Properties

Label 75.11.c.h
Level $75$
Weight $11$
Character orbit 75.c
Analytic conductor $47.652$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
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: \(47.6517939505\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 17880 x^{14} + 140656106 x^{12} - 568287997200 x^{10} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25}\cdot 3^{32}\cdot 5^{12} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + ( - \beta_{9} + \beta_{8}) q^{3} + (\beta_{2} - 136) q^{4} + ( - 3 \beta_{2} + \beta_1 + 1343) q^{6} + (\beta_{12} - 3 \beta_{9}) q^{7} + ( - \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - 32 \beta_{9} - 112 \beta_{8}) q^{8} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3937) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + ( - \beta_{9} + \beta_{8}) q^{3} + (\beta_{2} - 136) q^{4} + ( - 3 \beta_{2} + \beta_1 + 1343) q^{6} + (\beta_{12} - 3 \beta_{9}) q^{7} + ( - \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - 32 \beta_{9} - 112 \beta_{8}) q^{8} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3937) q^{9} + (\beta_{7} + \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{11} + (\beta_{14} + 7 \beta_{13} - 3 \beta_{12} - 9 \beta_{10} + 74 \beta_{9} - 2685 \beta_{8}) q^{12} + (\beta_{15} - \beta_{14} + 4 \beta_{12} + 2 \beta_{11} - 79 \beta_{9} - 11 \beta_{8}) q^{13} + ( - 3 \beta_{7} + 3 \beta_{5} - \beta_{3} + 3 \beta_{2} + 18 \beta_1 - 3) q^{14} + ( - 8 \beta_{6} + 8 \beta_{5} - 8 \beta_{4} + 398 \beta_{2} + \cdots - 263456) q^{16}+ \cdots + (28161 \beta_{7} + 114048 \beta_{6} - 120762 \beta_{5} + \cdots - 2683357818) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2184 q^{4} + 21516 q^{6} + 63000 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2184 q^{4} + 21516 q^{6} + 63000 q^{9} - 4218352 q^{16} - 1487600 q^{19} + 2444616 q^{21} - 28021368 q^{24} - 77667568 q^{31} - 251882368 q^{34} - 22344768 q^{36} + 70953984 q^{39} - 72018968 q^{46} - 89098816 q^{49} + 686556816 q^{51} - 441096084 q^{54} - 1671368368 q^{61} + 4172730848 q^{64} + 2368350000 q^{66} + 2906594616 q^{69} + 4079367264 q^{76} + 3904999600 q^{79} - 11376681984 q^{81} - 13119851016 q^{84} + 19275224832 q^{91} + 9366481832 q^{94} - 5999937552 q^{96} - 42957756000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 17880 x^{14} + 140656106 x^{12} - 568287997200 x^{10} + \cdots + 16\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 84\!\cdots\!87 \nu^{14} + \cdots - 29\!\cdots\!96 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 75\!\cdots\!49 \nu^{14} + \cdots + 26\!\cdots\!40 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!67 \nu^{14} + \cdots + 11\!\cdots\!28 ) / 97\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 48\!\cdots\!85 \nu^{14} + \cdots + 20\!\cdots\!16 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!45 \nu^{14} + \cdots + 82\!\cdots\!72 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 68\!\cdots\!41 \nu^{14} + \cdots + 25\!\cdots\!12 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 64\!\cdots\!59 \nu^{14} + \cdots - 14\!\cdots\!20 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35\!\cdots\!52 \nu^{15} + \cdots - 11\!\cdots\!68 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!19 \nu^{15} + \cdots + 16\!\cdots\!64 \nu ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!89 \nu^{15} + \cdots - 33\!\cdots\!44 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 88\!\cdots\!11 \nu^{15} + \cdots + 31\!\cdots\!96 \nu ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 66\!\cdots\!53 \nu^{15} + \cdots - 48\!\cdots\!48 \nu ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!84 \nu^{15} + \cdots + 11\!\cdots\!44 \nu ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 80\!\cdots\!90 \nu^{15} + \cdots - 17\!\cdots\!20 \nu ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 36\!\cdots\!79 \nu^{15} + \cdots - 14\!\cdots\!84 \nu ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -3\beta_{13} - 14\beta_{11} + 18\beta_{10} + 744\beta_{9} + 1042\beta_{8} ) / 3240 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 27\beta_{7} + 57\beta_{6} - 88\beta_{5} + 97\beta_{4} - 99\beta_{3} - 1908\beta_{2} + 54\beta _1 + 3619695 ) / 1620 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 540 \beta_{15} + 1260 \beta_{14} + 6015 \beta_{13} + 11610 \beta_{12} - 26300 \beta_{11} + 89784 \beta_{10} + 481722 \beta_{9} + 6679864 \beta_{8} ) / 1620 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 34614 \beta_{7} + 88419 \beta_{6} - 107636 \beta_{5} + 17534 \beta_{4} + 81834 \beta_{3} - 383526 \beta_{2} + 327918 \beta _1 + 1295094765 ) / 270 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 55755 \beta_{15} - 2148165 \beta_{14} + 42921321 \beta_{13} - 36386280 \beta_{12} - 66752 \beta_{11} + 357821271 \beta_{10} - 874749927 \beta_{9} + 22256239333 \beta_{8} ) / 810 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 857977758 \beta_{7} + 1173139323 \beta_{6} + 240197218 \beta_{5} - 773175382 \beta_{4} + 383632254 \beta_{3} + 8187534648 \beta_{2} + 6602920596 \beta _1 - 12532333084185 ) / 810 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3067699365 \beta_{15} + 13238055045 \beta_{14} + 60129377178 \beta_{13} - 493089495840 \beta_{12} + 1146785682404 \beta_{11} + 2148593955561 \beta_{10} + \cdots + 83846809361447 \beta_{8} ) / 810 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 936358332048 \beta_{7} + 1773491336583 \beta_{6} + 2960739015908 \beta_{5} - 3266614225172 \beta_{4} - 214433464272 \beta_{3} + \cdots - 20\!