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Properties

Label 375.2.g.c
Level $375$
Weight $2$
Character orbit 375.g
Analytic conductor $2.994$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [375,2,Mod(76,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.g (of order \(5\), degree \(4\), 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: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3x^{10} - 2x^{9} + 34x^{8} - 22x^{7} + 236x^{6} - 179x^{5} + 877x^{4} - 409x^{3} + 96x^{2} - 11x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{8} q^{3} + ( - \beta_{10} + \beta_{8} - \beta_{4} - 1) q^{4} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{6} + ( - \beta_{5} + 1) q^{7} + (\beta_{8} - \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{8} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{8} q^{3} + ( - \beta_{10} + \beta_{8} - \beta_{4} - 1) q^{4} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{6} + ( - \beta_{5} + 1) q^{7} + (\beta_{8} - \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{8} + \beta_{4} q^{9} + ( - \beta_{9} - \beta_{8} - \beta_{2} - 1) q^{11} + ( - \beta_{7} + \beta_{6} + 2 \beta_{4} + 1) q^{12} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 1) q^{13} + (\beta_{11} + 3 \beta_{8} - 4 \beta_{7} + 4 \beta_{4} - \beta_{2} + 3) q^{14} + (\beta_{11} + \beta_{10} - 3 \beta_{7} - \beta_{6} + 5 \beta_{4} + \beta_{2} - \beta_1 + 3) q^{16} + (\beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{17} + \beta_{5} q^{18} + ( - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{19} + (\beta_{8} - \beta_1) q^{21} + ( - \beta_{10} + 2 \beta_{8} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{22} + ( - \beta_{11} - \beta_{9} - 3 \beta_{8} + \beta_{7} - \beta_{4} - \beta_{2} - 3) q^{23} + ( - \beta_{8} + \beta_{7} + 2 \beta_{5} + \beta_{3} - \beta_{2}) q^{24} + ( - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} - 5) q^{26} + ( - \beta_{8} + \beta_{7} - \beta_{4} - 1) q^{27} + ( - \beta_{10} + \beta_{8} + \beta_{5} + \beta_{3} - 3 \beta_1) q^{28} + ( - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{29} + (\beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{5} - \beta_{4} - 2 \beta_{2} + \cdots - 1) q^{31}+ \cdots + (\beta_{11} - \beta_{6} + \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} - 10 q^{4} + 12 q^{7} - 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{3} - 10 q^{4} + 12 q^{7} - 9 q^{8} - 3 q^{9} - 4 q^{11} + 2 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 7 q^{19} - 3 q^{21} - 13 q^{22} - 19 q^{23} + 6 q^{24} - 56 q^{26} - 3 q^{27} - q^{28} - q^{29} + 13 q^{31} + 32 q^{32} + q^{33} - 25 q^{34} - 8 q^{37} + 22 q^{38} + 2 q^{39} + 8 q^{41} + 16 q^{42} + 4 q^{43} + 33 q^{44} - 22 q^{46} + 13 q^{47} + 16 q^{48} - 28 q^{49} + 26 q^{51} - 44 q^{52} - 44 q^{53} + 45 q^{56} + 22 q^{57} - 41 q^{58} - 22 q^{59} - 8 q^{61} - 41 q^{62} - 3 q^{63} + 49 q^{64} - 3 q^{66} + 6 q^{67} + 100 q^{68} + 6 q^{69} - 21 q^{71} - 9 q^{72} + 16 q^{73} - 44 q^{74} - 52 q^{76} - q^{77} + 19 q^{78} + 10 q^{79} - 