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Properties

Label 975.2.i.n
Level $975$
Weight $2$
Character orbit 975.i
Analytic conductor $7.785$
Analytic rank $0$
Dimension $12$
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] = [975,2,Mod(451,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("975.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.i (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 10x^{10} - 4x^{9} + 79x^{8} - 24x^{7} + 210x^{6} - 38x^{5} + 429x^{4} - 76x^{3} + 58x^{2} + 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} + ( - \beta_{6} - 1) q^{3} + ( - \beta_{9} + \beta_{6}) q^{4} + \beta_1 q^{6} + (\beta_{5} + \beta_1) q^{7} + (\beta_{10} - \beta_{8} + \cdots - \beta_{2}) q^{8}+ \cdots + \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{2} + ( - \beta_{6} - 1) q^{3} + ( - \beta_{9} + \beta_{6}) q^{4} + \beta_1 q^{6} + (\beta_{5} + \beta_1) q^{7} + (\beta_{10} - \beta_{8} + \cdots - \beta_{2}) q^{8}+ \cdots + (\beta_{11} - \beta_{10} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 8 q^{4} + q^{7} + 12 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} - 8 q^{4} + q^{7} + 12 q^{8} - 6 q^{9} - q^{11} + 16 q^{12} - 3 q^{13} + 30 q^{14} - 20 q^{16} + 8 q^{17} + 3 q^{19} - 2 q^{21} - 3 q^{22} - q^{23} - 6 q^{24} + 9 q^{26} + 12 q^{27} + 3 q^{28} - 12 q^{29} + 24 q^{31} - 32 q^{32} - q^{33} - 56 q^{34} - 8 q^{36} - 5 q^{37} + 14 q^{38} - 3 q^{39} + 8 q^{41} - 15 q^{42} - 15 q^{43} + 78 q^{44} - 5 q^{46} - 24 q^{47} - 20 q^{48} - 9 q^{49} - 16 q^{51} + 62 q^{52} - 16 q^{53} - 17 q^{56} - 6 q^{57} - 6 q^{58} + 2 q^{59} - 33 q^{61} - 2 q^{62} + q^{63} + 72 q^{64} + 6 q^{66} + 13 q^{67} + 36 q^{68} - q^{69} + 23 q^{71} - 6 q^{72} - 12 q^{73} - 59 q^{74} + 35 q^{76} - 36 q^{77} - 6 q^{78} + 22 q^{79} - 6 q^{81} + 28 q^{82} - 34 q^{83} + 3 q^{84} - 26 q^{86} - 12 q^{87} - 7 q^{88} - 8 q^{89} + 5 q^{91} + 38 q^{92} - 12 q^{93} - 8 q^{94} + 64 q^{96} - 19 q^{97} + 43 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 10x^{10} - 4x^{9} + 79x^{8} - 24x^{7} + 210x^{6} - 38x^{5} + 429x^{4} - 76x^{3} + 58x^{2} + 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 648578 \nu^{11} - 379348 \nu^{10} + 6324948 \nu^{9} - 5331730 \nu^{8} + 51332742 \nu^{7} + \cdots + 495704777 ) / 169238865 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6005909 \nu^{11} + 1297156 \nu^{10} + 59300394 \nu^{9} - 11373740 \nu^{8} + 463803351 \nu^{7} + \cdots + 50701736 ) / 338477730 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25270874 \nu^{11} - 46204489 \nu^{10} + 257990944 \nu^{9} - 574457815 \nu^{8} + \cdots - 4306765424 ) / 1015433190 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11790614 \nu^{11} + 2847689 \nu^{10} - 110190649 \nu^{9} + 81866840 \nu^{8} + \cdots + 170447984 ) / 338477730 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12675434 \nu^{11} - 6005909 \nu^{10} + 125457184 \nu^{9} - 110002130 \nu^{8} + 1012733026 \nu^{7} + \cdots - 253212554 ) / 338477730 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32118538 \nu^{11} + 13879607 \nu^{10} + 325731518 \nu^{9} + 10946045 \nu^{8} + 2523312407 \nu^{7} + \cdots + 295482512 ) / 338477730 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19994486 \nu^{11} + 4074691 \nu^{10} - 202244509 \nu^{9} + 121094095 \nu^{8} + \cdots - 301770310 ) / 203086638 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 36729146 \nu^{11} - 17259031 \nu^{10} + 363721656 \nu^{9} - 319342930 \nu^{8} + 2935533594 \nu^{7} + \cdots - 735614026 ) / 338477730 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 186169597 \nu^{11} - 1359973 \nu^{10} - 1862778767 \nu^{9} + 744086195 \nu^{8} + \cdots + 704851072 ) / 1015433190 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 212435956 \nu^{11} - 71185556 \nu^{10} + 2118812021 \nu^{9} - 1564069745 \nu^{8} + \cdots - 682424656 ) / 1015433190 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 3\beta_{6} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{8} + 5\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 8\beta_{9} + 16\beta_{6} - \beta_{5} + \beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{11} - 9 \beta_{10} + 12 \beta_{9} - \beta_{7} - 3 \beta_{6} + \beta_{4} - 29 \beta_{3} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{11} + 2 \beta_{10} - 2 \beta_{8} - 10 \beta_{7} + 10 \beta_{5} - 20 \beta_{4} + 6 \beta_{3} + \cdots + 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 81\beta_{11} - 111\beta_{9} + 69\beta_{8} + 50\beta_{6} + 8\beta_{5} + 12\beta_{4} + 280\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 196 \beta_{11} - 34 \beta_{10} + 464 \beta_{9} + 77 \beta_{7} - 666 \beta_{6} + 81 \beta_{4} + \cdots - 666 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 115 \beta_{11} + 507 \beta_{10} - 507 \beta_{8} + 43 \beta_{7} - 43 \beta_{5} - 230 \beta_{4} + \cdots + 571 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1022\beta_{11} - 3549\beta_{9} + 400\beta_{8} + 4701\beta_{6} - 550\beta_{5} + 622\beta_{4} + 2756\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5743 \beta_{11} - 3699 \beta_{10} + 7877 \beta_{9} - 150 \beta_{7} - 5602 \beta_{6} + 1022 \beta_{4} + \cdots - 5602 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\beta_{6}\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−1.40603 − 2.43531i
−0.813080 − 1.40830i
−0.115825 − 0.200615i
0.215564 + 0.373368i
0.892010 + 1.54501i
1.22736 + 2.12585i
−1.40603 + 2.43531i
−0.813080 + 1.40830i
−0.115825 + 0.200615i
0.215564 − 0.373368i
0.892010 − 1.54501i
1.22736 − 2.12585i
−1.40603 + 2.43531i −0.500000 + 0.866025i −2.95384 − 5.11620i 0 −1.40603 − 2.43531i −0.333570 − 0.577759i 10.9886 −0.500000 − 0.866025i 0
451.2 −0.813080 + 1.40830i −0.500000 + 0.866025i −0.322200 − 0.558066i 0 −0.813080 − 1.40830i −1.54239 − 2.67150i −2.20443 −0.500000 − 0.866025i 0
451.3 −0.115825 + 0.200615i −0.500000 + 0.866025i 0.973169 + 1.68558i 0 −0.115825 − 0.200615i −0.580982 − 1.00629i −0.914171 −0.500000 − 0.866025i 0
451.4 0.215564 − 0.373368i −0.500000 + 0.866025i 0.907064 + 1.57108i 0 0.215564 + 0.373368i 2.03980 + 3.53303i 1.64438 −0.500000 − 0.866025i 0
451.5 0.892010 − 1.54501i −0.500000 + 0.866025i −0.591364 − 1.02427i 0 0.892010 + 1.54501i −1.17557 − 2.03615i 1.45803 −0.500000 − 0.866025i 0
451.6 1.22736 − 2.12585i −0.500000 + 0.866025i −2.01283 − 3.48633i 0 1.22736 + 2.12585i 2.09272 + 3.62470i −4.97244 −0.500000 − 0.866025i 0
601.1 −1.40603 − 2.43531i −0.500000 − 0.866025i −2.95384 + 5.11620i 0 −1.40603 + 2.43531i −0.333570 + 0.577759i 10.9886 −0.500000 + 0.866025i 0