\cdots\!15 ) / 270 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 33467613351885 \beta_{15} + 222854962655565 \beta_{14} + 44394994952664 \beta_{13} + \cdots - 12\!\cdots\!07 \beta_{8} ) / 810 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 35\!\cdots\!64 \beta_{7} + \cdots - 81\!\cdots\!81 ) / 162 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 71\!\cdots\!35 \beta_{15} + \cdots - 62\!\cdots\!63 \beta_{8} ) / 810 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 34\!\cdots\!76 \beta_{7} + \cdots - 16\!\cdots\!15 ) / 90 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 24\!\cdots\!15 \beta_{15} + \cdots - 66\!\cdots\!97 \beta_{8} ) / 810 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 26\!\cdots\!48 \beta_{7} + \cdots + 30\!\cdots\!25 ) / 810 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 10\!\cdots\!95 \beta_{15} + \cdots - 41\!\cdots\!73 \beta_{8} ) / 810 \) Copy content Toggle raw display

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
50.0826 + 19.4690i
−50.0826 + 19.4690i
63.3725 6.53233i
−63.3725 6.53233i
80.7929 + 29.3450i
−80.7929 + 29.3450i
30.7435 61.4341i
−30.7435 61.4341i
30.7435 + 61.4341i
−30.7435 + 61.4341i
80.7929 29.3450i
−80.7929 29.3450i
63.3725 + 6.53233i
−63.3725 + 6.53233i
50.0826 19.4690i
−50.0826 19.4690i
50.9058i −150.248 + 190.983i −1567.41 0 9722.16 + 7648.49i −8178.10 27662.5i −13900.2 57389.6i 0
26.2 50.9058i 150.248 + 190.983i −1567.41 0 9722.16 7648.49i 8178.10 27662.5i −13900.2 + 57389.6i 0
26.3 36.7249i −190.117 151.342i −324.720 0 −5558.02 + 6982.05i −17185.2 25681.0i 13240.3 + 57545.5i 0
26.4 36.7249i 190.117 151.342i −324.720 0 −5558.02 6982.05i 17185.2 25681.0i 13240.3 57545.5i 0
26.5 26.2772i −242.379 + 17.3641i 333.511 0 456.279 + 6369.03i 22118.4 35671.5i 58446.0 8417.38i 0
26.6 26.2772i 242.379 + 17.3641i 333.511 0 456.279 6369.03i −22118.4 35671.5i 58446.0 + 8417.38i 0
26.7 3.37421i −92.2305 + 224.817i 1012.61 0 758.580 + 311.205i −16006.0 6871.97i −42036.1 41469.9i 0
26.8 3.37421i 92.2305 + 224.817i 1012.61 0 758.580 311.205i 16006.0 6871.97i −42036.1 + 41469.9i 0
26.9 3.37421i −92.2305 224.817i 1012.61 0 758.580 311.205i −16006.0 6871.97i −42036.1 + 41469.9i 0
26.10 3.37421i 92.2305 224.817i 1012.61 0 758.580 + 311.205i 16006.0 6871.97i −42036.1 41469.9i 0
26.11 26.2772i −242.379 17.3641i 333.511 0 456.279 6369.03i 22118.4 35671.5i 58446.0 + 8417.38i 0
26.12 26.2772i 242.379 17.3641i 333.511 0 456.279 + 6369.03i −22118.4 35671.5i 58446.0 8417.38i 0
26.13 36.7249i −190.117 + 151.342i −324.720 0 −5558.02 6982.05i −17185.2 25681.0i 13240.3 57545.5i 0
26.14 36.7249i 190.117 + 151.342i −324.720 0 −5558.02 + 6982.05i 17185.2 25681.0i 13240.3 + 57545.5i 0
26.15 50.9058i −150.248 190.983i −1567.41 0 9722.16 7648.49i −8178.10 27662.5i −13900.2 + 57389.6i 0
26.16 50.9058i 150.248 190.983i −1567.41 0 9722.16 + 7648.49i 8178.10 27662.5i −13900.2 57389.6i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.16
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 75.11.c.h 16
3.b odd 2 1 inner 75.11.c.h 16
5.b even 2 1 inner 75.11.c.h 16
5.c odd 4 2 15.11.d.c 16
15.d odd 2 1 inner 75.11.c.h 16
15.e even 4 2 15.11.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.11.d.c 16 5.c odd 4 2
15.11.d.c 16 15.e even 4 2
75.11.c.h 16 1.a even 1 1 trivial
75.11.c.h 16 3.b odd 2 1 inner
75.11.c.h 16 5.b even 2 1 inner
75.11.c.h 16 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{8} + 4642T_{2}^{6} + 6268416T_{2}^{4} + 2484083200T_{2}^{2} + 27476377600 \) Copy content Toggle raw display
\( T_{7}^{8} - 1107626292 T_{7}^{6} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 4642 T^{6} + \cdots + 27476377600)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 31500 T^{14} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 1107626292 T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 99459208800 T^{6} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 922071494592 T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 7373530731592 T^{6} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 371900 T^{3} + \cdots + 82\!\cdots\!16)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 118467520731292 T^{6} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 19416892 T^{3} + \cdots + 89\!\cdots\!16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 417842092 T^{3} + \cdots - 64\!\cdots\!84)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 976249900 T^{3} + \cdots - 30\!\cdots\!84)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
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