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Properties

Label 770.2.l.a
Level $770$
Weight $2$
Character orbit 770.l
Analytic conductor $6.148$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [770,2,Mod(573,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.573");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.l (of order \(4\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + \beta_1 q^{3} - \beta_{8} q^{4} + (\beta_{13} + \beta_{2}) q^{5} + ( - \beta_{13} - \beta_{5} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{12} + \beta_{9} + \beta_{8}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} + \beta_1 q^{3} - \beta_{8} q^{4} + (\beta_{13} + \beta_{2}) q^{5} + ( - \beta_{13} - \beta_{5} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{12} + \beta_{9} + \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{11} + 16 q^{15} - 16 q^{16} + 16 q^{18} - 16 q^{23} + 16 q^{25} - 16 q^{35} + 16 q^{36} + 16 q^{37} - 16 q^{43} + 32 q^{46} - 32 q^{50} + 32 q^{51} + 16 q^{53} + 48 q^{57} - 32 q^{58} + 16 q^{60} + 16 q^{65} + 80 q^{67} + 48 q^{70} + 16 q^{72} + 32 q^{78} + 48 q^{81} - 64 q^{85} - 32 q^{91} - 16 q^{92} + 64 q^{93} + 64 q^{95} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} - 3 \nu^{13} + 20 \nu^{12} - 69 \nu^{11} + 100 \nu^{10} - 506 \nu^{9} - 4 \nu^{8} - 1488 \nu^{7} + \cdots - 72 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{14} - 69\nu^{12} - 506\nu^{10} - 1488\nu^{8} - 1638\nu^{6} - 594\nu^{4} - 36\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{14} - 71\nu^{12} - 553\nu^{10} - 1848\nu^{8} - 2790\nu^{6} - 2110\nu^{4} - 770\nu^{2} - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{14} + 72\nu^{12} + 575\nu^{10} + 1994\nu^{8} + 3126\nu^{6} + 2232\nu^{4} + 630\nu^{2} + 44 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7 \nu^{14} - 3 \nu^{13} - 164 \nu^{12} - 69 \nu^{11} - 1250 \nu^{10} - 506 \nu^{9} - 3984 \nu^{8} + \cdots - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6\nu^{15} + 151\nu^{13} + 1312\nu^{11} + 5192\nu^{9} + 9900\nu^{7} + 8830\nu^{5} + 3288\nu^{3} + 352\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{15} - 260\nu^{13} - 2018\nu^{11} - 6672\nu^{9} - 9702\nu^{7} - 6544\nu^{5} - 2004\nu^{3} - 272\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\nu^{15} + 261\nu^{13} + 2042\nu^{11} + 6862\nu^{9} + 10334\nu^{7} + 7410\nu^{5} + 2412\nu^{3} + 252\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{15} + 12 \nu^{14} - 258 \nu^{13} + 282 \nu^{12} - 1971 \nu^{11} + 2164 \nu^{10} - 6311 \nu^{9} + \cdots + 136 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11 \nu^{15} - 8 \nu^{14} + 261 \nu^{13} - 192 \nu^{12} + 2040 \nu^{11} - 1534 \nu^{10} + 6817 \nu^{9} + \cdots - 164 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11 \nu^{15} + 8 \nu^{14} + 261 \nu^{13} + 192 \nu^{12} + 2040 \nu^{11} + 1534 \nu^{10} + 6817 \nu^{9} + \cdots + 164 ) / 16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11 \nu^{15} + 12 \nu^{14} + 258 \nu^{13} + 282 \nu^{12} + 1971 \nu^{11} + 2164 \nu^{10} + 6311 \nu^{9} + \cdots + 136 ) / 16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -19\nu^{15} - 448\nu^{13} - 3460\nu^{11} - 11326\nu^{9} - 16094\nu^{7} - 10328\nu^{5} - 2904\nu^{3} - 316\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38 \nu^{15} - 15 \nu^{14} + 896 \nu^{13} - 352 \nu^{12} + 6921 \nu^{11} - 2692 \nu^{10} + 22674 \nu^{9} + \cdots - 76 ) / 16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 38 \nu^{15} + 15 \nu^{14} + 896 \nu^{13} + 352 \nu^{12} + 6921 \nu^{11} + 2692 \nu^{10} + 22674 \nu^{9} + \cdots + 76 