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Properties

Label 75.7.c.e
Level $75$
Weight $7$
Character orbit 75.c
Analytic conductor $17.254$
Analytic rank $0$
Dimension $8$
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] = [75,7,Mod(26,75)]
 
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([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), 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: \(17.2540562715\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 216x^{6} + 10030x^{4} + 48600x^{2} + 50625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - 9) q^{4} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 77) q^{6} + (\beta_{6} - 3 \beta_{4} + 6 \beta_{2}) q^{7} + (8 \beta_{4} + 10 \beta_1) q^{8} + ( - 3 \beta_{7} + 3 \beta_{3} + 246) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - 9) q^{4} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 77) q^{6} + (\beta_{6} - 3 \beta_{4} + 6 \beta_{2}) q^{7} + (8 \beta_{4} + 10 \beta_1) q^{8} + ( - 3 \beta_{7} + 3 \beta_{3} + 246) q^{9} + ( - 6 \beta_{7} + \beta_{5} - 3 \beta_{3} - 3) q^{11} + ( - 3 \beta_{6} + 51 \beta_{4} + \cdots + 45 \beta_1) q^{12}+ \cdots + ( - 1134 \beta_{7} - 1845 \beta_{5} + \cdots - 790857) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{4} - 612 q^{6} + 1980 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{4} - 612 q^{6} + 1980 q^{9} + 784 q^{16} + 28520 q^{19} + 24228 q^{21} - 14904 q^{24} + 5896 q^{31} - 105344 q^{34} - 186624 q^{36} - 233568 q^{39} + 59336 q^{46} + 1041472 q^{49} + 66888 q^{51} - 917892 q^{54} + 1493416 q^{61} + 2063584 q^{64} - 1221840 q^{66} - 1502532 q^{69} - 219168 q^{76} + 1286360 q^{79} - 1752192 q^{81} - 6238728 q^{84} + 383616 q^{91} + 7641736 q^{94} - 151056 q^{96} - 6322320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 216x^{6} + 10030x^{4} + 48600x^{2} + 50625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{7} - 2331\nu^{5} - 101285\nu^{3} - 191925\nu ) / 27000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\nu^{7} - 27\nu^{6} + 7515\nu^{5} - 5427\nu^{4} + 342005\nu^{3} - 189405\nu^{2} + 1277325\nu + 718875 ) / 54000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 216\nu^{4} - 9805\nu^{2} - 24300 ) / 150 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{7} + 1503\nu^{5} + 68401\nu^{3} + 255465\nu ) / 5400 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 216\nu^{5} + 10030\nu^{3} + 51975\nu ) / 225 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -37\nu^{6} - 7812\nu^{4} - 340555\nu^{2} - 895500 ) / 750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -25\nu^{7} + 6\nu^{6} - 5355\nu^{5} + 1296\nu^{4} - 240355\nu^{3} + 58830\nu^{2} - 702225\nu + 144900 ) / 1800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 5\beta_{4} - 5\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - 16\beta_{4} + 5\beta_{3} + 32\beta_{2} - 810 ) / 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 40\beta_{7} - 91\beta_{5} + 805\beta_{4} + 20\beta_{3} + 205\beta _1 + 20 ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 186\beta_{6} + 1976\beta_{4} - 1080\beta_{3} - 3952\beta_{2} + 99735 ) / 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8040\beta_{7} + 11501\beta_{5} - 119605\beta_{4} - 4020\beta_{3} + 18995\beta _1 - 4020 ) / 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -30371\beta_{6} - 269936\beta_{4} + 182005\beta_{3} + 539872\beta_{2} - 13965210 ) / 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1335440\beta_{7} - 1616711\beta_{5} + 18020405\beta_{4} + 667720\beta_{3} - 5899195\beta _1 + 667720 ) / 30 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 7.64874i
1.96111i
12.3358i
− 1.21597i
− 12.3358i
1.21597i
7.64874i
− 1.96111i
− 11.8944i −25.3441 − 9.31012i −77.4763 0 −110.738 + 301.452i −553.145 160.292i 555.643 + 471.913i 0
26.2 − 11.8944i 25.3441 − 9.31012i −77.4763 0 −110.738 − 301.452i 553.145 160.292i 555.643 − 471.913i 0
26.3 − 2.12691i −18.2805 − 19.8701i 59.4763 0 −42.2619 + 38.8810i 435.542 − 262.622i −60.6432 + 726.473i 0
26.4 − 2.12691i 18.2805 − 19.8701i 59.4763 0 −42.2619 − 38.8810i −435.542 − 262.622i −60.6432 − 726.473i 0
26.5 2.12691i −18.2805 + 19.8701i 59.4763 0 −42.2619 − 38.8810i 435.542 262.622i −60.6432 − 726.473i 0
26.6 2.12691i 18.2805 + 19.8701i 59.4763 0 −42.2619 + 38.8810i −435.542 262.622i −60.6432 + 726.473i 0
26.7 11.8944i −25.3441 + 9.31012i −77.4763 0 −110.738 − 301.452i −553.145 − 160.292i 555.643 − 471.913i 0
26.8 11.8944i 25.3441 + 9.31012i −77.4763 0 −110.738 + 301.452i 553.145 − 160.292i 555.643 + 471.913i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.8
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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.7.c.e 8
3.b odd 2 1 inner 75.7.c.e 8
5.b even 2 1 inner 75.7.c.e 8
5.c odd 4 2 15.7.d.c ✓ 8
15.d odd 2 1 inner 75.7.c.e 8
15.e even 4 2 15.7.d.c ✓ 8
20.e even 4 2 240.7.c.c 8
60.l odd 4 2 240.7.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.7.d.c ✓ 8 5.c odd 4 2
15.7.d.c ✓ 8 15.e even 4 2
75.7.c.e 8 1.a even 1 1 trivial
75.7.c.e 8 3.b odd 2 1 inner
75.7.c.e 8 5.b even 2 1 inner
75.7.c.e 8 15.d odd 2 1 inner
240.7.c.c 8 20.e even 4 2
240.7.c.c 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{4} + 146T_{2}^{2} + 640 \) Copy content Toggle raw display
\( T_{7}^{4} - 495666T_{7}^{2} + 58041360000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 146 T^{2} + 640)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 495666 T^{2} + 58041360000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 1754154144000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 3379669401600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 28015260981760)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 7130 T + 12704536)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 1474 T - 159937856)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 56\!\cdots\!60)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 373354 T + 19740227104)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 321590 T - 331589130704)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 80\!\cdots\!60)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
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