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Properties

Label 75.6.e.f
Level $75$
Weight $6$
Character orbit 75.e
Analytic conductor $12.029$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \( x^{16} + 1642x^{12} + 327633x^{8} + 12558424x^{4} + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{12}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{6} + \beta_1) q^{3} + (2 \beta_{8} - 38 \beta_{3}) q^{4} + ( - \beta_{14} + 4 \beta_{11} - 5 \beta_{2}) q^{6} + ( - \beta_{13} + \beta_{10} + 18 \beta_{9}) q^{7} + (4 \beta_{7} - 20 \beta_{6} + \cdots - 4 \beta_1) q^{8}+ \cdots + ( - \beta_{15} - 5 \beta_{12} + \cdots + 37 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{6} + \beta_1) q^{3} + (2 \beta_{8} - 38 \beta_{3}) q^{4} + ( - \beta_{14} + 4 \beta_{11} - 5 \beta_{2}) q^{6} + ( - \beta_{13} + \beta_{10} + 18 \beta_{9}) q^{7} + (4 \beta_{7} - 20 \beta_{6} + \cdots - 4 \beta_1) q^{8}+ \cdots + ( - 1383 \beta_{15} + \cdots + 36120 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{6} - 22432 q^{16} + 13776 q^{21} - 31928 q^{31} - 127968 q^{36} + 418272 q^{46} - 12684 q^{51} - 168568 q^{61} + 420840 q^{66} - 364816 q^{76} - 404964 q^{81} + 665352 q^{91} - 508032 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1642x^{12} + 327633x^{8} + 12558424x^{4} + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -38201\nu^{13} - 63083334\nu^{9} - 13076166489\nu^{5} - 541384274636\nu ) / 10501107936 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 82\nu^{12} + 130071\nu^{8} + 20152872\nu^{4} + 357290240 ) / 14781965 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4637\nu^{14} - 7614534\nu^{10} - 1520222261\nu^{6} - 58512688268\nu^{2} ) / 1651117600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1201427 \nu^{14} - 98305 \nu^{12} + 1973774694 \nu^{10} - 153230490 \nu^{8} + \cdots + 431968798100 ) / 131263849200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1201427 \nu^{14} - 98305 \nu^{12} - 1973774694 \nu^{10} - 153230490 \nu^{8} + \cdots + 431968798100 ) / 131263849200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4086317 \nu^{14} + 526665 \nu^{13} - 130445 \nu^{12} + 6709956549 \nu^{10} + \cdots - 1446091727600 ) / 131263849200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8172634 \nu^{14} + 4311625 \nu^{13} + 260890 \nu^{12} - 13419913098 \nu^{10} + \cdots + 2892183455200 ) / 262527698400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -375486\nu^{14} - 616374657\nu^{10} - 122756538173\nu^{6} - 4684632124284\nu^{2} ) / 5469327050 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 76354501\nu^{15} + 125376056742\nu^{11} + 25019318835933\nu^{7} + 959266698314524\nu^{3} ) / 525055396800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35869899 \nu^{15} - 8172634 \nu^{14} - 260890 \nu^{12} + 58898187318 \nu^{11} + \cdots - 2892183455200 ) / 262527698400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39365309 \nu^{15} + 1296050 \nu^{13} + 485440 \nu^{12} + 64636325178 \nu^{11} + \cdots + 2115158220800 ) / 175018465600 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 39365309 \nu^{15} - 6007776 \nu^{14} - 1296050 \nu^{13} + 64636325178 \nu^{11} + \cdots - 13941444096200 \nu ) / 175018465600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 282344489 \nu^{15} - 16345268 \nu^{14} - 521780 \nu^{12} - 463605816438 \nu^{11} + \cdots - 5784366910400 ) / 525055396800 