Properties

Label 4730.2.a.bc
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} -\beta_{7} q^{7} - q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} -\beta_{7} q^{7} - q^{8} + ( 1 + \beta_{2} ) q^{9} + q^{10} + q^{11} + \beta_{1} q^{12} + ( \beta_{2} + \beta_{3} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{13} + \beta_{7} q^{14} -\beta_{1} q^{15} + q^{16} + ( -1 + \beta_{1} + \beta_{6} ) q^{17} + ( -1 - \beta_{2} ) q^{18} + ( 2 + \beta_{4} + \beta_{6} ) q^{19} - q^{20} + ( -\beta_{1} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{21} - q^{22} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{23} -\beta_{1} q^{24} + q^{25} + ( -\beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{26} + ( 1 + 2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{27} -\beta_{7} q^{28} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{29} + \beta_{1} q^{30} + ( 1 + \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{31} - q^{32} + \beta_{1} q^{33} + ( 1 - \beta_{1} - \beta_{6} ) q^{34} + \beta_{7} q^{35} + ( 1 + \beta_{2} ) q^{36} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{37} + ( -2 - \beta_{4} - \beta_{6} ) q^{38} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{10} ) q^{39} + q^{40} + ( -\beta_{2} + \beta_{4} + \beta_{8} + \beta_{10} ) q^{41} + ( \beta_{1} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{42} - q^{43} + q^{44} + ( -1 - \beta_{2} ) q^{45} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{46} + ( -2 + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{47} + \beta_{1} q^{48} + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} ) q^{49} - q^{50} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{51} + ( \beta_{2} + \beta_{3} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{52} + ( -1 + \beta_{1} - \beta_{8} + \beta_{10} ) q^{53} + ( -1 - 2 \beta_{1} + \beta_{5} - \beta_{6} ) q^{54} - q^{55} + \beta_{7} q^{56} + ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{57} + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{58} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} + \beta_{10} ) q^{59} -\beta_{1} q^{60} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{61} + ( -1 - \beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{62} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{63} + q^{64} + ( -\beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{65} -\beta_{1} q^{66} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{8} ) q^{67} + ( -1 + \beta_{1} + \beta_{6} ) q^{68} + ( -\beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{69} -\beta_{7} q^{70} + ( \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{71} + ( -1 - \beta_{2} ) q^{72} + ( 3 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{73} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{74} + \beta_{1} q^{75} + ( 2 + \beta_{4} + \beta_{6} ) q^{76} -\beta_{7} q^{77} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{10} ) q^{78} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{8} + \beta_{10} ) q^{79} - q^{80} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{81} + ( \beta_{2} - \beta_{4} - \beta_{8} - \beta_{10} ) q^{82} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{10} ) q^{83} + ( -\beta_{1} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{84} + ( 1 - \beta_{1} - \beta_{6} ) q^{85} + q^{86} + ( -2 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{87} - q^{88} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} ) q^{89} + ( 1 + \beta_{2} ) q^{90} + ( 4 + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{8} + \beta_{10} ) q^{91} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{92} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} + 3 \beta_{8} + 2 \beta_{9} ) q^{93} + ( 2 - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{94} + ( -2 - \beta_{4} - \beta_{6} ) q^{95} -\beta_{1} q^{96} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} + 2 \beta_{10} ) q^{97} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{98} + ( 1 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 11q^{2} + 11q^{4} - 11q^{5} - 11q^{8} + 15q^{9} + O(q^{10}) \) \( 11q - 11q^{2} + 11q^{4} - 11q^{5} - 11q^{8} + 15q^{9} + 11q^{10} + 11q^{11} - q^{13} + 11q^{16} - 15q^{17} - 15q^{18} + 14q^{19} - 11q^{20} + 7q^{21} - 11q^{22} - 2q^{23} + 11q^{25} + q^{26} + 3q^{27} + 6q^{29} + 13q^{31} - 11q^{32} + 15q^{34} + 15q^{36} + 16q^{37} - 14q^{38} + 4q^{39} + 11q^{40} - 7q^{41} - 7q^{42} - 11q^{43} + 11q^{44} - 15q^{45} + 2q^{46} - 19q^{47} + 23q^{49} - 11q^{50} + 32q^{51} - q^{52} - 16q^{53} - 3q^{54} - 11q^{55} - 2q^{57} - 6q^{58} + 7q^{59} + 20q^{61} - 13q^{62} + 11q^{64} + q^{65} + 9q^{67} - 15q^{68} + 10q^{69} + 13q^{71} - 15q^{72} + 20q^{73} - 16q^{74} + 14q^{76} - 4q^{78} + 13q^{79} - 11q^{80} + 19q^{81} + 7q^{82} - 6q^{83} + 7q^{84} + 15q^{85} + 11q^{86} - 23q^{87} - 11q^{88} + 10q^{89} + 15q^{90} + 43q^{91} - 2q^{92} + 22q^{93} + 19q^{94} - 14q^{95} + 3q^{97} - 23q^{98} + 15q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 24 x^{9} - x^{8} + 200 x^{7} + 14 x^{6} - 653 x^{5} - 26 x^{4} + 620 x^{3} - 177 x^{2} - 90 x + 32\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 52 \nu^{10} - 35 \nu^{9} - 1205 \nu^{8} + 947 \nu^{7} + 10132 \nu^{6} - 8658 \nu^{5} - 38070 \nu^{4} + 28329 \nu^{3} + 59983 \nu^{2} - 15093 \nu - 9602 \)\()/674\)
\(\beta_{4}\)\(=\)\((\)\( -75 \nu^{10} + 44 \nu^{9} + 1900 \nu^{8} - 815 \nu^{7} - 17128 \nu^{6} + 4568 \nu^{5} + 63891 \nu^{4} - 7710 \nu^{3} - 82282 \nu^{2} + 5003 \nu + 15884 \)\()/674\)
\(\beta_{5}\)\(=\)\((\)\( -91 \nu^{10} - 23 \nu^{9} + 2193 \nu^{8} + 786 \nu^{7} - 18068 \nu^{6} - 8270 \nu^{5} + 55333 \nu^{4} + 27513 \nu^{3} - 36475 \nu^{2} - 126 \nu + 3492 \)\()/674\)
\(\beta_{6}\)\(=\)\((\)\( -91 \nu^{10} - 23 \nu^{9} + 2193 \nu^{8} + 786 \nu^{7} - 18068 \nu^{6} - 8270 \nu^{5} + 55333 \nu^{4} + 28187 \nu^{3} - 36475 \nu^{2} - 5518 \nu + 2818 \)\()/674\)
\(\beta_{7}\)\(=\)\((\)\( 210 \nu^{10} + 79 \nu^{9} - 4983 \nu^{8} - 2099 \nu^{7} + 40814 \nu^{6} + 18618 \nu^{5} - 128884 \nu^{4} - 56259 \nu^{3} + 110485 \nu^{2} + 9649 \nu - 17650 \)\()/674\)
