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Properties

Label 270.3.h.a
Level $270$
Weight $3$
Character orbit 270.h
Analytic conductor $7.357$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,3,Mod(71,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.h (of order \(6\), degree \(2\), not 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: \(7.35696713773\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - 2 \beta_{10} q^{4} + \beta_{12} q^{5} + ( - \beta_{15} + \beta_{14} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - 2 \beta_{10} q^{4} + \beta_{12} q^{5} + ( - \beta_{15} + \beta_{14} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - 15 \beta_{15} + 8 \beta_{14} + \cdots + 22) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 4 q^{7} + 20 q^{13} + 36 q^{14} - 32 q^{16} + 80 q^{19} + 24 q^{22} - 108 q^{23} + 40 q^{25} - 16 q^{28} + 72 q^{29} - 16 q^{31} - 48 q^{34} - 88 q^{37} + 72 q^{38} - 108 q^{41} + 92 q^{43} + 24 q^{46} - 216 q^{47} - 84 q^{49} - 40 q^{52} + 72 q^{56} - 144 q^{59} - 76 q^{61} - 128 q^{64} - 180 q^{65} + 56 q^{67} - 72 q^{68} + 60 q^{70} + 416 q^{73} + 288 q^{74} + 80 q^{76} + 684 q^{77} + 80 q^{79} - 192 q^{82} - 396 q^{83} - 60 q^{85} + 216 q^{86} - 48 q^{88} - 656 q^{91} - 216 q^{92} - 84 q^{94} + 360 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 41 \nu^{14} + 225 \nu^{12} - 1333 \nu^{10} - 31 \nu^{8} + 114072 \nu^{6} - 798716 \nu^{4} + \cdots - 15616332 ) / 4341600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 44283 \nu^{15} - 22189559 \nu^{14} + 3460430 \nu^{13} + 234345795 \nu^{12} + \cdots + 1973100051612 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 596263 \nu^{15} + 10396956 \nu^{14} - 3172220 \nu^{13} - 130809690 \nu^{12} + \cdots - 1353106674648 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 175182 \nu^{15} + 1459405 \nu^{14} - 3362576 \nu^{13} - 12604772 \nu^{12} + 32750354 \nu^{11} + \cdots - 69677142768 ) / 18618517440 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1107259 \nu^{15} + 6091090 \nu^{14} - 14528775 \nu^{13} - 64140690 \nu^{12} + \cdots - 313671374280 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 72336 \nu^{15} - 137953 \nu^{14} + 727168 \nu^{13} + 1003727 \nu^{12} - 8534536 \nu^{11} + \cdots + 238211532 ) / 3723703488 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1764117 \nu^{15} - 25638384 \nu^{14} + 14718770 \nu^{13} + 259438970 \nu^{12} + \cdots + 1885962752712 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 72336 \nu^{15} - 137953 \nu^{14} - 727168 \nu^{13} + 1003727 \nu^{12} + 8534536 \nu^{11} + \cdots + 238211532 ) / 3723703488 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 436425 \nu^{15} + 1866377 \nu^{14} + 6505444 \nu^{13} - 24376288 \nu^{12} + \cdots - 219798997488 ) / 18618517440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 215 \nu^{15} + 2296 \nu^{13} - 25907 \nu^{11} + 183272 \nu^{9} - 859816 \nu^{7} + \cdots - 4110048 ) / 8220096 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 881988 \nu^{15} + 2003904 \nu^{14} - 8761925 \nu^{13} - 15591600 \nu^{12} + 105064844 \nu^{11} + \cdots + 1201238208 ) / 31030862400 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 881988 \nu^{15} + 2003904 \nu^{14} + 8761925 \nu^{13} - 15591600 \nu^{12} + \cdots + 1201238208 ) / 31030862400 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2684310 \nu^{15} + 6857464 \nu^{14} + 34992080 \nu^{13} - 60611635 \nu^{12} + \cdots - 515808409212 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5350534 \nu^{15} - 1763143 \nu^{14} + 29652625 \nu^{13} + 34223065 \nu^{12} + \cdots + 689107770324 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 48776 \nu^{15} - 2911 \nu^{14} + 509200 \nu^{13} + 15975 \nu^{12} - 5779688 \nu^{11} + \cdots - 1108759572 ) / 616507200 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{15} + 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \cdots - 2 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} + \cdots + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{15} + 4 \beta_{14} - 26 \beta_{12} + 26 \beta_{11} - 18 \beta_{10} + 8 \beta_{9} - 15 \beta_{8} + \cdots - 14 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{15} - 10 \beta_{13} - 16 \beta_{12} - 16 \beta_{11} - 6 \beta_{9} - 19 \beta_{8} - 7 \beta_{7} + \cdots - 100 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4 \beta_{15} + 24 \beta_{14} - 10 \beta_{13} + 58 \beta_{12} - 58 \beta_{11} + 50 \beta_{10} + \cdots + 42 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 19 \beta_{15} - 208 \beta_{14} + 204 \beta_{13} - 114 \beta_{12} - 114 \beta_{11} + 208 \beta_{10} + \cdots - 50 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 764 \beta_{15} - 46 \beta_{14} + 172 \beta_{13} + 904 \beta_{12} - 904 \beta_{11} + 2776 \beta_{10} + \cdots + 1394 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 17 \beta_{15} - 680 \beta_{14} + 1306 \beta_{13} + 712 \beta_{12} + 712 \beta_{11} + 680 \beta_{10} + \cdots + 5040 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1950 \beta_{15} - 1028 \beta_{14} + 1146 \beta_{13} - 3014 \beta_{12} + 3014 \beta_{11} + 3738 \beta_{10} + \cdots - 470 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2019 \beta_{15} + 7776 \beta_{14} - 10576 \beta_{13} + 8786 \beta_{12} + 8786 \beta_{11} - 7776 \beta_{10} + \cdots + 19106 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 54896 \beta_{15} - 246 \beta_{14} - 3412 \beta_{13} - 58604 \beta_{12} + 58604 \beta_{11} - 146236 \beta_{10} + \cdots - 74946 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12725 \beta_{15} + 55880 \beta_{14} - 96546 \beta_{13} - 44664 \beta_{12} - 44664 \beta_{11} + \cdots - 234248 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 13850 \beta_{15} + 54548 \beta_{14} - 82826 \beta_{13} - 43178 \beta_{12} + 43178 \beta_{11} + \cdots + 494 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 59291 \beta_{15} - 333104 \beta_{14} + 479800 \beta_{13} - 632402 \beta_{12} - 632402 \beta_{11} + \cdots - 2099970 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2955360 \beta_{15} + 115630 \beta_{14} - 221100 \beta_{13} + 2998180 \beta_{12} - 2998180 \beta_{11} + \cdots + 4618834 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(-\beta_{10}\) \(1\)

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
−1.42311 + 1.82514i
1.42311 + 1.82514i
1.42311 − 0.410927i
−1.42311 − 0.410927i
2.11536 + 0.410927i
−2.11536 + 0.410927i
2.11536 − 1.82514i
−2.11536 − 1.82514i
−1.42311 − 1.82514i
1.42311 − 1.82514i
1.42311 + 0.410927i
−1.42311 + 0.410927i
2.11536 − 0.410927i
−2.11536 − 0.410927i
2.11536 + 1.82514i
−2.11536 + 1.82514i
−1.22474 + 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 −0.496891 − 0.860641i 2.82843i 0 3.16228
71.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 0.531482 + 0.920554i 2.82843i 0 3.16228
71.3 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 −5.39499 − 9.34440i 2.82843i 0 −3.16228
71.4 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 0.686165 + 1.18847i 2.82843i 0 −3.16228
71.5 1.22474 − 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 −4.35057 − 7.53542i − 2.82843i 0 −3.16228
71.6 1.22474 − 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 3.31598 + 5.74345i − 2.82843i 0 −3.16228
71.7 1.22474 − 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 −3.24916 − 5.62771i − 2.82843i 0 3.16228
71.8 1.22474 − 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 6.95799 + 12.0516i − 2.82843i 0 3.16228
251.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 −0.496891 + 0.860641i − 2.82843i 0 3.16228
251.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 0.531482 − 0.920554i − 2.82843i 0 3.16228
251.3 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 −5.39499 + 9.34440i − 2.82843i 0 −3.16228
251.4 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 0.686165 − 1.18847i − 2.82843i 0 −3.16228
251.5 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 −4.35057 + 7.53542i 2.82843i 0 −3.16228
251.6 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 3.31598 − 5.74345i 2.82843i 0 −3.16228
251.7 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 −3.24916 + 5.62771i 2.82843i 0 3.16228
251.8 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 6.95799 − 12.0516i 2.82843i 0 3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.3.h.a 16
3.b odd 2 1 90.3.h.a ✓ 16
4.b odd 2 1 2160.3.bs.d 16
5.b even 2 1 1350.3.i.g 16
5.c odd 4 2 1350.3.k.b 32
9.c even 3 1 90.3.h.a ✓ 16
9.c even 3 1 810.3.d.c 16
9.d odd 6 1 inner 270.3.h.a 16
9.d odd 6 1 810.3.d.c 16
12.b even 2 1 720.3.bs.d 16
15.d odd 2 1 450.3.i.g 16
15.e even 4 2 450.3.k.c 32
36.f odd 6 1 720.3.bs.d 16
36.h even 6 1 2160.3.bs.d 16
45.h odd 6 1 1350.3.i.g 16
45.j even 6 1 450.3.i.g 16
45.k odd 12 2 450.3.k.c 32
45.l even 12 2 1350.3.k.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.h.a ✓ 16 3.b odd 2 1
90.3.h.a ✓ 16 9.c even 3 1
270.3.h.a 16 1.a even 1 1 trivial
270.3.h.a 16 9.d odd 6 1 inner
450.3.i.g 16 15.d odd 2 1
450.3.i.g 16 45.j even 6 1
450.3.k.c 32 15.e even 4 2
450.3.k.c 32 45.k odd 12 2
720.3.bs.d 16 12.b even 2 1
720.3.bs.d 16 36.f odd 6 1
810.3.d.c 16 9.c even 3 1
810.3.d.c 16 9.d odd 6 1
1350.3.i.g 16 5.b even 2 1
1350.3.i.g 16 45.h odd 6 1
1350.3.k.b 32 5.c odd 4 2
1350.3.k.b 32 45.l even 12 2
2160.3.bs.d 16 4.b odd 2 1
2160.3.bs.d 16 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(270, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 6662640625 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 7901855928576 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 991374241321216 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} - 40 T^{7} + \cdots + 4403341840)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 35\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{8} + 44 T^{7} + \cdots - 221671664)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 64\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 638114737319936)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 95\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
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