Properties

Label 315.2.bj.b
Level $315$
Weight $2$
Character orbit 315.bj
Analytic conductor $2.515$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(26,315)]
 
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([3, 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("315.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.bj (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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 312x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{4} + 1) q^{4} + \beta_{4} q^{5} + (\beta_{6} - \beta_1) q^{7} + ( - \beta_{9} - \beta_{7} - \beta_{5} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{4} + 1) q^{4} + \beta_{4} q^{5} + (\beta_{6} - \beta_1) q^{7} + ( - \beta_{9} - \beta_{7} - \beta_{5} + 2 \beta_1) q^{8} - \beta_{3} q^{10} + ( - \beta_{11} - \beta_{8} - \beta_{6} - \beta_{4} - 1) q^{11} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 1) q^{13} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{14}+ \cdots + (\beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} + 6 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} + 6 q^{5} - 2 q^{7} - 12 q^{11} + 12 q^{14} - 16 q^{16} + 6 q^{19} + 16 q^{20} + 32 q^{22} - 12 q^{23} - 6 q^{25} - 20 q^{28} + 6 q^{31} - 60 q^{32} + 2 q^{35} - 10 q^{37} + 36 q^{38} - 24 q^{41} - 4 q^{43} - 12 q^{44} - 4 q^{46} + 6 q^{49} - 12 q^{53} + 60 q^{56} + 20 q^{58} + 24 q^{59} - 24 q^{62} - 56 q^{64} - 18 q^{65} + 6 q^{67} + 60 q^{68} - 12 q^{70} - 42 q^{73} - 84 q^{74} + 36 q^{77} + 18 q^{79} + 16 q^{80} - 72 q^{82} + 24 q^{83} + 84 q^{86} + 4 q^{88} - 12 q^{89} - 18 q^{91} + 12 q^{94} + 6 q^{95} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 312x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 22\nu^{2} + 4\nu + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 20\nu^{9} + 144\nu^{7} + 446\nu^{5} + 544\nu^{3} + 180\nu + 24 ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} - 20\nu^{9} - 142\nu^{7} - 426\nu^{5} - 500\nu^{3} + 8\nu^{2} - 168\nu + 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{11} - 94 \nu^{9} - 606 \nu^{7} - 6 \nu^{6} - 1510 \nu^{5} - 84 \nu^{4} - 1064 \nu^{3} - 300 \nu^{2} + 72 \nu - 156 ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{11} - 94 \nu^{9} - 606 \nu^{7} + 6 \nu^{6} - 1510 \nu^{5} + 84 \nu^{4} - 1064 \nu^{3} + 300 \nu^{2} + 72 \nu + 156 ) / 48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{11} + 6 \nu^{10} - 94 \nu^{9} + 114 \nu^{8} - 612 \nu^{7} + 756 \nu^{6} - 1594 \nu^{5} + 2040 \nu^{4} - 1412 \nu^{3} + 1932 \nu^{2} - 324 \nu + 384 ) / 48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4 \nu^{11} + 77 \nu^{9} + 516 \nu^{7} - 3 \nu^{6} + 1394 \nu^{5} - 42 \nu^{4} + 1306 \nu^{3} - 162 \nu^{2} + 360 \nu - 114 ) / 24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5 \nu^{11} - 6 \nu^{10} - 94 \nu^{9} - 108 \nu^{8} - 612 \nu^{7} - 660 \nu^{6} - 1594 \nu^{5} - 1548 \nu^{4} - 1388 \nu^{3} - 1056 \nu^{2} - 180 \nu - 24 ) / 48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5 \nu^{11} + 6 \nu^{10} + 94 \nu^{9} + 114 \nu^{8} + 612 \nu^{7} + 756 \nu^{6} + 1594 \nu^{5} + 2040 \nu^{4} + 1412 \nu^{3} + 