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Properties

Label 1000.2.k.e
Level $1000$
Weight $2$
Character orbit 1000.k
Analytic conductor $7.985$
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] = [1000,2,Mod(307,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, 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("1000.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.k (of order \(4\), degree \(2\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} + 10 x^{13} - 109 x^{12} + 280 x^{11} - 198 x^{10} - 1168 x^{9} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{12} + \beta_{7} + \beta_{3}) q^{3} + (\beta_{14} + \beta_{9} - \beta_{2} - 1) q^{4} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{9}) q^{6}+ \cdots + (\beta_{15} + \beta_{14} + 2 \beta_{12} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{12} + \beta_{7} + \beta_{3}) q^{3} + (\beta_{14} + \beta_{9} - \beta_{2} - 1) q^{4} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{9}) q^{6}+ \cdots + (\beta_{15} - 2 \beta_{14} + 9 \beta_{12} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 4 q^{3} - 12 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 4 q^{3} - 12 q^{6} - 8 q^{8} + 8 q^{11} - 8 q^{12} + 16 q^{16} + 32 q^{17} + 8 q^{18} + 12 q^{22} + 20 q^{27} - 20 q^{28} + 64 q^{32} - 80 q^{33} - 16 q^{36} + 36 q^{38} + 8 q^{41} - 20 q^{42} + 4 q^{43} - 20 q^{46} - 24 q^{48} + 120 q^{51} + 40 q^{52} - 40 q^{56} + 80 q^{57} - 20 q^{58} - 20 q^{62} - 40 q^{66} - 48 q^{67} + 16 q^{68} + 16 q^{72} - 24 q^{73} + 8 q^{76} - 20 q^{78} - 8 q^{81} - 28 q^{82} - 24 q^{83} + 52 q^{86} + 16 q^{88} - 40 q^{91} + 20 q^{92} - 32 q^{96} + 48 q^{97} + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} + 10 x^{13} - 109 x^{12} + 280 x^{11} - 198 x^{10} - 1168 x^{9} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 63878 \nu^{15} + 3344167 \nu^{14} + 50634081 \nu^{13} - 236431440 \nu^{12} + \cdots - 2265390781250 ) / 651236953125 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 256701 \nu^{15} + 1327689 \nu^{14} + 6933752 \nu^{13} - 21099280 \nu^{12} + \cdots - 290143359375 ) / 651236953125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 173441 \nu^{15} + 13142014 \nu^{14} - 62145978 \nu^{13} + 45934140 \nu^{12} + \cdots + 1109367578125 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1086242 \nu^{15} + 8534838 \nu^{14} - 79403991 \nu^{13} + 117059340 \nu^{12} + \cdots - 1163691250000 ) / 651236953125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21238 \nu^{15} - 1031322 \nu^{14} + 5312684 \nu^{13} - 3514030 \nu^{12} - 30481492 \nu^{11} + \cdots - 99028437500 ) / 11840671875 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 564240 \nu^{15} + 224729 \nu^{14} + 12812329 \nu^{13} - 46797373 \nu^{12} + \cdots - 329552203125 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 74113 \nu^{15} + 616343 \nu^{14} - 1804126 \nu^{13} + 3246140 \nu^{12} + 12427708 \nu^{11} + \cdots + 27784296875 ) / 15883828125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2032231 \nu^{15} + 17326249 \nu^{14} - 41154973 \nu^{13} - 28289160 \nu^{12} + \cdots + 350653640625 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2195249 \nu^{15} - 4137243 \nu^{14} - 7364740 \nu^{13} + 83581919 \nu^{12} + \cdots - 83580000000 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11075084 \nu^{15} - 85118096 \nu^{14} + 177880162 \nu^{13} + 228226960 \nu^{12} + \cdots - 2351709140625 ) / 651236953125 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11349856 \nu^{15} + 32450499 \nu^{14} - 46325348 \nu^{13} - 102013035 \nu^{12} + \cdots + 2932434218750 ) / 651236953125 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21389 \nu^{15} + 15849 \nu^{14} - 192528 \nu^{13} + 559035 \nu^{12} + 343314 \nu^{11} + \cdots + 5258500000 ) / 1058921875 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16664587 \nu^{15} - 49117717 \nu^{14} + 294008639 \nu^{13} - 570647565 \nu^{12} + \cdots - 4269121796875 ) / 651236953125 