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Properties

Label 75.11.f.d
Level $75$
Weight $11$
Character orbit 75.f
Analytic conductor $47.652$
Analytic rank $0$
Dimension $20$
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] = [75,11,Mod(7,75)]
 
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([0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.7");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(47.6517939505\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 8 x^{18} - 57344 x^{17} + 18664853 x^{16} - 274248412 x^{15} + 2591841992 x^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{34}\cdot 5^{20} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{3} + \beta_1 + 3) q^{2} + \beta_{7} q^{3} + (\beta_{11} - 2 \beta_{7} + \cdots - 451 \beta_{3}) q^{4}+ \cdots + 19683 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_{3} + \beta_1 + 3) q^{2} + \beta_{7} q^{3} + (\beta_{11} - 2 \beta_{7} + \cdots - 451 \beta_{3}) q^{4}+ \cdots + (39366 \beta_{19} + \cdots - 17498187 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 64 q^{2} - 4860 q^{6} - 10604 q^{7} + 39948 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 64 q^{2} - 4860 q^{6} - 10604 q^{7} + 39948 q^{8} + 32080 q^{11} + 620136 q^{12} + 69352 q^{13} - 3111700 q^{16} - 347360 q^{17} + 1259712 q^{18} + 1448280 q^{21} + 27255980 q^{22} + 30995704 q^{23} + 80977840 q^{26} + 104897636 q^{28} - 8846480 q^{31} + 81249676 q^{32} - 86048244 q^{33} + 177540660 q^{36} + 27635896 q^{37} - 44187744 q^{38} + 149264920 q^{41} - 442105452 q^{42} - 675552392 q^{43} - 1916100680 q^{46} + 257112832 q^{47} - 1182772368 q^{48} - 711183240 q^{51} - 1397512520 q^{52} + 152646064 q^{53} + 1735516800 q^{56} - 342507528 q^{57} + 1947576252 q^{58} + 2582791000 q^{61} + 969372632 q^{62} - 208718532 q^{63} - 1968290280 q^{66} + 6731030200 q^{67} + 12869460704 q^{68} + 7511442640 q^{71} - 786296484 q^{72} - 1660222316 q^{73} + 9998646360 q^{76} + 13264676792 q^{77} + 5574993480 q^{78} - 7748409780 q^{81} - 27089146528 q^{82} - 30753878864 q^{83} - 46532117120 q^{86} - 3048661476 q^{87} - 5813201532 q^{88} + 14175275920 q^{91} - 30045377384 q^{92} + 6062778072 q^{93} - 18513718020 q^{96} + 32149992820 q^{97} - 54432471592 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 8 x^{18} - 57344 x^{17} + 18664853 x^{16} - 274248412 x^{15} + 2591841992 x^{14} + \cdots + 11\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 50\!\cdots\!97 \nu^{19} + \cdots + 24\!\cdots\!00 ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73\!\cdots\!39 \nu^{19} + \cdots + 29\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45\!\cdots\!63 \nu^{19} + \cdots - 79\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38\!\cdots\!09 \nu^{19} + \cdots + 20\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 81\!\cdots\!59 \nu^{19} + \cdots - 10\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 86\!\cdots\!64 \nu^{19} + \cdots - 69\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!63 \nu^{19} + \cdots + 69\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!73 \nu^{19} + \cdots - 68\!\cdots\!00 ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!55 \nu^{19} + \cdots + 44\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 45\!\cdots\!69 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 98\!