Properties

Label 416.2.u.b
Level $416$
Weight $2$
Character orbit 416.u
Analytic conductor $3.322$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(47,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.u (of order \(4\), 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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{15} q^{5} + ( - \beta_{9} + \beta_{4}) q^{7} + ( - \beta_{19} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{15} q^{5} + ( - \beta_{9} + \beta_{4}) q^{7} + ( - \beta_{19} + \beta_{2} + 1) q^{9} - \beta_{12} q^{11} + (\beta_{10} + \beta_{7} - \beta_{5}) q^{13} + (\beta_{18} + \beta_{7}) q^{15} + (\beta_{13} + \beta_{11}) q^{17} + (\beta_{16} - \beta_{14} + \beta_{13} + \cdots + 1) q^{19}+ \cdots + (2 \beta_{19} + \beta_{16} - \beta_{13} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{3} + 16 q^{9} + 8 q^{11} + 12 q^{19} + 52 q^{27} - 12 q^{33} - 44 q^{35} - 24 q^{41} + 8 q^{57} - 20 q^{59} - 8 q^{65} + 16 q^{67} + 24 q^{73} - 44 q^{81} - 16 q^{83} - 16 q^{89} - 32 q^{91} + 12 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3 \nu^{19} + 20 \nu^{18} + 58 \nu^{17} - 316 \nu^{16} + 125 \nu^{15} - 124 \nu^{14} + \cdots + 38912 ) / 31232 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{19} - 42 \nu^{18} - 42 \nu^{17} - 24 \nu^{16} - 147 \nu^{15} + 774 \nu^{14} - 186 \nu^{13} + \cdots - 24064 ) / 31232 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29 \nu^{19} + 112 \nu^{18} - 212 \nu^{17} - 696 \nu^{16} + 1371 \nu^{15} - 1280 \nu^{14} + \cdots + 167936 ) / 62464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 66 \nu^{19} + 81 \nu^{18} + 72 \nu^{17} - 118 \nu^{15} + 361 \nu^{14} - 232 \nu^{13} - 912 \nu^{12} + \cdots - 27136 ) / 62464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34 \nu^{19} + 105 \nu^{18} - 512 \nu^{17} + 24 \nu^{16} - 182 \nu^{15} - 927 \nu^{14} + \cdots + 186880 ) / 62464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 65 \nu^{19} + 183 \nu^{18} - 68 \nu^{17} + 208 \nu^{16} - 641 \nu^{15} - 273 \nu^{14} + \cdots + 48640 ) / 62464 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33 \nu^{19} + 111 \nu^{18} + 108 \nu^{17} - 196 \nu^{16} + 31 \nu^{15} - 777 \nu^{14} + \cdots - 33792 ) / 31232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 97 \nu^{19} + 118 \nu^{18} - 158 \nu^{17} + 112 \nu^{16} + 463 \nu^{15} - 634 \nu^{14} + \cdots + 29696 ) / 31232 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 72 \nu^{19} + 127 \nu^{18} + 8 \nu^{17} - 156 \nu^{16} + 192 \nu^{15} - 681 \nu^{14} + \cdots + 34816 ) / 31232 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 87 \nu^{19} - 246 \nu^{18} + 364 \nu^{17} - 280 \nu^{16} - 129 \nu^{15} + 2442 \nu^{14} + \cdots - 82944 ) / 62464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7 \nu^{19} - 67 \nu^{18} + 68 \nu^{17} + 46 \nu^{16} - 129 \nu^{15} + 269 \nu^{14} - 220 \nu^{13} + \cdots - 17920 ) / 15616 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 57 \nu^{19} + 79 \nu^{18} - 84 \nu^{17} + 26 \nu^{16} + 535 \nu^{15} - 641 \nu^{14} + 124 \nu^{13} + \cdots + 7424 ) / 15616 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 73 \nu^{19} + 80 \nu^{18} - 134 \nu^{17} - 44 \nu^{16} + 439 \nu^{15} - 496 \nu^{14} + \cdots - 31744 ) / 15616 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 81 \nu^{19} - 13 \nu^{18} + 68 \nu^{17} - 166 \nu^{16} + 335 \nu^{15} - 61 \nu^{14} + \cdots - 28928 ) / 15616 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 193 \nu^{19} + 359 \nu^{18} - 260 \nu^{17} - 120 \nu^{16} + 1343 \nu^{15} - 2113 \nu^{14} + \cdots + 133632 ) / 62464 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 47 \nu^{19} - 48 \nu^{18} + 74 \nu^{17} - 132 \nu^{16} + 137 \nu^{15} + 48 \nu^{14} - 230 \nu^{13} + \cdots + 8192 ) / 15616 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 14 \nu^{19} - 315 \nu^{18} + 104 \nu^{17} + 720 \nu^{16} - 358 \nu^{15} + 125 \nu^{14} + \cdots - 6656 ) / 62464 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 179 \nu^{19} + 541 \nu^{18} - 732 \nu^{17} - 312 \nu^{16} + 2125 \nu^{15} - 3035 \nu^{14} + \cdots + 259584 ) / 62464 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 237 \nu^{19} + 342 \nu^{18} + 98 \nu^{17} - 432 \nu^{16} + 1075 \nu^{15} - 1562 \nu^{14} + \cdots + 66560 ) / 31232 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{19} + \beta_{18} - 2\beta_{15} - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{18} + \beta_{17} + \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 2 \beta_{10} + \cdots - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{19} + \beta_{17} - 2\beta_{12} - 2\beta_{11} - \beta_{10} - 3\beta_{9} + \beta_{8} - \beta_{5} + \beta_{3} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{19} + \beta_{18} - \beta_{17} + 3 \beta_{16} + \beta_{15} - \beta_{14} + \beta_{12} - 3 \beta_{9} + \cdots + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{19} - \beta_{18} + 4 \beta_{15} - 2 \beta_{14} + 2 \beta_{11} - 3 \beta_{10} - \beta_{8} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{18} - 3 \beta_{17} + 9 \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + \cdots - 11 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{19} + 3 \beta_{17} + 4 \beta_{16} - 4 \beta_{13} - 2 \beta_{12} - 10 \beta_{11} - 7 \beta_{10} + \cdots - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7 \beta_{19} + 3 \beta_{18} - 3 \beta_{17} + 5 \beta_{16} + 11 \beta_{15} + 5 \beta_{14} - 5 \beta_{12} + \cdots - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{19} + 9 \beta_{18} - 16 \beta_{16} + 12 \beta_{15} + 18 \beta_{14} - 16 \beta_{13} + \cdots - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5 \beta_{18} - 5 \beta_{17} - 9 \beta_{15} + \beta_{14} - 7 \beta_{13} + \beta_{12} + 9 \beta_{11} + \cdots - 13 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( \beta_{19} - 11 \beta_{17} - 12 \beta_{16} + 12 \beta_{13} - 14 \beta_{12} + 66 \beta_{11} - 33 \beta_{10} + \cdots + 66 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - \beta_{19} + 45 \beta_{18} - 45 \beta_{17} + 3 \beta_{16} - 67 \beta_{15} + 3 \beta_{14} - 3 \beta_{12} + \cdots + 29 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 9 \beta_{19} + 79 \beta_{18} - 8 \beta_{16} - 28 \beta_{15} - 18 \beta_{14} - 8 \beta_{13} + 170 \beta_{11} + \cdots - 170 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 27 \beta_{18} - 27 \beta_{17} + 33 \beta_{15} - 41 \beta_{14} + 15 \beta_{13} - 41 \beta_{12} + \cdots + 37 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 31 \beta_{19} - 93 \beta_{17} + 36 \beta_{16} - 36 \beta_{13} + 110 \beta_{12} + 150 \beta_{11} + \cdots + 150 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 9 \beta_{19} + 19 \beta_{18} - 19 \beta_{17} + 85 \beta_{16} - 53 \beta_{15} - 187 \beta_{14} + \cdots + 235 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 247 \beta_{19} - 135 \beta_{18} - 96 \beta_{16} + 268 \beta_{15} - 78 \beta_{14} - 96 \beta_{13} + \cdots - 382 ) / 4 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 5 \beta_{18} - 5 \beta_{17} + 7 \beta_{15} - 47 \beta_{14} - 151 \beta_{13} - 47 \beta_{12} + \cdots - 445 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 127 \beta_{19} + 197 \beta_{17} + 68 \beta_{16} - 68 \beta_{13} - 14 \beta_{12} - 430 \beta_{11} + \cdots - 430 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−1.41123 0.0918109i
−0.0918109 1.41123i
1.15650 0.813947i
−0.813947 + 1.15650i
1.30315 + 0.549370i
0.549370 + 1.30315i
−0.209223 + 1.39865i
1.39865 0.209223i
0.456912 1.33837i
−1.33837 + 0.456912i
−1.41123 + 0.0918109i
−0.0918109 + 1.41123i
1.15650 + 0.813947i
−0.813947 1.15650i
1.30315 0.549370i
0.549370 1.30315i
−0.209223 1.39865i
1.39865 + 0.209223i
0.456912 + 1.33837i
−1.33837 0.456912i
0 −2.19371 0 −0.274863 + 0.274863i 0 −0.352378 0.352378i 0 1.81237 0
47.2 0 −2.19371 0 0.274863 0.274863i 0 0.352378 + 0.352378i 0 1.81237 0
47.3 0 −1.38809 0 −2.40872 + 2.40872i 0 0.127019 + 0.127019i 0 −1.07320 0
