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Properties

Label 770.2.n.h
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: \( x^{12} - 3 x^{11} + 7 x^{10} - 9 x^{9} + 55 x^{8} - 32 x^{7} + 287 x^{6} - 302 x^{5} + 1175 x^{4} + \cdots + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 7 x^{10} - 9 x^{9} + 55 x^{8} - 32 x^{7} + 287 x^{6} - 302 x^{5} + 1175 x^{4} + \cdots + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 66027685671987 \nu^{11} + 300911576247391 \nu^{10} - 495079952903319 \nu^{9} + \cdots + 13\!\cdots\!67 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 78194599018136 \nu^{11} - 583919559521768 \nu^{10} + \cdots - 25\!\cdots\!37 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13935225036164 \nu^{11} + 14741542109035 \nu^{10} + 14427641195505 \nu^{9} + \cdots + 70\!\cdots\!02 ) / 10\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12974065433470 \nu^{11} - 36758593588006 \nu^{10} + 79828823292525 \nu^{9} + \cdots + 231171353576831 ) / 63\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 274108974347116 \nu^{11} - 675019018801603 \nu^{10} + \cdots - 24\!\cdots\!25 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18386092761440 \nu^{11} + 38489443756396 \nu^{10} - 86670508680991 \nu^{9} + \cdots - 14\!\cdots\!84 ) / 63\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22736202141892 \nu^{11} - 64882405687006 \nu^{10} + 140403633386251 \nu^{9} + \cdots - 10\!\cdots\!65 ) / 63\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 519081844887701 \nu^{11} + \cdots + 90\!\cdots\!22 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 533612005537835 \nu^{11} + \cdots - 87\!\cdots\!70 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 99158490471442 \nu^{11} + 290212999539605 \nu^{10} - 612416220981491 \nu^{9} + \cdots + 46\!\cdots\!16 ) / 63\!\cdots\!29 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + 5\beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{10} - 7\beta_{7} + 2\beta_{6} - 7\beta_{5} - \beta_{4} + 2\beta_{3} + 3\beta_{2} - 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{11} + 35 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 8 \beta_{6} - 12 \beta_{5} + \cdots - 10 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{11} + 24 \beta_{10} + 14 \beta_{9} + 57 \beta_{8} + 14 \beta_{7} - 25 \beta_{5} + 14 \beta_{4} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 67 \beta_{9} + 130 \beta_{8} + 130 \beta_{7} - 67 \beta_{6} + 96 \beta_{4} + 145 \beta_{3} - 137 \beta_{2} + \cdots - 137 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 101 \beta_{11} - 323 \beta_{10} + 101 \beta_{9} + 259 \beta_{8} + 608 \beta_{7} - 259 \beta_{6} + \cdots - 323 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 608 \beta_{11} - 2486 \beta_{10} + 184 \beta_{8} + 1340 \beta_{7} - 932 \beta_{6} + 1340 \beta_{5} + \cdots - 1113 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1664 \beta_{11} - 6232 \beta_{10} - 1032 \beta_{9} - 3094 \beta_{8} + 1697 \beta_{7} + \cdots + 3361 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3361 \beta_{11} - 13610 \beta_{10} - 5790 \beta_{9} - 13686 \beta_{8} - 5790 \beta_{7} + 8077 \beta_{5} + \cdots + 9581 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 17047 \beta_{9} - 44524 \beta_{8} - 44524 \beta_{7} + 17047 \beta_{6} - 27553 \beta_{4} + \cdots + 39272 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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
2.55917 − 1.85934i
0.609000 − 0.442464i
−1.85915 + 1.35075i
2.55917 + 1.85934i
0.609000 + 0.442464i
−1.85915 − 1.35075i
0.603111 − 1.85618i
0.254744 − 0.784022i
−0.666872 + 2.05242i
0.603111 + 1.85618i
0.254744 + 0.784022i
−0.666872 − 2.05242i
0.309017 + 0.951057i −1.75015 − 1.27156i −0.809017 + 0.587785i −0.309017 + 0.951057i 0.668497 − 2.05742i −0.809017 + 0.587785i −0.809017 − 0.587785i 0.519111 + 1.59766i −1.00000
71.2 0.309017 + 0.951057i 0.200017 + 0.145321i −0.809017 + 0.587785i −0.309017 + 0.951057i −0.0763997 + 0.235134i −0.809017 + 0.587785i −0.809017 − 0.587785i −0.908162 − 2.79504i −1.00000
