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Properties

Label 930.2.bg.f
Level $930$
Weight $2$
Character orbit 930.bg
Analytic conductor $7.426$
Analytic rank $0$
Dimension $16$
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(121,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.bg (of order \(15\), degree \(8\), 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: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: 16.0.1669788916259765625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 16x^{14} + 90x^{12} + 239x^{10} + 329x^{8} + 239x^{6} + 90x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{13} - \beta_{11} + 1) q^{2} + ( - \beta_{15} - \beta_{3}) q^{3} - \beta_{13} q^{4} + ( - \beta_{14} - 1) q^{5} - \beta_{14} q^{6} + (\beta_{12} + 2 \beta_{10} + \beta_{8} + \cdots - 1) q^{7}+ \cdots - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} - \beta_{13} - \beta_{11} + 1) q^{2} + ( - \beta_{15} - \beta_{3}) q^{3} - \beta_{13} q^{4} + ( - \beta_{14} - 1) q^{5} - \beta_{14} q^{6} + (\beta_{12} + 2 \beta_{10} + \beta_{8} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{15} - \beta_{14} - \beta_{11} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 2 q^{3} - 4 q^{4} - 8 q^{5} + 8 q^{6} - 13 q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 2 q^{3} - 4 q^{4} - 8 q^{5} + 8 q^{6} - 13 q^{7} + 4 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{14} - 4 q^{15} - 4 q^{16} + 6 q^{17} - 2 q^{18} + 9 q^{19} + 2 q^{20} + 2 q^{21} - 13 q^{22} - 3 q^{23} - 2 q^{24} - 8 q^{25} + q^{26} - 4 q^{27} + 2 q^{28} - q^{29} - 16 q^{30} - 19 q^{31} - 16 q^{32} + 14 q^{33} + 4 q^{34} + 11 q^{35} - 8 q^{36} - 3 q^{37} + 6 q^{38} - 13 q^{39} - 2 q^{40} - 2 q^{42} + 2 q^{43} - 7 q^{44} + 2 q^{45} - 7 q^{46} - q^{47} + 2 q^{48} + 3 q^{49} - 2 q^{50} + 6 q^{51} - 6 q^{52} - 14 q^{53} + 4 q^{54} + 13 q^{55} - 7 q^{56} - 6 q^{57} - 4 q^{58} + 16 q^{59} - 4 q^{60} - 28 q^{61} - 21 q^{62} - 14 q^{63} - 4 q^{64} - 6 q^{65} + 11 q^{66} + 47 q^{67} - 4 q^{68} + 9 q^{69} - 11 q^{70} + 3 q^{71} - 2 q^{72} - 27 q^{73} + 8 q^{74} + 2 q^{75} - 6 q^{76} + 14 q^{77} - 12 q^{78} + 4 q^{79} + 2 q^{80} + 2 q^{81} + 30 q^{82} + 66 q^{83} - 13 q^{84} + 18 q^{85} + 13 q^{86} + 3 q^{87} - 3 q^{88} - 45 q^{89} - 2 q^{90} + 12 q^{91} - 8 q^{92} - 26 q^{93} + 36 q^{94} - 18 q^{95} - 2 q^{96} + 5 q^{97} - 13 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 16x^{14} + 90x^{12} + 239x^{10} + 329x^{8} + 239x^{6} + 90x^{4} + 16x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{14} + 29\nu^{12} + 135\nu^{10} + 253\nu^{8} + 166\nu^{6} - 16\nu^{4} - 33\nu^{2} + \nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 2 \nu^{14} + 15 \nu^{13} + 30 \nu^{12} + 75 \nu^{11} + 149 \nu^{10} + 164 \nu^{9} + 314 \nu^{8} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{15} + 2 \nu^{14} + 77 \nu^{13} + 33 \nu^{12} + 404 \nu^{11} + 194 \nu^{10} + 956 \nu^{9} + \cdots + 13 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{15} + 6 \nu^{14} + 46 \nu^{13} + 92 \nu^{12} + 239 \nu^{11} + 479 \nu^{10} + 553 \nu^{9} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8 \nu^{15} - \nu^{14} + 125 \nu^{13} - 15 \nu^{12} + 673 \nu^{11} - 75 \nu^{10} + 1658 \nu^{9} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{15} - 8 \nu^{14} + 31 \nu^{13} - 125 \nu^{12} + 165 \nu^{11} - 673 \nu^{10} + 403 \nu^{9} + \cdots - 21 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8 \nu^{15} + \nu^{14} + 125 \nu^{13} + 15 \nu^{12} + 673 \nu^{11} + 75 \nu^{10} + 1658 \nu^{9} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8 \nu^{15} + 3 \nu^{14} + 122 \nu^{13} + 46 \nu^{12} + 628 \nu^{11} + 239 \nu^{10} + 1433 \nu^{9} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\nu^{14} + 185\nu^{12} + 973\nu^{10} + 2313\nu^{8} + 2652\nu^{6} + 1410\nu^{4} + 317\nu^{2} + \nu + 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10 \nu^{15} + 11 \nu^{14} - 153 \nu^{13} + 171 \nu^{12} - 793 \nu^{11} + 912 \nu^{10} - 