↑

Properties

Label 315.2.bf.c
Level $315$
Weight $2$
Character orbit 315.bf
Analytic conductor $2.515$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(109,315)]
 
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([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.bf (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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8x^{14} + 26x^{12} - 96x^{10} - 781x^{8} - 2400x^{6} + 16250x^{4} + 125000x^{2} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + ( - \beta_{8} + \beta_{5} + 1) q^{4} + ( - \beta_{14} + \beta_{9} + \beta_1) q^{5} - \beta_{4} q^{7} + (\beta_{14} + \beta_{12} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + ( - \beta_{8} + \beta_{5} + 1) q^{4} + ( - \beta_{14} + \beta_{9} + \beta_1) q^{5} - \beta_{4} q^{7} + (\beta_{14} + \beta_{12} + \cdots - \beta_1) q^{8}+ \cdots + ( - \beta_{14} - 3 \beta_{12} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 8 q^{10} - 16 q^{16} + 16 q^{19} - 16 q^{25} - 40 q^{31} - 80 q^{34} + 8 q^{40} + 8 q^{46} - 24 q^{61} + 96 q^{64} - 56 q^{70} + 144 q^{76} - 16 q^{79} + 48 q^{85} + 112 q^{91} + 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8x^{14} + 26x^{12} - 96x^{10} - 781x^{8} - 2400x^{6} + 16250x^{4} + 125000x^{2} + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1547 \nu^{14} - 7449 \nu^{12} - 12128 \nu^{10} - 110837 \nu^{8} + 1254368 \nu^{6} + \cdots - 125000000 ) / 29812500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 8\nu^{13} + 26\nu^{11} - 96\nu^{9} - 781\nu^{7} - 2400\nu^{5} + 16250\nu^{3} + 125000\nu ) / 78125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2269 \nu^{14} - 10152 \nu^{12} + 11881 \nu^{10} + 183949 \nu^{8} + 1464089 \nu^{6} + \cdots - 197046875 ) / 29812500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3848 \nu^{14} - 43134 \nu^{12} - 8848 \nu^{10} + 365183 \nu^{8} + 4255888 \nu^{6} + \cdots - 679046875 ) / 22359375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 34369 \nu^{14} - 129048 \nu^{12} + 47219 \nu^{10} - 3327799 \nu^{8} + 12561811 \nu^{6} + \cdots - 2057359375 ) / 89437500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26311 \nu^{14} - 69888 \nu^{12} - 2411 \nu^{10} + 2573956 \nu^{8} + 6311291 \nu^{6} + \cdots - 1034187500 ) / 44718750 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28067 \nu^{14} + 78936 \nu^{12} + 16192 \nu^{10} - 2932532 \nu^{8} - 7788352 \nu^{6} + \cdots + 1201750000 ) / 44718750 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11438 \nu^{15} - 121929 \nu^{13} - 3913 \nu^{11} + 1158248 \nu^{9} + 11684503 \nu^{7} + \cdots - 1885759375 \nu ) / 89437500 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2267 \nu^{15} - 1911 \nu^{13} - 392 \nu^{11} + 210107 \nu^{9} + 188552 \nu^{7} + \cdots - 29093750 \nu ) / 16562500 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14233 \nu^{15} - 15576 \nu^{13} + 413 \nu^{11} - 1507933 \nu^{9} + 2048017 \nu^{7} + \cdots - 274553125 \nu ) / 89437500 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3848 \nu^{15} - 43134 \nu^{13} - 8848 \nu^{11} + 365183 \nu^{9} + 4255888 \nu^{7} + \cdots - 656687500 \nu ) / 22359375 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 98366 \nu^{14} - 511053 \nu^{12} - 31141 \nu^{10} + 9670886 \nu^{8} + 50252971 \nu^{6} + \cdots - 7873140625 ) / 89437500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 36974 \nu^{15} + 116817 \nu^{13} + 12649 \nu^{11} - 3744104 \nu^{9} - 12118219 \nu^{7} + \cdots + 1872296875 \nu ) / 149062500 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 122416 \nu^{15} - 333453 \nu^{13} - 86441 \nu^{11} + 