Properties

Label 725.6.a.a
Level $725$
Weight $6$
Character orbit 725.a
Self dual yes
Analytic conductor $116.278$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 725.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(116.278269364\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
Defining polynomial: \( x^{4} - 34x^{2} - 27x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 7) q^{3} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 2) q^{4} + (\beta_{3} + 5 \beta_{2} + 3 \beta_1 - 48) q^{6} + (12 \beta_{3} + 4 \beta_{2} + 58) q^{7} + (5 \beta_{2} - 14 \beta_1 + 126) q^{8} + ( - 20 \beta_{2} - 8 \beta_1 - 70) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 7) q^{3} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 2) q^{4} + (\beta_{3} + 5 \beta_{2} + 3 \beta_1 - 48) q^{6} + (12 \beta_{3} + 4 \beta_{2} + 58) q^{7} + (5 \beta_{2} - 14 \beta_1 + 126) q^{8} + ( - 20 \beta_{2} - 8 \beta_1 - 70) q^{9} + ( - 20 \beta_{3} - 29 \beta_{2} - 21 \beta_1 - 41) q^{11} + ( - 10 \beta_{3} - \beta_{2} + 16 \beta_1 - 10) q^{12} + ( - 60 \beta_{3} + 38 \beta_{2} + 10 \beta_1 + 85) q^{13} + ( - 64 \beta_{3} + 66 \beta_{2} - 24 \beta_1 + 160) q^{14} + (27 \beta_{3} + 33 \beta_{2} + 81 \beta_1 - 90) q^{16} + (4 \beta_{3} - 50 \beta_{2} - 14 \beta_1 - 44) q^{17} + (20 \beta_{3} - 146 \beta_{2} + 60 \beta_1 - 792) q^{18} + (116 \beta_{3} - 80 \beta_{2} + 168 \beta_1 - 540) q^{19} + (220 \beta_{3} - 38 \beta_{2} - 10 \beta_1 + 358) q^{21} + (129 \beta_{3} - 103 \beta_{2} + 107 \beta_1 - 1320) q^{22} + (140 \beta_{3} - 14 \beta_{2} - 146 \beta_1 + 368) q^{23} + (19 \beta_{3} - 213 \beta_{2} - 83 \beta_1 + 1706) q^{24} + (262 \beta_{3} + 305 \beta_{2} - 54 \beta_1 + 1312) q^{26} + ( - 12 \beta_{3} + 149 \beta_{2} + 269 \beta_1 - 623) q^{27} + ( - 130 \beta_{3} + 498 \beta_{2} - 134 \beta_1 - 76) q^{28} - 841 q^{29} + ( - 92 \beta_{3} - 319 \beta_{2} + 485 \beta_1 - 4849) q^{31} + ( - 168 \beta_{3} - 355 \beta_{2} + 322 \beta_1 - 1722) q^{32} + ( - 368 \beta_{3} - 272 \beta_{2} - 64 \beta_1 + 2461) q^{33} + (30 \beta_{3} - 256 \beta_{2} + 146 \beta_1 - 1888) q^{34} + (46 \beta_{3} - 1082 \beta_{2} + 674 \beta_1 - 1844) q^{36} + ( - 40 \beta_{3} - 544 \beta_{2} + 712 \beta_1 + 2712) q^{37} + ( - 500 \beta_{3} - 1560 \beta_{2} + 124 \beta_1 - 136) q^{38} + ( - 1052 \beta_{3} + 185 \beta_{2} - 171 \beta_1 - 2709) q^{39} + ( - 804 \beta_{3} + 1950 \beta_{2} - 366 \beta_1 - 682) q^{41} + ( - 1062 \beta_{3} - 22 \beta_{2} - 106 \beta_1 - 992) q^{42} + ( - 772 \beta_{3} - 1603 \beta_{2} + 341 \beta_1 + 4969) q^{43} + (98 \beta_{3} - 1357 \beta_{2} + 852 \beta_1 - 434) q^{44} + ( - 686 \beta_{3} + 596 \beta_{2} - 98 \beta_1 - 2240) q^{46} + ( - 72 \beta_{3} - 449 \beta_{2} + 1675 \beta_1 - 5979) q^{47} + (438 \beta_{3} + 903 \beta_{2} + 108 \beta_1 - 8046) q^{48} + (1040 \beta_{3} - 1424 \beta_{2} - 1152 \beta_1 + 3133) q^{49} + (36 \beta_{3} - 318 \beta_{2} - 66 \beta_1 + 3204) q^{51} + (305 \beta_{3} + 1521 \beta_{2} - 1497 \beta_1 + 7418) q^{52} + ( - 628 \beta_{3} + 2258 \beta_{2} - 1898 \beta_1 - 2529) q^{53} + ( - 89 \beta_{3} - 673 \beta_{2} - 435 \beta_1 + 8808) q^{54} + (2200 \beta_{3} + 834 \beta_{2} - 596 \beta_1 + 9676) q^{56} + (1840 \beta_{3} + 1484 \beta_{2} + 312 \beta_1 - 11316) q^{57} - 841 \beta_{2} q^{58} + ( - 140 \beta_{3} - 2466 \beta_{2} + 3098 \beta_1 - 2780) q^{59} + ( - 2396 \beta_{3} + 3582 \beta_{2} - 3142 \beta_1 + 11164) q^{61} + (779 \beta_{3} - 7807 \beta_{2} + 1049 \beta_1 - 4240) q^{62} + (1016 \beta_{3} - 1600 \beta_{2} + 208 \beta_1 - 6364) q^{63} + (331 \beta_{3} - 5351 \beta_{2} - 1359 \beta_1 - 5018) q^{64} + (2112 \beta_{3} + 1661 \beta_{2} + 1184 \beta_1 - 10880) q^{66} + (1032 \beta_{3} - 4676 \beta_{2} + 4428 \beta_1 + 2476) q^{67} + ( - 22 \beta_{3} - 2036 \beta_{2} + 1186 \beta_1 - 5192) q^{68} + (2652 \beta_{3} - 1606 \beta_{2} + 302 \beta_1 + 16024) q^{69} + ( - 96 \beta_{3} - 3598 \beta_{2} + 3250 \beta_1 - 12234) q^{71} + (212 \beta_{3} - 4650 \beta_{2} + 1280 \beta_1 - 1916) q^{72} + (1504 \beta_{3} + 2200 \beta_{2} - 3784 \beta_1 + 19500) q^{73} + (744 \beta_{3} - 2104 \beta_{2} + 1672 \beta_1 - 8608) q^{74} + (348 \beta_{3} - 5248 \beta_{2} - 196 \beta_1 - 35024) q^{76} + (1876 \beta_{3} + 1122 \beta_{2} + 1742 \beta_1 - 31226) q^{77} + (5075 \beta_{3} - 219 \beta_{2} + 497 \beta_1 + 1792) q^{78} + (2864 \beta_{3} - 5901 \beta_{2} - 1457 \beta_1 - 25087) q^{79} + ( - 336 \beta_{3} + 8380 \beta_{2} + 2440 \beta_1 - 15091) q^{81} + (2070 \beta_{3} + 10970 \beta_{2} - 5046 \beta_1 + 59568) q^{82} + (1108 \beta_{3} - 2566 \beta_{2} + 5038 \beta_1 - 15168) q^{83} + ( - 1708 \beta_{3} + 1494 \beta_{2} + 1448 \beta_1 - 15812) q^{84} + (5463 \beta_{3} - 3297 \beta_{2} + 5581 \beta_1 - 51272) q^{86} + (841 \beta_{2} + 841 \beta_1 - 5887) q^{87} + ( - 3261 \beta_{3} - 6577 \beta_{2} + 549 \beta_1 + 8226) q^{88} + (4380 \beta_{3} - 7382 \beta_{2} - 842 \beta_1 + 29082) q^{89} + ( - 5132 \beta_{3} + 12448 \beta_{2} + 4708 \beta_1 - 69790) q^{91} + ( - 1646 \beta_{3} + 2168 \beta_{2} + 3570 \beta_1 + 5744) q^{92} + ( - 2460 \beta_{3} + 7134 \beta_{2} + 2646 \beta_1 - 56595) q^{93} + (809 \beta_{3} - 13177 \beta_{2} + 1419 \beta_1 + 8040) q^{94} + ( - 3701 \beta_{3} + 2523 \beta_{2} - 491 \beta_1 - 21502) q^{96} + ( - 7180 \beta_{3} - 7150 \beta_{2} - 3962 \beta_1 + 8790) q^{97} + ( - 3776 \beta_{3} - 1571 \beta_{2} + 3232 \beta_1 - 62464) q^{98} + ( - 1972 \beta_{3} + 2714 \beta_{2} + 850 \beta_1 + 40694) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{3} + 10 q^{4} - 194 q^{6} + 208 q^{7} + 504 q^{8} - 280 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{3} + 10 q^{4} - 194 q^{6} + 208 q^{7} + 504 q^{8} - 280 q^{9} - 124 q^{11} - 20 q^{12} + 460 q^{13} + 768 q^{14} - 414 q^{16} - 184 q^{17} - 3208 q^{18} - 2392 q^{19} + 992 q^{21} - 5538 q^{22} + 1192 q^{23} + 6786 q^{24} + 4724 q^{26} - 2468 q^{27} - 44 q^{28} - 3364 q^{29} - 19212 q^{31} - 6552 q^{32} + 10580 q^{33} - 7612 q^{34} - 7468 q^{36} + 10928 q^{37} + 456 q^{38} - 8732 q^{39} - 1120 q^{41} - 1844 q^{42} + 21420 q^{43} - 1932 q^{44} - 7588 q^{46} - 23772 q^{47} - 33060 q^{48} + 10452 q^{49} + 12744 q^{51} + 29062 q^{52} - 8860 q^{53} + 35410 q^{54} + 34304 q^{56} - 48944 q^{57} - 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 27488 q^{63} - 20734 q^{64} - 47744 q^{66} + 7840 q^{67} - 20724 q^{68} + 58792 q^{69} - 48744 q^{71} - 8088 q^{72} + 74992 q^{73} - 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 2982 q^{78} - 106076 q^{79} - 59692 q^{81} + 234132 q^{82} - 62888 q^{83} - 59832 q^{84} - 216014 q^{86} - 23548 q^{87} + 39426 q^{88} + 107568 q^{89} - 268896 q^{91} + 26268 q^{92} - 221460 q^{93} + 30542 q^{94} - 78606 q^{96} + 49520 q^{97} - 242304 q^{98} + 166720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 34x^{2} - 27x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 3\nu - 17 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 17 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - \nu^{2} - 67\nu - 25 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 3\beta _1 + 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 32\beta_{2} + 35\beta _1 + 42 ) / 2 \) Copy content Toggle raw display

