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Properties

Label 4176.2.o.s
Level $4176$
Weight $2$
Character orbit 4176.o
Analytic conductor $33.346$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4176,2,Mod(289,4176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4176.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4176 = 2^{4} \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4176.o (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: \(33.3455278841\)
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: \( x^{16} - 8 x^{15} + 32 x^{14} - 64 x^{13} + 92 x^{12} - 184 x^{11} + 576 x^{10} - 1098 x^{9} + 1280 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 2088)
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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{5} + \beta_{12} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{5} + \beta_{12} q^{7} + \beta_{2} q^{11} - \beta_{8} q^{13} - \beta_{3} q^{17} - \beta_{10} q^{19} + \beta_{15} q^{23} + (\beta_{12} + \beta_{4} + 2) q^{25} + (\beta_{13} + \beta_{9} - \beta_{2}) q^{29} + ( - \beta_{10} + \beta_{7} - \beta_{5}) q^{31} + ( - \beta_{15} + \beta_{9} - \beta_1) q^{35} + ( - \beta_{11} - \beta_{7}) q^{37} + ( - \beta_{13} + \beta_{3}) q^{41} + (\beta_{11} + \beta_{10}) q^{43} + ( - \beta_{6} - \beta_{3}) q^{47} + (\beta_{12} - \beta_{8} + \beta_{4} + 2) q^{49} + ( - \beta_{14} + \beta_{9} - \beta_1) q^{53} + (\beta_{11} + 2 \beta_{7} - 2 \beta_{5}) q^{55} + (\beta_{13} + \beta_{9} - \beta_{6} + \cdots + \beta_1) q^{59}+ \cdots + ( - \beta_{11} - 2 \beta_{7} + \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{7} - 8 q^{13} + 36 q^{25} + 28 q^{49} - 48 q^{67} - 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 32 x^{14} - 64 x^{13} + 92 x^{12} - 184 x^{11} + 576 x^{10} - 1098 x^{9} + 1280 x^{8} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1855387484047 \nu^{15} + 27851413691231 \nu^{14} - 237663237938641 \nu^{13} + \cdots + 30\!\cdots\!44 ) / 160640475331864 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 260845329044 \nu^{15} - 7245524113792 \nu^{14} + 45631467362016 \nu^{13} + \cdots - 79405020032458 ) / 20080059416483 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3401517284393 \nu^{15} + 38629515497421 \nu^{14} - 191393571070275 \nu^{13} + \cdots + 224354854242760 ) / 80320237665932 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4183635349440 \nu^{15} + 31602641710120 \nu^{14} - 119689897072116 \nu^{13} + \cdots + 91796852261565 ) / 20080059416483 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5697800110451 \nu^{15} - 73329085135664 \nu^{14} + 383854701661250 \nu^{13} + \cdots - 538356587279142 ) / 20080059416483 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27373419865257 \nu^{15} - 210893696253899 \nu^{14} + 815056154355893 \nu^{13} + \cdots - 416341451243004 ) / 80320237665932 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18628505568759 \nu^{15} - 133728980685888 \nu^{14} + 483090519779994 \nu^{13} + \cdots - 139397635700326 ) / 40160118832966 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18571799423856 \nu^{15} + 140434578488552 \nu^{14} - 531970476459484 \nu^{13} + \cdots + 19495067297726 ) / 20080059416483 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 150983409940335 \nu^{15} + \cdots + 494373550642316 ) / 160640475331864 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50691895237555 \nu^{15} - 369334397097040 \nu^{14} + \cdots - 446898214019798 ) / 40160118832966 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25671573432948 \nu^{15} - 192802915666360 \nu^{14} + 728283670721012 \nu^{13} + \cdots - 307174175050724 ) / 20080059416483 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26825292124872 \nu^{15} + 202850134162120 \nu^{14} - 768407821180136 \nu^{13} + \cdots + 72904139970597 ) / 20080059416483 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 30502028544271 \nu^{15} - 243762151512730 \nu^{14} + 971304907499854 \nu^{13} + \cdots - 597125923949512 ) / 20080059416483 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 65190512068075 \nu^{15} - 500587254130613 \nu^{14} + \cdots - 264468755510306 ) / 40160118832966 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 39949018132045 \nu^{15} + 307414450353699 \nu^{14} + \cdots + 210211317074828 ) / 20080059416483 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} + 2\beta_{9} - \beta_{7} + \beta_{6} - \beta_{2} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{13} - \beta_{11} + \beta_{10} - \beta_{7} + 6\beta_{6} + 2\beta_{3} + 3\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{15} + 33 \beta_{14} - 21 \beta_{13} + 12 \beta_{12} - 12 \beta_{11} + 18 \beta_{10} + \cdots - 58 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5 \beta_{15} + 29 \beta_{14} - 13 \beta_{13} + 22 \beta_{12} - 23 \beta_{9} - 33 \beta_{8} + 13 \beta_{6} + \cdots - 81 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 77 \beta_{15} + 341 \beta_{14} - 73 \beta_{13} + 180 \beta_{12} + 104 \beta_{11} - 182 \beta_{10} + \cdots - 622 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 178 \beta_{13} + 216 \beta_{11} - 390 \beta_{10} + 484 \beta_{7} - 852 \beta_{6} - 23 \beta_{5} + \cdots - 678 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 506 \beta_{15} - 1924 \beta_{14} + 1157 \beta_{13} - 1148 \beta_{12} + 535 \beta_{11} - 995 \beta_{10} + \cdots + 3532 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1226 \beta_{15} - 4498 \beta_{14} + 1762 \beta_{13} - 2922 \beta_{12} + 2982 \beta_{9} + 4335 \beta_{8} + \cdots + 8723 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12517 \beta_{15} - 44729 \beta_{14} + 8349 \beta_{13} - 27876 \beta_{12} - 11900 \beta_{11} + \cdots + 81806 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 21804 \beta_{13} - 28196 \beta_{11} + 54258 \beta_{10} - 71700 \beta_{7} + 105096 \beta_{6} + \cdots + 90652 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 150813 \beta_{15} + 525157 \beta_{14} - 305181 \beta_{13} + 332508 \beta_{12} - 136972 \beta_{11} + \cdots - 956530 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 91772 \beta_{15} + 317474 \beta_{14} - 121086 \beta_{13} + 203136 \beta_{12} - 203462 \beta_{9} + \cdots - 581353 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 898099 \beta_{15} + 3093397 \beta_{14} - 566512 \beta_{13} + 1970072 \beta_{12} + 799881 \beta_{11} + \cdots - 5619454 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1482817 \beta_{13} + 1935643 \beta_{11} - 3767779 \beta_{10} + 5036007 \beta_{7} + \cdots - 6278783 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 21283493 \beta_{15} - 72970009 \beta_{14} + 42083721 \beta_{13} - 46573196 \beta_{12} + \cdots + 132366210 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4176\mathbb{Z}\right)^\times\).

