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Properties

Label 315.2.t.c
Level $315$
Weight $2$
Character orbit 315.t
Analytic conductor $2.515$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(101,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.t (of order \(6\), degree \(2\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{3} - 32 q^{4} - 16 q^{5} - 2 q^{6} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{3} - 32 q^{4} - 16 q^{5} - 2 q^{6} + q^{7} + q^{9} + 3 q^{11} + 12 q^{12} + 6 q^{13} - 15 q^{14} - q^{15} + 32 q^{16} - 3 q^{17} - 13 q^{18} + 16 q^{20} - q^{21} - 21 q^{22} - 9 q^{23} - 4 q^{24} - 16 q^{25} + 12 q^{26} + 23 q^{27} - 31 q^{28} + 18 q^{29} - 2 q^{30} + 19 q^{33} - 30 q^{34} + q^{35} + 18 q^{36} - q^{37} - 30 q^{38} + 21 q^{39} + 6 q^{41} + 19 q^{42} - 19 q^{43} + 21 q^{44} - 8 q^{45} + 6 q^{46} - 30 q^{47} - 35 q^{48} + 5 q^{49} + 36 q^{51} + 21 q^{52} - 24 q^{53} - 59 q^{54} + 30 q^{56} + 27 q^{57} + 30 q^{59} + 3 q^{60} - 32 q^{63} + 76 q^{64} + 26 q^{66} - 50 q^{67} - 3 q^{68} - 50 q^{69} + 9 q^{70} - 14 q^{72} + 12 q^{73} + 60 q^{74} + 2 q^{75} + 54 q^{76} - 27 q^{77} - 42 q^{78} + 4 q^{79} - 16 q^{80} - 23 q^{81} - 24 q^{82} - 42 q^{83} - 72 q^{84} - 3 q^{85} + 51 q^{86} + 34 q^{87} + 42 q^{88} + 30 q^{89} + 41 q^{90} - 57 q^{91} + 6 q^{92} - 33 q^{93} + 15 q^{96} - 42 q^{97} + 6 q^{98} - q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 − 2.62557i −1.73024 − 0.0792089i −4.89362 −0.500000 + 0.866025i −0.207969 + 4.54286i 1.43974 + 2.21972i 7.59741i 2.98745 + 0.274101i 2.27381 + 1.31279i
101.2 − 2.34793i −0.0139204 + 1.73199i −3.51278 −0.500000 + 0.866025i 4.06660 + 0.0326841i 2.22052 − 1.43851i 3.55190i −2.99961 − 0.0482201i 2.03337 + 1.17397i
101.3 − 2.21457i 1.62808 − 0.591077i −2.90433 −0.500000 + 0.866025i −1.30898 − 3.60549i −0.612213 − 2.57395i 2.00271i 2.30126 − 1.92463i 1.91788 + 1.10729i
101.4 − 1.93560i −1.06817 − 1.36345i −1.74654 −0.500000 + 0.866025i −2.63910 + 2.06755i −2.29351 − 1.31902i − 0.490599i −0.718011 + 2.91281i 1.67628 + 0.967799i
101.5 − 1.23655i 1.51263 + 0.843766i 0.470942 −0.500000 + 0.866025i 1.04336 − 1.87045i 2.44060 + 1.02150i − 3.05545i 1.57612 + 2.55262i 1.07088 + 0.618275i
101.6 − 1.09551i −1.60475 + 0.651761i 0.799850 −0.500000 + 0.866025i 0.714013 + 1.75802i −2.64482 − 0.0701258i − 3.06727i 2.15042 − 2.09182i 0.948743 + 0.547757i
101.7 − 0.524306i 1.63490 − 0.571919i 1.72510 −0.500000 + 0.866025i −0.299860 − 0.857190i −1.53835 + 2.15255i − 1.95309i 2.34582 − 1.87006i 0.454062 + 0.262153i
101.8 − 0.103991i 0.570662 − 1.63534i 1.98919 −0.500000 + 0.866025i −0.170061 − 0.0593436i −0.107447 − 2.64357i − 0.414839i −2.34869 − 1.86646i 0.0900587 + 0.0519954i
