Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [770,2,Mod(19,770)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(770, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 25, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("770.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 770.br (of order \(30\), degree \(8\), 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: | \(6.14848095564\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −0.104528 | − | 0.994522i | −2.23946 | + | 2.48717i | −0.978148 | + | 0.207912i | 1.85162 | − | 1.25359i | 2.70764 | + | 1.96721i | −1.36078 | + | 2.26898i | 0.309017 | + | 0.951057i | −0.857262 | − | 8.15630i | −1.44027 | − | 1.71044i |
19.2 | −0.104528 | − | 0.994522i | −2.15108 | + | 2.38901i | −0.978148 | + | 0.207912i | 0.451448 | + | 2.19002i | 2.60077 | + | 1.88957i | −0.368128 | − | 2.62002i | 0.309017 | + | 0.951057i | −0.766666 | − | 7.29434i | 2.13084 | − | 0.677894i |
19.3 | −0.104528 | − | 0.994522i | −1.71966 | + | 1.90987i | −0.978148 | + | 0.207912i | −0.866727 | − | 2.06126i | 2.07916 | + | 1.51060i | 2.56818 | − | 0.635956i | 0.309017 | + | 0.951057i | −0.376808 | − | 3.58509i | −1.95937 | + | 1.07744i |
19.4 | −0.104528 | − | 0.994522i | −1.36311 | + | 1.51389i | −0.978148 | + | 0.207912i | 1.17435 | + | 1.90287i | 1.64808 | + | 1.19740i | 2.37638 | + | 1.16310i | 0.309017 | + | 0.951057i | −0.120202 | − | 1.14364i | 1.76969 | − | 1.36682i |
19.5 | −0.104528 | − | 0.994522i | −1.34422 | + | 1.49291i | −0.978148 | + | 0.207912i | −1.50836 | + | 1.65071i | 1.62524 | + | 1.18081i | 0.569988 | + | 2.58362i | 0.309017 | + | 0.951057i | −0.108263 | − | 1.03005i | 1.79934 | + | 1.32755i |
19.6 | −0.104528 | − | 0.994522i | −1.24215 | + | 1.37955i | −0.978148 | + | 0.207912i | −1.69929 | + | 1.45342i | 1.50183 | + | 1.09114i | −2.64273 | − | 0.126351i | 0.309017 | + | 0.951057i | −0.0466281 | − | 0.443636i | 1.62308 | + | 1.53806i |
19.7 | −0.104528 | − | 0.994522i | −1.01305 | + | 1.12511i | −0.978148 | + | 0.207912i | 2.23518 | + | 0.0631719i | 1.22483 | + | 0.889894i | 2.63065 | − | 0.282317i | 0.309017 | + | 0.951057i | 0.0739924 | + | 0.703990i | −0.170814 | − | 2.22953i |
19.8 | −0.104528 | − | 0.994522i | −0.883875 | + | 0.981643i | −0.978148 | + | 0.207912i | 1.96039 | − | 1.07558i | 1.06866 | + | 0.776423i | −2.39824 | + | 1.11733i | 0.309017 | + | 0.951057i | 0.131198 | + | 1.24827i | −1.27460 | − | 1.83722i |
19.9 | −0.104528 | − | 0.994522i | −0.708055 | + | 0.786375i | −0.978148 | + | 0.207912i | −0.558846 | − | 2.16511i | 0.856079 | + | 0.621977i | −2.10929 | − | 1.59716i | 0.309017 | + | 0.951057i | 0.196542 | + | 1.86997i | −2.09483 | + | 0.782100i |
19.10 | −0.104528 | − | 0.994522i | −0.614654 | + | 0.682642i | −0.978148 | + | 0.207912i | −2.21688 | − | 0.292321i | 0.743152 | + | 0.539931i | 1.26118 | − | 2.32582i | 0.309017 | + | 0.951057i | 0.225384 | + | 2.14439i | −0.0589926 | + | 2.23529i |
19.11 | −0.104528 | − | 0.994522i | −0.164424 | + | 0.182611i | −0.978148 | + | 0.207912i | 0.163561 | − | 2.23008i | 0.198798 | + | 0.144435i | 0.305488 | + | 2.62806i | 0.309017 | + | 0.951057i | 0.307274 | + | 2.92351i | −2.23496 | + | 0.0704416i |
