LMFDB - Newform orbit 775.2.j.a

Newspace parameters

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.j (of order $5$, degree $4$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.18840615665$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$1$	$-\zeta_{10}^{3}$

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} $

156.1

−0.309017	+	0.951057i
0.809017	+	0.587785i
0.809017	−	0.587785i
−0.309017	−	0.951057i

0.618034

1.90211i

−2.61803

1.90211i

−1.61803

1.17557i

−0.690983

2.12663i

−5.23607

−

3.80423i

−0.236068

2.30902

−

7.10642i

−4.47214

311.1

−1.61803

1.17557i

−0.381966

1.17557i

0.618034

−

1.90211i

−1.80902

−

1.31433i

−0.763932

−

2.35114i

4.23607

1.19098

0.865300i

4.47214

466.1

−1.61803

−

1.17557i

−0.381966

−

1.17557i

0.618034

1.90211i

−1.80902

1.31433i

−0.763932

2.35114i

4.23607

1.19098

−

0.865300i

4.47214

621.1

0.618034

−

1.90211i

−2.61803

−

1.90211i

−1.61803

−

1.17557i

−0.690983

−

2.12663i

−5.23607

3.80423i

−0.236068

2.30902

7.10642i

−4.47214

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	775.2.j.a	✓	4
25.d	even	5	1	inner	775.2.j.a	✓	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
775.2.j.a	✓	4	1.a	even	1	1	trivial
775.2.j.a	✓	4	25.d	even	5	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 8T_{2} + 16 $ T2^4 + 2*T2^3 + 4*T2^2 + 8*T2 + 16 acting on $S_{2}^{\mathrm{new}}(775, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 $ T^4 + 2T^3 + 4T^2 + 8*T + 16
$3$	$ T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 $ T^4 + 6T^3 + 16T^2 + 16*T + 16
$5$	$ T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25 $ T^4 + 5T^3 + 15T^2 + 25*T + 25
$7$	$ (T^{2} - 4 T - 1)^{2} $ (T^2 - 4*T - 1)^2
$11$	$ T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121 $ T^4 + 7T^3 + 34T^2 + 88*T + 121
$13$	$ T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 $ T^4 + 6T^3 + 16T^2 + 11*T + 121
$17$	$ T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1 $ T^4 + 7T^3 + 19T^2 + 3*T + 1
$19$	$ T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 $ T^4 + 10T^3 + 60T^2 + 200*T + 400
$23$	$ T^{4} + 11 T^{3} + 51 T^{2} + \cdots + 961 $ T^4 + 11T^3 + 51T^2 + 31*T + 961
$29$	$ T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25 $ T^4 + 5T^3 + 15T^2 + 25*T + 25
$31$	$ T^{4} + T^{3} + T^{2} + T + 1 $ T^4 + T^3 + T^2 + T + 1
$37$	$ T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 $ T^4 + 17T^3 + 114T^2 + 88*T + 121
$41$	$ T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 $ T^4 + 17T^3 + 114T^2 + 88*T + 121
$43$	$ (T^{2} - 2 T - 19)^{2} $ (T^2 - 2*T - 19)^2
$47$	$ T^{4} + 22 T^{3} + 244 T^{2} + \cdots + 5776 $ T^4 + 22T^3 + 244T^2 + 1368*T + 5776
$53$	$ T^{4} - 24 T^{3} + 306 T^{2} + \cdots + 9801 $ T^4 - 24T^3 + 306T^2 - 2079*T + 9801
$59$	$ T^{4} - 15 T^{3} + 115 T^{2} + \cdots + 9025 $ T^4 - 15T^3 + 115T^2 - 475*T + 9025
$61$	$ T^{4} + 7 T^{3} + 199 T^{2} + \cdots + 361 $ T^4 + 7T^3 + 199T^2 - 437*T + 361
$67$	$ T^{4} + 22 T^{3} + 304 T^{2} + \cdots + 10201 $ T^4 + 22T^3 + 304T^2 + 2323*T + 10201
$71$	$ T^{4} + 2 T^{3} + 24 T^{2} + 133 T + 361 $ T^4 + 2T^3 + 24T^2 + 133*T + 361
$73$	$ T^{4} + 16 T^{3} + 106 T^{2} + \cdots + 3721 $ T^4 + 16T^3 + 106T^2 + 61*T + 3721
$79$	$ T^{4} + 15 T^{3} + 100 T^{2} + \cdots + 625 $ T^4 + 15T^3 + 100T^2 + 250*T + 625
$83$	$ T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 $ T^4 + 6T^3 + 16T^2 + 11*T + 121
$89$	$ T^{4} - 25 T^{3} + 235 T^{2} + \cdots + 3025 $ T^4 - 25T^3 + 235T^2 + 275*T + 3025