101.9 0.396951i −1.27680 − 1.17037i 1.84243 −0.500000 + 0.866025i 0.464581 − 0.506829i 2.43622 − 1.03190i 1.52526i 0.260457 + 2.98867i −0.343770 − 0.198476i
101.10 0.465802i −1.08628 + 1.34907i 1.78303 −0.500000 + 0.866025i −0.628402 − 0.505990i 1.73784 + 1.99497i 1.76214i −0.640003 − 2.93094i −0.403396 − 0.232901i
101.11 0.645959i 0.803605 + 1.53435i 1.58274 −0.500000 + 0.866025i −0.991125 + 0.519096i −2.64433 + 0.0866018i 2.31430i −1.70844 + 2.46602i −0.559417 − 0.322980i
101.12 1.57681i −1.21339 − 1.23600i −0.486323 −0.500000 + 0.866025i 1.94893 − 1.91328i −2.52230 + 0.798755i 2.38678i −0.0553747 + 2.99949i −1.36556 − 0.788404i
101.13 1.71080i 1.72161 + 0.189893i −0.926832 −0.500000 + 0.866025i −0.324870 + 2.94533i 1.44091 − 2.21896i 1.83597i 2.92788 + 0.653845i −1.48159 − 0.855399i
101.14 2.32447i −1.01455 + 1.40381i −3.40314 −0.500000 + 0.866025i −3.26312 − 2.35828i −1.17052 − 2.37274i − 3.26155i −0.941384 − 2.84847i −2.01305 − 1.16223i
101.15 2.44435i 0.807396 + 1.53235i −3.97484 −0.500000 + 0.866025i −3.74561 + 1.97356i 0.510801 + 2.59597i − 4.82720i −1.69622 + 2.47444i −2.11687 − 1.22217i
101.16 2.51890i −0.170788 − 1.72361i −4.34486 −0.500000 + 0.866025i 4.34160 − 0.430199i 1.80687 + 1.93267i − 5.90648i −2.94166 + 0.588745i −2.18143 − 1.25945i
131.1 − 2.51890i −0.170788 + 1.72361i −4.34486 −0.500000 − 0.866025i 4.34160 + 0.430199i 1.80687 − 1.93267i 5.90648i −2.94166 − 0.588745i −2.18143 + 1.25945i
131.2 − 2.44435i 0.807396 − 1.53235i −3.97484 −0.500000 − 0.866025i −3.74561 − 1.97356i 0.510801 − 2.59597i 4.82720i −1.69622 − 2.47444i −2.11687 + 1.22217i
131.3 − 2.32447i −1.01455 − 1.40381i −3.40314 −0.500000 − 0.866025i −3.26312 + 2.35828i −1.17052 + 2.37274i 3.26155i −0.941384 + 2.84847i −2.01305 + 1.16223i
131.4 − 1.71080i 1.72161 − 0.189893i −0.926832 −0.500000 − 0.866025i −0.324870 − 2.94533i 1.44091 + 2.21896i − 1.83597i 2.92788 − 0.653845i −1.48159 + 0.855399i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.t.c ✓ 32
3.b odd 2 1 945.2.t.c 32
7.d odd 6 1 315.2.be.c yes 32
9.c even 3 1 945.2.be.c 32
9.d odd 6 1 315.2.be.c yes 32
21.g even 6 1 945.2.be.c 32
63.i even 6 1 inner 315.2.t.c ✓ 32
63.t odd 6 1 945.2.t.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.c ✓ 32 1.a even 1 1 trivial
315.2.t.c ✓ 32 63.i even 6 1 inner
315.2.be.c yes 32 7.d odd 6 1
315.2.be.c yes 32 9.d odd 6 1
945.2.t.c 32 3.b odd 2 1
945.2.t.c 32 63.t odd 6 1
945.2.be.c 32 9.c even 3 1
945.2.be.c 32 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 48 T_{2}^{30} + 1032 T_{2}^{28} + 13118 T_{2}^{26} + 109596 T_{2}^{24} + 632922 T_{2}^{22} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display