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 624 \nu^{15} - 3266 \nu^{14} + 5072 \nu^{13} + 16955 \nu^{12} - 95466 \nu^{11} + 160150 \nu^{10} + \cdots - 37187500 ) / 23671875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 46414854 \nu^{15} + 177103336 \nu^{14} - 119704312 \nu^{13} - 1395099855 \nu^{12} + \cdots + 1334335000000 ) / 651236953125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{11} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} - 2\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{12} - 2\beta_{10} - 3\beta_{7} + 3\beta_{6} + 2\beta_{4} - 2\beta_{3} - 2\beta_{2} - 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{15} + 5 \beta_{14} + 3 \beta_{13} + 9 \beta_{12} + 2 \beta_{11} + 11 \beta_{9} + 3 \beta_{8} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 17 \beta_{14} + 18 \beta_{12} - 8 \beta_{11} + 8 \beta_{10} + 3 \beta_{9} + 12 \beta_{8} - 26 \beta_{7} + \cdots + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{14} - 2\beta_{12} + 18\beta_{10} + 12\beta_{7} - 3\beta_{5} + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 56 \beta_{15} - 105 \beta_{14} - 30 \beta_{13} - 34 \beta_{12} + 30 \beta_{10} - 105 \beta_{9} + \cdots + 105 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 30 \beta_{15} + 41 \beta_{14} - 15 \beta_{13} - 135 \beta_{12} + 64 \beta_{11} + 71 \beta_{9} + \cdots + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8 \beta_{15} + 462 \beta_{14} - 67 \beta_{12} - 8 \beta_{11} - 312 \beta_{9} + 408 \beta_{8} + \cdots - 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 27 \beta_{14} - 96 \beta_{13} - 78 \beta_{12} - 29 \beta_{11} + 58 \beta_{10} - 67 \beta_{9} + \cdots - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -9\beta_{15} - 127\beta_{12} + 9\beta_{10} + 471\beta_{7} - 924\beta_{5} - 924\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 706 \beta_{15} - 4973 \beta_{14} - 2310 \beta_{13} - 1694 \beta_{12} + 1957 \beta_{11} + 4883 \beta_{9} + \cdots + 4710 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 4620 \beta_{15} + 16786 \beta_{14} - 5769 \beta_{12} + 4004 \beta_{11} + 616 \beta_{10} + 7546 \beta_{9} + \cdots + 6930 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3317 \beta_{14} - 5769 \beta_{13} + 14251 \beta_{12} + 12166 \beta_{11} - 17935 \beta_{9} + \cdots + 1232 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 829 \beta_{14} + 32814 \beta_{13} + 59598 \beta_{12} - 34046 \beta_{10} + 829 \beta_{9} + 62062 \beta_{7} + \cdots - 829 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -26784\beta_{15} - 108819\beta_{14} + 111899\beta_{12} - 82035\beta_{7} - 61659\beta_{3} - 3080 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(-\beta_{12}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
2.23431 − 0.0885831i
−2.09758 − 0.774688i
−2.23122 − 0.147217i
1.19237 − 1.89162i
0.774688 + 2.09758i
0.0885831 − 2.23431i
1.89162 − 1.19237i
0.147217 + 2.23122i
2.23431 + 0.0885831i
−2.09758 + 0.774688i
−2.23122 + 0.147217i
1.19237 + 1.89162i
0.774688 − 2.09758i
0.0885831 + 2.23431i
1.89162 + 1.19237i
0.147217 − 2.23122i
−1.26007 + 0.642040i −1.24163 + 1.24163i 1.17557 − 1.61803i 0 0.767370 − 2.36172i 0.284054 − 0.284054i −0.442463 + 2.79360i − 0.0832940i 0
307.2 −1.26007 + 0.642040i 1.85966 − 1.85966i 1.17557 − 1.61803i 0 −1.14934 + 3.53729i −3.36174 + 3.36174i −0.442463 + 2.79360i − 3.91671i 0
307.3 0.221232 + 1.39680i −2.16751 + 2.16751i −1.90211 + 0.618034i 0 −3.50710 − 2.54806i −2.22073 + 2.22073i −1.28408 − 2.52015i − 6.39616i 0
307.4 0.221232 + 1.39680i 0.549472 − 0.549472i −1.90211 + 0.618034i 0 0.889064 + 0.645943i 2.94727 − 2.94727i −1.28408 − 2.52015i 2.39616i 0
307.5 0.642040 − 1.26007i −1.24163 + 1.24163i −1.17557 − 1.61803i 0 0.767370 + 2.36172i −0.284054 + 0.284054i −2.79360 + 0.442463i − 0.0832940i 0
307.6 0.642040 − 1.26007i 1.85966 − 1.85966i −1.17557 − 1.61803i 0 −1.14934 − 3.53729i 3.36174 − 3.36174i −2.79360 + 0.442463i − 3.91671i 0