11729686 \nu^{9} + 30344771 \nu^{7} + \cdots - 4889515625 \nu ) / 447187500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{8} - 2\beta_{5} - \beta_{4} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} - \beta_{14} - 2\beta_{12} + 2\beta_{11} - 2\beta_{10} + \beta_{9} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{13} - 8\beta_{7} - 2\beta_{6} - 3\beta_{5} - 2\beta_{4} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{15} - 17\beta_{12} + 6\beta_{11} + 26\beta_{10} + 26\beta_{9} - 22\beta_{3} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{13} - \beta_{8} + \beta_{7} - 3\beta_{6} + 32\beta_{4} + 29\beta_{2} + 50 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -28\beta_{15} - 89\beta_{14} + 56\beta_{11} - 43\beta_{9} + 77\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 44\beta_{13} - 96\beta_{8} + 104\beta_{6} - 289\beta_{5} - 148\beta_{4} + 148\beta_{2} - 289 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 52 \beta_{15} - 324 \beta_{14} - 325 \beta_{12} - 52 \beta_{11} - 468 \beta_{10} + 324 \beta_{9} + \cdots + 324 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 416\beta_{13} - 169\beta_{7} + 429\beta_{6} - 1438\beta_{5} + 429\beta_{4} - 416\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1196\beta_{15} - 1594\beta_{12} + 598\beta_{11} + 2678\beta_{10} + 2678\beta_{9} + 949\beta_{3} + 1594\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -486\beta_{13} - 200\beta_{8} + 200\beta_{7} - 486\beta_{6} + 3276\beta_{4} + 2790\beta_{2} - 10611 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -2590\beta_{15} - 8770\beta_{14} + 5180\beta_{11} - 5020\beta_{9} - 8107\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -11697\beta_{13} - 24647\beta_{8} + 11200\beta_{6} + 3694\beta_{5} + 497\beta_{4} - 497\beta_{2} + 3694 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 36344 \beta_{15} - 14203 \beta_{14} + 1944 \beta_{12} - 36344 \beta_{11} - 19656 \beta_{10} + \cdots + 14203 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.733576 − 2.11231i
1.46253 + 1.69145i
0.717291 − 2.11790i
2.19280 + 0.437757i
−2.19280 − 0.437757i
−0.717291 + 2.11790i
−1.46253 − 1.69145i
0.733576 + 2.11231i
−0.733576 + 2.11231i
1.46253 − 1.69145i
0.717291 + 2.11790i
2.19280 − 0.437757i
−2.19280 + 0.437757i
−0.717291 − 2.11790i
−1.46253 + 1.69145i
0.733576 − 2.11231i
−2.05774 + 1.18804i 0 1.82288 − 3.15731i −1.46253 − 1.69145i 0 −2.03996 − 1.68480i 3.91044i 0 5.01902 + 1.74303i
109.2 −2.05774 + 1.18804i 0 1.82288 − 3.15731i 0.733576 + 2.11231i 0 2.03996 + 1.68480i 3.91044i 0 −4.01902 − 3.47508i
109.3 −0.515448 + 0.297594i 0 −0.822876 + 1.42526i −2.19280 − 0.437757i 0 −1.68480 + 2.03996i − 2.16991i 0 1.26055 − 0.426923i
109.4 −0.515448 + 0.297594i 0 −0.822876 + 1.42526i −0.717291 + 2.11790i 0 1.68480 − 2.03996i − 2.16991i 0 −0.260548 − 1.30513i
109.5 0.515448 − 0.297594i 0 −0.822876 + 1.42526i 0.717291 − 2.11790i 0 1.68480 − 2.03996i 2.16991i 0 −0.260548 − 1.30513i
109.6 0.515448 − 0.297594i 0 −0.822876 + 1.42526i 2.19280 + 0.437757i 0 −1.68480 + 2.03996i 2.16991i 0 1.26055 − 0.426923i
109.7 2.05774 − 1.18804i 0 1.82288 − 3.15731i −0.733576 − 2.11231i 0 2.03996 + 1.68480i − 3.91044i 0 −4.01902 − 3.47508i
109.8 2.05774 − 1.18804i 0 1.82288 − 3.15731i 1.46253 + 1.69145i 0 −2.03996 − 1.68480i − 3.91044i 0 5.01902 + 1.74303i
289.1 −2.05774 − 1.18804i 0 1.82288 + 3.15731i −1.46253 + 1.69145i 0 −2.03996 + 1.68480i − 3.91044i 0 5.01902 − 1.74303i
289.2 −2.05774 − 1.18804i 0 1.82288 + 3.15731i 0.733576 − 2.11231i 0 2.03996 − 1.68480i − 3.91044i 0 −4.01902 + 3.47508i
