Properties

Label 30.4.e.a
Level $30$
Weight $4$
Character orbit 30.e
Analytic conductor $1.770$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [30,4,Mod(17,30)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(30, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 30.e (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: \(1.77005730017\)
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: \( x^{12} + 1577x^{8} + 284056x^{4} + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{9} - \beta_{3} + 1) q^{3} - 4 \beta_{3} q^{4} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{11} - \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{6} + (\beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{7} - 4 \beta_{5} q^{8} + (2 \beta_{11} - \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{5} + 7 \beta_{3} - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{9} - \beta_{3} + 1) q^{3} - 4 \beta_{3} q^{4} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{11} - \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{6} + (\beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{7} - 4 \beta_{5} q^{8} + (2 \beta_{11} - \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{5} + 7 \beta_{3} - 2 \beta_1 - 1) q^{9} + ( - \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} - 3 \beta_{3} + \beta_1 + 2) q^{10} + ( - \beta_{10} + 3 \beta_{9} + \beta_{6} + 7 \beta_{5} - 3 \beta_{2} - 7 \beta_1 - 2) q^{11} + ( - 4 \beta_{3} - 4 \beta_{2} - 4) q^{12} + ( - 5 \beta_{11} + 5 \beta_{10} + 10 \beta_{9} - 5 \beta_{8} + 5 \beta_{6} + \cdots - 15) q^{13}+ \cdots + ( - 5 \beta_{11} + 64 \beta_{10} - 53 \beta_{9} - 8 \beta_{8} + 2 \beta_{7} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{3} + 8 q^{6} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{3} + 8 q^{6} + 12 q^{7} + 24 q^{10} - 32 q^{12} - 120 q^{13} - 172 q^{15} - 192 q^{16} - 16 q^{18} + 464 q^{21} + 312 q^{22} + 504 q^{25} - 688 q^{27} + 48 q^{28} + 168 q^{30} - 504 q^{31} + 788 q^{33} + 368 q^{36} + 768 q^{37} - 192 q^{40} - 872 q^{42} - 1968 q^{43} - 1328 q^{45} - 1152 q^{46} - 128 q^{48} + 256 q^{51} + 480 q^{52} + 1572 q^{55} + 968 q^{57} + 2280 q^{58} + 896 q^{60} + 1848 q^{61} + 1268 q^{63} + 944 q^{66} - 1752 q^{67} - 2616 q^{70} + 64 q^{72} + 180 q^{73} - 1112 q^{75} - 1152 q^{76} - 4080 q^{78} - 4316 q^{81} + 2208 q^{82} - 1872 q^{85} + 3620 q^{87} + 1248 q^{88} + 5168 q^{90} + 4080 q^{91} + 584 q^{93} - 128 q^{96} - 7596 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1577x^{8} + 284056x^{4} + 810000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -83\nu^{9} - 150271\nu^{5} - 41313548\nu ) / 26668320 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2729 \nu^{10} - 32070 \nu^{9} + 12150 \nu^{8} + 4619533 \nu^{6} - 48423390 \nu^{5} + 12358350 \nu^{4} + 1096505924 \nu^{2} - 6506949720 \nu - 1808202600 ) / 1600099200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -53\nu^{10} - 84481\nu^{6} - 17028668\nu^{2} ) / 28573200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1484 \nu^{11} - 1245 \nu^{10} + 61650 \nu^{8} + 2365468 \nu^{7} - 2254065 \nu^{6} + 92338650 \nu^{4} + 476802704 \nu^{3} - 619703220 \nu^{2} + \cdots + 9374344200 ) / 800049600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 37339\nu^{11} + 59248103\nu^{7} + 10977117484\nu^{3} ) / 12000744000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 81859 \nu^{11} + 40935 \nu^{10} + 518400 \nu^{9} + 546750 \nu^{8} - 130212143 \nu^{7} + 69292995 \nu^{6} + 793972800 \nu^{5} + \cdots - 57367629000 ) / 24001488000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81859 \nu^{11} + 78285 \nu^{10} + 518400 \nu^{9} + 182250 \nu^{8} + 130212143 \nu^{7} + 136914945 \nu^{6} + 793972800 \nu^{5} + \cdots - 27123039000 ) / 24001488000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1484 \nu^{11} - 21493 \nu^{10} - 12150 \nu^{8} - 2365468 \nu^{7} - 33450761 \nu^{6} - 12358350 \nu^{4} - 476802704 \nu^{3} - 5446486708 \nu^{2} + \cdots + 1008153000 ) / 