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Properties

Label 315.2.i.e
Level $315$
Weight $2$
Character orbit 315.i
Analytic conductor $2.515$
Analytic rank $0$
Dimension $12$
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(106,315)]
 
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([4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.i (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 2 x^{10} - x^{9} - 4 x^{8} + 20 x^{7} - 38 x^{6} + 60 x^{5} - 36 x^{4} - 27 x^{3} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{2} + 1) q^{2} + \beta_{9} q^{3} + (\beta_{9} + \beta_{7} - 2 \beta_{4} - \beta_1) q^{4} + \beta_{4} q^{5} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1) q^{6}+ \cdots + (\beta_{11} + \beta_{6} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{2} + 1) q^{2} + \beta_{9} q^{3} + (\beta_{9} + \beta_{7} - 2 \beta_{4} - \beta_1) q^{4} + \beta_{4} q^{5} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1) q^{6}+ \cdots + ( - 2 \beta_{11} + 4 \beta_{10} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + q^{3} - 11 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + q^{3} - 11 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} - 3 q^{9} + 6 q^{10} + 7 q^{11} - 37 q^{12} - 10 q^{13} + 3 q^{14} - q^{15} - 13 q^{16} + 14 q^{17} + 24 q^{18} + 30 q^{19} + 11 q^{20} - 2 q^{21} + 7 q^{22} + 14 q^{23} + 6 q^{24} - 6 q^{25} - 26 q^{26} - 2 q^{27} + 22 q^{28} - 13 q^{29} - 9 q^{30} - 10 q^{31} - 18 q^{32} - 14 q^{33} - 15 q^{34} - 12 q^{35} - 4 q^{36} + 42 q^{37} - 4 q^{38} - 11 q^{39} + 3 q^{40} - 4 q^{41} - 3 q^{42} - 13 q^{43} - 78 q^{44} - 30 q^{46} + 8 q^{47} + 42 q^{48} - 6 q^{49} + 3 q^{50} - 19 q^{51} - 31 q^{52} + 20 q^{53} + 84 q^{54} + 14 q^{55} - 3 q^{56} - 22 q^{57} - 3 q^{58} + 21 q^{59} - 17 q^{60} - 2 q^{61} + 50 q^{62} + 3 q^{63} + 62 q^{64} + 10 q^{65} + 19 q^{66} - 6 q^{67} - 33 q^{68} - 19 q^{69} - 3 q^{70} - 58 q^{71} + 51 q^{72} + 16 q^{73} - 5 q^{74} - 2 q^{75} - 31 q^{76} + 7 q^{77} - 5 q^{78} + 22 q^{79} - 26 q^{80} + 21 q^{81} + 36 q^{82} + 5 q^{83} + 20 q^{84} + 7 q^{85} + 23 q^{86} - 19 q^{87} + 19 q^{88} + 2 q^{89} + 9 q^{90} + 20 q^{91} + 9 q^{92} + 31 q^{93} - 31 q^{94} + 15 q^{95} - 66 q^{96} - 32 q^{97} - 6 q^{98} + 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 2 x^{10} - x^{9} - 4 x^{8} + 20 x^{7} - 38 x^{6} + 60 x^{5} - 36 x^{4} - 27 x^{3} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25 \nu^{11} + 10 \nu^{10} - 152 \nu^{9} + 301 \nu^{8} - 389 \nu^{7} + 865 \nu^{6} + 281 \nu^{5} + \cdots - 7533 ) / 17496 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13 \nu^{11} - 25 \nu^{10} - 61 \nu^{9} - 46 \nu^{8} - 49 \nu^{7} + 110 \nu^{6} - 311 \nu^{5} + \cdots - 4860 ) / 8748 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{11} + 17 \nu^{10} - 49 \nu^{9} + 110 \nu^{8} - 181 \nu^{7} + 68 \nu^{6} + 79 \nu^{5} + \cdots - 1458 ) / 2916 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{11} + 11 \nu^{10} - 61 \nu^{9} + 116 \nu^{8} - 211 \nu^{7} + 272 \nu^{6} - 293 \nu^{5} + \cdots - 6318 ) / 2916 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8 \nu^{11} - 77 \nu^{10} + 157 \nu^{9} - 191 \nu^{8} + 316 \nu^{7} + 13 \nu^{6} - 946 \nu^{5} + \cdots - 243 ) / 4374 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2 \nu^{11} + 7 \nu^{10} - 21 \nu^{9} + 51 \nu^{8} - 102 \nu^{7} + 141 \nu^{6} + 8 \nu^{5} + \cdots - 2511 ) / 972 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14 \nu^{11} + 29 \nu^{10} - 61 \nu^{9} + 197 \nu^{8} - 292 \nu^{7} + 353 \nu^{6} - 284 \nu^{5} + \cdots - 7047 ) / 4374 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 2 \nu^{9} + \nu^{8} + 4 \nu^{7} - 20 \nu^{6} + 38 \nu^{5} - 60 \nu^{4} + \cdots + 243 ) / 243 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4 \nu^{11} - 13 \nu^{10} + 35 \nu^{9} - 67 \nu^{8} + 56 \nu^{7} - 37 \nu^{6} - 134 \nu^{5} + \cdots + 1215 ) / 972 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 65 \nu^{11} + 107 \nu^{10} - 217 \nu^{9} + 248 \nu^{8} - 331 \nu^{7} - 424 \nu^{6} + 1411 \nu^{5} + \cdots - 12150 ) / 8748 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 2\beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} - 2\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} - 2\beta_{10} - 2\beta_{9} - 3\beta_{8} + 2\beta_{7} + 2\beta_{6} - 4\beta_{2} - \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + 4\beta_{9} - 3\beta_{8} + 5\beta_{7} - 4\beta_{5} + \beta_{4} + 2\beta_{2} + \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + 7 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{11} - 18 \beta_{10} - 6 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 12 \beta_{5} - 27 \beta_{4} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 33 \beta_{10} - 21 \beta_{9} - 11 \beta_{8} - 6 \beta_{7} + 24 \beta_{6} - 19 \beta_{5} + \cdots - 30 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 28 \beta_{11} + 15 \beta_{10} + 32 \beta_{9} - 6 \beta_{8} + 25 \beta_{7} + 3 \beta_{6} + 10 \beta_{5} + \cdots - 63 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 56 \beta_{11} - 17 \beta_{10} - 113 \beta_{9} + 12 \beta_{8} - 22 \beta_{7} - 55 \beta_{6} + 36 \beta_{5} + \cdots + 96 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 92 \beta_{11} - 12 \beta_{10} - 137 \beta_{9} - 18 \beta_{8} - 124 \beta_{7} - 105 \beta_{6} + \cdots - 339 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
1.24736 + 1.20170i
0.478182 − 1.66473i
−0.764584 − 1.55416i
−1.73190 − 0.0231100i
1.65373 − 0.514941i
−0.382799 + 1.68922i
1.24736 − 1.20170i
0.478182 + 1.66473i
−0.764584 + 1.55416i
−1.73190 + 0.0231100i
1.65373 + 0.514941i
−0.382799 − 1.68922i
−1.35359 + 2.34448i 1.24736 − 1.20170i −2.66438 − 4.61485i 0.500000 + 0.866025i 1.12895 + 4.55102i −0.500000 + 0.866025i 9.01155 0.111834 − 2.99791i −2.70717
106.2 −0.631422 + 1.09366i 0.478182 + 1.66473i 0.202612 + 0.350934i 0.500000 + 0.866025i −2.12258 − 0.528185i −0.500000 + 0.866025i −3.03742 −2.54268 + 1.59209i −1.26284
106.3 0.368623 − 0.638475i −0.764584 + 1.55416i 0.728233 + 1.26134i 0.500000 + 0.866025i 0.710448 + 1.06107i −0.500000 + 0.866025i 2.54827 −1.83082 − 2.37657i 0.737247
106.4 0.746337 − 1.29269i −1.73190 + 0.0231100i −0.114038 − 0.197519i 0.500000 + 0.866025i −1.26270 + 2.25606i −0.500000 + 0.866025i 2.64491 2.99893 − 0.0800482i 1.49267
106.5 1.09108 − 1.88981i 1.65373 + 0.514941i −1.38091 − 2.39181i 0.500000 + 0.866025i 2.77750 − 2.56340i −0.500000 + 0.866025i −1.66244 2.46967 + 1.70315i 2.18216
106.6 1.27897 − 2.21523i −0.382799 − 1.68922i −2.27151 − 3.93437i 0.500000 + 0.866025i −4.23161 − 1.31247i −0.500000 + 0.866025i −6.50486 −2.70693 + 1.29326i 2.55793
