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Properties

Label 75.9.d.c
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: \( x^{20} + 943 x^{18} + 318815 x^{16} + 48938090 x^{14} + 3842259173 x^{12} + 159675554657 x^{10} + \cdots + 336685801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{22}\cdot 5^{32} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} - 2 \beta_1) q^{3} + ( - \beta_{2} + 79) q^{4} + ( - \beta_{16} + 2 \beta_{2} - 529) q^{6} + (\beta_{17} + \beta_{15} + \cdots + 3 \beta_1) q^{7}+ \cdots + (\beta_{18} + 2 \beta_{16} + \cdots - 392) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} - 2 \beta_1) q^{3} + ( - \beta_{2} + 79) q^{4} + ( - \beta_{16} + 2 \beta_{2} - 529) q^{6} + (\beta_{17} + \beta_{15} + \cdots + 3 \beta_1) q^{7}+ \cdots + (11750 \beta_{18} + 25429 \beta_{16} + \cdots + 3356693) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9} + 560772 q^{16} + 463032 q^{19} + 579144 q^{21} - 2272668 q^{24} + 1763240 q^{31} + 2222552 q^{34} - 1337324 q^{36} - 3653584 q^{39} - 50849208 q^{46} - 18708428 q^{49} - 55465384 q^{51} + 15959596 q^{54} + 44834040 q^{61} + 45870004 q^{64} - 54839600 q^{66} - 67125264 q^{69} + 397844872 q^{76} - 324621848 q^{79} - 187150780 q^{81} + 394693536 q^{84} + 888576928 q^{91} + 184100072 q^{94} - 721614812 q^{96} + 67930400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 943 x^{18} + 318815 x^{16} + 48938090 x^{14} + 3842259173 x^{12} + 159675554657 x^{10} + \cdots + 336685801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 47\!\cdots\!45 \nu^{18} + \cdots - 65\!\cdots\!87 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!80 \nu^{18} + \cdots - 30\!\cdots\!43 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28\!\cdots\!88 \nu^{19} + \cdots - 38\!\cdots\!39 \nu ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 37\!\cdots\!62 \nu^{19} + \cdots + 37\!\cdots\!07 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80\!\cdots\!82 \nu^{19} + \cdots + 15\!\cdots\!41 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!48 \nu^{19} + \cdots + 21\!\cdots\!85 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!54 \nu^{19} + \cdots - 44\!\cdots\!29 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!16 \nu^{19} + \cdots - 19\!\cdots\!88 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!24 \nu^{19} + \cdots - 81\!\cdots\!60 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 81\!\cdots\!28 \nu^{19} + \cdots - 84\!\cdots\!21 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26\!\cdots\!52 \nu^{19} + \cdots - 33\!\cdots\!79 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 54\!\cdots\!84 \nu^{19} + \cdots + 72\!\cdots\!95 \nu ) / 96\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 29\!\cdots\!20 \nu^{19} + \cdots + 13\!\cdots\!33 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!94 \nu^{19} + \cdots - 14\!\cdots\!31 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 72\!\cdots\!14 \nu^{19} + \cdots + 37\!\cdots\!07 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 43\!\cdots\!92 \nu^{19} + \cdots - 36\!\cdots\!90 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 58\!\cdots\!92 \nu^{19} + \cdots - 11\!\cdots\!39 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 63\!\cdots\!40 \nu^{19} + \cdots - 25\!\cdots\!63 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 39\!\cdots\!34 \nu^{19} + \cdots - 14\!\cdots\!83 ) / 50\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 33 \beta_{19} + 9 \beta_{18} + 144 \beta_{17} + 18 \beta_{16} + 56 \beta_{15} + 80 \beta_{14} + \cdots + 1340 \beta_1 ) / 67500 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 486 \beta_{18} - 1432 \beta_{16} + 83 \beta_{15} + 83 \beta_{14} + 1638 \beta_{13} + \cdots - 6373730 ) / 67500 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15795 \beta_{19} - 7587 \beta_{18} - 110160 \beta_{17} - 27024 \beta_{16} - 26190 \beta_{15} + \cdots - 2325 ) / 135000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 71016 \beta_{18} + 556192 \beta_{16} - 5507 \beta_{15} - 4657 \beta_{14} - 452328 \beta_{13} + \cdots + 1701715130 ) / 67500 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5029449 \beta_{19} + 4014405 \beta_{18} + 40684932 \beta_{17} + 16973060 \beta_{16} + \cdots + 2009125 ) / 135000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7755015 \beta_{18} - 100229780 \beta_{16} - 205668 \beta_{15} - 2078543 \beta_{14} + \cdots - 282224428450 ) / 33750 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 868816092 \beta_{19} - 938121672 \beta_{18} - 7275464406 \beta_{17} - 4154201894 \beta_{16} + \cdots - 517288975 ) / 67500 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4659901944 \beta_{18} + 71594191728 \beta_{16} + 436401407 \beta_{15} + 2695206757 \beta_{14} + \cdots + 197865004894170 ) / 67500 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 617117792445 \beta_{19} + 824255098209 \beta_{18} + 5210028399060 \beta_{17} + \cdots + 464602407675 ) / 135000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1595993368704 \beta_{18} - 25733393246248 \beta_{16} - 207929591855 \beta_{15} + \cdots - 70\!\cdots\!70 ) / 67500 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 222380240304261 \beta_{19} - 347804003712909 \beta_{18} + \cdots - 197038392273525 ) / 135000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 142923037562205 \beta_{18} + \cdots + 64\!\cdots\!25 ) / 16875 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 40\!\cdots\!53 \beta_{19} + \cdots + 40\!\cdots\!00 ) / 67500 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 20\!\cdots\!26 \beta_{18} + \cdots - 93\!\cdots\!30 ) / 67500 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 59\!\cdots\!75 \beta_{19} + \cdots - 65\!\cdots\!25 ) / 27000 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 76\!\cdots\!96 \beta_{18} + \cdots + 34\!\cdots\!30 ) / 67500 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 11\!\cdots\!89 \beta_{19} + \cdots + 12\!\cdots\!25 ) / 135000 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 14\!\cdots\!25 \beta_{18} + \cdots - 64\!\cdots\!50 ) / 33750 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 20\!\cdots\!32 \beta_{19} + \cdots - 25\!\cdots\!75 ) / 67500 \) 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
17.2930i
− 17.2930i
− 0.00158591i
0.00158591i
9.57665i
− 9.57665i
− 1.32388i
1.32388i
8.01205i
− 8.01205i
5.77599i
− 5.77599i
− 3.55995i
3.55995i
7.34058i
− 7.34058i
2.23448i
− 2.23448i
19.5290i
− 19.5290i
−29.5009 69.5124 − 41.5815i 614.301 0 −2050.68 + 1226.69i 3174.27i −10570.2 3102.96 − 5780.86i 0
74.2 −29.5009 69.5124 + 41.5815i 614.301 0 −2050.68 − 1226.69i − 3174.27i −10570.2 3102.96 + 5780.86i 0
74.3 −23.7888 10.0775 − 80.3707i 309.906 0 −239.732 + 1911.92i 692.753i −1282.36 −6357.89 − 1619.88i 0
74.4 −23.7888 10.0775 + 80.3707i 309.906 0 −239.732 − 1911.92i − 692.753i −1282.36 −6357.89 + 1619.88i 0
74.5 −10.2357 −56.7364 − 57.8099i −151.230 0 580.739 + 591.726i − 3448.05i 4168.30 −122.959 + 6559.85i 0
74.6 −10.2357 −56.7364 + 57.8099i −151.230 0 580.739 − 591.726i 3448.05i 4168.30 −122.959 − 6559.85i 0
74.7 −8.27106 39.9876 − 70.4414i −187.590 0 −330.740 + 582.625i 860.291i 3668.96 −3362.98 − 5633.57i 0
74.8 −8.27106 39.9876 + 70.4414i −187.590 0 −330.740 − 582.625i − 860.291i 3668.96 −3362.98 + 5633.57i 0
74.9 −7.97572 75.3023 − 29.8424i −192.388 0 −600.590 + 238.014i 3211.86i 3576.22 4779.87 − 4494.40i 0
74.10 −7.97572 75.3023 + 29.8424i −192.388 0 −600.590 − 238.014i − 3211.86i 3576.22 4779.87 + 4494.40i 0
74.11 7.97572 −75.3023 − 29.8424i −192.388 0 −600.590 − 238.014i 3211.86i −3576.22 4779.87 + 4494.40i 0
74.12 7.97572 −75.3023 + 29.8424i −192.388 0 −600.590 + 238.014i − 3211.86i −3576.22 4779.87 − 4494.40i 0
74.13 8.27106 −39.9876 − 70.4414i −187.590 0 −330.740 − 582.625i 860.291i −3668.96 −3362.98 + 5633.57i 0
74.14 8.27106 −39.9876 + 70.4414i −187.590 0 −330.740 + 582.625i − 860.291i −3668.96 −3362.98 − 5633.57i 0
74.15 10.2357 56.7364 − 57.8099i −151.230 0 580.739 − 591.726i − 3448.05i −4168.30 −122.959 − 6559.85i 0
74.16 10.2357 56.7364 + 57.8099i −151.230 0 580.739 + 591.726i 3448.05i −4168.30 −122.959 + 6559.85i 0
74.17 23.7888 −10.0775 − 80.3707i 309.906 0 −239.732 − 1911.92i 692.753i 1282.36 −6357.89 + 1619.88i 0
74.18 23.7888 −10.0775 + 80.3707i 309.906 0 −239.732 + 1911.92i − 692.753i 1282.36 −6357.89 − 1619.88i 0
74.19 29.5009 −69.5124 − 41.5815i 614.301 0 −2050.68 − 1226.69i 3174.27i 10570.2 3102.96 + 5780.86i 0
74.20 29.5009 −69.5124 + 41.5815i 614.301 0 −2050.68 + 1226.69i − 3174.27i 10570.2 3102.96 − 5780.86i 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.c 20
3.b odd 2 1 inner 75.9.d.c 20
5.b even 2 1 inner 75.9.d.c 20
5.c odd 4 1 15.9.c.a ✓ 10
5.c odd 4 1 75.9.c.g 10
15.d odd 2 1 inner 75.9.d.c 20
15.e even 4 1 15.9.c.a ✓ 10
15.e even 4 1 75.9.c.g 10
20.e even 4 1 240.9.l.b 10
60.l odd 4 1 240.9.l.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.c.a ✓ 10 5.c odd 4 1
15.9.c.a ✓ 10 15.e even 4 1
75.9.c.g 10 5.c odd 4 1
75.9.c.g 10 15.e even 4 1
75.9.d.c 20 1.a even 1 1 trivial
75.9.d.c 20 3.b odd 2 1 inner
75.9.d.c 20 5.b even 2 1 inner
75.9.d.c 20 15.d odd 2 1 inner
240.9.l.b 10 20.e even 4 1
240.9.l.b 10 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 1673T_{2}^{8} + 850776T_{2}^{6} - 143194100T_{2}^{4} + 9610475200T_{2}^{2} - 224550432000 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + \cdots - 224550432000)^{2} \) 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
$7$ \( (T^{10} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 11\!\cdots\!32)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 13\!\cdots\!68)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 10\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 23\!\cdots\!32)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
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