\(\beta_{8}\)\(=\)\((\)\( 374 \nu^{10} + 176 \nu^{9} - 8913 \nu^{8} - 4608 \nu^{7} + 73028 \nu^{6} + 40177 \nu^{5} - 227694 \nu^{4} - 118797 \nu^{3} + 180079 \nu^{2} + 17653 \nu - 26780 \)\()/337\)
\(\beta_{9}\)\(=\)\((\)\( -843 \nu^{10} - 476 \nu^{9} + 20008 \nu^{8} + 12003 \nu^{7} - 162674 \nu^{6} - 101438 \nu^{5} + 498303 \nu^{4} + 293206 \nu^{3} - 366562 \nu^{2} - 47077 \nu + 49236 \)\()/674\)
\(\beta_{10}\)\(=\)\((\)\( -487 \nu^{10} - 249 \nu^{9} + 11551 \nu^{8} + 6458 \nu^{7} - 93905 \nu^{6} - 55905 \nu^{5} + 288194 \nu^{4} + 164875 \nu^{3} - 215114 \nu^{2} - 28075 \nu + 30083 \)\()/337\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{5} + 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 8 \beta_{2} + \beta_{1} + 29\)
\(\nu^{5}\)\(=\)\(-\beta_{9} - 2 \beta_{8} + 3 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} - \beta_{4} - \beta_{3} + 65 \beta_{1} + 13\)
\(\nu^{6}\)\(=\)\(16 \beta_{10} + \beta_{9} + 17 \beta_{8} + 18 \beta_{7} - 12 \beta_{6} - \beta_{5} + 16 \beta_{4} - \beta_{3} + 61 \beta_{2} + 15 \beta_{1} + 229\)
\(\nu^{7}\)\(=\)\(2 \beta_{10} - 19 \beta_{9} - 33 \beta_{8} + 47 \beta_{7} + 105 \beta_{6} - 108 \beta_{5} - 16 \beta_{4} - 14 \beta_{3} + 3 \beta_{2} + 540 \beta_{1} + 128\)
\(\nu^{8}\)\(=\)\(188 \beta_{10} + 21 \beta_{9} + 207 \beta_{8} + 233 \beta_{7} - 116 \beta_{6} - 22 \beta_{5} + 195 \beta_{4} - 17 \beta_{3} + 469 \beta_{2} + 168 \beta_{1} + 1872\)
\(\nu^{9}\)\(=\)\(48 \beta_{10} - 256 \beta_{9} - 396 \beta_{8} + 556 \beta_{7} + 968 \beta_{6} - 1015 \beta_{5} - 189 \beta_{4} - 155 \beta_{3} + 53 \beta_{2} + 4569 \beta_{1} + 1177\)
\(\nu^{10}\)\(=\)\(1967 \beta_{10} + 299 \beta_{9} + 2218 \beta_{8} + 2642 \beta_{7} - 1056 \beta_{6} - 318 \beta_{5} + 2131 \beta_{4} - 202 \beta_{3} + 3667 \beta_{2} + 1698 \beta_{1} + 15643\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.00504
−2.74176
−2.26805
−1.44320
−0.419974
0.429703
0.510396
0.604446
2.60939
2.70832
3.01577
−1.00000 −3.00504 1.00000 −1.00000 3.00504 −1.41479 −1.00000 6.03028 1.00000
1.2 −1.00000 −2.74176 1.00000 −1.00000 2.74176 3.45531 −1.00000 4.51722 1.00000
1.3 −1.00000 −2.26805 1.00000 −1.00000 2.26805 −4.08168 −1.00000 2.14403 1.00000
1.4 −1.00000 −1.44320 1.00000 −1.00000 1.44320 −0.640387 −1.00000 −0.917168 1.00000
1.5 −1.00000 −0.419974 1.00000 −1.00000 0.419974 3.08102 −1.00000 −2.82362 1.00000
1.6 −1.00000 0.429703 1.00000 −1.00000 −0.429703 2.14075 −1.00000 −2.81536 1.00000
1.7 −1.00000 0.510396 1.00000 −1.00000 −0.510396 −1.71358 −1.00000 −2.73950 1.00000
1.8 −1.00000 0.604446 1.00000 −1.00000 −0.604446 −3.36004 −1.00000 −2.63465 1.00000
1.9 −1.00000 2.60939 1.00000 −1.00000 −2.60939 2.64901 −1.00000 3.80893 1.00000
1.10 −1.00000 2.70832 1.00000 −1.00000 −2.70832 4.04483 −1.00000 4.33498 1.00000
1.11 −1.00000 3.01577 1.00000 −1.00000 −3.01577 −4.16045 −1.00000 6.09484 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4730.2.a.bc 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.bc 11 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\(T_{3}^{11} - \cdots\)
\(T_{7}^{11} - \cdots\)
\(T_{13}^{11} + \cdots\)