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Properties

Label 315.2.be.b
Level $315$
Weight $2$
Character orbit 315.be
Analytic conductor $2.515$
Analytic rank $0$
Dimension $30$
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(236,315)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.236");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.be (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: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} - q^{3} + 15 q^{4} + 30 q^{5} + q^{6} + 6 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} - q^{3} + 15 q^{4} + 30 q^{5} + q^{6} + 6 q^{7} - 5 q^{9} + 3 q^{10} - 18 q^{12} + 12 q^{13} - 9 q^{14} - q^{15} - 21 q^{16} + 3 q^{17} - 22 q^{18} + 15 q^{20} - 10 q^{21} + 15 q^{22} + 2 q^{24} + 30 q^{25} - 24 q^{26} + 5 q^{27} + 27 q^{28} + q^{30} + 6 q^{31} + 9 q^{32} - 17 q^{33} - 48 q^{34} + 6 q^{35} + 21 q^{36} - 3 q^{37} - 60 q^{38} + 12 q^{39} + 18 q^{41} - 47 q^{42} + 12 q^{43} - 15 q^{44} - 5 q^{45} + 9 q^{46} - 30 q^{47} + 40 q^{48} - 24 q^{49} + 3 q^{50} + 33 q^{51} + 30 q^{53} + 13 q^{54} + 72 q^{56} - 21 q^{57} + 15 q^{59} - 18 q^{60} - 30 q^{61} - 12 q^{62} + 10 q^{63} - 138 q^{64} + 12 q^{65} + 44 q^{66} - 6 q^{67} - 42 q^{68} - 32 q^{69} - 9 q^{70} - 137 q^{72} + 6 q^{73} - q^{75} + 54 q^{76} - 21 q^{77} - 18 q^{78} - 12 q^{79} - 21 q^{80} - 17 q^{81} + 6 q^{82} + 6 q^{83} - 12 q^{84} + 3 q^{85} - 47 q^{87} + 96 q^{88} + 3 q^{89} - 22 q^{90} + 15 q^{91} - 3 q^{92} - 18 q^{93} + 3 q^{94} + 60 q^{96} - 36 q^{97} - 24 q^{98} - q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
236.1 −2.43144 + 1.40379i −1.45528 − 0.939239i 2.94126 − 5.09441i 1.00000 4.85691 + 0.240797i 0.979355 − 2.45782i 10.9005i 1.23566 + 2.73371i −2.43144 + 1.40379i
236.2 −2.02825 + 1.17101i 0.587320 + 1.62943i 1.74252 − 3.01813i 1.00000 −3.09931 − 2.61714i 2.45764 + 0.979787i 3.47799i −2.31011 + 1.91400i −2.02825 + 1.17101i
236.3 −1.57665 + 0.910280i −0.582275 − 1.63124i 0.657218 − 1.13833i 1.00000 2.40293 + 2.04187i −0.602558 + 2.57622i − 1.24811i −2.32191 + 1.89966i −1.57665 + 0.910280i
236.4 −1.16645 + 0.673451i −1.58477 + 0.698930i −0.0929278 + 0.160956i 1.00000 1.37786 − 1.88253i 0.696317 − 2.55248i − 2.94413i 2.02299 − 2.21529i −1.16645 + 0.673451i
236.5 −1.16637 + 0.673405i 1.48755 − 0.887234i −0.0930506 + 0.161168i 1.00000 −1.13757 + 2.03657i 2.59059 − 0.537423i − 2.94426i 1.42563 − 2.63962i −1.16637 + 0.673405i
236.6 −0.804799 + 0.464651i 1.68100 + 0.417423i −0.568200 + 0.984151i 1.00000 −1.54682 + 0.445136i −2.32336 + 1.26570i − 2.91466i 2.65152 + 1.40338i −0.804799 + 0.464651i
236.7 −0.0228655 + 0.0132014i −0.396980 − 1.68594i −0.999651 + 1.73145i 1.00000 0.0313339 + 0.0333092i −1.43459 − 2.22305i − 0.105593i −2.68481 + 1.33857i −0.0228655 + 0.0132014i
236.8 0.441540 − 0.254923i 1.05462 − 1.37396i −0.870028 + 1.50693i 1.00000 0.115404 − 0.875506i 2.05349 + 1.66829i 1.90685i −0.775539 − 2.89802i 0.441540 − 0.254923i