211.1 −1.35359 − 2.34448i 1.24736 + 1.20170i −2.66438 + 4.61485i 0.500000 − 0.866025i 1.12895 − 4.55102i −0.500000 − 0.866025i 9.01155 0.111834 + 2.99791i −2.70717
211.2 −0.631422 − 1.09366i 0.478182 − 1.66473i 0.202612 − 0.350934i 0.500000 − 0.866025i −2.12258 + 0.528185i −0.500000 − 0.866025i −3.03742 −2.54268 − 1.59209i −1.26284
211.3 0.368623 + 0.638475i −0.764584 − 1.55416i 0.728233 − 1.26134i 0.500000 − 0.866025i 0.710448 − 1.06107i −0.500000 − 0.866025i 2.54827 −1.83082 + 2.37657i 0.737247
211.4 0.746337 + 1.29269i −1.73190 − 0.0231100i −0.114038 + 0.197519i 0.500000 − 0.866025i −1.26270 − 2.25606i −0.500000 − 0.866025i 2.64491 2.99893 + 0.0800482i 1.49267
211.5 1.09108 + 1.88981i 1.65373 − 0.514941i −1.38091 + 2.39181i 0.500000 − 0.866025i 2.77750 + 2.56340i −0.500000 − 0.866025i −1.66244 2.46967 − 1.70315i 2.18216
211.6 1.27897 + 2.21523i −0.382799 + 1.68922i −2.27151 + 3.93437i 0.500000 − 0.866025i −4.23161 + 1.31247i −0.500000 − 0.866025i −6.50486 −2.70693 − 1.29326i 2.55793
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.i.e ✓ 12
3.b odd 2 1 945.2.i.e 12
9.c even 3 1 inner 315.2.i.e ✓ 12
9.c even 3 1 2835.2.a.u 6
9.d odd 6 1 945.2.i.e 12
9.d odd 6 1 2835.2.a.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.e ✓ 12 1.a even 1 1 trivial
315.2.i.e ✓ 12 9.c even 3 1 inner
945.2.i.e 12 3.b odd 2 1
945.2.i.e 12 9.d odd 6 1
2835.2.a.u 6 9.c even 3 1
2835.2.a.v 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 3 T_{2}^{11} + 16 T_{2}^{10} - 33 T_{2}^{9} + 135 T_{2}^{8} - 255 T_{2}^{7} + 628 T_{2}^{6} + \cdots + 441 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} + \cdots + 441 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} - 7 T^{11} + \cdots + 99225 \) Copy content Toggle raw display
$13$ \( T^{12} + 10 T^{11} + \cdots + 677329 \) Copy content Toggle raw display
$17$ \( (T^{6} - 7 T^{5} + \cdots - 501)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 15 T^{5} + \cdots + 400)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 14 T^{11} + \cdots + 46656 \) Copy content Toggle raw display
$29$ \( T^{12} + 13 T^{11} + \cdots + 3600 \) Copy content Toggle raw display
$31$ \( T^{12} + 10 T^{11} + \cdots + 19360000 \) Copy content Toggle raw display
$37$ \( (T^{6} - 21 T^{5} + \cdots + 42472)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 4 T^{11} + \cdots + 4410000 \) Copy content Toggle raw display
$43$ \( T^{12} + 13 T^{11} + \cdots + 41938576 \) Copy content Toggle raw display
$47$ \( T^{12} - 8 T^{11} + \cdots + 86545809 \) Copy content Toggle raw display
$53$ \( (T^{6} - 10 T^{5} + \cdots - 21480)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 5625000000 \) Copy content Toggle raw display
$61$ \( T^{12} + 2 T^{11} + \cdots + 21086464 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 71742551104 \) Copy content Toggle raw display
$71$ \( (T^{6} + 29 T^{5} + \cdots + 6219)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 8 T^{5} + \cdots + 86137)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 18108815761 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 375003225 \) Copy content Toggle raw display
$89$ \( (T^{6} - T^{5} + \cdots - 36492)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 32 T^{11} + \cdots + 95277121 \) Copy content Toggle raw display
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