# Properties

 Label 300.2.e.c Level $300$ Weight $2$ Character orbit 300.e 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.e (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} + 8 \nu^{3} - 10 \nu^{2} - 8 \nu - 8$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 3 \nu^{5} - 3 \nu^{4} + 2 \nu^{3} + 6 \nu^{2} - 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{5} - 3 \nu^{4} + 4 \nu^{3} - 10 \nu^{2} + 16 \nu - 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 4 \nu^{5} - 3 \nu^{4} - 4 \nu^{3} - 10 \nu^{2} - 16 \nu - 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} - 4 \nu^{3} - 4 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} - 3 \nu^{5} - 3 \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 8$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} + 2 \nu^{2}$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{7} - 3 \beta_{6} - \beta_{4} - \beta_{3} - 3 \beta_{2} - 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + \beta_{3} + 5 \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-12 \beta_{7} - \beta_{6} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 4$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{6} - 13 \beta_{5} + 10 \beta_{4} - \beta_{3} + 3 \beta_{2} - 9 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −1.17915 + 0.780776i −1.17915 − 0.780776i −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 1.51022 0.848071i 0.780776 + 1.84130i 0 −2.44293 0.179147i 3.02045i 0.516994 2.78078i 1.56155 2.56155i 0
251.2 −1.17915 + 0.780776i 1.51022 + 0.848071i 0.780776 1.84130i 0 −2.44293 + 0.179147i 3.02045i 0.516994 + 2.78078i 1.56155 + 2.56155i 0
251.3 −0.599676 1.28078i −0.468213 + 1.66757i −1.28078 + 1.53610i 0 2.41656 0.400324i 0.936426i 2.73546 + 0.719224i −2.56155 1.56155i 0
251.4 −0.599676 + 1.28078i −0.468213 1.66757i −1.28078 1.53610i 0 2.41656 + 0.400324i 0.936426i 2.73546 0.719224i −2.56155 + 1.56155i 0
251.5 0.599676 1.28078i 0.468213 1.66757i −1.28078 1.53610i 0 −1.85500 1.59968i 0.936426i −2.73546 + 0.719224i −2.56155 1.56155i 0
251.6 0.599676 + 1.28078i 0.468213 + 1.66757i −1.28078 + 1.53610i 0 −1.85500 + 1.59968i 0.936426i −2.73546 0.719224i −2.56155 + 1.56155i 0
251.7 1.17915 0.780776i −1.51022 + 0.848071i 0.780776 1.84130i 0 −1.11862 + 2.17915i 3.02045i −0.516994 2.78078i 1.56155 2.56155i 0
251.8 1.17915 + 0.780776i −1.51022 0.848071i 0.780776 + 1.84130i 0 −1.11862 2.17915i 3.02045i −0.516994 + 2.78078i 1.56155 + 2.56155i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b 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.e.c 8
3.b odd 2 1 inner 300.2.e.c 8
4.b odd 2 1 inner 300.2.e.c 8
5.b even 2 1 60.2.e.a 8
5.c odd 4 1 300.2.h.a 8
5.c odd 4 1 300.2.h.b 8
12.b even 2 1 inner 300.2.e.c 8
15.d odd 2 1 60.2.e.a 8
15.e even 4 1 300.2.h.a 8
15.e even 4 1 300.2.h.b 8
20.d odd 2 1 60.2.e.a 8
20.e even 4 1 300.2.h.a 8
20.e even 4 1 300.2.h.b 8
40.e odd 2 1 960.2.h.g 8
40.f even 2 1 960.2.h.g 8
60.h even 2 1 60.2.e.a 8
60.l odd 4 1 300.2.h.a 8
60.l odd 4 1 300.2.h.b 8
120.i odd 2 1 960.2.h.g 8
120.m even 2 1 960.2.h.g 8

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

## Hecke kernels

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

 $$T_{7}^{4} + 10 T_{7}^{2} + 8$$ $$T_{13}^{2} - 2 T_{13} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 4 T^{2} + 4 T^{4} + T^{6} + T^{8}$$
$3$ $$81 + 18 T^{2} + 2 T^{4} + 2 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 8 + 10 T^{2} + T^{4} )^{2}$$
$11$ $$( 32 - 20 T^{2} + T^{4} )^{2}$$
$13$ $$( -16 - 2 T + T^{2} )^{4}$$
$17$ $$( 4 + T^{2} )^{4}$$
$19$ $$( 32 + 20 T^{2} + T^{4} )^{2}$$
$23$ $$( 8 - 58 T^{2} + T^{4} )^{2}$$
$29$ $$( 256 + 36 T^{2} + T^{4} )^{2}$$
$31$ $$( 128 + 28 T^{2} + T^{4} )^{2}$$
$37$ $$( -16 + 2 T + T^{2} )^{4}$$
$41$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$43$ $$( 128 + 62 T^{2} + T^{4} )^{2}$$
$47$ $$( 8 - 10 T^{2} + T^{4} )^{2}$$
$53$ $$( 2704 + 168 T^{2} + T^{4} )^{2}$$
$59$ $$( 10368 - 252 T^{2} + T^{4} )^{2}$$
$61$ $$( -16 + 2 T + T^{2} )^{4}$$
$67$ $$( 512 + 46 T^{2} + T^{4} )^{2}$$
$71$ $$( 512 - 56 T^{2} + T^{4} )^{2}$$
$73$ $$( -68 + T^{2} )^{4}$$
$79$ $$( 5408 + 148 T^{2} + T^{4} )^{2}$$
$83$ $$( 5000 - 250 T^{2} + T^{4} )^{2}$$
$89$ $$( 4096 + 144 T^{2} + T^{4} )^{2}$$
$97$ $$( -6 + T )^{8}$$