# Properties

 Label 6300.2.d.f Level $6300$ Weight $2$ Character orbit 6300.d Analytic conductor $50.306$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6300.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$50.3057532734$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.101415451701035401216.7 Defining polynomial: $$x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{20}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 1260) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{7} +O(q^{10})$$ $$q + \beta_{7} q^{7} -\beta_{1} q^{11} + \beta_{2} q^{13} -\beta_{4} q^{17} -\beta_{3} q^{19} + \beta_{15} q^{23} + \beta_{13} q^{29} + ( \beta_{3} - \beta_{11} ) q^{31} + ( -\beta_{6} + \beta_{7} ) q^{37} + ( \beta_{5} + \beta_{10} ) q^{41} -\beta_{12} q^{43} + ( -3 \beta_{4} + \beta_{8} ) q^{47} + ( 1 + \beta_{11} + \beta_{14} ) q^{49} + ( -2 \beta_{9} - \beta_{15} ) q^{53} + ( -\beta_{5} + 2 \beta_{10} ) q^{59} + ( -\beta_{6} + \beta_{7} + \beta_{12} ) q^{67} -\beta_{1} q^{71} + ( -\beta_{2} - 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -2 \beta_{4} - \beta_{8} - 2 \beta_{9} - \beta_{15} ) q^{77} + ( -4 + 2 \beta_{14} ) q^{79} + ( \beta_{4} - 3 \beta_{8} ) q^{83} -\beta_{10} q^{89} + ( 2 + \beta_{3} - \beta_{11} ) q^{91} + ( -5 \beta_{2} - 4 \beta_{6} - 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + O(q^{10})$$ $$16 q + 16 q^{49} - 64 q^{79} + 32 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{12} - 18 \nu^{8} + 141 \nu^{4} - 36$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$27 \nu^{15} + 22 \nu^{13} - 474 \nu^{11} - 388 \nu^{9} + 3699 \nu^{7} + 3062 \nu^{5} - 156 \nu^{3} - 824 \nu$$$$)/288$$ $$\beta_{3}$$ $$=$$ $$($$$$143 \nu^{15} + 194 \nu^{13} - 2450 \nu^{11} - 3500 \nu^{9} + 18535 \nu^{7} + 27970 \nu^{5} + 6164 \nu^{3} - 10408 \nu$$$$)/1440$$ $$\beta_{4}$$ $$=$$ $$($$$$-27 \nu^{15} + 22 \nu^{13} + 474 \nu^{11} - 388 \nu^{9} - 3699 \nu^{7} + 3062 \nu^{5} + 156 \nu^{3} - 824 \nu$$$$)/288$$ $$\beta_{5}$$ $$=$$ $$($$$$-47 \nu^{15} + 118 \nu^{13} + 770 \nu^{11} - 2140 \nu^{9} - 5455 \nu^{7} + 17150 \nu^{5} - 6716 \nu^{3} - 7496 \nu$$$$)/720$$ $$\beta_{6}$$ $$=$$ $$($$$$35 \nu^{15} + 51 \nu^{14} - 30 \nu^{13} - 2 \nu^{12} - 650 \nu^{11} - 930 \nu^{10} + 540 \nu^{9} + 20 \nu^{8} + 5395 \nu^{7} + 7635 \nu^{6} - 4470 \nu^{5} - 10 \nu^{4} - 5140 \nu^{3} - 6252 \nu^{2} + 3240 \nu - 1016$$$$)/1440$$ $$\beta_{7}$$ $$=$$ $$($$$$35 \nu^{15} - 51 \nu^{14} - 30 \nu^{13} + 2 \nu^{12} - 650 \nu^{11} + 930 \nu^{10} + 540 \nu^{9} - 20 \nu^{8} + 5395 \nu^{7} - 7635 \nu^{6} - 4470 \nu^{5} + 10 \nu^{4} - 5140 \nu^{3} + 6252 \nu^{2} + 3240 \nu + 1016$$$$)/1440$$ $$\beta_{8}$$ $$=$$ $$($$$$-41 \nu^{15} + 10 \nu^{13} + 734 \nu^{11} - 172 \nu^{9} - 5857 \nu^{7} + 1274 \nu^{5} + 2212 \nu^{3} + 472 \nu$$$$)/288$$ $$\beta_{9}$$ $$=$$ $$($$$$39 \nu^{14} + 64 \nu^{12} - 690 \nu^{10} - 1120 \nu^{8} + 5415 \nu^{6} + 8960 \nu^{4} - 948 \nu^{2} - 2288$$$$)/720$$ $$\beta_{10}$$ $$=$$ $$($$$$-241 \nu^{15} + 110 \nu^{13} + 4270 \nu^{11} - 1940 \nu^{9} - 33785 \nu^{7} + 15310 \nu^{5} + 8372 \nu^{3} - 1240 \nu$$$$)/1440$$ $$\beta_{11}$$ $$=$$ $$($$$$-145 \nu^{15} - 34 \nu^{13} + 2590 \nu^{11} + 580 \nu^{9} - 20705 \nu^{7} - 4490 \nu^{5} + 7820 \nu^{3} - 1672 \nu$$$$)/720$$ $$\beta_{12}$$ $$=$$ $$($$$$-51 \nu^{14} - 10 \nu^{12} + 930 \nu^{10} + 100 \nu^{8} - 7635 \nu^{6} - 50 \nu^{4} + 6252 \nu^{2} - 5080$$$$)/720$$ $$\beta_{13}$$ $$=$$ $$($$$$-11 \nu^{14} + 194 \nu^{10} - 1531 \nu^{6} + 268 \nu^{2}$$$$)/144$$ $$\beta_{14}$$ $$=$$ $$($$$$-7 \nu^{14} + 130 \nu^{10} - 1079 \nu^{6} + 884 \nu^{2}$$$$)/72$$ $$\beta_{15}$$ $$=$$ $$($$$$195 \nu^{14} - 64 \nu^{12} - 3450 \nu^{10} + 1120 \nu^{8} + 27075 \nu^{6} - 8960 \nu^{4} - 4740 \nu^{2} + 2288$$$$)/720$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{11} + 3 \beta_{10} - 3 \beta_{4} + \beta_{3} - 3 \beta_{2}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} - 3 \beta_{14} - 6 \beta_{13} + \beta_{12} - \beta_{9} + 5 \beta_{7} - 5 \beta_{6}$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} + 3 \beta_{10} + 3 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 12 \beta_{4} - 5 \beta_{3} + 9 \beta_{2}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{12} + 5 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{1} + 18$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-21 \beta_{11} + 29 \beta_{10} - 27 \beta_{8} - 27 \beta_{7} - 27 \beta_{6} - 5 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 33 \beta_{2}$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$-35 \beta_{15} - 21 \beta_{14} - 150 \beta_{13} + 5 \beta_{12} - 35 \beta_{9} + 25 \beta_{7} - 25 \beta_{6}$$$$)/12$$ $$\nu^{7}$$ $$=$$ $$($$$$-39 \beta_{11} - 37 \beta_{10} + 87 \beta_{8} - 87 \beta_{7} - 87 \beta_{6} + 43 \beta_{5} - 96 \beta_{4} - 45 \beta_{3} + 9 \beta_{2}$$$$)/12$$ $$\nu^{8}$$ $$=$$ $$($$$$-17 \beta_{15} + 4 \beta_{12} + 85 \beta_{9} - 4 \beta_{7} + 4 \beta_{6} - 72 \beta_{1} + 34$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-179 \beta_{11} + 201 \beta_{10} - 333 \beta_{8} - 333 \beta_{7} - 333 \beta_{6} - 135 \beta_{5} + 240 \beta_{4} - 113 \beta_{3} - 93 \beta_{2}$$$$)/12$$ $$\nu^{10}$$ $$=$$ $$($$$$-451 \beta_{15} + 123 \beta_{14} - 1914 \beta_{13} - 29 \beta_{12} - 451 \beta_{9} - 145 \beta_{7} + 145 \beta_{6}$$$$)/12$$ $$\nu^{11}$$ $$=$$ $$($$$$-797 \beta_{11} - 1041 \beta_{10} + 1173 \beta_{8} - 1173 \beta_{7} - 1173 \beta_{6} + 309 \beta_{5} - 138 \beta_{4} - 65 \beta_{3} - 1035 \beta_{2}$$$$)/12$$ $$\nu^{12}$$ $$=$$ $$($$$$-165 \beta_{15} - 210 \beta_{12} + 825 \beta_{9} + 210 \beta_{7} - 210 \beta_{6} - 700 \beta_{1} - 1782$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-309 \beta_{11} - 379 \beta_{10} - 2115 \beta_{8} - 2115 \beta_{7} - 2115 \beta_{6} - 1685 \beta_{5} + 5034 \beta_{4} - 2373 \beta_{3} + 2919 \beta_{2}$$$$)/12$$ $$\nu^{14}$$ $$=$$ $$($$$$-3107 \beta_{15} + 5019 \beta_{14} - 13182 \beta_{13} - 1183 \beta_{12} - 3107 \beta_{9} - 5915 \beta_{7} + 5915 \beta_{6}$$$$)/12$$ $$\nu^{15}$$ $$=$$ $$($$$$-8643 \beta_{11} - 13189 \beta_{10} + 8691 \beta_{8} - 8691 \beta_{7} - 8691 \beta_{6} - 449 \beta_{5} + 10596 \beta_{4} + 4995 \beta_{3} - 19287 \beta_{2}$$$$)/12$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$$.

 $$n$$ $$2801$$ $$3151$$ $$3277$$ $$3601$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3401.1
 −0.332046 − 1.81768i 0.332046 − 1.81768i 0.332046 + 1.81768i −0.332046 + 1.81768i 0.137538 − 0.752908i −0.137538 − 0.752908i −0.137538 + 0.752908i 0.137538 + 0.752908i −1.81768 + 0.332046i 1.81768 + 0.332046i 1.81768 − 0.332046i −1.81768 − 0.332046i 0.752908 + 0.137538i −0.752908 + 0.137538i −0.752908 − 0.137538i 0.752908 − 0.137538i
0 0 0 0 0 −2.57794 0.595188i 0 0 0
3401.2 0 0 0 0 0 −2.57794 0.595188i 0 0 0
3401.3 0 0 0 0 0 −2.57794 + 0.595188i 0 0 0
3401.4 0 0 0 0 0 −2.57794 + 0.595188i 0 0 0
3401.5 0 0 0 0 0 −1.16372 2.37608i 0 0 0
3401.6 0 0 0 0 0 −1.16372 2.37608i 0 0 0
3401.7 0 0 0 0 0 −1.16372 + 2.37608i 0 0 0
3401.8 0 0 0 0 0 −1.16372 + 2.37608i 0 0 0
3401.9 0 0 0 0 0 1.16372 2.37608i 0 0 0
3401.10 0 0 0 0 0 1.16372 2.37608i 0 0 0
3401.11 0 0 0 0 0 1.16372 + 2.37608i 0 0 0
3401.12 0 0 0 0 0 1.16372 + 2.37608i 0 0 0
3401.13 0 0 0 0 0 2.57794 0.595188i 0 0 0
3401.14 0 0 0 0 0 2.57794 0.595188i 0 0 0
3401.15 0 0 0 0 0 2.57794 + 0.595188i 0 0 0
3401.16 0 0 0 0 0 2.57794 + 0.595188i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3401.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6300.2.d.f 16
