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Properties

Label 75.6.e.e
Level $75$
Weight $6$
Character orbit 75.e
Analytic conductor $12.029$
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] = [75,6,Mod(32,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, 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("75.32");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(12.0287864860\)
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: \( 2^{10}\cdot 3^{10}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + \beta_{5} q^{3} + ( - \beta_{7} - 13 \beta_{4}) q^{4} + (\beta_{14} + \beta_{10} + \beta_{8} + \cdots - 5) q^{6}+ \cdots + (2 \beta_{15} + \beta_{14} + \cdots - 8 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + \beta_{5} q^{3} + ( - \beta_{7} - 13 \beta_{4}) q^{4} + (\beta_{14} + \beta_{10} + \beta_{8} + \cdots - 5) q^{6}+ \cdots + (1043 \beta_{15} + \cdots - 968 \beta_{2}) q^{99}+O(q^{100}) \) 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 \( 16 q - 84 q^{6} + 80 q^{7} - 420 q^{12} + 2120 q^{13} - 452 q^{16} - 6000 q^{18} + 1416 q^{21} - 3380 q^{22} + 12960 q^{27} + 10900 q^{28} - 21568 q^{31} - 5760 q^{33} + 61932 q^{36} - 55720 q^{37} + 103020 q^{42} + 23360 q^{43} - 151168 q^{46} - 74820 q^{48} + 138816 q^{51} - 204160 q^{52} + 163800 q^{57} + 163500 q^{58} - 175168 q^{61} - 143040 q^{63} + 263400 q^{66} - 140320 q^{67} + 328680 q^{72} + 240320 q^{73} - 271536 q^{76} - 429960 q^{78} + 326016 q^{81} - 431120 q^{82} + 331680 q^{87} + 401700 q^{88} - 487168 q^{91} + 22320 q^{93} + 311748 q^{96} - 440800 q^{97}+O(q^{100}) \) 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
\(\beta_{3}\)\(=\) \( ( 1665491\nu^{13} + 17404336013\nu^{9} + 21906537631021\nu^{5} + 3477742368975475\nu ) / 175266961632000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65179\nu^{14} + 704329347\nu^{10} + 1093176906949\nu^{6} + 338344643638525\nu^{2} ) / 1156275094100000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17652037 \nu^{14} - 44517950 \nu^{13} - 182440375 \nu^{12} + 186662751291 \nu^{10} + \cdots - 16\!\cdots\!75 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17652037 \nu^{14} - 44517950 \nu^{13} + 182440375 \nu^{12} - 186662751291 \nu^{10} + \cdots + 16\!\cdots\!75 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 40605037 \nu^{14} + 433934370291 \nu^{10} + 626362399315347 \nu^{6} + 11\!\cdots\!25 \nu^{2} ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 809631139 \nu^{15} + 10061661090 \nu^{14} - 16305422375 \nu^{13} + 241916930000 \nu^{12} + \cdots + 11\!\cdots\!00 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 809631139 \nu^{15} - 89844586550 \nu^{14} + 24763832875 \nu^{13} + \cdots - 38\!\cdots\!25 \nu ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2501702961 \nu^{15} + 26901148843423 \nu^{11} + \cdots + 10\!\cdots\!25 \nu^{3} ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2077147966 \nu^{15} - 1676943515 \nu^{14} - 17331835625 \nu^{12} - 22414270741638 \nu^{11} + \cdots - 16\!\cdots\!25 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1847159129 \nu^{15} + 10230875375 \nu^{13} - 17331835625 \nu^{12} + 19827685429197 \nu^{11} + \cdots - 16\!\cdots\!25 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1847159129 \nu^{15} - 1676943515 \nu^{14} - 10230875375 \nu^{13} + \cdots - 22\!\cdots\!25 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12429557103 \nu^{15} - 132829575475729 \nu^{11} + \cdots - 42\!\cdots\!75 \nu^{3} ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2077147966 \nu^{15} - 1676943515 \nu^{14} - 17331835625 \nu^{12} + 22414270741638 \nu^{11} + \cdots - 16\!\cdots\!25 ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{13} - 3\beta_{12} - 5\beta_{6} - 9\beta_{5} + 3\beta_{3} + 21\beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - 3 \beta_{14} + 3 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} + 12 \beta_{7} + \cdots - 3 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 95\beta_{15} - 107\beta_{14} - 27\beta_{13} - 27\beta_{12} - 231\beta_{11} - 1229\beta_{10} ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{15} + 129 \beta_{14} - 264 \beta_{11} + 129 \beta_{10} + 258 \beta_{8} + 260 \beta_{6} + \cdots - 24045 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1707\beta_{13} + 1707\beta_{12} + 25195\beta_{6} + 72171\beta_{5} - 29637\beta_{3} - 383139\beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 685 \beta_{15} + 2535 \beta_{14} - 7125 \beta_{11} + 2535 \beta_{10} - 5070 \beta_{9} + \cdots + 2535 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2388515 \beta_{15} + 2718039 \beta_{14} - 15171 \beta_{13} - 