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Properties

Label 315.2.p.e
Level $315$
Weight $2$
Character orbit 315.p
Analytic conductor $2.515$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(118,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.p (of order \(4\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{13} - \beta_{7} - \beta_{6}) q^{4} + ( - \beta_{15} + \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{13} - \beta_{11} - \beta_{10} + \cdots - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{13} - \beta_{7} - \beta_{6}) q^{4} + ( - \beta_{15} + \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{13} - 2 \beta_{12} + \cdots + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} - 24 q^{8} + 16 q^{11} - 48 q^{16} - 16 q^{22} + 40 q^{23} + 24 q^{28} - 48 q^{32} + 8 q^{35} + 32 q^{37} - 16 q^{43} + 64 q^{46} + 72 q^{50} - 24 q^{53} - 24 q^{56} + 32 q^{58} - 40 q^{65} - 32 q^{67} - 40 q^{70} - 64 q^{71} + 24 q^{77} + 48 q^{85} - 64 q^{86} - 64 q^{88} - 48 q^{91} + 40 q^{92} + 72 q^{95} + 96 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13\nu^{15} - 168\nu^{13} + 174\nu^{11} - 84\nu^{9} - 3\nu^{7} + 732\nu^{5} - 336\nu^{3} - 5056\nu ) / 7680 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\nu^{14} - 72\nu^{12} + 6\nu^{10} - 36\nu^{8} + 513\nu^{6} - 1332\nu^{4} + 3696\nu^{2} - 4544 ) / 3840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\nu^{15} - 96\nu^{13} + 378\nu^{11} - 588\nu^{9} + 1359\nu^{7} - 2676\nu^{5} + 4368\nu^{3} - 2752\nu ) / 7680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\nu^{14} - 48\nu^{12} - 126\nu^{10} + 36\nu^{8} + 27\nu^{6} + 492\nu^{4} + 144\nu^{2} - 1216 ) / 3840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29 \nu^{15} + 16 \nu^{14} + 144 \nu^{13} - 336 \nu^{12} + 18 \nu^{11} + 288 \nu^{10} + \cdots - 17152 ) / 7680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29 \nu^{15} + 20 \nu^{14} + 144 \nu^{13} + 18 \nu^{11} + 120 \nu^{10} + 132 \nu^{9} + 240 \nu^{8} + \cdots - 1280 ) / 7680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -43\nu^{14} + 108\nu^{12} - 114\nu^{10} + 324\nu^{8} - 747\nu^{6} + 528\nu^{4} - 3024\nu^{2} + 6016 ) / 1920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29 \nu^{15} - 104 \nu^{14} + 144 \nu^{13} + 144 \nu^{12} + 18 \nu^{11} - 432 \nu^{10} + \cdots + 5888 ) / 7680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 83\nu^{15} - 48\nu^{13} + 114\nu^{11} - 444\nu^{9} + 387\nu^{7} - 2868\nu^{5} + 4944\nu^{3} - 7616\nu ) / 7680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - \nu^{13} + 2\nu^{11} - 10\nu^{9} + 13\nu^{7} - 13\nu^{5} + 56\nu^{3} - 80\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -89\nu^{15} + 204\nu^{13} - 102\nu^{11} + 732\nu^{9} - 1401\nu^{7} + 984\nu^{5} - 4752\nu^{3} + 9728\nu ) / 3840 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 83 \nu^{15} - 270 \nu^{14} - 48 \nu^{13} + 720 \nu^{12} + 114 \nu^{11} - 660 \nu^{10} - 444 \nu^{9} + \cdots + 44160 ) / 7680 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29 \nu^{15} - 440 \nu^{14} + 144 \nu^{13} + 960 \nu^{12} + 18 \nu^{11} - 720 \nu^{10} + \cdots + 66560 ) / 7680 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 122 \nu^{15} - 135 \nu^{14} - 312 \nu^{13} + 360 \nu^{12} + 156 \nu^{11} - 330 \nu^{10} + \cdots + 22080 ) / 3840 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 337 \nu^{15} - 270 \nu^{14} - 672 \nu^{13} + 720 \nu^{12} + 486 \nu^{11} - 660 \nu^{10} + \cdots + 44160 ) / 7680 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{15} + \beta_{12} - 2\beta_{11} - \beta_{10} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - 2 \beta_{8} - 2 \beta_{6} + \cdots - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{11} - \beta_{10} + 