Properties

Label 27.10.a.c
Level $27$
Weight $10$
Character orbit 27.a
Self dual yes
Analytic conductor $13.906$
Analytic rank $1$
Dimension $3$
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] = [27,10,Mod(1,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(13.9059675764\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.177113.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 118x + 136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 199) q^{4} + ( - 3 \beta_{2} - 7 \beta_1 - 661) q^{5} + ( - 5 \beta_{2} - 233 \beta_1 - 1231) q^{7} + (3 \beta_{2} + 32 \beta_1 - 1501) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 199) q^{4} + ( - 3 \beta_{2} - 7 \beta_1 - 661) q^{5} + ( - 5 \beta_{2} - 233 \beta_1 - 1231) q^{7} + (3 \beta_{2} + 32 \beta_1 - 1501) q^{8} + ( - 22 \beta_{2} - 1657 \beta_1 - 6327) q^{10} + (84 \beta_{2} + 124 \beta_1 - 5621) q^{11} + (208 \beta_{2} - 2360 \beta_1 + 38972) q^{13} + ( - 258 \beta_{2} - 2227 \beta_1 - 167821) q^{14} + ( - 465 \beta_{2} + 444 \beta_1 - 79973) q^{16} + (96 \beta_{2} + 8472 \beta_1 - 338016) q^{17} + (66 \beta_{2} + 23682 \beta_1 - 5074) q^{19} + ( - 231 \beta_{2} - 5230 \beta_1 - 849469) q^{20} + (544 \beta_{2} + 22483 \beta_1 + 101907) q^{22} + (2208 \beta_{2} - 44248 \beta_1 - 975706) q^{23} + (1132 \beta_{2} + 37588 \beta_1 + 711244) q^{25} + ( - 1320 \beta_{2} + 116564 \beta_1 - 1588372) q^{26} + ( - 957 \beta_{2} - 129306 \beta_1 - 1178575) q^{28} + ( - 3762 \beta_{2} - 44306 \beta_1 - 1922930) q^{29} + ( - 8483 \beta_{2} - 72215 \beta_1 - 2191741) q^{31} + ( - 3417 \beta_{2} - 255324 \beta_1 + 895899) q^{32} + (8952 \beta_{2} - 330888 \beta_1 + 5699376) q^{34} + (6480 \beta_{2} + 428752 \beta_1 + 5780179) q^{35} + (9624 \beta_{2} + 303456 \beta_1 - 3895342) q^{37} + (24012 \beta_{2} - 53746 \beta_1 + 16824458) q^{38} + (4879 \beta_{2} - 63704 \beta_1 - 1376937) q^{40} + ( - 16434 \beta_{2} - 420274 \beta_1 + 7404506) q^{41} + ( - 30674 \beta_{2} + 1100902 \beta_1 + 15128138) q^{43} + ( - 17805 \beta_{2} + 155386 \beta_1 + 19068997) q^{44} + ( - 33208 \beta_{2} - 94450 \beta_1 - 31879530) q^{46} + ( - 44328 \beta_{2} + 60952 \beta_1 - 4130678) q^{47} + (67444 \beta_{2} + 1186060 \beta_1 + 6311154) q^{49} + (43248 \beta_{2} + 982228 \beta_1 + 27661348) q^{50} + (3468 \beta_{2} - 1177224 \beta_1 + 60912164) q^{52} + (42939 \beta_{2} - 1534761 \beta_1 - 26859879) q^{53} + (42059 \beta_{2} - 764761 \beta_1 - 58245111) q^{55} + ( - 1995 \beta_{2} + 25144 \beta_1 - 7283507) q^{56} + ( - 63116 \beta_{2} - 3065330 \beta_1 - 34252974) q^{58} + (85632 \beta_{2} + 2420336 \beta_1 - 81342220) q^{59} + ( - 32696 \beta_{2} - 2098280 \beta_1 + 123243320) q^{61} + ( - 114630 \beta_{2} - 4850833 \beta_1 - 55432447) q^{62} + ( - 34329 \beta_{2} + 276180 \beta_1 - 140230709) q^{64} + ( - 3852 \beta_{2} + 2136700 \beta_1 - 164075228) q^{65} + ( - 227522 \beta_{2} - 235562 \beta_1 - 84204862) q^{67} + ( - 335280 \beta_{2} + 5389104 \beta_1 - 54090048) q^{68} + (461152 \beta_{2} + 6690643 \beta_1 + 311697459) q^{70} + (188880 \beta_{2} - 3010488 \beta_1 - 134397696) q^{71} + (194220 \beta_{2} + 2361348 \beta_1 - 135542239) q^{73} + (351576 \beta_{2} - 1543174 \beta_1 + 213791186) q^{74} + (32522 \beta_{2} + 13000580 \beta_1 - 13166530) q^{76} + ( - 42579 \beta_{2} - 6239431 \beta_1 - 120147733) q^{77} + ( - 673892 \beta_{2} - 4693196 \beta_1 + 88483952) q^{79} + (78963 \beta_{2} + 3145916 \beta_1 + 389453279) q^{80} + ( - 502444 \beta_{2} + 3094202 \beta_1 - 294802722) q^{82} + ( - 296820 \beta_{2} + 3456100 \beta_1 - 40541957) q^{83} + (884568 \beta_{2} - 12297840 \beta_1 + 105417072) q^{85} + (947532 \beta_{2} + 1426946 \beta_1 + 789652190) q^{86} + ( - 212167 \beta_{2} + 1055648 \beta_1 + 73085913) q^{88} + ( - 927582 \beta_{2} + 20852538 \beta_1 - 125968002) q^{89} + (304876 \beta_{2} - 22455452 \beta_1 + 81686380) q^{91} + ( - 1390986 \beta_{2} - 20198716 \beta_1 + 392918186) q^{92} + ( - 160688 \beta_{2} - 19340726 \beta_1 + 28861146) q^{94} + ( - 413088 \beta_{2} - 39510824 \beta_1 - 178959590) q^{95} + (606616 \beta_{2} - 7785896 \beta_1 - 146302513) q^{97} + (1523280 \beta_{2} + 25616490 \beta_1 + 864060762) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 597 q^{4} - 1983 q^{5} - 3693 q^{7} - 4503 