Properties

Label 4730.2.a.be
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 12
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 26 x^{10} + 79 x^{9} + 247 x^{8} - 766 x^{7} - 1023 x^{6} + 3281 x^{5} + 1634 x^{4} - 5557 x^{3} - 483 x^{2} + 1648 x + 392\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{11} - 22 \nu^{10} - 120 \nu^{9} + 589 \nu^{8} + 972 \nu^{7} - 5794 \nu^{6} - 2609 \nu^{5} + 24982 \nu^{4} - 2032 \nu^{3} - 41419 \nu^{2} + 12734 \nu + 8732 \)\()/26\)
\(\beta_{4}\)\(=\)\((\)\( -35 \nu^{11} + 115 \nu^{10} + 866 \nu^{9} - 3005 \nu^{8} - 7519 \nu^{7} + 28806 \nu^{6} + 25283 \nu^{5} - 120963 \nu^{4} - 14246 \nu^{3} + 195189 \nu^{2} - 50541 \nu - 39154 \)\()/26\)
\(\beta_{5}\)\(=\)\((\)\( -94 \nu^{11} + 359 \nu^{10} + 2308 \nu^{9} - 9456 \nu^{8} - 19675 \nu^{7} + 91398 \nu^{6} + 62382 \nu^{5} - 387005 \nu^{4} - 13528 \nu^{3} + 629874 \nu^{2} - 173679 \nu - 128968 \)\()/78\)
\(\beta_{6}\)\(=\)\((\)\( 341 \nu^{11} - 1165 \nu^{10} - 8444 \nu^{9} + 30477 \nu^{8} + 73313 \nu^{7} - 292404 \nu^{6} - 245409 \nu^{5} + 1228417 \nu^{4} + 128162 \nu^{3} - 1982277 \nu^{2} + 518361 \nu + 398822 \)\()/78\)
\(\beta_{7}\)\(=\)\((\)\( 505 \nu^{11} - 1676 \nu^{10} - 12562 \nu^{9} + 43863 \nu^{8} + 109846 \nu^{7} - 420978 \nu^{6} - 373437 \nu^{5} + 1768688 \nu^{4} + 222052 \nu^{3} - 2852223 \nu^{2} + 732594 \nu + 570634 \)\()/78\)
\(\beta_{8}\)\(=\)\((\)\( 329 \nu^{11} - 1081 \nu^{10} - 8195 \nu^{9} + 28299 \nu^{8} + 71807 \nu^{7} - 271689 \nu^{6} - 245169 \nu^{5} + 1141831 \nu^{4} + 150581 \nu^{3} - 1841703 \nu^{2} + 470967 \nu + 367967 \)\()/39\)
\(\beta_{9}\)\(=\)\((\)\( 752 \nu^{11} - 2521 \nu^{10} - 18698 \nu^{9} + 66054 \nu^{8} + 163289 \nu^{7} - 634776 \nu^{6} - 552720 \nu^{5} + 2670667 \nu^{4} + 314612 \nu^{3} - 4313280 \nu^{2} + 1116159 \nu + 864902 \)\()/78\)
\(\beta_{10}\)\(=\)\((\)\( 805 \nu^{11} - 2684 \nu^{10} - 19996 \nu^{9} + 70233 \nu^{8} + 174484 \nu^{7} - 674004 \nu^{6} - 590739 \nu^{5} + 2831822 \nu^{4} + 340918 \nu^{3} - 4567815 \nu^{2} + 1179096 \nu + 915154 \)\()/78\)
\(\beta_{11}\)\(=\)\((\)\( -1145 \nu^{11} + 3829 \nu^{10} + 28442 \nu^{9} - 100197 \nu^{8} - 248159 \nu^{7} + 961578 \nu^{6} + 839703 \nu^{5} - 4040173 \nu^{4} - 481358 \nu^{3} + 6517233 \nu^{2} - 1683699 \nu - 1305740 \)\()/78\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{9} + \beta_{8} - \beta_{5} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{4} + 9 \beta_{2} - \beta_{1} + 35\)
