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Properties

Label 975.2.i.o
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: \( x^{12} + 8x^{10} + 48x^{8} - 2x^{7} + 116x^{6} - 32x^{5} + 208x^{4} - 32x^{3} + 100x^{2} + 12x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3} - 1) q^{4} - \beta_1 q^{6} - \beta_{7} q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{4} + \cdots + 1) q^{8}+ \cdots + (\beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3} - 1) q^{4} - \beta_1 q^{6} - \beta_{7} q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{4} + \cdots + 1) q^{8}+ \cdots + (\beta_{11} - \beta_{10} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 4 q^{4} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} - 4 q^{4} + 2 q^{7} - 6 q^{9} + 2 q^{11} + 8 q^{12} + 8 q^{13} - 12 q^{14} + 8 q^{16} + 6 q^{17} + 8 q^{19} - 4 q^{21} + 14 q^{22} + 4 q^{23} - 24 q^{26} + 12 q^{27} + 8 q^{28} + 6 q^{29} + 4 q^{31} - 10 q^{32} + 2 q^{33} + 16 q^{34} - 4 q^{36} - 14 q^{37} - 4 q^{39} - 20 q^{41} + 6 q^{42} + 20 q^{43} - 20 q^{44} - 12 q^{46} + 12 q^{47} + 8 q^{48} + 8 q^{49} - 12 q^{51} - 8 q^{52} - 40 q^{53} - 12 q^{56} - 16 q^{57} - 12 q^{58} - 6 q^{59} + 10 q^{61} - 2 q^{62} + 2 q^{63} - 24 q^{64} - 28 q^{66} - 6 q^{67} + 12 q^{68} + 4 q^{69} + 2 q^{71} + 6 q^{74} + 4 q^{76} - 8 q^{77} + 12 q^{78} - 68 q^{79} - 6 q^{81} - 4 q^{82} - 12 q^{83} + 8 q^{84} + 24 q^{86} + 6 q^{87} + 32 q^{89} + 30 q^{91} - 44 q^{92} - 2 q^{93} + 24 q^{94} + 20 q^{96} - 20 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 8x^{10} + 48x^{8} - 2x^{7} + 116x^{6} - 32x^{5} + 208x^{4} - 32x^{3} + 100x^{2} + 12x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2368 \nu^{11} - 8684 \nu^{10} + 18496 \nu^{9} - 51512 \nu^{8} + 108805 \nu^{7} - 313920 \nu^{6} + \cdots + 3306786 ) / 1308086 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39163 \nu^{11} - 102912 \nu^{10} + 299096 \nu^{9} - 771192 \nu^{8} + 1768848 \nu^{7} + \cdots - 1384476 ) / 7848516 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32851 \nu^{11} + 15176 \nu^{10} - 217160 \nu^{9} + 246354 \nu^{8} - 1299166 \nu^{7} + \cdots + 2887776 ) / 2616172 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17152 \nu^{11} + 2368 \nu^{10} + 128532 \nu^{9} + 18496 \nu^{8} + 771784 \nu^{7} + 74501 \nu^{6} + \cdots + 234978 ) / 1308086 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 34427 \nu^{11} - 85544 \nu^{10} + 262104 \nu^{9} - 668168 \nu^{8} + 1551238 \nu^{7} + \cdots - 7998048 ) / 2616172 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 110567 \nu^{11} + 201936 \nu^{10} - 829624 \nu^{9} + 2001528 \nu^{8} - 4927260 \nu^{7} + \cdots + 23935968 ) / 7848516 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 47277 \nu^{11} - 80304 \nu^{10} - 378790 \nu^{9} - 657154 \nu^{8} - 2292816 \nu^{7} + \cdots - 1998600 ) / 2616172 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30912 \nu^{11} + 31776 \nu^{10} - 234649 \nu^{9} + 182928 \nu^{8} - 1399950 \nu^{7} + \cdots - 3383200 ) / 1308086 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 164186 \nu^{11} + 36279 \nu^{10} - 1024072 \nu^{9} + 503649 \nu^{8} - 5644830 \nu^{7} + \cdots + 5609664 ) / 3924258 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 203758 \nu^{11} + 206472 \nu^{10} + 1804997 \nu^{9} + 1616793 \nu^{8} + 10881600 \nu^{7} + \cdots + 6385278 ) / 3924258 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 3\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 4\beta_{5} + \beta_{4} + \beta_{2} - 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 6\beta_{6} + 13\beta_{3} - \beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - 6\beta_{9} - 7\beta_{8} - 6\beta_{7} - 7\beta_{6} - 18\beta_{5} + 8\beta_{3} - 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{9} - 10\beta_{5} - 32\beta_{2} + 10\beta _1 + 62 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{10} + 32\beta_{7} + 42\beta_{6} - 40\beta_{4} - 54\beta_{3} + 86\beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50\beta_{9} + 2\beta_{8} + 50\beta_{7} + 168\beta_{6} + 72\beta_{5} - 308\beta_{3} + 168\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 48 \beta_{11} - 48 \beta_{10} + 168 \beta_{9} + 214 \beta_{8} + 426 \beta_{5} + 214 \beta_{4} + \cdots - 336 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2\beta_{10} - 290\beta_{7} - 882\beta_{6} + 30\beta_{4} + 1566\beta_{3} - 458\beta _1 - 1566 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 262 \beta_{11} - 884 \beta_{9} - 1114 \beta_{8} - 884 \beta_{7} - 1370 \beta_{6} + \cdots - 1370 \beta_{2} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\) \(1\)