3 q^{81} - 26 q^{82} + 10 q^{83} - 6 q^{84} + 56 q^{86} + 4 q^{87} + 16 q^{88} + 57 q^{89} - 7 q^{91} - 3 q^{92} - 22 q^{93} - 23 q^{94} - 23 q^{96} - 4 q^{97} + 18 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3x^{10} - 2x^{9} + 34x^{8} - 22x^{7} + 236x^{6} - 179x^{5} + 877x^{4} - 409x^{3} + 96x^{2} - 11x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 78219134 \nu^{11} - 275969449 \nu^{10} - 322545241 \nu^{9} - 688323383 \nu^{8} - 2382459144 \nu^{7} - 7518729011 \nu^{6} + \cdots - 1397761481 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 444010163 \nu^{11} - 328887363 \nu^{10} - 1432734827 \nu^{9} - 101573726 \nu^{8} - 14512620193 \nu^{7} - 1187789257 \nu^{6} + \cdots - 6770591262 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1397761481 \nu^{11} + 78219134 \nu^{10} - 3917314994 \nu^{9} + 3118068203 \nu^{8} - 46835566971 \nu^{7} + 33133211726 \nu^{6} + \cdots - 20040639729 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1818269283 \nu^{11} + 240671723 \nu^{10} + 5444121987 \nu^{9} - 2938441244 \nu^{8} + 61232986233 \nu^{7} - 31823778603 \nu^{6} + \cdots - 3685345568 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1905700356 \nu^{11} - 2131145819 \nu^{10} + 4372551549 \nu^{9} - 10545703663 \nu^{8} + 64978959816 \nu^{7} - 112688230641 \nu^{6} + \cdots + 25111873119 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3685345568 \nu^{11} - 1818269283 \nu^{10} - 11296708427 \nu^{9} + 1926569149 \nu^{8} - 122363308068 \nu^{7} + 19844616263 \nu^{6} + \cdots - 24741863047 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6770591262 \nu^{11} - 444010163 \nu^{10} + 19982886423 \nu^{9} - 14973917351 \nu^{8} + 230098529182 \nu^{7} - 163465627957 \nu^{6} + \cdots - 51950796787 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3594254357 \nu^{11} + 648267081 \nu^{10} - 10484886687 \nu^{9} + 9161303434 \nu^{8} - 122610276713 \nu^{7} + 100643733429 \nu^{6} + \cdots + 13852270880 ) / 6061764550 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21468863544 \nu^{11} - 1399563761 \nu^{10} + 63167876941 \nu^{9} - 47451650867 \nu^{8} + 728953008894 \nu^{7} - 518817498679 \nu^{6} + \cdots - 164302304099 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29133222771 \nu^{11} - 6153218841 \nu^{10} - 88476638949 \nu^{9} + 39572030488 \nu^{8} - 981450539091 \nu^{7} + \cdots + 45677018676 ) / 30308822750 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} + 4\beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{5} - \beta_{4} - 6\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{11} - \beta_{10} - 6\beta_{9} + 3\beta_{8} - 25\beta_{7} - 5\beta_{6} - \beta_{5} + 9\beta_{4} + \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} - 9\beta_{8} + 10\beta_{7} + \beta_{6} + 37\beta_{5} + 11\beta_{3} - 11\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -37\beta_{10} - 11\beta_{9} + 79\beta_{8} - 11\beta_{6} - 12\beta_{5} - 71\beta_{4} - 12\beta_{3} + 2\beta _1 - 82 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{11} + 12\beta_{10} - 87\beta_{7} + 