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Properties

Label 770.2.n.i
Level $770$
Weight $2$
Character orbit 770.n
Analytic conductor $6.148$
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] = [770,2,Mod(71,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(10))
 
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("770.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.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: \(6.14848095564\)
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: \( x^{12} - 3 x^{11} + 11 x^{10} - 21 x^{9} + 61 x^{8} - 34 x^{7} + 141 x^{6} + 192 x^{5} + 289 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 11 x^{10} - 21 x^{9} + 61 x^{8} - 34 x^{7} + 141 x^{6} + 192 x^{5} + 289 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 27218910292 \nu^{11} - 98998119348 \nu^{10} + 577035333512 \nu^{9} - 1678425495360 \nu^{8} + \cdots - 8921537346326 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 57839182555 \nu^{11} + 25030916843 \nu^{10} - 944387310072 \nu^{9} + \cdots + 128492152255425 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 756356413694 \nu^{11} - 1988591540098 \nu^{10} + 7508583879638 \nu^{9} + \cdots - 8905487495815 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 758466237704 \nu^{11} - 2315285952880 \nu^{10} + 8258213919829 \nu^{9} + \cdots - 6099055983892 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6099055983892 \nu^{11} + 17538701713972 \nu^{10} - 64774329869932 \nu^{9} + \cdots - 384461798009 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6115105834403 \nu^{11} - 18315988768907 \nu^{10} + 66840471646303 \nu^{9} + \cdots - 135257465459680 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8905487495815 \nu^{11} + 25960106073751 \nu^{10} - 95971770913867 \nu^{9} + \cdots + 82450603826419 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13210691808625 \nu^{11} + 37908100871573 \nu^{10} - 140869867980898 \nu^{9} + \cdots + 127689837420424 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13325947510218 \nu^{11} + 40254045460811 \nu^{10} - 147445262229447 \nu^{9} + \cdots + 210612483311877 ) / 83041551747559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37373296056781 \nu^{11} - 110083091955905 \nu^{10} + 404321849282105 \nu^{9} + \cdots - 335824354765506 ) / 83041551747559 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 3\beta_{8} + 3\beta_{7} + 2\beta_{6} + \beta_{5} + 5\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{11} - 6 \beta_{9} - \beta_{8} + 17 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} - 6 \beta_{4} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{10} - 8 \beta_{9} + 2 \beta_{7} + 2 \beta_{6} + 38 \beta_{5} - 38 \beta_{4} + 10 \beta_{3} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36 \beta_{11} + 36 \beta_{10} - 5 \beta_{9} + 14 \beta_{8} - 36 \beta_{7} + 93 \beta_{6} - 28 \beta_{4} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 59\beta_{11} - 25\beta_{9} + 199\beta_{8} - 85\beta_{7} - 249\beta_{5} - 72\beta_{4} - 72\beta _1 - 85 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 72 \beta_{11} - 224 \beta_{10} + 558 \beta_{8} - 558 \beta_{7} - 557 \beta_{6} - 512 \beta_{5} + \cdots - 629 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 236 \beta_{11} - 440 \beta_{10} + 236 \beta_{9} + 504 \beta_{8} - 1300 \beta_{7} - 268 \beta_{6} + \cdots - 236 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 749 \beta_{10} + 749 \beta_{9} - 21 \beta_{7} - 21 \beta_{6} - 3462 \beta_{5} + 3462 \beta_{4} + \cdots + 3414 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2017 \beta_{11} - 2017 \beta_{10} + 3342 \beta_{9} - 4543 \beta_{8} + 2017 \beta_{7} - 5546 \beta_{6} + \cdots + 4543 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
0.0547712 + 0.0397936i
−1.00857 − 0.732772i
2.26282 + 1.64404i
0.0547712 − 0.0397936i
−1.00857 + 0.732772i
2.26282 − 1.64404i
0.701664 + 2.15950i
−0.748642 − 2.30408i
0.237961 + 0.732370i
0.701664 − 2.15950i
−0.748642 + 2.30408i
0.237961 − 0.732370i
0.309017 + 0.951057i −1.97811 − 1.43718i −0.809017 + 0.587785i 0.309017 − 0.951057i 0.755572 − 2.32541i −0.809017 + 0.587785i −0.809017 − 0.587785i 0.920387 + 2.83266i 1.00000
