Properties

Label 75.10.a.f
Level 75
Weight 10
Character orbit 75.a
Self dual yes
Analytic conductor 38.628
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Defining polynomial: \(x^{2} - x - 60\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 3\sqrt{241})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -16 - \beta ) q^{2} + 81 q^{3} + ( 286 + 31 \beta ) q^{4} + ( -1296 - 81 \beta ) q^{6} + ( -7168 - 224 \beta ) q^{7} + ( -13186 - 239 \beta ) q^{8} + 6561 q^{9} +O(q^{10})\) \( q + ( -16 - \beta ) q^{2} + 81 q^{3} + ( 286 + 31 \beta ) q^{4} + ( -1296 - 81 \beta ) q^{6} + ( -7168 - 224 \beta ) q^{7} + ( -13186 - 239 \beta ) q^{8} + 6561 q^{9} + ( -11940 - 2368 \beta ) q^{11} + ( 23166 + 2511 \beta ) q^{12} + ( -9470 + 5344 \beta ) q^{13} + ( 236096 + 10528 \beta ) q^{14} + ( 194082 + 899 \beta ) q^{16} + ( 74718 - 7520 \beta ) q^{17} + ( -104976 - 6561 \beta ) q^{18} + ( -50812 - 5728 \beta ) q^{19} + ( -580608 - 18144 \beta ) q^{21} + ( 1474496 + 47460 \beta ) q^{22} + ( 354496 - 26272 \beta ) q^{23} + ( -1068066 - 19359 \beta ) q^{24} + ( -2744928 - 70690 \beta ) q^{26} + 531441 q^{27} + ( -5813696 - 279328 \beta ) q^{28} + ( -1423394 - 168576 \beta ) q^{29} + ( 5314848 - 152736 \beta ) q^{31} + ( 3158662 - 85199 \beta ) q^{32} + ( -967140 - 191808 \beta ) q^{33} + ( 2880352 + 38082 \beta ) q^{34} + ( 1876446 + 203391 \beta ) q^{36} + ( -10884918 + 198496 \beta ) q^{37} + ( 3917568 + 136732 \beta ) q^{38} + ( -767070 + 432864 \beta ) q^{39} + ( 12784138 - 492096 \beta ) q^{41} + ( 19123776 + 852768 \beta ) q^{42} + ( 3946748 + 702336 \beta ) q^{43} + ( -43201976 - 973980 \beta ) q^{44} + ( 8567488 + 39584 \beta ) q^{46} + ( 14804856 - 1970528 \beta ) q^{47} + ( 15720642 + 72819 \beta ) q^{48} + ( 38222009 + 3161088 \beta ) q^{49} + ( 6052158 - 609120 \beta ) q^{51} + ( 87081468 + 1069150 \beta ) q^{52} + ( -1476694 + 177728 \beta ) q^{53} + ( -8503056 - 531441 \beta ) q^{54} + ( 123533760 + 4613280 \beta ) q^{56} + ( -4115772 - 463968 \beta ) q^{57} + ( 114142496 + 3952034 \beta ) q^{58} + ( -19921508 - 4348352 \beta ) q^{59} + ( 171223774 + 950208 \beta ) q^{61} + ( -2254656 - 3023808 \beta ) q^{62} + ( -47029248 - 1469664 \beta ) q^{63} + ( -103730718 - 2340965 \beta ) q^{64} + ( 119434176 + 3844260 \beta ) q^{66} + ( 143584628 - 1026560 \beta ) q^{67} + ( -104981692 + 398658 \beta ) q^{68} + ( 28714176 - 2128032 \beta ) q^{69} + ( 102870392 - 4545280 \beta ) q^{71} + ( -86513346 - 1568079 \beta ) q^{72} + ( 115747446 - 1168192 \beta ) q^{73} + ( 66573856 + 7907478 \beta ) q^{74} + ( -110774088 - 3035812 \beta ) q^{76} + ( 373080064 + 19117952 \beta ) q^{77} + ( -222339168 - 5725890 \beta ) q^{78} + ( -2852960 + 19049120 \beta ) q^{79} + 43046721 q^{81} + ( 62169824 - 5402698 \beta ) q^{82} + ( 182410932 - 7260288 \beta ) q^{83} + ( -470909376 - 22625568 \beta ) q^{84} + ( -443814080 - 14481788 \beta ) q^{86} + ( -115294914 - 13654656 \beta ) q^{87} + ( 464186824 + 33512156 \beta ) q^{88} + ( -209294982 + 9049152 \beta ) q^{89} + ( -580923392 - 34987456 \beta ) q^{91} + ( -340036288 + 4290016 \beta ) q^{92} + ( 430502688 - 12371616 \beta ) q^{93} + ( 831148480 + 14753064 \beta ) q^{94} + ( 255851622 - 6901119 \beta ) q^{96} + ( -879104002 + 13450880 \beta ) q^{97} + ( -2324861840 - 85638329 \beta ) q^{98} + ( -78338340 - 15536448 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 31q^{2} + 162q^{3} + 541q^{4} - 2511q^{6} - 14112q^{7} - 26133q^{8} + 13122q^{9} + O(q^{10}) \) \( 2q - 31q^{2} + 162q^{3} + 541q^{4} - 2511q^{6} - 14112q^{7} - 26133q^{8} + 13122q^{9} - 21512q^{11} + 43821q^{12} - 24284q^{13} + 461664q^{14} + 387265q^{16} + 156956q^{17} - 203391q^{18} - 95896q^{19} - 1143072q^{21} + 2901532q^{22} + 735264q^{23} - 2116773q^{24} - 5419166q^{26} + 1062882q^{27} - 11348064q^{28} - 2678212q^{29} + 10782432q^{31} + 6402523q^{32} - 1742472q^{33} + 5722622q^{34} + 3549501q^{36} - 21968332q^{37} + 7698404q^{38} - 1967004q^{39} + 26060372q^{41} + 37394784q^{42} + 7191160q^{43} - 85429972q^{44} + 17095392q^{46} + 31580240q^{47} + 31368465q^{48} + 73282930q^{49} + 12713436q^{51} + 173093786q^{52} - 3131116q^{53} - 16474671q^{54} + 242454240q^{56} - 7767576q^{57} + 224332958q^{58} - 35494664q^{59} + 341497340q^{61} - 1485504q^{62} - 92588832q^{63} - 205120471q^{64} + 235024092q^{66} + 288195816q^{67} - 210362042q^{68} + 59556384q^{69} + 210286064q^{71} - 171458613q^{72} + 232663084q^{73} + 125240234q^{74} - 218512364q^{76} + 727042176q^{77} - 438952446q^{78} - 24755040q^{79} + 86093442q^{81} + 129742346q^{82} + 372082152q^{83} - 919193184q^{84} - 873146372q^{86} - 216935172q^{87} + 894861492q^{88} - 427639116q^{89} - 1126859328q^{91} - 684362592q^{92} + 873376992q^{93} + 1647543896q^{94} + 518604363q^{96} - 1771658884q^{97} - 4564085351q^{98} - 141140232q^{99} + O(q^{100}) \)