$97$	$ T^{4} + 12 T^{3} + 94 T^{2} + \cdots + 7921 $ T^4 + 12T^3 + 94T^2 + 623*T + 7921
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.j (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{2} + (2 \zeta_{10} - 2) q^{3} - 2 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{6} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3) q^{7} + (4 \zeta_{10}^{2} - 5 \zeta_{10} + 4) q^{9} +O(q^{10}) \) q + (2z^3 - 2z^2 + 2z - 2) q^2 + (2z - 2) q^3 - 2z^3 q^4 + (-2z^2 + z - 2) q^5 + (-4z^3 + 4z^2 - 4z) q^6 + (-2z^3 + 2z^2 + 3) * q^7 + (4z^2 - 5z + 4) * q^9	\( q + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{2} + (2 \zeta_{10} - 2) q^{3} - 2 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{6} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3) q^{7} + (4 \zeta_{10}^{2} - 5 \zeta_{10} + 4) q^{9} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2) q^{10} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{11} + (4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{12} + ( - \zeta_{10}^{2} - 3 \zeta_{10} - 1) q^{13} + (6 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 6) q^{14} + ( - 4 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{15} + 4 \zeta_{10} q^{16} + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 2 \zeta_{10}) q^{17} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 2) q^{18} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{19} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{20} + (4 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{21} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{22} + (7 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 7) q^{23} + 5 \zeta_{10}^{2} q^{25} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 8) q^{26} + (8 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 8) q^{27} + ( - 6 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{28} + (\zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{29} + (8 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{30} + \zeta_{10}^{2} q^{31} - 8 q^{32} + (4 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 4 \zeta_{10}) q^{33} + (4 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{34} + ( - 4 \zeta_{10}^{2} - 3 \zeta_{10} - 4) q^{35} + (2 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 2) q^{36} + ( - 5 \zeta_{10}^{2} - 2 \zeta_{10} - 5) q^{37} + (8 \zeta_{10}^{2} - 4 \zeta_{10} + 8) q^{38} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{39} + ( - 5 \zeta_{10}^{2} - 2 \zeta_{10} - 5) q^{41} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{42} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 1) q^{43} + (6 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 6 \zeta_{10}) q^{44} + (6 \zeta_{10}^{3} - 13 \zeta_{10}^{2} + 6 \zeta_{10}) q^{45} + ( - 14 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{46} + (2 \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{47} + (8 \zeta_{10}^{2} - 8 \zeta_{10}) q^{48} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 6) q^{49} - 10 \zeta_{10} q^{50} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 4) q^{51} + (8 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 8) q^{52} + ( - 3 \zeta_{10}^{3} - 9 \zeta_{10} + 9) q^{53} + ( - 16 \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{54} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 7) q^{55} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 8) q^{57} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{58} + (2 \zeta_{10}^{2} + 9 \zeta_{10} + 