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
1.16230 + 2.01317i
0.672211 + 1.16430i
0.421315 + 0.729738i
−0.301021 − 0.521384i
−0.881436 − 1.52669i
−1.07337 − 1.85914i
1.16230 − 2.01317i
0.672211 − 1.16430i
0.421315 − 0.729738i
−0.301021 + 0.521384i
−0.881436 + 1.52669i
−1.07337 + 1.85914i
−1.16230 + 2.01317i −0.500000 + 0.866025i −1.70191 − 2.94779i 0 −1.16230 − 2.01317i 0.273150 + 0.473110i 3.26331 −0.500000 − 0.866025i 0
451.2 −0.672211 + 1.16430i −0.500000 + 0.866025i 0.0962645 + 0.166735i 0 −0.672211 − 1.16430i 1.96115 + 3.39681i −2.94768 −0.500000 − 0.866025i 0
451.3 −0.421315 + 0.729738i −0.500000 + 0.866025i 0.644988 + 1.11715i 0 −0.421315 − 0.729738i −0.200681 − 0.347589i −2.77223 −0.500000 − 0.866025i 0
451.4 0.301021 − 0.521384i −0.500000 + 0.866025i 0.818772 + 1.41816i 0 0.301021 + 0.521384i −1.77934 − 3.08191i 2.18996 −0.500000 − 0.866025i 0
451.5 0.881436 − 1.52669i −0.500000 + 0.866025i −0.553860 − 0.959313i 0 0.881436 + 1.52669i 1.11276 + 1.92736i 1.57298 −0.500000 − 0.866025i 0
451.6 1.07337 − 1.85914i −0.500000 + 0.866025i −1.30426 − 2.25904i 0 1.07337 + 1.85914i −0.367038 − 0.635729i −1.30633 −0.500000 − 0.866025i 0
601.1 −1.16230 − 2.01317i −0.500000 − 0.866025i −1.70191 + 2.94779i 0 −1.16230 + 2.01317i 0.273150 − 0.473110i 3.26331 −0.500000 + 0.866025i 0
601.2 −0.672211 − 1.16430i −0.500000 − 0.866025i 0.0962645 − 0.166735i 0 −0.672211 + 1.16430i 1.96115 − 3.39681i −2.94768 −0.500000 + 0.866025i 0
601.3 −0.421315 − 0.729738i −0.500000 − 0.866025i 0.644988 − 1.11715i 0 −0.421315 + 0.729738i −0.200681 + 0.347589i −2.77223 −0.500000 + 0.866025i 0
601.4 0.301021 + 0.521384i −0.500000 − 0.866025i 0.818772 − 1.41816i 0 0.301021 − 0.521384i −1.77934 + 3.08191i 2.18996 −0.500000 + 0.866025i 0
601.5 0.881436 + 1.52669i −0.500000 − 0.866025i −0.553860 + 0.959313i 0 0.881436 − 1.52669i 1.11276 − 1.92736i 1.57298 −0.500000 + 0.866025i 0
601.6 1.07337 + 1.85914i −0.500000 − 0.866025i −1.30426 + 2.25904i 0 1.07337 − 1.85914i −0.367038 + 0.635729i −1.30633 −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.o 12
5.b even 2 1 975.2.i.q 12
5.c odd 4 2 195.2.ba.a ✓ 24
13.c even 3 1 inner 975.2.i.o 12
15.e even 4 2 585.2.bs.b 24
65.n even 6 1 975.2.i.q 12
65.q odd 12 2 195.2.ba.a ✓ 24
195.bl even 12 2 585.2.bs.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.ba.a ✓ 24 5.c odd 4 2
195.2.ba.a ✓ 24 65.q odd 12 2
585.2.bs.b 24 15.e even 4 2
585.2.bs.b 24 195.bl even 12 2
975.2.i.o 12 1.a even 1 1 trivial
975.2.i.o 12 13.c even 3 1 inner
975.2.i.q 12 5.b even 2 1
975.2.i.q 12 65.n even 6 1

Hecke kernels

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

\( T_{2}^{12} + 8T_{2}^{10} + 48T_{2}^{8} + 2T_{2}^{7} + 116T_{2}^{6} + 32T_{2}^{5} + 208T_{2}^{4} + 32T_{2}^{3} + 100T_{2}^{2} - 12T_{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{12} - 2 T_{7}^{11} + 19 T_{7}^{10} - 18 T_{7}^{9} + 250 T_{7}^{8} - 276 T_{7}^{7} + 895 T_{7}^{6} + \cdots + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 8 T^{10} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{12} - 2 T^{11} + \cdots + 1600 \) Copy content Toggle raw display
$13$ \( T^{12} - 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} + \cdots + 98596 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + \cdots + 19891600 \) Copy content Toggle raw display
$23$ \( T^{12} - 4 T^{11} + \cdots + 419904 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + \cdots + 20884900 \) Copy content Toggle raw display
$31$ \( (T^{6} - 2 T^{5} + \cdots - 16255)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1398161664 \) Copy content Toggle raw display
$41$ \( T^{12} + 20 T^{11} + \cdots + 1232100 \) Copy content Toggle raw display
$43$ \( T^{12} - 20 T^{11} + \cdots + 3003289 \) Copy content Toggle raw display
$47$ \( (T^{6} - 6 T^{5} + \cdots + 158)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 20 T^{5} + \cdots + 57088)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 162052900 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 39170347225 \) Copy content Toggle raw display
$67$ \( T^{12} + 6 T^{11} + \cdots + 17147881 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 88559808100 \) Copy content Toggle raw display
$73$ \( (T^{6} - 115 T^{4} + \cdots + 415)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 34 T^{5} + \cdots + 1540129)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 6 T^{5} + \cdots + 80280)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 32 T^{11} + \cdots + 6749604 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 57682109241 \) Copy content Toggle raw display
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