\(\nu^{5}\)\(=\)\(\beta_{10} - 13 \beta_{9} + 14 \beta_{8} - 3 \beta_{7} + \beta_{6} - 12 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 54 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(16 \beta_{11} - 14 \beta_{10} - 2 \beta_{9} + 16 \beta_{8} + 17 \beta_{7} + 31 \beta_{6} - 13 \beta_{4} + 3 \beta_{3} + 81 \beta_{2} - 14 \beta_{1} + 264\)
\(\nu^{7}\)\(=\)\(5 \beta_{11} + 21 \beta_{10} - 138 \beta_{9} + 156 \beta_{8} - 49 \beta_{7} + 18 \beta_{6} - 125 \beta_{5} + 32 \beta_{4} + 11 \beta_{3} + 21 \beta_{2} + 438 \beta_{1} + 17\)
\(\nu^{8}\)\(=\)\(198 \beta_{11} - 148 \beta_{10} - 43 \beta_{9} + 196 \beta_{8} + 216 \beta_{7} + 365 \beta_{6} - 7 \beta_{5} - 135 \beta_{4} + 56 \beta_{3} + 745 \beta_{2} - 152 \beta_{1} + 2096\)
\(\nu^{9}\)\(=\)\(112 \beta_{11} + 302 \beta_{10} - 1377 \beta_{9} + 1610 \beta_{8} - 591 \beta_{7} + 239 \beta_{6} - 1245 \beta_{5} + 383 \beta_{4} + 93 \beta_{3} + 313 \beta_{2} + 3689 \beta_{1} + 225\)
\(\nu^{10}\)\(=\)\(2220 \beta_{11} - 1410 \beta_{10} - 628 \beta_{9} + 2183 \beta_{8} + 2416 \beta_{7} + 3896 \beta_{6} - 169 \beta_{5} - 1307 \beta_{4} + 737 \beta_{3} + 6964 \beta_{2} - 1492 \beta_{1} + 17350\)
\(\nu^{11}\)\(=\)\(1704 \beta_{11} + 3691 \beta_{10} - 13426 \beta_{9} + 16086 \beta_{8} - 6335 \beta_{7} + 2834 \beta_{6} - 12167 \beta_{5} + 4099 \beta_{4} + 743 \beta_{3} + 4011 \beta_{2} + 31978 \beta_{1} + 2770\)

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
−3.03435
−2.56242
−2.52230
−1.94615
−0.343172
−0.342987
0.769997
2.14541
2.38729
2.59957
2.72061
3.12850
1.00000 −3.03435 1.00000 1.00000 −3.03435 −3.44511 1.00000 6.20730 1.00000
1.2 1.00000 −2.56242 1.00000 1.00000 −2.56242 3.71438 1.00000 3.56600 1.00000
1.3 1.00000 −2.52230 1.00000 1.00000 −2.52230 4.63187 1.00000 3.36197 1.00000
1.4 1.00000 −1.94615 1.00000 1.00000 −1.94615 −1.80392 1.00000 0.787492 1.00000
1.5 1.00000 −0.343172 1.00000 1.00000 −0.343172 1.96292 1.00000 −2.88223 1.00000
1.6 1.00000 −0.342987 1.00000 1.00000 −0.342987 −3.87285 1.00000 −2.88236 1.00000
1.7 1.00000 0.769997 1.00000 1.00000 0.769997 4.71912 1.00000 −2.40710 1.00000
1.8 1.00000 2.14541 1.00000 1.00000 2.14541 −4.86538 1.00000 1.60277 1.00000
1.9 1.00000 2.38729 1.00000 1.00000 2.38729 3.92530 1.00000 2.69915 1.00000
1.10 1.00000 2.59957 1.00000 1.00000 2.59957 1.31461 1.00000 3.75779 1.00000
1.11 1.00000 2.72061 1.00000 1.00000 2.72061 1.91272 1.00000 4.40172 1.00000
1.12 1.00000 3.12850 1.00000 1.00000 3.12850 −0.193657 1.00000 6.78751 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.be 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.be 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\(T_{3}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)
\(T_{13}^{12} - \cdots\)