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Properties

Label 850.2.h.n
Level $850$
Weight $2$
Character orbit 850.h
Analytic conductor $6.787$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [850,2,Mod(251,850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(850, base_ring=CyclotomicField(4))
 
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("850.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.h (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: \(6.78728417181\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.23045668864.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 237x^{4} + 892x^{2} + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 170)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{6} q^{3} - q^{4} - \beta_{7} q^{6} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{7} + \beta_{3} q^{8} + (\beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_{6} q^{3} - q^{4} - \beta_{7} q^{6} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{7} + \beta_{3} q^{8} + (\beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_1) q^{9} + (2 \beta_{5} - \beta_{3} + 1) q^{11} - \beta_{6} q^{12} + ( - \beta_{2} + 2) q^{13} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{14} + q^{16} + (\beta_{6} + 2 \beta_{3} + \beta_1 + 1) q^{17} + ( - \beta_{5} + \beta_{4} + \beta_{2} + 3) q^{18} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{19}+ \cdots + ( - 4 \beta_{6} - 4 \beta_{4} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 4 q^{7} + 8 q^{11} + 12 q^{13} - 4 q^{14} + 8 q^{16} + 8 q^{17} + 28 q^{18} + 16 q^{21} - 8 q^{22} + 16 q^{23} - 12 q^{27} - 4 q^{28} + 24 q^{29} + 4 q^{31} - 16 q^{33} + 12 q^{34} + 20 q^{37} - 20 q^{38} - 4 q^{39} - 8 q^{44} - 16 q^{46} - 20 q^{47} + 4 q^{51} - 12 q^{52} + 12 q^{54} + 4 q^{56} - 40 q^{57} + 24 q^{58} - 16 q^{61} + 4 q^{62} + 64 q^{63} - 8 q^{64} + 16 q^{67} - 8 q^{68} + 8 q^{69} + 4 q^{71} - 28 q^{72} - 28 q^{73} + 20 q^{74} - 4 q^{78} + 8 q^{79} - 8 q^{81} - 16 q^{84} + 32 q^{86} + 8 q^{88} - 44 q^{89} - 44 q^{91} - 16 q^{92} - 20 q^{97} - 48 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 237x^{4} + 892x^{2} + 1156 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 8\nu^{5} + 307\nu^{3} + 1454\nu ) / 272 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} + 96\nu^{5} + 505\nu^{3} + 618\nu ) / 272 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 17\nu^{6} + 52\nu^{5} - 340\nu^{4} + 406\nu^{3} - 2057\nu^{2} + 900\nu - 3706 ) / 136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} + 17\nu^{6} + 52\nu^{5} + 340\nu^{4} + 406\nu^{3} + 2057\nu^{2} + 900\nu + 3706 ) / 136 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{7} - 17\nu^{6} + 392\nu^{5} - 408\nu^{4} + 2463\nu^{3} - 2941\nu^{2} + 4606\nu - 6018 ) / 272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} + 17\nu^{6} + 392\nu^{5} + 408\nu^{4} + 2463\nu^{3} + 2941\nu^{2} + 4606\nu + 6018 ) / 272 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + 5\beta_{5} + 5\beta_{4} - 26\beta_{3} - 14\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} - 13\beta_{2} + 57 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -30\beta_{7} - 30\beta_{6} - 17\beta_{5} - 17\beta_{4} + 278\beta_{3} + 114\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -40\beta_{7} + 40\beta_{6} + 24\beta_{5} - 24\beta_{4} + 139\beta_{2} - 511 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 374\beta_{7} + 374\beta_{6} - 55\beta_{5} - 55\beta_{4} - 2850\beta_{3} - 1022\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/850\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(751\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
− 1.69904i
3.26843i
2.69904i
− 2.26843i
1.69904i
− 3.26843i
− 2.69904i
2.26843i
− 1.00000i −1.90851 − 1.90851i −1.00000 0 −1.90851 + 1.90851i 2.69904 − 2.69904i 1.00000i 4.28483i 0
251.2 − 1.00000i −1.60402 − 1.60402i −1.00000 0 −1.60402 + 1.60402i −2.26843 + 2.26843i 1.00000i 2.14578i 0
251.3 − 1.00000i 1.20140 + 1.20140i −1.00000 0 1.20140 − 1.20140i −1.69904 + 1.69904i 1.00000i − 0.113256i 0
251.4 − 1.00000i 2.31113 + 2.31113i −1.00000 0 2.31113 − 2.31113i 3.26843 − 3.26843i 1.00000i 7.68265i 0
701.1 1.00000i −1.90851 + 1.90851i −1.00000 0 −1.90851 − 1.90851i 2.69904 + 2.69904i − 1.00000i − 4.28483i 0
701.2 1.00000i −1.60402 + 1.60402i −1.00000 0 −1.60402 − 1.60402i −2.26843 − 2.26843i − 1.00000i − 2.14578i 0
701.3 1.00000i 1.20140 − 1.20140i −1.00000 0 1.20140 + 1.20140i −1.69904 − 1.69904i − 1.00000i 0.113256i 0
701.4 1.00000i 2.31113 − 2.31113i −1.00000 0 2.31113 + 2.31113i 3.26843 + 3.26843i − 1.00000i − 7.68265i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 850.2.h.n 8
5.b even 2 1 170.2.h.b ✓ 8
5.c odd 4 1 850.2.g.i 8
5.c odd 4 1 850.2.g.l 8
15.d odd 2 1 1530.2.q.g 8
17.c even 4 1 inner 850.2.h.n 8
20.d odd 2 1 1360.2.bt.b 8
85.f odd 4 1 850.2.g.i 8
85.i odd 4 1 850.2.g.l 8
85.j even 4 1 170.2.h.b ✓ 8
85.m even 8 1 2890.2.a.bd 4
85.m even 8 1 2890.2.a.be 4
85.m even 8 2 2890.2.b.o 8
255.i odd 4 1 1530.2.q.g 8
340.n odd 4 1 1360.2.bt.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.h.b ✓ 8 5.b even 2 1
170.2.h.b ✓ 8 85.j even 4 1
850.2.g.i 8 5.c odd 4 1
850.2.g.i 8 85.f odd 4 1
850.2.g.l 8 5.c odd 4 1
850.2.g.l 8 85.i odd 4 1
850.2.h.n 8 1.a even 1 1 trivial
850.2.h.n 8 17.c even 4 1 inner
1360.2.bt.b 8 20.d odd 2 1
1360.2.bt.b 8 340.n odd 4 1
1530.2.q.g 8 15.d odd 2 1
1530.2.q.g 8 255.i odd 4 1
2890.2.a.bd 4 85.m even 8 1
2890.2.a.be 4 85.m even 8 1
2890.2.b.o 8 85.m even 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\):

