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Properties

Label 1710.2.f.a
Level $1710$
Weight $2$
Character orbit 1710.f
Analytic conductor $13.654$
Analytic rank $0$
Dimension $16$
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] = [1710,2,Mod(341,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.341");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.f (of order \(2\), degree \(1\), 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: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 34x^{14} + 459x^{12} + 3216x^{10} + 12647x^{8} + 27742x^{6} + 30865x^{4} + 12848x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - \beta_{9} q^{5} + \beta_{3} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - \beta_{9} q^{5} + \beta_{3} q^{7} - q^{8} + \beta_{9} q^{10} + \beta_{11} q^{11} + ( - \beta_{12} - \beta_{9}) q^{13} - \beta_{3} q^{14} + q^{16} + (\beta_{12} + \beta_{10} + \cdots + \beta_{4}) q^{17}+ \cdots + (\beta_{14} - \beta_{13} + 2 \beta_{3} - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} - 8 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 16 q^{4} - 8 q^{7} - 16 q^{8} + 8 q^{14} + 16 q^{16} - 4 q^{19} - 16 q^{25} - 8 q^{28} - 24 q^{29} - 16 q^{32} + 4 q^{38} - 24 q^{41} + 24 q^{43} + 24 q^{49} + 16 q^{50} - 16 q^{53} + 8 q^{56} + 24 q^{58} - 16 q^{59} + 16 q^{64} - 16 q^{65} + 16 q^{73} - 4 q^{76} + 24 q^{82} - 24 q^{86} - 8 q^{89} - 12 q^{95} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 34x^{14} + 459x^{12} + 3216x^{10} + 12647x^{8} + 27742x^{6} + 30865x^{4} + 12848x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{14} + 120\nu^{12} + 1803\nu^{10} + 12752\nu^{8} + 43865\nu^{6} + 66752\nu^{4} + 30533\nu^{2} - 928 ) / 280 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 383 \nu^{14} - 11386 \nu^{12} - 127129 \nu^{10} - 687784 \nu^{8} - 1896641 \nu^{6} - 2480758 \nu^{4} + \cdots - 12808 ) / 1680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30 \nu^{14} - 927 \nu^{12} - 10897 \nu^{10} - 62714 \nu^{8} - 185082 \nu^{6} - 258587 \nu^{4} + \cdots - 632 ) / 140 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 223 \nu^{15} + 372 \nu^{14} + 6806 \nu^{13} + 11016 \nu^{12} + 78149 \nu^{11} + 122268 \nu^{10} + \cdots + 25824 ) / 6720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 208 \nu^{15} + 279 \nu^{14} - 5996 \nu^{13} + 8262 \nu^{12} - 64304 \nu^{11} + 91701 \nu^{10} + \cdots + 19368 ) / 5040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 208 \nu^{15} + 279 \nu^{14} + 5996 \nu^{13} + 8262 \nu^{12} + 64304 \nu^{11} + 91701 \nu^{10} + \cdots + 19368 ) / 5040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 487 \nu^{14} - 14888 \nu^{12} - 172595 \nu^{10} - 977384 \nu^{8} - 2835613 \nu^{6} - 3898496 \nu^{4} + \cdots - 3896 ) / 840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 551 \nu^{14} - 16846 \nu^{12} - 195337 \nu^{10} - 1106776 \nu^{8} - 3215441 \nu^{6} - 4437370 \nu^{4} + \cdots - 15160 ) / 840 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43 \nu^{15} - 1310 \nu^{13} - 15113 \nu^{11} - 85016 \nu^{9} - 244237 \nu^{7} - 329570 \nu^{5} + \cdots + 7384 \nu ) / 576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 383 \nu^{15} + 11386 \nu^{13} + 127129 \nu^{11} + 687784 \nu^{9} + 1896641 \nu^{7} + 2480758 \nu^{5} + \cdots + 12808 \nu ) / 5040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 169 \nu^{15} - 5150 \nu^{13} - 59639 \nu^{11} - 339896 \nu^{9} - 1011655 \nu^{7} + \cdots - 153896 \nu ) / 1680 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3613 \nu^{15} - 112418 \nu^{13} - 1331231 \nu^{11} - 7700648 \nu^{9} - 22683403 \nu^{7} + \cdots + 1029160 \nu ) / 20160 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1745 \nu^{15} - 2160 \nu^{14} + 54106 \nu^{13} - 66744 \nu^{12} + 639091 \nu^{11} - 784584 \nu^{10} + \cdots - 85824 ) / 10080 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1745 \nu^{15} + 2160 \nu^{14} + 54106 \nu^{13} + 66744 \nu^{12} + 639091 \nu^{11} + 784584 \nu^{10} + \cdots + 85824 ) / 10080 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 237 \nu^{15} - 7338 \nu^{13} - 86535 \nu^{11} - 500664 \nu^{9} - 1492203 \nu^{7} + \cdots - 72456 \nu ) / 1120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{15} - 3\beta_{14} - 3\beta_{13} - 3\beta_{12} - 3\beta_{11} + 3\beta_{9} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - \beta_{13} + 2\beta_{3} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16 \beta_{15} + 18 \beta_{14} + 18 \beta_{13} + 15 \beta_{12} + 15 \beta_{11} - 12 \beta_{10} + \cdots + 3 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10 \beta_{14} + 10 \beta_{13} - \beta_{8} + 5 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - 26 \beta_{3} + \cdots + 53 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 158 \beta_{15} - 141 \beta_{14} - 141 \beta_{13} - 99 \beta_{12} - 105 \beta_{11} + \cdots - 30 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 104 \beta_{14} - 104 \beta_{13} + 14 \beta_{8} - 