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Properties

Label 1682.2.b.k
Level $1682$
Weight $2$
Character orbit 1682.b
Analytic conductor $13.431$
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] = [1682,2,Mod(1681,1682)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1682.1681");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.b (of order \(2\), degree \(1\), 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: \(13.4308376200\)
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: \( x^{16} + 37x^{14} + 548x^{12} + 4119x^{10} + 16415x^{8} + 33099x^{6} + 30128x^{4} + 10537x^{2} + 961 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - \beta_{13} q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{8} - \beta_{6} - \beta_{3} + \cdots - 1) q^{5}+ \cdots + (\beta_{5} + \beta_{4} + 2 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{8} - \beta_{6} - \beta_{3} + \cdots - 1) q^{5}+ \cdots + ( - 3 \beta_{15} + 2 \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 10 q^{5} - 2 q^{6} + 14 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 10 q^{5} - 2 q^{6} + 14 q^{7} - 26 q^{9} - 26 q^{13} + 16 q^{16} + 10 q^{20} + 14 q^{22} + 24 q^{23} + 2 q^{24} + 90 q^{25} - 14 q^{28} - 40 q^{30} + 8 q^{33} - 18 q^{34} + 26 q^{36} + 26 q^{38} - 68 q^{42} + 60 q^{45} + 54 q^{49} + 84 q^{51} + 26 q^{52} + 8 q^{53} - 4 q^{54} + 22 q^{57} + 16 q^{59} - 30 q^{62} - 24 q^{63} - 16 q^{64} - 30 q^{65} - 68 q^{67} + 22 q^{71} + 8 q^{74} - 18 q^{78} - 10 q^{80} + 24 q^{81} + 16 q^{82} + 20 q^{83} - 24 q^{86} - 14 q^{88} - 24 q^{91} - 24 q^{92} - 30 q^{93} - 26 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 37x^{14} + 548x^{12} + 4119x^{10} + 16415x^{8} + 33099x^{6} + 30128x^{4} + 10537x^{2} + 961 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1284 \nu^{14} + 2104 \nu^{12} - 607273 \nu^{10} - 8639916 \nu^{8} - 44272376 \nu^{6} + \cdots + 6710203 ) / 6652257 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27407 \nu^{14} + 991285 \nu^{12} + 14477445 \nu^{10} + 104632823 \nu^{8} + 356706384 \nu^{6} + \cdots - 465992667 ) / 73174827 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83062 \nu^{14} + 2028858 \nu^{12} + 15594884 \nu^{10} + 19187494 \nu^{8} - 260579504 \nu^{6} + \cdots - 272137079 ) / 73174827 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 111310 \nu^{14} - 2075146 \nu^{12} - 2234878 \nu^{10} + 170890658 \nu^{8} + 1234571776 \nu^{6} + \cdots + 563561575 ) / 73174827 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 97 \nu^{14} + 3198 \nu^{12} + 40629 \nu^{10} + 249235 \nu^{8} + 759007 \nu^{6} + 1071058 \nu^{4} + \cdots + 73501 ) / 35299 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 210476 \nu^{14} + 7093859 \nu^{12} + 95469962 \nu^{10} + 648806909 \nu^{8} + 2301318694 \nu^{6} + \cdots + 372298774 ) / 73174827 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 297522 \nu^{14} - 8735498 \nu^{12} - 96829366 \nu^{10} - 496888416 \nu^{8} - 1147603841 \nu^{6} + \cdots + 167012926 ) / 73174827 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1284 \nu^{15} + 2104 \nu^{13} - 607273 \nu^{11} - 8639916 \nu^{9} - 44272376 \nu^{7} + \cdots + 6710203 \nu ) / 