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Properties

Label 15.7.f.a
Level $15$
Weight $7$
Character orbit 15.f
Analytic conductor $3.451$
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] = [15,7,Mod(7,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.7");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 15.f (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: \(3.45081125430\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 420 x^{9} + 22793 x^{8} - 52752 x^{7} + 116864 x^{6} + 5895920 x^{5} + \cdots + 71571600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{10}\cdot 5^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1 + 1) q^{2} - \beta_{6} q^{3} + ( - \beta_{11} + \beta_{10} + \cdots + 18 \beta_1) q^{4}+ \cdots - 243 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1 + 1) q^{2} - \beta_{6} q^{3} + ( - \beta_{11} + \beta_{10} + \cdots + 18 \beta_1) q^{4}+ \cdots + ( - 729 \beta_{11} + 5832 \beta_{10} + \cdots + 486) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{2} + 136 q^{5} + 324 q^{6} - 696 q^{7} - 3468 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 16 q^{2} + 136 q^{5} + 324 q^{6} - 696 q^{7} - 3468 q^{8} + 5304 q^{10} + 3248 q^{11} + 1944 q^{12} + 108 q^{13} - 4536 q^{15} - 16020 q^{16} - 5540 q^{17} + 3888 q^{18} + 12564 q^{20} - 15552 q^{21} + 49620 q^{22} + 11776 q^{23} - 65124 q^{25} + 24464 q^{26} + 23004 q^{28} + 17496 q^{30} + 10992 q^{31} - 126476 q^{32} - 17496 q^{33} - 59360 q^{35} + 61236 q^{36} - 105516 q^{37} - 55776 q^{38} + 376248 q^{40} + 395312 q^{41} + 143532 q^{42} + 7392 q^{43} + 32076 q^{45} - 751368 q^{46} - 246752 q^{47} - 423792 q^{48} - 669776 q^{50} - 225504 q^{51} + 694920 q^{52} + 954196 q^{53} - 74664 q^{55} + 961920 q^{56} + 412128 q^{57} - 324732 q^{58} + 705996 q^{60} - 420240 q^{61} - 1459672 q^{62} - 169128 q^{63} - 843212 q^{65} - 1180008 q^{66} + 1300800 q^{67} + 761696 q^{68} + 560820 q^{70} + 831584 q^{71} + 842724 q^{72} - 1208004 q^{73} + 769824 q^{75} - 292584 q^{76} - 3205552 q^{77} - 897480 q^{78} - 1124876 q^{80} - 708588 q^{81} + 815328 q^{82} + 1244224 q^{83} + 2746812 q^{85} + 8173568 q^{86} + 3063096 q^{87} - 948228 q^{88} - 32076 q^{90} - 5171328 q^{91} - 7092056 q^{92} - 1823472 q^{93} - 4466016 q^{95} - 5713092 q^{96} + 1506780 q^{97} + 6386552 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 420 x^{9} + 22793 x^{8} - 52752 x^{7} + 116864 x^{6} + 5895920 x^{5} + \cdots + 71571600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15\!\cdots\!99 \nu^{11} + \cdots - 53\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 57\!\cdots\!83 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 88\!\cdots\!53 \nu^{11} + \cdots + 34\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!21 \nu^{11} + \cdots + 27\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!77 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24\!\cdots\!69 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 82\!\cdots\!98 \nu^{11} + \cdots + 22\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!09 \nu^{11} + \cdots - 72\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!98 \nu^{11} + \cdots + 10\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!79 \nu^{11} + \cdots + 43\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21\!\cdots\!19 \nu^{11} + \cdots + 70\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{10} + 3\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 3\beta_{5} - 3\beta_{4} - 3\beta_{2} + 13\beta _1 + 14 ) / 45 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 10 \beta_{11} - 9 \beta_{10} - 2 \beta_{8} - \beta_{7} - 12 \beta_{6} + 4 \beta_{5} - 192 \beta_{4} + \cdots + 1 ) / 45 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 139 \beta_{11} - 396 \beta_{10} - 396 \beta_{9} - 309 \beta_{8} + 36 \beta_{7} - 893 \beta_{6} + \cdots - 4597 ) / 45 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 219 \beta_{11} - 903 \beta_{9} - 876 \beta_{8} - 114 \beta_{7} - 1982 \beta_{6} - 438 \beta_{5} + \cdots - 349569 ) / 45 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4770 \beta_{11} + 54372 \beta_{10} - 54372 \beta_{9} - 11219 \beta_{8} + 15989 \beta_{7} + \cdots - 756376 ) / 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 126940 \beta_{11} + 124731 \beta_{10} + 