\(n\) \(929\) \(1045\) \(1567\) \(4033\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.399799 + 0.399799i
0.399799 − 0.399799i
0.797677 + 0.797677i
0.797677 − 0.797677i
2.42914 − 2.42914i
2.42914 + 2.42914i
−0.586168 + 0.586168i
−0.586168 − 0.586168i
1.58617 − 1.58617i
1.58617 + 1.58617i
−1.42914 + 1.42914i
−1.42914 − 1.42914i
0.202323 + 0.202323i
0.202323 − 0.202323i
0.600201 + 0.600201i
0.600201 − 0.600201i
0 0 0 −3.95440 0 4.04750 0 0 0
289.2 0 0 0 −3.95440 0 4.04750 0 0 0
289.3 0 0 0 −2.60463 0 −3.57952 0 0 0
289.4 0 0 0 −2.60463 0 −3.57952 0 0 0
289.5 0 0 0 −2.48771 0 −1.41674 0 0 0
289.6 0 0 0 −2.48771 0 −1.41674 0 0 0
289.7 0 0 0 −0.624444 0 1.94876 0 0 0
289.8 0 0 0 −0.624444 0 1.94876 0 0 0
289.9 0 0 0 0.624444 0 1.94876 0 0 0
289.10 0 0 0 0.624444 0 1.94876 0 0 0
289.11 0 0 0 2.48771 0 −1.41674 0 0 0
289.12 0 0 0 2.48771 0 −1.41674 0 0 0
289.13 0 0 0 2.60463 0 −3.57952 0 0 0
289.14 0 0 0 2.60463 0 −3.57952 0 0 0
289.15 0 0 0 3.95440 0 4.04750 0 0 0
289.16 0 0 0 3.95440 0 4.04750 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4176.2.o.s 16
3.b odd 2 1 inner 4176.2.o.s 16
4.b odd 2 1 2088.2.o.f ✓ 16
12.b even 2 1 2088.2.o.f ✓ 16
29.b even 2 1 inner 4176.2.o.s 16
87.d odd 2 1 inner 4176.2.o.s 16
116.d odd 2 1 2088.2.o.f ✓ 16
348.b even 2 1 2088.2.o.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2088.2.o.f ✓ 16 4.b odd 2 1
2088.2.o.f ✓ 16 12.b even 2 1
2088.2.o.f ✓ 16 116.d odd 2 1
2088.2.o.f ✓ 16 348.b even 2 1
4176.2.o.s 16 1.a even 1 1 trivial
4176.2.o.s 16 3.b odd 2 1 inner
4176.2.o.s 16 29.b even 2 1 inner
4176.2.o.s 16 87.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4176, [\chi])\):

\( T_{5}^{8} - 29T_{5}^{6} + 256T_{5}^{4} - 752T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{3} - 17T_{7}^{2} + 9T_{7} + 40 \) Copy content Toggle raw display
\( T_{11}^{8} + 54T_{11}^{6} + 617T_{11}^{4} + 900T_{11}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 29 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} - 17 T^{2} + \cdots + 40)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 54 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 27 T^{2} + 44)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 47 T^{6} + \cdots + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 93 T^{6} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 128 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 500246412961 \) Copy content Toggle raw display
$31$ \( (T^{8} + 164 T^{6} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 229 T^{6} + \cdots + 5234944)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 109 T^{6} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 277 T^{6} + \cdots + 4194304)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 75 T^{6} + \cdots + 29584)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 260 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 341 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 260 T^{6} + \cdots + 5161984)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 12 T^{3} + \cdots - 968)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 276 T^{6} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 340 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 384 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 320 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 546 T^{6} + \cdots + 38688400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 496 T^{6} + \cdots + 2768896)^{2} \) Copy content Toggle raw display
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