307.7 1.39680 + 0.221232i −2.16751 + 2.16751i 1.90211 + 0.618034i 0 −3.50710 + 2.54806i 2.22073 − 2.22073i 2.52015 + 1.28408i − 6.39616i 0
307.8 1.39680 + 0.221232i 0.549472 − 0.549472i 1.90211 + 0.618034i 0 0.889064 − 0.645943i −2.94727 + 2.94727i 2.52015 + 1.28408i 2.39616i 0
443.1 −1.26007 − 0.642040i −1.24163 − 1.24163i 1.17557 + 1.61803i 0 0.767370 + 2.36172i 0.284054 + 0.284054i −0.442463 − 2.79360i 0.0832940i 0
443.2 −1.26007 − 0.642040i 1.85966 + 1.85966i 1.17557 + 1.61803i 0 −1.14934 − 3.53729i −3.36174 − 3.36174i −0.442463 − 2.79360i 3.91671i 0
443.3 0.221232 − 1.39680i −2.16751 − 2.16751i −1.90211 − 0.618034i 0 −3.50710 + 2.54806i −2.22073 − 2.22073i −1.28408 + 2.52015i 6.39616i 0
443.4 0.221232 − 1.39680i 0.549472 + 0.549472i −1.90211 − 0.618034i 0 0.889064 − 0.645943i 2.94727 + 2.94727i −1.28408 + 2.52015i − 2.39616i 0
443.5 0.642040 + 1.26007i −1.24163 − 1.24163i −1.17557 + 1.61803i 0 0.767370 − 2.36172i −0.284054 − 0.284054i −2.79360 − 0.442463i 0.0832940i 0
443.6 0.642040 + 1.26007i 1.85966 + 1.85966i −1.17557 + 1.61803i 0 −1.14934 + 3.53729i 3.36174 + 3.36174i −2.79360 − 0.442463i 3.91671i 0
443.7 1.39680 − 0.221232i −2.16751 − 2.16751i 1.90211 − 0.618034i 0 −3.50710 − 2.54806i 2.22073 + 2.22073i 2.52015 − 1.28408i 6.39616i 0
443.8 1.39680 − 0.221232i 0.549472 + 0.549472i 1.90211 − 0.618034i 0 0.889064 + 0.645943i −2.94727 − 2.94727i 2.52015 − 1.28408i − 2.39616i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.k.e yes 16
5.b even 2 1 1000.2.k.c ✓ 16
5.c odd 4 1 1000.2.k.c ✓ 16
5.c odd 4 1 inner 1000.2.k.e yes 16
8.d odd 2 1 inner 1000.2.k.e yes 16
40.e odd 2 1 1000.2.k.c ✓ 16
40.k even 4 1 1000.2.k.c ✓ 16
40.k even 4 1 inner 1000.2.k.e yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.k.c ✓ 16 5.b even 2 1
1000.2.k.c ✓ 16 5.c odd 4 1
1000.2.k.c ✓ 16 40.e odd 2 1
1000.2.k.c ✓ 16 40.k even 4 1
1000.2.k.e yes 16 1.a even 1 1 trivial
1000.2.k.e yes 16 5.c odd 4 1 inner
1000.2.k.e yes 16 8.d odd 2 1 inner
1000.2.k.e yes 16 40.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\):

\( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} - 6T_{3}^{5} + 59T_{3}^{4} + 86T_{3}^{3} + 72T_{3}^{2} - 132T_{3} + 121 \) Copy content Toggle raw display
\( T_{7}^{16} + 910T_{7}^{12} + 233275T_{7}^{8} + 15006250T_{7}^{4} + 390625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 2 T^{7} + \cdots + 121)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 910 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 18 T^{2} + \cdots + 44)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 60 T^{4} + 400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 16 T^{7} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 140 T^{6} + \cdots + 633616)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 910 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$29$ \( (T^{8} - 90 T^{6} + \cdots + 60025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 160 T^{6} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 16649664160000 \) Copy content Toggle raw display
$41$ \( (T^{4} - 2 T^{3} + \cdots - 571)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1766100625 \) Copy content Toggle raw display
$53$ \( (T^{8} + 5760 T^{4} + 102400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 184 T^{6} + \cdots + 2298256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 410 T^{6} + \cdots + 3783025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 24 T^{7} + \cdots + 929296)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 180 T^{6} + \cdots + 672400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 12 T^{7} + \cdots + 3254416)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 220 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 12 T^{7} + \cdots + 290521)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 390 T^{6} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 24 T^{7} + \cdots + 19044496)^{2} \) Copy content Toggle raw display
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