2) q^{59} + (4 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 4 \zeta_{10}) q^{60} + ( - 5 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11 \zeta_{10} + 5) q^{61} - 2 \zeta_{10} q^{62} + (2 \zeta_{10}^{2} + 3 \zeta_{10} + 2) q^{63} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{64} + (7 \zeta_{10}^{3} - \zeta_{10}^{2} + 7 \zeta_{10}) q^{65} + ( - 8 \zeta_{10}^{2} + 20 \zeta_{10} - 8) q^{66} + ( - 9 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 9 \zeta_{10}) q^{67} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 2) q^{68} + ( - 10 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 10 \zeta_{10}) q^{69} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 14) q^{70} + ( - 5 \zeta_{10}^{3} - \zeta_{10} + 1) q^{71} + (10 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 10) q^{73} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 14) q^{74} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2}) q^{75} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 4) q^{76} + (9 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 9) q^{77} + (4 \zeta_{10}^{3} + 12 \zeta_{10} - 12) q^{78} + (5 \zeta_{10} - 5) q^{79} + ( - 8 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 8 \zeta_{10}) q^{80} + ( - 12 \zeta_{10}^{3} + 17 \zeta_{10}^{2} - 12 \zeta_{10}) q^{81} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 14) q^{82} + ( - \zeta_{10}^{3} + 4 \zeta_{10}^{2} - \zeta_{10}) q^{83} + (4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{84} + (3 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{85} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 2) q^{86} + (2 \zeta_{10}^{2} - 6 \zeta_{10} + 2) q^{87} + ( - 13 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 13) q^{89} + ( - 12 \zeta_{10}^{2} + 26 \zeta_{10} - 12) q^{90} + ( - 9 \zeta_{10}^{2} - 5 \zeta_{10} - 9) q^{91} + (4 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 4 \zeta_{10}) q^{92} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{93} + ( - 16 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 16 \zeta_{10}) q^{94} + 10 \zeta_{10}^{3} q^{95} + ( - 16 \zeta_{10} + 16) q^{96} + ( - 9 \zeta_{10}^{3} + \zeta_{10} - 1) q^{97} + (12 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 12) q^{98} + ( - 11 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11) q^{99} +O(q^{100}) \) q + (2z^3 - 2z^2 + 2z - 2) q^2 + (2z - 2) q^3 - 2z^3 q^4 + (-2z^2 + z - 2) q^5 + (-4z^3 + 4z^2 - 4z) q^6 + (-2z^3 + 2z^2 + 3) * q^7 + (4z^2 - 5z + 4) * q^9 + (-4z^3 + 4z^2 + 2) * q^10 + (z^3 + 2z^2 - 2z - 1) * q^11 + (4z^2 - 4z + 4) * q^12 + (-z^2 - 3z - 1) q^13 + (6z^3 - 2z^2 + 2z - 6) q^14 + (-4z^3 + 6z^2 - 6z + 4) q^15 + 4z q^16 + (-2z^3 + 3z^2 - 2z) q^17 + (8z^3 - 8z^2 + 2) * q^18 + (-4z^3 + 2z^2 - 4z) q^19 + (2z^3 + 2z^2 - 2z - 2) q^20 + (4z^3 + 2z - 2) * q^21 + (-2z^3 - 6z + 6) * q^22 + (7z^3 - 5z^2 + 5z - 7) q^23 + 5z^2 q^25 + (-2z^3 + 2z^2 + 8) * q^26 + (8z^3 - 12z^2 + 12z - 8) q^27 + (-6z^3 - 4z + 4) * q^28 + (z^3 + 2z - 2) q^29 + (8z^3 - 4z + 4) * q^30 + z^2 * q^31 - 8 * q^32 + (4z^3 - 10z^2 + 4z) q^33 + (4z^2 - 6z + 4) * q^34 + (-4z^2 - 3z - 4) * q^35 + (2z^3 - 10z^2 + 10z - 2) q^36 + (-5z^2 - 2z - 5) * q^37 + (8z^2 - 4z + 8) * q^38 + (-2z^3 - 4z^2 + 4z + 2) q^39 + (-5z^2 - 2z - 5) * q^41 + (-4z^3 - 4z^2 - 4z) q^42 + (4z^3 - 4z^2 - 1) * q^43 + (6z^3 - 4z^2 + 6z) q^44 + (6z^3 - 13z^2 + 6z) q^45 + (-14z^3 - 4z + 4) * q^46 + (2z^3 + 8z - 8) * q^47 + (8z^2 - 8z) * q^48 + (-8z^3 + 8z^2 + 6) * q^49 - 10z q^50 + (6z^3 - 6z^2 + 4) * q^51 + (8z^3 - 6z^2 + 6z - 8) q^52 + (-3z^3 - 9z + 9) * q^53 + (-16z^3 + 8z - 8) * q^54 + (z^3 - z^2 + 7) * q^55 + (4z^3 - 4z^2 + 8) * q^57 + (-4z^3 + 2z^2 - 4z) q^58 + (2z^2 + 9z + 2) * q^59 + (4z^3 - 12z^2 + 4z) q^60 + (-5z^3 + 11z^2 - 11z + 5) q^61 - 2z q^62 + (2z^2 + 3z + 2) * q^63 + (-8z^3 + 8z^2 - 8z + 8) q^64 + (7z^3 - z^2 + 7z) * q^65 + (-8z^2 + 20z - 8) * q^66 + (-9z^3 + 4z^2 - 9z) q^67 + (4z^3 - 4z^2 + 2) * q^68 + (-10z^3 + 6z^2 - 10z) q^69 + (-8z^3 + 8z^2 + 14) * q^70 + (-5z^3 - z + 1) q^71 + (10z^3 - 7z^2 + 7z - 10) q^73 + (-10z^3 + 10z^2 + 14) * q^74 + (10z^3 - 10z^2) * q^75 + (8z^3 - 8z^2 - 4) * q^76 + (9z^3 - 4z^2 + 4z - 9) q^77 + (4z^3 + 12z - 12) * q^78 + (5z - 5) q^79 + (-8z^3 + 4z^2 - 8z) q^80 + (-12z^3 + 17z^2 - 12z) q^81 + (-10z^3 + 10z^2 + 14) * q^82 + (-z^3 + 4z^2 - z) q^83 + (4z^2 + 4z + 4) * q^84 + (3z^3 - 4z + 4) * q^85 + (-2z^3 - 6z^2 + 6z + 2) q^86 + (2z^2 - 6z + 2) * q^87 + (-13z^3 + 7z^2 - 7z + 13) q^89 + (-12z^2 + 26z - 12) * q^90 + (-9z^2 - 5z - 9) * q^91 + (4z^3 + 10z^2 + 4z) q^92 + (2z^3 - 2z^2) * q^93 + (-16z^3 + 12z^2 - 16z) q^94 + 10z^3 q^95 + (-16z + 16) q^96 + (-9z^3 + z - 1) q^97 + (12z^3 + 4z^2 - 4z - 12) q^98 + (-11z^3 + 11z^2 - 11) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 6 q^{3} - 2 q^{4} - 5 q^{5} - 12 q^{6} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 6 * q^3 - 2 * q^4 - 5 * q^5 - 12 * q^6 + 8 * q^7 + 7 * q^9	\( 4 q - 2 q^{2} - 6 q^{3} - 2 q^{4} - 5 q^{5} - 12 q^{6} + 8 q^{7} + 7 q^{9} - 7 q^{11} + 8 q^{12} - 6 q^{13} - 14 q^{14} + 4 q^{16} - 7 q^{17} + 24 q^{18} - 10 q^{19} - 10 q^{20} - 2 q^{21} + 16 q^{22} - 11 q^{23} - 5 q^{25} + 28 q^{26} + 6 q^{28} - 5 q^{29} + 20 q^{30} - q^{31} - 32 q^{32} + 18 q^{33} + 6 q^{34} - 15 q^{35} + 14 q^{36} - 17 q^{37} + 20 q^{38} + 14 q^{39} - 17 q^{41} - 4 q^{42} + 4 q^{43} + 16 q^{44} + 25 q^{45} - 2 q^{46} - 22 q^{47} - 16 q^{48} + 8 q^{49} - 10 q^{50} + 28 q^{51} - 12 q^{52} + 24 q^{53} - 40 q^{54} + 30 q^{55} + 40 q^{57} - 10 q^{58} + 15 q^{59} + 20 q^{60} - 7 q^{61} - 2 q^{62} + 9 q^{63} + 8 q^{64} + 15 q^{65} - 4 q^{66} - 22 q^{67} + 16 q^{68} - 26 q^{69} + 40 q^{70} - 2 q^{71} - 16 q^{73} + 36 q^{74} + 20 q^{75} - 19 q^{77} - 32 q^{78} - 15 q^{79} - 20 q^{80} - 41 q^{81} + 36 q^{82} - 6 q^{83} + 16 q^{84} + 15 q^{85} + 18 q^{86} + 25 q^{89} - 10 q^{90} - 32 q^{91} - 2 q^{92} + 4 q^{93} - 44 q^{94} + 10 q^{95} + 48 q^{96} - 12 q^{97} - 44 q^{98} - 66 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 6 * q^3 - 2 * q^4 - 5 * q^5 - 12 * q^6 + 8 * q^7 + 7 * q^9 - 7 * q^11 + 8 * q^12 - 6 * q^13 - 14 * q^14 + 4 * q^16 - 7 * q^17 + 24 * q^18 - 10 * q^19 - 10 * q^20 - 2 * q^21 + 16 * q^22 - 11 * q^23 - 5 * q^25 + 28 * q^26 + 6 * q^28 - 5 * q^29 + 20 * q^30 - q^31 - 32 * q^32 + 18 * q^33 + 6 * q^34 - 15 * q^35 + 14 * q^36 - 17 * q^37 + 20 * q^38 + 14 * q^39 - 17 * q^41 - 4 * q^42 + 4 * q^43 + 16 * q^44 + 25 * q^45 - 2 * q^46 - 22 * q^47 - 16 * q^48 + 8 * q^49 - 10 * q^50 + 28 * q^51 - 12 * q^52 + 24 * q^53 - 40 * q^54 + 30 * q^55 + 40 * q^57 - 10 * q^58 + 15 * q^59 + 20 * q^60 - 7 * q^61 - 2 * q^62 + 9 * q^63 + 8 * q^64 + 15 * q^65 - 4 * q^66 - 22 * q^67 + 16 * q^68 - 26 * q^69 + 40 * q^70 - 2 * q^71 - 16 * q^73 + 36 * q^74 + 20 * q^75 - 19 * q^77 - 32 * q^78 - 15 * q^79 - 20 * q^80 - 41 * q^81 + 36 * q^82 - 6 * q^83 + 16 * q^84 + 15 * q^85 + 18 * q^86 + 25 * q^89 - 10 * q^90 - 32 * q^91 - 2 * q^92 + 4 * q^93 - 44 * q^94 + 10 * q^95 + 48 * q^96 - 12 * q^97 - 44 * q^98 - 66 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	775.2.j.a

