Properties

Label 930.2.n.f
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: \( x^{12} - 3 x^{11} + 10 x^{10} - 15 x^{9} + 61 x^{8} - 25 x^{7} + 316 x^{6} + 50 x^{5} + 1336 x^{4} - 1720 x^{3} + 1235 x^{2} - 462 x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - \beta_{6} q^{3} + (\beta_{8} - \beta_{6} - \beta_{3} - 1) q^{4} + q^{5} + q^{6} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{7} - \beta_{3} q^{8} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - \beta_{6} q^{3} + (\beta_{8} - \beta_{6} - \beta_{3} - 1) q^{4} + q^{5} + q^{6} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{7} - \beta_{3} q^{8} + \beta_{3} q^{9} + \beta_{8} q^{10} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{11} + \beta_{8} q^{12} + (\beta_{7} - \beta_{4}) q^{13} + ( - \beta_{9} - \beta_{4} - \beta_{3}) q^{14} - \beta_{6} q^{15} + \beta_{6} q^{16} + ( - \beta_{9} + 2 \beta_{4} - \beta_{2} + \beta_1) q^{17} - \beta_{6} q^{18} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{19} + (\beta_{8} - \beta_{6} - \beta_{3} - 1) q^{20} + ( - \beta_{11} + \beta_{4} + \beta_1) q^{21} + (2 \beta_{8} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{22} + ( - \beta_{11} + \beta_{5} + 2 \beta_{3}) q^{23} + (\beta_{8} - \beta_{6} - \beta_{3} - 1) q^{24} + q^{25} + (\beta_{7} - \beta_{4} + \beta_{2}) q^{26} + ( - \beta_{8} + \beta_{6} + \beta_{3} + 1) q^{27} - \beta_{10} q^{28} + ( - \beta_{11} - 3 \beta_{8} + \beta_{7} + 3 \beta_{6} + 3 \beta_{3} + \beta_{2}) q^{29} + q^{30} + (\beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_1 + 2) q^{31} - q^{32} + (\beta_{8} - \beta_{7} - 2 \beta_{6} - 2 \beta_{3} - \beta_{2}) q^{33} + ( - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{4} - 3 \beta_1) q^{34} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{35} + q^{36} + (\beta_{8} - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{2} + 3) q^{37} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{38} + ( - \beta_{2} + \beta_1) q^{39} - \beta_{3} q^{40} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1) q^{41} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{42} + ( - \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + \cdots + 6 \beta_1) q^{43}+ \cdots + (2 \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{2} - 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 \( 12 q + 3 q^{2} + 3 q^{3} - 3 q^{4} + 12 q^{5} + 12 q^{6} - q^{7} + 3 q^{8} - 3 q^{9} + 3 q^{10} + 14 q^{11} + 3 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} - 3 q^{16} - 7 q^{17} + 3 q^{18} + 12 q^{19} - 3 q^{20} + q^{21} - 14 q^{22} - 6 q^{23} - 3 q^{24} + 12 q^{25} + 6 q^{26} + 3 q^{27} - q^{28} - 23 q^{29} + 12 q^{30} + 34 q^{31} - 12 q^{32} + 11 q^{33} - 3 q^{34} - q^{35} + 12 q^{36} + 22 q^{37} + 8 q^{38} + q^{39} + 3 q^{40} + 15 q^{41} - q^{42} - 17 q^{43} - 11 q^{44} - 3 q^{45} + 6 q^{46} - 2 q^{47} + 3 q^{48} - 8 q^{49} + 3 q^{50} + 7 q^{51} + 4 q^{52} - 21 q^{53} - 3 q^{54} + 14 q^{55} - 4 q^{56} + 8 q^{57} - 17 q^{58} - 15 q^{59} + 3 q^{60} + 16 q^{61} + 26 q^{62} + 4 q^{63} - 3 q^{64} + 4 q^{65} + 14 q^{66} + 24 q^{67} + 8 q^{68} + 6 q^{69} + q^{70} + 23 q^{71} + 3 q^{72} + 56 q^{73} + 8 q^{74} + 3 q^{75} - 8 q^{76} - 8 q^{77} + 4 q^{78} - 9 q^{79} - 3 q^{80} - 3 q^{81} + 15 q^{82} - 24 q^{83} + q^{84} - 7 q^{85} - 18 q^{86} - 12 q^{87} + 6 q^{88} - 27 q^{89} + 3 q^{90} - 3 q^{91} + 24 q^{92} + 11 q^{93} - 28 q^{94} + 12 q^{95} - 3 q^{96} + 38 q^{97} - 52 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 10 x^{10} - 15 x^{9} + 61 x^{8} - 25 x^{7} + 316 x^{6} + 50 x^{5} + 1336 x^{4} - 1720 x^{3} + 1235 x^{2} - 462 x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2027942507365 \nu^{11} - 6741096197250 \nu^{10} + 8771348618837 \nu^{9} - 72535728446286 \nu^{8} + \cdots - 14\!