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Properties

Label 75.9.c.h
Level $75$
Weight $9$
Character orbit 75.c
Analytic conductor $30.553$
Analytic rank $0$
Dimension $12$
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] = [75,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(30.5533957546\)
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: \( x^{12} + 64 x^{10} + 385774 x^{8} - 323639784 x^{6} - 48708595080 x^{4} + 21531002169600 x^{2} + 82\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{18}\cdot 5^{8} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{3} + \beta_{2}) q^{3} + ( - \beta_{5} - 142) q^{4} + (\beta_{7} + \beta_{5} - 251) q^{6} + ( - \beta_{6} - 2 \beta_{4} + \cdots - 8 \beta_{2}) q^{7}+ \cdots + (\beta_{10} + \beta_{9} + 5 \beta_{5} + 1902) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{3} + \beta_{2}) q^{3} + ( - \beta_{5} - 142) q^{4} + (\beta_{7} + \beta_{5} - 251) q^{6} + ( - \beta_{6} - 2 \beta_{4} + \cdots - 8 \beta_{2}) q^{7}+ \cdots + ( - 1377 \beta_{10} - 6372 \beta_{9} + \cdots + 53363340) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 1704 q^{4} - 3012 q^{6} + 22824 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 1704 q^{4} - 3012 q^{6} + 22824 q^{9} + 413328 q^{16} - 276192 q^{19} - 604044 q^{21} - 1173336 q^{24} - 279216 q^{31} + 1225344 q^{34} - 10311840 q^{36} - 3780864 q^{39} + 37414536 q^{46} + 6222300 q^{49} + 3931248 q^{51} + 53281692 q^{54} - 91958256 q^{61} - 57497760 q^{64} + 111065040 q^{66} - 8138748 q^{69} - 232646880 q^{76} - 420402672 q^{79} + 98480772 q^{81} + 528357816 q^{84} - 100211328 q^{91} - 543022776 q^{94} + 333306864 q^{96} + 640360080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 64 x^{10} + 385774 x^{8} - 323639784 x^{6} - 48708595080 x^{4} + 21531002169600 x^{2} + 82\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!62 \nu^{11} + \cdots - 11\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!62 \nu^{11} + \cdots + 42\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!71 \nu^{10} + \cdots + 51\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!62 \nu^{11} + \cdots - 82\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77790690643 \nu^{10} + 10812365407831 \nu^{8} + \cdots + 37\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!81 \nu^{11} + \cdots + 26\!\cdots\!00 \nu ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 68\!\cdots\!65 \nu^{11} + \cdots - 31\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 24\!\cdots\!65 \nu^{11} + \cdots - 93\!\cdots\!00 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!77 \nu^{11} + \cdots + 19\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 68\!\cdots\!65 \nu^{11} + \cdots - 20\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!14 \nu^{11} + \cdots - 42\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 5 \beta_{11} + 15 \beta_{10} + 30 \beta_{9} - 18 \beta_{8} - 87 \beta_{7} + 15 \beta_{6} + \cdots + 1295 \beta_{2} ) / 10800 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 375 \beta_{10} - 375 \beta_{7} - 825 \beta_{5} - 4634 \beta_{4} + 41390 \beta_{3} + 8290 \beta_{2} + \cdots - 57600 ) / 5400 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 425 \beta_{11} + 630 \beta_{10} + 4680 \beta_{9} - 5382 \beta_{8} - 2448 \beta_{7} + \cdots - 465395 \beta_{2} ) / 5400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 15633 \beta_{10} - 15633 \beta_{7} + 153072 \beta_{5} + 158501 \beta_{4} - 446870 \beta_{3} + \cdots - 34535340 ) / 270 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 147565 \beta_{11} - 1527855 \beta_{10} - 14782710 \beta_{9} + 8703846 \beta_{8} + \cdots + 36091735 \beta_{2} ) / 5400 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 67371825 \beta_{10} + 67371825 \beta_{7} - 1284425025 \beta_{5} + 1244659958 \beta_{4} + \cdots + 