Level	$775$
Weight	$2$
Character orbit	775.j
Analytic conductor	$6.188$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.18840615665\)
Analytic rank:	\(1\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \) x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$p$	$F_p(T)$
$2$	\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) T^4 + 2T^3 + 4T^2 + 8*T + 16
$3$	\( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) T^4 + 6T^3 + 16T^2 + 16*T + 16
$5$	\( T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25 \) T^4 + 5T^3 + 15T^2 + 25*T + 25
$7$	\( (T^{2} - 4 T - 1)^{2} \) (T^2 - 4*T - 1)^2
$11$	\( T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121 \) T^4 + 7T^3 + 34T^2 + 88*T + 121
$13$	\( T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 \) T^4 + 6T^3 + 16T^2 + 11*T + 121
$17$	\( T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1 \) T^4 + 7T^3 + 19T^2 + 3*T + 1
$19$	\( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 \) T^4 + 10T^3 + 60T^2 + 200*T + 400
$23$	\( T^{4} + 11 T^{3} + 51 T^{2} + \cdots + 961 \) T^4 + 11T^3 + 51T^2 + 31*T + 961
$29$	\( T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25 \) T^4 + 5T^3 + 15T^2 + 25*T + 25
$31$	\( T^{4} + T^{3} + T^{2} + T + 1 \) T^4 + T^3 + T^2 + T + 1
$37$	\( T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 \) T^4 + 17T^3 + 114T^2 + 88*T + 121
$41$	\( T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 \) T^4 + 17T^3 + 114T^2 + 88*T + 121
$43$	\( (T^{2} - 2 T - 19)^{2} \) (T^2 - 2*T - 19)^2
$47$	\( T^{4} + 22 T^{3} + 244 T^{2} + \cdots + 5776 \) T^4 + 22T^3 + 244T^2 + 1368*T + 5776
$53$	\( T^{4} - 24 T^{3} + 306 T^{2} + \cdots + 9801 \) T^4 - 24T^3 + 306T^2 - 2079*T + 9801
$59$	\( T^{4} - 15 T^{3} + 115 T^{2} + \cdots + 9025 \) T^4 - 15T^3 + 115T^2 - 475*T + 9025
$61$	\( T^{4} + 7 T^{3} + 199 T^{2} + \cdots + 361 \) T^4 + 7T^3 + 199T^2 - 437*T + 361
$67$	\( T^{4} + 22 T^{3} + 304 T^{2} + \cdots + 10201 \) T^4 + 22T^3 + 304T^2 + 2323*T + 10201
$71$	\( T^{4} + 2 T^{3} + 24 T^{2} + 133 T + 361 \) T^4 + 2T^3 + 24T^2 + 133*T + 361
$73$	\( T^{4} + 16 T^{3} + 106 T^{2} + \cdots + 3721 \) T^4 + 16T^3 + 106T^2 + 61*T + 3721
$79$	\( T^{4} + 15 T^{3} + 100 T^{2} + \cdots + 625 \) T^4 + 15T^3 + 100T^2 + 250*T + 625
$83$	\( T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 \) T^4 + 6T^3 + 16T^2 + 11*T + 121
$89$	\( T^{4} - 25 T^{3} + 235 T^{2} + \cdots + 3025 \) T^4 - 25T^3 + 235T^2 + 275*T + 3025
$97$	\( T^{4} + 12 T^{3} + 94 T^{2} + \cdots + 7921 \) T^4 + 12T^3 + 94T^2 + 623*T + 7921
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\(n\)	\(251\)	\(652\)
\(\chi(n)\)	\(1\)	\(-\zeta_{10}^{3}\)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
Format: