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Properties

Label 930.2.n.c
Level $930$
Weight $2$
Character orbit 930.n
Analytic conductor $7.426$
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] = [930,2,Mod(481,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.481");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.n (of order \(5\), degree \(4\), 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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 25x^{10} + 205x^{8} + 675x^{6} + 795x^{4} + 230x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{4} q^{3} + \beta_{8} q^{4} - q^{5} + q^{6} + (\beta_{10} - \beta_{8}) q^{7} + ( - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{8} + ( - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{4} q^{3} + \beta_{8} q^{4} - q^{5} + q^{6} + (\beta_{10} - \beta_{8}) q^{7} + ( - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{8} + ( - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{9} + \beta_1 q^{10} + ( - \beta_{10} - \beta_{4} - 1) q^{11} - \beta_1 q^{12} + (\beta_{8} + \beta_{7} + \beta_{6} + \cdots + 1) q^{13}+ \cdots + (\beta_{4} - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 12 q^{5} + 12 q^{6} - q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 12 q^{5} + 12 q^{6} - q^{7} - 3 q^{8} - 3 q^{9} + 3 q^{10} - 5 q^{11} - 3 q^{12} - 3 q^{13} - q^{14} + 3 q^{15} - 3 q^{16} - 3 q^{17} - 3 q^{18} - 4 q^{19} + 3 q^{20} + 4 q^{21} - 5 q^{22} + 10 q^{23} - 3 q^{24} + 12 q^{25} + 2 q^{26} - 3 q^{27} + 4 q^{28} - 5 q^{29} - 12 q^{30} - 8 q^{31} + 12 q^{32} + 5 q^{33} + 2 q^{34} + q^{35} + 12 q^{36} + 2 q^{37} + 11 q^{38} + 2 q^{39} + 3 q^{40} + 13 q^{41} - q^{42} + 16 q^{43} + 5 q^{44} + 3 q^{45} - 5 q^{46} + q^{47} - 3 q^{48} + 4 q^{49} - 3 q^{50} - 3 q^{51} - 3 q^{52} + 3 q^{53} - 3 q^{54} + 5 q^{55} - 6 q^{56} - 14 q^{57} + 10 q^{58} + 23 q^{59} + 3 q^{60} - 46 q^{61} + 7 q^{62} - 6 q^{63} - 3 q^{64} + 3 q^{65} - 5 q^{66} - 66 q^{67} + 2 q^{68} + 10 q^{69} + q^{70} + 23 q^{71} - 3 q^{72} - 4 q^{73} + 7 q^{74} - 3 q^{75} + 11 q^{76} + 10 q^{77} - 3 q^{78} + 21 q^{79} + 3 q^{80} - 3 q^{81} + 13 q^{82} + 4 q^{83} - q^{84} + 3 q^{85} - 4 q^{86} - 10 q^{87} - 5 q^{89} + 3 q^{90} - 6 q^{91} - 10 q^{92} - 3 q^{93} + 6 q^{94} + 4 q^{95} - 3 q^{96} + 25 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 25x^{10} + 205x^{8} + 675x^{6} + 795x^{4} + 230x^{2} + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13 \nu^{11} + 21 \nu^{10} + 294 \nu^{9} + 458 \nu^{8} + 2079 \nu^{7} + 2959 \nu^{6} + 6146 \nu^{5} + \cdots + 907 ) / 968 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{10} - 166\nu^{8} - 2409\nu^{6} - 11122\nu^{4} - 14167\nu^{2} - 2317 ) / 484 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{10} - 160\nu^{8} - 1111\nu^{6} - 2734\nu^{4} - 1765\nu^{2} - 361 ) / 242 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13 \nu^{11} - 21 \nu^{10} + 294 \nu^{9} - 458 \nu^{8} + 2079 \nu^{7} - 2959 \nu^{6} + 6146 \nu^{5} + \cdots - 907 ) / 968 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{11} + 13 \nu^{10} + 41 \nu^{9} + 261 \nu^{8} + 220 \nu^{7} + 1397 \nu^{6} + 206 \nu^{5} + \cdots + 1 ) / 242 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{11} + 13 \nu^{10} - 41 \nu^{9} + 261 \nu^{8} - 220 \nu^{7} + 1397 \nu^{6} - 206 \nu^{5} + \cdots + 1 ) / 242 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{11} - 19 \nu^{10} + 146 \nu^{9} - 362 \nu^{8} + 275 \nu^{7} - 1683 \nu^{6} - 2076 \nu^{5} + \cdots + 117 ) / 484 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37 \nu^{11} + 21 \nu^{10} + 918 \nu^{9} + 458 \nu^{8} + 7447 \nu^{7} + 2959 \nu^{6} + 24238 \nu^{5} + \cdots + 423 ) / 968 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 59 \nu^{11} + 11 \nu^{10} + 1534 \nu^{9} + 330 \nu^{8} + 13277 \nu^{7} + 3289 \nu^{6} + 46150 \nu^{5} + \cdots - 671 ) / 968 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 59 \nu^{11} + 11 \nu^{10} - 1534 \nu^{9} + 330 \nu^{8} - 13277 \nu^{7} + 3289 \nu^{6} - 46150 \nu^{5} + \cdots - 671 ) / 968 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 133 \nu^{11} - 13 \nu^{10} - 3238 \nu^{9} - 294 \nu^{8} - 25443 \nu^{7} - 2079 \nu^{6} + \cdots - 3411 ) / 968 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{8} - 2\beta_{7} - 3\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 4\beta _1 - 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} - \beta_{9} - \beta_{6} - \beta_{5} - 3\beta_{4} + 3\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - 2\beta_{10} + 8\beta_{8} + 6\beta_{7} + 6\beta_{6} - 5\beta_{4} + 3\beta_{3} - \beta_{2} - 11\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{10} + 9\beta_{9} + 13\beta_{6} + 13\beta_{5} + 40\beta_{4} - 4\beta_{3} - 3\beta_{2} - 40\beta _1 + 67 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 34 \beta_{11} + 30 \beta_{10} + 4 \beta_{9} - 128 \beta_{8} - 76 \beta_{7} - 67 \beta_{6} - 9 \beta_{5} + \cdots - 85 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 96 \beta_{10} - 96 \beta_{9} - 161 \beta_{6} - 161 \beta_{5} - 507 \beta_{4} + 77 \beta_{3} + \cdots - 742 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 476 \beta_{11} - 399 \beta_{10} - 77 \beta_{9} + 1774 \beta_{8} + 954 \beta_{7} + 800 \beta_{6} + \cdots + 1126 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1123 \beta_{10} + 1123 \beta_{9} + 2004 \beta_{6} + 2004 \beta_{5} + 6386 \beta_{4} - 1118 \beta_{3} + \cdots + 8819 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6244 \beta_{11} + 5126 \beta_{10} + 1118 \beta_{9} - 23202 \beta_{8} - 11986 \beta_{7} - 9844 \beta_{6} + \cdots - 14472 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 13695 \beta_{10} - 13695 \beta_{9} - 25078 \beta_{6} - 25078 \beta_{5} - 80364 \beta_{4} + \cdots - 108266 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 79878 \beta_{11} - 65017 \beta_{10} - 14861 \beta_{9} + 296520 \beta_{8} + 150702 \beta_{7} + \cdots + 183672 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1} - \beta_{4} - \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
− 0.153723i
− 2.29476i
1.27291i
0.620070i
3.54652i
− 2.26448i
− 0.620070i
− 3.54652i
2.26448i
0.153723i
2.29476i
− 1.27291i
−0.809017 + 0.587785i −0.809017 − 0.587785i 0.309017 − 0.951057i −1.00000 1.00000 −1.26952 + 3.90719i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 − 0.587785i
481.2 −0.809017 + 0.587785i −0.809017 − 0.587785i 0.309017 − 0.951057i −1.00000 1.00000 0.0931451 − 0.286671i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 − 0.587785i
481.3 −0.809017 + 0.587785i −0.809017 − 0.587785i 0.309017 − 0.951057i −1.00000 1.00000 0.367362 − 1.13062i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 − 0.587785i
721.1 0.309017 − 0.951057i 0.309017 + 0.951057i −0.809017 − 0.587785i −1.00000 1.00000 −2.70650 − 1.96638i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
721.2 0.309017 − 0.951057i 0.309017 + 0.951057i −0.809017 − 0.587785i −1.00000 1.00000 0.606949 + 0.440975i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
721.3 0.309017 − 0.951057i 0.309017 + 0.951057i −0.809017 − 0.587785i −1.00000 1.00000 2.40856 + 1.74992i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
841.1 0.309017 + 0.951057i 0.309017 − 0.951057i −0.809017 + 0.587785i −1.00000 1.00000 −2.70650 + 1.96638i −0.809017 − 0.587785i −0.809017 − 0.587785i −0.309017 − 0.951057i
841.2 0.309017 + 0.951057i 0.309017 − 0.951057i −0.809017 + 0.587785i −1.00000 1.00000 0.606949 − 0.440975i −0.809017 − 0.587785i −0.809017 − 0.587785i −0.309017 − 0.951057i
841.3 0.309017 + 0.951057i 0.309017 − 0.951057i −0.809017 + 0.587785i −1.00000 1.00000 2.40856 − 1.74992i −0.809017 − 0.587785i −0.809017 − 0.587785i −0.309017 − 0.951057i
901.1 −0.809017 − 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 1.00000 −1.26952 − 3.90719i 0.309017 − 0.951057i 0.309017 − 0.951057i 0.809017 + 0.587785i
901.2 −0.809017 − 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 1.00000 0.0931451 + 0.286671i 0.309017 − 0.951057i 0.309017 − 0.951057i 0.809017 + 0.587785i
901.3 −0.809017 − 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 1.00000 0.367362 + 1.13062i 0.309017 − 0.951057i 0.309017 − 0.951057i 0.809017 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.c ✓ 12
31.d even 5 1 inner 930.2.n.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.c ✓ 12 1.a even 1 1 trivial
930.2.n.c ✓ 12 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + T_{7}^{11} + 9 T_{7}^{10} - 30 T_{7}^{9} + 40 T_{7}^{8} + 121 T_{7}^{7} + 1341 T_{7}^{6} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{12} + 5 T^{11} + \cdots + 32400 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{12} + 3 T^{11} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{12} + 4 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$23$ \( T^{12} - 10 T^{11} + \cdots + 32400 \) Copy content Toggle raw display
$29$ \( T^{12} + 5 T^{11} + \cdots + 20250000 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 887503681 \) Copy content Toggle raw display
$37$ \( (T^{6} - T^{5} - 60 T^{4} + \cdots - 7229)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 31768071696 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 114041041 \) Copy content Toggle raw display
$47$ \( T^{12} - T^{11} + \cdots + 13220496 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 102333456 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1628606736 \) Copy content Toggle raw display
$61$ \( (T^{6} + 23 T^{5} + \cdots + 251579)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 33 T^{5} + \cdots - 60731)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 22525207056 \) Copy content Toggle raw display
$73$ \( T^{12} + 4 T^{11} + \cdots + 68740681 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 78092861401 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 3794067216 \) Copy content Toggle raw display
$89$ \( T^{12} + 5 T^{11} + \cdots + 32400 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11471481025 \) Copy content Toggle raw display
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