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
8.26209
−7.26209
−38.7863 81.0000 992.374 0 −3141.69 −12272.1 −18631.9 6561.00 0
1.2 7.78626 81.0000 −451.374 0 630.687 −1839.88 −7501.08 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

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 75.10.a.f 2
3.b odd 2 1 225.10.a.k 2
5.b even 2 1 15.10.a.d 2
5.c odd 4 2 75.10.b.f 4
15.d odd 2 1 45.10.a.d 2
15.e even 4 2 225.10.b.i 4
20.d odd 2 1 240.10.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.d 2 5.b even 2 1
45.10.a.d 2 15.d odd 2 1
75.10.a.f 2 1.a even 1 1 trivial
75.10.b.f 4 5.c odd 4 2
225.10.a.k 2 3.b odd 2 1
225.10.b.i 4 15.e even 4 2
240.10.a.r 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 31 T_{2} - 302 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(75))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 31 T + 722 T^{2} + 15872 T^{3} + 262144 T^{4} \)
$3$ \( ( 1 - 81 T )^{2} \)
$5$ 1
$7$ \( 1 + 14112 T + 103286414 T^{2} + 569470101984 T^{3} + 1628413597910449 T^{4} \)
$11$ \( 1 + 21512 T + 1790961254 T^{2} + 50724170728792 T^{3} + 5559917313492231481 T^{4} \)
$13$ \( 1 + 24284 T + 5870669214 T^{2} + 257519662773932 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$17$ \( 1 - 156956 T + 212670095078 T^{2} - 18613078743463132 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \)
$19$ \( 1 + 95896 T + 629883192438 T^{2} + 30944459466214984 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \)
$23$ \( 1 - 735264 T + 3363187908526 T^{2} - 1324322710477931232 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \)
$29$ \( 1 + 2678212 T + 15397908029438 T^{2} + 38853212438324066228 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \)
$31$ \( 1 - 10782432 T + 69294691361342 T^{2} - \)\(28\!\cdots\!72\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \)
$37$ \( 1 + 21968332 T + 359210373327534 T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \)
$41$ \( 1 - 26060372 T + 693239183881142 T^{2} - \)\(85\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \)
$43$ \( 1 - 7191160 T + 750634586008230 T^{2} - \)\(36\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 31580240 T + 382042606129310 T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \)
$53$ \( 1 + 3131116 T + 6584849973489806 T^{2} + \)\(10\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \)
$59$ \( 1 + 35494664 T + 7388006896329158 T^{2} + \)\(30\!\cdots\!96\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \)
$61$ \( 1 - 341497340 T + 52053805546777278 T^{2} - \)\(39\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \)
$67$ \( 1 - 288195816 T + 74605839041196758 T^{2} - \)\(78\!\cdots\!52\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \)
$71$ \( 1 - 210286064 T + 91549406631588686 T^{2} - \)\(96\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \)
$73$ \( 1 - 232663084 T + 130536207391012086 T^{2} - \)\(13\!\cdots\!92\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \)
$79$ \( 1 + 24755040 T + 43090694479668638 T^{2} + \)\(29\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \)
$83$ \( 1 - 372082152 T + 379908828789982198 T^{2} - \)\(69\!\cdots\!56\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \)
$89$ \( 1 + 427639116 T + 702028302670151638 T^{2} + \)\(14\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \)
$97$ \( 1 + 1771658884 T + 2207048700436243398 T^{2} + \)\(13\!\cdots\!28\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \)
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