3\beta_{8} - 3\beta_{7} - \beta_{5} - \beta_{4} - 3\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} - 2 \beta_{13} + 3 \beta_{12} + 4 \beta_{11} + 3 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + \cdots + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2 \beta_{13} - 4 \beta_{11} - 4 \beta_{10} + 3 \beta_{8} + 3 \beta_{7} + 6 \beta_{6} + \beta_{5} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{15} - 6\beta_{14} - \beta_{12} - 3\beta_{11} - 6\beta_{10} - 8\beta_{9} + 5\beta_{3} - 9\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6 \beta_{13} - 9 \beta_{11} - 9 \beta_{10} + 5 \beta_{8} - 15 \beta_{7} + 10 \beta_{6} - 3 \beta_{5} + \cdots - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 13 \beta_{15} - 12 \beta_{14} - 8 \beta_{13} + 15 \beta_{12} + 2 \beta_{11} - 31 \beta_{10} + \cdots - 7 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12 \beta_{13} - 8 \beta_{11} - 8 \beta_{10} + \beta_{8} - 45 \beta_{7} + 12 \beta_{6} - 9 \beta_{5} + \cdots - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15 \beta_{15} - 14 \beta_{14} - 6 \beta_{13} + 11 \beta_{12} + 31 \beta_{11} + 12 \beta_{10} + \cdots + 31 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 12 \beta_{13} - 9 \beta_{11} - 9 \beta_{10} + 7 \beta_{8} + 13 \beta_{7} + 60 \beta_{6} - 37 \beta_{5} + \cdots - 71 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 49 \beta_{15} - 24 \beta_{14} - 6 \beta_{13} - 13 \beta_{12} + 88 \beta_{11} + 63 \beta_{10} + \cdots - 23 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 18 \beta_{13} - 12 \beta_{11} - 12 \beta_{10} - 33 \beta_{8} - 69 \beta_{7} + 90 \beta_{6} - 51 \beta_{5} + \cdots - 67 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 47 \beta_{15} - 150 \beta_{14} + 12 \beta_{13} + 79 \beta_{12} - 191 \beta_{11} - 106 \beta_{10} + \cdots - 49 \beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-\beta_{7}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
−0.944649 + 1.05244i
0.944649 − 1.05244i
−1.36166 + 0.381939i
1.36166 − 0.381939i
−1.40927 + 0.118126i
1.40927 − 0.118126i
0.517174 + 1.31626i
−0.517174 − 1.31626i
−0.944649 − 1.05244i
0.944649 + 1.05244i
−1.36166 − 0.381939i
1.36166 + 0.381939i
−1.40927 − 0.118126i
1.40927 + 0.118126i
0.517174 − 1.31626i
−0.517174 + 1.31626i
−1.48838 + 1.48838i 0 − 2.43055i −1.28999 + 1.82645i 0 −1.75993 + 1.97552i 0.640825 + 0.640825i 0 −0.798469 − 4.63845i
118.2 −1.48838 + 1.48838i 0 − 2.43055i 1.28999 − 1.82645i 0 −1.97552 + 1.75993i 0.640825 + 0.640825i 0 0.798469 + 4.63845i
118.3 −0.540143 + 0.540143i 0 1.41649i −1.03649 − 1.98133i 0 0.614060 − 2.57351i −1.84539 − 1.84539i 0 1.63006 + 0.510348i
118.4 −0.540143 + 0.540143i 0 1.41649i 1.03649 + 1.98133i 0 2.57351 − 0.614060i −1.84539 − 1.84539i 0 −1.63006 − 0.510348i
118.5 0.167056 − 0.167056i 0 1.94418i −2.23450 + 0.0836010i 0 −2.64501 + 0.0627175i 0.658899 + 0.658899i 0 −0.359321 + 0.387253i
118.6 0.167056 − 0.167056i 0 1.94418i 2.23450 − 0.0836010i 0 −0.0627175 + 2.64501i 0.658899 + 0.658899i 0 0.359321 − 0.387253i
118.7 1.86147 − 1.86147i 0 − 4.93012i −1.50619 − 1.65269i 0 1.46123 + 2.20563i −5.45433 − 5.45433i 0 −5.88016 − 0.272713i
118.8 1.86147 − 1.86147i 0 − 4.93012i 1.50619 + 1.65269i 0 −2.20563 − 1.46123i −5.45433 − 5.45433i 0 5.88016 + 0.272713i
307.1 −1.48838 − 1.48838i 0 2.43055i −1.28999 − 1.82645i 0 −1.75993 − 1.97552i 0.640825 − 0.640825i 0 −0.798469 + 4.63845i
307.2 −1.48838 − 1.48838i 0 2.43055i 1.28999 + 1.82645i 0 −1.97552 − 1.75993i 0.640825 − 0.640825i 0 0.798469 − 4.63845i
307.3 −0.540143 − 0.540143i 0 − 1.41649i −1.03649 + 1.98133i 0 0.614060 + 2.57351i −1.84539 + 1.84539i 0 1.63006 − 0.510348i
