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Properties

Label 1875.4.a.e
Level $1875$
Weight $4$
Character orbit 1875.a
Self dual yes
Analytic conductor $110.629$
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] = [1875,4,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(110.628581261\)
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: \( x^{14} - 6 x^{13} - 60 x^{12} + 359 x^{11} + 1384 x^{10} - 7950 x^{9} - 16010 x^{8} + 80765 x^{7} + \cdots - 178496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 75)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + (3 \beta_1 - 3) q^{6} + ( - \beta_{10} + \beta_{3} - \beta_{2} + \cdots - 2) q^{7}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + (3 \beta_1 - 3) q^{6} + ( - \beta_{10} + \beta_{3} - \beta_{2} + \cdots - 2) q^{7}+ \cdots + (9 \beta_{11} - 9 \beta_{10} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 8 q^{2} + 42 q^{3} + 46 q^{4} - 24 q^{6} - 29 q^{7} - 117 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 8 q^{2} + 42 q^{3} + 46 q^{4} - 24 q^{6} - 29 q^{7} - 117 q^{8} + 126 q^{9} + 5 q^{11} + 138 q^{12} - 20 q^{13} + 221 q^{14} + 138 q^{16} - 126 q^{17} - 72 q^{18} - 132 q^{19} - 87 q^{21} - 287 q^{22} - 308 q^{23} - 351 q^{24} - 57 q^{26} + 378 q^{27} - 1272 q^{28} - 384 q^{29} + 7 q^{31} - 1484 q^{32} + 15 q^{33} - 318 q^{34} + 414 q^{36} - 694 q^{37} + 561 q^{38} - 60 q^{39} - 268 q^{41} + 663 q^{42} - 464 q^{43} - 176 q^{44} - 323 q^{46} - 1012 q^{47} + 414 q^{48} + 249 q^{49} - 378 q^{51} - 76 q^{52} - 893 q^{53} - 216 q^{54} + 2410 q^{56} - 396 q^{57} - 623 q^{58} - 43 q^{59} - 1684 q^{61} - 128 q^{62} - 261 q^{63} + 3117 q^{64} - 861 q^{66} + 254 q^{67} - 609 q^{68} - 924 q^{69} - 2362 q^{71} - 1053 q^{72} - 3378 q^{73} + 941 q^{74} + 1288 q^{76} + 1002 q^{77} - 171 q^{78} - 2875 q^{79} + 1134 q^{81} - 1978 q^{82} + 2069 q^{83} - 3816 q^{84} - 2945 q^{86} - 1152 q^{87} - 1367 q^{88} + 808 q^{89} - 2198 q^{91} - 1079 q^{92} + 21 q^{93} - 5845 q^{94} - 4452 q^{96} - 4261 q^{97} - 5720 q^{98} + 45 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 60 x^{12} + 359 x^{11} + 1384 x^{10} - 7950 x^{9} - 16010 x^{8} + 80765 x^{7} + \cdots - 178496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 168648113 \nu^{13} - 1257081680 \nu^{12} - 8321207452 \nu^{11} + 72846611279 \nu^{10} + \cdots - 19440134117152 ) / 3801730229760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 122596501 \nu^{13} - 898839664 \nu^{12} - 6150969356 \nu^{11} + 52528074187 \nu^{10} + \cdots - 15051506789024 ) / 1900865114880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4670340185 \nu^{13} - 32616326504 \nu^{12} - 275941527628 \nu^{11} + 2019492966983 \nu^{10} + \cdots - 415803381190432 ) / 55125088331520 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12199687355 \nu^{13} - 78512560232 \nu^{12} - 561626684932 \nu^{11} + \cdots - 664882892044000 ) / 110250176663040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3289548779 \nu^{13} + 7568457776 \nu^{12} + 257170765540 \nu^{11} + \cdots + 984389599212832 ) / 27562544165760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5634156381 \nu^{13} - 40822019552 \nu^{12} - 306357293644 \nu^{11} + 2443337604131 \nu^{10} + \cdots - 623470463378592 ) / 36750058887680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 427804975 \nu^{13} - 3206859248 \nu^{12} - 20733160868 \nu^{11} + 184139682001 \nu^{10} + \cdots - 45390253798624 ) / 1267243409920 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21254380403 \nu^{13} + 132819476912 \nu^{12} + 1156373835988 \nu^{11} + \cdots + 845610570306016 ) / 55125088331520 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 108627117841 \nu^{13} - 812970022456 \nu^{12} - 5223227950796 \nu^{11} + \cdots - 97\!\cdots\!60 ) / 110250176663040 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 118097334571 \nu^{13} - 880476522232 \nu^{12} - 5648186770340 \nu^{11} + \cdots - 13\!\cdots\!36 ) / 110250176663040 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 161792143547 \nu^{13} - 1214233408880 \nu^{12} - 7833000080500 \nu^{11} + \cdots - 18\!\cdots\!28 ) / 110250176663040 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} + 19\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - 3 \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{5} + \cdots + 211 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} - 20 \beta_{9} - 24 \beta_{8} + 4 \beta_{7} + \beta_{6} + \cdots + 370 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 33 \beta_{13} + 55 \beta_{12} + 23 \beta_{11} - 37 \beta_{10} - 150 \beta_{9} + 31 \beta_{8} + \cdots + 4843 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 30 \beta_{13} + 264 \beta_{12} - 74 \beta_{11} - 180 \beta_{10} - 1007 \beta_{9} - 494 \beta_{8} + \cdots + 12839 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 930 \beta_{13} + 2119 \beta_{12} + 386 \beta_{11} - 1222 \beta_{10} - 5845 \beta_{9} + 767 \beta_{8} + \cdots + 122514 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1806 \beta_{13} + 10339 \beta_{12} - 2074 \beta_{11} - 6328 \beta_{10} - 38188 \beta_{9} + \cdots + 420952 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 26160 \beta_{13} + 72630 \beta_{12} + 5060 \beta_{11} - 38850 \beta_{10} - 206985 \beta_{9} + \cdots + 3302429 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 76235 \beta_{13} + 363590 \beta_{12} - 54045 \beta_{11} - 206355 \beta_{10} - 1308240 \beta_{9} + \cdots + 13432710 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 755925 \beta_{13} + 2367755 \beta_{12} + 30995 \beta_{11} - 1213055 \beta_{10} - 6969775 \beta_{9} + \cdots + 92991799 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2806925 \beta_{13} + 12143250 \beta_{12} - 1390535 \beta_{11} - 6535415 \beta_{10} - 42739205 \beta_{9} + \cdots + 422249876 \) 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
−4.58535
−4.25760
−2.83286
−2.60704
−1.56367
−1.49559
−0.296623
1.01184
1.58296
2.48789
3.84931
4.54270
4.61855
5.54547
−5.58535 3.00000 23.1962 0 −16.7561 −14.5931 −84.8760 9.00000 0
1.2 −5.25760 3.00000 19.6424 0 −15.7728 −36.6146 −61.2110 9.00000 0
1.3 −3.83286 3.00000 6.69082 0 −11.4986 20.3108 5.01791 9.00000 0
1.4 −3.60704 3.00000 5.01073 0 −10.8211 7.04529 10.7824 9.00000 0
1.5 −2.56367 3.00000 −1.42761 0 −7.69100 −25.1018 24.1693 9.00000 0
1.6 −2.49559 3.00000 −1.77205 0 −7.48676 −10.8656 24.3870 9.00000 0
1.7 −1.29662 3.00000 −6.31877 0 −3.88987 7.54047 18.5660 9.00000 0
1.8 0.0118428 3.00000 −7.99986 0 0.0355283 26.1506 −0.189482 9.00000 0
1.9 0.582965 3.00000 −7.66015 0 1.74889 −13.5827 −9.12931 9.00000 0
1.10 1.48789 3.00000 −5.78618 0 4.46367 27.0465 −20.5123 9.00000 0
1.11 2.84931 3.00000 0.118586 0 8.54794 −6.69745 −22.4566 9.00000 0
1.12 3.54270 3.00000 4.55073 0 10.6281 15.6925 −12.2197 9.00000 0
1.13 3.61855 3.00000 5.09391 0 10.8557 −15.7010 −10.5158 9.00000 0
1.14 4.54547 3.00000 12.6613 0 13.6364 −9.62995 21.1877 9.00000 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
\(3\) \(-1\)
\(5\) \(1\)

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 1875.4.a.e 14
5.b even 2 1 1875.4.a.h 14
25.e even 10 2 75.4.g.a ✓ 28
75.h odd 10 2 225.4.h.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.g.a ✓ 28 25.e even 10 2
225.4.h.c 28 75.h odd 10 2
1875.4.a.e 14 1.a even 1 1 trivial
1875.4.a.h 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 8 T_{2}^{13} - 47 T_{2}^{12} - 465 T_{2}^{11} + 658 T_{2}^{10} + 10147 T_{2}^{9} + \cdots - 5744 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 8 T^{13} + \cdots - 5744 \) Copy content Toggle raw display
$3$ \( (T - 3)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 67\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 92\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 46\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 60\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 35\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 86\!\cdots\!75 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 17\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 51\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 53\!\cdots\!31 \) Copy content Toggle raw display
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