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Properties

Label 930.2.bg.d
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: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 21x^{14} + 167x^{12} + 653x^{10} + 1350x^{8} + 1472x^{6} + 777x^{4} + 149x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{7} + \beta_{5}) q^{3} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1) q^{4}+ \cdots + (\beta_{8} + \beta_{7} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{7} + \beta_{5}) q^{3} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1) q^{4}+ \cdots + (\beta_{15} - 2 \beta_{14} - 3 \beta_{12} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 2 q^{3} - 4 q^{4} + 8 q^{5} - 8 q^{6} + 11 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 2 q^{3} - 4 q^{4} + 8 q^{5} - 8 q^{6} + 11 q^{7} - 4 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 20 q^{13} - 4 q^{14} + 4 q^{15} - 4 q^{16} - 12 q^{17} + 2 q^{18} - 7 q^{19} - 2 q^{20} - 4 q^{21} + 11 q^{22} + 11 q^{23} + 2 q^{24} - 8 q^{25} + 5 q^{26} - 4 q^{27} - 4 q^{28} + 23 q^{29} - 16 q^{30} - 17 q^{31} + 16 q^{32} - 2 q^{33} - 2 q^{34} + 7 q^{35} - 8 q^{36} + 17 q^{37} - 2 q^{38} - 5 q^{39} - 2 q^{40} + 8 q^{41} - 4 q^{42} + 32 q^{43} + q^{44} - 2 q^{45} + 11 q^{46} - 7 q^{47} + 2 q^{48} - q^{49} + 2 q^{50} - 12 q^{51} + 10 q^{52} + 54 q^{53} - 4 q^{54} - 11 q^{55} + q^{56} + 18 q^{57} - 22 q^{58} - 38 q^{59} + 4 q^{60} - 28 q^{61} - 27 q^{62} - 2 q^{63} - 4 q^{64} - 10 q^{65} - 7 q^{66} + q^{67} - 2 q^{68} - 13 q^{69} + 7 q^{70} - 11 q^{71} + 2 q^{72} + 53 q^{73} + 2 q^{74} + 2 q^{75} - 2 q^{76} + 44 q^{77} + 10 q^{78} + 48 q^{79} - 2 q^{80} + 2 q^{81} + 8 q^{82} + 56 q^{83} + 11 q^{84} - 24 q^{85} - 13 q^{86} + q^{87} - 9 q^{88} - 43 q^{89} - 2 q^{90} - 50 q^{91} - 44 q^{92} - 22 q^{93} - 72 q^{94} - 14 q^{95} + 2 q^{96} + 17 q^{97} + 29 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 21x^{14} + 167x^{12} + 653x^{10} + 1350x^{8} + 1472x^{6} + 777x^{4} + 149x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 831 \nu^{15} - 200 \nu^{14} + 16820 \nu^{13} - 3799 \nu^{12} + 125756 \nu^{11} - 27571 \nu^{10} + \cdots - 39009 ) / 37642 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 831 \nu^{15} + 200 \nu^{14} + 16820 \nu^{13} + 3799 \nu^{12} + 125756 \nu^{11} + 27571 \nu^{10} + \cdots + 39009 ) / 37642 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3182 \nu^{15} - 4831 \nu^{14} - 54984 \nu^{13} - 92800 \nu^{12} - 299191 \nu^{11} - 639534 \nu^{10} + \cdots - 48359 ) / 37642 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 88 \nu^{15} + 529 \nu^{14} + 1740 \nu^{13} + 10382 \nu^{12} + 12405 \nu^{11} + 74046 \nu^{10} + \cdots - 2484 ) / 3422 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 88 \nu^{15} - 529 \nu^{14} + 1740 \nu^{13} - 10382 \nu^{12} + 12405 \nu^{11} - 74046 \nu^{10} + \cdots + 2484 ) / 3422 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 262 \nu^{15} - 5100 \nu^{13} - 35800 \nu^{11} - 113585 \nu^{9} - 160570 \nu^{7} - 74527 \nu^{5} + \cdots - 649 ) / 1298 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 419 \nu^{15} + 8649 \nu^{14} + 11629 \nu^{13} + 170781 \nu^{12} + 126552 \nu^{11} + 1230044 \nu^{10} + \cdots + 9899 ) / 37642 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 419 \nu^{15} - 8649 \nu^{14} + 11629 \nu^{13} - 170781 \nu^{12} + 126552 \nu^{11} - 1230044 \nu^{10} + \cdots - 9899 ) / 37642 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4768 \nu^{15} + 10016 \nu^{14} + 91321 \nu^{13} + 199288 \nu^{12} + 622662 \nu^{11} + 1454534 \nu^{10} + \cdots - 9836 ) / 37642 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 402 \nu^{14} - 7954 \nu^{12} - 57501 \nu^{10} - 193130 \nu^{8} - 311137 \nu^{6} - 222689 \nu^{4} + \cdots - 262 ) / 1298 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 402 \nu^{14} - 7954 \nu^{12} - 57501 \nu^{10} - 193130 \nu^{8} - 311137 \nu^{6} - 222689 \nu^{4} + \cdots - 262 ) / 1298 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6019 \nu^{15} + 7817 \nu^{14} - 118001 \nu^{13} + 155730 \nu^{12} - 842077 \nu^{11} + \cdots + 10235 ) / 37642 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4768 \nu^{15} - 10016 \nu^{14} + 91321 \nu^{13} - 199288 \nu^{12} + 622662 \nu^{11} - 1454534 \nu^{10} + \cdots + 9836 ) / 37642 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2982 \nu^{15} - 13836 \nu^{14} - 51185 \nu^{13} - 273731 \nu^{12} - 271620 \nu^{11} + \cdots - 59562 ) / 37642 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 831 \nu^{15} + 15834 \nu^{14} - 16820 \nu^{13} + 315259 \nu^{12} - 125756 \nu^{11} + 2301933 \nu^{10} + \cdots + 97121 ) / 37642 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{10} + \beta_{9} + 2 \beta_{7} + 2 \beta_{5} - \beta_{4} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + 5 \beta_{11} - 5 \beta_{10} - \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} - 8 \beta_{12} + 2 \beta_{11} - 14 \beta_{10} - 9 \beta_{9} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{15} + 4 \beta_{14} + 9 \beta_{13} - 12 \beta_{12} - 30 \beta_{11} + 30 \beta_{10} + 9 \beta_{9} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 23 \beta_{15} + 62 \beta_{14} - 70 \beta_{13} + 62 \beta_{12} - 26 \beta_{11} + 98 \beta_{10} + \cdots - 50 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 60 \beta_{15} - 9 \beta_{14} - 77 \beta_{13} + 111 \beta_{12} + 208 \beta_{11} - 208 \beta_{10} + \cdots - 63 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 217 \beta_{15} - 490 \beta_{14} + 551 \beta_{13} - 490 \beta_{12} + 253 \beta_{11} - 727 \beta_{10} + \cdots + 368 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 461 \beta_{15} - 31 \beta_{14} + 650 \beta_{13} - 953 \beta_{12} - 1567 \beta_{11} + 1567 \beta_{10} + \cdots + 508 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1911 \beta_{15} + 3940 \beta_{14} - 4427 \beta_{13} + 3940 \beta_{12} - 2235 \beta_{11} + 5645 \beta_{10} + \cdots - 2862 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3632 \beta_{15} + 698 \beta_{14} - 5431 \beta_{13} + 7962 \beta_{12} + 12330 \beta_{11} - 12330 \beta_{10} + \cdots - 4147 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16295 \beta_{15} - 32013 \beta_{14} + 36004 \beta_{13} - 32013 \beta_{12} + 18982 \beta_{11} + \cdots + 22795 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 29111 \beta_{15} - 7594 \beta_{14} + 45064 \beta_{13} - 65816 \beta_{12} - 99171 \beta_{11} + \cdots + 34019 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 136628 \beta_{15} + 261566 \beta_{14} - 294541 \beta_{13} + 261566 \beta_{12} - 158424 \beta_{11} + \cdots - 183918 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 235818 \beta_{15} + 70191 \beta_{14} - 372339 \beta_{13} + 541827 \beta_{12} + 806355 \beta_{11} + \cdots - 279525 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
− 1.94768i
1.13420i
1.04753i
− 0.631711i
− 2.07247i
0.0834312i
2.07247i
− 0.0834312i
− 1.04753i
0.631711i
1.94768i
− 1.13420i
− 2.86648i
1.38019i
2.86648i
− 1.38019i
−0.809017 + 0.587785i −0.104528 + 0.994522i 0.309017 − 0.951057i 0.500000 − 0.866025i −0.500000 − 0.866025i 1.12135 − 0.238351i 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 2.37487 − 0.504794i 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 −0.195942 − 1.86427i −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 0.153192 + 1.45753i −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 −1.16565 + 1.29459i 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 0.978446 − 1.08667i 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 −1.16565 − 1.29459i 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 0.978446 + 1.08667i 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 −0.195942 + 1.86427i −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 0.153192 − 1.45753i −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 1.12135 + 0.238351i 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 2.37487 + 0.504794i 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 −0.461080 + 0.205286i −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 2.69481 − 1.19981i −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 −0.461080 − 0.205286i −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 2.69481 + 1.19981i −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.d ✓ 16
31.g even 15 1 inner 930.2.bg.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.d ✓ 16 1.a even 1 1 trivial
930.2.bg.d ✓ 16 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 11 T_{7}^{15} + 54 T_{7}^{14} - 166 T_{7}^{13} + 401 T_{7}^{12} - 848 T_{7}^{11} + \cdots + 841 \) 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
$7$ \( T^{16} - 11 T^{15} + \cdots + 841 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 132043081 \) Copy content Toggle raw display
$13$ \( T^{16} + 20 T^{15} + \cdots + 525625 \) Copy content Toggle raw display
$17$ \( T^{16} + 12 T^{15} + \cdots + 42784681 \) Copy content Toggle raw display
$19$ \( T^{16} + 7 T^{15} + \cdots + 7946761 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 259499881 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 2598042841 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 28425622801 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 419881081 \) Copy content Toggle raw display
$43$ \( T^{16} - 32 T^{15} + \cdots + 61763881 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 531836191441 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3799220807281 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 708454521342481 \) Copy content Toggle raw display
$61$ \( (T^{8} + 14 T^{7} + \cdots - 9157649)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 86869874800801 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 3171245078401 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 132043081 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 8570379205441 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 241031182982041 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 52098210332281 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 5282049589441 \) Copy content Toggle raw display
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