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Properties

Label 75.9.d.d
Level $75$
Weight $9$
Character orbit 75.d
Analytic conductor $30.553$
Analytic rank $0$
Dimension $20$
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(74,75)]
 
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, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.74");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.d (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: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 3278 x^{18} + 4245491 x^{16} + 2854629536 x^{14} + 1117319469691 x^{12} + 266849787342054 x^{10} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{22}\cdot 5^{26} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{2} + 155) q^{4} + ( - \beta_{10} - \beta_{2} + 225) q^{6} + ( - \beta_{15} - \beta_{7} + \cdots + 5 \beta_1) q^{7}+ \cdots + ( - \beta_{11} - \beta_{9} + \cdots + 1120) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{2} + 155) q^{4} + ( - \beta_{10} - \beta_{2} + 225) q^{6} + ( - \beta_{15} - \beta_{7} + \cdots + 5 \beta_1) q^{7}+ \cdots + ( - 3363 \beta_{19} - 4140 \beta_{18} + \cdots + 38305952) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9} + 89268 q^{16} + 287868 q^{19} + 1346856 q^{21} + 2033718 q^{24} - 6028120 q^{31} - 9954292 q^{34} + 9001054 q^{36} + 15026564 q^{39} - 27877272 q^{46} - 17907092 q^{49} + 12418574 q^{51} + 17772544 q^{54} + 3040440 q^{61} + 9073996 q^{64} + 20930590 q^{66} - 22789956 q^{69} - 59058092 q^{76} - 17099792 q^{79} + 45225890 q^{81} + 273766584 q^{84} - 214448528 q^{91} - 712237192 q^{94} + 1050848002 q^{96} + 764842670 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 3278 x^{18} + 4245491 x^{16} + 2854629536 x^{14} + 1117319469691 x^{12} + 266849787342054 x^{10} + \cdots + 89\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!27 \nu^{18} + \cdots - 49\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 38\!\cdots\!71 \nu^{18} + \cdots - 18\!\cdots\!00 ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\!\cdots\!80 \nu^{19} + \cdots + 79\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!60 \nu^{19} + \cdots + 68\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 89\!\cdots\!55 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!60 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!25 \nu^{19} + \cdots + 53\!\cdots\!00 \nu ) / 11\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51\!\cdots\!69 \nu^{19} + \cdots - 27\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!77 \nu^{19} + \cdots - 11\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!77 \nu^{19} + \cdots + 20\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 36\!\cdots\!11 \nu^{19} + \cdots - 17\!\cdots\!40 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33\!\cdots\!49 \nu^{19} + \cdots - 15\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 63\!\cdots\!40 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 30\!\cdots\!93 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\!\cdots\!00 \nu^{19} + \cdots - 26\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 65\!\cdots\!00 \nu^{19} + \cdots - 68\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 67\!\cdots\!00 \nu^{19} + \cdots - 79\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 60\!\cdots\!67 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10\!\cdots\!97 \nu^{19} + \cdots + 58\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 567 \beta_{19} - 225 \beta_{18} + 350 \beta_{14} - 333 \beta_{12} + 4379 \beta_{11} + \cdots - 2556 ) / 540000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{17} + 81 \beta_{16} + 3 \beta_{15} - 328 \beta_{14} - 63 \beta_{13} + 744 \beta_{12} + \cdots - 3542616 ) / 10800 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4836 \beta_{19} + 600 \beta_{18} - 425 \beta_{17} - 1025 \beta_{16} + 2850 \beta_{15} + 5400 \beta_{14} + \cdots + 56823 ) / 11250 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4391 \beta_{17} - 17775 \beta_{16} - 2085 \beta_{15} + 107482 \beta_{14} + 15345 \beta_{13} + \cdots + 609098094 ) / 2700 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8914932 \beta_{19} + 900000 \beta_{18} + 3601125 \beta_{17} + 4882125 \beta_{16} + \cdots - 168477201 ) / 33750 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1281988 \beta_{17} + 3080106 \beta_{16} + 568248 \beta_{15} - 18808457 \beta_{14} + \cdots - 91719638019 ) / 450 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7503510483 \beta_{19} - 1585725975 \beta_{18} - 5311332775 \beta_{17} - 6527312575 \beta_{16} + \cdots + 168151998069 ) / 33750 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5242790008 \beta_{17} - 11012344800 \beta_{16} - 2324381880 \beta_{15} + 57191094341 \beta_{14} + \cdots + 269394452114967 ) / 1350 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2389744860369 \beta_{19} + 590916637725 \beta_{18} + 2276324457375 \beta_{17} + \cdots - 56286920663067 ) / 11250 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6557476905556 \beta_{17} + 13334583120354 \beta_{16} + 2916098799672 \beta_{15} + \cdots - 26\!\cdots\!77 ) / 1350 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 71\!\cdots\!83 \beta_{19} + \cdots + 17\!\cdots\!69 ) / 33750 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 26\!\cdots\!88 \beta_{17} + \cdots + 90\!\cdots\!49 ) / 450 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 71\!\cdots\!07 \beta_{19} + \cdots - 17\!\cdots\!01 ) / 33750 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 91\!\cdots\!96 \beta_{17} + \cdots - 27\!\cdots\!17 ) / 1350 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 24\!\cdots\!61 \beta_{19} + \cdots + 58\!\cdots\!23 ) / 11250 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 10\!\cdots\!80 \beta_{17} + \cdots + 28\!\cdots\!47 ) / 1350 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 74\!\cdots\!07 \beta_{19} + \cdots - 17\!\cdots\!01 ) / 33750 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 39\!\cdots\!08 \beta_{17} + \cdots - 95\!\cdots\!79 ) / 450 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 76\!\cdots\!83 \beta_{19} + \cdots + 18\!\cdots\!69 ) / 33750 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
2.43232i
− 2.43232i
− 12.5890i
12.5890i
32.4704i
− 32.4704i
− 14.7835i
14.7835i
− 17.0371i
17.0371i
15.0371i
− 15.0371i
12.7835i
− 12.7835i
− 30.4704i
30.4704i
14.5890i
− 14.5890i
− 4.43232i
4.43232i
−29.4751 −75.3446 − 29.7353i 612.784 0 2220.79 + 876.451i 3164.62i −10516.3 4792.63 + 4480.79i 0
74.2 −29.4751 −75.3446 + 29.7353i 612.784 0 2220.79 − 876.451i − 3164.62i −10516.3 4792.63 − 4480.79i 0
74.3 −24.1370 70.6742 − 39.5746i 326.594 0 −1705.86 + 955.212i − 1042.22i −1703.92 3428.70 − 5593.81i 0
74.4 −24.1370 70.6742 + 39.5746i 326.594 0 −1705.86 − 955.212i 1042.22i −1703.92 3428.70 + 5593.81i 0
74.5 −18.2182 −40.7889 − 69.9805i 75.9021 0 743.099 + 1274.92i 4676.68i 3281.06 −3233.54 + 5708.85i 0
74.6 −18.2182 −40.7889 + 69.9805i 75.9021 0 743.099 − 1274.92i − 4676.68i 3281.06 −3233.54 − 5708.85i 0
74.7 −16.2639 22.8278 − 77.7167i 8.51375 0 −371.269 + 1263.98i − 569.895i 4025.09 −5518.78 − 3548.20i 0
74.8 −16.2639 22.8278 + 77.7167i 8.51375 0 −371.269 − 1263.98i 569.895i 4025.09 −5518.78 + 3548.20i 0
74.9 −3.03414 −79.6728 − 14.6031i −246.794 0 241.739 + 44.3080i 59.7348i 1525.55 6134.50 + 2326.94i 0
74.10 −3.03414 −79.6728 + 14.6031i −246.794 0 241.739 − 44.3080i − 59.7348i 1525.55 6134.50 − 2326.94i 0
74.11 3.03414 79.6728 − 14.6031i −246.794 0 241.739 − 44.3080i 59.7348i −1525.55 6134.50 − 2326.94i 0
74.12 3.03414 79.6728 + 14.6031i −246.794 0 241.739 + 44.3080i − 59.7348i −1525.55 6134.50 + 2326.94i 0
74.13 16.2639 −22.8278 − 77.7167i 8.51375 0 −371.269 − 1263.98i − 569.895i −4025.09 −5518.78 + 3548.20i 0
74.14 16.2639 −22.8278 + 77.7167i 8.51375 0 −371.269 + 1263.98i 569.895i −4025.09 −5518.78 − 3548.20i 0
74.15 18.2182 40.7889 − 69.9805i 75.9021 0 743.099 − 1274.92i 4676.68i −3281.06 −3233.54 − 5708.85i 0
74.16 18.2182 40.7889 + 69.9805i 75.9021 0 743.099 + 1274.92i − 4676.68i −3281.06 −3233.54 + 5708.85i 0
74.17 24.1370 −70.6742 − 39.5746i 326.594 0 −1705.86 − 955.212i − 1042.22i 1703.92 3428.70 + 5593.81i 0
74.18 24.1370 −70.6742 + 39.5746i 326.594 0 −1705.86 + 955.212i 1042.22i 1703.92 3428.70 − 5593.81i 0
74.19 29.4751 75.3446 − 29.7353i 612.784 0 2220.79 − 876.451i 3164.62i 10516.3 4792.63 − 4480.79i 0
74.20 29.4751 75.3446 + 29.7353i 612.784 0 2220.79 + 876.451i − 3164.62i 10516.3 4792.63 + 4480.79i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.20
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.d.d 20
3.b odd 2 1 inner 75.9.d.d 20
5.b even 2 1 inner 75.9.d.d 20
5.c odd 4 1 75.9.c.e ✓ 10
5.c odd 4 1 75.9.c.f yes 10
15.d odd 2 1 inner 75.9.d.d 20
15.e even 4 1 75.9.c.e ✓ 10
15.e even 4 1 75.9.c.f yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.c.e ✓ 10 5.c odd 4 1
75.9.c.e ✓ 10 15.e even 4 1
75.9.c.f yes 10 5.c odd 4 1
75.9.c.f yes 10 15.e even 4 1
75.9.d.d 20 1.a even 1 1 trivial
75.9.d.d 20 3.b odd 2 1 inner
75.9.d.d 20 5.b even 2 1 inner
75.9.d.d 20 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 2057T_{2}^{8} + 1478418T_{2}^{6} - 442732064T_{2}^{4} + 48388216960T_{2}^{2} - 409079808000 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + \cdots - 409079808000)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 52\!\cdots\!67)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 96\!\cdots\!08)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 41\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 23\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 23\!\cdots\!52)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
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