\cdots\!57 \nu^{19} + \cdots + 38\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!31 \nu^{19} + \cdots - 59\!\cdots\!00 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25\!\cdots\!21 \nu^{19} + \cdots - 31\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26\!\cdots\!16 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11\!\cdots\!57 \nu^{19} + \cdots + 14\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 21\!\cdots\!41 \nu^{19} + \cdots - 26\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 21\!\cdots\!76 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 50\!\cdots\!21 \nu^{19} + \cdots - 39\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 2\beta_{7} + 2\beta_{6} - 1457\beta_{3} - 6\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} + \beta_{18} + \beta_{13} - 6 \beta_{11} + \beta_{10} + \beta_{9} - 10 \beta_{8} + \cdots + 9105 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 36 \beta_{19} - 12 \beta_{18} + 24 \beta_{17} - 36 \beta_{16} + 12 \beta_{15} - 12 \beta_{14} + \cdots - 3700699 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 408 \beta_{17} + 5121 \beta_{16} + 1120 \beta_{14} - 3393 \beta_{13} + 610 \beta_{12} + \cdots + 51586013 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 205740 \beta_{19} + 91836 \beta_{18} + 113904 \beta_{17} - 205740 \beta_{16} - 77756 \beta_{15} + \cdots + 163574522 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 22379655 \beta_{19} - 18912159 \beta_{18} + 7518464 \beta_{15} - 9967511 \beta_{13} + \cdots - 246534062087 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 946221176 \beta_{19} + 486099528 \beta_{18} - 460121648 \beta_{17} + 946221176 \beta_{16} + \cdots + 40430749340987 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 19912010840 \beta_{17} - 94152995677 \beta_{16} - 36912929248 \beta_{14} + 28267996829 \beta_{13} + \cdots - 11\!\cdots\!25 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4089205695812 \beta_{19} - 2280519531444 \beta_{18} - 1808686164368 \beta_{17} + \cdots - 32\!\cdots\!66 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 391442287497431 \beta_{19} + 293075930759631 \beta_{18} - 163379738092864 \beta_{15} + \cdots + 48\!\cdots\!27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 17\!\cdots\!20 \beta_{19} + \cdots - 58\!\cdots\!55 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 45\!\cdots\!76 \beta_{17} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 72\!\cdots\!12 \beta_{19} + \cdots + 59\!\cdots\!94 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 66\!\cdots\!31 \beta_{19} + \cdots - 88\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 30\!\cdots\!56 \beta_{19} + \cdots + 92\!\cdots\!55 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 86\!\cdots\!84 \beta_{17} + \cdots - 37\!\cdots\!85 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 12\!\cdots\!48 \beta_{19} + \cdots - 10\!\cdots\!78 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 11\!\cdots\!39 \beta_{19} + \cdots + 15\!\cdots\!03 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−45.4637 − 45.4637i
−32.9381 − 32.9381i
−23.8144 − 23.8144i
−14.6866 − 14.6866i
−0.145441 − 0.145441i
9.55545 + 9.55545i
16.3738 + 16.3738i
25.0945 + 25.0945i
29.7993 + 29.7993i
38.2250 + 38.2250i
−45.4637 + 45.4637i
−32.9381 + 32.9381i
−23.8144 + 23.8144i
−14.6866 + 14.6866i
−0.145441 + 0.145441i
9.55545 − 9.55545i
