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Properties

Label 15.6.e.a
Level $15$
Weight $6$
Character orbit 15.e
Analytic conductor $2.406$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [15,6,Mod(2,15)]
 
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([2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.2");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 15.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: \(2.40575729719\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10768x^{12} + 16341006x^{8} + 4217167600x^{4} + 50906640625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{10}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - 13 \beta_{3}) q^{4} + ( - \beta_{14} + \beta_{9} + \beta_{2}) q^{5} + (\beta_{15} - \beta_{14} + \beta_{11} + \cdots - 5) q^{6}+ \cdots + (2 \beta_{15} + \beta_{14} + \cdots + 7 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - 13 \beta_{3}) q^{4} + ( - \beta_{14} + \beta_{9} + \beta_{2}) q^{5} + (\beta_{15} - \beta_{14} + \beta_{11} + \cdots - 5) q^{6}+ \cdots + ( - 865 \beta_{15} + \cdots + 2413 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 84 q^{6} - 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 84 q^{6} - 80 q^{7} + 1060 q^{10} + 420 q^{12} - 2120 q^{13} - 3000 q^{15} - 452 q^{16} + 6000 q^{18} + 1416 q^{21} + 3380 q^{22} + 9160 q^{25} - 12960 q^{27} - 10900 q^{28} - 13380 q^{30} - 21568 q^{31} + 5760 q^{33} + 61932 q^{36} + 55720 q^{37} + 21480 q^{40} - 103020 q^{42} - 23360 q^{43} - 21240 q^{45} - 151168 q^{46} + 74820 q^{48} + 138816 q^{51} + 204160 q^{52} + 89120 q^{55} - 163800 q^{57} - 163500 q^{58} - 217380 q^{60} - 175168 q^{61} + 143040 q^{63} + 263400 q^{66} + 140320 q^{67} + 385260 q^{70} - 328680 q^{72} - 240320 q^{73} - 199920 q^{75} - 271536 q^{76} + 429960 q^{78} + 326016 q^{81} + 431120 q^{82} + 2440 q^{85} - 331680 q^{87} - 401700 q^{88} - 348360 q^{90} - 487168 q^{91} - 22320 q^{93} + 311748 q^{96} + 440800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10768x^{12} + 16341006x^{8} + 4217167600x^{4} + 50906640625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -983\nu^{12} - 10530009\nu^{8} - 14985743913\nu^{4} - 1365328353575 ) / 52711868160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -193037\nu^{13} - 2063897891\nu^{9} - 3039554293747\nu^{5} - 800189026191325\nu ) / 175266961632000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65179\nu^{14} + 704329347\nu^{10} + 1093176906949\nu^{6} + 338344643638525\nu^{2} ) / 1156275094100000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17652037 \nu^{14} + 44517950 \nu^{13} - 182440375 \nu^{12} + 186662751291 \nu^{10} + \cdots - 16\!\cdots\!75 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17652037 \nu^{14} + 44517950 \nu^{13} + 182440375 \nu^{12} - 186662751291 \nu^{10} + \cdots + 16\!\cdots\!75 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 40605037 \nu^{14} + 433934370291 \nu^{10} + 626362399315347 \nu^{6} + 11\!\cdots\!25 \nu^{2} ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2501702961 \nu^{15} - 26901148843423 \nu^{11} + \cdots - 10\!\cdots\!25 \nu^{3} ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2077147966 \nu^{15} - 5030830545 \nu^{14} - 16232545500 \nu^{13} - 120958465000 \nu^{12} + \cdots - 57\!\cdots\!00 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2077147966 \nu^{15} + 1676943515 \nu^{14} + 17331835625 \nu^{12} - 22414270741638 \nu^{11} + \cdots + 16\!\cdots\!25 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2077147966 \nu^{15} + 39891462730 \nu^{14} - 16232545500 \nu^{13} + 51995506875 \nu^{12} + \cdots + 48\!