10\) −0.666644 + 3.26014i −0.210811 + 1.03095i
\(11\) −1.16219 − 2.80577i −0.350413 − 0.845971i −0.996569 − 0.0827662i \(-0.973625\pi\)
0.646156 − 0.763205i \(-0.276375\pi\)
\(12\) 0 0
\(13\) 5.29700 + 2.19409i 1.46912 + 0.608531i 0.966659 − 0.256066i \(-0.0824264\pi\)
0.502465 + 0.864597i \(0.332426\pi\)
\(14\) −0.432223 − 2.23541i −0.115516 − 0.597437i
\(15\) 0 0
\(16\) −2.88834 + 2.76722i −0.722085 + 0.691804i
\(17\) − 1.37204i − 0.332768i −0.986061 − 0.166384i \(-0.946791\pi\)
0.986061 − 0.166384i \(-0.0532092\pi\)
\(18\) 0 0
\(19\) −0.435446 − 0.180368i −0.0998982 − 0.0413792i 0.332175 − 0.943218i \(-0.392218\pi\)
−0.432073 + 0.901839i \(0.642218\pi\)
\(20\) −3.29178 − 3.36303i −0.736064 − 0.751997i
\(21\) 0 0
\(22\) 4.20782 + 0.860429i 0.897109 + 0.183444i
\(23\) −0.900399 − 0.900399i −0.187746 − 0.187746i 0.606975 − 0.794721i \(-0.292383\pi\)
−0.794721 + 0.606975i \(0.792383\pi\)
\(24\) 0 0
\(25\) 0.379330 − 0.379330i 0.0758661 − 0.0758661i
\(26\) −6.76583 + 4.46858i −1.32689 + 0.876360i
\(27\) 0 0
\(28\) 2.96143 + 1.26398i 0.559658 + 0.238870i
\(29\) 2.18893 − 5.28455i 0.406475 − 0.981317i −0.579583 − 0.814913i \(-0.696785\pi\)
0.986058 − 0.166404i \(-0.0532155\pi\)
\(30\) 0 0
\(31\) 3.64689 0.655001 0.327500 − 0.944851i \(-0.393794\pi\)
0.327500 + 0.944851i \(0.393794\pi\)
\(32\) −0.954714 − 5.57571i −0.168771 − 0.985655i
\(33\) 0 0
\(34\) 1.60755 + 1.08662i 0.275693 + 0.186355i
\(35\) −1.44966 + 3.49980i −0.245038 + 0.591573i
\(36\) 0 0
\(37\) 10.4431 − 4.32566i 1.71683 − 0.711134i 0.716927 − 0.697149i \(-0.245548\pi\)
0.999902 − 0.0139853i \(-0.00445180\pi\)
\(38\) 0.556192 − 0.367344i 0.0902263 − 0.0595911i
\(39\) 0 0
\(40\) 6.54733 − 1.19337i 1.03522 − 0.188689i
\(41\) 3.37936 + 3.37936i 0.527767 + 0.527767i 0.919906 − 0.392139i \(-0.128265\pi\)
−0.392139 + 0.919906i \(0.628265\pi\)
\(42\) 0 0
\(43\) −0.290826 − 0.702117i −0.0443506 − 0.107072i 0.900152 − 0.435576i \(-0.143455\pi\)
−0.944502 + 0.328505i \(0.893455\pi\)
\(44\) −4.34062 + 4.24866i −0.654374 + 0.640509i
\(45\) 0 0
\(46\) 1.76805 − 0.341858i 0.260685 − 0.0504043i
\(47\) − 9.15553i − 1.33547i −0.744398 − 0.667736i \(-0.767264\pi\)
0.744398 − 0.667736i \(-0.232736\pi\)
\(48\) 0 0
\(49\) 4.40807i 0.629725i
\(50\) 0.144022 + 0.744865i 0.0203678 + 0.105340i
\(51\) 0 0
\(52\) 0.122770 − 11.4662i 0.0170251 − 1.59008i
\(53\) 0.279738 + 0.675348i 0.0384250 + 0.0927662i 0.941927 − 0.335818i \(-0.109013\pi\)
−0.903502 + 0.428584i \(0.859013\pi\)
\(54\) 0 0
\(55\) −5.05286 − 5.05286i −0.681328 − 0.681328i
\(56\) −3.82633 + 2.46872i −0.511315 + 0.329897i
