Properties

 Label 605.2.g.f Level $605$ Weight $2$ Character orbit 605.g Analytic conductor $4.831$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.g (of order $$5$$, degree $$4$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.64000000.2 Defining polynomial: $$x^{8} + 2 x^{6} + 4 x^{4} + 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{2} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{3} + ( 2 \beta_{1} + \beta_{6} ) q^{4} -\beta_{2} q^{5} + ( -4 \beta_{2} - 2 \beta_{7} ) q^{6} -2 \beta_{6} q^{7} + ( -\beta_{1} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{8} + ( -5 - 5 \beta_{2} - 5 \beta_{4} - 5 \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{2} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{3} + ( 2 \beta_{1} + \beta_{6} ) q^{4} -\beta_{2} q^{5} + ( -4 \beta_{2} - 2 \beta_{7} ) q^{6} -2 \beta_{6} q^{7} + ( -\beta_{1} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{8} + ( -5 - 5 \beta_{2} - 5 \beta_{4} - 5 \beta_{6} ) q^{9} + ( -1 - \beta_{5} ) q^{10} + ( -8 - 2 \beta_{5} ) q^{12} + ( 4 + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} ) q^{13} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{14} + 2 \beta_{1} q^{15} + 3 \beta_{2} q^{16} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{17} + ( 5 \beta_{1} + 5 \beta_{6} ) q^{18} + ( 1 + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{20} + 4 \beta_{5} q^{21} -2 \beta_{5} q^{23} + ( 4 + 4 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} ) q^{24} + \beta_{4} q^{25} -2 \beta_{1} q^{26} -4 \beta_{7} q^{27} + ( -2 \beta_{2} - 4 \beta_{7} ) q^{28} + ( -4 \beta_{1} + 2 \beta_{6} ) q^{29} + ( -2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{30} + ( -3 + \beta_{5} ) q^{32} + ( 8 + 6 \beta_{5} ) q^{34} + ( -2 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{35} + ( -10 \beta_{1} - 10 \beta_{3} + 5 \beta_{4} - 10 \beta_{5} - 10 \beta_{7} ) q^{36} + ( -4 \beta_{1} - 2 \beta_{6} ) q^{37} + ( -8 \beta_{2} + 8 \beta_{7} ) q^{39} + ( -\beta_{1} - 3 \beta_{6} ) q^{40} + 6 \beta_{4} q^{41} + ( -8 - 8 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 8 \beta_{6} ) q^{42} -6 q^{43} -5 q^{45} + ( 4 + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{47} -6 \beta_{1} q^{48} -3 \beta_{2} q^{49} + ( \beta_{2} + \beta_{7} ) q^{50} + ( -8 \beta_{1} - 8 \beta_{6} ) q^{51} + ( 6 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} ) q^{52} + ( -6 - 6 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - 6 \beta_{6} ) q^{53} + ( -8 - 4 \beta_{5} ) q^{54} + ( -6 - 2 \beta_{5} ) q^{56} + ( 2 \beta_{1} + 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{58} + ( 4 \beta_{1} - 4 \beta_{6} ) q^{59} + ( 8 \beta_{2} + 2 \beta_{7} ) q^{60} + ( 2 \beta_{2} - 8 \beta_{7} ) q^{61} -10 \beta_{4} q^{63} + ( 7 + 7 \beta_{2} - 2 \beta_{3} + 7 \beta_{4} + 7 \beta_{6} ) q^{64} + ( 4 - 2 \beta_{5} ) q^{65} + ( 4 + 6 \beta_{5} ) q^{67} + ( -12 - 12 \beta_{2} + 10 \beta_{3} - 12 \beta_{4} - 12 \beta_{6} ) q^{68} + 8 \beta_{4} q^{69} + ( 2 \beta_{1} + 2 \beta_{6} ) q^{70} + 8 \beta_{7} q^{71} + ( 15 \beta_{2} + 5 \beta_{7} ) q^{72} + ( 2 \beta_{1} - 4 \beta_{6} ) q^{73} + ( 6 \beta_{1} + 6 \beta_{3} - 10 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} ) q^{74} -2 \beta_{3} q^{75} + 8 q^{78} + ( -4 - 4 \beta_{2} - 4 \beta_{4} - 4 \beta_{6} ) q^{79} -3 \beta_{4} q^{80} + \beta_{6} q^{81} + ( 6 \beta_{2} + 6 \beta_{7} ) q^{82} -6 \beta_{2} q^{83} + ( 4 \beta_{1} + 16 \beta_{6} ) q^{84} + ( 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{85} + ( 6 + 6 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 6 \beta_{6} ) q^{86} + ( 16 - 4 \beta_{5} ) q^{87} + ( -2 - 8 \beta_{5} ) q^{89} + ( 5 + 5 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 5 \beta_{6} ) q^{90} + ( 4 \beta_{1} + 4 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{91} + ( -2 \beta_{1} - 8 \beta_{6} ) q^{92} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{94} + ( -6 \beta_{1} - 6 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} - 6 \beta_{7} ) q^{96} + ( 2 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{97} + ( -3 - 3 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} - 2q^{4} + 2q^{5} + 8q^{6} + 4q^{7} - 6q^{8} - 10q^{9} + O(q^{10})$$ $$8q - 2q^{2} - 2q^{4} + 2q^{5} + 8q^{6} + 4q^{7} - 6q^{8} - 10q^{9} - 8q^{10} - 64q^{12} + 8q^{13} + 4q^{14} - 6q^{16} - 8q^{17} - 10q^{18} + 2q^{20} + 8q^{24} - 2q^{25} + 4q^{28} - 4q^{29} - 8q^{30} - 24q^{32} + 64q^{34} - 4q^{35} - 10q^{36} + 4q^{37} + 16q^{39} + 6q^{40} - 12q^{41} - 16q^{42} - 48q^{43} - 40q^{45} + 8q^{46} + 6q^{49} - 2q^{50} + 16q^{51} - 8q^{52} - 12q^{53} - 64q^{54} - 48q^{56} + 12q^{58} + 8q^{59} - 16q^{60} - 4q^{61} + 20q^{63} + 14q^{64} + 32q^{65} + 32q^{67} - 24q^{68} - 16q^{69} - 4q^{70} - 30q^{72} + 8q^{73} + 20q^{74} + 64q^{78} - 8q^{79} + 6q^{80} - 2q^{81} - 12q^{82} + 12q^{83} - 32q^{84} + 8q^{85} + 12q^{86} + 128q^{87} - 16q^{89} + 10q^{90} - 16q^{91} + 16q^{92} - 8q^{94} + 8q^{96} + 4q^{97} - 24q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/4$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{7}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$1$$ $$\beta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.437016 + 1.34500i −0.437016 − 1.34500i 1.14412 + 0.831254i −1.14412 − 0.831254i 0.437016 − 1.34500i −0.437016 + 1.34500i 1.14412 − 0.831254i −1.14412 + 0.831254i
−1.95314 1.41904i −0.874032 + 2.68999i 1.18305 + 3.64105i 0.809017 0.587785i 5.52431 4.01365i −0.618034 1.90211i 1.36407 4.19817i −4.04508 2.93893i −2.41421
81.2 0.335106 + 0.243469i 0.874032 2.68999i −0.565015 1.73894i 0.809017 0.587785i 0.947822 0.688633i −0.618034 1.90211i 0.490035 1.50817i −4.04508 2.93893i 0.414214
251.1 −0.127999 + 0.393941i −2.28825 + 1.66251i 1.47923 + 1.07472i −0.309017 0.951057i −0.362036 1.11423i 1.61803 + 1.17557i −1.28293 + 0.932102i 1.54508 4.75528i 0.414214
251.2 0.746033 2.29605i 2.28825 1.66251i −3.09726 2.25029i −0.309017 0.951057i −2.11010 6.49422i 1.61803 + 1.17557i −3.57117 + 2.59461i 1.54508 4.75528i −2.41421
366.1 −1.95314 + 1.41904i −0.874032 2.68999i 1.18305 3.64105i 0.809017 + 0.587785i 5.52431 + 4.01365i −0.618034 + 1.90211i 1.36407 + 4.19817i −4.04508 + 2.93893i −2.41421
366.2 0.335106 0.243469i 0.874032 + 2.68999i −0.565015 + 1.73894i 0.809017 + 0.587785i 0.947822 + 0.688633i −0.618034 + 1.90211i 0.490035 + 1.50817i −4.04508 + 2.93893i 0.414214
511.1 −0.127999 0.393941i −2.28825 1.66251i 1.47923 1.07472i −0.309017 + 0.951057i −0.362036 + 1.11423i 1.61803 1.17557i −1.28293 0.932102i 1.54508 + 4.75528i 0.414214