3.b odd 2 1 inner 6300.2.d.f 16
5.b even 2 1 inner 6300.2.d.f 16
5.c odd 4 1 1260.2.f.a 8
5.c odd 4 1 1260.2.f.b yes 8
7.b odd 2 1 inner 6300.2.d.f 16
15.d odd 2 1 inner 6300.2.d.f 16
15.e even 4 1 1260.2.f.a 8
15.e even 4 1 1260.2.f.b yes 8
20.e even 4 1 5040.2.k.e 8
20.e even 4 1 5040.2.k.f 8
21.c even 2 1 inner 6300.2.d.f 16
35.c odd 2 1 inner 6300.2.d.f 16
35.f even 4 1 1260.2.f.a 8
35.f even 4 1 1260.2.f.b yes 8
60.l odd 4 1 5040.2.k.e 8
60.l odd 4 1 5040.2.k.f 8
105.g even 2 1 inner 6300.2.d.f 16
105.k odd 4 1 1260.2.f.a 8
105.k odd 4 1 1260.2.f.b yes 8
140.j odd 4 1 5040.2.k.e 8
140.j odd 4 1 5040.2.k.f 8
420.w even 4 1 5040.2.k.e 8
420.w even 4 1 5040.2.k.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.f.a 8 5.c odd 4 1
1260.2.f.a 8 15.e even 4 1
1260.2.f.a 8 35.f even 4 1
1260.2.f.a 8 105.k odd 4 1
1260.2.f.b yes 8 5.c odd 4 1
1260.2.f.b yes 8 15.e even 4 1
1260.2.f.b yes 8 35.f even 4 1
1260.2.f.b yes 8 105.k odd 4 1
5040.2.k.e 8 20.e even 4 1
5040.2.k.e 8 60.l odd 4 1
5040.2.k.e 8 140.j odd 4 1
5040.2.k.e 8 420.w even 4 1
5040.2.k.f 8 20.e even 4 1
5040.2.k.f 8 60.l odd 4 1
5040.2.k.f 8 140.j odd 4 1
5040.2.k.f 8 420.w even 4 1
6300.2.d.f 16 1.a even 1 1 trivial
6300.2.d.f 16 3.b odd 2 1 inner
6300.2.d.f 16 5.b even 2 1 inner
6300.2.d.f 16 7.b odd 2 1 inner
6300.2.d.f 16 15.d odd 2 1 inner
6300.2.d.f 16 21.c even 2 1 inner
6300.2.d.f 16 35.c odd 2 1 inner
6300.2.d.f 16 105.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(6300, [\chi])$$:

 $$T_{11}^{2} + 14$$ $$T_{37}^{4} - 32 T_{37}^{2} + 144$$ $$T_{41}^{4} - 140 T_{41}^{2} + 3528$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 2401 - 196 T^{2} - 10 T^{4} - 4 T^{6} + T^{8} )^{2}$$
$11$ $$( 14 + T^{2} )^{8}$$
$13$ $$( 8 + 12 T^{2} + T^{4} )^{4}$$
$17$ $$( 8 - 12 T^{2} + T^{4} )^{4}$$
$19$ $$( 648 + 52 T^{2} + T^{4} )^{4}$$
$23$ $$( 324 + 64 T^{2} + T^{4} )^{4}$$
$29$ $$( 2 + T^{2} )^{8}$$
$31$ $$( 72 + 76 T^{2} + T^{4} )^{4}$$
$37$ $$( 144 - 32 T^{2} + T^{4} )^{4}$$
$41$ $$( 3528 - 140 T^{2} + T^{4} )^{4}$$
$43$ $$( 1296 - 128 T^{2} + T^{4} )^{4}$$
$47$ $$( 2592 - 104 T^{2} + T^{4} )^{4}$$
$53$ $$( 1764 + 112 T^{2} + T^{4} )^{4}$$
$59$ $$( 288 - 152 T^{2} + T^{4} )^{4}$$
$61$ $$T^{16}$$
$67$ $$( 576 - 176 T^{2} + T^{4} )^{4}$$
$71$ $$( 14 + T^{2} )^{8}$$
$73$ $$( 72 + 76 T^{2} + T^{4} )^{4}$$
$79$ $$( -96 + 8 T + T^{2} )^{8}$$
$83$ $$( 1568 - 168 T^{2} + T^{4} )^{4}$$
$89$ $$( 72 - 20 T^{2} + T^{4} )^{4}$$
$97$ $$( 31752 + 364 T^{2} + T^{4} )^{4}$$