Properties

Label 17.12.a.a
Level $17$
Weight $12$
Character orbit 17.a
Self dual yes
Analytic conductor $13.062$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [17,12,Mod(1,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 17.a (trivial)

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: yes
Analytic conductor: \(13.0618340695\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 8440x^{4} - 21100x^{3} + 19034528x^{2} + 24205632x - 12354600960 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + (\beta_{2} - 79) q^{3} + (\beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 767) q^{4} + ( - 2 \beta_{5} - 9 \beta_{4} - \beta_{3} - 11 \beta_{2} - 40 \beta_1 - 2129) q^{5} + ( - 19 \beta_{5} - \beta_{4} - 21 \beta_{3} - 12 \beta_{2} - 184 \beta_1 + 167) q^{6} + (30 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} - 46 \beta_{2} - 480 \beta_1 - 3670) q^{7} + (9 \beta_{5} + 14 \beta_{4} + 99 \beta_{3} - 97 \beta_{2} - 221 \beta_1 + 12577) q^{8} + ( - 32 \beta_{5} + 85 \beta_{4} + 93 \beta_{3} + 43 \beta_{2} + \cdots + 50010) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{2} + (\beta_{2} - 79) q^{3} + (\beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 767) q^{4} + ( - 2 \beta_{5} - 9 \beta_{4} - \beta_{3} - 11 \beta_{2} - 40 \beta_1 - 2129) q^{5} + ( - 19 \beta_{5} - \beta_{4} - 21 \beta_{3} - 12 \beta_{2} - 184 \beta_1 + 167) q^{6} + (30 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} - 46 \beta_{2} - 480 \beta_1 - 3670) q^{7} + (9 \beta_{5} + 14 \beta_{4} + 99 \beta_{3} - 97 \beta_{2} - 221 \beta_1 + 12577) q^{8} + ( - 32 \beta_{5} + 85 \beta_{4} + 93 \beta_{3} + 43 \beta_{2} + \cdots + 50010) q^{9}+ \cdots + (15339924 \beta_{5} + 13611870 \beta_{4} + \cdots - 1825489899) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{2} - 476 q^{3} + 4613 q^{4} - 12884 q^{5} + 552 q^{6} - 23436 q^{7} + 74805 q^{8} + 296398 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 9 q^{2} - 476 q^{3} + 4613 q^{4} - 12884 q^{5} + 552 q^{6} - 23436 q^{7} + 74805 q^{8} + 296398 q^{9} - 676038 q^{10} - 962060 q^{11} - 2352344 q^{12} - 435268 q^{13} - 7990948 q^{14} - 9450288 q^{15} - 12496671 q^{16} + 8519142 q^{17} - 20195421 q^{18} - 26398480 q^{19} - 47202914 q^{20} - 56428792 q^{21} - 51774200 q^{22} - 99172772 q^{23} - 110557128 q^{24} + 66085866 q^{25} + 74451914 q^{26} + 105183712 q^{27} + 102848900 q^{28} + 165683964 q^{29} + 401475744 q^{30} - 199133468 q^{31} + 465766501 q^{32} + 518429376 q^{33} - 12778713 q^{34} + 804442912 q^{35} + 627274777 q^{36} - 785778644 q^{37} + 2174484940 q^{38} + 627357728 q^{39} + 657666206 q^{40} + 166444428 q^{41} + 652753248 q^{42} - 1110947880 q^{43} - 997577064 q^{44} - 1706447988 q^{45} - 1891667412 q^{46} - 5828211928 q^{47} - 2359114472 q^{48} - 1968801674 q^{49} - 126509183 q^{50} - 675851932 q^{51} - 6633403554 q^{52} - 9889898636 q^{53} - 1961072736 q^{54} - 10730153984 q^{55} + 5703448884 q^{56} - 13522850128 q^{57} + 14316796258 q^{58} + 204095112 q^{59} + 22313648592 q^{60} - 15864546948 q^{61} - 1838602020 q^{62} + 504344540 q^{63} + 6177095465 q^{64} + 12794774792 q^{65} + 56255165136 q^{66} + 17196640232 q^{67} + 6549800341 q^{68} + 2949266904 q^{69} + 52391765944 q^{70} + 8751653884 q^{71} + 38682669705 q^{72} - 13704156916 q^{73} - 9383651494 q^{74} - 15917467268 q^{75} + 59548672452 q^{76} - 12012382872 q^{77} + 35038956192 q^{78} - 89923384436 q^{79} + 25877503334 q^{80} - 152313828506 q^{81} + 57834669670 q^{82} - 26042106648 q^{83} + 3397466240 q^{84} - 18293437588 q^{85} + 29108045060 q^{86} - 195382431072 q^{87} - 228837945880 q^{88} + 53269579420 q^{89} - 10044062046 q^{90} - 226028668544 q^{91} - 9212077436 q^{92} - 62709484936 q^{93} + 83948222448 q^{94} - 170219637424 q^{95} + 116885663928 q^{96} - 106272517044 q^{97} + 132039821039 q^{98} - 10550584980 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 8440x^{4} - 21100x^{3} + 19034528x^{2} + 24205632x - 12354600960 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -223\nu^{5} - 211643\nu^{4} + 4129848\nu^{3} + 1314400972\nu^{2} - 3188346096\nu - 1377136394880 ) / 