\( T_{3}^{8} + 4T_{3}^{5} + 101T_{3}^{4} + 52T_{3}^{3} + 8T_{3}^{2} - 136T_{3} + 1156 \) Copy content Toggle raw display
\( T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} + 40T_{7}^{5} + 212T_{7}^{4} - 512T_{7}^{3} + 1152T_{7}^{2} + 6528T_{7} + 18496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 18496 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} - 3 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( T^{8} + 86 T^{6} + \cdots + 50176 \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 24 T^{7} + \cdots + 4624 \) Copy content Toggle raw display
$31$ \( T^{8} - 4 T^{7} + \cdots + 4624 \) Copy content Toggle raw display
$37$ \( T^{8} - 20 T^{7} + \cdots + 506944 \) Copy content Toggle raw display
$41$ \( T^{8} + 5256 T^{4} + 5626384 \) Copy content Toggle raw display
$43$ \( T^{8} + 116 T^{6} + \cdots + 3136 \) Copy content Toggle raw display
$47$ \( (T^{4} + 10 T^{3} + \cdots - 392)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 58 T^{6} + \cdots + 784 \) Copy content Toggle raw display
$59$ \( T^{8} + 222 T^{6} + \cdots + 80656 \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + \cdots + 1547536 \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + \cdots + 752)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 4 T^{7} + \cdots + 1817104 \) Copy content Toggle raw display
$73$ \( T^{8} + 28 T^{7} + \cdots + 30316036 \) Copy content Toggle raw display
$79$ \( T^{8} - 8 T^{7} + \cdots + 3625216 \) Copy content Toggle raw display
$83$ \( T^{8} + 364 T^{6} + \cdots + 23116864 \) Copy content Toggle raw display
$89$ \( (T^{4} + 22 T^{3} + \cdots - 32368)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 20 T^{7} + \cdots + 15984004 \) Copy content Toggle raw display
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