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Properties

Label 325.2.e.d
Level $325$
Weight $2$
Character orbit 325.e
Analytic conductor $2.595$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,2,Mod(126,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.126");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.e (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 8x^{8} - 2x^{7} + 52x^{6} - 5x^{5} + 97x^{4} + 60x^{3} + 141x^{2} + 36x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{8} - \beta_{6} + \beta_{4} + 1) q^{3} + (\beta_{9} - \beta_{6}) q^{4} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - 2 \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{8} - \beta_{6} + \beta_{4} + 1) q^{3} + (\beta_{9} - \beta_{6}) q^{4} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{6}+ \cdots + (3 \beta_{5} + 5 \beta_{4} - \beta_{3} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{3} - 6 q^{4} - 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{3} - 6 q^{4} - 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} - 3 q^{11} - 4 q^{12} - 10 q^{13} - 16 q^{14} - 4 q^{16} + 4 q^{17} + 4 q^{18} + 4 q^{19} - 16 q^{21} + 8 q^{22} + 15 q^{23} + 14 q^{24} - 9 q^{26} - 42 q^{27} - 17 q^{28} + q^{29} + 31 q^{32} - 2 q^{33} + 54 q^{34} + 13 q^{36} + 17 q^{37} - 6 q^{38} - 10 q^{39} - 6 q^{41} + 32 q^{42} + 12 q^{43} - 16 q^{44} + 7 q^{46} + 24 q^{47} - 2 q^{48} - 7 q^{49} - 23 q^{52} - 16 q^{53} - 19 q^{54} + 17 q^{56} + 8 q^{57} + 38 q^{58} + 12 q^{59} - 5 q^{61} + 13 q^{62} + 26 q^{63} - 10 q^{64} + 86 q^{66} - 16 q^{67} + 25 q^{68} - 20 q^{69} - 19 q^{71} - 45 q^{72} + 16 q^{73} - 2 q^{74} - 24 q^{76} - 64 q^{77} - 42 q^{78} - 28 q^{79} - 29 q^{81} - 23 q^{82} + 14 q^{83} + 34 q^{84} - 84 q^{86} + 21 q^{87} + 2 q^{88} + 10 q^{89} + 17 q^{91} - 142 q^{92} - 33 q^{93} + 31 q^{94} + 34 q^{96} + 37 q^{97} - 21 q^{98} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 8x^{8} - 2x^{7} + 52x^{6} - 5x^{5} + 97x^{4} + 60x^{3} + 141x^{2} + 36x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1304 \nu^{9} + 1776 \nu^{8} - 1628 \nu^{7} + 8447 \nu^{6} - 14800 \nu^{5} + 93980 \nu^{4} + \cdots - 242496 ) / 983355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 592 \nu^{9} - 4020 \nu^{8} + 3685 \nu^{7} - 27536 \nu^{6} + 33500 \nu^{5} - 212725 \nu^{4} + \cdots - 987267 ) / 327785 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4744 \nu^{9} - 20940 \nu^{8} + 19195 \nu^{7} - 124843 \nu^{6} + 174500 \nu^{5} - 1108075 \nu^{4} + \cdots - 1749321 ) / 983355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3213 \nu^{9} - 12516 \nu^{8} + 11473 \nu^{7} - 129958 \nu^{6} + 104300 \nu^{5} - 662305 \nu^{4} + \cdots - 553661 ) / 327785 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + \cdots + 1012608 ) / 983355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47884 \nu^{9} + 55843 \nu^{8} - 351659 \nu^{7} + 476704 \nu^{6} - 2541330 \nu^{5} + 2272140 \nu^{4} + \cdots + 13443 ) / 983355 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 75292 \nu^{9} - 9639 \nu^{8} - 564788 \nu^{7} + 116165 \nu^{6} - 3525310 \nu^{5} + \cdots - 1809360 ) / 983355 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + \cdots + 1012608 ) / 327785 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 3\beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{9} - \beta_{8} - \beta_{7} + 14\beta_{6} + \beta_{4} - 6\beta_{3} + 3\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} + 8\beta_{8} - 6\beta_{6} + 28\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{5} - 9\beta_{4} + 37\beta_{3} - 24\beta_{2} + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -69\beta_{9} - 53\beta_{8} - \beta_{7} + 71\beta_{6} + 53\beta_{4} - 69\beta_{3} + 167\beta _1 - 71 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 236\beta_{9} + 71\beta_{8} + 52\beta_{7} - 446\beta_{6} + 52\beta_{5} + 211\beta_{2} - 263\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -19\beta_{5} - 340\beta_{4} + 499\beta_{3} - 1022\beta_{2} + 614 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
1.08066 − 1.87176i
0.904178 − 1.56608i
−0.134432 + 0.232843i
−0.547998 + 0.949161i
−1.30241 + 2.25583i