289.3 −0.515448 − 0.297594i 0 −0.822876 − 1.42526i −2.19280 + 0.437757i 0 −1.68480 − 2.03996i 2.16991i 0 1.26055 + 0.426923i
289.4 −0.515448 − 0.297594i 0 −0.822876 − 1.42526i −0.717291 − 2.11790i 0 1.68480 + 2.03996i 2.16991i 0 −0.260548 + 1.30513i
289.5 0.515448 + 0.297594i 0 −0.822876 − 1.42526i 0.717291 + 2.11790i 0 1.68480 + 2.03996i − 2.16991i 0 −0.260548 + 1.30513i
289.6 0.515448 + 0.297594i 0 −0.822876 − 1.42526i 2.19280 − 0.437757i 0 −1.68480 − 2.03996i − 2.16991i 0 1.26055 + 0.426923i
289.7 2.05774 + 1.18804i 0 1.82288 + 3.15731i −0.733576 + 2.11231i 0 2.03996 − 1.68480i 3.91044i 0 −4.01902 + 3.47508i
289.8 2.05774 + 1.18804i 0 1.82288 + 3.15731i 1.46253 − 1.69145i 0 −2.03996 + 1.68480i 3.91044i 0 5.01902 − 1.74303i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bf.c ✓ 16
3.b odd 2 1 inner 315.2.bf.c ✓ 16
5.b even 2 1 inner 315.2.bf.c ✓ 16
7.c even 3 1 inner 315.2.bf.c ✓ 16
7.c even 3 1 2205.2.d.p 8
7.d odd 6 1 2205.2.d.r 8
15.d odd 2 1 inner 315.2.bf.c ✓ 16
21.g even 6 1 2205.2.d.r 8
21.h odd 6 1 inner 315.2.bf.c ✓ 16
21.h odd 6 1 2205.2.d.p 8
35.i odd 6 1 2205.2.d.r 8
35.j even 6 1 inner 315.2.bf.c ✓ 16
35.j even 6 1 2205.2.d.p 8
105.o odd 6 1 inner 315.2.bf.c ✓ 16
105.o odd 6 1 2205.2.d.p 8
105.p even 6 1 2205.2.d.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bf.c ✓ 16 1.a even 1 1 trivial
315.2.bf.c ✓ 16 3.b odd 2 1 inner
315.2.bf.c ✓ 16 5.b even 2 1 inner
315.2.bf.c ✓ 16 7.c even 3 1 inner
315.2.bf.c ✓ 16 15.d odd 2 1 inner
315.2.bf.c ✓ 16 21.h odd 6 1 inner
315.2.bf.c ✓ 16 35.j even 6 1 inner
315.2.bf.c ✓ 16 105.o odd 6 1 inner
2205.2.d.p 8 7.c even 3 1
2205.2.d.p 8 21.h odd 6 1
2205.2.d.p 8 35.j even 6 1
2205.2.d.p 8 105.o odd 6 1
2205.2.d.r 8 7.d odd 6 1
2205.2.d.r 8 21.g even 6 1
2205.2.d.r 8 35.i odd 6 1
2205.2.d.r 8 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{6} + 34T_{2}^{4} - 12T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 6 T^{6} + 34 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 8 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} + 91 T^{4} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 42 T^{6} + \cdots + 142884)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 28 T^{2} + 189)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} - 54 T^{6} + \cdots + 521284)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + 19 T^{2} + \cdots + 9)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 38 T^{6} + \cdots + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 70 T^{2} + 378)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T + 25)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} - 28 T^{6} + \cdots + 441)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 154 T^{2} + 3402)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 196 T^{2} + 9261)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 20 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 104 T^{6} + \cdots + 6718464)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 70 T^{6} + \cdots + 1764)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + 34 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 196 T^{6} + \cdots + 1058841)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 70 T^{2} + 378)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 84 T^{6} + \cdots + 35721)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots + 3481)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 150 T^{2} + 1922)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 42 T^{6} + \cdots + 142884)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 196 T^{2} + 6804)^{4} \) Copy content Toggle raw display
show more
show less