800049600 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 213257 \nu^{11} + 40935 \nu^{10} - 182250 \nu^{8} + 333727789 \nu^{7} + 69292995 \nu^{6} - 185375250 \nu^{4} + 57065269892 \nu^{3} + \cdots + 27123039000 ) / 24001488000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 83532 \nu^{11} + 13645 \nu^{10} + 12450 \nu^{9} - 182250 \nu^{8} - 130991964 \nu^{7} + 23097665 \nu^{6} + 22540650 \nu^{5} - 185375250 \nu^{4} + \cdots + 19122543000 ) / 8000496000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 257777 \nu^{11} + 481050 \nu^{9} + 364500 \nu^{8} + 404691829 \nu^{7} + 726350850 \nu^{5} + 370750500 \nu^{4} + 71369351012 \nu^{3} + \cdots - 30244590000 ) / 24001488000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{11} + 3\beta_{10} - 2\beta_{7} + \beta_{6} + 3\beta_{5} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5 \beta_{11} - 5 \beta_{10} - 10 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 5 \beta_{6} - 3 \beta_{5} - 116 \beta_{3} - 4 \beta_{2} - 7 \beta _1 + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 31 \beta_{11} - 31 \beta_{10} - 50 \beta_{9} + 56 \beta_{7} - 87 \beta_{6} - 225 \beta_{5} - 56 \beta_{3} - 31 \beta _1 + 25 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 191 \beta_{11} + 191 \beta_{10} + 382 \beta_{9} - 29 \beta_{6} + 81 \beta_{5} + 162 \beta_{4} + 162 \beta_{3} - 220 \beta_{2} + 81 \beta _1 - 3345 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2943 \beta_{11} - 2943 \beta_{10} + 1796 \beta_{7} - 1147 \beta_{6} - 2943 \beta_{5} - 1796 \beta_{3} - 1298 \beta_{2} - 7561 \beta _1 + 1147 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7043 \beta_{11} + 7043 \beta_{10} + 14086 \beta_{9} - 4890 \beta_{8} + 9196 \beta_{7} + 7043 \beta_{6} + 2445 \beta_{5} + 121352 \beta_{3} + 9196 \beta_{2} + 11641 \beta _1 + 2153 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 42595 \beta_{11} + 42595 \beta_{10} + 39794 \beta_{9} - 62492 \beta_{7} + 105087 \beta_{6} + 360021 \beta_{5} + 62492 \beta_{3} + 42595 \beta _1 - 19897 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 260123 \beta_{11} - 260123 \beta_{10} - 520246 \beta_{9} + 95345 \beta_{6} - 82389 \beta_{5} - 164778 \beta_{4} - 164778 \beta_{3} + 355468 \beta_{2} - 82389 \beta _1 + 4097757 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3835023 \beta_{11} + 3835023 \beta_{10} - 2256140 \beta_{7} + 1578883 \beta_{6} + 3835023 \beta_{5} + 2256140 \beta_{3} + 1354514 \beta_{2} + 11263561 \beta _1 - 1578883 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9619931 \beta_{11} - 9619931 \beta_{10} - 19239862 \beta_{9} + 5866794 \beta_{8} - 13373068 \beta_{7} - 9619931 \beta_{6} - 2933397 \beta_{5} - 159397208 \beta_{3} + \cdots - 3753137 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 58474579 \beta_{11} - 58474579 \beta_{10} - 48444338 \beta_{9} + 82696748 \beta_{7} - 141171327 \beta_{6} - 503192517 \beta_{5} - 82696748 \beta_{3} - 58474579 \beta _1 + 24222169 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
2.67233 2.67233i
−4.30203 + 4.30203i
0.922599 0.922599i
−0.922599 + 0.922599i
−2.67233 + 2.67233i
4.30203 4.30203i
2.67233 + 2.67233i
−4.30203 4.30203i
0.922599 + 0.922599i
−0.922599 0.922599i
−2.67233 2.67233i
4.30203 + 4.30203i
−1.41421 + 1.41421i −4.79094 2.01170i 4.00000i −10.8454 2.71609i 9.62038 3.93044i −16.1789 16.1789i 5.65685 + 5.65685i 18.9062 + 19.2758i 19.1789 11.4966i
17.2 −1.41421 + 1.41421i 2.56513 4.51886i 4.00000i 10.0777 4.84155i 2.76299 + 10.0183i 10.4050 + 10.4050i 5.65685 + 5.65685i −13.8402 23.1830i −7.40500 + 21.0990i
17.3 −1.41421 + 1.41421i 3.51870 + 3.82345i 4.00000i −5.59622 + 9.67896i −10.3834 0.430985i 8.77386 + 8.77386i 5.65685 + 5.65685i −2.23754 + 26.9071i −5.77386 21.6024i
17.4 1.41421 1.41421i −3.82345 3.51870i 4.00000i 5.59622 9.67896i −10.3834 + 0.430985i 8.77386 + 8.77386i −5.65685 5.65685i 2.23754 + 26.9071i −5.77386 21.6024i
17.5 1.41421 1.41421i 2.01170 + 4.79094i 4.00000i 10.8454 + 2.71609i 9.62038 + 3.93044i −16.1789 16.1789i −5.65685 5.65685i −18.9062 + 19.2758i 19.1789 11.4966i
