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Properties

Label 2106.2.b.d
Level $2106$
Weight $2$
Character orbit 2106.b
Analytic conductor $16.816$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2106,2,Mod(649,2106)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2106, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2106.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2106.b (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: \(16.8164946657\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 34x^{12} + 435x^{10} + 2617x^{8} + 7651x^{6} + 10260x^{4} + 5589x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 234)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - q^{4} - \beta_{11} q^{5} + ( - \beta_{8} - \beta_1) q^{7} - \beta_{8} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - q^{4} - \beta_{11} q^{5} + ( - \beta_{8} - \beta_1) q^{7} - \beta_{8} q^{8} + \beta_{3} q^{10} + \beta_{12} q^{11} + ( - \beta_{10} - \beta_{6}) q^{13} + (\beta_{7} + 1) q^{14} + q^{16} + (\beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{17}+ \cdots + (\beta_{13} + \beta_{12} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{4} - 2 q^{13} + 8 q^{14} + 14 q^{16} - 8 q^{17} - 8 q^{23} - 14 q^{25} - 4 q^{26} - 16 q^{29} + 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} + 60 q^{53} - 8 q^{56} - 28 q^{61} - 34 q^{62} - 14 q^{64} - 8 q^{65} + 8 q^{68} + 16 q^{74} - 24 q^{77} - 28 q^{79} - 24 q^{82} + 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 34x^{12} + 435x^{10} + 2617x^{8} + 7651x^{6} + 10260x^{4} + 5589x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{13} + 256\nu^{11} + 3630\nu^{9} + 25015\nu^{7} + 84463\nu^{5} + 113931\nu^{3} + 15714\nu ) / 9234 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{12} - 130\nu^{10} - 1545\nu^{8} - 8236\nu^{6} - 19789\nu^{4} - 18795\nu^{2} - 3996 ) / 513 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 151\nu^{12} + 4936\nu^{10} + 59250\nu^{8} + 318433\nu^{6} + 747619\nu^{4} + 618183\nu^{2} + 116478 ) / 9234 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 86\nu^{12} + 2852\nu^{10} + 35070\nu^{8} + 196739\nu^{6} + 500504\nu^{4} + 484605\nu^{2} + 114129 ) / 4617 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -55\nu^{13} - 1645\nu^{11} - 16356\nu^{9} - 51001\nu^{7} + 82484\nu^{5} + 574866\nu^{3} + 424197\nu ) / 27702 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 67\nu^{12} + 2206\nu^{10} + 26634\nu^{8} + 143425\nu^{6} + 332563\nu^{4} + 248967\nu^{2} + 20250 ) / 3078 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -277\nu^{12} - 9031\nu^{10} - 108174\nu^{8} - 580945\nu^{6} - 1365586\nu^{4} - 1111986\nu^{2} - 163863 ) / 9234 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\nu^{13} + 331\nu^{11} + 4044\nu^{9} + 22336\nu^{7} + 54901\nu^{5} + 49293\nu^{3} + 10773\nu ) / 1458 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 419 \nu^{13} - 249 \nu^{12} + 13931 \nu^{11} - 8007 \nu^{10} + 172284 \nu^{9} - 92628 \nu^{8} + \cdots + 49815 ) / 55404 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 529 \nu^{13} - 249 \nu^{12} - 17221 \nu^{11} - 8007 \nu^{10} - 204996 \nu^{9} - 92628 \nu^{8} + \cdots + 49815 ) / 55404 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 356\nu^{13} + 11627\nu^{11} + 139614\nu^{9} + 754208\nu^{7} + 1809329\nu^{5} + 1616553\nu^{3} + 420795\nu ) / 27702 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -127\nu^{13} - 4156\nu^{11} - 49980\nu^{9} - 269017\nu^{7} - 625807\nu^{5} - 465399\nu^{3} - 18630\nu ) / 9234 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -8\nu^{13} - 260\nu^{11} - 3090\nu^{9} - 16301\nu^{7} - 36671\nu^{5} - 25620\nu^{3} + 729\nu ) / 513 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{11} - \beta_{5} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{10} - 3\beta_{9} + \beta_{7} - 2\beta_{6} + 3\beta_{5} + 2\beta_{4} + 2\beta_{3} + 3\beta_{2} - 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{13} - 3\beta_{12} - 7\beta_{11} + 3\beta_{10} - 3\beta_{9} + 3\beta_{8} + 10\beta_{5} - 13\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 39 \beta_{10} + 39 \beta_{9} - 7 \beta_{7} + 23 \beta_{6} - 39 \beta_{5} - 26 \beta_{4} - 11 \beta_{3} + \cdots + 132 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 70\beta_{13} + 42\beta_{12} + 61\beta_{11} - 51\beta_{10} + 51\beta_{9} - 42\beta_{8} - 103\beta_{5} + 127\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 453 \beta_{10} - 453 \beta_{9} + 55 \beta_{7} - 254 \beta_{6} + 453 \beta_{5} + 305 \beta_{4} + \cdots - 1320 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 760 \beta_{13} - 468 \beta_{12} - 562 \beta_{11} + 675 \beta_{10} - 675 \beta_{9} + 522 \beta_{8} + \cdots - 1246 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5106 \beta_{10} + 5106 \beta_{9} - 532 \beta_{7} + 2765 \beta_{6} - 5106 \beta_{5} - 3440 \beta_{4} + \cdots + 13785 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8440 \beta_{13} + 4863 \beta_{12} + 5416 \beta_{11} - 8211 \beta_{10} + 8211 \beta_{9} + \cdots + 12523 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 56766 \beta_{10} - 56766 \beta_{9} + 6118 \beta_{7} - 29687 \beta_{6} + 56766 \beta_{5} + \cdots - 146688 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 94282 \beta_{13} - 49047 \beta_{12} - 54205 \beta_{11} + 96306 \beta_{10} - 96306 \beta_{9} + \cdots - 128197 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 626583 \beta_{10} + 626583 \beta_{9} - 76663 \beta_{7} + 315443 \beta_{6} - 626583 \beta_{5} + \cdots + 1575231 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1054255 \beta_{13} + 486387 \beta_{12} + 557770 \beta_{11} - 1109673 \beta_{10} + 1109673 \beta_{9} + \cdots + 1327846 \beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times\).