1932 \nu^{2} + 324 \nu + 384 ) / 48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{5} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} - 2\beta_{3} - 7\beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 9\beta_{9} - \beta_{8} - 8\beta_{7} + \beta_{6} - 11\beta_{5} - 6\beta_{4} + \beta_{2} + 39\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{7} + 10\beta_{6} + 28\beta_{3} + 48\beta_{2} - 14\beta _1 - 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10 \beta_{11} + 68 \beta_{9} + 10 \beta_{8} + 58 \beta_{7} - 10 \beta_{6} + 96 \beta_{5} + 84 \beta_{4} - 14 \beta_{2} - 264 \beta _1 - 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} + 74 \beta_{7} - 78 \beta_{6} - 4 \beta_{5} - 284 \beta_{3} - 340 \beta_{2} + 142 \beta _1 + 666 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 70 \beta_{11} - 488 \beta_{9} - 70 \beta_{8} - 414 \beta_{7} + 74 \beta_{6} - 780 \beta_{5} - 836 \beta_{4} + 146 \beta_{2} + 1830 \beta _1 + 418 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 148 \beta_{11} - 152 \beta_{10} + 76 \beta_{9} + 4 \beta_{8} - 486 \beta_{7} + 562 \beta_{6} + 76 \beta_{5} + 2548 \beta_{3} + 2470 \beta_{2} - 1274 \beta _1 - 4592 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 406 \beta_{11} + 3438 \beta_{9} + 406 \beta_{8} + 2952 \beta_{7} - 486 \beta_{6} + 6138 \beta_{5} + 7348 \beta_{4} - 1350 \beta_{2} - 12894 \beta _1 - 3674 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
2.46680i
1.99567i
0.398211i
0.979668i
1.13965i
2.74137i
2.46680i
1.99567i
0.398211i
0.979668i
1.13965i
2.74137i
−2.13631 1.23340i 0 2.04255 + 3.53780i 0.500000 0.866025i 0 −1.85267 + 1.88881i 5.14351i 0 −2.13631 + 1.23340i
26.2 −1.72830 0.997835i 0 0.991350 + 1.71707i 0.500000 0.866025i 0 1.78020 + 1.95727i 0.0345244i 0 −1.72830 + 0.997835i
26.3 −0.344861 0.199105i 0 −0.920714 1.59472i 0.500000 0.866025i 0 −2.30243 1.30338i 1.52970i 0 −0.344861 + 0.199105i
26.4 0.848417 + 0.489834i 0 −0.520126 0.900884i 0.500000 0.866025i 0 1.24698 + 2.33346i 2.97844i 0 0.848417 0.489834i
26.5 0.986962 + 0.569823i 0 −0.350603 0.607263i 0.500000 0.866025i 0 2.18931 1.48558i 3.07842i 0 0.986962 0.569823i
26.6 2.37409 + 1.37068i 0 2.75754 + 4.77621i 0.500000 0.866025i 0 −2.06138 1.65853i 9.63615i 0 2.37409 1.37068i
206.1 −2.13631 + 1.23340i 0 2.04255 3.53780i 0.500000 + 0.866025i 0 −1.85267 1.88881i 5.14351i 0 −2.13631 1.23340i
206.2 −1.72830 + 0.997835i 0 0.991350 1.71707i 0.500000 + 0.866025i 0 1.78020 1.95727i 0.0345244i 0 −1.72830 0.997835i
206.3 −0.344861 + 0.199105i 0 −0.920714 + 1.59472i 0.500000 + 0.866025i 0 −2.30243 + 1.30338i 1.52970i 0 −0.344861 0.199105i
206.4 0.848417 0.489834i 0 −0.520126 + 0.900884i 0.500000 + 0.866025i 0 1.24698 2.33346i 2.97844i 0 0.848417 + 0.489834i
206.5 0.986962 0.569823i 0 −0.350603 + 0.607263i 0.500000 + 0.866025i 0 2.18931 + 1.48558i 3.07842i 0 0.986962 + 0.569823i
206.6 2.37409 1.37068i 0 2.75754 4.77621i 0.500000 + 0.866025i 0 −2.06138 + 1.65853i 9.63615i 0 2.37409 + 1.37068i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g 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.bj.b yes 12
3.b odd 2 1 315.2.bj.a 12