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{5} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + \beta_{14} + \beta_{12} + \beta_{9} + \beta_{4} + \beta_{3} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 3 \beta_{14} - 4 \beta_{13} + 8 \beta_{12} - 6 \beta_{11} - 6 \beta_{10} + \cdots - 11 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10 \beta_{15} - 10 \beta_{14} - 13 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 13 \beta_{9} - 4 \beta_{5} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25 \beta_{15} + 25 \beta_{14} + 31 \beta_{13} - 47 \beta_{12} + 27 \beta_{11} + 27 \beta_{10} + \cdots + 63 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 109 \beta_{15} + 109 \beta_{14} + 161 \beta_{12} - 67 \beta_{11} + 67 \beta_{10} + 161 \beta_{9} + \cdots - 252 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 336 \beta_{15} - 336 \beta_{14} - 413 \beta_{13} + 571 \beta_{12} - 292 \beta_{11} - 292 \beta_{10} + \cdots - 752 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1272 \beta_{15} - 1272 \beta_{14} - 1980 \beta_{12} + 900 \beta_{11} - 900 \beta_{10} - 1980 \beta_{9} + \cdots + 2910 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4248 \beta_{15} + 4248 \beta_{14} + 5220 \beta_{13} - 6971 \beta_{12} + 3407 \beta_{11} + \cdots + 9127 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 15294 \beta_{15} + 15294 \beta_{14} + 24294 \beta_{12} - 11376 \beta_{11} + 11376 \beta_{10} + \cdots - 34858 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 52613 \beta_{15} - 52613 \beta_{14} - 64692 \beta_{13} + 85244 \beta_{12} - 40966 \beta_{11} + \cdots - 111425 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 186020 \beta_{15} - 186020 \beta_{14} - 297758 \beta_{12} + 140896 \beta_{11} - 140896 \beta_{10} + \cdots + 423384 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 646906 \beta_{15} + 646906 \beta_{14} + 795794 \beta_{13} - 1043178 \beta_{12} + 498250 \beta_{11} + \cdots + 1363030 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2272238 \beta_{15} + 2272238 \beta_{14} + 3647582 \beta_{12} - 1732466 \beta_{11} + 1732466 \beta_{10} + \cdots - 5168936 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7933264 \beta_{15} - 7933264 \beta_{14} - 9761614 \beta_{13} + 12770386 \beta_{12} + \cdots - 16684816 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
573.1
2.08509i
− 0.886177i
− 0.528036i
− 3.49930i
0.357857i
2.13875i
− 0.724535i
1.05636i
− 2.08509i
0.886177i
0.528036i
3.49930i
− 0.357857i
− 2.13875i
0.724535i
− 1.05636i
−0.707107 + 0.707107i −1.30656 + 1.30656i − 1.00000i −2.10100 − 0.765367i − 1.84776i 0.348579 + 2.62269i 0.707107 + 0.707107i − 0.414214i 2.02683 − 0.944437i
573.2 −0.707107 + 0.707107i −1.30656 + 1.30656i − 1.00000i 2.10100 − 0.765367i − 1.84776i −0.348579 − 2.62269i 0.707107 + 0.707107i − 0.414214i −0.944437 + 2.02683i
573.3 −0.707107 + 0.707107i 1.30656 − 1.30656i − 1.00000i −2.10100 + 0.765367i 1.84776i 2.62269 + 0.348579i 0.707107 + 0.707107i − 0.414214i 0.944437 − 2.02683i
573.4 −0.707107 + 0.707107i 1.30656 − 1.30656i − 1.00000i 2.10100 + 0.765367i 1.84776i −2.62269 − 0.348579i 0.707107 + 0.707107i − 0.414214i −2.02683 + 0.944437i
573.5 0.707107 − 0.707107i −0.541196 + 0.541196i − 1.00000i −1.25928 − 1.84776i 0.765367i 0.754883 − 2.53577i −0.707107 − 0.707107i 2.41421i −2.19701 − 0.416117i
573.6 0.707107 − 0.707107i −0.541196 + 0.541196i − 1.00000i 1.25928 − 1.84776i 0.765367i −0.754883 + 2.53577i −0.707107 − 0.707107i 2.41421i −0.416117 − 2.19701i
573.7 0.707107 − 0.707107i 0.541196 − 0.541196i − 1.00000i −1.25928 + 1.84776i − 0.765367i −2.53577 + 0.754883i −0.707107 − 0.707107i 2.41421i 0.416117 + 2.19701i