307.4 −0.540143 − 0.540143i 0 − 1.41649i 1.03649 − 1.98133i 0 2.57351 + 0.614060i −1.84539 + 1.84539i 0 −1.63006 + 0.510348i
307.5 0.167056 + 0.167056i 0 − 1.94418i −2.23450 − 0.0836010i 0 −2.64501 − 0.0627175i 0.658899 − 0.658899i 0 −0.359321 − 0.387253i
307.6 0.167056 + 0.167056i 0 − 1.94418i 2.23450 + 0.0836010i 0 −0.0627175 − 2.64501i 0.658899 − 0.658899i 0 0.359321 + 0.387253i
307.7 1.86147 + 1.86147i 0 4.93012i −1.50619 + 1.65269i 0 1.46123 − 2.20563i −5.45433 + 5.45433i 0 −5.88016 + 0.272713i
307.8 1.86147 + 1.86147i 0 4.93012i 1.50619 − 1.65269i 0 −2.20563 + 1.46123i −5.45433 + 5.45433i 0 5.88016 − 0.272713i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.p.e 16
3.b odd 2 1 105.2.m.a ✓ 16
5.c odd 4 1 inner 315.2.p.e 16
7.b odd 2 1 inner 315.2.p.e 16
12.b even 2 1 1680.2.cz.d 16
15.d odd 2 1 525.2.m.b 16
15.e even 4 1 105.2.m.a ✓ 16
15.e even 4 1 525.2.m.b 16
21.c even 2 1 105.2.m.a ✓ 16
21.g even 6 2 735.2.v.a 32
21.h odd 6 2 735.2.v.a 32
35.f even 4 1 inner 315.2.p.e 16
60.l odd 4 1 1680.2.cz.d 16
84.h odd 2 1 1680.2.cz.d 16
105.g even 2 1 525.2.m.b 16
105.k odd 4 1 105.2.m.a ✓ 16
105.k odd 4 1 525.2.m.b 16
105.w odd 12 2 735.2.v.a 32
105.x even 12 2 735.2.v.a 32
420.w even 4 1 1680.2.cz.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.m.a ✓ 16 3.b odd 2 1
105.2.m.a ✓ 16 15.e even 4 1
105.2.m.a ✓ 16 21.c even 2 1
105.2.m.a ✓ 16 105.k odd 4 1
315.2.p.e 16 1.a even 1 1 trivial
315.2.p.e 16 5.c odd 4 1 inner
315.2.p.e 16 7.b odd 2 1 inner
315.2.p.e 16 35.f even 4 1 inner
525.2.m.b 16 15.d odd 2 1
525.2.m.b 16 15.e even 4 1
525.2.m.b 16 105.g even 2 1
525.2.m.b 16 105.k odd 4 1
735.2.v.a 32 21.g even 6 2
735.2.v.a 32 21.h odd 6 2
735.2.v.a 32 105.w odd 12 2
735.2.v.a 32 105.x even 12 2
1680.2.cz.d 16 12.b even 2 1
1680.2.cz.d 16 60.l odd 4 1
1680.2.cz.d 16 84.h odd 2 1
1680.2.cz.d 16 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{8} + 4T_{2}^{5} + 34T_{2}^{4} + 24T_{2}^{3} + 8T_{2}^{2} - 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{16} + 2160T_{17}^{12} + 268896T_{17}^{8} + 9541376T_{17}^{4} + 100000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 4 T^{5} + 34 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 28 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 8 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} - 12 T^{2} + \cdots - 60)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 736 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 100000000 \) Copy content Toggle raw display
$19$ \( (T^{8} - 104 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 20 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 48 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 120 T^{6} + \cdots + 274576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 16 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 184 T^{6} + \cdots + 129600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 8 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 25216 T^{12} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( (T^{8} + 12 T^{7} + \cdots + 129600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 160 T^{6} + \cdots + 1183744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 288 T^{6} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 16 T^{7} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 16 T^{3} + \cdots - 1132)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 169459576016896 \) Copy content Toggle raw display
$79$ \( (T^{8} + 320 T^{6} + \cdots + 18939904)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45137758519296 \) Copy content Toggle raw display
$89$ \( (T^{8} - 136 T^{6} + \cdots + 107584)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
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