5.b even 2 1 1575.2.bk.e 12
5.c odd 4 2 1575.2.bc.c 24
7.c even 3 1 2205.2.b.a 12
7.d odd 6 1 315.2.bj.a 12
7.d odd 6 1 2205.2.b.b 12
15.d odd 2 1 1575.2.bk.f 12
15.e even 4 2 1575.2.bc.d 24
21.g even 6 1 inner 315.2.bj.b yes 12
21.g even 6 1 2205.2.b.a 12
21.h odd 6 1 2205.2.b.b 12
35.i odd 6 1 1575.2.bk.f 12
35.k even 12 2 1575.2.bc.d 24
105.p even 6 1 1575.2.bk.e 12
105.w odd 12 2 1575.2.bc.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bj.a 12 3.b odd 2 1
315.2.bj.a 12 7.d odd 6 1
315.2.bj.b yes 12 1.a even 1 1 trivial
315.2.bj.b yes 12 21.g even 6 1 inner
1575.2.bc.c 24 5.c odd 4 2
1575.2.bc.c 24 105.w odd 12 2
1575.2.bc.d 24 15.e even 4 2
1575.2.bc.d 24 35.k even 12 2
1575.2.bk.e 12 5.b even 2 1
1575.2.bk.e 12 105.p even 6 1
1575.2.bk.f 12 15.d odd 2 1
1575.2.bk.f 12 35.i odd 6 1
2205.2.b.a 12 7.c even 3 1
2205.2.b.a 12 21.g even 6 1
2205.2.b.b 12 7.d odd 6 1
2205.2.b.b 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 10T_{2}^{10} + 78T_{2}^{8} + 12T_{2}^{7} - 208T_{2}^{6} + 424T_{2}^{4} - 264T_{2}^{3} - 84T_{2}^{2} + 72T_{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 10 T^{10} + 78 T^{8} + 12 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} - T^{10} + 6 T^{9} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} + 38 T^{10} + \cdots + 19044 \) Copy content Toggle raw display
$13$ \( T^{12} + 102 T^{10} + 3831 T^{8} + \cdots + 2025 \) Copy content Toggle raw display
$17$ \( T^{12} + 60 T^{10} - 120 T^{9} + \cdots + 2125764 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} - 39 T^{10} + \cdots + 227529 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 113422500 \) Copy content Toggle raw display
$29$ \( T^{12} + 164 T^{10} + 8988 T^{8} + \cdots + 4088484 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} - 111 T^{10} + \cdots + 1750329 \) Copy content Toggle raw display
$37$ \( T^{12} + 10 T^{11} + \cdots + 12130158769 \) Copy content Toggle raw display
$41$ \( (T^{6} + 12 T^{5} - 54 T^{4} - 756 T^{3} + \cdots - 36450)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} - 193 T^{4} + \cdots - 114167)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 120 T^{10} + \cdots + 466560000 \) Copy content Toggle raw display
$53$ \( T^{12} + 12 T^{11} - 28 T^{10} + \cdots + 43877376 \) Copy content Toggle raw display
$59$ \( T^{12} - 24 T^{11} + 438 T^{10} + \cdots + 72900 \) Copy content Toggle raw display
$61$ \( T^{12} - 228 T^{10} + \cdots + 593994384 \) Copy content Toggle raw display
$67$ \( T^{12} - 6 T^{11} + 183 T^{10} + \cdots + 235898881 \) Copy content Toggle raw display
$71$ \( T^{12} + 392 T^{10} + \cdots + 491508900 \) Copy content Toggle raw display
$73$ \( T^{12} + 42 T^{11} + 681 T^{10} + \cdots + 1565001 \) Copy content Toggle raw display
$79$ \( T^{12} - 18 T^{11} + \cdots + 1973580625 \) Copy content Toggle raw display
$83$ \( (T^{6} - 12 T^{5} - 198 T^{4} + \cdots - 287874)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 12 T^{11} + \cdots + 872493444 \) Copy content Toggle raw display
$97$ \( T^{12} + 732 T^{10} + \cdots + 71571600 \) Copy content Toggle raw display
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