15171 \beta_{12} + \cdots + 37442433 \beta_{10} ) / 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 419512 \beta_{15} - 1048344 \beta_{14} + 3355224 \beta_{11} - 1048344 \beta_{10} - 2096688 \beta_{8} + \cdots + 185581065 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2406879 \beta_{13} + 2406879 \beta_{12} - 75890135 \beta_{6} - 232484163 \beta_{5} + \cdots + 1197489767 \beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 89861071 \beta_{15} - 196307187 \beta_{14} + 662197587 \beta_{11} - 196307187 \beta_{10} + \cdots - 196307187 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 21662418685 \beta_{15} - 23820502401 \beta_{14} + 887812239 \beta_{13} + 887812239 \beta_{12} + \cdots - 342377495847 \beta_{10} ) / 90 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 495930106 \beta_{15} + 1029265377 \beta_{14} - 3546321072 \beta_{11} + 1029265377 \beta_{10} + \cdots - 181544707525 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 91643620533 \beta_{13} - 91643620533 \beta_{12} + 2058195285995 \beta_{6} + \cdots - 32545553533059 \beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 862455839813 \beta_{15} + 1754232431151 \beta_{14} - 6095832381741 \beta_{11} + \cdots + 1754232431151 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 65138846432705 \beta_{15} + 71144188770653 \beta_{14} - 2990689930017 \beta_{13} + \cdots + 10\!\cdots\!91 \beta_{10} ) / 30 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−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 15.1419 + 3.70445i − 59.9217i 0 −127.768 + 77.5396i −47.9711 − 47.9711i 189.294 + 189.294i 215.554 + 112.185i 0
32.2 −5.80199 + 5.80199i −13.8908 − 7.07428i − 35.3261i 0 121.639 − 39.5494i 140.332 + 140.332i 19.2981 + 19.2981i 142.909 + 196.535i 0
32.3 −3.17835 + 3.17835i −5.54767 + 14.5679i 11.7962i 0 −28.6694 − 63.9343i −83.6967 − 83.6967i −139.200 − 139.200i −181.447 − 161.636i 0
32.4 −0.879873 + 0.879873i −0.0935382 − 15.5882i 30.4516i 0 13.7979 + 13.6333i 11.3356 + 11.3356i −54.9495 − 54.9495i −242.983 + 2.91618i 0
32.5 0.879873 − 0.879873i 15.5882 + 0.0935382i 30.4516i 0 13.7979 − 13.6333i 11.3356 + 11.3356i 54.9495 + 54.9495i 242.983 + 2.91618i 0
32.6 3.17835 − 3.17835i −14.5679 + 5.54767i 11.7962i 0 −28.6694 + 63.9343i −83.6967 − 83.6967i 139.200 + 139.200i 181.447 − 161.636i 0
32.7 5.80199 − 5.80199i 7.07428 + 13.8908i − 35.3261i 0 121.639 + 39.5494i 140.332 + 140.332i −19.2981 − 19.2981i −142.909 + 196.535i 0
32.8 6.77944 − 6.77944i −3.70445 − 15.1419i − 59.9217i 0 −127.768 − 77.5396i −47.9711 − 47.9711i −189.294 − 189.294i −215.554 + 112.185i 0
68.1 −6.77944 − 6.77944i 15.1419 − 3.70445i 59.9217i 0 −127.768 − 77.5396i −47.9711 + 47.9711i 189.294 − 189.294i 215.554 − 112.185i 0
68.2 −5.80199 − 5.80199i −13.8908 + 7.07428i 35.3261i 0 121.639 + 39.5494i 140.332 − 140.332i 19.2981 − 19.2981i 142.909 − 196.535i 0
68.3 −3.17835 − 3.17835i −5.54767 − 14.5679i − 11.7962i 0 −28.6694 + 63.9343i −83.6967 + 83.6967i −139.200 + 139.200i −181.447 + 161.636i 0
68.4 −0.879873 − 0.879873i −0.0935382 + 15.5882i − 30.4516i 0 13.7979 − 13.6333i 11.3356 − 11.3356i −54.9495 + 54.9495i −242.983 − 2.91618i 0
68.5 0.879873 + 0.879873i 15.5882 − 0.0935382i − 30.4516i 0 13.7979 + 13.6333i 11.3356 − 11.3356i 54.9495 − 54.9495i 242.983 − 2.91618i 0
68.6 3.17835 + 3.17835i −14.5679 − 5.54767i − 11.7962i 0 −28.6694 − 63.9343i −83.6967 + 83.6967i 139.200 − 139.200i 181.447 + 161.636i 0
68.7 5.80199 + 5.80199i 7.07428 − 13.8908i 35.3261i 0 121.639 − 39.5494i 140.332 − 140.332i −19.2981 + 19.2981i −142.909 − 196.535i 0
68.8 6.77944 + 6.77944i −3.70445 + 15.1419i 59.9217i 0 −127.768 + 77.5396i −47.9711 + 47.9711i −189.294 + 189.294i −215.554 − 112.185i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.8
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 75.6.e.e 16
3.b odd 2 1 inner 75.6.e.e 16
5.b even 2 1 15.6.e.a ✓ 16
5.c odd 4 1 15.6.e.a ✓ 16
5.c odd 4 1 inner 75.6.e.e 16
15.d odd 2 1 15.6.e.a ✓ 16
15.e even 4 1 15.6.e.a ✓ 16
15.e even 4 1 inner 75.6.e.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.e.a ✓ 16 5.b even 2 1
15.6.e.a ✓ 16 5.c odd 4 1
15.6.e.a ✓ 16 15.d odd 2 1
15.6.e.a ✓ 16 15.e even 4 1
75.6.e.e 16 1.a even 1 1 trivial
75.6.e.e 16 3.b odd 2 1 inner
75.6.e.e 16 5.c odd 4 1 inner
75.6.e.e 16 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 13393T_{2}^{12} + 43631856T_{2}^{8} + 15738515200T_{2}^{4} + 37480960000 \) acting on \(S_{6}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

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
$5$ \( T^{16} \) 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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