\(n\) \(1379\) \(1783\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
2.42952i
− 1.97451i
− 3.24018i
1.04629i
3.32820i
− 0.430446i
− 1.15887i
1.15887i
0.430446i
− 3.32820i
− 1.04629i
3.24018i
1.97451i
− 2.42952i
− 1.00000i 0 −1.00000 − 3.15852i 0 − 1.57492i 1.00000i 0 −3.15852
649.2 − 1.00000i 0 −1.00000 − 2.93284i 0 2.20354i 1.00000i 0 −2.93284
649.3 − 1.00000i 0 −1.00000 − 0.633132i 0 2.48683i 1.00000i 0 −0.633132
649.4 − 1.00000i 0 −1.00000 − 0.484256i 0 5.05438i 1.00000i 0 −0.484256
649.5 − 1.00000i 0 −1.00000 0.594758i 0 − 1.67682i 1.00000i 0 0.594758
649.6 − 1.00000i 0 −1.00000 2.77754i 0 0.876431i 1.00000i 0 2.77754
649.7 − 1.00000i 0 −1.00000 3.83645i 0 − 3.36944i 1.00000i 0 3.83645
649.8 1.00000i 0 −1.00000 − 3.83645i 0 3.36944i − 1.00000i 0 3.83645
649.9 1.00000i 0 −1.00000 − 2.77754i 0 − 0.876431i − 1.00000i 0 2.77754
649.10 1.00000i 0 −1.00000 − 0.594758i 0 1.67682i − 1.00000i 0 0.594758
649.11 1.00000i 0 −1.00000 0.484256i 0 − 5.05438i − 1.00000i 0 −0.484256
649.12 1.00000i 0 −1.00000 0.633132i 0 − 2.48683i − 1.00000i 0 −0.633132
649.13 1.00000i 0 −1.00000 2.93284i 0 − 2.20354i − 1.00000i 0 −2.93284
649.14 1.00000i 0 −1.00000 3.15852i 0 1.57492i − 1.00000i 0 −3.15852
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2106.2.b.d 14
3.b odd 2 1 2106.2.b.c 14
9.c even 3 2 702.2.t.a 28
9.d odd 6 2 234.2.t.a ✓ 28
13.b even 2 1 inner 2106.2.b.d 14
39.d odd 2 1 2106.2.b.c 14
117.n odd 6 2 234.2.t.a ✓ 28
117.t even 6 2 702.2.t.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.t.a ✓ 28 9.d odd 6 2
234.2.t.a ✓ 28 117.n odd 6 2
702.2.t.a 28 9.c even 3 2
702.2.t.a 28 117.t even 6 2
2106.2.b.c 14 3.b odd 2 1
2106.2.b.c 14 39.d odd 2 1
2106.2.b.d 14 1.a even 1 1 trivial
2106.2.b.d 14 13.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 \(S_{2}^{\mathrm{new}}(2106, [\chi])\):

\( T_{5}^{14} + 42T_{5}^{12} + 657T_{5}^{10} + 4657T_{5}^{8} + 13932T_{5}^{6} + 10944T_{5}^{4} + 3240T_{5}^{2} + 324 \) Copy content Toggle raw display
\( T_{17}^{7} + 4T_{17}^{6} - 61T_{17}^{5} - 217T_{17}^{4} + 867T_{17}^{3} + 2886T_{17}^{2} - 2655T_{17} - 8937 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 42 T^{12} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( T^{14} + 54 T^{12} + \cdots + 46656 \) Copy content Toggle raw display
$11$ \( T^{14} + 84 T^{12} + \cdots + 419904 \) Copy content Toggle raw display
$13$ \( T^{14} + 2 T^{13} + \cdots + 62748517 \) Copy content Toggle raw display
$17$ \( (T^{7} + 4 T^{6} + \cdots - 8937)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + 84 T^{12} + \cdots + 419904 \) Copy content Toggle raw display
$23$ \( (T^{7} + 4 T^{6} + \cdots - 5616)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 8 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 345736836 \) Copy content Toggle raw display
$37$ \( T^{14} + 234 T^{12} + \cdots + 419904 \) Copy content Toggle raw display
$41$ \( T^{14} + 360 T^{12} + \cdots + 2742336 \) Copy content Toggle raw display
$43$ \( (T^{7} - 2 T^{6} + \cdots + 783341)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 1124529156 \) Copy content Toggle raw display
$53$ \( (T^{7} - 30 T^{6} + \cdots - 1737936)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 82612555776 \) Copy content Toggle raw display
$61$ \( (T^{7} + 14 T^{6} + \cdots + 38224)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 152680873536 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 13266893124 \) Copy content Toggle raw display
$73$ \( T^{14} + 336 T^{12} + \cdots + 26873856 \) Copy content Toggle raw display
$79$ \( (T^{7} + 14 T^{6} + \cdots + 177552)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 408 T^{12} + \cdots + 5308416 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 4897760256 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 104556515904 \) Copy content Toggle raw display
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