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Properties

Label 75.9.c.g
Level $75$
Weight $9$
Character orbit 75.c
Analytic conductor $30.553$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 433 x^{8} - 2220 x^{7} + 49747 x^{6} + 744964 x^{5} + 4580249 x^{4} + 16418988 x^{3} + \cdots + 53656344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{11}\cdot 5^{10} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2 \beta_1 + 11) q^{3} + ( - \beta_{3} - 79) q^{4} + ( - \beta_{6} - 2 \beta_{3} + \cdots - 529) q^{6}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \cdots + 385) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2 \beta_1 + 11) q^{3} + ( - \beta_{3} - 79) q^{4} + ( - \beta_{6} - 2 \beta_{3} + \cdots - 529) q^{6}+ \cdots + ( - 25421 \beta_{9} - 13675 \beta_{8} + \cdots - 3323878) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 112 q^{3} - 786 q^{4} - 5282 q^{6} - 7156 q^{7} + 3922 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 112 q^{3} - 786 q^{4} - 5282 q^{6} - 7156 q^{7} + 3922 q^{9} + 3812 q^{12} + 55464 q^{13} + 280386 q^{16} + 419800 q^{18} - 231516 q^{19} + 289572 q^{21} - 1129940 q^{22} + 1136334 q^{24} + 335512 q^{27} + 3340724 q^{28} + 881620 q^{31} - 1266460 q^{33} - 1111276 q^{34} - 668662 q^{36} - 4672616 q^{37} + 1826792 q^{39} + 5392860 q^{42} - 7731336 q^{43} - 25424604 q^{46} - 22413388 q^{48} + 9354214 q^{49} - 27732692 q^{51} - 21064016 q^{52} - 7979798 q^{54} + 2856304 q^{57} + 4351100 q^{58} + 22417020 q^{61} - 8830596 q^{63} - 22935002 q^{64} - 27419800 q^{66} + 46646024 q^{67} + 33562632 q^{69} - 54175560 q^{72} + 129964884 q^{73} + 198922436 q^{76} - 60388360 q^{78} + 162310924 q^{79} - 93575390 q^{81} - 202877560 q^{82} - 197346768 q^{84} + 168322540 q^{87} + 484775700 q^{88} + 444288464 q^{91} - 463412376 q^{93} - 92050036 q^{94} - 360807406 q^{96} + 258825724 q^{97} - 33965200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 433 x^{8} - 2220 x^{7} + 49747 x^{6} + 744964 x^{5} + 4580249 x^{4} + 16418988 x^{3} + \cdots + 53656344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 422181448349 \nu^{9} - 6737848279449 \nu^{8} - 139312194537578 \nu^{7} + \cdots - 41\!\cdots\!04 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 841473910807 \nu^{9} - 32087853683079 \nu^{8} + 747028733465068 \nu^{7} + \cdots - 99\!\cdots\!32 ) / 52\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6879197511740 \nu^{9} + 17194261561041 \nu^{8} + \cdots - 24\!\cdots\!12 ) / 26\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 44483873323468 \nu^{9} - 350484802552473 \nu^{8} + \cdots + 80\!\cdots\!02 ) / 65\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 109711893437129 \nu^{9} - 568404995899629 \nu^{8} + \cdots + 37\!\cdots\!16 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1878004954637 \nu^{9} + 19249058755332 \nu^{8} + 724171099736024 \nu^{7} + \cdots + 43\!\cdots\!47 ) / 17\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 840821834983 \nu^{9} + 6345736290088 \nu^{8} + 335214403336196 \nu^{7} + \cdots - 79\!\cdots\!72 ) / 76\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17345250624505 \nu^{9} + 145830337403353 \nu^{8} + \cdots - 26\!\cdots\!16 ) / 58\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 261825563207441 \nu^{9} + \cdots - 40\!