71.3 0.309017 + 0.951057i 2.66817 + 1.93854i −0.809017 + 0.587785i −0.309017 + 0.951057i −1.01915 + 3.13662i −0.809017 + 0.587785i −0.809017 − 0.587785i 2.43414 + 7.49150i −1.00000
141.1 0.309017 − 0.951057i −1.75015 + 1.27156i −0.809017 − 0.587785i −0.309017 − 0.951057i 0.668497 + 2.05742i −0.809017 − 0.587785i −0.809017 + 0.587785i 0.519111 − 1.59766i −1.00000
141.2 0.309017 − 0.951057i 0.200017 − 0.145321i −0.809017 − 0.587785i −0.309017 − 0.951057i −0.0763997 − 0.235134i −0.809017 − 0.587785i −0.809017 + 0.587785i −0.908162 + 2.79504i −1.00000
141.3 0.309017 − 0.951057i 2.66817 − 1.93854i −0.809017 − 0.587785i −0.309017 − 0.951057i −1.01915 − 3.13662i −0.809017 − 0.587785i −0.809017 + 0.587785i 2.43414 − 7.49150i −1.00000
421.1 −0.809017 + 0.587785i −0.912128 − 2.80724i 0.309017 − 0.951057i 0.809017 + 0.587785i 2.38798 + 1.73497i 0.309017 − 0.951057i 0.309017 + 0.951057i −4.62157 + 3.35777i −1.00000
421.2 −0.809017 + 0.587785i −0.563761 − 1.73508i 0.309017 − 0.951057i 0.809017 + 0.587785i 1.47595 + 1.07234i 0.309017 − 0.951057i 0.309017 + 0.951057i −0.265619 + 0.192984i −1.00000
421.3 −0.809017 + 0.587785i 0.357855 + 1.10136i 0.309017 − 0.951057i 0.809017 + 0.587785i −0.936877 − 0.680681i 0.309017 − 0.951057i 0.309017 + 0.951057i 1.34211 − 0.975098i −1.00000
631.1 −0.809017 − 0.587785i −0.912128 + 2.80724i 0.309017 + 0.951057i 0.809017 − 0.587785i 2.38798 − 1.73497i 0.309017 + 0.951057i 0.309017 − 0.951057i −4.62157 − 3.35777i −1.00000
631.2 −0.809017 − 0.587785i −0.563761 + 1.73508i 0.309017 + 0.951057i 0.809017 − 0.587785i 1.47595 − 1.07234i 0.309017 + 0.951057i 0.309017 − 0.951057i −0.265619 − 0.192984i −1.00000
631.3 −0.809017 − 0.587785i 0.357855 − 1.10136i 0.309017 + 0.951057i 0.809017 − 0.587785i −0.936877 + 0.680681i 0.309017 + 0.951057i 0.309017 − 0.951057i 1.34211 + 0.975098i −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.h ✓ 12
11.c even 5 1 inner 770.2.n.h ✓ 12
11.c even 5 1 8470.2.a.da 6
11.d odd 10 1 8470.2.a.cu 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.h ✓ 12 1.a even 1 1 trivial
770.2.n.h ✓ 12 11.c even 5 1 inner
8470.2.a.cu 6 11.d odd 10 1
8470.2.a.da 6 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 6 T_{3}^{10} - 11 T_{3}^{9} + 61 T_{3}^{8} + 168 T_{3}^{7} + 657 T_{3}^{6} + 622 T_{3}^{5} + \cdots + 121 \) 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
$3$ \( T^{12} + 6 T^{10} + \cdots + 121 \) 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
$13$ \( T^{12} + 4 T^{10} + \cdots + 309136 \) Copy content Toggle raw display
$17$ \( T^{12} + 32 T^{10} + \cdots + 297025 \) Copy content Toggle raw display
$19$ \( T^{12} + T^{11} + \cdots + 160801 \) Copy content Toggle raw display
$23$ \( (T^{6} + 2 T^{5} + \cdots + 3520)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 22 T^{11} + \cdots + 234256 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + \cdots + 5856400 \) Copy content Toggle raw display
$37$ \( T^{12} + 10 T^{11} + \cdots + 46840336 \) Copy content Toggle raw display
$41$ \( T^{12} - 16 T^{11} + \cdots + 12313081 \) Copy content Toggle raw display
$43$ \( (T^{6} - 15 T^{5} + \cdots - 29)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 296390656 \) Copy content Toggle raw display
$53$ \( T^{12} + 26 T^{11} + \cdots + 3254416 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2929082641 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 487703056 \) Copy content Toggle raw display
$67$ \( (T^{6} + 29 T^{5} + \cdots - 15679)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 14 T^{11} + \cdots + 1936 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 85539285841 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 2448666256 \) Copy content Toggle raw display
$83$ \( T^{12} - 35 T^{11} + \cdots + 2927521 \) Copy content Toggle raw display
$89$ \( (T^{6} + 29 T^{5} + \cdots + 370271)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 5679189441025 \) Copy content Toggle raw display
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