# Properties

 Label 3150.2.m.d Level 3150 Weight 2 Character orbit 3150.m Analytic conductor 25.153 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} + \zeta_{8}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} + \zeta_{8}^{3} q^{8} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{11} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{13} + q^{14} - q^{16} + ( 3 - 2 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{17} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{19} + ( -2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{22} + ( 2 - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{26} + \zeta_{8} q^{28} + ( -6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} -\zeta_{8} q^{32} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{34} + ( 1 - \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{37} + ( -5 + 5 \zeta_{8}^{2} ) q^{38} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{41} + ( 5 + 2 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{43} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{46} + ( 5 + 2 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{47} -\zeta_{8}^{2} q^{49} + ( 2 - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{52} + ( -2 + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{53} + \zeta_{8}^{2} q^{56} + ( 2 - 6 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{58} -8 q^{59} + ( 8 - \zeta_{8} + \zeta_{8}^{3} ) q^{61} + ( -2 - 6 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{62} -\zeta_{8}^{2} q^{64} + ( -1 + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{67} + ( -3 + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{68} + ( -3 \zeta_{8} - 8 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{71} + ( -6 - 6 \zeta_{8}^{2} ) q^{73} + ( -10 + \zeta_{8} - \zeta_{8}^{3} ) q^{74} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{76} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{77} + ( 8 \zeta_{8} + 2 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{79} + ( 2 - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{82} + ( -4 + 4 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{83} + ( 5 \zeta_{8} + 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{86} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{88} + ( -2 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{89} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{91} + ( 2 + 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{92} + ( 5 \zeta_{8} + 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{94} + ( -4 + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{97} -\zeta_{8}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 8q^{13} + 4q^{14} - 4q^{16} + 12q^{17} - 8q^{22} + 8q^{23} - 24q^{29} - 24q^{31} + 4q^{37} - 20q^{38} + 20q^{43} - 8q^{44} + 16q^{46} + 20q^{47} + 8q^{52} - 8q^{53} + 8q^{58} - 32q^{59} + 32q^{61} - 8q^{62} - 4q^{67} - 12q^{68} - 24q^{73} - 40q^{74} + 8q^{77} + 8q^{82} - 16q^{83} - 8q^{88} - 8q^{89} - 8q^{91} + 8q^{92} - 16q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1457.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i 0.707107 + 0.707107i 0 0
1457.2 0.707107 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i −0.707107 0.707107i 0 0
2843.1 −0.707107 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i 0.707107 0.707107i 0 0
2843.2 0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 0.707107i −0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.m.d yes 4
3.b odd 2 1 3150.2.m.c 4
5.b even 2 1 3150.2.m.f yes 4
5.c odd 4 1 3150.2.m.c 4
5.c odd 4 1 3150.2.m.e yes 4
15.d odd 2 1 3150.2.m.e yes 4
15.e even 4 1 inner 3150.2.m.d yes 4
15.e even 4 1 3150.2.m.f yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.2.m.c 4 3.b odd 2 1
3150.2.m.c 4 5.c odd 4 1
3150.2.m.d yes 4 1.a even 1 1 trivial
3150.2.m.d yes 4 15.e even 4 1 inner
3150.2.m.e yes 4 5.c odd 4 1
3150.2.m.e yes 4 15.d odd 2 1
3150.2.m.f yes 4 5.b even 2 1
3150.2.m.f yes 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3150, [\chi])$$:

 $$T_{11}^{4} + 24 T_{11}^{2} + 16$$ $$T_{13}^{4} + 8 T_{13}^{3} + 32 T_{13}^{2} + 32 T_{13} + 16$$ $$T_{17}^{4} - 12 T_{17}^{3} + 72 T_{17}^{2} - 168 T_{17} + 196$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ 1
$5$ 1
$7$ $$1 + T^{4}$$
$11$ $$1 - 20 T^{2} + 214 T^{4} - 2420 T^{6} + 14641 T^{8}$$
$13$ $$1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 1768 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8}$$
$17$ $$1 - 12 T + 72 T^{2} - 372 T^{3} + 1726 T^{4} - 6324 T^{5} + 20808 T^{6} - 58956 T^{7} + 83521 T^{8}$$
$19$ $$( 1 + 12 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 - 8 T + 32 T^{2} - 120 T^{3} + 386 T^{4} - 2760 T^{5} + 16928 T^{6} - 97336 T^{7} + 279841 T^{8}$$
$29$ $$( 1 + 12 T + 86 T^{2} + 348 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 12 T + 90 T^{2} + 372 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 - 4 T + 8 T^{2} + 244 T^{3} - 2162 T^{4} + 9028 T^{5} + 10952 T^{6} - 202612 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 140 T^{2} + 8134 T^{4} - 235340 T^{6} + 2825761 T^{8}$$
$43$ $$1 - 20 T + 200 T^{2} - 1780 T^{3} + 13726 T^{4} - 76540 T^{5} + 369800 T^{6} - 1590140 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 20 T + 200 T^{2} - 1860 T^{3} + 15182 T^{4} - 87420 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 8 T + 32 T^{2} + 360 T^{3} + 3986 T^{4} + 19080 T^{5} + 89888 T^{6} + 1191016 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 + 8 T + 59 T^{2} )^{4}$$
$61$ $$( 1 - 16 T + 184 T^{2} - 976 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 4 T + 8 T^{2} + 260 T^{3} + 8446 T^{4} + 17420 T^{5} + 35912 T^{6} + 1203052 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 120 T^{2} + 9074 T^{4} - 604920 T^{6} + 25411681 T^{8}$$
$73$ $$( 1 + 12 T + 72 T^{2} + 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 - 52 T^{2} + 11110 T^{4} - 324532 T^{6} + 38950081 T^{8}$$
$83$ $$1 + 16 T + 128 T^{2} + 816 T^{3} + 4178 T^{4} + 67728 T^{5} + 881792 T^{6} + 9148592 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 4 T + 54 T^{2} + 356 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 16 T + 128 T^{2} + 1488 T^{3} + 17282 T^{4} + 144336 T^{5} + 1204352 T^{6} + 14602768 T^{7} + 88529281 T^{8}$$