6652257 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 836497 \nu^{15} - 30535640 \nu^{13} - 456007590 \nu^{11} - 3538203628 \nu^{9} + \cdots - 7632738708 \nu ) / 2268419637 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1126355 \nu^{15} + 25028414 \nu^{13} + 126772847 \nu^{11} - 793255129 \nu^{9} + \cdots - 16490385368 \nu ) / 2268419637 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2276840 \nu^{15} + 95943751 \nu^{13} + 1564112238 \nu^{11} + 12441794609 \nu^{9} + \cdots + 4712034027 \nu ) / 2268419637 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2371 \nu^{15} + 84720 \nu^{13} + 1200170 \nu^{11} + 8506650 \nu^{9} + 31193680 \nu^{7} + \cdots + 7251940 \nu ) / 1094269 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5002886 \nu^{15} + 210639250 \nu^{13} + 3491805708 \nu^{11} + 28921496975 \nu^{9} + \cdots + 68484475212 \nu ) / 2268419637 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10751328 \nu^{15} + 371223239 \nu^{13} + 5113668292 \nu^{11} + 35594343402 \nu^{9} + \cdots + 45353098205 \nu ) / 2268419637 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 2\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2\beta_{13} + \beta_{12} - \beta_{11} + 3\beta_{10} + \beta_{9} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{5} - 8\beta_{4} - 2\beta_{3} - 20\beta_{2} + 47 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{15} + 23\beta_{13} - 14\beta_{12} + 8\beta_{11} - 27\beta_{10} - 12\beta_{9} + 56\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{8} - 2\beta_{7} - 2\beta_{6} + 78\beta_{5} + 57\beta_{4} + 26\beta_{3} + 182\beta_{2} - 381 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 88 \beta_{15} - 5 \beta_{14} - 214 \beta_{13} + 148 \beta_{12} - 75 \beta_{11} + 211 \beta_{10} + \cdots - 459 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 105\beta_{8} + 47\beta_{7} + 41\beta_{6} - 682\beta_{5} - 392\beta_{4} - 273\beta_{3} - 1612\beta_{2} + 3118 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 764 \beta_{15} + 105 \beta_{14} + 1905 \beta_{13} - 1454 \beta_{12} + 746 \beta_{11} + \cdots + 3800 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1476 \beta_{8} - 672 \beta_{7} - 596 \beta_{6} + 6000 \beta_{5} + 2608 \beta_{4} + 2680 \beta_{3} + \cdots - 25693 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6688 \beta_{15} - 1476 \beta_{14} - 16884 \beta_{13} + 13872 \beta_{12} - 7384 \beta_{11} + \cdots - 31693 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 17603 \beta_{8} + 7880 \beta_{7} + 7509 \beta_{6} - 52949 \beta_{5} - 16546 \beta_{4} - 25495 \beta_{3} + \cdots + 213109 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 59273 \beta_{15} + 17603 \beta_{14} + 150533 \beta_{13} - 130248 \beta_{12} + 71737 \beta_{11} + \cdots + 266058 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 192790 \beta_{8} - 83439 \beta_{7} - 87260 \beta_{6} + 468043 \beta_{5} + 96772 \beta_{4} + \cdots - 1778929 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 531490 \beta_{15} - 192790 \beta_{14} - 1351947 \beta_{13} + 1209998 \beta_{12} - 684646 \beta_{11} + \cdots - 2246972 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times\).