\(57\) 0 0
\(58\) 4.45807 + 6.74992i 0.585374 + 0.886308i
\(59\) 9.51130 − 3.93971i 1.23827 − 0.512906i 0.335094 − 0.942185i \(-0.391232\pi\)
0.903172 + 0.429278i \(0.141232\pi\)
\(60\) 0 0
\(61\) −3.14636 + 7.59599i −0.402850 + 0.972566i 0.584121 + 0.811667i \(0.301440\pi\)
−0.986971 + 0.160899i \(0.948560\pi\)
\(62\) −2.88826 + 4.27289i −0.366809 + 0.542657i
\(63\) 0 0
\(64\) 7.28891 + 3.29725i 0.911113 + 0.412156i
\(65\) 13.4906 1.67330
\(66\) 0 0
\(67\) 2.00791 − 4.84753i 0.245306 − 0.592221i −0.752488 − 0.658606i \(-0.771147\pi\)
0.997794 + 0.0663851i \(0.0211466\pi\)
\(68\) −2.54629 + 1.02291i −0.308783 + 0.124046i
\(69\) 0 0
\(70\) −2.95244 − 4.47026i −0.352884 − 0.534299i
\(71\) 3.53207 − 3.53207i 0.419180 − 0.419180i −0.465741 − 0.884921i \(-0.654212\pi\)
0.884921 + 0.465741i \(0.154212\pi\)
\(72\) 0 0
\(73\) −4.79474 − 4.79474i −0.561182 − 0.561182i 0.368461 − 0.929643i \(-0.379885\pi\)
−0.929643 + 0.368461i \(0.879885\pi\)
\(74\) −3.20251 + 15.6615i −0.372285 + 1.82061i
\(75\) 0 0
\(76\) −0.0100924 + 0.942593i −0.00115768 + 0.108123i
\(77\) 4.51714 + 1.87106i 0.514776 + 0.213227i
\(78\) 0 0
\(79\) 14.3012i 1.60902i 0.593942 + 0.804508i \(0.297571\pi\)
−0.593942 + 0.804508i \(0.702429\pi\)
\(80\) −3.78712 + 8.61632i −0.423413 + 0.963334i
\(81\) 0 0
\(82\) −6.63581 + 1.28305i −0.732803 + 0.141690i
\(83\) 7.87490 + 3.26189i 0.864383 + 0.358039i 0.770420 − 0.637537i \(-0.220047\pi\)
0.0939626 + 0.995576i \(0.470047\pi\)
\(84\) 0 0
\(85\) −1.23544 − 2.98262i −0.134002 − 0.323510i
\(86\) 1.05296 + 0.215314i 0.113544 + 0.0232179i
\(87\) 0 0
\(88\) −1.54027 − 8.45055i −0.164193 − 0.900831i
\(89\) −1.44497 + 1.44497i −0.153166 + 0.153166i −0.779531 − 0.626364i \(-0.784542\pi\)
0.626364 + 0.779531i \(0.284542\pi\)
\(90\) 0 0
\(91\) −8.52789 + 3.53237i −0.893966 + 0.370293i
\(92\) −0.999720 + 2.34229i −0.104228 + 0.244200i
\(93\) 0 0
\(94\) 10.7271 + 7.25099i 1.10642 + 0.747882i
\(95\) −1.10901 −0.113782
\(96\) 0 0
\(97\) 8.99231 0.913031 0.456515 − 0.889716i \(-0.349097\pi\)
0.456515 + 0.889716i \(0.349097\pi\)
\(98\) −5.16473 − 3.49110i −0.521716 − 0.352654i
\(99\) 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 864.2.v.a.109.11 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.22 yes 128
32.5 even 8 inner 864.2.v.a.325.11 yes 128
96.5 odd 8 inner 864.2.v.a.325.22 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.11 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.22 yes 128 3.2 odd 2 inner
864.2.v.a.325.11 yes 128 32.5 even 8 inner
864.2.v.a.325.22 yes 128 96.5 odd 8 inner