\cdots\!75 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4963927071 \nu^{15} + 3353887030 \nu^{14} - 82856539500 \nu^{13} - 34663671250 \nu^{12} + \cdots - 32\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4963927071 \nu^{15} - 3353887030 \nu^{14} + 82856539500 \nu^{13} + 34663671250 \nu^{12} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2077147966 \nu^{15} + 1676943515 \nu^{14} + 17331835625 \nu^{12} + 22414270741638 \nu^{11} + \cdots + 16\!\cdots\!25 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14854266548 \nu^{15} + 3353887030 \nu^{14} - 33222243375 \nu^{13} + 34663671250 \nu^{12} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14854266548 \nu^{15} + 3353887030 \nu^{14} + 33222243375 \nu^{13} + 34663671250 \nu^{12} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6\beta_{15} - 6\beta_{14} - 3\beta_{12} + 3\beta_{11} + 5\beta_{5} + 9\beta_{4} + 12\beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} - 6 \beta_{10} + \cdots - 6 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -27\beta_{15} - 27\beta_{14} + 95\beta_{13} + 134\beta_{12} + 134\beta_{11} - 231\beta_{9} + 1068\beta_{7} ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 129 \beta_{15} - 129 \beta_{14} + 2 \beta_{13} - 129 \beta_{12} + 129 \beta_{11} + 264 \beta_{9} + \cdots - 24045 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 31344 \beta_{15} + 31344 \beta_{14} + 1707 \beta_{12} - 1707 \beta_{11} - 25195 \beta_{5} + \cdots - 350088 \beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2535 \beta_{15} + 2535 \beta_{14} + 685 \beta_{13} + 2535 \beta_{12} - 2535 \beta_{11} + \cdots + 5070 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 15171 \beta_{15} - 15171 \beta_{14} - 2388515 \beta_{13} - 2702868 \beta_{12} + \cdots - 34754736 \beta_{7} ) / 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1048344 \beta_{15} + 1048344 \beta_{14} - 419512 \beta_{13} + 1048344 \beta_{12} - 1048344 \beta_{11} + \cdots + 185581065 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 81977282 \beta_{15} - 81977282 \beta_{14} + 2406879 \beta_{12} - 2406879 \beta_{11} + \cdots + 1117919364 \beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 196307187 \beta_{15} - 196307187 \beta_{14} - 89861071 \beta_{13} - 196307187 \beta_{12} + \cdots - 392614374 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 887812239 \beta_{15} + 887812239 \beta_{14} + 21662418685 \beta_{13} + 22932690162 \beta_{12} + \cdots + 320332617924 \beta_{7} ) / 90 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1029265377 \beta_{15} - 1029265377 \beta_{14} + 495930106 \beta_{13} - 1029265377 \beta_{12} + \cdots - 181544707525 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2160764841264 \beta_{15} + 2160764841264 \beta_{14} - 91643620533 \beta_{12} + \cdots - 30476432312328 \beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1754232431151 \beta_{15} + 1754232431151 \beta_{14} + 862455839813 \beta_{13} + \cdots + 3508464862302 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2990689930017 \beta_{15} - 2990689930017 \beta_{14} - 65138846432705 \beta_{13} + \cdots - 965018230513872 \beta_{7} ) / 30 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−6.88827 − 6.88827i
1.33460 + 1.33460i
4.35647 + 4.35647i
2.96509 + 2.96509i
−2.96509 − 2.96509i
−4.35647 − 4.35647i
−1.33460 − 1.33460i
6.88827 + 6.88827i
−6.88827 + 6.88827i
1.33460 − 1.33460i
4.35647 − 4.35647i
2.96509 − 2.96509i
−2.96509 + 2.96509i
−4.35647 + 4.35647i
−1.33460 + 1.33460i
6.88827 − 6.88827i
−6.77944 + 6.77944i 3.70445 + 15.1419i − 59.9217i −52.1700 − 20.0821i −127.768 − 77.5396i 47.9711 + 47.9711i 189.294 + 189.294i −215.554 + 112.185i 489.829 − 217.538i