\(n\) \(843\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1681.1
− 2.88972i
− 2.79588i
− 1.70955i
− 0.710020i
0.371284i
1.06263i
2.66631i
3.00493i
− 3.00493i
− 2.66631i
− 1.06263i
− 0.371284i
0.710020i
1.70955i
2.79588i
2.88972i
− 1.00000i − 2.88972i −1.00000 2.81297 −2.88972 3.85101 1.00000i −5.35050 − 2.81297i
1681.2 − 1.00000i − 2.79588i −1.00000 −0.869361 −2.79588 2.55676 1.00000i −4.81692 0.869361i
1681.3 − 1.00000i − 1.70955i −1.00000 −4.23180 −1.70955 1.69226 1.00000i 0.0774536 4.23180i
1681.4 − 1.00000i − 0.710020i −1.00000 3.33469 −0.710020 4.40617 1.00000i 2.49587 − 3.33469i
1681.5 − 1.00000i 0.371284i −1.00000 3.64078 0.371284 −3.75542 1.00000i 2.86215 − 3.64078i
1681.6 − 1.00000i 1.06263i −1.00000 −4.09274 1.06263 2.34135 1.00000i 1.87081 4.09274i
1681.7 − 1.00000i 2.66631i −1.00000 −1.88957 2.66631 −4.43319 1.00000i −4.10924 1.88957i
1681.8 − 1.00000i 3.00493i −1.00000 −3.70497 3.00493 0.341047 1.00000i −6.02962 3.70497i
1681.9 1.00000i − 3.00493i −1.00000 −3.70497 3.00493 0.341047 − 1.00000i −6.02962 − 3.70497i
1681.10 1.00000i − 2.66631i −1.00000 −1.88957 2.66631 −4.43319 − 1.00000i −4.10924 − 1.88957i
1681.11 1.00000i − 1.06263i −1.00000 −4.09274 1.06263 2.34135 − 1.00000i 1.87081 − 4.09274i
1681.12 1.00000i − 0.371284i −1.00000 3.64078 0.371284 −3.75542 − 1.00000i 2.86215 3.64078i
1681.13 1.00000i 0.710020i −1.00000 3.33469 −0.710020 4.40617 − 1.00000i 2.49587 3.33469i
1681.14 1.00000i 1.70955i −1.00000 −4.23180 −1.70955 1.69226 − 1.00000i 0.0774536 − 4.23180i
1681.15 1.00000i 2.79588i −1.00000 −0.869361 −2.79588 2.55676 − 1.00000i −4.81692 − 0.869361i
1681.16 1.00000i 2.88972i −1.00000 2.81297 −2.88972 3.85101 − 1.00000i −5.35050 2.81297i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1681.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1682.2.b.k 16
29.b even 2 1 inner 1682.2.b.k 16
29.c odd 4 1 1682.2.a.u ✓ 8
29.c odd 4 1 1682.2.a.v yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1682.2.a.u ✓ 8 29.c odd 4 1
1682.2.a.v yes 8 29.c odd 4 1
1682.2.b.k 16 1.a even 1 1 trivial
1682.2.b.k 16 29.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1682, [\chi])\):

\( T_{3}^{16} + 37T_{3}^{14} + 548T_{3}^{12} + 4119T_{3}^{10} + 16415T_{3}^{8} + 33099T_{3}^{6} + 30128T_{3}^{4} + 10537T_{3}^{2} + 961 \) Copy content Toggle raw display
\( T_{5}^{8} + 5T_{5}^{7} - 30T_{5}^{6} - 160T_{5}^{5} + 265T_{5}^{4} + 1650T_{5}^{3} - 300T_{5}^{2} - 5400T_{5} - 3600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 37 T^{14} + \cdots + 961 \) Copy content Toggle raw display
$5$ \( (T^{8} + 5 T^{7} + \cdots - 3600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 7 T^{7} + \cdots + 976)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 93 T^{14} + \cdots + 2653641 \) Copy content Toggle raw display
$13$ \( (T^{8} + 13 T^{7} + \cdots - 464)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 147 T^{14} + \cdots + 6305121 \) Copy content Toggle raw display
$19$ \( T^{16} + 93 T^{14} + \cdots + 2595321 \) Copy content Toggle raw display
$23$ \( (T^{8} - 12 T^{7} + \cdots - 144)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 701190400 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 2590402816 \) Copy content Toggle raw display
$41$ \( T^{16} + 168 T^{14} + \cdots + 77841 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 13610668861696 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 6106750361856 \) Copy content Toggle raw display
$53$ \( (T^{8} - 4 T^{7} + \cdots - 924624)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 8 T^{7} + \cdots + 5024961)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 173602222336 \) Copy content Toggle raw display
$67$ \( (T^{8} + 34 T^{7} + \cdots + 967471)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 11 T^{7} + \cdots + 6717456)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 293 T^{14} + \cdots + 1745041 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 12606362695936 \) Copy content Toggle raw display
$83$ \( (T^{8} - 10 T^{7} + \cdots + 110386845)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 6276723504921 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
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