80 \beta_{7} + 77 \beta_{6} + 77 \beta_{5} + 306 \beta_{3} + \cdots - 464 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1684 \beta_{15} + 1314 \beta_{14} + 1314 \beta_{13} + 813 \beta_{12} + 963 \beta_{11} + \cdots + 279 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1139 \beta_{14} + 1139 \beta_{13} - 181 \beta_{8} + 1047 \beta_{7} - 1111 \beta_{6} - 1111 \beta_{5} + \cdots + 4779 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 18722 \beta_{15} - 13641 \beta_{14} - 13641 \beta_{13} - 7863 \beta_{12} - 10251 \beta_{11} + \cdots - 2766 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12903 \beta_{14} - 12903 \beta_{13} + 2278 \beta_{8} - 12922 \beta_{7} + 14452 \beta_{6} + 14452 \beta_{5} + \cdots - 53186 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 213556 \beta_{15} + 150540 \beta_{14} + 150540 \beta_{13} + 84015 \beta_{12} + 116307 \beta_{11} + \cdots + 29397 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 149034 \beta_{14} + 149034 \beta_{13} - 28059 \beta_{8} + 155903 \beta_{7} - 179398 \beta_{6} + \cdots + 612167 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2473706 \beta_{15} - 1716543 \beta_{14} - 1716543 \beta_{13} - 945615 \beta_{12} + \cdots - 327744 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1739842 \beta_{14} - 1739842 \beta_{13} + 340200 \beta_{8} - 1861966 \beta_{7} + 2175893 \beta_{6} + \cdots - 7148622 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 28911136 \beta_{15} + 19910592 \beta_{14} + 19910592 \beta_{13} + 10915545 \beta_{12} + \cdots + 3756489 \beta_{4} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
3.43811i
− 2.52778i
− 1.75478i
0.964990i
0.0710079i
1.77759i
− 2.05989i
2.09075i
− 3.43811i
2.52778i
1.75478i
− 0.964990i
− 0.0710079i
− 1.77759i
2.05989i
− 2.09075i
−1.00000 0 1.00000 − 1.00000i 0 −4.86222 −1.00000 0 1.00000i
341.2 −1.00000 0 1.00000 − 1.00000i 0 −3.57482 −1.00000 0 1.00000i
341.3 −1.00000 0 1.00000 − 1.00000i 0 −2.48163 −1.00000 0 1.00000i
341.4 −1.00000 0 1.00000 − 1.00000i 0 −1.36470 −1.00000 0 1.00000i
341.5 −1.00000 0 1.00000 − 1.00000i 0 −0.100420 −1.00000 0 1.00000i
341.6 −1.00000 0 1.00000 − 1.00000i 0 2.51390 −1.00000 0 1.00000i
341.7 −1.00000 0 1.00000 − 1.00000i 0 2.91313 −1.00000 0 1.00000i
341.8 −1.00000 0 1.00000 − 1.00000i 0 2.95677 −1.00000 0 1.00000i
341.9 −1.00000 0 1.00000 1.00000i 0 −4.86222 −1.00000 0 − 1.00000i
341.10 −1.00000 0 1.00000 1.00000i 0 −3.57482 −1.00000 0 − 1.00000i
341.11 −1.00000 0 1.00000 1.00000i 0 −2.48163 −1.00000 0 − 1.00000i
341.12 −1.00000 0 1.00000 1.00000i 0 −1.36470 −1.00000 0 − 1.00000i
341.13 −1.00000 0 1.00000 1.00000i 0 −0.100420 −1.00000 0 − 1.00000i
341.14 −1.00000 0 1.00000 1.00000i 0 2.51390 −1.00000 0 − 1.00000i
341.15 −1.00000 0 1.00000 1.00000i 0 2.91313 −1.00000 0 − 1.00000i
341.16 −1.00000 0 1.00000 1.00000i 0 2.95677 −1.00000 0 − 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.f.a ✓ 16
3.b odd 2 1 1710.2.f.b yes 16
19.b odd 2 1 1710.2.f.b yes 16
57.d even 2 1 inner 1710.2.f.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.f.a ✓ 16 1.a even 1 1 trivial
1710.2.f.a ✓ 16 57.d even 2 1 inner
1710.2.f.b yes 16 3.b odd 2 1
1710.2.f.b yes 16 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{29}^{8} + 12T_{29}^{7} - 70T_{29}^{6} - 1040T_{29}^{5} + 48T_{29}^{4} + 11648T_{29}^{3} - 20608T_{29}^{2} + 12288T_{29} - 2304 \) acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + \cdots - 128)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 124 T^{14} + \cdots + 331776 \) Copy content Toggle raw display
$13$ \( T^{16} + 124 T^{14} + \cdots + 23970816 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 10511990784 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1719926784 \) Copy content Toggle raw display
$29$ \( (T^{8} + 12 T^{7} + \cdots - 2304)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 256 T^{14} + \cdots + 95883264 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 35802694656 \) Copy content Toggle raw display
$41$ \( (T^{8} + 12 T^{7} + \cdots + 1260000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 12 T^{7} + \cdots - 61312)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3317760000 \) Copy content Toggle raw display
$53$ \( (T^{8} + 8 T^{7} + \cdots - 9900288)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 8 T^{7} + \cdots + 119952)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 216 T^{6} + \cdots - 55552)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 210691031040000 \) Copy content Toggle raw display
$71$ \( (T^{8} - 200 T^{6} + \cdots - 230400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 8 T^{7} + \cdots - 1679616)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 1336409985024 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1858837745664 \) Copy content Toggle raw display
$89$ \( (T^{8} + 4 T^{7} + \cdots - 1871136)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
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