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Properties

Label 770.2.n.j
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: \( x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} + \cdots + 25 \) 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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 12925652272 \nu^{11} - 689411783229 \nu^{10} + 1158393636547 \nu^{9} + \cdots - 68957262745775 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 171694051509 \nu^{11} - 3538073194043 \nu^{10} + 6710004699384 \nu^{9} + \cdots - 872744346462385 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1093010781921 \nu^{11} - 1619787283007 \nu^{10} + 10879795136331 \nu^{9} + \cdots - 79252706903350 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1093010781921 \nu^{11} + 1619787283007 \nu^{10} - 10879795136331 \nu^{9} + \cdots + 79252706903350 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1185075482262 \nu^{11} - 2450405323424 \nu^{10} + 12479616226722 \nu^{9} + \cdots - 68337530631950 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2733501225278 \nu^{11} - 4281926968294 \nu^{10} + 27618108154634 \nu^{9} + \cdots + 2642224095290 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2758290509831 \nu^{11} - 5503655367390 \nu^{10} + 29651783824912 \nu^{9} + \cdots - 366378456407095 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3170108276134 \nu^{11} + 5247205770347 \nu^{10} - 33251403754467 \nu^{9} + \cdots + 203977836612425 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6759751764174 \nu^{11} - 11030588267832 \nu^{10} + 70986300515617 \nu^{9} + \cdots + 7410594838995 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6923728370080 \nu^{11} + 13776035137694 \nu^{10} - 76812432137873 \nu^{9} + \cdots + 13\!\cdots\!10 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7100336146293 \nu^{11} + 10864290641274 \nu^{10} - 70862768140539 \nu^{9} + \cdots - 7554307496840 ) / 206669284189945 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 3\beta_{8} - \beta_{7} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{6} - 5\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{11} - 2\beta_{10} + 9\beta_{8} + 14\beta_{7} + 9\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{10} - 11\beta_{9} + 15\beta_{7} - 15\beta_{6} + 11\beta_{5} + 11\beta_{4} - 20\beta_{2} + 19\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -50\beta_{11} - 22\beta_{9} - 74\beta_{8} - 151\beta_{6} - 24\beta_{5} + 30\beta_{4} + 30\beta_{3} + 24\beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 74 \beta_{11} + 102 \beta_{10} + 74 \beta_{9} - 64 \beta_{8} - 152 \beta_{7} - 201 \beta_{5} + \cdots - 152 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 198 \beta_{11} + 198 \beta_{10} + 377 \beta_{9} + 485 \beta_{8} - 485 \beta_{7} + 1076 \beta_{6} + \cdots - 1076 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 893 \beta_{11} - 603 \beta_{10} + 1373 \beta_{8} + 700 \beta_{7} + 1373 \beta_{6} + 1453 \beta_{5} + \cdots - 664 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2949 \beta_{10} - 2949 \beta_{9} + 4713 \beta_{7} - 4713 \beta_{6} + 1976 \beta_{5} + 1976 \beta_{4} + \cdots + 8097 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7597 \beta_{11} - 4925 \beta_{9} - 11843 \beta_{8} - 18471 \beta_{6} - 6395 \beta_{5} + 11046 \beta_{4} + \cdots + 11843 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(1\)

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
−0.674672 − 2.07643i
0.289142 + 0.889888i
0.885530 + 2.72538i
−0.674672 + 2.07643i
0.289142 − 0.889888i
0.885530 − 2.72538i
−1.36475 + 0.991547i
0.198931 − 0.144532i
1.66582 − 1.21029i
−1.36475 − 0.991547i
0.198931 + 0.144532i
1.66582 + 1.21029i
−0.309017 − 0.951057i −1.76631 − 1.28330i −0.809017 + 0.587785i 0.309017 − 0.951057i −0.674672 + 2.07643i −0.809017 + 0.587785i 0.809017 + 0.587785i 0.545950 + 1.68026i −1.00000
71.2 −0.309017 − 0.951057i 0.756984 + 0.549981i −0.809017 + 0.587785i 0.309017 − 0.951057i 0.289142 − 0.889888i −0.809017 + 0.587785i 0.809017 + 0.587785i −0.656505 − 2.02052i −1.00000