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 162457949 \nu^{15} + 4502150 \nu^{13} + 266755816158 \nu^{11} + 7384814700 \nu^{9} + \cdots + 57577378908200 \nu ) / 175018465600 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 162457949 \nu^{15} + 4502150 \nu^{13} - 266755816158 \nu^{11} + 7384814700 \nu^{9} + \cdots + 57577378908200 \nu ) / 175018465600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + 2\beta_{12} - 2\beta_{11} - \beta_{8} + \beta_{2} + 6\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + 3\beta_{5} - 3\beta_{4} - 44\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 11 \beta_{15} + 11 \beta_{14} - 15 \beta_{13} - 32 \beta_{12} - 32 \beta_{11} + 15 \beta_{10} + \cdots + 16 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{13} + 20 \beta_{10} - 4 \beta_{9} - 4 \beta_{7} + 20 \beta_{6} + 120 \beta_{5} + 120 \beta_{4} + \cdots - 1252 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 332 \beta_{15} - 332 \beta_{14} - 1194 \beta_{12} + 1194 \beta_{11} + 597 \beta_{8} + \cdots - 3907 \beta_1 ) / 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 66 \beta_{13} + 330 \beta_{10} - 66 \beta_{9} - 543 \beta_{8} + 66 \beta_{7} - 330 \beta_{6} + \cdots - 66 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11534 \beta_{15} - 11534 \beta_{14} + 29085 \beta_{13} + 45298 \beta_{12} + 45298 \beta_{11} + \cdots - 22649 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8040 \beta_{13} - 40200 \beta_{10} + 8040 \beta_{9} + 8040 \beta_{7} - 40200 \beta_{6} + \cdots + 1562048 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 422318 \beta_{15} + 422318 \beta_{14} + 1713326 \beta_{12} - 1713326 \beta_{11} - 856663 \beta_{8} + \cdots + 5267763 \beta_1 ) / 15 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 310114 \beta_{13} - 1550570 \beta_{10} + 310114 \beta_{9} + 2383061 \beta_{8} - 310114 \beta_{7} + \cdots + 310114 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 15747146 \beta_{15} + 15747146 \beta_{14} - 42284165 \beta_{13} - 64628322 \beta_{12} + \cdots + 32314161 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3923412 \beta_{13} + 19617060 \beta_{10} - 3923412 \beta_{9} - 3923412 \beta_{7} + 19617060 \beta_{6} + \cdots - 727713904 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 590838782 \beta_{15} - 590838782 \beta_{14} - 2434772854 \beta_{12} + 2434772854 \beta_{11} + \cdots - 7408212867 \beta_1 ) / 15 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 444332454 \beta_{13} + 2221662270 \beta_{10} - 444332454 \beta_{9} - 3391841629 \beta_{8} + \cdots - 444332454 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 22216033874 \beta_{15} - 22216033874 \beta_{14} + 60089764485 \beta_{13} + 91680457658 \beta_{12} + \cdots - 45840228829 \beta_{2} ) / 15 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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
−4.33841 + 4.33841i
4.33841 − 4.33841i
−2.56827 + 2.56827i
2.56827 − 2.56827i
0.118783 − 0.118783i
−0.118783 + 0.118783i
1.88892 − 1.88892i
−1.88892 + 1.88892i
−4.33841 − 4.33841i
4.33841 + 4.33841i
−2.56827 − 2.56827i
2.56827 + 2.56827i
0.118783 + 0.118783i
−0.118783 − 0.118783i
1.88892 + 1.88892i
−1.88892 − 1.88892i
−7.62690 + 7.62690i −14.5935 + 5.47988i − 84.3392i 0 69.5088 − 153.098i −80.6254 − 80.6254i 399.186 + 399.186i 182.942 − 159.942i 0
32.2 −7.62690 + 7.62690i 5.47988 − 14.5935i − 84.3392i 0 69.5088 + 153.098i 80.6254 + 80.6254i 399.186 + 399.186i −182.942 − 159.942i 0
