Properties

Label 1875.4.a.a
Level $1875$
Weight $4$
Character orbit 1875.a
Self dual yes
Analytic conductor $110.629$
Analytic rank $1$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 61 x^{8} + 54 x^{7} + 1219 x^{6} - 1069 x^{5} - 8923 x^{4} + 9432 x^{3} + 18852 x^{2} - 30528 x + 10944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{4} - \beta_{3} + \beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} + (\beta_{9} + \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{7} + (\beta_{9} - \beta_{8} - \beta_{6} + 2 \beta_{5} + 9 \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{4} - \beta_{3} + \beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} + (\beta_{9} + \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{7} + (\beta_{9} - \beta_{8} - \beta_{6} + 2 \beta_{5} + 9 \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{8} + 9 q^{9} + (\beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_1 - 5) q^{11} + ( - 3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 12) q^{12} + ( - 3 \beta_{9} + \beta_{4} + \beta_{2} + 9 \beta_1 + 4) q^{13} + (\beta_{9} - \beta_{8} - 3 \beta_{6} + 2 \beta_{5} + 11 \beta_{3} - 4 \beta_{2} - 16 \beta_1 - 2) q^{14} + ( - \beta_{9} + 3 \beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \cdots + 27) q^{16}+ \cdots + (9 \beta_{9} - 18 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} - 27 \beta_{3} + \cdots - 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - 30 q^{3} + 43 q^{4} + 3 q^{6} + 51 q^{7} - 6 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - 30 q^{3} + 43 q^{4} + 3 q^{6} + 51 q^{7} - 6 q^{8} + 90 q^{9} - 36 q^{11} - 129 q^{12} + 41 q^{13} - 106 q^{14} + 283 q^{16} + 70 q^{17} - 9 q^{18} - 109 q^{19} - 153 q^{21} + 115 q^{22} - 148 q^{23} + 18 q^{24} - 1235 q^{26} - 270 q^{27} + 1488 q^{28} - 1018 q^{29} - 47 q^{31} - 909 q^{32} + 108 q^{33} - 909 q^{34} + 387 q^{36} + 391 q^{37} - 357 q^{38} - 123 q^{39} - 492 q^{41} + 318 q^{42} + 919 q^{43} - 405 q^{44} - 212 q^{46} - 88 q^{47} - 849 q^{48} + 599 q^{49} - 210 q^{51} + 915 q^{52} + 462 q^{53} + 27 q^{54} - 660 q^{56} + 327 q^{57} - 299 q^{58} - 596 q^{59} - 1199 q^{61} + 727 q^{62} + 459 q^{63} + 1510 q^{64} - 345 q^{66} - 405 q^{67} - 483 q^{68} + 444 q^{69} + 540 q^{71} - 54 q^{72} + 177 q^{73} + 734 q^{74} - 6169 q^{76} - 318 q^{77} + 3705 q^{78} - 4065 q^{79} + 810 q^{81} - 848 q^{82} - 732 q^{83} - 4464 q^{84} - 186 q^{86} + 3054 q^{87} + 554 q^{88} + 1866 q^{89} - 4352 q^{91} + 2541 q^{92} + 141 q^{93} + 702 q^{94} + 2727 q^{96} + 895 q^{97} - 981 q^{98} - 324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 61 x^{8} + 54 x^{7} + 1219 x^{6} - 1069 x^{5} - 8923 x^{4} + 9432 x^{3} + 18852 x^{2} - 30528 x + 10944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 79 \nu^{9} - 39 \nu^{8} + 6221 \nu^{7} + 4296 \nu^{6} - 175309 \nu^{5} - 85407 \nu^{4} + 1995983 \nu^{3} - 103842 \nu^{2} - 7286832 \nu + 4591008 ) / 692160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 839 \nu^{9} - 681 \nu^{8} - 51101 \nu^{7} + 32864 \nu^{6} + 1014149 \nu^{5} - 546273 \nu^{4} - 7315583 \nu^{3} + 3921482 \nu^{2} + 16024512 \nu - 12423648 ) / 1384320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 997 \nu^{9} - 603 \nu^{8} - 63543 \nu^{7} + 24272 \nu^{6} + 1364767 \nu^{5} - 375459 \nu^{4} - 11307549 \nu^{3} + 5513486 \nu^{2} + 30598176 \nu - 38217504 ) / 1384320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10671 \nu^{9} + 6049 \nu^{8} + 647189 \nu^{7} - 269216 \nu^{6} - 12844221 \nu^{5} + 4519417 \nu^{4} + 93820007 \nu^{3} - 40768538 \nu^{2} + \cdots + 161162592 ) / 1384320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10421 \nu^{9} + 879 \nu^{8} + 622939 \nu^{7} + 48484 \nu^{6} - 11974031 \nu^{5} - 1715553 \nu^{4} + 81530257 \nu^{3} + 1514062 \nu^{2} - 162565368 \nu + 81838272 ) / 692160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26193 \nu^{9} + 2767 \nu^{8} + 1578907 \nu^{7} + 30192 \nu^{6} - 30747363 \nu^{5} - 910889 \nu^{4} + 214279081 \nu^{3} - 35300214 \nu^{2} + \cdots + 292159776 ) / 1384320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13863 \nu^{9} - 2463 \nu^{8} + 858757 \nu^{7} + 209192 \nu^{6} - 17509413 \nu^{5} - 3117639 \nu^{4} + 133454791 \nu^{3} - 13634914 \nu^{2} + \cdots + 204857376 ) / 692160 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 34619 \nu^{9} - 9059 \nu^{8} + 2116481 \nu^{7} + 749416 \nu^{6} - 42055169 \nu^{5} - 13617947 \nu^{4} + 302794043 \nu^{3} + 22209718 \nu^{2} + \cdots + 348159648 ) / 1384320 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} - 9\beta_{3} - \beta_{2} + 20\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 3\beta_{7} - \beta_{6} - 2\beta_{5} + 26\beta_{4} - 22\beta_{3} + 36\beta_{2} + 11\beta _1 + 251 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 33 \beta_{9} + 32 \beta_{8} + 5 \beta_{7} + 31 \beta_{6} - 80 \beta_{5} - 7 \beta_{4} - 393 \beta_{3} - 31 \beta_{2} + 480 \beta _1 - 93 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 48 \beta_{9} + 3 \beta_{8} + 117 \beta_{7} - 18 \beta_{6} - 114 \beta_{5} + 650 \beta_{4} - 677 \beta_{3} + 1175 \beta_{2} + 552 \beta _1 + 5967 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 920 \beta_{9} + 884 \beta_{8} + 234 \beta_{7} + 824 \beta_{6} - 2614 \beta_{5} - 264 \beta_{4} - 14004 \beta_{3} - 518 \beta_{2} + 12457 \beta _1 - 2276 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1820 \beta_{9} + 204 \beta_{8} + 3732 \beta_{7} - 20 \beta_{6} - 4672 \beta_{5} + 16465 \beta_{4} - 24449 \beta_{3} + 36537 \beta_{2} + 21172 \beta _1 + 148792 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 25113 \beta_{9} + 23929 \beta_{8} + 8608 \beta_{7} + 21473 \beta_{6} - 80578 \beta_{5} - 7460 \beta_{4} - 457701 \beta_{3} - 401 \beta_{2} + 336528 \beta _1 - 51900 \) Copy content Toggle raw display

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
5.46982
4.54955
2.87444
1.35653
1.04884
0.658275
−2.44553
−2.91436
−4.47014
−5.12743
−5.46982 −3.00000 21.9189 0 16.4094 31.1300 −76.1337 9.00000 0
1.2 −4.54955 −3.00000 12.6984 0 13.6487 8.49133 −21.3758 9.00000 0
1.3 −2.87444 −3.00000 0.262378 0 8.62331 8.99984 22.2413 9.00000 0
1.4 −1.35653 −3.00000 −6.15983 0 4.06959 19.2974 19.2082 9.00000 0
1.5 −1.04884 −3.00000 −6.89993 0 3.14653 −32.0322 15.6277 9.00000 0
1.6 −0.658275 −3.00000 −7.56667 0 1.97482 0.266664 10.2471 9.00000 0
1.7 2.44553 −3.00000 −2.01939 0 −7.33659 −8.35285 −24.5027 9.00000 0
1.8 2.91436 −3.00000 0.493465 0 −8.74307 −19.3569 −21.8767 9.00000 0
1.9 4.47014 −3.00000 11.9821 0 −13.4104 30.1752 17.8006 9.00000 0
1.10 5.12743 −3.00000 18.2905 0 −15.3823 12.3816 52.7640 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.a 10
5.b even 2 1 1875.4.a.d yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.4.a.a 10 1.a even 1 1 trivial
1875.4.a.d yes 10 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + T_{2}^{9} - 61 T_{2}^{8} - 54 T_{2}^{7} + 1219 T_{2}^{6} + 1069 T_{2}^{5} - 8923 T_{2}^{4} - 9432 T_{2}^{3} + 18852 T_{2}^{2} + 30528 T_{2} + 10944 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} - 61 T^{8} - 54 T^{7} + \cdots + 10944 \) Copy content Toggle raw display
$3$ \( (T + 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 51 T^{9} + \cdots - 23688338000 \) Copy content Toggle raw display
$11$ \( T^{10} + 36 T^{9} + \cdots + 43932717994176 \) Copy content Toggle raw display
$13$ \( T^{10} - 41 T^{9} + \cdots - 13\!\cdots\!89 \) Copy content Toggle raw display
$17$ \( T^{10} - 70 T^{9} + \cdots - 30\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 128742355320975 \) Copy content Toggle raw display
$23$ \( T^{10} + 148 T^{9} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + 1018 T^{9} + \cdots - 65\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{10} + 47 T^{9} + \cdots - 85\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{10} - 391 T^{9} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} + 492 T^{9} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} - 919 T^{9} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{10} + 88 T^{9} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{10} - 462 T^{9} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + 596 T^{9} + \cdots - 94\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{10} + 1199 T^{9} + \cdots - 20\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{10} + 405 T^{9} + \cdots - 28\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{10} - 540 T^{9} + \cdots - 29\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{10} - 177 T^{9} + \cdots - 82\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + 4065 T^{9} + \cdots - 47\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{10} + 732 T^{9} + \cdots - 19\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{10} - 1866 T^{9} + \cdots + 45\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{10} - 895 T^{9} + \cdots + 49\!\cdots\!11 \) Copy content Toggle raw display
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