470126617200 ) / 2700 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 596001125 \beta_{11} + 353551890 \beta_{10} + 1276191240 \beta_{9} - 1214393706 \beta_{8} + \cdots + 170853162215 \beta_{2} ) / 2700 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1353345546 \beta_{10} + 1353345546 \beta_{7} - 26408339889 \beta_{5} - 61827855779 \beta_{4} + \cdots + 9074745219780 ) / 135 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 220686746905 \beta_{11} + 641229229935 \beta_{10} + 4068490340070 \beta_{9} + \cdots - 73553853374395 \beta_{2} ) / 2700 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 39432674685825 \beta_{10} - 39432674685825 \beta_{7} + 553741516895625 \beta_{5} + \cdots - 17\!\cdots\!00 ) / 1350 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 122551381231175 \beta_{11} - 198410905505580 \beta_{10} + \cdots - 39\!\cdots\!05 \beta_{2} ) / 1350 \) 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
−4.76228 − 15.2142i
4.76228 + 15.2142i
−20.6227 − 2.01276i
20.6227 + 2.01276i
−17.1763 − 23.2266i
17.1763 + 23.2266i
−17.1763 + 23.2266i
17.1763 − 23.2266i
−20.6227 + 2.01276i
20.6227 − 2.01276i
−4.76228 + 15.2142i
4.76228 − 15.2142i
− 29.2008i −80.0515 + 12.3596i −596.686 0 360.909 + 2337.57i 2074.09 9948.30i 6255.48 − 1978.80i 0
26.2 − 29.2008i 80.0515 + 12.3596i −596.686 0 360.909 − 2337.57i −2074.09 9948.30i 6255.48 + 1978.80i 0
26.3 − 17.5844i −31.2287 − 74.7380i −53.2123 0 −1314.22 + 549.140i 3426.92 − 3565.91i −4610.53 + 4667.95i 0
26.4 − 17.5844i 31.2287 − 74.7380i −53.2123 0 −1314.22 − 549.140i −3426.92 − 3565.91i −4610.53 − 4667.95i 0
26.5 − 5.66585i −72.8768 + 35.3550i 223.898 0 200.316 + 412.909i −1674.62 − 2719.03i 4061.05 − 5153.12i 0
26.6 − 5.66585i 72.8768 + 35.3550i 223.898 0 200.316 − 412.909i 1674.62 − 2719.03i 4061.05 + 5153.12i 0
26.7 5.66585i −72.8768 − 35.3550i 223.898 0 200.316 − 412.909i −1674.62 2719.03i 4061.05 + 5153.12i 0
26.8 5.66585i 72.8768 − 35.3550i 223.898 0 200.316 + 412.909i 1674.62 2719.03i 4061.05 − 5153.12i 0
26.9 17.5844i −31.2287 + 74.7380i −53.2123 0 −1314.22 − 549.140i 3426.92 3565.91i −4610.53 − 4667.95i 0
26.10 17.5844i 31.2287 + 74.7380i −53.2123 0 −1314.22 + 549.140i −3426.92 3565.91i −4610.53 + 4667.95i 0
26.11 29.2008i −80.0515 − 12.3596i −596.686 0 360.909 − 2337.57i 2074.09 − 9948.30i 6255.48 + 1978.80i 0
26.12 29.2008i 80.0515 − 12.3596i −596.686 0 360.909 + 2337.57i −2074.09 − 9948.30i 6255.48 − 1978.80i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.12
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 75.9.c.h 12
3.b odd 2 1 inner 75.9.c.h 12
5.b even 2 1 inner 75.9.c.h 12
5.c odd 4 2 15.9.d.c ✓ 12
15.d odd 2 1 inner 75.9.c.h 12
15.e even 4 2 15.9.d.c ✓ 12
20.e even 4 2 240.9.c.c 12
60.l odd 4 2 240.9.c.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.d.c ✓ 12 5.c odd 4 2
15.9.d.c ✓ 12 15.e even 4 2
75.9.c.h 12 1.a even 1 1 trivial
75.9.c.h 12 3.b odd 2 1 inner
75.9.c.h 12 5.b even 2 1 inner
75.9.c.h 12 15.d odd 2 1 inner
240.9.c.c 12 20.e even 4 2
240.9.c.c 12 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 \(S_{9}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{6} + 1194T_{2}^{4} + 300960T_{2}^{2} + 8464000 \) Copy content Toggle raw display
\( T_{7}^{6} - 18849978T_{7}^{4} + 95517523696896T_{7}^{2} - 141675830049600000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 1194 T^{4} + \cdots + 8464000)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 79\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots + 466181174169488)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 37\!\cdots\!32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 34\!\cdots\!28)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 37\!\cdots\!68)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
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