71.3 −0.309017 − 0.951057i 2.31835 + 1.68438i −0.809017 + 0.587785i 0.309017 − 0.951057i 0.885530 − 2.72538i −0.809017 + 0.587785i 0.809017 + 0.587785i 1.61056 + 4.95678i −1.00000
141.1 −0.309017 + 0.951057i −1.76631 + 1.28330i −0.809017 − 0.587785i 0.309017 + 0.951057i −0.674672 − 2.07643i −0.809017 − 0.587785i 0.809017 − 0.587785i 0.545950 − 1.68026i −1.00000
141.2 −0.309017 + 0.951057i 0.756984 − 0.549981i −0.809017 − 0.587785i 0.309017 + 0.951057i 0.289142 + 0.889888i −0.809017 − 0.587785i 0.809017 − 0.587785i −0.656505 + 2.02052i −1.00000
141.3 −0.309017 + 0.951057i 2.31835 − 1.68438i −0.809017 − 0.587785i 0.309017 + 0.951057i 0.885530 + 2.72538i −0.809017 − 0.587785i 0.809017 − 0.587785i 1.61056 − 4.95678i −1.00000
421.1 0.809017 − 0.587785i −0.521287 − 1.60436i 0.309017 − 0.951057i −0.809017 − 0.587785i −1.36475 − 0.991547i 0.309017 − 0.951057i −0.309017 − 0.951057i 0.124831 − 0.0906951i −1.00000
421.2 0.809017 − 0.587785i 0.0759851 + 0.233858i 0.309017 − 0.951057i −0.809017 − 0.587785i 0.198931 + 0.144532i 0.309017 − 0.951057i −0.309017 − 0.951057i 2.37814 − 1.72782i −1.00000
421.3 0.809017 − 0.587785i 0.636285 + 1.95828i 0.309017 − 0.951057i −0.809017 − 0.587785i 1.66582 + 1.21029i 0.309017 − 0.951057i −0.309017 − 0.951057i −1.00297 + 0.728698i −1.00000
631.1 0.809017 + 0.587785i −0.521287 + 1.60436i 0.309017 + 0.951057i −0.809017 + 0.587785i −1.36475 + 0.991547i 0.309017 + 0.951057i −0.309017 + 0.951057i 0.124831 + 0.0906951i −1.00000
631.2 0.809017 + 0.587785i 0.0759851 − 0.233858i 0.309017 + 0.951057i −0.809017 + 0.587785i 0.198931 − 0.144532i 0.309017 + 0.951057i −0.309017 + 0.951057i 2.37814 + 1.72782i −1.00000
631.3 0.809017 + 0.587785i 0.636285 − 1.95828i 0.309017 + 0.951057i −0.809017 + 0.587785i 1.66582 − 1.21029i 0.309017 + 0.951057i −0.309017 + 0.951057i −1.00297 − 0.728698i −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.j ✓ 12
11.c even 5 1 inner 770.2.n.j ✓ 12
11.c even 5 1 8470.2.a.cw 6
11.d odd 10 1 8470.2.a.dc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.j ✓ 12 1.a even 1 1 trivial
770.2.n.j ✓ 12 11.c even 5 1 inner
8470.2.a.cw 6 11.c even 5 1
8470.2.a.dc 6 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 3 T_{3}^{11} + 6 T_{3}^{10} - 4 T_{3}^{9} + 29 T_{3}^{8} - 32 T_{3}^{7} + 199 T_{3}^{6} + \cdots + 25 \) 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
$3$ \( T^{12} - 3 T^{11} + \cdots + 25 \) 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
$13$ \( T^{12} - 2 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 208253761 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} + \cdots + 126736 \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} + \cdots - 2624)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 20 T^{11} + \cdots + 3225616 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + \cdots + 6400 \) Copy content Toggle raw display
$37$ \( T^{12} - 22 T^{11} + \cdots + 5837056 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 531579136 \) Copy content Toggle raw display
$43$ \( (T^{6} + 30 T^{5} + \cdots - 45004)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 4 T^{11} + \cdots + 19360000 \) Copy content Toggle raw display
$53$ \( T^{12} - 18 T^{11} + \cdots + 1478656 \) Copy content Toggle raw display
$59$ \( T^{12} + 32 T^{11} + \cdots + 80656 \) Copy content Toggle raw display
$61$ \( T^{12} - 8 T^{11} + \cdots + 41165056 \) Copy content Toggle raw display
$67$ \( (T^{6} - 18 T^{5} + \cdots - 44)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 161900176 \) Copy content Toggle raw display
$73$ \( T^{12} - 14 T^{11} + \cdots + 3025 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 694638736 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 6456283201 \) Copy content Toggle raw display
$89$ \( (T^{6} + 18 T^{5} + \cdots - 2420)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 7943978641 \) Copy content Toggle raw display
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