2\beta_{6} + 116\beta_{4} + 142\beta_{3} + 93\beta_{2} - 93\beta _1 + 87 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 235 \beta_{11} + 93 \beta_{9} - 438 \beta_{8} + 1059 \beta_{7} + 111 \beta_{5} - 1059 \beta_{4} - 144 \beta_{2} + 111 \beta _1 - 438 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 144 \beta_{11} - 111 \beta_{10} - 144 \beta_{9} + 568 \beta_{8} - 1015 \beta_{7} - 33 \beta_{6} - 1529 \beta_{5} + 712 \beta_{4} - 815 \beta_{3} + 1529 \beta_{2} - 815 \beta _1 - 144 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 714 \beta_{11} + 1529 \beta_{10} + 1529 \beta_{9} - 2286 \beta_{8} + 3815 \beta_{7} + 714 \beta_{6} + 1303 \beta_{5} + 934 \beta_{3} - 934 \beta_{2} + 4799 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1303 \beta_{10} - 934 \beta_{9} + 1821 \beta_{8} - 934 \beta_{6} - 5243 \beta_{5} - 5634 \beta_{4} - 5243 \beta_{3} + 4900 \beta _1 - 6568 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(\beta_{8}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.623865 + 1.92006i
0.0437845 + 0.134755i
−0.667650 − 2.05481i
1.97423 − 1.43436i
0.199632 − 0.145041i
−2.17386 + 1.57940i
1.97423 + 1.43436i
0.199632 + 0.145041i
−2.17386 − 1.57940i
0.623865 − 1.92006i
0.0437845 − 0.134755i
−0.667650 + 2.05481i
−1.63330 − 1.18666i 0.309017 + 0.951057i 0.641469 + 1.97424i 0 0.623865 − 1.92006i −1.01887 0.0473123 − 0.145612i −0.809017 + 0.587785i 0
76.2 −0.114629 − 0.0832830i 0.309017 + 0.951057i −0.611830 − 1.88302i 0 0.0437845 − 0.134755i 0.858311 −0.174259 + 0.536314i −0.809017 + 0.587785i 0
76.3 1.74793 + 1.26995i 0.309017 + 0.951057i 0.824463 + 2.53744i 0 −0.667650 + 2.05481i 3.16056 −0.446002 + 1.37265i −0.809017 + 0.587785i 0
151.1 −0.754089 − 2.32085i −0.809017 + 0.587785i −3.19965 + 2.32468i 0 1.97423 + 1.43436i 3.44028 3.85959 + 2.80415i 0.309017 − 0.951057i 0
151.2 −0.0762527 − 0.234682i −0.809017 + 0.587785i 1.56877 − 1.13978i 0 0.199632 + 0.145041i 1.24676 −0.786373 − 0.571334i 0.309017 − 0.951057i 0
151.3 0.830342 + 2.55553i −0.809017 + 0.587785i −4.22323 + 3.06835i 0 −2.17386 − 1.57940i −1.68704 −7.00026 − 5.08599i 0.309017 − 0.951057i 0
226.1 −0.754089 + 2.32085i −0.809017 − 0.587785i −3.19965 − 2.32468i 0 1.97423 − 1.43436i 3.44028 3.85959 − 2.80415i 0.309017 + 0.951057i 0
226.2 −0.0762527 + 0.234682i −0.809017 − 0.587785i 1.56877 + 1.13978i 0 0.199632 − 0.145041i 1.24676 −0.786373 + 0.571334i 0.309017 + 0.951057i 0
226.3 0.830342 − 2.55553i −0.809017 − 0.587785i −4.22323 − 3.06835i 0 −2.17386 + 1.57940i −1.68704 −7.00026 + 5.08599i 0.309017 + 0.951057i 0
301.1 −1.63330 + 1.18666i 0.309017 − 0.951057i 0.641469 − 1.97424i 0 0.623865 + 1.92006i −1.01887 0.0473123 + 0.145612i −0.809017 − 0.587785i 0
301.2 −0.114629 + 0.0832830i 0.309017 − 0.951057i −0.611830 + 1.88302i 0 0.0437845 + 0.134755i 0.858311 −0.174259 − 0.536314i −0.809017 − 0.587785i 0
301.3 1.74793 − 1.26995i 0.309017 − 0.951057i 0.824463 − 2.53744i 0 −0.667650 − 2.05481i 3.16056 −0.446002 − 1.37265i −0.809017 − 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.g.c 12