71.2 0.309017 + 0.951057i 0.338879 + 0.246210i −0.809017 + 0.587785i 0.309017 − 0.951057i −0.129440 + 0.398376i −0.809017 + 0.587785i −0.809017 − 0.587785i −0.872831 − 2.68630i 1.00000
71.3 0.309017 + 0.951057i 2.13924 + 1.55425i −0.809017 + 0.587785i 0.309017 − 0.951057i −0.817115 + 2.51482i −0.809017 + 0.587785i −0.809017 − 0.587785i 1.23360 + 3.79662i 1.00000
141.1 0.309017 − 0.951057i −1.97811 + 1.43718i −0.809017 − 0.587785i 0.309017 + 0.951057i 0.755572 + 2.32541i −0.809017 − 0.587785i −0.809017 + 0.587785i 0.920387 − 2.83266i 1.00000
141.2 0.309017 − 0.951057i 0.338879 − 0.246210i −0.809017 − 0.587785i 0.309017 + 0.951057i −0.129440 − 0.398376i −0.809017 − 0.587785i −0.809017 + 0.587785i −0.872831 + 2.68630i 1.00000
141.3 0.309017 − 0.951057i 2.13924 − 1.55425i −0.809017 − 0.587785i 0.309017 + 0.951057i −0.817115 − 2.51482i −0.809017 − 0.587785i −0.809017 + 0.587785i 1.23360 − 3.79662i 1.00000
421.1 −0.809017 + 0.587785i −0.867835 − 2.67092i 0.309017 − 0.951057i −0.809017 − 0.587785i 2.27202 + 1.65072i 0.309017 − 0.951057i 0.309017 + 0.951057i −3.95363 + 2.87248i 1.00000
421.2 −0.809017 + 0.587785i 0.361989 + 1.11409i 0.309017 − 0.951057i −0.809017 − 0.587785i −0.947700 − 0.688544i 0.309017 − 0.951057i 0.309017 + 0.951057i 1.31690 − 0.956780i 1.00000
421.3 −0.809017 + 0.587785i 1.00585 + 3.09567i 0.309017 − 0.951057i −0.809017 − 0.587785i −2.63334 − 1.91323i 0.309017 − 0.951057i 0.309017 + 0.951057i −6.14442 + 4.46418i 1.00000
631.1 −0.809017 − 0.587785i −0.867835 + 2.67092i 0.309017 + 0.951057i −0.809017 + 0.587785i 2.27202 − 1.65072i 0.309017 + 0.951057i 0.309017 − 0.951057i −3.95363 − 2.87248i 1.00000
631.2 −0.809017 − 0.587785i 0.361989 − 1.11409i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.947700 + 0.688544i 0.309017 + 0.951057i 0.309017 − 0.951057i 1.31690 + 0.956780i 1.00000
631.3 −0.809017 − 0.587785i 1.00585 − 3.09567i 0.309017 + 0.951057i −0.809017 + 0.587785i −2.63334 + 1.91323i 0.309017 + 0.951057i 0.309017 − 0.951057i −6.14442 − 4.46418i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.n.i ✓ 12
11.c even 5 1 inner 770.2.n.i ✓ 12
11.c even 5 1 8470.2.a.de 6
11.d odd 10 1 8470.2.a.cy 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.i ✓ 12 1.a even 1 1 trivial
770.2.n.i ✓ 12 11.c even 5 1 inner
8470.2.a.cy 6 11.d odd 10 1
8470.2.a.de 6 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 2 T_{3}^{11} + 14 T_{3}^{10} - 17 T_{3}^{9} + 87 T_{3}^{8} - 118 T_{3}^{7} + 459 T_{3}^{6} + \cdots + 841 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). 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
$3$ \( T^{12} - 2 T^{11} + \cdots + 841 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} - T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + \cdots + 234256 \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{11} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{12} + 13 T^{11} + \cdots + 525625 \) Copy content Toggle raw display
$23$ \( (T^{6} + 6 T^{5} + \cdots - 1856)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 26 T^{11} + \cdots + 9120400 \) Copy content Toggle raw display
$31$ \( T^{12} + 46 T^{10} + \cdots + 80656 \) Copy content Toggle raw display
$37$ \( T^{12} + 18 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( T^{12} - 16 T^{11} + \cdots + 13315201 \) Copy content Toggle raw display
$43$ \( (T^{6} - 19 T^{5} + \cdots - 20719)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 2485620736 \) Copy content Toggle raw display
$53$ \( T^{12} + 100 T^{10} + \cdots + 59536 \) Copy content Toggle raw display
$59$ \( T^{12} - 21 T^{11} + \cdots + 9025 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 12578968336 \) Copy content Toggle raw display
$67$ \( (T^{6} - 5 T^{5} + \cdots - 5821)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 169937296 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 5152224841 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 719137920400 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2876069641 \) Copy content Toggle raw display
$89$ \( (T^{6} - T^{5} + \cdots - 362975)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 19 T^{11} + \cdots + 21613201 \) Copy content Toggle raw display
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