16.3738 − 16.3738i
25.0945 − 25.0945i
29.7993 − 29.7993i
38.2250 − 38.2250i
−42.4637 − 42.4637i 99.2043 − 99.2043i 2582.33i 0 −8425.16 3582.02 + 3582.02i 66172.3 − 66172.3i − 19683.0i 0
7.2 −29.9381 − 29.9381i −99.2043 + 99.2043i 768.579i 0 5939.98 −15970.0 − 15970.0i −7646.81 + 7646.81i − 19683.0i 0
7.3 −20.8144 − 20.8144i 99.2043 − 99.2043i − 157.526i 0 −4129.75 6932.43 + 6932.43i −24592.7 + 24592.7i − 19683.0i 0
7.4 −11.6866 − 11.6866i −99.2043 + 99.2043i − 750.847i 0 2318.72 21416.5 + 21416.5i −20741.9 + 20741.9i − 19683.0i 0
7.5 2.85456 + 2.85456i −99.2043 + 99.2043i − 1007.70i 0 −566.369 −5862.00 − 5862.00i 5799.62 − 5799.62i − 19683.0i 0
7.6 12.5555 + 12.5555i 99.2043 − 99.2043i − 708.721i 0 2491.11 8580.16 + 8580.16i 21755.1 − 21755.1i − 19683.0i 0
7.7 19.3738 + 19.3738i 99.2043 − 99.2043i − 273.309i 0 3843.94 −3127.99 − 3127.99i 25133.9 − 25133.9i − 19683.0i 0
7.8 28.0945 + 28.0945i −99.2043 + 99.2043i 554.607i 0 −5574.20 −20119.5 − 20119.5i 13187.4 − 13187.4i − 19683.0i 0
7.9 32.7993 + 32.7993i −99.2043 + 99.2043i 1127.59i 0 −6507.67 16059.1 + 16059.1i −3397.64 + 3397.64i − 19683.0i 0
7.10 41.2250 + 41.2250i 99.2043 − 99.2043i 2375.00i 0 8179.40 −16792.7 − 16792.7i −55695.2 + 55695.2i − 19683.0i 0
43.1 −42.4637 + 42.4637i 99.2043 + 99.2043i − 2582.33i 0 −8425.16 3582.02 − 3582.02i 66172.3 + 66172.3i 19683.0i 0
43.2 −29.9381 + 29.9381i −99.2043 − 99.2043i − 768.579i 0 5939.98 −15970.0 + 15970.0i −7646.81 − 7646.81i 19683.0i 0
43.3 −20.8144 + 20.8144i 99.2043 + 99.2043i 157.526i 0 −4129.75 6932.43 − 6932.43i −24592.7 − 24592.7i 19683.0i 0
43.4 −11.6866 + 11.6866i −99.2043 − 99.2043i 750.847i 0 2318.72 21416.5 − 21416.5i −20741.9 − 20741.9i 19683.0i 0
43.5 2.85456 − 2.85456i −99.2043 − 99.2043i 1007.70i 0 −566.369 −5862.00 + 5862.00i 5799.62 + 5799.62i 19683.0i 0
43.6 12.5555 − 12.5555i 99.2043 + 99.2043i 708.721i 0 2491.11 8580.16 − 8580.16i 21755.1 + 21755.1i 19683.0i 0
43.7 19.3738 − 19.3738i 99.2043 + 99.2043i 273.309i 0 3843.94 −3127.99 + 3127.99i 25133.9 + 25133.9i 19683.0i 0
43.8 28.0945 − 28.0945i −99.2043 − 99.2043i − 554.607i 0 −5574.20 −20119.5 + 20119.5i 13187.4 + 13187.4i 19683.0i 0
43.9 32.7993 − 32.7993i −99.2043 − 99.2043i − 1127.59i 0 −6507.67 16059.1 − 16059.1i −3397.64 − 3397.64i 19683.0i 0
43.10 41.2250 − 41.2250i 99.2043 + 99.2043i − 2375.00i 0 8179.40 −16792.7 + 16792.7i −55695.2 − 55695.2i 19683.0i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
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 75.11.f.d 20
5.b even 2 1 15.11.f.a ✓ 20
5.c odd 4 1 15.11.f.a ✓ 20
5.c odd 4 1 inner 75.11.f.d 20
15.d odd 2 1 45.11.g.c 20
15.e even 4 1 45.11.g.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.11.f.a ✓ 20 5.b even 2 1
15.11.f.a ✓ 20 5.c odd 4 1
45.11.g.c 20 15.d odd 2 1
45.11.g.c 20 15.e even 4 1
75.11.f.d 20 1.a even 1 1 trivial
75.11.f.d 20 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 64 T_{2}^{19} + 2048 T_{2}^{18} - 13316 T_{2}^{17} + 16946117 T_{2}^{16} + \cdots + 68\!\cdots\!76 \) acting on \(S_{11}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( (T^{4} + 387420489)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 18\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 71\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 79\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
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