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
0.275208
−1.10057
6.17343
−5.34807
−5.91663 18.2828 3.00648 0 −108.173 −139.558 171.544 91.2623 0
1.2 −4.16235 17.5258 −14.6748 0 −72.9487 220.793 194.277 64.1549 0
1.3 0.863638 −7.07413 −31.2541 0 −6.10949 36.7447 −54.6287 −192.957 0
1.4 9.21534 −0.734546 52.9225 0 −6.76909 90.0205 192.808 −242.460 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(29\) \(1\)

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 725.6.a.a 4
5.b even 2 1 29.6.a.a 4
15.d odd 2 1 261.6.a.a 4
20.d odd 2 1 464.6.a.i 4
145.d even 2 1 841.6.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.a 4 5.b even 2 1
261.6.a.a 4 15.d odd 2 1
464.6.a.i 4 20.d odd 2 1
725.6.a.a 4 1.a even 1 1 trivial
841.6.a.a 4 145.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 69T_{2}^{2} - 168T_{2} + 196 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(725))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 69 T^{2} - 168 T + 196 \) Copy content Toggle raw display
$3$ \( T^{4} - 28 T^{3} + 46 T^{2} + \cdots + 1665 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 208 T^{3} + \cdots - 101924272 \) Copy content Toggle raw display
$11$ \( T^{4} + 124 T^{3} + \cdots - 3717303119 \) Copy content Toggle raw display
$13$ \( T^{4} - 460 T^{3} + \cdots - 116053863479 \) Copy content Toggle raw display
$17$ \( T^{4} + 184 T^{3} + \cdots + 11464717824 \) Copy content Toggle raw display
$19$ \( T^{4} + 2392 T^{3} + \cdots + 2620094791680 \) Copy content Toggle raw display
$23$ \( T^{4} - 1192 T^{3} + \cdots + 2033080361984 \) Copy content Toggle raw display
$29$ \( (T + 841)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 421127233952247 \) Copy content Toggle raw display
$37$ \( T^{4} - 10928 T^{3} + \cdots + 16079593861120 \) Copy content Toggle raw display
$41$ \( T^{4} + 1120 T^{3} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{4} - 21420 T^{3} + \cdots + 47\!\cdots\!37 \) Copy content Toggle raw display
$47$ \( T^{4} + 23772 T^{3} + \cdots - 35\!\cdots\!87 \) Copy content Toggle raw display
$53$ \( T^{4} + 8860 T^{3} + \cdots + 16\!\cdots\!53 \) Copy content Toggle raw display
$59$ \( T^{4} + 10840 T^{3} + \cdots - 43\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{4} - 49448 T^{3} + \cdots + 90\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} - 7840 T^{3} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{4} + 48744 T^{3} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{4} - 74992 T^{3} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + 106076 T^{3} + \cdots - 53\!\cdots\!75 \) Copy content Toggle raw display
$83$ \( T^{4} + 62888 T^{3} + \cdots - 18\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} - 107568 T^{3} + \cdots - 18\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{4} - 49520 T^{3} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
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