511.2 0.746033 + 2.29605i 2.28825 + 1.66251i −3.09726 + 2.25029i −0.309017 + 0.951057i −2.11010 + 6.49422i 1.61803 1.17557i −3.57117 2.59461i 1.54508 + 4.75528i −2.41421
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 511.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.f 8
11.b odd 2 1 605.2.g.l 8
11.c even 5 1 55.2.a.b 2
11.c even 5 3 inner 605.2.g.f 8
11.d odd 10 1 605.2.a.d 2
11.d odd 10 3 605.2.g.l 8
33.f even 10 1 5445.2.a.y 2
33.h odd 10 1 495.2.a.b 2
44.g even 10 1 9680.2.a.bn 2
44.h odd 10 1 880.2.a.m 2
55.h odd 10 1 3025.2.a.o 2
55.j even 10 1 275.2.a.c 2
55.k odd 20 2 275.2.b.d 4
77.j odd 10 1 2695.2.a.f 2
88.l odd 10 1 3520.2.a.bo 2
88.o even 10 1 3520.2.a.bn 2
132.o even 10 1 7920.2.a.ch 2
143.n even 10 1 9295.2.a.g 2
165.o odd 10 1 2475.2.a.x 2
165.v even 20 2 2475.2.c.l 4
220.n odd 10 1 4400.2.a.bn 2
220.v even 20 2 4400.2.b.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 11.c even 5 1
275.2.a.c 2 55.j even 10 1
275.2.b.d 4 55.k odd 20 2
495.2.a.b 2 33.h odd 10 1
605.2.a.d 2 11.d odd 10 1
605.2.g.f 8 1.a even 1 1 trivial
605.2.g.f 8 11.c even 5 3 inner
605.2.g.l 8 11.b odd 2 1
605.2.g.l 8 11.d odd 10 3
880.2.a.m 2 44.h odd 10 1
2475.2.a.x 2 165.o odd 10 1
2475.2.c.l 4 165.v even 20 2
2695.2.a.f 2 77.j odd 10 1
3025.2.a.o 2 55.h odd 10 1
3520.2.a.bn 2 88.o even 10 1
3520.2.a.bo 2 88.l odd 10 1
4400.2.a.bn 2 220.n odd 10 1
4400.2.b.q 4 220.v even 20 2
5445.2.a.y 2 33.f even 10 1
7920.2.a.ch 2 132.o even 10 1
9295.2.a.g 2 143.n even 10 1
9680.2.a.bn 2 44.g even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{8} + \cdots$$ $$T_{3}^{8} + 8 T_{3}^{6} + 64 T_{3}^{4} + 512 T_{3}^{2} + 4096$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 5 T^{2} - 12 T^{3} + 29 T^{4} + 12 T^{5} + 5 T^{6} + 2 T^{7} + T^{8}$$
$3$ $$4096 + 512 T^{2} + 64 T^{4} + 8 T^{6} + T^{8}$$
$5$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$7$ $$( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$4096 - 4096 T + 3584 T^{2} - 3072 T^{3} + 2624 T^{4} - 384 T^{5} + 56 T^{6} - 8 T^{7} + T^{8}$$
$17$ $$4096 + 4096 T + 3584 T^{2} + 3072 T^{3} + 2624 T^{4} + 384 T^{5} + 56 T^{6} + 8 T^{7} + T^{8}$$
$19$ $$T^{8}$$
$23$ $$( -8 + T^{2} )^{4}$$
$29$ $$614656 - 87808 T + 34496 T^{2} - 8064 T^{3} + 2384 T^{4} + 288 T^{5} + 44 T^{6} + 4 T^{7} + T^{8}$$
$31$ $$T^{8}$$
$37$ $$614656 + 87808 T + 34496 T^{2} + 8064 T^{3} + 2384 T^{4} - 288 T^{5} + 44 T^{6} - 4 T^{7} + T^{8}$$
$41$ $$( 1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$43$ $$( 6 + T )^{8}$$
$47$ $$4096 + 512 T^{2} + 64 T^{4} + 8 T^{6} + T^{8}$$
$53$ $$256 + 768 T + 2240 T^{2} + 6528 T^{3} + 19024 T^{4} + 1632 T^{5} + 140 T^{6} + 12 T^{7} + T^{8}$$
$59$ $$65536 + 32768 T + 20480 T^{2} + 12288 T^{3} + 7424 T^{4} - 768 T^{5} + 80 T^{6} - 8 T^{7} + T^{8}$$
$61$ $$236421376 - 7626496 T + 2152640 T^{2} - 130944 T^{3} + 21584 T^{4} + 1056 T^{5} + 140 T^{6} + 4 T^{7} + T^{8}$$
$67$ $$( -56 - 8 T + T^{2} )^{4}$$
$71$ $$268435456 + 2097152 T^{2} + 16384 T^{4} + 128 T^{6} + T^{8}$$
$73$ $$4096 - 4096 T + 3584 T^{2} - 3072 T^{3} + 2624 T^{4} - 384 T^{5} + 56 T^{6} - 8 T^{7} + T^{8}$$
$79$ $$( 256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$83$ $$( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$89$ $$( -124 + 4 T + T^{2} )^{4}$$
$97$ $$614656 + 87808 T + 34496 T^{2} + 8064 T^{3} + 2384 T^{4} - 288 T^{5} + 44 T^{6} - 4 T^{7} + T^{8}$$