\cdots\!64 ) / 87\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{9} - 8 \beta_{8} - 9 \beta_{7} + 9 \beta_{6} + 8 \beta_{5} + 11 \beta_{4} + 174 \beta_{3} + \cdots + 2822 ) / 6750 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 39 \beta_{9} + 13 \beta_{8} - 576 \beta_{7} + 176 \beta_{6} + 87 \beta_{5} - 146 \beta_{4} + \cdots + 596558 ) / 6750 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 489 \beta_{9} - 767 \beta_{8} - 6966 \beta_{7} + 4046 \beta_{6} + 2463 \beta_{5} + 1306 \beta_{4} + \cdots + 8075888 ) / 6750 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 20223 \beta_{9} + 851 \beta_{8} - 148302 \beta_{7} + 96892 \beta_{6} + 68337 \beta_{5} + \cdots + 162093346 ) / 6750 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 373149 \beta_{9} + 120383 \beta_{8} - 2524536 \beta_{7} + 2028166 \beta_{6} + 1539711 \beta_{5} + \cdots + 2811642448 ) / 6750 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3035925 \beta_{9} + 1566725 \beta_{8} - 14754330 \beta_{7} + 14657200 \beta_{6} + \cdots + 16339448230 ) / 2250 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 61951659 \beta_{9} + 46546793 \beta_{8} - 251838456 \beta_{7} + 299747246 \beta_{6} + \cdots + 278884286128 ) / 2250 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3949175943 \beta_{9} + 3464763091 \beta_{8} - 12398581422 \beta_{7} + 18364352372 \beta_{6} + \cdots + 13744423936826 ) / 6750 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 79947602973 \beta_{9} + 81748755271 \beta_{8} - 197196436872 \beta_{7} + 365611545062 \beta_{6} + \cdots + 218198417696456 ) / 6750 \) 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
18.9110 − 2.23607i
−0.616448 + 2.23607i
−7.95862 − 2.23607i
−1.94191 + 2.23607i
−6.39402 − 2.23607i
−6.39402 + 2.23607i
−1.94191 − 2.23607i
−7.95862 + 2.23607i
−0.616448 − 2.23607i
18.9110 + 2.23607i
− 29.5009i −41.5815 − 69.5124i −614.301 0 −2050.68 + 1226.69i −3174.27 10570.2i −3102.96 + 5780.86i 0
26.2 − 23.7888i 80.3707 − 10.0775i −309.906 0 −239.732 − 1911.92i 692.753 1282.36i 6357.89 − 1619.88i 0
26.3 − 10.2357i 57.8099 + 56.7364i 151.230 0 580.739 − 591.726i −3448.05 − 4168.30i 122.959 + 6559.85i 0
26.4 − 8.27106i −70.4414 − 39.9876i 187.590 0 −330.740 + 582.625i −860.291 − 3668.96i 3362.98 + 5633.57i 0
26.5 − 7.97572i 29.8424 − 75.3023i 192.388 0 −600.590 − 238.014i 3211.86 − 3576.22i −4779.87 − 4494.40i 0
26.6 7.97572i 29.8424 + 75.3023i 192.388 0 −600.590 + 238.014i 3211.86 3576.22i −4779.87 + 4494.40i 0
26.7 8.27106i −70.4414 + 39.9876i 187.590 0 −330.740 − 582.625i −860.291 3668.96i 3362.98 − 5633.57i 0
26.8 10.2357i 57.8099 − 56.7364i 151.230 0 580.739 + 591.726i −3448.05 4168.30i 122.959 − 6559.85i 0
26.9 23.7888i 80.3707 + 10.0775i −309.906 0 −239.732 + 1911.92i 692.753 − 1282.36i 6357.89 + 1619.88i 0
26.10 29.5009i −41.5815 + 69.5124i −614.301 0 −2050.68 − 1226.69i −3174.27 − 10570.2i −3102.96 − 5780.86i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.g 10
3.b odd 2 1 inner 75.9.c.g 10
5.b even 2 1 15.9.c.a ✓ 10
5.c odd 4 2 75.9.d.c 20
15.d odd 2 1 15.9.c.a ✓ 10
15.e even 4 2 75.9.d.c 20
20.d odd 2 1 240.9.l.b 10
60.h even 2 1 240.9.l.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.c.a ✓ 10 5.b even 2 1
15.9.c.a ✓ 10 15.d odd 2 1
75.9.c.g 10 1.a even 1 1 trivial
75.9.c.g 10 3.b odd 2 1 inner
75.9.d.c 20 5.c odd 4 2
75.9.d.c 20 15.e even 4 2
240.9.l.b 10 20.d odd 2 1
240.9.l.b 10 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{10} + 1673T_{2}^{8} + 850776T_{2}^{6} + 143194100T_{2}^{4} + 9610475200T_{2}^{2} + 224550432000 \) Copy content Toggle raw display
\( T_{7}^{5} + 3578T_{7}^{4} - 10349514T_{7}^{3} - 38916314892T_{7}^{2} + 263729201400T_{7} + 20950685488530000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 224550432000 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
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