↑

Properties

Label 575.2.p.d
Level $575$
Weight $2$
Character orbit 575.p
Analytic conductor $4.591$
Analytic rank $0$
Dimension $100$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,2,Mod(49,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.p (of order \(22\), degree \(10\), not 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 22 q^{4} - 22 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 22 q^{4} - 22 q^{6} - 6 q^{9} - 32 q^{11} + 24 q^{14} + 54 q^{16} + 10 q^{19} - 18 q^{21} - 204 q^{24} - 38 q^{26} + 76 q^{29} + 4 q^{31} + 62 q^{34} - 118 q^{36} - 64 q^{39} - 22 q^{41} + 110 q^{44} + 306 q^{46} - 40 q^{51} - 38 q^{54} + 154 q^{56} + 100 q^{59} + 8 q^{61} + 88 q^{64} - 288 q^{66} + 100 q^{69} - 50 q^{71} + 94 q^{74} - 52 q^{76} - 144 q^{79} - 142 q^{81} - 200 q^{84} - 40 q^{86} + 48 q^{89} + 76 q^{91} - 300 q^{94} - 338 q^{96} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −2.75058 + 0.395474i −0.0520830 − 0.0237855i 5.49030 − 1.61210i 0 0.152665 + 0.0448264i 0.743614 + 1.15709i −9.40848 + 4.29671i −1.96244 − 2.26477i 0
49.2 −2.52189 + 0.362593i 1.76116 + 0.804293i 4.30948 − 1.26538i 0 −4.73308 − 1.38976i −0.362741 − 0.564436i −5.77406 + 2.63693i 0.490203 + 0.565724i 0
49.3 −1.29320 + 0.185934i 0.932839 + 0.426013i −0.281191 + 0.0825651i 0 −1.28556 − 0.377474i −0.663422 − 1.03230i 2.72515 − 1.24453i −1.27588 − 1.47244i 0
49.4 −1.16108 + 0.166938i −1.15375 − 0.526900i −0.598752 + 0.175809i 0 1.42755 + 0.419168i −2.72045 − 4.23310i 2.79988 − 1.27866i −0.911065 − 1.05142i 0
49.5 −0.273550 + 0.0393306i 2.09261 + 0.955665i −1.84570 + 0.541947i 0 −0.610022 − 0.179119i 1.69534 + 2.63800i 0.986355 − 0.450453i 1.50116 + 1.73243i 0
49.6 0.273550 − 0.0393306i −2.09261 − 0.955665i −1.84570 + 0.541947i 0 −0.610022 − 0.179119i −1.69534 − 2.63800i −0.986355 + 0.450453i 1.50116 + 1.73243i 0
49.7 1.16108 − 0.166938i 1.15375 + 0.526900i −0.598752 + 0.175809i 0 1.42755 + 0.419168i 2.72045 + 4.23310i −2.79988 + 1.27866i −0.911065 − 1.05142i 0
49.8 1.29320 − 0.185934i −0.932839 − 0.426013i −0.281191 + 0.0825651i 0 −1.28556 − 0.377474i 0.663422 + 1.03230i −2.72515 + 1.24453i −1.27588 − 1.47244i 0
49.9 2.52189 − 0.362593i −1.76116 − 0.804293i 4.30948 − 1.26538i 0 −4.73308 − 1.38976i 0.362741 + 0.564436i 5.77406 − 2.63693i 0.490203 + 0.565724i 0
49.10 2.75058 − 0.395474i 0.0520830 + 0.0237855i 5.49030 − 1.61210i 0 0.152665 + 0.0448264i −0.743614 − 1.15709i 9.40848 − 4.29671i −1.96244 − 2.26477i 0
124.1 −0.762237 − 2.59594i 0.894513 − 0.775100i −4.47539 + 2.87616i 0 −2.69394 − 1.73129i 2.77933 + 1.26928i 6.78823 + 5.88204i −0.227571 + 1.58279i 0
124.2 −0.591987 − 2.01612i −2.41265 + 2.09058i −2.03179 + 1.30575i 0 5.64312 + 3.62661i −3.17325 − 1.44918i 0.659333 + 0.571316i 1.02344 − 7.11821i 0
124.3 −0.561816 − 1.91337i 0.0141367 − 0.0122495i −1.66284 + 1.06864i 0 −0.0313800 − 0.0201667i 3.50714 + 1.60166i −0.0352332 − 0.0305297i −0.426895 + 2.96912i 0
124.4 −0.380775 − 1.29680i 2.16478 − 1.87580i 0.145804 − 0.0937028i 0 −3.25683 − 2.09304i −3.64647 − 1.66529i −2.21990 − 1.92355i 0.740735 − 5.15193i 0
124.5 −0.0248738 − 0.0847124i −2.18556 + 1.89380i 1.67595 − 1.07707i 0 0.214792 + 0.138038i −1.23793 − 0.565342i −0.266377 − 0.230817i 0.763258 − 5.30858i 0
124.6 0.0248738 + 0.0847124i 2.18556 − 1.89380i 1.67595 − 1.07707i 0 0.214792 + 0.138038i 1.23793 + 0.565342i 0.266377 + 0.230817i 0.763258 − 5.30858i 0
124.7 0.380775 + 1.29680i −2.16478 + 1.87580i 0.145804 − 0.0937028i 0 −3.25683 − 2.09304i 3.64647 + 1.66529i 2.21990 + 1.92355i 0.740735 − 5.15193i 0
124.8 0.561816 + 1.91337i −0.0141367 + 0.0122495i −1.66284 + 1.06864i 0 −0.0313800 − 0.0201667i −3.50714 − 1.60166i 0.0352332 + 0.0305297i −0.426895 + 2.96912i 0
124.9 0.591987 + 2.01612i 2.41265 − 2.09058i −2.03179 + 1.30575i 0 5.64312 + 3.62661i 3.17325 + 1.44918i −0.659333 − 0.571316i 1.02344 − 7.11821i 0
124.10 0.762237 + 2.59594i −0.894513 + 0.775100i −4.47539 + 2.87616i 0 −2.69394 − 1.73129i −2.77933 − 1.26928i −6.78823 − 5.88204i −0.227571 + 1.58279i 0
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.p.d 100
5.b even 2 1 inner 575.2.p.d 100
5.c odd 4 1 115.2.g.c ✓ 50
5.c odd 4 1 575.2.k.d 50
23.c even 11 1 inner 575.2.p.d 100
115.j even 22 1 inner 575.2.p.d 100
115.k odd 44 1 115.2.g.c ✓ 50
115.k odd 44 1 575.2.k.d 50
115.k odd 44 1 2645.2.a.y 25
115.l even 44 1 2645.2.a.x 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.c ✓ 50 5.c odd 4 1
115.2.g.c ✓ 50 115.k odd 44 1
575.2.k.d 50 5.c odd 4 1
575.2.k.d 50 115.k odd 44 1
575.2.p.d 100 1.a even 1 1 trivial
575.2.p.d 100 5.b even 2 1 inner
575.2.p.d 100 23.c even 11 1 inner
575.2.p.d 100 115.j even 22 1 inner
2645.2.a.x 25 115.l even 44 1
2645.2.a.y 25 115.k odd 44 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{100} - 21 T_{2}^{98} + 227 T_{2}^{96} - 1818 T_{2}^{94} + 12421 T_{2}^{92} - 87547 T_{2}^{90} + \cdots + 279841 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display