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Properties

Label 490.2.k.d
Level $490$
Weight $2$
Character orbit 490.k
Analytic conductor $3.913$
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] = [490,2,Mod(71,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.k (of order \(7\), degree \(6\), 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: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{11} + 11x^{10} - 20x^{9} + 44x^{8} - 80x^{7} + 106x^{6} - 91x^{5} + 84x^{3} - 98x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{11} + 11x^{10} - 20x^{9} + 44x^{8} - 80x^{7} + 106x^{6} - 91x^{5} + 84x^{3} - 98x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 955643 \nu^{11} - 4006406 \nu^{10} + 8661961 \nu^{9} - 17022173 \nu^{8} + 35733187 \nu^{7} + \cdots - 20436801 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1383748 \nu^{11} - 3243810 \nu^{10} + 2545364 \nu^{9} - 7191605 \nu^{8} + 18532190 \nu^{7} + \cdots + 10431708 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13136080 \nu^{11} - 55455866 \nu^{10} + 100934164 \nu^{9} - 182761547 \nu^{8} + 433864838 \nu^{7} + \cdots - 819697935 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14865209 \nu^{11} + 60416479 \nu^{10} - 107107226 \nu^{9} + 198858915 \nu^{8} + \cdots + 748832042 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21173116 \nu^{11} - 87511401 \nu^{10} + 158903388 \nu^{9} - 291912677 \nu^{8} + 685453832 \nu^{7} + \cdots - 1237927754 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23754204 \nu^{11} - 98553547 \nu^{10} + 177791249 \nu^{9} - 324842653 \nu^{8} + 770294472 \nu^{7} + \cdots - 1388236360 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24526013 \nu^{11} - 100403659 \nu^{10} + 179881936 \nu^{9} - 331157758 \nu^{8} + \cdots - 1327992953 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25841360 \nu^{11} + 107915183 \nu^{10} - 193914998 \nu^{9} + 351859423 \nu^{8} + \cdots + 1471816444 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29917613 \nu^{11} - 123888920 \nu^{10} + 223542099 \nu^{9} - 407235891 \nu^{8} + \cdots - 1737833587 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29955515 \nu^{11} + 124654994 \nu^{10} - 224933126 \nu^{9} + 410477064 \nu^{8} + \cdots + 1701647521 ) / 53534957 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 32783075 \nu^{11} + 133422022 \nu^{10} - 236829219 \nu^{9} + 435987054 \nu^{8} + \cdots + 1786569722 ) / 53534957 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + \cdots + 5 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} - 9 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 4 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} - 26 \beta_{7} + 32 \beta_{6} + 9 \beta_{5} + \cdots + 6 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17 \beta_{11} - \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 3 \beta_{7} + \beta_{6} + 32 \beta_{5} + \cdots + 12 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{11} - 93 \beta_{10} - 5 \beta_{9} + 9 \beta_{8} - 48 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} + \cdots - 53 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 110 \beta_{11} - 74 \beta_{10} + 53 \beta_{9} - 24 \beta_{8} - 243 \beta_{7} + 151 \beta_{6} + 86 \beta_{5} + \cdots - 8 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 150 \beta_{11} - 8 \beta_{10} + 8 \beta_{9} - 90 \beta_{8} + 53 \beta_{7} - 132 \beta_{6} + 158 \beta_{5} + \cdots - 65 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 85 \beta_{11} - 387 \beta_{10} + 23 \beta_{9} + 23 \beta_{8} + 85 \beta_{7} - 215 \beta_{6} + \cdots - 368 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 558 \beta_{11} + 213 \beta_{10} + 368 \beta_{9} - 136 \beta_{8} - 481 \beta_{7} + 417 \beta_{6} + \cdots + 461 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 234 \beta_{11} + 902 \beta_{10} + 78 \beta_{9} - 83 \beta_{8} + 2531 \beta_{7} - 2365 \beta_{6} + \cdots + 117 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1989 \beta_{11} - 667 \beta_{10} + 457 \beta_{9} + 1325 \beta_{8} + 3242 \beta_{7} - 2420 \beta_{6} + \cdots - 1992 ) / 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(\beta_{7}\) \(1\)

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
1.83316 + 0.664424i
−1.05568 − 1.63935i
0.867238 + 0.0303020i
−0.768207 + 0.403582i
1.36625 − 0.760527i
0.257239 + 1.54236i
1.36625 + 0.760527i
0.257239 − 1.54236i
0.867238 − 0.0303020i
−0.768207 − 0.403582i
1.83316 − 0.664424i