65098 \beta_{8} + 32549 \beta_{7} + 477228 \beta_{6} + \cdots - 32549 ) / 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1646111 \beta_{11} + 7280814 \beta_{10} + 7280814 \beta_{9} + 3927021 \beta_{8} - 337104 \beta_{7} + \cdots + 108182753 ) / 45 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4417851 \beta_{11} + 20215227 \beta_{9} + 17671404 \beta_{8} + 26270286 \beta_{7} + \cdots + 5315013261 ) / 45 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5998410 \beta_{11} - 968241348 \beta_{10} + 968241348 \beta_{9} + 159752731 \beta_{8} + \cdots + 15519827504 ) / 45 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4361407880 \beta_{11} - 3361720779 \beta_{10} - 1179862802 \beta_{8} - 589931401 \beta_{7} + \cdots + 589931401 ) / 45 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12438220939 \beta_{11} - 128827301826 \beta_{10} - 128827301826 \beta_{9} - 60261037149 \beta_{8} + \cdots - 2184889627297 ) / 45 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.880576 − 0.880576i
0.478553 + 0.478553i
−5.28673 − 5.28673i
7.17737 + 7.17737i
8.39205 + 8.39205i
−7.88066 − 7.88066i
−0.880576 + 0.880576i
0.478553 − 0.478553i
−5.28673 + 5.28673i
7.17737 − 7.17737i
8.39205 − 8.39205i
−7.88066 + 7.88066i
−6.41935 − 6.41935i −11.0227 + 11.0227i 18.4160i −7.31458 + 124.786i 141.517 364.812 + 364.812i −292.620 + 292.620i − 243.000i 847.998 − 754.088i
7.2 −5.23228 − 5.23228i 11.0227 − 11.0227i − 9.24645i −111.899 + 55.7110i −115.348 −316.711 − 316.711i −383.246 + 383.246i − 243.000i 876.980 + 293.989i
7.3 −1.18257 − 1.18257i −11.0227 + 11.0227i − 61.2031i 11.1041 − 124.506i 26.0702 −253.950 − 253.950i −148.061 + 148.061i − 243.000i −160.368 + 134.106i
7.4 2.04420 + 2.04420i 11.0227 − 11.0227i − 55.6425i 123.145 + 21.4538i 45.0653 106.360 + 106.360i 244.573 − 244.573i − 243.000i 207.878 + 295.590i
7.5 7.92768 + 7.92768i −11.0227 + 11.0227i 61.6963i 52.2559 + 113.553i −174.769 −108.499 − 108.499i 18.2634 − 18.2634i − 243.000i −485.945 + 1314.48i
7.6 10.8623 + 10.8623i 11.0227 − 11.0227i 171.980i 0.707971 − 124.998i 239.464 −140.012 − 140.012i −1172.91 + 1172.91i − 243.000i 1365.46 − 1350.08i
13.1 −6.41935 + 6.41935i −11.0227 − 11.0227i − 18.4160i −7.31458 − 124.786i 141.517 364.812 − 364.812i −292.620 − 292.620i 243.000i 847.998 + 754.088i
13.2 −5.23228 + 5.23228i 11.0227 + 11.0227i 9.24645i −111.899 − 55.7110i −115.348 −316.711 + 316.711i −383.246 − 383.246i 243.000i 876.980 − 293.989i
13.3 −1.18257 + 1.18257i −11.0227 − 11.0227i 61.2031i 11.1041 + 124.506i 26.0702 −253.950 + 253.950i −148.061 − 148.061i 243.000i −160.368 − 134.106i
13.4 2.04420 − 2.04420i 11.0227 + 11.0227i 55.6425i 123.145 − 21.4538i 45.0653 106.360 − 106.360i 244.573 + 244.573i 243.000i 207.878 − 295.590i
13.5 7.92768 − 7.92768i −11.0227 − 11.0227i − 61.6963i 52.2559 − 113.553i −174.769 −108.499 + 108.499i 18.2634 + 18.2634i 243.000i −485.945 − 1314.48i
13.6 10.8623 − 10.8623i 11.0227 + 11.0227i − 171.980i 0.707971 + 124.998i 239.464 −140.012 + 140.012i −1172.91 − 1172.91i 243.000i 1365.46 + 1350.08i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.7.f.a ✓ 12
3.b odd 2 1 45.7.g.b 12
4.b odd 2 1 240.7.bg.c 12
5.b even 2 1 75.7.f.d 12
5.c odd 4 1 inner 15.7.f.a ✓ 12
5.c odd 4 1 75.7.f.d 12
15.d odd 2 1 225.7.g.k 12
15.e even 4 1 45.7.g.b 12
15.e even 4 1 225.7.g.k 12
20.e even 4 1 240.7.bg.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.7.f.a ✓ 12 1.a even 1 1 trivial
15.7.f.a ✓ 12 5.c odd 4 1 inner
45.7.g.b 12 3.b odd 2 1
45.7.g.b 12 15.e even 4 1
75.7.f.d 12 5.b even 2 1
75.7.f.d 12 5.c odd 4 1
225.7.g.k 12 15.d odd 2 1
225.7.g.k 12 15.e even 4 1
240.7.bg.c 12 4.b odd 2 1
240.7.bg.c 12 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 3128836096 \) Copy content Toggle raw display
$3$ \( (T^{4} + 59049)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 88\!\cdots\!08)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 89\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 77\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 33\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 76\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 31\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 61\!\cdots\!04 \) Copy content Toggle raw display
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