236.9 0.600029 − 0.346427i −0.431152 + 1.67753i −0.759977 + 1.31632i 1.00000 0.322438 + 1.15593i 2.55145 − 0.700092i 2.43881i −2.62822 − 1.44654i 0.600029 − 0.346427i
236.10 0.794909 − 0.458941i −1.63482 − 0.572153i −0.578746 + 1.00242i 1.00000 −1.56212 + 0.295477i −2.55889 + 0.672387i 2.89821i 2.34528 + 1.87074i 0.794909 − 0.458941i
236.11 1.07014 − 0.617846i 1.67621 + 0.436249i −0.236533 + 0.409687i 1.00000 2.06332 − 0.568792i −0.432012 − 2.61024i 3.05595i 2.61937 + 1.46249i 1.07014 − 0.617846i
236.12 1.52284 − 0.879212i 0.575251 + 1.63373i 0.546028 − 0.945749i 1.00000 2.31241 + 1.98215i −1.21730 + 2.34908i 1.59655i −2.33817 + 1.87962i 1.52284 − 0.879212i
236.13 1.87276 − 1.08124i 0.865496 − 1.50031i 1.33817 − 2.31777i 1.00000 −0.00132154 − 3.74553i −1.75124 + 1.98322i − 1.46255i −1.50183 − 2.59702i 1.87276 − 1.08124i
236.14 2.03201 − 1.17318i −0.909209 − 1.47423i 1.75271 − 3.03578i 1.00000 −3.57705 − 1.92897i 0.498797 − 2.59831i − 3.53226i −1.34668 + 2.68076i 2.03201 − 1.17318i
236.15 2.36259 − 1.36404i −1.43297 + 0.972928i 2.72121 − 4.71328i 1.00000 −2.05841 + 4.25326i 1.49231 + 2.18472i − 9.39122i 1.10682 − 2.78836i 2.36259 − 1.36404i
311.1 −2.43144 − 1.40379i −1.45528 + 0.939239i 2.94126 + 5.09441i 1.00000 4.85691 − 0.240797i 0.979355 + 2.45782i − 10.9005i 1.23566 − 2.73371i −2.43144 − 1.40379i
311.2 −2.02825 − 1.17101i 0.587320 − 1.62943i 1.74252 + 3.01813i 1.00000 −3.09931 + 2.61714i 2.45764 − 0.979787i − 3.47799i −2.31011 − 1.91400i −2.02825 − 1.17101i
311.3 −1.57665 − 0.910280i −0.582275 + 1.63124i 0.657218 + 1.13833i 1.00000 2.40293 − 2.04187i −0.602558 − 2.57622i 1.24811i −2.32191 − 1.89966i −1.57665 − 0.910280i
311.4 −1.16645 − 0.673451i −1.58477 − 0.698930i −0.0929278 − 0.160956i 1.00000 1.37786 + 1.88253i 0.696317 + 2.55248i 2.94413i 2.02299 + 2.21529i −1.16645 − 0.673451i
311.5 −1.16637 − 0.673405i 1.48755 + 0.887234i −0.0930506 − 0.161168i 1.00000 −1.13757 − 2.03657i 2.59059 + 0.537423i 2.94426i 1.42563 + 2.63962i −1.16637 − 0.673405i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 236.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s 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.be.b yes 30
3.b odd 2 1 945.2.be.b 30
7.d odd 6 1 315.2.t.b ✓ 30
9.c even 3 1 945.2.t.b 30
9.d odd 6 1 315.2.t.b ✓ 30
21.g even 6 1 945.2.t.b 30
63.k odd 6 1 945.2.be.b 30
63.s even 6 1 inner 315.2.be.b yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.b ✓ 30 7.d odd 6 1
315.2.t.b ✓ 30 9.d odd 6 1
315.2.be.b yes 30 1.a even 1 1 trivial
315.2.be.b yes 30 63.s even 6 1 inner
945.2.t.b 30 9.c even 3 1
945.2.t.b 30 21.g even 6 1
945.2.be.b 30 3.b odd 2 1
945.2.be.b 30 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 3 T_{2}^{29} - 18 T_{2}^{28} + 63 T_{2}^{27} + 213 T_{2}^{26} - 828 T_{2}^{25} - 1417 T_{2}^{24} + \cdots + 27 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display