47.4 0 −1.38809 0 2.40872 2.40872i 0 −0.127019 0.127019i 0 −1.07320 0
47.5 0 −0.238218 0 −0.328612 + 0.328612i 0 2.68064 + 2.68064i 0 −2.94325 0
47.6 0 −0.238218 0 0.328612 0.328612i 0 −2.68064 2.68064i 0 −2.94325 0
47.7 0 1.86790 0 −1.55756 + 1.55756i 0 1.24165 + 1.24165i 0 0.489042 0
47.8 0 1.86790 0 1.55756 1.55756i 0 −1.24165 1.24165i 0 0.489042 0
47.9 0 2.95212 0 −1.04334 + 1.04334i 0 2.37322 + 2.37322i 0 5.71504 0
47.10 0 2.95212 0 1.04334 1.04334i 0 −2.37322 2.37322i 0 5.71504 0
239.1 0 −2.19371 0 −0.274863 0.274863i 0 −0.352378 + 0.352378i 0 1.81237 0
239.2 0 −2.19371 0 0.274863 + 0.274863i 0 0.352378 0.352378i 0 1.81237 0
239.3 0 −1.38809 0 −2.40872 2.40872i 0 0.127019 0.127019i 0 −1.07320 0
239.4 0 −1.38809 0 2.40872 + 2.40872i 0 −0.127019 + 0.127019i 0 −1.07320 0
239.5 0 −0.238218 0 −0.328612 0.328612i 0 2.68064 2.68064i 0 −2.94325 0
239.6 0 −0.238218 0 0.328612 + 0.328612i 0 −2.68064 + 2.68064i 0 −2.94325 0
239.7 0 1.86790 0 −1.55756 1.55756i 0 1.24165 1.24165i 0 0.489042 0
239.8 0 1.86790 0 1.55756 + 1.55756i 0 −1.24165 + 1.24165i 0 0.489042 0
239.9 0 2.95212 0 −1.04334 1.04334i 0 2.37322 2.37322i 0 5.71504 0
239.10 0 2.95212 0 1.04334 + 1.04334i 0 −2.37322 + 2.37322i 0 5.71504 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
13.d odd 4 1 inner
104.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.u.b 20
4.b odd 2 1 104.2.m.b 20
8.b even 2 1 104.2.m.b 20
8.d odd 2 1 inner 416.2.u.b 20
12.b even 2 1 936.2.w.h 20
13.d odd 4 1 inner 416.2.u.b 20
24.h odd 2 1 936.2.w.h 20
52.f even 4 1 104.2.m.b 20
104.j odd 4 1 104.2.m.b 20
104.m even 4 1 inner 416.2.u.b 20
156.l odd 4 1 936.2.w.h 20
312.y even 4 1 936.2.w.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.m.b 20 4.b odd 2 1
104.2.m.b 20 8.b even 2 1
104.2.m.b 20 52.f even 4 1
104.2.m.b 20 104.j odd 4 1
416.2.u.b 20 1.a even 1 1 trivial
416.2.u.b 20 8.d odd 2 1 inner
416.2.u.b 20 13.d odd 4 1 inner
416.2.u.b 20 104.m even 4 1 inner
936.2.w.h 20 12.b even 2 1
936.2.w.h 20 24.h odd 2 1
936.2.w.h 20 156.l odd 4 1
936.2.w.h 20 312.y even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - T_{3}^{4} - 9T_{3}^{3} + 3T_{3}^{2} + 18T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{5} - T^{4} - 9 T^{3} + \cdots + 4)^{4} \) Copy content Toggle raw display
$5$ \( T^{20} + 163 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{20} + 343 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{10} - 4 T^{9} + \cdots + 5408)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( (T^{10} + 71 T^{8} + \cdots + 355216)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} - 6 T^{9} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 94 T^{8} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 186 T^{8} + \cdots + 21632)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 92829679353856 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 911076029249296 \) Copy content Toggle raw display
$41$ \( (T^{10} + 12 T^{9} + \cdots + 512)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 111 T^{8} + \cdots + 795664)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 1416468496 \) Copy content Toggle raw display
$53$ \( (T^{10} + 230 T^{8} + \cdots + 3442688)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 10 T^{9} + \cdots + 1384448)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 392 T^{8} + \cdots + 8388608)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 8 T^{9} + \cdots + 78575648)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 97459622042896 \) Copy content Toggle raw display
$73$ \( (T^{10} - 12 T^{9} + \cdots + 512)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 442 T^{8} + \cdots + 336338048)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 8 T^{9} + \cdots + 414950432)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 117546549248)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 6 T^{9} + \cdots + 7270250528)^{2} \) Copy content Toggle raw display
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