19.12 | −0.104528 | − | 0.994522i | −0.131298 | + | 0.145822i | −0.978148 | + | 0.207912i | −1.81100 | + | 1.31160i | 0.158747 | + | 0.115337i | 1.50133 | − | 2.17853i | 0.309017 | + | 0.951057i | 0.309561 | + | 2.94527i | 1.49371 | + | 1.66398i |
19.13 | −0.104528 | − | 0.994522i | −0.0728546 | + | 0.0809133i | −0.978148 | + | 0.207912i | 2.21334 | − | 0.317988i | 0.0880854 | + | 0.0639978i | −1.41085 | − | 2.23819i | 0.309017 | + | 0.951057i | 0.312346 | + | 2.97178i | −0.547603 | − | 2.16798i |
19.14 | −0.104528 | − | 0.994522i | 0.343389 | − | 0.381372i | −0.978148 | + | 0.207912i | −1.93351 | − | 1.12318i | −0.415177 | − | 0.301644i | −2.06024 | + | 1.65994i | 0.309017 | + | 0.951057i | 0.286057 | + | 2.72165i | −0.914919 | + | 2.04032i |
19.15 | −0.104528 | − | 0.994522i | 0.380444 | − | 0.422526i | −0.978148 | + | 0.207912i | 0.813911 | − | 2.08268i | −0.459979 | − | 0.334194i | 1.92359 | + | 1.81653i | 0.309017 | + | 0.951057i | 0.279795 | + | 2.66207i | −2.15635 | − | 0.591753i |
19.16 | −0.104528 | − | 0.994522i | 0.494228 | − | 0.548896i | −0.978148 | + | 0.207912i | 0.356709 | + | 2.20743i | −0.597550 | − | 0.434146i | −2.19539 | − | 1.47657i | 0.309017 | + | 0.951057i | 0.256560 | + | 2.44101i | 2.15805 | − | 0.585494i |
19.17 | −0.104528 | − | 0.994522i | 0.912264 | − | 1.01317i | −0.978148 | + | 0.207912i | 1.85934 | + | 1.24212i | −1.10298 | − | 0.801361i | 1.15841 | − | 2.37867i | 0.309017 | + | 0.951057i | 0.119294 | + | 1.13501i | 1.04097 | − | 1.97899i |
19.18 | −0.104528 | − | 0.994522i | 1.05781 | − | 1.17482i | −0.978148 | + | 0.207912i | 1.96806 | + | 1.06148i | −1.27896 | − | 0.929215i | −0.347398 | + | 2.62284i | 0.309017 | + | 0.951057i | 0.0523512 | + | 0.498088i | 0.849949 | − | 2.06823i |
19.19 | −0.104528 | − | 0.994522i | 1.38819 | − | 1.54174i | −0.978148 | + | 0.207912i | −1.25777 | + | 1.84879i | −1.67840 | − | 1.21943i | −1.06120 | + | 2.42360i | 0.309017 | + | 0.951057i | −0.136310 | − | 1.29690i | 1.97013 | + | 1.05763i |
19.20 | −0.104528 | − | 0.994522i | 1.51064 | − | 1.67774i | −0.978148 | + | 0.207912i | −0.966180 | − | 2.01656i | −1.82645 | − | 1.32699i | −2.49542 | − | 0.879132i | 0.309017 | + | 0.951057i | −0.219179 | − | 2.08534i | −1.90452 | + | 1.17167i |
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
55.h | odd | 10 | 1 | inner |
385.br | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 770.2.br.b | yes | 192 |
5.b | even | 2 | 1 | 770.2.br.a | ✓ | 192 | |
7.d | odd | 6 | 1 | inner | 770.2.br.b | yes | 192 |
11.d | odd | 10 | 1 | 770.2.br.a | ✓ | 192 | |
35.i | odd | 6 | 1 | 770.2.br.a | ✓ | 192 | |
55.h | odd | 10 | 1 | inner | 770.2.br.b | yes | 192 |
77.n | even | 30 | 1 | 770.2.br.a | ✓ | 192 | |
385.br | even | 30 | 1 | inner | 770.2.br.b | yes | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
770.2.br.a | ✓ | 192 | 5.b | even | 2 | 1 | |
770.2.br.a | ✓ | 192 | 11.d | odd | 10 | 1 | |
770.2.br.a | ✓ | 192 | 35.i | odd | 6 | 1 | |
770.2.br.a | ✓ | 192 | 77.n | even | 30 | 1 | |
770.2.br.b | yes | 192 | 1.a | even | 1 | 1 | trivial |
770.2.br.b | yes | 192 | 7.d | odd | 6 | 1 | inner |
770.2.br.b | yes | 192 | 55.h | odd | 10 | 1 | inner |
770.2.br.b | yes | 192 | 385.br | even | 30 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{192} - 45 T_{3}^{190} + 846 T_{3}^{188} - 6015 T_{3}^{186} + 150 T_{3}^{185} + \cdots + 21\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\).