32.3 −3.29096 + 3.29096i 10.6427 + 11.3900i 10.3392i 0 −72.5088 − 2.45919i −138.604 − 138.604i −139.336 − 139.336i −16.4641 + 242.442i 0
32.4 −3.29096 + 3.29096i 11.3900 + 10.6427i 10.3392i 0 −72.5088 + 2.45919i 138.604 + 138.604i −139.336 − 139.336i 16.4641 + 242.442i 0
32.5 3.29096 − 3.29096i −11.3900 − 10.6427i 10.3392i 0 −72.5088 + 2.45919i −138.604 − 138.604i 139.336 + 139.336i 16.4641 + 242.442i 0
32.6 3.29096 − 3.29096i −10.6427 − 11.3900i 10.3392i 0 −72.5088 − 2.45919i 138.604 + 138.604i 139.336 + 139.336i −16.4641 + 242.442i 0
32.7 7.62690 − 7.62690i −5.47988 + 14.5935i − 84.3392i 0 69.5088 + 153.098i −80.6254 − 80.6254i −399.186 − 399.186i −182.942 − 159.942i 0
32.8 7.62690 − 7.62690i 14.5935 − 5.47988i − 84.3392i 0 69.5088 − 153.098i 80.6254 + 80.6254i −399.186 − 399.186i 182.942 − 159.942i 0
68.1 −7.62690 − 7.62690i −14.5935 − 5.47988i 84.3392i 0 69.5088 + 153.098i −80.6254 + 80.6254i 399.186 − 399.186i 182.942 + 159.942i 0
68.2 −7.62690 − 7.62690i 5.47988 + 14.5935i 84.3392i 0 69.5088 − 153.098i 80.6254 − 80.6254i 399.186 − 399.186i −182.942 + 159.942i 0
68.3 −3.29096 − 3.29096i 10.6427 − 11.3900i − 10.3392i 0 −72.5088 + 2.45919i −138.604 + 138.604i −139.336 + 139.336i −16.4641 − 242.442i 0
68.4 −3.29096 − 3.29096i 11.3900 − 10.6427i − 10.3392i 0 −72.5088 − 2.45919i 138.604 − 138.604i −139.336 + 139.336i 16.4641 − 242.442i 0
68.5 3.29096 + 3.29096i −11.3900 + 10.6427i − 10.3392i 0 −72.5088 − 2.45919i −138.604 + 138.604i 139.336 − 139.336i 16.4641 − 242.442i 0
68.6 3.29096 + 3.29096i −10.6427 + 11.3900i − 10.3392i 0 −72.5088 + 2.45919i 138.604 − 138.604i 139.336 − 139.336i −16.4641 − 242.442i 0
68.7 7.62690 + 7.62690i −5.47988 − 14.5935i 84.3392i 0 69.5088 − 153.098i −80.6254 + 80.6254i −399.186 + 399.186i −182.942 + 159.942i 0
68.8 7.62690 + 7.62690i 14.5935 + 5.47988i 84.3392i 0 69.5088 + 153.098i 80.6254 − 80.6254i −399.186 + 399.186i 182.942 + 159.942i 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.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.6.e.f ✓ 16
3.b odd 2 1 inner 75.6.e.f ✓ 16
5.b even 2 1 inner 75.6.e.f ✓ 16
5.c odd 4 2 inner 75.6.e.f ✓ 16
15.d odd 2 1 inner 75.6.e.f ✓ 16
15.e even 4 2 inner 75.6.e.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.e.f ✓ 16 1.a even 1 1 trivial
75.6.e.f ✓ 16 3.b odd 2 1 inner
75.6.e.f ✓ 16 5.b even 2 1 inner
75.6.e.f ✓ 16 5.c odd 4 2 inner
75.6.e.f ✓ 16 15.d odd 2 1 inner
75.6.e.f ✓ 16 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 14004T_{2}^{4} + 6350400 \) acting on \(S_{6}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 14004 T^{4} + 6350400)^{2} \) 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
$7$ \( (T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 300195 T^{2} + 21781872000)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 547322 T^{2} + 72178732921)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 8641769472000)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3991 T - 65786)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 21071 T + 77367754)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 54\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 29\!\cdots\!76)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 33\!\cdots\!00)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
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