Properties

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

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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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 7 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + 7 i q^{7} -9 q^{9} + 4 q^{11} + 54 i q^{13} + 14 i q^{17} -92 q^{19} -21 q^{21} -152 i q^{23} -27 i q^{27} + 106 q^{29} -144 q^{31} + 12 i q^{33} -158 i q^{37} -162 q^{39} -390 q^{41} -508 i q^{43} + 528 i q^{47} -49 q^{49} -42 q^{51} + 606 i q^{53} -276 i q^{57} + 364 q^{59} + 678 q^{61} -63 i q^{63} -844 i q^{67} + 456 q^{69} -8 q^{71} -422 i q^{73} + 28 i q^{77} -384 q^{79} + 81 q^{81} -548 i q^{83} + 318 i q^{87} -1194 q^{89} -378 q^{91} -432 i q^{93} + 1502 i q^{97} -36 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{9} + O(q^{10}) \) \( 2q - 18q^{9} + 8q^{11} - 184q^{19} - 42q^{21} + 212q^{29} - 288q^{31} - 324q^{39} - 780q^{41} - 98q^{49} - 84q^{51} + 728q^{59} + 1356q^{61} + 912q^{69} - 16q^{71} - 768q^{79} + 162q^{81} - 2388q^{89} - 756q^{91} - 72q^{99} + O(q^{100}) \)

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.g 2
5.b even 2 1 inner 2100.4.k.g 2
5.c odd 4 1 84.4.a.b 1
5.c odd 4 1 2100.4.a.g 1
15.e even 4 1 252.4.a.a 1
20.e even 4 1 336.4.a.e 1
35.f even 4 1 588.4.a.a 1
35.k even 12 2 588.4.i.h 2
35.l odd 12 2 588.4.i.a 2
40.i odd 4 1 1344.4.a.b 1
40.k even 4 1 1344.4.a.p 1
60.l odd 4 1 1008.4.a.d 1
105.k odd 4 1 1764.4.a.l 1
105.w odd 12 2 1764.4.k.c 2
105.x even 12 2 1764.4.k.n 2
140.j odd 4 1 2352.4.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 5.c odd 4 1
252.4.a.a 1 15.e even 4 1
336.4.a.e 1 20.e even 4 1
588.4.a.a 1 35.f even 4 1
588.4.i.a 2 35.l odd 12 2
588.4.i.h 2 35.k even 12 2
1008.4.a.d 1 60.l odd 4 1
1344.4.a.b 1 40.i odd 4 1
1344.4.a.p 1 40.k even 4 1
1764.4.a.l 1 105.k odd 4 1
1764.4.k.c 2 105.w odd 12 2
1764.4.k.n 2 105.x even 12 2
2100.4.a.g 1 5.c odd 4 1
2100.4.k.g 2 1.a even 1 1 trivial
2100.4.k.g 2 5.b even 2 1 inner
2352.4.a.v 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 \(S_{4}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11} - 4 \)
\( T_{13}^{2} + 2916 \)
\( T_{17}^{2} + 196 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 49 + T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( 2916 + T^{2} \)
$17$ \( 196 + T^{2} \)
$19$ \( ( 92 + T )^{2} \)
$23$ \( 23104 + T^{2} \)
$29$ \( ( -106 + T )^{2} \)
$31$ \( ( 144 + T )^{2} \)
$37$ \( 24964 + T^{2} \)
$41$ \( ( 390 + T )^{2} \)
$43$ \( 258064 + T^{2} \)
$47$ \( 278784 + T^{2} \)
$53$ \( 367236 + T^{2} \)
$59$ \( ( -364 + T )^{2} \)
$61$ \( ( -678 + T )^{2} \)
$67$ \( 712336 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 178084 + T^{2} \)
$79$ \( ( 384 + T )^{2} \)
$83$ \( 300304 + T^{2} \)
$89$ \( ( 1194 + T )^{2} \)
$97$ \( 2256004 + T^{2} \)
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