\cdots\!98 ) / 69\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 134293628406918 \nu^{11} + 425188252801769 \nu^{10} + \cdots - 15\!\cdots\!67 ) / 76\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15260073350092 \nu^{11} + 29181641218473 \nu^{10} - 109621640737688 \nu^{9} + 75584589358371 \nu^{8} + \cdots - 37\!\cdots\!03 ) / 69\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15433725086754 \nu^{11} + 40585830735791 \nu^{10} - 146367463122124 \nu^{9} + 223757360866823 \nu^{8} + \cdots + 32\!\cdots\!83 ) / 69\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 340012095451573 \nu^{11} + \cdots - 16\!\cdots\!79 ) / 76\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33528790708398 \nu^{11} + 91949782233819 \nu^{10} - 305918373930930 \nu^{9} + 411412294337441 \nu^{8} + \cdots + 48\!\cdots\!17 ) / 69\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 441758859332147 \nu^{11} + 956459880204063 \nu^{10} + \cdots - 20\!\cdots\!13 ) / 76\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 671468142034590 \nu^{11} + \cdots + 78\!\cdots\!35 ) / 76\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!67 \nu^{11} + \cdots - 60\!\cdots\!78 ) / 76\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!93 \nu^{11} + \cdots - 23\!\cdots\!79 ) / 38\!\cdots\!09 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + 3\beta_{3} - 7\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{11} - \beta_{10} + \beta_{9} - 27\beta_{8} - 12\beta_{7} + 2\beta_{6} + \beta_{4} + 3\beta_{3} - 12\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{11} - 11 \beta_{10} - 21 \beta_{8} - 56 \beta_{7} + 21 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} - 17 \beta_{2} - 11 \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 50 \beta_{10} - 17 \beta_{9} - 60 \beta_{7} + 169 \beta_{6} + 17 \beta_{5} + 60 \beta_{4} - 57 \beta_{3} - 17 \beta_{2} - 66 \beta _1 - 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 110 \beta_{11} - 143 \beta_{9} + 256 \beta_{8} + 110 \beta_{5} + 379 \beta_{4} - 530 \beta_{3} + 74 \beta_{2} - 74 \beta _1 - 256 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 596 \beta_{11} + 379 \beta_{10} - 596 \beta_{9} + 1768 \beta_{8} + 693 \beta_{7} - 1219 \beta_{6} + 379 \beta_{5} + 379 \beta_{4} - 2364 \beta_{3} + 890 \beta_{2} - 97 \beta _1 - 1219 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1486 \beta_{11} + 1072 \beta_{10} - 1072 \beta_{9} + 4101 \beta_{8} + 4446 \beta_{7} - 2717 \beta_{6} - 1977 \beta_{4} - 3789 \beta_{3} + 4446 \beta_{2} - 905 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2469 \beta_{11} + 2469 \beta_{10} + 6490 \beta_{8} + 12591 \beta_{7} - 6490 \beta_{6} - 3049 \beta_{5} - 7910 \beta_{4} + 10379 \beta_{2} + 2469 \beta _1 + 9394 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4681 \beta_{10} + 10379 \beta_{9} + 21807 \beta_{7} - 14366 \beta_{6} - 10379 \beta_{5} - 21807 \beta_{4} + 37810 \beta_{3} + 10379 \beta_{2} + 20173 \beta _1 + 27431 \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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
0.472164 + 0.343047i
−1.70225 1.23676i
2.53910 + 1.84477i
0.784484 + 2.41439i
−0.738812 2.27383i
0.145311 + 0.447222i
0.784484 2.41439i
−0.738812 + 2.27383i
