# Properties

 Label 300.2.h.a Level $300$ Weight $2$ Character orbit 300.h Analytic conductor $2.396$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.342102016.5 Defining polynomial: $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{4} q^{3} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{4} + ( -1 + \beta_{2} - \beta_{6} ) q^{6} + ( -\beta_{5} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} ) q^{8} + ( -\beta_{1} - \beta_{3} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{4} q^{3} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{4} + ( -1 + \beta_{2} - \beta_{6} ) q^{6} + ( -\beta_{5} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} ) q^{8} + ( -\beta_{1} - \beta_{3} - \beta_{6} ) q^{9} + ( -\beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{11} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{12} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{14} + ( -2 - 3 \beta_{1} - \beta_{5} + \beta_{6} ) q^{16} + 2 q^{17} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{18} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{21} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{22} + ( -4 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{23} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{24} + ( -2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{26} + ( -2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{27} + ( -\beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{28} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{29} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{31} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{32} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{33} + 2 \beta_{1} q^{34} + ( 2 - 3 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{36} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} ) q^{38} + ( 2 \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{39} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{41} + ( 2 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{42} + ( -\beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{43} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{44} + ( 2 + 4 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{46} + ( \beta_{2} - \beta_{4} ) q^{47} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{48} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{49} -2 \beta_{4} q^{51} + ( -2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{52} + ( 6 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{53} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{54} + ( -\beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{56} + ( -2 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{57} + ( -\beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{58} + ( 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} ) q^{59} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{61} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{62} + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{63} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{64} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{66} + ( \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{68} + ( -1 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{69} + ( 2 \beta_{2} + 2 \beta_{4} ) q^{71} + ( 2 + 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{72} + ( -2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{73} + ( -2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{74} + ( -4 - 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} ) q^{76} -4 q^{77} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} ) q^{78} + ( 6 \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{79} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{81} + ( -2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{82} + ( -5 \beta_{2} + 5 \beta_{4} ) q^{83} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 4 \beta_{5} - 2 \beta_{7} ) q^{84} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{86} + ( 4 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{87} + ( 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{88} + ( 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( -4 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{91} + ( -4 - 2 \beta_{1} - 4 \beta_{5} + 4 \beta_{6} ) q^{92} + ( -4 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{93} + ( -2 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{94} + ( 6 + \beta_{1} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{96} + ( -3 \beta_{6} - 3 \beta_{7} ) q^{97} + ( 4 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{98} + ( -4 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} + 2q^{4} - 6q^{6} - 14q^{8} + 4q^{9} + O(q^{10})$$ $$8q - 2q^{2} + 