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Properties

Label 325.4.d.c
Level $325$
Weight $4$
Character orbit 325.d
Analytic conductor $19.176$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [325,4,Mod(324,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.324"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 325.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.1756207519\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 84x^{12} + 2674x^{10} + 40048x^{8} + 278769x^{6} + 727552x^{4} + 339456x^{2} + 9216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{6} q^{3} + ( - \beta_{2} + 4) q^{4} + ( - \beta_{12} - \beta_{8}) q^{6} + (\beta_{7} - \beta_{5} + \beta_{3} + 8) q^{7} + ( - \beta_{4} + 4 \beta_{3} + \beta_{2} - 2) q^{8}+ \cdots + (43 \beta_{13} - 3 \beta_{12} + \cdots - 27 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{2} + 56 q^{4} + 108 q^{7} - 48 q^{8} - 158 q^{9} + 6 q^{13} + 152 q^{14} + 280 q^{16} - 272 q^{18} - 344 q^{26} + 572 q^{28} - 588 q^{29} - 1788 q^{32} + 248 q^{33} + 496 q^{36} + 940 q^{37}+ \cdots + 7364 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 84x^{12} + 2674x^{10} + 40048x^{8} + 278769x^{6} + 727552x^{4} + 339456x^{2} + 9216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 331 \nu^{12} + 17100 \nu^{10} + 135334 \nu^{8} - 4575008 \nu^{6} - 70189989 \nu^{4} + \cdots - 19036416 ) / 81415200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1021 \nu^{12} - 101940 \nu^{10} - 3781054 \nu^{8} - 63490672 \nu^{6} - 456526881 \nu^{4} + \cdots + 107593296 ) / 20353800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1069 \nu^{12} + 72444 \nu^{10} + 1716106 \nu^{8} + 17134688 \nu^{6} + 67868029 \nu^{4} + \cdots + 20876288 ) / 2171072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 85961 \nu^{13} - 10663860 \nu^{11} - 455553794 \nu^{9} - 8486131952 \nu^{7} + \cdots - 102549673344 \nu ) / 1953964800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9121 \nu^{12} - 552170 \nu^{10} - 10325274 \nu^{8} - 49992972 \nu^{6} + 270744159 \nu^{4} + \cdots + 500690176 ) / 13569200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24787 \nu^{13} + 2086080 \nu^{11} + 66485638 \nu^{9} + 994293784 \nu^{7} + 6854947107 \nu^{5} + \cdots + 6236239488 \nu ) / 488491200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28759 \nu^{12} - 1884204 \nu^{10} - 42056782 \nu^{8} - 373558048 \nu^{6} - 1106587287 \nu^{4} + \cdots + 59258208 ) / 16283040 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 125921 \nu^{13} - 10533000 \nu^{11} - 333240194 \nu^{9} - 4944018872 \nu^{7} + \cdots - 21297154944 \nu ) / 488491200 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12559 \nu^{13} - 1051590 \nu^{11} - 33310486 \nu^{9} - 494859388 \nu^{7} + \cdots - 3149309136 \nu ) / 40707600 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66328 \nu^{13} - 4596495 \nu^{11} - 113152372 \nu^{9} - 1212399646 \nu^{7} + \cdots - 2733351072 \nu ) / 122122800 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1372879 \nu^{13} - 108533580 \nu^{11} - 3229476526 \nu^{9} - 45256907728 \nu^{7} + \cdots - 407606424576 \nu ) / 1953964800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + 2\beta_{10} - 2\beta_{8} - 21\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 4\beta_{5} + 3\beta_{4} - 13\beta_{3} - 25\beta_{2} + 239 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{13} - 12\beta_{12} + 104\beta_{11} - 64\beta_{10} + 196\beta_{8} - 44\beta_{6} + 505\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -52\beta_{9} + 12\beta_{7} - 188\beta_{5} - 120\beta_{4} + 682\beta_{3} + 639\beta_{2} - 5320 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -208\beta_{13} + 560\beta_{12} - 3834\beta_{11} + 1838\beta_{10} - 9126\beta_{8} + 2000\beta_{6} - 12933\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1847\beta_{9} - 560\beta_{7} + 6476\beta_{5} + 3931\beta_{4} - 26011\beta_{3} - 16889\beta_{2} + 127477 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7388 \beta_{13} - 19360 \beta_{12} + 125756 \beta_{11} - 52076 \beta_{10} + 335844 \beta_{8} + \cdots + 344885 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 57940 \beta_{9} + 19360 \beta_{7} - 200952 \beta_{5} - 120940 \beta_{4} + 875400 \beta_{3} + \cdots - 3231832 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 231760 \beta_{13} + 605344 \beta_{12} - 3906234 \beta_{11} + 1478354 \beta_{10} + \cdots - 9454325 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1731425 \beta_{9} - 605344 \beta_{7} + 5983428 \beta_{5} + 3618263 \beta_{4} - 27673917 \beta_{3} + \cdots + 85467975 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 6925700 \beta_{13} - 18161908 \beta_{12} + 117857736 \beta_{11} - 42152624 \beta_{10} + \cdots + 264042385 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
5.37688i
− 5.37688i
− 3.65500i
3.65500i
2.10835i
− 2.10835i
0.170066i
− 0.170066i
0.742335i
− 0.742335i
− 4.27643i
4.27643i
4.29153i
− 4.29153i
−5.37688 − 0.936724i 20.9108 0 5.03665i 0.201771 −69.4197 26.1225 0
324.2 −5.37688 0.936724i 20.9108 0 − 5.03665i 0.201771 −69.4197 26.1225 0
324.3 −3.65500 − 8.59394i 5.35901 0 31.4108i 19.4590 9.65282 −46.8559 0
324.4 −3.65500 8.59394i 5.35901 0 − 31.4108i 19.4590 9.65282 −46.8559 0
324.5 −2.10835 − 3.08338i −3.55484 0 6.50086i −17.4384 24.3617 17.4928 0
324.6 −2.10835 3.08338i −3.55484 0 − 6.50086i −17.4384 24.3617 17.4928 0
324.7 −0.170066 − 8.43460i −7.97108 0 1.43444i 32.7758 2.71614 −44.1424 0
324.8 −0.170066 8.43460i −7.97108 0 − 1.43444i 32.7758 2.71614 −44.1424 0
324.9 0.742335 − 5.13862i −7.44894 0 − 3.81458i −9.97127 −11.4683 0.594575 0
324.10 0.742335 5.13862i −7.44894 0 3.81458i −9.97127 −11.4683 0.594575 0
324.11 4.27643 − 7.17327i 10.2878 0 − 30.6760i −5.20875 9.78383 −24.4558 0
324.12 4.27643 7.17327i 10.2878 0 30.6760i −5.20875 9.78383 −24.4558 0
324.13 4.29153 − 5.89541i 10.4172 0 − 25.3003i 34.1818 10.3735 −7.75582 0
324.14 4.29153 5.89541i 10.4172 0 25.3003i 34.1818 10.3735 −7.75582 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.4.d.c 14
5.b even 2 1 325.4.d.d 14
5.c odd 4 1 65.4.c.a ✓ 14
5.c odd 4 1 325.4.c.e 14
13.b even 2 1 325.4.d.d 14
15.e even 4 1 585.4.b.e 14
20.e even 4 1 1040.4.k.d 14
65.d even 2 1 inner 325.4.d.c 14
65.f even 4 1 845.4.a.l 7
65.h odd 4 1 65.4.c.a ✓ 14
65.h odd 4 1 325.4.c.e 14
65.k even 4 1 845.4.a.i 7
195.s even 4 1 585.4.b.e 14
260.p even 4 1 1040.4.k.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.c.a ✓ 14 5.c odd 4 1
65.4.c.a ✓ 14 65.h odd 4 1
325.4.c.e 14 5.c odd 4 1
325.4.c.e 14 65.h odd 4 1
325.4.d.c 14 1.a even 1 1 trivial
325.4.d.c 14 65.d even 2 1 inner
325.4.d.d 14 5.b even 2 1
325.4.d.d 14 13.b even 2 1
585.4.b.e 14 15.e even 4 1
585.4.b.e 14 195.s even 4 1
845.4.a.i 7 65.k even 4 1
845.4.a.l 7 65.f even 4 1
1040.4.k.d 14 20.e even 4 1
1040.4.k.d 14 260.p even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 2T_{2}^{6} - 40T_{2}^{5} - 64T_{2}^{4} + 409T_{2}^{3} + 568T_{2}^{2} - 480T_{2} - 96 \) acting on \(S_{4}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{7} + 2 T^{6} - 40 T^{5} + \cdots - 96)^{2} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 2069886016 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T^{7} - 54 T^{6} + \cdots + 3984000)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 24\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 440579141474400)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 877152531948576)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 94\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 71\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 35\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots + 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 65\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 34\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 13\!\cdots\!72)^{2} \) Copy content Toggle raw display
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