601.2 −0.813080 − 1.40830i −0.500000 − 0.866025i −0.322200 + 0.558066i 0 −0.813080 + 1.40830i −1.54239 + 2.67150i −2.20443 −0.500000 + 0.866025i 0
601.3 −0.115825 − 0.200615i −0.500000 − 0.866025i 0.973169 − 1.68558i 0 −0.115825 + 0.200615i −0.580982 + 1.00629i −0.914171 −0.500000 + 0.866025i 0
601.4 0.215564 + 0.373368i −0.500000 − 0.866025i 0.907064 − 1.57108i 0 0.215564 − 0.373368i 2.03980 − 3.53303i 1.64438 −0.500000 + 0.866025i 0
601.5 0.892010 + 1.54501i −0.500000 − 0.866025i −0.591364 + 1.02427i 0 0.892010 − 1.54501i −1.17557 + 2.03615i 1.45803 −0.500000 + 0.866025i 0
601.6 1.22736 + 2.12585i −0.500000 − 0.866025i −2.01283 + 3.48633i 0 1.22736 − 2.12585i 2.09272 − 3.62470i −4.97244 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.6
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 975.2.i.n ✓ 12
5.b even 2 1 975.2.i.p yes 12
5.c odd 4 2 975.2.bb.l 24
13.c even 3 1 inner 975.2.i.n ✓ 12
65.n even 6 1 975.2.i.p yes 12
65.q odd 12 2 975.2.bb.l 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.i.n ✓ 12 1.a even 1 1 trivial
975.2.i.n ✓ 12 13.c even 3 1 inner
975.2.i.p yes 12 5.b even 2 1
975.2.i.p yes 12 65.n even 6 1
975.2.bb.l 24 5.c odd 4 2
975.2.bb.l 24 65.q odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{12} + 10 T_{2}^{10} - 4 T_{2}^{9} + 79 T_{2}^{8} - 24 T_{2}^{7} + 210 T_{2}^{6} - 38 T_{2}^{5} + \cdots + 4 \) Copy content Toggle raw display
\( T_{7}^{12} - T_{7}^{11} + 26 T_{7}^{10} + 39 T_{7}^{9} + 450 T_{7}^{8} + 763 T_{7}^{7} + 4693 T_{7}^{6} + \cdots + 9216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 10 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - T^{11} + \cdots + 9216 \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + \cdots + 20502784 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + \cdots + 861184 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} + \cdots + 2696164 \) Copy content Toggle raw display
$23$ \( T^{12} + T^{11} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{12} + 12 T^{11} + \cdots + 90326016 \) Copy content Toggle raw display
$31$ \( (T^{6} - 12 T^{5} + \cdots - 3648)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2347208704 \) Copy content Toggle raw display
$41$ \( T^{12} - 8 T^{11} + \cdots + 861184 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 149426176 \) Copy content Toggle raw display
$47$ \( (T^{6} + 12 T^{5} + \cdots + 61696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 8 T^{5} + \cdots - 20768)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 2 T^{11} + \cdots + 262144 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 446392384 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 20694548736 \) Copy content Toggle raw display
$71$ \( T^{12} - 23 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + \cdots - 17803)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 11 T^{5} + \cdots - 70818)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 17 T^{5} + \cdots + 116448)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1900692366336 \) Copy content Toggle raw display
$97$ \( T^{12} + 19 T^{11} + \cdots + 11888704 \) Copy content Toggle raw display
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