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Properties

Label 1875.4.a.g
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: \( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} + (\beta_{8} + \beta_{5} + \beta_{3} - 2 \beta_1 - 1) q^{7} + (\beta_{6} + 3 \beta_{3} + 3 \beta_1 + 3) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} + (\beta_{8} + \beta_{5} + \beta_{3} - 2 \beta_1 - 1) q^{7} + (\beta_{6} + 3 \beta_{3} + 3 \beta_1 + 3) q^{8} + 9 q^{9} + ( - \beta_{13} - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 4) q^{11} + (3 \beta_{2} + 12) q^{12} + (\beta_{13} - 2 \beta_{12} + 2 \beta_{7} + \beta_{6} - \beta_{4} - 5 \beta_{3} + \beta_1 - 15) q^{13} + ( - \beta_{13} + 3 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - 3 \beta_{5} + \cdots - 26) q^{14}+ \cdots + ( - 9 \beta_{13} - 18 \beta_{9} - 9 \beta_{8} + 9 \beta_{7} - 9 \beta_{6} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} - 33 q^{11} + 150 q^{12} - 188 q^{13} - 287 q^{14} + 82 q^{16} - 146 q^{17} - 184 q^{19} - 81 q^{21} - 277 q^{22} - 164 q^{23} + 63 q^{24} + 103 q^{26} + 378 q^{27} + 224 q^{28} + 252 q^{29} - 889 q^{31} - 422 q^{32} - 99 q^{33} - 230 q^{34} + 450 q^{36} - 642 q^{37} - 1833 q^{38} - 564 q^{39} - 164 q^{41} - 861 q^{42} - 696 q^{43} - 6 q^{44} - 419 q^{46} + 92 q^{47} + 246 q^{48} + 81 q^{49} - 438 q^{51} - 1956 q^{52} - 949 q^{53} - 2380 q^{56} - 552 q^{57} - 1041 q^{58} - 81 q^{59} - 496 q^{61} - 1454 q^{62} - 243 q^{63} - 3903 q^{64} - 831 q^{66} - 1926 q^{67} - 685 q^{68} - 492 q^{69} + 2498 q^{71} + 189 q^{72} + 1026 q^{73} - 707 q^{74} - 5704 q^{76} - 5434 q^{77} + 309 q^{78} - 695 q^{79} + 1134 q^{81} + 886 q^{82} - 5315 q^{83} + 672 q^{84} + 3997 q^{86} + 756 q^{87} - 2969 q^{88} - 1424 q^{89} - 1194 q^{91} - 1607 q^{92} - 2667 q^{93} - 629 q^{94} - 1266 q^{96} - 291 q^{97} + 1018 q^{98} - 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 154777191 \nu^{13} + 214678293 \nu^{12} + 12768694032 \nu^{11} - 14915595999 \nu^{10} - 404069408515 \nu^{9} + \cdots - 29392108806288 ) / 9573422384000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 39842017881 \nu^{13} + 18793858963 \nu^{12} + 3381917381312 \nu^{11} + 617153824991 \nu^{10} - 107039813741365 \nu^{9} + \cdots - 61\!\cdots\!08 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 214678293 \nu^{13} - 231741561 \nu^{12} + 15999036336 \nu^{11} + 15269104723 \nu^{10} - 450292207345 \nu^{9} + \cdots - 6510547762224 ) / 9573422384000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 464331573 \nu^{13} - 644034879 \nu^{12} - 38306082096 \nu^{11} + 44746787997 \nu^{10} + 1212208225545 \nu^{9} + \cdots + 59456059266864 ) / 9573422384000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 52774069869 \nu^{13} - 146787187287 \nu^{12} - 4038749648088 \nu^{11} + 10590731904141 \nu^{10} + 114153083031385 \nu^{9} + \cdots - 24\!\cdots\!08 ) / 679712989264000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 198268707541 \nu^{13} - 330612632143 \nu^{12} - 15446080755832 \nu^{11} + 24130842178949 \nu^{10} + 453486628821265 \nu^{9} + \cdots - 30\!\cdots\!12 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24723383429 \nu^{13} - 19698303673 \nu^{12} + 1994905817228 \nu^{11} + 1654834981079 \nu^{10} - 61648333032485 \nu^{9} + \cdots - 91\!\cdots\!52 ) / 203913896779200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8146109293 \nu^{13} + 6363639039 \nu^{12} + 663666222536 \nu^{11} - 440450615677 \nu^{10} - 20765197179345 \nu^{9} + \cdots - 212680826375024 ) / 53661551784000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2089067853 \nu^{13} + 1186234119 \nu^{12} + 166990685056 \nu^{11} - 90007487317 \nu^{10} - 5108559447745 \nu^{9} + \cdots - 228794009350704 ) / 9573422384000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 268898217 \nu^{13} - 60162011 \nu^{12} - 21662421664 \nu^{11} + 3229526593 \nu^{10} + 662126520605 \nu^{9} - 31650082226 \nu^{8} + \cdots + 16216865165616 ) / 957342238400 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 629711814493 \nu^{13} + 211929067839 \nu^{12} + 51141259110536 \nu^{11} - 12148429797277 \nu^{10} + \cdots - 10\!\cdots\!24 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 3\beta_{3} + 19\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} - \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 23 \beta_{2} - 3 \beta _1 + 229 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - 5 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 4 \beta_{7} + 34 \beta_{6} - 15 \beta_{5} - 4 \beta_{4} + 78 \beta_{3} - 2 \beta_{2} + 400 \beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 41 \beta_{13} - 38 \beta_{11} + 28 \beta_{10} + 76 \beta_{9} - 34 \beta_{8} + \beta_{7} - 74 \beta_{6} - 9 \beta_{5} - 72 \beta_{4} + 141 \beta_{3} + 515 \beta_{2} - 164 \beta _1 + 4850 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 122 \beta_{13} + 49 \beta_{12} + 146 \beta_{11} - 202 \beta_{10} + 106 \beta_{9} + 128 \beta_{8} - 180 \beta_{7} + 931 \beta_{6} - 724 \beta_{5} - 174 \beta_{4} + 1565 \beta_{3} - 104 \beta_{2} + 8783 \beta _1 + 636 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1257 \beta_{13} - 9 \beta_{12} - 1152 \beta_{11} + 672 \beta_{10} + 2204 \beta_{9} - 872 \beta_{8} + 23 \beta_{7} - 2125 \beta_{6} + 421 \beta_{5} - 2046 \beta_{4} + 6944 \beta_{3} + 11581 \beta_{2} + \cdots + 107016 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3623 \beta_{13} + 1534 \beta_{12} + 5048 \beta_{11} - 6120 \beta_{10} + 3642 \beta_{9} + 4046 \beta_{8} - 5813 \beta_{7} + 23756 \beta_{6} - 24641 \beta_{5} - 5368 \beta_{4} + 27815 \beta_{3} - 3748 \beta_{2} + \cdots - 5982 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 34186 \beta_{13} - 247 \beta_{12} - 32506 \beta_{11} + 15550 \beta_{10} + 58426 \beta_{9} - 20316 \beta_{8} - 212 \beta_{7} - 56078 \beta_{6} + 26260 \beta_{5} - 53402 \beta_{4} + 239954 \beta_{3} + \cdots + 2413393 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 95312 \beta_{13} + 39355 \beta_{12} + 152581 \beta_{11} - 166261 \beta_{10} + 104498 \beta_{9} + 114853 \beta_{8} - 163757 \beta_{7} + 587178 \beta_{6} - 732762 \beta_{5} - 144636 \beta_{4} + \cdots - 620070 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 873549 \beta_{13} + 1297 \beta_{12} - 886792 \beta_{11} + 355354 \beta_{10} + 1487098 \beta_{9} - 455066 \beta_{8} - 37205 \beta_{7} - 1425812 \beta_{6} + 998693 \beta_{5} + \cdots + 55193599 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2359274 \beta_{13} + 891798 \beta_{12} + 4304259 \beta_{11} - 4273639 \beta_{10} + 2722752 \beta_{9} + 3092949 \beta_{8} - 4288261 \beta_{7} + 14271698 \beta_{6} - 20388722 \beta_{5} + \cdots - 25133913 \) 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
−4.95189
−4.79301
−4.01755
−2.93781
−2.33957
−0.955230
−0.574607
−0.134967
1.33976
1.73740
3.90684
4.24928
4.65416
4.81720
−4.95189 3.00000 16.5212 0 −14.8557 28.2766 −42.1962 9.00000 0
1.2 −4.79301 3.00000 14.9730 0 −14.3790 0.140520 −33.4216 9.00000 0
1.3 −4.01755 3.00000 8.14069 0 −12.0526 1.75849 −0.565245 9.00000 0
1.4 −2.93781 3.00000 0.630743 0 −8.81344 18.9115 21.6495 9.00000 0
1.5 −2.33957 3.00000 −2.52642 0 −7.01870 −32.9322 24.6273 9.00000 0
1.6 −0.955230 3.00000 −7.08754 0 −2.86569 12.4836 14.4121 9.00000 0
1.7 −0.574607 3.00000 −7.66983 0 −1.72382 −2.67744 9.00399 9.00000 0
1.8 −0.134967 3.00000 −7.98178 0 −0.404902 −17.3099 2.15702 9.00000 0
1.9 1.33976 3.00000 −6.20504 0 4.01928 −12.2101 −19.0313 9.00000 0
1.10 1.73740 3.00000 −4.98145 0 5.21219 7.66213 −22.5539 9.00000 0
1.11 3.90684 3.00000 7.26339 0 11.7205 22.0918 −2.87782 9.00000 0
1.12 4.24928 3.00000 10.0564 0 12.7478 −28.2853 8.73812 9.00000 0
1.13 4.65416 3.00000 13.6612 0 13.9625 −26.0445 26.3483 9.00000 0
1.14 4.81720 3.00000 15.2054 0 14.4516 1.13492 34.7100 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.g 14
5.b even 2 1 1875.4.a.f 14
25.d even 5 2 75.4.g.b ✓ 28
75.j odd 10 2 225.4.h.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.g.b ✓ 28 25.d even 5 2
225.4.h.a 28 75.j odd 10 2
1875.4.a.f 14 5.b even 2 1
1875.4.a.g 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 81 T_{2}^{12} - 7 T_{2}^{11} + 2512 T_{2}^{10} + 517 T_{2}^{9} - 36970 T_{2}^{8} - 12987 T_{2}^{7} + 257291 T_{2}^{6} + 125779 T_{2}^{5} - 718713 T_{2}^{4} - 371750 T_{2}^{3} + 579848 T_{2}^{2} + \cdots + 42064 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 81 T^{12} - 7 T^{11} + \cdots + 42064 \) Copy content Toggle raw display
$3$ \( (T - 3)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + 27 T^{13} + \cdots + 4350562367600 \) Copy content Toggle raw display
$11$ \( T^{14} + 33 T^{13} + \cdots + 57\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{14} + 188 T^{13} + \cdots - 21\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{14} + 146 T^{13} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{14} + 184 T^{13} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + 164 T^{13} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} - 252 T^{13} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + 889 T^{13} + \cdots - 98\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + 642 T^{13} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} + 164 T^{13} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + 696 T^{13} + \cdots - 55\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{14} - 92 T^{13} + \cdots - 35\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{14} + 949 T^{13} + \cdots - 14\!\cdots\!75 \) Copy content Toggle raw display
$59$ \( T^{14} + 81 T^{13} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + 496 T^{13} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{14} + 1926 T^{13} + \cdots - 22\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{14} - 2498 T^{13} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{14} - 1026 T^{13} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + 695 T^{13} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + 5315 T^{13} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{14} + 1424 T^{13} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + 291 T^{13} + \cdots + 17\!\cdots\!71 \) Copy content Toggle raw display
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