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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 = [128,0,0,0,0,0,0,0,0,16] 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.b.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)
 
\(f(q)\) \(=\) \(q+(-0.679990 + 1.24001i) q^{2} +(-1.07523 - 1.68638i) q^{4} +(-3.54476 + 1.46829i) q^{5} +(1.60611 - 1.60611i) q^{7} +(2.82227 - 0.186565i) q^{8} +(0.589716 - 5.39394i) q^{10} +(0.347249 + 0.838334i) q^{11} +(-5.66583 - 2.34686i) q^{13} +(0.899444 + 3.08372i) q^{14} +(-1.68777 + 3.62649i) q^{16} -0.593837i q^{17} +(5.16462 + 2.13925i) q^{19} +(6.28751 + 4.39907i) q^{20} +(-1.27566 - 0.139468i) q^{22} +(0.0201157 + 0.0201157i) q^{23} +(6.87390 - 6.87390i) q^{25} +(6.76283 - 5.42982i) q^{26} +(-4.43544 - 0.981580i) q^{28} +(2.80250 - 6.76583i) q^{29} +6.90566 q^{31} +(-3.34920 - 4.55882i) q^{32} +(0.736362 + 0.403803i) q^{34} +(-3.33503 + 8.05148i) q^{35} +(1.45330 - 0.601978i) q^{37} +(-6.16457 + 4.94948i) q^{38} +(-9.73032 + 4.80522i) q^{40} +(8.95213 + 8.95213i) q^{41} +(2.56146 + 6.18392i) q^{43} +(1.04038 - 1.48699i) q^{44} +(-0.0386221 + 0.0112651i) q^{46} -2.63414i q^{47} +1.84085i q^{49} +(3.84949 + 13.1979i) q^{50} +(2.13435 + 12.0782i) q^{52} +(2.47496 + 5.97509i) q^{53} +(-2.46183 - 2.46183i) q^{55} +(4.23322 - 4.83250i) q^{56} +(6.48400 + 8.07581i) q^{58} +(-7.46732 + 3.09307i) q^{59} +(-2.06985 + 4.99706i) q^{61} +(-4.69578 + 8.56306i) q^{62} +(7.93039 - 1.05307i) q^{64} +23.5299 q^{65} +(3.91146 - 9.44309i) q^{67} +(-1.00144 + 0.638510i) q^{68} +(-7.71609 - 9.61038i) q^{70} +(10.7650 - 10.7650i) q^{71} +(-4.07812 - 4.07812i) q^{73} +(-0.241775 + 2.21144i) q^{74} +(-1.94554 - 11.0097i) q^{76} +(1.90417 + 0.788734i) q^{77} +7.64704i q^{79} +(0.658016 - 15.3332i) q^{80} +(-17.1880 + 5.01333i) q^{82} +(2.67059 + 1.10620i) q^{83} +(0.871923 + 2.10501i) q^{85} +(-9.40986 - 1.02877i) q^{86} +(1.13643 + 2.30122i) q^{88} +(-3.74752 + 3.74752i) q^{89} +(-12.8692 + 5.33061i) q^{91} +(0.0122938 - 0.0555518i) q^{92} +(3.26635 + 1.79119i) q^{94} -21.4483 q^{95} +14.4996 q^{97} +(-2.28266 - 1.25176i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.679990 + 1.24001i −0.480825 + 0.876816i
\(3\) 0 0
\(4\) −1.07523 − 1.68638i −0.537614 − 0.843191i
\(5\) −3.54476 + 1.46829i −1.58526 + 0.656638i −0.989236 − 0.146327i \(-0.953255\pi\)
−0.596027 + 0.802964i \(0.703255\pi\)
\(6\) 0 0
\(7\) 1.60611 − 1.60611i 0.607051 − 0.607051i −0.335123 − 0.942174i \(-0.608778\pi\)
0.942174 + 0.335123i \(0.108778\pi\)
\(8\) 2.82227 − 0.186565i 0.997822 − 0.0659606i
\(9\) 0 0
\(10\) 0.589716 − 5.39394i 0.186484 − 1.70571i
\(11\) 0.347249 + 0.838334i 0.104700 + 0.252767i 0.967544 − 0.252702i \(-0.0813193\pi\)
−0.862845 + 0.505469i \(0.831319\pi\)
\(12\) 0 0
\(13\) −5.66583 − 2.34686i −1.57142 − 0.650903i −0.584394 − 0.811470i \(-0.698668\pi\)
−0.987025 + 0.160567i \(0.948668\pi\)
\(14\) 0.899444 + 3.08372i 0.240387 + 0.824158i
\(15\) 0 0
\(16\) −1.68777 + 3.62649i −0.421943 + 0.906622i
\(17\) − 0.593837i − 0.144027i −0.997404 − 0.0720134i \(-0.977058\pi\)
0.997404 − 0.0720134i \(-0.0229424\pi\)
\(18\) 0 0
\(19\) 5.16462 + 2.13925i 1.18484 + 0.490779i 0.886073 − 0.463546i \(-0.153423\pi\)
0.298772 + 0.954325i \(0.403423\pi\)
\(20\) 6.28751 + 4.39907i 1.40593 + 0.983663i
\(21\) 0 0
\(22\) −1.27566 − 0.139468i −0.271973 − 0.0297346i
\(23\) 0.0201157 + 0.0201157i 0.00419442 + 0.00419442i 0.709201 − 0.705006i \(-0.249056\pi\)
−0.705006 + 0.709201i \(0.749056\pi\)
\(24\) 0 0
\(25\) 6.87390 − 6.87390i 1.37478 − 1.37478i
\(26\) 6.76283 − 5.42982i 1.32630 − 1.06487i
\(27\) 0 0
\(28\) −4.43544 − 0.981580i −0.838219 − 0.185501i
\(29\) 2.80250 − 6.76583i 0.520411 − 1.25638i −0.417237 − 0.908798i \(-0.637002\pi\)
0.937648 − 0.347586i \(-0.112998\pi\)
\(30\) 0 0
\(31\) 6.90566 1.24029 0.620147 − 0.784486i \(-0.287073\pi\)
0.620147 + 0.784486i \(0.287073\pi\)
\(32\) −3.34920 − 4.55882i −0.592060 − 0.805894i
\(33\) 0 0
\(34\) 0.736362 + 0.403803i 0.126285 + 0.0692517i
\(35\) −3.33503 + 8.05148i −0.563723 + 1.36095i
\(36\) 0 0
\(37\) 1.45330 − 0.601978i 0.238921 − 0.0989645i −0.260010 − 0.965606i \(-0.583726\pi\)
0.498932 + 0.866641i \(0.333726\pi\)
\(38\) −6.16457 + 4.94948i −1.00003 + 0.802912i
\(39\) 0 0
\(40\) −9.73032 + 4.80522i −1.53850 + 0.759772i
\(41\) 8.95213 + 8.95213i 1.39809 + 1.39809i 0.805524 + 0.592563i \(0.201884\pi\)
0.592563 + 0.805524i \(0.298116\pi\)
\(42\) 0 0
\(43\) 2.56146 + 6.18392i 0.390619 + 0.943039i 0.989805 + 0.142429i \(0.0454912\pi\)
−0.599186 + 0.800610i \(0.704509\pi\)
\(44\) 1.04038 − 1.48699i 0.156843 − 0.224173i
\(45\) 0 0
\(46\) −0.0386221 + 0.0112651i −0.00569452 + 0.00166095i
\(47\) − 2.63414i − 0.384229i −0.981373 − 0.192115i \(-0.938465\pi\)
0.981373 − 0.192115i \(-0.0615345\pi\)
\(48\) 0 0
\(49\) 1.84085i 0.262979i
\(50\) 3.84949 + 13.1979i 0.544400 + 1.86646i
\(51\) 0 0
\(52\) 2.13435 + 12.0782i 0.295981 + 1.67494i
\(53\) 2.47496 + 5.97509i 0.339962 + 0.820741i 0.997719 + 0.0675106i \(0.0215056\pi\)
−0.657756 + 0.753231i \(0.728494\pi\)
\(54\) 0 0
\(55\) −2.46183 − 2.46183i −0.331953 − 0.331953i
\(56\) 4.23322 − 4.83250i 0.565687 − 0.645770i
\(57\) 0 0
\(58\) 6.48400 + 8.07581i 0.851390 + 1.06041i
\(59\) −7.46732 + 3.09307i −0.972163 + 0.402683i −0.811517 − 0.584329i \(-0.801358\pi\)
−0.160646 + 0.987012i \(0.551358\pi\)
\(60\) 0 0
\(61\) −2.06985 + 4.99706i −0.265017 + 0.639808i −0.999235 − 0.0391060i \(-0.987549\pi\)
0.734218 + 0.678914i \(0.237549\pi\)
\(62\) −4.69578 + 8.56306i −0.596365 + 1.08751i
\(63\) 0 0
\(64\) 7.93039 − 1.05307i 0.991298 − 0.131634i
\(65\) 23.5299 2.91852
\(66\) 0 0
\(67\) 3.91146 − 9.44309i 0.477860 − 1.15366i −0.482750 − 0.875758i \(-0.660362\pi\)
0.960610 − 0.277899i \(-0.0896380\pi\)
\(68\) −1.00144 + 0.638510i −0.121442 + 0.0774307i
\(69\) 0 0
\(70\) −7.71609 − 9.61038i −0.922249 − 1.14866i
\(71\) 10.7650 − 10.7650i 1.27757 − 1.27757i 0.335549 − 0.942023i \(-0.391078\pi\)
0.942023 − 0.335549i \(-0.108922\pi\)
\(72\) 0 0
\(73\) −4.07812 − 4.07812i −0.477308 − 0.477308i 0.426962 − 0.904270i \(-0.359584\pi\)
−0.904270 + 0.426962i \(0.859584\pi\)
\(74\) −0.241775 + 2.21144i −0.0281058 + 0.257075i
\(75\) 0 0
\(76\) −1.94554 − 11.0097i −0.223168 − 1.26290i
\(77\) 1.90417 + 0.788734i 0.217000 + 0.0898845i
\(78\) 0 0
\(79\) 7.64704i 0.860359i 0.902743 + 0.430180i \(0.141550\pi\)
−0.902743 + 0.430180i \(0.858450\pi\)
\(80\) 0.658016 − 15.3332i 0.0735684 − 1.71430i
\(81\) 0 0
\(82\) −17.1880 + 5.01333i −1.89810 + 0.553630i
\(83\) 2.67059 + 1.10620i 0.293135 + 0.121421i 0.524405 − 0.851469i \(-0.324288\pi\)
−0.231269 + 0.972890i \(0.574288\pi\)
\(84\) 0 0
\(85\) 0.871923 + 2.10501i 0.0945734 + 0.228320i
\(86\) −9.40986 − 1.02877i −1.01469 − 0.110936i
\(87\) 0 0
\(88\) 1.13643 + 2.30122i 0.121144 + 0.245311i
\(89\) −3.74752 + 3.74752i −0.397236 + 0.397236i −0.877257 − 0.480021i \(-0.840629\pi\)
0.480021 + 0.877257i \(0.340629\pi\)
\(90\) 0 0
\(91\) −12.8692 + 5.33061i −1.34906 + 0.558800i
\(92\) 0.0122938 − 0.0555518i 0.00128172 − 0.00579168i
\(93\) 0 0
\(94\) 3.26635 + 1.79119i 0.336898 + 0.184747i
\(95\) −21.4483 −2.20055
\(96\) 0 0
\(97\) 14.4996 1.47221 0.736106 − 0.676866i \(-0.236662\pi\)
0.736106 + 0.676866i \(0.236662\pi\)
\(98\) −2.28266 − 1.25176i −0.230584 − 0.126447i
\(99\) 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.b.109.11 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.22 yes 128
32.5 even 8 inner 864.2.v.b.325.11 yes 128
96.5 odd 8 inner 864.2.v.b.325.22 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.11 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.22 yes 128 3.2 odd 2 inner
864.2.v.b.325.11 yes 128 32.5 even 8 inner
864.2.v.b.325.22 yes 128 96.5 odd 8 inner