Properties

Label 2100.4.k.j
Level $2100$
Weight $4$
Character orbit 2100.k
Analytic conductor $123.904$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2100.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(123.904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 7 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + 7 i q^{7} -9 q^{9} + 36 q^{11} -62 i q^{13} + 114 i q^{17} + 76 q^{19} -21 q^{21} + 24 i q^{23} -27 i q^{27} -54 q^{29} -112 q^{31} + 108 i q^{33} -178 i q^{37} + 186 q^{39} + 378 q^{41} + 172 i q^{43} -192 i q^{47} -49 q^{49} -342 q^{51} + 402 i q^{53} + 228 i q^{57} -396 q^{59} + 254 q^{61} -63 i q^{63} -1012 i q^{67} -72 q^{69} + 840 q^{71} -890 i q^{73} + 252 i q^{77} -80 q^{79} + 81 q^{81} + 108 i q^{83} -162 i q^{87} + 1638 q^{89} + 434 q^{91} -336 i q^{93} + 1010 i q^{97} -324 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{9} + O(q^{10}) \) \( 2q - 18q^{9} + 72q^{11} + 152q^{19} - 42q^{21} - 108q^{29} - 224q^{31} + 372q^{39} + 756q^{41} - 98q^{49} - 684q^{51} - 792q^{59} + 508q^{61} - 144q^{69} + 1680q^{71} - 160q^{79} + 162q^{81} + 3276q^{89} + 868q^{91} - 648q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.00000i
1.00000i
0 3.00000i 0 0 0 7.00000i 0 −9.00000 0
1849.2 0 3.00000i 0 0 0 7.00000i 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.4.k.j 2
5.b even 2 1 inner 2100.4.k.j 2
5.c odd 4 1 84.4.a.a 1
5.c odd 4 1 2100.4.a.l 1
15.e even 4 1 252.4.a.b 1
20.e even 4 1 336.4.a.k 1
35.f even 4 1 588.4.a.d 1
35.k even 12 2 588.4.i.c 2
35.l odd 12 2 588.4.i.f 2
40.i odd 4 1 1344.4.a.q 1
40.k even 4 1 1344.4.a.d 1
60.l odd 4 1 1008.4.a.h 1
105.k odd 4 1 1764.4.a.j 1
105.w odd 12 2 1764.4.k.f 2
105.x even 12 2 1764.4.k.l 2
140.j odd 4 1 2352.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 5.c odd 4 1
252.4.a.b 1 15.e even 4 1
336.4.a.k 1 20.e even 4 1
588.4.a.d 1 35.f even 4 1
588.4.i.c 2 35.k even 12 2
588.4.i.f 2 35.l odd 12 2
1008.4.a.h 1 60.l odd 4 1
1344.4.a.d 1 40.k even 4 1
1344.4.a.q 1 40.i odd 4 1
1764.4.a.j 1 105.k odd 4 1
1764.4.k.f 2 105.w odd 12 2
1764.4.k.l 2 105.x even 12 2
2100.4.a.l 1 5.c odd 4 1
2100.4.k.j 2 1.a even 1 1 trivial
2100.4.k.j 2 5.b even 2 1 inner
2352.4.a.d 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11} - 36 \)
\( T_{13}^{2} + 3844 \)
\( T_{17}^{2} + 12996 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 49 + T^{2} \)
$11$ \( ( -36 + T )^{2} \)
$13$ \( 3844 + T^{2} \)
$17$ \( 12996 + T^{2} \)
$19$ \( ( -76 + T )^{2} \)
$23$ \( 576 + T^{2} \)
$29$ \( ( 54 + T )^{2} \)
$31$ \( ( 112 + T )^{2} \)
$37$ \( 31684 + T^{2} \)
$41$ \( ( -378 + T )^{2} \)
$43$ \( 29584 + T^{2} \)
$47$ \( 36864 + T^{2} \)
$53$ \( 161604 + T^{2} \)
$59$ \( ( 396 + T )^{2} \)
$61$ \( ( -254 + T )^{2} \)
$67$ \( 1024144 + T^{2} \)
$71$ \( ( -840 + T )^{2} \)
$73$ \( 792100 + T^{2} \)
$79$ \( ( 80 + T )^{2} \)
$83$ \( 11664 + T^{2} \)
$89$ \( ( -1638 + T )^{2} \)
$97$ \( 1020100 + T^{2} \)
show more
show less