5.b even 2 1 75.2.g.c ✓ 12
5.c odd 4 2 375.2.i.d 24
15.d odd 2 1 225.2.h.d 12
25.d even 5 1 inner 375.2.g.c 12
25.d even 5 1 1875.2.a.k 6
25.e even 10 1 75.2.g.c ✓ 12
25.e even 10 1 1875.2.a.j 6
25.f odd 20 2 375.2.i.d 24
25.f odd 20 2 1875.2.b.f 12
75.h odd 10 1 225.2.h.d 12
75.h odd 10 1 5625.2.a.p 6
75.j odd 10 1 5625.2.a.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c ✓ 12 5.b even 2 1
75.2.g.c ✓ 12 25.e even 10 1
225.2.h.d 12 15.d odd 2 1
225.2.h.d 12 75.h odd 10 1
375.2.g.c 12 1.a even 1 1 trivial
375.2.g.c 12 25.d even 5 1 inner
375.2.i.d 24 5.c odd 4 2
375.2.i.d 24 25.f odd 20 2
1875.2.a.j 6 25.e even 10 1
1875.2.a.k 6 25.d even 5 1
1875.2.b.f 12 25.f odd 20 2
5625.2.a.p 6 75.h odd 10 1
5625.2.a.q 6 75.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 8 T_{2}^{10} + 3 T_{2}^{9} + 34 T_{2}^{8} + 8 T_{2}^{7} + 91 T_{2}^{6} + 96 T_{2}^{5} + 852 T_{2}^{4} + 321 T_{2}^{3} + 96 T_{2}^{2} + 14 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 8 T^{10} + 3 T^{9} + 34 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 6 T^{5} + 4 T^{4} + 25 T^{3} + \cdots + 20)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + 11 T^{10} + \cdots + 59536 \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + 19 T^{10} + \cdots + 10201 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + 29 T^{10} + \cdots + 21520321 \) Copy content Toggle raw display
$19$ \( T^{12} - 7 T^{11} + 73 T^{10} + \cdots + 144400 \) Copy content Toggle raw display
$23$ \( T^{12} + 19 T^{11} + 156 T^{10} + \cdots + 4080400 \) Copy content Toggle raw display
$29$ \( T^{12} + T^{11} + 12 T^{10} + \cdots + 4431025 \) Copy content Toggle raw display
$31$ \( T^{12} - 13 T^{11} + 44 T^{10} + \cdots + 8410000 \) Copy content Toggle raw display
$37$ \( T^{12} + 8 T^{11} + 44 T^{10} + \cdots + 36300625 \) Copy content Toggle raw display
$41$ \( T^{12} - 8 T^{11} + \cdots + 6831849025 \) Copy content Toggle raw display
$43$ \( (T^{6} - 2 T^{5} - 91 T^{4} + 174 T^{3} + \cdots - 6284)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 13 T^{11} + 178 T^{10} + \cdots + 5216656 \) Copy content Toggle raw display
$53$ \( T^{12} + 44 T^{11} + \cdots + 1189905025 \) Copy content Toggle raw display
$59$ \( T^{12} + 22 T^{11} + 213 T^{10} + \cdots + 15366400 \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + \cdots + 28314456361 \) Copy content Toggle raw display
$67$ \( T^{12} - 6 T^{11} + 29 T^{10} + \cdots + 215619856 \) Copy content Toggle raw display
$71$ \( T^{12} + 21 T^{11} + 259 T^{10} + \cdots + 38416 \) Copy content Toggle raw display
$73$ \( T^{12} - 16 T^{11} + \cdots + 493062025 \) Copy content Toggle raw display
$79$ \( T^{12} - 10 T^{11} - 35 T^{10} + \cdots + 64000000 \) Copy content Toggle raw display
$83$ \( T^{12} - 10 T^{11} + \cdots + 1683953296 \) Copy content Toggle raw display
$89$ \( T^{12} - 57 T^{11} + \cdots + 142170473025 \) Copy content Toggle raw display
$97$ \( T^{12} + 4 T^{11} + 139 T^{10} + \cdots + 5755201 \) Copy content Toggle raw display
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