17.6 1.41421 1.41421i 4.51886 2.56513i 4.00000i −10.0777 + 4.84155i 2.76299 10.0183i 10.4050 + 10.4050i −5.65685 5.65685i 13.8402 23.1830i −7.40500 + 21.0990i
23.1 −1.41421 1.41421i −4.79094 + 2.01170i 4.00000i −10.8454 + 2.71609i 9.62038 + 3.93044i −16.1789 + 16.1789i 5.65685 5.65685i 18.9062 19.2758i 19.1789 + 11.4966i
23.2 −1.41421 1.41421i 2.56513 + 4.51886i 4.00000i 10.0777 + 4.84155i 2.76299 10.0183i 10.4050 10.4050i 5.65685 5.65685i −13.8402 + 23.1830i −7.40500 21.0990i
23.3 −1.41421 1.41421i 3.51870 3.82345i 4.00000i −5.59622 9.67896i −10.3834 + 0.430985i 8.77386 8.77386i 5.65685 5.65685i −2.23754 26.9071i −5.77386 + 21.6024i
23.4 1.41421 + 1.41421i −3.82345 + 3.51870i 4.00000i 5.59622 + 9.67896i −10.3834 0.430985i 8.77386 8.77386i −5.65685 + 5.65685i 2.23754 26.9071i −5.77386 + 21.6024i
23.5 1.41421 + 1.41421i 2.01170 4.79094i 4.00000i 10.8454 2.71609i 9.62038 3.93044i −16.1789 + 16.1789i −5.65685 + 5.65685i −18.9062 19.2758i 19.1789 + 11.4966i
23.6 1.41421 + 1.41421i 4.51886 + 2.56513i 4.00000i −10.0777 4.84155i 2.76299 + 10.0183i 10.4050 10.4050i −5.65685 + 5.65685i 13.8402 + 23.1830i −7.40500 21.0990i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.4.e.a 12
3.b odd 2 1 inner 30.4.e.a 12
4.b odd 2 1 240.4.v.d 12
5.b even 2 1 150.4.e.c 12
5.c odd 4 1 inner 30.4.e.a 12
5.c odd 4 1 150.4.e.c 12
12.b even 2 1 240.4.v.d 12
15.d odd 2 1 150.4.e.c 12
15.e even 4 1 inner 30.4.e.a 12
15.e even 4 1 150.4.e.c 12
20.e even 4 1 240.4.v.d 12
60.l odd 4 1 240.4.v.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.e.a 12 1.a even 1 1 trivial
30.4.e.a 12 3.b odd 2 1 inner
30.4.e.a 12 5.c odd 4 1 inner
30.4.e.a 12 15.e even 4 1 inner
150.4.e.c 12 5.b even 2 1
150.4.e.c 12 5.c odd 4 1
150.4.e.c 12 15.d odd 2 1
150.4.e.c 12 15.e even 4 1
240.4.v.d 12 4.b odd 2 1
240.4.v.d 12 12.b even 2 1
240.4.v.d 12 20.e even 4 1
240.4.v.d 12 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} - 8 T^{11} + 32 T^{10} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{12} - 252 T^{10} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 6 T^{5} + 18 T^{4} + \cdots + 17452232)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 2826 T^{4} + 1382592 T^{2} + \cdots + 154457888)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 60 T^{5} + 1800 T^{4} + \cdots + 64800000000)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 11627928 T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{6} + 21288 T^{4} + \cdots + 12078889216)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 2216644488 T^{8} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} - 82170 T^{4} + \cdots - 8528180000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 126 T^{2} - 7008 T + 39488)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 384 T^{5} + \cdots + 33414920101152)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 176736 T^{4} + \cdots + 165740576768)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 984 T^{5} + \cdots + 40492872313632)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 51730245000 T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + 50932685448 T^{8} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{6} - 594834 T^{4} + \cdots - 13\!\cdots\!92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 462 T^{2} - 319332 T + 123755256)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 876 T^{5} + \cdots + 122924426333312)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 182880 T^{4} + \cdots + 5009245520000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 90 T^{5} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 1271868 T^{4} + \cdots + 23\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 1176370071048 T^{8} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{6} - 543504 T^{4} + \cdots - 53\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 3798 T^{5} + \cdots + 13\!\cdots\!28)^{2} \) Copy content Toggle raw display
show more
show less