2q^{4} - 6q^{6} - 14q^{8} + 4q^{9} - 14q^{12} - 14q^{16} + 16q^{17} - 18q^{18} + 4q^{21} + 2q^{24} + 18q^{32} + 24q^{33} - 4q^{34} + 18q^{36} + 4q^{38} + 16q^{42} + 20q^{46} + 10q^{48} - 16q^{49} + 32q^{53} + 10q^{54} - 16q^{57} - 8q^{61} - 28q^{62} + 2q^{64} - 40q^{66} + 4q^{68} - 12q^{69} + 10q^{72} - 36q^{76} - 32q^{77} + 8q^{78} - 16q^{84} - 44q^{92} - 24q^{93} - 12q^{94} + 42q^{96} + 38q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} - 2 \nu^{2} - 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - \nu^{5} - 3 \nu^{4} + 6 \nu^{3} - 10 \nu^{2} - 8 \nu - 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} + 24 \nu + 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} - 8 \nu + 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{6} + 4 \nu^{5} + \nu^{4} + 4 \nu^{3} + 6 \nu^{2} + 8$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{6} + 4 \nu^{5} - \nu^{4} + 4 \nu^{3} - 6 \nu^{2} - 8$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{7} + 3 \nu^{6} - 2 \nu^{5} + \nu^{4} + 6 \nu^{2} + 8$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} - \beta_{5} - 3 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - \beta_{1} - 2$$ $$\nu^{7}$$ $$=$$ $$($$$$-14 \beta_{7} - 10 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 −0.599676 + 1.28078i 0.599676 − 1.28078i −0.599676 − 1.28078i 0.599676 + 1.28078i −1.17915 − 0.780776i 1.17915 + 0.780776i −1.17915 + 0.780776i 1.17915 − 0.780776i
−1.28078 0.599676i −1.66757 + 0.468213i 1.28078 + 1.53610i 0 2.41656 + 0.400324i 0.936426 −0.719224 2.73546i 2.56155 1.56155i 0
299.2 −1.28078 0.599676i 1.66757 + 0.468213i 1.28078 + 1.53610i 0 −1.85500 1.59968i −0.936426 −0.719224 2.73546i 2.56155 + 1.56155i 0
299.3 −1.28078 + 0.599676i −1.66757 0.468213i 1.28078 1.53610i 0 2.41656 0.400324i 0.936426 −0.719224 + 2.73546i 2.56155 + 1.56155i 0
299.4 −1.28078 + 0.599676i 1.66757 0.468213i 1.28078 1.53610i 0 −1.85500 + 1.59968i −0.936426 −0.719224 + 2.73546i 2.56155 1.56155i 0
299.5 0.780776 1.17915i −0.848071 1.51022i −0.780776 1.84130i 0 −2.44293 0.179147i −3.02045 −2.78078 0.516994i −1.56155 + 2.56155i 0
299.6 0.780776 1.17915i 0.848071 1.51022i −0.780776 1.84130i 0 −1.11862 2.17915i 3.02045 −2.78078 0.516994i −1.56155 2.56155i 0
299.7 0.780776 + 1.17915i −0.848071 + 1.51022i −0.780776 + 1.84130i 0 −2.44293 + 0.179147i −3.02045 −2.78078 + 0.516994i −1.56155 2.56155i 0
299.8 0.780776 + 1.17915i 0.848071 + 1.51022i −0.780776 + 1.84130i 0 −1.11862 + 2.17915i 3.02045 −2.78078 + 0.516994i −1.56155 + 2.56155i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 299.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.h.a 8
3.b odd 2 1 300.2.h.b 8
4.b odd 2 1 inner 300.2.h.a 8
5.b even 2 1 300.2.h.b 8
5.c odd 4 1 60.2.e.a 8
5.c odd 4 1 300.2.e.c 8
12.b even 2 1 300.2.h.b 8
15.d odd 2 1 inner 300.2.h.a 8
15.e even 4 1 60.2.e.a 8
15.e even 4 1 300.2.e.c 8
20.d odd 2 1 300.2.h.b 8
20.e even 4 1 60.2.e.a 8
20.e even 4 1 300.2.e.c 8
40.i odd 4 1 960.2.h.g 8
40.k even 4 1 960.2.h.g 8
60.h even 2 1 inner 300.2.h.a 8
60.l odd 4 1 60.2.e.a 8
60.l odd 4 1 300.2.e.c 8
120.q odd 4 1 960.2.h.g 8
120.w even 4 1 960.2.h.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.e.a 8 5.c odd 4 1
60.2.e.a 8 15.e even 4 1
60.2.e.a 8 20.e even 4 1
60.2.e.a 8 60.l odd 4 1
300.2.e.c 8 5.c odd 4 1
300.2.e.c 8 15.e even 4 1
300.2.e.c 8 20.e even 4 1
300.2.e.c 8 60.l odd 4 1
300.2.h.a 8 1.a even 1 1 trivial
300.2.h.a 8 4.b odd 2 1 inner
300.2.h.a 8 15.d odd 2 1 inner
300.2.h.a 8 60.h even 2 1 inner
300.2.h.b 8 3.b odd 2 1
300.2.h.b 8 5.b even 2 1
300.2.h.b 8 12.b even 2 1
300.2.h.b 8 20.d odd 2 1
960.2.h.g 8 40.i odd 4 1
960.2.h.g 8 40.k even 4 1
960.2.h.g 8 120.q odd 4 1
960.2.h.g 8 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{4} - 10 T_{7}^{2} + 8$$ $$T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{3} + T^{4} )^{2}$$
$3$ $$81 - 18 T^{2} + 2 T^{4} - 2 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 8 - 10 T^{2} + T^{4} )^{2}$$
$11$ $$( 32 - 20 T^{2} + T^{4} )^{2}$$
$13$ $$( 256 + 36 T^{2} + T^{4} )^{2}$$
$17$ $$( -2 + T )^{8}$$
$19$ $$( 32 + 20 T^{2} + T^{4} )^{2}$$
$23$ $$( 8 + 58 T^{2} + T^{4} )^{2}$$
$29$ $$( 256 + 36 T^{2} + T^{4} )^{2}$$
$31$ $$( 128 + 28 T^{2} + T^{4} )^{2}$$
$37$ $$( 256 + 36 T^{2} + T^{4} )^{2}$$
$41$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$43$ $$( 128 - 62 T^{2} + T^{4} )^{2}$$
$47$ $$( 8 + 10 T^{2} + T^{4} )^{2}$$
$53$ $$( -52 - 8 T + T^{2} )^{4}$$
$59$ $$( 10368 - 252 T^{2} + T^{4} )^{2}$$
$61$ $$( -16 + 2 T + T^{2} )^{4}$$
$67$ $$( 512 - 46 T^{2} + T^{4} )^{2}$$
$71$ $$( 512 - 56 T^{2} + T^{4} )^{2}$$
$73$ $$( 68 + T^{2} )^{4}$$
$79$ $$( 5408 + 148 T^{2} + T^{4} )^{2}$$
$83$ $$( 5000 + 250 T^{2} + T^{4} )^{2}$$
$89$ $$( 4096 + 144 T^{2} + T^{4} )^{2}$$
$97$ $$( 36 + T^{2} )^{4}$$