573.8 0.707107 − 0.707107i 0.541196 − 0.541196i − 1.00000i 1.25928 + 1.84776i − 0.765367i 2.53577 − 0.754883i −0.707107 − 0.707107i 2.41421i 2.19701 + 0.416117i
727.1 −0.707107 − 0.707107i −1.30656 − 1.30656i 1.00000i −2.10100 + 0.765367i 1.84776i 0.348579 − 2.62269i 0.707107 − 0.707107i 0.414214i 2.02683 + 0.944437i
727.2 −0.707107 − 0.707107i −1.30656 − 1.30656i 1.00000i 2.10100 + 0.765367i 1.84776i −0.348579 + 2.62269i 0.707107 − 0.707107i 0.414214i −0.944437 − 2.02683i
727.3 −0.707107 − 0.707107i 1.30656 + 1.30656i 1.00000i −2.10100 − 0.765367i − 1.84776i 2.62269 − 0.348579i 0.707107 − 0.707107i 0.414214i 0.944437 + 2.02683i
727.4 −0.707107 − 0.707107i 1.30656 + 1.30656i 1.00000i 2.10100 − 0.765367i − 1.84776i −2.62269 + 0.348579i 0.707107 − 0.707107i 0.414214i −2.02683 − 0.944437i
727.5 0.707107 + 0.707107i −0.541196 − 0.541196i 1.00000i −1.25928 + 1.84776i − 0.765367i 0.754883 + 2.53577i −0.707107 + 0.707107i − 2.41421i −2.19701 + 0.416117i
727.6 0.707107 + 0.707107i −0.541196 − 0.541196i 1.00000i 1.25928 + 1.84776i − 0.765367i −0.754883 − 2.53577i −0.707107 + 0.707107i − 2.41421i −0.416117 + 2.19701i
727.7 0.707107 + 0.707107i 0.541196 + 0.541196i 1.00000i −1.25928 − 1.84776i 0.765367i −2.53577 − 0.754883i −0.707107 + 0.707107i − 2.41421i 0.416117 − 2.19701i
727.8 0.707107 + 0.707107i 0.541196 + 0.541196i 1.00000i 1.25928 − 1.84776i 0.765367i 2.53577 + 0.754883i −0.707107 + 0.707107i − 2.41421i 2.19701 − 0.416117i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 573.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.l.a ✓ 16
5.c odd 4 1 inner 770.2.l.a ✓ 16
7.b odd 2 1 inner 770.2.l.a ✓ 16
35.f even 4 1 inner 770.2.l.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.l.a ✓ 16 1.a even 1 1 trivial
770.2.l.a ✓ 16 5.c odd 4 1 inner
770.2.l.a ✓ 16 7.b odd 2 1 inner
770.2.l.a ✓ 16 35.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 12T_{3}^{4} + 4 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 12 T^{4} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 124 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T - 1)^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + 1328 T^{12} + \cdots + 21381376 \) Copy content Toggle raw display
$17$ \( T^{16} + 1568 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( (T^{8} - 112 T^{6} + \cdots + 148996)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 8 T^{7} + \cdots + 315844)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 128 T^{6} + \cdots + 391876)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 120 T^{6} + \cdots + 490000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots + 669124)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 112 T^{6} + \cdots + 148996)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 8 T^{7} + \cdots + 6210064)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 754507653376 \) Copy content Toggle raw display
$53$ \( (T^{8} - 8 T^{7} + \cdots + 196)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 160 T^{6} + \cdots + 40000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 272 T^{6} + \cdots + 6635776)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 20 T^{3} + \cdots + 196)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 64 T^{2} + 224)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 37120 T^{12} + \cdots + 16777216 \) Copy content Toggle raw display
$79$ \( (T^{8} + 544 T^{6} + \cdots + 93122500)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1665379926016 \) Copy content Toggle raw display
$89$ \( (T^{8} - 512 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
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