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Properties

Label 75.16.b.h
Level $75$
Weight $16$
Character orbit 75.b
Analytic conductor $107.020$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,16,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not 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: \(107.020128825\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} + 499268 x^{9} + 2957430986 x^{8} + 50610157076 x^{7} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{8}\cdot 5^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 39 \beta_1) q^{2} - 2187 \beta_1 q^{3} + (\beta_{4} + 64 \beta_{3} - 15519) q^{4} + (2187 \beta_{3} - 85293) q^{6} + (\beta_{8} - 19 \beta_{5} + \cdots + 431710 \beta_1) q^{7}+ \cdots - 4782969 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 39 \beta_1) q^{2} - 2187 \beta_1 q^{3} + (\beta_{4} + 64 \beta_{3} - 15519) q^{4} + (2187 \beta_{3} - 85293) q^{6} + (\beta_{8} - 19 \beta_{5} + \cdots + 431710 \beta_1) q^{7}+ \cdots + ( - 23914845 \beta_{11} + \cdots - 85686162623712) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 186224 q^{4} - 1023516 q^{6} - 57395628 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 186224 q^{4} - 1023516 q^{6} - 57395628 q^{9} + 214978248 q^{11} + 1127968884 q^{14} + 7245659120 q^{16} + 3204681884 q^{19} + 11329631028 q^{21} + 61290185112 q^{24} - 51431789556 q^{26} - 253788937992 q^{29} + 303521683292 q^{31} - 2863839213368 q^{34} + 890703619056 q^{36} - 480622494564 q^{39} + 2182563425232 q^{41} - 2826689156352 q^{44} - 10961724740088 q^{46} - 39047692107160 q^{49} - 15628489539624 q^{51} + 4895445299004 q^{54} - 150504551659080 q^{56} - 17593208910504 q^{59} - 13919330811500 q^{61} - 100608483700416 q^{64} - 17603867266488 q^{66} - 28675151623656 q^{69} + 209268325435824 q^{71} + 90010555935624 q^{74} + 152377077052656 q^{76} - 371028048732320 q^{79} + 274521509459532 q^{81} - 11\!\cdots\!04 q^{84}+ \cdots - 10\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} + 499268 x^{9} + 2957430986 x^{8} + 50610157076 x^{7} + \cdots + 60\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13\!\cdots\!56 \nu^{11} + \cdots - 31\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!53 \nu^{11} + \cdots + 13\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!23 \nu^{11} + \cdots - 38\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30\!\cdots\!83 \nu^{11} + \cdots + 69\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!24 \nu^{11} + \cdots - 46\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 60\!\cdots\!09 \nu^{11} + \cdots + 45\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33\!\cdots\!43 \nu^{11} + \cdots + 52\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!09 \nu^{11} + \cdots + 52\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43\!\cdots\!19 \nu^{11} + \cdots - 45\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24\!\cdots\!43 \nu^{11} + \cdots - 10\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!37 \nu^{11} + \cdots + 94\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{3} + 14\beta_{2} + 46765\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - 74 \beta_{5} + 68 \beta_{4} - 84389 \beta_{3} + \cdots - 499327 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18 \beta_{11} + 77 \beta_{9} - 2 \beta_{8} - 15 \beta_{7} - 2 \beta_{6} - 142 \beta_{5} + \cdots - 1973637198 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2212 \beta_{11} + 2392 \beta_{10} - 70881 \beta_{9} + 71651 \beta_{8} - 61545 \beta_{7} + \cdots - 113941568127 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 6906 \beta_{11} - 1262538 \beta_{10} - 213808 \beta_{9} - 5081057 \beta_{8} - 184420 \beta_{7} + \cdots - 312225139677 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 158783272 \beta_{11} + 141106480 \beta_{10} + 4005038875 \beta_{9} + 3933898687 \beta_{8} + \cdots + 80\!\cdots\!05 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 70179186570 \beta_{11} + 599843960 \beta_{10} - 249349931455 \beta_{9} + 15879870684 \beta_{8} + \cdots + 50\!\cdots\!52 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6505523059476 \beta_{11} - 7768960542264 \beta_{10} + 204192262857023 \beta_{9} - 208681415318405 \beta_{8} + \cdots + 65\!\cdots\!65 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 35695206857118 \beta_{11} + \cdots + 30\!\cdots\!21 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 33\!\cdots\!16 \beta_{11} + \cdots - 36\!\cdots\!75 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−153.105 − 154.105i
168.382 − 167.382i
−100.114 − 101.114i
−32.6844 − 33.6844i
61.3960 − 60.3960i
59.1247 − 58.1247i
59.1247 + 58.1247i
61.3960 + 60.3960i
−32.6844 + 33.6844i
−100.114 + 101.114i
168.382 + 167.382i
−153.105 + 154.105i
− 346.210i − 2187.00i −87093.0 0 −757160. 4.02969e6i 1.88078e7i −4.78297e6 0
49.2 − 296.765i 2187.00i −55301.4 0 649025. − 491831.i 6.68712e6i −4.78297e6 0
49.3 − 240.228i − 2187.00i −24941.5 0 −525379. − 3.11015e6i − 1.88015e6i −4.78297e6 0
49.4 − 105.369i − 2187.00i 21665.4 0 −230442. 764473.i − 5.73558e6i −4.78297e6 0
49.5 − 82.7921i 2187.00i 25913.5 0 181066. 3.04747e6i − 4.85836e6i −4.78297e6 0
49.6 − 78.2494i 2187.00i 26645.0 0 171131. − 3.46184e6i − 4.64903e6i −4.78297e6 0
49.7 78.2494i − 2187.00i 26645.0 0 171131. 3.46184e6i 4.64903e6i −4.78297e6 0
49.8 82.7921i − 2187.00i 25913.5 0 181066. − 3.04747e6i 4.85836e6i −4.78297e6 0
49.9 105.369i 2187.00i 21665.4 0 −230442. − 764473.i 5.73558e6i −4.78297e6 0
49.10 240.228i 2187.00i −24941.5 0 −525379. 3.11015e6i 1.88015e6i −4.78297e6 0
49.11 296.765i − 2187.00i −55301.4 0 649025. 491831.i − 6.68712e6i −4.78297e6 0
49.12 346.210i 2187.00i −87093.0 0 −757160. − 4.02969e6i − 1.88078e7i −4.78297e6 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.16.b.h 12
5.b even 2 1 inner 75.16.b.h 12
5.c odd 4 1 75.16.a.i ✓ 6
5.c odd 4 1 75.16.a.j yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.16.a.i ✓ 6 5.c odd 4 1
75.16.a.j yes 6 5.c odd 4 1
75.16.b.h 12 1.a even 1 1 trivial
75.16.b.h 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 289720 T_{2}^{10} + 29138332452 T_{2}^{8} + \cdots + 28\!\cdots\!04 \) acting on \(S_{16}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( (T^{2} + 4782969)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 41\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 53\!\cdots\!89 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 85\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 24\!\cdots\!75)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 40\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 91\!\cdots\!81)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 65\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 27\!\cdots\!81 \) Copy content Toggle raw display
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