1836 \nu^{9} + \cdots + 27 ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13 \nu^{15} - 6 \nu^{14} - 201 \nu^{13} - 93 \nu^{12} - 1061 \nu^{11} - 494 \nu^{10} - 2525 \nu^{9} + \cdots - 15 ) / 2 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10 \nu^{15} + 11 \nu^{14} + 153 \nu^{13} + 171 \nu^{12} + 793 \nu^{11} + 912 \nu^{10} + 1836 \nu^{9} + \cdots + 27 ) / 2 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 22\nu^{15} + 340\nu^{13} + 1795\nu^{11} + 4285\nu^{9} + 4925\nu^{7} + 2606\nu^{5} + 570\nu^{3} + 35\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14 \nu^{15} + 11 \nu^{14} + 219 \nu^{13} + 171 \nu^{12} + 1182 \nu^{11} + 912 \nu^{10} + 2927 \nu^{9} + \cdots + 26 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + \beta_{14} - 4 \beta_{13} + 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - 8 \beta_{8} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{15} + 5 \beta_{14} - 20 \beta_{13} + 20 \beta_{12} + 5 \beta_{11} + 10 \beta_{10} + 22 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{15} - 10 \beta_{14} + 19 \beta_{13} - 19 \beta_{11} - 29 \beta_{10} - 28 \beta_{9} + 62 \beta_{8} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 53 \beta_{15} - 29 \beta_{14} + 163 \beta_{13} - 166 \beta_{12} - 30 \beta_{11} - 83 \beta_{10} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 83 \beta_{15} + 83 \beta_{14} - 112 \beta_{13} + 154 \beta_{11} + 194 \beta_{10} + 183 \beta_{9} + \cdots + 107 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 410 \beta_{15} + 194 \beta_{14} - 1269 \beta_{13} + 1308 \beta_{12} + 208 \beta_{11} + 654 \beta_{10} + \cdots - 108 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 654 \beta_{15} - 654 \beta_{14} + 759 \beta_{13} - 1202 \beta_{11} - 1399 \beta_{10} - 1307 \beta_{9} + \cdots - 778 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3165 \beta_{15} - 1399 \beta_{14} + 9762 \beta_{13} - 10122 \beta_{12} - 1534 \beta_{11} - 5061 \beta_{10} + \cdots + 883 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5061 \beta_{15} + 5061 \beta_{14} - 5512 \beta_{13} + 9268 \beta_{11} + 10438 \beta_{10} + \cdots + 5836 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 24334 \beta_{15} + 10438 \beta_{14} - 74836 \beta_{13} + 77802 \beta_{12} + 11577 \beta_{11} + \cdots - 6948 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 38901 \beta_{15} - 38901 \beta_{14} + 41267 \beta_{13} - 71139 \beta_{11} - 79029 \beta_{10} + \cdots - 44305 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 186631 \beta_{15} - 79029 \beta_{14} + 573060 \beta_{13} - 596450 \beta_{12} - 88120 \beta_{11} + \cdots + 53801 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.559146i
− 1.78844i
− 0.361487i
2.76635i
0.779734i
− 1.28249i
− 0.779734i
1.28249i
0.361487i
− 2.76635i
− 0.559146i
1.78844i
− 1.39148i
0.718661i
1.39148i
− 0.718661i
0.809017 − 0.587785i −0.104528 + 0.994522i 0.309017 − 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i −2.95386 + 0.627863i −0.309017 − 0.951057i −0.978148 − 0.207912i 0.104528 + 0.994522i
121.2 0.809017 − 0.587785i −0.104528 + 0.994522i 0.309017 − 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.41394 − 0.300541i −0.309017 − 0.951057i −0.978148 − 0.207912i 0.104528 + 0.994522i
361.1 −0.309017 − 0.951057i 0.669131 + 0.743145i −0.809017 + 0.587785i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.0662721 − 0.630537i 0.809017 + 0.587785i −0.104528 + 0.994522i −0.669131 + 0.743145i
361.2 −0.309017 − 0.951057i 0.669131 + 0.743145i −0.809017 + 0.587785i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.318079 + 3.02632i 0.809017 + 0.587785i −0.104528 + 0.994522i −0.669131 + 0.743145i
391.1 0.809017 + 0.587785i 0.913545 + 0.406737i 0.309017 + 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.88789 + 2.09671i −0.309017 + 0.951057i 0.669131 + 0.743145i −0.913545 + 0.406737i
391.2 0.809017 + 0.587785i 0.913545 + 0.406737i 0.309017 + 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.736831 − 0.818334i −0.309017 + 0.951057i 0.669131 + 0.743145i −0.913545 + 0.406737i