2.2 −5.80199 + 5.80199i −7.07428 − 13.8908i − 35.3261i 42.8643 − 35.8839i 121.639 + 39.5494i −140.332 − 140.332i 19.2981 + 19.2981i −142.909 + 196.535i −40.5000 + 456.896i
2.3 −3.17835 + 3.17835i 14.5679 − 5.54767i 11.7962i 11.9990 + 54.5987i −28.6694 + 63.9343i 83.6967 + 83.6967i −139.200 − 139.200i 181.447 − 161.636i −211.671 − 135.397i
2.4 −0.879873 + 0.879873i −15.5882 − 0.0935382i 30.4516i −51.8842 + 20.8094i 13.7979 − 13.6333i −11.3356 − 11.3356i −54.9495 − 54.9495i 242.983 + 2.91618i 27.3419 − 63.9612i
2.5 0.879873 − 0.879873i 0.0935382 + 15.5882i 30.4516i 51.8842 − 20.8094i 13.7979 + 13.6333i −11.3356 − 11.3356i 54.9495 + 54.9495i −242.983 + 2.91618i 27.3419 − 63.9612i
2.6 3.17835 − 3.17835i 5.54767 − 14.5679i 11.7962i −11.9990 − 54.5987i −28.6694 − 63.9343i 83.6967 + 83.6967i 139.200 + 139.200i −181.447 − 161.636i −211.671 − 135.397i
2.7 5.80199 − 5.80199i 13.8908 + 7.07428i − 35.3261i −42.8643 + 35.8839i 121.639 − 39.5494i −140.332 − 140.332i −19.2981 − 19.2981i 142.909 + 196.535i −40.5000 + 456.896i
2.8 6.77944 − 6.77944i −15.1419 − 3.70445i − 59.9217i 52.1700 + 20.0821i −127.768 + 77.5396i 47.9711 + 47.9711i −189.294 − 189.294i 215.554 + 112.185i 489.829 − 217.538i
8.1 −6.77944 − 6.77944i 3.70445 − 15.1419i 59.9217i −52.1700 + 20.0821i −127.768 + 77.5396i 47.9711 − 47.9711i 189.294 − 189.294i −215.554 − 112.185i 489.829 + 217.538i
8.2 −5.80199 − 5.80199i −7.07428 + 13.8908i 35.3261i 42.8643 + 35.8839i 121.639 − 39.5494i −140.332 + 140.332i 19.2981 − 19.2981i −142.909 − 196.535i −40.5000 − 456.896i
8.3 −3.17835 − 3.17835i 14.5679 + 5.54767i − 11.7962i 11.9990 − 54.5987i −28.6694 − 63.9343i 83.6967 − 83.6967i −139.200 + 139.200i 181.447 + 161.636i −211.671 + 135.397i
8.4 −0.879873 − 0.879873i −15.5882 + 0.0935382i − 30.4516i −51.8842 − 20.8094i 13.7979 + 13.6333i −11.3356 + 11.3356i −54.9495 + 54.9495i 242.983 − 2.91618i 27.3419 + 63.9612i
8.5 0.879873 + 0.879873i 0.0935382 − 15.5882i − 30.4516i 51.8842 + 20.8094i 13.7979 − 13.6333i −11.3356 + 11.3356i 54.9495 − 54.9495i −242.983 − 2.91618i 27.3419 + 63.9612i
8.6 3.17835 + 3.17835i 5.54767 + 14.5679i − 11.7962i −11.9990 + 54.5987i −28.6694 + 63.9343i 83.6967 − 83.6967i 139.200 − 139.200i −181.447 + 161.636i −211.671 + 135.397i
8.7 5.80199 + 5.80199i 13.8908 − 7.07428i 35.3261i −42.8643 − 35.8839i 121.639 + 39.5494i −140.332 + 140.332i −19.2981 + 19.2981i 142.909 − 196.535i −40.5000 − 456.896i
8.8 6.77944 + 6.77944i −15.1419 + 3.70445i 59.9217i 52.1700 − 20.0821i −127.768 − 77.5396i 47.9711 − 47.9711i −189.294 + 189.294i 215.554 − 112.185i 489.829 + 217.538i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
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 15.6.e.a ✓ 16
3.b odd 2 1 inner 15.6.e.a ✓ 16
5.b even 2 1 75.6.e.e 16
5.c odd 4 1 inner 15.6.e.a ✓ 16
5.c odd 4 1 75.6.e.e 16
15.d odd 2 1 75.6.e.e 16
15.e even 4 1 inner 15.6.e.a ✓ 16
15.e even 4 1 75.6.e.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.e.a ✓ 16 1.a even 1 1 trivial
15.6.e.a ✓ 16 3.b odd 2 1 inner
15.6.e.a ✓ 16 5.c odd 4 1 inner
15.6.e.a ✓ 16 15.e even 4 1 inner
75.6.e.e 16 5.b even 2 1
75.6.e.e 16 5.c odd 4 1
75.6.e.e 16 15.d odd 2 1
75.6.e.e 16 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 37480960000 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 652674756250000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 12072937186816)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 18\!\cdots\!84)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 33\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
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