1.08066 + 1.87176i
0.904178 + 1.56608i
−0.134432 − 0.232843i
−0.547998 − 0.949161i
−1.30241 − 2.25583i
−1.08066 + 1.87176i 0.980433 − 1.69816i −1.33565 − 2.31341i 0 2.11903 + 3.67026i −1.53837 − 2.66453i 1.45089 −0.422497 − 0.731787i 0
126.2 −0.904178 + 1.56608i −0.929015 + 1.60910i −0.635076 − 1.09998i 0 −1.67999 − 2.90983i 2.08417 + 3.60988i −1.31983 −0.226138 − 0.391682i 0
126.3 0.134432 − 0.232843i −0.301414 + 0.522064i 0.963856 + 1.66945i 0 0.0810394 + 0.140364i −0.715471 − 1.23923i 1.05602 1.31830 + 2.28336i 0
126.4 0.547998 − 0.949161i 1.68234 − 2.91389i 0.399395 + 0.691773i 0 −1.84383 − 3.19362i 0.795836 + 1.37843i 3.06747 −4.16051 − 7.20621i 0
126.5 1.30241 − 2.25583i 0.0676602 − 0.117191i −2.39253 − 4.14398i 0 −0.176242 − 0.305261i −1.62616 − 2.81660i −7.25455 1.49084 + 2.58222i 0
276.1 −1.08066 − 1.87176i 0.980433 + 1.69816i −1.33565 + 2.31341i 0 2.11903 − 3.67026i −1.53837 + 2.66453i 1.45089 −0.422497 + 0.731787i 0
276.2 −0.904178 − 1.56608i −0.929015 − 1.60910i −0.635076 + 1.09998i 0 −1.67999 + 2.90983i 2.08417 − 3.60988i −1.31983 −0.226138 + 0.391682i 0
276.3 0.134432 + 0.232843i −0.301414 − 0.522064i 0.963856 − 1.66945i 0 0.0810394 − 0.140364i −0.715471 + 1.23923i 1.05602 1.31830 − 2.28336i 0
276.4 0.547998 + 0.949161i 1.68234 + 2.91389i 0.399395 − 0.691773i 0 −1.84383 + 3.19362i 0.795836 − 1.37843i 3.06747 −4.16051 + 7.20621i 0
276.5 1.30241 + 2.25583i 0.0676602 + 0.117191i −2.39253 + 4.14398i 0 −0.176242 + 0.305261i −1.62616 + 2.81660i −7.25455 1.49084 − 2.58222i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 126.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.e.d yes 10
5.b even 2 1 325.2.e.c ✓ 10
5.c odd 4 2 325.2.o.c 20
13.c even 3 1 inner 325.2.e.d yes 10
13.c even 3 1 4225.2.a.bn 5
13.e even 6 1 4225.2.a.bm 5
65.l even 6 1 4225.2.a.bo 5
65.n even 6 1 325.2.e.c ✓ 10
65.n even 6 1 4225.2.a.bp 5
65.q odd 12 2 325.2.o.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.e.c ✓ 10 5.b even 2 1
325.2.e.c ✓ 10 65.n even 6 1
325.2.e.d yes 10 1.a even 1 1 trivial
325.2.e.d yes 10 13.c even 3 1 inner
325.2.o.c 20 5.c odd 4 2
325.2.o.c 20 65.q odd 12 2
4225.2.a.bm 5 13.e even 6 1
4225.2.a.bn 5 13.c even 3 1
4225.2.a.bo 5 65.l even 6 1
4225.2.a.bp 5 65.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 8T_{2}^{8} + 2T_{2}^{7} + 52T_{2}^{6} + 5T_{2}^{5} + 97T_{2}^{4} - 60T_{2}^{3} + 141T_{2}^{2} - 36T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 8 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 2 T^{9} + \cdots + 9025 \) Copy content Toggle raw display
$11$ \( T^{10} + 3 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{10} + 10 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} - 4 T^{9} + \cdots + 239121 \) Copy content Toggle raw display
$19$ \( T^{10} - 4 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$23$ \( T^{10} - 15 T^{9} + \cdots + 281961 \) Copy content Toggle raw display
$29$ \( T^{10} - T^{9} + \cdots + 881721 \) Copy content Toggle raw display
$31$ \( (T^{5} - 57 T^{3} + \cdots + 225)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 17 T^{9} + \cdots + 826281 \) Copy content Toggle raw display
$41$ \( T^{10} + 6 T^{9} + \cdots + 50625 \) Copy content Toggle raw display
$43$ \( T^{10} - 12 T^{9} + \cdots + 14190289 \) Copy content Toggle raw display
$47$ \( (T^{5} - 12 T^{4} + \cdots - 2025)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 8 T^{4} + \cdots - 6075)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 12 T^{9} + \cdots + 4782969 \) Copy content Toggle raw display
$61$ \( T^{10} + 5 T^{9} + \cdots + 403225 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 2567550241 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 101787921 \) Copy content Toggle raw display
$73$ \( (T^{5} - 8 T^{4} + \cdots - 10125)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 14 T^{4} + \cdots + 16875)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 7 T^{4} + \cdots + 53775)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 10 T^{9} + \cdots + 2537649 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1683378841 \) Copy content Toggle raw display
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