437669760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -241\nu^{5} - 106061\nu^{4} + 5321856\nu^{3} + 636174244\nu^{2} - 11713342032\nu - 663710838720 ) / 218834880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5939 \nu^{5} - 116599 \nu^{4} + 39209424 \nu^{3} + 1352281196 \nu^{2} - 41170505328 \nu - 1857468055680 ) / 437669760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 229\nu^{5} + 4409\nu^{4} - 1602504\nu^{3} - 41978116\nu^{2} + 1877653968\nu + 45942107520 ) / 8257920 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + 7\beta _1 + 2811 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 15\beta_{5} + 26\beta_{4} + 105\beta_{3} - 103\beta_{2} + 3905\beta _1 + 21259 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6289\beta_{5} + 12606\beta_{4} + 7443\beta_{3} - 9529\beta_{2} + 84823\beta _1 + 11102489 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 203255\beta_{5} + 305858\beta_{4} + 774781\beta_{3} - 720623\beta_{2} + 18777233\beta _1 + 249673079 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−64.1259
−52.1049
−33.8139
35.7479
39.1370
78.1598
−66.1259 −521.571 2324.63 −3767.16 34489.3 49367.9 −18292.4 94889.1 249107.
1.2 −54.1049 604.032 879.341 −6505.38 −32681.1 8946.74 63230.2 187708. 351973.
1.3 −35.8139 −519.883 −765.364 12105.2 18619.1 7435.72 100758. 93131.8 −433535.
1.4 33.7479 −36.5133 −909.079 2444.10 −1232.25 10334.3 −99795.2 −175814. 82483.3
1.5 37.1370 473.718 −668.845 −9760.84 17592.4 −83403.0 −100895. 47261.5 −362488.
1.6 76.1598 −475.782 3752.32 −7399.94 −36235.5 −16117.6 129800. 49221.7 −563578.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.12.a.a 6
3.b odd 2 1 153.12.a.a 6
4.b odd 2 1 272.12.a.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.12.a.a 6 1.a even 1 1 trivial
153.12.a.a 6 3.b odd 2 1
272.12.a.f 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 9T_{2}^{5} - 8410T_{2}^{4} - 88580T_{2}^{3} + 18705368T_{2}^{2} + 99820416T_{2} - 12230355456 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(17))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 9 T^{5} + \cdots - 12230355456 \) Copy content Toggle raw display
$3$ \( T^{6} + 476 T^{5} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$5$ \( T^{6} + 12884 T^{5} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + 23436 T^{5} + \cdots + 45\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{6} + 962060 T^{5} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{6} + 435268 T^{5} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T - 1419857)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 26398480 T^{5} + \cdots - 87\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{6} + 99172772 T^{5} + \cdots + 75\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{6} - 165683964 T^{5} + \cdots - 81\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{6} + 199133468 T^{5} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + 785778644 T^{5} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{6} - 166444428 T^{5} + \cdots + 44\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{6} + 1110947880 T^{5} + \cdots - 31\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{6} + 5828211928 T^{5} + \cdots - 30\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{6} + 9889898636 T^{5} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} - 204095112 T^{5} + \cdots - 16\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{6} + 15864546948 T^{5} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} - 17196640232 T^{5} + \cdots - 81\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{6} - 8751653884 T^{5} + \cdots - 47\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{6} + 13704156916 T^{5} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{6} + 89923384436 T^{5} + \cdots - 20\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{6} + 26042106648 T^{5} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{6} - 53269579420 T^{5} + \cdots - 59\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{6} + 106272517044 T^{5} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
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