0.145311 0.447222i
0.472164 0.343047i
−1.70225 + 1.23676i
2.53910 1.84477i
0.809017 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i 1.00000 1.00000 −1.31116 + 4.03534i −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 0.782218 2.40742i −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 0.837960 2.57897i −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 −4.03161 2.92914i 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 0.545848 + 0.396582i 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 2.67675 + 1.94477i 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 −4.03161 + 2.92914i 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 0.545848 0.396582i 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 2.67675 1.94477i 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 −1.31116 4.03534i −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 0.782218 + 2.40742i −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 0.837960 + 2.57897i −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.f 12
31.d even 5 1 inner 930.2.n.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.f 12 1.a even 1 1 trivial
930.2.n.f 12 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + T_{7}^{11} + 15 T_{7}^{10} - 25 T_{7}^{9} + 306 T_{7}^{8} - 875 T_{7}^{7} + 7459 T_{7}^{6} - 26150 T_{7}^{5} + 106561 T_{7}^{4} - 212490 T_{7}^{3} + 392040 T_{7}^{2} - 303264 T_{7} + 104976 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + 15 T^{10} + \cdots + 104976 \) Copy content Toggle raw display
$11$ \( T^{12} - 14 T^{11} + 125 T^{10} + \cdots + 22801 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} + 40 T^{9} + 226 T^{8} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{12} + 7 T^{11} + 37 T^{10} + \cdots + 64883025 \) Copy content Toggle raw display
$19$ \( T^{12} - 12 T^{11} + 69 T^{10} + \cdots + 4000000 \) Copy content Toggle raw display
$23$ \( T^{12} + 6 T^{11} + 13 T^{10} + \cdots + 93025 \) Copy content Toggle raw display
$29$ \( T^{12} + 23 T^{11} + 264 T^{10} + \cdots + 1229881 \) Copy content Toggle raw display
$31$ \( T^{12} - 34 T^{11} + \cdots + 887503681 \) Copy content Toggle raw display
$37$ \( (T^{6} - 11 T^{5} - 198 T^{4} + \cdots + 472249)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 15 T^{11} + 271 T^{10} + \cdots + 5856400 \) Copy content Toggle raw display
$43$ \( T^{12} + 17 T^{11} + \cdots + 22817915136 \) Copy content Toggle raw display
$47$ \( T^{12} + 2 T^{11} + 134 T^{10} + \cdots + 4405801 \) Copy content Toggle raw display
$53$ \( T^{12} + 21 T^{11} + 407 T^{10} + \cdots + 8294400 \) Copy content Toggle raw display
$59$ \( T^{12} + 15 T^{11} + \cdots + 5935315681 \) Copy content Toggle raw display
$61$ \( (T^{6} - 8 T^{5} - 141 T^{4} + 918 T^{3} + \cdots + 32400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 12 T^{5} - 134 T^{4} + \cdots + 14400)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 23 T^{11} + 385 T^{10} + \cdots + 8737936 \) Copy content Toggle raw display
$73$ \( T^{12} - 56 T^{11} + \cdots + 9591460096 \) Copy content Toggle raw display
$79$ \( T^{12} + 9 T^{11} + 35 T^{10} + \cdots + 96412761 \) Copy content Toggle raw display
$83$ \( T^{12} + 24 T^{11} + 427 T^{10} + \cdots + 467856 \) Copy content Toggle raw display
$89$ \( T^{12} + 27 T^{11} + \cdots + 725386076416 \) Copy content Toggle raw display
$97$ \( T^{12} - 38 T^{11} + \cdots + 1957531536 \) Copy content Toggle raw display
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