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Properties

Label 4375.2.a.g
Level $4375$
Weight $2$
Character orbit 4375.a
Self dual yes
Analytic conductor $34.935$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4375,2,Mod(1,4375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4375, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4375.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4375 = 5^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4375.a (trivial)

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: yes
Analytic conductor: \(34.9345508843\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 127 x^{10} - 226 x^{9} - 441 x^{8} + 745 x^{7} + 761 x^{6} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{13} - \beta_{12} - \beta_{10} + \cdots + 2) q^{6}+ \cdots + (\beta_{13} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{13} - \beta_{12} - \beta_{10} + \cdots + 2) q^{6}+ \cdots + (\beta_{13} + 6 \beta_{12} - 4 \beta_{9} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 3 q^{3} + 12 q^{4} + 15 q^{6} - 14 q^{7} - 6 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 3 q^{3} + 12 q^{4} + 15 q^{6} - 14 q^{7} - 6 q^{8} + 15 q^{9} + 6 q^{11} - 14 q^{12} + q^{13} + 2 q^{14} - 12 q^{16} - 23 q^{17} - 31 q^{18} - 4 q^{19} + 3 q^{21} - 19 q^{22} - 22 q^{23} + 19 q^{24} - 4 q^{26} - 3 q^{27} - 12 q^{28} + 14 q^{29} - 8 q^{31} - 24 q^{32} - 9 q^{33} + 35 q^{34} + 27 q^{36} - 15 q^{37} - 5 q^{38} - 38 q^{39} - 15 q^{41} - 15 q^{42} - 9 q^{43} - 55 q^{44} - 2 q^{46} - 38 q^{47} - 28 q^{48} + 14 q^{49} + q^{51} + 18 q^{52} - 27 q^{53} + 28 q^{54} + 6 q^{56} - 31 q^{57} - 29 q^{58} + 3 q^{59} - 9 q^{61} - 11 q^{62} - 15 q^{63} - 10 q^{64} - 87 q^{66} + 3 q^{67} - 57 q^{68} + 3 q^{69} + 31 q^{71} - 25 q^{72} - 41 q^{73} + 35 q^{74} - 3 q^{76} - 6 q^{77} - 13 q^{78} + 59 q^{79} - 50 q^{81} - 13 q^{82} + 8 q^{83} + 14 q^{84} + 25 q^{86} - 40 q^{87} + 26 q^{88} - 31 q^{89} - q^{91} + 5 q^{92} - 48 q^{93} - 29 q^{94} + 21 q^{96} - 34 q^{97} - 2 q^{98} + 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 127 x^{10} - 226 x^{9} - 441 x^{8} + 745 x^{7} + 761 x^{6} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25 \nu^{13} + 8 \nu^{12} - 517 \nu^{11} - 159 \nu^{10} + 4084 \nu^{9} + 1098 \nu^{8} - 15408 \nu^{7} + \cdots - 107 ) / 69 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32 \nu^{13} + 13 \nu^{12} - 659 \nu^{11} - 267 \nu^{10} + 5222 \nu^{9} + 1905 \nu^{8} - 19932 \nu^{7} + \cdots - 79 ) / 69 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80 \nu^{13} + 209 \nu^{12} + 1130 \nu^{11} - 3024 \nu^{10} - 5534 \nu^{9} + 15903 \nu^{8} + \cdots + 94 ) / 69 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10 \nu^{13} + 17 \nu^{12} - 239 \nu^{11} - 266 \nu^{10} + 2089 \nu^{9} + 1534 \nu^{8} - 8500 \nu^{7} + \cdots - 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77 \nu^{13} - 83 \nu^{12} - 1355 \nu^{11} + 1158 \nu^{10} + 9137 \nu^{9} - 5820 \nu^{8} - 29340 \nu^{7} + \cdots + 32 ) / 69 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 27 \nu^{13} + 7 \nu^{12} + 528 \nu^{11} - 73 \nu^{10} - 3982 \nu^{9} + 219 \nu^{8} + 14463 \nu^{7} + \cdots + 30 ) / 23 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74 \nu^{13} - 302 \nu^{12} - 821 \nu^{11} + 4467 \nu^{10} + 1907 \nu^{9} - 24165 \nu^{8} + 7434 \nu^{7} + \cdots - 187 ) / 69 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 62 \nu^{13} + 64 \nu^{12} - 1376 \nu^{11} - 1065 \nu^{10} + 11489 \nu^{9} + 6507 \nu^{8} - 45432 \nu^{7} + \cdots + 248 ) / 69 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 176 \nu^{13} + 308 \nu^{12} + 2831 \nu^{11} - 4362 \nu^{10} - 17198 \nu^{9} + 22332 \nu^{8} + \cdots - 290 ) / 69 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{13} - 73 \nu^{12} + 178 \nu^{11} + 1103 \nu^{10} - 2410 \nu^{9} - 6115 \nu^{8} + 12258 \nu^{7} + \cdots - 119 ) / 23 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 133 \nu^{13} + 43 \nu^{12} + 2560 \nu^{11} - 432 \nu^{10} - 19069 \nu^{9} + 1089 \nu^{8} + 68706 \nu^{7} + \cdots + 20 ) / 69 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + \beta_{9} - \beta_{7} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - \beta_{11} - \beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{12} - \beta_{10} + 8\beta_{9} - \beta_{8} - 9\beta_{7} - \beta_{5} - \beta_{3} + 9\beta_{2} + 27\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{12} - 9 \beta_{11} - 11 \beta_{10} - \beta_{8} - \beta_{7} - 8 \beta_{6} + 8 \beta_{5} + \cdots + 66 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{13} - 64 \beta_{12} - 2 \beta_{11} - 13 \beta_{10} + 52 \beta_{9} - 12 \beta_{8} - 63 \beta_{7} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{13} - 78 \beta_{12} - 64 \beta_{11} - 88 \beta_{10} + 2 \beta_{9} - 13 \beta_{8} - 15 \beta_{7} + \cdots + 369 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 25 \beta_{13} - 421 \beta_{12} - 28 \beta_{11} - 118 \beta_{10} + 318 \beta_{9} - 101 \beta_{8} + \cdots + 626 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 16 \beta_{13} - 559 \beta_{12} - 420 \beta_{11} - 628 \beta_{10} + 36 \beta_{9} - 120 \beta_{8} + \cdots + 2182 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 216 \beta_{13} - 2683 \beta_{12} - 267 \beta_{11} - 932 \beta_{10} + 1905 \beta_{9} - 740 \beta_{8} + \cdots + 4227 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 173 \beta_{13} - 3851 \beta_{12} - 2664 \beta_{11} - 4258 \beta_{10} + 408 \beta_{9} - 964 \beta_{8} + \cdots + 13313 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1618 \beta_{13} - 16866 \beta_{12} - 2175 \beta_{11} - 6871 \beta_{10} + 11344 \beta_{9} - 5073 \beta_{8} + \cdots + 27835 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.54121
2.48148
1.96895
1.48564
1.46528
1.15426
0.555932
−0.00648097
−0.442950
−1.59048
−1.65425
−1.74012
−1.86020
−2.35828
−2.54121 −2.52498 4.45775 0 6.41650 −1.00000 −6.24565 3.37552 0
1.2 −2.48148 −2.21797 4.15774 0 5.50384 −1.00000 −5.35438 1.91937 0
1.3 −1.96895 2.51047 1.87675 0 −4.94297 −1.00000 0.242679 3.30245 0
1.4 −1.48564 −3.10195 0.207141 0 4.60839 −1.00000 2.66355 6.62209 0
1.5 −1.46528 1.53682 0.147044 0 −2.25188 −1.00000 2.71510 −0.638169 0
1.6 −1.15426 −2.28456 −0.667692 0 2.63696 −1.00000 3.07920 2.21920 0
1.7 −0.555932 −0.0140226 −1.69094 0 0.00779563 −1.00000 2.05191 −2.99980 0
1.8 0.00648097 2.40903 −1.99996 0 0.0156129 −1.00000 −0.0259236 2.80345 0
1.9 0.442950 −0.925988 −1.80380 0 −0.410167 −1.00000 −1.68489 −2.14255 0
1.10 1.59048 −0.991912 0.529618 0 −1.57761 −1.00000 −2.33861 −2.01611 0
1.11 1.65425 −2.00495 0.736547 0 −3.31668 −1.00000 −2.09007 1.01981 0
1.12 1.74012 2.29188 1.02801 0 3.98814 −1.00000 −1.69138 2.25272 0
1.13 1.86020 2.29818 1.46033 0 4.27506 −1.00000 −1.00389 2.28162 0
1.14 2.35828 0.0199316 3.56146 0 0.0470042 −1.00000 3.68236 −2.99960 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4375.2.a.g 14
5.b even 2 1 4375.2.a.h 14
25.d even 5 2 175.2.h.b ✓ 28
25.e even 10 2 875.2.h.b 28
25.f odd 20 4 875.2.n.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.h.b ✓ 28 25.d even 5 2
875.2.h.b 28 25.e even 10 2
875.2.n.b 56 25.f odd 20 4
4375.2.a.g 14 1.a even 1 1 trivial
4375.2.a.h 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4375))\):

\( T_{2}^{14} + 2 T_{2}^{13} - 18 T_{2}^{12} - 34 T_{2}^{11} + 127 T_{2}^{10} + 226 T_{2}^{9} - 441 T_{2}^{8} + \cdots - 1 \) Copy content Toggle raw display
\( T_{3}^{14} + 3 T_{3}^{13} - 24 T_{3}^{12} - 74 T_{3}^{11} + 218 T_{3}^{10} + 716 T_{3}^{9} - 884 T_{3}^{8} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{14} + 3 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 6 T^{13} + \cdots - 522661 \) Copy content Toggle raw display
$13$ \( T^{14} - T^{13} + \cdots + 25150131 \) Copy content Toggle raw display
$17$ \( T^{14} + 23 T^{13} + \cdots + 614309 \) Copy content Toggle raw display
$19$ \( T^{14} + 4 T^{13} + \cdots + 5960905 \) Copy content Toggle raw display
$23$ \( T^{14} + 22 T^{13} + \cdots - 84969 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 49409388225 \) Copy content Toggle raw display
$31$ \( T^{14} + 8 T^{13} + \cdots + 63892225 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 2962275761 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 253762501 \) Copy content Toggle raw display
$43$ \( T^{14} + 9 T^{13} + \cdots + 25825536 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 26751771981 \) Copy content Toggle raw display
$53$ \( T^{14} + 27 T^{13} + \cdots - 7110189 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 2481342805 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 6779645559 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 108616792559 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 89916705811 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 35738227861 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 26221865725 \) Copy content Toggle raw display
$83$ \( T^{14} - 8 T^{13} + \cdots + 87255971 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 12229803295 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 135282092429 \) Copy content Toggle raw display
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