−1.05568 + 1.63935i
−0.900969 + 0.433884i −0.277479 + 1.21572i 0.623490 − 0.781831i −0.222521 + 0.974928i −0.277479 − 1.21572i −1.84991 − 1.89152i −0.222521 + 0.974928i 1.30194 + 0.626980i −0.222521 − 0.974928i
71.2 −0.900969 + 0.433884i −0.277479 + 1.21572i 0.623490 − 0.781831i −0.222521 + 0.974928i −0.277479 − 1.21572i 0.325449 + 2.62566i −0.222521 + 0.974928i 1.30194 + 0.626980i −0.222521 − 0.974928i
141.1 0.623490 − 0.781831i 0.400969 + 0.193096i −0.222521 − 0.974928i −0.900969 − 0.433884i 0.400969 − 0.193096i −1.57636 − 2.12487i −0.900969 − 0.433884i −1.74698 − 2.19064i −0.900969 + 0.433884i
141.2 0.623490 − 0.781831i 0.400969 + 0.193096i −0.222521 − 0.974928i −0.900969 − 0.433884i 0.400969 − 0.193096i 2.42237 + 1.06401i −0.900969 − 0.433884i −1.74698 − 2.19064i −0.900969 + 0.433884i
211.1 −0.222521 + 0.974928i −1.12349 + 1.40881i −0.900969 − 0.433884i 0.623490 − 0.781831i −1.12349 − 1.40881i −1.76897 − 1.96742i 0.623490 − 0.781831i −0.0549581 − 0.240787i 0.623490 + 0.781831i
211.2 −0.222521 + 0.974928i −1.12349 + 1.40881i −0.900969 − 0.433884i 0.623490 − 0.781831i −1.12349 − 1.40881i 2.44742 − 1.00506i 0.623490 − 0.781831i −0.0549581 − 0.240787i 0.623490 + 0.781831i
281.1 −0.222521 − 0.974928i −1.12349 − 1.40881i −0.900969 + 0.433884i 0.623490 + 0.781831i −1.12349 + 1.40881i −1.76897 + 1.96742i 0.623490 + 0.781831i −0.0549581 + 0.240787i 0.623490 − 0.781831i
281.2 −0.222521 − 0.974928i −1.12349 − 1.40881i −0.900969 + 0.433884i 0.623490 + 0.781831i −1.12349 + 1.40881i 2.44742 + 1.00506i 0.623490 + 0.781831i −0.0549581 + 0.240787i 0.623490 − 0.781831i
351.1 0.623490 + 0.781831i 0.400969 − 0.193096i −0.222521 + 0.974928i −0.900969 + 0.433884i 0.400969 + 0.193096i −1.57636 + 2.12487i −0.900969 + 0.433884i −1.74698 + 2.19064i −0.900969 − 0.433884i
351.2 0.623490 + 0.781831i 0.400969 − 0.193096i −0.222521 + 0.974928i −0.900969 + 0.433884i 0.400969 + 0.193096i 2.42237 − 1.06401i −0.900969 + 0.433884i −1.74698 + 2.19064i −0.900969 − 0.433884i
421.1 −0.900969 − 0.433884i −0.277479 − 1.21572i 0.623490 + 0.781831i −0.222521 − 0.974928i −0.277479 + 1.21572i −1.84991 + 1.89152i −0.222521 − 0.974928i 1.30194 − 0.626980i −0.222521 + 0.974928i
421.2 −0.900969 − 0.433884i −0.277479 − 1.21572i 0.623490 + 0.781831i −0.222521 − 0.974928i −0.277479 + 1.21572i 0.325449 − 2.62566i −0.222521 − 0.974928i 1.30194 − 0.626980i −0.222521 + 0.974928i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.k.d ✓ 12
49.e even 7 1 inner 490.2.k.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.k.d ✓ 12 1.a even 1 1 trivial
490.2.k.d ✓ 12 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{6} + 2T_{3}^{5} + 4T_{3}^{4} + T_{3}^{3} + 2T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{12} - T_{11}^{11} - 8 T_{11}^{10} + 24 T_{11}^{9} + 62 T_{11}^{8} - 537 T_{11}^{7} + 2261 T_{11}^{6} + \cdots + 5041 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 35 T^{9} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - T^{11} + \cdots + 5041 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( T^{12} - 5 T^{11} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( (T^{6} + T^{5} - 13 T^{4} + \cdots - 13)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 8 T^{11} + \cdots + 13756681 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 277588921 \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} + \cdots + 6539)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 183250369 \) Copy content Toggle raw display
$41$ \( T^{12} - 13 T^{11} + \cdots + 32296489 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1205547841 \) Copy content Toggle raw display
$47$ \( T^{12} + 8 T^{11} + \cdots + 3265249 \) Copy content Toggle raw display
$53$ \( T^{12} - 22 T^{11} + \cdots + 9406489 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2925619921 \) Copy content Toggle raw display
$61$ \( T^{12} + T^{11} + \cdots + 9066121 \) Copy content Toggle raw display
$67$ \( (T^{6} + 3 T^{5} + \cdots - 83341)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 276590161 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 928754310961 \) Copy content Toggle raw display
$79$ \( (T^{6} + 4 T^{5} + \cdots + 11999)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2818441921 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 97268510641 \) Copy content Toggle raw display
$97$ \( (T^{6} - 21 T^{5} + \cdots + 388969)^{2} \) Copy content Toggle raw display
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