421.1 0.809017 − 0.587785i 0.913545 − 0.406737i 0.309017 − 0.951057i −0.500000 − 0.866025i 0.500000 − 0.866025i −1.88789 − 2.09671i −0.309017 − 0.951057i 0.669131 − 0.743145i −0.913545 − 0.406737i
421.2 0.809017 − 0.587785i 0.913545 − 0.406737i 0.309017 − 0.951057i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.736831 + 0.818334i −0.309017 − 0.951057i 0.669131 − 0.743145i −0.913545 − 0.406737i
541.1 −0.309017 + 0.951057i 0.669131 − 0.743145i −0.809017 − 0.587785i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.0662721 + 0.630537i 0.809017 − 0.587785i −0.104528 − 0.994522i −0.669131 − 0.743145i
541.2 −0.309017 + 0.951057i 0.669131 − 0.743145i −0.809017 − 0.587785i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.318079 − 3.02632i 0.809017 − 0.587785i −0.104528 − 0.994522i −0.669131 − 0.743145i
661.1 0.809017 + 0.587785i −0.104528 − 0.994522i 0.309017 + 0.951057i −0.500000 − 0.866025i 0.500000 − 0.866025i −2.95386 − 0.627863i −0.309017 + 0.951057i −0.978148 + 0.207912i 0.104528 − 0.994522i
661.2 0.809017 + 0.587785i −0.104528 − 0.994522i 0.309017 + 0.951057i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.41394 + 0.300541i −0.309017 + 0.951057i −0.978148 + 0.207912i 0.104528 − 0.994522i
691.1 −0.309017 + 0.951057i −0.978148 − 0.207912i −0.809017 − 0.587785i −0.500000 − 0.866025i 0.500000 − 0.866025i −3.16349 + 1.40848i 0.809017 − 0.587785i 0.913545 + 0.406737i 0.978148 − 0.207912i
691.2 −0.309017 + 0.951057i −0.978148 − 0.207912i −0.809017 − 0.587785i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.897334 + 0.399519i 0.809017 − 0.587785i 0.913545 + 0.406737i 0.978148 − 0.207912i
751.1 −0.309017 − 0.951057i −0.978148 + 0.207912i −0.809017 + 0.587785i −0.500000 + 0.866025i 0.500000 + 0.866025i −3.16349 − 1.40848i 0.809017 + 0.587785i 0.913545 − 0.406737i 0.978148 + 0.207912i
751.2 −0.309017 − 0.951057i −0.978148 + 0.207912i −0.809017 + 0.587785i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.897334 − 0.399519i 0.809017 + 0.587785i 0.913545 − 0.406737i 0.978148 + 0.207912i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bg.f ✓ 16
31.g even 15 1 inner 930.2.bg.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.f ✓ 16 1.a even 1 1 trivial
930.2.bg.f ✓ 16 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 13 T_{7}^{15} + 76 T_{7}^{14} + 272 T_{7}^{13} + 691 T_{7}^{12} + 1174 T_{7}^{11} + \cdots + 7921 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 13 T^{15} + \cdots + 7921 \) Copy content Toggle raw display
$11$ \( T^{16} + 2 T^{15} + \cdots + 29474041 \) Copy content Toggle raw display
$13$ \( T^{16} + 6 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 208253761 \) Copy content Toggle raw display
$19$ \( T^{16} - 9 T^{15} + \cdots + 5517801 \) Copy content Toggle raw display
$23$ \( T^{16} + 3 T^{15} + \cdots + 8288641 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 407821454881 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 41383358041 \) Copy content Toggle raw display
$41$ \( T^{16} - 83 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 679697041 \) Copy content Toggle raw display
$47$ \( T^{16} + T^{15} + \cdots + 1907161 \) Copy content Toggle raw display
$53$ \( T^{16} + 14 T^{15} + \cdots + 12952801 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 253796761 \) Copy content Toggle raw display
$61$ \( (T^{8} + 14 T^{7} + \cdots + 15601)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 6132304369801 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 9771124801 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 96448721530561 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 37181713577761 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 199218502921 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 5042562121 \) Copy content Toggle raw display
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