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 597 q^{4} - 1983 q^{5} - 3693 q^{7} - 4503 q^{8} - 18981 q^{10} - 16863 q^{11} + 116916 q^{13} - 503463 q^{14} - 239919 q^{16} - 1014048 q^{17} - 15222 q^{19} - 2548407 q^{20} + 305721 q^{22} - 2927118 q^{23} + 2133732 q^{25} - 4765116 q^{26} - 3535725 q^{28} - 5768790 q^{29} - 6575223 q^{31} + 2687697 q^{32} + 17098128 q^{34} + 17340537 q^{35} - 11686026 q^{37} + 50473374 q^{38} - 4130811 q^{40} + 22213518 q^{41} + 45384414 q^{43} + 57206991 q^{44} - 95638590 q^{46} - 12392034 q^{47} + 18933462 q^{49} + 82984044 q^{50} + 182736492 q^{52} - 80579637 q^{53} - 174735333 q^{55} - 21850521 q^{56} - 102758922 q^{58} - 244026660 q^{59} + 369729960 q^{61} - 166297341 q^{62} - 420692127 q^{64} - 492225684 q^{65} - 252614586 q^{67} - 162270144 q^{68} + 935092377 q^{70} - 403193088 q^{71} - 406626717 q^{73} + 641373558 q^{74} - 39499590 q^{76} - 360443199 q^{77} + 265451856 q^{79} + 1168359837 q^{80} - 884408166 q^{82} - 121625871 q^{83} + 316251216 q^{85} + 2368956570 q^{86} + 219257739 q^{88} - 377904006 q^{89} + 245059140 q^{91} + 1178754558 q^{92} + 86583438 q^{94} - 536878770 q^{95} - 438907539 q^{97} + 2592182286 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 118x + 136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 9\nu^{2} + 6\nu - 713 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 2\beta _1 + 711 ) / 9 \) 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
−10.9320
1.15428
10.7777
−32.7960 0 563.577 −1315.38 0 5158.54 −1691.52 0 43139.3
1.2 3.46285 0 −500.009 1404.01 0 1665.57 −3504.44 0 4861.88
1.3 32.3331 0 533.432 −2071.63 0 −10517.1 692.954 0 −66982.2
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

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 27.10.a.c yes 3
3.b odd 2 1 27.10.a.b 3
9.c even 3 2 81.10.c.g 6
9.d odd 6 2 81.10.c.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.10.a.b 3 3.b odd 2 1
27.10.a.c yes 3 1.a even 1 1 trivial
81.10.c.g 6 9.c even 3 2
81.10.c.h 6 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2}^{2} - 1062T_{2} + 3672 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(27))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3 T^{2} - 1062 T + 3672 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 1983 T^{2} + \cdots - 3825897975 \) Copy content Toggle raw display
$7$ \( T^{3} + 3693 T^{2} + \cdots + 90362069875 \) Copy content Toggle raw display
$11$ \( T^{3} + 16863 T^{2} + \cdots + 30446445345165 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 955953747392320 \) Copy content Toggle raw display
$17$ \( T^{3} + 1014048 T^{2} + \cdots + 78\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{3} + 15222 T^{2} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{3} + 2927118 T^{2} + \cdots - 45\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{3} + 5768790 T^{2} + \cdots - 42\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{3} + 6575223 T^{2} + \cdots - 62\!\cdots\!31 \) Copy content Toggle raw display
$37$ \( T^{3} + 11686026 T^{2} + \cdots - 10\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{3} - 22213518 T^{2} + \cdots + 36\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{3} - 45384414 T^{2} + \cdots + 45\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{3} + 12392034 T^{2} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{3} + 80579637 T^{2} + \cdots - 13\!\cdots\!97 \) Copy content Toggle raw display
$59$ \( T^{3} + 244026660 T^{2} + \cdots - 52\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{3} - 369729960 T^{2} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{3} + 252614586 T^{2} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{3} + 403193088 T^{2} + \cdots - 98\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{3} + 406626717 T^{2} + \cdots - 64\!\cdots\!45 \) Copy content Toggle raw display
$79$ \( T^{3} - 265451856 T^{2} + \cdots + 83\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{3} + 121625871 T^{2} + \cdots - 21\!\cdots\!07 \) Copy content Toggle raw display
$89$ \( T^{3} + 377904006 T^{2} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + 438907539 T^{2} + \cdots - 26\!\cdots\!55 \) Copy content Toggle raw display
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